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\begin{document} \def\mathbb N{\mathbb N} \def\mathbb C{\mathbb C} \def\mathbb Q{\mathbb Q} \def\mathbb R{\mathbb R} \def\mathbb T{\mathbb T} \def\mathbb A{\mathbb A} \def\mathbb Z{\mathbb Z} \def\frac{1}{2}{\frac{1}{2}} \begin{titlepage} \author{Abed Bounemoura~\footnote{A.Bounemoura@warwick.ac.uk, Mathematics Institute, University of Warwick}} \title{\LARGE{\textbf{An example of instability in high-dimensional Hamiltonian systems}}} \end{titlepage} \maketitle \begin{abstract} In this article, we use a mechanism first introduced by Herman, Marco, and Sauzin to show that if a Gevrey or analytic perturbation of a quasi-convex integrable Hamiltonian system is not too small with respect to the number of degrees of freedom, then the classical exponential stability estimates do not hold. Indeed, we construct an unstable solution whose drifting time is polynomial with respect to the inverse of the size of the perturbation. A different example was already given by Bourgain and Kaloshin, with a linear time of drift but with a perturbation which is larger than ours. As a consequence, we obtain a better upper bound on the threshold of validity of exponential stability estimates. \end{abstract} \section{Introduction} \paraga Consider a near-integrable Hamiltonian system of the form \begin{equation*} \begin{cases} H(\theta,I)=h(I)+f(\theta,I) \\ |f| < \varepsilon \end{cases} \end{equation*} with angle-action coordinates $(\theta,I) \in \mathbb T^n \times \mathbb R^n$, and where $f$ is a small perturbation, of size $\varepsilon$, in some suitable topology defined by a norm $|\,.\,|$. If the system is analytic and $h$ satisfies a generic condition, it is a remarkable result due to Nekhoroshev (\cite{Nek77}, \cite{Nek79}) that the action variables are stable for an exponentially long interval of time with respect to the inverse of the size of the perturbation: one has \[ |I(t)-I_0| \leq c_1\varepsilon^b, \quad |t|\leq c_2\exp(c_3\varepsilon^{-a}), \] for some positive constants $c_1,c_2,c_3,a,b$ and provided that the size of the perturbation $\varepsilon$ is smaller than a threshold $\varepsilon_0$. Of course, all these constants strongly depend on the number of degrees of freedom $n$, and when the latter goes to infinity, the threshold $\varepsilon_0$ and the exponent of stability $a$ go to zero. More precisely, in the case where $h$ is quasi-convex and the system is analytic or even Gevrey, then we know that the exponent $a$ is of order $n^{-1}$ and this is essentially optimal (see \cite{LN92}, \cite{Pos93}, \cite{MS02}, \cite{LM05} \cite{KZ09} and \cite{BM10} for more information on the optimality of the stability exponent). \paraga This fact was used by Bourgain and Kaloshin in \cite{BK05} to show that if the size of the perturbation is \[ \varepsilon_n \sim e^{-n}, \] then there is no exponential stability: they constructed unstable solutions for which the time of drift is linear with respect to the inverse of the size of the perturbation, that is \[ |I(\tau_n)-I_0|\sim 1, \quad \tau_n \sim \varepsilon_n^{-1}. \] More precisely, in the first part of \cite{BK05}, Bourgain proved this result for a specific example of Gevrey non-analytic perturbation of a quasi-convex system, then for an analytic perturbation he obtained a time $\tau_n \sim \varepsilon_n^{-1-c}$, for any $c>0$. In the second part of \cite{BK05}, using much more elaborated techniques (especially Mather theory), Kaloshin proved the above result for both Gevrey and analytic perturbation and for a wider class of integrable Hamiltonians, including convex and quasi-convex systems. Their motivation was the implementation of stability estimates in the context of Hamiltonian partial differential equations, which requires to understand the relative dependence between the size of the perturbation and the number of degrees of freedom. Their result indicates that for infinite dimensional Hamiltonian systems, Nekhoroshev's mechanism does not survive and that ``fast diffusion" should prevail. Of course, in their example, one cannot simply take $n=\infty$ as the size of the perturbation $\varepsilon_n \sim e^{-n}$ goes to zero and the time of instability $\tau_n \sim e^n$ goes to infinity exponentially fast with respect to $n$. A more precise interpretation concerns the threshold of validity $\varepsilon_0$ in Nekhoroshev's theorem: it has to satisfy \[ \varepsilon_0 <\!\!< e^{-n}, \] and so it deteriorates faster than exponentially with respect to $n$. \paraga As was noticed by the authors in \cite{BK05}, their examples share some similarities with the mechanism introduced by Herman, Marco and Sauzin in \cite{MS02} (see also \cite{LM05} for the analytic case). In this present article, we use the approach of \cite{MS02} and \cite{LM05} to show, using simpler arguments than those contained in \cite{BK05}, that if the size of the perturbation is \[ \varepsilon_n \sim e^{-n\ln (n\ln n)}, \] then it is still too large to have exponential stability: we will show that one can find an unstable solution where the time of drift is polynomial, more precisely \[ |I(\tau_n)-I_0|\sim 1, \quad \tau_n \sim \varepsilon_n^{-n}. \] As in the first part of \cite{BK05}, we will construct specific examples of Gevrey and analytic perturbations of a quasi-convex system. We refer to Theorem~\ref{thmnonpert} and Theorem~\ref{thmnonpertana} below for precise statements. Hence one can infer that the threshold $\varepsilon_0$ in Nekhoroshev's theorem further satisfies \[ \varepsilon_0 <\!\!< e^{-n\ln (n\ln n)} <\!\!< e^{-n}, \] and this gives another evidence that the finite dimensional mechanism of stability cannot extend so easily to infinite dimensional systems. Let us point out that our time of drift is worst than the one obtained in \cite{BK05}, but this stems from the fact that the size of our perturbation is smaller than theirs and so it is natural for the time of instability to be larger. Moreover, our exponent $n$ in the time of drift can be a bit misleading since in any cases, that is even for a linear time of drift, $\tau_n$ goes to infinity exponentially fast with $n$, so such results do not apply at all to infinite dimensional Hamiltonian systems. A natural question, which was raised by Marco, is the following. \begin{question} Given $\varepsilon>0$ arbitrarily small, construct an $\varepsilon_n$-perturbation of an integrable system having an unstable orbit with a time of instability $\tau_n$ such that \[ \lim_{n\rightarrow +\infty}\varepsilon_n=\varepsilon, \quad \lim_{n\rightarrow +\infty}\tau_n<+\infty. \] \end{question} We believe that one can give a positive answer to this question, by using a more clever construction. However, even if one can formally let $n$ goes to infinity, by no means this implies the existence of an unstable solution for an infinite-dimensional Hamiltonian systems, which is a very difficult problem (see \cite{CKSTT} and \cite{GG10} for related results in some examples of Hamiltonian partial differential equations). \section{Main results} \subsection{The Gevrey case} \paraga Let us first state our result in the Gevrey case. Let $n\geq 3$ be the number of degrees of freedom, and given $R>0$, let $B=B_R$ be the open ball of $\mathbb R^n$ around the origin, of radius $R>0$ with respect to the supremum norm $|\,.\,|$, and $\overline{B}$ its closure. The phase space is $\mathbb T^n \times B$, and we consider a Hamiltonian system of the form \[ H(\theta,I)=h(I)+f(\theta,I), \quad (\theta,I)\in \mathbb T^n \times B. \] Our quasi-convex integrable Hamiltonian $h$ is the simplest one, namely \[ h(I)=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+I_n, \quad I=(I_1,\dots,I_n)\in B. \] Let us recall that, given $\alpha \geq 1$ and $L>0$, a function $f\in C^{\infty}(\mathbb T^n \times \overline{B})$ is $(\alpha,L)$-Gevrey if, using the standard multi-index notation, \[ |f|_{\alpha,L}=\sum_{l\in \mathbb N^{2n}}L^{|l|\alpha}(l!)^{-\alpha}|\partial^l f|_{C^0(\mathbb T^n \times \overline{B})} < \infty. \] The space of such functions, with the above norm, is a Banach space that we denote by $G^{\alpha,L}(\mathbb T^n \times \overline{B})$. One can see that analytic functions correspond exactly to $\alpha=1$. \paraga Now we can state our theorem. \begin{theorem}\label{thmnonpert} Let $n\geq 3$, $R>1$, $\alpha>1$ and $L>0$. Then there exist positive constants $c,\gamma,C$ and $n_0\in\mathbb N^*$ depending only on $R,\alpha$ and $L$ such that for any $n\geq n_0$, the following holds: there exists a function $f_n \in G^{\alpha,L}(\mathbb T^n \times \overline{B})$ with $\varepsilon_n=|f_n|_{\alpha,L}$ satisfying \[e^{-2(n-2)\ln (4n\ln 2n)}\leq \varepsilon_n \leq c\, e^{-2(n-2)\ln (n\ln 2n)},\] such that the Hamiltonian system $H_n=h+f_n$ has an orbit $(\theta(t),I(t))$ for which the estimates \[ |I(\tau_n)-I_0|\geq 1, \quad \tau_n\leq C\left(\frac{c}{\varepsilon_n}\right)^{n\gamma}, \] hold true. \end{theorem} As we have already explained, this statement gives an upper bound on the threshold of applicability of Nekhoroshev's estimates, which is an important issue when trying to use abstract stability results for ``realistic" problems, for instance for the so-called planetary problem (see \cite{Nie96}). So let us consider the set of Gevrey quasi-convex integrable Hamiltonians $\mathcal{H}=\mathcal{H}(n,R,\alpha,L,M,m)$ defined as follows: $h\in\mathcal{H}$ if $h\in G^{\alpha,L}(\overline{B})$ and satisfies both \[ \forall I\in B, \quad |\partial ^k h(I)|\leq M, \quad 1\leq |k_1|+\cdots+|k_n|\leq 3, \] and \[ \forall I\in B, \forall v\in\mathbb R^n, \quad \nabla h(I).v=0 \Longrightarrow \nabla^2 h(I)v.v \geq m|v|^2.\] From Nekhoroshev's theorem (see \cite{MS02} for a statement in Gevrey classes), we know that there exists a positive constant $\varepsilon_0(\mathcal{H})=\varepsilon_0(n,R,\alpha,L,M,m)$ such that the following holds: for any $h\in\mathcal{H}$, there exist positive constants $c_1,c_2,c_3,a$ and $b$ such that if \[ f\in G^{\alpha,L}(\mathbb T^n \times \overline{B}), \quad |f|_{\alpha,L}<\varepsilon_0(\mathcal{H}),\] then any solution $(\theta(t),I(t))$ of the system $H=h+f$, with $I(0)\in B_{R/2}$, satisfies \[ |I(t)-I_0| \leq c_1\varepsilon^b, \quad |t|\leq c_2\exp(c_3\varepsilon^{-a}). \] Then we can state the following corollary of our Theorem~\ref{thmnonpert}. \begin{corollary} With the previous notations, one has the upper bound \[ \varepsilon_0(\mathcal{H})<e^{-2(n-2)\ln (4n\ln 2n)}. \] \end{corollary} This improves the upper bound $\varepsilon_0(\mathcal{H})<e^{-n}$ obtained in \cite{BK05} for Gevrey functions. \subsection{The analytic case} \paraga Let us now state our result in the analytic case. Here $B=B_R$ is still the open ball of $\mathbb R^n$ around the origin, of radius $R>0$ with respect to the supremum norm, and we will also consider a Hamiltonian system of the form \[ H(\theta,I)=h(I)+f(\theta,I), \quad (\theta,I)\in \mathbb T^n \times B, \] where \[ h(I)=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+I_n, \quad I=(I_1,\dots,I_n)\in B. \] Given $\rho>0$, let us introduce the space $\mathcal{A}_\rho(\mathbb T^n \times B)$ of bounded real-analytic functions on $\mathbb T^n \times B$ admitting a bounded holomorphic extension to the complex neighbourhood \[ V_\rho=V_\rho(\mathbb T^n \times B)=\{(\theta,I)\in(\mathbb C^n/\mathbb Z^n)\times \mathbb C^{n} \; | \; |\mathcal{I}(\theta)|<\rho,\;d(I,B)< \rho\}, \] where $\mathcal{I}(\theta)$ is the imaginary part of $\theta$ and the distance $d$ is associated to the supremum norm on $\mathbb C^n$. Such a space $\mathcal{A}_\rho(\mathbb T^n \times B)$ is obviously a Banach space with the norm \[ |f|_\rho=|f|_{C^0(V_\rho)}=\sup_{z\in V_\rho}|f(z)|, \quad f\in\mathcal{A}_\rho(\mathbb T^n \times B). \] Furthermore, for bounded real-analytic vector-valued functions defined on $\mathbb T^n \times B$ admitting a bounded holomorphic extension to $V_\rho$, we shall extend this norm componentwise (in particular, this applies to Hamiltonian vector fields and their time-one maps). \paraga Now we can state our theorem. \begin{theorem}\label{thmnonpertana} Let $n\geq 4$, $R>1$, and $\sigma>0$. Then there exist positive constants $\rho,\gamma,C$ and $n_0\in\mathbb N^*$ depending only on $R$ and $\sigma$, and a constant $c_n$ that may also depends on $n$, such that for any $n\geq n_0$, the following holds: there exists a function $f_n \in \mathcal{A}_\rho(\mathbb T^n \times B)$ with $\varepsilon_n=|f_n|_{\rho}$ satisfying \[e^{-2(n-3)\ln (4n\ln 2n)}\leq \varepsilon_n \leq c_n\, e^{-2(n-3)\ln (n\ln 2n)},\] such that the Hamiltonian system $H_n=h+f_n$ has an orbit $(\theta(t),I(t))$ for which the estimates \[ |I(\tau_n)-I_0|\geq 1, \quad \tau_n\leq C\left(\frac{c_n}{\varepsilon_n}\right)^{n\gamma}, \] hold true. \end{theorem} In the above statement, the constant $\sigma$ has to be chosen sufficiently small but independently of the choice of $n$ and $R$ (see Proposition~\ref{LocMar}). Moreover, the theorem is slightly different than the one in the Gevrey case, since there is a constant $c_n$ depending also on $n$: this comes from the use of suspension arguments due to Kuksin and Pöschel (\cite{Kuk93}, \cite{KP94}) in the analytic case, which are more difficult than in the Gevrey case. Here we can also define a threshold of validity in the Nekhoroshev theorem $\varepsilon_0(\mathcal{H})=\varepsilon_0(n,R,\rho,M,m)$ and state the following corollary of our Theorem~\ref{thmnonpertana}. \begin{corollary} With the previous notations, one has the upper bound \[ \varepsilon_0(\mathcal{H})<e^{-2(n-3)\ln (4n\ln 2n)}. \] \end{corollary} This improves the upper bound $\varepsilon_0(\mathcal{H})<e^{-n}$ obtained in \cite{BK05} for analytic functions. For concrete Hamiltonians like in the planetary problem, the actual distance to the integrable system is essentially of order $10^{-3}$, hence the above corollary yields the impossibility to apply Nekhoroshev's estimates for $n>3$. \subsection{Some remarks} \paraga Theorem~\ref{thmnonpert} and Theorem~\ref{thmnonpertana} are obtained from the constructions in \cite{MS02} and \cite{LM05}, but one has to choose properly the dependence with respect to $n$ of the various parameters involved. As the reader will see, we will use only rough estimates leading to the factor $n$ in the time of instability: this can be easily improved but we do not know if it is possible in our case (that is with a perturbation of size $e^{-n\ln (n\ln n)}$) to obtain a linear time of drift. Let us note also we have restricted the perturbation to a compact subset of $\mathbb A^n=\mathbb T^n \times \mathbb R^n$ just in order to evaluate Gevrey or analytic norms. In fact, in both theorems the Hamiltonian vector field generated by $H_n=h+f_n$ is complete and the unstable solution $(\theta(t),I(t))$ satisfies \[ \lim_{t\rightarrow \pm \infty}|I(t)-I_0|=+\infty, \] which means that it is bi-asymptotic to infinity. \paraga Ii is important to note that our approach leads, as in the first part of \cite{BK05}, to results for an autonomous perturbation of a quasi-convex integrable system (or, equivalently, for a time-dependent time-periodic perturbation of a convex integrable system). In the second part of \cite{BK05}, for a class of convex integrable systems, Kaloshin was able to reduce the case of an autonomous perturbation to the case of a time-dependent time periodic perturbation, partly because of his more general (but more involved) approach. We could have tried to apply his general arguments to our case, but for simplicity we decided not to pursue this further. \paraga In this text, we will have to deal with time-one maps associated to Hamiltonian flows. So given a function $H$, we will denote by $\Phi_t^H$ the time-$t$ map of its Hamiltonian flow and by $\Phi^H=\Phi^H_1$ the time-one map. We shall use the same notation for time-dependent functions $H$, that is $\Phi^H$ will be the time-one map of the Hamiltonian isotopy (the flow between $t=0$ and $t=1$) generated by $H$. \section{Proof of Theorem~\ref{thmnonpert}} The proof of Theorem~\ref{thmnonpert} is contained in section~\ref{sectnonpert}, but first in section~\ref{mechanism} we recall the mechanism of instability presented in the paper \cite{MS02} (see also \cite{MS04}). This mechanism has two main features. The first one is that it deals with perturbation of integrable maps rather than perturbation of integrable flows, and then the latter is recovered by a suspension process. This point of view, which is only of technical matter, was already used for example in \cite{Dou88} and offers more flexibility in the construction. The second feature, which is the most important one, is that instead of trying to detect instability in a map close to integrable by means of the usual splitting estimates, we will start with a map having already ``unstable" orbits and try to embed it in a near-integrable map. This will be realized through a ``coupling lemma", which is really the heart of the mechanism. As we will see, the construction offers an easy and very efficient way of computing the drifting time of unstable solutions, therefore avoiding all the technicalities that are usually required for such a task. \subsection{The mechanism} \label{mechanism} \paraga Given a potential function $U : \mathbb T \rightarrow \mathbb R$, we consider the following family of maps $\psi_q : \mathbb A \rightarrow \mathbb A$ defined by \begin{equation}\label{maps1} \psi_q(\theta,I)=\left(\theta +qI, I-q^{-1}U'(\theta + qI)\right), \quad (\theta,I)\in\mathbb A, \end{equation} for $q \in \mathbb N^*$. If we require $U'(0)=-1$, for example if we choose \[ U(\theta)=-(2\pi)^{-1}\sin (2\pi\theta),\quad U'(\theta)=-\cos(2\pi\theta),\] then it is easy to see that $\psi_q(0,0)=(0,q^{-1})$ and by induction \begin{equation}\label{driftstand} \psi_q^k(0,0)=(0,kq^{-1}) \end{equation} for any $k \in \mathbb Z$ (see figure~\ref{drift}). After $q$ iterations, the point $(0,0)$ drifts from the circle $I=0$ to the circle $I=1$ and it is bi-asymptotic to infinity, in the sense that the sequence $\left(\psi_q^k(0,0)\right)_{k \in \mathbb Z}$ is not contained in any semi-infinite annulus of $\mathbb A$. \begin{figure} \caption{Drifting point for the map $\psi_q$} \label{drift} \end{figure} Clearly these maps are exact-symplectic, but obviously they have no invariant circles and so they cannot be ``close to integrable". However, we will use the fact that they can be written as a composition of time-one maps, \begin{equation}\label{driftstand2} \psi_q=\Phi^{q^{-1}U} \circ \left(\Phi^{\frac{1}{2}I^2} \circ \cdots \circ \Phi^{\frac{1}{2}I^2}\right) =\Phi^{q^{-1}U} \circ \left(\Phi^{\frac{1}{2}I^2}\right)^q, \end{equation} to embed $\psi_q$ in the $q^{th}$-iterate of a near-integrable map of $\mathbb A^n$, for $n \geq 2$. To do so, we will use the following ``coupling lemma", which is easy but very clever. \begin{lemma}[Herman-Marco-Sauzin] \label{coupling} Let $m,m' \geq 1$, $F : \mathbb A^m \rightarrow \mathbb A^m$ and $G : \mathbb A^{m'} \rightarrow \mathbb A^{m'}$ two maps, and $f : \mathbb A^m \rightarrow \mathbb R$ and $g : \mathbb A^{m'} \rightarrow \mathbb R$ two Hamiltonian functions generating complete vector fields. Suppose there is a point $a \in \mathbb A^{m'}$ which is $q$-periodic for $G$ and such that the following ``synchronisation" conditions hold: \begin{equation}\label{sync} g(a)=1, \quad dg(a)=0, \quad g(G^k(a))=0, \quad dg(G^k(a))=0, \tag{S} \end{equation} for $1 \leq k \leq q-1$. Then the mapping \[ \Psi=\Phi^{f \otimes g} \circ (F \times G) : \mathbb A^{m+m'} \longrightarrow \mathbb A^{m+m'} \] is well-defined and for all $x \in \mathbb A^m$, \[ \Psi^q(x,a)=(\Phi^f \circ F^q(x),a). \] \end{lemma} The product of functions acting on separate variables was denoted by $\otimes$, \textit{i.e.} \[ f \otimes g(x,x')=f(x)g(x') \quad x \in \mathbb A^m , \; x' \in \mathbb A^{m'}. \] Let us give an elementary proof of this lemma since it is a crucial ingredient. \begin{proof} First note that since the Hamiltonian vector fields $X_f$ and $X_g$ are complete, an easy calculation shows that for all $x\in\mathbb A^m$, $x'\in\mathbb A^{m'}$ and $t\in \mathbb R$, one has \begin{equation}\label{coupl} \Phi_{t}^{f \otimes g}(x,x')=\left(\Phi_{t}^{g(x')f}(x),\Phi_{t}^{f(x)g}(x')\right) \end{equation} and therefore $X_{f \otimes g}$ is also complete. Using the above formula and condition~(\ref{sync}), the points $(F^{k}(x),G^{k}(a))$, for $1\leq k \leq q-1$, are fixed by $\Phi^{f \otimes g}$ and hence \[ \Psi^{q-1}(x,a)=(F^{q-1}(x),G^{q-1}(a)). \] Since $a$ is $q$-periodic for $G$ this gives \[ \Psi^{q}(x,a)=\Phi^{f \otimes g}(F^{q}(x),a), \] and we end up with \[ \Psi^{q}(x,a)=(\Phi^{f}(F^{q}(x)),a) \] using once again~(\ref{sync}) and~(\ref{coupl}). \end{proof} Therefore, if we set $m=1$, $F=\Phi^{\frac{1}{2}I_1^2}$ and $f=q^{-1}U$ in the coupling lemma, the $q^{th}$-iterate $\Psi^q$ will leave the submanifold $\mathbb A \times \{a\}$ invariant, and its restriction to this annulus will coincide with our ``unstable map" $\psi_q$. Hence, after $q^2$ iterations of $\Psi$, the $I_1$-component of the point $((0,0),a) \in \mathbb A^2$ will move from $0$ to $1$. \paraga The difficult part is then to find what kind of dynamics we can put on the second factor to apply this coupling lemma. In order to have a continuous system with $n$ degrees of freedom at the end, we may already choose $m'=n-2$ so the coupling lemma will give us a discrete system with $n-1$ degrees of freedom. First, a natural attempt would be to try \[ G=G_n=\Phi^{\frac{1}{2}I_2^2+\cdots+\frac{1}{2}I_{n-1}^2}.\] Indeed, in this case \[ F \times G_n=\Phi^{\frac{1}{2}I_2^2+\cdots+\frac{1}{2}I_{n-1}^2}=\Phi^{\tilde{h}} \] where \[ \tilde{h}(I_1,\dots,I_{n-1})=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2) \] and the unstable map $\Psi$ given by the coupling lemma appears as a perturbation of the form $\Psi=\Phi^u \circ \Phi^{\tilde{h}}$, with $u=f\otimes g$. However, this cannot work. Indeed, for $j\in\{2,\dots,n-1\}$, one can choose a $p_j$-periodic point $a^{(j)}\in\mathbb A$ for the map $\Phi^{\frac{1}{2}I_j^2}$, and then setting \[ a_n=(a^{(2)},\dots,a^{(n-1)})\in\mathbb A^{n-2}, \quad q_n=p_2 \cdots p_{n-1},\] the point $a_n$ is $q_n$-periodic for $G_n$ provided that the numbers $p_j$ are mutually prime. One can easily see that the latter condition will force the product $q_n$ to converge to infinity when $n$ goes to infinity. So necessarily the point $a_n$ gets arbitrarily close to its first iterate $G_n(a_n)$ when $n$ (and therefore $q_n$) is large: this is because $q_n$-periodic points for $G_n$ are equi-distributed on $q_n$-periodic tori. As a consequence, a function $g_n$ with the property \[ g_n(a_n)=1, \quad g_n(G_n(a_n))=0, \] will necessarily have very large derivatives at $a_n$ if $q_n$ is large. Then as the size of the perturbation is essentially given by \[ |f\otimes g_n|=|q_{n}^{-1}U\otimes g_n|=|q_{n}^{-1}||g_n|,\] one can check that it is impossible to make this quantity converge to zero when the number of degrees of freedom $n$ goes to infinity. \paraga As in \cite{MS02}, the idea to overcome this problem is the following one. We introduce a new sequence of ``large" parameters $N_n \in \mathbb N^*$ and in the second factor we consider a family of suitably rescaled penduli on $\mathbb A$ given by \[ P_n(\theta_2,I_2)=\frac{1}{2} I_2^2 +N_n^{-2}V(\theta_2), \] where $V(\theta)=-\cos 2\pi\theta$. The other factors remain unchanged, so \[ G_n=\Phi^{\frac{1}{2}(I_2^2+I_3^2+\cdots+I_{n-1}^2)+N_{n}^{-2}V(\theta_2)}. \] In this case, the map $\Psi$ given by the coupling lemma is also a perturbation of $\Phi^{\tilde{h}}$ but of the form $\Psi=\Phi^u \circ \Psi^{\tilde{h}+v}$, with $v=N_n^{-2}V$. But now for the map $G_n$, due to the presence of the pendulum factor, it is now possible to find a periodic orbit with an irregular distribution: more precisely, a $q_n$-periodic point $a_n$ such that its distance to the rest of its orbit is of order $N_{n}^{-1}$, no matter how large $q_n$ is. \paraga Let us denote by $(p_j)_{j\geq 0}$ the ordered sequence of prime numbers and let us choose $N_n$ as the product of the $n-2$ prime numbers $\{p_{n+3},\dots,p_{2n}\}$, that is \[ N_n=p_{n+3}p_{n+4}\cdots p_{2n}\in\mathbb N^*.\] Our goal is to prove the following proposition. \begin{proposition}\label{pertu} Let $n\geq 3$, $\alpha>1$ and $L_1>0$. Then there exist a function $g_n\in G^{\alpha,L_1}(\mathbb A^{n-2})$, a point $a_n\in\mathbb A^{n-2}$ and positive constants $c_1$ and $c_2$ depending only on $\alpha$ and $L_1$ such that if \[ M_n=2\left[c_1N_ne ^{c_2(n-2)p_{2n}^{\frac{1}{\alpha}}}\right], \quad q_n=N_nM_n, \] then $a_n$ is $q_n$-periodic for $G_n$ and $(g_n, G_n, a_n, q_n)$ satisfy the synchronization conditions (\ref{sync}): \begin{equation*} g_n(a_n)=1, \quad dg_n(a_n)=0, \quad g_n(G_n^k(a_n))=0, \quad dg_n(G_n^k(a_n))=0, \end{equation*} for $1 \leq k \leq q_n-1$. Moreover, the estimate \begin{equation}\label{estimgn} q_n^{-1}|g_n|_{\alpha,L_1}\leq N_{n}^{-2}, \end{equation} holds true. \end{proposition} The rest of this section is devoted to the proof of the above proposition. Note that together with the coupling lemma (Lemma~\ref{coupling}), this proposition easily gives a result of instability (analogous to Proposition 2.1 in \cite{MS02}) for a discrete system which is a perturbation of the map $\Phi^{\tilde{h}}$, but we prefer not to state such a result in order to focus on the continuous case. \paraga We first consider the simple pendulum \[ P(\theta,I)=\frac{1}{2} I^2 + V(\theta), \quad (\theta,I)\in\mathbb A. \] With our convention, the stable equilibrium is at $(0,0)$ and the unstable one is at $(0,1/2)$. Given any $M\in\mathbb N^*$, there is a unique point $b^M=(0,I_M)$ which is $M$-periodic for $\Phi^P$ (this is just the intersection between the vertical line $\{0\} \times \mathbb R$ and the closed orbit for the pendulum of period $M$). One can check that $I_M \in\, ]2,3[$ and as $M$ goes to infinity, $(0,I_M)$ tends to the point $(0,2)$ which belongs to the upper separatrix. Since $P_n(\theta,I)=\frac{1}{2} I^2 +N_n^{-2}V(\theta)$, then one can see that \[ \Phi^{P_{n}}=(S_n)^{-1} \circ \Phi^{N_{n}^{-1}P} \circ S_{n}, \] where $S_{n}(\theta,I)=(\theta,N_n I)$ is the rescaling by $N_n$ in the action components. Therefore the point $b_n^M=(0,N_{n}^{-1}I_{M})$ is $q_n$-periodic for $\Phi^{P_{n}}$, for $q_n=N_nM$. Let $(\Phi_t^{P})_{t \in \mathbb R}$ be the flow of the pendulum, and \[ \Phi_t^{P}(0,I_{M})=(\theta_{M}(t),I_{M}(t)). \] The function $\theta_{M}(t)$ is analytic. The crucial observation is the following simple property of the pendulum (see Lemma 2.2 in \cite{MS02} for a proof). \begin{figure} \caption{The point $b^M$ and its iterates} \label{bN} \end{figure} \begin{lemma} Let $\sigma=-\frac{1}{2} +\frac{2}{\pi}\arctan e^{\pi} < \frac{1}{2}$. For any $M \in \mathbb N^*$, \[ \theta_{M}(t) \notin [-\sigma,\sigma], \] for $t\in[1/2,M-1/2]$. \end{lemma} Hence no matter how large $M$ is, most of the points of the orbit of $b^M\in\mathbb A$ will be outside the set $\{-\sigma \leq \theta \leq \sigma\}\times \mathbb R$ (see figure~\ref{bN}). The construction of a function that vanishes, as well as its first derivative, at these points, will be easily arranged by means of a function, depending only on the angle variables, with support in $\{-\sigma \leq \theta \leq \sigma\}$. As for the other points, it is convenient to introduce the function \[ \tau_{M} : [-\sigma,\sigma] \longrightarrow \, ]-1/2,1/2[ \] which is the analytic inverse of $\theta_{M}$. One can give an explicit formula for this map: \[ \tau_{M}(\theta)=\int_{0}^{\theta}\frac{d\varphi}{\sqrt{I_M^2-4\sin^2 \pi\varphi}}. \] In particular, it is analytic and therefore it belongs to $G^{\alpha,L_1}([-\sigma,\sigma])$ for $\alpha\geq 1$ and $L_1>0$, and one can obtain the following estimate (see Lemma 2.3 in \cite{MS02} for a proof). \begin{lemma}\label{lemmelambda} For $\alpha>1$ and $L_1>0$, \[ \Lambda=\sup_{M\in\mathbb N^*}|\tau_M|_{\alpha,L_1}<+\infty. \] \end{lemma} Note that $\Lambda$ depends only on $\alpha$ and $L_1$. Under the action of $\tau_{M}$, the points of the orbit of $b^M$ whose projection onto $\mathbb T$ belongs to $\{-\sigma \leq \theta \leq \sigma\}$ get equi-distributed, and we can use the following elementary lemma. \begin{lemma}\label{funct} For $p \in \mathbb N^*$, the analytic function $\eta_p : \mathbb T \rightarrow \mathbb R$ defined by \[ \eta_p(\theta)=\left(\frac{1}{p}\sum_{l=0}^{p-1}\cos 2\pi l\theta \right)^2 \] satisfies \[ \eta_p(0)=1, \quad \eta_p'(0)=0, \quad \eta_p(k/p)=\eta_p'(k/p)=0, \] for $1 \leq k \leq p-1$, and \[ |\eta_p|_{\alpha,L_1}\leq e^{2\alpha L_1(2\pi p)^{\frac{1}{\alpha}}}. \] \end{lemma} The proof is trivial (see \cite{MS02}, Lemma 2.4). \paraga We can now pass to the proof of Proposition~\ref{pertu}. \begin{proof}[Proof of Proposition~\ref{pertu}] For $\alpha>1$ and $L_1>0$, consider the bump function $\varphi_{\alpha,L_1}\in G^{\alpha,L_1}(\mathbb T)$ given by Lemma~\ref{lemmeGev1} (see Appendix~\ref{Gev}). We choose our function $g_n\in G^{\alpha,L_1}(\mathbb A^{n-2})$, depending only on the angle variables, of the form \[ g_n=g_n^{(2)}\otimes \cdots \otimes g_n^{(n-1)}, \] where \[ g_n^{(2)}(\theta_2)=\eta_{p_{n+3}}(\tau_{M_n}(\theta_2))\varphi_{\alpha,(4\sigma)^{-\frac{1}{\alpha}}L_1}((4\sigma)^{-1}\theta_2), \] and \[ g_n^{(i)}(\theta_i)=\eta_{p_{n+1+i}}(\theta_i), \quad 3\leq i \leq n-1. \] Let us write \[ c_1=\left|\varphi_{\alpha,(4\sigma)^{-\frac{1}{\alpha}}L_1}\right|_{\alpha,(4\sigma)^{-\frac{1}{\alpha}}L_1}. \] Now we choose our point $a_n=(a_n^{(2)},\dots,a_n^{(n-1)})\in\mathbb A^{n-2}$. We set \[ a_n^{(2)}=b_n^{M_n}=(0,N_{n}^{-1}I_{M_n}), \] and \[ a_n^{(i)}=(0,p_{n+1+i}^{-1}), \quad 3\leq i \leq n-1. \] Let us prove that $a_n$ is $q_n$-periodic for $G_n$. We can write \[ G_n=\Phi^{\frac{1}{2}I_2^2+N_{n}^{-2}V(\theta_2)}\times \Phi^{\frac{1}{2}(I_3^2+I_4^2+\cdots+I_{n-1}^2)}=\Phi^{P_n}\times \widehat{G}. \] Since $p_{n+4}, \dots, p_{2n}$ are mutually prime, the point $(a_n^{(3)},\dots,a_n^{(n-1)})\in\mathbb A^{n-3}$ is periodic for $\widehat{G}$, with period \[N_n'=p_{n+4} \cdots p_{2n}.\] By construction, the point $a_n^{(2)}=b_n^{M_n}\in\mathbb A$ is periodic for $\Phi^{P_n}$, with period $q_n=N_nM_n$, where \[ N_n=p_{n+3}p_{n+4}\cdots p_{2n}.\] This means that $a_n$ is periodic for the product map $G_n$, and the exact period is given by the least common multiple of $q_n$ and $N_n'$. Since $N_n'$ divides $q_n$, the period of $a_n$ is $q_n$. Now let us show that the synchronization conditions~(\ref{sync}) hold true, that is \[ g_n(a_n)=1, \quad dg_n(a_n)=0, \quad g_n(G_n^k(a_n))=0, \quad dg_n(G_n^k(a_n))=0, \] for $1\leq k\leq q_n-1$. Since $\varphi_{\alpha,L_1}(0)=1$, then \[ g_n(a_n)=g_n^{(2)}(0)\cdots g_n^{(n-1)}(0)=1 \] and as $\varphi_{\alpha,L_1}'(0)=0$, then \[ dg_n(a_n)=0. \] To prove the other conditions, let us write $G_n^k(a_n)=(\theta_k,I_k)\in\mathbb A^{n-2}$, for $1\leq k\leq q_n-1$. If $\theta_k^{(2)}$ does not belong to $]-\sigma,\sigma[$, then $g_n^{(2)}$ and its first derivative vanish at $\theta_k^{(2)}$ because it is the case for $\varphi_{\alpha,(4\sigma)^{-\frac{1}{\alpha}}L}$, so \[ g_n(\theta_k)=dg_n(\theta_k)=0. \] Otherwise, if $-\sigma<\theta_k^{(2)}<\sigma$, one can easily check that \[ - \frac{N_n-1}{2}\leq k \leq \frac{N_n-1}{2} \] and therefore \[ \tau_{M_n}(\theta_k^{(2)})=\frac{k}{N_n}, \] while \[ \theta_k^{(i)}=\frac{k}{p_{n+i+1}}, \quad 3\leq i \leq n-1. \] If $N_n'=p_{n+4} \cdots p_{2n}$ divides $k$, that is $k=k'N_n'$ for some $k'\in\mathbb Z$, then \[ \tau_{M_n}(\theta_k^{(2)})=\frac{k}{N_n}=\frac{k'}{p_{n+3}} \] and therefore, by Lemma~\ref{funct}, $\eta_{p_{n+3}}$ vanishes with its differential at $\theta_k^{(2)}$, and so does $g_n^{(2)}$. Otherwise, $N_n'$ does not divide $k$ and then, for $3\leq i \leq n-1$, at least one of the functions $\eta_{p_{n+1+i}}$ vanishes with its differential at $\theta_k^{(2)}$, and so does $g_n^{(i)}$. Hence in any case \[ g_n(\theta_k)=dg_n(\theta_k)=0, \quad 1\leq k\leq q_n-1, \] and the synchronization conditions~(\ref{sync}) are satisfied. Now it remains to estimate the norm of the function $g_n$. First, using Lemma~\ref{lemmeGev2}, one finds \[ |g_n|_{\alpha,L_1} \leq \left|\varphi_{\alpha,(4\sigma)^{-\frac{1}{\alpha}}L_1}\right|_{\alpha,(4\sigma)^{-\frac{1}{\alpha}}L_1} |\eta_{p_{n+3}}\circ\tau_{M_n}|_{\alpha,L_1}|\eta_{p_{n+4}}|_{\alpha,L_1}|\eta_{p_{2n}}|_{\alpha,L_1}, \] which by definition of $c_1$ gives \[ |g_n|_{\alpha,L_1} \leq c_1|\eta_{p_{n+3}}\circ\tau_{M_n}|_{\alpha,L_1}|\eta_{p_{n+4}}|_{\alpha,L_1}|\eta_{p_{2n}}|_{\alpha,L_1}. \] Then, by definition of $\Lambda$ (Lemma~\ref{lemmelambda}) and using Lemma~\ref{lemmeGev3} (with $\Lambda_1=\Lambda^{\frac{1}{\alpha}}$), \[ |\eta_{p_{n+3}}\circ\tau_{M_n}|_{\alpha,L_1} \leq |\eta_{p_{n+3}}|_{\alpha,\Lambda^{\frac{1}{\alpha}}}, \] so using Lemma~\ref{funct} and setting $c_2=2\alpha\sup\left\{\Lambda^{\frac{1}{\alpha}},L_1\right\}(2\pi)^{\frac{1}{\alpha}}$, this gives \begin{eqnarray*} |g_n|_{\alpha,L_1} & \leq & c_1 e^{2\alpha(\Lambda^{\frac{1}{\alpha}}+(n-3)L_1)(2\pi p_{2n})^{\frac{1}{\alpha}}} \\ & \leq & c_1 e^{2\alpha\sup\left\{\Lambda^{\frac{1}{\alpha}},L_1\right\}(n-2)(2\pi p_{2n})^{\frac{1}{\alpha}}} \\ & \leq & c_1 e^{c_2(n-2)p_{2n}^{\frac{1}{\alpha}}}. \end{eqnarray*} Finally, by definition of $M_n$ we obtain \begin{equation*} |g_n|_{\alpha,L_1}\leq M_nN_n^{-1}, \end{equation*} and as $q_n=N_nM_n$, we end up with \[ q_n^{-1}|g_n|_{\alpha,L_1}\leq N_n^{-2}. \] This concludes the proof. \end{proof} \subsection{Proof of Theorem~\ref{thmnonpert}} \label{sectnonpert} \paraga In the previous section, we were concerned with a perturbation of the integrable diffeomorphism $\Phi^{\tilde{h}}$, which can be written as $\Phi^u \circ \Phi^{\tilde{h}+v}$. So now we will briefly describe a suspension argument to go from this discrete case to a continuous case (we refer once again to \cite{MS02} for the details). Here we will make use of bump functions, however the process is still valid, though more difficult, in the analytic category, (see for example \cite{Dou88} or \cite{KP94}). The basic idea is to find a time-dependent Hamiltonian function on $\mathbb A^n$ such that the time-one map of its isotopy is $\Phi^u \circ \Phi^{\tilde{h}+v}$, or, equivalently, an autonomous Hamiltonian function on $\mathbb A^{n+1}$ such that its first return map to some $2n$-dimensional Poincaré section coincides with our map $\Phi^u \circ \Phi^{\tilde{h}+v}$. Given $\alpha>1$ and $L>1$, let us define the function \[ \phi_{\alpha,L}=\left(\int_{\mathbb T}\varphi_{\alpha,L}\right)^{-1}\varphi_{\alpha,L}, \] where $\varphi_{\alpha,L}$ is the bump function given by Lemma~\ref{lemmeGev1}. If $\phi_0(t)=\phi_{\alpha,L}\big(t-\frac{1}{4}\big)$ and $\phi_1(t)=\phi_{\alpha,L}\big(t-\frac{3}{4}\big)$, the time-dependent Hamiltonian \[ H^*(\theta,I,t)=(\tilde{h}(I)+v(\theta)) \otimes \phi_0(t) + u(\theta) \otimes \phi_1(t) \] clearly satisfies \[ \Phi^{H^*}=\Phi^u \circ \Phi^{\tilde{h}+v}. \] But as $u$ and $v$ go to zero, $H^*$ converges to $\tilde{h} \otimes \phi_0$ rather than $\tilde{h}$. However, using classical generating functions, it is not difficult to modify the Hamiltonian in order to prove the following proposition (see Lemma 2.5 in \cite{MS02}). \begin{proposition}[Marco-Sauzin]\label{sus} Let $n\geq 1$, $R>1$, $\alpha>1$, $L_1>0$ and $L>0$ satisfying \begin{equation}\label{LL1} L_1^\alpha=L^\alpha(1+(L^\alpha+R+1/2)|\phi_{\alpha,L}|_{\alpha,L}). \end{equation} If $u_n,v_n\in G^{\alpha,L_1}(\mathbb T^{n-1})$, there exists $f_n\in G^{\alpha,L}(\mathbb T^n\times \overline{B})$, independent of the variable $I_n$, such that if \[ H_n(\theta,I)=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+I_n+f_n(\theta,I), \quad (\theta,I)\in\mathbb A^n, \] for any energy $e\in\mathbb R$, the Poincaré map induced by the Hamiltonian flow of $H_n$ on the section $\{\theta_n=0\}\cap H_{n}^{-1}(e)$ coincides with the diffeomorphism \[ \Phi^{u_n}\circ\Phi^{\tilde{h}+v_n}.\] Moreover, one has \begin{equation}\label{taillesus} \sup\{|u_n|_{C^0},|v_n|_{C^0}\}\leq |f_n|_{\alpha,L} \leq c_3\sup\{|u_n|_{\alpha,L_1},|v_n|_{\alpha,L_1}\}, \end{equation} where $c_3=2|\phi_{\alpha,L}|_{\alpha,L}$ depends only on $\alpha$ and $L$. \end{proposition} \paraga Now we can finally prove our theorem. \begin{proof}[Proof of Theorem~\ref{thmnonpert}] Let $R>1$, $\alpha>1$ and $L>0$, and choose $L_1$ satisfying the relation~(\ref{LL1}). The constants $c_1$ and $c_2$ of Proposition~\ref{pertu} depend only on $\alpha$ and $L_1$, hence they depend only on $R$, $\alpha$ and $L$. We can define $u_n, v_n\in G^{\alpha,L_1}(\mathbb T^{n-1})$ by \[ u_n=q_{n}^{-1}U\otimes g_n,\quad v_n=N_{n}^{-2}V, \] where $U(\theta_1)=-(2\pi)^{-1}\sin 2\pi\theta_1$, $V(\theta_2)=-\cos 2\pi\theta_2$ (so $v_n$ is formally defined on $\mathbb T$ but we identify it with a function on $\mathbb T^{n-1}$) and $g_n$ is the function given by Proposition~\ref{pertu}. Let us apply Proposition~\ref{sus}: there exists $f_n\in G^{\alpha,L}(\mathbb T^n\times \overline{B})$, independent of the variable $I_n$, such that if \[ H_n(\theta,I)=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+I_n+f_n(\theta,I), \quad (\theta,I)\in\mathbb A^n, \] for any energy $e\in\mathbb R$, the Poincaré map induced by the Hamiltonian flow of $H$ on the section $\{\theta_n=0\}\cap H^{-1}(e)$ coincides with the diffeomorphism \[ \Phi^{u_n}\circ\Phi^{\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+v_n}=\Phi^{u_n}\circ\Phi^{\tilde{h}+v_n}.\] Let us show that our system $H_n$ has a drifting orbit. First consider its Poincaré section defined by \[ \Psi_n=\Phi^{u_n}\circ\Phi^{\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+v_n}=\Phi^{f_n \otimes g_n} \circ (F \times G_n),\] with \[ f_n=q_n^{-1}U, \quad F=\Phi^{\frac{1}{2} I_1^2}, \quad G_n=\Phi^{\frac{1}{2}(I_2^2+I_3^2+\cdots+I_{n-1}^2)+N_{n}^{-2}V(\theta_2)}. \] By Proposition~\ref{pertu}, we can apply the coupling lemma (Lemma~\ref{coupling}), so \[ \Psi_n^{q_n}((0,0),a_n)=(\Phi^f_n \circ F^{q_n}(0,0),a_n). \] Then, using~(\ref{driftstand2}), observe that \[ \Phi^{f_n} \circ F^{q_n}=\Phi^{q_n^{-1}U} \circ \left(\Phi^{\frac{1}{2}I_1^2}\right)^{q_n} =\psi_{q_n}, \] so \begin{eqnarray*} \Psi_n^{q_n^2}((0,0),a_n) & = & ((\Phi^{f_n} \circ F^{q_n})^{q_n}(0,0),a_n) \\ & = & (\psi_{q_n}^{q_n}(0,0),a_n) \\ & = & ((0,1),a_n), \end{eqnarray*} where the last equality follows from~(\ref{driftstand}). Hence, after $q_n^2$ iterations, the $I_1$-component of the point $x_n=((0,0),a_n)\in\mathbb A^{n-1}$ drifts from $0$ to $1$. Then, for the continuous system, the initial condition $(x_n,t=0,I_n=0)$ in $\mathbb A^n$ gives rise to a solution $(x(t),t,I_n(t))=(x(t),\theta_n(t),I_n(t))$ of the Hamiltonian vector field generated by $H_n$ such that \[ x(k)=\Psi_n^k(x_n), \quad k\in\mathbb Z. \] So after a time $\tau_n=q_n^2$, the point $(x_n,(0,0))$ drifts from $0$ to $1$ in the $I_1$-direction, and this gives our drifting orbit. Now let $\varepsilon_n=|f_n|_{\alpha,L_1}$ be the size of our perturbation. Using the estimate~(\ref{estimgn}) and~(\ref{taillesus}) one finds \begin{equation}\label{estNeps} N_{n}^{-2}\leq \varepsilon_n \leq c_3 N_{n}^{-2}. \end{equation} By the prime number theorem, $p_n$ is equivalent to $n\ln n$, so there exists $n_0\in\mathbb N^*$ such that for $n\geq n_0$, one can ensure that \[ p_{2n}/4 \leq p_{n+i} \leq p_{2n}, \quad 3\leq i\leq n, \] which gives \begin{equation}\label{estimPN} (p_{2n}/4)^{n-2} \leq N_n \leq p_{2n}^{n-2}, \quad N_n^{\frac{1}{n-2}}\leq p_{2n} \leq 4N_n^{\frac{1}{n-2}}. \end{equation} We can also assume by the prime number theorem that for $n\geq n_0$, one has \begin{equation}\label{nompremier} 2n\ln 2n \leq p_{2n} \leq 2(2n \ln 2n)=4n\ln 2n. \end{equation} From the above estimates~(\ref{estimPN}) and~(\ref{nompremier}) one easily obtains \begin{equation}\label{estN} e^{(n-2)\ln (2^{-1}n\ln 2n)}\leq N_n\leq e^{(n-2)\ln (4n\ln 2n)}, \end{equation} and, together with~(\ref{estNeps}), one finds \begin{equation}\label{estpert} e^{-2(n-2)\ln (4n\ln 2n)}\leq \varepsilon_n \leq c_3e^{-2(n-2)\ln (2^{-1}n\ln 2n)}. \end{equation} Finally it remains to estimate the time $\tau_n$. First recall that \[ M_n=2\left[c_1N_n e^{c_2(n-2)p_{2n}^{\frac{1}{\alpha}}}\right], \] and with~(\ref{estimPN}) \[ q_n=N_nM_n \leq 3c_1N_n^2 e^{4c_2(n-2)N_n^{\frac{1}{\alpha(n-2)}}}. \] Hence \[ q_n^2\leq 9c_1^2N_n^4 e^{8c_2(n-2)N_n^{\frac{1}{\alpha(n-2)}}}. \] Then using~(\ref{estN}) we have \[ N_n^{\frac{1}{\alpha(n-2)}}\leq (4n \ln 2n)^{\frac{1}{\alpha}} \] and from~(\ref{estNeps}) we know that \[ N_n^4 \leq \left(\frac{c_3}{\varepsilon_n}\right)^2, \] so we obtain \[ q_n^2\leq 9c_1^2\left(\frac{c_3}{\varepsilon_n}\right)^2 e^{8c_2(n-2)(4n \ln 2n)^{\frac{1}{\alpha}}}. \] Now taking $n_0$ larger if necessary, as $\alpha>1$, one can ensure that for $n\geq n_0$, \[ (4n)^{\frac{1}{\alpha}} \leq n, \quad (\ln 2n)^{\frac{1}{\alpha}}\leq \ln (2^{-1}n\ln 2n), \] so \[ 8c_2(n-2)(4n \ln 2n)^{\frac{1}{\alpha}} \leq 8c_2(n-2)n\ln (2^{-1}n\ln 2n). \] Therefore \begin{eqnarray*} q_n^2 & \leq & 9c_1^2\left(\frac{c_3}{\varepsilon_n}\right)^2 e^{8c_2n(n-2)\ln (2^{-1}n\ln 2n)} \\ & \leq & 9c_1^2\left(\frac{c_3}{\varepsilon_n}\right)^2\left(e^{2(n-2)\ln (2^{-1}n\ln 2n))}\right)^{4c_2n}. \end{eqnarray*} Finally by~(\ref{estpert}) we obtain \begin{eqnarray*} q_n^2 & \leq & 9c_1^2\left(\frac{c_3}{\varepsilon_n}\right)^2\left(\frac{c_3}{\varepsilon_n}\right)^{4c_2n} \\ & \leq & C\left(\frac{c}{\varepsilon_n}\right)^{n\gamma} \end{eqnarray*} with $C=9c_1^2$, $c=c_3$ and $\gamma=2+4c_2$. This ends the proof. \end{proof} \section{Proof of Theorem~\ref{thmnonpertana}} The proof of Theorem~\ref{thmnonpertana} will be presented in section~\ref{sectnonpertana}, but first in section~\ref{mechanismana}, following \cite{LM05}, we will explain how the mechanism of instability that we explained in the Gevrey context can be (partly) generalized to an analytic context. Recall that the first feature of the mechanism is to study perturbations of integrable maps and to obtain a result for perturbations of integrable flows by a ``quantitative" suspension argument. In the Gevrey case, this was particularly easy using compactly-supported functions. In the analytic case, this is more difficult but such a result exists, and here we will use a version due to Kuskin and Pöschel (\cite{KP94}). The second and main feature of the mechanism is the use of a coupling lemma, which enables us to embed a low-dimensional map having unstable orbits into a multi-dimensional near-integrable map. In the Gevrey case, we simply used the family of maps $\psi_q : \mathbb A \rightarrow \mathbb A$ defined as in~(\ref{maps1}) and the difficult part was the choice of the coupling, where we made an important use of the existence of compactly-supported functions. We do not know if this approach can be easily extended to the analytic case. However, by a result of Lochak and Marco (\cite{LM05}), one can still follow this path by using instead a suitable family of maps $\mathcal{F}_q : \mathbb A^2 \rightarrow \mathbb A^2$ having a well-controlled unstable orbit. \subsection{The modified mechanism}\label{mechanismana} \paraga So let us describe this family of maps $\mathcal{F}_q : \mathbb A^2 \rightarrow \mathbb A^2$, $q\in\mathbb N^*$. We fix a width of analyticity $\sigma>0$ (to be chosen small enough in Proposition~\ref{LocMar} below). For $q$ large enough, $\mathcal{F}_q$ will appear as a perturbation of the following \textit{a priori} unstable map \[ \mathcal{F}_*=\Phi^{\frac{1}{2} (I_1^2+I_2^2)+\cos 2\pi\theta_1} : \mathbb A^2 \rightarrow \mathbb A^2. \] More precisely, for $q\in\mathbb N^*$, let us define an analytic function $f_q : \mathbb A^2 \rightarrow \mathbb R$, depending only on the angle variables, by \[ f_q(\theta_1,\theta_2)=f_{q}^{(1)}(\theta_1)f^{(2)}(\theta_2), \] where \[ f_{q}^{(1)}(\theta_1)=(\sin\pi\theta_1)^{\nu(q,\sigma)}, \quad f^{(2)}(\theta_2)=-\pi^{-1}(2+\sin 2\pi(\theta_2+6^{-1})). \] We still have to define the exponent $\nu(q,\sigma)$ in the above expression for $f_{q}^{(1)}$. Let us denote by $[\,.\,]$ the integer part of a real number. Given $\sigma>0$, let $q_\sigma$ be the smallest positive integer such that \[ \left[\frac{\ln q_\sigma}{4\pi\sigma}+1\right]=1, \] then we set \[ \nu(q,\sigma)=2\left[\frac{\ln q_\sigma}{4\pi\sigma}+1\right], \quad q\geq q_\sigma. \] In particular, for $q\geq q_\sigma$, $\nu(q,\sigma) \geq 2$ and it is always an even integer, hence $f^{(q)}$ is a well-defined $1$-periodic function. The reasons for the choice of this function $f^{(q)}$ are explained at length in \cite{Mar05} and \cite{LM05}, so we refer to these papers for some motivations. Finally, for $q\geq q_\sigma$, we can define \begin{equation}\label{Fq} \mathcal{F}_q=\Phi^{q^{-1}f_q}\circ\mathcal{F}_* : \mathbb A^2 \rightarrow \mathbb A^2. \end{equation} Let us also define the family of points $(\xi_{q,k})_{k\in\mathbb Z}$ of $\mathbb A^2$ by their coordinates \[ \xi_{q,k} : (\theta_1=1/2, I_1=2, \theta_2=0, I_2=q^{-1}(k+1)). \] Clearly, the $I_2$-component of the point $\xi_{q,k}$ converges to $\pm \infty$ when $k$ goes to $\pm \infty$, hence the sequence $(\xi_{q,k})_{k\in\mathbb Z}$ is wandering in $\mathbb A^2$. \paraga The following result was proved in \cite{LM05}, Proposition 2.1. \begin{proposition}[Lochak-Marco]\label{LocMar} There exist a width $\sigma>0$, an integer $q_0$ and a constant $0<d<1$ such that for any $q\geq q_0$, the diffeomorphism $\mathcal{F}_q : \mathbb A^2 \rightarrow \mathbb A^2$ has a point $\zeta_q \in \mathbb A^2$ which satisfies \begin{equation*} |\mathcal{F}_{q}^{kq}(\zeta_q)-\xi_{q,k}|\leq d^{\nu(q,\sigma)}. \end{equation*} \end{proposition} As a consequence, the orbit of the point $\zeta_q \in \mathbb A^2$ under the map $\mathcal{F}_{q}$ is also wandering in $\mathbb A^2$. In particular, for $k=0$ and $k=3q$ the above estimate yields \[ |\zeta_q-\xi_{q,0}|\leq d^{\nu(q,\sigma)}, \quad |\mathcal{F}_{q}^{3q^2}(\zeta_q)-\xi_{q,3q}|\leq d^{\nu(q,\sigma)}, \] and as \[ |\xi_{q,3q}-\xi_{q,0}|=3, \] one obtains \begin{equation}\label{timeana} |\mathcal{F}_{q}^{3q^2}(\zeta_q)-\zeta_q| \geq 3-2d^{\nu(q,\sigma)} \geq 1. \end{equation} The proof of the above proposition is rather difficult and it would be too long to explain it. We just mention that crucial ingredients are on the one hand a conjugacy result for normally hyperbolic manifolds (in the spirit of Sternberg) adapted to this analytic and symplectic context, and on the other hand the classical method of correctly aligned windows introduced by Easton. \paraga Now this family of maps $\mathcal{F}_q : \mathbb A^2 \rightarrow \mathbb A^2$ will be used in the coupling lemma. More precisely, recalling the notations of the coupling lemma~\ref{coupling}, in the following we shall take $m=2$, \[ F=F_n=\Phi^{\frac{1}{2} (I_1^2+I_2^2)+N_n^{-2}\cos 2\pi\theta_1},\] which is just a rescaled version of the map $\mathcal{F}_*$ we introduced before, and $f=f_n=q_{n}^{-1}f_{q_n}$, for some positive integer parameters $N_n$ and $q_n$ to be defined below. It remains to choose the dynamics on the second factor, and here it will be an easy task. In order to have a result for a continuous system with $n$ degrees of freedom, we set $m'=n-3$, and it will be just fine to take \[ G=G_n=\Phi^{\frac{1}{2}(I_3^2+\cdots+I_{n-1}^{2})}. \] If $(p_j)_{j\geq 0}$ is the ordered sequence of prime numbers, now we let $N_n$ be the product of the $n-3$ prime numbers $\{p_{n+4},\dots,p_{2n}\}$, that is \begin{equation}\label{Nn} N_n=p_{n+4}p_{n+5}\dots p_{2n}. \end{equation} The next proposition is the analytic analogue of Proposition~\ref{pertu}, and its proof is even simpler. \begin{proposition}\label{pertuana} Let $n\geq 4$ and $\sigma >0$. Then there exist a function $g_n\in \mathcal{A}_{\sigma}(\mathbb T^{n-3})$ and a point $a_n\in\mathbb A^{n-3}$ such $a_n$ is $N_n$-periodic for $G_n$ and $(g_n, G_n, a_n, N_n)$ satisfy the synchronization conditions (\ref{sync}): \begin{equation*} g_n(a_n)=1, \quad dg_n(a_n)=0, \quad g_n(G_n^k(a_n))=0, \quad dg_n(G_n^k(a_n))=0, \end{equation*} for $1 \leq k \leq N_n-1$. Moreover, there exists a positive constant $c$ depending only on $\sigma$ such that if \begin{equation}\label{qn} q_n=2N_n^4[e^{c(n-3)p_{2n}}], \end{equation} the estimate \begin{equation}\label{estimgnana} q_n^{-1/2}|g_n|_{\sigma}\leq N_{n}^{-2}, \end{equation} holds true. \end{proposition} The function $g_n$ belongs to $\mathcal{A}_{\sigma}(\mathbb T^{n-3})$, but it can also be considered as a function in $\mathcal{A}_{\sigma}(\mathbb A^{n-3})$ depending only on the angle variables. As in the previous section, one can easily see that the coupling lemma, together with both Proposition~\ref{LocMar} and Proposition~\ref{pertuana}, already give us a result of instability for a perturbation of an integrable map, but we shall not state it. \begin{proof} Recall that for $p \in \mathbb N^*$, we have defined in Lemma~\ref{funct} an analytic function $\eta_p : \mathbb T \rightarrow \mathbb R$ by \[ \eta_p(\theta)=\left(\frac{1}{p}\sum_{l=0}^{p-1}\cos 2\pi l\theta \right)^2. \] We choose our function $g_n\in \mathcal{A}_{\sigma}(\mathbb T^{n-3})$ of the form \[ g_n=g_n^{(3)}\otimes \cdots \otimes g_n^{(n-1)}, \] where \[ g_n^{(i)}(\theta_i)=\eta_{p_{n+1+i}}(\theta_i), \quad 3\leq i \leq n-1, \] and our point $a_n=(a_n^{(3)},\dots,a_n^{(n-1)})\in\mathbb A^{n-3}$ where \[ a_n^{(i)}=(0,p_{n+1+i}^{-1}), \quad 3\leq i \leq n-1. \] Recalling the definition of $G_n$ and $N_n$, it is obvious that $a_n$ is $N_n$-periodic for $G_n$. Moreover, by Lemma~\ref{funct}, the function $\eta_p$ satisfies \[ \eta_p(0)=1, \quad \eta_p'(0)=0, \quad \eta_p(k/p)=\eta_p'(k/p)=0, \] for $1 \leq k \leq p-1$, from which one can easily deduce that $(g_n, G_n, a_n, N_n)$ satisfy the synchronization conditions (\ref{sync}). Concerning the estimate, first note that \[ |\eta_p|_{\sigma}\leq e^{4\pi\sigma p} \] so that \[ |g_n|_{\sigma} \leq |\eta_{p_{n+4}}|_{\sigma}\cdots |\eta_{p_{2n}}|_{\sigma} \leq e^{4\pi \sigma(n-3)p_{2n}}. \] Therefore, if we set $c=8\pi\sigma$, then by definition of $q_n$ one has \[ q_n^{1/2} \geq N_n^2 e^{4\pi \sigma(n-3)p_{2n}}\] and this eventually gives us \[ q_n^{-1/2}|g_n|_{\sigma}\leq N_{n}^{-2}, \] which is the desired estimate. \end{proof} \subsection{Proof of Theorem~\ref{thmnonpertana}}\label{sectnonpertana} \paraga First we shall recall the following result of Kuksin-Pöschel (\cite{KP94}, see also \cite{Kuk93}). \begin{proposition}[Kuksin-Pöschel]\label{susana} Let $\Psi_n : \mathbb A^{n-1} \rightarrow \mathbb A^{n-1}$ be a bounded real-analytic exact-symplectic diffeomorphism, which has a bounded holomorphic extension to some complex neighbourhood $V_{\varrho}$, for some width $\varrho>0$ independent of $n\in\mathbb N^*$. Assume also that $|\Psi_n-\Phi^{\tilde{h}}|_{\varrho}$ goes to zero when $n$ goes to infinity, where $\tilde{h}(I_1,\dots,I_{n-1})=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)$. Then there exist $n_0\in\mathbb N^*$, $\rho<\varrho$ such that for any $n\geq n_0$, there exists $f_n\in \mathcal{A}_\rho(\mathbb T^n \times B)$, independent of the variable $I_n$, such that if \[ H_n(\theta,I)=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+I_n+f_n(\theta,I), \quad (\theta,I)\in\mathbb A^n, \] for any energy $e\in\mathbb R$, the Poincaré map induced by the Hamiltonian flow of $H_n$ on the section $\{\theta_n=0\}\cap H_{n}^{-1}(e)$ coincides with $\Psi_n$. Moreover, the estimate \begin{equation}\label{taillesusana} |\Psi_n-\Phi^{\tilde{h}}|_{\varrho} \leq |f_n|_\rho \leq \delta_n |\Psi_n-\Phi^{\tilde{h}}|_{\varrho}, \end{equation} holds true for some constant $\delta_n$ that may depends on $n\in\mathbb N^*$. \end{proposition} This suspension result is slightly less accurate (since more difficult) than Proposition~\ref{sus}, as there is a constant $\delta_n$ depending on $n$. However, what really matters is that the resulting width of analyticity $\rho$ depends only on $\varrho$ and $R$, but not on $n$. \paraga Now we can finally prove the theorem. \begin{proof}[Proof of Theorem~\ref{thmnonpertana}] Let $n\geq 4$, $R>1$ and $\sigma>0$ given by the Proposition~\ref{LocMar}, and let $N_n$ and $q_n$ defined as in (\ref{Nn}) and (\ref{qn}) respectively. We will first construct a map $\Psi_n$ with a well-controlled wandering point. To this end, by Proposition~\ref{pertuana} we can apply the coupling lemma~\ref{coupling} with the following data: \[ F_n=\Phi^{\frac{1}{2} (I_1^2+I_2^2)+N_n^{-2}\cos 2\pi\theta_1}, \quad f_n=q_{n}^{-1}f_{q_n}, \quad G_n=\Phi^{\frac{1}{2}(I_3^2+\cdots+I_{n-1}^{2})}, \] and with the function $g_n$ and the point $a_n$ given by the aforementioned proposition. This gives us the following: if \[ u_n=q_{n}^{-1} f_{q_n}\otimes g_n , \quad v_n=N_n^{-2}V \] where $V(\theta_1)=\cos 2\pi\theta_1$, then the $N_n$-iterates of the map \[ \Psi_n=\Phi^{u_n} \circ \Phi^{\tilde{h}+v_n} : \mathbb A^{n-1} \rightarrow \mathbb A^{n-1}\] satisfies the following relation: \begin{equation}\label{coup} \Psi_n^{N_n}(x,a_n)=(\Phi^{q_{n}^{-1}f_{q_n}}\circ F_n^{N_n}(x),a_n), \quad x\in \mathbb A^2. \end{equation} Now let us look at the map \[ \Phi^{q_{n}^{-1}f_{q_n}}\circ F_n^{N_n}=\Phi^{q_{n}^{-1}f_{q_n}}\circ\left(\Phi^{\frac{1}{2} (I_1^2+I_2^2)+N_n^{-2}\cos 2\pi\theta_1}\right)^{N_n}. \] If $S_n(\theta_1,\theta_2,I_1,I_2)=(\theta_1,\theta_2,N_nI_1,N_nI_2)$ is the rescaling by $N_n$ in the action components, one sees that \[ \Phi^{q_{n}^{-1}f_{q_n}}\circ F_n^{N_n}=S_n^{-1}\circ \mathcal{F}_{N_n^{-1}q_n}\circ S_n \] where $\mathcal{F}_{N_n^{-1}q_n}$ is defined in~(\ref{Fq}). Now by Proposition~\ref{LocMar}, choosing $n$ large enough so that $N_n^{-1}q_n\geq q_0$, this map has a wandering point $\zeta_{N_n^{-1}q_n}\in\mathbb A^2$, which by~(\ref{timeana}) satisfies \[ \left|\mathcal{F}_{N_n^{-1}q_n}^{3N_n^{-2}q_n^2} \left(\zeta_{N_n^{-1}q_n}\right)-\zeta_{N_n^{-1}q_n}\right| \geq 1. \] Using the above conjugacy relation, one finds that the point \[ \chi_n=S_n^{-1}(\zeta_{N_n^{-1}q_n})\in\mathbb A^2 \] wanders under the iteration of $\Phi^{q_{n}^{-1}f_{q_n}}\circ F_n^{N_n}$, and that its drift is bigger than one after $N_n(3N_n^{-2}q_n^2)=3N_n^{-1}q_n^2$ iterations, that is \[ \left|(\Phi^{q_{n}^{-1}f_{q_n}}\circ F_n^{N_n})^{3N_n^{-1}q_n^2}(\chi_n)-\chi_n\right|\geq 1. \] By the relation~(\ref{coup}) this gives a wandering point $x_n=(\chi_n,a_n)\in\mathbb A^{n-1}$ for the map $\Psi_n$, satisfying the estimate \begin{equation}\label{estimpsi} |\Psi_n^{3q_n^2}(x_n)-x_n|\geq 1. \end{equation} Next let us estimate the distance between $\Psi_n$ and the integrable diffeomorphism $\Phi^{\tilde{h}}$. First note that since $u_n,v_n\in \mathcal{A}_\sigma(\mathbb T^{n-1})$, $\Psi_n$ extends holomorphically to a complex neighbourhood of size $\sigma$. Let us now estimate the norms of $u_n$ and $v_n$. Obviously, one has \[ N_n^{-2}\leq|v_n|_\sigma\leq e^{2\pi\sigma}N_n^{-2}. \] By definition of $f_q$ and the exponent $\nu(q,\sigma)$, one easily obtains \[ |q_n^{-1}f_{q_n}|_\sigma \leq q_n^{-1/2}|f^{(2)}|_\sigma, \] and hence \[ |u_n|_\sigma \leq |q_n^{-1}f_{q_n}|_\sigma |g_n|_\sigma \leq q_n^{-1/2} |g_n|_\sigma |f^{(2)}|_\sigma \leq N_n^{2} |f^{(2)}|_\sigma, \] where the last inequality follows from the estimate~(\ref{estimgnana}). Then by using Cauchy estimates and general inequalities on time-one maps, we obtain \begin{equation}\label{estimdis} N_n^{-2} \leq |\Psi_n-\Phi^{\tilde{h}}|_{\varrho} \leq c_\sigma N_n^{-2}, \end{equation} for $n$ large enough, and for some constants $c_\sigma$ and $\varrho>0$ depending only on $\sigma$ (for instance, one can choose $\varrho=6^{-1}\sigma$). Now we can eventually apply Proposition~\ref{susana}: there exist $n_0\in\mathbb N^*$, $\rho<\varrho$ such that for any $n\geq n_0$, there exists $f_n\in \mathcal{A}_\rho(\mathbb T^n \times B)$, independent of the variable $I_n$, such that if \[ H_n(\theta,I)=\frac{1}{2}(I_1^2+\cdots+I_{n-1}^2)+I_n+f_n(\theta,I), \quad (\theta,I)\in\mathbb A^n, \] for any energy $e\in\mathbb R$, the Poincaré map induced by the Hamiltonian flow of $H_n$ on the section $\{\theta_n=0\}\cap H_{n}^{-1}(e)$ coincides with $\Psi_n$. Clearly, the wandering point $x_n$ for $\Psi_n$ gives us a wandering orbit $(x(t),t,I_n(t))=(x(t),\theta_n(t),I_n(t))$ for the Hamiltonian vector field generated by $H_n$, such that \[ x(k)=\Psi_n^k(x_n), \quad k\in\mathbb Z. \] In particular, after a time $\tau_n=3q_n^2$, by the above equality and the relation~(\ref{estimpsi}) this orbit drifts from $0$ to $1$. Now it remains to estimate the size of the perturbation $\varepsilon_n=|f_n|_\rho$ and the time of drift $\tau_n$ in terms of the number of degrees of freedom $n$. First, by~(\ref{taillesusana}) and~(\ref{estimdis}), \begin{equation}\label{estNepsana} N_{n}^{-2}\leq \varepsilon_n \leq c_n N_{n}^{-2}, \end{equation} with $c_n=c_\sigma \delta_n$. Then, by the prime number theorem, taking $n_0$ large enough, one can ensure that \[ p_{2n}/4 \leq p_{n+i} \leq p_{2n}, \quad 4\leq i\leq n, \] which gives \begin{equation}\label{estimPNana} (p_{2n}/4)^{n-3} \leq N_n \leq p_{2n}^{n-3}, \quad N_n^{\frac{1}{n-3}}\leq p_{2n} \leq 4N_n^{\frac{1}{n-3}}. \end{equation} We can also assume by the prime number theorem that for $n\geq n_0$, one has \begin{equation}\label{nompremierana} 2n\ln 2n \leq p_{2n} \leq 2(2n \ln 2n)=4n\ln 2n. \end{equation} From the above estimates~(\ref{estimPNana}) and~(\ref{nompremierana}) one easily obtains \begin{equation}\label{estNana} e^{(n-3)\ln (2^{-1}n\ln 2n)}\leq N_n\leq e^{(n-3)\ln (4n\ln 2n)}, \end{equation} and, together with~(\ref{estNepsana}), one finds \begin{equation}\label{estpertana} e^{-2(n-3)\ln (4n\ln 2n)}\leq \varepsilon_n \leq c_n e^{-2(n-3)\ln (2^{-1}n\ln 2n)}. \end{equation} Concerning the time $\tau_n$, we have \[ \tau_n=3q_n^2 \leq 12 N_n^{8} e^{2c(n-3)p_{2n}} \leq 12 N_n^{8} e^{8c(n-3)N_n^{\frac{1}{n-3}}} , \] where the last inequality follows from~(\ref{estimPNana}). Then using~(\ref{estNana}) we have \[ N_n^{\frac{1}{(n-3)}}\leq 4n \ln 2n \] and from~(\ref{estNepsana}) we know that \[ N_n^8 \leq \left(\frac{c_n}{\varepsilon_n}\right)^4, \] so we obtain \[ q_n^2\leq 12\left(\frac{c_3}{\varepsilon_n}\right)^4 e^{32c(n-3)n \ln 2n}. \] Then one can ensure that for $n\geq n_0$, \[ \ln 2n \leq \ln (2^{-1}n\ln 2n), \] so \[ 32c(n-3)n \ln 2n \leq 32c(n-3)n \ln (2^{-1}n\ln 2n). \] Therefore \begin{eqnarray*} q_n^2 & \leq & 12\left(\frac{c_n}{\varepsilon_n}\right)^4 e^{32c(n-3)n \ln (2^{-1}n\ln 2n)} \\ & \leq & 12\left(\frac{c_n}{\varepsilon_n}\right)^4\left(e^{2(n-3)\ln (2^{-1}n\ln 2n))}\right)^{16 cn}. \end{eqnarray*} Finally by~(\ref{estpertana}) we obtain \begin{eqnarray*} q_n^2 & \leq & 12\left(\frac{c_n}{\varepsilon_n}\right)^4\left(\frac{c_n}{\varepsilon_n}\right)^{16cn} \\ & \leq & C\left(\frac{c_n}{\varepsilon_n}\right)^{n\gamma} \end{eqnarray*} with $C=12$ and $\gamma=4+16c$. This concludes the proof. \end{proof} \appendix \section{Gevrey functions}\label{Gev} In this very short appendix, we recall some facts about Gevrey functions that we used in the text. We refer to~\cite{MS02}, Appendix A, for more details. The most important property of $\alpha$-Gevrey functions is the existence, for $\alpha>1$, of bump functions. \begin{lemma}\label{lemmeGev1} Let $\alpha>1$ and $L>0$. There exists a non-negative $1$-periodic function $\varphi_{\alpha,L}\in G^{\alpha,L}\left([-\frac{1}{2},\frac{1}{2}]\right)$ whose support is included in $[-\frac{1}{4},\frac{1}{4}]$ and such that $\varphi_{\alpha,L}(0)=1$ and $\varphi_{\alpha,L}'(0)=0$. \end{lemma} The following estimate on the product of Gevrey functions follows easily from the Leibniz formula. \begin{lemma}\label{lemmeGev2} Let $L>0$, and $f,g\in G^{\alpha,L}(\mathbb T^n \times \overline{B})$. Then \[ |fg|_{\alpha,L}\leq |f|_{\alpha,L}|g|_{\alpha,L}. \] \end{lemma} Finally, estimates on the composition of Gevrey functions are much more difficult (see Proposition A.1 in \cite{MS02}), but here we shall only need the following statement. \begin{lemma}\label{lemmeGev3} Let $\alpha\geq 1$, $\Lambda_1>0, L_1>0$, and $I,J$ be compact intervals of $\mathbb R$. Let $f\in G^{\alpha,\Lambda_1}(I)$, $g\in G^{\alpha,L_1}(J)$ and assume $g(J)\subseteq I$. If \[ |g|_{\alpha,L_1}\leq \Lambda_{1}^{\alpha},\] then $f\circ g \in G^{\alpha,L_1}(J)$ and \[ |f\circ g|_{\alpha,L_1} \leq |f|_{\alpha,\Lambda_1}. \] \end{lemma} {\it Acknowledgments.} The author is indebted to Jean-Pierre Marco for suggesting him this problem to work on, for helpful discussions, comments and corrections on a first version of this paper. He also thanks the anonymous referee for several interesting suggestions. Finally, the author thanks the University of Warwick where this work has been finished while he was a Research Fellow through the Marie Curie training network ``Conformal Structures and Dynamics (CODY)". \addcontentsline{toc}{section}{References} \end{document}

\begin{document} \title{Monads for framed torsion-free sheaves on multi-blow-ups of the projective plane } \footnotetext[1]{henni@ime.unicamp.br} \begin{abstract} We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we construct monads for families of such sheaves parameterized by a noetherian scheme $S$ of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine. \end{abstract} \section{Introduction} In this paper we are concerned with the construction of the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a},k}$ of framed torsion-free sheaves of a fixed Chern character on a multi-blow-up of the complex projective plane: $\pi:\tilde{X}de{\mathbb{P}}\longrightarrow\mathbb{P}^{2}$, by using monadic descriptions which lead to a parametrization of $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a},k}$ in terms of linear data of the ADHM (Atiyah-Drinfel'd-Hitchin-Manin) type \cite{ADHM}. The ADHM data will be useful, at a first step, to give a presentation of the moduli space as a quotient $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}=P/G$ where $P$ is a space of some matrices satisfying certain conditions and which will be described below. At a second step the monadic description is used to prove that the space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a},k}$ is a smooth algebraic variety. This is done by generalizing Buchdahl's construction for holomorphic bundles \cite{Buch}, in order to extend it to torsion-free sheaves. An additional result is the construction of a monad corresponding to an $S-$flat family $\mathcal{F}$ on a product $\tilde{X}de{\mathbb{P}}\times S$, where $S$ is a noetherian scheme of finite type. In particular, there is a universal monad on $\tilde{X}de{\mathbb{P}}\times\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$. Using the ADHM presentation of the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ and the properties of the universal monads constructed, we prove that the scheme $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is a fine moduli space. Another way of treating the moduli space is to show that one can choose a polarization on $\tilde{X}de{\mathbb{P}}$ such that a framed sheaf $(\mathcal{E}, \Phi)$ is stable in the sense of Huybrechts and Lehn (Nakajima \cite{Naka1}, Bruzzo and Markushevich \cite{bruzzo}), and using their result that the moduli space of such objects is a quasi-projective scheme \cite[Theorem 0.1]{Huy1},\cite{Huy2}. Moreover this moduli space is fine \cite[Theorem 0.1]{Huy1} and its smoothness follows from the vanishing of the obstruction in \cite[Section 4]{Huy1}, while in the present work, the smoothness proof is based on the ADHM construction, where the moduli space is a quotient of an affine space by a non-reductive group. The equivalence between the two approaches is established by the fact that in both cases the moduli space is fine, as we prove for the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ in this work. This generalizes the result by Nakajima \cite{Naka} and Okonek et. al. \cite{Okonek} in the cases of Hilbert schemes of points on the projective plane, and rank-2 stable bundles on the projective plane, respectively. This comparison also implies that $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is quasi-projective. In the present paper all the varieties (or schemes) are over the field $k=\mathbb{C}.$ For every given coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ over a variety $X$, we denote by $\mathcal{G}\longrightarrow\Hom_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$ the functor from the category of coherent sheaves of $\mathcal{O}_{X}$-modules to the category of abelian groups (then $\Hom_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$ is the group of homomorphisms of sheaves of $\mathcal{O}_{X}$-modules), and by $\mathcal{G}\longrightarrow\ext^{i}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$ its $i$-th right derived functor. We also denote by $\mathcal{G}\longrightarrow\mathcal{H}om_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$ the functor from the category of coherent sheaves of $\mathcal{O}_{X}$-modules to itself (then $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$ is the sheaf of local homomorphism groups of sheaves of $\mathcal{O}_{X}$-modules defined as the sheaf associated to the pre-sheaf $U\longrightarrow\Hom_{\mathcal{O}_{X}|_U}(\mathcal{F}|_{U},\mathcal{G}|_{U})$), and by $\mathcal{G}\longrightarrow\mathcal{E}xt^{i}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$ its $i$-th right derived functor. For more details see \cite{Godement,Grothendieck0,Hart}. To avoid overloading the text we omit the subscript $\mathcal{O}_{X}$ except if needed; for example on $\mathbb{P}^{2}$ we just write $\ext^{i}(\mathcal{F},\mathcal{G})$ instead of $\ext^{i}_{\mathcal{O}_{\mathbb{P}^{2}}}(\mathcal{F},\mathcal{G})$ and $\mathcal{E}xt^{i}(\mathcal{F},\mathcal{G})$ instead of $\mathcal{E}xt^{i}_{\mathcal{O}_{\mathbb{P}^{2}}}(\mathcal{F},\mathcal{G}).$ \paragraph{Acknowledgement} I would like to thank first my supervisor Ugo Bruzzo, to whom I am very grateful. Thanks a lot to Claudio Rava for useful discussions, remarks and suggestions. Many thanks to the Department of Mathematics of the University of Genova for helping me during my brief visits. I would also like to thank Professor Vladimir Rubtsov for his interest in my work and for help while he was visiting SISSA in summer 2008. \section{The construction of the monad} Before starting our construction we introduce some definitions about the objects we shall study in this paper and will state some of their properties. \begin{definition}[Monad]\label{monad} A \underline{monad} $M$ on a scheme $X$ is a complex $\xymatrix@C-1.2pc{M: &0\ar[r]& \mathcal{U}\ar[r]^A & \mathcal{W}\ar[r]^B & \mathcal{V}\ar[r]&0}$ of vector bundles $\mathcal{U}$, $\mathcal{W}$ and $\mathcal{V}$ on $X$ which is exact except at $\mathcal{W}$; i.e., the bundle map $A$ is injective, the bundle map $B$ is surjective and their composition $B\circ A$ is zero. The vector bundle $\mathcal{E}:=KerB/ImA$ is the cohomology of the monad $M.$ \end{definition} \begin{definition}[The monad display] Each monad $M$ comes with an associated commutative diagram of exact sequences: $$\xymatrix@C-1pc@R-1pc{ & 0\ar[d] & 0\ar[d] & & \\ & \mathcal{U}\ar@{=}[r]\ar[d] & \mathcal{U}\ar[d]^A & & \\ 0\ar[r]& \mathcal{X} \ar[r]\ar[d] & \mathcal{W}\ar[r]^B\ar[d] & \mathcal{V}\ar@{=}[d]\ar[r] & 0\\ 0\ar[r]& \mathcal{E}\ar[r]\ar[d] & Q \ar[r]\ar[d] & \mathcal{V}\ar[r] & 0\\ & 0 & 0 & & }$$ called the \underline{display} of $M$, where $\mathcal{X}=KerB$ and $Q=CokerA$. The conditions on the injectivity of the map $A$ and the surjectivity of the map $B$ are called the non-degeneracy conditions for $M.$ \end{definition} \begin{rmk} \begin{itemize} \item[(i)] \textnormal{Let $X$ be a $2-$dimensional scheme. Recall that if the morphism $\mathcal{U}\stackrel{A}{\longrightarrow}\mathcal{V}$ is a morphism in the category of bundles, then $kerA$ is a subbundle of $\mathcal{U}$ and $ImA$ is a subbundle of $\mathcal{V}.$ If the first map $A$ is only injective as a sheaf map, i.e., $A(x)$ fails to be injective, as a bundle map, at a finite number of points $x\in X$, then the cohomology $\mathcal{E}:=KerB/ImA$ of the monad is no longer a vector bundle: the sheaves $\mathcal{U}$ and $KerB$ are locally free thus their quotient $\mathcal{E}$ is locally free if and only if $A(x)$ is injective as a bundle map. Moreover the singularity locus of $\mathcal{E}$ is supported exactly on the points on which $A(x)$ fails to be injective. This locus has codimension $2,$ thus $\mathcal{E}$ is torsion-free. The non-degeneracy conditions are reduced to the surjectivity of $B$ and the injectivity of $A$ at all $X$ except at finitely many points.} \item[(ii)]\textnormal{Allowing for generic surjectivity of the map $B$ will not be considered here; permitting this would introduce a cohomology $\mathcal{J}:=Coker(B)$ in the last term of the monad. Hence the monad in that case would parameterize a couple of sheaves: $(KerB/ImA, \mathcal{J}).$ This can be thought of as a particular case of a perverse sheaf. Examples of this kind of monads on $\mathbb{P}^{3}$ are treated in \cite{HL}. } \end{itemize} \end{rmk} Now assume that $X$ is a nonsingular complex projective surface. \begin{definition}[Framing] We say that a torsion-free sheaf $\mathcal{F}$, of rank $r$, is \underline{framed} on the divisor $D\subset X$ if $\mathcal{F}|_{D}$ is trivial and there is a fixed holomorphic trivialization $\Phi:\mathcal{F}|_{D}\longrightarrow\mathcal{O}|_{D}^{\oplus r}$ which is an isomorphism called the \underline{framing} of the sheaf $\mathcal{F}$ on $D$. \end{definition} In what follows, we take $X=\tilde{X}de{\mathbb{P}}$ where $\pi:\tilde{X}de{\mathbb{P}}\longrightarrow\mathbb{P}^{2}$ is the blow-up of the projective plane at $n$ distinct points. $\tilde{X}de{\mathbb{P}}$ is regular ($\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{O})=0$) and its Picard group (which in our case can be identified with $\h^{2}(\tilde{X}de{\mathbb{P}},\mathbb{Z})$) is generated by $n+1$ elements, namely: $Pic(\tilde{X}de{\mathbb{P}})=\oplus_{i=1}^{n}E_{i}\mathbb{Z}\oplus H\mathbb{Z},$ where every $E_{i}$ is an exceptional divisor with the following intersection numbers: $E_{i}^{2}=-1$, $E_{i}\cdot E_{j}=0$ for $i\neq j$, $E_{i}\cdot H=0$ and where $H$ is the divisor given by the generic line in $\mathbb{P}^{2}$ and satisfying $H^{2}=1.$ The canonical divisor of the surface $\tilde{X}de{\mathbb{P}}$ is given by $K_{\tilde{X}de{\mathbb{P}}}=-3H+\Sigma_{i=1}^{n}E_{i}.$ The Poincaré duals of these divisors are given by $(h,e_{1}, \cdots e_{n})$ where $<h,H>=1$, $<e_{i},E_{j}>=\delta_{ij}$ and also $e_{i}\cdot e_{j}=-\delta_{ij}.$ In terms of line bundles, a divisor of the form $D=pH+\Sigma_{i=1}^{n}q_{i}E_{i}$ has the associated line bundle $\mathcal{O}(D)=\mathcal{O}(p,\overrightarrow{q})=\mathcal{O}(pH)\otimes\mathcal{O}(q_{1}E_{1})\otimes\cdots\otimes\mathcal{O}(q_{n}E_{n})$ where $\overrightarrow{q}=(q_{1},\cdots , q_{n}).$ Then the canonical bundle is given by $\mathcal{O}(K_{\tilde{X}de{\mathbb{P}}})=\mathcal{O}(-3H+\Sigma_{i=1}^{n}E_{i})=\mathcal{O}(-3,\overrightarrow{1}).$ The Riemann-Roch formula for a line bundle $\mathcal{O}(p,\overrightarrow{q})$ is given by: $$\chi(\mathcal{O}(p,\overrightarrow{q}))=\frac{1}{2}[(p+1)(p+2)-|\overrightarrow{q}|^{2}+\Sigma_{i=1}^{n}q_{i}].$$ where $|\overrightarrow{q}|^{2}=\Sigma_{i=1}^{n}q_{i}^{2}.$ We also use the fact that a line bundle $\mathcal{O}(p,\overrightarrow{q})$ restricts to $\mathcal{O}(p)$ on the divisors in the linear system $|\mathcal{O}(H)|$ and restricts to $\mathcal{O}(-q_{i})$ on the divisors in the linear system $|\mathcal{O}(E_{i})|.$ Let $\omega\in \h^{4}(\tilde{X}de{\mathbb{P}},\mathbb{Z})$ be the fundamental class of $\tilde{X}de{\mathbb{P}}$. For a torsion-free sheaf $\mathcal{E}$, of Chern character $ch(\mathcal{E})=r+(aH+\Sigma_{i=1}^{n}a_{i}E_{i})-(k-\frac{a^{2}-|\overrightarrow{a}|^{2}}{2})\omega$, twisted by a line bundle $\mathcal{O}(p,\overrightarrow{q})$ the Riemann-Roch formula is given by: $$\chi(\mathcal{E}(p,\overrightarrow{q}))=-[k-\frac{a}{2}(a+3)+\frac{1}{2}\Sigma_{i=1}^{n}a_{i}(a_{i}-1)]+ \frac{r}{2}[(p+1)(p+2)-\Sigma_{i=1}^{n}q_{i}(q_{i}-1)]+[ap-\Sigma_{i=1}^{n}a_{i}q_{i}].$$ General results about surfaces and their blow-ups can be found in details in \cite{beauville, barth}, so that one can reconstruct the above description. We remind the reader that, in \cite{Buch}, Buchdahl has a different notation for line bundles; he uses a notation involving the Poincaré duals $h,e_{1}, \cdots e_{n}$ of the divisors $H,E_{1}, \cdots E_{n}$ respectively. Now we restrict ourselves to the case of torsion-free sheaves $\mathcal{E}$ with Chern character $ch(\mathcal{E})=r+(\Sigma_{i=1}^{n}a_{i}E_{i})-(k+\frac{|\overrightarrow{a}|^{2}}{2})\omega$ which are framed on a fixed rational curve $l_{\infty}$ in the linear system $|H|,$ i.e., we have a fixed trivialization $\Phi:\mathcal{E}|_{l_{\infty}}\longrightarrow\mathcal{O}|_{l_{\infty}}^{\oplus r}$. The direct image $\pi_{\ast}\mathcal{E}$ of $\mathcal{E}$ is a normalized torsion-free sheaf since it is framed on $\mathbb{P}^{2},$ so that $c_{1}(\pi_{\ast}\mathcal{E})=0$ (a rank $r$ sheaf $\mathcal{F}$ on $\mathbb{P}^{2}$ is normalized if its first Chern class $c_{1}(\mathcal{F})$ satisfies $|c_{1}(\mathcal{F})|<r$). From the natural injection of the sheaf $\mathcal{E}$ in its double dual $\mathcal{E}^{\ast\ast}$, we have the following exact sequence: \begin{equation}\label{doubledual} 0\longrightarrow\mathcal{E}\longrightarrow\mathcal{E}^{\ast\ast}\longrightarrow\Delta\longrightarrow0 \end{equation} where $\mathcal{E}^{\ast\ast}$ has Chern character $ch(\mathcal{E}^{\ast\ast})=r+(\Sigma_{i=1}^{n}a_{i}E_{i})-(k-l+\frac{|\overrightarrow{a}|^{2}}{2})\omega$, and $l$ is the length of the quotient sheaf $\Delta$ supported on finitely many points with $Supp(\Delta)\cap l_{\infty}=\emptyset$. \begin{pr} $\h^{0}(\tilde{X}de{\mathbb{P}}, \mathcal{E}^{\ast\ast}(p,\overrightarrow{q}))=\h^{0}(\tilde{X}de{\mathbb{P}}, \mathcal{E}^{\ast}(p,\overrightarrow{q}))=0$ $\qquad\forall \overrightarrow{q}$ if $p<0$ \quad and \hspace{2.5cm}$\h^{2}(\tilde{X}de{\mathbb{P}}, \mathcal{E}^{\ast\ast}(p,\overrightarrow{q}))=\h^{2}(\tilde{X}de{\mathbb{P}}, \mathcal{E}^{\ast}(p,\overrightarrow{q}))=0$ $\qquad\forall \overrightarrow{q}$ if $p=-1,-2.$ \end{pr} \begin{proof} The first vanishing follows by taking the direct image $\pi_{\ast}(\mathcal{E}^{\ast\ast})$ of $\mathcal{E}^{\ast\ast}$. By using the framing condition one can easily verify that, on $\mathbb{P}^{2}$, the group $\h^{0}(\mathbb{P}^{2}, \pi_{\ast}(\mathcal{E}^{\ast\ast})(p))$ vanishes for $p<0.$ But $\h^{0}(\mathbb{P}^{2}, \pi_{\ast}(\mathcal{E}^{\ast\ast})(p))\cong\h^{0}(\tilde{X}de{\mathbb{P}}, \mathcal{E}^{\ast\ast}(p,\overrightarrow{q})),$ thus the latter is zero for $p<0$. This also holds for the dual $\mathcal{E}^{\ast}$ since it is also framed. Finally by Serre duality the last two conditions follow easily. \end{proof} \begin{cor}\label{cor} $\h^{0}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(p,\overrightarrow{q}))=0$ $\qquad\forall \overrightarrow{q}$ if $p<0$ \quad and \hspace{2cm} $\h^{2}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(p,\overrightarrow{q}))=0$ $\qquad\forall \overrightarrow{q}$ if $p=-1,-2.$ \end{cor} We also use the following form of Serre-Grothendieck duality for coherent sheaves \cite{Grothendieck0,Hart1}: \begin{thm} On a smooth algebraic projective variety $X$ of dimension $n$ over an algebraically closed field $k$, and for every two coherent sheaves $\mathcal{F}$ and $\mathcal{J}$, the following formula holds: $$\ext^{i}(\mathcal{F},\mathcal{J})=\ext^{n-i}(\mathcal{J},\mathcal{F}\otimes\omega_{X})^{\ast}$$ where $\omega_{X}$ is the canonical sheaf. \end{thm} \begin{definition}[Non-locally free monad] A \underline{non-locally free monad} is a complex $$\xymatrix@C-1.2pc{M: &0\ar[r]& \mathcal{U}\ar[r]^A & \mathcal{W}\ar[r]^B & \mathcal{V}\ar[r]&0}$$ as in definition \ref{monad} in which $\mathcal{U}$, $\mathcal{V}$ and $\mathcal{W}$ are coherent sheaves (not necessarily locally free sheaves). \end{definition} We start our program by showing that a framed torsion-free sheaf $\mathcal{E}$ can be described as the cohomology of a non-locally free monad with a torsion-free sheaf in its middle term, and locally free sheaves in the first and the third terms. Since the maps involved are sheaf maps, instead of being bundle maps, one cannot obtain an ADHM description. This problem will be solved later by constructing a monad, out of the obtained non-locally free monad, and proving that its cohomology is the starting sheaf $\mathcal{E}.$ This construction is mainly a generalization of \cite[Section 1]{Buch} to the case of framed torsion-free sheaves. First define the spaces $B_{i}:=\Hom(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))^{\ast}$ and let $\mathcal{B}_{1}=\oplus_{i=1}^{n} B_{i}(1,-E_{i})$. Then the extensions of the form $$0\longrightarrow\mathcal{E}\longrightarrow Q_{1}\longrightarrow\mathcal{B}_{1}\longrightarrow0$$ are classified by the group $\ext^{1}(\mathcal{B}_{1},\mathcal{E})\cong\oplus_{i=1}^{n} B_{i}^{\ast}\otimes \ext^{1}(\mathcal{E}, \mathcal{O}(-2,\overrightarrow{1}-E_{i}))^{\ast}$. Applying the functor $\Hom(\mathcal{E},\cdot)$ to the sequence \begin{equation}\label{rest2} 0\longrightarrow\mathcal{O}(0,-E_{i})\longrightarrow\mathcal{O}\longrightarrow\mathcal{O}|_{E_{i}}\longrightarrow0 \end{equation} after twisting by $\mathcal{O}(-2,\overrightarrow{1})$ one obtains $$\xymatrix{\Hom(\mathcal{E},\mathcal{O}(-2,\overrightarrow{1}))\ar[r]& \Hom(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))\ar[r]& \ext^{1}(\mathcal{E},\mathcal{O}(-2,\overrightarrow{1}-E_{i}))}$$ but $\Hom(\mathcal{E},\mathcal{O}(-2,\overrightarrow{1}))=\h^{2}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,0))^{\ast}=0$ by the corollary above. Then the map $$\xymatrix{B_{i}^{\ast}\ar@{^{(}->}[r]& \ext^{1}(\mathcal{E},\mathcal{O}(-2,\overrightarrow{1}-E_{i}))}$$ is injective. This implies that the map $$\xymatrix{\ext^{1}(\mathcal{E},\mathcal{O}(-2,\overrightarrow{1}-E_{i}))^{\ast}\ar[r]^{\qquad\qquad r}& B_{i}}$$ is surjective. Thus there exists an extension $Q_{1}$ which is mapped to the identity in $\End(B_{i})$ for each $i$ under the composition of the projection on the $i-$th factor and the map $r$ above. Let us now define $A_{i}:=\ext^{1}(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))^{\ast}$ and $\mathcal{A}_{1}:=\oplus_{i=1}^{n}A_{i}(-1,E_{i}).$ The extensions of the form $$0\longrightarrow\mathcal{A}_{1}\longrightarrow X_{1}\longrightarrow\mathcal{E}\longrightarrow0$$ are classified by $\ext^{1}(\mathcal{E},\mathcal{A}_{1})\cong\oplus_{i=1}^{n} A_{i}\otimes \ext^{1}(\mathcal{E},\mathcal{O}(-1,E_{i})).$ Applying the functor $\Hom(\mathcal{E}, \cdot)$ to \eqref{rest2} twisted by $\mathcal{O}(-1,E_{i})$ one has the following exact sequence: \begin{equation}\label{Adual} \ext^{1}(\mathcal{E},\mathcal{O}(-1,E_{i}))\longrightarrow \underbrace{\ext^{1}(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))}_{A_{i}^{\ast}}\longrightarrow \ext^{2}(\mathcal{E},\mathcal{O}(-1,0)) \end{equation} where $\ext^{2}(\mathcal{E},\mathcal{O}(-1,0))=\Hom(\mathcal{O}(-1,0),\mathcal{E}(-3,\overrightarrow{1}))^{\ast} =\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\overrightarrow{1}))^{\ast}=0$ also by the corollary above. Thus there exists an extension in $\ext^{1}(\mathcal{E},\mathcal{A}_{1})$ which maps to the identity in $\End(A_{i})$ for every $i=1,...,n.$ To construct a display of a non-locally free monad one may apply the following: \begin{pr}\cite[Proposition \textbf{2.2.3}]{King}\label{King223} Suppose we are given two extensions \begin{align}&\xymatrix@C-1pc{0\ar[r] & \mathcal{U}\ar[r]^{i_{1}} & \mathcal{X}\ar[r]^{j_{1}} & \mathcal{E}\ar[r] & 0} \label{ext1}\\ &\xymatrix@C-1pc{0\ar[r] & \mathcal{E}\ar[r]^{i_{4}} & \mathcal{Q}\ar[r]^{j_{4}} & \mathcal{V}\ar[r] & 0} \label{ext2}. \end{align} of torsion free sheaves. Then we can fit them into a non-locally free monad display if and only if the double extension \begin{equation}\label{doublext} \xymatrix@C-1pc{0\ar[r] & \mathcal{U}\ar[r] & \mathcal{X}\ar[r] & \mathcal{Q}\ar[r] & \mathcal{V}\ar[r] & 0}, \end{equation} i.e., their $\ext$-product in $\ext^{2}(\mathcal{V},\mathcal{U}),$ is trivial. Furthermore any two way of completing the display differ by an action of $\ext^{1}(\mathcal{V},\mathcal{U}).$ \end{pr} \begin{proof} By applying the contravariant functor $\Hom(\bullet,\mathcal{U})$ on the second sequence one gets $$\xymatrix@C-1pc{\ext^{1}(\mathcal{V},\mathcal{U})\ar[r]^{\tilde{X}de{j_{4}}} & \ext^{1}(\mathcal{Q},\mathcal{U})\ar[r]^{\tilde{X}de{i_{4}}} & \ext^{1}(\mathcal{E},\mathcal{U}) \ar[r]^{\delta} & \ext^{2}(\mathcal{V},\mathcal{U})\ar[r] & \cdots}.$$ The map $\delta$ sends the extension \eqref{ext1} to the extension \eqref{doublext} in $\ext^{2}(\mathcal{V},\mathcal{U}).$ Conversely when the double extension \eqref{doublext} is trivial one can find an extension $$\xymatrix@C-1pc{0\ar[r] & \mathcal{U}\ar[r]^{\alpha} & \mathcal{W}\ar[r]^{j_{2}} & \mathcal{Q}\ar[r] & 0}$$ in $\ext^{1}(\mathcal{Q},\mathcal{U})$ which is mapped to \eqref{ext1} by $\tilde{X}de{i_{4}}.$ This implies that there is a uniquely determined map $i_{3}:\mathcal{X}\longrightarrow\mathcal{W}$ such that $$\xymatrix@C-1pc@R-1pc{0\ar[r]& \mathcal{U}\ar[r]\ar@{=}[d]&\mathcal{X}\ar[r]\ar[d]^{i_{3}}&\mathcal{E}\ar[r]\ar[d]^{i_{4}}&0 \\ 0\ar[r]&\mathcal{U}\ar[r]&\mathcal{W}\ar[r]&\mathcal{Q}\ar[r]&0}$$ commutes. Finally, putting $\beta=j_{4}\circ j_{2}$ one has the sequence $$\xymatrix@C-1pc{0\ar[r] & \mathcal{X}\ar[r]^{i_{3}} & \mathcal{W}\ar[r]^{\beta} & \mathcal{V}\ar[r] & 0}$$ required to complete the display. One can verify that $\ext^{1}(\mathcal{V},\mathcal{U})$ acts naturally on the space of all monads with ends $\mathcal{U}$ and $\mathcal{V}.$ The monad we obtained in this way belongs to the orbit of this action. \end{proof} In our case one has $\ext^{2}(\mathcal{B}_{1},\mathcal{A}_{1})\cong\oplus_{i,j}^{n}A_{i}\otimes B^{\ast}_{j}\otimes \h^{2}(\tilde{X}de{\mathbb{P}},\mathcal{O}(-2,E_{i}+E_{j}))=0$, thus there exists a sheaf $W_{1}$ and exact sequences $$0\longrightarrow\mathcal{A}_{1}\longrightarrow W_{1}\longrightarrow Q_{1}\longrightarrow0$$ $$0\longrightarrow X_{1}\longrightarrow W_{1}\longrightarrow\mathcal{B}_{1}\longrightarrow0$$ which fit into the following commutative diagram: \begin{equation}\label{display} \xymatrix@R-1pc@C-1pc{ & 0 \ar[d] & 0 \ar[d] & & \\ & \mathcal{A}_{1} \ar@{=}[r]\ar[d] & \mathcal{A}_{1} \ar[d] & & \\ 0 \ar[r] & X_{1} \ar[r]\ar[d] & W_{1} \ar[r] \ar[d] & \mathcal{B}_{1} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{E} \ar[r]\ar[d] & Q_{1} \ar[r] \ar[d] & \mathcal{B}_{1} \ar[r] & 0 \\ & 0 & 0 & & \\ } \end{equation} and thus one has a non-locally free monad \begin{equation}\label{M1} M_{1}:\quad 0\longrightarrow\mathcal{A}_{1}\longrightarrow W_{1}\longrightarrow\mathcal{B}_{1}\longrightarrow0 \end{equation} with cohomology the torsion-free sheaf $\mathcal{E}.$ Note that when the extension $Q_{1}$ was constructed, we also required that the induced maps $B_{i}\longrightarrow B_{i},$ in cohomology, are all isomorphisms. For further computations one needs to know the Chern characters of the sheaves involved in the display. By the Riemann-Roch theorem one, first, has $\chi(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))=a_{i}$, where the Euler characteristic of a pair of coherent sheaves $(\mathcal{F}, \mathcal{G})$ on an algebraic variety $X$ is $\chi(\mathcal{F},\mathcal{G}):=\Sigma_{i=0}^{dimX}(-1)^{i}dim\ext^{i}(\mathcal{F},\mathcal{G})$, see \cite[Definition \textbf{6.1.1}]{Huy}. If we put $d_{i}=dim\ext^{1}(\mathcal{E}, \mathcal{O}|_{E_{i}}(-1))=dimA_{i}$ and $d'_{i}=dim\Hom(\mathcal{E}, \mathcal{O}|_{E_{i}}(-1))=dimB_{i}$ then $d_{i}-d'_{i}=-a_{i}$. We also put $D=\Sigma_{i=1}^{n}d_{i}=rk\mathcal{A}_{1}$ and $D'=rk\mathcal{B}_{1}=\Sigma_{i=1}^{n}d'_{i}$, then $D-D'=-\Sigma_{i=1}^{n}a_{i}:=-\bar{a}$. It follows that: $$ch(\mathcal{A}_{1})=D-[DH-\Sigma_{i=1}^{n}d_{i}E_{i}]$$ $$ch(\mathcal{B}_{1})=D'+[D'H-\Sigma_{i=1}^{n}d'_{i}E_{i}]$$ By the additivity of the Chern character on exact sequences one has the following: $$ch(X_{1})=ch(\mathcal{A}_{1})+ch(\mathcal{E})=(r+D)-[DH-\Sigma_{i=1}^{n}(d_{i}+a_{i})E_{i}]-(k+\frac{|\overrightarrow{a}|^{2}}{2})\omega$$ $$ch(Q_{1})=ch(\mathcal{B}_{1})+ch(\mathcal{E})=(r+D')+[D'H-\Sigma_{i=1}^{n}(d'_{i}-a_{i})E_{i}]-(k+\frac{|\overrightarrow{a}|^{2}}{2})\omega$$ $$ch(W_{1})=ch(\mathcal{A}_{1})+ch(Q_{1})=ch(\mathcal{B}_{1})+ch(X_{1})=(r+D+D')+\bar{a}H-(k+\frac{|\overrightarrow{a}|^{2}}{2})\omega$$ From the lower row of the display \eqref{display}, it follows that $Q_{1}$ is torsion-free since $\mathcal{E}$ is torsion-free and $\mathcal{B}_{1}$ is locally free. Similarly from the middle column of the display it follows that $W_{1}$ is also torsion-free. This implies from the middle row that $X_{1}$ is a torsion-free sheaf. Furthermore, by dualizing twice the exact sequences in the display one can show that $\Delta_{X}\cong\Delta_{W}$ and $\Delta_{Q}\cong\Delta,$ where we define $\Delta_{W}:=W_{1}^{\ast\ast}/W_{1}$, $\Delta_{X}:=X_{1}^{\ast\ast}/X_{1}$ and $\Delta_{Q}:=Q_{1}^{\ast\ast}/Q_{1}.$ So the complex \eqref{M1} has the disadvantage of containing a non-locally free sheaf in its middle term. To solve this problem we need to construct a monad according to the definition \textbf{\ref{monad}}, i.e., with holomorphic bundles in all its terms. This will be done in few steps, but first we need the following: \begin{pr}\label{push&pull} Let $\mathcal{F}$ be a torsion-free sheaf on $\tilde{X}de{\mathbb{P}}$, satisfying $$\h^{0}(E_{i}, \mathcal{F}|_{E_{i}}(-1))=0,$$ $$\h^{1}(E_{i}, \mathcal{F}|_{E_{i}}(-1))=0,$$ for some $i$, and let $\Delta_{\mathcal{F}}$ be the quotient sheaf $(\mathcal{F}^{\ast\ast}/\mathcal{F})$. Then $Supp(\Delta_{\mathcal{F}})\cap E_{i}=\emptyset$ and $\mathcal{F}$ is trivial on the divisor $E_{i}$. Moreover if this holds for every $i$, then $\mathcal{F}\cong \pi^{\ast}(\pi_{\ast}\mathcal{F})$. \end{pr} \begin{proof} Let us consider the sequence $0\longrightarrow\mathcal{F}\longrightarrow\mathcal{F}^{\ast\ast}\longrightarrow\Delta_{\mathcal{F}}\longrightarrow0.$ Restricting to $E_{i}$ we have: $$\xymatrix@R-1.5pc@C-1pc{0\ar[r]&T\ar[r]&\mathcal{F}|_{E_{i}}\ar[rr]\ar@{-->}[rd]&&\mathcal{F}^{\ast\ast}|_{E_{i}}\ar[r] &\Delta_{\mathcal{F}}\otimes\mathcal{O}|_{E_{i}}\ar[r]&0\\ &&&\mathcal{L}\ar@{-->}[rd]\ar@{-->}[ru]&&&\\ &&0\ar@{-->}[ru]&&0&& }$$ where $T:=\tor^{1}(\mathcal{O}|_{E_{i}}, \Delta_{\mathcal{F}})$ and $\mathcal{L}$ is locally free. Since $E_{i}$ is a curve, then $\mathcal{F}|_{E_{i}}\cong\mathcal{L}\oplus T.$ The first condition means that $\h^{0}(E_{i}, \mathcal{L}(-1))\oplus \h^{0}(E_{i}, T(-1))=0$, thus $T=\tor^{1}(\mathcal{O}|_{E_{i}}, \Delta_{\mathcal{F}})=0$. It follows that $Supp(\Delta_{\mathcal{F}})\cap E_{i}=\emptyset$. Then $\Delta_{\mathcal{F}}\otimes\mathcal{O}|_{E_{i}}=0$, and the sheaf $\mathcal{F}|_{E_{i}}$ is isomorphic to $\mathcal{F}^{\ast\ast}|_{E_{i}}.$ The second condition means that $\mathcal{F}|_{E_{i}}$ is trivial on $E_{i}$. Now if the conditions hold for every divisor $E_{i}$, then $\mathcal{F}|_{E_{i}}$ is trivial on every $E_{i}$ and $Supp(\Delta_{\mathcal{F}})\cap E_{i}=\emptyset\quad\forall i$. Hence $\mathcal{F}$ is the pull-back of its direct image on $\mathbb{P}^{2}$. \end{proof} \begin{lem} $W_{1}\cong\pi^{\ast}(\pi_{\ast}W_{1})$. \end{lem} \begin{proof} Twisting the monad by $\mathcal{O}(E_{i})$ and restricting to the exceptional divisor $E_{i}$ one has, from the middle column of the display: $$\xymatrix@C-1.3pc{0\ar[r]&\tor^{1}(\mathcal{E}(-1), \mathcal{O}|_{E_{i}})\ar[r]&\tor^{1}(X_{1}(-1), \mathcal{O}|_{E_{i}})\ar[r]&\oplus_{j\neq i}A_{j}(-1)\oplus A_{i}(-2)}$$ $$\xymatrix@C-1.2pc{\ar[r]&W_{1}|_{E_{i}}(-1)\ar[r]&Q_{1}|_{E_{i}}(-1)\ar[r]&0}$$ which can be split as $$\xymatrix@R-1.6pc@C-1.4pc{0\ar[r]&\tor^{1}(\mathcal{E}(-1), \mathcal{O}|_{E_{i}})\ar[r] & \tor^{1}(X_{1}(-1), \mathcal{O}|_{E_{i}})\ar[r]&\mathcal{F}\ar[r]&0 \\ 0\ar[r]&\mathcal{F}\ar[r]&\oplus_{j\neq i}A_{j}(-1)\oplus A_{i}(-2)\ar[r]&\mathcal{G}\ar[r]&0 \\ 0\ar[r]&\mathcal{G}\ar[r]&W_{1}|_{E_{i}}(-1)\ar[r]&Q_{1}|_{E_{i}}(-1)\ar[r]&0 }$$ The $\tor-$sheaves are supported on points lying on the curve $E_{i}$, and so is the sheaf $\mathcal{F}.$ Thus $\h^{1}(E_{i},\mathcal{F})=0.$ On the other hand $\h^{0}(E_{i},\oplus_{j\neq i}A_{j}(-1)\oplus A_{i}(-2))=0$ implying that $\h^{0}(E_{i},\mathcal{F})=0,$ then $\mathcal{F}$ is the zero sheaf. This means that we have an isomorphism of sheaves $\tor^{1}(\mathcal{E}(-1) \mathcal{O}|_{E_{i}})\cong\tor^{1}(X_{1}(-1), \mathcal{O}|_{E_{i}}).$ Moreover the sequence $$\xymatrix@C-1.4pc{0\ar[r]&\oplus_{j\neq i}A_{j}(-1)\oplus A_{i}(-2)\ar[r]&W_{1}|_{E_{i}}(-1)\ar[r]&Q_{1}|_{E_{i}}(-1)\ar[r]&0}$$ is exact. Its long exact sequence in cohomology gives: \begin{align}\label{trivW} 0&\longrightarrow\underbrace{\h^{0}(E_{i},\oplus_{j\neq i}A_{j}(-1)\oplus A_{i}(-2))}_{0}\longrightarrow \h^{0}(E_{i},W_{1}|_{E_{i}}(-1))\longrightarrow \h^{0}(E_{i},Q_{1}|_{E_{i}}(-1)) \notag \\ & \\ &\longrightarrow\underbrace{\h^{1}(E_{i},\oplus_{j\neq i}A_{j}(-1)\oplus A_{i}(-2))}_{A_{i}}\longrightarrow \h^{1}(E_{i},W_{1}|_{E_{i}}(-1))\longrightarrow \h^{1}(E_{i},Q_{1}|_{E_{i}}(-1))\longrightarrow0 \notag \end{align} but from the last row of the display, i.e. $\xymatrix@C-1.3pc{0 \ar[r] & \mathcal{E}|_{E_{i}}(-1) \ar[r]&Q_{1}|_{E_{i}}(-1)\ar[r] & \oplus_{j\neq i}B_{j}(-1)\oplus B_{i} \ar[r] & 0}$, one has $$\xymatrix{0\ar[r]& A_{i}\ar[r]& \h^{0}(E_{i},Q_{1}|_{E_{i}}(-1))\ar[r]& B_{i}\ar[r]^\sim & B_{i}\ar[r]& \h^{1}(E_{i},Q_{1}|_{E_{i}}(-1))\ar[r]&0}$$ which means that $$\h^{0}(E_{i},Q_{1}|_{E_{i}}(-1))\cong A_{i},\quad\quad \h^{1}(E_{i},Q_{1}|_{E_{i}}(-1))=0$$ and $$\quad\quad\quad \h^{0}(E_{i},W_{1}|_{E_{i}}(-1))=0, \quad\quad \h^{1}(E_{i},W_{1}|_{E_{i}}(-1))=0 \quad \forall i=1,n.$$ Thus the lemma follows from proposition \textbf{\ref{push&pull}}. We used the fact that the complex $M_{1}$ was constructed so that all the induced maps $B_{i}=\h^{0}(E_{i},\mathcal{B}_{1}\otimes\mathcal{O}|_{E_{i}}(-1))\longrightarrow B_{i}=\h^{0}(E_{i},\mathcal{E}\otimes\mathcal{O}|_{E_{i}}(-1))$ are isomorphisms for each $i.$ \end{proof} The next step will be the construction of a monad on $\mathbb{P}^{2}$ which describes the sheaf $\pi_{\ast}W_{1}$. For this we need the following: \begin{thm}\label{thm1} A torsion-free sheaf $\mathcal{F}$ on $\mathbb{P}^{2}$ is given by the cohomology of a monad with trivial middle term if $$\h^{0}(\mathbb{P}^{2},\mathcal{F}(-1))=0, \qquad \textrm{and}\qquad \h^{0}(\mathbb{P}^{2},\mathcal{F}^{\ast}(-1))=0$$ \end{thm} \begin{proof} Beilinson's theorem (\cite{Okonek} \textbf{3.1.3} and \textbf{3.1.4}) extends to the case of a torsion-free sheaf (\cite{Naka} \textbf{2.1}, \cite{Ancona1} and \cite{Ancona} for applications), hence there exists a spectral sequence $E_{r}^{p,q}$ with first term: $E_{1}^{p,q}=\h^{q}(\mathbb{P}^{2},\mathcal{F}\otimes \Omega^{-p}(-p))\otimes\mathcal{O}(p)$ which converges to : $$E_{\infty}^{p,q}=\left\{\begin{array}{ll}\mathcal{F} & \textrm{for }p+q=0\\0& \textrm{otherwise} \end{array}\right.$$ We apply this to the sheaf $\mathcal{F}(-1)$ and use the vanishing conditions. This leads to a monad, with cohomology $\mathcal{F}(-1)$, given by $$\xymatrix{0\ar[r]&E_{1}^{-2,1}\ar[r]^{d_{1}^{-2,1}}&E_{1}^{-1,1}\ar[r]^{d_{1}^{-1,1}}&E_{1}^{0,1}\ar[r]&0}$$ Twisting the complex by $\mathcal{O}(-1)$ one has the monad: $$\xymatrix@C-0.3pc@R-1.4pc{0\ar[r]&\h^{1}(\mathbb{P}^{2},\mathcal{F}(-2))\otimes\mathcal{O}(-1)\ar[r]^{\quad d_{1}^{-2,1}}&\h^{1}(\mathbb{P}^{2},\mathcal{F}\otimes\Omega^{1})\otimes\mathcal{O} &&\\ &\qquad\quad\ar[r]^{d_{1}^{-1,1}\quad}&\h^{1}(\mathbb{P}^{2},\mathcal{F}(-1))\otimes\mathcal{O}(1)\ar[r]&0&&}$$ with cohomology the sheaf $\mathcal{F}.$ \end{proof} \begin{lem}\label{dualdirectimage} $\h^{0}(\mathbb{P}^{2},\pi_{\ast}(W_{1}^{\ast})(-1))=\h^{0}(\mathbb{P}^{2},(\pi_{\ast}W_{1})^{\ast}(-1))$ \end{lem} \begin{proof} First one has $\h^{0}(\mathbb{P}^{2},\pi_{\ast}(W_{1}^{\ast})(-1))= \h^{0}(\tilde{X}de{\mathbb{P}},W_{1}^{\ast}(-1,0))$ which by Serre duality is equal to $\h^{2}(\tilde{X}de{\mathbb{P}},W_{1}^{\ast\ast}(-2,\vec{1}))^{\ast}.$ From the natural injection of a torsion-free sheaf in its double dual, one has \begin{align} \h^{2}(\tilde{X}de{\mathbb{P}},W_{1}^{\ast\ast}(-2,\vec{1}))^{\ast}&=\h^{2}(\tilde{X}de{\mathbb{P}},W_{1}(-2,\vec{1}))^{\ast}\notag \\ &=\ext^{2}_{\mathcal{O}_{\tilde{X}de{\mathbb{P}}}}(\mathcal{O}(-2,\vec{1}),W_{1})^{\ast} \notag \\ &=\Hom_{\mathcal{O}_{\tilde{X}de{\mathbb{P}}}}(W_{1},\mathcal{O}(-1,0)). \notag \end{align} Moreover for any sheaf of $\mathcal{O}_{\tilde{X}de{\mathbb{P}}}$-modules $\mathcal{F}$ and for any sheaf of $\mathcal{O}_{\mathbb{P}^{2}}$-modules $\mathcal{G}$ one has the formula (\cite{Hart},II. 5 page 110): $\Hom_{\tilde{X}de{\mathbb{P}}}(\pi^{\ast}\mathcal{G},\mathcal{F})=\Hom_{\mathbb{P}^{2}}(\mathcal{G},\pi_{\ast}\mathcal{F})$ since $\pi_{\ast}$ and $\pi^{\ast}$ are adjoint functors. Then using the fact that $W_{1}$ is the pull-back of its direct image on $\mathbb{P}^{2}$, and the fact that $\pi_{\ast}\mathcal{O}(-1,0)\cong\mathcal{O}(-1)$ we have the canonical isomorphisms \begin{align} \Hom_{\mathcal{O}_{\tilde{X}de{\mathbb{P}}}}(\pi^{\ast}(\pi_{\ast}W_{1}),\mathcal{O}(-1,0))&= \Hom_{\mathcal{O}_{\mathbb{P}^{2}}}(\pi_{\ast}W_{1},\mathcal{O}(-1))\notag \\ &=\ext^{2}_{\mathcal{O}_{\mathbb{P}^{2}}}(\mathcal{O}(-1),\pi_{\ast}W_{1}(-3))^{\ast}\notag \\ &=\h^{2}(\mathbb{P}^{2},\pi_{\ast}W_{1}(-2))^{\ast}. \notag \end{align} Again, by the natural injection of a torsion-free sheaf in its double dual, one has \par\noindent $\h^{2}(\mathbb{P}^{2},\pi_{\ast}W_{1}(-2))^{\ast}= \h^{2}(\mathbb{P}^{2},(\pi_{\ast}W_{1})^{\ast\ast}(-2))^{\ast}= \h^{0}(\mathbb{P}^{2},(\pi_{\ast}W_{1})^{\ast}(-1))$ from which the claim follows. \end{proof} \begin{pr} The direct image $\pi_{\ast}W_{1}$ of the sheaf $W_{1}$ is given by the cohomology of a monad on $\mathbb{P}^{2}$ with trivial middle term. \end{pr} \begin{proof} It suffices to verify the vanishing given in theorem \textbf{\ref{thm1}}. \underline{$\h^{0}(\mathbb{P}^{2},(\pi_{\ast}W_{1})^{\ast}(-1))=0$:} From lemma \textbf{\ref{dualdirectimage}} we have $\h^{0}(\mathbb{P}^{2},(\pi_{\ast}W_{1})^{\ast}(-1))=\h^{0}(\mathbb{P}^{2},\pi_{\ast}(W_{1}^{\ast})(-1)).$ On the other hand we have $\h^{0}(\mathbb{P}^{2},\pi_{\ast}(W_{1}^{\ast})(-1))=\h^{0}(\tilde{X}de{\mathbb{P}},W_{1}^{\ast}(-1,0))$ since $W_{1}^{\ast}(-1,0)$ is the pullback of $\pi_{\ast}(W_{1}^{\ast})(-1)$ under the blow-down map. Then it suffices to show that $\h^{0}(\tilde{X}de{\mathbb{P}},W_{1}^{\ast}(-1,0))=0$ in order to prove the first vanishing: dualizing the first row of the display \eqref{display} and twisting the resulting sequence by $\mathcal{O}(-1,0),$ one has the following sequence in cohomology $$0\longrightarrow\oplus_{i=1}^{n}B^{\ast}_{i}\otimes\underbrace{\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(-2,-E_{i}))}_{0}\longrightarrow \h^{0}(\tilde{X}de{\mathbb{P}},W_{1}^{\ast}(-1,0))\longrightarrow \h^{0}(\tilde{X}de{\mathbb{P}},X_{1}^{\ast}(-1,0))$$ Taking the dual sequence of the left column of the display \eqref{display} and twisting by $\mathcal{O}(-1,0),$ one has the following induced exact sequence in cohomology $$0\longrightarrow\underbrace{\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{E}^{\ast}(-1,0))}_{0}\longrightarrow \h^{0}(\tilde{X}de{\mathbb{P}},X^{\ast}(-1,0))\longrightarrow \h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{A}^{\ast}_{1}(-1,0))$$ but $\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{A}_{1}^{\ast}(-1,0))\cong\oplus_{i=1}^{n}A_{i}^{\ast}\otimes \h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(0,-E_{i}))$ where the group $\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(0,-E_{i}))$ is zero for any $i$. Thus $\h^{0}(\tilde{X}de{\mathbb{P}},X_{1}^{\ast}(-1,0))=0$ which implies that $W^{\ast}_{1}(-1)$ has no global sections. \underline{$\h^{0}(\mathbb{P}^{2},\pi_{\ast}W_{1}(-1))=0$:} $\quad$ Again it suffices to show that $\h^{0}(\tilde{X}de{\mathbb{P}},W_{1}(-1,0))=0.$ If one twists the display \eqref{display} by $\mathcal{O}(-1,0),$ then the vanishing follows by the same argument as above. Hence the torsion-free sheaf $\pi_{\ast}W_{1}$ is described as the cohomology of a monad on $\mathbb{P}^{2}$ with trivial middle term. \end{proof} The monad which has cohomology the sheaf $\pi_{\ast}W_{1}$ is given by \begin{equation*} M'_{0}:\quad 0\longrightarrow K_{0}(-1)\longrightarrow W\longrightarrow L_{0}(1)\longrightarrow0 \end{equation*} where $K_{0}=\h^{1}(\tilde{X}de{\mathbb{P}}, W_{1}(-2,\overrightarrow{1}))$, $L_{0}=\h^{1}(\tilde{X}de{\mathbb{P}}, W_{1}(-1,0))$ and $W$ is a trivial bundle. To see that the spaces $K_{0}$ and $L_{0}$ yield the Chern character $ch(W_{1})$ we shall compute their dimensions: \begin{pr} $$\h^{0}(\tilde{X}de{\mathbb{P}}, W_{1}(-2,\overrightarrow{1}))=\h^{0}(\tilde{X}de{\mathbb{P}}, W_{1}(-1,0))=0$$ and $$\h^{2}(\tilde{X}de{\mathbb{P}}, W_{1}(-2,\overrightarrow{1}))=\h^{2}(\tilde{X}de{\mathbb{P}}, W_{1}(-1,0))=0$$ \end{pr} \begin{proof} The proof is given by using the display \eqref{display} twisted by $\mathcal{O}(-1,0)$ and taking the induced long exact sequences in cohomology. \end{proof} \begin{cor} The spaces $K_{0}$ and $L_{0}$ have dimension $k+\frac{|\overrightarrow{a}|^{2}-\bar{a}}{2}$ and $k+\frac{|\overrightarrow{a}|^{2}+\bar{a}}{2},$ respectively. \end{cor} \begin{proof} Using the Riemann-Roch formula we compute the Euler characteristics of $W_{1}(-2,\overrightarrow{1})$ and $W_{1}(-1,0)$. This gives \begin{equation} \chi(W_{1}(-2,\overrightarrow{1}))=-(k+\frac{|\overrightarrow{a}|^{2}+\bar{a}}{2})\quad\textrm{and}\quad \chi(W_{1}(-1,0))=-(k+\frac{|\overrightarrow{a}|^{2}-\bar{a}}{2}) \end{equation} The corollary follows from the vanishing of the groups in the proposition above. \end{proof} Now we want to construct an intermediate monad with trivial middle term and with cohomology the original sheaf $\mathcal{E}.$ First we have to pull-back the monad $M'_{0}$ to a monad $M_{0}$ on $\tilde{X}de{\mathbb{P}}$: \begin{equation*} M_{0}:\quad 0\longrightarrow K_{0}(-1,0)\longrightarrow W\longrightarrow L_{0}(1,0)\longrightarrow0 \end{equation*} Indeed the first map in the monad $M'_{0}$ vanishes on the singularity set $Sing(\pi_{\ast}W_{1})$ of the torsion-free sheaf $\pi_{\ast}W_{1}.$ On the other hand $Sing(W_{1})\cap E_{i}=\emptyset$ for all $i.$ Moreover $W_{1}=\pi^{\ast}\pi_{\ast}W_{1}.$ This implies that $Sing(\pi_{\ast}W_{1})\cap p_{i}=\emptyset$ for all $i$, where $p_{i}\in\mathbb{P}^{2}$ are the blow-up points. Now the first map in $M'_{0}$ has maximal rank at $p_{i}$ for all $i$, as well as the second map. Consequently the locus on which the first map in $M_{0}$ is not of maximal rank is zero dimensional. Thus the pull-back $M_{0}$ of $M'_{0}$ is a monad. Then we should lift the morphism $\mathcal{A}_{1}\longrightarrow W_{1}$ to a morphism $\mathcal{A}_{1}\longrightarrow X'_{0}$ where $X'_{0}=ker(W\longrightarrow L_{0}(1,0))$ i.e. \begin{equation*} \xymatrix@C-0.5pc@R-0.5pc{&X'_{0}\ar[d] \\ \mathcal{A}_{1}\ar@{-->}[ru]\ar@{^{(}->}[r]&W_{1} } \end{equation*} so we want a surjective morphism $\Hom(\mathcal{A}_{1},X'_{0})\longrightarrow \Hom(\mathcal{A}_{1},W_{1}).$ The obstruction for such a lifting lies in the group $\ext^{1}(\mathcal{A}_{1},K_{0}(-1,0))$ which is zero since $\ext^{1}(\mathcal{A}_{1},K_{0}(-1,0))\cong\oplus_{i=1}^{n}A^{\ast}_{i}\otimes K_{0}\otimes \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{O}(0,-E_{i}))=0.$ This means that all the extensions $$0\longrightarrow K_{0}(-1,0)\longrightarrow \mathcal{A}\longrightarrow \mathcal{A}_{1}\longrightarrow0$$ split, hence $\mathcal{A}\cong K_{0}(-1,0)\oplus\mathcal{A}_{1}.$ Furthermore we have a sheaf monomorphism $\mathcal{A}\longrightarrow W.$ Dually, we want to lift the morphism $W_{1}\longrightarrow\mathcal{B}_{1}$ to a morphism $Q'_{0}\longrightarrow \mathcal{B}_{1}$, where $Q'_{0}=coker(K_{0}(-1,0)\longrightarrow W)$ i.e. \begin{equation*} \xymatrix@C-0.5pc@R-0.5pc{W_{1}\ar@{^{(}->}[d]\ar@{->>}[r]&\mathcal{B}_{1} \\ Q'_{0}\ar@{-->}[ru]& } \end{equation*} We also want a surjective morphism $\Hom(Q'_{0},\mathcal{B}_{1})\longrightarrow \Hom(W_{1},\mathcal{B}_{1})$ in order to do the lift. In this case the obstruction is in the group $\ext^{1}(L_{0}(1,0),\mathcal{B}_{1})$ which also vanishes since $\ext^{1}(L_{0}(1,0),\mathcal{B}_{1})\cong\oplus_{i=1}^{n}B_{i}\otimes L^{\ast}_{0}\otimes \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{O}(0,-E_{i}))=0.$ This means that all the extensions $$0\longrightarrow\mathcal{B}_{1} \longrightarrow \mathcal{B}\longrightarrow L_{0}(1,0)\longrightarrow0$$ split, hence $\mathcal{B}\cong L_{0}(1,0)\oplus\mathcal{B}_{1}.$ Furthermore we have an epimorphism $W\longrightarrow \mathcal{B}$. Consequently we have a monad \begin{equation*} M:\quad 0\longrightarrow K_{0}(-1,0)\oplus\mathcal{A}_{1}\longrightarrow W\longrightarrow L_{0}(1,0)\oplus\mathcal{B}_{1}\longrightarrow0 \end{equation*} with cohomology $\mathcal{F}$ and the following associated display \begin{equation}\label{M'0} \xymatrix@C-1.2pc@R-1.3pc{ & 0 \ar[d] & 0 \ar[d] & & \\ & K_{0}(-1,0)\oplus\mathcal{A}_{1} \ar@{=}[r]\ar[d] & K_{0}(-1,0)\oplus\mathcal{A}_{1} \ar[d] & & \\ 0 \ar[r] & X \ar[r]\ar[d] & W \ar[r] \ar[d] & L_{0}(1,0)\oplus\mathcal{B}_{1} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r]\ar[d] & Q \ar[r] \ar[d] & L_{0}(1,0)\oplus\mathcal{B}_{1} \ar[r] & 0 \\ & 0 & 0 & & \\ } \end{equation} from which we compute the Chern character of the cohomology $\mathcal{F};$ \begin{align} ch(\mathcal{F})&=ch(Q)-ch(L_{0}(1,0))-ch(\mathcal{B}_{1})=ch(X)-ch(K_{0}(-1,0))-ch(\mathcal{A}_{1})\notag \\ &=rk(W)-D-D'-k_{0}-l_{0}-[(D'-D-k_{0}+l_{0})H-\Sigma_{i=1}^{n}(d_{i}+d'_{i})E_{i}]-\frac{(l_{0}+k_{0})}{2}\omega\notag \end{align} and by using the relations: \begin{align} &rk(W)=r+D+D'+k_{0}+l_{0}, \qquad &k_{0}+l_{0}=2k+|\overrightarrow{a}|^{2}\notag \\ &\Sigma_{i=1}^{n}(d_{i}-d'_{i})=-\Sigma_{i=1}^{n}a_{i}=-\bar{a} \qquad &k_{0}-l_{0}=\bar{a}=-(D-D') \end{align} we get $$ch(\mathcal{F})=r+\Sigma_{i=1}^{n}a_{i}E_{i}-(k+\frac{|\overrightarrow{a}|^{2}}{2})\omega=ch(\mathcal{E})$$ One can also use the three displays of $M_{1}$, $M_{0}$ and $M$ to see that $\mathcal{F}\cong\mathcal{E}.$ Thus, a similar result to \cite[Proposition 1.8]{Buch} is given by the following : \begin{thm}\label{thm2} Let $\mathcal{E}$ be a framed torsion-free sheaf with Chern character $ch(\mathcal{E})=r+\Sigma_{i=1}^{n}a_{i}E_{i}-(k+\frac{|\overrightarrow{a}|^{2}}{2})\omega$ on a multi-blow-up, $\pi:\tilde{X}de{\mathbb{P}}\longrightarrow\mathbb{P}^{2}$, of $\mathbb{P}^{2}$ at $n$ distinct points. Then $\mathcal{E}$ is given by the cohomology of a monad: $$M:\quad 0\longrightarrow K_{0}(-1,0)\oplus\mathcal{A}_{1}\longrightarrow W\longrightarrow L_{0}(1,0)\oplus\mathcal{B}_{1}\longrightarrow0$$ where the first map in $M$ is injective as a sheaf map and the second map is surjective, and where \begin{align} &A_{i}:=\ext^{1}(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))^{\ast},\qquad B_{i}:=\Hom(\mathcal{E},\mathcal{O}|_{E_{i}}(-1))^{\ast} \notag \\ &\mathcal{A}_{1}:=\oplus_{i=1}^{n} A_{i}(-1,E_{i}), \qquad\quad \mathcal{B}_{1}:=\oplus_{i=1}^{n} B_{i}(1,-E_{i})\notag\\ &K_{0}:=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1})),\qquad\quad L_{0}:=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,0)). \notag \end{align} \end{thm} \begin{rmk} \begin{itemize} \item[(i)] \textnormal{Using the display of the monad $M_{1}$ we can write \begin{align} &K_{0}:=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1}))\oplus \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{A}_{1}(-2,\overrightarrow{1}))\oplus \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{B}_{1}(-2,\overrightarrow{1})) \notag \\ &L_{0}:=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,0))\oplus \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{A}_{1}(-1,0))\oplus \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{B}_{1}(-1,0)). \notag \end{align} From the Riemann-Roch formula one has $\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{A}_{1}(-2,\overrightarrow{1}))\cong\oplus_{i=1}^{n}A_{i}\otimes \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{O}(-3,\overrightarrow{1}+E_{i}))=0$ and also $\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{B}_{1}(-2,\overrightarrow{1}))\cong\oplus_{i=1}^{n}B_{i}\otimes \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{O}(-1,\overrightarrow{1}-E_{i}))=0.$ The vanishing holds also for $\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{A}_{1}(-1,0))$ and $\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{B}_{1}(-1,0)).$ Hence we have the forms of $K_{0}$ and $L_{0}$ given in the theorem above.} \item[(ii)]\textnormal{The fact that we restrict to blow-ups of $\mathbb{P}^{2}$ made at distinct points, excluding the case of iterated blow-ups, is a technical assumption, and not a conceptual one. Without it the construction of explicit ADHM type data would be considerably more complicated. In this general case, one can have a monad, associated to $\mathcal{E}$ with the same conditions as in the theorem above, in which the first and last terms are non-trivial extensions of the form $$\xymatrix@C-0.5pc@R-2pc{0\ar[r]&K_{0}(-1,0)\ar[r]&\mathcal{A}\ar[r]&\mathcal{A}_{1}\ar[r]&0 \\ 0\ar[r]&\mathcal{A}_{1}\ar[r]&\mathcal{B}\ar[r]&L_{0}(1,0)\ar[r]&0}$$ rather then being direct sums, as given by \cite[Proposition 1.5]{Buch} in the case of bundles. This is because, for any $i,$ one has $\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{O}(0,-E_{i}))\neq0.$ So in what follows we will still be considering only the case of multiple blow-ups at distinct points. } \end{itemize} \end{rmk} In order to study the moduli space of framed torsion-free sheaves on $\tilde{X}de{\mathbb{P}}$ through monads, we need to know if families of monads of the type we are considering behave well in describing families of torsion-free sheaves; we start by reminding the following: \begin{pr} Let $M:0\longrightarrow\mathcal{A}\longrightarrow \mathcal{W}\longrightarrow \mathcal{B}\longrightarrow0$ and $M':0\longrightarrow\mathcal{A}'\longrightarrow \mathcal{W}'\longrightarrow \mathcal{B}'\longrightarrow0$ be two monads on a surface $\mathcal{S}$ with cohomologies $\mathcal{E}$ and $\mathcal{E}'$ respectively. The morphism $H:\Hom(M,M')\longrightarrow \Hom(\mathcal{E},\mathcal{E}')$ is surjective if $$\ext^{1}(\mathcal{B},\mathcal{W}')=\ext^{1}(\mathcal{W},\mathcal{A}')=\ext^{2}(\mathcal{B},\mathcal{A}')=0.$$ Furthermore its kernel is identified with $\ext^{1}(\mathcal{B},\mathcal{A}')$ if $\Hom(\mathcal{B},\mathcal{W}')=\Hom(\mathcal{W},\mathcal{A}')=0.$ \end{pr} \begin{proof} The proof for the locally-free case can be found in \cite[Chapter \textbf{II}, Lemma \textbf{4.1.3}]{Okonek}. In the torsion-free case, the steps of the proof can be repeated provided that one replaces suitably the global section functor $\Gamma$ by the $\Hom$ functor. \end{proof} In our case, one has \begin{align} &\ext^{1}(\mathcal{B},W')=0,\quad\quad \ext^{1}(W,\mathcal{A}')=0,\quad\quad \ext^{2}(\mathcal{B},\mathcal{A}')=0, \notag\\ &\Hom(\mathcal{B},W')=0, \quad\quad \Hom(W,\mathcal{A}')=0,\notag \end{align} by using the Riemann-Roch theorem. Hence $H:\Hom(M,M')\longrightarrow \Hom(\mathcal{E},\mathcal{E}')$ is surjective with kernel $\ext^{1}(\mathcal{B},\mathcal{A}')\cong\oplus_{i=1}^{n} B^{\ast}_{i}\otimes A_{i}'$. This means that the functor \begin{equation}\label{functor} \mathfrak{H}:\mathfrak{M}on\longrightarrow\mathfrak{F}ram, \end{equation} from the category of monads given by theorem ${\bf\ref{thm2}}$ to the category of framed torsion-free sheaves on $\tilde{X}de{\mathbb{P}},$ is full. In the next section we will get rid of the kernel of $H:=\mathfrak{H}(M,M')$ by reducing the monad, so that the corresponding reduced functor is fully faithfull. We also remark that since the operation of taking cohomology is functorial, then the morphisms of the monad of theorem ${\bf\ref{thm2}}$ are natural in the family of framed torsion-free sheaves. The monad we constructed describes well families of framed torsion-free sheaves and one can talk about a moduli space of such objects, but we still have to fix the problem of the control on the dimensions of $A_{i}$ and $B_{i}$; their difference is constant $dimA_{i}-dimB_{i}=a_{i}$, but each dimension can, a priori, jump. Proceeding as in \cite[Secion {\bf 1}]{Buch}, we apply the functor $\Hom(\mathcal{E},\cdot)$ to the sequence \eqref{rest2} twisted by $\mathcal{O}(-1,E_{i})$ : $$0\longrightarrow B_{i}^{\ast}\longrightarrow \ext^{1}(\mathcal{E},\mathcal{O}(-1,0))\longrightarrow \ext^{1}(\mathcal{E},\mathcal{O}(-1,E_{i}))\longrightarrow A_{i}^{\ast}\longrightarrow0.$$ We split this sequence by defining $V_{i}^{\ast}=ker(\ext^{1}(\mathcal{E},\mathcal{O}(-1,E_{i}))\longrightarrow A_{i}^{\ast}),$ thus getting $$0\longrightarrow B_{i}^{\ast}\longrightarrow \ext^{1}(\mathcal{E},\mathcal{O}(-1,0))\longrightarrow V_{i}^{\ast}\longrightarrow0$$ $$0\longrightarrow V_{i}^{\ast}\longrightarrow \ext^{1}(\mathcal{E},\mathcal{O}(-1,E_{i}))\longrightarrow A_{i}^{\ast}\longrightarrow0$$ dualizing the sequences and using the fact that $$\ext^{1}(\mathcal{E},\mathcal{O}(-1,0))^{\ast}=\ext^{1}(\mathcal{O}(-1,0),\mathcal{E}(-3,\overrightarrow{1})) =\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1}))\quad\textrm{and}$$ $$\ext^{1}(\mathcal{E},\mathcal{O}(-1,E_{i}))^{\ast}=\ext^{1}(\mathcal{O}(-1,E_{i}),\mathcal{E}(-3,\overrightarrow{1})) =\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1}-E_{i}))$$ which have dimensions, respectively, $k+\frac{1}{2}\Sigma_{j=1}^{n}a_{j}(a_{j}+1)$ and $k+\frac{1}{2}\Sigma_{j=1}^{n}a_{j}(a_{j}+1)-a_{i}$, one has the isomorphisms : \begin{equation}\label{isom} A_{i}\oplus V_{i}\cong \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1}-E_{i})),\qquad B_{i}\oplus V_{i}\cong \h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1})). \end{equation} On the other hand consider, for each $i,$ the following extension \begin{equation}\label{W0} 0\longrightarrow \mathcal{O}(-1,E_{i})\longrightarrow W_{0}\longrightarrow\mathcal{O}(1,-E_{i})\longrightarrow0 \end{equation} The group classifying such extensions is $\ext^{1}(\mathcal{O}(1,-E_{i}),\mathcal{O}(-1,E_{i}))\cong\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{O}(-2, 2E_{i}))\cong\mathbb{C}$ whose dimension is 1. Taking the non-trivial extension in this group we show that $W_{0}$ is a trivial 2-bundle: the extension obtained after restricting to the generic line $H$ corresponds to a non zero element in $\h^{1}(H,\mathcal{O}_{H}(-2))=\mathbb{C}$, since there is an isomorphism $\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{O}(-2, 2E_{i}))\cong\h^{1}(H,\mathcal{O}_{H}(-2)),$ furthermore it is a twisted Euler sequence $0\longrightarrow\mathcal{O}_{H}(-1)\longrightarrow\mathcal{O}_{H}^{\oplus2}\longrightarrow T_{H}(-1)\longrightarrow0$ associated to the line $H.$ The twisting is given by $\mathcal{O}_{H}(-1)$ and the twisted tangent space $T_{H}(-1)$ is identified with $\mathcal{O}_{H}(1).$ Thus $W_{0}|_{H}$ is the trivial bundle $\mathcal{O}_{H}^{\oplus2}$. Restricting the extension \eqref{W0} to the exceptional divisor $E_{j}$, one concludes, in the same way as above, that if $i=j$ then $W_{0}|_{E_{j}}$ corresponds to a twisted Euler sequence associated to the line $E_{i},$ where the twisting is given by $\mathcal{O}_{E_{i}}(-1),$ hence $W_{0}|_{E_{j}}$ is the trivial 2-bundle $\mathcal{O}_{E_{j}}^{\oplus2}$. When $i\neq j$ it is obvious that the restriction is a trivial 2-bundle. Consequently $W_{0}$ is a trivial 2-bundle on $\tilde{X}de{\mathbb{P}}$ since its Chern classes are both zero. Now we have to twist the extension \eqref{W0} by $V_{i}$; \begin{equation} 0\longrightarrow V_{i}(-1,E_{i})\longrightarrow V_{i}\otimes W_{0}\longrightarrow V_{i}(1,-E_{i})\longrightarrow0 \end{equation} but since adding such exact sequences, for every $i$, to the monad in the theorem \textbf{\ref{thm2}} will not change the cohomology $\mathcal{E},$ and moreover, by using the isomorphisms \eqref{isom} one has a generalization of \cite[Prposition 1.10]{Buch} given by the following \begin{pr}\label{prop} Let $\mathcal{E}$ be a torsion-free sheaf satisfying the conditions of theorem \textbf{\ref{thm2}}. Then there exists a monad $M$ of the form: $$\xymatrix@R-1pc{M:& 0\ar[r]& \oplus_{i=0}^{n}K_{i}(-1,E_{i})\ar[r]^{\qquad\quad\alpha}& W \ar[r]^{\beta\qquad\quad}&\oplus_{i=0}^{n}L_{i}(1,-E_{i})\ar[r]&0}$$ whose cohomology is $\mathcal{E},$ and in which $\alpha$ is injective as a sheaf morphism and $\beta$ is surjective. Moreover $$\left\{\begin{array}{l} K_{i}=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1}-E_{i})) \\ L_{i}=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1})) \end{array}\right. i\neq0$$ $$\left\{\begin{array}{l} K_{0}:=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-2,\overrightarrow{1})) \\ L_{0}:=\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,0)). \end{array}\right.\qquad\quad$$ and $E_{0}:=0.$ \end{pr} \begin{rmk}\label{remark} \begin{itemize} \item[(i)] \textnormal{In the proposition above we have used the same notation as in theorem \textbf{\ref{thm2}}, while the spaces and the monads are different.} \item[(ii)] \textnormal{Since the dimensions of our spaces are as follows: \begin{align} &dim K_{0}=k+\frac{|a|^{2}+\bar{a}}{2},\qquad dim L_{0}=k+\frac{|a|^{2}-\bar{a}}{2}\notag\\ &dim K_{i}=k+\frac{|a|^{2}+\bar{a}}{2}-a_{i},\qquad dim L_{i}=k+\frac{|a|^{2}+\bar{a}}{2}\notag \end{align} then $$dim K_{0}+\Sigma_{i=1}^{n}dim K_{i}=(n+1)k+\frac{n+1}{2}(|a|^{2}+\bar{a})-\bar{a}$$ and $$dim L_{0}+\Sigma_{i=1}^{n}dim L_{i}=(n+1)k+\frac{n+1}{2}|a|^{2}+\frac{n-1}{2}\bar{a}=(n+1)k+\frac{n+1}{2}(|a|^{2}+\bar{a})-\bar{a},$$ thus the left and right hand terms in the monad have the same rank $dim L_{0}+\Sigma_{i=1}^{n}dim L_{i}.$ Hence $rkW=2(dim L_{0}+\Sigma_{i=1}^{n}dim L_{i})+r.$} \item[(iii)] \textnormal{Using the display of the monad above one can see that the kernel $X$ of the map $\beta$ is a locally free sheaf (since it is the kernel of a bundle map). The fact that the cohomology is a torsion-free sheaf implies that the first map $\alpha$ fails to be of maximal rank on finitely many points. These are the singularity set of the torsion-free sheaf that we are describing. Conversely if the map $\alpha$ is not of maximal rank on some finite set of points, then the cohomology of the monad would be a torsion-free sheaf with singularity set, exactly, the set of the points where $\alpha$ vanishes.} \end{itemize} \end{rmk} \section{The ADHM data}\label{ADHM} In this section we describe briefly the ADHM data associated to the monad of proposition \textbf{\ref{prop}}. This will lead to a presentation of the moduli space of framed torsion-free sheaves under study as quotient of a space of some matrices satisfying some constraints by the action of an algebraic group. We now summarize some notations due to Penrose and Buchdahl (see \cite[Section \textbf{2}]{Buch}). We denote the homogeneous coordinates on $\mathbb{P}^{2}$ by $(z^{0},z^{1},z^{2})$. The line at infinity $l_{\infty}$ is given by the equation $z^{2}=0$. Let $p_{i}$ be one of the points in $\mathbb{P}^{2}$ at which we perform a blow-up, and assume $p_{i}\notin l_{\infty}$ for all $i$. So all these points are in the chart $[z^{0},z^{1},1]$ and denote their inhomogeneous coordinates there by $p^{a}_{i}=(p^{A}_{i},1)$ where $A=0,1$ and $a=0,1,2$. Locally, near every blow-up point, $\tilde{X}de{\mathbb{P}}$ can be described by $\{([z^{0},z^{1},z^{2}],[w_{i}^{0},w_{i}^{1}])\in\mathbb{P}^{2}\times\mathbb{P}^{1} \mid (z^{0}-p^{0}_{i}z^{2})w^{1}_{i}=(z^{1}-p^{1}_{i}z^{2})w^{0}_{i}\}.$ Each blow-up equation \begin{equation}\label{blowup} (z^{0}-p^{0}_{i}z^{2})w^{1}_{i}=(z^{1}-p^{1}_{i}z^{2})w^{0}_{i}, \qquad\qquad (w^{0},w^{1})\in\mathbb{C}^{2}\setminus\{0\}, \end{equation} is satisfied if and only if there exists a unique scalar $\lambda_{i}$ such that \begin{equation}\label{blowup1} \lambda_{i}w_{i}^{1}=z^{1}-p^{1}_{i}z^{2}\quad\textnormal{and}\quad\lambda_{i}w_{i}^{0}=z^{0}-p^{0}_{i}z^{2}. \end{equation} Moreover $z^{0}-p^{0}_{i}z^{2}$ and $z^{1}-p^{1}_{i}z^{2}$ are both sections of $\mathcal{O}(H)$ vanishing on $E_{i},$ while $w_{i}^{0}$ and $w_{i}^{1}$ can be viewed as sections of $\mathcal{O}(H-E_{i}).$ Thus the homogeneous equation \eqref{blowup} uniquely determines a section $\lambda_{i}$ of a line bundle $\mathcal{O}(E_{i})$ over the blow-up such that \eqref{blowup1} is satisfied. Note also that the restriction of $w^{A}_{i}$ to the exceptional divisor $E_{i}$ gives the homogeneous coordinates $(w^{0}_{i},w^{1}_{i})$ of $E_{i}$. Finally we denote $$\left\{\begin{array}{ll}w_{i0}:=-w^{1}_{i} & \\ w_{i1}:=w^{0}_{i}&\end{array}\right.$$ Assuming the convention of summing over repeated upper and lower indices, it follows that $w^{A}_{i}w_{iA}=0.$ More generally if we have a pair of matrices as $m_{A}=(m_{0},m_{1})$ we use the same two index notations as above, and we write $m^{A}m_{A}=m_{0}m_{1}-m_{1}m_{0}.$ For the direct sum $V\oplus V$ and the morphism $(m_{0},m_{1}):V\longrightarrow V\oplus V$ we use the notation $m_{A}:V\longrightarrow V_{A}$. Given a morphism $V\oplus V\longrightarrow V$ where $(v_{0},v_{1})\longrightarrow m^{0}v_{0}+m^{1}v_{1}$, this would be written as $m^{A}:V_{A}\longrightarrow V$. Let us consider a monad $M$ as in Proposition \textbf{\ref{prop}} $$\xymatrix{M:&0\ar[r]&\oplus_{i=0}^{n}K_{i}(-1,E_{i})\ar[r]^{\quad\quad\alpha}& W\ar[r]^{\beta\quad\quad}& \oplus_{i=0}^{n}L_{i}(1,-E_{i})\ar[r]&0}$$ with cohomology a framed torsion-free sheaf $\mathcal{E}$. The map $\beta$ is surjective everywhere on $\tilde{X}de{\mathbb{P}},$ while the map $\alpha$ is injective except at some finite set of points in $\tilde{X}de{\mathbb{P}}.$ The condition $\beta\circ\alpha=0$ will give a set of matrix equations. At this point we can reduce the linear data, associated to the monad above and the automorphism group acting on it, in a similar way to reference \cite[section \textbf{3}]{Buch}, to which we refer for more details, since the difference between the above monad and the one given in \cite{Buch} resides only in the condition imposed on the rank of the map $\alpha$; in our case the rank of $\alpha$ is maximal except at finitely many points. It follows that the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ of framed torsion-free sheaves $\mathcal{E}$, on $\tilde{X}de{\mathbb{P}}$, with Chern character $ch(\mathcal{E})=r+\Sigma_{i=1}^{n}a_{i}E_{i}-(k+\frac{|\vec{a}|^{2}}{2})\omega$, is identified with the quotient $P'/G' $, where $P'$ is the space of configurations $(a,a_{00}^{A},c,d)$ defined as follows: $a$ is a non-singular matrix of the form \begin{equation} a=\left[\begin{array}{lllll}a_{00}&a_{01}&a_{02}&\cdots&a_{0n}\\ a_{10}&a_{11}&0&\cdots&0\\ a_{20}&0&a_{22}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ a_{n0}&0&0&\cdots&a_{nn}\\ \end{array}\right] \end{equation} where $a_{ij}\in \Hom(K_{j},L_{i}),$ $c\in \Hom(K_{0},\mathbb{C}^{r})$, $d\in \Hom(\mathbb{C}^{r},L_{0})$ and $a_{00A}\in \Hom(K_{0},L_{0}),$ for $A=0,1$. The configuration $(a,a_{00}^{A},c,d)$ satisfies the equation \begin{equation}\label{constr} (q^{A}a^{-1}q_{A})^{00}+dc=0 \end{equation} where the $00$ subscript means the $00$ entry and the matrix $q^{A}$ is defined as \begin{equation} q^{A}=\left[\begin{array}{lllll}-a_{00}^{A}&p^{A}_{1}a_{01}&p^{A}_{2}a_{02}&\cdots&p^{A}_{n}a_{0n} \\ p^{A}_{1}a_{10}&p^{A}_{1}a_{11}&0 &\cdots&0 \\ p^{A}_{2}a_{20}&0&p^{A}_{2}a_{22}&\cdots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ p^{A}_{n}a_{n0}&0&0&\cdots&p^{A}_{n}a_{nn} \end{array}\right] \end{equation} The group $G'$ consists of the non-singular transformations of the form \begin{equation} g=\left[\begin{array}{lllll}g_{00}&g_{01}&g_{02}&\cdots&g_{0n}\\ 0&g_{11}&0&\cdots&0\\ 0&0&g_{22}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&g_{nn}\\ \end{array}\right], \quad h=\left[\begin{array}{lllll}h_{00}&0&0&\cdots&0\\ h_{10}&h_{11}&0&\cdots&0\\ h_{20}&0&h_{22}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ h_{n0}&0&0&\cdots&h_{nn}\\ \end{array}\right] \end{equation} The set of transformations that fix the configuration $(a,q^{A},c,d)$ are of the form $g=1+am$, $h=(1+ma)^{-1}$, where the matrix $m$ is of the form \begin{equation} m=\left[\begin{array}{lllll}0&0&0&\cdots&0\\ 0&m_{11}&0&\cdots&0\\ 0&0&m_{22}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&m_{nn}\\ \end{array}\right]\in \Hom(\oplus_{i=0}^{n}L_{i}, \oplus_{i=0}^{n}K_{i}) \end{equation} Fixing an isomorphism $L_{i}\cong K_{0}$, one can choose the matrix $a$ so that $a_{0i}=1 \quad \forall i>0$. To preserve this form of $a$, the transformation $g$ must be of the above form and such that $g_{ii}=(h_{00}+a_{ii}h_{i0})^{-1}$. One checks that for every matrix $h$ such that $h_{00}+a_{ii}h_{i0}$ is non-singular, there exists a matrix $D$ of the form $D=diag(d_{00},d_{11},\cdots,d_{nn})$ such that $h=(1+ma)^{-1}D,$ where $a$ of the above form. We remark that the space of matrices $m,$ as above, is exactly the kernel of $H:\End(M)\longrightarrow \End(\mathcal{E}).$ Moreover if we consider only actions of $(g,h)$ having the form \begin{equation}\label{group-form} h=diag(h_{00},h_{11},\cdots,h_{nn}),\quad\quad g=\left[\begin{array}{lllll}g_{00}&g_{01}&g_{02}&\cdots&g_{0n}\\ 0&h^{-1}_{00}&0&\cdots&0\\ 0&0&h^{-1}_{00}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&h^{-1}_{00}\\ \end{array}\right]. \end{equation} then one has a free action since the isotropy subgroup, for a given configuration, is essentially the discarded transformations $m$. Finally the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ has another presentation as the quotient $P/G $, where \begin{equation}\label{spaceP} P=\left\{\begin{array}{l}\textnormal{configurations } (a,a_{00}^{A},c,d) \\ \textnormal{ as above} \end{array}| \begin{array}{l}a_{i0} \textnormal{ is equal to the identity matrix } \mathbb{I}_{K_{0}} \textnormal{ on } K_{0} \\ \textnormal{ for each } i>0 \textnormal{ and satisfying the equation \eqref{constr}}\end{array}\right\} \end{equation} The group $G$ consists of the non-singular transformations of the form \eqref{group-form} acting explicitly as follows \begin{align}\label{P and G} &a_{00}\longrightarrow g_{00}(a_{00}+\Sigma_{i=1}^{n}g^{-1}_{00}g_{0i})h_{00}, \quad a^{A}_{00}\longrightarrow g_{00}(a^{A}_{00}-\Sigma_{i=1}^{n}g^{-1}_{00}g_{0i}p_{i}^{A})h_{00} \\ &a_{ii}\longrightarrow h^{-1}_{00}a_{ii}h_{ii}, \quad a_{0i}\longrightarrow g_{00}(a_{0i}+g^{-1}_{00}g_{0i}a_{ii})h_{ii}, \quad i>0, \quad c\longrightarrow ch_{00}, \quad d\longrightarrow g_{00}d.\notag \end{align} The above free action of the group, obtained from the reduction of the monad, has isotropy subgroup $G_{\mathcal{C}}=\{Id\},$ for any configuration $\mathcal{C}=(a,a_{00}^{A},c,d)$. This will give a nonsingular quotient as we shall prove later, and one can easily check that the dimension of this moduli space is $dim\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}=2r(k+\frac{|\vec{a}|^{2}}{2})-|\vec{a}|^{2}$. One can reconstruct the monad from the data above by the following; \begin{equation}\label{alpha} \alpha=\left[\begin{array}{lllll}a_{00}z_{A}+a_{00A}z^{2}&a_{01}w_{1A}&a_{02}w_{2A}&\cdots&a_{0n}w_{nA}\\ \mathbb{I}_{K_{0}}\lambda_{1} w_{1A}&a_{11}w_{1A}&\quad0&\cdots&\quad0\\ \mathbb{I}_{K_{0}}\lambda_{2} w_{2A}&\quad0&a_{22}w_{2A}&\cdots&\quad0\\ \quad\vdots&\quad\vdots&\quad\vdots&\ddots&\quad\vdots\\ \mathbb{I}_{K_{0}}\lambda_{n} w_{nA}&\quad0&\quad0&\cdots&a_{nn}w_{nA}\\ cz^{2}&\quad0&\quad0&\cdots&\quad0 \end{array}\right] \end{equation} \begin{equation}\label{beta} \beta=\left[\begin{array}{llllll}z^{A}+b^{A}_{00}z^{2}&b^{A}_{01}z^{1}&b^{A}_{02}z^{2}&\cdots& b^{A}_{0n}z^{n}& dz^{2}\\ \quad0&w^{1A}&0&\cdots&\quad0&0\\ \quad0&0&w^{2A}&\cdots&\quad0&0\\ \quad\vdots&\vdots&\vdots&\ddots&\quad\vdots&\vdots\\ \quad0&0&0&\cdots&w^{nA}&0\end{array}\right] \end{equation} Here the $b^{A}_{0i}$ are given by the following: we define the matrix $b^{A}=\tilde{X}de{a}^{A}a^{-1},$ where $\tilde{X}de{a}^{A}:=\left[\begin{array}{lllll}a^{A}_{00}&-p^{A}_{1}&-p^{A}_{2}&\cdots&-p^{A}_{n}\end{array}\right].$ We would like to remind that $p^{A}_{i}$'s are the inhomogeneous coordinates, in $\mathbb{P}^{2},$ of the blow-up points. Then $b^{A}$ is of the form $b^{A}:=\left[\begin{array}{llll}b^{A}_{00}&b^{A}_{01}&\cdots&b^{A}_{0n}\end{array}\right].$ Recall that the non-degeneracy conditions imposed on the data are the surjectivity of $\beta$ everywhere on $\tilde{X}de{\mathbb{P}}$, and injectivity of $\alpha$ generically, i.e., $\alpha$ can be non-injective only at finitely many points in $\tilde{X}de{\mathbb{P}}.$ Finally we remark that Since the isotropy subgroups were discarded when we reduced the monad, and since that was exactly the kernel of the map $H:\End(M,M)\longrightarrow\End(\mathcal{E}),$ then the functor $\mathfrak{H}$, in \eqref{functor}, from the category of torsion-free sheaves on $\tilde{X}de{\mathbb{P}}$ to the category of monads obtained from \eqref{alpha} and \eqref{beta} is fully faithfull. \section{Moduli functor and universal monads}\label{unversal} In the following we want to introduce some necessary material which will enable us to prove, in the last section of this paper, that the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is fine. As a first step we consider families of framed torsion-free sheaves, parameterized by some scheme $S.$ For each family we shall build a monad $\mathcal{M}$ on the product $\tilde{X}de{\mathbb{P}}\times S$ whose cohomology is the family we started with. But before starting this program let us give some definitions and set some notations for our moduli functor problem. Let $\mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}: \mathfrak{S}ch \longrightarrow \mathfrak{S}et$ be the contravariant functor from the category of noetherian schemes of finite type to the category of sets, which is defined as follows; to every such scheme $S$ we associate the set $$\mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(S)=\left\{[\mathcal{F}] \Large{\textbf{/}}\small{ \begin{array}{l} \quad\mathcal{F} \textrm{ is a coherent sheaf on }\tilde{X}de{\mathbb{P}}\times S,\hspace{0.1cm}\textrm{flat on } S, \textrm{ and such that}\\ \mathcal{F}\otimes k(s)\cong\mathcal{E}\textrm{ is a framed torsion-free sheaf on } \tilde{X}de{\mathbb{P}}\textrm{ with }\\ \quad\quad\quad ch(\mathcal{E})=r+\Sigma_{i=1}^{n}a_{i}E_{i}-(k-\frac{|\vec{a}|^{2}}{2})\omega \end{array}} \right\}$$ where $[\mathcal{F}]$ stands for the class of the sheaf $\mathcal{F}$ under the following equivalence: $\mathcal{F}'\in[\mathcal{F}]$ if there is a line bundle $L$ on $S$ such that $\mathcal{F}'\cong\mathcal{F}\otimes\tilde{X}de{q}^{\ast}L.$ The morphism $\tilde{X}de{q}$ is the projection $\tilde{X}de{\mathbb{P}}\times S\longrightarrow S.$ We also denote by $k(s)$ the residue field of a point $s\in S.$ \begin{definition}\cite[Chapter \textbf{I}, $\S$ \textbf{2}]{newstead} A scheme ${\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}}$ is called a coarse moduli space if the following conditions are satisfied: \begin{itemize} \item There is a natural transformation $$\Phi: \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)\longrightarrow \Hom(\bullet,\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k})$$ which is a bijection for every closed point $s\in S$. \item For every scheme $\mathcal{R}$ and every natural transformation $$\Psi: \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)\longrightarrow \Hom(\bullet,\mathcal{R})$$ there is a unique morphism of schemes $f:\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}\longrightarrow \mathcal{R}$ such that the diagram of natural transformations \begin{equation} \xymatrix@C-0.8pc@R-0.9pc{\mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)\ar[r]\ar[rd]& \Hom(\bullet,\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k})\ar[d]^{f_{\ast}} \\ &\Hom(\bullet,\mathcal{R}) } \end{equation} commutes. \end{itemize} \end{definition} \begin{definition}\cite[Chapter \textbf{I}, $\S$ \textbf{2}]{newstead} A scheme $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is called a fine moduli space for the functor $\mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)$ if the above natural transformation $\Phi$ is an isomorphism. \end{definition} Given a noetherian scheme of finite type $S$, let $\mathcal{F}$ be a coherent sheaf on $\tilde{X}de{\mathbb{P}}\times S$ which is flat on $S.$ For every point $s\in S$, $\mathcal{F}_{s}:=\mathcal{F}\otimes k(s)$ is a framed torsion-free sheaf $\mathcal{E}$ on $\tilde{X}de{\mathbb{P}}$ with Chern character $ch(\mathcal{E})=r+\Sigma_{i=1}^{n}a_{i}E_{i}-(k-\frac{|\vec{a}|^{2}}{2})\omega .$ We consider the following diagram: \begin{equation} \xymatrix@C-0.5pc@R-0.5pc{& \mathcal{F} \ar[d]& \\ &\tilde{X}de{\mathbb{P}}\times S\ar[ld]_{\tilde{X}de{p}}\ar[rd]^{\tilde{X}de{q}}& \\ \tilde{X}de{\mathbb{P}}&& S } \end{equation} where $\tilde{X}de{p}$ and $\tilde{X}de{q}$ are the natural projections onto the two factors. The aim is to construct a monad $\mathcal{M}$ on $\tilde{X}de{\mathbb{P}}\times S$ which is associated to the sheaf $\mathcal{F}$, i.e., $Coh(\mathcal{M})=\mathcal{F}$. For every two sheaves, $\mathcal{F}$ on $\tilde{X}de{\mathbb{P}}$ and $\mathcal{G}$ on $S$, we use the following notation: $\mathcal{F}\boxtimes \mathcal{G}:=\tilde{X}de{p}^{\ast}\mathcal{F}\otimes\tilde{X}de{q}^{\ast}\mathcal{G}.$ We consider also, for a morphism $f:X\longrightarrow Y$ of noetherian schemes of finite type, the functor $\mathcal{H}om_{f}(\mathcal{G}, \bullet):= f_{\ast}\circ\mathcal{H}om(\mathcal{G},\bullet)$ called the relative $\mathcal{H}om$-functor, and we denote its $i$-th right derived functors by $\mathcal{E}xt^{i}_{f}(\mathcal{G}, \bullet)$. For more details see for example \cite{Kleiman}. We remind the reader that a sheaf $\mathcal{F}$ on a topological space $X$ is \emph{flasque} (or \emph{flabby}) if for every inclusion $V\subseteq U$ of open sets, the restriction map $\mathcal{F}(U)\longrightarrow\mathcal{F}(V)$ is surjective. Before proceeding we want to introduce the following useful result, which is a kind of "relative local-to-global" spectral sequence: \begin{pr}\label{loctoglob} Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes of finite type, and $\mathcal{G}$ a coherent sheaf on $X$ flat on $Y.$ Then for any coherent sheaf $\mathcal{J}$ on $X$ flat on $Y$, there exists a spectral sequence $E_{r}^{p,q}$ with $E_{2}$-term $E_{2}^{p,q}=\h^{p}(Y,\mathcal{E}xt_{f}^{q}(\mathcal{G},\mathcal{J}))$ which converges to $E_{\infty}^{p+q}=\ext^{p+q}_{X}(\mathcal{G},\mathcal{J}).$ \end{pr} \begin{proof} Let $\mathcal{J}\longrightarrow I^{\bullet}$ be an injective resolution of the sheaf $\mathcal{J}$, and consider a Cartan-Eilenberg resolution given by the \v Cech complex associated to the complex of sheaves $\mathcal{H}om_{f}(\mathcal{G}, I^{\bullet})$ for a suitable open cover $\mathcal{Y}$ of $Y$. This defines a double complex $C^{\bullet}(\mathcal{Y}, \mathcal{H}om_{f}(\mathcal{G}, I^{\bullet}))$ with differentials $$\left\{\begin{array}{l}\delta_{1}=d \textrm{ the differential associated to the injective resolution of }\mathcal{J} \\ \delta_{2}=\delta \textrm{ the differential associated to the }\check{C}\textrm{ech complex.} \end{array}\right.$$ There are two spectral sequences associated with this double complex \cite{Grothendieck}, with $E_{2}$-terms: $$'E_{2}^{p,q}=\h^{p}_{d}[\h^{q}_{\delta}(C^{\bullet}(\mathcal{Y}, \mathcal{H}om_{f}(\mathcal{G}, I^{\bullet})))]$$ $$''E_{2}^{p,q}=\h^{p}_{\delta}[\h^{q}_{d}(C^{\bullet}(\mathcal{Y}, \mathcal{H}om_{f}(\mathcal{G}, I^{\bullet})))]$$ The $E_{1}$-term of the first spectral sequence is given by: \begin{align} 'E_{1}^{p,q}&=\h^{q}_{\delta}(C^{\bullet}(\mathcal{Y}, \mathcal{H}om_{f}(\mathcal{G}, I^{p}))) \notag\\ &=\h^{q}_{\delta}(C^{\bullet}(\mathcal{Y}, f_{\ast}\circ\mathcal{H}om(\mathcal{G}, I^{p}))) \notag \end{align} By \cite[Lemma \textbf{II. 7. 3}]{Godement} the sheaf $\mathcal{H}om(\mathcal{G}, I^{p})$ is flasque for every term $I^{p}$ of the injective resolution. Moreover the direct image of a flasque sheaf is flasque \cite[\textbf{II} Theorem \textbf{3.1.1}]{Godement}. Hence $$'E_{1}^{p,q}=\left\{\begin{array}{ll} \h^{0}(Y,f_{\ast}\circ\mathcal{H}om(\mathcal{G}, I^{p}))& q=0 \\ \quad 0 & q\neq0\end{array}\right.$$ On the other hand we have \begin{align} \h^{0}(Y,f_{\ast}\circ\mathcal{H}om(\mathcal{G}, I^{p}))&=\h^{0}(X,\mathcal{H}om(\mathcal{G}, I^{p})) \notag \\ &=\Hom(\mathcal{G}, I^{p}) \notag \end{align} Thus $$'E_{1}^{p,q}=\left\{\begin{array}{ll} \Hom(\mathcal{G}, I^{p})& q=0\\ \quad 0 & q\neq0\end{array}\right.$$ This spectral sequence degenerates at the second step and converges to $'E^{p+q}_{\infty}=\ext^{p+q}_{X}(\mathcal{G}, \mathcal{J})$. The second spectral sequence has $E_{2}$-term: \begin{align} ''E_{2}^{p,q}&=\h^{p}_{\delta}(C^{\bullet}(\mathcal{Y}, \mathcal{H}^{q}_{d}(\mathcal{H}om_{f}(\mathcal{G}, I^{\bullet})))) \notag\\ &=\h^{p}_{\delta}(C^{\bullet}(\mathcal{Y}, \mathcal{E}xt^{q}_{f}(\mathcal{G}, \mathcal{J}))) \notag \end{align} Then \begin{equation*} ''E_{2}^{p,q}=\h^{p}(Y, \mathcal{E}xt^{q}_{f}(\mathcal{G}, \mathcal{J})). \end{equation*} \end{proof} This proposition will be used in the next step to prove the existence of a display of a monad on $\tilde{X}de{\mathbb{P}}\times S$ associated to a family of framed torsion-free sheaves on $\tilde{X}de{\mathbb{P}}$. Consider the following extensions: \begin{align} &0\longrightarrow\mathcal{U}\longrightarrow\mathcal{K}\longrightarrow\mathcal{F}\longrightarrow0 \in\ext^{1}(\mathcal{F},\mathcal{U}) \\ &0\longrightarrow\mathcal{F}\longrightarrow\mathcal{Q}\longrightarrow\mathcal{V}\longrightarrow0 \in\ext^{1}(\mathcal{F},\mathcal{V}) \end{align} where $\mathcal{U}=\oplus_{i=0}^{n}\mathcal{O}(-1,E_{i})\boxtimes\mathcal{U}_{i}$, with $$\left\{\begin{array}{l}\mathcal{U}_{0}=\mathcal{R}^{1}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1})) \\ \mathcal{U}_{i}=\mathcal{R}^{1}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}-E_{i})) \end{array}\right.$$ and $\mathcal{V}=\oplus_{i=0}^{n}\mathcal{O}(1,-E_{i})\boxtimes\mathcal{V}_{i}$ with $$\left\{\begin{array}{l}\mathcal{V}_{0}=\mathcal{R}^{1}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,0)) \\ \mathcal{V}_{i}=\mathcal{R}^{1}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1})). \end{array}\right.$$ We denoted by $\mathcal{R}^{i}\tilde{X}de{q}_{\ast}$ the i-th right derived functor of the direct image functor $\tilde{X}de{q}_{\ast}$ from the category of coherent sheaves of $\mathcal{O}_{\tilde{X}de{\mathbb{P}}\times S}$-modules to the category of coherent sheaves of $\mathcal{O}_{S}$-modules. We also remind the reader that we defined $E_{0}:=0.$ We remark that the extensions \eqref{A} and \eqref{B} exist since the groups $\ext^{1}(\mathcal{F},\mathcal{U})$ and $\ext^{1}(\mathcal{V},\mathcal{F})$ classifying them respectively are non trivial, as it will be proved in proposition ${\bf\ref{universal-existence}}$ below. The sheaves $\mathcal{U}$ and $\mathcal{V}$ are locally free on $\tilde{X}de{\mathbb{P}}\times S$; let us consider, for the moment, the sheaves $\mathcal{U}_{0}$ and $\mathcal{U}_{i}$: by Grauert's theorem (\cite[\textbf{III. 12. 9}]{Hart}), for every point $s\in S$, there are maps: \begin{align} &\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}))\otimes k(s)\longrightarrow \h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}))=0 \notag\\ &\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}-E_{i}))\otimes k(s)\longrightarrow \h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i}))=0 \notag\\ &\mathcal{R}^{2}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}))\otimes k(s)\longrightarrow \h^{2}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}))=0 \notag\\ &\mathcal{R}^{2}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}-E_{i}))\otimes k(s)\longrightarrow \h^{2}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i}))=0 \notag. \end{align} Since all of these maps go to zero, the sheaves in the left-hand side have rank zero at all $s\in S$, thus they vanish identically (\cite[\textbf{III.12.5}]{Hart}). By the Riemann-Roch theorem, the dimensions of $\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}))$ and $\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i}))$ are constant, hence $\mathcal{U}_{0}$ and $\mathcal{U}_{i}$ are locally free. In the same way one can show that $\mathcal{V}_{0}$ and $\mathcal{V}_{i}$ are locally free. \begin{pr}\label{universal-existence} There exist non trivial extensions \begin{align} &0\longrightarrow\mathcal{U}\longrightarrow\mathcal{K}\longrightarrow\mathcal{F}\longrightarrow0 \in\ext^{1}(\mathcal{F},\mathcal{U}) \label{A}\\ &0\longrightarrow\mathcal{F}\longrightarrow\mathcal{Q}\longrightarrow\mathcal{V}\longrightarrow0 \in\ext^{1}(\mathcal{V},\mathcal{F}) \label{B} \end{align} \end{pr} \begin{proof} The statement of this proposition will be proved by showing that neither $\ext^{1}(\mathcal{F},\mathcal{U})$ nor $\ext^{1}(\mathcal{V},\mathcal{F})$ is trivial. \underline{$\ext^{1}(\mathcal{V},\mathcal{F})$}: One has $\ext^{1}(\mathcal{V},\mathcal{F})=\oplus_{i=0}^{n}\ext^{1}(\mathcal{O}(1,-E_{i})\boxtimes\mathcal{V}_{i}, \mathcal{F}).$ By the result of theorem \textbf{\ref{loctoglob}}, it follows that the terms contributing to each component $\ext^{1}(\mathcal{O}(1,-E_{i})\boxtimes\mathcal{V}_{i}, \mathcal{F})$ are given by $$\left\{\begin{array}{l} \h^{0}(S,\mathcal{E}xt_{\tilde{X}de{q}}^{1}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))) \\ \h^{1}(S,\mathcal{H}om_{\tilde{X}de{q}}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))) \end{array}\right.$$ The second term $\h^{1}(S,\mathcal{H}om_{\tilde{X}de{q}}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i})))$ is trivial; this is because the sheaf $\mathcal{H}om_{\tilde{X}de{q}}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))$ is identically zero. Indeed one has \begin{align} \mathcal{H}om_{\tilde{X}de{q}}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))&= \tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))] \notag \\ &=\mathcal{V}^{\ast}_{i}\otimes \tilde{X}de{q}_{\ast}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i})).\notag \end{align} The second equality is obtained from the fact that $\mathcal{V}_{i}$ is locally free and by using the projection formula. Moreover, from the natural morphism $$\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))\otimes k(s)\longrightarrow\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-1,E_{i}))=0$$ at any closed point $s\in S,$ the claim follows. Recall that the Grothendieck spectral sequence associated to the composition of the direct image functor $\tilde{X}de{q}_{\ast}:\mathcal{C}oh(\tilde{X}de{\mathbb{P}}\times S)\longrightarrow\mathcal{C}oh(S)$ with the local Hom functor $\mathcal{H}om(\mathcal{G},-):\mathcal{C}oh(\tilde{X}de{\mathbb{P}}\times S)\longrightarrow\mathcal{C}oh(\tilde{X}de{\mathbb{P}}\times S)$ for a fixed coherent sheaf $\mathcal{G}$ on $\tilde{X}de{\mathbb{P}}\times S$ has a second term $$E_{2}^{s,t}=\mathcal{R}^{t}\tilde{X}de{q}_{\ast}[\mathcal{E}xt^{s}(\mathcal{G},\mathcal{H})].$$ and converges to $E_{\infty}^{s,t}=\mathcal{E}xt_{\tilde{X}de{q}}^{s+t}(\mathcal{G},\mathcal{H}).$ It follows that the contributing terms to $\h^{0}(S,\mathcal{E}xt_{\tilde{X}de{q}}^{1}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))),$ from the Grothendieck spectral sequence $E_{r}^{s,t}$ (we take the sheaf $\mathcal{G}$ to be $\tilde{X}de{q}^{\ast}\mathcal{V}_{i}$), are given by $$\left\{\begin{array}{l} \tilde{X}de{q}_{\ast}[\mathcal{E}xt^{1}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))] \\ \mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))] \end{array}\right.$$ Since $\mathcal{V}^{i}$ is locally free then $\mathcal{E}xt^{1}(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))=0.$ The other term can be written as \begin{align} \mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{V}_{i},\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))]&=\mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\tilde{X}de{q}^{\ast}\mathcal{V}^{\ast}_{i}\otimes\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))] \notag \\ &=\mathcal{V}^{\ast}_{i}\otimes\mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))].\notag \end{align} The natural morphism, at a point $s\in S,$ given by $$\mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i}))]\otimes k(s)\longrightarrow\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,E_{i}))$$ shows that the fibre of the sheaf is not zero. By using the Riemann-Roch formula, it follows that the dimension of the space $\h^{1}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,E_{i}))$ is non zero and independent from the point $s,$ since $\mathcal{F}$ is $S-$flat and $\h^{0,2}(\tilde{X}de{\mathbb{P}}, \mathcal{E}(-1,E_{i}))=0.$ Thus, each component of the extension $\ext^{1}(\mathcal{V},\mathcal{F})$ is given by $$\ext^{1}(\mathcal{O}(-1,E_{i})\boxtimes\mathcal{V}_{i}, \mathcal{F})=\h^{0}(\tilde{X}de{\mathbb{P}}\times S,\mathcal{H}om(\mathcal{V}_{i},\mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,E_{i})]))$$ Now for $i=0$ one has $\mathcal{H}om(\mathcal{V}_{0},\mathcal{R}^{1}\tilde{X}de{q}_{\ast}[\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-1,0)])=\mathcal{E}nd(\mathcal{V}_{0}).$ Moreover we have a surjective morphism $\ext^{1}(\mathcal{V},\mathcal{F})\to \mathcal{E}nd(\mathcal{V}_{0}),$ which is the projection on the zero-th factor. Then there exist at least one non-trivial extension which is sent to the identity element $1\in\End(\mathcal{V}_{0}).$ Hence at least one non-trivial extension \eqref{B} exists. \underline{$\ext^{1}(\mathcal{F},\mathcal{U})$}: One has $\ext^{1}(\mathcal{F},\mathcal{U})=\oplus_{i=0}^{n}\ext^{1}(\mathcal{F},\mathcal{U}_{i}\boxtimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i})).$ By using the spectral sequence in \textbf{\ref{loctoglob}} one shows that $$\ext^{1}(\mathcal{F},\mathcal{U}_{i}\boxtimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i}))=\h^{0}(S,\mathcal{E}xt_{\tilde{X}de{q}}^{1}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i}),\tilde{X}de{q}^{\ast}\mathcal{U}_{i}))$$ since the sheaf $\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i}),\tilde{X}de{q}^{\ast}\mathcal{U}_{i})$ vanishes identically; indeed one has $$\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i}),\tilde{X}de{q}^{\ast}\mathcal{U}_{i})\otimes k(s)\to\Hom(\mathcal{E}(1,-E_{i}),\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i})))$$ but $$\Hom(\mathcal{E}(1,-E_{i}),\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i})))=\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i})) \otimes\underbrace{\h^{2}(\tilde{X}de{\mathbb{P}},\mathcal{E}(-2,\vec{1}-E_{i}))^{\ast}}_{0}=0.$$ On the other hand we have $\mathcal{E}xt_{\tilde{X}de{q}}^{1}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i}),\tilde{X}de{q}^{\ast}\mathcal{U}_{i})=\mathcal{E}xt_{\tilde{X}de{q}}^{1}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}-E_{i}),\tilde{X}de{p}^{\ast}\mathcal{O}(-3,\vec{1})\otimes \tilde{X}de{q}^{\ast}\mathcal{U}_{i}).$ By using the relative duality morphism, \cite[Main Theorem]{Kleiman}, one has $$\mathcal{E}xt_{\tilde{X}de{q}}^{1}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(1,-E_{i}),\tilde{X}de{q}^{\ast}\mathcal{U}_{i})\cong \mathcal{H}om_{S}(\mathcal{R}^{1}\tilde{X}de{q}_{\ast}(\mathcal{F}\otimes \tilde{X}de{p}^{\ast}\mathcal{O}(-2,\vec{1}-E_{i})),\mathcal{U}_{i})=\mathcal{E}nd(\mathcal{U}_{i}).$$ Notice that we used the fact that the relative dualizing sheaf is isomorphic to $\tilde{X}de{p}^{\ast}\mathcal{O}(-3,\vec{1}).$ Thus $\ext^{1}(\mathcal{F},\mathcal{U})=\oplus_{i=0}^{n}\End(\mathcal{U}_{i}).$ Moreover the morphism $\ext^{1}(\mathcal{F},\mathcal{U})\to\End(\mathcal{U}_{i})$ is surjective, $\forall i=0,\cdots,n,$ since it is just the natural projection on the $i-$th factor. Hence for every $i$ there exists at least one non-trivial projection which is sent to $1\in\End(\mathcal{U}_{i}).$ It follows that $\ext^{1}(\mathcal{F},\mathcal{U})$ is not trivial. \end{proof} \begin{cor} There exist two extensions \begin{align} &0\longrightarrow\mathcal{U}\longrightarrow\mathcal{W}\longrightarrow\mathcal{Q}\longrightarrow0 \label{C}\\ &0\longrightarrow\mathcal{K}\longrightarrow\mathcal{W}\longrightarrow\mathcal{V}\longrightarrow0 \label{D} \end{align} such that the extension \eqref{A} and \eqref{B} fit in the following display of a monad \begin{equation} \xymatrix@R-1pc@C-1pc{&0\ar[d]&0\ar[d]&& \\ &\mathcal{U}\ar[d]\ar@{=}[r]&\mathcal{U}\ar[d]&& \\ 0\ar[r]&\mathcal{K}\ar[d]\ar[r]&\mathcal{W}\ar[d]\ar[r]&\mathcal{V}\ar@{=}[d]\ar[r]&0 \\ 0\ar[r]&\mathcal{F}\ar[d]\ar[r]&\mathcal{Q}\ar[d]\ar[r]&\mathcal{V}\ar[r]&0\\ &0&0&& } \end{equation} \end{cor} \begin{proof} By proposition \textbf{\ref{King223}}, the two extensions, \eqref{A} and \eqref{B}, fit into the display of a monad on $\tilde{X}de{\mathbb{P}}\times S$ if and only if their $\ext$-product in $\ext^{2}(\mathcal{V},\mathcal{U})$ is trivial. To show this vanishing property we use the "relative local to global" spectral sequence that we constructed above; we have $$E_{2}^{p,q}=\h^{p}(S, \mathcal{E}xt^{\tilde{X}de{q}}_{\tilde{X}de{\tilde{X}de{q}}}(\mathcal{V}, \mathcal{U}))\Longrightarrow \ext^{p+q}_{\tilde{X}de{\mathbb{P}}\times S}(\mathcal{V}, \mathcal{U})$$ The spectral sequence terms which contributes to $\ext^{2}(\mathcal{V},\mathcal{U})$ are $$\h^{0}(S, \mathcal{E}xt^{2}_{\tilde{X}de{\tilde{X}de{q}}}(\mathcal{V}, \mathcal{U})),\quad \h^{1}(S, \mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))\quad \textrm{ and }\quad \h^{2}(S, \mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})).$$ Next we will prove that both $\h^{2}(S, \mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$ and $\h^{0}(S, \mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$ vanish, hence $\ext^{2}(\mathcal{V},\mathcal{U})=\h^{1}(S, \mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$. \underline{$\h^{2}(S, \mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$}: $$\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})\otimes k(s)=\oplus_{i,j=0}^{n}\tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{V}_{i}, \tilde{X}de{q}^{\ast}\mathcal{U}_{j}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,E_{i}+E_{j}))]\otimes k(s),$$ There is a natural map $$\oplus_{i,j=0}^{n}\tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{V}_{i}, \tilde{X}de{q}^{\ast}\mathcal{U}_{j}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-2,E_{i}+E_{j}))]\otimes k(s)\qquad\qquad\qquad$$ $$\qquad\qquad\qquad\qquad\qquad\longrightarrow \oplus_{i,j=0}^{n}\Hom(\tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s), \tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s)\otimes\mathcal{O}(-2,E_{i}+E_{j}))$$ which is an isomorphism if it is surjective. Then the pull-back $\tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s)$, of the stalk $\mathcal{V}_{i}(s)$ at a point $s$, is just the pull-back associated to the map $\tilde{X}de{\mathbb{P}}\longrightarrow s=\spec k(s)$, hence $\tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s)$ is constant on $\tilde{X}de{\mathbb{P}}$. This is also true for the pull-back $\tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s)$ of $\mathcal{U}_{j}(s)$. Thus $$\oplus_{i,j=0}^{n}\Hom(\tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s), \tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s)\otimes\mathcal{O}(-2,E_{i}+E_{j}))= \oplus_{i,j=0}^{n} [\tilde{X}de{q}^{\ast}\mathcal{V}_{j}(s)]^{\ast}\otimes\tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s)\otimes \underbrace{\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(-2,E_{i}+E_{j}))}_{0}$$ every $i,j=0,...,n,$ and the above natural map is obviously surjective. Furthermore this implies the vanishing of $\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})$ identically. Hence $\h^{2}(S, \mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))=0.$ \underline{$\h^{0}(S, \mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$}: By the relative Serre duality \cite{Kleiman} we have $$\mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})\otimes k(s)=\oplus_{i,j=0}^{n}\{\tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{U}_{j}, \tilde{X}de{q}^{\ast}\mathcal{V}_{i}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-1,\vec{1}-E_{i}-E_{j}))]\}^{\ast}\otimes k(s)$$ In the same way as above, there is a map $$\oplus_{i,j=0}^{n}\{\tilde{X}de{q}_{\ast}[\mathcal{H}om(\tilde{X}de{q}^{\ast}\mathcal{U}_{j}, \tilde{X}de{q}^{\ast}\mathcal{V}_{i}\otimes\tilde{X}de{p}^{\ast}\mathcal{O}(-1,\vec{1}-E_{i}-E_{j}))]\}^{\ast}\otimes k(s)\qquad\qquad\qquad\qquad$$ $$\qquad\qquad\qquad\qquad\longrightarrow \oplus_{i,j=0}^{n}\Hom(\tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s), \tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s)\otimes\mathcal{O}(-1,\vec{1}-E_{i}-E_{j}))^{\ast}$$ which is an isomorphism if it is surjective. This map is surjective since for every $i,j=0,...,n$ in the direct sum we have $$\Hom(\tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s), \tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s)\otimes\mathcal{O}(-1,\vec{1}-E_{i}-E_{j}))^{\ast}= \tilde{X}de{q}^{\ast}\mathcal{U}_{j}(s)\otimes[\tilde{X}de{q}^{\ast}\mathcal{V}_{i}(s)]^{\ast}\otimes \underbrace{\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(-1,\vec{1}-E_{i}-E_{j}))^{\ast}}_{0}$$ which implies that $\mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})=0$ and hence $\h^{0}(S, \mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))=0.$ Computing the fiber of the sheaf $\mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})$, over a point $s\in S$, one can check that it is not zero, hence $\ext^{2}(\mathcal{V},\mathcal{U})=\h^{1}(S, \mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$ may not vanish in general. However one can overcome this obstruction as follows: the sequences \eqref{A} and \eqref{B} are classified by $\ext^{1}(\mathcal{F}, \mathcal{U})$ and $\ext^{1}(\mathcal{V}, \mathcal{F}),$ respectively, and one can compute their Yoneda product. If this product is zero then one can apply proposition \textbf{\ref{King223}} to construct a display of a monad. So we shall prove that the Yoneda product \begin{equation}\label{pairing} \ext^{1}(\mathcal{V}, \mathcal{F})\times\ext^{1}(\mathcal{F}, \mathcal{U})\longrightarrow\ext^{2}(\mathcal{V}, \mathcal{U}) \end{equation} is zero. As for the sheaf $\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})$, one can prove that the sheaves $\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{F}, \mathcal{U})$ and $\mathcal{H}om_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{F})$ vanish identically by computing their associated fibers at a given point $s\in S.$ By the spectral sequence in proposition \textbf{\ref{loctoglob}}, it follows that $$\ext^{1}(\mathcal{F}, \mathcal{U})=\h^{0}(S,\mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{F}, \mathcal{U}))\quad\textrm{ and }\quad\ext^{1}(\mathcal{V}, \mathcal{F})=\h^{0}(S, \mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{F})),$$ and the pairing \eqref{pairing} can be written as $$\h^{0}(S,\mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{F})\times\mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{F}, \mathcal{U}))\longrightarrow \ext^{2}(\mathcal{V}, \mathcal{U}).$$ On the other hand this product is inherited from the relative local version of the Yoneda product \cite[Section \textbf{10.1.7}]{Huy} $$\mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{F})\times\mathcal{E}xt^{1}_{\tilde{X}de{q}}(\mathcal{F}, \mathcal{U})\longrightarrow \mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U})$$ Thus the global Yoneda product takes values in the $\h^{0}(S,\mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))$ part of the group $\ext^{2}(\mathcal{V}, \mathcal{U})$. But we already proved that $\h^{0}(S,\mathcal{E}xt^{2}_{\tilde{X}de{q}}(\mathcal{V}, \mathcal{U}))=0$, consequently the Yoneda product of the extensions \eqref{A} and \eqref{B} is zero, implying the existence of sequences \begin{align} &0\longrightarrow\mathcal{U}\longrightarrow\mathcal{W}\longrightarrow\mathcal{Q}\longrightarrow0 \notag\\ &0\longrightarrow\mathcal{K}\longrightarrow\mathcal{W}\longrightarrow\mathcal{V}\longrightarrow0 \notag \end{align} such that the diagram \begin{equation} \xymatrix@R-1pc@C-1pc{&0\ar[d]&0\ar[d]&& \\ &\mathcal{U}\ar[d]\ar@{=}[r]&\mathcal{U}\ar[d]&& \\ 0\ar[r]&\mathcal{K}\ar[d]\ar[r]&\mathcal{W}\ar[d]\ar[r]&\mathcal{V}\ar@{=}[d]\ar[r]&0 \\ 0\ar[r]&\mathcal{F}\ar[d]\ar[r]&\mathcal{Q}\ar[d]\ar[r]&\mathcal{V}\ar[r]&0\\ &0&0&& } \end{equation} commutes. \end{proof} The display existence shows that there exists a monad \begin{equation}\label{S-monad} \mathcal{M}: \quad\oplus_{i=0}^{n}\mathcal{O}_{\tilde{X}de{\mathbb{P}}}(-1,E_{i})\boxtimes\mathcal{U}_{i}\longrightarrow \mathcal{W} \longrightarrow\oplus_{i=0}^{n}\mathcal{O}_{\tilde{X}de{\mathbb{P}}}(1,-E_{i})\boxtimes\mathcal{V}_{i} \end{equation} associated to the family $\mathcal{F}$ on $\tilde{X}de{\mathbb{P}}$ parameterized by $S$. One can show that the restriction to the fibers of $\tilde{X}de{q}$ gives a monad isomorphic to the one in Proposition \textbf{2.5} and, using the display, one can show that the second term $\mathcal{W}$ of the monad $\mathcal{M}$ is trivial along the fibers of $\tilde{X}de{q}.$ Another useful monad we now introduce is the universal monad on $\tilde{X}de{\mathbb{P}}\times P$ : \begin{equation}\label{univ-monad} \xymatrix@C-1pc{\mathbb{M}:& \oplus_{i=0}^{n}\mathcal{O}_{\tilde{X}de{\mathbb{P}}}(-1,E_{i})\boxtimes K_{i}\otimes\mathcal{O}_{P}\ar[r]& \mathcal{O}_{\tilde{X}de{\mathbb{P}}}\boxtimes W\otimes\mathcal{O}_{P}\ar[r]& \oplus_{i=0}^{n}\mathcal{O}_{\tilde{X}de{\mathbb{P}}}(1,-E_{i})\boxtimes L_{i}\otimes\mathcal{O}_{P}} \end{equation} We denote its cohomology by $\mathfrak{F}.$ The vector spaces $K_{i}$ and $L_{i}$ are given by Proposition \textbf{\ref{prop}}. Note that at each point $p$ of the space $P,$ defined in \eqref{spaceP}, the maps $\alpha_{p}$ and $\beta_{p},$ as in \eqref{alpha} and \eqref{beta}, gives a monad $M(p)$ whose cohomology, $\mathcal{E}(p),$ is a framed torsion-free sheaf. By varying the point $p$ in on an open neighborhood $U\subset P,$ the maps $\alpha$ and $\beta$ will depend on the sheaf $\mathcal{O}_{U}.$ The action of the group $G,$ defined in \eqref{P and G}, provides the gluing of the maps $\alpha$ and $\beta$ in the intersection of two open sets. This allows the construction of the canonical monad \eqref{univ-monad} on $\tilde{X}de{\mathbb{P}}\times P$. By construction, the Chern character of cohomology $\mathfrak{F}\otimes k(p)\cong\mathcal{E}(p)$ is independent of the point $p\in P.$ Hence, by the Riemann-Roch formula, the Hilbert polynomial is independent from $p.$ Thus $\mathfrak{F}$ is flat over $P.$ \section{Smoothness of the space $P$} A needed step, in proving smoothness of the moduli space, is to show the smoothness of the space $P$ of ADHM type configurations subject to the monad condition and defined in \eqref{spaceP}. In this section, we follow the argument given by Okonek et. al. in \cite[Chapter\textbf{II}, \textbf{4.1}]{Okonek}, provided that one makes suitable changes in order for the proofs to work in the coherent case. We need first to prove the following \begin{thm}\label{Kuneth} Let $L_{\bullet}$ and $M^{\bullet}$ be two bounded complexes of coherent sheaves of $\mathcal{O}_{X}$-modules over an algebraic variety $X$, and suppose that $L_{i}$ is locally-free or $M^{i}$ is injective, for every $i$ in the complexes. Then there is a spectral sequence $E_{r}^{pq}$ with second term $$E_{2}^{pq}=\oplus_{i+j=q}\mathcal{E}xt^{p}(H_{i}(L_{\bullet}),H^{j}(M^{\bullet}))$$ for which the $E_{\infty}$-term is the bi-graded group associated to a suitable filtration of the cohomology of the complex $\mathcal{H}om(L_{\bullet},M^{\bullet}).$ \end{thm} \begin{proof} The proof is similar to \cite[Theorem \textbf{5.4.1}]{Godement}, replacing projectives by locally-free. \end{proof} \begin{lem}\label{obstruction} On a non-singular algebraic surface $X,$ let $\xymatrix{M:0\ar[r]&A\ar[r]^{a_{0}}&B\ar[r]^{b_{0}}&C\ar[r]&0}$ be a monad whose cohomology is a torsion-free sheaf $\mathcal{E}.$ Let $d_{0}$ be the following homomorphism $$\xymatrix@R-1pc@C-1pc{d_{0}:&\Hom(A,B)\oplus\Hom(B,C)\ar[r]&\Hom(A,C)\\ &(a,b)\ar[r]&d_{0}(a,b)=ba_{0}+b_{0}a}.$$ Then the map $$\h^{2}(X, \mathcal{E}nd(\mathcal{E}))\longrightarrow Coker d_{0}$$ is surjective if the following groups vanish: $$\xymatrix@R-1.6pc@C-2pc{\h^{1}(X,\mathcal{H}om(B,C)),&\h^{1}(X,\mathcal{H}om(A,B)),&\h^{1}(X,\mathcal{E}nd(B)),&\h^{1}(X,\mathcal{E}nd(C))\\ \h^{1}(X,\mathcal{E}nd(A))\quad,&\h^{2}(X,\mathcal{E}nd(A))\quad,&\h^{2}(X,\mathcal{E}nd(B))\quad,&\h^{2}(X,\mathcal{E}nd(C))\\ \h^{2}(X,\mathcal{H}om(B,A)),&\h^{2}(X,\mathcal{H}om(C,B))&&}$$ \end{lem} \begin{proof} The proof is a generalization of \cite[Chapter \textbf{II}, \textbf{4.1}]{Okonek} replacing tesor product in the case of bundles by suitable sheaves of local homomorphisms in the coherent case. Consider the double complex \begin{displaymath} \xymatrix@R-1.4pc@C-1pc{\mathcal{H}om(C,A)\ar[r]\ar[d]&\mathcal{H}om(C,B)\ar[r]\ar[d]&\mathcal{E}nd(C)\ar[d]\\ \mathcal{H}om(B,A)\ar[r]\ar[d]&\mathcal{E}nd(B)\ar[r]\ar[d]&\mathcal{H}om(B,C)\ar[d] \\ \mathcal{E}nd(A)\ar[r]&\mathcal{H}om(A,B)\ar[r]&\mathcal{H}om(A,C) } \end{displaymath} The associated total complex is given by: \begin{displaymath} \xymatrix@C-1pc{K^{\bullet}:&0\ar[r]&K^{-2}\ar[r]&K^{-1}\ar[r]^S&K^{0}\ar[r]^T&K^{1}\ar[r]^U&K^{2}\ar[r]&0} \end{displaymath} where \begin{displaymath} \xymatrix@R-2pc@C-2pc{K^{-2}=\mathcal{H}om(C,A),& K^{-1}=\mathcal{H}om(C,B)\oplus\mathcal{H}om(B,A),\\ K^{0}=\mathcal{E}nd(A)\oplus\mathcal{E}nd(B)\oplus\mathcal{E}nd(C), &K^{1}=\mathcal{H}om(A,B)\oplus\mathcal{H}om(B,C), \\ K^{2}=\mathcal{H}om(A,C).& } \end{displaymath} Using theorem \textbf{\ref{Kuneth}} one has $$E_{2}^{pq}=\oplus_{i+j=q}\mathcal{E}xt^{p}(\h_{i}(L_{\bullet}),\h^{j}(M^{\bullet}))\Longrightarrow E_{\infty}^{p+q}=\h^{p+q}(\mathcal{H}om(L_{\bullet},M^{\bullet})).$$ This allows one to compute the cohomology of the complex $K^{\bullet}$: \begin{equation} \xymatrix@R-1.6pc@C-1pc{\h^{-2}(K^{\bullet})=0,&\h^{-1}(K^{\bullet})=0,&\h^{2}(K^{\bullet})=0, \\ \h^{0}(K^{\bullet})=\mathcal{E}nd(\mathcal{E}),&\h^{1}(K^{\bullet})=\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E}).& } \end{equation} So one can write the following exact sequences: \begin{align} &\xymatrix@C-1pc{0\ar[r]&ImS\ar[r]&KerT\ar[r]&\mathcal{E}nd(\mathcal{E})\ar[r]&0}\label{S1} \\ &\xymatrix@C-1pc{0\ar[r]&KerT\ar[r]&K^{0}\ar[r]&ImT\ar[r]&0}\label{S2} \\ &\xymatrix@C-1pc{0\ar[r]&K^{-2}\ar[r]&K^{-1}\ar[r]&ImS\ar[r]&0}\label{S3} \\ &\xymatrix@C-1pc{0\ar[r]&KerU\ar[r]&K^{1}\ar[r]&K^{2}\ar[r]&0}\label{S4} \\ &\xymatrix@C-1pc{0\ar[r]&ImT\ar[r]&KerU\ar[r]& \mathcal{E}xt^{1}(\mathcal{E},\mathcal{E})\ar[r]&0}\label{S5} \end{align} Now $d_{0}$ in the lemma is the map $d_{0}:\h^{0}(X,K^{1})\longrightarrow\h^{0}(X,K^{2})$ where \begin{displaymath} \xymatrix@C-1pc{\h^{0}(X,K^{1})=\h^{0}(X,\mathcal{H}om(A,B)\oplus\mathcal{H}om(B,C)),&\h^{0}(X,K^{2})=\mathcal{H}om(A,C).} \end{displaymath} From \eqref{S1}, \eqref{S2}, \eqref{S3} and the vanishing of the cohomology groups in the assumptions of the lemma, one has \begin{align} \h^{2}(X,\mathcal{E}nd(\mathcal{E}))&=\h^{2}(X,KerT)\notag \\ &=\h^{1}(X,ImT).\notag \end{align} From \eqref{S4} one has $Cokerd_{0}=\h^{1}(X,KerU).$ Combining all this with the long sequence in cohomology induced by \eqref{S5} one has \begin{equation} \cdots\longrightarrow\h^{2}(X,\mathcal{E}nd(\mathcal{E}))\longrightarrow Cokerd_{0}\longrightarrow\h^{1}(X,\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E})) \end{equation} To prove that $\h^{1}(X,\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E}))=0$ one uses first the natural sequence \eqref{doubledual}. Applying $\mathcal{H}om(\bullet,\mathcal{E})$ to \eqref{doubledual} it follows that $\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E})=\mathcal{E}xt^{2}(\Delta,\mathcal{E})$ and $\mathcal{E}xt^{2}(\mathcal{E},\mathcal{E})=0$ The restriction to any open subset $V\subset X$ gives $\mathcal{E}xt^{2}(\Delta,\mathcal{E})|_{V}=\ext^{2}(\Delta|_{V},\mathcal{E}|_{V})$ but $\Delta|_{V}=0$ for $Supp\Delta\cap V=\emptyset.$ Hence $$\mathcal{E}xt^{2}(\Delta,\mathcal{E})|_{V}=\left\{\begin{array}{ll}\ext^{2}(\Delta|{V},\mathcal{E}|_{V}) & Supp\Delta\cap V\neq\emptyset.\\ 0 &Supp\Delta\cap V=\emptyset.\end{array}\right.$$ It follows that $\mathcal{E}xt^{2}(\Delta,\mathcal{E})$ is also supported on points, namely $Supp\Delta.$ Thus $\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E})$ is supported on $\Delta.$ Consequently $\h^{1}(X,\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E}))=0.$ \end{proof} \begin{pr} The space $P,$ of ADHM type configurations, is smooth. \end{pr} \begin{proof} Consider the following monad $$\xymatrix{M:&0\ar[r]&\oplus_{i=0}^{n}K_{i}(-1,E_{i})\ar[r]^{\quad\quad\alpha}& W\ar[r]^{\beta\quad\quad}& \oplus_{i=0}^{n}L_{i}(1,-E_{i})\ar[r]&0}$$ on $\tilde{X}de{\mathbb{P}}$ and the mapping \begin{displaymath} \xymatrix@R-1pc@C-1pc{g:&\mathbb{H}\oplus\mathbb{F}\ar[r]&\mathbb{G} \\ &(\alpha,\beta)\ar[r]&\beta\circ\alpha } \end{displaymath} The spaces $\mathbb{H}$, $\mathbb{F}$ and $\mathbb{G}$ are defined by $$\left\{\begin{array}{l}\mathbb{H}=\oplus_{i=0}^{n}\Hom(V_{i}, \Hom(K_{i},W))\\ \mathbb{F}=\oplus_{i=0}^{n}\Hom(V_{i}, \Hom(W,L_{i})) \\ \mathbb{G}=\oplus_{i,j=0}^{n}\Hom(V_{ij}, \Hom(K_{i}),L_{j}))\end{array}\right.$$ The spaces $V_{i}$ and $V_{ij}$ are defined so that $V_{i}^{\ast}=\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(1,-E_{i})),$ $V_{ij}^{\ast}=\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(2,-E_{i}-E_{j})).$ The differential of $g$ at a point $(\alpha_{0},\beta_{0})$ is $$dg_{(\alpha_{0},\beta_{0})}=\beta_{0}\circ\alpha+\beta\circ\alpha_{0}.$$ Since $P\subset g^{-1}(0)$ is a Zariski-open subset, it suffices to show that $dg_{(\alpha_{0},\beta_{0})}$ is surjective for any point $(\alpha_{0},\beta_{0}).$ Using lemma \textbf{\ref{obstruction}} and verifying the vanishing of its hypothesis in our case, it follows that $$\h^{2}(\tilde{X}de{\mathbb{P}},\mathcal{E}nd(\mathcal{E}))\longrightarrow Coker(dg_{(\alpha_{0},\beta_{0})})\longrightarrow0$$ is exact. Finally we show that $\h^{2}(\tilde{X}de{\mathbb{P}},\mathcal{E}nd(\mathcal{E}))=0:$ from the local to global spectral sequence \cite[Section \textbf{II.7.4}]{Godement} one has \begin{displaymath} E_{2}^{pq}=\h^{p}(\tilde{X}de{\mathbb{P}},\mathcal{E}xt^{q}(\mathcal{E},\mathcal{E}))\Longrightarrow\ext^{p+q}(\mathcal{E},\mathcal{E}) \end{displaymath} and since $\h^{1}(\tilde{X}de{\mathbb{P}},\mathcal{E}xt^{1}(\mathcal{E},\mathcal{E}))=0$ and $\mathcal{E}xt^{2}(\mathcal{E},\mathcal{E})$ is the zero sheaf, then we have $\h^{2}(\mathcal{E}nd(\mathcal{E},\mathcal{E}))=\ext^{2}(\mathcal{E},\mathcal{E}).$ On the other hand $\ext^{2}(\mathcal{E},\mathcal{E})=\Hom(\mathcal{E},\mathcal{E}(-3,\overrightarrow{1}))^{\ast}.$ By using \begin{displaymath} \xymatrix@C-1pc{0\ar[r]&\mathcal{E}(-3,\overrightarrow{1})\ar[r]&\mathcal{E}(-2,\overrightarrow{1})\ar[r]&\mathcal{E}|_{l_{\infty}}(-2)\ar[r]&0} \end{displaymath} it follows \begin{displaymath} \xymatrix@C-1pc{0\ar[r]&\Hom(\mathcal{E},\mathcal{E}(-3,\overrightarrow{1}))\ar[r]&\Hom(\mathcal{E},\mathcal{E}(-2,\overrightarrow{1})) \ar[r]&\Hom(\mathcal{E},\mathcal{E}|_{l_{\infty}}(-2))\ar[r]&0} \end{displaymath} But $\Hom(\mathcal{E},\mathcal{E}(-2,\overrightarrow{1}))=\ext^{2}(\mathcal{E},\mathcal{E}(-1,0))^{\ast}$ which also vanish because of the following; twisting the sequence by $\mathcal{O}(-1,0)$, and applying the functor $\Hom(\mathcal{E}^{\ast\ast},\bullet)$ one gets the long exact sequence: $$\longrightarrow \ext^{1}(\mathcal{E}^{\ast\ast},\Delta))\longrightarrow \ext^{2}(\mathcal{E}^{\ast\ast},\mathcal{E}(-1,0))\longrightarrow \ext^{2}(\mathcal{E}^{\ast\ast},\mathcal{E}^{\ast\ast}(-1,0))\longrightarrow0.$$ But $\ext^{1}(\mathcal{E}^{\ast\ast},\Delta))\cong\h^{1}(\Sigma,\Delta))^{\oplus r}$ vanishes since $Supp(\Delta)$ is zero dimensional. $\ext^{2}(\mathcal{E}^{\ast\ast},\mathcal{E}^{\ast\ast}(-1,0))$ $=\h^{2}(\tilde{X}de{\mathbb{P}}, \mathcal{E}^{\ast}\otimes\mathcal{E}^{\ast\ast}(-1,0))$ vanishes since for a framed sheaf $\mathcal{F}$, $\h^{2}(\tilde{X}de{\mathbb{P}}, \mathcal{F}(-1,\vec{q})=0 \quad\forall \vec{q}$. Hence $\ext^{2}(\mathcal{E}^{\ast\ast},\mathcal{E}(-1,0))=0$. Applying again the functor $\Hom(\bullet,\mathcal{E}(-1,0))$ on the sequence \eqref{doubledual}, one has $$\longrightarrow \ext^{2}(\mathcal{E}^{\ast\ast},\mathcal{E}(-1,0))\longrightarrow \ext^{2}(\mathcal{E},\mathcal{E}(-1,0))\longrightarrow0.$$ Thus $\ext^{2}(\mathcal{E},\mathcal{E}(-1,0))=0$ \end{proof} \section{Smoothness of the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$} In this section we will prove the smoothness of the moduli space $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}=P/G$ where $P$ is the space of the ADHM data $\rho:=(a,q^{A},c,d)$ and $G$ is the symmetry group acting on this data as described in section \textbf{\ref{ADHM}}. First let $\mathcal{E}$ and $\mathcal{E}'$ be two framed torsion-free sheaves with the same fixed Chern class, and let $\alpha:\mathcal{E}\longrightarrow\mathcal{E}'$ be a morphism preserving the framing up to a homothety, i.e., the diagram \begin{equation} \xymatrix@C-0.5pc@R-0.5pc{\mathcal{E}\ar[r]\ar[d]_{\alpha}&\mathcal{E}|_{l_{\infty}}\ar[r]^{\Phi}\ar[d]_{\alpha|_{l_{\infty}}}& \mathcal{O}|_{l_{\infty}}^{\oplus r}\ar[d]^{\lambda}\\ \mathcal{E}' \ar[r]&\mathcal{E}'|_{l_{\infty}}\ar[r]_{\Phi'}&\mathcal{O}|_{l_{\infty}}^{\oplus r} } \end{equation} commutes. $\lambda\in\mathcal{O}^{\ast}|_{l_{\infty}}$ and $r$ is the rank of the sheaves $\mathcal{E}$ and $\mathcal{E}'$. Since $\Phi$ and $\Phi'$ are isomorphisms, one gets the following relation \begin{equation}\label{relation} \alpha|_{l_{\infty}}=\lambda\Phi'^{-1}\Phi. \end{equation} Since $Supp(|H|)$ is open dense in $\tilde{X}de{\mathbb{P}}$, and the fixed line $l_{\infty}$ is linearly equivalent to $H$, then the morphism $\alpha$ is completely determined by its restriction $\alpha|_{l_{\infty}}.$ Define $\Hom^{\Phi}(\mathcal{E},\mathcal{E}')$ to be the subgroup of $\Hom(\mathcal{E},\mathcal{E}')$ which contains the morphisms preserving the framing up to a homothety. $\Phi'^{-1}\Phi$ being a fixed element of $\End(\mathcal{O}|_{l_{\infty}}^{\oplus r})$ and $\lambda\in\mathcal{O}|_{l_{\infty}}$, the dimension of $\Hom^{\Phi}(\mathcal{E},\mathcal{E}')$ as a subspace of $\Hom(\mathcal{E},\mathcal{E}')$ is $1$. Let us consider the universal monad on $\tilde{X}de{\mathbb{P}}\times P$ given by \eqref{univ-monad}; its cohomology is the $P$-flat family $\mathfrak{F}$. For every point $\rho\in P$, the fiber $\mathfrak{F}_{\rho}\cong\mathcal{E}(\rho)$ is a framed torsion-free sheaf. On $P\times P\times\tilde{X}de{\mathbb{P}}$ one has the following natural projections: $$\xymatrix@R-0.5pc{P\times P\times\tilde{X}de{\mathbb{P}}\ar[d]_{pr_{12}}\ar@<1ex>[r]^{pr_{13}}\ar@<-1ex>[r]_{pr_{23}}& P\times\tilde{X}de{\mathbb{P}}\\ P\times P& }$$ where $pr_{12}$ projects on the first and the second factors, $pr_{13}$ projects on the first and the third factors and $pr_{23}$ projects on the second and the third factors. Consider the sheaf $\mathcal{H}om(pr_{13}^{\ast}\mathfrak{F},pr_{23}^{\ast}\mathfrak{F})$ on $P\times P\times\tilde{X}de{\mathbb{P}}$. The sheaves $pr_{23}^{\ast}\mathfrak{F}$ and $pr_{23}^{\ast}\mathfrak{F}$ are flat on $P\times P$, and then $\mathcal{H}om(pr_{13}^{\ast}\mathfrak{F},pr_{23}^{\ast}\mathfrak{F})$ is flat on $P\times P$. Let us use the following notation: first we omit the pull-back symbols in order to avoid overloading text and formulas. Then we put $(\mathfrak{F}, \phi):=\xymatrix{\mathfrak{F}\ar[r]^\phi & \mathcal{O}^{\oplus r}_{l_{\infty}}}$ where the morphism $\phi$ is given by the triangle $$\xymatrix@C-0.5pc@R-0.5pc{\mathfrak{F}\ar[r]^r \ar[dr]_\phi & \mathfrak{F}|_{l_{\infty}}\ar[d]^\Phi \\ & \mathcal{O}^{\oplus r}_{l_{\infty}} }$$ and define \begin{displaymath} \Hom((\mathfrak{F}, \phi),(\mathfrak{F}', \psi)):=\Bigg\{\begin{array}{ll}\alpha:\mathfrak{F}\longrightarrow\mathfrak{F}'& \\ \lambda\in\mathcal{O}_{l_{\infty}}& \end{array} {\Large\Bigg{|}} \small{\textrm{ such that }} \quad\begin{array}{l} \\ \xymatrix@R-0.5pc@C-0.5pc{\ar@{}[dr] |{\circlearrowright}\mathfrak{F}\ar[r]^\alpha \ar[d]^\phi & \mathfrak{F}'\ar[d]^\psi \\ \mathcal{O}^{\oplus r}_{l_{\infty}}\ar[r]^\lambda & \mathcal{O}^{\oplus r}_{l_{\infty}}} \\ \\ \end{array}\Bigg\} \end{displaymath} Now define the pre-sheaf $\mathcal{H}om((\mathfrak{F}, \phi),(\mathfrak{F}', \psi))$ given by \begin{equation}\label{sheaf} U\longrightarrow \Hom((\mathfrak{F}|_{U}, \phi|_{U}),(\mathfrak{F}'|_{U}, \psi|_{U})), \end{equation} i.e., to every open subset $U$ such that $U\cap l_{\infty}\neq\emptyset$ (for example $\tilde{X}de{\mathbb{P}}\backslash E_{i},$ where $E_{i}$ is one of the exceptional divisors for example) we associate the diagram \begin{equation} \xymatrix@C-0.5pc@R-0.5pc{\ar@{}[dr] |{\circlearrowright}\mathfrak{F}|_{U}\ar[r]^{\alpha|_{U}} \ar[d]_{\phi|_{U}} & \mathfrak{F}'|_{U}\ar[d]^{\psi|_{U}} \\ \mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\ar[r]^\lambda & \mathcal{O}^{\oplus r}_{U\cap l_{\infty}}.} \end{equation} and to every open $U$ such that $U\cap l_{\infty}=\emptyset$ (for instance $\tilde{X}de{\mathbb{P}}\backslash l_{\infty}$) we just associate $\Hom(\mathfrak{F}|_{U},\mathfrak{F}'|_{U}).$ Moreover this is a sheaf; the sheaf axioms are inherited from the sheaf properties of $\mathfrak{F}$, $\mathfrak{F}'$ and the commutation of the square diagrams. \begin{pr} The sheaf $\mathcal{H}om((\mathfrak{F}, \phi),(\mathfrak{F}', \psi))$ is flat over $P\times P$ \end{pr} \begin {proof} Since the question is local on $P\times P,$ one can consider an open affine $W=\spec A\subset P\times P$ and work with $A$-modules, where $A$ is a commutative noetherian ring. Then one can show that $\mathcal{H}om((\mathfrak{F}, \phi),(\mathfrak{F}', \psi))$ is $A$-flat: let \begin{equation}\label{modules} 0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow0 \end{equation} be an exact sequence of $A$-modules, and let $\mathfrak{F}$ be an $A$-flat family of framed torsion-free sheaves on $\tilde{X}de{\mathbb{P}}$. One has the short exact sequence $$0\longrightarrow\mathfrak{F}\otimes_{A}M'\longrightarrow\mathfrak{F}\otimes_{A}M\longrightarrow\mathfrak{F}\otimes_{A}M''\longrightarrow0.$$ The restriction of $\mathfrak{F}$ to any open $U$ is also an $A$-flat module, then one has the sequence $$0\longrightarrow\mathfrak{F}|_{U}\otimes_{A}M'\longrightarrow\mathfrak{F}|_{U}\otimes_{A}M\longrightarrow \mathfrak{F}|_{U}\otimes_{A}M''\longrightarrow0$$ The same situation is true for any other such family $\mathfrak{F}'$, and if we consider any morphism $\alpha|_{U}:\mathfrak{F}|_{U}\longrightarrow\mathfrak{F}'|_{U}$ in $\Hom((\mathfrak{F}|_{U}, \phi|_{U}),(\mathfrak{F}'|_{U}, \psi|_{U}))$, we get the following diagram: {\scriptsize \begin{displaymath} \xymatrix@C-1.6pc@R-0.9pc{&0\ar[rr]&&\mathfrak{F}|_{U}\otimes_{A}M'\ar[rr]\ar@{->}'[d][ddd]|{\alpha|_{U}\otimes I_{M'}}\ar[dl]|{\phi|_{U}\otimes_{A}I_{M'}}&&\mathfrak{F}|_{U}\otimes_{A}M\ar[rr]\ar@{->}'[d][ddd]|{\alpha|_{U}\otimes I_{M}}\ar[dl]|{\phi|_{U}\otimes_{A}I_{M}}&& \mathfrak{F}|_{U}\otimes_{A}M''\ar[rr]\ar@{->}'[d][ddd]|{\alpha|_{U}\otimes I_{M''}}\ar[dl]|{\phi|_{U}\otimes_{A}I_{M''}}&&0\\ 0\ar[rr]&&\mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\otimes_{A}M'\ar[rr]\ar@{-}[d]&&\mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\otimes_{A}M\ar[rr]\ar@{-}[d]&&\mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\otimes_{A}M''\ar[rr]\ar@{-}[d]&&0&\\ &&\lambda\otimes I_{M'}\ar[dd]&&\lambda\otimes I_{M}\ar[dd]&&\lambda\otimes I_{M''}\ar[dd]&&&\\ &0\ar@{->}'[r][rr]&&\mathfrak{F}'|_{U}\otimes_{A}M'\ar@{->}'[r][rr]\ar[dl]|{\psi|_{U}\otimes_{A}I_{M'}}& &\mathfrak{F}'|_{U}\otimes_{A}M\ar@{->}'[r][rr]\ar[dl]|{\psi|_{U}\otimes_{A}I_{M}}& &\mathfrak{F}'|_{U}\otimes_{A}M''\ar[rr]\ar[dl]|{\psi|_{U}\otimes_{A}I_{M''}}&&0\\ 0\ar[rr]&&\mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\otimes_{A}M'\ar[rr]&&\mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\otimes_{A}M\ar[rr]&&\mathcal{O}^{\oplus r}_{U\cap l_{\infty}}\otimes_{A}M''\ar[rr]&&0&\\ } \end{displaymath} } where $I_{M},$ $I_{M'}$ and $I_{M''}$ are the identity morphisms on $M$, $M'$ and $M''$ respectively. On the other hand, if we twist the sequence \eqref{modules} by the $A-$module $\Hom((\mathfrak{F}|_{U},\phi|_{U}),(\mathfrak{F}'|_{U},\psi|_{U}))$, then we get the sequence {\small $$\Hom((\mathfrak{F}|_{U},\phi|_{U}),(\mathfrak{F}'|_{U},\psi|_{U}))\otimes_{A}M'\stackrel{\Xi}{ \longrightarrow} \Hom((\mathfrak{F}|_{U},\phi|_{U}),(\mathfrak{F}'|_{U},\psi|_{U}))\otimes_{A}M \qquad\qquad$$ $$\qquad\qquad\qquad\longrightarrow \Hom((\mathfrak{F}|_{U},\phi|_{U}),(\mathfrak{F}'|_{U},\psi|_{U}))\otimes_{A}M''\longrightarrow0.$$ } \underline{{\bf Claim:}} The map $\Xi,$ in the sequence above, is injective. If we denote the first map in \eqref{modules} by $\theta$, then $\Xi:\Sigma_{i}\alpha_{i}\otimes_{A}m_{i}\longrightarrow\Sigma_{i}\alpha_{i}\otimes_{A}\theta(m_{i})$ for all $\alpha_{i}$'s as in the three-dimensional commutative diagram above. By the commutativity of the following diagram \begin{displaymath} \xymatrix@C-0.6pc@R-0.6pc{0\ar[r]&\mathfrak{F}|_{U}\otimes_{A}M'\ar[r]\ar[d]_{\alpha_{i}\otimes_{A}I_{M'}}&\mathfrak{F}|_{U}\otimes_{A}M \ar[r]\ar[d]^{\alpha_{i}\otimes_{A}I_{M}}&\mathfrak{F}|_{U}\otimes_{A}M''\ar[r]\ar[d]&0 \\ 0\ar[r]&\mathfrak{F}'|_{U}\otimes_{A}M'\ar[r]^{\tilde{X}de{\theta}}&\mathfrak{F}'|_{U}\otimes_{A}M\ar[r]&\mathfrak{F}'|_{U}\otimes_{A}M''\ar[r]&0 } \end{displaymath} it follows that if $\alpha_{i}\otimes_{A}I_{M}=0,$ then $\tilde{X}de{\theta}\circ\alpha_{i}\otimes_{A}I_{M'}=0.$ Moreover $\tilde{X}de{\theta}$ is injective, hence $\alpha_{i}\otimes_{A}I_{M'}=0$. Thus $\Xi$ is injective. After gluing sections of the involved sheaves, one gets an exact sequence {\small $$\xymatrix@1{0\ar[r]&\mathcal{H}om((\mathfrak{F},\phi),(\mathfrak{F}',\psi))\otimes_{A}M'\ar[r]& \mathcal{H}om((\mathfrak{F},\phi),(\mathfrak{F}',\psi))\otimes_{A}M&}$$ $$\xymatrix@1{\ar[r]&\mathcal{H}om((\mathfrak{F},\phi),(\mathfrak{F}',\psi))\otimes_{A}M''\ar[r]&0}$$} Hence the sheaf $\mathcal{H}om((\mathfrak{F},\phi),(\mathfrak{F}',\psi))$ is flat over $P\times P$. \end{proof} Let us remark that since the group $G$ acts freely on the non-singular space $P$, with trivial stabilizer for any point $\rho\in P$, then the quotient is smooth if the graph of the group action is closed, that is, the image $\Gamma:=Im\gamma$ of the morphism $$\gamma:G\times P\longrightarrow P\times P$$ is closed \cite[Chapter \textbf{I}, Theorem \textbf{1}]{Sesha}. In our case the pair $(\rho,\sigma)$ is in the graph $\Gamma$ if and only if $dim \Hom^{\Phi}(\mathcal{E}(\rho),\mathcal{E}(\sigma))=1$, in other words \begin{equation} \Gamma =\left\{ (\rho,\sigma)\in P\times P \textbf{/} h^{0}(pr_{12}^{-1}(\rho,\sigma),\mathcal{H}om^{\Phi}(pr_{13}^{\ast}\mathfrak{F}(\rho),pr_{23}^{\ast}\mathfrak{F}(\sigma)))>0 \right\} \end{equation} Since the sheaf $\mathcal{H}om^{\Phi}(pr_{13}^{\ast}\mathfrak{F},pr_{23}^{\ast}\mathfrak{F})$ is flat on $P\times P$, then it follows from the semi-continuity theorem that $\Gamma$ is a closed subscheme. Hence the quotient $P/G$ is smooth. \section{Fineness of the Moduli space} In this section we shall prove that $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is a fine moduli space by using the monads obtained in section \textbf{\ref{unversal}}. We start by showing that it is a coarse moduli space. The construction of the following natural transformation $$\Phi: \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)\longrightarrow \Hom(\bullet,\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k});$$ goes as the following: for any classifying scheme $S$, $\xi\in \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(S)$, and a family $\mathcal{F}$ of torsion-free sheaves on $\tilde{X}de{\mathbb{P}}$, with Chern character $r+\Sigma_{i=1}^{n}a_{i}E_{i}-(k-\frac{|\vec{a}|^{2}}{2})\omega$, classified by $S$, one has a monad given as in \eqref{S-monad} which is canonically associated to $\mathcal{F}$. If we consider an open covering $\{S_{j}\}_{j\in J}$ of $S$, then on every open affine $S_{j}$ the restriction $\mathcal{M}|_{S_{j}}$ is isomorphic to a monad of the form: $$\xymatrix@C-0.8pc{\mathbb{M}(\alpha_{j},\beta_{j}):& \oplus_{i=0}^{n}\mathcal{O}_{\tilde{X}de{\mathbb{P}}}(-1,E_{i})\boxtimes K_{i}\otimes\mathcal{O}_{S_{j}}\ar[r]^{\quad\quad\quad\alpha_{j}}& \mathcal{O}_{\tilde{X}de{\mathbb{P}}}\boxtimes W\otimes\mathcal{O}_{S_{j}}\ar[r]^{\beta_{j}\quad\quad\quad}& \oplus_{i=0}^{n}\mathcal{O}_{\tilde{X}de{\mathbb{P}}}(1,-E_{i})\boxtimes L_{i}\otimes\mathcal{O}_{S_{j}}}$$ where $$\alpha_{j}:S_{j}\longrightarrow \mathbb{H}, \quad\quad\quad\beta_{j}:S_{j}\longrightarrow \mathbb{F}.$$ The spaces $\mathbb{H}$, $\mathbb{F}$ are defined by $$\left\{\begin{array}{l}\mathbb{H}=\oplus_{i=0}^{n}\Hom(V_{i}, \Hom(K_{i},W))\\ \mathbb{F}=\oplus_{i=0}^{n}\Hom(V_{i}, \Hom(W,L_{i}))\end{array}\right.$$ and $V_{i}$ is such that $V_{i}^{\ast}=\h^{0}(\tilde{X}de{\mathbb{P}},\mathcal{O}(1,-E_{i})).$ From the monad condition, $\beta_{i}\circ\alpha_{i}=0,$ we have regular maps $f_{j}=(\alpha_{j},\beta_{j}):S_{j}\longrightarrow P.$ By construction these morphisms satisfy $$f_{i}(s)\sim_{G} f_{j}(s)$$ for $s$ in the intersection $S_{i}\cap S_{j}$, where $G$ is the group defined in \eqref{group-form} with the action given by \eqref{P and G}. The maps $f_{j}$ glue to form a global morphism $$f:S\longrightarrow \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}.$$ This defines the desired natural transformation: \begin{equation}\label{natural-transform} \xymatrix@R-1pc{\Phi:& \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)\ar[r]& \Hom(\bullet,\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k})} \end{equation} by the association $\xymatrix{\xi\ar[r]&\Phi(\xi):=f}$ for every class $\xi=[\mathcal{F}]$ on a given scheme $S.$ Obviously $f$ depends only on the class $\xi=[\mathcal{F}].$ By using the monad, on $\tilde{X}de{\mathbb{P}},$ associated to a closed point $s\in S$ (obtained by restricting the monad $\mathcal{M}$ to the point $s\in S$) one deduces that $\Phi: \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\spec k(s))\longrightarrow \Hom(\spec k(s),\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k})$ is a bijection. Now let $\mathcal{R}$ be another parameterizing scheme such that there is a natural transformation $$\Psi: \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(\bullet)\longrightarrow \Hom(\bullet,\mathcal{R}).$$ If $\mathfrak{F}$ is a universal family on $\tilde{X}de{\mathbb{P}}\times P$ parameterized by $P$ such that $\Phi(\mathfrak{F})=\pi:P\longrightarrow \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$, then, for the natural transformation $\Psi$, we have $\Psi(\mathfrak{F}):P\longrightarrow \mathcal{R}$. \begin{pr} $\Psi(\mathfrak{F})$ is constant along the fibers of the projection $\pi:P\longrightarrow \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}.$ \end{pr} \begin{proof} Let $p=\spec k(p)\in P$ and let $\rho_{1}$, $\rho_{2}\in \Hom(p,P)$ such that $\pi(\rho_{1})=\pi(\rho_{2})$; if we consider the pull-back \begin{equation} \xymatrix@C-0.4pc@R-0.5pc{ &\rho_{i}^{\ast}\mathfrak{F}\ar[r]\ar[d]& \mathfrak{F}\ar[d]\\ &\tilde{X}de{\mathbb{P}}\times p\ar[r]^{(Id_{\tilde{X}de{\mathbb{P}}}\times \rho_{i})}& \tilde{X}de{\mathbb{P}}\times P } \end{equation} then $\Phi(\rho_{1}^{\ast}\mathfrak{F})=\Phi(\mathfrak{F})(\rho_{1})$, and by definition $\Phi(\mathfrak{F})(\rho_{1})=\pi(\rho_{1})$. By assumption, we have $\pi(\rho_{1})=\pi(\rho_{2})=\Phi(\mathfrak{F})(\rho_{2})=\Phi(\rho_{2}^{\ast}\mathfrak{F})$, and since the natural transformation $\Phi$ is a bijection for every closed point, it follows that $$\rho_{1}^{\ast}\mathfrak{F}=\rho_{2}^{\ast}\mathfrak{F}$$ On the other hand we have $$\Psi(\mathfrak{F})(\rho_{1})=\Psi(\rho_{1}^{\ast}\mathfrak{F})=\Psi(\rho_{2}^{\ast}\mathfrak{F})=\Psi(\mathfrak{F})(\rho_{2}).$$ Thus $\Psi(\mathfrak{F})$ is constant along the fibers of $\pi:P\longrightarrow \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ \end{proof} The projection $\pi:P\longrightarrow \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ locally has sections, so one can construct local mappings $\bar{\phi}:\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}\longrightarrow \mathcal{R}$, but since $\Psi(\mathfrak{F})$ is constant along the fibers of $\pi$, then the map $\bar{\phi}$ can be lifted to a global map $\phi$ such that the following diagram commutes: \begin{equation} \xymatrix@C-0.5pc@R-0.5pc{P \ar[r]^{\Psi(\mathfrak{F})}\ar[d]_{\Phi(\mathfrak{F})=\pi}& \mathcal{R}\\ \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}\ar[ru]_{\phi}& } \end{equation} Now for any parameterizing scheme $S$ and a family $\xi$ on $\tilde{X}de{\mathbb{P}}\times S$, one has $\Psi(\xi):S\longrightarrow\mathcal{R}$. Let $\{S_{i}\}_{i\in I}$ be an open cover of $S$. The diagram \begin{displaymath} \xymatrix@C-0.5pc@R-0.5pc{S \ar[dr]\ar@/^/@{.>}[drr]^{\Psi(\xi)}&& \\ &P\ar[r]& \mathcal{R}\\ S_{i}\ar@{^{(}->}[uu]^{g_{i}}\ar@{-->}[ur]\ar[urr]&& } \end{displaymath} commutes, and we have $\xi|_{S_{i}}=g_{i}^{\ast}(\mathfrak{F})$. Consequently \begin{align} \Psi(\xi)|_{S_{i}}&=\Psi(\xi|_{S_{i}})\notag \\ &=\Psi(g_{i}^{\ast}(\mathfrak{F})) \notag \\ &=g_{i}^{\ast}\Psi(\mathfrak{F}) \notag \end{align} On the other hand $\Psi(\mathfrak{F})=\phi\circ\Phi(\mathfrak{F})$, hence \begin{align} \Psi(\xi)|_{S_{i}}&=g_{i}^{\ast}(\phi\circ\Phi(\mathfrak{F}))\notag \\ &=\phi\circ\Phi(g_{i}^{\ast}(\mathfrak{F})) \notag \\ &=\phi\circ\Phi(\xi|_{S_{i}}) \notag \\ \Psi(\xi)|_{S_{i}}&=[\phi\circ\Phi(\xi)]|_{S_{i}} \notag \end{align} These maps glue together to form a global map $$\Psi(\xi)=\phi\circ\Phi(\xi)$$ on $S$. Since $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is reduced (this follows from the smoothness), then the map $\phi$ is uniquely determined. By this we showed the following: \begin{thm} The scheme $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is a coarse moduli space. \end{thm} The final step is to show that $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is fine. To do this we shall descend the universal monadic description on $\tilde{X}de{\mathbb{P}}\times P$ to a well behaved monadic description on $\tilde{X}de{\mathbb{P}}\times\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$. This can be realized, in our case, because the space $\tilde{X}de{\mathbb{P}}\times P$ is a $G$-space: indeed there is a natural action \begin{equation*} \xymatrix@R-1.5pc{G\times\tilde{X}de{\mathbb{P}}\times P \ar[r]&\tilde{X}de{\mathbb{P}}\times P \\ (g,(x,\mathcal{C}))\ar[r]&(x,g\cdot\mathcal{C})} \end{equation*} which induces a $G$-action on the universal monad $\mathbb{M}$, in \eqref{univ-monad}, and which descends to an action on its cohomology $\mathfrak{F}.$ Since the action is free and the isotropy subgroup is trivial at all points, we have a well defined family $\mathfrak{F}/G\longrightarrow\tilde{X}de{\mathbb{P}}\times P/G$. We put $\mathfrak{U}:=\mathfrak{F}/G$ which is a canonical family $$\mathfrak{U}\longrightarrow\tilde{X}de{\mathbb{P}}\times\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$$ parameterized by $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$. \begin{pr} For any noetherian scheme $S$ of finite type, the mapping $$\begin{array}{cccc}\Hom(S,\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}) & \longrightarrow & \mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}(S)\\ \phi & \longrightarrow & \phi^{\ast}[\mathfrak{U}]=[(Id_{\tilde{X}de{\mathbb{P}}}\times\phi)^{\ast}\mathfrak{U}]\end{array}$$ is bijective. \end{pr} \begin{proof} \underline{Injectivity}: Let $\phi_{1}, \phi_{2}: S\longrightarrow\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ such that $(Id_{\tilde{X}de{\mathbb{P}}}\times\phi_{1})^{\ast}\mathfrak{U}\cong(Id_{\tilde{X}de{\mathbb{P}}}\times\phi_{2})^{\ast}\mathfrak{U}$ then for every point $s\in S$, one has $\mathfrak{U}(\phi_{1}(s))=\mathfrak{U}(\phi_{2}(s))$. Since the torsion-free sheaf $\mathfrak{U}(\phi_{i}(s))$ is the one given by the ADHM data associated to the point $\phi_{i}(s)\in \mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$, then $\phi_{1}(s)=\phi_{2}(s)$ for every point $s\in S$, thus $\phi_{1}=\phi_{2}.$ \underline{Surjectivity}: Given a family $\mathcal{F}$ parameterized by $S,$ one has the morphism $\phi=\Phi(\mathcal{F})$ given by the natural transformation \eqref{natural-transform}. Then $\mathcal{F}$ is the pull-back of the family $\mathfrak{U}$ parameterized by $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$. \end{proof} This finishes the proof of the following \begin{thm} The scheme $\mathcal{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}$ is a fine moduli space for the moduli functor $\mathfrak{M}^{\tilde{X}de{\mathbb{P}}}_{\vec{a}, k}.$ \end{thm} \end{document}

\begin{document} \newtheorem{thm}{Theorem} \newtheorem{cor}{Corollary} \newproof{pf}{Proof} \begin{frontmatter} \title{Two stage design for estimating the product of means with cost in the case of the exponential family} \author[1]{Zohra BENKAMRA} \author[2]{Mekki TERBECHE} \author[1]{Mounir TLEMCANI\corref{*}} \address[1]{Department of Mathematics. University of Oran, Algeria} \address[2]{Department of Physics, L.A.A.R University Mohamed Boudiaf, Oran, Algeria} \cortext[*]{Corresponding author : mounir.tlemcani@univ-pau.fr (M.Tlemcani)} \begin{abstract} We investigate the problem of estimating the product of means of independent populations from the one parameter exponential family in a Bayesian framework. We give a random design which allocates $m_{i}$ the number of observations from population $P_{i}$ such that the Bayes risk associated with squared error loss and cost per unit observation is as small as possible. The design is shown to be asymptotically optimal. \end{abstract} \begin{keyword} Two stage design, product of means, exponential family, Bayes risk, cost, asymptotic optimality. \end{keyword} \end{frontmatter} \section{Introduction} Assume that for $i=1,...,n$; a random variable $X_{i}$ whose distribution belongs to the one parameter exponential family is observable from population $P_{i}$ with cost $c_{i}$ per unit observation. The problem of estimating several means in the case of the exponential family distributions with linear combination of losses was addressed by \cite{cohen}. The problem of interest in this paper is to estimate the product of means using a Bayesian approach associated with squared error loss and cost. Since a Bayesian framework is considered; see, e.g., \cite{page,shapiro}, then typically optimal estimators are Bayesian estimators and the problem turns to design a sequential allocation scheme; see, e.g., \cite{woodroofe}, to select $m_{i}$ the number of observations from population $P_{i}$ such that the Bayes risk plus the corresponding budget $B=\sum_{i=1}^{n}c_{i}m_{i}$ is as small as possible. \cite{terbeche aas}, have defined a sequential design to estimate the difference between means of two populations from the exponential family with associated cost. The random allocation was shown to be the best from numerical considerations; see, e.g., \cite{terbeche phd}. Similarly, the problem of estimating the product of several means of independent populations, subject to the constraint of a total number of observations $M$ fixed, was addressed by \cite{rekab}, using a two stage approach. The allocation of $m_{i}$ was nonrandom and the first order optimality was shown for large $M$. Suppose that $X_{i}$ has the distribution of the form \[ f_{\theta _{i}}(x_{i})\varpropto e^{\theta _{i}x_{i}-\psi (\theta _{i})},~x_{i}\in \mathbb{R} ,~\theta _{i}\in \Omega \] where $\Omega $ is a bounded open interval in $\mathbb{R}$. It follows that $E_{\theta _{i}}\left[ x_{i}\right] =\psi ^{\prime }\left( \theta _{i}\right) $ and $Var_{\theta _{i}}\left[ x_{i}\right] =\psi ^{\prime \prime }\left( \theta _{i}\right)$. One assumes that prior distribution for each $\theta _{i}$ is given by \[ \pi _{i}\left( \theta _{i}\right)\varpropto e^{r_{i}\left( \mu _{i}\theta _{i}-\psi \left( \theta _{i}\right)\right) } \] where $r_{i}$ and $\mu _{i}$ are reals and $r_{i}>0$, $i=1,...,n$. Here we treat $\theta _{i}$ as a realization of a random variable and assume that for each population, $x_{i1},...,x_{im_{i}}$ are conditionally independent and that $\theta _{1},...,\theta _{n}$ are a priori independent. Our aim is to estimate the product $\theta =\prod_{i=1}^{n}\psi ^{\prime }\left( \theta _{i}\right)$, subject to squared error loss and linear cost. \section{The Bayes risk} Let $\mathcal{F}_{m_{1},...,m_{n}}$ the $\sigma $-Field generated by $\left( X_{1},...,X_{n}\right) $ where $X_{i}=\left( x_{i1},...,x_{im_{i}}\right) $ and let $\mathcal{F}_{m_{i}}=\sigma \left( X_{i}\right) =\sigma \left( x_{i1},...,x_{im_{i}}\right) $. It was shown that \begin{eqnarray} E\left[ \psi ^{\prime }\left( \theta _{i}\right) /\mathcal{F}_{m_{i}} \right] &=&=\frac{\mu_{i}r_{i}+\sum_{j=1}^{m_{i}}x_{ij}}{m_{i}+r_{i}} \label{th 2.2.1-1} \\ Var\left[ \psi ^{\prime }\left( \theta _{i}\right) /\mathcal{F}_{m_{i}} \right] &=&E\left[ \frac{\psi ^{\prime \prime }\left( \theta _{i}\right) }{ m_{i}+r_{i}}/\mathcal{F}_{m_{i}}\right], \label{th 2.2.1-2} \end{eqnarray} (see \cite{terbeche sort}). Using independence across populations, the Bayes estimator of $\theta $ is \[ \hat{\theta}=E\left[ \theta /\mathcal{F}_{m_{1},...,m_{n}}\right] =\prod\limits_{i=1}^{n}E\left[ \psi ^{\prime }\left( \theta _{i}\right) / \mathcal{F}_{m_{i}}\right] \] Assume that there exists $p\geq 1$ such that \begin{equation} E\left[ \left( \psi ^{\prime \prime }\left( \theta _{i}\right) \right) ^{p} \right] <+\infty ~ and ~ E\left[ \left( \psi ^{\prime }\left( \theta _{i}\right) \right) ^{2p}\right] <+\infty, \label{cond} \end{equation} for all $i=1,...,n$; then the corresponding Bayes risk associated with quadratic loss and cost can be written as follows, \begin{equation} \label{r} R\left( m_{1},...,m_{n}\right) =E\left[ \sum\limits_{i=1}^{n}\frac{U_{im_{i}} }{m_{i}+r_{i}}+\sum\limits_{i=1}^{n}c_{i}m_{i}\right] +\sum \limits_{i=1}^{n}o\left( \frac{1}{m_{i}}\right) \end{equation} and by the way, it can be approximated for large samples by \begin{equation} \label{rtilde} \tilde{R}\left( P\right) =\tilde{R}\left( m_{1},...,m_{n}\right) =E\left[ \sum\limits_{i=1}^{n}\frac{U_{im_{i}}}{m_{i}+r_{i}}+\sum \limits_{i=1}^{n}c_{i}m_{i}\right] \end{equation} where $U_{im_{i}}=E\left[ V_{i}/\mathcal{F}_{m_{1},...,m_{n}}\right] $ and $V_{i}=\psi ^{\prime \prime }\left( \theta _{i}\right) \prod_{j\neq i}\psi ^{\prime 2}\left( \theta _{j}\right)$ \section{Lower bound for the scaled Bayes risk} From now on, the notation $c\rightarrow 0$ means that $c_{j}\rightarrow 0$, for all $j=1,...,n$. Assume that for all $i$, \begin{equation} \label{ci} \frac{c_{i}}{\sum\limits_{j=1}^{n}c_{j}}\rightarrow \lambda _{i}\in \left] 0,1\right[ ,\ as \ c\rightarrow 0. \end{equation} \begin{thm} \label{th 233}For any random design (P) satisfying \begin{equation} \label{ai} m_{i}\sqrt{c_{i}}\rightarrow a_{i}\neq 0~,~a.s., \ as\ c\rightarrow 0; \end{equation} then \begin{equation} \label{res} \liminf_{c\rightarrow 0}\frac{R(P)}{\sqrt{ \sum\limits_{j=1}^{n}c_{j}}}\geq 2E \left[ \sum\limits_{i=1}^{n}\sqrt{\lambda _{i}}\sqrt{V_{i}}\right] \end{equation} \end{thm} \begin{pf} Expressions (\ref{r}) and (\ref{rtilde}) with the help of (\ref{ci}) and (\ref{ai}), give \begin{equation} \label{rrtilde} \liminf_{c\rightarrow 0 }\frac{R(P)}{\sqrt{ \sum\limits_{j=1}^{n}c_{j}}}=\liminf_{c\rightarrow 0}\frac{\tilde{R}(P)}{\sqrt{\sum\limits_{j=1}^{n}c_{j}}} \end{equation} and the scaled approximated Bayes risk satisfies the following inequality: \[ \frac{\tilde{R}(P)}{\sqrt{\sum\limits_{j=1}^{n}c_{j}}}\geq 2E\left[ \sum\limits_{i=1}^{n}\sqrt{\frac{c_{i}}{\sum\limits_{j=1}^{n}c_{j}}}\sqrt{ U_{im_{i}}}\right] -\sum\limits_{i=1}^{n}\sqrt{\frac{c_{i}}{ \sum\limits_{j=1}^{n}c_{j}}}\sqrt{c_{i}}r_{i}, \] since for all $i$, \begin{eqnarray*} \frac{U_{im_{i}}}{m_{i}+r_{i}}+c_{i}m_{i} &=&\left( \frac{\sqrt{U_{im_{i}}}}{\sqrt{m_{i}+r_{i}}}+\sqrt{c_{i}}\sqrt{ m_{i}+r_{i}}\right) ^{2}+2\sqrt{c_{i}}\sqrt{U_{im_{i}}}-c_{i}r_{i}\\ &\geq &2\sqrt{c_{i}}\sqrt{U_{im_{i}}}-c_{i}r_{i}. \end{eqnarray*} Finally, Fatou's lemma and condition (\ref{ci}) give \[ \liminf_{c\rightarrow 0}\frac{\tilde{R}(P)}{ \sqrt{\sum\limits_{j=1}^{n}c_{j}}}\geq 2E\left[ \liminf_{c\rightarrow 0}\sum\limits_{i=1}^{n}\sqrt{\frac{c_{i}}{ \sum\limits_{j=1}^{n}c_{j}}}\sqrt{U_{im_{i}}}\right] =2E\left[ \sum\limits_{i=1}^{n}\sqrt{\lambda _{i}}\sqrt{V_{i}}\right] \] and the proof follows. \end{pf} \section{First order optimal design} According to condition (\ref{ai}) and identity (\ref{rrtilde}) a first order optimal design with respect to $\sum_{i=1}^{n}m_{i}=m$ must satisfy \begin{equation} \label{conver} \frac{\tilde{R}(P)}{\sqrt{\sum\limits_{j=1}^{n}c_{j}}}-2E\left[ \sum\limits_{i=1}^{n}\sqrt{\lambda _{i}}\sqrt{V_{i}}\right] \rightarrow 0, \ as\ c\rightarrow 0, \end{equation} It should be pointed that condition (\ref{conver}) is actually similar to the first order efficiency property for A.P.O rules in Bayes sequential estimation; see, e.g., \cite{leng,hwang}, for one-parameter exponential families, which involves a sequential allocation procedure and a stopping time. In our approach, condition (\ref{conver}) is handled by the following expansion. \begin{eqnarray*} \frac{\tilde{R}(P)}{\sqrt{\sum\limits_{j=1}^{n}c_{j}}} &=& \frac{2E\left[ \sum\limits_{i=1}^{n}\sqrt{c_{i}}\sqrt{U_{im_{i}}}\right] }{ \sqrt{\sum\limits_{j=1}^{n}c_{j}}} +\frac{E\left[ \sum\limits_{i=1}^{n}\frac{\left( \sqrt{U_{im_{i}}}-\left( m_{i}+r_{i}\right) \sqrt{c_{i}}\right) ^{2}}{m_{i}+r_{i}}\right] }{\sqrt{ \sum\limits_{j=1}^{n}c_{j}}} \\ &-&\sum\limits_{i=1}^{n}\sqrt{c_{i}}\sqrt{\frac{ c_{i}}{\sum\limits_{j=1}^{n}c_{j}}}r_{i} \end{eqnarray*} The last term goes to zero as $c\rightarrow 0$, thanks to condition (\ref{ci}). Hence, sufficient conditions for a design to satisfy (\ref{conver}) are \begin{eqnarray} E\left[ \sum\limits_{i=1}^{n}\sqrt{\frac{c_{i}}{\sum\limits_{j=1}^{n}c_{j}}} \sqrt{U_{im_{i}}}\right] -E\left[ \sum\limits_{i=1}^{n}\sqrt{\lambda _{i}} \sqrt{V_{i}}\right] &\rightarrow &0 \label{cs1} \\ E\left[ \frac{\left( \sqrt{U_{im_{i}}}-\left( m_{i}+r_{i}\right) \sqrt{c_{i}} \right) ^{2}}{\left( m_{i}+r_{i}\right) \sqrt{\sum\limits_{j=1}^{n}c_{j}}} \right] &\rightarrow &0,~\forall i \label{cs2} \end{eqnarray} as $c\rightarrow 0$. \begin{thm} \label{th 241}Let $P$ a random policy satisfying $m_{i}\rightarrow +\infty ,~a.s.$, and suppose that condition (\ref{cond}) is true, then \[ E\left[ \sum\limits_{i=1}^{n}\sqrt{\frac{c_{i}}{\sum\limits_{j=1}^{n}c_{j}}} \sqrt{U_{im_{i}}}\right] -E\left[ \sum\limits_{i=1}^{n}\sqrt{\lambda _{i}} \sqrt{V_{i}}\right] \rightarrow 0, \ as\ c\rightarrow 0. \] \end{thm} \begin{pf} Remark that \begin{equation} \lim_{m_{1},...,m_{n}\rightarrow +\infty }\sqrt{U_{im_{i}}}=\sqrt{V_{i}} ,~a.s. \label{conv} \end{equation} Now \begin{eqnarray*} \sup_{m_{1},...,m_{n}}E\left[ \left( \sqrt{U_{im_{i}}}\right) ^{2}\right] &=&\sup_{m_{1},...,m_{n}}E\left[ U_{im_{i}}\right] \\ &=&E\left[ \psi ^{^{\prime \prime }}\left( \theta _{i}\right) \prod\limits_{j\neq i}\psi ^{\prime ^{2}}\left( \theta _{j}\right) \right] \\ &=&E\left[ \psi ^{^{\prime \prime }}\left( \theta _{i}\right) \right] \prod\limits_{j\neq i}E\left[ \psi ^{\prime ^{2}}\left( \theta _{j}\right) \right] <+\infty \end{eqnarray*} hence, the uniform integrability of $\sqrt{U_{im_{i}}}$ follows from condition (\ref{cond}) and martingales properties. Therefore, the convergence in (\ref{conv}) holds in $L^{1}$ and consequently : \[ \sqrt{\frac{c_{i}}{\sum\limits_{j=1}^{n}c_{j}}}\sqrt{U_{im_{i}}}\rightarrow \sqrt{\lambda _{i}}\sqrt{V_{i}} ~ in ~ L^{1}, \ as\ c\rightarrow 0, \] which achieves the proof. \end{pf} \section{The two stage procedure} Following the previous section, our strategy now is to satisfy condition ( \ref{cs2}). Then, we define the two stage sequential scheme as follows. \begin{description} \item[Stage one] proceed for $k_{i}$ observation from population $P_{i}$ for $i=1,...,n$; such that $k_{i}\sqrt{c_{i}}\rightarrow 0$ and $ k_{i}\rightarrow +\infty $ as $c_{i}\rightarrow 0$. \item[Stage two] for $i=1,...,n$; select $m_{i}$ integer as follows : \[ m_{i}=\max \left\{ k_{i},\left[ \frac{\sqrt{U_{ik_{i}}}}{\sqrt{c_{i}}}-r_{i}\right]\right\} \] where $\left[ x\right]$ denotes the integer part of $x$ and \[ U_{ik_{i}}=E\left[ \psi ^{^{\prime \prime }}\left( \theta _{i}\right) \prod\limits_{j\neq i}\psi ^{\prime ^{2}}\left( \theta _{j}\right) /\mathcal{ F}_{k_{1},...,k_{n}}\right] \] \end{description} We give now the main result of the paper. \begin{thm} \label{th 251}Assume condition (\ref{cond}) satisfied for a $p\geq 1$, then the two stage design is first order optimal. \end{thm} \begin{pf} The $m_{i}$, as defined by the two stage, satisfies \[ \lim_{c_{i}\rightarrow 0}\left( m_{i}+r_{i}\right) \sqrt{c_{i}}=\sqrt{V_{i}} \] and since \[ \sqrt{\sum\limits_{j=1}^{n}c_{j}}\left( m_{i}+r_{i}\right) =\frac{\sqrt{c_{i} }\left( m_{i}+r_{i}\right) }{\sqrt{\frac{c_{i}}{\sum\limits_{j=1}^{n}c_{j}}}} \rightarrow \sqrt{\frac{V_{i}}{\lambda _{i}}}, \ as\ c\rightarrow 0, \] then \begin{equation} \frac{\left( \sqrt{U_{im_{i}}}-\left( m_{i}+r_{i}\right) \sqrt{c_{i}}\right) ^{2}}{\sqrt{\sum\limits_{j=1}^{n}c_{j}}\left( m_{i}+r_{i}\right) } \rightarrow 0,~a.s.,\ as\ c\rightarrow 0 \label{ae} \end{equation} To show the convergence in $L^{1}$, it will be sufficient to show the uniform integrability of the left hand side of (\ref{ae}). So, observe that \begin{eqnarray*} \frac{\left( \sqrt{U_{im_{i}}}-\left( m_{i}+r_{i}\right) \sqrt{c_{i}}\right) ^{2}}{\sqrt{\sum\limits_{j=1}^{n}c_{j}}\left( m_{i}+r_{i}\right) } &\leq & \frac{U_{im_{i}}+\left( m_{i}+r_{i}\right) ^{2}c_{i}}{\sqrt{ \sum\limits_{j=1}^{n}c_{j}}\left( m_{i}+r_{i}\right) } \\ &\leq &\frac{U_{im_{i}}}{\sqrt{U_{ik_{i}}}}+\sqrt{U_{ik_{i}}} \end{eqnarray*} $\sqrt{U_{ik_{i}}}$ is uniformly integrable, as a result of martingales L$^{p}$ convergence properties with $p=2$. Now, remark that \[ \frac{U_{im_{i}}}{\sqrt{U_{ik_{i}}}}\leq \max_{k^{\prime }}\sqrt{ U_{ik^{\prime }}} \] and for the remainder of the proof, we use Doob's inequality to show that $E\left[ \max_{k^{\prime }}\sqrt{U_{ik^{\prime }}}\right] <+\infty$. We have, \[ E\left[ \max_{k^{\prime }}\left( \sqrt{U_{ik^{\prime }}}\right) ^{2p}\right] \leq \left( \frac{2p}{2p-1}\right) ^{2p}E\left[ \left( \sqrt{V_{i}}\right) ^{2p}\right] <+\infty \] hence, since $p\geq 1,$ $\max_{k^{\prime }}\sqrt{U_{ik^{\prime }}}$ is integrable and the proof follows. \end{pf} \section{Conclusion} The proof of the first order asymptotic optimality for the two stage design has been obtained mainly through an adequate scaling of the approximated Bayes risk associated with squared error loss and cost, a lower bound for the scaled Bayes risk, martingales properties and Doob's inequality. \section*{References} \end{document}

\begin{document} \title{THE DISTANCE BETWEEN THE WEIGHTS OF THE NEURAL NETWORK IS MEANINGFUL\} \begin{abstract} In the application of neural networks, we need to select a suitable model based on the problem complexity and the dataset scale. To analyze the network's capacity, quantifying the information learned by the network is necessary. This paper proves that the distance between the neural network weights in different training stages can be used to estimate the information accumulated by the network in the training process directly. The experiment results verify the utility of this method. An application of this method related to the label corruption is shown at the end. \end{abstract} \section{Introduction} Since Dr. Hebb opened the door of machine learning in \cite{hebb1949organization}, people have obtained endless wealth from it. At the beginning of this century, neural networks' potential in machine learning tasks was discovered in many fields. With more and more people noticing this delicate model's power, applications based on the neural network develop rapidly and change the world gradually \cite{rumelhart1986learning,hochreiter1997long,fukushima1980neocognitron,lecun2015deep,hinton1994autoencoders,sajjadi2017enhancenet,he2017mask,an2015variational,arjovsky2017wasserstein}, which makes people eager to reveal the essence of the neural network. As \cite{hornik1991approximation} proves, the network exists that is capable of arbitrarily accurate approximation to a specific function and its derivatives. Therefore, We need to find a suitable network structure for a specific task. To explain it, we denote one network's simulation capability as $I_0$, the information quantity of the relationship between two variables as $I_1$, and the information quantity of the training dataset as $I_2$. To get a trustworthy model, the basic requirement is $I_2 \geq I_0 >> I_1$. If $I_0<I1$, the model cannot simulate the relationship, which will make it hard to train to fine-tuned (underfitting). Else if $I_1 > I_2$, the dataset cannot express the relationship between the variables, which will mislead the model. Else if $I_0 > I_2$, there are too many options to simulate the relationship, which will make the model overfitted in most cases. In practice, $I_1$ is the exploration target, which can be estimated based on the background research, and $I_2$ can be calculated directly. Now, the question is reduced to \textbf{how to estimate the capability of one model.} There is no doubt that we can use the network's scale to estimate its capacity, but it is just a theoretical estimation. The theoretical upper limit of the human brain far exceeds what we can use. As a system with a similar structure, the network's real capacity is far less than its theoretical estimation. That is the reason why some small models can perform better than the bigger ones. What we take care of is the part we can use. Therefore, the question is reduced to \textbf{how to quantifying the information learned by the neural network in training.} In \cite{tishby2000information}, the author uses the mutual information to analyze the changing of the information quantity in the neural network and put up with the concept called "information bottleneck." The information bottleneck theory describes the neural network's behavior and defines the optimal target, preserving the relevant information about another variable (maximize the bottleneck). One step forward, in \cite{tishby2015deep}, the author develops this method and puts it up with a tool, information plain, to visualize the neural network's behavior. Series of methods related to it are put forward \cite{alemi2016deep,higgins2016beta,yu2020understanding,achille2019information}, and people start to open the black box of the neural network. \section{Motivation} The similarity of the methods mentioned above is that they analyze the network's information by analyzing the data's changes. These methods have their rationality, and the weakness is also apparent. Generally, errors in statistics cannot be eliminated. Comparing with the scale of the domain of the possible networks' input, the scale of the test case is too small, which will further magnify statistical errors. Specifically, let $I(X;\tilde{X})$ be the mutual information between the input data $X$ and the compressed representation (like the output of one layer) $\tilde{X}$. \begin{equation} I(X ; \tilde{X})=\sum_{x \in X} \sum_{\dot{x} \in \tilde{X}} p(x, \tilde{x}) \log \left[\frac{p(\tilde{x} \mid x)}{p(\tilde{x})}\right] \label{eq_mutual_information} \end{equation} Based on Eq. \ref{eq_mutual_information}, we need to know the joint distribution $P(X,\tilde{X})$ and the distribution $P(\tilde{X})$, which two need to be counted in the experiment. Theoretically, for the neural network, $\tilde{X}$ is a continuous space. Discretization is necessary to count the probability distributions mentioned above. Different discretization functions will impact the result of observation significantly. Moreover, $\tilde{X}$ is tightly related to the layer's activate function based on the definition. If we just discretize $\tilde{X}$ evenly without further discussion, the activate function's feature will impact the observation result and mislead us. For example, in \cite{shwartz2017opening}, the author explains one behavior of the network. As mentioned in that paper, the experiment result indicates that the network's goal is to optimize the Information Bottleneck (IB) trade-off between compression and prediction, successively, for each layer. However, the related conclusion is challenged by other researchers. In \cite{saxe2019information}, the author proves that the two phases are just a special case caused by the non-linear activate function. In a word, based on analyzing the relationship between input and output, the results will be impacted by the experiment's bias. Whereas directly analyzing the weights of the neural network can avoid the errors mentioned above. In this paper, we will provide proof to show that the difference between the initial weights and the training's output weight can be used to estimate the quantity of information of the network accumulating in training. Moreover, we apply this method to analyze the impact of the label corruption. The corresponding experiment is shown at the end of this paper. \section{The uncertainty of weights} As Shannon mentioned in \cite{shannon1948mathematical}, information can be thought of as the resolution of uncertainty. Generally, we can use $H(X)$ to represent the chaos of the variable $X$ ($H(X)=\mathrm{E}[I(X)]$). The increase in information or energy will lead to a decrease in system entropy in the view of physics. Oppositely, if the entropy decrease is quantified, we can quantify the quantity of information accumulated by the system in this process. \begin{equation} I = H_{P_0}(X) - H_p(X) \label{eq_information} \end{equation} Letting $I$ be the quantified information, which equal to the chaos reduction, we can use Eq. \ref{eq_information} to calculate it, where $H_{P_0}(X)$ and $H_{P}(X)$ is the system's entropy before and after receiving the information. To measure the information stored in the weights, we need to understand the uncertainty of the weights. Our viewpoint is the weights of the network are a variable in the training process. The appearance of a specific value of weights is uncertain because of the random factors in the training process, like the randomized initialization, optimizer (e.g., SGD), order of training data, etc. A certain training process is that in which there are no random factors. And it can be viewed as a special case of the uncertain ones. There is another view to understanding the uncertainty of the weights. Generally, the calculation in the layer (including the activate function) is irreversible, which gives the capability of generalization to the network. Otherwise, for a network with specific weights, we can use the output to recover the input, which means the relationship between the input data $X$ and its corresponding compress representation $T$ is bijective, and the network degenerates to a codebook of input $X$. Therefore, for the external observer (only the network's input and output are visible), the network's weights cannot be calculated, which is the same as we cannot make sure the status of the Erwin Schrödinger's Cat \cite{marshall1997s}. The weights' value is hidden in an unknown wave function, like the cat's status is unknown after closing the box. Once we open the black box and observe the weights, the wave function collapses into a constant \cite{von2018mathematical}, which is the same as taking a sample from the current wave function. In the view of information theory, the happening of an uncertain event (probability less than 100\%) provides the information to the receiver, called the variable's self-information \cite{jones1979elementary}. Respectively, we can receive the information through the appearance of specific weights. For a specific training stage, the appearance of a specific weight has a probability. With the network being trained continuously, the possibility distribution of the weights' appearance is changing respectively. Therefore, the information increase before and after the training can evaluate the information quantity provided by the training process. \section{Probability space of neural networks weights} For a specific neural network architecture, the status of one neural network can be identified by its weight uniquely. The set of all its possible weights is denoted as $\Omega$. $\mathbb{P}$ is the corresponding probability mass function. For any $\Sigma \in \mathbb{F}$, $p(\Sigma)$ is the corresponding probability. $(\Omega,\mathbb{F},\mathbb{P})$ is a probability space. The following discussion is based on this space. As we all know, the backpropagation (BP) algorithm, which is the basic method for network optimization, is a method working on Euclidean space. Therefore, essentially, this space is a Euclidean space and has two features. \begin{enumerate} \item The dimension of elements in the space is high, which means there are too many elements in the space to enumerate. \item The possibility of one event happening $p(\{\omega\}) (\omega \in \Omega)$ is low, which means a single event occurs is almost impossible to observe. \end{enumerate} Moreover, two adjacent elements in this space might have different appearance probabilities in the specific training stage because of the limitation of computers' precision. For example, there are two weights $\omega_1$ and $\omega_2$, $\omega_1$ equals to $\omega_2 + \mathbb{\epsilon}$, where $\mathbb{\epsilon}$ is a vector whose components in each dimension are almost equal to the lowest precision error. For the same input, the output of these two layers might be different. This error will be magnified as the network depth increases, and the loss of these two weights can be different. The one with higher accuracy has a higher probability of appearing at the end of the training process. Therefore, the original space's discretization is hard to calculate directly, which means the prior probability $\mathbb{P}$ cannot be counted by the traditional method. \subsection{The weights distribution visualization} As Eq. \ref{eq_information} shows, to quantify the information accumulated by the network in the training process, we need to know the probability measure $\mathbb{P}_0$ and $\mathbb{P}$, where $\mathbb{P}_0$ and $\mathbb{P}$ are the probability mass function of the appearance of the weight before and after training respectively. The initialization of the weights is randomized, and $\mathbb{P}_0$ is an even distribution in a range defined by the initialization function. Now, the question is \textbf{how to estimate $\mathbb{P}$.} To observe the distribution of $\mathbb{P}$, we use the same training configuration to repeatedly train a specific network and collect the input weight (randomized) and output weights. Then, we use multiple dimensional scaling (MDS) to visualizing the level of similarity of individual weights. MDS is a method used to translate "information about the pairwise 'distances' among a set of n objects or individuals" into a configuration of $n$ points mapped into an abstract Cartesian space \cite{mead1992review}. The most important feature of MDS is that it can keep the Euclidean distance after dimensional reduction. The classic MDS algorithm \cite{wickelmaier2003introduction} is shown in Alg. \ref{MDS_fig} \begin{algorithm}[htbp] \caption*{\textbf{Algorithm 1} Multidimensional Scaling} \begin{algorithmic}[1] \STATE Set up the squared proximity matrix $D^{(2)}=\left[d_{i j}^{2}\right]$, where $d_{ij}$ is the Euclidean distance between $i^{th}$ and $j^{th}$ elements. \STATE Apply double centering \cite{marden1996analyzing}: \\${\textstyle B=-{\frac {1}{2}}JD^{(2)}J}$ using the centering matrix ${\textstyle J=I-{\frac {1}{n}}11'}$, where ${\textstyle n}{\textstyle n}$ is the number of objects, $1$ being an $N ×1$ column vector of all ones. \STATE Determine the ${\textstyle m}$ largest eigenvalues ${\textstyle \lambda _{1},\lambda _{2},...,\lambda _{m}}$ and corresponding eigenvectors ${\textstyle e_{1},e_{2},...,e_{m}}$ of ${\textstyle B}$(where ${\textstyle m}$ is the number of dimensions desired for the output). \STATE Now, ${\textstyle X=E_{m}\Lambda _{m}^{1/2}}$, where ${\textstyle E_{m}}$ is the matrix of ${\textstyle m}$ eigenvectors and ${\textstyle \Lambda _{m}}$ is the diagonal matrix of ${\textstyle m}$ eigenvalues of ${\textstyle B}$. \label{MDS_alg} \end{algorithmic} \end{algorithm} Specifically, in this experiment, we use a TensorFlow CNN Demo \cite{TensorflowDemo} to identify the images in CIFAR-10 \cite{Krizhevsky09learningmultiple}. We fix all the super parameters in the training process and train the network from different scratches repeatedly. The experiment process is shown in Exp. \ref{Randomizedinit_exp} and the training configuration is shown in Tab. \ref{Tab_traininginfo}. \begin{algorithm} \caption{Input and output weights distribution by randomized training from scratches \label{Randomizedinit_exp}} \begin{algorithmic}[1] \STATE Fix the training configuration, including all super parameters. \STATE Initialize 1000 networks randomly and save their initial value of weights. \STATE Train the networks to fine-tuned and save their value of weights at the end of the training. \STATE Use the MDS algorithm to visualize the input and output weights. \end{algorithmic} \end{algorithm} As Fig. \ref{MDS_init} ($\Bar{r} = 1.28, std = 0.007$) and Fig \ref{MDS_end} ($\Bar{r} = 1.34, std = 0.12$) shows, all the points are distributed on the spherical surface evenly, which means their source vectors are also distributed evenly in the corresponding high-dimensional space. \begin{figure} \caption{The left shows the MDS result of the weights at the beginning. The right shows the MDS result of the weights at the end.} \label{MDS_init} \label{MDS_end} \label{MDS_fig} \end{figure} As the reference, we add a constraint that limits the initial weights into a small range $\{w_0', w_0" \}$. Then train the network repeatedly. The experiment process is shown in Exp. \ref{Twoinit_exp}. \begin{algorithm} \caption{Input and output Weights distribution by randomized training from 2 specific scratches\label{Twoinit_exp}} \begin{algorithmic}[1] \STATE Fix the training configuration, including all super parameters. \STATE Initialize 200 networks. Half of them use $w_0'$ as the initial weights, and the others use the $w_0''$. \STATE Train the networks to fine-tuned and save their value of weights at the end of the training. \STATE Use the MDS algorithm \ref{MDS_alg} to visualize the input and output weights. \end{algorithmic} \end{algorithm} \begin{figure} \caption{The initial weights are mapped to these two red points, and the output weights are mapped to these blue points.} \label{MDS_fig_polarized} \end{figure} We use the MDS algorithm to reduce the dimension of the initial weights and the output weights together. The output is shown in Fig. \ref{MDS_fig_polarized}. It shows that the output weights' mapping points (shown as the blue points) are distributed near their initial weights' mapping points (shown as the red points), which means the weights have higher appearance probability if its mapping point is in the region with more points. We can infer that if the mapping points distribute evenly in a region, their source's appearance probability is similar. Based on the experiments mentioned above, we have Thm. \ref{even_distribution} \begin{theorem} For a randomized training process, all the weights have the same appearance probability if they can appear. \label{even_distribution} \end{theorem} Letting $supp(p_0)$ and $supp(p)$ be the support of $p_0$ and $p$ respectively ($|supp(p_0)| \geq |supp(p_0)| > 0$), based on Thm. \ref{even_distribution}, we have \begin{align*} H_{p_0}(X) &= \log(\frac{1}{|supp(p_0)|}) \\ H_{p}(X) &= \log(\frac{1}{|supp(p)|}). \\ \end{align*} And we have \begin{align*} I &= \log(\frac{1}{|supp(p_0)|}) - \log(\frac{1}{|supp(p)|}) \\ &= \log (\frac{|supp(p)|}{|supp(p_0)|}). \end{align*} \begin{theorem} \label{support_set_size} The information provided by training can be measured by the ratio between the support size before and after training. \end{theorem} Generally, if one training process cannot make the network converge into a stable status, the training fails. In this paper, we only discuss the successful training process ($supp(p) \subset Supp(p_0)$ and $I > 0$), and we have Thm. \ref{support_set_size}. Letting $r(p_0,p) = \frac{|supp(p)|}{|supp(p_0)|}$, the question is reduced to \textbf{how to estimate the ratio $r$ between the scale of support before and after the training.} \section{Quasi-Monte Carlo method to estimate the set scale ratio} Generally, the Monte Carlo method \cite{kroese2014monte} (MCM) can be used to estimate the scale shrink between one set and its subset. However, when one set's scale is much smaller than the other and elements are in a high dimensional space, the traditional MCM is not helpful. We provide a new quasi-Monte Carlo method (QMCM) to estimate the scale differences between two sets $ X $ and $X'$ when $X' \subset X$. Briefly, \textbf{the QMCM uses the expectation of the shortest distance to estimate the ratio between $X$ and $X'$.} For more details, we show the derivation below. \subsection{Basic assumption} For a set $X$ which can be embedded into a measurable space and its non-empty proper subset $X'$, we define $d_{X'}(x)$ as the distance between $x$ and its closest element in $X'$ as Fig. \ref{dm_definition} shows, and we have $d_{X'}(x) = 0$ if $x \in X'$. Letting $D_{X'}(X)$ denote the sum of shortest distance for $x \in X$, we have \begin{align*} D_{X'}(X) &= \sum_{x\in X}d_{X'}(x). \\ \Bar{d}_{X'}(X) &= \frac{D_{X'}(X)}{|X|}(x \in X) \end{align*} Abbreviating $\Bar{d}_{X'}(X) $ as $\Bar{d}_{X'}$, for specific set $X$, $|X|$ is a constant. If we want to use $\Bar{d}_{X'}$ to estimate $r$, we need to prove Thm. \ref{estimate_distance}. \begin{theorem} Letting $r(X') = \frac{|X|}{|X'|}$, $\Bar{d}_{X'}$ is a monotonically increasing function of $r$ on for any subset $X'$ of $X$. \label{estimate_distance} \end{theorem} \textbf{Proof.} Letting $X''$ be the union of set $X'$ and the $\{x''\}$ ($x''\in X-X'$), we have \begin{equation*} \frac{|X|}{|X''|} < \frac{|X|}{|X'|}. \end{equation*} $d_{X'}(x'')$ is always positive, we have \begin{equation*} D_{X'}(X) - d_{X'}(x'') < D_{X'}(X), \end{equation*} which leads to \begin{equation*} D_{X'}(X-\{x''\}) < D(X,X'). \end{equation*} Adding a new element $x''$ to $X'$ will update the distance from some elements to $X'$, noted as $\sum \Delta d$, and we have \begin{equation*} D_{X''}(X) = D_{X'}(X-\{x''\})-\sum \Delta d. \end{equation*} Therefore, we have \begin{equation*} D_{X'}(X-\{x''\}) - \sum \Delta d < D_{X'}(X-\{x''\}), \end{equation*} and we have \begin{equation*} D_{X''}(X) < D_{X'}(X-\{x''\}), \end{equation*} which leads to \begin{equation*} D_{X''}(X) < D_{X'}(X). \end{equation*} And we have \begin{equation*} \frac{D_{X''}(X)}{|X|} < \frac{D_{X'}(X)}{|X|}, \end{equation*} \begin{equation*} \Bar{d}_{X''} < \Bar{d}_{X'}. \end{equation*} And we have \begin{equation*} r(X'') < r(X') \Rightarrow \Bar{d}_{X''} < \Bar{d}_{X'}. \end{equation*} The proof of other side is similar, and we have \begin{equation*} r(X'') < r(X') \Leftrightarrow \Bar{d}_{X''} < \Bar{d}_{X'} \square \end{equation*} \begin{figure} \caption{$d_{X'} \label{dm_definition} \end{figure} Denoting the support of function $d_{X'}$ as $supp(d)$ and the expectation of $d_{X'}$ on $supp(d)$ as $\Bar{d}(supp(d))$. Based on the assumption, \begin{align*} |supp(d)| = (1-r)|X|,\\ \end{align*} we have \begin{equation*} \Bar{d}(supp(d)) = (1-r)\Bar{d}(X) \end{equation*} For a specific $X'$, $r$ is a constant. When $1>>r$, we have \begin{equation*} \Bar{d}(supp(d)) \approx \Bar{d}(X). \end{equation*} Now, the question is reduced to \textbf{how to estimate $\Bar{d}(supp(d))$.} \subsection{The distribution of element-wised shortest distance} Generally, we can use repeated random sampling to get a numerical approximation of $\Bar{d}(supp(d))$. If the cost of one sampling is high, we cannot take enough samples to prove the estimation accuracy. However, for a particular case, \textbf{when the mode of the distribution is equal to its mean, we can estimate the mean of the population with very few samples.} To prove this, we need to analyze the distribution of $d_{X'}(x) (x \in supp(d))$. We use numerical simulation to analyze this function. \begin{enumerate} \item Generate a set $X$ randomly with 10,000 elements, which are 100-dimensional normalized vectors. \item Select $r\%$ elements from $X$ randomly as the subset $X'$. \item Calculate the shortest distance from $x (x \in X-X')$ to $X'$ and count its distribution. \end{enumerate} As an example, Fig. \ref{Distribution_sample} shows the result of the distribution when $r = 10\%$, and it is similar to the corresponding normal distribution (with the same mean value and standard difference value). Moreover, we change $r$ and calculate the KL divergence \cite{kullback1951information} between the shortest distance probability distribution ($P$) and the corresponding normal distribution ($Q$) as Eq. \ref{eq_KLdiv} shows. The result is shown in Fig. \ref{KL_curve}. Except for the cases when $r \geq 80\% $, the distribution trend is similar to a normal distribution, and we have Thm. \ref{support_distance_theorom}. \begin{equation} D_{\mathrm{KL}}(P \| Q)=-\sum_{i} P(i) \ln \frac{Q(i)}{P(i)} \label{eq_KLdiv} \end{equation} \begin{figure} \caption{The shortest distance distribution is the distribution when $r = 10\%$, and the normal distribution is a randomized normal distribution with the same mean and standard difference value.} \label{Distribution_sample} \end{figure} \begin{figure} \caption{The curve of the KL divergence between the shortest distance distribution and the corresponding normal distribution.} \label{KL_curve} \end{figure} \begin{theorem} For the elements in $supp(d)$, the distribution of their function value can be viewed as a normal distribution. \label{support_distance_theorom} \end{theorem} As a normal distribution, the mode of this distribution is equal to its mean. Letting $Y$ be the subset of $X$, when $c_v(d(y)) \leq t$ where $t$ is a threshold, we have \begin{equation*} \Bar{d}(Y) \approx \Bar{d}(X) \end{equation*} And the QMCM method can be described as Alg. 2. \begin{algorithm}[htbp] \caption*{\textbf{Algorithm 2} QMCM for Subset scale estimation} \begin{algorithmic}[1] \STATE Set the threshold $t$ and amount of samples $n$. \STATE Take $n$ samples from $X$ as $Y$ and calculate $Var(d(y))$. \WHILE {$c_v(d(y)) > t$} \STATE Remove the sample from $Y$ with the largest deviation from the mean. \STATE Resample and add the sample to $Y$. \ENDWHILE \STATE Output $\Bar{d}(Y)$. \end{algorithmic} \end{algorithm} We can adjust the accuracy of the estimation by adjusting the value of $t$ and $n$. In the following experiment, we set $t = 0.3, n = 200$, Unlike the traditional MCM, the novelty of this method uses the distance between related two points to estimate the ratio. Abbreviating $\Bar{d}(Y)$ as $\hat{d}$, if we have two training process $t_1$ and $t_2$, we have \begin{equation} \hat{d}_{t_1} < \hat{d}_{t_2} \Leftrightarrow I_{t_1} < I_{t_2}. \label{information_comparing} \end{equation} If $ \hat{d}_{t_1} < \hat{d}_{t_2}$, we have $I_{t_1} < I_{t_2}$. Oppositely, if $I_{t_1} < I_{t_2}$, we have $\hat{d}_{t_1} < \hat{d}_{t_2}$, which can be used to verify the correctness of our method. \subsection{Apply on the neural network} Correspondingly, for the network's training process, we have its initial weights and the output weights. The condition to implement QMCM to estimate the information is that the output weight is the closest one of the initial weights in $supp(P)$. We select seven network models from simple to complex to verify this, TensorFlow MNIST classification Demo (classical version) \cite{TensorFlowMNIST}, TensorFlow CNN Demo \cite{TensorflowDemo}, GoogleNet \cite{szegedy2015going}, AlexNet \cite{krizhevsky2017imagenet}, ResNet \cite{he2016deep}, VGG \cite{simonyan2014very} and Yolo v3 \cite{redmon2018yolov3}. To ensure that these networks are used in scenarios that adapt to them, we select 4 dataset with different input scale and complexity, MNIST \cite{lecun1998gradient}, CIFAR-10 \cite{Krizhevsky09learningmultiple}, TensorFlow Flowers \cite{tfflowers} and Pascal VOC \cite{pascal-voc-2007}. We train each kind of model from scratch to fine-tune it with the same configuration 1000 times repeatedly. The basic information of the training is shown in Tab. \ref{Tab_traininginfo}. And then, we calculate the distance of arbitrary pairs of initial and end states. The result shows that all the output weight is the closest one of the initial weights in $supp(P)$. Therefore, we can use Eq. \ref{information_comparing} to compare the influence of the two training processes to the same model. Moreover, we calculate the mean, standard difference, and coefficient of variation ($c_v$) value of the distance (see Tab. \ref{Tab_distancestatistic}). It shows that although the difference in the mean value among models is big, the coefficient of variation is still at a low level, which reflects the stability of this estimation. \begin{table*}[] \centering \begin{tabularx}{\textwidth}{|c|X|X|X|X|X|X|X|X|X|X|} \hline \multicolumn{2}{|c|}{Model} & TF CNN & TF MNIST & GoogleNet & ResNet & VGG & AlexNet & Yolo v3\\ \hline \multicolumn{2}{|c|}{Dataset} & CIFAR-10 & MNIST & TF Flower & TF Flower & TF Flower & TF Flower & Pascal\\ \hline \multirow{2}{*}{Super} & LR & 0.01 & 0.01 & 0.01 & 0.01(decay) &0.01(decay) & 0.01(decay) & 0.01(decay)\\ \cline{2-9} \multirow{2}{*}{params}& Epoch & 10 & 10 & 20 & 40 & 20 & 20 & 20\\ \cline{2-9} & Batch Size & 128 &128 & 32 & 32&32 &32 &64 \\ \cline{2-9} & Optimizer & \multicolumn{7}{c|}{Stochastic Gradient Descent}\\ \hline & Normlize & \multicolumn{7}{c|}{Yes} \\ \cline{2-9} Pre-& Mean Sub & \multicolumn{2}{c|}{No} &Yes &Yes &Yes &Yes & No \\ \cline{2-9} process & Rescale & \multicolumn{2}{c|}{Yes} &Yes &Yes &Yes &Yes & Yes\\\cline{2-9} & Standardize & \multicolumn{2}{c|}{Yes}&No &No &No &No & Yes\\\cline{2-9} \hline \end{tabularx} \caption{Training configuration of experiments.} \label{Tab_traininginfo} \end{table*} \begin{table}[] \centering \begin{tabular}{|c|c|c|c|} \hline Network & Mean & STD & $c_v$(\%) \\ \hline TF CNN Demo & 4.026 & 0.053 & 1.316 \\ TF MNIST Demo & 1.358 & 0.031 & 2.282 \\ GoogleNet & 28.084 & 0.482 & 1.718\\ ResNet & 8785.243 & 1172.836 & 13.350 \\ VGG & 0.556 & 0.012 & 2.158 \\ AlexNet & 4.849 & 0.589 & 12.163\\ Yolo v3 & 26.863 & 6.286 & 23.400 \\ \hline \end{tabular} \caption{The statistic result (mean value, standard difference and coefficient of variation) about the distance between initial and end states of neural networks.} \label{Tab_distancestatistic} \end{table} \section{Verification} As mentioned in Eq. \ref{information_comparing}, we have $I_{t_1} < I_{t_2} \Rightarrow \hat{d}_{t_1} < \hat{d}_{t_2}$, which can be used to verify the correctness of our method. We can construct two training process $t_1$ and $t_2$ such that $I(t_1) < I(t_2)$. Then calculate $\hat{d}_{t_1}$ and $\hat{d}_{t_2}$ respectively. Now, we need to construct these two training process $t_1$ and $t_2$. Based on the theory of information \cite{shannon1948mathematical}, higher entropy means more information. If we control all the other random factors and make $t_1$ contains more kinds of samples than $t_2$, we have $I(t_1)>I(t_2)$. Specifically, we use the same models to verify our method (see Tab. \ref{Tab_traininginfo}). \begin{algorithm}[ht] \caption{Training with different amount of labels\label{labels_exp}} \begin{algorithmic}[1] \STATE Fix the training configuration, including all super parameters. \STATE Initialize 5 networks $\{m_i\} (i \in [0,4])$. \STATE Use data with $(20\times i)\%$ labels to train the network separately and record the weights changing. \STATE Calculate $d(w,w')$ of each weight, where $w$ is the current weights, $w'$ is the output weights of the same training process. \end{algorithmic} \end{algorithm} In \cite{wang2020generalizing}, the author proves that few samples can also be used to guide the network to complete a complex task. Because of that, to ensure $I(t_1)>I(t_2)$, we control the numbers of labels. There are $[20\%,40\%,60\%,80\%,100\%]$ kinds of samples are used in the training. To eliminate the impact of the weights update numbers, we train the same amount of steps in all the model training process. Finally, we calculate the $\hat{d_{t_i}}$ for each training process (see Exp. \ref{labels_exp}). \begin{figure*} \caption{The summary of the Exp. \ref{labels_exp} \label{ed_1} \end{figure*} As Fig. \ref{ed_1} shows, in all experiments, as the $I$ increases, $\hat{d_{t_i}}$ increases, which verifies the utility of our method. \section{Application: impact of label corruption} The most significant application is to evaluate the training process. In most cases, the trained network's performance is the only measure in the evaluation of the training process. However, as the author mentioned in \cite{geirhos2020shortcut}, some of the information in the dataset can be the "shortcut" to complete tasks, which leads to the training's failure. If the test dataset has similar "shortcuts," this failure will be unnoticeable without complex analysis. Moreover, the Clever Hans effect has been observed in the early version of BERT \cite{devlin2018bert} when it completes the argument reasoning comprehension task. The network makes the correct judgment based on a hidden trick but not the logic we want it to learn, which Niven and Kao noticed in \cite{niven2019probing} . By analyzing the quantity of information, we can reveal the essence of the networks' learning process and avoid being misled by the errors mentioned above. Here, we use the impact of the label corruption as an example to show our method's application. For the impact of the label corruption, as the author mentions in \cite{zhang2016understanding}, some networks can build the relationship between the label and the data even the label is randomized. Their experiment shows that the time of overfitting of their models increases with the error label rate. In other words, the network needs more time to learn the noisy dataset. Based on this phenomenon, there are two assumptions based on our intuition. \begin{enumerate} \item The label corruption makes the relationship between the data and the label more complex. It makes the cost to describe the relationship increases, which means the network can accumulate more information in training. \item The label corruption makes the dataset contains more conflict information, which decreases the quantity of information accumulated by the network. \end{enumerate} The performance-based evaluation is not useful to answer these questions, and we use our method to verify these two assumptions. Briefly, if the first assumption is correct, the network accumulates more information in the same length of time. Otherwise, if the second assumption is correct, the network accumulates less information within the same time. We use TensorFlow CNN Demo as the test model and CIFAR-10 as the test dataset with the same training configuration shown in Tab. \ref{Tab_traininginfo}. The experiment process is shown in Exp. \ref{labelcorruption_exp}. \begin{algorithm}[ht] \caption{Impact of label corruption \label{labelcorruption_exp}} \begin{algorithmic}[1] \STATE Create eleven datasets based on CIFAR-10, $\{A_i\} (i\in [0,100])$. In $A_i$, $(i\times 10)\%$ labels are shuffled randomly. \STATE Initialize 10 networks by the same initial weights. \STATE Use these datasets to train the model separately within the same time. \STATE calculate the distance $d(w_i, w)$, where $w_i$ is the current weights, and $w$ is the final weights after training. \end{algorithmic} \end{algorithm} The result is shown in Fig. \ref{Label_corruption}. The intercept of the curve shows that the amount of the information accumulated by the network with the label error rate increasing, which means the label corruption hinders the network's learning. \begin{figure} \caption{The label corruption can decrease the quantity of the information provided by the dataset.} \label{Label_corruption} \end{figure} \section{Discussion} \textbf{Question about information effectiveness}. This paper provides a tool to analyze the information accumulated by the network in the training process. The precondition of this method is that most of the information learned by the network needs to be useful, which means the training process's output needs to be fine-tuned. Otherwise, we cannot guarantee that the nearest points of the initial weights in $supp(P)$ might not be the output weights. However, in practice, this precondition is not always satisfied. The best counterexample is the overfitting phenomenon, which shows that the network can perform well on the training dataset but performs badly on the test dataset. One of the accepted explanations is that the network learned too much knowledge from the training dataset, not the commonality between the training dataset and the test dataset. It indicates that not all the information learned by the network is useful and meaningful. Therefore, to analyze such failed training cases, we still need further research to quantify the information's effectiveness. \textbf{Question about cross-modal verification}. We use four datasets from simple to complex (MNIST, CIFAR-10, TensorFlow Flowers, Pascal VOC). And we use seven models to accept the information from these four from simple to complex (TF MNIST Demo, TF CNN Demo, AlexNet, VGG, ResNet, GoogleNet, Yolo). Although our method is verified by all of these models when the model's structure is fixed, the size order of the information quantity is not consistent with our expectations when the network structure is different (see Tab. \ref{Tab_distanceorder}). On the one hand, to train the model to fine-tuned, we use different training configuration to train the model as Tab. \ref{Tab_traininginfo} shows, which might impact the information accumulation. On the other hand, for different models, their capacity of representation is different. It means to store a specific piece of information, the bigger ones' weights need less change than the smaller ones', which means the presented information quantity is less than the others. Even though we do not deny that this reflects our research's limitations, it indicates that we need further study to provide a more general model based on the current achievement. \begin{table}[] \centering \scalebox{0.85}{ \begin{tabular}{|c|c|c|} \hline Rank & Expected order & Actual order \\ \hline 1& TF MNIST Demo (MNIST) & VGG (TF Flower) \\ 2& TF CNN Demo (cifar-10) & TF MNIST Demo (MNIST)\\ 3& AlexNet (cifar-10) & TF CNN Demo (cifar-10)\\ 4& VGG (TF Flower)& AlexNet (cifar-10)\\ 5& ResNet (TF Flower)& Yolo (Pascal VOC)\\ 6& GoogleNet (TF Flower) & GoogleNet (TF Flower)\\ 7& Yolo v3 (Pascal VOC) & ResNet (TF Flower)\\ \hline \end{tabular}} \caption{The information quantity rank in the experiments.} \label{Tab_distanceorder} \end{table} \section{Future Works} Even though this work still gives us a new view to analyze the essence of the neural network. Quantify the information is great progress to answer the questions about the network explanation. We will apply it in the following fields to solve the related questions. \begin{enumerate} \item By quantifying the information in a different part of the network, we can target the data's key feature with more confidence. \item By quantifying the network's information, we can evaluate the training process and optimize it. \item By analyzing the information quantity changing, we can reveal the essence of the double decent phenomenon \cite{nakkiran2019deep}. \end{enumerate} Moreover, we will keep working in this field for a more general model. \input{Formatting-Instructions-LaTeX-2021.bbl} \bibstyle{aaai21} \end{document}

\begin{document} \title{Interor and $\mathfrak h$ Operators on the Category of Locales} \author{Joaqu\'in Luna-Torres } \thanks{Programa de Matem\'aticas, Universidad Distrital Francisco Jos\'e de Caldas, Bogot\'a D. C., Colombia (retired professor)} \email{jlunator@fundacionhaiko.org} \subjclass{06D22; 18B35; 18F70} \keywords{ Frame, Locale, Sublocale, Interior operator, $\mathfrak h$ operator, Topological category} \begin{abstract} We present the concept of interior operator $I$ on the category $\mathbf{Loc}$ of locales and then we construct a topological category \linebreak $\big(\mathbf{I\text{-}Loc},\ U\big)$, where $U:\mathbf{I\text{-}Loc}\rightarrow \mathbf{Loc}$ is a forgetful functor; and we also introduce the notion of $\mathfrak h$ operator on the category $\mathbf{Loc}$ and discuss some of their properties for constructing the topological category $\big(\mathbf{\mathfrak h\text{-}Loc},\ U\big)$ associated to the forgetful functor $U:\mathbf{\mathfrak h\text{-}Loc}\rightarrow \mathbf{Loc}$. \end{abstract} \maketitle \baselineskip=1.7\baselineskip \section*{0. Introduction} Kuratowski operators (closure, interior, exterior, boundary and others) have been used intensively in General Topology (\cite{Du}, \cite{K1}, \cite{K2}). For a topological space it is well-known that, for example, the associated closure and interior operators provide equivalent descriptions of the topology; but this is not always true in other categories, consequently it makes sense to define and study separately these operators. In this context, we study an interior operator $I$ on the the coframe $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ of sublocales of every object $L$ in the category $\mathbf{Loc}$. On the other hand, a new topological operador $\mathfrak h$ was introduced by M. Suarez \cite{MSM} in order to complete a Boolean algebra with all topological operators in General Topology. Following his ideas, we study an operator $\mathfrak h$ on the colection $\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$ of all complemented sublocales of every object $L$ in the category $\mathbf{Loc}$. The paper is organized as follows, we begin presenting, in section 1, the basic concepts of Heyting algebras, Frames, locales, sublocales, images and preimages of sublocales for the morphisms of $\mathbf{Loc}$ and the notions of closed and open sublocales; these notions can be found in Picado and Pultr \cite{PP} and A. L. Suarez \cite{ALS}, In section 2, we present the concept of interior operator $I$ on the category $\mathbf{Loc}$ and then we construct a topological category $\big(\mathbf{I\text{-}Loc},\ U\big)$, where $U:\mathbf{I\text{-}Loc}\rightarrow \mathbf{Loc}$ is a forgetful functor. Finally in section 3 we present the notion of $\mathfrak h$ operator on the category $\mathbf{Loc}$ and discuss some of their properties for constructing the topological category $\big(\mathbf{\mathfrak h\text{-}Loc},\ U\big)$ associated to the forgetful functor $U:\mathbf{\mathfrak h\text{-}Loc}\rightarrow \mathbf{Loc}$. \section{Preliminaries} For a comprehensive account on the the categories of frames and locales we refer to Picado and Pultr \cite{PP} and A. L. Suarez \cite{ALS}, from whom we take the following useful facts. \subsection{Heyting algebras} A bounded lattice $L$ is called a Heyting algebra if there is a binary operation $x \rightarrow y$ (the Heyting operation) such that for all $a, b, c$ in $L$, \[ a \land b \leqslant c\,\ \text{iff}\,\ a \leqslant b \rightarrow c. \] Thus for every $b \in L$ the mapping $b \rightarrow (-) : L \rightarrow L$ is a right adjoint to $(-) \land b : L \rightarrow L$ and hence, if it exists, is uniquely determined. In a complete Heyting algebra we have $(\bigvee A) \land b = \bigvee_{a\in A}(a \land b)$ for any $A\subseteq L$, $b\rightarrow (\bigvee A) = \bigvee_{a\in A}(b \rightarrow a)$, and $(\bigvee A) \rightarrow b = \bigvee{a\in A}(a \rightarrow b)$. \subsection{Frames} A {\it frame} is a complete lattice $L$ satisfying the distributive law \[ \big(\bigvee A\big)\land b =\bigvee \{ a\land b\mid a\in A\} \] for all $A \subseteq L$ and $b\in L$ (hence a complete Heyting algebra); a {\it frame homomorphism} preserves all joins and all finite meets. The lattice $\Omega(X)$ of all open subsets of a topological space $X$ is an example of a frame, and if $f: X \rightarrow Y$ is continuous we obtain a frame homomorphism $\Omega( f ): \Omega(Y )\rightarrow \Omega (X)$ by setting $\Omega(f)(U) = f^{-1}[U]$. Thus we have a contravariant functor $\Omega : \mathbf{Top} \rightarrow \mathbf{Frm}^{op} $ from the category of topological spaces into that of frames. \subsection{Locales} The adjunction $\Omega : \mathbf{Top} \rightarrow \mathbf{Frm}^{op} $, \,\ $\mathbf{pt}:\mathbf{Frm}^{op}\rightarrow \mathbf{Top}$ with $\Omega \dashv \mathbf{pt}$, connects the categories of frames with that of topological spaces. The functor $\Omega$ assigns to each space its lattice of opens, and $\mathbf{pt}$ assigns to a frame $L$ the collection of the frame maps $f : L \rightarrow 2$, topologized by setting the opens to be exactly the sets of the form $ \{f: L \rightarrow 2 \mid f(a) = 1\}$ for some $a\in L$. A frame $L$ is {\it spatial} if for $a, b \in L$ whenever $a\nleqslant b$ there is some frame map $f: L \rightarrow 2$ such that $f(a) = 1\ne f(b)$. Spatial frames are exactly those of the form $\Omega(X)$ for some space $X$. A space is {\it sober} if every irreducible closed set is the closure of a unique point. Sober spaces are exactly those of the form $\mathbf{pt}(L)$ for some frame $L$. The adjunction $\Omega\dashv \mathbf{pt}$ restricts to a dual equivalence of categories between spatial frames and sober spaces The category of sober spaces is a full reflective subcategory of $\mathbf{Top}$. For each space $X$ we have a sobrification map $N : X\rightarrow \mathbf{pt}(\Omega(X))$ mapping each point $x\in X$ to the map $f_x :(X) \rightarrow 2$ defined as $f(U) = 1$ if and only if $x\in U$. The category of {\it spatial frames} is a full reflective subcategory of $\mathbf{Frm}$. For each frame we have a spatialization map $\phi : L \rightarrow \Omega(\mathbf{pt}(L))$ which sends each $a\in L$ to $\{f : L \rightarrow 2 \mid f(a) = 1\}$. This justifies to view the dual category $\mathbf{Loc} =\mathbf{Frm}^{op}$ as an extended category of spaces; one speaks of the category of {\it locales}. Maps in the category of locales have a concrete description: they can be characterized as the right adjoints of frame maps (since frame maps preserve all joins, they always have right adjoints). \subsubsection{\bf{Sublocales}} A {\it sublocale} of a locale $L$ is a subset $S\subseteq L$ such that it is closed under arbitrary meets, and such that $s\in S$ implies $x\rightarrow s \in S$ for every $x \in L$. This is equivalent to $S\subseteq L$ being a locale in the inherited order, and the subset inclusion being a map in $\mathbf{Loc}$. Sublocales of $L$ are closed under arbitrary intersections, and so the collection $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ of all sublocales of $L$, ordered under set inclusion, is a complete lattice. The join of sublocales is (of course) not the union, but we have a very simple formula $\bigvee_{i} S_{i} = \{\bigvee M \mid M \subseteq\bigcup_{i} S_{i}\}$. In the coframe $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ the bottom element is the sublocale $\{1\}$ and the top element is $L$. \subsubsection{\bf{ Images and Preimages of sublocales}} Let $f: L\rightarrow M$ be a localic map and if $S \subseteq L$ is a sublocale then the standard set-theoretical image $f [S]$ is also a sublocale The set-theoretic preimage $f^{-1}[T]$ of a sublocale $T\subseteq M$ is not necessarily a sublocale of $L$. To obtain a concept of a preimage suitable for our purposes we will, first, make the following observation: ``Let $A\subseteq L$ be a subset closed under meets. Then $\{1\} \subseteq A$ and if $S_i \subseteq A$ for $i \in J$ then $\bigwedge_{i\in J} S_i\subseteq A$''. Consequently there exists the largest sublocale contained in $A$. It will be denoted by $A_{sloc}$. The set-theoretic preimage $f^{-1}[T]$ of a sublocale $T$ is closed under meets \big(indeed, $f(1) = 1$, and if $x_i \in f^{-1}[T])$ then $f(x_i) \in T$, and hence $ f(\bigwedge_{i\in J} x_i)=\bigwedge_{i\in J} f(x_i)$ belongs to $T$ and $\bigwedge_{i\in J} x_i\in f^{-1}[T]$ \big) and we have the sublocale $f_{-1}[T]:= f^{-1}[T]_{sloc}$. It will be referred to as {\it the preimage} of $T$, and we shall sat that $f_{-1}[-]$ is {\it the preimage function} of $f$. For every localic map $f: L \rightarrow M$, the preimage function $f_{-1}[-] $ is a right Galois adjoint of the image function $f [-] :\mathcal{S}_\text{{\bsifamily{l}}}(L)\rightarrow \mathcal{S}_\text{{\bsifamily{l}}}(M)$. \subsubsection{\bf{ Closed and Open sublocales}}\label{open} Embedded in $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ we have the coframe of {\it closed sublocales} which is isomorphic to $L^{op}$. The closed sublocale $\mathfrak c(a) \subseteq L$ is defined to be $\uparrow a$ for $a \in L$. Embedded in $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ we also have the frame of open sublocales which is isomorphic to $L$. The open sublocale is defined to be $\{a \rightarrow x \mid x \in L\}$ for $a \in L$. The sublocales $\mathfrak o(a)$ and $\mathfrak c(a)$ are complements of one another in the coframe $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ for any element $a\in L$. Furthermore, open and closed sublocales generate the coframe $\mathcal{S}_\text{{\bsifamily{l}}}(L)$ in the sense that for each \linebreak $S \in \mathcal{S}_\text{{\bsifamily{l}}}(L)$ we have $S = \bigcap\{\mathfrak o(x) \cup \mathfrak c(y) \mid S \subseteq \mathfrak o(x) \cup \mathfrak c(y)\}$. A pseudocomplement of an element $a$ in a meet-semilattice $L$ with $0$ is the largest element $b$ such that $b\land a = 0$, if it exists. It is usually denoted by $\neg a$. Recall that in a Heyting algebra $H$ the pseudocomplement can be expressed as $\neg x= x\rightarrow 0$. \section{Interior Operators} We shall be conserned in this section with a version on locales of the interior operator studied in \cite{LO}. Before stating the next definition, we need to observe that since for localic maps $f: L \rightarrow M$ and $g:M\rightarrow N$: \begin{itemize} \item the preimage function $f_{-1}[-] $ is a right Galois adjoint of the image function $f [-] :\mathcal{S}_\text{{\bsifamily{l}}}(L)\rightarrow \mathcal{S}_\text{{\bsifamily{l}}}(M)$; \item $ g [-]\circ f [-]=(g\circ f) [-]$. \end{itemize} Therefore $g_{-1}[-]\circ f_{-1}[-]= (g\circ f)_{-1}[-]$ because given two adjunctions the composite functors yield an adjunction. \begin{defi} An interior operator $I$ of the category $\mathbf{Loc}$ is given by a family $I =(i_{\text{\tiny{$L$}}})_{\text{$L\in \mathbf{Loc}$}}$ of maps $i_{\text{\tiny{$L$}}}:\mathcal{S}_\text{{\bsifamily{l}}}(L)\rightarrow \mathcal{S}_\text{{\bsifamily{l}}}(L)$ such that \begin{itemize} \item[($I_1)$] $\left(\text{Contraction}\right)$\,\ $i_{\text{\tiny{$L$}}}(S)\subseteq S$ for all $S \in \mathcal{S}_\text{{\bsifamily{l}}}(L)$; \item[($I_2)$] $\left(\text{Monotonicity}\right)$\,\ If $S\subseteq T$ in $\mathcal{S}_\text{{\bsifamily{l}}}(L)$, then $i_{\text{\tiny{$L$}}}(S)\subseteq i_{\text{\tiny{$L$}}}(T)$ \item[($I_3)$] $\left(\text{Upper bound}\right)$\,\ $i_{\text{\tiny{$L$}}}(L)=L$. \end{itemize} \end{defi} \begin{defi} An $I$-space is a pair $(L,i_{\text{\tiny{$L$}}})$ where $L$ is an object of $\mathbf{Loc}$ and $i_{\text{\tiny{$L$}}}$ is an interior operator on $L$. \end{defi} \begin{defi} A morphism $f:L\rightarrow M$ of $\mathbf{Loc}$ is said to be $I$-continuous if \begin{equation}\label{conti} f_{-1}\left[ i_{\text{\tiny{$M$}}}(T)\right]\subseteq i_{\text{\tiny{$L$}}}\left( f_{-1}[T]\right) \end{equation} for all $T\in \mathcal{S}_\text{{\bsifamily{l}}}(M)$. Where $f_{-1}[-]$ is the preimage of $f[-]$. \end{defi} \begin{prop} Let $f:L\rightarrow M$ and $g:M\rightarrow N$ be two $I$-continuous morphisms of $\mathbf{Loc}$ then $g\centerdot f$ is an $I$-continuous morphism of $\mathbf{Loc}$. \end{prop} \begin{proof} Since $g:M\rightarrow N$ is $I$-continuous, we have $$g_{-1}\big[ i_{\text{\tiny{$N$}}}(S)\big]\subseteq i_{\text{\tiny{$M$}}}\big( g_{-1}[S]\big)$$ for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}(N)$, it fallows that $$f_{-1}\Big[g_{-1}\big[( i_{\text{\tiny{$N$}}}(S)\big]\Big]\subseteq f_{-1}\Big[ i_{\text{\tiny{$M$}}}\big( g_{-1}[S]\big)\Big];$$ now, by the $I$-continuity of $f$,$$ f_{-1}\Big[ i_{\text{\tiny{$M$}}} \big( g_{-1}[S]\big)\Big]\subseteq i_{\text{\tiny{$L$}}}\Big( f_{-1}\big[g_{-1}[S]\big]\Big),$$ therefore $$f_{-1}\Big[g_{-1}\big[ i_{\text{\tiny{$N$}}}(S)\big]\Big]\subseteq i_{\text{\tiny{$L$}}}\Big( f_{-1}\big[g_{-1}[S]\big]\Big),$$ that is to say $$(g\centerdot f)_{-1}\big[ i_{\text{\tiny{$N$}}}(S)\Big]\subseteq i_{\text{\tiny{$L$}}}\Big( (g\centerdot f)_{-1}[S]\Big)$$ \end{proof} As a consequence we obtain \begin{defi} The category $\mathbf{I\text{-}Loc}$ of $I$-spaces comprises the following data: \begin{enumerate} \item {\bf Objects}: Pairs $(L,i_{\text{\tiny{$L$}}})$ where $L$ is an object of $\mathbf{Loc}$ and $i_{\text{\tiny{$L$}}}$ is an interior operator on $L$. \item {\bf Morphisms}: Morphisms of $\mathbf{Loc}$ which are $I$-continuous. \end{enumerate} \end{defi} \subsection{The lattice structure of all interior operators} For the category $\mathbf{Loc}$ we consider the collection \[ Int(\mathbf{Loc}) \] of all interior operators on $\mathbf{Loc}$. It is ordered by \[ I\leqslant J \Leftrightarrow i_{\text{\tiny{$L$}}}(S)\subseteq j_{\text{\tiny{$L$}}}(S), \,\,\ \text{for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}$ and all $L$ object of $\mathbf{Loc}$}. \] This way $Int(\mathbf{Loc})$ inherits a lattice structure from $\mathcal{S}_\text{{\bsifamily{l}}}$: \begin{prop} Every family $(I_{\text{\tiny{$\lambda$}}})_{\text{\tiny{$\lambda\in \Lambda$}}}$ in $Int(\mathbf{Loc})$ has a join $\bigvee\limits_{\text{\tiny{$\lambda\in \Lambda $}}}I_{\text{\tiny{$\lambda $}}}$ and a meet $\bigwedge\limits_{\text{\tiny{$\lambda\in \Lambda $}}}I_{\text{\tiny{$\lambda $}}}$ in $Int(\mathbf{Loc})$. The discrete interior operator \[ I_{\text{\tiny{$D$}}}=({i_{\text{\tiny{$D$}}}}_{\text{\tiny{$L$}}})_{\text{$L\in \mathbf{Loc}$}}\,\,\ \text{with}\,\,\ {i_{\text{\tiny{$D$}}}}_{\text{\tiny{$L$}}}(S)=S\,\,\ \text{for all}\,\ S\in \mathcal{S}_\text{{\bsifamily{l}}} \] is the largest element in $Int(\mathbf{Loc})$, and the trivial interior operator \[ I_{\text{\tiny{$T$}}}=({i_{\text{\tiny{$T$}}}}_\text{\tiny{$L$}})_{\text{$L\in \mathbf{Loc}$}}\,\,\ \text{with}\,\,\ {i_{\text{\tiny{$T$}}}}_{\text{\tiny{$L$}}}(S)= \begin{cases} \{1\}& \text{for all}\,\ S\in \mathcal{S}_\text{{\bsifamily{l}}},\,\ S\ne L\\ L&\text {if}\,\ S=L \end{cases} \] is the least one. \end{prop} \begin{proof} For $\Lambda\ne\emptyset$, let $\widehat{I}=\bigvee\limits_{\text{\tiny{$\lambda\in\Lambda $}}}I_{\text{\tiny{$\lambda $}}}$, then \[ \widehat{i_{\text{\tiny{$L$}}}}=\bigvee\limits_{\text{\tiny{$\lambda\in \Lambda$}}} {i_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}, \] for all $L$ object of $\mathbf{Loc}$, satisfies \begin{itemize} \item $ \widehat{i_{\text{\tiny{$L$}}}}(S)\subseteq S$, because ${i_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(S)\subseteq S$ for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}$ and for all $\lambda \in \Lambda$. \item If $S\leqslant T$ in $\mathcal{S}_\text{{\bsifamily{l}}}$ then ${i_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(S)\subseteq {i_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(T)$ for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}$ and for all $\lambda \in \Lambda$, therefore $ \widehat{i_{\text{\tiny{$S$}}}}(S)\subseteq \widehat{i_{\text{\tiny{$L$}}}}(T)$. \item Since ${i_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(L)=L $ for all $\lambda \in \Lambda$, we have that $ \widehat{i_{\text{\tiny{$L$}}}}(L)=L$. \end{itemize} Similarly $\bigwedge\limits_{\text{\tiny{$\lambda\in \Lambda $}}}I_{\text{\tiny{$\lambda $}}}$,\,\ $ I_{\text{\tiny{$D$}}}$ and $I_{\text{\tiny{$T$}}}$ are interior operators. \end{proof} \begin{coro}\label{complete} For every object $L$ of $\mathbf{Loc}$ \[ Int(L) = \{i_{\text{\tiny{$L$}}}\mid i_{\text{\tiny{$L$}}}\,\ \text{ is an interior operator on}\,\ L\} \] is a complete lattice. \end{coro} \subsection{Initial interior operators} Let $\mathbf{I\text{-}Loc}$ be the ctegory of $I$-spaces. Let $(M,i_{\text{\tiny{$M$}}})$ be an object of $\mathbf{I\text{-}Loc}$ and let $L$ be an object of $\mathbf{Loc}$. For each morphism $f:L\rightarrow M$ in $\mathbf{Loc}$ we define on $L$ the operotor \begin{equation} \label{initial} i_{\text{\tiny{$L_{f}$}}}:=f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*}. \end{equation} \begin{prop}\label{ini-cont} The operator (\ref{initial}) is an interior operator on $L$ for which the morphism $f$ is $I$-continuous. \end{prop} \begin{proof}\ \begin{enumerate} \item[($I_1)$] $\left(\text{Contraction}\right)$\,\ $i_{\text{\tiny{$L_{f}$}}}(S)= f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*}[S]\subseteq f_{-1}\centerdot f_{*}[S]\subseteq S$ for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}$; \item[($I_2)$] $\left(\text{Monotonicity}\right)$\,\ $S\subseteq T$ in $\mathcal{S}_\text{{\bsifamily{l}}}$, implies $f_{*}[S]\subseteq f_{*}[T]$, then $i_{\text{\tiny{$M$}}}\centerdot f_{*}[S]\subseteq i_{\text{\tiny{$M$}}}\centerdot f_{*}[T]$, consequently $ f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*}[S]\subseteq f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*}[T]$; \item[($I_3)$] $\left(\text{Upper bound}\right)$\,\ $i_{\text{\tiny{$L_{f}$}}}(L)=f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*}[L]=L$. \end{enumerate} Finally, \begin{align*} f_{-1}\big(i_{\text{\tiny{$M$}}}(T)\big)&\subseteq f_{-1}\big(i_{\text{\tiny{$M$}}}\centerdot f_{*}\centerdot f^{-1}(T)\big)=(f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*})\big(f^{-1}(T)\big)\\ &= i_{\text{\tiny{$L_{f}$}}}\big(f^{-1}(T)\big), \end{align*} for all $T\in \mathcal{S}_\text{{\bsifamily{l}}}$. \end{proof} It is clear that $ i_{\text{\tiny{$L_{f}$}}}$ is the coarsest interior operator on $L$ for which the morphism $f$ is $I$-continuous; more precisaly \begin{prop}\label{unique} Let $(L,i_{\text{\tiny{$L$}}})$ and $(M,i_{\text{\tiny{$M$}}})$ be objects of $\mathbf{I\text{-}Loc}$, and let $N$ be an object of $\mathbf{Loc}$. For each morphism $g:N\rightarrow L$ in $\mathbf{Loc}$ and for\linebreak $f:(L,i_{\text{\tiny{$L_{f}$}}})\rightarrow (M,i_{\text{\tiny{$N$}}})$ an $I$-continuous morphism, $g$ is $I$-continuous if and only if $f\centerdot g$ is $I$-continuous. \end{prop} \begin{proof} Suppose that $g\centerdot f$ is $I$-continuous, i. e. $$(f\centerdot g)_{-1}\big(i_{\text{\tiny{$M$}}}(T)\big)\subseteq i_{\text{\tiny{$N$}}}\big( (f\centerdot g)_{-1}(T) \big)$$ for all $T\in \mathbf{S(N)}$. Then, for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}$, we have \begin{align*} g_{-1}\big(i_{\text{\tiny{$L_{f}$}}}(S)\big)&=g_{-1}\big(f_{-1}\centerdot i_{\text{\tiny{$M$}}}\centerdot f_{*}(S)\big)=(f\centerdot g)_{-1}\big( i_{\text{\tiny{$M$}}}( f_{*}(S)) \big)\\ &\subseteq i_{\text{\tiny{$N$}}}\big( (f\centerdot g)_{-1}(f_{*}(S) ) \big)=i_{\text{\tiny{$N$}}}\big( g_{-1}\centerdot f_{-1}\centerdot f_{*} (S)\big)\\ &\subseteq i_{\text{\tiny{$N$}}}\big( g_{-1}(S)\big),\\ \end{align*} i.e. $g$ is $I$-continuous. \end{proof} As a consequence of corollary(\ref{complete}), proposition(\ref{ini-cont}) and proposition (\ref{unique}) (cf. \cite{AHS} or \cite{JM}), we obtain \begin{theorem} The forgetful functor $U:\mathbf{I\text{-}Loc}\rightarrow \mathbf{Loc}$ is topological, i.e. the concrete category $\big(\mathbf{I\text{-}Loc},\ U\big)$ is topological. \end{theorem} \subsection{Open subobjects} We introduce a notion of open subobjects different from the one alluded in \ref{open}. \begin{defi} An sublocale $S$ of a locale $L$ is called $I$-open \big(in $L$\big) if it is isomorphic to its $I$-interior, that is: if $i_{\text{\tiny{$L$}}}(S)= S$. \end{defi} The $I$-continuity condition (\ref{conti}) implies that $I$-openness is preserve by inverse images: \begin{prop} Let $f:L\rightarrow M$ be a morphism in $\mathbf{Loc}$. If $T$ is $I$-open in $M$, then $f_{-1}(T)$ is $I$-open in $L$. \end{prop} \begin{proof} If $T= i_{\text{\tiny{$M$}}}(T)$ then $f_{-1}(T)=f_{-1}\big(i_{\text{\tiny{$M$}}}(T)\big)\subseteq i_{\text{\tiny{$L$}}}\big(f_{-1}(T)\big)$, so \linebreak $i_{\text{\tiny{$L$}}}\big(f_{-1}(T)\big)=f_{-1}(T)$. \end{proof} \section{$\mathfrak h$ Operators} In this section we shall be conserned with a weak categorical version of a topological function studied by M, Suarez M. in \cite{MSM}. For that purpose we will use the colection $\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$ of all complemented sublocales of a locale $L$ (See P, T. Johnston \cite{PJ1}, for example). \begin{defi} An $\mathfrak h$ operator of the category $\mathbf{Loc}$ is given by a family $\mathfrak h =(h_{\text{\tiny{$L$}}})_{\text{$L\in \mathbf{Loc}$}}$ of maps $h_{\text{\tiny{$L$}}}:\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)\rightarrow \mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$ such that \begin{itemize} \item [($h_1$)] $S\cap h_{\text{\tiny{$L$}}}(S)\subseteq S$, for all $S \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$; \item [($h_2$)] If $S\subseteq T$ then $S\cap h_{\text{\tiny{$L$}}}(S)\subseteq T\cap h_{\text{\tiny{$L$}}}(T)$, for all $S,T \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$; \item [($h_3$)] $ h_{\text{\tiny{$L$}}}(L)=L$. \end{itemize} \end{defi} \begin{defi} An $\mathfrak h$-space is a pair $(L,h_{\text{\tiny{$L$}}})$ where $L$ is an object of $\mathbf{Loc}$ and $h_{\text{\tiny{$L$}}}$ is an $\mathfrak h$ operator on $L$. \end{defi} \begin{defi} A morphism $f:L\rightarrow M$ of $\mathbf{Loc}$ is said to be $\mathfrak h$-continuous if \begin{equation}\label{h-conti} f_{-1}\left[T\cap h_{\text{\tiny{$M$}}}(T)\right]\subseteq f_{-1}[T]\cap h_{\text{\tiny{$L$}}}\left( f_{-1}[T]\right) \end{equation} for all $T\in \mathcal{S}_\text{{\bsifamily{l}}}^{c}(M)$. Where $f_{-1}[-]$ is the inverse image of $f[-]$. \end{defi} \begin{prop} Let $f:L\rightarrow M$ and $g:M\rightarrow N$ be two $\mathfrak h$-continuous morphisms of $\mathbf{Loc}$ then $g\centerdot f$ is an $\mathfrak h$-continuous morphism of $\mathbf{Loc}$. \end{prop} \begin{proof} Since $g:M\rightarrow N$ is $I$-continuous, we have $$ g_{-1}\left[V\cap h_{\text{\tiny{$N$}}}(V)\right]\subseteq g_{-1}[V]\cap h_{\text{\tiny{$M$}}}\left( g_{-1}[V]\right) $$ for all $V\in \mathcal{S}_\text{{\bsifamily{l}}}^{c}(N)$, it fallows that $$ f_{-1}\big[ g_{-1}\left[V\cap h_{\text{\tiny{$N$}}}(V)\right]\big]\subseteq f_{-1}\big[g_{-1}[V]\cap h_{\text{\tiny{$M$}}}\left( g_{-1}[V]\right)\big] $$ now, by the $\mathfrak h$-continuity of $f$, $$ f_{-1}\big[g_{-1}[V]\cap h_{\text{\tiny{$M$}}}(g_{-1}[V])\big]\subseteq f_{-1}\big[g_{-1}[V]\big]\cap h_{\text{\tiny{$L$}}}\left( f_{-1}\big[g_{-1}[V]\big]\right)$$ therefore $$(g\centerdot f)_{-1}\big[V\cap h_{\text{\tiny{$N$}}}(V)\big]\subseteq (g\centerdot f)_{-1}\cap h_{\text{\tiny{$L$}}}\big((g\centerdot f)_{-1}[V] \big).$$ This complete the proof. \end{proof} As a consequence we obtain \begin{defi} The category $\mathbf{\mathfrak h\text{-}Loc}$ of $\mathfrak h$-spaces comprises the following data: \begin{enumerate} \item {\bf Objects}: Pairs $(L,h_{\text{\tiny{$L$}}})$ where $L$ is an object of $\mathbf{Loc}$ and $h_{\text{\tiny{$L$}}}$ is an $\mathfrak h$-operator on $L$. \item {\bf Morphisms}: Morphisms of $\mathbf{Loc}$ which are $\mathfrak h$-continuous. \end{enumerate} \end{defi} \subsection{The lattice structure of all $\mathfrak h$ operators} For the category $\mathbf{Loc}$ we consider the collection \[ \mathfrak h(\mathbf{Loc}) \] of all $\mathfrak h$ operators on $\mathbf{Loc}$. It is ordered by \[ \mathfrak h\leqslant \mathfrak{h^{'}} \Leftrightarrow h_{\text{\tiny{$L$}}}(S)\subseteq h^{'}_{\text{\tiny{$L$}}}(S), \,\,\ \text{for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$ and all $L$ object of $\mathbf{Loc}$}. \] This way $\mathfrak h(\mathbf{Loc})$ inherits a lattice structure from $ \mathcal{S}_\text{{\bsifamily{l}}}^{c}$. \begin{prop} Every family $( \mathfrak h_{\text{\tiny{$\lambda$}}})_{\text{\tiny{$\lambda\in \Lambda$}}}$ in $\mathfrak h(\mathbf{Loc})$ has a join $\bigvee\limits_{\text{\tiny{$\lambda\in \Lambda $}}} \mathfrak h_{\text{\tiny{$\lambda $}}}$ and a meet $\bigwedge\limits_{\text{\tiny{$\lambda\in \Lambda $}}} \mathfrak h_{\text{\tiny{$\lambda $}}}$ in $Int(\mathbf{Loc})$. The discrete $\mathfrak h$ operator \[ \mathfrak h_{\text{\tiny{$D$}}}=({h_{\text{\tiny{$D$}}}}_{\text{\tiny{$L$}}})_{\text{$L\in \mathbf{Loc}$}}\,\,\ \text{with}\,\,\ {h_{\text{\tiny{$D$}}}}_{\text{\tiny{$L$}}}(S)=S\,\,\ \text{for all}\,\ S\in \mathcal{S}_\text{{\bsifamily{l}}}^{c}(L) \] is the largest element in $\mathfrak h(\mathbf{Loc})$, and the trivial $\mathfrak h$ operator \[ \mathfrak h_{\text{\tiny{$T$}}}=({h_{\text{\tiny{$T$}}}}_\text{\tiny{$L$}})_{\text{$L\in \mathbf{Loc}$}}\,\,\ \text{with}\,\,\ {h_{\text{\tiny{$T$}}}}_{\text{\tiny{$L$}}}(S)= \begin{cases} \{1\}& \text{for all}\,\ S\in \mathcal{S}_\text{{\bsifamily{l}}}^{c}(L),\,\ S\ne L\\ L&\text {if}\,\ S=L \end{cases} \] is the least one. \end{prop} \begin{proof} For $\Lambda\ne\emptyset$, let $\widehat{\mathfrak h}=\bigvee\limits_{\text{\tiny{$\lambda\in\Lambda $}}}\mathfrak h_{\text{\tiny{$\lambda $}}}$, then \[ \widehat{h_{\text{\tiny{$L$}}}}=\bigvee\limits_{\text{\tiny{$\lambda\in \Lambda$}}} {h_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}, \] for all $L$ object of $\mathbf{Loc}$, satisfies \begin{itemize} \item $S\cap \widehat{h_{\text{\tiny{$L$}}}}(S)\subseteq S$, because $S\cap {h_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$S$}}}(L)\subseteq S$, for all $S \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$ and for all $\lambda \in \Lambda$. \item If $S\subseteq T$ then $S\cap \widehat{h_{\text{\tiny{$L$}}}}(S)\subseteq T\cap \widehat{h_{\text{\tiny{$L$}}}}(T)$, since $S \cup {h_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(S)\subseteq T \cup {h_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(T)$, for all $S,T \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$ and for all $\lambda \in \Lambda$. \item $L\cap \widehat{h_{\text{\tiny{$L$}}}}(L)= L$, because $L\cap {h_{\text{\tiny{$\lambda $}}}}_{\text{\tiny{$L$}}}(L)= L$ for all $\lambda \in \Lambda$. \end{itemize} Similarly $\bigwedge\limits_{\text{\tiny{$\lambda\in \Lambda $}}}\mathfrak h_{\text{\tiny{$\lambda $}}}$,\,\ $ \mathfrak h_{\text{\tiny{$D$}}}$ and $\mathfrak h _{\text{\tiny{$T$}}}$ are $\mathfrak h$ operators. \end{proof} \begin{coro}\label{h-complete} For every object $L$ of $\mathbf{Loc}$ \[ \mathfrak h(L) = \{h_{\text{\tiny{$L$}}}\mid h_{\text{\tiny{$L$}}}\,\ \text{ is an $\mathfrak h$ operator on}\,\ L\} \] is a complete lattice. \end{coro} \subsection{Initial $\mathfrak h$ operators} Let $\mathbf{\mathfrak h\text{-}Loc}$ be the category of $\mathfrak h$-spaces. Let $(M,h_{\text{\tiny{$M$}}})$ be an object of $\mathbf{\mathfrak h\text{-}Loc}$ and let $L$ be an object of $\mathbf{Loc}$. For each morphism $f:L\rightarrow M$ in $\mathbf{Loc}$ we define on $L$ the operotor \begin{equation} \label{h-initial} h_{\text{\tiny{$L_{f}$}}}:=f_{-1}\centerdot h_{\text{\tiny{$M$}}}\centerdot f_{*}. \end{equation} \begin{prop}\label{h-ini-cont} The operator (\ref{h-initial}) is an $\mathfrak h$ operator on $L$ for which the morphism $f$ is $\mathfrak h$-continuous. \end{prop} \begin{proof}\ \begin{enumerate} \item[($h_1)$] $S\cap h_{\text{\tiny{$L_{f}$}}}(S)= f_{-1} \Big[f_{*}[S]\cap h_{\text{\tiny{$M$}}}\big[ f_{*}[S]\big]\Big]\subseteq f_{-1}\big[f_{*}[s]\big]\subseteq S$, for all $S\in \mathcal{S}_\text{{\bsifamily{l}}}^c(L)$. \item[($h_2)$] $S\subseteq T$ in $\mathcal{S}_\text{{\bsifamily{l}}}^c(L)$, implies $f_{*}[S]\subseteq f_{*}[T]$, then $f_{*}[S]\cap h_{\text{\tiny{$M$}}}\big( f_{*}[S]\big)\subseteq f_{*}[T]\cap h_{\text{\tiny{$M$}}}\big( f_{*}[T]\big)$, therefore $f_{-1}\Big(f_{*}[S]\cap h_{\text{\tiny{$M$}}}\big( f_{*}[S]\big)\Big)\subseteq f_{-1}\Big(f_{*}[T]\cap h_{\text{\tiny{$M$}}}\big( f_{*}[T]\big)\Big)$, consequently $S\cap h_{\text{\tiny{$L_{f}$}}}(S)\subseteq T\cap h_{\text{\tiny{$L_{f}$}}}(T)$, for all $S,T \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$; \item[($h_3)$] $L\cap h_{\text{\tiny{$L_{f}$}}}(L)= f_{-1} \Big[f_{*}[L]\cap h_{\text{\tiny{$M$}}}\big[ f_{*}[L]\big]\Big]=L$. \end{enumerate} \end{proof} It is clear that $h_{\text{\tiny{$L_{f}$}}}(L)$ is the coarsest $\mathfrak h$ operator on $L$ for which the morphism $f$ is $\mathfrak h$-continuous; more precisaly \begin{prop}\label{h-unique} Let $(L,h_{\text{\tiny{$L$}}})$ and $(M,h_{\text{\tiny{$M$}}})$ be objects of $\mathbf{\mathfrak h\text{-}Loc}$,and let $N$ be an object of $\mathbf{Loc}$. For each morphism $g:N\rightarrow L$ in $\mathbf{Loc}$ and for\linebreak $f:(L,h_{\text{\tiny{$L_{f}$}}})\rightarrow (M,h_{\text{\tiny{$N$}}})$ an $\mathfrak h$-continuous morphism, $g$ is $\mathfrak h$-continuous if and only if $f\centerdot g$ is $\mathfrak h$-continuous. \end{prop} \begin{proof} Suppose that $g\centerdot f$ is $I$-continuous, i. e. $$(f\centerdot g)_{-1}\big(T\cap h_{\text{\tiny{$M$}}}(T)\big)\subseteq T\cap h_{\text{\tiny{$N$}}}\big( (f\centerdot g)_{-1}(T) \big)$$ for all $T\in T \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(N)$. Then, for all $S\in T \in\mathcal{S}_\text{{\bsifamily{l}}}^{c}(L)$, we have \begin{align*} g_{-1}\Big(S\cap \big(h_{\text{\tiny{$L_{f}$}}}(S)\big)\Big)&=g_{-1}\Big(f_{-1}\big( f_{*}(S)\cap h_{\text{\tiny{$M$}}}\centerdot f_{*}(S)\big)=(f\centerdot g)_{-1}\big( f_{*}(S)\cap h_{\text{\tiny{$M$}}}( f_{*}(S)) \big)\\ &\subseteq f\centerdot g)_{-1}(f_{*}(S) \cap \Big(h_{\text{\tiny{$N$}}}\big( (f\centerdot g)_{-1}(f_{*}(S) ) \big)\Big)\\ &=(f\centerdot g)_{-1}(f_{*}(S) ) \cap h_{\text{\tiny{$N$}}}\big( g_{-1}\centerdot f_{-1}\centerdot f_{*} (S)\big)\\ &\subseteq g_{-1}(S)\cap h_{\text{\tiny{$N$}}}\big( g_{-1}(S)\big),\\ \end{align*} i.e. $g$ is $I$-continuous. \end{proof} As a consequence of corollary(\ref{h-complete}), proposition(\ref{h-ini-cont}) and proposition (\ref{h-unique}) (cf. \cite{AHS} or \cite{JM}), we obtain \begin{theorem} The forgetful functor $U:\mathbf{\mathfrak h\text{-}Loc}\rightarrow \mathbf{Loc}$ is topological, i.e. the concrete category $\big(\mathbf{\mathfrak h\text{-}Loc},\ U\big)$ is topological. \end{theorem} \end{document}

\begin{document} \begin{abstract} We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account ``memory''. The precise model is \[ \mathcal{D}_t^{\alpha} u - \text{div}(u(-\mathcal{D}elta)^{-\sigma} u) = f, \quad 0<\sigma <1/2. \] We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\gammaeq 0 $ as well as right hand side $f\gammaeq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is H\"older continuity for such weak solutions. \epsilonnd{abstract} \maketitle \section{Introduction} In this paper we study both existence and regularity for solutions to a porous medium equation. The pressure is related to the density via a nonlocal operator. This diffusion takes into account long-range effects. The time derivative is nonlocal and fractional and therefore takes into account the past. In the typical derivation of the porous medium equation (see \cite{v07}) the equation one considers is \[ \partial_t u + \text{div} (\textbf{v}u)=0, \] with $u(t,x)\gammaeq 0$. By Darcy's law in a porous medium $\textbf{v}= -\nabla p$ arises as a potential where $p$ is the pressure. According to a state law $p=f(u)$. In our case we consider a potential which takes into account long range interactions, namely $p=(-\mathcal{D}elta)^{-\sigma}u$. A porous medium equation with a pressure of this type \begin{equation} \label{e:tloc} \partial_t u = \text{div}(u(-\mathcal{D}elta)^{-\sigma}u) \epsilonnd{equation} has been recently studied. For $0<\sigma<1$ with $\sigma \neq 1/2$, existence of solutions was shown in \cite{cv11} while regularity and further existence properites were studied in \cite{cfv13}. Uniqueness for the range $1/2\leq \sigma <1$ was shown in \cite{zxc14}. Another model of the porous medium equation \[ D_t^{\alpha} u - \text{div}(\kappa (u)Du)=f \] was introduced by Caputo in \cite{c99}. In the above equation $D_t^{\alpha}$ is the Caputo derivative and the diffusion is local. Solvability for a more general equation was recently studied in \cite{z12}. The fractional derivative takes into account models in which there is ``memory''. The Caputo derivative has also been recently shown (see \cite{d04}, \cite{d05} ) to be effective in modeling problems in plasma transport. See also \cite{mk00} and \cite{z02} for further models that utilize fractional equations in both space and time to account for long-range interactions as well as the past. The specific equation we study is \begin{equation} \label{e:e} \mathcal{D}_t^{\alpha} u(t,x) - \text{div} \left(u \nabla (-\mathcal{D}elta)^{-\sigma} u \right)=f(t,x). \epsilonnd{equation} The operator $\mathcal{D}_t^{\alpha}$ is of Caputo-type and is defined by \[ \mathcal{D}_t^{\alpha} u := \frac{\alpha}{\Gamma(1-\alpha)}\int_{-\infty}^t [u(t,x)-u(s,x)]K(t,s,x) \ ds. \] When $K(t,s,x)=(t-s)^{-1-\alpha}$ this is exactly the Caputo derivative - see Section \ref{s:caputo} - which we denote by $D_t^{\alpha}$. We assume the following bounds on the kernel $K$ \begin{equation} \label{e:kernelb} \frac{1}{\Lambda(t-s)^{1+\alpha}} \leq \frac{\alpha}{\Gamma(1-\alpha)} K(t,s,x) \leq \frac{\Lambda}{(t-s)^{1+\alpha}} \epsilonnd{equation} Our kernel in time then can be thought of as having ``bounded, measureable coefficients''. We also require the following relation on the kernel \begin{equation} \label{e:kernel} K(t,t-s)=K(t+s,t). \epsilonnd{equation} The relation \epsilonqref{e:kernel} allows us to give a weak - in space and in time - formulation of \epsilonqref{e:e}. This weak formulation is given in Section \ref{s:caputo}. In this paper we also restrict ourselves to the range $0<\sigma<1/2$. In \cite{cfv13} use of a transport term was made to work in the range $1/2<\sigma<1$. We have not yet found the correct manner in which to prove our results for $1/2<\sigma<1$ when dealing with the nonlocal fractional time derivative $\mathcal{D}_t^{\alpha}$. \subsection{Accounting for the Past.} Nonlocal equations are effective in taking into account long-range interactions and taking into account the past. However, the nonlocal aspect of the equation provides both advantages and disadvantages in studying local aspects of the equation. One advantage is that there is a relation between two points built into the equation. Indeed, we utilize two nonlocal terms that are not present in the classical porous medium equation to prove Lemma \ref{l:2down}. One disadvantage of nonlocal equations is that when rescaling of the form $v(t,x)=Au(Bt,Cx)$, the far away portions of $u$ cannot be discarded, and hence $v$ begins to build up a ``tail''. Consequently, the usual test function $(u-k)_+$ or $F((u-k)_+)$ for some function $F$ and a constant $k$ is often insufficient. One must instead consider $F((u-\phi)_+)$ where $\phi$ is constant close by but has some ``tail'' growth at infinity. This difficulty of course presents itself with the Caputo derivative. One issue becomes immediately apparent. If we choose $F((u-\phi)_+)$ as a test function, then \[ \begin{aligned} \int_a^T F((u-\phi)_+) D_t^{\alpha} u \ dt &= \int_a^T F((u-\phi)_+) D_t^{\alpha} ((u-\phi)_+ - (u-\phi)_- ) \ dt \\ &\quad + \int_a^T F((u-\phi)_+) D_t^{\alpha} \phi \ dt. \epsilonnd{aligned} \] The second term will no longer be identically zero if $\phi$ is not constant. When using energy methods, this second term can be treated as part of the right hand side, and hence it becomes natural to consider an equation of the form \epsilonqref{e:e} with a right hand side. The main challenge with accomodating a nonzero right hand side is that the natural test function $\ln u$ used in \cite{cv11} and \cite{cfv13} is no longer available since the function $u$ can evaluate zero. Indeed if the initial data for a solution is compactly supported, then the solution is compactly supported for every time $t>0$ (see Remark \ref{r:finite}). We choose as our basic test function $u^{\gammaamma}$ for $\gammaamma>0$. For $\sigma$ small we will have to choose $\gammaamma$ small. We then can accomodate a right hand side as well as avoiding delicate integrability issues involved when using $\ln u$ as a test function. Using careful analysis, it is still most likely possible to utilize $\ln u$ as a test function for our equation \epsilonqref{e:e} with zero right hand side, but we find it more convenient to use $u^{\gammaamma}$ and prove the stronger result that includes a right hand side. Our method using $u^{\gamma}$ should also work for the equation \epsilonqref{e:tloc} to be able to prove existence and regularity with a right hand side. One benefit of accomodating a right hand side in $L^{\infty}$ is that we obtain immediately regularity up to the initial time for smooth initial data, see Theorem \ref{t:continuity}. \subsection{Overview of the Main results} We will prove our results for a class of weak solutions \epsilonqref{e:main} later formulated in Section \ref{s:caputo}. Our first main result is existence. We use an approximating scheme as in \cite{cv11} as well as discretizing in time as in \cite{acv15}. We prove \begin{theorem} \label{t:existence} Let $0\leq u_0(x), f(t,x) \leq Ae^{-|x|}$ for some $A \gammaeq 0$. Then there exists a solution $u$ to \epsilonqref{e:main} in $(0,\infty)$ that has initial data $u(0,x)=u_0(x)$. \epsilonnd{theorem} \begin{remark} \label{r:unique} Our constructions are made via recursion over a finite time interval $(0,T_1)$. Since our constructions are made via recursion, if $T_2 =mT_1$ for $m \in {\mathbb N}$, it is immediate that if $u_i$ is the solution constructed on $(0,T_i)$, then $u_2 = u_1$ on $(0,T_1)$. \epsilonnd{remark} \begin{remark} \label{r:n1} For technical reasons seen in the proof of Lemma \ref{l:triebel}, when $n=1$ we make the further restriction $0<\sigma<1/4$. \epsilonnd{remark} The main result of the paper is an interior H\"older regularity result. As expected the H\"older norm will depend on the distance from the interior domain to the initial time $t_0$. However, if we assume the intial data $u_0$ is regular enough - say for intance $C^2$ - then we obtain regularity up to the initial time. This is a benefit of allowing a right hand side. By extending the values of our solution $u(t,x)=u(0,x)$ for $t<0$, we satisfy \epsilonqref{e:main} on $(-\infty, \infty) \times {\mathbb R}^n$ with a right hand side in $L^{\infty}$. The right hand side $f$ for $t \leq 0$ will not necessarily satisfy $f \gammaeq 0$; however, this nonnegativity assumption on $f$ was only necessary to guarantee the existence of a solution $u \gammaeq 0$. It is not a necessary assumption to prove regularity. From Remark \ref{r:unique} the solution constructed on $(-\infty, T)$ will agree with the original solution over the interval $(0,T)$. \begin{theorem} \label{t:continuity} Let $u$ be a solution to \epsilonqref{e:main} obtained via approximation from Theorem \ref{t:existence} on $[0,T] \times {\mathbb R}^n$ with $0\leq u_0(x), f(t,x) \leq Ae^{-|x|}$. Assume also $u_0 \in C^2$. Then $u$ is $C^{\beta}$ continuous on $[0,T]\times {\mathbb R}^n$- for some exponent $\beta$ depending on $\alpha, \Lambda, n, \sigma$ - with a constant that depends on the $L^{\infty}$ norm of $u$ and $f$ and $C^2$ norm of $u_0$. \epsilonnd{theorem} \subsection{Future Directions} We prove existence and regularity for solutions obtained via limiting approximations. In this paper we do not address the issue of uniqueness. As mentioned earlier, uniqueness for \epsilonqref{e:tloc} for the range $1/2\leq \sigma <1$ was shown in \cite{zxc14}. The issue of uniqueness for \epsilonqref{e:e} is not trivial because of the nonlinear aspect of the equation as well as the lack of a comparison principle. The equation \epsilonqref{e:main} which we consider also should present new difficulties because of the weak/very-weak formulation in time as well as the minimal ``bounded, measurable'' assumption \epsilonqref{e:kernelb} on the kernel $K(t,s,x)$. An interesting problem would be to then address the issue of uniqueness for solutions of \epsilonqref{e:main}. Theorems \ref{t:existence} and \ref{t:continuity} can most likely be further refined by making less assumptions on $u(0,x)$, assuming a right hand side $f \in L^p$ as was done for a similar problem in \cite{z12}, and proving the estimates uniform as $\sigma \to 0$ and recoving H\"older continuity for the local diffusion problem. Also, as mentioned earlier the Theorems can be improved to include the range $1/2\leq \sigma <1$. Finally, just like in the local porous medium equation \cite{v07} as well as for \epsilonqref{e:tloc}, the equation \epsilonqref{e:main} has the property of finit propagation, see Remark \ref{r:finite}. Therefore, it is of interest to study the free boundary $\partial \{u(t,x)>0\}$. \subsection{Outline} The outline of this paper will be as follows. In Section \ref{s:caputo} we state basic results for the Caputo derivative. We also give the weak formulation of the equation we study. In Section \ref{s:discretize} we state some results for the discretized version of $\mathcal{D}_t^{\alpha}$ that we will use to prove the existence of solutions. In Section \ref{s:existence} we follow the method approximation method and use the estimates from \cite{cv11} combined with the method of discretization and the estimates presented in \cite{acv15} to prove existence. In Section \ref{s:lemmas} we state the main Lemmas that we will need to be able to prove H\"older regularity. In Section \ref{s:pullup} we prove the most technically difficult Lemma \ref{l:pullup} of the paper. This Lemma \ref{l:pullup} most directly handles the degenerate nature of the problem. In Section \ref{s:pulldown} we prove an analogue of \ref{l:pullup}. In Section \ref{s:oscillation} we prove the final Lemmas we need which give a one-sided decrease in oscillation from above. The one-sided decrease in oscillation combined with Lemma \ref{l:pullup} is enough to prove the H\"older regularity and this is explained in Section \ref{s:regularity}. \subsection{Notation} We list here the notation that will be used consistently throughout the paper. The following letters are fixed throughout the paper and always refer to: \begin{itemize} \item $\alpha$ - the order of the Caputo derivative. \item $\sigma$ - the order of inverse fractional Laplacian $(-\mathcal{D}elta)^{-\sigma}$. We use $\sigma$ for the order because $s$ will always be a variable for time. \item $a$ - the initial time for which our equation is defined. \item $D_t^{\alpha}$ - the Caputo derivative as defined in Section \ref{s:caputo}. \item $\mathcal{D}_t^{\alpha}$ - the Caputo-type fractional derivative with ``bounded, measurable'' coefficients with bounds \epsilonqref{e:kernelb} and relation \epsilonqref{e:kernel}. \item $\Lambda$ the constant appearing in \epsilonqref{e:kernelb}. \item $\mathcal{D}_{\epsilon}^{\alpha}$ - the discretized version of $\mathcal{D}_t^{\alpha}$ as defined in \epsilonqref{e:dc} \item $\epsilon$ - will always refer to the time length of the discrete approximations as defined in Section \ref{s:discretize} \item $n$ - will always refer to the space dimension. \item $\Gamma_m$ - the parabolic cylinder $(-m,0)\times B_m$. \item $W^{\beta,p}$ - the fractional Sobolev space as defined in \cite{dpv12}. \item $u_{\pm}$ - the positive and negative parts respectively so that $u = u_+ - u_-$. \item $\tilde{u}$ - the extension $\tilde{u}(t)=u(\epsilon j)$ for $\epsilon j-1 < t, \epsilon j$. \epsilonnd{itemize} \section{Caputo Derivative} \label{s:caputo} In this section we state various properties of the Caputo derivative that will be useful. The Caputo derivative for $0<\alpha<1$ is defined by \[ _{a}D_{t}^{\alpha} u(t) := \frac{1}{\Gamma(1-\alpha)} \int_{a}^{t} \frac{u'(s)}{(t-s)^{\alpha}} \ ds \] By using integration by parts we have \begin{equation} \label{e:fracalt} \Gamma(1-\alpha) \ _{a}D_{t}^{\alpha} u(t) = \frac{u(t)-u(a)}{(t-a)^{\alpha}} + \alpha \int_{a}^t \frac{u(t)-u(s)}{(t-s)^{\alpha + 1}} \ ds. \epsilonnd{equation} For the remainder of the paper we will drop the subscript $a$ when the initial point is understood. We now recall some properties of the Caputo derivative that were proven in \cite{acv15}. For a function $g(t)$ defined on $[a,t]$, it is advantageous to define $g(t)$ for $t<a$. Then we have the formulation \[ _a D_t^{\alpha} g(t) = \ _{-\infty}D_t^{\alpha} g(t) = \frac{\alpha}{\Gamma(1-\alpha)} \int_{-\infty}^t \frac{g(t)-g(s)}{(t-s)^{1+\alpha}}, \] utilized in \cite{acv15}. (See also \cite{bmst15} for properties of this one-sided nonlocal derivative.) This looks very similar to $(-\mathcal{D}elta)^{\alpha}$ except the integration only occurs for $s<t$. In this manner the Caputo derivative retains directional derivative behavior while at the same time sharing certain properties with $(-\mathcal{D}elta)^{\alpha}$. This is perhaps best illustrated by the following integration by parts formula for the Caputo derivative \begin{proposition} \label{p:changev} Let $g,h\in C^1(a,T)$. Then \begin{equation} \label{e:changev} \begin{aligned} \int_{a}^T g D_t^{\alpha}h + h D_t^{\alpha} g &= \int_{a}^T {g(t)h(t) \left[\frac{1}{(T-t)^{\alpha}} + \frac{1}{(t-a)^{\alpha}} \right]} \ dt \\ & \quad + \alpha \int_{a}^T \int_{a}^t \frac{[g(t)-g(s)][h(t)-h(s)]}{(t-s)^{1+\alpha}} \ ds \ dt \\ & \quad - \int_{a}^T \frac{g(t)h(a)+h(t)g(a)}{(t-a)^{\alpha}} \ dt. \epsilonnd{aligned} \epsilonnd{equation} \epsilonnd{proposition} Formula \epsilonqref{e:fracalt} is based on the following formal computation \[ \begin{aligned} \int_a^T \int_a^t \frac{g(t)-g(s)}{(t-s)^{1+\alpha}} \ ds \ dt &= \int_a^T \int_a^t \frac{g(t)}{(t-s)^{1+\alpha}} \ ds \ dt - \int_a^T \int_a^t \frac{g(s)}{(t-s)^{1+\alpha}} \ ds \ dt \\ &= \int_a^T \int_a^t \frac{g(t)}{(t-s)^{1+\alpha}} \ ds \ dt - \int_a^T \int_s^T \frac{g(s)}{(t-s)^{1+\alpha}} \ dt \ ds \\ &= \int_a^T \int_a^t \frac{g(t)}{(t-s)^{1+\alpha}} \ ds \ dt - \int_a^T \int_t^T \frac{g(t)}{(s-t)^{1+\alpha}} \ ds \ dt \\ &= \int_a^T g(t)\left( \int_0^{t-a} \frac{ds}{s^{1+\alpha}} - \int_{0}^{T-t} \frac{ds}{s^{1+\alpha}} \right) \ dt \\ &= \alpha^{-1}\int_a^T g(t) \left(\frac{1}{(T-t)^{\alpha}} - \frac{1}{(t-a)^{\alpha}} \right) \ dt. \epsilonnd{aligned} \] In the above computation to utilize the cancellation we only need a kernel $K(t,s)$ satsifying \[ K(t,t-s)=K(t+s,t). \] To make the above computation rigorous we will use the discretization in Section \ref{s:discretize}. An alternative, equivalent integration by parts formula is to extend $g(t)=g(a)$ for $t<a$. Then for any $h\in C^1$ with $h(t)=0$ for any $t<b$ for some $b$ we have \begin{equation} \label{e:crhs} \begin{aligned} \int_{-\infty}^T h(t) D_t^{\alpha} g(t) \ dt &= c_{\alpha} \int_{-\infty}^T \int_{-\infty}^{2t-T} \frac{[g(t)-g(s)][h(t)-h(s)]}{(t-s)^{1+\alpha}} \ ds \ dt \\ &\quad + c_{\alpha} \int_{-\infty}^T \int_{-\infty}^t \frac{g(t)h(t)}{(t-s)^{1+\alpha}} \ ds \ dt \\ &\quad - \int_{-\infty}^T g(t) D_t^{\alpha} h(t) \ dt, \epsilonnd{aligned} \epsilonnd{equation} with $c_{\alpha}=\alpha \Gamma(1-\alpha)^{-1}$. Now \epsilonqref{e:changev} and \epsilonqref{e:crhs} will both imply each other, so \epsilonqref{e:crhs} is an alternative way of handling the initial condition $g(a)$. Furthermore, both \epsilonqref{e:changev} and \epsilonqref{e:crhs} are weak formulations for the Caputo derivative that only require that $g \in H^{\alpha/2}((a,T))$. \epsilonqref{e:crhs} will work for any kernel $K(t,s)$ satisfying the relation \epsilonqref{e:kernel}. In view of \epsilonqref{e:crhs} we now give the exact formulation of our weak solutions. We assume the bounds \epsilonqref{e:kernelb} and the relation \epsilonqref{e:kernel} on the kernel $K(t,s,x)$. For smooth initial data $u_0 \in C^2$, we assign $u(t,x)=u(a,x)$ for $t<a$. Then as stated earlier in the Introduction, for $t\leq a$, a solution $u$ will have right hand side \[ \text{div}(u_0 (-\mathcal{D}elta)^{-\sigma} u_0) \in L^{\infty}. \] We say that $u$ is a weak solution if for any $\phi \in C_0^{\infty} (-\infty,T)\times {\mathbb R}^n$ we have \begin{equation} \label{e:main} \begin{aligned} &\int_{{\mathbb R}^n} \int_{-\infty}^T \int_{-\infty}^t [u(t,x)-u(s,x)][\phi(t,x)-\phi(s,x)]K(t,s,x) \ ds \ dt \ dx \\ &\quad + \int_{{\mathbb R}^n} \int_{-\infty}^T \int_{-\infty}^{2t-T} u(t,x)\phi(t,x) K(t,s,x) \ ds \ dt \ dx \\ &\quad -\int_{{\mathbb R}^n} \int_{-\infty}^T u(t,x) \mathcal{D}_t^{\alpha} \phi(t,x) \ dt \ dx \\ &\quad +\int_{-\infty}^T \int_{{\mathbb R}^n} \nabla \phi(t,x) u(t,x) \nabla (-\mathcal{D}elta)^{-\sigma} u \ dx \ dt\\ &= \int_{-\infty}^T \int_{B_R} f(t,x) \phi(t,x). \epsilonnd{aligned} \epsilonnd{equation} We will also utilize a fractional Sobolev norm that arises from the fractional derivative. \begin{lemma} \label{l:ext} Let $u$ be defined on $[a,T]$. We have for two constants $c_1,c_2$ depending on $\alpha,|T-a|$ \[ \begin{aligned} \| u \|_{L^{\frac{2}{1-\alpha}}}(a,T) & \leq c_1 \|u \|_{H^{\alpha/2}}^2(a,T) \\ &\leq c_2 \left( \alpha \int_{a}^T \int_{a}^t \frac{|u(t)-u(s)|^2}{|t-s|^{1+\alpha}} \ ds \ dt + \int_a^T \frac{u^2(t)}{(T-a)^{\alpha}} \right) \epsilonnd{aligned} \] \epsilonnd{lemma} The following estimate will be needed for our choice of cut-off functions. \begin{lemma} \label{l:fracbound} Let \[ h(t):= \max \{ |t|^{\nu} -1 , 0 \} \\ \] with $\nu < \alpha$. Then \[ \mathcal{D}_t^\alpha h \gammaeq -c_{\nu,\alpha, \Lambda} \] for $t \in {\mathbb R}$. Here, $c_{\nu,\alpha,\Lambda}$ is a constant depending only on $\alpha , \nu, \Lambda$. \epsilonnd{lemma} Finally, we point out that if $g=g_+ - g_-$ the positive and negative parts respectively, then \begin{equation} \label{e:posneg} \int_a^T g_{\pm}(t)\mathcal{D}_t^{\alpha} g_{\mp}(t) \gammaeq 0. \epsilonnd{equation} \section{Discretization in time} \label{s:discretize} To prove existence of solutions to \epsilonqref{e:main} we will discretize in time. The discretization also allows us to make the computations involving the fractional derivative rigorous. This section contains properties of a discrete fractional derivative which we will utilize. For future reference we denote the discrete Fractional derivative with kernel $K$ as \begin{equation} \label{e:dc} \mathcal{D}_{\epsilon}^{\alpha} u(a+\epsilon j):= \epsilonpsilon \sum_{-\infty<i <j} [u(a+\epsilon j) - u(a+\epsilon i)]K(\epsilon j, \epsilon i) \epsilonnd{equation} The following is the discretized argument for the cancellation that appears in the formal computation of the version of \epsilonqref{e:fracalt} that we will need. \begin{lemma} \label{l:cancel} Assume $g(a)=0$ and define $g(t):=0$ for $t<a$. Assume relation \epsilonqref{e:kernel} on $K$. Then \[ \epsilon^2 \sum_{j\leq k} \sum_{i<j} [g(a+\epsilon j)-g(a+\epsilon i)]K(a+\epsilon j, a+ \epsilon i) = \epsilon^2 \sum_{j\leq k} \ \sum_{i<2j-k-1} g(\epsilon j)K(\epsilon j, \epsilon i). \\ \] Let $\tilde{g}(t):=g(\epsilon j)$ for $\epsilon j-1< t \leq \epsilon j$. If $g\gammaeq 0$, then there exists $c$ depending only on $\alpha, \Lambda$ such that if $\epsilon < 1$, then \[ \epsilon^2 \sum_{j\leq k} \sum_{i<j} [g(a+\epsilon j)-g(a+\epsilon i)]K(a+\epsilon j, a+ \epsilon i) \gammaeq c \int_{-\infty}^T \frac{\tilde{g}(t)}{(T-t)^{\alpha}} \ dt. \] \epsilonnd{lemma} \begin{proof} For notational simplicity we assume $a=0$. \[ \begin{aligned} &\epsilon^2 \sum_{j\leq k} \sum_{i<j} [g(\epsilon j)-g(\epsilon i)]K(\epsilon j, \epsilon i) \\ &= \epsilon^2 \sum_{j\leq k} \sum_{i<j} g(\epsilon j)K(\epsilon j, \epsilon i) - \epsilon^2 \sum_{j\leq k} \sum_{i<j} g(\epsilon i)K(\epsilon j, \epsilon i) \\ &= \epsilon^2 \sum_{j\leq k} \sum_{i<j} g(\epsilon j)K(\epsilon j, \epsilon i) - \epsilon^2 \sum_{i< k} \sum_{i<j\leq k} g(\epsilon i)K(\epsilon j, \epsilon i) \\ &= \epsilon^2 \sum_{j\leq k} \sum_{i<j} g(\epsilon j)K(\epsilon j, \epsilon i) - \epsilon^2 \sum_{j< k} \sum_{j<i\leq k} g(\epsilon j)K(\epsilon i, \epsilon j) \\ &= \epsilon^2 \sum_{j\leq k} \sum_{i<2j-k-1} g(\epsilon j)K(\epsilon j, \epsilon i) \epsilonnd{aligned} \] The second inequality follows from the estimates in \cite{acv15}. \epsilonnd{proof} Lemma \ref{l:cancel} combined with the estimates in \cite{acv15} can be used to show. \begin{lemma} \label{l:1byparts} Let $u(0)=0$ and assume $u\gammaeq 0$. For fixed $0<\epsilon<1$, let $\tilde{u}$ be the extension defined as in Lemma \ref{l:cancel}. Let $K$ satisfy conditions \epsilonqref{e:kernelb} and \epsilonqref{e:kernel}. Then there exists two constants $c_1,c_2$ depending only on $\alpha$ such that \[ \epsilon^2 \sum_{j\leq k} \sum_{i<j} u(\epsilon j)[u(\epsilon j)-u(\epsilon i)]K(\epsilon j, \epsilon i) \gammaeq c \| u \|_{H^{\alpha/2}}^2 \gammaeq c \| u \|_{H^{\alpha/2}}^2c \left(\int_{-\infty}^T u^{\frac{2}{(1-\alpha)}} \right)^{1-\alpha} = \] where $c$ depends on $\alpha$ and $\Lambda$. \epsilonnd{lemma} This next lemma is analogous to Lemma \ref{l:fracbound} and was shown in \cite{acv15}. \begin{lemma} \label{l:dfracbound} Let $h$ be as in Lemma \ref{l:fracbound}. Then for $0<\epsilonpsilon < 1$ there exists $c_{\nu,\alpha}$ depending on $\alpha$ and $\nu$ but independent of $a$ such that \[ \mathcal{D}_{\epsilonpsilon}^{\alpha} h(t) \gammaeq -c_{\nu,\alpha} \] for $t\in \epsilonpsilon {\mathbb Z}$ and $a<t<0$. \epsilonnd{lemma} This last estimate we will use often \begin{lemma} \label{l:discconvex} Let $F$ be a convex function with $F'' \gammaeq \gammaamma, F' \gammaeq 0, F(0)=0$. Assume $g\gammaeq 0, g(a)=0$. Then there exists $c$ depending on $\alpha, \Lambda$ such that \[ \epsilon \sum_{j\leq k} F(g(\epsilon j)) \mathcal{D}_{\epsilon}^{\alpha}g(\epsilon j) \gammaeq c\epsilon \sum_{j\leq k} \frac{F(g(\epsilon j))}{(\epsilon(j-i))^{1+\alpha}} + c \frac{\gamma}{2}\epsilon^2 \sum_{j\leq k}\sum_{e<j} \frac{[g(\epsilon j)-g(\epsilon i)]^2}{(\epsilon(j-i))^{1+\alpha}} \] \epsilonnd{lemma} \begin{proof} Since $F$ is convex, \[ F(g(\epsilon j))[g(\epsilon j)-g(e i)] \gammaeq F(g(\epsilon j))- F(g(\epsilon i)) + \frac{\gamma}{2} [g(\epsilon j)- g(\epsilon i)]^2. \] The result then follows from applying Lemma \ref{l:cancel}. \epsilonnd{proof} Finally we point out that if $g$ is a limit of $\tilde{g}_{\epsilon}$ which are discretized problems with the assumptions in Lemma \ref{l:discconvex} it follows that \begin{equation} \label{e:convex} \int_a^T F(g(t)) \mathcal{D}_t^{\alpha} g(t) \ dt \gammaeq c \int_a^T \frac{F(g(t))}{(T-a)^{\alpha}} + c \frac{\gamma}{2}\int_a^T \int_a^t \frac{[g(t)-g(s)]^2}{(t-s)^{1+\alpha}} \ ds \ dt. \epsilonnd{equation} \section{Existence} \label{s:existence} In this section we prove the existence of weak solutions following the construction given in \cite{cv11}. We will also discretize in time. We first consider a smooth approximation of the kernel $(-\mathcal{D}elta)^{-\sigma}$ as $K_{\zeta}$. We start with the smooth classical solution to the elliptic problem \begin{equation} \label{e:aprox} g u - \delta \text{div}((u+d) \nabla u) - \text{div}((u + d) \nabla K_{\zeta} u ) = f \text{ on } B_R \epsilonnd{equation} with $u \epsilonquiv 0$ on $\partial B_R$. $g,f\gammaeq 0$ and smooth. The sign of $f,g$ guarantees the solution is nonnegative. $\delta, d>0$ are constants. The nonlocal part is computed in the expected way by extending $u=0$ on $B_R^c$. To find such a solution we first consider the linear problem \[ g u - \delta\text{div}((v+d)\nabla u) - \text{div}((v + d) \nabla K_{\zeta} u ) = f \text{ on } B_R \] for $v \in C_0^{0,\beta}$ with $v\gammaeq 0$. With fixed $d,\delta, R, \zeta >0$ one can apply Schauder estimate theory to conclude \[ \| u \|_{C_0^{1,\beta}} \leq C \| v \|_{C_0^{0,\beta}} \] with $u \gammaeq 0$. The map $T: v \to u$ is then a compact map. The set $\{v\}$ with $v \gammaeq 0$ and $v \in C_0^{0,\beta}$ is a closed convex set, and hence we can apply the fixed point theorem (Corollary 11.2 in \cite{gt01}), to conclude there is a solution to \epsilonqref{e:aprox}. By bootstrapping we conclude $u$ is smooth. Now we use the existence of solutions to \epsilonqref{e:aprox} to obtain - via recursion - solutions to the discretized problem \begin{equation} \label{e:daprox} \mathcal{D}_{\epsilon}^{\alpha} u - \delta \text{div}((u+d)\nabla u) - \text{div}((u + d) \nabla K_{\zeta} u ) = f \text{ on } [0,T]\times B_R. \epsilonnd{equation} with $u(0,x)=u_0(x)$ an initially defined smooth function with compact support. $\epsilon=T/k$ for some $k\in {\mathbb N}$. We will eventually let $k \to \infty$, so that $\epsilon \to 0$. For the next two Lemmas we will utilize the solution to \begin{equation} \label{e:ode} D_t^{\alpha} Y(t) = cY(t)+h(t) \epsilonnd{equation} which as in \cite{d10} is given by \[ Y(t)= Y(0)E_{\alpha}(ct^{\alpha})+\alpha\int_{0}^t(t-s)^{\alpha-1}E_{\alpha}'(c(t-s)^{\alpha})h(t) \ dt, \] where $E_{\alpha}$ is the Mittag-Leffler function of order $\alpha$. We will utilize in the next Lemmas two specific instances of \epsilonqref{e:ode}. We define $Y_1(t)$ to be the solution to \epsilonqref{e:ode} with $Y(0)=\sup u(0,x), c=0, h=2\Lambda f$. We define $Y_2(t)$ to be the solution to \epsilonqref{e:ode} with $c=C\Lambda^{-1},Y_{2}(0)=2$. and $h=0$. The constant $C$ will be chosen later. \begin{lemma} \label{l:boundab} Let $u$ be a solution to \epsilonqref{e:daprox}. Let $Y_1(t)$ be defined as above. Then there exists $\epsilon_0$ depending only on $T,\alpha,\|f\|_{L^{\infty}}$ such that if $\epsilon \leq \epsilon_0$, then \[ u(\epsilon j) \leq Y_1(\epsilon j) \] \epsilonnd{lemma} \begin{proof} Since $Y_1(t)$ is an increasing function \[ \mathcal{D}_{\epsilon}^{\alpha} Y(\epsilon j) \gammaeq \Lambda^{-1} D_{\epsilon}^{\alpha} Y(\epsilon j). \] Depending on $T$ and $\|f\|_{L^{\infty}}$, there exists $\epsilon_0$ such that if $\epsilon\leq \epsilon_0$, then \[ \mathcal{D}_{\epsilon}^{\alpha} Y(\epsilon j) \gammaeq \Lambda^{-1} D_{\epsilon}^{\alpha} Y(\epsilon j) \gammaeq \frac{2}{3\Lambda} D_{t}^{\alpha} Y(\epsilon j)= \frac{4}{3}f \] We use $(u(t,x)-Y(t))_+$ as a test function. Since $(u-Y)_+(0)=0$ it follows from \epsilonqref{e:posneg} and Lemma \ref{l:discconvex} that \[ \epsilon \sum_{j<k} (u-Y)_+ \mathcal{D}_{\epsilon}^{\alpha} [(u-Y)_+ -(u-Y)_-] \gammaeq 0. \] We define \[ \mathcal{B}_{\zeta}(u,v) = \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} H_{\zeta}(x,y)[u(x)-u(y)][v(x)-v(y)] \ dx \ dy, \] where $H_{\zeta}(x,y)=\mathcal{D}elta K_{\zeta}$. We have the identity \cite{cfv13} \[ \mathcal{B}_{\zeta}(u,v) = \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \nabla u(x) \nabla K_{\zeta} v(y) \ dx \ dy. \] Then for $\epsilon$ small enough and $j>0$ \[ \begin{aligned} &\epsilon\sum_{j\leq k}\int_{B_R} f(\epsilon j,x)(u(\epsilon j, x)-Y(\epsilon j))_+ \\ &=\int_{B_R} \epsilon \sum_{j\leq k}(u-Y)_+\mathcal{D}_{\epsilon}^{\alpha}[(u-Y)_+ - (u-Y)_-+Y] \\ &\quad + \epsilon\sum_{j\leq k} \int_{B_R} (u+d) \nabla (u-Y)_+ \nabla u \\ &\quad +\epsilon\sum_{j\leq k} \int_{B_R}(u+d) \nabla (u-Y)_+ \nabla K_{\zeta} u \\ &\gammaeq \int_{B_r}\epsilon \sum_{j\leq k}\frac{4}{3}f(u-Y)_+ + \int_{{\mathbb R}^n}(u+d) \chi_{\{u>Y\}}\nabla u \nabla K_{\zeta} u \\ &=\int_{B_R}\epsilon \sum_{j\leq k}\frac{4}{3}f(u-Y)_+ \\ &\quad + \frac{\epsilon}{2}\sum_{j\leq k} \mathcal{B}_{\zeta}(\chi_{\{u>Y\}} (u+d)^2, u) \\ &\gammaeq \frac{4}{3} \epsilon \sum_{j\leq k}\int_{B_R}f(\epsilon j,x)(u(\epsilon j,x)-Y(\epsilon j))_+. \\ \epsilonnd{aligned} \] Thus $(u-Y)_+ \epsilonquiv 0$. \epsilonnd{proof} \begin{lemma} \label{l:tail} Let $u$ be a solution to \epsilonqref{e:daprox} in $[0,T]\times B_R$. Assume that \begin{equation} \label{e:exponent} 0\leq u(0,x)\leq Ae^{-|x|}, f \leq Ae^{-|x|}. \epsilonnd{equation} If $A$ is large there exists constants \begin{gather*} \mu_0 , \delta_0, \zeta_0 \text{ depending only on } R,n,\sigma \\ \epsilon_0 \text{ depending only on } T,\alpha, \Lambda \\ C \text{ depending only on } n,\sigma, \| u(0,x)\|_{L^{\infty}}, \| f\|_{L^{\infty}} \epsilonnd{gather*} such that if $\mu < \mu_0, \delta < \delta_0, \zeta < \zeta_0, \epsilon<\epsilon_0$, and $Y_2(t)$ is the solution defined earlier with constant $C$ given above, then \[ 0 \leq u(\epsilon j,x) \leq AY_2(\epsilon j)e^{-|x|}, \] for any $t=\epsilon j$. \epsilonnd{lemma} \begin{proof} As before there exists $\epsilon_0$ depending only on $T, \alpha$ such that for $\epsilon \leq \epsilon_0$ we have \[ \mathcal{D}_{\epsilon}^{\alpha} Y_2(\epsilon j) \gammaeq \frac{2}{3\Lambda} D_{t}^{\alpha} Y_2(\epsilon j). \] Since $u$ is smooth and hence continuous, $u \leq LY_2(\epsilon j)e^{-|x|}$ for some $L>A$. We lower $L\gammaeq A$ until it touches $u$ for the first time. Since $u=0$ on $\partial B_R$ this cannot happen on the boundary. Since $u$ is smooth this cannot happen at a point $(\epsilon j, 0)$. Also, $LY_2 \gammaeq 2A\gammaeq 2u(0,x)$, so this cannot occur at the initial time. We label a point of touching as $(t_c,r_c)$. We compute the operator in nondivergence form and write $K_{\zeta}(u)=p$ and use the estimates in \cite{cv11} to conclude for $\epsilon$ small enough that \[ \begin{aligned} \frac{2}{3}CLY_2(t_c)e^{-r_c} &= \frac{\Lambda}{3}L D_t^{\alpha} Y_2(t_c )e^{-r_c} \\ &\leq \mathcal{D}_{\epsilon}^{\alpha} LY_2(t_c)e^{-r_c} \\ &\leq \mathcal{D}_{\epsilon}^{\alpha} u(t_c) \\ &= \delta \text{div}((u+d)\nabla u) + \text{div}((u+d) \nabla K_{\zeta} u) +f(t_c,r_c)\\ &= \delta 2[LY_2(t_c)e^{-r_c}]^2 + \delta d LY_2(t_c)e^{-r_c} \\ &\quad -LY_2(t_c)e^{-r_c} \overline{\partial_r p} +(LY_2(t_c)e^{-r_c}+d) \overline{\mathcal{D}elta p} + f(t_c,r_c), \epsilonnd{aligned} \] where in the equation the bar above means evaluation at $r_c$ Then using again the estimates from \cite{cv11}, for small enough $\zeta$ we have a universal constant $M$ depending only on $n,\sigma$ such that \[ |\overline{\partial_r p}|, |\overline{\mathcal{D}elta p}| \leq Y_1(T)M. \] Now recalling also that $LY_2(t_c)e^{-r_c}\leq Y_1(T)$ \[ \begin{aligned} \frac{2}{3}C &\leq \delta(2+d)Y_1(T) + MY_1(T) + \left(1+\frac{d}{LY_2(t_c)} e^{r_c} \right) MY_1(T) + \frac{f(r_c)e^{r_c}}{LY_2(t_c)} \\ &\leq \delta(2+d)Y_1(T) + 2MY_1{T}\left(1+\frac{d}{L} e^{r_c} \right) +\frac{A}{L}. \epsilonnd{aligned} \] Choosing $\delta,d$ small enough the above inequality implies \[ C \leq 4MY_1(T) +4. \] If we choose now $C>2MY_1(T)+4$ we obtain a contradiction. We note that $C$ will only depend on $n,\sigma, \|u(0,x) \|_{L^{\infty}},\|f \|_{L^{\infty}} $. \epsilonnd{proof} We now give some Sobolev estimates. Because we have a right hand side we choose to not use $\ln(u)$ as the test function. For $0<\gammaamma<1$, we use $(u+d)^\gammaamma-d^\gammaamma$ as a test function. The function \[ F(t)=\frac{1}{\gamma +1} (t+d)^{\gamma+1} -d^\gamma t \] will satisfy the conditions in Lemma \ref{l:discconvex}. We now assume $u$ is a solution to \epsilonqref{e:daprox} with assumptions as in Lemma \ref{l:tail}, so that $|u|\leq Me^{-|x|}$ for some large $M$. As discussed in the introduction we can extend $u(\epsilon j,x)=u(0,x)$ for $j<0$, and $u$ will be a solution to \epsilonqref{e:daprox} on $(-\infty,T)\times {\mathbb R}^n$ with right hand side \[ \delta \text{div}((u(0,x)+d)\nabla u(0,x)) + \text{div}((u(0,x)+d)\nabla (-\mathcal{D}elta)^{-\sigma}u(0,x)). \] for $j \leq 0$. This right hand side is not necessarily nonnegative; however, we only required the nonnegativity of the right hand side to guarantee that our solution is nonnegative. In this case we already know our solution is nonnegative. We fix a smooth cut-off $\phi(t)$ with $\phi(t)\gammaeq M$ for $t \leq -2$ and $\phi(t)=0$ for $t\gammaeq -1$. We now take our test function as $\epsilon F'([u(t,x)-\phi(t)]_+)$. We define \[ u= (u-\phi)_+ - (u-\phi)_- + \phi =: u_{\phi}^+ - u_{\phi}^- + \phi. \] We define $\tilde{u}=u(t)$ for $\epsilon j-1<t\leq \epsilon j$. From Lemma \ref{l:discconvex} and the estimates in Section \ref{s:discretize}, there exist two constans $c,C$ depending on $\alpha,T, \Lambda$ such that for $\epsilon <1$, \[ \begin{aligned} &\epsilon \sum_{j \leq k} F'(u_{\phi}^+(\epsilon j))\mathcal{D}_{\epsilon}^{\alpha} u(\epsilon j) \\ &\gammaeq c\int_{-\infty}^T \int_{-\infty}^t \frac{[\tilde{u}_{\phi}^+(t) - \tilde{u}_{\phi}^+(s)]^2}{(t-s)^{1+\alpha}} \\ &\quad + c\int_{-\infty}^T \frac{F(\tilde{u}_{\phi}^+(t))}{(T-t)^{\alpha}} \ dt \\ &\quad - C\int_{-\infty}^T F'(\tilde{u}_{\phi}^+(t)) D_t^{\alpha} \phi(t) \ dt \epsilonnd{aligned} \] We now consider the nonlocal spatial term. We will also use the following property: For an increasing function $V$ and a constant $l$ \[ \mathcal{B}_{\zeta}(V((u-l)_+),u) \gammaeq \mathcal{B}_{\zeta}(V((u-l)_+),(u-l)_+) \gammaeq 0. \] We have for the nonlocal spatial terms \[ \begin{aligned} &\epsilon \sum_{j\leq k} \int_{B_R} \nabla F'(u_{\phi}^+(\epsilon j,x)) (u+d) \nabla K_{\zeta} u \\ &= \epsilon \sum_{j\leq k} \int_{B_R} \nabla F'(u_{\phi}^+(\epsilon j,x)) [(u_{\phi}^+)+ d + \phi] \nabla K_{\zeta} u \\ &= \frac{\gamma}{\gamma+1}\int_{-2}^{T} \mathcal{B}_{\zeta}((\tilde{u}_{\phi}^+ +d)^{\gamma+1}, u) \\ &\quad + \int_{-2}^{T} \phi(t)\mathcal{B}_{\zeta}((\tilde{u}_{\phi}^+ +d)^{\gamma}, u) \\ &\gammaeq \frac{\gamma}{\gamma+1}\int_{-2}^{T} \mathcal{B}_{\zeta}((\tilde{u}_{\phi}^+ +d)^{\gamma+1}, u) \\ &\gammaeq \frac{\gamma}{\gamma+1}\int_{-2}^{T} \mathcal{B}_{\zeta}((\tilde{u}_{\phi}^+ +d)^{\gamma+1}, u_{\phi}^+) \epsilonnd{aligned} \] From Proposition \ref{p:gamma}, if $u_{\phi}^+(x)-u_{\phi}^+(y)\gammaeq0$, then \[ (u_{\phi}^+ +d)^{\gamma+1}(x) - (u_{\phi}^+ +d)^{\gamma+1}(y) \gammaeq (u_{\phi}^+(x) -u_{\phi}^+(y))^{\gamma+1}. \] Then \[ \epsilon \sum_{j\leq k} \int_{B_R} \nabla F'(u_{\phi}^+(\epsilon j,x)) (u+d) \nabla K_{\zeta} u \gammaeq \frac{c\gamma}{\gamma+1}\int_{0}^{T} \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} H_{\zeta}(x,y)|\tilde{u}(x)-\tilde{u}(y)|^{2+\gamma} \ dx \ dy \ dt. \] For the local spatial term we have \[ \epsilon \sum_{j\leq k} \int_{B_R} \nabla F'(u_{\phi}^+(\epsilon j,x)) (u+d) \nabla u \gammaeq \gamma \int_{0}^T \int_{B_R} (\tilde{u}+d)^{\gamma}|\nabla \tilde{u}|^2. \] Now combining the previous estimates with the right hand side term $f$ we have for a certain constant $C$ depending on $n,\sigma, \alpha, \Lambda, \gammaamma, M, T$ which can change line by line. \[ \begin{aligned} &\delta \gamma \int_{0}^T \int_{B_R} (\tilde{u}+d)^{\gamma}|\nabla \tilde{u}|^2 \\ &\quad + \frac{c\gamma}{\gamma+1}\int_{0}^{T} \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} H_{\zeta}(x,y)|\tilde{u}(x)-\tilde{u}(y)|^{2+\gamma} \ dx \ dy \ dt \\ &\quad + c\int_{B_R}\int_{0}^T \int_{0}^t \frac{[\tilde{u}(t) - \tilde{u}(s)]^2}{(t-s)^{1+\alpha}} \\ &\quad + c\int_{B_R}\int_{0}^T \frac{F(\tilde{u}(t))}{(T-t)^{\alpha}} \ dt \\ &\leq C\int_{B_R} \int_{-2}^T F'(\tilde{u}_{\phi}^+(t)) D_t^{\alpha} \phi(t) \ dt \\ &\quad + \int_{-2}^T \int_{B_R} f(t,x) F'(\tilde{u}_{\phi}^+) \\ &\leq C \int_{-2}^T \int_{B_R} (\tilde{u}_{\phi}^+ +d)^{\gamma}-d^{\gamma} \\ & \leq C \int_{-2}^T \int_{B_R} (Me^{-|x|}+d)^{\gamma}-d^{\gamma} \leq C \int_{-2}^T \int_{B_R} M^{\gamma} e^{-\gamma|x|} \leq C. \epsilonnd{aligned} \] The second to last inequality comes from Proposition \ref{p:exp}. The value $C$ is independent of $\zeta, d, R, \epsilon, \delta$ if \[ \zeta, \epsilonpsilon, \delta , d< 1 \text{ and } R>1. \] Then as $\zeta, d \to 0$ we have uniform control and obtain the estimate \begin{equation} \label{e:almost} \begin{aligned} &\delta \gamma \int_{0}^T \int_{B_R} (\tilde{u}+d)^{\gamma}|\nabla \tilde{u}|^2 + \int_{0}^T \| \tilde{u}\|_{W^{(2-2\sigma)/(2+\gamma), 2+\gamma}(B_R)}^{2+\gamma} \\ &\quad + \int_{B_R} \|\tilde{u} \|_{W^{\alpha/2,2}(0,T)}^2 \leq C \epsilonnd{aligned} \epsilonnd{equation} Notice that the constant $C$ only depends on the exponential decay of $f, u_0$ and on $\sigma, \alpha, n, T$, but not on $R,\delta$. Letting $d, \zeta \to 0$ we obtain \begin{equation} \label{e:d2aprox} \mathcal{D}_{\epsilon}^{\alpha} u - \delta \text{div}(u\nabla u) - \text{div}(u \nabla (-\mathcal{D}elta)^{-\sigma} u ) = f \text{ on } [0,T]\times B_R. \epsilonnd{equation} We now give a compactness result. \begin{lemma} \label{l:compact} Assume for any $v \in \mathcal{F}$, \begin{equation} \label{e:compact} \int_{0}^T \| v(t,x)\|_{W^{(2-2\sigma)/(2+\gamma), 2+\gamma}(B_R)}^{2+\gamma} + \int_{B_r} \|v(t,x) \|_{W^{\alpha/2,2}(0,T)}^2 \leq C. \epsilonnd{equation} Then $\mathcal{F}$ is totally bounded in $L^p([0,T]\times B_R)$ for $1\leq p\leq 2$. \epsilonnd{lemma} \begin{proof} We utilize the proof provided in \cite{dpv12} for compactness in fractional Sobolev spaces. We will show the result for $p=2$, and it will follow for $p<2$ since $B_R$ is a bounded set. We divide $T$ into $k$ increments. (This $k$ is unrelated to the number $k$ for the $\epsilon$ approximations). Let $l = T/k$. We define \[ v_l(x,t) := \frac{1}{l} \int_{jl}^{j(l+1)}v(x,s) \ ds. \] From \cite{dpv12} \begin{equation} \label{e:vl} \int_{B_R} \int_{0}^T [v_l(x,t)-v(x,t)]^2 \leq c_{\alpha} l^{\alpha} \int_{B_R} \|v(x,\cdot) \|_{W^{\alpha/2,2}(0,T)}^2 \leq Cl^{\alpha}. \epsilonnd{equation} The above estimate is uniform for any $v_j$. We now utilize that $[0,T]$ is a finite measure space as well as Minkowski's inequality: the norm of the sum is less than or equal to the sum of the norm. \[ \begin{aligned} C &\gammaeq \sum_{j=0}^{k-1} \int_{lj}^{l(j+1)} \| v(\cdot, t) \|_{W^{(2-2\sigma)/(2+\gamma), 2+\gamma}(B_R)}^{2+\gamma} \\ &\gammaeq \sum_{j=0}^{k-1} \frac{1}{l^{1+\gamma}} \left( \int_{lj}^{l(j+1)} \| v(\cdot, t) \|_{W^{(2-2\sigma)/(2+\gamma), 2+\gamma}(B_R)} \right)^{2+\gamma} \\ &\gammaeq \sum_{j=0}^{k-1} \frac{1}{l^{1+\gamma}} \left( \left\| \int_{lj}^{l(j+1)} v(x, t) \right\|_{W^{(2-2\sigma)/(2+\gamma), 2+\gamma}(B_R)} \right)^{2+\gamma} \\ &= l \sum_{j=0}^{k-1} \left( \left\| \frac{1}{l}\int_{lj}^{l(j+1)} v(x, t) \right\|_{W^{(2-2\sigma)/(2+\gamma), 2+\gamma}(B_R)} \right)^{2+\gamma} \\ \epsilonnd{aligned} \] It then follows from the result in \cite{dpv12} that for every $j$ and $\lambda>0$ there exists finitely many $\{\beta_1, \ldots, \beta_{M_j} \}$ such that for any fixed $j$ and $v\in \mathcal{F}$ there exists $\beta_i \in \{\beta_1, \ldots, \beta_{M_j} \} $ such that \[ \int_{B_R} \left|\beta_i - \frac{1}{l}\int_{lj}^{l(j+1)} v(x, t) \right|^2 \leq \lambda. \] Then combining the above estimate with \epsilonqref{e:vl} we obtain that \[ \begin{aligned} \int_{B_R} \int_{0}^T |v-\beta_{i,j}|^2 &\leq \int_{B_R} \int_{0}^T |v-v_l|^2+ \int_{B_R} \int_{0}^T |v_l-\beta_{i,j}|^2 \\ &= \int_{B_R} \int_{0}^T |v-v_l|^2+ l\sum_{j=0}^{k-1} \int_{B_R}|v_l-\beta_{i,j}|^2 \\ & Cl^{\alpha} + T\lambda. \epsilonnd{aligned} \] Since $l,\lambda$ can be chosen arbitrarily small, $\mathcal{F}$ is totally bounded. \epsilonnd{proof} The following result will guarantee that $\nabla (-\mathcal{D}elta)^{-\sigma} u \in L^p$ as $\delta \to 0, R \to \infty$. \begin{lemma} \label{l:triebel} Let $u$ be a solution to \epsilonqref{e:d2aprox} with right hand side $f$ and $u_0$ both satisfying the exponential bound \epsilonqref{e:exponent}. Then \[ \int_{0}^T \| (-\mathcal{D}elta)^{-\sigma} u(t,\cdot) \|_{W^{(2-2\sigma)/(2+\gamma) + 2\sigma, 2+\gamma}}^{2+\gammaamma} \ dt \leq C \] with the constant $C$ depending only on the exponential bounds in \epsilonqref{e:exponent}, $n, \gammaamma,T$. \epsilonnd{lemma} \begin{proof} $u$ is extended to be zero outside of $B_R$. The proof is a consequence of the following results found in \cite{t10}. \[ W^{\beta,p}({\mathbb R}^n) = B_{p,p}^{\beta}({\mathbb R}^n) = L^p({\mathbb R}^n) \cap \dot{B}_{p,p}^{\beta}. \] We also have the lifting property of the Riesz potential for the homogeneous Besov spaces \[ \| (-\mathcal{D}elta)^{-\sigma} u\|_{\dot{B}_{p,p}^{\beta+2\sigma}} \leq C \| u\|_{\dot{B}_{p,p}^{\beta}} \] To bound $u$ in the nonhomogeneous Besov space we recall \[ \| (-\mathcal{D}elta)^{-\sigma} u\|_{L^{nq/(n-2\sigma q)}({\mathbb R}^n)} \leq C \| u\|_{L^q({\mathbb R}^n)}. \] for any $1\leq q<n/(2\sigma)$. From the exponential bounds \epsilonqref{e:exponent} and growth we have that $u$ is uniformly in $L^q$ for all $1\leq q \leq \infty$. Letting \[ q = \frac{(2+\gamma)n}{n+2\sigma(2+\gamma)} > 1 \text{ for } \sigma < 1/2 \text{ and } n\gammaeq2, \] (or let $\sigma <1/4$ for $n=1$), we obtain by the finite length of $T$ \[ \int_0^T \| (-\mathcal{D}elta)^{-\sigma} u\|_{L^{2+\gamma}({\mathbb R}^n)}^{2+\gamma}\leq C. \] Using again the characterization of homogeneous besov spaces we obtain the result. \epsilonnd{proof} \begin{corollary} Let $u_k$ be a sequence of solutions to \epsilonqref{e:d2aprox} with $R \to \infty$ and $\delta \to 0$. For fixed $\rho>0$, there exists a subsequence and limit with \[ u_k \to u_0 \in L^p(B_{\rho}) \text{ for } 1\leq p\leq 2 \text{ and } u_k \rightharpoonup u_0 \in W^{(2-2\sigma)/(2+\gamma), 2+\gammaamma }. \] Furthermore, for any compactly supported $\phi$ \begin{equation} \label{e:recurse} \epsilon \sum_{j\leq k} \int_{{\mathbb R}^n} \left[ \phi(x, \epsilon j)\mathcal{D}_{\epsilon}^{\alpha} u_0(\epsilon j, x) + u_0\nabla \phi \nabla (-\mathcal{D}elta)^{-\sigma} u_0 \right] = \epsilon \sum_{j\leq k}\int_{{\mathbb R}^n}f\phi \epsilonnd{equation} \epsilonnd{corollary} \begin{proof} The strong and weak convergence is an immediate result of the bound \epsilonqref{e:almost} and Lemma \ref{l:compact}. For $\gammaamma$ small enough depending on $\sigma$, then \[ \frac{2-2\sigma}{2+\gammaamma} + 2\sigma >1. \] Then from Lemma \ref{l:triebel} we have that \[ \nabla (-\mathcal{D}elta)^{-\sigma} u_k \rightharpoonup \nabla (-\mathcal{D}elta)^{-\sigma} u_0 \in W^{(2-2\sigma)/(2+\gamma)+2\sigma-1, 2+\gammaamma }, \] And in particular \begin{equation} \label{e:pconv} \nabla (-\mathcal{D}elta)^{-\sigma} u_k \rightharpoonup \nabla (-\mathcal{D}elta)^{-\sigma} u_0 \in L^{2+\gamma}({\mathbb R}^n). \epsilonnd{equation} Then it is immediate from the weak and strong convergence that $u_0$ is a solution. \epsilonnd{proof} We now show the \begin{proof}[Proof of Theorem \ref{t:existence}] We first assume $f, u_0$ smooth and satisfying the exponential bounds \epsilonqref{e:exponent}. Consider solutions $u_{\epsilon}$ to \epsilonqref{e:recurse} over a finite interval $(0,T)$. As before, as $\epsilon \to 0$ there exists a subsequence and a limit $u_{\epsilon} \to u_0$ with the weak convergence as in \epsilonqref{e:pconv} and strong convergence over compact sets for $1\leq p \leq 2$ just as in Lemma \ref{l:compact}. Then for fixed $\phi\in C_0^{\infty}$, that $u_0$ is a solution follows from this convergence. The spatial piece and right hand side is straightforward to show, and the nonlocal time piece is taken care of as in \cite{acv15}. We now consider a sequence of solution $\{u_j\}$ with $\{f_j\}, \{(u_0)_j\} \in C^{\infty}$ with $f_j \to f$ and $ (u_0)_j \to u_0$ in weak$^*$ $L^{\infty}$. Then again there exists a limit solution $u$ with right hand side $f$. From Remark \ref{r:unique} and Lemma \ref{l:boundab} we can let $T \to \infty$. \epsilonnd{proof} \begin{remark} \label{r:finite} In this Section we have shown how the estimates in \cite{cv11} work for equations of the form \epsilonqref{e:main}. In the same way one can show that the method of ``true (exaggerated) supersolutions'' as shown in \cite{cv11} for $\sigma <1/2$ will also work to prove the property of finite propagation for solutions to \epsilonqref{e:main}. As the main result of this paper is H\"older regularity of solutions we will not make this presentation here. \epsilonnd{remark} \section{Continuity: Method and Lemmas} \label{s:lemmas} In this Section we outline the method used to prove H\"older regularity of solutions to \epsilonqref{e:main}. We follow the method used in \cite{cfv13} which is an adaption of the ideas originally used by De Giorgi. We prove a decrease in oscillation on smaller cylinders and then utilize the scaling property that if $u$ is a solution to \epsilonqref{e:main} , then $v(t,x)=A(Bt,Cx)$ is also a solution to \epsilonqref{e:main} if $A=B^{\alpha}C^{2-2\sigma}$. Because of the degenerate nature of the problem the decrease in oscillation will only occur from above. Since we do not have a decrease in oscillation from below we will need a Lemma that says in essence that if the solution $u$ is above $1/2$ on most of the space time, then $u$ is a distance from zero on a smaller cylinder. To prove the Lemmas in this section we will use energy methods, and thus we will want to use as a test function $F(u)$ for some $F$. If $u$ is a solution to \epsilonqref{e:main}, then \[ u \in W^{(2-2\sigma)/(2+\gamma),2+\gamma} \] and it is not clear that $\nabla F(u)$ will be a valid test function. We therefore prove the Lemmas for the approximate problems \begin{equation} \label{e:1delaprox} \mathcal{D}_t^{\alpha} u - \delta \text{div}(D(u) \nabla u) - \text{div}(u\nabla (-\mathcal{D}elta)^{-\sigma} u) =f \text{ on } B_R, \epsilonnd{equation} for some large $R>0$ and small $\delta>0$ with $u \epsilonquiv 0$ on $\partial B_R$. It is actually only necessary to prove the energy inequalities that we will utilize with constants uniform as $\delta \to 0$ and $R \to \infty$. We could also prove the Lemmas for the approximate problems \epsilonqref{e:d2aprox}; however, for notational convenience and to make the proofs more transparent we have chosen to let $\epsilon \to 0$. Because our solution is a limit of discretized solutions we then are allowed to make the formal computations involved with $\mathcal{D}_t^{\alpha} u$ even though $u$ may not be regular enough for $\mathcal{D}_t^{\alpha} u$ to be defined. One simply proves the energy inequalities (and hence the Lemmas) for the discretized solutions as was done in \cite{acv15}. Because of the one-sided nature of our problem we prove the Lemmas for solutions to the equation with the modified term div$(D(u)\nabla (-\mathcal{D}elta)^{-\sigma}u)$, where $D(u)=d_1u +d_2$. We assume $0\leq d_1, d_2 \leq 2$ and either $d_1=1$ or $d_2 \gammaeq 1/2$. As will be seen later, when $d_2 \gammaeq 1/2$, the proofs are simpler because the problem is no longer degenerate. We now define the exact class of solutions for which we prove the Lemmas of this section. $u$ is a solution if $u \epsilonquiv 0$ on $\partial B_r$ and for every $\phi \in C_0^{\infty}((-\infty,T)\times B_R) $, we have \begin{equation} \label{e:2delaprox} \begin{aligned} &\int_{B_R} \int_{-\infty}^T \int_{-\infty}^t [u(t,x)-u(s,x)][\phi(t,x)-\phi(s,x)]K(t,s) \ ds \ dt \ dx \\ &\quad + \int_{B_R} \int_{-\infty}^T \int_{-\infty}^{2t-T} u(t,x)\phi(t,x) K(t,s) \ ds \ dt \ dx \\ &\quad -\int_{B_R} \int_{-\infty}^T u(t,x) \mathcal{D}_t^{\alpha} \phi(t,x) \ dt \ dx \\ &\quad +\int_{-\infty}^T \int_{B_R} \nabla \phi(t,x) D(u)\nabla u(t,x) \ dx \ dt \\ &\quad +\int_{-\infty}^T \int_{B_R} \nabla \phi(t,x) D(u) \nabla (-\mathcal{D}elta)^{-\sigma} u \ dx \ dt\\ &= \int_{-\infty}^T \int_{B_R} f(t,x) \phi(t,x), \epsilonnd{aligned} \epsilonnd{equation} By Lemmas \ref{l:compact} and \ref{l:triebel}, the Lemmas stated in this section will be true when $R \to \infty$ and $\delta \to 0$. Before stating the Lemmas we define the following function for small $0<\tau <1/4$. \[ \overline{\Psi}(x,t) := 1+ (|x|^{\tau} -2)_+ + (|t|^{\tau} -2)_+. \] We now state the Lemmas we will need. \begin{lemma} \label{l:pullup} Let $u$ be a solution to \epsilonqref{e:2delaprox} with $R>4$ and assume \[ 1-\overline{\Psi}\leq u \leq \overline{\Psi} \text{ for } \tau < \tau_0 \] Given $\mu_0 \in (0,1/2)$ and $\tau_0<1/4$, there exists $\kappa>0$ depending on $\mu_0, \tau_0, \sigma, \alpha, n$ such that if \[ |\{u \gammaeq 1/2\} \cap \Gamma_4| \gammaeq (1-\kappa) |\Gamma_4| \] then $u \gammaeq \mu_0$ on the smaller cylinder $\Gamma_1$. \epsilonnd{lemma} We have a similar Lemma from above \begin{lemma} \label{l:pulldown} Under the same assumptions as Lemma \ref{l:pullup}, given $\mu_1 \in (0,1/2)$ and $\tau_0<1/4$, there exists $\kappa>0$ depending on $\mu_1, \tau_0, \sigma, \alpha, n$ such that if \[ |\{u > 1/2\} \cap \Gamma_2| \leq \kappa |\Gamma_2| \] then $u \leq 1-\mu_1$ on the smaller cylinder $\Gamma_1$. \epsilonnd{lemma} Lemma \ref{l:pulldown} is not sufficient. We need the stronger \begin{lemma} \label{l:2down} Under the same assumptions as Lemma \ref{l:pullup}, assume further for fixed $k_0$ \begin{equation} \label{e:2down} |\{u<1/2\}\cap \Gamma_4| \gammaeq (1-\kappa_0)|\Gamma_4|, \epsilonnd{equation} then $u \leq 1-\mu_2$ on $\Gamma_1$ for some $\mu_2$ depending on $\kappa_0$. \epsilonnd{lemma} We will choose $\kappa_0$ to equal the $\kappa$ in Lemma \ref{l:pullup}. \section{Pull-up} \label{s:pullup} In this section we provide the proof of Lemma \ref{l:pullup}. This Lemma is the most technical to prove. We first prove the Lemma in the most difficult case when $D(u)=u+d$ with $0\leq d \leq 2$. Afterwards, we show how the proof is much simpler when $D(u)=d_1 u +d_2$ with $d_2 \gammaeq 1/2$ and $0\leq d_1 \leq 2$. We will need the following technical Lemma. The proof is found in the appendix. \begin{lemma} \label{l:control} Let $u,\phi$ be two functions such that $0\leq u \leq \phi \leq 1$. Let $0<\gammaamma<1$ be a constant. If $|u(x)-u(y)|\gammaeq 4|\phi(x)-\phi(y)|$, then \begin{equation} \label{e:control1} \frac{2}{5}\left(\frac{4}{5} \right)^{\gamma} |u_{\phi}^-(x)-u_{\phi}^-(y)|^{1+\gamma} \leq \left| \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \right| \leq \frac{14}{3} |u_{\phi}^-(x)-u_{\phi}^-(y)|. \epsilonnd{equation} Also, if \[ 0 \leq \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \] then $0\leq u_{\phi}^-(x)-u_{\phi}^-(y)$. If instead we assume $|u(x)-u(y)|\leq 4|\phi(x)-\phi(y)|$, then \begin{equation} \label{e:control2} \left| \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \right| \leq 14 |\phi(x)-\phi(y)|. \epsilonnd{equation} \epsilonnd{lemma} \begin{remark} \label{r:2} When $0\leq u \leq \phi \leq 3$, Lemma \ref{l:control} will hold with new constants by applying the Lemma to $u/3,\phi/3 $. \epsilonnd{remark} We will use a sequence of cut-off functions $\{\phi_k\}$ which will be chosen to be smooth cut-off functions in space, and smooth increasing cut-off functions in time. We recall that for small $0<\tau <1/2$, \[ \overline{\Psi}(x,t) := 1+ (|x|^{\tau} -2)_+ + (|t|^{\tau} -2)_+. \] We now recall the construction of a sequence of smooth radial cut-offs $\theta_k$ from \cite{cfv13} that satisfy \begin{itemize} \item $\theta_k(x)\leq \theta_{k-1}(x)\leq \ldots \leq \theta_0(x) $, \item $|\nabla \theta_k|/\theta_k \leq C^k \theta_k^{-1/m}$, with $m \gammaeq 2$ \item $\theta_{k-1}-\theta_k\gammaeq (1-\mu_0)2^{-k}$ in the support of $\theta_k$, \item $\theta_k \to \mu_0 \chi_{B_2}$ as $k \to \infty$, \item the support of $\theta_k$ is contained in the set where $\theta_{k-1}$ achieves its maximum. \epsilonnd{itemize} We also have $\theta_0 \epsilonquiv 1$ on $B_3$ and the support of $\theta_0$ is contained in $B_4$. As a cut-off in time we consider a sequence $\{\xi_k\}$ satisfying \begin{itemize} \item $\xi_k(t) \leq \xi_{k-1}(t)$, \item $\xi_k'(t) \leq C^k $, \item $\xi_k \to \chi_{\{t>-2\}}$ as $k \to \infty$, \item $\xi_k =\max \xi_k = 1$ on the interval $[-2-2^{-k},0]$. \item the support of $\xi_k$ is contained in the set where $\xi_{k-1}$ achieves its maximum. \epsilonnd{itemize} We now define \[ \phi_k(x,t) := 1-\overline{\Psi}(x,t) + \frac{1}{2}\xi_k(t)\theta_k(x). \] We use the convention for negative part that $u=u_+ - u_-$. We also write $u_{\phi_k}^- := (u-\phi_k)_-$. We now consider the convex function \begin{equation} \label{e:ftest} F(x):= \frac{1}{\gamma + 1}(1-x)^{\gamma +1} +x -\frac{1}{\gamma +1}. \epsilonnd{equation} Because of the degenerate nature of our equation we will want to utilize the test function \begin{equation} \label{e:test} -F'(u_{\phi}^-/(\phi+d))= \left(1- \frac{(u-\phi)_-}{\phi+d} \right)^{\gamma} -1 = -\left[\left(\frac{u+d}{\phi+d}\right)^{\gamma}-1\right]_-. \epsilonnd{equation} \begin{proof}[Proof of Lemma \ref{l:pullup}] \textbf{First Step: Obtaining an energy in time.} We note that for $0\leq x \leq 1$, $F(x)$ is convex,$F'(x)\gammaeq 0$, and $F''(x)\gammaeq \gamma$. From the convexity and second derivative estimate we also conclude for $0\leq x,y\leq 1$ \begin{gather} \label{e:convexity} F'(x)(x-y) \gammaeq F(x)-F(y) + (\gamma/2) (x-y)^2 \\ \label{e:xsquared}F(x) \approx x^2 \epsilonnd{gather} We now consider $- F'(u_{\phi_k}^-/(\phi_k+d)) \mathcal{D}_t^{\alpha} u$, and rewrite $u = u_{\phi_k}^+ - u_{\phi_k}^- + \phi_k$. To obtain an energy in time we first consider \[ \begin{aligned} &\int_{-\infty}^0 F' \left(\frac{u_{\phi}^-(t)}{\phi(t)+d} \right) \mathcal{D}_t^{\alpha} u_{\phi}^- (t) \\ &= \int_{-\infty}^0 \int_{-\infty}^0 F' \left(\frac{u_{\phi}^-(t)}{\phi(t)+d} \right) \left[u_{\phi}^-(t)-u_{\phi}^-(s)\right] K(t,s) \\ &= \int_{-\infty}^0 \int_{-\infty}^0(\phi(t)+d) F' \left(\frac{u_{\phi}^-(t) }{\phi(t)+d} \right) \frac{u_{\phi}^-(t)-u_{\phi}^-(s)}{\phi(t)+d} K(t,s) \\ &\gammaeq \int_{-\infty}^0 \int_{-\infty}^0 (\phi(t)+d)F' \left(\frac{u_{\phi}^-(t)}{\phi(t)+d} \right) \left[ \frac{u_{\phi}^-(t)}{\phi(t)+d}-\frac{u_{\phi}^-(s)}{\phi(s)+d} \right]K(t,s) \\ &\gammaeq \int_{-\infty}^0 \int_{-\infty}^0 (\phi(t)+d)\left[F\left(\frac{u_{\phi}^-(t)}{\phi(t)+d} \right) - F\left(\frac{u_{\phi}^-(s)}{\phi(s)+d} \right)\right] K(t,s) \\ & \ + \int_{-\infty}^0 \int_{-\infty}^0 (\phi(t)+d) \frac{\gammaamma}{2} \left[\frac{u_{\phi}^- (t)}{\phi(t)+d} - \frac{u_{\phi}^-(s)}{\phi(s)+d} \right]^2 K(t,s)\\ &\gammaeq \int_{-\infty}^0 \int_{-\infty}^0 (\phi(t)+d)\left[F\left(\frac{u_{\phi}^-(t)}{\phi(t)+d} \right) - F\left(\frac{u_{\phi}^-(s)}{\phi(s)+d} \right)\right] K(t,s) \\ & \ + \int_{-2-2^{-k}}^0 \int_{-2-2^{-k}}^0 \frac{\gammaamma}{2} \left[u_{\phi}^- (t) - u_{\phi}^-(s)\right]^2 K(t,s)\\ &= (1)+(2). \epsilonnd{aligned} \] In the first inequality we used that $\phi$ is increasing in $t$ and positive for $t\gammaeq -4$ as well as $u_{\phi}^-(s)=0$ for $s\leq -4$, and in the second inequality we used \epsilonqref{e:convexity}. Term $(2)$ is half of what we will need for the Sobolev embedding (see Lemma \ref{l:ext}). To gain the other half we consider term $(1)$. For $c,C^k$ depending on $\Lambda,\alpha$ and the Lipschitz constant of $\phi_k$ we have \[ \begin{aligned} & \int_{-\infty}^0 \int_{-\infty}^0(\phi_k(t)+d)\left[F\left(\frac{u_{\phi_k}^-(t)}{\phi_k(t)+d} \right) - F\left(\frac{u_{\phi_k}^-(s)}{\phi_k(s)+d} \right)\right] K(t,s) \\ &= \int_{-\infty}^0 \int_{-\infty}^0 \left[(\phi_k(t)+d)F\left(\frac{u_{\phi_k}^-(t)}{\phi_k(t)+d} \right) - (\phi_k(s)+d)F\left(\frac{u_{\phi_k}^-(s)}{\phi_k(s)+d} \right)\right] K(t,s) \\ &\quad + \int_{-\infty}^0 \int_{-\infty}^0 [\phi_k(s)-\phi_k(t)]F\left(\frac{u_{\phi_k}^-(s)}{\phi_k(t)+d} \right) K(t,s) \\ &\gammaeq c \int_{-\infty}^0 (\phi(t)+d) F\left(\frac{u_{\phi_k}^-(t)}{\phi_k(t)+d} \right) \frac{1}{(0-t)^{\alpha}} \\ &\quad - C^k \int_{-\infty}^0 \chi_{\{u(t)<\phi_k(t)\}} \ dt \\ &\gammaeq c \int_{-2-2^{-k}}^0 \frac{(u_{\phi_k}^-)^2(t)}{(\phi_k(t)+d)(0-t)^{\alpha}} \\ &\quad - C^k \int_{-\infty}^T \chi_{\{u(t)<\phi_k(t)\}} \ dt \\ \epsilonnd{aligned} \] The last inequality coming from \epsilonqref{e:xsquared}. Now \[ -\int_{-\infty}^0 F' \left(\frac{u_{\phi}^-(t)}{\phi(t)+d} \right) \mathcal{D}_t^{\alpha} u_{\phi}^+ (t) \gammaeq 0, \] and in this proof we ignore this term which will be on the left hand side. Now for the term involving $\phi_k$ we have \[ -\int_{-\infty}^0 F' \left(\frac{u_{\phi_k}^-(t)}{\phi_k(t)+d} \right) \mathcal{D}_t^{\alpha} \phi_k (t) \gammaeq -C^k \int_{-\infty}^0 \chi_{\{u<\phi_k\}}. \] Then utilizing the embedding theorem for fractional Sobolev spaces \cite{dpv12} combined with the above inequalities we obtain \begin{equation} \label{e:timeenergy} \begin{aligned} \int_{-\infty}^0 F'\left(\frac{u_{\phi_k}^-}{\phi_k+d} \right) \mathcal{D}_t^{\alpha}u(t) &\gammaeq c \int_{-2-2^{-k}}^0 \frac{(u_{\phi_k}^-)^2(t)}{(0-t)^{\alpha}} \\ & \quad + c \int_{-2-2^{-k}}^0 \int_{-2-2^{-k}}^t \frac{\left[u_{\phi_k}^-(t) - u_{\phi_k}^-(s)\right]^2}{(t-s)^{1+\alpha}} \\ & \quad - C^k \int_{-\infty}^0 \chi_{\{u<\phi_k\}} \ dt \\ &\gammaeq c \left( \int_{-2-2^{-k}}^0 \left(u_{\phi_k}^-\right)^{\frac{2}{1-\alpha}}\right)^{1-\alpha} - C^k \int_{-\infty}^0 \chi_{\{u<\phi_k\}} \ dt. \epsilonnd{aligned} \epsilonnd{equation} After integrating in the spatial variable we have \[ \begin{aligned} &c\int_{{\mathbb R}^n} \left( \int_{-2-2^{-k}}^T (u-\phi_k)_-^{\frac{2}{1-\alpha}}\right)^{1-\alpha} - \int_{-\infty}^0 \int_{{\mathbb R}^n} (u+d) \nabla F'\left(\frac{u_{\phi_k}^-}{\phi_k+d} \right) \nabla (-\mathcal{D}elta)^{-\sigma} u \\ &\quad - \int_{-\infty}^0 \int_{{\mathbb R}^n} (u+d) \nabla F'\left(\frac{u_{\phi_k}^-}{\phi_k+d} \right) \nabla u \\ &\leq -\int_{-\infty}^0 \int_{{\mathbb R}^n} f F'\left(\frac{u_{\phi_k}^-}{\phi_k+d} \right) + C^k\int_{{\mathbb R}^n} \int_{-\infty}^0 \chi_{\{u<\phi_k\}} \ dt \ dx \\ &\leq C^k\int_{-\infty}^0 \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \ dt \ dx \\ \epsilonnd{aligned} \] \textbf{Second Step: Obtaining an energy in space.} We now turn our attention to the elliptic portion of the problem. We recall from \cite{cfv13} the identity \[ \mathcal{B}(v,w) = \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \frac{[v(x)-v(y)][w(x)-w(y)]}{|x-y|^{n+2-2\sigma}} dx dy = c_{n,\sigma} \int_{{\mathbb R}^n} \nabla v \nabla (-\mathcal{D}elta)^{-\sigma}u . \] We multiply by our test function \epsilonqref{e:test} and integrate by parts. On the left hand side of the equation we have \[ \begin{aligned} &\chi_{\{u<\phi\}}\nabla\left[\left(\frac{u+d}{\phi+d}\right)^{\gamma} -1\right] (u+d) \nabla (-\mathcal{D}elta)^{-\sigma}u \\ &= \chi_{\{u<\phi\}}\gamma \left(\frac{u+d}{\phi+d}\right)^{\gamma-1} \nabla((u+d)/(\phi+d)) (u+d) \nabla (-\mathcal{D}elta)^{-\sigma} u \\ &= \chi_{\{u<\phi\}}\frac{\gamma}{\gamma+1} \nabla \left[\left(\frac{u+d}{\phi+d}\right)^{\gamma+1} -1\right](\phi+d) \nabla (-\mathcal{D}elta)^{-\sigma} u \\ &= \chi_{\{u<\phi\}}\frac{\gamma}{\gamma+1} \nabla \left[\frac{(u+d)^{\gamma+1}}{(\phi+d)^{\gamma}}-(\phi+d) \right] \nabla (-\mathcal{D}elta)^{-\sigma} u \\ &\qquad -\frac{\gamma}{\gamma+1} \chi_{\{u<\phi\}} \left[\left({\frac{u+d}{\phi+d}}\right)^{\gamma+1}-1\right]\nabla \phi \nabla (-\mathcal{D}elta)^{-\sigma} u\\ &:= (1)+(2). \epsilonnd{aligned} \] We now focus on $(1)$ which will give us the energy term we need. For the term $(-\mathcal{D}elta)^{-\sigma} u$, we rewrite $u=(u-\phi_k)_+ - (u-\phi_k)_- + \phi_k := u_{\phi_k}^+ - u_{\phi_k}^- + \phi_k$. Then we rewrite $(1)=(1a)+(1b)+(1c)$. We focus on the term $(1b)$. We rewrite \[ \begin{aligned} (1b)&=(1bi)+(1bii) \\ &:= -\chi_{\{u<\phi\}}\frac{\gamma}{\gamma+1} \nabla \left[\frac{(u+d)^{\gamma+1}}{(\phi+d)^{\gamma}} \right] \nabla (-\mathcal{D}elta)^{-\sigma} u_{\phi}^- \\ &\quad + \chi_{\{u<\phi\}}\frac{\gamma}{\gamma+1} \nabla \phi \nabla (-\mathcal{D}elta)^{-\sigma} u_{\phi}^-. \epsilonnd{aligned} \] The term $(1bi)$ will give us the energy term in space that we will need. \[ \begin{aligned} (1bi)&=\int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \frac{\gamma}{\gamma+1} \nabla \left[\frac{(u+d)^{\gamma+1}}{(\phi+d)^{\gamma}}(x) \right] \frac{1}{|x-y|^{n-2\sigma}} \nabla u_{\phi}^-(y) dx dy \\ &= c_{n,\sigma}\frac{\gamma}{\gamma+1} \mathcal{B}(\chi_{\{u<\phi\}}(u+d)^{\gamma+1} / (\phi+d)^{\gamma}, -u_{\phi}^-). \epsilonnd{aligned} \] We define the set \[ A_k:= \{|u(x)-u(y)| \gammaeq 4 |\phi_k(x)-\phi_k(y)|\}. \] It is clear that $A_k$ contains the set $V_k \times V_k$ where we define $V_k$ as the set on which $\theta_k$ achieves its maximum. From Lemma \ref{l:control} and Remark \ref{r:2} we have \[ \iint\limits_{A_k} \left[ \frac{(u+d)^{\gamma+1}(y)}{(\phi+d)^{\gamma}(y)} - \frac{(u+d)^{\gamma+1}(x)}{(\phi+d)^{\gamma}(x)}\right] \frac{[u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]}{|x-y|^{n+2-2\sigma}} \gammaeq c \iint\limits_{A_k} \frac{|u_{\phi_k}^-(x)-u_{\phi_k}^-(y)|^{2+\gamma}}{|x-y|^{n+2-2\sigma}} \] We now label $U_k$ as the set where $\phi_k$ achieves its maximum. Notice that $U_k = [-2-2^{-K}, 0] \times V_k$. To utilize the fractional Sobolev embedding on $V_k \times V_k$, we also will need an $L^p$ norm of $u_{\phi_k}^-$ on $V_k$. We utilize half of the integral of $u_{\phi_k}^-$ that we gained from the fractional time term: \[ \begin{aligned} & \int_{{\mathbb R}^n} \int_{-2-2^{-k}}^0 \frac{(u_{\phi_k}^-)^2(t)}{(0-t)^{\alpha}} \\ &\gammaeq \frac{1}{2} \iint\limits_{U_k} \frac{(u_{\phi_k}^-)^2(t)}{(0-t)^{\alpha}} + \frac{1}{2} \iint\limits_{U_k} \frac{(u_{\phi_k}^-)^{2+\gamma}(t)}{(0-t)^{\alpha}}. \epsilonnd{aligned} \] The inequality comes from the fact that $0\leq u_{\phi}^-\leq 1$. Now from the fractional sobolev embedding \cite{dpv12}, \begin{equation} \label{e:leftene} \begin{aligned} & \int_{-2-2^{-k}}^T \iint\limits_{V_k \times V_k} \frac{|u_{\phi_k}^-(x)-u_{\phi_k}^-(y)|^{2+\gamma}}{|x-y|^{n+2-2\sigma}} + \frac{1}{2} \iint\limits_{U_k} (u-\phi_k)_-^{2+\gamma}(t) \\ &\gammaeq c_{n,\sigma, \gammaamma} \int_{-2-2^{-k}}^T \left( \int_{V_k} (u_{\phi_k}^-)^{n(2+\gamma)/(n-2+2\sigma)} \right)^{(n-2+2\sigma)/n} \epsilonnd{aligned} \epsilonnd{equation} This is the helpful spatial term on the left hand side that we will return to later. \textbf{Third Step: Bounding the remaining terms.} We will now show that everything left in our equation can be bounded by \begin{equation} \label{e:rhs} C^k \int_{-\infty}^T\int_{{\mathbb R}^n} \chi_{\{u < \phi_k\}} \ dx. \epsilonnd{equation} We will denote \[ X_k(x,y):= \chi_{\{u(x)<\phi_k(x)\}} + \chi_{\{u(y)<\phi_k(y)\}} \] For the remainder of term $(1bi)$ we have \[ \begin{aligned} &\left |\iint\limits_{A_k^c} \left[ \frac{(u+d)^{\gamma+1}(y)}{(\phi+d)^{\gamma}(y)} - \frac{(u+d)^{\gamma+1}(x)}{(\phi+d)^{\gamma}(x)}\right] \frac{[u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]}{|x-y|^{n+2-2\sigma}} \right| \\ &\leq C \iint\limits_{A_k^c} X_k(x,y)\frac{|\phi_k(x)-\phi_k(y)|^{2}}{|x-y|^{n+2-2\sigma}} \\ &\leq C^k \int_{{\mathbb R}^n} \chi_{\{u <\phi_k\}}. \epsilonnd{aligned} \] The last inequality is due to the Lipschitz constant of $\phi_k$ when $x,y$ are close, and the tail growth of $\phi_k$ when $x,y$ are far apart. We now control the term $(1bii)$. Again, we split the region of integration over $A_k$ and $A_k^c$. Using H\"older's inequality (provided $2\sigma>\gamma/(1+\gamma)$ and therefore we must choose $\gammaamma$ small when $\sigma$ is small) as well as the Lipschitz and $\sup$ bounds on $\phi_k$ we have \[ \begin{aligned} (1bii) &= c_{n,\sigma}\iint\limits_{A_k} \frac{[\phi_k(x)-\phi_k(y)][u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]}{|x-y|^{n+2-2\sigma}} dx dy. \\ &\leq \epsilonta \iint\limits_{A_k} \frac{[u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]^{2+\gamma}}{|x-y|^{n+2-2\sigma}} dx dy. \\ &\quad + C \iint\limits_{A_k} \frac{[\phi_k(x)-\phi_k(y)]^{(2+\gamma)/(1+\gamma)}}{|x-y|^{n+2-2\sigma}} X_k(x,y)dx dy. \\ &\leq \epsilonta \iint\limits_{A_k} \frac{[u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]^{2+\gamma}}{|x-y|^{n+2-2\sigma}} dx dy. \\ &\quad + C^k \int_{{\mathbb R}^n}\chi_{\{u<\phi_k\}} \ dx. \epsilonnd{aligned} \] The first term is absorbed into the left hand side and the second term is controlled exactly as before. We now consider the integration over $A_k^c$. \[ \begin{aligned} (1bii) &= \iint\limits_{A_k} \frac{[\phi_k(x)-\phi_k(y)][u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]}{|x-y|^{n+2-2\sigma}} dx dy. \\ &\leq \iint\limits_{A_k} \frac{[\phi_k(x)-\phi_k(y)]^{2}}{|x-y|^{n+2-2\sigma}} X_k(x,y)dx dy. \\ &\leq C^k \int_{{\mathbb R}^n}\chi_{\{u<\phi_k\}} \ dx. \epsilonnd{aligned} \] We now turn our attention to the term $(1c)$. By Lemma \ref{l:control} we have \[ \begin{aligned} &\left| \mathcal{B}(\chi_{\{u<\phi\}} ((u+d)^{\gamma+1}/(\phi_k+d)^{\gamma}-\phi_k),\phi_k) \right| \\ &\leq \left| \mathcal{B}(\chi_{\{u<\phi\}} (u+d)^{\gamma+1}/(\phi_k+d)^{\gamma},\phi_k) \right| + \left| \mathcal{B}(\chi_{\{u<\phi\}} \phi_k),\phi_k) \right| \\ \epsilonnd{aligned} \] Both of the above terms are handled exactly as before by using Lemma \ref{l:control} and splitting the region of integration over $A_k$ and $A_k^c$. The term $(1a)$ is \begin{equation} \label{e:posi} \begin{aligned} (1a)&=\mathcal{B}(\chi_{\{u<\phi_k\}} u^{\gamma+1}/\phi_k^{\gamma}-(\phi_k+d),u_{\phi_k}^+) \\ &= 2\int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \chi_{\{u(x)<\phi_k(x)\}}\left[\phi_k(x)+d-\frac{(u+d)^{\gamma+1}(x)}{(\phi_k+d)^{\gamma}(x)} \right] \frac{u_{\phi_k}^+(y)}{|x-y|^{n+2-2\sigma}} \ dx \ dy \gammaeq 0 \epsilonnd{aligned} \epsilonnd{equation} The factor of $2$ comes form the symmetry of the kernel. We will utilize this nonnegative term shortly. We now consider the term $(2)$ which we recall as \[ -\int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{\gamma}{\gamma+1} \chi_{\{u<\phi_k\}} (((u+d)/(\phi_k+d))^{\gamma+1}-1)\nabla \phi_k \nabla L(x-y)[u(y)-u(x)] \ dx \ dy. \] In the above $L = \nabla (-\mathcal{D}elta)^{-\sigma}$ and we have \[ \nabla L(x-y) \approx |x-y|^{-(n+1-2\sigma)}. \] We again write $u=u_{\phi_k}^+ -u_{\phi_k}^- + \phi_k$. To control the term involving $\phi_k$ we integrate over the two sets $\{|x-y|\leq 8\}$ and $\{|x-y|>8\}$. We use that $|\phi_k(x)-\phi_k(y)|\leq C^k|x-y|$ when $|x-y|\leq 8$ and $|\phi_k(x)-\phi_k(y)|\leq |x-y|^\tau$ when $|x-y|>8$ as well as the bound $|\nabla \phi_k|\leq C^k$ to obtain \[ \begin{aligned} &\left| \int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{\gamma}{\gamma+1} \chi_{\{u<\phi_k\}} \left(\left(\frac{u+d}{\phi_k+d}\right)^{\gamma+1}-1\right) \nabla \phi_k \nabla L(x-y)[\phi_k(y)-\phi_k(x)] \ dx \ dy \right| \\ &\leq C^k\iint\limits_{|x-y|\leq 8} \chi_{\{u<\phi_k\}} |x-y|^{-(n-2\sigma)} \ dx \ dy \\ & \quad + C^k \iint\limits_{|x-y| > 8} \chi_{\{u<\phi_k\}} |x-y|^{-(n+1-2\sigma-\tau)}\ dx \ dy \\ &\leq C^k \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \ dx. \epsilonnd{aligned} \] We now use the same set decomposition with $-u_{\phi_k}^-$, the inequality $|u_{\phi_k}^-|\leq 1$ as well as H\"older's inequality \[ \begin{aligned} &\left| \int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{\gamma}{\gamma+1} \chi_{\{u<\phi_k\}} \left(\left(\frac{u+d}{\phi_k+d}\right)^{\gamma+1}-1\right)\nabla \phi_k \nabla L(x-y)[u_{\phi_k}^-(y)- u_{\phi_k}^-(x)] \ dx \ dy \right| \\ &\leq C^k \iint\limits_{|x-y|\leq 8} \chi_{\{u<\phi_k\}} |x-y|^{-(n-2\sigma+\gamma/(1+\gamma))} \ dx \ dy \\ &\quad + \zeta \iint\limits_{|x-y|\leq 8} \frac{[u_{\phi_k}^-(y)-u_{\phi_k}^-(x)]^{2+\gamma}}{|x-y|^{n+2-2\sigma}} \ dx \ dy \\ & C^k \quad + \iint\limits_{|x-y| > 8} \chi_{\{u<\phi_k\}} |x-y|^{-(n+1-2\sigma-\tau)}\ dx \ dy \\ \epsilonnd{aligned} \] The third term is bounded by \[ C \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \ dx \] provided $\tau <1-2\sigma$ as well as the first term provided again that $2\sigma > \gamma/(1+\gamma)$. The second term can be bounded as before by splitting the region of integration over $A_k$ and $A_k^c$ and absorbing the region over $A_k$ into the left hand side. We now turn our attention to the last term involving $u_{\phi_k}^+$. We first remark that the integral becomes \[ \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \chi_{\{u(x)<\phi_k(x)\}} (((u+d)/(\phi_k+d))^{\gamma+1}-1)\nabla \phi_k \nabla L(x-y)u_{\phi_k}^+(y) \ dx \ dy. \] We first consider the set $|x-y|>8$. Since $u_{\phi_k}^+ \leq \overline{\Psi}$, \[ \begin{aligned} &\left|\quad \iint\limits_{|x-y|> 8} \chi_{\{u(x)<\phi_k(x)\}} (((u+d)/(\phi_k+d))^{\gamma+1}-1)\nabla \phi_k \nabla L(x-y)u_{\phi_k}^+(y) \ dx \ dy \right| \\ &\leq C^k \left| \quad \iint\limits_{|x-y|> 8} \chi_{\{u(x)<\phi_k(x)\}} |x-y|^{-(n+1-2\sigma+\tau)} \ dx \ dy \right| \\ &\leq C^k \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \ dx. \epsilonnd{aligned} \] When $|x-y|<8$, we make the further decomposition \[ \frac{|\nabla \phi_k(x)|}{\phi_k(x)} |x-y| \leq \epsilonta \] to absorb the integral by the nonnegative quantity \epsilonqref{e:posi}. In the complement when \[ \frac{|\nabla \phi_k(x)|}{\phi_k(x)} |x-y| > \epsilonta \] we use $\phi_k^{-1/m}C^k \gammaeq |\nabla \phi_k|/\phi_k$ and integrate in $y$ \[ \left| \int_{B_8} \nabla L(x-y)u_{\phi_k}^+(y) dy \right|\leq \int_{\epsilonta \phi_k^{1/m}C^{-k}}^8 \frac{r^{n-1}}{r^{n+1-2\sigma}} \leq \max\{C, (\epsilonta C^k)^{2\sigma -1} \phi_k^{(2\sigma-1)/m}\}. \] The remainder of the terms are bounded by $|\nabla \phi_k|\leq C^k \phi_k^{1-1/m}$ By multiplying by the term $\chi_{\{u<\phi\}}$ and integrating, we end up in the worst case with \[ C^k \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \phi_k^{1-1/m +(2\sigma -1)/m} \ dx \leq C^k \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \ dx, \] since $m \gammaeq 2$. The last term to consider is the local spatial term. We use Cauchy-Schwarz \[ \begin{aligned} &\delta\int_{B_R} \chi_{\{u<\phi_k\}}\nabla \left[\left(\frac{u+d}{\phi_k+d} \right)^{\gamma}-1 \right] (u+d)\nabla u\\ &= \delta \int_{B_R} \gamma \left( \frac{u+d}{\phi_k+d} \right)^{\gamma} \frac{|\nabla u|^2}{\phi_k+d} \\ &\quad - \delta \int_{B_R} \gamma \left( \frac{u+d}{\phi_k+d} \right)^{\gamma+1} \nabla u \nabla \phi_k \\ &\gammaeq (1-\epsilonta) \delta \gamma \int_{B_R} \left( \frac{u+d}{\phi_k+d} \right)^{\gamma} |\nabla u|^2 \\ &- C\delta\int_{B_R}\left( \frac{u+d}{\phi_k+d} \right)^{\gamma+2} |\nabla \phi_k|^2 \chi_{\{u < \phi_k\}} \\ &\gammaeq (1-\epsilonta) \delta \gamma \int_{B_R} \left( \frac{u+d}{\phi_k+d} \right)^{\gamma} |\nabla u|^2 - C^k\delta\int_{B_R} \chi_{\{u < \phi_k\}} \ dx \\ \epsilonnd{aligned} \] Retaining the energy from \epsilonqref{e:leftene} on the left hand side and moving everything else to the right hand side which is bounded by \epsilonqref{e:rhs}, our energy inequality becomes \begin{equation} \label{e:energy1} \begin{aligned} & c \int_{V_k} \left( \int_{-2-2^{-k}}^T (u-\phi)_-^{\frac{2}{1-\alpha}}\right)^{1-\alpha} \\ &\quad + c \int_{-2-2^{-k}}^T \left( \int_{V_k} (u_{\phi_k}^-)^{n(2+\gamma)/(n-2+2\sigma)} \right)^{(n-2+2\sigma)/n} \\ &\leq C^k\int_{{\mathbb R}^n} \int_{-\infty}^T \chi_{\{u<\phi_k\}} \leq C^k\iint\limits_{U_{k-1}} \chi_{\{u<\phi_k\}} \epsilonnd{aligned} \epsilonnd{equation} \textbf{Fourth Step: The nonlinear recursion relation.} We now (as in \cite{acv15}) use H\"older's inequality twice with the relations \[ \frac{\beta}{p_1} + \frac{1-\beta}{p_2}= \frac{1}{p}= \frac{\beta}{p_3}+ \frac{1-\beta}{p_4}. \] for a function $v$ to obtain \[ \begin{aligned} \int \int v^p & \leq \int \int v^{p \beta} v^{p (1-\beta)} \\ & \leq \int \left(\int v^{p_1} \right)^{p\beta/p_1} \left(\int v^{p_2} \right)^{p(1-\beta)/p_2} \\ & \leq \left( \int \left(\int v^{p_1} \right)^{p_3/p_1}\right)^{\beta p / p_3} \left( \int \left(\int v^{p_2} \right)^{p_4/p_2}\right)^{(1-\beta) p / p_4} \epsilonnd{aligned} \] We now choose \[ p_1 =2, \quad p_2 = \frac{n(2+\gamma)}{n-(2-2\sigma)}, \quad p_3 = \frac{2}{1-\alpha}, \quad p_4 = 2+\gamma, \] so that if $r=2-2\sigma$ \[ p = 2\frac{r+\alpha n (2+\gamma)/2}{(1-\alpha)r + \alpha n}\quad \text{and}\quad \beta = \frac{r}{r+\alpha n (2+\gamma)/2}. \] We now use H\"older's inequality one more time to obtain \[ \begin{aligned} \left( \int \int v^p \right)^{b/p} &\leq \left( \int \left(\int v^{p_1} \right)^{p_3/p_1}\right)^{\beta b / p_3} \left( \int \left(\int v^{p_2} \right)^{p_4/p_2}\right)^{(1-\beta) b / p_4} \\ &\leq \frac{1}{\omega}\left( \int \left(\int v^{p_1} \right)^{p_3/p_1}\right)^{\beta b \omega / p_3} + \frac{\omega-1}{\omega}\left( \int \left(\int v^{p_2} \right)^{p_4/p_2}\right)^{\frac{(1-\beta) b \omega}{ p_4(\omega-1)}} \epsilonnd{aligned} \] We choose \[ \omega = \frac{(2+\gamma \beta)}{(2+\gamma)\beta} \quad \text{and} \quad b = \frac{2(2+\gamma)}{2+\gamma \beta} < p \] so that \begin{equation} \label{e:minkowski} \begin{aligned} & \left( \int \int v^p \right)^{b/p} \\ &\leq \frac{1}{\omega}\left( \int \left(\int v^{2} \right)^{1/(1-\alpha)}\right)^{1-\alpha} + \frac{\omega-1}{\omega}\left( \int \left(\int v^{\frac{n(2+\gamma)}{n-r}} \right)^{\frac{n-r}{n}}\right) \\ &\leq \frac{1}{\omega}\int \left( \int v^{\frac{2}{1-\alpha}} \right)^{1-\alpha} + \frac{\omega-1}{\omega}\left( \int \left(\int v^{\frac{n(2+\gamma)}{n-r}} \right)^{\frac{n-r}{n}}\right), \epsilonnd{aligned} \epsilonnd{equation} where we used Minkowski's inequality in the last inequality. Substituting $u_{\phi_k}^-$ for $v$ in \epsilonqref{e:minkowski} and utilizing \epsilonqref{e:leftene} we obtain \begin{equation} \label{e:iteration} \left( \iint\limits_{U_k} (u-\phi_k)_-^p \right)^{b/p} \leq C^k\iint\limits_{U_{k-1}} \chi_{\{u<\phi_k\}}. \epsilonnd{equation} We first recall that $\phi_{k-1}\gammaeq \phi_k + (1-\mu_0)2^{-k}$. We now utilize Tchebychev's inequality \[ \iint\limits_{U_k} \chi_{\{u<\phi_k\}} \leq (2^k/(1-\mu_0))^p \iint\limits_{U_{k-1}} (u-\phi_{k-1})_-^p. \] Combining the above inequality with \epsilonqref{e:iteration} we conclude \[ \iint\limits_{U_k} (u-\phi_k)_-^p \leq C^k\left( \iint\limits_{U_{k-1}} (u-\phi_{k-1})_-^p \right)^{p/b}. \] If we define \[ M_k := \iint\limits_{U_k} (u-\phi_k)_-^p, \] Then \[ M_k \leq C^k M_{k-1}^{p/b}. \] Since $p>b$, if $M_0$ is sufficiently small - depending on $C$ and $p/b$ - we obtain that $U_k \to 0$, and hence $u\gammaeq \mu_0$. \epsilonnd{proof} We now prove Lemma \ref{l:pullup} in the case when $D(x)=(d_1 x + d_2)$ with $d_1 \leq 2$ and $d_2 \gammaeq 1/2$. This is actually much simpler because we can utilize the test function $-(u-\phi_k)_-$ as when dealing with a linear equation. \begin{proof} We choose as the test function $-(u-\phi_k)_-$. Notice that \[ \nabla u_{\phi_k}^- (d_1 u + d_2) = \nabla u_{\phi_k}^- d_1 u + \nabla u_{\phi_k}^- d_2. \] The fact that $d_2\gammaeq 1/2$ gives a nondegenerate linear term which we utilize. From the computations in \cite{acv15} we then have \[ \begin{aligned} &\int_{{\mathbb R}^n} \int_{-\infty}^0 u_{\phi_k}^- \mathcal{D}_t^{\alpha} u \\ &\gammaeq c \int_{{\mathbb R}^n} \left( \int_{-\infty }^0 \left(u_{\phi_k}^-\right)^{\frac{2}{1-\alpha}}\right)^{1-\alpha} \\ &\quad - C^k \int_{{\mathbb R}^n}\int_{-\infty}^0 \chi_{\{u<\phi_k\}} \ dt, \epsilonnd{aligned} \] and even more importantly \begin{equation} \label{e:better} \begin{aligned} &\int_{-\infty}^0 \int_{{\mathbb R}^n} \nabla u_{\phi_k}^- \nabla (-\mathcal{D}elta)^{-\sigma} u \\ &\gammaeq c\int_{-\infty}^0 \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} \frac{[u_{\phi_k}^-(x)-u_{\phi_k}^-(y)]^2}{|x-y|^{n+2-2\sigma}} \ dx \ dy \ dt \\ &- C^k \int_{-\infty}^0 \int_{{\mathbb R}^n} \chi_{\{u<\phi_k\}} \ dx \ dt. \epsilonnd{aligned} \epsilonnd{equation} Notice that in the above inequality we have the power $|\cdot|^2$ rather than $|\cdot|^{2+\gamma}$. We now show how to bound the terms invovling $d_1 u$. \[ -\nabla u_{\phi_k}^-[u_{\phi_k}^+ - u_{\phi_k}^- + \phi_k] = \nabla[(u_{\phi_k}^-)^2/2 - u_{\phi_k}^- \phi_k] + u_{\phi_k}^- \nabla \phi_k = (1)+(2) \] Then multiplying $(1)$ by $\nabla (-\mathcal{D}elta)^{-\sigma} u$ and integrating over ${\mathbb R}^n$ we have \[ \mathcal{B}((u_{\phi_k}^-)^2 /2 -u_{\phi_k}^- \phi_k, u_{\phi_k}^+ - u_{\phi_k}^- + \phi_k ). \] Now \[ \begin{aligned} &[(u_{\phi_k}^-)^2 /2 -u_{\phi_k}^- \phi_k](x) - [(u_{\phi_k}^-)^2 /2 -u_{\phi_k}^- \phi_k](y) \\ &= \frac{1}{2}[u_{\phi_k}^- (y) -u_{\phi_k^-}(x)][\phi_k(x)+\phi_k(y)-u_{\phi_k}^- (x)- u_{\phi_k}^- (y)]\\ &\quad - \frac{1}{2}[u_{\phi_k}^- (y) +u_{\phi_k^-}(x)][\phi_k(x)-\phi_k(y)] \\ &=(1a)+(1b) \epsilonnd{aligned} \] We write $u=u_{\phi_k}^+ -u_{\phi_k}$. We break up our set into the two regions \[ F_k:= \{(x,y): |u_\phi^-(x)-u_\phi^-(y)| \gammaeq 2|\phi(x)-\phi(y)| \} \\ \] We notice that on the set $F$ we have that \[ \phi(x)+\phi(y)-u_\phi^-(x)-u_\phi^-(y) \gammaeq |u_\phi^-(x)-u_\phi^-(y)|/2. \] Then integrating over $F$ we have for the term $(1a)$ with right term $-u_{\phi_k}^-$ \[ \begin{aligned} &-\iint\limits_{F_k} \frac{([(u_{\phi_k}^-)^2 /2 -u_{\phi_k}^- \phi_k](x) - [(u_{\phi_k}^-)^2 /2 -u_{\phi_k}^- \phi_k](y))(u_{\phi_k^-(x)}-u_{\phi_k}^-(y))}{|x-y|^{n+2-2\sigma}} \\ &\gammaeq \iint\limits_{F_k} \frac{|u_{\phi_k^-(x)}-u_{\phi_k}^-(y)|^3}{|x-y|^{n+2-2\sigma}} \gammaeq 0. \epsilonnd{aligned} \] This is the nonnegative energy piece which we actually do not need having obtained a better piece in \epsilonqref{e:better}. All of the remaining terms in $(1)$ can be bounded by breaking up the region of integration over $F_k,F_k^c$. Over $F_k$ we use H\"older's inequality with $p=2$ rather than with $p=2+\gamma$ and absorb the small pieces by the term in \epsilonqref{e:better}. We use the same methods as before to bound the integration over $F_k^c$. Bounding the term $(2)$ is done as before with slightly easier computations. The local spatial term is bounded in the usual manner. \epsilonnd{proof} \section{Pull-down} \label{s:pulldown} In this section we prove Lemma \ref{l:pulldown}. We will need the following estimate that is analogous to Lemma \ref{l:control}. \begin{lemma} \label{l:pulldest} Let $u,\phi$ be two functions such that $1/2\leq \phi \leq u \leq 1$. Let $0<\gammaamma<1$ be a constant. If $|u(x)-u(y)|\gammaeq 4|\phi(x)-\phi(y)|$, then \begin{equation} \label{e:pcontrol1} c_1 |u_{\phi}^+(x)-u_{\phi}^+(y)|^{1+\gamma} \leq \left| \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \right| \leq c_2 |u_{\phi}^+(x)-u_{\phi}^+(y)|. \epsilonnd{equation} Also, if \[ 0 \leq \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \] then $0\leq (u-\phi)_-(x)-(u-\phi)_-(y)$. If instead we assume $|u(x)-u(y)|\leq 4|\phi(x)-\phi(y)|$, then \begin{equation} \left| \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \right| \leq 14 |\phi(x)-\phi(y)|. \epsilonnd{equation} \epsilonnd{lemma} The proof is similar to the proof of Lemma \ref{l:control}. In this case since $u>\phi$ one uses the bound above on $u$ and the fact that $\phi$ is bounded by below. \begin{proof}[Proof of Lemma \ref{l:pulldown}] The proof is nearly identical. We mention the differences. We only consider $D(u)=u$ since the modifications for handling $D(u)=d_1 u +d_2$ have already been shown in the proof of Lemma \ref{l:pullup}. We consider a similar test function \[ F(x)= \frac{1}{\gamma+1}(1+x)^{\gamma+1} -x + \frac{1}{\gamma+1} \] We then utilize \begin{equation} \label{e:test2} F'(u_{\phi_k}^+/\phi_k) = \left(1+ \frac{(u-\phi_k)_+}{\phi} \right)^{\gamma} -1 \epsilonnd{equation} This time we consider the same test functions $\theta_k(x)$ in the space variable, but this time we multiply only by a single cut-off in time $\xi_0(t)$. We define our $\phi_k$ as \[ \phi_k:= \overline{\Psi}(x,t) -\xi_0(t)\theta_k(x/2). \] To obtain the same estimate in time we only need to recognize that $\phi_k$ is now decreasing in time and bounded by below by $1/2$. \begin{equation} \label{e:pultime} \begin{aligned} &F' \left(\frac{u_{\phi}^+(t)}{\phi(t)} \right) \left[u_{\phi}^+(t)-u_{\phi}^+(s)\right] \\ &= \phi(t)F' \left(\frac{u_{\phi}^+(t)}{\phi(t)} \right) \frac{u_{\phi}^+(t)-u_{\phi}^+(s)}{\phi(t)} \\ &= \phi(t)F' \left(\frac{u_{\phi}^+(t)}{\phi(t)} \right) \left[\frac{u_{\phi}^+(t)}{\phi(t)}-\frac{u_{\phi}^+(s)}{\phi(s)}\right] \\ &\quad + \phi(t)u_{\phi}^s F' \left(\frac{u_{\phi}^+(t)}{\phi(t)} \right) \left[\frac{1}{\phi(t)}-\frac{1}{\phi(s)}\right] \\ &\gammaeq \frac{1}{2}\left[F(u_{\phi}^+(t)/\phi(t)) - F(u_{\phi}^+(s)/\phi(s)) \right] \\ &\quad + \frac{\gamma}{2}[u_{\phi}^+(t)-u_{\phi}^+(s)]^2 \\ &\quad - C u_{\phi}^+(s) (t-s) \epsilonnd{aligned} \epsilonnd{equation} The negative constant comes from the fact that $\phi^{-1}$ is Lipschitz. Then everything proceeds as before. Since our cut-off is bounded by below our $L^p$ norm in time occurs over all of $(-\infty,0)$. We obtain as before \[ \begin{aligned} &c\int_{{\mathbb R}^n} \left( \int_{-\infty}^0 (u-\phi)_+^{\frac{2}{1-\alpha}}\right)^{1-\alpha} +\int_{-\infty}^0 \int_{{\mathbb R}^n} u \nabla F'\left(\frac{u_{\phi}^+}{\phi} \right) \nabla (-\mathcal{D}elta)^{-\sigma} u \\ &\leq \int_{{\mathbb R}^n} \int_{-\infty}^T \chi_{\{u>\phi\}} \ dt \ dx \epsilonnd{aligned} \] The spatial portion of the problem is handled exactly as before. \epsilonnd{proof} \section{Decrease in Oscillation} \label{s:oscillation} We define \[ F(t,x):= \frac{1}{4}\sup\{-1, \inf\{0,|x|^2-9\}\} +\frac{1}{4}\sup\{-1, \inf\{0,|t|^2-9\}\}. \] We point out that $F$ is Lipschitz, compactly supported in $[-3,0]\times B_3$ and equal to $-1/2$ in $[-2,0]\times B_2$. We also define for $0<\lambda<1/4$, \[ \psi_{\lambda}(t,x):= ((|x|-\lambda^{-1/\nu})^{\nu}-1)_+ + ((|t|-\lambda^{-1/\nu})^{\nu}-1)_+ \text{ for } |t|,|x|\gammaeq \lambda^{-1,\nu} \] and zero otherwise. The value of $\nu$ will be determined later. Finally, we define for $i \in \{0,1,2,3,4\}$ \[ \phi_i = 1 + \psi_{\lambda^3} + \lambda^i F. \] Then $1/2\leq \phi_0 \leq \ldots \leq \phi_4 \leq 1$ in $\Gamma_4$. \begin{lemma} \label{l:decintime} Let $\kappa$ be the constant defined in Lemma \ref{l:pulldown}. Let $u$ be a solution to \epsilonqref{e:2delaprox}. There exists a small constant $\rho>0$ depending only on $n,\sigma, \alpha$ and $\lambda_0$ depending only on $n,\sigma, \alpha, \rho, \delta$ such that for any solution $u$ defined in $(a,0)\times {\mathbb R}^n$ with $a<-4$ and \[ - \psi_{\lambda^3} \leq u(t,x) \leq 1 + \psi_{\lambda^3} \text{ in } (a,0)\times {\mathbb R}^n \] with $\lambda \leq \lambda_0$, and $f\leq \lambda^3$, then if \[ |\{u<\phi_0\}\cap (B_1 \times (-4,-2))| \gammaeq \rho, \] then \[ |\{u>\phi_4\} \cap ({\mathbb R}^n \times (-2,0))| \leq \kappa. \] \epsilonnd{lemma} \begin{proof} We will show the computations for $D(u)=u$. The general situation is handled as before as in Lemma \ref{l:pullup}. \textbf{First Step: Revisiting the energy inequality.} We return again to the energy inequality. This time, however, we will make use of the nonnegative terms. We seek to obtain a bounde on the right hand side of the form $C\lambda^{(2+\gamma)/(1+\gamma)}$. We now consider the test function as in \epsilonqref{e:test2}, but with cut-off $\phi_1$. If $u>\phi_i$ , then $1/2\leq \phi_i \leq u\leq 1$, and so \[ F'(u/\phi_i) = \chi_{\{u>\phi_i\}}\frac{u^{\gamma}-\phi^{\gamma}}{\phi^{\gamma}}\leq 2\gammaamma u_{\phi_i}^+ \leq 2\gammaamma \lambda^{2i}. \] To take care of the piece in time we first note that $\phi_1$ is Lipschitz in time for $t \in [0,4]$ with Lipschitz constant $2\lambda$. Then as before \[ \int F'(u/\phi_1) \mathcal{D}_t^{\alpha} u \ dt = \int F'(u/\phi_1) \mathcal{D}_t^{\alpha} (u_{\phi_1}^+ - u_{\phi_1}^- + \phi) = (T1)+(T2)+(T3). \] For $(T1)$ we return to the inequality \epsilonqref{e:pultime} and utilize the Lipschitz nature of $\phi_1^{-1}$, to obtain \[ \begin{aligned} \iint F'(u/\phi_1) \mathcal{D}_t^{\alpha} u_{\phi_1}^+ &\gammaeq c \left( \int_{-\infty}^0 (u-\phi_1)_+^{\frac{2}{1-\alpha}}\right)^{1-\alpha} \\ &\quad +\int_{-\infty}^0 \int_{-\infty}^t \phi_1(t) u_{\phi_1}^{s} F'(u_{\phi_1}^+(t)/\phi(t))(\phi_1^{-1}(t)-\phi_1^{-1}(s)) \ dt \\ &\gammaeq c \left( \int_{-\infty}^0 (u-\phi_1)_+^{\frac{2}{1-\alpha}}\right)^{1-\alpha}\\ &\quad - C \int_{-4}^0 \lambda^2. \epsilonnd{aligned} \] The nonnegative piece (T2) will be utilized in the second step of this proof. For $(T3)$ we note that since $\phi_i$ is decreasing, we have \[ 0 \leq - \mathcal{D}_t^{\alpha} \phi_i \leq -\Lambda^{-1} D_t^{\alpha} \phi_i = -D_t^{\alpha} \psi_{\lambda^3} -D_t^{\alpha} \lambda^i F. \] Clearly, $-D_t^{\alpha} \lambda^i F \leq C\lambda^i$ for $t\leq 0$ from the Lipschitz nature of $F$. For $-4\leq t\leq 0$ we have \[ -D_t^{\alpha} \psi_{\lambda^3} \leq \int_{-\infty}^{\lambda^{-1/\nu}} \frac{|s|}{|s|^{1+\alpha}} \leq C_{\alpha} \lambda^{(\alpha - \nu)/(\nu)}. \] We therefore pick $\nu$ small enough that $(\alpha-\nu)/\nu>2$. \[ \int_{-\infty}^0 F'(u/\phi_1) \mathcal{D}_t^{\alpha} \phi_1 \leq C \int_{-4}^0 \lambda u_{\phi}^+ \leq C\lambda^2. \] Our energy inequality becomes \[ \begin{aligned} &c \left( \int_{-\infty}^0 (u-\phi)_+^{\frac{2}{1-\alpha}}\right)^{1-\alpha} \\ &\quad + c\int_{{\mathbb R}^n} \iint \frac{[u_{\phi_1}^+(t)-u_{\phi_1}^+(s)][u_{\phi_1}^-(t)- u_{\phi_1}^-(s)]}{(t-s)^{1+\alpha}} \\ &\quad + \text{ ``Spatial Terms'' } \leq C \lambda^2 + \iint f u_{\phi_1}^+. \epsilonnd{aligned} \] Since $f\leq \lambda^3$, everything is bounded on the right hand side by $C\lambda^2$. We now turn our attention to the elliptic portion. We consider the terms $(1a),(1bi),(1bii),(1c),(2)$ as the analogous terms for those defined in the proof of Lemma \ref{l:pullup}. As before we obtain a nonnegative energy from the term $(1bi)$. Everything else we will absorb into this energy or bound by $C\lambda^{(2+\gamma)/(1+\gamma)}$. The term from $(1bi)$ over $A_1^c$ is bounded as follows: \[ \begin{aligned} &c_{n,\gamma,\sigma}\iint_{A_1^c} \frac{[\phi_1(x)-\phi_1(y)][u_{\phi_1}^+(x)-u_{\phi_1}^+(y)]}{|x-y|^{n+2-2\sigma}} \ dx \ dy \\ &\leq C \iint_{A_1^c} \frac{[\phi_1(x)-\phi_1(y)]^2 X_{x,y}}{|x-y|^{n+2-2\sigma}} \ dx \ dy \epsilonnd{aligned} \] We have the following inequality from the computations given in \cite{ccv11}: \begin{equation} \label{e:wingbound} \iint \frac{[\phi_1(x)-\phi_1(y)]^{(2+\gamma)/(1+\gamma)} X_{x,y}}{|x-y|^{n+2-2\sigma}} \ dx \ dy \leq C \lambda^{(2+\gamma)/(1+\gamma)}, \epsilonnd{equation} with $(2-2\sigma-2\nu)/\nu\gammaeq 2$. In particular, \epsilonqref{e:wingbound} will hold for $\gamma=0$. For the term $(1bii)$ we break up the region of integration into $A_1$ and $A_1^c$. On $A_1$ we use H\"older's inequality as before \[ \begin{aligned} &c_{n,\gamma,\sigma}\iint_{A_1} \frac{[\phi_1(x)-\phi_1(y)][u_{\phi_1}(x)-u_{\phi_1}(y)]}{|x-y|^{n+2-2\sigma}} \ dx \ dy \\ &\leq C \iint_{A_1} \frac{[\phi_1(x)-\phi_1(y)]^{(2+\gamma)/(1+\gamma)}}{|x-y|^{n+2-2\sigma}}X_{x,y} \ dx \ dy \\ &\quad + \epsilonta \iint_{A_1} \frac{[u_{\phi_1}(x)-u_{\phi_1}(y)]^2}{|x-y|^{n+2-2\sigma}} \ dx \ dy \\ &\leq C\lambda^{(2+\gamma)/(1+\gamma)} + \epsilonta \iint_{A_1} \frac{[u_{\phi_1}(x)-u_{\phi_1}(y)]^2}{|x-y|^{n+2-2\sigma}} \ dx \ dy \\ \epsilonnd{aligned} \] The last term is absorbed into the left hand side. The other term is bounded again from \epsilonqref{e:wingbound}. The term $(1c)$ is bounded in exactly the same way. $(1a)$ is nonnegative and will be utilized later. We now turn our attention to the term $(2)$. We rewrite $u=u_{\phi_1}^+ - u_{\phi_1}^- + \phi_1$. The term involving $u_{\phi_1}^+$ with $|x-y|\leq \epsilonta$ is absorbed by the nonnegative term $(1a)$ on the left hand side. We now utilize the inequalities: \begin{itemize} \item $u_{\phi_1}^+ \leq \lambda $ \item $|\nabla L(x-y)| \approx 1/|x-y|^{n+1-2\sigma}$ \item $|\nabla \phi_1| \leq C $ for all $x$ \item $|\nabla \phi_1| \leq C\lambda $ in the support of $u_{\phi_1}^+$ \item $\chi_{\{u>\phi_1\}} [(u/\phi_1)^{1+\gamma}-1] \leq 4 u_{\phi_1}^+ \leq 4 \lambda. $ \epsilonnd{itemize} Then \[ \begin{aligned} &\left| \int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{\gamma}{\gamma+1} \chi_{\{u<\phi\}} ((u/\phi)^{\gamma+1}-1) \nabla \phi_1 \nabla L(x-y)[u(y)-u(x)] \ dx \ dy \right| \\ &\leq C \lambda^2 \left| \int_{{\mathbb R}^n}\int_{{\mathbb R}^n} \chi_{\{u<\phi\}} \nabla L(x-y)[u(y)-u(x)] \ dx \ dy \right| \\ \epsilonnd{aligned} \] These terms are all bounded as before. Notice that we have $\lambda^2$ on the outside. Then all nonloacl terms on the right hand side are bounded by $C\lambda^{(2+\gamma)/(1+\gamma)}$. The local term div$(D(u)\nabla u)$ is handled in the usual manner by use of Cauchy-Schwarz. \textbf{Second Step: Using the ``good'' spatial piece.} We now utilize the two nonnegative pieces. From Proposition \ref{p:gamma} we have \[ \left[ \left(\frac{u}{\phi_1}\right)^{\gamma+1}-\phi_1 \right]_+ \gammaeq 4 (u_{\phi_1}^+)^{1+\gamma}. \] Then we conclude that \begin{equation} \label{e:phi1} \begin{aligned} &\int_{{\mathbb R}^n} \int_{-4}^0 \int_{-4}^0\frac{u_{\phi_1}^+(t)u_{\phi_1}^-(s)}{t-s}^{1+\alpha} \\ &\quad + \int_{-4}^0 \int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{(u_{\phi_1}^+)^{\gamma+1}(x)u_{\phi_1}^-(y)}{|x-y|^{n+2-2\sigma}}\\ &\leq C \lambda^{(2+\gamma)/(1+\gamma)}. \epsilonnd{aligned} \epsilonnd{equation} Since we used $\Psi_{\lambda^3}$, replacing $\phi_1$ with $\phi_3$ we have the same inequality but with the bound $C\lambda^{3(2+\gamma)/(1+\gamma)}$. We now show how the inequality \epsilonqref{e:phi1} and its analogue for $\phi_3$ are enough to prove the remainder of the Lemma as in \cite{acv15}. We note that for the proof as written in \cite{acv15} to work we need $3(2+\gamma)/(1+\gamma)>5$ which is achieved for $\gammaamma$ small enough. We first utilize \[ \int_{-4}^0 \int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{(u_{\phi_1}^+)^{\gamma+1}(x)u_{\phi_1}^-(y)}{|x-y|^{n+2-2\sigma}}\\ \leq C \lambda^{(2+\gamma)/(1+\gamma)}. \] From our hypothesis \[ |\{w<\phi_0\} \cap ((-4,-2)\times B_1)| \gammaeq \rho. \] Then the set of times $\Sigma \in (-4,-2)$ for which $|\{u(t,\cdot)<\phi_0\} \cap B_1| \gammaeq \rho/4$ has atleast measure $\rho/(2|B_1|)$. And so \[ C\lambda^{(2+\gamma)/(1+\gamma)} \gammaeq c\rho \int_{\Sigma} \int_{{\mathbb R}^n} (u_{\phi_1}^+)^{1+\gamma} \ dx \ dt \] Now $(\{u-\phi_2>0\}\cap (\Sigma \times B_2)) \subset (\{u-\phi_1>\lambda/2\}\cap (\Sigma \times B_2))$, and so from Tchebychev's inequality \[ |\{u-\phi_2>0\}\cap (\Sigma \times B_2)| \leq \frac{C}{\rho} \lambda^{\frac{2+\gamma}{1+\gamma}-(1+\gamma)}. \] The exponent on $\lambda$ is positive for $\gamma$ small enough. We write this as \[ |\{u\leq \phi_2\}\cap (\Sigma \times B_2)| \gammaeq |\Sigma \times B_2| - \frac{C}{\rho} \lambda^{\frac{2+\gamma}{1+\gamma}-(1+\gamma)} \gammaeq \rho/2 - \frac{C}{\rho} \lambda^{\frac{2+\gamma}{1+\gamma}-(1+\gamma)}. \] This will be positive for $\lambda$ small enough depending on $n,\sigma, \alpha, \gammaamma, \rho$. The proof then proceeds just as in \cite{acv15} where we then utilize $3(2+\gamma)/(1+\gamma)>5$ as well as the analogue of \epsilonqref{e:phi1} for $\phi_3$. \epsilonnd{proof} This next Lemma will imply Lemma \ref{l:2down}. For this next lemma we define \[ \psi_{\tau, \lambda} = ((|x|-1/\lambda^{4/\sigma})^{\tau}-1)_+ + ((|t|-1/\lambda^{4/\alpha})^{\tau}-1)_+ \] \begin{lemma} \label{l:oscdec} Given $\rho>0$ there exist $\tau>0$ and $\mu_1$ such that for any solution to \epsilonqref{e:2delaprox} in ${\mathbb R}^n \times (a,0)$ with $a<-4$ and $|f|\leq \lambda^3$ satisfying \[ - \psi_{\tau, \lambda} \leq u \leq 1 + \psi_{\tau, \lambda}, \] If \[ |\{u<\phi_0\} \cap (B_1 \times (-4,-2))| > \rho, \] then \[ \sup_{B_1 \times (-1,0)} u \leq 1- \mu_1. \] \epsilonnd{lemma} \begin{proof} We consider the rescaled function $w(t,x)= (u-(1-\lambda^4))/\lambda^4$. We fix $\tau$ small enough such that \[ \frac{(|x|^{\tau}-1)_+}{\lambda^4} \leq (|x|^{\sigma/4}-1)_+ \text{ and } \frac{(|t|^{\tau}-1)_+}{\lambda^4} \leq (|t|^{\sigma/4}-1)_+ \] Then $w$ satisfies equation \epsilonqref{e:2delaprox} with $D_2(x) = D_1(\lambda^4 x + (1-\lambda^4))$ where $D_1$ is the coefficient for the equation $u$ satisfies. From our hypothesis and Lemma \ref{l:decintime} \[ |\{w>1/2\} \cap (B_2 \times [-2,0])| = |\{u>\phi_4\} \cap (B_2 \times [-2,0])| \leq \rho. \] Then from Lemma \ref{l:pulldown} we conclude that $w\leq 1-\mu_1$ on $(-1,0)\times B_1$, and so $u\leq 1-\lambda^4 \mu_1= 1-\mu_2$. \epsilonnd{proof} \section{Proof of Regularity} \label{s:regularity} With Lemmas \ref{l:pullup}, \ref{l:pulldown}, and \ref{l:2down} we are ready to finish the proof of Theorem \ref{t:continuity}. We first mention that solutions of \epsilonqref{e:main} satisfy the following scaling property: If $u$ is a solution on $(a,0)\times {\mathbb R}^n$, then $v(t,x)=Au(Bt,Cx)$ is a solution on $(a/B,0)\times {\mathbb R}^n$ if $A=B^{\alpha}C^{2\sigma-2}$. The method of proof is given in \cite{cfv13} which we now briefly outline. We take any point $p=(x,t)\in {\mathbb R}^n \times (a, T) $ and prove that $u$ is H\"older continuous around $p$. The H\"older continuity exponent will depend only on $\alpha, \sigma, n$. The constant will depend on the $L^{\infty}$ norm of $u,f$ and on the $C^2$ norm of $u(a,x)$. By translation we assume that $p=(0,0)$. By scaling we assume that $0\leq u(t,x)\leq \overline{\Psi}(t,x)$ and $|f|\leq \lambda^3$ for $\lambda$ as defined in We now take a positive constant $M<1/4$ such that for $0<K\leq M$ \[ \frac{1}{1-\mu_2/2} \psi_{\tau,\lambda^3}(Kt,Mx). \] $M$ will depend only on $\lambda, \mu_2$ and $\tau>0$. During the iteration we have the following alternative. \textit{Alternative 1.} Suppose that we can apply Lemma \ref{l:2down} repeatedly. We then consider the rescaled functions \[ u_{j+1}(t,x)= \frac{1}{1-\mu_2/4} u_j(M_1t, Mx), \quad M_1 = \left(\frac{M^{2-2\sigma}}{1-\mu_2/4} \right)^{1/\alpha}. \] Notice that $M_1 < M$. All the $u_j$ satisfy the same equation. If we can apply Lemma \ref{l:2down} at every step, then $u_j\leq 1-\mu_2$ on the cylinder $\Gamma_1$. This implies H\"older regularity around $p$ and also implies $u(p)=0$ . \textit{Alternative 2.} If at some point the assumption \epsilonqref{e:2down} fails, then we are in the situation of Lemma \ref{l:pullup} and \[ 0<\mu_0 \leq u_j(t,x)\leq 1. \] Scaling the above situation our equations will have $D(u)=d_1 u + d_2$ with $d_2>0$. We may then repeat the procedure since Lemma \ref{l:pullup} and \ref{l:2down} apply also in this situation. \section{appendix} \begin{proof}[Proof of Lemma \ref{l:control}] Since throughout the paper we only require $\gammaamma$ small when $\sigma$ is small, we will prove the Lemma for $\gammaamma=1/k$ for $k\in {\mathbb N}$. Now we assume without loss of generality that \[ \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \gammaeq 0. \] We first assume that $|u(x)-u(y)|\leq 4|\phi(x)-\phi(y)|$. We need to bound \begin{equation} \label{e:quotient} \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)}. \epsilonnd{equation} We first notice that term above in \epsilonqref{e:quotient} will be larger if we assume that $u(y)\gammaeq u(x)$ and $\phi(x)\gammaeq \phi(y)$ without changing $|u(x)-u(y)|$ and $|\phi(x)-\phi(y)|$. Furthermore, the term in \epsilonqref{e:quotient} will still be greater if $u(y)=\phi(y)$ and not changing $|u(y)-u(x)|$. We are then looking for the bound \[ u(y)- \frac{u^{1+\gamma}(x)}{\phi^{\gamma}(x)} \leq c_2|\phi(x)-\phi(y)| =c_2(\phi(x)-u(y)). \] Thus, for a constant $l$ we need the bound \begin{equation} \label{e:mubound} l - \frac{(l-\upsilon)^{1+\gamma}}{(l+ \mu)^{\gamma}} \leq C \mu. \epsilonnd{equation} Recalling that we are assuming $4|\phi(x)-\phi(y)|\gammaeq |u(x)-u(y)|$ (or $\upsilon \leq 4\mu$) the above term is maximized when $\upsilon$ is largest or when $\upsilon=4\mu$. Now \[ \begin{aligned} &l - \frac{(l-4\mu)^{1+\gamma}}{(l+ \mu)^{\gamma}} \\ &= \frac{l[(l+\mu)^\gamma -(l-4\mu)^{\gamma}]}{(l+\mu)^{\gamma}} + 4\frac{\mu(l-4\mu)^{\gamma}}{(l+\mu)^{-\gamma}} \\ &:= L_1 + L_2 \epsilonnd{aligned} \] It is clear that \[ L_2 \leq 4\mu. \] To control $L_1$ we first consider when $l-4\mu \leq l/2$. Then $l\leq 8\mu$ and it is clearly true that \[ L_1 \leq 8 \mu. \] Now when $l-4\mu \gammaeq l/2$, from the concavity of $x^\gammaamma$ we have \[ L_1 \leq \frac{l}{l-4\mu}\frac{(l-4\mu)^{\gamma}}{(l+\mu)^{\gamma}}5\mu \leq 10 \mu. \] Then \begin{equation} \label{e:mu1} l - \frac{(l-\upsilon)^{1+\gamma}}{(l+ \mu)^{\gamma}} \leq 14 \mu, \epsilonnd{equation} and \epsilonqref{e:control2} is proven with constant $c_2=14$. We now assume $4|\phi(x)-\phi(y)|\leq |u(x)-u(y)|$, and the left hand side of \epsilonqref{e:mubound} is maximized again when $4\mu=\upsilon$ and so we have \[ l - \frac{(l-\upsilon)^{1+\gamma}}{(l+ \upsilon/4)^{\gamma}} \leq \frac{14}{4}\upsilon, \] which is just \epsilonqref{e:mu1} rewritten with the substitution $\mu=\upsilon/4$. Then \begin{equation} \label{e:Abound1} \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} \leq \frac{14}{4}|u(x)-u(y)| \leq \frac{14}{3} |u_{\phi}^-(x)-u_{\phi}^-(y)|. \epsilonnd{equation} and the right hand side of \epsilonqref{e:control1} is shown. Now \[ \frac{u^{\gamma+1}(y)}{\phi^{\gamma}(y)} - \frac{u^{\gamma+1}(x)}{\phi^{\gamma}(x)} = \frac{u^{\gamma+1}(y)-u^{\gamma+1}(x)}{\phi^{\gamma}(y)} - u^{\gamma+1}(x)\frac{\phi^{\gamma}(x)-\phi^{\gamma}(y)}{\phi^{\gamma}(x)\phi^{\gamma}(y)} := M_1 + M_2 \] We suppose $\gammaamma = 1/k$. By factoring we have \[ \begin{aligned} |M_1| &= \frac{|u(y)-u(x)|}{\phi^{\gamma}(y)} \frac{\sum_{j=0}^k u^{(k-j)/k}(y)u^{j/k}(x)}{\sum_{j=0}^{k-1} u^{(k-1-j)/k}(y)u^{j/k}(x)} \\ &\gammaeq \frac{u(x)}{\phi^{\gamma}(y)} \frac{4|\phi(x)-\phi(y)|}{\sum_{j=0}^{k-1} u^{(k-1-j)/k}(y)u^{j/k}(x)} \\ |M_2| &= \frac{u^{1+1/k}(x)}{\phi^{1/k}(x)\phi^{1/k}(y)} \frac{|\phi(x)-\phi(y)|}{\sum_{j=0}^{k-1} \phi^{(k-1-j)/k}(y)\phi^{j/k}(x)} \\ &\leq \frac{u(x)}{\phi^{\gamma}(y)} \frac{|\phi(x)-\phi(y)|}{\sum_{j=0}^{k-1} u^{(k-1-j)/k}(y)u^{j/k}(x)} \\ \epsilonnd{aligned} \] Thus $M_2 \leq M_1 / 4$. Thus, $M_1$ is the dominant term. We then have from the convexity of $x^{\gamma+1}$ \[ \begin{aligned} M_1 + M_2 \gammaeq M_1/2 &\gammaeq \frac{1}{2} \frac{u^{\gamma+1}(y)-u^{\gamma+1}(x)}{\phi^{\gamma}(y)} \\ &\gammaeq \frac{1}{2} \frac{(u(y)-u(x))^{1+\gamma}}{\phi^{\gamma}(y)} \\ &\gammaeq (4/5)^{1+\gamma} \frac{1}{2}\frac{[u_{\phi}^-(y)-u_{\phi}^-(x)]^{1+\gamma}}{\phi^{\gamma}(y)} \\ &\gammaeq \frac{2}{5}(4/5)^{\gamma}\frac{[u_{\phi}^-(y)-u_{\phi}^-(x)]^{1+\gamma}}{\phi^{\gamma}(y)}. \epsilonnd{aligned} \] \epsilonnd{proof} \begin{proposition} \label{p:gamma} Let $F$ be a function satisfying $F,F'' \gammaeq 0$ for $x \gammaeq 0$. Assume also $F(0)=0$. If $y \gammaeq x \gammaeq 0$, then \[ F(y)-F(x) \gammaeq F(y-x). \] \epsilonnd{proposition} \begin{proof} For fixed $h>0$, \[ \frac{d}{dx} \frac{F(x+h)-F(x)}{h} = \frac{F'(x+h)-F'(x)}{h} \gammaeq 0. \] Then for $x,h\gammaeq 0$ \[ \frac{F(x+h)-F(x)}{h} \gammaeq \frac{F(0+h)-F(0)}{h} = \frac{F(h)}{h}. \] Let $h=y-x$, and multiply both sides of the equation by $y-x$. \epsilonnd{proof} \begin{proposition} \label{p:exp} Let $0<\gammaamma<1$. Let $x,d\gammaeq 0$. Then \[ (x+d)^{\gamma} -d^{\gamma} \leq 2^{\gamma} x^{\gamma} \] \epsilonnd{proposition} \begin{proof} First assume $x\leq d$. From the concavity of $x^{\gamma}$ we have \[ (x+d)^{\gamma}-d^{\gamma}\leq \gamma d^{\gamma-1}x = \gamma d^{\gamma-1}x^{1-\gamma}x^{\gamma} \leq \gamma x^{\gamma}. \] If on the other hand $x >d$, then \[ (x+d)^{\gamma}-d^{\gamma}\leq (x+d)^{\gamma}\leq (2x)^{\gamma}. \] \epsilonnd{proof} \epsilonnd{document}

\begin{document} \title{On the Chv\'atal-Gomory Closure of a Compact Convex Set} \author{Daniel Dadush\thanks{H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA 30332-0205, USA {\tt dndadush@gatech.edu}} \and Santanu S. Dey\thanks{H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA 30332-0205, USA {\tt santanu.dey@isye.gatech.edu}} \and Juan Pablo Vielma\thanks{Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA 15261, USA {\tt jvielma@pitt.edu}} } \maketitle \thispagestyle{empty} $\quad$ \begin{abstract} In this paper, we show that the Chv\'atal-Gomory closure of a compact convex set is a rational polytope. This resolves an open question discussed in Schrijver~\cite{Schrijver80} and generalizes the same result for the case of rational polytopes~\cite{Schrijver80}, rational ellipsoids~\cite{Dey09} and strictly convex sets~\cite{Dadush:de:vi:10}. In particular, it shows that the CG closure of an irrational polytope is a rational polytope, which was the open question in~\cite{Schrijver80}. \end{abstract} \pagebreak \setcounter{page}{1} \section{Introduction} Gomory~\cite{Gomory58} introduced the Gomory fractional cuts, also known as Chv\'atal-Gomory (CG) cuts, to design the first finite cutting plane algorithm for integer linear programs. Since then, many important classes of facet-defining inequalities for combinatorial optimization problems have been identified as CG cuts. For example, the matching polytope can be obtained using Chv\'atal-Gomory cuts~\cite{edmonds:1965}. CG cuts have also been effective from a computational perspective; see for example~\cite{Bonami08b}, \cite{Fischetti07}. Although traditionally CG cuts have been defined for rational polyhedron for solving integer linear programs, they can be defined for general convex sets so as to be useful in solving convex integer programs, i.e. discrete optimization problems where the continuous relaxation is a convex optimization problem. CG cuts for non-polyhedral sets were considered implicitly in \cite{Chvatal73,Schrijver80} and more explicitly in \cite{CezikIyengar05,Dey09}. Let $K\subseteq \mathbb{R}^n$ be a closed convex set and let $h_K$ represent its support function, i.e. $h_K(a) = \textup{sup}\{ \langle a,x \rangle\,:\, x \in K\}$. Then given $a \in \mathbb{Z}^n$ such that $h_K(a) < +\infty$, the CG cut corresponding to $a$ is derived as, \begin{eqnarray}\label{CGdefine} \langle a, x \rangle \leq \lfloor h_K(a) \rfloor. \end{eqnarray} The CG closure is defined as the convex set obtained by the intersection of all viable CG cuts. A classical result of Chv\'atal~\cite{Chvatal73} and Schriver~\cite{Schrijver80} states that the CG closure of a rational polyhedron is a rational polyhedron. This is a crucial property, since it is a mathematical guarantee that there exists a `relatively important' finite subset of CG cuts that defines the CG closure. Recently, we were able to verify that the CG closure of a compact convex set obtained as the intersection of a strictly convex set and a rational polyhedron is a rational polyhedron~\cite{Dey09,Dadush:de:vi:10}. The proof involved using significantly different techniques to the ones used in~\cite{Schrijver80}. While intersection of strictly convex sets and rational polyhedron is an important class of convex sets, they do not capture the whole gamut of interesting convex sets that appear in convex IPs. The barrier in extending our understanding of the CG closure from the setting of the intersection of a strictly convex set and a rational polyhedra to the setting of a general convex set, is in dealing with irrationality. When working with integer linear programs, it is reasonable to assume that the set is defined by rational data and all the extreme points and rays of the feasible set are rational. However, when dealing with general convex IPs, this assumption breaks down in a natural way. For example, the Lorentz cone~\cite{nemibook} has irrational extreme rays and second order representable sets naturally (not always) inherit irrational generators. One way to design tools to deal with irrationality is to perhaps work with irrational polytopes. Schrijver~\cite{Schrijver80} considered this question. In a discussion section at the end of the paper, he writes that\footnote{Theorem 1 in~\cite{Schrijver80} is the result that the CG clsoure is a polyhedron. $P'$ is the notation used for CG closure in~\cite{Schrijver80}}: \begin{quote} ``We do not know whether the analogue of Theorem 1 is true in real spaces. We were able to show only that if $P$ is a bounded polyhedron in real space, and $P'$ has empty intersection with the boundary of $P$, then $P'$ is a (rational) polyhedron." \end{quote} In this paper, we are able to prove the CG closure of any compact convex set\footnote{If the convex hull of a set of integer points is not a polyhedron, then the CG closure cannot be expected to be a rational polyhedron. Since we do not understand well when the convex hull of integer points in general convex sets is a polyhedron, we consider the question of CG closure only for compact convex sets here.} is a rational polytope, thus also resolving the question raised in~\cite{Schrijver80} for any polytope. We note here that while the intersection of the CG closure of a convex set $K$ and the boundary of $K$ is not always empty, this intersection identified in~\cite{Schrijver80} plays a crucial role in our proof. As discussed above, proving that the compact convex set involved understanding and developing tools to handle irrationality. Therefore, while the proof presented in this paper is similar in parts to the proof in~\cite{Dadush:de:vi:10}, major components of the proof are new. New connections with diophantine approximations were necessary for the proof here. Moreover, we have been able to unearth some interesting new properties of CG closures and convex sets in general, and also design new techniques that we believe are important on their own. This paper is organized as follows. In Section~\ref{def} we give some notation, formally state our main result and give an overview of the proof which is presented in Sections~\ref{sec:long}--\ref{sec:final}. \section{Definitions, main result and proof idea}\label{def} \begin{defn}[CG Closure]For a convex set $K\subseteq \mathbb{R}^n$ and $S\subseteq \mathbb{Z}^n$ let $\mathbb{C}C(K,S) :=\bigcap_{a\in S} \set{x\in \mathbb{R}^n\,:\, \pr{x}{y} \leq \floor{h_K(y)}}$. The CG Closure of $K$ is defined to be the set $\mathbb{C}C(K):=\mathbb{C}C(K,\mathbb{Z}^n)$. \end{defn} In this paper, we are able to establish the following result. \begin{thm}\label{paperresult} If $K \subseteq \mathbb{R}^n$ be a non-empty compact convex set, then $\mathbb{C}C(K)$ is finitely generated. That is, there exists $S\subseteq \mathbb{Z}^n$ such that $|S|< \infty$ and $\mathbb{C}C(K)=\mathbb{C}C(K,S)$. In particular $\mathbb{C}C(K)$ is a rational polyhedron. \end{thm} We will use the following notation in our proof: \begin{itemize} \item Let $B^n = \set{x\in\mathbb{R}^n\,: \|x\| \leq 1}$ and $S^{n-1} = \mathrm{bd}(B^n)$. ($\mathrm{bd}$ stands for boundary) \item For a convex set $K$ and $v\in S^{n-1}$ we let $H_v(K) := \set{x\in \mathbb{R}^n\,:\, h_K(v) = \pr{v}{x}}$ be the hyperplane defined by $v$ and the support function of $K$. We also let $F_v(K) := K\cap H_v(K)$ be the face of $K$ exposed by $v$. If $F_v(K)\neq K$ we say that $F_v(K)$ is a proper exposed face and if the context is clear we regularly drop $K$ from the notation and simply write $H_v$ and $F_v$. \item For $A \subseteq \mathbb{R}^n$, let $\aff(A)$ denote the smallest affine subspace containing $A$. Furthermore denote $\aff_I(A) = \aff(\aff(A) \cap \mathbb{Z}^n)$, i.e. the largest integer subspace in $\aff(A)$. \end{itemize} This notation is fairly standard with the exception of $\aff_I(A)$. Understanding the properties of $\aff_I(A)$ when $\aff(A)$ is not a rational affine space will be crucial for the proof of Theorem~\ref{paperresult}. In particular, we will repeatedly use the fact that if $K$ is a compact convex set, then we can obtain a inner approximation of $K\cap\aff_I(K)$ using a finite number of CG cuts. The outline of the main steps in our proof of Theorem \ref{paperresult} is as follows: \begin{enumerate} \item (Section \ref{sec:long}) For $v \in \mathbb{R}^n$ and $S\subseteq \mathbb{Z}^n$, show that $\exists S'\subseteq \mathbb{Z}^n$ such that $\mathbb{C}C(F_v,S) = H_v \cap \mathbb{C}C(K,S')$ and $|S|<\infty \mathbb{R}ightarrow |S'|< \infty$ by proving the following: \begin{enumerate} \item (Section \ref{sec:lift}) CG cuts for $F_v$ can be rotated or ``lifted'' to become CG cuts for $K$ such that points in $F_v \cap \aff_I(H_v)$ separated by the original CG cut for $F_v$ are separated by the new ``lifted'' one. \item (Section \ref{sec:sepirrational}) A finite number of CG cuts for $K$ separate all points in $F_v \setminus \aff_I(H_v)$. \end{enumerate} \item (Section \ref{sec:approx}) Assuming $\mathbb{C}C(F_v)$ is finitely generated for any proper exposed face $F_v$ create an approximation $\mathbb{C}C(K,S)$ of $\mathbb{C}C(K)$ such that (i) $|S|<\infty$, (ii) $\mathbb{C}C(K,S)\subseteq K \cap \textup{aff}_I(K)$ (iii) $\mathbb{C}C(K,S) \cap \mathrm{relbd}(K) = \mathbb{C}C(K) \cap \mathrm{relbd}(K)$. This is done in the following two steps: \begin{enumerate} \item(Section \ref{sec:approx1}) Using the assumption, $\mathbb{C}C(F_v,S) = H_v \cap \mathbb{C}C(K,S')$ and a compactness argument create a first approximation satisfying (i) and (ii). \item(Section \ref{sec:approx2}) Using the assumption and noting that a polytope $P\subseteq K$ intersects $\mathrm{relbd}(K)$ along a finite number of faces of $P$ refine the approximation to satisfy (iii). \end{enumerate} \item (Section \ref{sec:final}) Finally, we are able to establish the result of the Theorem by induction on the dimension of $K$. The key observation is that there are only finitely many CG cuts that separate at least one vertex of the second approximation of the CG closure. \end{enumerate} \section{$\mathbb{C}C(F_v,S) = H_v \cap \mathbb{C}C(K,S')$}\label{sec:long} In the case of a rational polyhedra $K$, a key property of the CG closure is that, if $F$ is a face of $K$, then $\mathbb{C}C(F) = F \cap \mathbb{C}C(K)$. Using an induction argument this property can be used to construct the second approximation in the outline of our proof for the case in which $K$ is a rational polyhedron. However, this property is not enough for general convex sets. For instance, when $K$ is a strictly convex set all proper faces of $K$ are single points and property $\mathbb{C}C(F) = F \cap \mathbb{C}C(K)$ (or even $\mathbb{C}C(F) = F \cap \mathbb{C}C(K,S')$ for $|S'|<\infty$) only tells us that every non-integral point in $\mathrm{bd}(K)$ can be separated with CG cuts, but it does not tell us anything about the neighborhood of integral points. For this reason we need the stronger property $\mathbb{C}C(F_v,S) = H_v \cap \mathbb{C}C(K,S')$. In particular, this property implies that if $K$ is a full dimensional compact strictly convex and $\mathbb{C}C(F_v)$ is finitely generated for every $v$ then for each integer point $x\in \mathrm{bd}(K)$ there exists a finite number of CG cuts that separate a neighborhood of $\mathrm{bd}(K)$ around $x$ which is exactly what is needed in \cite{Dadush:de:vi:10}. We finally note here that the proof of the fact that $\mathbb{C}C(F_v,S) = H_v \cap \mathbb{C}C(K,S')$ for the case of general compact convex set is significantly more involved than for the case where $K$ is a rational polyhedron or a strictly convex set. \subsection{Lifting CG cuts}\label{sec:lift} $\mathbb{C}C(F) = F \cap \mathbb{C}C(K)$ is usually proven using a `lifting approach', i.e., given a CG for $F$ of the form $\pr{w}{x} \leq \floor{h_{F}(w)}$ where $w \in \mathbb{Z}^n$, it is shown that there exists $w' \in \mathbb{Z}^n$ such that \begin{eqnarray}\label{toprovelifting} \set{x: \pr{w'}{x} \leq \floor{h_K(w')}} \cap \aff(F) \subseteq \set{x: \pr{w}{x} \leq \floor{h_{F}(w)}} \cap \aff(F). \end{eqnarray} In order to prove (\ref{toprovelifting}) (in the case of a rational polyhedron) we typically appeal to the rational description of $K$ and Farka's Lemma. The appropriate version of \eqref{toprovelifting} for strictly convex sets is proven in \cite{Dadush:de:vi:10} by approximating the left hand side of CG cuts for $F$ using Dirichlet's diophantine approximation theorem. The appropriate version of (\ref{toprovelifting}) for the case of general compact convex sets simply replaces $\aff(F)$ with $\aff_I(H_v)$ and generalizes both (\ref{toprovelifting}) and the version in \cite{Dadush:de:vi:10}. This general version is given in Proposition \ref{lem:lift2} with a proof that is similar to that in \cite{Dadush:de:vi:10} and for which Dirichlet's theorem again plays an important role. Lemmas \ref{lem:nopt-conv}- \ref{lem:2step-conv} are technical results that are needed for proving Proposition \ref{lem:lift2}. \begin{lem} Let $K$ be a compact convex set in $\mathbb{R}^n$. Let $v \in \mathbb{R}^n$, and let $(x_i)_{i=1}^\infty$, $x_i \in K$, be a sequence such that $\lim_{i \rightarrow \infty} \pr{v}{x_i} = h_K(v)$. Then \[ \lim_{i \rightarrow \infty} d(F_v(K), x_i) = 0. \] \label{lem:nopt-conv} \end{lem} \begin{proof} Let us assume that $\lim_{i \rightarrow \infty} d(F_v(K), x_i) \neq 0$. Then there exists an $\epsilon > 0$ such that for some subsequence $(x_{\alpha_i})_{i=1}^\infty$ of $(x_i)_{i=1}^\infty$ we have that $d(F_v(K), x_{\alpha_i}) \geq \epsilon$. Since $(x_{\alpha_i})_{i=1}^\infty$ is an infinite sequence on a compact set $K$, there exists a convergent subsequence $(x_{\beta_i})_{i=1}^\infty$ where $\lim_{i \rightarrow \infty} x_{\beta_i} = x$ and $x \in K$. Now we note that $d(F_v(K), x) = \lim_{i \rightarrow \infty} d(F_v(K), x_{\beta_i}) \geq \epsilon$, where the first equality follows from the continuity of $d(F_v(K), \cdot)$. Since $d(F_v(K),x) > 0$ we have that $x \notin F_v(K)$. On the other hand, \[ h_K(v) = \lim_{i \rightarrow \infty} \pr{v}{x_i} = \lim_{i \rightarrow \infty} \pr{v}{x_{\beta_i}} = \pr{v}{x} \] and hence $x \in F_v(K)$, a contradiction. \end{proof} \begin{lem} Let $K$ be a compact convex set in $\mathbb{R}^n$. Let $v \in \mathbb{R}^n$, and let $(v_i)_{i=1}^\infty$, $v_i \in \mathbb{R}^n$, be a sequence such that $\lim_{i \rightarrow \infty} v_i = v$. Then for any sequence $(x_i)_{i=1}^\infty$, $x_i \in F_{v_i}(K)$, we have that \[ \lim_{i \rightarrow \infty} d(F_v(K), x_i) = 0. \] \label{lem:dir-conv} \end{lem} \begin{proof} We claim that $\lim_{i \rightarrow \infty} \pr{x_i}{v} = h_K(v)$. Since $K$ is compact, there exists $R \geq 0$ such that $K \subseteq RB^n$. Hence we get that \begin{align*} h_K(v) &= \lim_{i \rightarrow \infty} h_K(v_i) = \lim_{i \rightarrow \infty} \pr{v_i}{x_i} \\ &= \lim_{i \rightarrow \infty} \pr{v}{x_i} + \pr{v_i-v}{x_i} \leq \lim_{i \rightarrow \infty} \pr{v}{x_i} + \|v_i-v\|R = \lim_{i \rightarrow \infty} \pr{v}{x_i}, \end{align*} where the first equality follows by continuity of $h_K$ ($h_K$ is convex on $\mathbb{R}^n$ and finite valued). Since each $x_i \in K$, we get the opposite inequality $\lim_{i \rightarrow \infty} \pr{v}{x_i} \leq h_K(v)$ and hence we get equality throughout. Now by lemma \ref{lem:nopt-conv} we get that $\lim_{i \rightarrow \infty} d(F_v(K),x_i) = 0$ as needed. \end{proof} In the next lemma, vector $w$ will eventually represent the left-hand-side of the CG cut for $F_v$ that we want to lift and vectors $(s_i)_{i = 1}^{\infty}$ will represent a sequence of left-hand-side vectors that will be used to derive ``lifted" CG cuts for $K$. The conditions given in Lemma \ref{lem:2step-conv} on $s_i$ will be achieved as a consequence of Dirichlet's approximation theorem applied to $v$ and the result of the lemma will allow the original and lifted CG cuts to separate the same points in $\aff_I(F_v)$. \begin{lem} Let $K \subseteq \mathbb{R}^n$ be a compact convex set. Take $v, w \in \mathbb{R}^n$, $v \neq 0$. Let $(s_i,t_i)_{i=1}^\infty$, $s_i \in \mathbb{R}^n, t_i \in \mathbb{R}_+$ be a sequence such that \begin{equation} \label{eq:2step-cd} \begin{split} a.&~~ \lim_{i \rightarrow \infty} t_i = \infty, \\ b.&~~ \lim_{i \rightarrow \infty} s_i - t_iv = w. \\ \end{split} \end{equation} Then for every $\epsilon > 0$ there exists $N_\epsilon \geq 0$ such that for all $i \geq N_{\epsilon}$ \begin{equation} \label{four} h_K(s_i) + \epsilon \geq t_ih_{K}(v) + h_{F_v(K)}(w) \geq h_K(s_i) - \epsilon. \end{equation} \label{lem:2step-conv} \end{lem} \begin{proof} By \ref{eq:2step-cd} (a,b) we have that \begin{equation} \label{eq:2step-dir} \lim_{i \rightarrow \infty} \frac{s_i}{t_i} = v \end{equation} and that we may pick $N_1 \geq 0$ such that \begin{equation} \label{eq:2step-pd} \|s_i - t_iv\| \leq \|w\|+1 \leq C \quad \text{ for } i \geq N_1. \end{equation} Let $(x_i)_{i=1}^\infty$ be any sequence such that $x_i \in F_{s_i}(K)=F_{s_i/t_i}(K)$. For each $i \geq 1$, let $\tilde{x}_i = \argmin_{y \in F_v(K)} \|x_i-y\|$. By (\ref{eq:2step-dir}) and Lemma \ref{lem:dir-conv}, we may pick $N_2 \geq 0$ such that \begin{equation} \label{eq:2step-gc} d(F_v(K), x_i) = \|x_i-\tilde{x}_i\| \leq \frac{\epsilon}{2C} \quad \text{ for } i \geq N_2. \end{equation} Since $h_{F_v(K)}$ is a continuous function, we may pick $N_3 \geq 0$ such that \begin{equation} \label{eq:2step-of} |h_{F_v(K)}(s_i - t_iv) - h_{F_v(K)}(w)| \leq \frac{\epsilon}{2} \quad \text{ for }i \geq N_3. \end{equation} Let $N_\epsilon = \max \set{N_1,N_2,N_3}$. Now since $x_i \in F_{s_i}(K)$ and $\tilde{x}_i \in F_v(K)$ we have that \begin{equation} \label{eq:2step-optc} \pr{x_i}{s_i} \geq \pr{\tilde{x}_i}{s_i} \quad \text{ and } \pr{\tilde{x}_i}{t_iv} \geq \pr{x_i}{t_iv}. \end{equation} From (\ref{eq:2step-pd}), (\ref{eq:2step-gc}), (\ref{eq:2step-optc}) we get that for $i \geq N_\epsilon$ \begin{equation} \label{eq:2step-nopt} \begin{split} \pr{x_i}{s_i} - \pr{\tilde{x}_i}{s_i} &\leq \pr{x_i}{s_i} - \pr{\tilde{x}_i}{s_i} + \pr{\tilde{x}_i}{t_iv} - \pr{x_i}{t_iv} = \pr{x_i-\tilde{x}_i}{s_i-t_iv} \\ &\leq \|x_i-\tilde{x}_i\|\|s_i-t_iv\| \leq \left(\frac{\epsilon}{2C}\right)C = \frac{\epsilon}{2}. \end{split} \end{equation} From (\ref{eq:2step-nopt}) we see that for $i \geq N_\epsilon$ \begin{equation} \label{eq:2step-nopt2} h_K(s_i) \geq h_{F_v(K)}(s_i) \geq \pr{s_i}{\tilde{x}_i} \geq \pr{s_i}{x_i} - \frac{\epsilon}{2} = h_K(s_i) - \frac{\epsilon}{2}. \end{equation} Since $\pr{v}{\cdot}$ is constant on $F_v(K)$, we have that \begin{equation} \label{eq:2step-od} h_{F_v(K)}(s_i) = h_{F_v(K)}(s_i-t_iv+t_iv) = h_{F_v(K)}(s_i-t_iv) + t_ih_{F_v(K)}(v) = h_{F_v(K)}(s_i-t_iv) + t_ih_K(v) \end{equation} Combining (\ref{eq:2step-of}), (\ref{eq:2step-nopt2}) and (\ref{eq:2step-od}) we get that for $i \geq N_\epsilon$, \[ h_K(s_i) + \epsilon \geq t_ih_K(v) + h_{F_v(K)}(w) \geq h_K(s_i) - \epsilon \] as needed. \end{proof} \begin{thm}[Dirichlet's Approximation Theorem] Let $(\alpha_1,\ldots,\alpha_l) \in \mathbb{R}^l$. Then for every positive integer $N$, there exists $1 \leq n \leq N$ such that $ \max_{1 \leq i \leq l} |n\alpha_i - \round{n\alpha_i}| \leq 1/N^{1/l}$ \label{lem:dir-approx} \end{thm} \begin{prop} Let $K \subseteq \mathbb{R}^n$ be a compact and convex set, $v \in \mathbb{R}^n$ and $w \in \mathbb{Z}^n$. Then $\exists w' \in \mathbb{Z}^n$ such that $\set{x: \pr{w'}{x} \leq \floor{h_K(w')}} \cap \aff_I(H_v(K)) \subseteq \set{x: \pr{w}{x} \leq \floor{h_{F_v(K)}(w)}}\cap \aff_I(H_v(K))$. \label{lem:lift2} \end{prop} \begin{proof} First, by possibly multiplying $v$ by a positive scalar we may assume that $h_K(v) \in \mathbb{Z}$. Let $S = \aff_I(H_v(K))$ . We may assume that $S \neq \emptyset$, since otherwise the statement is trivially true. From Theorem~\ref{lem:dir-approx} for any $v \in \mathbb{R}^n$ there exists $(s_i,t_i)_{i=1}^\infty$, $s_i \in \mathbb{Z}^n$, $t_i \in \mathbb{N}$ such that (a.) $t_i \rightarrow \infty$ and (b.) $\|s_i - t_iv\| \rightarrow 0$. Now define the sequence $(w_i,t_i)_{i=1}^\infty$, where $w_i = w + s_i$, $i \geq 1$. Note that the sequence $(w_i,t_i)$ satisfies \eqref{eq:2step-cd} and hence by Lemma \ref{lem:2step-conv} for any $\epsilon > 0$, there exists $N_\epsilon$ such that \eqref{four} holds. Let $\epsilon = \frac{1}{2}\bigl(1-(h_{F_v(K)}(w)-\lfloor h_{F_v(K)}(w)\rfloor)\bigr)$, and let $N_1 = N_\epsilon$. Note that $\floor{h_{F_v(K)}(w) + \epsilon} = \floor{h_{F_v(K)}(w)}$. Hence, since $h_K(v) \in \mathbb{Z}$ by assumption, for all $i \geq N_1$ we have that \begin{equation*} \floor{h_K(w_i)} \leq \floor{t_ih_K(v) + h_{F_v(K)}(w) + \epsilon} = t_ih_K(v) + \floor{h_{F_v(K)}(w) + \epsilon} = t_ih_K(v) + \floor{h_{F_v(K)}(w)}. \end{equation*} Now pick $z_1,\dots,z_k \in S$ such that $\aff(z_1,\dots,z_k) = S$ and let $R = \max \set{\|z_j\|: 1 \leq j \leq k}$. Choose $N_2$ such that $\|w_i - t_i v - w\| \leq \frac{1}{2R}$ for $i \geq N_2$. Now note that for $i \geq N_2$, \begin{equation*} |\pr{z_j}{w_i} - \pr{z_j}{t_iv + w}| = |\pr{z_j}{w_i - t_iv - w}| \leq \|z_j\|\|w_i - t_iv-w\| \leq R \frac{1}{2R} = \frac{1}{2} \quad \forall j \in\{1,\dots,k\}. \end{equation*} Next note that since $z_j, w_i \in \mathbb{Z}^n$, $\pr{z_j}{w_i} \in \mathbb{Z}$. Furthermore, $t_i \in \mathbb{N}$, $\pr{v}{z_i} = h_K(v) \in \mathbb{Z}$ and $w \in \mathbb{Z}^n$ implies that $\pr{z_j}{t_i v + w} \in \mathbb{Z}$. Given this, we must have $\pr{z_j}{w_i} = \pr{z_j}{t_i v + w} \quad \forall j \in [k],\, i\geq 1$ and hence we get $\pr{x}{w_i} = \pr{x}{t_i v + w} \quad \forall x \in S,\, i\geq 1$. Let $w' = w_i$ where $i = \max \set{N_1,N_2}$. Now examine the set $L = \set{x: \pr{x}{w'} \leq \floor{h_K(w')}} \cap S$. Here we get that $\pr{x}{w_i} \leq t_ih_K(v) + \lfloor h_{F_v(K)}(w) \rfloor$ and $ \pr{x}{v} = h_K(v)$ for all $x \in L$ Hence, we see that $\pr{x}{w_i-t_iv} \leq \lfloor h_{F_v(K)}(w)\rfloor$ for all $x \in L$. Furthermore, since $\pr{x}{w_i-t_iv} = \pr{x}{w}$ for all $x \in L \subseteq \aff(S)$, we have that $\pr{x}{w} \leq \floor{h_{F_v(K)}(w)}$ for all $x \in L$, as needed. \end{proof} \subsection{Separating all point in $F_v \setminus \aff_I(H_v)$}\label{sec:sepirrational} Replacing $\aff(F)$ by $\aff_I(H_v)$ in the generalization of \eqref{toprovelifting} strengthens property by replacing $F$ with $H_v$, but weakens it by replacing $\aff(\cdot)$ by $\aff_I(\cdot)$. Because of this we need to explicitly deal with the points in $F_v\setminus \aff_I(H_v)$. In this section, we show that points in $F_v\setminus \aff_I(H_v)$ can be separated by using a finite number of CG cuts in Proposition \ref{prop:killirr}. In prove this, we need the Kronecker simultaneous approximation theorem that is stated next. See Niven~\cite{niven1963} or Cassels~\cite{cassels1972} for a proof. \begin{thm} Let $(x_1,\dots,x_n) \in \mathbb{R}^n$ be such that the numbers $x_1,\dots,x_n,1$ are linearly independent over $\mathbb{Q}$. Then the set $\set{(nx_1 \pmod 1, \dots, nx_n \pmod 1): n \in \mathbb{N}}$ is dense in $[0,1)^n$. \label{thm:dense} \end{thm} The following lemmas conveniently normalize vector $v$ defining $F_v$ and $H_v$. \begin{lem} Let $K \subseteq \mathbb{R}^n$ be a closed convex set, and let $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be an invertible linear transformation. Then $h_K(v) = h_{TK}(T^{-t}v)$ and $F_v(K) = T^{-1}(F_{T^{-t} v}(TK))$ for all $v \in \mathbb{R}^n$. Furthermore, if $T$ is a unimodular transformation, then $ CC(K) = T^{-1}(CC(TK))$. \label{lem:unimod} \end{lem} \begin{proof} Observe that \[ h_{TK}(T^{-t}v) = \sup_{x \in TK} \pr{T^{-t}v}{x} = \sup_{x \in K} \pr{T^{-t}v}{Tx} = \sup_{x \in K} \pr{v}{x} = h_K(v). \] Now note that \begin{align*} T^{-1}(F_{T^{-t} v}(TK)) &= T^{-1} \left(~\set{x: ~x \in TK, ~h_{TK}(T^{-t}v) = \pr{T^{-t}v}{x}}~\right) \\ &= \set{x: ~Tx \in TK,~ h_{TK}(T^{-t}v) = \pr{T^{-t}v}{Tx}} = \set{x: ~x \in K, ~h_K(v) = \pr{v}{x}} \\ &= F_v(K). \end{align*} Finally, \begin{align*} T^{-1}(CC(TK)) &= T^{-1}\left(\set{x: x \in TK, ~\pr{v}{x} \leq \floor{h_{TK}(v)} ~ \forall ~ v \in \mathbb{Z}^n}\right) \\ &= \set{x: Tx \in TK, ~\pr{v}{Tx} \leq \floor{h_{TK}(v)} ~ \forall ~ v \in \mathbb{Z}^n} \\ &= \set{x: Tx \in TK, ~\pr{T^{-t}v}{Tx} \leq \floor{h_{TK}(T^{-t}v)} ~ \forall v \in \mathbb{Z}^n} \\ &= \set{x: x \in K, ~\pr{v}{x} \leq \floor{h_{K}(v)} ~ \forall v \in \mathbb{Z}^n} = CC(K). \end{align*} \end{proof} \begin{lem} Take $v \in \mathbb{R}^n$. Then there exists an unimodular transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\lambda \in \mathbb{Q}_{> 0}$ such that for $v' = \lambda T v$ we get that \begin{equation} \label{eq:irr-red} v' = \left(~\underbrace{0,\dots,0}_{t ~ \mathrm{times}}, ~\underbrace{1}_{s ~ \mathrm{times}},~\alpha_1,\dots,\alpha_{r}\right), \end{equation} where $t,r \in \mathbb{Z}_+$, $s \in \set{0,1}$, and $\{1,\alpha_1,\dots,\alpha_r\}$ are linearly independent over $\mathbb{Q}$. Furthermore, we have that $\mathcal{D}(v) = \inf \set{\dim(W): v \in W, W = \set{x \in \mathbb{R}^n: Ax = 0}, A \in \mathbb{Q}^{m \times n}} = s + r$. \label{lem:irr-red} \end{lem} \begin{proof} Choose a permutation matrix $P$ such that the rational entries of $Pa$ form a contiguous block starting from the first entry of $Pa$, i.e. let $k \in \set{0,\dots,n}$ such that $(Pa)_1,\dots,(Pa)_k \in \mathbb{Q}$ and $(Pa)_{k+1},\dots,(Pa)_n \in \mathbb{R} \setminus \mathbb{Q}$. Now we set our initial transformation $T \leftarrow P$, $\lambda \leftarrow 1$, and working vector $a' \leftarrow Pa$. In what follows, we will apply successive updates to $T$,$\lambda$ and $a'$ such that we maintain that $T$ is unimodular, $\lambda \in \mathbb{Q}_{> 0}$, and $a' = \lambda T a$. First consider a vector $a' \in \mathbb{R}^n$ such that $a'_1,\dots,a'_k$ are rational and $(1,a'_{k+1},\dots,a'_n)$ are linearly independent over $\mathbb{Q}$. If $k=0$, i.e. $(1, a'_1,\dots,a'_n)$ are linearly independent over $\mathbb{Q}$, then we are done. We may therefore assume that $k \geq 1$. Similarly, if $(a'_1,\dots,a'_k) = 0^k$, then again we are done. Now let $a'_R = (a'_1,\dots,a'_k)$ and $a'_I = (a'_{k+1},\dots, a'_n)$. By our assumptions, we note that $a'_R \neq 0$. Via an appropriate scaling $\lambda' \in \mathbb{Q}_{> 0}$, we may achieve $\lambda' a'_R \in \mathbb{Z}^k$ and $\mathrm{gcd}(\lambda' a'_1,\dots, \lambda' a_k) = 1$. Since $\lambda' \in \mathbb{Q}$, note that $(1,a'_{k+1},\dots,a'_n)$ are linearly independent over $\mathbb{Q}$ iff $(1, \lambda' a'_{k+1},\dots, \lambda' a'_n)$ are. Set $\lambda \leftarrow \lambda' \lambda$ and $a' \leftarrow \lambda' a'$. Next, applying the Euclidean algorithm on the vector $a'_R$, we get a unimodular transformation $E$ such that \[ E a'_R = (0^{k-1}, \mathrm{gcd}(a'_1,\dots,a'_k)) = (0^{k-1}, 1). \] Now define the unimodular transformation $T'$, where \[ T'(x) = (E(x_1,\dots,x_k), x_{k+1},\dots,x_n). \] By construction, note that $((Ta')_1,\dots,(Ta')_k) = E a'_R = (0^{k-1}, 1)$. Next note that $((Ta')_{k+1},\dots,(Ta')_n)$ are linearly independent over $\mathbb{Q}$. Letting $T \leftarrow T' T$ and $a' \leftarrow T' a'$, we have that $a' = \lambda T a$ satisfies the required form. Given the above case analysis, we are left with the case where $a'_R = (a'_1,\dots,a'_k) \in \mathbb{Q}^k$, $a'_I = (a'_{k+1},\dots,a'_{n}) \in (\mathbb{R} \setminus \mathbb{Q})^{n-k}$ and where $(1,a'_{k+1},\dots,a'_{n})$ have a linear dependency over $\mathbb{Q}$. Now after an appropriate scaling of this dependency, we get numbers $c_0 \in \mathbb{Q}, c \in \mathbb{Z}^{n-k} \setminus \set{0}$, $\mathrm{gcd}(c_1,\dots,c_{n-k}) = 1$, and where \[ \pr{a_I}{c} = c_0 \] Applying the Euclidean algorithm on $c$, we get a unimodular matrix $E$ such that \[ E c = (\mathrm{gcd}(c_1,\dots,c_{n-k}), 0^{n-k-1}) = (1, 0^{n-k-1}) \] Let $\hat{a} = E^{-t} a'_I$. Note that $E$ is unimodular iff $E^{-t}$ is unimodular. We get that \[ \pr{a_I}{c} = c_0 \mathbb{R}ightarrow \pr{E^{-t} a_I}{E c} = c_0 \mathbb{R}ightarrow \hat{a}_1 = c_0 \] Hence we see that $\hat{a}_1 = c_0 \in \mathbb{Q}$. Let $T'$ be the unimodular transformation defined by \[ T'(x) = (x_1,\dots,x_k, E^{-t}(x_{k+1},\dots,x_n)) \] Here $T'$ is the identity on the first $k$ coordinates, and acts like $E^{-t}$ on the last $n-k$ coordinates. Note that $((T'a')_1,\dots,(T'a')_k) = (a'_1,\dots,a'_k) \in \mathbb{Q}^k$. Next $((T'a')_{k+1},\dots,(T'a')_n) = E^{-t} a'_I = \hat{a}$, and $\hat{a}_1 \in \mathbb{Q}$. Hence $T'a'$ has at least one more rational coefficient than $a'$. By repeating the above operation suitable number of times, we obtain a vector $a' \in \mathbb{R}^n$ such that $a'_1,\dots,a'_k$ are rational and $(1,a'_{k+1},\dots,a'_n)$ are linearly independent over $\mathbb{Q}$. By the previous analysis, there exists unimodular transformation $T''$, $\lambda' \in \mathbb{Q}$ such that $\lambda'T''T'a'$ satisfies the required form. Letting $T \leftarrow T''T'T$, $\lambda \leftarrow \lambda'\lambda$, and $a' \leftarrow \lambda'T''T'a'$, we get the desired result. For proving the second part of the result, we first claim that $\mathcal{D}(a') = \mathcal{D}(a)$. To see this, note that \[ Aa' = 0 \Leftrightarrow A(\lambda Ta) = 0 \Leftrightarrow ATa = 0 \quad \text{ and } \quad Aa = 0 \Leftrightarrow A\left(\frac{1}{\lambda} T^{-1} a'\right) = 0 \Leftrightarrow AT^{-1}a' = 0 \] since $T$ is invertible and $\lambda \neq 0$. Since both $AT,AT^{-1}$ are rational, this gives that $\mathcal{D}(a') = \mathcal{D}(a)$ as needed. Hence we need only show that $\mathcal{D}(a') = s+t$. Take $y \in \mathbb{Q}^n$ such that $\pr{y}{a'} = 0$. Note that $a' = (0^t, 1^s, \alpha_1,\dots,\alpha_r)$ where $(1,\alpha_1,\dots,\alpha_r)$ are linearly independent over $\mathbb{Q}$. If $s=0$, then $\sum_{i=1}^r y_{t+i}\alpha_i = 0$. Since $y \in \mathbb{Q}^n$, this gives a linear dependence of $(\alpha_1,\dots,\alpha_r)$ over $\mathbb{Q}$, and hence by assumption we must have that $y_{t+i} = 0$ for $1 \leq i \leq r$. Otherwise if $s=1$, we get $y_{t+1} + \sum_{i=1}^r y_{t+i+1}\alpha_i = 0$, which gives a linear dependence of $(1,\alpha_1,\dots,\alpha_r)$ over $\mathbb{Q}$. Therefore $y_{t+i} = 0$ for $1 \leq i \leq t+1$. Hence in both cases, we get that $y_{t+i} = 0$ for $1 \leq i \leq r+s$. Next note that for $y \in \mathbb{Q}^t \times 0^{n-t}$, we have that $\pr{y}{a'} = 0$ since $a'_1,\dots,a'_r = 0$ by assumption. By the previous observations, we obtain that \[ L := \set{y \in \mathbb{Q}^n: \pr{y}{a'} = 0} = \mathbb{Q}^t \times 0^{n-t} = \mathbb{Q}^t \times 0^{s+r}. \] Now let $W \subseteq \mathbb{R}^n$ denote the linear subspace $W = \set{x \in \mathbb{R}^n: x_i = 0, 1 \leq i \leq t}$. Note that $a' \in W$, and hence $\mathcal{D}(a') \leq \dim(W) = s+r$. Now take any $M = \set{x \in \mathbb{R}^n: Ax = 0}$, such that $a' \in M$ and $A \in \mathbb{Q}^{m \times n}$. We claim that $W \subseteq M$. Let $a_1,\dots,a_m \in \mathbb{Q}^n$ denote the rows of $A$. Since $a' \in M$, we have $\pr{a_i}{a'} = 0 ~ \forall ~ i \in \{1,\dots,m\}$. Hence we must have that $a_i \in L = \mathbb{Q}^t \times 0$. Since $W = 0^t \times \mathbb{R}^{s+r}$, we have that for all $x \in W$, $\pr{a_i}{x} = 0$, and hence $W \subseteq L$. Hence \[ \dim(L) \geq \dim(W) = s+r, \] from which conclude that $\mathcal{D}(a') = s+r$ as needed. \end{proof} We now show that points belonging to $F_v\setminus \aff_I(H_v)$ can be separated by using a finite number of CG cuts. The proof can be viewed as follows: We select $\mathcal{D}(v)+1$ vectors whose conic span is the linear subspace corresponding to the irrational components. Using each of these directions as guides, we scale the vector $v$ (corresponding to the face $F_v$) by integers and use the Kronecker theorem to compute a tiny ``correction vector" to be added to the scaled version of $v$. In this way we produce $\mathcal{D}(v)+1$ integer vectors that are very close in angle to $v$. These integer vectors have the property that the CG cuts corresponding to them separate points in $F_v\setminus \aff_I(H_v)$. In all this, Lemma \ref{lem:unimod} is crucial as it allows to simplify the choice of case analysis. \begin{prop}\label{prop:killirr} Let $K \subseteq \mathbb{R}^n$ be a compact convex set and $v \in \mathbb{R}^n$. Then there exists $C \subseteq \mathbb{Z}^n$, $|C| \leq \mathcal{D}(v)+1$, such that \begin{alignat*}{3} \mathbb{C}C(K,C) \cap H_v(K) &\subseteq \aff_I(H_v(K))\\ \mathbb{C}C(K,C)&\subseteq \set{x: \pr{v}{x} \leq h_K(v)}. \end{alignat*} \end{prop} \begin{proof} By scaling $v$ by a positive scalar if necessary, we may assume that $h_K(v) \in \set{0,1,-1}$. Let $T$ and $\lambda$ denote the transformation and scaling promised for $v$ in Lemma \ref{lem:irr-red}. Note that \[ T^{-t} \set{x \in \mathbb{R}^n: \pr{v}{x} = h_K(v)} = \set{x \in \mathbb{R}^n: \pr{v}{T^tx} = h_K(v)} = \set{x \in \mathbb{R}^n: \pr{\lambda Tv}{x} = h_{T^{-t}K}(\lambda Tv)}. \] Now let $v' = \lambda Tv$ and $b' = h_{T^{-t}K}(\lambda Tv)$. By Lemma \ref{lem:unimod}, it suffices to prove the statement for $v'$ and $K' = T^{-t} K$. Now $v'$ has the form \eqref{eq:irr-red} where $t,r \in \mathbb{Z}_+$, $s \in \set{0,1}$, and $(1,\alpha_1,\dots,\alpha_r)$ are linearly independent over $\mathbb{Q}$. For convenience, let $k = s+t$, where we note that $v'_{k+1},\dots,v'_{k+r} = (\alpha_1,\dots,\alpha_r)$. \paragraph{Claim 1:} Let $S = \set{x \in \mathbb{Z}^n: \pr{v'}{x} = b'}$. Then $S$ satisfies one of the following \begin{enumerate} \item $S = \mathbb{Z}^t \times b' \times 0^r$: $s=1, b' \in \mathbb{Z}$. \item $S = \mathbb{Z}^t \times 0^r$: $s=0, b' = 0$. \item $S = \emptyset$: $s=0, b' \neq 0$ or $s=1, b' \notin \mathbb{Z}$. \end{enumerate} Note that $b' = h_{T^{-t}K}(\lambda Tv) = \lambda h_K(v) \in \set{0, \pm \lambda} \subseteq \mathbb{Q}$. We first see that \[ (s=1):~ b'= \pr{v'}{x} = x_k + \sum_{i=1}^r x_{k+i}\alpha_i, \quad (s=0):~ b'=\pr{v'}{x} = \sum_{i=1}^r x_{k+i}\alpha_i. \] Now if $x \in S$, then \[ (s=1):~ (x_k-b') + \sum_{i=1}^r x_{k+i}\alpha_i = 0, \quad (s=0): (-b') + \sum_{i=1}^r x_{k+i}\alpha_i = 0. \] Since $b' \in \mathbb{Q}$, and $x \in \mathbb{Z}^n$, in both cases the above equations give us a linear dependence of $(1,\alpha_1,\dots,\alpha_r)$ over $\mathbb{Q}$. Since by assumption $(1,\alpha_1,\dots,\alpha_r)$ are linearly independent over $\mathbb{Q}$, we have that \[ (s=0,1):~ x_{k+i} = 0, 1 \leq i \leq r \quad (s=1):~ x_k = b' \quad (s=0):~ b' = 0. \] If $s=1$, then we must have that $b' \in \mathbb{Z}$, since $x_k = b'$ and $x \in \mathbb{Z}^n$. From this we immediately recover case $(1)$. If $s=0$, then the conditions $b' = 0$ and $x_{k+i} = 0$, $1 \leq i \leq r$, verify case $(2)$. If we are neither in case $(1)$ or $(2)$, then by the above analysis $S$ must be empty, and so we are done. \paragraph{Claim 2:} Let $I = \set{n v' \pmod 1: n \in N}$. Then Theorem \ref{thm:dense} implies that $I$ is dense in $0^k \times [0,1)^r$.\linebreak We first note that $v'_1,\dots,v'_k \in \mathbb{Z}$ and hence $v'_1,\dots,v'_k \equiv 0 \pmod 1$. Next note that $(1,\alpha_1,\dots,\alpha_r)$ are linearly independent over $\mathbb{Q}$, and hence by Theorem \ref{thm:dense} we have that $\set{n (\alpha_1,\dots,\alpha_r): n \in N}$ is dense over $[0,1)^r$. Putting the last two statements together immediately yields the claim. \paragraph{Claim 3:} There exists $a_1,\dots,a_{r+1} \subseteq \mathbb{Z}^n$ and $\lambda_1,\dots,\lambda_{r+1} \geq 0$ such that $\sum_{i=1}^{r+1} \lambda_i a_i = v'$ and $\sum_{i=1}^{r+1} \lambda_i\floor{h_K'(a_i)} \leq b'$. \quad\linebreak \indent Since $K'$ is compact, there exists $R > 0$ such that $K' \subseteq RB^n$. Take the subspace $W = 0^k \times \mathbb{R}^r$. Let $w_1,\dots,w_{r+1} \in W \cap S^{n-1}$, be any vectors such that for some $0 < \epsilon < 1$ we have $\sup_{1 \leq i \leq r+1} \pr{w_i}{d} \geq \epsilon$ for all $d \in S^{n-1} \cap W$ (e.g. $w_1,\dots,w_{r+1}$ are the vertices of a scaled isotropic $r$-dimensional simplex). Let $a = \frac{1}{8} \min \set{\frac{1}{R}, \epsilon}$, and $b = \frac{1}{2}\epsilon a$. Now, for $1 \leq i \leq r+1$ define $E_i = \set{x: x \in a w_i + b (B^n \cap W) \pmod 1}$. Since $W = 0^k \times \mathbb{R}^r$, note that $E_i \subseteq 0^k \times [0,1)^r$. By Claim 2 the set $I$ is dense in $0^k \times [0,1)^r$. Furthermore each set $E_i$ has non-empty interior with respect to the subspace topology on $0^k \times [0,1)^r$. Hence for all $i$, $1 \leq i \leq r + 1$, we can find $n_i \in \mathbb{N}$ such that $n_iv' \pmod 1 \in E_i$. Now $n_iv' \pmod 1 \in E_i$, implies that for some $\delta'_i \in E_i$, $n_iv' - \delta'_i \in \mathbb{Z}^n$. Furthermore $\delta'_i \in E_i$ implies that there exists $\delta_i \in a w_i + b(B^n \cap W)$ such that $\delta_i' - \delta_i \in \mathbb{Z}^n$. Hence $(n_iv' - \delta'_i) + (\delta'_i - \delta_i) = n_iv' - \delta_i \in \mathbb{Z}^n$. Let $a_i = n_iv' - \delta_i$. Note that $ \|a_i - n_iv'\| = \|-\delta_i\| \leq a + b \leq 2a \leq 1/(4R)$. We claim that $\floor{h_{K'}(a_i)} \leq h_{K'}(n_iv')$. First note that $h_{K'}(n_iv') = n_ib'$. Since we assume that $S \neq \emptyset$, we must have that $b' \in \mathbb{Z}$ and hence $n_ib' \in \mathbb{Z}$. Now note that \begin{align*} h_{K'}(a_i) &= h_{K'}((a_i-n_iv')+n_iv') \leq h_{K'}(n_iv') + h_{K'}(a_i-n_iv') = n_ib' + h_{K'}(-\delta_i) \\ &\leq n_ib' + h_{RB^n}(-\delta_i) \leq n_ib' + R\|\delta_i\| \leq n_ib' + R\left(\frac{1}{4R}\right) = n_ib' + \frac{1}{4}. \end{align*} Therefore we have that $\floor{h_{K'}(a_i)} \leq \floor{n_ib' + \frac{1}{4}} = n_ib' = h_{K'}(n_iv')$, since $n_ib' \in \mathbb{Z}$. We claim that $\frac{a\epsilon}{4}B^n \cap W \subseteq \mathrm{conv} \set{\delta_1,\dots,\delta_{r+1}}$. First note that by construction, $\mathrm{conv}\set{\delta_1,\dots,\delta_{r+1}} \subseteq W$. Hence if the conclusion is false, then by the separator theorem there exists $d \in W \cap S^{n-1}$ such that $h_{\frac{a\epsilon}{4}B^n \cap W}(d) = \frac{a\epsilon}{4} > \sup_{1 \leq i \leq r+1} \pr{d}{\delta_i}$. For each $i$, $1 \leq i \leq r+1$, we write $\delta_i = a w_i + bz_i$ where $\|z_i\| \leq 1$. Now note that \begin{align*} \sup_{1 \leq i \leq r+1} \pr{d}{\delta_i} &= \sup_{1 \leq i \leq r+1} \pr{d}{a w_i + bz_i} = \sup_{1 \leq i \leq r+1} a \pr{d}{w_i} + b\pr{d}{z_i} \\ &\geq \sup_{1 \leq i \leq r+1} a\pr{d}{w_i} - b\|d\|\|z_i\| \geq a \epsilon - b = \frac{a\epsilon}{2} > \frac{a\epsilon}{4}, \end{align*} a contradiction. Hence there exists $\lambda_1,\dots,\lambda_{r+1} \geq 0$ and $\sum_{i=1}^{r+1} \lambda_in_i = 1$ such that $\sum_{i=1}^{r+1} \lambda_i \delta_i = 0$. Now we see that \begin{equation} \sum_{i=1}^{r+1} \lambda_ia_i = \sum_{i=1}^{r+1} \lambda_in_iv' + \sum_{i=1}^{r+1} \lambda_i(a_i-n_iv') = \left(\sum_{i=1}^{r+1} \lambda_in_i\right) v' - \sum_{i=1}^{r+1} \lambda_i \delta_i = \left(\sum_{i=1}^{r+1} \lambda_in_i\right) v'. \end{equation} Next note that \begin{equation} \sum_{i=1}^{r+1} \lambda_i\floor{h_{K'}(a_i)} \leq \sum_{i=1}^{r+1} \lambda_ih_{K'}(n_iv') = h_{K'}\left(\left(\sum_{i=1}^{r+1} \lambda_in_i\right) v'\right). \end{equation} \paragraph{Case 2: $S = \emptyset$.} \hspace{1em} \noindent The proof here shall proceed very similarly to the one above, with the exception that we need to do some extra work to guarantee a strict inequality. If $s=0$, then since $S = \emptyset$ we must have that $b' \neq 0$. Let $v^z = \frac{1}{|b'|}v'$ and $b^z = \mathrm{sign}(b')$, and $v^f = \frac{1}{2|b'|}v'$ and $b^f = \frac{1}{2}\mathrm{sign}(b')$. Note that $h_{K'}(v^z) = b^z \in \set{\pm1}$ and $h_{K'}(v^f) = b^f \in \set{\pm 1/2}$. Furthermore, since $b' \in \mathbb{Q}$, we see that \[ (1,v^z_{k+1},\dots,v^z_{k+r}) = (1,\frac{1}{2|b'|}\alpha_1,\dots,\frac{1}{2|b'|}\alpha_r) \] are still linearly independent over $\mathbb{Q}$, and that $v^z_1,\dots,v^z_k = v'_1,\dots,v'_k= 0 \in \mathbb{Z}$. Next if $s = 1$, then $b' \in \mathbb{Q} \setminus \mathbb{Z}$. Let $c_1 \in \mathbb{Z}$ denote the least positive integer such that $c_1b' \in \mathbb{Z}$ and let $c_2 \in \mathbb{Z}$ denote the least positive integer such that $\frac{1}{3} \leq c_2b' \pmod 1 \leq \frac{2}{3}$ (always exists since $b' \neq 0$). Let $v^z = c_1v'$ and $b^z = c_1b'$, and let $v^f= c_2v'$ and $b^f = c_2b'$. Again we have that $h_{K'}(v^z) = b^z \in \mathbb{Z}$, and $h_{K'}(v^f) = b^f$ (since $c_1,c_2 \geq 0$). Lastly, since $c_1,c_2 \in \mathbb{Z}$, we note that $v^z_1,\dots,v^z_{k-1} = v^f_1,\dots, v^f_{k-1} = 0 \in \mathbb{Z}$, $v^z_k = c_1, v^f_k = c_2 \in \mathbb{Z}$, and $(1,v^z_{k+1},\dots,v^z_{k+r}) = (1,c_1\alpha_1,\dots,c_1\alpha_r)$ are still linearly independent over $\mathbb{Q}$. Now let $I' = \set{nv^z \pmod 1: n \in \mathbb{N}}$. Using the proof of Claim $2$, we see that $I'$ is dense in $0^k \times [0,1)^r$. Furthermore since $v^f \mod 1 \in 0^k \times [0,1)^r$, we have that $I' + v^f \pmod 1$ is also dense in $0^k \times [0,1)^r$. Note that $I' + v^f \pmod 1 = \set{(nc_1 + c_2)v' \pmod 1: n \in \mathbb{N}}$. Let $w_1,\dots,w_{l+1}$, $E_1,\dots,E_{l+1}$ be defined identically as in Case 1. Via the same density argument as in case $1$, we may pick $n_i \in \mathbb{N}$, such that $(n_ic_1 + c_2)v' \in E_i$. Again we define $a_1,\dots,a_{r+1}$ in exactly the same way as in Case $1$. To conclude the proof of the claim, we need only show that $\floor{h_{K'}(a_i)} \leq \floor{n_ib' + \frac{1}{4}} = n_ib' = h_{K'}(n_iv')$ holds with a strict inequality in this case. The exact same argument gives us now that \begin{equation} h_{K'}(a_i) \leq (n_ic_1+c_2)b' + \frac{1}{4}. \end{equation} Now $n_ic_1b' = n_ib^z \in \mathbb{Z}$ and $\frac{1}{3} \leq c_2b' \pmod 1 \leq \frac{2}{3}$. Therefore \begin{equation} \floor{h_{K'}(a_i)} < (n_ic_1+c_2)b', \end{equation} as needed. \paragraph{Claim 4:} Let $C = \{a_i\}_{i=1}^{r+1}$ for the $a_i$'s from Claim 3. Then $ \mathbb{C}C(K,C) \cap \set{x: \pr{v'}{x} = b'} \subseteq \aff(S)$. If $S = \emptyset$, note that by the Claim $3$, we have that \[ \sup \set{\pr{v'}{x}: x \in \mathbb{R}^n, \pr{a_i}{x} \leq \floor{h_{K'}(a_i)}, 1 \leq i \leq r+1} < b', \] and hence $\mathbb{C}C(K,C) \cap \set{x: \pr{v'}{x} = b'} = \emptyset$ as needed. If $S \neq \emptyset$, examine the set \quad\linebreak \indent Examine the set $P = \set{x: \pr{v'}{x} = b', \pr{a_i}{x} \leq \floor{h_{K'}(a_i)}, 1 \leq i \leq l+1}$. From the proof of Claim $3$, we know that for each $i$, $1 \leq i \leq r+1$, we have $\floor{h_{K'}(a_i)} \leq h_{K'}(n_iv') = n_ib'$ and hence $\pr{n_iv' - a_i}{x} = \pr{\delta_i}{x} \geq 0$, is a valid inequality for $P$. Now, from the proof of Claim $3$, we have \begin{equation} \frac{a\epsilon}{4}B^n \cap W \subseteq \mathrm{conv} \set{\delta_1,\dots,\delta_{r+1}}. \label{eq:lift1-bc} \end{equation} We claim that for all $H \subseteq \set{1,\dots,r+1}$, $|H| = r$, the set $\set{\delta_i: i \in H}$ is linearly independent. Assume not, then WLOG we may assume that $\delta_1,\dots,\delta_r$ are not linearly independent. Hence there exists $d \in S^{n-1} \cap W$, such that $\pr{d}{\delta_i} = 0$ for all $1 \leq i \leq n$. Now by possibly switching $d$ to $-d$, we may assume that $\pr{d}{\delta_{r+1}} \leq 0$. Hence we get that $\sup_{1 \leq i \leq r+1} \pr{d}{\delta_i} \leq 0$ in contradiction to (\ref{eq:lift1-bc}). Now let $\lambda_1,\dots,\lambda_{r+1} \geq 0$, $\sum_{i=1}^{r+1} \lambda_in_i = 1$ be a combination such that $\sum_{i=1}^{r+1} \lambda_i \delta_i = 0$. Note that $\lambda_1,\dots,\lambda_{r+1}$ forms a linear dependency on $\delta_1,\dots,\delta_{r+1}$, and hence by the previous claim we must have that $\lambda_i > 0$ for all $1 \leq i \leq r+1$. We claim for $P\subseteq W^\perp$. To see this, note that $ 0 = \pr{x}{0} = \pr{x}{\sum_{i=1}^{r+1} \lambda_i \delta_i} = \sum_{i=1}^{r+1} \lambda_i \pr{x}{\delta_i}$ for every $x\in P$. Now since $\mathrm{span}(\delta_1,\dots,\delta_{r+1}) = W$, we see that $\pr{x}{\delta_i} = 0$ for all $1 \leq i \leq r+1$ iff $x \in W^\perp$. Hence if $x \notin W^\perp$, then by the above equation and the fact that $\lambda_i > 0$ for all $i \in \set{1,\dots,r + 1}$, there exists $i,j \in \set{1,\dots,r + 1}$ such that $\pr{x}{\delta_i} > 0$ and $\pr{x}{\delta_j} < 0$. But then $x \notin P$, since $\pr{x}{\delta_j} < 0$, a contradiction. Now $W = 0^k \times \mathbb{R}^r$, hence $W^\perp = \mathbb{R}^k \times 0^r$. To complete the proof we see that $P \subseteq \set{x: x \in \mathbb{R}^k \times 0^r, \pr{v'}{x} = b'} = \aff(S)$. \end{proof} \subsection{Combining results of Section \ref{sec:lift} and \ref{sec:sepirrational} to show $\mathbb{C}C(F_v,S) = H_v \cap \mathbb{C}C(K,S')$}\label{sec:facehomogenity} \begin{prop}\label{PPPP}\label{thm:cg-lift} Let $K \subseteq \mathbb{R}^n$ be a compact convex set. Take $v \in \mathbb{R}^n$. Assume that $\mathbb{C}C(F_v(K))$ is finitely generated. Then $\exists~ S \subseteq \mathbb{Z}^n$, $|S| < \infty$, such that $\mathbb{C}C(K,S)$ is a polytope and \begin{alignat}{3} \label{331} \mathbb{C}C(K,S) \cap H_v(K) &= \mathbb{C}C(F_v(K))\\ \label{332} \mathbb{C}C(K,S) &\subseteq \set{x: \pr{v}{x} \leq h_K(v)}. \end{alignat} \ \end{prop} \begin{proof} The right to left containment in \eqref{331} is direct from $\mathbb{C}C(F_v(K))\subseteq \mathbb{C}C(K,S)$ as every CG cut for $K$ is a CG cut for $F_v(K)$. For the reverse containment and for \eqref{332} we proceed as follows. Using Proposition \ref{prop:killirr} there exists $S_1 \subseteq \mathbb{Z}^n$ such that $\mathbb{C}C(K,S_1) \cap H_v(K) \subseteq \aff_I(H_v(K))$ and $ \mathbb{C}C(K,S_1) \subseteq \set{x: \pr{v}{x} \leq h_K(v)}$. Next let $G \subseteq \mathbb{Z}^n$ be such that $\mathbb{C}C(F_v(K), G) = \mathbb{C}C(F_v(K))$. For each $w \in G$, by Proposition \ref{lem:lift2} there exists $w' \in \mathbb{Z}^n$ such that \begin{equation*} \mathbb{C}C(K, w') \cap \aff_I(H_v(K)) \subseteq \mathbb{C}C(F_v(K), w) \cap \aff_I(H_v(K)). \end{equation*} For each $w \in G$, add $w'$ above to $S_2$. Now note that \begin{align*} \mathbb{C}C(K,S_1 \cup S_2) \cap H_v(K) &= \mathbb{C}C(K,S_1) \cap \mathbb{C}C(K,S_2) \cap H_v(K) \\ &\subseteq \mathbb{C}C(K,S_2) \cap \aff_I(H_v(K)) = \mathbb{C}C(F_v(K), G) \cap \aff(A) \subset \mathbb{C}C(F_v(K)). \end{align*} Now let $S_3 = \set{\pm e_i: 1 \leq i \leq n}$. Note that since $K$ is compact $\mathbb{C}C(K,S_3)$ is a cuboid with bounded side lengths, and hence is a polytope. Letting $S = S_1 \cup S_2 \cup S_3$, yields the desired result. \end{proof} We also obtain a generalization of the classical result known for rational polyhedra. \begin{cor} If $F$ is an exposed face of $K$ then $\mathbb{C}C(F)=\mathbb{C}C(K)\cap F$. \end{cor} \section{Approximation of the CG closure}\label{sec:approx} \subsection{Approximation 1 of the CG closure}\label{sec:approx1} In this section, we construct our first approximation of the CG closure. Under the assumption that the CG closure of every proper exposed face of $K$ is defined by a finite number of CG cuts and by the use of Proposition~\ref{PPPP} and a compactness argument we construct a first approximation of the CG closure that uses a finite number of CG cuts. The main properties of this approximation are that it is a polytope and it is contained in $K \cap \textup{aff}_I(K)$. For this we will need the following lemma that describes integer affine subspaces. \begin{lem} Take $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$. Then there exists $\lambda \in \mathbb{R}^m$ such that for $a' = \lambda A$, $b' = \lambda b$, we have that $\set{x \in \mathbb{Z}^n: Ax = b} = \set{x \in \mathbb{Z}^n: a'x = b'}$. \label{lem:eqsys-red} \end{lem} \begin{proof} If $\set{x \in \mathbb{R}^n: Ax = b} = \emptyset$, then by Farka's Lemma there exists $\lambda \in \mathbb{R}^m$ such that $\lambda A =0$ and $\lambda b = 1$. Hence $\set{x \in \mathbb{R}^n: Ax = b} = \set{x \in \mathbb{R}^n: 0x = 1} = \emptyset$ as needed. We may therefore assume that $\set{x \in \mathbb{R}^n: Ax = b} \neq \emptyset$. Therefore we may also assume that the rows of the augmented matrix $[A\,|\,b]$ are linearly independent. Let $T = \mathrm{span}(a_1,\dots,a_m)$, where $a_1,\dots,a_m$ are the rows of $A$. Define $r: T \rightarrow \mathbb{R}$ where for $w \in T$ we let $r(w) = \lambda b$ for $\lambda \in \mathbb{R}^m$ where $\lambda A = w$. Since the rows of $A$ are linearly independent we obtain that $r$ is well defined and is a linear operator. Let $S = \set{x \in \mathbb{Z}^n: Ax = b}$. For $z \in \mathbb{Z}^n$, examine $T_z = \set{w \in T: \langle w, z \rangle = r(w)}$. By linearity of $r$, we see that $T_z$ is a linear subspace of $T$. Note that for $z \in \mathbb{Z}^n$, $T_z = T$ iff $z \in S$. Therefore $\forall ~ z \in \mathbb{Z}^n \setminus S$, we must have that $T_z \neq T$, and hence $\dim(T_z) \leq \dim(T) - 1$. Let $m_T$ denote the Lebesgue measure on $T$. Since $\dim(T_z) < \dim(T)$, we see that $m_T(T_z) = 0$. Let $T' = \bigcup_{z \in \mathbb{Z}^n \setminus S} T_z$. Since $\mathbb{Z}^n \setminus S$ is countable, by the countable subadditivity of $m_T$ we have that $m_T\left(T' \right) \leq \sum_{z \in \mathbb{Z}^n \setminus S} m_T(T_z) = 0$. Since $m_T(T) = \infty$, we must have that $T \setminus T' \neq \emptyset$. Hence we may pick $a' \in T \setminus T'$. Letting $b' = r(a')$, we note that by construction there $\exists ~ \lambda \in \mathbb{R}^m$ such that $\lambda A = a'$ and $\lambda b = b'$. Hence for all $z \in S$, $\lambda A z = \lambda b \mathbb{R}ightarrow a' x = b'$. Now take $z \in \mathbb{Z}^n \setminus S$. Now since $a' \in T \setminus T'$, we have that $a' \notin T_z$. Hence $a' z \neq b'$. Therefore we see that $\set{x \in \mathbb{Z}^n: a' x = b'} = \set{x \in \mathbb{Z}^n: A x = b}$ as needed. \end{proof} \begin{prop} Let $\emptyset\neq K \subseteq \mathbb{R}^n$ be a compact convex set. If $\mathbb{C}C(F_v(K))$ is finitely generated for any proper exposed face $F_v(K)$ then $\exists~ S \subseteq \mathbb{Z}^n$, $|S| < \infty$, such that $\mathbb{C}C(K,S) \subseteq K \cap \aff_I(K)$ and $\mathbb{C}C(K,S)$ is a polytope. \label{lem:move-inside} \end{prop} \begin{proof} Let us express $\aff(K)$ as $\set{x \in \mathbb{R}^n: Ax = b}$. Note that $\aff(K) \neq \emptyset$ since $K \neq \emptyset$. By Lemma \ref{lem:eqsys-red} there exists $\lambda$, $c = \lambda A$ and $d = \lambda b$, and such that $\aff(K) \cap \mathbb{Z}^n = \set{x \in \mathbb{Z}^n: \pr{c}{x} = b}$. Since $h_K(c) = b$ and $h_K(-c) = -b$, using Proposition \ref{prop:killirr} on $c$ and $-c$, we can find $S_A \subseteq \mathbb{Z}^n$ such that $\mathbb{C}C(K,S_A) \subseteq \aff(\set{x \in \mathbb{Z}^n: \pr{c}{x} = b}) = \aff_I(K)$. Express $\aff(K)$ as $W + a$, where $W \subseteq \mathbb{R}^n$ is a linear subspace and $a \in \mathbb{R}^n$. Now take $v \in W \cap S^{n-1}$. Note that $F_v(K)$ is a proper exposed face and hence, by assumption, $\mathbb{C}C(F_v(K))$ is finitely generated. Hence by Proposition \ref{thm:cg-lift} there exists $S_v \subseteq \mathbb{Z}^n$ such that $\mathbb{C}C(K,S_v)$ is a polytope, $\mathbb{C}C(K,S_v) \cap H_v(K) = \mathbb{C}C(F_v(K))$ and $\mathbb{C}C(K,S_v) \subseteq \set{x: \pr{x}{v} \leq h_K(v)}$. Let $K_v = \mathbb{C}C(K,S_v)$, then we have the following claim. \quad\linebreak \textbf{Claim:} $\,\exists$ open neighborhood $N_v$ of $v$ in $W \cap S^{n-1}$ such that $ v' \in N_v \mathbb{R}ightarrow h_{K_v}(v') \leq h_K(v')$. \quad\linebreak Since $K_v$ is a polytope, there exists $C \subseteq \mathbb{R}^n$, $|C| < \infty$, such that $K_v = \mathrm{conv} (C)$. Then note that $h_{K_v}(w) = \sup_{c \in C} \pr{c}{w}$. Now let $H = \set{c: h_K(v) = \pr{v}{c}, c \in C}$. By construction, we have that $\mathrm{conv} (H) = \mathbb{C}C(F_v(K))$. First assume that $\mathbb{C}C(F_v(K)) = \emptyset$. Then $H = \emptyset$, and hence $h_{K_v}(v) < h_K(v)$. Since $K_v,K$ are compact convex sets, we have that $h_{K_v},h_K$ are both continuous functions on $\mathbb{R}^n$ and hence $h_K - h_{K_v}$ is continuous. Therefore there exists $\epsilon > 0$ such that $h_{K_v}(v') < h_K(v')$ for $\|v-v'\| \leq \epsilon$ as needed. Now assume that $\mathbb{C}C(F_v(K)) \neq \emptyset$. Let $R = \max_{c \in C} \|c\|$, and let \[ \delta = h_K(v) - \sup \set{\pr{v}{c}: c \in C \setminus H}. \] Now let $\epsilon = \frac{\delta}{2R}$. Now take any $v'$ such that $\|v'-v\| < \epsilon$. Now for all $c \in H$, we have that \[ \pr{c}{v'} = \pr{c}{v} + \pr{c}{v'-v} = h_K(v) + \pr{c}{v'-v} \geq h_K(v) - \|c\|\|v'-v\| > h_K(v) - R\frac{\delta}{2R} = h_K(v) - \frac{\delta}{2}, \] and that for all $c \in C \setminus H$, we have that \[ \pr{c}{v'} = \pr{c}{v} + \pr{c}{v'-v} \leq h_K(v) - \delta + \pr{c}{v'-v} \leq h_K(v) -\delta + \|c\|\|v'-v\| < h_K(v) - \delta + \frac{\delta}{2} = h_K(v) - \frac{\delta}{2}. \] Therefore we have that $\pr{c}{v'}>\pr{c'}{v'}$ for all $c\in H$, $c'\in C \setminus H$ and hence \begin{equation} h_{K_v}(v') = \sup_{c \in C} \pr{c}{v'} = \sup_{c \in H} \pr{c}{v'} = h_{\mathbb{C}C(F_v(K))}(v') \leq h_K(v'), \end{equation} since $\mathbb{C}C(F_v(K)) \subseteq F_v(K) \subseteq K$. The statement thus holds by letting $N_v=\{v' \in S^{n-1}\,:\,\|v'-v\| \leq \epsilon\}$. Note that $\set{N_v: v \in W \cap S^{n-1}}$ forms an open cover of $W \cap S^{n-1}$, and since $W \cap S^{n-1}$ is compact, there exists a finite subcover $N_{v_1},\dots,N_{v_k}$ such that $\bigcup_{i=1}^k N_{v_i} = W \cap S^{n-1}$. Now let $S = S_A ~\cup ~ \cup_{i=1}^k S_{v_i}$. We claim that $\mathbb{C}C(K,S) \subseteq K$. Assume not, then there exists $x \in \mathbb{C}C(K,S) \setminus K$. Since $\mathbb{C}C(K,S) \subseteq \mathbb{C}C(K,S_A) \subseteq W+a$ and $K \subseteq W+a$, by the separator theorem there exists $w \in W \cap S^{n-1}$ such that $h_K(w) = \sup_{y \in K} \pr{y}{w} < \pr{x}{w} \leq h_{\mathbb{C}C(K,S)}(w)$. Since $w \in W \cap S^{n-1}$, there exists $i$, $1 \leq i \leq k,$ such that $w \in N_{v_i}$. Note then we obtain that \begin{equation*} h_{\mathbb{C}C(K,S)}(w) \leq h_{\mathbb{C}C(K,S_{v_i})}(w) = h_{K_{v_i}}(w) \leq h_K(w), \end{equation*} a contradiction. Hence $\mathbb{C}C(K,S) \subseteq K$ as claimed. $\mathbb{C}C(K,S)$ is a polytope because it is the intersection of polyhedra or which at least one is a polytope. \end{proof} \subsection{Approximation 2 of the CG closure}\label{sec:approx2} In this section, we augment the first approximation of $\mathbb{C}C(K)$ by finitely more CG cuts to construct a better approximation of $\mathbb{C}C(K)$. Apart from satisfying the condition that this approximation is contained in $K \cap \textup{aff}_I(K)$, it also satisfies the condition that its intersection with the relative boundary of $K$ is equal to the intersection of $\mathbb{C}C(K)$ with the relative boundary of $K$. To achieve this approximation, the key observation is that since the first approximation of the CG closure was a polytope, therefore its intersection with relative boundary of $K$ is the union of a finite numbers of faces of the first approximation of the CG closure. This implies that there are a finite number of faces of $K$ such that if we apply Proposition \ref{thm:cg-lift} to them (i.e. separates points in $F_v \setminus \aff_I(H_v)$ and add lifted version of the CG cuts for $F_v$), we are able to achieve the second approximation of the CG closure. \begin{lem} Let $K \subseteq \mathbb{R}^n$ be a convex set and $P \subseteq K$ be a polytope. Then there exists $F_{v_1},\dots,F_{v_k} \subseteq K$, proper exposed faces of $K$, such that $ P \cap \mathrm{relbd}(K) ~\subseteq~ \bigcup_{i=1}^k F_{v_i}$ \label{lem:finite-bd} \end{lem} \begin{proof} Let $\mathcal{F} = \set{F: F \subseteq P, F \text{ a face of P }, \mathrm{relint}(F) \cap \mathrm{relbd}(K) \neq \emptyset}$. Since $P$ is polytope, note that the total number of faces of $P$ is finite, and hence $|\mathcal{F}| < \infty$. We claim that \begin{equation} P \cap \mathrm{relbd}(K) \subseteq \bigcup_{F \in \mathcal{F}} F. \end{equation} Take $x \in P \cap \mathrm{relbd}(K)$. Let $F_x$ denote the minimal face of $P$ containing $x$ (note that $P$ is a face of itself). By minimality of $F_x$, we have that $x \in \mathrm{relint}(F_x)$. Since $x \in \mathrm{relbd}(K)$, we have that $F_x \in \mathcal{F}$, as needed. Take $F \in \mathcal{F}$. We claim that there exists $H_F \subseteq K$, $H_F$ a proper exposed face of $K$, such that $F \subseteq H_F$. Take $x \in \mathrm{relint}(F) \cap \mathrm{relbd}(K)$. Let $\aff(K) = W + a$, where $W$ is a linear subspace and $a \in \mathbb{R}^n$. Since $x \notin \mathrm{relint}(K)$, by the separator theorem, there exists $v \in W \cap S^{n-1}$ such that $h_K(v) = \pr{x}{v}$. Let $H_F = F_v(K)$. Note that since $v \in W \cap S^{n-1}$, $F_v(K)$ is a proper exposed face of $K$. We claim that $F \subseteq H_F$. Since $F$ is a polytope, we have that $F = \mathrm{conv}(\mathrm{ext}(F))$. Write $\mathrm{ext}(F) = \set{c_1,\dots,c_k}$. Now since $x \in \mathrm{relint}(F)$, there exists $\lambda_1,\dots,\lambda_k > 0$, $\sum_{i=1}^k \lambda_i = 1$, such that $\sum_{i=1}^k \lambda_i c_i = x$. Now since $c_i \in K$, we have that $\pr{c_i}{v} \leq h_K(v)$. Therefore, we note that \begin{equation} \pr{x}{v} = \pr{\sum_{i=1}^k \lambda_i c_i}{v} = \sum_{i=1}^k \lambda_i \pr{c_i}{v} \leq \sum_{i=1}^k \lambda_i h_K(v) = h_K(v) \end{equation} Since $\pr{x}{v} = h_K(v)$, we must have equality throughout. To maintain equality, since $\lambda_i > 0$, $1 \leq i \leq k$, we must have that $\pr{c_i}{v} = h_K(v)$, $1 \leq i \leq k$. Therefore $c_i \in H_F$, $1 \leq i \leq k$, and hence $F = \mathrm{conv}(c_1,\dots,c_k) \subseteq H_F$, as needed. To conclude the proof, we note that the set $\set{H_F: F \in \mathcal{F}}$ satisfies the conditions of the lemma. \end{proof} \begin{prop} \label{lem:inter-body} Let $K \subseteq \mathbb{R}^n$ be a compact convex set. If $\mathbb{C}C(F_v)$ is finitely generated for any proper exposed face $F_v$ then $\exists ~ S \subseteq \mathbb{Z}^n$, $|S| < \infty$, such that \begin{alignat}{4} \label{eqn56}\mathbb{C}C(K,S) &\subseteq K \cap \aff_I(K)\\ \label{eqn56b}\mathbb{C}C(K,S) \cap \mathrm{relbd}(K) &= \mathbb{C}C(K) \cap \mathrm{relbd}(K) \end{alignat} \end{prop} \begin{proof} By Proposition \ref{lem:move-inside}, there exists $S_I \subseteq \mathbb{Z}^n$, $|S_I| < \infty$, such that $\mathbb{C}C(K,S_I) \subseteq K \cap \aff_I(K)$ and $\mathbb{C}C(K,S_I)$ is a polytope. Since $\mathbb{C}C(K,S_I) \subseteq K$ is a polytope, let $F_{v_1},\dots,F_{v_k}$ be the proper exposed faces of $K$ given by Lemma~\ref{lem:finite-bd}. By Proposition \ref{thm:cg-lift}, there exists $S_i \subseteq \mathbb{Z}^n$, $|S_i| < \infty$, such that $\mathbb{C}C(K,S_i) \cap H_{v_i} = \mathbb{C}C(F_{v_i})$. Let $S = S_I \cup \cup_{i=1}^k S_i$. We claim that $\mathbb{C}C(K,S) \cap \mathrm{relbd}(K) \subseteq \mathbb{C}C(K) \cap \mathrm{relbd}(K)$. For this note that $x \in \mathbb{C}C(K,S) \cap \mathrm{relbd}(K)$ implies $x \in \mathbb{C}C(K,S_I) \cap \mathrm{relbd}(K)$, and hence there exists $i$, $1 \leq i \leq k$, such that $x \in F_{v_i}$. Then \begin{align*} x \in \mathbb{C}C(K,S) \cap H_{v_i} \subseteq \mathbb{C}C(K,S_i) \cap H_{v_i} = \mathbb{C}C(F_{v_i}) \subseteq \mathbb{C}C(K) \cap \mathrm{relbd}(K). \end{align*} The reverse inclusion is direct. \end{proof} \section{Proof of Theorem}\label{sec:final} Finally, we have all the ingredients to prove the main result of this paper. The proof is by induction on the dimension of $K$. Trivially, the result holds for zero dimensional convex body. Now by the induction hypothesis, we are able to construct the second approximation of $\mathbb{C}C(K)$ described in Section \ref{sec:approx2} (since it assumes that the CG closure of every exposed face is a polytope). Now the key observation is that any CG cut that is not dominated by those already considered in the second approximation of the CG closure must separate a vertex of this second approximation that additionally lies in the relative interior of $K$. Then it is not difficult to show that there can exist only a finite number of such CG cuts, showing that the CG closure is a polytope. This proof idea is similar to a proof idea used in the case strictly convex sets. \begin{thm} Let $K \subseteq \mathbb{R}^n$ be a non-empty compact convex set. Then $\mathbb{C}C(K)$ is finitely generated. \end{thm} \begin{proof} We proceed by induction on the affine dimension of $K$. For the base case, $\dim(\aff(K)) = 0$, i.e. $K = \set{x}$ is a single point. Here it is easy to see that setting $S = \set{\pm e_i: i \in [n]}$, we get that $\mathbb{C}C(K,S) = \mathbb{C}C(K)$. The base case thus holds. Now for the inductive step let $0 \leq k < n$ let $K$ be a compact convex set where $\dim(\aff(K)) = k+1$ and assume the result holds for sets of lower dimension. By the induction hypothesis, we know that $\mathbb{C}C(F_v)$ is finitely generated for every proper exposed face $F_v$ of $K$, since $\dim(F_v)\leq k$. By Proposition \ref{lem:inter-body}, there exists a set $S \subseteq \mathbb{Z}^n$, $|S| < \infty$, such that \eqref{eqn56} and \eqref{eqn56b} hold. If $\mathbb{C}C(K,S) = \emptyset$, then we are done. So assume that $\mathbb{C}C(K,S) \neq \emptyset$. Let $A = \aff_I(K)$. Since $\mathbb{C}C(K,S) \neq \emptyset$, we have that $A \neq \emptyset$ (by (\ref{eqn56})), and so we may pick $t \in A \cap \mathbb{Z}^n$. Note that $A - t = W$, where $W$ is a linear subspace of $\mathbb{R}^n$ satisfying $W = \mathrm{span}(W \cap \mathbb{Z}^n)$. Let $L = W \cap \mathbb{Z}^n$. Since $t \in \mathbb{Z}^n$, we easily see that $\mathbb{C}C(K-t,T) = \mathbb{C}C(K,T) - t$ for all $T \subseteq \mathbb{Z}^n$. Therefore $\mathbb{C}C(K)$ is finitely generated iff $\mathbb{C}C(K-t)$ is. Hence replacing $K$ by $K-t$, we may assume that $\aff_I(K) = W$. Let $\pi_W$ denote the orthogonal projection onto $W$. Note that for all $x \in W$, and $z \in \mathbb{Z}^n$, we have that $\pr{z}{x} = \pr{\pi_W(z)}{x}$. Now since $\mathbb{C}C(K,S) \subseteq K \cap W$, we see that for all $z \in \mathbb{Z}^n$ \begin{equation*} \mathbb{C}C(K,S \cup \set{z}) = \mathbb{C}C(K,S) \cap \set{x: \pr{z}{x} \leq \floor{h_K(z)}} = \mathbb{C}C(K,S) \cap \set{x: \pr{\pi_W(z)}{x} \leq \floor{h_K(z)}}. \end{equation*} Let $L^* = \pi_W(\mathbb{Z}^n)$. Since $W$ is a rational subspace, we have that $L^*$ is full dimensional lattice in $W$. Now fix an element of $w \in L^*$ and examine $V_w := \set{\floor{h_K(z)}: \pi_W(z) = w, z \in \mathbb{Z}^n}$. Note that $V_w \subseteq \mathbb{Z}$. We claim that $\inf(V_w) \geq -\infty$. To see this, note that \begin{align} \inf \set{\floor{h_K(z)}: \pi_W(z) = w, z \in \mathbb{Z}^n} &\geq \inf \set{\floor{h_{K \cap W}(z)}: \pi_W(z)=w, z \in \mathbb{Z}^n} \\ &= \inf \set{\floor{h_{K \cap W}(\pi_W(z))}: \pi_W(z)=w, z \in \mathbb{Z}^n} \\ &= \floor{h_{K \cap W}(w)} > -\infty. \end{align} Now since $V_w$ is a lower bounded set of integers, there exists $z_w \in \pi^{-1}_W(w) \cap \mathbb{Z}^n$ such that $\inf(V_w) = \floor{h_K(z_w)}$. From the above reasoning, we see that $ \mathbb{C}C(K,S \cup \pi^{-1}_W(z) \cap \mathbb{Z}^n) = \mathbb{C}C(K, S \cup \set{z_w})$. Now examine the set $C = \set{w: w \in L^*, \mathbb{C}C(K,S \cup \set{z_w}) \subsetneq \mathbb{C}C(K,S)}$. Here we get that \begin{equation*} \mathbb{C}C(K) = \mathbb{C}C(K, S \cup \mathbb{Z}^n) = \mathbb{C}C(K, S \cup \set{z_w: w \in L^*}) = \mathbb{C}C(K, S \cup \set{z_w: w \in C}). \end{equation*} From the above equation, if we show that $|C| < \infty$, then $\mathbb{C}C(K)$ is finitely generated. To do this, we will show that there exists $R > 0$, such that $C \subseteq RB_n$, and hence $C \subseteq L^* \cap RB_n$. Since $L^*$ is a lattice, $|L^* \cap RB_n| < \infty$ for any fixed $R$, and so we are done. Now let $P = \mathbb{C}C(K,S)$. Since $P$ is a polytope, we have that $P = \mathrm{conv}(\mathrm{ext}(P))$. Let $I = \set{v: v \in \mathrm{ext}(P), v \in \mathrm{relint}(K)}$, and let $B = \set{v: v \in \mathrm{ext}(P), v \in \mathrm{relbd}(K)}$. Hence $\mathrm{ext}(P) = I \cup B$. By assumption on $\mathbb{C}C(K,S)$, we know that for all $v \in B$, we have that $v \in \mathbb{C}C(K)$. Hence for all $z \in \mathbb{Z}^n$, we must have that $\pr{z}{v} \leq \floor{h_K(z)}$ for all $v \in B$. Now assume that for some $z \in \mathbb{Z}^n$, $\mathbb{C}C(K,S \cup \set{z}) \subsetneq \mathbb{C}C(K,S) = P$. We claim that $\pr{z}{v} > \floor{h_K(z)}$ for some $v \in I$. If not, then $\pr{v}{z} \leq \floor{h_K(z)}$ for all $v \in \mathrm{ext}(P)$, and hence $\mathbb{C}C(K,S \cup \set{z}) = \mathbb{C}C(K,S)$, a contradiction. Hence such a $v \in I$ must exist. For $z \in \mathbb{Z}^n$, note that $h_K(z) \geq h_{K \cap W}(z) = h_{K \cap W}(\pi_W(z))$. Hence $\pr{z}{v} > \floor{h_K(z)}$ for $v \in I$ only if $\pr{\pi_W(z)}{v} = \pr{z}{v} > \floor{h_{K \cap W}(\pi_W(z))}$. Let $C' := \set{ w \in L^*,:\, \exists v \in I, \pr{v}{w} > \floor{h_{K \cap W}}(w)}$. From the previous discussion, we see that $C \subseteq C'$. Since $I \subseteq \mathrm{relint}(K) \cap W = \mathrm{relint}(K \cap W)$ we have $\delta_v = \sup \set{r \geq 0: rB_n \cap W + v \subseteq K \cap W} > 0$ for all $v \in I$. Let $\delta = \inf_{v \in I} \delta_v$. Since $|I| < \infty$, we see that $\delta > 0$. Now let $R = \frac{1}{\delta}$. Take $w \in L^*$, $\|w\| \geq R$. Note that $\forall v \in I$, \begin{equation} \floor{h_{K \cap W}(w)} \geq h_{K \cap W}(w) - 1 \geq h_{(v + \delta B_n) \cap W}(w) - 1 = \pr{v}{w} + \delta\|w\|-1 \geq \pr{v}{w}. \end{equation} Hence $w \notin C'$. Therefore $C \subseteq C' \subseteq RB_n$ and $\mathbb{C}C(K)$ is finitely generated. \end{proof} \end{document}

\begin{document} \title{Matrix Product State Pre-Training for Quantum Machine Learning} \begin{abstract} Hybrid Quantum-Classical algorithms are a promising candidate for developing uses for NISQ devices. In particular, Parametrised Quantum Circuits (PQCs) paired with classical optimizers have been used as a basis for quantum chemistry and quantum optimization problems. Training PQCs relies on methods to overcome the fact that the gradients of PQCs vanish exponentially in the size of the circuits used. Tensor network methods are being increasingly used as a classical machine learning tool, as well as a tool for studying quantum systems. We introduce a circuit pre-training method based on matrix product state machine learning methods, and demonstrate that it accelerates training of PQCs for both supervised learning, energy minimization, and combinatorial optimization. \end{abstract} \begin{multicols}{2} \section{Introduction} Parametrised Quantum Circuits (PQCs) have been the focus of attempts to demonstrate quantum computational advantage on NISQ devices, for problems of both scientific and commercial interest. Typically these efforts involve parametrising a quantum circuit with a series of rotation angles, and using a classical optimizer to find a set of angles which minimizes a given cost function. The quantum device is used to estimate the cost function associated with a particular set of parameters, where it is assumed to be hard to calculate the cost function classically. Algorithms based on these methods are often called hybrid quantum-classical algorithms. A major hurdle in developing useful hybrid algorithms is the problem of vanishing gradients. It has been shown that the size of initial gradients decrease exponentially towards zero as the number of qubits and the depth of the circuits increases, when parameters are randomly initialised \cite{BarrenPlateau}. Other circuit metrics have been demonstrated to produce these so-called barren plateaus, such as ansatz expressibility \cite{ansatz_plateau}, the entanglement between hidden and visible nodes in the circuit \cite{EntanglementPlateaus, entanglementMitigation}, and circuit noise~\cite{noise_barren_plateau}. The existence of the barren plateau has motivated attempts to improve PQC learning algorithms to avoid training costs growing exponentially. These include developing gradient free algorithms~\cite{rotosolve, analytic_gradient}, defining local cost functions as targets for learning algorithms~\cite{cost_function_dependant_plateaus}, and initialisation schemes~\cite{block_iden}. Here we introduce a novel initialisation scheme based on tensor network algorithms. Tensor network based methods are the state of the art for numerical simulations of 1D and 2D spin systems~\cite{OrusTN}, and also for the simulation of quantum circuits~\cite{TensorNetwork}. Tensor networks have been used to solve optimization problems such as portofolio optimization, and are found to be competitive with commercial solvers~\cite{PortfolioOptTN,QEnhancedOpt}. Recently, tensor network based methods have also been used as the basis for machine learning algorithms. Matrix Product States (MPS) have been trained as a classifier for several machine learning tasks \cite{SupervisedMPSLearning, Unsupervised_Modelling_Samples}. The 2D generalisation of MPS, Projected Entangled Pair States (PEPS), have also been used for image classification\cite{PEPS_CNN}. Both Tree Tensor Networks (TTN) and MERA networks have also been used as image classifiers \cite{towards_qml, HierarchicalQC}. \end{multicols} \begin{figure} \caption{\textbf{Outline of the MPS initialisation procedure} \label{fig:AlgorithmOutline} \end{figure} \begin{multicols}{2} In this work, we optimize tensor networks as candidate ground states, as classifiers, and as solutions to combinatorial optimization problems. We use these networks to seed a PQC with an effective set of starting parameters, before continuing to train the quantum circuit. By beginning training in a part of the parameter space that is close to the target state, the number of steps to reach the desired minima can be reduced. In this way the impact of the barren plateau is reduced and limited quantum resources can be used more effectively during training. From here onwards this procedure will be known as \emph{MPS Pretraining}. \section{MPS Pretraining} MPS pretraining involves three steps: \begin{itemize} \item[1.] Train a tensor network \item[2.] Compile the tensor network into gates \item[3.] Initialise a circuit with these gates \end{itemize} Fig.~\ref{fig:AlgorithmOutline} outlines this procedure. \subsection{MPS Optimization} Given a problem instance, such as a Hamiltonian or a labeled data set, and a cost function where the minimum corresponds to the state of interest, the first step is to produce a MPS that minimizes the given cost function. The bond dimension of a MPS, denoted $\chi$, is a parameter that determines the complexity of the MPS model. Higher bond dimension models tend to produce better results, at the cost of greater training complexity. This training is done purely classically, and software exists both to efficiently contract and optimize these states, for example we refer the reader to Refs.~ \cite{itensor,TensorNetwork, tenpy, quimb}. In the following we discuss MPS algorithms, but the insights apply to other tensor network architectures. There are a variety of algorithms that can be used to optimize MPS. To find the ground states of Hamiltonians the DMRG algorithm is known to be very effective~\cite{SchollwoeckDMRG}. Similar optimisation algorithms include imaginary-time TEBD and TDVP, both of which are used in this work~\cite{TDVP,TEBD}. In Ref.~\cite{SupervisedMPSLearning} a DMRG-inspired machine learning algorithm is introduced which we adapt for this work. Note that in principle gradient descent can always be performed directly on all the tensors in the tensor network. This more closely reflects classical machine learning algorithms. However these quantum-inspired learning algorithms tend to perform as well in practice, and often perform better for quantum-based problems such as energy minimisation. \subsection{MPS Compilation} Having trained a MPS to minimize a cost function, the next step is to represent the MPS as a set of rotation angles in a PQC. As opposed to other classical machine learning methods, low bond dimension tensor networks permit efficient representations on quantum circuits \cite{QMPS}. MPS have a gauge freedom which means that any MPS can be brought into \emph{canonical form} in which each tensor is an isometry~\cite{mps_representations}. These can then easily be embedded in unitary matrices. Bond dimension 2 MPS can be expressed as a staircase of 2 qubit unitary gates~\cite{QMPS}. The rotation angles are extracted from these unitaries with some compilation scheme. For 2 qubit unitaries we use the KAK Decomposition \cite{KAK_Decomp}, which decomposes 2 qubit unitaries into 4 single qubit gates with 3 rotation angles each, and 3 different 2-qubit interaction gates. If necessary these gates can be further decomposed into hardware efficient gates. Restricted gate sets can be used as compilation targets, and the rotation angles can be found variationally~\cite{QuantumCompiling}. Later we suggest an approximate compilation scheme, based on insights from Ref.~\cite{reverse_stair_ansatz}, which ensures that the largest unitary matrices that need to be compiled are 2-qubit gates. Often brick wall circuits are used as ans\:atze for PQC research, being the most general way to parametrise a quantum circuit limited to nearest neighbour interactions. To initialise a brick wall circuit with a pretrained MPS, the gates on the diagonal of the brick wall must be initialised with the angles extracted in the compilation of the MPS. All of the off-diagonal gates are initialised to the identity. This ensures that before quantum training begins, the circuit represents exactly as good a candidate for the target state as the MPS is after compilation is completed. The optimizer is then free to vary the angles of all of the gates. We demonstrate that starting with this initialisation accelerates training, requires fewer gradient updates, and avoids local minima. \section{Results} \subsection{Combinatorial Optimization}\label{sec:qaoa} We test the initialisation scheme on the Max Cut optimisation problem which is often used as a benchmark for QAOA algorithms~\cite{qaoa_farhi}. In the Max Cut problem a graph, $G(E,V)$, is provided along with weights, $w_{ij}$, on each edge. The task is to find a set of vertices, $S$, such that the total weight of the edges connecting $S$ to their complement is maximized. This is equivalent to finding the ground state of the Hamiltonian, \begin{equation} H = \sum_{\langle i,j\rangle} w_{ij}(1 - Z_{i}Z_{j}) \end{equation} where $Z$ is the Pauli Z matrix. We use imaginary time TEBD to find a bond dimension 2 approximation to the ground state for an instance of the Max Cut problem on a graph with 6 vertices. In Fig.~\ref{fig:QAOARes} we show the results of training with this initialisation for depth 6, 9, and 12 circuits. Optimization was performed for an imaginary time of $3\times 10^{-2}$, with a time step of $1\times 10^{-3}$. We only optimize the MPS for short periods of time because for small problem instances low bond dimension MPS can often get close to the optimal answer. To more closely match the performance on large problem instances which require quantum circuits that cannot be easily simulated we do not fully optimize the MPS. \end{multicols} \begin{figure} \caption{\textbf{Max Cut optimization} \label{fig:QAOARes} \end{figure} \begin{multicols}{2} We compare training starting from MPS pretrained circuits to that starting from random initialisations of the same ansatz. Instead of the standard QAOA scheme, we use a brick wall circuit made up of independent 2-qubit gates. These sorts of circuits have been explored for optimisation problems and their performance is competitive with QAOA \cite{VQEvQAOA}. The MPS initialised circuits start off at a better energy than the random counterparts, which is to be expected. However the initial steps from the MPS initialised states are noticeably larger than those taken from randomly initialised circuits. These circuits reach a minimum with fewer gradient descent updates, and reach better minima than the any of the randomly initialised states. In the random state there is a notable decrease in performance between the depth 9 and depth 12 ansatz, where the depth 12 gets stuck in a local minimum. No drop in performance is observed in the MPS initialised circuits. \subsection{Finding Ground States} We also implement MPS pretraining for the purpose of finding ground states of electronic Hamiltonians, a common benchmark in hybrid quantum classical algorithm research. We use imaginary time TDVP to estimate the ground state of the electronic Hamiltonian of $H_{2}$ and $LiH$. The Hamiltonians are constructed using the OpenFermion package~\cite{openFermion}. The same is done for the Transverse Field Ising Model (TFIM) which is a widely studied Hamiltonian in condensed matter physics. In all these cases imaginary time TDVP was used to construct a MPS approximation to the ground state, and this MPS is used to initialise a quantum circuit. For the $H_2$ and $LiH$ problems, the MPS converges to the same ground state as the optimized quantum circuit, suggesting either low bond dimension MPS are good approximations to these ground states, or the brick wall circuits do not offer significantly more expressivity in this problem. To demonstrate that even in cases where low bond dimension MPS are not as effective, we run the TDVP algorithm for a shorter time, meaning a worse approximation is used to initialise the circuits. These results are given in Fig.~\ref{fig:VQEH2}. In all three cases we find significant decreases in the number of iterations needed to reach the lowest energy states, compared to the randomly initialised circuits. We also compared circuits initialised to the identity, with all rotation angles set to 0. In each case the zero initialised circuit failed to optimized, and quickly ended up in a local minima. Fig.~\ref{fig:VQEH2}b compares the performance of the MPS pretrained circuit with a randomly initialised circuit in finding the ground state of the $H_{2}$ electronic Hamiltonian. Optimisation of the MPS initialised circuit required fewer function evaluations than the randomly initialised circuits, while both found effectively the same ground state, differing by less that $1\times 10^{-8}$ at depth 10. The number of function evaluations needed for the MPS initialised circuit grows more slowly as a function of depth than for the randomly initialised circuit, implying that even relatively poor approximations of the ground state still initialise close to the target state. We see similar results for the LiH and TFIM Hamiltonians, Fig.~\ref{fig:VQEH2}c and d resprectively. \end{multicols} \begin{figure} \caption{\textbf{Initialising VQE with MPS} \label{fig:VQEH2} \end{figure} \begin{multicols}{2} \subsection{Machine Learning} Finally we pretrain quantum circuits to classify clothing labels using the Fashion MNIST dataset. The images were compressed so small circuits could be used to classify each image. This compression is done using principle component analysis on the training data set, and projecting the training images onto the principle components. The set of training images is collected into a matrix, $\bf{X}$. We compute the covariance matrix of the training dataset, given by \begin{equation} \bf{\Sigma} = \bf{X}^{T}\bf{X} \end{equation} Then we perform an SVD on the matrix $\bf{\Sigma}$ \begin{equation} \bf{\Sigma} = \bf{U}\bf{\Lambda}\bf{U^{\dagger}} \end{equation} The principle components are identified as the columns of $\bf{U}$. To compress an image so that an $N$ qubit circuit can be used to classify the image we take the $N$ top principle components, $\{\vec{u}_{1},\vec{u}_{2},\cdots ,\vec{u}_{N}\}$, and take the inner product between each image (reshaped into a vector) and the principle component. The input to the $i^{th}$ qubit from the $j^{th}$ image in the training set is given by \begin{equation} \tilde{x}_{i,j} = \langle\vec{x}_{j},\vec{u}_{i}\rangle \end{equation} Each projection was used as the input to a single qubit, so to use $N$ qubits, we projected an image onto the $N$ most significant principle components. The individual $\tilde{x}_{i}$ values are used as rotation angles in parametrised Y gates, Fig.~\ref{fig:MNIST}a. This method, as opposed to other compression methods, such as pooling, gives greater flexibility in the number of qubits that can be used. The circuits used here were trained as binary classifiers. They were trained to distinguish between t-shirts and trousers. In Fig.~\ref{fig:MNIST}b we show the training set loss and accuracy at each epoch during training. The MPS pretrained circuit is compared to both a random initialisation and an initialisation such that the circuit evaluates to the identity matrix. Ref.~\cite{block_iden} suggests the identity initialisation increases the size of initial gradients and is commonly used in quantum chemistry research. In this case all angles were set to zero, as all gates used were parameterised rotations. For an ansatz with static gates, more care must be taken so that the entire circuit evaluates to the identity. The MPS initialised circuit has a lower loss and higher accuracy on the training set during training than either the identity initialisation or the random initialisation. \end{multicols} \begin{figure} \caption{\textbf{Fashion MNIST Results.} \label{fig:MNIST} \end{figure} \begin{multicols}{2} \section{Compilation} For situations where bond dimension 2 MPS ansatz are not able to optimize a cost function, higher bond dimensions need to be used to capture a wider range of quantum states. Higher bond dimension MPS tensors are compiled to unitaries over more than 2 qubits, each doubling of the bond dimension requires another qubit in the representation of each tensor. This raises the problem of compiling high bond dimension MPS. An approximate compilation scheme has been proposed which has been demonstrated to effectively represent states of physical interest in tensor network simulations \cite{reverse_stair_ansatz}. Many-qubit unitary matrices are decomposed into a reverse stair-case of nearest neighbour unitary matrices, Fig.~\ref{fig:ApproxComp}a. When compiling an MPS into a circuit in this way, the diagonals adjacent to the central diagonal in a brick wall circuit are also initialised, and not set to the identity, Fig.~\ref{fig:ApproxComp}b. Training higher bond dimension MPS can then be viewed as a way to initialise more of the initial brick wall circuit. This compilation process is not exact, and hence errors will accumulate throughout this process and worsen the initialisation of the circuit. \begin{figure} \caption{\textbf{Approximate high bond dimension compilation.} \label{fig:ApproxComp} \end{figure} \section{Discussion}\label{sec:outlook} \subsection{Comparison to other methods} Below we compare conceptually similar ideas which have been proposed to overcome barren plateaus and accelerate the training of variational quantum circuits. This is not an exhaustive list but highlighting these examples helps highlight the benefits of MPS pre-training. \subsubsection{Warm-Start QAOA} The MPS initialisation method introduced here is one of many proposed methods to improve the training of PQCs. This solution is very similar in methodology to the warm-start QAOA algorithm \cite{warmStartQAOA}. Relaxations are applied to the optimization problem which make the problem efficiently solvable, and this is used to initialise the QAOA algorithm. The relative merits of these two methods ultimately depends on the relative effectiveness of relaxation methods and MPS based methods to approximate the optimal solution for optimization problems. There is currently no way to concretely answer this question, but there have been results demonstrating that tensor network based methods are competitive with state of the art commercial solvers for optimization problems \cite{PortfolioOptTN, QEnhancedOpt}. \subsubsection{Layer-Wise Learning} Another procedure that has shown promising results is layer wise learning\cite{layer_wise_training, VQEvQAOA}. Layers in a quantum circuit are trained one at a time, keeping all other layers fixed. Finally layers are grouped together and trained simultaneously before training the entire circuit. The reasoning behind this method is very similar to that proposed here, to initialise the entire circuit with an approximation to the optimal solution, but instead of using MPS, the approximation is generated by sequential optimisation of layers in a PQC. Once again the question remains as to whether the layer wise trained approximation is better than MPS based approximations. Consider the fact that depth 1 or 2 nearest neighbour circuits, which are often considered in layer wise training regimes, can be faithfully represented with a bond dimension 2 MPS, and in fact this set of states is a restriction on the set of states that can be represented with a bond dimension 2 MPS. Layer wise training could be reformulated as sequential training of finite correlation length MPS, with the bond dimension of the restricted MPS growing with each trained layer. Seeing as the MPS initialisation scheme that we have introduced requires no restriction on the set of accessible states it should be the case that better initialisation states are accessible with the methods introduced here. It has been demonstrated that layer-wise learning suffers from abrupt transitions in trainability~\cite{layer_wise_learning_failure}; there exist circuit ansatz and cost functions where, below a threshold depth, piece wise training fails to minimize the cost function. It has not escaped our attention that insights from tensor network optimisation techniques could be used to augment this initialisation method with a training scheme similar to layer wise learning. In the DMRG algorithm, the bond dimension of the MPS is gradually increased. A similar approach with brick wall circuits would involve sequentially training diagonals either side of the central diagonal. It remains to be seen if this training scheme could be as effective as layer wise learning in training a circuit combined with MPS pretraining. \subsubsection{Entanglement Restriction} The authors of~\cite{entanglementMitigation} note that the impacts of vanishing gradients can be mitigated by restricting entanglement between hidden and visible qubits in a PQC. A qubit is visible if the output of that qubit is used to calculate a cost function, and it is hidden if it is ignored. They propose a number of schemes, including starting with no entanglement between hidden and visible nodes, having a fixed entanglement, and learning circuits which have low entanglement. The quantum circuit MPS formalism used in this work can easily be extended to the regime with hidden and visible qubits. In this case our initialisation scheme would resemble a fixed entanglement initialisation scheme, where the entanglement between the hidden and visible nodes is fixed by the bond dimension of the tensor network. They additionally propose including additional terms to the cost function, one to restrict entanglement growth between hidden and visible nodes, and one to add Langevin noise which acts to mitigate the effect of vanishing gradients. Both these schemes could be implementing on top of the initialisation scheme proposed here. \subsection{Alternative Tensor Network Structures} There are many tensor network geometries that are used to represent states with properties that are not effectively captured by MPS. Many of these have circuit counterparts which could be initialised in the same way. For example MERA networks~\cite{MERA} have been proposed as a basis for quantum machine learning MPS have a freedom in the location of the \emph{othogonality centre} when put in canonical form. In all results above, the MPS are put into left or right canonical form. The circuits to represent these states have a depth at least as large as the number of qubits. However choosing a \emph{mixed canonical} form actually reduces the circuit depth needed to initialise the MPS, Fig.~\ref{fig:TNCircuit}. A circuit initialised in this way may have larger initial gradients, as gradients vanish as a function of circuit depth. \begin{figure} \caption{\textbf{Gauge freedom and circuit depth.} \label{fig:TNCircuit} \end{figure} \subsection{Consequences for tensor network simulations} It is interesting to note that the bond dimension of the MPS needed to represent the circuit increases as training proceeds. Deep brick wall circuits represent a restricted class of high bond dimension tensor networks. Ordinarily any variational calculations with these extremely large bond dimension tensor networks would be impractical on quantum devices because of the difficulty in optimizing these circuits. To simulate spin systems with high bond dimension tensor networks it would be possible to simulate up to the classically feasible limit, translate the tensor network into circuits, and then seed a higher bond dimension simulation with the classical tensor network as is done here. This could open up the possibility of very large bond dimension tensor network simulations of spin systems on NISQ devices. \section{Acknowledgements And Contributions} JD, FB, and AGG were supported by the EPSRC through grants EP/L015242/1, EP/L015854/1 and EP/S005021/1. This work is supported by the Engineering and Physical Sciences Research Council grant number EP/L015242/1. VM is supported by ESPRC Prosperity Partnership grant EP/S516090/1. LW and FB are supported by the EPSRC CDT in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES) via grant number EP/L015854/1. The project was conceived in group discussions. JD and FB wrote code to translate MPS into circuits, and to continue training. VM wrote classical MPS training methods inspired by DMRG. The manuscript was written by JD and FB. \end{multicols} \printbibliography \end{document}

\begin{equation}gin{document} \newcommand{\begin{equation}}{\begin{equation}gin{equation}} \newcommand{\end{equation}}{\end{equation}} \newcommand{\begin{equation}a}{\begin{equation}gin{eqnarray}} \newcommand{\end{equation}a}{\end{eqnarray}} \newcommand{\displaystyle}{\displaystylelaystyle} \newcommand{\langle}{\langlengle} \newcommand{\rangle}{\ranglengle} \newtheorem{thm}{Theorem}[subsection] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{definition}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{prf}[thm]{Proof} \title{Golden Binomials and Carlitz Characteristic Polynomials} \begin{equation}gin{abstract} The golden binomials, introduced in the golden quantum calculus, have expansion determined by Fibonomial coefficients and the set of simple zeros given by powers of Golden ratio. We show that these golden binomials are equivalent to Carlitz characteristic polynomials of certain matrices of binomial coefficients. It is shown that trace invariants for powers of these matrices are determined by Fibonacci divisors, quantum calculus of which was developed very recently. \end{abstract} \section{Introduction} The golden quantum calculus, based on the Binet formula for Fibonacci numbers $F_n$ as $q$-numbers, was introduced in \cite{golden}. In this calculus, the finite-difference $q$-derivative operator is determined by two Golden ratio bases $\varphi$ and $\varphi'$, while the golden binomial expansion, by Fibonomial coefficients. The coefficients are expressed in terms of Fibonacci numbers, while zeros of these binomials are given by powers of Golden ratio $\varphi$ and $\varphi'$. It was observed that similar polynomials were introduced by Carlitz in 1965 from different reason, as characteristic polynomials of certain matrices of binomial coefficients \cite{Carlitz}. The goal of the present paper is to show equivalence of Carlitz characterisitc polynomials with golden binomials. In addition, the proof and interpretation of main formulas for trace of powers of the matrix $A_{n+1}$ in terms of Fibonacci divisors and corresponding quantum calculus, developed recently in \cite{FNK} would be given. \section{Golden Binomials} \subsection{Fibonomials and Golden Pascal Triangle} The binomial coefficients defined by \begin{equation} { n \brack k}_F= \frac{[n]_{F}!}{[n-k]_{F}! [k]_{F}!}= \frac{F_n!}{F_{n-k}! F_k!}, \langlebel{goldenbinom}\end{equation} with $n$ and $k$ being non-negative integers, $n\geq k$, are called the Fibonomials. Using the addition formula for Fibonacci numbers \cite{golden}, \begin{equation} F_{n+m} = \varphi^n F_m + {\varphi'}^m F_n \end{equation} we have following expression \begin{equation} F_n=F_{n-k+k}=\left(-\frac{1}{\varphi}\right)^k F_{n-k}+\varphi^{n-k} F_k. \end{equation} By using \begin{equation} \varphi^n = \varphi F_n + F_{n-1}, \,\,\,\,{\varphi'}^n = \varphi' F_n + F_{n-1},\langlebel{phin} \end{equation} it can be rewritten as follows \begin{equation}a F_n &=& F_{n-k-1} F_k + F_{n-k} F_{k+1} \nonumber \\ &=& F_{n-k} F_{k-1}+ F_{n-k+1} F_k. \end{equation}a With the above definition (\ref{goldenbinom}) it gives recursion formula for Fibonomials in two forms, \begin{equation}a { n \brack k}_{F}&=& \frac{(-\frac{1}{\varphi})^k [n-1]_{F}!}{[k]_{F}! [n-k-1]_{F}!} + \frac{\varphi^{n-k} [n-1]_{F}!}{[n-k]_{F}![k-1]_{F}!} \nonumber \\ &=& \left(-\frac{1}{\varphi}\right)^k { n-1 \brack k}_{F} + \varphi^{n-k} { n-1 \brack k-1}_{F} \langlebel{goldenpascal1}\\ &=& \varphi^k { n-1 \brack k}_{F} + \left(-\frac{1}{\varphi}\right)^{n-k} { n-1 \brack k-1}_{F} .\langlebel{goldenpascal2} \end{equation}a These formulas, for $1\leq k\leq n-1$, determine the Golden Pascal triangle for Fibonomials \cite{golden}. \subsection{Golden Binomial} The Golden Binomial is defined as \cite{golden}, \begin{equation} (x+y)_F^n = (x+\varphi^{n-1} y)(x+\varphi^{n-2} \varphi' y)...(x+ \varphi{\varphi'}^{n-2} y) (x+ {\varphi'}^{n-1} y)\end{equation} or due to $\varphi \varphi' = -1$ it is \begin{equation} (x+y)_F^n = (x+\varphi^{n-1} y)(x-\varphi^{n-3} y)...(x+ (-1)^{n-1}\varphi^{-n+1} y).\end{equation} It has n-zeros at powers of the Golden ratio $$\frac{x}{y}=-\varphi^{n-1},\,\,\,\, \frac{x}{y}=-\varphi^{n-3},\,\,\,\,...,\frac{x}{y}=-\varphi^{-n+1}.$$ For Golden binomial the following expansion in terms of Fibonomials is valid \cite{golden} \begin{equation}a (x+y)_F^n &=&\sum^{n}_{k=0}{ n \brack k}_{F} (-1)^{\frac{k(k-1)}{2}} x^{n-k} y^k \nonumber \\ &=& \sum^{n}_{k=0} \frac{F_n!}{F_{n-k}! F_k!}(-1)^{\frac{k(k-1)}{2}} x^{n-k} y^k. \langlebel{goldenbinomexpansion}\end{equation}a The proof is easy by induction and using recursion formulas (\ref{goldenpascal1}), (\ref{goldenpascal2}) . In terms of Golden binomials we introduce the Golden polynomials \begin{equation} P_n (x) = \frac{(x-a)_F^n}{F_n!},\end{equation} where $n=1,2,...$, and $P_0(x) =1$. These polynomials satisfy relations \begin{equation} D_F^x P_n(x) = P_{n-1}(x), \end{equation} where the Golden derivative is defined as \begin{equation} D_F^x P_n(x) = \frac{P_n (\varphi x) - P_n (\varphi' x) }{(\varphi - \varphi') x}.\end{equation} For even and odd polynomials we have different products \begin{equation} P_{2n} (x) = \frac{1}{F_{2n}!} \prod^n_{k=1} (x- (-1)^{n+k}\varphi^{2k-1} a) (x + (-1)^{n+k}\varphi^{-2k +1} a) ,\end{equation} \begin{equation} P_{2n+1} (x) = \frac{(x - (-1)^n a)}{F_{2n+1}!} \prod^n_{k=1} (x- (-1)^{n+k}\varphi^{2k} a) (x - (-1)^{n+k}\varphi^{-2k} a) .\end{equation} By using (\ref{phin}) it is easy to find \begin{equation} \varphi^{2k} + \frac{1}{\varphi^{2k}} = F_{2k} + 2 F_{2k-1} ,\end{equation} \begin{equation} \varphi^{2k+1} - \frac{1}{\varphi^{2k+1}} = F_{2k+1} + 2 F_{2k} .\end{equation} Then we can rewrite our polynomials in terms of Fibonacci numbers \begin{equation} P_{2n} (x) = \frac{1}{F_{2n}!} \prod^n_{k=1} (x^2 - (-1)^{n+k} (F_{2k-1} + 2 F_{2k-2})x a - a^2) ,\end{equation} \begin{equation} P_{2n+1} (x) = \frac{(x - (-1)^n a)}{F_{2n+1}!} \prod^n_{k=1} (x^2- (-1)^{n+k}(F_{2k} + 2 F_{2k-1})x a + a^2) .\end{equation} The first few odd polynomials are \begin{equation} P_1(x) = (x-a),\end{equation} \begin{equation} P_3 (x) = \frac{1}{2} (x+a)(x^2 - 3 x a + a^2 ),\end{equation} \begin{equation} P_5 (x) = \frac{1}{2\cdot 3 \cdot 5} (x-a)(x^2 + 3 x a + a^2 )(x^2 - 7 x a + a^2 ),\end{equation} \begin{equation} P_7 (x) = \frac{1}{2\cdot 3 \cdot 5 \cdot 8 \cdot 13} (x+a)(x^2 - 3 x a + a^2 )(x^2 + 7 x a + a^2 )(x^2 - 18 x a + a^2 ),\end{equation} and the even ones \begin{equation} P_2 (x) = (x^2 - x a - a^2 ),\end{equation} \begin{equation} P_4 (x) = \frac{1}{2\cdot 3} (x^2 + x a - a^2 )(x^2 - 4 x a - a^2),\end{equation} \begin{equation} P_6 (x) = \frac{1}{2\cdot 3\cdot 5 \cdot 8} (x^2 - x a - a^2 )(x^2 + 4 x a - a^2)(x^2 - 11 x a - a^2).\end{equation} \subsection{Golden analytic function} By golden binomials in complex domain, the golden analytic function can be derived, which is complex valued function of complex argument, not analytic in usual sense \cite{Eskisehir}. The complex golden binomial is defined as \begin{equation}a (x+iy)^n_F &=& (x + i\varphi^{n-1}y)(x - i\varphi^{n-3} y)... (x + i(-1)^{n-1}\varphi^{1-n}y) \\ &=&\sum^n_{k=0}\left[\begin{equation}gin{array}{c}n \\ k \end{array}\right]_{F} (-1)^{\frac{k(k-1)}{2}}x^{n-k}i^k y^k.\end{equation}a It can be generated by the golden translation $$ E^{iy D^x_F}_F x^n = (x+iy)^n_F,$$ where $$ E^x_F = \sum^\infty_{n=0} (-1)^{\frac{n(n-1)}{2}} \frac{x^n}{F_n!}.$$ The binomials determine the golden analytic function $$ f(z, F) = E^{iy D^x_F}_F f(x) = \sum^\infty_{n=0} a_n \frac{(x+iy)^n_F}{F_n!},$$ satisfying the golden $\bar\partial_F$ equation \begin{equation} \frac{1}{2}(D^x_{F} + i D^y_{-F}) f(z;F) = 0,\end{equation} where $D^x_{-F} = (-1)^{x\frac{d}{dx}} D^x_F$. For $u(x,y) = Cos_{F} (y D^x_{F}) f(x)$ and $v(x,y) = Sin_{F} (y D^x_{F}) f(x)$, the golden Cauchy-Riemann equations are \begin{equation} D^x_{F} u(x,y) = D^y_{-F} v(x,y),\,\,\,\,D^y_{-F} u(x,y) = -D^x_{F} v(x,y),\end{equation} and the golden-Laplace equation is \begin{equation} (D^x_{F})^2 u(x,y) + (D^y_{-F})^2 u(x,y) = 0. \end{equation} \subsection{Particular Case} The golden binomial $(x -a)^n_F$ can be also generated by the golden translation \begin{equation} E^{-a D^x_F}_F x^n = (x-a)^n_F. \end{equation} In particular case $a = 1$ we have \begin{equation} (x-1)^m_F = (x - \varphi^{m-1}) (x + \varphi^{m-3})... (x - (-1)^{m-1}\varphi^{-m+1}) .\langlebel{x1} \end{equation} First few binomials are \begin{equation}a (x-1)^1_F &=& x -1 ,\\ (x-1)^2_F& =& (x -\varphi) (x - \varphi'), \\ (x-1)^3_F& =& (x -\varphi^2)(x+1) (x - {\varphi'}^2), \\ (x-1)^4_F& =& (x -\varphi^3) (x +\varphi) (x + \varphi') (x - {\varphi'}^3) , \end{equation}a and corresponding zeros \begin{equation}a m = 1 &\Rightarrow & x =1 \\ m = 2 & \Rightarrow& x =\varphi, x = \varphi'\\ m=3 & \Rightarrow & x =\varphi^2, x = -1, x = {\varphi'}^2 \\ m=4& \Rightarrow& x =\varphi^3, x = -\varphi, x = -\varphi', x = {\varphi'}^3. \end{equation}a For arbitrary even and odd $n$ we have following zeros of Golden binomials \begin{equation}a n = 2k & \Rightarrow & (x-1)^{2k}_F : \varphi^{n-1}, {\varphi'}^{n-1}, -\varphi^{n-3}, -{\varphi'}^{n-3}, ..., \pm\varphi, \pm {\varphi'};\langlebel{even}\\ n = 2k+1 & \Rightarrow & (x-1)^{2k+1}_F : \varphi^{n-1}, {\varphi'}^{n-1}, -\varphi^{n-3}, -{\varphi'}^{n-3}, ..., \pm 1.\langlebel{odd} \end{equation}a \section{Carlitz Polynomials} In Section 2 we have introduced the Golden binomials. Now we are going to relate these binomials with characteristic equations for some matrices, constructed from binomial coefficients by Carlitz \cite{Carlitz}. \begin{equation}gin{definition} We define an $(n+1) \times (n+1)$ matrix $A_{n+1}$ with binomial coefficients, \begin{equation}a A_{n+1}=\left[{r \choose n-s} \right], \end{equation}a where $r,s=0,1,2,...,n .$ Here, \begin{equation}gin{eqnarray} {n \choose k} =\left\{ \begin{equation}gin{array}{ll} \frac{n!}{(n-k)! \phantom{.} k!}, & \hbox{if k $\leq$ n;} \\ 0, & \hbox{k $>$ n.} \end{array} \right. \end{eqnarray} \end{definition} First few matrices are, \begin{equation}a &&{n=0} \phantom{a} \Rightarrow r=s=0 \Rightarrow \phantom{a} A_{1}=\left[{0 \choose 0} \right]=(1) \nonumber \\ &&{n=1} \phantom{a} \Rightarrow r,s=0,1 \Rightarrow \phantom{a} A_{2}=\left[{r \choose 1-s} \right]=\left( \begin{equation}gin{array}{cc} {0 \choose 1} & {0 \choose 0} \\ {1 \choose 1} & {1 \choose 0} \\ \end{array} \right)=\left( \begin{equation}gin{array}{cc} 0 & 1 \\ 1 & 1 \\ \end{array} \right)\nonumber \\ &&{n=2} \Rightarrow r,s=0,1,2 \Rightarrow A_{3}=\left[{r \choose 2-s} \right]=\left( \begin{equation}gin{array}{ccc} {0 \choose 2} & {0 \choose 1} & {0 \choose 0} \\ {1 \choose 2} & {1 \choose 1} & {1 \choose 0} \\ {2 \choose 2} & {2 \choose 1} & {2 \choose 0} \\ \end{array} \right)=\left( \begin{equation}gin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 1 \\ \end{array} \right) \nonumber \end{equation}a Continuing, the general matrix $A_{n+1}$ of order $(n+1)$ can be written as, \begin{equation}gin{eqnarray} A_{n+1}=\left( \begin{equation}gin{array}{ccccc} \ldots \phantom{.} 0 & 0 & 0 & 0 & 1 \\ \ldots \phantom{.} 0 & 0 & 0 & 1 & 1 \\ \ldots \phantom{.} 0 & 0 & 1 & 2 & 1 \\ \ldots \phantom{.} 0 & 1 & 3 & 3 & 1 \\ \ldots \phantom{.} 1 & 4 & 6 & 4 & 1 \\ \phantom{.....} \vdots & \vdots & \vdots & \vdots & \vdots \\ \end{array} \right)_{(n+1)\times(n+1)}, \nonumber \end{eqnarray} where the lower triangular matrix is build from Pascal's triangle. We notice that trace of first few matrices $A_{n+1}$ gives Fibonacci numbers. As would be shown, it is valid for any n (Theorem $(\ref{invarianttheorem})$ equation $(\ref{invarianttheoremequation1})$) . \begin{equation}gin{definition} Characteristic polynomial of matrix $A_{n+1}$ is determined by, \begin{equation}a Q_{n+1}(x)=\mbox{det}(x I-A_{n+1}). \langlebel{characteristicequation} \end{equation}a \end{definition} First few polynomials explicitly are \begin{equation}a &&{n=0:}\phantom{abc} Q_{1}(x)=x - 1 ,\nonumber \\ &&{n=1:}\phantom{abc} Q_{2}(x)=\mbox{det}(x I-A_{2})=\left| \begin{equation}gin{array}{cc} x & -1 \\ -1 & x-1 \\ \end{array} \right| =x^2-x-1, \nonumber \\ &&{n=2:}\phantom{abc} Q_{3}(x)=\mbox{det}(x I-A_{3})=\left| \begin{equation}gin{array}{ccc} x & 0 & -1 \\ 0 & x-1 & -1 \\ -1 & -2 & x-1 \\ \end{array} \right|=x^3-2 x^2-2 x+1 ,\nonumber \\ &&{n=3:}\phantom{abc} \nonumber \end{equation}a \begin{equation}a Q_{4}(x)=\mbox{det}(x I-A_{4})&=&\left| \begin{equation}gin{array}{cccc} x & 0 & 0 & -1 \\ 0 & x & -1 & -1 \\ 0 & -1 & x-2 & -1 \\ -1 & -3 & -3 & x-1 \\ \end{array} \right| \nonumber \\ &=&-x^4+3 x^3+6 x^2-3 x-1 . \nonumber \end{equation}a Corresponding eigenvalues are represented by powers of $\varphi$ and $\varphi'$; ${n=0}$\phantom{abc} $\Rightarrow$ \phantom{a} $x_1=1$, ${n=1}$\phantom{abc}$\Rightarrow$ \phantom{a} $x_1=\varphi,\quad x_2=\varphi'$, ${n=2}$\phantom{abc}$\Rightarrow$ \phantom{a} $x_1=\varphi^2,\quad x_2=-1,\quad x_3={\varphi'}^2$, ${n=3}$\phantom{abc}$\Rightarrow$ \phantom{a} $x_1=\varphi^3, x_2=-\varphi, x_3=-\varphi',x_4=\varphi'^3$. Comparing zeros of first few characteristic polynomials, with zeros of Golden Binomial $(\ref{x1})$, we notice that they coincide. According to this, we have following conjecture. \textbf{Conjecture:} The characteristic equation $(\ref{characteristicequation})$ of matrix $A_{n+1}$ coincides with Golden Binomial; \begin{equation}a Q_{n+1}(x)=\mbox{det}(x I-A_{n+1})=(x-1)^{n+1}_{F}. \end{equation}a To prove this conjecture, firstly we represent Golden binomials in the product form. \begin{equation}gin{prop} The Golden binomial can be written as a product, \begin{equation}gin{eqnarray} (x-1)^{n+1}_{F}=\prod_{j=0}^{n} \left(x-\varphi^j \varphi'^{n-j}\right). \end{eqnarray} \end{prop} \begin{equation}gin{prf} Starting from Golden binomial in product representation \begin{equation}a (x+y)^{n}_{F} \equiv \prod_{j=0}^{n-1} \left(x-(-1)^{j-1}\phantom{.} \varphi^{n-1}\phantom{.} \varphi^{-2j} y \right) \end{equation}a by using \begin{equation}a \varphi^{-2j}=\left(\frac{1}{\varphi} \right)^{2j}=\left(-\frac{1}{\varphi} \right)^{2j}=\varphi'^{2j}, \end{equation}a after substitution $y=-1$ we have \begin{equation}a (x-1)^{n}_{F} \equiv \prod_{j=0}^{n-1} \left(x-(-1)^{j} \phantom{.} \varphi^{n-1}\phantom{.} \varphi'^{2j} \right) .\nonumber \end{equation}a By shifting $n \rightarrow n+1$, \begin{equation}a (x-1)^{n+1}_{F}&=&\prod_{j=0}^{n} \left(x-(-1)^{j} \phantom{.} \varphi^{n}\phantom{.} \varphi'^{2j} \right) \nonumber \\ &=&\prod_{j=0}^{n} \left(x-(-1)^{j} \phantom{.} \varphi^{n}\phantom{.} \frac{(-1)^{2j}}{\varphi^{j} \varphi^{j}} \right) \nonumber \\ &=&\prod_{j=0}^{n} \left(x- \varphi^{n} \left(-\frac{1}{\varphi}\right)^{j} \phantom{.} \frac{1}{\varphi^{j}} \right) \nonumber \\ &=&\prod_{j=0}^{n} \left(x- \varphi^{n-j} \varphi'^{j} \phantom{.} \right) \nonumber \end{equation}a and substituting $j=n-m$ we get, \begin{equation}a (x-1)^{n+1}_{F}=\prod_{m=0}^{n} \left(x- \varphi^{m} \varphi'\phantom{.}^{n-m} \phantom{.} \right). \nonumber \end{equation}a The formula shows explicitly that zeros of Golden binomial in $(\ref{even})$ and $(\ref{odd})$ are given by powers of $\varphi$ and $\varphi'$. \end{prf} \begin{equation}gin{cor} The eigenvalues of matrix $A_{n+1}$ are the numbers, \begin{equation}a \varphi^n, \varphi^{n-1}\varphi', \varphi^{n-2}\varphi'^{2}, \ldots ,\varphi \phantom{.}\varphi'^{n-1}, \varphi'^{n} \langlebel{eigenvaluesofmatrixAn+1}. \end{equation}a \end{cor} As it was shown by Carlitz \cite{Carlitz}, this product formula is just characteristic equation $(\ref{characteristicequation})$ for matrix $A_{n+1}$. Since zeros of two polynomials $\mbox{det}(x I -A_{n+1})$ and $(x-1)^{n+1}_{F}$ coincide, then the conjecture is correct and we have following theorem. \begin{equation}gin{thm} Characteristic equation for combinatorial matrix $A_{n+1}$ is given by Golden binomial: \begin{equation}a Q_{n+1}(x)=\mbox{det}(x I-A_{n+1})=(x-1)^{n+1}_{F}. \end{equation}a \end{thm} \section{Powers of $A_{n+1}$ and Fibonacci Divisors} \begin{equation}gin{prop} Arbitrary $n^{th}$ power of $A_{2}$ matrix is written in terms of Fibonacci numbers, \begin{equation}a A^{n}_{2}=\left( \begin{equation}gin{array}{cc} F_{n-1} & F_{n} \\ F_{n} & F_{n+1} \\ \end{array} \right). \end{equation}a \end{prop} \begin{equation}gin{prf} Proof will be done by induction. For $n=1$, \begin{equation}a A_{2}=\left( \begin{equation}gin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right) =\left( \begin{equation}gin{array}{cc} F_{0} & F_{1} \\ F_{1} & F_{2} \end{array}\right), \nonumber \end{equation}a and for $n=2$, \begin{equation}a A^2_{2}=\left( \begin{equation}gin{array}{cc} 1 & 1 \\ 1 & 2 \end{array} \right) =\left( \begin{equation}gin{array}{cc} F_{1} & F_{2} \\ F_{2} & F_{3} \end{array}\right). \nonumber \end{equation}a Suppose for $n=k$, \begin{equation}a A^{k}_{2}=\left( \begin{equation}gin{array}{cc} F_{k-1} & F_{k} \\ F_{k} & F_{k+1} \\ \end{array} \right) , \nonumber \end{equation}a then \begin{equation}a A^{k+1}_{2}&=&A^{k}_{2}\phantom{.} A_{2}=\left( \begin{equation}gin{array}{cc} F_{k-1} & F_{k} \\ F_{k} & F_{k+1} \\ \end{array} \right)\phantom{.} \left( \begin{equation}gin{array}{cc} 0 & 1 \\ 1 & 1 \\ \end{array} \right) \\ &=& \left( \begin{equation}gin{array}{cc} F_{k} & F_{k}+F_{k-1} \\ F_{k+1} & F_{k}+F_{k+1} \\ \end{array} \right)=\left( \begin{equation}gin{array}{cc} F_{k} & F_{k+1} \\ F_{k+1} & F_{k+2} \\ \end{array} \right).\nonumber \end{equation}a This result can be understood from observation that eigenvalues of matrix $A_2$ are $\varphi$ and $\varphi'$, and eigenvalues of $A^{n}_2$ are powers $\varphi^n$, $\varphi'^n$ related with Fibonacci numbers. \end{prf} As we have seen, eigenvalues of matrix $A_3$ are $\varphi^2, \varphi'^2, -1$. It implies that for $A^{n}_3$, eigenvalues are $\varphi^{2n}, \varphi'^{2n}, (-1)^n$, and the matrix can be expressed by Fibonacci divisor $F^{(2)}_n$ conjugate to $F_2$, due to \cite{FNK}, \begin{equation}a (\varphi^k)^n = \varphi^k F^{(k)}_n + (-1)^{k+1} F^{(k)}_{n-1},\langlebel{phikn1}\\ ({\varphi'}^k)^n = {\varphi'}^k F^{(k)}_n + (-1)^{k+1} F^{(k)}_{n-1},\langlebel{phikn2} \end{equation}a where $F^{(k)}_n = F_{nk}/F_k$. \begin{equation}gin{prop} \langlebel{A3powern} Arbitrary $n^{th}$ power of $A_{3}$ matrix can be expressed in terms of Fibonacci divisors $F_{n}^{(2)}$, \begin{equation}gin{eqnarray} A^{n}_3= \frac{1}{5}\left( \begin{equation}gin{array}{ccc} (2F_{n}^{(2)}-3F_{n-1}^{(2)}+2(-1)^n) & (2F_{n}^{(2)}+2F_{n-1}^{(2)}+2(-1)^n) & (3F_{n}^{(2)}-2F_{n-1}^{(2)}-2(-1)^n) \\ (F_{n}^{(2)}+F_{n-1}^{(2)}+(-1)^n) & (6F_{n}^{(2)}-4F_{n-1}^{(2)}+(-1)^n) & (4F_{n}^{(2)}-F_{n-1}^{(2)}-(-1)^n) \\ (3F_{n}^{(2)}-2F_{n-1}^{(2)}-2(-1)^n) & (8F_{n}^{(2)}-2F_{n-1}^{(2)}-2(-1)^n) & (7F_{n}^{(2)}-3F_{n-1}^{(2)}+2(-1)^n) \\ \end{array} \right) \nonumber \end{eqnarray} \end{prop} \begin{equation}gin{prf} Let's diagonalize the matrix $A_{3}$, \begin{equation}a \phi_{3}=\sigma^{-1}_{3}\phantom{a} A_{3} \phantom{a} \sigma_{3}, \nonumber \end{equation}a where $\phi_{3}$ is the diagonal matrix and \begin{equation}a A_{3}=\sigma_{3} \phantom{.} \phi_{3} \phantom{.} \sigma^{-1}_{3}. \nonumber \end{equation}a Taking the $n^{th}$ power of both sides gives, \begin{equation}a A^{n}_{3}=(\sigma_{3} \phantom{.} \phi_{3} \phantom{.} \underbrace{\sigma^{-1}_{3})\phantom{a}(\sigma_{3}}_{I} \phantom{.} \phi_{3} \phantom{.} \sigma^{-1}_{3})\phantom{a}...\phantom{a}(\sigma_{3} \phantom{.} \phi_{3} \phantom{.} \underbrace{ \sigma^{-1}_{3})\phantom{a}(\sigma_{3}}_{I} \phantom{.} \phi_{3} \phantom{.} \sigma^{-1}_{3}) \nonumber \end{equation}a Therefore, \begin{equation}a {A^{n}_{3}=\sigma_{3} \phantom{.} \phi^{n}_{3} \phantom{.} \sigma^{-1}_{3}}. \langlebel{A3powerintermsofphiandsigma} \end{equation}a By using the diagonalization principle, $\sigma_{3}$ and $\sigma^{-1}_{3}$ matrices can be obtained as, \begin{equation}a \sigma_{3}=\frac{1}{2}\left( \begin{equation}gin{array}{ccc} -\varphi' & \frac{4}{3} & -\varphi \\ 1 & \frac{2}{3} & 1 \\ \varphi & -\frac{4}{3} & \varphi' \end{array} \nonumber \right) \end{equation}a and, \begin{equation}a \sigma^{-1}_{3}=\left( \begin{equation}gin{array}{ccc} \frac{2(\varphi'+2)}{5\left(\varphi-\varphi'\right)} & -\frac{4(\varphi'+2)}{5 \varphi'\left(\varphi-\varphi'\right)} & \frac{2(2\varphi'-1)}{5 \varphi' \left(\varphi-\varphi'\right)} \\ \frac{3}{5} & \frac{3}{5} & -\frac{3}{5}\\ -\frac{2(\varphi+2)}{5\left(\varphi-\varphi'\right)} & \frac{4(\varphi+2)}{5 \varphi\left(\varphi-\varphi'\right)} & \frac{2(1-2\varphi)}{5 \varphi \left(\varphi-\varphi'\right)} \end{array} \right)=\frac{2}{5\sqrt{5}}\left( \begin{equation}gin{array}{ccc} \varphi'+2 & -2(1-2\varphi) & (2+\varphi) \\ \frac{3 \sqrt{5}}{2} & \frac{3\sqrt{5}}{2} & -\frac{3\sqrt{5}}{2}\\ -(\varphi+2) & 2 (1-2\varphi') & -(2+\varphi') \end{array} \right) \nonumber \end{equation}a Since eigenvalues of matrix $A_{3}$ are $\varphi^2,-1,\varphi'^2$, the diagonal matrix $\phi_{3}$ is, \begin{equation}a \phi_{3} =\left( \begin{equation}gin{array}{ccc} \varphi'^2 & 0 & 0 \\ 0 & -1 & 0\\ 0 & 0 & \varphi'^2 \end{array} \right), \end{equation}a and an arbitrary $n^{th}$ power of this matrix is, \begin{equation}a \phi^n_{3} =\left( \begin{equation}gin{array}{ccc} (\varphi'^2)^n & 0 & 0 \\ 0 & (-1)^n & 0\\ 0 & 0 & (\varphi'^2)^n \end{array} \right). \end{equation}a Finally by using $(\ref{A3powerintermsofphiandsigma})$, $A^{n}_3=$ \begin{equation}gin{eqnarray} \frac{1}{5}\left( \begin{equation}gin{array}{ccc} (2F_{n}^{(2)}-3F_{n-1}^{(2)}+2(-1)^n) & (2F_{n}^{(2)}+2F_{n-1}^{(2)}+2(-1)^n) & (3F_{n}^{(2)}-2F_{n-1}^{(2)}-2(-1)^n) \\ (F_{n}^{(2)}+F_{n-1}^{(2)}+(-1)^n) & (6F_{n}^{(2)}-4F_{n-1}^{(2)}+(-1)^n) & (4F_{n}^{(2)}-F_{n-1}^{(2)}-(-1)^n) \\ (3F_{n}^{(2)}-2F_{n-1}^{(2)}-2(-1)^n) & (8F_{n}^{(2)}-2F_{n-1}^{(2)}-2(-1)^n) & (7F_{n}^{(2)}-3F_{n-1}^{(2)}+2(-1)^n) \\ \end{array} \right) \nonumber \end{eqnarray} is obtained. \end{prf} As we can expect, these results can be generalized to arbitrary matrix $A_{n+1}$. Since eigenvalues of $A_{n+1}$ are powers $\varphi^n$,$\varphi'^n$, $\ldots$, for $A^{N}_{n+1}$ eigenvalues are $\varphi^{nN}$,$\varphi'^{nN}$, \ldots But these powers can be written in terms of Fibonacci divisors as in $(\ref{phikn1})$, $(\ref{phikn2})$, and the matrix $A^{N}_{n+1}$ itself can be represented by Fibonacci divisors $F^{(n)}_{N}$. For powers of matrix $A_{n+1}$ we have the following identities. \begin{equation}gin{thm} \langlebel{invarianttheorem} Invariants of $A^{k}_{n+1}$ matrix are found as, \begin{equation}a Tr\left( A^k_{n+1} \right)&=&\frac{F_{kn+k}}{F_{k}}=F^{(k)}_{n+1}, \langlebel{invarianttheoremequation1} \\ {det}\left(A^k_{n+1} \right)&=&(-1)^{k \phantom{.} \frac{n(n+1)}{2}} . \langlebel{invarianttheoremequation2} \end{equation}a For $k=1$, it gives \begin{equation}a Tr\left( A_{n+1} \right)&=& F_{n+1}, \nonumber \\ {det}\left(A_{n+1} \right)&=&(-1)^{\frac{n(n+1)}{2}} . \nonumber \end{equation}a \end{thm} \begin{equation}gin{prf} Let's diagonalize the general matrix $A_{n+1}$ as, \begin{equation}a \phi_{n+1}=\sigma^{-1}_{n+1}\phantom{a} A_{n+1} \phantom{a} \sigma_{n+1} \nonumber \end{equation}a where $\phi_{n+1}$ is diagonal and \begin{equation}a A_{n+1}=\sigma_{n+1} \phantom{.} \phi_{n+1} \phantom{.} \sigma^{-1}_{n+1}. \nonumber \end{equation}a Taking the $k^{th}$ power of both sides gives, \begin{equation}a A^{k}_{n+1}=(\sigma_{n+1} \phantom{.} \phi_{n+1} \phantom{.} \underbrace{\sigma^{-1}_{n+1})\phantom{a}(\sigma_{n+1}}_{I} \phantom{.} \phi_{n+1} \phantom{.} \sigma^{-1}_{n+1})\phantom{a}...\phantom{a}(\sigma_{n+1} \phantom{.} \phi_{n+1} \phantom{.} \underbrace{ \sigma^{-1}_{n+1})\phantom{a}(\sigma_{n+1}}_{I} \phantom{.} \phi_{n+1} \phantom{.} \sigma^{-1}_{n+1}) \nonumber \end{equation}a and \begin{equation}a A^{k}_{n+1}=\sigma_{n+1} \phantom{.} \phi^{k}_{n+1} \phantom{.} \sigma^{-1}_{n+1}. \langlebel{An+1powerk} \end{equation}a By taking trace from both sides and using the cyclic permutation property of trace, \begin{equation}a Tr(A^{k}_{n+1})=Tr\phantom{.}(\sigma_{n+1} \phantom{.} \phi^{k}_{n+1} \phantom{.} \sigma^{-1}_{n+1})=Tr\phantom{.}(\sigma^{-1}_{n+1} \phantom{.} \sigma_{n+1} \phantom{.} \phi^{k}_{n+1})=Tr(I \phantom{a} \phi^{k}_{n+1})=Tr\phantom{.}( \phi^{k}_{n+1}) \nonumber \end{equation}a we get \begin{equation}a {Tr(A^{k}_{n+1})=Tr\phantom{.}( \phi^{k}_{n+1})}. \nonumber \end{equation}a The eigenvalues of matrix $A_{n+1}$ in $(\ref{eigenvaluesofmatrixAn+1})$, allows one to construct the diagonal matrix $\phi_{n+1}$ and calculate \begin{equation}a Tr(A^{k}_{n+1})=Tr \phantom{..} \left( \begin{equation}gin{array}{ccccccccc} \varphi^n & 0 & 0 & .& . & . & 0 & 0 & 0 \\ 0 & \varphi^{n-1} \varphi'& 0 & . & . & . & 0 & 0 & 0 \\ 0 & 0 & \varphi^{n-2} \varphi'^2 & . & . & . & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & . & . & . & \varphi^{2} \varphi'^{n-2} & 0 & 0 \\ 0 & 0 & 0 & . & . & . & 0 & \varphi \varphi'^{n-1} & 0 \\ 0 & 0 & 0 & . & . & . & 0 & 0 & \varphi'^{n} \\ \end{array} \right)^{k}. \nonumber \end{equation}a It gives \begin{equation}a Tr(A^{k}_{n+1})=Tr \phantom{..}\left( \begin{equation}gin{array}{ccccccccc} (\varphi^n)^k & 0 & 0 & . & 0 & 0 & 0 \\ 0 & (\varphi^{n-1} \varphi')^k & 0 & . & 0 & 0 & 0 \\ 0 & 0 & (\varphi^{n-2} \varphi'^2)^k & . & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & . & (\varphi^{2} \varphi'^{n-2})^k & 0 & 0 \\ 0 & 0 & 0 &. & 0 & (\varphi \varphi'^{n-1})^k & 0 \\ 0 & 0 & 0 &. & 0 & 0 & (\varphi'^{n})^k \\ \end{array} \right) \nonumber \langlebel{matrixfipowerk} \end{equation}a and \begin{equation} Tr(A^{k}_{n+1})=(\varphi^n)^k+(\varphi^{n-1} \varphi')^k+\ldots +(\varphi \varphi'^{n-1})^k+(\varphi'^{n})^k, \nonumber \end{equation} or \begin{equation} Tr(A^{k}_{n+1})=(\varphi^k)^n+(\varphi^{k})^{n-1} \varphi'^k+\ldots +\varphi^k (\varphi'^k)^{n-1}+(\varphi'^{k})^n . \nonumber \end{equation} The powers $(\varphi^k)^n$ and $(\varphi'^{k})^n$ substituted from equations $(\ref{phikn1})$ and $(\ref{phikn2})$ give $Tr(A^{k}_{n+1}) =$ \begin{equation}a &=&\left(\varphi^k \phantom{.} F^{(k)}_n + (-1)^{k+1} \phantom{.} F^{(k)}_{n-1}\right)+\left(\varphi^k \phantom{.} F^{(k)}_{n-1} + (-1)^{k+1} \phantom{.} F^{(k)}_{n-2}\right)\varphi'^{k}+\ldots \nonumber \\ &&+\left(\varphi^k \phantom{.} F^{(k)}_1 + (-1)^{k+1} \phantom{.} F^{(k)}_{0}\right)(\varphi'^{k})^{n-1}+(\varphi'^{k})^{n} \nonumber \\ &=& \varphi^k \left(F^{(k)}_n+F^{(k)}_{n-1}(\varphi'^k)+F^{(k)}_{n-2}(\varphi'^k)^2+\ldots +F^{(k)}_{1}(\varphi'^k)^{n-1}\right) \nonumber \\ &&+(-1)^{k+1}\left(F^{(k)}_{n-1}+F^{(k)}_{n-2}(\varphi'^k)+F^{(k)}_{n-3}(\varphi'^k)^2+\ldots +F^{(k)}_{0}(\varphi'^k)^{n-1}\right) \nonumber \\ &&+(\varphi'^k)^n \nonumber \\ &=&\varphi^k \left(\frac{F_{kn}}{F_{k}}+\frac{F_{(n-1)k}}{F_{k}}(\varphi'^k)+\frac{F_{(n-2)k}}{F_{k}}(\varphi'^k)^2+\ldots +\frac{F_{k}}{F_{k}}(\varphi'^k)^{n-1}\right) \nonumber \\ &&+(-1)^{k+1}\left(\frac{F_{(n-1)k}}{F_{k}}+\frac{F_{(n-2)k}}{F_{k}}(\varphi'^k)+\frac{F_{(n-3)k}}{F_{k}}(\varphi'^k)^2+\ldots +\frac{F_{0}}{F_{k}}(\varphi'^k)^{n-1}\right)\nonumber\\ &&+(\varphi'^k)^n \nonumber \\ &=&\frac{F_{kn}}{F_{k}}\phantom{..} \varphi^k + \frac{F_{(n-1)k}}{F_{k}} (-1)^{k} + \frac{F_{(n-2)k}}{F_{k}} (-1)^{k} (\varphi'^k)+\ldots +\frac{F_{k}}{F_{k}}(\varphi^k) (\varphi'^k)^{n-1}\nonumber \\ &&+ \frac{F_{(n-1)k}}{F_{k}} (-1)^{k+1} + \frac{F_{(n-2)k}}{F_{k}} (-1)^{k+1}(\varphi'^k) + \frac{F_{(n-3)k}}{F_{k}} (-1)^{k+1}(\varphi'^k)^2 \nonumber \\ &&+\ldots +\frac{F_{0}}{F_{k}} (-1)^{k+1}\varphi'^{n-1} + (\varphi'^k)^n \nonumber \\ &=&\frac{F_{kn}}{F_{k}}\phantom{..} \varphi^k + \frac{F_{(n-1)k}}{F_{k}} (-1)^{k} + \frac{F_{(n-2)k}}{F_{k}} (-1)^{k} (\varphi'^k)+\ldots \nonumber \\ &&+ \frac{F_{(n-(n-1))k}}{F_{k}} (-1)^{k}(\varphi'^k)^{n-2} + \frac{F_{(n-1)k}}{F_{k}} (-1)^{k+1}+ \frac{F_{(n-2)k}}{F_{k}} (-1)^{k+1}(\varphi'^k) \nonumber \\ &&+ \frac{F_{(n-3)k}}{F_{k}} (-1)^{k+1}(\varphi'^k)^{2}+\ldots +\frac{F_{k}}{F_{k}} (-1)^{k+1}(\varphi'^k)^{n-2} + (\varphi'^k)^{n} \nonumber \\ &=& \frac{F_{kn}}{F_{k}}\phantom{..} \varphi^k + \frac{F_{(n-1)k}}{F_{k}} \left( (-1)^k + (-1)^{k+1} \right) \nonumber \\ &&+ \frac{F_{(n-2)k}}{F_{k}} \left( (-1)^k \varphi'^k +(-1)^{k+1} \varphi'^k \right) + \frac{F_{(n-3)k}}{F_{k}} \left((-1)^k (\varphi'^k)^2 + (-1)^{k+1} (\varphi'^k)^2 \right)\nonumber \\ &&+\ldots +\frac{F_{k}}{F_{k}} \left( (-1)^k (\varphi'^k)^{n-2} + (-1)^{k+1} (\varphi'^k)^{n-2} \right) + (\varphi'^k)^{n} \nonumber \\ &=&\frac{F_{kn}}{F_{k}}\phantom{..} \varphi^k + \frac{F_{(n-1)k}}{F_{k}} (-1)^{k} (1+(-1)) + \frac{F_{(n-2)k}}{F_{k}} (-1)^{k} \varphi'^k (1+(-1))\nonumber \\ &&+\frac{F_{(n-3)k}}{F_{k}} (-1)^{k} (\varphi'^k)^2 (1+(-1))+\ldots +\frac{F_{k}}{F_{k}} (-1)^{k} (\varphi'^k)^{n+2} (1+(-1))\nonumber \\ &&+(\varphi'^k)^{n} \nonumber \\ &=&\frac{F_{kn}}{F_{k}} \phantom{..} \varphi^k + (\varphi'^k)^{n} \nonumber \\ &{(\ref{phikn2})}{=}&\frac{F_{kn}}{F_{k}}\phantom{..} \varphi^k + \varphi'^k \phantom{.} F^{(k)}_{n} + (-1)^{k+1} \phantom{.} F^{(k)}_{n-1} \nonumber \\ &=&\frac{F_{kn}}{F_{k}} \phantom{..} \varphi^k + \varphi'^k \phantom{.} \frac{F_{kn}}{F_{k}} + (-1)^{k+1} \phantom{.} \frac{F_{k(n-1)}}{F_{k}} \nonumber \end{equation}a \begin{equation}a &=&\frac{1}{F_{k}} \left( F_{kn} \varphi^k + \varphi'^k \phantom{.} F_{kn} + (-1)^{k+1} \phantom{.} F_{k(n-1)} \right)\nonumber \\ &=&\frac{1}{F_{k}} \frac{1}{\varphi-\varphi'} \left[ \left(\varphi^{kn}-\varphi'^{kn}\right)\varphi^{k}+\varphi'^{k} \left(\varphi^{kn}-\varphi'^{kn}\right)+ (-1)^{k+1}\left(\varphi^{(n-1)k}-\varphi'^{(n-1)k}\right) \right] \nonumber \\ &=&\frac{1}{F_{k}} \frac{1}{\varphi-\varphi'} \left[ \varphi^{k(n+1)}-\varphi'^{kn} \varphi^{k}+\varphi'^{k} \varphi^{kn}-\varphi'^{k+kn}+ (-1)^{k+1} \varphi^{(n-1)k}-(-1)^{k+1} \varphi'^{(n-1)k} \right] \nonumber \\ &=&\frac{1}{F_{k}} \frac{1}{\varphi-\varphi'} \bigg[ \varphi^{k(n+1)}-\varphi'^{k(n+1)}-\left(-\frac{1}{\varphi}\right)^{kn} \varphi^{k}+\left(-\frac{1}{\varphi}\right)^{k} \varphi^{kn}+(-1)^{k+1} \varphi^{(n-1)k} \nonumber \\ &&-(-1)^{k+1} \left(-\frac{1}{\varphi}\right)^{(n-1)k} \bigg] \nonumber \\ &=&\frac{1}{F_{k}} \frac{1}{\varphi-\varphi'} \bigg[ \varphi^{k(n+1)}-\varphi'^{k(n+1)}-(-1)^{kn} \varphi^{k(1-n)}+(-1)^{k} \varphi^{k(n-1)}-(-1)^{k} \varphi^{k(n-1)} \nonumber \\ &&+(-1)^k (-1)^{k(n-1)} \varphi^{k(1-n)} \bigg] \nonumber \\ &=&\frac{1}{F_{k}} \frac{1}{\varphi-\varphi'} \left[ \varphi^{k(n+1)}-\varphi'^{k(n+1)}-(-1)^{kn} \varphi^{k(1-n)}+(-1)^k (-1)^{kn}(-1)^{-k} \varphi^{k(1-n)} \right] \nonumber \\ &=&\frac{1}{F_{k}} \frac{1}{\varphi-\varphi'} \left[ \varphi^{k(n+1)}-\varphi'^{k(n+1)}-(-1)^{kn} \varphi^{k(1-n)}+(-1)^{kn} \varphi^{k(1-n)} \right] \nonumber \\ &=&\frac{1}{F_{k}} \phantom{.} \frac{1}{\varphi-\varphi'} \left[ \varphi^{k(n+1)}-\varphi'^{k(n+1)} \right] \nonumber \\ &=&\frac{1}{F_{k}}\phantom{.} \frac{\varphi^{k(n+1)}-\varphi'^{k(n+1)}}{\varphi-\varphi'} \nonumber \\ &=&\frac{1}{F_{k}} F_{k(n+1)} \nonumber \\ &=&\frac{F_{k(n+1)}}{F_{k}} . \nonumber \end{equation}a To prove the relation for ${det}\left(A^k_{n+1} \right)$, we take the determinant from both sides in $(\ref{An+1powerk})$, \begin{equation}a {\det}\left(A^k_{n+1} \right)={\det}\left( \sigma_{n+1} \phantom{.} \phi^{k}_{n+1} \phantom{.} \sigma^{-1}_{n+1}\right). \end{equation}a By using property of determinants, \begin{equation}a \det(AB)=\det(A) \det(B) \end{equation}a we obtain, \begin{equation}a {\det}\left(A^k_{n+1} \right)&=&{\det}\left( \sigma_{n+1}\right) \phantom{.} {\det} \left( \phi^{k}_{n+1} \right) \phantom{.} {\det} \left( \sigma^{-1}_{n+1} \right) \Rightarrow \nonumber \end{equation}a \begin{equation}a {\det}\left(A^k_{n+1} \right)&=&{\det}\left( \sigma_{n+1}\right) \phantom{.} {\det} \left( \sigma^{-1}_{n+1} \right) \phantom{.}{\det} \left( \phi^{k}_{n+1} \right) \nonumber \Rightarrow \\ {\det}\left(A^k_{n+1} \right)&=&{\det}\left( \sigma_{n+1} \phantom{.}\sigma^{-1}_{n+1} \right) \phantom{.}{\det} \left( \phi^{k}_{n+1} \right) \nonumber \Rightarrow \\ {\det}\left(A^k_{n+1} \right)&=&{\det}\left(I\right) \phantom{.}{\det} \left( \phi^{k}_{n+1} \right) \nonumber \Rightarrow\\ {\det}\left(A^k_{n+1} \right)&=&{\det} \left( \phi^{k}_{n+1} \right) .\nonumber \end{equation}a Since the matrix $\phi^{k}_{n+1}$ is known, the above equation becomes, \begin{equation}a {\det}\left(A^k_{n+1} \right)&=& (\varphi^n)^k \phantom{.} (\varphi^{n-1} \varphi')^k \phantom{.} (\varphi^{n-2} \varphi'^2)^k \ldots (\varphi^{2} \varphi'^{n-2})^k \phantom{.} (\varphi \varphi'^{n-1})^k \phantom{.}(\varphi'^n)^k \nonumber \\ &=&\left(\varphi^{nk}\phantom{.}\varphi^{(n-1)k}\phantom{.}\varphi^{(n-2)k} \ldots \varphi^{2k}\phantom{.}\varphi^{k}\right)\left(\varphi'^{k}\phantom{.}\varphi'^{2k} \ldots \varphi'^{(n-2)k}\phantom{.}\varphi'^{(n-1)k}\phantom{.}\varphi'^{nk}\right) \nonumber \\ &=&\left(\varphi^{nk+(n-1)k+(n-2)k+\ldots +2k+k}\right)\left(\varphi'\phantom{.}^{k+2k+ \ldots +(n-2)k+(n-1)k+nk}\right) \nonumber \\ &=&\varphi^{k\left[n+(n-1)+(n-2)+\ldots +2+1\right]} \phantom{.} \varphi'\phantom{.}^{k\left[1+2+ \ldots +(n-2)+(n-1)+n\right]} \nonumber \\ &=& \varphi^{k\left(\frac{n(n+1)}{2}\right)} \phantom{.} \varphi'^{k\left(\frac{n(n+1)}{2}\right)} \nonumber \\ &=& \left(\varphi^{\frac{n(n+1)}{2}}\right)^k \phantom{.} \left(\varphi'^ {\frac{n(n+1)}{2}}\right)^k \nonumber \\ &=& \left[(\varphi \varphi')^{\frac{n(n+1)}{2}}\right]^k \nonumber \\ &{\left(\varphi \varphi'=-1\right)}{=}& (-1)^{k \phantom{.} \frac{n\phantom{.}(n+1)}{2}}. \nonumber \end{equation}a \end{prf} The above Theorem represnts Fibonacci divisors $F^{(k)}_{n+1}$ in terms of combinatorial matrix $A_{n+1}$. Quantum calculus for such divisors was constracted recently in \cite{FNK}. As was shown, it is related with several problems from hydrodynamics, quantum integrable systems and quantum information theory. This is why results of the present paper can be useful in the studies of this calculus and its applications. \begin{equation}gin{thebibliography}{99} \bibitem{golden} Pashaev O K and Nalci S 2012. Golden quantum oscillator and Binet-Fibonacci calculus \textit{J Phys A:Math Theor} \textbf{45} 015303 \bibitem{Eskisehir} Pashaev O K 2016. Quantum calculus of classical vortex images, integrable models and quantum states \textit{J of Phys: Conf Series} \textbf{766} 012015 \bibitem{Carlitz} Carlitz L 1965. The characteristic polynomial of a certain matix of binomial coefficients \textit{Fibonacci Quarterly} \textbf{3} pp.81-89 \bibitem{FNK} Pashaev O K 2020. Quantum Calculus of Fibonacci Divisors and Infinite Hierarchy of Bosonic-Fermionic Golden Quantum Oscillators. \textit{arXiv:} \textbf{2010.12386}[math-ph] 20 October 2020 \bibitem{Kac} Kac, V. and Cheung, P.,2002. Quantum Calculus, Springer, New York. \end{thebibliography} \end{document}

\begin{document} \title{Hadamard type operations for qubits} \author{Arpita Maitra and Preeti Parashar} \affiliation{Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700 108, India, Email: \{arpita\_r, parashar\}@isical.ac.in} \begin{abstract} We obtain the most general ensemble of qubits, for which it is possible to design a universal Hadamard gate. These states when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard `type' of operations and find ensembles of states for which such transformations hold. Unequal superposition of a qubit and its orthogonal complement is also investigated. \end{abstract} \maketitle \newcommand{\qed}{ \rule{2mm}{2mm}} \newcommand{{\bf Proof : }}{{\bf Proof : }} \newtheorem{definition}{Definition} \newtheorem{algorithm}{Algorithm} \newtheorem{construction}{Construction} \newtheorem{theorem}{Theorem} \newtheorem{question}{Question} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \newtheorem{remark}{Remark} \newtheorem{corollary}{Corollary} \newtheorem{example}{Example} \newcommand{\binom}[2] {\mbox{$\left( { #1 \atop #2 } \right)$}} {\bf Keywords:} Unitary operations, Hadamard Gate, Bloch Sphere, Qubits. {\bf PACS:} 03.67.Lx \section{I. Introduction} Qubits and quantum gates are the two basic building blocks of quantum computers which are believed to be computationally stronger than their classical counterparts. One such important gate is the Hadamard gate which has found wide applications in computer and communication science ~\cite{qNC02}. There are a number of seminal papers in quantum computation and information theory where Hadamard transform has been used~\cite{qDJ92,BV93,BV97,qGR96, DD89}. Shor's fast algorithm for factoring and discrete logarithm~\cite{Sh94} are based on Fourier transform which is a generalization of the Hadamard transform in higher dimensions. Furthermore, the Toffoli and Hadamard gates comprise the simplest quantum universal set of gates \cite{Shi02, DA03}. So, in order to achieve the full power of quantum computation, one needs to add only the Hadamard gate to the classical set. Thus, the role played by the Hadamard gate in quantum algorithms is indeed significant. Of late, Pati~\cite{PT02} has shown that one can not design a universal Hadamard gate for an arbitrary unknown qubit. Linearity, which is at the heart of quantum mechanics, does not allow linear superposition of an unknown state $|\psi\rangle$ with its orthogonal complement $|\psi_{\perp}\rangle$. However, if one considers qubit states from the polar or equatorial great circles on a Bloch sphere, then it is possible to design Hadamard type of gates. By a Hadamard `type' gate we mean a unitary matrix that is not exactly a Hadamard matrix. However, it still creates an equal superposition (up to a sign or a phase) of a qubit and its complement to produce two orthogonal states. Very recently, Song et. al.~\cite{Son04} have tried to implement the Hadamard gate in a probabilistic manner for any unknown state chosen from a set of linearly independent states. Motivated by Pati's work, our primary aim in this paper is to construct the most general class of qubit states, for which the Hadamard gate can be designed in a deterministic way. This is achieved in Sec. II, by imposing restrictions ( due to linearity ) on a completely arbitrary unknown quantum state. States from this set are geometrically represented on the three - dimensional unit sphere known as the Bloch sphere. In Sec. III, we show that certain Hadamard `type' transformations are indeed possible for arbitrary states when partial information is available. A Hadamard type gate is obtained for qubits chosen from, not only the polar great circle but also from any polar circle. We also demonstrate with an example, that there is a unique class of states (up to isomorphism) associated with a particular gate, satisfying a fixed transformation. As for the second Hadamard type of transformation, which is related to the states lying on the equatorial great circle, a new ensemble of states is found. In Sec. IV, unequal superposition of a qubit with its orthogonal complement is investigated. This is a generalization of the usual Hadamard transformation when the two amplitudes are not equal. In this context, many new classes of quantum states are found for which the unequal superposition works.Summary and Concluding remarks are made in Sec. V. \section{II. Hadamard Transform for Special Qubits} The Hadamard transform $H$, which is a one qubit gate, rotates the two computational basis vectors $|0\rangle$ and $|1\rangle$ to two other orthogonal vectors $\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)$ and $\frac{1}{\sqrt{2}} (|0\rangle - |1\rangle)$, respectively. Thus it creates an equal superposition of the amplitudes of the state and its orthogonal. The matrix representation of the Hadamard gate is given by $H = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array}\right]$. The question we ask in this paper is: What is the most general set of qubit states $\{|\psi\rangle, |\psi_{\perp}\rangle\}$, such that the application of the Hadamard gate $H$ takes them to two other orthogonal states $\frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle)$ and $\frac{1}{\sqrt{2}} (|\psi\rangle - |\psi_{\perp}\rangle)$ respectively? If $|\psi\rangle$ is completely arbitrary and unknown, then such a universal Hadamard gate does not exist \cite{PT02}. So, we shall obtain a special class of qubit states such that \begin{equation} \label{eq1} H(|\psi\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle), H(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle - |\psi_{\perp}\rangle). \end{equation} \begin{figure*} \caption{Points on Bloch sphere in reference to Theorem~\ref{th1} \label{fig1} \end{figure*} We start by considering a completely arbitrary, unknown qubit state $|\psi\rangle = a|0\rangle + b |1\rangle$ and its orthogonal complement $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Here $a, b$ are complex numbers obeying the normalization condition $|a|^2 + |b|^2 = 1$. The operator $H$ acts linearly, i.e., $H(|\psi\rangle) = a H(|0\rangle) + b H(|1\rangle)$. In what follows, we show that this restricts the form of $|\psi\rangle$ to $(\alpha + i\beta) |0\rangle + \alpha |1\rangle$, with $2\alpha^2 + \beta^2 = 1$, where $\alpha$ and $\beta$ are real. Now we substantiate the assertions made herein above. Ideally, from the Hadamard transformation we obtain \begin{eqnarray} H(|\psi\rangle) &=& \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle) \nonumber \\ \ &=& \frac{1}{\sqrt{2}} (a |0\rangle + b |1\rangle + b^* |0\rangle - a^* |1\rangle) \nonumber \\ \ &=& \frac{(a+b^*) |0\rangle - (a^*-b) |1\rangle}{\sqrt{2}}, \nonumber \end{eqnarray} and from linearity we get \begin{eqnarray} H(|\psi\rangle) &=& a H(|0\rangle) + b H(|1\rangle) \nonumber \\ \ &=& a \frac{|0\rangle + |1\rangle}{\sqrt{2}} + b \frac{|0\rangle - |1\rangle}{\sqrt{2}} \nonumber \\ \ &=& \frac{(a+b) |0\rangle + (a-b) |1\rangle}{\sqrt{2}}. \nonumber \end{eqnarray} These two expressions should be equal. Hence $a + b^* = a + b$, i.e., $b = b^*$. Thus $b$ is real. Let $b = \alpha$, where $\alpha$ is a real number. Moreover, $-(a^* - b) = (a - b)$, i.e., $a+a^* = 2b$. So the real part of $a$ is $\alpha$. Let $a = \alpha + i\beta$, where $\beta$ is a real number. Thus $|\psi\rangle$ is of the form $(\alpha + i\beta) |0\rangle + \alpha |1\rangle$. Clearly $-\frac{1}{\sqrt{2}} \leq \alpha \leq \frac{1}{\sqrt{2}}$, since $\beta^2 = 1 - 2 \alpha^2$. Therefore, $|\psi\rangle$ has complex as well as real amplitudes when expressed in the computational basis $\{|0\rangle, |1\rangle \}$; the real parts of which are equal. It can be easily checked, in a similar fashion, that if we consider $H(|\psi_{\perp}\rangle)$, we get $b = b^*$ (i.e., $b$ is real) and $a+a^* = 2b^* = 2b$ (as $b$ is real) leading to the same result. Thus $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$ is restricted to the form $\alpha |0\rangle - (\alpha - i\beta) |1\rangle$. Our next task is to map the states from this ensemble to points on the Bloch sphere. But before attempting to do this, we give a brief pedagogical description of how to geometrically represent a general qubit state on the Bloch sphere. Consider $|\psi\rangle = a |0\rangle + b |1\rangle$. Since $a$ and $b$ are complex, assume $a = r_1 e^{i\gamma}, b = r_2 e^{i(\gamma+\phi)}$. Then $|a| = r_1, |b| = r_2$. Let $r_1 = \cos{\frac{\theta}{2}}, r_2 = \sin{\frac{\theta}{2}}$. Hence, $a = \cos{\frac{\theta}{2}} e^{i\gamma}, b = \sin{\frac{\theta}{2}} e^{i(\gamma+\phi)}$, where $\theta, \phi$ and $\gamma$ are real. Thus, any qubit $|\psi\rangle = a |0\rangle + b |1\rangle$ can be written as $e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle)$. Further, two qubits $e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle)$ and $(\cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle)$ are treated on equal footing under measurement, since they differ only by an overall phase factor which has no observable effect. The qubit $(\cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle)$ is mapped to a point $(1, \theta, \phi)$ on the unit Bloch sphere. Here $\theta$ and $\phi$ are the usual polar and azimuthal angles respectively, and they are related to the cartesian coordinates $(x, y, z)$ through the usual relations $x = \cos{\phi} \sin{\theta}$, $y = \sin{\phi} \sin{\theta}$, $z = \cos{\theta}$. If we fix $\phi = 0$, then we obtain states of the form $\cos{\frac{\theta}{2}} |0\rangle + \sin{\frac{\theta}{2}} |1\rangle$ and $\cos{\frac{\theta}{2}} |1\rangle - \sin{\frac{\theta}{2}} |0\rangle$ for $0 \leq \theta \leq \pi$. These lie on the polar great circle of the Bloch sphere. On the other hand, for $\theta = \pi /2$, one obtains states of the form $\frac{1}{\sqrt{2}}(|0\rangle + e^{i\phi} |1\rangle)$ and $\frac{1}{\sqrt{2}}(|1\rangle - e^{-i\phi} |0\rangle)$ for $0 \leq \phi \leq 2\pi$, which lie on the equatorial great circle. Now we can conveniently plot the states from our special ensemble $|\psi\rangle = (\alpha + i\beta) |0\rangle + \alpha |1\rangle$. Bringing it to the desired form, it is clear that $e^{i\gamma} = \frac{\alpha + i\beta}{\sqrt{\alpha^2 + \beta^2}}, \cos{\frac{\theta}{2}} = \sqrt{\alpha^2 + \beta^2}, \sin{\frac{\theta}{2}} = \alpha, e^{i\phi} = \frac{\sqrt{\alpha^2 + \beta^2}}{\alpha + i\beta}$. We thus arrive at the following identification: \begin{eqnarray} \label{al1} x &=& \cos{\phi}\sin{\theta} = 2\alpha^2, \nonumber \\ y &=& \sin{\phi} \sin{\theta} = - 2\alpha \sqrt{1-2\alpha^2}, \nonumber \\ z &=& \cos{\theta} = 1 - 2\alpha^2. \end{eqnarray} These points are represented by curve 1 on the Bloch sphere. Next, we consider $|\psi\rangle = a|0\rangle + b |1\rangle$ and the other orthogonal complement $|\psi_{\perp}\rangle = - b^* |0\rangle + a^* |1\rangle$, which differs from the first one just by an overall negative sign. This yields that $|\psi\rangle$ must be of the form $(\alpha + i\beta) |0\rangle + i\beta |1\rangle$, with $\alpha^2 = 1 - 2\beta^2$ where $-\frac{1}{\sqrt{2}} \leq \beta \leq \frac{1}{\sqrt{2}}$, since $\alpha$ is real. Therefore, in the computational basis, the qubit state $|\psi\rangle$ has complex and imaginary amplitudes; the imaginary parts of which are equal. As for $|\psi_{\perp}\rangle = -b^* |0\rangle + a^* |1\rangle$, it assumes the form $i\beta |0\rangle + (\alpha - i\beta) |1\rangle$. For the qubits of the form $(\alpha + i\beta) |0\rangle + i\beta |1\rangle$, we have $e^{i\gamma} = \frac{\alpha + i\beta}{\sqrt{\alpha^2 + \beta^2}}, \cos{\frac{\theta}{2}} = \sqrt{\alpha^2 + \beta^2}, \sin{\frac{\theta}{2}} = \beta, e^{i\phi} = i\frac{\sqrt{\alpha^2 + \beta^2}}{\alpha + i\beta}$. Thus on the Bloch sphere: \begin{eqnarray} \label{al2} x &=& \cos{\phi}\sin{\theta} = 2\beta^2, \nonumber \\ y &=& \sin{\phi} \sin{\theta} = 2\beta \sqrt{1-2\beta^2}, \nonumber \\ z &=& \cos{\theta} = 1 - 2\beta^2. \end{eqnarray} It is immediately clear that a point represented by Eq(\ref{al1}), for a particular value of $\alpha$, is exactly equal to the one obtained from Eq(\ref{al2}) for the same value of $(- \beta)$. This implies that these two ensembles give the same trajectory on the Bloch sphere. Hence, we shall consider them to be isomorphic to each other. Our result is thus summarized in the following theorem. \begin{theorem} \label{th1} The most general qubit states for which it is possible to design a universal Hadamard gate satisfying Eq(\ref{eq1}) are given by $\{{|\psi\rangle, |\psi_{\perp}\rangle} || |\psi\rangle = (\alpha + i\beta) |0\rangle + \alpha |1\rangle ; |\psi_{\perp}\rangle = \alpha |0\rangle - (\alpha - i\beta) |1\rangle\}$ where $\alpha, \beta$ are real such that $2\alpha^2 + \beta^2 = 1$ and $-\frac{1}{\sqrt{2}} \leq \alpha \leq \frac{1}{\sqrt{2}}$. \end{theorem} Note that if we choose $\alpha = 0$, then from Eq(\ref{al1}), we obtain the point $(0, 0, 1)$ on the Bloch sphere (i.e., north pole), which can be identified with the computational basis state $|0\rangle$ on curve 1. In a similar fashion, the trajectory of $|\psi_\perp\rangle$ can be sketched, which would lie on the other side of the Bloch sphere (not visible in the figure). It can be checked that the orthogonal state $|1\rangle$ would be one of its points $(0, 0, -1)$ (i.e., south pole). We demonstrate that this trajectory has some intersection points with the equatorial great circle also. To this end, for $\theta = \frac{\pi}{2}$, $z = \cos{\theta} = 0 = 1 - 2\alpha^2$, i.e., $\alpha = \pm \frac{1}{\sqrt{2}}$, and $2\alpha^2 + \beta^2 = 1$, i.e., $\beta = 0$. Substituting these values we get $|\psi\rangle = (\alpha + i \beta) |0\rangle + \alpha |1\rangle = \pm \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|\psi_\perp\rangle = \alpha |0\rangle + (\alpha - i\beta) |1\rangle = \pm \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$. For the second case, $\beta = \pm \frac{1}{\sqrt{2}}$ and $\alpha = 0$. This yields $|\psi\rangle = = \pm \frac{i}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|\psi_\perp\rangle = = \pm \frac{i}{\sqrt{2}}(|0\rangle - |1\rangle)$. Therefore,if one chooses any qubit $|\psi\rangle$ from curve 1 in the figure, and takes its orthogonal complement, then the Hadamard transformation works perfectly well to generate the superposition. To see it explicitly, take $|\psi\rangle = (\alpha + i\beta) |0\rangle + \alpha |1\rangle$ and $|\psi_\perp\rangle = \alpha |0\rangle - (\alpha - i\beta) |1\rangle$. $H$ rotates $|\psi\rangle$ to \begin{eqnarray} H|\psi\rangle &=& \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array}\right] \left[\begin{array}{c} \alpha + i\beta \\ \alpha \\ \end{array}\right] = \frac{1}{\sqrt{2}} \left[\begin{array}{c} \alpha + i\beta + \alpha\\ \alpha + i\beta - \alpha\\ \end{array}\right] \nonumber \\ \ &=& \frac{1}{\sqrt{2}} \left[\begin{array}{c} (\alpha + i\beta) + (\alpha)\\ (\alpha) - (\alpha - i\beta)\\ \end{array}\right] = \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle), \nonumber \end{eqnarray} while it acts on $|\psi_{\perp}\rangle$ to give \begin{eqnarray} H|\psi_{\perp}\rangle &=& \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & 1 \\ 1 & -1 \\ \end{array}\right] \left[\begin{array}{c} \alpha \\ -(\alpha - i\beta) \\ \end{array}\right] \nonumber \\ \ &=& \frac{1}{\sqrt{2}} \left[\begin{array}{c} \alpha - (\alpha - i\beta)\\ \alpha + (\alpha - i\beta)\\ \end{array}\right] \nonumber \\ \ &=& \frac{1}{\sqrt{2}} \left[\begin{array}{c} (\alpha + i\beta) - (\alpha)\\ (\alpha) + (\alpha - i\beta)\\ \end{array}\right] = \frac{1}{\sqrt{2}} (|\psi\rangle - |\psi_{\perp}\rangle).\nonumber \end{eqnarray} Alternatively, one can also prove the above theorem by unitarity, as was done in~\cite{PT02} for the general case. Take any two quantum states $\{|\psi^{(k)}\rangle, |\psi^{(l)}\rangle\}$ from this special ensemble, and their complement states $\{|\psi_{\perp}^{(k)}\rangle, |\psi_{\perp}^{(l)}\rangle\}$. Applying the Hadamard transformation (\ref{eq1}) on them and taking inner product, we get \begin{eqnarray} \label{eqip1} \langle\psi^{(k)} |\psi^{(l)}\rangle &=& \frac{1}{2}(\langle\psi^{(k)} |\psi^{(l)}\rangle + \langle\psi^{(k)} |\psi_{\perp}^{(l)}\rangle + \langle\psi_{\perp}^{(k)} |\psi^{(l)}\rangle \nonumber \\ \ &+& \langle\psi_{\perp}^{(k)}|\psi_{\perp}^{(l)}\rangle) \end{eqnarray} \begin{eqnarray} \label{eqip2} \langle\psi_{\perp}^{(k)} |\psi_{\perp}^{(l)}\rangle &=& \frac{1}{2}(\langle\psi^{(k)} |\psi^{(l)}\rangle - \langle\psi^{(k)} |\psi_{\perp}^{(l)}\rangle - \langle\psi_{\perp}^{(k)} |\psi^{(l)}\rangle \nonumber \\ \ &+& \langle\psi_{\perp}^{(k)}|\psi_{\perp}^{(l)}\rangle). \end{eqnarray} Any two qubits from this ensemble obey the conjugation rules \begin{eqnarray} \label{eqip3} \langle\psi^{(k)}| \psi_{\perp}^{(l)}\rangle &=& - \langle\psi_{\perp}^{(k)}| \psi^{(l)}{\rangle}^* = \langle\psi_{\perp}^{(k)}| \psi^{(l)}\rangle, \nonumber \\ \langle\psi^{(k)}| \psi^{(l)}\rangle &=& \langle\psi_{\perp}^{(k)}| \psi_{\perp}^{(l)}{\rangle}^*. \end{eqnarray} Substituting these conditions in the above inner product relations, it is straightforward to check that the inner product is preserved. Hence, a universal Hadamard gate exists for any qubit chosen from this special class. \section{III. Hadamard Type Transforms} In this section, we consider some operations which are not exactly Hadamard transforms, but similar, in the sense that they produce equal superposition of the amplitudes up to a sign or a phase. These have been discussed by Pati, in the context of qubits from the polar and equatorial great circles. Here, we elaborate more on these transformations and present some general results. \subsection{Polar Type Transformation} Any two orthogonal vectors on the polar great circle, $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + \sin{\frac{\theta}{2}} |1\rangle$ and $|\psi_{\perp}\rangle = \cos{\frac{\theta}{2}} |1\rangle - \sin{\frac{\theta}{2}} |0\rangle$, can be shown to transform as \cite{PT02} \begin{equation} \label{eq1a} U(|\psi\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle), U(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}} (|\psi_{\perp}\rangle - |\psi\rangle). \end{equation} This differs from the usual Hadamard transformation by an overall sign in the second part. The appropriate unitary operator $U$ which does the job is denoted by $H_P = \sigma_x H = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & -1 \\ 1 & 1 \\ \end{array}\right]$, where $\sigma_x$ is the Pauli flip matrix $\left[\begin{array}{cr} 0 & 1 \\ 1 & 0 \\ \end{array}\right]$. We now extend this result to vectors from any polar circle. Take $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle$ on any of the polar circles, and its orthogonal complement $|\psi_{\perp}\rangle = -\sin{\frac{\theta}{2}} |0\rangle + e^{i\phi} \cos{\frac{\theta}{2}} |1\rangle$. Then, for any $\phi$, we can construct a unitary operator $H_G^{\phi} = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & -e^{-i\phi} \\ e^{i\phi} & 1 \\ \end{array}\right]$, such that Eq(\ref{eq1a}) is satisfied, i.e., $H_G^{\phi} |\psi\rangle = \frac{1}{\sqrt{2}} (|\psi\rangle + |\psi_{\perp}\rangle)$ and $H_G^{\phi} |\psi_\perp\rangle = \frac{1}{\sqrt{2}} (|\psi_\perp\rangle - |\psi\rangle)$. For $\phi = 0$, $H_G^{\phi}$ reduces to $H_P$, thereby covering the polar great circle case. \begin{theorem} \label{th2} For any state $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle$ and its orthogonal state $|\psi_{\perp}\rangle = -\sin{\frac{\theta}{2}} |0\rangle + e^{i\phi} \cos{\frac{\theta}{2}} |1\rangle$, it is possible to design a Hadamard type gate $H_G^{\phi}$ that satisfies the transformation (\ref{eq1a}), once $\phi$ is known. \end{theorem} It is clear that the unitary operator $U$, satisfying Eq(\ref{eq1a}), depends on the type of states chosen. For instance, we get two different gates above, depending on whether $\{|\psi\rangle, |\psi_{\perp}\rangle\}$ belongs to the polar great circle or some other polar circle. Therefore, if one fixes the operator $U$, then one can show that there is a unique ensemble of states satisfying the transformation (\ref{eq1a}). For this purpose, let us consider the gate $H_P$, and find the associated class of states for which it works. We follow our previous procedure of taking $|\psi\rangle = a|0\rangle + b |1\rangle$ and its orthogonal complement $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Linearity yields, \begin{eqnarray} H_P(|\psi\rangle) &=& aH_P(|0\rangle) + bH_P(|1\rangle)\\ \ &=& \frac{a}{\sqrt{2}} (|0\rangle + |1\rangle) + \frac{b}{\sqrt{2}} (|1\rangle - |0\rangle) \nonumber \\ \ &=& \frac{(a-b) |0\rangle - (a+b) |1\rangle}{\sqrt{2}}. \nonumber \end{eqnarray} On the other hand, \begin{eqnarray} H(|\psi\rangle) &=& \frac{1}{\sqrt{2}}( |\psi\rangle+|\psi_\perp\rangle) \nonumber \\ \ &=& \frac{a|0\rangle + b|1\rangle}{\sqrt{2}} + \frac{b^*|0\rangle - a^*|1\rangle}{\sqrt{2}} \nonumber \\ \ &=& \frac{(a+b^*) |0\rangle + (b-a^*) |1\rangle}{\sqrt{2}} \nonumber \end{eqnarray} Thus, $a - b = a + b^*$, i.e., $b+b^* = 0$. So $b$ is imaginary. Moreover, $a + b = b - a^*$, i.e., $a+a^* = 0$. Therefore, $a$ is also imaginary. Hence, given $\alpha, \beta$ real, we get $|\psi\rangle = i\alpha |0\rangle + i \beta |1\rangle$ and $|\psi_{\perp}\rangle = -i\beta |0\rangle + i \alpha |1\rangle$. Rewriting $|\psi\rangle = i(\alpha |0\rangle + \beta |1\rangle)$ and representing it on the Bloch sphere, one can readily check that $e^{i\gamma} = i$, $\cos{\frac{\theta}{2}} = \alpha$, $\sin{\frac{\theta}{2}} = \beta$, $e^{i\phi} = 1$. The resulting trajectory is that of the polar great circle. However, if we had taken $|\psi_{\perp}\rangle = -b^* |0\rangle + a^* |1\rangle$, we would have got $|\psi\rangle$ of the form $\alpha |0\rangle + \beta |1\rangle$, which again are the states on the polar great circle. We thus conclude that, up to isomorphism, this is the only class of qubit states which transforms according to Eq(\ref{eq1a}), under the action of the gate $H_P$. Alternatively, one can also fix the states and determine the corresponding gate uniquely. \subsection{Equatorial Type Transformation} The second kind of operation discussed in~\cite{PT02} is that of an equal superposition of amplitudes up to a phase such that \begin{equation} \label{eq2} U(|\psi\rangle) = \frac{1}{\sqrt{2}} (|\psi\rangle + i|\psi_{\perp}\rangle), U(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}} (i|\psi\rangle + |\psi_{\perp}\rangle). \end{equation} This alternative universal definition of a Hadamard type gate, has the advantage that it is invariant under the interchange of $|\psi\rangle$ and $|\psi_{\perp}\rangle$. Vectors of the form $|\psi(\phi)\rangle = H(\cos\frac{\phi}{2} |0\rangle) - i \sin\frac{\phi}{2} |1\rangle) = \frac{1}{\sqrt{2}} e^{-i\phi /2} (|0\rangle + e^{i\phi} |1\rangle)$ and the corresponding orthogonal $|\psi_{\perp}(\phi)\rangle = H(i\sin\frac{\phi}{2} |0\rangle) - \cos\frac{\phi}{2} |1\rangle) = \frac{1}{\sqrt{2}} e^{i\phi /2} (|1\rangle - e^{-i\phi} |0\rangle)$ chosen from the equatorial great circle satisfy this transformation provided the unitary matrix is $H_E = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1-i & 0 \\ 0 & 1+i \\ \end{array}\right]$. We wish to clarify here that the matrix $H_E$ presented in \cite{PT02} does not work for the states considered, and the correct form of $H_E$ should essentially be what we have given above. Our next task is to find the most general class of states satisfying the phase dependent transformation (\ref{eq2}), provided the computational basis vectors $\{|0\rangle, |1\rangle\}$ also transform in the same fashion, i.e., to $\frac{1}{\sqrt{2}}(|0\rangle + i |1\rangle)$ and $\frac{1}{\sqrt{2}}(i |0\rangle + |1\rangle)$, respectively. Thus fixing the unitary operator as $U = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & i \\ i & 1 \\ \end{array}\right]$, we obtain conditions on the form of $|\psi\rangle$ and $|\psi_{\perp}\rangle$. Following the earlier procedure, we assume that $|\psi\rangle = a|0\rangle + b |1\rangle$ and $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Then using linearity of the operation, we find that $|\psi\rangle$ must be of the form $i\alpha |0\rangle + \beta |1\rangle$. The complement $|\psi_{\perp}\rangle$ is restricted to $\beta |0\rangle + i\alpha |1\rangle$. Now $|\psi\rangle$ can be written as $i\sqrt{\alpha^2+\beta^2} (\frac{\alpha}{\sqrt{\alpha^2+\beta^2}} |0\rangle - i\frac{\beta}{\sqrt{\alpha^2+\beta^2}} |1\rangle)$, i.e., $e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\frac{3\pi}{2}} \sin{\frac{\theta}{2}} |1\rangle)$. On the Bloch sphere, $x = \cos{\phi}\sin{\theta} = 0, y = \sin{\phi} \sin{\theta} = 2\alpha \sqrt{1-\alpha^2}, z = \cos{\theta} = 2\alpha^2 - 1$, where $-1 \leq \alpha \leq 1$. Similarly considering the second complement $|\psi_{\perp}\rangle = -b^* |0\rangle + a^* |1\rangle$, and using linearity of the operation, we get $|\psi\rangle = \alpha |0\rangle + i\beta |1\rangle$ and $|\psi_{\perp}\rangle = i\beta |0\rangle + \alpha |1\rangle$. Identifying with the Bloch sphere picture, $|\psi\rangle$ can be written as $\sqrt{\alpha^2+\beta^2} (\frac{\alpha}{\sqrt{\alpha^2+\beta^2}} |0\rangle + i\frac{\beta}{\sqrt{\alpha^2+\beta^2}} |1\rangle)$, i.e., $e^{i\gamma}(\cos{\frac{\theta}{2}} |0\rangle + e^{i\frac{\pi}{2}} \sin{\frac{\theta}{2}} |1\rangle)$. Therefore, on the Bloch sphere, $x = \cos{\phi}\sin{\theta} = 0, y = \sin{\phi} \sin{\theta} = 2\alpha \sqrt{1-\alpha^2}, z = \cos{\theta} = 2\alpha^2 - 1$, where $-1 \leq \alpha \leq 1$. Hence $|\psi\rangle$, when expressed in computational basis, is made up of one real and one imaginary amplitude. As expected, the above two ensembles give the same trajectory, represented by curve 2 in the figure. We thus have the following result. \begin{theorem} \label{th3} It is possible to design a universal Hadamard type gate $U$, satisfying the transformation (\ref{eq2}), for any state of the form $|\psi\rangle = i\alpha |0\rangle + \beta |1\rangle$ and its orthogonal complement $|\psi_{\perp}\rangle = \beta |0\rangle + i\alpha |1\rangle$, where $\alpha, \beta$ are real such that $\alpha^2 + \beta^2 = 1$ and $-1 \leq \alpha \leq 1$. \end{theorem} One can check explicitly that $U(|\psi\rangle) = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & i \\ i & 1 \\ \end{array}\right]$ $\left[\begin{array}{c} i\alpha \\ \beta \\ \end{array}\right]$ $= \frac{1}{\sqrt{2}} \left[\begin{array}{c} i(\alpha + \beta)\\ -\alpha + \beta\\ \end{array}\right] = \frac{1}{\sqrt{2}} (|\psi\rangle + i|\psi_{\perp}\rangle)$. Similarly, $U(|\psi_{\perp}\rangle) = \frac{1}{\sqrt{2}} \left[\begin{array}{cr} 1 & i \\ i & 1 \\ \end{array}\right]$ $\left[\begin{array}{c} \beta \\ i\alpha \\ \end{array}\right]$ $= \frac{1}{\sqrt{2}} \left[\begin{array}{c} \beta - \alpha\\ i(\beta + \alpha)\\ \end{array}\right] = \frac{1}{\sqrt{2}} (i|\psi\rangle + |\psi_{\perp}\rangle)$. Interestingly, this trajectory cuts the equatorial great circle when $z = 0$, i.e., $\alpha = \pm \frac{1}{\sqrt{2}}$. These intersection points imply that there are quantum states (and their orthogonals ) from this ensemble which also belong to the equatorial great circle. Let us find out these states. For $\theta = \frac{\pi}{2}$, $\alpha = \pm \frac{1}{\sqrt{2}}$, and from normalization condition, $\beta = \pm \frac{1}{\sqrt{2}}$. Substituting these values in $|\psi\rangle$ and $|\psi_\perp\rangle$ of Theorem~\ref{th3}, we get $|\psi\rangle = \pm \frac{1}{\sqrt{2}} (i |0\rangle \pm |1\rangle)$ and $|\psi_\perp\rangle = \pm \frac{1}{\sqrt{2}} (|0\rangle \pm i|1\rangle)$. One can similarly find the states corresponding to the second orthogonal complement. \section{IV. Unequal Superposition} We shall now focus our attention on unequal superposition of the amplitudes of a qubit state. Like the equal superposition case, it is impossible to create unequal superposition of an arbitrary unknown qubit with its complement state \cite{PT02}. Our task therefore, is to obtain special classes of states for which such a superposition would be possible. This can be regarded as a generalized version of the usual Hadamard transformation and is given by \begin{equation} \label{equ1} U(|\psi\rangle) = p|\psi\rangle + q|\psi_{\perp}\rangle, U(|\psi_{\perp}\rangle) = q^*|\psi\rangle - p^*|\psi_{\perp}\rangle. \end{equation} Here, $p, q$ are known complex numbers with $|p|^2 + |q|^2 = 1$. We again demand that $|\psi\rangle$ and $|\psi_{\perp}\rangle$ transform like the computational basis vectors $|0\rangle$ and $|1\rangle$ respectively. Thus $U$ can be fixed as $\left[\begin{array}{cr} p & q^* \\ q & -p^* \\ \end{array}\right]$. Repeating the linearity procedure, take $|\psi\rangle = a|0\rangle + b |1\rangle$ and $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Ideally we should have \begin{eqnarray} U(|\psi\rangle) &=& p|\psi\rangle + q|\psi_{\perp}\rangle \nonumber \\ \ &=& p(a |0\rangle + b |1\rangle) + q(b^* |0\rangle - a^* |1\rangle) \nonumber \\ \ &=& (pa+qb^*) |0\rangle + (pb - qa^*) |1\rangle. \nonumber \end{eqnarray} On the other hand, from linearity we get \begin{eqnarray} U(|\psi\rangle) &=& a U(|0\rangle) + b U(|1\rangle) \nonumber \\ \ &=& a (p|0\rangle + q|1\rangle) + b (q^*|0\rangle - p^*|1\rangle) \nonumber \\ \ &=& (ap+bq^*) |0\rangle + (aq-bp^*) |1\rangle \nonumber \end{eqnarray} Hence $pa + qb^* = pa + q^*b$, i.e., $qb^* = q^*b = (qb^*)^*$. Thus $qb^*$ is real, which implies that \begin{enumerate} \item both $q, b$ are real, or \item both $q, b$ are imaginary, or \item both $q, b$ are complex, with the constraint $\frac{q_1}{q_2} = \frac{b_1}{b_2}$. (Here, any complex number $z = (a, b, q, p)$ has been written as $z= z_1 + i z_2$). \end{enumerate} Further, $pb - qa^* = qa - p^*b$, i.e., $q(a + a^*) = b(p + p^*)$, so, $q \cdot Re(a) = b \cdot Re(p)$, i.e., $Re(a) = \frac{b}{q} \cdot Re(p)$. Therefore, $|\psi\rangle$ and $|\psi_{\perp}\rangle$ are restricted to the form $|\psi\rangle = (\frac{b}{q} \cdot Re(p) + ia_2) |0\rangle + b |1\rangle$ and $|\psi_{\perp}\rangle = b^* |0\rangle - (\frac{b}{q} Re(p) - i a_2) |1\rangle$. Depending on whether $q$ and $b$ are both real, or imaginary or complex ( with $\frac{q_1}{q_2} = \frac{b_1}{b_2}$), we get different classes of states for which the unequal superposition transformation (\ref{equ1}) holds. For the special value of $p = q = \frac{1}{\sqrt{2}}$, this ensemble goes over to the set of states $|\psi\rangle = (b + ia_2) |0\rangle + b |1\rangle$ (i.e., complex and real amplitudes such that real parts are the same) obtained in Sec. II. Also, the associated Hadamard matrix $H$, satisfying the transformation (\ref{eq1}), can be recovered for these values of $p,q$ from the above $U$. Now, analogous to the previous section, we concentrate below, on two specific unequal superposition transformations. \subsection{Unequal Polar Type Transformation} According to the prescription outlined in~\cite{PT02}, for the vectors on the polar great circle, one can find a unitary gate $U_P = \left[\begin{array}{cr} p & -q \\ q & p \\ \end{array}\right]$, where $p^2 + q^2 = 1$ and $p, q$ are now real. In this case $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + \sin{\frac{\theta}{2}} |1\rangle$ and $|\psi_{\perp}\rangle = \cos{\frac{\theta}{2}} |1\rangle - \sin{\frac{\theta}{2}} |0\rangle$ and they transform as \begin{equation} \label{equ2} U_P (|\psi\rangle) = q|\psi_{\perp}\rangle + p|\psi\rangle, U_P (|\psi_{\perp}\rangle) = p|\psi_{\perp}\rangle - q|\psi\rangle. \end{equation} This is almost similar to Eq(\ref{equ1}), up to an overall sign (when $p, q$ are real). Now we present a generalization of this result. Take a qubit $|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + e^{i\phi} \sin{\frac{\theta}{2}} |1\rangle$ on any polar circle, and the orthogonal qubit $|\psi_{\perp}\rangle = -\sin{\frac{\theta}{2}} |0\rangle + e^{i\phi} \cos{\frac{\theta}{2}} |1\rangle$. Then, for any $\phi$, one can construct a corresponding unitary matrix $U_G^{\phi} = \left[\begin{array}{cr} p & -q e^{-i\phi} \\ q e^{i\phi} & p \\ \end{array}\right]$, such that $U_G^{\phi} |\psi\rangle = p |\psi\rangle + q |\psi_\perp\rangle$ and $U_G^{\phi} |\psi_\perp\rangle = p |\psi_\perp\rangle - q |\psi\rangle$. In the limit when $\phi = 0$, we recover the polar great circle case since $U_G^0 = U_P$. Thus if partial information $(\phi)$ is known, given any arbitrary state, it is possible to design a generalized Hadamard type gate for unequal superposition. Note that for $p = q = \frac{1}{\sqrt{2}}$, $U_G^{\phi} = H_G^{\phi}$, thereby yielding the result of Theorem~\ref{th2}. \subsection{Unequal Equatorial Type Transformation} The generalized version of the phase dependent Hadamard type of transformation can be written as \begin{equation} \label{equ1x} U(|\psi\rangle) = p|\psi\rangle + iq|\psi_{\perp}\rangle, U(|\psi_{\perp}\rangle) = iq^*|\psi\rangle + p^*|\psi_{\perp}\rangle. \end{equation} Here again $p, q$ are known complex numbers with $|p|^2 + |q|^2 = 1$. Under the assumption that $\{|\psi\rangle, |\psi_{\perp}\rangle\}$ transform in the same way as $\{|0\rangle, |1\rangle\}$, $U$ is fixed to be $\left[\begin{array}{cr} p & iq^* \\ iq & p^* \\ \end{array}\right]$. In order to obtain classes of states obeying this transformation under the action of $U$, take $|\psi\rangle = a|0\rangle + b |1\rangle$ and $|\psi_{\perp}\rangle = b^* |0\rangle - a^* |1\rangle$. Ideally we should have \begin{eqnarray} U(|\psi\rangle) &=& p|\psi\rangle + iq|\psi_{\perp}\rangle \nonumber \\ \ &=& p(a |0\rangle + b |1\rangle) + iq(b^* |0\rangle - a^* |1\rangle) \nonumber \\ \ &=& (ap+iqb^*) |0\rangle + (bp - iqa^*) |1\rangle. \nonumber \end{eqnarray} Then from linearity we get \begin{eqnarray} U(|\psi\rangle) &=& a U(|0\rangle) + b U(|1\rangle) \nonumber \\ \ &=& a (p|0\rangle + iq|1\rangle) + b (iq^*|0\rangle + p^*|1\rangle) \nonumber \\ \ &=& (ap+ibq^*) |0\rangle + (iaq+bp^*) |1\rangle \nonumber \end{eqnarray} Hence $pa + iqb^* = pa + iq^*b$, i.e., $qb^* = q^*b = (qb^*)^*$. Thus $qb^*$ is real, i.e., \begin{enumerate} \item both $q, b$ are real or \item both $q, b$ are imaginary or \item both $q, b$ are complex, with the constraint $\frac{b_1}{b_2} = \frac{q_1}{q_2}$. \end{enumerate} Further, $pb - iqa^* = iqa + p^*b$, i.e., $iq(a + a^*) = b(p - p^*)$, so $a_1 = \frac{b}{q} \cdot p_2$. Hence we get a general class of states, $|\psi\rangle = (\frac{b}{q} p_2 + i \cdot a_2) |0\rangle + b |1\rangle$ and $|\psi_\perp\rangle = b^* |0\rangle - (\frac{b}{q} \cdot p_2 - i \cdot a_2) |1\rangle)$. Depending on the above three possible solutions, i.e., whether $q, b$ are both real, or imaginary or complex, we get different classes of states for which the unequal superposition transformation (\ref{equ1x}) holds. Again, it is easy to see that for the special case $p = q = \frac{1}{\sqrt{2}}$, this reduces to the class of states obtained in Theorem~\ref{th3}. \section{V. Summary and Conclusions} In this paper we have found the most general class of qubits (up to isomorphisms) for which the Hadamard gate can be designed. This was achieved using one of the fundamental axioms of quantum mechanics, namely linearity. If expressed in the computational basis, the qubit state assumes a specific form: one complex and one pure real (imaginary) amplitude; the real (imaginary) parts of which are equal. The Hadamard gate is universal for this class of ensemble, i.e., it works for any state belonging to this particular ensemble. When represented on a Bloch sphere, these states give a new trajectory.Interestingly, it has some intersection points with the polar and equatorial great circles. Equal superposition of $|0\rangle$ and $|1\rangle$ states has played a very crucial role in quantum algorithms to study various problems, e.g., distinguishing between constant and balanced Boolean functions~\cite{qDJ92}, database search~\cite{qGR96} etc. It would be interesting to construct specific computational problems, which one can study by exploiting the superposition of $|\psi\rangle$ and $|\psi_\perp\rangle$ from the class of states obtained in this paper. We have also considered some Hadamard type transformations which hold for polar and equatorial qubits, and have obtained some new results. The situation becomes more general when the superposition of the two amplitudes is not equal. Many new classes of states have been found and all the results of the equal superposition case can be recovered by letting the parameters to be equal. The next step would be to generalize these results to higher dimensions,where the analogue of the Hadamard transform would be the discrete Fourier transform. In this direction we have obtained partial results so far and further work is in progress. \ \\ \noindent{\bf Acknowledgments:} We thank G. Kar and P. Mukhopadhyay for useful discussions. PP acknowledges financial assistance from DST under the SERC Fast Track Proposal scheme for young scientists. \end{document}

\text{b}egin{equation}gin{document} \preprint{APS/123-QED} \title{Classes of exactly solvable Generalized semi-classical Rabi Systems} \author{R. Grimaudo} \address{Dipartimento di Fisica e Chimica dell'Universit\`a di Palermo, Via Archirafi, 36, I-90123 Palermo, Italy.} \address{I.N.F.N., Sezione di Catania, Catania, Italy.} \author{A. S. M. de Castro} \affiliation{Universidade Estadual de Ponta Grossa, Departamento de F\'{\text{i}}sica, CEP 84030-900, Ponta Grossa, PR, Brazil.} \author{H. Nakazato} \affiliation{Department of Physics, Waseda University, Tokyo 169-8555, Japan.} \author{A. Messina} \address{I.N.F.N., Sezione di Catania, Catania, Italy.} \address{ Dipartimento di Matematica ed Informatica dell'Universit\`a di Palermo, Via Archirafi, 34, I-90123 Palermo, Italy.} \date{\today} \text{b}egin{equation}gin{abstract} The exact quantum dynamics of a single spin-1/2 in a generic time-dependent classical magnetic field is investigated and compared with the quantum motion of a spin-1/2 studied by Rabi and Schwinger. The possibility of regarding the scenario studied in this paper as a generalization of that considered by Rabi and Schwinger is discussed and a notion of time-dependent resonance condition is introduced and carefully legitimated and analysed. Several examples help to disclose analogies and departures of the quantum motion induced in a generalized Rabi system with respect to that exhibited by the spin-1/2 in a magnetic field precessing around the $z$-axis. We find that, under generalized resonance condition, the time evolution of the transition probability $P_+^-(t)$ between the two eigenstates of ${\hat{S}}^z$ may be dominated by a regime of distorted oscillations, or may even exhibit a monotonic behaviour. At the same time we succeed in predicting no oscillations in the behaviour of $P_+^-(t)$ under general conditions. New scenarios of experimental interest originating a Landau-Zener transition is brought to light. Finally, the usefulness of our results is emphasized by showing their applicability in a classical guided wave optics scenario. \end{abstract} \pacs{Valid PACS appear here} leywords{Exactly solvable time dependent models; Exact single qubit dynamics; Semiclassical Rabi model.Suggested keywords} -aketitle +ection{Introduction} Spin language is transversal to every physical scenario wherein, by definition, the states of the specific system under scrutiny live in a finite-dimensional Hilbert space. When the generally time-dependent Hamiltonian of the system may in particular be expressed as linear combination of the three generators of su(2), the corresponding quantum dynamical problem coincides with that of a spin $j$ in a time-dependent magnetic field. It is well known that the time evolution of the spin $j$ in this case is fully recoverable from that of a spin $j=1/2$ subjected to the same magnetic field \cite{Hioe}. On the other hand, however, a closed general form of the 2x2 unitary evolution operator is still unavailable. As a consequence, to bring to light new dynamical scenarios of a spin-1/2 represents a target of basic and applicative importance in many contexts such as quantum optics \cite{Haroche}, quantum control \cite{Daems, Greilich}, quantum information, and quantum computing \cite{NC,Braak,Oliveira}. Rabi \cite{Rabi 1937, Rabi 1954} and Schwinger \cite{Schwinger 1937} exactly solved the quantum dynamics of a spin-1/2 in the now called Rabi scenario, that is, subjected to a static magnetic field $B_z$ along the $z$-axis and an r.f. magnetic field rotating in the $x$-$y$ plane with frequency $\omega_{xy}$, namely \text{b}egin{equation}gin{equation} \label{Magn Field Rabi} \textbf{B}_R(t)= B_\perp[\cos(\omega_{xy} t) \textbf{c}_1-+in(\omega_{xy} t) \textbf{c}_2]+B_0 \textbf{c}_3, \end{equation} $\textbf{c}_1,\textbf{c}_2$ and $\textbf{c}_3$ being fixed unit vectors in the laboratory frame. Their seminal papers show that the probability of transition between the two Zeeman states generated by $-athbf{B}_z=B_0-athbf{c}_3$ is dominated by periodic oscillations reaching maximum amplitude under the so-called resonance condition $\Delta=\omega_{xy}-\omega_L=0$. Here $\omega_L$ is the spin Larmor frequency. The exact treatment of this basic problem provides the robust platform for the NMR technology implementation \cite{Vandersypen}. The recently published special issue on semiclassical and quantum Rabi models \cite{Issue} witnesses the evergreen attractiveness of this problem. Searching new exactly solvable time-dependent scenarios of a single spin-1/2 could be very interesting and worth both from basic and applicative points of view, especially in the quantum control context. To pursue this target, over the last years, new methods have been developed to face the problem with an original strategy \cite{Bagrov, KunaNaudts, Das Sarma, Mess-Nak,MGMN}. We stress that progresses along this direction might be of relevance even for treating time-dependent Hamiltonians describing interacting qubits or qudits \cite{GMN, GMIV, GBNM}. In turn exact treatments of such scenarios might stimulate the interpretation of experimental results in fields from condensed matter physics \cite{Calvo, Borozdina} to quantum information and quantum computing \cite{Petta, Anderlini, Foletti, Das Sarma Nat}. In this paper we construct the exact time evolution of a spin-1/2 subjected to time-dependent magnetic fields never considered before. Our investigation introduces, in a very natural way, three different classes of Generalized Rabi Systems (GRSs) wherein Rabi oscillations of maximum amplitude still survive. We succeed indeed in identifying generalized resonance conditions which are, as in the Rabi scenario, at the origin of the complete population transfer between the two Zeeman levels of the spin. We bring to light that, even at resonance, these oscillations might loose its periodic character, significantly differing, thus, from the sinusoidal behaviour occurring in the Rabi scenario. In this paper we have also considered time-dependent magnetic fields giving rise to exactly solvable models not satisfying the resonance condition. In this way, we are able to write down a special link among all the time-dependent parameters appearing in the Hamiltonian of the system even if the resonance condition is not met, for which, however, the dynamical problem is exactly solvable. The departure of the time evolution of the transition probability out of generalized resonance from the corresponding behaviour in the Rabi scenario is illustrated with the help of two exemplary cases. The paper is organized as follows: in Sec. \text{r}ef{MN Res} two sets of parametrized solutions of the dynamical problem of a single spin-1/2 in a time-dependent magnetic field are given. The generalized time-dependent resonance condition and the generalized out of resonance case are introduced and physically legitimated In Sec. \text{r}ef{R and OOR Cases}. In the subsequent section (Sec. \text{r}ef{Examples}) several examples illustrate the occurrence of effects on the Rabi transition probability due to the magnetic field time-dependence. Finally, in the Conclusion section, a summary of the results with possible applications and future outlooks are briefly discussed. +ection{Resolution of 2x2 su(2) quantum dynamical problems} \label{MN Res} The problem of a single spin-1/2 subjected to a generic time-dependent magnetic field $\textbf{B}(t) \equiv [B_x(t),B_y(t),B_z(t)]$ is investigated by assuming the su(2) Hamiltonian model \text{b}egin{equation}gin{equation}\label{SU(2) Hamiltonian} H(t)= \text{b}egin{equation}gin{pmatrix} \Omega(t) & \omega(t) \\ \omega^*(t) & -\Omega(t) \end{pmatrix}, \end{equation} with \text{b}egin{equation}gin{subequations} \label{Def Magn Field} \text{b}egin{equation}gin{align} &\Omega(t)={\hbar -u_0 g \over 2} B_z(t), \\ &\omega(t)={\hbar -u_0 g \over 2} [B_x(t)-iB_y(t)] \equiv \omega_x-i \omega_y \equiv |\omega(t)|e^{i\phi_\omega(t)}. \end{align} \end{subequations} Here $-u_0 g$ is the magnetic moment associated to the spin-1/2, $g$ and $-u_0$ being the appropriate Land\'e factor and the Bohr magneton, respectively. The entries $a(t)\equiv|a(t)|\exp \{i \phi_a(t)\}$ and $b(t)\equiv|b(t)|\exp \{i\phi_b(t)\}$ of the unitary time evolution operator \text{b}egin{equation}gin{equation} \label{Time Ev Op SU2} U (t)= \text{b}egin{equation}gin{pmatrix} a(t) & b(t) \\ -b^*(t) & a^*(t) \end{pmatrix}, \quad |a(t)|^2+|b(t)|^2=1, \end{equation} generated by $H(t)$, must satisfy the Cauchy-Liouville problem $i\hbar\dot{U}(t)=H(t)U(t)$, $U(0)=-athbb{1}$, which originates the following system of linear differential equations \text{b}egin{equation}gin{equation}\label{LC problem} \left\{ \text{b}egin{equation}gin{aligned} &\dot{a}(t)=\text{f}rac{\Omega}{i\hbar}a(t)-\text{f}rac{\omega}{i\hbar}b^{*}(t),\\ &\dot{b}(t)=\text{f}rac{\omega}{i\hbar}a^{*}(t)+\text{f}rac{\Omega}{i\hbar}b(t),\\ &a(0)=1,\quad b(0)=0. \end{aligned} \text{r}ight. \end{equation} It is possible to demonstrate \cite{Mess-Nak} that if $Theta(t)$ is a complex-valued $C1$ function of $t$ satisfying the nonlinear integral-differential Cauchy problem \text{b}egin{equation}gin{subequations}\label{Cauchy problem Theta} \text{b}egin{equation}gin{align} &{1\over2}\dot{Theta}(t) + {|\omega(t)| \over \hbar}+inTheta(t) \cot\Bigl[{2\over\hbar} \text{i}nt_0^t|\omega(t')| \cosTheta(t') dt'\Bigr]= \nonumber \\ &={\Omega(t) \over \hbar}+{\dot{\phi}_\omega(t) \over 2}, \label{Rel exactly solvable scenario gen} \\ &Theta(0)=0, \end{align} \end{subequations} then the solutions of the Cauchy problem \eqref{LC problem} can be represented as follows \text{b}egin{equation}gin{subequations}\label{a and b} \text{b}egin{equation}gin{align} a(t) =& \cos\text{b}egin{equation}gin{itemize}ggl[{ 1 \over \hbar}\text{i}nt_0^t |\omega(t')|\cos \text{b}egin{equation}gin{itemize}gl[ Theta(t') \text{b}egin{equation}gin{itemize}gr] dt' \text{b}egin{equation}gin{itemize}ggr] \times \nonumber \\ &\exp\left\{i\left(\dfrac{\phi_\omega(t)-\phi_\omega(0)}{2} - {Theta(t) \over 2} - -athcal{R}(t)\text{r}ight) \text{r}ight\}, \label{Gen a} \\ b(t) =& +in\text{b}egin{equation}gin{itemize}ggl[{1\over\hbar}\text{i}nt_0^t|\omega(t')|\cos \text{b}egin{equation}gin{itemize}gl[ Theta(t') \text{b}egin{equation}gin{itemize}gr] dt' \text{b}egin{equation}gin{itemize}ggr] \times \nonumber \\ &\exp\left\{ i\left(\dfrac{\phi_\omega(t)+\phi_\omega(0)}{2} - {Theta(t) \over 2} + -athcal{R}(t) - \dfrac{\pi}{2}\text{r}ight) \text{r}ight\}, \label{Gen b} \end{align} \end{subequations} with \text{b}egin{equation}gin{equation}\label{Integral R} -athcal{R}(t)=\text{i}nt_0^t \dfrac{|\omega(t')|+in[Theta(t')]}{+in\left[ 2 \text{i}nt_0^{t'} |\omega(t'')|\cos[Theta(t'')]dt'' \text{r}ight]}dt'. \end{equation} \textit{Vice versa}, if $a(t)$ and $b(t)$ are solutions of the Cauchy problem \eqref{LC problem}, then the representations given in Eqs. \eqref{a and b} are still valid and $Theta(t)$ satisfies Eqs. \eqref{Cauchy problem Theta}. Generally speaking, solving Eq. \eqref{Cauchy problem Theta} is a difficult task. This equation however may be exploited in a different way, giving rise to a strategy \cite{Mess-Nak} aimed at singling out exactly solvable dynamical problems represented by Eq. \eqref{LC problem}. Fixing, indeed, at will the function $Theta(t)$ in Eqs. \eqref{Cauchy problem Theta}, that is, $Theta(t)$ regarded now as a parameter (function) rather than an unknown, determines a link between $\Omega (t)$ and $\omega (t)$ under which the corresponding dynamical problem may be exactly solved in view of Eqs. \eqref{a and b}. We emphasize that if we knew the solution of the Cauchy problem given in Eq. \eqref{Cauchy problem Theta}, whatever $\Omega(t)$, $|\omega(t)|$ and $\dot{\phi}_\omega(t)$ are, then we would be in condition to solve in general the corresponding Cauchy dynamical problem expressed by Eqs. \eqref{LC problem}. Another useful way of parametrizing the expressions of $a(t)$ and $b(t)$ is \text{b}egin{equation}gin{subequations}\label{a and b particular} \text{b}egin{equation}gin{align} a(t)=&+qrt{\hbar^2+c^2\cos^2[\Phi(t)]\over\hbar^2+c^2} \times \nonumber \\ &\exp\left\{ i\left( {\phi_\omega(t)\over 2}-\tan^{-1}\left[{\hbar\over+qrt{\hbar^2+c^2}}\tan[\Phi(t)]\text{r}ight] \text{r}ight) \text{r}ight\},\\ b(t)=&{c\over+qrt{\hbar^2+c^2}}+in[\Phi(t)] \exp\left\{ i\left( {\phi_\omega(t) \over 2}-{\pi \over 2} \text{r}ight) \text{r}ight\}, \label{b special cond} \end{align} \end{subequations} with \text{b}egin{equation}gin{equation} \Phi(t)={+qrt{\hbar^2+c^2}\over \hbar c} \text{i}nt_0^t|\omega(t')|dt', \end{equation} $c$ being an arbitrary real number and having put, without loss of generality, $\phi_\omega(0)=0$. In this case, it is possible to check that they solve the system \eqref{LC problem} if the following condition holds \text{b}egin{equation}gin{equation}\label{Relation exactly solvable scenario} {|\omega(t)| \over c}={\Omega(t) \over \hbar}+{\dot{\phi}_\omega(t) \over 2}. \end{equation} It is stressed that this last equation does only express the condition under which, whatever $c$ is, the representations \eqref{a and b particular} satisfy the Cauchy problem \eqref{LC problem}. This means that the real number $c$ plays in this case the role of parameter. When Eq. \eqref{Relation exactly solvable scenario} cannot be satisfied for any $c$, of course the solution of the dynamical problem exists but cannot be represented using Eqs. \eqref{a and b particular}. In this case there certainly exists a function $Theta(t)$ enabling the representation of the solutions by using Eqs. \eqref{a and b}. Finally, it is interesting to underline that Eq. \eqref{Rel exactly solvable scenario gen} turns into the simpler condition \eqref{Relation exactly solvable scenario} on $\textbf{B}(t)$ under an appropriate choice of the parameter function $Theta(t)$ \cite{Mess-Nak}. +ection{Generalized Resonance Condition and out of Resonance Cases}\label{R and OOR Cases} The experimental set-up considered by Rabi, as described in the introduction, leads to the Hamiltonian model \eqref{SU(2) Hamiltonian} where \text{b}egin{equation}gin{subequations} \label{Def Magn Field} \text{b}egin{equation}gin{align} &\Omega(t)={\hbar -u_0 g \over 2} B_0 \equiv \Omega_0, \\ &|\omega(t)|={\hbar -u_0 g \over 2} +qrt{B_x^2(t)+B_y^2(t)}={\hbar -u_0 g \over 2}B_\perp \equiv |\omega_0|,\\ &\phi_\omega(t)=\nu_0 t \equiv \dot{\phi}_0 t. \end{align} \end{subequations} Then it is characterized by the three time-independent parameters: $\Omega_0$, $|\omega_0|$ and $\dot{\phi}_0$. In this paper we generalize this Rabi scenario by making some out of or all these parameters time-dependent: $\Omega \text{r}ightarrow \Omega(t)$, $|\omega| \text{r}ightarrow |\omega(t)|$ and $\dot{\phi}_\omega \text{r}ightarrow \dot{\phi}_\omega(t)$. Firstly, we rewrite the general Hamiltonian \eqref{SU(2) Hamiltonian} as follows \text{b}egin{equation}gin{equation} H=\Omega(t) \hat{+igma}^z + \omega_x(t) \hat{+igma}^x + \omega_y(t) \hat{+igma}^y, \end{equation} with $\omega_{x/y}(t)=\hbar-u_0gB_{x/y}(t)/2$ and $\hat+igma^{x/y/z}$ being Pauli matrices. Generalizing the approach in Ref. \cite{Rabi 1954}, we pass from the laboratory frame to the time-dependent one tuned with $\phi_\omega(t)$, where the time-dependent Schr\"odinger equation for the transformed state, \text{b}egin{equation}gin{equation} let{\psi(t)}=\exp\{i \phi_\omega(t) \hat{+igma}^z/2\} let{\tilde{\psi}(t)}, \end{equation} is governed by the following effective time-dependent transformed Hamiltonian \text{b}egin{equation}gin{equation}\label{H eff} H_{GR}(t)=\left( \Omega(t)+{\hbar \over 2}\dot{\phi}_\omega(t) \text{r}ight) \hat{+igma}^z + |\omega(t)| \hat{+igma}^x. \end{equation} It is worth noticing its strict similarity with the analogous one got in Ref. \cite{Rabi 1954} where the unitary transformation is indeed a uniform rotation around the $z$-axis. In fact, it is enough to make $\Omega(t)$, $|\omega(t)|$ and $\dot{\phi}_\omega(t)$ time-independent in $H_{GR}$ ($GR$ stands for Generalized Rabi) to immediately recover the transformed Hamiltonian got by Rabi \cite{Rabi 1954}. On the basis of this observation it then appears natural to refer to the following condition \text{b}egin{equation}gin{equation}\label{Gen Res Rabi Cond} {\Omega(t)}+{\hbar \over 2}\dot{\phi}_\omega(t)=0, \end{equation} as a generalized resonance condition, in accordance with the corresponding static resonance condition ${\Omega_0}+{\hbar\dot{\phi}_0 / 2}=0$ brought to light by Rabi in Ref. \cite{Rabi 1937}. We underline that the generalized resonance condition does not lead to a time-independent transformed dynamical problem (as it happens in the Rabi scenario), but, whatever $H$ is, it easily enables the explicit construction of the time evolution operator describing the quantum motion of the spin in the laboratory frame. In view of Eq. \eqref{Relation exactly solvable scenario}, the entries of such an operator are indeed exactly given by Eqs. \eqref{a and b particular} in the limit $c \text{r}ightarrow \text{i}nfty$, namely \text{b}egin{equation}gin{subequations}\label{a and b c inf} \text{b}egin{equation}gin{align} a(t)=&\cos\text{b}egin{equation}gin{itemize}ggl[ \text{i}nt_0^t {|\omega| \over \hbar} dt' \text{b}egin{equation}gin{itemize}ggr] \text{exp}\Bigl\{i{\phi_\omega(t)\over 2} \Bigr\}, \\ b(t)=&+in\text{b}egin{equation}gin{itemize}ggl[ \text{i}nt_0^t {|\omega| \over \hbar} dt' \text{b}egin{equation}gin{itemize}ggr]\text{exp}\left\{i{\phi_\omega(t)\over 2}-i{\pi \over 2}\text{r}ight\}. \end{align} \end{subequations} By definition, we say to be in generalized out of resonance when the left hand side of Eq. \eqref{Gen Res Rabi Cond} is non-vanishing, namely \text{b}egin{equation}gin{equation}\label{OoRC} {\Omega(t) \over \hbar}+{\dot{\phi}_\omega(t) \over 2}=\Delta(t) \neq 0, \end{equation} where $\Delta(t)$ is an arbitrary energy-dimensioned well-behaved function of time. Generally speaking, to find the exact quantum dynamics of the spin in the generic out of resonance case, is a very complicated mathematical problem. Let us observe that, on the basis of the structure of $H_{GR}$ in Eq. \eqref{H eff}, when $\Delta(t)$ is proportional to $|\omega(t)|$, the dynamical problem may be exactly solved. Indeed, this condition coincides with that expressed by Eq. \eqref{Relation exactly solvable scenario} which in turn enables one to write down exact solutions of the Cauchy problem \eqref{LC problem} in the form given by Eqs. \eqref{a and b particular}. In this paper we report the exact solutions of special non trivial out of resonance dynamical problems. Our aim is to illustrate the occurrence of analogies and differences in the time behaviour on the Rabi transition probability \text{b}egin{equation}gin{equation} P_+^-(t)=|\average{-|U(t)|+}|^2=|b(t)|^2, \end{equation} ($\hat{+igma}^zlet{\pm}=\pmlet{\pm}$), when the time evolution of the magnetic field acting upon the spin cannot be described as a perfect precession around the $z$-axis. +ection{Examples of Generalized Rabi models} \label{Examples} This section is aimed at showing how the Rabi transition probability $P_+^-(t)=|\average{-|U(t)|+}|^2$ is only slightly as well as strongly affected by different choices of the time-dependent magnetic fields under general conditions. The following examples are reported to illustrate such behaviour. +ubsection{Examples of GRSs Dynamics under Generalized Resonance Condition}\label{Var One Par} Let us consider, firstly, the generalized resonance condition in Eq. \eqref{Gen Res Rabi Cond}. We know that, in this instance, the time evolution operator is characterized by the time behaviour of its two entries given in Eq. \eqref{a and b c inf}, so that the transition probability reads \text{b}egin{equation}gin{equation}\label{P+- Res Cond} P_+^-(t)=+in^2\text{b}egin{equation}gin{itemize}ggl[ \text{i}nt_0^t {|\omega| \over \hbar} dt' \text{b}egin{equation}gin{itemize}ggr]. \end{equation} It is immediately evident that $P_+^-(t)$ could or could not be periodic. Indeed, e.g., setting $|\omega(t)|=|\omega_0|+ech({|\omega_0|t/\hbar})$, obtainable by an $x$-$y$ magnetic field varying over time as \text{b}egin{equation}gin{equation} \label{Transverse mag field sech decreasing} \text{b}egin{equation}gin{aligned} \textbf{B}_{tr} =& B_x(t) \textbf{c}_1 + B_y(t) \textbf{c}_2 \\ =& B_\perp +ech({|\omega_0|t/\hbar}) [\cos\left(\dot{\phi}_0 t\text{r}ight) \textbf{c}_1 - +in\left(\dot{\phi}_0 t\text{r}ight)\textbf{c}_2], \end{aligned} \end{equation} we get \text{b}egin{equation}gin{equation}\label{Trans Prob sech decreasing} P_+^-(t)=\tanh^2(|\omega_0|t/\hbar), \end{equation} resulting in a Landau-Zener-like transition \cite{LZ}, that is an asymptotic aperiodic inversion of population. Figures \text{r}ef{fig:MFMP} and \text{r}ef{fig:MP} represent the transverse magnetic field in Eq. \eqref{Transverse mag field sech decreasing} and the resulting transition probability in Eq. \eqref{Trans Prob sech decreasing}, respectively, plotted against the dimensionless time $\tilde{\tau}=|\omega_0|t/\hbar$ with $\dot{\phi}_0/\hbar|\omega_0|=10$. However, of course, it is easy to understand that it is possible to make choices either resulting in a oscillating but not periodic transition probability or exhibiting a periodic behaviour, even if not coincident with that characterizing the Rabi scenario. If we consider, for example, \text{b}egin{equation}gin{equation}\label{Mod omega exp} |\omega(t)|=|\omega_0|e^{-\gamma t}, \end{equation} reproducible by engineering the transverse magnetic field as \text{b}egin{equation}gin{equation} \label{Transverse mag field exp decreasing} \text{b}egin{equation}gin{aligned} \textbf{B}_{tr} =& B_x(t) \textbf{c}_1 + B_y(t) \textbf{c}_2 \\ =& B_\perp e^{-\gamma t} [\cos\left(\dot{\phi}_0 t\text{r}ight) \textbf{c}_1 - +in\left(\dot{\phi}_0 t\text{r}ight)\textbf{c}_2], \end{aligned} \end{equation} the resulting transition probability yields \text{b}egin{equation}gin{equation}\label{Trans Prob alfa} P_+^-(t)=+in^2[\alpha(1-e^{-\gamma t})], \end{equation} with $\alpha={|\omega_0| / \hbar \gamma}$. We point out that, for the sake of simplicity, in Eqs. \eqref{Transverse mag field sech decreasing} and \eqref{Transverse mag field exp decreasing} we have put $\phi_\omega(t)=\dot{\phi}_0 t$, even if, in general, the expression of the probability in Eq. \eqref{Trans Prob alfa} holds whatever $\phi_\omega(t)$ is, provided that Eq. \eqref{Gen Res Rabi Cond} is satisfied. Figure \text{r}ef{fig:MFDXY} shows the time behaviour of the magnetic field in the $x$-$y$ plane, against the dimensionless parameter $\gamma t$, when $\alpha={9 \pi/ 2}$ and $\dot{\phi}_0/\gamma=10$. \text{b}egin{equation}gin{figure}[htp] \centering +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{MFMonotonicP.eps}\label{fig:MFMP}} \qquad +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{MonotonicP.eps}\label{fig:MP}} \\ +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{MagFDampedxy.eps}\label{fig:MFDXY}} \qquad +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{P+-MFD.eps}\label{fig:PMFDXY}} \\ +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{TransFieldSomm.eps}\label{fig:TFS}} \qquad +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{Modb2TFS.eps}\label{fig:PTFS}} \captionsetup{justification=justified,format=plain,skip=4pt} \caption{(Color online) a) The normalized magnetic field in Eq. \eqref{Transverse mag field sech decreasing}, parametrically represented in the $x$-$y$ plane and the joined b) transition probability in Eq. \eqref{Trans Prob sech decreasing} against the dimensionless parameter $\tilde{\tau}=|\omega_0|t/\hbar$ with $\dot{\phi}_0/\hbar|\omega_0|=10$; c) the normalized magnetic field in Eq. \eqref{Transverse mag field exp decreasing}, parametrically represented in the $x$-$y$ plane and the related d) transition probability in Eq. \eqref{Trans Prob alfa} against the dimensionless parameter $\gamma t$ with ${|\omega_0| / \hbar \gamma}={9 \pi/ 2}$ and $\dot{\phi}_0/\gamma=10$; e) the normalized transverse magnetic field in Eq. \eqref{Mag Field Somm}, parametrically represented in the $x$-$y$ plane in terms of $\tilde{\tau}=\dot{\phi}_0 t$ with ${A' / B_\perp}=1$ and $\lambda=10\dot{\phi}_0$ and the related f) transition probability in Eq. \eqref{Trans Prob TFS Parametrized} for $k=1$ and $n=10$ and $C=1$.} \end{figure} The time behaviour of $P_+^-(t)$ as given in Eq. \eqref{Trans Prob alfa} is reported in Fig. \text{r}ef{fig:PMFDXY} for $\alpha=9\pi/2 $. We recognize the existence of a transient wherein $P_+^-(t)$ exhibits aperiodic oscillations of maximum amplitude which, after a finite interval of time, turn into a monotonic increase that asymptotically approaches 1. We emphasize that the number of complete oscillations, preceding the asymptotic behaviour of $P_+^-(t)$ as well as $P_+^-(\text{i}nfty)$ itself, are $\alpha$-dependent. Equation \eqref{Trans Prob alfa}, indeed, predicts \text{b}egin{equation}gin{equation} P_+^-(\text{i}nfty)=+in^2(\alpha), \end{equation} which immediately leads to \text{b}egin{equation}gin{equation} \left\{ \text{b}egin{equation}gin{aligned} &P_+^-(\text{i}nfty)=0, \quad \alpha=n\pi, \\ &P_+^-(\text{i}nfty)=1, \quad \alpha={2n+1\over 2}\pi, \\ &P_+^-(\text{i}nfty)=+in^2(\alpha), \quad \text{otherwise}. \end{aligned} \text{r}ight. \end{equation} As our third example, we consider the following modulation of $|\omega(t)|$ \text{b}egin{equation}gin{equation}\label{Choice of mod omega} |\omega(t)|=|\omega_0|+A\cos(\lambda t), \end{equation} realizable by engineering the transverse magnetic field as \text{b}egin{equation}gin{equation}\label{Mag Field Somm} \text{b}egin{equation}gin{aligned} B_x(t)&=[B_\perp+A'\cos(\lambda t)]\cos(\dot{\phi}_0 t), \\ B_y(t)&=-[B_\perp+A'\cos(\lambda t)]+in(\dot{\phi}_0 t). \end{aligned} \end{equation} Here $A={\hbar-u_0gA' / 2}$, $A'>0$ and $\lambda={n\dot{\phi}_0}$ with $n \text{i}n -athbb{N}^*$. The transverse field is represented in Fig. \text{r}ef{fig:TFS} against the adimensional time parameter $\tilde{\tau}=\dot{\phi}_0 t$, once more supposing for simplicity $\phi_\omega(t)=\dot{\phi}_0 t$. In this case, the Rabi's transition probability results \text{b}egin{equation}gin{equation}\label{Trans Prob TFS Parametrized} P_+^-(t)=+in^2\Bigl[ C\Bigl(\tilde{\tau} + {k \over n} +in(n\tilde{\tau}) \Bigr) \Bigr], \end{equation} with \text{b}egin{equation}gin{equation} C={|\omega_0| \over \hbar \dot{\phi}_0}, \quad k={A' \over B_\perp}, \quad \tilde{\tau}=\dot{\phi}_0 t, \quad n={\lambda \over \dot{\phi}_0}. \end{equation} The behaviour of $P_+^-(t)$ in Eq. \eqref{Trans Prob TFS Parametrized} is shown in Fig. \text{r}ef{fig:PTFS}, having put $k=1$, $n=10$ and $C=1$. Differently from the previous example, we see that, in this case, the characteristic sinusoidal behaviour of the Rabi transition probability turns into a periodic population transfer, still of maximum amplitude, between the two energy levels of the spin. We emphasize that, in view of Eq. \eqref{P+- Res Cond}, different time evolutions of $P_+^-(t)$ require different choices of $|\omega(t)|$ only, then regardless of $\Omega(t)$ and $\phi_\omega(t)$ which are constrained by the generalized resonance condition \eqref{Gen Res Rabi Cond} only. We stress, however, that distinct realizations of the resonance condition, keeping the same $|\omega(t)|$, introduce significant changes in the dynamical behaviour of the GRS with respect to the Rabi system. It is enough to consider, for example, that \text{b}egin{equation}gin{equation} \average{+|U^\dagger(t)\hat{+igma}^{x/y}U(t)|+}=-p 2 \hbar |a(t)||b(t)| \cos[\phi_a(t)+\phi_b(t)], \end{equation} depend on both $\phi_\omega(t)$ and $|\omega(t)|$, in view of Eqs. \eqref{a and b c inf}. +ubsection{Examples of GRSs Dynamics in Generalized out of Resonance Cases}\label{Var Two Par} In this subsection we analyse the generalized out of resonance case, defined in Eq. \eqref{OoRC}. Since it appears hopeless to have an exact closed treatment of the Cauchy problem in Eq. \eqref{Cauchy problem Theta} with an arbitrary $\Delta(t)$, we confine ourselves to the following specific forms \text{b}egin{equation}gin{equation} \hbar\Delta(t)= \left\{ \text{b}egin{equation}gin{aligned} &\text{b}egin{equation}ta_0 |\omega(t)|, \\ &\text{b}egin{equation}ta(t) |\omega(t)|. \end{aligned} \text{r}ight. \end{equation} where $\text{b}egin{equation}ta_0$ and $\text{b}egin{equation}ta(t)$ are non-negative adimensional functions. The first form coincides with the condition in Eq. \eqref{Relation exactly solvable scenario} with $\text{b}egin{equation}ta_0=\hbar/c$. In this case, the solutions $a(t)$ and $b(t)$ of the system in Eq. \eqref{LC problem} may be cast as reported in Eqs. \eqref{a and b particular} so that \text{b}egin{equation}gin{equation}\label{P+- OoR c} P_+^-(t)={1\over1+\text{b}egin{equation}ta_0^2}+in^2\left[ +qrt{1+\text{b}egin{equation}ta_0^2} \text{i}nt_0^t{|\omega(t')|\over \hbar}dt' \text{r}ight]. \end{equation} In the limit $\text{b}egin{equation}ta_0\text{r}ightarrow 0$ we recover Eq. \eqref{P+- Res Cond} from this equation. Thus, we may compare $P_+^-(t)$ in the resonant and this non-resonant cases when $|\omega(t)|$ is fixed in the same way. It is easy to convince oneself that the main effect of a positive value of the parameter $\text{b}egin{equation}ta_0$ on $P_+^-(t)$ is nothing but a scale effect determined by the ratio $1/(1+\text{b}egin{equation}ta_0^2)$. \Ignore{ We observe that this solution holds when $|\omega(t)|$, $\dot{\phi}_\omega(t)$ and $\Omega(t)$ are such to make time-independent the expression \text{b}egin{equation}gin{equation}\label{c} \text{b}egin{equation}ta_0={\Omega(t)+{\hbar \over 2} \dot{\phi}_\omega(t) \over |\omega(t)|}={\Omega(0)+{\hbar \over 2} \dot{\phi}_\omega(0) \over |\omega(0)|}. \end{equation} It is remarkable that from the knowledge of a specific magnetic field, $\tilde{\omega}(t)$ and $\tilde{\Omega}(t)$, satisfying Eq. \eqref{c}, infinitely-many other different time-dependent magnetic fields may be easily found under the same condition. To this end it is enough to consider the following identities \text{b}egin{equation}gin{equation}\label{c Rabi Gen} \tilde{\text{b}egin{equation}ta_0}={{\tilde{\Omega}(t)} + {\hbar\dot{\tilde{\phi}}_\omega(t) \over 2} \over |\tilde{\omega}(t)|}= {{\tilde{\Omega}(t)}f(t)+ \epsilon g(t) + {\hbar\dot{\tilde{\phi}}_{\omega}(t) \over 2} f(t)- \epsilon g(t) \over |\tilde{\omega}(t)|f(t)}, \end{equation} where $\epsilon$ is a positive energy-dimensioned time-independent parameter and $f(t)$ and $g(t)$ are arbitrary adimensional positive functions of class $C^1$. We observe that the assumption \text{b}egin{equation}gin{equation} \tilde{\Omega}(t)=\Omega_0, \quad \tilde{\omega}(t)=|\omega_0|\exp \{i \dot{\phi}_0 t\}, \quad \dot{\phi}_0=\dot{\tilde{\phi}}_\omega(0) \end{equation} is compatible with Eq. \eqref{Relation exactly solvable scenario} if \text{b}egin{equation}gin{equation}\label{c Rabi} \tilde{\text{b}egin{equation}ta_0}={{\Omega_0} + {\hbar\dot{\phi}_0 \over 2} \over |\omega_0|}\equiv \text{b}egin{equation}ta_R. \end{equation} The subscript $R$ has been used since the magnetic field coincides with that adopted by Rabi, written in Eq. \eqref{Magn Field Rabi}. The entries of the evolution operator $U_R(t)$ under $-athbf{B}_R$, then, are given by Eqs. \eqref{a and b particular} with $c=c_R$ in Eq. \eqref{c Rabi}. In view of the approach reported in Ref. \cite{Mess-Nak}, the entries of the evolution operator $U_R(t)$ under $-athbf{B}_R$ are given by Eqs. \eqref{a and b particular} with $c=c_R$ in Eq. \eqref{c Rabi}. In consideration of Eq. \eqref{c Rabi Gen}, the evolution operator of the spin-1/2 when the magnetic field $-athbf{B}_R$ is substituted by the following one \text{b}egin{equation}gin{equation} \label{Magn Field Rabi Generalized} \text{b}egin{equation}gin{aligned} \textbf{B} = & B_\perp f(t) \cos\left[\dot{\phi}_0 F(t) - \nu G(t)\text{r}ight] \textbf{c}_1 - \\ & -B_\perp f(t) +in\left[\dot{\phi}_0 F(t) - \nu G(t)\text{r}ight]\textbf{c}_2 + \\ & +\left[ B_0 f(t) + B_\nu g(t) \text{r}ight] \textbf{c}_3, \end{aligned} \end{equation} with $\nu=-u_0 g B_\nu$ and $F(t)[G(t)] \equiv \text{i}nt_0^t f(t')[g(t')] dt'$, may be given as \text{b}egin{equation}gin{subequations}\label{Gab} \text{b}egin{equation}gin{align} |a(t)|=&+qrt{\hbar^2+c_R^2\cos^2\Phi(t)\over\hbar^2+c_R^2}, \\ \phi_a(t)=& \exp\left\{-i\text{b}egin{equation}gin{itemize}ggl[{\Omega_0\over\hbar} F(t) + \nu G(t) -{|\omega_0|\over c_R}F(t)\text{b}egin{equation}gin{itemize}ggr]\text{r}ight. \nonumber \\ & \left. -i\tan^{-1}\text{b}egin{equation}gin{itemize}ggl[{\hbar\over+qrt{\hbar^2+c_R^2}}\tan\Phi(t) \text{b}egin{equation}gin{itemize}ggr] \text{r}ight\}, \\ |b(t)|=&{c_R \over +qrt{\hbar^2+c_R^2}}\text{b}egin{equation}gin{itemize}gl|+in\Phi(t)\text{b}egin{equation}gin{itemize}gr|, \\ \phi_b(t)=& \exp \left\{-i\text{b}egin{equation}gin{itemize}ggl[{\Omega_0\over\hbar} F(t) + \nu G(t) - {|\omega_0|\over c_R}F(t) + {\pi \over 2}\text{b}egin{equation}gin{itemize}ggr]\text{r}ight\}, \end{align} \end{subequations} with $\Phi(t)$ dependent on the function $F(t)$ according to \text{b}egin{equation}gin{equation} \Phi(t)={+qrt{\hbar^2+c_R^2}\over\hbar c_R}|\omega_0|F(t). \end{equation} We see, then, that by putting $f(t)=1$ and $g(t)=0$ we get the standard Rabi scenario and the related time evolution operator. } We wish now to bring to light and to discuss some exactly solvable scenarios of generalized, out of resonance, Rabi problems wherein $\Delta(t)=\text{b}egin{equation}ta(t)|\omega(t)|$. To this end, it is useful to observe that postulating $Theta(t)$ as function of $t$ through \text{b}egin{equation}gin{equation}\label{tau} \tau(t) = \text{i}nt_0^t{|\omega(t')| \over \hbar}dt', \end{equation} we would get, by Eq. \eqref{Cauchy problem Theta}, the desired form of $\Delta(t)$, by construction. We stress however that the corresponding function $\text{b}egin{equation}ta(t)$ would be functionally-dependent on $|\omega(t)|$, that is determined by the knowledge of $|\omega(t)|$. We emphasize that this aspect, however, does not spoil of interest such a particular procedure. In the following examples we indeed report two applications of the general strategy \cite{Mess-Nak} exposed after Eq. \eqref{Integral R} in Sec. \eqref{MN Res}, where those $\text{b}egin{equation}ta(t)$s that make Eqs.\ (6) solvable are fixed. +ubsubsection{CASE 1}\label{Case 1} Assuming the solution of the Cauchy problem \eqref{Cauchy problem Theta} as \text{b}egin{equation}gin{align}\label{Theta 1} Theta(t) = 2 \tan^{-1}\left({2\tau \over +qrt{2+4\tau^2}}\text{r}ight), \end{align} it is straightforward to show that \text{b}egin{equation}gin{equation} \label{Position 3} \text{i}nt_0^t{|\omega(t')|\over\hbar}\cos[Theta(t')] dt' = {1 \over 2} \tan^{-1}(2\tau). \end{equation} Equation \eqref{Rel exactly solvable scenario gen} immediately yields \text{b}egin{equation}gin{equation} \label{Omega case 3} \Delta(t)= {4(1+\tau^2) \over (1+4\tau^2) +qrt{2+4\tau^2}} {|\omega(t)| \over \hbar} \equiv \tilde{\text{b}egin{equation}ta}(\tau)|\omega(t)|=\text{b}egin{equation}ta(t)|\omega(t)|. \end{equation} From this point on, we are ready to specialize Eqs. \eqref{a and b} getting \text{b}egin{equation}gin{equation}\label{a b case 3} |a(t)| = +qrt{{+qrt{1+4\tau^2} + 1 \over 2 +qrt{1+4\tau^2}}}, \quad |b(t)| = +qrt{{+qrt{1+4\tau^2} - 1 \over 2 +qrt{1+4\tau^2}}}, \end{equation} and \text{b}egin{equation}gin{subequations} \text{b}egin{equation}gin{align} \phi_{a}(t) =& \dfrac{\phi_\omega(t)-\phi_\omega(0)}{2} - \tan^{-1}\left({2\tau \over +qrt{2+4\tau^2}}\text{r}ight) \nonumber \\ &+ {i \over +qrt{2}} \text{EllipticE}[i +inh^{-1}(2\tau), 1/2], \\ \phi_{b}(t) =& \dfrac{\phi_\omega(t)+\phi_\omega(0)}{2} - \tan^{-1}\left({2\tau \over +qrt{2+4\tau^2}}\text{r}ight) \nonumber \\ &- {i \over +qrt{2}} \text{EllipticE}[i +inh^{-1}(2\tau), 1/2] - \dfrac{\pi}{2}, \end{align} \end{subequations} with $\text{EllipticE}(\phi,m)=\text{i}nt_0^\phi [1-m+in^2(\theta)]^{1/2} d\theta$. It is interesting to consider a simple case in which $|\omega(t)|=const.=|\omega_0|$. In this instance we have such a situation that $P_+^-(t)=|b(t)|^2$ $(P_+^+(t)=|a(t)|^2)$ goes from 0 (1), at $t=0$, to ${1 / 2}$ $\text{b}egin{equation}gin{itemize}gl( {1 / 2} \text{b}egin{equation}gin{itemize}gr)$, when $t \text{r}ightarrow \text{i}nfty$, as it can be appreciated by their plots in Fig. \text{r}ef{fig:TPB1}: full blue and dashed red lines, respectively. In Fig. \text{r}ef{fig:DC1} we may appreciate the time behaviour of $\hbar\Delta(t)/|\omega_0|$ related to this specific physical scenario. This specific out of resonance time-dependent scenario, then, asymptotically evolves the initial state $let{+}$ towards an equal-weighted superposition of the two eigenstates of $\hat{+igma}^z$. One can convince oneself that this circumstance is intimately related to the fact that, in this case, the ``detuning'' $\Delta(t)$ vanishes asymptotically (see Fig. \text{r}ef{fig:DC1}), getting then established a dynamical reply of the system as if it were under the generalized resonance condition. \text{b}egin{equation}gin{figure}[htp] \centering +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{DeltaCase1.eps}\label{fig:DC1}} \qquad +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{P+-Beta1.eps}\label{fig:TPB1}} \\ +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{DeltaCase2.eps}\label{fig:DC2}} \qquad +ubfloat[][]{\text{i}ncludegraphics[scale=.4]{P+-Beta2.eps}\label{fig:TPB2}} \captionsetup{justification=justified,format=plain,skip=4pt} \caption{(Color online) Time-dependence of the ``detuning'' $\hbar\Delta(\tau)/|\omega_0|=\text{b}egin{equation}ta(\tau)$ against the adimensional time $\tau=|\omega_0|t/\hbar$ related to the example a) \text{r}ef{Case 1} and c) \text{r}ef{Case 2}; transition probabilities $P_+^+(\tau)=|a(\tau)|^2$ (full blue) and $P_+^-(\tau)=|b(\tau)|^2$ (dashed red) for the time-dependent scenario b) \text{r}ef{Case 1} and d) \text{r}ef{Case 2}. } \end{figure} +ubsubsection{CASE 2}\label{Case 2} The second scenario is based on the following assumption \text{b}egin{equation}gin{align} Theta(t) = 2 \tan^{-1}\left({\tau \over +qrt{2+\tau^2}}\text{r}ight), \end{align} which, notwithstanding its apparent similarity with the previous case given in Eq. \eqref{Theta 1}, leads however to a remarkable different temporal behaviour of the correspondent generalized Rabi system. This time it results in \text{b}egin{equation}gin{equation} \label{Position 4} \text{i}nt_0^t{|\omega(t')|\over\hbar}\cos[Theta(t')] dt'= \tan^{-1}(\tau), \end{equation} so that the solutions of \eqref{LC problem} read \text{b}egin{equation}gin{equation}\label{a b case 4} \text{b}egin{equation}gin{aligned} |a(t)| = {1 \over +qrt{1+\tau^2}}, \qquad |b(t)| = {\tau \over +qrt{1+\tau^2}} = |a(t)|\tau, \end{aligned} \end{equation} and \text{b}egin{equation}gin{subequations} \text{b}egin{equation}gin{align} \phi_{a}(t) =& \dfrac{\phi_\omega(t)-\phi_\omega(0)}{2} - \tan^{-1}\left({\tau \over +qrt{2+\tau^2}}\text{r}ight) \nonumber \\ &- {1 \over 2} \text{b}egin{equation}gin{itemize}ggl[ {\tau +qrt{2+\tau^2} \over 2} + +inh^{-1}\left({\tau \over +qrt{2}}\text{r}ight) \text{b}egin{equation}gin{itemize}ggr], \\ \phi_{b}(t) =& \dfrac{\phi_\omega(t)+\phi_\omega(0)}{2} - \tan^{-1}\left({\tau \over +qrt{2+\tau^2}}\text{r}ight) \nonumber \\ &+ {1 \over 2} \text{b}egin{equation}gin{itemize}ggl[ {\tau +qrt{2+\tau^2} \over 2} + +inh^{-1}\left({\tau \over +qrt{2}}\text{r}ight) \text{b}egin{equation}gin{itemize}ggr] - \dfrac{\pi}{2}. \end{align} \end{subequations} Finally, the special form of $\Delta(t)$ underlying this specific scenario is \text{b}egin{equation}gin{equation} \label{Omega case 4} \Delta(t) = \left[ {2+(1-\tau^2)(2+\tau^2) \over 2(1+\tau^2)+qrt{2+\tau^2}}\text{r}ight] {|\omega(t)| \over \hbar}. \end{equation} In this case, it is easy to see that if $|\omega|=const.=|\omega_0|$, $P_+^-(t)=|b(t)|^2$ ($P_+^+(t)=|a(t)|^2$) goes from 0 (1) to 1 (0) asymptotically. These behaviours, evoking the transition probabilities in the Landau-Zener scenario \cite{LZ}, are illustrated by full blue and dashed red lines, respectively, in Fig. \text{r}ef{fig:TPB2}. In this case, the time behaviour of the ``detuning'' $\hbar\Delta(t)/|\omega_0|$ is characterized by an asymptotic linear dependence on $t$, as shown in Fig. \text{r}ef{fig:DC2}. As in the resonant scenario, even here different time-dependences of the magnetic field may give rise to qualitatively different time evolutions of $P_+^-(t)$ with respect to the Rabi scenario. As a final remark we want to emphasize that it could be very hard to get analytical expressions for $a(t)$ and $b(t)$, in Eq. \eqref{Gen a} and \eqref{Gen b}, respectively, depending on the choice of $Theta(t)$ and two of the three Hamiltonian parameters. Nevertheless, such a bottleneck does not influence our capability to predict the Rabi transition probability and the expression of $\Delta(t)$ in order to know how to engineer the magnetic fields to get the desired time evolution. Indeed, we would be always able to find accordingly the expressions of $|a(t)|$ and $|b(t)|$. Thus, as a consequence, when $\Omega(t)$, $|\omega(t)|$ and $\dot{\phi}_\omega(t)$ are chosen in such a way to generate the same detuning $\Delta(t)$, the related different physical scenarios share the same analytical expressions of $|a(t)|$ and $|b(t)|$ and then of all physical observables depending only on these quantities, e.g. $P_+^-(t)$ or \text{b}egin{equation}gin{equation} \average{\pm|U^\dagger(t)\hat{+igma}^zU(t)|\pm}= \pm {\hbar} (|a(t)|^2-|b(t)|^2). \end{equation} +ection{An exotic application} In this section we present a possible intriguing application of our results in the classical context of guided wave optics. Let us consider two electromagnetic modes propagating in the same direction (say the $z$ direction) and characterized by the two complex amplitudes $A$ and $B$ \cite{Yariv}. These may be defined in such a way that their squared modulus, $|A|^2$ and $|B|^2$, represent the power carried by the two modes. The amplitudes $A$ and $B$ does not depend on the coordinate $z$ if the medium through which the modes propagate is unperturbed. However, in a more realistic and experimental scenario several causes may perturb the medium, e.g. an electric field, a sound wave, surface corrugations, etc.. In this case, power is exchanged by the two modes and the amplitudes result mutually coupled in accordance with the following equations \cite{Yariv} \text{b}egin{equation}gin{equation}\label{Problem electric modes} \text{b}egin{equation}gin{aligned} {dA(z) \over dz}&=k_{ab}(z)e^{-i\Delta z}B(z), \\ {dB(z) \over dz}&=k_{ba}(z)e^{i\Delta z}A(z). \end{aligned} \end{equation} Here $\Delta$ is the phase-mismatch constant and $k_{ab}(z)$ and $k_{ba}(z)$ are complex coupling coefficients determined by the particular physical situation we analyse. The power conservation may be written as $\dfrac{d}{dz}(|A(z)|^2+|B(z)|^2)=0$ and it is accomplished by the condition $k_{ab}(z)=-k_{ba}^*(z)=k(z)$ \cite{Yariv}. It is easy to see that, after straightforward algebraic manipulations, the system may be cast in the following form \text{b}egin{equation}gin{equation}\label{su(1,1) problem z dependent} i{dV(z) \over dz}=H(z)V(z), \end{equation} where $V(z)=[\tilde{A}(z),\tilde{B}(z)]^T$, with \text{b}egin{equation}gin{equation} \tilde{A}(z)=A(z)e^{i\Delta z/2}, \quad \tilde{B}(z)=B(z)e^{-i\Delta z/2}, \end{equation} and \text{b}egin{equation}gin{equation}\label{Hamiltonian z dependent} H(z)= \text{b}egin{equation}gin{pmatrix} -\Delta/2 & \gamma(z) \\ \gamma^*(z) & \Delta/2 \end{pmatrix}, \end{equation} having put $\gamma(z) \equiv ik(z)$. We see immediately that the problem under scrutiny is mathematically equivalent to a Schr\"odinger dynamical problem based on a $su$(2) Hamiltonian. The physical relevance of our results in connection with this classical guided wave optics scenario may be easily clarified. Writing $V(z)=-athcal{U}(z)V(0)$, then the system reads \text{b}egin{equation}gin{equation}\label{U(z) problem} i{d-athcal{U}(z)\over dz}=H(z)-athcal{U}(z), \quad -athcal{U}(0)=-athbb{1}, \end{equation} being nothing but the problem for which we have sets of exact solutions related to specific relations between the Hamiltonian parameters exposed in Sec. \text{r}ef{MN Res}. Therefore, our solvability conditions and the specific cases reported in Sec \text{r}ef{Examples}, adapted to the context under scrutiny, furnish special links between $\Delta$ and $k(z)$ turning out in exactly solvable scenarios for the classical problem of two propagating modes in a perturbed medium. +ection{Conclusions} The Rabi scenario consists in a spin-1/2 subjected to a time-dependent magnetic field precessing around the quantized axis ($\hat{z}$) \cite{Rabi 1937} and is characterized by three time-independent parameters: $\Omega_0$, $|\omega_0|$ and $\dot{\phi}_0$. Rabi shows that when $\Omega_0+\hbar\dot{\phi}_0/2=\Delta=0$ the transverse magnetic field acts as a probe of the energy separation $2\Omega_0$ due to the longitudinal field alone. The measurable physical quantity revealing $\Omega_0$ is the transition probability $P_+^-(t)=\average{-|U(t)|+}$ which, at resonance, oscillates between 0 and 1 with frequency now referred to as Rabi frequency. In this work we generalize this Rabi scenario by assuming an SU(2) general time-dependent Hamiltonian model where then $\Omega_0$, $|\omega_0|$ and $\dot{\phi}_0$ are now replaced with time-dependent counterparts. Along the lines of the Rabi approach \cite{Rabi 1954}, we firstly show that, in the rotating frame with the time-dependent angular frequency $\dot{\phi}_\omega(t)$, the condition $\Omega(t)+\hbar\dot{\phi}_\omega(t)/2=\Delta(t)=0$ plays the same role of the Rabi resonance condition in the Rabi scenario. Such an occurrence makes of basic interest a direct comparison between the Rabi scenario and its generalized version on both time-dependent resonance and out of resonance ($\Delta(t) \neq 0$) cases. To bring to light the occurrence of analogies and differences, we have focussed our attention on the study of the transition probability $P_+^-(t)$ between the two eigenstates of $\hat{S}^z$. We show that, on resonance, $P_+^-(t)$ depends only on the integral of $|\omega(t)|$. Our examples illustrate that this circumstance determines a transition probability characterized by three possible different regimes: oscillatory (the only one dominating the Rabi scenario), monotonic and mixed which means an initial oscillatory transient followed by an asymptotic monotonic behaviour. To capture significant dynamical consequences stemming from the detuning time dependence, we have constructed \cite{Mess-Nak} exactly solvable problems and analysed the corresponding quantum dynamics of the spin-1/2. We have thus highlighted that when $\Delta(t)$ is proportional to $|\omega(t)|$, the main effect emerging in the time behaviour of $P_+^-(t)$ is a scale effect both in amplitude and in frequency (like in the Rabi scenario). We have further investigated two specific exactly solvable scenarios of experimental interest for which $\Delta(t)/|\omega(t)|$ varies over time. One of them predicts a Landau-Zener transition, while the other an equal weighted superposition of the two states of the system. It is important to underline that our examples illustrate exactly solvable cases for which, then, the corresponding system dynamics may be fully disclosed. We highlighted, however, that when one is interested in the Rabi transition probability $P_+^-(t)$ only, it is enough the knowledge of $|a(t)|$ and $|b(t)|$. We point out that this circumstance leads us to wider and richer classes of physical scenarios, since it gets us rid of possible analytical difficulties stemming from Eq. \eqref{Integral R}. We underline that the knowledge of the exactly solvable problems reported in this paper provide stimulating ideas for technological applications with single qubit devices. In addition it furnishes ready-to-use tools for interacting qudits systems \cite{GMN, GMIV, GBNM}, being of relevance in several fields, from condensed matter physics \cite{Calvo, Borozdina} to quantum information and quantum computing \cite{Petta, Anderlini, Foletti, Das Sarma Nat}. As a conclusive remark, we emphasize the applicability of our approach to the propagation of two electromagnetic modes in a perturbed medium. We have shown the mathematical equivalence of this problem to that of a 2x2 $su(2)$ dynamical problem. 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\begin{document} \title{\Large{\bf{Birational geometry of hypersurfaces in products of projective spaces} \thispagestyle{empty} ^{\vee}space{-0.8cm} \begin{abstract} We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in ${\mathbb P}^m \times {\mathbb P}^n$, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample. {\mathbf e}nd{abstract} \section{Introduction} Let $X$ be a hypersurface of ${\mathbb P}^m\times {\mathbb P}^n$ defined by a bihomogeneous polynomial. If the dimension of $X$ is at least three, the Lefschetz theorem says that the inclusion induces an isomorphism of Picard groups $\Pic(X)\simeq \Pic({\mathbb P}^m\times {\mathbb P}^n)\simeq {\mathbb Z}^2$. It is therefore natural to ask how the various cones of divisors of $X$ are related to those of the ambient space ${\mathbb P}^m\times {\mathbb P}^n$. In general this relation is not obvious, as examples of Hassett--Lin--Wang \cite{HLW02} and Szendr\~{o}i \cite{Sze03} show that the nef cone of $X$ can be strictly greater than that of the ambient variety. The purpose of this paper is to give a complete picture describing the birational structure of such hypersurfaces. In particular, we compute the cones of effective, movable or nef divisors on $X$ and determine its birational models. In the last section we also consider hypersurfaces in products of more than two projective spaces. Recall that a normal ${\mathbb Q}$-factorial projective variety $X$ is a {\mathbf e}mph{Mori dream space} if the following three conditions are fulfilled: (i) $\Pic(X)$ is finitely generated; (ii) the nef cone $\operatorname{Nef}(X)$ is generated by the classes of finitely many semiample divisors; and (iii) there is a finite collection of small ${\mathbb Q}$-factorial modifications $\phi_i\colon X\dashrightarrow X_i$ such that each $X_i$ satisfies (ii) and the movable cone of $X$ decomposes as $\operatorname{Mov}(X)=\bigcup \phi_i^*\operatorname{Nef}(X_i)$. Mori dream spaces were introduced by Hu and Keel in \cite{HK00} as a class of varieties with good birational geometry properties. For example, the condition of being a Mori dream space is equivalent to having a finitely generated Cox ring \cite[Theorem 2]{HK00} (see section |ef{notations}). Moreover, choosing a presentation for the Cox ring gives an embedding of $X$ into a simplicial toric variety $Y$ such that each of the modifications $\phi_i$ above is induced from a modification of the ambient toric variety $Y$ (see \cite[Proposition 2.11]{HK00}). From this one shows that the Minimal Model Program can be carried out for any divisor and has a combinatorial structure as in the case of toric varieties. Being a Mori dream space is a relatively strong condition and there are classical examples of varieties that are not. Perhaps the most famous of these is Nagata's counterexample to Hilbert's 14th problem, in which he proves that the blow-up of ${\mathbb P}^2$ along the base-locus of a general cubic pencil has infinitely many $(-1)$-curves \cite{Nag60}. This blow-up is clearly not a Mori dream space since each of the $(-1)$-curves would require a generator of the Cox ring. The same phenomenon happens for a K3 surface with large Picard number, where one typically expects infinitely many $(-2)$-curves. There are also other obstructions to being a Mori dream space than the non-polyhedrality of the effective/nef cones. Indeed, there might be integral classes in the boundary of the nef cone which are not semiample. In this paper we will construct concrete examples of this phenomenon even for Picard number 2; In fact `most' hypersurfaces in ${\mathbb P}^1\times {\mathbb P}^n$ have an extremal divisor of the nef cone which does not even have an effective multiple, so they are not Mori dream spaces. Thus these hypersurfaces provide simple counterparts to Nagata's examples above. The following theorem summarizes the geometry of such hypersurfaces: \begin{theorem}\label{decones} Let $X$ be a ${\mathbb Q}$-factorial, normal hypersurface of bidegree $(d,e)$ in ${\mathbb P}^m\times {\mathbb P}^n$ of dimension at least three and let $H_i=\operatorname{p}_i^*{\mathscr O}(1)$. If $m,n\ge 2$, $X$ is a Mori dream space and the Cox ring is isomorphic to $k[x_0,\ldots,x_m,y_0,\ldots,y_n]/(f)$. In particular, $${{\mathbf e}ff(X)}=\operatorname{Mov}(X)=\operatorname{Nef}(X)={\mathbb R}_{\ge 0} H_1+{\mathbb R}_{\ge 0}H_2.$$ \noindent When $X$ is a general hypersurface in ${\mathbb P}^1\times {\mathbb P}^n$, we have the following: \begin{enumerate}[(i)] ^{-1}tem If $d= 1$, the second projection realizes $X$ as the blow-up of ${\mathbb P}^n$ along $\{f_0=f_1=0\}$, and the exceptional divisor is linearly equivalent to $E=eH_2-H_1$, and $$ {{\mathbf e}ff}(X)={\mathbb R}_{\ge 0} H_1+{\mathbb R}_{\ge 0} E \mbox{ and }\operatorname{Mov}(X)=\operatorname{Nef}(X)={\mathbb R}_{\ge 0}H_1+{\mathbb R}_{\ge 0}H_2. $$ ^{-1}tem $1<d< n$. There is a variety $X^+$ and a small birational modification $\phi:X\dashrightarrow X^+$, which induces a decomposition $\operatorname{Mov}(X)=\operatorname{Nef}(X)\cup \phi^*\operatorname{Nef}(X^+)$. Also, $$ {{\mathbf e}ff}(X)=\operatorname{Mov}(X)={\mathbb R}_{\ge 0} H_1+{\mathbb R}_{\ge 0} (eH_2-H_1),\mbox{ and }{\operatorname{Nef}}(X)={\mathbb R}_{\ge 0}H_1+{\mathbb R}_{\ge 0}H_2. $$ ^{-1}tem $d=n$. The divisor $eH_2-H_1$ is base-point free and defines a contraction to ${\mathbb P}^{n-1}$. Also, $${\mathbf e}ff(X)=\operatorname{Mov}(X)=\operatorname{Nef}(X)={\mathbb R}_{\ge 0} H_1+{\mathbb R}_{\ge 0} (eH_2-H_1).$$^{-1}tem If $e=1$, $X$ is a ${\mathbb P}^{n-1}$-bundle over ${\mathbb P}^1$. {\mathbf e}nd{enumerate} In these cases $X$ is a Mori dream space and the Cox ring has the following presentation \begin{equation}\label{coxringpres} \operatorname{\mathcal R}(X)=k[x_0,x_1,y_0,\ldots,y_n,z_1,\ldots,z_d]/I {\mathbf e}nd{equation}where $I=(f_0+x_1z_1,f_1-x_0z_1+x_1z_2,\ldots,f_{d-1}-x_0z_{d-1}+x_1z_d,f_d-x_0z_d)$. For very general hypersurfaces in ${\mathbb P}^1\times {\mathbb P}^n$ of degree $(d,e)$ with $d\ge n+1$ and $e\ge 2$ however, $X$ is {{\mathbf e}m not} a Mori dream space. Here $$\overline{{\mathbf e}ff(X)}=\overline{\operatorname{Mov}(X)}=\operatorname{Nef}(X)={\mathbb R}_{\ge 0} H_1+{\mathbb R}_{\ge 0} (neH_2-dH_1).$$ but the divisor $neH_2-dH_1$ has no effective multiple. {\mathbf e}nd{theorem} Note that Mori dream hypersurfaces in ${\mathbb P}^1\times {\mathbb P}^n$ have bidegrees $(d,e)$ lying in the L-shaped region given by $\{1\le d\le n \mbox{ or } e=1\}$. Hence it is essentially the value of $d$, rather than the anticanonical divisor, that determines whether a general hypersurface of degree $(d,e)$ is a Mori dream space or not. In particular, it is not true that a sufficiently ample hypersurface in a Mori dream space is again a Mori dream space. In this case that there are only a few bidegrees $(d,e)$ where $X$ is Fano, in which case it is well-known that $X$ is a Mori dream space. The case $(2,n+1)$ corresponds to a Calabi-Yau manifold. On the other hand, for general type varieties, when $K_X$ is ample, most hypersurfaces are not Mori dream spaces, although there are some that are (e.g., bidegree $(n,n+1)$ is a Mori dream space, $(n+1,n+1)$ is not). It is also interesting to note that all the cones involved are rational polyhedral for any bidegree. \section*{Relation to an example of Mumford} There is a corresponding result for surfaces in ${\mathbb P}^1\times {\mathbb P}^2$, but this requires a slightly modified argument, as the Picard group of $X$ might be larger than that of the ambient space. Nevertheless, using the Noether-Lefschetz theorem, we prove the following analogue of Theorem |ef{decones} for surfaces: \begin{proposition}\label{desurfaces} Let $X$ be a very general surface in ${\mathbb P}^1\times {\mathbb P}^2$ of bidegree $(d,e)$. \begin{enumerate}[(i)] ^{-1}tem If $d=1$, $X$ is the blow-up of ${\mathbb P}^2$ along the intersection of two very general degree $e$ curves. It is a Mori dream space if and only if $e\le 2$, in which case $X$ is a del Pezzo surface. In the other case, $X$ is a rational surface with infinitely many $(-1)$-curves. ^{-1}tem If $d=2$, $X$ is a double cover of ${\mathbb P}^2$ branched along a smooth curve of degree $2e$ and is always a Mori dream space. ^{-1}tem If $e=1$, $X$ is a Hirzebruch surface. ^{-1}tem If $d\ge 3$ and $e\ge 2$, then the effective cone of $X$ is not closed. Hence $X$ is not a Mori dream space. {\mathbf e}nd{enumerate} {\mathbf e}nd{proposition} Proposition |ef{desurfaces} gives a simple example of a line bundle $L$ which has positive intersection with every curve $C$, but is not ample. More precisely, a very general surface of bidegree $(3,3)$ in ${\mathbb P}^1\times {\mathbb P}^2$ has the line bundle $L={\mathscr O}(2H_2-H_1)$ which satisfies this condition. Of course here $L$ has top-self-intersection 0. In section 5 we also construct higher dimensional analogues of this. Examples of line bundles with such properties were first constructed by Mumford using certain projective bundles ${\mathbb P}({\mathscr E})$ over curves of genus $\ge 2$. Our geometric construction in Theorem |ef{decones} was inspired by Mumford's example, but we use only projective bundles over elliptic curves. \section*{The Lefschetz theorem for Mori dream spaces} Let $X$ be a ${\mathbb Q}$-factorial, normal hypersurface of ${\mathbb P}^m\times {\mathbb P}^n$ where $m,n\ge 2$. In this case it is easy to show that $X$ is a Mori dream space, by computing the Cox ring of $X$ directly. Indeed, let $D$ be any divisor on ${\mathbb P}^m\times {\mathbb P}^n$ and consider the sequence $$ 0\to H^0({\mathbb P}^m\times {\mathbb P}^n,{\mathscr O}(D-X))\to H^0({\mathbb P}^m\times {\mathbb P}^n,{\mathscr O}(D))\to H^0(X,{\mathscr O}_X(D))\to 0 $$This is exact on the right because $H^1({\mathbb P}^m\times {\mathbb P}^n,L)=0$ for any line bundle $L$ when $m,n\ge 2$. This shows that the Cox ring of $X$ is a quotient of that of ${\mathbb P}^m\times {\mathbb P}^n$: $$\operatorname{\mathcal R}(X)= k[x_0,\ldots,x_m,y_0,\ldots,y_n]/(f),$$ where the grading is $\deg x_i=H_1$ and $\deg y_j=H_2$. Moreover, the only contractions of $X$ are given by the two projections. In particular, this means that the questions mentioned in the introduction are only interesting for hypersurfaces in ${\mathbb P}^1\times {\mathbb P}^n$. Here the situation becomes more complicated because the presence of higher cohomology makes it difficult to compute $H^0(X,{\mathscr O}_X(D))$. In particular, we will see that the Cox ring of $X$ is {{\mathbf e}m not} a quotient of that of ${\mathbb P}^1\times {\mathbb P}^n$ in these cases. In general it is an interesting question when a sufficiently ample hypersurface in a Mori dream space is again a Mori dream space. This is not always the case, even for arbitrarily ample hypersurfaces, as shown by Theorem |ef{decones}. In the positive direction, Hausen \cite{Hau08}, Jow \cite{Jow11} and Artebani-Laface \cite{AL12} give criteria for when the Cox ring of the hypersurface is a quotient of that of the ambient variety. A necessary condition for this to hold is that $X$ and $Y$ have isomorphic Picard groups. In \cite{Jow11}, Jow proves that $\operatorname{\mathcal R}(X)\simeq \operatorname{\mathcal R}(Y)/(f)$ for any smooth ample divisor $X$ on $Y$, provided $Y$ is smooth of dimension $\ge 4$, and $I_{\text{irr}}(Y)$ has codimension at least 3 in $\operatorname{\mathcal R}(Y)$. Here $I_{\text{irr}}(Y)$ denotes the so-called {\mathbf e}mph{irrelevant ideal} of $Y$, which describes the unstable locus of the action of the Picard torus on $\operatorname{\mathcal R}(Y)$. In the case $Y={\mathbb P}^m\times {\mathbb P}^n$, the irrelevant ideal is given by $I_{\text{irr}}=(x_0,\ldots,x_m)\cap (y_0\ldots,y_n)$, which has codimension at least $3$ if and only if $m,n\ge 2$. Generalizing Jow's result, Artebani and Laface show that the above conclusion holds also under considerably weaker assumptions on $Y$ provided $X$ is ample and general in its linear system \cite{AL12}. \section*{Notation}\label{notations} Throughout the paper we will be working over an uncountable algebraically closed field of characteristic 0. The main reason for this is that some of the arguments used in section 5 requires working with {\mathbf e}mph{very general} hypersurfaces over {{\mathbf e}m general} ones. Here the latter will refer to the hypersurface being chosen outside a finite union of closed algebraic subsets of the parameter space, whereas `very general' means outside a countable union of closed algebraic subsets. We will also use properties of semistable vector bundles (such as the fact that a symmetric power of a semistable vector bundle is again semistable), which are known to be false in positive characteristic. On the other hand it is likely that many of the results in section 2 and 3 can be extended to positive characteristic using the Grothendieck--Lefschetz theorem, which is known to hold in all characteristics. We let $N^1(X)$ denote the N\'eron-Severi group of $X$, i.e., the ${\mathbb R}$-vector space of divisors modulo numerical equivalence. For most of the varieties in this paper numerical and linear equivalence coincide, so that $N^1(X)=\Pic(X)\otimes {\mathbb R}$. Inside $N^1(X)$ we define the effective cone ${\mathbf e}ff(X)$ to be the cone of all effective divisors. Similarly, we denote by $\operatorname{Nef}(X)$ (resp. $\operatorname{Mov}(X)$) the cone of nef divisors (resp. movable divisors). Here we call a divisor $D$ {\mathbf e}mph{nef} (resp. {\mathbf e}mph{movable}) if $D\operatorname{cd}ot C\ge 0$ for every curve $C$ (resp. if the linear system $|mD|$ has no fixed components for $m>0$ sufficiently large). Note that the nef cone is always closed, whereas the other two need not be. When $\Pic(X)$ is a free abelian group, the Cox ring of $X$ is defined as the ring $$\operatorname{\mathcal R}(X)=\bigoplus_{\mathbf m^{-1}n {\mathbb Z}^|ho}H^0(X,{\mathscr O}_X(m_1D_1+\ldots+m_|ho D_|ho))$$ for a chosen basis $D_1,\ldots,D_|ho$ for $\Pic(X)$. As usual, we consider this ring with its $\Pic(X)$-grading. (The Cox ring can more generally be defined using the class group as in \cite{CoxBook}, but this is not necessary for the varieties considered in this paper). \subseteqsection*{Acknowledgements}Thanks to Burt Totaro for his advice and encouragement. Also thanks to Laurent Gruson, Antonio Laface, Victor Lozovanu, Christian Peskine, Diane MacLagan and Kenji Oguiso for useful discussions and comments. After this paper was written, we learned that some of the hypersurfaces in Theorem |ef{decones} were considered by Ito in \cite{Ito13}, who shows that they are Mori dream spaces using a different argument. \section{Mori dream hypersurfaces in ${\mathbb P}^1\times {\mathbb P}^n$} Consider a hypersurface $X\subseteqset\pn{n}$ defined by a bihomogeneous form \begin{equation}\label{deform} f=x_0^df_0+x_0^{d-1}x_1f_1+\ldots+x_1^df_d=0 {\mathbf e}nd{equation}where $x_0,x_1$ coordinates on ${\mathbb P}^1$ and the $f_i$ are homogenous forms of degree $e$ in the coordinates $y_0,\ldots,y_n$ on ${\mathbb P}^n$. We will let ${\mathscr O}_{\pn{n}}(a,b)$ denote the line bundle $\operatorname{p}_1^*{\mathscr O}(a)\otimes \operatorname{p}_2^*{\mathscr O}(b)$ on $\pn{n}$. We will in this section assume that $n\ge 3$ and that $1<d\le n$. For the remaining cases, see sections 3 and 4. We will also assume that $X$ is general in the sense that it is smooth and that the $f_i$ generate a regular sequence. (However many of the arguments go through under weaker assumptions, e.g., normal and ${\mathbb Q}$-factorial). In this case, we have by the Grothendieck--Lefschetz theorem that $\Pic(X)= {\mathbb Z} H_1\oplus{\mathbb Z} H_2$ where $H_1={\mathscr O}_{\pn{n}}(1,0)|_X$ and $H_2={\mathscr O}_{\pn{n}}(0,1)|_X$. \def{\mathscr A}A{\mathbb A} The hypersurface $X$ admits an interesting birational map which can be seen if we write $f$ as the determinant of the {{\mathbf e}m companion matrix} \begin{equation}\label{detM} A=\begin{pmatrix} x_1 & 0 &\operatorname{cd}ots & 0 & f_0\\ -x_0 & x_1 & \ddots & ^{\vee}dots & f_1 \\ 0 & \ddots & \ddots &0 & ^{\vee}dots\\ & \ddots & -x_0 & x_1 &f_{d-1}\\ 0 & \operatorname{cd}ots & 0 & -x_0 & f_d\\ {\mathbf e}nd{pmatrix} {\mathbf e}nd{equation}Let $Y\subseteqset {\mathscr A}A={\mathscr A}A^2\times {\mathscr A}A^{n+1}$ be the affine hypersurface defined by $f=\det A$. Note that if there is a $z=(z_1,\ldots,z_d,1)^{-1}n {\mathbb C}^{d+1}$ with $A\operatorname{cd}ot z^t=0$, then also $B\operatorname{cd}ot (x_0,x_1,1)^t=0$ where \begin{equation}\label{yzmatrix} B=\begin{pmatrix} 0 & z_1 & f_0\\ -z_1 & z_2 & f_1\\ ^{\vee}dots & ^{\vee}dots & ^{\vee}dots\\ -z_{d-1} & z_d & f_{d-1}\\ -z_d & 0 & f_d\\ {\mathbf e}nd{pmatrix} {\mathbf e}nd{equation}Let $Y^+$ be the subvariety of ${\mathscr A}A^{d}\times {\mathscr A}A^{n+1}$ defined by the maximal minors of $B$. Note that for fixed $(x_0,x_1)^{-1}n {\mathscr A}A^2-0$, the kernel of $A$ is at most 1-dimensional. Hence we get a well-defined rational map $\psi: Y\dashrightarrow Y^+$ by defining $$\psi(x_0,x_1,y_0,\ldots,y_n)=(z_1,\ldots,z_d,y_0,\ldots,y_n).$$Similarly, given $(z_1,\ldots,z_d,y_0,\ldots,y_n)^{-1}n Y^+$ with at least one $z_i$ non-zero, the matrix $B$ also has a kernel which is 1-dimensional, giving a well-defined inverse of $\psi$. So the map $\psi$ is birational. Everything here is compatible with the various ${\mathbb C}^*$-actions, so we get a birational map $\phi:X\dashrightarrow X^+$, where $X^+\subseteqset {\mathbb P}^{d-1}\times {\mathbb P}^n$ is defined by the minors of $B$. The map $\phi$ is a morphism outside the locus where $f_0=\operatorname{cd}ots=f_d=0$ in $X$. Indeed, in this case at least one $z_i$ is non-zero, so the corresponding point in ${\mathbb P}^{d-1}$ is well-defined. The corresponding statement also holds for $\phi^{-1}$. In particular, when the $f_i$ form a regular sequence, $\phi$ is an isomorphism in codimension $d$ (thus an isomorphism for $d=n$). The variety $X^+$ will usually be singular, even for $f_0,\ldots,f_d$ general, but it will follow from the computation below and \cite[Proposition 1.11]{HK00} that the singularities are ${\mathbb Q}$-factorial and terminal when $X$ is smooth. To distinguish between $X$ and $X^+$ we use ${\mathscr O}_{X^+}(1,0)$ and ${\mathscr O}_{X^+}(0,1)$ to denote the line bundles on $X^+$ coming from the two projections. From the construction of $\phi$ we have that $$\phi^*({\mathscr O}_{X^+}(1,0))=eH_2-H_1 \mbox{ and }\phi^*({\mathscr O}_{X^+}(0,1))=H_2$$ Here the line bundle ${\mathscr O}_{{\mathbb P}^{d-1}\times {\mathbb P}^n}(1,0)$ is not big on $X^+$, since it gives the contraction to the lower-dimensional variety ${\mathbb P}^{d-1}$ (here we are using $d\le n$). It follows that $eH_2-H_1$ is in the boundary of the effective cone on $X$. From this, we see that the movable cone decomposes as $\operatorname{Nef}(X)\cup \operatorname{Nef}(X^+)$. In particular, $X$ is a Mori dream space. This decomposition is illustrated in the figure below in the case $2\le d<n$. \begin{figure}[h!] \centering ^{-1}ncludegraphics[width=0.3\textwidth]{mov13} \label{weight}\label{decompic} {\mathbf e}nd{figure} When $d=n$, $\phi$ is an isomorphism, so the three cones are equal to ${\mathbb R}_{\ge 0}H_1+{\mathbb R}_{\ge 0}(eH_2-H_1)$. The variety $X^+$ and $\phi$ still make sense for $d>n$, but we can not conclude that the effective cone decomposes as above, since $eH_2-H_1$ {{\mathbf e}m is} big in this case (cf. section 5). To compute the Cox ring, we proceed as in \cite{Ott13}. We will assume that $X$ is general (so in particular, $X$ is smooth, and the polynomials $f_0,\ldots,f_d$ form a regular sequence). We will use induction to write any element of $\operatorname{\mathcal R}(X)$ as a polynomial in the above sections $x_i,y_j,z_k$. Let $D$ be an effective divisor on $X$ and let $L$ be a base-point free line bundle. By a result of Mumford \cite{mumford}, the multiplication map $$ H^0(X,{\mathscr O}_X(D-L))\otimes H^0(X,{\mathscr O}_X(L))\to H^0(X,{\mathscr O}_X(D)) $$is surjective provided $H^i(X,{\mathscr O}_X(D-iL))=H^{i}(X,{\mathscr O}_X(D-(i+1)L))=0$ for $i=1,\ldots,m-1$, where $m=h^0(X,{\mathscr O}_X(L))$. Mumford states the result with the assumption that the latter cohomology groups vanish for all $i>0$, but the proof shows that this is not necessary. (In fact, one can give a quick proof of this result by writing out the Koszul complex of $m$ sections generating $H^0(X,{\mathscr O}_X(L))$ and taking its cohomology). Now, if this map is surjective, we see that sections of ${\mathscr O}_X(D)$ are generated by products of sections coming from ${\mathscr O}_X(L)$ and ${\mathscr O}_X(D-L)$, so by induction it follows that $x_i,y_j,z_k$ generate the ring. To show that the above multiplication map is surjective, suppose $D=aH_1+bH_2$ is an effective line bundle on $X$. If $a\ge 1,b\ge 0$, we may take $L=H_1$ above and note that $H^1(X,{\mathscr O}_X(D-H_1))=0$. When $a=0$, we use instead $L=H_2$ and find that $H^i(X,bH_2)=0$ for all $i=1,\ldots,n-1$ and any $b\ge 0$: This follows by taking cohomology of the exact sequence $$ 0\to {\mathscr O}_{\pn{n}}(-d,b-e)\to {\mathscr O}_{\pn{n}}(0,b)\to {\mathscr O}_{X}(bH_2)\to 0. $$Similarly, for $a\ge 1,b\ge 0$, the multiplication map $$ H^0(X^+,{\mathscr O}_{X^+}(a-1,b))\otimes H^0(X^+,{\mathscr O}_{X^+}(1,0))\to H^0(X^+,{\mathscr O}_{X^+}(a,b)) $$is surjective. That the above cohomology groups vanish, follows by resolving ${\mathscr O}_{X^+}$ as an ${\mathscr O}_{{\mathbb P}^{d-1}\times {\mathbb P}^{n}}$-module, using the Eagon--Northcott complex of $B$ \cite[Appendix B]{Laz04}. It follows that $H^0(X^+,{\mathscr O}_{X^+}(a,b))$ and hence $H^0(X,{\mathscr O}_X(a(eH_2-H_1)+bH_2))$ is spanned by polynomials in $x_i,y_j,z_k$. In all, this means that the Cox ring of $X$ is generated by the sections $x_i,y_j,z_k$. Let now $R$ denote the polynomial ring on the right hand side of {\mathbf e}qref{coxringpres}. When the $f_0,\ldots,f_d$ are general, $I$ is a complete intersection and a prime ideal (e.g., it is true for $f_i=y_i^d$ and these are open conditions). In particular, both $R/I$ and $\operatorname{\mathcal R}(X)$ are integral domains. By \cite[Proposition 2.9]{HK00}, the Krull dimension of $\operatorname{\mathcal R}(X)$ equals $\operatorname{rank} \Pic(X)+\dim X=n+2$. Similarly, since $I$ is a complete intersection, the Krull dimension of $R/I$ is $(2+n+1+d)-(d+1)=n+2$. If follows that the surjection $R/I \to \operatorname{\mathcal R}(X)$ is in fact an isomorphism. This completes the proof of the part about $\operatorname{\mathcal R}(X)$ in Theorem |ef{decones}. \begin{remark}The above birational map $\phi$ has the following interpretation in terms of geometric invariant theory. Consider the ${\mathbb Z}^2$-graded ring $R=k[x_0,x_1,y_0,\ldots,y_n,z_1,\ldots,z_d]$ where the grading of the variables is given by columns of the matrix $$ \left[ \begin{array}{rrrrrrrr} 1 & 1 & 0 & \operatorname{cd}ots & 0 & -1 & \operatorname{cd}ots &-1\\ 0 & 0 & 1 & \operatorname{cd}ots & 1 & e & \operatorname{cd}ots &e\\ {\mathbf e}nd{array} |ight] $$The torus $G=({\mathbb C}^*)^2$ acts on ${\mathscr A}A=\operatorname{Spec} R$ via these weights. We wish to study the various GIT quotients ${\mathscr A}A/\!\! / G$. To do this, we consider the trivial line bundle ${ L}\to {\mathscr A}A$, with coordinate $t$, defined by the embedding $R\subseteqset R[t]$. We extend the action of $G$ to ${ L}$ by choosing a character $\chi: G\to {\mathbb C}^*$, and defining for $g^{-1}n G$, $g^*(t)=\chi(g)^{-1}\operatorname{cd}ot t$. As shown in \cite{HK00}, the set of semistable points of the action is ${\mathscr A}A-V(B_\chi)$ where $B_\chi$ is the irrelevant ideal of $R$, defined as the radical of the ideal generated by the subring of $R$ with degrees multiples of $\chi$. This defines the GIT quotient of ${\mathscr A}A$ by $G$ associated to $\chi$ as $({\mathscr A}A-V(B_\chi))/ G$. With our grading, there are essentially three different GIT quotients $Y,Y^+,Z$, corresponding to the characters in the three chambers ${\mathbb R}_{>0}(\begin{smallmatrix}1\\0{\mathbf e}nd{smallmatrix})+{\mathbb R}_{>0}(\begin{smallmatrix}0\\1{\mathbf e}nd{smallmatrix})$, ${\mathbb R}_{>0}(\begin{smallmatrix}0\\1{\mathbf e}nd{smallmatrix})$, ${\mathbb R}_{>0}(\begin{smallmatrix}1\\0{\mathbf e}nd{smallmatrix})+{\mathbb R}_{>0}(\begin{smallmatrix}-1\{\mathbf e}{\mathbf e}nd{smallmatrix})$ respectively. These correspond to the three irrelevant ideals $B=(x_0,x_1)\cap (y_0,\ldots,y_n,z_1,\ldots,z_d)$, $B^+=(x_0,x_1,y_0,\ldots,y_n)\cap (z_1,\ldots,z_d)$ and $B\cap B^+$ and fit into the following diagram: \begin{equation}\label{flipdiagram} \timesymatrix{ & Y \ar@{->}[rd]_{\nu} \ar@{->}[ld]_x \ar@{-->}[rr]^\phi& &Y^+\ar@{->}[ld]^{\nu^+} \ar@{->}[rd]^z\\ {\mathbb P}^1& & Z & & {\mathbb P}^{d-1}} {\mathbf e}nd{equation}The hypersurface $X$ (resp. $X^+$) can be embedded in $Y$ (resp. $Y^+$) as a complete intersection defined by the $d+1$ equations \begin{equation}\label{equationsforX} f_0+x_1z_1=0,f_1-x_0z_1+x_1z_2=0,\ldots, f_{d-1}-x_0z_{d-1}+x_1z_d=0,f_d-x_0z_d=0.{\mathbf e}nd{equation}It is straightforward to check that this $\phi$ restricts to the birational map constructed earlier.{\mathbf e}nd{remark} \section{Examples} \subseteqsection{Bidegree $(d,1)$.} Let $X\subseteqset {\mathbb P}^1\times {\mathbb P}^n$ be defined by a bihomogeneous form of degree $(d,1)$. The first projection gives $X$ the structure of a ${\mathbb P}^{n-1}$-bundle over ${\mathbb P}^1$. In this case, $X$ is a toric variety (hence a Mori dream space) and its birational geometry is well-known \cite{miles}. $X$ has two contractions: One given by the first projection and the other either flipping or contracting depending on $d$. Let us consider the case where $d\le n$ and $X$ is general. Then it is straightforward to check that the hypersurface is isomorphic to ${\mathbb P}({\mathscr E})$ where ${\mathscr E}= {\mathscr O}_{{\mathbb P}^1}^{n-d}\oplus {\mathscr O}_{{\mathbb P}^1}^{d}(1)$. Moreover, the toric variety $Y$ coincides with the projective bundle ${\mathbb P}({\mathscr O}^n\oplus{\mathscr O}(1)^d)$. Under this identification we have $H_1\sim \pi^*{\mathscr O}_{{\mathbb P}^1}(1)$ and $H_2\sim {\mathscr O}_{{\mathbb P}({\mathscr E})}(1)$, where $\pi:{\mathbb P}({\mathscr E})\to {\mathbb P}^1$ is the projection map. The vector bundle ${\mathscr E}$ is generated by the $n+d$ sections $u_{ij}=x_iz_j$ for $i=0,1, j=1,\ldots, d$ and $u_1=z_{d+1}\ldots,u_{n-d}=z_n$ and these sections define an embedding of $X$ inside ${\mathbb P}^1\times {\mathbb P}^{n+d-1}$ with defining equations given by the minors of the matrix $$ \begin{pmatrix} x_0 &u_{01} & \ldots & u_{0d}\\ x_1 &u_{11} & \ldots & u_{1d}\\ {\mathbf e}nd{pmatrix}. $$From this we see that the second projection is birational and the image $Z$ is a cone over the Segre embedding of ${\mathbb P}^1\times {\mathbb P}^{d-1}$ in ${\mathbb P}^{2d-1}$. In fact, from the defining equations we find that $X$ is the blow-up of $Z$ along the ideal $(u_{01},u_{11})$. Blowing up $Z$ along the other ruling gives the other birational model $X^+$. For $d=2$ and $n=3$, this construction gives the Atiyah flop, where the variety $Z$ above is the quadric cone in ${\mathbb P}^4$ and $X\to Z$ and $X^+\to Z$ are its two small resolutions. \subseteqsection{Bidegree $(1,e)$.} If $X$ is defined by a form $f=x_0f_0+x_1f_1$, the second projection contracts the divisor $eH_2-H_1$ and realizes $X$ as a blow-up of ${\mathbb P}^n$ along the codimension 2 subvariety $Z=\{f_0=f_1=0\}$. The Cox ring of $X$ is isomorphic to $k[x_0,x_1,y_0,\ldots,y_n,z]/(zx_0+f_1,zx_1-f_0)$ and the nef cone is spanned by $H_1$ and $H_2$. On $X$ every movable divisor is nef, as the exceptional divisor is the only possible base-locus of an effective divisor. \subseteqsection{Bidegree $(2,e)$.} For hypersurfaces $X$ of bidegree $(2,e)$ in $\pn{n}$, the symmetry of the matrices $A$ and $B$ above show that the birational model $X^+$ is actually isomorphic to $X$. We explain this fact as follows. Suppose that $X$ is defined by $f=x_0^2f_0+x_0x_1f_1+x_1^2f_2=0$ in ${\mathbb P}^1\times {\mathbb P}^n$. The second projection $\operatorname{p}_2:X\to {\mathbb P}^n$ is generically 2:1, but it contracts the codimension $2$ locus given by $W=\{f_0=f_1=f_2=0\}$ which is a union of rational curves. Let $\tau:X\to Z$ be the Stein factorization of $\operatorname{p}_2$. Explicitly, $Z$ is the double cover of ${\mathbb P}^n$ branched over the divisor given by $D=\{f_1^2-4f_0f_2=0\}\subseteqset {\mathbb P}^n$. Let $\sigma:Z\to Z$ be the involution that interchanges the sheets of the double cover. $\sigma$ induces a birational pseudoautomorphism of $X$ defined outside $W$. Using this description, it is easy to show that $\sigma^*H_1+H_1=eH_2$ and $\sigma$ is the $-H_1$-flip of $\tau$. This recovers the decomposition of the movable cone from section 2. \section{Surfaces in $\pn{2}$}\label{surfaces} Let $X$ be a very general surface in ${\mathbb P}^1\times {\mathbb P}^2$ of bidegree $(d,e)$. Much of the theory from the previous sections can be used to study the birational structure of $X$, but some care must be taken because the Picard group of $X$ might be larger than ${\mathbb Z} H_1\oplus{\mathbb Z} H_2$. However, the Noether-Lefschetz theorem of \cite{RS09} says that when $X$ is very general in its linear system and $K_{\pn{2}}\otimes {\mathscr O}_{\pn{2}}(X)$ is globally generated, then we have $\Pic(X)=\Pic(\pn{2})$. This is the case if and only if $d\ge 2$ and $e\ge 3$. For the remaining cases, we can proceed by a case-by-case analysis. $d=1$. Here the situation is drastically different than that of hypersurfaces in higher dimension, because hypersurfaces of bidegree $(1,e)$ hypersurfaces are usually {\mathbf e}mph{not} Mori dream spaces. In fact, very general hypersurfaces of bidegree $(1,e)$ can be described as the blow-up of ${\mathbb P}^2$ along the $e^2$ intersection points of two very general degree $d$ curves. This is known to have infinitely many $(-1)$-curves for $e\ge 3$, so their effective cones of divisors are not rational polyhedral. In these cases the rank of the Picard group is $e^2+1$. For $e=1,2$, they are del Pezzo surfaces and hence Mori dream spaces. $d=2$. Very general hypersurfaces $X$ of bidegree $(2,e)$ in $\pn{2}$ are Mori dream spaces. Indeed, if $e=2$, $X$ is a del Pezzo surface of degree $4$, which is a Mori dream space with Picard number 6. When $e\ge 3$, the Noether-Lefschetz theorem quoted above gives that $\Pic(X)\simeq \Pic(\pn{2})$. In this case an analysis similar to that in section 2 gives that the nef cone is spanned by $H_1$ and $eH_2-H_1$ and equals the effective cone. Moreover, $\operatorname{\mathcal R}(X)$ has a presentation as in {\mathbf e}qref{coxringpres}. $e=1$. As before, $X$ is a projective bundle over ${\mathbb P}^1$, that is, $X$ is a Hirzebruch surface. When $d\ge 3$ and $e\ge 2$, a very general surface of bidegree $(d,e)$ is not a Mori dream space. We postpone the proof of this claim to the next section. \section{Non-Mori dream space hypersurfaces}\label{nonMDS} In this section we give examples of bidegree $(d,e)$ hypersurfaces in ${\mathbb P}^1\times {\mathbb P}^n$ which do not have a closed effective cone and hence are not Mori dream spaces. From the the previous sections we may restrict to the cases where $d\ge n+1$ and $e\ge 2$. Consider a `hypersurface' $C$ of bidegree $(2,2)$ in ${\mathbb P}^1\times {\mathbb P}^1$. $C$ is an elliptic curve, so $\Pic(C)$ is too big for $C$ to be a Mori dream space. However, it still makes sense to ask whether the subalgebra of $\operatorname{\mathcal R}(C)$ given by $$ \operatorname{\mathcal R}(H_1,H_2)=\bigoplus_{a,b}H^0(C,aH_1+bH_2) $$is finitely generated. It turns out that it is not, at least for $C$ very general in its linear system. Here is the reason: Pick two degree two line bundles $D_1,D_2$ on the elliptic curve $C$ such that $D_1-D_2$ represents a non-torsion point on $\Pic^0(C)$. Then $D_1$ and $D_2$ determine two morphisms $f,g:C\to {\mathbb P}^1$, hence a morphism $F=(f\times g):C\to {\mathbb P}^1\times {\mathbb P}^1$. This is an embedding and the image has bidegree $(2,2)$. However, the line bundle $L=H_2-H_1$ has no effective multiple by our choice of $D_1,D_2$. However, the line bundle $mL+H_1$ has positive degree and so {{\mathbf e}m is} effective for any $m\ge 0$. Hence $\operatorname{\mathcal R}(H_1,H_2)$ is not finitely generated. For the proof of the last part of Theorem |ef{decones}, we will use a variation on this idea. We will consider a projective bundle $Y={\mathbb P}({\mathscr E})$ of a semistable vector bundle over an elliptic curve $C$ and construct a generically finite morphism from $Y$ to a bidegree $(d,e)$-hypersurface $X_0$ in ${\mathbb P}^1\times {\mathbb P}^n$. We will do this in a way, so that the line bundle on ${\mathbb P}^1\times {\mathbb P}^n$ restricting to an extremal ray on a very general hypersurface pulls back to a line bundle with no effective multiple on $Y$. By semicontinuity, this will imply that the extremal ray of the nef cone of a general divisor is not semiample, since it has no effective multiple. Hence a very general bidegree $(d,e)$ hypersurface is not a Mori dream space. We first need the following lemma which gives a bound for the effective cone of a very general hypersurface in $\pn{n}$. It essentially says that the class $neH_2-dH_1$, which has top-self-intersection 0, is pseudoeffective. \begin{lemma}\label{subconesss} Let $X$ be a hypersurface of bidegree $(d,e)$ in ${\mathbb P}^1\times {\mathbb P}^n$. Then the ${\mathbf e}ff(X)$ contains the subcone \begin{equation}\label{constraints} {\mathbb R}_{> 0}H_1+{\mathbb R}_{>0}(neH_2-dH_1) {\mathbf e}nd{equation} {\mathbf e}nd{lemma} \begin{proof} Let $L={\mathscr O}_X(aH_1+bH_2)$ be a line bundle in {\mathbf e}qref{constraints}. We have $L^n=b^{n-1}(bd+aen)>0$. It suffices to show that $L$ is big in the case $a<0$ and $b>0$. By K\"unneth, $H^i({\mathscr O}_{\pn{n}}(-x,y))=0$ for all $i>1$ and $x,y>0$. So from the exact sequence $$0\to {\mathscr O}_{\pn{n}}(a-d,b-e)\to {\mathscr O}_{\pn{n}}(a,b) \to { L}\to 0$$we see that $H^i(Y,mL)=0$ for $i>1$ for $m$ large. Hence $h^0(X,{\mathscr O}_X(mL))\ge \chi({\mathscr O}_X(mL))=m^nL^n/n!+\ldots$ which is positive for $m$ large. Hence $L$ is big.{\mathbf e}nd{proof} The line bundles in this cone correspond exactly to the line bundles $L$ such that $L^n>0$ and $L^{n-1}\operatorname{cd}ot H_1>0$. The two divisors $H_1$ and $neH_2-dH_1$ will in fact turn out to be extremal in the effective cone, so generically there exist no divisors of negative top self-intersection on $X$. \subseteqsection{Construction of the special hypersurfaces.} Let $C$ be a smooth elliptic curve. By results of Atiyah \cite{Ati57}, $C$ has a semistable rank $r=n$ vector bundle ${\mathscr E}$ of degree $d>n$. Let $Y={\mathbb P}({\mathscr E})$ denote the variety of hyperplanes in ${\mathscr E}$ with its projection $\pi:Y\to C$. An essential point of the construction is defining two morphisms $f:Y\to {\mathbb P}^1$ and $g:Y\to {\mathbb P}^n$, so that the induced map $F=f\times g:Y\to {\mathbb P}^1\times {\mathbb P}^n$ is birational onto its image, which is a hypersurface of bidegree $(d,e)$. Choose a degree $e$ morphism $p:C\to {\mathbb P}^1$ (here we are using the fact that $e\ge 2$). Let $f:Y\to {\mathbb P}^1$ be the composition $f=p\circ \pi$. The generic fiber of $f:Y\to{\mathbb P}^1$ consists of $e$ distinct fibers of $\pi$. Let $L_1=f^*{\mathscr O}_{{\mathbb P}^1}(1)$ and $L_2={\mathscr O}_Y(1)$. Here $L_1\operatorname{cd}ot L_2^{n-1}=e$ and $L_2^n=d$. Note that the choice of the morphism $p:C\to {\mathbb P}^1$ amounts to choosing a degree $e$ line bundle on $C$ along with two global sections. In this sense, we may talk about $p$ being `general' and `very general' with respect to these data. \begin{lemma} The line bundles $L_1$ and $L_2$ are base-point free and $L_2$ is ample. {\mathbf e}nd{lemma} \begin{proof} $L_1$ is the pullback of a base-point free divisor on $C$. $L_2$ is base-point free because ${\mathscr E}$ is generated by sections: Indeed, this is true for any semistable vector bundle of degree $d> r(2g(C)-1)=n$. When ${\mathscr E}$ is semistable, any effective divisor on ${\mathbb P}({\mathscr E})$ is nef \cite[1.5.A]{Laz04}. Moreover, $L_2$ is big, since it is nef and $L_2^2=d$, so it lies in the interior of the nef cone and hence is ample. {\mathbf e}nd{proof} When ${\mathscr E}$ is semistable of degree $d> n(2g-1)=n$, we have $h^1(C,{\mathscr E})=0$ and so by Riemann-Roch, $h^0(C,{\mathscr E})=d$. We are assuming that $d\ge n+1$, so a choice of $n+1$ generic sections of $L_2$ defines a finite morphism $g:Y\to {\mathbb P}^n$ of degree $d$. \begin{lemma}\label{imagebidegree} For $p:C\to {\mathbb P}^1$ general, the image $X_0$ of the morphism $F=f\times g:Y\to {\mathbb P}^1\times {\mathbb P}^n$ is a hypersurface of bidegree $(d,e)$. {\mathbf e}nd{lemma} \begin{proof} First of all, $F$ is finite, since $g$ is finite, and so the image is a (possibly singular) hypersurface in ${\mathbb P}^1\times {\mathbb P}^n$. We will show that $F$ has degree one below. Granting this for the moment, it follows that the image $X_0$ has bidegree $(d,e)$. Indeed, note that the projections $\operatorname{p}_1:Y\to {\mathbb P}^1$ and $\operatorname{p}_2:Y\to {\mathbb P}^n$ factor through $f\times g$ and determine the bidegree uniquely: If $X_0$ has bidegree $(a,b)$, we have $a=X_0\operatorname{cd}ot \operatorname{p}_2^*{\mathscr O}_{{\mathbb P}^n}(1)^n=L_2^n=d$ and $b=X_0\operatorname{cd}ot \operatorname{p}_2^*{\mathscr O}_{{\mathbb P}^1}(1)\operatorname{cd}ot \operatorname{p}_2^*{\mathscr O}_{{\mathbb P}^n}(1)^{n-1}=L_1\operatorname{cd}ot L_2^{n-1}=e$. We now show that $F$ is birational onto its image. Recall that the generic fiber of $f$ consists of $e$ disjoint fibers of $\pi$ and $g$ is finite of degree $d$. Let $y^{-1}n Y$ be a general point and let $y'^{-1}n Y$ be a point so that $y\neq y'$ and $g(y)=g(y')$. First we note that the $y$ and $y'$ lie in different fibers of $\pi$; this is because generically the preimage $g^{-1}(l)$ of a line $l\subseteqset {\mathbb P}^n$ through $g(y)$ is a section of $\pi$ (because $L_2^{n-1}\operatorname{cd}ot \pi^{*}{\mathscr O}_C(p)=1$ for a fiber over $p^{-1}n C$). Suppose now that $y'\neq y$ is a point in $Y$ so that $F(y)=F(y')$. By the above, we must have $\pi(y)\neq \pi(y')$. Now, we are choosing the degree $e$ map $p:C\to {\mathbb P}^1$ generically, so we may assume that $\pi(y)$ and $\pi(y')$ map to different points on ${\mathbb P}^1$. But $f=p\circ \pi$, so $y$ and $y'$ are separated by $f$, and consequently by $F$, a contradiction. In particular, $F$ is injective in a neighbourhood of $y$, and so it is birational onto its image. {\mathbf e}nd{proof} \begin{lemma}\label{noeffectivemultiple} If the morphism $p:C\to {\mathbb P}^1$ is very general, the divisor $D=enL_2-dL_1$ on $Y={\mathbb P}({\mathscr E})$ does not have a positive integral multiple which is effective. {\mathbf e}nd{lemma} \begin{proof} We need to show that for each $m>0$, $$ H^0(Y,{\mathscr O}_Y(mD))=H^0(C,\pi_*{\mathscr O}_Y(mD))=H^0(C,S^{enm}{\mathscr E}\otimes {\mathscr O}_C(-dm p^*{\mathscr O}_{{\mathbb P}^1}(1)))=0. $$Note that $S^{emn}{\mathscr E}$ is a semistable vector bundle of rank $r={emn+n-1 \choose n-1}$ and degree $edmr$ and so $S^{emn}{\mathscr E}\otimes {\mathscr O}_C(-dm p^*{\mathscr O}_{{\mathbb P}^1}(1)))$ is semistable of degree 0. If $p$ is very general, then these vector bundles do not have any global sections by the lemma below. {\mathbf e}nd{proof} \begin{lemma}\label{nosections} Let ${\mathscr E}$ be a semistable vector bundle of rank $r$ and degree $0$ on a curve of positive genus. Then for a line bundle $L$ defining a general point in $Pic^0(C)$, we have $H^0({\mathscr E}\otimes L)=0$. {\mathbf e}nd{lemma} \begin{proof} If ${\mathscr E}$ is a line bundle, then the statement holds, since the only effective line bundle of degree 0 is the trivial bundle and $\Pic^0(C)$ has dimension $>0$. So we may assume that rank ${\mathscr E}\ge 2$. The statement is also true if ${\mathscr E}$ is stable, because in that case so is ${\mathscr E}\otimes L$, and if ${\mathscr E} \otimes L$ has a section, then ${\mathscr O}$ is a subsheaf, contradicting the stability condition. If ${\mathscr E}$ is strictly semistable, then the Jordan-H\"older filtration says that there is a semistable subbundle ${\mathscr E}'\subseteqset {\mathscr E}$ such that ${\mathscr E}/{\mathscr E}'$ is a stable vector bundle and both ${\mathscr E}$ and ${\mathscr E}/{\mathscr E}'$ have the same slope as ${\mathscr E}$ (that is, 0). From this we get an exact sequence of degree $0$ vector bundles $$ 0\to {\mathscr E}'\to {\mathscr E}\to {\mathscr E}/{\mathscr E}'\to 0. $$Tensoring this with $L$ and taking cohomology, the result follows by induction on the rank. {\mathbf e}nd{proof} We are ready to prove the main theorem of this section: \begin{theorem} Let $X$ be a very general hypersurface of $ {\mathbb P}^1\times {\mathbb P}^n$ of bidegree $(d,e)$ with $d\ge n+1$ and $e\ge 2$. Then the effective cone of $X$ is not closed. In particular, $X$ is not a Mori dream space. {\mathbf e}nd{theorem} \begin{proof} To prove this it is sufficient by semi-continuity of $\dim H^0$ to exhibit a single hypersurface $X_0$ of bidegree $(d,e)$ such that no multiple of the line bundle $L:={\mathscr O}_{\pn{n}}(-d,ne)$ restricts to an effective divisor on $X_0$; then the same conclusion holds for a very general deformation of it. Since this line bundle is pseudoeffective on any hypersurface, the result follows. We will let $X_0$ be the image of $Y={\mathbb P}({\mathscr E})$ under the morphism $F=f\times g$ defined earlier. By construction, the image $X_0$ is a hypersurface of bidegree $(d,e)$ such that the line bundle $F^*\left(L|_{X_0}|ight)={\mathscr O}_Y(enL_2-dL_1)$ has no effective multiple on $Y$ (Lemma |ef{noeffectivemultiple}). Note that ${X_0}$ is reduced (although it may be singular), hence the natural map ${\mathscr O}_{X_0}\to F_*{\mathscr O}_{Y}$ is injective, and we have for $m\ge 1$ \[ H^0({X_0},\left(mL|_{X_0}|ight))\subseteqseteq H^0(X_0,\left(m{ L}|_{X_0}|ight)\otimes F_*{\mathscr O}_Y) = H^0(Y,F^*\left(mL|_{X_0}|ight))=0\]Hence no multiple of $L$ is effective on ${X_0}$ and the proof is complete. {\mathbf e}nd{proof} \begin{corollary} Let $X$ be a very general hypersurface $X\subseteqset {\mathbb P}^1\times {\mathbb P}^n$ of bidegree $(d,e)$ with $d\ge n+1$ and $e\ge 2$ (and $d,e\ge 3$ in the case $n=2$). Then \begin{equation}\label{effectiveconesss} \overline {\mathbf e}ff(X)=\overline \operatorname{Mov}(X)=\operatorname{Nef}(X)={\mathbb R}_{\ge 0}H_1+{\mathbb R}_{\ge 0}(neH_2-dH_1). {\mathbf e}nd{equation} {\mathbf e}nd{corollary} \begin{proof} By Lemma |ef{subconesss} we have that the pseudoeffective cone contains the cone on the right hand side of {\mathbf e}qref{effectiveconesss}, and so two cones coincide by the theorem. Moreover, $neH_2-dH_1$ is nef on the special hypersurface $X_0$ used in the proof of the theorem, since it pulls back to a nef divisor on ${\mathbb P}({\mathscr E})$ via a finite surjective morphism. Moreover, on $X_0$, the pseudoeffective cone and the nef cone coincide, so by the argument of \cite[Lemma 4.1]{Mou12}, the same conclusion holds for a very general deformation $X$ of $X_0$. {\mathbf e}nd{proof} \begin{example}In dimension 2, $(3,2)$ is the first bidegree for which a very general hypersurface is not a Mori dream space. This variety is rational surface that is isomorphic to a blow-up of a Hirzebruch surface in 9 general points, so the Picard number is in fact 11. {\mathbf e}nd{example} \begin{example}\label{Mumford} By the Noether-Lefschetz theorem, a very general surface of bidegree $(3,3)$ in ${\mathbb P}^1\times {\mathbb P}^2$ has Picard number 2. By the theorem, the line bundle $L=2H_2-H_1$ is nef, but is not semiample. In fact, $L$ is {\mathbf e}mph{strictly nef}, in the sense that $\deg L|_C>0$ for every curve $C$ (since $L\operatorname{cd}ot C=0$ implies $C\sim_{\mathbb Q} L$, and $L$ is not ${\mathbb Q}$-linearly equivalent to an effective divisor). This gives a simple counterpart of Mumford's example mentioned in the introduction. {\mathbf e}nd{example} \begin{example} A very general hypersurface of bidegree $(4,2)$ in ${\mathbb P}^1\times {\mathbb P}^3$ can be viewed as a quadric surface bundle over ${\mathbb P}^1$. It is therefore a rational threefold with Picard number 2, which is not a Mori dream space. {\mathbf e}nd{example} \begin{remark}[Relation to a conjecture of Keel] Consider a surface $S$ defined over the field $k=\overline{\mathbb F_p}$ and a line bundle $L$ on $S$. In \cite{keel} Keel posed the problem whether $L$ pseudoeffective implies that it is ${\mathbb Q}$-effective, that is, that some multiple of $L$ has a section. This is of course false over ${\mathbb C}$, as we have seen. However, the proof using projective bundles over an elliptic curve fails over $\overline{\mathbb F_p}$, since every degree 0 line bundle is in fact torsion on $\Pic^0(E)$ (and thus is ${\mathbb Q}$-effective). This raises the question \noindent {\bf Question. }Let $k=\overline{\mathbb F_p}$ and consider a smooth hypersurface $S$ in ${\mathbb P}_k^1\times {\mathbb P}_k^2$ of large bidegree. Does every pseudoeffective line bundle on $S$ have an effective multiple? {\mathbf e}nd{remark} \section{Hypersurfaces in products of several projective spaces} It is not surprising that the picture does not become simpler when considering high-degree hypersurfaces in products of more projective spaces. For example, a variation of the previous argument using projective bundles over elliptic curves produces non-Mori dream hypersurfaces multidegree $(d_1,\ldots,d_k,e)$ in $({\mathbb P}^1)^k\times {\mathbb P}^r$ for $d_i\ge r+1,e\ge 2$. In this section, we remark that for hypersurfaces $X$ in products of projective spaces with more than one ${\mathbb P}^1$-factor, the situation becomes even more complicated. In particular, we don't expect Mori dream spaces, even for low degree hypersurfaces. The following example, which appears in the work of Kawamata \cite[Example 3.8]{Kaw97}, illustrates this already for a Calabi-Yau threefold. Consider a smooth hypersurface of tridegree $(2,2,3)$ in ${\mathbb P}={\mathbb P}^1\times {\mathbb P}^1\times {\mathbb P}^2$ defined by an equation $$f(x_i,y_i,z_i)=x_0^2f_0+x_0x_1f_1+x_1^2f_2$$ where $f_0,f_1,f_2$ are forms of tridegree $(0,2,3)$. The projection $(\operatorname{p}_1\times \operatorname{p}_3):X\to {\mathbb P}^1\times {\mathbb P}^2$ contracts the codimension 2 locus $W=\{f_0=f_1=f_2=0\}$ which is a union of 54 rational curves. Taking the Stein factorization gives a small contraction $\phi:X\to Z$ where $Z$ is the double cover of ${\mathbb P}^1\times {\mathbb P}^2$ ramified over the divisor defined by $f_1^2-4f_0f_2=0$. Note that $Z$ has a natural involution $\sigma':Z\to Z$, which switches the sheets of the covering. This determines a birational pseudoautomorphism $\sigma: X\dashrightarrow X$ defined outside $W$. In terms of $H_1,H_2,H_3$ it is not hard to show that $\sigma^*H_1+H_1=2H_2+3H_3$ and that $\sigma:X\dashrightarrow X$ is the $(-H_1)$-flip of $\phi$. One can repeat the argument with the other contraction $(\operatorname{p}_2\times \operatorname{p}_3):X\to \pn{2}$ to get another pseudoautomorphism $\sigma'$ of $X$. Moreover, $\sigma$ and $\sigma'$ generate an infinite subgroup of the group of pseudoautomorphisms of $X$, $\PsAut(X)$. In fact, also the group $\PsAut(X)^*=\mbox{im}(\PsAut(X)\to GL(N^1(X))$ is infinite. In particular, $X$ is not a Mori dream space, because there are infinitely many non-isomorphic marked small ${\mathbb Q}$-factorial modifications. Using essentially the same method, one can show the following result: \begin{proposition}\label{CY} Let $X$ be a smooth Calabi-Yau hypersurface of dimension $\ge 3$ in ${\mathbb P}=({\mathbb P}^1)^m \times {\mathbb P}^{n_1}\times \operatorname{cd}ots \times {\mathbb P}^{n_k}$ where $n_1,\ldots,n_k\ge 2$ and let $H_i=\operatorname{p}_i^*{\mathscr O}(1)$. Then the nef cone is given by $\operatorname{Nef}(X)={\mathbb R}_{\ge 0} H_1+{\mathbb R}_{\ge 0} H_2+\ldots+{\mathbb R}_{\ge 0} H_{k+m}$. Moreover, the following hold: \begin{enumerate}[(i)] ^{-1}tem If $m=0$, then ${\mathbf e}ff(X)=\operatorname{Mov}(X)=\operatorname{Nef}(X)={\mathbf e}ff({\mathbb P})$ and $X$ is a Mori dream space. ^{-1}tem If $m=1$, then the effective cone is strictly larger than that of ${\mathbb P}$, and $X$ is a Mori dream space. ^{-1}tem If $m>1$, then $X$ is not a Mori dream space. In fact the group $\PsAut(X)^*$ is infinite and the movable cone is not rational polyhedral. {\mathbf e}nd{enumerate} {\mathbf e}nd{proposition} \begin{proof} The description of the nef cone follows from a more general theorem of Koll\'ar \cite{Bor89}, which states that if $Y$ is a smooth Fano variety of dimension at least $4$ and $X^{-1}n |-K_Y|$ is a smooth divisor, then the inclusion induces an isomorphism of the cones of curves $i_*: \overline{NE}_1(X)\to \overline{NE}_1(Y)$. Taking duals gives that the nef cone is the first quadrant $\sum_{j}{\mathbb R}_{\ge 0}H_j$ in $N^1(X)$. The case $(i)$ follows as in the first part of Theorem 1.1. For $(ii)$, let $\sigma: X\dasharrow X$ be the pseudoautomorphism of $X$ obtained by viewing $X$ as a double cover of ${\mathbb P}^{n_1}\times \operatorname{cd}ots \times {\mathbb P}^{n_k}$. It is not hard to show that any divisor $D$ lying in the `coordinate planes' ${\mathbf e}_i^\perp=\sum_{j\neq i}{\mathbb R}_{\ge 0}H_j$ is not big for any $i>1$. This follows for example by taking cohomology of the ideal sheaf sequence $0\to {\mathscr O}_{\mathbb P}(D-X)\to {\mathscr O}_{\mathbb P}(D)\to {\mathscr O}_X(D)\to 0$ and using the vanishing of higher cohomology of line bundles on ${\mathbb P}$. Hence the only supporting hyperplane of $\operatorname{Nef}(X)$ containing big divisors is $e_1^\perp$. This hyperplane is fixed under the involution $\sigma^*$, and so by applying $\sigma^*$ to $\operatorname{Nef}(X)$, we see that any divisor in the boundary of $\operatorname{Nef}(X)\cup\sigma^*\operatorname{Nef}(X)$ is not big. Hence ${\mathbf e}ff(X)=\operatorname{Mov}(X)=\operatorname{Nef}(X)\cup\sigma^*\operatorname{Nef}(X)$ and so $X$ is a Mori dream space. $(iii)$ follows as in the example above, noting that two ${\mathbb P}^1$-factors give rise to two pseudoautomorphisms $\sigma,\sigma'$ generating an infinite subgroup of $\PsAut(X)^*$. {\mathbf e}nd{proof} In \cite{CO11}, Cantat and Oguiso give a detailed description of the cones of effective, movable, and nef divisors on hypersurfaces in $({\mathbb P}^1)^m$ of multidegree $(2,\ldots,2)$. In particular, they verify the Morrison-Kawamata cone conjecture for these hypersurfaces, which means that even though the movable cone itself fails to be polyhedral, it has a rational polyhedral fundamental domain under the action of $\PsAut(X)$ on $N^1(X)$. In fact, they show that $\PsAut(X)\simeq {\mathbb Z}/2{\mathbb Z} *\operatorname{cd}ots *{\mathbb Z}/2{\mathbb Z}$, generated by the birational involutions above and that the fundamental domain is in fact the nef cone of $X$. It is likely that these statements generalize to the hypersurfaces in Proposition |ef{CY}. \begin{thebibliography}{1} { \bibitem{ahm13} H. Ahmadinezhad. On pliabillity of del Pezzo fibrations and Cox rings. arXiv:1304.4357, 2013. \bibitem{CoxBook} I. Arzhantsev, U. Derenthal, J. Hausen and A. Laface. {{\mathbf e}m Cox rings}. Cambridge University Press, 2014. \bibitem{AL12} M. Artebani, A. Laface. Hypersurfaces in Mori dream spaces. {{\mathbf e}m J. Algebra 371} (2012), 26--37. \bibitem{Ati57} M. F. Atiyah. 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On the ample cone of an ample hypersurface. {{\mathbf e}m Asian Journal of Mathematics} 7 (2003), 1--6. } {\mathbf e}nd{thebibliography} \textsc{Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK} {{^{-1}t Email:} ^{\vee}erb"J.C.Ottem@dpmms.cam.ac.uk"} {\mathbf e}nd{document}

\begin{document} \begin{abstract} In this paper we are concerned with finite soluble groups $G$ admitting a factorisation $G=AB$, with $A$ and $B$ proper subgroups having coprime order. We are interested in bounding the Fitting height of $G$ in terms of some group-invariants of $A$ and $B$: including the Fitting heights and the derived lengths. \end{abstract} \maketitle \section{Introduction}\label{intro} In this paper, all groups considered are finite and soluble, and hence the word ``group'' should always be understood as ``finite soluble group''. We investigate groups $G$ in which a \textit{factorisation} $$G=AB=\{ab\mid a\in A,\,b\in B\}$$ with $A$ and $B$ subgroups of $G$ of coprime order is given. We are interested in obtaining some upper bounds on the \textit{Fitting height} $h(G)$ of $G$, in terms of the Fitting heights ($h(A)$ and $h(B)$) and of the \textit{derived lengths} ($d(A)$ and $d(B)$) of $A$ and $B$. (Our notation is standard, see Section~\ref{1.1} for undefined terminology.) \begin{theorem}\label{thrmA} Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. If $|B|$ is odd, then \begin{equation}\label{eq:1} h(G)\leq h(A)+h(B)+2d(B)-1. \end{equation} If $B$ is nilpotent, then \begin{equation}\label{eq:2} h(G)\leq h(A)+2d(B). \end{equation} \end{theorem} Before continuing with our discussion we need to introduce some notation. Given a group $G$, we write \begin{equation*} \delta(G):=\max\{d(S)\mid S \textrm{ Sylow subgroup of }G\}, \end{equation*} that is, $\delta(G)$ is the maximal derived length of the Sylow subgroups of $G$. We also bound the Fitting height of $G$ in terms of the group-invariants $\delta(A)$ and $\delta(B)$. \begin{theorem}\label{thrmB} Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. Then \begin{equation*} h(G)\leq h(A)+(2\delta(B)+1)h(B)-1. \end{equation*} \end{theorem} Both Theorems~\ref{thrmA} and~\ref{thrmB} extend and generalise some well-known results on groups admitting a factorisation with subgroups of coprime order, see for example the two monographs~\cite[Chapter~$2$]{AFD} and~\cite[pages~$133$--$135$]{BB}. Observe that when $A$ and $B$ are both nilpotent, we have $h(A)=h(B)=1$ and the inequality in Theorem~\ref{thrmB} specialises to the inequality of the main result in~\cite{Gemma}. When $B$ is nilpotent, we have $\delta(B)=d(B)$ and $h(B)=1$, and thus Theorem~\ref{thrmA}~\eqref{eq:2} follows immediately from Theorem~\ref{thrmB}. The hypothesis of $|B|$ being odd in Theorem~\ref{thrmA}~\eqref{eq:1} is important in our proof because at a critical juncture we apply a remarkable theorem of Kazarin~\cite{Ka} (which requires $B$ having odd order). However, we believe that our hypothesis is only factitious and in fact we pose the following: \begin{conjecture} Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. Then $$ h(G)\leq h(A)+h(B)+2d(B)-1.$$ \end{conjecture} We also prove: \begin{theorem}\label{thrmC} Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. Then \begin{equation*} h(G)\leq h(A)\delta(A)+h(B)\delta(B). \end{equation*} \end{theorem} Finally, with an immediate application of Theorem~\ref{thrmA} and of the machinery developed in Section~\ref{section3}, we prove: \begin{corollary}\label{corcor} Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. For each $p\in \pi(B)$, let $B_p$ be a Sylow $p$-subgroup of $B$. Then $$h(G)\leq h(A)+2\sum_{p\in\pi(B)}d(B_p).$$ In particular, $h(G)\leq h(A)+2|\pi(B)|\delta(B)$. \end{corollary} In Section~\ref{1.1} we introduce some basic notation and some preliminary results that we use throughout the whole paper. In Section~\ref{section3} we present our main tool (the \textit{towers} as defined by Turull~\cite{Tu}) and we prove some auxiliary results. Section~\ref{section4} is dedicated to the proof of Theorems~\ref{thrmA} and~\ref{thrmB} and of Corollary~\ref{corcor}. The proof of Theorem~\ref{thrmC} (which requires a slightly different machinery) is postponed to Section~\ref{section5}. \section{Notation and preliminary results}\label{1.1} Given a group $G$, we denote by $\F G$ the \textit{Fitting subgroup} of $G$ (that is, the largest normal nilpotent subgroup of $G$). Moreover, the Fitting series of $G$ is defined inductively by $\FF 0 G:=1$ and $\FF {i+1}G/\FF i G:=\F{G/\FF i G}$, for every $i\ge 0$. Clearly, $\FF i G<\FF {i+1} G$ when $\FF i G<G$, and the minimum natural number $h$ with $\FF h G=G$ is called the \textit{Fitting height} (or Fitting length) of $G$ and is denoted by $h(G)$. Similarly, the \textit{derived length} of $G$ is indicated by $d(G)$. We let $|G|$ denote the order of $G$ and we let $\pi(G)$ denote the set of prime divisors of $|G|$. Given a prime number $p$, we write $G_p$ for a Sylow $p$-subgroup of $G$. A \textit{Sylow basis} of $G$ is a family $\{G_p\}_{p\in \pi(G)}$ of Sylow subgroups of $G$ such that $G_{p}G_q=G_qG_p$ for any $p,q\in \pi(G)$. By a pioneering result of Philip Hall~\cite[$9.1.7$,~$9.1.8$ and $9.2.1$~(ii)]{Rob}, every (finite soluble) group has a Sylow basis. In particular, for every set of primes $\pi$, $G$ contains a Hall $\pi$-subgroup, which will be denoted by $G_{\pi}$. Given a set $\pi$ of prime numbers, we set $\pi':=\{p\textrm{ prime}\mid p\notin \pi\}$. Moreover, when $\pi=\{p\}$, for simplicity we write $p'$ for $\pi'$. As usual, $\O \pi G$ is the largest normal $\pi$-subgroup of $G$ and the upper $\pi'\pi$-\textit{series} of $G$ is generated by applying ${\bf O}_{\pi'}$ and ${\bf O}_\pi$ (in this order) repeatedly to $G$, that is, the series $1=P_0\leq N_0\leq P_1\leq N_1\leq \cdots \leq P_i\leq N_i\leq \cdots $ defined by \[ N_i/P_i:=\O{\pi'}{G/P_i}\quad\textrm{and}\quad P_{i+1}/N_i:=\O \pi {G/N_i}. \] This is a series of characteristic subgroups having factor groups $\pi'$- and $\pi$-groups, alternately. The minimum natural number $\ell$ such that the $\pi'\pi$-series terminates is named the $\pi$-\textit{length} of $G$ and denoted by $\ell_\pi(G)$. When $\pi=\{p\}$, we write simply $\O p G$ and $\ell_p(G)$. We first state a basic elementary result which will be used repeatedly and without comment. \begin{lemma}\label{lemma:2.1} Let $G=AB$ be a group factorised by $A$ and $B$ with $\gcd(|A|,|B|)=1$. Then there exists a Sylow basis $\{G_p\}_{p\in \pi(G)}$ with $A=\prod_{p\in \pi(A)}G_p$ and $B=\prod_{p\in \pi(B)}G_p$. \end{lemma} \begin{proof} From~\cite[Lemma~$1.3.2$]{AFD}, we see that for every $p\in \pi(G)$ there exists a Hall $p'$-subgroup $A_{p'}$ of $A$ and a Hall $p'$-subgroup $B_{p'}$ of $B$ such that $A_{p'}B_{p'}$ is a Hall $p'$-subgroup of $G$. Now, for each $p\in \pi(G)$, define $G_p:=\bigcap_{q\in \pi(G)\setminus\{p\}}A_{q'}B_{q'}$. A computation shows that $\{G_p\}_{p\in \pi(G)}$ is a Sylow basis of $G$ (see for example~\cite[$9.2.1$]{Rob}). Moreover, $A=\prod_{p\in \pi(A)}G_p$ and $B=\prod_{p\in \pi(B)}G_p$. \end{proof} The next two results are crucial for our proofs of Theorems~\ref{thrmA} and~\ref{thrmB}. \begin{theorem}\label{thrm2.2} Let $G$ be a group and let $p$ be a prime. Then $\ell_p(G)\leq d(G_p)$. \end{theorem} \begin{proof} When $p$ is odd, this is~\cite[Theorem~$A$~(i)]{HH}. The analogous result for $p=2$ is proved in~\cite{Br}. \end{proof} Kazarin~\cite{Ka} has proved Theorem~\ref{thrm2.2} for arbitrary sets of primes $\pi$ with $2\notin \pi$. We state this generalisation in a form tailored to our needs. \begin{theorem}\label{thrm2.3} Let $G$ be a group and let $\pi$ be a set of primes. If $2\notin \pi$ or if $G_\pi$ is nilpotent, then $\ell_\pi(G)\leq d(G_\pi)$. \end{theorem} \begin{proof} When $2\notin \pi$, this is the main result of~\cite{Ka} (see also~\cite[Theorem~$1.7.20$]{BB}). When $G_\pi$ is nilpotent, the proof follows from Theorem~\ref{thrm2.2}. \end{proof} \section{Our toolkit: towers}\label{section3} We start this section with a pivotal definition introduced by Turull~\cite{Tu}. (The definition of $B$-\textit{tower} in~\cite[Definition~$1.1$]{Tu} is actually more general then the one we give here and coincides with ours when $B=1$.) \begin{definition}\label{Ttower}{\rm Let $G$ be a group. A family $\mathfrak{T} := (P_i\mid i\in \{1,\ldots,h\})$ is said to be a \textit{tower of length} $h$ of $G$ if the following are satisfied. \begin{enumerate} \item $P_i$ is a $p_i$-subgroup of $G$ and $p_i\in \pi(G)$. \item If $1\leq i\leq j\leq h$, then $P_i$ normalises $P_j$. \item Define inductively $\overline{P_h}:=P_h$, and $\overline{P_i}:=P_i/\cent {P_i}{\overline{P_{i+1}}}$ for $i\in \{1,\ldots,h-1\}$. Then $\overline{P_i}\neq 1$, for every $i\in \{1,\ldots,h\}$. \item $p_i\neq p_{i+1}$, for every $i\in\{1,\ldots,h-1\}$. \end{enumerate} } \end{definition} A concept that resembles the definition of tower was orinigally introduced by Dade in~\cite{Dade} for investigating the Fitting height of a group. The relationship between Fitting height and towers was uncovered by Turull. \begin{lemma}[{{\cite[Lemma~$1.9$]{Tu}}}]\label{lemma31}Let $G$ be a group. Then \[h(G)=\max\{h\mid G \textrm{ admits a tower of length }h\}.\] \end{lemma} In view of Lemma~\ref{lemma31} we give the following: \begin{definition}\label{Fittingtower}{\rm We say that the tower $\mathfrak{T}$ of $G$ a \textit{Fitting tower} if $\mathfrak{T}$ has length $h(G)$.} \end{definition} The following is an easy consequence of~\cite[Lemma~$1.5$]{Tu}. For simplifying the notation, given a $p$-group $P$, we write $\pi^*(P)=p$ when $P\neq 1$, and $\pi^*(P)=1$ when $P=1$. Observe that when $P\neq 1$ we have $\pi(P)=\{\pi^*(P)\}$. \begin{lemma}\label{lemma33} Let $G$ be a group, let $\mathfrak{T}=(P_i\mid i\in \{1,\ldots,h\})$ be a tower of $G$, let $j\in \{1,\ldots,h\}$, let $s\geq 0$ be an integer and let $\mathfrak{T}'=(P_i\mid i\in \{1,\ldots,h\}\setminus\{j,j+1,\ldots,j+s-1,j+s\})$. Then either $\mathfrak{T}'$ is a tower of $G$, or $1<j\leq j+s<h$ and $\pi^*(P_{j-1})=\pi^*(P_{j+s+1})$. \end{lemma} \begin{proof} Lemma~$1.5$ in~\cite{Tu} says that, for every $h_0$ with $1\leq h_0\leq h$ and for every increasing function $f:\{1,\ldots,h_0\}\to \{1,\ldots,h\}$, the family $(P_{f(i)}\mid i\in \{1,\ldots,h_0\})$ satisfies the conditions ~$(1)$,~$(2)$ and~$(3)$ in Definition~$\ref{Ttower}$. Applying this with $h_0:=h-s-1$ and with $f:\{1,\ldots,h_0\}\to \{1,\ldots,h\}$ defined by \[ f(i)=\begin{cases} i&\textrm{if }1\leq i< j,\\ i+s+1&\textrm{if }j\leq i\leq h_0, \end{cases} \] we obtain that $\mathfrak{T}'$ satisfies the conditions~$(1)$,~$(2)$ and~$(3)$ of Definition~\ref{Ttower}. As $\mathfrak{T}$ satisfies Definition~\ref{Ttower}~$(4)$, we immediately get that either $\mathfrak{T'}$ satisfies also~$(4)$ (and hence is a tower of $G$), or $1<j\leq j+s<h$ and $\pi^*(P_{j-1})=\pi^*(P_{j+s+1})$. \end{proof} \begin{definition}{\rm Let $G$ be a group, let $\mathfrak{T}=(P_i\mid i\in \{1,\ldots,h\})$ be a tower of $G$ and let $\sigma$ be a set of primes. We set $$\nu_\sigma(\mathfrak{T}):=|\{i\in \{1,\ldots,h\}\mid \pi^*(P_i)\in \sigma\}|.$$ Clearly, $\nu_\sigma(\mathfrak{T})=0$ when $\sigma$ has no element in common with $\{\pi^*(P_1),\ldots,\pi^*(P_h)\}$. Now, set $P_0:=1$ and $P_{h+1}:=1$. For $i,j\in \{1,\ldots,h\}$ with $i\leq j$, the sequence $(P_\ell\mid i\leq \ell\leq j)$ of consecutive elements of $\mathfrak{T}$ is said to be a $\sigma$-\textit{block} if \begin{itemize} \item $\pi^*(P_{i+s})\in \sigma$ for every $s$ with $0\leq s\leq j-i$, and \item $\pi^*(P_{i-1})\notin\sigma$, $\pi^*(P_{j+1})\notin \sigma$. \end{itemize} Moreover, we denote by $\beta_\sigma(\mathfrak{T})$ the number of $\sigma$-blocks of $\mathfrak{T}$.} \end{definition} The main result of this section is Lemma~\ref{lemma34}: before proceeding to its proof we single out two basic observations. \begin{lemma}\label{elementary} Let $\mathfrak{T}=(P_i\mid i\in \{1,\ldots,h\})$ be a tower of $G$. Then, for $j\in \{1,\ldots,h-1\}$, we have $\cent {P_j}{P_h}\le \cent {P_j}{\overline{P_{j+1}}}$. \end{lemma} \begin{proof} We argue by induction on $h-j$. If $j=h-1$, then $\overline{P_h}=P_h$ and hence there is nothing to prove. Suppose $h-j>1$ and set $R:=\cent {P_j}{P_h}$. We have $[R,P_h,P_{j+1}]=1$, and also $[P_h,P_{j+1},R]\leq [P_h,R]=1$ by Definition~\ref{Ttower}~$(2)$. Thus the Three Subgroups Lemma yields $[P_{j+1},R,P_h]=1$, that is, $[P_{j+1},R]\leq \cent {P_{j+1}}{P_h}$. Now the inductive hypothesis gives $[P_{j+1},R]\leq \cent {P_{j+1}}{\overline{P_{j+2}}}$, and hence $[\overline{P_{j+1}},R]=1$. Therefore $\cent {P_j}{P_h}=R\leq \cent {P_j}{\overline{P_{j+1}}}$. \end{proof} \begin{lemma}\label{elementaryy} Let $\mathfrak{T}=(P_i\mid i\in \{1,\ldots,h\})$ be a tower of $G$ and let $N$ be a normal subgroup of $G$ with \begin{equation}\label{eq1} P_j\cap N\leq \cent {P_j}{P_h}, \end{equation} for every $j\in \{1,\ldots,h-1\}$. Then $\mathfrak{T}':=(P_iN/N\mid i\in 1,\ldots,h-1\})$ is a tower of $G/N$. \end{lemma} \begin{proof} From~\eqref{eq1} and Lemma~\ref{elementary}, we have $P_j\cap N\leq \cent {P_j}{\overline{P_{j+1}}}$ for $j<h$. Set $R_h:=1$, and set $R_j:=\cent {P_j}{\overline{P_{j+1}}}$ for $j<h$. Thus $\overline{P_j}=P_j/R_j$, for every $j$. Now, for $j<h$, we have $$ P_j\cap N= R_j\cap N$$ and hence \begin{equation}\label{Pjbar} \frac{P_jN}{R_jN}=\frac{P_j(R_jN)}{R_jN}\cong \frac{P_j}{P_j\cap R_jN}=\frac{P_j}{R_j(P_j\cap N)}=\frac{P_j}{R_j(R_j\cap N)}=\frac{P_j}{R_j}=\overline{P_j}. \end{equation} For each $j\in \{1,\ldots,h-1\}$, set $Q_j:=P_jN/N$, and define $\overline{Q_{h-1}}:=Q_{h-1}$, and $\overline{Q_j}:=Q_j/\cent {Q_j}{\overline{Q_{j+1}}}$ for $j<h-1$. In particular, for each $j\in \{1,\ldots,h-2\}$, there exists $L_j\leq P_j$ with $\cent {Q_j}{\overline{Q_{j+1}}}=L_jN/N$. Moreover, set $L_{h-1}:=1$. We show (by induction on $h-j$) that $L_j\leq R_j$, for each $j\in \{1,\ldots,h-1\}$. If $h-j=1$, then $L_j=L_{h-1}=1\leq R_{h-1}=R_j$. Assume then that $h-j>1$ and let $x\in L_{j}$. As $[xN,\overline{Q_{j+1}}]=1$, we get $[xN,Q_{j+1}]\leq \cent {Q_{j+1}}{\overline{Q_{j+2}}}=L_{j+1}N/N$ when $h-j>2$, and $[xN,Q_{j+1}]=1$ when $h-j=2$. In both cases, applying the inductive hypothesis, we obtain $$[x,Q_{j+1}]\leq \frac{L_{j+1}N}{N}\leq \frac{R_{j+1}N}{N}.$$ This gives $$[x,P_{j+1}]\leq P_{j+1}\cap R_{j+1}N=R_{j+1}(P_{j+1}\cap N).$$ Combining~\eqref{eq1}, Lemma~\ref{elementary} and the definition of $R_{j+1}$, we have $P_{j+1}\cap N\leq \cent {P_{j+1}}{P_h}\leq \cent {P_{j+1}}{\overline{P_{j+2}}}=R_{j+1}$. Therefore $[x,P_{j+1}]\leq R_{j+1}$ and hence $x\in \cent {P_{j}}{P_{j+1}/R_{j+1}}=\cent {P_j}{\overline{P_{j+1}}}=R_j$. Thus $L_j\leq R_j$ and the induction is proved. Observe that \begin{equation}\label{Qjbar} \overline{Q_j}=\frac{P_jN/N}{L_{j}N/N}\cong \frac{P_jN}{L_jN}. \end{equation} As $L_jN\le R_jN \le P_jN$, from~\eqref{Pjbar} and~\eqref{Qjbar}, we see that $\overline{P_j}$ is an epimorphic image of $\overline Q_j$. Finally, since $\mathfrak{T}$ is a tower of $G$, it follows immediately that $\mathfrak{T}'$ is a tower of $G/N$. \end{proof} Given a tower $\mathfrak{T}=(P_i\mid i\in \{1,\ldots,h\})$ and $j\in \{1,\ldots,h\}$, we set $T_j:=P_hP_{h-1}\cdots P_j$. Observe that from Definition~\ref{Ttower}~(2), we have $T_j\unlhd T_1$. We are now ready to prove one of the main tools of our paper. \begin{lemma}\label{lemma34} Let $G$ be a group, let $\sigma$ be a non-empty subset of $\pi(G)$, let $A$ be a Hall $\sigma$-subgroup of $G$ and let $\mathfrak{T}:=(P_i\mid i\in \{1,\ldots,h\})$ be a tower of $G$. Then \begin{enumerate} \item $h(A)\geq \nu_\sigma(\mathfrak{T})-\beta_\sigma(\mathfrak{T})+1$, and \item $\ell_\sigma(G)\geq\beta_\sigma(\mathfrak{T})$. \end{enumerate} \end{lemma} \begin{proof}Observe that $h(A),\ell_\sigma(G)\geq 1$ because $\emptyset\neq\sigma\subseteq \pi(G)$. In particular, we may assume that $\nu_\sigma(\mathfrak{T}),\beta_{\sigma}(\mathfrak{T})\neq 0$ and hence $\sigma_0:=\sigma\cap \{\pi^*(P_i)\mid 1\leq i\leq h\}\neq\emptyset$. Let $A_0$ be a Hall $\sigma_0$-subgroup of $T_1$. Observe that $\mathfrak{T}$ is a tower of $T_1$ and that the hypothesis of this lemma are satisfied with $(G,\sigma,A)$ replaced by $(T_1,\sigma_0,A_0)$. As $h(A_0)\leq h(A)$ and $\ell_\sigma(T_1)\leq \ell_\sigma(G)$, for proving parts~$(1)$ and~$(2)$ we may assume that $G=T_1$, $\sigma=\sigma_0$ and $A=A_0$. Part~(1): We argue by induction on $h+|G|$. If $h=1$, then $\nu_\sigma(\mathfrak{T})=\beta_\sigma(\mathfrak{T})=1$ and the proof follows. Assume that $\pi^*(P_h)\notin \sigma$. Write $\mathfrak{T}':=(P_i\mid i\in\{1,\ldots,h-1\})$. From Lemma~\ref{lemma33}, the family $\mathfrak{T}'$ is a tower of $G$. As $\nu_{\sigma}(\mathfrak{T}')=\nu_{\sigma}(\mathfrak{T})$, $\beta_\sigma(\mathfrak{T}')=\beta_\sigma(\mathfrak{T})$ and $\mathfrak{T}'$ has length $h-1$, the proof follows by induction. Assume that $\pi^*(P_h)\in \sigma$. Let $t\in\{1,\ldots, h\}$ with $T_t=P_hP_{h-1}\cdots P_t$ a $\sigma$-block of $\mathfrak{T}$. Suppose that $T_t$ is the only $\sigma$-block of $\mathfrak{T}$. Thus $\nu_\sigma(\mathfrak{T})=h-t+1$, $\beta_\sigma(\mathfrak{T})=1$ and $T_t$ is a Hall $\sigma$-subgroup of $G$. Moreover, since $T_t\unlhd T_1=G$, we have $A=T_t$. Write $\mathfrak{T}':=(P_i\mid i\in \{t,\ldots,h\})$. From Lemma~\ref{lemma33}, the family $\mathfrak{T}'$ is a tower of $G$ and hence a tower of $A$. As $\mathfrak{T}'$ has length $h-t+1$, from Lemma~\ref{lemma31}, we get $h(A)\geq h-t+1$ and the proof follows. Suppose that $T_t$ is not the only $\sigma$-block of $G$, and let $j\in \{1,\ldots,t-1\}$ be maximal with $\pi^*(P_j)\in\sigma$. Suppose $\pi^*(P_j)\neq \pi^*(P_t)$. Then Lemma~\ref{lemma33} yields that $\mathfrak{T}':=(P_i\mid i\in \{1,\ldots,h\}\setminus\{j+1,\ldots,t-1\})$ is a tower of $G$. Since $\mathfrak{T}'$ has length $h-(t-j-1)<h$, from our induction we deduce $$h(A)\geq \nu_\sigma(\mathfrak{T}')-\beta_\sigma(\mathfrak{T}')+1=\nu_\sigma(\mathfrak{T})- (\beta_\sigma(\mathfrak{T})-1)+1=\nu_\sigma(\mathfrak{T})-\beta_\sigma(\mathfrak{T})+2.$$ Finally, suppose that $\pi^*(P_j)=\pi^*(P_t)$. In particular, either $\pi^*(P_{j-1})\neq \pi^*(P_t)$ or $j=1$. Now, Lemma~\ref{lemma33} gives that $\mathfrak{T}':=(P_i\mid i\in \{1,\ldots,h\}\setminus\{j,\ldots,t-1\})$ is a tower of $G$. As $\mathfrak{T}'$ has length $h-(t-j)<h$, the inductive hypothesis yields $$h(A)\geq \nu_\sigma(\mathfrak{T}')-\beta_\sigma(\mathfrak{T}')+1=(\nu_\sigma(\mathfrak{T})-1)- (\beta_\sigma(\mathfrak{T})-1)+1=\nu_\sigma(\mathfrak{T})-\beta_\sigma(\mathfrak{T})+1.$$ Part~(2): As in Part~$(1)$, we proceed by induction on $h+|G|$. Assume $\pi^*(P_h)\notin\sigma$. Then $\mathfrak{T}':=(P_i\mid i\in \{1,\ldots,h-1\})$ is a tower of $G$ of length $h-1$ with $\beta_\sigma(\mathfrak{T}')=\beta_\sigma(\mathfrak{T})$. Thus the proof follows by induction. Assume that $\pi^*(P_{h})\in \sigma$. Write $N:=\O {\sigma'} G$ and assume first that $N\neq 1$. For $j\in \{1,\ldots,h-1\}$, we have $[P_j\cap N,P_h]\leq N\cap P_h=1$ and hence $P_j\cap N\leq \cent {P_j}{P_h}$. In particular, by Lemma~\ref{elementaryy}, $\mathfrak{T}':=(P_iN/N\mid i\in \{1,\ldots,h-1\})$ is a tower of $G/N$ and, by induction, $\beta_\sigma(\mathfrak{T}')\leq \ell_{\sigma}(G/N)$. Since $\beta_\sigma(\mathfrak{T})=\beta_\sigma(\mathfrak{T}')$ and $\ell_\sigma(G)\geq \ell_{\sigma}(G/N)$, we get $\beta_\sigma(\mathfrak{T})\leq \ell_{\sigma}(G)$. Assume then that $N=1$. Write $\mathfrak{T}':=(P_i\mid i\in \{1,\ldots,h-1\})$. By Lemma~\ref{lemma33}, $\mathfrak{T}'$ is a tower of $G$ of length $h-1$. If $P_h$ is a not $\sigma$-block, then $\beta_\sigma(\mathfrak{T}')=\beta_\sigma(\mathfrak{T})$ and, by induction, $\beta_\sigma(\mathfrak{T})\leq \ell_\sigma(G)$. Suppose that $P_h$ is a $\sigma$-block, that is, $\pi^*(P_{h-1})\notin\sigma$. Clearly, $ \beta_\sigma(\mathfrak{T})=\beta_{\sigma}(\mathfrak{T}')+1$. Write $M:=\O {\sigma}G$ and observe that $M\neq 1$ and $\ell_\sigma(G)=\ell_{\sigma}(G/M)+1$ because $\O {\sigma'} G=N=1$. For $j\in \{1,\ldots,h-2\}$, we have $[ P_j\cap M,P_{h-1}]\leq M\cap P_{h-1}=1$ and hence $P_j\cap M\leq \cent {P_j}{P_{h-1}}$. In particular, by Lemma~\ref{elementaryy} (applied to $\mathfrak{T}'$), $\mathfrak{T}{''}:=(P_jM/M\mid j\in \{1,\ldots,h-2\})$ is a tower of $G/M$. Now, by induction, $\ell_\sigma(G/M)\geq \beta_\sigma(\mathfrak{T}^{''})=\beta_\sigma(\mathfrak{T}')$ from which it follows that $\ell_\sigma(G)\geq\beta_\sigma(\mathfrak{T})$. \end{proof} \section{Factorisations: Proofs of Theorems~\ref{thrmA} and~\ref{thrmB} and Corollary~\ref{corcor}}\label{section4} We start by proving the following. \begin{lemma}\label{lemma41} Let $G$ be a group, let $\sigma$ be a non-empty proper subset of $\pi(G)$ and let $G=AB$ be a factorisation, with $A$ a $\sigma$-subgroup of $G$ and $B$ a $\sigma'$-subgroup of $G$. Then \[ h(G)\leq h(A)+h(B)+\ell_\sigma(G)+\ell_{\sigma'}(G)-2 \] and \[ h(G)\leq h(A)+h(B)+2\min\{\ell_\sigma(G),\ell_{\sigma'}(G)\}-1. \] \end{lemma} \begin{proof} Let $\mathfrak{T}$ be a Fitting tower of $G$ (see Definition~\ref{Fittingtower}). Using first Lemma~\ref{lemma34} part~(1) and then part~(2), we have \begin{eqnarray*} (\dag)\qquad h(G)&=&\nu_\sigma(\mathfrak{T})+\nu_{\sigma'}(\mathfrak{T})\leq (h(A)+\beta_\sigma(\mathfrak{T})-1)+(h(B)+\beta_{\sigma'}(\mathfrak{T})-1)\\ &=&h(A)+h(B)+\beta_{\sigma}(\mathfrak{T})+\beta_{\sigma'}(\mathfrak{T})-2\\ &\leq &h(A)+h(B)+\ell_{\sigma}(G)+\ell_{\sigma'}(G)-2. \end{eqnarray*} Observe that, for each set of prime numbers $\pi$, from the definition of $\pi'\pi$-series we have $\ell_{\pi'}(G)\leq \ell_\pi(G)+1$. Applying this remark with $\pi=\sigma$ and with $\pi=\sigma'$, from $(\dag)$ we get $h(G)\leq h(A)+h(B)+2\min\{\ell_\sigma(G),\ell_{\sigma'}(G)\}-1.$ \end{proof} \begin{proof}[Proof of Theorem~$\ref{thrmA}$] Write $\sigma:=\pi(A)$ and $\sigma':=\pi(B)$. If $|B|$ is odd or if $B$ is nilpotent, then Theorem~\ref{thrm2.3} yields $\ell_{\sigma'}(G)\leq d(B)$. In the first case, Eq.~\eqref{eq:1} follows directly from Lemma~\ref{lemma41}. In the second case, $h(B)=1$ and now Eq.~\eqref{eq:2} follows again from Lemma~\ref{lemma41}. \end{proof} We now show that the bounds in Theorem~\ref{thrmA} are (in some cases) best possible. (We denote by $C_n$ a cyclic group of order $n$.) \begin{example}\label{ex1}{\rm Let $p,q,r$ and $t$ be distinct primes and let $n\geq 1$. Define $H_0:=C_p\mathop{\mathrm{wr}} C_q$ and $H_1:=(H_0\mathop{\mathrm{wr}} C_r)\mathop{\mathrm{wr}} (C_q\mathop{\mathrm{wr}} C_p)$. Now, for each $i\geq 1$, define inductively $H_{2i}:=(H_{2i-1}\mathop{\mathrm{wr}} C_r)\mathop{\mathrm{wr}} (C_p\mathop{\mathrm{wr}} C_q)$ and $H_{2i+1}:=(H_{2i}\mathop{\mathrm{wr}} C_r)\mathop{\mathrm{wr}} (C_q\mathop{\mathrm{wr}} C_p)$. We let $H:=H_{n}$ and $G:=C_t\mathop{\mathrm{wr}} H$. Let $A$ be a Hall $\{p,q\}$-subgroup of $G$ and let $B$ be a Hall $\{r,t\}$-subgroup of $G$. A computation shows that $h(A)=n+2$, $h(B)=2$, $h(G)=3n+3$ and $d(B)=n+1$. Theorem~\ref{thrmA}~\eqref{eq:1} predicts $h(G)\leq h(A)+h(B)+2d(B)-1$, and in fact in this example the equality is met. } \end{example} \begin{example}\label{ex2}{\rm Let $p$ and $q$ be distinct primes and let $n\geq 0$. Define $G_0:=C_p$ and $G_1:=G_0\mathop{\mathrm{wr}} C_q$. Now, for each $i\geq 1$, define inductively $G_{2i}:=G_{2i-1}\mathop{\mathrm{wr}} C_p$ and $G_{2i+1}:=G_{2i}\mathop{\mathrm{wr}} C_q$. Let $G:=G_{2n}$, let $A$ be a Sylow $p$-subgroup of $G$ and let $B$ be a Sylow $q$-subgroup of $G$. A computation shows that $h(A)=1$, $d(B)=n$ and $h(G)=2n+1$. Theorem~\ref{thrmA}~\eqref{eq:2} predicts $h(G)\leq h(A)+2d(B)$, and in fact in this example the equality is met.} \end{example} \begin{proof}[Proof of Corollary~$\ref{corcor}$] From Lemma~\ref{lemma:2.1}, there exists a Sylow basis $\{G_p\}_{p\in\pi(G)}$ of $G$ with $A=\prod_{p\in\pi(A)}G_p$ and $B=\prod_{p\in \pi(B)}G_p$. Now, we argue by induction on $|\pi(B)|$. If $|\pi(B)|=1$, then $B$ is nilpotent and hence the proof follows from Theorem~\ref{thrmA}~\eqref{eq:2}. Suppose that $|\pi(B)|>1$. Fix $q\in \pi(B)$ and write $B_{q'}:=\prod_{p\in \pi(B)\setminus\{q\}}G_p$. Clearly, $G=AB=(AG_q)B_{q'}$ and hence, by induction, \begin{eqnarray*} h(G)&\leq& h(AG_q)+2\sum_{p\in \pi(B_{q'})}d(G_p)\leq (h(A)+2d(G_q))+2\sum_{p\in\pi(B_{q'})}d(G_p)\\ &=&h(A)+2\sum_{p\in \pi(B)}d(G_p). \end{eqnarray*} \end{proof} The proof of Theorem~\ref{thrmB} will follow at once from the following lemma, which (in our opinion) is of independent interest. \begin{lemma}\label{lemma4.4} Let $G$ be a group, let $\sigma$ be a non-empty subset of $\pi(G)$ and let $H$ be a Hall $\sigma$-subgroup of $G$. Then $\ell_\sigma(G)\leq \delta(H)h(H)$. \end{lemma} \begin{proof} When $|\sigma|=1$, the proof follows immediately from Theorem~\ref{thrm2.2}. In particular, we may assume that $|\sigma|>1$. Now we proceed by induction on $|G|+|\sigma|$. Clearly, $\ell_\sigma(G)=\ell_\sigma(G/\O{\sigma'}G)$ and $H\O{\sigma'} G/\O{\sigma'}G\cong H$ is a Hall $\sigma$-subgroup of $G/\O{\sigma'}G$. When $\O{\sigma'}G\neq 1$, the proof follows by induction, and hence we may assume that $\O{\sigma'}G=1$. Suppose that $G$ contains two distinct minimal normal subgroups $N$ and $M$. Clearly, $\pi(N),\pi(M)\subseteq \sigma$. As $\O{\sigma'}G=1$, we deduce that $\ell_\sigma(G/N)=\ell_\sigma(G)=\ell_\sigma(G/M)$. Moreover, by induction, $\ell_\sigma(G/N)\leq \delta(H/N)h(H/N)\leq \delta(H)h(H)$. This gives $\ell_\sigma(G)\leq\delta(H)h(H)$, and hence we may assume that $G$ contains a unique minimal normal subgroup. This yields $\F G=\O p G$, for some $p\in \sigma$. As $\cent G{\O p G}\leq \O p G$ and $\O p G\leq H$, we have $\F H=\O p H$. Write $\tau:=\sigma\setminus\{p\}$. Observe that $\ell_\sigma(G)\leq \ell_p(G)+\ell_\tau(G)$. As $G_\tau$ is isomorphic to a subgroup of $H/\F H$, we get $h(G_\tau)\leq h(H/\F H)=h(H)-1$. Now, from the inductive hypothesis, we obtain \begin{eqnarray*} \ell_\sigma(G)&\leq& \ell_p(G)+\ell_\tau(G)\leq \delta(G_p)h(G_p)+\delta(G_\tau)h(G_\tau)\\ &\leq& \delta(H)+\delta(H)(h(H)-1)\leq \delta(H)h(H). \end{eqnarray*} \end{proof} \begin{proof}[Proof of Theorem~$\ref{thrmB}$] Write $\sigma:=\pi(A)$ and $\sigma':=\pi(B)$. From Lemma~\ref{lemma4.4}, we get $\ell_\sigma(G)\leq \delta(A)h(A)$ and $\ell_{\sigma'}(G)\leq \delta(B)h(B)$. Now the proof follows from the second inequality in Lemma~\ref{lemma41}. \end{proof} \section{Factorisations: Proof of Theorem~\ref{thrmC}}\label{section5} Before proceeding with the proof of Theorem~\ref{thrmC} we need to introduce some auxiliary notation. Given a group $G$, we denote with $\R G $ the \textit{nilpotent residual} of $G$, that is, the smallest (with respect to inclusion) normal subgroup $N$ of $G$ with $G/N$ nilpotent. Then, we define inductively the descending normal series $\{\RR i G\}_i$ by $\RR 0 G:=G$ and $\RR {i+1} G:=\R{\RR i G}$, for every $i\geq 0$. Observe that if $h=h(G)$, then for every $i\in \{0,\ldots,h\}$ we have $\RR {h-i}G\leq \FF i G$. Now, let $A$ be a Hall subgroup of $G$ and, for $i\in \{1,\ldots,h\}$, define \[ \ell^i(G,A):=\max\{\ell_p(G)\mid p\in \pi(\RR{i-1} A /\RR i A)\}\quad\textrm{and}\quad \Lambda_G(A):=\sum_{i=1}^{h(A)}\ell^i(G,A). \] It is clear that, for every normal subgroup $N$ of $G$, $\Lambda_{G/N}(AN/N)\leq \Lambda_G(A)$. \begin{lemma}\label{thrm48}Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. Then $h(G)\leq \Lambda_G(A)+\Lambda_G(B)$. \end{lemma} \begin{proof} We argue by induction on $|G|$. Suppose that $G$ contains two distinct minimal normal subgroups $N$ and $M$. Clearly, $h(G/N)=h(G)=h(G/M)$ and hence by induction $h(G)\leq \Lambda_{G/N}(AN/N)+\Lambda_{G/N}(BN/N)\leq \Lambda_G(A)+\Lambda_G(B)$. In particular, we may assume that $G$ contains a unique minimal normal subgroup $N$ and, replacing $A$ by $B$ if necessary, that $\{p\}=\pi(N)\subseteq \pi(A)$. This yields $\F G=\O p G$. As $\cent G{\O p G}\leq \O p G$ and $\O p G\leq A$, we have $\F A=\O p A$. Write $h:=h(A)$. Now, $\RR{h-1}A\leq \FF 1 A=\F A$ and hence $\RR {h-1}A$ is a $p$-group. Thus $$\Lambda_G(A)=\ell_p(G)+\sum_{i=1}^{h-1}\ell^i(G,A).$$ Since $\ell_p(G/\F G)=\ell_p(G)-1$, we get $\Lambda_{G/\F G}(A/\F G)\leq \Lambda_G(A)-1$. Moreover, since $p\notin \pi(B)$, we have $\Lambda_{G/\F G}(B\F G/\F G)=\Lambda_{G}(B)$. Therefore the inductive hypothesis gives \begin{eqnarray*} h(G)&=&h(G/\F G)+1\\ &\leq& \Lambda_{G/\F G}(A\F G/\F G)+\Lambda_{G/\F G}(B\F G/\F G)+1\\ &\leq& \Lambda_G(A)+\Lambda_G(B), \end{eqnarray*} and the proof is complete. \end{proof} \begin{proof}[Proof of Theorem~$\ref{thrmC}$] For each $p\in \pi(A)$, Theorem~\ref{thrm2.2} yields $\ell_p(G)\leq d(G_p)$ and hence $\ell_p(G)\leq \delta(A)$. It follows that $\Lambda_G(A)\leq \delta(A)h(A)$. The same argument applied to $B$ gives $\Lambda_G(B)\leq \delta(B)h(B)$. Now the proof follows from Lemma~\ref{thrm48}. \end{proof} A weaker form of Theorem~\ref{thrmC} can be deduced from the results in Section~\ref{section4}. Indeed, from the first inequality in Lemma~\ref{lemma41} and from Lemma~\ref{lemma4.4}, we get \begin{eqnarray*} h(G)&\leq &h(A)+h(B)+\ell_{\sigma}(G)+\ell_{\sigma'}(G)-2\\ &\leq& h(A)+h(B)+\delta(A)h(A)+\delta(B)h(B)-2\\ &=&(\delta(A)+1)h(A)+(\delta(B)+1)h(B)-2. \end{eqnarray*} Clearly Theorem~\ref{thrmC} always offer a better estimate on $h(G)$. \thebibliography{10} \bibitem{AFD}B.~Amberg, S.~Franciosi, F.~de Giovanni, \textit{Products of groups}, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1992. \bibitem{BB}A.~Ballester-Bolinches, R.~Esteban-Romero, M.~Asaad, \textit{Products of finite groups}, Expositions in Mathematics 53, Walter de Gruyter, Berlin, 2010. \bibitem{Br}E.~G.~Brjuhanova, The relation between $2$-length and derived length of a Sylow $2$-subgroup of a finite soluble group. (Russian), \textit{Mat. Zametki} \textbf{29} (1981), 161--170, 316. \bibitem{Dade} E.~C.~Dade, Carter subgroups and Fitting heights of finite solvable groups, \textit{Illinois J. 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\begin{document} \title{The ample cone of moduli spaces of sheaves on the plane} \date{\today} \author[I. Coskun]{Izzet Coskun} \author[J. Huizenga]{Jack Huizenga} \address{Department of Mathematics, Statistics and CS \\University of Illinois at Chicago, Chicago, IL 60607} \email{coskun@math.uic.edu} \email{huizenga@math.uic.edu} \thanks{During the preparation of this article the first author was partially supported by the NSF CAREER grant DMS-0950951535, and the second author was partially supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship} \subjclass[2010]{Primary: 14J60. Secondary: 14E30, 14J26, 14D20, 13D02} \keywords{Moduli space of stable vector bundles, Minimal Model Program, Bridgeland Stability Conditions, ample cone} \begin{abstract} Let $\xi$ be the Chern character of a stable sheaf on $\mathbb{P}^2$. Assume either $\rk(\xi)\leq 6$ or $\rk(\xi)$ and $c_1(\xi)$ are coprime and the discriminant $\Delta(\xi)$ is sufficiently large. We use recent results of Bayer and Macr\`i \cite{BayerMacri2} on Bridgeland stability to compute the ample cone of the moduli space $M(\xi)$ of Gieseker semistable sheaves on $\mathbb{P}^2$. We recover earlier results, such as those by Str\o mme \cite{Stromme} and Yoshioka \cite{Yoshioka}, as special cases. \end{abstract} \maketitle \setcounter{tocdepth}{1} \tableofcontents \section{Introduction} Let $\xi$ be the Chern character of a stable sheaf on $\mathbb{P}^2$. The moduli space $M(\xi)$ parameterizes $S$-equivalence classes of Gieseker semistable sheaves with Chern character $\xi$. It is an irreducible, normal, factorial, projective variety \cite{LePotierLectures}. In this paper, we determine the ample cone of $M(\xi)$ when either $\rk(\xi)\leq 6$ or $\rk(\xi)$ and $c_1(\xi)$ are coprime and the discriminant $\Delta(\xi)$ is sufficiently large. The {\em ample cone} $\Amp(X)$ of a projective variety $X$ is the open convex cone in the N\'{e}ron-Severi space spanned by the classes of ample divisors. It controls embeddings of $X$ into projective space and is among the most important invariants of $X$. Its closure, the {\em nef cone} $\mathbb{N}ef(X)$, is spanned by divisor classes that have non-negative intersection with every curve on $X$ and is dual to the Mori cone of curves on $X$ under the intersection pairing (see \cite{Lazarsfeld}). We now describe our results on $\Amp(M(\xi))$ in greater detail. Let $\xi$ be an integral Chern character with rank $r>0$. We record such a character as a triple $(r,\mu,\Delta)$, where $$\mu = \frac{\ch_1}{r} \qquad \otimesxtrm{and} \qquad \Delta = \frac{1}{2}\mu^2 - \frac{\ch_2}{r}$$ are the \emph{slope} and \emph{discriminant}, respectively. We call the character $\xi$ (semi)stable if there exists a (semi)stable sheaf of character $\xi$. Dr\'{e}zet and Le Potier give an explicit curve $\delta(\mu)$ in the $(\mu, \Delta)$-plane such that the moduli space $M(\xi)$ is positive dimensional if and only if $\Delta \geq \delta(\mu)$ \cite{DLP}, \cite{LePotierLectures}. The vector bundles whose Chern characters satisfy $\Delta = \delta(\mu)$ are called {\em height zero bundles}. Their moduli spaces have Picard group isomorphic to $\mathbb{Z}Z$. Hence, their ample cone is spanned by the positive generator and there is nothing further to discuss. Therefore, we will assume $\Delta>\delta(\mu)$, and say $\xi$ has \emph{positive height}. There is a nondegenerate symmetric bilinear form on the $K$-group $K(\mathbb{P}^2)$ sending a pair of Chern characters $\xi, \zeta$ to the Euler Characteristic $\chi(\xi^*, \zeta)$. When $\xi$ has positive height, the Picard group of the moduli space $M(\xi)$ is naturally identified with the orthogonal complement $\xi^\perp$ and is isomorphic to $\mathbb{Z}Z \oplus \mathbb{Z}Z$ \cite{LePotierLectures}. Correspondingly, the N\'{e}ron-Severi space is a two-dimensional vector space. In order to describe $\Amp(M(\xi))$, it suffices to specify its two extremal rays. The moduli space $M(\xi)$ admits a surjective, birational morphism $j: M(\xi)\rightarrow M^{DUY}(\xi)$ to the Donaldson-Uhlenbeck-Yau compactification $M^{DUY}(\xi)$ of the moduli space of stable vector bundles (see \cite{JunLiDonaldson} and \cite{HuybrechtsLehn}). As long as the locus of singular (i.e., non-locally-free) sheaves in $M(\xi)$ is nonempty (see Theorem \ref{thm-singular}), the morphism $j$ is not an isomorphism and contracts curves (see Proposition \ref{prop-DUY}). Consequently, the line bundle $\mathcal{L}_1$ defining $j$ is base-point-free but not ample (see \cite{HuybrechtsLehn}). It corresponds to a Chern character $u_1\in \xi^\perp \cong \mathbb{P}ic M(\xi)$ and spans an extremal ray of $\Amp(M(\xi))$. For all the characters $\xi$ that we will consider in this paper there are singular sheaves in $M(\xi)$, so one edge of $\Amp(M(\xi))$ is always spanned by $u_1$. We must compute the other edge of the cone, which we call the {\em primary edge}. We now state our results. Let $\xi = (r, \mu, \Delta)$ be a stable Chern character. Let $\xi'= (r', \mu', \Delta')$ be the stable Chern character satisfying the following defining properties: \begin{itemize} \item $0< r' \leq r$ and $\mu'< \mu$, \item Every rational number in the interval $(\mu', \mu)$ has denominator greater than $r$, \item The discriminant of any stable bundle of slope $\mu'$ and rank at most $r$ is at least $\Delta'$, \item The minimal rank of a stable Chern character with slope $\mu'$ and discriminant $\Delta'$ is $r'$. \end{itemize} The character $\xi'$ is easily computed using Dr\'ezet and Le Potier's classification of stable bundles. \begin{theorem}\label{thm-asymptotic} Let $\xi = (r,\mu,\Delta)$ be a positive height Chern character such that $r$ and $c_1$ are coprime. Suppose $\Delta$ is sufficiently large, depending on $r$ and $\mu$. The cone $\Amp(M(\xi))$ is spanned by $u_1$ and a negative rank character in $(\xi')^\perp$. \end{theorem} The required lower bound on $\Delta$ can be made explicit; see Remark \ref{rem-explicit}. Our second result computes the ample cone of small rank moduli spaces. \begin{theorem}\label{thm-smallRank} Let $\xi = (r,\mu,\Delta)$ be a positive height Chern character with $r\leq 6$. \begin{enumerate} \item If $\xi$ is not a twist of $(6,\frac{1}{3},\frac{13}{18})$, then $\Amp(M(\xi))$ is spanned by $u_1$ and a negative rank character in $(\xi')^\perp$. \item If $\xi = (6,\frac{1}{3},\frac{13}{18})$, then $\Amp(M(\xi))$ is spanned by $u_1$ and a negative rank character in $(\ch \mathcal{O}_{\mathbb{P}^2})^\perp$. \end{enumerate} \end{theorem} The new ingredient that allows us to calculate $\Amp(M(\xi))$ is Bridgeland stability. Bridgeland \cite{bridgeland:stable}, \cite{Bridgeland} and Arcara, Bertram \cite{ArcaraBertram} construct Bridgeland stability conditions on the bounded derived category of coherent sheaves on a projective surface. On $\mathbb{P}^2$, these stability conditions $\sigma_{s,t} = (\mathcal A_s, Z_{s,t})$ are parameterized by a half plane $H := \{ (s,t) | s,t \in \mathbb{R}, t>0\}$ (see \cite{ABCH} and \S \ref{sec-prelim}). Given a Chern character $\xi$, $H$ admits a finite wall and chamber decomposition, where in each chamber the collection of $\sigma_{s,t}$-semistable objects with Chern character $\xi$ remains constant. These walls are disjoint and consist of a vertical line $s = \mu$ and nested semicircles with center along $t=0$ \cite{ABCH}. In particular, there is a largest semicircular wall $W_{\max}$ to the left of the vertical wall. We will call this wall the {\em Gieseker wall}. Outside this wall, the moduli space $M_{\sigma_{s,t}}(\xi)$ of $\sigma_{s,t}$-semistable objects is isomorphic to the Gieseker moduli space $M(\xi)$ \cite{ABCH}. Let $\sigma_0$ be a stability condition on the Gieseker wall for $M(\xi)$. Bayer and Macr\`i \cite{BayerMacri2} construct a nef divisor $\ell_{\sigma_0}$ on $M(\xi)$ corresponding to $\sigma_0$. They also characterize the curves $C$ in $M(\xi)$ which have intersection number $0$ with $\ell_{\sigma_0}$, as follows: $\ell_{\sigma_0}.C=0$ if and only if two general sheaves parameterized by $C$ are $S$-equivalent with respect to $\sigma_0$ (that is, their Jordan-H\"older factors with respect to the stability condition $\sigma_0$ are the same). The divisor $\ell_{\sigma_0}$ is therefore an extremal nef divisor if and only if such a curve in $M(\xi)$ exists. This divisor can also be constructed via the GIT methods of Li and Zhao \cite{LiZhao}. In light of the results of Bayer and Macr\`i, our proofs of Theorems \ref{thm-asymptotic} and \ref{thm-smallRank} amount to the computation of the Gieseker wall. For simplicity, we describe our approach to proving Theorem \ref{thm-asymptotic}; the basic strategy for the proof of Theorem \ref{thm-smallRank} is similar. \begin{theorem}\label{thm-introWall} Let $\xi$ be as in Theorem \ref{thm-asymptotic}. The Gieseker wall for $M(\xi)$ is the wall $W(\xi',\xi)$ where $\xi$ and $\xi'$ have the same Bridgeland slope. \end{theorem} There are two parts to the proof of this theorem. First, we show that $W_{\max}$ is no larger than $W(\xi',\xi)$. This is a numerical computation based on Bridgeland stability. The key technical result (Theorem \ref{thm-excludeHighRank}) is that if a wall is larger than $W(\xi',\xi)$, then the rank of a destabilizing subobject corresponding to the wall is at most $\rk(\xi)$. We then find that the extremality properties defining $\xi'$ guarantee that $W(\xi',\xi)$ is at least as large as any wall for $M(\xi)$ (Theorem \ref{thm-main}). In the other direction, we must show that $W(\xi',\xi)$ is an actual wall for $M(\xi)$. Define a character $\xi'' = \xi-\xi'$. Our next theorem produces a sheaf $E\in M(\xi)$ which is destabilized along $W(\xi',\xi)$. \begin{theorem}\label{thm-introSt} Let $\xi$ be as in Theorem \ref{thm-asymptotic}. Fix general sheaves $F\in M(\xi')$ and $Q\in M(\xi'')$. Then the general sheaf $E$ given by an extension $$0\to F\to E\to Q\to 0$$ is Gieseker stable. Furthermore, we obtain curves in $M(\xi)$ by varying the extension class. \end{theorem} If $E$ is a Gieseker stable extension as in the theorem, then $E$ is strictly semistable with respect to a stability condition $\sigma_0$ on $W(\xi',\xi)$, and not semistable with respect to a stability condition below $W(\xi',\xi)$. Thus $W(\xi',\xi)$ is an actual wall for $M(\xi)$, and it is the Gieseker wall. Any two Gieseker stable extensions of $Q$ by $F$ are $S$-equivalent with respect to $\sigma_0$, so any curve $C$ in $M(\xi)$ obtained by varying the extension class satisfies $\ell_{\sigma_0}.C=0$. Therefore, $\ell_{\sigma_0}$ spans an edge of the ample cone. Dually, $C$ spans an edge of the Mori cone of curves. The natural analogs of Theorems \ref{thm-introWall} and \ref{thm-introSt} are almost true when instead $\rk(\xi)\leq 6$ as in Theorem \ref{thm-smallRank}; some minor adjustments to the statements need to be made for certain small discriminant cases. See Theorems \ref{thm-smallRankCurves}, \ref{thm-mainsmall}, \ref{thm-mainSporadic}, and Propositions \ref{prop-sporadic} and \ref{prop-sporadic2} for precise statements. As the rank increases beyond $6$, these exceptions become more common, and many more ad hoc arguments are required when using current techniques. Bridgeland stability conditions were effectively used to study the birational geometry of Hilbert schemes of points on $\mathbb{P}^2$ in \cite{ABCH} and moduli spaces of rank 0 semistable sheaves in \cite{Woolf}. The ample cone of $M(\xi)$ was computed earlier for some special Chern characters. The ample cone of the Hilbert scheme of points on $\mathbb{P}^2$ was computed in \cite{li} (see also \cite{ABCH}, \cite{Ohkawa}). Str\o mme computed $\Amp(M(\xi))$ when the rank of $\xi$ is two and either $c_1$ or $c_2 - \frac{1}{4}c_1^2$ is odd \cite{Stromme}. Similarly, when the slope is $\frac{1}{r}$, Yoshioka \cite{Yoshioka} computed the ample cone of $M(\xi)$ and described the first flip. Our results contain these as special cases. Bridgeland stability has also been effectively used to compute ample cones of moduli spaces of sheaves on other surfaces. For example, see \cite{ArcaraBertram}, \cite{BayerMacri2}, \cite{BayerMacri3}, \cite{MYY1}, \cite{MYY2} for K3 surfaces, \cite{MM}, \cite{Yoshioka2}, \cite{YanagidaYoshioka} for abelian surfaces, \cite{Nuer} for Enriques surfaces, and \cite{BertramCoskun} for the Hilbert scheme of points on Hirzebruch surfaces and del Pezzo surfaces. \subsection*{Organization of the paper} In \S \ref{sec-prelim}, we will introduce the necessary background on $M(\xi)$ and Bridgeland stability conditions on $\mathbb{P}^2$. In \S \ref{sec-hp} and \S \ref{sec-extremal}, we study the stability of extensions of sheaves and prove the first statement in Theorem \ref{thm-introSt}. In \S \ref{sec-elementaryMod} and \ref{sec-smallRank}, we prove the analogue of the first assertion in Theorem \ref{thm-introSt} for $\rk(\xi) \leq 6$. In \S \ref{sec-curves}, we complete the proof of Theorem \ref{thm-introSt} (and its small-rank analogue) by constructing the desired curves of extensions. Finally, in \S \ref{sec-ample}, we compute the Gieseker wall, completing the proofs of Theorems \ref{thm-asymptotic} and \ref{thm-smallRank}. \section{Preliminaries}\label{sec-prelim} In this section, we recall basic facts concerning the classification of stable vector bundles on $\mathbb{P}^2$ and Bridgeland stability. \subsection{Stable sheaves on $\mathbb{P}^2$} Let $\xi$ be the Chern character of a (semi)stable sheaf on $\mathbb{P}^2$. We will call such characters {\em (semi)stable characters}. The classification of stable characters on $\mathbb{P}^2$ due to Dr\'{e}zet and Le Potier is best stated in terms of the slope $\mu$ and the discriminant $\Delta$. Let $$P(m)= \frac{1}{2}(m^2 + 3m +2)$$ denote the Hilbert polynomial of $\mathcal{O}_{\mathbb{P}^2}$. In terms of these invariants, the Riemann-Roch formula reads $$\chi(E,F) = \rk(E) \rk(F) ( P( \mu(F) - \mu(E)) - \Delta(E) - \Delta(F)).$$ An {\em exceptional bundle} $E$ on $\mathbb{P}^2$ is a stable bundle such that $\Ext^1(E,E)=0$. The exceptional bundles are rigid; their moduli spaces consist of a single reduced point \cite[Corollary 16.1.5]{LePotierLectures}. They are the stable bundles $E$ on $\mathbb{P}^2$ with $\Delta(E) < \frac{1}{2}$ \cite[Proposition 16.1.1]{LePotierLectures}. Examples of exceptional bundles include line bundles $\mathcal{O}_{\mathbb{P}^2}(n)$ and the tangent bundle $T_{\mathbb{P}^2}$. All exceptional bundles can be obtained from line bundles via a sequence of mutations \cite{DrezetBeilinson}. An {\em exceptional slope} $\alpha\in \mathbb{Q}$ is the slope of an exceptional bundle. If $\alpha$ is an exceptional slope, there is a unique exceptional bundle $E_\alpha$ with slope $\alpha$. The rank of the exceptional bundle is the smallest positive integer $r_{\alpha}$ such that $r_{\alpha} \alpha$ is an integer. The discriminant $\Delta_{\alpha}$ is then given by $$\Delta_{\alpha} = \frac{1}{2} \left( 1 - \frac{1}{r_{\alpha}^2}\right).$$ The set $\F E$ of exceptional slopes is well-understood (see \cite{DLP} and \cite{CoskunHuizengaWoolf}). The classification of positive dimensional moduli spaces of stable vector bundles on $\mathbb{P}^2$ is expressed in terms of a fractal-like curve $\delta$ in the $(\mu, \Delta)$-plane. For each exceptional slope $\alpha \in \F E$, there is an interval $I_{\alpha} = ( \alpha-x_{\alpha}, \alpha + x_{\alpha})$ with $$x_{\alpha} = \frac{3- \sqrt{5+8 \Delta_{\alpha}}}{2}$$ such that the function $\delta(\mu)$ is defined on $I_{\alpha}$ by the function $$\delta(\mu) = P(-|\mu - \alpha|) - \Delta_{\alpha}, \ \ \mbox{if} \ \ \alpha - x_{\alpha} < \mu < \alpha + x_{\alpha}.$$ The graph of $\delta(\mu)$ is an increasing concave up parabola on the interval $[\alpha - x_{\alpha}, \alpha]$ and a decreasing concave up parabola on the interval $[\alpha, \alpha + x_{\alpha}]$. The function $\delta$ is invariant under translation by integers. The main classification theorem of Dr\'{e}zet and Le Potier is as follows. \begin{theorem}[\cite{DLP}, \cite{LePotierLectures}] There exists a positive dimensional moduli space of Gieseker semistable sheaves $M(\xi)$ with integral Chern character $\xi$ if and only if $\Delta \geq \delta(\mu)$. In this case, $M(\xi)$ is a normal, irreducible, factorial projective variety of dimension $r^2(2 \Delta -1) + 1$. \end{theorem} \subsection{Singular sheaves on $\mathbb{P}^2$} For studying one extremal edge of the ample cone, we need to understand the locus of singular sheaves in $M(\xi)$. The following theorem, which is likely well-known to experts, characterizes the Chern characters where the locus of singular sheaves in $M(\xi)$ is nonempty. We include a proof for lack of a convenient reference. \begin{theorem}\label{thm-singular} Let $\xi= (r, \mu, \Delta)$ be an integral Chern character with $r>0$ and $\Delta \geq \delta(\mu)$. The locus of singular sheaves in $M(\xi)$ is empty if and only if $\Delta - \delta(\mu) < \frac{1}{r}$ and $\mu$ is not an exceptional slope. \end{theorem} \begin{proof} Let $F$ be a singular sheaf in $M(\xi)$. Then $F^{**}$ is a $\mu$-semistable, locally free sheaf \cite[\S 8]{HuybrechtsLehn} with invariants $$\rk(F^{**}) = r, \ \ \mu(F^{**}) = \mu, \ \ \mbox{and} \ \ \Delta(F^{**}) \leq \Delta(F) - \frac{1}{r}.$$ Since the set $\mathbb{R} - \cup_{\alpha \in \F{E}} I_{\alpha}$ does not contain any rational numbers \cite{DLP}, \cite[Theorem 4.1]{CoskunHuizengaWoolf}, $\mu \in I_{\alpha}$ for some exceptional slope $\alpha$. Let $E_{\alpha}$ with invariants $(r_{\alpha}, \alpha, \Delta_{\alpha})$ be the corresponding exceptional bundle. If $\Delta - \delta(\mu)< \frac{1}{r}$ and $F$ is a singular sheaf in $M(\xi)$, then $\Delta(F^{**})< \delta(\mu)$. If $\alpha > \mu$, then $\hom(E_{\alpha}, F^{**})>0$. If $\alpha< \mu$, then $\hom(F^{**}, E_{\alpha})>0$. In either case, these homomorphisms violate the $\mu$-semistability of $F^{**}$, leading to a contradiction. Therefore, if $\Delta - \delta(\mu) < \frac{1}{r}$ and $\mu$ is not an exceptional slope, then the locus of singular sheaves in $M(\xi)$ is empty. To prove the converse, we construct singular sheaves using elementary modifications. If $\Delta - \delta(\mu) \geq \frac{1}{r}$, then $\zeta = (r, \mu, \Delta - \frac{1}{r})$ is a stable Chern character. Let $G$ be a $\mu$-stable bundle in $M(\zeta)$, which exists by \cite[Corollary 4.12]{DLP}. Choose a point $p\in \mathbb{P}^2$ and let $G \rightarrow \mathcal{O}_p$ be a general surjection. Then the kernel sheaf $F$ defined by $$0 \rightarrow F \rightarrow G \rightarrow \mathcal{O}_p \rightarrow 0$$ is a $\mu$-stable, singular sheaf with Chern character $\xi$ (see \S \ref{sec-elementaryMod} for more details on elementary modifications). We are reduced to showing that if $\mu = \alpha$ and $\Delta - \delta(\alpha) < \frac{1}{r}$, then the locus of singular sheaves in $M(\xi)$ is nonempty. Since $c_1(E_{\alpha})$ and $r_{\alpha}$ are coprime, the rank of any bundle with slope $\alpha$ is a multiple of $r_{\alpha}$. Write $$r= k r_{\alpha}^2 + m r_{\alpha}, \ \ 0 \leq k, \ 0 < m \leq r_{\alpha}.$$ By integrality, there exists an integer $N$ such that $\Delta - \frac{N}{r} = \Delta_{\alpha}$. Our choice of $k$ implies $$ \Delta= \Delta_{\alpha} + \frac{k+1}{r}.$$ First, assume $k=0$. If $r'<r$, then $\Delta_{\alpha} + \frac{1}{r'} > \Delta_{\alpha} + \frac{1}{r}$. Consequently, the only Gieseker semistable sheaves of character $(r',\mu,\Delta')$ with $r'<r$ and $\Delta'<\Delta$ are semi-exceptional sheaves $E_{\alpha}^{\oplus \ell}$ with $\ell < m$. Let $F$ be a general elementary modification of the form $$0 \rightarrow F \rightarrow E_{\alpha}^{\oplus m} \rightarrow \mathcal{O}_p \rightarrow 0.$$ Then $F$ is a $\mu$-semistable singular sheaf with Chern character $\xi$. If $F$ were not Gieseker semistable, then it would admit an injective map $\phi: E_{\alpha} \rightarrow F$. By Lemma \ref{lem-Segre} below, for a general surjection $\psi: E_{\alpha}^{\oplus m} \rightarrow \mathcal{O}_p$, there does not exist an injection $E_{\alpha} \rightarrow E_{\alpha}^{\oplus m}$ which maps to $0$ under $\psi$. Composing $\phi$ with the maps in the exact sequence defining $F$, we get a contradiction. We conclude that $F$ is Gieseker semistable. This constructs singular sheaves when $k=0$. Next assume $k>0$. If $m= r_{\alpha}$, then we can construct a singular sheaf in $M(\xi)$ as a $(k+1)$-fold direct sum of a semistable singular sheaf constructed in the case $k=0$, $m=r_\alpha$. Hence, we may assume that $m< r_{\alpha}$. Let $G$ be a $\mu$-stable vector bundle with Chern character $$\zeta= \left(kr_{\alpha}^2, \alpha, \delta(\alpha)=\Delta_{\alpha} + \frac{1}{r_{\alpha}^2}\right).$$ Note that $(\mu(\zeta),\Delta(\zeta))$ lies on the curve $\delta$, hence $\chi(E_{\alpha}, G) = \chi(G, E_{\alpha}) =0$. Every locally free sheaf in $M(\zeta)$ has a two-step resolution in terms of exceptional bundles orthogonal to $E_{\alpha}$ \cite{DLP}. Consequently, $\hom(G, E_{\alpha})=0$. We also have $\hom(E_\alpha, G)=0$ by stability. Let $ \phi: E_{\alpha}^{\oplus m} \oplus G \rightarrow \mathcal{O}_p$ be a general surjection and let $F$ be defined as the corresponding elementary modification $$0 \rightarrow F \rightarrow E_{\alpha}^{\oplus m} \oplus G \rightarrow \mathcal{O}_p \rightarrow 0.$$ We first check that the Chern character of $F$ is $\xi$. Clearly, $\rk(F)=r$ and $\mu(F)= \alpha$. The discriminant equals $$ \Delta(F) = \frac{1}{r} \left( m r_{\alpha} \Delta_{\alpha} + k r_{\alpha}^2 \left(\Delta_{\alpha} + \frac{1}{r_{\alpha}^2}\right)\right) + \frac{1}{r} = \Delta_{\alpha} + \frac{k+1}{r} = \Delta.$$ Hence, $F$ is a singular sheaf with the correct invariants. It remains to check that it is Gieseker semistable. Note that $F$ is at least $\mu$-semistable. Suppose $\psi: U \rightarrow F$ is an injection from a Gieseker stable sheaf $U$ that destabilizes $F$. Since $F$ is $\mu$-semistable, $\mu(U) = \alpha$ and $\Delta(U) < \Delta$. Then we claim that either $U = E_{\alpha}$ or $\rk(U) > m r_{\alpha}$. Suppose $U \not= E_{\alpha}$ and $\rk(U) = s r_{\alpha}$. Then $$\Delta = \Delta_{\alpha} + \frac{k+1}{r}> \Delta(U) \geq \Delta_{\alpha} + \frac{1}{s r_{\alpha}}.$$ Hence, $$ s > \frac{k r_{\alpha} + m}{k+1} > \frac{km + m} {k+1} = m.$$ If $U \neq E_{\alpha}$, composing $\psi$ with the inclusion to $E_{\alpha}^{\oplus m} \oplus G$ gives an injection $\psi' : U \rightarrow E_{\alpha}^{\oplus m} \oplus G$. Since $\rk (U) > m r_{\alpha}$, the projection to $G$ cannot be zero. Hence, we get a nonzero map $\vartheta : U \rightarrow G$. Let $V = \im \vartheta$. We have $\rk V = \rk G$ by the $\mu$-stability of $G$. We claim that $\vartheta$ is in fact surjective. The quotient $G/V$ is $0$-dimensional by stability, and, if it is nonzero, then $$\Delta(U) < \Delta < \delta(\alpha) + \frac{1}{r}< \delta(\alpha) + \frac{1}{kr_\alpha^2} \leq \Delta(V).$$ This violates the stability of $U$, so $V = G$ and $\vartheta$ is surjective. If $\rk(U) = \rk(G)$, then $U\cong G$ and $\psi'$ maps $U$ isomorphically onto $G \subset E_{\alpha}^{\oplus m} \oplus G$. A general hyperplane in the fiber $(E_\alpha^{\oplus m} \oplus G)_p$ is transverse to $G_p$, so this contradicts the fact that $\phi \circ \psi' = 0$ and $\phi$ is general. Suppose $\rk(U)>\rk(G)$, and write $$\rk(U) = kr_{\alpha}^2+nr_\alpha$$ with $0< n<m$. Then we find $$\Delta_\alpha+\frac{k+1}{r} =\Delta>\Delta(U) \geq \Delta_\alpha + \frac{k+1}{\rk(U)},$$ contradicting $\rk(U) < r$. We conclude that if $U\neq E_\alpha$, then $U$ cannot destabilize $F$. On the other hand, if $U = E_{\alpha}$, then by the semistability of $G$ the composition of $\psi'$ with the projection to $G$ must be $0$. A general hyperplane in the fiber $(E_\alpha^{\oplus m} \oplus G)_p$ intersects $(E_{\alpha}^{\oplus m})_p$ in a hyperplane $H\subset (E_\alpha^{\oplus m})_p$. Since $m\leq r_\alpha$, Lemma \ref{lem-Segre} shows the composition of $\psi'$ with $\phi$ is nonzero, a contradiction. We conclude that $F$ is Gieseker semistable. \end{proof} \begin{lemma}\label{lem-Segre} Let $E_{\alpha}$ be an exceptional bundle of rank $r_{\alpha}$. Let $H$ be a general codimension $c$ subspace of the fiber of $E_{\alpha}^{\oplus m}$ over a point $p$. Then there exists an injection $\phi: E_{\alpha} \rightarrow E_\alpha^{\oplus m}$ such that $\phi_p(E_{\alpha}) \subset H$ if and only if $c r_{\alpha} \leq m-1$. \end{lemma} \begin{proof} For simplicity set $E=E_\alpha $ and $r =r_\alpha$. Let $S$ denote the Segre embedding of $\mathbb{P}^{r-1} \times \mathbb{P}^{m-1}$ in $\mathbb{P}^{rm-1}$. Let $q_1, q_2$ denote the two projections from $S$ to $\mathbb{P}^{r-1}$ and $\mathbb{P}^{m-1}$, respectively. We will call a linear $\mathbb{P}^{r-1}$ in $S$ contracted by $q_2$ a {\em $\mathbb{P}^{r-1}$ fiber}. Let $\phi: E \rightarrow E^{\oplus m}$ be an injection. Composing $\phi$ with the $m$ projections, we get $m$ morphisms $E\rightarrow E$. Since $E$ is simple, the resulting maps are all homotheties. Let $M = (\lambda_1 I \ \ \lambda_2 I \ \ \dots \ \ \lambda_m I)$ be the $r \times rm$ matrix, where $I$ is the $r\times r$ identity matrix and $\lambda_i$ are scalars. Let $\vec{x} = (x_1, \dots, x_r)^T$. Hence, $\phi_p(E)$ has the form $$M \vec{x} = (\lambda_1 x_1, \lambda_1 x_2, \dots, \lambda_1 x_r, \dots, \lambda_m x_1, \dots, \lambda_m x_r)^T.$$ If we projectivize, we see that the fibers $\mathbb{P} (\phi_p(E))$ are $\mathbb{P}^{r -1}$ fibers contained in the Segre embedding of $\mathbb{P}^{r -1} \times \mathbb{P}^{m-1}$ in $\mathbb{P}((E^{\oplus m})_p)$. Conversely, every $\mathbb{P}^{r-1}$ fiber in $S$ is obtained by fixing a point $(\lambda_1, \dots, \lambda_m) \in \mathbb{P}^{m-1}$ and, hence, is the fiber of an injection $E \rightarrow E^{\oplus m}$. The lemma thus reduces to the statement that a general codimension $c$ linear subspace of $\mathbb{P}^{m r -1}$ contains a $\mathbb{P}^{r -1}$ fiber in $S$ if and only if $c r \leq m-1$. Consider the incidence correspondence $$J = \{ (A, H) : H \cong \mathbb{P}^{m r - 1 -c}, A \subset H \cap S \ \ \mbox{is a } \ \mathbb{P}^{r-1} \ \mbox{fiber}\}.$$ Then the first projection $\pi_1$ maps $J$ onto $\mathbb{P}^{m-1}$. The fiber of $\pi_1$ over a linear space $A$ is the set of codimension $c$ linear spaces that contain $A$, hence it is isomorphic to the Grassmannian $G((m-1)r - c, (m-1)r)$. By the theorem on the dimension of fibers, $J$ is irreducible of dimension $(cr+1)(m-1) - c^2$. The second projection cannot dominate $G(mr-c, mr)$ if $\dim(J) < \dim(G(mr-c, mr)= c(mr-c)$. Comparing the two inequalities, we conclude that if $c r > m-1$, the second projection is not dominant. Hence, the general codimension $c$ linear space does not contain a $\mathbb{P}^{r-1}$ fiber in $S$. To see the converse, we check that if $r \leq m-1$, then a general hyperplane contains a codimension $r$ locus of linear $\mathbb{P}^{r-1}$ fibers of $S$. Consider the hyperplane $ H$ defined by $\sum_{i=1}^{r} Z_{(i-1)r + i} =0.$ Substituting the equations of the Segre embedding, we see that $\sum_{i=1}^{r} \alpha_i x_i = 0$. Since this equation must hold for every choice of $x_i$, we conclude that $\alpha_1 = \cdots = \alpha_r =0$. Hence, the locus of $\mathbb{P}^{r-1}$ fibers of $S$ contained in $H$ is a codimension $r$ linear space in $\mathbb{P}^{m-1}$. A codimension $c$ linear space is the intersection of $c$ hyperplanes. Moreover, the intersection of $c$ codimension $r$ subvarieties of $\mathbb{P}^{m-1}$ is nonempty if $c r \leq m-1$. Hence, if $c r \leq m-1$, every codimension $c$ linear space contains a $\mathbb{P}^{r-1}$ fiber of $S$. This suffices to prove the converse. \end{proof} \subsection{The Picard group and Donaldson-Uhlenbeck-Yau compactification} Stable vector bundles with $\Delta = \delta(\mu)$ are called {\em height zero} bundles. Their moduli spaces have Picard rank one. The ample generator spans the ample cone and there is nothing further to discuss. For the rest of the subsection, suppose $\xi=(r,\mu,\Delta)$ is a \emph{positive height} Chern character, meaning $\Delta> \delta(\mu)$. There is a pairing on $K(\mathbb{P}^2)$ given by $(\xi, \zeta) = \chi(\xi^*, \zeta)$. When $\Delta > \delta(\mu)$, Dr\'{e}zet proves that the Picard group of $M(\xi)$ is a free abelian group on two generators naturally identified with $\xi^{\perp}$ in $K(\mathbb{P}^2)$ \cite{LePotierLectures}. In $M(\xi)$, linear equivalence and numerical equivalence coincide and the N\'{e}ron-Severi space $\mathbb{N}S(M(\xi)) = \mathbb{P}ic(M(\xi)) \otimes \mathbb{R}$ is a two-dimensional vector space. In order to specify the ample cone, it suffices to determine its two extremal rays. In $\xi^\perp \cong \mathbb{P}ic(M(\xi))$ there is a unique character $u_1$ with $\rk(u_1) = 0$ and $c_1(u_1) = -r$. The corresponding line bundle $\mathcal L_1$ is base-point-free and defines the Jun Li morphism $j: M(\xi) \to M^{DUY}(\xi)$ to the Donaldson-Uhlenbeck-Yau compactification \cite[\S 8]{HuybrechtsLehn}. \begin{proposition}\label{prop-DUY} Let $\xi= (r,\mu,\Delta)$ be a positive height character, and suppose that there are singular sheaves in $M(\xi)$. Then $u_1$ spans an extremal edge of $\Amp(M(\xi))$. \end{proposition} \begin{proof} We show that $j$ contracts a curve in $M(\xi)$. Two stable sheaves $E,E'\in M(\xi)$ are identified by $j$ if $E^{**} \cong (E')^{**}$ and the sets of singularities of $E$ and $E'$ are the same (counting multiplicity). The proof of Theorem \ref{thm-singular} constructs singular sheaves via an elementary modification that arises from a surjection $E= E_{\alpha}^{\oplus m} \oplus G \rightarrow \mathcal{O}_p$. Here $m=0$ if $\Delta - \delta(\mu) \geq \frac{1}{r}$ or $\mu$ is not exceptional. Otherwise, $1 \leq m < r_{\alpha}$. Note that $\hom(E, \mathcal{O}_p)= r$ and $\dim (\Aut(E)) = m^2 +1$ if $G\not= 0$ and $\dim (\Aut(E)) = m^2$ if $G=0$. Hence, if $r>1$, varying the surjection $E \rightarrow \mathcal{O}_p$ gives a positive dimensional family of nonisomorphic Gieseker semistable sheaves with the same singular support and double dual. If instead $r = 1$ and $\Delta\geq 2$, then (up to a twist) $j$ is the Hilbert-Chow morphism to the symmetric product, and the result is still true. \end{proof} \begin{corollary}\label{cor-DUY} Let $\xi = (r,\mu,\Delta)$ be a positive height character. If $\Delta$ is sufficiently large or if $r\leq 6$ then $u_1$ spans an edge of $\Amp(M(\xi))$. \end{corollary} \begin{proof} In either case, this follows from Theorem \ref{thm-singular} and Proposition \ref{prop-DUY}. \end{proof} \subsection{Bridgeland stability conditions on $\mathbb{P}^2$} We now recall basic facts concerning Bridgeland stability conditions on $\mathbb{P}^2$ developed in \cite{ABCH}, \cite{CoskunHuizenga} and \cite{HuizengaPaper2}. A {\em Bridgeland stability condition} $\sigma$ on the bounded derived category $\mathcal{D}^b(X)$ of coherent sheaves on a smooth projective variety $X$ is a pair $\sigma = (\mathcal A, Z)$, where $\mathcal A$ is the heart of a bounded $t$-structure and $Z$ is a group homomorphism $$Z: K(\mathcal{D}^b(\mathbb{P}^2)) \rightarrow \mathbb{C}$$ satisfying the following two properties. \begin{enumerate} \item (Positivity) For every object $0 \not= E \in \mathcal A$, $Z(E) \in \{r e^{i \pi \theta} | r> 0, 0 < \theta \leq 1\}$. Positivity allows one to define the slope of a non-zero object in $\mathcal A$ by setting $$\mu_Z(E) = - \frac{\mathbb{R}e(Z(E))}{\F{I}m(Z(E))}.$$ An object $E$ of $\mathcal A$ is called {\em (semi)stable} if for every proper subobject $F\subset E$ in $\mathcal A$ we have $\mu_Z(F) < (\leq) \mu_Z (E)$. \item (Harder-Narasimhan Property) Every object of $\mathcal A$ has a finite Harder-Narasimhan filtration. \end{enumerate} Bridgeland \cite{Bridgeland} and Arcara and Bertram \cite{ArcaraBertram} have constructed Bridgeland stability conditions on projective surfaces. In the case of $\mathbb{P}^2$, the relevant Bridgeland stability conditions have the following form. Any torsion-free coherent sheaf $E$ on $\mathbb{P}^2$ has a Harder-Narasimhan filtration $$0= E_0 \subset E_1 \subset \cdots \subset E_n = E$$ with respect to the Mumford slope with semistable factors $\mathrm{gr}_i = E_i / E_{i-1}$ such that $$\mu_{\max}(E) = \mu(\mathrm{gr}_1) > \cdots > \mu(\mathrm{gr}_n) = \mu_{\min}(E) .$$ Given $s\in \mathbb{R}$, let $\mathcal Q_s$ be the full subcategory of $\coh(\mathbb{P}^2)$ consisting of sheaves such that their quotient by their torsion subsheaf have $\mu_{\min}(Q)> s$. Similarly, let $\mathcal F_s$ be the full subcategory of $\coh(\mathbb{P}^2)$ consisting of torsion free sheaves $F$ with $\mu_{\max}(F) \leq s$. Then the abelian category $$\mathcal A_s := \{ E \in \mathcal{D}^b(\mathbb{P}^2) : \mbox{H}^{-1}(E) \in \mathcal F_s, \mbox{H}^0(E) \in \mathcal Q_s, H^i(E) = 0 \ \mbox{for} \ i \not= -1, 0 \}$$ obtained by tilting the category of coherent sheaves with respect to the torsion pair $(\mathcal F_s, \mathcal Q_s)$ is the heart of a bounded $t$-structure. Let $$Z_{s,t} (E) = - \int_{\mathbb{P}^2} e^{-(s+it)H} \ch(E),$$ where $H$ is the hyperplane class on $\mathbb{P}^2$. The pair $(\mathcal A_s, Z_{s,t})$ is a Bridgeland stability condition for every $s > 0$ and $t \in \mathbb{R}$. We thus obtain a half plane of Bridgeland stability conditions on $\mathbb{P}^2$ parameterized by $(s, t)$, $t>0$. \subsection{Bridgeland walls} If we fix a Chern character $\xi\in K(\mathbb{P}^2)$, the $(s,t)$-plane of stability conditions for $\mathbb{P}^2$ admits a finite wall and chamber structure where the objects in $\mathcal A_s$ with Chern character $\xi$ that are stable with respect to the stability condition $(\mathcal A_s, Z_{s,t})$ remain unchanged within the interior of a chamber (\cite{ABCH}, \cite{Bridgeland}, \cite{BayerMacri}, \cite{BayerMacri2}). An object $E$ is destabilized along a wall $W(E, F)$ by $F$ if $E$ is semistable on one side of the wall but $F \subset E$ in the category $\mathcal A_s$ with $\mu_{s,t} (F) > \mu_{s,t}(E)$ on the other side of the wall. We call these walls {\em Bridgeland walls}. The equations of the wall $W(E,F)$ can be computed using the relation $\mu_{s,t} (F) = \mu_{s,t}(E)$ along the wall. Suppose $\xi,\zeta\in K(\mathbb{P}^2)\otimes \mathbb{R}$ are two linearly independent real Chern characters. A \emph{potential Bridgeland wall} is a set in the $(s,t)$-half-plane of the form $$W(\xi,\zeta) = \{(s,t):\mu_{s,t}(\xi) = \mu_{s,t}(\zeta)\},$$ where $\mu_{s,t}$ is the slope associated to $\mathcal{Z}_{s,t}$. Bridgeland walls are always potential Bridgeland walls. The \emph{potential Bridgeland walls for $\xi$} are all the potential walls $W(\xi,\zeta)$ as $\zeta$ varies in $K(\mathbb{P}^2)\otimes \mathbb{R}$. If $E,F\in D^b(\mathbb{P}^2)$, we also write $W(E,F)$ as a shorthand for $W(\ch(E),\ch(F))$. The potential walls $W(\xi,\zeta)$ can be easily computed in terms of the Chern characters $\xi$ and $\zeta $. \begin{enumerate} \item If $\mu(\xi) = \mu(\zeta)$ (where the Mumford slope is interpreted as $\infty$ if the rank is $0$), then the wall $W(\xi,\zeta)$ is the vertical line $s= \mu(\xi)$ (interpreted as the empty set when the slope is infinite). \item Otherwise, without loss of generality assume $\mu(\xi)$ is finite, so that $r\neq 0$. The walls $W(\xi,\zeta)$ and $W(\xi,\xi+\zeta)$ are equal, so we may further reduce to the case where both $\xi$ and $\zeta$ have nonzero rank. Then we may encode $\xi = (r_1,\mu_1,\Delta_1)$ and $\zeta = (r_2,\mu_2,\Delta_2)$ in terms of slope and discriminant instead of $\ch_1$ and $\ch_2$. The wall $W(\xi,\zeta)$ is the semicircle centered at the point $(s,0)$ with $$s = \frac{1}{2}(\mu_1+\mu_2)-\frac{\Delta_1-\Delta_2}{\mu_1-\mu_2}$$ and having radius $\rho$ given by $$\rho^2 = (s-\mu_1)^2-2\Delta_1.$$ \end{enumerate} In the principal case of interest, the Chern character $\xi = (r,\mu,\Delta)$ has nonzero rank $r\neq 0$ and nonnegative discriminant $\Delta\geq 0$. In this case, the potential walls for $\xi$ consist of a vertical wall $s=\mu$ together with two disjoint nested families of semicircles on either side of this line \cite{ABCH}. Specifically, for any $s$ with $|s-\mu| > \sqrt{2\Delta}$, there is a unique semicircular potential wall with center $(s,0)$ and radius $\rho$ satisfying $$\rho^2 = (s-\mu)^2 - 2\Delta.$$ The semicircles are centered along the $s$-axis, with smaller semicircles having centers closer to the vertical wall. Every point in the $(s,t)$-half-plane lies on a unique potential wall for $\xi$. When $r>0$, only the family of semicircles left of the vertical wall is interesting, since an object $E$ with Chern character $\xi$ can only be in categories $\mathcal A_s$ with $s<\mu$. Since the number of Bridgeland walls is finite, there exists a largest semicircular Bridgeland wall $W_{\max}$ to the left of the vertical line $s = \mu$ that contains all other semicircular walls. Furthermore, for every $(s,t)$ with $s< \mu$ and contained outside $W_{\max}$, the moduli space of Bridgeland stable objects in $\mathcal A_s$ with respect to $Z_{s,t}$ and Chern character $\xi$ is isomorphic to the moduli space $M(\xi)$ \cite{ABCH}. We call $W_{\max}$ the \emph{Gieseker wall}. \subsection{A nef divisor on $M(\xi)$}\label{ssec-BayerMacriPlan} Let $(\mathcal A,Z) = \sigma_0 \in W_{\max}$ be a stability condition on the Gieseker wall. Bayer and Macr\`{i} \cite{BayerMacri2} construct a nef divisor $\ell_{\sigma_0}$ on $M(\xi)$ corresponding to $\sigma_0$. They also compute its class and describe geometrically the curves $C\subset M(\xi)$ with $C \cdot \ell_{\sigma_0} = 0$. To describe the class of $\ell_{\sigma_0}$ in $\xi^\perp \cong \mathbb{P}ic M(\xi)$, consider the functional \begin{align*}N^1(M(\xi)) & \to \mathbb{R}\\ \xi' &\mapsto \F{I}m\left( -\frac{Z(\xi')}{Z(\xi)}\right).\end{align*} Since the pairing $(\xi,\zeta) = \chi(\xi\otimes \zeta)$ is nondegenerate, we can write this functional as $(\zeta,-)$ for some unique $\zeta\in \xi^\perp$. In terms of the isomorphism $\xi^\perp \cong \mathbb{P}ic M(\xi)$, we have $\zeta = [\ell_{\sigma_0}].$ Considering $(\zeta,\ch \mathcal{O}_p)$ shows that $\zeta$ has negative rank. Furthermore, if $W_{\max} = W(\xi',\xi)$ (so that $Z(\xi')$ and $Z(\xi)$ are real multiples of one another), then $\zeta$ is a negative rank character in $(\xi')^\perp$. The ray in $N^1(M(\xi))$ determined by $\sigma_0$ depends only on $W_{\max}$, and not the particular choice of $\sigma_0$. A curve $C\subset M(\xi)$ is orthogonal to $\ell_{\sigma_0}$ if and only if two general sheaves parameterized by $C$ are $S$-equivalent with respect to $\sigma_0$. This gives an effective criterion for determining when the Bayer-Macr\`i divisor $\ell_{\sigma_0}$ is an extremal nef divisor. In every case where we compute the ample cone of $M(\xi)$, the divisor $\ell_{\sigma_0}$ is in fact extremal. \section{Admissible decompositions}\label{sec-hp} In this section, we introduce the notion of an admissible decomposition of a Chern character of positive rank. Each such decomposition corresponds to a potential Bridgeland wall. In the cases when we can compute the ample cone, the Gieseker wall will correspond to a certain admissible decomposition. \begin{definition}\label{def-admissible} Let $\xi$ be a stable Chern character of positive rank. A \emph{decomposition} of $\xi$ is a triple $\Xi = (\xi',\xi,\xi'')$ such that $\xi = \xi'+\xi''$. We say $\Xi$ is an \emph{admissible decomposition} if furthermore \begin{enumerate}[label=(D\arabic*)] \item \label{cond-Fstable} $\xi'$ is semistable, \item \label{cond-Qstable} $\xi''$ is stable, \item\label{cond-rank} $0 < \rk(\xi') \leq \rk(\xi)$, \item \label{cond-Fslope} $\mu(\xi') < \mu(\xi)$, and \item \label{cond-slopeDiff} if $\rk(\xi'')>0$, then $\mu(\xi'')-\mu(\xi')<3$. \end{enumerate} \end{definition} \begin{remark} The Chern characters in an admissible decomposition $\Xi$ span a $2$-plane in $K(\mathbb{P}^2)$. We write $W(\Xi)$ for the potential Bridgeland wall where characters in this plane have the same slope. Condition \ref{cond-Fstable} means that $\xi'$ is either semiexceptional or stable. We require $\xi''$ to be stable since this holds in all our examples and makes admissibility work better with respect to elementary modifications; see \S\ref{sec-elementaryMod}. \end{remark} There are a couple numerical properties of decompositions which will frequently arise. \begin{definition}\label{def-numericalProps} Let $\Xi = (\xi',\xi,\xi'')$ be a decomposition. \begin{enumerate} \item $\Xi$ is \emph{coprime} if $\rk(\xi)$ and $c_1(\xi)$ are coprime. \item $\Xi$ is \emph{torsion} if $\rk(\xi'')=0$, and \emph{torsion-free} otherwise. \end{enumerate} \end{definition} The conditions in the definition of an admissible decomposition ensure that there is a well-behaved space of extensions of the form $$0\to F\to E\to Q\to 0$$ with $F\in M(\xi')$ and $Q\in M(\xi'')$. \begin{lemma}\label{existenceOfExtensions} Let $\Xi = (\xi',\xi,\xi'')$ be an admissible torsion-free decomposition. We have $\chi(\xi'',\xi')<0$. In particular, for any $F\in M(\xi')$ and $Q\in M(\xi'')$ there are non-split extensions $$0\to F\to E \to Q \to 0.$$ Furthermore, $\Ext^1(Q,F)$ has the expected dimension $-\chi(\xi'',\xi')$ for any $F\in M(\xi')$ and $Q\in M(\xi'')$. \end{lemma} \begin{proof} From \ref{cond-Fslope} and the torsion-free hypothesis, we have $\mu(\xi)<\mu(\xi'')$. Let $F\in M(\xi')$ and $Q\in M(\xi'')$. By stability, $\Hom(Q,F) = 0$. Using Serre duality with condition \ref{cond-slopeDiff}, we have $\Ext^2(Q,F)=0$. Therefore $\ext^1(Q,F) = -\chi(\xi'',\xi')$ and $\chi(\xi'',\xi')\leq 0$. To prove $\chi(\xi'',\xi')<0$, first suppose $\xi'$ is semiexceptional. Then $$\chi(\xi'',\xi')=\chi(\xi,\xi')-\chi(\xi',\xi')<\chi(\xi,\xi').$$ As in the previous paragraph, $\chi(\xi,\xi')\leq 0$, hence $\chi(\xi'',\xi')<0$. A similar argument works if $\xi''$ is semiexceptional. Assume neither $\xi'$ or $\xi''$ is semiexceptional. Then $-3<\mu(\xi')-\mu(\xi'')<0$ and $\Delta(\xi')+\Delta(\xi'')>1$. Since $P(x)<1$ for $-3< x < 0 $, we conclude $\chi(\xi'',\xi')<0$ by the Riemann-Roch formula. \end{proof} We now introduce a notion of stability for an admissible decomposition $\Xi$. Let $F_{s'}/S'$ (resp. $Q_{s''}/S''$) be a complete flat family of semistable sheaves with Chern character $\xi'$ (resp. $\xi''$), parameterized by a smooth and irreducible base variety. Since $\ext^1(Q_{s''},F_{s'})$ does not depend on $(s',s'')\in S'\times S''$, there is a projective bundle $S$ over $S'\times S''$ such that the fiber over a point $(s',s'')$ is $\mathbb{P}\Ext^1(Q_{s''},F_{s'})$. Then $S$ is smooth, irreducible, and it carries a universal extension sheaf $E_s/S$. We wish to examine the stability properties of the general extension $E_s/S$. If $E_s$ is \mbox{$(\mu$-)(semi)stable} for some $s\in S$, then the general $E_s$ has the same stability property. Since the moduli spaces $M(\xi')$ and $M(\xi'')$ are irreducible, the general $E_s$ will be $(\mu$-)(semi)stable if and only if there exists some extension $$0\to F \to E\to Q\to 0$$ where $F\in M(\xi')$, $Q\in M(\xi'')$, and $E$ is $(\mu$-)(semi)stable. Since $S$ is complete, we do not need to know that $E$ is parameterized by a point of $S$. \begin{definition}\label{def-stableTriple} Let $\Xi$ be an admissible decomposition. We say that $\Xi$ is \emph{generically} \mbox{$(\mu$-)}(semi)stable if there is some extension $$0\to F\to E\to Q\to 0$$ where $F\in M(\xi')$, $Q\in M(\xi'')$, and $E$ is $(\mu$-)(semi)stable. \end{definition} \section{Extremal triples}\label{sec-extremal} We now introduce the decomposition of a Chern character $\xi$ which frequently corresponds to the primary edge of the ample cone of $M(\xi)$. \begin{definition}\label{def-extremal} We call a triple $\Xi=(\xi',\xi,\xi'')$ of Chern characters \emph{extremal} if it is an admissible decomposition of $\xi$ with the following additional properties: \begin{enumerate}[label=(E\arabic*)] \item \label{cond-slopeClose} $\xi'$ and $\xi$ are \emph{slope-close}: we have $\mu(\xi') < \mu(\xi)$, and every rational number in the interval $(\mu(\xi'),\mu(\xi))$ has denominator larger than $\rk(\xi)$. \item \label{cond-discMinimal} $\xi'$ is \emph{discriminant-minimal}: if $\theta'$ is a stable Chern character with $0<\rk(\theta')\leq \rk(\xi)$ and $\mu(\theta') = \mu(\xi')$, then $\Delta(\theta')\geq \Delta(\xi')$. \item \label{cond-rankMinimal} $\xi'$ is \emph{rank-minimal}: if $\theta'$ is a stable Chern character with $\mu(\theta')=\mu(\xi')$ and $\Delta(\theta') = \Delta(\xi')$, then $\rk(\theta')\geq \rk(\xi')$. \end{enumerate} \end{definition} \begin{remark}\label{rem-extremalRemark} If $\Xi$ is an extremal triple, then it is uniquely determined by $\xi$. The wall $W(\Xi)$ thus also only depends on $\xi$. Not every stable character $\xi$ can be decomposed into an extremal triple $\Xi = (\xi',\xi,\xi'')$, but the vast majority can; see Lemma \ref{lem-extremalExist}. Condition \ref{cond-discMinimal} in Definition \ref{def-admissible} is motivated by the formula for the center $(s,0)$ of $W(\Xi)$: $$s = \frac{\mu(\xi')+\mu(\xi)}{2}-\frac{\Delta(\xi')-\Delta(\xi)}{\mu(\xi')-\mu(\xi)}.$$ If $\Delta(\xi')$ decreases while the other invariants are held fixed, then the center of $W(\Xi)$ moves left. Correspondingly, the wall becomes larger. As we are searching for the largest walls, intuitively we should restrict our attention to triples with minimal $\Delta(\xi')$. Similarly, condition \ref{cond-slopeClose} typically helps make the wall $W(\Xi)$ large. In the formula for $s$, the term $$-\frac{\Delta(\xi')-\Delta(\xi)}{\mu(\xi')-\mu(\xi)}$$ will dominate the expression if $\Delta(\xi)$ is sufficiently large and $\mu(\xi')$ is sufficiently close to $\mu(\xi)$. Condition \ref{cond-rankMinimal} forces $\xi'$ to be stable, since semiexceptional characters are multiples of exceptional characters. \end{remark} The next lemma shows the definition of an extremal triple is not vacuous. \begin{lemma}\label{lem-extremalExist} Let $\xi = (r,\mu,\Delta)$ be a stable Chern character, and suppose either \begin{enumerate} \item $\Delta$ is sufficiently large (depending on $r$ and $\mu$) or \item $r\leq 6$. \end{enumerate} Then there is a unique extremal triple $\Xi = (\xi',\xi,\xi'')$. \end{lemma} \begin{proof} Let $(r^\bullet, \mu^\bullet, \Delta^\bullet)$ denote the rank, slope and discriminant of $\xi^\bullet$. The Chern character $\xi'$ is uniquely determined by conditions \ref{cond-Fstable}, \ref{cond-rank}, and \ref{cond-slopeClose}-\ref{cond-rankMinimal}; it depends only on $r$ and $\mu$, and not $\Delta$. Set $\xi''=\xi-\xi'$, and observe that $r''$ and $\mu''$ depend only on $r$ and $\mu$. We must check that $\xi''$ is stable and $\mu''-\mu'<3$ if $r''>0$. If $r''=0$, then $c_1(\xi'')>0$, so stability is automatic. Suppose $r''>0$. Let us show $\mu''-\mu'<3$. By \ref{cond-slopeClose} we have $\mu'\geq\mu-\frac{1}{r}$, so $$r''\mu''=r\mu - r'\mu'\leq (r-r')\mu+\frac{r'}{r}<r''\mu+1$$ and $$\mu''-\mu' < \mu+\frac{1}{r''}-\mu+\frac{1}{r} = \frac{1}{r''}+\frac{1}{r}\leq \frac{3}{2}.$$ If $r\leq 6$, we will see that $\xi''$ is stable in \S\ref{sec-smallRank}. Suppose $\Delta$ is sufficiently large. We have a relation $$r\Delta = r'\Delta'+r''\Delta''-\frac{r'r''}{r}(\mu'-\mu'')^2.$$ The invariants $r',\mu',\Delta',r'',\mu''$ depend only on $r$ and $\mu$. By making $\Delta$ large, we can make $\Delta''$ as large as we want, and thus we can make $\xi''$ stable. \end{proof} It is easy to prove a weak stability result for extremal triples. \begin{proposition}\label{prop-slopeSemistable} Let $\Xi=(\xi',\xi,\xi'')$ be an extremal torsion-free triple. Then $\Xi$ is generically $\mu$-semistable. \end{proposition} \begin{proof} By Lemma \ref{existenceOfExtensions}, there is a non-split extension $$0\to F \to E \to Q\to 0$$ with $F\in M^s(\xi')$ stable and $Q\in M(\xi)$. We will show $E$ is $\mu$-semistable. Since $F$ and $Q$ are torsion-free, $E$ is torsion-free as well. Suppose $E$ is not $\mu$-semistable. Then there is some surjection $E\to C$ with $\mu(C)<\mu(E)$ and $\rk(C)<\rk(E)$. By passing to a suitable quotient of $C$, we may assume $C$ is stable. Using slope-closeness \ref{cond-slopeClose}, we find $\mu(C)\leq \mu(F)$. First assume $\mu(C)<\mu(F)$. By stability, the composition $F\to E\to C$ is zero, and thus $E\to C$ induces a map $Q\to C$. This map is zero by stability, from which we conclude $E\to C$ is zero, a contradiction. Next assume $\mu(C)= \mu(F)$. If $\Delta(C)>\Delta(F)$, then we have an inequality $p_C<p_F$ of reduced Hilbert polynomials, so $F\to C$ is zero by stability and we conclude as in the previous paragraph. On the other hand, $\Delta(C)<\Delta(F)$ cannot occur by the minimality condition \ref{cond-discMinimal}. Finally, suppose $\mu(C) = \mu(F)$ and $\Delta(C) = \Delta(F)$. Since $C$ and $F$ are both stable, any nonzero map $F\to C$ is an isomorphism. Then the composition $E\to C\to F$ with the inverse isomorphism splits the sequence. \end{proof} The following corollary gives the first statement of Theorem \ref{thm-introSt} in the torsion-free case. \begin{corollary}\label{cor-slopeStable} If $\Xi$ is a coprime, torsion-free, extremal triple, then it is generically $\mu$-stable. \end{corollary} \section{Elementary modifications}\label{sec-elementaryMod} Many stability properties of an admissible decomposition $\Xi=(\xi',\xi,\xi'')$ are easier to understand when the discriminant $\Delta(\xi)$ is small. Elementary modifications allow us to reduce to the small discriminant case. \begin{definition} Let $G$ be a coherent sheaf and let $G\to \mathcal{O}_p$ be a surjective homomorphism. Then the kernel $$0\to G'\to G\to \mathcal{O}_p\to 0$$ is called an \emph{elementary modification} of $G$. \end{definition} If $G$ has positive rank, we observe the equalities $$\rk(G') = \rk(G) \qquad \mu(G') = \mu(G) \qquad \Delta(G') = \Delta(G) + \frac{1}{\rk(G)} \qquad \chi(G') =\chi(G)-1.$$ The next lemma is immediate. \begin{lemma} If $G$ is $\mu$-(semi)stable, then any elementary modification of $G$ is $\mu$-(semi)stable. \end{lemma} \begin{warning} Elementary modifications do not generally preserve Gieseker (semi)stability. This is our reason for focusing on $\mu$-stability of extensions. \end{warning} Given a short exact sequence of sheaves, there is a natural induced sequence involving compatible elementary modifications. \begin{proposition-definition}\label{def-elementaryModSequence} Suppose $$0\to F \to E \to Q\to 0$$ is a short exact sequence of sheaves. Let $Q'$ be the elementary modification of $Q$ corresponding to a homomorphism $Q\to \mathcal{O}_p$, and let $E'$ be the elementary modification of $E$ corresponding to the composition $E\to Q\to \mathcal{O}_p$. Then there is a natural short exact sequence $$0\to F \to E'\to Q'\to 0.$$ This sequence is called an \emph{elementary modification} of the original sequence. \end{proposition-definition} \begin{proof} A straightforward argument shows that there is a natural commuting diagram $$\xymatrix{ &&0\ar[d]&0\ar[d]&\\ 0\ar[r]&F\ar@{=}[d]\ar[r]&E'\ar[d]\ar[r]&Q'\ar[d]\ar[r]&0\\ 0\ar[r]&F\ar[r]&E\ar[d]\ar[r]&Q\ar[d]\ar[r]&0\\ &&\mathcal{O}_p\ar[d]\ar@{=}[r]&\mathcal{O}_p\ar[d]&\\ &&0&0& }$$ with exact rows and columns. \end{proof} We similarly extend the notion of elementary modifications to decompositions of Chern characters. \begin{definition} Let $\Xi = (\xi',\xi,\xi'')$ be a decomposition. Let $\Theta = (\theta',\theta,\theta'')$ be the decomposition such that \begin{enumerate} \item $\theta'=\xi'$, \item $\theta$ and $\xi$ have the same rank and slope, and \item $\Delta(\theta) = \Delta(\xi) + \frac{1}{\rk(\xi)}$. \end{enumerate} We call $\Theta$ the \emph{elementary modification} of $\Xi$. If $\Xi$ is admissible, then $\Theta$ is admissible as well. If $\Xi$ and $\Theta$ are admissible decompositions, we say $\Theta$ lies \emph{above} $\Xi$, and write $\Xi\preceq \Theta$, if conditions (1)-(3) are satisfied and $\Delta(\xi)\leq \Delta(\theta)$. Finally, $\Xi$ is \emph{minimal} if it is a minimal admissible decomposition with respect to $\preceq$. \end{definition} The next result follows from the integrality of the Euler characteristic and the Riemann-Roch formula. \begin{lemma} Let $\Xi$ and $\Theta$ be admissible decompositions. Then $\Xi\preceq \Theta$ if and only if $\Theta$ is an iterated elementary modification of $\Xi$. \end{lemma} Extremality is preserved by elementary modifications. \begin{lemma} Suppose $\Xi$ and $\Theta$ are admissible decompositions with $\Xi\preceq \Theta$. If one decomposition is extremal, then the other is as well. \end{lemma} Combining our results so far in this subsection, we obtain the following tool for proving results on generic $\mu$-stability of triples. \begin{proposition}\label{prop-minimalReduction} Suppose $\Xi$ is a minimal admissible decomposition and that $\Xi$ is generically $\mu$-stable. Then any $\Theta$ which lies above $\Xi$ is also generically $\mu$-stable. \end{proposition} \section{Stability of small rank extremal triples}\label{sec-smallRank} The goal of this subsection is to prove the following theorem. \begin{theorem}\label{thm-slopeCloseStable} Let $\Xi = (\xi',\xi,\xi'')$ be an extremal triple with $\rk(\xi)\leq 6$. Then $\Xi$ is generically $\mu$-stable. \end{theorem} By Proposition \ref{prop-minimalReduction}, we only need to consider cases where $\Xi$ is minimal. We also assume $\Xi$ is torsion-free and defer to \S\ref{ssec-torsion} for the torsion case. By twisting, we may assume $0<\mu(\xi)\leq 1$. After these reductions, there are a relatively small number of triples to consider, which we list in Table \ref{table-slopeClose}. For each triple, we also indicate the strategy we will use to prove the triple is generically $\mu$-stable. \begin{center} \renewcommand*{\arraystretch}{1.3} \begin{longtable}{cccccccccccc} \caption[]{The minimal, extremal, torsion-free triples $\Xi = (\xi',\xi,\xi'') = ((r',\mu',\Delta'),(r,\mu,\Delta),(r'',\mu'',\Delta''))$ which must be considered in Theorem \ref{thm-slopeCloseStable}.}\label{table-slopeClose}\\ \toprule $\xi'$ & $\xi$ & $\xi''$ && Strategy &$\qquad$ & $\xi'$ & $\xi$ & $\xi''$ && Strategy\\\midrule \endfirsthead \multicolumn{12}{l}{{\setminusall \it continued from previous page}}\\ \toprule $\xi'$ & $\xi$ & $\xi''$ && Strategy &$\qquad$ & $\xi'$ & $\xi$ & $\xi''$ && Strategy\\\midrule \endhead \bottomrule \multicolumn{12}{r}{{\setminusall \it continued on next page}} \\ \endfoot \bottomrule \endlastfoot $(1,0,0)$ & $(2,\frac{1}{2},\frac{3}{8})$ & $(1,1,1)$ && Coprime &&$(2,\frac{1}{2},\frac{3}{8})$&$(5,\frac{3}{5},\frac{12}{25})$&$(3,\frac{2}{3},\frac{5}{9})$ &&Coprime\\ $(1,0,0)$ & $(3,\frac{1}{3},\frac{5}{9})$ & $(2,\frac{1}{2},\frac{7}{8})$ && Coprime &&$(4,\frac{3}{4},\frac{21}{32})$ & $(5,\frac{4}{5},\frac{18}{25})$ & (1,1,1) && Coprime\\ $(2,\frac{1}{2},\frac{3}{8})$ & $(3,\frac{2}{3},\frac{5}{9})$ & $(1,1,1)$ && Coprime &&$(1,0,0)$ & $(6,\frac{1}{6},\frac{55}{72})$ & $(5,\frac{1}{5},\frac{23}{25})$ && Coprime\\ $(1,0,0)$ & $(4,\frac{1}{4},\frac{21}{32})$ & $(3,\frac{1}{3},\frac{8}{9})$ && Coprime && $(4,\frac{1}{4},\frac{21}{32})$ & $(6,\frac{1}{3},\frac{5}{9})$ & $(2,\frac{1}{2},\frac{3}{8})$ && Complete\\ $(3,\frac{1}{3},\frac{5}{9})$ & $(4,\frac{1}{2},\frac{5}{8})$ & $(1,1,1)$ && Complete && $(5,\frac{2}{5},\frac{12}{25})$ & $(6,\frac{1}{2},\frac{17}{24})$ & $(1,1,2)$ && Prop. \ref{prop-rank6adHoc}\\ $(3,\frac{2}{3},\frac{5}{9})$ & $(4,\frac{3}{4},\frac{21}{32})$ & $(1,1,1)$ && Coprime && $(5,\frac{3}{5},\frac{12}{25})$ & $(6,\frac{2}{3},\frac{5}{9})$ & $(1,1,1)$ && Complete\\ $(1,0,0)$ & $(5,\frac{1}{5},\frac{18}{25})$ & $(4,\frac{1}{4},\frac{29}{32})$ && Coprime && $(5,\frac{4}{5},\frac{18}{25})$ & $(6,\frac{5}{6},\frac{55}{72})$ & $(1,1,1)$ && Coprime\\ $(3,\frac{1}{3},\frac{5}{9})$ &$(5,\frac{2}{5},\frac{12}{25})$ & $(2,\frac{1}{2},\frac{3}{8})$ && Coprime \\ \end{longtable} \end{center} Observing that $\xi''$ is always stable in Table \ref{table-slopeClose} completes the proof of Lemma \ref{lem-extremalExist} as promised. The triples labelled ``Coprime'' are all generically $\mu$-stable by Corollary \ref{cor-slopeStable}. We turn next to the triples labelled ``Complete.'' \begin{definition} An admissible decomposition $\Xi$ is called \emph{complete} if the general $E\in M(\xi)$ can be expressed as an extension $$0\to F \to E \to Q\to 0$$ with $F\in M(\xi')$ and $Q\in M(\xi'')$. \end{definition} \begin{remark} Suppose $\Xi$ is admissible and generically semistable. Recall the universal extension sheaf $E_s/S$ discussed preceding Definition \ref{def-stableTriple}. If $U\subset S$ is the open subset parameterizing semistable sheaves, then $\Xi$ is complete if and only if the moduli map $U\to M(\xi)$ is dominant. By generic smoothness, $E_s/U$ is a complete family of semistable sheaves over a potentially smaller dense open subset. \end{remark} If $\xi$ is stable, then the general sheaf in $M(\xi)$ is $\mu$-stable by a result of Dr\'{e}zet and Le Potier \cite[4.12]{DLP}. Thus if $\Xi$ is complete, then $\Xi$ is generically $\mu$-stable. \begin{proposition}\label{prop-completeTriples} Let $\Xi$ be one of the three triples in Table \ref{table-slopeClose} labelled ``Complete.'' Then $\Xi$ is complete, and in particular generically $\mu$-stable. \end{proposition} \begin{proof} First suppose $\Xi=(\xi',\xi,\xi'')$ is one of $$((3,\tfrac{1}{3},\tfrac{5}{9}),(4,\tfrac{1}{2},\tfrac{5}{8}),(1,1,1)) \qquad \otimesxtrm{or} \qquad ((4,\tfrac{1}{4},\tfrac{21}{32}),(6,\tfrac{1}{3},\tfrac{5}{9}),(2,\tfrac{1}{2},\tfrac{3}{8})).$$ Let $E$ be a $\mu$-stable sheaf of character $\xi$, and let $Q\in M(\xi'')$ be semistable. We have $\chi(E,Q)>0$ in either case, which implies $\hom(E,Q)>0$ by stability. Pick a nonzero homomorphism $f:E\to Q$, and let $R\subset Q$ be the image of $f$. By stability considerations, $R$ must have the same rank and slope as $Q$, and $\Delta(R)\geq \Delta(Q)$. Letting $F\subset E$ be the kernel of $f$, we find that $\rk(F) = \rk(\xi')$, $\mu(F)=\mu(\xi')$, and $\Delta(F)\leq \Delta(\xi')$, with equality if and only if $f$ is surjective. Furthermore, $F$ is $\mu$-semistable. Indeed, if there is a subsheaf $G\subset F$ with $\mu(G)>\mu(F)$, then $\mu(G)\geq \mu(E)$ by slope-closeness \ref{cond-slopeClose}, so $G\subset E$ violates $\mu$-stability of $E$. Then discriminant minimality \ref{cond-discMinimal} forces $\ch F = \xi'$. Furthermore, since $\rk(\xi')$ and $c_1(\xi')$ are coprime, $F$ is actually semistable. Thus $E$ is expressed as an extension $$0\to F\to E\to Q\to 0$$ of semistable sheaves as required. For the final triple $((5,\frac{3}{5},\frac{12}{25}),(6,\frac{2}{3},\frac{5}{9}),(1,1,1))$ a slight modification to the previous argument is needed. Fix a $\mu$-stable sheaf $E$ of character $\xi$. This time $\chi(\xi,\xi'')=0$, so the expectation is that if $Q\in M(\xi'')$ is general, then there is no nonzero map $E\to Q$. Consider the locus $$D_E = \{Q\in M(\xi''):\hom(E,Q)\neq 0\}.$$ Then either $D_E = M(\xi'')$ or $D_E$ is an effective divisor, in which case we can compute its class to show $D_E$ is nonempty. Either way, there is some $Q\in M(\xi'')$ which admits a nonzero homomorphism $E\to Q$. The argument can now proceed as in the previous cases. \end{proof} The next proposition treats the last remaining case, completing the proof of Theorem \ref{thm-slopeCloseStable}. \begin{proposition}\label{prop-rank6adHoc} The triple $\Xi =(\xi',\xi,\xi'')=((5,\frac{2}{5},\frac{12}{25}),(6,\frac{1}{2},\frac{17}{24}), (1,1,2))$ is generically $\mu$-stable \end{proposition} \begin{proof} Observe that $\xi'$ is the Chern character of the exceptional bundle $F = E_{2/5}$ and $\xi''$ is the Chern character of an ideal sheaf $Q=I_Z(1)$, where $Z$ has degree $2$. Let $Q_{s''}/M(\xi'')$ be the universal family. Then the projective bundle $S$ over $M(\xi'')$ with fibers $\mathbb{P}\Ext^1(Q_{s''},F)$ is smooth and irreducible of dimension $$\dim S = \dim M(\xi'')-\chi(\xi'',\xi')-1=14,$$ and there is a universal extension $E_s/S$. Every $E_s$ is $\mu$-semistable by Proposition \ref{prop-slopeSemistable}. A simple computation shows $\Hom(F,Q_{s''})=0$ for every $Q_{s''}$. If $E$ is any sheaf which sits as an extension $$0\to F\to E\to Q_{s''}\to 0,$$ then we apply $\Hom(F,-)$ to see $\Hom(F,E) \cong \Hom(F,F) = \mathbb{C}$. Thus the homomorphism $F\to E$ is unique up to scalars, the sheaf $Q_{s''}$ is determined as the cokernel, and since $Q_{s''}$ is simple the corresponding extension class in $\Ext^1(Q_{s''},F)$ is determined up to scalars. We find that distinct points of $S$ parameterize non-isomorphic sheaves. A straightforward computation further shows that the Kodaira-Spencer map $T_sS \to \Ext^1(E_s,E_s)$ is injective for every $s\in S$. We now proceed to show that the general $E_s$ also satisfies stronger notions of stability. \emph{Step 1: the general $E_s$ is semistable}. If $E_s$ is not semistable, it has a Harder-Narasimhan filtration of length $\ell\geq 2$, and all factors have slope $\frac{1}{2}$. For each potential set of numerical invariants of a Harder-Narasimhan filtration, we check that the corresponding Shatz stratum of $s\in S$ such that the Harder-Narasimhan filtration has that form has positive codimension. There are only a handful of potential numerical invariants of the filtration. A non-semistable $E_s$ has a semistable subsheaf $G$ with $\mu(G) = \mu(E) = \frac{1}{2}$ and $\Delta(G) < \Delta(E) = \frac{17}{24}$. Then the Chern character of $G$ must be one of \begin{equation}\tag{$\ast$} (2,\tfrac{1}{2},\tfrac{3}{8}), \qquad (4,\tfrac{1}{2},\tfrac{3}{8}), \qquad \otimesxtrm{or} \qquad (4,\tfrac{1}{2},\tfrac{5}{8}).\end{equation} We can rule out the first two cases immediately by an ad hoc argument. In either of these cases $E$ has a subsheaf isomorphic to $T_{\mathbb{P}^2}(-1)$. Then there is a sequence $$0\to T_{\mathbb{P}^2}(-1)\to E\to R\to 0.$$ Applying $\Hom(F,-)$, we see $\Hom(F,T_{\mathbb{P}^2}(-1))$ injects into $\Hom(F,E)=\mathbb{C}$. But $\chi(F,T_{\mathbb{P}^2}(-1))=3$, so this is absurd. Thus the only Shatz stratum we must consider is the locus of sheaves with a filtration $$0 \subset G_1 \subset G_2 = E_s$$ having $\ch \mathrm{gr}_1 = \zeta_1 := (4,\frac{1}{2},\frac{5}{8})$ and $\ch \mathrm{gr}_2 := \zeta_2 = (2,\frac{1}{2},\frac{7}{8})$. Let $$\Sigma = \Flag(E/S;\zeta_1,\zeta_2) \xrightarrow{\pi} S$$ be the relative flag variety parameterizing sheaves with a filtration of this form. By the uniqueness of the Harder-Narasimhan filtration, $\pi$ is injective, and its image is the Shatz stratum. The differential of $\pi$ at a point $t = (s,G_1)\in \Sigma$ can be analyzed via the exact sequence $$0 \to \Ext^0_+(E_s,E_s)\to T_t \Sigma\xrightarrow{T_t\pi} T_s S\xrightarrow{\omega_+} \Ext_+^1(E_s,E_s).$$ We have $\Ext_+^0(E_s,E_s)=0$ by \cite[Proposition 15.3.3]{LePotierLectures}, so $T_t\pi$ is injective and $\pi$ is an immersion. The codimension of the Shatz stratum near $s$ is at least $\rk \omega_+$. The map $\omega_+$ is the composition $T_sS\to \Ext^1(E_s,E_s)\to \Ext^1_+(E_s,E_s)$ of the Kodaira-Spencer map with the canonical map from the long exact sequence of $\Ext_{\pm}$. The Kodaira-Spencer map is injective, and $\Ext^1(E_s,E_s)\to \Ext_+^1(E_s,E_s)$ is surjective since $\Ext_-^2(E_s,E_s)=0$. We have $$\dim T_sS = 14, \qquad \ext^1(E_s,E_s)=16, \qquad \otimesxtrm{and} \qquad \ext^1_+(E_s,E_s)=-\chi(\mathrm{gr}_1,\mathrm{gr}_2) = 4,$$ so we conclude $\rk \omega_+\geq 2$. Therefore the Shatz stratum is a proper subvariety of $S$. We conclude $\Xi$ is generically semistable. \emph{Step 2: the general $E_s$ is $\mu$-stable}. Note that a semistable sheaf in $M(\xi)$ is automatically stable. Then the moduli map $S\to M^s(\xi)$ is injective, and its image has codimension $2$ in $M^s(\xi)$. If a sheaf $E\in M^s(\xi)$ is not $\mu$-stable, then there is a filtration $$0\subset G_1 \subset G_2 = E$$ such that the quotients $\mathrm{gr}_i$ are semistable of slope $\frac{1}{2}$ and $\Delta(\mathrm{gr}_1) > \Delta(E) > \Delta(\mathrm{gr}_2)$ (see the proof of \cite[Theorem 4.11]{DLP}). Then as in the previous step $\zeta_2 = \ch(\mathrm{gr}_2)$ is one of the characters $(\ast)$, and $\zeta_1 = \ch(\mathrm{gr}_1)$ is determined by $\zeta_2$. For each of the three possible filtrations, the Shatz stratum in $M^s(\xi)$ of sheaves with a filtration of the given form has codimension at least $\ext^1_+(E,E) = -\chi(\mathrm{gr}_1,\mathrm{gr}_2)$. When $\zeta_2 = (4,\frac{1}{2},\frac{3}{8})$ we compute $-\chi(\mathrm{gr}_1,\mathrm{gr}_2) = 6$, and when $\zeta_2 = (4,\frac{1}{2},\frac{5}{8})$ we have $-\chi(\mathrm{gr}_1,\mathrm{gr}_2) = 4$. In particular, the corresponding Shatz strata have codimension bigger than $2$. On the other hand, for $\zeta_2 = (2,\frac{1}{2},\frac{3}{8})$ we only find the stratum has codimension at least $2$, and it is a priori possible that it contains the image of $S\to M^s(\xi)$. To get around this final problem, we must show that the general sheaf $E_s$ parameterized by $S$ does not admit a nonzero map $E_s\to T_{\mathbb{P}^2}(-1)$. This can be done by an explicit calculation. Put $Q = I_Z(1)$, where $Z = V(x,y^2)$. By stability, $\Hom(Q,T(-1))=0$, so there is an exact sequence $$\xymatrix{ 0 \ar[r]& \Hom(E_s,T_{\mathbb{P}^2}(-1))\ar[r] & \Hom(F,T_{\mathbb{P}^2}(-1))\ar[r]^{f} \ar@{=}[d] & \Ext^1(Q,T_{\mathbb{P}^2}(-1))\ar@{=}[d]\\ &&\mathbb{C}^3&\mathbb{C}^{4}&}$$ and we must see $f$ is injective. The map $f$ is the contraction of the canonical map $$\Ext^1(Q,F) \otimes \Hom(F,T_{\mathbb{P}^2}(-1))\to \Ext^1(Q,T_{\mathbb{P}^2}(-1))$$ corresponding to the extension class of $E$ in $\Ext^1(Q,F)$. This canonical map can be explicitly computed using the standard resolutions $$\xymatrix@R=1mm{ 0 \ar[r] &\mathcal{O}_{\mathbb{P}^2}(-2)\ar[r] &\mathcal{O}_{\mathbb{P}^2}^6 \ar[r] &F \ar[r] &0\\ 0\ar[r] &\mathcal{O}_{\mathbb{P}^2}(-2) \ar[r]&\mathcal{O}_{\mathbb{P}^2}(-1)\oplus \mathcal{O}_{\mathbb{P}^2} \ar[r] & Q\ar[r]& 0\\ 0\ar[r]& \mathcal{O}_{\mathbb{P}^2}(-1)\ar[r]& \mathcal{O}_{\mathbb{P}^2}^3 \ar[r]& T_{\mathbb{P}^2}(-1)\ar[r]& 0}$$ with the special form of $Q$ simplifying the calculation. Injectivity of $f$ for a general $E_s$ follows easily from this computation. \end{proof} \section{Curves of extensions}\label{sec-curves} \subsection{General results} Let $F$ and $Q$ be sheaves, and suppose the general extension $E$ of $Q$ by $F$ is semistable of Chern character $\xi$. In this section, we study the moduli map $$\mathbb{P} \Ext^1(Q,F)\dashrightarrow M(\xi).$$ In particular, we would like to be able to show this map is nonconstant. \begin{definition} Let $\Xi$ be a generically semistable admissible decomposition. We say $\Xi$ \emph{gives curves} if for a general $F\in M(\xi')$ and $Q\in M(\xi'')$, the map $\mathbb{P} \Ext^1(Q,F)\dashrightarrow M(\xi)$ is nonconstant. \end{definition} There are three essential ways that $\Xi$ could fail to give curves. \begin{enumerate} \item If $-\chi(\xi'',\xi') = 1$, then $\mathbb{P}\Ext^1(Q,F)$ is a point. \item Sheaves parameterized by $\mathbb{P} \Ext^1(Q,F)$ might all be strictly semistable and $S$-equivalent. \item The sheaves parameterized by $\mathbb{P}\Ext^1(Q,F)$ might all be isomorphic. \end{enumerate} Possibility (1) is easy to check for any given triple. If $\Xi$ is generically stable, then possibility (2) cannot arise when $F$ and $Q$ are general, so this is also easy to rule out. The third case requires the most work to deal with. \begin{lemma}\label{lem-nonIsoExtensions} Let $F$ and $Q$ be simple sheaves with $\Hom(F,Q) = 0$. Then distinct points of $\mathbb{P}\Ext^1(Q,F)$ parameterize nonisomorphic sheaves. \end{lemma} \begin{proof} Suppose $E$ is a sheaf which can be realized as an extension $$0\to F\to E\to Q\to 0.$$ Since $F$ is simple and $\Hom(F,Q) = 0$ we find $\hom(F,E) = 1$. Similarly, since $Q$ is simple and $\Hom(F,Q)=0$, we have $\hom(E,Q)=1$. This means that the corresponding class in $\mathbb{P}\Ext^1(Q,F)$ depends only on the isomorphism class of $E$. \end{proof} The lemma gives us a simple criterion for proving a triple $\Xi$ gives curves. \begin{proposition}\label{prop-curveCriterion} Let $\Xi=(\xi',\xi,\xi'')$ be an admissible, generically stable triple, and assume $\xi'$ is stable. Suppose either \begin{enumerate} \item $\Xi$ is not minimal, or \item $-\chi(\xi'',\xi') \geq 2$. \end{enumerate} If $\Hom(F,Q)=0$ for a general $F\in M(\xi')$ and $Q\in M(\xi'')$, then $\Xi$ gives curves. \end{proposition} \begin{proof} If $\Xi$ is not minimal, then $-\chi(\xi'',\xi')\geq 2$ holds automatically. Indeed, since $\Xi$ is not minimal it is an elementary modification of another admissible triple $\Theta = (\theta',\theta,\theta'')$. Then $\chi(\xi'',\xi')<\chi(\theta'',\theta')<0$ by the Riemann-Roch formula and Lemma \ref{existenceOfExtensions}. Since $\xi'$ and $\xi''$ are stable, we can choose stable sheaves $F\in M(\xi')$ and $Q\in M(\xi'')$ such that $\Hom(F,Q) = 0$ and the general extension of $Q$ by $F$ is stable. By Lemma \ref{lem-nonIsoExtensions}, $\Xi$ gives curves. \end{proof} We also observe that elementary modifications behave well with respect to the notion of giving curves. \begin{lemma}\label{lem-curveBump} Suppose $\Xi$ is admissible and generically $\mu$-stable. If $\Xi \preceq \Theta$ and $\Xi$ gives curves, then $\Theta$ gives curves. \end{lemma} \begin{proof} Suppose $\Theta$ is obtained from $\Xi$ by a single elementary modification. Let $F\in M(\xi')$ and $Q\in M(\xi'')$ be general. Take $U \subset \mathbb{P}\Ext^1(Q,F)$ to be the dense open subset parameterizing $\mu$-stable sheaves $E$ which are locally free at a general fixed point $p\in \Supp Q$. Let $Q\to \mathcal{O}_p$ be a surjective homomorphism. Given any extension $$0\to F \to E \to Q \to 0$$ corresponding to a point of $U$, we get an exact sequence of compatible elementary modifications $$0\to F\to E'\to Q'\to 0$$ as in Definition \ref{def-elementaryModSequence}. As $E$ can be recovered from $E'$ and the map $U\to M(\xi)$ is nonconstant, we conclude that the map $\mathbb{P}\Ext^1(Q',F)\dashrightarrow M(\theta)$ is nonconstant. Thus $\Theta$ gives curves. \end{proof} \subsection{Curves from coprime triples with large discriminant} Our next result provides the dual curves we will need to prove Theorem \ref{thm-asymptotic}. \begin{theorem}\label{thm-curves} Let $\Xi = (\xi',\xi,\xi'')$ be a coprime extremal triple, and suppose $\Delta(\xi)$ is sufficiently large, depending on $\rk(\xi)$ and $\mu(\xi)$. Then $\Xi$ gives curves. \end{theorem} \begin{proof} The triple $\Xi$ is not minimal since $\Delta(\xi)$ is large. By Proposition \ref{prop-curveCriterion}, we only need to show that if $F\in M(\xi')$ and $Q\in M(\xi'')$ are general and $\Delta(\xi)$ is sufficiently large, then $\Hom(F,Q) = 0$. Fix a general $F\in M(\xi')$ an $Q\in M(\xi'')$. If $\Hom(F,Q)\neq 0$, choose a nonzero homomorphism $F\to Q$. We can find a surjective homomorphism $Q\to \mathcal{O}_p$ such that $F\to Q\to \mathcal{O}_p$ is also surjective. Then applying $\Hom(F,-)$ to the elementary modification sequence $$0\to Q'\to Q\to \mathcal{O}_p\to 0$$ we find that $\Hom(F,Q')$ is a proper subspace of $\Hom(F,Q)$. Repeating this process, we can find some $\Theta \succeq \Xi$ such that $\Hom(F,Q)=0$ for general $F\in M(\theta')$ and $Q\in M(\theta'')$. Then $\Theta$ gives curves, and by Lemma \ref{lem-curveBump} any $\Lambda \succeq \Theta$ also gives curves. \end{proof} \subsection{Curves from small rank triples} We now discuss extremal curves in the moduli space $M(\xi)$ when the rank is small. For all but a handful of characters $\xi$ we can apply the next theorem. \begin{theorem}\label{thm-smallRankCurves}\label{thm-curvessmall} Let $\Xi = (\xi',\xi,\xi'')$ be an extremal triple with $\rk(\xi)\leq 6$. Suppose $\chi(\xi',\xi'')\leq 0$. Then $\Xi$ gives curves. \end{theorem} \begin{proof} As with the proof of Theorem \ref{thm-slopeCloseStable}, we assume $0<\mu(\xi)\leq 1$. By Lemma \ref{lem-curveBump}, it is enough to consider triples $\Xi$ such that any admissible $\Theta$ with $\Theta \prec \Xi$ has $\chi(\theta',\theta'')>0$. We also assume $\Xi$ is torsion-free, and handle the torsion case in \S\ref{ssec-torsion}. We list the relevant triples together with $\chi(\xi',\xi'')$ in Table \ref{table-curves}. \begin{center} \renewcommand*{\arraystretch}{1.3} \begin{longtable}{cccccccccccc} \caption[]{Triples to be considered for the proof of Theorem \ref{thm-smallRankCurves}.}\label{table-curves}\\ \toprule $\xi'$ & $\xi$ & $\xi''$ && $\chi(\xi',\xi'')$ &$\qquad$ & $\xi'$ & $\xi$ & $\xi''$ && $\chi(\xi',\xi'')$\\\midrule \endfirsthead \multicolumn{12}{l}{{\setminusall \it continued from previous page}}\\ \toprule $\xi'$ & $\xi$ & $\xi''$ && $\chi(\xi',\xi'')$ &$\qquad$ & $\xi'$ & $\xi$ & $\xi''$ && $\chi(\xi',\xi'')$\\\midrule \endhead \bottomrule \multicolumn{12}{r}{{\setminusall \it continued on next page}} \\ \endfoot \bottomrule \endlastfoot $(1,0,0)$ & $(2,\frac{1}{2},\frac{11}{8})$ & $(1,1,3)$ && $0$ &&$(2,\frac{1}{2},\frac{3}{8})$&$(5,\frac{3}{5},\frac{17}{25})$&$(3,\frac{2}{3},\frac{8}{9})$ &&0\\ $(1,0,0)$ & $(3,\frac{1}{3},\frac{11}{9})$ & $(2,\frac{1}{2},\frac{15}{8})$ && $0$ &&$(4,\frac{3}{4},\frac{21}{32})$ & $(5,\frac{4}{5},\frac{18}{25})$ & (1,1,1) && $-1$\\ $(2,\frac{1}{2},\frac{3}{8})$ & $(3,\frac{2}{3},\frac{8}{9})$ & $(1,1,2)$ && $-1$ &&$(1,0,0)$ & $(6,\frac{1}{6},\frac{79}{72})$ & $(5,\frac{1}{5},\frac{33}{25})$ && 0\\ $(1,0,0)$ & $(4,\frac{1}{4},\frac{37}{32})$ & $(3,\frac{1}{3},\frac{14}{9})$ && 0 && $(4,\frac{1}{4},\frac{21}{32})$ & $(6,\frac{1}{3},\frac{13}{18})$ & $(2,\frac{1}{2},\frac{7}{8})$ && $-1$\\ $(3,\frac{1}{3},\frac{5}{9})$ & $(4,\frac{1}{2},\frac{7}{8})$ & $(1,1,2)$ && $-1$ && $(5,\frac{2}{5},\frac{12}{25})$ & $(6,\frac{1}{2},\frac{17}{24})$ & $(1,1,2)$ && $-2$\\ $(3,\frac{2}{3},\frac{5}{9})$ & $(4,\frac{3}{4},\frac{21}{32})$ & $(1,1,1)$ && 0 && $(5,\frac{3}{5},\frac{12}{25})$ & $(6,\frac{2}{3},\frac{13}{18})$ & $(1,1,2)$ && $-4$\\ $(1,0,0)$ & $(5,\frac{1}{5},\frac{28}{25})$ & $(4,\frac{1}{4},\frac{45}{32})$ && 0 && $(5,\frac{4}{5},\frac{18}{25})$ & $(6,\frac{5}{6},\frac{55}{72})$ & $(1,1,1)$ && $-2$\\ $(3,\frac{1}{3},\frac{5}{9})$ &$(5,\frac{2}{5},\frac{17}{25})$ & $(2,\frac{1}{2},\frac{7}{8})$ && $-1$ \\ \end{longtable} \end{center} Every triple in the table satisfies $\chi(\xi'',\xi')\leq -2$. To apply Proposition \ref{prop-curveCriterion}, we need to see that if $F\in M(\xi')$ and $Q\in M(\xi'')$ are general, then $\Hom(F,Q)= 0$. This can easily be checked case by case via standard sequences or by Macaulay2. We omit the details.\end{proof} \subsection{Sporadic small rank triples}\label{ssec-sporadic} Here we discuss the few sporadic Chern characters not addressed by Theorem \ref{thm-smallRankCurves}. Suppose $\xi$ has positive height and $\rk(\xi)\leq 6$, and let $\Xi = (\xi',\xi,\xi'')$ be extremal. If $\Xi$ is torsion-free and $\chi(\xi',\xi'')>0$, then $\xi' = (1,0,0)$ and $$\xi\in \{(2,\tfrac{1}{2},\tfrac{7}{8}),(3,\tfrac{1}{3},\tfrac{8}{9}),(4,\tfrac{1}{4},\tfrac{29}{32}),(5,\tfrac{1}{5},\tfrac{23}{25}),(6,\tfrac{1}{6},\tfrac{67}{72})\}.$$ That is, $$\xi = (r,\tfrac{1}{r},P(-\tfrac{1}{r})+\tfrac{1}{r})$$ for some $r$ with $2\leq r \leq 6$. In this case, we have $\chi(\xi',\xi) = 2$, so the general sheaf $E\in M(\xi)$ admits \emph{two} maps from $\mathcal{O}_{\mathbb{P}^2}$. Along the wall $W(\Xi)$, the destabilizing subobject of $E$ should therefore be $\mathcal{O}_{\mathbb{P}^2}^2$ instead of $\mathcal{O}_{\mathbb{P}^2}$. Furthermore, assuming there are sheaves $E\in M(\xi)$ which are Bridgeland stable just outside $W(\Xi)$, the wall $W(\Xi)$ must be the collapsing wall. We will show in the next section that $W(\Xi)$ is actually the Gieseker wall. This implies that the primary edges of the ample and effective cones of divisors on $M(\xi)$ coincide. Our study of the effective cone in \cite{CoskunHuizengaWoolf} easily implies the next result. \begin{proposition}\label{prop-sporadic} Let $r\geq 2$, and let $\Xi=(\xi',\xi,\xi'')$ be the admissible decomposition with $\xi' = (2,0,0)$ and $\xi = (r,\tfrac{1}{r},P(-\tfrac{1}{r})+\tfrac{1}{r})$. Then $\Xi$ is complete, and it gives curves. \end{proposition} While we had to modify the original extremal triple $\Xi$ in order to make it give curves, note that the wall $W(\Xi)$ is unchanged by this modification. When showing that $W(\Xi)$ is the largest wall in \S\ref{sec-ample}, we will not need to handle these cases separately. In the case of the Chern character $\xi = (6,\frac{1}{3},\frac{13}{18})$, Theorem \ref{thm-smallRankCurves} shows that the corresponding extremal triple $\Xi$ gives curves. However, the wall $W(\Xi)$ is actually empty. In this case $\chi(\mathcal{O}_{\mathbb{P}^2},E) = 5$ for $E\in M(\xi)$ and the Chern character $\xi' = (5,0,0)$ will correspond to the primary edges of both the effective and ample cones. Again, curves dual to this edge of the effective cone are given by \cite{CoskunHuizengaWoolf}. \begin{proposition}\label{prop-sporadic2} The admissible decomposition $\Xi = ((5,0,0),(6,\frac{1}{3},\frac{13}{18}),(1,2,6))$ is complete and gives curves. \end{proposition} \subsection{Torsion triples}\label{ssec-torsion} Let $\Xi=(\xi',\xi,\xi'')$ be an extremal torsion triple, and let $r=\rk(\xi) = \rk(\xi')$. We have $\mu(\xi)-\mu(\xi') = \frac{1}{r}$ by the slope-closeness condition \ref{cond-slopeClose}. Write $$\mu(\xi') = \frac{a}{b} \qquad \otimesxtrm{and}\qquad \mu(\xi) = \frac{c}{d}$$ in lowest terms. The numbers $\mu(\xi')$ and $\mu(\xi)$ are consecutive terms in the \emph{Farey sequence} of order $r$, so $\mu(\xi) - \mu(\xi') = \frac{1}{bd}$ and we deduce that $bd=r$. Also, the \emph{mediant} $$\frac{a+c}{b+d}$$ must have denominator $b+d$ larger than $r$. The two conditions $bd = r$ and $b+d>r$ together imply that either $b=1$ or $d=1$. That is, either $\mu(\xi')$ or $\mu(\xi)$ is an integer. If $\mu(\xi')$ is an integer, then by discriminant minimality \ref{cond-discMinimal} and rank minimality \ref{cond-rankMinimal} we have $\xi' = (1,\mu(\xi'),0)$. Thus $r=1$, and in every case $\mu(\xi)$ is also an integer. We may assume $\mu(\xi) = 1$. Consider the triple $\Xi$ with $$\xi' = (r,1-\tfrac{1}{r},P(-\tfrac{1}{r})) \qquad \otimesxtrm{and} \qquad \xi = (r,1,1).$$ The character $\xi''$ is then $\ch \mathcal{O}_L$, where $L\subset \mathbb{P}^2$ is a line. We have $\chi(\xi,\xi'')=0$, and $\Xi$ is complete (hence generically $\mu$-stable) by a similar argument to Proposition \ref{prop-completeTriples}. If $r\geq 2$, then $$\dim M(\xi')+\dim M(\xi'')=(r^2-3r+2)+2< r^2+1 = \dim M(\xi),$$ so $\Xi$ must give curves. When $r=1$, for any $p\in L$ there is a sequence $$0\to \mathcal{O}_{\mathbb{P}^2}\to I_p(1) \to \mathcal{O}_L\to 0,$$ so $\Xi$ gives curves when $r=1$ as well. Applying elementary modifications, we conclude our discussion with the following. \begin{proposition} Let $\Xi = (\xi',\xi,\xi'')$ be a torsion extremal triple. If $\xi$ is not the Chern character of a line bundle, then $\Xi$ is generically $\mu$-stable and gives curves. \end{proposition} \section{The ample cone}\label{sec-ample} \subsection{Notation} We begin by fixing notation for the rest of the paper. Let $\xi$ be a stable Chern character of positive height. We assume one of the following three hypotheses hold: \begin{enumerate}[label=(H\arabic*)] \item \label{hyp-asymptotic} $\rk(\xi)$ and $c_1(\xi)$ are coprime and $\Delta(\xi)$ is sufficiently large, \item \label{hyp-smallRank} $\rk(\xi) \leq 6$ and $\xi$ is not a twist of $(6,\frac{1}{3},\frac{13}{18})$, or \item \label{hyp-exceptionalCase} $\xi = (6,\frac{1}{3},\frac{13}{18})$. \end{enumerate} Suppose we are in case \ref{hyp-asymptotic} or \ref{hyp-smallRank}. There is an extremal triple $\Xi = (\xi',\xi,\xi'')$. Either $\Xi$ gives curves or we are in one of the cases of Proposition \ref{prop-sporadic}, in which case there is a decomposition of $\xi$ which gives curves and has the same corresponding wall. As discussed in \S\ref{ssec-BayerMacriPlan}, to show the primary edge of the ample cone corresponds to $W(\Xi)$ it will be enough to show that $W(\Xi)$ is the Gieseker wall $W_{\max}$. Note that $W_{\max}$ cannot be strictly nested inside $W(\Xi)$, since then by our work so far there are sheaves $E\in M(\xi)$ destabilized along $W(\Xi)$. We must show $W_{\max}$ is not larger than $W(\Xi)$. Let $E\in M(\xi)$ be a sheaf which is destabilized along some wall. For any $(s,t)$ on the wall we have an exact sequence $$0\to F\to E\to Q\to 0$$ of $\sigma_{s,t}$-semistable objects of the same slope, where the sequence is exact in any of the corresponding categories $\mathcal A_s$ along the wall. Above the wall, $\mu_{s,t}(F)<\mu_{s,t}(E)$, and below the wall the inequality is reversed. Let $\Theta = (\theta',\theta,\theta'') = (\ch F,\ch E,\ch Q)$ be the corresponding decomposition of $\xi = \theta$, so that the wall is $W(\Theta)$. Our job is to show that $W(\Theta)$ is no larger than $W(\Xi)$ by imposing numerical restrictions on $\theta'$. We begin by imposing some easy restrictions on $\theta'$. \begin{lemma}\label{lem-boundsTrivial} The object $F$ is a nonzero torsion-free sheaf, so $\rk(F)\geq 1$. We have $\mu(F) < \mu(E)$, and every Harder-Narasimhan factor of $F$ has slope at most $\mu(E)$. \end{lemma} \begin{proof} Fix a category $\mathcal A_s$ along $W(\Theta)$. Taking cohomology sheaves of the destabilizing sequence of $E$, we get a long exact sequence $$0\to {\rm H}^{-1}(F)\to 0 \to {\rm H}^{-1}(Q) \to {\rm H}^0(F)\to {\rm H}^0(E)\to {\rm H}^0(Q)\to 0$$ since $E\in \mathcal Q_s$. Thus $F$ is a sheaf in $\mathcal Q_s$. We write $K = {\rm H}^{-1}(Q)$ and $C = {\rm H}^{0}(Q)$, so $K\in \mathcal F_s$, $C\in \mathcal Q_s$, and we have an exact sequence of sheaves $$0\to K\to F \to E \to C\to 0.$$ Since $E$ is torsion-free, the torsion subsheaf of $F$ is contained in $K$. Since $K\in \mathcal F_s$ is torsion-free, we conclude $F$ is torsion-free. Clearly also $F$ is nonzero, for otherwise $(\theta',\theta,\theta'')$ wouldn't span a $2$-plane in $K(\mathbb{P}^2)$. We conclude $\rk(F)\geq 1$. Let $$\{0\} \subset F_1\subset \cdots \subset F_\ell = F$$ be the Harder-Narasimhan filtration of $F$. If $\mu(F_1) > \mu(E)$ then $F_1\to E$ is zero and $F_1\subset K$. Since $K\in \mathcal F_s$ for any $s$ along $W(\Theta)$, this is absurd. Therefore $\mu(F_1)\leq \mu(E)$. We can't have $\mu(F) = \mu(E)$ since then $W(\Theta)$ would be the vertical wall, so we conclude $\mu(F)<\mu(E)$. \end{proof} \subsection{Excluding higher rank walls} In this subsection, we bound the rank of $F$ under the assumption that $W(\Theta)$ is larger than $W(\Xi)$. In general, there will be walls corresponding to ``higher rank'' subobjects. We show that the Gieseker wall cannot correspond to such a subobject. \begin{theorem}\label{thm-excludeHighRank} Keep the notation and hypotheses from above. \begin{enumerate} \item Suppose hypothesis \ref{hyp-asymptotic} or \ref{hyp-smallRank} holds. If $W(\Theta)$ is larger than $W(\Xi)$, then $1\leq \rk(\theta') \leq \rk(\xi)$. \item If $\xi = (6,\frac{1}{3},\frac{13}{18})$, then the same result holds for the decomposition $\Xi = (\xi',\xi,\xi'')$ with $\xi' = (5,0,0)$. \end{enumerate} \end{theorem} The next inequality is our main tool for proving the theorem. \begin{proposition}\label{prop-highRank} If $\rk(F)>\rk(E)$, then the radius $\rho_\Theta$ of $W(\Theta)$ satisfies $$\rho_\Theta^2 \leq \frac{\rk(E)^2}{2(\rk(E)+1)}\Delta(E).$$ \end{proposition} \begin{proof} Consider the exact sequence of sheaves $$0\to K^k\to F^f\to E^e\to C^c\to 0,$$ with the superscripts denoting the ranks of the sheaves. Since $F$ is in the categories $\mathcal Q_s$ along $W(\Theta)$, we have $$f(s_\Theta+\rho_\Theta)\leq f\mu(F) = c_1(F) = c_1(K)+c_1(E) - c_1(C)= k\mu(K)+e\mu(E)-c_1(C).$$ Next, since $K$ is nonzero and in $\mathcal F_s$ along $W(\Theta)$, we have $\mu(K)\leq s_\Theta-\rho_\Theta$, and thus $$f(s_\Theta+\rho_\Theta)\leq k(s_\Theta-\rho_\Theta)+e\mu(E)-c_1(C).$$ Rearranging, $$(k+f)\rho_\Theta\leq (k-f)s_\Theta+e\mu(E) - c_1(C).$$ If $C$ is zero or torsion, then $k-f=-e$ and $c_1(C)\geq 0$, from which we get \begin{equation}\label{eqn1} (k+f)\rho_\Theta\leq (k-f)(s_\Theta-\mu(E))\end{equation} This inequality also holds if $C$ is not torsion. In that case, we have $k-f=c-e$ and since $E$ is semistable $c_1(C) = c\mu(C) \geq c\mu(E)$, from which the inequality follows. Both sides of Inequality (\ref{eqn1}) are positive, and squaring both sides gives $$(k+f)^2\rho_\Theta^2\leq (k-f)^2(\rho_{\Theta}^2+2\Delta(E)).$$ We conclude $$\rho_\Theta^2 \leq \frac{(k-f)^2}{2kf} \Delta(E).$$ This inequality is as weak as possible when the coefficient $(k-f)^2/(2kf)$ is maximized. Viewing $e$ as fixed, $k$ and $f$ are integers satisfying $f\geq e+1$ and $f-e\leq k\leq f$. It is easy to see the coefficient is maximized when $f = e+1$ and $k=1$, which corresponds to the inequality we wanted to prove. \end{proof} \begin{proof}[Proof of Theorem \ref{thm-excludeHighRank}] We recall $$\rho_\Xi^2 = \left(\frac{\mu(\xi')-\mu(\xi)}{2}-\frac{\Delta(\xi)-\Delta(\xi')}{\mu(\xi)-\mu(\xi')}\right)^2-2\Delta(\xi).$$ If we view $\rho_\Xi^2$ as a function of $\Delta(\xi)$, then it grows quadratically as $\Delta(\xi)$ increases. Suppose $\Delta(\xi)$ is large enough so that $$\rho_\Xi^2 \geq \frac{\rk(\xi)^2}{2(\rk(\xi)+1)}\Delta(\xi).$$ Then if $\rk(\theta')>\rk(\xi)$, we have $\rho^2_\Theta\leq \rho^2_\Xi$. This proves the theorem if hypothesis \ref{hyp-asymptotic} holds. Next suppose \ref{hyp-smallRank} holds, and write $\xi = (r,\mu,\Delta)$. View $\Delta$ as variable, and consider the quadratic equation in $\Delta$ $$\rho_\Xi^2 = \frac{r^2}{2(r+1)}\Delta;$$ this equation depends only on $r$ and $\mu$. Assuming this equation has roots, let $\Delta_1(r,\mu)$ be the larger of the two roots. Then the theorem is true for $\xi$ if $\Delta\geq \Delta_1(r,\mu)$. Let $\Delta_0(r,\mu)$ be the minimal discriminant of a rank $r$, slope $\mu$ sheaf satisfying \ref{hyp-smallRank}. We record the values of $\Delta_0(r,\mu)$ and $\Delta_1(r,\mu)$ for all pairs $(r,\mu)$ with $1\leq r\leq 6$ and $0 < \mu \leq 1$ in Table \ref{table-discValues}. For later use, we also record the value of the right endpoint $(x^+(r,\mu),0)$ of the wall $W(\Xi)$ corresponding to the character $(r,\mu,\Delta_0)$. \begin{center} \renewcommand*{\arraystretch}{1.3} \begin{longtable}{ccccccccccccccccc} \caption[]{Computation of $\Delta_0(r,\mu)$, $\Delta_1(r,\mu)$, and $x^+(r,\mu)$.}\label{table-discValues}\\ \toprule $r$ & $\mu$ & $\Delta_0$ & $\Delta_1 $ &$x^+$ &$\qquad$& $r$ & $\mu$ & $\Delta_0$ & $\Delta_1$ &$x^+$&$\qquad$&$r$ & $\mu$ & $\Delta_0$ & $\Delta_1$ &$x^+$\\\midrule \endfirsthead \multicolumn{17}{l}{{\setminusall \it continued from previous page}}\\ \toprule $r$ & $\mu$ & $\Delta_0$ & $\Delta_1 $ &$x^+$ &$\qquad$& $r$ & $\mu$ & $\Delta_0$ & $\Delta_1$ &$x^+$&$\qquad$&$r$ & $\mu$ & $\Delta_0$ & $\Delta_1$ &$x^+$\\\midrule \endhead \bottomrule \multicolumn{17}{r}{{\setminusall \it continued on next page}} \\ \endfoot \bottomrule \endlastfoot 1 & 1 & 2 & 1.00 & 0 && 4 & $\frac{1}{2}$ & $\frac{7}{8}$ & 0.81 & 0 && 5 & 1 & $\frac{6}{5}$ & 1.13 &0.46 \\ 2 & $\frac{1}{2}$ & $\frac{7}{8}$ & 0.25 & 0 && 4 & $\frac{3}{4}$ & $\frac{29}{32}$ & 0.67 & 0.53 && 6 & $\frac{1}{6}$ & $\frac{67}{72}$ & 0.03 & 0 \\ 2 & 1 & $\frac{3}{2}$ & 1.11 & 0.30 && 4 & 1 & $\frac{5}{4}$ & 1.13 & 0.44 && 6 & $\frac{1}{3}$ & $\frac{8}{9}$ & 0.79 & 0 \\ 3 & $\frac{1}{3}$ & $\frac{8}{9}$ & 0.11 & 0 && 5 & $\frac{1}{5}$ & $\frac{23}{25}$ & 0.04 &0 && 6 & $\frac{1}{2}$ & $\frac{17}{24}$ & 0.64 &0.17 \\ 3 & $\frac{2}{3}$ & $\frac{8}{9}$ & 0.57 & 0.37 && 5 & $\frac{2}{5}$ & $\frac{17}{25}$ & 0.65 &0 && 6 & $\frac{2}{3}$ & $\frac{13}{18}$ & 0.58 & 0.46 \\ 3 & 1 & $\frac{4}{3}$ & 1.13 &0.39 && 5 & $\frac{3}{5}$ & $\frac{17}{25}$ & 0.48 &0.37 && 6 & $\frac{5}{6}$ & $\frac{67}{72}$ & 0.78 &0.68 \\ 4 & $\frac{1}{4}$ & $\frac{29}{32}$ & 0.06 & 0 && 5 & $\frac{4}{5}$ & $\frac{23}{25}$ & 0.73 &0.62 && 6 & 1 & $\frac{7}{6}$ & 1.13 & 0.48 \\ \end{longtable} \end{center} In every case, we find that $\Delta_0(r,\mu) \geq \Delta_1(r,\mu)$ as required. We note that $\Delta_1(6,\frac{1}{3})> \frac{13}{18}$, so the proof does not apply to $\xi = (6,\frac{1}{3},\frac{13}{18})$. When $\xi = (6,\frac{1}{3},\frac{13}{18})$, we put $\Xi = ((5,0,0),(6,\frac{1}{3},\frac{13}{18}),(1,2,6))$ and compute $\rho_\Xi^2=4$. If $\rk(\theta')> 6$ then Proposition \ref{prop-highRank} gives $\rho_{\Theta}^2\leq \frac{13}{7}$, so $W(\Xi)$ is not nested in $W(\Theta)$ in this case either. \end{proof} The proof of the theorem also gives the following nonemptiness result. \begin{corollary} If hypothesis \ref{hyp-asymptotic} or \ref{hyp-smallRank} holds, then $W(\Xi)$ is nonempty. If $\xi = (6,\frac{1}{3},\frac{13}{18})$, the wall corresponding to $\xi'=(5,0,0)$ is nonempty. \end{corollary} \subsection{The ample cone, large discriminant case} Here we finish the proof that $W(\Xi)$ is the Gieseker wall if $\xi$ satisfies hypothesis \ref{hyp-asymptotic}. View $\xi = \xi(\Delta) = (\rk(\xi),\mu(\xi),\Delta)$ as having fixed rank and slope and variable $\Delta$, so that the extremal triple $\Xi=\Xi(\Delta)$ decomposing $\xi(\Delta)$ depends on $\Delta$. We begin with the following lemma that will also be useful in the small rank case \ref{hyp-smallRank}. \begin{lemma}\label{lem-rightPoint} The right endpoint $x^+_{\Xi(\Delta)} = s_{\Xi(\Delta)} + \rho_{\Xi(\Delta)}$ of $W(\Xi(\Delta))$ is a strictly increasing function of $\Delta$, and $$\lim_{\Delta\to \infty} x_{\Xi(\Delta)}^+ = \mu(\xi').$$ \end{lemma} \begin{proof} The statement that the function is increasing follows as in the second paragraph of Remark \ref{rem-extremalRemark}. The walls $W(\Xi(\Delta))$ are all potential walls for the Chern character $\xi'$, so they form a nested family of semicircles foliating the quadrant left of the vertical wall $s = \mu(\xi')$. If the radius of such a wall is arbitrarily large, then its right endpoint is arbitrarily close to the vertical wall. We saw in the proof of Theorem \ref{thm-excludeHighRank} that if $\Delta$ is arbitrarily large, then the radius of $W(\Xi(\Delta))$ is arbitrarily large. \end{proof} \begin{theorem}\label{thm-main} Suppose $\xi$ satisfies \ref{hyp-asymptotic}, and let $\Xi$ be the extremal triple decomposing $\xi$. Then $W(\Xi) = W_{\max}$, and the primary edge of the ample cone of $M(\xi)$ corresponds to $W(\Xi)$. \end{theorem} \begin{proof} Let $\Delta(\xi)$ be large enough that there is an extremal $\Xi$ that gives curves. Also assume $\Delta(\xi)$ is large enough that Theorem \ref{thm-excludeHighRank} holds. If necessary, further increase $\Delta(\xi)$ so that no rational numbers with denominator at most $\rk(\xi)$ lie in the interval $[x^+_\Xi,\mu(\xi'))$. Suppose the decomposition $\Theta$ corresponds to an actual wall $W(\Theta)$ which is at least as large as $W(\Xi)$, and let $$0\to F\to E \to Q\to 0$$ be a destabilizing sequence along $W(\Theta)$. Since $F\in \mathcal Q_s$ along $W(\Xi)$, we have $\mu(F) \geq x_\Xi^+$. Now $\rk(F) \leq \rk(\xi)$, so by the choice of $\Delta(\xi)$ and the slope-closeness condition \ref{cond-slopeClose} we conclude $\mu(F) = \mu(\xi')$. Furthermore, $F$ is $\mu$-semistable. If it were not, by Lemma \ref{lem-boundsTrivial} the only possibility would be that $F$ has a subsheaf of slope $\mu(\xi)$. Then $F$ must have a Harder-Narasimhan factor of slope smaller than $\mu(\xi')$, and this violates that $F\in \mathcal Q_s$ for all $s$ along $W(\Theta)$ by our choice of $\Delta(\xi)$. Finally, the $\mu$-semistability of $F$ implies $\Delta(F) \geq \Delta(\xi')$ by the discriminant minimality condition \ref{cond-discMinimal}. If $\Delta(F)>\Delta(\xi')$, then $W(\Theta)$ is nested inside $W(\Xi)$. We conclude $\Delta(F) = \Delta(\xi')$, and $W(\Theta) = W(\Xi)$. Therefore $W(\Xi)$ is the Gieseker wall. \end{proof} \begin{remark}\label{rem-explicit} The lower bound on $\Delta$ needed for our proof of Theorem \ref{thm-main} can be made explicit. We have increased $\Delta$ on several occasions throughout the paper. If $r$ and $\mu$ are fixed, then $\Delta$ needs to be large enough that the following statements hold. \begin{enumerate} \item $\xi''$ is stable (Lemma \ref{lem-extremalExist}). \item $\Xi$ gives curves. Alternately, it is enough to know that if $F\in M(\xi')$ and $Q\in M(\xi'')$ are general, then $\Hom(F,Q)=0$. The proof of Theorem \ref{thm-curves} allows us to give a lower bound for $\Delta$ if $\hom(F,Q)$ can be computed for some extremal triple $\Xi$ decomposing a character $\xi$ with rank $r$ and slope $\mu$. \item The wall $W(\Xi)$ is large enough to imply the destabilizing subobject along $W_{\max}$ has rank at most $r$ (Proposition \ref{prop-highRank}). \item The right endpoint $x_\Xi^+$ of $W(\Xi)$ is close enough to $\mu(\xi')$ that every rational number in $[x_{\Xi}^+,\mu(\xi'))$ has denominator larger than $r$. \end{enumerate} \end{remark} \begin{remark}\label{rem-Yoshioka} As an application of Theorem \ref{thm-main} and the discussion in the preceding remark, we explain how our results recover Yoshioka's computation \cite{Yoshioka} of the ample cone of $M(\xi)$ in case $c_1(\xi) = 1$ and $r\geq 2$. Let $\theta' = (2,0,0)$ and $\theta = (r,\frac{1}{r},P(-\frac{1}{r})+\frac{1}{r})$, and let $\Theta$ be the corresponding admissible triple. By Proposition \ref{prop-sporadic}, $\Theta$ is complete and gives curves. Any triple $\Lambda \succeq \Theta$ also gives curves by Lemma \ref{lem-curveBump}. Now suppose $\xi$ has positive height and $c_1(\xi) = 1$. Then either $\xi=\theta$ or $\xi$ is an elementary modification of $\theta$. Let $\Xi$ be the extremal triple decomposing $\xi$. If $W(\Xi)$ is the Gieseker wall, then the curves in $M(\xi)$ constructed in the previous paragraph are orthogonal to the divisor class on $M(\xi)$ coming from $W(\Xi)$, so we only need to check that $W(\Xi)$ is the Gieseker wall. To do this, we verify that if $\Delta \geq P(-\frac{1}{r})+\frac{1}{r}$, then statements (1), (3), and (4) in Remark \ref{rem-explicit} hold. It is clear that $\xi''$ is stable, so (1) holds. To check (3) and (4), it is enough to verify they hold for the decomposition $\Theta$. For (3), by Proposition \ref{prop-highRank} we must show $$\left(r-\frac{1}{2}\right)^2=\rho_\Theta^2 \geq \frac{r^2}{2(r+1)}\Delta(\theta)=\frac{2r^2-r+1}{4r+4},$$ which is clear for $r\geq 2$. For (4), we need $x_\Theta^+>-\frac{1}{r}$; in fact, $x_\Theta^+ = 0$ holds. \end{remark} \subsection{The ample cone, small rank case} We next compute the Gieseker wall in the small rank case. \begin{theorem}\label{thm-mainsmall} Suppose $\xi$ satisfies \ref{hyp-smallRank}, and let $\Xi$ be the extremal triple decomposing $\xi$. Then $W(\Xi) = W_{\max}$, and the primary edge of the ample cone of $M(\xi)$ corresponds to $W(\Xi)$. \end{theorem} \begin{proof} We may assume $0 < \mu(\xi) \leq 1$. Suppose $\Theta = (\theta',\theta,\theta'')$ is a decomposition of $\xi$ corresponding to an actual wall $W(\Theta)$ which is larger than $W(\Xi)$. Let $F\to E$ be a destabilizing inclusion corresponding $W(\Theta)$. We will show that $\mu(\theta')>\mu(\xi')$. Combining this with Lemma \ref{lem-boundsTrivial}, Theorem \ref{thm-excludeHighRank}, and slope-closeness \ref{cond-slopeClose} then gives a contradiction. To prove $\mu(\theta')>\mu(\xi')$, we first derive two auxiliary inequalities. We will make use of the nondegenerate symmetric bilinear form $(\xi,\zeta) = \chi(\xi \otimes \zeta)$ on $K(\mathbb{P}^2)$. Let $\gamma$ be a Chern character of positive rank such that $\gamma^\perp = \langle \xi',\xi\rangle$. \emph{First inequality:} Since $\mu(\theta')<\mu(\xi)$, the assumption that $W(\Theta)$ is bigger than $W(\Xi)$ means $(\theta',\gamma) > 0$. Indeed, $W(\Theta) = W(\Xi)$ if and only if $\theta'\in \gamma^\perp$. If $\Delta(\theta')$ is decreased starting from a character on $\gamma^\perp$, then $(\theta',\gamma)$ increases and the wall $W(\Theta)$ gets bigger. \emph{Second inequality:} Put $\zeta_1 = \ch \mathcal{O}_{\mathbb{P}^2}(-1)$ and $\zeta_2 = \ch \mathcal{O}_{\mathbb{P}^2}(-3).$ We observe that $\xi'$ lies in either $\zeta_1^\perp$ or $\zeta_2^\perp$. Let $i$ be such that $\xi'\in \zeta_i^\perp$; we will show that $(\theta',\zeta_i)\leq 0$ in either case. \emph{Case 1: $(\xi',\zeta_1)=0$.} If $(\theta',\zeta_1)>0$, then $\chi(\mathcal{O}_{\mathbb{P}^2}(1),F)>0$. Suppose $\Ext^2(\mathcal{O}_{\mathbb{P}^2}(1),F)=0$. Then there is a nonzero homomorphism $\mathcal{O}_{\mathbb{P}^2}(1)\to F$, and composing with the inclusion $F\to E$ gives a nonzero homomorphism $\mathcal{O}_{\mathbb{P}^2}(1)\to E$. Since $\xi$ has slope at most $1$ and positive height this contradicts semistability of $E$. It remains to prove $\Ext^2(\mathcal{O}_{\mathbb{P}^2}(1),F)=0$. Dually, we must show $\Hom(F,\mathcal{O}_{\mathbb{P}^2}(-2))=0$. If $x^+_\Xi \geq -2$, then since $W(\Theta)$ is larger than $W(\Xi)$ we will have $F\in \mathcal Q_{-2}$, proving this vanishing. By Lemma \ref{lem-rightPoint}, we only have to check this inequality when $\Delta(\xi)$ is minimal subject to satisfying \ref{hyp-smallRank}. We carried out this computation in Table \ref{table-discValues}. \emph{Case 2: $(\xi',\zeta_2)=0$.} If $(\theta',\zeta_2)>0$, then either $\Hom(\mathcal{O}_{\mathbb{P}^2}(3),F)$ or $\Ext^2(\mathcal{O}_{\mathbb{P}^2}(3),F)=\Hom(F,\mathcal{O}_{\mathbb{P}^2})^*$ is nonzero. Clearly $\Hom(\mathcal{O}_{\mathbb{P}^2}(3),F)=0$ by Lemma \ref{lem-boundsTrivial}. We must show $\Hom(F,\mathcal{O}_{\mathbb{P}^2})=0$. This follows from $x_{\Xi}^+\geq 0$, which is again true. Now we use the inequalities $(\theta',\gamma)>0$ and $(\theta',\zeta_i)\leq 0$ to prove $\mu(\theta')>\mu(\xi')$. There is a character $\nu$ such that if $\eta$ has positive rank, then $(\eta,\nu)\geq 0$ (resp. $>$) if and only if $\mu(\eta) \geq \mu(\xi')$ (resp. $>$). We summarize the known information about the signs of various pairs of characters here, noting that $(\xi,\zeta_i)<0$ since $\xi$ has positive height. $$ \begin{array}{c|ccc} (-,-)& \gamma & \zeta_i & \nu\\ \hline \xi & 0 & <0 & >0 \\ \xi' & 0 & 0 & 0 \\ \theta' & >0 & \leq 0 & \\ \end{array} $$ The character $\nu$ is in $(\xi')^\perp$, and $\gamma$ and $\zeta_i$ form a basis for $(\xi')^\perp$. Write $\nu = a\gamma+b \zeta_i$ as a linear combination. Since $0<(\xi,\nu) = b(\xi,\zeta_i)$, we find $b<0$. The character $\nu$ has rank $0$, so this forces $a>0$. We conclude $$(\theta',\nu) = a(\theta',\gamma)+b(\theta',\zeta_i)>0,$$ so $\mu(\theta')>\mu(\xi')$. \end{proof} We finish the paper by considering the last remaining case. \begin{theorem}\label{thm-mainSporadic} Let $\xi = (6,\frac{1}{3},\frac{13}{18})$, and let $\Xi$ be the decomposition of $\xi$ with $\xi' = (5,0,0)$. Then $W(\Xi) = W_{\max}$, and the primary edge of the ample cone corresponds to $W(\Xi)$. \end{theorem} \begin{proof} We use the same notation as in the proof of the previous theorem. We compute $x_{\Xi}^+ = 0$, so since $W(\Theta)$ is larger than $W(\Xi)$ we have $F\in \mathcal Q_\epsilon$ for some small $\epsilon>0$. Consider the Harder-Narasimhan filtration $$0\subset F_1\subset \cdots \subset F_\ell = F.$$ Every quotient $\mathrm{gr}_i$ of this filtration satisfies $0<\mu(\mathrm{gr}_i)\leq \frac{1}{3}$. Since $\mathrm{gr}_i$ is semistable, we find $\chi(\mathrm{gr}_i,\mathcal{O}_{\mathbb{P}^2})\leq 0$ for all $i$. It follows that $\chi(\theta',\ch(\mathcal{O}_{\mathbb{P}^2}))\leq 0$ as well. A straightforward computation using this inequality and $\chi(\theta',\gamma)>0$ shows $\mu(\theta')> \frac{2}{7}$. This contradicts Theorem \ref{thm-excludeHighRank} since every rational number in the interval $(\frac{2}{7},\frac{1}{3})$ has denominator larger than $6$. \end{proof} \end{document}

\begin{document} \draft \title{Nonlocality of Hardy type in experiments using independent particle sources} \author{Xu-Bo Zou \footnote{Present address: Electromagnetic Theory Group at THT, Department of Electrical Engineering, University of Hannover, Germany} and Hai-Woong Lee} \address{ Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea \\} \author{ Jaewan Kim } \address{ School of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea} \author{ Jae-Weon Lee and Eok Kyun Lee \\} \address{ Department of Chemistry, School of Molecular Science (BK 21), Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea.} \maketitle \begin{abstract} {\normalsize By applying Hardy's argument, we demonstrate the violation of local realism in a gedanken experiment using independent and separated particle sources. } \end{abstract} \pacs{PACS:03.65.BZ, 42.50.Dv,42.50.Ar} The nonlocal nature of quantum systems arising from entanglement has played a central role in quantum information science. Discussions about quantum nonlocality were initiated by Einstein, Podolsky, and Rosen\cite{epr} and extended by Bell\cite{bell}. Although the violation of Bell's inequality predicted by quantum mechanics has been experimentally verified\cite{exp}, there have been arguments about the detection loopholes\cite{loophole,hwang}. Greenberger, Horne, and Zeilinger(GHZ) \cite{ghz} demonstrated quantum mechanical violation of local realism without using the Bell's inequality for more than three particles. Hardy proved the nonlocality without using the Bell's inequality for all entangled states (except maximally entangled states) of two spin-$\frac{1}{2}$ particles\cite{hardy}. Considerable theoretical and experimental effort has been devoted to testing this Hardy type nonlocality\cite{goldstein,hardy2,hwang,wu,yurke}. An attempt to extend Hardy's theorem to cover maximally entangled states was made by Wu et al. using a quantum optical setting\cite{wu}. Recently Yurke and Stoler demonstrated violation of local realism in an experimental configuration involving independent sources\cite{yurke,yurke2,yurke3}. Specifically, they showed that: (1) in the fermion case the Pauli exclusion principle can be exploited in a local realism experiment of the Hardy type\cite{yurke} ; (2) GHZ type nonlocality can arise even when the particles come from independent widely separated sources\cite{yurke2}; and (3) violation of the Bell's inequality can be demonstrated by a quantum optical setting using independent particle sources \cite{yurke3}. \\ \indent The aim of our paper is to demonstrate nonlocality of Hardy type in experiments using independent particle sources. A schematic of the apparatus for the Gedanken experiment is shown in Fig. 1, which is similar to the setup proposed by Yurke and Stoler \cite{yurke3}, except that the four beam splitters $B_i$ have transmittance $T_i$ and reflectivity $R_i=1-T_i$ where $T_i \neq R_i$ and $i=1..4$. We parameterize $T_i$ and $R_i$ as ${T_i}=sin^2(\theta_i)\equiv S^2_i$ and ${R_i}=cos^2(\theta_i)\equiv C^2_i$, respectively. \begin{figure}\label{fig1} \end{figure} Two independent particles radiated from the source $ S_1$ and $S_2$ are incident on the input ports of the beam splitters $B_1$ and $B_2$, respectively. Vacuum ($|0\rangle$) enters the other input ports of $B_1$ and $B_2$. The outputs of these beam splitters propagate to two detectors. Detector 1 consists of the phase shifter $\phi_3$, the beam splitter $B_3$, and the particle counters E and F. Similarly, detector 2 consists of the phase shifter $\phi_4$, the beam splitter $B_4$, and the particle counters G and H. The beam path labels appearing in Fig. 1 will also be used to denote the annihilation operators for modes propagating along these beam paths. \indent The analysis of the firing statistics at each particle counter is carried out as follows. The beam splitters $B_1$, $B_2$, $B_3$, and $B_4$ perform the mode transformation \beqa \label{trans} \left ( \begin{array}{cc} a'\\ b' \end{array} \right )&=&\left ( \begin{array}{cc} S_{1}&iC_{1}\\ iC_{1}&{S_1} \end{array} \right ) \left ( \begin{array}{cc} a\\ b \end{array} \right ), \no \left ( \begin{array}{cc} c'\\ d' \end{array} \right )&=&\left ( \begin{array}{cc} S_{2}&iC_{2}\\ iC_{2}&S_{2} \end{array} \right )\left ( \begin{array}{cc} c\\ d \end{array} \right ), \no \left ( \begin{array}{cc} e\\ f \end{array} \right )&=&\left ( \begin{array}{cc} S_{3}&iC_{3}\\ iC_{3}&S_{3} \end{array} \right )\left ( \begin{array}{cc} e^{-i{\phi_3}}b'\\ c' \end{array} \right ), \no \left ( \begin{array}{cc} g\\ h \end{array} \right )&=&\left ( \begin{array}{cc} S_{4}&iC_{4}\\ iC_{4}&S_{4} \end{array} \right )\left ( \begin{array}{cc} a'\\ e^{-i\phi_4}d' \end{array} \right ). \eeqa From the mode transformation shown in Eq. (1), it follows that the annihilation operators for the modes $a,b,c$ and $d$ can be expressed in terms of those for $e,f,g$ and $h$ as follows: \beqa \label{a} a&=&S_1 S_4g-iS_1 C_4h - ie^{i\phi_3}{C_1 S_3}e -e^{i\phi_3}{C_1 C_3}f \no b&=&-i{C_1 S_4}g-{C_1 C_4}h + e^{i\phi_3}{S_1 S_3}e -ie^{i\phi_3}{S_1 C_3}f \no c&=&-i{S_2 C_3}e+{S_2 S_3}f -e^{i\phi_4}{C_2 C_4}g -ie^{i\phi_4}{C_2 S_4}h \no d&=&-{C_2 C_3}e-i{C_2 S_3}f -ie^{i\phi_4}{S_2 C_4}g +e^{i\phi_4}{S_2 S_4}h. \eeqa The state vector for two identical bosons injected into the interferometer can be expressed as the direct product of the individual state vectors. In second quantized notation, the input state vector is therefore given by \beq \label{psi} |\psi\rangle=a^\dagger c^\dagger |0\rangle. \eeq By substituting Eq. (\ref{a}) into Eq. (\ref{psi}), we obtain the output state vector. This vector can be divided into two parts \beq |\psi\rangle=|\psi_1\rangle+|\psi_2\rangle, \eeq where \beqa |\psi_1\rangle&=& i[{S_1 S_2 C_3 S_4} -e^{-i\phi_3-i\phi_4}{C_1 C_2 S_3 C_4}]|1\rangle_e|1\rangle_g \no &+&[{S_1 S_2 S_3 S_4} +e^{-i\phi_3-i\phi_4} {C_1 C_2 C_3 C_4}]|1\rangle_f|1\rangle_g \no &-&[{S_1 S_2 C_3 C_4} +e^{-i\phi_3-i\phi_4}{C_1 C_2 S_3 S_4}]|1\rangle_e|1\rangle_h \no &+&i[{S_1 S_2 S_3 C_4} -e^{-i\phi_3-i\phi_4}{C_1 C_2 C_3 S_4}]|1\rangle_f|1\rangle_h, \no |\psi_2\rangle&=&ie^{-i\phi_3}{C_1 S_2}(S_3^2 -C_3^2) |1\rangle_e|1\rangle_f \no &+&ie^{-i\phi_4}{S_1 C_2}(S_4^2-C_4^2)|1\rangle_g|1\rangle_h \no &-&{2}e^{-i\phi_3} {C_1 S_2S_3C_3}[|2\rangle_e+|2\rangle_f] \no &-&{2}e^{-i\phi_4} {S_1C_2S_4C_4}[|2\rangle_g+|2\rangle_h]. \eeqa Here $|n\rangle_e$ denotes the $n$ particle state of the mode $e$. Now consider detector 1 and let $\bar{E}~$($\bar{F}$) denote the event in which the counter E (F) counts a single particle and the counter F (E) counts no particle. Similarly for detector 2, we define events $\bar{G}~(\bar{H})$ in which the counter G (H) counts one particle and the counter H (G) counts no particle. Among all the possible events, we are interested only in the events $\bar{E},\bar{F},\bar{G}$ and $\bar{H}$. Thus, we do not need to pay attention to the evolution of the state $|\psi_2\rangle$, because $\bar{E},\bar{F},\bar{G}$ and $\bar{H}$ are not reflected in $|\psi_2\rangle$. Hence, below we consider only the evolution of the state $|\psi_1\rangle$. Let us consider the following four cases.\\ a) Set $\phi_3 =\phi_4=\pi/2$ and choose \beq S_3=S_4=\sqrt{\frac{{C_1C_2}}{{C_1C_2}+{S_1S_2}}} \equiv \tau. \eeq Then we obtain \beqa \label{psi1} |\psi_1\rangle&=& i\sqrt{C_1C_2S_1S_2} (|1\rangle_e|1\rangle_g +|1\rangle_f|1\rangle_h) \no &+&[{C_1C_2} -{S_1S_2}]|1\rangle_e|1\rangle_h. \eeqa Let $FG(S_3=\tau,S_4=\tau,\phi_3=\frac{\pi}{2},\phi_4=\frac{\pi}{2})$ denote the probability of the simultaneous appearance of events $\bar{F}$ and $\bar{G}$ for the following experimental settings. The transmittance of $B_3$ is set to $S_3^2=\tau^2$ and that of $B_4$ is set to $S^2_4=\tau^2$. Since there is no $|1\rangle_f|1\rangle_g$ term in Eq. (\ref{psi1}), we obtain \beq FG(S_3=\tau,S_4=\tau,\phi_3=\frac{\pi}{2},\phi_4=\frac{\pi}{2})=0. \label{fg1} \eeq \\ b) If $\phi_3=\frac{\pi}{2}$, $\phi_4=\frac{3\pi}{2}$, and \beqa S_3&=&\tau, \no S_4&=&\sqrt{\frac{(C_1C_2)^3}{(S_1S_2)^3 + (C_1C_2)^3}}\equiv \tau', \label{r4t4} \eeqa we have \beqa |\psi_1\rangle&=& \frac{{S_1S_2C_1C_2}} {\sqrt{(C_1C_2)^2 +(S_1S_2)^2-{S_1S_2C_1C_2}}} |1\rangle_f|1\rangle_g \no &-&\sqrt{(C_1C_2)^2+(S_1S_2)^2-{S_1S_2C_1C_2}} |1\rangle_e|1\rangle_h \no &+&i\frac{\sqrt{S_1S_2C_1C_2}({S_1S_2}-{C_1C_2})} {\sqrt{(C_1C_2)^2+(S_1S_2)^2-{S_1S_2C_1C_2}}} |1\rangle_f|1\rangle_h. \eeqa Thus we have the following quantum prediction:\\ $F(S_3=\tau,S_4=\tau',\phi_3=\frac{\pi}{2}, \phi_4=\frac{3\pi}{2})=1,$ if \beq \label{gf1} G(S_3=\tau,S_4=\tau',\phi_3=\frac{\pi}{2}, \phi_4=\frac{3\pi}{2})=1, \eeq since there is only one term $|1\rangle_f|1\rangle_g$ containing $|1\rangle_g$.\\ \\ c) Setting $\phi_3=\frac{3\pi}{2}$, $\phi_4=\frac{\pi}{2}$ and \beqa \label{r3t3} S_3&=&\tau', \no S_4&=&\tau, \eeqa we obtain \beqa |\psi_1\rangle&=& \frac{{S_1S_2C_1C_2}} {\sqrt{(C_1C_2)^2+(S_1S_2)^2-{S_1S_2C_1C_2}}} (|1\rangle_f|1\rangle_g) \no &-& \sqrt{(S_1S_2)^2 + (C_1C_2)^2 -{S_1S_2C_1C_2}} { (|1\rangle_e|1\rangle_h) }\no &+&i \frac{ \sqrt{S_1S_2C_1C_2}({S_1S_2}-{C_1C_2})} {\sqrt{(C_1C_2)^2 + (S_1S_2)^2-{S_1S_2C_1C_2}}} { (|1\rangle_e|1\rangle_g)}. \eeqa Thus, if $$F(S_3=\tau',S_4=\tau ,\phi_3=\frac{3\pi}{2},\phi_4=\frac{\pi}{2})=1,$$ then \beq G(S_3=\tau',S_4=\tau,\phi_3=\frac{3\pi}{2}, \phi_4=\frac{\pi}{2})=1. \label{fg2} \eeq \\ d) Setting $\phi_3=3\pi/2, \phi_4=3\pi/2$, and choosing $S_3=S_4=\tau'$, we obtain \beqa |\psi_1\rangle&=& i\frac{\sqrt{(C_1C_2S_1S_2)^3}} {(S_1S_2)^2+(C_1C_2)^2-{S_1S_2C_1C_2}} (|1\rangle_e|1\rangle_g \no &+& |1\rangle_f|1\rangle_h ) + \frac{ {S_1S_2C_1C_2}({C_1C_2}-{S_1S_2})} {(S_1S_2)^2 + (C_1C_2)^2-{S_1S_2C_1C_2}} { (|1\rangle_f |1\rangle_g)}\no &+& \frac{ [(C_1C_2)^2+(S_1S_2)^2]({C_1C_2}-{S_1S_2})} {(S_1S_2)^2+(C_1C_2)^2-{S_1S_2C_1C_2}} { (|1\rangle_e |1\rangle_h)}. \eeqa Thus the following quantum prediction is obtained \beq FG(S_3=\tau',S_4=\tau',\phi_3=\frac{3\pi}{2},\phi_4=\frac{3\pi}{2})=1 \label{fg3} \eeq with a nonzero probability \beq P=\frac{ (S_1S_2C_1C_2)^2({S_1S_2}-{C_1C_2})^2} {[(S_1S_2)^2+(C_1C_2)^2 - {S_1S_2C_1C_2}]^2}, \eeq where $C_1C_2 \neq S_1S_2$. \\ \indent Finally, we demonstrate that, following Hardy's\cite{hardy} and Wu et al.'s argument\cite{wu}, local realism and quantum mechanics are incompatible using an experimental setting with independent and separated particle sources. The notion of local realism is introduced by assuming that there exist some hidden variables $\lambda$ that describe the state of individual particles. According to the assumption of locality, the choice of the measurement at detector 1 would not influence the outcome of the measurement at detector 2, which means that, for a specified $\lambda$ , the probability of the event $\bar{F}$ is uniquely determined by the transmittance of $B_3$ and $\phi_3$, whereas that of $\bar{G}$ is determined solely by the transmittance of $B_4$ and $\phi_4$. Let us denote the probabilities of the events $\bar{F}$ and $\bar{G}$ for a value of hidden variable $\lambda$ by $F(\lambda,S_3,\phi_3)$ and $G(\lambda,S_4,\phi_4)$, respectively. Using Eq. (\ref{fg3}), for some values of hidden variable $\lambda$, we expect simultaneous occurrence of events $\bar{F}$ and $\bar{G}$ when $S_3=\tau',S_4=\tau',\phi_3=\frac{3\pi}{2}$ and $\phi_4=\frac{3\pi}{2}$ and thus obtain $F(\lambda,S_3=\tau',\phi_3=\frac{3\pi}{2}) =G(\lambda,S_4=\tau',\phi_4=\frac{3\pi}{2})=1$. On the other hand, from Eqs. (\ref{gf1}) and (\ref{fg2}), we have $G(\lambda,S_4=\tau,\phi_4=\frac{\pi}{2})=1$, since $F(\lambda,S_3=\tau',\phi_3=\frac{3\pi}{2})=1$; and $F(\lambda,S_3=\tau,\phi_3=\frac{\pi}{2})=1$ since $G(\lambda,S_4=\tau',\phi_4=\frac{3\pi}{2})=1$ for the same values of $\lambda$. Therefore, we should have $F(\lambda,S_3=\tau,\phi_3=\frac{\pi}{2}) =G(\lambda,S_4=\tau,\phi_4=\frac{\pi}{2})=1$. But this contradicts the quantum prediction of Eq. (8) that the probability of simultaneous occurrence of the events $\bar{F}$ and $\bar{G}$ is zero when $S_3=\tau,S_4=\tau,\phi_3=\frac{\pi}{2}$ and $\phi_4=\frac{\pi}{2}$.\\ \indent In summary, we have shown the violation of local realism of EPR type without using Bell's inequality for the case of two particles originating from independent sources. \newline \\ We acknowledge the support of the Brain Korea 21 Project of the Korean Ministry of Education. \end{document}

\begin} \def\beq{\beg} \def\F{\scr Fin{document} \begin} \def\beq{\beg} \def\F{\scr Fin{frontmatter} \title{Logarithmic heat kernel estimates without curvature restrictions} \runtitle{A sample running head title} \begin} \def\beq{\beg} \def\F{\scr Fin{aug} \author[A]{\fracnms{Xin}~\snm{Chen}\varepsilonad[label=e1]{chenxin217@sjtu.edu.cn}}, \author[B]{\fracnms{Xue-Mei}~\snm{Li}\varepsilonad[label=e2]{xue-mei.li@epfl.ch}} \and \author[C]{\fracnms{Bo}~\snm{Wu}\varepsilonad[label=e3]{wubo@fudan.edu.cn}} \address[A]{School of Mathematical Sciences, Shanghai Jiao Tong University\printead[presep={,\ }]{e1}} \address[B]{ Imperial College London and EPFL\printead[presep={,\ }]{e2}} \address[C]{School of Mathematical Sciences, Fudan University\printead[presep={,\ }]{e3}} \varepsilonnd{aug} \begin} \def\beq{\beg} \def\F{\scr Fin{abstract} The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques. A basic tool developed here is intrinsic stochastic variations with prescribed second order covariant differentials, allowing to obtain a path integration representation for the second order derivatives of the heat semigroup $P_t$ on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an $\varepsilonpsilon^2$ term in the variation allowing greater control. We also construct a family of cut-off stochastic processes adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time, furthermore the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to by-pass the lack of continuity of the exit time of the Brownian motions on its initial position. \varepsilonnd{abstract} \begin} \def\beq{\beg} \def\F{\scr Fin{keyword}[class=MSC] \kwd[Primary ]{60Gxx} \kwd{60Hxx} \kwd[; secondary ]{58J65, 58J70} \varepsilonnd{keyword} \begin} \def\beq{\beg} \def\F{\scr Fin{keyword} \kwd{Riemannian manifold} \kwd{Heat kernel estimate} \kwd{Curvature} \kwd{Gradient formula} \varepsilonnd{keyword} \varepsilonnd{frontmatter} \text{\rm sect}ion{Introduction} Let $(M,g)$ be an $n$-dimensional connected and complete Riemannian manifold endowed with the Levi-Civita connection $\nabla$. Let $\Delta$ denote the Laplace-Beltrami operator, and let $p(t,x,y)$ denote its heat kernel, by which we mean the minimal positive fundamental solution to the equation $\frac \partial {\partial t} =\frac 12 \Delta $. The objective of this article is to provide estimates on the first and the second order gradients of $\log p(t,x,\cdot)$, without imposing any curvature conditions on $M$. For a fixed $x\in M$, we use the abbreviation $\log p$ for the logarithmic heat kernel $\log p(t,x,\cdot)$ and use $\nabla \log p$ and $\nabla^2 \log p$ for its first and second order derivatives respectively. We begin with explaining some of the motivations and potential applications. Let $o\in M$ be fixed, we denote $$P_o(M):=\{\gamma\in C([0,1];M): \gamma(0)=o\}$$ the based path space over $M$. Likewise, let $L_o(M)$ denote the based loop space over $M$, $$L_o(M):=\{\gamma\in P_o(M):\ \ \gamma(0)=\gamma(1)=o\}.$$ A classical problem is to seek a suitable probability measure on $P_o(M)$ or $L_o(M)$, with which analysis on these infinite dimensional non-linear spaces can be made and understanding of the path spaces can be furthered. If $M$ is compact or more generally with bounded geometry, a natural candidate for the probability measure on $L_o(M)$ is the probability distribution of the diffusion process with the infinitesimal operator $$L:=\frac 12\Delta + \nabla \log p(1-t,\cdot, o)$$ and the initial value $o$. This is the Brownian bridge measure. Since there is no analogue of a Lebesque measure, translation invariant, on $L_o(M)$, the Brownian bridge measure is essentially the canonical measure to use. Indeed, for $M=\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ the Brownian bridge measure is a Gaussian measure and it is quasi-invariant under translations of Cameron-Martin vectors. To construct such a diffusion process, which is usually called the Ornstein-Uhlenbeck process, we define a pre-Dirichlet form. This form will be called the Ornstein-Uhlenbeck (O-U) Dirichlet form. To verify that the pre-Dirichlet form yields a Markov process, it is necessary to show it is closed -- a property following readily once we have an integration by parts (IBP) formula. The key ingredient for such an IBP formula is suitable short time estimates on $\nabla \log p$ and $\nabla^2 \log p$. We refer the reader to Aida \cite{A,A3}, Airault and Malliavin \cite{Airault-Malliavin}, Driver \cite{D2}, Hsu \cite{Hsu2} and Li \cite{Li2} for more detail. Another interesting problem is to establish functional inequalities for the O-U Dirichlet form. This includes the Poincar\'e inequality and logarithmic Sobolev inequality. They describe the long time behaviours of the associated diffusion process. The logarithmic Sobolev inequality for Gaussian measures was obtained by Gross in the celebrated paper \cite{Gro75}. However, this is not known to hold for loop space over a general manifold $M$. When $M$ was the hyperbolic space, Poincar\'e inequality was shown to hold on $L_0(M)$ by the authors of the article \cite{CLW1} and Aida \cite{A4}. If $M$ was compact simply connected with strictly positive Ricci curvature, a weak Poincar\'e inequality with explicit rate function was also established by the authors of the article \cite{CLW2}. It was shown in Gross \cite{Gross-91} that the Poincar\'e inequality for O-U Dirichlet form did not hold on $L_o(M)$ when $M$ was not simply connected. Soon after, Eberle \cite{E1} constructed a simply connected compact manifold for which the Poincar\'e inequality for O-U Dirichlet form did not hold on $L_o(M)$. When the based manifold $M$ was compact, Aida \cite{A}, Eberle \cite{E}, Gong and Ma \cite{GM}, Gong, R\"ockner and Wu \cite{Gong-Rockner-Wu} and Gross \cite{Gross-91} have obtained weighted log-Sobolev inequalities or other different versions of modified log-Sobolev inequalities on $L_o(M)$. In all the results mentioned above, the crucial ingredient was again the asymptotic estimates for $\nabla \log p$ and $\nabla^2\log p$. We want to stress that all the results mentioned above have been established for the base manifold $M$ compact or with some bounded geometry conditions, since the short time or asymptotic estimates for $\nabla \log p$ and $\nabla^2\log p$ were only known for manifolds with such restrictions. Our immediate concern is to study the construction of diffusion processes and functional inequalities on $L_o(M)$ without any bounded geometry conditions on $M$. We will obtain short time or asymptotic estimates for $\nabla \log p$ and $\nabla^2\log p$ in this paper. These estimates will be applied to study several problems on $L_o(M)$ in a forthcoming paper \cite{CLW}. It is intriguing that estimates for $\nabla \log p $ and $\nabla^2 \log p$ are also main tools for proving the continuous counterpart of Talagrand's conjecture for the hypercube $\Omega_n=\{-1, 1\}^n$ which we explain below. Let $\sigma^i$ denote the configuration with the ith coordinate of $\sigma$ flipped and let $\sigma_i$ denote the i-th component of $\sigma\in \Omega_n$. Let $\mu_n\varepsilonquiv 2^{-n}$ be the uniform measure on $\Omega_n$ which is reversible associated with the generator $Lf(\sigma):=\frac 12 \sum_{i=1}^n (f(\sigma^i)-f(\sigma))$ where $\sigma\in\Omega_n$. Setting $T_sf(\sigma):=\int_{\Omega_n} f(\varepsilonta) \Pi_{i=1}^n (1+e^{-s} \sigma_i \varepsilonta_i) d\mu_n(\varepsilonta)$, then Talagrad's conjecture states that for any $s>0$ there exists a constant $c_s$ independent of the dimension $n$ such that $\mu_n\Big(\Big\{\sigma: T_sf(\sigma)\ge t\Big\}\Big)\le c_s \frac 1 {t\sqrt { \log t}}$ for $ t>1$. The value $c_s$ is uniformly in the function $f$ with $\|f\|_{L^1(\mu_n)}=1$ and in the dimension. The continuous counter-part of the conjecture is for the Ornstein-Uhlenbeck semi-group $T_t$ with generator $\Delta-x\cdot \nabla$ $$ \sup_{f \ge 0, \|f\|_{L^1(\gamma_n)}=1} \gamma_n \left(\Big\{\sigma: T_sf(\sigma) \ge t\Big\}\right) \le c_s \frac 1 {t\sqrt { \log t}}, \qquad t \ge 2,$$ where $\gamma_n\sim N(0, I_{n\times n})$ is the standard $n$-dimensional normal distribution. This was proven to be affirmative in Ball, Barthe, Bednorz, Oleszkiewicz and Wolff \cite{BBBOW13}. The dimension free best constants were given in Eldan and Lee \cite{EL15:focs} and Lehec \cite{Leh16} where the key ingredients are: \begin} \def\beq{\beg} \def\F{\scr Fin{itemize} \item [(1)] For any $g\in L^1(\gamma_n)$ and any $s>0$, $\nabla^2 (\log T_s g)\ge -c_s^2\, \mathrm {Id}$, \item [(2)] For any $g\in L^1(\gamma_n)$ non-negative and with $\nabla^2 (\log g)\ge - \beta \, \mathrm {Id}$ with a $\beta>0$, one has $\gamma_n (g\ge t)\le \frac {C_\beta}{t \sqrt{\log t}}$ for any $ t>1$. \varepsilonnd{itemize} Here $\mathrm {Id}$ is the identity operator. Such estimates for non-Gaussian measures and also for the $M/M/\infty$ queue on ${\mathbb N}$ were obtained by Gozlan, Li, Madiman, Roberto and Samson \cite{GLMRS}. \text{\rm sect}ion{Main Results} The short time and asymptotic estimates are presented in \varepsilonqref{e1-1}--\varepsilonqref{e1-3b} below. To the best of our knowledge, such estimates were obtained only for a Riemannian manifold with bounded geometry including a compact Riemannian manifold. Gradient and Hessian estimates of the form (\ref{e1-1}-\ref{e1-1a}) were proved by Sheu \cite{Sh} for $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ with a non-trivial Riemannian metrics where the objective was a non-degenerate parabolic PDEs with bounded derivatives up to order three, and (\ref{e1-1}) for a compact Riemannian manifold can be found in Driver \cite{D2}, obtained using a result of Hamilton \cite{Hamilton93}, Corollary 1.3 and the Gaussian bounds on heat kernels, see e.g. Li and Yau \cite{Li-Yau}, Cheeger and Yau \cite{Cheeger-Yau}, Davies \cite{Davies}, Setti \cite{Setti}, and Varopoulos \cite{Varopoulos1,Varopoulos2}. The estimate (\ref{e1-1a}) was shown in Hsu \cite{HsuEstimates} again for the compact case. For a non-compact Riemannian manifold with non-negative Ricci curvature, \varepsilonqref{e1-1} was obtained by Kotschwar \cite{Kotschwar}. Under a bounded geometry condition together with a volume non-collapsing condition, similar estimates were obtained by Souplet and Zhang \cite{SZ} and Engoulatov \cite{En}. For the heat kernel associated with the Witten Laplacian operator, these estimates were proved by X.D. Li \cite{XLi} under a bounded geometry condition on the Bakry-Emery Ricci curvature. In addition, in all the references mentioned above, suitable bounded geometry conditions were required. Likewise, the bounded geometry restrictions are used to derive differential Harnack inequalities and global heat kernel estimates, by Cheeger, Gromov and Taylor \cite{Cheeger-Gromov-Taylor}, Cheng, P. Li and Yau \cite{CLY}, Hamilton \cite{Hamilton93}, P. Li and Yau \cite{Li-Yau}, they provide an important step toward \varepsilonqref{e1-1}--\varepsilonqref{e1-1a}. Meanwhile, the asymptotic gradient estimate \varepsilonqref{e1-2b} was first shown in Bismut \cite{Bis} for a compact Riemannian manifold. It was extended to the hypo-elliptic heat kernel and the heat kernel on a vector bundle, for $M$ with bounded geometry, respectively by Ben Arous \cite{Ben}, Ben Arous and L\'eandre \cite{Ben-Le} and Norris \cite{Norris}, c.f. also Azencott \cite{Azencott-as}. The asymptotic second order gradient estimate \varepsilonqref{e1-3b} was established by Malliavin and Stroock \cite{MS} for a compact Riemannian manifold. For `asymptotically flat' Riemannian manifolds with poles and bounded geometry this can be found in Aida \cite{A3}. On cut-locus estimates was studied by Neel \cite{Neel07}. A natural question is then whether the estimates \varepsilonqref{e1-1}--\varepsilonqref{e1-3b} still hold for a general non-compact Riemannian manifold? Note that in Azencott \cite{Azencott}, it was illustrated that Gaussian type heat kernel estimates could not be automatically extended to an arbitrary manifold and may fail if the completeness of the Riemannian metric was removed. We state the main estimate. For any $y\in M$, let Cut(y) be the cut-locus of $y$ and $i(y)$ be the injectivity radius of $y$. \begin} \def\beq{\beg} \def\F{\scr Fin{theorem}[Theorems \ref{thm6.7} and \ref{thm6.10}] \label{main} Suppose that $M$ is a complete Riemannian manifold with Riemannian distance $d$. \begin} \def\beq{\beg} \def\F{\scr Fin{itemize} \item [(1)] For every compact subset $K$ of $ M$, the following statements hold. \begin} \def\beq{\beg} \def\F{\scr Fin{itemize} \item [(a)] There exists a positive constant $C(K)$, which may depend on $K$, such that \begin} \def\beq{\beg} \def\F{\scr Fin{eqnarray} \label{e1-1} \quad \left|\nabla_x \log p(t,x,y)\right|_{T_x M} &\le& C(K)\left(\fracrac{1}{\sqrt{t}}+\fracrac{d(x,y)}{t}\right),\\ \left|\nabla_x^2 \log p(t,x,y)\right|_{T_x M \otimes T_x M}&\le& C(K)\left(\fracrac{d^2(x,y)}{t^2}+\fracrac{1}{t}\right)\label{e1-1a} \varepsilonnd{eqnarray} for any $ x,y \in K $ and for any $\ t\in (0,1]$. \item [(b)] For each $y\in M$ and $\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Delta<i(y)$ there exist positive constants $t_0$ and $C_1$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.29}\aligned &\left|t\nabla_x^2 \log p(t,x,y)+\textbf{I}_{T_x M}\right|_{T_x M \otimes T_x M}\\ &\le C_1\left(d(x,y)+ \sqrt{t}\right),\quad \quad \ x\in B_y(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Delta),\ t \in (0,t_0],\varepsilonndaligned \varepsilonnd{equation} where $\textbf{I}_{T_x M}$ is the identical map on $T_x M$. \varepsilonnd{itemize} \item [(2)] Let $y\in M$ and assume that $\tilde K \subset M\setminus \text{Cut}(y)$ is a compact set. Then \begin} \def\beq{\beg} \def\F{\scr Fin{eqnarray} \label{e1-2b} &\lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}\sup_{x \in \tilde K} \left|t\nabla_x\log p(t,x,y)+\nabla_x\left(\fracrac{d^2(x,y)}{2}\right) \right|_{T_x M}=0,\\ &\lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}\sup_{x \in \tilde K} \left|t\nabla_x^2\log p(t,x,y)+\nabla_x^2\left(\fracrac{d^2(x,y)}{2}\right) \right|_{T_x M \otimes T_x M}=0. \label{e1-3b} \varepsilonnd{eqnarray} \varepsilonnd{itemize} \varepsilonnd{theorem} {\bf Remarks on the main theorem.} As explained in Section 1, these estimates are crucial for the stochastic analysis of the loop space $L_o(M)$. Despite of the collective efforts, so far, these type of results have been largely proved only for based manifolds with bounded geometry. While in this paper, we only need to assume that the based manifold $M$ is complete and stochastically complete. For analysis on the path space $P_o(M)$ over a general complete Riemanian manifold without curvature conditions, some work have already been done by Chen and Wu \cite{CW} and Hsu and Ouyang \cite{Hsu-Ouyang}. For $P_o(M)$, the content of Theorem \ref{main} is not essential. In a forthcoming paper \cite{CLW}, we shall apply these to obtain integration by parts formula and construct of O-U Dirichlet form on $L_o(M)$, and to prove several functional inequalities on $L_o(M)$. Our main idea is to obtain localised asymptotic comparison theorems for the first and the second order gradients of logarithmic heat kernel (see Proposition \ref{prp6.6} and \ref{prp6.9} below). One novelty is a new second order derivative formula via a new type of (second order) stochastic variation for Brownian paths on the orthonormal frame bundles, which is in particular different from that used by Bismut \cite{Bis} or Stroock \cite{S}. The idea of stochastic variation was initiated in \cite{Bis} for obtaining an integration by part formula. While the choice of the variation in \cite{S} will produce a term with (the time reverse of) a non-random vector field on $L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$, see also Malliavin and Stroock \cite[(1.5)]{MS}, it seems not possible to replace the non-random vector field in their paper by a random one (otherwise the time reversed field is not adapted, hence It\^o's integral is not well defined), which prevents the extension of the formula in \cite{MS} to a general non-compact $M$ by a suitable localisation argument. We shall choose a variation (see Section \ref{section2} below) with desired properties, which in particular ensures that the formula for the second order gradient of heat semigroup can take a random vector fields. This is the key step for us to extend the new formula to a general complete $M$ (see e.g. Theorem \ref{thm3.1} below). The expression we obtain for the second order gradient of heat semigroup is different from that by Elworthy and Li \cite{EL}, Li \cite{Li-doubly-damped, Li18}, or from that in Arnaudon, Plank and Thalmaier \cite{APT} or that in Thompson \cite{Thompson}. We prove the formula by combining the second order stochastic variation (shown to hold for a compact manifold) and approximation arguments (for a non-compact manifold), which is totally different from that in \cite{APT,Thompson}. This new method is adapted for both the proof of Proposition \ref{prp6.9} here and the integration by parts formula in our forthcoming paper~\cite{CLW}. \text{\rm sect}ion{Expression for the second order gradient of heat semigroup} Throughout the paper, $(\Omega, \mathscr{F}, \mathscr{F}_t, \mathbb{P})$ denotes a filtered probability space satisfying the standard assumptions, and $B_t=(B_t^1,B_t^2,\cdots, B_t^n)$ is a standard $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued Brownian motion. Let $L(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ denote the collection of all stochastic processes $h: \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt_+\times\Omega \rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ which are $\mathscr{F}_t$-adapted. Let $h'(\cdot, \omega)$ denote the time derivative of $h(\cdot, \omega)$. We define the Cameron-Martin space on the Wiener space as follows \begin} \def\beq{\beg} \def\F{\scr Fin{equation*}\aligned L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n):&=\bigg\{ h\in L(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n): \; h(\cdot,\omega)~\text{is absolutely continuous for a.s.}~\omega\in \Omega,\\& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{and}~\mathbb{E}\Big[\int_0^1 |h'(s, \omega)|^2\, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\Big] <\infty\bigg\}. \varepsilonndaligned\varepsilonnd{equation*} Elements of $L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ are usually called (random) Cameron-Martin vectors. Let $C_b(M)$ and $C_c(M)$ denote the collection of all real valued bounded and continuous functions on $M$ and continuous functions with compact supports in $M$ respectively. Let ${\mathfrak {so}}(n)$ denote the set of of anti-symmetric $n\times n$ matrices and let $SO(n)$ denote the collection of orthonormal $n\times n$ matrices. {\bf The curvature.} Let ${\bf R}_x$ denote the sectional curvature tensor and let $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic_x$ denote the Ricci curvature tensor at $x\in M$ respectively. Thus both ${\bf R}_x:T_x M\times T_x M \rightarrow T_x M \times T_x M$ and $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic^{\sharp}_x:T_x M \rightarrow T_xM$ are linear map, the latter is given by the duality: $$\big\langle \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic^{\sharp}_x(v_1), v_2\big\rangle_{T_x M} =\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic_x(v_1,v_2),\quad\ \fracorall\ v_1,v_2\in T_x M.$$ {\bf The horizontal Brownian motion.} Given a point $x\in M$, let $O_xM$ denote the space of linear isometries from $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ to $T_xM$. Let $OM:=\cup_{x\in M} O_xM$, which is the orthonormal frame bundle over $M$, and let $\pi: OM \rightarrow M$ denote the canonical projection which takes a frame $u\in O_x M$ to its base point $x$. For every $u\in OM$, we define ${\rm R}_u:\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n\times \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n \rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n\times \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ and $ {\rm ric}_u:\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n \rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ by $$\begin} \def\beq{\beg} \def\F{\scr Fin{aligned}{\rm R}_u(e_1,e_2):&=u^{-1}\big({\bf R}_{\pi(u)}(ue_1,ue_2)\big), \\ \text{\rm ric}_u(e_1):&=u^{-1}\big({\bf \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic^{\sharp}}_{\pi(u)}(ue_1)\big) \varepsilonnd{aligned}$$ for every $ e_1,e_2\in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$. Given a vector $e\in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$, we denote by $H_{e}$ the associated canonical horizontal vector field on $OM$ with the property that $(T\pi)_{u}(H_{e})=u e\in T_{\pi(u)}M$. Thus the solution of the ODE $$u'(t)=H_e(u(t))$$ projects to the geodesic on $M$ with the initial position $x$ and the initial speed $u(0)(e)$. We choose an orthonormal basis $\{e_i\}_{i=1}^n$ of $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$. Suppose $\{U_t\}_{t\ge 0}$ is the solution of following $OM$-valued Stratonovich stochastic differential equation \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{sde1} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D U_t=\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Displaystyle\sum^n_{i=1}H_{e_i}\left(U_t \right)\circ\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^i, \varepsilonnd{equation} where the initial value $U_0$ is a fixed orthonormal basis of $T_xM$. We usually call $\{U_t\}_{0\le t<\zeta}$ the canonical horizontal Brownian motion, where $\zeta:\Omega \to \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt_+$ is the life time for $U_t$. Let $X_t^x:=\pi(U_t)$, $0\le t<\zeta(x)$, then $X_t^x$ is a Brownian motion on $M$ with initial value $x$ and life time $\zeta(x)$. This is the celebrated intrinsic construction of $M$-valued Brownian motion by Eells and Elworthy \cite{EE} and Elworthy \cite{Elworthy}, see also Malliavin \cite{Malliavin}. It is well known that the Brownian motion on $M$ does not explode if and only if the horizontal Brownian motion $U_t$ on $OM$ does not explode. In particular, it does not rely on the choice of an isometrically embedding from $M$ to an ambient Euclidean space. Let $$P_tf(x):=\mathbb{E}\left[f\left(X_t^x\right){\bf 1}_{\{t<\zeta(x)\}}\right]$$ be the heat semigroup associated to Brownian motion $X_{\cdot}$. \varepsilonmph{The superscript $x$ may be omitted if there is no risk of confusion.} \subsection{Second order gradient of the heat semigroup} Let $\{U_t\}_{0\le t <\zeta(x)}$ denote the horizontal Brownian motion on $M$ and $\{X_t^x=\pi(U_t)\}_{0\le t<\zeta(x)}$ is the Brownian motion on $M$ with initial value $x$ and life time $\zeta(x)$. For any $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$, we set \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e3-2} \Gamma^h_t:=\int_0^t {\rm R}_{U_s} \left( \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, h(s)\right),\qquad \scr Theta^h_t:=h'(t)+\fracrac{1}{2}\text{\rm ric}_{U_t}(h(t)), \varepsilonnd{equation} It is easy to see that $\Gamma^h_t$ is an ${\mathfrak {so}}(n)$-valued process. For $ t\geq0$, we define \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e3-2a} \begin} \def\beq{\beg} \def\F{\scr Fin{split} {\mathcal L}ambda_t^h:=\Gamma_t^h h'(t)+\fracrac{1}{2}U_t^{-1} \; \nabla \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic^{\sharp}_{X_t}\big(U_th(t), U_t h(t)\big) -\frac 12\Gamma_t^h \;\text{\rm ric}_{U_t}(h(t))+\frac 12 \text{\rm ric}_{U_t}\left(\Gamma_t^h h(t)\right). \varepsilonnd{split} \varepsilonnd{equation} We are now ready to state one of our main tools, the second order gradient formula on a general complete $M$. \begin} \def\beq{\beg} \def\F{\scr Fin{thm}\label{thm3.1} Suppose that $M$ is a complete Riemannian manifold. Let $\{D_m\}_{m=1}^\infty$ denote the increasing family of exhaustive relatively compact open sets of $M$ and let $\{l_m\}_{m=1}^\infty$ denote the cut-off vector fields as constructed in Lemma \ref{lem5.1}. Let $x\in m$, and there exists $m_0\in {\mathbb N}$ such that $x\in D_{m_0+1}$. For every $m> m_0$, $v \in T_x M$, and $t\in (0,1]$, we define $$h(s):=\Big(\fracrac{t-2s}{t}\Big)^+\cdot l_m\left(s,X_{\cdot}^x\right)\cdot U_0^{-1}v, \quad s\geq0.$$ Then $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$. Furthermore, for any $f\in C_b(M)$ we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{t3-1-1} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\big\langle \nabla^2 P_t f(x), v\otimes v\big\rangle_{T_x M \otimes T_x M}\\ &=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[\left( \left(\int_0^t\langle \scr Theta^h_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle\right)^2- \int_0^t \langle {\mathcal L}ambda^h_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle-\int_0^t \left|\scr Theta^h_s\right|^2 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right)f(X_t^x){\bf 1}_{\{t<\zeta(x)\}}\right]. \varepsilonnd{split} \varepsilonnd{equation} \varepsilonnd{thm} In particular, the processes $l_m(t, \gamma)$ equals to $1$ at any time before $\gamma$ exits $D_{m-1}$ and equals to zero after it exits $D_m$ for the first time. So it is obvious to see that $h(t, \gamma)=U_0^{-1}v$ at $t=0$ and vanishes after the first exit time of $\gamma$ from $D_m$. \subsection{Comments} The main idea for proving the second order gradient of the heat semigroup $P_t$ is to approximate the formula on $M$ by those for a family of specific compact manifolds. We first use a result of Greene and Wu \cite{GW} to construct a family of relatively compact exhausting open subsets $\{D_m\}_{m=1}^\infty$, which is valid for a complete Riemannian manifold $M$. This allows to construct a series of random cut-off vector fields $l_m\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ vanishing, as soon as the sample path exits $D_m$ for the first time, with the necessary quantitative estimates needed for the localisation. See Lemma \ref{lem5.1} below for details. The lemma is partly inspired by the work of Thalmaier \cite{T} and Thalmaier and Wang \cite{TW}, where geodesic balls are used. For the purpose of embedding into compact manifolds, we make sure that each $D_m$ having a smooth boundary which, because of the cut locus, cannot be taken as granted of geodesic balls on arbitrary Riemannian manifolds. We want to remark that this offers a more powerful (and also a more reliable) alternative to localisation with stopping times, the latter has been commonly used in stochastic calculus and occasionally incorrectly used. The stopping time argument relies on a continuity assumption on the Brownian motion with respect to the initial value. Such continuity condition seems not easy to verify (for stopping times), and ought not be applied casually, see e.g. Elworthy \cite{Elworthy78}, Li and Sheutzow \cite{Li-Scheutzow}, and Li \cite{Li-flow} for more details. Note, however, that exit times from regular domains do have good regularity properties in the sense of Malliavin calculus, we refer the reader to the work of Airault, Maillian, and Ren \cite{Airault-Malliavin-Ren} for more details. Cut-off vector fields have been previously applied by Arnaudon, Plank and Thalmaier \cite{APT}, Thompson \cite{Thompson}, Thalmaier \cite{T}, and Thalmaier and Wang \cite{TW} to provide a {\it localised} differential formula for heat semigroups. As explained earlier, we use a new type of (second order) stochastic variation argument to construct the global second order gradient formula given below. In particular, the expression here is different from that of Elworthy and Li \cite{EL}, Arnaudon, Plank, and Thalmaier \cite{APT}, Li \cite{Li-doubly-damped, Li18} and Thompson \cite{Thompson} and particularly we do not use the doubly parallel translation operators used in \cite{Li-doubly-damped, Li18}. \subsection{Comparison theorems} The outline of the proof is as follows. We first show that the formula holds for a compact Riemannian manifold, this proof is given in Section \ref{section2} using a new stochastic variation. To pass from a compact manifold to a non-compact manifold, we use a suitable isometric embedding from $D_m$ into a compact Riemannian manifold $\tilde M_m$, as well as the quantitative cut-off process $l_m$ constructed by Lemma \ref{lem7.1} and Lemma \ref{lem5.1} respectively. Denote by $p_{\tilde M_m}(t,x,y)$ the heat kernel on $\tilde M_m$. Although the heat kernel of a Riemannian manifold is determined in a global manner by the Riemannian metric, we obtain, below, short time comparison theorems between $\nabla \log p_{\tilde M_m}$, $\nabla^2 \log p_{\tilde M_m}$ and $\nabla \log p$, $\nabla^2 \log p$. These are used for proving \varepsilonqref{e1-1}--\varepsilonqref{e1-3b}. The comparison theorem below allows us to obtain estimates for $\nabla \log p$ and $\nabla^2 \log p$, with the successive applications of first order and second order gradient formula as well as comparison estimates for functionals of the Brownian motions on $M$ and that on $\tilde M_m$. \begin} \def\beq{\beg} \def\F{\scr Fin{prp} (Propositions \ref{prp6.6} and \ref{prp6.9})\label{prp3.1} Suppose $K$ is a compact subset of $M$. For any constant $L>1$, there exists a $m_0=m_0(K,L)\in {\mathbb N}$, which may depend on $K$ and $L$, such that for all $m\ge m_0$ we could find a positive time $t_0=t_0(K,L,m)$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{eqnarray*} &\sup_{x,y\in K} \left|\nabla_x\log p(t,x,y)- \nabla_x \log p_{\tilde M_{m}}(t,x,y)\right|_{T_x M} \le C(m)e^{-\fracrac{L}{t}},\quad \fracorall\ t\in (0,t_0],\\ &\sup_{x,y\in K} e^{\fracrac{L}{t}} \left|\nabla_x^2\log p(t,x,y)-\nabla_x^2 \log p_{\tilde M_{m}}(t,x,y) \right|_{T_x M\otimes T_x M}\le C(m)e^{-\fracrac{L}{t}},\quad \fracorall\ t\in (0,t_0], \varepsilonnd{eqnarray*} where $C(m)$ is a positive constant depending on $m$. \varepsilonnd{prp} \text{\rm sect}ion{Second Order Variation on a Compact Manifold}\label{section2} \quad \quad \varepsilonmph{Throughout this section, $M$ is an $n$-dimensional compact Riemannian manifold.} In Proposition \ref{prp4.5} below, we shall establish (\ref{t3-1-1}) for a compact manifold, which is a fundamental step toward Theorem \ref{thm3.1}. The first second order differential formula for the heat semigroup $P_t$ was obtained by Elworthy and Li \cite{EL1} for a non-compact manifold, however with restrictions on their curvature. Another disadvantage of the formula was its involvement of a non-intrinsic curvature which was due to the application of the derivative flow of gradient stochastic differential equations, as well as a martingale approach developed in Li \cite{Li-thesis}. An intrinsic formula for $\nabla^2 P_t f$ was given by Stroock \cite{S} for a compact Riemannian manifold, while a localised intrinsic formula was obtained by Arnaudon, Plank and Thalmaier \cite{APT} with the martingale approach. The study of the second order gradient of the Feynman-Kac semigroup of an operator $\Delta+V,$ with a potential function, was pioneered by Li \cite{Li18, Li-doubly-damped}, where a path integration formula was obtained with the help of doubly damped stochastic parallel transport equation. (The first order gradient formula was previously obtained in Li and Thompson \cite{Li-Thompson}, c.f. \cite{EL1, EL-Vilnius}.) A localised version of the Hessian formula (still with doubly stochastic damped parallel translations) for the Feynman-Kac semigroup was derived by Thompson \cite{Thompson}. However, all the expressions mentioned earlier do not seem to lead to our application, such as the proof of Proposition \ref{prp3.1}. To overcome this problem we introduce a quantitative localisation procedure and obtain a second order gradient formula to which this localisation method can be applied. One of our main tools is to extend Bismut's idea to perturb the $M$-valued Brownian motion with initial value $\xi(\varepsilon)$ (where $\xi(\varepsilon)$ is a smooth curve in $M$), they will be constructed as solutions of a family of SDEs with the driving Brownian motion $\{B_t\}_{t\ge 0}$ rotated and translated appropriately. The rotation and translation exerted on $\{B_t\}_{t\ge 0}$ transmits the variation in the initial value of the Brownian motion on the manifold to variations, in the same parameter, of the Radon Nikodym derivatives of a family of probability measures, with respect to which the solutions are Brownian motions on $M$. This simple and elegant idea was applied in Bismut \cite{Bis} for deducing an integration by parts formula. Incidentally, such integration by parts formula and the first order gradient formula of the heat semigroup were proved to be equivalent on a compact manifold by Elworthy and Li \cite{EL}. In Stroock \cite{S}, by calculating the concrete form of the second variation introduced by Bismut, this idea was adapted for obtaining the second order derivative formula for the heat semigroup on a compact manifold. As explained earlier, the choice of stochastic variation in \cite{S} (see also Malliavin and Stroock \cite[(1.5)]{MS}) will produce a term coming from the time reverse of a non-random vector field on $L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$, and it seems not possible to replace the non-random vector field by a random one (otherwise the time reversed field is not adapted, hence It\^o's integral is not well defined). Therefore the formula obtained in Stroock \cite{S} may not be extended to the one with a random vector field and so is not suitable to for extension to non-compact manifolds with the localisation technique we introduce shortly. One crucial ingredient for our choice of the stochastic variation is that it ensures \varepsilonqref{l3-3-1}, which implies that the second variation vanishes at time $t$ when we choose a vector field $h$ in the translated part satisfying $h(t)\varepsilonquiv 0$. This allows us to derive a second order gradient formula with localised vector fields and to extend it to a general (non-compact) complete Riemannian manifold. \subsection{A novel stochastic variation with a second order term} As before, $\{U_t\}_{0\le t<\zeta(x)}$ is the solution of equation \varepsilonqref{sde1} with initial point $U_0$ and $\pi(U_0)=x$. In Bismut \cite{Bis} the following classical perturbation for the driving force $B_t$ was used: $$\mathfrak hat B_t^\varepsilon=\int_0^t e^{-\varepsilon \Gamma^h_s}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s+\varepsilon \int_0^t \left(h'(s) +\frac 12 \text{\rm ric}_{U_s} h(s)\right) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s.$$ where $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ is a chosen Cameron-Martin vector and $\Gamma^h_t:=\int_0^t {\rm R}_{U_s} \left( \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, h(s)\right)$. This perturbation of the noise works well with the first variation for which one needs to ensure that $\frac {\partial} {\partial\varepsilon}|_{\varepsilon =0} \pi (U_t^\varepsilon) = U_th(t)$ and has been the popular and standard perturbation, as used also in Driver \cite{D1}, Fang and Malliavin \cite{FM}. Other variation of the noise are also of first order perturbations. However, with the above mentioned variation, $\frac {\partial^2} {\partial\varepsilon^2}|_{\varepsilon =0} \pi (U_t^\varepsilon) \not =0$ as long as $h(t)\not \varepsilonquiv 0$. To solve this problem, we will introduce a second order variation (such perturbation is not unique and we may find a slightly different choice). Unlike the case with the classical perturbation, this time we cannot avoid differentiating the structure equation so have to choose a connection on the frame bundle. Our approach is inspired by the theory of linear connections induced by a SDE developed by Elworthy, LeJan and Li \cite{ELL}. We believe that the same method can also be used for higher order variations. For any $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$, we have defined an ${\mathfrak {so}}(n)$-valued process $\Gamma_t^h$ and $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued process $\scr Theta_t^h$, ${\mathcal L}ambda_t^h$ by \varepsilonqref{e3-2} and \varepsilonqref{e3-2a} respectively. We first introduce the translation and define the $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued process $B_t^{\varepsilon,h}$ as follows \begin} \def\beq{\beg} \def\F{\scr Fin{equation} B_t^{\varepsilon,h}:=B_t+\varepsilon\int_0^t h'(s)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s+\fracrac{\varepsilon^2}{2}\int_0^t \Phi^h_s\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s, \varepsilonnd{equation} where $\Phi^h_t:=\Gamma^h_t h'(t)$. We then introduce a rotation for $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued Brownian motion. Let us first set \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e3-3} \begin} \def\beq{\beg} \def\F{\scr Fin{aligned} \Gamma_t^{(2),h}:=&\int_0^t U_s^{-1}\nabla {\rm {\bf R}}_{\pi(U_s)}\big(U_s h(s), U_s\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, U_s h(s)\big)-\int_0^t \Gamma^h_s{\rm R}_{U_s}(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, h(s))\\ &+\int_0^t {\rm R}_{U_s}(h'(s),h(s))\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s+\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, \Gamma^h_s h(s)\right). \varepsilonnd{aligned} \varepsilonnd{equation} It is easy to see that $\Gamma_t^{(2),h}$ is an ${\mathfrak {so}}(n)$-valued process. Then for every $\varepsilon>0$, we define $SO(n)$-valued process $G_t^{\varepsilon,h}$ as follows \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{aligned} G_t^{\varepsilon,h}&:=\varepsilonxp\left({-\varepsilon \Gamma^h_t -\fracrac{\varepsilon^2}{2} \Gamma_t^{(2),h} }\right), \varepsilonnd{aligned} \varepsilonnd{equation*} where $\varepsilonxp:{\mathfrak {so}}(n)\to SO(n)$ is the exponential map in the Lie algebra ${\mathfrak {so}}(n)$ of $SO(n)$. We can now introduce $\tilde B_t^{\varepsilon,h}$, the variation of $B_t$, as well as the corresponding equation on~$OM$. \begin} \def\beq{\beg} \def\F{\scr Fin{defn}\label{def4.1} Let $\xi(\varepsilon)$, $\varepsilon\in(-1,1)$, be a geodesic with $\xi(0)=x$. Let $\{U_0^{\varepsilon,h}: \varepsilon\in(-1,1)\}$ be a parallel orthonormal frame along $\xi(\varepsilon)$ with $\pi(U_0^{\varepsilon,h})=\xi(\varepsilon)$. Let $U_t^\varepsilonpsilon$ denote the solution of the following equation with initial condition $U_0^{\varepsilon,h}$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{sde2} \begin} \def\beq{\beg} \def\F{\scr Fin{aligned} &\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D U_t^{\varepsilon,h}=\sum_{i=1}^n H_{e_i}(U_t^{\varepsilon,h})\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \tilde B_t^{\varepsilon,h,i},\\& \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\tilde B_t^{\varepsilon,h} =G_t^{\varepsilon,h}\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^{\varepsilon,h},\quad \tilde B_0^{\varepsilon,h}=0. \varepsilonnd{aligned} \varepsilonnd{equation} We define $X_t^{\varepsilon,\xi(\varepsilon),h}= \pi(U_t^{\varepsilon,h})$. If $\varepsilon=0$, then $X_t^{0,x,h}=X_t^x$ with $X_t^x=\pi(U_t)$. \varepsilonnd{defn} We remark that the perturbation in $U_t^{\varepsilon,h}$ has a translation part $B_t^{\varepsilon,h}$, and a rotation part $G_t^{\varepsilon,h}$. The rotation $G_t^{\varepsilon,h}$ is chosen to offset precisely the twisting effects induced by the second order stochastic variation. \varepsilonmph{For simplicity we omit the subscript $h$, in $\scr Theta^h_t,{\mathcal L}ambda^h_t$, $X_t^{\varepsilon,h},\Gamma_t^{h},\Gamma_t^{(2),h},G_t^{\varepsilon,h}$, $B_t^{\varepsilon,h}$ and $U_t^{\varepsilon,h}$, from time to time.} Let $\varpi$ and $\theta$ denote respectively the ${\mathfrak {so}}(n)$-valued connection $1$-form and the $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued solder $1$-form respectively. Set $$\varpi_t^\varepsilon:= \varpi\left( \frac \partial {\partial \varepsilon}U_t^\varepsilon\right), \qquad \theta_t^\varepsilon:= \theta\left( \frac \partial {\partial \varepsilon} U_t^\varepsilon\right).$$ Through this paper, we use $D_t$, $\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t$ to denote the stochastic covariant differential for vector fields and stochastic differential on $M$ along a semi-martingale respectively and $\fracrac{D}{\partial \varepsilon}$ denotes the covariant derivative for vector fields on $M$ with respect to the variable $\varepsilon$. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem 4.1} If we choose $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ such that $h(0)=U_0^{-1}\left(\fracrac{\partial}{\partial \varepsilon}\Big|_{\varepsilon=0}\xi(\varepsilon)\right)$, then \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{varpi} \varpi_t^\varepsilon=\int_0^t {\rm R}_{U_s^\varepsilon} (G_s^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^\varepsilon, \theta_s^\varepsilon). \varepsilonnd{equation} And $\theta_t^\varepsilon$ satisfy the following equation, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l3-1-1} \begin} \def\beq{\beg} \def\F{\scr Fin{cases} & \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\theta_t^\varepsilon=-\big(\Gamma_t+\varepsilon\Gamma^{(2)}_t\big)G_t^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon+\varpi_t^\varepsilon G_t^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon+ G_t^\varepsilon\big(h'(t)+\varepsilon\Phi_t\big)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\\ & \theta_0^\varepsilon=(U_0^\varepsilon)^{-1}\fracrac{d \xi(\varepsilon)}{d \varepsilon}. \varepsilonnd{cases} \varepsilonnd{equation} In particular, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation} \left\{\begin} \def\beq{\beg} \def\F{\scr Fin{aligned} &\theta_t^0:=\theta\left( \frac \partial {\partial \varepsilon} \Big|_{\varepsilon=0}U_t^\varepsilon\right)=h(t), \label{l3-1-2} \\ &\varpi^0_t=\Gamma_t,\\ &\fracrac{D}{\partial \varepsilon}\Big|_{\varepsilon=0}\left(U_t^\varepsilon G_t^\varepsilon e\right)=0,\quad \ \fracorall\ e\in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^d,\\ &\fracrac{\partial X_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}=U_t h(t). \varepsilonnd{aligned}\right. \varepsilonnd{equation} \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} We first use the structure equation \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\varpi\Big(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon, \frac \partial {\partial \varepsilon}U_t^\varepsilon\Big)&=-\varpi\varpiedge \varpi \left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon, \frac \partial {\partial \varepsilon}U_t^\varepsilon \right)+{\rm R}_{U_t^\varepsilon}\left(\theta\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon\right), \theta\left( \frac \partial {\partial \varepsilon}U_t^\varepsilon\right)\right)\\ &=-\sum_{i=1}^n\varpi\varpiedge \varpi\left(H_{e_i}(U_t^{\varepsilon})\circ d\tilde B^{\varepsilon,i}_t, \frac \partial {\partial \varepsilon}U_t^\varepsilon\right)+{\rm R}_{U_t^\varepsilon}\left(\theta\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon\right),\theta_t^\varepsilon\right)\\ &={\rm R}_{U_t^\varepsilon}\left(\theta\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon\right),\theta_t^\varepsilon\right) \varepsilonnd{align*} to obtain \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \varpi_t^\varepsilon &=\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\varpi\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon, \frac \partial {\partial \varepsilon}U_t^\varepsilon\right)={\rm R}_{U_t^\varepsilon}\left(\theta\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t U_t^\varepsilon\right), \theta_t^\varepsilon\right)={\rm R}_{U_t^\varepsilon}\left(G_t^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon, \theta_t^\varepsilon\right). \varepsilonnd{align*} Since at time $0$, the variation $\{U_0^\varepsilon;\varepsilon\in (-1,1)\}$ is parallel along the geodesic $\xi$, $\varpi_0^\varepsilon=0$. Then \varepsilonqref{varpi} follows immediately. Here we have used the Transfer Principle: on the compact manifold $M$ we could \varepsilonmph{treat the Stratonovich integral as the ordinary derivative} (with respect to time variable) in the computation. Crucially we could \varepsilonmph{exchange the order of differentiations and integrations. } The transfer principle is well known for \varepsilonmph{compact} manifolds, see e.g. \cite{FM} or \cite{MalliavinMon}, but not automatically apply to non-compact manifolds nor automatically to the less smooth case nor to the derivative processes. This is used in similar computations later in the article without further comment. Due to the torsion free property, the time derivative and the derivative for $\varepsilon$ could commute: $D_t\fracrac{\partial }{\partial \varepsilon}= \fracrac{D}{\partial \varepsilon}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t$. Also note that $\theta_t^\varepsilon=(U_t^{\varepsilon})^{-1}T\pi ( \frac \partial {\partial \varepsilon} U_t^\varepsilon)$, so we have, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{1derivative} \begin} \def\beq{\beg} \def\F{\scr Fin{split} {\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D } \theta_t^\varepsilon&=(U_t^\varepsilon)^{-1}\left(D_t \left(\fracrac{\partial}{\partial \varepsilon}X_t^\varepsilon\right) \right)=(U_t^\varepsilon)^{-1}\left(\frac D {\partial \varepsilon}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_t X_t^\varepsilon\right)\\ &=(U_t^\varepsilon)^{-1}\left(\fracrac{D}{\partial \varepsilon}\left(U_t^\varepsilon G_t^\varepsilon \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon\right)\right)\\ &=\varpi_t^\varepsilon G_t^\varepsilon \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon +\fracrac{\partial G_t^\varepsilon}{\partial \varepsilon}\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon+G_t^\varepsilon \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\left(\fracrac{\partial}{\partial \varepsilon}B_t^\varepsilon\right)\\ &=\varpi_t^\varepsilon G_t^\varepsilon \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon-\big(\Gamma_t+\varepsilon\Gamma^{(2)}_t\big)G_t^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon+ G_t^\varepsilon\big(h'(t)+\varepsilon\Phi_t\big)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t, \varepsilonnd{split} \varepsilonnd{equation} where the fourth equality is due to \begin} \def\beq{\beg} \def\F{\scr Fin{equation} \label{proof-1-4} \begin} \def\beq{\beg} \def\F{\scr Fin{aligned} \frac D {\partial \varepsilon} \left( U_t^\varepsilon G_t^\varepsilon \right) &=U_t^\varepsilon \left( \varpi_t^\varepsilon G_t^\varepsilon + \frac {\partial}{\partial\varepsilon} G_t^\varepsilon \right).\varepsilonnd{aligned} \varepsilonnd{equation} So we have obtained the first equation in \varepsilonqref{l3-1-1}. The initial condition in \varepsilonqref{l3-1-1} follows trivially from the fact $ \theta_0^\varepsilon=(U_0^{\varepsilon})^{-1}\pi ( \frac \partial {\partial \varepsilon} U_0^\varepsilon))$, $\{U_0^\varepsilon; \varepsilon\in (-1,1)\}$ is a parallel orthonormal frame bundle along $\xi(\cdot)$ and $X_0^\varepsilon=\xi(\varepsilon)$. Based on the fact that $$\varpi_t^0=\int_0^t {\rm R}_{U_s} \left( \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, \theta_s^0\right), \qquad \Gamma_t=\int_0^t {\rm R}_{U_s} \left( \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, h(s)\right),$$ and taking $\varepsilon=0$ in \varepsilonqref{l3-1-1} we arrive at \begin} \def\beq{\beg} \def\F{\scr Fin{align*} d\theta^0_t=\left( \int_0^t {\rm R}_{U_s} \left( \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, \theta_s^0\right)-\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s,h(s)\right) \right)\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t+ h'(t)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t, \quad \ \theta_0^0=h(0). \varepsilonnd{align*} It is easy to verify that $\theta^0_t=h(t)$ is the unique solution to above equation, proving the first line of \varepsilonqref{l3-1-2}. Then plugging in $\theta_t^0=h(t)$ into \varepsilonqref{varpi} to see that $\varpi^0_t=\Gamma_t$, so we have \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \frac D {\partial \varepsilon}\Big|_{\varepsilon=0}\left(U_t^\varepsilon G_t^\varepsilon e\right)= U_t\left(\varpi_t^0 e+\fracrac{\partial}{\partial \varepsilon}\Big|_{\varepsilon=0}G_t^\varepsilon e\right)=U_t\left(\Gamma_t e-\Gamma_t e\right)=0, \varepsilonnd{align*} which is the third line of \varepsilonqref{l3-1-2}. Finally, $D_t\Big(\fracrac{\partial X_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\Big)=U_t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \theta_t^0=U_t h'(t)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t$, giving $ \fracrac{\partial X_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}=U_th(t)$. This completes the proof. \varepsilonnd{proof} In particular, we obtain the following lemma: \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem 4.2} For every $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ with $h(0)\varepsilonquiv v=U_0^{-1}\left(\fracrac{\partial}{\partial \varepsilon}\Big|_{\varepsilon=0}\xi(\varepsilon)\right)$, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l3-2-1} \fracrac{\partial }{\partial \varepsilon}\Big|_{\varepsilon=0}\varpi_t^\varepsilon=\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s,\varepsilonta_s\right)+\Gamma_t^{(2)}, \varepsilonnd{equation} where $\varepsilonta_s:=\frac {\partial \theta_s^\varepsilon} {\partial \varepsilon} \Big|_{\varepsilon=0}$ and $\Gamma_t^{(2)}$ is defined by \varepsilonqref{e3-3}. \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} By the first line of \varepsilonqref{l3-1-2} we have $\theta_t^0=h(t)$. We differentiate the integral expression \varepsilonqref{varpi} for $\varpi_t^\varepsilon$ and apply the third line of \varepsilonqref{l3-1-2} to obtain \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \fracrac{\partial }{\partial \varepsilon}\Big|_{\varepsilon=0}\varpi_t^\varepsilon =&\fracrac{\partial}{\partial\varepsilon}\Big|_{\varepsilon=0} \int_0^t (U_s^{\varepsilon})^{-1} {\rm { R}}_{X_s^\varepsilon}\left( U_s^\varepsilon G_s^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^\varepsilon, U_s^\varepsilon \theta_s^\varepsilon\right) \\ =& \int_0^t \fracrac{\partial}{\partial\varepsilon}\Big|_{\varepsilon=0}\left(G_s^\varepsilon ( U_s^{\varepsilon}G_s^\varepsilon )^{-1} {\rm { R}}_{X_s^\varepsilon} \left(U_s^\varepsilon G_s^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^\varepsilon, U_s^\varepsilon G_s^{\varepsilon}(G_s^{\varepsilon})^{-1}\theta_s^\varepsilon\right) \right)\\ =&\int_0^t \left(\fracrac{\partial G_s^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\right) {\rm R}_{U_s} \left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s,\,\theta_s^0\right)+ \int_0^t U_s ^{-1}\nabla {\rm { R}}_{X_s}\left(U_s\theta_s^0, U_s\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, U_s \,\theta_s^0\right) \\ &+\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \fracrac{\partial B_s^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}, \theta_s^0\right)+ \int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, \left(\fracrac{\partial (G_s^\varepsilon)^{-1}}{\partial \varepsilon}\Big|_{\varepsilon=0}\right)\theta_s^0\right)\\ &+\int_0^t \, {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, \frac {\partial \theta_s^\varepsilon} {\partial \varepsilon} \Big|_{\varepsilon=0}\right). \varepsilonnd{align*} Here the last term is $\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s,\varepsilonta_s\right)$, while the sum of the rest is $\Gamma_t^{(2)}$, so we have completed the proof. \varepsilonnd{proof} We observe that $\varepsilonta_s=\frac {\partial \theta_s^\varepsilon} {\partial \varepsilon} \Big|_{\varepsilon=0}$ is essentially the second variation of $\pi(U_s^\varepsilonpsilon)$. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem 4.3} For every $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ with $h(0)\varepsilonquiv v=U_0^{-1}\left(\fracrac{\partial}{\partial \varepsilon}\Big|_{\varepsilon=0}\xi(\varepsilon)\right)$, we have $\varepsilonta_t\varepsilonquiv 0$ for all $t\in [0,1]$ and \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l3-3-1} \fracrac{D}{\partial \varepsilon}\Big|_{\varepsilon=0}\left(\fracrac{\partial X_t^\varepsilon}{\partial \varepsilon}\right)=U_t\Gamma_th(t). \varepsilonnd{equation} \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} We recall the first equation of \varepsilonqref{l3-1-1} $$d\theta_t^\varepsilon=-\big(\Gamma_t+\varepsilon\Gamma^{(2)}_t\big)G_t^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon+\varpi_t^\varepsilon G_t^\varepsilon\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^\varepsilon+ G_t^\varepsilon\big(h'(t)+\varepsilon \Gamma_th'(t))\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t.$$ Differentiating it at $\varepsilon=0$, using \varepsilonqref{l3-2-1} and the following fact \begin} \def\beq{\beg} \def\F{\scr Fin{equation*}\label{summarise} \varpi_t^0=\Gamma_t, \quad \fracrac{\partial B_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}=h'(t), \quad \fracrac{\partial G_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}=-\Gamma_t, \quad \Phi_t=\Gamma_t h'(t), \varepsilonnd{equation*} we could obtain \begin} \def\beq{\beg} \def\F{\scr Fin{equation*}\label{l3-3-2} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\varepsilonta_t=&-\left(\Gamma_t^{(2)}+\Gamma_t\fracrac{\partial G_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\right)\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t -\Gamma_t \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\left(\fracrac{\partial B_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\right)+\left(\varpi_t^0\fracrac{\partial G_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0} +\fracrac{\partial \varpi_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\right)\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t\\ &+\varpi_t^0 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\left(\fracrac{\partial B_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\right)+\fracrac{\partial G_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}h'(t)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t + \Gamma_th'(t)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\\ =&\left(\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, \varepsilonta_s\right)\right)\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t. \varepsilonnd{split} \varepsilonnd{equation*} At the same time, since $X_0^\varepsilon=\xi(\varepsilon)$, $\xi(\cdot)$ is a geodesic, and also $\{U_0^\varepsilon, \varepsilon\in (-1,1)\}$ is a parallel orthonormal frame bundle along $\xi(\cdot)$, we could verify that $$\varepsilonta_0=\fracrac{\partial \theta_0^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}=U_0^{-1}\left(\fracrac{D}{\partial \varepsilon}\Big|_{\varepsilon=0} \left(\fracrac{\partial \xi(\varepsilon)}{\partial \varepsilon}\right)\right)=0.$$ Observe that the unique solution to following equation is $v_t\varepsilonquiv 0$ \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D v_t=\left(\int_0^t {\rm R}_{U_s}\left(\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s, v_s \right)\right)\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t,\quad \ v_0=0. \varepsilonnd{align*} Then we derive that $\varepsilonta_t\varepsilonquiv 0$ for all $t\in [0,1]$. Moreover, note that by definition we have $\fracrac{\partial X_t^\varepsilon}{\partial \varepsilon}=U_t^\varepsilon \theta_t^\varepsilon$, due to the fact $\varepsilonta_t=\fracrac{\partial \theta_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\varepsilonquiv 0$ we obtain \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \fracrac{D}{\partial \varepsilon}\Big|_{\varepsilon=0}\left(\fracrac{\partial X_t^\varepsilon}{\partial \varepsilon}\right) =\fracrac{D}{\partial \varepsilon}\Big|_{\varepsilon=0}\left(U_t^\varepsilon \theta_t^\varepsilon\right)= U_t\left(\varpi_t^0\theta_t^0+\fracrac{\partial \theta_t^\varepsilon}{\partial \varepsilon}\Big|_{\varepsilon=0}\right) =U_t\Gamma_th(t). \varepsilonnd{align*} Now we have obtained \varepsilonqref{l3-3-1}. \varepsilonnd{proof} \subsection{Proof for the 2nd order gradient formula on a compact manifold} \begin} \def\beq{\beg} \def\F{\scr Fin{prp}\label{prp4.5} Let $t>0$, $x\in M$ and $v\in T_x M$. Then for any $f\in C_b(M)$ and $h\in L^{2,1}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ satisfying that $h(0)=U_0^{-1}v$ and $h(t)=0~a.s.$, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-0} \big\langle \nabla P_tf(x), v\big\rangle_{T_x M}=-\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[ f(X_t^x)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^h, dB_s\>\right], \varepsilonnd{equation} where $ \scr Theta^h_t:=h'(t)+\fracrac{1}{2}{\rm ric}_{U_t}(h(t))$. Furthermore, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-1} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\quad \big\langle \nabla^2 P_t f(x), v\otimes v\big\rangle_{T_x M \otimes T_x M}\\ &=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[f(X_t^x)\left( \left(\int_0^t\big\langle \scr Theta_s^h, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\big\rangle\right)^2- \int_0^t \big\langle {\mathcal L}ambda_s^h, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\big\rangle-\int_0^t \left|\scr Theta_s^h\right|^2 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right)\right]. \varepsilonnd{split} \varepsilonnd{equation} \varepsilonnd{prp} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} We take $\xi(\cdot)$ to be a geodesic with initial value $\xi(0)=x$ and initial velocity $\fracrac{\partial \xi(\varepsilon)}{\partial \varepsilon}\Big|_{\varepsilon=0}=v$. Let $\{U_0^\varepsilon\in (-1,1)\}$ denote the parallel orthonormal frame bundle along $\xi(\cdot)$ with $U_0^\varepsilon\Big|_{\varepsilon=0}=U_0$. In particular, it holds that $\pi(U_0^\varepsilon)=\xi(\varepsilon)$. Recall that $U_t^\varepsilon$ is the solution to \varepsilonqref{sde2} with initial value $U_0^\varepsilon$ chosen above. It holds that \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \int_0^t G_s^\varepsilon \circ dB_s^\varepsilon&= \int_0^t G_s^\varepsilon \circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s+\int_0^t G_s^\varepsilon \left(\varepsilon h'(s)+\fracrac{\varepsilon^2}{2}\Gamma_s\,h'(s)\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\\ &=\int_0^t G_s^\varepsilon \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s+\int_0^t \fracrac{1}{2}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \langle G_{\cdot}^\varepsilon, B_{\cdot}\rangle_s+ \int_0^t G_s^\varepsilon \left(\varepsilon h'(s)+\fracrac{\varepsilon^2}{2} \Gamma_s\,h'(s)\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\\ &=\int_0^t G_s^\varepsilon \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s+\varepsilon\int_0^t G_s^\varepsilon \scr Theta_s \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s +\fracrac{\varepsilon^2}{2}\int_0^t G_s^\varepsilon {\mathcal L}ambda_s \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s. \varepsilonnd{align*} Here we have used that \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \langle G_{\cdot}^\varepsilon, B_{\cdot}\rangle_t&= -\varepsilon G_t^\varepsilon \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle\Gamma_\cdot , B_{\cdot}\rangle_t-\fracrac{\varepsilon^2}{2} G_t^\varepsilon \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle\Gamma_\cdot^{(2)} , B_{\cdot}\rangle_t\\ &=\fracrac{\varepsilon}{2}G_t^\varepsilon\;\text{\rm ric}_{U_t}(h(t))\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t+\fracrac{\varepsilon^2}{2}G_t^\varepsilon\Big[U_t^{-1}\nabla {\rm Ric}^\sharp_{\pi(U_t)}\left(U_th(t), U_t h(t)\right)\\ &\ +\Gamma_t \; \text{\rm ric}_{U_t}(h(t))-\; \text{\rm ric}_{U_t}\left(\Gamma_t h(t)\right)\Big]. \varepsilonnd{align*} Note that $W_t^\varepsilon:=\int_0^t G_s^\varepsilon \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s$ is still an $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued Brownian motion, so we have \begin} \def\beq{\beg} \def\F{\scr Fin{align*} dU_t^\varepsilon&=H(U_t^\varepsilon)\circ \left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D W_t^\varepsilon+G_s^\varepsilon\left(\varepsilon\scr Theta_t+\fracrac{\varepsilon^2}{2}{\mathcal L}ambda_t\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\right). \varepsilonnd{align*} Let \begin} \def\beq{\beg} \def\F{\scr Fin{align*} M_t^\varepsilon:=\varepsilonxp\left(-\int_0^t \left\langle \varepsilon\scr Theta_s+\fracrac{\varepsilon^2}{2}{\mathcal L}ambda_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle- \int_0^t \left(\fracrac{\varepsilon^2}{2}\left|\scr Theta_s+\fracrac{\varepsilon}{2}{\mathcal L}ambda_s\right|^2\right) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \right). \varepsilonnd{align*} Then by the Girsanov theorem, the distribution of $\{U_{s}^\varepsilon; s\in [0,t]\}$ under $d\mathbb{Q}^\varepsilon:=M_t^\varepsilon d\mathbb{P}$ is the same as that of $\{U_s^{0,\varepsilon}; s\in [0,t]\}$, where $U_\cdot^{0,\varepsilon}$ is the solution to equation \varepsilonqref{sde1} with initial value $U_0^{0,\varepsilon}=U_0^\varepsilon$. Therefore we obtain \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-1a} P_t f(\xi(\varepsilon))=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[f(X_t^{\xi(\varepsilon)})\right]=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[f(X_t^{\varepsilon,\xi(\varepsilon)})M_t^\varepsilon\right], \varepsilonnd{equation} where $X_t^{\varepsilon,\xi(\varepsilon)}=\pi(U_t^\varepsilon)$, $X_t^{\xi(\varepsilon)}=\pi(U_t^{0,\varepsilon})$. We first assume $f\in C_b^2(M)$, differentiating \varepsilonqref{p3-1-1a} with respect to $\varepsilon$ yields that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-2b} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \big\langle\nabla P_t f (x), v\big\rangle_{T_x M}&=\fracrac{\partial}{\partial \varepsilon}\bigg|_{\varepsilon=0} P_t f\big(\xi(\varepsilon)\big)\\ &=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[\fracrac{\partial }{\partial \varepsilon}\bigg|_{\varepsilon=0}f\big(X_t^{\varepsilon,\xi(\varepsilon)}\big)\right]+\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[f(X_t)\left(\fracrac{\partial }{\partial \varepsilon}\bigg|_{\varepsilon=0} M_t^\varepsilon\right)\right], \varepsilonnd{split} \varepsilonnd{equation} Another round of differentiation gives: \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-3} \aligned &\left\langle \nabla^2 P_t f(x), v\otimes v\right\rangle_{T_x M \otimes T_x M}=\fracrac{\partial^2}{\partial \varepsilon^2}\bigg|_{\varepsilon=0} P_t f(\xi(\varepsilon)) \\&=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[\fracrac{\partial^2 }{\partial \varepsilon^2}\bigg|_{\varepsilon=0}f\big(X_t^{\varepsilon,\xi(\varepsilon)}\big)\right]+2 \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[\left(\fracrac{\partial }{\partial \varepsilon}\bigg|_{\varepsilon=0}f\big(X_t^{\varepsilon,\xi(\varepsilon)}\big)\right) \left(\fracrac{\partial }{\partial \varepsilon}\bigg|_{\varepsilon=0} M_t^\varepsilon\right)\right]\\&+\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[f\big(X_t^x\big)\fracrac{\partial^2 }{\partial \varepsilon^2}\bigg|_{\varepsilon=0}M_t^\varepsilon\right]. \varepsilonndaligned \varepsilonnd{equation} According to the last line of \varepsilonqref{l3-1-2}, \varepsilonqref{l3-3-1}, the definition of $M_t^\varepsilon$ and the fact that $h(t)\varepsilonquiv 0$ we derive $$\fracrac{\partial }{\partial \varepsilon}\bigg|_{\varepsilon=0}f\Big(X_t^{\varepsilon,\xi(\varepsilon)}\Big)=\langle \nabla f(X_t^x),U_th(t)\rangle_{T_{X_t^x}M}=0$$ and also, $$ \fracrac{\partial }{\partial \varepsilon}\bigg|_{\varepsilon=0} M_t^\varepsilon=-\int_0^t\langle \scr Theta_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle.$$ Furthermore, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} &\fracrac{\partial^2 }{\partial \varepsilon^2}\bigg|_{\varepsilon=0}f\Big(X_t^{\varepsilon,\xi(\varepsilon)}\Big)=\left \langle \nabla^2 f\big(X_t^x\big), \fracrac{\partial X_t^{\varepsilon,\xi(\varepsilon)}}{\partial \varepsilon}\bigg|_{\varepsilon=0}\bigotimes\fracrac{\partial X_t^{\varepsilon,\xi(\varepsilon)}}{\partial \varepsilon}\bigg|_{\varepsilon=0}\right\rangle_{T_{X_t^x}M\otimes T_{X_t^x}M}\\&\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad +\left\langle \nabla f\big(X_t^x\big), \fracrac{D}{\partial \varepsilon}\bigg|_{\varepsilon=0}\left(\fracrac{\partial X_t^{\varepsilon,\xi(\varepsilon)}}{\partial \varepsilon}\right)\right\rangle_{T_{X_t^x}M}\\ &\quad\quad=\left\langle\nabla^2 f(X_t^x), U_t h(t)\otimes U_t h(t)\right\rangle_{T_{X_t^x}M\otimes T_{X_t^x}M}+\left\langle \nabla f\big(X_t^x\big),U_t\Gamma_t h(t)\right\rangle_{T_{X_t^x}M}=0,\\ &\fracrac{\partial^2 }{\partial \varepsilon^2}\Big|_{\varepsilon=0} M_t^\varepsilon=\left(\int_0^t\langle \scr Theta_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle\right)^2- \int_0^t \langle {\mathcal L}ambda_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle-\int_0^t \left|\scr Theta_s\right|^2 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s. \varepsilonnd{align*} Crucially this special choice of variation ensures that $\fracrac{\partial^2 }{\partial \varepsilon^2}\bigg|_{\varepsilon=0}f\Big(X_t^{\varepsilon,\xi(\varepsilon)}\Big)$ depends only on $h(t)$, not on the history of the process $h$. Putting these back to \varepsilonqref{p3-1-2b} and \varepsilonqref{p3-1-3} yields \varepsilonqref{p3-1-0}, \varepsilonqref{p3-1-1} for $f\in C_b^2(M)$. By standard approximation procedure and the compact property of $M$ we see that these equalities still hold for any $f\in C_b(M)$. \varepsilonnd{proof} \text{\rm sect}ion{Quantitative Cut-off Processes}\label{s2-1} \label{cut-off} \varepsilonmph{From now on, we assume that $M$ is an $n$-dimensional general complete Riemannian manifold, not necessarily compact.} In this section we introduce a class of cut-off processes satisfying estimates crucial for the localisation procedures, which we shall apply later to (\ref{p3-1-1}) and to obtain the asymptotic gradient estimates for the logarithmic heat kernel. Since geodesic balls have typically non-regular boundary, we firstly construct a family of relatively compact open sets $\{D_m\}_{m=1}^\infty$ with smooth boundary which plays the roles of geodesic balls and such that $\cup_{m=1}^\infty D_m=M$. Our localisation procedure crucially relies on $ D_m$ has smooth boundaries, see Lemma \ref{lem7.1}. We first use a result in Greene and Wu \cite{GW} on the existence of a smooth approximate distance function, which is valid for complete manifold, and then construct a family of cut off vector fields adapted to $\{D_m\}_{m=1}^\infty$. Fixing an $o\in M$, denote by $d$ the Riemannian distance function on $M$ from $o$. Since $M$ is complete, according to \cite{GW} there exists a non-negative smooth function $\mathfrak hat d:M\rightarrow\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt_+$ with the property that $0<|\nabla \mathfrak hat d|\le 1$ and $$\left|\mathfrak hat d(x)-\fracrac{1}{2}d(x)\right|<1,\quad \fracorall\; x\in M.$$ For every non-negative $m$, define $D_m:=\mathfrak hat d^{-1}((-\infty,m)):=\{z \in M; \mathfrak hat d(z)<m\}$, then it is easy to verify $B_o(2m-2)\subset D_m\subset B_o(2m+2)$, where $B_o(r):=\{z \in M; d(z)<r\}$ is the geodesic ball centred at $o$ with radius $r$. Let $\phi:\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt\rightarrow [0,1]$ be a smooth function such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq2.1} \phi(r)=\left\{ \begin} \def\beq{\beg} \def\F{\scr Fin{array}{ll} 1, \qquad &r\leq1\\ \in (0,1), &r\in (1,2)\\ 0, &r\geq2.\varepsilonnd{array}\right. \varepsilonnd{equation} Setting \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq2.2} f_m(z):=\phi\Big(\mathfrak hat d(z)-m+2\Big),\quad z\in M, \varepsilonnd{equation} then it is easy to see that \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} f_m(z)=\left\{ \begin} \def\beq{\beg} \def\F{\scr Fin{array}{ll} 1, \qquad &\text{if}~~z \in \overline D_{m-1}\\ 0, \qquad &\text{if}~~z \in D_m^c\\ \in (0,1), \qquad &\text{orthewise}\varepsilonnd{array}\right. \varepsilonnd{equation*} and $D_m=\{z \in M; f_m(z)>0\}$. Without loss of generality we can assume that $D_m$ is a bounded connected open set (otherwise we could take the connected component of $D_m$ containing $B_o(2m-2)$). Moreover, since $\partial D_m=\{z\in M; \mathfrak hat d (z)=m\}$ and $|\nabla \mathfrak hat d(z)|\neq 0$ for all $z\in M$, we know $\partial D_m$ is a smooth $n-1$ dimensional submanifold of $M$. As before we suppose that $\{U_t\}_{0\le t<\zeta(x)}$ is the solution to the canonical horizontal equation \varepsilonqref{sde1} with $\zeta(x)$ denoting its explosion time, and $\{X_t^x:=\pi(U_t)\}_{0\le t<\zeta(x)}$ is a Brownian motion on $M$ with initial value $x:=\pi(U_0)$. Let $\partial$ denote the cemetery state for $M$ and set $\bar M=M\cup \{\partial\}$. Given a $x\in M$ we let $$P_x(\bar M):=\{\gamma\in C([0,1];\bar M): \; \gamma(0)=x\}$$ denote the collection of all $\bar M$-valued continuous paths with initial vale $x$. Let $\mu_x$ denote the Brownian motion measure on $P_x(\bar M)$. We also refer the natural filtration of the canonical process $\gamma(\cdot)$ as the canonical filtration on $P_x(\bar M)$, which is augmented to be complete and right continuous as usual. It is well known that the distribution of $\{X_{t}^x\}_{0\le t<\zeta(x)}$ and $\{U_t\}_{0\le t<\zeta(x)}$ under $\mathbb{P}$ is the same as that of the canonical process $\{\gamma(t)\}_{0\le t<\zeta(\gamma)}$ and its horizontal lift $\{U_t(\gamma)\}_{0\le t<\zeta(\gamma)}$ under $\mu_x$, where $\zeta(\gamma)$ denotes the explosion time of $\gamma(\cdot)$. Set $$\tau_m(\gamma)=\tau_{D_m}(\gamma):=\inf\left\{s \ge 0:\ \gamma(s) \notin D_m\right\}.$$ \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem5.1} For any $m \in \mathbb{N}$ there exists a stochastic process (vector field) $l_m: [0,1]\times P_x(\bar M) \rightarrow [0,1]$, such that \begin} \def\beq{\beg} \def\F{\scr Fin{enumerate} \item [(1)] $l_m(t,\gamma)=\left\{ \begin} \def\beq{\beg} \def\F{\scr Fin{array}{ll} 1, \qquad &t \le \tau_{m-1}(\gamma)\varpiedge 1\\ 0, &t > \tau_{m}(\gamma)\varepsilonnd{array}\right..$ \item[(2)] {\bf Absolute continuity:} $l_m(t,\cdot)$ is adapted to the canonical filtration and $l_m(\cdot,\gamma)$ is absolutely continuous for $\mu_x$-a.s. $\gamma \in P_x(\bar M)$. \item[(3)] {\bf Local uniform moment estimates:} For every positive integer $k \in {\mathbb N} $, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq2.4} \sup_{x \in D_{m-1}}\int_{P_x(\bar M)} \int^1_0|l_m'(s, \gamma)|^k \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\; \mu_x(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\gamma)\le C_1(m,k) \varepsilonnd{equation} for some positive constant $C_1(m,k)$ (which may depends on $m$ and $k$). \varepsilonnd{enumerate} \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} In the proof, the constant $C$ (which may depend on $m$) will change in different lines. The main idea of the proof is inspired by the article of Thalmaier \cite{T} and Thalmaier and Wang \cite{TW}. $(1)$ Since for any $m\geq1$, $D_m\subset D_{m+1}\uparrow M$, there exists a $m_0\in \mathbb{N}$ such that $$~~~~~~~~\left\{ \begin} \def\beq{\beg} \def\F{\scr Fin{array}{ll} x\in D_m, \qquad &\text{when}~m\geq m_0\\ x\notin D_m, &\text{when}~1\leq m<m_0\varepsilonnd{array}\right..$$ When $x\notin D_m$, let $l_m(t,\gamma)\varepsilonquiv0$. In the following, we will consider the case of $x\in D_m$ (which implies that $\tau_m(\gamma)>0$) without loss of generality. Let $f_m:M \rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt_+$ be the function given by \varepsilonqref{eq2.2}, we define a sequence of functions: $$T_m(t,\gamma):=\left\{ \begin} \def\beq{\beg} \def\F{\scr Fin{aligned}\int^t_0\frac {\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s}{ \left[ f_m\left(\gamma(s)\right)\right]^2},\quad\quad\quad\quad t< \tau_m(\gamma)\\ \infty, \quad\quad\quad\quad \quad\quad\quad t\ge\tau_m(\gamma).\varepsilonnd{aligned}\right. $$ Then each $T_m(\cdot, \gamma)$ is an increasing right continuous function of $t$. For any $t\ge 0$, set $$A_m(t,\gamma):=\inf\left\{s \ge 0:\ T_m(s,\gamma)\ge t\right\}.$$ We may omit the parameter $\gamma$ in the notation of $T_m(t,\gamma)$, $A_m(t,\gamma)$ for simplicity in the proof. Since $\inf_{s\in [0,t]}f_m(\gamma(t))>0$ for $t<\tau_m(\gamma)$, then $T_m(t)<\infty$ for every $t<\tau_m$ and $T_m(\cdot)$ is strictly increasing and continuous in $[0,\tau_m)$ (with respect to the variable $t$). Therefore $A_m(\cdot)$ is continuous on $[0,T_m(\tau_m))$ and $T_m(A_m(t))=t$ for every $0\le \tau_m<T_m(\tau_m)$. Furthermore we have $T_m(\tau_m)=\infty$. To see this we only need to observe that \begin} \def\beq{\beg} \def\F{\scr Fin{align*} f_m(\gamma(s))&=f_m(\gamma(s))-f_m\left(\gamma(\tau_m)\right) \le \fracrac{1}{2}\sup_{x\in D_m}|\nabla^2 f_m(x)|d\left(\gamma(s),\gamma\left(\tau_m\right)\right)^2\\ &\le C_m(\gamma)\sqrt{|s-\tau_m|},\qquad \fracorall\ s<\tau_m, \varepsilonnd{align*} where $C_m(\gamma)$ is a constant, and we applied the property that $d\left(\gamma(s),\gamma\left(\tau_m\right)\right)\le C_m(\gamma)|s-\tau_m|^{1/4}$ which is easy to prove by the Kolmogorov criterion. Combing the fact $T_m(\tau_m)=\infty$ with $T_m(t)<\infty$ for all $0\le t<\tau_m$ immediately yields that $A_m(T_m(t))=t$ for every $0\le t\le \tau_m$ and $\tau_m>A_m(t)$ for every $0\le t<\infty$. Next, we use the truncation function $\phi:\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt \rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt$ in \varepsilonqref{eq2.1} to define \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq2.5} l_m(t,\gamma)=\phi\left(\int_0^{t} \frac {\phi\big(T_m(s)-2\big)}{f_m^{2}\left(\gamma(s)\right)}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right), \varepsilonnd{equation} which is clearly adapted to the canonical filtration. Suppose that $t\geq \tau_m>A_m(3)$, then \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \int_0^t \frac {\phi\big(T_m(s)-2\big)}{f_m^{2}\left(\gamma(s)\right)}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s &\geq \int_0^t {\bf 1}_{\{T_m(s)\le 3\}}f_m^{-2}\left(\gamma(s)\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\\ & =\int_0^t {\bf 1}_{\{s\le A_m(3)\}}f_m^{-2}\left(\gamma(s)\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\\ &=T_m(A_m(3))=3, \varepsilonnd{split} \varepsilonnd{equation*} which implies $l_m(t,\gamma)=0$ for $t \geq \tau_m$ by the definition of $\phi$. If $s\le \tau_{m-1}(\gamma)$ then $f_m(\gamma(s))=1$ and so $T_m(s)=s$. Consequently, $\phi\big(T_m(s)-2\big)=1$ for every $s\le \tau_{m-1}\varpiedge 1$. Hence we obtain $$l_m(t,\gamma)=\phi( t\varpiedge 1)=1,\qquad \ \fracorall\; t\le \tau_{m-1}\varpiedge 1, $$ concluding the proof of part (1). $(2)$ Still by the expression of \varepsilonqref{eq2.5} we know the conclusion of part (2) holds. $(3)$ Now it only remains to verify the estimates \varepsilonqref{eq2.4}. Firstly, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} |l_m'(t)|&=\left|\phi'\left(\int_0^t \frac {\phi\big(T_m(s)-2\big)} {f_m^{2}\left(\gamma(s)\right)}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right)\right| \frac {\phi\big(T_m(t)-2\big)} {f_m^{2}\left(\gamma(t)\right)}\\ &\leq \|\phi'\|_\infty f_m^{-2}\left(\gamma(t)\right){\bf 1}_{\{\phi(T_m(t)-2) \neq 0\}}\leq C f_m^{-2}\left(\gamma(t)\right){\bf 1}_{\{T_m(t)\leq 4\}}. \varepsilonnd{split} \varepsilonnd{equation*} Then for every $k \in {\mathbb N} $, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-2a} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \int_0^1|l_m'(s)|^k\,\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s&\leq C\int_0^1 f_m^{-2k}\left(\gamma(s)\right){\bf 1}_{\{T_m(s)\leq 4\}}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\\ &\leq C\int^{1}_0 f_m^{-2k+2}\bigl(\gamma(s) \bigl) {\bf 1}_{\{s\leq A_m(4)\}}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D T_m(s)\\ &=C\int^{4\varpiedge T_m(1)}_0 f_m^{-2k+2}\left(\gamma\left(A_m(r)\right)\right) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\\ &\le C\int^{4}_0 f_m^{-2k+2}\left(\gamma\left(A_m(r)\right)\right) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r. \varepsilonnd{split} \varepsilonnd{equation} Observe that the distribution of $X_{\cdot}^x$ under $\mathbb{P}$ is the same as that of $\gamma(\cdot)$ under $\mu_x$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq2.6} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\sup_{x \in D_{m-1}}\int_{P_x(\bar M)}\int^{4}_0 f_m^{-2k+2}\left(\gamma\left(A_m(s)\right)\right) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \mu_x(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \gamma)\\ &=\sup_{x \in D_{m-1}}\mathbb{E}\left[\int_0^{4} f_m^{-2k+2}\left(X_{A_m(s,X_{\cdot}^x)}^x\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right]. \varepsilonnd{split} \varepsilonnd{equation} Let $S_{j,m}(\gamma):=\inf\left\{t>0; f_m\left(\gamma(t)\right)\le \fracrac{1}{j}\right\}$. According to It\^o's formula we obtain for all $j,k \in \mathbb{N}$ and $x\in D_{m-1}$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p3-1-2} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E \left[f_m^{-k} \left(X_{A_m(t)\varpiedge S_{j,m}} \right)\right] &=f_m^{-k}(x) +\fracrac{1}{2} \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E \left[\int_0^{A_m(t)\varpiedge S_{j,m}} \Delta \left(f_m^{-k}\right) (X_s) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right]\\ &=1 + \fracrac{1}{2}\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[\int_0^{A_m(t)\varpiedge S_{j,m}} \left(f_m^2 \Delta \left(f_m^{-k}\right)\right) \left(X_{A_m(T_m(s))}\right) \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D T_m(s)\right], \varepsilonnd{split} \varepsilonnd{equation} we have applied the fact that ${A_m\left(T_m(s)\right)=s}$ for every $0\le s<S_{j,m}$ and $f_m(x)=1$ for all $x\in D_{m-1}$. Meanwhile we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} f_m^2\Delta(f_m^{-k})&=k(k+1)f_m^{-k}|\nabla f_m|^2-k f_m^{-k+1}\Delta f_m\\ &=k(k+1)f_m^{-k}\left|\phi'\left(\mathfrak hat d-m+2\right)\right|^2 \left|\nabla \mathfrak hat d\right|^2\\ &-k f_m^{-k}\left(f_m\phi''\left(\mathfrak hat d-m+2\right)\left|\nabla \mathfrak hat d\right|^2+\phi' \left(\mathfrak hat d-m+2\right)f_m\Delta \mathfrak hat d\right)\\ &\leq k(k+1)f_m^{-k}\left(\|\phi'\|_{\infty}+\|\phi''\|_{\infty}+ \|\phi'\|_{\infty}\sup_{z \in D_m}|\Delta \mathfrak hat d(z)|\right)\\ &\le Cf_m^{-k}. \varepsilonnd{split} \varepsilonnd{equation*} Putting this into \varepsilonqref{p3-1-2} we arrive at \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E \left[f_m^{-k} \left(X_{A_m(t)\varpiedge S_{j,m}} \right)\right]& \le 1 +C\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[\int_0^{A_m(t)\varpiedge S_{j,m}}f_m^{-k}\bigl(X_{A_m(T_m(s))}\bigr)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D T_m(s)\right]\\ &\le 1+C\int_0^t \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E\left[f_m^{-k}\bigl(X_{A_m(r)\varpiedge S_{j,m}}\bigr)\right]\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r, \varepsilonnd{align*} where the last step follows from the procedure of change of variable $u=T_m(s)$ and the fact $A_m(t)\le t$. Hence by Grownwall's inequality we arrive at for all $k,j\in \mathbb{N}$, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E \left[f_m^{-k} \bigl(X_{A_m(t)\varpiedge S_{j,m}} \bigr)\right]\le Ce^{Ct}. \varepsilonnd{align*} Then letting $j \rightarrow \infty$ and observing that $A_m(t)\le \tau_m=\lim_{j \rightarrow \infty}S_{j,m}$ we obtain for all $k\in \mathbb{N}$, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E \left[f_m^{-k} \left(X_{A_m(t)} \right)\right]\le Ce^{Ct}, \varepsilonnd{align*} combing this with \varepsilonqref{p3-1-2a} yields \varepsilonqref{eq2.4}. This completes the proof for Lemma \ref{lem5.1}. \varepsilonnd{proof} \text{\rm sect}ion{Proof of the Main Estimates}\label{section5} In this section, we shall apply the cut-off procedures, using the quantitative localised vector fields introduced in Section \ref{cut-off}, to obtain short time as well as asymptotic first and second order gradient estimates for the logarithmic heat kernel of a complete Riemannian manifold without imposing on it any curvature bounds. Let $\{D_m\}_{m=1}^{\infty}$ and $\{f_m\}_{m=1}^{\infty}$ be the sequences of domains and functions constructed in Section \ref{cut-off}. Recall that for every $m$, $D_m=\{x \in M: f_m(x)>0\}$ is a bounded connected open set. By Lemma \ref{lem7.1} from the Appendix there exists a compact Riemannian manifold $\tilde M_m$ such that $D_m$ is isometrically embedded into $\tilde M_m$ as an open set. We could and will view $D_m \subset \tilde M_m$ as an open subset of $\tilde M_m$. In particular, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.5} d_{\tilde M_m}(x,y)=d(x,y),\quad \quad \fracorall\ x,y \in B_{o}(2m-2), \varepsilonnd{equation} where $d$ and $d_{\tilde M_m}$ are the Riemannian distance function on $M$ and $\tilde M_m$. We denote the heat kernel on $M$ and $\tilde M_m$ by $p(t,x,y)$ and $p_{\tilde M_m}(t,x,y)$ respectively. For every $e\in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ we also let $H_{e}^m$ denote the horizontal lift of $ue$ on $TO(\tilde M_m)$. Let us fix a probability space $(\Omega, \mathscr{F},\mathbb{P})$. Let $\{B_t\}_{t\ge 0}$ be the standard $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$-valued Brownian motion with $B_t=(B_t^1, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Dots, B_t^n)$, and we denote by $\mathscr{F}_t$ the filtration generated by it. Now we fix an orthonormal basis $\{e_i\}_{i=1}^n$ of $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$. For $x\in D_{m}\subset \tilde M_m$ and $U_0$ a frame at $x$ so that $U_0 \in O_xM=O_x\tilde M_m$, let $U_t^m$ denote the solution to the following $O(\tilde M_m)$-valued stochastic differential equation \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e4-1} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D U_t^m=\sum_{i=1}^n H_{e_i}^m\left(U_t^m\right)\circ \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t^i,\quad \ U_0^m=U_0. \varepsilonnd{equation} Set $X_t^{m,x}:=\pi(U_t^m)$. This is a $\tilde M_m$-valued Brownian motion. Recall that $X_t^x:=\pi(U_t)$, where $U_t$ is the solution to \varepsilonqref{sde1}, with the same driving Brownian motion $B_t$ and the same initial value $U_0$ as in \varepsilonqref{e4-1}. Throughout this section, for every $m,k\in \mathbb{N}$ with $k\ge m$, we define $$\tau_m:=\inf\{t>0; X_t^x\notin D_m\},\quad \quad \tau_{m}^k:=\inf\{t>0; X_t^{k,x}\notin D_m\}.$$ Note that for every $k>m$, $H_{e_i}^k=H_{e_i}$ on $\pi^{-1}(D_m)$. It is easy to verify that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e4-2} \begin} \def\beq{\beg} \def\F{\scr Fin{aligned}\tau_m=& \tau_{m}^k, \quad X_t^x=& X_t^{k,x}, \qquad \fracorall\ k\ge m>1,\ 0\le t\le \tau_m. \varepsilonnd{aligned} \varepsilonnd{equation} As before, the superscript $x$ may be omitted from time to time when there is no risk of confusion. \varepsilonmph{The probability and the expectation for the functional generated by $X_{\cdot}^x$ or $X_{\cdot}^{m,x}$ (with respect to $\mathbb{P}$) are denoted by $\mathbb{P}_x$ and $\mathbb{E}_x$ respectively in this section.} \varepsilonmph{If $M$ is compact, then when $m$ is large enough we have $D_m=M$ and we can take $\tilde M_m=M$ (we do not have to apply Lemma \ref{lem7.1} when $M$ is compact), then all the conclusions in this section automatically hold. Hence in this section we always assume that $M$ is non-compact.} We shall use the following estimates which are crucial for our proof. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}[\cite{Azencott, Molchanov, Va1,Va2}]\label{lem6.1} For any $x, y \in M$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.1} \lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}t\log p(t,x,y)=-\fracrac{d(x,y)^2}{2}. \varepsilonnd{equation} and the convergence is uniformly in $(x,y)$ on $ K\times K$ for any compact subset $K$. Moreover, for every connected bounded open set $D\supseteq K$ with smooth boundary, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.2} \lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}t\log \mathbb{P}_x\left(\tau_D<t\right)=-\fracrac{d(x,\partial D)^2}{2},\quad \ \fracorall\ x\in K. \varepsilonnd{equation} Here $\tau_D:=\inf\{t>0; X_t \notin D\}$ is the first exit time from $D$ and $d(x,\partial D):=\inf_{z \in \partial D}d(x,z)$. And the convergence is also uniform in $x$ on $K$. \varepsilonnd{lem} The asymptotic estimates \varepsilonqref{eq3.1} and \varepsilonqref{eq3.2} were firstly shown to hold for $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ in Varadhan \cite{Va1,Va2}, extension to a complete Riemannian manifold was given in Molchanov \cite{Molchanov}. In addition, Azencott \cite{Azencott-as} and \cite{Hsu90} indicated that these statements may fail for an incomplete Riemannian manifold. We shall also use the following statement, which follows readily from the small time asymptotics and the Gaussian heat kernel upper bounds. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}( \cite{Azencott-as}, \cite[Lemma 2.2]{Hsu90})\label{lem6.2} For any compact subset $K$ of $M$ and any positive number~$r$, Then there exists a positive number $t_0$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.3} \sup_{t\in (0,t_0]} \sup_{ d(z,y) \ge r, \;y \in K}p(t,z,y)\le 1. \varepsilonnd{equation} \varepsilonnd{lem} \subsection{Comparison theorem for functional integrals involving approximate heat kernels} Let $D_m$ dentoe the relatively compact subset and let $l_m:[0,1]\times P_x(M)\rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt$ be the cut-off processes adapted to $D_m$, as constructed by Lemma \ref{lem5.1}. Let $p^{D_m}(t,x,y)$ denote the Dirichlet heat kernel on $D_m$. Let $K$ be a compact set and $x,y\in K$ be such that $d(x,y)<d(x, \partial D^m)\vee d(y, \partial D^m)$, then $p(t,x,y)$ and $p^{D_m}(t,x,y)$ are asymptotically the same for small $t$. See \cite{Azencott-as}, Lemma 2.3 on page 156. Below we give a quantitative estimate on $p$ and $p^{D_m}$ on a compact set $K\times K$, for sufficiently large $m$. By sufficiently large, we mean that $m\ge m_0$ for a natural number $m_0$ and $m_0$ may depend on other data. In all the results below, it depends on the compact set $K$ and the prescribed exponential factor $L>0$. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem6.3} Suppose that $K$ is a compact subset of $M$ and $L>1$ is a positive number. Then for sufficiently large $m$, there exists a positive number $t_0=t_0(K,L,m)$ such that for every $t\in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.6} \begin} \def\beq{\beg} \def\F{\scr Fin{aligned} \sup_{x,y\in K}\left|p(t,x,y)-p^{D_{m}}(t,x,y)\right| &\le \, e^{-\fracrac{2L}{t}}, \\\sup_{x,y\in K} \left|p_{\tilde M_{m}}(t,x,y)-p^{D_{m}}(t,x,y)\right| &\le e^{-\fracrac{2L}{t}}. \varepsilonnd{aligned} \varepsilonnd{equation} In particular, for every $t\in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.7a} \sup_{x,y\in K}\left|p(t,x,y)-p_{\tilde M_{m}}(t,x,y)\right| \leq e^{-\fracrac{L}{t}}. \varepsilonnd{equation} \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} The estimates in \varepsilonqref{eq3.6} could be found in Azencott \cite[section 4.2]{Azencott-as}, \and also in Bismut \cite[Section III.a]{Bis} and Hsu \cite[The proof of Theorem 5.1.1]{Hsu2}. Here we include a proof for the convenience of the reader. The technique and the intermediate estimates will be used later. By the strong Markovian property, $$P_tf(x)=\mathbb{E}_x \left[f\left(X_t\right){\bf 1}_{\{t\le \tau_{m}\}}\right]+\mathbb{E}_x \left[\mathbb{E}_{X_{\tau_m}} \left[f\left(X_{t-\tau_m}\right)\right]{\bf 1}_{\{\tau_{m}<t<\zeta\}}\right]$$ and so for any $x,y \in K$ and $ t>0$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.8} \begin} \def\beq{\beg} \def\F{\scr Fin{split} p(t,x,y)=p^{D_m}(t,x,y)+\mathbb{E}_x\left[ p\left(t-\tau_{m},X_{\tau_{m}},y\right) {\bf 1}_{\{\tau_{m}<t<\zeta\}}\right]. \varepsilonnd{split} \varepsilonnd{equation} Since $M$ is non-compact, given any number $L>1$, there exists a natural number $m_0$ such that $$K\subset B_o(2m_0-2), \quad d(K,\partial D_{m_0})\ge d(K,\partial B_o(2m_0-2))>4L.$$ Then, according to \varepsilonqref{eq3.2} and \varepsilonqref{eq3.3}, for every $m\ge m_0$, we could find a positive number $t_0(K,L,m)$ such that for any $t\in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \mathbb{P}_x\left(t>\tau_{m}\right)\le \varepsilonxp\left(-\fracrac{d(x,\partial D_{m})^2-1}{2t}\right)\le e^{-\fracrac{2L}{t}}, \qquad \fracorall x\in K,\\ & p(t,z,y)\le 1, \quad \mathfrak hbox{ for all } \; z\in \partial D_m, \mathfrak hbox{ and } \; y\in K.\varepsilonnd{split} \varepsilonnd{equation*} By these estimates we obtain that, for all $m\ge m_0$ and all $t \in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \mathbb{E}_x\left[p\left(t-\tau_{m},X_{\tau_m},y\right) {\bf 1}_{\{t> \tau_{D_{m}}\}}\right] &\le \sup_{t \in (0,t_0)}\sup_{ z\in \partial D_{m}, y \in K}p(t,z,y)\cdot \mathbb{P}_x\left(t>\tau_{m}\right)\\ &\le e^{-\fracrac{2L}{t}}. \varepsilonnd{split} \varepsilonnd{equation*} Putting this into \varepsilonqref{eq3.8} we arrive at that for all $m\ge m_0$, all $x,y\in K$, and for all $ t \in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.9} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & |p(t,x,y)-p^{D_{m}}(t,x,y)|\le e^{-\fracrac{2L}{t}}. \varepsilonnd{split} \varepsilonnd{equation} Note that for every $m\ge m_0$, $D_{m}\subset \tilde M_{m}$ and $x\in K$, $$d_{\tilde M_{m}}(x, \partial D_{m})\ge d_{\tilde M_{m}}(x, \partial B_o(2m-2)) =d(x, \partial B_o(2m-2))$$ which is due to \varepsilonqref{eq3.5}. By the same argument for \varepsilonqref{eq3.9} and changing the constant $t_0$ if necessary we could find a $t_0(K,L,m)$ such that for all $m\ge m_0$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \left|p_{\tilde M_{m}}(t,x,y)-p^{D_{m}}(t,x,y)\right|\le e^{-\fracrac{2L}{t}},\quad \ x,y\in K,\ t \in (0,t_0]. \varepsilonnd{split} \varepsilonnd{equation*} This, together with \varepsilonqref{eq3.9}, yields \varepsilonqref{eq3.6} and \varepsilonqref{eq3.7a}. \varepsilonnd{proof} \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem6.4} Suppose that $K$ is a compact subset of $M$ and $L>1$ is a positive number. \begin} \def\beq{\beg} \def\F{\scr Fin{itemize} \item[$(1)$] For $m_0$ sufficiently large and any $m>m_0$, there exists a $t_0(K,L,m)$ such that for every $0<s\le \fracrac{t}{2}$ and $0<t\le t_0$, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l4-1-7} \sup_{x, y\in K}\sup_{z\in D_{m_0}} \left| \frac {p(t-s, x,z)}{p(t,x,y)}- \frac {p_{\tilde M_m}(t-s, x,z)}{p_{\tilde M_m}(t,x,y)} \right| \le 2e^{-\fracrac{4L}{t}}. \varepsilonnd{equation} \item[$(2)$] Suppose $\Upsilon_t$ is an $\mathscr{F}_t$ adapted process, and for any $q>0$ and $m\geq1$ we set $$F^q_m(\Upsilon, X_\cdot)=\Bigl( \int_0^s \Upsilon_r l_m'\Bigl(r,X_{\cdot}\Bigr)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\Bigr)^q,\quad F_m^q(\Upsilon, X_\cdot^m)= \Bigl(\int_0^s \Upsilon_r l_m'\Bigl(r,X_{\cdot}^m\Bigr)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\Bigr)^q.$$ We also assume that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l4-1-0} \sup_{x\in K}\mathbb{E}_x\left[\int_0^{1\varpiedge \tau_m} |\Upsilon_s|^{2q}\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right]<\infty, \quad \fracorall m\geq1 \varepsilonnd{equation} for some $q\in \mathbb{N}$. Then for every sufficiently large $m$ (any $m$ greater than some number $m_0(K,L)$), we can find a positive number $t_0(K,L,m)$ with the property that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l4-1-1} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\sup_{x,y\in K}\Bigg |\mathbb{E}_x \Bigl[ F_m^q(\Upsilon, X_\cdot) \; \frac {p(t-s, X_s,y)}{p(t,x,y)}\; \Bigr] -\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\Bigl[F_m^q(\Upsilon, X_\cdot^m)\; \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)} \Bigr]\Bigg|\\ & \le C(m)\;e^{-\fracrac{L}{t}} \varepsilonnd{split} \varepsilonnd{equation} for any $0<t\le t_0$, $0<s\le \fracrac{t}{2}$. Here the positive constant $C(m)$ may depend on $m$ and on $\alpha_m:=\sup_{x\in K}\mathbb{E}_x\left[\int_0^{1}\left|\Upsilon_r l_m'\left(r,X_{\cdot}\right)\right|^{q} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\right]$. (Note that $l_m'\left(r,X_{\cdot}\right)\neq 0$ only for $r<\tau_m=\tau_{m}^m$ so the quantity is well defined.) \varepsilonnd{itemize} \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} In the proof, the constant $C$ may represent different constants in different lines. Let $r_0:=\sup_{x,y\in K}d_M(x,y)$ denote the diameter of $K$. Since $M$ is non-compact, we can choose a natural number $\tilde m_0$ (which may depend on $K$ and $L$) such that $$K\subset B_o(2\tilde m_0-2)\subset D_{\tilde m_0}$$ and for all $m>\tilde m_0$, $$d\left(K, \partial B_o(2\tilde m_0-2)\right)=d_{\tilde M_m} \left(K, \partial B_o(2\tilde m_0-2)\right)> 4(L+ r_0+1).$$ Also, by the heat kernel comparison \varepsilonqref{eq3.6} and \varepsilonqref{eq3.7a}, we can find a $m_0>\tilde m_0$ so that for all $m>m_0$, there exists a constant $t_2(K,L,m)>0$ such that, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.12} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \left|p(t,z,y)-p_{\tilde M_{m}}(t,z,y)\right| \le e^{-\fracrac{4(L+r_0+1)^2}{t}},\quad\quad \ \fracorall\ t\in (0,t_2],\ z,y\in D_{m_0}. \varepsilonnd{split} \varepsilonnd{equation} According to the asymptotic relations \varepsilonqref{eq3.1} and \varepsilonqref{eq3.2}, for every $m>m_0$( taking $m_0$ larger as is necessary) we could find a constant $0<t_1(K,L,m)\le t_2$ such that for all $t\in (0,t_1]$, \begin} \def\beq{\beg} \def\F{\scr Fin{eqnarray} \label{l4-1-3} &&p(t,z,y)\le e^{\fracrac{1}{t}}, \qquad \ \qquad p_{\tilde M_m}(t,z,y)\le e^{\fracrac{1}{t}},\quad \qquad \quad \fracorall z,y\in D_{m_0}, \\ \label{l4-1-4} &&p(t,z,y)\ge e^{-\fracrac{r_0^2+1}{t}},\qquad \; p_{\tilde M_m}(t,z,y)\ge e^{-\fracrac{r_0^2+1}{t}},\; ~\qquad \fracorall z,y\in K, \\ \label{l4-1-5} &&\mathbb{P}_z\left(\tau_{m_0}<t\right)\le e^{-\fracrac{4(L+r_0+1)^2}{t}}, \qquad\qquad\qquad\qquad \qquad ~~~\fracorall\,\ \ z\in K. \varepsilonnd{eqnarray} By the small time locally uniform heat kernel bound \varepsilonqref{eq3.3}, for every $m>m_0$ there exists a number $0<t_0(K,L,m)\le t_1$ such that for all $ t\in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l4-1-6} p(t,z_1,y)\vee\ p_{\tilde M_m}(t,z_2,y)\le 1, ~~~~~~\quad \ \fracorall z_1\in M\cap D_{m_0}^c,\; z_2\in \tilde M_m\cap D_{m_0}^c, \; y\in K. \varepsilonnd{equation} Therefore for every $m>m_0$ and for every $0<s\le \fracrac{t}{2}$, every $0<t\le t_0$, and for all $x,y\in K$ and $ z\in D_{m_0}$, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\left| \frac {p(t-s, x,z)}{p(t,x,y)}- \frac {p_{\tilde M_m}(t-s, x,z)}{p_{\tilde M_m}(t,x,y)} \right| \\ &\le \fracrac{ p(t-s,x,z)\left|p_{\tilde M_m}(t,x,y) -p(t,x,y)\right|+ p(t,x,y) \left|p_{\tilde M_m}(t-s,x,z) -p(t-s,x,z) \right| }{p(t,x,y)p_{\tilde M_m}(t,x,y)}\\ &\le 2e^{\fracrac{2(1+r_0^2)}{t}} e^{\fracrac{2}{t}} e^{-\fracrac{4(L+r_0+1)^2}{t}} \le 2e^{-\fracrac{4L}{t}}. \varepsilonnd{split} \varepsilonnd{equation} Here the second step above is due to \varepsilonqref{eq3.12}--\varepsilonqref{l4-1-4}. Thus, we finish the proof of $(1)$. For all $m>m_0$, let us split the terms as follows, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} &\mathbb{E}_x \left[ F^q_m(\Upsilon, X_\cdot) \frac {p(t-s, X_s,y)}{p(t,x,y)} \right]\\ =&\mathbb{E}_x \left[ F^q_m(\Upsilon, X_\cdot) \frac {p(t-s, X_s,y)}{p(t,x,y)}{\bf 1}_{\{t\le \tau_{m_0}\}}\right] +\mathbb{E}_x \left[ F^q_m(\Upsilon, X_\cdot) \frac {p(t-s, X_s,y)}{p(t,x,y)}{\bf 1}_{\{ t>\tau_{m_0}\}}\right]\\ =&:I_1^m(s,t)+I_2^m(s,t). \varepsilonnd{align*} Since $l_m'\left(r,X_{\cdot}\right)\neq 0$ if only if $t<\tau_m$, then $l_m'\left(r,X_{\cdot}^m\right)=l_m'\left(r, X_{\cdot}\right)$ and we have \begin} \def\beq{\beg} \def\F{\scr Fin{align*} F_m^q(\Upsilon,X_\cdot^m)&=\left(\int_0^{s} \Upsilon_r l_m'\left(r,X^m_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\right)^q= \left(\int_0^{s\varpiedge \tau_m} \Upsilon_r l_m'\left(r,X^m_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\right)^q\\ &=\left(\int_0^{s} \Upsilon_r l_m'\left(r,X_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\right)^q=F_m^q(\Upsilon,X_\cdot). \varepsilonnd{align*} Note also, $X_s^m=X_s$ for every $s\le \fracrac{t}{2}<\tau_m$. It holds that \begin} \def\beq{\beg} \def\F{\scr Fin{align*} &\mathbb{E}_x \left[ F_m^q(\Upsilon,X_\cdot^m) \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)}\right]\\ =&\mathbb{E}_x \left[ F^q_m(\Upsilon, X_\cdot) \frac {p_{\tilde M_m}(t-s, X_s,y)}{p_{\tilde M_m}(t,x,y)}{\bf 1}_{\{t\le \tau_{m_0}\}}\right] +\mathbb{E}_x \left[ F^q_m(\Upsilon, X_\cdot) \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)}{\bf 1}_{\{t> \tau_{m_0}\}}\right]\\ =&:J_1^m(s,t)+J_2^m(s,t),\quad 0<s<\fracrac{t}{2}, \varepsilonnd{align*} Note that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{14-1-11}\alpha_m=\sup_{x\in K}\mathbb{E}_x\left[\int_0^1 \left|\Upsilon_r l_m'\left(r,X_{\cdot}\right)\right|^q \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \right]<\infty.\varepsilonnd{equation} This follows from the moment estimates on $l_m$, \varepsilonqref{eq2.4}, the assumption \varepsilonqref{l4-1-0}, and also $$ \alpha_m \le \sup_{x\in K}\mathbb{E}_x\left[\int_0^{1\varpiedge \tau_m} \left|\Upsilon_r\right|^{2q} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \right]^{1/2} \sup_{x\in K}\mathbb{E}_x\left[\int_0^{1\varpiedge \tau_m} \left|l_m'\left(r,X_{\cdot}\right)\right|^{2q} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \right]^{1/2}. $$ For all $m>m_0$, $x,y\in K$, $0<s\le \fracrac{t}{2}$, and $0<t\le t_0$, we may assume that $t_0\le 2$, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} &|I_1^m(s,t)-J_1^m(s,t)|\\ &\le \sup_{z\in D_{m_0}}\left| \frac {p(t-s, z,y)}{p(t,x,y)} - \frac {p_{\tilde M_m}(t-s, z,y)}{p_{\tilde M_m}(t,x,y)} \right| \mathbb{E}_x\left[\left|\int_0^{s} \Upsilon_r l_m'\left(r,X_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\right|^q \right]\\ &\le Ce^{-\fracrac{4L}{t}}\sup_{x\in K}\mathbb{E}_x\left[\int_0^{1\varpiedge \tau_m} \left|\Upsilon_r l_m'\left(r,X_{\cdot}\right)\right|^q \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \right]=C \alpha_m\;e^{-\fracrac{4L}{t}}. \varepsilonnd{align*} In the penultimate step, we have applied Burkholder-Davies-Gundy inequality and \varepsilonqref{l4-1-7}. According to \varepsilonqref{l4-1-3} and \varepsilonqref{l4-1-6} we also have $$\sup_{z\in M, y\in K}p(t,z,y)\le e^{\fracrac{1}{t}},\ \ \fracorall\ 0<t\le t_0.$$ Combining this with \varepsilonqref{l4-1-4}--\varepsilonqref{l4-1-5}, Cauchy-Schwartz inequality, and Burkholder-Davies-Gundy inequality, we obtain that for every $m>m_0$, $x,y\in K$, $0<s\le \fracrac{t}{2}$, and $0<t\le t_0$, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} &|I_2^m(s,t)|\\ &\le C\; e^{\fracrac{r_0^2+1}{t}}\;\sup_{r\in [\fracrac{t}{2},t],z\in M,y\in K}p(r,z,y)\; \mathbb{E}_x\left[\left|\int_0^{s} \Upsilon_r l_m'\left(r,X_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\right|^{2q} \right]^{1/2} \mathbb{P}_x\left(\tau_{m_0}<t\right)^{1/2}\\ &\le Ce^{\fracrac{r_0^2+1}{t}}e^{\fracrac{2}{t}}e^{-\fracrac{2(r_0+L+1)^2}{t}}\mathbb{E}_x\left[\int_0^{1\varpiedge \tau_m} \left|\Upsilon_r l_m'\left(r,X_{\cdot}\right)\right|^{2q} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r \right]^{1/2}\le C\alpha_m \;e^{-\fracrac{2L}{t}}. \varepsilonnd{align*} Here in the last step we used (\ref{14-1-11}). Similarly, we obtain that for every $m>m_0$, $x,y\in K$, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} |J_2^m(s,t)|\le C \alpha_m \;e^{-\fracrac{2L}{t}}, \qquad \mathfrak hbox{ for all $0<s\le \fracrac{t}{2}$ and $0<t\le t_0$.} \varepsilonnd{align*} Combing the above estimates for $I_1^m,I_2^m,J_1^m,J_2^m$ we see that, for every $m>m_0$, $x,y\in K$, $0<s\le \fracrac{t}{2}$ and $0<t\le t_0$, \begin} \def\beq{\beg} \def\F{\scr Fin{align*} &\Bigg |\mathbb{E}_x \left[ F_m^q(\Upsilon, X_\cdot) \frac {p(t-s, X_s,y)}{p(t,x,y)} \right] - \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[F_m^q(\Upsilon, X_\cdot^m) \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)} \right]\Bigg|\\ &\le |I_1^m(s,t)-J_1^m(s,t)|+|I_2^m(s,t)|+|J_2^m(s,t)|\le C \alpha_m e^{-\fracrac{L}{t}}, \varepsilonnd{align*} which is \varepsilonqref{l4-1-1} and we have finished the proof. \varepsilonnd{proof} \begin} \def\beq{\beg} \def\F{\scr Fin{remark}\label{rem6.1} By the same arguments in the proof for \varepsilonqref{l4-1-1} we could obtain the following under the conditions of Lemma \ref{lem6.4}: For sufficiently large $m$ we could find a positive number $t_0(K,L,m)$ so that for every $x,y\in K$, $0<s\le \fracrac{t}{2}$, and $0<t\le t_0$ the following estimates hold, replacing $l_m$ by $l_{m'}$, or $dB_r$ by $dr$. \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{r4-1-1} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\Bigg |\mathbb{E}_x \left[ \left(\int_0^{s} \Upsilon_r \,l_m (r,X_{\cdot})\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\right)^q \left( \frac {p(t-s, X_s,y)}{p(t,x,y)} - \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)}\right) \right] \Bigg| \le C(m)e^{-\fracrac{L}{t}}, \varepsilonnd{split} \varepsilonnd{equation} \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{r4-1-2} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\Bigg |\mathbb{E}_x \left[ \left(\int_0^{s} \Upsilon_r l_m'\left(r,X_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\right)^q \left( \frac {p(t-s, X_s,y)}{p(t,x,y)} - \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)} \right) \right]\Bigg| \le C(m)e^{-\fracrac{L}{t}}, \varepsilonnd{split} \varepsilonnd{equation} \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{r4-1-3} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \Bigg |\mathbb{E}_x \left[ \left(\int_0^{s} \Upsilon_r l_m\left(r,X_{\cdot}\right)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\right)^q \left( \frac {p(t-s, X_s,y)}{p(t,x,y)} - \frac {p_{\tilde M_m}(t-s, X_s^m,y)}{p_{\tilde M_m}(t,x,y)}\right) \right]\Bigg| \le C(m)e^{-\fracrac{L}{t}}. \varepsilonnd{split} \varepsilonnd{equation} These estimates will also be used in the proof for Proposition \ref{prp6.6}. \varepsilonnd{remark} \subsection{Proof of the main theorem: gradient estimates} \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem6.5} Let $t>0$, $x\in M$ and $v\in T_x M$. Suppose that $m$ is a natural number such that $x\in D_m$. Let $h\in L^{1,2}(\Omega;\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n)$ be given by $$h(s)=\left(\fracrac{t-2s}{t}\right)^+\times l_m(s,X_{\cdot})\times U_0^{-1}v.$$ Then for any $f\in C_b(M)$ we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l5-2-1} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \langle \nabla P_tf(x), v\rangle_{T_x M}&=-\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[ f(X_t)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^h, dB_s\> {\bf 1}_{\{t<\zeta\}}\right], \varepsilonnd{split} \varepsilonnd{equation} where $\scr Theta_s^h$ defined by \varepsilonqref{e3-2} with the $h$ chosen above. \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} When $M$ is compact, \varepsilonqref{l5-2-1} is just \varepsilonqref{p3-1-0} established in Proposition \ref{prp4.5}. For general non-compact complete $M$, we will use the arguments based on truncation and approximation. For each $k>m$, let $\{U_t^k\}_{t\ge 0}$ be the horizontal Brownian motion on compact manifold $\tilde M_k$ as defined in (\ref{e4-1}) with $\pi(U_0^k)=x\in D_m$. Set $X_t^k=\pi(U_t^k)$ and $P_t^k f(x)=\mathbb{E}_x\left[f(X_t^k)\right]$. Let \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l5-2-1a} h(s)=\left(\fracrac{t-2s}{t}\right)^+ \cdot l_m(s,X_\cdot)\cdot U_0^{-1}v. \varepsilonnd{equation} According to \varepsilonqref{e4-2}, precisely $\tau_m= \tau_{m}^k$ and $h(s)\neq 0$ if and only if when $s\le \frac t 2\varpiedge \tau_m$. So furthermore $$h(s)=\left(\fracrac{t-2s}{t}\right)^+ \cdot l_m(s,X_\cdot^k){\bf 1}_{\{s<\frac t 2 \varpiedge \tau_m\}}\cdot U_0^{-1}v,\quad \fracorall\ k>m,$$ $$h'(s)=\left(-\frac 2 tl_m(s,X_\cdot^k) + l_m'(s,X_\cdot^k)\left(\fracrac{t-2s}{t}\right)^+\right){\bf 1}_{\{s<\frac t 2 \varpiedge \tau_m\}}\cdot U_0^{-1}v,\quad \fracorall\ k>m,$$ which means that we can replace $X_\cdot$ by $X_\cdot^k$ in the expression of $h(s)$ and $h'(s)$. Let $\scr Theta_s^{h,k}$ be given by \varepsilonqref{e3-2}, with the manifold $M$ replaced by $\tilde M_k$ (associated with $X_\cdot^k$). Therefore by \varepsilonqref{e3-2} we have the following expression \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{add1} \scr Theta_s^{h,k}=h'(s)+\text{\rm ric}^{\tilde M_k}_{U_s^k}(h(s))=h'(s)+\text{\rm ric}_{U_s}(h(s))=\scr Theta_s^h,\quad \fracorall\ k>m. \varepsilonnd{equation} Here both sides of \varepsilonqref{add1} vanish for $s>\tau_m$, meanwhile we have used the fact that $U_s=U_s^k$ when $s<\tau_m$ and $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic^{\tilde M_k}_z=\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic_z$ for every $z\in D_m$ and \begin} \def\beq{\beg} \def\F{\scr Fin{align*} \text{\rm ric}^{\tilde M_k}_{U_s^k}(h(s))=\text{\rm ric}^{\tilde M_k}_{U_s^k}(h(s)){\bf 1}_{\{s<\tau_m\}}= \text{\rm ric}_{U_s}(h(s)),\quad \fracorall\ k>m. \varepsilonnd{align*} Moreover, we observe that for any compact set $K\subset D_m$ and $q>0$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.13} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \sup_{x\in K}\mathbb{E}_x\left[ \int_0^{1\varpiedge \tau_m} \left| \text{\rm ric}_{U_s}(U_0^{-1}v)\right|^q \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right] &\le |v|^q \sup_{x\in K}\mathbb{E}_x\left[\int_0^{1\varpiedge \tau_m}\left|\text{\rm ric}_{U_s}{\bf 1}_{\{s<\tau_m\}}\right|^q \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right]\\ &\le |v|^q \sup_{z\in D_m}\|{\color{blue} \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrtic_z}\|^q<\infty. \varepsilonnd{split} \varepsilonnd{equation} Combining this with \varepsilonqref{eq2.4} and the fact that $h(s)\neq 0$ only if $s\le t\varpiedge \tau_m=t\varpiedge \tau_{m}^k$ yields immediately that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{add4} \sup_{x\in D_m, v\in T_x M, |v|=1}\mathbb{E}_x\left[\int^t_0|\scr Theta_s^{h}|^2ds \right]<\infty,\quad \ \fracorall\ t>0. \varepsilonnd{equation} Thus, applying Proposition \ref{prp4.5} to $P_t^kf$ (note that $\tilde M_k$ is compact) and using \varepsilonqref{add1} we obtain that for all $v\in T_x M$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{add2} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \langle \nabla P^k_tf(x), v\rangle_{T_x M}&=-\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[ f(X^k_t)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h, k}, dB_s\> \right]=-\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[ f(X^k_t)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h}, dB_s\>\right]. \varepsilonnd{split} \varepsilonnd{equation} For any function $\psi \in C_c^{\infty}(M)$ and vector field $V \in C_c^{\infty}(M;TM)$ with supports in $ D_m$ satisfying that $|V(x)|\le 1$ for all $x\in D_m$, we can use $\nabla$ and $\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x$ for the the gradient operator and the Riemannian volume measure on both manifolds $M$ and $\tilde M_k$, so we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{add3} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\int_{M} \scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[ f(X^k_t)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\> \right]\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x \\&=\int_{M} \langle \nabla P^k_tf(x), V(x)\rangle_{T_x M}\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x\\ &=\int_{\tilde M_k} \langle \nabla P^k_tf(x), V(x)\rangle_{T_x M}\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x\\ &=-\int_{\tilde M_k} \mathbb{E}_x\left[f\left(X_t^{k}\right)\right]\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(V\psi)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x=-\int_{M} \mathbb{E}_x\left[f\left(X_t^{k}\right)\right]\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(V\psi)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x. \varepsilonnd{split} \varepsilonnd{equation} Here $h(x)$ is defined by \varepsilonqref{l5-2-1a} with $v=V(x)$. Meanwhile note that $X_t=X_t^k$ if $t<\tau_k$, for every $x\in D_m$ it holds \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\lim_{k \rightarrow \infty} \left|\mathbb{E}_x\left[f(X_t^{k})\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>\right] -\mathbb{E}_x\left[f(X_t)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>{\bf 1}_{\{t<\zeta\}}\right]\right|\\ &\le \lim_{k \rightarrow \infty} \mathbb{E}_x\left[\left|f(X_t^{k})-f(X_t){\bf 1}_{\{t<\zeta\}}\right| \left|\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>\right|\right]\\ &\le \lim_{k \rightarrow \infty}\sqrt{\mathbb{E}_x\left[\left|f(X_t^{k})-f(X_t){\bf 1}_{\{t<\zeta\}}\right|^2\right]} \sqrt{\mathbb{E}_x\left[\left|\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>\right|^2\right]}\\ &\le \lim_{k \rightarrow \infty}\sqrt{2}C\|f\|_{\infty}\sqrt{\mathbb{P}_x\left(\tau_k\le t<\zeta\right)}=0, \varepsilonnd{split} \varepsilonnd{equation*} where $$C:=\sup_{x\in D_m, v\in T_x M, |v|=1}\mathbb{E}_x\left[\int^t_0|\scr Theta_s^{h}|^2ds \right]$$ is finite for every $t>0$, which is due to \varepsilonqref{add4}. With this we may take $k\to \infty$ in \varepsilonqref{add3}, then $$\mathbb{E}_x\left[f(X_t^{k})\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>\right]\to \mathbb{E}_x\left[f(X_t)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>{\bf 1}_{\{t<\zeta\}}\right]$$ and consequently \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \int_{D_m} \mathbb{E}_x\left[f\left(X_t\right)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>{\bf 1}_{\{t<\zeta\}}\right]\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x =\int_{D_m} \mathbb{E}_x\left[f\left(X_t\right){\bf 1}_{\{t<\zeta\}}\right]\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(V\psi)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x. \varepsilonnd{split} \varepsilonnd{equation*} Since $m$ is arbitrary, so it follows that for all test vector fields $V \in C_c^{\infty}(M;TM)$ and test functions $\psi \in C_c^{\infty}(M)$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \int_{M} \mathbb{E}_x\left[f\left(X_t\right)\int_0^t \langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>{\bf 1}_{\{t<\zeta\}}\right]\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x =\int_{M} P_tf(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(V\psi)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x, \varepsilonnd{split} \varepsilonnd{equation*} which means that the weak (distributional) gradient $\nabla P_t f$ exists \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \left\langle \nabla P_t f(x), V(x)\right\rangle_{T_x M}=\mathbb{E}_x\left[f\left(X_t\right) \int^t_0\langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>{\bf 1}_{\{t<\zeta\}}\right],\quad \ x\in M. \varepsilonnd{split} \varepsilonnd{equation*} According to the same arguments in the proof of Lemma \ref{lem7.2} in the Appendix, the functional $x \mapsto \mathbb{E}_x\Big[f\left(X_t\right)$ $\int^t_0\langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gamma\scr Theta_s^{h(x)}, dB_s\>{\bf 1}_{\{t<\zeta\}}\Big]$ is continuous. So we have verified that the distributional derivative $\nabla P_t f$ exists and is continuous, then $\nabla P_t f$ is the classical gradient and expression \varepsilonqref{l5-2-1} holds. \varepsilonnd{proof} Now we present an estimate for the difference between the gradients of logarithmic heat kernels. Note that for every $x, y \in K \subset B_o(2m-2)\subset D_m$, we could view $\nabla_x\log p(t,x,y)$ and $\nabla_x\log p_{\tilde M_{m}}(t,x,y)$ as vectors in $T_x M$, so that $$|\nabla_x\log p(t,x,y)-\nabla_x \log p_{\tilde M_{m}}(t,x,y)|_{T_x M}$$ is well defined. Let $\partial$ be the cemetery point. We make the convention that $p(t,\partial,y)=0$ for all $t$. \begin} \def\beq{\beg} \def\F{\scr Fin{prp}\label{prp6.6} Suppose that $K$ is a compact subset of $M$ and $L>1$ is a positive number. Then for every suficiently large $m$, we could find a number $t_0(K,L,m)$, depending on $K,L, m$, such that for every $0<t\le t_0$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.7} \sup_{x,y \in K} \left|\nabla_x\log p(t,x,y)- \nabla_x \log p_{\tilde M_{m}}(t,x,y)\right|_{T_x M} \le C(m)e^{-\fracrac{L}{t}}, \varepsilonnd{equation} where $C(m)$ is a positive constant, which may depend on $m$. \varepsilonnd{prp} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} Let us fix points $x,y \in K$ and a unit vector $v \in T_x M$. Let $m $ be a natural number such that $B_o(2m-2)\supset K$. Let $t>0$ be fixed By \varepsilonqref{l5-2-1} where $\scr Theta^h=h'(t)+\frac 12 \text{\rm ric}_{U_t}(h(t))$, we have, for every $f \in C_c(M)$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \langle \nabla P_t f(x),v\rangle_{T_x M} =&\fracrac{2}{t}\mathbb{E}_x\Bigl[\int_0^{\fracrac{t}{2}} \left\langle l_m(s)U_0^{-1}v,\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s \right\rangle \; f\bigl(X_t\bigr){\bf 1}_{\{t<\zeta\}}\Bigr]\\ &-\mathbb{E}_x\Bigl[\int_0^{\fracrac{t}{2}} \Bigl\langle \bigl(\fracrac{t-2s}{t}\bigr)l_m'(s)U_0^{-1}v, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s \Bigr\rangle\;f\left(X_t\right){\bf 1}_{\{t<\zeta\}}\Bigr]\\ &-\fracrac{1}{2}\mathbb{E}_x\Bigl[\int_0^{\fracrac{t}{2}}\Bigl\langle \, \text{\rm ric}_{U_t}\Bigl(\bigl(\fracrac{t-2s}{t}\bigr)l_m(s)U_0^{-1}v\Bigr), \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\Bigr\rangle \;f\left(X_t\right){\bf 1}_{\{t<\zeta\}}\Bigr] \varepsilonnd{split} \varepsilonnd{equation*} Since $f$ has compact support, the indicator function ${\bf 1}_{\{t<\zeta\}}$ can be removed. Taking the conditional expectation on $\sigma(X_t)$, we obtain, for all $x \in M$ and almost everywhere $y \in M$ (with respect to volume measure on $M$), \begin} \def\beq{\beg} \def\F{\scr Fin{equation} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &t\bigl\langle \nabla_x \log p(t,x,y),v\bigr\rangle_{T_x M}\\ & =t\fracrac{\langle \nabla_x p(t,x,y), v \rangle_{T_x M}}{p(t,x,y)} =2\mathbb{E}_x\Bigl[{\bf 1}_{\{t<\zeta\}}\int_0^{\fracrac{t}{2}} \left\langle l_m(s)U_0^{-1}v, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s \right\rangle\; |\; X_t=y\Bigr]\\ &\qquad \qquad \qquad -t\mathbb{E}_x\left[{\bf 1}_{\{t<\zeta\}}\int_0^{\fracrac{t}{2}} \Bigl\langle \bigl(\fracrac{t-2s}{t}\bigr)l_m'(s)U_0^{-1}v, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\Bigr\rangle \; \Big|X_t=y\right]\\ &\qquad \qquad \qquad -\fracrac{t}{2}\mathbb{E}_x\left[{\bf 1}_{\{t<\zeta\}}\int_0^{\fracrac{t}{2}}\Bigl\langle \, \text{\rm ric}_{U_t}\bigl(\fracrac{t-2s}{t}\bigr)l_m(s) U_0^{-1}v, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\Bigr\rangle \;\Big|\;X_t=y\right]\\ &=\mathbb{E}_x\Bigl[{\bf 1}_{\{t<\zeta\}}\int_0^{\fracrac{t}{2}} g_m(s)\bigl\langle U_0^{-1}v, dB_s \bigr\rangle\; |\; X_t=y\Bigr].\varepsilonnd{split} \varepsilonnd{equation} where $$g_m(s):=2l_m(s) -t \left(\fracrac{t-2s}{t}\right)l_m'(s) -\fracrac{t}{2}\;\text{\rm ric}_{U_t}\left(\fracrac{t-2s}{t}\right)l_m(s).$$ Thus, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e3-8} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &t\bigl\langle \nabla_x \log p(t,x,y),v\bigr\rangle_{T_x M} =\mathbb{E}_x\Bigl[{\bf 1}_{\{t<\zeta\}}\int_0^{\fracrac{t}{2}} g_m(s)\bigl\langle U_0^{-1}v, dB_s \bigr\rangle\; |\; X_t=y\Bigr]\\ &=\mathbb{E}_x\left[\int_0^{\fracrac{t}{2}} g_m(s)\bigl\langle U_0^{-1}v, dB_s \bigr\rangle\;\fracrac{p\left(\fracrac{t}{2},X_{\fracrac{t}{2}},y\right)}{p(t,x,y)}{\bf 1}_{\{\frac t 2<\zeta\}} \right]\\ &=\mathbb{E}_x\left[\int_0^{\fracrac{t}{2}} g_m(s)\bigl\langle U_0^{-1}v, dB_s \bigr\rangle\;\fracrac{p\left(\fracrac{t}{2},X_{\fracrac{t}{2}},y\right)}{p(t,x,y)} \right]. \varepsilonnd{split} \varepsilonnd{equation} We have used the property that for $p(\fracrac{t}{2},X_{\fracrac{t}{2}},y)=0$ whenever $\frac t 2\ge\zeta(x)$. Based on the heat kernel estimates in the previous lemmas, by the proof of Lemma \ref{lem7.2} we know immediately $$x\mapsto\mathbb{E}_x\left[\int_0^{\fracrac{t}{2}} g_m(s)\bigl\langle U_0^{-1}v, dB_s \bigr\rangle\;\fracrac{p\left(\fracrac{t}{2},X_{\fracrac{t}{2}},y\right)}{p(t,x,y)}\right]$$ is continuous. So the expression above is true for all $x,y \in M$. Since $l_m'\left(s,X_{\cdot}^m\right)= l_m'\left(s, X_{\cdot}\right)$ and $l_m\left(s,X_{\cdot}^m\right)= l_m\left(s, X_{\cdot}\right)$ for $s<\tau_m$ and $X_s=X_s^m$, applying the same arguments above to $\tilde M_{m}$ we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{e3-9} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\langle t\nabla \log p_{\tilde M_{m}}(t,x,y),v \rangle_{T_x M} =\mathbb{E}_x\left[\int_0^{\fracrac{t}{2}} g_m(s)\bigl\langle U_0^{-1}v, dB_s \bigr\rangle\;\fracrac{p_{\tilde M_m}\left(\fracrac{t}{2},X_{\fracrac{t}{2}}^m,y\right)}{p_{\tilde M_m}(t,x,y)}\right]. \varepsilonnd{split} \varepsilonnd{equation} To apply Lemma \ref{lem6.4} it remains to make moment estimates for $\int_0^t g(s)\langle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\>{\rangle} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\GG{\Gamma} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\gg{\gammav , U_0dB_s\>$. For any $m\in {\mathbb N}$ large enough and $q>0$, \varepsilonqref{eq3.13} implies that condition \varepsilonqref{l4-1-0} in Lemma \ref{lem6.4} holds for the process $\Upsilon_t=\text{\rm ric}_{U_t}$ and we could apply \varepsilonqref{r4-1-1} and \varepsilonqref{l4-1-1} to conclude the estimates. \varepsilonnd{proof} We are now in a position to proceed to prove the gradient estimates for $\log p(t,x,y)$. \begin} \def\beq{\beg} \def\F{\scr Fin{thm}\label{thm6.7} The following statements hold. \begin} \def\beq{\beg} \def\F{\scr Fin{enumerate} \item [(1)] Suppose $x,y \in M$ and $x\notin \text{Cut}_M(y)$, then \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.14} \lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}t\nabla_x\log p(t,x,y)=-\nabla_x\left(\fracrac{d^2(x,y)}{2}\right). \varepsilonnd{equation} Here the convergence is uniformly in $x $ on any compact subset of $M\setminus \text{Cut}_M(y)$. \item[(2)] Let $K$ be a compact subset of $M$. Then there exists a positive constant $C(K)$, which may depend on $K$, such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.15} \left|\nabla_x \log p(t,x,y)\right|_{T_x M}\le C(K)\left(\fracrac{d(x,y)}{t}+\fracrac{1}{\sqrt{t}}\right),\quad x,y\in K, \ t \in (0,1]. \varepsilonnd{equation} \varepsilonnd{enumerate} \varepsilonnd{thm} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} In the proof, the constant $C$ (which depends on $\tilde K$ or $K$) may change from line to line. For every $m \in {\mathbb N} $ with $K\subset B_o(2m-2)\subset D_{m}$, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.16} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &t\nabla_x \log p(t,x,y)\\&= t\nabla_x \log p_{\tilde M_m}(t,x,y)+\left(t\nabla_x \log p(t,x,y)-t\nabla_x \log p_{\tilde M_m}(t,x,y)\right). \varepsilonnd{split} \varepsilonnd{equation} For each compact set $\tilde K \subset M\setminus \text{Cut}_M(y)$, by \varepsilonqref{eq3.7}\varepsilonmph{} we could choose a $m_0 \in {\mathbb N} $ large enough such that $\tilde K \subset B_o(2m_0-2)\subset D_{m_0}$ and \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}\sup_{x \in \tilde K} |t\nabla_x \log p(t,x,y)-t\nabla_x \log p_{\tilde M_{m_0}}(t,x,y)|_{T_x M}=0. \varepsilonnd{equation*} At the same time, since $\tilde M_{m_0}$ is compact and $\tilde K$ is outside of the cut locus $\text{Cut}_{\tilde M_m}(y)$, we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}\sup_{x \in \tilde K} \left|t\nabla_x \log p_{\tilde M_{m_0}}(t,x,y) +\nabla_x \left(\fracrac{d^2(x,y)}{2}\right)\right|_{T_x M}\\ &= \lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}\sup_{x \in \tilde K} \left|t\nabla_x \log p_{\tilde M_{m_0}}(t,x,y) +\nabla_x \left(\fracrac{d_{\tilde M_{m_0}}^2(x,y)}{2}\right)\right|_{T_x \tilde M_{m_0}}= 0. \varepsilonnd{split} \varepsilonnd{equation*} In the first step, we used that $d_{\tilde M_{m_0}}(x,y)=d(x,y)$ for $x,y \in \tilde K$, while the second step is due to Corollary 2.29 from Malliavin and Stroock \cite{MS}, (see also Bismut \cite{Bis} and Norris \cite{Norris}). Plugging this into \varepsilonqref{eq3.16} with $m=m_0$, then we have shown \varepsilonqref{eq3.14}. Given a compact set $K \subset M$ and a constant $L>1$, based on \varepsilonqref{eq3.7} there exists a sufficiently large natural number $m_0$ such that $K \subset B_o(2m_0-2)\subset D_{m_0}$ and and $t_0\in (0,1)$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{t3-5-1} \sup_{x,y \in K}|I^{m_0}(t,x,y)|_{T_x M}\le Ce^{-\fracrac{L}{t}},\ \quad \fracorall \; t\in (0,t_0]. \varepsilonnd{equation} Since $\tilde M_{m_0}$ is compact, we can apply Hsu \cite[Theorem 5.5.3]{Hsu2} or Sheu \cite{Sh} to show that for all $x,y\in K$ and $t\in (0,1]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{t3-5-2} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \left|\nabla_x \log p_{\tilde M_{m_0}}\left(t,x,y\right)\right|_{T_x M}&\le C(K)\left(\fracrac{d_{\tilde M_{m_0}}(x,y)}{t}+\fracrac{1}{\sqrt{t}}\right)\\ &=C(K)\left(\fracrac{d(x,y)}{t}+\fracrac{1}{\sqrt{t}}\right). \varepsilonnd{split} \varepsilonnd{equation} Combing \varepsilonqref{t3-5-1} and \varepsilonqref{t3-5-2} into \varepsilonqref{eq3.16} with $m=m_0$ we immediately find \varepsilonqref{eq3.15} holds for all $t \in (0,t_0]$. Also note that for all $ x,y\in K$ and for all $\ t\in [t_0,1]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \left|\nabla_x \log p\left(t,x,y\right)\right|_{T_x M}\le C(K,t_0) \le C\left(\fracrac{d(x,y)}{t}+\fracrac{1}{\sqrt{t}}\right). \varepsilonnd{split} \varepsilonnd{equation*} By now we have completed the proof of \varepsilonqref{eq3.15}. \varepsilonnd{proof} {\bf Remark:} \begin} \def\beq{\beg} \def\F{\scr Fin{itemize} \item [(1)] The gradient estimate \varepsilonqref{eq3.15} was proved in \cite{S,Sh,Hsu2} for a complete manifold with Ricci curvature bounded from below by a constant $C_0$. In that case, the constant $C(K)$ in \varepsilonqref{eq3.15} in uniform and only depends on $C_0$, see also \cite{XLi} for the case of the estimates for heat kernel associated with the Witten Laplacian operator. \item [(2)] By carefully tracking the proof, we know the constant $C(K)$ from \varepsilonqref{eq3.15} depends only on $C_1(m_0)$, $\inf_{x\in D_{m_0}}\|{\rm Ric}_x\|$, and $\sup_{x\in D_{m_0}}\mathbb{E}_x \int_0^{1}\left|l'_{m_0}(s)\right|^2\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s$, where $C_1(m_0)$ is the positive constant such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} |\nabla_x \log p_{\tilde M_{m_0}}(t,x,y)|_{T_x M}&\le C_1(m_0)\left(\fracrac{d_{\tilde M_{m_0}}(x,y)}{t}+\fracrac{1}{\sqrt{t}}\right)\\ &=C_1(m_0)\left(\fracrac{d(x,y)}{t}+\fracrac{1}{\sqrt{t}}\right). \varepsilonnd{split} \varepsilonnd{equation*} \varepsilonnd{itemize} \subsection{Proof of Theorem \ref{thm3.1} and the main theorem: Hessian estimates} \label{proof-second-order} Now we can prove the claim for the second order gradient of logarithmic heat kernel. In Proposition \ref{prp4.5}, we have established a second order gradient formula for $P_t f$ on a compact manifold. In its proof we exchanged the differential and the integral operators several times, which may not hold if $M$ is not compact. So it is not trivial to extend Proposition \ref{prp4.5} to a non-compact manifold. To prove Theorem \ref{thm3.1}, we begin with comparing the terms in $\nabla^2 P_t^k$ and $\nabla^2P_t$. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lemma-I} Given a point $x\in M$ and a vector $v\in T_x M$, suppose that $m$ is sufficiently large so $x\in D_m$ and $k>m$. Let $\{U_t^k\}_{t\ge 0}$ be the horizontal Brownian motion on $\tilde M_k$ as defined in (\ref{e4-1}). Set $X_t^k=\pi(U_t^k)$ and $P_t^k f(x)=\mathbb{E}_x\left[f(X_t^k)\right]$. Let $h(s)=\left(\fracrac{t-2s}{t}\right)^+ \cdot l_m(s,X_\cdot) \cdot U_0^{-1}v$ and define \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p4-1-2} I \left(\fracrac{t}{2},X_{\cdot},v\right) :=\left(\int_0^{\fracrac{t}{2}}\langle \scr Theta^h_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle\right)^2- \int_0^{\fracrac{t}{2}} \langle {\mathcal L}ambda^h_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle-\int_0^{\fracrac{t}{2}} \left|\scr Theta_s^h\right|^2 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s. \varepsilonnd{equation} Let $I\bigl(\fracrac{t}{2},X_{\cdot}^k,v\bigr)$ be defined with the corresponding terms in $\tilde M_k$. Then we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p4-1-2a} I\left(\fracrac{t}{2},X_{\cdot}^k,v\right)=I\left(\fracrac{t}{2},X_{\cdot},v\right) =\left(\int_0^t\langle \scr Theta_s^h, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle\right)^2-\int_0^t \langle {\mathcal L}ambda_s^h, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle-\int_0^t \left|\scr Theta_s^h\right|^2 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s. \varepsilonnd{equation} Furthermore it holds that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p4-1-4} \sup_{x\in D_m, v\in T_x M, |v|=1}\mathbb{E}_x\left[\left|I\left(\fracrac{t}{2},X_{\cdot},v\right)\right|^2\right]<\infty,\quad \ \fracorall\ t>0. \varepsilonnd{equation} \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} Let $\scr Theta_s^{h,k}$, $\Gamma_s^{h,k}$, ${\mathcal L}ambda_s^{h,k}$ be the corresponding terms of $\scr Theta_s^h$, $\Gamma_s^h$, ${\mathcal L}ambda_s^h$ defined on $\tilde M_k$. By \varepsilonqref{add1} we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p4-1-3} \scr Theta_s^{h,k}=h'(s)+\text{\rm ric}^{\tilde M_k}_{U_s^k}(h(s))=h'(s)+\text{\rm ric}_{U_s}(h(s))=\scr Theta_s^h,\quad \fracorall\ k>m. \varepsilonnd{equation} Still based on \varepsilonqref{e3-2}, \varepsilonqref{e3-2a} and the same arguments for \varepsilonqref{p4-1-3} we can obtain that \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \Gamma_s^{h,k}=\Gamma_s^h,\quad {\mathcal L}ambda_s^{h,k}={\mathcal L}ambda_s^h,\qquad \fracorall\ k>m. \varepsilonnd{equation*} Therefore the term $I\left(\fracrac{t}{2},X_{\cdot}^k,v\right)$ in \varepsilonqref{p4-1-2} is independent of $k$ and the required identity \varepsilonqref{p4-1-2a} holds. Finally, (\ref{p4-1-4}) immediately follows from the moment estimates \varepsilonqref{eq2.4} for $l_m'$ and the same arguments for \varepsilonqref{eq3.13}. \varepsilonnd{proof} We can now begin the \ \newline\varepsilonmph{Proof of Theorem \ref{thm3.1}.} The idea of the proof is similar to that of Lemma \ref{lem6.5}. For convenience of the reader, here we provide a detailed proof. Let $m_0\in {\mathbb N}$ satisfy that $x\in D_{m_0+1}$, then for every $k>m>m_0$ it holds that $B_o(2m-2)\subset D_m \subset D_k$. Let $h(s)=\left(\fracrac{t-2s}{t}\right)^+ \cdot l_m(s,X_\cdot)\cdot U_0^{-1}v=\left(\fracrac{t-2s}{t}\right)^+ \cdot l_m(s,X_\cdot^k)\cdot U_0^{-1}v$. We can apply \varepsilonqref{p3-1-1} in Proposition \ref{prp4.5} to the compact manifold $\tilde M_k$ to obtain that for every $k>m$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p4-1-2-d} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\quad \left\langle \nabla^2 P_t^k f(x), v\otimes v\right\rangle_{T_x M \otimes T_x M}\\ &=\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E_x\left[f(X_t^k)\left( \left(\int_0^t\langle \scr Theta_s^{h,k}, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle\right)^2- \int_0^t \langle {\mathcal L}ambda^{h,k}_s, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\rangle-\int_0^t \left|\scr Theta^{h,k}_s\right|^2 \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\right)\right]\\ &=\mathbb{E}_x\left[f(X_t^k)I\left(\fracrac{t}{2},X_{\cdot}^k,v\right)\right]= \mathbb{E}_x\left[f(X_t^k)I\left(\fracrac{t}{2},X_{\cdot},v\right)\right], \varepsilonnd{split} \varepsilonnd{equation} where the process $\scr Theta_s^{h,k}$, ${\mathcal L}ambda_s^{h,k}$ are defined by \varepsilonqref{e3-2}, \varepsilonqref{e3-2a} on $\tilde M_k$, and in the last step we have applied \varepsilonqref{p4-1-2a}. According to \varepsilonqref{p4-1-2} and integration by parts formula (on compact manifold $\tilde M_k$), for any $\psi \in C_c^{\infty}(M)$, $V \in C_c^{\infty}(M;TM)$ with $\text{\rm supp}{\psi}\cup\text{\rm supp}{V}\subset D_m$ we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.26} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\int_{M} \mathbb{E}_x\left[f\left(X_t^{k}\right)I\left(\fracrac{t}{2},X_{\cdot},V(x)\right)\right]\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x \\&=\int_{M} \left\langle \nabla^2 P_t^{k} f(x), V(x)\otimes V(x)\right\rangle_{T_x M \otimes T_x M}\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x\\ &=\int_{\tilde M_k} \left\langle \nabla^2 P_t^{k} f(x), V(x)\otimes V(x)\right\rangle_{T_x M \otimes T_x M}\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x\\ &=\int_{\tilde M_k} \mathbb{E}_x\left[f\left(X_t^{k}\right)\right]\Psi(\psi,V)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x=\int_{M} \mathbb{E}_x\left[f\left(X_t^{k}\right)\right]\Psi(\psi,V)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x. \varepsilonnd{split} \varepsilonnd{equation} Here we denote the gradient operator and Riemannian volume measure on both $M$ and $\tilde M_k$ by $\nabla$ and $\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x$, and we set \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \Psi(\psi,V)(x):=&\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(V\psi) V)(x)+\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div(\psi\nabla_V V)(x)\\ =&\psi(x)\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div\left(\nabla_V V\right)+\left(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div V\right)^2+ \left\langle V, \nabla \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div V\right\rangle_{T_x M}\right)(x)\\ &+2\left\langle \nabla \psi, \nabla_V V+(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Div V) V\right\rangle_{T_x M}(x)+ \left\langle \nabla^2 \psi(x), V(x)\otimes V(x)\right\rangle_{T_xM \otimes T_x M}.\\ \varepsilonnd{split} \varepsilonnd{equation*} The second and last step above follow from the properties that Riemannian volume measure $\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x$ and the second order gradient operator $\nabla^2$ on $M$ are the same as that on $\tilde M_k$, when they are restricted on $D_m$, the third equality is due to the integration by parts formula. Meanwhile note that $X_t=X_t^k$ if $t<\tau_k$, for every $x\in D_m$ it holds \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\lim_{k \rightarrow \infty} \left|\mathbb{E}_x\left[f(X_t^{k})I\left(\fracrac{t}{2},X_{\cdot},V(x)\right)\right] -\mathbb{E}_x\left[f(X_t)I\left(\fracrac{t}{2},X_{\cdot},V(x)\right){\bf 1}_{\{t<\zeta\}}\right]\right|\\ &\le \lim_{k \rightarrow \infty} \mathbb{E}_x\left[\left|f(X_t^{k})-f(X_t){\bf 1}_{\{t<\zeta\}}\right| \left|I\left(\frac t 2,X_{\cdot},V(x)\right)\right|\right]\\ &\le \lim_{k \rightarrow \infty}\sqrt{\mathbb{E}_x\left[\left|f(X_t^{k})-f(X_t){\bf 1}_{\{t<\zeta\}}\right|^2\right]} \sqrt{\mathbb{E}_x\left[\left|I\left(\frac t 2,X_{\cdot},V(x)\right)\right|^2\right]}\\ &\le \lim_{k \rightarrow \infty}\sqrt{2}C\|f\|_{\infty}\sqrt{\mathbb{P}_x\left(\tau_k\le t<\zeta\right)}=0, \varepsilonnd{split} \varepsilonnd{equation*} where the last inequality is due to \varepsilonqref{p4-1-4}. Putting this into \varepsilonqref{eq3.26}, letting $k \rightarrow \infty$ we see that for every $\psi \in C_c^{\infty}(M)$ and $V \in C^{\infty}(M;TM)$ with $\text{\rm supp}{\psi}\cup\text{\rm supp}{V}\subset D_m$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \int_{D_m} \mathbb{E}_x\left[f\left(X_t\right)I\left(\fracrac{t}{2},X_{\cdot},V(x)\right){\bf 1}_{\{t<\zeta\}}\right]\psi(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x =\int_{D_m} \mathbb{E}_x\left[f\left(X_t\right){\bf 1}_{\{t<\zeta\}}\right]\Psi(\psi,V)(x)\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D x, \varepsilonnd{split} \varepsilonnd{equation*} which implies the weak (distributional) second order gradient $\nabla^2 P_t f$ exists on $D_m$ and \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.27} \begin} \def\beq{\beg} \def\F{\scr Fin{split} \left\langle \nabla^2 P_t f(x), v\otimes v\right\rangle_{T_x M \otimes T_x M}=\mathbb{E}_x\left[f\left(X_t\right) I\left(\fracrac{t}{2},X_{\cdot},v\right){\bf 1}_{\{t<\zeta\}}\right],\ \ x\in D_m, v \in T_x M. \varepsilonnd{split} \varepsilonnd{equation} As shown by Lemma \ref{lem7.2} in Appendix, the functional $x \mapsto \mathbb{E}_x\left[f\left(X_t\right) I\left(\fracrac{t}{2},X_{\cdot},V(x)\right){\bf 1}_{\{t<\zeta\}}\right]$ is continuous. Now the distributional derivative $\nabla^2 P_t f$ exists and is continuous, then $\nabla^2 P_t f$ is the classical second order gradient on $D_m$ and expression \varepsilonqref{t3-1-1} holds. \begin} \def\beq{\beg} \def\F{\scr Fin{prp}\label{prp6.9} Suppose that $K$ is a compact subset of $M$ and $L>1$ is a positive constant. Then, for any sufficiently large $m$, we could find a $t_0(K,L,m)$ such that for any $t\in (0,t_0]$, \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.17} \sup_{x,y\in K}e^{\fracrac{L}{t}}\left|t\nabla_x^2\log p(t,x,y)-t \nabla_x^2 \log p_{\tilde M_{m}}(t,x,y) \right|_{T_x M\otimes T_x M}\le C(m)e^{-\fracrac{L}{t}} \varepsilonnd{equation} where $C(m)$ is a positive constant which may depend on $m$. \varepsilonnd{prp} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} Let us fix $x,y \in K$ and a unit vector $v \in T_x M$ with $|v|=1$. Suppose that $m \in {\mathbb N} $ such that $K \subset B_o(2m-2)\subset D_m$. Then by \varepsilonqref{t3-1-1} we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \left\langle \nabla^2 P_t f(x), v\otimes v\right\rangle_{T_x M \otimes T_x M}=\mathbb{E}_x\left[f\left(X_t\right) I\left(\fracrac{t}{2},X_{\cdot},v\right){\bf 1}_{\{t<\zeta\}}\right], \varepsilonnd{equation*} where $I\left(\fracrac{t}{2},X_{\cdot},v\right)$ is defined by \varepsilonqref{p4-1-2} with $h(s):=\Big(\fracrac{t-2s}{t}\Big)^+\cdot l_m\left(s,X_{\cdot}\right)\cdot U_0^{-1}v$. By this representation and following the same arguments of \varepsilonqref{e3-8} and \varepsilonqref{e3-9} we obtain \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} & \fracrac{\left\langle \nabla^2_x p(t,x,y), v\otimes v\right\rangle_{T_x M \otimes T_x M}}{p(t,x,y)}= \mathbb{E}_x\left[I\left(\fracrac{t}{2},X_{\cdot},v\right) \fracrac{p\left(\fracrac{t}{2},X_{\fracrac{t}{2}},y\right)}{p(t,x,y)} \right],\\ & \fracrac{\left\langle \nabla^2_x p_{\tilde M_m}(t,x,y), v\otimes v\right\rangle_{T_x M\otimes T_x M}}{p_{\tilde M_m}(t,x,y)}= \mathbb{E}_x\left[I\left(\fracrac{t}{2},X_{\cdot},v\right) \fracrac{p_{\tilde M_m}\left(\fracrac{t}{2},X_{\fracrac{t}{2}}^m,y\right)}{p_{\tilde M_m}(t,x,y)}\right]. \varepsilonnd{split} \varepsilonnd{equation*} Based on above expression and following the same arguments in the proof of Proposition \ref{prp6.6} (especially applying \varepsilonqref{l4-1-1} and \varepsilonqref{r4-1-1}--\varepsilonqref{r4-1-3}) we could find a $m_0(K,L)\in {\mathbb N}$ such that for all $m\ge m_0$, there exists a $t_0(K,L,m)>0$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{p4-3-1} \sup_{x,y\in K}\left|\fracrac{\nabla^2_x p(t,x,y)}{p(t,x,y)}-\fracrac{\nabla^2_x p_{\tilde M_m}(t,x,y)}{p_{\tilde M_m}(t,x,y)}\right|_{T_x M \otimes T_x M} \le C(m)e^{-\fracrac{L}{t}},\quad \ t\in (0,t_0]. \varepsilonnd{equation} Meanwhile we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\left\langle\nabla^2_x \log p(t,x,y),v\otimes v\right\rangle_{T_x M \otimes T_x M} =\fracrac{\left\langle \nabla_x^2 p(t,x,y), v\otimes v\right\rangle_{T_x M \otimes T_x M}}{p(t,x,y)} -\left(\fracrac{\langle \nabla_x p(t,x,y),v\rangle_{T_x M}}{p(t,x,y)}\right)^2,\\ \varepsilonnd{split} \varepsilonnd{equation*} and the similar expression holds for $\left\langle\nabla^2_x \log p_{\tilde M_m}(t,x,y), v\otimes v\right\rangle_{T_x M \otimes T_x M}$. Together with \varepsilonqref{eq3.7} and \varepsilonqref{p4-3-1}, this yields \varepsilonqref{eq3.17} and concludes the proof. \varepsilonnd{proof} With \varepsilonqref{eq3.17} we are in the position to prove the second part of the main theorem on the short time and asymptotic second order gradient estimates. \begin} \def\beq{\beg} \def\F{\scr Fin{thm}\label{thm6.10} The following statements hold. \begin} \def\beq{\beg} \def\F{\scr Fin{enumerate} \item [(1)] Suppose $y \in M$ and $\tilde K \subset M\setminus \text{Cut}_M(y)$ is a compact set, then \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.28} \lim_{t \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Downarrow 0}\sup_{x \in \tilde K} \left|t\nabla_x^2\log p(t,x,y)+\nabla_x^2\left(\fracrac{d^2(x,y)}{2}\right) \right|_{T_x M \otimes T_x M}=0. \varepsilonnd{equation} \item[(2)] For each $y\in M$ and $\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Delta<i(y)$ there exist positive constants $t_0$ and $C_1$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.29}\aligned &\left|t\nabla_x^2 \log p(t,x,y)+\textbf{I}_{T_x M}\right|_{T_x M \otimes T_x M}\\ &\le C_1\left(d(x,y)+ \sqrt{t}\right),\quad \quad \ x\in B_y(\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Delta),\ t \in (0,t_0],\varepsilonndaligned \varepsilonnd{equation} where $\textbf{I}_{T_x M}$ is the identical map on $T_x M$. \item[(3)] Suppose $K \subset M$ is a compact subset of $M$, then there exists a positive constant $C_2(K)$, such that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{eq3.30} \left|\nabla_x^2 \log p(t,x,y)\right|_{T_x M \otimes T_x M}\le C_2\left(\fracrac{d^2(x,y)}{t^2}+\fracrac{1}{t}\right),\quad x,y\in K, \ t \in (0,1]. \varepsilonnd{equation} \varepsilonnd{enumerate} \varepsilonnd{thm} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} By Malliavin and Stroock \cite[Corollary 2.29]{MS}, Gong and Ma \cite[Theorem 3.1]{GM} and Stroock \cite{S} (or Sheu \cite{Sh}), we know \varepsilonqref{eq3.28}--\varepsilonqref{eq3.30} hold when $M$ is compact. Then using the estimates \varepsilonqref{eq3.17} and following the same procedure as in the proof of Theorem \ref{thm6.7} we can verify that \varepsilonqref{eq3.28}-\varepsilonqref{eq3.30} hold for any complete Riemannian manifold. \varepsilonnd{proof} \begin} \def\beq{\beg} \def\F{\scr Fin{appendix} \text{\rm sect}ion*{Approximation procedure}\label{appn} Let $(M,g)$ and $D_m\subset M$ be the same terms in Section \ref{cut-off}. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem7.1} For every $m \in \mathbb{Z}_+$, there exists a (smooth) compact Riemannian manifold $(\tilde M_m,\tilde g_m)$, such that $(D_m,g)$ is isometrically embedded into $(\tilde M_m,\tilde g_m)$ as an open set. In particular, if $y,x \in D_m$ and $x\not \in \text{\rm cut}_y(M)$, then $x\not \in \text{\rm cut}_y(\tilde M_m)$. \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} Let $G_m =D_{m+1}$, recall that $\partial G_m$ is a connected smooth $n-1$-dimensional sub-manifold of $M$. Hence $\overline{G_m}$ is an $n$-dimensional manifold with smooth boundary, then there exist an atlas of local charts $\{(V_i,\psi_i)\}_{i=1}^N$ of $\overline{G_m}$ such that \begin} \def\beq{\beg} \def\F{\scr Fin{enumerate} \item [(1)] $\bigcup_{i=1}^N V_i=\overline{G_m}$ ; \item [(2)] For $i=1, \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Dots, N_1\le N$, these are charts for the interior. So $V_i\cap \partial G_m=\varepsilonmptyset$ and $\psi_i: V_i \rightarrow \mathbf{B}^n:=\{z \in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n; |z|< 1\}$ is a smooth diffeomorphsim for all $1\le i\le N_1$; \item [(3)] For all $i>N_1$, $V_i\cap \partial G_m\neq\varepsilonmptyset$, $$\psi_i: V_i \rightarrow \mathbf{B}^{n,+}:=\{z=(z_1,\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Dots,z_n) \in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n; |z|< 1,z_1\ge 0\}$$ is a smooth diffeomorphsim and $\psi_i\big(V_i\cap \partial G_m\big)=\partial \mathbf{B}^{n,+}$. \varepsilonnd{enumerate} By the Whitney embedding theorem, we could embed $M$ into a (ambient) Euclidean space $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^p$. Let $\mathfrak hat G_m$ be an identical copy of $G_m$ in $\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^p$ endowed with the local charts $\{(\mathfrak hat V_i,\mathfrak hat \psi_i)\}_{i=1}^N$ (which is also an identical copy of $\{(V_i,\psi_i)\}_{i=1}^N$). We define $h:\partial G_m \rightarrow \partial \mathfrak hat G_m$ by $h(x):=\mathfrak hat \psi_i^{-1}\left(\psi_i(x)\right)$, if $x \in V_i\cap \partial G_m$, $h$ is well defined and is a smooth diffeomorphism. We glue the boundary of $G_m$ and $\tilde G_m$ together to get $\tilde M_m:=(G_m\sqcup\mathfrak hat G_m)/\thicksim$, where $\thicksim$ is an equivalent relation such that $x \thicksim y$ if and only if $h(x)=y$, $x \in \partial G_m$, $y \in \partial \mathfrak hat G_m$. Then $\tilde M_m$ is a smooth compact manifold without boundary. In fact, $\{(U_i,\phi_i)\}_{i=1}^{N+N_1}=\{(V_i,\psi_i)\}_{i=1}^{N_1}\bigcup \{(\mathfrak hat V_i,\mathfrak hat \psi_i)\}_{i=1}^{N_1}$ $\bigcup \{(\tilde V_i,\tilde \psi_i)\}_{i=N_1+1}^N$ is a local charts of $\tilde M_m$. Here $\tilde V_i=(V_i\sqcup\mathfrak hat V_i)/\thicksim$ for every $N_1<i\le N$ and \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \tilde \psi_i(x)= \begin} \def\beq{\beg} \def\F{\scr Fin{cases} &\psi_i(x),\ \ \ \ \ \text{if}\ \ x\in V_i,\\ &\mathbf{S}\big(\mathfrak hat \psi_i(x)\big), \ \ \ \text{if}\ \ x\in \mathfrak hat V_i, \varepsilonnd{cases} \varepsilonnd{equation*} where $\mathbf{S}:\mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n\rightarrow \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$ is a map such that $\mathbf{S}x=(-x_1,x_2,\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Dots,x_n)$ for all $x=(x_1,x_2,\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Dots,x_n)\in \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt^n$. It is easy to see $\tilde \psi_i: \tilde V_i \rightarrow \mathbf{B}^n$, $N_1<i\le N$ is a smooth diffeomorphsim, and the transition map between different local charts on $\{(U_i,\phi_i)\}_{i=1}^{N+N_1}$ is smooth. We construct a smooth Riemannian metric $\tilde g_m$ on $\tilde M_m$ to ensure that $\tilde g_m(z)=g(z)$ for every $z \in D_m$. For the open set $D_m\subset G_m\subset \tilde M_m$, by the standard procedure (via the finite local charts) we could construct a function $\chi_m:\tilde M_m \rightarrow [0,1]$ such that $\text{\rm supp}\chi_m \subset G_m$ and $\chi_m(x)=1$ for every $x \in D_m$. Note that $G_m$ could also be viewed as an open subset of $\tilde M_m$, so $\mathfrak hat g_m(x):=g(x)\chi_m(x)$, $x \in \tilde M_m$ is well defined on $\tilde M_m$. Fixing a smooth Riemannian metric $g_m^0$ on $\tilde M_m$, which exists, we set \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \tilde g_m(x):=g(x)\chi_m(x)+g_m^0(x)\big(1-\chi_m(x)\big),\quad x\in \tilde M_m. \varepsilonnd{equation*} It is easy to see $\tilde g_m$ is a smooth Riemannian metric on $\tilde M_m$ and $\tilde g_m(x)=g(x)$ for each $x \in D_m$. By now we have completed the proof. \varepsilonnd{proof} Let $I(t,X_{\cdot},v)$ be as defined in \varepsilonqref{p4-1-2}. \begin} \def\beq{\beg} \def\F{\scr Fin{lem}\label{lem7.2} For every fixed $f \in C_c^{\infty}(M)$, $V \in C^{\infty}(M;TM)$ with compact supports and $t>0$, the function \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} F(x):=\mathbb{E}\left[f\left(X_t^x\right)I\left(\fracrac{t}{2},X_{\cdot}^{x},V(x)\right){\bf 1}_{\{t<\zeta(x)\}}\right],\quad x\in M. \varepsilonnd{equation*} is continuous. \varepsilonnd{lem} \begin} \def\beq{\beg} \def\F{\scr Fin{proof} Let $\zeta(x)$ denotes the explosion time of the solution $X_t^x$ to \varepsilonqref{sde1} with the initial value $x$. Let $U$ be a frame at $x$. Then the explosion time of the horizontal Brownian motion agree with $\xi(x)$ almost surely. So we use $\xi$ for the explosion time of both. Furthermore, by Elworthy \cite{Elworthy}, there exist a maximal solution flow $\{U_t(\cdot,\omega)\}_{0\le t <\zeta(\cdot,\omega)}$ to \varepsilonqref{sde1} such that $U_t(u,\omega)$ is the solution of \varepsilonqref{sde1} with initial value $u\in OM$, and there is a null set $\Delta$ such that for all $\omega \notin \Delta$, \begin} \def\beq{\beg} \def\F{\scr Fin{itemize} \item [(1)] For each $t>0$, set $\scr Xi_t(\omega):=\{u\in OM:\ t<\zeta(u,\omega)\}$, Then $\scr Xi_t$ is open in $OM$ (i.e. $\zeta(\cdot,\omega):OM \to \mathbb R} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\scr E} \def\si{\sigma} \def\ess{\text{\rm{ess}}E{\mathbb E} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\Z{\mathbb Z} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\fracf{\fracrac} \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr Def\ss{\sqrt_+$ is lower semi-continuous) and $U_t(.,\omega): \scr Xi_t(\omega)\rightarrow OM$ is a $C^{\infty}$ diffeomorphism onto its image. \item[(2)] For each fixed $u \in OM$ with $\pi(u)=x$, there exists a null set $\Delta(u)$ depending on $u$, such that $\zeta(u,\omega)=\zeta(X_{\cdot}^{x})$ for each $\omega \notin \Delta(u) \bigcup \Delta$. \varepsilonnd{itemize} Fix a point $x_0 \in M$. For each sequence $\{x_k\}_{k=1}^{\infty}$ with $\lim_{k \rightarrow \infty}x_k=x_0$, we take a sequence $\{u_k\}_{k=1}^{\infty}$ and $U_0$ in $OM$, such that $\pi(u_k)=x_k$, $\pi(U_0)=x_0$ and $\lim_{k \rightarrow \infty}u_k=u_0$ in $OM$. Set $\tilde \Delta:=\left(\bigcup_{k=0}^{\infty}\Delta(u_k)\right)\bigcup \Delta$. For each $k$ and $\omega \notin \tilde \Delta$, $\zeta(U_k,\omega)=\zeta\left({x_k},\omega\right)$. By the lower semi-continuity of $\zeta$, $\zeta({x_0})\le \liminf_{k \rightarrow \infty}\zeta({x_k})$, hence $u_k \in \scr Xi_t(\omega)$ for each $t<\zeta({x_0})$ when $k$ is large enough. By the property (1) above, we have immediately $$\lim_{k \rightarrow \infty} U_t(u_k, \omega){\bf 1}_{\{t<\zeta({x_k})\}}=U_t(u_0,\omega){\bf 1}_{\{t<\zeta({x_0})\}}, \; \quad \omega \notin \tilde \Delta,\ t>0.$$ Combing this with the definition $\scr Theta(s,X_{\cdot},v)$ and the expression \varepsilonqref{eq2.5} of $l_m$, we see that \begin} \def\beq{\beg} \def\F{\scr Fin{equation}\label{l5-1-1} \lim_{k \rightarrow \infty}\scr Theta\left(s,X_{\cdot}^{x_k},V(x_k)\right)= \scr Theta\left(s,X_{\cdot}^{x_0},V(x_0)\right),\quad \ s>0. \varepsilonnd{equation} Let $h(s, X_{\cdot},V(x)):=\Big(\fracrac{t-2s}{t}\vee 0\Big)\cdot l_m(s,X_{\cdot})\cdot u_0(x)^{-1}V(x)$, where $u_0(\cdot)$ is a smooth section of $OM$ with $\pi(u_0(x))=x$. We only need to demonstrate the proof for one of the term in $I(t,X_{\cdot}^{x},v)$, for this we set \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} w(x)&:= \mathbb{E}\left[f\left(X_t^x\right)\left(\int_0^{\fracrac{t}{2}} \left\langle \scr Theta\left(s,X_{\cdot}^x,V(x)\right), \text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle\right){\bf 1}_{\{t<\zeta(x)\}}\right]\\ &=:\mathbb{E}\left[f\left(X_t^x\right)\left(\int_0^{\fracrac{t}{2}} \left\langle\left(h'(s)+\fracrac{1}{2}\text{\rm ric}_{U_s}h(s)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle\right){\bf 1}_{\{t<\zeta(x)\}}\right], \varepsilonnd{split} \varepsilonnd{equation*} For simplicity, we only prove the continuity for the function $x\rightarrow w(x)$, the continuity property for the other terms in $F(x)$ could be verified similarly. According to \varepsilonqref{eq2.4} we obtain \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \sup_{k>0}\mathbb{E}\left[\left|\int_0^{\fracrac{t}{2}}\left\langle \scr Theta\left(s,X_{\cdot}^{x_k},V(x_k)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle\right|^4\right]<\infty. \varepsilonnd{equation*} Based on this and \varepsilonqref{l5-1-1} we have \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \lim_{k \rightarrow \infty}\mathbb{E}\left[\left|\int_0^{\fracrac{t}{2}}\left\langle \scr Theta\left(s,X_{\cdot}^{x_k},V(x_k)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle- \int_0^{\fracrac{t}{2}}\left\langle \scr Theta\left(s,X_{\cdot}^{x_0},V(x_0)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle \right|^2\right]=0. \varepsilonnd{equation*} Similarly from \varepsilonqref{l5-1-1} we arrive at \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \lim_{k \rightarrow \infty}\mathbb{E}\left[\left|f(X_t^{x_k}){\bf 1}_{\{t<\zeta({x_k})\}}-f(X_t^{x_0}){\bf 1}_{\{t<\zeta({x_0})\}}\right|^2\right]=0. \varepsilonnd{equation*} Therefore by Cauchy-Schwartz inequality \begin} \def\beq{\beg} \def\F{\scr Fin{equation*} \begin} \def\beq{\beg} \def\F{\scr Fin{split} &\lim_{k \rightarrow \infty}|w(x_k)-w(x_0)|^2\\ &\le 2\|f\|_{\infty}^2\lim_{k \rightarrow \infty}\mathbb{E}\left[\left|\int_0^{\fracrac{t}{2}}\left\langle \scr Theta\left(s,X_{\cdot}^{x_k},V(x_k)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle- \int_0^{\fracrac{t}{2}}\left\langle \scr Theta\left(s,X_{\cdot}^{x_0},V(x_0)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle \right|^2\right]\\ &+ 2\sup_{k>0}\mathbb{E}\left[\left|\int_0^{\fracrac{t}{2}}\left\langle \scr Theta\left(s,X_{\cdot}^{x_k},V(x_k)\right),\text{\rm{d}}} \def\loc{\text{\rm{loc}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s\right\rangle\right|^4\right]\cdot\lim_{k \rightarrow \infty}\mathbb{E}\left[\left|f(X_t^{x_k}){\bf 1}_{\{t<\zeta({x_k})\}}-f(X_t^{x_0}){\bf 1}_{\{t<\zeta({x_0})\}}\right|^2\right]\\ &=0 \varepsilonnd{split} \varepsilonnd{equation*} Since $\{x_k\}_{k=1}^{\infty}$ is arbitrarily chosen, $w(\cdot)$ is continuous at $x_0\in M$. Again $x_0$ is arbitrary, so $w(\cdot)$ is continuous on $M$. This completes the proof for the lemma. \varepsilonnd{proof} \varepsilonnd{appendix} \begin} \def\beq{\beg} \def\F{\scr Fin{acks}[Acknowledgments] We would like to thank Christian B\"ar and Robert Neel for helpful comments and the referees for their valuable comments and suggestions. \varepsilonnd{acks} \begin} \def\beq{\beg} \def\F{\scr Fin{funding} The research of Xin Chen is supported by the National Natural Science Foundation of China (No. 12122111). The research of Xue-Mei Li is supported by EPSRC(Nos. EP/E058124/1, S023925/1, and EP/V026100/1). The research of Bo Wu is supported by the National Natural Science Foundation of China (No. 12071085). \varepsilonnd{funding} \varepsilonnd{document}

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B}_H(\betalag)}{{\betacal B}_H(\betalangleg)} \betanewcommand{\betaalpha}{\betaalpha} \betanewcommand{\betabeta}{\betabegin{eqnarray*}ta} \betanewcommand{\betalanglem}{\betalanglembda} \betanewcommand{\betaLambda}{\betaLambda} \betanewcommand{\betaDelta}{\betaDeltaelta} \betanewcommand{\beta}{\betabegin{eqnarray*}ta} \betanewcommand{\betaה}{\betaalpha} \betanewcommand{\betaW}{\betaOmega} \betabegin{itemize}bliographystyle{alpha} \betatitle{Towards a classification of Lorentzian holonomy groups} \betaauthor{Thomas Leistner } \betamaketitle \betabegin{eqnarray*}gin{abstract} If the holonomy representation of an $(n+2)$--dimensional simply-connected Lorentzian manifold $(M,h)$ admits a degenerate invariant subspace its holonomy group is contained in the parabolic group $( \betamathbb{R} \betatimes SO(n) )\betaltimes \betamathbb{R}^n$. The main ingredient of such a holonomy group is the $SO(n)$--projection $G:=pr_{SO(n)}(Hol_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is the case if $G\betasigmaubsetset U(n/2)$ or if the irreducible acting components of $G$ are simple. \betaend{abstract} \betasigmaetcounter{tocdepth}{1} \betatableofcontents \betasigmaection*{Introduction} The very first step in a classification of the holonomy groups of semi-Riemannian manifolds is the decomposition theorem of de Rham and Wu (\betacite{derham52} for Riemannian manifolds and \betacite{wu64} for general semi-Riemannian manifolds). It asserts that every simply-connected, complete semi-Riemannian manifold is isometric to a product of simply-connected, complete {semi-Riemannian} manifolds, of which one can be flat and all other are indecomposable (often called ``weakly-irreducible'', i.e. with no non-degenerate invariant subspace under holonomy representation). For a Riemannian manifold this theorem asserts that the holonomy representation is completely reducible, i.e. decomposes into factors which are trivial or irreducible, and are again Riemannian holonomy representations. For pseudo-Riemannian manifolds indecomposability is not the same as irreducibility. We can have degenerate invariant subspaces under holonomy representation. On the other hand all irreducible factors are known by the Berger classification of possible irreducible semi-Riemannian holonomy groups (\betacite{berger55}, \betacite{simons62}, \betacite {alekseevskii68}, \betacite{brown-gray72} and \betacite{bryant87}). This classification uses an algebraic condition which has to be satisfied by every holonomy group of a torsionfree connection. It follows from the first Bianchi identity and the Amrose-Singer holonomy theorem \betacite{as} and can be formulated very easily: If $\betamathfrak{h}$ is the Lie algebra of the holonomy group of a torsionfree connection, acting on the vector space $E\betasigmaimeq T_p M$, then it obeys $\betamathfrak{h}=\betaleft\beta{ R(u,v)\beta |\beta u,v\betain V, R\betain{\betacal K}(\betamathfrak{h})\betaright\beta}$, where \beta[{\betacal K}(\betamathfrak{h})\beta :=\beta \betaleft\beta{R\betain\betaomegaedge^2V^*\betaotimes \betamathfrak{h}\beta |\beta R(u,v)w+R(v,w)u+R(w,u)v=0\betamathbbox{ for all }u,v,w\betain V \betaright\beta}\beta] is the space of curvature endomorphisms. Lie algebras satisfying this conditions are called Berger algebras. All irreducible Berger algebras are classified in \betacite{schwachhoefer1} and \betacite{schwachhoefer2}. For non-irreducible, indecomposable holonomy representations (resp. Berger algebras) such a classification is missing. For a Lorentzian manifold $(M,h)$ of dimension $m>2$ the de Rham/Wu--decomposition yields the following two cases: \betabegin{eqnarray*}gin{description} \betaitem{Completely reducible:} Here $(M,h)$ decomposes into irreducible or flat Riemannian manifolds and a manifold which is an irreducible or flat Lorentzian manifold or $(\betamathbb{R},-dt)$. The irreducible Riemannian holonomies are known, as well as the irreducible Lorentzian one, which has to be the whole $SO(1,m-1)$. (The latter follows from the Berger list but was directly proved by \betacite{olmos-discala01}.) \betaitem{Not completely reducible:} This is equivalent to the existence of a degenerate invariant subspace and entails the existence of exactly one holonomy invariant lightlike subspace. The Lorentzian manifold decomposes into irreducible or flat Riemannian manifolds and a Lorentzian manifold with indecomposable, but non-irreducible holonomy representation, i.e. with invariant lightlike (i.e. {one-dimensional}) subspace. \betaend{description} Thus in order to classify holonomy groups of simply-connected Lorentzian manifolds one has to find the possible holonomy groups of indecomposable, but non-irreducible Lorentzian manifolds. The holonomy algebra of such a manifold of dimension $m:=n+2>2$ is contained in $ ( \betamathbb{R} \betaoplus \betamathfrak{so}(n) )\betaltimes \betamathbb{R}^n$. L. Berard-Bergery and A. Ikemakhen studied in \betacite{bb-ike93} the projections of such a holonomy algebra and achieved two important results. The first gives a classification into four types based on the possible projections on $\betamathbb{R}$ and $\betamathbb{R}^n$. For two of these types the projections are coupled and for the remaining two uncoupled to the $\betalangleson$--component. The second result is a decomposition property for the $\betamathfrak{so}(n)$--projection (see theorem \betaref{theoI}), i.e. there is a decomposition of the representation space into irreducible components and of the Lie algebra into ideals which act irreducible on the components. The relation between the $\betalangleson$--part and the $\betamathbb{R}$-- and $\betamathbb{R}n$--parts is understood quite well (\betacite{boubel00}, or very recently \betacite{galaev}): If one has a simply-connected, indecomposable, non-irreducible Lorentzian manifold with holonomy of uncoupled type, then, under certain conditions, one can construct a Lorentzian manifold with coupled type holonomy. Now one may ask: Which algebras can occur as $\betamathfrak{so}(n)$-projection of an indecomposable, but not-irreducible Lorentzian manifold? Of course it has to satisfy the decomposition property. Riemannian holonomy algebras are the first examples, because there is a method to construct from a given Riemannian manifold an indecomposable Lorentzian manifold with holonomy of uncoupled type for which the $\betalangleson$--projection equals to the Riemannian holonomy. Furthermore one can show that the Lorentzian manifold is a $pp$-wave if and only if the $\betamathbb{R}$-- and the $SO(n)$--component vanish \betacite{leistner01}. In \betacite{leistner02} we derived an algebraic criterion on the $\betalangleson$--component of an indecomposable, non-irreducible, simply-connected Lorentzian manifold $(M,h)$, in analogy to the well known Berger criterion for holonomy algebras. If $\betalangleg$ is the $\betalangleson$-component of an indecomposable, non-irreducible, simply-connected Lorentzian manifold, acting on an $n$--dimensional Euclidean vector space $(E,h)$ then it obeys $\betamathfrak{g}=\betaleft\beta{ Q(u) | Q\betain {\betacal B}_{h}(\betalangleg), u\betain E\betaright\beta}$ where ${\betacal B}_{h}(\betalangleg)$ is defined as follows \beta[ {\betacal B}_{h}(\betalangleg) = \betaleft\beta{ Q\betain E^*\betaotimes \betamathfrak{g}\beta |\beta h( Q(u)v,w)+h( Q(v)w,u)+h( Q(w)u,v)=0,\beta \betaforall\beta u,v,w\betain E\betaright\beta}.\beta] Since orthogonal Berger algebras do satisfy this criterion we called these algebras weak-Berger algebras. Furthermore we showed that every irreducible weak-Berger algebra, which is contained in $\betamathfrak{u}(n/2)$ is a Berger algebra, in particular a Riemannian holonomy algebra. This, together with the decomposition property implies that $\betalangleg:=pr_{\betalangleson}\betamathfrak{hol}_p(M,h)$ is a Riemannian holonomy algebra if it is contained in $\betamathfrak{u}(n/2)$. In the present paper we prove the following: If $\betalangleg$ is a simple weak-Berger algebra, not contained in $\betamathfrak{u}(n/2)$, which acts irreducible on $\betamathbb{R}n$, then it is a Berger algebra, and in particular a Riemannian holonomy algebra. This of course applies to the irreducible components of the $\betalangleson$--projection of $\betamathfrak{hol}_p(M,h)$. In the proof we proceed analogously to \betacite{schwachhoefer2}, where the holonomy groups of torsion free connections are classified. This will be the main part of this paper and is contained in section \betaref{sectionzwei}. In the first section we recall the results of \betacite{bb-ike93} and our results from \betacite{leistner02} introducing the notion of weak-Berger algebras. The third section presents again for sake of completeness the proof of the fact that weak-Berger algebras in $\betamathfrak{u}(n/2)$ are Berger algebras. In the appendix we recall facts about representations of real Lie algebras. These results leave open the question: Are there semisimple, non simple, irreducible acting Lie algebras, not contained in $\betamathfrak{u}(n/2)$, which are weak-Berger, but not Berger? We guess that this is not the case, and we intent to apply the methods of the present paper also in the semisimple case. Up to dimension eleven this was proved very recently by \betacite{galaev} also for algebras not contained in $\betamathfrak{u}(n/2)$. In his paper he studied the space of curvature endomorphisms for subalgebras in $( \betamathbb{R} \betaoplus \betamathfrak{so}(n) )\betaltimes \betamathbb{R}^n$ which are of the types found in \betacite{bb-ike93}. Reducing everything to one uncoupled type he proved the other direction of our result: a subalgebra of one of these types is a Berger algebra, if its $\betalangleson$--projection is a weak-Berger algebra (in our terms). We are aware that the proofs we will present here are a cumbersome case-by-case analysis using the methods of representation theory. It is very desirable to get a direct and more geometric proof of the proposition that every $SO(n)$--projection of an indecomposable, non-irreducible Lorentzian holonomy group is a Riemannian holonomy group, which includes the remaining semisimple case of course. We want to remark that the starting point of this investigation was the question for the existence of parallel spinors on Lorentzian manifolds. Such a spinor defines a parallel vector field which can be light like. Hence the manifold has an indecomposable, non-irreducible factor. But the existence of parallel spinors on indecomposable Lorentzian manifolds with parallel lightlike vector field depends only on the $SO(n)$--projection. Thus a complete list of the latter would answer this question. In the physically important dimensions below twelve the question for the maximal indecomposable Lorentzian holonomy groups admitting parallel spinors is answered \betacite{bryant00}, \betacite{farrill99mw}. \betasigmaection{Indecomposable Lorentzian holonomy and weak-Berger algebras} \betasigmaubsetsection{Basic properties} Let $(M,h)$ be an indecomposable, non-irreducible Lorentzian manifold with $dim \beta M =n+2>2$. The holonomy group in a point $p\betain M$ acting on $T_p M$ --- defined as the group of parallel displacements along loops starting at $p$ --- then has a lightlike, one-dimensional invariant subspace $\betaXi_p$ which is the fibre of a parallel distribution $\betaXi$. This is equivalent to the existence of a recurrent lightlike vector field. The subspace $\betaXi_p^\betabot$ also is holonomy invariant and the fibre of a parallel distribution $\betaXi^\betabot$. (We call a distribution parallel if it is closed under $\betanabla_U$ for every $U\betain TM$.) With respect to a basis \betabegin{eqnarray*}gin{equation} \betalanglebel{basis} \betabegin{eqnarray*}gin{array}{l} (X, E_1, \betaldots E_n, Z) \betamathbbox{ adapted to $\betaXi_p\betasigmaubsetset \betaXi_p^\betabot$, i.e. $X\betain \betaXi_p, E_i\betain \betaXi_p^\betabot$}\beta\beta \betamathbbox{with $h(E_i, E_j)= \betadelta_{ij}, h(Z,Z)=h(Z,E_i)=h(X,E_i)=0$ and $h(X,Z)=1$} \betaend{array} \betaend{equation} the holonomy algebra is contained in the following Lie algebra \betabegin{eqnarray*}gin{equation} \betalanglebel{holform} \betamathfrak{hol}_p(M,h)\betasigmaubsetset \betaleft\beta{ \betaleft( \betaleft. \betabegin{eqnarray*}gin{array}{ccc} a & u^t&0 \beta\beta 0 & A &-u \beta\beta 0 &0^t&-a \betaend{array} \betaright) \betaright| a\betain \betamathbb{R}, u\betain \betamathbb{R}^n, A \betain \betamathfrak{so}(n) \betaright\beta}\beta = \beta (\betamathbb{R}\betaoplus\betalangleson )\betaltimes \betamathbb{R}n. \betaend{equation} Choosing a different basis of type (\betaref{basis}) corresponds to conjugation with an element in $O(1,n+1)$ which respects the form (\betaref{basis}). Hence the $\betamathfrak{so}(n)$--component is uniquely defined with respect to conjugation in $O(n)$. The projections of $ \betamathfrak{hol}_p(M,h)$ on the $\betamathbb{R}$-- and on the $ \betamathbb{R}^n$--component are well understood. With respect to these projections there exist four different types (see \betacite{ike90}, \betacite{bb-ike93} and \betacite{ike96}). For the types $I$ and $II$ the holonomy is equal to $(\betamathbb{R} \betaoplus \betamathfrak{g}) \betaltimes \betamathbb{R}^n$ resp. $\betamathfrak{g}\betaltimes\betamathbb{R}^n$. In case of types $II$ and $IV$ the projection on $\betamathbb{R}$ is zero, which implies the existence not only of a recurrent lightlike vector field but also of a parallel one. In case of types $III$ and $IV$ the $\betamathbb{R}$-- respectively the $\betamathbb{R}n$-- components are coupled to the $\betalangleson$--component, or more precisely to its center. In the following shall be ${\betafrak g}:= pr_{{\betafrak s}{\betafrak o}(n)} (\betamathfrak{hol}_p(M,h) )$. About ${\betafrak g}$ the in \betacite{bb-ike93} is proved \betabegin{eqnarray*}gin{theo} \betalanglebel{theoI} \betacite{bb-ike93} Let ${\betafrak g}:= pr_{\betamathfrak{so}(n)} \betaleft(\betamathfrak{hol}_p(M,h) \betaright)$ be the projection of the holonomy algebra of an indecomposable, non-irreducible, $n+2$--dimensional Lorentzian manifold onto ${\betafrak s}{\betafrak o}(n)$. Then $ {\betafrak g}$ satisfies the following decomposition property: There exists a decomposition of $\betamathbb{R}^n$ into orthogonal subspaces and of ${\betafrak g}$ into ideals \beta[ \betamathbb{R}^n = E_0 \betaoplus E_1 \betaoplus \betaldots \betaoplus E_r\beta \betamathbbox{ and }\beta {\betafrak g} = {\betafrak g}_1 \betaoplus \betaldots \betaoplus {\betafrak g}_r,\beta] such that ${\betafrak g}$ acts trivial on $E_0$, ${\betafrak g}_i$ acts irreducible on $E_i$ and trivial on $E_j$ for $i\betanot=j$. \betaend{theo} This theorem has two important consequences making a further algebraic investigation of ${\betafrak g}$ possible. Irreducible acting, connected subgroups of $SO(n)$ are are closed and therefore compact. Now by the theorem the group $G:=pr_{SO(n)}Hol^0_p(M,h)$ decomposes in such irreducible acting subgroups. Thus we have as first consequence that $G$ is compact, although the whole holonomy group must not be compact (for such examples see also \betacite{bb-ike93}). The second is, that it suffices to study irreducible acting groups or algebras ${\betafrak g}$, a fact which is necessary for trying a classification. We will see this in detail in the following section. We will describe the local situation briefly. Locally there are $n$-dimensional Riemannian submanifolds defined via special coordinates respecting the foliation $\betaXi\betasigmaubsetset \betaXi^{\betabot}$, denoted by $(x, y_1, \betaldots , y_n, z)$ with $ \betafrac{\betapartial}{\betapartial x}\betain \betaXi$, $\betafrac{\betapartial}{\betapartial y_i}\betain \betaXi^\betabot$. The restriction of $h$ to these submanifolds defined by $y_1, \betaldots , y_n$ gives a family of Riemannian metrics $g_z$ on it, depending only on the coordinate $z$ (since $ \betafrac{\betapartial}{\betapartial x}\betain \betaXi$, see \betacite{brinkmann25}, also \betacite{ike96}). Although these coordinates are unique under certain conditions (see \betacite{boubel00}) it is not clear how the Lie algebra ${\betafrak g}$ can be obtained by the holonomies $\betamathfrak{hol}_{p(z)}(g_z)$ of the family of metrics $g_z$. The only known point is, that all these $\betamathfrak{hol}_{p(z)}(h_z)$ are contained in ${\betafrak g}$ \betacite{ike96}. If the dependence on $z$ is trivial --- i.e. $g_z\betaequiv g$ or $g_z\betaequiv f(z) g$ --- then ${\betafrak g}$ is equal to the holonomy of the Riemannian metric $g$. In particular this gives a way to construct indecomposable, non-irreducible Lorentzian manifolds with holonomy equal to $ (\betamathbb{R} \betaoplus $Riemannian holonomy$)\betaltimes \betamathbb{R}^n$: Let $(N,g)$ be an $n$-dimensional Riemannian manifold, $\betatheta $ a closed form on $N$ and $q$ a function on $N\betatimes \betamathbb{R}^2$, the latter sufficiently general. Then \beta[(M=N\betatimes \betamathbb{R}^2, h=dxdz + q dz^2 + \betatheta dz+ f (z) g)\beta] is a Lorentzian manifold with holonomy \beta[ \betamathfrak{hol}_{(x,z,p)}(M,h)=(\betamathbb{R} \betaoplus \betamathfrak{hol}_p(N,g))\betaltimes \betamathbb{R}^n.\beta] In case of Riemannian K\beta"ahler- and hyper-K\beta"ahler manifolds $(N,g)$ these conditions can be weakened \betacite{leistner01}. Furthermore there is a method to construct manifolds with coupled holonomy from manifolds with uncoupled holonomy \betacite{boubel00}: If $(M,h)$ is a simply-connected, indecomposable, non-irreducible Lorentzian manifold with uncoupled holonomy $\betalangleg\betaltimes\betamathbb{R}n$ or $(\betamathbb{R}\betaoplus\betalangleg)\betaltimes\betamathbb{R}n$ such that $\betalangleg$ has non-trivial center (and further conditions), then there is a metric $\betatilde{h}$ on $M$ such that $(M,\betatilde{h})$ has holonomy of coupled type and with $\betalangleson$--projection $\betalangleg$. In the following we will go an algebraic way, in order to classify the possible algebras ${\betafrak g}$. This algebraic way uses the Bianchi--identity, restricted to $\betamathbb{R}^n$ as representation space of ${\betafrak g}$. This is the aim of the next sections. \betasigmaubsetsection{Berger and weak-Berger algebras} Here we will introduce the notion of weak-Berger algebras in comparison to Berger algebras. We present some basic properties, in particular a decomposition property and the behavior under complexification. For the details to this section see \betacite{leistner02} Let $E$ be a vector space over the field $\betamathbb{K}$ and let ${\betafrak g}\betasigmaubsetset \betamathfrak{gl}(E) $ be a Lie algebra. Then one defines \betabegin{eqnarray*}gin{align*} {\betacal K}({\betafrak g})&\beta :=\beta \beta{ R\betain \betaLambda^2 E^* \betaotimes {\betafrak g} \beta |\beta R(x,y)z + R(y,z)x + R(z,x)y=0\beta}\beta\beta \betaunderline{{\betafrak g}}&\beta :=\beta span \beta{ R(x,y)\beta |\beta x,y\betain E, R\betain {\betacal K}({\betafrak g})\beta},\beta\beta \betaintertext{and for ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(E,h)$:} {\betacal B}_h({\betafrak g})&\beta :=\beta \beta{ Q\betain E^* \betaotimes {\betafrak g} \beta |\beta h(Q(x)y,z) + h(Q(y)z,x) + h(Q(z)x,y)=0\beta}\beta\beta {\betafrak g}_h&\beta :=\beta span\beta{ Q(x)\beta |\beta x\betain E, Q\betain {\betacal B}_h({\betafrak g})\beta}. \betaend{align*} Then we have the following basic properties. \betabegin{eqnarray*}gin{satz}\betalanglebel{module} $ {\betacal K}({\betafrak g})\betasigmaubsetset \betaLambda^2E^* \betaotimes {\betafrak g}$ and ${\betacal B}_h({\betafrak g}) \betasigmaubsetset E^* \betaotimes {\betafrak g}$ are $ {\betafrak g}$-modules. $ \betaunderline{{\betafrak g}}$ and ${\betafrak g}_h$ are ideals in $\betalangleg$. \betaend{satz} The representation of ${\betafrak g}$ on $ {\betacal B}_h({\betafrak g})$ and $ {\betacal K}({\betafrak g})$ is given by the standard and the adjoint representation \betabegin{eqnarray*}gin{eqnarray} (A \betacdot Q) (x) &=& - Q(Ax) + [A, Q(x)]\betalanglebel{weakaction}\beta\beta (A \betacdot R) (x,y) &=& - R(Ax,y) - R(x,Ay) + [A, R(x,y)]. \betaend{eqnarray} \betabegin{eqnarray*}gin{de}\betalanglebel{weakdef} Let ${\betafrak g}\betasigmaubsetset \betamathfrak{gl}( E)$ be a Lie algebra. Then ${\betafrak g}$ is is called {\betabf Berger algebra} if $\betaunderline{\betafrak g} = {\betafrak g}$. If ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(E,h)$ is an orthogonal Lie algebra with $ {\betafrak g}_h={\betafrak g}$, then we call it {\betabf weak-Berger algebra}. \betaend{de} Equivalent to the (weak-)Berger property is the fact that there is no ideal $\betamathfrak{h}$ in $\betalangleg$ such that ${\betacal K}(\betamathfrak{h})={\betacal K}(\betamathfrak{\betalangleg})$ (resp. ${\betacal B}_h(\betamathfrak{h})= {\betacal B}_h(\betamathfrak{g})$). The notion ``weak-Berger'' is satisfied by the following \betabegin{eqnarray*}gin{satz} Every Berger algebra which is orthogonal is a weak-Berger algebra. \betaend{satz} This proposition has a \betabegin{eqnarray*}gin{folg}\betalanglebel{corollar} Let ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(E,h)$ be an orthogonal Lie algebra. Then \betabegin{eqnarray*}gin{equation} \betalanglebel{folgerung} span \beta{R(x,y)+Q(z)|R\betain {\betacal K}({\betafrak g}), Q\betain {\betacal B}_h({\betafrak g}),x,y,z\betain E \beta} \betasigmaubsetset {\betafrak g}_h. \betaend{equation} \betaend{folg} Concerning the decomposition of Berger and weak-Berger algebras the following proposition holds. \betabegin{eqnarray*}gin{satz} If ${\betafrak g}_1\betasigmaubsetset \betamathfrak{gl}(V_1) $ , ${\betafrak g}_2 \betasigmaubsetset \betamathfrak{gl}(V_2)$ and ${\betafrak g}:= {\betafrak g}_1 \betaoplus {\betafrak g}_2 \betasigmaubsetset \betamathfrak{gl}(V:=V_1 \betaoplus V_2)$, then it holds: \betabegin{eqnarray*}gin{enumerate} \betaitem If ${\betafrak g}_1$ and $ {\betafrak g}_2$ are Berger algebras, then ${\betafrak g}$ is a Berger algebra. \betaitem If in addition ${\betafrak g}_1\betasigmaubsetset \betamathfrak{so}(V_1, h_1) $, ${\betafrak g}_2 \betasigmaubsetset \betamathfrak{so}(V_2,h_2)$ and ${\betafrak g}:= {\betafrak g}_1 \betaoplus {\betafrak g}_2 \betasigmaubsetset \betamathfrak{so}(V:=V_1 \betaoplus V_2, h:= h_1 \betaoplus h_2)$, then holds: ${\betafrak g}_1$ and ${\betafrak g}_2$ are weak-Berger algebras if and only if ${\betafrak g}$ is a weak-Berger algebra. \betaend{enumerate} \betalanglebel{zerlegung} \betaend{satz} The Ambrose-Singer holonomy theorem \betacite{as} then implies that holonomy algebras of torsion free connections --- in particular of a Levi-Civita-connection --- are Berger algebras. The list of all irreducible Berger algebras is known (\betacite{berger55} for orthogonal, non-symmetric Berger algebras, \betacite{berger57} for orthogonal symmetric ones, and \betacite{schwachhoefer1} in the general affine case). We should mention that in our notation Berger algebras are not only non-symmetric Berger algebras, as it is sometimes defined. For us only the possibility of being the holonomy algebra of a Riemannian manifold is of interest, symmetric or non symmetric. The $ {\betafrak s}{\betafrak o}(n)$--projection of an indecomposable, non-irreducible Lorentzian manifold is no holonomy algebra, and therefore not necessarily a Berger algebra. But the following statement, which we proved in \betacite{leistner02}, asserts that it is a weak-Berger algebra. \betabegin{eqnarray*}gin{theo}\betalanglebel{weak-Berger} Let $(M,h)$ be an indecomposable, but non-irreducible, simply connected Lorentzian manifold and ${\betafrak g}=pr_{{\betafrak s}{\betafrak o}(n)}(\betamathfrak{hol}_p(M,h))$. Then $ {\betafrak g}$ is a weak-Berger algebra. \betaend{theo} From point two of proposition \betaref{zerlegung} we get the following \betabegin{eqnarray*}gin{folg} Let $(M,h)$ be an indecomposable, but non-irreducible Lorentzian manifold and ${\betafrak g}=pr_{{\betafrak s}{\betafrak o}(n)}(\betamathfrak{hol}_p(M,h)) \betasigmaubsetset E^* \betaotimes E$ and $ {\betafrak g} = {\betafrak g}_1 \betaoplus \betaldots \betaoplus {\betafrak g}_r$ with $ {\betafrak g}_i\betain \betamathfrak{so}(E_i,h_i)$ the decomposition in irreducible acting ideals from theorem \betaref{theoI}. Then these $ {\betafrak g}_i$ are irreducible weak-Berger algebras. \betaend{folg} This corollary ensures that we are at a similar point as in the Riemannian situation, but reaching it by a different way. This is shown schematically in the following diagram: \beta[ \betabegin{eqnarray*}gin{array}{rccc} \betamathbbox{ geometric level:}& \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(3.1,0.7) \betaput(0,-0.4){\betaframebox(3,1)[tr]{ \betaput(-0.1,-0.1){\betamakebox(0,0)[tr]{{\betasigmacriptsize $\betamathfrak{g}=\betamathfrak{hol}$}}}}} \betaput(0,-0.4){\betaframebox(3,1)[bl]{ \betaput(0.1,0.5){\betamakebox(0,0)[l]{{\betasigmacriptsize $\betamathfrak{g}=$}}} \betaput(0.1,0){\betamakebox(0,0)[bl]{{\betasigmacriptsize $pr_{\betamathfrak{so}(n)}\betamathfrak{hol}$}}}}} \betaput(0,0.6){\betaline(3,-1){3}} \betaend{picture} & \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(2.8,1) \betaput(0,0){\betavarepsilonctor(1,0){2.9}} \betaput(0.2,0.2){\betamakebox(0,0)[l]{{\betasigmacriptsize deRham splitting}}} \betaend{picture} &\betaframebox[3cm]{{\betasigmacriptsize $\betabegin{eqnarray*}gin{array}{l} {\betafrak g}={\betafrak g}_1 \betaoplus \betaldots \betaoplus {\betafrak g}_r,\beta\beta \betamathbbox{with }{\betafrak g}_i=\betamathfrak{hol}_i \betaend{array}$}} \beta\beta \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(1,1) \betaput(0.2,0.4){\betamakebox(0,0)[r]{{\betasigmacriptsize 1. Bianchi identity}}} \betaput(0.4,0.7){\betavarepsilonctor(0,-1){0.8}} \betaend{picture} & \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(1,1) \betaput(1,0.7){\betavarepsilonctor(0,-1){0.8}} \betaput(0.8,0.4){\betamakebox(0,0)[r]{{\betasigmacriptsize theorem \betaref{weak-Berger}}}} \betaend{picture} && \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(1,1) \betaput(0,0.7){\betavarepsilonctor(0,-1){0.8}} \betaput(0.2,0.4){\betamakebox(0,0)[l]{{\betasigmacriptsize Ambrose-Singer}}} \betaend{picture} \beta\beta \betamathbbox{algebraic level:}&\betaframebox[3cm]{{\betasigmacriptsize $\betabegin{eqnarray*}gin{array}{c}\beta\beta{\betafrak g}\betamathbbox{ weak- Berger}\beta\beta \betaend{array}$}}& \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(2.8,1) \betaput(0,0){\betavarepsilonctor(1,0){2.9}} \betaput(0,0.2){\betamakebox(0,0)[l]{{\betasigmacriptsize thm. \betaref{theoI} $+$ prop. \betaref{zerlegung}}}} \betaend{picture} & \betasigmaetlength{\betaunitlength}{1cm} \betabegin{eqnarray*}gin{picture}(3.1,0.6) \betaput(0,-0.4){ \betaframebox(3,1)[tr]{ \betaput(-0.1,-0.1){\betamakebox(0,0)[tr]{{\betasigmacriptsize $\betamathfrak{g}_i$ irreducible}}} \betaput(-0.1,-0.5){\betamakebox(0,0)[r]{{\betasigmacriptsize Berger}}}}} \betaput(0,-0.4){ \betaframebox(3,1)[bl]{ \betaput(0.1,0.4){\betamakebox(0,0)[l]{{\betasigmacriptsize $\betamathfrak{g}_i$ irred.}}} \betaput(0.1,0){\betamakebox(0,0)[bl]{{\betasigmacriptsize weak-Berger}}}}} \betaput(0.1,0.6){\betaline(3,-1){3}} \betaend{picture}\beta\beta &&& \betaend{array}\beta] The proof of the theorem gives another \betabegin{eqnarray*}gin{folg} Let $(M,h)$ be an indecomposable, non-irreducible Lorentzian manifold and ${\betafrak g}= pr_{ \betamathfrak{so}(n)} \betamathfrak{hol}_p(M,h)$. If there exists coordinates $(x, y_1, \betaldots , y_n,z)$ of the above form (i.e. respecting the foliation $\betaXi\betasigmaubsetset \betaXi^\betaperp$), with the property that everywhere holds ${\betacal R}(\betafrac{\betapartial}{\betapartial z}, \betafrac{\betapartial}{\betapartial y_i}, \betafrac{\betapartial}{\betapartial y_j}, \betafrac{\betapartial}{\betapartial y_k})=0$, then ${\betafrak g}$ is a Berger-algebra. \betaend{folg} The aim of the following sections will be to classify all weak-Berger algebras. Before we do this we have to say a word about real and complex (weak-) Berger algebras. \betasigmaubsetsection{Real and complex weak-Berger algebras} Because of the above result we have to classify the real weak-Berger algebras. Since we will use the representation theory of complex semisimple Lie algebras we have to describe the transition of a real weak-Berger algebra to its complexification. First we note that the spaces $ {\betacal K}({\betafrak g})$ and ${\betacal B}_h({\betafrak g})$ for ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(E,h)$ can be described by the following exact sequences: \beta[ \betabegin{eqnarray*}gin{array}{rcccccl} 0 &\betarightarrow & {\betacal K}({\betafrak g}) & \betahookrightarrow& \betaomegaedge^2 E^* \betaotimes {\betafrak g} &\betasigmatackrel{\betalanglembda}{\betatwoheadrightarrow} &\betaomegaedge^3 E^* \betaotimes E\beta\beta 0 &\betarightarrow & {\betacal B}_h({\betafrak g}) & \betahookrightarrow & E^* \betaotimes {\betafrak g} &\betasigmatackrel{\betalanglembda_h}{\betatwoheadrightarrow}& \betaomegaedge^3 E^*, \betaend{array}\beta] where the map $ \betalanglembda$ is the skew-symmetrization and $\betalanglembda_h$ the dualization by $h$ and the skew-symmetrization. If we now consider a real Lie algebra ${\betafrak g}$ acting orthogonal on a real vector space $E$, i.e. ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(E,h)$, then $h$ extends by complexification (linear in both components) to a non-degenerate complex-bilinear form $h^\betamathbb{C}$ which is invariant under ${\betafrak g}^\betamathbb{C}$, i.e. ${\betafrak g}^\betamathbb{C}\betasigmaubsetset \betamathfrak{so}( E^\betamathbb{C}, h^\betamathbb{C})$. Then the complexification of the above exact sequences gives \betabegin{eqnarray*}gin{eqnarray} \betalanglebel{complex1} {\betacal K}({\betafrak g})^\betamathbb{C}&=& {\betacal K}({\betafrak g^\betamathbb{C}})\beta\beta \betalanglebel{complex2} \betaleft({\betacal B}_h({\betafrak g})\betaright)^\betamathbb{C}&=& {\betacal B}_{h^\betamathbb{C}}({\betafrak g^\betamathbb{C}}). \betaend{eqnarray} This implies \betabegin{eqnarray*}gin{satz} \betalanglebel{real-complex} ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(E,h)$ is a (weak-) Berger algebra if and only if ${\betafrak g}^\betamathbb{C}\betasigmaubsetset \betamathfrak{so}(E^\betamathbb{C}, h^\betamathbb{C})$ is a (weak-) Berger algebra. \betaend{satz} I.e. complexification preserves the weak-Berger as well as the Berger property. Because of proposition \betaref{zerlegung} it suffices to classify the real weak-Berger algebras which are irreducible. Now irreducibility is a property which is not preserved under complexification. We have to deal with this problem. At a first step one recalls the following definition, distinguishing two cases for a module of a real Lie algebras. \betabegin{eqnarray*}gin{de} Let $\betalangleg$ be a real Lie algebra. Irreducible real $\betalangleg$-modules $E$ for which $E^\betamathbb{C}$ is an irreducible $\betalangleg$-module and irreducible complex modules $V$ for which $V_\betamathbb{R}$ is a reducible $\betalangleg$-module are called of {\betabf real type}. Irreducible real $\betalangleg$-modules $E$ for which $E^\betamathbb{C}$ is a reducible $\betalangleg$-module and irreducible complex modules $V$ for which $V_\betamathbb{R}$ is a irreducible $\betalangleg$-module are called of {\betabf non-real type}. \betaend{de} This notation corresponds to the distinction of complex irreducible $\betalangleg$-modules into real, complex and quaternionic ones. It makes sense because the complexification of a module of real type is of real type --- recall that $(E^\betamathbb{C})_\betamathbb{R}$ is a reducible $\betalangleg$-module --- and the reellification of a module of non-real type is of non-real type. These relations are described in the appendix \betaref{real representations}. In the original papers of Cartan \betacite{cartan1914} and Iwahori \betacite{iwahori59}, see also \betacite {goto78}, where these distinction is introduced, a representation of real type is called as representation of {\betabf first type} and a representation of non-real type is called of {\betabf second type}. If one now complexifies the Lie algebra $ {\betafrak g}$ too, then $E^\betamathbb{C}$ becomes a $\betalangleg^\betamathbb{C}$--module. This transition preserves irreducibility. \betabegin{eqnarray*}gin{lem} Let $ {\betafrak g}^\betamathbb{C}\betasigmaubsetset \betamathfrak{gl}(V)$ be the complexification of ${\betafrak g}\betasigmaubsetset \betafrak{gl}(V)$ with a complex $\betalangleg$-module $V$. Then it holds: \betabegin{eqnarray*}gin{enumerate} \betaitem ${\betafrak g}$ is irreducible if and only if ${\betafrak g}^\betamathbb{C}$ is irreducible. \betaitem ${\betafrak g}\betasigmaubsetset \betamathfrak{so}(V,H)$ if and only if ${\betafrak g}^\betamathbb{C}\betasigmaubsetset \betamathfrak{so}(V, H)$, where $H$ is a symmetric bilinear form. \betaend{enumerate} \betaend{lem} In the following sections we will describe the weak-Berger property for real and non-real modules of a real Lie algebra $\betalangleg$. \betasigmaection{Weak-Berger algebras of real type} \betalanglebel{sectionzwei} \betalanglebel{complex weak berger} In this section we will make efforts to classify weak-Berger algebras of real type, at least the simple ones. The argumentation in this section is analogously to the reasoning in \betacite{schwachhoefer1}. \betabegin{itemize}gskip $\betalangleg_0$ shall be a real Lie algebra and $E$ a real irreducible module of real type. Furthermore we suppose $\betalangleg_0\betain \betalangleso(E,h)$ with $h$ positive definite. Then $E^\betamathbb{C}$ is an irreducible $\betalangleg_0$-module (also of real type). If $h^\betamathbb{C}$ denotes the complexification of $h$, bilinear in both components we have that $\betalangleg_0\betasigmaubsetset \betalangleso(E^\betamathbb{C},h^\betamathbb{C})$. Now we can extend $h$ also sesqui-linear on $E^\betamathbb{C}$ and get a hermitian form $\betatheta^h$ on $V$ which is invariant under $\betalangleg_0$. Thus we have $\betalangleg_0\betasigmaubsetset \betamathfrak{u}(V,\betatheta^h).$ $\betatheta^h$ has the same index as $h$ (see appendix \betaref{real representations}). Since the bilinear form $h$ we start with is positive definite we can make another simplification. Subalgebras of $\betalangleso(E,h)$ with positive definite $h$ are compact and therefore reductive. I.e. its Levi-decomposition is $\betalangleg_0=\betamathfrak{z}_0\betaoplus\betamathfrak{d}_0$, with center $\betamathfrak{z}_0$ and semisimple derived algebra $\betamathfrak{d}_0$. Thus $\betalangleg_0^\betamathbb{C}=\betamathfrak{z}\betaoplus \betamathfrak{d}$ is also reductive. But since it is irreducible by assumption, the Schur lemma implies that the center $\betamathfrak{z}$ is $\betamathbb{C}\beta Id$ or zero. But $\betamathbb{C} \beta Id$ is not contained in $\betalangleso(V,H)$. Thus the center has to be zero and $\betalangleg$ is semisimple. Proposition \betaref{real-complex} gives the following. \betabegin{eqnarray*}gin{satz} If ${\betafrak g}_0\betasigmaubsetset \betamathfrak{so}(E,h)$ is a weak-Berger algebra of real type then, $ {\betafrak g}_0^\betamathbb{C}\betasigmaubsetset \betamathfrak{so}(E^\betamathbb{C}, h^\betamathbb{C})$ is an irreducible weak-Berger algebra. $E^\betamathbb{C}$ is a $\betalangleg_0$-module of real type and if $h$ is positive definite then $\betalangleg_0^\betamathbb{C} $ is semisimple. If $ {\betafrak g}\betasigmaubsetset \betamathfrak{so}(V, H)$ is an irreducible complex weak-Berger which is semisimple. Then $\betalangleg$ has a compact real form $\betalangleg_0$ and if $V$ is a $\betalangleg_0$-module of real type, then $V=E^\betamathbb{C}$, $\betalangleg_0$ is unitary with respect to a hermitian form $\betatheta$ and ${\betafrak g}_0\betasigmaubsetset \betamathfrak{so}(E,h)$ is a weak-Berger algebra of real type. The indices of $h$ and $\betatheta$ are equal. \betaend{satz} \betabegin{eqnarray*}gin{proof} The first direction follows obviously from proposition \betaref{real-complex}. That $E^\betamathbb{C}$ is a module of real type holds because of $(E^\betamathbb{C})_\betamathbb{R}$ is reducible (see appendix proposition \betaref{realtype1}). Since $\betalangleg$ is semisimple it has a compact real form $\betalangleg_0$. If $V$ is a $\betalangleg_0$-module of real type, then $\betalangleg_0$ is unitary since it is orthogonal (see proposition \betaref{dualtype}) and it is $V=E^\betamathbb{C}$ (proposition \betaref{realtype1}). By proposition \betaref{realtypeorthogonal} follows that $\betalangleg_0$ is orthogonal w.r.t. $h$ which has the same index as $\betatheta$. Then the proposition follows by proposition \betaref{real-complex}. \betaend{proof} The main point of this proposition is the implication that if $\betalangleg_0\betasigmaubsetset\betalangleso(E,h)$ is weak-Berger of real type, then $\betalangleg_0^\betamathbb{C}\betasigmaubsetset\betalangleso(E^\betamathbb{C}, h^\betamathbb{C})$ is an irreducible acting, complex semisimple weak-Berger algebra. These we have to classify. \betabegin{bem} Before we start we have to make a remark about definition of holonomy up to conjugation. The $SO(n)$--component of an indecomposable, non-irreducible Lorentzian manifold was defined modulo conjugation in $O(n)$. Hence we shall not distinct between subalgebras of $\betamathfrak{gl}(n,\betamathbb{C})$ which are isomorphic under $Ad_\betavarphi$ where $\betavarphi$ is an element from $O(n,\betamathbb{C})$ and $Ad$ the adjoint action in of $Gl(n,\betamathbb{C})$ on $\betamathfrak{gl}(n,\betamathbb{C})$. We say that an orthogonal representation $\betakappa_1$ of a complex semisimple Lie algebra $\betalangleg$ is {\betabf congruent} to an orthogonal representation $\betakappa_2$ if there is an element $\betavarphi\betain O(n,\betamathbb{C})$ such that the following equivalence of $\betalangleg$--representations is valid: $\betakappa_1\betasigmaim Ad_\betavarphi\betacirc\betakappa_2$. Hence we have to classify semisimple, orthogonal, irreducible acting, complex weak-Berger algebras of real type up to this congruence of representations. If the automorphism $Ad_\betavarphi$ is inner, then the representations are equivalent, if it is outer then only congruent. For semisimple Lie algebras it holds that $Out(\betalangleg):=Aut(\betalangleg)/Inn(\betalangleg)$ counts the connection components of $Aut(\betalangleg)$ and (see for example \betacite{onishchik-vinberg3}) $Out(\betalangleg)$ is isomorphic to the automorphism of the fundamental system, i.e. symmetries of the Dynkin diagram. \beta\beta[.2cm] \betabegin{eqnarray*}gin{minipage}[b]{10.5cm}{ For us this becomes relevant in case of $\betalangleso(8,\betamathbb{C})$. In the picture one sees that the symmetries of the Dynkin diagram generate the symmetric group ${\betacal S}_3$, i.e. $Out(\betalangleso(8,\betamathbb{C}))={\betacal S}_3$ and it contains the so-called ``triality automorphism'' which interchanges vector and spin representations of $ \betalangleso(8,\betamathbb{C})$ without fixing one. } \betaend{minipage} \betahfill \betabegin{eqnarray*}gin{minipage}[b]{3.5cm}{ \betabegin{eqnarray*}gin{picture}(0,0) \betaincludegraphics{trias.pstex} \betaend{picture} \betasigmaetlength{\betaunitlength}{4144sp} \betabegin{eqnarray*}gingroup\betamakeatletter\betaifx\betaSetFigFont\betaundefined \betagdef\betaSetFigFont#1#2#3#4#5{ \betareset@font\betafontsize{#1}{#2pt} \betafontfamily{#3}\betafontseries{#4}\betafontshape{#5} \betasigmaelectfont} \betafi\betaendgroup \betabegin{eqnarray*}gin{picture}(1355,1110)(86,-556) \betaput(1396,389){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{10}{12.0}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaDeltaelta_8^+$}}} \betaput(1441,-556){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{10}{12.0}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaDeltaelta_8^-$}}} \betaput( 86,-26){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{10}{12.0}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betamathbb{C}^8$}}} \betaend{picture} } \betaend{minipage}\beta\beta[.1cm] We will use that the automorphism which interchanges the vector representation with one spinor representation and fixes the second spinor representation resp. interchanges the spinor representations and fixes the vector representation comes from $Ad_\betavarphi$ with $\betavarphi\betain O(n,\betamathbb{C})$. Hence the vector and the spinor representations of $\betalangleso(8,\betamathbb{C})$ are congruent to each other. Finally we should remark that compact real forms equivalent to a given one correspond to inner automorphism of $\betalangleg$. Hence the corresponding representations are equivalent \betaend{bem} \betasigmaubsetsection{Irreducible, complex, orthogonal, semisimple Lie algebras} In the following $V$ will be a complex vector space equipped with a non-degenerate symmetric bilinear 2--form $H$. $\betalangleg$ shall be an irreducible acting, complex, semisimple subalgebra of $\betalangleso(V,H)$. Thus all the tools of root space decomposition and representation theory will apply. Let $\betamathfrak{t}$ be the Cartan subalgebra of $\betalangleg$. We denote by $\betaDeltaelta\betasigmaubsetset \betamathfrak{t}^*$ the roots of $\betalangleg$ and we set $\betaDeltaelta_0:= \betaDeltaelta\betacup \beta{0\beta}$. Then $\betalangleg $ decomposes into its root spaces $\betalangleg_\betaalpha:=\beta{ A\betain \betalangleg| [T, A]=\betaalpha(T) \betacdot A\betamathbbox{ for all } T\betain \betamathfrak{t}\beta}\betanot=\beta{0\beta}$. It is \beta[\betalangleg =\betabegin{itemize}goplus_{\betaalpha\betain \betaDeltaelta_0} \betalangleg_\betaalpha \beta;\beta;\beta;\betamathbbox{ where $\betalangleg_0=\betamathfrak{t}$.}\beta] By $\betaOmega\betasigmaubsetset \betamathfrak{t}^*$ we denote the weights of $\betalangleg\betasigmaubsetset \betalangleso (V,H)$. Then $V$ decomposes into the weight spaces $V_\betamu:=\beta{ v\betain V| T(v)=\betamu(T)\betacdot v\betamathbbox{ for all } T\betain \betamathfrak{t}\beta}\betanot=\beta{0\beta}$, i.e. \beta[V=\betabegin{itemize}goplus_{\betamu\betain \betaOmega} V_\betamu.\beta] Now the following holds. \betabegin{eqnarray*}gin{satz} \betalanglebel{ortogonal} Let $ {\betafrak g}\betasigmaubsetset \betalangleso (V,H)$ be a complex, semisimple Lie algebra with weight space decomposition. Then \beta[ V( \betamu)\betabot V(\betalanglembda)\betamathbbox{ if and only if } \betalanglembda\betanot= - \betamu.\beta] In particular if $\betamu$ is a weight, then $-\betamu$ too. \betaend{satz} \betabegin{eqnarray*}gin{proof} For any $T\betain \betamathfrak{t}$, $u\betain V_\betamu $ and $v\betain V_\betalanglembda$ we have \beta[0= H(Tu,v)+H(u,Tv)=\betaleft(\betamu(T) + \betalanglembda(T)\betaright) H(u,v).\beta] Now if $\betalanglembda\betanot=-\betamu$ there is a $T$ such that $\betamu(T) + \betalanglembda(T)\betanot=0$. But this implies $V_\betalanglembda \betabot V_\betamu$. On the other hand $V_\betamu\betabot V_{-\betamu}$ would imply $V_\betamu\betabot V$ which contradicts the non-degeneracy of $H$. Its non-degeneracy also implies that $\betamu\betain \betamathfrak{t}^*$ is a weight if and only if $- \betamu$ is a weight. \betaend{proof} \betasigmaubsetsection{Irreducible complex weak-Berger algebras} We will now draw consequences from the weak-Berger property. Therefore we consider the space ${\betacal B}_H(\betalangleg)$ defined by the Bianchi identity. If $\betalangleg$ is weak-Berger it has to be non-zero, i.e. by proposition \betaref{module} it is a non-zero $\betalangleg$--module. If we denote by $\betaPi$ all its weights then it decomposes into weight spaces \beta[ {\betacal B}_H(\betalag)\beta = \beta \betabegin{itemize}goplus_{\betaphi\betain \betaPi}{\betacal B}_\betaphi.\beta] If $\betaOmega$ are the weights of $V$ then we define the following set \beta[\betaGamma := \betaleft\beta{\beta \betamu+\betaphi\beta \betaleft| \betabegin{eqnarray*}gin{array}{l} \betamu\betain \betaOmega,\beta \betaphi \betain \betaPi\beta \betamathbbox{ and there is an $u\betain V_\betamu$}\beta\beta \betamathbbox{and a $Q\betain {\betacal B}_\betaphi$ such that $Q(u)\betanot=0$} \betaend{array}\betaright.\betaright\beta}\betasigmaubsetset \betamathfrak{t}^*.\beta] Then one proves a \betabegin{eqnarray*}gin{lem} $\betaGamma\betasigmaubsetset \betaDeltaelta_0$. \betaend{lem} \betabegin{eqnarray*}gin{proof} We have to show that every $\betamu+\betaphi\betain \betaGamma$ is a root of $\betalangleg$. Therefore we consider weight elements $Q_\betaphi\betain {\betacal B}_\betaphi$ and $u_\betamu\betain V_\betamu$ with $0\betanot=Q_\betaphi(u_\betamu)$. Then for every $T\betain \betamathfrak{t}$ holds (because of the definition of the $\betalangleg$-module ${\betacal B}_H(\betalag)$): \betabegin{eqnarray*} \betaleft[ T, Q_\betaphi(u_\betamu)\betaright]&=& (TQ_\betaphi)(u_\betamu)+ Q_\betaphi(T(u_\betamu))\beta\beta &=&\betaleft(\betaphi(T) + \betamu(T)\betaright) Q_\betaphi(u_\betamu) \betaend{eqnarray*} I.e. $\betaphi+\betamu$ is a root or zero. \betaend{proof} For weak-Berger algebras now the other inclusion is true. \betabegin{eqnarray*}gin{satz} If $\betalangleg\betasigmaubsetset \betalangleso(V,h)$ is irreducible, semisimple Lie algebra. If it is weak-Berger then $\betaGamma =\betaDeltaelta_0$. If $\betaGamma =\betaDeltaelta_0$ and $span\beta{Q_{\beta-\betamu}(u_\betamu)\beta |\beta \betamu\betain\betaOmega\beta}=\betamathfrak{t}$ then it is weak-Berger. \betaend{satz} \betabegin{eqnarray*}gin{proof} The decomposition of ${\betacal B}_H(\betalag)$ and $V$ into weight spaces and the fact that $Q_\betaphi(u_\betamu)\betain \betalangleg_{\betaphi+\betamu}$ imply the following inclusion \beta[\betalangleg_H\beta =\beta span\beta{Q_\betaphi(u_\betamu)\beta |\beta \betaphi+ \betamu\betain \betaGamma\beta}\betasigmaubsetset \betabegin{itemize}goplus_{\betabegin{eqnarray*}ta\betain \betaGamma}\betalangleg_\betabegin{eqnarray*}ta.\beta] But if $\betalangleg=\betabegin{itemize}goplus_{\betaalpha\betain \betaDeltaelta_0}\betalangleg_\betaalpha$ is weak-Berger it holds that $\betalangleg \betasigmaubsetset \betalangleg_H$ and thus \beta[\betabegin{itemize}goplus_{\betaalpha\betain \betaDeltaelta_0}\betalangleg_\betaalpha\betasigmaubsetset \betabegin{itemize}goplus_{\betabegin{eqnarray*}ta\betain \betaGamma}\betalangleg_\betabegin{eqnarray*}ta \betasigmaubsetset \betabegin{itemize}goplus_{\betaalpha\betain \betaDeltaelta_0}\betalangleg_\betaalpha. \beta] But this implies $\betaGamma = \betaDeltaelta_0$. If now $\betaGamma = \betaDeltaelta_0$ and $span\beta{Q_{\beta-\betamu}(u_\betamu)\beta |\beta \betamu\betain\betaOmega\beta}=\betamathfrak{t}$ we have that \beta[\betalangleg_H\beta =\beta span\beta{Q_\betaphi(u_\betamu)\beta |\beta \betaphi+ \betamu\betain \betaGamma\beta}= \betabegin{itemize}goplus_{\betabegin{eqnarray*}ta\betain \betaGamma}\betalangleg_\betabegin{eqnarray*}ta= \betamathfrak{t}\betaoplus\betabegin{itemize}goplus_{\betabegin{eqnarray*}ta\betain \betaDeltaelta}\betalangleg_\betabegin{eqnarray*}ta =\betalangleg .\beta] This completes the proof.\betaend{proof} To derive necessary conditions for the weak Berger property we have to fix a notation. Let $\betaalpha\betain \betaDeltaelta$ be a root. Then we denote by $\betaOmega_\betaalpha$ the following subset of $\betaOmega$: \beta[\betaOmega_\betaalpha:= \betaleft\beta{ \betalanglembda\betain \betaOmega\beta |\beta \betalanglembda+\betaalpha\betain \betaOmega\betaright\beta}.\beta] Then of course $\betaalpha+\betaOmega_\betaalpha$ are the weights of $\betalangleg_\betaalpha V$. \betabegin{eqnarray*}gin{satz}\betalanglebel{musatz} Let $\betalangleg$ be a semisimple Lie algebra with roots $\betaDeltaelta$ and $\betaDeltaelta_0=\betaDeltaelta\betacup\beta{0\beta}$. Let $\betalangleg\betasigmaubsetset \betalangleso(V,H)$ irreducible, weak-Berger with weights $\betaOmega$. Then the following properties are satisfied: \betabegin{eqnarray*}gin{description} \betaitem[(PI)] There is a $\betamu\betain \betaOmega$ and a hyperplane $U\betasigmaubsetset \betamathfrak{t}^*$ such that \betabegin{eqnarray*}q \betalanglebel{mu} \betaOmega\betasigmaubsetset \betaleft\beta{ \betamu+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup U\betacup \betaleft\beta{- \betamu+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}. \betaend{eqnarray*}q \betaitem[(PII)] For every $\betaalpha\betain \betaDeltaelta$ there is a $\betamu_\betaalpha\betain \betaOmega$ such that \betabegin{eqnarray*}q \betalanglebel{mua} \betaOmega_\betaalpha\betasigmaubsetset \betaleft\beta{ \betamu_\betaalpha-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{ -\betamu_\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}. \betaend{eqnarray*}q \betaend{description} \betaend{satz} \betabegin{eqnarray*}gin{proof} If $\betalangleg$ is weak-Berger we have $\betaGamma=\betaDeltaelta_0$. We will use this property for $0\betain\betaDeltaelta_0$ as well as for every $\betaalpha\betain \betaDeltaelta$. {\betabf (PI)} $\betaGamma=\betaDeltaelta_0$ implies that there is $\betaphi\betain \betaPi$ and $\betamu\betain \betaOmega$ such that $0=\betaphi+\betamu$ with $Q\betain {\betacal B}_\betaphi$ and $ u\betain V_\betamu$ such that $0\betanot= Q(u)\betain \betamathfrak{t}$, i.e. $\betaphi=-\betamu \betain \betaPi$. We fix such $u, Q $ and $\betamu$. For arbitrary $\betalanglembda\betain \betaOmega$ then occur the following cases: \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1:}} There is a $v_+\betain V_\betalanglembda$ such that $Q(v_+)\betanot=0$ or a $v_-\betain V_{-\betalanglembda}$ such that $Q(v_-)\betanot=0$. This implies $-\betamu+\betalanglembda \betain \betaDeltaelta_0$ or $-\betamu-\betalanglembda \betain \betaDeltaelta_0$, i.e. $\betalanglembda\betain \betaleft\beta{\betamu+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain\betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{-\betamu+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain\betaDeltaelta_0\betaright\beta}$. \betaitem{{\betaem Case 2:}} For all $v\betain V_\betalanglembda \betaoplus V_{-\betalanglembda}$ holds $Q(v)=0$. Then the Bianchi identity implies for $v_+\betain V_\betalanglembda$ and $v_-\betain V_{-\betalanglembda}$ that $0=\betalanglembda( Q(u))H(v_+,v_-).$ Now one can choose $v_+$ and $v_-$ such that $H(v_+,v_-)\betanot=0$. This implies $\betalanglembda\betain Q(u)^\betabot=:U$ and we get (PI). \betaend{description} {\betabf (PII)} Let $\betaalpha\betain \betaDeltaelta$. $\betaGamma=\betaDeltaelta_0$ implies the existence of $\betaphi\betain \betaPi$ and $\betamu_\betaalpha\betain \betaOmega$ such that $\betaalpha=\betaphi+\betamu_\betaalpha$ with $Q\betain {\betacal B}_\betaphi$ and $ u\betain V_{\betamu_\betaalpha}$ such that $0\betanot = Q(u)\betain \betalangleg_\betaalpha$. We fix $Q$ and $u$ for $\betaalpha$ and have that $\betaalpha-\betamu_\betaalpha=\betaphi\betain \betaPi$ a weight of ${\betacal B}_H$. Let now $\betalanglembda$ be a weight in $ \betaOmega_\betaalpha$, i.e. $\betalanglembda+\betaalpha$ is also a weight. Thus $-\betalanglembda-\betaalpha$ is a weight. If $v\betain V_\betalanglembda$ then $Q(u)v\betain V_{\betalanglembda+\betaalpha}$. Since $H$ is non-degenerate there is a $w\betain V_{-\betalanglembda-\betaalpha}$ such that $H(Q(u)v,w )\betanot=0$. Since $Q\betain {\betacal B}_H(\betalangleg)$ the Bianchi identity then gives \beta[0=H(Q(u)v,w )+H(Q(v)w,u)+H(Q(w)u,v),\beta] i.e. at least one of $Q(v)$ or $Q(w)$ has to be non-zero. Hence we have two cases for $\betalanglembda\betain \betaOmega_\betaalpha$: \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1:}} $Q(v)\betanot=0$. This implies $-\betamu_\betaalpha+\betaalpha+\betalanglembda \betain \betaDeltaelta_0$, i.e. $\betalanglembda\betain \betaleft\beta{\betamu_\betaalpha-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain\betaDeltaelta_0\betaright\beta}$. \betaitem{{\betaem Case 2:}} $Q(w)\betanot=0$. This implies $-\betamu_\betaalpha+\betaalpha-\betalanglembda-\betaalpha=-\betamu_\betaalpha-\betalanglembda \betain \betaDeltaelta_0$, i.e. $\betalanglembda\betain \betaleft\beta{-\betamu_\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain\betaDeltaelta_0\betaright\beta}$. \betaend{description} But this is (PII). \betaend{proof} Of course it is desirable to find weights $\betamu$ and $\betamu_\betaalpha$ which are extremal in order to handle criteria (PI) and (PII). To show in which sense this is possible we need a \betabegin{eqnarray*}gin{lem} \betalanglebel{nulllemma} Let $\betalangleg\betasigmaubsetset \betalangleso(V,H)$ an irreducible acting, complex semisimple Lie algebra. For an extremal weight vector $u\betain V_\betaLambda$ there is a weight element $Q\betain {\betacal B}_H(\betalangleg)$ such that $Q(u)\betanot=0$. \betaend{lem} \betabegin{eqnarray*}gin{proof} Let $u\betain V_\betaLambda$ be extremal with $Q(u)=0$ for every weight element $Q$. Since ${\betacal B}_H(\betalag)\beta = \beta \betabegin{itemize}goplus_{\betaphi\betain \betaPi}{\betacal B}_\betaphi$ the assumption implies $Q(u)=0$ for all $Q\betain {\betacal B}_H(\betalag)$. But this gives for every $A\betain \betalangleg$ and every weight element $Q$ that \beta[Q(A u)\beta = \beta \betaleft[A,Q(u)\betaright]- \betaunderbrace{(A\betacdot Q)}_{\betain {\betacal B}_H(\betalag)} (u) \beta =\beta 0.\beta] On the other hand $V$ is irreducible and thats why generated as vector space by elements of the form $A_1\betacdot \betaldots \betacdot A_k\betacdot u$ with $A_i\betain\betalangleg$ and $k\betain\betamathbb{N}$ (see for example \betacite{serre87}). By successive application of $\betalangleg$ to $u$ we get that $Q(v)=0$ for every weight element $Q$ and every weight vector $v$. But this gives $Q(v)=0$ for all $Q\betain {\betacal B}_H(\betalag)$ and every $v\betain V$, hence ${\betacal B}_H(\betalag)=0$. \betaend{proof} \betabegin{eqnarray*}gin{satz}\betalanglebel{gewichtssatz} Let $\betalangleg$ be a semisimple Lie algebra with roots $\betaDeltaelta$ and $\betaDeltaelta_0=\betaDeltaelta\betacup\beta{0\beta}$. Let $\betalangleg\betasigmaubsetset \betalangleso(V,H)$ irreducible, weak-Berger with weights $\betaOmega$. Then there is an ordering of $\betaDeltaelta$ such that the following holds: If $\betaLambda $ is the highest weight of $\betalangleg\betasigmaubsetset\betalangleso(V,H)$ with respect to that ordering, then the following properties are satisfied: \betabegin{eqnarray*}gin{description} \betaitem[(QI)] There is a $\betadelta\betain \betaDeltaelta_+\betacup\beta{0\beta}$ and a hyperplane $U\betasigmaubsetset \betamathfrak{t}^*$ such that \betabegin{eqnarray*}q \betalanglebel{highest} \betaOmega\betasigmaubsetset \betaleft\beta{ \betaLambda-\betadelta+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup U\betacup \betaleft\beta{ -\betaLambda+\betadelta+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}. \betaend{eqnarray*}q \betaend{description} If $\betadelta$ can not be chosen to be zero, then holds \betabegin{eqnarray*}gin{description} \betaitem[(QII)] There is an $\betaalpha\betain \betaDeltaelta$ such that \betabegin{eqnarray*}q \betalanglebel{qii} \betaOmega_\betaalpha\betasigmaubsetset \betaleft\beta{ \betaLambda-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{ -\betaLambda+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}. \betaend{eqnarray*}q \betaend{description} \betaend{satz} \betabegin{eqnarray*}gin{proof} First we consider the extremal weights of the representation, i.e. the images of the highest weight under the Weyl group. These do not lie in one hyper plane (because this would imply that all roots lie in one hyperplane). Thus by proposition \betaref{musatz} --- fixing $\betamu\betain \betaOmega$ --- there is an extremal weight $\betaLambda$ with $\betaLambda+\betamu\betain \betaDeltaelta_0$ or $\betaLambda-\betamu\betain \betaDeltaelta_0$. This one we fix. Since the Weyl group acts transitively on the extremal weights we can find a fundamental root system, i.e. an ordering on the roots, such that $\betaLambda$ is the highest weight. With respect to this fundamental root system the roots splits into positive and negative roots $\betaDeltaelta=\betaDeltaelta_+ \betacup\betaDeltaelta_-$. This implies \betabegin{eqnarray*}q\betalanglebel{lambda-mu} \betamu=\betavarepsilon (\betaLambda - \betadelta) \betaend{eqnarray*}q with $\betadelta\betain \betaDeltaelta_+$ and $\betavarepsilon=\betapm 1$. For an arbitrary $\betalanglembda\betain \betaOmega$ then holds $\betalanglembda\betain U=Q(u)^\betabot$ or $\betalanglembda+\betamu\betain \betaDeltaelta_0 $ or $\betalanglembda-\betamu\betain \betaDeltaelta_0$. But with (\betaref{lambda-mu}) this implies that we find an $\betabegin{eqnarray*}ta\betain \betaDeltaelta_0$ such that $\betalanglembda = \betapm(\betaLambda-\betadelta) + \betabegin{eqnarray*}ta$ with $\betabegin{eqnarray*}ta \betain \betaDeltaelta_0$. This is (QI). Note that we are still free to choose $\betaLambda$ or $-\betaLambda$ as highest weight. Now we suppose that $\betadelta$ can not be chosen to be zero. Let $v\betain V_\betaLambda$ be a highest weight vector or $v\betain V_{-\betaLambda}$. Looking at the proof of proposition \betaref{musatz} one has that for all weight elements $Q\betain {\betacal B}_h(\betalangleg)$ holds $Q(v)\betain \betalangleg_\betaalpha$ for a $\betaalpha\betain \betaDeltaelta$. Since $\betalangleg$ is weak-Berger ${\betacal B}_H(\betalag)$ is non-zero. Thus we get by lemma \betaref{nulllemma} that there is a weight element $Q$ such that $0\betanot=Q(v)\betain \betalangleg_\betaalpha$ and we are done (possibly by making $-\betaLambda$ to the highest weight). \betaend{proof} \betaparagraph{Representations of $\betamathfrak{sl}(2,\betamathbb{C})$} To illustrate how these criteria shall work we apply them to irreducible representations of $\betamathfrak{sl}(2,\betamathbb{C})$. \betabegin{eqnarray*}gin{satz} Let $V$ be an irreducible, complex, orthogonal $\betamathfrak{sl}(2,\betamathbb{C})$--module of highest weight $\betaLambda$. If it is weak-Berger then $\betaLambda \betain \beta{2,4\beta}$. \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betamathfrak{sl}(2,\betamathbb{C})\betasigmaubsetset \betalangleso (N, \betamathbb{C})$ be an irreducible representation of highest weight $\betaLambda$. I.e. $\betaLambda(H)=l\betain \betamathbb{N}$ for $\betamathfrak{sl}(2,\betamathbb{C})=span (H,X,Y)$ where $X$ has the root $\betaalpha$. Since the representation is orthogonal, $l$ must be even and $0$ is a weight. The hypersurface $U$ is the point $0$. Now property (\betaref{mu}) ensures that $l\betain \beta{2,4,6\beta}$. If $\betamu=\betaLambda$ we obtain $l\betain\beta{2,4\beta}$. If $\betamu\betanot=\betaLambda$ we can apply (QII): We have that $\betaOmega_\betaalpha=\betaOmega\betasigmaetminus \beta{\betaLambda\beta}$ and $\betaOmega_{-\betaalpha}=\betaOmega\betasigmaetminus \beta{-\betaLambda\beta}$. Then (QII) implies that $l\betain\beta{2,4\beta}$. \betaend{proof} So we get the first result. \betabegin{eqnarray*}gin{folg} Let $\betamathfrak{su}(2)\betasigmaubsetset \betalangleso(E,h)$ be a real irreducible weak-Berger algebra of real type. Then it is a Berger algebra. In particular it is equivalent to the Riemannian holonomy representations of $\betalangleso(3,\betamathbb{R})$ on $\betamathbb{R}^3$ or of the symmetric space of type $AI$, i.e. $\betamathfrak{su}(3)/\betalangleso(3,\betamathbb{R})$ in the compact case or $\betamathfrak{sl}(3,\betamathbb{R})/\betalangleso(3,\betamathbb{R})$ in the non-compact case. \betaend{folg} \betasigmaubsetsection{Berger algebras, weak Berger algebras and spanning triples} In this section we will describe a result of \betacite{schwachhoefer1} and \betacite{schwachhoefer2}, where holonomy groups of torsionfree connections, i.e. Berger algebras, are classified. We will describe our results in their language such that we can use a partial result of \betacite{schwachhoefer2}. For a Berger algebra holds that for every $\betaalpha \betain \betaDeltaelta_0$ there is a weight element $R\betain {\betacal K}(\betalangleg)$ and weight vectors $u_1\betain V_{\betamu_1}$ and $u_2\betain V_{\betamu_2}$ such that $0\betanot= R(u_1,u_2)\betain \betalangleg_\betaalpha$. The Bianchi identity then gives for an arbitrary $v\betain V$ \beta[R(u_1,u_2)v\beta =\beta R(v,u_2)u_1 + R(u_1,v)u_2.\beta] Choosing now $u_1,u_2$ such that $0\betanot=R(u_1,u_2)\betain \betamathfrak{t}$ one gets for any $\betalanglembda\betain \betaOmega$ and $v\betain V_\betalanglembda$ that \beta[\betalanglembda(R(u_1,u_2))v\beta =\beta R(v,u_2)u_1 + R(u_1,v)u_2.\beta] This implies $\betalanglembda\betain \betaleft(R(u_1,u_2)\betaright)^\betabot\betasigmaubsetset \betamathfrak{t}^*$ or $V_\betalanglembda\betasigmaubsetset \betalangleg V_{\betamu_1}\betaoplus \betalangleg V_{\betamu_2}$. This gives property \betabegin{eqnarray*}gin{description} \betaitem[(RI)] There are weights $\betamu_1,\betamu_2\betain \betaOmega$ such that \beta[\betaOmega\betasigmaubsetset \beta{\betamu_1+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}\betacup U\betacup \beta{\betamu_2+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}.\beta] \betaend{description} If one chooses $u_1,u_2$ such that $0\betanot=R(u_1,u_2)=A_\betaalpha\betain \betalangleg_\betaalpha$ with $\betaalpha\betain \betaDeltaelta$ then one gets for $\betalanglembda\betain \betaOmega$ that $A_\betaalpha V_\betalanglembda\betasigmaubsetset \betalangleg V_{\betamu_1}\betaoplus\betalangleg V_{\betamu_2}$. This means that the weights of $A_\betaalpha V_\betalanglembda$ are contained in $\beta{\betamu_1+\betabegin{eqnarray*}ta|\betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}\betacup \beta{\betamu_2+\betabegin{eqnarray*}ta|\betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}$. But this is property \betabegin{eqnarray*}gin{description} \betaitem[(RII)] For every $\betaalpha\betain \betaDeltaelta$ there are weights $\betamu_1,\betamu_2\betain \betaOmega$ such that \beta[\betaOmega_\betaalpha\betasigmaubsetset \beta{\betamu_1-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta} \betacup \beta{\betamu_2-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}.\beta] \betaend{description} Of course our (PI) is a special case of (RI) with $\betamu_1=-\betamu_2$. (PII) is not a special case of (RII) since $\betamu_\betaה+\betaה$ is not a weight apriori. To describe this situation further in \betacite{schwachhoefer2} the following definitions are made. We point out that here $\betaOmega_\betaalpha$ does not denote the weights of $\betalangleg_\betaalpha V$ but the weights $\betalanglembda$ of $V$ such that $\betalanglembda+\betaalpha$ is a weight. \betabegin{eqnarray*}gin{de} Let $\betalangleg\betasigmaubsetset End(V) $ be an irreducible acting complex Lie algebra, $\betaDeltaelta_0$ be the roots and zero of the semisimple part of $\betalangleg$, $\betaOmega$ the weights of $\betalangleg$ and $\betaOmega_\betaalpha$ as above. \betabegin{enumerate} \betaitem A triple $(\betamu_1,\betamu_2,\betaalpha)\betain \betaOmega\betatimes\betaOmega\betatimes\betaDeltaelta$ is called {\betabf spanning triple} if \beta[\betaOmega_\betaalpha \betasigmaubsetset \betaleft\beta{ \betamu_1-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{ \betamu_2-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}.\beta] \betaitem A spanning triple $(\betamu_1,\betamu_2,\betaalpha)$ is called {\betabf extremal} if $\betamu_1$ and $\betamu_2$ are extremal. \betaitem A triple $(\betamu_1,\betamu_2, U)$ with $\betamu_1,\betamu_2$ extremal weights and $U$ an affine hyperplane in $\betamathfrak{t}^*$ is called {\betabf planar spanning triple} if every extremal weight different from $\betamu_1$ and $\betamu_2$ is contained in $U$ and $\betaOmega\betasigmaubsetset \betaleft\beta{ \betamu_1+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup U \betacup \betaleft\beta{ \betamu_2+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}$. \betaend{enumerate} \betaend{de} From (RI) and (RII) in \betacite{schwachhoefer2} the following proposition is deduced. \betabegin{eqnarray*}gin{satz} \betacite{schwachhoefer2} Let $\betalangleg\betasigmaubsetset End(V)$ be an irreducible complex Berger algebra. Then for every root $\betaalpha\betain \betaDeltaelta$ there is a spanning triple. Furthermore there is an extremal spanning triple or a planar spanning triple. \betaend{satz} If we return to the weak-Berger case we can reformulate proposition \betaref{gewichtssatz} as follows. \betabegin{eqnarray*}gin{satz}\betalanglebel{triplesatz} Let $\betalangleg\betasigmaubsetset \betalangleso(V,H)$ be an irreducible complex weak-Berger algebra. Then there is an extremal weight $\betaLambda$ such that one of the following properties is satisfied: \betabegin{eqnarray*}gin{description} \betaitem[(SI)] There is a planar spanning triple of the form $(\betaLambda,-\betaLambda, U)$. \betaitem[(SII)] There is an $\betaה\betain \betaDeltaelta$ such that $ \betaOmega_\betaalpha\betasigmaubsetset \betaleft\beta{ \betaLambda-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{ -\betaLambda+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}. $ \betaend{description} There is a fundamental system such that the extremal weight in (SI) and (SII) is the highest weight. \betaend{satz} \betabegin{eqnarray*}gin{proof} The proof is analogous the the one of proposition \betaref{gewichtssatz}. If there is an $\betaalpha\betain \betaDeltaelta$ such that the corresponding $\betamu_\betaalpha$ is extremal we are done. If not, then for every extremal weight vector $u\betain V_\betaLambda$ and every weight element $Q\betain {\betacal B}_\betaphi$ holds that $Q(u)\betain\betamathfrak{t}^*$. Then by lemma \betaref{nulllemma} there is a $Q$ such that $0\betanot=Q(u)\betain \betamathfrak{t}^*$. As before this implies \beta[\betaOmega\betasigmaubsetset \beta{\betaLambda+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}\betacup U\betacup \beta{-\betaLambda+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}.\beta] To ensure that $(\betaLambda,-\betaLambda,U)$ is a planar spanning triple we have to show that every extremal weight $\betalanglembda$ different from $\betaLambda$ and $-\betaLambda$ is contained in $U=Q(u)^\betabot$. Let $\betalanglembda$ be extremal and different from $\betaLambda$ and $- \betaLambda$, $v_\betapm\betain V_{\betapm\betalanglembda}$ and $u\betain V_\betaLambda$. Since $Q(v_\betapm)\betain \betamathfrak{t}$ the Bianchi identity gives \betabegin{eqnarray*} 0 & =& H(Q(u)v_+,v_- )+H(Q(v_+)v_-,u)+H(Q(v_-)u,v_+) \beta\beta&=&\betalanglembda\betaleft(Q(u)\betaright)\betaunderbrace{H(v_+,v_- )}_{\betanot=0}- \betaunderbrace{\betalanglembda\betaleft(Q(v_+)\betaright)H(v_-,u)+\betaLambda\betaleft((Q(v_-)\betaright)H(u,v_+)}_{\betamathbbox{$=0$ since $u$ is neither in $V_\betalanglembda$ nor in $V_{-\betalanglembda}$}}. \betaend{eqnarray*} Hence $\betalanglembda\betain U$. \betaend{proof} Obviously we are in a slightly different situation as in the Berger case since $- \betaLambda+\betaalpha$ is not necessarily a weight and in case it is a weight, it is not extremal in general. \betasigmaubsetsection{Properties of root systems} In this section we will recall the properties of abstract root systems. Let $(E,\betalangle.,.\betarangle)$ be a euclidian vector space. A finite set of vectors $\betaDeltaelta$ is called root system if it satisfies the following properties \betabegin{enumerate} \betaitem $\betaDeltaelta$ spans $E$. \betaitem For every $\betaalpha \betain \betaDeltaelta$ the reflection on the hyperplane perpendicular to $\betaalpha$ defined by \beta[s_\betaalpha (\betavarphi):= \betavarphi - \betafrac{2\betalangle\betavarphi, \betaalpha\betarangle}{\beta|\betaalpha\beta|^2}\betaalpha\beta] maps $\betaDeltaelta$ onto itself. \betaitem For $\betaalpha,\betabegin{eqnarray*}ta\betain \betaDeltaelta$ the number $\betafrac{2\betalangle\betabegin{eqnarray*}ta, \betaalpha\betarangle}{\beta|\betaalpha\beta|^2}$ is an integer. \betaend{enumerate} A root system is called indecomposable if it does not split into orthogonal subsets. It is called reduced if $2\betaalpha$ is not a root if $\betaalpha$ is a root. The indecomposable, reduced root systems corresponds to the roots of simple Lie algebras. They are classified in a finite list: $A_n,\beta B_n,\beta C_n,\beta D_n,\beta E_6,\beta E_7,\beta E_8,\beta F_4$ and $G_2$. The index designates the dimension of $E$. We will cite some basic properties of root systems, which can be found for example in \betacite{knapp96}. \betabegin{eqnarray*}gin{satz}(See for example \betacite{knapp96}, pp. 149) Let $\betaDeltaelta$ be an abstract reduced root system in $(E,\betalangle.,.\betarangle)$.\betalanglebel{knappsatz} \betabegin{enumerate} \betaitem \betalanglebel{ks1} If $\betaalpha\betain \betaDeltaelta$, then the only root which is proportional to $\betaalpha$ is $-\betaalpha$. \betaitem \betalanglebel{ks2} If $\betaalpha,\betabegin{eqnarray*}ta\betain \betaDeltaelta$, then $\betafrac{2\betalangle\betabegin{eqnarray*}ta, \betaalpha\betarangle}{\beta|\betaalpha\beta|^2}\betain \beta{0,\betapm1,\betapm 2,\betapm 3\beta}$. If $\betaDeltaelta$ is one of the indecomposable root systems $\betapm 3$ occurs only for the root system $G_2$. If both roots are non proportional then $\betapm 2$ only occurs for $B_n, C_n, F_4$ or $G_2$. \betaitem \betalanglebel{ks3} If $\betaalpha$ and $\betabegin{eqnarray*}ta$ are nonproportional in $\betaDeltaelta$ and $\beta|\betabegin{eqnarray*}ta\beta|\betale\beta|\betaalpha\beta|$, then $\betafrac{2\betalangle\betabegin{eqnarray*}ta, \betaalpha\betarangle}{\beta|\betaalpha\beta|^2}\betain \beta{0,\betapm1\beta}$. \betaitem \betalanglebel{ks4} Let be $\betaalpha,\betabegin{eqnarray*}ta\betain \betaDeltaelta$. If $\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle>0$, then $\betaalpha-\betabegin{eqnarray*}ta\betain \betaDeltaelta$. If $\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle<0$, then $\betaalpha+\betabegin{eqnarray*}ta\betain \betaDeltaelta$. I.e. if neither $\betaalpha-\betabegin{eqnarray*}ta\betain \betaDeltaelta$ nor $\betaalpha+\betabegin{eqnarray*}ta\betain \betaDeltaelta$, then $\betalangle \betaalpha,\betabegin{eqnarray*}ta\betarangle=0$. \betaitem \betalanglebel{ks5} The subset of $\betaDeltaelta$ defined by $\betaleft\beta{\betabegin{eqnarray*}ta+ k\betaalpha\betain \betaDeltaelta\betacup\beta{0\beta}|k\betain \betamathbb{Z}\betaright\beta}$ is called $\betaalpha$--string through $\betabegin{eqnarray*}ta$. It has no gaps, i.e. $\betabegin{eqnarray*}ta + k\betaalpha \betain \betaDeltaelta$ for $ -p\betale k\betale q$ with $p,q\betage 0$ and it holds $p-q=\betafrac{2\betalangle\betabegin{eqnarray*}ta, \betaalpha\betarangle}{\beta|\betaalpha\beta|^2}$. The maximal length of such string is given by $\betamax_{\betaalpha,\betabegin{eqnarray*}ta\betain\betaDeltaelta} \betafrac{2\betalangle\betabegin{eqnarray*}ta, \betaalpha\betarangle}{\beta|\betaalpha\beta|^2}+1$, i.e. it contains at most four roots. \betaend{enumerate} \betaend{satz} As a consequence of that proposition we get the following lemmata. In these we will refer to long and short roots. This notion is evident because in the indecomposable reduced root systems of type $B_n,\beta C_n,\beta F_4$ and $G_2$ the roots have two different lengths. \betabegin{eqnarray*}gin{lem} \betalanglebel{wurzellemma0} Let $\betaDelta$ be an indecomposable, reduced root system. Then it holds: \betabegin{enumerate} \betaitem If $a\betaה+\betabeta\betain \betaDelta$ for $a\betain \betamathbb{N}$ and $a>1$, then $\betalangle\betaה,\betabeta\betarangle<0$ and $\betaה$ is a short root. \betaitem If $\betaDeltaelta$ is a root system, where the roots have equal length or if $\betaה$ is a long root, then $\betaה+\betabeta\betain\betaDelta$ implies $\betalangle\betaה,\betabeta\betarangle<0$. \betaitem Let $\betaה$ and $\betabeta$ be two short roots. If $\betaה+\betabeta$ is a long root then $\betalangle\betaה,\betabeta\betarangle=0$, if it is a short one then $\betalangle\betaה,\betabeta\betarangle<0$. The sum of a short and a long root is a short one \betaitem If $\betabeta$ is a long root in $\betaDelta\betanot=G_2$, then there are orthogonal roots $\betaה$ and $\betagamma$ such that $\betabeta=\betaה+\betagamma$. \betaend{enumerate} \betaend{lem} \betabegin{eqnarray*}gin{proof} The proof follows directly from proposition \betaref{knappsatz}.\betaend{proof} \betabegin{eqnarray*}gin{lem}\betalanglebel{wurzellemma1} Let $\betaalpha$ and $\betabegin{eqnarray*}ta$ be two nonproportional roots and $a\betain \betamathbb{N}$. If $a(\betaalpha+\betabegin{eqnarray*}ta)\betain \betaDeltaelta$ then $a=1$. \betaend{lem} \betabegin{eqnarray*}gin{proof} If $a>1$ then $\betaalpha+\betabegin{eqnarray*}ta$ is not a root. This implies $\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle\betage 0$ and yields for $a(\betaalpha+\betabegin{eqnarray*}ta)=\betagamma\betain \betaDeltaelta$: \beta[\betabegin{array}{rcccl} 0&<& a\betaleft(\beta|\betaalpha\beta|^2 +\betalangle \betaalpha,\betabegin{eqnarray*}ta\betarangle \betaright)&=&\betalangle\betaalpha,\betagamma\betarangle\beta\beta 0&<& a\betaleft(\betalangle \betaalpha,\betabegin{eqnarray*}ta\betarangle+\beta|\betabegin{eqnarray*}ta\beta|^2 \betaright)&=&\betalangle\betagamma,\betabegin{eqnarray*}ta\betarangle. \betaend{array}\beta] On the other hand we have \beta[\beta|\betagamma\beta|^2=a\betaleft(\betalangle \betaalpha,\betagamma\betarangle + \betalangle \betabegin{eqnarray*}ta,\betagamma\betarangle \betaright).\beta] But this gives \beta[1=\betafrac{a}{2} \betaBig(\betaunderbrace{ \betaunderbrace{\betafrac{2\betalangle\betagamma, \betaalpha\betarangle}{\beta|\betagamma\beta|^2}}_{\betamathbbox{$>0$ in $\betamathbb{N}$}} + \betaunderbrace{\betafrac{2\betalangle\betagamma, \betabegin{eqnarray*}ta\betarangle}{\beta|\betagamma\beta|^2}}_{\betamathbbox{$>0$ in $\betamathbb{N}$ }}}_{\betamathbbox{$\betage 2$ in $\betamathbb{N}$}}\betaBig) .\beta] This is a contradiction. Hence $a=1$. \betaend{proof} The next lemma is a little more general. \betabegin{eqnarray*}gin{lem}\betalanglebel{wurzellemma2} Let $\betaalpha$ and $\betabegin{eqnarray*}ta$ be two non-proportional roots in an indecomposable root system and $a,b\betain \betamathbb{N}$ with $a\betale b$ such that $a\betaalpha+b\betabegin{eqnarray*}ta\betain \betaDeltaelta$. \betabegin{enumerate} \betaitem If $\betaDeltaelta$ is not $G_2$ then $a=1$. If $\betaDeltaelta= A_n, D_n, E_6,E_7, E_8$ then $b=1$ too. If $\betaDeltaelta=B_n, C_n , F_4$ then $b\betale 2$. \betaitem If $\betaDeltaelta=G_2$ then $a\betale 2$ and $b\betale 3$. \betaend{enumerate} \betaend{lem} \betabegin{eqnarray*}gin{proof} We suppose $a\betaalpha + b \betabegin{eqnarray*}ta =\betagamma\betain \betaDeltaelta$. First we consider the case $\betalangle \betaalpha, \betabegin{eqnarray*}ta\betarangle\betage 0$. This gives \beta[\betabegin{array}{rcccl} 0&<& a\beta|\betaalpha\beta|^2 +b\betalangle \betaalpha,\betabegin{eqnarray*}ta\betarangle &=&\betalangle\betaalpha,\betagamma\betarangle\beta\beta 0&<& a(\betalangle \betaalpha,\betabegin{eqnarray*}ta\betarangle+b\beta|\betabegin{eqnarray*}ta\beta|^2 &=&\betalangle\betagamma,\betabegin{eqnarray*}ta\betarangle. \betaend{array}\beta] On the other hand we have $\beta|\betagamma\beta|^2=a\betalangle \betaalpha,\betagamma\betarangle + b\betalangle \betabegin{eqnarray*}ta,\betagamma\betarangle $ and thus \betabegin{eqnarray*} 1&=&\betafrac{a}{2}\betaoverbrace{\betafrac{2\betalangle\betagamma, \betaalpha\betarangle}{\beta|\betagamma\beta|^2}}^{>0} +\betafrac{b}{2}\betaoverbrace{\betafrac{2\betalangle\betagamma, \betabegin{eqnarray*}ta\betarangle}{\beta|\betagamma\beta|^2}}^{>0} \beta\beta&\betage& \betafrac{a}{2} \betaBig(\betaunderbrace{ \betafrac{2\betalangle\betagamma, \betaalpha\betarangle}{\beta|\betagamma\beta|^2} + \betafrac{2\betalangle\betagamma, \betabegin{eqnarray*}ta\betarangle}{\beta|\betagamma\beta|^2}}_{\betamathbbox{$\betage 2$ in $\betamathbb{N}$}}\betaBig) .\betaend{eqnarray*} Hence $a=1$. Let now be $\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle<0$. This implies, that $\betaalpha+\betabegin{eqnarray*}ta=:\betadelta$ is a root with the property $\betadelta-\betabegin{eqnarray*}ta=\betaalpha\betain \betaDeltaelta$ Although the above proposition does not assert that this implies $\betalangle \betadelta,\betabegin{eqnarray*}ta\betarangle\betage 0$ we can show this. Suppose that $\betalangle\betadelta,\betabegin{eqnarray*}ta\betarangle<0$. Hence $\betadelta+\betabegin{eqnarray*}ta=\betaalpha+2\betabegin{eqnarray*}ta$ is a root. If we exclude the root system $G_2$ point \betaref{ks5} of proposition \betaref{knappsatz} implies $\betafrac{2\betalangle\betaalpha, \betabegin{eqnarray*}ta\betarangle}{\beta|\betabegin{eqnarray*}ta\beta|^2}=-2$, i.e. $\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle=-\beta|\betabegin{eqnarray*}ta\beta|^2$ and finally $\betalangle\betadelta,\betabegin{eqnarray*}ta\betarangle=0$, which was excluded. Thus we have that $\betalangle\betadelta,\betabegin{eqnarray*}ta\betarangle\betage 0$. Analogously to the first case we get \beta[ 1=\betafrac{a}{2}\beta \betafrac{2\betalangle\betagamma, \betadelta\betarangle}{\beta|\betagamma\beta|^2} +\betafrac{(b-a)}{2}\beta \betafrac{2\betalangle\betagamma, \betabegin{eqnarray*}ta\betarangle}{\beta|\betagamma\beta|^2}. \beta] In case that $a\betale b-a$ we get again that $a=1$. Otherwise we get $b-a=1$, i.e. $a\betadelta+\betabegin{eqnarray*}ta=\betagamma$. Again by point \betaref{ks5} of proposition \betaref{knappsatz} we get $p-q=\betafrac{2\betalangle\betagamma, \betadelta\betarangle}{\beta|\betadelta\beta|^2}\betage 0$. But this implies $a\betale 1$. The possible values for $b$ follow also by proposition \betaref{knappsatz}. For $G_2$ the possible values of $a$ and $b$ can be calculated analogously. \betaend{proof} \betabegin{eqnarray*}gin{lem}\betalanglebel{wurzellemma3} Let $\betaeta$ be a long root of an indecomposable root system. \betabegin{enumerate} \betaitem Let $a,b\betain\betamathbb{N}$ and $\betaalpha\betain \betaDeltaelta$ not proportional to $\betaeta$ such that $a\betaeta+b\betaalpha\betain \betaDeltaelta$. Then $a\betale b$, i.e. $a=1$ if $\betaDeltaelta$ not equal to $G_2$ and $a\betale 2$ otherwise. \betaitem Let $\betaalpha,\betabegin{eqnarray*}ta$ in $\betaDeltaelta$ not proportional to $\betaeta$ and $a\betain \betamathbb{N}$ such that $a\betaeta +\betaalpha +\betabegin{eqnarray*}ta\betain \betaDeltaelta$. Then $a\betale2$. \betaend{enumerate} \betaend{lem} \betabegin{eqnarray*}gin{proof} 1.) First we exclude $G_2$ and suppose that $b=1$, i.e. $a\betaeta+\betaalpha=\betagamma\betain¸\betaDeltaelta$. Hence $-p\betale a\betale q$ and \beta[ \betaleft| p-q \betaright|=\betafrac{2|\betalangle\betaeta, \betaalpha\betarangle|}{\beta|\betaeta\beta|^2} < 2\betafrac{\beta|\betaeta\beta|\betacdot \beta|\betaeta\beta|}{\beta|\betaeta\beta|^2} \betale 2, \beta] i.e. $|p-q|\betale 1$. But since we have excluded $G_2$ we have that $a=1$. For $G_2$ a long root $\betaeta$ is given by $2e_3-e_1-e_2$ with the notations of appendix C of \betacite{knapp96}. For this we get the wanted result. 2.) Let $a\betaeta+ \betaalpha +\betabegin{eqnarray*}ta =\betagamma$. First we consider the case that $\betaalpha+\betabegin{eqnarray*}ta$ or $\betaalpha-\betagamma$ or $\betabegin{eqnarray*}ta-\betagamma$ is a root. If this root is not proportional to $\betaeta$ we have by the first point that $a\betale 1$. If it is proportional to $\betaeta$ we get that $a\betale 2$ and we are done. Now we suppose that neither $\betaalpha+\betabegin{eqnarray*}ta$ nor $\betaalpha-\betagamma$ nor $\betabegin{eqnarray*}ta-\betagamma$ is a root. This implies $\betalangle \betaalpha,\betabegin{eqnarray*}ta\betarangle\betage 0$, $\betalangle \betaalpha,\betagamma\betarangle\betale 0$ and $\betalangle \betabegin{eqnarray*}ta,\betagamma\betarangle\betale 0$. We consider the equations \beta[\betabegin{array}{rcccccccl} a\betalangle\betaeta,\betaalpha\betarangle &+& \beta|\betaalpha\beta|^2 &+& \betaunderbrace{\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle}_{\betage 0}&=&\betalangle\betagamma,\betaalpha\betarangle&\betale&0\beta\beta a\betalangle\betaeta,\betabegin{eqnarray*}ta\betarangle &+& \beta|\betabegin{eqnarray*}ta\beta|^2 &+& \betaunderbrace{\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle}_{\betage 0}&=&\betalangle\betagamma,\betabegin{eqnarray*}ta\betarangle&\betale&0\beta\beta a\betalangle\betaeta,\betagamma\betarangle &+& \betaunderbrace{\betalangle\betaalpha,\betagamma\betarangle}_{\betale 0} &+& \betaunderbrace{\betalangle\betabegin{eqnarray*}ta,\betagamma\betarangle}_{\betale 0}&=&\beta|\betagamma\beta|^2&>&0. \betaend{array}\beta] Hence we have that $\betalangle\betaeta,\betaalpha\betarangle<0$, $\betalangle\betaeta,\betabegin{eqnarray*}ta\betarangle<0$ and $\betalangle\betaeta,\betagamma\betarangle>0$. But since $\betaeta$ is long, not proportional neither to $\betaalpha$ nor to $\betabegin{eqnarray*}ta$ we have that \betabegin{eqnarray*} \beta|\betaeta\beta|^2\beta \betage\beta \betalangle \betagamma,\betaeta\betarangle &=& a \beta|\betaeta\beta|^2 +\betalangle \betaalpha,\betaeta\betarangle+\betalangle\betabegin{eqnarray*}ta,\betaeta\betarangle\beta\beta &=& a \beta|\betaeta\beta|^2 - \betaunderbrace{|\betalangle \betaalpha,\betaeta\betarangle|}_{<\beta|\betaalpha\beta|\betacdot\beta|\betaeta\beta|\betale \beta|\betaeta\beta|^2} -\betaunderbrace{|\betalangle\betabegin{eqnarray*}ta,\betaeta\betarangle|}_{<\beta|\betabegin{eqnarray*}ta\beta|\betacdot\beta|\betaeta\beta|\betale \beta|\betaeta\beta|^2}\beta\beta &>&(a-2)\beta|\betaeta\beta|^2. \betaend{eqnarray*} This gives $a-2<1$ which is the proposition. \betaend{proof} \betabegin{eqnarray*}gin{lem}\betalanglebel{wurzellemma5} Let $\betaalpha$ be a long root and $\betaeta$ be a short one with $\betalangle\betaalpha,\betaeta\betarangle>0$, i.e. $\betafrac{2\betalangle\betaalpha,\betaeta\betarangle}{\beta|\betaeta\beta|^2}\betage 2$. Then there is a short root $\betabegin{eqnarray*}ta$ with $\betabeta\betanot\betasigmaim\betaeta$, $ \betalangle\betabegin{eqnarray*}ta,\betaalpha\betarangle <0$ and $\betalangle\betabeta,\betaeta\betarangle\betale 0$. If the rank of the root system is greater than $2$ or if $\betafrac{2\betalangle\betaalpha,\betaeta\betarangle}{\beta|\betaeta\beta|^2}= 3$ (which can only occur for $G_2$), $\betabeta$ can be chosen such that $\betalangle\betabeta,\betaeta\betarangle<0$. \betaend{lem} \betabegin{eqnarray*}gin{proof} $\betalangle\betaalpha,\betaeta\betarangle>0$ implies that $\betaeta-\betaalpha $ is a root, in particular a short one. For the inner product we get \beta[ \betalangle\betaalpha,\betaeta-\betaalpha\betarangle=\betalangle\betaalpha,\betaeta\betarangle-\beta|\betaalpha\beta|^2<\beta|\betaalpha\beta|\beta|\betaeta\beta|-\beta|\betaalpha\beta|^2<\beta|\betaalpha\beta|^2-\beta|\betaalpha\beta|^2=0 \beta] and \beta[ \betalangle\betaeta,\betaeta-\betaalpha\betarangle=\beta|\betaeta\beta|^2-\betalangle\betaalpha,\betaeta\betarangle\betale 0.\beta] In case of $\betafrac{2\betalangle\betaalpha,\betaeta\betarangle}{\beta|\betaeta\beta|^2}= 3$ the last $\betale$ is a $<$, and we are done with the second point in case of $G_2$. If the rank of the root system is greater than $2$ this can be seen with the help of the definitions of the reduced, indecomposable root systems (see appendix C of \betacite{knapp96}). \betaend{proof} \betasigmaection{Simple weak-Berger algebras of real type} In this section we will apply the result of proposition \betaref{triplesatz} to simple complex irreducible acting Lie algebras. We will do this step by step under the following special conditions: \betabegin{enumerate} \betaitem The highest weight of the representation is a root. \betaitem The representation satisfies (SI), i.e. admits a planar spanning triple $(\betaLambda,-\betaLambda,U)$. \betaitem The representation satisfies (SII) and has weight zero. \betaitem The representation satisfies (SII) and does not have weight zero. \betaend{enumerate} Throughout this section the considered Lie algebra is supposed to be different from $\betamathfrak{sl}(2,\betamathbb{C})$. \betasigmaubsetsection{Representations with roots as highest weight} \betabegin{eqnarray*}gin{satz} \betalanglebel{wurzelgewicht} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$ and satisfying (SI) or (SII). If we suppose in addition that there is an extremal weight $\betaLambda$ with $\betaLambda = a \betaeta$ for a root $\betaeta\betain \betaDeltaelta$ and $a>0$, then holds the following: \betabegin{enumerate} \betaitem If $\betaeta $ is a long root, then $a=1$ and the representation is the adjoint one. \betaitem If $\betaeta $ is a short root, then holds the following for $a$: \betabegin{enumerate} \betaitem If $\betaDelta=B_n$ or $G_2$ then $a=1,2$. \betaitem If $\betaDelta=C_n$ or $F_4$ then $a=1$. \betaend{enumerate} \betaend{enumerate} \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betaLambda = a\betaeta$ with $ \betaeta\betain \betaDeltaelta$, $a\betain \betamathbb{N}$. W.l.o.g. we may suppose that $\betaLambda $ is the extremal weight in the properties (SI) and (SII). (If not then there is an element of the Weyl group $\betasigmaigma$ mapping $\betaLambda$ to the extremal weight of (SI) and (SII) $\betaLambda^pr_{\betalason}ime$. Then $\betaLambda^pr_{\betalason}ime = a \betasigmaigma \betaeta$ and $\betasigmaigma \betaeta \betain \betaDeltaelta$.) First we show that $a\betain \betamathbb{N}$. If we chose an fundamental system $(\betapi_1,\betaldots, \betapi_n)$ such that $\betaLambda=a\betaeta$ is the highest weight we get that $\betalangle\betaLambda,\betapi_i\betarangle=a\betalangle\betaeta,\betapi_i\betarangle\betain \betamathbb{N}$ for all $i$. $a\betanot\betain\betamathbb{N}$ would imply that $\betalangle\betaeta,\betapi_i\betarangle\betage 2$ for all $i$ with $\betalangle\betaeta,\betapi_i\betarangle\betanot=0$. This holds only for the root system $C_n$ where $\betaLambda=\betaomega_1= \betaend{itemize}nhalb \betaeta$. But this representation is symplectic but not orthogonal. (For an explicit formulation of this criterion see \betacite{onishchik-vinberg3}.) So we get $a\betain \betamathbb{N}$. Now we consider two cases. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: $\betaeta$ is a long root:}} In this case the root system of long roots, denoted by $\betaDeltaelta_l$ is the orbit of $\betaeta$ under the Weyl group. Hence $a\betacdot \betaDelta_l$ are the extremal weights and $\betaDelta\betasigmaubsetset\betaOmega$. This implies $0\betain \betaOmega_\betaalpha$ for every $\betaalpha\betain \betaDeltaelta$. Furthermore for all roots holds that $a\betacdot \betaDeltaelta\betasigmaubsetset \betaOmega$. This is true because we can find a short root such that $\betalangle\betaeta,\betabegin{eqnarray*}ta\betarangle>0$. This implies $\betaeta-\betabegin{eqnarray*}ta\betain \betaDelta_s$. On the other hand it is $\betafrac{2\betalangle a\betaeta,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabegin{eqnarray*}ta\beta|^2}\betage a$. Hence $a \betaeta-a\betabegin{eqnarray*}ta=a(\betaeta-\betabegin{eqnarray*}ta)\betain \betaOmega$. Applying the Weyl group to this weight we get the property for all short roots. \betabegin{eqnarray*}gin{description} \betaitem{(SI)} Let $\betaLambda$ satisfy (SI), i.e. $\betaLambda$ and $-\betaLambda$ define a planar spanning triple $(\betaLambda,-\betaLambda,U)$. This would imply that every long root different from $\betaeta$ lies in the hyperplane $U$. This is only possible for the the root system $C_n$, because all other root systems have an indecomposable system of long roots. For $C_n$ holds that $\betaDelta_l=A_1\betatimes\betaldots\betatimes A_1$. But we have still a root $\betabegin{eqnarray*}ta$ --- possibly a short one --- such that $\betabegin{eqnarray*}ta\betanot\betain U$ and $\betabegin{eqnarray*}ta$ not proportional to $\betaeta$. This implies $\betaOmega\betani a\betabegin{eqnarray*}ta=\betaLambda+\betagamma=a\betaeta+\betagamma$ or $\betaOmega\betani a\betabegin{eqnarray*}ta=-\betaLambda+\betagamma=-a\betaeta+\betagamma$ with $\betagamma\betain \betaDeltaelta_0$. Then Lemma \betaref{wurzellemma1} implies $a=1$. \betaitem{(SII)} Lets suppose that $\betaLambda$ satisfies (SII), i.e. there is an $\betaalpha\betain \betaDeltaelta$ such that $\betaOmega_\betaalpha\betasigmaubsetset \beta{\betaLambda-\betaalpha+\betabegin{eqnarray*}ta|\betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}\betacup\beta{-\betaLambda+\betabegin{eqnarray*}ta|\betabegin{eqnarray*}ta\betain \betaDeltaelta_0\beta}$. $0\betain \betaOmega_\betaalpha$ implies $0=\betaLambda-\betaalpha+\betabegin{eqnarray*}ta=a\betaeta-\betaalpha+\betabegin{eqnarray*}ta$ or $0= -\betaLambda+\betabegin{eqnarray*}ta=-a\betaeta+\betabegin{eqnarray*}ta$ with $\betabegin{eqnarray*}ta\betain \betaDeltaelta_0$. The second is not possible and the first implies by lemma \betaref{wurzellemma3} that $a=1$ or $a=2$ and $\betaeta=\betaה$. In the second case we find a root $\betagamma\betanot\betasigmaim\betaה$ such that $\betalangle\betagamma,\betaה\betarangle<0$, hence $2\betagamma\betain\betaW_\betaה$. Since $2\betagamma-2\betaה\betanot\betain\betaDelta$ it has to be $2\betagamma=\betaה+\betabeta$, but this is prevented by $\betalangle\betagamma,\betaה\betarangle<0$ and lemma \betaref{wurzellemma0}. \betaend{description} Of course if $\betaeta $ is a long root the representation is the adjoint one. \betaitem{{\betaem Case 2: $\betaeta$ is a short root:}} Lets denote by $\betaDelta_s$ the root system of short roots. It equals to the orbit of $\betaeta $ under the Weyl group. It is a root system of the same rank as $\betaDelta$ and all roots have the same length. Clearly $\betaDelta_s\betasigmaubsetset \betaOmega$ and $a\betacdot\betaDelta_s$ are the extremal weights in $\betaOmega$. For the root system $B_n$ the root system of short roots $\betaDelta_s$ equals to $A_1\betatimes \betaldots \betatimes A_1$, otherwise it is indecomposable. Furthermore holds the following: If $a\betage2$ then $\betaDelta\betasigmaubsetset \betaOmega$. To verify this, we consider a long root $\betabegin{eqnarray*}ta\betain \betaDeltaelta_l$ with the property that $\betalangle\betabeta,\betaeta\betarangle >0$. Such a $ \betabeta$ always exists. Then we have $\betafrac{2\betalangle \betaeta,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaeta\beta|^2}> \betafrac{2\betalangle \betaeta,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabegin{eqnarray*}ta\beta|^2}\betage1$. This implies $2\betaeta-\betabegin{eqnarray*}ta\betain \betaDelta$ (see proposition \betaref{knappsatz}). On the other hand $a\betage 2$ ensures that $\betaOmega\betani s_\betabeta(2\betaeta)=2\betaleft(\betaeta-\betafrac{2\betalangle \betaeta,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabegin{eqnarray*}ta\beta|^2}\betabeta\betaright)$. This implies that the long root $2\betaeta-\betabeta $ is a weight. Now applying the Weyl group to $\betabeta$ shows that every long root is a weight. \betabegin{eqnarray*}gin{description} \betaitem{(SI)} We suppose that there is a planar spanning triple $(\betaLambda,-\betaLambda,U)$. This implies that $a\betabegin{eqnarray*}ta$ lies in the hyperplane $U$ if $\betabegin{eqnarray*}ta$ is a short root. But this is only possible for $B_n$ because the short roots of all other root systems are indecomposable. In case of $B_n$ we can at least find a long root $\betaalpha$ which is not in $U$. Since the long roots are weights, we have $\betaalpha=a\betaeta +\betagamma$ or $\betaalpha=-a\betaeta+\betagamma$ with $\betagamma\betain \betaDelta_0$. But this implies for $B_n$ that $a\betale 2$. \betaitem{(SII)} Suppose that there is an $\betaה\betain \betaDeltaelta$ such that $ \betaOmega_\betaalpha\betasigmaubsetset \betaleft\beta{ \betaLambda-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{ -\betaLambda+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}$. $\betaDelta\betasigmaubsetset \betaOmega$ implies $0\betain \betaOmega_\betaalpha$ for all $\betaalpha$. $0=-a\betaeta+\betagamma$ with $\betagamma\betain \betaDelta_0$ implies $a=1$. Hence if we suppose $a\betage 2$ we must have \betabegin{eqnarray*}q \betalanglebel{nulleqn} 0=a\betaeta-\betaalpha+\betagamma \betaend{eqnarray*}q Thus we have to deal with the following cases: \betabegin{enumerate} \betaitem[(a)] $\betaalpha=\betaeta$ and $a=2$. \betaitem[(b)] $\betaalpha\betanot\betasigmaim\betaeta$ and by \betaref{ks5} of proposition \betaref{knappsatz} $a\betale \betafrac{2\betalangle \betaeta,\betaalpha\betarangle}{\beta|\betaeta\beta|^2}\betale 3$. I.e. if $a\betage2$, $\betaalpha$ is a long root. \betaend{enumerate} We exclude the first case for any root system different from $B_n$. Set $a=2$ and $\betaalpha=\betaeta$. If $\betaDelta\betanot=B_n$ the short roots are indecomposable, i.e. there is a short root $\betabeta$ such that $\betabeta\betanot\betasigmaim \betaeta$ and $\betalangle\betabeta,\betaeta\betarangle<0$. Hence $2\betabeta\betain \betaOmega_{\betaeta}$ and $\betabeta+\betaeta\betain \betaDeltaelta$. The existence of a spanning triple implies then $2\betabegin{eqnarray*}ta=\betaeta+\betagamma$ or $2\betabegin{eqnarray*}ta=-2\betaeta+\betagamma$ with $\betagamma\betain \betaDeltaelta_0$. The second case is impossible because of lemma \betaref{wurzellemma1}. The first implies $2\betabegin{eqnarray*}ta-\betaeta\betain\betaDeltaelta$. Again this is not possible by \betaref{wurzellemma0} and $\betalangle\betabeta,\betaeta\betarangle<0$. Hence the case (a) is excluded. Now we consider the case (b). First we show that $a=3$ is not possible. Set $a=3$. We notice that $\betalangle\betaeta,\betaalpha\betarangle>0 $ implies $\betafrac{2\betalangle \betaeta,\betaalpha\betarangle}{\beta|\betaalpha\beta|^2}\betage 1$ and hence $3\betaeta-3\betaalpha\betain\betaOmega_\betaalpha$. Thus we have the alternative $3\betaeta-3\betaalpha=3\betaeta-\betaalpha+\betagamma$ or $3\betaeta-3\betaalpha=-3\betaeta+\betagamma$ with $\betagamma\betain \betaDelta_0$. The first implies $2\betaalpha\betain \betaDeltaelta$ and the second $6\betaeta-3\betaalpha\betain \betaDeltaelta$. Both are not true, hence $a=3$ is impossible. We continue with case (b) and have that $\betaalpha $ is a long root with \beta[ \betafrac{2\betalangle \betaeta,\betaalpha\betarangle}{\beta|\betaeta\beta|^2}\betage 2\beta , \beta;\beta;\betamathbbox{ i.e. $2\betaeta-\betaalpha\betain\betaDelta$.}\beta] From now on we suppose, that the root system is different from $G_2$. Then we have \betabegin{eqnarray*}q\betalanglebel{alphaeta} \betafrac{2\betalangle \betaeta,\betaalpha\betarangle}{\beta|\betaeta\beta|^2}= 2. \betaend{eqnarray*}q In a next step we will show that under these conditions there is no short root $\betabegin{eqnarray*}ta$ with \betabegin{eqnarray*}q \betalanglebel{f4} \betabeta\betain\betaDeltaelta_s\betamathbbox{ with }\betalangle\betaalpha,\betabegin{eqnarray*}ta\betarangle<0\beta ,\beta; \betalangle\betaalpha,\betaeta\betarangle<0 \betamathbbox{ and }\beta; \betabeta\betanot\betasigmaim\betaeta. \betaend{eqnarray*}q Suppose that there is such a $\betabegin{eqnarray*}ta$. Then the first condition implies that $2\betabegin{eqnarray*}ta\betain \betaOmega_\betaalpha$ and hence $2\betabeta=2\betaeta-\betaalpha +\betagamma$ or $2\betabeta=-2\betaeta+\betagamma$ with $\betagamma\betain\betaDeltaelta_0$. The latter is not possible. The second implies the following using (\betaref{alphaeta}): \beta[ -2\beta \betage\beta 2\betacdot\betafrac{2\betalangle \betabegin{eqnarray*}ta,\betaeta\betarangle}{\beta|\betaeta\beta|^2} \beta =\beta \betafrac{2\betalangle2\betaeta-\betaalpha,\betaeta\betarangle}{\beta|\betaeta\beta|^2}+ \betafrac{2\betalangle \betagamma,\betaeta\betarangle}{\beta|\betaeta\beta|^2} \beta =\beta 2+ \betafrac{2\betalangle \betagamma,\betaeta\betarangle}{\beta|\betaeta\beta|^2}. \beta] Hence $-4\betage \betafrac{2\betalangle \betagamma,\betaeta\betarangle}{\beta|\betaeta\beta|^2}$ which is impossible. Now by the lemma \betaref{wurzellemma5} there is such a $\betabegin{eqnarray*}ta$. Hence for any remaining root systems different from $G_2$ and different from $B_n$ we have that $a=1$. \betaend{description} \betaend{description} All in all we have shown, that for a long root holds $a=1$ and for a short root $a=2$ implies $\betaDeltaelta=B_n$ or $G_2$. \betaend{proof} \betabegin{eqnarray*}gin{folg}\betalanglebel{wurzelgewichtfolge} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible complex simple weak Berger algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$ and with the additional property that the highest weight is of the form $\betaLambda = a \betaeta$ for a root $\betaeta\betain \betaDeltaelta$. Then $\betalangleg$ is the complexification of a holonomy algebra of a Riemannian manifold or the representation of $G_2$ with highest weight $2\betaomega_1$. \betaend{folg} \betabegin{eqnarray*}gin{proof} Clearly if $\betaeta$ is a long root the representation is the adjoint one, i.e. the complexification of a holonomy representation of a Lie group with positive definite bi-invariant metric. For a short root $\betaeta$ we get the following: \betabegin{eqnarray*}gin{description} \betaitem[$B_n,\beta a=1:$] This is the representation of highest weight $\betaomega_1$, i.e. the standard representation of $\betalangleso(2n+1,\betamathbb{C})$ on $\betamathbb{C}^{2n+1}$. Of course this is the complexification of the generic Riemannian holonomy representation. \betaitem[$B_n,\beta a=2:$] This is the representation of highest weight $2\betaomega_1$. A further analysis shows that this is the complexified representation of the Riemannian symmetric space of type $AI$, i.e. of the symmetric spaces $SU(2n+1)/SO(2n+1,\betamathbb{R})$, respectively $SL(2n+1,\betamathbb{R})/SO(2n+1,\betamathbb{R})$. \betaitem[$C_n,\beta a=1:$] (for $n\betage 3$) This is the representation of highest weight $\betaomega_2$. It is the complexified representation of the Riemannian symmetric space of type $AII$, i.e. of the symmetric spaces $SU(2n)/Sp(n)$, respectively $SL(2n,\betamathbb{R})/Sp(n)$. \betaitem[$F_4,\beta a=1:$] This is the representation of highest weight $\betaomega_1$. It is the complexified representation of the Riemannian symmetric space of type $EIV$, i.e. of the symmetric spaces $E_6/F_4$, respectively $E_{6(-26)}/F_4$. \betaitem[$G_2,\beta a=1:$] This is the representation of highest weight $\betaomega_1$. It is the representation of $G_2$ on $\betamathbb{C}^7$, i.e. the complexification of the holonomy of a Riemannian $G_2$--manifold. \betaitem[$G_2,\beta a=2:$] This is the representation $2\betaomega_1$ of $G_2$. It is a $27$-dimensional representation of $G_2$ isomorphic to $Sym_0^2\betamathbb{C}^7$, where $\betamathbb{C}^7$ denotes the standard module of $G_2$ and $Sym^2_0\betamathbb{C}^7$ its symmetric, trace free $(2,0)$--tensors. This is the exception, because there is no Riemannian manifold with this complexified holonomy representation. \betaend{description} \betaend{proof} \betasigmaubsetsection{Representations with planar spanning triples} Now we consider representations of a simple Lie algebra under the condition that there is a planar spanning triple. The proof of this proposition is a copy of the proof in \betacite{schwachhoefer2} adding the additional properties of our planar spanning triple. \betabegin{eqnarray*}gin{satz}\betalanglebel{planarsatz} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$ and satisfying (SI), i.e. with a planar spanning triple of the form $(\betaLambda,-\betaLambda,U)$. If there is no root $\betaalpha$ such that $\betaLambda = a \betaalpha$ then $\betalangleg$ is of type $D_n$ with $n\betage3$ and the representation is congruent to the one with highest weight $\betaomega_1$ or $2\betaomega_1$. \betaend{satz} \betabegin{eqnarray*}gin{proof} The condition $\betaLambda \betanot= a \betaalpha$ implies that there is no root such that $-\betaLambda=s_\betaalpha (\betaLambda)$. The existence of a planar spanning triple then gives that for any $\betaalpha\betain \betaDelta$ with $\betalangle\betaLambda,\betaalpha\betarangle\betanot=0$ the image of the reflection lies in $U$. If we set $U=T^\betabot$ this gives \betabegin{eqnarray*}q\betalanglebel{teqn} \betamathbbox{For $\betaalpha\betain \betaDeltaelta$ with $\betalangle\betaalpha,\betaLambda\betarangle\betanot=0$ holds }\beta;\beta;\beta; \betalangle \betaalpha,T\betarangle\beta =\beta \betafrac{\beta|\betaalpha\beta|^2}{2\betalangle\betaLambda,\betaalpha\betarangle}\betalangle\betaLambda,T\betarangle\beta \betanot=\beta 0. \betaend{eqnarray*}q In the following we prove various claims to get the wanted result. We follow completely the lines of reasoning in \betacite{schwachhoefer2}. \betabegin{eqnarray*}gin{description} \betaitem{Claim 1:} {\betaem For any non-proportional $ \betaalpha,\betabegin{eqnarray*}ta\betain \betaDeltaelta$ with $\betalangle\betaLambda,\betaalpha\betarangle\betanot=0$ and $\betalangle\betaLambda,\betabeta\betarangle\betanot=0$ holds that $\betalangle \betaalpha,\betabeta\betarangle=0 $ or both have the same length.} To show this we prove that for two such roots hold that they are orthogonal or that $\betalangle\betaLambda,s_\betaalpha\betabeta\betarangle=\betalangle\betaLambda,s_\betabeta\betaalpha\betarangle=0$. Suppose that $\betalangle\betaLambda,s_\betaalpha\betabeta\betarangle\betanot=0$. Then (\betaref{teqn}) gives the following \betabegin{eqnarray*} \beta|\betabeta\beta|^2&=&\beta|s_\betaalpha \betabeta\beta|^2 \beta\beta &=& \betafrac{2}{\betalangle\betaLambda,T\betarangle}\betacdot \betalangle\betaLambda,s_\betaalpha\betabeta\betarangle\betacdot \betalangle s_\betaalpha\betabeta,T\betarangle \beta\beta&=& \betafrac{2}{\betalangle\betaLambda,T\betarangle}\betacdot \betaleft( \betalangle\betaLambda,\betabeta\betarangle-\betafrac{2 \betalangle \betaalpha,\betabeta\betarangle}{\beta|\betaalpha\beta|^2}\betalangle\betaLambda,\betaalpha\betarangle \betaright)\betacdot \betaleft( \betalangle\betabeta,T\betarangle -\betafrac{2 \betalangle \betaalpha,\betabeta\betarangle}{\beta|\betaalpha\beta|^2}\betalangle\betaalpha,T\betarangle \betaright) \beta\beta&=& 2\betacdot \betaleft( \betalangle\betaLambda,\betabeta\betarangle-\betafrac{2 \betalangle \betaalpha,\betabeta\betarangle}{\beta|\betaalpha\beta|^2}\betalangle\betaLambda,\betaalpha\betarangle \betaright)\betacdot \betaleft( \betafrac{\beta|\betabeta\beta|^2}{2\betalangle\betaLambda,\betabeta\betarangle} -\betafrac{2 \betalangle \betaalpha,\betabeta\betarangle}{\betalangle\betaLambda,\betaalpha\betarangle} \betaright) \beta\beta &=& 2\betacdot \betaleft( \betafrac{\beta|\betabeta\beta|^2}{2}- 2 \betalangle \betaalpha,\betabeta\betarangle\beta \betafrac{\betalangle\betaLambda,\betabeta\betarangle}{\betalangle\betaLambda,\betaalpha\betarangle} - 2 \betalangle \betaalpha,\betabeta\betarangle\beta \betafrac{\betalangle\betaLambda,\betaalpha\betarangle}{\betalangle\betaLambda,\betabeta\betarangle}\betafrac{\beta|\betabeta\beta|^2}{\beta|\betaalpha\beta|^2} + 4\beta \betafrac{2 \betalangle \betaalpha,\betabeta\betarangle^2}{\beta|\betaalpha\beta|^2} \betaright). \betaend{eqnarray*} Subtracting $\beta|\betabeta\beta|^2$ and multiplying by the denominators gives \beta[ 0\beta =\beta \betalangle\betaalpha,\betabeta\betarangle \betaleft( \beta|\betabeta\beta|^2\betalangle\betaLambda,\betaalpha\betarangle^2 + \beta|\betaalpha\beta|^2\betalangle\betaLambda,\betabeta\betarangle^2 -2\betalangle\betabeta,\betaalpha\betarangle\betalangle\betaLambda,\betaalpha\betarangle\betalangle\betaLambda,\betabeta\betarangle\betaright). \beta] But this gives the following pair of equations \betabegin{eqnarray*} 0& =& \betalangle\betaalpha,\betabeta\betarangle \betaBig( \betaunderbrace{\betaleft(\beta|\betabeta\beta|\betalangle\betaLambda,\betaalpha\betarangle + \beta|\betaalpha\beta|\betalangle\betaLambda,\betabeta\betarangle\betaright)^2}_{>0} -2\betaunderbrace{\betaleft( \beta|\betaalpha\beta|\beta|\betabeta\beta|+\betalangle\betabeta,\betaalpha\betarangle\betaright)}_{>0}\betalangle\betaLambda,\betaalpha\betarangle\betalangle\betaLambda,\betabeta\betarangle\betaBig)\beta\beta 0& =& \betalangle\betaalpha,\betabeta\betarangle \betaBig( \betaunderbrace{\betaleft(\beta|\betabeta\beta|\betalangle\betaLambda,\betaalpha\betarangle - \beta|\betaalpha\beta|\betalangle\betaLambda,\betabeta\betarangle\betaright)^2}_{>0} +2\betaunderbrace{\betaleft(\beta|\betaalpha\beta|\beta|\betabeta\beta|-\betalangle\betabeta,\betaalpha\betarangle\betaright)}_{>0}\betalangle\betaLambda,\betaalpha\betarangle\betalangle\betaLambda,\betabeta\betarangle\betaBig). \betaend{eqnarray*} This implies $\betalangle\betaalpha,\betabeta\betarangle=0$ or $\betalangle\betaLambda,\betaalpha\betarangle\betalangle\betaLambda,\betabeta\betarangle=0$, but this was excluded. This argument is symmetric in $\betaalpha$ and $\betabeta$ hence we get the same result for $s_\betabeta\betaalpha$. Thus we have proved that $\betalangle\betaLambda,s_\betaalpha\betabeta\betarangle=\betalangle\betaLambda,s_\betabeta \betaalpha\betarangle=0$ or $\betalangle\betaalpha,\betabeta\betarangle=0$. Now $\betalangle\betaLambda,s_\betaalpha\betabeta\betarangle=\betalangle\betaLambda,s_\betabeta \betaalpha\betarangle=0$ implies $\betalangle\betaLambda,\betaalpha\betarangle=\betafrac{2\betalangle\betaalpha,\betabeta\betarangle}{\beta|\betaalpha\beta|^2}\betacdot \betafrac{2\betalangle\betaalpha,\betabeta\betarangle}{\beta|\betabeta\beta|^2} \betacdot \betalangle\betaLambda,\betaalpha\betarangle$. Since $\betalangle\betaLambda,\betaalpha\betarangle $ was supposed to be non zero we have that $\betafrac{2\betalangle\betaalpha,\betabeta\betarangle}{\beta|\betaalpha\beta|^2}\betacdot \betafrac{2\betalangle\betaalpha,\betabeta\betarangle}{\beta|\betabeta\beta|^2}=1$ which implies --- since both factors are in $\betamathbb{Z}$ --- that $\beta|\betaalpha\beta|^2=\beta|\betabeta\beta|^2$. This holds if $\betalangle\betaalpha,\betabeta\betarangle\betanot=0$. \betaitem{Claim 2:} {\betaem All roots in $\betaDeltaelta$ have the same length.} Suppose we have short and long roots. Then we can write a long root $\betaalpha$ as the sum of two short ones, lets say $\betaalpha=\betabegin{eqnarray*}ta+\betagamma$. This implies $\betalangle\betaalpha,\betabeta\betarangle\betanot=0$ and $\betalangle\betaalpha,\betagamma\betarangle\betanot=0$. Since $\betaalpha$ is long and $\betabegin{eqnarray*}ta$ and $\betagamma$ are short we have by the first claim that $\betalangle\betaLambda,\betaalpha\betarangle\betacdot \betalangle\betaLambda,\betabegin{eqnarray*}ta\betarangle=0$ and $\betalangle\betaLambda,\betaalpha\betarangle\betacdot \betalangle\betaLambda,\betagamma\betarangle=0$. Now $\betalangle\betaLambda,\betaalpha\betarangle=\betalangle\betaLambda,\betabegin{eqnarray*}ta\betarangle+\betalangle\betaLambda,\betagamma\betarangle$ gives that $\betalangle\betaLambda,\betaalpha\betarangle=0$ for every long root. But this is impossible. Hence all roots have the same length and in particular holds for non-proportional roots \betabegin{eqnarray*}q\betafrac{2\betalangle\betaalpha,\betabeta\betarangle}{\beta|\betaalpha\beta|^2}=\betapm1. \betalanglebel{gleichlang}\betaend{eqnarray*}q \betaitem{Claim 3:} {\betaem There is an $a\betain\betamathbb{N}$ such that for every root $\betaalpha$ holds $\betalangle\betaLambda,\betaalpha\betarangle\betain\beta{0,\betapm a\beta}$. Furthermore $a$ is less or equal than the length of the roots.} We consider $\betaalpha\betain \betaDelta$ with $\betalangle\betaLambda,\betaalpha\betarangle\betanot=0$ and set $a:=\betalangle\betaLambda,\betaalpha\betarangle$. Then we define the vector space $A:= span \beta{\betabegin{eqnarray*}ta\betain\betaDelta\beta |\beta \betalangle\betaLambda,\betabegin{eqnarray*}ta\betarangle=\betapm a\beta}\betasigmaubsetset \betamathfrak{t}^*$. We show that $A=\betamathfrak{t}^*$ and that every root $\betagamma$ with $\betalangle\betaLambda,\betagamma\betarangle\betanot\betain\beta{0,\betapm a\beta}$ is orthogonal to $A$. To verify $A=\betamathfrak{t}^*$ we show that every root is either in $A$ or in $A^\betabot$. First consider $\betagamma\betain\betaDelta$ with $\betalangle\betaLambda,\betagamma\betarangle=0$. If it is not in $A^\betabot$ then there is a root $\betabegin{eqnarray*}ta\betain A$ and a $\betadelta\betanot\betain A$ such that $\betagamma=\betabegin{eqnarray*}ta+\betadelta$. But this implies $0=\betalangle\betaLambda,\betagamma\betarangle=\betalangle\betaLambda,\betabegin{eqnarray*}ta\betarangle+\betalangle\betaLambda,\betadelta\betarangle=\betapm a+\betalangle\betaLambda,\betadelta\betarangle$. Hence $\betadelta\betain A$ and therefore $\betagamma\betain A$ which is a contradiction. Thus $\betagamma\betain A^\betabot$. Now we consider a root $\betagamma$ with $\betalangle\betaLambda,\betagamma\betarangle\betanot\betain \beta{0,\betapm a\beta}$. For any $\betabeta$ with $\betalangle\betaLambda,\betabeta\betarangle=\betapm a$ then we have because of (\betaref{gleichlang}) that $\betalangle\betaLambda,s_\betabegin{eqnarray*}ta\betagamma\betarangle=\betalangle\betaLambda,\betagamma\betarangle\betapm a\betanot=0$. Because of the proof of claim 1 this gives $\betalangle\betabegin{eqnarray*}ta,\betagamma\betarangle=0$. Hence $\betagamma\betain A^\betabot$. Since the root system is indecomposable we have that $A=\betamathfrak{t}^*$. Furthermore we have shown that any root with $\betalangle\betaLambda,\betagamma\betarangle\betanot\betain \beta{0,\betapm a\beta}$ is orthogonal to $A=\betamathfrak{t}^*$. Thus the first part of claim 3 is proved. Now we suppose that $a > c$ where $c$ denotes the length of the roots. We consider an $\betaה\betain\betaDelta$ with $\betalangle\betaLambda,\betaה\betarangle=a$. $s_\betaה(\betaLambda)=\betaLambda-\betafrac{2a}{c}\betaה$ is an extremal weight in $ U$. Then $a>c$ implies $\betaLambda-2\betaה\betain \betaOmega$ but not in $U$. Then the existence of the planar spanning triple $(\betaLambda,-\betaLambda,U)$ implies $\betaLambda-2\betaה=-\betaLambda+\betabegin{eqnarray*}ta$ for a $\betabegin{eqnarray*}ta\betain \betaDelta$. Hence \beta[\betafrac{2\betalangle\betaLambda,\betagamma\betarangle}{c}\beta =\beta 1+\betafrac{2\betalangle\betaה,\betagamma\betarangle}{c}\beta =\beta 2\beta] and therefore $\betalangle\betaLambda,\betagamma\betarangle=a$ and $a=c$ which is a contradiction. \betaend{description} Now we consider for any $\betaalpha\betain \betaDelta$ the set $\betaDelta_\betaalpha^\betabot:=\beta{\betabegin{eqnarray*}ta\betain \betaDelta\beta |\beta \betalangle\betaalpha,\betabeta\betarangle=0\beta}\betasigmaubsetset \betaDelta$. This set is a root system, reduced but not necessarily indecomposable. But we can make the following claim. \betabegin{eqnarray*}gin{description} \betaitem{Claim 4:} {\betaem Let $\betaalpha\betain\betaDelta$ with $\betalangle\betaLambda,\betaה\betarangle\betanot=0$. Then one of the following cases holds: \betabegin{enumerate} \betaitem $\betaDelta_\betaalpha^\betabot$ is orthogonal to $\betaLambda$ or \betaitem there is a unique $\betabegin{eqnarray*}ta\betain\betaDelta_\betaalpha^\betabot$ with $\betalangle\betaLambda,\betabeta\betarangle\betanot=0$ such that \betabegin{enumerate} \betaitem $\betaLambda=\betapm \betafrac{a}{c}(\betaalpha+\betabeta)$ where $c$ is the lengths of the roots, and \betaitem $\betaDelta_\betaalpha^\betabot$ is decomposable with a direct summand $A_1=\beta{\betapm\betabegin{eqnarray*}ta\beta}$. \betaend{enumerate} \betaend{enumerate} } Suppose that there is a $\betabegin{eqnarray*}ta\betain\betaDelta_\betaalpha^\betabot$ with $\betalangle\betaLambda,\betabegin{eqnarray*}ta\betarangle\betanot=0$. W.l.o.g. we can suppose that $\betalangle\betaLambda,\betabegin{eqnarray*}ta\betarangle=\betalangle\betaLambda,\betaalpha\betarangle=\betapm a$. $\betalangle\betaalpha,\betabeta\betarangle$ implies then \beta[s_\betaalpha s_\betabeta(\betaLambda)\beta =\beta \betaLambda\betamp \betafrac{2a}{c}(\betaalpha+\betabegin{eqnarray*}ta).\beta] Now we show with the help of (\betaref{teqn}) that $s_\betaalpha s_\betabeta (\betaLambda) $ is not in $U$: \betabegin{eqnarray*} \betalangle s_\betaalpha s_\betabeta (\betaLambda),T\betarangle &=& \betalangle\betaLambda,T\betarangle - \betafrac{2\betalangle\betaLambda,\betaalpha\betarangle }{\beta|\betaalpha\beta|^2}\betalangle\betaalpha,T\betarangle -\betafrac{2\betalangle\betaLambda,\betabeta\betarangle }{\beta|\betabeta\beta|^2}\betalangle\betabeta,T\betarangle \beta\beta &=&-\betalangle\betaLambda,T\betarangle\betanot=0. \betaend{eqnarray*} But this implies $-\betaLambda=s_\betaalpha s_\betabeta (\betaLambda)=\betaLambda\betapm \betafrac{2a}{c}(\betaalpha+\betabeta)$. By this equation $\betaalpha$ determines $\betabeta$ uniquely. We still have to show that such $\betabegin{eqnarray*}ta$ is orthogonal to all other roots in $\betaDelta_\betaalpha$. For $\betagamma\betanot\betasigmaim\betabegin{eqnarray*}ta$ in $\betaDelta_\betaalpha$ we have \beta[\betalangle\betaLambda,s_\betabegin{eqnarray*}ta\betagamma\betarangle= \betaunderbrace{\betalangle \betaLambda,\betagamma\betarangle}_{=0} - \betafrac{2\betalangle\betabeta,\betagamma\betarangle}{\beta|\betabeta\beta|^2}\betalangle\betaLambda,\betabeta\betarangle.\beta] The uniqueness of $\betabeta$ implies that $\betabeta$ is orthogonal to $\betaDelta_\betaalpha$. \betaitem{Claim 5:} {\betaem The root system of $\betalangleg$ is of type $A_n$ or $D_n$.} The only root system with roots of equal length where the root system $\betaDelta_\betaה^\betabot$ is decomposable for a root $\betaה$ is $D_n$. Hence for every root system different from $D_n$ we have that $\betaDeltaelta_\betaה^\betabot \betabot\beta \betaLambda$ by claim 4. Any root system different from $A_n$ satisfies that $span(\betaDelta_\betaה^\betabot)=\betaה^\betabot$. Both together imply that for any root system different from $D_n$ and $A_n$ we have that $\betaה=\betaLambda$ but this was excluded. \betaend{description} To find the representations of $A_n$ and $D_n$ which obey the above claims we introduce a fundamental system $\betaPi=(\betapi_1,\betaldots, \betapi_n)$ which makes $\betaLambda$ to the highest weight of the representation. Then we have that $\betaLambda=\betasigmaum_{k=1}^n m_k\betaomega_k$ with $m_k\betain\betamathbb{N}\betacup\beta{0\beta}$ and $\betaomega_k$ the fundamental representations. $\betalangle\betaomega_i,\betapi_j\betarangle=\betadelta_{ij}$ implies $m_i=\betalangle\betaLambda,\betapi_i\betarangle\betain\beta{0,a\beta}$. Then we get \betabegin{eqnarray*}gin{description} \betaitem{Claim 6.} {\betaem The root system is of type $D_n$ and the representation is the $a$-th power of a fundamental representation, i.e. $\betaLambda=a\betaomega_i$.} Applying $\betaLambda$ to the root $\betasigmaum_{k=1}^n \betapi_k$ gives $\betasigmaum_{k=1}^n m_k=a$. Applying $\betaLambda$ to any of the $\betapi_i$ gives that $\betasigmaum_{k=1}^n m_k=m_i$ for one $i$. Now we consider the root system $A_n$. $n=1$ was excluded from the beginning. Recalling $A_3\betasigmaimeq D_3$ we can also exclude $A_3$. Now we impose the condition that the representation is orthogonal. This forces $n$ to be odd and $\betaLambda=a \betaomega_{\betafrac{n+1}{2}}$ where $a $ has to be $2$ when $\betafrac{n+1}{2}$ is odd. Thus we can suppose that $n>3$. Using the usual notation we consider now the root $\betasigmaum_{k=1}^n \betapi_k=e_1-e_{n+1}$ for which holds that $\betalangle\betaLambda,\betaeta\betarangle=a$. Hence by claim 4 we have that $\betaDelta_\betaeta^\betabot $ is orthogonal to $\betaLambda$. On the other hand $\betaDelta_\betaeta^\betabot=\beta{\betapm (e_i-e_j)\beta |\beta 2\betale i<j\betale n\beta}$ with $n>3$ is not orthogonal to $a \betaomega_{\betafrac{n+1}{2}}=a\betaleft(e_1+\betaldots +e_{\betafrac{n+1}{2}}\betaright)$. This yields a contradiction. \betaend{description} Finally we show that only the representations of $D_n$ given in the proposition satisfy the derived properties. The fundamental representations of $D_n$ are given by $\betaomega_i=e_1+\betaldots +e_i$ for $i=1\betaldots n-2$ and $\betaomega_i=\betaend{itemize}nhalb (e_1+\betaldots +e_{n-1}\betapm e_n)$ for $i=n-1, n$. Then $\betalangle a\betaomega_i,\betapi_i\betarangle=a$. On the other hand for the largest root $\betaeta=e_1+e_2$ holds \beta[ \betalangle a\betaomega_i,\betaeta\betarangle=\betaleft\beta{ \betabegin{eqnarray*}gin{array}{rcl} a&:&i=1,n-1,n\beta\beta 2a&:&2\betale i\betale n-2. \betaend{array} \betaright.\beta] Hence the representation of $a\betaomega_i$ with $2\betale i\betale n-2$ does not satisfy claim 3. Now we consider for $n>4$ the representations $\betaLambda=\betaend{itemize}nhalb(e_1+\betaldots +e_{n-1}\betapm e_n)$. For the root $\betaה=e_{n-1}\betapm e_n$ holds that $\betalangle\betaLambda,\betaה\betarangle=a\betanot=0$. The roots $\betabeta_1:=e_1-e_2$ and $\betagamma:=e_1+e_3$ both satisfy $\betalangle\betaLambda,\betabeta\betarangle=\betalangle\betaLambda,\betagamma\betarangle=a$ and $\betalangle \betaה,\betabeta\betarangle=\betalangle\betaה,\betagamma\betarangle=0$. But this is a violation of the uniqueness property in claim 4. Hence $n=4$. For $D_4$ it holds that, $\betaomega_3$ and $w_4$ are congruent to $\betaomega_1$, i.e. there is an involutive automorphism of the Dynkin diagram which interchanges $\betaomega_1$ with $\betaomega_3$ respectively $\betaomega_1$ with $\betaomega_4$. For $D_3\betasigmaimeq A_3$ only the representations $\betaomega_2$ and $2\betaomega_2$ are orthogonal. \betaend{proof} Again we get a \betabegin{eqnarray*}gin{folg}\betalanglebel{planarfolge} Every representation of a Lie algebra which satisfies the conditions of proposition \betaref{planarsatz} is the complexification of a Riemannian holonomy representation. \betaend{folg} \betabegin{eqnarray*}gin{proof} The representation with highest weight $\betaomega_1$ of $D_n$ is the standard representation of $\betalangleso(2n,\betamathbb{C})$ in $\betamathbb{C}^{2n}$. Hence it is the holonomy representation of a generic Riemannian manifold. The representation with highest weight $2\betaomega_1$ is the complexified holonomy representation of a symmetric space of type $AI$ for even dimensions, i.e. of $SU(2n)/SO(2n,\betamathbb{R})$ respectively $Sl(2n,\betamathbb{R})/SO(2n,\betamathbb{R})$. \betaend{proof} \betasigmaubsetsection{Representations with the property (SII) and weight zero} Now we will study the property (SII) for representation for which zero is a weight. For this we need a lemma. \betabegin{eqnarray*}gin{lem} Let $\betalangleg\betasigmaubsetset \betalangleso(N,H)$ the irreducible representation of a simple Lie algebra with weights $\betaOmega$. If $0\betain \betaOmega$ then \betabegin{enumerate} \betaitem $ \betaDelta\betasigmaubsetset \betaOmega$ or \betaitem the extremal weights are short roots or \betaitem $\betaDelta=C_n$ and the representation is a fundamental one with highest weight $\betaomega_{2k}$ for $k\betage 2$. \betaend{enumerate} \betaend{lem} \betabegin{eqnarray*}gin{proof} $0\betain\betaOmega $ implies that there is a $\betalanglem\betain \betaOmega$ and an $\betaeta\betain \betaDelta$ such that $0=\betalanglem-\betaeta$, i.e $\betalanglem=\betaeta$. Now we consider two cases. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: $\betaeta$ is a long root.}} Of course we have that the root system of long roots is contained in $\betaW$. We have to show that the short roots are in $\betaW$. This is the case if one short root is in $\betaW$. For this we write $\betaeta=\betaה+\betabeta$ where $\betaה$ and $ \betabeta$ are short roots. If $\betaDelta\betanot= G_2$ we have that $\betalangle\betaה,\betabeta\betarangle=0$. In this case we have that $\betafrac{2\betalangle \betaeta,\betaה\betarangle}{\beta|\betaה\beta|^2}=2$, i.e. $\betaeta-\betaה=\betabeta\betain \betaW$. For $\betaDelta=G_2$ we have that $\betafrac{2\betalangle\betaה,\betabeta\betarangle}{\beta|\betaה\beta|^2}=\betafrac{2\betalangle\betaה,\betabeta\betarangle}{\beta|\betabeta\beta|^2}=1$ and therefore $\betafrac{2\betalangle \betaeta,\betaה\betarangle}{\beta|\betaה\beta|^2}=3$, i.e. $\betaeta-\betaה=\betabeta\betain \betaW$ too. Hence also the short roots are weights and we have $\betaDelta\betasigmaubsetset \betaW$. \betaitem{{\betaem Case 2: $\betaeta$ is a short root.}} Again the short roots are weights. We have to show that one long root is a weight if $\betaeta$ is not extremal or that we are in the case of the $C_n$ with the above representations. If $\betaeta $ is not extremal then exists an $\betaalpha\betain \betaDelta$ such that $\betaeta+\betaalpha\betain \betaW$ and $¸\betaeta-\betaה\betain \betaW$. This $\betaה$ we fix and consider the following cases. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case A: $\betaalpha=\betaeta$, i.e. $2\betaeta\betain\betaW$.}} If $\betaDelta\betanot= G_2$ we find a long root $\betabeta$ such that $\betafrac{2\betalangle \betaeta,\betabeta\betarangle}{\beta|\betaeta\beta|^2}=-2$. This implies that $\betabeta+2\betaeta $ is a long root but also a weight. In case of $G_2$ we find a short root $\betabeta$ with $\betalangle\betaeta,\betabeta\betarangle<0$ and such that $2\betaeta+\betabeta\betain \betaDelta$ a long root. This long root is also in $\betaW$ since $\betalangle\betaeta,\betabeta\betarangle<0$. \betaitem{{\betaem Case B: $\betaה\betanot\betasigmaim\betaeta$ and $\betalangle\betaה,\betaeta\betarangle\betanot= 0$.}} First we consider the case where $\betaה$ is a short root. W.l.o.g. let be $\betalangle\betaה,\betaeta\betarangle< 0$ Then $\betaה+\betaeta$ is a root and a weight. If $\betaDelta$ is different from $C_n$ it is a long root and we are ready. For $C_n$ we have to analyze the situation in detail (see the appendix of \betacite{knapp96}): Let $\betaeta=e_i+e_j$ and $\betaה=e_k-e_j$ with $i\betanot=k$ be the two short roots. Since $\betaW \betani \betaeta-\betaה=2e_j+e_i-e_k$ we have that $\betafrac{2\betalangle \betaeta-\betaה,e_i-e_k\betarangle}{\beta|e_i-e_k\beta|^2}=2$. Hence $\betaeta-\betaה-(e_i-e_k)=2e_j\betain\betaW$. But $2e_j$ is a long root of $C_n$ and we are ready. If $\betaה$ is a long root we proceed as follows. For $G_2$ one of $\betaeta\betapm\betaה$ is a short root, lets say $\betaeta-\betaה$. Then we have that $\betalangle \betaeta+\betaה,\betaeta-\betaה\betarangle<0$ hence $2\betaeta$ is a weight and we may argue as in the first case A. If $\betaDelta$ is different from $G_2$ we write $\betaה=\betaה_1+\betaה_2$ with two orthogonal short roots $\betaה_1$ and $\betaה_2$. For one of these is $\betalangle\betaeta,\betaה_i\betarangle\betanot=0$ and hence $\betaeta\betapm \betaה_i$ a long root, but also a weight. \betaitem{{\betaem Case C: $\betalangle\betaה,\betaeta\betarangle=0$ and $\betaDeltaelta\betanot=C_n$.}} For $G_2$ this case implies that $\betaה$ is a long root and that $\betaeta+\betaה$ is two times a short root. Hence for $G_2$ we can proceed as above to get the result. If $\betaDelta$ is different from $G_2$ we consider the root system $\betaDelta_\betaeta^\betabot$ of roots orthogonal to $\betaeta$, which contains $\betaה$. In case of $C_n$ this root system is equal to $A_1\betatimes C_{n-2}$ and in the remaining cases --- $B_n$ and $F_4$ --- equal to $B_{n-1}$ resp. $B_3$. Now we show that there is a short root $\betaה_1$ in $\betaDelta_\betaeta^\betabot$ such that $\betaeta+\betaה_1\betain\betaW$. If $\betaה$ is short this is trivial and if $\betaה$ is long we write $\betaה=\betaה_1+\betaה_2$ with two orthogonal short roots from $\betaDelta_\betaeta^\betabot$. Then $\betalangle\betaeta+\betaה,\betaה_2\betarangle>0$ and thus $\betaeta+\betaה_1\betain\betaOmega$. On the other hand there is a short root $\betagamma\betain \betaDelta_\betaeta^\betabot$ with $\betaeta+\betagamma $ is a long root. Applying now the Weyl group of $\betaDelta_\betaeta^\betabot$ on $\betaeta+\betagamma$ we get that $\betaeta+\betaה_1 $ is a long root. In case of $C_n$ this argument does not apply since $\betagamma$ spans the $A_1$ factor of $\betaDelta_\betaeta^\betabot$. \betaend{description} Hence we have verified $\betaDelta\betasigmaubsetset \betaW$ in the cases A, B and C. It remains to show that in the situation where $\betalangle\betaה,\betaeta\betarangle=0$, $\betaDeltaelta=C_n$ and neither case A nor case B applies, it holds that $\betaDelta\betasigmaubsetset \betaW$ or the representation of $C_n$ is the one with highest weight $\betaomega_{2k}$ with $k\betage2$. We suppose that $\betaDelta\betanot\betasigmaubsetset \betaW$. Hence no long root can be a weight. First of all we show that under these conditions $\betaה$ has to be a short root. This is true because $\betafrac{2\betalangle \betaeta\betapm\betaה,\betaeta\betarangle}{\beta|\betaeta\beta|^2}=2$ implies $\betaW\betani \betaeta\betapm\betaה-\betaeta=\betapm\betaה$. Hence $\betaה$ has to be short. Secondly we note that neither $\betaeta+\betaה$ nor $\betaeta- \betaה$ can be a root because it would be a long root and a weight. This implies $n\betage 4$. In a third step we show that there is no long root $\betabeta$ such that $\betaeta+\betaה+\betabeta\betain \betaW$ and $\betaeta+\betaה-\betabeta\betain \betaW$. We consider the number \betabegin{eqnarray*}q\betalanglebel{pmbeta} \betafrac{2\betalangle \betaeta+\betaה\betapm\betabeta,\betaה\betarangle}{\beta|\betaה\beta|^2}=2 \betapm \betafrac{2\betalangle \betaה,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaה\beta|^2}. \betaend{eqnarray*}q If $\betalangle\betaה,\betabeta\betarangle=0$ we have that $\betaeta+\betaה\betapm\betabeta-\betaה=\betaeta\betapm\betabegin{eqnarray*}ta\betain \betaW$. But this was excluded (First step or case B). Hence we suppose that $\betafrac{2\betalangle \betaה,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaה\beta|^2}=2$. We still have that $\betaeta+\betabeta\betain \betaOmega$. We consider the number $\betafrac{2\betalangle \betaeta+\betabeta,\betaeta\betarangle}{\beta|\betaeta\beta|^2}=2 \betapm \betafrac{2\betalangle \betaeta,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaeta\beta|^2}\betage 0.$ If this is not zero we have that $\betaW\betani\betaeta+\betabegin{eqnarray*}ta-\betaeta=\betabeta$ which was excluded. Hence $\betafrac{2\betalangle \betaeta,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaeta\beta|^2}=-2$. But this together with $\betafrac{2\betalangle \betaה,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaה\beta|^2}=2$ is a contradiction since the long roots of $C_n$ are of the form $\betapm 2e_i$ and the short ones of the form $\betapm(e_i\betapm e_j)$. Hence if there is a root such that $\betaeta+\betaה+\betabeta\betain \betaW$ and $\betaeta+\betaה-\betabeta\betain \betaW$, it has to be a short one. If there is no such $\betabeta$ then $\betaeta+\betaה$ is extremal. Considering the fundamental weights of $C_n$ this gives easily that the highest weight of the representation is $\betaomega_4$. Finally we suppose that there is such a short root $\betabeta$. Since $\betabeta$ is short equation (\betaref{pmbeta}) implies $\betaeta\betapm\betabeta\betain \betaW$. Since we have excluded case A and B it must hold $\betalangle\betaeta,\betabeta\betarangle=0$ and neither $\betaeta+\betabeta$ nor $\betaeta-\betabeta$ is a root. On the other hand the same holds for $\betaה$ and $\betabegin{eqnarray*}ta$ since any other would imply that $\betaה\betapm\betabeta$ is a long root which was excluded or a short root $\betagamma$ orthogonal to $\betaeta$ and with $\betaeta\betapm\betagamma\betain\betaW$. This way we go on attaining that any extremal wight is the sum of orthogonal short roots whose pairwise sum is no long root. But this is nothing else than the fact that the highest weight of the representation is $\betaomega_{2k}$ for $k\betage 2$. \betaend{description}All in all we have shown the proposition. \betaend{proof} \betabegin{eqnarray*}gin{satz}\betalanglebel{s2satz0} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$ and satisfying (SII). If $\beta 0\betain\betaW$ then there is a root $\betaה$ such that for the extremal weight from property (SII) holds $\betaLambda = a \betaalpha$ or the representation is congruent to one of the following: \betabegin{enumerate} \betaitem $\betaDelta=C_4$ with highest weight $\betaomega_4$. \betaitem $\betaDelta=D_n$ with highest weight $2\betaomega_1$. \betaend{enumerate} \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betaLambda$ and $\betaה$ be the extremal weight and the root from property (SII). We suppose that $\betaLambda$ is not the multiple of a root. First of all we consider the case where $0\betain\betaW_\betaה$. By the previous lemma this is true in the following cases: \betabegin{enumerate} \betaitem[(a)] $\betaDelta\betanot=C_n$, because in this case $\betaDeltaelta\betasigmaubsetset \betaW$. \betaitem[(b)] $\betaDelta=C_n$ but the highest weight of the representation is not equal to $\betaomega_{2k}$ with $k\betage 2$, because this again implies $\betaDelta\betasigmaubsetset \betaW$. \betaitem[(c)] $\betaDelta=C_n$ and $\betaalpha$ is a short root, because for representations with $0\betain\betaW$ holds that the short roots are weights. \betaend{enumerate} For $0\betain\betaW_\betaה$ property (SII) gives $0=\betaLambda -\betaה -\betabeta$ or $0=-\betaLambda+\betabeta$. The second case was excluded thus we have to consider the first case. Suppose that $\betaLambda=\betaה+\betabeta$ where $\betaה+\betabeta\betanot\betasigmaim \betagamma\betain\betaDeltaelta$. In particular $\betaה+\betabeta$ is not a root which implies that $\betalangle\betaה,\betabeta\betarangle\betage 0$. We consider three cases. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: $\betaDelta=G_2$.}} In this case $\betaה+\betabegin{eqnarray*}ta\betanot\betasigmaim\betagamma\betain\betaDelta$ implies $\betalangle\betaה,\betabegin{eqnarray*}ta\betarangle>0$ and $ \betaה$ and $\betabeta$ must have different length. Thus we can chose a long root $\betagamma$ not proportional neither to $\betaה$ nor to $\betabeta$ and such that $\betalangle\betaה,\betagamma\betarangle<0$ and $\betalangle\betabeta,\betagamma\betarangle<0$ which implies $\betagamma\betain\betaW_\betaה$ as well as $\betagamma\betain \betaW_\betabeta$. (SII) implies then $\betagamma-\betabeta\betain\betaDelta$ or $\betagamma-\betaה\betain\betaDelta$ or $\betagamma+\betaה+\betabeta\betain\betaDelta$. The first two cases are not possible because of lemma \betaref{wurzellemma0}. For the third case we suppose that $\betaה$ is the long root and consider $\betafrac{2\betalangle\betagamma+\betabeta,\betaה\betarangle}{\beta|\betaה\beta|^2}=0$ because $\betaה$ is long and both terms have opposite sign. Hence $\betagamma+\betaה+\betabeta$ can not be a root. \betaitem{{\betaem Case 2: $\betaDelta\betanot=G_2$ and $\betalangle\betaה,\betabeta\betarangle>0$.}} This implies $\betaה-\betabeta\betain\betaDeltaelta$. We consider the number $k:= \betafrac{2\betalangle\betaה,\betaה+\betabeta\betarangle}{\beta|\betaה\beta|^2}=2+ \betafrac{2\betalangle\betaה,\betabeta\betarangle}{\beta|\betaה\beta|^2}\betage 3$. Since $G_2$ was excluded we have that $k\betain\beta{3,4\beta}$. Hence $\betaה+\betabeta-k\betaה=\betabeta-(k-1)\betaה\betain\betaW_\betaה$. Then property (SII) implies $\betabeta-(k-1)\betaה=-\betaה-\betabeta+\betagamma$ with $\betagamma \betain \betaDelta_0$, i.e. $2\betabeta-(k-2)\betaה\betain\betaDeltaelta$. At first this implies $k=3$ and thus $\betafrac{2\betalangle\betaה,\betabeta\betarangle}{\beta|\betaה\beta|^2}=1$. Secondly we must have $\betafrac{2\betalangle\betaה,\betabeta\betarangle}{\beta|\betabeta\beta|^2}=2$, therefore $\beta|\betaה\beta|^2=2\beta|\betabeta\beta|^2$, i.e. $\betaה$ as well as $2\betabeta-\betaה$ are long roots and $\betabeta$ and $\betabeta-\betaה$ are short ones. This implies $\betafrac{2\betalangle\betabeta-\betaה,\betaה+\betabeta\betarangle}{\beta|\betabeta-\betaה\beta|^2}= \betafrac{2(\beta|\betabeta\beta|^2-\beta|\betaה\beta|^2)}{\beta|\betabeta\beta|^2}=-2$. Hence $\betaה+\betabeta+2(\betabeta-\betaה)=3\betabeta-\betaה\betain\betaW$ and since $\betafrac{2\betalangle\betaה,\betaה-3\betabeta\betarangle}{\beta|\betaה\beta|^2}=2-3=-1$ holds $\betaה-3\betabeta\betain\betaW_\betaה$. (SII) then gives $\betaה-3\betabeta=\betabeta-\betagamma$ or $\betaה-3\betabeta=-\betabeta-\betaה+\betagamma$ with $\betagamma\betain\betaDelta_0$. But none of these equations can be true. \betaitem{{\betaem Case 3: $\betalangle\betaה,\betabeta\betarangle=0$ and $\betaDelta\betanot=G_2$.}} Since $\betaה+\betabeta\betanot\betasigmaim\betagamma \betain\betaDelta$ the rank of $\betaDelta$ has to be greater than $3$ or it is $\betaDelta=D_n$ and $\betaLambda=2e_i$, i.e. $\betaLambda=2w_1$. In the second case we are ready and we exclude this representation in the following. We can suppose $rk \betaDelta\betage 4$. In this situation we prove the following lemma. \betabegin{eqnarray*}gin{lem} Let $rk\betaDelta\betage 4$ and let $\betaLambda=\betaה+\betabeta$ be an extremal weight of a representation satisfying property (SII) for $(\betaLambda,-\betaLambda+\betaה,\betaה)$ with $\betabeta\betain\betaDelta$ satisfying $\betalangle\betaה,\betabeta\betarangle=0$ and $\betaה+\betabeta\betanot\betasigmaim\betagamma\betain\betaDelta$. Then $\betaDelta$ is a root system with roots of the same length or $\betaDelta=C_n$ and $\betaה$ and $\betabeta$ are two short roots. \betalanglebel{lemma1} \betaend{lem} \betabegin{eqnarray*}gin{proof} Suppose that $\betaDelta$ has roots of different length. First we assume that $\betabeta$ is a long root. We consider the root system $\betaDelta_\betaה^\betabot$ which contains $\betabeta$. We notice that $\betabeta$ lies not in an $A_1$ factor of $\betaDelta_\betaה^\betabot$ because otherwise $\betaה+\betabeta$ would be the multiple of a root. Since $\betabeta$ is long we find a short root $\betagamma\betain\betaDelta_\betaה^\betabot$ such that $\betafrac{2\betalangle\betabeta,\betagamma\betarangle}{\beta|\betagamma\beta|^2}=-2$. Hence $\betaה+\betabeta+2\betagamma\betain\betaW$ and --- since $\betafrac{2\betalangle\betaה,\betaה+\betabeta+2\betagamma\betarangle}{\beta|\betaה\beta|^2}=2$ --- it is $-\betaה-\betabeta-2\betagamma\betain\betaW_\betaה$. But this contradicts property (SII). Now we suppose that $\betaה$ is a long root. Here we consider the root system $\betaDelta_\betabeta^\betabot$ containing $\betaה$. Again $\betaה$ lies not in an $A_1$ factor of $\betaDelta_\betabeta^\betabot$ because otherwise $\betaה+\betabeta$ would be the multiple of a root. Since $\betaה$ is long we find a short root $\betagamma\betain\betaDelta_\betabeta^\betabot$ such that $\betafrac{2\betalangle\betaה,\betagamma\betarangle}{\beta|\betagamma\beta|^2}=-2$. Hence $\betaה+\betabeta+2\betagamma\betain\betaW$. Now we have that $\betafrac{2\betalangle\betaה,\betagamma\betarangle}{\beta|\betaה\beta|^2}=-1$ and therefore $\betafrac{2\betalangle\betaה,\betaה+\betabeta+2\betagamma\betarangle}{\beta|\betaה\beta|^2}=2-1=1$. Thus $-\betaה-\betabeta-2\betagamma\betain\betaW_\betaה$. Again this contradicts (SII). If $\betaה$ and $\betabeta$ are short and orthogonal and the root system is not $C_n$, i.e. it is $B_n$ or $F_4$, then the sum of two orthogonal short roots is the multiple of a root. \betaend{proof} Now we prove a second \betabegin{eqnarray*}gin{lem}\betalanglebel{lemma2} The assumptions of the previous lemma imply that there is no $\betagamma\betain\betaDelta$ such that \betabegin{eqnarray*}q \betalangle\betaה,\betagamma\betarangle=0\betamathbbox{ and }\betafrac{2\betalangle\betabeta,\betagamma\betarangle}{\beta|\betagamma\beta|^2}=1. \betalanglebel{gamma}\betaend{eqnarray*}q \betaend{lem} \betabegin{eqnarray*}gin{proof} Lets suppose that there is a $\betagamma\betain \betaDelta$ such that $\betalangle\betaה,\betagamma\betarangle=0$ and $\betafrac{2\betalangle\betabeta,\betagamma\betarangle}{\beta|\betagamma\beta|^2}=1$. In case of $C_n$ $\betagamma $ is a short root. We note that both together imply that neither $\betaה+\betagamma$ nor $\betaה-\betagamma$ is a root. But $\betagamma-\betabeta$ is a root, in case of $C_n$ a short one. Furthermore $\betaLambda-\betagamma\betain \betaW$ Hence \beta[ \betafrac{2\betalangle\betaLambda-\betagamma,\betagamma-\betabeta\betarangle}{\beta|\betagamma-\betabeta\beta|^2}= \betafrac{2\betalangle\betaה+\betabeta-\betagamma,\betagamma-\betabeta\betarangle}{\beta|\betagamma-\betabeta\beta|^2} =-2 .\beta] Hence $\betaLambda-\betagamma+2(\betagamma-\betabeta)=\betaה-\betabeta+\betagamma\betain \betaW$. Now $\betafrac{2\betalangle\betaה-\betabeta+\betagamma,\betaה\betarangle}{\beta|\betaה\beta|^2}=2$, i.e. $-\betaה+\betabeta-\betagamma\betain \betaW_\betaה$. (SII) implies now that $-\betaה+\betabeta-\betagamma=\betabeta+\betadelta$ or $-\betaה+\betabeta-\betagamma=-\betaה-\betabeta+\betadelta$ for $\betadelta\betain \betaDelta_0$. But both options are not possible since $\betaה+\betagamma$ is not a root and because $\betagamma$ is short. \betaend{proof} We conclude that lemma \betaref{lemma1} left us with representations of $A_n$, $D_n$, $E_6,\beta E_7,\beta E_8$ or $C_n$ where $\betaLambda$ is the sum of two orthogonal (short) roots but not a root. Now one easily verifies that lemma \betaref{lemma2} implies $n\betale 4$ and $\betaDelta\betanot=A_4$. Hence the remaining representations are $2\betaomega_1$, $2\betaomega_3$ and $2\betaomega_4$ of $D_4$, which are congruent to each other, and $w_4$ of $C_4$. \betaend{description} To finish the proof we have to consider the representation of highest weight $\betaomega_{2k}$ (with $k\betage 2$) of $C_n$ supposing $\betaה$ is a long root. $0\betain\betaW$ implies that the short roots are weights. Let $\betabeta$ be a short root with $\betalangle\betaה,\betabeta\betarangle<0$, i.e. $\betabegin{eqnarray*}ta\betain\betaW_\betaalpha$. (SII) then gives $\betabeta=\betaomega_{2k}-\betaה+\betadelta$ or $\betabeta=\betaomega_{2k}-\betadelta$ for a $\betadelta \betain\betaDeltaelta_0$. Analyzing roots and fundamental weights of $C_n$ we get that (SII) implies $k=2$ and $\betaalpha=2e_i$ for $1\betale i\betale 4$. But for $n> 4$ lemma \betaref{lemma2} applies analogously. The remaining representation is $\betaomega_4$ of $C_4$. \betaend{proof} \betabegin{eqnarray*}gin{folg} Let $\betalangleg\betasigmaubsetset\betalangleso(N,\betamathbb{C})$ be an orthogonal algebra of real type different from $\betamathfrak{sl}(2,\betamathbb{C})$ and satisfying (SII). If $0\betain\betaW$, in particular if $\betaDelta =G_2,F_4$ or $E_8$ then it is the complexification of a Riemannian holonomy representation with the exception of $G_2$ in corollary \betaref{wurzelgewichtfolge}. \betaend{folg} \betabegin{eqnarray*}gin{proof} If $\betaLambda$ is the multiple of a root then we are in the situation of corollary \betaref{wurzelgewichtfolge}. For $D_n$ the remaining representations are those which appear in corollary \betaref{planarfolge}. The representation of highest weight $\betaomega_4$ of $C_4$ is the complexification of the holonomy representation of the Riemannian symmetric space of type $EI$, i.e. of $E_6/Sp(4)$ resp. $E_{6(6)}/Sp(4)$. Furthermore analyzing the roots and fundamental representations of the exceptional algebras we notice that every representation of $G_2$, $F_4$ and $E_8$ contains zero as weight. \betaend{proof} \betasigmaubsetsection{Representations with the property (SII) where zero is no weight} First we need a \betabegin{eqnarray*}gin{lem} Let $0\betanot\betain \betaOmega$. Then there is a weight $\betalanglem\betain \betaW$, such that for every root holds $\betaleft|\betafrac{2\betalangle\betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2}\betaright|\betale 1$. \betaend{lem} \betabegin{eqnarray*}gin{proof} Let $\betalanglem$ be a weight and $\betaה$ a root such that $\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2}=:k\betage 2$. If $k$ is even we have that $0\betanot=\betalanglem-\betafrac{k}{2}\betaה\betain\betaW$. But for this weight holds $\betafrac{2\betalangle\betalanglem-\betafrac{k}{2}\betaה,\betaה\betarangle}{\beta|\betaה\beta|^2}=k-k=0$. If $k$ is odd we have that $0\betanot=\betalanglem-\betafrac{k-1}{2}\betaה\betain\betaW$ and $\betafrac{2\betalangle\betalanglem-\betafrac{k-1}{2}\betaה,\betaה\betarangle}{\beta|\betaה\beta|^2}=1$. \betaend{proof} \betabegin{eqnarray*}gin{satz}\betalanglebel{s2satz1} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$, with $0\betanot\betain\betaW$ and satisfying (SII). Then $\betaleft|\betafrac{2\betalangle \betaLambda,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 3$ for all roots $\betabeta\betain\betaDelta$. \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betaה\betain \betaDelta$ with the property (SII), i.e. $\betaOmega_\betaalpha\betasigmaubsetset \betaleft\beta{ \betaLambda-\betaalpha+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}\betacup \betaleft\beta{ -\betaLambda+\betabegin{eqnarray*}ta\beta |\beta \betabegin{eqnarray*}ta\betain \betaDeltaelta_0\betaright\beta}$. By the previous lemma there is a $\betalanglem\betain\betaOmega$ such that $\betaleft|\betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 1$ for all roots $\betabeta\betain\betaDelta$. Applying the Weyl group one can choose $\betalanglem$ such that $\betalangle\betalanglem,\betaה\betarangle<0$. $\betalangle\betalanglem,\betaה\betarangle<0$ implies $\betalanglem\betain\betaOmega_\betaה$. Hence (SII) gives $\betalanglem=\betaLambda-\betaה-\betagamma$ or $\betalanglem=-\betaLambda+\betagamma$ with $\betagamma\betain \betaDelta_0$. The second case gives for every $\betabeta\betain\betaDelta$ \beta[ \betaleft|\betafrac{2\betalangle \betaLambda,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale \betaleft|\betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|+ \betaleft|\betafrac{2\betalangle \betagamma,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 3, \beta] because we have excluded $G_2$. Thus we have to consider the first case $\betaLambda=\betalanglem+\betaה+\betagamma$ with $\betagamma\betain\betaDelta_0$ and have to verify that \betabegin{eqnarray*}q\betalanglebel{kleiner3} \betaleft|\betafrac{2\betalangle \betaLambda,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|= \betaleft|\betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta\beta|^2}+ \betafrac{2\betalangle \betaה,\betabeta\betarangle}{\beta|\betabeta\beta|^2}+ \betafrac{2\betalangle \betagamma,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 3. \betaend{eqnarray*}q for all roots $\betabeta\betain\betaDelta$. For $\betabegin{eqnarray*}ta=\betapm\betaה$ this is satisfied: \beta[ \betafrac{2\betalangle \betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}= \betapm\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2}\betapm 2+ \betafrac{2\betalangle \betagamma,\betaהה\betarangle}{\beta|\betaה\beta|^2} =\betamp 1\betapm 2 +\betafrac{2\betalangle \betagamma,\betaהה\betarangle}{\beta|\betaה\beta|^2}\betale 3. \beta] Now we have to show (\betaref{kleiner3}) for all $\betabeta\betain\betaDelta$ with $\betabeta\betanot\betasigmaim \betaה$. For this we consider three cases. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: All roots have the same length.}} This implies $\betaleft|\betafrac{2\betalangle \betagamma,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 1$ for all roots which are not proportional to each other. Thus we have (\betaref{kleiner3}) for all $\betabeta\betanot\betasigmaim\betagamma $: \beta[ \betaleft|\betafrac{2\betalangle \betaLambda,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale \betaleft|\betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|+ \betaleft|\betafrac{2\betalangle \betaה,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|+ \betaleft|\betafrac{2\betalangle \betagamma,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 3. \beta] For $\betabegin{eqnarray*}ta=\betapm\betagamma$ we have \beta[ \betafrac{2\betalangle \betaLambda,\betagamma\betarangle}{\beta|\betagamma\beta|^2}= \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma\beta|^2} +\betafrac{2\betalangle \betaה,\betagamma\betarangle}{\beta|\betagamma\beta|^2} +2. \beta] This has absolute value $\betage 4 $ only if $\betalangle\betalanglem,\betagamma\betarangle >0$ and $\betalangle \betaה,\betagamma\betarangle>0$. This implies that $\betaה-\betagamma$ is a root. But for this one holds $\betafrac{2\betalangle \betalanglem,\betagamma-\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma-\betaה\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}=2$ since all roots have the same length. This is a contradiction to the choice of $\betalanglem$. \betaitem{{\betaem Case 2: There are long and short roots and $\betabeta$ is a long root.}} This implies again $\betaleft|\betafrac{2\betalangle \betagamma,\betabeta\betarangle}{\beta|\betabeta\beta|^2}\betaright|\betale 1$ for all $\betabeta$ which are not proportional to $\betagamma$. This implies (\betaref{kleiner3}) in this case. For $\betabeta=\betapm \betagamma$ we argue as above, recalling that $\betagamma-\betaalpha$ and $\betaה$ have to be short roots in this case. Hence $\betafrac{2\betalangle \betalanglem,\betagamma-\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma-\betaה\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}\betage \betafrac{2\betalangle \betaLambda,\betagamma\betarangle}{\beta|\betagamma\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2} \betage 2$ which is a contradiction. \betaitem{{\betaem Case 3: There are long and short roots and $\betabeta$ is a short root.}} First we consider the case where $\betabeta=\betapm\betagamma$. Again (\betaref{kleiner3}) is not satisfied only if $\betalangle\betalanglem,\betagamma\betarangle $ and $\betalangle \betaה,\betagamma\betarangle$ are non zero and have the same sign, lets say $+$. If $\betaה$ is a short root too, then because of $\betalangle \betaה,\betagamma\betarangle\betanot=0$ lemma \betaref{wurzellemma0} gives that $\betaה-\betagamma$ is also a short root. Hence $\betafrac{2\betalangle \betalanglem,\betagamma-\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma-\betaה\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2} = 2$ yields a contradiction. If $\betaה$ is a long root, then $\betagamma-\betaה$ has to be a short one and we get again a contradiction: $\betafrac{2\betalangle \betalanglem,\betagamma-\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma-\betaה\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betagamma-\betaה\beta|^2}\betage \betafrac{2\betalangle \betalanglem,\betagamma\betarangle}{\beta|\betagamma\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2} \betage 2$. Now suppose that $\betabegin{eqnarray*}ta\betanot\betasigmaim \betagamma$. Then $\betafrac{2\betalangle \betaLambda,\betabeta\betarangle}{\beta|\betabeta\beta|^2}= \betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta\beta|^2}+ \betafrac{2\betalangle \betaה,\betabeta\betarangle}{\beta|\betabeta\beta|^2}+ \betafrac{2\betalangle \betagamma,\betabeta\betarangle}{\beta|\betabeta\beta|^2}$ has absolute value $\betage 4$ only if all three right hand side terms have the same sign --- lets say they are positive --- and at least one of the last two terms has absolute value greater than one, i.e. $\betagamma$ or $\betaalpha$ is a long root. If $\betaalpha $ is a long root then $\betaalpha-\betabegin{eqnarray*}ta $ is a short one and arguing as above gives the contradiction. If $\betaה$ is a short root then $\betalangle\betaה,\betabeta\betarangle>0$ implies by lemma \betaref{wurzellemma0} that $\betabeta-\betaalpha$ is a short root. Again we have a contradiction: $\betafrac{2\betalangle \betalanglem,\betabeta-\betaה\betarangle}{\beta|\betabeta-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta-\betaה\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betabeta-\betaה\beta|^2}= \betafrac{2\betalangle \betalanglem,\betabeta\betarangle}{\beta|\betabeta\beta|^2} -\betafrac{2\betalangle \betalanglem,\betaה\betarangle}{\beta|\betaה\beta|^2} = 2$. \betaend{description} \betaend{proof} \betabegin{eqnarray*}gin{satz} \betalanglebel{s2satz2} Under the same assumptions as in the previous proposition holds that $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta\betarangle}{\beta|\betaeta\beta|^2}\betaright|\betale 2$ for all long roots $\betaeta$. \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betaLambda$ and $\betaה$ be the extremal weight and the root from property (SII). We suppose that there is a long root $\betaeta$ with \betabegin{eqnarray*}q\betalanglebel{indirekt}\betafrac{2\betalangle\betaLambda,\betaeta\betarangle}{\beta|\betaeta\beta|^2}=-3\betaend{eqnarray*}q and derive a contradiction considering different cases. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: All roots have the same length.}} By applying the Weyl group we find an extremal weight $\betaLambda^pr_{\betalason}ime$ such that $a:=\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=-3$. First we find a root $\betabeta$ with \beta[ \betafrac{2\betalangle\betaה,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabeta\beta|^2}=1\betamathbbox{ and } \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabeta\beta|^2}\betale-2.\beta] This is obvious: We find a $\betabeta$ such that $\betafrac{2\betalangle\betaה,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabeta\beta|^2}=1$. If $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betabegin{eqnarray*}ta\betarangle}{\beta|\betabeta\beta|^2}\betage-1$ then we consider the root $\betaה-\betabeta$. It satisfies $ \betafrac{2\betalangle\betaה,\betaה-\betabegin{eqnarray*}ta\betarangle}{\beta|\betaה-\betabeta\beta|^2}=1$ and we have \beta[\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה-\betabegin{eqnarray*}ta\betarangle}{\beta|\betaה-\betabeta\beta|^2}=-3- \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betabegin{eqnarray*}ta\betarangle}{\beta|\betaה-\betabeta\beta|^2}\betale -2.\beta] Hence we have $\betaLambda^pr_{\betalason}ime+k\betabegin{eqnarray*}ta\betain \betaW$ for $0\betale k\betale 2$ and $\betaLambda^pr_{\betalason}ime+k\betaה\betain \betaW$ for $0\betale k\betale 3$. Furthermore \beta[\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime+l\betabegin{eqnarray*}ta,\betaה\betarangle}{\beta|\betaה\beta|^2}=-3- \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=-3+l.\beta] But this gives \beta[\betaLambda^pr_{\betalason}ime + k \betaה +l\betabeta\betain \betaW_\betaה \betamathbbox{ for } 0\betale k\betale 2, 0\betale k+l\betale 2.\beta] Among others (SII) implies the existence of $\betagamma_i$ and $\betadelta_i$ from $\betaDelta_0$ for $i=0,1, 2$ such that that the following alternatives must hold \betabegin{eqnarray*}gin{align} \betaLambda^pr_{\betalason}ime+\betaה&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or } &&\betaLambda^pr_{\betalason}ime+2\betaה&&=&&-\betaLambda+\betadelta_1 \betalanglebel{zweitens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+2\betabeta&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betabeta&&=&&-\betaLambda+\betadelta_2. \betalanglebel{fuenftens} \betaend{align} First we suppose that the first alternative of (\betaref{nulltens}) holds, i.e $\betaLambda^pr_{\betalason}ime+\betaה=\betaLambda+\betagamma_0$. Since $a=-3$ and both $\betaLambda$ and $\betaLambda^pr_{\betalason}ime$ are extremal we have that $\betaה\betanot=-\betagamma_0$. Hence the first case of (\betaref{zweitens}) can not be true and we have $\betaLambda^pr_{\betalason}ime + 2\betaה=-\betaLambda+\betadelta_1$. We consider now (\betaref{fuenftens}): The left side of (\betaref{nulltens}) gives $\betaLambda^pr_{\betalason}ime+2\betabeta+\betaה=\betaLambda+\betagamma_0+2\betabeta$. If the left side of (\betaref{fuenftens}) holds, we would have $\betagamma_0=-\betabegin{eqnarray*}ta$. Hence $\betaLambda+\betabeta\betain \betaW$ and on the other hand $\betaW\betani \betaLambda^pr_{\betalason}ime+\betaה=\betaLambda-\betabegin{eqnarray*}ta$ which contradicts the extremality of $\betaLambda$. Thus the right hand side of (\betaref{fuenftens}) must be satisfied. From $\betaLambda^pr_{\betalason}ime + 2\betaה=-\betaLambda+\betadelta_1$ follows $\betaLambda^pr_{\betalason}ime +2\betabeta=-\betaLambda+\betadelta_1 +2 (\betabeta-\betaה)$ and therefore $\betadelta_1=-(\betabeta-\betaה)$. Again we have $-\betaLambda+(\betabeta-\betaה)\betain \betaW$ and $-\betaLambda-(\betabeta-\betaה)\betain \betaW$ which contradicts the extremality of $\betaLambda$. If one starts with the right hand side of (\betaref{nulltens}) we can proceed analogously and get a contradiction in the case where all roots have the same length. \betaitem{{\betaem Case 2. The roots have different length and $\betaalpha$ is a short root.}} On one hand we find a short root $\betabeta$ which is orthogonal to $\betaה$ and $\betaה+\betabeta$ is a long root, and on the other we can find an extremal weight $\betaLambda^pr_{\betalason}ime$ such that \beta[\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה+\betabeta\betarangle}{\beta|\betaה+\betabeta\beta|^2}=-3.\beta] Since $\betaה\betabot\betabeta$ we have \beta[ -3\beta =\beta \betafrac{2(\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle+\betalangle\betaLambda^pr_{\betalason}ime,\betabeta\betarangle)}{\beta|\betaה\beta|^2+\beta|\betabeta\beta|^2}\beta\beta \beta =\beta \betafrac{1}{2} \betaleft( \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}+ \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betabeta\betarangle}{\beta|\betabeta\beta|^2} \betaright). \beta] Because of the previous proposition we get \beta[\betafrac{2(\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}\beta =\beta \betafrac{2(\betalangle\betaLambda^pr_{\betalason}ime,\betabeta\betarangle}{\beta|\betabeta\beta|^2} \beta = \beta -3.\beta] Hence $\betaLambda^pr_{\betalason}ime +k\betaה+l\betabeta\betain\betaW$ for $0\betale k,l\betale 3$ and therefore $\betaLambda^pr_{\betalason}ime +k\betaה+l\betabeta\betain\betaW_\betaה$ for $0\betale k\betale 2 $ and $0\betale l\betale 3 $. (SII) implies the following alternatives \betabegin{eqnarray*}gin{align} \betaLambda^pr_{\betalason}ime+\betaה&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{1nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+3\betabeta&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or } &&\betaLambda^pr_{\betalason}ime+3\betabeta&&=&&-\betaLambda+\betadelta_1 \betalanglebel{1erstens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה+3\betabeta&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or } &&\betaLambda^pr_{\betalason}ime+\betaה+3\betabeta&&=&&-\betaLambda+\betadelta_2 \betalanglebel{1zweitens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה+2\betabeta&&=&&\betaLambda+\betagamma_3&&\betamathbbox{ or } &&\betaLambda^pr_{\betalason}ime+2(\betaה+\betabeta)&&=&&-\betaLambda+\betadelta_3 \betalanglebel{1drittens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה+3\betabeta&&=&&\betaLambda+\betagamma_4&&\betamathbbox{ or } &&\betaLambda^pr_{\betalason}ime+2\betaה+3\betabeta&&=&&-\betaLambda+\betadelta_4. \betalanglebel{1viertens} \betaend{align} If the left hand side of the first alternative is valid then the left hand sides of the remaining four can not be satisfied: For (\betaref{1erstens}) we would have $3\betabeta=\betagamma_1-\betagamma_0$ which is not possible. (\betaref{1zweitens}) would imply $3\betabeta+\betaה=\betagamma_2-\betagamma_0$ which is by lemma \betaref{wurzellemma3} a contradiction since $\betaה\betanot=-\betabeta$ and $\betagamma_0\betanot=-\betaה$. (\betaref{1drittens}) would imply $2(\betaה+\betabeta)=\betagamma_3-\betagamma_0$. Since $\betaה+\betabeta$ is a long root this would give $\betagamma_0=-\betaה+\betabeta$ and $\betagamma_3=\betaה+\betabeta$ which is a contradiction to the extremality of $\betaLambda$. (\betaref{1viertens}) would give $2\betaה+3\betabeta=\betagamma_4-\betagamma_0$ which also is not possible. Thus for the last four equations the right hand side must hold. Taking everything together we would get $\betaה=\betadelta_2-\betadelta_1=\betadelta_4-\betadelta_2$ and $\betabeta= \betadelta_4-\betadelta_3$. This gives $2 \betaה= \betadelta_4-\betadelta_1$ and thus \beta[\betafrac{2\betalangle\betadelta_4,\betaה\betarangle}{\beta|\betaה\beta|^2}- \betafrac{2\betalangle\betadelta_1,\betaה\betarangle}{\beta|\betaה\beta|^2}= \betafrac{4 \beta|\betaה\beta|^2}{\beta|\betaה\beta|^2}=4.\beta] The extremality of $\betaLambda$ prevents that $\betaה=\betadelta_4=-\betadelta_1$. Hence $\betadelta_1$ and $\betadelta_4$ are long roots, in particular \beta[ \betafrac{2\betalangle\betadelta_4,\betaה\betarangle}{\beta|\betaה\beta|^2}=- \betafrac{2\betalangle\betadelta_1,\betaה\betarangle}{\beta|\betaה\beta|^2}=2.\beta] For $\betabeta$ again $\betabeta= \betadelta_4=-\betadelta_3$ can not hold by the extremality of $\betaLambda$ and we have \beta[0= \betafrac{2\betalangle\betabeta,\betaה\betarangle}{\beta|\betaה\beta|^2}= \betafrac{2\betalangle\betadelta_4,\betaה\betarangle}{\beta|\betaה\beta|^2}- \betafrac{2\betalangle\betadelta_3,\betaה\betarangle}{\beta|\betaה\beta|^2} =2-\betafrac{2\betalangle\betadelta_3,\betaה\betarangle}{\beta|\betaה\beta|^2}\beta] which forces $\betadelta_3$ to be a long root too. Now we have a contradiction because the short root $\betabeta$ is the sum of two long roots. This is impossible. If we start with the right hand side of the first alternative one proceeds analogously. \betaitem{{\betaem Case 3. The roots have different length and $\betaalpha$ is a long root.}} In this case we find an extremal weight $\betaLambda^pr_{\betalason}ime$ such that $ \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=-3$. Now we can write $\betaה=\betaה_1+\betaה_2$ with $\betaה_1\betabot \betaה_2$ two short roots. As above we get \betabegin{eqnarray*}q \betafrac{2(\betalangle\betaLambda^pr_{\betalason}ime,\betaה_1\betarangle}{\beta|\betaה_1\beta|^2}\beta =\beta \betafrac{2(\betalangle\betaLambda^pr_{\betalason}ime,\betaה_2\betarangle}{\beta|\betaה_2\beta|^2} \beta = \beta -3.\betalanglebel{extremal}\betaend{eqnarray*}q Again this implies $\betaLambda^pr_{\betalason}ime +k\betaה+l\betabeta\betain\betaW$ for $0\betale k,l\betale 3$ and therefore $\betaLambda^pr_{\betalason}ime +k\betaה+l\betabeta\betain\betaW_\betaה$ for $0\betale k,l\betale 2$. Now (SII) implies the existence of $\betagamma_i$ and $\betadelta_i$ from $\betaDelta_0$ for $i=0,\betaldots, 8$ such that that the following alternatives must hold \betabegin{eqnarray*}gin{align} &&(L)&&&&&&&&(R)&&\betanonumber\beta\beta \betaLambda^pr_{\betalason}ime+\betaה_1+\betaה_2&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{2nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה_1+\betaה_2&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה_1&&=&&-\betaLambda+\betadelta_1 \betalanglebel{2erstens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה_1+\betaה_2&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betaה_1&&=&&-\betaLambda+\betadelta_2 \betalanglebel{2zweitens}\beta\beta \betaLambda^pr_{\betalason}ime+ \betaה_1+2\betaה_2&&=&&\betaLambda+\betagamma_3&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה_2&&=&&-\betaLambda+\betadelta_3 \betalanglebel{2drittens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה_1+3\betaה_2&&=&&\betaLambda+\betagamma_4&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betaה_2&&=&&-\betaLambda+\betadelta_4 \betalanglebel{2viertens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה_1+2\betaה_2&&=&&\betaLambda+\betagamma_5&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה_1+\betaה_2&&=&&-\betaLambda+\betadelta_5 \betalanglebel{2fuenftens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה_1+3\betaה_2&&=&&\betaLambda+\betagamma_6&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה_1+2\betaה_2&&=&&-\betaLambda+\betadelta_6 \betalanglebel{2sechstens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה_1+2\betaה_2&&=&&\betaLambda+\betagamma_7&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betaה_1+\betaה_2&&=&&-\betaLambda+\betadelta_7 \betalanglebel{2siebentens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה_1+3\betaה_2&&=&&\betaLambda+\betagamma_8&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betaה_1+2\betaה_2&&=&&-\betaLambda+\betadelta_8. \betalanglebel{2achtens} \betaend{align} In the following we denote the left hand side formulas with an .L and the right hand side formulas with an .R. Again we suppose that (\betaref{2nulltens}.L) is satisfied, i.e. $\betaLambda^pr_{\betalason}ime+\betaה_1+\betaה_2=\betaLambda+\betagamma_0$. Then (\betaref{extremal}) and the extremality of $\betaLambda$ implies that $\betagamma_0$ does not equal to $\betaה_i$. Now (\betaref{2achtens}.L) would imply that $2(\betaה_1+\betaה_2)=2\betaה=\betagamma_8-\betagamma_0$. Since $\betaה$ is a long root this is not possible and we have (\betaref{2achtens}.R), i.e. $\betaLambda^pr_{\betalason}ime+2\betaה_1+2\betaה_2=-\betaLambda+\betadelta_8$. Thinking for a moment gives that (\betaref{2siebentens}.L) implies $\betagamma_0=-\betaה_1$ and (\betaref{2sechstens}.L) implies $\betagamma_0=-\betaה_2$. On the other hand (\betaref{2viertens}.L) implies $\betagamma_0\betanot=-\betaה_1$ and (\betaref{2zweitens}.L) implies $\betagamma_0\betanot=-\betaה_2$. Hence (\betaref{2siebentens}.L) entails (\betaref{2sechstens}.R) and (\betaref{2viertens}.R), as well as (\betaref{2sechstens}.L) entails (\betaref{2siebentens}.R) and (\betaref{2zweitens}.R). Now we suppose that (\betaref{2siebentens}.L) is satisfied. Then we have (\betaref{2achtens}.R), (\betaref{2sechstens}.R) and (\betaref{2viertens}.R), i.e. \beta[\betaה_2=\betadelta_8-\betadelta_7\betamathbbox{ and } 2\betaה_1=\betadelta_8-\betadelta_4.\beta] Again because of the extremality of $\betaLambda$ these roots are not proportional. It implies $2\betaה_1-\betaה_2=\betadelta_7-\betadelta_4$. Now $\betaה_1\betabot\betaה_2$ and $\betadelta_7\betanot=\betaה_1,\betanot=\betaה_1-\betaה_2$ (Extremality of $\betaLambda$) gives a contradiction. In the same way we argue supposing that (\betaref{2sechstens}.L) holds. Hence we have shown that neither (\betaref{2sechstens}.L) nor (\betaref{2siebentens}.L) can be satisfied. Thus we have (\betaref{2sechstens}.R) and (\betaref{2siebentens}.R). These together with (\betaref{2achtens}.L) are no contradiction, but if one supposes one of the remaining (\betaref{2erstens}.R), (\betaref{2drittens}.R) or (\betaref{2fuenftens}.R) we get a contradiction. Hence (\betaref{2erstens}.L), (\betaref{2drittens}.L) and (\betaref{2fuenftens}.L) must be valid. But from these together with (\betaref{2nulltens}.L) we derive as above a contradiction. If we start with the right hand side of the first alternative one proceeds analogously. \betaend{description} All in all we have shown, that the assumption of a long root with (\betaref{indirekt}) leads to a contradiction. \betaend{proof} Now we are in a position that we can use results of \betacite{schwachhoefer2} explicitly. First we will cite them. \betabegin{eqnarray*}gin{satz}\betacite{schwachhoefer2} \betalanglebel{s2schwachhoefer} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$. Then it holds: \betabegin{enumerate} \betaitem If there is an extremal spanning $(\betaLambda_1,\betaLambda_2,\betaה)$ triple then there is no weight $\betalanglem$ for which exists a pair of orthogonal long roots $\betaeta_1$ and $\betaeta_2$ such that $\betaleft|\betafrac{2\betalangle\betalanglem,\betaeta_i\betarangle}{\beta|\betaeta_i\beta|^2}\betaright|= 2$. \betaitem If furthermore all roots have the same length, then there is no weight $\betalanglem$ for which exists a triple of orthogonal roots $\betaeta_1\betabot\betaeta_2\betabot\betaeta_3\betabot\betaeta_1$ such that $\betaleft|\betafrac{2\betalangle\betalanglem,\betaeta_1\betarangle}{\beta|\betaeta_1\beta|^2}\betaright|= 2$ and $\betaleft|\betafrac{2\betalangle\betalanglem,\betaeta_2\betarangle}{\beta|\betaeta_2\beta|^2}\betaright|= \betaleft|\betafrac{2\betalangle\betalanglem,\betaeta_3\betarangle}{\beta|\betaeta_3\beta|^2}\betaright|= 1$. \betaend{enumerate} \betaend{satz} We will now show that existence of such a pair or triple of roots implies that (SII) defines an extremal spanning pair. \betabegin{eqnarray*}gin{satz} \betalanglebel{s2satz3} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$, with $0\betanot\betain\betaW$ and satisfying (SII). Then it holds: If there is a pair of orthogonal long roots $\betaeta_1$ and $\betaeta_2$ such that $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_i\betarangle}{\beta|\betaeta_i\beta|^2}\betaright|= 2$ for the extremal weight $\betaLambda$ from the property $(SII)$, then $\betaLambda-\betaה$ is an extremal weight, i.e. (SII) defines an extremal spanning triple. \betaend{satz} \betabegin{eqnarray*}gin{proof} Again we argue indirectly considering three different cases for the root $\betaה$ from the property (SII) \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: All roots have the same length or $\betaה$ is a long root.}} Again by applying the Weyl group the indirect assumption implies that there is an extremal weight $\betaLambda^pr_{\betalason}ime$ and a root long $\betabeta$ orthogonal to $\betaה$ such that $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betabeta\betarangle}{\beta|\betabeta\beta|^2}=-2$. This gives $\betaLambda^pr_{\betalason}ime +k\betaה+l\betabeta\betain \betaW$ for $0\betale k,l\betale 2$ and hence $ \betaLambda^pr_{\betalason}ime + k \betaה +l\betabeta\betain \betaW_\betaה$ for $ 0\betale k\betale 1, 0\betale l\betale 2.$ Among others (SII) implies the existence of $\betagamma_i$ and $\betadelta_i$ from $\betaDelta_0$ for $i=0,\betaldots, 3$ such that that the following alternatives must hold \betabegin{eqnarray*}gin{align} &&(L)&&&&&&&&(R)&&\betanonumber\beta\beta \betaLambda^pr_{\betalason}ime+\betaה&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{3nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה&&=&&-\betaLambda+\betadelta_1 \betalanglebel{3erstens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+2\betabeta&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betabeta&&=&&-\betaLambda+\betadelta_2 \betalanglebel{3zweitens}\beta\beta \betaLambda^pr_{\betalason}ime+ה_1+2\betaה+2\betabeta&&=&&\betaLambda+\betagamma_3&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה+2\betabeta&&=&&-\betaLambda+\betadelta_3 .\betalanglebel{3drittens} \betaend{align} Supposing again (\betaref{3nulltens}.L) we conclude that (\betaref{3zweitens}.L) and (\betaref{3drittens}.L) can not hold because $\betabeta$ is long and the extremality of $\betaLambda$. Hence (\betaref{3zweitens}.R) and (\betaref{3drittens}.R) must be satisfied. Again the extremality of $\betaLambda$ prevents that (\betaref{3erstens}.R) can be valid. Hence we have (\betaref{3erstens}.L). Now (\betaref{3nulltens}.L) gives \beta[\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}= \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}+2 -\betafrac{2\betalangle\betagamma_0,\betaה\betarangle}{\beta|\betaה\beta|^2} =-\betafrac{2\betalangle\betagamma_0,\betaה\betarangle}{\beta|\betaה\beta|^2}\beta] by assumption. On the other hand (\betaref{3nulltens}.L) together with (\betaref{3erstens}.L) and (\betaref{3zweitens}.R) and (\betaref{3drittens}.R) implies that $\betaה=\betagamma_1-\betagamma_0=\betadelta_3-\betadelta_2$. We note that $\betagamma_0$ can not be equal to $0$ and $\betagamma_1$ not equal to $\betaה$. If $\betagamma_0=-\betaה$ and $\betagamma_1=0$ then $\betaLambda =\betaLambda^pr_{\betalason}ime+2\betaה$. Then (\betaref{3zweitens}.R) and (\betaref{3drittens}.R) imply \beta[\betabegin{array}{rcccl} \betalangle\betadelta_2,\betaה\betarangle&=& 2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle +2\beta|\betaה\beta|^2&=&0\beta;\betamathbbox{ and}\beta\beta \betalangle\betadelta_3,\betaה\betarangle&=& 2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle +3\beta|\betaה\beta|^2&=&\beta|\betaה\beta|^2. \betaend{array}\beta] Since $\betaה$ is long this entails $\betadelta_2=0$ and $\betadelta_3=\betaה$. Taking now (\betaref{3nulltens}.L) and (\betaref{3zweitens}.R) together we get $\betaLambda=\betaה-\betabeta$. But this forces $0\betain\betaW$ which was excluded. Thus we have $\betaה=\betagamma_1-\betagamma_0$ with non-proportionality. But this implies, since $\betaה$ is long, that $\betafrac{2\betalangle\betagamma_0,\betaה\betarangle}{\beta|\betaה\beta|^2}=-1$ and hence $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}= 1$. But this means that $\betaLambda-\betaה$ is an extremal weight. \betaitem{{\betaem Case 2: There are roots with different length and $\betaה$ is a short root.}} By assumption there is a short root $\betagamma$ such that $\betagamma\betabot \betaה$ and $\betaeta:=\betaה+\betagamma$ is a long root and an extremal weight $\betaLambda^pr_{\betalason}ime$ and a long root $\betabeta$ such that $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaeta\betarangle}{\beta|\betaeta\beta|^2}=\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betabeta\betarangle}{\beta|\betabeta\beta|^2}=-2$. Analogously to the previous theorem the orthogonality of $\betaה$ and $\betagamma$ gives \beta[-2=\betaend{itemize}nhalb\betaleft( \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}+\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betagamma\betarangle}{\beta|\betagamma\beta|^2}\betaright).\beta] Hence we have to consider three cases: \betabegin{enumerate} \betaitem[(a)] $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betagamma\betarangle}{\beta|\betagamma\beta|^2}-2$, \betaitem[(b)] $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=-3$ and $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betagamma\betarangle}{\beta|\betagamma\beta|^2}-1$, \betaitem[(c)] $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=-1$ and $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betagamma\betarangle}{\beta|\betagamma\beta|^2}-3$. \betaend{enumerate} Then an easy calculation shows that $\betalangle\betaה,\betabegin{eqnarray*}ta\betarangle=\betalangle\betagamma,\betabeta\betarangle=0$ in each case. We shall consider the case (a),(b) and (c) separately. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case (a):}} Here we can proceed completely analogously to the first case 1. We have that $ \betaLambda^pr_{\betalason}ime + k \betaה +l\betabeta\betain \betaW_\betaה$ for $ 0\betale k\betale 1, 0\betale l\betale 2$ leading to the same set of equations (\betaref{3nulltens}) --- (\betaref{3drittens}) and the same implications since $\betabeta$ is long again. The proportional case is excluded as above and we get that $\betaה=\betagamma_1-\betagamma_0$ non proportional. At least one has to be a short root and $\betalangle \betagamma_0,\betaה\betarangle<0$ and $\betalangle \betagamma_1,\betaה\betarangle>0$. On the other hand we have $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}=-\betafrac{2\betalangle\betagamma_0,\betaה\betarangle}{\beta|\betaה\beta|^2}$ and $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}=-\betafrac{2\betalangle\betagamma_1,\betaה\betarangle}{\beta|\betaה\beta|^2}+2$ by (\betaref{3nulltens}.L) and (\betaref{3erstens}.L). But this implies that both are short and $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}=1$ which is the proposition. \betaitem{{\betaem Case (b):}} $\betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}=-3$ implies $\betaLambda^pr_{\betalason}ime + k \betaה +l\betabeta\betain \betaW_\betaה$ for $0\betale k,l\betale 2$. (SII) then implies \betabegin{eqnarray*}gin{align} &&(L)&&&&&&&&(R)&&\betanonumber\beta\beta \betaLambda^pr_{\betalason}ime+\betaה&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{4nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה&&=&&-\betaLambda+\betadelta_1 \betalanglebel{4erstens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betaה&&=&&-\betaLambda+\betadelta_2 \betalanglebel{4zweitens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה+2\betabeta&&=&&\betaLambda+\betagamma_3&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה+2\betabeta&&=&&-\betaLambda+\betadelta_3 \betalanglebel{4drittens}\beta\beta \betaLambda^pr_{\betalason}ime+3\betaה+2\betabeta&&=&&\betaLambda+\betagamma_4&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betaה+2\betabeta&&=&&-\betaLambda+\betadelta_3. \betalanglebel{4viertens} \betaend{align} Supposing (\betaref{4nulltens}.L) excludes (\betaref{4drittens}.L) and (\betaref{4viertens}.L) because $\betabeta$ is long. Hence (\betaref{4drittens}.R) and (\betaref{4viertens}.R) are valid and exclude (\betaref{4erstens}.R) and (\betaref{4zweitens}.L). Hence (\betaref{4erstens}.L) and (\betaref{4zweitens}.L) are satisfied. This gives $\betaה=\betagamma_2-\betagamma_1=\betagamma_1-\betagamma_0$ with $\betagamma_0$ different from $0$ and $-\betaה$, $\betagamma_1$ different from $0$ and $\betaה$ and $\betagamma_2$ different from $\betapm\betaה$. Hence $\betaה+\betapm\betadelta_1$ is a long root with $\betaה\betabot\betadelta_1$. But this gives $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}= \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}+4=1$, i.e. $\betaLambda-\betaה$ is an extremal weight. \betaitem{{\betaem Case (c):}} Here we have that $\betaLambda^pr_{\betalason}ime +k\betagamma +l\betabeta\betain\betaW_\betaה$ for $0\betale k\betale 3$ and $0\betale l\betale 2$ since $\betafrac{2\betalangle \betaLambda^pr_{\betalason}ime +k\betagamma +l\betabeta,\betaהה\betarangle}{\beta|\betaה\beta|^2}=-1$. The equations implied by (SII) lead easily to a contradiction: \betabegin{eqnarray*}gin{align} &&(L)&&&&&&&&(R)&&\betanonumber\beta\beta \betaLambda^pr_{\betalason}ime+\betaה&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{5nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+3\betagamma&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+3\betagamma&&=&&-\betaLambda+\betadelta_1 \betalanglebel{5erstens}\beta\beta \betamakebox[2.5cm][r]{$\betaLambda^pr_{\betalason}ime+\betaה+2\betabegin{eqnarray*}ta+3\betagamma$}&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+2\betabeta+3\betagamma&&=&&-\betaLambda+\betadelta_2. \betalanglebel{5zweitens} \betaend{align} Supposing (\betaref{5nulltens}.L) excludes (\betaref{5erstens}.L) and (\betaref{5zweitens}.L). Hence (\betaref{5erstens}.R) and (\betaref{5zweitens}.R) are valid but contradict to each other because $\betabeta$ is long. \betaend{description} \betaend{description} \betaend{proof} \betabegin{eqnarray*}gin{satz} \betalanglebel{s2satz3a} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$, with $0\betanot\betain\betaW$ and satisfying (SII). If furthermore all roots have the same length, and if there is a triple of orthogonal roots $\betaeta_1\betabot\betaeta_2\betabot\betaeta_3\betabot\betaeta_1$ such that $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_1\betarangle}{\beta|\betaeta_1\beta|^2}\betaright|= 2$ and $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_2\betarangle}{\beta|\betaeta_2\beta|^2}\betaright|= \betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_3\betarangle}{\beta|\betaeta_3\beta|^2}\betaright|= 1$ then holds one of the cases \betabegin{enumerate} \betaitem $\betaLambda-\betaה$ is an extremal weight, i.e. (SII) defines an extremal spanning triple, or \betaitem $\betaLambda=\betaה+\betaend{itemize}nhalb(\betabeta+\betagamma)$ with roots $\betaה\betabot\betabeta\betabot\betagamma\betabot\betaה$. \betaend{enumerate} \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betaה$ be the root determined by (SII). The assumption implies that there is an extremal weight $\betaLambda^pr_{\betalason}ime$ and roots $\betabeta$ and $\betagamma$ such that \beta[ \betafrac{2\betalangle\betaLambda^pr_{\betalason}ime,\betaה\betarangle}{\beta|\betaה\beta|^2}= -2\beta;\betamathbbox{ and }\beta; \betafrac{2\betalangle\betaLambda,\betabeta\betarangle}{\beta|\betabeta\beta|^2}= \betafrac{2\betalangle\betaLambda,\betagamma\betarangle}{\beta|\betagamma\beta|^2}= -1.\beta] Then $\betaLambda^pr_{\betalason}ime +k\betaה+l\betabeta+m\betagamma\betain \betaW_ה$ for $k,l,m=0,1$. Hence (SII) implies again \betabegin{eqnarray*}gin{align} &&(L)&&&&&&&&(R)&&\betanonumber\beta\beta \betaLambda^pr_{\betalason}ime+\betaה&&=&&\betaLambda+\betagamma_0&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime&&=&&-\betaLambda+\betadelta_0 \betalanglebel{6nulltens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה&&=&&\betaLambda+\betagamma_1&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה&&=&&-\betaLambda+\betadelta_1 \betalanglebel{6erstens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+\betabeta&&=&&\betaLambda+\betagamma_2&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betabeta&&=&&-\betaLambda+\betadelta_2 \betalanglebel{6zweitens}\beta\beta \betaLambda^pr_{\betalason}ime+ 2\betaה+\betabeta&&=&&\betaLambda+\betagamma_3&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה+\betabeta&&=&&-\betaLambda+\betadelta_3 \betalanglebel{6drittens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+\betagamma&&=&&\betaLambda+\betagamma_4&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betagamma&&=&&-\betaLambda+\betadelta_4 \betalanglebel{6viertens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה+\betagamma&&=&&\betaLambda+\betagamma_5&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה+\betagamma&&=&&-\betaLambda+\betadelta_5 \betalanglebel{6fuenftens}\beta\beta \betaLambda^pr_{\betalason}ime+\betaה+\betabeta+\betagamma&&=&&\betaLambda+\betagamma_6&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betabeta+\betagamma&&=&&-\betaLambda+\betadelta_6 \betalanglebel{6sechstens}\beta\beta \betaLambda^pr_{\betalason}ime+2\betaה+\betabeta+\betagamma&&=&&\betaLambda+\betagamma_7&&\betamathbbox{ or }&&\betaLambda^pr_{\betalason}ime+\betaה+\betabeta+\betagamma&&=&&-\betaLambda+\betadelta_7. \betalanglebel{6siebentens} \betaend{align} Supposing (\betaref{6nulltens}.L) excludes (\betaref{6siebentens}.R) because of the orthogonality of the roots. Thus it must hold (\betaref{6siebentens}.R). Now we consider two cases: \betabegin{eqnarray*}gin{description} \betaitem{{\betaem Case 1: $\betalangle\betagamma_0,\betabeta\betarangle=\betalangle\betagamma_0,\betagamma\betarangle=0$.}} This excludes (\betaref{6zweitens}.L), (\betaref{6viertens}.L) and (\betaref{6sechstens}.L) and implies therefore (\betaref{6zweitens}.R), (\betaref{6viertens}.R) and (\betaref{6sechstens}.R). The latter together with (\betaref{6siebentens}.R) gives $\betaה=\betadelta_7-\betadelta_6$. Since $\betadelta_7\betanot=0$ this implies $\betalangle\betaה,\betadelta_7\betarangle >0$. On the other hand (\betaref{6siebentens}.R) and the assumption gives $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}=\betafrac{2\betalangle\betadelta_7,\betaה\betarangle}{\beta|\betaה\beta|^2}>0$. If $\betaה\betanot=\betadelta_7$ we are done. But $\betadelta_7=\betaה$ implies $\betaLambda^pr_{\betalason}ime+\betabeta+\betagamma=-\betaLambda=-\betaLambda^pr_{\betalason}ime-\betaה-\betagamma_0$ and hence $-2=2-2-\betafrac{2\betalangle\betaה,\betagamma_0\betarangle}{\beta|\betaה\beta|^2}$, i.e. $\betagamma_0=-\betaה$. Taking everything together we get $2\betaLambda=2\betaה-(\betabeta+\betagamma)$. \betaitem{{\betaem Case 2: $\betalangle\betagamma_0,\betabeta\betarangle$ or $\betalangle\betagamma_0,\betagamma\betarangle$ not equal to zero.}} This implies $\betagamma_0\betanot=\betapm \betaה$ and thus $\betafrac{2\betalangle\betaLambda,\betaה\betarangle}{\beta|\betaה\beta|^2}=-\betafrac{2\betalangle\betagamma_0,\betaה\betarangle}{\beta|\betaה\beta|^2}=\betapm 1$ or zero. Now (\betaref{6erstens}.L) would imply $\betaה=\betagamma_1-\betagamma_0$, i.e. $\betalangle\betaה,\betagamma_0\betarangle< 0$. This would be the proposition. Hence we suppose (\betaref{6erstens}.R). This together with the starting point (\betaref{6nulltens}.L) gives \betabegin{eqnarray*} \betaLambda&=&\betaend{itemize}nhalb\betaleft(\betadelta_1-\betagamma_0\betaright)\beta;\beta;\betamathbbox{ and}\beta\beta \betaLambda^pr_{\betalason}ime&=&-\betaה +\betaend{itemize}nhalb (\betadelta_1+\betagamma_0). \betaend{eqnarray*} The second equation implies using the assumption that $\betalangle\betaה,\betadelta_1+\betagamma_0\betarangle=0$. For the length of both extremal weights then holds \betabegin{eqnarray*} \beta|\betaLambda\beta|^2&=& \betafrac{1}{4}\betaleft( \beta|\betadelta_1\beta|^2+\beta|\betagamma_0\beta|^2 - 2\betalangle\betadelta_1,\betagamma_0\betarangle\betaright)\beta\beta \beta|\betaLambda^pr_{\betalason}ime\beta|^2&=& \beta|\betaה\beta|^2 - \betaunderbrace{\betalangle\betaה,\betadelta_1+\betagamma_0\betarangle}_{=0}+ \betafrac{1}{4}\betaleft( \beta|\betadelta_1\beta|^2+\beta|\betagamma_0\beta|^2 + 2\betalangle\betadelta_1,\betagamma_0\betarangle\betaright). \betaend{eqnarray*} This gives $0=\beta|\betaה\beta|^2+\betalangle\betadelta_1,\betagamma_0\betarangle$. Since all roots have the same length this implies $ \betadelta_1=-\betagamma_0$. Hence $\betaLambda$ is a root. But this was excluded. \betaend{description} \betaend{proof} Now using the proposition \betaref{s2schwachhoefer} of Schwachh\beta"ofer we get a corollary. \betabegin{eqnarray*}gin{folg} \betalanglebel{sf} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$, with $0\betanot\betain\betaW$ and satisfying (SII). Then it holds: \betabegin{enumerate} \betaitem \betalanglebel{sf1} There is no a pair of orthogonal long roots $\betaeta_1$ and $\betaeta_2$ such that $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_i\betarangle}{\beta|\betaeta_i\beta|^2}\betaright|= 2$ for the extremal weight $\betaLambda$ from the property $(SII)$. \betaitem\betalanglebel{sf2} If furthermore all roots have the same length, and if there is a triple of orthogonal roots $\betaeta_1\betabot\betaeta_2\betabot\betaeta_3\betabot\betaeta_1$ such that $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_1\betarangle}{\beta|\betaeta_1\beta|^2}\betaright|= 2$ and $\betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_2\betarangle}{\beta|\betaeta_2\beta|^2}\betaright|= \betaleft|\betafrac{2\betalangle\betaLambda,\betaeta_3\betarangle}{\beta|\betaeta_3\beta|^2}\betaright|= 1$ then $\betaLambda=\betaה+\betaend{itemize}nhalb(\betabeta+\betagamma)$ with roots $\betaה\betabot\betabeta\betabot\betagamma\betabot\betaה$. \betaend{enumerate} \betaend{folg} Before we apply this corollary we have to deal with the remaining exception in the second point. \betabegin{eqnarray*}gin{lem}\betalanglebel{ausnahmelemma} If the representation of a simple Lie algebra with roots of the same length has an extremal weight $\betaLambda$ such that $\betaLambda=\betaה+\betaend{itemize}nhalb(\betabeta+\betagamma)$ with roots $\betaה\betabot\betabeta\betabot\betagamma\betabot\betaה$. Then it holds \betabegin{enumerate} \betaitem There is no root $\betadelta$ such that $\betalangle\betadelta,\betabeta\betarangle=0$, $\betalangle\betadelta,\betagamma\betarangle\betanot=0$ and $\betadelta\betanot\betasigmaim\betagamma$. \betaitem The root system is $D_n$ and the representation has one of the following highest weights: $\betaomega_3$ for arbitrary $n$, $\betaomega_1+\betaomega_3$ or $\betaomega_1+\betaomega_4$ for $n=4$ and $\betaomega_2$ for $n=3$. \betaend{enumerate}\betaend{lem} \betabegin{eqnarray*}gin{proof} The first point is easy to see: If there is such a $\betadelta$ then we have \beta[\betafrac{2\betalangle\betaLambda,\betadelta\betarangle}{\beta|\betadelta\beta|^2}= \betafrac{2\betalangle\betaה,\betadelta\betarangle}{\beta|\betadelta\beta|^2}+\betaend{itemize}nhalb \betafrac{2\betalangle\betagamma,\betadelta\betarangle}{\beta|\betadelta\beta|^2}= \betafrac{2\betalangle\betaה,\betadelta\betarangle}{\beta|\betadelta\beta|^2}\betapm\betaend{itemize}nhalb\betanot\betain\betamathbb{Z}.\beta] This is a contradiction. Now we consider the different root systems with roots of constant length. \betabegin{eqnarray*}gin{description} \betaitem{$A_n$:} Here the assumption means that $\betaLambda=e_i-e_j+\betaend{itemize}nhalb \betaleft(e_p-e_q+e_r-e_s\betaright)$ with all indices different from each other. But then $\betafrac{2\betalangle\betaLambda,e_i-e_p\betarangle}{\beta|e_i-e_p\beta|^2}$ is not an integer. \betaitem{$D_n$:} If $\betaה=e_i\betapm e_j$, $\betabeta=e_p\betapm e_q$ and $\betagamma=e_r\betapm e_s$ with all indices different we get the same contradiction as in the $A_n$ case. Thus we are left with two cases. The first is $\betabeta+\betagamma=e_p+e_q+e_p-e_q=2e_p$ and hence $\betaLambda=e_i\betapm e_j +e_p$. This leads to $\betaLambda=\betaomega_3$ or for $n=3$ to $\betaLambda=\betaomega_2$. The second is $\betaה=e_i+e_j$, $\betabeta=e_i-e_j$ and $\betagamma=e_p\betapm e_q$. For $n>4$ we found a root $e_p+e_s$ which leads to a contradiction by applying the first point. For $n=4$ we have $\betaLambda=\betafrac{3}{2}e_i+\betaend{itemize}nhalb\betaleft(e_j+e_p\betapm e_q\betaright)$. But this yields the remaining representations. \betaitem{$E_6$:} $E_6 $ has two different types of roots: \beta[ e_i\betapm e_j\beta;\beta;\betamathbbox{ and }\beta;\beta; \betaend{itemize}nhalb\betabegin{itemize}g( e_8-e_7-e_6 \beta \betaunderbrace{\betapm e_5\betapm e_4\betapm e_3\betapm e_2\betapm e_1}_{\betamakebox[2cm][c]{even number of minus signs}}\betabegin{itemize}g). \beta] The only possibility for $\betabeta$ and $\betagamma$ for which the first point yields no contradiction is $\betabeta=e_i+e_j$ and $\betagamma=e_i-e_j$. Hence $ \betaLambda=\betaה+e_i$. $\betaה\betabot\betabeta$ and $\betaה\betabot\betagamma $ implies $\betaה=e_p+e_q$. But then $\betalangle\betaLambda, \betaend{itemize}nhalb( \betaldots )\betarangle\betanot\betain \betamathbb{Z}$. \betaend{description} Proceeding analogously for $E_7$ and $E_8$ we prove the second assertion. \betaend{proof} Now using all these properties we can find the representations without weight zero and satisfying (SII). \betabegin{eqnarray*}gin{satz} \betalanglebel{s2satz4} Let $\betalangleg \betasigmaubsetset \betalangleso(N, \betamathbb{C} ) $ be an irreducible representation of real type of a complex simple Lie algebra different from $\betamathfrak{sl}(2,\betamathbb{C})$, with $0\betanot\betain\betaW$ and satisfying (SII). Then the roots system and the highest weight of the representation is is one of the following (modulo congruence): \betabegin{eqnarray*}gin{description} \betaitem{$A_n$:} $\betaomega_4$ for $n=7$. \betaitem{$B_n$:} $\betaomega_n$ for $n=3,4,7$. \betaitem{$D_n$:} $\betaomega_1$, $2\betaomega_1$ for arbitrary $n$ and $\betaomega_8$ for $n=8$. \betaend{description} \betaend{satz} \betabegin{eqnarray*}gin{proof} We apply proposition \betaref{s2satz2} and corollary \betaref{sf} to the remaining representations with $0\betanot\betain\betaW$, i.e. representations of $A_n,\beta B_n,\beta C_n,\beta D_n,\beta E_6$ and $E_7$. Therefore we use a fundamental system such that the extremal weight $\betaLambda$ determined by (SII) is the highest weight. It can be written in the fundamental representations $\betaLambda=\betasigmaum_{k=1}^n m_k\betaomega_k$ with $m_k\betain\betamathbb{N}\betacup\beta{0\beta}$. \betabegin{eqnarray*}gin{description} \betaitem{$A_n$:} Proposition \betaref{s2satz2} gives for the largest root \beta[2\beta \betage\beta \betafrac{2\betalangle\betaLambda,e_1-e_{n+1}\betarangle}{\beta|e_1-e_{n+1}\beta|^2}\beta =\beta \betasigmaum_{k=1}^n m_k\betalangle\betaomega_k,e_1-e_{n+1}\betarangle\beta =\beta \betasigmaum_{k=1}^n m_k.\beta] Since the representation has to be self dual we have that $m_i=m_{n+1-i}$. First we consider the case that $\betaLambda=\betaomega_i+\betaomega_{n+1-i}$. For $n>2$ we get in case $i>1$ that $\betalangle\betaLambda,e_2-e_n\betarangle=2$. But $(e_2-e_n)\betabot(e_1-e_{n+1})$ gives a contradiction to \betaref{sf1} of corollary \betaref{sf}. For $n\betage 2$ it has to be \beta[\betaLambda=\betaomega_1+\betaomega_n=2e_1 + e_2 +\betaldots e_n=e_1-e_{n+1}\beta] recalling that for $A_n$ holds that $e_1=-(e_2+\betaldots +e_{n+1})$. Thus the representation is the adjoint one with $0\betain \betaW$. Now we consider the case that $n+1$ is even and $\betaLambda=2\betaomega_{\betafrac{n+1}{2}}$. This representation is orthogonal but again we have $\betalangle\betaLambda,e_2-e_n\betarangle=2$ for $n>2$. But this is impossible because of point \betaref{sf1} of corollary \betaref{sf}. For $n+1$ even we have to study the case $\betaLambda=\betaomega_{\betafrac{n+1}{2}}$. This representation is orthogonal if $\betafrac{n+1}{2}$ is even. The weights of this representation are given by $\betapm e_{k_1}\betapm \betaldots \betapm e_{k_{\betafrac{n+1}{2}}}$ where the $\betapm$'s are meant to be independent of each other. We will show that (SII) implies $n\betale 7$. Hence suppose that there is a root $\betaה$ such that (SII) with $\betaLambda$. We have to consider two cases for $\betaה$. The first is that $\betaה=e_i-e_j$ with $1\betale i\betale\betafrac{n+1}{2}<j\betale n+1$. W.l.o.g. we take $\betaה=e_{\betafrac{n+1}{2}}-e_{\betafrac{n+1}{2}+1}$ and consider the weight \beta[\betalanglem:=e_1+\betaldots e_{\betafrac{n+1}{2}-3}+e_{\betafrac{n+1}{2}+1}+e_{\betafrac{n+1}{2}+2}+e_{\betafrac{n+1}{2}+3}.\beta] $\betalangle\betalanglem,\betaה\betarangle<0$ implies $\betalanglem\betain\betaW_\betaה$. Then $\betalanglem-(\betaLambda-\betaה)\betain \betaDeltaelta_0$ or $\betalanglem+\betaLambda\betain\betaDeltaelta_0$. We check the first alternative: $\betaLambda-\betaה=e_1+\betaldots e_{\betafrac{n+1}{2}-1}+e_{\betafrac{n+1}{2}+1}$ implies \beta[ \betalanglem-(\betaLambda-\betaה) =e_{\betafrac{n+1}{2}-3}+e_{\betafrac{n+1}{2}-2}+e_{\betafrac{n+1}{2}+2}+e_{\betafrac{n+1}{2}+3}.\beta] But this is not a root. For the second alternative we get, recalling that $-e_1=e_2+\betaldots + e_{n+1}$, \beta[ \betalanglem+\betaLambda =e_1+\betaldots +e_{\betafrac{n+1}{2}-3}-e_{\betafrac{n+1}{2}+4}-\betaldots -e_{n+1}.\beta] This is not a root if $\betafrac{n+1}{2}>4$, i.e. $n>7$. For the second type of root $\betaה=e_i-e_j$ with $1\betale i< j\betale \betafrac{n+1}{2}$ and $\betafrac{n+1}{2}<i<j\betale n+1$ one derives analogously that $n\betale 5$. Hence for $\betaLambda=\betaomega_{\betafrac{n+1}{2}}$ the property (SII) can only be fulfilled if $n\betale 7$. These representations are orthogonal for $n=7$ and $n=3$. $A_3 $ is isomorphic to $D_3$ and the representation with highest weight $\betaomega_2$ of $A_3$ is equivalent to the one with $\betaomega_1$ of $D_3$. \betaitem{$B_n$:} Again proposition \betaref{s2satz2} gives for the largest root \beta[2\beta \betage\beta \betafrac{2\betalangle\betaLambda,e_1+e_2\betarangle}{\beta|e_1+e_2\beta|^2}\beta =\beta \betasigmaum_{k=1}^n m_k\betalangle\betaomega_k,e_1+e_2\betarangle\beta =\beta m_1+2m_2+\betaldots 2m_{n-1}+m_n.\beta] The only representations with $0\betanot\betain \betaW$ are these with $\betaLambda=\betaomega_1+\betaomega_n$ and the spin representation $\betaLambda=\betaomega_n$. There is no possibility to apply the first point of corollary \betaref{sf}. But we verify that for $\betaLambda=\betaomega_1+\betaomega_n$ (SII) implies $n\betale 2$ and for the spin representation $\betaLambda=\betaomega_n$ (SII) implies $n\betale 7$. \betabegin{eqnarray*}gin{description} \betaitem{{\betaem The spin representations:}} For these we show that (SII) implies $n\betale 7$ The spin representation of highest weight $\betaLambda=\betaend{itemize}nhalb(e_1+\betaldots +e_n)$ has weights $\betaW=\betaleft\beta{\betaend{itemize}nhalb(\betavarepsilon_1 e_1 +\betaldots +\betavarepsilon_n e_n)|\betavarepsilon_i=\betapm 1 \betaright\beta}$. We have to consider three types for the root $\betaה$: $\betaה=e_i$, $\betaה=e_i+e_j$ and $\betaה=e_i-e_j$. For the first we can assume w.l.o.g. that $\betaה=e_1$. Then $\betaW_\betaה=\beta{\betaend{itemize}nhalb(-e_1+\betavarepsilon_2 e_2 +\betaldots +\betavarepsilon_n e_n)|\betavarepsilon=\betapm 1\beta}$. It is $\betaLambda-\betaה=\betaend{itemize}nhalb(-e_1+e_2+\betaldots +e_n)$. Hence for $\betalanglem\betain \betaW_\betaה$ we have \betabegin{eqnarray*} \betaLambda-\betaה-\betalanglem&=&\betaend{itemize}nhalb((1-\betavarepsilon_2)e_2 + \betaldots + (1-\betavarepsilon_n)e_n )\beta;\betamathbbox{ and }\beta\beta \betaLambda+\betalanglem&=&\betaend{itemize}nhalb((1+\betavarepsilon_2)e_2 + \betaldots +(1+\betavarepsilon_n)e_n ) \betaend{eqnarray*} If (SII) is satisfied at least one of these expression has to be a root. But if $n\betage 7$ we can choose $(\betavarepsilon_2, \betaldots \betavarepsilon_n)$ such that non of them is a root. The second type of root shall be w.l.o.g. $\betaה=e_1-e_2$. In this case $\betaW_\betaה=\beta{\betaend{itemize}nhalb(-e_1+e_2+\betavarepsilon_3 e_3 +\betaldots +\betavarepsilon_n e_n)|\betavarepsilon_i=\betapm 1\beta}$ and $\betaLambda-\betaה=\betaend{itemize}nhalb(-e_1+2e_2+e_3+\betaldots +e_n)$. Hence for $\betalanglem\betain\betaW_\betaה$ \betabegin{eqnarray*} \betaLambda-\betaה-\betalanglem&=&\betaend{itemize}nhalb(e_2+(1-\betavarepsilon_3)e_3 + \betaldots +(1-\betavarepsilon_n)e_n )\beta;\betamathbbox{ and }\beta\beta \betaLambda+\betalanglem&=&\betaend{itemize}nhalb(2e_2+(1+\betavarepsilon_3)e_3 + \betaldots +(1+\betavarepsilon_n)e_n ) \betaend{eqnarray*} We can choose $\betalanglem$ such that none of them is a roots if $n\betage 4$. Now we consider the last type of root, $\betaה=e_1+e_2$. $\betaW_\betaה=\beta{\betaend{itemize}nhalb(-e_1-e_2+\betavarepsilon_3 e_3 +\betavarepsilon_n e_n)|\betavarepsilon_i=\betapm 1\beta}$ and $\betaLambda-\betaה=\betaend{itemize}nhalb(-e_1-e_2+e_3+\betaldots +\betaldots +e_n)$. Hence for $\betalanglem\betain\betaW_\betaה$ \betabegin{eqnarray*} \betaLambda-\betaה-\betalanglem&=&\betaend{itemize}nhalb((1-\betavarepsilon_3)e_3 + \betaldots +(1-\betavarepsilon_n)e_n )\beta;\betamathbbox{ and }\beta\beta \betaLambda+\betalanglem&=&\betaend{itemize}nhalb((1+\betavarepsilon_3)e_3 + \betaldots +(1+\betavarepsilon_n)e_n ) \betaend{eqnarray*} We can choose $\betalanglem$ such that none of them is a roots if $n\betage 8$. Hence if (SII) is satisfied it has to be $n\betale 7$ and for $n=7$ the pair of property (SII) is of the shape $(\betaLambda, e_1+e_2)$. Now for $n=2$, $n=5$ and $n=6$ the spin representations are symplectic but not orthogonal. \betaitem{{\betaem The representations of $\betaLambda=\betaomega_1+\betaomega_n=\betafrac{3}{2}e_1 +\betaend{itemize}nhalb(e_2+\betaldots +e_n)$.}} Then the weights are given by $\betaend{itemize}nhalb( a e_{k_1} + \betavarepsilon_2 e_{k_2} + \betaldots +\betavarepsilon_n e_{k_n})$ with $a\betain\beta{\betapm1,\betapm3\beta}$ and $\betavarepsilon_i=\betapm 1$. For these one shows analogously that (SII) implies $n\betale 2$. For $n=2$ this representation is symplectic. \betaend{description} \betaitem{$C_n$:} For the largest root we get \beta[2\beta \betage\beta \betafrac{2\betalangle\betaLambda,2e_1\betarangle}{\beta|2e_1\beta|^2}\beta =\beta \betasigmaum_{k=1}^n m_k\betalangle\betaomega_k,e_1\betarangle\beta =\beta \betasigmaum_{k=1}^n m_k.\beta] In case that one $m_i=2$ and all others zero we have that $0\betain \betaW$. Hence we suppose that $\betaLambda=\betaomega_i+\betaomega_j$ for $i\betanot=j$. If $i>1$ we get for the root $2e_2$ which is orthogonal to $2e_1$ that $\betafrac{2\betalangle\betaLambda,2e_2}{\beta|2e_2\beta|^2}=2$. Thus by \betaref{sf1} of corollary \betaref{sf} we have $i=1$. But $\betaLambda=\betaomega_1 +\betaomega_i$ is only orthogonal if $i$ is odd, but if $i$ is odd we have that $0\betain \betaW$. Hence we have to deal with the case $\betaLambda=\betaomega_i$. This is orthogonal if $i$ is even, but in this case $0\betain \betaW$. \betaitem{$D_n$:} Here we get for the largest root \beta[2\beta \betage\beta \betafrac{2\betalangle\betaLambda,e_1+e_2\betarangle}{\beta|e_1+e_2\beta|^2}\beta =\beta \betasigmaum_{k=1}^n m_k\betalangle\betaomega_k,e_1+e_2\betarangle\beta =\beta m_1 +2m_2+\betaldots +2m_{n-2}+m_{n-1}+m_{n} .\beta] First we consider the representation where this number is equal to $2$. For the representations $2\betaomega_n$ and $2\betaomega_{n-1}$ we have that $0\betain \betaW$. For the representations $\betaLambda=\betaomega_1+\betaomega_n$ and $\betaLambda=\betaomega_1+\betaomega_{n-1}$ we get that $n=4$ or there is no triple as in the second point of proposition \betaref{sf}. Thus suppose in this case $n>4$. We have that$\betalangle\betaLambda,e_1+e_2\betarangle=2$ and for the orthogonal roots $\betalangle\betaLambda,e_1-e_2\betarangle=\betalangle\betaLambda ,e_3\betapm e_4\betarangle=1$. But this contradicts proposition \betaref{sf},\betaref{sf1}. For $\betaLambda=\betaomega_{n-1}+\betaomega_n=e_1+\betaldots +e_{n-1}$ we have that $0\betanot\betain \betaW$ implies $n-1$ even. The first point of corollary \betaref{sf} then gives for $n>4$ that $2=\betalangle\betaLambda,e_3+e_4\betarangle$ which is impossible. Hence $n\betale 4$. Then $1=\betalangle \betaLambda,e_3\betapm e_4\betarangle$ and the second point of corollary \betaref{sf} imply $n\betale3$. Now suppose that $\betaLambda=\betaomega_i$ for $2\betale i\betale n-2$. We apply the first point of corollary \betaref{sf}. If $n\betage 4$ we get that $\betalangle\betaomega_i,e_3+e_4\betarangle=2$ for $i\betage 4$ but this was excluded. Hence $i\betale 3$. In the case $n=3$ we have that only $\betaomega_2$ is an orthogonal representation. But for this holds that $0\betain \betaW$. Thus, to get the assertion of the proposition we have to show that \betabegin{enumerate} \betaitem For the spin representations $\betaLambda=\betaomega_{n-1}$ and $\betaLambda=\betaomega_n$ (SII) implies $n\betale 8$ \betaitem $\betaLambda=\betaomega_3$ does not satisfy (SII), \betaitem $\betaLambda=\betaomega_1+\betaomega_3$ and $\betaomega_1+\betaomega_4$ for $n=4$ do not satisfy (SII). \betaend{enumerate} \betabegin{eqnarray*}gin{description} \betaitem{{\betaem The spin representations:}} For these we show that (SII) implies first $n\betale 8$. Because we are interested in the representations modulo congruence it suffices to consider the spin representation of highest weight $\betaLambda=\betaend{itemize}nhalb(e_1+\betaldots +e_n)$ with weights $\betaW=\betaleft\beta{\betaend{itemize}nhalb(\betavarepsilon_1 e_1 +\betavarepsilon_n e_n)|\betavarepsilon_i=\betapm 1 \betamathbbox{ and $\betavarepsilon_i=-1$ for an even number}\betaright\beta}$. Analogously as for $B_n$ we get for two types of roots $\betaה=e_i+e_j$ and $\betaה=e_i-e_j$ that (SII) implies $n\betale 8$ (We have to admit one dimension higher because of the sign restriction of the weights). Now for $n$ odd the spin representation is not self dual, and for $n=6$ not orthogonal. For $n=4$ it is congruent to $\betaomega_1$. \betaitem{$\betaLambda=\betaomega_3=e_1+e_2+e_3$:} Here it is $\betaW=\beta{(\betavarepsilon_1e_{k_1}+\betavarepsilon_2e_{k_2}+\betavarepsilon_3e_{k_3}|\betavarepsilon_i=\betapm 1\beta}\betacup\beta{\betapm e_i\beta}$. For $n=3$ and $n=4$ this is a spin representation. Hence suppose $n\betage 5$. For $\betaה=e_1+e_2$ we get $\betaLambda-\betaה=e_3$. Set $\betalanglem:=-e_1+e_4+e_5\betain\betaW_\betaה$. Hence $\betaLambda-\betaה-\betalanglem=e_3+e_1-e_4-e_5$ and $\betaLambda+\betalanglem=e_2+e_3+e_4+e_5$. None is a root, i.e. $\betaomega_3$ for $n\betage5$ does not satisfy (SII). For $\betaה=e_1-e_2$ we get the same. \betaitem{{\betaem $\betaLambda=\betaomega_1+\betaomega_3$ and $\betaomega_1+\betaomega_4$ for $n=4$.}} These are congruent to each other and as above it can be shown that they do not satisfy (SII). \betaend{description} \betaitem{$E_6$ and $E_7$:} For these we refer to \betacite{schwachhoefer2}. There is shown that under the conclusions of proposition \betaref{s2satz2} and \betaref{s2schwachhoefer} --- which is our situation because of lemma \betaref{ausnahmelemma} --- the only remaining representations are the standard representations of $E_6$ and $E_7$. But the first is not self dual and the latter symplectic but not orthogonal. \betaend{description} \betaend{proof} We get the following \betabegin{eqnarray*}gin{folg} Let $\betalangleg\betasigmaubsetset\betalangleso(N,\betamathbb{C})$ be an orthogonal algebra of real type different from $\betamathfrak{sl}(2,\betamathbb{C})$. If $0\betanot\betain\betaW$ and (SII) is satisfied, then it is the complexification of a Riemannian holonomy representation or the spin representation of $\betalangleso(15,\betamathbb{C})$. \betaend{folg} \betabegin{eqnarray*}gin{proof} We give the Riemannian manifolds the complexified holonomy representation of which is one of the representations of proposition \betaref{s2satz4}. The representation with highest weight $\betaomega_4$ of $A_7$ is the complexified holonomy representation of the symmetric space of type $EV$, i.e. of $E_7/SU(8)$ resp. $E_{7(7)}/SU(8)$. The spin representations of $B_n$ for $n=3,4$ are the holonomy representations of a non-symmetric $Spin(7)$--manifold and of the symmetric space of type $FII$, i.e. of $F_4/Spin(9)$ resp. $F_{4(-20)}/Spin(9)$. For $n=7$ we have an exception. For $D_n$ first we have the standard representation, i.e. the complexified holonomy representation of a generic manifold. The representation with highest weight $2\betaomega_1$ is the complexified holonomy representation of the symmetric space of type $AI$, i.e. of $SU(2n)/SO(2n,\betamathbb{R})$, resp. $Sl(2n,\betamathbb{R})/SO(2n,\betamathbb{R})$. The remaining representation of $Spin(16)$ is the complexified holonomy representation of the symmetric space of type $EVIII$, i.e. of $E_8/Spin(16)$, resp. $E_{8(8)}/Spin(16)$. \betaend{proof} \betasigmaubsetsection{Consequences for simple weak-Berger algebras of real type} Before we conclude the result we need a lemma to exclude both exceptions. \betabegin{eqnarray*}gin{lem} The spin representation of $B_7$ and the representation of $G_2$ with two times a short root as highest weight are not weak-Berger. \betaend{lem} \betabegin{eqnarray*}gin{proof} 1.) Suppose that the spin representation of $B_7$ is weak-Berger. We have shown that it does not satisfy the property (SI). Hence it obeys (SII). Let $(\betaLambda,\betaה)$ be the pair of (SII). We choose a fundamental system such that $\betaLambda=\betaomega_7$ is the highest weight. In the proof of proposition \betaref{s2satz4} we have shown that in this case $\betaה=e_i+e_j$. Let now $Q_\betaphi$ be the weight element from ${\betacal B}_H(\betalag)$ and $u_\betaLambda\betain V_\betaLambda$ such that $Q_\betaphi(u_\betaLambda)=A_{e_i+e_j}\betain\betalangleg_{e_i+e_j}$. Since $Q_\betaphi(u_\betaLambda)\betain \betalangleg_{\betaphi+\betaLambda}$ this implies that $\betaphi=e_i+e_j-\betaLambda$ is a weight of ${\betacal B}_H(\betalag)$. Hence $\betaphi=-\betaend{itemize}nhalb(e_1+\betaldots +e_{i-1}-e_i+e_{i+1}+\betaldots +e_{j-1}-e_j+e_{j+1}+\betaldots+e_7)$ is also an extremal weight of $V$ and we can consider a weight vector $u_{-\betaphi}\betain V_{-\betaphi}$. For this we get $Q_\betaphi(u_{-\betaphi})\betain\betamathfrak{t}$. In case it does not vanish it would define a planar spanning triple $(\betaphi,-\betaphi,\betaleft(Q_\betaphi(u_{-\betaphi})\betaright)^\betabot)$, i.e. (SI) would be satisfied. But this was not possible, and thus $Q_\betaphi(u_{-\betaphi})=0$. On the other hand we have that $0\betanot=Q_\betaphi(u_\betaLambda)u_{-\betaphi}\betain V_\betaLambda$ and thus there is a $v\betain V_{-\betaLambda}$ such that $H(Q_\betaphi(u_\betaLambda)u_{-\betaphi},v)\betanot=0$. Now the Bianchi identity gives \beta[0= H(Q_\betaphi(u_\betaLambda)u_{-\betaphi},v) +\betaunderbrace{H(Q_\betaphi(u_{-\betaphi})v,u_\betaLambda))}_{=0} + H(Q_\betaphi(v)u_\betaLambda, u_{-\betaphi},v) .\beta] Hence $0\betanot=Q_\betaphi(v)\betain \betalangleg_{\betaphi-\betaLambda}$. But $\betaphi-\betaLambda=-(e_1+\betaldots +e_{i-1}+e_{i+1}+\betaldots +e_{j-1}+e_{j+1}+\betaldots+e_7)$ is not a root, hence $\betalangleg_{\betaphi-\betaLambda}=\beta{0\beta}$. This is a contradiction. 2.) Suppose that the representation of $G_2$ with two times a short root as highest weight is weak-Berger. We will argue analogously as for $B_n$. \beta\beta[.2cm] \betabegin{eqnarray*}gin{minipage}[b]{8cm}{ In the picture we see the weight lattice of this representation (the arrows represent the roots). Obviously there is no planar spanning triple, because there is no hypersurface which contains all but two extremal weight (see also proof of proposition \betaref{wurzelgewicht}).} The weak-Berger property implies that there is a pair $(\betaLambda,\betaה)$ such that (SII) is satisfied. We choose a fundamental system such that $\betaLambda=2\betaeta$ is the maximal weight. \betaend{minipage} \betahfill \betabegin{eqnarray*}gin{minipage}[b]{5cm}{ \betabegin{eqnarray*}gin{picture}(0,0) \betaincludegraphics{g2.pstex} \betaend{picture} \betasigmaetlength{\betaunitlength}{4144sp} \betabegin{eqnarray*}gingroup\betamakeatletter\betaifx\betaSetFigFont\betaundefined \betagdef\betaSetFigFont#1#2#3#4#5{ \betareset@font\betafontsize{#1}{#2pt} \betafontfamily{#3}\betafontseries{#4}\betafontshape{#5} \betasigmaelectfont} \betafi\betaendgroup \betabegin{eqnarray*}gin{picture}(2360,2165)(299,-1643) \betaput(1856,-90){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaeta$}}} \betaput(2157,417){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaLambda=2\betaeta$}}} \betaend{picture}} \betaend{minipage}\beta\beta Using the realization of $G_2$ from the appendix of \betacite{knapp96} we have that $\betaeta=e_3-e_2$. Now we have to determine the roots for which (SII) is satisfied.\beta\beta \betabegin{eqnarray*}gin{minipage}[t]{5cm}{ \betabegin{eqnarray*}gin{picture}(0,0) \betaincludegraphics{2g2.pstex} \betaend{picture} \betasigmaetlength{\betaunitlength}{4144sp} \betabegin{eqnarray*}gingroup\betamakeatletter\betaifx\betaSetFigFont\betaundefined \betagdef\betaSetFigFont#1#2#3#4#5{ \betareset@font\betafontsize{#1}{#2pt} \betafontfamily{#3}\betafontseries{#4}\betafontshape{#5} \betasigmaelectfont} \betafi\betaendgroup \betabegin{eqnarray*}gin{picture}(2437,2601)(267,-1950) \betaput(2611,-1591){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaOmega_\betaalpha$}}} \betaput(2108,387){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$2\betaeta$}}} \betaput(2668,-409){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$2\betaeta-\betaalpha$}}} \betaput(2108,387){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$2\betaeta$}}} \betaput(624,-1805){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$-2\betaeta$}}} \betaput(1424,-838){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{12}{14.4}{\betarmdefault}{\betamddefault}{\betaupdefault}$0$}}} \betaput(1744,-79){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{8}{9.6}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaeta$}}} \betaput(1395,495){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{12}{14.4}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betaalpha$}}} \betaput(2425,-155){\betamakebox(0,0)[lb]{\betasigmamash{\betaSetFigFont{12}{14.4}{\betarmdefault}{\betamddefault}{\betaupdefault}$\betabegin{eqnarray*}ta$}}} \betaend{picture}} \betaend{minipage} \betahfill \betabegin{eqnarray*}gin{minipage}[b]{8cm}{ In the picture one can see that the long roots $\betaה$ and $\betabeta$ satisfy (SII). (We illustrate the situation in detail only for $\betaה$.) Contemplate the picture for a moment one sees that there are no short roots and no other long root for which (SII) can be valid. Now $\betaה$ and $\betabeta$ are the only roots with $\betalangle\betaLambda,\betaה\betarangle>0$ and $\betalangle\betaLambda,\betabeta\betarangle>0$. Hence $\betaה=2e_3-e_1-e_2$ and $\betabeta=-2e_2+e_1+e_3$.} \betaend{minipage}\beta\beta We consider the case where $(\betaLambda,\betaה)$ satisfies (SII). There is a weight element $Q_\betaphi$ from ${\betacal B}_H(\betalag)$ such that $Q_\betaphi(u_\betaLambda)=A_{2e_3-e_1-e_2}$, i.e. $\betaphi= 2e_3-e_1-e_2-\betaLambda=e_2-e_1$. But this is a short root and therefore a weight. Thus we consider $u_{-\betaphi}\betain =V_{-\betaphi}$. Then $Q_\betaphi(u_{-\betaphi})\betain\betamathfrak{t}$. Since there is no planar spanning triple it has to be zero. As above the Bianchi identity gives that $\betaphi-\betaLambda $ has to be a root. But $\betaphi-\betaLambda=e_2-e_1-2e_3+2e_2= 3e_2-2e_3-e_1$ is no root. For $\betabeta$ one proceeds analogously. \betaend{proof} Now we can draw the conclusions from the previous sections. If a Lie algebra acts irreducible of real type the it is semi-simple and obeys the properties (SI) or (SII). The simple Lie algebras with (SI) or (SII) we have listed above. Thus we get \betabegin{eqnarray*}gin{theo} Let $\betalangleg\betasigmaubsetset \betalangleso(N,\betamathbb{R})$ be a irreducible weak-Berger algebra of real type. Then it is the holonomy representation of a Riemannian manifold. The conclusion holds in particular if $\betalangleg$ is simple, of real type and the irreducible component of the $\betalangleson$-projection of an indecomposable, non-irreducible simply connected Lorentzian manifold. \betaend{theo} \betabegin{bem} {\betabf Quaternionic symmetric spaces.}\betalanglebel{quatsym1} With the result of course we have covered all simple irreducible acting Riemannian holonomy groups of real type. If one considers a quaternionic symmetric space $G/Sp(1)\betacdot H$ with $H\betasigmaubsetset Sp(n)$ then of course $\betamathfrak{sp}(1) \betaoplus \betamathfrak{h}\betasigmaubsetset \betalangleso(4n,\betamathbb{R}) $ is a real Berger algebra of real type and thus its complexification is a complex Berger algebra of real type. Then the restriction of this representation to $\betamathfrak{h}$ is of quaternionic, i.e. of non-real type, its complexification decomposes into two irreducible components $\betamathbb{C}^{2n}\betaoplus\betaoverline{\betamathbb{C}^{2n}}$. For this situation in \betacite{schwachhoefer2} is proved that $\betamathfrak{h}^\betamathbb{C}_{\betabegin{itemize}g|\betamathbb{C}^{2n}}$ is a complex Berger algebra. This result does not collide with our list because this representation is not of real type and hence not orthogonal. $ \betamathfrak{h}\betasigmaubsetset \betalangleso(4n,\betamathbb{R})$ is not a real Berger algebra. \betaend{bem} \betasigmaection{Weak-Berger algebras of non-real type} In this section we will classify weak-Berger algebras of non-real type, and we will show that these are Berger algebras. For the classification we will use the classification of first prolongations of irreducible complex Lie algebras. We will show that the complexification of the space ${\betacal B}_h(\betalangleg_0)$ is isomorphic to the first prolongation of the complexified Lie algebra. \betabegin{itemize}gskip In this section $\betalangleg_0$ is a real Lie algebra and $E$ a $\betalangleg_0$-module of non-real type, i.e. $E^\betamathbb{C}$ is not irreducible. Thus the situation is a little bit more puzzling then in the real case. Since $\betalangleg_0\betasigmaubsetset \betalangleso(E,h)$ with $h$ positive definite, $\betalangleg_0$ is compact. For a compact real Lie algebra with module of non-real type the corresponding complex representation of non-real type is not orthogonal but unitary (See appendix \betaref{real representations}, in particular proposition \betaref{main1}). But if we switch to the complexified algebra the $(\betalangleg^\betamathbb{C},V)$ irreducible remains, but it can no longer be unitary of course. We have to handle this situation. With the same notations as in appendix \betaref{real representations} the complex representations space $W=E^\betamathbb{C}$ splits into the irreducible modules $W=V \betaoplus \betaoverline{V}$ under ${\betafrak g}_0$. This splitting is of course ${\betafrak g}_0^\betamathbb{C}$ invariant. Now we define the complex Lie algebra \betabegin{eqnarray*}gin{equation} \betalanglebel{defg} {\betafrak g}\beta :=\beta \betaleft\beta{ A_{|V}\betaleft|\beta A\betain {\betafrak g}_0^\betamathbb{C}\betasigmaubsetset \betamathfrak{so}(W=V \betaoplus \betaoverline{V}, H)\betaright.\betaright\beta}\betasigmaubsetset \betamathfrak{gl}(V). \betaend{equation} Here $H$ denotes again $h^\betamathbb{C}$. Since the symmetric bilinear form we start with is positive definite the appendix \betaref{real representations} gives two important results (see proposition \betaref{main1}): \betabegin{enumerate} \betaitem \betalanglebel{unitaer} Since $\betalangleg_0$ is compact there is a positive definite hermitian form $\betatheta^h$ on $V$ which is the the restriction of the sesqui-linear extension of $h$ on $V$, such that $({\betafrak g}_0)_{|V}\betasigmaubsetset{\betafrak u}(V, \betatheta^h)$. \betaitem\betalanglebel{nichtorthogonal} $\betalangleg$ is not orthogonal, in particular $H_{\betabegin{itemize}g|V\betatimes V}=0$. This is the case since modules of non-real type are symplectic if they are self-dual. Thus they can not be orthogonal. \betaend{enumerate} In ${\betafrak g}_0^\betamathbb{C}$ as well as in ${\betafrak g}$ we have a conjugation $\betaoverline{{ }^{\betaleft.\beta \betaright.}}$ with respect to $ {\betafrak g}_0$ and $({\betafrak g}_0)_{|V}$ respectively. Since an $A\betain {\betafrak g}_0$ acts on $V\betaoplus \betaoverline{V}$ by $A(v+ \betaoverline{w})= Av + \betaoverline{Aw}$ we have for $iA\betain {\betafrak g}_0^\betamathbb{C}$ that \beta[ iA(v+ \betaoverline{w})\beta =\beta i(Av + \betaoverline{Aw})\beta =\beta (iAv+ \betaoverline{-iAw}).\beta] So we write the action of $A\betain {\betafrak g}_0^\betamathbb{C}$ with the help of the conjugation in ${\betafrak g}$ as follows \betabegin{eqnarray*}gin{equation} \betalanglebel{g0c} A(v+ \betaoverline{w})\beta =\beta Av + \betaoverline{\betaoverline{A} w}. \betaend{equation} This gives the following Lie algebra isomorphism \beta[ \betabegin{eqnarray*}gin{array}{rcrcl} \betavarphi&:& {\betafrak g}_0^\betamathbb{C} &\betasigmaimeq & {\betafrak g}\beta\beta &&A&\betamapsto& A_{|V}. \betaend{array}\beta] This is clearly a Lie algebra homomorphism. It is injective because for $A_{|V}=B_{|V}$ holds that $A(v+ \betaoverline{w}) = Av + \betaoverline{\betaoverline{A}w}=Bv + \betaoverline{\betaoverline{B}w}=B(v+\betaoverline{w}) $ for all $v,w\betain V$, i.e. $A=B$. By definition it is surjective and $\betavarphi^{-1}$ is given by \betabegin{eqnarray*}gin{equation} \betalanglebel{phi-1} \betavarphi^{-1}(A)\beta :\beta v + \betaoverline{w} \beta \betalongmapsto\beta Av + \betaoverline{\betaoverline{A}w}\betamakebox[2cm][c]{for all} A\betain {\betafrak g}. \betaend{equation} These notations are needed to show the relation to the first prolongation. \betasigmaubsetsection{The first prolongation of a Lie algebra of non-real type} Now we define the first prolongation of an arbitrary Lie algebra ${\betafrak g}\betasigmaubsetset \betamathfrak{gl}(V)$. \betabegin{eqnarray*}gin{de} The ${\betafrak g}$-module \betabegin{eqnarray*}gin{align} {\betafrak g}^{(1)} &\beta :=\beta \beta{ Q \betain V^* \betaotimes {\betafrak g}\beta |\beta Q(u)v=Q(v)u\beta}. \betalanglebel{hg}\beta\beta \betaintertext{is called {\betabf first prolongation} of ${\betafrak g}\betasigmaubsetset \betamathfrak{gl}(V)$. Furthermore we set} \betatilde{ {\betafrak g}}&\beta :=\beta span\beta{Q(u)\betain {\betafrak g}\beta |\beta Q\betain {\betafrak g}^{(1)}, u\betain V\beta}\betasigmaubsetset {\betafrak g}, \betanonumber\beta\beta \betaintertext{and if in ${\betafrak g}$ a conjugation $\betaoverline{{ }^{\betaleft.\beta \betaright.}}$ is given:} {\betafrak g}^{[1,1]}&\beta :=\beta \beta{ R\betain \betaoverline{V}^* \betaotimes {\betafrak g}^{(1)}\beta |\beta \betaoverline{R(\betaoverline{u},v)}=- R(\betaoverline{v},u)\beta},\betanonumber\beta\beta \betatilde{\betatilde{{\betafrak g}}}&\beta :=\beta span \beta{ R(\betaoverline{u},v)\beta |\beta R\betain {\betafrak g}^{[1,1]}, \betaoverline{u}\betain \betaoverline{V}, v\betain V\beta}\betasigmaubsetset {\betafrak g}\betanonumber. \betaend{align} \betaend{de} We will now describe the spaces ${\betacal B}_{H}({\betafrak g}_0^\betamathbb{C})$ and ${\betacal K}({\betafrak g}_0^\betamathbb{C})$ --- which are essential for the Berger and the weak-Berger property --- with the help of the first prolongation of $ {\betafrak g}$. In the setting of the above notations we can now prove the following. \betabegin{eqnarray*}gin{satz} \betalanglebel{bhg-isom} Let $E$ be a non-real type module of ${\betafrak g}_0$, orthogonal with respect to a positive definite scalar product $h$, and $E^\betamathbb{C}=V \betaoplus \betaoverline{V}$ the corresponding $ {\betafrak g}_0^\betamathbb{C}$ invariant decomposition, ${\betafrak g}$ defined as in (\betaref{defg}). Then there is an isomorphism \beta[\betabegin{eqnarray*}gin{array}{rcrcl} \betaphi&:& {\betacal B}_{H}({\betafrak g}_0^\betamathbb{C}) & \betasigmaimeq & {\betafrak g}^{(1)} \beta\beta &&Q&\betamapsto & Q_{|V\betatimes V}. \betaend{array}\beta] \betaend{satz} \betabegin{eqnarray*}gin{proof} For the prove we will use the $\betalangleg_0$--invariant hermitian form $\betatheta$ on $V$ which is given by $\betatheta(u,v)=h^\betamathbb{C}(u,\betaoverline{v})$, where $\betaoverline{{ }^{\betaleft.\beta \betaright.}}$ is the conjugation in $E^\betamathbb{C}=V\betaoplus\betaoverline{V}$ with respect to $E$. The linearity of $\betaphi$ mapping is clear. we have to show the following: 1.) The definition of $\betaphi$ is correct, i.e. for $Q\betain {\betacal B}_{H}({\betafrak g}_0^\betamathbb{C})$ it is $Q_{|V\betatimes V}\betain {\betafrak g}^{(1)}$. We have for every $u,v,w \betain V$ and $H= h^\betamathbb{C}$ that \betabegin{eqnarray*}gin{eqnarray*} \betatheta(Q(u)v,w)&=& h^\betamathbb{C}(Q(u)v,\betaoverline{w}) \beta\beta &=& -h^\betamathbb{C}(Q(v)\betaoverline{w},u)\beta;\beta; - \betaunderbrace{h^\betamathbb{C}(Q(\betaoverline{w})u,v)}_{\betamakebox[3cm][c]{ \betabegin{eqnarray*}gin{minipage}{4.5cm}{\betasigmacriptsize \betacenterline{$=0$} since $h^\betamathbb{C}_{V\betatimes V}=0$ (proposition \betaref{main1})} \betaend{minipage}}} \beta\beta &\betasigmatackrel{\betamathbbox{{\betasigmacriptsize $h^\betamathbb{C}$ invariant}}}{=}& h^\betamathbb{C}(Q(v)u, \betaoverline{w}) \beta\beta &=& \betatheta(Q(v)u,w), \betaend{eqnarray*} i.e. $Q(u)v=Q(v)u$ which means that $Q_{|V\betatimes V} \betain {\betafrak g}^{(1)}$. 2.) The homomorphism $\betaphi$ is injective. Let $Q_1$ and $Q_2$ be in $ {\betacal B}_{H}({\betafrak g}_0^\betamathbb{C})$ with $(Q_1)_{|V\betatimes V} = (Q_2)_{|V\betatimes V}$. Then it is \betabegin{eqnarray*}gin{enumerate} \betaitem[a)] $(Q_1)_{|\betaoverline{V}\betatimes \betaoverline{V}} = (Q_2)_{|\betaoverline{V}\betatimes \betaoverline{V}}$, since $Q_1(\betaoverline{u})\betaoverline{v} = \betaoverline{ Q_1(u)v}= \betaoverline{ Q_2(u)v} = Q_2(\betaoverline{u})\betaoverline{v},$ \betaitem[b)] $(Q_1)_{|\betaoverline{V}\betatimes V} = (Q_2)_{|\betaoverline{V}\betatimes V}$, since \beta[\betabegin{eqnarray*}gin{array}{rcccl} \betatheta(Q_1(\betaoverline{u})v,w)& =& h^\betamathbb{C}(Q_1(\betaoverline{u})v,\betaoverline{w}) &=& -h^\betamathbb{C}(v,Q_1(\betaoverline{u})\betaoverline{w})\beta = \beta\beta =\beta h^\betamathbb{C}(v,Q_2(\betaoverline{u})\betaoverline{w}) &=& h^\betamathbb{C}(Q_2(\betaoverline{u})v,\betaoverline{w}) &=& \betatheta(Q_2(\betaoverline{u})v,w). \betaend{array}\beta] \betaitem[c)] $(Q_1)_{|V\betatimes \betaoverline{V}} = (Q_2)_{|V\betatimes \betaoverline{V}}$ because of b) with the same argument as in a). \betaend{enumerate} 3.) The homomorphism $\betaphi$ is surjective. For $Q\betain {\betafrak g}^{(1)}$ we define $\betaphi^{-1}$ using $\betavarphi$: \betabegin{eqnarray*}gin{eqnarray*} ( \betaphi ^{-1}Q)(u):= \betavarphi^{-1}(Q(u))&\betamathbbox{and}& (\betaphi ^{-1}Q)(\betaoverline{u}):= \betavarphi^{-1}(\betaoverline{Q(u)})\betain \betamathfrak{gl} (E^\betamathbb{C}),\beta\beta \betamathbbox{i.e. } ( \betaphi ^{-1}Q)(u,v)=Q(u)v&,& (\betaphi ^{-1}Q)(\betaoverline{u})=\betaoverline{Q(u)}v\beta ,\beta\beta (\betaphi ^{-1}Q)(u, \betaoverline{v})= \betaoverline{\betaoverline{Q(u)}v}&,& ( \betaphi ^{-1}Q)(\betaoverline{u}, \betaoverline{v})= \betaoverline{Q(u)v}. \betaend{eqnarray*} It is $(\betaphi ^{-1}Q)(\betaoverline{u}, \betaoverline{v})= \betaoverline{(\betaphi ^{- 1}Q)(u,v)}$. Then obviously $\betaphi \betacirc \betaphi^{-1} = id$, since $\betaphi \betaleft( \betaphi^{-1}(Q)\betaright)=\betaphi^{-1}(Q)_{|V\betatimes V}=Q$. Because of the symmetry of $Q$ we have also that $(\betaphi ^{-1}Q)\betain {\betacal B}_{H}({\betafrak g}_0^\betamathbb{C})$: \betabegin{eqnarray*}gin{align*} \betaintertext{$\betabullet$ For $u,v\betain V, \betaoverline{w}\betain \betaoverline{V}$:} H( (\betaphi^{-1}Q)(u)v,\betaoverline{w})+ H( (\betaphi^{-1}Q)(v)\betaoverline{w},u)+ \betaoverbrace{H( (\betaphi^{-1}Q)(\betaoverline{w})u,v)}^{\betamakebox[2cm][l]{{\betasigmacriptsize $=0$ because $H=0$ on $V\betatimes V$}}} &= \beta\beta H( (\betavarphi^{-1}(Q(u))v,\betaoverline{w})+ \betaunderbrace{H( (\betavarphi^{-1}(Q(v))\betaoverline{w},u)}_{= -H( (\betaoverline{w},\betavarphi^{-1}(Q(v))u)}&= H( (Q(u)v - Q(v)u ,\betaoverline{w})\beta\beta &=0. \beta\beta \betaintertext{$\betabullet$ For $u\betain V,\betaoverline{v}, \betaoverline{w}\betain \betaoverline{V}$:} \betaoverbrace{H( (\betaphi^{- 1}Q)(u)\betaoverline{v},\betaoverline{w})}^{=0}+ H( (\betaphi^{-1}Q)(\betaoverline{v})\betaoverline{w},u)+ H( (\betaphi^{-1}Q)(\betaoverline{w})u,\betaoverline{v})& = \beta\beta H( (\betavarphi^{-1}(\betaoverline{Q(v)})\betaoverline{w},u)+ \betaunderbrace{H( \betavarphi^{-1}(\betaoverline{Q(w)})u,\betaoverline{w})}_{= -H( (u,\betavarphi^{-1}(\betaoverline{Q(w)})\betaoverline{v})} &= H(\betaoverline{ (Q(v)w - Q(w)v} ,u)\beta\beta&=0. \betaend{align*} Terms with entries only from $V$ or only from $\betaoverline{V}$ are zero. \betaend{proof} Furthermore we show for the space $ {\betacal K}({\betafrak g})$ an analogous result. \betabegin{eqnarray*}gin{satz} \betalanglebel{kg-isom} Let $E$ be an orthogonal non-real type module of ${\betafrak g}_0$ and $E^\betamathbb{C}=V \betaoplus \betaoverline{V}$ the corresponding $ {\betafrak g}_0^\betamathbb{C}$ invariant decomposition, ${\betafrak g}$ defined as in (\betaref{defg}). Suppose that $\betatheta:=\betatheta^h$ is non-degenerate. Then there is an isomorphism \beta[\betabegin{eqnarray*}gin{array}{rcrcl} \betapsi&:& {\betacal K}({\betafrak g}_0^\betamathbb{C}) & \betasigmaimeq & {\betafrak g}^{[1,1]} \beta\beta &&R&\betamapsto & R_{|\betaoverline{V}\betatimes V\betatimes V}. \betaend{array}\beta] \betaend{satz} \betabegin{eqnarray*}gin{proof} The proof is completely analogous to the previous one. 1.) The definition is correct. We have for $u,v,w\betain V$ and $R\betain {\betacal K}( {\betafrak g}_0^\betamathbb{C} )$ that \beta[ \betaunderbrace{R(u,v)\betaoverline{w}}_{\betain \betaoverline{V}}\beta =\beta \betaunderbrace{R(\betaoverline{w},v)u}_{\betain V}- \betaunderbrace{R(\betaoverline{w},u)v}_{\betain V}\beta =\beta 0. \beta] but this means that $R(\betaoverline{u},.)_{|V\betatimes V}\betain {\betafrak g}^{(1)}$. Further $R(u,v) \betaoverline{w}=0$ implies $R(u,v)w=0$ because \beta[ \betatheta(R(u,v)w,z)\beta =\beta h^\betamathbb{C}(R(u,v)w, \betaoverline{z})\beta =\beta - h^\betamathbb{C}(w,R(u,v) \betaoverline{z})\beta =\beta 0.\beta] This implies $R( \betaoverline{u}, \betaoverline{v}) \betaoverline{w}=R( \betaoverline{u}, \betaoverline{v}) w=0$ too. For a $ R \betain {\betacal K}( {\betafrak g}_0^\betamathbb{C})$ we have due to the skew symmetry \beta[\betaoverline{R(\betaoverline{u},v)}\betasigmatackrel{\betamathbbox{ easy calculation }}{=} R(u, \betaoverline{v})\betasigmatackrel{\betamathbbox{skew-symm.}}{=}-R( \betaoverline{v},u),\beta] i.e. the restriction of $R$ on $\betaoverline{V}\betatimes V \betatimes V$ is in $ {\betafrak g}^{[1,1]}$. 2.) The homomorphism $\betapsi$ is injective. Let $R_1$ and $R_2$ be in $ {\betacal K}({\betafrak g}_0^\betamathbb{C})$ with $(R_1)_{\betaoverline{V}\betatimes V\betatimes V}=(R_2)_{\betaoverline{V}\betatimes V\betatimes V}$. Then again via $\betatheta$ the remaining non zero terms $R_i( \betaoverline{u}, v)\betaoverline{w}$ are determined by $R_i( \betaoverline{u},v)w$ which are equal for $i=1,2$ and by the skew symmetry of $R$. 3.) The homomorphism $\betapsi$ is surjective. We set \betabegin{eqnarray*}gin{eqnarray*} (\betapsi ^{-1}R) (\betaoverline{u},v))\beta :=\beta \betavarphi^{-1}(R( \betaoverline{u},v)&\betamathbbox{ , }& (\betapsi ^{-1}R) (u,\betaoverline{v})\beta :=\beta \betavarphi^{-1}(\betaoverline{R( \betaoverline{u},v)})\betamathbbox{ and}\beta\beta (\betapsi ^{-1}R)(u,v)&:=& (\betapsi ^{-1}R)(\betaoverline{u}, \betaoverline{v})\beta :=\beta 0 \betaend{eqnarray*} So we have the skew symmetry, i.e. $\betapsi^{-1}R\betain \betaomegaedge^2 E^\betamathbb{C}\betaotimes {\betafrak g}_0^\betamathbb{C}$, because \beta[(\betapsi ^{-1}R) (u,\betaoverline{v})= \betavarphi^{-1}(\betaoverline{R( \betaoverline{u},v)}) =-\betavarphi^{-1}(R( \betaoverline{v},u)) =- (\betapsi^{-1}R)(\betaoverline{v},u).\beta] The Bianchi identity is also satisfied: \betabegin{eqnarray*}gin{align*} \betaintertext{$\betabullet$ For $u\betain \betaoverline{V},v,w\betain V$:} (\betapsi^{-1}R)(\betaoverline{u},v)w+ \betaoverbrace{(\betapsi^{-1}R)(v,w)\betaoverline{u}}^{=0} +(\betapsi^{-1}R)(w,\betaoverline{u})v&=\beta\beta \betavarphi^{-1}\betaleft(R(\betaoverline{u},v)\betaright)w+ \betavarphi^{-1}\betaleft(\betaoverline{R(\betaoverline{w},u)}\betaright)v &=\beta\beta \betavarphi^{-1}\betaleft(R(\betaoverline{u},v)\betaright)w- \betavarphi^{-1}\betaleft(R(\betaoverline{u},w)\betaright)v &=\beta\beta R(\betaoverline{u},v)w-R(\betaoverline{u},w)v &=0 \beta\beta \betaintertext{$\betabullet$ For $\betaoverline{u}, \betaoverline{v}\betain \betaoverline{V}, w\betain V$:} \betaoverbrace{(\betapsi^{-1}R)(\betaoverline{u},\betaoverline{v})w}^{=0}+ (\betapsi^{-1}R)(\betaoverline{v},w)\betaoverline{u} +(\betapsi^{-1}R)(w,\betaoverline{u})\betaoverline{v} &=\beta\beta \betavarphi^{-1}\betaleft(R(\betaoverline{v},w)\betaright)\betaoverline{u}+ \betavarphi^{-1}\betaleft(\betaoverline{R(\betaoverline{w},u)}\betaright)\betaoverline{v} &=\beta\beta -\betavarphi^{-1}\betaleft(\betaoverline{R(\betaoverline{w},v)}\betaright)\betaoverline{u}+ \betavarphi^{-1}\betaleft(\betaoverline{R(\betaoverline{w},u)}\betaright)\betaoverline{v} &=\beta\beta-\betaoverline{R(\betaoverline{w},v)u} + \betaoverline{R(\betaoverline{w},u)v} &=0 \betaend{align*} Terms with entries only from $V$ or only from $\betaoverline{V}$ are zero. \betaend{proof} In contrary to the previous proof, in this proof we only supposed the fact that $\betatheta^h$ is non-degenerate and not that $h^\betamathbb{C}_{|V\betatimes V}=0$. If we assume $h$ to be positive definite, then both facts are satisfied. \betasigmaubsetsection{Consequences for Berger and weak-Berger algebras} Both propositions give three important corollaries. \betabegin{eqnarray*}gin{folg}\betalanglebel{wichtig} Let $ {\betafrak h}_0\betasigmaubsetset {\betafrak g}_0\betasigmaubsetset {\betamathfrak so}(E^\betamathbb{C}, H)$ be subalgebras of non-real type, ${\betafrak h}$ and ${\betafrak g}$ defined as above. If \beta[ {\betafrak h}^{(1)}= {\betafrak g}^{(1)},\beta] then $( {\betafrak h}_0^\betamathbb{C})_{H}=({\betafrak g}_0^\betamathbb{C})_{H}$. I.e. if in ${\betafrak g}$ exists a proper subalgebra which has the same first prolongation and a compact real form in ${\betafrak g}_0$ of non-real type, then ${\betafrak g}_0^{\betamathbb{C}}$ and therefore ${\betafrak g}_0$ can not be weak-Berger algebras. \betaend{folg} \betabegin{eqnarray*}gin{proof} Because of $Q\betain {\betacal B}_{H}({\betafrak h}_0^\betamathbb{C})\betasigmaimeq {\betafrak h}^{(1)}={\betafrak g}^{(1)}\betasigmaimeq {\betacal B}_{H}({\betafrak g}_0^\betamathbb{C})$ we have $Q(u)\betain ({\betafrak g}_0^\betamathbb{C})_{H}$ if and only if $Q(u)\betain ({\betafrak h}_0^\betamathbb{C})_{H}$. \betaend{proof} \betabegin{eqnarray*}gin{folg}\betalanglebel{wichtiger} Let ${\betafrak g}_0\betasigmaubsetset \betamathfrak{so}(E^\betamathbb{C}, H) $ be a Lie algebra of non-real type, and ${\betafrak g}$ defined as above. Then \betabegin{eqnarray*}gin{enumerate} \betaitem $({\betafrak g}_0^\betamathbb{C})_H = {\betafrak g}_0^\betamathbb{C}$ (i.e. ${\betafrak g}_0^\betamathbb{C}$ is a weak-Berger-algebra) if and only if $ {\betafrak g}= \betatilde{ {\betafrak g}}$. \betaitem $\betaunderline{{\betafrak g}_0^\betamathbb{C}}= {\betafrak g}_0^\betamathbb{C}$ (i.e. ${\betafrak g}_0^\betamathbb{C}$ is a Berger-algebra) if and only if ${\betafrak g}=\betatilde{\betatilde{ {\betafrak g}}}$. \betaend{enumerate} \betaend{folg} \betabegin{eqnarray*}gin{proof} 1.) First we show the sufficiency: Let $A\betain {\betafrak g}_0^\betamathbb{C}$ be arbitrary. The assumption ${\betafrak g} = \betatilde{ {\betafrak g}}$ gives w.l.o.g. that $\betavarphi(A)= Q(u)$ with $Q\betain {\betafrak g}^{(1)}$ and $u\betain V$. But then we have \beta[(\betaphi^{-1}Q)(u)\betasigmatackrel{\betamathbbox{ per def. }} {=} \betavarphi^{-1}(Q(u))\beta =\beta \betavarphi^{-1}( \betavarphi(A))\beta =\beta A,\beta] with $(\betaphi^{-1}Q)\betain {\betacal B}_H( {\betafrak g}_0^\betamathbb{C})$, i.e. $A\betain ({\betafrak g}_0^\betamathbb{C})_H$. Now we show the necessity: If $A\betain {\betafrak g}$, then the assumption $ {\betafrak g}_0^\betamathbb{C}= ( {\betafrak g}_0^\betamathbb{C})_H$ gives w.l.o.g. that $ \betavarphi^{-1}(A)=\betahat{Q}(u+ \betaoverline{v})$ with $\betahat{Q}\betain {\betacal B}_H({\betafrak g}_0^\betamathbb{C})$, $u\betain V $ and $\betaoverline{v}\betain \betaoverline{V}$. But by the isomorphism of the proposition \betaref{bhg-isom} there is a $Q\betain {\betafrak g}^{(1)}$ such that \beta[ \betavarphi^{-1}(A)=\betahat{Q}(u+ \betaoverline{v})=( \betaphi ^{-1} Q)(u+\betaoverline{v}) = \betavarphi^{-1}(Q(u)) + \betavarphi^{-1}(\betaoverline{Q(v)}).\beta] But this means that \beta[A\beta = \beta \betaunderbrace{Q(u)}_{\betain \betatilde{{\betafrak g}}} + \betaunderbrace{\betaoverline{Q(v)}}_{\betain \betatilde{{\betafrak g}}}\beta \betain \beta \betatilde{ {\betafrak g}},\beta] i.e. $ {\betafrak g}\betasigmaubsetset \betatilde{ {\betafrak g}}$. 2.) Both directions are proved completely analogous to 1.) Suppose that ${\betafrak g}=\betatilde{\betatilde{g}}$. Then for $A\betain {\betafrak g}_0^\betamathbb{C}$ one has that $\betavarphi(A)= R( \betaoverline{u},v)$ and \beta[(\betapsi^{-1}R)( \betaoverline{u},v)\beta =\beta \betavarphi^{-1}(R( \betaoverline{u},v)\beta =\beta A.\beta] On the other hand we have for $A\betain {\betafrak g}$ that $\betavarphi^{-1}(A)= \betahat{R}(z+\betaoverline{u}, v+\betaoverline{w})$. This gives \beta[\betabegin{eqnarray*}gin{array}{rcccl} \betavarphi^{-1}(A)&=&\betahat{R}(z,\betaoverline{w})+ \betahat{R}(\betaoverline{u},v)&=& ( \betapsi ^{-1} R)(z,\betaoverline{w})+=( \betapsi ^{-1} R)(\betaoverline{u},v)\beta\beta &=& \betavarphi^{-1}(\betaoverline{R(\betaoverline{z},w)}) + \betavarphi^{-1}(R(\betaoverline{u},v))&&. \betaend{array}\beta] and therefore $A\betain \betatilde{\betatilde{{\betafrak g}}}$. \betaend{proof} As a result of the previous and this section we have to investigate complex irreducible representations of complex Lie algebras with non-vanishing first prolongation. Fortunately these are classified by Cartan \betacite{cartan09}, Kobayashi and Nagano \betacite{ko-na65} in a rather short list. In the next section we will present this list and check for the entries with the help of the previous corollaries whether they are Berger or weak-Berger algebras. \betasigmaubsetsection{Lie algebras with non-trivial first prolongation and the result} There are only a few complex Lie algebras ${\betafrak g}$ contained irreducibly in $\betamathfrak{gl}(V)$which have non vanishing first prolongation. The classification is due to \betacite{cartan09} and \betacite{ko-na65}. We will cite them following \betacite{schwachhoefer1} in two tables. \betabegin{eqnarray*}gin{center} \betabegin{eqnarray*}gin{tabular}{r|c|c|cl|c} \betamulticolumn{6}{l}{{\betabf Table 1} Complex Lie-groups and algebras with ${\betafrak g}^{(1)}\betanot=0 $ and ${\betafrak g}^{(1)}\betanot=V^*$:}\beta\beta \betamulticolumn{6}{l}{ }\beta\beta &$G$&${\betafrak g}$& \betamulticolumn{2}{|c|}{$V$} &${\betafrak g}^{(1)} $\beta\beta \betacline{1-6} 1.& $Sl(n, \betamathbb{C})$&$\betamathfrak{sl}(n, \betamathbb{C})$&$ \betamathbb{C}^n,$&$n\betage2$&$(V \betaotimes \betaodot^2 V^*)_0$ \beta\beta 2.& $Gl(n, \betamathbb{C})$&$\betamathfrak{gl}(n, \betamathbb{C})$&$ \betamathbb{C}^n$,&$n\betage 1$&$V \betaotimes \betaodot^2 V^*$ \beta\beta 3.&$Sp(n, \betamathbb{C})$& $\betamathfrak{sp}(n, \betamathbb{C})$&$ \betamathbb{C}^{2n}$,&$n\betage 2$&$ \betaodot^3 V^*$ \beta\beta 4.& $\betamathbb{C}^* \betatimes Sp(n, \betamathbb{C})$&$\betamathbb{C} \betaoplus\betamathfrak{sp}(n, \betamathbb{C})$,&$ \betamathbb{C}^{2n}$,&$n\betage 2$&$\betaodot^3 V^*$ \betaend{tabular} \betaend{center} \betavspace{.3cm} \betabegin{eqnarray*}gin{center}\betalanglebel{table2} \betabegin{eqnarray*}gin{tabular}{r|c|c|cl} \betamulticolumn{5}{l}{{\betabf Table 2} Complex Lie-groups and algebras with first prolongation ${\betafrak g}^{(1)}=V^*$:}\beta\beta \betamulticolumn{5}{l}{ }\beta\beta &$G$&${\betafrak g}$&\betamulticolumn{2}{|c}{$ V$} \beta\beta \betacline{1-5} 1.&$CO(n, \betamathbb{C})$&$ \betamathfrak{co}(n, \betamathbb{C})$&$ \betamathbb{C}^{n}$,&$n\betage 3$ \beta\beta 2.&$Gl(n, \betamathbb{C})$&$ \betamathfrak{gl}(n, \betamathbb{C})$&$ \betaodot^2 \betamathbb{C}^n$,&$n\betage 2$ \beta\beta 3.&$Gl(n, \betamathbb{C})$&$ \betamathfrak{gl}(n, \betamathbb{C})$&$ \betaomegaedge^2\betamathbb{C}^n$,&$n\betage 5$ \beta\beta 4.&$Gl(n, \betamathbb{C})\betacdot Gl(m, \betamathbb{C})$&$ \betamathfrak{sl}(\betamathfrak{gl}(n, \betamathbb{C}) \betaoplus \betamathfrak{gl}(m, \betamathbb{C}))$&$ \betamathbb{C}^n \betaotimes \betamathbb{C}^m$,&$m,n\betage2$ \beta\beta 5.&$\betamathbb{C}^*\betacdot Spin(10, \betamathbb{C})$&$ \betamathbb{C} \betaoplus\betamathfrak{spin}(10, \betamathbb{C})$&$\betaDeltaelta^+_{10}\betasigmaimeq\betamathbb{C}^{16}$& \beta\beta 6.&$\betamathbb{C}^* \betacdot E_6$&$ \betamathbb{C} \betaoplus {\betafrak e}_6$&$ \betamathbb{C}^{27}$& \betaend{tabular} \betaend{center} We have to make two remarks about the second table: The fourth Lie algebra is defined as \betabegin{eqnarray*}gin{eqnarray*} \betamathfrak{sl}(\betamathfrak{gl}(n, \betamathbb{C}) \betaoplus \betamathfrak{gl}(m, \betamathbb{C}))&=&\beta{(X,Y)\betain \betamathfrak{gl}(n, \betamathbb{C}) \betaoplus \betamathfrak{gl}(m, \betamathbb{C})|tr\beta X + tr\beta Y=0\beta} \beta\beta &=& \betaleft(\betamathfrak{gl}(n, \betamathbb{C}) \betaoplus \betamathfrak{gl}(m, \betamathbb{C})\betaright)\betacap \betamathfrak{sl}(n+m, \betamathbb{C}). \betaend{eqnarray*} The identification with the Lie algebra of the group is given as follows \betabegin{eqnarray*}gin{eqnarray*} \betamathfrak{sl}(\betamathfrak{gl}(n, \betamathbb{C}) \betaoplus \betamathfrak{gl}(m, \betamathbb{C}))&\betasigmaimeq&LA(Gl(n,\betamathbb{C})\betacdot GL(m,\betamathbb{C}))\betasigmaubsetset \betamathfrak{gl}(n\betacdot m, \betamathbb{C})\beta\beta (A,B)&\betalongmapsto& (x \betaotimes u\betamapsto Ax \betaotimes u - x \betaotimes Bu). \betaend{eqnarray*} In entry 5. $\betaDeltaelta^+_{10}$ denotes the irreducible $Spin(10, \betamathbb{C})$ spinor module. The representation in 6. is one of the two $27$-dimensional, irreducible ${\betafrak e}_6$ representations, which are conjugate to each other as representations of the compact real form of ${\betafrak e}_6$. \betaparagraph{The algebras of table 1} The first three entries of table 1 are all complexifications of Riemannian holonomy algebras $ \betamathfrak{su} (n)$, $ \betamathfrak{u} (n)$ acting on $\betamathbb{R}^{2n}$ and $\betamathfrak{sp}(n)$ acting on $\betamathbb{R}^{4n}$ and therefore Berger algebras. The fourth has the compact real form $i \betamathbb{R} \betaoplus \betamathfrak{sp}(n)\betasigmaimeq \betamathfrak{so}(2) \betaoplus \betamathfrak{sp}(n)$ acting irreducible on $ \betamathbb{R}^{4n}$ where $i\beta id$ corresponds to the element $J\betain \betamathfrak{u} (2n)$. Since the representation of $\betamathfrak{sp} (n) $ on $ \betamathbb{R}^{4n}$ is of non-real type we are in the situation of corollary \betaref{wichtig}, because $( \betamathbb{C} Id \betaoplus \betamathfrak{sp}(n, \betamathbb{C}))^{(1)}= \betamathfrak{sp}(n, \betamathbb{C})^{(1)}$. Hence $\betamathbb{C} \betaoplus {\betamathfrak sp}(2n, \betamathbb{C})$ is not a weak-Berger algebra. \betaparagraph{The algebras of table 2} If one looks at the unique (up to inner automorphisms) compact real form and the reellification of the Lie algebras and representations in table 2 one sees that they correspond to the holonomy representation of Riemannian symmetric spaces which are K\beta"ahlerian. This gives the following proposition. \betabegin{eqnarray*}gin{samepage} \betabegin{eqnarray*}gin{satz}\betalanglebel{unbewiesen} The compact real forms of the algebras in table 2 and the reellification of the representations are equivalent to the holonomy representations of the following Riemannian, K\beta"ahlerian symmetric spaces (see \betacite{helgason78}): \betabegin{eqnarray*}gin{center} \betabegin{eqnarray*}gin{tabular}{r|l|c|c|l} &Type&non-compact&compact&dim.\beta\beta \betacline{1-5} 1.& $BD \beta I$&$SO_0(2,n)\betabegin{itemize}g/SO(2)\betatimes SO(n)$&$ SO(2+n)\betabegin{itemize}g/SO(2)\betatimes SO(n)$&$2n$\beta\beta 2.&$C\beta I$&$Sp(n, \betamathbb{R})\betabegin{itemize}g/U(n)$&$Sp(n)\betabegin{itemize}g/U(n)$&$n(n+1)$\beta\beta 3.&$D\beta III$&$SO^*(2n)\betabegin{itemize}g/U(n)$&$SO(2n)\betabegin{itemize}g/U(n)$&$n(n-1)$\beta\beta 4.&$A\beta III$&$SU(n,m)\betabegin{itemize}g/U(n)\betacdot U(m)$&$SU(n+m)\betabegin{itemize}g/U(n)\betacdot U(m)$&$2nm$\beta\beta 5.&$E\beta III$&$\betaleft( {\betafrak e}_{6(-14)}, \betamathfrak{so}(2) \betaoplus \betamathfrak{so}(10) \betaright)$&$\betaleft( {\betafrak e}_{6(-78)}, \betamathfrak{so}(2) \betaoplus \betamathfrak{so}(10) \betaright)$&$32$\beta\beta 6.&$E\beta VII$&$\betaleft( {\betafrak e}_{7(-25)}, \betamathfrak{so}(2) \betaoplus \betamathfrak{e}_6 \betaright)$&$\betaleft( {\betafrak e}_{7(-133)}, \betamathfrak{so}(2) \betaoplus \betamathfrak{e}_6 \betaright)$&$54$\beta\beta \betamulticolumn{5}{l}{ }\beta\beta \betamulticolumn{5}{l}{{\betabf Table 3} Riemannian, K\beta"ahlerian symmetric spaces corresponding to table 2 } \betaend{tabular} \betaend{center} \betaend{satz} \betaend{samepage} So we obtain that all algebras corresponding to table 2 are Berger algebras and therefore also weak-Berger algebras. \betabegin{eqnarray*}gin{theo} Let $\betalangleg$ be a Lie algebra and $E$ an irreducible $\betalangleg$--module of non-real type. If $\betalangleg\betasigmaubsetset \betalangleso(E,h)$ is a weak-Berger algebra then it is a Berger algebra. \betaend{theo} \betaparagraph{Consequences for Lorentzian holonomy} All in all we have shown, that every real Lie algebra ${\betafrak g}_0$ of non-real type, i.e. contained in $\betamathfrak{u} (n)$, that can be weak-Berger is a Berger algebra. Further each of these Lie algebras is the holonomy algebra of a Riemannian manifold, the remaining entries of table 1 of non-symmetric ones, and the entries of table 2 of symmetric ones. Before we we apply this to the irreducible components of the $\betamathfrak{so}(n)$-projection of the holonomy algebra of an indecomposable Lorentzian manifold with light like invariant subspace, we prove a lemma to get the result in full generality. \betabegin{eqnarray*}gin{lem} Let ${\betafrak g}\betasigmaubsetset {\betafrak u}(n)\betasigmaubsetset \betamathfrak{so}(2n)$ be a Lie algebra with the decomposition property of theorem \betaref{theoI}, ie. there exists decompositions of $\betamathbb{R}^{2n}$ into orthogonal subspaces and of ${\betafrak g}$ into ideals \beta[ \betamathbb{R}^{2n} = E_0 \betaoplus E_1 \betaoplus \betaldots \betaoplus E_r\beta \betamathbbox{ and }\beta {\betafrak g} = {\betafrak g}_1 \betaoplus \betaldots \betaoplus {\betafrak g}_r\beta] where ${\betafrak g}$ acts trivial on $E_0$, ${\betafrak g}_i$ acts irreducible on $E_i$ and ${\betafrak g}_i (E_j)=\beta{0\beta}$ for $i\betanot=j$. Then ${\betafrak g}\betasigmaubsetset {\betafrak u}(n)$ implies $dim\beta E_i= 2 k_i$ and ${\betafrak g}_i \betasigmaubsetset {\betafrak u}(k_i)$ for $i=1, \betaldots ,r$. \betaend{lem} \betabegin{eqnarray*}gin{proof} Let $\betamathbb{R}^{2n}= \betamathbb{C}^n$ and $ \betatheta$ be the positive definite hermitian form on $\betamathbb{C}^n$. Let $E_i $ be an invariant subspace on which ${\betafrak g}$ acts irreducible. If $E_i= V^i_\betamathbb{R}$ for a complex vector space $V^i$, then we can restrict $\betatheta$ to $V^i$. Because $\betatheta$ is positive definite it is non-degenerate on $V^i$ --- since $\betatheta(v,v)>0$ for $v \betanot=0$ --- we get that $ {\betafrak g}_i\betasigmaubsetset {\betafrak u}(V^i, \betatheta)$, i.e. ${\betafrak g}\betasigmaubsetset {\betafrak u}(k_i)$. Hence we have to consider a subspace $E_i$ which is not the reellification of a complex vector space. Let $J$ be the complex structure on $\betamathbb{R}^{2n}$. We consider the real vector space $JE_i$, which is invariant under $ {\betafrak g}$, since $J$ commutes with ${\betafrak g}$. Then the space $JE_i \betacap E_i$ is contained in $E_i$ as well as in $JE_i$ and invariant under $ {\betafrak g}$. Because ${\betafrak g}$ acts irreducible on $E_i$ we get two cases. The first is $E_i \betacap JE_i = E_i = JE_i$, but this was excluded since $E_i$ was not a reellification. The second is $E_i\betacap JE_i=\beta{0\beta}$. So we have two invariant irreducible subspaces on which $ {\betafrak g}$ acts simultaneously, i.e. $A(x, Jy)=(Ax, AJy)$, but this is not possible because of the Borel-Lichnerowicz decomposition property from theorem \betaref{theoI}. \betaend{proof} \betabegin{eqnarray*}gin{theo}\betalanglebel{theoun} Let $(M,h)$ be an indecomposable $n+2$-dimensional Lorentzian manifold with light like holonomy-invariant subspace. Set ${\betafrak g}:= pr_{\betamathfrak{so}(n)} \betamathfrak{hol}_p(M,h)$ and suppose ${\betafrak g}\betasigmaubsetset \betamathfrak{u} (n)$. Then ${\betafrak g}$ is the holonomy algebra of a Riemannian manifold. \betaend{theo} \betabegin{eqnarray*}gin{proof} ${\betafrak g}\betasigmaubsetset \betamathfrak{u} (n)$ is a weak-Berger algebra. Then all the ${\betafrak g}_i$ of the decomposition of theorem \betaref{theoI} are unitary because of the lemma and weak-Berger because of corollary \betaref{zerlegung}. Hence they are weak-Berger of non-real type. Then ${\betafrak g}_i$ corresponds to a compact real form of the entries of table 1 or 2. But these are all Riemannian holonomy algebras, and therefore $\betamathfrak{g}$ is a Riemannian holonomy algebra. \betaend{proof} \betabegin{bem} {\betabf Quaternionic symmetric spaces.}\betalanglebel{quatsym2} Again we have to make a remark about quaternionic symmetric spaces (see remark \betaref{quatsym1}). If $G/Sp(1)\betacdot H$ with $H\betasigmaubsetset Sp(n)$ is a quaternionic symmetric space then the corresponding complex irreducible representation of $H$ is of quaternionic, i.e. of non real type, and it is Berger \betacite{schwachhoefer2}. But the real representation of $H$, i.e. the reellification of the complex one, is not. Thats why it does not occur in the above list. The place of $Sp(1)\betacdot H$ would be in a list of real semisimple, but non-simple, weak-Berger algebras of real type. \betaend{bem} \betabegin{eqnarray*}gin{appendix} \betasigmaection{Representations of real Lie algebras} \betalanglebel{real representations} In this appendix we will collect and illustrate some standard facts about representations of real Lie algebras. Because of the theorem \betaref{theoI} and proposition \betaref{zerlegung} we are interested in irreducible real representations of real Lie algebras which are orthogonal. First we will recall some facts about irreducible complex representations of real Lie algebras, in particular orthogonal or unitary ones. Then we will use the results of E. Cartan (\betacite{cartan1914}, see also \betacite{goto78}, pp.363 and \betacite{iwahori59}), in order to reduce the study of real representations to that of complex ones. Throughout the whole section ${\betafrak g}$ is a real Lie algebra. \betasigmaubsetsection{Preliminaries} First of all we recall the Schur-lemma. \betabegin{eqnarray*}gin{satz}[Schur-lemma] Let $\betakappa_1$, $\betakappa_2$ be irreducible representations of ${\betafrak g}$ on $\betamathbb{K}$-vector spaces $V_1$ and $V_2$. Let $f\betain Hom_{\betafrak g}(V_1, V_2)$ be an invariant homomorphism, i.e. \beta[ f \betacirc \betakappa_1(A) \beta = \beta \betakappa_2(A) \betacirc f\betamakebox[4cm][r]{for all $A\betain {\betafrak g}$.}\beta] Then holds \betabegin{eqnarray*}gin{enumerate} \betaitem $f$ is zero or an isomorphism, i.e. $V_1 \betanot\betasigmaimeq V_2$ implies $Hom_{\betafrak g}(V_1, V_2)=0$. \betaitem If $V_1=V_2=:V$ and if $f$ has an eigenvalue $\betalanglembda\betain \betamathbb{K}$, then $f= \betalanglembda\beta id_V$. I.e. if $\betamathbb{K}= \betamathbb{C}$ and $V_1=V_2$ we have always $f=\betalanglembda\beta id$ with $\betalanglembda \betain \betamathbb{C}$. \betaend{enumerate} \betaend{satz} For invariant bilinear forms, i.e. forms $\betabegin{eqnarray*}ta$ which satisfy \betabegin{eqnarray*}gin{equation} \betalanglebel{invarianz} \betabegin{eqnarray*}ta( \betakappa(A)u,v) + \betabegin{eqnarray*}ta(u, \betakappa(A)v)=0 \betamakebox[3cm][r]{for all $A\betain {\betafrak g}$} \betaend{equation} this gives the following consequence. \betabegin{eqnarray*}gin{folg} Let $\betakappa$ be an irreducible representation of $ {\betafrak g} $ on a $\betamathbb{K}$--vector space $V$ and $ \betabegin{eqnarray*}ta$ be the invariant bilinear form. Then $ \betabegin{eqnarray*}ta$ is zero or non-degenerate. If $\betamathbb{K}= \betamathbb{C}$, then the space of invariant bilinear forms is zero or one-dimensional. It is one-dimensional if and only if $V\betasigmaimeq_\betakappa V^*$. Then it is generated by a symmetric or an anti-symmetric bilinear form. \betaend{folg} This consequence is obvious by applying the Schur-lemma to the endomorphism of $V$, which is induced by two invariant bilinear forms. For complex representations and invariant sesqui-linear forms, i.e. forms $\betatheta$ with \beta[\betatheta (\betalanglembda u,v)= \betalanglembda \betatheta (u,v)\betamakebox[2cm][c]{ and } \betatheta (u,\betalanglembda v)= \betaoverline{\betalanglembda} \betatheta (u,v),\beta] one has an analogous result. \betabegin{eqnarray*}gin{folg} Let $\betakappa $ be an irreducible representation of $ {\betafrak g}$ on a $ \betamathbb{C}$-vector space $V$. Every invariant sesqui-linear form is zero or non-degenerate, and the space of invariant sesqui-linear-forms is zero or one-dimensional. It is one dimensional if and only if $\betaoverline{V}\betasigmaimeq_\betakappa V^*$. In this case it is generated by a hermitian or an anti-hermitian form, and the spaces of invariant hermitian and invariant anti-hermitian forms are one-dimensional real subspaces, identified by the multiplication with $i$. \betaend{folg} In these corollaries we refer to the dual and the conjugate representations, which are defined as follows: \betabegin{eqnarray*}gin{eqnarray*} (\betakappa^*(A) \betaalpha)v&=& - \betaalpha (\betakappa(A)v)\beta\beta \betaoverline{\betakappa}(A)\betaoverline{v}&=&\betaoverline{\betakappa(A)v}. \betaend{eqnarray*} \betabegin{eqnarray*}gin{de} Let $\betakappa$ be an arbitrary representation of a Lie algebra ${\betafrak g}$ on a $\betamathbb{K}$-vector space $V$. \betabegin{eqnarray*}gin{enumerate} \betaitem Then $\betakappa$ is called {\betabf self-dual} if there is an invariant isomorphism between $V$ and $V^*$. This is equivalent to the existence of an invariant bilinear form $\betabegin{eqnarray*}ta$. \betaitem If $ \betamathbb{K}=\betamathbb{C}$, then $\betakappa$ is called {\betabf self-conjugate} if there is an invariant isomorphism from $V$ to $\betaoverline{V}$, i.e. there exists an anti-linear bijective mapping $J:V\betalongrightarrow V$ which is invariant, i.e. $J \betacirc \betakappa(A)= \betakappa(A) \betacirc J\beta \betamathbbox{ for all }A\betain {\betafrak g}$. \betaend{enumerate} \betaend{de} It is evident that the existence of an invariant hermitian form or a self-conjugate representation is only possible for {\betabf real Lie algebras}. \betasigmaubsetsection{Irreducible complex representations of real Lie algebras} \betabegin{eqnarray*}gin{de} Let $\betakappa$ be an {\betabf irreducible} complex representation of a real Lie algebra ${\betafrak g}$ on $V$. $\betakappa$ is called \betabegin{eqnarray*}gin{description} \betaitem[of real type] if $\betakappa$ is self-conjugate with $J^2=1$, \betaitem[of quaternionic type] if $\betakappa$ is self-conjugate with $J^2=-1$ and \betaitem[of complex type] if $\betakappa$ is not self-conjugate.\betaend{description} \betaend{de} From the Schur-lemma it is clear that every complex irreducible representation is either real, complex or quaternionic: If $\betakappa$ is self-conjugate, then $J^2$ is a linear automorphism of $\betakappa$ so that $J^2= \betalanglembda \beta id$. Furthermore $\betalanglembda$ must be real because of \beta[\betalanglembda Jv= J^2 Jv = J J^2v = J \betalanglembda v = \betaoverline{\betalanglembda} Jv.\beta] Dividing $J$ by $\betasigmaqrt{|\betalanglembda|}$ one gets $\betalanglembda=\betapm1$. Now it holds \betabegin{eqnarray*}gin{satz} \betalanglebel{realtype} Let $\betakappa$ be a complex irreducible representation of a real Lie algebra ${\betafrak g}$. Then $\betakappa$ is of real type if and only if the reellification $\betakappa_\betamathbb{R}$ is reducible. \betaend{satz} \betabegin{eqnarray*}gin{proof} ($\betaLongrightarrow$) Let $\betakappa $ be of real type, i.e. there is an anti-linear, invariant automorphism of the complex representation space $V$ with $J^2= $id. Then $J$ is $\betamathbb{R}$-linear, and $V_{\betamathbb{R}}$ splits into invariant, real vector spaces \betabegin{eqnarray*}gin{eqnarray*} V_{\betapm }&=& \beta{ v\betain V\beta |\beta Jv=\betapm v\beta \beta}\beta\beta V_{\betamathbb{R}}&=& V_+\betaoplus V_-. \betaend{eqnarray*} So $\betakappa_\betamathbb{R}$ is reducible. ($\betaLongleftarrow$) Let $W$ be a real, $\betakappa_\betamathbb{R}$-invariant subspace of $ V_\betamathbb{R}$. On $V_\betamathbb{R}$ the multiplication with $i$ gives an $\betamathbb{R}$-automorphism, which defines two subspaces of $V_\betamathbb{R}$: $W\betacap iW$ and $W+iW$. Then both are complex vector spaces in an obvious way, such that they are complex subspaces of $V$. Since $W$ is $\betakappa_\betamathbb{R}$ invariant, both are $\betakappa$ invariant. Since $\betakappa$ is irreducible, it remains the case that $W\betacap iW=\beta{0\beta}$ and $W\betaoplus $i$W=V$. But this means that $V=W^\betamathbb{C}$ such that $W$ defines a conjugation $J$ in $V$ with the desired properties. \betaend{proof} \betaparagraph{Orthogonal and unitary representations} \betabegin{eqnarray*}gin{satz}\betalanglebel{dualtype} Let $\betakappa$ be an irreducible representation of ${\betafrak g}$. If $\betakappa$ is of complex type, then it can not be both, unitary and self-dual. If $\betakappa$ is not of complex type, then it is unitary if and only if it is self-dual. In particular one has for real and quaternionic representations ($J$ denotes the automorphism): \betabegin{eqnarray*}gin{enumerate} \betaitem If $\betakappa$ is of real type, then it is orthogonal if and only if it is unitary with respect to $\betatheta$ for which holds $J^*\betatheta=\betaoverline{\betatheta}$. It is symplectic if and only if it is unitary with respect to $\betatheta$ satisfying $J^*\betatheta=-\betaoverline{\betatheta}$. \betaitem If $\betakappa$ is of quaternionic type, then it is orthogonal if and only if it is unitary with respect to $\betatheta$ with $J^*\betatheta=- \betaoverline{\betatheta}$. It is symplectic if and only if it is unitary with respect to $\betatheta$ satisfying $J^*\betatheta=\betaoverline{\betatheta}$. \betaend{enumerate} \betaend{satz} \betabegin{eqnarray*}gin{proof} Unitary is equivalent to $V^*\betasigmaimeq_\betakappa \betaoverline{V}$ and therefore self-dual is the same as $V\betasigmaimeq_\betakappa \betaoverline{V}$. This gives the proposition. For the remaining single points we get: 1.) Let $\betakappa$ be of real type with respect to a real structure $J$. By this $J$ one gets from an invariant bilinear form $\betabegin{eqnarray*}ta$ an invariant sesqui-linear form $\betabegin{eqnarray*}ta(.,J.)$ which is the complex multiple of an invariant hermitian form $\betatheta$ and vice versa. Then one gets for $\betabegin{eqnarray*}ta$ symmetric/anti-symmetric: \betabegin{eqnarray*}gin{eqnarray*} J^*\betatheta(u,v)&=& \betatheta (Ju,Jv) \beta =\beta \betalanglembda\betabegin{eqnarray*}ta(Ju, J^2v)\beta \betasigmatackrel{J^2=id}{=}\beta \betalanglembda\betabegin{eqnarray*}ta(Ju, v) \beta\beta &=& \betapm \betalanglembda\betabegin{eqnarray*}ta(v Ju) \beta =\beta \betapm\betatheta (v,u)\beta =\beta \betapm\betaoverline{\betatheta (u,v)}. \betaend{eqnarray*} 2.) analogous with $J^2=-id$. \betaend{proof} \betabegin{eqnarray*}gin{folg}\betalanglebel{+unitary} If $\betakappa$ is positive definite unitary, then it is \betabegin{eqnarray*}gin{enumerate} \betaitem of real type if and only if it is orthogonal, \betaitem of complex type if and only it is not self-dual, \betaitem of quaternionic type if and only if it is symplectic. \betaend{enumerate} \betaend{folg} \betabegin{eqnarray*}gin{proof} If $\betatheta $ is positive definite it can not be $J^*\betatheta = - \betatheta$. \betaend{proof} \betasigmaubsetsection{Irreducible real representations} For a real irreducible representation $\betarho$ of a real Lie algebra $ {\betafrak g}$ on a real vector space $E$ two cases are possible: $\betarho^{\betamathbb{C}}$ is irreducible or reducible. We will describe these cases due to results of E. Cartan (\betacite{cartan1914}, see also \betacite{goto78}, pp.363 and \betacite{iwahori59}), in order to reduce the study of real representations to that of complex ones. \betasigmaubsetsubsection{Representations of real type} \betabegin{eqnarray*}gin{satz}\betalanglebel{realtype1} Let ${\betafrak g} $ be a real Lie algebra and $\betarho$ a representation of $ {\betafrak g}$ on a real vector space $E$ such that $ \betarho^{\betamathbb{C}}$ is {\betabf irreducible} on $ E^\betamathbb{C}$. Then the complex representation $ \betarho^{\betamathbb{C}}$ is of real type. If otherwise $\betakappa$ is a complex representation of $ {\betafrak g} $ of {\betabf real type} on $V$, then $\betakappa$ is the complexification of a real irreducible representation of ${\betafrak g}$. \betaend{satz} \betabegin{eqnarray*}gin{proof} 1.) We show the existence of a $ \betarho^{\betamathbb{C}}$-invariant anti-linear isomorphism $J$ with $J^2=id$. If we denote by $J$ the conjugation in $ E^\betamathbb{C}$ with respect to $E$, then it is $J^2=1$ and we have \beta[J \betaleft( \betarho^{\betamathbb{C}}(A)(u+iv) \betaright) = \betarho^{\betamathbb{C}}(A)(u)- i \betarho^{\betamathbb{C}}(A)(v) = \betarho^{\betamathbb{C}}(A)\betaleft( J(u+iv) \betaright)\beta] i.e. $J$ is $ \betarho^{\betamathbb{C}}$-invariant. 2.) In the proof of proposition \betaref{realtype} we had already shown that for complex representations of real type holds that $V=W^\betamathbb{C}$. \betaend{proof} So the following definition makes sense. \betabegin{eqnarray*}gin{de} Irreducible real representations with irreducible complexification and irreducible complex representations with reducible reellification (i.e. of real type) are called representations of {\betabf real type}. \betaend{de} We have the following correspondence: \betabegin{eqnarray*}gin{eqnarray} \betaleft\beta{ \betamathbbox{real representation of real type}\betaright\beta}_{/\betasigmaim}&\betaleftrightarrow& \betaleft\beta{ \betamathbbox{complex representations of real type} \betaright\beta}_{/\betasigmaim}\betalanglebel{type1}\beta\beta\betanonumber \betarho&\betamapsto& \betarho^\betamathbb{C}\beta\beta\betanonumber (\betakappa_\betamathbb{R})_{|\betamathbbox{maximal invariant subspace}}&\betaleftarrowtail& \betakappa. \betaend{eqnarray} Here $\betasigmaim$ denotes the equivalence of representations. \betasigmaubsetsubsection{Representations of non-real type} The situation in this case is described by the following \betabegin{eqnarray*}gin{satz} Let ${\betafrak g} $ be a real Lie algebra and $\betarho$ be an irreducible representation of $ {\betafrak g}$ on a real vector space $E$ such that $ \betarho^{\betamathbb{C}}$ is {\betabf reducible} on $ E^\betamathbb{C}$. \betabegin{eqnarray*}gin{enumerate} \betaitem If $V\betasigmaubsetset E^\betamathbb{C}$ is any invariant subspace of $ \betarho^{\betamathbb{C}}$. Then holds \beta[E^\betamathbb{C}= V \betaoplus \betaoverline{V},\beta] where $\betaoverline{{ }^{\betaleft.\beta \betaright.}}$ is the conjugation in $E^\betamathbb{C}$ with respect to $E$. $V$ and $\betaoverline{V}$ are irreducible and unique as maximal invariant proper subspaces. The representations on $V$ and $\betaoverline{V}$ are conjugate to each other. \betaitem The irreducible representations of $ {\betafrak g}$ on $V$ and on $\betaoverline{V}$ are of complex or of quaternionic type, its reellifications are equivalent to $\betarho$. \betaend{enumerate} If otherwise $\betakappa$ is a complex irreducible representation of complex or quaternionic type, then $\betakappa$ is the restriction on the maximal invariant proper subspace of the complexification of $\betakappa_\betamathbb{R}$. If $\betakappa$ is of complex type, then exists and $\betakappa_\betamathbb{R}$--invariant complex structure $J$ on $V_\betamathbb{R}$. $\betakappa$ is of quaternionic type if and only if there exists and $\betakappa_\betamathbb{R}$--invariant quaternionic structure $(I,J,K)$ on $V_\betamathbb{R}$. \betaend{satz} \betabegin{eqnarray*}gin{proof} 1.) Let $V\betasigmaubsetset E^\betamathbb{C}$ be any invariant, proper subspace of $ \betarho^{\betamathbb{C}}$. Lets denote by $\betaoverline{{ }^{\betaleft.\beta \betaright.}}$ the conjugation in $E^\betamathbb{C}$ with respect to $E$. We consider $W:=V + \betaoverline{V}$. Now it is $\betaoverline{W}=W$ which is equivalent to $W= F^\betamathbb{C}$, where $F=W\betacap E$ is a real subspace of $E$. Since $W$ is invariant under $ \betarho^{\betamathbb{C}}$, $F$ is invariant under $\betarho$. Now $\betarho$ is irreducible and therefore $F=E$, i.e. $V + \betaoverline{V}= E^\betamathbb{C}$. Analogously one shows that $V \betacap \betaoverline{V}=\beta{0\beta}$, so that one gets \beta[V \betaoplus \betaoverline{V} =E^\betamathbb{C}.\beta] It remains to show that $V $ is irreducible: This is clear since every invariant subspace $U\betasigmaubsetset V$ is invariant in $ E^\betamathbb{C}$, but then holds that $U \betaoplus \betaoverline{U} = E^\betamathbb{C}$ which implies $U=V$. $\betaoverline{V}$ is irreducible too. Hence we have two irreducible representations of ${\betafrak g}$, one on $V$ and one on $ \betaoverline{V}$, which are conjugate to each other: \beta[ \betarho^{\betamathbb{C}}(A) \betaoverline{v} = \betaoverline{ \betarho^{\betamathbb{C}}v}.\beta] So we will denote it by $\betakappa$ and $\betaoverline{\betakappa}$. 2.) In order to show that $\betakappa$ and $ \betaoverline{\betakappa}$ are of complex or quaternionic type, we verify that $ \betakappa_\betamathbb{R}$ and $ \betaoverline{\betakappa}_\betamathbb{R}$ are irreducible. For this we show that $\betakappa_\betamathbb{R}$ and $\betaoverline{\betakappa}_\betamathbb{R}$ are isomorphic to $\betarho$. The isomorphism between $V$ and $E$ is given by \beta[\betabegin{eqnarray*}gin{array}{rcrcl} \betapsi&:& V_\betamathbb{R}&\betalongrightarrow& E\beta\beta &&v&\betalongmapsto & \betafrac{1}{2}(v+\betaoverline{v}). \betaend{array}\beta] This is obviously an isomorphism of real vector spaces. (Of course this is also an isomorphism between $\betaoverline{V}_\betamathbb{R}$ and $E$.) It is also invariant since \beta[\betapsi \betacirc \betakappa_\betamathbb{R}(A)(x+iy) = \betapsi (\betarho(A)x + i\betarho(A)y)= \betarho(A)x = \betarho(A)(\betapsi (x+iy) \beta] for all $x+iy\betain V_\betamathbb{R}$. The existence of the complex and the quaternionic structure on $V_\betamathbb{R}$ is clear. \betaend{proof} Again on defines: \betabegin{eqnarray*}gin{de} Irreducible real representations with reducible complexification and irreducible complex representations with irreducible reellification are called representations of {\betabf non-real type} (of {\betabf complex} or {\betabf quaternionic type} respectively). \betaend{de} Again we have the correspondence \betabegin{eqnarray*}gin{eqnarray}\betalanglebel{type2} \betaleft\beta{ \betabegin{eqnarray*}gin{array}{l}\betamathbbox{real representations}\beta\beta \betamathbbox{of non-real type} \betaend{array}\betaright\beta}_{/\betasigmaim}&\betaleftrightarrow& \betaleft\beta{ \betabegin{eqnarray*}gin{array}{l}\betamathbbox{complex representations}\beta\beta \betamathbbox{of non-real type} \betaend{array}\betaright\beta}_{/\betaapprox}\beta\beta \betanonumber \betarho&\betamapsto & \betarho^\betamathbb{C}_{|\betamathbbox{maximal invariant subspace}}\beta\beta\betanonumber \betakappa_\betamathbb{R}&\betaleftarrowtail&\betakappa. \betaend{eqnarray} Here $\betasigmaim$ denotes the equivalence of representation and $\betaapprox$ the equivalence \betabegin{eqnarray*}gin{eqnarray*} \betakappa_1\betaapprox \betakappa_2 &\betaLeftrightarrow& \betakappa_1 \betasigmaim \betakappa_2 \betamathbbox{ or } \betakappa_1 \betasigmaim \betaoverline{\betakappa_2}. \betaend{eqnarray*} On the real space $E\betasigmaimeq V_\betamathbb{R}$ we have the complex structure $J$, i.e. an $\betamathbb{R}$--automorphism with $J^2=-1$ given by the multiplication with $i$: $Jv=iv$. $J$ commutes with $\betarho$ since \beta[\betarho(A)(Jv)= \betakappa_\betamathbb{R} (A)(Jv)= \betakappa (A)iv=i \betakappa(A)v= J( \betakappa(A)v).\beta] One describes the complex vector space $V$ as a subspace in $E^\betamathbb{C}$ as follows. One extends the complex structure to an automorphism of $E^\betamathbb{C}$ also denoted by $J$ and with the property $J^2=-1$. Then one defines \beta[ V_{\betapm}:= \beta{v\betain E^\betamathbb{C}|Jv=\betapm i\beta v\beta}\betasigmaubsetset E^\betamathbb{C}\beta] and gets $E^\betamathbb{C}= V_+ \betaoplus V_-$. Furthermore it is \betabegin{eqnarray*}gin{equation} \betalanglebel{v+form} V_{\betapm}= \beta{x\betamp iJx|x\betain E \beta}\betamathbbox{ and therefore }V_{\betapm}=\betaoverline{V}_{\betamp}. \betaend{equation} Then one has the following isomorphisms, invariant under the corresponding representations: \betabegin{eqnarray*}gin{equation} \betalanglebel{isomorphisms} \betabegin{eqnarray*}gin{array}{rcccccl} E&\betasigmaimeq_\betamathbb{R}&V&\betasigmaimeq_\betamathbb{C}&V_+&\betasigmaimeq_\betamathbb{R}& V_- =\betaoverline{V_+}\beta\beta \betafrac{1}{2}(v+\betaoverline{v})&\betaleftarrowtail &v&\betamapsto&\betafrac{1}{2}(v-iJv)&\betamapsto&\betafrac{1}{2}(v+iJv). \betaend{array} \betaend{equation} \betasigmaubsetsection{Orthogonal real representations} Let now $\betarho$ be a real representation of ${\betafrak g}$ on $E$ which should be orthogonal (or symplectic) with respect to a (anti-)symmetric bilinear form $h$. On $E^\betamathbb{C}$ $h$ defines a (anti-)symmetric bilinear form by bilinear extension, denoted by $h^\betamathbb{C}$ and a (anti-)hermitian form by conjugate linear extension in the second component, denoted by $h^pr_{\betalason}ime$. Both are invariant under $ \betarho^{\betamathbb{C}}({\betafrak g})$. The hermitian form has the same signature as the symmetric form $h$. The existence of an invariant {anti-hermitian} form is equivalent to the existence of an invariant hermitian form. For the conjugation in $E^\betamathbb{C}$ we have the following relations \beta[ \betabegin{eqnarray*}gin{array} {rcccccl} h^pr_{\betalason}ime(u,v)&=&h^\betamathbb{C}(u,\betabar{v})&=&\betaoverline{h^\betamathbb{C} (\betaoverline{u},v)} &=&\betaoverline{h^pr_{\betalason}ime(v,u)} \betaend{array}.\beta] \betasigmaubsetsubsection{Orthogonal or symplectic representations of real type} From these introductory remarks we obtain the following proposition for real type representations which can be found in \betacite{berger55}(for the orthogonal case). \betabegin{eqnarray*}gin{satz}\betalanglebel{prop1} \betacite{berger55} Let $\betarho$ be a real representation of real type of a real Lie algebra ${\betafrak g}$ on a real vector space $E$, orthogonal or symplectic with respect to $h$. Let $\betabegin{eqnarray*}ta^h$ denote the complex linear and $\betatheta^h $ the hermitian extension of $h$ on $V=E^\betamathbb{C}$. Then both are non-degenerate and $\betarho^\betamathbb{C}$ is orthogonal/symplectic with respect to $\betabegin{eqnarray*}ta^h$ and unitary with respect to $\betatheta^h$. $\betatheta^h$ has the same index as $h$ in case $h$ is orthogonal. \betaend{satz} This gives a \betabegin{eqnarray*}gin{folg} If $\betarho$ is of real type, then the space of invariant bilinear form is one- dimensional and generated by a symmetric or an anti-symmetric form.\betaend{folg} The {\betait proof} is clear because the irreducibility of $\betarho^{\betamathbb{C}}$ gives that $h_1^\betamathbb{C}=h_2^\betamathbb{C}$, which implies $h_1=h_2$. \betahfill $\betaBox$\beta\beta We will now prove the other direction of proposition \betaref{prop1}. \betabegin{eqnarray*}gin{satz}\betalanglebel{realtypeorthogonal} Let ${\betafrak g}$ be a real Lie algebra and $\betakappa$ an irreducible, complex representation of real type on $V$, which decomposes $\betakappa_\betamathbb{R}$-invariant into $V=E \betaoplus i E $, and set $\betarho = (\betakappa_\betamathbb{R})_{|E}$ the corresponding irreducible real representation. If $\betakappa$ is unitary (and therefore self-dual), then $\betarho$ is self-dual, i.e. orthogonal or symplectic and we have two cases: \betabegin{eqnarray*}gin{enumerate} \betaitem If $ \betakappa$ is orthogonal, then $\betarho $ is orthogonal. \betaitem If $\betakappa$ is symplectic, then $\betarho $ is symplectic \betaend{enumerate} \betaend{satz} \betabegin{eqnarray*}gin{proof} Let $\betakappa$ be unitary with respect to $\betatheta$, which defines two bilinear mappings on $E$ \betabegin{eqnarray*}gin{eqnarray*} h_1(x,y)&=& Re \betaleft( \betatheta(x,y) \betaright)\beta \betamathbbox{ symmetric}\beta\beta h_2(x,y)&=& Im \betaleft( \betatheta(x,y) \betaright)\beta \betamathbbox{ anti-symmetric.} \betaend{eqnarray*} Both are $\betarho$ invariant. If both are degenerate, then both are zero by the Schur-lemma and so $\betatheta$ must be zero, which is a contradiction. 1.) If in addition $\betakappa$ is orthogonal, then for $\betatheta$ holds by proposition \betaref{dualtype} that $J^*\betatheta =\betaoverline{\betatheta}$, where $J$ is the conjugation of $E$ in $ E^\betamathbb{C}$. But in this case $h_2$ is zero, because $E=\beta{v\betain V|Jv=v\beta}$: \beta[ h_2(x,y)=Im\betatheta(x,y)=Im\betatheta(Jx,Jy)=Im\betaoverline{\betatheta(x,y)}=- Im\betatheta(x,y)=-h_2(x,y).\beta] Hence $h_1$ must be non degenerate and therefore $\betarho$ orthogonal. 2.) If $\betakappa$ is symplectic one shows analogously with proposition \betaref{dualtype} that $h_1=0$ and therefore $\betarho$ symplectic. \betaend{proof} Both results give the following equivalence: \betabegin{eqnarray*}gin{eqnarray} \betalanglebel{typ1equivalenz} \betaleft\beta{ \betarho\betamathbbox{ real, real type, self-dual} \betaright\beta}_{/\betasigmaim}&\betaleftrightarrow& \betaleft\beta{ \betabegin{eqnarray*}gin{array}{l} \betakappa \betamathbbox{ complex, real type,}\beta\beta \betamathbbox{self-dual $\betahat{=}$ unitary} \betaend{array} \betaright\beta}_{/\betasigmaim}\beta\beta \betalanglebel{typ1equivalenz2} \betaleft\beta{ \betabegin{eqnarray*}gin{array}{l} \betarho\betamathbbox{ real, real type,}\beta\beta \betamathbbox{orthogonal/symplectic} \betaend{array} \betaright\beta}_{/\betasigmaim}&\betaleftrightarrow& \betaleft\beta{ \betabegin{eqnarray*}gin{array}{l} \betakappa \betamathbbox{ complex, real type,}\beta\beta \betamathbbox{orthogonal/symplectic}\betaend{array} \betaright\beta}_{/\betasigmaim}. \betaend{eqnarray} \betasigmaubsetsubsection{Orthogonal representations of non-real type} For non-real type representations we have the $ \betarho^{\betamathbb{C}}$-invariant decomposition $E^\betamathbb{C}=V \betaoplus\betaoverline{V}$. In a basis, adapted to this decomposition $h^\betamathbb{C}$ and $h^pr_{\betalason}ime$ are given as follows \betabegin{eqnarray*}gin{eqnarray*} h^\betamathbb{C}= \betaleft( \betabegin{eqnarray*}gin{array}{cc} A & B \beta\beta B^t & \betaoverline{A} \beta\beta \betaend{array} \betaright) &\betamathbbox{and}& h^pr_{\betalason}ime= \betaleft( \betabegin{eqnarray*}gin{array}{cc} B & A \beta\beta \betaoverline{A} & B^t \beta\beta \betaend{array} \betaright) \betaend{eqnarray*} where $A=A^t$ and $B^t=\betaoverline{B}$ are quadratic matrices with the dimension of $V$. Now one defines a bilinear and a sesqui-linear form on $V$ resp. on $\betaoverline{V}$: \beta[\betabegin{eqnarray*}gin{array}{rcccl} \betabegin{eqnarray*}ta^h(u,v)&:=& h^\betamathbb{C}(u,v)&=& h^pr_{\betalason}ime(u,\betaoverline{v})\betamakebox[5cm][r]{symmetric/anti-symmetric}\beta\beta \betatheta^h(u,v)&:=& h^\betamathbb{C}(u,\betaoverline{v})&=& h^pr_{\betalason}ime(u, v)\betamakebox[5cm][r]{hermitian/anti-hermitian} \betaend{array}\beta] for $u,v\betain V$ resp. $\betaoverline{V}$. Both are invariant under $\betakappa= \betarho^{\betamathbb{C}}_{|V}({\betafrak g})$. From the Schur-lemma it is clear that at least one of them is non-degenerate, since $h^\betamathbb{C}$ is non-degenerate. Using the isomorphisms of (\betaref{isomorphisms}) we can give $ \betatheta^h$ and $\betabegin{eqnarray*}ta^h$ explicitly: \betabegin{eqnarray*}gin{eqnarray} \betabegin{eqnarray*}ta^h(x-iJx, y-iJy) &=&\betalanglebel{h-omega} \betafrac {1}{4}\betaleft( h(x,y)- h(Jx,Jy) -i \betaleft(h(J x,y) + h(x,Jy) \betaright) \betaright)\beta\beta \betatheta^h (x-iJx,y-iJy) &=&\betalanglebel{h-theta} \betafrac {1}{4}\betaleft( h(x,y) + h(Jx,Jy) +i \betaleft(h(x,Jy) - h(Jx,y) \betaright) \betaright). \betaend{eqnarray} Again we have the proposition of Berger (for the orthogonal case). \betabegin{eqnarray*}gin{satz}\betacite{berger55} Let $\betarho $ be a real orthogonal/symplectic representation of non-real type, i.e. $(E,\betarho)=(V_\betamathbb{R},\betakappa_\betamathbb{R})$. Then $\betakappa$ is invariant under $ \betabegin{eqnarray*}ta^h$ and $ \betatheta^h$ and at least one of them is non-degenerate, i.e. $ \betakappa$ is orthogonal/symplectic or unitary/anti-unitary with respect to $ \betabegin{eqnarray*}ta^h$ or $ \betatheta^h$. Furthermore holds: If $ {\betafrak g}$ contains a real sub-algebra ${\betafrak h}\betanot=0$ such that ${\betafrak h}= {\betafrak p}_\betamathbb{R}$ where ${\betafrak p}$ is a complex Lie algebra, then $\betatheta^h=0$ i.e. $\betabegin{eqnarray*}ta^h$ non-degenerate. \betaend{satz} \betabegin{eqnarray*}gin{proof} We only have to prove the second assertion. By assumption we have a complex Lie structure on ${\betafrak h}$, i.e. a automorphism $J$ with $J^2=-1$ and $J \betacirc ad_X = ad_X \betacirc J$. As above for vector spaces we have here a Lie algebra decomposition \beta[ {\betafrak g}^\betamathbb{C}\betasigmaupset {\betafrak h}^\betamathbb{C}= {\betafrak p}_+ \betaoplus {\betafrak p}_-\betamathbbox{ with }{\betafrak p}_{\betapm}= \beta{v\betain {\betafrak h}^\betamathbb{C}| Jv=\betapm i v\beta}.\beta] Then ${\betafrak p}\betasigmaimeq_\betamathbb{C} {\betafrak p}_+$. Let now $\betarho^{\betamathbb{C}}$ be extended to ${\betafrak g}^\betamathbb{C}$. Then because of its linearity $h^\betamathbb{C}$ is invariant under $ \betarho^{\betamathbb{C}}({\betafrak g}^\betamathbb{C})$. But if we suppose that $\betatheta^h$ is invariant under ${\betafrak g}$ we have for a $H\betain {\betafrak h}$ and $\betakappa= \betarho^{\betamathbb{C}}_{|V}$ as above \betabegin{eqnarray*}gin{eqnarray*} 0&=& \betatheta^h (\betakappa(JH)v,w) + \betatheta^h (v, \betakappa(JH)w) \beta\beta &\betasigmatackrel{p.d.}{=}& h^\betamathbb{C}( \betakappa(JH)v,\betaoverline{w}) + h^\betamathbb{C}(v, \betaoverline{\betakappa(JH)w}) \beta\beta &=& h^\betamathbb{C}( \betarho^{\betamathbb{C}} (JH)v,\betaoverline{w}) + h^\betamathbb{C}(v, \betaoverline{ \betarho^{\betamathbb{C}} (JH)w}) \beta\beta &\betasigmatackrel{H\betain {\betafrak p}_+}{=}& i \betaleft(h^\betamathbb{C}( \betarho^{\betamathbb{C}} (H)v,\betaoverline{w}) - h^\betamathbb{C}(v, \betaoverline{ \betarho^{\betamathbb{C}} (H)w}) \betaright) \beta\beta &=& i \betaleft(\betatheta^h (\betakappa(H)v,w) - \betatheta^h (v, \betakappa(H)w) \betaright) \beta\beta &\betasigmatackrel{\betatheta^h\betamathbbox{ invariant}}{=}&2i \betatheta^h (\betakappa(H)v,w) \betaend{eqnarray*} for all $H\betain {\betafrak h}$, $v,w\betain V$. This means ${\betafrak h}\betasigmaubsetset ker\beta \betakappa=0$ . \betaend{proof} We also can show the other direction. \betabegin{eqnarray*}gin{satz} Let $ {\betafrak g} $ be a real Lie algebra, $\betakappa$ be a complex representation of non-real type (of complex or quaternionic type), i.e. $\betarho=\betakappa_\betamathbb{R}$ is irreducible. Then holds: \betabegin{eqnarray*}gin{enumerate} \betaitem If $ \betakappa$ is unitary with respect to $\betatheta$ or orthogonal with respect to $ \betabegin{eqnarray*}ta$, then $\betarho$ is orthogonal with respect to $h$ and $ \betatheta^h= \betatheta$ or $ \betabegin{eqnarray*}ta^h= \betabegin{eqnarray*}ta$. \betaitem If $ \betakappa$ is anti-unitary with respect to $\betatheta$ or symplectic with respect to $ \betabegin{eqnarray*}ta$, then $\betarho$ is symplectic with respect to $h$ and $ \betatheta^h= \betatheta$ or $ \betabegin{eqnarray*}ta^h= \betabegin{eqnarray*}ta$. \betaend{enumerate} \betaend{satz} \betabegin{eqnarray*}gin{proof} We define a bilinear form on $E=V_\betamathbb{R}$ by \beta[h(x,y):= Re\beta \betatheta(x-iJx, y-iJy)\betamakebox[1cm][c]{ or }h(x,y):= Re\beta \betabegin{eqnarray*}ta(x- iJx, y-iJy).\beta] This form is invariant and --- since $Re\beta i z = -Im\beta z$ --- also non-degenerate. (The difference to real type is that here the arguments in $\betatheta/ \betabegin{eqnarray*}ta$ run over the whole complex vector space $V$.) $h$ is symmetric if $\betakappa$ is unitary or orthogonal and anti-symmetric if $ \betabegin{eqnarray*}ta$ is anti-symmetric or anti-unitary. The fact that the extensions are equal to $\betatheta$ resp. $ \betabegin{eqnarray*}ta$ follows from the formulas (\betaref{h-theta}) and (\betaref{h-omega}). \betaend{proof} Again we have the following correspondence: \betabegin{eqnarray*}gin{eqnarray}\betalanglebel{typ2equivalenz} \betaleft\beta{ \betarho\betamathbbox{ real, non-real type, orthogonal} \betaright\beta}_{/\betasigmaim}&\betaleftrightarrow& \betaleft\beta{\betabegin{eqnarray*}gin{array}{l} \betakappa \betamathbbox{ complex, non-real type,}\beta\beta \betamathbbox{unitary or orthogonal} \betaend{array} \betaright\beta}_{/\betaapprox}\beta\beta \betalanglebel{typ2equivalenz2} \betaleft\beta{ \betarho\betamathbbox{ real, non-real type, symplectic} \betaright\beta}_{/\betasigmaim}&\betaleftrightarrow& \betaleft\beta{\betabegin{eqnarray*}gin{array}{l} \betakappa \betamathbbox{ complex, non-real type,}\beta\beta \betamathbbox{symplectic or anti-unitary} \betaend{array} \betaright\beta}_{/\betaapprox} \betaend{eqnarray} The fact that a complex representation is unitary if and only if it is anti-unitary (the anti-hermitian form is $i\betatheta$) implies that a real, orthogonal representation of non-real type with non-degenerate $\betatheta^h$ on the corresponding complex representation is also symplectic. This corresponds to the equality of real matrix algebras: \betabegin{eqnarray*}gin{eqnarray*} \betamathfrak{u} (n)&=& \betamathfrak{so}(2n) \betacap \betamathfrak{sp}(2n)\beta\beta &=& \betaleft\beta{X\betain \betamathfrak{gl}(2n)\betaleft|X^t=-X\betaright.\betaright\beta} \betacap \betaleft\beta{\betaleft.\betaleft(\betabegin{eqnarray*}gin{array}{cc}A&B\beta\betaC&-A^t\betaend{array}\betaright) \betaright| A,B,C\betain \betamathfrak{gl}(n), B^t=B, C^t=C \betaright\beta}\beta\beta &=& \betaleft\beta{\betaleft.\betaleft(\betabegin{eqnarray*}gin{array}{cc}A&B\beta\beta-B&A\betaend{array}\betaright) \betaright| A,B\betain \betamathfrak{gl}(n), A^t=-A, B^t=B, \betaright\beta}. \betaend{eqnarray*} I.e. if a complex representation $\betakappa$ of non-real type is unitary, then $\betakappa_\betamathbb{R}$ is orthogonal and symplectic. Furthermore one proves the following \betabegin{eqnarray*}gin{lem} Let $h$ be symmetric, $\betabegin{eqnarray*}ta^h$, $\betatheta^h$ as above and $J$ the complex structure on $E$. Then holds \betabegin{eqnarray*}gin{enumerate} \betaitem $ \betabegin{eqnarray*}ta^h=0$ if and only if $h(x,y)= h(Jx,Jy)$ for all $x,y\betain E$. \betaitem $ \betatheta^h=0$ if and only if $h(x,y)= -h(Jx,Jy)$ for all $x,y\betain E$. \betaend{enumerate}\betalanglebel{jh} \betaend{lem} \betabegin{eqnarray*}gin{proof} If we write every element of $V=V_+$ in the form (\betaref{v+form}) we get the proposition due to formulas (\betaref{h-omega}) and (\betaref{h-theta}). \betaend{proof} We will now prove the main result for the case that $h$ is positive definite. \betabegin{eqnarray*}gin{satz}\betalanglebel{main1} Let $\betarho$ be irreducible of non-real type and orthogonal with respect to $h$ where $h$ is {\betabf positive definite}. Then the corresponding complex representation $\betakappa$ of non-real type is unitary, with respect to a positive definite hermitian form, which is the standard hermitian form for representations of compact Lie groups/Lie algebras. $\betakappa$ is not orthogonal, i.e. the linear extension $ \betabegin{eqnarray*}ta^h$ of $h$ vanishes on $V\betatimes V$. \betaend{satz} \betabegin{eqnarray*}gin{proof} We can prove this in two ways. If $\betatheta^h$ is degenerate, then it is zero and we have by lemma \betaref{jh} that $h(x,x)=-h(JxJx)$. But this is not possible if $h$ is positive definite. So $ \betatheta^h$ is non degenerate and by formula (\betaref{h-theta}) positive definite, since $h$ is positive definite. But the existence of a positive definite hermitian form entails by corollary \betaref{+unitary} for non-real type representations, i.e. of complex or quaternionic type, that the representation can not be orthogonal. So $\betabegin{eqnarray*}ta^h=0$. An easier way to argue is that representations of compact Lie algebras are unitary with respect to a standard positive definite hermitian form. This form is unique and thats why equal to $\betatheta^h$ and by corollary \betaref{+unitary} the representation can not be orthogonal. \betaend{proof} \betaend{appendix} \betabegin{itemize}bliography{GEOBIB,SPINBIB,HOLBIB,ALGBIB,thomas} \betaend{document}

\begin{document} \maketitle \begin{abstract} In this paper, we prove that the small energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper, we construct Tao's caloric gauge in the case when nontrivial harmonic map occurs. With the ``dynamic separation" the master equation of the heat tension field appears as a semilinear magnetic wave equation. By the endpoint and weighted Strichartz estimates for magnetic wave equations obtained by the first author \cite{Lize1}, the asymptotic stability follows by a bootstrap argument. \end{abstract} \maketitle \tableofcontents \section{Introduction} Let $(M,h)$ and $(N,g)$ be two Riemannian manifolds without boundary. A wave map is a map from the Lorentz manifold $\Bbb R \times M$ into $N$, $$u:\Bbb R\times M\to N, $$ which is locally a critical point for the functional \begin{align}\label{w1} F(u) = \int_{\Bbb R \times M} {\left(- {{{\left\langle {{\partial _t}u,{\partial _t}u} \right\rangle }_{{u^*}g}} + {h^{ij}}{{\left\langle {{\partial _{{x_i}}}u,{\partial _{{x_j}}}u} \right\rangle }_{{u^*}g}}} \right)} {\rm{dtdvo{l_h}}}. \end{align} Here ${h_{ij}}dx^idx^j$ is the metric tension under a local coordinate $(x^1,...,x^m)$ for $M$. In a coordinate free expression, the integrand in the functional $F(u)$ is the energy density of $u$ under the Lorentz metric of $\Bbb R\times M$, $$\mathbf{\eta}=-dt\otimes dt+h_{ij}dx^i\otimes dx^j.$$ Given a local coordinate $(y^1,...,y^n)$ for $N$, the Euler-Lagrange equation for (\ref{w1}) is given by \begin{align}\label{wmap1} \Box u^k +{\eta ^{\alpha \beta }}\overline{\Gamma}_{ij}^k(u){\partial _\alpha }{u^i}{\partial _\beta }{u^j}= 0, \end{align} where $\Box=-\partial_t^2+\Delta_M$ is the D'Alembertian on $\Bbb R\times M$, $\overline{\Gamma}^k_{ij}(u)$ are the Christoffel symbols at the point $u(t,x)\in N$. In this paper, we consider the case $M=\Bbb H^2$, $N=\Bbb H^2$. The wave map equation on a flat spacetime, which is sometimes known as the nonlinear $\sigma$-model, arises as a model problem in general relativity and particle physics, see for instance \cite{MS}. The wave map equation on curved spacetime is related to the wave map-Einstein system and the Kerr Ernst potential, see \cite{AGS,IK,GL}. We remark that the case where the background manifold is the hyperbolic space is of particular interest. Indeed, the anti-de Sitter space (AdSn), which is the exact solution of Einstein's field equation for an empty universe with a negative cosmological constant, is asymptotically hyperbolic. There exist plenty of works on the Cauchy problem, the long dynamics and blow up for wave maps on $\Bbb R^{1+m}$. We first recall the non-exhaustive lists of results on equivariant maps. The critical well-posedness theory was initially considered by Christodoulou, Tahvildar-Zadeh \cite{CT} for radial wave maps and Shatah, Tahvildar-Zadeh \cite{STZ2} for equivariant wave maps. The global well-posedness result of \cite{CT} was recently improved to scattering by Chiodaroli, Krieger, Luhrmann \cite{CKL}. The bubbling theorem of wave maps was proved by Struwe \cite{S}. The explicit construction of blow up solutions behaving as a perturbation of the rescaling harmonic map was achieved by Krieger, Schlag, Tataru \cite{KST}, Raphael, Rodnianski \cite{RR}, and Rodnianski, Sterbenz \cite{RS} for the $\Bbb S^2$ target in the equivariant class. And the ill-posedness theory was studied in D'Ancona, Georgiev \cite{DG} and Tao \cite{Tao10}. Without equivariant assumptions on the initial data the sharp subcritical well-posedness theory was developed by Klainerman, Machedon \cite{KM1,KM2} and Klainerman, Selberg \cite{KS2}. The small data critical case was started by Tataru \cite{Tataru2} in the critical Besov space, and then completed by Tao \cite{Tao1,Tao2} for wave maps from $\Bbb R^{1+d}$ to $\Bbb S^m$ in the critical Sobolev space. The small data theory in critical Sobolev space for general targets was considered by Krieger \cite{J1,J2}, Klainerman, Rodnianski \cite{KR3}, Shatah, Struwe \cite{SS}, Nahmod, Stefanov, Uhlenbeck \cite{NSU}, and Tataru \cite{Tataru3}. The dynamic behavior for wave maps on $\Bbb R^{1+2}$ with general data was obtained by Krieger, Schlag \cite{KS} for the $\Bbb H^2$ targets, Sterbenz, Tataru \cite{ST1,ST2} for compact Riemann manifolds and initial data below the threshold, and Tao \cite{Tao7} for the $\Bbb H^n$ targets. In fact, Sterbenz, Tataru \cite{ST1,ST2} proved that for any initial data with energy less than that of the minimal energy nontrivial harmonic map evolves to a global and scattering solution. The works on the wave map equations on curved spacetime were relatively less. The existence and orbital stability of equivariant time periodic wave maps from $\Bbb R\times \Bbb S^2$ to $\Bbb S^2$ were proved by Shatah, Tahvildar-Zadeh \cite{STZ1}, see Shahshahani \cite{S} for an generalization of $\Bbb S^2$. The critical small data Cauchy problem for wave maps on small asymptotically flat perturbations of $\Bbb R^4$ to compact Riemann manifolds was studied by Lawrie \cite{LA}. The soliton resolution and asymptotic stability of harmonic maps under wave maps on $\Bbb H^2$ to $\Bbb S^2$ or $\Bbb H^2$ in the 1-equivariant case were established by Lawrie, Oh, Shahshahani \cite{LOS1,LOS2,LOS4,LOS5}, see also \cite{LOS} for critical global well-posedness for wave maps from $\Bbb R\times \Bbb H^d$ to compact Riemann manifolds with $d\ge4$. In this paper, we study the asymptotic stability of harmonic maps to (\ref{w1}). The motivation is the so called soliton resolution conjecture in dispersive PDEs which claims that every global bounded solution splits into the superposition of divergent solitons with a radiation part plus an asymptotically vanishing remainder term as $t\to\infty$. The version for wave maps and hyperbolic Yang-Mills has been verified by Cote \cite{C} and Jia, Kenig \cite{JK} for equivariant maps along a time sequence, see also \cite{KLLS1,KLLS2} for exotic-ball wave maps and \cite{Gy} for wormholes. Recently Duyckaerts, Jia, Kenig, Merle \cite{DJKM} obtained the universal blow up profile for type II blow up solutions to wave maps $u:\Bbb R\times\Bbb R^2\to \Bbb S^2$ with initial data of energy slightly above the ground state. For wave maps from $\Bbb R\times \Bbb H^2$ to $\Bbb H^2$, Lawrie, Oh, Shahshahani \cite{LOS,LOS4} raised the following soliton resolution conjecture,\\ {\bf Conjecture 1.1} Consider the Cauchy problem for wave map $u:\Bbb R\times \Bbb H^2\to \Bbb H^2$ with finite energy initial data $(u_0,u_1)$. Suppose that outside some compact subset $\mathcal{K}$ of $\Bbb H^2$ for some harmonic map $Q:\Bbb H^2\to \Bbb H^2$ we have $$ u_0(x)=Q(x), \mbox{ }{\rm{for}}\mbox{ }x\in \Bbb H^2\backslash\mathcal{K}. $$ Then the unique solution $(u(t),\partial_tu(t))$ to the wave map scatters to $(Q(x),0)$ as $t\to\infty$. In this paper, we consider the easiest case of Conjecture 1.1, i.e., when the initial data is a small perturbation of harmonic maps with small energy. In order to state our main result, we introduce the notion of admissible harmonic maps. \begin{definition}\label{2as} Let $D=\{z:|z|<1\}$ with the hyperbolic metric be the Poincare disk. We say the harmonic map $Q:D\to D$ is admissible if $Q(D)$ is a compact subset of $D$ covered by a geodesic ball centered at 0 of radius $R_0$, $\|\nabla^kdQ\|_{L^2}<\infty$ for $k=0,1,2$, and there exists some $\varrho>0$ such that $e^{\varrho r}|dQ|\in L^{\infty}$, where $r$ is the distance between $x\in D$ and the origin point in $D$. \end{definition} For any given admissible harmonic map $Q$, we define the space $\bf{H}^k\times\bf{H}^{k-1}$ by (\ref{h897}). Our main theorem is as follows. \begin{theorem}\label{a1} Fix any $R_0>0$. Assume the given admissible harmonic map $Q$ in Definition \ref{2as} satisfies \begin{align}\label{as4} \|dQ\|_{L^2_x}<\mu_1, \mbox{ }\|e^{\varrho r}|dQ|\|_{L^{\infty}_x}<\mu_1, \mbox{ }\|\nabla^2dQ\|_{L^{\infty}_x}+\|\nabla dQ\|_{L^{\infty}_x}<\mu_1. \end{align} And assume that the initial data $(u_0,u_1)\in {\bf{H}^3\times\bf{H}^2}$ to (\ref{wmap1}) with $u_0:\Bbb H^2\to\Bbb H^2$, $u_1(x)\in T_{u_0(x)}N$ for each $x\in \Bbb H^2$ satisfy \begin{align}\label{as3} \|(u_0,u_1)-(Q,0)\|_{{\bf{H}}^2\times {\bf{H}^1}}<\mu_2. \end{align} Then if $\mu_1>0$ and $\mu_2>0$ are sufficiently small depending only on $R_0$, (\ref{wmap1}) has a global solution $(u(t),\partial_tu(t))$ which converges to the harmonic map $Q:\Bbb H^2\to \Bbb H^2$ as $t\to\infty$, i.e., $$ \mathop {\lim }\limits_{t\to\infty }\mathop {\sup }\limits_{x \in {\mathbb{H}^2}} {d_{{\mathbb{H}^2}}}\left( {u(t,x),Q(x)} \right) = 0. $$ \end{theorem} The initial data considered in this paper are perturbations of harmonic maps in the $\bf H^2$ norm. If one considers perturbations in the energy critical norm $H^1$, the $S_k$ v.s. $N_k$ norm constructed by Tataru \cite{Tataru2} and Tao \cite{Tao2} should be built for the hyperbolic background. \noindent{\bf{Remark 1.1}} Notice that the limit harmonic map coincides with the unperturbed harmonic map in Theorem 1.1. The reason for this coincidence is that the $\bf H^2\times\bf H^1$ norm assume the initial data coincide with $Q$ at the infinity, then the uniqueness of harmonic maps with prescribed boundary map shows the limit harmonic map is exactly the unperturbed one. \noindent{\bf{Remark 1.2}}{\bf(Examples for the admissible harmonic maps)}\\ \noindent Denote $D=\{z:|z|<1\}$ to be the Poincare disk. Then any holomorphic map $f:D \to D$ is a harmonic map. If we assume that $f(z)$ can be analytically extended into a larger disk than the unit disk, then $\mu_1f:D\to D$ satisfies all the conditions in Definition 1.1 and Theorem 1.1 if $0<\mu_1\ll1$. Hence the harmonic maps involved in Theorem 1.1 are relatively rich. See [Appendix,\cite{LZ}] for the proof of these facts. It is important to see in theses examples that the dependence of $\mu_1$ on $R_0$ is neglectable. \noindent{\bf{Remark 1.3}}(Examples for the perturbations of admissible harmonic maps) Since we have global coordinates for $\Bbb H^2$ given by (\ref{vg}), the perturbation in the sense of (\ref{as3}) is nothing but perturbations of $\Bbb R^2$-valued functions. Since we are dealing with non-equivariant data where the linearization method seems to be hard to apply, we use the caloric gauge technique introduced by Tao \cite{Tao4} to prove Theorem 1.1. The caloric gauge of Tao was applied to solve the global regularity of wave maps from $\Bbb R^{2+1}$ to $\Bbb H^n$ in the heat-wave project. We briefly recall the main idea of the caloric gauge. Given a solution to the wave map $u(t,x):\Bbb R^{1+2}\to \Bbb H^n$, suppose that $\widetilde u(s,t,x)$ solves the heat flow equation with initial data $u(t,x)$ $$ \left\{ \begin{array}{l} \partial_s \widetilde{u}(s,t,x)=\sum^2_{i=1}\nabla_i\partial_i\widetilde{u}\\ \widetilde{u}(s,t,x) \upharpoonright_{s=0}= u(t,x). \\ \end{array} \right. $$ Since there exists no nontrivial finite energy harmonic map from $\Bbb R^2$ to $\Bbb H^n$, one can expect that the corresponding heat flow $\widetilde{u}(s,t,x)$ converges to a fixed point $Q$ as $s\to\infty$. For any given orthonormal frame at the point $Q$, one can pullback the orthonormal frame parallel with respect to $s$ along the heat flow to obtain the frame at $\widetilde{u}(s,t,x)$, particularly $u(t,x)$ when $s=0$. Then rewriting (\ref{wmap1}) under the constructed frame will give us a scalar system for the differential fields and connection coefficients. Despite the fact that the caloric gauge can be viewed as a nonlinear Littlewood-Paley decomposition, the essential advantage of the caloric gauge is that it removes some troublesome frequency interactions, which is of fundamental importance for critical problems in low dimensions. Generally the caloric gauge was used in the case where no harmonic map occurs, for instance energy critical geometric wave equations with energy below the threshold. In our case nontrivial harmonic exists no mater how small the data one considers. However, as observed in our work \cite{LZ}, the caloric gauge is still extraordinarily powerful. In fact, denoting the solution of the heat flow with initial data $u(0,x)$ by $U(s,x)$, it is known that $U(s,x)$ converges to some harmonic map $Q(x)$ as $s\to\infty$. And one can expect that the solution $u(t,x)$ of (\ref{wmap1}) also converges to the same harmonic map $Q(x)$ as $t\to\infty$. This heuristic idea combined with the caloric gauge reduces the convergence of solutions to (\ref{wmap1}) to proving the decay of the heat tension filed. There are three main ingredients in our proof. The first is to guarantee that all the heat flows initiated from $u(t,x)$ for different $t$ converge to the same harmonic map. This enables us to construct the caloric gauge. The second is to derive the master equation for the heat tension field, which finally reduces to a linear wave equation with a small magnetic potential. The third is to design a suitable closed bootstrap program. All these ingredients are used to overcome the difficulty that no integrability with respect to $t$ is available for the energy density because the harmonic maps prevent the energy from decaying to zero as $t\to\infty$. The key for the first ingredient is using the decay of $\partial_tu$ along the heat flow. In order to construct the caloric gauge, one has to prove the heat flow initiated from $u(t,x)$ converges to the same harmonic map independent of $t$. If one only considers $t$ as a smooth parameter, i.e., in the homotopy class, the corresponding limit harmonic map yielded by the heat flow initiated from $u(t,x)$ can be different when $t$ varies. Indeed, there exist a family of harmonic maps $\{Q_{\lambda}\}$ which depend smoothly with respect to $\lambda\in(0,1)$. Therefore the heat flow with initial data $Q_{\lambda}$ remains to be $Q_{\lambda}$, which changes according to the variation of $\lambda$. This tells us the structure of (\ref{wmap1}) should be considered. The essential observation is $\partial_t u$ decays fast along the heat flow as $s\to\infty$. By a monotonous property observed initially by Hartman \cite{Hh} and the decay estimates of the heat semigroup, we can prove the distance between the heat flows initiated from $u(t_1)$ and $u(t_2)$ goes to zero as $s\to\infty$. Therefore the limit harmonic map for the heat flow generated from $u(x,t)$ are all the same for different $t$. Similar idea works for the Landau-Lifshitz flow, see our paper \cite{LZ}. And we remark that this part can be adapted to energy critical wave maps form $\Bbb R\times\Bbb H^2$ to $\Bbb H^2$ since essentially we only use the $L^2_x$ norm of $\partial_tu$ in the arguments which is bounded by the energy. Different from the usual papers on the asymptotic stability, we will not use the linearization arguments involving spectrum analysis of the linearized operator and modulation equations. But the master equation appears naturally as a semilinear wave equation with a small magnetic potential. Indeed, the main equation we need to consider is the nonlinear wave equation for the heat tension filed. The point is that although the nonlinear part of this equation is not controllable, one can separate part of them to be a magnetic potential with a remainder likely to be controllable. This is why we need the Strichartz estimates for magnetic wave equations. The second ingredient is to control the remained terms in the nonlinear part of the master equation after we separate the magnetic potential away. In fact, the terms involving one order derivatives of the heat tension filed can not be controlled only by Strichartz estimates, even if we are working in the subcritical regularity. In this paper, the one order derivative terms are controlled by the weighted Strichartz estimates and the exotic Strichartz estimates owned only by hyperbolic backgrounds compared with the flat case. These estimates were obtained in the first author's work \cite{Lize1}. The third ingredient is to close the bootstrap, by which the global spacetime norm bounds of the heat tension field follows. The caloric gauge yields the gauged equation for the corresponding differential fields $\phi_{x,t}$, connection coefficients $A_{x,t}$ and the heat tension filed. It has been discovered in Tao \cite{Tao7} that the key field one needs to study is the heat tension field which satisfies a semilinear wave equation. And for the small data Cauchy problem of wave maps on $\Bbb R\times\Bbb H^4$, Lawrie, Oh, Shahshahani \cite{LOS} shows in order to close the bootstrap arguments it suffices to firstly proving a global spacetime bound for the heat tension filed $\phi_s$. In our case, since the energy will not decay, one has to get rid of the inhomogeneous terms which involve only the differential fields $\phi_x$ in the master equation. Furthermore, these troublesome terms involving only $\phi_x$ are much more serious in the study of the equation of wave map tension filed. This difficulty is overcome by using identities from intrinsic geometry to gain some cancelation and adding a space-time bound for $|\partial_tu|$ on the basis of the bootstrap arguments of \cite{LOS,Tao7}. This paper is organized as follows. In Section 2, we recall some notations and notions and prove an equivalence between the intrinsic and extrinsic Sobolev norms in some sense. In Section 3, we construct the caloric gauge and obtain the estimates of the connection coefficients. In Section 4, we we derive the master equation. In Section 5, we first recall the non-endpoint and endpoint Strichartz estimates, Morawetz inequality, and weighted Strichartz estimates for the linear magnetic wave equation. Then we close the bootstrap and deduce the global spacetime bounds for the heat tension field. In Section 6, we finish the proof of Theorem 1.1. In Section 7, we prove some remaining claims in the previous sections. We denote the constants by $C(M)$ and they can change from line to line. Small constants are usually denoted by $\delta$ and it may vary in different lemmas. $A\lesssim B$ means there exists some constant $C$ such that $A\le CB$. \section{Preliminaries} Some standard preliminaries on the geometric notions of the hyperbolic spaces, Sobolev embedding inequalities and an equivalence relationship for the intrinsic and extrinsic formulations of the Sobolev spaces are recalled first. As a corollary we prove the local well-posedness for initial data $(u_0,u_1)$ in the $\bf{H}^3\times \bf{H}^2$ regularity and a conditional global well-posedness proposition. In addition, the smoothing effect of heat semigroup is recalled. \subsection{The global coordinates and definitions of the function spaces} The covariant derivative in $TN$ is denoted by $\widetilde{\nabla}$, the covariant derivative induced by $u$ in $u^*(TN)$ is denoted by $\nabla$. We denote the Riemann curvature tension of $N$ by $\mathbf{R}$. The components of Riemann metric are denoted by $h_{ij}$ for M and $g_{ij}$ for N respectively. The Christoffel symbols on $M$ and $N$ are denoted by $\Gamma^{k}_{ij}$ and $\overline{\Gamma}^{k}_{ij}$ respectively. We recall some facts on hyperbolic spaces. Let $\Bbb R^{1+2}$ be the Minkowski space with Minkowski metric $-(dx^0)^2+(dx^1)^2+(dx^2)^2$. Define a bilinear form on $\Bbb R^{1+2}\times \Bbb R^{1+2}$, $$ [x,y]=x^0y^0-x^1y^1-x^2y^2. $$ The hyperbolic space $\mathbb{H}^2$ is defined by $$\mathbb{H}^2=\{x\in \Bbb R^{2+1}: [x,x]=1 \mbox{ }{\rm{and}}\mbox{ }x^0>0\},$$ with a Riemannian metric being the pullback of the Minkowski metric by the inclusion map $\iota:\mathbb{H}^2\to \Bbb R^{1+2}.$ By Iwasawa decomposition we have a global system of coordinates. Indeed, the diffeomorphism $\Psi:\Bbb R\times \Bbb R\to \mathbb{H}^2$ is given by \begin{align}\label{vg} \Psi(x_1,x_2)=({\rm{cosh}} x_2+e^{-x_2}|x_1|^2/2, {\rm{sinh}} x_2+e^{-x_2}|x_1|^2/2, e^{-x_2}x_1). \end{align} The Riemannian metric with respect to this coordinate system is given by $$ e^{-2x_2}(dx_1)^2+(dx_2)^2. $$ The corresponding Christoffel symbols are \begin{align}\label{christ} \Gamma^1_{2,2}=\Gamma^2_{2,1}=\Gamma^2_{2,2}=\Gamma^1_{1,1}=0; \mbox{ }\Gamma^1_{2,1}=-1, \mbox{ }\Gamma^2_{1,1}=e^{-2x_2}. \end{align} For any $(t,x)$ and $u:[0,T]\times\mathbb{H}^2\to \Bbb H^2$, we define an orthonormal frame at $u(t,x)$ by \begin{align}\label{frame} \Theta_1(u(t,x))=e^{u^2(t,x)}\frac{\partial}{\partial y_1}; \mbox{ }\Theta_2(u(t,x))=\frac{\partial}{\partial y_2}. \end{align} where $(u^1,u^2)$ denotes the coordinate of $u$ given by (\ref{vg}). {\bf Throughout this paper we will use coordinates (\ref{vg}) for both the target manifold $N=\Bbb H^2$ and the starting manifold $M=\Bbb H^2$.} Recall also the identity for Riemannian curvature on $N=\Bbb H^2$ \begin{align}\label{2.best} {\bf R}(X,Y)Z={\widetilde{\nabla} _X}{\widetilde{\nabla} _Y}Z - {\widetilde{\nabla }_Y}{\widetilde{\nabla} _X}Z - {\widetilde{\nabla}_{[X,Y]}}Z = \left\langle {X,Z} \right\rangle Y - \left\langle {Y,Z} \right\rangle X. \end{align} We have a useful identity for $X,Y,Z\in u^*(TN)$ \begin{align}\label{2.4best} {\nabla _i }\left( {{\bf R}\left( {X,Y} \right)Z} \right) = {\bf R}\left( {X,{\nabla _i }Y} \right)Z + {\bf R}\left( {{\nabla _i }X,Y} \right)Z + {\bf R}\left( {X,Y} \right){\nabla _i }Z. \end{align} For simplicity, denote $(X\wedge Y)Z=\left\langle {X,Z} \right\rangle Y - \left\langle {Y,Z} \right\rangle X$. Let $H^k(\mathbb{H}^2;\Bbb R)$ be the usual Sobolev space for scalar functions defined on manifolds. We also recall the norm of $H^k$: $$ \|f\|^2_{H^k}=\sum^k_{l=1}\|\nabla^l f\|^2_{L^2_x}, $$ where $\nabla^l f$ is the covariant derivative. For maps $u:\mathbb{H}^2\to \mathbb{H}^2$, we define the intrinsic Sobolev semi-norm $\mathfrak{H}^k$ by $$ \|u\|^2_{{\mathfrak{H}}^k}=\sum^k_{i=1}\int_{\mathbb{H}^2} |\nabla^{i-1} du|^2 {\rm{dvol_h}}. $$ The map $u:\mathbb{H}^2\to\mathbb{H}^2$ is associated with a vector-valued function $u:\mathbb{H}^2\to \Bbb R^2$ by (\ref{vg}). Indeed, the vector $(u^1(x),u^2(x))$ is defined by $\Psi(u^1(x),u^2(x))=u(x)$ for any $x\in \mathbb{H}^2$ . Let $Q:\Bbb H^2\to\Bbb H^2$ be an admissible harmonic map in Definition 1.1. Then the extrinsic Sobolev space is defined by \begin{align}\label{h897} {\bf{H}^k_{Q}}=\{u: u^1-Q^1(x), u^2-Q^2(x)\in H^k(\mathbb{H}^2;\Bbb R)\}, \end{align} where $(Q^1(x),Q^2(x))\in \Bbb R^2$ is the corresponding components of $Q(x)$ under the coordinate (\ref{vg}). Denote the set of smooth maps which coincide with $Q$ outside of some compact subset of $M=\Bbb H^2$ by $\mathcal{D}$. Let $\mathcal{H}^k_{Q}$ be the completion of $\mathcal{D}$ under the metric given by \begin{align}\label{h897} {\rm{dist}}_{k,Q}(u,w)=\sum^2_{j=1}\|u^j-w^j\|_{H^k(\mathbb{H}^2;\Bbb R)}, \end{align} where $u,w\in \mathcal{H}^k_{Q}$. Since $C^{\infty}_c(\mathbb{H}^2;\Bbb R)$ is dense in $H^k(\mathbb{H}^2;\Bbb R)$ (see Hebey \cite{Hebey}), $\mathcal{H}^k_{Q}$ coincides with ${\bf{H}^k_{Q}}$. And for simplicity, we write $\bf{H}^k$ without confusions. If $u$ is a map from $\Bbb R\times\Bbb H^2$ to $\Bbb H^2$, we define the space $\bf{H}^k\times\bf{H}^{k-1}$ by \begin{align}\label{h897} {\bf{H}^k\times\bf{H}^{k-1}}=\left\{u:\sum^2_{j=1}\|u^j-Q^j\|_{H^k(\mathbb{H}^2;\Bbb R)}+\|\partial_tu^j\|_{H^{k-1}(\mathbb{H}^2;\Bbb R)}<\infty\right\}. \end{align} The distance in ${\bf{H}^k\times\bf{H}^{k-1}}$ is given by \begin{align}\label{zvh897} {\rm{dist}}_{\bf{H}^k\times\bf{H}^{k-1}}(u,w)=\sum^2_{j=1}\|u^j-w^j(x)\|_{H^k}+\|\partial_tu^j-\partial_t w^j\|_{H^{k-1}}. \end{align} \subsection{Sobolev embedding and Equivalence lemma} The Fourier transform on hyperbolic spaces takes proper functions defined on $\mathbb{H}^2$ to functions defined on $\Bbb R\times \Bbb S^1$, see Helgason \cite{Hel} for details. The operator $(-\Delta)^{\frac{s}{2}}$ is defined by the Fourier multiplier $\lambda\to (\frac{1}{4}+\lambda^2)^{\frac{s}{2}}$. We now recall the Sobolev inequalities of functions in $H^k$. \begin{lemma}\label{wusijue} If $f\in C^{\infty}_c(\mathbb{H}^2;\Bbb R)$, then for $1<p<\infty,$ $p\le q\le \infty$, $0<\theta<1$, $1<r<2$, $r\le l<\infty$, $\alpha>1$, the following inequalities hold \begin{align} {\left\| f \right\|_{{L^2}}} &\lesssim {\left\| {\nabla f} \right\|_{{L^2}}} \label{{uv111}}\\ {\left\| f \right\|_{{L^q}}} &\lesssim \left\| {\nabla f} \right\|_{{L^2}}^\theta \left\| f \right\|_{{L^p}}^{1 - \theta }\mbox{ }{\rm{when}}\mbox{ }\frac{1}{p} - \frac{\theta }{2} = \frac{1}{q} \label{uv211}\\ {\left\| f \right\|_{{L^l}}} &\lesssim {\left\| {\nabla f} \right\|_{{L^r}}}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }{\rm{when}}\mbox{ }\frac{1}{r} - \frac{1}{2} = \frac{1}{l} \label{uv311}\\ {\left\| f \right\|_{{L^\infty }}} &\lesssim {\left\| {{{\left( { - \Delta } \right)}^{\frac{\alpha }{2}}}f} \right\|_{{L^2}}}\mbox{ }\mbox{ }\mbox{ }{\rm{when}}\mbox{ }\alpha>1 \label{uv4}\\ {\left\| {\nabla f} \right\|_{{L^p}}} &\sim{\left\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}f} \right\|_{{L^p}}} \label{uv5}. \end{align} \end{lemma} For the proof, we refer to Bray \cite{B} for (\ref{uv211}), Ionescu, Pausader, Staffilani \cite{IPS} for (\ref{uv311}), Hebey \cite{Hebey} for (\ref{uv4}), see also Lawrie, Oh, Shahshahani \cite{LOS}. (\ref{uv5}) is obtained in \cite{Stri}. We also recall the diamagnetic inequality which sometimes refers to Kato's inequality (see \cite{LOS}) and a more generalized Sobolev inequality (see [Proposition 2.2,\cite{AP}]). \begin{lemma} $(a)$ If $T$ is a tension field defined on $\Bbb H^2$, then in the distribution sense, one has the diamagnetic inequality \begin{align}\label{wusijue3} |\nabla|T||\le |\nabla T|. \end{align} $(b)$ Let $1<p,q<\infty$ and $\sigma_1,\sigma_2\in\Bbb R$ such that $\sigma_1-\sigma_2\ge n/p-n/q\ge0$. Then for all $f\in C^{\infty}_c(\Bbb H^n;\Bbb R)$ \begin{align*} \|(-\Delta)^{\sigma_2}f\|_{L^q}\lesssim \|(-\Delta)^{\sigma_1}f\|_{L^p}. \end{align*} \end{lemma} \begin{remark} Lemma \ref{wusijue} and (\ref{wusijue3}) have several useful corollaries, for instance for $f\in H^2$ \begin{align} \|f\|_{L_x^{\infty}}&\lesssim \|\nabla^2f\|_{L^2_x}\label{{uv111}6}\\ \|f\|_{L_x^{2}}&\lesssim \|\nabla^2f\|_{L^2_x}. \end{align} \end{remark} The intrinsic and extrinsic formulations are equivalent in the following sense, see [Section 2, \cite{LZ}]. \begin{lemma}\label{new} Suppose that $Q$ is an admissible harmonic map in Definition 1.1. If $u\in \bf{H}_{Q}^k$ then for $k=2,3$ \begin{align} \|u\|_{{\bf H}_{Q}^k}\thicksim\|u\|_{\mathfrak{H}^k}, \end{align} in the sense that there exist continuous functions $\mathcal{P},\mathcal{Q}$ such that \begin{align} \|u\|_{{\bf H}^k_Q}&\le \mathcal{P}(\|u\|_{\mathfrak{H}^k})C(R_0,\|u\|_{\mathfrak{H}^2})\label{jia8}\\ \|u\|_{\mathfrak{H}^k}&\le \mathcal{Q}(\|u\|_{\bf{H}^k_Q})C(R_0,\|u\|_{{\bf{H}}_{Q}^2})\label{jia9}. \end{align} \end{lemma} Lemma \ref{new} and its proof imply the following corollary, by which we can view Theorem 1.1 as a small data problem in the intrinsic sense. The proof of Corollary \ref{new2} is presented in Section 7. \begin{corollary}\label{new2} If $(u_0,u_1)$ belongs to ${\bf{H}^3}\times {\bf{H}^2}$ satisfying (\ref{as3}) then for $0<\mu_1\le 1,0<\mu_2\le 1$ \begin{align} \|\nabla du_0\|_{L^2}+\|\nabla u_1\|_{L^2}+\|du\|_{L^2}+\|u_1\|_{L^2}\le C(R_0)\mu_2+C(R_0)\mu_1. \end{align} \end{corollary} \begin{lemma}\label{8.5} We have the decay estimates for heat equations on $\Bbb H^2$: \begin{align} \|e^{s\Delta_{\Bbb H^2}}f\|_{L^{\infty}_x}&\lesssim e^{-\frac{s}{4}}s^{-1}\|f\|_{L^{1}_x}\label{huhu899}\\ \|e^{s\Delta_{\Bbb H^2}}f\|_{L^{2}_x}&\lesssim e^{-\frac{s}{4}}\|f\|_{L^{2}_x}\label{m8}\\ \|e^{s\Delta_{\Bbb H^2}}f\|_{L^p_x}&\lesssim s^{\frac{1}{p}-\frac{1}{r}}\|f\|_{L^{r}_x},\label{huhu89}\\ \|e^{s\Delta_{\Bbb H^2}}(-\Delta_{\Bbb H^2})^{\alpha} f\|_{L^{q}_x}&\lesssim s^{-\alpha}e^{-\delta s}\|f\|_{L^{q}_x},\label{mm8} \end{align} where $1\le r\le p\le\infty$, $\alpha\in[0,1]$, $1<q<\infty$, $0<\delta\ll1$. \end{lemma} \begin{proof} (\ref{huhu899}) and (\ref{huhu89}) are known in the literature, see \cite{LZ,Coding}. (\ref{m8}) is a corollary of the spectral gap of $\frac{1}{4}$ for $-\Delta_{\Bbb H^2}$. The $s^{-\alpha}$ part of (\ref{mm8}) follows by interpolation between the three estimates of [Lemma 2.11,\cite{LOS}]. Thus it suffices to prove (\ref{mm8}) for $s$ large. The case of (\ref{mm8}) when $\alpha=0$ follows by directly estimating the heat kernel given in \cite{BM}. Since one has $e^{s\Delta}(-\Delta)^{\alpha}f=e^{\frac{s}{2}\Delta}e^{\frac{s}{2}\Delta}(-\Delta)^{\alpha}f$, by applying the exponential decay $L^p-L^p$ estimate to the first $e^{\frac{s}{2}\Delta}$ and the $s^{-\alpha}$ decay of $L^p\to (-\Delta)^{\alpha}L^p$ for the second $e^{\frac{s}{2}\Delta}$ proved just now, we obtain the full (\ref{mm8}). \end{proof} The $\Bbb R^2$ version of the following lemma was proved in [Lemma 2.5,\cite{Tao4}]. We remark that the same arguments work in the $\Bbb H^2$ case, because the proof in \cite{Tao4} only uses the decay estimate (\ref{huhu89}) and the self-ajointness of $e^{t\Delta_{\Bbb R^2}}$, which are also satisfied by $e^{t\Delta_{\Bbb H^2}}$. \begin{lemma}\label{ktao1} For $f\in L^2_x$ defined on $\Bbb H^2$, one has $$\int^{\infty}_0\|e^{s\Delta_{\Bbb H^2}}f\|^2_{L^{\infty}_x}ds\lesssim \|f\|^2_{L^2_x}. $$ \end{lemma} Without confusion, we will always use $\Delta$ instead of $\Delta_{\Bbb H^2}$. \subsection{The Local and conditional global well-posedness} We quickly sketch the local well-posedness and conditional global well-posedness for (\ref{wmap1}). The local well-posedness of (\ref{wmap1}) for $(u_0,u_1)\in{\bf{H}^3\times\bf{H}^2}$ is standard by fixed point argument. Thus we present the following lemma with a rough proof. \begin{lemma}\label{local} For any initial data $(u_0,u_1)\in \bf{H}^3\times\bf{H}^2$, there exists $T>0$ depending only on $\|(u_0,u_1)\|_{\bf{H}^3\times\bf{H}^2}$ such that (\ref{wmap1}) has a unique local solution $(u,\partial_tu)\in C([0,T];\bf{H}^3\times\bf{H}^2)$. \end{lemma} \begin{proof} In the coordinates (\ref{vg}), (\ref{wmap1}) can be written as the following semilinear wave equation \begin{align}\label{XV12} \frac{{{\partial ^2}{u^k}}}{{\partial {t^2}}} - {\Delta}{u^k} + {\overline\Gamma}_{ij}^k\frac{{\partial {u^i}}}{{\partial t}}\frac{{\partial {u^j}}}{{\partial t}} - {h^{ij}}{\overline\Gamma} _{mn}^k\frac{{\partial {u^m}}}{{\partial {x^i}}}\frac{{\partial {u^n}}}{{\partial {x^j}}} = 0. \end{align} Notice that $\bf{H}^3$ and $\bf{H}^2$ are embedded to $L^{\infty}$ as illustrated in Remark 9.1, we can prove the local well-posedness of (\ref{XV12}) by the standard contradiction mapping argument in the complete metric space $\bf{H}^3\times\bf{H}^2$ with the metric given by \begin{align*} {\rm{dist}}(u,w)=\sum^2_{j=1}\|u^j-w^j\|_{H^3}+\sum^2_{j=1}\|\partial_tu^j-\partial_tw^j\|_{H^2}. \end{align*} Moreover we can obtain the blow-up criterion: $T_*>0$ is the lifespan of $(\ref{XV12})$ if and only if \begin{align}\label{09ijn} \mathop {\lim }\limits_{t \to T_*} {\left\| {(u(t,x),\partial_tu(t,x))} \right\|_{{\bf{H}^3\times\bf{H}^2}}} = \infty. \end{align} \end{proof} The conditional global well-posedness is given by the following proposition. We remark that in the flat case $M=\Bbb R^d$, $1\le d\le 3$, Theorem 7.1 of Shatah, Struwe \cite{SStruwe} gave a local theory for Cauchy problem in $H^2\times H^1$. \begin{proposition}\label{global} Let $(u_0,u_1)\in\bf{H}^3\times \bf{H}^2$ be the initial data of (\ref{wmap1}), $T_*$ is the maximal lifespan determined by Lemma \ref{local}. If the solution $(u,\partial_tu)$ satisfies uniformly for all $t\in[0,T_*)$ \begin{align}\label{hcxp} \|\nabla du\|_{L^2_x}+\|du\|_{L^2_x}+\|\nabla\partial_t u\|_{L^2_x}+\|\partial_t u\|_{L^2_x}\le C_1, \end{align} for some $C_1>0$ independent of $t\in[0,T_*)$ then $T_*=\infty$. \end{proposition} \begin{proof} By the local well-posedness in Lemma \ref{local}, it suffices to obtain a uniform bound for $\|(u,\partial_t u)\|_{\bf{H}^3\times\bf{H}^2 }$ with respect to $t\in[0,T]$. By Lemma \ref{new}, it suffices to prove the intrinsic norms are uniformly bounded up to order three. We first point out a useful inequality which can be verified by integration by parts \begin{align}\label{V6} \|\nabla^2 du\|^2_{L^2_x}\lesssim \|\nabla \tau(u)\|^2_{L^2_x}+ \|du\|^6_{L^6_x}+\|\nabla d u\|^2_{L^4_x}\|d u\|^2_{L^4_x}+C(\|u \|^2_{\mathfrak{H}^2}), \end{align} where $\tau(u)$ denotes the tension field which in the local coordinates is written as \begin{align*} \tau(u)=\left(\Delta u^k+h^{pq}\overline{\Gamma}^k_{ij}\frac{\partial u^{i}}{\partial x^p}\frac{\partial u^{j}}{\partial x^q}\right)\frac{\partial}{\partial y^k}. \end{align*} Thus (\ref{V6}), Gagliardo-Nirenberg inequality and Young inequality further yield \begin{align}\label{equil} \|\nabla^2 du\|^2_{L^2_x}\lesssim \mathcal{P}(\|u \|^2_{\mathfrak{H}^2})+\|\nabla \tau(u)\|^2_{L^2_x}, \end{align} where $\mathcal{P}(x)$ is some polynomial. Define \begin{align*} {E_3}(u,{\partial _t}u) = \frac{1}{2}\int_{{\Bbb H^2}} {{{\left| {\nabla \tau (u)} \right|}^2}} {\rm{dvo}}{{\rm{l}}_{\rm{h}}} + \frac{1}{2}{\int_{{\Bbb H^2}} {\left| {{\nabla ^2}{\partial _t}u} \right|} ^2}{\rm{dvo}}{{\rm{l}}_{\rm{h}}}. \end{align*} Then integration by parts yields \begin{align*} \frac{d}{{dt}}{E_3}(u,{\partial _t}u) &= \int_{{\Bbb H^2}} {{h^{ii}}\left\langle {{\nabla _t}{\nabla _i}\tau (u),{\nabla _i}\tau (u)} \right\rangle } {\rm{dvo}}{{\rm{l}}_{\rm{h}}}\\ &+ \int_{{\Bbb H^2}} {{h^{ii}h^{jj}}\left\langle {{\nabla _t}{\nabla _i}{\nabla _j}{\partial _t}u - \Gamma _{ij}^k{\nabla _k}{\partial _t}u,{\nabla _i}{\nabla _j}{\partial _t}u}-\Gamma_{ij}^k{\nabla _k}{\partial _t}u \right\rangle {\rm{dvo}}{{\rm{l}}_{\rm{h}}}}. \end{align*} Furthermore we have \begin{align*} &\int_{{\Bbb H^2}} {{h^{ii}h^{jj}}\left\langle {{\nabla _t}{\nabla _i}{\nabla _j}{\partial _t}u - \Gamma _{ij}^k{\nabla _k}{\partial _t}u,{\nabla _i}{\nabla _j}{\partial _t}u}-\Gamma_{ij}^k{\nabla _k}{\partial _t}u \right\rangle {\rm{dvo}}{{\rm{l}}_{\rm{h}}}} \\ &= \int_{{\Bbb H^2}} {{h^{ii}h^{jj}}\left\langle {{\nabla _i}{\nabla _j}{\nabla _t}{\partial _t}u - \Gamma _{ij}^k{\nabla _k}{\partial _t}u,{\nabla _i}{\nabla _j}{\partial _t}u}-\Gamma_{ij}^k{\nabla _k}{\partial _t}u \right\rangle {\rm{dvo}}{{\rm{l}}_{\rm{h}}}} \\ &+ \int_{\Bbb H^2}O\big(\left| {\nabla {\partial _t}u} \right|\left| {{\nabla ^2}{\partial _t}u} \right|\big){\rm{dvol_h}} + O\big(\int_{\Bbb H^2}\left| {du} \right|\left| {{\partial _t}u} \right|\left| {\nabla {\partial _t}u} \right|\left| {{\nabla ^2}{\partial _t}u} \right|{\rm{dvol_h}} \big)\\ &+ \int_{\Bbb H^2}O\big(\left| {\nabla {\partial _t}u} \right|\left| {{\nabla ^2}{\partial _t}u} \right|\big){\rm{dvol_h}}+ \int_{\Bbb H^2}O\big({\left| {du} \right|^2}{\left| {{\partial _t}u} \right|^2}\left| {{\nabla ^2}{\partial _t}u} \right|\big){\rm{dvol_h}}\\ &+ \int_{\Bbb H^2}O\big({\left| {{\partial _t}u} \right|^2}\left| {\nabla du} \right|\left| {{\nabla ^2}{\partial _t}u} \right|\big){\rm{dvol_h}} \end{align*} Since $u$ solves (\ref{wmap1}), $\nabla_t\partial_tu=\tau(u)$. Then by integration by parts the leading term can be expanded as \begin{align*} &\int_{{\Bbb H^2}} {{h^{ii}h^{jj}}}\left\langle {{\nabla _i}{\nabla _j}{\nabla _t}{\partial _t}u,{\nabla _i}{\nabla _j}{\partial _t}u}-\Gamma_{ij}^k{\nabla _k}{\partial _t}u \right\rangle {\rm{dvol_h}}\\ &= \int_{{\Bbb H^2}}{h^{ii}h^{jj}}\left\langle {{\nabla _i}{\nabla _j}\tau (u),{\nabla _i}{\nabla _j}{\partial _t}u}-\Gamma_{ij}^k{\nabla _k}{\partial _t}u \right\rangle {\rm{dvol_h}} \\ &=-\int_{{\Bbb H^2}} {{h^{ii}}\left\langle {{\nabla _i}\tau (u),{\nabla _t}{\nabla _i}\tau (u)} \right\rangle {\rm{dvo}}{{\rm{l}}_{\rm{h}}}} + \int_{\Bbb H^2}O\big(\left| {\nabla \tau (u)} \right|\left| {du} \right|\left| {{\partial _t}u} \right|\left| {\tau (u)} \right|\big){\rm{dvol_h}}\\ &+ \int_{\Bbb H^2}O\big(\left| {\nabla \tau (u)} \right|\left| {{\nabla ^2}u} \right|\left| {du} \right|\left| {{\partial _t}u} \right|\big){\rm{dvol_h}}+ \int_{\Bbb H^2}O\big(\left| {\nabla \tau (u)} \right|\left| {{\partial _t}u} \right|{\left| {du} \right|^2}\big){\rm{dvol_h}} \\ &+ \int_{\Bbb H^2}O\big(\left| {\nabla \tau (u)} \right|\left| {\nabla {\partial _t}u} \right|{\left| {du} \right|^2}\big){\rm{dvol_h}} +\int_{\Bbb H^2}O\big( \left| {\nabla \tau (u)} \right|\left| {{\partial _t}u} \right|{\left| {du} \right|^3}\big){\rm{dvol_h}}\\ &+ \int_{\Bbb H^2}O\big(\left| {\nabla {\partial _t}u} \right|\left| {\nabla \tau (u)} \right|\big){\rm{dvol_h}}. \end{align*} Thus we conclude \begin{align*} &\frac{d}{{dt}}{E_3}(u,{\partial _t}u)\\ &\le {\left\| {du} \right\|_{L_x^8}}{\left\| {{\partial _t}u} \right\|_{L_x^4}}{\left\| {\nabla {\partial _t}u} \right\|_{L_x^8}}{\left\| {{\nabla ^2}{\partial _t}u} \right\|_{L_x^2}} + {\left\| {\nabla {\partial _t}u} \right\|_{L_x^2}}{\left\| {{\nabla ^2}{\partial _t}u} \right\|_{L_x^2}} \\ &+ {\left\| {{\nabla ^2}{\partial _t}u} \right\|_{L_x^2}}\left\| {du} \right\|_{L_x^8}^2\left\| {{\partial _t}u} \right\|_{L_x^8}^2 + {\left\| {\nabla du} \right\|_{L_x^6}}\left\| {{\partial _t}u} \right\|_{L_x^6}^2{\left\| {\nabla^2 {\partial _t}u} \right\|_{L_x^2}} \\ &+ {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}{\left\| {\nabla du} \right\|_{L_x^6}}{\left\| {du} \right\|_{L_x^6}}{\left\| {{\partial _t}u} \right\|_{L_x^6}} + {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}{\left\| {\nabla du} \right\|_{L_x^4}}\left\| {du} \right\|_{L_x^8}^2 \\ &+ {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}\left\| {du} \right\|_{L_x^{12}}^3{\left\| {{\partial _t}u} \right\|_{L_x^4}} + {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}{\left\| {{\partial _t}u} \right\|_{L_x^6}}{\left\| {\tau (u)} \right\|_{L_x^6}}{\left\| {du} \right\|_{L_x^6}} \\ &+ {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}{\left\| {{\partial _t}u} \right\|_{L_x^6}}\left\| {du} \right\|_{L_x^8}^2 + {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}{\left\| {\nabla {\partial _t}u} \right\|_{L_x^2}} + {\left\| {\nabla \tau (u)} \right\|_{L_x^2}}{\left\| {\nabla^2{\partial _t}u} \right\|_{L_x^2}}. \end{align*} Hence Young's inequality, Sobolev embedding and (\ref{V6}), (\ref{equil}) give \begin{align*} \frac{d}{{dt}}{E_3}(u,{\partial _t}u) \le C{E_3}(u,{\partial _t}u)+C. \end{align*} where $C$ depends only on $C_1$ in (\ref{hcxp}). Thus Gronwall shows \begin{align*} {E_3}(u,{\partial _t}u) \le e^{Ct}({E_3}({u_0},{u_1}) + C) . \end{align*} If $T_*<\infty$ this contradicts with (\ref{09ijn}). \end{proof} \subsection{Geometric identities related to Gauges} Let $\{e_1(t,x),e_2(t,x)\}$ be an orthonormal frame for $u^*(T\mathbb{H}^2)$. Let $\phi_\alpha=(\psi^1_\alpha,\psi^2_\alpha)$ for $\alpha=0,1,2$ be the components of $\partial_{t,x}u$ in the frame $\{e_1,e_2\}$, i.e., $$ \phi_\alpha^j = \left\langle {{\partial _\alpha}u,{e_j}} \right\rangle. $$ For given $\Bbb R^2$-valued function $\phi$ defined on $[0,T]\times\Bbb H^2$, associate $\phi$ with a tangent filed $e\phi$ on $u^*(TN)$ by \begin{align}\label{poill} \phi\leftrightarrow e\phi=\sum^2_{j=1}\phi^je_j, \end{align} The map $u$ induces a covariant derivative on the trivial boundle $([0,T]\times\Bbb H^2,\Bbb R^2)$ defined by $$D_\alpha\phi=\partial_\alpha \phi+[A_\alpha]\phi, $$ where the coefficient matrix is defined by \begin{align*} [{A_\alpha}]^k_j = \left\langle {{\nabla_\alpha}{e_j},{e_k}} \right\rangle. \end{align*} It is easy to check the torsion free identity \begin{align}\label{pknb} D_\alpha\phi_\beta=D_\beta\phi_\alpha, \end{align} and the commutator identity \begin{align}\label{commut1} e[D_\alpha,D_\beta]\phi=e(\partial_\alpha A_\beta-\partial_\beta A_\alpha)\phi+e[A_\alpha,A_\beta]\phi=\mathbf{R}(u)(\partial_\alpha u, \partial_\beta u)(e\phi). \end{align} In the two dimensional case, (\ref{commut1}) can be further simplified to \begin{align}\label{commut} e[D_\alpha,D_\beta]\phi=e(\partial_\alpha A_\beta-\partial_\beta A_\alpha)\phi=\mathbf{R}(u)(\partial_\alpha u, \partial_\beta u)(e\phi). \end{align} \noindent{\bf{Remark 2.1}} Sometimes in the same line, we will use both the intrinsic quantities such as ${\bf{R}}(\partial_tu,\partial_su)$ and frame dependent quantities such as $\phi_i$. This will not cause trouble by remembering the correspondence (\ref{poill}). And we define a matrix valued function $\bf{a}\wedge\bf{b}$ by \begin{align}\label{nb890km} (\bf{a}\wedge\bf{b})\bf{c}=\left\langle {\bf{a},\bf{c}} \right\rangle\bf{b} - \left\langle {\bf{b},\bf{c}} \right\rangle \bf{a}, \end{align} where $\bf{a},\bf{b},\bf{c}$ are vectors on $\Bbb R^2$. It is easy to see (\ref{nb890km}) coincide with (\ref{2.best}) by letting $X=a_1e_1+a_2e_2$, $Y=b_1e_1+b_2e_2$, $Z=c_1e_1+c_2e_2$. Hence (\ref{commut}) can be written as \begin{align}\label{nb90km} [D_\alpha,D_\beta]\phi=(\phi_\alpha\wedge\phi_{\beta})\phi \end{align} \begin{lemma} With the notions and notations given above, (\ref{wmap1}) can be written as \begin{align}\label{jnk} D_t\phi_t-h^{ij}D_i\phi_j+h^{ij}\Gamma^k_{ij}\phi_k=0¡£ \end{align} \end{lemma} \begin{proof} In the intrinsic formulation, (\ref{wmap1}) can be written as \begin{align*} {\nabla _t}{\partial _t}u - \left( {{\nabla _{{x_i}}}{\partial _{{x_j}}}u - {u_*}({\nabla _{\frac{\partial}{\partial x_i}}}\frac{\partial}{\partial {x_j}})} \right){h^{ij}} = 0. \end{align*} Expanding $\nabla_i\partial_j u$ and $u_*(\nabla_i\partial_j)$ by the frame $\{e_i\}^2_{i=1}$ yields \begin{align*} &{h^{ij}}{\nabla _i}{\partial _j}u - {h^{ij}}{u_*}({\nabla _{\frac{\partial }{{\partial {x_i}}}}}\frac{\partial }{{\partial {x_j}}}) = \sum\nolimits_{l =1}^2{{h^{ij}}} {\nabla _i}\left( {\left\langle {{\partial _j}u,{e_l}} \right\rangle {e_l}} \right) - \Gamma _{i,j}^k{h^{ij}}{\partial _k}u \\ &= {h^{ij}}\left( {{\partial _i}\psi _j^p{e_p} + [{A_i}]_l^p\psi _j^l{e_p}} \right) - \Gamma _{i,j}^k{h^{ij}}\psi _k^l{e_l} = e{h^{ij}}\left( {{D_i}{\phi _j}} \right) - e\Gamma _{i,j}^k{h^{ij}}{\phi _k}¡£ \end{align*} And $\nabla_t\partial_tu$ is expanded as \begin{align*} {\nabla _t}{\partial _t}u = \sum\nolimits_{l = 1}^2 {{\nabla _t}\left( {\left\langle {{\partial _t}u,{e_l}} \right\rangle {e_l}} \right)} = \left( {{\partial _t}\phi _0^p{e_p} + [{A_0}]_l^p\phi _0^l{e_p}} \right) = e\left( {{D_0}{\phi _0}} \right). \end{align*} Hence (\ref{jnk}) follows. \end{proof} \section{Caloric Gauge} Denote the space $C([0,T];\bf{H}^3\times\bf{H}^2)$ by $\mathcal{X}_T$. The caloric gauge was first introduced by Tao \cite{Tao4} for the wave maps from $\Bbb R^{2+1}$ to $\mathbb{H}^n$. We give the definition of the caloric gauge in our setting. \begin{definition}\label{pp} Let $u(t,x):[0,T]\times \mathbb{H}^2\to \mathbb{H}^2$ be a solution of (\ref{wmap1}) in $\mathcal{X}_T$. Suppose that the heat flow initiated from $u_0$ converges to a harmonic map $Q:\mathbb{H}^2\to \mathbb{H}^2$. Then for a given orthonormal frame $\Xi(x)\triangleq\{\Xi_j(Q(x))\}^2_{j=1}$ which spans the tangent space $T_{Q(x)}\mathbb{H}^2$ for any $x\in \mathbb{H}^2$, by saying a caloric gauge we mean a tuple consisting of a map $\widetilde{u}:\Bbb R^+\times [0,T]\times\mathbb{H}^2\to\Bbb H^2$ and an orthonormal frame $\Omega\triangleq\{\Omega_j(\widetilde{u}(s,t,x))\}^2_{j=1}$ such that \begin{align}\label{muqi} \left\{ \begin{array}{l} {\partial _s}\widetilde{u}= \tau (\widetilde{u}) \\ {\nabla _s}{\Omega _j} = 0 \\ \mathop {\lim }\limits_{s \to \infty } {\Omega _j} = {\Xi _j} \\ \end{array} \right. \end{align} where the convergence of frames is defined by \begin{align}\label{convergence} \left\{ \begin{array}{l} \mathop {\lim }\limits_{s \to \infty } \widetilde{u}(s,t,x) = Q(x) \\ \mathop {\lim }\limits_{s \to \infty } \left\langle {{\Omega _i}(s,t,x),{\Theta _j}(\widetilde{u}(s,t,x))} \right\rangle = \left\langle {{\Xi _i}(Q(x)),{\Theta _j}(Q(x))} \right\rangle \\ \end{array} \right. \end{align} \end{definition} The remaining part of this section is devoted to the existence of the caloric gauge. \subsection{Warming up for the heat flows} In this subsection, we prove the estimates needed for the existence of the caloric gauge and the bounds for connection coefficients. The equation of the heat flow is given by \begin{align}\label{8.29.1} \left\{ \begin{array}{l} {\partial _s}u = \tau (u) \\ u(0,x)= v(x) \\ \end{array} \right. \end{align} The energy density $e$ is defined by $$ e(u)=\frac{1}{2}|du|^2. $$ The following lemma is due to Li, Tam \cite{LT}. (\ref{VI4}), (\ref{uu}) are proved in \cite{LZ}. \begin{lemma}\label{8.44} Given initial data $v:\Bbb H^2\to\Bbb H^2$ with bounded energy density, suppose that $\tau(v)\in L^p_x$ for some $p>2$ and the image of $\Bbb H^2$ under the map $v$ is contained in a compact subset of $\Bbb H^2$. Then the heat flow equation (\ref{8.29.1}) has a global solution $u$. Moreover for some $K,C>0$, we have \begin{align} (\partial_s-\Delta)|du|^2+2|\nabla du|^2&\le K|du|^2\label{8.4}\\ (\partial_s-\Delta)|\partial_s u|^2+2|\nabla \partial_su|^2&\le 0\label{8.3}\\ (\partial_s-\Delta)|\partial_s u|&\le 0\label{VI4}\\ (\partial_s-\Delta)(|du|e^{-Cs})&\le 0.\label{uu} \end{align} \end{lemma} Consider the heat flow from $\mathbb{H}^2$ to $\mathbb{H}^2$ with a parameter \begin{align}\label{8.29.2} \left\{ \begin{array}{l} {\partial _s}\widetilde{u} = \tau (\widetilde{u}) \\ \widetilde{u}(s,t,x) \upharpoonright_{s=0}= u(t,x) \\ \end{array} \right. \end{align} We will give two types of estimates of $\nabla^k\partial_s\widetilde{u},\nabla^k\partial_x\widetilde{u}$ in the following. One is the decay of $\|\nabla^k\partial_s\widetilde{u}\|_{L^{2}_x}$ as $s\to\infty$ which can be easily proved via energy arguments. The other is the global boundedness of $\|\partial_x\widetilde{u}\|_{L^{\infty}_x}$ away from $s=0$ and the decay of $\|\nabla^k\partial_s\widetilde{u}\|_{L^{\infty}_x}$ as $s\to\infty$, both of which need additional efforts. And we will prove the decay estimates with respect to $s$ for $\|\partial_t\widetilde{u}\|_{{L^{\infty}_x}\bigcap L^2_x}$, which is the key integrability gain to compensate the loss of decay of $\partial_x\widetilde{u}$. We start with the estimate of $\|d\widetilde{u}\|_{L^{\infty}_x}$ which is the cornerstone for all other estimates. \begin{remark}\label{ki78} The following inequality which can be verified by Moser iteration is known in the heat flow literature: If $v$ is a nonnegative function satisfying \begin{align*} \partial_t v-\Delta v\le 0, \end{align*} then for $t\ge1$, \begin{align*} v(x,t)\le \int^t_{t-1}\int_{B(x,1)}v(y,s){\rm{dvol_y}}ds. \end{align*} \end{remark} Introduce the norm: \begin{align}\label{ytgvfre} \|u(t,x)\|_{\mathcal{X}_T}=&\|\nabla du\|_{C([0,T];L^2_x)}+\|\nabla \partial_tu\|_{C([0,T];L^2_x)}\nonumber\\ &+\| du\|_{C([0,T];L^2_x)}+\| \partial_tu\|_{C([0,T];L^2_x)}. \end{align} Trivial applications of Remark \ref{ki78}, (\ref{uu}) and the non-increasing of the energy along the heat flow give the bounds for $\|d\widetilde{u}\|_{L^{\infty}_x}$. See also \cite{LZ} for another proof. \begin{lemma}\label{density} Let $(u,\partial_tu)$ solve $(\ref{wmap1})$ in $\mathcal{X}_T$ (see (\ref{ytgvfre}) ) with $\|u\|_{\mathcal{X}_T}\le M$. If $\widetilde{u}$ is the solution to $(\ref{8.29.2})$ with initial data $u(t,x)$, then we have uniformly for $t\in[0,T]$, $s\in[1,\infty)$ \begin{align} \left\| d\widetilde{u}(s,t,x) \right\| _{L_x^\infty }&\lesssim \left\| d u(t,x) \right\|_{L_x^2},\label{3.14a} \end{align} \end{lemma} The decay of $\|\nabla^k\partial_s\widetilde{u}\|_{L^2_x}$ follows from an energy argument and the bound of the energy density provided by (\ref{3.14a}). \begin{lemma}\label{chen2222} Let $(u,\partial_tu)\in \mathcal{X}_T$ with $\|u\|_{\mathcal{X}_T}\le M$, then for some universal constant $\delta>0$ the solution $\widetilde{u}(s,t,x)$ to heat flow (\ref{8.29.2}) satisfies \begin{align} \|\partial_s\widetilde{u}(s,t,x)\|_{L^2_x}&\lesssim e^{-\delta s}MC(M), \mbox{ }{\rm{for}}\mbox{ }s>0\label{f41}\\ \|\nabla\partial_s\widetilde{u}(s,t,x)\|_{L^2_x}&\lesssim e^{-\delta s}MC(M), \mbox{ }{\rm{for}}\mbox{ }s\ge2\label{f42}\\ \int^{\infty}_0\|\nabla\partial_s\widetilde{u}(s,t,x)\|^2_{L^2_x}ds&\lesssim MC(M).\label{f40} \end{align} for all $t\in[0,T]$. The constant $C(M)$ grows polynomially as $M$ grows. \end{lemma} \begin{proof} First we notice that (\ref{8.3}), (\ref{huhu899}) and maximum principle yield \begin{align} \|\partial_s\widetilde{u}(s,t,x)\|_{L^{2}_x}&\lesssim e^{-\frac{s}{4}}\|\partial_s\widetilde{u}(0,t,x)\|_{L^{2}_x}\label{lao1}\\ \|\partial_s\widetilde{u}(s,t,x)\|_{L^{\infty}_x}&\lesssim s^{-1}e^{-\frac{s}{4}}\|\partial_s\widetilde{u}(0,t,x)\|_{L^{2}_x}.\label{lao2} \end{align} We introduce three energy functionals: \begin{align*} {\mathcal{E}_1}(\widetilde{u}) = \frac{1}{2}\int_{{\Bbb H^2}} {{{\left| {\nabla \widetilde{u}} \right|}^2}} dx,\mbox{ }{\mathcal{E}_2}(u) = \frac{1}{2}\int_{{\Bbb H^2}} {{{\left| {{\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_h}},\mbox{ }{\mathcal{E}_3}(u) = \frac{1}{2}\int_{{\Bbb H^2}} {{{\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_h}}. \end{align*} By integration by parts and (\ref{8.29.2}), we have \begin{align*} \frac{d}{{ds}}{\mathcal{E}_1}(\widetilde{u}) = - \int_{{\Bbb H^2}} {{{\left| {\tau (\widetilde{u})} \right|}^2}} {\rm{dvol_h}}. \end{align*} Thus the energy is decreasing with respect to $s$ and \begin{align}\label{V1} \|d\widetilde{u}\|^2_{L^2_x}+\int^s_0\|\partial_s \widetilde{u}\|^2_{L^2_x}\le \mathcal{E}_1(u_0). \end{align} The non-positive sectional curvature assumption with integration by parts yields $$\|\nabla d\widetilde{u}(s)\|^2_{L^2_x}\le \|\tau(\widetilde{u}(s))\|^2_{L^2_x}+\|d\widetilde{u}\|^2_{L^2_x} $$ Hence by (\ref{V1}), (\ref{8.29.2}) we conclude \begin{align}\label{V5} \|\widetilde{u}\|^2_{\mathfrak{H}^2}+\int^s_0\|\partial_s \widetilde{u}\|^2_{L^2_x}ds\lesssim\|u_0\|^2_{\mathfrak{H}^2}. \end{align} Again by (\ref{8.29.2}) and integration by parts, one has \begin{align} \frac{d}{{ds}}{\mathcal{E}_2}(\widetilde{u}) &= \int_{{\Bbb H^2}} {\left\langle {{\nabla _s}{\partial _s}\widetilde{u},{\partial _s}\widetilde{u}} \right\rangle } {\rm{dvol_h}}=\int_{{\Bbb H^2}} {\left\langle {{\nabla _s}\tau (\widetilde{u}),{\partial _s}\widetilde{u}} \right\rangle } {\rm{dvol_h}} \nonumber\\ &\le -\int_{{\Bbb H^2}} {\left\langle {\nabla {\partial _s}(\widetilde{u}),\nabla {\partial _s}\widetilde{u}} \right\rangle } {\rm{dvol_h}}+ C\int_{{\Bbb H^2}} {{{\left| {d\widetilde{u}} \right|}^2}{{\left| {{\partial _s}\widetilde{u}} \right|}^2}{\rm{dvol_h}}}.\label{j089} \end{align} Integrating (\ref{j089}) with respect to $s$ in $(s_1,s_2)$ for any $1<s_1<s_2$, we infer from (\ref{lao2}) and (\ref{V1}) that \begin{align} \mathcal{E}_2(\widetilde{u}(s_2))-{\mathcal{E}_2}(\widetilde{u}(s_1))+\int^{s_2}_{s_1}\mathcal{E}_3(\widetilde{u}(s))ds &\lesssim \left\| { d\widetilde{u}} \right\|^2_{L^{\infty}_s{L_x^2}}\int^{s_2}_{s_1}\left\| {{\partial _s}\widetilde{u}} \right\|^2_{L_x^\infty}ds\lesssim M^4e^{-\delta s_1}. \end{align} Then by (\ref{lao1}) we have for $1<s<s_1<s_2$ and any $t\in [0,T]$ \begin{align}\label{lao3} \int^{s_2}_{s_1}\|\nabla\partial_s\widetilde{u}(\tau,t,x)\|^2_{L^2_x}d\tau\lesssim M^2e^{-\delta s}. \end{align} Integration by parts and (\ref{8.29.2}) yield \begin{align} &\frac{d}{{ds}}{\mathcal{E}_3}(\widetilde{u}(s))\nonumber\\ &\le -\int_{{\Bbb H^2}} \big({{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}} + C\left| {d \widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|{\left| {{\partial _s}\widetilde{u}} \right|^2} + C\left| {d\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|{\left| {{\partial _s}\widetilde{u}} \right|^2} \big){\rm{dvol_h}}\nonumber\\ &+ C\int_{\Bbb H^2}\big({\left| {d\widetilde{u}} \right|^3}\left| {{\partial _t}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right| + C\left| {{\nabla ^2}\widetilde{u}} \right|\left| {d\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right| + C{\left| {{\partial _t}\widetilde{u}} \right|^2}{\left| {d \widetilde{u}} \right|^4}\big){\rm{dvol_h}} \nonumber\\ &+ C\int_{\Bbb H^2}\big({\left| {\nabla {\partial _t}\widetilde{u}} \right|^2}{\left| {d\widetilde{ u}} \right|^2} + C{\left| {d\widetilde{u}} \right|^2}\left| {{\nabla ^2}{\partial _t}\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|\big){\rm{dvol_h}}.\label{9xian} \end{align} By H\"older, (\ref{lao3}), (\ref{lao2}), we see for $1<s<s_1<s_2$ and any $t\in [0,T]$ \begin{align*} &\int_{{s_1}}^{{s_2}} {\int_{{\Bbb H^2}} {\left| {d\widetilde{u}} \right|} \left| {\nabla {\partial _s}\widetilde{u}} \right|{{\left| {{\partial _s}\widetilde{u}} \right|}^2}{\rm{dvol_h}}} ds\\ &\lesssim \left\| {{\partial _s}\widetilde{u}} \right\|_{L_s^4L_x^\infty ([{s_1},{s_2}] \times {\Bbb H^2})}^2{\left\| {d\widetilde{u}} \right\|_{L_s^\infty L_x^2([{s_1},{s_2}] \times {\Bbb H^2})}}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2([{s_1},{s_2}] \times {\Bbb H^2})}}\\ &\lesssim M^4{e^{ - \delta {s_1}}}. \end{align*} Similarly we have from (\ref{lao3}), (\ref{lao2}), (\ref{3.14a}) that for $1<s<s_1<s_2$ and any $t\in [0,T]$ \begin{align*} &\int_{{s_1}}^{{s_2}} \int_{{\Bbb H^2}} {{{\left| {d\widetilde{u}} \right|}^3}} \left| {\nabla {\partial _s}\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|{\rm{dvol_hds}}\\ &\lesssim \left\| {d\widetilde{u}} \right\|_{L_s^\infty L_x^\infty ([{s_1},{s_2}] \times {\Bbb H^2})}^3{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2([{s_1},{s_2}] \times {\Bbb H^2})}}{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2([{s_1},{s_2}] \times {\Bbb H^2})}} \\ &\lesssim {M^5}{e^{ - \delta {s_1}}}. \end{align*} And similarly we obtain for $1<s<s_1<s_2$ and all $t\in [0,T]$ \begin{align*} &\int_{{s_1}}^{{s_2}} \int_{{\Bbb H^2}} {\left| {\nabla d\widetilde{u}} \right|\left| {d\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|} {\rm{dvol_hds}}\\ &\lesssim {\left\| {d\widetilde{u}} \right\|_{L_s^\infty L_x^\infty }}{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^\infty }}{\left\| {\nabla d\widetilde{u}} \right\|_{L_s^\infty L_x^2}}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2}} \\ &\lesssim {M^4}{e^{ - \delta {s_1}}}, \end{align*} where the integrand domains are $[{s_1},{s_2}] \times {\Bbb H^2}$. The remaining three terms in (\ref{9xian}) are easier to bound. In fact, Sobolev embedding, (\ref{lao2}) and (\ref{V5}) show \begin{align*} \int_{{s_1}}^{{s_2}} {\int_{{\Bbb H^2}} {{{\left| {\nabla\widetilde{ u}} \right|}^4}} {{\left| {{\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_hds}} \le \left\| {{\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^\infty}^2\left\| {{\nabla d}\widetilde{u}} \right\|_{L_s^\infty L_x^2}^4 \le {M^6}{e^{ - \delta {s_1}}}. \end{align*} Similarly we obtain \begin{align*} \int_{{s_1}}^{{s_2}} {\int_{{\Bbb H^2}} {{{\left| {d\widetilde{u}} \right|}^2}} {{\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_hds}} \le \left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2}^2\left\| {d\widetilde{u}} \right\|_{L_s^\infty L_x^\infty }^2 \le {M^4}{e^{ - \delta {s_1}}}. \end{align*} The last remaining term in (\ref{9xian}) is absorbed by the negative term on the left. Indeed, for sufficiently small $\eta>0$ \begin{align*} &\int_{{s_1}}^{{s_2}} {\int_{{\Bbb H^2}} {{{\left| {d\widetilde{u}} \right|}^2}} \left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|} {\rm{dvol_hds}} \\ &\lesssim \eta \left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2([{s_1},{s_2}] \times {\Bbb H^2})}^2+ {\eta ^{ - 1}}\left\| {d\widetilde{u}} \right\|_{L_s^\infty L_x^\infty}^4\left\| {{\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2}^2 \\ &\lesssim \eta \left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_s^2L_x^2}^2 + {\eta ^{ - 1}}{M^6}{e^{ - \delta {s_1}}}. \end{align*} (\ref{V5}) implies that there exists $s_0\in(1,2)$ such that \begin{align} \int_{{\Bbb H^2}} {{\left| {\nabla {\partial _s}u} \right|}^2}({s_0},t,x) {\rm{dvol_h}} \le {M^2}. \end{align} Hence applying (\ref{{uv111}}) and Gronwall inequality to (\ref{9xian}), we have for $s>s_0$ \begin{align} &\int_{{\Bbb H^2}} {\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2(s,t,x){\rm{dvol_h}}\nonumber\\ &\lesssim e^{ - \delta (s - {s_0})}\int_{{\Bbb H^2}} {{\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2}({s_0},t,x) {\rm{dvol_h}} + MC(M)\int_{s_0}^s e^{ - \delta (s - \tau )}e^{ - \delta \tau }d\tau \nonumber\\ &\lesssim MC(M)\left( e^{ - \delta (s - {s_0})}+ e^{ - \delta s}(s - {s_0})\right).\label{7si} \end{align} Since $s_0\in(1,2)$, we have verified $(\ref{f42})$ for $s\ge 2$. (\ref{f40}) follows directly from (\ref{f42}), (\ref{V5}) and integrating (\ref{j089}) with respect to $s$. \end{proof} We then prove the pointwise decay of $|\nabla\partial_s\widetilde{u}|$ with respect to $s$. First we need the Bochner formula for high derivatives of $\widetilde{u}$ along the heat flow. The proof of following four lemmas is direct calculations with the Bochner technique. Considering that the proof is quite standard, we state the results without detailed calculations. \begin{lemma}\label{9tian} Let $\widetilde{u}$ be a solution to heat flow equation. Then $|\nabla\partial_s\widetilde{u}|^2$ satisfies \begin{align} {\partial _s}{\left| {\nabla {\partial _s}\widetilde{u}} \right|^2} - {\Delta}{\left| {\nabla {\partial _s}\widetilde{u}} \right|^2} + 2{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|^2} &\lesssim {\left| {\nabla {\partial _s}\widetilde{u}} \right|^2}+ \left| {{\partial _s}\widetilde{u}} \right|{\left| {d\widetilde{u}} \right|^3}\left| {\nabla {\partial _s}\widetilde{u}} \right| \nonumber\\ &+{\left| {\nabla {\partial _s}\widetilde{u}} \right|^2}{\left| {d\widetilde{u}} \right|^2} + \left| {{\partial _s}\widetilde{u}} \right|\left| {{\nabla }d\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|\left| {d \widetilde{u}} \right|.\label{9tian1} \end{align} \end{lemma} \begin{lemma}\label{B1} Let $\widetilde{u}$ be a solution to heat flow equation, then we have \begin{align} &{\partial _s}{\left| {\nabla d\widetilde{u}} \right|^2} - {\Delta}{\left| {\nabla d\widetilde{u}} \right|^2} + 2{\left| {{\nabla ^2}d\widetilde{u}} \right|^2} \lesssim {\left| {\nabla d\widetilde{u}} \right|^2} + {\left| {d\widetilde{u}} \right|^2}{\left| {\nabla d\widetilde{u}} \right|^2} \nonumber\\ &+ {\left| {d\widetilde{u}} \right|}^2 + {\left| {d\widetilde{u}} \right|^4}\left| {\nabla d\widetilde{u}} \right|.\label{ion1} \end{align} \end{lemma} \begin{lemma}\label{8zu} Let $\widetilde{u}$ be a solution to heat flow equation. Then $|\nabla\partial_t\widetilde{u}|^2$ satisfies \begin{align} &(\partial_s-\Delta)|\partial_t \widetilde{u}|^2=-2|\nabla \partial_t \widetilde{u}|^2-\mathbf{R}(\widetilde{u})(\nabla \widetilde{u},\partial_t \widetilde{u},\nabla \widetilde{u}, \partial_t \widetilde{u})\le 0.\label{10.127}\\ &{\partial _s}{\left| {\nabla {\partial _t}\widetilde{u}} \right|^2} - {\Delta}{\left| {\nabla {\partial _t}\widetilde{u}} \right|^2} + 2{\left| {{\nabla ^2}{\partial _t}\widetilde{u}} \right|^2} \lesssim {\left| {\nabla {\partial _t}\widetilde{u}} \right|^2} + {\left| {{\partial _s}\widetilde{u}} \right|{{\left| {{d}\widetilde{u}} \right|}^2}\left| {\nabla {\partial _t}\widetilde{u}} \right|}\nonumber \\ &+ {{{\left| {d\widetilde{u}} \right|}^3}\left| {{\partial _t}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right|} + {\left| {d\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|\left| {\nabla d\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right|}+ {{{\left| {\nabla {\partial _t}\widetilde{u}} \right|}^2}{{\left| {d\widetilde{u}} \right|}^2}} . \end{align} \end{lemma} \begin{lemma} Let $\widetilde{u}$ be a solution to heat flow equation, then \begin{align} &{\partial _s}{\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|^2} - \Delta {\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|^2} + 2{\left| {\nabla {\nabla _t}{\partial _s}\widetilde{u}} \right|^2} \lesssim {\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|^2}+\left| {\nabla d\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|\left|{\nabla _t}{\partial _s}\widetilde{u}\right|\left| {{\partial _s}\widetilde{u}} \right| \nonumber\\ &+ \left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|\left| {d\widetilde{u}} \right| + \left| {\nabla {\partial _t}\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|\left|{\nabla _t}{\partial _s}\widetilde{u}\right|\left| {d\widetilde{u}} \right|+{\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|^2}{\left| {d\widetilde{u}} \right|^2} .\label{iconm} \end{align} \end{lemma} We have previously seen that the bound of $\|d\widetilde{u}\|_{L^{\infty}}$ is useful for bounding $\|\nabla\partial_s\widetilde{u}\|_{L^{2}}$. In order to bound $\|\nabla\partial_s\widetilde{u}\|_{L^{\infty}}$, it is convenient if one has a bound for $\|\nabla d\widetilde{u}\|_{L^{\infty}}$ firstly. \begin{lemma} If $(u(t,x),\partial_tu(t,x))$ is a solution to (\ref{wmap1}) with $\|u(t,x)\|_{\mathcal{X}_T}\le M$. Then for $s\ge 2$ \begin{align} \|\nabla d\widetilde{u}\|_{L^{\infty}_x}\lesssim MC(M).\label{chen2} \end{align} \end{lemma} \begin{proof} The proof of (\ref{chen2}) is also based on Remark \ref{ki78}. One can rewrite (\ref{ion1}) by Young inequality in the following form \begin{align*} {\partial _s}{\left| {\nabla d\widetilde{u}} \right|^2} - {\Delta}{\left| {\nabla d\widetilde{u}} \right|^2} + 2{\left| {{\nabla ^2}d\widetilde{u}} \right|^2} \le C\left( {1 + {{\left| {d\widetilde{u}} \right|}^2}} \right){\left| {\nabla d\widetilde{u}} \right|^2} + {\left| {d\widetilde{u}} \right|^2}+ {\left| {d\widetilde{u}} \right|^8}. \end{align*} Since for $s\ge1$, $\|d\widetilde{u}\|_{L^{\infty}}\lesssim M$, $\|\partial_s\widetilde{u}\|_{L^{\infty}}\lesssim M$, let $r(s,t,x) = {\left| {\nabla d\widetilde{u}} \right|^2} + M^2 + {M^8}$, then we have $${\partial _s}r - {\Delta}r \le C\left( {{M^2} + 1} \right)r.$$ Let $v = {e^{ - C\left( {{M^2} + 1} \right)s}}r$. For $s\ge2$, it is obvious that $v$ satisfies $$ {\partial _s}v - {\Delta}v \le 0. $$ By Remark \ref{ki78}, we deduce for $d(x,y)\le 1$, $s\ge2$ \begin{align*} v(s,t,x)\lesssim \int^{s}_{s-1}\int_{B(x,1)} v(\tau,t,y)d\tau{\rm{dvol_y}}. \end{align*} Thus by $\|u\|_{\mathfrak{H}^2}\le M$ and (\ref{V5}), we conclude $${\left| {\nabla d\widetilde{u}} \right|^2}\left( {{s},t,x} \right)\lesssim MC(M). $$ Hence (\ref{chen2}) follows. \end{proof} Now we prove the decay of $\|\nabla\partial_s\widetilde{u}\|_{L^{\infty}_x}$ as $s\to\infty$. \begin{lemma}\label{decayingt} If $(u,\partial_tu)$ is a solution to (\ref{wmap1}) in $\mathcal{X}_T$ with $\|u(t,x)\|_{\mathcal{X}_T}\le M$. Then for some universal constant $\delta>0$ \begin{align} \|\nabla\partial_s\widetilde{u}\|_{L^{\infty}_x}\lesssim MC(M)e^{-\delta s}, \mbox{ }{\rm{for}} \mbox{ }s\ge1.\label{koo4} \end{align} \end{lemma} \begin{proof} By (\ref{lao2}), for $s\ge1$ \begin{align}\label{kongkong} \|\partial_s\widetilde{u}\|_{L^{\infty}_x}\lesssim e^{-\delta s}M. \end{align} We can rewrite (\ref{9tian1}) by Young inequality as \begin{align} {\partial _s}{\left| {\nabla {\partial _s}\widetilde{u}} \right|^2} - {\Delta}{\left| {\nabla {\partial _s}\widetilde{u}} \right|^2} \le (1 + {\left| {d\widetilde{u}} \right|^2}){\left| {\nabla {\partial _s}\widetilde{u}} \right|^2} + {\left| {d\widetilde{u}} \right|^6}{\left| {{\partial _s}\widetilde{u}} \right|^2} + {\left| {d\widetilde{u}} \right|^2}{\left| {\nabla d\widetilde{u}} \right|^2}{\left| {{\partial _s}\widetilde{u}} \right|^2}. \end{align} Let $g(s,t,x) = {\left| {d\widetilde{u}} \right|^6}{\left| {{\partial _s}\widetilde{u}} \right|^2} + {\left| {d\widetilde{u}} \right|^2}{\left| {\nabla d\widetilde{u}} \right|^2}{\left| {{\partial _s}\widetilde{u}} \right|^2}$, then by Lemma \ref{density}, Lemma \ref{chen2222} and (\ref{kongkong}), $g(s,t,x) \le C(M)M{e^{ - \delta s}}$ for $s\ge1$. Let $f(s,t,x) = {\left| {\nabla \partial_s \widetilde{u}} \right|^2}\left( {s,t,x} \right) + \frac{1}{\delta }C(M)M{e^{ - \delta s}}$, then $${\partial _s}f - {\Delta}f \le C\left( {{M^2} + 1} \right)f.$$ Then $\bar v=e^{-C( {{M^2} + 1})s}f$ satisfies $$ {\partial _s}\bar v - {\Delta}\bar v \le0. $$ Applying Remark \ref{ki78} to $\bar{v}$ as before implies $${\left| {\nabla {\partial _s}\widetilde{u}} \right|^2}\left( {{s},t,x} \right) + \frac{1}{\delta }C(M)M{e^{ - \delta {s}}} \le \int_{{s} - 1}^{{s}} {\int_{{\Bbb H^2}} {{{\left| {\nabla {\partial _s}\widetilde{u}\left( {\tau,t,y} \right)} \right|}^2}{\rm{dvol_hd\tau}}}} +C(M)M{e^{ - \delta {s}}}. $$ Therefore, (\ref{koo4}) follows from \begin{align}\label{ingjh} \int_{{s} - 1}^{{s}} \int_{{\Bbb H^2}} {{{\left| {\nabla {\partial _s}\widetilde{u}\left( {\tau,t,y} \right)} \right|}^2}{\rm{dvol_h}d\tau}}\lesssim MC(M)e^{-\delta s}, \end{align} which arises from (\ref{f42}). \end{proof} We move to the decay for $|\partial_t\widetilde{u}|$ with respect to $s$. \begin{lemma} If $(u,\partial_tu)$ is a solution to (\ref{wmap1}) in $\mathcal{X}_T$ with $\|u(t,x)\|_{\mathcal{X}_T}\le M$. Then \begin{align} \|\partial_t\widetilde{u}\|_{L^{2}_x}&\lesssim MC(M)e^{-\delta s},\mbox{ }{\rm{for}} \mbox{ }s>0\label{xiu1}\\ \|\partial_t\widetilde{u}\|_{L^{\infty}_x}&\lesssim MC(M)e^{-\delta s},\mbox{ }{\rm{for}} \mbox{ }s\ge1\label{xiu2}\\ \int^{\infty}_0\|\nabla\partial_t\widetilde{u}\|^2_{L^{2}_x}ds&\lesssim MC(M), \label{xiu3}\\ \|\nabla\partial_t\widetilde{u}\|_{L^{\infty}_x}&\lesssim MC(M)e^{-\delta s}, \mbox{ }{\rm{for}} \mbox{ }s\ge1.\label{xiu4} \end{align} \end{lemma} \begin{proof} The maximum principle and (\ref{huhu899}) imply \begin{align}\label{sdf8uj} \| {{\partial _t}\widetilde{u}(s,t,x)} \|^2_{L^{\infty}_x} \le {s^{- 2}}{e^{ - \delta s}} \left\| {{\partial _t}\widetilde{u}(0,t,x)} \right\|^2_{L^2_x}. \end{align} Moreover further calculations with (\ref{10.127}) show \begin{align*} (\partial_s-\Delta)|\partial_t \widetilde{u}|\le 0. \end{align*} Thus maximum principle and (\ref{m8}) give \begin{align}\label{10.26} \|{{\partial _t}\widetilde{u}(s,t,x)} \|_{L^2_x}\lesssim e^{-\frac{1}{4}s}\|\partial_tu\|_{L^2_x}\le M. \end{align} Therefore, (\ref{xiu1}) and (\ref{xiu2}) follow from (\ref{sdf8uj}) and (\ref{10.26}) respectively. Second, we prove (\ref{xiu3}) by energy arguments. Introduce the energy functionals $$ \mathcal{E}_4(\widetilde{u})=\frac{1}{2}\int_{\Bbb H^2}|\partial_t\widetilde{u}|^2{\rm{dvol_h}},\mbox{ }\mathcal{E}_5(\widetilde{u})=\int_{\Bbb H^2}|\nabla\partial_t\widetilde{u}|^2{\rm{dvol_h}}. $$ Then integration by parts gives \begin{align}\label{muxc6zb8} \frac{d}{{ds}}{\mathcal{E}_4}\left( \widetilde{u} \right) + {\mathcal{E}_5}\left( \widetilde{u} \right) \le \int_{\Bbb H^2}{\left| {d\widetilde{u}} \right|^2}{\left| {{\partial _t}\widetilde{u}} \right|^2}{\rm{dvol_h}}. \end{align} Integrating this formula with respect to $s$ in $[0.\kappa)$ with $\kappa>1$ shows $$\int_0^{\kappa} {\left\| {\nabla \partial {}_t\widetilde{u}} \right\|_{L_x^2}^2ds} \le \left\| {\partial_t\widetilde{u}(\kappa)} \right\|_{L_x^2}^2 + \int_0^1 {\left\| {{\partial _t}\widetilde{u}} \right\|_{{L^4}}^2\left\| {d\widetilde{u}} \right\|_{{L^4}}^2} ds + {\mathcal{E}_1}\left( \widetilde{u} \right)M\int_1^{\kappa} {{e^{ - 2\delta s}}ds}, $$ where we have used (\ref{xiu1}), (\ref{xiu2}) and H\"older. By Sobolev embedding and letting $\kappa\to\infty$, we obtain \begin{align}\label{chuyunyi} \int_0^\infty {\left\| {\nabla \partial_t\widetilde{u}} \right\|_{L_x^2}^2ds} \le {M^4} + {M^2}. \end{align} Finally, the proof of (\ref{xiu4}) follows by the same arguments as (\ref{koo4}) illustrated in Lemma \ref{decayingt}. \end{proof} \begin{lemma}\label{zhangqiling} Let $(u,\partial_tu)$ be a solution to (\ref{wmap1}) in $\mathcal{X}_T$ with $\|u(t,x)\|_{\mathcal{X}_T}\le M$. Then \begin{align} {\left\| {{s^{\frac{1}{2}}}{\nabla _t}{\partial _s}\widetilde{u}} \right\|_{L_s^\infty L_x^2}} &\lesssim MC(M)\mbox{ }{\rm{for}}\mbox{ }s\in[0,1]\label{haorenhao9}. \end{align} Moreover, for $s\in[1,\infty)$ and some $0<\delta\ll1$ \begin{align} {\left\| {{\nabla _t}{\partial _s}\widetilde{u}} \right\|_{L_s^\infty L_x^\infty }} &\lesssim e^{-\delta s}MC(M).\label{haorenhao10} \end{align} \end{lemma} \begin{proof} It is easy to see $\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right| \le \left| {\nabla {\partial _t}\widetilde{u}} \right|+ \left| {{h^{ii}}{\nabla _i}{\nabla _t}{\partial _i}\widetilde{u}} \right| + \left| {{\partial _t}\widetilde{u}} \right|{\left| {d\widetilde{u}} \right|^2}$, then \begin{align}\label{facik} \left| {{\nabla _t}{\partial _s}\widetilde{u}} \right| \le \left| {{\nabla ^2}{\partial _t}\widetilde{u}} \right| + \left| {{\partial _t}\widetilde{u}} \right|{\left| {d\widetilde{u}} \right|^2} + \left| {\nabla {\partial _t}\widetilde{u}} \right|. \end{align} Integration by parts gives \begin{align} &\frac{d}{{ds}}\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2\nonumber\\ &\le - {\int_{{\Bbb H^2}} {\left| {{\nabla ^2}{\partial _t}\widetilde{u}} \right|} ^2}{\rm{dvol_h}} + \int_{{\Bbb H^2}} {\left| {\nabla {\partial _t}\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|} \left| {d\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|{\rm{dvol_h}} \nonumber\\ &+ \int_{{\Bbb H^2}} {\left| {{\nabla ^2}{\partial _t}\widetilde{u}} \right|} {\left| {d\widetilde{u}} \right|^2}\left| {{\partial _s}\widetilde{u}} \right|{\rm{dvol_h}} + {\int_{{\Bbb H^2}} {\left| {\nabla {\partial _t}\widetilde{u}} \right|} ^2}{\rm{dvol_h}}+ {\int_{{\Bbb H^2}} {\left| {\nabla {\partial _t}\widetilde{u}} \right|} ^2}{\left| {d\widetilde{u}} \right|^2}{\rm{dvol_h}}.\label{haoren6} \end{align} By Sobolev embedding, we obtain $$\frac{d}{{ds}}\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2 \le C\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2\left( {1 + \left\| {{\partial _s}\widetilde{u}} \right\|_{{L^\infty }}^2 + \left\| {d\widetilde{u}} \right\|_{{L^\infty }}^2} \right) + \left\| {\nabla d\widetilde{u}} \right\|_{{L^2}}^4\left\| {{\partial _s}\widetilde{u}} \right\|_{{L^\infty }}^2. $$ Thus we get $$\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2 \le \left\| {\nabla {\partial _t}\widetilde{u}(0,t,x)} \right\|_{{L^2}}^2 + {e^{\int^s_0V(\tau)d\tau}}\int_0^s {{e^{ - \int_0^{\kappa} {V(\tau )} d\tau }}} \left\| {\nabla d\widetilde{u}} \right\|_{{L^2}}^4\left\| {{\partial _s}\widetilde{u}} \right\|_{{L^\infty }}^2d\kappa, $$ where $V(s)=Cs+C\|d\widetilde{u}\|^2_{L^{\infty}}+C\|\partial_s\widetilde{u}\|^2_{L^{\infty}}$. By Lemma \ref{ktao1} and Lemma \ref{8.44} \begin{align}\label{haorenh7} \int^{1}_0\|d\widetilde{u}\|^2_{L^{\infty}}ds+\int^1_0\|\partial_s\widetilde{u}\|^2_{L^{\infty}}ds\le M^2. \end{align} Hence we conclude for $s\in[0,1]$, \begin{align}\label{jidujihao1} \left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2 \le \left\| {\nabla {\partial _t} \widetilde{u}(0,t,x)} \right\|_{{L^2}}^2 + {e^{MC(M)s}}MC(M). \end{align} With (\ref{haoren6}), we further deduce that \begin{align}\label{jidujihao} \int_0^1 {{{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|}_{{L^2}}}ds} \lesssim MC(M). \end{align} Integration by parts shows, \begin{align*} &\frac{d}{{ds}}\left( {\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}^2s} \right) \\ &\le - s{\int_{{\Bbb H^2}} {\left| {{\nabla ^3}{\partial _t}\widetilde{u}} \right|} ^2}{\rm{dvol_hdt}} + \left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}^2 + s{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^4}}{\left\| {d\widetilde{u}} \right\|_{L_x^4}}{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}} \\ &+ s{\left\| {{\partial _t}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}}{\left\| {d\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}} + s\left\| {d\widetilde{u}} \right\|_{{L^8}}^2{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^4}}{\left\| {{\nabla ^3}{\partial _t}\widetilde{u}} \right\|_{L_x^2}} \\ &+ s{\left\| {d\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}} + s\left\| {d\widetilde{u}} \right\|_{{L^\infty }}^2{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}} \\ &+ s\left\| {d\widetilde{u}} \right\|_{{L^\infty }}^2{\left\| {\nabla d\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}} + s{\left\| {{\partial _t}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {\nabla d\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}}\\ &+ s{\left\| {\nabla d\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\partial _t}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {{\nabla ^3}{\partial _t}\widetilde{u}} \right\|_{L_x^2}}{\left\| {d\widetilde{u}} \right\|_{L_x^\infty }} + s\left\| {d\widetilde{u}} \right\|_{{L^\infty }}^2\left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{{L^2}}^2. \end{align*} Then Gronwall with (\ref{V5}), (\ref{haorenh7}) yields for all $s\in[0,1]$ \begin{align}\label{haoren8} \left\| {{\nabla ^2}{\partial _t}\widetilde{u}} \right\|_{L_x^2}^2s + \int_0^s {{\int_{{\Bbb H^2}} {\left| {{\nabla ^3}{\partial _t}\widetilde{u}} \right|}^2}\tau} {\rm{dvol_h}}d\tau \le MC(M). \end{align} Thus by (\ref{haoren8}), (\ref{jidujihao}), (\ref{jidujihao1}), and (\ref{facik}), we conclude \begin{align} {\left\| {{s^{\frac{1}{2}}}{\nabla _t}{\partial _s}\widetilde{u}} \right\|_{L_s^\infty[0,1] L_x^2}} \le MC(M). \end{align} (\ref{haorenhao10}) follows by the same path as Lemma \ref{decayingt} with the help of (\ref{iconm}). The essential ingredient is to prove for $s_1\ge2$ \begin{align}\label{huaqian} \int^{s_1+1}_{s_1}\|\nabla^2\partial_t\widetilde{u}\|^2_{L^2_x}ds\lesssim MC(M)e^{-\delta s_1}. \end{align} The remaining proof is devoted to verifying (\ref{huaqian}). By (\ref{{uv111}}) and (\ref{haoren6}), we obtain for any $0<c\ll1$ \begin{align} &\frac{d}{{ds}}\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2 +c {\int_{{\Bbb H^2}} {\left| {{\nabla}{\partial _t}\widetilde{u}} \right|} ^2}{\rm{dvol_h}}+c {\int_{{\Bbb H^2}} {\left| {{\nabla^2}{\partial _t}\widetilde{u}} \right|} ^2}{\rm{dvol_h}}\nonumber\\ &\lesssim \int_{{\Bbb H^2}} {\left| {\nabla {\partial _t}\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|} \left| {d\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|{\rm{dvol_h}}+ \int_{{\Bbb H^2}} {\left| {{\nabla ^2}{\partial _t}\widetilde{u}} \right|} {\left| {d\widetilde{u}} \right|^2}\left| {{\partial _s}\widetilde{u}} \right|{\rm{dvol_h}}\nonumber\\ &+ {\int_{{\Bbb H^2}} {\left| {\nabla {\partial _t}\widetilde{u}} \right|} ^2}{\left| {d\widetilde{u}} \right|^2}{\rm{dvol_h}} +\frac{1}{c} {\int_{{\Bbb H^2}} {\left| {\nabla {\partial _t}\widetilde{u}} \right|} ^2}{\rm{dvol_h}}.\label{haoren68} \end{align} By Lemma 3.3 and (\ref{lao2}), we have for $s\ge1$ \begin{align} &\left\| {d\widetilde{u}} \right\|_{L_x^{\infty}}\lesssim \|du\|_{L^2_x}\lesssim M\label{g8uiknlo}\\ &\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^{\infty}}\lesssim e^{-\delta s}\|\partial_s\widetilde{u}(0,t)\|_{L^2_x}\lesssim e^{-\delta s}M.\label{g8uiknlp} \end{align} Then by Sobolev embedding and Gronwall inequality, for $s\ge1$ \begin{align} \left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}^2 &\lesssim {e^{-cs}}\left\| {\nabla {\partial _t}\widetilde{u}(1,t,x)} \right\|_{L_x^2}^2 + {e^{ - cs}}\int_1^s {{e^{c\kappa}}} \left\| {\nabla {\partial _t}\widetilde{u}(\kappa)} \right\|_{L_x^2}^2d\kappa\nonumber\\ &+ {e^{ - cs}}\int_1^s {{e^{c\kappa}}} \left\| {\nabla d\widetilde{u}(\kappa,t)} \right\|_{L_x^2}^4\left\| {{\partial _s}\widetilde{u}(\kappa,t)} \right\|_{L_x^\infty }^2d\kappa.\label{ua5vvz} \end{align} Hence (\ref{chuyunyi}), (\ref{V5}), (\ref{g8uiknlp}) and (\ref{ua5vvz}) give for $s\in[0,\infty)$ \begin{align}\label{fanlin} \left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}^2 \le MC(M), \end{align} where (\ref{fanlin}) when $s\in[0,1]$ follows by (\ref{jidujihao1}). Integrating (\ref{muxc6zb8}) with respect to $s$ in $[s_1,s_2]$ for $1\le s_1\le s_2<\infty$ yields \begin{align}\label{chanjiang1} \int_{{s_1}}^{{s_2}} {\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}^2} ds \le \left\| {{\partial _t}\widetilde{u}} \right\|_{L_x^2}^2({s_2}) - \left\| {{\partial _t}\widetilde{u}} \right\|_{L_x^2}^2({s_1}) + \int_{{s_1}}^{{s_2}} {\left\| {{\partial _t}\widetilde{u}} \right\|_{L_x^2}^2} \left\| {d\widetilde{u}} \right\|_{L_x^\infty }^2ds. \end{align} By (\ref{xiu1}), Lemma \ref{density}, \begin{align}\label{chanjiang2} \int_{{s_1}}^{{s_2}} {\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}^2} ds \lesssim MC(M){e^{ - \delta {s_1}}}. \end{align} Thus in any interval $[s_*,s_*+1]$ there exists $s^0_*\in[s_*,s_*+1]$ such that \begin{align}\label{chanjiang21} {\left\| {\nabla {\partial _t}\widetilde{u}}(s^0_*) \right\|_{L_x^2}^2} ds \lesssim MC(M){e^{ - \delta {s_*}}}. \end{align} Fix $s_*\ge 1$, applying Gronwall to (\ref{haoren68}) in $[s^0_*,a]$ with $a\in[s^0_*,s_*+2]$ gives \begin{align} \left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}^2(a,t) &\le {e^{ - ca}}\left\| {\nabla {\partial _t}\widetilde{u}(s_*^0,t,x)} \right\|_{L_x^2}^2+ {e^{ - ca}}\int_{s^0_*}^a {{e^{cs}}}\left\| {\nabla {\partial _t}\widetilde{u}(s)} \right\|_{L_x^2}^2ds\nonumber\\ &+ {e^{ - ca}}\int_{s^0_*}^a {{e^{cs}}} \left\| {\nabla d\widetilde{u}(s)} \right\|_{L_x^2}^4\left\| {{\partial _s}\widetilde{u}(s)} \right\|_{L_x^\infty }^2ds.\label{ua6vvz} \end{align} Thus by (\ref{xiu1}), Lemma \ref{density}, (\ref{chanjiang21}) and the fact that $a$ at leat ranges over all $[s_*+1,s_*+2]$, we have for $s\ge2$, \begin{align}\label{chanjiang3} \left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}^2\le MC(M) e^{-\delta s}. \end{align} Integrating (\ref{haoren68}) with respect to $s$ again in $[s_1,s_1+1]$, we obtain (\ref{huaqian}) by (\ref{chanjiang3}) and (\ref{xiu1}), Lemma 3.3. Finally using maximum principle and Remark \ref{ki78} as Lemma \ref{decayingt}, we get (\ref{haorenhao10}) from (\ref{ua5vvz}), (\ref{chanjiang3}), (\ref{facik}) and Lemma \ref{decayingt}. \end{proof} In the remaining part of this subsection, we consider the short time behaviors of the differential fields under the heat flow. Since the energy of the solution to the heat flow in our case will not decay to zero, we can not expect that it behaves as a solution to the linear heat equation in the large time scale. However, one can still expect that the solution to the heat flow is almost governed by the linear equation in the short time scale. We summarize these useful estimates in the following proposition. \begin{proposition}\label{lize1} Let $u:[0,T]\times\Bbb H^2\to \Bbb H^2$ be a solution to (\ref{wmap1}) satisfying \begin{align*} \|(\nabla du,\nabla\partial_t u)\|_{L^2\times L^2}+\|(du,\partial_t u)\|_{L^2\times L^2}\le M. \end{align*} If $\widetilde{u}:\Bbb R^+\times[0,T]\times\Bbb H^2\to \Bbb H^2$ is the solution to (\ref{8.29.2}) with initial data $u(t,x)$, then for any $\eta>0$, it holds uniformly for $(s,t)\in(0,1)\times[0,T]$ that \begin{align*} &s^{\frac{1}{2}}{\left\| {\nabla d\widetilde{u}} \right\|_{{L^\infty_x }}} + s^{\frac{1}{2}}{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^\infty_x }}} + {s}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{{L^\infty_x }}} + s^{\frac{1}{2}}{\left\| {{\partial _s}\widetilde{u}} \right\|_{{L^\infty_x }}}\\ &+ s^{\frac{1}{2}}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{{L^2_x}}} + {\left\| {s^{\frac{1}{2}}{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_{s,x}^2}} +s{\left\| {\nabla_t {\partial _s}\widetilde{u}} \right\|_{{L^{\infty}_x}}}+s^{\eta}\|d\widetilde{u}\|_{L^{\infty}_x}\le MC(M). \end{align*} \end{proposition} \begin{proof} Since $\|\nabla d\widetilde{u}\|_{L^2_x}\le M$ shown by (\ref{V5}), Sobolev embedding implies $\|d\widetilde{u}\|_{L^p_x}\le M$ for any $p\in(2,\infty)$. Then (\ref{uu}) and (\ref{huhu89}) yield $s^{\eta}\|d\widetilde{u}\|_{L^{\infty}_x}\le M$ for any $\eta>0$ and all $(t,s)\in[0,T]\times(0,1)$. By (\ref{ion1}), one has $${\partial _s}\left| {\nabla d\widetilde{u}} \right| - {\Delta}\left| {\nabla d\widetilde{u}} \right|{\rm{ }} \le K\left| {\nabla d\widetilde{u}} \right| + {\left| {d\widetilde{u}} \right|^2}\left| {\nabla d\widetilde{u}} \right| + {\left| {d\widetilde{u}} \right|} + {\left| {d\widetilde{u}} \right|^4}. $$ Furthermore we obtain $$\left( {\partial _s} - {\Delta} \right)\left( {{e^{ - sK}}{e^{ - \int_0^s {\left\| {d\widetilde{u}(\tau )} \right\|_{L_x^\infty }^2d\tau } }}\left| {\nabla d\widetilde{u}} \right|} \right) \le {e^{ - sK}}{e^{ - \int_0^s {\left\| {d\widetilde{u}(\tau )} \right\|_{L_x^\infty }^2d\tau } }}\left( {{{\left| {d\widetilde{u}} \right|}} + {{\left| {d\widetilde{u}} \right|}^4}} \right). $$ Then maximum principle implies for $s\in[0,2]$ \begin{align*} {\left\| { {\nabla d\widetilde{u}}(s)} \right\|_{L_x^\infty }} &\lesssim {\big\| {{e^{{\Delta}\frac{s}{2}}}}( {{e^{ - \frac{{sK}}{2}}}{e^{ - \int_0^{\frac{s}{2}} {\left\| {d\widetilde{u}(\tau )} \right\|_{L_x^\infty }^2d\tau } }}\left| {\nabla d\widetilde{u}} \right|(\frac{s}{2})})\big\|_{L_x^\infty }} \\ &+ {\big\| {\int_{\frac{s}{2}}^s {{e^{{\Delta}(s - \tau )}} {{e^{ - K\tau }}{e^{ - \int_0^\tau {\| {d\widetilde{u}(\tau_1)} \|_{L_x^\infty }^2d\tau_1 } }}( {{{| {d\widetilde{u}}|}}+ {{| {d\widetilde{u}} |}^4}} )(\tau )} d\tau } }\big \|_{L_x^\infty }}. \end{align*} By the smoothing effect of the heat semigroup, we obtain for $s\in[0,1]$ \begin{align*} {\left\| { {\nabla d\widetilde{u}}(s)} \right\|_{L_x^\infty }} \lesssim {s^{ - \frac{1}{2}}}{\big\| { {\nabla d\widetilde{u}} (\frac{s}{2})} \big\|_{L_x^2}} + {\int_{\frac{s}{2}}^s {{{\big\| {{{| {d\widetilde{u}}|}^4}(\tau )} \big\|}_{L_x^\infty }} + \big\| {{{| {d\widetilde{u}}|}}|(\tau )} \big\|} _{L_x^\infty }}d\tau. \end{align*} Then Lemma \ref{8.5} and Lemma \ref{8.44} show for $s\in(0,1)$ $${\left\| \nabla d\widetilde{u}(s) \right\|_{L_x^\infty }} \le {s^{ - \frac{1}{2}}}{\big\| { {\nabla d\widetilde{u}}(\frac{s}{2})} \big\|_{L_x^2}} + \int_{\frac{s}{2}}^s {{\tau ^{ - 3/2}}( \big\| {du} \big\|^4_{L_x^{\frac{8}{3}}} + \big\| {du} \big\|_{L_x^2})} d\tau. $$ Therefore by Sobolev inequality we conclude \begin{align*} {\left\| {\left| {\nabla d\widetilde{u}} \right|(s)} \right\|_{L_x^\infty }} &\le {s^{ -\frac{1}{2}}}\mathop {\sup }\limits_{t \in [0,T]} \left( \left\| {\nabla du}(t) \right\|_{L_x^2}^4 + \left\| {du}(t) \right\|_{L_x^2} \right)\nonumber\\ &+{s^{ - \frac{1}{2}}}\mathop {\sup }\limits_{s \in [0,1]} {\left\| {\left| {\nabla d\widetilde{u}} \right|(s)} \right\|_{L_x^2}}. \end{align*} Thus by (\ref{V5}), we obtain for $s\in[0,1]$ \begin{align}\label{qian1} s^{\frac{1}{2}}{\left\| {\left| {\nabla d\widetilde{u}} \right|(s)} \right\|_{L_x^\infty }} \le MC(M). \end{align} By (\ref{9xian}) we have \begin{align} &\frac{d}{{ds}}\left( {s{\mathcal{E}_3}(\widetilde{u}(s))} \right) \nonumber\\ &\lesssim {\mathcal{E}_3}(\widetilde{u}(s)) - \int_{{\Bbb H^2}} {s{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}{\rm{dvol_h}}}+ \int_{\Bbb H^2}s\left( \left| {d\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|{{\left| {{\partial _s}\widetilde{u}} \right|}^2} \right) {\rm{dvol_h}}\nonumber \\ &+ \int_{{\Bbb H^2}} s\left( \left| {{\nabla d}\widetilde{u}} \right|\left| {d\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right| + {{\left| {{\partial _s}\widetilde{u}} \right|}^2}{{\left| {d\widetilde{u}} \right|}^4} + {{\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2}{{\left| {d\widetilde{u}} \right|}^2} \right){\rm{dvol_h}} \nonumber\\ &+\int_{{\Bbb H^2}}s\big( {{\left| {d\widetilde{u}} \right|}^3}\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|+ {{\left| {d\widetilde{u}} \right|}^2}\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|\big){\rm{dvol_h}}.\label{hu897} \end{align} The terms in the right hand side can be bounded by Sobolev and H\"older as follows \begin{align*} \int_{{\Bbb H^2}} {s\left| {d\widetilde{u}} \right|} \left| {\nabla {\partial _s}\widetilde{u}} \right|{\left| {{\partial _s}\widetilde{u}} \right|^2}{\rm{dvol_h}}&\le {s^{\frac{1}{2}}}{\left\| {d\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}}{s^{\frac{1}{2}}}\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^4}^2 \\ \int_{{\Bbb H^2}} {s{{\left| {d\widetilde{u}} \right|}^3}\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|} {\rm{dvol_h}} &\le s\left\| {d\widetilde{u}} \right\|_{L_x^{12}}^3{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^4}} \\ \int_{{\Bbb H^2}} s\left| {{\nabla d}\widetilde{u}} \right|\left| {d\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _s}\widetilde{u}} \right|{\rm{dvol_h}} &\le {{\left\| {s{\nabla d}\widetilde{u}} \right\|}_{L_x^\infty }}{{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|}_{L_x^2}}{{\left\| {d\widetilde{u}} \right\|}_{L_x^4}}{{\left\| {{\partial _s}\widetilde{u}} \right\|}_{L_x^4}} \\ \int_{{\Bbb H^2}} s{{\left| {{\partial _s}\widetilde{u}} \right|}^2}{{\left| {d\widetilde{u}} \right|}^4}{\rm{dvol_h}} &\le \left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^2}^2s\left\| {d\widetilde{u}} \right\|_{L_x^{\infty}}^4 \\ \int_{{\Bbb H^2}} s{{\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2}{{\left| {d\widetilde{u}} \right|}^2}{\rm{dvol_h}} &\le \left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}^2s\left\| {d\widetilde{u}} \right\|_{L_x^\infty }^2. \end{align*} The highest order term can be absorbed by the negative term, indeed we have \begin{align*} \int_{{\Bbb H^2}} s{{\left| {d\widetilde{u}} \right|}^2}\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|\left| {{\partial _s}\widetilde{u}} \right|{\rm{dvol_h}} &\le \frac{s}{{2C}}\int_{\Bbb H^2} {{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_h} + C\int_{{\Bbb H^2}} {s{{\left| {{\partial _s}\widetilde{u}} \right|}^2}{{\left| {d\widetilde{u}} \right|}^4}{\rm{dvol_h}}}} \\ &\le \frac{s}{{2C}}\int_{{\Bbb H^2}} {{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_h}} + C\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^2}^2s\left\| {d\widetilde{u}} \right\|_{L_x^{\infty}}^4. \end{align*} Recall the fact $\left| {d\widetilde{u}} \right|(s) \le e^{{\Delta s}}\left| {d{u}} \right|$ when $s\in[0,1]$, $\left| {{\partial _s}\widetilde{u}} \right|(s) \le {e^{{\Delta s}}}\left| {\tau({u})} \right|$, the terms involved above are bounded by smoothing effect \begin{align} s\left\| {d\widetilde{u}} \right\|_{L_x^\infty }^2 + {s^{\frac{1}{4}}}{\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^4}} \le \left\| {d{u}} \right\|_{L_x^2}^2 + {\left\| {\tau({u})} \right\|_{L_x^2}}. \end{align} Thus integrating (\ref{hu897}) with respect to $s$ in $[0,s]$ with (\ref{qian1}) gives for $s\in[0,1]$ \begin{align}\label{ki6ll1} s{\mathcal{E}_3}(\widetilde{u}(s)) + \int_0^s {\int_{\Bbb H^2}} {s{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}} {\rm{dvol_h}} ds \lesssim \int_0^s {\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}^2ds'}. \end{align} Therefore by (\ref{f40}), we conclude for $s\in[0,1]$ \begin{align}\label{qian2} \int_{{\Bbb H^2}} s{{\left| {\nabla {\partial _s}\widetilde{u}} \right|}^2}{\rm{dvo}}{{\rm{l}}_{\rm{h}}} \le MC(M). \end{align} By (\ref{9tian1}), we deduce $$ {\partial _s}\left| {\nabla {\partial _s}\widetilde{u}} \right| - {\Delta}\left| {\nabla {\partial _s}\widetilde{u}} \right|\le \left| {\nabla {\partial _s}\widetilde{u}} \right|{\left| {d\widetilde{u}} \right|^2} + \left| {{\partial _s}\widetilde{u}} \right|{\left| {d\widetilde{u}} \right|^3} + \left| {{\partial _s}\widetilde{u}} \right|\left| {{\nabla d}\widetilde{u}} \right|\left| {d \widetilde{u}} \right|. $$ Then as above considering the equation of ${e^{ - \int_0^s {\| {d\widetilde{u}(\tau )}\|_{L_x^\infty }^2d\tau } }}\left| {\nabla {\partial _s}\widetilde{u}} \right|$, we obtain by maximum principle that \begin{align*} &{\| { {\nabla {\partial _s}\widetilde{u}}(s)} \|_{L_x^\infty }}\\ &\le {s^{ - \frac{1}{2}}}{\| {{\nabla {\partial _s}\widetilde{u}}(\frac{s}{2})} \|_{L_x^2}} + {\int_{\frac{s}{2}}^s {{{\| {\left| {{\partial _s}\widetilde{u}} \right|{{\left| {d\widetilde{u}} \right|}^3}(\tau )} \|}_{L_x^\infty }} + \|{\left| {{\partial _s}\widetilde{u}} \right|\left| {{\nabla d}\widetilde{u}} \right|\left| {d\widetilde{u}} \right|(\tau )} \|} _{L_x^\infty }}d\tau. \end{align*} Hence (\ref{qian1}) and (\ref{qian2}) give \begin{align*} {\left\| {\left| {\nabla {\partial _s}\widetilde{u}} \right|(s)} \right\|_{L_x^\infty }} \le& {s^{ - 1}}M + \left( {\mathop {\sup }\limits_{s \in [0,1]} s{{\left\| {{\partial _s}\widetilde{u}} \right\|}_{L_x^\infty }}{{\left\| {d\widetilde{u}} \right\|}_{L_x^\infty }}} \right)\int_{\frac{s}{2}}^s {{\tau ^{ - 1}}\left\| {d\widetilde{u}} \right\|_{L_x^\infty }^2} d\tau\\ &+ \left( {\mathop {\sup }\limits_{s \in [0,1]} s{{\left\| {\nabla d\widetilde{u}} \right\|}_{L_x^\infty }}{{\left\| {d\widetilde{u}} \right\|}_{L_x^\infty }}} \right)\int_{\frac{s}{2}}^s {{\tau ^{ - 1}}{{\left\| {d\widetilde{u}} \right\|}_{L_x^\infty }}} d\tau\\ &\le {s^{ - 1}}M + {s^{ - 1}}M^2\int_{\frac{s}{2}}^s {\left( {\left\| {d\widetilde{u}} \right\|_{L_x^\infty }^2 + {{\left\| {d\widetilde{u}} \right\|}_{L_x^\infty }}} \right)} d\tau. \end{align*} Consequently, we have by Lemma \ref{ktao1}, \begin{align}\label{qian3} {\left\| {\left| {\nabla \partial_s \widetilde{u}} \right|(s)} \right\|_{L_x^\infty }} \le MC(M)s^{- 1}. \end{align} By Lemma \ref{8zu}, one deduces \begin{align*} {\partial _s}\left| {\nabla {\partial _t}\widetilde{u}} \right| - {\Delta}\left| {\nabla {\partial _t}\widetilde{u}} \right| &\le K\left| {\nabla {\partial _t}\widetilde{u}} \right|+\left| {\nabla {\partial _t}\widetilde{u}} \right||d\widetilde{u}|^2+|\partial_s\widetilde{u}||d\widetilde{u}|^2\\ &+ {\left| {d\widetilde{u}} \right|^3}\left| {{\partial _t}\widetilde{u}} \right| + \left| {d\widetilde{u}} \right|\left| {{\partial _t}\widetilde{u}} \right|\left| {\nabla d\widetilde{u}} \right|. \end{align*} Considering the equation of ${e^{ - \int_0^s \big({\left\| {d\widetilde{u}(\tau )} \right\|_{L_x^\infty }^2-K\big)d\tau } }}\left| {\nabla {\partial _t}\widetilde{u}} \right|$, we have by maximum principle that \begin{align*} \left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L^{\infty}_x} &\le {s^{ - \frac{1}{2}}}{\| {\nabla {\partial _t}\widetilde{u}} \|_{L_x^2}} +\int_{\frac{s}{2}}^s {{{(s - \tau )}^{ - \frac{1}{2}}}} \| {{\left| {d\widetilde{u}} \right|}^3}\left| {{\partial _t}\widetilde{u}} \right|(\tau )\|_{{L^2_x}}d\tau\\ &+ \int_{\frac{s}{2}}^s\||d\widetilde{u}|\left| {{\partial _t}\widetilde{u}} \right|\left| {\nabla d\widetilde{u}} \right|(\tau )\|_{L^{\infty}_x}+\||\partial_s\widetilde{u}||d\widetilde{u}|^2\|_{L^{\infty}_x}d\tau \\ &\le {s^{ - \frac{1}{2}}}{\left\| {\nabla \partial_t\widetilde{u}} \right\|_{L_x^2}} + \mathop {\sup }\limits_{s \in [0,1]} \left( {{{\left\| {{\partial _t}\widetilde{u}} \right\|}_{L_x^4}}\left\| {d\widetilde{u}} \right\|_{L_x^{12}}^3} \right)\int_{\frac{s}{2}}^s {{{(s - \tau )}^{ - \frac{1}{2}}}d\tau } \\ &+ \mathop {\sup }\limits_{s \in [0,1]} \left( {{s^{\frac{1}{2}}}{{\left\| {\nabla d\widetilde{u}} \right\|}_{L_x^\infty }}} \right)\int_{\frac{s}{2}}^s {{\tau ^{ - \frac{1}{2}}}{{\left\| {{\partial _t}\widetilde{u}} \right\|}_{L_x^\infty }}{{\left\| {d\widetilde{u}} \right\|}_{L_x^\infty }}d\tau }\\ &+ \mathop {\sup }\limits_{s \in [0,1]} \left( s\left\| d\widetilde{u}\right\|^2_{L_x^{\infty} } s^{\frac{1}{2}}\left\| \partial_s\widetilde{u} \right\|_{L_x^{\infty} } \right)\int_{\frac{s}{2}}^s \tau ^{ - \frac{3}{2}}d\tau. \end{align*} Hence we deduce by Lemma \ref{ktao1} \begin{align}\label{qian4} {\left\| {\left| {\nabla \partial_t u} \right|(s)} \right\|_{L_x^\infty }} \le MC(M){s^{ - \frac{1}{2}}}. \end{align} The bounds for $|\nabla_t\partial_s\widetilde{u}|$ follows by the same arguments as (\ref{qian1}) with help of Lemma \ref{zhangqiling} and (\ref{iconm}). \end{proof} We summarize the long time and short time behaviors as a proposition. \begin{proposition}\label{sl} Let $u:[0,T]\times\Bbb H^2\to \Bbb H^2$ be a solution to (\ref{wmap1}) satisfying \begin{align*} \|(\nabla du,\nabla\partial_t u)\|_{L^2\times L^2}+\|(du,\partial_t u)\|_{L^2\times L^2}\le M, \end{align*} If $\widetilde{u}:\Bbb R^+\times[0,T]\times\Bbb H^2\to \Bbb H^2$ is the solution to (\ref{8.29.2}) with initial data $u(t,x)$, then for any $\eta>0$, it holds uniformly for $t\in[0,T]$ that \begin{align*} &\left\| {d\widetilde{u}} \right\|_{L_s^\infty[1,\infty) L_x^{\infty}}+\left\| {\nabla d\widetilde{u}} \right\|_{L_s^\infty[1,\infty) L_x^{\infty}}+\left\| {\nabla d\widetilde{u}} \right\|_{L_s^\infty L_x^2} + {\left\| {{{\nabla\partial _t}\widetilde{u}}} \right\|_{L_s^\infty L_x^2 }}\\ &+{\| {{s^{\frac{1}{2}}}\left| {\nabla d\widetilde{u}} \right|}\|_{L_s^\infty[0,1] L_x^\infty }} + {\| {e^{\delta s}\left| {{\partial _s}\widetilde{u}} \right|}\|_{L_s^\infty L_x^2 }}+ {\left\| {{s}\left| {{\nabla_t}{\partial _s}\widetilde{u}} \right|} \right\|_{L^{\infty}_s[0,1]L_x^{\infty}}}\\ &+ {\| {{s^{\frac{1}{2}}}\left| {\nabla {\partial _t}\widetilde{u}} \right|} \|_{L_s^\infty[0,1] L_x^\infty }}+ {\left\| {s\left| {\nabla {\partial _s}\widetilde{u}} \right|} \right\|_{L_s^\infty[0,1] L_x^\infty }} + {\| {{s^{\frac{1}{2}}}e^{\delta s}\left| {{\partial _s}\widetilde{u}} \right|}\|_{L_s^\infty L_x^\infty }}\\ &+{\| {{s^{\frac{1}{2}}}\left| {\nabla {\partial _s}\widetilde{u}} \right|}\|_{L_s^\infty[0,1] L_x^2}} +{\| {{s^{\frac{1}{2}}}\left| {\nabla_t {\partial _s}\widetilde{u}} \right|}\|_{L_s^\infty[0,1] L_x^2}}+\left\|s^{\eta} {d\widetilde{u}} \right\|_{L_s^\infty(0,1) L_x^{\infty}}\\ &{\left\| {{s^{\frac{1}{2}}}{e^{\delta s}}\left| {\nabla {\partial _t}\widetilde{u}} \right|} \right\|_{L_s^\infty L_x^\infty }} + {\left\| {s{e^{\delta s}}\left| {\nabla {\partial _s}\widetilde{u}} \right|} \right\|_{L_s^\infty L_x^\infty }}+ {\left\| se^{\delta s}{\left| {{\nabla_t\partial _s}\widetilde{u}} \right|} \right\|_{L_s^\infty L_x^\infty }}\\ &{\left\| {{e^{\delta s}}\left| {\nabla {\partial _t}\widetilde{u}} \right|} \right\|_{L_s^\infty L_x^2 }} + {\left\| {s^{\frac{1}{2}}{e^{\delta s}}\left| {\nabla {\partial _s}\widetilde{u}} \right|}\right\|_{L_s^\infty L_x^2}}+ {\left\|s^{\frac{1}{2}}e^{\delta s} { {{\nabla_t\partial _s}\widetilde{u}}} \right\|_{L_s^\infty L_x^2 }}\le MC(M). \end{align*} \end{proposition} \begin{lemma}\label{fotuo1} If $(u,\partial_tu)$ solves (\ref{wmap1}) and $\|u(t,x)\|_{\mathcal{X}_T}\le M$, then we have \begin{align} \left\| {{\nabla}^2d\widetilde{u}} \right\|_{L^2_x}&\le \max(s^{-\frac{1}{2}},1)MC(M)\label{fotuo2}\\ \left\| {{\nabla}^2d\widetilde{u}} \right\|_{L^{\infty}_x}&\le \max(s^{-1},1)MC(M)\label{fotuo3}\\ se^{\delta' s}\|\nabla ^2\partial _s\widetilde{u}\|_{L^2_x}&\lesssim MC(M)\label{fotuo4}\\ s^{\frac{3}{2}}e^{\delta' s}\|\nabla ^2\partial _s\widetilde{u}\|_{L^\infty_x}&\lesssim MC(M)\label{fotuo5} \end{align} \end{lemma} \begin{proof} The Bochner formula for $\left| {{\nabla}^2d\widetilde{u}} \right|^2$ is as follows \begin{align} &{\partial _s}{| {{\nabla}^2d\widetilde{u}}|^2} - \Delta {| {{\nabla}^2d\widetilde{u}}|^2} + 2{| { {\nabla^3 }d\widetilde{u}}|^2} \lesssim |\nabla^2d\widetilde{u}|^2(|d\widetilde{u}|^2+1)+|\nabla d\widetilde{u}|^2|\nabla^2d\widetilde{u}||d\widetilde{u}|\nonumber\\ &+|d\widetilde{u}|^3|\nabla d\widetilde{u}||\nabla^2d\widetilde{u}|+|\nabla d\widetilde{u}||\nabla^2 d\widetilde{u}|^2.\label{piiguuuu7} \end{align} Interpolation by parts and $\tau(\widetilde{u})=\partial_s\widetilde{u}$ give \begin{align}\label{hupke3} \left\| {{\nabla}^2d\widetilde{u}} \right\|^2_{L^2_x}\lesssim \left\| \nabla\partial_s\widetilde{u} \right\|^2_{L^2_x}+\left\| \nabla d\widetilde{u} \right\|^3_{L^2_x} +\left\| \nabla d\widetilde{u} \right\|^2_{L^2_x}\|du\|^2_{L^{\infty}_x} \end{align} Then Proposition \ref{sl} yields (\ref{fotuo2}). (\ref{piiguuuu7}) shows $| {{\nabla}^2d\widetilde{u}}|$ satisfies \begin{align} &{\partial _s}{| {{\nabla}^2d\widetilde{u}}|} - \Delta {| {{\nabla}^2d\widetilde{u}}|}\lesssim |\nabla^2d\widetilde{u}|(|d\widetilde{u}|^2+1)+|\nabla d\widetilde{u}|^2|d\widetilde{u}|+|d\widetilde{u}|^3|\nabla d\widetilde{u}|+|\nabla d\widetilde{u}||\nabla^2 d\widetilde{u}|. \end{align} Let $f={| {{\nabla}^2d\widetilde{u}}|}e^{-\int^{s}_0(\|d\widetilde{u}\|^2_{L^{\infty}}+\|\nabla d\widetilde{u}\|_{L^{\infty}}+1)d\kappa}$. Then for $s\in[0,1]$, by Duhamel principle and smoothing effect, Lemma \ref{ktao1}, \begin{align} &\|f(s,x)\|_{L^{\infty}_x}\lesssim s^{-\frac{1}{2}}\|f(\frac{s}{2},x)\|_{L^{2}_x}+\int^s_{\frac{s}{2}}(s-\tau)^{-\frac{1}{2}}\||\nabla d\widetilde{u}|^2|d\widetilde{u}|+|d\widetilde{u}|^3|\nabla d\widetilde{u}|\|_{L^2_x}d\tau. \end{align} Then (\ref{fotuo3}) when $s\in[0,1]$ follows by Lemma \ref{ktao1} and Proposition \ref{sl}. (\ref{fotuo2}) gives $\| {{\nabla}^2d\widetilde{u}}\|_{L^2_x}\le MC(M)$ for all $s\ge1$. Meanwhile Proposition \ref{sl} shows $\|\nabla d\widetilde{u}\|_{L^{\infty}}+\| d\widetilde{u}\|_{L^{\infty}}\le MC(M)$ when $s\ge1$. Then if let $Z\triangleq e^{-C_1(M)s}(e^{-C_1(M) s}|\nabla^2 d\widetilde{u}|+C_1(M))$, then $(\partial_s-\Delta)Z\le0$. Applying Remark \ref{ki78} to $Z$ gives \begin{align} \|\nabla^2 d\widetilde{u}(s,x)\|^2_{L^{\infty}_x}\lesssim \int^{s}_{s-1}\|\nabla^2 d\widetilde{u}(\tau,x)\|^2_{L^{2}_x}d\tau+MC(M). \end{align} Then (\ref{fotuo3}) when $s\ge1$ follows by (\ref{hupke3}), (\ref{ingjh}) and Proposition \ref{sl}. The Bochner formula for $\left| {{\nabla}^2\partial_s{\widetilde{u}}} \right|^2$ is as follows \begin{align} &{\partial _s}{\left| {{\nabla}^2{\partial _s}\widetilde{u}} \right|^2} - \Delta {\left| {{\nabla}^2{\partial _s}\widetilde{u}} \right|^2} + 2{\left| { {\nabla^3 }{\partial _s}\widetilde{u}} \right|^2} \lesssim |\nabla^2\partial_s\widetilde{u}|^2(|d\widetilde{u}|^2+1)+|\partial_s\widetilde{u}|^2|\nabla^2\partial_s\widetilde{u}||\nabla d\widetilde{u}|\nonumber\\ &+|\partial_s\widetilde{u}||d\widetilde{u}||\nabla \partial_s\widetilde{u}||\nabla^2\partial_s\widetilde{u}|+|\nabla^2\partial_s\widetilde{u}|^2|d\widetilde{u}||\partial_s\widetilde{u}|+ |\nabla^2d\widetilde{u}||d\widetilde{u}||\partial_s\widetilde{u}||\nabla^2\partial_s\widetilde{u}|\nonumber\\ &+|\nabla\partial_s\widetilde{u}||\nabla^2\partial_s\widetilde{u}||d\widetilde{u}||\nabla d\widetilde{u}| +|\nabla\partial_s\widetilde{u}||\nabla^2\partial_s\widetilde{u}||d\widetilde{u}||\nabla d\widetilde{u}|+|\nabla d\widetilde{u}||\nabla^2 \partial_s\widetilde{u}|^2.\label{ftuo4} \end{align} Then one has \begin{align*} &\frac{d}{{ds}}\int_{{\Bbb H^2}} {{s^2}{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}{\rm{dvol_h}}} \\ &\le \int_{{\Bbb H^2}} 2{s{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}} - 2{s^2}{\left| {{\nabla ^3}{\partial _s}\widetilde{u}} \right|^2} + {s^2}{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|^2}|d\widetilde{u}{|^2}{\rm{dvol_h}} \\ &+ \int_{{\Bbb H^2}} {{s^2}} |{\nabla ^2}d\widetilde{u}||d\widetilde{u}||{\partial _s}\widetilde{u}||{\nabla ^2}{\partial _s}\widetilde{u}| + {s^2}|\nabla {\partial _s}\widetilde{u}||{\nabla ^2}{\partial _s}\widetilde{u}||d\widetilde{u}||\nabla d\widetilde{u}|{\rm{dvol_h}} \\ &+ \int_{{\Bbb H^2}} {{s^2}|{\partial _s}\widetilde{u}||d\widetilde{u}||\nabla {\partial _s}\widetilde{u}||{\nabla ^2}{\partial _s}\widetilde{u}|} + {s^2}|{\nabla ^2}{\partial _s}u{|^2}|d\widetilde{u}||{\partial _s}\widetilde{u}|{\rm{dvol_h}} \\ &+ \int_{{\Bbb H^2}} {{s^2}|{\partial _s}\widetilde{u}{|^2}|{\nabla ^2}{\partial _s}\widetilde{u}||\nabla d\widetilde{u}| + } {s^2}|{\nabla ^2}{\partial _s}\widetilde{u}|^2|\nabla d\widetilde{u}|{\rm{dvol_h}} \\ &+ \int_{{\Bbb H^2}} {{s^2}} |\nabla {\partial _s}\widetilde{u}||{\nabla ^2}{\partial _s}\widetilde{u}||d\widetilde{u}||\nabla d\widetilde{u}|{\rm{dvol_h}} +\int_{{\Bbb H^2}} {{s^2}{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}{\rm{dvol_h}}}. \end{align*} Integrating the above formula in $s\in[s_1,\tau]$ with any $0<s_1<\tau<2$, by Sobolev embedding, Gagliardo-Nirenberg and Young inequality, we obtain \begin{align*} &\int_{{\Bbb H^2}} {{\tau ^2}{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}(\tau ,t){\rm{dvol_h}}} - \int_{{\Bbb H^2}} {s_1^2{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}({s_1},t){\rm{dvol_h}}} \\ &\lesssim \int_{{s_1}}^\tau {\int_{{\Bbb H^2}} {s{{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|}^2}} } - {s^2}{\left| {{\nabla ^3}{\partial _s}\widetilde{u}} \right|^2} + \left\| {d\widetilde{u}} \right\|_{L_s^\infty L_x^4}^2{s^2}{\left| {{\nabla ^2}{\partial _s}\widetilde{u}} \right|^2}{\rm{dvol_h}}ds \\ &+ \int_{{s_1}}^\tau {\int_{{\Bbb H^2}} {{s^3}|d\widetilde{u}|^2|{\partial _s}\widetilde{u}|^2} } {\left| {{\nabla ^2}d\widetilde{u}} \right|^2} + {s^3}|\nabla {\partial _s}\widetilde{u}|^2|d\widetilde{u}|^2|\nabla d\widetilde{u}|^2{\rm{dvol_h}}ds \\ &+ \int_{{s_1}}^\tau {\int_{{\Bbb H^2}} {{s^3}|{\partial _s}\widetilde{u}|^2|d\widetilde{u}|^2|\nabla {\partial _s}\widetilde{u}|^2} } + {s^2}|d\widetilde{u}|^2|{\partial _s}\widetilde{u}|^2{\rm{dvol_h}}ds \\ &+ \int_{{s_1}}^\tau {\int_{{\Bbb H^2}} {{s^3}|\nabla d\widetilde{u}|^2|{\partial _s}\widetilde{u}|^2} } + {s^2}|\nabla d\widetilde{u}|^2{\rm{dvol_h}}ds \\ &+ \int_{{s_1}}^\tau {\int_{{\Bbb H^2}} {{s^3}} |\nabla {\partial _s}\widetilde{u}|^2|d\widetilde{u}|^2|\nabla d\widetilde{u}|^2{\rm{dvol_h}}} ds. \end{align*} Thus letting $s_1\to0$, for $\tau\in(0,2)$, we deduce from (\ref{ki6ll1}), (\ref{fotuo3}) and Proposition \ref{sl} that \begin{align*} \|s\nabla ^2\partial _s\widetilde{u}\|_{L^2_x}\lesssim MC(M), \end{align*} from which (\ref{fotuo4}) when $s\in(0,1)$ follows. Integrating (\ref{ftuo4}) with respect to $x$ in $\Bbb H^2$, one obtains by (\ref{{uv111}}) and Proposition \ref{sl} especially the $L^{\infty}_x$ bounds for $|d\widetilde{u}|+|\nabla d\widetilde{u}|$ that for $s\ge1$ and any $0<c\ll 1$ \begin{align} &\frac{d}{{ds}}\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}^2+c\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}^2\lesssim {\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^\infty }}{\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}}+\frac{1}{c}\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}^2\nonumber\\ &+ {\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}} + {\left\| {\nabla {\partial _s}\widetilde{u}} \right\|_{L_x^2}}{\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}}+{\left\| {{\nabla ^2}{\partial _s}\widetilde{u}} \right\|_{L_x^2}}\left\| {{\partial _s}\widetilde{u}} \right\|_{L_x^4}^2.\label{kioplmmnn} \end{align} Meanwhile integrating (\ref{hu897}) with respect to $s$ in $(s',\infty)$, we obtain from the exponential decay of $|\partial_s\widetilde{u}|+|\nabla\partial_s\widetilde{u}|$ in Proposition \ref{sl} that for $s'\ge1$ \begin{align}\label{oi98nhbgfg} \int^{\infty}_{s'}\|\nabla ^2\partial _s\widetilde{u}\|_{L^2_x}d\tau\lesssim e^{-\delta s'}MC(M). \end{align} Hence Gronwall inequality gives if choosing $0<c<\delta$ then for $s\ge1$ one has \begin{align} \left\| {{\nabla ^2}{\partial _s}\widetilde{u}} (s)\right\|_{L_x^2}^2&\le e^{-cs}MC(M)+e^{-cs}\int^{s}_{1}e^{c\tau}\|\nabla ^2\partial _s\widetilde{u}\|^2_{L^2_x}d\tau +e^{-cs}\int^{s}_{1}e^{c\tau}e^{-\delta \tau}d\tau\nonumber\\ &\lesssim MC(M).\label{po67gvg} \end{align} Applying Gronwall inequality to (\ref{kioplmmnn}) again in $(\frac{s}{2},s)$, we deduce from (\ref{po67gvg}) that \begin{align} \left\| {{\nabla ^2}{\partial _s}\widetilde{u}} (s)\right\|_{L_x^2}^2\le e^{-\frac{c}{2}s}MC(M)+e^{-cs}\int^{s}_{\frac{s}{2}}e^{c\tau}\|\nabla ^2\partial _s\widetilde{u}\|^2_{L^2_x}d\tau +e^{-cs}\int^{s}_{\frac{s}{2}}e^{c\tau}e^{-\delta \tau}d\tau. \end{align} Thus (\ref{fotuo4}) follows by (\ref{oi98nhbgfg}). Finally (\ref{fotuo5}) follows by (\ref{fotuo4}) and applying Remark \ref{ki78} to (\ref{ftuo4}) as before. \end{proof} \subsection{The existence of caloric gauge} As a preparation for the existence of the caloric gauge, we prove that the heat flows initiated from $u(t,x)$ with different $t$ converge to the same harmonic map as $u_0$. \begin{lemma}\label{lhu880} If $(u,\partial_tu)$ is a solution to (\ref{wmap1}) in $\mathcal{X}_T$, then there exists a harmonic map $\widetilde{Q}$ such that as $s\to\infty$, $$ \mathop {\lim }\limits_{s \to \infty } \mathop {\sup }\limits_{(x,t) \in {\mathbb{H}^2} \times [0,T]} dist_{\Bbb H^2}(\widetilde{u}(s,x,t),\widetilde{Q}(x))=0. $$ \end{lemma} \begin{proof} The global existence of $\widetilde{u}$ is due to Lemma \ref{8.44}, the embedding ${\bf{H}^2}\hookrightarrow L^{\infty}$, ${\bf{H}^1}\hookrightarrow L^{p}$ for $p\in[2,\infty)$ and diamagnetic inequality. Then (\ref{8.3}), maximum principle and (\ref{huhu899}) show \begin{align}\label{10.112} \left\| {{\partial _s}\widetilde{u}(s,t,x)} \right\|^2_{L^{\infty}_x} \le {s^{ - 1}}{e^{ - \frac{1}{4}s}}\int_{{\mathbb{H}^2}} {{{\left| {{\partial _s}\widetilde{u}(0,t,x)} \right|}^2}{\rm{dvol_h}}}. \end{align} Thus (\ref{8.29.2}) yields $$ \mathop {\sup }\limits_{(x,t) \in {\mathbb{H}^2}\times[0,T]} \left| {{\partial _s}\widetilde{u}(s,t,x)} \right| \le {s^{ -\frac{1}{2}}}{e^{ - \frac{1}{8}s}}\int_{{\mathbb{H}^2}} {{{\left| {{\partial _t}u(t,x)} \right|}^2}{\rm{dvol_h}}}\le C{s^{ - 1}}{e^{ - \frac{1}{8}s}}. $$ Therefore for any $1<s_0<s_1<\infty$ it holds $${d_{{\mathbb{H}^2}}}(\widetilde{u}({s_0},t,x),\widetilde{u}({s_1},t,x)) \lesssim \int_{{s_0}}^{{s_1}} {{e^{ - \frac{1}{8}s}}ds},$$ which implies $\widetilde{u}(s,t,x)$ converges to some map $\widetilde{Q}(t,x)$ uniformly on $(t,x)\in[0,T]\times \mathbb{H}^2$. By [Theorem 5.2,\cite{LT}], for any fixed $t$, $\widetilde{Q}(t,x)$ is a harmonic map form $\Bbb H^2\to\Bbb H^2$. It suffices to verify $\widetilde{Q}(t,x)$ is indeed independent of $t$. By (\ref{10.127}), maximum principle and (\ref{huhu899}), \begin{align}\label{sdf} \mathop {\sup }\limits_{x \in {\mathbb{H}^2}} {\left| {{\partial _t}\widetilde{u}(s,t,x)} \right|^2} \le {s^{ - 1}}{e^{ - \frac{1}{4}s}}\int_{{\Bbb H^2}} {{{\left| {{\partial _t}\widetilde{u}(0,t,x)} \right|}^2}{\rm{dvol_h}}}. \end{align} As a consequence, for $0\le t_1<t_2\le T$ one has $${d_{{\Bbb H^2}}}(\widetilde{u}(s,{t_1},x),\widetilde{u}(s,{t_2},x)) \le \int_{{t_1}}^{{t_2}} {\left| {{\partial _t}\widetilde{u}(s,t,x)} \right|} dt \le C{s^{ - \frac{1}{2}}}{e^{ - s/8}}({t_2} - {t_1}). $$ Let $s\to\infty$, we get ${d_{{\mathbb{H}^2}}}(\widetilde{Q}({t_1},x),\widetilde{Q}({t_2},x)) = 0$, thus finishing the proof. \end{proof} \begin{lemma}\label{z8vcxzvb} Let $Q$ be an admissible harmonic map in Definition 1.1, and $\mu_1,\mu_2$ be sufficiently small. If $(u,\partial_tu)$ is a solution to (\ref{wmap1}) in $\mathcal{X}_T$, then $\widetilde{u}(s,t,x)$ uniformly converges to $Q$ as $s\to\infty$. \end{lemma} \begin{proof} By Lemma \ref{lhu880}, it suffices to prove $Q=\widetilde{Q}$. In the coordinate (\ref{vg}), the harmonic map equation can be written as \begin{align} \Delta {\widetilde{Q}^l} + {h^{ij}}\overline \Gamma _{pq}^l\frac{{\partial {\widetilde{Q}^p}}}{{\partial {x_i}}}\frac{{\partial {\widetilde{Q}^q}}}{{\partial {x_j}}} &= 0\label{cxvgn} \\ \Delta {Q^l} + {h^{ij}}\overline \Gamma _{pq}^l\frac{{\partial {Q^p}}}{{\partial {x_i}}}\frac{{\partial {Q^q}}}{{\partial {x_j}}} &= 0. \label{cxvgn2} \end{align} Denote the heat flow initiated from $u_0$ by $U(s,x)$, then by (\ref{{uv111}}), \begin{align*} &\|U^1(s,x)-Q^1(x)\|_{L^2}+\|U^2(s,x)-Q^2(x)\|_{L^2}\\ &\lesssim \|\nabla(U^1-Q^1)\|_{L^2} +\|\nabla(U^2-Q^2)\|_{L^2}. \end{align*} [Lemma 2.3,\cite{LZ}] shows that for $k=1,2,l=1,2,$ \begin{align*} \|\nabla^l U^k\|_{L^2}\lesssim C(\|U\|_{\mathfrak{H}^2},R_0,\|Q\|_{\mathfrak{H}^2})\|\nabla^{l-1}dU\|_{L^2}. \end{align*} By energy arguments, one obtains $\|\nabla dU\|_{L^2}\le C(\|\nabla du_0\|_{L^2},\|du_0\|_{L^2})$ and the energy decreases along the heat flow, see (\ref{V5}). Thus we have by Sobolev embedding and Corollary \ref{new2} that \begin{align} &\|U^1(s,x)-Q^1(x)\|_{L^2}+\|U^2(s,x)-Q^2(x)\|_{L^2}+ \|dU\|_{L^2}\nonumber\\ &\le C(R_0)\mu_2+C(R_0)\mu_1\label{hbvcjin}\\ &\|U^1(s,x)\|_{L^{\infty}}+\|U^2(s,x)\|_{L^{\infty}}\le C(R_0).\label{pokeryu} \end{align} Hence letting $s\to\infty$, we have for some constant $C(R_0)$ \begin{align} \|\widetilde{Q}^1\|_{L^{\infty}}+\|\widetilde{Q}^2\|_{L^{\infty}}\le C,\mbox{ }\|\nabla\widetilde{Q}^1\|_{L^{2}}+\|\nabla\widetilde{Q}^2\|_{L^{2}}\le \mu_1C(R_0)\label{ktu8n2} \end{align} Multiplying the difference between (\ref{cxvgn}) and (\ref{cxvgn2}) with $-{Q^l} + {\widetilde{Q}^l}$, we have by integration by parts that \begin{align*} &{\left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}} \le \left\langle {{h^{ij}}\left( {\overline \Gamma _{pq}^l(Q) - \overline \Gamma _{pq}^l(\widetilde Q)} \right)\frac{{\partial {Q^p}}}{{\partial {x_i}}}\frac{{\partial {Q^q}}}{{\partial {x_j}}}, - {Q^l} + {{\widetilde Q}^l}} \right\rangle \\ &+ \left\langle {h^{ij}}\overline \Gamma _{pq}^l(\widetilde Q)\left( \frac{\partial {Q^p}}{\partial {x_i}} - \frac{\partial {\widetilde Q}^p}{\partial {x_i}} \right)\frac{{\partial {Q^q}}}{{\partial {x_j}}}, - {Q^l} + {\widetilde{Q}^l}\right\rangle \\ &+ \left\langle {{h^{ij}}\overline \Gamma _{pq}^l(\widetilde{Q})\frac{{\partial {\widetilde{Q}^p}}}{{\partial {x_i}}}\left( {\frac{{\partial {Q^q}}}{{\partial {x_i}}} - \frac{{\partial {\widetilde{Q}^q}}}{{\partial {x_j}}}} \right), - {Q^l} + {\widetilde{Q}^l}} \right\rangle. \end{align*} Thus using the explicit formula for ${\overline \Gamma _{pq}^l}$, by (\ref{hbvcjin}), (\ref{pokeryu}), (\ref{ktu8n2}) we get \begin{align*} &\left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}^2\\ &\lesssim \left( {\left\| {{Q^l} - {\widetilde{Q}^l}} \right\|_{{L^2}}^2 + \left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}^2} \right)\left( {\sum\limits_{k = 1}^2 {\left\| {\nabla {\widetilde{Q}^k}} \right\|_{{L^2}}^2 + \left\| {\nabla {Q^k}} \right\|_{{L^2}}^2} } \right). \end{align*} Therefore, we conclude for some constant $C(R_0)$ which is independent of $\mu_1,\mu_2$ provided $0\le \mu_1,\mu_2\le1$ \begin{align*} &\sum\limits_{l = 1}^2 {\left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}^2}\\ &\le C(R_0)\left( {\left\| {dQ} \right\|_{{L^2}}^2 + \left\| {d\widetilde{Q}} \right\|_{{L^2}}^2} \right)\left( {\sum\limits_{l = 1}^2 {\left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}^2 + \left\| {{Q^l} - {\widetilde{Q}^l}} \right\|_{{L^2}}^2} } \right). \end{align*} Let $\mu_1$, $\mu_2$ be sufficiently small, (\ref{{uv111}}) gives \begin{align*} &\sum\limits_{l = 1}^2 {\left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}^2} + \left\| {{Q^l} - {\widetilde{Q}^l}} \right\|_{{L^2}}^2\\ &\le \left( {{\mu _1} + {\mu _2}} \right)\left( {\sum\limits_{l = 1}^2 {\left\| {\nabla \left( {{Q^l} - {\widetilde{Q}^l}} \right)} \right\|_{{L^2}}^2 + \left\| {{Q^l} - {\widetilde{Q}^l}} \right\|_{{L^2}}^2} } \right). \end{align*} Hence $\widetilde{Q}=Q$. \end{proof} Now we are ready to prove the existence of the caloric gauge in Definition \ref{pp}. \begin{proposition}\label{3.3} Given any solution $(u,\partial_tu)$ of (\ref{wmap1}) in $\mathcal{X}_T$ with $(u_0,u_1)\in \bf H_Q^3\times\bf H_Q^2$. For any fixed frame $\Xi\triangleq\{\Xi_1(Q(x)),\Xi_2(Q(x))\}$, there exists a unique corresponding caloric gauge defined in Definition \ref{pp}. \end{proposition} \begin{proof} We first show the existence part. Choose an arbitrary orthonormal frame $E_0(t,x)\triangleq\{\texttt{e}_i(t,x)\}^2_{i=1}$ such that $E_0(t,x)$ spans the tangent space $T_{u(t,x)}{\mathbb{H}^2}$ for each $(t,x)\in [0,T]\times \mathbb{H}^2$. The desired frame does exist, in fact we have a global orthonormal frame for $\mathbb{H}^2$ defined by (\ref{frame}). Then evolving (\ref{8.29.2}) with initial data $u(t,x)$, we have from Lemma \ref{z8vcxzvb} that $\widetilde{u}(s,t,x)$ converges to $Q$ uniformly for $(t,x)\in[0,T]\times \mathbb{H}^2$ as $s\to\infty$. Meanwhile, we evolve $E_0$ in $s$ according to \begin{align}\label{11.2} \left\{ \begin{array}{l} {\nabla _s}{\Omega _i}(s,t,x) = 0 \\ {\Omega _i}(s,t,x)\upharpoonright_{s=0} = {\texttt{e}_i}(t,x) \\ \end{array} \right. \end{align} Denote the evolved frame as $E_s\triangleq \{\Omega_i(s,t,x)\}^2_{i=1}$. We claim that there exists some orthonormal frame $E_{\infty}\triangleq\{\texttt{e}_i(\infty,t,x)\}^2_{i=1}$ which spans $T_{Q(x)}\Bbb H^2$ for each $(t,x)\in [0,T]\times\Bbb H^2$ such that \begin{align}\label{pl} \mathop {\lim }\limits_{s \to \infty }{\Omega_i(s,t,x)} = \texttt{e}_i(\infty,t,x). \end{align} Indeed, by the definition of the convergence of frames given in (\ref{convergence}) and the fact $\widetilde{u}(s,t,x)$ converges to $Q(x)$, it suffices to show for some scalar function $c_i:[0,T]\times \mathbb{H}^2\to \Bbb R$ \begin{align}\label{aw} \mathop {\lim }\limits_{s \to \infty } \left\langle {{\Omega_i}(s,t,x),{\Theta _i}(\widetilde{u}(s,t,x))} \right\rangle = c_i(t,x). \end{align} By direct calculations, $$ \left| \nabla _s \Theta_i(\widetilde{u}(s,t,x))\right| \lesssim \left| {{\partial _s}\widetilde{u}} \right|. $$ then (\ref{10.112}) and $\nabla_s\Omega=0$ imply that for $s>1$ $$\big|{\partial _s}\left\langle {{\Omega_i}(s,t,x),{\Theta _i}(\widetilde{u}(s,t,x))} \right\rangle \big|\lesssim Me^{-\delta s}. $$ Hence $(\ref{aw})$ holds for some $c_i(t,x)$, thus verifying (\ref{pl}). It remains to adjust the initial frame $E_0$ to make the limit frame $E_{\infty}$ coincide with the given frame $\Xi$. This can be achieved by the gauge transform invariance illustrated in Section 2.1. Indeed, since for any $U:[0,T]\times\mathbb{H}^2\to SO(2)$, and the solution $\widetilde{u}(s,t,x)$ to (\ref{8.29.2}), one has $\nabla_s U(t,x)\Omega(s,t,x)=U(t,x)\nabla_s\Omega(s,t,x)$, then the following gauge symmetry holds \begin{align*} {E_0}\triangleq\left\{ {\texttt{e}_i(t,x)} \right\}^2_{i=1} &\mapsto {{E'}_0}\triangleq\left\{ {U(t,x){\texttt{e}_i}(t,x)} \right\}^2_{i=1} \\ {E_s}\triangleq\left\{ {{\Omega_i}(s,t,x)} \right\}^2_{i=1} &\mapsto {{E'}_s}\triangleq\left\{ {U(t,x){\Omega_i}(s,t,x)} \right\}^2_{i=1}. \end{align*} Therefore choosing $U(t,x)$ such that $U(t,x)E_{\infty}=\Xi$, where $E_{\infty}$ is the limit frame obtained by (\ref{pl}), suffices for our purpose. The uniqueness of the gauge follows from the identity $$ \frac{d}{{ds}}\left\langle {{\Phi _1} - {\Phi_2},{\Phi_1} - {\Phi_2}} \right\rangle = 0, $$ where $(\Phi _1)$ and $(\Phi _2)$ are two caloric gauges satisfying (\ref{muqi}). \end{proof} \subsection{Expressions for the connection coefficients} The following lemma gives the expressions for the connection coefficients matrix $A_{x,t}$ by differential fields. The proof of Lemma \ref{po87bg} is almost the same as [Lemma 3.6,\cite{LZ}], thus we omit it. \begin{lemma}\label{po87bg} Suppose that $\Omega(s,t,x)$ is the caloric gauge constructed in Proposition \ref{3.3}, then we have for $i=1,2$ \begin{align} &\mathop {\lim }\limits_{s \to \infty } [{A_i}]^j_k(s,t,x) =\left\langle {{\nabla _i}{\Xi_k }(x),{\Xi_j}(x)} \right\rangle\label{kji}\\ &\mathop {\lim }\limits_{s \to \infty } {A_t}(s,t,x) = 0\label{kji22} \end{align} Particularly let $\Xi(x)=\Theta(Q(x))$ in Proposition \ref{3.3}, denote $A^{\infty}_i$ the limit coefficient matrix, i.e., $[A^{\infty}_i]^k_j=\left\langle {{\nabla _i}{\Xi_k }(Q(x)),{\Xi_j}(Q(x))} \right\rangle$, then we have for $i=1,2$, $s>0$, \begin{align} &{A_i}(s,t,x)\sqrt{h^{ii}(x)} = \int_s^\infty \sqrt{ h^{ii}(x)}{\mathbf{R}(\widetilde{u}(\kappa))\left( {{\partial _s}\widetilde{u}(\kappa),{\partial _i}\widetilde{u}(\kappa)} \right)} d\kappa + { \sqrt{h^{ii}(x)}}A^{\infty}_i.\label{edf}\\ &{A_t}(s,t,x)=\int^{\infty}_s\phi_s\wedge\phi_td\kappa,\label{edf22} \end{align} \end{lemma} \begin{remark}\label{3sect} For convenience, we rewrite (\ref{edf}) as $A_i(s,t,x)=A^{\infty}_i(s,t,x)+A^{con}_i(s,t,x),$ where $A^{\infty}_i$ denotes the limit part, and $A^{con}_i$ denotes the controllable part, i.e., \begin{align*} A^{con}_i=\int^{\infty}_s\phi_s\wedge\phi_id\kappa. \end{align*} Similarly, we split $\phi_i$ into $\phi_i=\phi^{\infty}_i+\phi^{con}_i$, where $\phi^{con}_i=\int^{\infty}_s\partial_s\phi_id\kappa,$ and $$\phi^{\infty}_i={\left( {\left\langle {{\partial _i}Q(x),{\Xi _1}(Q(x))} \right\rangle ,\left\langle {{\partial _i}Q(x),{\Xi _2}(Q(x))} \right\rangle } \right)^t}.$$ \end{remark} \section{Derivation of the master equation for the heat tension field} Recall that the heat tension filed $\phi_s$ satisfies \begin{align}\label{heat} \phi_s=h^{ij}D_i\phi_j-h^{ij}\Gamma^k_{ij}\phi_k. \end{align} And we define the wave tension filed as Tao by \begin{align}\label{wm} \mathfrak{W} = {D_t}{\phi _t} - {h^{ij}}{D_i}{\phi _j} + {h^{ij}}\Gamma _{ij}^k{\phi _k}. \end{align} In fact (\ref{heat}) is the gauged equation for the heat flow equation, and (\ref{wm}) is the gauged equation for the wave map (\ref{wmap1}), see Lemma 2.7. The evolution of $\phi_s$ with respect to $t$ is given by the following lemma. \begin{lemma}\label{asdf} The heat tension field $\phi_s$ satisfies \begin{align} {D_t}{D_t}{\phi _s} - {h^{ij}}{D_i}{D_j}{\phi _s} + {h^{ij}}\Gamma _{ij}^k{D_k}{\phi _s} &= {\partial _s}\mathfrak{W} + {h^{ij}}\mathbf{R}({\partial _s}\widetilde{u},{\partial _i}\widetilde{u})\left( {\partial_j\widetilde{u}} \right) \nonumber\\ &+ \mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})\left( {\partial_t\widetilde{u}} \right).\label{heating} \end{align} \end{lemma} \begin{proof} By the torsion free identity and the commutator identity, we have \begin{align*} &{D_t}{D_t}{\phi _s} = {D_t}{D_s}{\phi _t} = {D_s}{D_t}{\phi _t} + \mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})\left( { \partial_t\widetilde{u}} \right) \\ &= {D_s}\left( {\mathfrak{W} + {h^{ij}}{D_i}{\phi _j} - {h^{ij}}\Gamma _{ij}^k{\phi _k}} \right) +\mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})\left( {\partial_t\widetilde{u}} \right) \\ &= {\partial _s}\mathfrak{W} + {h^{ij}}{D_s}{D_i}{\phi _j} - {h^{ij}}\Gamma _{ij}^k{D_s}{\phi _k} + \mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})\left( {\partial_t\widetilde{u}} \right) \\ &= {\partial _s}\mathfrak{W} + {h^{ij}}{D_i}{D_j}{\phi _s} - {h^{ij}}\Gamma _{ij}^k{D_k}{\phi _s} + {h^{ij}}\mathbf{R}({\partial _s}\widetilde{u},{\partial _i}\widetilde{u})\left( {{\partial_j\widetilde{u}}} \right) + \mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})\left( {\partial_t\widetilde{u}} \right). \end{align*} Thus (\ref{heating}) is verified. \end{proof} The evolution of $\mathfrak{W}$ with respect to $s$ is given by the following lemma. \begin{lemma}\label{ab1} Under orthogonal coordinates, the wave tension field $\mathfrak{W}$ satisfies \begin{align*} {\partial _s}\mathfrak{W} =& \Delta \mathfrak{W} + 2h^{ii}{A_i}{\partial _i}\mathfrak{W} +h^{ii} {A_i}{A_i}\mathfrak{W} + h^{ii}{\partial _i}{A_i}\mathfrak{W} - {h^{ii}}\Gamma _{ii}^k{A_k}\mathfrak{W} + {h^{ii}}\left( {\mathfrak{W} \wedge {\phi _i}} \right){\phi _i}\\ & + 3{h^{ii}}({\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\nabla _t}{\partial _i}\widetilde{u}. \end{align*} \end{lemma} \begin{proof} In the following calculations, we always use the convention in Remark 2.1. By $\mathfrak{W}=D_t\phi_t-\phi_s$, we have from commutator equality that $${\partial _s}\mathfrak{W}= {D_s}({D_t}{\phi _t} - {\phi _s}) = {D_t}{D_t}{\phi _s} - {D_s}{\phi _s} + {\bf R}({\partial _s}\widetilde{u},{\partial _t}\widetilde{u})\left( {{\partial_t\widetilde{u}}} \right). $$ Further applications of the torsion free identity and commutator identity show \begin{align*} &{D_t}{D_t}{\phi _s} - {D_s}{\phi _s} \\ &= {D_t}{D_t}\left( {{h^{ij}}{D_i}{\phi _j} - {h^{ij}}\Gamma _{ij}^k{\phi _k}} \right) - {D_s}\left( {{h^{ij}}{D_i}{\phi _j} - {h^{ij}}\Gamma _{ij}^k{\phi _k}} \right) \\ &= {h^{ij}}{D_t}{D_t}{D_i}{\phi _j} - {h^{ij}}\Gamma _{ij}^k{D_t}{D_t}{\phi _k} - \left( {{h^{ij}}{D_s}{D_i}{\phi _j} - {h^{ij}}\Gamma _{ij}^k{D_s}{\phi _k}} \right) \\ &= {h^{ij}}{D_t}\left( {{D_i}{D_j}{\phi _t} + \mathbf{R}({\partial _t}\widetilde{u},{\partial _i}\widetilde{u})({\partial _j}\widetilde{u})} \right) - {h^{ij}}\Gamma _{ij}^k\left( {{D_k}{D_t}{\phi _t} + \mathbf{R}({\partial _t}\widetilde{u},{\partial _k}\widetilde{u})({\partial _t}\widetilde{u})} \right) \\ &- \left( {{h^{ij}}{D_i}{D_j}{\phi _s} - {h^{ij}}\Gamma _{ij}^k{D_k}{\phi _s} + {h^{ij}}\mathbf{R}({\partial _s}\widetilde{u},{\partial _i}\widetilde{u})({\partial _j}\widetilde{u})} \right) \\ &= {h^{ij}}{D_t}{D_i}{D_j}{\phi _t} - {h^{ij}}\Gamma _{ij}^k{D_k}{D_t}{\phi _t} - {h^{ij}}{D_i}{D_j}{\phi _s} + {h^{ij}}\Gamma _{ij}^k{D_k}{\phi _s} - {h^{ij}}\mathbf{R}({\partial _s}\widetilde{u},{\partial _i}\widetilde{u})(\partial_j\widetilde{u}) \\ &+ {h^{ij}}{\nabla _t}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _i}\widetilde{u})({\partial _j}\widetilde{u})} \right) - {h^{ij}}\Gamma _{ij}^k\mathbf{R}({\partial _t}\widetilde{u},{\partial _k}\widetilde{u})\left( {{\partial _t}\widetilde{u}} \right). \end{align*} The leading term can be written as \begin{align*} {h^{ij}}{D_t}{D_i}{D_j}{\phi _t}& = {h^{ij}}{D_i}{D_t}{D_j}{\phi _t} + {h^{ij}}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _i}\widetilde{u})e\left( {{D_j}{\phi _t}} \right)} \right) \\ &= {h^{ij}}{D_i}{D_j}{D_t}{\phi _t} + {h^{ij}}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _i}\widetilde{u}){\nabla _j}{\partial _t}\widetilde{u} } \right) + {h^{ij}}{\nabla _i}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _j}\widetilde{u}){\partial _t}\widetilde{u}} \right). \end{align*} Thus we conclude as \begin{align*} {\partial _s}\mathfrak{W} &={h^{ij}}{D_i}{D_j}({D_t}{\phi _t} - {\phi _s}) - {h^{ij}}\Gamma _{ij}^k{D_k}({D_t}{\phi _t} - {\phi _s}) + {h^{ij}}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _i}\widetilde{u}){\nabla _j}{\partial _t}\widetilde{u}} \right) \\ &+ {h^{ij}}{\nabla _i}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _j}\widetilde{u}){\partial _t}\widetilde{u}} \right) - {h^{ij}}\mathbf{R}({\partial _s}\widetilde{u},{\partial _i}\widetilde{u})(\partial_j\widetilde{u}) + {h^{ij}}{\nabla _t}\left( {\mathbf{R}({\partial _t}\widetilde{u},{\partial _i}\widetilde{u})({\partial _j}\widetilde{u})} \right) \\ &- {h^{ij}}\Gamma _{ij}^k\mathbf{R}({\partial _t}\widetilde{u},{\partial _k}\widetilde{u})\left( {{\partial _t}\widetilde{u}} \right) + \mathbf{R}({\partial _s}\widetilde{u},{\partial _t}\widetilde{u}){\partial _t}\widetilde{u}. \end{align*} Using $\mathfrak{W}=D_t\phi_t-\phi_s$ and (\ref{2.4best}) yields \begin{align} {\partial _s}\mathfrak{W}& = \Delta \mathfrak{W} + 2h^{ii}{A_i}{\partial _i}\mathfrak{W}+ h^{ii}{A_i}{A_i}\mathfrak{W} + h^{ii}{\partial _i}{A_i}\mathfrak{W} - {h^{ij}}\Gamma _{ij}^k{A_k}\mathfrak{W}\nonumber\\ &+ \left\{ { - {h^{ii}}\left( {{\partial _s}\widetilde{u} \wedge {\partial _i}\widetilde{u}} \right){\partial _i}\widetilde{u} + {h^{ii}}({\nabla _t}{\partial _t}\widetilde{u } \wedge {\partial _i}\widetilde{u}){\partial _i}\widetilde{u}} \right\}\nonumber \\ &+ {h^{ii}}({\partial _t}\widetilde{u} \wedge {\nabla _t}{\partial _i}\widetilde{u}){\partial _i}\widetilde{u} + {h^{ii}}({\partial _t}\widetilde{u }\wedge {\partial _i}\widetilde{u}){\nabla _t}{\partial _i}\widetilde{u}\nonumber \\ &+ {h^{ii}}({\nabla _i}{\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\partial _t}\widetilde{u} + {h^{ii}}({\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\nabla _i}{\partial _t}\widetilde{u}\nonumber\\ &+ \left\{ {{h^{ii}}({\partial _t}\widetilde{u} \wedge {\nabla _i}{\partial _i}\widetilde{u}){\partial _t}\widetilde{u} - {h^{ii}}\Gamma _{ii}^k({\partial _t}\widetilde{u} \wedge {\partial _k}\widetilde{u}){\partial _t}\widetilde{u} + ({\partial _s}\widetilde{u} \wedge {\partial _t}\widetilde{u}){\partial _t}\widetilde{u}} \right\}.\label{wanxiao4} \end{align} Recalling the facts that $\mathfrak{W}$ is the gauged field for $\nabla_t\partial_t \widetilde{u}-\tau(\widetilde{u})$ and $\partial_s\widetilde{u}=\tau(\widetilde{u})$, we have \begin{align} - {h^{ii}}\left( {{\partial _s}\widetilde{u} \wedge {\partial _i}\widetilde{u}} \right){\partial _i}\widetilde{u} + {h^{ii}}({\nabla _t}{\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\partial _i}\widetilde{u} &= {h^{ii}}\left( {({\nabla _t}{\partial _t}\widetilde{u} - {\partial _s}\widetilde{u}) \wedge {\partial _i}\widetilde{u}} \right){\partial _i}\widetilde{u}\nonumber\\ &= {h^{ii}}\left( {\mathfrak{W} \wedge {\phi _i}} \right){\phi _i}.\label{wanxiao1} \end{align} Meanwhile, $\partial_s\widetilde{u}=\tau(\widetilde{u})$ also implies \begin{align} &{h^{ii}}({\partial _t}\widetilde{u} \wedge {\nabla _i}{\partial _i}\widetilde{u}){\partial _t}\widetilde{u} - {h^{ii}}\Gamma _{ii}^k({\partial _t}\widetilde{u} \wedge {\partial _k}\widetilde{u}){\partial _t}\widetilde{u} + ({\partial _s}\widetilde{u} \wedge {\partial _t}\widetilde{u}){\partial _t}\widetilde{u }\nonumber\\ &= {h^{ii}}\left( {{\partial _t}\widetilde{u} \wedge \left( {\tau (\widetilde{u}) - {\partial _s}\widetilde{u}} \right)} \right){\partial _t}\widetilde{u} = 0.\label{wanxiao2} \end{align} Bianchi identity gives \begin{align}\label{wanxiao3} {h^{ii}}({\partial _t}\widetilde{u} \wedge {\nabla _t}{\partial _i}\widetilde{u}){\partial _i}\widetilde{u} + {h^{ii}}({\nabla _i}{\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\partial _t}\widetilde{u} &= - {h^{ii}}({\partial _i}\widetilde{u }\wedge {\partial _t}\widetilde{u}){\nabla _t}{\partial _i}\widetilde{u}\nonumber\\ &= {h^{ii}}({\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\nabla _t}{\partial _i}\widetilde{u}. \end{align} By (\ref{wanxiao1}), (\ref{wanxiao2}) and (\ref{wanxiao3}), (\ref{wanxiao4}) can be further simplified as \begin{align*} {\partial _s}\mathfrak{W} =& \Delta \mathfrak{W} + 2h^{ii}{A_i}{\partial _i}\mathfrak{W} + h^{ii}{A_i}{A_i}\mathfrak{W} +h^{ii} {\partial _i}{A_i}\mathfrak{W} - {h^{ii}}\Gamma _{ii}^k{A_k}\mathfrak{W}\\ & + {h^{ii}}\left( {\mathfrak{W} \wedge {\phi _i}} \right){\phi _i}+ 3{h^{ii}}({\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}){\nabla _t}{\partial _i}\widetilde{u}. \end{align*} \end{proof} \begin{lemma}\label{xuejin} Let $Q$ be an admissible harmonic map in Definition 1.1. Fix the frame $\Xi$ in Remark \ref{3sect} by taking $\Xi(Q(x))=\Theta(Q(x))$ given by (\ref{vg}). Recall the definitions of $A^{\infty}_i$ in Lemma \ref{po87bg}. Then \begin{align} &|A_i^{\infty}|\lesssim|dQ|, |\sqrt{h^{ii}}\phi_i^{\infty}|\lesssim |dQ|\label{kulun1}\\ &|{h^{ii}}\left( {{\partial _i}{A^{\infty}_i} - \Gamma _{ii}^k{A^{\infty}_k}} \right)|\lesssim |dQ|^2.\label{kulun} \end{align} \end{lemma} \begin{proof} Recall the definition $$[A_i^\infty ]_k^j = \left\langle {{\nabla _i}{\Theta _k},{\Theta _j}} \right\rangle ,{\Theta _1} = {e^{{Q^2}(x)}}\frac{\partial }{{\partial {y_1}}},{\Theta _2} = \frac{\partial }{{\partial {y_2}}}. $$ Since $A_i$ is skew-symmetric, it suffices to consider the $[A_i]^1_{2}$ terms. Direct calculation gives \begin{align*} [A_1^\infty ]_2^1 = \left\langle {{\nabla _1}{\Theta _2},{\Theta _1}} \right\rangle& = {e^{{Q^2}(x)}}\frac{{\partial {Q^{_k}}}}{{\partial {x_1}}}\left\langle {{\nabla _{\frac{\partial }{{\partial {y_k}}}}}\frac{\partial }{{\partial {y_2}}},\frac{\partial }{{\partial {y_1}}}} \right\rangle = {e^{ - {Q^2}(x)}}\frac{{\partial {Q^{_k}}}}{{\partial {x_1}}}\overline{\Gamma}_{k2}^1 \\ &= - {e^{ - {Q^2}(x)}}\frac{{\partial {Q^1}}}{{\partial {x_1}}}, \end{align*} and similarly we obtain \begin{align*} [A_1^\infty ]_1^2 ={e^{ - {Q^2}(x)}}\frac{{\partial {Q^1}}}{{\partial {x_1}}};\mbox{ } [A_2^\infty ]_1^2 = - [A_2^\infty ]_2^1 = {e^{ - {Q^2}(x)}}\frac{{\partial {Q^1}}}{{\partial {x_2}}}. \end{align*} Thus one has \begin{align} &{h^{ii}}\left( {{\partial _i}[A_i^\infty ]_2^1 - \Gamma _{ii}^k[A_k^\infty ]_2^1} \right)\nonumber \\ &= - \left( {\frac{{{\partial ^2}{Q^1}}}{{\partial {x_2}^2}}{e^{2{x_2}}} + \frac{{{\partial ^2}{Q^1}}}{{\partial {x_1}^2}} - \frac{{\partial {Q^1}}}{{\partial {x_1}}}\frac{{\partial {Q^2}}}{{\partial {x_1}}}{e^{2{x_2}}} - \frac{{\partial {Q^1}}}{{\partial {x_2}}}\frac{{\partial {Q^2}}}{{\partial {x_2}}} - \frac{{\partial {Q^1}}}{{\partial {x_2}}}} \right){e^{ - {Q^2}(x)}} \label{chumen} \end{align} Writing the harmonic map equation for $Q$ in the coordinate (\ref{vg}) shows for $l=1,2$ $${h^{ii}}\frac{{{\partial ^2}{Q^l}}}{{\partial {x_i}^2}} - {h^{ii}}\Gamma _{ii}^k{\partial _k}{Q^l} + {h^{ii}}\bar \Gamma _{pq}^l\frac{{\partial {Q^p}}}{{\partial {x_i}}}\frac{{\partial {Q^q}}}{{\partial {x_i}}} = 0. $$ Let $l=1$ in the above equation, we have $${e^{2{x_2}}}\frac{{{\partial ^2}{Q^1}}}{{\partial {x_1}^2}} + \frac{{{\partial ^2}{Q^1}}}{{\partial {x_2}^2}} - \frac{{\partial {Q^1}}}{{\partial {x_2}}} - 2{e^{2{x_2}}}\frac{{\partial {Q^1}}}{{\partial {x_1}}}\frac{{\partial {Q^2}}}{{\partial {x_1}}} - 2\frac{{\partial {Q^1}}}{{\partial {x_2}}}\frac{{\partial {Q^2}}}{{\partial {x_2}}} = 0, $$ which combined with (\ref{chumen}) yields \begin{align}\label{chunmen2} {h^{ii}}\left( {{\partial _i}A_i^\infty - \Gamma _{ii}^kA_k^\infty } \right) = \left( {{e^{2{x_2}}}\frac{{\partial {Q^1}}}{{\partial {x_1}}}\frac{{\partial {Q^2}}}{{\partial {x_1}}} + \frac{{\partial {Q^1}}}{{\partial {x_2}}}\frac{{\partial {Q^2}}}{{\partial {x_2}}}} \right){e^{ - {Q^2}(x)}}. \end{align} Writing the energy density in coordinates (\ref{vg}), we obtain \begin{align*} {\left| {dQ} \right|^2} &= {h^{ij}}\left\langle {\frac{{\partial {Q^k}}}{{\partial {x_i}}}\frac{\partial }{{\partial {y_k}}},\frac{{\partial {Q^k}}}{{\partial {x_j}}}\frac{\partial }{{\partial {y_k}}}} \right\rangle \\ &= {e^{2{x_2}}}{\left| {\frac{{\partial {Q^1}}}{{\partial {x_1}}}} \right|^2}{e^{ - 2{Q_2}}} + {e^{2{x_2}}}{\left| {\frac{{\partial {Q^2}}}{{\partial {x_1}}}} \right|^2} + {\left| {\frac{{\partial {Q^1}}}{{\partial {x_2}}}} \right|^2}{e^{ - 2{Q_2}}} + {\left| {\frac{{\partial {Q^2}}}{{\partial {x_2}}}} \right|^2}. \end{align*} Thus (\ref{kulun}) follows by (\ref{chunmen2}) and Young inequality. (\ref{kulun1}) is much easier and follows immediately by the same arguments. \end{proof} Now we separate the main term in the equation of $\phi_s$. Recall the limit of $A_{s,t,x}$ given in (\ref{kji}), (\ref{kji22}), one can easily see the main term of (\ref{heating}) is a magnetic wave equation. Precisely, we have the following lemma. \begin{lemma}\label{hushuo} Fix the frame $\Xi$ in Proposition 3.3 by letting $\Xi_i(x)=\Theta_i(Q(x))$, $i=1,2$. Then the heat tension filed $\phi_s$ satisfies \begin{align*} &(\partial^2_t-\Delta){\phi _s} + W{\phi _s}\\ &= - 2{A_t}{\partial _t}{\phi _s} - {A_t}{A_t}{\phi _s} - {\partial _t}{A_t}{\phi _s} + {\partial _s}w + \mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})({\partial _t}\widetilde{u}) + 2{h^{ii}}A_i^{con}{\partial _i}{\phi _s} \\ &+ {h^{ii}}A_i^{con}A_i^\infty {\phi _s} + {h^{ii}}A_i^\infty A_i^{con}{\phi _s} + {h^{ii}}A_i^{con}A_i^{con}{\phi _s} + {h^{ii}}\left( {{\partial _i}A_i^{con} - \Gamma _{ii}^kA_k^{con}} \right){\phi _s} \\ &+ {h^{ii}}\left( {{\phi _s} \wedge \phi _i^\infty } \right)\phi _i^{con} + {h^{ii}}\left( {{\phi _s} \wedge \phi _i^{con}} \right)\phi _i^\infty + {h^{ii}}\left( {{\phi _s} \wedge \phi _i^{con}} \right)\phi _i^{con}, \end{align*} where $A_{x}^{\infty}$, $A_{x}^{con}$ are defined in Remark \ref{3sect}, and $W$ is given by \begin{align}\label{iuo9} W\varphi = -2 {h^{ii}}A_i^\infty {\partial _i}\varphi -{h^{ii}}A_i^\infty A_i^\infty \varphi - {h^{ii}}\left( {\varphi \wedge \phi _i^\infty } \right)\phi _i^\infty-h^{ii}(\partial_iA^{\infty}_{i}-\Gamma^k_{ii}A^{\infty}_k). \end{align} Furthermore, $-\Delta+W$ is a self-adjoint operator in $L^2(\Bbb H^2;\Bbb C^2)$. And it is strictly positive if $0<\mu_1\ll1$. \end{lemma} \begin{proof} By (\ref{heating}), expanding $D_{x,t}$ as $\partial_{t,x}+A_{t,x}$ implies \begin{align} &\partial _t^2{\phi _s} - {\Delta}{\phi _s}\nonumber\\ & = - 2{A_t}{\partial _t}{\phi _s} - {A_t}{A_t}{\phi _s} - {\partial _t}{A_t}{\phi _s} + {h^{ii}}{A_i}{A_i}{\phi _s} + {h^{ii}}\left( {{\partial _i}{A_i} - \Gamma _{ii}^k{A_k}} \right){\phi _s} \nonumber\\ &+2 {h^{ii}}{A_i}{\partial _i}{\phi _s}+ {\partial _s}\mathfrak{W} + {h^{ii}}\mathbf{R}({\partial _s}\widetilde{u},{\partial _i}\widetilde{u})({\partial _i}\widetilde{u}) + \mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})({\partial _t}\widetilde{u}).\label{huta} \end{align} By Remark \ref{3sect}, $A_i=A^{\infty}_i+A^{con}_i$, $\phi_i=\phi^{\infty}_{i}+\phi^{con}_i$. Then fixing $\Xi$ to be $(\Theta_1(Q),\Theta_2(Q))$, we have (\ref{huta}) reduces to \begin{align*} &\partial _t^2{\phi _s} - {\Delta}{\phi _s} - 2{h^{ii}}A_i^\infty {\partial _i}{\phi _s} - {h^{ii}}A_i^\infty A_i^\infty {\phi _s} - {h^{ii}}\left( {{\phi _s} \wedge \phi _i^\infty } \right)\phi _i^\infty-h^{ii}(\partial_iA^{\infty}_i-\Gamma^k_{ii}A^{\infty}_k)\phi_s \\ &= - 2{A_t}{\partial _t}{\phi _s} - {A_t}{A_t}{\phi _s} - {\partial _t}{A_t}{\phi _s} + {\partial _s}w +(\phi _t\wedge\phi_s)\phi_t + {h^{ii}}A_i^{con}{\partial _i}{\phi _s} + {h^{ii}}A_i^{con}A_i^\infty {\phi _s}\\ &+ {h^{ii}}A_i^\infty A_i^{con}{\phi _s}+ {h^{ii}}A_i^{con}A_i^{con}{\phi _s} + {h^{ii}}\left( {{\partial _i}A_i^{con} - \Gamma _{ii}^kA_k^{con}} \right){\phi _s} + {h^{ii}}\left( {{\phi _s} \wedge \phi _i^\infty } \right)\phi _i^{con}\\ &+ {h^{ii}}\left( {{\phi _s} \wedge \phi _i^{con}} \right)\phi _i^\infty + {h^{ii}}\left( {{\phi _s} \wedge \phi _i^{con}} \right)\phi _i^{con}. \end{align*} Then from the non-negativeness of the sectional curvature for the target $N=\Bbb H^2$ and the skew-symmetry of the connection matrix $A^{\infty}_i$, we have $W$ is a nonnegative symmetric operator in $L^2(\Bbb H^2;\Bbb C^2)$ by direct calculations, see Lemma \ref{symm} in Section 7. The self-adjointness of $W$ follows from Kato's perturbation theorem. In fact, there exists a self-adjoint realization denote by $((\Delta_{col}),D(\Delta_{col}))$ of $(\Delta,C^{\infty}_c(\Bbb H^2,\Bbb C^2))$. It is known that $D(\Delta_{col})$ consists of functions $f\in L^2$ whose Laplacian $\Delta f$ in distribution sense belong to $L^2$, see for instance \cite{Stri}. Write $W$ as $W=V_1+V_2\nabla$, then $V_1$ and $V_2$ are of exponential decay as $d(x,0)\to\infty$ by Lemma \ref{xuejin} and Definition 1.1. For any fixed $\varepsilon>0$, take $R>0$ sufficiently large such that $${\left\| {{V_1}(x)} \right\|_{L_{d(x,0) \ge R}^\infty }} \le \varepsilon ,\mbox{ }{\left\| {{V_2}(x)} \right\|_{L_{d(x,0) \ge R}^\infty }} \le \varepsilon, $$ then for any $f\in C^{\infty}_c(\Bbb H^2,\Bbb C^2)$, \begin{align}\label{huli} {\left\| {{V_1}(x)f + {V_2}\nabla f} \right\|_{L_{d(x,0) \ge R}^2}} \le \varepsilon {\left\| f \right\|_{{L^2}}} + \varepsilon {\left\| {\nabla f} \right\|_{{L^2}}}. \end{align} For this $R$, the compactness of Sobolev embedding in bounded domains implies there exists $C(\varepsilon, R)$ such that \begin{align}\label{huli2} {\left\| {{V_1}(x)f + {V_2}\nabla f} \right\|_{L_{d(x,0) \le R}^2}} \le C(\varepsilon ,R){\left\| f \right\|_{{L^2}}} + \varepsilon {\left\| {\Delta f} \right\|_{{L^2}}}. \end{align} Hence by (\ref{huli}) and (\ref{huli2}), one has for any $\varepsilon>0$ there exists $C(\varepsilon)$ such that \begin{align} {\left\| {{V_1}(x)f + {V_2}\nabla f} \right\|_{{L^2}}} \le C(\varepsilon){\left\| f \right\|_{{L^2}}} + \varepsilon {\left\| {\Delta f} \right\|_{{L^2}}}. \end{align} Since $C^{\infty}_c(\Bbb H^2,\Bbb C^2)$ is a core of $\Delta_{col}$, Kato's compact perturbation theorem shows $-\Delta+W$ is self-adjoint in $L^2$ with domain $D(\Delta_{col})$. \end{proof} \section{Bootstrap for the heat tension filed} \subsection{Strichartz estimates for wave equation with magnetic potential} Theorem 5.2 and Remark 5.5 of Anker, Pierfelice \cite{AP} obtained the Strichartz estimates for linear wave/Klein-Gordon equation: Let $((p,q),(\widetilde{p},\widetilde{q}))$ be a $(\sigma,\tilde{\sigma})$ admissible couple, i.e., \begin{align*} &\left\{ {({p^{ - 1}},{q^{ - 1}}) \in (0,\frac{1}{2}] \times (0,\frac{1}{2}):\frac{1}{p} > \frac{1}{2}(\frac{1}{2} - \frac{1}{q})} \right\} \cup \left\{ {\left( {0,\frac{1}{2}} \right)} \right\}\\ &\sigma \ge \frac{3}{2}\left( {\frac{1}{2} - \frac{1}{q}} \right),\tilde \sigma \ge \frac{3}{2}\left( {\frac{1}{2} - \frac{1}{{\tilde q}}} \right). \end{align*} If $u$ solves $\partial_t^2u-\Delta u=g$ with initial data $(f_0,f_1)$, then $$ {\left\| {\widetilde D_x^{ - \sigma + \frac{1}{2}}u} \right\|_{L_t^pL_x^q}} + {\left\| {\widetilde D_x^{ - \sigma - \frac{1}{2}}{\partial _t}u} \right\|_{L_t^pL_x^q}} \lesssim {\left\| {\widetilde D_x^{\frac{1}{2}}f_0} \right\|_{{L^2}}} + {\left\| {\widetilde D_x^{ - \frac{1}{2}}f_1} \right\|_{{L^2}}} + {\left\| {\widetilde D_x^{\tilde \sigma - \frac{1}{2}}g} \right\|_{L_t^{\tilde p'}L_x^{\tilde q'}}}. $$ where $\widetilde D=(-\Delta-\frac{1}{4}+\kappa^2)$ for some $\kappa>\frac{1}{2}$. Let $\rho(x)=e^{-d(x,0)}$. The endpoint and non-endpoint Strichartz estimates for magnetic wave equations in the small potential case were obtained in the first author's work [Corollary. 1.1. Proposition 3.1 \cite{Lize1}]. We recall this for reader's convenience. Consider the magnetic wave equation on $\Bbb H^2$, \begin{align}\label{wavem} \left\{ \begin{array}{l} \partial _t^2f - \Delta f + {B_0}(x)f + \sum^2_{i=1}{h^{ii}}{B_i}(x){\partial _i}f = F \\ f(0,x) = {f_0}(x),{\partial _t}f(0,x) = {f_1}(x) \\ \end{array} \right. \end{align} \begin{lemma}[\cite{Lize1}]\label{poi9nmb} Assume that $B_0,B_1,B_2$ in (\ref{wavem}) satisfy for some $\varrho>0$ \begin{align} \|B_0\|_{L^2\cap e^{-r\varrho}L^{\infty}}+\sum^2_{i=1}\|\sqrt{h^{ii}}B_i\|_{L^2\cap e^{-r\varrho}L^{\infty}}\le \mu_1. \end{align} And assume that the Schr\"odinger operator $H=-\Delta+B_0+h^{ii}B_i\partial_i$ is symmetric. If $0<\mu_1\ll 1$, $u$ solves (\ref{wavem}), then for any $0<\sigma\ll\varrho$, $p\in(2,6)$ \begin{align*} &{\left\| {{\rho ^{\sigma}}\nabla f} \right\|_{L_t^2L_x^2}}+{\left\| (-\Delta)^{\frac{1}{4}}f \right\|_{L_t^2L_x^{p}}} + {\left\| {{\partial _t}f} \right\|_{L_t^\infty L_x^2}} + {\left\| {\nabla f} \right\|_{L_t^\infty L_x^2}}\\ &\lesssim {\left\| {\nabla {f_0}} \right\|_{{L^2}}} + {\left\| {{f_1}} \right\|_{{L^2}}} + {\left\| F \right\|_{L_t^1L_x^2}}. \end{align*} \end{lemma} Hence by Lemma \ref{xuejin}, Lemma \ref{hushuo} and Lemma \ref{poi9nmb}, we have: \begin{proposition}\label{tianxia} Let $W$ be defined above and $0<\mu_1\ll 1$, $0<\sigma\ll \varrho\ll 1$, then we have the weighted and endpoint Strichartz estimates for the magnetic wave equation: If $f$ solves the equation \begin{align*} \left\{ \begin{array}{l} \partial _t^2f - {\Delta}f + Wf = F \\ f(0,x) = {f_0},{\partial _t}f(0,x) = {f_1} \\ \end{array} \right. \end{align*} then it holds for any $p\in(2,6)$, $0<\sigma\ll \varrho$ \begin{align} &{\left\| {{{\left| D \right|}^{\frac{1}{2}}}f} \right\|_{L_t^2L_x^{p}}} +{\left\| {{\rho ^{\sigma}}\nabla f} \right\|_{L_t^2L_x^2}}+ {\left\| {{\partial _t}f} \right\|_{L_t^\infty L_x^2}} + {\left\| {\nabla f} \right\|_{L_t^\infty L_x^2}}+{\left\| {{\rho ^{\sigma }}\nabla f} \right\|_{L_t^2L_x^2}}\nonumber\\ &\lesssim {\left\| {\nabla {f_0}} \right\|_{{L^2}}} + {\left\| {{f_1}} \right\|_{{L^2}}} + {\left\| F \right\|_{L_t^1L_x^2}}.\label{gutu4} \end{align} \end{proposition} \begin{remark}\label{tataru0} For all $\sigma\in \Bbb R$, $p\in(1,\infty)$, $\|\widetilde D^{\sigma} f\|_{p}$ is equivalent to $\|(-\Delta)^{\sigma/2}f\|_{p}$. Tataru \cite{Tataru4} shows for all $p\in(1,\infty)$, $\|\Delta f\|_{p}$ is equivalent to $\|\nabla^2f\|_{p}+\|\nabla f\|_{p}+\|f\|_{p}$. \end{remark} \subsection{Setting of Bootstrap} We fix the constants $\mu_1,\varepsilon_1, \varrho, \sigma$ to be \begin{align}\label{dozuoki} 0<\mu_2<\mu_1\ll\varepsilon_1\ll 1,\mbox{ }0<\sigma\ll\varrho\ll 1. \end{align} Let $L>0$ be sufficiently large say $L=100$. Define $\omega:\Bbb R^+\to \Bbb R^+$ and $a:\Bbb R^+\to \Bbb R^+$ by $$\omega(s) = \left\{ \begin{array}{l} {s^{\frac{1}{2}}}\mbox{ }{\rm{when}}\mbox{ }0 \le s \le 1 \\ {s^L}\mbox{ }\mbox{ }{\rm{when}}\mbox{ }s \ge 1 \\ \end{array} \right.,a(s) = \left\{ \begin{array}{l} s^{\frac{3}{4}}\mbox{ }\mbox{ }{\rm{when}}\mbox{ }0 \le s \le 1 \\ {s^L}\mbox{ }\mbox{ }{\rm{when}}\mbox{ }s \ge 1 \\ \end{array} \right. $$ \begin{proposition}\label{oures} Assume that $\mathcal{A}$ is the set of $T\in[0,T_*)$ such that for any $2<q<6+2\gamma$, $p\in(2,6)$ with some fixed $0<\gamma\ll1$, \begin{align} {\left\| {(du,\partial_tu)} \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}} + {\left\| (\nabla{\partial _t}u,\nabla du) \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}}\nonumber\\ +{\left\| {{\partial _t}u} \right\|_{L_t^2L^q_x([0,T] \times {\Bbb H^2})}} &\le {\varepsilon _1}.\label{boot2}\\ {\left\| {\omega (s)|D{|^{ - \frac{1}{2}}}{\partial _t}{\phi _s}} \right\|_{L_s^\infty L_t^2L_x^p}} + {\left\| {\omega (s){\partial _t}{\phi _s}} \right\|_{L_s^\infty L_t^\infty L_x^2}}\nonumber\\ +{\left\| {\omega (s)\nabla {\phi _s}} \right\|_{L_s^\infty L_t^\infty L_x^2}} + {\left\| {\omega (s){{\left| D \right|}^{\frac{1}{2}}}{\phi _s}} \right\|_{L_s^\infty L_t^2L_x^p}} &\le {\varepsilon _1}.\label{boot5} \end{align} Then for all $T\in \mathcal{A}$ we have \begin{align} &{\left\| {\omega (s){{\left| D \right|}^{ - \frac{1}{2}}}{\partial _t}{\phi _s}} \right\|_{L_s^\infty L_t^2L_x^p([0,T] \times {\Bbb H^2})}} + {\left\| {\omega (s){{\left| D \right|}^{\frac{1}{2}}}{\phi _s}} \right\|_{L^{\infty}_sL_t^2L_x^p([0,T] \times {\Bbb H^2})}}\nonumber\\ &+ {\left\| {\omega (s){\partial _t}{\phi _s}} \right\|_{L_s^\infty L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}} + {\left\| {\omega (s)\nabla {\phi _s}} \right\|_{L_s^\infty L_t^\infty L_x^2([0,T] \times {H^2})}} \le \varepsilon _1^2. \label{huojiq} \end{align} and for any $r\in(2,6+2\gamma]$ it holds that \begin{align} {\left\| {(du,\partial_tu)} \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}} + {\left\| {(\nabla{\partial _t}u,\nabla du)} \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}}&\le {\varepsilon^2 _1}\label{boot8q}\\ {\left\| {{\partial _t}u} \right\|_{L_t^2L_x^r([0,T] \times {\Bbb H^2})}} &\le {\varepsilon^2_1}.\label{boot9q} \end{align} Moreover we have \begin{align} {\left\| {du} \right\|_{L_t^\infty L_x^2([0,{T}] \times {\Bbb H^2})}}& + {\left\| {{\partial _t}u} \right\|_{L_t^\infty L_x^2([ 0,{T}] \times {\Bbb H^2})}} + {\left\| {\nabla du} \right\|_{L_t^\infty L_x^2([0,{T}] \times {\Bbb H^2})}} \nonumber\\ &+ {\left\| {\nabla {\partial _t}u} \right\|_{L_t^\infty L_x^2([0,{T}] \times {\Bbb H^2})}} + {\left\| {{\partial _t}u} \right\|_{L_t^2L_x^6([ 0,{T}] \times {\Bbb H^2})}} \le \varepsilon _1^2.\label{hua12xde} \end{align} \end{proposition} The proof of Proposition \ref{oures} will be divided into several lemmas below. (\ref{huojiq}) is proved in Proposition 5.11. (\ref{boot8q}), (\ref{boot9q}) and (\ref{hua12xde}) are proved in Proposition 5.13 and Corollary 5.15 respectively. The bootstrap programm we apply here is based on the design of \cite{LOS,Tao7}. The essential refinement is we add a spacetime bound $\|\partial_tu\|_{L^2_tL^p_x}$ to the primitive bootstrap assumption. The most important original ingredient in this part is we use the weighted Strichartz estimates in Section 5.1 to control the one order derivative terms of $\phi_s$. \begin{proposition}\label{aaop} Assume (\ref{boot2}) holds, then we have for any $\eta>0$ \begin{align} {\big\| {{A_t}} \big\|_{L_t^\infty L_x^\infty }} &\le \varepsilon_1\label{aaaw811}\\ {\big\| {{h^{ii}} {\partial _i}{A_i}(s)} \big\|_{L_t^\infty L_x^\infty }} &\le \varepsilon_1 \max(1,{s^{ -\eta}})\label{butterfly}\\ {\big\| {\sqrt {{h^{ii}}} {\partial _t}{A_i}(s)} \big\|_{L_t^\infty L_x^\infty }} &\le \varepsilon_1 {s^{ - \frac{1}{2}}}\label{u82}\\ {\big\| {\sqrt {{h^{ii}}} {A_i}(s)} \big\|_{L_t^\infty L_x^\infty }} &\le \varepsilon_1 \label{u81}. \end{align} \end{proposition} \begin{proof} By the commutator identity and the facts $|\partial_t\widetilde{u}|\le e^{s\Delta}|\partial_tu|$, $|\partial_s\widetilde{u}|\le e^{s\Delta}|\partial_su|$, (\ref{aaaw811}) is bounded by Lemma \ref{ktao1}, \begin{align*} {\left\| {{A_t}} \right\|_{L_t^\infty L_x^\infty }} &\le {\big\| {\int_s^\infty {{{\big\| {{\phi _t}} \big\|}_{L_x^\infty }}{{\big\| {{\phi _s}} \big\|}_{L_x^\infty }}} d\kappa} \big\|_{L_t^\infty }} \le \mathop {\sup }\limits_{t \in [0,T]} {\big\| {{\phi _t}} \big\|_{L_s^2L_x^\infty }}{\big\| {{\phi _s}} \big\|_{L_s^2L_x^\infty }} \\ &\le \mathop {\sup }\limits_{t \in [0,T]} {\big\| {{\partial _t}u} \big\|_{L_x^2}}{\big\| {{\partial _s}u} \big\|_{L_x^2}} \le {\varepsilon _1}. \end{align*} By the commutator identity, \begin{align*} &{\big\| {\sqrt {{h^{ii}}} {\partial _t}{A_i}} \big\|_{L_t^\infty L_x^\infty }} \le \int_s^\infty {{{\big\| {\sqrt {{h^{ii}}} {\partial _t}\left( {{\phi _i} \wedge {\phi _s}} \right)} \big\|}_{L_t^\infty L_x^\infty }}} d\kappa\\ & \le \int_s^\infty {{{\big\| {\sqrt {{h^{ii}}} {\partial _t}{\phi _i}} \big\|}_{L_t^\infty L_x^\infty }}\big\| {{\phi _s}} \big\|_{L_t^\infty L^{\infty}_x }} d\kappa+ \int_s^\infty {{{\big\| {\sqrt {{h^{ii}}} {\phi _i}} \big\|}_{L_t^\infty L_x^\infty }}\big\| {{\partial _t}{\phi _s}} \big\|_{L_t^\infty L^{\infty}_x}} d\kappa. \end{align*} Using the relation between the induced derivative $D_{i,t}$ and the covariant derivative on $u^*(TN)$, one obtains $| {\sqrt {{h^{ii}}} {\partial _t}{\phi _i}}| \le | {\nabla {\partial _t}\widetilde{u}}| + | {\sqrt {{h^{ii}}} {A_t}{\phi _i}}|+ | {\sqrt {{h^{ii}}} {A_i}{\phi _t}}|$ and similarly $| {{\partial _t}{\phi _s}}| \le | {{\nabla _t}{\partial _s}\widetilde{u}}| + | {A_t}{\phi _s}|$. Hence it suffices to prove \begin{align} {\int_s^\infty {\left\| {\left| {d\widetilde{u}} \right|\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|} \right\|} _{L_t^\infty L_x^\infty }}d\kappa + {\int_s^\infty {\left\| {\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right|} \right\|} _{L_t^\infty L_x^\infty }}d\kappa&\le \varepsilon_1s^{-\frac{1}{2}}\label{po987}\\ \int_s^\infty {{{\| {\sqrt {{h^{ii}}} {A_i}{\phi _t}{\phi _s}}\|}_{L_t^\infty L_x^\infty }}}d\kappa+\int_s^\infty {{{\| {\sqrt {{h^{ii}}} {A_t}{\phi _i}{\phi _s}}\|}_{L_t^\infty L_x^\infty }}}d\kappa&\le \varepsilon_1s^{-\frac{1}{2}} \label{pojn89} \end{align} For $s\in(0,1]$, Proposition \ref{sl} and $|d\widetilde{u}|\le e^{s\Delta}|du|$ give \begin{align}\label{0918} {\left\| {\left| {d\widetilde{u}} \right|\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|} \right\|_{L_t^\infty L_x^\infty }} + {\left\| {\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right|} \right\|_{L_t^\infty L_x^\infty }} \le \varepsilon_1 {s^{ - \frac{1}{2}}}{s^{ -1}} + \varepsilon_1 {s^{ -\frac{1}{2}}}{s^{ - 1}}. \end{align} For $s\ge1$, we have by Proposition \ref{sl} \begin{align}\label{01918} {\left\| {\left| {d\widetilde{u}} \right|\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|} \right\|_{L_t^\infty L_x^\infty }} + {\left\| {\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right|} \right\|_{L_t^\infty L_x^\infty }} \le \varepsilon_1 {e^{ - \delta s}}. \end{align} Therefore (\ref{01918}) and (\ref{0918}) yield for all $s\in(0,\infty)$ $${\left\| {\left| {d\widetilde{u}} \right|\left| {{\nabla _t}{\partial _s}\widetilde{u}} \right|} \right\|_{L_t^\infty L_x^\infty }} + {\left\| {\left| {{\partial _s}\widetilde{u}} \right|\left| {\nabla {\partial _t}\widetilde{u}} \right|} \right\|_{L_t^\infty L_x^\infty }} \le \varepsilon_1 {s^{ -3/2}}. $$ Hence we obtain (\ref{po987}). (\ref{pojn89}) and (\ref{u81}) can be proved similarly. By (\ref{christ}) and direct calculations similar to Lemma \ref{xuejin}, \begin{align}\label{ipu83} |h^{ii}\partial_iA^{\infty}_i|\lesssim |\nabla dQ|+|dQ|. \end{align} And the same route as (\ref{u82}) shows for any $\eta>0$ \begin{align}\label{ipu82} |h^{ii}\partial_iA^{con}_i|\le \varepsilon_1 s^{-\eta}. \end{align} Thus (\ref{butterfly}) follows by (\ref{ipu82}), (\ref{ipu83}) \end{proof} \begin{lemma}\label{aoao1} Assume (\ref{boot2}) and (\ref{boot5}) hold, then we have \begin{align} \left\| \sqrt{h^{pp}}|\partial_p(h^{ii} {\partial _i}{A_i}(s))| \right\|_{L_t^\infty L_x^{\infty}} &\le \varepsilon_1 \max(s^{ -1},1)\label{1q2} \end{align} \end{lemma} \begin{proof} By Remark \ref{3sect}, it suffices to bound $A^{\infty}$ and $A^{con}$ part separately. Direct calculations as Lemma \ref{xuejin} and (\ref{christ}) yield the bound for the $A^{\infty}$ part is \begin{align*} \sqrt{h^{pp}}|\partial_p(h^{ii} {\partial _i}{A^{\infty}_i}(s))| \le |\nabla^2 dQ|+ |\nabla dQ|+|dQ|. \end{align*} Thus (\ref{as4}) shows the $A^{\infty}$ part is bounded by \begin{align*} \|\sqrt{h^{pp}}|\partial_p(h^{ii} {\partial _i}{A^{\infty}_i}(s))|\|_{L^{\infty}_x}\le \varepsilon_1. \end{align*} By (\ref{christ}) and direct calculations, \begin{align*} &\sqrt{h^{pp}}|\partial_p(h^{ii} {\partial _i}({\phi_i\wedge\phi_s})(s))|\\ &\le |\nabla^2\partial_s\widetilde{u}| |d\widetilde{u}|+\sqrt{h^{ii}h^{pp}}|A_iA_p||\partial_s\widetilde{u}| |d\widetilde{u}| +|\partial_s\widetilde{u}| |\nabla^2d\widetilde{u}|+ |\nabla\partial_s\widetilde{u}||\nabla d\widetilde{u}|\\ &+\sqrt{h^{ii}}|A_i||\nabla\partial_s\widetilde{u}||d\widetilde{u}|+ \sqrt{h^{ii}}|A_i||\partial_s\widetilde{u}||\nabla d\widetilde{u}|+ \sqrt{h^{ii}}|A_i\|\nabla\partial_s\widetilde{u}| |d\widetilde{u}|\\ &+\sqrt{h^{pp}h^{ii}}|\partial_pA_i||\nabla\partial_s\widetilde{u}| |d\widetilde{u}|+\sqrt{h^{pp}h^{ii}}|\partial_pA_i||\partial_s\widetilde{u}| |\nabla d\widetilde{u}|. \end{align*} Thus the $A^{con}$ part follows by Lemma \ref{fotuo1} and interpolation. \end{proof} \begin{proposition}\label{bootstrap} Suppose that (\ref{boot2}), (\ref{boot5}) hold. Then we have for $p\in(2,6)$ \begin{align} {\big\| {a(s){{\left\| {{\partial _t}{\phi _s}} \right\|}_{L_t^2L_x^p}}} \big\|_{L_s^\infty}}&\le {\varepsilon _1}\label{huojikn1} \\ {\big\| {a(s){{\left\| {\nabla {\phi _s}} \right\|}_{L_t^2L_x^p}}} \big\|_{L_s^\infty}} &\le {\varepsilon _1}\label{huojikn}. \end{align} Generally we have for $\theta\in[0,2]$ \begin{align} {\big\| {\omega_\theta (s){{\left( { - \Delta } \right)}^\theta }{\phi _s}} \big\|_{L_s^\infty L_t^2L_x^p}}&\le {\varepsilon _1}\label{uojiknmpx3}\\ \big\| \omega_1(s)|D|{\partial _t}{\phi _s} \big\|_{L_s^\infty L_t^2L_x^p} &\le {\varepsilon _1},\label{9o0o} \end{align} where $\omega_\theta (s)=s^{\theta+\frac{1}{4}}$ when $s\in[0,1]$ and $\omega_\theta (s)=s^{L}$ when $s\ge1$. \end{proposition} \begin{proof} By (\ref{991}) and Duhamel principle we have \begin{align} {\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{\phi _s}(s)}\|_{L_t^2L_x^p}} &\le {\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{\frac{s}{2}\Delta }}{\phi _s}(\frac{s}{2})} \|_{L_t^2L_x^p}}\nonumber\\ &+ {\big\| {\int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}{h^{ii}}{A_i}{\partial _i}{\phi _s}(\tau )} d\tau }\big\|_{L_t^2L_x^p}} \label{wulaso}\\ &+ {\big\| {\int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}G(\tau)}d\tau } \big\|_{L_t^2L_x^p}}.\label{wulasoo} \end{align} where $G(\tau)={{h^{ii}}\left( {{\partial _i}{A_i}} \right){\phi _s} - {h^{ii}}\Gamma _{ii}^k{A_k}{\phi _s} + {h^{ii}}{A_i}{A_i}{\phi _s} + {h^{ii}}\left( {{\phi _s} \wedge {\phi _i}} \right){\phi _i}}.$ For (\ref{wulaso}), the smoothing effect and (\ref{u81}) show \begin{align*} &s^{\frac{3}{4}}{\big\| {\int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}{h^{ii}}{A_i}{\partial _i}{\phi _s}(\tau )} d\tau } \big\|_{L_t^2L_x^p}} \\ &\lesssim s^{\frac{3}{4}}\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}{{\left\| {{h^{ii}}{A_i}{\partial _i}{\phi _s}(\tau )} \right\|}_{L_t^2L_x^p}}} d\tau \\ &\lesssim s^{\frac{3}{4}}\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}{{\left\| {\nabla {\phi _s}(\tau )} \right\|}_{L_t^2L_x^p}}{{\big\| {\sqrt {{h^{ii}}} {A_i}} \big\|}_{L_t^\infty L_x^\infty }}d} \tau \\ &\lesssim s^{\frac{3}{4}}\varepsilon_1\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}{{\left\| {\nabla {\phi _s}(\tau )} \right\|}_{L_t^2L_x^p}}d} \tau. \end{align*} Thus we conclude when $s\in[0,1]$ \begin{align}\label{w2} &s^{\frac{3}{4}}{\big\| {\int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}{h^{ii}}{A_i}{\partial _i}{\phi _s}(\tau )} d\tau } \big\|_{L_t^2L_x^p}}\nonumber\\ &\le \varepsilon_1{\left\| {s^{\frac{3}{4}}{{\left\| {\nabla {\phi _s}(s)} \right\|}_{L_t^2L_x^p}}} \right\|_{L_s^\infty }}. \end{align} Similarly we have for (\ref{wulasoo}) that \begin{align*} &s^{\frac{3}{4}}\int_{\frac{s}{2}}^s (s - \tau )^{-\frac{1}{2}}\|G(\tau)\|_{L_t^2L_x^p} d\tau \\ &\le s^{\frac{3}{4}}\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}{{\left\| {{h^{ii}}{\partial _i}{A_i}} \right\|}_{L_t^\infty L_x^\infty }}{{\left\| {{\phi _s}} \right\|}_{L_t^2L_x^p}}} d\tau + s^{\frac{3}{4}}\int_{\frac{s}{2}}^s {{{\left\| {{A_2}} \right\|}_{L_t^\infty L_x^\infty }}{{\left\| {{\phi _s}} \right\|}_{L_t^2L_x^p}}} d\tau \\ &+ s^{\frac{3}{4}}\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}\left( {{{\left\| {{h^{ii}}{A_i}{A_i}} \right\|}_{L_t^\infty L_x^\infty }} + {{\left\| {{h^{ii}}{\phi _i}{\phi _i}} \right\|}_{L_t^\infty L_x^\infty }}} \right)} {\left\| {{\phi _s}} \right\|_{L_t^2L_x^p}}d\tau. \end{align*} Thus by Proposition \ref{aaop} and Proposition \ref{sl}, we have for all $s\in[0,1]$ \begin{align}\label{wulaso2} (\ref{wulasoo})\lesssim {\left\| {{s^{\frac{1}{2}}}{{\left\| {{\phi _s}(s)} \right\|}_{L_t^2L_x^p}}} \right\|_{L_s^\infty }}. \end{align} For $s\ge1$, we also have by Duhamel principle \begin{align*} &{s^L}{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{\phi _s}(s)} \big\|_{L_t^2L_x^p}}\\ &\le {s^L}{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{\frac{s}{2}\Delta }}{\phi _s}(\frac{s}{2})} \big\|_{L_t^2L_x^6}} + {s^L}{\big\| {\int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}G_1(\tau )} d\tau } \big\|_{L_t^2L_x^p}}, \end{align*} where $G_1$ is the inhomogeneous term. The linear term is bounded by \begin{align*} {s^L}{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{\frac{s}{2}\Delta }}{\phi _s}(\frac{s}{2})} \big\|_{L_t^2L_x^p}} \le {s^L}{e^{ - \frac{1}{16} s}}{\big\| {{\phi _s}(\frac{s}{2})} \big\|_{L_t^2L_x^p}}. \end{align*} By Proposition \ref{aaop} and smoothing effect, the first term in $G_1$ is bounded as \begin{align*} &{s^L}{\big\| {\int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}{h^{ii}}{A_i}{\partial _i}{\phi _s}(\tau )} d\tau } \big\|_{L_t^2L_x^p}}\\ &\le {s^L}\int_{\frac{s}{2}}^s {{{{{\left( {s - \tau } \right)}^{-\frac{1}{2}}}}}{e^{ -\delta (s - \tau )}}{{\big\| {\nabla {\phi _s}(\tau )} \big\|}_{L_t^2L_x^p}}{{\big\| {\sqrt {{h^{ii}}} {A_i}} \big\|}_{L_t^\infty L_x^\infty }}} d\tau \\ &\le \varepsilon_1{s^L}\int_{\frac{s}{2}}^s {{e^{ -\delta (s - \tau )}}{\tau ^{ - L}}{{{{\left( {s - \tau } \right)}^{-\frac{1}{2}}}}}{{\big\| {{\tau ^L}\nabla {\phi _s}(\tau )} \big\|}_{L_t^2L_x^p}}d} \tau. \end{align*} The other terms in $G_1$ can be estimated similarly, thus we obtain for $s\ge1$ \begin{align}\label{wulaso3} {s^L}{\left\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{\phi _s}(s)} \right\|_{L_t^2L_x^p}} \le \varepsilon_1{\left\| {{s^L}{{\left\| {\nabla {\phi _s}(s)} \right\|}_{L_t^2L_x^p}}} \right\|_{L_s^\infty (s \ge 1)}} + {\left\| {{s^L}{{\left\| {{\phi _s}(\tau )} \right\|}_{L_t^2L_x^p}}} \right\|_{L_s^\infty (s \ge 1)}}. \end{align} Combing (\ref{wulaso}), (\ref{wulasoo}), with (\ref{wulaso3}) gives corresponding estimates in (\ref{huojikn}) for $\nabla\phi_s$. It suffices to prove the remaining estimates in (\ref{huojikn}) for $\partial_t\phi_s$. Denote the inhomogeneous term in (\ref{9923}) by $G_3$, then Duhamel principle gives \begin{align*} s^{\frac{3}{4}}{\left\| {{\partial _t}{\phi _s}(s)} \right\|_{L_t^2L_x^p}} \le s^{\frac{3}{4}}{\big\| {{e^{\Delta \frac{s}{2}}}{\partial _t}{\phi _s}(\frac{s}{2})} \big\|_{L_t^2L_x^p}} + s^{\frac{3}{4}}{\big\| {\int_{\frac{s}{2}}^s {{e^{\Delta (s - \tau )}}G_3(\tau )d\tau } } \big\|_{L_t^2L_x^p}}. \end{align*} The first term of $G_3$ is bounded by \begin{align*} s^{\frac{3}{4}}{\int_{\frac{s}{2}}^s {\big\| {{e^{\Delta (s - \tau )}}{h^{ii}}\left( {{\partial _t}{A_i}} \right){\partial _i}{\phi _s}(\tau )} \big\|} _{L_t^2L_x^p}}d\tau\le s^{\frac{3}{4}}\int_{\frac{s}{2}}^s {\big\| {\sqrt {{h^{ii}}}{\partial _t}{A_i}} \big\|_{L_t^\infty L_x^\infty }}\big\|{\nabla}{\phi_s} \big\|_{L_t^2 L_x^p } d\tau. \end{align*} This is acceptable by Proposition \ref{aaop}. The second term in $G_3$ is bounded as \begin{align} &s^{\frac{3}{4}}{\int_{\frac{s}{2}}^s {\big\| {{e^{\Delta (s - \tau )}}2{h^{ii}}{A_i}{\partial _i}{\partial _t}{\phi _s}(\tau )} \big\|} _{L_t^2L_x^p}}d\tau\nonumber\\ &\le s^{\frac{3}{4}}{\int_{\frac{s}{2}}^s {\big\| {{e^{\Delta (s - \tau )}}\sqrt {{h^{ii}}} {\partial _i}\left( {\sqrt {{h^{ii}}} {A_i}{\partial _t}{\phi _s}} \right)} \big\|} _{L_t^2L_x^p}}d\tau+ s^{\frac{3}{4}}{\int_{\frac{s}{2}}^s {\big\| {{e^{\Delta (s - \tau )}}{h^{ii}}{\partial _i}{A_i}{\partial _t}{\phi _s}} \big\|} _{L_t^2L_x^p}}d\tau\nonumber\\ &\triangleq I+II. \end{align} $I$ is bounded by the smoothing effect, boundedness of Riesz transform and Proposition \ref{aaop} \begin{align*} I &\le {s^{\frac{3}{4}}}\int_{\frac{s}{2}}^s {{{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{\Delta (s - \tau )}}\big( {\sqrt {{h^{ii}}} {A_i}{\partial _t}{\phi _s}} \big)} \big\|}_{L_t^2L_x^p}}} d\tau \\ &\le {s^{\frac{3}{4}}}\int_{\frac{s}{2}}^s {{{{{\big( {s - \tau } \big)}^{-\frac{1}{2}}}}}{{\big\| {\sqrt {{h^{ii}}} {A_i}} \big\|}_{L_t^\infty L_x^\infty }}{{\big\| {{\partial _t}{\phi _s}} \big\|}_{L_t^2L_x^p}}} d\tau\\ &\le {s^{\frac{3}{4}}}\int_{\frac{s}{2}}^s {{{{{\left( {s - \tau } \right)}^{-\frac{1}{2}}}}}} {\varepsilon _1}{\big\| {{\partial _t}{\phi _s}} \big\|_{L_t^2L_x^p}}d\tau. \end{align*} $II$ is estimated as the first term of $G_3$ above. The third term of $G_3$ is bounded as \begin{align} &{s^{\frac{3}{4}}}\int_{\frac{s}{2}}^s {{{\big\| {{e^{\Delta (s - \tau )}}{h^{ii}}\left( {{\partial _i}{\partial _t}{A_i}} \right){\phi _s}} \big\|}_{L_t^2L_x^p}}} d\tau \nonumber \\ &\le {s^{\frac{3}{4}}}{\int_{\frac{s}{2}}^s {\big\| {{e^{\Delta (s - \tau )}}\sqrt {{h^{ii}}} {\partial _i}\big( {\sqrt {{h^{ii}}} {\partial _t}{A_i}{\phi _s}} \big)} \big\|} _{L_t^2L_x^p}}d\tau\nonumber \\ &+ {s^{\frac{3}{4}}}\int_{\frac{s}{2}}^s {{{\big\| {{e^{\Delta (s - \tau )}}{h^{ii}}{\partial _t}{A_i}{\partial _i}{\phi _s}} \big\|}_{L_t^2L_x^p}}} d\tau.\label{woshi} \end{align} The remaining arguments are almost the same as $I$ and $II$. And the rest nine terms in $G_3$ can be estimated as above as well. Hence the desired estimates in (\ref{huojikn1}) for $\partial_t\phi_s$ when $s\in(0,1]$ is verified. It suffices to prove (\ref{huojikn1}) for $\partial_t\phi_s$ when $s\ge1$. The proof for this part is exactly close to the estimates of $\nabla\phi_s$ when $s\ge1$ and that of $I,II$. (\ref{9o0o}) follows by the same arguments as (\ref{huojikn1}) by applying smoothing effect of the heat semigroup. By interpolation, in order to verify (\ref{uojiknmpx3}), it suffices to prove \begin{align} \left\| \omega_{1}(s) (-\Delta){\phi _s} \right\|_{L_s^\infty L_t^2L_x^p}\le {\varepsilon _1}. \end{align} By (\ref{991}), Duhamel principle and the smoothing effect we have \begin{align*} &\|(- \Delta){\phi _s}(s)\|_{L_t^2L_x^p} \le s^{-1}e^{-\frac{\delta}{2}s}\|{\phi _s}(\frac{s}{2})\|_{L_t^2L_x^p}\\ &+ \int^{s}_{\frac{s}{2}} (s-\tau)^{-\frac{1}{2}}e^{-\delta(s-\tau)}\big(\|\nabla (h^{ii}A_i\partial_i\phi_s)\|_{{L_t^2L_x^p}}+\|\nabla G\|_{{L_t^2L_x^p}}\big)d\tau. \end{align*} Then by Lemma \ref{aoao1}, Proposition \ref{aaop}, (\ref{huojikn}), (\ref{boot2}), (\ref{boot5}), one obtains \begin{align*} \left\| \omega_{1}(s) (-\Delta){\phi _s} \right\|_{L_s^\infty L_t^2L_x^p}\le {\varepsilon _1}\left\| \omega_{1} (s) \nabla^2{\phi _s} \right\|_{L_s^\infty L_t^2L_x^p}+\varepsilon_1. \end{align*} Thus (\ref{uojiknmpx3}) follows by Remark \ref{tataru0}. \end{proof} \begin{lemma} Assume that (\ref{boot2}), (\ref{boot5}) hold, then for $q\in(2,6+2\gamma]$ \begin{align} {\left\| {{\phi _t}(s)} \right\|_{L_s^\infty L_t^2L_x^q}} &\le {\varepsilon _1}\label{time2}\\ {\left\| {{A_t}} \right\|_{L_t^1L_x^\infty }} &\le \varepsilon _1^2\label{aaop11} \end{align} \end{lemma} \begin{proof} First notice that $\phi_t$ satisfies $(\partial_s-\Delta)|\phi_t|\le 0$, thus for any fixed $(t,s,x)$ one has the pointwise estimate \begin{align*} |\phi_t(s,t,x)|\le |\phi_t(0,t,x)|=|\partial_t u(t,x)|. \end{align*} Hence (\ref{time2}) follows by (\ref{boot2}). From commutator identity we have \begin{align}\label{qx1} {\left\| {{A_t}} \right\|_{L_t^1L_x^\infty }} \le \int_0^\infty {{{\left\| {{\partial _t}u} \right\|}_{L_t^2L_x^\infty }}{{\left\| {{\partial _s}u} \right\|}_{L_t^2L_x^\infty }}} ds. \end{align} Sobolev inequality implies for $p_*$ slightly less than 6 \begin{align}\label{yu0cv} \|\phi_s\|_{L^{\infty}_x}\le \||D|^{\frac{1}{2}}\phi_s\|_{L^{p_*}_x}. \end{align} And since $|\partial_t \widetilde{u}|$ satisfies $(\partial_s-\Delta)|\partial_t \widetilde{u}|\le 0$, then \begin{align}\label{0yu0cv} \|\phi_t(s)\|_{L^{\infty}_x}\lesssim s^{-1/{p_*}}e^{-\delta s}\|\phi(\frac{s}{2})\|_{L^{p_*}_x}. \end{align} By (\ref{0yu0cv}), (\ref{yu0cv}) and (\ref{boot5}), \begin{align}\label{0yu0cv9} \int^{1}_{0}\|\phi_t(s)\|_{L^2_tL^{\infty}_x}\|\phi_s(s)\|_{L^2_tL^{\infty}_x}&\lesssim \int^1_0s^{-\frac{1}{2}-\frac{1}{p_*}}\|\phi_t(\frac{s}{2})\|_{L^2_tL^{p_*}_x}s^{\frac{1}{2}}\||D|^{\frac{1}{2}}\phi_s(s)\|_{L^2_tL^{p_*}_x} ds.\\ \int^{\infty}_{1}\|\phi_t(s)\|_{L^2_tL^{\infty}_x}\|\phi_s(s)\|_{L^2_tL^{\infty}_x}&\lesssim \int^{\infty}_1s^{-4L}\|\phi_t(\frac{s}{2})\|_{L^2_tL^{p_*}_x}\||D|^{\frac{1}{2}}\phi_s(s)\|_{L^2_tL^{p_*}_x}ds. \end{align} Thus (\ref{aaop11}) is obtained by (\ref{boot2}) and (\ref{boot5}). \end{proof} \begin{lemma}\label{gdie1} Assume that (\ref{boot2}) and (\ref{boot5}) hold, then for $p\in(2,6+2\gamma]$ with $0<\gamma\ll1$, $\phi_t$ satisfies \begin{align} {\left\| {\omega (s)|D|{\phi _t}(s)} \right\|_{L_s^\infty L_t^2L_x^p}} &\le {\varepsilon _1}\label{time}\\ {\left\| {\omega_{\frac{3}{4}} (s)\Delta{\phi _t}(s)} \right\|_{L_s^\infty L_t^2L_x^p}} &\le {\varepsilon _1}\label{timeling} \end{align} \end{lemma} \begin{proof} By Duhamel principle and (\ref{yfcvbn}) \begin{align*} {s^{\frac{1}{2}}}{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{\phi _t}(s)} \big\|_{L_t^2L_x^p}} &\le {s^{\frac{1}{2}}}{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{\frac{s}{2}\Delta }}{\phi _t}(\frac{s}{2})} \big\|_{L_t^2L_x^p}} \\ &+ {s^{\frac{1}{2}}}\int_{\frac{s}{2}}^s {{{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}\mathcal{G}(\tau )} \big\|}_{L_t^2L_x^p}}} d\tau, \end{align*} where $\mathcal{G}$ denotes the inhomogeneous terms. By smoothing effect and Proposition \ref{aaop}, the first term in $\mathcal{G}$ is bounded by \begin{align*} &{s^{\frac{1}{2}}}\int_{\frac{s}{2}}^s {{{\big\| {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}{h^{ii}}{A_i}{\partial _i}{\phi _t}} \big\|}_{L_t^2L_x^p}}} d\tau \\ &\le {s^{\frac{1}{2}}}\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}{{\big\| {\nabla {\phi _t}} \big\|}_{L_t^2L_x^p}}} {\big\| {\sqrt {{h^{ii}}} {A_i}} \big\|_{L_t^\infty L_x^\infty }}d\tau \\ &\le \varepsilon_1{s^{\frac{1}{2}}}\int_{\frac{s}{2}}^s {{{{{(s - \tau )}^{-\frac{1}{2}}}}}{{\big\| {\nabla {\phi _t}} \big\|}_{L_t^2L_x^p}}} d\tau. \end{align*} The large time estimates follow by the same route. Similar estimates for the rest terms in $\mathcal{G}$ and (\ref{time2}) yield (\ref{time}). By Duhamel principle and smoothing effect, we have \begin{align*} \| \Delta\phi_t\|_{L^2_tL^p_x} \lesssim s^{-\frac{1}{2}}e^{-\delta \frac{s}{2}}\|\nabla\phi_t\|_{L^2_tL^p_x}+\int_{\frac{s}{2}}^s (s-\tau)^{-\frac{1}{2}}e^{-\delta(s-\tau)}\|\nabla \mathcal{G} \|_{L_t^2L_x^p} d\tau. \end{align*} Then Lemma \ref{aoao1}, Proposition \ref{aaop}, (\ref{time}), (\ref{boot2}), (\ref{boot5}) give \begin{align*} \| \omega_{\frac{3}{4}}(s)\Delta\phi_t\|_{L^{\infty}_sL^2_tL^p_x} \lesssim \epsilon_1+\epsilon_1\| \omega_{\frac{3}{4}}(s)\nabla^2\phi_t\|_{L^{\infty}_sL^2_tL^p_x} \end{align*} Thus (\ref{timeling}) follows by Remark \ref{tataru0}. \end{proof} \begin{lemma} Suppose that (\ref{boot2}) and (\ref{boot5}) hold, then the wave map tension field satisfies \begin{align} {\left\|s^{-\frac{1}{2}} \mathfrak{W}(s) \right\|_{L^\infty_s L_t^1L_x^2}}\le \varepsilon^2_1\label{time5}\\ {\left\| \nabla \mathfrak{W}(s) \right\|_{L^\infty_s L_t^1L_x^2}}\le \varepsilon^2_1\label{time6}\\ {\left\| s^{\frac{1}{2}}\Delta \mathfrak{W}(s) \right\|_{L^\infty_s L_t^1L_x^2}}\le \varepsilon^2_1\label{time7}\\ {\left\| \omega(s)\partial_s \mathfrak{W}(s) \right\|_{L^\infty_s L_t^1L_x^2}}\le \varepsilon^2_1.\label{time8} \end{align} \end{lemma} \begin{proof} Recall the equation for $\mathfrak{W}$ evolving along $s$: \begin{align} {\partial _s}\mathfrak{W} &= \Delta \mathfrak{W} + 2{h^{ii}}{A_i}{\partial _i}\mathfrak{W} + {h^{ii}}{A_i}{A_i}\mathfrak{W} + {h^{ii}}{\partial _i}{A_i}\mathfrak{W} - {h^{ii}}\Gamma _{ii}^k{A_k}\mathfrak{W} + {h^{ii}}\left( {\mathfrak{W} \wedge {\phi _i}} \right){\phi _i}\nonumber\\ &+ 3{h^{ii}}({\partial _t}\widetilde{u} \wedge {\partial _i}{\widetilde{u}}){\nabla _t}{\partial _i}\widetilde{u}.\label{gurenjim} \end{align} Since $\mathfrak{W}(0,s,x)$=0 for all $(s,x)\in\Bbb R^+\times \Bbb H^2$, Duhamel principle gives $$(-\Delta)^k\mathfrak{W}(s,t,x)= \int_0^s {{e^{(s - \tau )\Delta }}{{\left( { - \Delta } \right)}^k}G_2(\tau )d\tau }, $$ where $G_2$ denotes the inhomogeneous term. \\ {\bf{Step One.}} In this step, we consider short time behavior, and all the integrand domain of $L^{\infty}_s$ is restricted in $s\in[0,1]$. By (\ref{u81}), (\ref{butterfly}), \begin{align*} \int_0^s {{{\left\| {h^{ii}{A_i}{A_i}\mathfrak{W}} \right\|}_{L_t^1L_x^2}}} d\kappa &\le \int_0^s {{{\left\| {h^{ii}{A_i}{A_i}} \right\|}_{L_t^\infty L_x^\infty }}} {\left\| \mathfrak{W} \right\|_{L_t^1L_x^2}}d\kappa \le {s^{\frac{3}{2}}}\varepsilon _1^2{\left\| {\mathfrak{W}{s^{ - \frac{1}{2}}}} \right\|_{L_s^\infty L_t^1L_x^2}} \\ \int_0^s {{{\left\| {{h^{ii}}{\partial _i}{A_i}\mathfrak{W}} \right\|}_{L_t^1L_x^2}}} d\kappa &\le \int_0^s {{{\left\| {{h^{ii}}{\partial _i}{A_i}} \right\|}_{L_t^\infty L_x^\infty }}} {\left\| \mathfrak{W} \right\|_{L_t^1L_x^2}}d\kappa \le s^{\frac{1}{2}}\varepsilon _1^2{\left\| {\mathfrak{W}{s^{ - \frac{1}{2}}}} \right\|_{L_s^\infty L_t^1L_x^2}}. \end{align*} By Proposition \ref{sl}, \begin{align} \int_0^s {{{\left\| {{h^{ii}}\left( {\mathfrak{W} \wedge {\phi _i}} \right){\phi _i}} \right\|}_{L_t^1L_x^2}}} d\kappa \le \int_0^s {{{\left\| {{h^{ii}}{\phi _i}{\phi _i}} \right\|}_{L_t^\infty L_x^\infty }}} {\left\|\mathfrak{W} \right\|_{L_t^1L_x^2}}d\kappa \le s\varepsilon _1^2{\left\| {\mathfrak{W}{s^{ - \frac{1}{2}}}} \right\|_{L_s^\infty L_t^1L_x^2}}. \end{align} By (\ref{time}), (\ref{u81}) and Proposition \ref{sl}, \begin{align*} &\int_0^s {{{\left\| {{h^{ii}}\left( {{\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}} \right){\nabla _i}{\partial _t}\widetilde{u}} \right\|}_{L_t^1L_x^2}}} d\kappa \\ &\le \int_0^s {{{\left\| {d\widetilde{u}} \right\|}_{L_t^\infty L_x^6}}{{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|}_{L_t^2L_x^6}}} {\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^2L_x^6}}d\kappa \\ &\le \int_0^s {{{\left\| {d\widetilde{u}} \right\|}_{L_t^\infty L_x^6}}{{\left\| {\nabla {\phi _t}} \right\|}_{L_t^2L_x^6}}} {\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^2L_x^6}}d\kappa + \int_0^s {{{\left\| {d\widetilde{u}} \right\|}_{L_t^\infty L_x^6}}{{\left\| {\sqrt {{h^{ii}}} {A_i}{\phi _t}} \right\|}_{L_t^2L_x^6}}} {\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^2L_x^6}}d\kappa \\ &\le {s^{\frac{1}{2}}}\varepsilon _1^2. \end{align*} By the smoothing effect and the boundedness of Riesz transform, we have \begin{align*} &\int_0^s {{{\left\| {{e^{(s - \kappa)\Delta }}{h^{ii}}{A_i}{\partial _i}\mathfrak{W}} \right\|}_{L_t^1L_x^2}}} d\kappa \\ &\le \int_0^s {{{\left\| {{e^{(s - \kappa)\Delta }}{h^{ii}}{\partial _i}\left( {{A_i}\mathfrak{W}} \right)} \right\|}_{L_t^1L_x^2}}} d\kappa + \int_0^s {{{\left\| {{e^{(s - \kappa)\Delta }}{h^{ii}}{\partial _i}{A_i}\mathfrak{W}} \right\|}_{L_t^1L_x^2}}} d\kappa \\ &\le \int_0^s {{{(s - \kappa)}^{ - \frac{1}{2}}}{{\left\| {\sqrt {{h^{ii}}} {A_i}\mathfrak{W}} \right\|}_{L_t^1L_x^2}}} d\kappa + \int_0^s {{{\left\| {{h^{ii}}{\partial _i}{A_i}\mathfrak{W}} \right\|}_{L_t^1L_x^2}}} d\kappa \\ &\le {s^{\frac{1}{2}}}\varepsilon _1^2{\left\| {\mathfrak{W}{s^{ - \frac{1}{2}}}} \right\|_{L_s^\infty L_t^1L_x^2}}. \end{align*} Hence we conclude (\ref{time5}) for $s\in[0,1]$ by choosing $\varepsilon_1$ sufficiently small. In order to prove (\ref{time6}), we use the following Duhamel principle instead to apply (\ref{time5}), $${\left( { - \Delta } \right)^{\frac{1}{2}}}\mathfrak{W}(s) = {\left( { - \Delta } \right)^{\frac{1}{2}}}{e^{\frac{s}{2}\Delta }}\mathfrak{W}(\frac{s}{2}) + \int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^{\frac{1}{2}}}{e^{(s - \tau )\Delta }}{G_2}(\tau )d\tau }. $$ Then (\ref{time6}) follows by (\ref{time5}) and the smoothing effect. Again by Duhamel principle and the smoothing effect, $$\|\left( { - \Delta } \right)\mathfrak{W}(s)\|_{L^2_x} \le \|\left( { - \Delta } \right){e^{\frac{s}{2}\Delta }}\mathfrak{W}(\frac{s}{2})\|_{L^2_x} + \int_{\frac{s}{2}}^s (s-\tau)^{-\frac{1}{2}}e^{-\delta (s-\tau)}\|\left( { - \Delta } \right)^{\frac{1}{2}}{G_2}(\tau )\|_{L^2_x}d\tau. $$ Thus Lemma \ref{aoao1}, Proposition \ref{aaop}, (\ref{boot2}), (\ref{boot5}), Remark \ref{tataru0} and Lemma \ref{gdie1} give (\ref{time7}) for $s\in[0,1]$. For $s\in[0,1]$, (\ref{time8}) now arises from (\ref{time5})-(\ref{time7}).\\ {\bf{Step Two.}} We prove (\ref{time5})-(\ref{time7}) for $s\ge1$. This can be easily obtained by the same arguments as above with the help of $s^{-L}$ decay in the long time case.\\ {\bf{Step Three.}} We prove the large time behavior. The Duhamel principle we use is also $${\left( { - \Delta } \right)^k}\mathfrak{W}(s) = {\left( { - \Delta } \right)^k}{e^{\frac{s}{2}\Delta }}\mathfrak{W}(\frac{s}{2}) + \int_{\frac{s}{2}}^s {{{\left( { - \Delta } \right)}^k}{e^{(s - \tau )\Delta }}{G_2}(\tau )d\tau }. $$ Let $s\ge1$, applying smoothing effect we obtain $${s^L}{\left\| {\mathfrak{W}(s)} \right\|_{L_t^1L_x^2}} \le {s^L}{e^{ - s/8}}{\left\| {\mathfrak{W}(\frac{s}{2})} \right\|_{L_t^1L_x^2}} + {s^L}\int_{\frac{s}{2}}^s {{e^{ - (s - \tau )/8}}{{\left\| {{G_2}(\tau )} \right\|}_{L_t^1L_x^2}}d\tau }. $$ Then by Hausdorff-Young and (\ref{time5})-(\ref{time7}), for $s\ge1$ \begin{align}\label{1wuhusd} \|\mathfrak{W}\|_{L^1_tL^2_x}\le \varepsilon^2_1s^{-L}. \end{align} Similarly, we have for $s\in[1,\infty)$ \begin{align}\label{wuhusd} \|\nabla \mathfrak{W}\|_{L^1_tL^2_x}+\|\Delta \mathfrak{W}\|_{L^1_tL^2_x}\le \varepsilon^2_1s^{-L}. \end{align} Thus the longtime part of (\ref{time8}) now results from (\ref{1wuhusd}), (\ref{wuhusd}). \end{proof} \begin{lemma} Suppose that (\ref{boot2}) and (\ref{boot5}) hold, then for $0<\gamma\ll1$ \begin{align} \left\| s^{-\frac{1}{2}}\mathfrak{W}(s)\right\|_{L_s^\infty L_t^2L_x^{3+\gamma}}+\left\| \omega(s)\mathfrak{W}(s)\right\|_{L_s^\infty L_t^2L_x^{3+\gamma}}& \le {\varepsilon _1}\label{time90}\\ {\left\|{\omega (s){\partial _t}{\phi _t}(s)} \right\|_{L_s^\infty L_t^2L_x^{3+\gamma}}} &\le \varepsilon_1.\label{time3}\\ {\left\|{{\partial _t}{A_t}(s)} \right\|_{L_s^\infty L_t^2L_x^{3+\gamma}}} &\le \varepsilon_1.\label{time37} \end{align} \end{lemma} \begin{proof} (\ref{time3}) is a direct corollary of (\ref{time90}). In fact, the definition of the wave map tension field gives \begin{align*} D_t\phi_t=\phi_s+\mathfrak{W}(s). \end{align*} Hence $\partial_t\phi_t$ is bounded by $|\phi_s|+|A_t\phi_t|+|\mathfrak{W}|$, then (\ref{time3}) follows by (\ref{boot5}), (\ref{time90}), (\ref{time2}) and (\ref{aaaw811}). (\ref{time90}) follows by the same arguments as (\ref{time8}). The only difference is to use \begin{align*} {\left\| {{h^{ii}}\left( {{\partial _t}\widetilde{u} \wedge {\partial _i}\widetilde{u}} \right){\nabla _i}{\partial _t}\widetilde{u}} \right\|_{L_t^2L_x^{3+\gamma}}} \le {\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_t^2L_x^{6+2\gamma}}}{\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^\infty L_x^{12+4\gamma}}}{\left\| {d\widetilde{u}} \right\|_{L_t^\infty L_x^{12+4\gamma}}}, \end{align*} where the term ${\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^\infty L_x^{12+4\gamma}}}{\left\| {d\widetilde{u}} \right\|_{L_t^\infty L_x^{12+4\gamma}}}$ is bounded by Soboelv embedding and Proposition \ref{sl}. It remains to prove (\ref{time37}). By the definition of $D_t$ and $A_t$, we have \begin{align*} \left| {{\partial _t}{A_t}(s)} \right| &\le \int_s^\infty {\left| {{\partial _t}{\phi _t}} \right|\left| {{\phi _s}} \right|} d\kappa + \int_s^\infty {\left| {{\partial _t}{\phi _s}} \right|\left| {{\phi _t}} \right|d\kappa} \\ &\le \int_s^\infty {\left| {{D_t}{\phi _t}} \right|\left| {{\phi _s}} \right|} d\kappa + \int_s^\infty {\left| {{A_t}} \right|\left| {{\phi _s}} \right|d\kappa} + \int_s^\infty {\left| {{\partial _t}{\phi _s}} \right|\left| {{\phi _t}} \right|d\kappa}, \end{align*} By $\mathfrak{W}=D_t\phi_t-\phi_s$ and H\"older, \begin{align} &{\left\| {\int_s^\infty {\left| {{D_t}{\phi _t}} \right|\left| {{\phi _s}} \right|} d\kappa} \right\|_{L_t^2L_x^{3+\gamma}}}\nonumber\\ &\le \int_s^\infty {{{\left\| w \right\|}_{L_t^2L_x^{3+\gamma}}}{{\left\| {{\partial _s}\widetilde{u}} \right\|}_{L_t^\infty L_x^\infty }}} d\kappa + \int_s^\infty {{{\left\| {{\phi _s}} \right\|}_{L_t^\infty L_x^{6+2\gamma}}}} {\left\| {{\phi _s}} \right\|_{L_t^2L_x^{6+2\gamma}}}d\kappa.\label{huqiancvb} \end{align} Since ${\| {{\phi _s}}\|_{L_x^{6+2\gamma}}} \le {\| {{{\left| D \right|}^{\frac{1}{2}}}{\phi _s}} \|_{L_x^p}}$ for $p\in(4,6)$, then (\ref{huqiancvb}) is acceptable by Proposition \ref{sl} and Proposition \ref{bootstrap}. Again by H\"older and Sobolev embedding, for $\frac{1}{m}+\frac{1}{4}=\frac{1}{3+\gamma}$ \begin{align*} {\left\| {\int_s^\infty {\left| {{\partial _t}{\phi _s}} \right|\left| {{\phi _t}} \right|d\kappa} } \right\|_{L_t^2L_x^{3+\gamma}}} &\le \int_s^\infty {{{\left\| {{\partial _s}{\phi _t}} \right\|}_{L_t^2L_x^4}}} {\left\| {{\phi _t}} \right\|_{L_t^\infty L_x^m}}d\kappa \\ &\le \int_s^\infty {{{\left\| {{\partial _s}{\phi _t}} \right\|}_{L_t^2L_x^4}}} {\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{L_x^2}}d\kappa. \end{align*} Since $|\partial_s\phi_t|\le|\partial_t\phi_s|+|A_t\phi_s|$, this is also acceptable by Proposition \ref{sl}, Proposition \ref{bootstrap}, (\ref{aaop11}), and (\ref{aaaw811}). Thus (\ref{time37}) follows. \end{proof} \begin{proposition}\label{cuihua} Suppose that (\ref{boot2}) and (\ref{boot5}) hold. Then we have for $p\in(2,6)$ \begin{align} &{\left\| {\omega (s){{\left| D \right|}^{ - \frac{1}{2}}}{\partial _t}{\phi _s}} \right\|_{L_s^\infty L_t^2L_x^p([0,T] \times {\Bbb H^2})}} + {\left\| {\omega (s){{\left| D \right|}^{\frac{1}{2}}}{\phi _s}} \right\|_{L^{\infty}_sL_t^2L_x^p([0,T] \times {\Bbb H^2})}}\nonumber\\ &+ {\left\| {\omega (s){\partial _t}{\phi _s}} \right\|_{L_s^\infty L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}} + {\left\| {\omega (s)\nabla {\phi _s}} \right\|_{L_s^\infty L_t^\infty L_x^2([0,T] \times {H^2})}} \le \varepsilon _1^2. \label{huoji} \end{align} \end{proposition} \noindent{\bf{Proof}} By Lemma \ref{hushuo} and Proposition \ref{tianxia}, we obtain for any $p\in(2,6)$ \begin{align} &{\omega(s)}{\left\| {{\partial _t}{\phi _s}} \right\|_{L_t^\infty L_x^2}} + {\omega(s)}{\left\| {\nabla {\phi _s}} \right\|_{L_t^\infty L_x^2}} + {\omega(s)}{\left\| {{{\left| \nabla \right|}^{\frac{1}{2}}}{\phi _s}} \right\|_{L_t^2L_x^p}}\nonumber\\ &+ {\omega(s)}{\left\| {{{\left| D \right|}^{ - \frac{1}{2}}}{\partial _t}{\phi _s}} \right\|_{L_t^2L_x^p}} +{\omega(s)}{\left\| {{\rho ^\sigma }\nabla {\phi _s}} \right\|_{L_t^2L_x^2}} \nonumber\\ &\lesssim {\omega(s)}{\left\| {{\partial _t}{\phi _s}(0,s,x)} \right\|_{L_x^2}} + {\omega(s)}{\left\| {\nabla {\phi _s}(0,s,x)} \right\|_{L_x^2}} + {\omega(s)}{\left\| G_4 \right\|_{L_t^1L_x^2}}.\label{chunjie1} \end{align} where $G_4$ denotes the inhomogeneous term. First, the $\phi_s(0,s,x)$ term is acceptable by Proposition \ref{sl}, $\mu_2\ll \varepsilon_1$ and $$ |\nabla_{t,x}\phi_s(0,s,x)|\le |\nabla_{t,x}\partial_sU|+\sqrt{h^{\gamma\gamma}}|A_{\gamma}||\partial_sU|, $$ where $U(s,x)$ is the heat flow initiated from $u_0$. Second, the three terms involved with $A_t$ are bounded by \begin{align*} {\omega(s)}{\left\| {{A_t}{\partial _t}{\phi _s}} \right\|_{L_t^1L_x^2}} &\le {\left\| {{A_t}} \right\|_{L_t^1L_x^\infty }}{\omega(s)}{\left\| {{\partial _t}{\phi _s}} \right\|_{L_t^\infty L_x^2}} \\ {\omega(s)}{\left\| {{A_t}{A_t}{\phi _s}} \right\|_{L_t^1L_x^2}} &\le {\left\| {{A_t}} \right\|_{L_t^1L_x^\infty }}{\left\| {{A_t}} \right\|_{L_t^\infty L_x^\infty }}{\omega(s)}{\left\| {{\phi _s}} \right\|_{L_t^\infty L_x^2}} \\ {\omega(s)}{\left\| {{\partial _t}{A_t}{\phi _s}} \right\|_{L_t^1L_x^2}} &\le {\left\| {{\partial _t}{A_t}} \right\|_{L_t^2L_x^{3+\gamma}}}{\omega(s)}{\left\| {{\phi _s}} \right\|_{L_t^2L_x^k}}, \end{align*} where $\frac{1}{k}+\frac{1}{3+\gamma}=\frac{1}{2},$ and $k\in(2,6)$. They are admissible by (\ref{aaaw811}), (\ref{aaop11}) and (\ref{time37}). The $\partial_t\widetilde{u}$ term is bounded by \begin{align*} {\omega(s)}{\left\| {\mathbf{R}({\partial _t}\widetilde{u},{\partial _s}\widetilde{u})({\partial _t}\widetilde{u})} \right\|_{L_t^1L_x^2}} \le {\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^2L_x^{6+2\gamma}}}{\left\| {{\partial _t}\widetilde{u}} \right\|_{L_t^\infty L_x^{6+2\gamma}}}{\omega(s)}{\left\| {{\phi _s}} \right\|_{L_t^2L_x^k}}, \end{align*} where $\frac{1}{k}+\frac{1}{3+\gamma}=\frac{1}{2},$ and $k\in(2,6)$. The $\partial_s\mathfrak{W}$ term is bounded by (\ref{time8}). The $A^{con}_i$ terms should be dealt with separately. We present the estimates for these terms as a lemma. \begin{lemma}[Continuation of Proof of Proposition \ref{cuihua}] Under the assumption of Proposition \ref{cuihua}, we have \begin{align} {\omega(s)}{\left\| {{h^{ii}}A_i^{con}{\partial _i}{\phi _s}} \right\|_{L_t^1L_x^2}} &\le {\varepsilon _1}{\omega(s)}{\left\| {{\rho ^\sigma }\nabla {\phi _s}} \right\|_{L_t^2L_x^2}} + \varepsilon _1^2\label{cuihua1}\\ {\omega(s)}{\left\| {{h^{ii}}A_i^{con}A_i^\infty {\phi _s}} \right\|_{L_t^1L_x^2}}&\le \varepsilon _1^2\label{cuihua2}\\ {\omega(s)}{\left\| {{h^{ii}}A_i^{con}A_i^{con} {\phi _s}} \right\|_{L_t^1L_x^2}}&\le \varepsilon _1^2\label{cuihua3}\\ {\omega(s)}{\left\| {{h^{ii}}\partial_iA_i^{con}{\phi _s}} \right\|_{L_t^1L_x^2}}&\le \varepsilon _1^2\label{cuihua4}\\ {\omega(s)}{\left\| {{h^{ii}}\Gamma _{ii}^kA_k^{con}} \phi_s\right\|_{L_t^1L_x^2}}&\le \varepsilon _1^2\label{cuihua5}. \end{align} \end{lemma} \begin{proof} Expanding $\phi_i$ as $\phi^{\infty}_i+\int^{\infty}_s\partial_s \phi_id\kappa$ yields \begin{align} A_i^{con} = \int_s^\infty {{\phi _i} \wedge {\phi _s}} d\kappa = \int_s^\infty {\left( {\int_{\kappa}^\infty {{\partial _s}} {\phi _i}(\tau )d\tau + \phi _i^\infty } \right) \wedge {\phi _s}} (\kappa)d\kappa. \end{align} Hence we get \begin{align*} &{\omega(s)}{\left\| {{h^{ii}}A_i^{con}{\partial _i}{\phi _s}} \right\|_{L_t^1L_x^2}} \\ &\le {\omega(s)}{\left\| {{h^{ii}}(\int_s^\infty \phi _i^\infty\wedge {{\phi _s}(\kappa)d\kappa} }) {\partial _i}{\phi _s}\right\|_{L_t^1L_x^2}}\\ &+ {\omega(s)}{\left\| {{h^{ii}} \left( {\int_s^\infty {{\phi _s}(\kappa) \wedge \left( {\int_{\kappa}^\infty {{\partial _s}} {\phi _i}(\tau )d\tau } \right)d\kappa} } \right)} {\partial _i}{\phi _s}\right\|_{L_t^1L_x^2}} \\ &\buildrel \Delta \over = B_1 + B_2 \end{align*} The $B_1$ term is bounded by \begin{align} B_1 &\lesssim {\omega(s)}{\left\| {{\rho ^\sigma }\nabla {\phi _s}} \right\|_{L_t^2L_x^2}}{\left\| {\int_s^\infty {{\rho ^{ - \sigma }}\phi _i^\infty\sqrt{h^{ii}} {\phi _s}(\kappa)d\kappa} } \right\|_{L_t^2L_x^{\infty}}}\nonumber \\ &\le {\omega(s)}{\left\| {{\rho ^\sigma }\nabla {\phi _s}} \right\|_{L_t^2L_x^2}}{\left\| {{\rho ^{ - \sigma }}\phi _i^\infty }\sqrt{h^{ii}} \right\|_{L_x^{\infty}}}\int_s^\infty {{{\left\| {{\phi _s}(\kappa)} \right\|}_{L_t^2L_x^{\infty}}}d\kappa} \nonumber \\ &\lesssim {\omega(s)}{\left\| {{\rho ^\sigma }\nabla {\phi _s}} \right\|_{L_t^2L_x^2}}{\left\| {{\rho ^{ - \sigma }}\sqrt{h^{ii}}\phi _i^\infty } \right\|_{L_x^{\infty}}}{\left\| {a(s){{\left\| {\nabla{\phi _s}(s)} \right\|}_{L_t^2L_x^{4}}}} \right\|_{L_s^\infty }},\label{guihua} \end{align} where we have used the Sobolev embedding in the last step. Hence Proposition \ref{bootstrap} gives an acceptable bound, \begin{align*} {B_1} \le C\mu_1\varepsilon_1{\omega(s)}{\left\| {{\rho ^\sigma }\nabla {\phi _s}} \right\|_{L_t^2L_x^2}}. \end{align*} The $B_2$ term is bounded by \begin{align*} {B_2} \lesssim {\omega(s)}{\left\| {\nabla {\phi _s}} \right\|_{L_t^\infty L_x^2}}\int_s^\infty {{{\left\| {{\phi _s}(\kappa)} \right\|}_{L_t^2L_x^\infty }}\left( {\int_{\kappa}^\infty {{{\left\| {\nabla {\phi _s}(\tau )} \right\|}_{L_t^2L_x^\infty }}} d\tau } \right)d\kappa}. \end{align*} Meanwhile, Sobolev embedding and Proposition \ref{bootstrap} give when $\tau \in (0,1)$ \begin{align*} {\left\| {\nabla {\phi _s}(\tau )} \right\|_{L_t^2L_x^\infty }} &\le {\left( {{\tau ^{\frac{3}{4}}}{{\left\| {\nabla {\phi _s}(\tau )} \right\|}_{L_t^2L_x^5}}} \right)^{3/5}}{\left( {{\tau ^{5/4}}{{\left\| {{\nabla ^2}{\phi _s}(\tau )} \right\|}_{L_t^2L_x^5}}} \right)^{2/5}}{\tau ^{ - \frac{1}{2} -9/20}} \\ &\le {\varepsilon _1}{\tau ^{ -19/20}}, \end{align*} and when $\tau \in [1,\infty)$ \begin{align*} {\left\| {\nabla {\phi _s}(\tau )} \right\|_{L_t^2L_x^\infty }} &\le {\left( {{\tau ^L}{{\left\| {\nabla {\phi _s}(\tau )} \right\|}_{L_t^2L_x^5}}} \right)^{3/5}}{\left( {{\tau ^L}{{\left\| {{\nabla ^2}{\phi _s}(\tau )} \right\|}_{L_t^2L_x^5}}} \right)^{2/5}}{\tau ^{ - L}} \le {\varepsilon _1}{\tau ^{ - L}}. \end{align*} Similarly we deduce by Sobolev embedding $\|f\|_{L^{\infty}}\le \||D|^{\frac{1}{2}}f\|_{L^5}$ that \begin{align*} {\left\| {{\phi _s}(\tau )} \right\|_{L_t^2L_x^\infty }} \le {\varepsilon _1}{\tau ^{ - \frac{1}{2}}},\mbox{ }{\rm{when}}\mbox{ }\tau \in (0,1);\mbox{ }{\left\| {{\phi _s}(\tau )} \right\|_{L_t^2L_x^\infty }} \le {\varepsilon _1}{\tau ^{ - L}},\mbox{ }{\rm{when}}\mbox{ }\tau \in [1,\infty ). \end{align*} Therefore we conclude \begin{align} B_2 \le \varepsilon _1^2{\omega(s)}{\left\| {\nabla {\phi _s}} \right\|_{L_t^\infty L_x^2}}\label{guihua1}. \end{align} Proposition \ref{bootstrap} together with (\ref{guihua}), (\ref{guihua1}) yields (\ref{cuihua1}). Next we prove (\ref{cuihua2}). H\"older yields \begin{align*} {\omega(s)}{\| {{h^{ii}}A_i^{con}A_i^\infty {\phi _s}}\|_{L_t^1L_x^2}} \le {\| {\sqrt {{h^{ii}}} A_i^{con}} \|_{L_t^2L_x^{\frac{10}{3}}}}{\omega(s)}{\| {{\phi _s}} \|_{L_t^2L_x^5}}. \end{align*} Using the expression $A_i^{con} = \int_s^\infty {{\phi _i} \wedge {\phi _s}} d\kappa$, we obtain \begin{align} {\left\| {\sqrt {{h^{ii}}} A_i^{con}} \right\|_{L_t^2L_x^{\frac{10}{3}}}} &\lesssim {\left\| {\int_s^\infty {\sqrt {{h^{ii}}} {\phi _i} \wedge {\phi _s}} d\kappa} \right\|_{L_t^2L_x^{\frac{10}{3}}}} \le {\| {d\widetilde{u}}\|_{L_t^\infty L_x^{10}}}\int_s^\infty {{{\left\| {{\phi _s}} \right\|}_{L_t^2L_x^5}}} d\kappa \nonumber\\ &\lesssim {\| {\nabla d\widetilde{u}}\|_{L_t^\infty L_x^2}}{\| {\omega(s){{\| {{\phi _s}(s)} \|}_{L_t^2L_x^5}}}\|_{L_s^\infty }}.\label{guihua7} \end{align} Therefore Proposition \ref{bootstrap} gives (\ref{cuihua2}). Third, we verify (\ref{cuihua3}). H\"older yields \begin{align*} {\omega(s)}{\| {{h^{ii}}A_i^{con}A_i^{con}{\phi _s}} \|_{L_t^1L_x^2}} \le {\| {\sqrt {{h^{ii}}} A_i^{con}} \|_{L_t^2L_x^{\frac{10}{3}}}}{\| {\sqrt {{h^{ii}}} A_i^{con}} \|_{L_t^\infty L_x^\infty }}{\omega(s)}{\| {{\phi _s}} \|_{L_t^2L_x^5}}. \end{align*} The term ${\| {\sqrt {{h^{ii}}} A_i^{con}}\|_{L_t^2L_x^{\frac{10}{3}}}}$ has been estimated in (\ref{guihua7}). The ${\| {\sqrt {{h^{ii}}} A_i^{con}} \|_{L_t^{\infty}L_x^\infty}}$ term is bounded by \begin{align*} {\| {\sqrt {{h^{ii}}} A_i^{con}}\|_{L_t^{\infty}L_x^\infty }} \lesssim \left\| {\int_s^\infty {{{\| {d\widetilde{u}} \|}_{L^\infty_{x} }}{{\| {{\phi _s}} \|}_{L^\infty _{x}}}} d\kappa} \right\|_{L^{\infty}_t}. \end{align*} This is acceptable by Proposition \ref{sl} and Lemma \ref{ktao1}. Forth, we prove (\ref{cuihua4}). H\"older yields \begin{align*} {\omega(s)}{\left\| {{h^{ii}}\left( {{\partial _i}A_i^{con}} \right){\phi _s}} \right\|_{L_t^1L_x^2}} \le {\left\| {{h^{ii}}{\partial _i}A_i^{con}} \right\|_{L_t^2L_x^4}}{\omega(s)}{\left\| {{\phi _s}} \right\|_{L_t^2L_x^4}}. \end{align*} The $h^{ii}\partial_iA_i$ term is bounded by \begin{align*} & {\left\| {{h^{ii}}{\partial _i}A_i^{con}} \right\|_{L_t^2L_x^4}} = {\left\| {\int_s^\infty {{h^{ii}}{\partial _i}} {\phi _i}{\phi _s}d\kappa + \int_s^\infty {{h^{ii}}} {\phi _i}{\partial _i}{\phi _s}d\kappa} \right\|_{L_t^2L_x^4}} \\ &\le \int_s^\infty {{{\left\| {{h^{ii}}{\partial _i}{\phi _i}} \right\|}_{L_t^\infty L_x^{20}}}} {\left\| {{\phi _s}} \right\|_{L_t^2L_x^5}}d\kappa + {\int_s^\infty {\left\| {d\widetilde{u}} \right\|} _{L_t^{\infty}L_x^\infty }}{\left\| {\nabla {\phi _s}} \right\|_{L_t^2L_x^4}}d\kappa \\ &\le \int_s^\infty {\left( {{{\left\| {\nabla d\widetilde{u}} \right\|}_{L_t^\infty L_x^{20}}} + {{\left\| {{h^{ii}}{A_i}{\phi _i}} \right\|}_{L_t^\infty L_x^{20}}}} \right)} {\left\| {{\phi _s}} \right\|_{L_t^2L_x^5}}d\kappa \\ &+ {\int_s^\infty {\left\| {d\widetilde{u}} \right\|} _{L_t^{\infty}L_x^\infty }}{\left\| {\nabla {\phi _s}} \right\|_{L_t^2L_x^4}}d\kappa. \end{align*} Thus this is acceptable by Proposition \ref{aaop} and interpolation between the $\|\nabla d\widetilde{u}\|_{L^{\infty}}$ bound and the $\|\nabla d\widetilde{u}\|_{L^{2}}$ bound in Proposition \ref{sl}. Finally we notice that (\ref{cuihua5}) is a consequence of (\ref{guihua7}) and $${\omega(s)}{\left\| {{h^{ii}}\Gamma _{ii}^kA_k^{con}} \phi_s\right\|_{L_t^1L_x^2}} \le {\left\| {A_2^{con}} \right\|_{L_t^2L_x^{\frac{10}{3}}}}{\omega(s)}{\left\| {{\phi _s}} \right\|_{L_t^2L_x^5}}. $$ \end{proof} $\blacksquare$ Proposition \ref{bootstrap} with Proposition \ref{cuihua} yields \begin{proposition}\label{xiaozi} Assume that the solution to (\ref{wmap1}) satisfies (\ref{boot5}) and (\ref{boot2}), then for any $p\in(2,6)$, $\theta\in[0,2]$ \begin{align*} {\left\| \omega(s){\nabla\phi_s} \right\|_{L^{\infty}_sL_t^\infty L_x^2}} + {\left\|\omega(s) |D|^{\frac{1}{2}}{\phi_s} \right\|_{L^{\infty}_sL_t^2 L_x^p}} &\le {\varepsilon^2 _1}\\ {\left\| \omega_1(s)|D|{\partial _t}\phi_s \right\|_{L^{\infty}_sL_t^2L_x^p}} +\left\|\omega_{\theta}(s) (-\Delta)^{\theta}{\phi_s} \right\|_{L^{\infty}_sL_t^2L_x^p}&\le {\varepsilon^2 _1}. \end{align*} \end{proposition} \ \subsection{Close all the bootstrap} \begin{lemma} Assume that the solution to (\ref{wmap1}) satisfies (\ref{boot5}) and (\ref{boot2}), then for any $p\in(2,6+2\gamma]$ \begin{align} {\left\| {(du,\partial_tu)} \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}} + {\left\| {(\nabla{\partial _t}u,\nabla du)} \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}}&\le {\varepsilon^2 _1}\label{boot8}\\ {\left\| {{\partial _t}u} \right\|_{L_t^2L_x^p([0,T] \times {\Bbb H^2})}} &\le {\varepsilon^2_1}.\label{boot9} \end{align} \end{lemma} \begin{proof} First we prove (\ref{boot9}). By $D_s\phi_t=D_t\phi_s$, $A_s=0$, one has \begin{align} {\left\| {{\phi _t}(0,t,x)} \right\|_{L_t^2L_x^p}} \le {\left\| {\int_0^\infty {\left| {{\partial _s}{\phi _t}} \right|} ds} \right\|_{L_t^2L_x^p}} \le {\left\| {{\partial _t}{\phi _s}} \right\|_{L_s^1L_t^2L_x^p}}+{\left\| {{A _t}{\phi _s}} \right\|_{L_s^1L_t^2L_x^p}}. \end{align} Sobolev embedding gives $${\left\| {{\partial _t}{\phi _s}} \right\|_{L_x^{6 + 2\gamma }}} + {\left\| {{\phi _s}} \right\|_{L_x^{6 + 2\gamma }}} \le {\left\| {{{\left( { - \Delta } \right)}^\vartheta }{\partial _t}{\phi _s}} \right\|_{L_x^{6 - \eta }}} + {\left\| {{{\left( { - \Delta } \right)}^\vartheta }{\phi _s}} \right\|_{L_x^{6 - \eta }}}, $$ where $\frac{\vartheta }{2} = \frac{1}{{6 - \eta }} - \frac{1}{{6 + 2\gamma }}$, $0<\eta\ll1,0<\gamma\ll1$. Thus (\ref{boot9}) follows by Proposition \ref{xiaozi} and Proposition \ref{aaop}. Second, we prove (\ref{boot8}). By Remark \ref{3sect}, ${\phi _i}(0,t,x) = \phi _i^\infty + \int_0^\infty {{\partial _s}{\phi _i}d\kappa}.$ Since $|d\widetilde{u}|\le \sqrt{h^{ii}}|\phi_i|$, $\|\sqrt{h^{ii}}\phi _i^\infty\|_{L^2_x}\le \|dQ\|_{L^2}\le \mu_1$, it suffices to verify for any $t,x\in[0,T]\times\Bbb H^2$ $$\int_0^\infty \|\sqrt{h^{ii}}{\partial _s}{\phi _i}\|_{L^2_x}d\kappa\le \varepsilon^2_1. $$ This is acceptable by Proposition \ref{aaop}, Proposition \ref{xiaozi} and $|\sqrt{h^{ii}}\partial_s\phi_i|\le |\nabla \phi_s|+\sqrt{h^{ii}}|A_i||\phi_s|$. Recalling (\ref{poijn}) for the equation of $\phi_s$ evolving along the heat flow, we have by integration by parts, \begin{align*} \frac{d}{{ds}}\left\| {\tau (\widetilde{u})} \right\|_{L_x^2}^2 &= \frac{d}{{ds}}\left\langle {{\phi _s},{\phi _s}} \right\rangle = 2\left\langle {{D_s}{\phi _s},{\phi _s}} \right\rangle \\ &= 2{h^{ii}}\left\langle {{D_i}{D_i}{\phi _s} - \Gamma _{ii}^k{D_k}{\phi _s},{\phi _s}} \right\rangle +\left\langle h^{ij}(\phi_s\wedge\phi_i)\phi_j,\phi_s\right\rangle\\ &= - 2{h^{ii}}\left\langle {{D_i}{\phi _s},{D_i}{\phi _s}} \right\rangle+\left\langle h^{ij}(\phi_s\wedge\phi_i)\phi_j,\phi_s\right\rangle. \end{align*} Hence $\|\partial_s\widetilde{u}\|_{L^2_x}\le e^{-\delta s}$ shows \begin{align} &\left\| {\tau (\widetilde{u}(0,t,x))} \right\|_{L_x^2}^2\lesssim \int_0^\infty {{h^{ii}}\left\langle {{D_i}{\phi _s},{D_i}{\phi _s}} \right\rangle } ds \nonumber\\ &\lesssim \int_0^\infty {\left\langle {\nabla {\phi _s},\nabla {\phi _s}} \right\rangle } ds + \int_0^\infty {{h^{ii}}\left\langle {{A_i}{\phi _s},{A_i}{\phi _s}} \right\rangle } ds+\int^{\infty}_0|d\widetilde{u}|^2|\phi_s|^2ds.\label{muijbnc} \end{align} The nonnegative sectional curvature property of $N=\Bbb H^2$ with integration by parts implies $$\left\| {\nabla d\widetilde{u}} \right\|_{L_x^2}^2 \lesssim \left\| {\tau (\widetilde{u})} \right\|_{L_x^2}^2 + \left\| {d\widetilde{u}} \right\|_{L_x^2}^2. $$ Hence (\ref{muijbnc}) gives \begin{align} &\left\| {\nabla d\widetilde{u}(0,t,x)} \right\|_{L_x^2}^2 \nonumber\\ &\lesssim \int_0^\infty {\left\langle {\nabla {\phi _s},\nabla {\phi _s}} \right\rangle } ds +\int^{\infty}_0|d\widetilde{u}|^2|\phi_s|^2ds+\int_0^\infty {{h^{ii}}\left\langle {{A_i}{\phi _s},{A_i}{\phi _s}} \right\rangle } ds+\|d\widetilde{u}(0,t,x)\|^2_{L^2_x}.\label{hjbn7v89} \end{align} Since the $|d\widetilde{u}|$ term has been estimated, by Proposition \ref{xiaozi} , Proposition \ref{aaop} and (\ref{hjbn7v89}), $$ {\left\| {\nabla d\widetilde{u}} \right\|_{L_t^\infty L_x^2([0,T] \times {\Bbb H^2})}}\le \varepsilon^2_1. $$ Finally we prove the desired estimates for $|\nabla\partial_t\widetilde{u}|$. Integration by parts yields, \begin{align*} &\frac{d}{{ds}}\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2 = \frac{d}{{ds}}{h^{ii}}\left\langle {{D_i}{\phi _t},{D_i}{\phi _t}} \right\rangle = 2{h^{ii}}\left\langle {{D_s}{D_i}{\phi _t},{D_i}{\phi _t}} \right\rangle \\ &= 2{h^{ii}}\left\langle {{D_i}{D_t}{\phi _s},{D_i}{\phi _t}} \right\rangle + 2{h^{ii}}\left\langle {({\phi _s} \wedge {\phi _i}){\phi _t},{D_i}{\phi _t}} \right\rangle \\ &= - 2{h^{ii}}\left\langle {{D_t}{\phi _s},{D_i}{D_i}{\phi _t}} \right\rangle + 2\left\langle {{D_t}{\phi _s},{D_2}{\phi _t}} \right\rangle + 2{h^{ii}}\left\langle {({\phi _s} \wedge {\phi _i}){\phi _t},{D_i}{\phi _t}} \right\rangle \\ &= - 2\left\langle {{D_t}{\phi _s},{h^{ii}}{D_i}{D_i}{\phi _t} - {h^{ii}}\Gamma _{ii}^k{D_k}{\phi _t}} \right\rangle + 2{h^{ii}}\left\langle {({\phi _s} \wedge {\phi _i}){\phi _t},{D_i}{\phi _t}} \right\rangle. \end{align*} Recall (\ref{yfcvbn}), the parabolic equation of $\phi_t$ along heat flow, then $$\frac{d}{{ds}}\left\| {\nabla {\partial _t}\widetilde{u}} \right\|_{{L^2}}^2 = - 2\left\langle {{D_t}{\phi _s},{D_s}{\phi _t}} \right\rangle + 2{h^{ii}}\left\langle {({\phi _s} \wedge {\phi _i}){\phi _t},{D_i}{\phi _t}} \right\rangle + 2{h^{ii}}\left\langle {{D_t}{\phi _s},({\phi _t} \wedge {\phi _i}){\phi _i}} \right\rangle. $$ Hence we conclude, \begin{align*} &\left\| {\nabla {\partial _t}\widetilde{u}(0,t,x)} \right\|_{{L^2}}^2\\ &\lesssim \int_0^\infty {\left\langle {{\partial _t}{\phi _s},{\partial _t}{\phi _s}} \right\rangle d\kappa} + \int_0^\infty {\left\langle {{A_t}{\phi _s},{A_t}{\phi _s}} \right\rangle d\kappa} \\ &+ \int_0^\infty {{{\left\| {{\phi _s}} \right\|}_{L_x^2}}{{\left\| {d\widetilde{u}} \right\|}_{L_x^\infty }}{{\left\| {{\partial _t}\widetilde{u}} \right\|}_{L_x^\infty }}{{\left\| {\nabla {\partial _t}\widetilde{u}} \right\|}_{L_x^2}}d\kappa} + {\int_0^\infty {\left\| {{\partial _t}\widetilde{u}} \right\|} _{L_x^2}}\left\| {d\widetilde{u}} \right\|_{L_x^\infty }^2{\left\| {{D_t}{\phi _s}} \right\|_{L_x^2}}d\kappa. \end{align*} Thus by Proposition \ref{sl}, Proposition \ref{xiaozi} and Proposition \ref{aaop}, we have $$\left\| {\nabla {\partial _t}\widetilde{u}(0,t,x)} \right\|_{{L^2}}^2 \le \varepsilon _1^4. $$ Therefore, we have proved all estimates in (\ref{boot8}) and (\ref{boot9}). \end{proof} We summarize what we have proved in the following corollary. \begin{corollary} Assume $(-T^*,T_*)$ is the lifespan of solution to (\ref{wmap1}). And let $\mu_1,\mu_2$ be sufficiently small, then we have \begin{align*} {\left\| {du} \right\|_{L_t^\infty L_x^2([0,{T_*}] \times {\Bbb H^2})}}& + {\left\| {{\partial _t}u} \right\|_{L_t^\infty L_x^2([ 0,{T_*}] \times {\Bbb H^2})}} + {\left\| {\nabla du} \right\|_{L_t^\infty L_x^2([0,{T_*}] \times {\Bbb H^2})}} \\ &+ {\left\| {\nabla {\partial _t}u} \right\|_{L_t^\infty L_x^2([0,{T_*}] \times {\Bbb H^2})}} + {\left\| {{\partial _t}u} \right\|_{L_t^2L_x^6([ 0,{T_*}] \times {\Bbb H^2})}} \le \varepsilon _1^2. \end{align*} Thus by Proposition \ref{global}, we have $(u,\partial_tu)$ is a global solution to (\ref{wmap1}). \end{corollary} \section{Proof of Theorem 1.1} Finally, we prove Theorem 1.1 based on Proposition \ref{xiaozi}. \begin{proposition} Let $u$ be the solution to (\ref{wmap1}) in $\mathcal{X}_{[0,\infty)}$. Then as $t\to \infty$, $u(t,x)$ converges to a harmonic map, namely $$ \mathop {\lim }\limits_{t \to \infty } \mathop {\lim }\limits_{x\in\Bbb H^2 }{\rm{dist}}_{\Bbb H^2}(u(t,x),Q(x))= 0, $$ where $Q(x):\Bbb H^2\to \Bbb H^2$ is the unperturbed harmonic map. \end{proposition} \begin{proof} For $u(t,x)$, by Proposition \ref{3.3}, we have the corresponding heat flow converges to some harmonic map uniformly for $x\in\Bbb H^2$. Then by the definition of the distance on complete manifolds, we have \begin{align}\label{ppo0} {\rm{dist}_{{\Bbb H^2}}}(u(t,x),Q(x)) \le \int_0^\infty {{{\left\| {{\partial _s}\widetilde{u}} \right\|}_{L_x^\infty }}ds}. \end{align} For any $T>0$, $\mu>0$, since $|\partial_s\widetilde{u}|$ satisfies $(\partial_s-\Delta)|\partial_s\widetilde{u}|\le 0$, one has \begin{align} \int_T^\infty {{{\left\| {{\partial _s}\widetilde{u}(s,t,x)} \right\|}_{L_x^\infty }}ds} &\lesssim \int_T^\infty {{e^{ - \frac{1}{8}s}}{{\left\| {\tau(u(t,x))} \right\|}_{L_x^2}}} ds \lesssim {e^{ - T/8}}{\left\| {\nabla du(t,x)} \right\|_{L_x^2}} \label{mu1}\\ \int_0^\mu {{{\left\| {{\partial _s}\widetilde{u}(s,t,x)} \right\|}_{L_x^\infty }}ds} &\lesssim \int_0^\mu {{{\left\| {{e^{s{\Delta _{{\Bbb H^2}}}}}\tau(u(t,x))} \right\|}_{L_x^\infty }}} ds \le \int_0^\mu {{s^{ - \frac{1}{2}}}{{\left\| {\nabla du(t,x)} \right\|}_{L_x^2}}} ds \nonumber\\ &\lesssim {\mu ^{\frac{1}{2}}}{\left\| {\nabla du(t,x)} \right\|_{L_x^2}} \label{mu2} \end{align} Similarly, we have \begin{align} \int_\mu ^T {{{\left\| {{\partial _s}\widetilde{u}(s,t,x)} \right\|}_{L_x^\infty }}ds} &\lesssim \int_\mu ^T {{{\left\| {{e^{(s - \frac{\mu }{2}){\Delta _{{\Bbb H^2}}}}}{\partial _s}\widetilde{u}(\frac{\mu}{2},t,x)} \right\|}_{L_x^\infty }}} ds \nonumber\\ &\lesssim \int_\mu ^T {{{(s - \frac{\mu}{2})}^{ - \frac{1}{4}}}{{\left\| {{\partial _s}\widetilde{u}(\frac{\mu}{2},t,x)} \right\|}_{L_x^4}}} ds \nonumber\\ &\lesssim {\mu ^{ - \frac{1}{4}}}\int_\mu ^T {{{\left\| {{\phi _s}(\frac{\mu}{2},t,x)} \right\|}_{L_x^4}}} ds.\label{mu3} \end{align} Therefore it suffices to prove for a fixed $\mu>0$ \begin{align}\label{8ding} \mathop {\lim }\limits_{t \to \infty } {\left\| {{\phi _s}(\mu )} \right\|_{L_x^4}} = 0. \end{align} Proposition \ref{xiaozi} implies ${\mu ^{\frac{1}{2}}}{\left\| {{\phi _s}(\mu )} \right\|_{L_t^2L_x^4}}+{\mu ^{\frac{1}{2}}}{\left\| \partial_t{{\phi _s}(\mu )} \right\|_{L_t^2L_x^4}}< \infty $, thus for any $\epsilon>0$ there exists a $T_0$ such that \begin{align}\label{7vgj} {\left\| {{\phi _s}(\mu )} \right\|_{L_t^2L_x^4([{T_0},\infty ) \times {\Bbb H^2})}}+ {\left\|\partial_t {{\phi _s}(\mu )} \right\|_{L_t^2L_x^4([{T_0},\infty ) \times {\Bbb H^2})}}< \epsilon. \end{align} Particularly, for any interval $[a,a+1]$ of length one with $a\ge T_0$, there exists some $t_{a}\in[a,a+1]$ such that \begin{align}\label{shipo} {\left\| {{\phi _s}(\mu ,{t_{a}})} \right\|_{L_x^4}} \le \epsilon/2. \end{align} Then by fundamental theorem of calculus for any $t'\in[a,a+1]$ \begin{align}\label{gouq9} \left| {{{\left\| {{\phi _s}(\mu ,t')} \right\|}_{L_x^4}} - {{\left\| {{\phi _s}(\mu ,{t_a})} \right\|}_{L_x^4}}} \right| \le \int_{{t_a}}^{t'} {\left| {{\partial _t}{{\left\| {{\phi _s}(\mu ,t)} \right\|}_{L_x^4}}} \right|} dt. \end{align} Since $|{\partial _t}{\left\| {{\phi _s}(\mu ,t)} \right\|_{L_x^4}}|\le {\left\| {{\partial _t}{\phi _s}(\mu ,t)} \right\|_{L_x^4}}$, by H\"older, (\ref{gouq9}) and (\ref{7vgj}) show \begin{align*} \left| {{{\left\| {{\phi _s}(\mu ,t')} \right\|}_{L_x^4}} - {{\left\| {{\phi _s}(\mu ,{t_a})} \right\|}_{L_x^4}}} \right| \le {\left\| {{\partial _t}{\phi _s}(\mu ,t)} \right\|_{L_t^2L_x^4}}{(t' - a)^{\frac{1}{2}}} \le {\left\| {{\partial _t}{\phi _s}(\mu ,t)} \right\|_{L_t^2L_x^4}}. \end{align*} Thus we have by (\ref{shipo}) that for any $t\in[a,a+1]$, \begin{align*} {\left\| {{\phi _s}(\mu ,t)} \right\|_{L_x^4}}\le \epsilon. \end{align*} Since $a$ is arbitrary chosen, we obtain (\ref{8ding}). Therefore, Theorem 1.1 is proved, \end{proof} \section{Proof of remaining lemmas and claims} We first collect some useful inequalities for the harmonic maps. \begin{lemma} Suppose that $Q$ is an admissible harmonic map in Theorem 1.1. If $0<\mu_1\ll 1$, then \begin{align} \|\nabla dQ\|_{L^2}&\lesssim \mu_1\label{shubai4}\\ \|\nabla^2 dQ\|_{L^2}&\lesssim \mu_1.\label{tianren78} \end{align} \end{lemma} \begin{proof} By integration by parts and the non-positive sectional curvature of $N=\Bbb H^2$, \begin{align*} \|\nabla dQ\|^2_{L^2}&\lesssim \|dQ\|^2_{L^2}+\|\tau(Q)\|^2_{L^2}\\ \|\nabla^2 dQ\|^2_{L^2}&\lesssim \|\nabla\tau(Q)\|^2_{L^2}+\|\nabla dQ\|^3_{L^2}+\|\nabla dQ\|^2_{L^4}\| dQ\|^2_{L^4}+\| dQ\|^6_{L^2}. \end{align*} Hence by $\tau(Q)=0$, we have (\ref{shubai4}). And then (\ref{tianren78}) follows from (\ref{as4}), Gagliardo-Nirenberg inequality and Sobolev embedding. \end{proof} Now we prove Corollary 2.1. \begin{lemma} Fix $R_0>0$, let $0<\mu_1,\mu_2\ll\mu_3\ll1,$ then the initial data $(u_0,u_1)$ in Theorem 1.1 satisfy \begin{align} \|du_0\|_{L^2}+\|u_1\|_{L^2}+\|\nabla du_0\|_{L^2}+\|\nabla u_1\|_{L^2}\le \mu_3.\label{ojvbhuy} \end{align} \end{lemma} \begin{proof} First by (\ref{shubai4}), the harmonic map $Q$ satisfies \begin{align}\label{kijncvbn} \|\nabla dQ\|_{L^2}+\|dQ\|_{L^2}\le \mu_1. \end{align} By (1.4) and Sobolev embedding, \begin{align} \|u^k_0-Q^k\|_{L^{\infty}}\lesssim \|u^k_0-Q^k\|_{H^2}\le \mu_2. \end{align} Hence $|u^1_0|+|u^2_0|\lesssim R_0+\mu_2.$ Then choosing $R=CR_0+C\mu_2$ in [Lemma 2.3,\cite{LZ}], we have \begin{align}\label{zxzxsdu} \|du_0\|_{L^2}+\|\nabla du_0\|_{L^2}\le Ce^{8(CR_0+C\mu_2)}\big(\|\nabla^2 u^k_0\|_{L^2}+\|\nabla^2 u^k_0\|_{L^2}^2\big). \end{align} Again by [Lemma 2.3,\cite{LZ}] and (\ref{kijncvbn}), \begin{align}\label{zw34vbg} \|\nabla^2Q^k\|_{L^2}\le Ce^{8(R_0)}\big(\|\nabla dQ\|_{L^2}+\|\nabla dQ\|_{L^2}^2\big)\le Ce^{8(R_0)}\mu_1. \end{align} Therefore, (1.4), (\ref{zw34vbg}) and (\ref{zxzxsdu}) give \begin{align} \|du_0\|_{L^2}+\|\nabla du_0\|_{L^2}\le Ce^{8(CR_0+C\mu_2)}(\mu_1+\mu_2) \end{align} Let $\mu_1$ and $\mu_2$ be sufficiently small depending on $R_0$, we obtain \begin{align} \|du_0\|_{L^2}+\|\nabla du_0\|_{L^2}\le \mu_3. \end{align} \end{proof} \begin{lemma}\label{symm} Let $W$ be the magnetic operator defined in Lemma \ref{hushuo} as \begin{align} W\varphi = - 2 {h^{ii}}A_i^\infty {\partial _i}\varphi -{h^{ii}}A_i^\infty A_i^\infty \varphi -{h^{ii}}\left( {\varphi \wedge \phi _i^\infty } \right)\phi _i^\infty-h^{ii}(\partial_iA^{\infty}_{i}-\Gamma^k_{ii}A^{\infty}_k), \end{align} Then $W$ is symmetric with domain $C^{\infty}_c(\Bbb H^2,\Bbb C^2)$. And $-\Delta+W$ is strictly positive if $\mu_1$ is sufficiently small. \end{lemma} \begin{proof} Since we work with complex valued functions here, the wedge operator $\wedge$ should be first extended to the complex number field by taking the inner product in (\ref{nb890km}) to be the complex inner product. By the explicit formula for $\Gamma^{k}_{ii}$ and $h^{ii}$, one has \begin{align}\label{kjc6789} h^{ii}\Gamma^k_{ii}A^{\infty}_k=h^{11}\Gamma^2_{11}A^{\infty}_2=A^{\infty}_2. \end{align} It is easy to see by the non-positiveness and symmetry of the sectional curvature that $\varphi\longmapsto -{h^{ii}}\left( {\varphi \wedge \phi _i^\infty } \right)\phi _i^\infty$ is a non-negative and symmetric operator on $L^2(\Bbb H^2,\Bbb C^2)$. And by the skew-symmetry of $A^{\infty}_i$, $\varphi\longmapsto -{h^{ii}}\left( {\varphi \wedge A _i^\infty } \right)A_i^\infty$ is a non-negative and symmetric symmetric operator on $L^2(\Bbb H^2,\Bbb C^2)$. We claim that $$\varphi\longmapsto 2 {h^{ii}}A_i^\infty {\partial _i}\varphi +h^{ii}(\partial_iA^{\infty}_{i}-\Gamma^k_{ii}A^{\infty}_k) $$ is a symmetric operator on $L^2(\Bbb H^2,\Bbb C^2)$ as well. Indeed, by the skew-symmetry of $A^{\infty}_i$, $\partial_iA^{\infty}_i$, integration by parts and (\ref{kjc6789}), \begin{align*} &\left\langle {2{h^{ii}}A_i^\infty {\partial _i}f + {h^{ii}}({\partial _i}A_i^\infty - \Gamma _{ii}^kA_k^\infty )f,g} \right\rangle \\ &= \left\langle {2{h^{ii}}A_i^\infty {\partial _i}f + {h^{ii}}{\partial _i}A_i^\infty f - A_2^\infty f,g} \right\rangle \\ &= \left\langle {{h^{ii}}{\partial _i}A_i^\infty f - A_2^\infty f,g} \right\rangle - \left\langle {2{h^{ii}}{\partial _i}A_i^\infty f,g} \right\rangle - \left\langle {2{h^{ii}}A_i^\infty f,{\partial _i}g} \right\rangle + \left\langle {2{h^{22}}A_2^\infty f,g} \right\rangle \\ &= \left\langle { - {h^{ii}}{\partial _i}A_i^\infty f + A_2^\infty f,g} \right\rangle - \left\langle {2{h^{ii}}A_i^\infty f,{\partial _i}g} \right\rangle \\ &= \left\langle {f,{h^{ii}}{\partial _i}A_i^\infty g - A_2^\infty g} \right\rangle + \left\langle {f,2{h^{ii}}A_i^\infty {\partial _i}g} \right\rangle \\ &= \left\langle {f,2{h^{ii}}A_i^\infty {\partial _i}g + {h^{ii}}{\partial _i}A_i^\infty g - A_2^\infty g} \right\rangle. \end{align*} It remains to prove $-\Delta+W$ is positive. Since we have shown $\varphi\longmapsto -{h^{ii}}\left( {\varphi \wedge \phi _i^\infty } \right)\phi _i^\infty$ and $\varphi\longmapsto -{h^{ii}}\left( {\varphi \wedge A _i^\infty } \right)A_i^\infty$ are nonnegative, it suffices to prove for some $\delta>0$ $$\left\langle { - \Delta f + 2{h^{ii}}A_i^\infty {\partial _i}f + {h^{ii}}({\partial _i}A_i^\infty - \Gamma _{ii}^kA_k^\infty )f,f} \right\rangle \ge \delta \left\langle {f,f} \right\rangle. $$ By the skew-symmetry of $A^{\infty}_i$ and $\partial_iA^{\infty}_i$, it reduces to \begin{align*} \left\langle { - \Delta f + 2{h^{ii}}A_i^\infty {\partial _i}f,f} \right\rangle \ge \delta \left\langle {f,f} \right\rangle. \end{align*} H\"older, (\ref{{uv111}}) and (\ref{xuejin}) imply for some universal constant $c>0$ \begin{align*} & \left\langle { - \Delta f + 2{h^{ii}}A_i^\infty {\partial _i}f,f} \right\rangle \ge \left\| {\nabla f} \right\|_2^2 - 2{\left\| {\sqrt {{h^{ii}}} A_i^\infty } \right\|_\infty }{\left\| {\nabla f} \right\|_2}{\left\| f \right\|_2} \\ &\ge \frac{1}{2}\left\| {\nabla f} \right\|_2^2 + c\left\| f \right\|_2^2 - 2{\left\| {\sqrt {{h^{ii}}} A_i^\infty } \right\|_\infty }{\left\| {\nabla f} \right\|_2}{\left\| f \right\|_2} \\ &\ge \frac{1}{2}\left\| {\nabla f} \right\|_2^2 + c\left\| f \right\|_2^2 - 2{\mu _1}{\left\| {\nabla f} \right\|_2}{\left\| f \right\|_2}. \end{align*} Let $\mu_1$ be sufficiently small, then \begin{align*} \left\langle { - \Delta f + 2{h^{ii}}A_i^\infty {\partial _i}f,f} \right\rangle\ge \delta \left\langle {f,f} \right\rangle. \end{align*} \end{proof} Recall the equation of the tension field $\phi_s$: \begin{lemma} The evolution of differential fields and the heat tension filed along the heat flow are given by the following: \begin{align} &\partial_s\phi_s=h^{ii}D_iD_i\phi_s-h^{ii}\Gamma^k_{ii}D_k\phi_s+h^{ii}(\phi_s\wedge\phi_i)\phi_i \label{poijn}\\ &{\partial_s}{\phi _s}-\Delta {\phi _s}= 2{h^{ii}}{A_i}{\partial _i}{\phi _s} + {h^{ii}}\left( {{\partial _i}{A_i}} \right){\phi _s} - {h^{ii}}\Gamma _{ii}^k{A_k}{\phi _s} + {h^{ii}}{A_i}{A_i}{\phi _s}\nonumber \\ &+ {h^{ii}}\left( {{\phi _s} \wedge {\phi _i}} \right){\phi _i}\label{991}\\ &{\partial _s}{\phi _t} - \Delta {\phi _t} = 2h^{ii}{A_i}{\partial _i}{\phi _t} + h^{ii}{A_i}{A_i}{\phi _t} + h^{ii}{\partial _i}{A_i}{\phi _t} - {h^{ii}}\Gamma _{ii}^k{A_k}{\phi _t} \nonumber \\ &+ {h^{ii}}\left( {{\phi _t} \wedge {\phi _i}} \right){\phi _i}.\label{yfcvbn}\\ &{\partial _s}{\partial _t}{\phi _s}= \Delta {\partial _t}{\phi _s} + 2{h^{ii}}\left( {{\partial _t}{A_i}} \right){\partial _i}{\phi _s} + 2{h^{ii}}{A_i}{\partial _i}{\partial _t}{\phi _s} + {h^{ii}}\left( {{\partial _i}{\partial _t}{A_i}} \right){\phi _s}\nonumber\\ &+ {h^{ii}}\left( {{\partial _i}{A_i}} \right){\partial _t}{\phi _s} - {h^{ii}}\Gamma _{ii}^k\left( {{\partial _t}{A_k}} \right){\phi _s} - {h^{ii}}\Gamma _{ii}^k{A_k}{\partial _t}{\phi _s} + {h^{ii}}\left( {{\partial _t}{A_i}} \right){A_i}{\phi _s}\nonumber\\ &+ {h^{ii}}{A_i}\left( {{\partial _t}{A_i}} \right){\phi _s} + {h^{ii}}{A_i}{A_i}{\partial _t}{\phi _s} + {h^{ii}}\left( {{\partial _t}{\phi _s} \wedge {\phi _i}} \right){\phi _i}+ {h^{ii}}\left( {{\phi _s} \wedge {\partial _t}{\phi _i}} \right){\phi _i}\nonumber\\ & + {h^{ii}}\left( {{\phi _s} \wedge {\phi _i}} \right){\partial _t}{\phi _i}.\label{9923} \end{align} \end{lemma} \begin{proof} Recall that we use the orthogonal coordinates (\ref{vg}) throughout the paper. Recall the equation of $\phi_s$: \begin{align}\label{who} \phi_s=h^{ii}D_i\phi_i-h^{ii}\Gamma^k_{ii}\phi_k. \end{align} Applying $D_s$ to (\ref{who}) yields \begin{align*} &{D_s}{\phi _s} = {h^{ii}}{D_s}{D_i}{\phi _i} - {h^{ii}}\Gamma _{ii}^k{D_s}{\phi _k} = {h^{ii}}{D_i}{D_i}{\phi _s} - {h^{ii}}\Gamma _{ii}^k{D_k}{\phi _s} + {h^{ii}}\left( {{\phi _s} \wedge {\phi _i}} \right){\phi _i}\\ &= \Delta {\phi _s} + 2{h^{ii}}{A_i}{\partial _i}{\phi _s} + {h^{ii}}\left( {{\partial _i}{A_i}} \right){\phi _s} - {h^{ii}}\Gamma _{ii}^k{A_k}{\phi _s} + {h^{ii}}{A_i}{A_i}{\phi _s} + {h^{ii}}\left( {{\phi _s} \wedge {\phi _i}} \right){\phi _i}. \end{align*} The tension free identity and commutator identity give \begin{align*} {D_s}{\phi _t} &= {D_t}{\phi _s} = {D_t}\left( {{h^{ii}}{D_i}{\phi _j} - {h^{ii}}\Gamma _{ii}^k{\phi _k}} \right) = {h^{ii}}{D_t}{D_i}{\phi _i} - {h^{ii}}\Gamma _{ii}^k{D_t}{\phi _k} \\ &= {h^{ii}}{D_i}{D_i}{\phi _t} - {h^{ii}}\Gamma _{ii}^k{D_t}{\phi _k} + {h^{ii}}\left( {{\partial _t}u \wedge {\partial _i}u} \right){\partial _i}u. \end{align*} Therefore the differential filed $\phi_t$ satisfies \begin{align*} {\partial _s}{\phi _t} - \Delta {\phi _t} = 2h^{ii}{A_i}{\partial _i}{\phi _t} + h^{ii}{A_i}{A_i}{\phi _t} + h^{ii}{\partial _i}{A_i}{\phi _t} - {h^{ii}}\Gamma _{ii}^k{A_k}{\phi _t} + {h^{ii}}\left( {{\phi _t} \wedge {\phi _i}} \right){\phi _i}. \end{align*} Applying $\partial_t$ to (\ref{991}) gives (\ref{9923}). \end{proof} \section{Acknowledgments} We owe our thanks to the anonymous referee for helpful comments which greatly improved this paper. We thank Prof. Daniel Tataru for helpful comments on our work and especially the necessity of adding Remark 1.1. The first version of this paper is Chapter 3 of Li's thesis. We divide it into two parts for publication. Li owes gratitude to Prof. Youde Wang, Hao Yin, Cong Song for guidance on geometric PDEs and geometric analysis. \end{document}

\begin{document} \title{On the exterior Dirichlet problem for Hessian quotient equations\footnotemark[1]} \author{Dongsheng Li \footnotemark[3] \and Zhisu Li (\Letter)\footnotemark[2]~\footnotemark[3]} \renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \footnotetext[1]{This research is supported by NSFC.11671316.} \footnotetext[2]{Corresponding author.} \footnotetext[3]{School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China;\\ Dongsheng Li: \href{mailto:lidsh@mail.xjtu.edu.cn}{lidsh@mail.xjtu.edu.cn}; Zhisu Li: \href{mailto:lizhisu@stu.xjtu.edu.cn}{lizhisu@stu.xjtu.edu.cn}.} \date{} \maketitle \begin{abstract} In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Amp\`{e}re equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation. \end{abstract} \noindent\emph{Keywords:} {Dirichlet problem, existence and uniqueness, exterior domain, Hessian quotient equation, Perron's method, prescribed asymptotic behavior, viscosity solution} \noindent\emph{2010 MSC:} {35D40, 35J15, 35J25, 35J60, 35J96} \section{Introduction} In this paper, we consider the Dirichlet problem for the Hessian quotient equation \begin{equation}\label{eqn.hqe} \frac{\sigma_k(\lambda(D^2u))}{\sigma_l(\lambda(D^2u))}=1 \end{equation} in the exterior domain $\mathds R^n\setminus\overline{D}$, where $D$ is a bounded domain in $\mathds R^n$, $n\geq3$, $0\leq l<k\leq n$, $\lambda(D^2u)$ denotes the eigenvalue vector $\lambda:=(\lambda_1,\lambda_2,...,\lambda_n)$ of the Hessian matrix $D^2u$ of the function $u$, and \[\sigma_0(\lambda)\equiv1 \quad\text{and}\quad \sigma_j(\lambda):=\sum_{1\leq s_1<s_2<...<s_j\leq n} \lambda_{s_1}\lambda_{s_2}...\lambda_{s_j} ~(\forall 1\leq j\leq n)\] are the elementary symmetric functions of the $n$-vector $\lambda$. Note that when $l=0$, \eref{eqn.hqe} is the Hessian equation $\sigma_k(\lambda(D^2u))=1$; when $l=0$, $k=1$, it is the Poisson equation $\Delta u=1$, a linear elliptic equation; when $l=0$, $k=n$, it is the famous Monge-Amp\`{e}re equation $\det(D^2u)=1$; and when $l=1$, $k=3$, $n=3$ or $4$, it is the special Lagrangian equation $\sigma_1(\lambda(D^2u))=\sigma_3(\lambda(D^2u))$ in three or four dimension (in three dimension, this is $\det(D^2u)=\Delta u$ indeed) which arises from the special Lagrangian geometry \cite{HL82}. For linear elliptic equations of second order, there have been much extensive studies on the exterior Dirichlet problem, see \cite{MS60} and the references therein. For the Monge-Amp\`{e}re equation, a classical theorem of J\"{o}rgens \cite{Jor54}, Calabi \cite{Cal58} and Pogorelov \cite{Pog72} states that any convex classical solution of $\det(D^2u)=1$ in $\mathds R^n$ must be a quadratic polynomial. Related results was also given by \cite{CY86}, \cite{Caf95}, \cite{TW00} and \cite{JX01}. Caffarelli and Li \cite{CL03} extended the J\"{o}rgens-Calabi-Pogorelov theorem to exterior domains. They proved that if $u$ is a convex viscosity solution of $\det(D^2u)=1$ in the exterior domain $\mathds R^n\setminus\overline{D}$, where $D$ is a bounded domain in $\mathds R^n$, $n\geq3$, then there exist $A\in\mathds R^{n\times n},b\in\mathds R^n$ and $c\in\mathds R$ such that \begin{equation}\label{eqn.abc-n} \limsup_{|x|\rightarrow+\infty}|x|^{n-2} \left|u(x)-\left(\frac{1}{2}x^{T}Ax+b^Tx+c\right)\right|<\infty. \end{equation} With such prescribed asymptotic behavior at infinity, they also established an existence and uniqueness theorem for solutions of the Dirichlet problem of the Monge-Amp\`{e}re equation in the exterior domain of $\mathds R^n$, $n\geq3$. See \cite{FMM99}, \cite{FMM00} or \cite{Del92} for similar problems in two dimension. Recently, J.-G. Bao, H.-G. Li and Y.-Y. Li \cite{BLL14} extended the above existence and uniqueness theorem of the exterior Dirichlet problem in \cite{CL03} for the Monge-Amp\`{e}re equation to the Hessian equation $\sigma_k(\lambda(D^2u))=1$ with $2\leq k\leq n$ and with some appropriate prescribed asymptotic behavior at infinity which is modified from \eref{eqn.abc-n}. Before them, for the special case that $A=c_0 I$ with $c_0:=({C_n^k})^{-1/k}$ and $C_n^k:={n!}/(k!(n-k)!)$, the exterior Dirichlet problem for the Hessian equation has been investigated by Dai and Bao in \cite{DB11}. At the same time, Dai \cite{Dai11} proved the existence theorem of the exterior Dirichlet problem for the Hessian quotient equation \eref{eqn.hqe} with $k-l\geq3$, and with the prescribed asymptotic behavior at infinity of the special case that $A=c_\ast I$, that is, \begin{equation}\label{eqn.cc-kl} \limsup_{|x|\rightarrow+\infty}|x|^{k-l-2} \left|u(x)-\left(\frac{c_\ast}{2}|x|^2+c\right)\right|<\infty, \end{equation} where \begin{equation}\label{eqn.cst} c_\ast:=\left(\frac{C_n^l}{C_n^k}\right)^{\frac{1}{k-l}} ~\text{with}~C_n^i:=\frac{n!}{i!(n-i)!} ~\text{for}~i=k,l. \end{equation} As they pointed out in \cite{LB14} that the restriction $k-l\geq3$ rules out an important example, the special Lagrangian equation $\det(D^2u)=\Delta u$ in three dimension. Later, \cite{LD12} improve the result in \cite{Dai11} for \eref{eqn.hqe} with $k-l\geq3$ to that for \eref{eqn.hqe} with $0\leq l<k\leq(n+1)/2$. More recently, Li and Bao \cite{LB14} established the existence theorem of the exterior Dirichlet problem for a class of fully nonlinear elliptic equations related to the eigenvalues of the Hessian which include the Monge-Amp\`{e}re equations, Hessian equations, Hessian quotient equations and the special Lagrangian equations in dimension equal and lager than three, but with the prescribed asymptotic behavior at infinity only in the special case of \eref{eqn.abc-n} that $A=c^\ast I$ with $c^\ast$ some appropriate constant, like \eref{eqn.cc-kl} and \eref{eqn.cst}. In this paper, we focus our attention on the Hessian quotient equation \eref{eqn.hqe} and establish the existence and uniqueness theorem for the exterior Dirichlet problem of it with prescribed asymptotic behavior at infinity of the type similar to \eref{eqn.abc-n}. This extends the previous corresponding results on the Monge-Amp\`{e}re equations \cite{CL03} and on the Hessian equations \cite{BLL14} to Hessian quotient equations, and also extends those results on the Hessian quotient equations in \cite{Dai11}, \cite{LD12} and \cite{LB14} to be valid for general prescribed asymptotic behavior condition at infinity. Since we do not restrict ourselves to the case $k-l\geq3$ or $0\leq l<k\leq(n+1)/2$ only, our theorems also apply to the special Lagrangian equations $\det(D^2u)=\Delta u$ in three dimension and $\sigma_1(\lambda(D^2u))=\sigma_3(\lambda(D^2u))$ in four dimension. Indeed, we will show in our forthcoming paper \cite{LL16} that our method still works very well for the special Lagrangian equations with higher dimension and with general phase. We would like to remark that, for the interior Dirichlet problems there have been much extensive studies, see for example \cite{CIL92}, \cite{CNS85}, \cite{Ivo85}, \cite{Kry83}, \cite{Urb90}, \cite{Tru90} and \cite{Tru95}; see \cite{BCGJ03} and the references given there for more on the Hessian quotient equations; and for more on the special Lagrangian equations, we refer the reader to \cite{HL82}, \cite{Fu98}, \cite{Yuan02}, \cite{CWY09} and the references therein. For the reader's convenience, we give the following definitions related to Hessian quotient equation (see also \cite{CIL92}, \cite{CC95}, \cite{CNS85}, \cite{Tru90}, \cite{Tru95} and the references therein). We say that a function $u\in C^2\left(\mathds R^n\setminus\overline{D}\right)$ is \emph{$k$-convex}, if $\lambda(D^2u)\in\overline{\Gamma_k}$ in $\mathds R^n\setminus\overline{D}$, where $\Gamma_k$ is the connected component of $\set{\lambda\in\mathds R^n|\sigma_k(\lambda)>0}$ containing the \emph{positive cone} \[\Gamma^+:=\set{\lambda\in\mathds R^n|\lambda_i>0,\forall i=1,2,...,n}.\] It is well known that $\Gamma_k$ is an open convex symmetric cone with its vertex at the origin and that \[\Gamma_k=\set{\lambda\in\mathds R^n|\sigma_j(\lambda)>0, \forall j=1,2,...,k},\] which implies \[\set{\lambda\in\mathds R^n|\lambda_1+\lambda_2+...+\lambda_n>0} =\Gamma_1\supset...\supset\Gamma_k\supset\Gamma_{k+1} \supset...\supset\Gamma_n=\Gamma^+\] with the first term $\Gamma_1$ the half space and with the last term $\Gamma_n$ the positive cone $\Gamma^+$. Furthermore, we also know that \begin{equation}\label{eqn.pisjp} \partial_{\lambda_i}\sigma_{j}(\lambda)>0, ~\forall 1\leq i\leq n, ~\forall 1\leq j\leq k, ~\forall\lambda\in\Gamma_k, ~\forall 1\leq k\leq n \end{equation} (see \cite{CNS85} or \cite{Urb90} for more details). Let $\Omega$ be an open domain in $\mathds R^n$ and let $f\in C^0(\Omega)$ be nonnegative. Suppose $0\leq l<k\leq n$. A function $u\in C^0(\Omega)$ is said to be a \emph{viscosity subsolution} of \begin{equation}\label{eqn.hqe-f} \frac{\sigma_k(\lambda(D^2u))}{\sigma_l(\lambda(D^2u))}=f \quad\text{in}~\Omega \end{equation} \big{(}or say that $u$ satisfies \[\frac{\sigma_k(\lambda(D^2u))}{\sigma_l(\lambda(D^2u))}\geq f \quad\text{in}~\Omega\] in the viscosity sense, similarly hereinafter\big{)}, if for any function $v\in C^2(\Omega)$ and any point $x^\ast\in\Omega$ satisfying \[v(x)\geq u(x),~\forall x\in\Omega \quad\text{and}\quad v(x^\ast)=u(x^\ast),\] we have \[\frac{\sigma_k(\lambda(D^2v))}{\sigma_l(\lambda(D^2v))}\geq f \quad\text{in}~\Omega.\] A function $u\in C^0(\Omega)$ is said to be a \emph{viscosity supersolution} of \eref{eqn.hqe-f} if for any \textit{$k$-convex function} $v\in C^2(\Omega)$ and any point $x^\ast\in\Omega$ satisfying \[v(x)\leq u(x),~\forall x\in\Omega \quad\text{and}\quad v(x^\ast)=u(x^\ast),\] we have \[\frac{\sigma_k(\lambda(D^2v))}{\sigma_l(\lambda(D^2v))}\leq f \quad\text{in}~\Omega.\] A function $u\in C^0(\Omega)$ is said to be a \emph{viscosity solution} of \eref{eqn.hqe-f}, if it is both a viscosity subsolution and a viscosity supersolution of \eref{eqn.hqe-f}. A function $u\in C^0(\Omega)$ is said to be a \emph{viscosity subsolution} (respectively, \emph{supersolution, solution}) of \eref{eqn.hqe-f} and $u=\varphi$ on $\partial\Omega$ with some $\varphi\in C^0(\partial\Omega)$, if $u$ is a viscosity subsolution (respectively, supersolution, solution) of \eref{eqn.hqe-f} and $u\leq$(respectively, $\geq,=$)$\varphi$ on $\partial\Omega$. Note that in the definitions of viscosity solution above, we have used the ellipticity of the Hessian quotient equations indeed. For completeness and convenience, this will be proved in the end of \ssref{subsec.prelem}. See also \cite{CC95}, \cite{CIL92}, \cite{CNS85}, \cite{Urb90} and the references therein. Define \[\mathscr{A}_{k,l}:=\left\{A\in S(n)\big{|}\lambda(A)\in\Gamma^+, \sigma_k(\lambda(A))=\sigma_l(\lambda(A))\right\}.\] Note that there are plenty of elements in $\mathscr{A}_{k,l}$. In fact, for any $A\in S(n)$ with $\lambda(A)\in\Gamma^+$, if we set \[\varrho:=\left(\frac{\sigma_k(\lambda(A))}{\sigma_l(\lambda(A))}\right) ^{-\frac{1}{k-l}},\] we then have $\varrho A\in\mathscr{A}_{k,l}$. Let \[\mathscr{\widetilde A}_{k,l} :=\set{A\in\mathscr{A}_{k,l}\big{|}m_{k,l}(\lambda(A))>2},\] where $m_{k,l}(\lambda)$ is a quantity which plays an important role in this paper. We will give the specific definition of $m_{k,l}(\lambda)$ in \eref{eqn.mkla} in \ssref{subsec.xi}, and verify there that $\mathscr{\widetilde A}_{k,l}$ possesses the following fine properties. \begin{proposition}\label{prop.wtakl} Suppose $0\leq l<k\leq n$ and $n\geq3$. \begin{enumerate}[\quad(1)] \item If $k-l\geq2$, then $\mathscr{\widetilde A}_{k,l}=\mathscr{A}_{k,l}$. \item $\mathscr{\widetilde A}_{n,0}=\mathscr{A}_{n,0}$ and $m_{n,0}\equiv n$. \item $c_\ast I\in\mathscr{\widetilde A}_{k,l}$ and $m_{k,l}(c_\ast(1,1,...,1))=n$, where $c_\ast$ is the one defined in \eref{eqn.cst}. \end{enumerate} \end{proposition} The main result of this paper now can be stated as below. \begin{theorem}\label{thm.hqe} Let $D$ be a bounded strictly convex domain in $\mathds R^n$, $n\geq 3$, $\partial D\in C^2$ and let $\varphi\in C^2(\partial D)$. Then for any given $A\in\mathscr{\widetilde A}_{k,l}$ with $0\leq l<k\leq n$, and any given $b\in \mathds R^n$, there exists a constant $\tilde{c}$ depending only on $n,D,k,l,A,b$ and $\norm{\varphi}_{C^2(\partial D)}$, such that for every $c\geq\tilde{c}$, there exists a unique viscosity solution $u\in C^0(\mathds R^n\setminus D)$ of \begin{equation}\label{eqn.hqe-abc} \left\{ \begin{aligned} \displaystyle \frac{\sigma_k(\lambda(D^2u))}{\sigma_l(\lambda(D^2u))}&=1 \quad\text{in}~\mathds R^n\setminus\overline{D},\\%[0.05cm] \displaystyle u&=\varphi\quad\text{on}~\partial D,\\%[0.05cm] \displaystyle \limsup_{|x|\rightarrow+\infty}|x|^{m-2} &\left|u(x)-\left(\frac{1}{2}x^{T}Ax+b^Tx+c\right)\right|<\infty, \end{aligned} \right. \end{equation} where $m\in(2,n]$ is a constant depending only on $n,k,l$ and $\lambda(A)$, which actually can be taken as $m_{k,l}(\lambda(A))$. \end{theorem} \begin{remark} \begin{enumerate}[(1)] \item One can easily see that \tref{thm.hqe} still holds with $A\in\mathscr{\widetilde A}_{k,l}$ replaced by $A\in\mathscr{\widetilde A}^\ast_{k,l}$ and $\lambda(A)$ replaced by $\lambda(A^\ast)$, where \[\mathscr{\widetilde A}^\ast_{k,l} :=\left\{A\in\mathds R^{n\times n}\big{|} \lambda(A^\ast)\in\Gamma^+, \sigma_k(\lambda(A^\ast)) =\sigma_l(\lambda(A^\ast)), m_{k,l}(\lambda(A^\ast))>2\right\}\] and $A^\ast:=(A+A^T)/2$. This is to say that the above theorem can be adapted to a slightly more general form by modifying the meaning of $\mathscr{\widetilde A}_{k,l}$. \item For the special cases that $l=0$ (i.e., the Hessian equation $\sigma_k(\lambda(D^2u))=1$) and that $l=0$ and $k=n$ (i.e., the Monge-Amp\`{e}re equation $\det(D^2u)=1$), in view of \partialref{prop.wtakl}-\textsl{(2)}, our \tref{thm.hqe} recovers the corresponding results \cite[Theorem 1.1]{BLL14} and \cite[Theorem 1.5]{CL03}, respectively. \item For $A=c_\ast I$ with $c_\ast$ defined in \eref{eqn.cst}, by \partialref{prop.wtakl}-\textsl{(1),(3)}, our results improve those in \cite{Dai11} and \cite{LD12}. Indeed, by \partialref{prop.wtakl}-\textsl{(3)}, the main results in \cite{Dai11}, \cite{LD12} and those parts concerning the Hessian quotient equations in \cite{LB14} can all be recovered by \tref{thm.hqe} as special cases. Furthermore, our results also apply to the special Lagrangian equation $\det(D^2u)=\Delta u$ in three dimension (respectively, $\sigma_1(\lambda(D^2u))=\sigma_3(\lambda(D^2u))$ in four dimension), not only for $A=\sqrt{3}I$ (respectively, $A=I$), but also for any $A\in\mathscr{A}_{3,1}$. \qed \end{enumerate} \end{remark} The paper is organized as follows. In \sref{sec.prel}, after giving some basic notations in \ssref{subsec.nott}, we introduce the definitions of $\Xi_k,\underline\xi_k,\overline\xi_k$ and $m_{k,l}$, and investigate their properties in \ssref{subsec.xi}. Then we collect in \ssref{subsec.prelem} some preliminary lemmas which will be used in this paper. \sref{sec.pmaint} is devoted to the proof of the main theorem (\tref{thm.hqe}). To do this, we start in \ssref{subsec.csubsol} to construct some appropriate subsolutions of the Hessian quotient equation \eref{eqn.hqe}, by taking advantages of the properties of $\Xi_k,\underline\xi_k,\overline\xi_k$ and $m_{k,l}$ explored in \ssref{subsec.xi}. Then in \ssref{subsec.pmaint}, after reducing \tref{thm.hqe} to \lref{lem.hqe} by simplification and normalization, we prove \lref{lem.hqe} by applying the Perron's method to the subsolutions we constructed in \ssref{subsec.csubsol}. \section{Preliminary}\label{sec.prel} \subsection{Notation}\label{subsec.nott} \quad In this paper, $S(n)$ denotes the linear space of symmetric $n\times n$ real matrices, and $I$ denotes the identity matrix. For any $M\in S(n)$, if $m_1,m_2,...,m_n$ are the eigenvalues of $M$ (usually, the assumption $m_1\leq m_2\leq...\leq m_n$ is added for convenience), we will denote this fact briefly by $\lambda(M)=(m_1,m_2,...,m_n)$ and call $\lambda(M)$ the eigenvalue vector of $M$. For $A\in S(n)$ and $\rho>0$, we denote by \[ E_\rho:=\left\{x\in\mathds R^n\big{|}x^TAx<\rho^2\right\} =\left\{x\in\mathds R^n\big{|}r_A(x)<\rho\right\} \] the ellipsoid of size $\rho$ with respect to $A$, where we set $r_A(x):=\sqrt{x^TAx}$. For any $p\in\mathds R^n$, we write \[\sigma_k(p):=\sum_{1\leq s_1<s_2<...<s_k\leq n} p_{s_1}p_{s_2}...p_{s_k}\quad(\forall 1\leq k\leq n)\] as the $k$-th elementary symmetric function of $p$. Meanwhile, we will adopt the conventions that $\sigma_{-1}(p)\equiv0$, $\sigma_0(p)\equiv1$ and $\sigma_k(p)\equiv0$, $\forall k\geq n+1$; and we will also define \[\sigma_{k;i}(p):=\left(\sigma_{k}(\lambda) \big{|}_{\lambda_i=0}\right)\Big{|}_{\lambda=p} =\sigma_{k}\left(p_1,p_2,...,\widehat{p_i},...,p_n\right)\] for any $-1\leq k\leq n$ and any $1\leq i\leq n$, and similarly \[\sigma_{k;i,j}(p):=\left(\sigma_{k}(\lambda) \big{|}_{\lambda_i=\lambda_j=0}\right)\Big{|}_{\lambda=p} =\sigma_{k}\left(p_1,p_2,...,\widehat{p_i}, ...,\widehat{p_j},...,p_n\right)\] for any $-1\leq k\leq n$ and any $1\leq i,j\leq n$, $i\neq j$, for convenience. \subsection{Definitions and properties of $\Xi_k,\underline\xi_k,\overline\xi_k$ and $m_{k,l}$}\label{subsec.xi} \quad To establish the existence of the solution of \eref{eqn.hqe}, by the Perron's method, the key point is to find some appropriate subsolutions of the equation. Since the Hessian quotient equation \eref{eqn.hqe} is a highly fully nonlinear equation which including polynomials of the eigenvalues of the the Hessian matrix $D^2u$, $\sigma_k(\lambda)$ and $\sigma_l(\lambda)$, of different order of homogeneities, to solve it we need to strike a balance between them. It will turn out to be clear that the quantities $\Xi_k,\underline\xi_k,\overline\xi_k$ and $m_{k,l}$, which we shall introduce below, are very natural and perfectly fit for this purpose. \begin{definition} For any $0\leq k\leq n$ and any $a\in\mathds R^n\setminus\{0\}$, let \[\Xi_k:=\Xi_k(a,x) :=\frac{\sum_{i=1}^n\sigma_{k-1;i}(a)a_i^2x_i^2} {\sigma_k(a)\sum_{i=1}^{n}a_ix_i^2}, ~\forall x\in\mathds R^n\setminus\{0\},\] and define \[\overline{\xi}_k:=\overline{\xi}_k(a) :=\sup_{x\in\mathds R^n\setminus\{0\}}\Xi_k(a,x)\] and \[\underline{\xi}_k:=\underline{\xi}_k(a) :=\inf_{x\in\mathds R^n\setminus\{0\}}\Xi_k(a,x).\] \end{definition} \begin{definition} For any $0\leq l<k\leq n$ and any $a\in\mathds R^n\setminus\{0\}$, let \begin{equation}\label{eqn.mkla} \displaystyle m_{k,l}:=m_{k,l}(a):=\frac{k-l}{\overline{\xi}_k(a)-\underline{\xi}_l(a)}. \end{equation} \end{definition} We remark, for the reader's convenience, that $\Xi_k$ originates from the computation of $\sigma_k(D^2\Phi(x))$ where $\Phi(x)$ is a generalized radially symmetric function (see \lref{lem.skm} and the proof of \lref{lem.Phi-subsol}), that $\underline\xi_k$ and $\overline\xi_k$ result from the comparison between $\sigma_k(\lambda)$ and $\sigma_l(\lambda)$ in the attempt to derive an ordinary differential equation from the original equation (see the last part of the proof of \lref{lem.Phi-subsol}), and that $m_{k,l}$ arises in the process of solving this ordinary differential equation (see \eref{eqn.mdlnr} in the proof of \lref{lem.psi}). By $\Xi_k,\underline\xi_k$ and $\overline\xi_k$, we get a good balance between $\sigma_k(\lambda)$ and $\sigma_l(\lambda)$, which can be measured by $m_{k,l}$. Furthermore, we will find that $m_{k,l}$ has also some special meaning related to the decay and asymptotic behavior of the solution (see \lref{lem.psi}\textsl{-(iii)}, \cref{cor.mub} and \tref{thm.hqe}). It is easy to see that \[\overline{\xi}_k(\varrho a)=\overline{\xi}_k(a), ~\underline{\xi}_k(\varrho a)=\underline{\xi}_k(a), ~\forall\varrho\neq 0, ~\forall a\in\mathds R^n\setminus\{0\}, ~\forall 0\leq k\leq n,\] and \[\underline\xi_k(C(1,1,...,1))=\frac{k}{n} =\overline\xi_k(C(1,1,...,1)), ~\forall C>0,~\forall 0\leq k\leq n.\] Furthermore, we have the following lemma. \begin{lemma}\label{lem.xik} Suppose $a=(a_1,a_2,...,a_n)$ with $0<a_1\leq a_2\leq...\leq a_n$. Then \begin{equation}\label{eqn.uxknox} 0<\frac{a_1\sigma_{k-1;1}(a)}{\sigma_k(a)}=\underline{\xi}_k(a) \leq\frac{k}{n}\leq\overline{\xi}_k(a) =\frac{a_n\sigma_{k-1;n}(a)}{\sigma_k(a)}\leq 1, ~\forall 1\leq k\leq n; \end{equation} \begin{equation}\label{eqn.olxiin} 0=\overline{\xi}_0(a)<\frac{1}{n}\leq\frac{a_n}{\sigma_1(a)} =\overline{\xi}_1(a)\leq\overline{\xi}_2(a) \leq...\leq\overline{\xi}_{n-1}(a)<\overline{\xi}_n(a)=1; \end{equation} and \begin{equation}\label{eqn.ulxiin} 0=\underline{\xi}_0(a)<\frac{a_1}{\sigma_1(a)} =\underline{\xi}_1(a)\leq\underline{\xi}_2(a) \leq...\leq\underline{\xi}_{n-1}(a)<\underline{\xi}_n(a)=1. \end{equation} Moreover, \begin{equation}\label{eqn.xkkn} \underline\xi_k(a)=\frac{k}{n}=\overline\xi_k(a) \end{equation} for some $1\leq k\leq n-1$, if and only if $a=C(1,1,...,1)$ for some $C>0$. \end{lemma} \begin{proof} ($1^\circ$) By the definitions of $\sigma_k(a)$ and $\sigma_{k;i}(a)$, we see that \begin{equation}\label{eqn.sk} \sigma_k(a)=\sigma_{k;i}(a)+a_i\sigma_{k-1;i}(a), ~\forall 1\leq i\leq n; \end{equation} and \[\sum_{i=1}^{n}\sigma_{k;i}(a) =\frac{nC_{n-1}^k}{C_n^k}\sigma_k(a) =(n-k)\sigma_k(a).\] Hence we obtain \begin{equation}\label{eqn.ksk} \sum_{i=1}^{n}a_i\sigma_{k-1;i}(a)=k\sigma_k(a). \end{equation} Now we show that \begin{equation}\label{eqn.aiski} a_1\sigma_{k-1;1}(a) \leq a_2\sigma_{k-1;2}(a)\leq... \leq a_n\sigma_{k-1;n}(a). \end{equation} In fact, for any $i\neq j$, similar to \eref{eqn.sk}, we have \[a_i\sigma_{k-1;i}(a) =a_i\left(\sigma_{k-1;i,j}(a)+a_j\sigma_{k-2;i,j}(a)\right)\] and \[a_j\sigma_{k-1;j}(a) =a_j\left(\sigma_{k-1;i,j}(a)+a_i\sigma_{k-2;i,j}(a)\right),\] thus \[a_i\sigma_{k-1;i}(a)-a_j\sigma_{k-1;j}(a) =(a_i-a_j)\sigma_{k-1;i,j}(a).\] Hence if $a_i\lessgtr a_j$, then \begin{equation}\label{eqn.aslgeq} a_i\sigma_{k-1;i}(a)\lessgtr a_j\sigma_{k-1;j}(a). \end{equation} By the definition of $\overline\xi_k$, we have \begin{eqnarray*} \displaystyle \overline{\xi}_k(a) &=&\sup_{x\neq 0}\frac{\sum_{i=1}^n\sigma_{k-1;i}(a)a_i^2x_i^2} {\sigma_k(a)\sum_{i=1}^{n}a_ix_i^2}\\ \displaystyle &\geq& \sup_{\substack{x_1=...=x_{n-1}=0,\\x_n\neq 0}} \frac{\sum_{i=1}^n\sigma_{k-1;i}(a)a_i^2x_i^2} {\sigma_k(a)\sum_{i=1}^{n}a_ix_i^2}\\ \displaystyle &=&\sup_{x_n\neq 0}\frac{\sigma_{k-1;n}(a)a_n^2x_n^2} {\sigma_k(a)a_n x_n^2}\\ \displaystyle &=&\frac{a_n\sigma_{k-1;n}(a)}{\sigma_k(a)} \end{eqnarray*} and \begin{eqnarray*} \displaystyle \overline{\xi}_k(a) &=&\sup_{x\neq 0}\frac{\sum_{i=1}^n\sigma_{k-1;i}(a)a_i^2x_i^2} {\sigma_k(a)\sum_{i=1}^{n}a_ix_i^2}\\ \displaystyle &\leq& \sup_{x\neq 0}\frac{a_n\sigma_{k-1;n}(a)\sum_{i=1}^na_ix_i^2} {\sigma_k(a)\sum_{i=1}^{n}a_ix_i^2}\qquad\text{by \eref{eqn.aiski}}\\ \displaystyle &=&\frac{a_n\sigma_{k-1;n}(a)}{\sigma_k(a)}. \end{eqnarray*} Hence we obtain \begin{equation}\label{eqn.olxik} \overline\xi_k(a)=\frac{a_n\sigma_{k-1;n}(a)}{\sigma_k(a)}. \end{equation} Similarly \begin{equation}\label{eqn.ulxik} \underline\xi_k(a)=\frac{a_1\sigma_{k-1;1}(a)}{\sigma_k(a)}. \end{equation} From \eref{eqn.ksk}, we have \[\sum_{i=1}^{n}\frac{a_i\sigma_{k-1;i}(a)}{\sigma_k(a)}=k.\] Combining this with \eref{eqn.aiski}, \eref{eqn.olxik} and \eref{eqn.ulxik}, we deduce that \[\underline\xi_k(a)\leq\frac{k}{n}\leq\overline\xi_k(a).\] Thus the proof of \eref{eqn.uxknox} is complete, and \eref{eqn.xkkn} is also clear in view of \eref{eqn.aslgeq}. ($2^\circ$) Since it follows from \eref{eqn.sk} that \[a_i\sigma_{k-1;i}(a)<\sigma_k(a), ~\forall 1\leq i\leq n, ~\forall 1\leq k\leq n-1,\] we obtain \[\underline\xi_k(a)\leq\overline\xi_k(a)<1,~\forall 0\leq k\leq n-1.\] On the other hand, we have $\overline{\xi}_n(a)=\underline{\xi}_n(a)=1$ which follows from \[a_i\sigma_{n-1;i}(a)=\sigma_n(a),~\forall 1\leq i\leq n.\] Combining \eref{eqn.olxik} and \eref{eqn.sk}, we discover that \begin{eqnarray*} \overline\xi_k(a)&=&\frac{a_n\sigma_{k-1;n}(a)}{\sigma_k(a)} =\frac{a_n\sigma_{k-1;n}(a)}{\sigma_{k;n}(a)+a_n\sigma_{k-1;n}(a)}\\ &\leq&\frac{a_n\sigma_{k;n}(a)}{\sigma_{k+1;n}(a)+a_n\sigma_{k;n}(a)} =\frac{a_n\sigma_{k;n}(a)}{\sigma_{k+1}(a)} =\overline\xi_{k+1}(a), \end{eqnarray*} where we used the inequality \[\frac{\sigma_{k-1;n}(a)}{\sigma_{k;n}(a)} \leq\frac{\sigma_{k;n}(a)}{\sigma_{k+1;n}(a)}\] which is a variation of the famous Newton inequality(see \cite{HLP34}) \[\sigma_{k-1}(\lambda)\sigma_{k+1}(\lambda) \leq\left(\sigma_{k}(\lambda)\right)^2, ~\forall\lambda\in\mathds R^n.\] Thus the proof of \eref{eqn.olxiin}, and similarly of \eref{eqn.ulxiin}, is complete. \end{proof} Since it follows from \eref{eqn.uxknox} that \[\frac{k-l}{n}\leq\overline{\xi}_k(a)-\underline{\xi}_l(a) <\overline{\xi}_k(a)\leq 1,\] we obtain \begin{corollary}\label{cor.m} If $0\leq l<k\leq n$ and $a\in\Gamma^+$, then \[1\leq k-l<m_{k,l}(a)\overline\xi_k(a)\leq m_{k,l}(a)\leq n.\] \end{corollary} As an application of \cref{cor.m} and \lref{lem.xik}, we now verify \partialref{prop.wtakl}. \begin{proof}[\textbf{Proof of \partialref{prop.wtakl}}] \textsl{(1)} and \textsl{(2)} are clear. For \textsl{(3)}, we only need to note that $c_\ast I\in\mathscr{A}_{k,l}$ and $m_{k,l}(c_\ast(1,1,...,1))=n>2$. \end{proof} To help the reader to become familiar with these new quantities, it is worth to give the following examples which are also the applications of the above lemma. \begin{example} Note that, for $a=(a_1,a_2,a_3)\in\mathds R^3$ with $0<a_1\leq a_2\leq a_3$, by \lref{lem.xik}, we have \[\overline\xi_3(a)\equiv1\equiv\underline\xi_3(a),\] \[\overline\xi_2(a)=\frac{a_3(a_1+a_2)}{a_1a_2+a_1a_3+a_2a_3}, \quad \underline\xi_2(a)=\frac{a_1(a_2+a_3)}{a_1a_2+a_1a_3+a_2a_3},\] \[\overline\xi_1(a)=\frac{a_3}{a_1+a_2+a_3}, \quad \underline\xi_1(a)=\frac{a_1}{a_1+a_2+a_3},\] and \[\overline\xi_0(a)\equiv0\equiv\underline\xi_0(a).\] Thus we can compute, for $a=(1,2,3)$, that \[\overline\xi_2=\frac{9}{11},~\underline\xi_2=\frac{5}{11}, ~\overline\xi_1=\frac{1}{2},~\underline\xi_1=\frac{1}{6},\] \[m_{3,2}=\frac{11}{6}<2, ~m_{3,1}=\frac{12}{5}>2, ~m_{3,0}\equiv3>2,\] \[m_{2,1}=\frac{66}{43}<2, ~m_{2,0}=\frac{22}{9}>2 ~\text{and}~m_{1,0}=2,\] and, for $a=(11,12,13)$, that \[\overline\xi_2=\frac{299}{431},~\underline\xi_2=\frac{275}{431}, ~\overline\xi_1=\frac{13}{36},~\underline\xi_1=\frac{11}{36},\] \[m_{3,2}=\frac{431}{156}>2, ~m_{3,1}=\frac{72}{25}>2, ~m_{3,0}\equiv3>2,\] \[m_{2,1}=\frac{15516}{6023}>2, ~m_{2,0}=\frac{862}{299}>2 ~\text{and}~m_{1,0}=\frac{36}{13}>2.\] \end{example} \begin{remark} \begin{enumerate}[(1)] \item By definition of $m_{k,l}$, we can easily check that for any $1<k\leq n$, $m_{k,k-1}(a)>2$ if and only if $\underline\xi_{k-1}(a)\leq\overline\xi_{k}(a)\leq\underline\xi_{k-1}(a)+1/2$. This will show us how $m_{k,l}$ plays a role in the making of a balance between different order of homogeneities as we stated in the beginning of this subsection. \item \partialref{prop.wtakl}-\textsl{(1)} states that $\mathscr{\widetilde A}_{k,l}=\mathscr{A}_{k,l}$ provided $k-l\geq2$. Note that this is the best case we can expect, since in general $\mathscr{\widetilde A}_{k,k-1}\subsetneqq\mathscr{A}_{k,k-1}$, which is evident by the fact stated in the first item of this remark (and also by the above examples). For example, in $\mathds R^3$ we have \[m_{3,2}(a)>2 \Leftrightarrow \overline\xi_{3}(a)\leq\underline\xi_{2}(a)+1/2 \Leftrightarrow a_1>\frac{a_2a_3}{a_2+a_3},\] where the last inequality is not always true.\qed \end{enumerate} \end{remark} \subsection{Some preliminary lemmas}\label{subsec.prelem} \quad In this subsection, we collect some preliminary lemmas which will be mainly used in \sref{sec.pmaint}. We first give a lemma to compute $\sigma_k(\lambda(M))$ with $M$ of certain type. ~If $\Phi(x):=\partialhi(r)$ with $\partialhi\in C^2$, $r=\sqrt{x^TAx}$, $A\in S(n)\cap\Gamma^+$ and $a=\lambda(A)$ (we may call $\Phi$ a \emph{generalized radially symmetric function} with respect to $A$, according to \cite{BLL14}), one can conclude that \[\partial_{ij}\Phi(x) =\frac{\partialhi'(r)}{r}a_i\delta_{ij}+ \frac{\partialhi''(r)-\frac{\partialhi'(r)}{r}}{r^2}(a_ix_i)(a_jx_j), ~\forall 1\leq i,j\leq n,\] provided $A$ is normalized to a diagonal matrix (see the first part of \ssref{subsec.pmaint} and the proof of \lref{lem.Phi-subsol} for details). As far as we know, generally there is no explicit formula for $\lambda(D^2\Phi(x))$ of this type, but luckily we have a method to calculate $\sigma_k\left(\lambda(D^2\Phi(x))\right)$ for each $1\leq k\leq n$, which can be presented as follows. \begin{lemma}\label{lem.skm} If $M=\left(p_i\delta_{ij}+s q_iq_j\right)_{n\times n}$ with $p,q\in\mathds R^n$ and $s\in\mathds R$, then \[\sigma_k\left(\lambda(M)\right) =\sigma_k(p)+s\sum_{i=1}^n\sigma_{k-1;i}(p)q_i^2, ~\forall 1\leq k\leq n.\] \end{lemma} \begin{proof} See \cite{BLL14}. \end{proof} To process information on the boundary we need the following lemma. \begin{lemma}\label{lem.Qxi} Let $D$ be a bounded strictly convex domain of $\mathds R^n$, $n\geq 2$, $\partial D\in C^2$, $\varphi\in C^0(D)\cap C^2(\partial{D})$ and let $A\in S(n)$, $\det{A}\neq0$. Then there exists a constant $K>0$ depending only on $n$, $\mbox{\emph{diam}}\,D$, the convexity of $D$, $\norm{\varphi}_{C^2(\overline{D})}$, the $C^2$ norm of $\partial D$ and the upper bound of $A$, such that for any $\xi\in\partial D$, there exists $\bar{x}(\xi)\in\mathds R^n$ satisfying \[\left|\bar{x}(\xi)\right|\leq K \quad \mbox{and} \quad Q_\xi(x)<\varphi(x), ~\forall x\in \overline{D}\setminus\{\xi\},\] where \[Q_\xi(x):=\frac{1}{2}\left(x-\bar{x}(\xi)\right)^TA\left(x-\bar{x}(\xi)\right) -\frac{1}{2}\left(\xi-\bar{x}(\xi)\right)^TA\left(\xi-\bar{x}(\xi)\right) +\varphi(\xi),~\forall x\in\mathds R^n.\] \end{lemma} \begin{proof} See \cite{CL03} or \cite{BLL14}. \end{proof} \begin{remark}\label{rmk.Qxi} It is easy to check that $Q_\xi$ satisfy the following properties. \begin{enumerate}[\quad(1)] \item $Q_\xi\leq\varphi$ on $\overline{D}$ and $Q_\xi(\xi)=\varphi(\xi)$. \item If $A\in\mathscr{A}_{k,l}$, then \[\frac{\sigma_k(\lambda(D^2Q_\xi))}{\sigma_l(\lambda(D^2Q_\xi))}=1 \quad\mbox{in}~\mathds R^n.\] \item There exists $\bar{c}=\bar{c}(D,A,K)>0$ such that \[Q_\xi(x)\leq\frac{1}{2}x^TAx+\bar{c}, \quad \forall x\in\partial D,~\forall\xi\in\partial D.\] \end{enumerate} \end{remark} Now we introduce the following well known lemmas about the comparison principle and Perron's method which will be applied to the Hessian quotient equations but stated in a slightly more general setting. These lemmas are adaptions of those appeared in \cite{CNS85} \cite{Jen88} \cite{Ish89} \cite{Urb90} and \cite{CIL92}. For specific proof of them one may also consult \cite{BLL14} and \cite{LB14}. \begin{lemma}[Comparison principle]\label{lem.cp} Assume $\Gamma^+\subset\Gamma\subset\mathds R^n$ is an open convex symmetric cone with its vertex at the origin, and suppose $f\in C^1(\Gamma)$ and $f_{\lambda_i}(\lambda)>0$, $\forall \lambda\in\Gamma$, $\forall i=1,2,...,n$. Let $\Omega\subset\mathds R^n$ be a domain and let $\underline{u},\overline{u}\in C^0(\overline\Omega)$ satisfying \[f\left(\lambda\left(D^2\underline{u}\right)\right)\geq 1 \geq f\left(\lambda\left(D^2\overline{u}\right)\right)\] in $\Omega$ in the viscosity sense. Suppose $\underline{u}\leq \overline{u}$ on $\partial\Omega$ (and additionally \[\lim_{|x|\rightarrow+\infty}\left(\underline{u}-\overline{u}\right)(x)=0\] provided $\Omega$ is unbounded). Then $\underline{u}\leq \overline{u}$ in $\Omega$. \end{lemma} \begin{lemma}[Perron's method]\label{lem.pm} Assume that $\Gamma^+\subset\Gamma\subset\mathds R^n$ is an open convex symmetric cone with its vertex at the origin, and suppose $f\in C^1(\Gamma)$ and $f_{\lambda_i}(\lambda)>0$, $\forall \lambda\in\Gamma$, $\forall i=1,2,...,n$. Let $\Omega\subset\mathds R^n$ be a domain, $\varphi\in C^0(\partial\Omega)$ and let $\underline{u},\overline{u}\in C^0(\overline\Omega)$ satisfying \[f\left(\lambda\left(D^2\underline{u}\right)\right)\geq 1 \geq f\left(\lambda\left(D^2\overline{u}\right)\right)\] in $\Omega$ in the viscosity sense. Suppose $\underline{u}\leq \overline{u}$ in $\Omega$, $\underline{u}=\varphi$ on $\partial\Omega$ (and additionally \[\lim_{|x|\rightarrow+\infty}\left(\underline{u}-\overline{u}\right)(x)=0\] provided $\Omega$ is unbounded). Then \begin{eqnarray*} u(x)&:=&\sup\Big\{v(x)\big{|}v\in C^0(\Omega),~ \underline{u}\leq v\leq \overline{u}~\mbox{in}~\Omega,~ f\left(\lambda\left(D^2v\right)\right)\geq 1~\mbox{in}~\Omega\\ &~&\qquad\mbox{in the viscosity sense},~ v=\varphi~\mbox{on}~\partial\Omega\Big\} \end{eqnarray*} is the unique viscosity solution of the Dirichlet problem \[\left\{ \begin{aligned} f\left(\lambda\left(D^2u\right)\right)=1 \qquad &\mbox{in}& \Omega,\\ u=\varphi \qquad &\mbox{on}& \partial\Omega. \end{aligned}\right. \] \end{lemma} \begin{remark} In order to apply the above lemmas to the Hessian quotient operator \[f(\lambda):=\frac{\sigma_k(\lambda)}{\sigma_l(\lambda)}\] in the cone $\Gamma:=\Gamma_k$, we need to show that \begin{equation}\label{eqn.pskl} \partial_{\lambda_i}\left(\frac{\sigma_k(\lambda)}{\sigma_l(\lambda)}\right) >0,~\forall 1\leq i\leq n, ~\forall 0\leq l<k\leq n, ~\forall\lambda\in\Gamma_k, \end{equation} which indeed indicates that the Hessian quotient equations \eref{eqn.hqe} are elliptic equations with respect to its $k$-convex solution $u$. Indeed, for $l=0$, \eref{eqn.pskl} is clear in light of \eref{eqn.pisjp}. For $1\leq l<k\leq n$, since \[\partial_{\lambda_i}\sigma_k(\lambda) =\frac{\sigma_k(\lambda)-\sigma_{k;i}(\lambda)}{\lambda_i} =\sigma_{k-1;i}(\lambda)\] according to \eref{eqn.sk}, we have \[\partial_{\lambda_i}\left(\frac{\sigma_k(\lambda)} {\sigma_l(\lambda)}\right) =\frac{\sigma_{k-1;i}(\lambda)\sigma_l(\lambda) -\sigma_k(\lambda)\sigma_{l-1;i}(\lambda)} {(\sigma_l(\lambda))^2}.\] Thus to prove \eref{eqn.pskl}, it remains to verify \[\sigma_{k-1;i}(\lambda)\sigma_l(\lambda) \geq\sigma_k(\lambda)\sigma_{l-1;i}(\lambda).\] In view of \eref{eqn.sk}, this is equivalent to \[\sigma_{k-1;i}(\lambda)\sigma_{l;i}(\lambda) \geq\sigma_{k;i}(\lambda)\sigma_{l-1;i}(\lambda),\] which in turn is equivalent to \[\frac{\sigma_{l;i}(\lambda)}{\sigma_{l-1;i}(\lambda)} \geq\frac{\sigma_{k;i}(\lambda)}{\sigma_{k-1;i}(\lambda)},\] since $\sigma_{j;i}(\lambda)=\partial_{\lambda_i}\sigma_{j+1}(\lambda)>0$, $\forall 1\leq i\leq n$, $\forall 0\leq j\leq k-1$, $\forall\lambda\in\Gamma_k$, according to \eref{eqn.pisjp}. For the proof of the latter, we only need to note that \[\frac{\sigma_{j;i}(\lambda)}{\sigma_{j-1;i}(\lambda)} \geq\frac{\sigma_{j+1;i}(\lambda)}{\sigma_{j;i}(\lambda)},\] which is the variation of the Newton inequality(see \cite{HLP34}) \[\sigma_{j-1}(\lambda)\sigma_{j+1}(\lambda) \leq\left(\sigma_{j}(\lambda)\right)^2, ~\forall\lambda\in\mathds R^n,\] as we met in the proof of \lref{lem.xik}.\qed \end{remark} \section{Proof of the main theorem}\label{sec.pmaint} \subsection{Construction of the subsolutions}\label{subsec.csubsol} \quad The purpose of this subsection is to prove the following key lemma and then use it to construct subsolutions of \eref{eqn.hqe}. We remark that for the generalized radially symmetric subsolution $\Phi(x)=\partialhi(r)$ that we intend to construct, the solution $\partialsi(r)$ discussed in the the following lemma actually is equivalent to $\partialhi'(r)/r$ (see the proof of the \lref{lem.Phi-subsol}). \begin{lemma}\label{lem.psi} Let $0\leq l<k\leq n$, $n\geq3$, $A\in\mathscr{\widetilde A}_{k,l}$, $a:=(a_1,a_2,...,a_n):=\lambda(A)$, $0<a_1\leq a_2\leq...\leq a_n$ and $\beta\geq 1$. Then the problem \begin{equation}\label{eqn.psi} \left\{ \begin{aligned} \partialsi(r)^k+\overline{\xi}_k(a)r\partialsi(r)^{k-1}\partialsi'(r)\quad&\\ -\partialsi(r)^l-\underline{\xi}_l(a)r\partialsi(r)^{l-1}\partialsi'(r)&=0,~r>1,\\[0.1cm] \partialsi(1)&=\beta, \end{aligned} \right. \end{equation} has a unique smooth solution $\partialsi(r)=\partialsi(r,\beta)$ on $[1,+\infty)$, which satisfies \begin{enumerate}[\quad(i)] \item[(i)] $1\leq\partialsi(r,\beta)\leq\beta$, $\partial_r\partialsi(r,\beta)\leq0$, $\forall r\geq1$, $\forall\beta\geq1$. More specifically, $\partialsi(r,1)\equiv 1$, $\partialsi(1,\beta)\equiv\beta$; and $1<\partialsi(r,\beta)<\beta$, $\forall r>1$, $\forall\beta>1$. \item[(ii)] $\partialsi(r,\beta)$ is continuous and strictly increasing with respect to $\beta$ and \[\lim_{\beta\rightarrow+\infty}\partialsi(r,\beta)=+\infty,~\forall r\geq 1.\] \item[(iii)] $\partialsi(r,\beta)=1+O(r^{-m})~(r\rightarrow+\infty)$, where $m=m_{k,l}(a)\in(2,n]$ and the $O(\cdot)$ depends only on $k$, $l$, $\lambda(A)$ and $\beta$. \end{enumerate} \end{lemma} \begin{proof} For brevity, we will often write $\partialsi(r)$ or $\partialsi(r,\beta)$ (respectively, $\underline\xi(a),\overline\xi(a)$) simply as $\partialsi$ (respectively, $\underline\xi,\overline\xi$), when there is no confusion. The proof of this lemma now will be divided into three steps. \emph{Step 1.}\quad We deduce from \eref{eqn.psi} that \begin{equation}\label{eqn.psi-kl} \partialsi^k-\partialsi^l=-\frac{r}{dr}\left(\overline{\xi}_k\partialsi^{k-1} -\underline{\xi}_l\partialsi^{l-1}\right)d\partialsi \end{equation} and \begin{equation}\label{eqn.psid} \frac{d\partialsi}{dr}=-\frac{1}{r}\cdot\frac{\partialsi^k-\partialsi^l} {\overline{\xi}_k\partialsi^{k-1}-\underline{\xi}_l\partialsi^{l-1}} =-\frac{1}{r}\cdot\frac{\partialsi}{\overline{\xi}_k}\cdot \frac{\partialsi^{k-l}-1}{\partialsi^{k-l}-\frac{\underline{\xi}_l}{\overline{\xi}_k}} =:\frac{g(\partialsi)}{r}, \end{equation} where we set \[g(\nu):=-\frac{\nu}{\overline{\xi}_k}\cdot \frac{\nu^{k-l}-1}{\nu^{k-l}-\frac{\underline{\xi}_l}{\overline{\xi}_k}}.\] Hence the problem \eref{eqn.psi} is equivalent to the following problem \begin{equation}\label{eqn.psi-g} \left\{ \begin{aligned} \partialsi'(r)&=\frac{g(\partialsi(r))}{r},~r>1,\\ \partialsi(1)&=\beta. \end{aligned} \right. \end{equation} If $\beta=1$, then $\partialsi(r)\equiv 1$ is a solution of the problem \eref{eqn.psi-g} since $g(1)=0$. Thus, by the uniqueness theorem for the solution of the ordinary differential equation, we know that $\partialsi(r,1)\equiv 1$ is the unique solution satisfies the problem \eref{eqn.psi-g}. Now if $\beta>1$, since \[h(r,\nu):=\frac{g(\nu)}{r} \in C^{\infty}((1,+\infty)\times(\nu_0,+\infty)),\] where \[\frac{\underline{\xi}_l}{\overline{\xi}_k}<\nu_0<1\] (note that $\nu_0$ exists, since we have $\underline{\xi}_l\leq l/n<k/n\leq\overline{\xi}_k$ by \lref{lem.xik}), by the existence theorem (the Picard-Lindel\"{o}f theorem) and the theorem of the maximal interval of existence for the solution of the initial value problem of the ordinary differential equation, we know that the problem \eref{eqn.psi-g} has a unique smooth solution $\partialsi(r)=\partialsi(r,\beta)$ locally around the initial point and can be extended to a maximal interval $[1,\zeta)$ in which $\zeta$ can only be one of the following cases: \begin{enumerate}[\qquad($1^\circ$)] \item $\zeta=+\infty$; \item $\zeta<+\infty$, $\partialsi(r)$ is unbounded on $[1,\zeta)$; \item $\zeta<+\infty$, $\partialsi(r)$ converges to some point on $\{\nu=\nu_0\}$ as $r\rightarrow\zeta-$. \end{enumerate} Since \[\frac{g(\partialsi(r))}{r}<0,~\forall \partialsi(r)>1,\] we see that $\partialsi(r)=\partialsi(r,\beta)$ is strictly decreasing with respect to $r$ which exclude the case ($2^\circ$) above. We claim now that the case ($3^\circ$) can also be excluded. Otherwise, the solution curve must intersect with $\{\nu=1\}$ at some point $(r_0,\partialsi(r_0))$ on it and then tends to $\{\nu=\nu_0\}$ after crossing it. But $\partialsi(r)\equiv 1$ is also a solution through $(r_0,\partialsi(r_0))$ which contradicts the uniqueness theorem for the solution of the initial value problem of the ordinary differential equation. Thus we complete the proof of the existence and uniqueness of the solution $\partialsi(r)=\partialsi(r,\beta)$ of the problem \eref{eqn.psi} on $[1,+\infty)$. Due to the same reason, i.e., $\partialsi(r,\beta)$ is strictly decreasing with respect to $r$ and the solution curve can not cross $\{\nu=1\}$ provided $\beta>1$, assertion $\emph{(i)}$ of the lemma is also clear now, that is, $1<\partialsi(r,\beta)<\beta$, $\forall r>1$, $\forall\beta>1$. \emph{Step 2.}\quad By the theorem of the differentiability of the solution with respect to the initial value, we can differentiate $\partialsi(r,\beta)$ with respect to $\beta$ as blew: \[\left\{ \begin{aligned} &\frac{\partial\partialsi(r,\beta)}{\partial r}=\frac{g(\partialsi(r,\beta))}{r},&\\ &\partialsi(1,\beta)=\beta;& \end{aligned}\right.\] \[\mathds Ra\left\{ \begin{aligned} &\frac{\partial^2 \partialsi(r,\beta)}{\partial \beta\partial r} =\frac{g'(\partialsi(r,\beta))}{r}\cdot\frac{\partial\partialsi(r,\beta)}{\partial\beta},&\\ &\frac{\partial\partialsi(1,\beta)}{\partial\beta}=1.& \end{aligned}\right.\] Let \[v(r):=\frac{\partial\partialsi(r,\beta)}{\partial\beta}.\] We have \[\left\{ \begin{aligned} &\frac{dv}{dr} =\frac{g'(\partialsi(r,\beta))}{r}\cdot v,&\\ &v(1)=1.& \end{aligned}\right.\] Therefore we can deduce that \[\frac{dv}{v} =\frac{g'(\partialsi(r,\beta))}{r}dr,\] and hence \[\frac{\partial\partialsi(r,\beta)}{\partial\beta}=v(r) =\exp\int_1^r{\frac{g'(\partialsi(\tau,\beta))}{\tau}d\tau}.\] Since \[g'(\nu)=-\frac{\nu^{k-l}-1}{\overline{\xi}_k\nu^{k-l}-\underline{\xi}_l} -\frac{\nu}{\overline{\xi}_k}\cdot \frac{-\left(\frac{\underline{\xi}_l}{\overline{\xi}_k}-1\right)(k-l)\nu^{k-l-1}} {\left(\nu^{k-l}-\frac{\underline{\xi}_l}{\overline{\xi}_k}\right)^2}\] and \begin{equation}\label{eqn.onepsibeta} 0<\frac{\underline{\xi}_l}{\overline{\xi}_k} <1\leq\partialsi(r,\beta)\leq\beta=\partialsi(1),~\forall r\geq 1, \end{equation} we have \[g'(\partialsi(r,\beta))\leq -\frac{(k-l)\left(1-\frac{\underline{\xi}_l}{\overline{\xi}_k}\right)} {\overline{\xi}_k \left(\beta^{k-l}-\frac{\underline{\xi}_l}{\overline{\xi}_k}\right)^2} =-C\left(k,l,\lambda(A),\beta\right)<0,\] and hence \[0<\frac{\partial\partialsi(r,\beta)}{\partial\beta} \leq r^{-C}\leq 1,~\forall r\geq 1.\] Thus $\partialsi(r,\beta)$ is strictly increasing with respect to $\beta$. \emph{Step 3.}\quad By \eref{eqn.psi-kl}, we have \begin{eqnarray*} -d\ln r=-\frac{dr}{r}&=&\frac{\overline{\xi}_k\partialsi^{k-1} -\underline{\xi}_l\partialsi^{l-1}}{\partialsi^k-\partialsi^l}d\partialsi =\frac{\overline{\xi}_k}{\partialsi}\cdot \frac{\partialsi^{k-l}-\frac{\underline{\xi}_l}{\overline{\xi}_k}}{\partialsi^{k-l}-1}d\partialsi\\ &=&\frac{\overline{\xi}_k}{\partialsi}\left(1 +\frac{1-\frac{\underline{\xi}_l}{\overline{\xi}_k}}{\partialsi^{k-l}-1}\right)d\partialsi =\left(\frac{\overline{\xi}_k}{\partialsi} +\frac{\overline{\xi}_k-\underline{\xi}_l}{\partialsi(\partialsi^{k-l}-1)}\right)d\partialsi\\ &=&\overline{\xi}_kd\ln\partialsi-\frac{\overline{\xi}_k-\underline{\xi}_l}{k-l} d\ln\frac{\partialsi^{k-l}}{\partialsi^{k-l}-1}\\ &=&d\ln\left(\partialsi^{\overline{\xi}_k}\left(1-\partialsi^{-k+l}\right) ^{\frac{\overline{\xi}_k-\underline{\xi}_l}{k-l}}\right). \end{eqnarray*} Hence \begin{equation}\label{eqn.mdlnr} -md\ln r=d\ln\left(\partialsi(r)^{m\overline{\xi}_k} \left(1-\partialsi(r)^{-k+l}\right)\right), \end{equation} where \[m:=m_{k,l}(a):=\frac{k-l}{\overline{\xi}_k-\underline{\xi}_l},\] which has been already defined in \eref{eqn.mkla} in \ssref{subsec.xi}. Note that, by the assumptions on $A$ and \cref{cor.m}, we have $2<m\leq n$ and $m\overline{\xi}_k>k-l$. Integrating \eref{eqn.mdlnr} from $1$ to $r$ and recalling $\partialsi(1)=\beta\geq1$, we get \[\ln\left(\partialsi(r)^{m\overline{\xi}_k} \left(1-\partialsi(r)^{-k+l}\right)\right) =\ln\left(\beta^{m\overline{\xi}_k} \left(1-\beta^{-k+l}\right)\right)+\ln r^{-m},\] and hence \[\partialsi(r)^{m\overline{\xi}_k} \left(1-\partialsi(r)^{-k+l}\right) =\beta^{m\overline{\xi}_k} \left(1-\beta^{-k+l}\right)r^{-m}:=B(\beta)r^{-m},\] where we set \[B(\beta):=\beta^{m\overline{\xi}_k}\left(1-\beta^{-k+l}\right) =\beta^{m\overline{\xi}_k-k+l}\left(\beta^{k-l}-1\right).\] Since \begin{eqnarray*} &~&\partialsi(r)^{m\overline{\xi}_k} \left(1-\partialsi(r)^{-k+l}\right) =\partialsi(r)^{m\overline{\xi}_k-k+l} \left(\partialsi(r)^{k-l}-1\right)\\ &=&\partialsi(r)^{m\overline{\xi}_k-k+l}\left(\partialsi(r)-1\right) \left(\partialsi(r)^{k-l-1}+\partialsi(r)^{k-l-2}+...+\partialsi(r)+1\right), \end{eqnarray*} we thus conclude that \begin{equation}\label{eqn.psimoar} \frac{\partialsi(r)-1}{r^{-m}} =\left(\partialsi(r)^{m\overline{\xi}_k-k+l} \left(\partialsi(r)^{k-l-1}+\partialsi(r)^{k-l-2}+ ...+\partialsi(r)+1\right)\right)^{-1}B(\beta). \end{equation} Note that $m\overline{\xi}_k-k+l>0$ and \[\beta-1=\left(\beta^{m\overline{\xi}_k-k+l} \left(\beta^{k-l-1}+\beta^{k-l-2}+ ...+b+1\right)\right)^{-1}B(\beta).\] Recalling \eref{eqn.onepsibeta}, we obtain \begin{equation}\label{eqn.psi-b} \beta-1\leq\frac{\partialsi(r,\beta)-1}{r^{-m}} \leq\frac{B(\beta)}{k-l}, ~\forall r\geq 1. \end{equation} Thus we have \[\lim_{\beta\rightarrow+\infty}\partialsi(r,\beta)=+\infty, ~\forall r\geq 1,\] and \[\partialsi(r,\beta)\rightarrow 1~(r\rightarrow+\infty), ~\forall \beta\geq 1.\] Substituting the latter to \eref{eqn.psimoar}, we get \[\frac{\partialsi(r,\beta)-1}{r^{-m}} \rightarrow\frac{B(\beta)}{k-l}~(r\rightarrow+\infty), ~\forall \beta\geq 1.\] Therefore \[\partialsi(r,\beta)=1+\frac{B(\beta)}{k-l}r^{-m}+o(r^{-m}) =1+O(r^{-m})~(r\rightarrow+\infty),\] where $o(\cdot)$ and $O(\cdot)$ depend only on $k$, $l$, $\lambda(A)$ and $\beta$. This completes the proof of the lemma. \end{proof} \begin{remark} For $l=0$, i.e., the Hessian equation $\sigma_k(\lambda)=1$, we have an easy proof. Consider the problem \begin{equation}\label{eqn.psi-he} \left\{ \begin{aligned} \partialsi(r)^k+\overline{\xi}_k(a)r\partialsi(r)^{k-1}\partialsi'(r)&=1,~r>1,\\[0.1cm] \partialsi(1)&=\beta. \end{aligned} \right. \end{equation} Set $m:=m_{k,0}(a)=k/\overline\xi_k$. We have \[\partialsi^k-1=-r\overline\xi_k\partialsi^{k-1}\frac{d\partialsi}{dr} =-\frac{1}{m}\cdot r\cdot\frac{d\left(\partialsi^k-1\right)}{dr},\] \[\frac{d\left(\partialsi^k-1\right)}{\partialsi^k-1} =-m\frac{dr}{r}\] and \[d\ln\left(\partialsi(r)^k-1\right)=-md\ln r=d\ln r^{-m}.\] Integrating it from $1$ to $r$ and recalling $\partialsi(1)=\beta\geq1$, we get \[\partialsi(r)^k-1=\left(\partialsi(1)^k-1\right)r^{-m} =(\beta^k-1)r^{-m}\] and \begin{eqnarray} \nonumber\partialsi(r)&=&\left(1+(\beta^k-1)r^{-m}\right)^{\frac{1}{k}}\\ &=&\left(1+(\beta^k-1)r^{-\frac{k}{\overline\xi_k}}\right)^{\frac{1}{k}} \label{eqn.compare}\\ \nonumber&=&1+\frac{\beta^k-1}{k}r^{-m}+o(r^{-m})=1+O(r^{-m})~(r\rightarrow+\infty). \end{eqnarray} It is obvious that the $\partialsi(r)$ that we here solved from \eref{eqn.psi-he} for $l=0$ satisfies all the conclusions of \lref{lem.psi}. Moreover, comparing \eref{eqn.compare} with the corresponding ones in \cite{BLL14} and in \cite{CL03}, we observe that our method actually provides a systematic way for construction of the subsolutions, which gives results containing the previous ones as special cases. \qed \end{remark} Set \[\mu_R(\beta):=\int_R^{+\infty}\tau\big(\partialsi(\tau,\beta)-1\big)d\tau, \quad\forall R\geq 1,~\forall\beta\geq 1.\] Note that the integral on the right hand side is convergent in view of \lref{lem.psi}-\textsl{(iii)}. Moreover, as an application of \lref{lem.psi}, we have the following. \begin{corollary}\label{cor.mub} $\mu_R(\beta)$ is nonnegative, continuous and strictly increasing with respect to $\beta$. Furthermore, \[\mu_R(\beta)\geq\int_R^{+\infty}(\beta-1)r^{-m+1}d\tau \rightarrow+\infty~(\beta\rightarrow+\infty),~\forall R\geq 1;\] and \[\mu_R(\beta)=O(R^{-m+2})~(R\rightarrow+\infty),~\forall\beta\geq 1.\] \end{corollary} \begin{proof} By \lref{lem.psi}-\textsl{(ii),(iii)} and the above property \eref{eqn.psi-b} of $\partialsi(r,\beta)$. \end{proof} For any $\alpha,\beta,\gamma\in\mathds R$, $\beta,\gamma\geq1$ and for any diagonal matrix $A\in\mathscr{\widetilde A}_{k,l}$, let \[\partialhi(r):=\partialhi_{\alpha,\beta,\gamma}(r) :=\alpha+\int_\gamma^r\tau\partialsi(\tau,\beta)d\tau, ~\forall r\geq\gamma,\] and \[\Phi(x):=\Phi_{\alpha,\beta,\gamma,A}(x):=\partialhi(r) :=\partialhi_{\alpha,\beta,\gamma}(r_A(x)), ~\forall x\in\mathds R^n\setminus{E_\gamma},\] where $r=r_A(x)=\sqrt{x^TAx}$. Then we have \begin{eqnarray} \partialhi_{\alpha,\beta,\gamma}(r) &=&\nonumber\int_\gamma^r\tau\big(\partialsi(\tau,\beta)-1\big)d\tau +\frac{1}{2}r^2-\frac{1}{2}\gamma^2+\alpha\\ &=&\frac{1}{2}r^2+\left(\mu_\gamma(\beta)+\alpha -\frac{1}{2}\gamma^2\right)-\mu_r(\beta)\label{eqn.phi-mu}\\ &=&\frac{1}{2}r^2+\left(\mu_\gamma(\beta)+\alpha -\frac{1}{2}\gamma^2\right)+O(r^{-m+2}) ~(r\rightarrow+\infty),\quad\label{eqn.phi-O} \end{eqnarray} according to \cref{cor.mub}, and now we can assert that \begin{lemma}\label{lem.Phi-subsol} $\Phi$ is a smooth $k$-convex subsolution of \eref{eqn.hqe} in $\mathds R^n\setminus\overline{E_\gamma}$, that is, \[\sigma_j\left(\lambda\left(D^2{\Phi(x)}\right)\right)\geq0, ~\forall 1\leq j\leq k,~\forall x\in\mathds R^n\setminus\overline{E_\gamma},\] and \[\frac{\sigma_k\left(\lambda\left(D^2{\Phi(x)}\right)\right)} {\sigma_l\left(\lambda\left(D^2{\Phi(x)}\right)\right)}\geq 1, ~\forall x\in\mathds R^n\setminus\overline{E_\gamma}.\] \end{lemma} \begin{proof} By definition we have $\partialhi'(r)=r\partialsi(r)$ and $\partialhi''(r)=\partialsi(r)+r\partialsi'(r)$. Since \[r^2=x^TAx=\sum_{i=1}^{n}a_ix_i^2,\] we deduce that \[2r\partial_{x_i}r=\partial_{x_i}\left(r^2\right)=2a_ix_i \quad\text{and}\quad \partial_{x_i}r=\frac{a_ix_i}{r}.\] Consequently \[\partial_{x_i}\Phi(x)=\partialhi'(r)\partial_{x_i}r =\frac{\partialhi'(r)}{r}a_ix_i,\] \begin{eqnarray*} \partial_{x_ix_j}\Phi(x) &=&\frac{\partialhi'(r)}{r}a_i\delta_{ij}+ \frac{\partialhi''(r)-\frac{\partialhi'(r)}{r}}{r^2}(a_ix_i)(a_jx_j)\\ &=&\partialsi(r)a_i\delta_{ij}+\frac{\partialsi'(r)}{r}(a_ix_i)(a_jx_j), \end{eqnarray*} and therefore \[D^2\Phi=\left(\partialsi(r)a_i\delta_{ij} +\frac{\partialsi'(r)}{r}(a_ix_i)(a_jx_j)\right)_{n\times n}.\] So we can conclude from \lref{lem.skm} that \begin{eqnarray*} \sigma_j\left(\lambda\left(D^2\Phi\right)\right) &=&\sigma_j(a)\partialsi(r)^j+\frac{\partialsi'(r)}{r}\partialsi(r)^{j-1} \sum_{i=1}^n\sigma_{j-1;i}(a)a_i^2x_i^2\\ &=&\sigma_j(a)\partialsi^j+\Xi_j(a,x)\sigma_j(a)r\partialsi^{j-1}\partialsi'\\ &\geq&\sigma_j(a)\partialsi^j+\overline{\xi}_j(a)\sigma_j(a)r\partialsi^{j-1}\partialsi'\\ &=&\sigma_j(a)\partialsi^{j-1}\left(\partialsi+\overline{\xi}_j(a)r\partialsi'\right), ~\forall 1\leq j\leq n, \end{eqnarray*} where we have used the facts that $\partialsi(r)\geq1>0$ and $\partialsi'(r)\leq0$ for all $r\geq1$, according to \lref{lem.psi}-\textsl{(i)}. For any fixed $1\leq j\leq k$, in view of \lref{lem.xik} and \lref{lem.psi}-\textsl{(i)}, we have \[0\leq\frac{\partialsi^{k-l}-1}{\partialsi^{k-l}-\frac{\underline{\xi}_l(a)}{\overline{\xi}_k(a)}} <1\leq\frac{\overline{\xi}_k(a)}{\overline{\xi}_j(a)}.\] Hence it follows from \eref{eqn.psid} that \[\partialsi'=-\frac{1}{r}\cdot\frac{\partialsi}{\overline{\xi}_k(a)}\cdot \frac{\partialsi^{k-l}-1}{\partialsi^{k-l}-\frac{\underline{\xi}_l(a)}{\overline{\xi}_k(a)}} >-\frac{1}{r}\cdot\frac{\partialsi}{\overline{\xi}_j(a)},\] which yields $\partialsi+\overline{\xi}_j(a)r\partialsi'>0$. Since $A\in\mathscr{\widetilde A}_{k,l}$ implies $a\in\Gamma^+$, that is, $\sigma_i(a)>0$ for all $1\leq i\leq n$, we thus conclude that \[\sigma_j\left(\lambda\left(D^2\Phi\right)\right)>0, ~\forall 1\leq j\leq k.\] In particular, we have \[\sigma_k\left(\lambda\left(D^2\Phi\right)\right)>0 \quad\text{and}\quad \sigma_l\left(\lambda\left(D^2\Phi\right)\right)>0.\] On the other hand, \begin{eqnarray*} &~&\sigma_k\left(\lambda\left(D^2\Phi\right)\right) -\sigma_l\left(\lambda\left(D^2\Phi\right)\right)\\ &=&\sigma_k(a)\partialsi^k+\Xi_k(a,x)\sigma_k(a)r\partialsi^{k-1}\partialsi' -\sigma_l(a)\partialsi^l-\Xi_l(a,x)\sigma_l(a)r\partialsi^{l-1}\partialsi'\\ &\geq&\sigma_k(a)\partialsi^k+\overline{\xi}_k(a)\sigma_k(a)r\partialsi^{k-1}\partialsi' -\sigma_l(a)\partialsi^l-\underline{\xi}_l(a)\sigma_l(a)r\partialsi^{l-1}\partialsi'\\ &=&\sigma_k(a)\left(\partialsi^k+\overline{\xi}_k(a)r\partialsi^{k-1}\partialsi' -\partialsi^l-\underline{\xi}_l(a)r\partialsi^{l-1}\partialsi'\right)\\ &=&0. \end{eqnarray*} Therefore \[\frac{\sigma_k\left(\lambda\left(D^2{\Phi}\right)\right)} {\sigma_l\left(\lambda\left(D^2{\Phi}\right)\right)}\geq 1.\] This completes the proof of \lref{lem.Phi-subsol}. \end{proof} \subsection{Proof of \tref{thm.hqe}}\label{subsec.pmaint} \quad We first introduce the following lemma which is a special and simple case of \tref{thm.hqe} with the additional condition that the matrix $A$ is diagonal and the vector $b$ vanishes. \begin{lemma}\label{lem.hqe} Let $D$ be a bounded strictly convex domain in $\mathds R^n$, $n\geq 3$, $\partial D\in C^2$ and let $\varphi\in C^2(\partial D)$. Then for any given \textsl{diagonal matrix} $A\in\mathscr{\widetilde A}_{k,l}$ with $0\leq l<k\leq n$, there exists a constant $\tilde{c}$ depending only on $n,D,k,l,A$ and $\norm{\varphi}_{C^2(\partial D)}$, such that for every $c\geq\tilde{c}$, there exists a unique viscosity solution $u\in C^0(\mathds R^n\setminus D)$ of \begin{equation}\label{eqn.hqe-ac} \left\{ \begin{aligned} \displaystyle \frac{\sigma_k(\lambda(D^2u))}{\sigma_l(\lambda(D^2u))}&=1 \quad\text{in}~\mathds R^n\setminus\overline{D},\\%[0.05cm] \displaystyle u&=\varphi\quad\text{on}~\partial D,\\%[0.05cm] \displaystyle \limsup_{|x|\rightarrow+\infty}|x|^{m-2} &\left|u(x)-\left(\frac{1}{2}x^{T}Ax+c\right)\right|<\infty, \end{aligned} \right. \end{equation} where $m=m_{k,l}(\lambda(A))\in(2,n]$. \end{lemma} To prove \tref{thm.hqe}, it suffices to prove \lref{lem.hqe}. Indeed, suppose that $D,\varphi,A$ and $b$ satisfy the hypothesis of \tref{thm.hqe}. Consider the decomposition $A=Q^TNQ$, where $Q$ is an orthogonal matrix and $N$ is a diagonal matrix which satisfies $\lambda(N)=\lambda(A)$. Let \[\tilde{x}:=Qx,\quad\widetilde{D}:=\left\{Qx|x\in D\right\}\] and \[\tilde{\varphi}(\tilde{x}):=\varphi(x)-b^Tx =\varphi(Q^T\tilde{x})-b^TQ^T\tilde{x}.\] By \lref{lem.hqe}, we conclude that there exists a constant $\tilde{c}$ depending only on $n,\widetilde{D},k,l,N$ and $\norm{\tilde{\varphi}}_{C^2(\partial\widetilde{D})}$, such that for every $c\geq\tilde{c}$, there exists a unique viscosity solution $\tilde{u}\in C^0(\mathds R^n\setminus\widetilde{D})$ of \begin{equation}\label{eqn.hqe-nc} \left\{ \begin{aligned} \displaystyle \frac{\sigma_k(\lambda(D^2\tilde{u}))} {\sigma_l(\lambda(D^2\tilde{u}))}&=1 \quad\text{in}~\mathds R^n\setminus\overline{\widetilde{D}},\\%[0.05cm] \displaystyle \tilde{u}&=\tilde{\varphi} \quad\text{on}~\partial\widetilde{D},\\%[0.05cm] \displaystyle \limsup_{|\tilde{x}|\rightarrow+\infty}|\tilde{x}|^{m-2} &\left|\tilde{u}(\tilde{x})-\left(\frac{1}{2}\tilde{x}^{T}N\tilde{x} +c\right)\right|<\infty, \end{aligned} \right. \end{equation} where $m=m_{k,l}(\lambda(N))=m_{k,l}(\lambda(A))\in(2,n]$. Let \[u(x):=\tilde{u}(\tilde{x})+b^Tx=\tilde{u}(Qx)+b^Tx =\tilde{u}(\tilde{x})+b^TQ^T\tilde{x}.\] We claim that $u$ is the solution of \eref{eqn.hqe-abc} in \tref{thm.hqe}. To show this, we only need to note that \[D^2u(x)=Q^TD^2\tilde{u}(\tilde{x})Q,\quad \lambda\left(D^2u(x)\right)=\lambda\left(D^2\tilde{u}(\tilde{x})\right);\] \[u=\varphi\quad\text{on}~\partial D;\] and \begin{eqnarray*} &&\displaystyle |\tilde{x}|^{m-2}\left|\tilde{u}(\tilde{x}) -\left(\frac{1}{2}\tilde{x}^{T}N\tilde{x}+c\right)\right|\\ &=&\displaystyle \left(x^TQ^TQx\right)^{(m-2)/2}\left|u(x)-b^Tx -\left(\frac{1}{2}x^TQ^TNQx+c\right)\right|\\ &=&\displaystyle |x|^{m-2} \left|u(x)-\left(\frac{1}{2}x^{T}Ax+b^Tx+c\right)\right|. \end{eqnarray*} Thus we have proved that \tref{thm.hqe} can be established by \lref{lem.hqe}. \begin{remark} \begin{enumerate}[(1)] \item We may see from the above demonstration that the lower bound $\tilde{c}$ of $c$ in \tref{thm.hqe} can not be discarded generally. Indeed, for the radial solutions of the Hessian equation $\sigma_k(\lambda(D^2u))=1$ in $\mathds R^n\setminus\overline{B_1}$, \cite[Theorem 2]{WB13} states that there is no solution when $c$ is too small. \item Unlike the Poisson equation and the Monge-Amp\`{e}re equation, generally, for the Hessian quotient equation, the matrix $A$ in \tref{thm.hqe} can only be normalized to a diagonal matrix, and can not be normalized to $I$ multiplied by some constant. This is the reason why we study the generalized radially symmetric solutions, rather than the radial solutions, of the original equation \eref{eqn.hqe}. See also \cite{BLL14}.\qed \end{enumerate} \end{remark} Now we use the Perron's method to prove \lref{lem.hqe}. \begin{proof}[\textbf{Proof of \lref{lem.hqe}}] We may assume without loss of generality that $E_1\subset\subset D\subset\subset E_{\bar{r}} \subset\subset E_{\hat{r}}$ and $a:=(a_1,a_2,...,a_n):=\lambda(A)$ with $0<a_1\leq a_2\leq...\leq a_n$. The proof now will be divided into three steps. \emph{Step 1.}\quad Let \[\eta:=\inf_{\substack{x\in \overline{E_{\bar{r}}}\setminus D\\ \xi\in\partial D}}Q_\xi(x), \quad Q(x):=\sup_{\xi\in\partial D}Q_\xi(x)\] and \[\Phi_\beta(x):=\eta+\int_{\bar{r}}^{r_A(x)}\tau\partialsi(\tau,\beta)d\tau, \quad\forall r_A(x)\geq 1,~\forall \beta\geq 1,\] where $Q_{\xi}(x)$ and $\partialsi(r,\beta)$ are given by \lref{lem.Qxi} and \lref{lem.psi}, respectively. Then we have \begin{enumerate}[\quad(1)] \item Since $Q$ is the supremum of a collection of smooth solutions $\{Q_\xi\}$ of \eref{eqn.hqe}, it is a continuous subsolution of \eref{eqn.hqe}, i.e., \[\frac{\sigma_k(\lambda(D^2Q))}{\sigma_l(\lambda(D^2{Q}))}\geq 1\] in $\mathds R^n\setminus \overline{D}$ in the viscosity sense (see \cite[Proposition 2.2]{Ish89}). \item $Q=\varphi$ on $\partial D$. To prove this we only need to show that for any $\xi\in\partial D$, $Q(\xi)=\varphi(\xi)$. This is obvious since $Q_\xi\leq\varphi$ on $\overline{D}$ and $Q_\xi(\xi)=\varphi(\xi)$, according to \rref{rmk.Qxi}-\textsl{(1)}. \item By \lref{lem.Phi-subsol}, $\Phi_\beta$ is a smooth subsolution of \eref{eqn.hqe} in $\mathds R^n\setminus\overline{D}$. \item $\Phi_\beta\leq\varphi$ on $\partial D$ and $\Phi_\beta\leq Q$ on $\overline{E_{\bar{r}}}\setminus D$. To show them we first note that $\Phi_\beta(x)$ is strictly increasing with respect to $r_A(x)$ since $\partialsi(r,\beta)\geq 1>0$ by \lref{lem.psi}-\textsl{(i)}. Invoking $\Phi_\beta=\eta$ on $\partial E_{\bar{r}}$ and $\eta\leq Q$ on $\overline{E_{\bar{r}}}\setminus D$ by their definitions, we have $\Phi_\beta\leq\eta\leq Q$ on $\overline{E_{\bar{r}}}\setminus D$. On the other hand, according to \rref{rmk.Qxi}-\textsl{(1)}, we have $Q_\xi\leq\varphi$ on $\overline{D}$ which implies that $\eta\leq\varphi$ on $\overline{D}$. Combining these two aspects we deduce that $\Phi_\beta\leq\eta\leq\varphi$ on $\partial D$. \item $\Phi_\beta(x)$ is strictly increasing with respect to $\beta$ and \begin{equation}\label{eqn.Phib} \lim_{\beta\rightarrow+\infty}\Phi_\beta(x)=+\infty, ~\forall r_A(x)\geq 1, \end{equation} by the definition of $\Phi_\beta(x)$ and \lref{lem.psi}-\textsl{(ii)}. \item As we showed in \eref{eqn.phi-mu} and \eref{eqn.phi-O}, for any $\beta\geq 1$, we have \begin{eqnarray*} \Phi_\beta(x) &=&\eta+\int_{\bar{r}}^{r_A(x)}\tau \partialsi(\tau,\beta)d\tau\\ &=&\eta+\frac{1}{2}(r_A(x)^2-\bar{r}^2) +\int_{\bar{r}}^{r_A(x)}\tau\big(\partialsi(\tau,\beta)-1\big)d\tau\\ &=&\frac{1}{2}r_A(x)^2+\left(\eta-\frac{1}{2}\bar{r}^2 +\mu_{\bar{r}}(\beta)\right)-\mu_{r_A(x)}(\beta)\\ &=&\frac{1}{2}r_A(x)^2+\mu(\beta)-\mu_{r_A(x)}(\beta)\\ &=&\frac{1}{2}x^TAx+\mu(\beta)+O\left(|x|^{-m+2}\right) ~(|x|\rightarrow+\infty), \end{eqnarray*} where we set \[\mu(\beta):=\eta-\frac{1}{2}\bar{r}^2+\mu_{\bar{r}}(\beta),\] and used the fact that $x^TAx=O(|x|^2)~(|x|\rightarrow+\infty)$ since $\lambda(A)\in\Gamma^+$. \end{enumerate} \emph{Step 2.}\quad For fixed $\hat{r}>\bar{r}$, there exists $\hat{\beta}>1$ such that \[\min_{\partial E_{\hat{r}}}\Phi_{\hat{\beta}} >\max_{\partial E_{\hat{r}}}Q,\] in light of \eref{eqn.Phib}. Thus we obtain \begin{equation}\label{eqn.PhiQ} \Phi_{\hat{\beta}}>Q\quad\mbox{on}~\partial E_{\hat{r}}. \end{equation} Let \[\tilde{c}:=\max\left\{\eta,\mu(\hat{\beta}),\bar{c}\right\},\] where the $\bar{c}$ comes from \rref{rmk.Qxi}-\textsl{(3)}, and hereafter fix $c\geq\tilde{c}$. By \lref{lem.psi} and \cref{cor.mub} we deduce that \[\partialsi(r,1)\equiv 1\mathds Ra\mu_{\bar{r}}(1)=0 \mathds Ra\mu(1)=\eta-\frac{1}{2}\bar{r}^2<\eta\leq\tilde{c}\leq c,\] and \[\lim_{\beta\rightarrow+\infty}\mu_{\bar{r}}(\beta)=+\infty \mathds Ra\lim_{\beta\rightarrow+\infty}\mu(\beta)=+\infty.\] On the other hand, it follows from \cref{cor.mub} that $\mu(\beta)$ is continuous and strictly increasing with respect to $\beta$ \big{(}which indicates that the inverse of $\mu(\beta)$ exists and $\mu^{-1}$ is strictly increasing\big{)}. Thus there exists a unique $\beta(c)$ such that $\mu(\beta(c))=c$. Then we have \[\Phi_{\beta(c)}(x)=\frac{1}{2}r_A(x)^2+c-\mu_{r_A(x)}(\beta(c)) =\frac{1}{2}x^TAx+c+O\left(|x|^{-m+2}\right)~(|x|\rightarrow+\infty),\] and \[\beta(c)=\mu^{-1}(c)\geq\mu^{-1}(\tilde{c})\geq\hat{\beta}.\] Invoking the monotonicity of $\Phi_\beta$ with respect to $\beta$ and \eref{eqn.PhiQ}, we obtain \begin{equation}\label{eqn.PhigQ} \Phi_{\beta(c)}\geq\Phi_{\hat{\beta}}>Q\quad\mbox{on} ~\partial E_{\hat{r}}. \end{equation} Note that we already know \[\Phi_{\beta(c)}\leq Q\quad\mbox{on} ~\overline{E_{\bar{r}}}\setminus D,\] from (4) of \emph{Step 1}. Let \begin{equation*} \underline{u}(x):= \begin{cases} \max\left\{\Phi_{\beta(c)}(x),Q(x)\right\},& x\in E_{\hat{r}}\setminus D,\\ \Phi_{\beta(c)}(x),& x\in \mathds R^n\setminus E_{\hat{r}}. \end{cases} \end{equation*} Then we have \begin{enumerate}[\quad(1)] \item $\underline{u}$ is continuous and satisfies \[\frac{\sigma_k(\lambda(D^2{\underline{u}}))} {\sigma_l(\lambda(D^2{\underline{u}}))}\geq1\] in $\mathds R^n\setminus\overline{D}$ in the viscosity sense, by (1) and (3) of \emph{Step 1}. \item $\underline{u}=Q=\varphi$ on $\partial D$, by (2) of \emph{Step 1}. \item If $r_A(x)$ is large enough, then \[\underline{u}(x)=\Phi_{\beta(c)}(x) =\frac{1}{2}x^TAx+c+O\left(|x|^{-m+2}\right)~(|x|\rightarrow+\infty).\] \end{enumerate} \emph{Step 3.}\quad Let \[\overline{u}(x):=\frac{1}{2}x^TAx+c, ~\forall x\in\mathds R^n.\] Then $\overline{u}$ is obviously a supersolution and \[\lim_{|x|\rightarrow+\infty}\left(\underline{u}-\overline{u}\right)(x)=0.\] To use the Perron's method to establish \lref{lem.hqe}, we now only need to prove that \[\underline{u}\leq \overline{u}\quad\mbox{in}~\mathds R^n\setminus D.\] In fact, since \[\mu_{r_A(x)}(\beta)\geq 0, \quad\forall x\in\mathds R^n\setminus E_1, ~\forall\beta\geq 1,\] according to \cref{cor.mub}, we have \begin{equation}\label{eqn.Phiolu} \Phi_{\beta(c)}(x)=\frac{1}{2}x^TAx+c-\mu_{r_A(x)}(\beta(c)) \leq\frac{1}{2}x^TAx+c=\overline{u}(x),~\forall x\in\mathds R^n\setminus D. \end{equation} \Big{(}We remark that this \eref{eqn.Phiolu} can also be proved by using the comparison principle, in view of \[\Phi_{\beta(c)}\leq\eta\leq\tilde{c}\leq c\leq\overline{u} \quad\mbox{on}~\partial D,\] and \[\lim_{|x|\rightarrow+\infty}\left(\Phi_{\beta(c)}-\overline{u}\right)(x)=0.\Big{)}\] On the other hand, for every $\xi\in\partial D$, since \[Q_\xi(x)\leq\frac{1}{2}x^TAx+\bar{c} \leq\frac{1}{2}x^TAx+\tilde{c} \leq\frac{1}{2}x^TAx+c=\overline{u}(x), ~\forall x\in\partial D,\] and \[Q_\xi\leq Q<\Phi_{\beta(c)}\leq\overline{u} \quad\mbox{on}~\partial E_{\hat{r}}\] follows from \eref{eqn.PhigQ} and \eref{eqn.Phiolu}, we obtain \[Q_\xi\leq\overline{u}\quad\mbox{on} ~\partial\left(E_{\hat{r}}\setminus D\right).\] In view of \[\frac{\sigma_k(\lambda(D^2{Q_\xi}))}{\sigma_l(\lambda(D^2{Q_\xi}))} =1=\frac{\sigma_k(\lambda(D^2{\overline{u}}))} {\sigma_l(\lambda(D^2{\overline{u}}))} \quad\mbox{in}~E_{\hat{r}}\setminus D,\] we deduce from the comparison principle that \[Q_\xi\leq\overline{u}\quad\mbox{in}~E_{\hat{r}}\setminus D.\] Hence \begin{equation}\label{eqn.Qu} Q\leq\overline{u}\quad\mbox{in}~E_{\hat{r}}\setminus D. \end{equation} Combining \eref{eqn.Phiolu} and \eref{eqn.Qu}, by the definition of $\underline{u}$, we get \[\underline{u}\leq \overline{u}\quad\mbox{in}~\mathds R^n\setminus D.\] This finishes the proof of \lref{lem.hqe}. \end{proof} \begin{remark} To prove \lref{lem.hqe} we have used above \lref{lem.cp} and \lref{lem.pm} presented in \ssref{subsec.prelem}. In fact, one can follow the techniques in \cite{CL03} (see also \cite{DB11} \cite{Dai11} and \cite{LD12}) instead of \lref{lem.pm} to rewrite the whole proof. These two kinds of presentation look a little different but are essentially the same. \end{remark} \end{document}

\begin{document} \title{Purifying GHZ States Using Degenerate Quantum Codes} \author{K. H. Ho and H. F. Chau} \date{\today} \affiliation{Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China} \begin{abstract} Degenerate quantum codes are codes that do not reveal the complete error syndrome. Their ability to conceal the complete error syndrome makes them powerful resources in certain quantum information processing tasks. In particular, the most error-tolerant way to purify depolarized Bell states using one-way communication known to date involves degenerate quantum codes. Here we study three closely related purification schemes for depolarized GHZ states shared among $m \geq 3$ players by means of degenerate quantum codes and one-way classical communications. We find that our schemes tolerate more noise than all other one-way schemes known to date, further demonstrating the effectiveness of degenerate quantum codes in quantum information processing. \end{abstract} \pacs{03.67.Hk, 03.67.Pp, 89.70.Kn} \maketitle \section{Introduction} Quantum error correcting codes, unlike their classical counterparts, may not reveal the complete error syndrome. Codes with this property are known as degenerate codes~\cite{SS96,DSS98}. In a sense, degenerate codes pack more information than non-degenerate ones because different quantum errors may not take the code space to orthogonal spaces. By carefully utilizing the degenerate property, degenerate codes are useful resources in quantum information processing. Examples showing their usefulness were provided by Shor and his co-workers~\cite{SS96,DSS98}. In particular, they showed that a carefully constructed degenerate code is able to purify Bell states passing through a depolarizing channel with fidelity greater than 0.80944~\cite{DSS98}. Their scheme is more error-tolerant than all the known one-way depolarized Bell state purification schemes involving non-degenerate codes to date. It is instructive to ask if the degenerate codes can be used to improve the error-tolerant level of existing one-way multipartite purification protocols. Here we provide such an example by considering the purification of shared GHZ states. Specifically, suppose that a player prepares many copies of perfect GHZ state in the form \begin{equation} \ket{\Phi^{m+}} \equiv \frac{1}{\sqrt{2}} \left( \ket{0^{\otimes m}} + \ket{1^{\otimes m}} \right) ~. \label{eq:GHZ} \end{equation} For each perfect GHZ state, he/she keeps one of the qubit and sends the other to the remaining players through a depolarizing channel so that upon reception of their qubits, these $m$ players share copies of Werner state \begin{equation} W_F = F \ket{\Phi^{m+}} \bra{\Phi^{m+}} + \frac{1-F}{2^m - 1} \left( I - \ket{\Phi^{m+}} \bra{\Phi^{m+}} \right) ~, \label{eq:Wf} \end{equation} where $F$ is the fidelity of the channel and $I$ is the identity operator. Now, the players wanted to distill shared perfect GHZ states using an one-way purification scheme that works for as small a channel fidelity as possible. Clearly, this task is a generalization of the Bell state distillation problem investigated by Shor and his co-workers~\cite{SS96,DSS98}. We begin our study by defining a few notations and reviewing prior arts in Sec.~\ref{sec:GHZ_re}. Then we introduce three closely related one-way multipartite purification protocols involving concatenated degenerate codes and analyze their performances in Sec.~\ref{sec:our_pro}. Actually, all three protocols use the same repetition code as their inner codes. Moreover, in the case of $m = 2$, one of the our protocols is a generalization of the scheme proposed by DiVincenzo \emph{et al.}~\cite{DSS98}. Most importantly, for $m\geq 3$, our protocols are the most error tolerant ones discovered so far in the sense that ours can distill shared GHZ states from copies of Werner state in the form of Eq.~(\ref{eq:Wf}) with a fidelity $F$ so low that no other one-way purification schemes known to date can. Our schemes can also be generalized to the case when the Hilbert space dimension of each quantum particle is greater than 2. We briefly discuss this issue in Sec.~\ref{sec:high_spin}. Finally, we summarize our findings in Sec.~\ref{sec:sum}. \section{Prior Arts} \label{sec:GHZ_re} \subsection{Some notations} Given that $m\geq 2$ players share $N$ noisy GHZ states in the form of Eq.~(\ref{eq:GHZ}). Clearly, the GHZ state is stabilized by its stabilizer generators, namely, \begin{eqnarray} \label{eq:GHZ_gen} S_0 &=& X_0 X_1 \cdots X_{m-1} ~, \nonumber \\ S_i &=& Z_0 Z_i \end{eqnarray} for $1\leq i \leq m - 1$, where \begin{equation} X_i = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) ~, Z_i = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{equation} denote the spin flip and phase shift operation acting on the $i$th qubit respectively. For simplicity, we use the shorthand notation $(\beta, \boldsymbol{\alpha}) \equiv (\beta, \alpha_1, \alpha_2, \ldots, \alpha_{m-1}) \in GF(2) \times GF(2)^{m-1}$ to denote the eigenvalues of stabilizer generators. Here $\beta \in GF(2)$ is the eigenvalue of the operator $S_0$, namely, the phase error detected; and $\alpha_i \in GF(2)$ is the eigenvalue of the operator $S_i$, namely, the bit flip error detected, for $1\leq i\leq m - 1$. We sometimes abuse the notation to denote a state by $(\beta, \boldsymbol{\alpha})$. That is, we denote the states $(|0^{\otimes m}\rangle + |1^{\otimes m}\rangle)/\sqrt{2}$ and $(|0^{\otimes m}\rangle - |1^{\otimes m}\rangle)/\sqrt{2}$ by $(\beta, \boldsymbol{\alpha}) = (0, \boldsymbol{0})$ and $(\beta, \boldsymbol{\alpha}) = (1, \boldsymbol{0})$ respectively. \subsection{Depolarization to the GHZ-basis diagonal states} The $m$ players can depolarize each copy of their shared GHZ state into the GHZ diagonal basis using local operation and classical communication (LOCC) in the following way~\cite{DCT99}. A player randomly chooses an operator from the span of the set of stabilizer generators of the GHZ state and broadcast his/her choice to the other players. Then they collectively apply the chosen operator to the GHZ state. Since all stabilizer generators of the GHZ state in Eq.~(\ref{eq:GHZ_gen}) are tensor products of local unitary operators $X_i$ or $Z_i$, the players can apply the operator chosen above to the state locally using LOCC. Then, they forget which operator they have chosen. The resultant state is diagonal in the GHZ basis. Moreover, the error rate of the GHZ state is unchanged in this process. So, we can always assume that each state shared among the players are diagonal in the GHZ basis. Among all GHZ-basis diagonal states with a fixed error rate (and hence also among all states with a fixed error rate), Werner state is the most difficult to work on as far as distillation of GHZ states is concerned. This is because one can turn any state into a Werner state with the same error rate via a depolarizing channel. Hence, to study the worst case performance of the distillation of GHZ states, we suffices to investigate the case in which the input states are Werner states. \subsection{Maneva and Smolin's multi-party hashing protocol and its generalization by Chen and Lo} \label{sec:MH_pro} Maneva and Smolin~\cite{MS00} proposed a multi-party hashing protocol by generalizing the bilateral quantum XOR (BXOR) operation~\cite{BDSW96a} to the multipartite case. In their scheme, the $m$ players carefully choose two (classical) random hashing codes, one to correct spin flip errors and the other to correct phase errors, and apply them to their shared noisy GHZ states. This can be done by using local operation plus classical communication with the help of a few multi-lateral quantum XOR (MXOR) operations. Recall that MXOR is a linear map transforming the state $\bigotimes_{i=0}^{m-1} |j_i,k_i\rangle_i$ to $\bigotimes_{i=0}^{m-1} |j_i,j_i+k_i\rangle_i$ for all $j_i,k_i\in GF(2)$. Here, quantum particles with subscript $i$ belong to the $i$th player. Suppose the source and target states are eigenstates of the stabilizer generator in Eq.~(\ref{eq:GHZ_gen}) with eigenvalues $(\beta_1, \boldsymbol{\alpha}_1)$ and $(\beta_2, \boldsymbol{\alpha}_2)$ respectively. Then after the MXOR operation, the resultant state is also an eigenstate of the stabilizer generator with eigenvalues~\cite{MS00} \begin{equation}\label{eq:MXOR_Def} \text{MXOR}[(\beta_1, \boldsymbol{\alpha}_1), (\beta_2, \boldsymbol{\alpha}_2)] =[(\beta_1 + \beta_2, \boldsymbol{\alpha}_1), (\beta_2, \boldsymbol{\alpha}_1 - \boldsymbol{\alpha}_2)] ~. \end{equation} Using the observation that spin flip error occurred in different qubit of a GHZ state can be detected and corrected in parallel, Maneva and Smolin showed that the asymptotic yield of their hashing protocol in the limit of large number of shared noisy GHZ states is given by~\cite{MS00} \begin{equation}\label{eq:D1} D_1 = 1 - \max_{1 \leq i \leq m - 1 }[{H(b_i)}] - H(b_0) ~, \end{equation} where $H(x) \equiv - \sum_j p_j \log_2 p_j$ is the classical Shannon entropy function. Here the $n$-bit string $b_0$ represents the random choice of $\beta_1, \ldots, \beta_N$ where $\beta_\ell$ corresponds to the eigenvalue of the operator $S_0$ of the $\ell$th GHZ-state $\ket{\Phi^{m+}}$'s and the $N$-bit string $b_i$ represents the random choice of $\alpha_{1i}, \ldots, \alpha_{Ni}$ where $\alpha_{\ell i}$ is the eigenvalue of the operator $S_i$ of the $\ell$th GHZ state for $1\leq i\leq m - 1$. That is to say, $H(b_0)$ is the averaged phase error rate and $H(b_i)$ is the averaged bit flip rate corresponding to the stabilizer generator $S_i$ (for $i=1,2,\ldots ,m-1$) over the $N$ GHZ states respectively. Recently, Chen and Lo improved the above random hashing protocol by exploiting the correlation between the string $b_i$. Specifically, they replaced the spin flip error-correction random hashing code used in Maneva and Smolin protocol by the following scheme. (Actually, they only considered the case of three players. What we report below is a straight-forward generalization to the case of $m$ players as we need to use this generalization later on.) Player~1 applies (classical) random hashing to correct spin flip error occurred in his/her share of the GHZ states. He/She then broadcasts his/her hashing code used and his measurement results. For $i = 2,\ldots , m-1$, the $i$th player carefully picks his/her (classical) random hashing code to correct spin flip error occurred in his/her share of the GHZ states based on the broadcast information of players~$1,2,\ldots , i-1$. Then, the $i$th player broadcasts his/her code used and his/her measurement results. In this way, the yield of Maneva and Smolin scheme can be increased to~\cite{CL07} \begin{eqnarray}\label{eq:D2} D_2 & = & 1 - \max \{ H(b_1), H(b_2 | b_1), H(b_3 | b_2, b_1), \cdots , H(b_{m-1} | b_{m-2},b_{m-3}, \ldots , b_1) \} \nonumber \\ & & ~- H(b_0) + I(b_0; b_{m-1}, b_{m-2}, \ldots , b_1) \end{eqnarray} where the function $I(\,;\,)$ is the mutual information between the two classical random variables appear in its arguments. Applying the random hashing method of Maneva and Smolin to a collection of identical tripartite (that is, $m = 3$) Werner states in Eq.~(\ref{eq:Wf}), one can obtain perfect GHZ state with non-zero yield whenever the fidelity $F\geq 0.8075$~\cite{MS00}. Using the Chen and Lo's formula in Eq.~(\ref{eq:D2}), one can push this threshold fidelity down to $0.7554$~\cite{CL07}. \subsection{Shor-Smolin concatenation procedure and its generalization to the multipartite situation} \label{sec:SS_pro} Built on an earlier work by Shor and Smolin~\cite{SS96}, DiVincenzo \emph{et al.} introduced a highly error-tolerant way of distilling shared Bell states by means of a concatenation procedure~\cite{DSS98}. This procedure can be generalized to distill shared GHZ states in a straight-forward manner. We report this generalization below since we have to use a few related equations later on. Suppose $m$ players share $N n$ copies of imperfect GHZ states for $N\gg 1$. They perform the following two level decoding procedure. First, the players randomly divide these GHZ states into $N$ equal parts. Then each player applies a decoding transformation associated with an additive $[n,k_1,d_1]$ code to his/her own qubits followed by the error syndrome measurements. Surely, this can be done with the help of a few MXOR operations. By comparing the difference in player's measurement results, they obtain the syndrome $\vec{\boldsymbol{s}} \in GF(2)^{(m-1)(n-k_1)}$. To continue, each party applies another decoding transformation corresponding to a (classical) random hashing code $[N k_1, k_2, d_2]$ to correct errors in the GHZ diagonal basis and broadcast the measurement results. Finally, they apply the necessary unitary transformation according to the measured error syndrome of this random hashing code to get the purified GHZ states. Suppose that an additive code $[n, k_1, d_1]$ is applied and the remaining states after the decoding transformation and measurements are denoted by $(\delta,\boldsymbol{\gamma}) \equiv \text{TRAN}[(\beta_1, \boldsymbol{\alpha}_1), (\beta_2,\boldsymbol{\alpha}_2), \ldots, (\beta_{k_1}, \boldsymbol{\alpha}_{k_1})]$. Then, the yield of this concatenated scheme is given by the so-called Shor-Smolin capacity~\cite{SS96,DSS98} \begin{equation} \label{eq:D_SS} D_\textrm{SS} = \frac{1}{n} ( 1 - S_X) ~, \end{equation} where \begin{equation} \label{eq:Sx} S_X = \sum_{\vec{\boldsymbol{s}} \in GF(2)^{(m - 1) ( n - k_1)}} \text{Pr} (\vec{\boldsymbol{s}}) \enspace h(\{\text{Pr} ((\delta, \boldsymbol{\gamma}) | \vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \}) \end{equation} is the average of the von Neumann entropies of the quantum states conditional on the measurement outcomes. Note that in Eq.~(\ref{eq:Sx}), $\text{Pr} (\vec{\boldsymbol{s}})$ is the probability that the measurement outcome is $\vec{\boldsymbol{s}}$, \begin{equation} h(\{ p_i \} ) \equiv - \sum_i p_i \log_2 p_i ~, \end{equation} and \begin{equation} \sum_i p_i = 1 ~. \end{equation} By applying the above procedure to depolarized Bell states using a 5-qubit cat code as the inner code and a random hashing code as the outer code (that is, the case of $m = 2$ and $n = 5$), DiVincenzo \emph{et al.} found that one can attain a non-zero capacity whenever the channel fidelity $F > 0.80944$~\cite{DSS98}. Since the performance of this scheme exceeds that of quantum random hashing code and that the 5-qubit cat code is degenerate, the power of using degenerate quantum code in quantum information processing is demonstrated. \subsection{Other hashing and breeding schemes} Several other multipartite hashing schemes have been studied~\cite{MS00,HDM06}. In particular, Maneva and Smolin's hashing scheme can distill shared GHZ states from copies of Werner states with fidelity $F \geq 0.7798$ in the limit of arbitrarily large number of players (that is, when $m \rightarrow \infty$)~\cite{MS00}. Another approach is to use the so-called stabilizer breeding. In particular, Hostens \emph{et al.} showed that stabilizer breeding is able to purify depolarized $5$-qubit ring state with fidelity $F\geq 0.756$~\cite{HDM06b}. A few authors also studied the distillation of graph state subjected to local $Z$-noise~\cite{KPDB06} and bicolorable graph state.~\cite{GMR06} Furthermore, Glancy \emph{et al.} generalized the hashing protocol of Maneva and Smolin to purify a much larger class of output state.~\cite{GKV06} The second column in Table~\ref{tb:pw} summarizes the state-of-the-art one-way purification schemes to distill depolarized GHZ states before our work. \begin{center} \begin{table} \begin{tabular}{||c|c|c|c||} \hline \hline ~~$m$~~ & prior art & our best protocol & lower bound \\ \hline 2 & 0.8094 & 0.8094 & 0.7500 \\ 3 & 0.7554 & 0.7074 & 0.6111 \\ 4 & 0.7917 & 0.6601 & 0.5500 \\ \hline \hline \end{tabular} \caption{The threshold fidelity of the depolarizing channel above which a GHZ state can be distilled by prior art and by our protocols. Also listed is the lower bound of the fidelity below which no one-way protocol can distill shared GHZ state using Eq.~(\ref{eq:F_min_lower_bound}) in Sec.~\ref{subsec:limit}. As for prior art, the threshold fidelity for $m = 2$ is given by the 5-qubit cat code~\cite{DSS98}. For $m = 3$ case, the threshold is computed by the Chen and Lo's formula~\cite{CL07} in Eq.~(\ref{eq:D2}). For $m = 4$, the threshold is given by the Maneva and Smolin's hashing protocol~\cite{MS00} in Eq.~(\ref{eq:D1}). \label{tb:pw} } \end{table} \end{center} \section{Our protocols involving degenerate code and their performances} \label{sec:our_pro} \subsection{Our protocols} Our three protocols are natural extensions of the Shor-Smolin concatenation procedure to the case of purifying GHZ states. They all use the same degenerate quantum code as the inner code. Specifically, suppose the $m$ players share $N n$ copies of Werner state with $N\gg 1$. As shown in Fig.~\ref{fig:our_protocol}, to distill perfect GHZ state, each player applies the (classical) $[n, 1, n]$ repetition code, whose stabilizer generators are \begin{equation} \label{eq:GHZ_ZZ} Z_0 Z_1, Z_0 Z_2, \ldots, Z_0 Z_{n-1} ~, \end{equation} to his/her own $n$ qubits. That is to say, they randomly partition the $N n$ shared noisy GHZ states into $N$ sets, each containing $n$ noisy GHZ states. In each set ${\mathfrak S}$, they randomly assign one of the noisy GHZ state as the source (and call it the $0$th copy of $\ket{\Phi^{m+}}$ in the set) and the remaining $(n-1)$ noisy GHZ states as the targets (and call them the $j$th copy of $\ket{\Phi^{m+}}$ in the set for $j = 1, 2, \ldots, n - 1$). They apply the MXOR operation to copies of $\ket{\Phi^{m+}}$ in each set and then measure all the target GHZ states in the standard computational basis while leaving all the source GHZ states un-measured. We denote the syndrome and the remaining state in each set by $\vec{\boldsymbol{s}}_{\mathfrak S} \in GF(2)^{(n - 1) (m - 1)}$ and $(\delta_{\mathfrak S}, \boldsymbol{\gamma}_{\mathfrak S}) \in GF(2)^m$ respectively. (Since the partition into $N$ sets is arbitrarily chosen and our subsequent analysis only makes use of the statistical properties of the states in each set, we drop the set label ${\mathfrak S}$ in all quantities to be analyzed from now on.) Our three protocols differ in the use of outer codes. For the first protocol, each player applies a (classical) random hashing code $[N, k_2, d_2]$ that corrects GHZ diagonal basis errors to the $N$ remaining states (each coming from a different set ${\mathfrak S}$) and exchanges the measurement results. Clearly, this protocol is reduced to the Shor-Smolin concatenation procedure~\cite{SS96} when $m = 2$. \begin{figure} \caption{Our three GHZ state distillation protocols for $m = n = 3$. They all use the $[n,1,n]$ repetition code as their inner codes; and they differ by the kind of random hashing outer code used. \label{fig:our_protocol} \label{fig:our_protocol} \end{figure} For the second protocol, the players follow Maneva and Smolin's idea~\cite{MS00} by using two (classical) random hashing codes, one to correct spin flip error and the other to correct phase shift. In this sense, the outer code used in our second protocol is a random asymmetric Calderbank-Shor-Steane (CSS) code. Whereas players in our third protocol use the Chen and Lo's generalization~\cite{CL07} as their outer code. That is, the outer code is a random asymmetric CSS code whose decoding circuit is carefully designed to exploit the correlation between the bit string $b_i$. In all the above three protocols, the players have to apply the corresponding unitary transformation for the outer code to obtain the purified GHZ states. (See Fig.~\ref{fig:our_protocol}.) Clearly, the yield of the first protocol is the Shor-Smolin capacity given by Eqs.~(\ref{eq:D_SS}) and~(\ref{eq:Sx}). And by applying Eq.~(\ref{eq:D1}) to the noisy GHZ state to be fed into the outer code of the second protocol, we conclude that the yield of the second protocol equals \begin{equation} D_\textrm{MS} = \frac{1}{n} \left( 1 - \sum_{\vec{\boldsymbol{s}} \in GF(2)^{(m-1)(n-1)}} \textrm{Pr}(\vec{\boldsymbol{s}}) \left\{ \max_{1\leq i \leq m-1} \left[ H(\gamma_i | \vec{\boldsymbol{s}} ) \right] + H(\delta | \vec{\boldsymbol{s}}) \right\} \right) \label{eq:D_MS} \end{equation} where \begin{equation} H ( \gamma_i | \vec{\boldsymbol{s}} ) = h ( \{ \sum_{\delta, \gamma_1, \ldots , \gamma_{i-1},\gamma_{i+1}, \ldots ,\gamma_{m-1}\in GF(2)} \textrm{Pr} ((\delta, \boldsymbol{\gamma}) | \vec{\boldsymbol{s}}) : \gamma_i \in GF(2) \} ) \label{eq:D_hDef1} \end{equation} for $i = 1,2,\ldots ,m-1$, and \begin{equation} H ( \delta | \vec{\boldsymbol{s}} ) = h ( \{ \sum_{\boldsymbol{\gamma}\in GF(2)^{m-1}} \textrm{Pr} ((\delta, \boldsymbol{\gamma}) | \vec{\boldsymbol{s}}) : \delta \in GF(2) \} ) . \label{eq:D_hDef2} \end{equation} Similarly, from Eq.~(\ref{eq:D2}), the yield of the third protocol is \begin{equation} D_\textrm{CL} = \frac{1}{n} \left( 1 - \sum_{\vec{\boldsymbol{s}} \in GF(2)^{(m-1)(n-1)}} \textrm{Pr}(\vec{\boldsymbol{s}}) \left\{ \max_{1\leq i \leq m-1} \left[ H(\gamma_i | \gamma_{i-1}, \gamma_{i-2}, \ldots, \gamma_1, \vec{\boldsymbol{s}} ) \right] + H(\delta | \vec{\boldsymbol{s}}) - I (\delta;\boldsymbol{\gamma} | \vec{\boldsymbol{s}}) \right\} \right) \label{eq:D_CL} \end{equation} where \begin{equation} H (\gamma_i | \gamma_{i-1},\gamma_{i-1},\ldots , \gamma_1, \vec{\boldsymbol{s}}) = h ( \{ \sum_{\delta,\gamma_{i+1},\gamma_{i+2}, \ldots , \gamma_{m-1}\in GF(2)} \textrm{Pr} ((\delta,\boldsymbol{\gamma}) | \gamma_{i-1}, \gamma_{i-2} , \ldots , \gamma_1, \vec{\boldsymbol{s}}) : \gamma_i \in GF(2) \} ) \label{eq:D_hDef3} \end{equation} for $i=1,2,\ldots ,m-1$ and $I (\delta;\boldsymbol{\gamma} | \vec{\boldsymbol{s}})$ is the conditional mutual information between $\delta$, and $\boldsymbol{\gamma}$ given $\vec{\boldsymbol{s}}$. \subsection{Evaluating the yields for Werner states for our protocols} \label{subsec:evaluation} To analyze the performance of our three protocols when applied to Werner states, we first have to calculate the distribution of outcomes after passing the Werner states through the inner repetition code. For an arbitrary but fixed set ${\mathfrak S}$, using the compact notation introduced in Sec.\ref{sec:GHZ_re}, we denote the error experienced by the $j$th copy of $\ket{\Phi^{m+}}$ in this set by $(\beta_j, \boldsymbol{\alpha}_j)$ for $j = 0, 1, \ldots , n - 1$. After decoding the inner code, namely, the $[n, 1, n]$ repetition code whose generators of the stabilizer are written down in Eq.~(\ref{eq:GHZ_ZZ}), the syndrome $\vec{\boldsymbol{s}} \equiv (\boldsymbol{s}_1,\boldsymbol{s}_2, \ldots \boldsymbol{s}_{n-1}) \in GF(2)^{(m-1)(n-1)}$ obtained obeys \begin{equation} \boldsymbol{s}_j \equiv \boldsymbol{\alpha}_j - \boldsymbol{\alpha}_0 \in GF(2)^{m-1} \label{eq:alpha_j_relation} \end{equation} for all $1\leq j\leq n-1$. Furthermore, the remaining state shared among the $m$ players is \begin{equation} (\delta, \boldsymbol{\gamma}) = (\sum_{j = 0}^{n - 1} \beta_j, \boldsymbol{\alpha}_0) ~. \end{equation} To simplify notation in our subsequent discussions, we define \begin{equation} \boldsymbol{s}_0 = \boldsymbol{0} \end{equation} so that Eq.~(\ref{eq:alpha_j_relation}) is also valid for $j = 0$. To evaluate the capacity for each of our three protocols, we first have to calculate the conditional probabilities $\text{Pr}( (\delta, \boldsymbol{\gamma}) | \vec{\boldsymbol{s}})$ and $\textrm{Pr}( (\delta,\boldsymbol{\gamma}) | \gamma_1,\gamma_2, \ldots, \gamma_{i-1},\vec{\boldsymbol{s}})$ in Eqs.~(\ref{eq:Sx}) and~(\ref{eq:D_CL}), respectively. We begin by computing the probability $\text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}})$ that the source state has experienced the error $(\delta,\boldsymbol{\gamma}) = (\sum_{j=0}^{n-1} \beta_j,\boldsymbol{\alpha}_0)$ after the decoding transformation of the repetition code in Eq.~(\ref{eq:GHZ_ZZ}) and that the error syndrome for the repetition code is $\vec{\boldsymbol{s}} \in GF(2)^{(m - 1) (n - 1) }$. Clearly, \begin{equation} \text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}}) = \text{Pr}({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})) \label{eq:pr_a} \end{equation} where \begin{equation} {\mathfrak E}(\vec{\boldsymbol{s}}, \delta, \boldsymbol{\gamma}) \equiv \{ ( \beta_j, \boldsymbol{\alpha}_j)_{j=0}^{n-1} \in GF(2)^{m n} : \delta = \sum_{j=0}^{n - 1} \beta_j , \boldsymbol{\gamma} = \boldsymbol{\alpha}_0, \boldsymbol{\alpha}_\ell = \boldsymbol{s}_\ell + \boldsymbol{\alpha}_0 \enspace \text{for} \enspace \ell = 1, 2, \ldots, n -1\} ~. \label{eq:E_set_def} \end{equation} Since the repetition code and our decoding transformation are highly symmetric in the sense that they are invariant under relabeling of qubits, it is not surprising that the set ${\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$ is invariant under permutation of phase errors. That is to say, $(\beta_j,\boldsymbol{\alpha}_j)_{j=0}^{n-1} \in {\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$ if and only if $(\beta_{\pi(j)},\boldsymbol{\alpha}_j)_{j=0}^{n-1} \in {\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$ where $\pi$ is a permutation of $\{ 0,1,\ldots ,n-1 \}$. \subsubsection{Finding $\text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}})$} We proceed by introducing the concepts of depolarization weight and depolarization weight enumerator similar to the ones proposed by DiVincenzo \emph{et al.}~\cite{DSS98}. Let $(\beta_j, \boldsymbol{\alpha}_j)$ be the state of the $j$th noisy GHZ state shared among the $m$ players. The {\bf depolarization weight} of the order $n$-tuple $(\beta_j,\boldsymbol{\alpha}_j)_{j=0}^{n-1} \in GF(2)^{m n}$ is defined as its Hamming weight by regarding this $n$-tuple as a vector of elements in $GF(2)^m$. In other words, \begin{equation} \text{wt}\left((\beta_j,\boldsymbol{\alpha}_j)_{j=0}^{n-1} \right) = | \{ j \in \{ 0,1, \ldots , n-1 \} : (\beta_j, \boldsymbol{\alpha}_j) \neq (0, \boldsymbol{0}) \}| ~. \end{equation} Physically, the depolarization weight measures the number of shared GHZ states that experienced an error; thus, it is invariant under permutation of the $n$ possibly imperfect GHZ states. Since a GHZ state has equal probability of having each type of error after passing through a depolarizing channel, there is an equal probability for the $n$ depolarized GHZ states to experience errors with the same depolarization weight. Thus, we may find the probability $\text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}})$ by studying the {\bf depolarization weight enumerator} $w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y)$ where \begin{equation} w({\mathfrak A};x,y) = \sum_{\vec{\boldsymbol{a}}\in {\mathfrak A}} x^{\text{wt}(\vec{\boldsymbol{a}})} y^{n - \text{wt}(\vec{\boldsymbol{a}})} ~. \end{equation} The depolarization weight enumerator of a set is a natural generalization of the concept of weight enumerator of a code. Finding an explicit expression for the above depolarization weight enumerator for an arbitrary set or coset is a very difficult task. It is the high degree of symmetry in the repetition code that makes this task possible. In fact, one may transform a state in ${\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$ to another state in the same set by applying phase shifts to a few qubits. By counting the number of different possible combinations of $(\beta_j, \boldsymbol{\alpha}_j)$'s subjected to the constraint that $(\beta_j,\boldsymbol{\alpha}_j)_{j=0}^{n-1} \in {\mathfrak E} (\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$, we have \begin{equation} \label{eq:wx1} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = \mathop{{\sum}'} \frac{n!}{ \displaystyle \prod_{\genfrac{}{}{0pt}{}{i \in GF(2),}{\boldsymbol{t} \in GF(2)^{m-1}} } a_{i, \boldsymbol{t}}!} \enspace x^{n - a_{0, \boldsymbol{0}}} y^{a_{0, \boldsymbol{0}}} \end{equation} where the primed sum is over all $a_{i,\boldsymbol{t}}$'s satisfying the constraints \begin{equation} a_{i, \boldsymbol{t}} \geq 0 \enspace \forall i, \boldsymbol{t} ~, \label{eq:con_a} \end{equation} \begin{equation} \sum_{\genfrac{}{}{0pt}{}{i \in GF(2),}{\boldsymbol{t} \in GF(2)^{m-1}}} a_{i, \boldsymbol{t}} = n ~, \label{eq:con_b} \end{equation} \begin{equation} \sum_{i\in GF(2)} a_{i,\boldsymbol{t}} = | \{ j \in \{ 0,1,\ldots, n-1 \} : \boldsymbol{s}_j + \boldsymbol{\gamma} = \boldsymbol{t} \} | \enspace \forall \boldsymbol{t} \label{eq:con_c} \end{equation} and \begin{equation}\label{eq:con1} \sum_{\genfrac{}{}{0pt}{}{i \in GF(2),}{\boldsymbol{t} \in GF(2)^{m-1}}} i \, a_{i,\boldsymbol{t}} = \delta ~. \end{equation} Note that the symbol $a_{i,\boldsymbol{t}}$ in the above equations can be interpreted as the number of GHZ states that experienced the error $(i,\boldsymbol{t})$ before the commencement of our distillation protocol. Let \begin{equation} k \equiv k(\vec{\boldsymbol{s}},\boldsymbol{\gamma}) = |\{ j \in \{ 0,1,\ldots , n-1 \} : \boldsymbol{s}_j + \boldsymbol{\gamma} \neq \boldsymbol{0} \} | \label{eq:k_def} \end{equation} be the number of qubits having spin flip for each element in ${\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$. We have two cases to consider. \par\noindent Case~(a) $k > 0$: That is, there exists $\ell$ such that $\boldsymbol{s}_\ell + \boldsymbol{\gamma} \neq \boldsymbol{0}$. Hence, $\text{wt}((\beta_j, \boldsymbol{s}_j + \boldsymbol{\gamma})_{j=0}^{n-1})$ is independent of the value of $\beta_\ell \in GF(2)$. In addition, by regarding the equation $\sum_{j=0}^{n-1} \beta_j = \delta$ as a bijection relating $\beta_\ell \in GF(2)$ and $\delta\in GF(2)$, we conclude that the depolarization weight enumerator $w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y)$ is independent of the value of $\delta\in GF(2)$. Hence, \begin{equation} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = \frac{1}{2} \mathop{{\sum}''} \frac{n!}{\displaystyle \prod_{\genfrac{}{}{0pt}{}{i \in GF(2),}{\boldsymbol{t} \in GF(2)^{m-1}}} a_{i, \boldsymbol{t}}!} \enspace x^{n - a_{0, \boldsymbol{0}}} y^{a_{0,\boldsymbol{0}}} \end{equation} where the double primed sum is over all $a_{i,\boldsymbol{t}}$'s satisfying constraints Eq.~(\ref{eq:con_a})--(\ref{eq:con_c}) only. Consequently, \begin{eqnarray} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) &=& \frac{1}{2} \sum_{ \{ a_{i, \boldsymbol{t}} \} } \frac{k!}{\displaystyle \prod_{\genfrac{}{}{0pt}{}{i\in GF(2),}{\boldsymbol{t} \in {GF(2)^{m-1}}\setminus \{ \boldsymbol{0} \}}} a_{i,\boldsymbol{t}}} \enspace {n - k \choose a_{0, \boldsymbol{0}}} \enspace x^{n -a_{0, \boldsymbol{0}}} y^{a_{0,\boldsymbol{0}}} \nonumber \\ &=& 2^{k -1} \sum_{a_{0, \boldsymbol{0}}} {n - k \choose a_{0, \boldsymbol{0}}} \enspace x^{n - a_{0, \boldsymbol{0}}} y^{a_{0,\boldsymbol{0}}} \nonumber \\ &=& 2^{k-1} x^k (x+y)^{n-k} ~. \label{eq:dweight1} \end{eqnarray} \par\noindent Case~(b) $k = 0$: That is, $\boldsymbol{s}_j + \boldsymbol{\gamma} = \boldsymbol{0}$ for all $j$ so that phase shift is the only type of error a GHZ state may experience. In this case, the union of disjoint sets $\bigcup_{\delta\in GF(2)} {\mathfrak E}(\vec{\boldsymbol{s}},\delta, \boldsymbol{\gamma})$ is equal to the set of all possible phase errors experienced by the $n$ shared GHZ states. As a result, \begin{equation} \sum_{\delta\in GF(2)} w({\mathfrak E}( \vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = w(\bigcup_{\delta\in GF(2)} {\mathfrak E}(\vec{\boldsymbol{s}},\delta, \boldsymbol{\gamma});x,y) = \sum_i {n \choose i} \enspace x^i y^{n-i} = (x+y)^n ~. \label{eq:dweight2} \end{equation} Similarly, \begin{equation} \sum_{\delta\in GF(2)} (-1)^\delta w({\mathfrak E}( \vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = w(\bigcup_{\delta\in GF(2)} {\mathfrak E}(\vec{\boldsymbol{s}},\delta, \boldsymbol{\gamma});-x,y) = \sum_i {n \choose i} \enspace (-x)^i y^{n-i} = (y-x)^n ~. \label{eq:dweight3} \end{equation} (Note that the validity of the above equation follows from the observation that $\delta = 1$ if and only if the number of qubits having phase shift error before the commencement of our protocol is odd. Moreover, the coefficient of $w(\{ \vec{\boldsymbol{a}} \};-x,y)$ is negative if and only if $wt(\vec{\boldsymbol{a}})$ is odd.) From Eqs.~(\ref{eq:dweight1})--(\ref{eq:dweight3}), we conclude that \begin{equation} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) \equiv w(k,\delta;x,y) = \left\{ \begin{array}{ll} \displaystyle 2^{k - 1} x^k (x+y)^{n-k} & \enspace\text{if} \enspace 0 < k \leq n, \\ \\ \displaystyle \frac{1}{2} \left[ (x + y)^n + (y - x)^n \right] & \enspace\text{if} \enspace k = 0 \enspace \text{and} \enspace \delta = 0, \\ \\ \displaystyle \frac{1}{2} \left[ (x + y)^n - (y - x)^n \right] & \enspace\text{if} \enspace k = 0 \enspace \text{and} \enspace \delta \neq 0, \end{array} \right. \label{eq:w_k_delta_x} \end{equation} where $k = k(\vec{\boldsymbol{s}},\boldsymbol{\gamma})$ is the number of GHZ states that experienced some kind of spin flip for each of the state in ${\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$ as defined by Eq.~(\ref{eq:k_def}). Recall that ${\mathfrak E}( \vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma})$ is invariant under permutation of phase errors among the $n$ GHZ states. Moreover, both the depolarization weight and the value of $k(\vec{\boldsymbol{s}}, \boldsymbol{\gamma})$ are invariant under permutation of the $n$ GHZ states. So, it is not surprising that the depolarizing weight enumerator $w({\mathfrak E} (\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y)$ depends only on the values of $k$ and $\delta$. Therefore, our shorthand notation $w(k,\delta;x)$ makes sense. From Eq.~(\ref{eq:pr_a}) and by substituting $x = (1-F)/(2^m - 1)$, $y = F$ into Eq.~(\ref{eq:w_k_delta_x}), we find that \begin{equation} \text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}}) = \left\{ \begin{array}{ll} \displaystyle \frac{2^{k-1} (1-F)^k (2^m F - 2F + 1)^{n-k}}{(2^m - 1)^n} & \enspace \text{if} \enspace 0 < k \leq n , \\ \\ \displaystyle \frac{(2^m F - 2F + 1)^n + (2^m F - 1)^n}{2(2^m - 1)^n} & \enspace \text{if} \enspace k = 0 \enspace \text{and} \enspace \delta = 0 , \\ \\ \displaystyle \frac{(2^m F - 2F + 1)^n - (2^m F - 1)^n}{2(2^m - 1)^n} & \enspace \text{if} \enspace k = 0 \enspace \text{and} \enspace \delta \neq 0 . \end{array} \right. \label{eq:pr_k_delta} \end{equation} for a depolarizing channel with fidelity $F$. Note that by fixing the number of players $m$, the number of noisy GHZ states shared between the players $n$ and the fidelity of the depolarizing channel $F$, the probability $\text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}})$ can take on at most $(n+2)$ different values. \subsubsection{Finding $\text{Pr}((\delta,\boldsymbol{\gamma})| \vec{\boldsymbol{s}})$} \label{subsubsec:basic_probability} Clearly \begin{equation} \text{Pr} (\vec{\boldsymbol{s}}) = \sum_{\boldsymbol{t}\in GF(2)^{m-1}} \text{Pr} (\vec{\boldsymbol{s}} \wedge \boldsymbol{t}) \end{equation} where $\text{Pr}(\vec{\boldsymbol{s}} \wedge \boldsymbol{t})$ is the probability that the error experienced by the $n$ noisy GHZ states is $(\beta_j,\boldsymbol{s}_j + \boldsymbol{t})_{j=0}^{n-1}$ for $\beta_j \in GF(2)$. For a depolarizing channel with fidelity $F$, \begin{eqnarray} \text{Pr} (\vec{\boldsymbol{s}}) & = & \sum_{\boldsymbol{t}\in GF(2)^{m-1}} \left[ \frac{2(1-F)}{2^m - 1} \right]^{k(\vec{\boldsymbol{s}},\boldsymbol{t})} \enspace \left( F + \frac{1-F}{2^m - 1} \right)^{n - k(\vec{\boldsymbol{s}},\boldsymbol{t})} \nonumber \\ & = & \frac{1}{(2^m - 1)^n} \sum_{i=0}^n f_{\vec{\boldsymbol{s}}} (i) 2^i (1-F)^i (2^m F - 2F + 1)^{n-i} \label{eq:pr_s_sum} \end{eqnarray} where \begin{equation} f_{\vec{\boldsymbol{s}}} (i) = |\{ \boldsymbol{t} \in GF(2)^{m - 1} : k(\vec{\boldsymbol{s}},\boldsymbol{t}) = i \}| ~. \end{equation} Therefore, \begin{equation} \text{Pr}((\delta,\boldsymbol{\gamma}) | \vec{\boldsymbol{s}}) = \left\{ \begin{array}{ll} \displaystyle \frac{2^{k-1}(1-F)^k (2^m F - 2F + 1)^{n-k}}{\sum_i f_{\vec{\boldsymbol{s}}} (i) 2^i (1-F)^i (2^m F - 2F + 1)^{n-i}} & \enspace \text{if} \enspace 0 < k \leq n , \\ \\ \displaystyle \frac{(2^m F - 2F + 1)^n + (2^m F - 1)^n}{2 \sum_i f_{\vec{\boldsymbol{s}}} (i) 2^i (1-F)^i (2^m F - 2F + 1)^{n-i}} & \enspace \text{if} \enspace k = 0 \enspace \text{and} \enspace \delta = 0 , \\ \\ \displaystyle \frac{(2^m F - 2F + 1)^n - (2^m F - 1)^n}{2 \sum_i f_{\vec{\boldsymbol{s}}} (i) 2^i (1-F)^i (2^m F - 2F + 1)^{n-i}} & \enspace \text{if} \enspace k = 0 \enspace \text{and} \enspace \delta \neq 0 . \end{array} \right. \label{eq:p_delta_gamma_cond} \end{equation} So combined with Eqs.~(\ref{eq:D_SS}) and~(\ref{eq:Sx}), we have a working expression for $h(\{\text{Pr}((\delta,\boldsymbol{\gamma})| \vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \})$ and hence $D_\textrm{SS}$. While combined with Eqs.~(\ref{eq:D_MS})--(\ref{eq:D_hDef2}), we have a working expression for $h(\gamma_i | \vec{\boldsymbol{s}})$, $h(\delta | \vec{\boldsymbol{s}})$ and hence $D_\textrm{MS}$. In the calculation of $h(\gamma_i | \gamma_{i-1}, \gamma_{i-2}, \ldots , \gamma_1 , \vec{\boldsymbol{s}})$, we need to first compute the probability $\textrm{Pr}( (\delta, \boldsymbol{\gamma}) | \gamma_{i-1},\gamma_{i-2}, \ldots , \gamma_1, \vec{\boldsymbol{s}})$. As for repetition code, Eq.~(\ref{eq:alpha_j_relation}) tells us that the kind of spin flip error experienced by the $j$th GHZ state $\boldsymbol{\alpha}_j$ is known once $\boldsymbol{\gamma}$ and $\vec{\boldsymbol{s}}$ are fixed. Hence, $\textrm{Pr}( (\delta, \boldsymbol{\gamma}) | \gamma_{i-1},\gamma_{i-2}, \ldots , \gamma_1, \vec{\boldsymbol{s}})$ is also given by Eq.~(\ref{eq:p_delta_gamma_cond}). Consequently, using Eqs.~(\ref{eq:D_hDef2})--(\ref{eq:D_hDef3}), we get a working expression for $h(\gamma_i | \gamma_{i-1},\ldots ,\gamma_1, \vec{\boldsymbol{s}})$, $h(\delta | \vec{\boldsymbol{s}})$, $I(\delta; \boldsymbol{\gamma} | \vec{\boldsymbol{s}})$ and hence $D_\textrm{CL}$. \subsubsection{Complexity issue on the computation of $D_\textrm{SS}$, $D_\textrm{MS}$ and $D_\textrm{CL}$} \label{subsubsec:complexity} Apparently computing $D_\textrm{SS}$, $D_\textrm{MS}$ and $D_\textrm{CL}$ using Eqs.~(\ref{eq:D_SS}), (\ref{eq:Sx}), (\ref{eq:D_MS})--(\ref{eq:D_hDef3}), (\ref{eq:pr_s_sum})--(\ref{eq:p_delta_gamma_cond}) are extremely inefficient as the sum on $\vec{\boldsymbol{s}}$ may take on $2^{m(n-1)}$ possible values. Nonetheless, the numerical values of many terms in the R.H.S. of Eq.~(\ref{eq:Sx}) are the same because the $\vec{\boldsymbol{s}}$ dependence of $\text{Pr} (\vec{\boldsymbol{s}})$ and $\text{Pr}((\delta,\boldsymbol{\gamma}) | \vec{\boldsymbol{s}})$ come indirectly from the distribution of $\{ k(\vec{\boldsymbol{s}},\boldsymbol{t}) : \boldsymbol{t} \in GF(2)^{m-1} \}$. Note that there are at most $\sum_{i=0}^{2^{m - 1}} {\mathcal P}_i (n)$ different possible distributions for $\{ k(\vec{\boldsymbol{s}},\boldsymbol{t}) : \boldsymbol{t} \in GF(2)^{m-1} \}$ where ${\mathcal P}_i (n)$ denotes the number of ways to express $n$ as a sum of exactly $i$ positive integers. Moreover, $\mathcal{P}_{i}(n)$ scales as $\exp(\pi \sqrt{2n/3}) / 4n\sqrt{3}$ in the large $n$ limit~\cite{GE76}. Consequently, for a fixed $m$, we may regroup the sum Eq.~(\ref{eq:Sx}) so as to compute $S_X$ by summing only sub-exponential in $n$ terms. Although this is not a polynomial time in $n$ algorithm, it is good enough to obtain the numerical values for $S_X$ and hence the yield of our first protocol, namely, the Shor-Smolin capacity $D_\textrm{SS}$ for a reasonably large number of $n$. By the same token, the yields of our second and third protocols, namely $D_\textrm{MS}$ and $D_\textrm{CL}$ respectively, can also be computed in sub-exponential time in $n$. \subsection{Performance of our three schemes} \label{subsec:performance} We study the performance of our three protocols by studying the yield as a function of the channel fidelity $F$. In particular, we would like to find the threshold fidelity, namely, the minimum fidelity above which $D > 0$, as a function of the number of players $m$ and the repetition codeword size $n$. And we denote the threshold fidelities for our first, second and third protocols by $F_\textrm{min}^\textrm{SS} (m,n)$, $F_\textrm{min}^\textrm{MS} (m,n)$ and $F_\textrm{min}^\textrm{CL} (m,n)$, respectively. \subsubsection{Subtlety in the computation of threshold fidelities} \label{subsubsec:threshold_fidelities_computation} Finding the values of $F_\textrm{min}^\textrm{SS} (m,n)$, $F_\textrm{min}^\textrm{MS} (m,n)$ and $F_\textrm{min}^\textrm{CL} (m,n)$ requires extra care. Let us explain why for the case of $F_\textrm{min}^\textrm{SS} (m,n)$. And the reason for the other two cases are similar. Since $S_X$ is a continuous function of the channel fidelity $F$, Eq.~(\ref{eq:D_SS}) implies that $F_\text{min}^\textrm{SS}(m,n)$ is the root of the equation $S_X = 1$. Note that \begin{eqnarray} 1 - S_X & = & \text{Pr}(\vec{\boldsymbol{0}}) [ 1 - h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{0}}) : (\delta,\boldsymbol{\gamma}) \in GF(2)^m \}) ] \nonumber \\ & & ~~ - \sum_{\vec{\boldsymbol{s}}\neq \vec{\boldsymbol{0}}} \text{Pr} (\vec{\boldsymbol{s}}) [ h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{s}}) : (\delta,\boldsymbol{\gamma}) \in GF(2)^m \}) - 1 ] ~. \label{eq:Sx_alt} \end{eqnarray} From Eq.~(\ref{eq:p_delta_gamma_cond}), we know that for $F\gg 1/2$, $h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \})$ is less (greater) than $1$ if $\vec{\boldsymbol{s}} = \vec{\boldsymbol{0}}$ ($\vec{\boldsymbol{s}} \neq \vec{\boldsymbol{0}}$). More importantly, for a fixed $m$, $\lim_{n\rightarrow\infty} h(\{ \text{Pr}((\delta, \boldsymbol{\gamma})| \vec{\boldsymbol{s}}) : (\delta,\boldsymbol{\gamma}) \in GF(2)^m \}) = 1^- (1^+)$ for $\vec{\boldsymbol{s}} = \vec{\boldsymbol{0}}$ ($\vec{\boldsymbol{s}} \neq \vec{\boldsymbol{0}}$). Thus, Eq.~(\ref{eq:Sx_alt}) shows that $1 - S_X$ is the difference between two small positive terms. This makes the computation of $F_\text{min}^\textrm{SS}(m,n)$ together with the analysis of its trend as a function of $m$ and $n$, particularly for a large $n$, difficult. Even worse, for $F<1$ and for a sufficiently large $n$, the errors experienced by the noisy GHZ states $(\delta,\boldsymbol{\gamma})_{i=0}^{n-1}$ satisfying $\vec{\boldsymbol{s}} = \vec{\boldsymbol{0}}$ are not in the typical set. Actually, we found that for $F$ close to $F_\text{min}^\textrm{SS}(m,n)$, the dominant terms in the R.H.S. of Eq.~(\ref{eq:Sx_alt}) almost always correspond to atypical errors experienced by the GHZ states. In spite of these difficulties, we are able to accurately compute the yield of our first protocol $D_\textrm{SS}$, namely, the Shor-Smolin capacity, as a function of the channel fidelity $F$ for the classical $[n,1,n]$ repetition code acting on the $\ket{\Phi^{m+}}$'s. And from this, we can deduce the correct threshold fidelity for our first protocol $F_\textrm{min}^\textrm{SS} (m,n)$ as a function of the number of players $m$ and the number of shared noisy GHZ states $n$. The trick is to use rational number arithmetic to obtain an expression for $S_X$ for a given rational number $F$ before converting this expression to an approximate real number. The same trick also enables us to obtain accurate values for $F_\textrm{min}^\textrm{MS}(m,n)$ and $F_\textrm{min}^\textrm{CL}(m,n)$, namely, the threshold fidelities of our second and third protocols. \subsubsection{The superior performances of our three protocols} \label{subsubsec:performances} The yields of our three protocols are shown in Figs.~\ref{fig:SS_m2-m5}--\ref{fig:CL_m2-m5}; and the corresponding threshold fidelities are tabulated in Tables~\ref{tb:SS_m2-m6}--\ref{tb:CL_m2-m6}. By comparing these tables with the second column of Table~\ref{tb:pw}, we make the most important conclusion of this paper: for the multipartite case ($m \geq 3$) and for any number of shared GHZ states $n$, the error-tolerant capability of our third protocol is strictly better than our second, which is in turn strictly better than that of our first. And under the same conditions, the error-tolerant capability of our first protocol is already better than the best scheme in literature before this work. So once again, we show the powerfulness and usefulness of degenerate codes in one-way entanglement distillation. \begin{table}[t] \begin{tabular}{||c|c|ccccccccc||} \hline\hline \multicolumn{2}{||c|}{} & \multicolumn{9}{c||}{$m$} \\ \cline{3-11} \multicolumn{2}{||c|}{$F_\textrm{min}^\textrm{SS}(m, n)$} & 2 & ~~~~ & 3 & ~~~~ & 4 & ~~~~ & 5 & ~~~~ & 6 \\ \hline & 2 & 0.8113 & & 0.8109 & & 0.8103 & & 0.8115 & & 0.8142 \\ & 3 & 0.8099 & & 0.7870 & & 0.7699 & & 0.7593 & & 0.7536 \\ & 4 & 0.8102 & & 0.7753 & & 0.7486 & & 0.7301 & & 0.7184 \\ & 5 & 0.8097 & & 0.7675 & & 0.7351 & & 0.7118 & & 0.6961 \\ & 6 & 0.8100 & & 0.7622 & & 0.7256 & & 0.6992 & & 0.6808 \\ ~~$n$~~ & 7 & 0.8098 & & 0.7582 & & 0.7185 & & 0.6898 & & 0.6696 \\ & 11 & 0.8104 & & 0.7492 & & 0.7021 & & 0.6677 & & 0.6435 \\ & 15 & 0.8110 & & 0.7449 & & 0.6938 & & 0.6565 & & 0.6301 \\ & 21 & 0.8118 & & 0.7416 & & 0.6870 & & 0.6471 & & 0.6188 \\ & 31 & 0.8128 & & 0.7391 & & 0.6814 & & 0.6390 & & 0.6089 \\ \hline\hline \end{tabular} \caption{The threshold fidelity $F_\textrm{min}^\textrm{SS}$ as a function of $m$ and $n$. \label{tb:SS_m2-m6} } \end{table} \begin{table}[t] \begin{tabular}{||c|c|ccccccccc||} \hline\hline \multicolumn{2}{||c|}{} & \multicolumn{9}{c||}{$m$} \\ \cline{3-11} \multicolumn{2}{||c|}{$F_\textrm{min}^\textrm{MS}(m, n)$} & 2 & ~~~~ & 3 & ~~~~ & 4 & ~~~~ & 5 & ~~~~ & 6 \\ \hline & 2 & 0.8137 & & 0.7788 & & 0.7541 & & 0.7369 & & 0.7253 \\ & 3 & 0.8101 & & 0.7631 & & 0.7261 & & 0.6991 & & 0.6781 \\ & 4 & 0.8102 & & 0.7551 & & 0.7091 & & 0.6781 & & 0.6571 \\ ~~$n$~~ & 5 & 0.8095 & & 0.7566 & & 0.7111 & & 0.6771 & & 0.6521 \\ & 6 & 0.8100 & & 0.7522 & & 0.7081 & & 0.6721 & & 0.6421 \\ & 7 & 0.8098 & & 0.7501 & & 0.7051 & & 0.6711 & & 0.6441 \\ & 11 & 0.8104 & & 0.7475 & & 0.6951 & & 0.6581 & & 0.6311 \\ & 15 & 0.8110 & & 0.7446 & & 0.6901 & & 0.6511 & & 0.6221 \\ \hline\hline \end{tabular} \caption{The threshold fidelity $F_\textrm{min}^\textrm{MS}$ as a function of $m$ and $n$. \label{tb:MS_m2-m6} } \end{table} \begin{table}[t] \begin{tabular}{||c|c|ccccccccc||} \hline\hline \multicolumn{2}{||c|}{} & \multicolumn{9}{c||}{$m$} \\ \cline{3-11} \multicolumn{2}{||c|}{$F_\textrm{min}^\textrm{CL}(m, n)$} & 2 & ~~~~ & 3 & ~~~~ & 4 & ~~~~ & 5 & ~~~~ & 6 \\ \hline & 2 & 0.8137 & & 0.7084 & & 0.6655 & & 0.6378 & & 0.6204 \\ & 3 & 0.8101 & & 0.7122 & & 0.6793 & & 0.6584 & & 0.6501 \\ & 4 & 0.8102 & & 0.7165 & & 0.6906 & & 0.6680 & & 0.6532 \\ ~~$n$~~ & 5 & 0.8095 & & 0.7111 & & 0.6776 & & 0.6582 & & 0.6357 \\ & 6 & 0.8100 & & 0.7099 & & 0.6684 & & 0.6551 & & 0.6217 \\ & 7 & 0.8098 & & 0.7086 & & 0.6650 & & 0.6480 & & 0.6133 \\ & 11 & 0.8104 & & 0.7081 & & 0.6642 & & 0.6372 & & 0.6062 \\ & 15 & 0.8110 & & 0.7074 & & 0.6601 & & 0.6284 & & 0.6036 \\ \hline\hline \end{tabular} \caption{The threshold fidelity $F_\textrm{min}^\textrm{CL}$ as a function of $m$ and $n$. \label{tb:CL_m2-m6} } \end{table} Whereas for the bipartite case ($m = 2$), Tables~\ref{tb:SS_m2-m6}--\ref{tb:CL_m2-m6} show that all our three protocols can tolerate almost the same level of error. It means that the use of random asymmetric CSS outer code does not give any significant advantage here. (Actually, we find that using random asymmetric CSS outer code decreases the error-tolerant capability for $n \leq 3$.) Interestingly, the threshold fidelities for our second and third protocols agree to at least four significant figures for $m = 2$. This finding can be understood as follows. As we have discussed in Sec.~\ref{subsec:evaluation} and particularly in Eq.~(\ref{eq:p_delta_gamma_cond}), the probability of $\delta = 0$ equals the probability of $\delta = 1$ provided that $\vec{\boldsymbol{s}} \neq \vec{\boldsymbol{0}}$ irrespective of the value of $\boldsymbol{\gamma}$. That is to say, $I(\delta;\gamma|\vec{\boldsymbol{s}}) = 0$ whenever $\vec{\boldsymbol{s}} \neq \vec{\boldsymbol{0}}$. So, it is not surprising to find that the weighted mutual information $\sum_{\vec{\boldsymbol{s}}} \textrm{Pr} (\vec{\boldsymbol{s}}) I(\delta;\boldsymbol{\gamma}| \vec{\boldsymbol{s}})$ becomes negligibly small when the fidelity $F$ is close to its threshold value. Combined with Eqs.~(\ref{eq:D_MS}) and~(\ref{eq:D_CL}), it is not unnatural to find that $F_\textrm{min}^\textrm{MS}(2,n) \approx F_\textrm{min}^\textrm{CL}(2,n)$. \begin{figure} \caption{The yield $D_\textrm{SS} \label{fig:SS_m2-m5} \end{figure} \begin{figure} \caption{The yield $D_\textrm{MS} \label{fig:MS_m2-m5} \end{figure} \begin{figure} \caption{The yield $D_\textrm{CL} \label{fig:CL_m2-m5} \end{figure} Our numerical computation shows that $\lim_{n\rightarrow\infty} F_\text{min}^\textrm{SS}(m,n)$, $\lim_{n\rightarrow\infty} F_\text{min}^\textrm{MS}(m,n)$ and $\lim_{n\rightarrow\infty} F_\text{min}^\textrm{CL}(m,n)$ are decreasing functions of $m$. Besides, $\lim_{n\rightarrow\infty} F_\text{min}^\textrm{SS}(4,n)$ is smaller than 0.7798, the fidelity threshold of the Maneva and Smolin's hashing scheme in the large $m$ limit~\cite{MS00}. So, for $m\geq 3$, our three protocols all tolerate a higher noise level than all other one-way schemes known to date. Figs.~\ref{fig:SS_m2-m5}--\ref{fig:CL_m2-m5} further depict that the yields of our protocols $D_\textrm{SS}$, $D_\textrm{MS}$ and $D_\textrm{CL}$ are very steep functions of $F$ around their corresponding threshold fidelities. Thus, a reasonable yield can be obtained when $F$ is equal to, say, $0.02$ higher than the threshold. Another interesting feature found in Tables~\ref{tb:SS_m2-m6}--\ref{tb:CL_m2-m6} and Figs.~\ref{fig:SS_m2-m5}--\ref{fig:CL_m2-m5} is that the threshold fidelities $F_\text{min}^\textrm{SS}(m,n)$, $F_\textrm{min}^\textrm{MS}(m,n)$ and $F_\textrm{min}^\textrm{CL}(m,n)$ are all decreasing functions of $n$ for $m \geq 3$. That is, our protocols attain a higher capacity if players use a longer repetition code whenever $m\geq 3$. In contrast, DiVincenzo \emph{et al.} found that $F_\text{min}^\textrm{SS}(2,n)$ attains global minimum when $n = 5$. Besides, $F_\text{min}^\textrm{SS} (2,n) > F_\text{min}^\textrm{SS}(2,n\pm 1)$ for a small even integer $n$~\cite{DSS98}. Interestingly, Tables~\ref{tb:MS_m2-m6} and~\ref{tb:CL_m2-m6} show that $F_\textrm{min}^\textrm{MS}(2,n)$ and $F_\textrm{min}^\textrm{CL}(2,n)$ behave in the same way, too. Lastly, we remark that for $m\geq 3$, the improvement in the error-tolerant capability for increasing $n$ comes with a price. For a fixed $m\geq 3$, the yields of our protocols decrease as $n$ increases provided that the channel fidelity $F$ is close to $1$ as depicted in Figs.~\ref{fig:SS_m2-m5}--\ref{fig:CL_m2-m5}. This is because as $n$ increases, more shared GHZ states must be wasted in order to obtain the error syndrome $\vec{\boldsymbol{s}}$ even if the channel is noiseless. \subsection{Understanding the trend of the threshold fidelities} \label{subsec:explain_trend_q-2} Although the discussions in this subsection focuses on the trend of the threshold fidelity of our first protocol, namely, $F_\textrm{min}^\textrm{SS} (m,n)$, the essential ideas also apply to the cases of our second and third protocols, that is, $F_\textrm{min}^\textrm{MS}(m,n)$ and $F_\textrm{min}^\textrm{CL}(m,n)$. The reason why $F_\text{min}^\textrm{SS}(2,n)$ is a sawtooth-shaped function of $n$ for $n\lesssim 8$ is related to the behavior of $h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \})$. It is easy to check that for $m=2$, $h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \})$ is equal to (much less than) $2$ provided that the depolarization weight $\text{wt} ( \vec{\boldsymbol{s}}) = n / 2$ ($\text{wt} (\vec{\boldsymbol{s}}) \neq n/2$). For a small even $n$, there is a non-negligible probability of finding $\vec{\boldsymbol{s}}$ with $\text{wt}(\vec{\boldsymbol{s}}) = n/2$ so that the root of $S_X = 1$ and hence the value of $F_\text{min}^\textrm{SS}$ are determined mainly by the summing only over those $\vec{\boldsymbol{s}}$'s with depolarization weight $0$ or $n/2$ in Eq.~(\ref{eq:Sx_alt}). In contrast, for a small odd $n$, all entropies in the R.H.S. of Eq.~(\ref{eq:Sx_alt}) are much less than $2$. Hence, the corresponding value of $F_\text{min}^\textrm{SS}(2,n)$ is lower than $F_\text{min}^\textrm{SS}(2,n\pm 1)$. In other words, the reason for $F_\text{min}^\textrm{SS}(2,n) > F_\text{min}^\textrm{SS}(2,n\pm 1)$ for a small even $n$ is that there is a non-negligible chance that exactly half of Bell states used by the inner repetition code have spin flip error so that players have absolutely no idea what kind of error the remaining unmeasured Bell state has experienced. However, the situation is very different when $m\geq 3$. In this case, the condition for $h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \}) \geq 2$ is that one can find an integer $i$ such that $f_{\vec{\boldsymbol{s}}} (i) \geq 2$ and $f_{\vec{\boldsymbol{s}}}(j) = 0$ for all $j < i$. More importantly, for a depolarizing channel with $F > 1/2$, the probability $\text{Pr}(\vec{\boldsymbol{s}})$ of finding this kind of $\vec{\boldsymbol{s}}$ with $h(\{ \text{Pr}((\delta,\boldsymbol{\gamma})|\vec{\boldsymbol{s}}) : (\delta, \boldsymbol{\gamma}) \in GF(2)^m \}) \geq 2$ is much less than that in the situation of $m = 2$. Thus, the contribution of terms with entropy greater than or equal to $2$ in Eq.~(\ref{eq:Sx_alt}) becomes much less significant when $m \geq 3$. So, it is not surprising to find that for a fixed $m \geq 3$, $F_\text{min}^\textrm{SS}(m,n)$ is not a sawtooth-shaped function of $n$ when $n$ is small. It is also easy to understand why $\lim_{n\rightarrow\infty} F_\text{min}^\textrm{SS} (m,n)$ is a decreasing function of $m$. One simply check by Taylor's series expansion that in the limit of large $n$ and for a fixed $1/2 < F < 1$. Then we find that the first term in the R.H.S. of Eq.~(\ref{eq:Sx_alt}) is an increasing function of $m$; and that the summand in the second term in the R.H.S. of Eq.~(\ref{eq:Sx_alt}) is almost surely a decreasing function of $m$ in the large $n$ limit. As we have pointed out that the value of $F_\text{min}^\textrm{SS}(m,n)$ depends on the entropy of a few atypical set of errors experienced by the GHZ states. We do not have a good explanation why $F_\text{min}^\textrm{SS}(m,n)$ is a decreasing function of $n$ for $m\geq 3$. \subsection{Breaking the $F > 0.75$ limit?} \label{subsec:limit} No $t$ error correcting quantum code of codeword size $4t$ exists~\cite{KL97a,BDSW96a}. Hence, it is impossible to distill Bell states using an one-way scheme provided that the fidelity of the depolarizing channel is less than or equal to $0.75$~\cite{BDSW96a}. That is why $F_\text{min}^\textrm{SS}(2,n), F_\textrm{min}^\textrm{MS}(2,n), F_\textrm{min}^\textrm{CL}(2,n) > 0.75$. Interestingly, a few $F_\text{min}^\textrm{SS}(m,n)$'s, $F_\text{min}^\textrm{MS}(m,n)$'s and $F_\text{min}^\textrm{CL}(m,n)$'s listed in Tables~\ref{tb:SS_m2-m6}--\ref{tb:CL_m2-m6} are less than 0.75. Does it make sense? To solve this paradox, let us recall that the Pauli errors experienced by a GHZ state shared among $m$ players can always be regarded as taken place in $(m-1)$ of the $m$ qubits. From Eq.~(\ref{eq:GHZ_gen}), we may regard that at most one of the $(m-1)$ qubits may experience a phase error. So, the probability that a depolarized GHZ state has experienced phase error but not spin flip is $(1-F)/(2^m - 1)$, where $F$ is the channel fidelity. And the number of erroneous qubits equals $1$ in this case. Besides, the probability that exactly $i$ out of the $(m-1)$ qubits have experienced spin flip is $2(1-F) {m-1 \choose i} / (2^m - 1)$ for $i=1,2,\ldots , m-1$, where the extra factor of $2$ comes from the fact that the spin-flipped GHZ state may experience phase shift as well. Hence, the average number of erroneous qubits divided by $(m-1)$ is given by \begin{equation} \bar{e} = \frac{1}{m-1} \left[ \frac{1-F}{2^m - 1} + \frac{2(1-F)}{2^m - 1} \sum_{i=1}^{m-1} i \enspace {m-1 \choose i} \right] = \frac{1-F}{2^m - 1} \left( 2^{m-1} + \frac{1}{m-1} \right) ~. \label{eq:average_rate} \end{equation} Since no $t$ error correcting quantum code has codeword size less than or equal to $4t$~\cite{KL97a,BDSW96a}, $\bar{e} < 1/4$. Consequently, a lower bound for the threshold fidelities $F_\text{min}^\textrm{x}(m,n)$ (for $\textrm{x} = \textrm{SS}, \textrm{MS}, \textrm{CL}$) is given by \begin{equation} F_\text{min}^\textrm{x}(m,n) > F_\text{bound} = 1 - \frac{2^m - 1}{4} \left( 2^{m-1} + \frac{1}{m-1} \right)^{-1} ~. \label{eq:F_min_lower_bound} \end{equation} A quick look at the third and the fourth columns in Table~\ref{tb:pw} convinces us that our protocols do not violate this general limit. Actually, one of the reasons why we can distill shared GHZ states when $F<0.75$ for $m \geq 3$ is that the average qubit error rate for a depolarized GHZ state is given by Eq.~(\ref{eq:average_rate}), which is smaller than $(1-F)$. Note in particular that in the large $m$ limit, the average qubit error rate for a depolarized GHZ state is close to $1/2$. So, it is not surprising that the bound $F_\text{bound}$ approaches $1/2$ in this case. \section{Generalization to higher dimensional spin}\label{sec:high_spin} \subsection{Our extended protocols} Our three protocols can be generalized to the case when the Hilbert space dimension of each quantum particle is greater than 2. That is to say, the $m$ players wanted to share the state \begin{equation}\label{eq:GHZ_q} \ket{\Phi^{m+}_q} = \frac{1}{\sqrt{q}} \sum_{i = 0}^{q-1} \ket{i^{\otimes m}} \end{equation} through a depolarizing channel by one-way entanglement distillation. The quantum codes used in the three generalized protocols are extensions of their corresponding binary codes to the $q$-nary ones. In particular, their common inner code becomes classical $[n,1,n]_q$ repetition code. We have the following two cases to consider. \begin{enumerate} \item For $q = p^m$ where $p$ is a prime number, we may impose a finite field structure $GF(q)$ to the system by defining \begin{equation} X_j: \ket{i} \longmapsto \ket{i + j} \end{equation} and \begin{equation} Z_j: \ket{i} \longmapsto \omega_p^{\Tr{(ij)}} \ket{i} \end{equation} for all $j \in GF(q)$ where $\omega_p$ is a primitive $p$th root of unity, $\Tr$ is the absolute trace and all arithmetic are performed in the finite field $GF(q)$. \item Alternatively, for any integer $q\geq 2$, we may impose a ring structure ${\mathbb Z}/q{\mathbb Z}$ to the system by defining \begin{equation} X_j: \ket{i} \longmapsto \ket{i + j} \end{equation} and \begin{equation} Z_j: \ket{i} \longmapsto \omega_q^{ij} \ket{i} \end{equation} for all $j \in \mathbb{Z}/q \mathbb{Z}$ where $\omega_q$ is a primitive $q$th root of unity and all arithmetic are performed in the ring ${\mathbb Z}/q{\mathbb Z}$. \end{enumerate} From now on, we use the symbol ${\mathbb K}$ to denote either the finite field $GF(q)$ or the ring ${\mathbb Z}/q {\mathbb Z}$. Similar to the case of $q = 2$, we use the compact notation $(\beta,\boldsymbol{\alpha}) \equiv (\beta, \alpha_1, \alpha_2, \ldots, \alpha_{m-1})$ to denote the eigenvalue of the stabilizer generators where $\beta \in {\mathbb K}$ and $\boldsymbol{\alpha} \in {\mathbb K}^{m-1}$. In the qubit case (that is, $q = 2$), the error syndrome measurement is performed with the aid of CNOT gates. In the case of $q > 2$, this can be done via the operator $|i,j\rangle \longrightarrow |i,i-j\rangle$ for all $i,j\in {\mathbb K}$. Suppose the error experienced by the $j$th copy of $\ket{\Phi^{m+}_q}$ is $(\beta_j, \boldsymbol{\alpha}_j)$ for $j = 0, \ldots , n - 1$. Then after measuring the error syndrome for the classical $[n,1,n]_q$ repetition code, we get $\vec{\boldsymbol{s}} \equiv (\boldsymbol{s}_1, \ldots , \boldsymbol{s}_{n-1})$ where \begin{equation} \boldsymbol{s}_j \equiv \boldsymbol{\alpha}_j - \boldsymbol{\alpha}_0 \end{equation} for $1 \leq j \leq n-1$. Furthermore, the remaining state shared among the players becomes \begin{equation} (\delta, \boldsymbol{\gamma}) = (\sum_{j = 0}^{n - 1} \beta_j, \boldsymbol{\alpha}_0) ~. \label{eq:resultant_state_q} \end{equation} \subsection{Finding the capacities of our three generalized protocols} The analysis in Sec.~\ref{subsec:evaluation} can be easily generalized to the case of qudits (that is, $q > 3$). In particular, we prove in the Appendix that \begin{equation} \text{Pr}((\delta,\boldsymbol{\gamma}) | \vec{\boldsymbol{s}}) = \left\{ \begin{array}{ll} \displaystyle \frac{q^{k-1} (1-F)^k (q^m F - qF + q - 1)^{n-k}}{\sum_i f_{\vec{\boldsymbol{s}}} (i) \left[ q (1-F) \right]^i \left( q^m F - q F + q - 1 \right)^{n-i}} & \enspace \text{if} \enspace k > 0 , \\ \\ \displaystyle \frac{( q^m F - q F + q - 1 )^n + (q-1) \left( q^m F - 1 \right)^n}{q\sum_i f_{\vec{\boldsymbol{s}}} (i) \left[ q (1-F) \right]^i \left( q^m F - q F + q - 1 \right)^{n-i}} & \enspace \text{if} \enspace k = 0 \enspace \text{and} \enspace \delta = 0 , \\ \\ \displaystyle \frac{( q^m F - q F + q - 1 )^n - \left( q^m F - 1 \right)^n}{q\sum_i f_{\vec{\boldsymbol{s}}} (i) \left[ q (1-F) \right]^i \left( q^m F - q F + q - 1 \right)^{n-i}} & \enspace \text{if} \enspace k = 0 \enspace \text{and} \enspace \delta \neq 0 , \end{array} \right. \label{eq:qudit_pr} \end{equation} where $k$ and $f_{\vec{\boldsymbol{s}}}(i)$ are given by Eq.~(\ref{eq:k_def}) and Eq.~(\ref{eq:f_s_qudit}) respectively. The yields of our three generalized protocols can be computed using Eqs.~(\ref{eq:D_SS})--(\ref{eq:Sx}) and~(\ref{eq:D_MS})--(\ref{eq:D_hDef3}) just like the case of $q=2$. Nevertheless, there is an important subtlety. Since the players can make full use of each of the $q$ possible error syndrome measurement outcomes to distill the generalized GHZ state $|\Phi_q^{m+}\rangle$, the entropies and conditional entropies in Eqs.~(\ref{eq:Sx}), (\ref{eq:D_hDef1}), (\ref{eq:D_hDef2}) and~(\ref{eq:D_hDef3}) should be measured in the unit of dit rather than bit. That is to say, the base of the logarithm used in these entropies should be $q$ instead of $2$. And since the dimension of each information carrier $q$ has changed, one should not directly compare the yields of the qudit-based protocols with those of the standard qubit-based ones. Note further that similar to the original qubit-based protocols, we can compute these yields in a time sub-exponential in $n$. \subsection{Performance of our generalized protocols} Figs.~\ref{fig:SS_q-3-m2-m5}--\ref{fig:CL_q-3-m2-m5} depict the yields of our three generalized protocols in the case of $q = 3$. And Tables~\ref{tb:SS_q-3-m2-m6}--\ref{tb:CL_q-3-m2-m6} list the corresponding threshold fidelities. Clearly, the trend of the threshold fidelities of our three generalized protocols are very similar to their corresponding cases of $q = 2$. In particular, for $q = 3$, the threshold fidelities $F_\textrm{min}^\textrm{SS} (m,n)$, $F_\textrm{min}^\textrm{MS} (m,n)$ and $F_\textrm{min}^\textrm{CL} (m,n)$ are decreasing function of $n$ for any fixed integer $m\geq 3$; while they reach global minima at $n = 7$ provided that $m = 2$. These findings are not completely surprising because the arguments we have used to explain the trends of the yields and threshold fidelities for our three protocols in the case of $q = 2$ reported in Sec.~\ref{subsec:explain_trend_q-2} are also applicable here after minor adjustments. \begin{table}[t] \begin{tabular}{||c|c|ccccccccc||} \hline\hline \multicolumn{2}{||c|}{} & \multicolumn{9}{c||}{$m$} \\ \cline{3-11} \multicolumn{2}{||c|}{$F_\textrm{min}(m, n)$} & 2 & ~~~~ & 3 & ~~~~ & 4 & ~~~~ & 5 & ~~~~ & 6\\ \hline & 2 & 0.7462 & & 0.7538 & & 0.7609 & & 0.7693 & & 0.7778 \\ & 3 & 0.7445 & & 0.7243 & &0.7145 & & 0.7127 & & 0.7148 \\ & 4 & 0.7445 & & 0.7089& & 0.6885 & & 0.6799 & & 0.6779 \\ ~~$n$~~ & 5 & 0.7442 & & 0.6994 & & 0.6722 & & 0.6588 & & 0.6539 \\ & 6 & 0.7441 & & 0.6927 & & 0.6611 & & 0.6443 & & 0.6370 \\ & 7 & 0.7441 & & 0.6877 & & 0.6530 & & 0.6337 & & 0.6246 \\ & 11 & 0.7444 & & 0.6759 & & 0.6338 & & 0.6096 && 0.5962 \\ & 15 & 0.7449 & & 0.6700 & & 0.6238 & & 0.5974 & & 0.5823 \\ \hline\hline \end{tabular} \caption{The threshold fidelity $F_\textrm{min}^\textrm{SS}$ as a function of $m$ and $n$ for $q = 3$. \label{tb:SS_q-3-m2-m6} } \end{table} \begin{table}[t] \begin{tabular}{||c|c|ccccccccc||} \hline\hline \multicolumn{2}{||c|}{} & \multicolumn{9}{c||}{$m$} \\ \cline{3-11} \multicolumn{2}{||c|}{$F_\textrm{min}^\textrm{MS}(m,n)$} & 2 & ~~~~ & 3 & ~~~~ & 4 & ~~~~ & 5 & ~~~~ & 6 \\ \hline & 2 & 0.7499 & & 0.7114 & & 0.6892 & & 0.6780 & & 0.6728 \\ & 3 & 0.7450 & & 0.7034 & & 0.6776 & & 0.6640 & & 0.6575 \\ & 4 & 0.7448 & & 0.6944 & & 0.6591 & & 0.6389 & & 0.6289 \\ ~~$n$~~ & 5 & 0.7444 & & 0.6849 & & 0.6452 & & 0.6234 & & 0.6127 \\ & 6 & 0.7443 & & 0.6829 & & 0.6418& & 0.6172 & & 0.6041 \\ & 7 & 0.7443 & & 0.6810& & 0.6390 & & 0.6121 & & 0.5967 \\ & 11 & 0.7446 & & 0.6738 & & 0.6289& & 0.5997 & & 0.5808 \\ & 15 & 0.7451 & & 0.6692& & 0.6212 & & 0.5927 & & 0.5740 \\ \hline\hline \end{tabular} \caption{The threshold fidelity $F_\textrm{min}^\textrm{MS}$ as a function of $m$ and $n$ for $q = 3$. \label{tb:MS_q-3-m2-m6} } \end{table} \begin{table}[t] \begin{tabular}{||c|c|ccccccccc||} \hline\hline \multicolumn{2}{||c|}{} & \multicolumn{9}{c||}{$m$} \\ \cline{3-11} \multicolumn{2}{||c|}{$F_\textrm{min}^\textrm{CL}(m,n)$} & 2 & ~~~~ & 3 & ~~~~ & 4 & ~~~~ & 5 & ~~~~ & 6 \\ \hline & 2 & 0.7499 & & 0.6419& & 0.6120 & & 0.5981 & & 0.5921 \\ & 3 & 0.7450 & & 0.6222& & 0.5908 & & 0.5762 & & 0.5697 \\ & 4 & 0.7448 & & 0.6282 & & 0.6016& & 0.5821 & & 0.5868 \\ ~~$n$~~ & 5 & 0.7444 & & 0.6337 & & 0.6104 & & 0.5916 & & 0.5895 \\ & 6 & 0.7443 & & 0.6334 & & 0.6061& & 0.5896 & & 0.5855 \\ & 7 & 0.7443 & & 0.6304& & 0.6023 & & 0.5877 & & 0.5829 \\ & 11 & 0.7446 & & 0.6249 & & 0.5910& & 0.5790 & & 0.5670 \\ & 15 & 0.7451 & & 0.6235& & 0.5865 & & 0.5701 & & 0.5560 \\ \hline\hline \end{tabular} \caption{The threshold fidelity $F_\textrm{min}^\textrm{CL}$ as a function of $m$ and $n$ for $q = 3$. \label{tb:CL_q-3-m2-m6} } \end{table} \begin{figure} \caption{The yield $D_\textrm{SS} \label{fig:SS_q-3-m2-m5} \end{figure} \begin{figure} \caption{The yield $D_\textrm{MS} \label{fig:MS_q-3-m2-m5} \end{figure} \begin{figure} \caption{The yield $D_\textrm{CL} \label{fig:CL_q-3-m2-m5} \end{figure} \subsection{Lower bound for the three threshold fidelities} The proof that no $t$ error correcting quantum code with codeword size $4t$ is also applicable to qudits~\cite{R99}. We may use this fact to establish a lower bound for the threshold fidelities of our three generalized protocols when qudits are used as information carriers. Since the proof is also the same as that of the qubit case reported in Sec.~\ref{subsec:limit}, here we only write down the bound without giving the details of the proof: \begin{equation} F_\text{min}^\textrm{SS} (m,n), F_\text{min}^\textrm{MS} (m,n), F_\text{min}^\textrm{CL} (m,n) > F_\text{bound} = 1 - \frac{q^m - 1}{4(q-1)} \left( q^{m-1} + \frac{1}{m-1} \right)^{-1} ~. \end{equation} \section{Summary and Discussion}\label{sec:sum} In summary, we have introduced three one-way GHZ state purification protocols using degenerate codes by extending the works of DiVincenzo \emph{et al.} on one-way Bell state purification via degenerate codes~\cite{SS96,DSS98} as well as the works of Maneva and Shor~\cite{MS00} and its generalization by Chen and Lo~\cite{CL07} on multipartite entanglement purification using random asymmetric CSS codes. Then, we calculate the yields of our three protocols when the inputs are Werner states. The method we used to calculate these yields is divided into two steps. The first step is to calculate entropies or conditional entropies such as $h(\{ \textrm{Pr}((\delta,\boldsymbol{\gamma}) : (\delta,\boldsymbol{\gamma}) \in GF(2)^m \})$ by means of the so-called depolarization weight enumerator. Actually, the first step can be easily extended to the case of an arbitrary stabilizer inner code, an arbitrary un-correlated noise model and an arbitrary stabilizer output state. The second step involves the computation of a weighted sum of the entropies or conditional entropies obtained in the first step. Nonetheless, for a general stabilizer inner code, a general un-correlated noise model and a general output state, this sum may not be practical as it involves up to about $2^{m n}$ number of terms. Fortunately, as the inner code used in our purification scheme is the highly symmetrical classical repetition code, we are able to greatly simplify the sum, making the computation of the yields in a time which is sub-exponential in $n$ when the GHZ states are subjected to depolarization errors. In this way, we can calculate the corresponding threshold fidelities accurately and reasonably fast. This is quite an accomplishment because finding the threshold fidelities involves the accurate determination of the sign of the difference between two small positive numbers provided that the number of players $m\geq 3$ and the codeword size of the inner repetition code $n$ is large. (See, for example, Eq.~(\ref{eq:Sx_alt}).) Just like the Bell state case, we discover that the threshold fidelities of our three protocols are better than all known one-way GHZ state purification schemes to date. So, once again, the power of using degenerate codes to combat quantum errors is demonstrated. We also extended our scheme to tackle the case when the information carriers are qudits instead of qubits. We find that the performance trend of these generalized schemes are quite similar to those of the qubit cases. There are a few un-answered questions, however. Here we list some of them. The reason why the threshold fidelities $F_\text{min}^\textrm{SS}$ $F_\textrm{min}^\textrm{MS}$ and $F_\textrm{min}^\textrm{CL}$ decrease with $n$ for $m\geq 3$ is not apparent. And apart from the general statement that degenerate codes pack more information than non-degenerate ones making them powerful in one-way purification of GHZ states, can we specifically understand why using classical repetition code concatenated with a random hashing quantum code is more error-tolerant than a few other choices of degenerate codes?~\cite{HC08} Along a different line, it is important to find out the value of $\lim_{m\rightarrow\infty} \lim_{n\rightarrow\infty} F_\text{min}^\textrm{CL} (m,n)$ and compare it with the $1/2$ lower bound. Finally, it is instructive to extend our study to the case of using a different degenerate code to distill another type of entangled state subjected to another noise model, such as the Pauli channel~\cite{SS06}. \acknowledgments Useful discussions with C.-H.~F. Fung and H.-K. Lo are gratefully acknowledged. This work is supported by the RGC grants No.~HKU~7010/04P and No.~HKU~701007P of the HKSAR Government. \appendix \section{Proof Of Eq.~(\ref{eq:qudit_pr})} We prove the validity of Eq.~(\ref{eq:qudit_pr}) by following the analysis in Sec.~\ref{subsec:evaluation}. (And we follow the same notations as used in Sec.~\ref{subsec:evaluation} after possibly some straight-forward extension to the case of qudits.) First, we extend the definition of depolarization weight as follows. Let $(\beta_j,\boldsymbol{\alpha}_j)_{j=0}^{n-1} \in {\mathbb K}^{m n}$ be a ordered $n$-tuple. Then its depolarization weight is defined as the Hamming weight by regarding this $n$-tuple as a vector of elements in ${\mathbb K}$. Clearly, $\text{Pr}((\delta,\boldsymbol{\gamma})\wedge\vec{\boldsymbol{s}}) = \text{Pr}({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma}))$ where \begin{equation} {\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma}) = \{ (\beta_j,\boldsymbol{\alpha}_j)_{j=0}^{n-1} \in {\mathbb K}^{m n} : \delta = \sum_{j=0}^{n-1} \beta_j , \boldsymbol{\gamma} = \boldsymbol{\alpha}_0, \boldsymbol{\alpha}_\ell = \boldsymbol{s}_\ell + \boldsymbol{\alpha}_0 \enspace \text{for} \enspace \ell = 1,2,\ldots,n-1 \} ~. \end{equation} Using the same argument as in the case of $q = 2$, we conclude that for $k > 0$, \begin{eqnarray} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) &=& \frac{1}{q} \sum_{ \{ a_{i, \boldsymbol{t}} \} } \frac{k!}{\displaystyle \prod_{\genfrac{}{}{0pt}{}{i\in {\mathbb K},}{\boldsymbol{t} \in {{\mathbb K}^{m-1}}\setminus \{ \boldsymbol{0} \}}} a_{i,\boldsymbol{t}}!} \enspace \frac{(n-k)!}{\displaystyle \prod_{i\in {\mathbb K}^*} a_{i,\boldsymbol{0}}!} \enspace x^{n -a_{0, \boldsymbol{0}}} y^{a_{0,\boldsymbol{0}}} \nonumber \\ &=& q^{k -1} \sum_{a_{0, \boldsymbol{0}}} {n - k \choose a_{0, \boldsymbol{0}}} \enspace (q-1)^{n-k-a_{0,\boldsymbol{0}}} x^{n - a_{0, \boldsymbol{0}}} y^{a_{0,\boldsymbol{0}}} \nonumber \\ &=& q^{k-1} x^k [(q-1)x+y]^{n-k} ~. \end{eqnarray} For $k = 0$, we have to use a slightly different method to compute the depolarization weight enumerator. By substituting $x_0 = y$ and $x_\eta = x$ for all $\eta\neq 0$ into the identity \begin{equation} \left( \sum_{\eta\in {\mathbb K}} x_\eta \right)^n = \sum_{\{ a_\eta \}} \left( \frac{n!}{\prod_{\eta\in {\mathbb K}} a_\eta!} \prod_{\eta\in {\mathbb K}} x_\eta^{a_\eta} \right) ~, \label{eq:multisum} \end{equation} we have \begin{equation} \sum_{\delta\in{\mathbb K}} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta, \boldsymbol{\gamma});x,y) = [ (q-1)x + y ]^n ~. \end{equation} If ${\mathbb K} = GF(q)$, then by putting $x_0 = y$ and $x_\eta = \omega_p^{\Tr (\eta \rho)} x$ for all $\eta\neq 0$ where $\rho\in GF(q)^*$ into Eq.~(\ref{eq:multisum}), we have \begin{equation} \sum_{\delta\in GF(q)} \omega_p^{\Tr (\delta \rho)} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = (y-x)^n ~. \end{equation} If ${\mathbb K} = {\mathbb Z}/q {\mathbb Z}$, then we put $x_0 = y$ and $x_\eta = \omega_q^{\eta\rho)} x$ for all $\eta\neq 0$ where $\rho\in ({\mathbb Z}/q {\mathbb Z})^*$ into Eq.~(\ref{eq:multisum}), we arrive at \begin{equation} \sum_{\delta\in {\mathbb Z}/q {\mathbb Z}} \omega_q^{\delta \rho} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = (y-x)^n ~. \end{equation} Consequently, we conclude that for ${\mathbb K} = GF(q)$ or ${\mathbb Z}/q {\mathbb Z}$, \begin{equation} w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma});x,y) = \left\{ \begin{array}{ll} \displaystyle q^{k - 1} x^k \left[ (q-1)x+y \right]^{n-k} & \enspace \text{if} \enspace k > 0 , \\ \\ \displaystyle \frac{1}{q} \left\{ \left[ (q-1) x + y \right]^n + (q-1) ( y - x)^n \right\} & \enspace \text{if} \enspace k = 0 \enspace \text{and}\enspace \delta = 0, \\ \\ \displaystyle \frac{1}{q} \left\{ \left[ (q-1) x + y \right]^n - (y-x)^n \right\} & \enspace \text{if} \enspace k = 0 \enspace \text{and}\enspace \delta \neq 0 . \end{array} \right. \label{eq:weight_qudit} \end{equation} Surely, for depolarizing channel, $\text{Pr}((\delta,\boldsymbol{\gamma}) \wedge \vec{\boldsymbol{s}}) = w({\mathfrak E}(\vec{\boldsymbol{s}},\delta,\boldsymbol{\gamma}); (1-F)/(q^m - 1),F)$ and \begin{eqnarray} \text{Pr}(\vec{\boldsymbol{s}}) & = & \sum_{\boldsymbol{t}} \left[ \frac{q(1-F)}{q^m - 1} \right]^{k(\vec{\boldsymbol{s}},\boldsymbol{t})} \left[ F + \frac{(q-1)(1-F)}{q^m - 1} \right]^{n-k(\vec{\boldsymbol{s}}, \boldsymbol{t})} \nonumber \\ & = & \frac{1}{(q^m - 1)^n} \sum_{i=0}^n f_{\vec{\boldsymbol{s}}} (i) \left[ q (1-F) \right]^i \left( q^m F - q F + q - 1 \right)^{n-i} ~, \label{eq:prs_qudit} \end{eqnarray} where \begin{equation} f_{\vec{\boldsymbol{s}}} (i) = |\{ \boldsymbol{t} \in {\mathbb K}^{m - 1} : k(\vec{\boldsymbol{s}},\boldsymbol{t}) = i \} | ~. \label{eq:f_s_qudit} \end{equation} Combining Eqs.~(\ref{eq:weight_qudit})--(\ref{eq:f_s_qudit}), we arrive at Eq.~(\ref{eq:qudit_pr}). $\Box$ \end{document}

\begin{document} \title{ Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states} \author{Naeem Akhtar} \email{naeem\_abbasi@zjnu.edu.cn} \affiliation{Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China} \author{Barry C. Sanders} \email{sandersb@ucalgary.ca} \affiliation{Institute for Quantum Science and Technology, University of Calgary, Alberta, Canada T2N 1N4} \author{Gao Xianlong} \email{ gaoxl@zjnu.edu.cn} \affiliation{Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China} \date{\today} \begin{abstract} We investigate the sub-Planck-scale structures associated with the SU(1,1) group by establishing that the Planck scale on the hyperbolic plane can be considered as the inverse of the Bargmann index $k$. Our discussion involves SU(1,1) versions of Wigner functions, and quantum-interference effect is easily visualized through plots of these Wigner functions. Specifically, the superpositions of four Perelomov SU(1,1) coherent states (compass state) yields nearly isotropic sub-Planck structures in phase space scaling as $\nicefrac1{k}$ compared with $\nicefrac1{\sqrt{k}}$ scaling for individual SU(1,1) coherent states and anisotropic quadratically improved scaling for superpositions of two SU(1,1) coherent states (cat state). We show that displacement sensitivity exhibits the same quadratic improvement to scaling. \end{abstract} \maketitle \section{Introduction}\label{sec:introduction} The quantum uncertainty principle~\cite{Robertson1929,wheeler2014}, arising from commutator relations such as the position-momentum case $[\hat{x},\hat{p}]=\text{i}\hbar$, limits the size of a phase-space structure~\cite{wheeler2014}, for example, represented by the Wigner function~\cite{Wig32} for Heisenberg-Weyl (HW) symmetry~\cite{Wey50} generated by the HW algebra~$\mathfrak{hw}$(1), and, more generally, by Moyal symbols for other symmetries~\cite{Moy49}. The uncertainty principle does not actually mean that the displacement sensitivity, with ``displacement'' referring to group action on the state, is limited by this Planck scale because quantum interference in phase space~\cite{Gerry05book,Sch01,drummond2004quantum} yields finer-scale properties. For example, the compass states (superposition of four coherent states) have shown sub-Planck spotty structures in Wigner functions over phase space~\cite{Zurek2001}. Such sub-Planck structures are highly sensitive to environmental decoherence~\cite{Zurek2003} and play a crucial role in the sensitivity of a quantum state against phase-space displacements~\cite{Toscano06,Eff4}. Sub-Planck structures have been explored in various contexts~\cite{Eff1,Eff2,Eff3,Eff5,Eff6,Eff7,Eff8,Eff9,Eff10,Eff11,Eff12,Eff13,Eff14,Eff15}, and both theoretical~\cite{Prop1,Prop2,Prop3,Prop5,Prop6} and experimental~\cite{Exp1,Exp2,Exp3,Exp4,Exp5} studies of sub-Planck structure have been undertaken. Coherent states of the harmonic oscillator belong to the dynamical symmetry group, the so-called Heisenberg-Weyl (HW) group~\cite{weyl1950theory}. These coherent states were first introduced by Schr\"{o}dinger in 1926~\cite{Sch26,MN09} and then, for quantum optics, by Glauber in 1963~\cite{Gla63}. These states can be visualized in phase space by the Wigner function~\cite{Sch01,Gerry05book}. The concept of the coherent states has been extended to other dynamical group actions~\cite{Per86,Gaz09}. Coherent states exhibit the Planck limit to phase space, known as the standard quantum limit or shot-noise limit for HW symmetry. The coherent-state superpositions have been extensively studied for the harmonic oscillator~\cite{Gerry05book,PhysRevA.99.063813}, whose position and momentum operators obey $\mathfrak{hm}$(1) algebra for a single degree of freedom, and act on an infinite-dimensional Hilbert space. The phase-space features of a cat state (superposition of two distinct coherent states~\cite{Mil86,YS86}, parameterized by mean particle or photon number~$\bar{n}$ corresponding to phase-space distance from the phase-space origin) are not limited in all phase-space directions, and hence, cannot be considered as sub-Planck. Compared to coherent states, cat states as a superposition of two coherent states with the same~$\bar{n}$ but opposite phases have $\sqrt{\bar{n}}$-enhanced sensitivity against displacements, with respect to specific directions in phase space. Contrariwise, for a compass state, this enhanced sensitivity to displacements is independent of phase-space directions. These same sub-Planck structures present in compass states appear in cat mixtures~\cite{Eff12}. However, similar to cat states, the cat mixtures have shown this enhanced sensitivity along a specific direction in phase space. These results have been generalized to the case of SU(2) dynamics~\cite{Naeem2021}. Another symmetry of special interest for physicists is the Lie SU(1,1) group generated by the~$\mathfrak{su}$(1,1) algebra, which is associated with displacement-like operators and involves three generators as its basis elements, and acts on infinite-dimensional Hilbert space~\cite{Per86}. The SU(1,1) coherent states focused on in this paper are those of Perelomov~\cite{Gerry1986,Ojeda2014}. SU(1,1) symmetry is connected with many quantum optical systems~\cite{Wodkiewicz1985,Chaturvedi1991,Ban1992,Ban1993,ban1993b,Jing2011,Ojeda2014,Hudelist2014,Berrada2013,Yurke1896,Dong2016,Szigeti2017,barry2020,duan2022a,duan2022b}. The bosonic or Schwinger realization of the~$\mathfrak{su}$(1,1) algebra has a connection to the squeezing properties of boson fields~\cite{Lo1993}. For example, the single- and two-mode bosonic representations of the~$\mathfrak{su}$(1,1) algebra have immediate relevance to the single and two-mode squeezed states~\cite{Brif1996,Gerry1986,Gerry1991,Gerry1995,stoler1971,Yuen1976,Gerry01,yazdi2008,luc1992}, respectively. Similar to other dynamical groups, the SU(1,1) quasiprobability distributions defined over the hyperboloid surface are obtained through the Wigner function~\cite{Seyfarth2020,Klimov2021,Russel2021}. The SU(1,1) Wigner distribution can be visualized on the Poincar\'e\ disk by using the stereographic projection. The superposition of SU(1,1) coherent states has been extensively studied in different contexts~\cite{ban1994,Gerry1997,Miry2012}. Moreover, the SU(1,1) coherent-state superpositions have shown strong nonclassical properties~\cite{Gerry1997}. Multiple proposals suggest ways to realize experimentally SU(1,1) coherent-state superpositions ~\cite{Gerry1997,Miry2012,zheng2002generation}. Entangled coherent states~\cite{San92,San92E,San12} have been extended to superpositions of SU(2) and SU(1,1) coherent states~\cite{WSP00}. Here, we discuss the phase-space representations of interesting SU(1,1) coherent-state superpositions using the SU(1,1) Wigner function. We show that by considering SU(1,1) coherent-state superpositions on the hyperboloid surface, one can build cat states, compass states, and cat-state mixtures. These quantum states have similar phase-space features as their counterparts of the HW and the SU(2) groups when represented on the Poincar\'e\ disk. For a coherent state, the Wigner distribution appears as a lobe at the location where it is pinned on the Poincar\'e\ disk. The effective support of this lobe decreases as~$k$ ($k$ is the Bargmann index) grows, which scales as $\nicefrac1{k}$. We show that the concept of sub-Planck structures is extended to the SU(1,1) group by associating the support area of the coherent states as the SU(1,1) counterpart of the Planck action. In particular, we show that SU(1,1) compass state and cat-state mixtures have phase-space structures with extension proportional to $\nicefrac1{k}$ along any direction, which is a factor $\nicefrac1{\sqrt{k}}$ smaller than the extension found for coherent states. Two-mode bosonic realization of the $\mathfrak{su}$(1,1) algebra relates the Bargmann index $k$ to the asymmetry in photon numbers of correlated modes of two-mode squeezed number states. We then show that the existence of the sub-Planck structures are connected to larger values of this asymmetry in photon numbers of two correlated modes of the squeezed-number states. Moreover, we verify that the SU(1,1) coherent-state superpositions of the present work have exactly the same enhanced sensitivity to displacements found for their counterparts of the HW and the SU(2) groups, with the role of~$\bar{n}$ in the HW group and $j$ in the SU(2) group played by~$k$ for the SU(1,1) groups. Our paper is organized as follows.\;In~\S\ref{sec:background}, we review the basic concepts of sub-Planck structures associated with the~$\mathfrak{hw}(1)$ algebra. In~\S\ref{sec:su(1,1)_group}, we introduce specific SU(1,1) coherent-state superpositions and discuss their phase-space representation by the Wigner function. Here we explore various aspects associated with these states such as sub-Planck structures present in their Wigner function, two-photon realizations, and sensitivity against displacements. Furthermore, in this section we also compare the properties of these SU(1,1) states with their HW and SU(2) counterparts. Finally, we summarize our discussion in~\S\ref{sec:summary}. \begin{figure*} \caption{For $x_0 = 4$, (a)~Wigner function of the (Heisenberg-Weyl) compass state and (b)~overlap between compass state and its displaced versions. The quantities are in arbitrary units.} \label{fig:figure1} \end{figure*} \section{Background}\label{sec:background} In this section we present the background of the main concepts, including phase-space representations of quantum states, sub-Planck structures, and the sensitivity of a quantum state against phase-space displacements. We explain these concepts by means of the compass state of the harmonic oscillator or~the $\mathfrak{hw}(1)$ algebra. \subsection{Sub-Planck structures}\label{subsec:compass} Let us start to explain some basics of the~$\mathfrak{hw}(1)$ algebra. This algebra is defined through the creation $\hat{a}^\dagger$ operator and anhilation operator $\hat{a}$ that satisfy the commutation relation $[\hat{a},\hat{a}^\dagger]=1$. The dimensionless versions of the position and momentum operators, \begin{align} \hat{x}:=\frac{\hat{a}^\dagger+\hat{a}}{\sqrt{2}}\;\text{and}\; \hat{p}:=\frac{\text{i}(\hat{a}^\dagger-\hat{a})}{\sqrt{2}}, \end{align} respectively, obey the uncertainty relation $\Delta x\Delta p\geq1/2$, where \begin{equation} \label{eq:DeltaA2} \Delta C^2:=\braket{\hat{C}^2}-\braket{\hat{C}}^2 \end{equation} is the uncertainty of any operator~$\hat{C}$~\cite{Robertson1929, wheeler2014}. A Schr\"{o}dinger coherent state is a nonspreading wave packet of the quantum harmonic oscillator~\cite{Sch35} and is defined as an eigenstate of the annihilation operator: $\hat a\ket{\alpha}=\alpha\ket{\alpha}$ for $\alpha\in\mathbb{C}$~\cite{Gla63,Gerry05book}. The coherent state is generated by displacing the vacuum state by $\ket{\alpha}=\hat{D}(\alpha)\ket{0}$~\cite{Gaz09} for $\hat{D}(\alpha):=\exp\left(\alpha\hat{a}^\dagger-\alpha^*\hat{a}\right)$. In the Fock basis, \begin{align} \ket{\alpha}=\text{e}^{-\frac{|\alpha|^2}{2}}\sum^{\infty}_{n=0}\frac{\alpha^n}{\sqrt{n!}}\ket{n},\, \{\ket{n};n\in\mathbb{N}\}, \end{align} which yields a Poisson distribution of the particle number and mean~$\bar{n}=|\alpha|^2$. The Wigner function for any arbitrary quantum state $\hat{\rho}$ is written as an expectation value of the parity kernel as~\cite{CarlosNB15,Russel2021} \begin{align} W_{\hat{\rho}}\left(\bm{r}\right) :=\text{tr}\left[\hat{\rho}\hat{\Pi}(\alpha)\right],\, \bm{r}:=(x,p)^\top, \label{eq:wigner_general} \end{align} with \begin{align} \hat{\Pi}(\alpha):=2\hat{D}(\alpha)\hat{\Pi}\hat{D}^\dagger(\alpha),\, \hat{\Pi} :=\exp\left(\text{i}\pi\hat{a}^\dagger\hat{a}\right), \end{align} being the displaced parity operator. Zurek~\cite{Zurek2001} showed that the compass state leads to sub-Planck structures in phase space, and, importantly, these structures play a crucial role in enhancing its sensitivity towards phase-space displacements~\cite{Toscano06,Eff4,Naeem2021}. The superposition of four coherent states with equal amplitude and maximally-spaced phases leading to the Zurek compass state (we omit state normalization throughout) is as follows: \begin{align}\label{eq:compass_state} \ket{\psi}:=\ket{\nicefrac{x_0}{\sqrt{2}}}+\ket{\nicefrac{-x_0}{\sqrt{2}}}+\ket{\nicefrac{\text{i}x_0}{\sqrt{2}}}+\ket{\nicefrac{-\text{i}x_0}{\sqrt{2}}}, \end{align} with $x_0 \in \mathbb{R}$. The Wigner function of this compass state is shown in Fig.~\ref{fig:figure1}(a). Throughout we normalize the Wigner functions with their maximum amplitudes $W_{\rho}(0)$. The Wigner function of the compass state (\ref{eq:compass_state}) is written as a sum of the Wigner functions of the underlying coherent states plus the interferences between them, that is, \begin{align} W_{\ket{\psi_{\text{C}}}}(\bm{r})=W_{\text{coh}}(\bm{r})+I_\text{osc}(\bm{r})+I_\boxplus(\bm{r}), \label{eq8} \end{align} where \begin{equation} W_{\text{coh}}(\bm{r}) :=\frac14\left[\text{e}^{-p^2}G(x;x_0) +\text{e}^{-x^2}G(p;x_0)\right], \end{equation} with \begin{equation} \label{eq:doublepeak} G(x;x_0) :=\text{e}^{-\left(x-x_0\right)^2} +\text{e}^{-\left(x+x_0\right)^2}, \end{equation} represents the Wigner functions of four coherent states underlying the compass state. The locations of these coherent states in phase space are understood as the geographical points (east, west, north and south). The second term in Eq.~(\ref{eq8}) is \begin{equation} I_\text{osc}(\bm{r}):=\frac12\sum_{m_1,m_2=\pm 1}V(m_1 x,m_2 p), \end{equation} with \begin{equation} V(x,p):=\text{e}^{-\left[\left(x-\frac{x_0}{2}\right)^2+\left(p-\frac{x_0}{2}\right)^2\right]}\cos\Big[x_0\left(x+p-\frac{x_0}{2}\right)\Big], \end{equation} and represents the quantum interference between northeast, northwest, southeast, and southwest pairs of the coherent states. For our purpose, we focus on the central chessboardlike pattern \begin{align} I_\boxplus(x,p):=\frac12\text{e}^{-(x^2+p^2)}\Big[\cos(2x_0 x)+\cos(2x_0 p)\Big]. \end{align} This pattern contains tiles of alternating sign (denoted by different colors in the Fig.~\ref{fig:figure1}(a)) with areas proportional to $x_0^{-2}$, and, hence, below than the Planck scale for $x_0\gg1$. The same sub-Planck structures appear for the cat-state mixture~\cite{Eff12,Naeem2021}. Cat states do not have sub-Planck structures because the effective support of their phase-space structures appearing in the interference pattern is limited only in the specific direction. The sensitivity of the compass state to displacements is discussed next. \subsection{Sensitivity to displacements}\label{subsec:hw_sensitivity} The sensitivity of any arbitrary quantum state $\hat{\rho}$ to displacements is obtained by calculating its overlap with their displaced versions $\hat{D}(\delta\alpha)\hat{\rho}\hat{D}^\dagger(\delta\alpha)$. This overlap is \begin{equation} \label{eq:overlap_HW} \mathcal{F}_{\hat{\rho}}(\delta\alpha) :=\text{tr}\big[\hat{\rho}\hat{D}(\delta\alpha)\hat{\rho}\hat{D}^\dagger(\delta\alpha)\big] =\left|\braket{\psi|\hat{D}(\delta\alpha)|\psi}\right|^2, \end{equation} where $\delta\alpha\in\mathbb{C}$ is an arbitrary displacement and the last equality holds when the states are pure, $\hat{\rho}=\ket{\psi}\bra{\psi}$. This quantity provides a measure~\cite{Audenaert14} for the distinguishability of the state and its displaced version. The smaller the displacement $\delta \alpha$ needs to be in order to bring the overlap to zero, the more sensitive the state is said to be against displacements. For a coherent state $\ket{\alpha}$ the overlap (\ref{eq:overlap_HW}) is \begin{align} \mathcal{F}_{\ket{\alpha}}(\delta \alpha)=\text{e}^{-\frac12\left|\delta \alpha\right|^2}, \end{align} where the smallest noticeable displacement that vanishes this overlap is above the Planck scale, $|\delta \alpha|>1$. This inequality implies that the sensitivity of a given coherent state is at the standard quantum limit. Consider the compass state, for which the overlap (\ref{eq:overlap_HW}) under the assumption $x_0\gg1$ and $|\delta \alpha|\ll1$ is \begin{align} \mathcal{F}_{\ket{\psi}}(\delta\alpha)=\frac14 \mathrm{e}^{-\frac12|\delta\alpha|^2}\big[\cos\left(x_0 \delta_x\right)+\cos\left(x_0 \delta_p\right)\big]^2, \end{align} with \begin{equation} \delta\alpha =\delta_x+\text{i}\delta_p,\hspace{3mm} \delta_j\in\mathbb{R}. \end{equation} We plot this overlap in Fig.~\ref{fig:figure1}(b), and it vanishes when either of the conditions \begin{equation}\label{eq:compassov} \delta_x\pm\delta_p=\frac{2m+1}{x_0}\pi,~m\in\mathbb{Z}, \end{equation} are fulfilled. As illustrated in Fig.~\ref{fig:figure1}(b), the overlap vanishes for the displacements $|\delta \alpha|\sim x^{-1}_0$ and the arbitrary phase. Hence, as compared to coherent states, a compass state with excitations $\bar{n}$ (here $\bar{n}=\nicefrac{x^2_0}{2}$) has shown $\sqrt{\bar{n}}$-enhanced sensitivity against displacements of any arbitrary directions in phase space. For cat states and their mixtures this enhancement only occurs for displacements of specific directions. Having the same sub-Planck structures in phase-space, cat-state mixtures are inferior for metrology of compass states, for which the quantum superposition play a crucial role. Hence, sub-Planck structures are not the sole reason for the remarkable sensibility of compass states against displacements~\cite{Toscano06}. The coherent-state superpositions of the SU(2) group are well discussed~\cite{San89,SG14,davis2020,Huang15,Huang18,Maleki}. SU(2) quasiprobability distributions defined over the unit sphere are obtained through the Wigner function~\cite{davis2020,Varilly89,Heiss00,Klimov17,Glaser20}. The concept of sub-Planck structures has been extended to the SU(2) group~\cite{Naeem2021}. Specifically, by restricting SU(2) coherent-state superpositions along the equator, one can build cat states, compass states, and cat-state mixtures that have similar Wigner interference patterns as their HW counterparts when represented in the stereographic plane, with the role of~$x_0$ being played by $\sqrt{j}$~\cite{Naeem2021,davis2020}. The compass state and cat-state mixtures of the SU(2) group have shown the structures limited in all directions of the stereographic plane. These structures have an extension proportional to $\nicefrac1{j}$ in any direction, which is a factor $\nicefrac1{\sqrt{j}}$ smaller than the extension found for coherent states. Furthermore, SU(2) cat states have shown an interference pattern with structures limited only in one direction, just like their HW counterparts. These states have exactly the same enhanced sensitivity to displacements found for the~$\mathfrak{hw}$(1) algebra, where $j$ has played the role of~$\bar{n}$~\cite{Naeem2021}. \section{Sub-Planck structures of SU(1,1)}\label{sec:su(1,1)_group} Different algebras are associated with different systems~\cite{Per86}.\;For example, the~$\mathfrak{hw}(1)$ algebra, discussed in the previous section, acts on an infinite-dimensional Hilbert space and is typically associated with one-dimensional mechanical systems. Most recently SU(1,1) has achieved special attention for metrology~\cite{Jing2011,Hudelist2014,Berrada2013,Yurke1896,Dong2016,Szigeti2017,barry2020}. The SU(1,1) representation is associated with the Hamiltonian involving squeezing~\cite{Knight1987}. This is because the SU(1,1) displacement operator is considered as a squeezing operator, resulting in the SU(1,1) coherent state actually being the squeezed state~\cite{Brif1996,Gerry1986,Gerry1991,Gerry1995,stoler1971,Yuen1976,Gerry01,yazdi2008,luc1992}. In fact, squeezing of quantum states represents the leading strategy for enhanced quantum metrology~\cite{Maccone2004,drummond2004quantum}. It is interesting to extend the results found for one algebra to different ones. It may provide a way to devise experimental implementations of essentially any algebra we are interested in. We have found it useful to generalize the concept of sub-Planck structures from the harmonic oscillator to the SU(2) group~\citep{Naeem2021}. In this spirit, in the following we generalize the concept of sub-Planck structures to the SU(1,1) group. In~\S\ref{subsec:preliminary}, we review the main properties of the~$\mathfrak{su}$(1,1) algebra. In~\S\ref{subsec:su(1,1)_wigner}, we evaluate the Wigner function of the SU(1,1) coherent states. In~\S\ref{subsec:su(1,1)_superpositions}, we introduce specific coherent-state superpositions and show how the concept of sub-Planck structures can be extended to the SU(1,1) case. In~\S\ref{subsec:two_photon}, we explore two-photon representations of these superpositions, and then in ~\S\ref{subsec:su(1,1)orthogonality}, we discuss their sensitivity against phase-space displacements. In~\S\ref{subsec:comparision}, we compare the properties of these SU(1,1) coherent-state superpositions with their HW and SU(2) counterparts. \begin{figure*} \caption{Plots of the SU(1,1) Wigner function of the reference state $\ket{k,0} \label{fig:figure2} \end{figure*} \subsection{Basics of SU(1,1)}\label{subsec:preliminary} The~$\mathfrak{su}$(1,1) algebra is spanned by the generators $\hat{K}_+$, $\hat{K}_-$, and $\hat{K}_0$, which satisfy the commutation relations~\cite{Per86} \begin{align}\label{eq:commutations_su(1,1)} \big[\hat{K}_0,\hat{K}_{\pm}\big]=\pm \hat{K}_{\pm},\,\big[\hat{K}_-,\hat{K}_+\big]=2\hat{K}_0. \end{align} These generators are written in terms of the Hermitian operators $\hat{K_1}$ and $\hat{K_2}$ as \begin{align} \hat{K}_{\pm}=\pm \text{i} (\hat{K}_1\pm \text{i}\hat{K}_2). \end{align} The action of the SU(1,1) generators on the Fock space states~$\{\ket{k,n};n\in\mathbb{N}\}$ satisfies \begin{align} \label{eq:commu_su(1,1)} &\hat{K}_0\ket{k,n}=(k+n)\ket{k,n},\\ &\hat{K}_{+}\ket{k,n}=\sqrt{(n+1)(2k+n)}\ket{k,n+1},\\ &\hat{K}_{-}\ket{k,n}=\sqrt{n(2k+n+1)}\ket{k,n-1}, \end{align} where $\ket{k,0}$ is the normalized reference state. For any irreducible representation the Casimir operator satisfies \begin{align} \hat{K}^2=&\hat{K}_0-\frac12(\hat{K}_{+}\hat{K}_{-}+\hat{K}_{-}\hat{K}_{+}),\\=&\hat{K}^2_0-\hat{K}^2_1-\hat{K}^2_2=k(k-1)\mathbb{1}, \label{eq:casimor_su(1,1)} \end{align} where~$k$ is the Bargmann index which separates different irreducible representations. We restrict to positive discrete series for which $k>0$. The SU(1,1) displacement operator admits either of the forms~\cite{mufti1993,fujii2001introduction,ban1993b} \begin{equation} \hat{D}(\zeta):=\text{e}^{\xi \hat{K}_+-\xi^* \hat{K}_-} =\text{e}^{\zeta \hat{K}_+}\text{e}^{\ln(1-\left|\zeta\right|^2) \hat{K}_0}\text{e}^{-\zeta^* \hat{K}_-}, \label{eq:displacement_su11_dis} \end{equation} where \begin{equation} \xi=\frac{\tau}{2}\text{e}^{\text{i}\varphi},\,\zeta=\text{e}^{\text{i}\varphi}\tanh\left(\frac{\tau}{2}\right), \end{equation} with $-\infty<\tau<\infty$ and $0<\varphi<2\pi$. The parameter $|\zeta|$ is restricted to the interior of the Poincar\'e\ disk, $0\le|\zeta|<1$, whereas $\xi$ is defined on the upper sheet of the two-sheet hyperboloid surface. Multiplication of two operators obey~\cite{Per86} \begin{equation} \hat{D}(\zeta_ 1)\hat{D}(\zeta_ 2) =\hat{D}(\zeta_ 3)\text{e}^{\text{i}\phi \hat{K}_0}, \label{eq:product_operator} \end{equation} where \begin{equation} \zeta_3=\frac{\zeta_1+\zeta_2}{1+\zeta^*_1\zeta_2},\phi=-2\arg(1+\zeta^*_1\zeta_2). \end{equation} As Eq.~(\ref{eq:product_operator}) is independent of~$k$, this is proved by setting \begin{equation} \hat{K}_0=\frac{\sigma_3}2,\, \hat{K}_{1,2}=\frac{\text{i}\sigma_{1,2}}2, \end{equation} for the Pauli matrices $\{\sigma_j\}$. \subsection{SU(1,1) coherent states and the Wigner function}\label{subsec:su(1,1)_wigner} SU(1,1) coherent states are obtained by displacing~(\ref{eq:displacement_su11_dis}) the reference state $\ket{k,0}$ according to~\cite{Gerry1991,Gerry1986} \begin{align} \ket{\zeta}=\hat{D}(\zeta)\ket{k,0}=(1-\left|\zeta\right|^2)^k\sum^{\infty}_{n=0}\sqrt{\frac{\Gamma(n+2k)}{n!\Gamma(2k)}}\zeta^n\ket{k,n}. \label{eq:coherentstate_su11} \end{align} Equivalently, SU(1,1) coherent states can be associated with points on the two-sheeted hyperboloid surface through the hyperbolic version of the Bloch vector, namely, \begin{equation} \bm{n}=(\cosh\tau,\sinh\tau\cos\varphi,\sinh\tau\sin\varphi). \end{equation} Using Eqs.~(\ref{eq:product_operator}) and~(\ref{eq:coherentstate_su11}), the overlap between any two arbitrary coherent states is \begin{equation}\label{eq:overlap_SU(11)} \braket{\zeta_1|\zeta_2}=\left[\frac{(1-|\zeta_1|^2)(1-|\zeta_2|^2)}{\left(1-\zeta^*_1 \zeta_2\right)^2}\right]^k. \end{equation} The overlap between a reference state $\ket{k,0}$ and any state $\ket{\zeta}$ is approximately Gaussian as a function of~$\zeta$: \begin{align} \braket{k,0|\zeta}=(1-|\zeta|^2)^k\approx \mathrm{e}^{-k|\zeta|^2},\,k\gg1. \end{align} The width of this overlap is proportional to $\nicefrac1{\sqrt{k}}$, which decreases as~$k$ grows. Similarly, the overlap between two coherent states located at different points of the hyperboloid surface decreases with~$k$ according to \begin{align} |\braket{\zeta_1|\zeta_2}|=\cosh^{-k}(\chi/2), \end{align} where \begin{equation} \cosh\chi=\cosh\tau_1\cosh\tau_2-\cos(\varphi_1-\varphi_2)\sinh \tau_1 \sinh \tau_2, \end{equation} with \begin{equation} \zeta_j=\exp\left(\text{i}\varphi_j\right)\tanh(\tau_j/2). \end{equation} The SU(1,1) Wigner function of operator~$\hat{\rho}$ is evaluated via the expectation value of the displaced parity operator~\cite{Seyfarth2020,Klimov2021,Russel2021}, \begin{align} W_{\hat{\rho}}\left(\zeta\right):=\text{tr}\big[\hat{\rho}\hat{\Pi}(\zeta)\big], \label{eq:wigner_su(1,1)} \end{align} where \begin{equation} \hat{\Pi}(\zeta):=\hat{D}(\zeta) \hat{\Pi}\hat{D}^{\dagger}(\zeta), \end{equation} and $\hat{\Pi}:=\exp\left[\text{i}\pi(\hat{K}_0-k)\right]$ is the parity operator for SU(1,1)~\cite{edwin2018}. The SU(1,1) Wigner distribution is visualized on the surface of the Poincar\'e\ disk via stereographic projection \begin{align} \zeta:=x+\text{i}p. \end{align} The unit disk can be lifted to the upper sheet of the two-sheeted hyperboloid by the inverse stereographic map. Using Eq.~(\ref{eq:wigner_su(1,1)}), the Wigner function of the operator $\ket{\zeta_1}\bra{\zeta_2}$ is easily found to be \begin{widetext} \begin{align}\label{eq:su(1,1)_coh_general} W_{\ket{\zeta_2}\bra{\zeta_1}}(\zeta)=\Bigg[\frac{\left(|\zeta|^2-1\right)^2\left(|\zeta_1|^2-1\right)\left(|\zeta_2|^2-1\right)\left(\zeta_2\zeta^*-1\right)\left(\zeta_1^*\zeta-1\right)}{(\zeta_1\zeta^*-1)(\zeta_2^*\zeta-1)\left(1-2\zeta\zeta_1^*+\zeta_2\zeta_1^*+\zeta_2\zeta_1^*+|\zeta|^2-2\zeta^*\zeta_2+|\zeta|^2\zeta_1^*\zeta_2\right)^2}\Bigg]^k \text{e}^{2 \text{i}k \text{arg}\left(\frac{1-\zeta_1 \zeta^*}{1-\zeta_2 \zeta^*}\right)}. \end{align} \end{widetext} The detailed derivation of this Wigner function is provided in the Appendix. In the particular case of the reference state $\ket{0}=\ket{k,0}$, the Wigner function is obtained \begin{equation} W_{\ket{0}}(\zeta)=\left(\frac{|\zeta|^2-1}{|\zeta|^2+1}\right)^{2k}. \end{equation} The corresponding Wigner distribution, which we show in Fig.~\ref{fig:figure2}, appears as a lobe. The support area of this lobe decreases isotropically as $k$ grows. We can approximate this Wigner function as a Gaussian of the form $\exp\left(-4k |\zeta|^2\right)$ for $k\gg1$. Hence, the extension of the SU(1,1) coherent state along any direction in phase space is proportional to $\nicefrac1{\sqrt{k}}$, which is precisely the same scaling that we found for the width of the overlap between coherent states. In the following, we show that the notion of sub-Planck structures is extended to the SU(1,1) group by associating the effective support of the SU(1,1) coherent state as a counter part of the SU(1,1) Planck action. \begin{figure*} \caption{Plots of the SU(1,1) Wigner function of the cat state on the Poincar\'e\ disk: (a)~$k=6$, (b)~$k=10$, and (c) $k=14$. Insets represent the interference pattern of each case. In all cases $\zeta_0=0.8$. The quantities are in arbitrary units.} \label{fig:figure3} \end{figure*} \subsection{SU(1,1) coherent-state superpositions} \label{subsec:su(1,1)_superpositions} The SU(1,1) cat states (superposition of two distinct coherent states) have been discussed ~\cite{ban1994,Gerry1997,Miry2012,Klimov2021}. In particular, the ‘horizontal’ cat typically refers to the superposition of coherent states along the horizontal axis of the Poincar\'e\ disk: \begin{equation}\label{eq:horizontalcat_SU(1,1)} \ket{\psi_\text{H}}:=\ket{\zeta_0}+\ket{-\zeta_0}, \end{equation} where $\zeta_0\in \mathbb{R}$. The corresponding Wigner function of this cat state is \begin{align} W_{\ket{\psi_{\text{H}}}}(\zeta)=W_{\ket{\zeta_0}}(\zeta)+W_{\ket{-\zeta_0}}(\zeta)+I_{\text{H}}(\zeta), \end{align} where each term is fairly easy to obtain using Eq.~(\ref{eq:su(1,1)_coh_general}). The first two terms represent the Wigner functions of the coherent states, \begin{equation} W_{\ket{\pm\zeta_0}}(\zeta)=\frac12\left[\frac{(\zeta_0^2-1)(|\zeta|^2-1)}{(\zeta_0^2+1)(|\zeta|^2+1)\pm 2\zeta_0 (\zeta+\zeta^*)}\right]^{2k}, \end{equation} \begin{figure*} \caption{Plots of the SU(1,1) Wigner function of the compass state on the Poincar\'e\ disk: (a)~$k=6$, (b)~$k=10$, and (c) $k=14$. Insets represent the central interference pattern of each case. In all cases $\zeta_0=0.8$. The quantities are in arbitrary units.} \label{fig:figure4} \end{figure*} and the last term provides the interference between the underlying coherent states \begin{widetext} \begin{align}\label{eq:catinter} I_{\text{H}}(\zeta):=\left[\frac{(\zeta^2_0-1)^2(|\zeta|^2-1)^2}{1-2(2\zeta^2+1)\zeta^2_0+\zeta_0^4+(\zeta^2_0-1)^2|\zeta|^4+2\zeta (\zeta^2_0+1)^2\zeta^*-4\zeta^2_0\zeta^*}\right]^k \cos\left[2k\arg(\Theta)\right], \end{align} \end{widetext} with \begin{align} \Theta=\frac{(1-\zeta_0\zeta^*)(\zeta_0\zeta-1)}{(1+\zeta_0\zeta^*)(\zeta_0\zeta-1)+(\zeta+\zeta_0)(\zeta_0-\zeta^*)}. \end{align} In Fig.~\ref{fig:figure3} we plot the corresponding Wigner function on the Poincar\'e\ disk. Two lobes appear on the unit disk at the locations $(\pm\zeta_0,0)$ are representing the coherent states. In addition, interference appears as an oscillatory pattern directed along the $p$ direction of the stereographic plane. As illustrated in Fig.~\ref{fig:figure3} this interference pattern becomes pronounced (i.e., the number of oscillation increases) as the representation index~$k$ increases. Along the $p$ axis ($x = 0$) the interference (\ref{eq:catinter}) becomes \begin{align} I_{\text{H}}(p)=\left[\frac{(\zeta^2_0-1)^2(p^2-1)^2}{(\zeta^2_0-1)^2(1+p^4)+2p^2(\zeta^4_0+6\zeta^2_0+1)}\right]^k\cos\left(\theta^\prime \right), \end{align} where \begin{align}\label{eq:thetalabel} \theta^\prime=2k\tan^{-1}\left(\frac{4\zeta_0 p}{\zeta^2_0-1}\right). \end{align} The zeros of the interference pattern $I_{\text{H}}(\zeta)$ occur when \begin{align} p=\pm \frac{(\zeta^2_0-1)}{4\zeta_0}\tan\left[\frac{(2m+1)\pi}{4k}\right],\,m\in\mathbb{Z}. \end{align} This means that the first zeros are located at \begin{equation} p=\pm \frac{\zeta^2_0-1}{4\zeta_0}\tan\left[\frac{\pi}{4k}\right]\approx \pm \frac{\zeta^2_0-1}{16\zeta_0 k}\pi,\,k\gg1. \end{equation} Hence, the extension of the interference patches along the $p$ direction is proportional to $\nicefrac1{k}$ for $k\gg1$. In contrast, along the $x$ axis ($p=0$), interference is simply approximated by \begin{align} I_{\text{H}}(x)=&\left(\frac{x^2-1}{x^2+1}\right)^{2k}\approx \text{exp}(-4kx^2),\,k\gg1. \end{align} Therefore, along the $x$ direction the extension of the interference pattern is proportional to $\nicefrac1{\sqrt{k}}$. This is precisely the same extension that we found for a coherent state along the $x$ direction. Support of interference structures of the SU(1,1) horizontal cat state is limited only along the vertical direction of the phase space. Similarly, we can build cat states along the vertical axis of the stereographic plane as \begin{align} \ket{\psi_{\text{V}}}:=\ket{\text{i}\zeta_0}+\ket{-\text{i}\zeta_0}, \end{align} whose Wigner function appears to be the same as one of the horizontal cat states, but rotated by $\nicefrac{\pi}{2}$ in the Poincar\'e\ disk, that is, \begin{align} W_{\ket{\psi_\text{V}}}(\zeta)=&\nonumber W_{\ket{\text{i}\zeta_0}}(\zeta)+W_{\ket{-\text{i}\zeta_0}}(\zeta)+I_\text{V}(\zeta), \\ =& W_{\ket{\psi_\text{H}}}(p+\text{i}x). \end{align} Here, $W_{\ket{\pm\text{i}\zeta_0}}(\zeta)=W_{\ket{\pm \zeta_0}}(p+\text{i}x)$ represents the Wigner function of underlying coherent states, and $I_\text{V}(\zeta)=I_\text{H}(p+\text{i}x)$ is the interference. \begin{figure*} \caption{Plots of the SU(1,1) Wigner function of cat-state mixtures on the Poincar\'e\ disk: (a)~$k=6$, (b)~$k=10$, and (c) $k=14$. Insets represent the central interference pattern of each case. In all cases $\zeta_0=0.8$. The quantities are in arbitrary units.} \label{fig:figure5} \end{figure*} Let us consider now the superposition of horizontal and vertical cat states, leading to the SU(1,1) compass state \begin{align}\label{eq:compass_su(1,1)} \ket{\psi_{\text{C}}}:=\ket{\psi_{\text{H}}}+\ket{\psi_{\text{V}}}. \end{align} The corresponding Wigner function is shown in Fig.~\ref{fig:figure4}. This Wigner function is written as a sum of the Wigner functions of individual cat states plus the interference between these (cat-like interference patterns located at the northeast, northwest, southeast, and southwest positions). We can clearly see four lobes centered at positions $\left(\pm\zeta_0,0\right)$ and $\left(0,\pm\zeta_0\right)$ on the the Poincar\'e\ disk, which correspond to the coherent states. Note that, for $k\gg1$, the chessboardlike pattern around the origin of the Poincar\'e\ disk is evident. The support area of a fundamental tile appears in a chessboardlike pattern that decreases isotropically in phase space as $k$ increases. We focus on this central interference pattern, which is written as the sum of the interferences of the horizontal and vertical cat states, that is, \begin{align} I_\boxplus(\zeta)=I_\text{H}(\zeta)+I_\text{V}(\zeta). \end{align} The extension of each tile in this pattern is proportional to $\nicefrac1{k}$ along any arbitrary direction in phase space, which is a factor $\nicefrac1{\sqrt{k}}$ smaller than the extension found for coherent states. These results show that, as promised, the concept of sub-Planck structures is generalized to the SU(1,1) group. These sub-Planck structures present by the mixture of two cat states. In particular, we consider the incoherent mixture of horizontal and vertical cat states \begin{align} \label{eq:catmixture_SU(1,1)} \hat{\rho}_\text{M}:=\ket{\psi_{\text{H}}}\bra{\psi_{\text{H}}}+\ket{\psi_{\text{V}}}\bra{\psi_{\text{V}}}, \end{align} whose Wigner function is equal to the sum of the Wigner functions of horizontal and vertical cat states, that is, \begin{equation} W_{\hat{\rho}_\text{M}}(\zeta)=W_{\ket{\psi_{\text{H}}}}(\zeta)+W_{\ket{\psi_{\text{V}}}}(\zeta). \end{equation} This Wigner function is shown in Fig.~\ref{fig:figure5}, where the chessboardlike pattern appears around the origin of the Poincar\'e\ disk. Hence, the same chessboardlike pattern with sub-Planck structures appears for the SU(1,1) cat-state mixtures. \begin{figure*} \caption{The Wigner functions of the SU(1,1) coherent-state superpositions considered in this work are shown on the Poincar\'e\ disk. Panels (a)~and (b)~represent the Wigner functions of the cat state with $k=\nicefrac14$ and $k=\nicefrac{3} \label{fig:figure6} \end{figure*} \subsection{Correlated coherent states of the SU(1,1) group} \label{subsec:two_photon} In this subsection, we review the relation between a few well known quantum states and coherent states associated with SU(1,1) group. The relevance of the~$\mathfrak{su}(1,1)$ algebra to the physical system can be obtained through the realization of the generators in terms of the operators of the underlying physical system. Here we focus on the bosonic realizations of the~$\mathfrak{su}(1,1)$ algebra corresponding to one and two modes~\cite{Brif1996,Gerry1986,Gerry1991,Gerry1995,stoler1971,Yuen1976,Gerry01,yazdi2008,luc1992}. As a first example, we consider a single boson-mode system in which the elements of the~$\mathfrak{su}(1,1)$ algebra are expressed as a single set of boson annihilation and creation operators as \begin{align}\label{eq:onemode_su11} \hat{K}_+=\frac12\hat{a}^{\dagger2},\, \hat{K}_-=\frac12\hat{a}^2,\,\hat{K}_0=\frac14\left(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}\right). \end{align} The Casimir operator is $\hat{K}^2=-\nicefrac{3}{16}$, which leads to the Bargmann indices $k=\nicefrac14$ and $k=\nicefrac34$. For these irreducible representations, the SU(1,1) displacement operator~(\ref{eq:displacement_su11_dis}) is identified as a one-mode squeezed operator. The representation associated with the Bargmann index $k=\nicefrac{1}{4}$ is the even-numbered Fock states, while for $k=\nicefrac{3}{4}$ only the odd-numbered Fock states~\cite{Gerry01}. Hence, for $k=\nicefrac14$, the SU(1,1) Perelomov coherent state can be considered as an ordinary squeezed-vacuum state \begin{align} \ket{\zeta,\nicefrac14}=(1-\left|\zeta\right|^2)^{\nicefrac14}\sum^{\infty}_{n=0}\frac{\sqrt{2n!}}{2^n n!}\zeta^n\ket{2n}. \label{eq:coherentstate_su11squeezed} \end{align} For $k=\nicefrac{3}{4}$, the corresponding SU(1,1) Perelomov coherent state is just the squeezed one-photon state~\cite{Gerry01} \begin{align} \ket{\zeta,\nicefrac{3}{4}}=(1-\left|\zeta\right|^2)^{\nicefrac34}\sum^{\infty}_{n=0}\frac{\sqrt{(2n+1)!}}{2^n n!}\zeta^n\ket{2n+1}. \label{eq:coherentstate_su11squeezed2} \end{align} Hence, for these irreducible representations, the SU(1,1) coherent-state superpositions of the present work can be taken as the superpositions of the ordinary squeezed vacuum and squeezed one-photon states. In Fig.~\ref{fig:figure6} we plot the corresponding Wigner functions of each case. For the given parameters these Wigner functions appear as positive peaked distributions. The superpositions of ordinary squeezed-vacuum states have been investigated~\cite{TANG201586,barry1989,quantumrep2021,Happ2018,quesene}. Now we briefly review the realization two-mode standard case. Let ($\hat{a}_1$, $\hat{a}_2$) and ($\hat{a}_1^\dagger$, $\hat{a}_2^\dagger$) be, respectively, the annihilation and creation operators of modes 1 and 2. Furthermore, let $\ket{n_1}$ and $\ket{n_2}$ represent the number states of these two modes, and the complete number-state basis of the two-mode field is \begin{align} \ket{n_1,n_2}=\ket{n_1}\otimes \ket{n_2}. \end{align} The $\mathfrak{su}(1,1)$ algebra can be realized by two-mode annihilation and creation operators as \begin{align} & \hat{K}_+=\hat{a}_1^\dagger \hat{a}_2^\dagger,~\hat{K}_-=\hat{a}_1 \hat{a}_2,\nonumber\\ &\hat{K}_0=\frac12\left(\hat{a}_1^\dagger \hat{a}_1+\hat{a}_2^\dagger \hat{a}_2+1\right). \end{align} These SU(1,1) operators obey the commutation relations~(\ref{eq:commutations_su(1,1)}), and their action on the two-mode states can be described as \begin{align} &\hat{K}_0\ket{n_1,n_2}=\frac12\left(n_1+n_2+1\right)\ket{n_1,n_2},\\&\hat{K}_+\ket{n_1,n_2}=\sqrt{(n_1+1)(n_2+1)}\ket{n_1+1,n_2+1},\\&\hat{K}_-\ket{n_1,n_2}=\sqrt{n_1 n_2}\ket{n_1-1,n_2-1}. \label{eq:commu_twomode} \end{align} The Casimir operator (\ref{eq:casimor_su(1,1)}) in this case becomes \begin{align} \hat{K}^2_0=\frac14\left(\Delta^2-1\right), \end{align} where \begin{align} \Delta= \hat{a}_1^\dagger \hat{a}_1-\hat{a}_2^\dagger \hat{a}_2, \end{align} whose eigenvalue is equal to the difference between the number of quanta in modes 1 and 2, i.e., $n_1-n_2$. The representations that we obtain are those for which this difference is constant. The $\mathfrak{su}$(1,1) basis~$\ket{k,n}$ can be identified by \begin{align}\label{eq:irr} k=\frac12(q+1),\, n=\frac12\left(n_1+n_2-q\right),~q=0,1,2,\dots \end{align} where~$q$ is the degeneracy parameter representing the eigenvalue of~$|\Delta|$, and it measures asymmetry in the photon number of two correlated modes. We assume that mode~1 has~$q$ more photons than mode~2, so that $n_1=n_2+q$ and $n=n_2=0,1,2,\dots$. Thus, the weight states of SU(1,1) becomes $\ket{k,n}=\ket{n_2,n_2+q}$, with $n=0,1,2,\dots$ or, more conveniently, it can be just simply written as $\ket{n_2,n_2+q}=\ket{n,n+q}$ (with $n_2=n$). Therefore, SU(1,1) Perelomov coherent states (\ref{eq:coherentstate_su11}) can be written in terms of the two-mode squeezed number states as \begin{align} \ket{\zeta,q}=(1-|\zeta|^2)^{\nicefrac{1+q}{2}}\sum^\infty_{n=0}\sqrt{\frac{(n+q)!}{n! q!}}\zeta^n \ket{n,n+q}. \label{eq:coherent_state_twophotons} \end{align} Note that the state $\ket{q,0}$ will be interpreted as the ground state of the relevant unitary irreducible representations of SU(1,1). For $q=0$ we just have the familiar two-mode squeezed vacuum state. For $q>0$ it is the state obtained by the action of the two-mode squeezed vacuum operator on the number state $\ket{q,0}$. Hence, SU(1,1) coherent-state superpositions considered in this work can be just superpositions of ordinary two-mode squeezed number states. Superpositions of ordinary two-mode squeezed vacuum states have been investigated in Ref.~\cite{Cardoso2021}. The photon-number distribution of the SU(1,1) coherent states (\ref{eq:coherent_state_twophotons}) appears as a Poissonian distribution for $q=0$ (zero fluctuations in system)~\cite{Gerry1991,Gerry91b}. However, as~$q$ grows, the distribution has a peak value at $n>0$. Higher values of~$q$ inject more photons in the system. Hence, this distribution is sub-Poissonian as~$q$ increases. As mentioned earlier, larger values of the Bargmann index~$k$ yields the sub-Planck structures in the phase space of compass states. Note that $k$ relates to the degeneracy parameter~$q$ by Eq.~(\ref{eq:irr}). In other words, we can say that the sub-Planck structures of the compass states are associated with $q\gg1$ for two-boson-mode standard case of the SU(1,1). This can be understood in a way similar to that for compass states of the harmonic oscillator, i.e., injecting more photons in the states brings more sub-Planckness in the phase space. The influence of squeezing on the quantum decoherence that occurs in a two-qubit system has been investigated~\cite{Basit2021}. The $\mathfrak{su}$(1,1) algebra can also be associated with the four-mode boson field by a four-boson realization of SU(1,1)~\cite{PhysRevA.63.042310}. \begin{figure*} \caption{Overlap between SU(1,1) states considered in this work and their slightly displaced versions. The left column corresponds to $k=6$, the middle column to $k=10$, and the right column to $k=14$. Panels (a)-(c) represent cat-state overlaps, panels (d)-(f) represent the overlaps of the compass state, and panels (g)-(i) are the overlaps of the cat-state mixtures. In all cases $\zeta_0=0.8$. The quantities are in arbitrary units.} \label{fig:figure7} \end{figure*} \subsection{Sensitivity against SU(1,1) displacements}\label{subsec:su(1,1)orthogonality} In this subsection, we discuss the sensitivity against phase-space displacements of SU(1,1) coherent-state superpositions presented in the preceding section. We compute the overlap between the states and their $\delta\zeta$-displaced versions, as given by Eq.~(\ref{eq:overlap_HW}). Let us first consider SU(1,1) coherent states. We already discussed in~\S\ref{subsec:preliminary}, for $k\gg1$, this overlap is approximated in Gaussian form as $\exp\left(-k |\delta\zeta|^2\right)$. Hence, the sensitivity to displacements for SU(1,1) coherent states scales as $\nicefrac1{\sqrt{k}}$. In the following we have to compare the sensitivity of SU(1,1) coherent-state superpositions with this scaling. Consider next the horizontal cat state (\ref{eq:horizontalcat_SU(1,1)}). The overlap (\ref{eq:overlap_HW}) for this state under the approximation $k\gg1$ leads to \begin{align}\label{eq:overlap_horizontalcat_su(1,1)} \mathcal{F}_{\ket{\psi_\text{H}}}\left(\delta\zeta\right)=\frac12\left[\frac{(\zeta_0^2-1)^2(|\delta\zeta|^2-1)}{1-2\zeta_0^2+4\zeta_0^2\delta_p^2+\zeta_0^4}\right]^{2k}\cos^2(2k\theta), \end{align} where \begin{align} \theta=\tan^{-1}\left(\frac{2\zeta_0 \delta_p}{\zeta^2_0-1}\right) \end{align} and \begin{equation} \delta\zeta:=\delta _x+\text{i}\delta_ p, \end{equation} with $\delta_j\in\mathbb{R}$. Note, for $k\gg1$ and $|\delta\zeta|\ll1$, the contribution of the cross terms between the coherent states to the overlap, e.g., $\bra{\zeta_0}\hat{D}(\delta\zeta)\ket{-\zeta_0}$ and $\bra{-\zeta_0}\hat{D}(\delta\zeta)\ket{\zeta_0}$, is negligible. The condition to make this overlap equal to zero is \begin{align} \delta _p=&\frac{\zeta^2_0-1}{2\zeta_0} \tan\left[\frac{(2m+1)\pi}{4k}\right]\\\approx&\frac{(\zeta^2_0-1)(2m+1)}{8\zeta_0k}\pi,\,m\in\mathbb{Z}, k\gg1. \end{align} Thus, for large~$k$, the displacement $\delta_ p \sim \nicefrac1{k}$ along the vertical direction of the stereographic plane can make the horizontal cat state orthogonal. This overlap is plotted for different values of~$k$ in Figs.~\ref{fig:figure7}(a)-\ref{fig:figure7}(c). As compared to the coherent state, the horizontal cat state shows $\nicefrac1{\sqrt{k}}$ times higher sensitivity against displacements. This occurs for displacements in the vertical direction in the stereographic plane. However, for the horizontal displacement it does not show the enhanced sensitivity compared to coherent states. For $k\gg1$, the overlap (\ref{eq:overlap_HW}) of the compass state (\ref{eq:compass_su(1,1)}) is \begin{align}\label{eq:overlap_compass_su(1,1)} \mathcal{F}_{\ket{\psi_\text{C}}}(\delta\zeta)=\frac12\left[\sqrt{\mathcal{F}_{\ket{\psi_\text{H}}}\left(\delta\zeta\right)}+\sqrt{\mathcal{F}_{\ket{\psi_\text{V}}}\left(\delta\zeta\right)}\right]^2, \end{align} with \begin{equation} \mathcal{F}_{\ket{\psi_\text{V}}}\left(\delta\zeta\right)=\mathcal{F}_{\ket{\psi_\text{H}}}\left(\delta\zeta_p+\text{i}\delta\zeta_x\right). \end{equation} We plot this overlap in Figs.~\ref{fig:figure7}(d)-\ref{fig:figure7}(f) with different values of~$k$. This result shows that the SU(1,1) compass state has $\nicefrac1{\sqrt{k}}$ times higher sensitivity against displacements compared to the coherent states, but now this enhanced sensitivity is independent of the displacement directions. Finally, we consider the cat-state mixture~(\ref{eq:catmixture_SU(1,1)}). The overlap~(\ref{eq:overlap_HW}) for this state leads to be \begin{align} \mathcal{F}_{\hat{\rho}_{\text{M}}}\left(\delta\zeta\right)=\mathcal{F}_{\ket{\psi_\text{H}}}\left(\delta\zeta\right)+\mathcal{F}_{\ket{\psi_\text{V}}}\left(\delta\zeta\right). \end{align} This overlap is plotted for different values of~$k$ in Figs.~\ref{fig:figure7}(g)-\ref{fig:figure7}(i). Now, the $\sqrt{k}$-enhanced sensitivity is present for displacements directed along the $\delta _x=\pm \delta _p$ directions. \subsection{Analogies: SU(1,1) versus HW and SU(2)} \label{subsec:comparision} In this subsection, we compare properties (phase-space features and sensitivity against the displacements) of the SU(1,1) coherent-state superpositions of the present work with their HW and SU(2) counterparts. SU(1,1) compass states and cat-state mixtures have structures of extensions proportional to $\nicefrac1{k}$ ($k$ is the Bargmann index) in all phase-space directions, which is~$\nicefrac1{\sqrt{k}}$ times smaller than the extension found for coherent states. Interference features of SU(1,1) cat states are limited only in one direction, just like their HW and SU(2) counterparts. This shows that the Wigner functions of these states have exactly the same phase-space features as their HW and SU(2) counterparts when plotted on the Poincar\'e\ disk, with the role of~$x_0$ (distance of the coherent states from the origin) in the HW case and $\sqrt{j}$ ($j$ the angular momentum) in the SU(2) case now played by $\sqrt{k}$. These SU(1,1) coherent-state superpositions are shown be sensitive against displacements that are lower than the sensitivity found for the coherent states by a factor of $\nicefrac1{\sqrt{k}}$. This enhanced sensitivity for the SU(1,1) compass state is independent of the displacements directions in phase space. Whereas, for cat states and their mixtures, this enhancement always occurs in specific directions. This shows that these SU(1,1) states have shown exactly the same behavior against the displacements as their HW and SU(2) counterparts, with the role of~$\bar{n}$ in the HW case and $j$ in the SU(2) being played by~$k$ for SU(1,1). \section{Summary}\label{sec:summary} We have shown that by considering coherent-state superpositions on the hyperboloid surface, one can build SU(1,1) cat states, compass states, and cat-state mixtures with phase-space features similar to those of their HW and SU(2) counterparts when their Wigner functions are represented on the Poincar\'e\ disk. In particular, both SU(1,1) cat-state superpositions (compass state) and mixtures have sub-Planck structures in phase space, but interference structures of cat states are not considered to be sub-Planck (since they are not limited in all phase-space directions). Moreover, these SU(1,1) coherent state superpositions also behave similarly to their HW and SU(2) counterparts regarding their sensitivity against displacements. This generalizes sub-Planck structures found in the HW and SU(2) cases to the SU(1,1) group. We have reviewed the two-mode bosonic realization of the $\mathfrak{su}$(1,1) algebra, which relates the Bargmann index $k$ to the asymmetry in photons numbers of correlated modes of two-mode squeezed number states. Then we have shown that the existence of the sub-Planck structures is associated with larger asymmetry in photon numbers of two correlated modes of the squeezed number state. In a similar way, the enhanced sensitivity of these superpositions can also be connected with this asymmetry in photon numbers of two modes i.e., higher asymmetry in photon numbers of these two modes corresponds to better enhanced sensitivity against the displacements. A number of interesting schemes have been presented for the implementation of SU(1,1) cat states~\cite{Gerry1997,Miry2012,zheng2002generation}. Another future direction will concern how to generate the SU(1,1) compass states introduced in our work. Some of these schemes can be adapted to achieve the generation of SU(1,1) compass states, which otherwise will require a complete new proposal to generate superpositions of four coherent states. \section{SU(1,1) Wigner function}\label{appendix:appendixA} In this section, we provide the more detailed derivations of the Wigner function for SU(1,1) coherent states. For the operator $\hat{\rho}=\ket{\zeta_2}\bra{\zeta_1}$, we rewrite its Wigner function as \begin{align}\label{eq:appendix_su(1,1)wig} W_{\ket{\zeta_2}\bra{\zeta_1}}(\zeta)=&\nonumber\bra{\zeta_1}\hat{D}(\zeta) \hat{\Pi}\hat{D}^{\dagger}(\zeta)\ket{\zeta_2},\\=&\bra{k,0}\hat{D}^{\dagger}(\zeta_1)\hat{D}(\zeta) \hat{\Pi}\hat{D}^{\dagger}(\zeta)\hat{D}(\zeta_2)\ket{k,0}. \end{align} In some cases simpler expressions are found using the alternative form of the composition property of displacement operators, which we rewrite here as \begin{align} \hat{D}(\zeta_ 1)\hat{D}(\zeta_ 2)=\text{e}^{\text{-i}\phi \hat{K}_0}\hat{D}(\zeta_ 3), \label{eq:product2_operator} \end{align} with \begin{equation} \zeta_3=\frac{\zeta_1+\zeta_2}{1+\zeta_1\zeta^*_2},\, \phi=-2\arg(1+\zeta_1\zeta^*_2). \end{equation} Using composition laws given by Eqs.~(\ref{eq:product_operator}) and~(\ref{eq:product2_operator}), we simplify Eq.~(\ref{eq:appendix_su(1,1)wig}) as \begin{equation} W_{\ket{\zeta_2}\bra{\zeta_1}}(\zeta)=\text{e}^{2\text{i}k\text{arg}\left(\frac{1-\zeta_1\zeta^*}{1-\zeta_2\zeta^*}\right)}\bra{\zeta^{\prime}_1}\hat{\Pi}\ket{\zeta_2{^\prime}}, \end{equation} with~$\zeta^{\prime}_1=\frac{\zeta_1-\zeta}{1-\zeta_1\zeta^*},\, \zeta^{\prime}_2=\frac{\zeta_2-\zeta}{1-\zeta_2\zeta^*}.$ This expression is easily simplified to obtain Eq.~(\ref{eq:su(1,1)_coh_general}). \end{document}

\begin{document} \title{Voronoi-based estimation of Minkowski tensors from finite point samples} \begin{abstract} Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image. \end{abstract} \section{Introduction} Intrinsic volumes, such as volume, surface area, and Euler characteristic, are widely-used tools to capture geometric features of an object; see, for instance, \cite{meckeEtAl,OM,milesSerra}. Minkowski tensors are tensor valued generalizations of the intrinsic volumes, associating with every sufficiently regular compact set in $\mathbb{R}^d$ a symmetric tensor, rather than a scalar. They carry information about geometric features of the set such as position, orientation, and eccentricity. For instance, the volume tensor -- defined formally in Section \ref{minkowski} -- of rank $0$ is just the volume of the set, while the volume tensors of rank $1$ and $2$ are closely related to the center of gravity and the tensor of inertia, respectively. For this reason, Minkowski tensors are used as shape descriptors in materials science \cite{mickel,aste}, physics \cite{kapfer}, and biology \cite{beisbart,ziegel}. The main purpose of this paper is to present estimators that approximate all the Min\-kow\-ski tensors of a set $K$ when only weak information on $K$ is available. More precisely, we assume that a finite set $K_0$ which is close to $K$ in the Hausdorff metric is known. The estimators are based on the Voronoi decomposition of $\mathbb{R}^d$ associated with the finite set $K_0$, following an idea of M\'{e}rigot et al.\ \cite{merigot}. What makes these estimators so interesting is that they are consistent; that is, they converge to the respective Minkowski tensors of $K$ when applied to a sequence of finite approximations converging to $K$ in the Hausdorff metric. We emphasize that the notion of `estimator' is used here in the sense of digital geometry \cite{digital} meaning `approximation of the true value based on discrete input' and should not be confused with the statistical concept related to the inference from data with random noise. The main application we have in mind is the case where $K_0$ is a digitization of $K$. This is detailed in the following. As data is often only available in digital form, there is a need for estimators that allow us to approximate the {Minkowski} tensors from digital images. In a black-and-white image of a compact geometric object $K\subseteq \mathbb{R}^d$, each pixel (or voxel) is colored black if {its} midpoint belongs to $K$ and white otherwise. Thus, the information about $K$ contained in the image is the set of black pixel (voxel) midpoints $K_0=K\cap a\mathbb{L}$, where $\mathbb{L}$ is the lattice formed by {all} pixel (voxel) midpoints and $a^{-1}$ is the resolution. A natural criterion for {the reliability of} a digital estimator is that it yields the correct tensor when $a\to 0_+$. If this property holds for all objects in a given family of sets, for instance, for all sets with smooth boundary, then the estimator is called \emph{multigrid convergent} for this class. Digital estimators for the scalar Minkowski tensors, that is, for the intrinsic volumes, are widespread in the digital geometry literature; see, e.g.,~\cite{digital,OM,OS} and the references therein. For Minkowski tensors up to rank two, estimators based on binary images are given in \cite{turk} for the two-dimensional and in \cite{mecke} for the three-dimensional case. Even for the class of convex sets, multigrid convergence has not been proven for any of the above mentioned estimators. The only exception are volume related quantities. Most of the above mentioned estimators are \emph{$n$-local} for some given fixed $n\in \mathbb{N}$. We call an estimator $n$-local if it depends on the image only through the histogram of all $n\times \dotsm \times n$ configurations of black and white points. For instance, a natural surface area estimator \cite{lindblad} in three-dimensional space scans the image with a voxel cube of size $2\times 2\times2$ and assigns a surface contribution to each observed configuration. The sum of all contributions is then the surface area estimator, which is clearly $2$-local. The advantage of $n$-local estimators is that they are intuitive, easy to implement, and the computation time is linear in the number of pixels or voxels. However, many {$n$-local} estimators are not multigrid convergent for convex sets; see \cite{am3} and the detailed discussion in Section \ref{known}. This implies that many established estimators, like the mentioned one in \cite{lindblad} cannot be multigrid convergent for convex sets. All the estimators of 2D-Minkowski tensors in \cite{turk} are $2$-local. By the results in \cite{am3}, the estimators for the perimeter and the Euler characteristic can thus not be multigrid convergent for convex sets. The multigrid convergence of the other estimators has not been investigated. The algorithms for 3D-Minkowski tensors in \cite{mecke} have as input a triangulation of the object's boundary, and the way this triangulation is obtained determines whether the resulting estimators are $n$-local or not. There are no known results on multigrid convergence for these estimators either. Summarizing, to the best of our knowledge, this paper presents for the first time estimators of all Minkowski tensors of arbitrary rank that come with a multigrid convergence proof for a class of sets that is considerably larger than the class of convex sets. The present work is inspired by \cite{merigot}, and we therefore start by recalling some basic notions from this paper. For a nonempty compact set $K$, the authors of \cite{merigot} define a tensor valued measure, which they call the \emph{Voronoi covariance measure}, defined on a Borel set $A\subseteq \mathbb{R}^d$ by \begin{equation*} \mathcal{V}_R(K;A) = \int_{ K^R }\mathds{1}_A(p_K(y)) (y-p_K(y))(y-p_K(y))^\top\,dy. \end{equation*} Here, $K^R$ is the set of points at distance at most $R>0$ from $K$ and $p_K$ is the \emph{metric projection} on $K$: the point $p_K(x)$ is the point in $K$ closest to $x$, provided that this closest point is unique. The metric projection of $K$ is well-defined on $\mathbb{R}^d$ with the possible exception of a set of Lebesgue-measure zero; see, e.g., \cite{fremlin}. The paper \cite{merigot} uses the Voronoi covariance measure to determine local features of surfaces. It is proved there that if $K \subseteq \mathbb{R}^3$ is a smooth surface, then \begin{equation}\label{eigen} \mathcal{V}_R(K;B(x,r)) \approx \frac{2\pi}{3}R^3r^2\bigg(u(x)u(x)^\top + \frac{r^2}{4}\sum_{i=1,2}k_i(x)^2P_i(x)P_i(x)^\top\bigg), \end{equation} where $B(x,r)$ is the Euclidean ball with midpoint $x\in K$ and radius $r$, $u(x)$ is one of the two surface unit normals at $x\in K$, $P_1(x),P_2(x)$ are the principal directions and $k_1(x),k_2(x)$ the corresponding principal curvatures. Hence, the eigenvalues and -directions of the Voronoi covariance measure carry information about local curvatures and normal directions. Assuming that a compact set $K_0$ approximates $K$, \cite{merigot} suggests to estimate $\mathcal{V}_R(K;\cdot) $ by $\mathcal{V}_R(K_0;\cdot)$. It is shown in that paper that $\mathcal{V}_R(K_0;\cdot)$ converges to $\mathcal{V}_R(K;\cdot)$ in the bounded Lipschitz metric when $K_0 \to K$ in the Hausdorff metric. Moreover, if $K_0$ is a finite set, then the Voronoi covariance measure can be expressed in the form \begin{equation*} \mathcal{V}_R(K_0;A) = \sum_{x\in K_0 \cap A} \int_{B(x,R)\cap V_x(K_0) } (y-x)(y-x)^\top \,dy. \end{equation*} Here, $V_x(K_0)$ is the Voronoi cell of $x$ in the Voronoi decomposition of $\mathbb{R}^d$ associated with $K_0$. Thus, the estimator which is used to approximate $\mathcal{V}_R(K;A)$ is easily computed. Given the Voronoi cells of $K_0$, each Voronoi cell contributes with a simple integral. Figure \ref{fig} (a) shows the Voronoi cells of a finite set of points on an ellipse. The Voronoi cells are elongated in the normal direction. This is the intuitive reason why they can be used to approximate \eqref{eigen}. The Voronoi covariance measure $\mathcal{V}_R(K;A) $ can be identified with a symmetric 2-tensor. In the present work, we explore how natural extensions of the Voronoi covariance measure can be used to estimate general Minkowski tensors. The generalizations of the Voronoi covariance measure, which we will introduce, will be called \emph{Voronoi tensor measures}. {We will then show how the Minkowski tensors can be recovered from these}. When we apply the results to digital images, we will work with full-dimensional sets $K$, and the finite point sample $K_0$ is obtained from the representation $K_0=K\cap a\mathbb{L}$ of a digital image of $K$. The Voronoi cells associated with $K_0=K\cap a\mathbb{L}$ are sketched in Figure~\ref{fig}~(b). Taking point samples from $K$ with increasing resolution, convergence results will follow from an easy generalization of the convergence proof in \cite{merigot}. \begin{figure} \caption{(a). The Voronoi cells of a finite set of points on a surface. (b). A digital image and the associated Voronoi cells.} \label{fig} \end{figure} The paper is structured as follows: In Section~\ref{minkowski}, we recall the definition of Minkowski tensors and the classical as well as a local Steiner formula for sets of positive reach. In Section~\ref{construction}, we define the Voronoi tensor measures, discuss how they can be estimated from finite point samples, and explain how the Steiner formula can be used to connect the Voronoi tensor measures with the Minkowski tensors. Section \ref{convergence} is concerned with the convergence of the estimator. The results are specialized to digital images in Section \ref{DI}. Finally, the estimator is compared with existing approaches in Section \ref{known}. \section{Minkowski tensors}\label{minkowski} We work in Euclidean space $\mathbb{R}^d$ with scalar product $\langle\cdot\,,\cdot\rangle$ and norm $|\cdot|$. The Euclidean ball with center $x\in\mathbb{R}^d$ and radius $r\ge 0$ is denoted by $B(x,r)$, and we write $S^{d-1}$ for the unit sphere in $\mathbb{R}^d$. Let $\partial A$ and $\text{int}A$ be the boundary and the interior of a set $A\subseteq{\mathbb R}^d$, respectively. The $k$-dimensional Hausdorff-measure in $\mathbb{R}^d$ is denoted by ${\mathcal H}^k$, $0\le k\le d$. Let ${\mathcal C}^d$ be the family of nonempty compact subsets of $\mathbb{R}^d$ and ${\mathcal K}^d\subseteq \mathcal{C}^d$ the subset of nonemtpy compact convex sets. For two compact sets $K,M \in{\mathcal C}^d$, we define their \emph{Hausdorff distance} by \begin{equation*} d_H(K,M) = \inf\{\varepsilon>0\mid K\subseteq M^\varepsilon, M \subseteq K^\varepsilon\}. \end{equation*} Let $\mathbb{T}^p$ denote the space of symmetric $p$-tensors (tensors of rank $p$) over $\mathbb{R}^d$. Identifying $\mathbb{R}^d$ with its dual (via the scalar product), a symmetric $p$-tensor defines a symmetric multilinear map $(\mathbb{R}^d)^p\to \mathbb{R}$. Letting $e_1,\dots,e_d$ be the standard basis in $\mathbb{R}^d$, a tensor $T\in \mathbb{T}^p$ is determined by its coordinates \begin{equation*} T_{i_1\dots i_p}=T(e_{i_1},\dots,e_{i_p}) \end{equation*} with respect to the standard basis, for all choices of ${i_1},\dots,{i_p} \in \{1,\dots,d\}$. We use the norm on $\mathbb{T}^p$ given by \begin{equation*} |T|=\sup\big\{|T(v_1,\dots,v_p)| \,\mid \, |v_1|=\dots =|v_p|=1\big\} \end{equation*} for $T\in \mathbb{T}^p$. The same definition is used for arbitrary tensors of rank $p$. The symmetric tensor product of $y_1,\ldots, y_m\in \mathbb{R}^{d}$ is given by the symmetrization $y_1\odot\cdots\odot y_m=(m!)^{-1}\sum \otimes_{i=1}^m y_{\sigma(i)}$, where the sum extends over all permutations $\sigma$ of $\{1,\ldots,m\}$ and $\otimes$ is the usual tensor product. We write $x^r$ for the $r$-fold tensor product of $x\in \mathbb{R}^d$. For two symmetric tensors of the form $T_1=y_1 \odot \cdots \odot y_r$ and $T_2=y_{r+1} \odot \cdots \odot y_{r+s}$, where $y_1, \ldots , y_{r+s} \in\mathbb{R}^d$, the symmetric tensor product $T_1\odot T_2$ of $T_1$ and $T_2$, which we often abbreviate by $T_1T_2$, is the symmetric tensor product of $y_1, \ldots, y_{r+s} $. This is extended to general symmetric tensors $T_1$ and $T_2$ by linearity. Moreover, it follows from the preceding definitions that $$ |y_1\odot\cdots\odot y_m|\le |y_1|\cdots |y_m|, $$ $y_1,\ldots, y_m\in \mathbb{R}^{d}$. For any compact set $K\subseteq \mathbb{R}^d$, we can define an element of $\mathbb{T}^r$ called the \emph{$r$th volume tensor} \begin{equation*} \Phi_{d}^{r,0}(K) = \frac{1}{r!} \int_{K} x^r \,dx. \end{equation*} For $s\geq 1$ we define $\Phi_{d}^{r,s}(K)=0$. Some of the volume tensors have well-known physical interpretations. For instance, $\Phi_{d}^{0,0}(K)$ is the usual volume of $K$, $\Phi_{d}^{1,0}(K)$ is up to normalization the center of gravity, and $\Phi_{d}^{2,0}(K)$ is closely related to the tensor of inertia. All three tensors together can be used to find the best approximating ellipsoid of a particle \cite{ziegel}. The sequence of all volume tensors $(\Phi_{d}^{r,0}(K))_{r=0}^\infty$ determines the compact set $K$ uniquely. For convex sets in the plane even the following stability result \cite[Remark 4.4.]{JuliaAstrid} holds: If $K, L\in {\mathcal K}^2$ are contained in the unit square and have coinciding volume tensors up to rank $r$, then their distance, measured in the symmetric difference metric ${\mathcal H}^2\big((K\setminus L) \cup (L\setminus K)\big)$, is of order $O(r^{-1/2})$ as $r\to \infty$. We will now define \emph{Minkowski surface tensors}. These can also be used to characterize the shape of an object or the structure of a material as in \cite{beisbart,kapfer}. They require stronger regularity assumptions on $K$. Usually, like in \cite[Section 5.4.2]{schneider}, the set $K$ is assumed to be convex. However, as Minkowski tensors are tensor-valued integrals with respect to the generalized curvature measures (also called support measures) of $K$, they can be defined whenever the latter are available. We will use this to define Minkowski tensors for sets of positive reach. First, we recall the definition of a set of positive reach and explain how curvature measures of such sets are determined (see \cite{Federer59,zahle}). For a compact set $K\in {\mathcal C}^d$, we let $d_K(x)$ denote the distance from $x\in \mathbb{R}^d$ to $K$. Then, for $R\ge 0$, $K^R=\{x\in \mathbb{R}^d \mid d_K(x)\leq R\}$ is the $R$-parallel set of $K$. The \emph{reach} $\reach(K)$ of $K$ is defined as the supremum over all $R\geq 0$ such that for all $x\in \mathbb{R}^d$ with $d_K(x)<R$ there is a unique closest point $p_K(x)$ in $K$. We say that $K$ has positive reach if $\reach(K)>0$. Smooth surfaces (of class $C^{1,1}$) are examples of sets of positive reach, and compact convex sets are characterized by having infinite reach. By definition, the map $p_K$ is defined everywhere on $K^R$ if $R<\reach(K)$. Let $K\subseteq \mathbb{R}^d$ be a (compact) set of positive reach. The (global) Steiner formula for sets with positive reach states that for all $R<\reach(K)$ the $R$-parallel volume of $K$ is a polynomial, that is, \begin{align}\label{gloSt} \mathcal{H}^d( K^R){}&= \sum_{k=0}^d \kappa_{d-k} R^{d-k} \Phi_{k}^{0,0}(K). \end{align} Here $\kappa_j$ is the volume of the unit ball in $\mathbb{R}^j$ and the numbers $\Phi^{0,0}_0(K),\ldots, \allowbreak \Phi_d^{0,0}(K)$ are the so-called \emph{intrinsic volumes} of $K$. They are special cases of the Minkowski tensors to be defined below. Some of them have well-known interpretations. As mentioned, $\Phi^{0,0}_d(K)$ is the volume of $K$. Moreover, $2\Phi^{0,0}_{d-1}(K)$ is the surface area, $\Phi^{0,0}_{d-2}(K)$ is proportional to the total mean curvature, and $\Phi^{0,0}_0(K)$ is the Euler characteristic of $K$. For convex sets, \eqref{gloSt} is the classical Steiner formula which holds for all $R\ge 0$. Z\"ahle \cite{zahle} showed that a local version of \eqref{gloSt} can be established giving rise to the \emph{generalized curvature measures} $\mathbb{L}mbda_k(K;\cdot)$ of $K$, for $k=0,\dots,d-1$. An extension to general closed sets is considered in \cite{last}. The generalized curvature measures (also called support measures) are measures on $\Sigma = \mathbb{R}^d\times S^{d-1}$. They are determined by the following {\em local} Steiner formula which holds for all $R < \reach(K)$ and all Borel set $B\subseteq \Sigma$: \begin{equation}\label{clasSteiner} \mathcal{H}^d\left(\left\{x\in K^R \backslash K \mid \Big(p_K(x), \tfrac{x-p_K(x)}{|x-p_K(x)|}\Big)\in B\right\}\right) = \sum_{k=0}^{d-1} R^{d-k} \kappa_{d-k} \mathbb{L}mbda_k(K;B). \end{equation} The coefficients $\mathbb{L}mbda_k(K;B)$ on the right side of \eqref{clasSteiner} are signed Borel measures $\mathbb{L}mbda_k(K;\cdot)$ evaluated on $B\subseteq\Sigma$. These measures are called the {\em generalized curvature measures} of $K$. Since the pairs of points in $B$ on the left side of \eqref{clasSteiner} always consist of a boundary point of $K$ and an outer unit normal of $K$ at that point, each of the measures $\mathbb{L}mbda_k(K,\cdot)$ is concentrated on the set of all such pairs. For this reason, the generalized curvature measures $\mathbb{L}mbda_k(K;\cdot)$, $k\in\{0,\ldots,d-1\}$, are also called {\em support measures}. They describe the local boundary behavior of the part of $\partial K$ that consists of points $x$ with an outer unit normal $u$ such that $(x,u)\in B$. A description of the generalized curvature measures $\mathbb{L}mbda_k(K,\cdot)$ by means of generalized curvatures living on the normal bundle of $K$ was first given in \cite{zahle} (see also \cite[\S 2.5 and p.~217]{schneider} and the references given there). The total measures $\mathbb{L}mbda_k(K,\Sigma)$ are the intrinsic volumes. Based on the generalized curvature measures, for every $k\in\{0,\dots,d-1\}$, $r,s\geq 0$ and every set $K\subseteq\mathbb{R}^d$ with positive reach, we define the {\em Minkowski tensor} \begin{equation*} \Phi_{k}^{r,s}(K) = \frac{1}{r!s!}\frac{\omega_{d-k}}{\omega_{d-k+s}}\int_{\Sigma} x^r u^{s} \mathbb{L}mbda_k(K;d(x,u)) \end{equation*} in $\mathbb{T}^{r+s}$. Here $\omega_k$ is the surface area of the unit sphere $S^{k-1}$ in $\mathbb{R}^k$. More information on Minkowski tensors can for instance be found in \cite{hug,mcmullen,schuster,KVJLNM}. As in the case of volume tensors, the Minkowski tensors carry strong information on the underlying set. For instance, already the sequence $(\Phi_{1}^{0,s}(K))_{s=0}^\infty$ determines any $K\in {\mathcal K}^d$ up to a translation. A stability result also holds: if $K$ and $L$ are both contained in a fixed ball and have the same tensors $\Phi_{1}^{0,s}$ of rank $s\le s_0$, then a translation of $K$ is close to $L$ in the Hausdorff metric and the distance is $O(s_0^{-\beta})$ as $s_0\to \infty$ for any $0<\beta<3/(n+1)$; see \cite[Theorem 4.9]{AstridMarkus}. One can define \emph{local Minkowski tensors} in a similar way (see \cite{HS14}). For a Borel set $B\subseteq \Sigma$, for $k\in\{0,\dots,d-1\}$, $r,s\geq 0$ and a set $K\subseteq\mathbb{R}^d$ with positive reach, we put \begin{equation*} \Phi_{k}^{r,s}(K;B) = \frac{1}{r!s!}\frac{\omega_{d-k}}{\omega_{d-k+s}}\int_{B} x^r u^{s} \,\mathbb{L}mbda_k(K;d(x,u)) \end{equation*} and, for a Borel set $A \subseteq \mathbb{R}^d$, \begin{equation*} \Phi_{d}^{r,0}(K;A) = \frac{1}{r!} \int_{K\cap A} x^r \,dx. \end{equation*} In order to avoid a distinction of cases, we also write $\Phi_{d}^{r,0}(K;A\times S^{d-1})$ instead of $\Phi_{d}^{r,0}(K;A)$. Moreover, we define $\Phi_{d}^{r,s}(K;\cdot)=0$ if $s\ge 1$. The local Minkowski tensors can be used to describe local boundary properties. For instance, local 1- and 2-tensors are used for the detection of sharp edges and corners on surfaces in \cite{clarenz}. They also carry information about normal directions and principal curvatures as explained in the introduction. We conclude this section with a general remark on continuity properties of the Minkowski tensors. Although the functions $K\mapsto \Phi_{k}^{r,s}(K)$ are continuous when considered in the metric space $(\mathcal{K}^d,d_H)$, they are not continuous on ${\mathcal C}^d$. (For instance, the volume tensors of a finite set are always vanishing, but finite sets can be used to approximate any compact set in the Hausdorff metric.) This is the reason why our approach requires an approximation argument with parallel sets as outlined below. The consistency of our estimator is mainly based on a continuity result for the metric projection map. We quote this result \cite[Theorem 3.2]{chazal} in a slightly different formulation which is symmetric in the two bodies involved. Let $\|f\|_{L^1(E)}$ be the usual $L^1$-norm of the restriction of $f$ to a Borel set $E\subseteq \mathbb{R}^d$. \begin{proposition}\label{CHAZProp} Let $\rho>0$ and let $E\subseteq \mathbb{R}^d$ be a bounded measurable set. Then there is a constant $C_1=C_1\left(d,\diam(E\cup\{0\}),\rho\right)>0$ such that \[ \|p_K-p_{K_0}\|_{L^1(E)} \le C_1 d_H(K,K_0)^{\frac 12} \] for all $K,K_0\in {\mathcal C}^d$ with $K,K_0\subseteq B(0,\rho)$. \end{proposition} \begin{proof} Let $E'$ be the convex hull of $E$ and observe that \begin{equation*} \|p_K-p_{K_0}\|_{L^1(E)} \leq \|p_K-p_{K_0}\|_{L^1(E')}. \end{equation*} It is shown in \cite[Lemma 3.3]{chazal} (see also \cite[Theorem 4.8]{Federer59}) that the map $v_K:\mathbb{R}^d\to\mathbb{R}$ given by $v_K(x)=|x|^2-d_K^2(x)$ is convex and that its gradient coincides almost everywhere with $2p_K$. Since $E'$ has rectifiable boundary, \cite[Theorem~3.5]{chazal} implies that \begin{align*} \|p_K-p_{K_0}\|_{L^1(E')} \le {}& c_1(d) ({\mathcal H}^d(E')+(c_2+\|d_K^2-d_{K_0}^2\|_{\infty,E'}^{\frac 12}){\mathcal H}^{d-1}(\partial E'))\\ &\times \|d_K^2-d_{K_0}^2\|_{\infty,E'}^{\frac 12}. \end{align*} Here $c_2=\diam(2p_K(E')\cup 2p_{K_0}(E'))\le 2\diam (K\cup K_0)\le 4\rho$ and the supremum-norm $\|\cdot\|_{\infty,E'}$ on $E'$ can be estimated by \begin{align*} \|d_K^2-d_{K_0}^2\|_{\infty,E'}&\le 2\diam(E'\cup K\cup K_0) \|d_K-d_{K_0}\|_{\infty,E'} \\&\le 2\left[\diam(E'\cup\{0\})+2\rho\right]d_H(K,K_0). \end{align*} Moreover, intrinsic volumes are increasing on the class of convex sets, so \begin{align*} \mathcal{H}^d(E'){}&\leq \mathcal{H}^d(B(0, \diam(E'\cup \{0\})))\\ {\mathcal H}^{d-1}(\partial E') {}&\leq \mathcal{H}^{d-1}(\partial B(0, \diam(E'\cup \{0\}))). \end{align*} Together with the trivial estimate $d_H(K,K_0)\le2\rho$ and with the equality $\diam(E\cup\{0\})=\diam(E'\cup\{0\})$, this yields the claim. \end{proof} The authors of \cite{chazal} argue that the exponent $1/2$ in Proposition \ref{CHAZProp} is best possible. \section{Construction of the estimator} \label{construction} In Section \ref{VTM} below, we define the Voronoi tensor measures and show how the Minkowski tensors can be obtained from these. We then explain in Section \ref{finite} how the Voronoi tensor measures can be estimated from finite point samples. As a special case, we obtain estimators for all intrinsic volumes. This is detailed in Section \ref{intvol}. \subsection{The Voronoi tensor measures} \label{VTM} Let $K$ be a compact set. Here and in the following subsections, we let $r,s\in\mathbb{N}_0$ and $R\ge 0$. Define the $ \mathbb{T}^{r+s}$-valued measures $\mathcal{V}_{R}^{r,s}(K;\cdot)$ given on a Borel set $A\subseteq \mathbb{R}^d$ by \begin{equation}\label{star} \mathcal{V}_{R}^{r,s}(K;A) = \int_{K^R }\mathds{1}_A(p_K(x)) \,p_K(x)^r(x-p_K(x))^s \, dx. \end{equation} When $K$ is a smooth surface, $\mathcal{V}_{R}^{0,2}(K;\cdot)$ {corresponds to} the Voronoi covariance measure in \cite{merigot}. We will refer to the measures defined in \eqref{star} as the \emph{Voronoi tensor measures}. Note that if $f:\mathbb{R}^d \to \mathbb{R}$ is a bounded Borel function, then \begin{equation}\label{integralf} \int_{\mathbb{R}^d} f(x) \,\mathcal{V}_{R}^{r,s}(K;dx) = \int_{K^R}f(p_K(x))\,p_K(x)^r(x-p_K(x))^s \, dx \in \mathbb{T}^{r+s}. \end{equation} Suppose now that $K$ has positive reach with $\reach(K)>R$. Then a special case of the generalized Steiner formula derived in \cite{last} (or an extension of \eqref{clasSteiner}) implies the following version of the local Steiner formula for the Voronoi tensor measures: \begin{align}\nonumber \mathcal{V}_{R}^{r,s}(K;A) {}&= \sum_{k=1}^{d} \omega_{k} \int_{\Sigma} \int_{0}^R \mathds{1}_{A}(x) t^{s+k-1} x^r u^s \, dt\, \mathbb{L}mbda_{d-k}(K;d(x,u))\nonumber\\ &\qquad +\mathds{1}_{\{s = 0\}}\int_{K\cap A} x^r \,dx\nonumber\\ &= r!s! \sum_{k=0}^d \kappa_{k+s} R^{s+k} \Phi_{d-k}^{r,s}(K;A\times S^{d-1}),\label{steiner} \end{align} where $A\subseteq {\mathbb R}^d$ is a Borel set. In particular, the total measure is \begin{equation*} \mathcal{V}_{R}^{r,s}(K)=\mathcal{V}_{R}^{r,s}(K;\mathbb{R}^d) = r!s!\sum_{k=0}^d \kappa_{k+s} R^{s+k} \Phi_{d-k}^{r,s}(K) . \end{equation*} Note that the special case $r=s=0$ is the Steiner formula \eqref{gloSt} for sets with positive reach. Equation \eqref{steiner}, used for different parallel distances $R$, can be solved for the Minkowski tensors. More precisely, choosing $d+1$ different values $0<R_0<\ldots <R_d<\reach(K)$ for $R$, we obtain a system of $d+1$ linear equations: \begin{align}\label{matrixeq} \begin{pmatrix} \mathcal{V}_{R_0}^{r,s}(K;A)\\ \vdots \\ \mathcal{V}_{R_d}^{r,s}(K;A) \end{pmatrix} =r!s! \begin{pmatrix}\kappa_s R_0^{s} & \dots & \kappa_{s+d}R_0^{s+d} \\ \vdots & & \vdots \\ \kappa_sR_{d}^{s} & \dots & \kappa_{s+d}R_{d}^{s+d} \end{pmatrix} \begin{pmatrix}\Phi_{d}^{r,s}(K;{A\times S^{d-1}})\\ \vdots \\ \Phi_{0}^{r,s}(K;{A\times S^{d-1}}) \end{pmatrix}. \end{align} Since the Vandermonde-type matrix \begin{align}\label{matrixA} A_{R_0,\ldots,R_d}^{r,s} = {r!s!} \begin{pmatrix} \kappa_s R_0^{s} & \dots & \kappa_{s+d}R_0^{s+d} \\ \vdots & & \vdots \\ \kappa_s R_{d}^{s} & \dots & \kappa_{s+d}R_{d}^{s+d} \end{pmatrix}\in \mathbb{R}^{(d+1)\times(d+1)} \end{align} in \eqref{matrixeq} is invertible, the system can be solved for the tensors, and thus we get \begin{align}\label{matrix} \begin{pmatrix}{\Phi}_{d}^{r,s}(K;A{\times S^{d-1}})\\ \vdots \\ {\Phi}_{0}^{r,s}(K;{A{\times S^{d-1}}}) \end{pmatrix} =\left(A_{R_0,\ldots,R_d}^{r,s}\right)^{-1} \begin{pmatrix} \mathcal{V}_{R_0}^{r,s}(K;A)\\ \vdots \\ \mathcal{V}_{R_d}^{r,s}(K;A) \end{pmatrix}. \end{align} If $s>0$, then ${\Phi}_{d}^{r,s}(K;A\times S^{d-1})=0$ by definition, so we may omit one of the equations in the system \eqref{matrixeq}. \subsection{Estimation of Minkowski tensors}\label{finite} Let $K$ be a compact set of positive reach. Suppose that we are given a compact set $K_0$ that is close to $K$ in the Hausdorff metric. In the applications we have in mind, $K_0$ is a finite subset of $K$, but this is not necessary for the algorithm to work. Based on $K_0$, we want to estimate the local Minkowski tensors of $K$. We do this by approximating $\mathcal{V}_{R_k}^{r,s}(K;A)$ in Formula \eqref{matrix} by $\mathcal{V}_{R_k}^{r,s}(K_0;A)$, for $k=0,\dots,d$ and $A\subseteq \mathbb{R}^d$ a Borel set. This leads to the following set of estimators for $\Phi_k^{r,s}(K;A\times S^{d-1})$, $k\in\{0,\ldots,d\}$: \begin{align} \begin{pmatrix}\hat{\Phi}_{d}^{r,s}(K_0;A\times S^{d-1})\\ \vdots \\ \hat{\Phi}_{0}^{r,s}(K_0;A\times S^{d-1}) \end{pmatrix} =\left(A_{R_0,\ldots,R_d}^{r,s}\right)^{-1} \begin{pmatrix} \mathcal{V}_{R_0}^{r,s}(K_0;A)\\ \vdots \\ \mathcal{V}_{R_d}^{r,s}(K_0;A) \end{pmatrix}\label{defEst} \end{align} with $A_{R_0,\ldots,R_d}^{r,s}$ given by \eqref{matrixA}. Setting $A=\mathbb{R}^d$ in \eqref{defEst}, we obtain estimators \[ \hat{\Phi}_{k}^{r,s}(K_0)=\hat{\Phi}_{k}^{r,s}(K_0;\mathbb{R}^d\times S^{d-1}) \] of the {intrinsic volumes}. Note that this approach requires an estimate for the reach of $K$ because we need to choose $0<R_0<\dots<R_d <\reach(K)$. The idea to invert the Steiner formula is not new. It was used in \cite{chazal} to approximate curvature measures of sets of positive reach. In \cite{spodarev} and \cite{jan} it was used to estimate intrinsic volumes but without proving convergence for the resulting estimator. We now consider the case where $K_0$ is finite. Let \begin{equation*} V_x(K_0)=\{y\in \mathbb{R}^d \mid p_{K_0}(y)=x\} \end{equation*} denote the Voronoi cell of $x\in K_0$ with respect to the set $K_0$. Since $\mathbb{R}^d$ is the union of the finitely many Voronoi cells of $K_0$, it follows that $K^R_0$ is the union of the $R$-bounded parts $B(x,R)\cap V_x(K_0)$, $x\in K_0$, of the Voronoi cells $V_x(K_0)$, $x\in K_0$, which have pairwise disjoint interiors. Thus \eqref{star} simplifies to \begin{equation}\label{algorithm} \mathcal{V}_{R}^{r,s}(K_0;A)= \sum_{x\in K_0\cap A } x^r \int_{B(x,R)\cap V_x(K_0)} (y-x)^s \, dy. \end{equation} Like the Voronoi covariance measure, the Voronoi tensor measure $\mathcal{V}_{R}^{r,s}(K_0;A)$ is a sum of simple contributions from the individual Voronoi cells. An example of a Voronoi decomposition associated with a digital image is sketched in Figure~\ref{redblue}. The original set $K$ is the disk bounded by the inner black circle, and the disk bounded by the outer black circle is its $R$-parallel set $K^R$. The finite point sample is $K_0 = K \cap \mathbb{Z}^2$, which is shown as the set of red dots in the picture, and the red curve is the boundary of its $R$-parallel set. The Voronoi cells of $K_0$ are indicated by blue lines. The $R$-bounded part of one of the Voronoi cells is the part that is cut off by the red arc. \begin{figure} \caption{The Voronoi decomposition (blue lines) and $R$-parallel set (red curve) associated with a digital image.} \label{redblue} \end{figure} \subsection{The case of intrinsic volumes}\label{intvol} Recall that $\Phi_k^{0,0}(K)=\mathbb{L}mbda_k(K;\mathbb{R}^d)$ is the $k$th intrinsic volume. Thus, Section \ref{finite} provides estimators for all intrinsic volumes as a special case. This case is particularly simple. The measure $\mathcal{V}_{R}^{0,0}(K;A)$ is simply the volume of a local parallel set \begin{align*} \mathcal{V}_{R}^{0,0}(K;A){}&=\mathcal{H}^d\left(\{ x \in K^R\mid p_K(x) \in A\}\right),\\ \mathcal{V}_{R}^{0,0}(K){}&=\mathcal{H}^d( K^R). \end{align*} In particular, if $K\subseteq \mathbb{R}^d$ is a compact set with $\reach(K)>R$, then Equation~\eqref{steiner} reduces to the usual local Steiner formula \begin{align*} \mathcal{H}^d( \{ x \in K^R\mid p_K(x) \in A\})&= \sum_{k=0}^d \kappa_{k} R^{k} \mathbb{L}mbda_{d-k}(K;A\times S^{d-1}), \end{align*} and to the (global) Steiner formula \eqref{gloSt} if $A=\mathbb{R}^d$. In this case, our algorithm approximates the parallel volume $\mathcal{H}^d(K^R)$ by $\mathcal{H}^d(K_0^R)$. In the example in Figure \ref{redblue}, this corresponds to approximating the volume of the larger black disk by the volume of the region bounded by the red curve. This volume is again the sum of the volumes of the regions bounded by the red and blue curves. In other words, it is the sum of volumes of the $R$-bounded Voronoi cells on the right-hand side of the equation \begin{equation*} \mathcal{V}_{R}^{0,0}(K_0;A)= \sum_{x\in K_0\cap A } \mathcal{H}^d(B(x,R) \cap V_x(K_0)). \end{equation*} \subsection{Estimators for general local Minkowski tensors}\label{general1} In Section \ref{finite} we have only considered estimators for local tensors of the form $\Phi_k^{r,s}(K;A \times S^{d-1})$, where $K\subseteq\mathbb{R}^d$ is a set with positive reach. The natural way to estimate $\Phi_k^{r,s}(K;B)$, for a measurable set $B\subseteq \Sigma $, would be to copy the idea in Section \ref{finite} with $\mathcal{V}_{R}^{r,s}(K;A)$ replaced by the following generalization of the Voronoi tensor measures, \begin{equation}\label{baddef} \mathcal{W}_{R}^{r,s}(K;B) = \int_{K^R \backslash K}\mathds{1}_B(p_K(x),u_K(x))p_K(x)^r (x-p_K(x))^s \, dx, \end{equation} where $u_K(x) = ({x-p_K(x)})/{|x-p_K(x)|}$ estimates the normal direction. Of course, this definition works for any $K\in\mathcal{C}^d$. Moreover, we could define estimators related to \eqref{baddef} whenever we have a set $K_0$ which approximates $K$. However, even if $K$ has positive reach, the map $x\mapsto u_K(x)$ is not Lipschitz on $K^R\backslash K$, and therefore the convergence results in Section \ref{convergence} will not work with this definition. Since the map $x\mapsto u_K(x)$ is Lipschitz on $K^R\backslash K^{R/2}$, it is natural to proceed as follows. For any $K\in\mathcal{C}^d$, we define \begin{align}\label{modify} \overline{\mathcal{V}}_{R}^{r,s}(K;B) {}&= \int_{K^R \backslash K^{R/2}}\mathds{1}_B(p_K(x),u_K(x))p_K(x)^r (x-p_K(x))^s \, dx. \end{align} Note that \begin{equation}\label{Wdifference} \overline{\mathcal{V}}_{R}^{r,s}(K;\cdot)=\mathcal{W}_{R}^{r,s}(K;\cdot)-\mathcal{W}_{R/2}^{r,s}(K;\cdot), \end{equation} where $\mathcal{W}_{R}^{r,s}(K;\cdot)$ is defined as in \eqref{baddef}. We will not use the notation $\mathcal{W}_{R}^{r,s}(K;\cdot)$ in the following. If $K$ has positive reach and $0<R<\text{reach}(K)$, then the generalized Steiner formula yields \begin{align*} {\overline{\mathcal{V}}_{R}^{r,s}(K;B)} &= r!s! \sum_{k=1}^d \kappa_{s+k} R^{s+k} (1-2^{-(s+k)}){ {\Phi}_{d-k}^{r,s}(K;B)}. \end{align*} Again, choosing $0<R_1<\ldots<R_d<\text{reach}(K)$, we can recover the Minkowski tensors from \begin{align*} \begin{pmatrix}{\Phi}_{d-1}^{r,s}(K;B)\\ \vdots \\ {\Phi}_{0}^{r,s}(K;B) \end{pmatrix} = \left( \overline{A}_{R_1,\ldots,R_d}^{r,s} \right)^{-1} \begin{pmatrix} \overline{\mathcal{V}}_{R_1}^{r,s}(K;B)\\ \vdots \\ \overline{\mathcal{V}}_{R_d}^{r,s}(K;B) \end{pmatrix} \end{align*} where \[ \overline {A}_{R_1,\ldots,R_d}^{r,s}= \frac{1}{r!s!} \begin{pmatrix} \kappa_{s+1} (1-2^{-(s+1)}) R_1^{s+1} & \dots & \kappa_{s+d}(1-2^{-(s+d)})R_1^{s+d} \\ \vdots & & \vdots \\ \kappa_{s+1} (1-2^{-(s+1)}) R_{d}^{s+1} & \dots & \kappa_{s+d}(1-2^{-(s+d)})R_{d}^{s+d} \end{pmatrix} \] is a regular matrix. Using this, we can define estimators for ${\Phi}_{k}^{r,s}(K;B)$, for $0\le k\leq d-1$, by \begin{align*} \begin{pmatrix}\overline{\Phi}_{d-1}^{r,s}(K_0;B)\\ \vdots \\ \overline{\Phi}_{0}^{r,s}(K_0;B) \end{pmatrix} = \left( \overline{A}_{R_1,\ldots,R_d}^{r,s} \right)^{-1} \begin{pmatrix} \overline{\mathcal{V}}_{R_1}^{r,s}(K_0;B)\\ \vdots \\ \overline{\mathcal{V}}_{R_d}^{r,s}(K_0;B) \end{pmatrix}, \end{align*} where $K_0$ is a compact set which approximates $K$. Convergence of these modified estimators will be discussed in Section \ref{convergence}. The estimators $\overline{\Phi}_{k}^{r,s}$ can be used to approximate local tensors of the form $\Phi_k^{r,s}(K;B)$ where the set $B\subseteq \Sigma$ involves normal directions. Thus, they are more general than $\hat{\Phi}_{k}^{r,s}$. However, \eqref{Wdifference} shows that estimating $\overline{\mathcal{V}}_{R}^{r,s}(K;B)$ requires an approximation of two parallel sets, rather than one. We therefore expect more severe numerical errors for $\overline{\Phi}_{k}^{r,s}$. \section{Convergence properties}\label{convergence} In this section we prove the main convergence results. This is an immediate generalization of \cite[Theorem 5.1]{merigot}. \subsection{The convergence theorem} For a bounded Lipschitz function $f:\mathbb{R}^d \to \mathbb{R}$, we let $|f|_\infty$ denote the usual supremum norm, \begin{equation*} |f|_L = \sup \bigg\{ \frac{|f(x)-f(y)|}{|x-y|} \mid x\neq y\bigg\} \end{equation*} the Lipschitz semi-norm, and \begin{equation*} |f|_{bL}=|f|_L + |f|_\infty \end{equation*} the bounded Lipschitz norm. Let $d_{bL} $ be the bounded Lipschitz metric on the space of bounded $\mathbb{T}^p$-valued Borel measures on $\mathbb{R}^d$. For any two such measures $\mu$ and $\nu$ on $\mathbb{R}^d$, the distance with respect to $d_{bL}$ is defined by \begin{equation*} d_{bL}(\mu,\nu) = \sup \bigg\{\bigg|\int f \, d\mu - \int f \, d\nu\bigg| \mid |f|_{bL} \leq 1\bigg\}, \end{equation*} where the supremum extends over all bounded Lipschitz functions $f:\mathbb{R}^d\to \mathbb{R}$ with $ |f|_{bL} \leq 1$. The following theorem shows that the map \begin{equation*} K \mapsto \mathcal{V}_{R}^{r,s}(K;\cdot) \end{equation*} is H\"{o}lder continuous with exponent $ \frac{1}{2}$ with respect to the Hausdorff metric on $\mathcal{C}^d$ (restricted to compact subsets of a fixed ball) and the bounded Lipschitz metric. In the proof, we use the symmetric difference $A\Delta B=(A\setminus B)\cup(B\setminus A)$ of sets $A,B\subseteq \mathbb{R}^d$. \begin{thm}\label{converge} Let $R,\rho>0$ and $r,s\in {\mathbb N}_0$ be given. Then there is a positive constant $C_2=C_2(d,R,\rho,r,s)$ such that \begin{align*} d_{bL}(\mathcal{V}_{R}^{r,s}(K;\cdot),\mathcal{V}_{R}^{r,s}(K_0;\cdot )) \leq C_2 d_H(K,K_0)^{\frac{1}{2}} \end{align*} for all compact sets $K,K_0\subseteq B(0,\rho)$. \end{thm} \begin{proof} Let $f$ with $|f|_{bL} \leq 1$ be given. Then \eqref{integralf} yields \begin{align}\nonumber &\bigg|\int_{\mathbb{R}^d} f(x)\, \mathcal{V}_{R}^{r,s}(K;dx)-\int_{\mathbb{R}^d} f(x)\, \mathcal{V}_{R}^{r,s}(K_0;dx)\bigg|\\ &=\bigg|\int_{K^R }f(p_K(x))\,p_K(x)^r(x-p_K(x))^s \, dx\nonumber\\ &\qquad\qquad\qquad\qquad-\int_{K_0^R }f(p_{K_0}(x))p_{K_0}(x)^r(x-p_{K_0}(x))^s \, dx\bigg| \nonumber \\ &\leq {I}+{II},\label{AB} \end{align} where $I$ is the integral \begin{align*} \int_{K^R \cap K_0^R }|f(p_K(x))p_K(x)^r(x-p_K(x))^s-f(p_{K_0}(x))\,p_{K_0}(x)^r(x-p_{K_0}(x))^s |\,dx \end{align*} and \begin{align*} II={{\rho}^r}R^s \mathcal{H}^{d}(K^R \Delta K_0^R). \end{align*} By \cite[Corollary 4.4]{chazal}, there is a constant $c_1=c_1(d,R,\rho)>0$ such that \begin{equation}\label{symdif} \mathcal{H}^{d}(K^R \Delta K_0^R) \leq c_1\,d_H(K,K_0) \end{equation} when $d_H(K,K_0)\leq {R}/{2}$. Replacing $c_1$ by a possibly even bigger constant, we can ensure that \eqref{symdif} also holds when $R/2\le d_H(K,K_0)\leq 2 \rho$. Hence, \begin{equation}\label{B} II \leq {c_2}\,d_H(K,K_0)^{\frac 12} \end{equation} with some constant $c_2=c_2(d,R,\rho,r,s)>0$. Using the inequalities (and interpreting empty products as 1) \begin{align}\label{product} \bigg|\bigodot_{i=1}^m y_i - \bigodot_{i=1}^m z_i\bigg|\leq \bigg|\bigotimes_{i=1}^m y_i - \bigotimes_{i=1}^m z_i\bigg|\leq \sum_{j=1}^m |y_j-z_j|\textrm{pr}od_{i=1}^{j-1} |y_i| \textrm{pr}od_{i=j+1}^m |z_i|, \end{align} with $m=r+s$ and the rank-one tensors \[ \begin{array}{lcl} y_1=\ldots=y_r=p_K(x), &\qquad& y_{r+1}=\ldots=y_{r+s}=x-p_K(x),\\ z_1=\ldots=z_r=p_{K_0}(x), &\qquad& z_{r+1}=\ldots=z_{r+s}=x-p_{K_0}(x), \end{array} \] we get \begin{align*} |f{}&(p_K(x))\, p_K(x)^r(x-p_K(x))^s-f(p_{K_0}(x))\,p_{K_0}(x)^r(x-p_{K_0}(x))^s | \\ &\leq |f(p_K(x))-f(p_{K_0}(x)) | |p_K(x)|^{r} |x-p_{K}(x)|^s \\ &+|f(p_{K_0}(x))| \sum_{j=1}^r |p_K(x)-p_{K_0}(x)||p_K(x)|^{j-1} |p_{K_0}(x)|^{r-j}|x-p_{K_0}(x)|^s + \\ &+|f(p_{K_0}(x))| \sum_{j=1}^s |p_K(x)-p_{K_0}(x)||p_K(x)|^{r} |x-p_{K}(x)|^{j-1}|x-p_{K_0}(x)|^{s-j}. \end{align*} Since we assumed that $|f|_{bL}\le 1$, we get \begin{align}\nonumber I&\leq (r+s+1)\max\{\rho,1\}^r\max\{R,1\}^s\int_{K^R \cap K_0^R } |p_K(x)-p_{K_0}(x)|\, dx\\ &\leq c_3\, d_H(K,K_0)^{\frac{1}{2}}.\label{A} \end{align} The existence of the constant $c_3=c_3(d,R,\rho,r,s)$ in the last inequality is guaranteed by Proposition \ref{CHAZProp} with $K^R \cap K_0^R$ as the set $E$, because this choice of $E$ satisfies $\diam(E \cup \{0\})\leq 2(\rho + R)$. \end{proof} When $r=s=0$ and $f=1$, the above proof simplifies to Inequality \eqref{symdif} as $I$ vanishes. Hence we obtain the following strengthening of the theorem, which is relevant for the estimation of intrinsic volumes. \begin{thm}\label{IVconverge} Let $R,\rho>0$. Then there is a constant $C_3=C_3(d,R,\rho)>0$ such that \begin{equation*} \Big|\mathcal{V}_{R}^{0,0}(K)-\mathcal{V}_{R}^{0,0}(K_0) \Big|\leq C_3\, d_H(K,K_0) \end{equation*} for all compact sets $K,K_0\subseteq B(0,\rho)$. \end{thm} For local tensors, the proof of Theorem \ref{converge} can also be adapted to show a convergence result. \begin{thm}\label{locallip} Let $r,s\in\mathbb{N}_0$ and $R>0$. If $K_i \to K$ with respect to the Hausdorff metric on ${\mathcal C}^d$, as $i\to \infty$, then $\mathcal{V}_{R}^{r,s}(K_i;A)\to \mathcal{V}_{R}^{r,s}(K;A)$ in the tensor norm, for every Borel set $A\subseteq\mathbb{R}^d$ which satisfies \begin{equation}\label{4.3exceptional} \mathcal{H}^d(p_K^{-1}(\partial A)\cap K^R)=0. \end{equation} \end{thm} \begin{proof} Convergence of tensors is equivalent to coordinate-wise convergence. Hence, it is enough to show that the coordinates satisfy $$\mathcal{V}_{R}^{r,s}(K_i;A)_{i_1\dots i_{r+s}}\to \mathcal{V}_{R}^{r,s}(K;A)_{i_1\dots i_{r+s}}\qquad\text{as $i\to\infty$},$$ for all choices of indices ${i_1\dots i_{r+s}}$; see the notation at the beginning of Section~\ref{minkowski}. We write $T_K(x)=p_K(x)^r(x-p_K(x))^s$. Then \begin{equation*} \mathcal{V}_{R}^{r,s}(K;A)_{i_1\dots i_{r+s}}=\int_{K^R} \mathds{1}_A(p_K(x))T_K(x)_{i_1\dots i_{r+s}}\, dx \end{equation*} is a signed measure. Let $T_K(x)_{i_1\dots i_{r+s}}^+$ and $T_K(x)_{i_1\dots i_{r+s}}^-$ denote the positive and negative part of $T_K(x)_{i_1\dots i_{r+s}}$, respectively. Then \begin{equation*} \mathcal{V}_{R}^{r,s}(K;A)^{\pm}_{i_1\dots i_{r+s}}=\int_{K^R} \mathds{1}_A(p_K(x))T_K(x)_{i_1\dots i_{r+s}}^{\pm}\,dx \end{equation*} are non-negative measures such that \begin{equation*} \mathcal{V}_{R}^{r,s}(K;\cdot)_{i_1\dots i_{r+s}}=\mathcal{V}_{R}^{r,s}(K;\cdot)_{i_1\dots i_{r+s}}^+-\mathcal{V}_{R}^{r,s}(K;\cdot)_{i_1\dots i_{r+s}}^-. \end{equation*} The proof of Theorem \ref{converge} can immediately be generalized to show that $\mathcal{V}_{R}^{r,s}(K_i;\cdot)^{\pm}_{i_1\dots i_{r+s}}$ converges to $\mathcal{V}_{R}^{r,s}(K;\cdot)^{\pm}_{i_1\dots i_{r+s}}$ in the bounded Lipschitz norm (as $i\to\infty$), and hence the measures converge weakly. In particular, they converge on every continuity set of $\mathcal{V}_{R}^{r,s}(K;\cdot)^{\pm}_{i_1\dots i_{r+s}}$. If $\mathcal{H}^d(p_K^{-1}(\partial A)\cap K^R)=0$, then $A$ is such a continuity set. \end{proof} \begin{remark} Though relatively mild, the condition $\mathcal{H}^d(p_K^{-1}(\partial A)\cap K^R)=0$ can be hard to control if $K$ is unknown. It is satisfied if, for instance, $K$ and $A$ are smooth and their boundaries intersect transversely. A special case of this is when $K$ is a smooth surface and $A$ is a small ball centered on the boundary of $K$. This is the case in the application from \cite{merigot} that was described in the introduction. Examples where it is not satisfied are when $A=K$ or when $K$ is a polytope intersecting $\partial A$ at a vertex. \end{remark} \begin{remark}\label{Rem4.6new} Let $f:\mathbb{R}^d\to\mathbb{R}$ be a bounded measurable function. We define $$ \mathcal{V}_{R}^{r,s}(K;f):=\int_{\mathbb{R}^d} f(x)\, \mathcal{V}_{R}^{r,s}(K;dx). $$ Hence $\mathcal{V}_{R}^{r,s}(K;A)=\mathcal{V}_{R}^{r,s}(K;\mathds{1}_A)$ for every Borel set $A\subseteq\mathbb{R}^d$. Then, Theorem \ref{locallip} is equivalent to saying that, for all continuous test functions $f:\mathbb{R}^d\to\mathbb{R}$, $$ \mathcal{V}_{R}^{r,s}(K_i;f)\to \mathcal{V}_{R}^{r,s}(K;f),\quad \text{as }i\to\infty, $$ in the tensor norm, whenever $K_i \to K$ with respect to the Hausdorff metric on ${\mathcal C}^d$, as $i\to \infty$. Thus, if one is interested in the local behaviour of $\Phi^{r,s}_k(K; \cdot)$ at a neighborhood $A$, like in \cite{merigot}, then one can study $$ \Phi^{r,s}_k(K;f):=\int_{\Sigma} f(x)x^ru^s\, \mathbb{L}mbda_k(K;d(x,u)), $$ where $f$ is a continuous function with support in $A$. This avoids the extra condition \eqref{4.3exceptional}. \end{remark} As the matrix $A_{R_0,\ldots,R_d}^{r,s}$ in the definition \eqref{defEst} of $\hat \Phi_k^{r,s}(K_0;A\times S^{d-1})$ does not depend on the set $K_0$, the above results immediately yield a consistency result for the estimation of the Minkowski tensors. We formulate this only for $A=\mathbb{R}^d$. \begin{corollary}\label{corNew} Let $\rho>0$ and $K$ be a compact subset of $B(0,\rho)$ of positive reach such that $\mathrm{Reach}(K)>R_d>\ldots>R_0>0$. Let $K_0\subseteq B(0,\rho)$ be a compact set. Then there is a constant $C_4=C_4(d,R_0,\ldots,R_d,\rho)$ such that \[ \left| \hat{\Phi}^{0,0}_k(K_0)-\Phi^{0,0}_k(K)\right|\le C_4\, d_H(K_0,K), \] for all $k\in\{0,\ldots,d\}$. For $r,s\in {\mathbb N}_0$ there is a constant $C_5=C_5(d,R_0,\ldots,R_d,\rho,r,s)$ such that \[ \left| \hat{\Phi}^{r,s}_k(K_0)-\Phi^{r,s}_k(K)\right|\le C_5\, d_H(K_0,K)^{\frac12}, \] for all $k\in\{0,\ldots,d-1\}$. \end{corollary} Finally, we state the convergence results for the modified estimators for $\Phi_k^{r,s}(K;B)$, where $B\subseteq \Sigma$ is a Borel set, that were defined in Section \ref{general1}. The map $x\mapsto {x}/{|x|}$ is Lipschitz on $\mathbb{R}^d \backslash \indre({B(0,{R}/{2})})$ with Lipschitz constant ${4}/{R}$, and therefore the mapping $u_K$, which was defined after \eqref{baddef}, satisfies \begin{equation*} |u_K(x)-u_{K_0}(x)|\leq \tfrac{4}{R}|p_K(x)-p_{K_0}(x)|, \end{equation*} for $x\in (K^R \backslash K^{R/2}) \cap(K_0^R \backslash K_0^{R/2})$. Moreover, \begin{equation*} \left(K^R \backslash K^{R/2}\right) \Delta \left(K_0^R \backslash K_0^{R/2}\right) \subseteq \left(K^R \Delta K_0^{R}\right) \cup \left(K^{R/2} \Delta K_0^{R/2}\right). \end{equation*} Using this, it is straightforward to generalize the proofs of Theorems \ref{converge} and \ref{locallip} to obtain the following result. \begin{thm}\label{convergeloc2} Let $R,\rho>0$ and $r,s\in {\mathbb N}_0$ be given. Then there is a positive constant $C_6=C_6(d,R,\rho,r,s)$ such that \begin{align*} d_{bL}(\overline{\mathcal{V}}_{R}^{r,s}(K;\cdot),\overline{\mathcal{V}}_{R}^{r,s}(K_0;\cdot )) \leq C_6 d_H(K,K_0)^{\frac{1}{2}} \end{align*} for all compact sets $K,K_0\subseteq B(0,\rho)$. \end{thm} This in turn leads to the next convergence result. \begin{thm} Let $r,s\in\mathbb{N}_0$ and $R>0$. If $K,K_i\in \mathcal{C}^d$ are compact sets such that $K_i\to K$ in the Hausdorff metric, as $i\to\infty$, then $\overline{\mathcal{V}}_{R}^{r,s}(K_i;B)$ converges to $\overline{\mathcal{V}}_{R}^{r,s}(K;B)$ in the tensor norm, for any measurable set $B\subseteq \Sigma$ satisfying \begin{equation*} \mathcal{H}^d(\{x\in K^R \mid (p_K(x), u_K(x))\in \partial B\})=0. \end{equation*} Here $\partial B$ is the boundary of $B$ as a subset of $\Sigma$. If $B$ satisfies this condition and $\text{Reach}(K)>R_d$, then \begin{equation*} \lim_{i \to 0} \overline{\Phi}_{k}^{r,s}(K_i;B) = {\Phi_{k}^{r,s}(K;B)} . \end{equation*} \end{thm} \begin{remark} We can argue as in Remark \ref{Rem4.6new} to see that if $K,K_i\in \mathcal{C}^d$ are compact sets such that $K_i\to K$ in the Hausdorff metric, as $i\to\infty$, then $$ \overline{\mathcal{V}}_{R}^{r,s}(K_i;g)\to \overline{\mathcal{V}}_{R}^{r,s}(K;g),\quad \text{as }i\to\infty, $$ whenever $g:\Sigma\to\mathbb{R}$ is a continuous test function and $\overline{\mathcal{V}}_{R}^{r,s}(K;g)$ is defined similarly as before. If $K$ satisfies $\text{Reach}(K)>R_d$, we get $\overline{\Phi}_{k}^{r,s}(K_i;g) \to {\Phi_{k}^{r,s}(K;g)}$, as $i\to\infty$. \end{remark} \section{Application to digital images}\label{DI} Our main motivation for this paper is the estimation of Minkowski tensors from digital images. Recall that we model a black-and-white digital image of $K\subseteq \mathbb{R}^d$ as the set $K\cap a\mathbb{L}$, where $\mathbb{L}\subseteq \mathbb{R}^d$ is a fixed lattice and $a>0$. We refer to \cite{barvinok02} for basic information about lattices. The lower dimensional parts of $K$ are generally invisible in the digital image. When dealing with digital images, we will therefore always assume that the underlying set is topologically regular, which means that it is the closure of its own interior. In digital stereology, the underlying object $K$ is often assumed to belong to one of the following two set classes: \begin{itemize} \item $K$ is called \emph{$\delta$-regular} if it is topologically regular and the reach of its closed complement ${\rm cl}({\mathbb{R}^d \backslash K})$ and the reach of $K$ itself are both at least $\delta>0$. This is a kind of smoothness condition on the boundary, ensuring in particular that $\partial K$ is a $C^1$ manifold (see the discussion after Definition 1 in \cite{svane15b}). \item $K$ is called \emph{polyconvex} if it is a finite union of compact convex sets. While convex sets have infinite reach, note that polyconvex sets do generally not have positive reach. Also note that for a compact convex set $K\subseteq\mathbb{R}^d$, the set ${\rm cl}({\mathbb{R}^d \backslash K})$ need not have positive reach. \end{itemize} It should be observed that for a compact set $K\subseteq \mathbb{R}^d$ both assumptions imply that the boundary of $K$ is a $(d-1)$-rectifiable set in the sense of \cite{Federer69} (i.e., $\partial K$ is the image of a bounded subset of $\mathbb{R}^{d-1}$ under a Lipschitz map), which is a much weaker property that will be sufficient for the analysis in Section \ref{volten}. \subsection{The volume tensors}\label{volten} Simple and efficient estimators for the volume tensors $\Phi_d^{r,0}(K)$ of a (topologically regular) compact set $K$ are already known and are usually based on the approximation of $K$ by the union of all pixels (voxels) with midpoint in $K$. This leads to the estimator \begin{equation*} \phi_d^{r,0}(K\cap a\mathbb{L} ) = \frac1 {r!} \sum_{z \in K\cap a\mathbb{L}} \int_{z+aV_0(\mathbb{L})}x^r\,dx, \end{equation*} where $V_0(\mathbb{L})$ is the Voronoi cell of 0 in the Voronoi decomposition generated by $\mathbb{L}$. This, in turn, can be approximated by \begin{equation*} \hat{\phi}_d^{r,0}(K\cap a\mathbb{L} ) = \frac{a^{d}}{r!} \mathcal{H}^d\left(V_0(\mathbb{L})\right) \sum_{z \in K\cap a\mathbb{L}} z^r. \end{equation*} When $r\in \{0,1\}$, we even have ${\phi}_d^{r,0}(K\cap a\mathbb{L} )=\hat{\phi}_d^{r,0}(K\cap a\mathbb{L} )$. Choose $C>0$ such that $V_0(\mathbb{L}) \subseteq B(0,C)$. Then $$ K\Delta \bigcup_{z\in K\cap a\mathbb{L}} (z+aV_0(\mathbb{L}))\subseteq (\partial K)^{ aC}. $$ In fact, if $x\in \left[\bigcup_{z\in K\cap a\mathbb{L}} (z+aV_0(\mathbb{L}))\right]\setminus K$, then there is some $z\in K\cap a\mathbb{L}$ such that $x\in z+aV_0(\mathbb{L})$ and $x\notin K$. Since $z\in K$ and $x\notin K$, we have $[x,z]\cap\partial K\neq\emptyset$. Moreover, $x-z\in aV_0(\mathbb{L})\subseteq B(0,aC)$, and hence $|x-z|\le aC$. This shows that $x\in(\partial K)^{aC}$. Now assume that $x\in K$ and $x\notin (\partial K)^{aC}$. Then $B(x,\rho)\subseteq K$ for some $\rho>aC$. Since $\bigcup_{z\in a\mathbb{L}}(z+aV_0(\mathbb{L}))=\mathbb{R}^d$, there is some $z\in a\mathbb{L}$ such that $x\in z+aV_0(\mathbb{L})$. Hence $x-z\in aV_0(\mathbb{L})\subseteq B(0,aC)$. We conclude that $z\in B(x,aC)\subseteq K$, therefore $z\in K\cap a\mathbb{L}$ and thus $x\in \bigcup_{z\in K\cap a\mathbb{L}} (z+aV_0(\mathbb{L}))$. Hence \begin{equation}\label{Oabound} |{\phi}_d^{r,0}(K\cap a\mathbb{L} ) - {\Phi}_d^{r,0}(K)| \leq \frac{1} {r!} \int_{(\partial K)^{ aC}}|x|^r \, dx. \end{equation} If $\mathcal{H}^{d}(\partial K)=0$, then the integral on the right-hand side goes to zero by monotone convergence, so \begin{equation}\label{convzero} \lim_{a\to 0_+}{\phi}_d^{r,0}(K\cap a\mathbb{L} ) ={\Phi}_d^{r,0}(K). \end{equation} If $\partial K$ is $(d-1)$-rectifiable in the sense of \cite[Section 3.2.14]{Federer69}, that is, $\partial K$ is the image of a bounded subset of $\mathbb{R}^{d-1}$ under a Lipschitz map, then $\mathcal{H}^{d}(\partial K)=0$. Since $\partial K$ is compact, \cite[Theorem 3.2.39]{Federer69} implies that $\lim_{a\to 0_+}\mathcal{H}^d((\partial K)^{ aC})/a $ exists and equals a fixed multiple of $\mathcal{H}^{d-1}(\partial K)$ which is finite. Hence, \eqref{Oabound} shows that the speed of convergence in \eqref{convzero} is $O(a)$ as $a\to 0_+$. Inequality \eqref{product} yields that $|x^r-z^{r}|\leq aC r(|x|+aC)^{r-1}$ whenever $x\in z+ aV_0(\mathbb{L})$ and $r\ge 1$. Therefore, \begin{align*} |\hat{\phi}_d^{r,0}(K\cap a\mathbb{L} ) - \phi_d^{r,0}(K\cap a \mathbb{L})|{}& \leq \frac{aC } {(r-1)!} \sum_{z \in K\cap a\mathbb{L}} \int_{z+aV_0(\mathbb{L})}(|x|+aC)^{r-1}\, dx\\ & \leq \frac{aC } {(r-1)!} \int_{K^{aC}} (|x|+aC)^{r-1} \,dx, \end{align*} which shows that \begin{equation*} \lim_{a\to 0_+}\hat{\phi}_d^{r,0}(K\cap a\mathbb{L} ) ={\Phi}_d^{r,0}(K), \end{equation*} provided that $\mathcal{H}^d(\partial K)=0$. If $\partial K$ is $(d-1)$-rectifiable, then the speed of convergence is of the order $O(a)$. Hence, we suggest to simply use the estimators $\hat{\phi}_d^{r,0}(K\cap a\mathbb{L} )$ for the volume tensors. This estimator can be computed much faster and more directly than $\hat{\Phi}_d^{r,0}(K\cap a\mathbb{L} )$. Moreover, it does not require an estimate for the reach of $K$, and it converges for a much larger class of sets than those of positive reach. \subsection{Convergence for digital images} For the estimation of the remaining tensors we suggest to use the Voronoi tensor measures. Choosing $K_0=K \cap a\mathbb{L}$ in \eqref{algorithm}, we obtain \begin{equation}\label{algorithm2} \mathcal{V}_{R}^{r,s}(K\cap a\mathbb{L} ;A)= \sum_{x\in K \cap a\mathbb{L} \cap A } x^r \int_{B(x,R)\cap V_x(K\cap a\mathbb{L})} (y-x)^s \,dy, \end{equation} where $A\subseteq\mathbb{R}^d$ is a Borel set. To show some convergence results in Corollary \ref{convercor} below, we first note that the digital image converges to the original set in the Hausdorff metric. \begin{lemma}\label{dHbounds} If $K$ is compact and topologically regular, then \begin{equation*} \lim_{a\to 0_+} d_H(K,K\cap a\mathbb{L}) = 0. \end{equation*} If $K $ is $\delta$-regular, then $d_H(K,K\cap a\mathbb{L})$ is of order $O(a)$. The same holds if $K$ is topologically regular and polyconvex. \end{lemma} \begin{proof} Recall from \cite[p.~311]{barvinok02} that $ \mu(\mathbb{L})=\max_{x\in\mathbb{R}^d}\text{dist}(x,\mathbb{L}) $ is well defined and denotes the covering radius of $\mathbb{L}$. Let $\varepsilon>0$ be given. Since $K$ is compact, there are points $x_1,\ldots,x_m\in K$ such that $$ K\subseteq\bigcup_{i=1}^m B(x_i,\varepsilon). $$ Using the fact that $K$ is topologically regular, we conclude that there are points $y_i\in\text{int}(K)\cap \text{int}(B(x_i,2\varepsilon))$ for $i=1,\ldots,m$. Hence, there are $\varepsilon_i\in (0,2\varepsilon)$ such that $ B(y_i,\varepsilon_i)\subseteq K\cap B(x_i,2\varepsilon)$ for $i=1,\ldots,m$. Let $0<a<\min\{\varepsilon_i/\mu(\mathbb{L}) \mid i=1,\ldots,m\}$. Since $\varepsilon_i/a>\mu(\mathbb{L})$ it follows that $a\mathbb{L} \cap B(y_i,\varepsilon_i)\neq\emptyset$, for $i=1,\ldots,m$. Thus we can choose $z_i\in a\mathbb{L} \cap B(y_i,\varepsilon_i)\subseteq a\mathbb{L}\cap K$ for $i=1,\ldots,m$. By the triangle inequality, we have $|z_i-x_i|\le \varepsilon_i+2\varepsilon\le 4\varepsilon$, and hence $x_i\in (K\cap a\mathbb{L})+B(0,4\varepsilon)$, for $i=1,\ldots,m$. Therefore, $K\subseteq (K\cap a\mathbb{L}) +B(0,5\varepsilon)$ if $a>0$ is sufficiently small. Assume that $K$ is $\delta$-regular, for some $\delta>0$. We choose $0<a<\delta/(2\mu(\mathbb{L}))$. Since $a\mu(\mathbb{L})<\delta/2$, for any $x\in K$ there is a ball $B(y,a\mu(\mathbb{L}))$ of radius $a\mu(\mathbb{L})$ such that $x\in B(y,a\mu(\mathbb{L}))\subseteq K$. From $a\mathbb{L}\cap B(y,a\mu(\mathbb{L}))\neq\emptyset$ we conclude that there is a point $z\in K\cap a\mathbb{L}$ with $|x-z|\le 2a\mu(\mathbb{L})$. Hence $x\in (K\cap a\mathbb{L}) +B(0,2a\mu(\mathbb{L}))$, and therefore $d_H(K,K\cap a\mathbb{L})\le 2a\mu(\mathbb{L})$. Finally, we assume that $K$ is topologically regular and polyconvex. Then $K$ is the union of finitely many compact convex sets with interior points. Hence, for the proof we may assume that $K$ is convex with $B(0,\rho)\subseteq K$ for a fixed $\rho>0$. Choose $0<a<\rho/(2\mu(\mathbb{L}))$ and put $r=2a\mu(\mathbb{L})<\rho$. If $x\in K$, then $B((1-r/\rho)x,r)\subseteq K$ and $B((1-r/\rho)x,r)$ contains a point $z\in a\mathbb{L}$. Since $$ |x-z|\le r+({r}/{\rho})|x|\le 2a\mu(\mathbb{L})\left(1+\text{diam}(K)/\rho\right), $$ we get $$ K\subseteq (K\cap a\mathbb{L}) +B\big(0,2a\mu(\mathbb{L})\left(1+\text{diam}(K)/\rho \right)\big), $$ which completes the argument. \end{proof} Thus Theorems \ref{converge} and \ref{IVconverge} and Corollary \ref{corNew} together with Lemma \ref{dHbounds} yield the following result. \begin{corollary}\label{convercor} If $K $ is compact and topologically regular, then \begin{align*} &\lim_{a\to 0_+} d_{bL}(\mathcal{V}_{R}^{r,s}(K;\cdot),\mathcal{V}_{R}^{r,s}(K\cap a\mathbb{L};\cdot)) = 0,\\ &\lim_{a\to 0_+} \mathcal{V}_{R}^{r,s}(K\cap a\mathbb{L}) = \mathcal{V}_{R}^{r,s}(K). \end{align*} If, in addition, $K$ has positive reach, then \begin{align}\label{multigrid} &\lim_{a\to 0_+} \hat{\Phi}^{r,s}_k(K\cap a\mathbb{L}) = {\Phi}^{r,s}_k(K). \end{align} If $K$ is $\delta$-regular or a topologically regular convex set, then the speed of convergence is $O(a)$ when $r=s=0$ and $O(\sqrt{a})$ otherwise. \end{corollary} The property \eqref{multigrid} means that $\hat{\Phi}^{r,s}_k(K\cap a\mathbb{L})$ is multigrid convergent for the class of sets of positive reach as defined in the introduction. A similar statement about local tensors, but without the speed of convergence, can be made. We omit this here. \subsection{Possible refinements of the algorithm for digital images}\label{refinement} We first describe how the number of necessary radii $R_0<R_1<\ldots<R_d$ in \eqref{defEst} can be reduced by one if $s=0$ and $A=\mathbb{R}^d$. Setting $s=0$ and $A=\mathbb{R}^d$ and subtracting $(r!)\Phi_d^{r,0}(K)$ on both sides of Equation \eqref{steiner} yields \begin{align}\label{modstein} \int_{K^R\backslash K} p_K(x)^r \,dx = \mathcal{V}_{R}^{r,0}(K)-(r!)\Phi_d^{r,0}(K) = (r!) \sum_{k=1}^d \kappa_{k} R^{k} \Phi_{d-k}^{r,0}(K). \end{align} As mentioned in Section \ref{volten}, the volume tensor $\Phi_d^{r,0}(K)$ can be estimated by $\hat{\phi}_d^{r,0}(K\cap a\mathbb{L})$. We may take $\mathcal{V}_{R}^{r,0}(K\cap a\mathbb{L})-(r!)\hat{\phi}_d^{r,0}(K\cap a\mathbb{L})$ as an improved estimator for \eqref{modstein}. This corresponds to replacing the integration domains $B(x,R)\cap V_x(K\cap a\mathbb{L})$ in \eqref{algorithm2} by \[ (B(x,R)\cap V_x(K\cap a\mathbb{L}))\backslash V_x(a\mathbb{L}). \] This makes sense since $V_x(a\mathbb{L})$ is likely to be contained in $K$ while the left-hand side of \eqref{modstein} is an integral over $K^R\backslash K$. The Minkowski tensors can now be isolated from only $d$ equations of the form \eqref{modstein} with $d$ different values of $R$. We now suggest a slightly modified estimator for the Minkowski tensors satisfying the same convergence results as $\hat{\Phi}_k^{r,s}(K\cap a\mathbb{L})$ but where the number of summands in \eqref{algorithm2} is considerably reduced. As the volume tensors can easily be estimated with the estimators in Section \ref{volten}, we focus on the tensors with $k<d$. Let $K$ be a compact set. We define the {\em Voronoi neighborhood} $N_\mathbb{L}(0)$ of $0$ to be the set of points $y\in \mathbb{L}$ such that the Voronoi cells $V_0(\mathbb{L})$ and $V_y(\mathbb{L})$ of $0$ and $y$, respectively, have exactly one common $(d-1)$-dimensional face. Similarly, for $z\in \mathbb{L}$ the Voronoi neighborhood $N_\mathbb{L}(z)$ of $z$ is defined, and thus clearly $N_\mathbb{L}(z)=z+N_\mathbb{L}(0)$. When $\mathbb{L}\subset \mathbb{R}^2$ is the standard lattice, $N_\mathbb{L}(z)$ consists of the four points in $\mathbb{L}$ that are neighbors of $z$ in the usual $4$-neighborhood \cite{OM}. Define $I(K\cap a\mathbb{L})$ to be the set of points $z\in K\cap a\mathbb{L}$ such that $N_{a\mathbb{L}}(z)\subseteq K\cap a\mathbb{L}$. The relative complement $B(K\cap a\mathbb{L})=(K\cap a\mathbb{L})\setminus I(K\cap a\mathbb{L})$ of $I(K\cap a\mathbb{L})$ can be considered as the set of lattice points in $K\cap a\mathbb{L}$ that are close to the boundary of the given set $K$. We modify \eqref{algorithm2} by removing contributions from $I(K\cap a\mathbb{L})$ and define \begin{equation}\label{algorithm3} \tilde{\mathcal{V}}_{R}^{r,s}(K\cap a\mathbb{L} ;A)= \sum_{x\in B(K \cap a\mathbb{L}) \cap A } x^r \int_{B(x,R)\cap V_x(K\cap a\mathbb{L})} (y-x)^s\, dy. \end{equation} Assuming that $K$ has positive reach, let $0<R_0<R_1<\ldots<R_d< \textrm{Reach}(K)$. We write again $K_0$ for $K\cap a\mathbb{L}$. Then we obtain the estimators \begin{align} \begin{pmatrix} {\tilde{\Phi}}_{d}^{r,s}(K_0;A\times S^{d-1})\\ \vdots \\ {\tilde{\Phi}}_{0}^{r,s}(K_0;A\times S^{d-1}) \end{pmatrix} =\left(A_{R_0,\ldots,R_d}^{r,s}\right)^{-1} \begin{pmatrix} \tilde{\mathcal{V}}_{R_0}^{r,s}(K_0;A)\\ \vdots \\ \tilde{\mathcal{V}}_{R_d}^{r,s}(K_0;A) \end{pmatrix}\label{defEstcheck} \end{align} with $A_{R_0,\ldots,R_d}^{r,s}$ given by \eqref{matrixA}. Working with $\tilde{\mathcal{V}}_{R}^{r,s}(K\cap a\mathbb{L};A)$ reduces the workload considerably. For instance, when $K$ is $\delta$-regular or polyconvex and topologically regular, the number of elements in $I(K\cap a\mathbb{L})$ increases with $a^{-d}$, whereas the number of elements in $B(K \cap a\mathbb{L})$ only increases with $a^{-(d-1)}$ as $a\to 0_+$. The set $I(K\cap a\mathbb{L})$ can be obtained from the digital image of $K$ in linear time using a linear filter. Moreover, we have the following convergence result. \begin{proposition} Let $K$ be a topologically regular compact set with positive reach and let $C$ be such that $V_0(\mathbb{L})\subseteq B(0,C)$. If $A$ is a Borel set in $\mathbb{R}^d$ and $aC<R_0<R_1<\ldots<R_d<\mathrm{Reach}(K)$ and $K_0=K\cap a\mathbb{L}$, then \[ \tilde{\Phi}_{k}^{r,s}(K_0;A\times S^{d-1})=\hat{\Phi}_{k}^{r,s}(K_0;A\times S^{d-1}) \] for all $k\in\{0,\ldots,d-1\}$, whenever $s=0$ or $s$ is odd. If $s$ is even and $k\in\{0,\ldots,d-1\}$, then \begin{equation*} \lim_{a\to 0_+} \tilde{\Phi}_{k}^{r,s}(K_0;A\times S^{d-1})=\lim_{a\to 0_+}\hat{\Phi}_{k}^{r,s}(K_0;A\times S^{d-1}). \end{equation*} \end{proposition} \begin{proof} Let $aC<R<\mathrm{Reach}(K)$. For $x\in I(K\cap a\mathbb{L})$, we have \[ B(x,R)\cap V_{x}(K\cap a\mathbb{L})=V_{x}(a\mathbb{L}), \] so the contribution of $x$ to the sum in \eqref{algorithm2} is $(s!)x^r\Phi^{s,0}_d(V_{0}(a\mathbb{L}))$. It follows that \begin{align}\label{Vred} {\mathcal{V}}_{R}^{r,s}(K\cap a\mathbb{L} ;A)-\tilde{\mathcal{V}}_{R}^{r,s}(K\cap a\mathbb{L} ;A)= (s!)\Phi^{s,0}_d(V_{0}(a\mathbb{L}))\sum_{x\in I(K\cap a\mathbb{L})\cap A}x^r. \end{align} For odd $s$ we have $\Phi^{s,0}_d(V_{0}(a\mathbb{L}))=0$, so the claim follows. For $s=0$ the right-hand side of \eqref{Vred} does not vanish, but it is independent of $R$. A combination of \[ \left(A_{R_0,\ldots,R_d}^{r,0}\right)^{-1} \begin{pmatrix} 1\\1\\ \vdots \\ 1 \end{pmatrix}= \begin{pmatrix} (r!)^{-1}\\ 0\\ \vdots \\ 0 \end{pmatrix}, \] with \eqref{Vred}, \eqref{defEst} and \eqref{defEstcheck} gives the claim. For even $s>0$, we have that $\Phi^{s,0}_d(V_{0}(a\mathbb{L}))=a^{d+s}\Phi^{s,0}_d(V_{0}(\mathbb{L}))$, while \begin{align*} \left|\sum_{x\in I(K\cap a\mathbb{L})\cap A}x^r \right| &\leq \sum_{x\in I(K\cap a\mathbb{L})}|x|^r \\ &\leq \sup_{x\in K}|x|^r\sum_{x\in I(K\cap a\mathbb{L})} \left(a^{d}{\mathcal H}^d(V_{0}(\mathbb{L}))\right)^{-1}{\mathcal H}^d(V_{x}(a\mathbb{L}))\\ &\leq \sup_{x\in K}|x|^r \cdot a^{-d}\cdot {\mathcal H}^d(V_0(\mathbb{L}))^{-1}\cdot \mathcal{H}^d(K^{aC}). \end{align*} Therefore, the expression on the right-hand side of \eqref{Vred} converges to $0$. \end{proof} It should be noted that a similar modification for $\overline \Phi_k^{r,s}$ is not necessary. In fact the modified Voronoi tensor measure \eqref{modify} with $K=K_0$ has the advantage that small Voronoi cells that are completely contained in the $R_0/2 $-parallel set of $K\cap a\mathbb{L}$ do not contribute. In particular, contributions from $I(K\cap a\mathbb{L})$ are automatically ignored when $a$ is sufficiently small. \section{Comparison to known estimators}\label{known} Most {existing} estimators of intrinsic volumes \cite{digital,lindblad,OM} and Minkowski tensors \cite{turk,mecke} are $n$-local for some $n\in \mathbb{N}$. The idea is to look at all $n\times \dotsm \times n$ pixel blocks in the image and count how many times each of the $2^{n^d}$ possible configurations of black and white points occur. Each configuration is weighted by an element of $\mathbb{T}^{r+s}$ and $\Phi^{r,s}_k(K)$ is estimated as a weighted sum of the configuration counts. It is known that estimators of this type for intrinsic volumes other than ordinary volume are not multigrid convergent, even when $K$ is known to be a convex polytope; see \cite{am3}. {It is not difficult to see that there cannot be a multigrid convergent $n$-local estimator for the (even rank) tensors $\Phi_k^{0,2s}(K)$ with $k=0,\ldots,d-1$, $s\in\mathbb{N}$, for polytopes $K$, either. In fact, repeatedly taking the trace of such an estimator would lead to a multigrid convergent $n$-local estimator of the $k$th intrinsic volume, in contradiction to \cite{am3}.} The algorithm presented in this paper is not $n$-local for any $n\in \mathbb{N}$. It is required in the convergence proof that the parallel radius $R$ is fixed while the resolution $a^{-1}$ goes to infinity. {The non-local operation in the definition of our estimator is the calculation of the Voronoi diagram.} The computation time for Voronoi diagrams of $k$ points is $O(k\log k + k^{\lfloor d/2\rfloor})$, see \cite{chazelle}, which is somewhat slower than $n$-local algorithms for which the computation time for $k$ data points is $O(k)$. The computation time can be improved by ignoring interior points as discussed in Section \ref{refinement}. The idea to base digital estimators for intrinsic volumes on an inversion of the Steiner formula as in \eqref{matrix} has occurred before in \cite{spodarev,jan}. In both references, the authors define estimators for polyconvex sets which are not necessarily of positive reach. This more ambitious aim leads to problems with the convergence. In \cite{spodarev}, the authors use a version of the Steiner formula for polyconvex sets given in terms of the Schneider index, see \cite{schneider}. Since its definition is, however, $n$-local in nature, the authors choose an $n$-local algorithm to estimate it. As already mentioned, such algorithms are not multigrid convergent. In \cite{jan}, it is used that the intrinsic volumes of a polyconvex set can, on the one hand, be approximated by those of a parallel set with small parallel radius, and on the other hand, the closed complement of this parallel set has positive reach, so that its intrinsic volumes can be computed via the Steiner formula. The authors employ a discretization of the parallel volumes of digital images, but without showing that the convergence is preserved. It is likely that the ideas of the present paper combined with the ones of \cite{jan} could be used to construct {multigrid} convergent digital algorithms for polyconvex sets. The price for this is that the notion of convergence in \cite{jan} is slightly artificial for practical purposes, requiring very small parallel radii in order to get good approximations and at the same time large radii compared to resolution. In \cite{svane}, $n$-local algorithms based on grey-valued images are suggested. They are shown to converge to the true value when the resolution {tends} to infinity. However, they only apply to surface and certain mean curvature tensors. Moreover, they are hard to apply in practice, since they require detailed information about the underlying point spread function {which specifies the representation of the object as grey-value image. If grey-value images are given, the} algorithm of the present paper could be applied to thresholded images, but there may be more efficient ways to exploit the additional information of the grey-values. \end{document}

\begin{document} \title{Avoiding observability singularities in output feedback\linebreak bilinear systems} \begin{abstract} Control-affine output systems generically present observability singularities, \mathrm emph{i.e.~} inputs that make the system unobservable. This proves to be a difficulty in the context of output feedback stabilization, where this issue is usually discarded by uniform observability assumptions for state feedback stabilizable systems. Focusing on state feedback stabilizable bilinear control systems with linear output, we use a transversality approach to provide perturbations of the stabilizing state feedback law, in order to make our system observable in any time even in the presence of singular inputs. \mathrm end{abstract} \noindent {\bf Keywords:} \begin{minipage}[t]{.8\linewidth} \flushleft Observability, Transversality theory, Output feedback, Stabilization \mathrm end{minipage} \section{Introduction} Stabilizing the state of a dynamical system to a target point is a classical problem in control theory. However, in many physical problems, only part of the state is known. Hence a state feedback can not be directly implemented. When a stabilizing state feedback exists, a commonly used idea is to apply this feedback to an estimation of the state, relying on a dynamical system called the \mathrm emph{observer}, which learns the state of the system from its dynamics and the measured output. This strategy belongs to the family of \mathrm emph{dynamic output feedback stabilization} techniques. In the deterministic setting, output feedback stabilization has been extensively studied (see \mathrm emph{e.g.} \cite{ AndrieuPraly2009,AtassiKhalil1999, Coron1994,EsfandiariKhalil1992,GauthierKupka1992, EsfandiariKhalil1993, MarconiPralyIsidori2007, TeelPraly1994, TeelPraly1995}). The \mathrm emph{observability} of a controlled system for some fixed input qualifies the ability to estimate the state using its output, and characterizes the fact that two trajectories of the system can be distinguished by their respective outputs over a given time interval. This crucial notion constitutes a field of study in itself (see \mathrm emph{e.g.} \cite{AndrieuPraly2009, Bernard-etal.2017, Gauthier_book, TW2009}). A commonly used hypothesis to achieve output feedback stabilization is the \mathrm emph{uniform} observability of the system, that is the system is observable for all possible inputs. It is well-known that a globally state feedback stabilizable system that is uniformly observable is also semi-globally output feedback stabilizable (see \mathrm emph{e.g.} \cite{EsfandiariKhalil1992, EsfandiariKhalil1993, TeelPraly1994, TeelPraly1995}). However, as shown in \cite{Gauthier_book}, it is not generic for a dynamical system to be uniformly observable. There may exist singular inputs for the system, that are inputs that make the system unobservable on any time interval, and the output feedback may produce such singular inputs. This defeats the purpose of output feedback stabilization, which is still an open problem when such inputs exist. Investigating this issue, some authors propose a different approach by allowing time-varying (either periodic as in \cite{Coron1994} or ``sample and hold'' as in \cite{ShimTeel2002}) output feedback. Doing so, the authors use a separation principle to show output feedback stabilization. Adopting another point of view and in line with \cite{MarcAurele}, we are interested in smooth time-invariant output feedback. In this work, we restrict ourselves to the class of \begingroup single-input single-output \endgroup bilinear systems with linear observation that are state feedback stabilizable at some target point, which, with no loss of generality, is chosen to be $0$. We also assume the system to be observable at the target, that is, the constant input obtained by evaluation of the feedback at 0 is not singular. This class of systems is a natural choice of study for two reasons. First, the uniform observability hypothesis is still not generic in this case. In particular, one can easily check that there generically exists constant inputs that make the system unobservable in any time. Secondly, according to \cite{fliess}, any control-affine system with finite dimensional observation space may be immersed in such a system. In this context, a natural question to ask is: ``Can we ensure that only observable inputs are produced by the dynamics when the output feedback is obtained as a combination of an observer and a stabilizing state feedback?'' This question falls within the more general and unsolved problem of building a smooth separation principle for systems with observability singularities. \begingroup One cannot hope for generic bilinear systems that all stabilizing state feedback laws ensure the observability of the closed-loop system. However, we show that for any stabilizing state feedback law, there exist small additive perturbations to this feedback that satisfy this observability property and conserve its locally stabilizing property. Transversality theory is used to prove the existence of such an open and dense class of perturbations. In particular, for almost all considered systems, almost any locally stabilizing feedback law ensures observability of the closed-loop system. Stabilization by output feedback is beyond the scope of this paper, which focuses only on the observability issue. Yet, the obtained results may pave the way to the construction of a ``generic'' separation principle. \endgroup For our results to hold, some properties of the dynamical observer are needed. The problem is tackled with a general observer design, and it is shown in a closing section that the classical Luenberger and Kalman observers fit our hypotheses. \subsubsection*{Organization of the paper} In Section~\ref{sec:state}, we state the main results of this paper. We begin this section with some definitions and notations, and we emphasise the precise issue. In particular, we define the system and the class of feedback perturbations we are interested in. We then state our main results on observability properties of the perturbed system, and assert that the classical Kalman and Luenberger observers fit our hypotheses. In Section~\ref{sec:proof} the reader may find a proof of our main results in three subsections. We rely on a transversality approach, which requires some technical preliminary results (Section~\ref{sec:prel}). Sections \ref{Subsec:part2} and \ref{sec:target} are then focused on the proof of our first main theorem and its corollary, respectively. Lastly, we prove in Section~\ref{sec:appli} that the Luenberger and Kalman observers fit our hypotheses, so that we can apply our previous theorems to these observers. In order to do so, we prove that their dynamics are somehow compatible with the Kalman observability decomposition. \subsubsection*{Notations} Let $\mathbb{N}$ be the set of non-negative integers. For any subset $\mathcal{I}\subset\mathbb{N}$, $|\mathcal{I}|$ denotes its cardinality. Let $n, m$ be positive integers. Let $\leqslantslantft\langle\cdot,\cdot\right\rangle$ be the canonical scalar product on $\mathbb{R}^n$, $|\cdot|$ the induced Euclidean norm, $B(x,r)$ the open ball centered at $x$ of radius $r$ for this norm, and $\mathbb S^{n-1} \subset \mathbb{R}^n$ the unit sphere. Let $\mathcal{L}(\mathbb{R}^n, \mathbb{R}^m)$ be the set of \begingroup linear maps \endgroup from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $\mathrm{End}(\mathbb{R}^n) = \mathcal{L}(\mathbb{R}^n, \mathbb{R}^n)$. For any endomorphism $A\in\mathrm{End}(\mathbb{R}^n)$, denote by $A^*$ its adjoint operator. If $f$ is a function from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, the notation $D f(x)[v]$ stands for the differential at $x\in\mathbb{R}^{n}$ applied to the vector $v\in\mathbb{R}^{n}$ of the function $f$. The partial differential of $f$ at $x$ with respect to the variable $y$ is denoted by $D_{y}f(x)$. In particular, for any function $t \mapsto v(t)$ defined on a real interval containing zero, we use the shorthand notation $v^{(i)} = \frac{{|\I|}ff^i v}{{|\I|}ff t^i}(0)$ for all $i \in \mathbb{N}$. Let $k\in\mathbb{N}$. The set of all $k$-jets from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is denoted by $J^k(\mathbb{R}^{n}, \mathbb{R}^{m})$ (see, for instance, \cite[Chapter II]{GG}). The mapping \( \sigma : J^k_x(\mathbb{R}^{n}, \mathbb{R}^{m}) \to \mathbb{R}^{n} \) given by \( \sigma : j^k f \mapsto \sigma \big(j^k f\big)=x \) is called the source map and the mapping \( \tau : J^k_x(\mathbb{R}^{n}, \mathbb{R}^{m}) \to \mathbb{R}^{m} \) given by \( \tau : j^k f \mapsto \tau \big(j^k f\big)=f(x) \) is called the target map. Put $J^k_x(\mathbb{R}^{n}, \mathbb{R}^{m}) = \sigma^{-1}(x)$, $J^k(\mathbb{R}^{n}, \mathbb{R}^{m})_y = \tau^{-1}(y)$ and $J^k_x(\mathbb{R}^{n}, \mathbb{R}^{m})_y = \sigma^{-1}(x) \bigcap \tau^{-1}(y)$. We have \( J^k(\mathbb{R}^{n}, \mathbb{R}^{m}) = \coprod_{x\in\mathbb{R}^{n}} J^k_x(\mathbb{R}^{n}, \mathbb{R}^{m}) = \mathbb{R}^{n} \times J^k_x(\mathbb{R}^{n}, \mathbb{R}^{m}). \) \section{Statement of the results}\label{sec:state} \subsection{Problem statement} Let $n$ be a positive integer, $A,B\in \mathrm{End}(\mathbb{R}^n)$, $C\in \mathcal{L}(\mathbb{R}^n,\mathbb{R})$, $b\in \mathbb{R}^n$ and $u\in C^\infty(\mathbb{R}_+, \mathbb{R})$. \begingroup Set ${C^\infty(\R^n, \R)}u{u} = A +u B$. In the present article, we focus on the following observed bilinear control system: \endgroup \begin{equation}\label{E:observation_system} \leqslantslantft\{ \begin{aligned} &\dot{x}={C^\infty(\R^n, \R)}u{u} x+ bu \\ &y= C x. \mathrm end{aligned} \right. \mathrm end{equation} System~\mathrm eqref{E:observation_system} is said to be \mathrm emph{observable} in time $T>0$ and for the control function $u$ if and only if, for all pair of solutions $\big((x_1, y_1), (x_2, y_2)\big)$ of \mathrm eqref{E:observation_system}, $(y_1-y_2)|_{[0, T]} \mathrm equiv 0$ implies $(x_1-x_2)|_{[0, T]} \mathrm equiv 0$. \begingroup For bilinear control systems of the form \mathrm eqref{E:observation_system}, we have the following characterization. \begin{proposition}\label{prop:obs} System~\mathrm eqref{E:observation_system} is observable in time $T$ for the control $u$ if and only if for every $\omega_0 \in \mathbb S^{n-1}$ the solution of $\dot \omega = {C^\infty(\R^n, \R)}u{u(t)}\omega$ initiated from $\omega_0$ satisfies $C\omega|_{[0, T]} \not\mathrm equiv 0$. \mathrm end{proposition} \endgroup If \mathrm eqref{E:observation_system} is observable for $u=0$ in some time $T>0$, then it is also observable in any time $T>0$, and we say that the pair $(C, A)$ is \mathrm emph{observable}. According to the Kalman rank condition, $(C, A)$ is observable if and only if the rank of the following observability matrix \begin{align}\label{E:Kalman-matrix} \mathcal{O}(C, A) = \begin{pmatrix} C\\ CA\\ \vdots\\ CA^{n-1} \mathrm end{pmatrix} \mathrm end{align} is equal to $n$. Let $\xis$ be a finite dimensional manifold and let $\mathcal{L}L:\xis\to \mathcal{L}(\mathbb{R}, \mathbb{R}^n)$. For all $u\in \mathbb{R}$, let $f(\cdot,u)$ be a vector field over $\xis$. Denoting $\varepsilon= \hat{x}-x$, we introduce a dynamical observer system depending on the pair $(f, \mathcal{L}L)$: \begin{equation}\label{E:Observer_system} \leqslantslantft\{ \begin{aligned} \dot{\hat{x}} &= {C^\infty(\R^n, \R)}u{u} \hat{x} + b u - \mathcal{L}L(\xi)C\varepsilon \\ \dot{\varepsilon} &=\leqslantslantft( {C^\infty(\R^n, \R)}u{u} -\mathcal{L}L(\xi)C \right) \varepsilon \\ \dot{\xi} &= f(\xi,u). \mathrm end{aligned} \right. \mathrm end{equation} Let $\lambda\in C^\infty(\mathbb{R}^n, \mathbb{R})$ be such that $0$ is an asymptotically stable equilibrium point of the vector field $x \mapsto {C^\infty(\R^n, \R)}u{\lambda(x)}x + b\lambda(x)$ for some open domain of attraction $D(\lambda)$. We will further assume that $\lambda(0)=0$, which is true up to a substitution of $A$ with $A+\lambda(0) B$. As stated in the introduction, our goal is to make system~\mathrm eqref{E:observation_system} observable in time $T$ for the control $u = \lambda \circ \hat{x}$, where $\hat{x}$ follows \mathrm eqref{E:Observer_system} with initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$. Since the stabilizing feedback $\lambda$ does not guarantee this property, we consider a small perturbation $\lambda + \delta$ of it. For all $\delta\in{C^\infty(\R^n, \R)}$, we consider the coupled system \begin{equation}\label{E:kalman_coupled} \leqslantslantft\{ \begin{aligned} \dot{\hat{x}} &= {C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x})} \hat{x} + b (\lambda+\delta)(\hat{x}) - \mathcal{L}L(\xi)C \varepsilon \\ \dot{\varepsilon} &= \leqslantslantft({C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x})} - \mathcal{L}L(\xi) C \right) \varepsilon \\ \dot{\xi} &=f(\xi,(\lambda+\delta)(\hat{x}))\\ \dot{\omega} &= {C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x})} \omega. \mathrm end{aligned} \right. \mathrm end{equation} \begingroup \begin{remark} In system \mathrm eqref{E:kalman_coupled}, the dynamics of $(\hat{x}, \varepsilon, \xi)$ do not depend on $\omega$. However, the dynamics of $\omega$ are included in \mathrm eqref{E:kalman_coupled} as they are crucial for the observability analysis of \mathrm eqref{E:observation_system} with input $u=\lambda(\hat{x})$, as stated in Proposition~\ref{prop:obs}. We will sometimes consider $(\hat{x}, \varepsilon, \xi)$ to be the first coordinates of a solution of \mathrm eqref{E:kalman_coupled} without fixing any initial condition for $\omega$. \mathrm end{remark} \endgroup \begingroup From now on, we denote by $\mathcal{K}=\mathcal{K}x\times \mathcal{K}eps \times \mathcal{K}P$ a semi-algebraic compact subset of $D(\lambda)\times \mathbb{R}^n\times\Prics$, which stands for a subset of the space of initial conditions of system~\mathrm eqref{E:Observer_system}. For all $R>0$, let \[ VV_R = \leqslantslantft\{ \delta \in {C^\infty(\R^n, \R)} \;:\; \forall x\in B(0, R),\quad \delta(x) = 0 \right\}. \] \endgroup We ask the observer given by $(f, \mathcal{L}L)$ to satisfy the following important properties: \noindent \begin{minipage}[t]{.1\linewidth} \setword{\mathscr{F}C}{FC} \mathrm end{minipage} \begin{minipage}[t]{.85\linewidth} (Forward completeness.) For all $u\in C^{\infty}(\mathbb{R}_+,\mathbb{R})$, the time-varying vector field $f(\cdot ,u)$ is forward complete. Moreover, for all $(\hat{x}_0,\varepsilon_0,\xi_0,\omega_0)\in \mathcal{K} \times \mathbb S^{n-1}$ and for all $\delta\in{C^\infty(\R^n, \R)}$ bounded over $D(\lambda)$, the coupled system~\mathrm eqref{E:kalman_coupled} has a unique solution $(\hat{x}, \varepsilon, \xi, \omega)\in C^\infty(\mathbb{R}_+, \mathbb{R}^n\times\mathbb{R}^n\times\Prics\times\mathbb S^{n-1})$ defined on $[0, +\infty)$. \mathrm end{minipage} \noindent \begin{minipage}[t]{.1\linewidth} \setword{\mathbb{N}FOT}{NFOT} \mathrm end{minipage} \begin{minipage}[t]{.85\linewidth} (No flat observer trajectories.) For all $R>0$, there exists $\mathrm eta>0$ such that for all $\delta\inVV_R$ satisfying $\sup\{\abs{\delta(x)}\;:\; x\in \mathcal{K}x\} < \mathrm eta$, for all $(\hat{x}_0,\varepsilon_0,\xi_0,\omega_0)\in \mathcal{K}\times\mathbb S^{n-1}$ such that $(\hat{x}_0,\varepsilon_0)\neq (0,0)$, there exists a positive integer $k$ such that the solution of~\mathrm eqref{E:kalman_coupled} with initial condition $(\hat{x}_0,\varepsilon_0,\xi_0,\omega_0)$ satisfies $\hat{x}^{(k)}(0)\neq 0$. \mathrm end{minipage} \noindent These properties are investigated in the last section of the paper. There, we show that the classical Luenberger and Kalman observers fit these hypotheses so that the main results may be applied to these observers. For all $k\in\mathbb{N}$, $K\subset\mathbb{R}^n$ and $\delta \in {C^\infty(\R^n, \R)}$, let \begin{align*} \norm{\delta}_{k, K} &= \sup \leqslantslantft\{\abs{ \frac{\partial^\mathrm ell \delta}{\partial x_{i_1} \cdots \partial x_{i_\mathrm ell}}(x)} \;:\; 0\leqslantslantq\mathrm ell\leqslantslantq k,\quad 1\leqslantslantq i_1\leqslantslantq \cdots\leqslantslantq i_\mathrm ell \leqslantslantq n,\quad x\in K\right\}. \mathrm end{align*} For any $k \in \mathbb{N}$, any compact subset $K \subset \mathbb{R}^n$ and any $\mathrm eta>0$, $k \in \mathbb{N}$, let \[ \mathbb{N}NN(k, K, \mathrm eta) = \leqslantslantft\{ \delta \in {C^\infty(\R^n, \R)} \;:\; \norm{\delta}_{k, K} < \mathrm eta \right\}. \] \begingroup \begin{remark} One can check that for any open subset $UUU \subset D(\lambda)$ relatively compact in $D(\lambda)$, for all $\mathbb{R}RR>0$, there exists $\mathrm eta>0$ such that for all $\delta\inVV_\mathbb{R}RR$ satisfying $|\delta|<\mathrm eta$, the feedback $\lambda+\delta$ is such that 0 is asymptotically stable with domain of attraction containing $UUU$. Hence in the following we focus only on the observability properties of the stabilizing feedback $\lambda+\delta$. \mathrm end{remark} \endgroup \begin{issue} Let $T>0$. Under genericity assumptions on $(A, B, C)$, does there exist $\mathbb{R}RR, \mathrm eta>0$, a positive integer $k$ and a residual set $\mathcal{O}O\subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ such that we have the following property. For all $\delta\in\mathcal{O}O\capVV_\mathbb{R}RR$ and for all initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)\in\mathcal{K}$, system~\mathrm eqref{E:observation_system} is observable in time $T$ for the control $u = (\lambda + \delta) \circ \hat{x}$, where $\hat{x}$ follows \mathrm eqref{E:kalman_coupled} with initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$ and feedback perturbation $\delta$? \mathrm end{issue} \subsection{Main results}\label{sec:main_results} In this section, we state the main results of the paper whose proofs are postponed to the upcoming sections. We first state our main theorem, that deals with the observability of system~\mathrm eqref{E:kalman_coupled}. Its proof is the most technical part of the paper, and heavily relies on transversality theory. \begin{theorem}\label{Thm:main} Assume that the pairs $(C, A)$ and $(C, B)$ are observable. Assume that $0\notin \mathcal{K}x$. Then there exist $\mathrm eta>0$, a positive integer $k$ and a dense open (in the Whitney $C^\infty$ topology) subset $\mathcal{O}O \subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ such that the solution to \mathrm eqref{E:kalman_coupled} with $\delta\in\mathcal{O}O$ and initial condition $(\hat x(0), \varepsilon(0), \xi(0), \omega(0))\in\mathcal{K}\times\mathbb S^{n-1}$ satisfies \begin{equation}\label{main_eq} \mathrm exists k_0 \in \{ 0,\dots,k\}\quad\;:\;\quad \leqslantslantft.\frac{{|\I|}ff^{k_0}}{{|\I|}ff t^{k_0}}\right|_{t=0} C\omega(t) \neq 0. \mathrm end{equation} \mathrm end{theorem} The proof of this theorem can be found in Section~\ref{Subsec:part2}. \begin{remark}\label{rk:obs1} Property \mathrm eqref{main_eq} is stronger than observability of \mathrm eqref{E:kalman_coupled} in any time $T>0$. This implication is shown in Corollary~\ref{obs}. Pay attention to the assumption $0\notin \mathcal{K}x$. In Section~\ref{sec:target}, this assumption is removed, while only slightly weakening our observability result. \mathrm end{remark} Theorem~\ref{Thm:main} leads to the following corollary which states that under genericity assumptions on the system, there exists a generic class of perturbations $\delta$ such that the feedback $\lambda+\delta$ makes \mathrm eqref{E:kalman_coupled} observable. \begin{corollary}\label{Cor:main} Assume that the pairs $(C, A)$ and $(C, B)$ are observable. Assume that $0$ is in the interior of $\mathcal{K}x$. Let $T>0$. Then there exist $\mathbb{R}RR, \mathrm eta>0$, a positive integer $k$ and a dense open subset \begingroup $\mathcal{O}O \subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta) \cap VV_\mathbb{R}RR$ such that the solution to \mathrm eqref{E:kalman_coupled} with $\delta\in\mathcal{O}O$ \endgroup and initial condition $(\hat{x}_0, \varepsilon_0, \xi_0, \omega_0)\in\mathcal{K}\times\mathbb S^{n-1}$ satisfies \[ \mathrm exists t\in[0, T] \quad\;:\;\quad C \omega(t) \neq 0, \] that is system~\mathrm eqref{E:observation_system} is observable in time $T$ for the control $u = (\lambda + \delta) \circ \hat{x}$, where $\hat{x}$ follows \mathrm eqref{E:kalman_coupled} with initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$ and feedback perturbation $\delta$. \mathrm end{corollary} \begingroup This result also implies a generic observability property directly on the stabilizing state feedback law $\lambda$. \begin{corollary}\label{cor:feedback} Assume that the pairs $(C, A)$ and $(C, B)$ are observable. Assume that $0$ is in the interior of $\mathcal{K}x$. Denote by $\Lambda$ the set of feedbacks $\lambda\in C^\infty(\mathbb{R}^n, \mathbb{R})$ such that $0$ is a locally asymptotically stable equilibrium point of the vector field $x \mapsto {C^\infty(\R^n, \R)}u{\lambda(x)}x + b\lambda(x)$. Let $T>0$ and $FF_T\subset\Lambda$ be the set of feedbacks $\lambda\in\Lambda$ such that \mathrm eqref{E:observation_system} is observable in time $T$ for the control $u = \lambda \circ \hat{x}$, where $\hat{x}$ follows \mathrm eqref{E:kalman_coupled} with $\delta\mathrm equiv0$ and initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$ in $\mathcal{K}$. Then $FF_T$ is a dense open subset of $\Lambda$. \mathrm end{corollary} The proof of these two corollaries can be found in Section~\ref{sec:target}. \endgroup \begin{remark} \begingroup Because $VV_\mathbb{R}RR$ is not open in the Whitney $C^\infty$ topology, the set $\mathcal{O}O$ defined in Corollary~\ref{Cor:main} is not open in the Whitney $C^\infty$ topology, but it is open in the induced topology on $\mathbb{N}NN(k, \mathcal{K}x, \mathrm eta) \cap VV_\mathbb{R}RR$. \endgroup Also, the set of matrices $(A, B, C)\in \mathrm{End}(\mathbb{R}^n)^2\times\mathcal{L}(\mathbb{R}^n, \mathbb{R})$ such that $(C, A)$ and $(C, B)$ are both observable is open and dense. As a consequence, ``$(C, A)$ and $(C, B)$ are observable'' is a generic hypothesis. \begingroup Contrarily to the strategy followed in \cite{MarcAurele} on some specific example, the results of this paper do not explicitly design any perturbation $\delta\in\mathcal{O}O$, but rather state that for almost all bilinear system, almost all perturbation $\delta\in\mathbb{N}NN(k, \mathcal{K}x, \mathrm eta) \capVV_\mathbb{R}RR$ belongs to $\mathcal{O}O$ (in a topological sense). \endgroup \mathrm end{remark} Finally, the next theorem shows that the classical Luenberger and Kalman observers fit hypotheses~\ref{FC} and \ref{NFOT}. Hence, our results may be applied to these well-known observers. \begin{theorem}\label{thm:appli} Assume that $(C, A)$ is observable. Assume that $\lambda$ is bounded over $D(\lambda)$. Let $Q\in\Prics_n$. For all $\xi\in\Prics_n$ and all $u\in\mathbb{R}$, consider the following well-known observers: \begin{align} &f^\mathrm{Luenberger}(\xi, u) = 0\tag{Luenberger observer}\\ &f^\mathrm{Kalman}_{Q}(\xi, u) = \xi {C^\infty(\R^n, \R)}u{u}^* + {C^\infty(\R^n, \R)}u{u} \xi + Q - \xi C^*C\xi \tag{Kalman observer} \mathrm end{align} and $\mathcal{L}L(\xi) = \xi C^*$. Then the coupled system~\mathrm eqref{E:kalman_coupled} given by $(f, \mathcal{L}L)$ satisfies the hypotheses~\ref{FC} and \ref{NFOT} for any $f\in\{f^\mathrm{Luenberger}, f^\mathrm{Kalman}_{Q}\}$. \mathrm end{theorem} The proof of this theorem can be found in Section~\ref{sec:appli}. \begin{remark} If $\lambda$ is unbounded over $D(\lambda)$, then for any open subset $UUU$ relatively compact in $D(\lambda)$, we can obtain by smooth saturation of $\lambda$ a new bounded feedback law $\lambda_{\textrm{sat}}$ such that ${\lambda_{\textrm{sat}}}_{|U}={\lambda}_{|U}$, for which the previous statement holds. (In particular $U\subsetD(\lambda_{\textrm{sat}})$.) \mathrm end{remark} \section{Proofs of the observability statements}\label{sec:proof} In order to prove our main Theorem~\ref{Thm:main} and its Corollary~\ref{Cor:main}, we need a series of preliminary results that we state and prove below. The main results will appear as corollaries of these subsequent lemmas. Before we start the more technical elements of the paper, let us present the method we follow in order to prove the main results. Theorem~\ref{Thm:main} is an application of transversality theory to our particular problem (see \cite{GMP} for the statements we rely on; see also \cite{abraham1967transversal,GG}). Consider a solution to \mathrm eqref{E:kalman_coupled} for a given perturbation $\delta$ of the feedback law, and a set of initial conditions in $\mathcal{K}\times \mathbb S^{n-1}$. We set $h:C^{\infty}(\mathbb{R}^n,\mathbb{R})\times (\mathcal{K}\times \mathbb S^{n-1})\times \mathbb{R}^+\to \mathbb{R}$ to be the smooth map given by $$ h(\delta,(\hat{x}_0, \varepsilon_0, \xi_0,\omega_0),t)=C\omega(t). $$ As stated in Section~\ref{sec:state}, to get observability after perturbation of the feedback, we would like to show that there exists $\delta$, preferably small, such that \begin{equation}\label{E:obser_f} t\mapsto h(\delta,z_0,t)\neq0,\qquad \forall z_0=(\hat{x}_0, \varepsilon_0, \xi_0,\omega_0)\in \mathcal{K}\times \mathbb S^{n-1}. \mathrm end{equation} \begingroup A sufficient condition for $\delta$ to satisfy \mathrm eqref{E:obser_f} is that for each $z_0\in \mathcal{K}\times \mathbb S^{n-1}$, there exists an integer $k$ such that \( \leqslantslantft.\frac{{|\I|}ff^k }{{|\I|}ff t^k}\right|_{t=0}(h(\delta ,z_0,t))\neq 0. \) \endgroup In other words, our goal will be achieved if we can prove that there exists $\delta$ and a finite set $\mathcal{I}\subset \mathbb{N}$ such that the map $H:C^{\infty}(\mathbb{R}^n,\mathbb{R})\times (\mathcal{K}\times \mathbb S^{n-1})\to \mathbb{R}^{|\mathcal{I}|}$ given by \[ H(\delta,z_0)=\leqslantslantft(\leqslantslantft.\frac{{|\I|}ff^k }{{|\I|}ff t^k}\right|_{t=0}h(\delta,z_0,t)\right)_{k\in \mathcal{I}}, \] never vanishes. This is where transversality theory comes into play. Let $N$ denote the dimension of the surrounding space of $\mathcal{K}\times \mathbb S^{n-1}$. We can ensure that there exists $\delta$ satisfying \mathrm eqref{E:obser_f} if we can prove that for some choice of $\mathcal{I}$, with $|\mathcal{I}|>N$, $H$ is \mathrm emph{transversal} to $\{0\}$ at $\delta =0$. That is to say, if we can prove that the rank of the map $H(0,\cdot)$ is maximal, equal to $|\mathcal{I}|>N$, at any of its vanishing points (at which point $H(0,\cdot)$ is then a submersion). Now it should be noted that in general, proving that a map is transversal to a point is a major hurdle, especially if the dimensions $n$ and $N$ of the spaces are unspecified. As a general rule, considering more orders of derivation of $h$ greatly increases the degrees of freedom of the map $H$ (by including higher order derivatives of $v$, as jet spaces grow exponentially in dimension), while only slightly increasing the size of the target space. This points towards an augmentation of the rank of $H$, making a proof of transversality achievable. The difficulty lies however in producing a ``rank increasing property'' on $H$ as $|\mathcal{I}|$ increases. That is, finding a symmetry in the successive derivatives of $h$ that proves that for any dimension, a set $\mathcal{I}$ can be found by differentiating $h$ sufficiently many times. The symmetry we use to prove the rank condition on the map $H$ can be described as follows. For $k\in \mathbb{N}$, let $$ h^k(\delta,z_0,t)=CB^k\omega(t). $$ It turns out that if $h^{k+1}(0,z_0,\cdot)$ has a non-zero derivative of any order (including order 0), then we automatically get the rank condition for $h^{k}(0,z_0,\cdot)$ (this statement will be made precise in Corollary \ref{C:rank_N}). Here the hypothesis that $(C,B)$ is an observable pair becomes crucial. Indeed, observe that $h^{k}(0,z_0,0)=CB^k\omega_0$. Hence, for any $\omega_0\in \mathbb S^{n-1}$ there exists a $k\in\{0,\dots,n-1\}$ such that $$ h^{k}(0, z_0, 0)\neq 0. $$ This in turns induces a partition of $\mathcal{K}\times \mathbb S^{n-1}$ into $n$ subsets on each of which at least one of the maps $h^0,\dots ,h^{n-1}$ never vanishes. Since $h^{k+1}(0,z_0,\cdot)$ not vanishing implies that the rank condition is satisfied for $h^{k}(0,z_0,\cdot)$, we chain-apply $n$ successive transversality theorems to prove the existence of a $\delta$ such that $h(\delta,z_0,\cdot)$ has always at least one non-zero time derivative at any point $z_0\in \mathcal{K}\times \mathbb S^{n-1}$. Section~\ref{sec:prel} is aimed at making explicit the connection between the rank condition and the family of maps $(h^k)_{k\in\mathbb{N}}$. Section~\ref{Subsec:part2} is dedicated to the effective application of the principles presented in this introduction, which leads to the proof of Theorem~\ref{Thm:main}. Section~\ref{sec:target} concludes the proof of the observability statements by taking into account the behavior of the system near the target $0$. \subsection{Preliminary results}\label{sec:prel} Let $u\in C^\infty(\mathbb{R}_+,\mathbb{R})$ and consider the ordinary differential equation \begin{equation}\label{E:omega} \dot{\omega}=\leqslantslantft(A+u(t)B\right) \omega. \mathrm end{equation} For all $k,m\in \mathbb{N}$, let $F_k^m: C^\infty(\mathbb{R}_+,\mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}$ be the function such that $$ F_k^m(u, \omega_0)=CB^m\omega^{(k)}(0) $$ where $t\mapsto \omega(t) $ is the solution of \mathrm eqref{E:omega} with initial condition $\omega_0$. Let us introduce the $n \times n$ matrix valued polynomials in the indeterminates $X_0, \dots, X_{k-1}$ by: \begin{equation*} \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{k-1}] = \begin{cases} \mathrm{End}(\mathbb{R}^n) &\mbox{if } k=0 \\ \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{k-2}][X_{k-1}] &\mbox{otherwise}, \mathrm end{cases} \mathrm end{equation*} and set \[ \mathrm{End}(\mathbb{R}^n)\leqslantslantft[ (X_k)_{k \in \mathbb{N}} \right] = \bigcup_{k \in \mathbb{N}}\mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{k-1}]. \] \begingroup Let $\Psi : \mathrm{End}(\mathbb{R}^n)\leqslantslantft[ (X_k)_{k \in \mathbb{N}} \right] \to \mathrm{End}(\mathbb{R}^n)\leqslantslantft[ (X_k)_{k \in \mathbb{N}} \right]$ be the linear map defined by \[ \Psi (P)(X_0,\dots ,X_{k}) = P(X_0,\dots, X_{k-1})(A+X_0B)+\sum_{i=0}^{k-1}\frac{\partial P}{ \partial X_i}\leqslantslantft(X_0,\dots, X_{k-1}\right) X_{i+1}, \] where $k = \min \leqslantslantft\{ \mathrm ell \in \mathbb{N} \;:\; P \in \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{\mathrm ell-1}] \right\}$. \endgroup Finally, let us define the family $(P_k)_{k\in \mathbb{N}}$ of matrix valued polynomials such that $P_0 \in \mathrm{End}(\mathbb{R}^n)$ and $P_k\in \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{k-1}]$, for all $k \geqslantslant 1$, by \begin{equation}\label{Eq:recurrence_relation} P_0=\mathbbm{I}, \qquad P_{k+1} = \Psi (P_{k}), \qquad \forall k\in \mathbb{N}. \mathrm end{equation} It is clear\footnote{ Note that, for $k \neq 0$, the function $F_k^m$ actually acts on $(k-1)$-jets at zero of functions and not on functions themselves. Consequently, the restriction \( \leqslantslantft. F_k^m \right|_{J^\mathrm ell_0(\mathbb{R}, \mathbb{R}) \times \mathbb{R}^n} \) is well-defined as soon as $\mathrm ell \geqslantslant k-1$. Of course, for $k=0$, the restriction \( \leqslantslantft. F_0^m \right|_{J^\mathrm ell_0(\mathbb{R}, \mathbb{R}) \times \mathbb{R}^n} \) makes sense only if $\mathrm ell \geqslantslant 0$. In summary, the restriction \( \leqslantslantft. F_k^m \right|_{J^\mathrm ell_0(\mathbb{R}, \mathbb{R}) \times \mathbb{R}^n} \) is well-defined as soon as $\mathrm ell \geqslantslant k$.} that for all $m\in \mathbb{N}$, \[ F_k^m(u, \omega_0) = \begin{cases} CB^m\omega_0 &\mbox{if~} k=0 \\ CB^mP_k\leqslantslantft(u^{(0)},u^{(1)},\dots , u^{(k-1)}\right)\omega_0 &\mbox{otherwise}, \mathrm end{cases} \] where $u^{(i)}$ is shorthand for $\frac{{|\I|}ff^i u}{{|\I|}ff t^i}(0)$ for all $i\in \mathbb{N}$. For all $k\in \mathbb{N}$ and $i\in \mathbb{N}$, $1\leqslantslantq i \leqslantslantq k$, let $Q_i^k=\dfrac{\partial P_k}{\partial X_{k-i}}$. \begin{lemma}\label{lemma:polynomial} For all $i\in \mathbb{N}\setminus\{0\}$, there exist $R_i^0,\dots, R_i^{i-1}\in \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{i-1}]$ such that \footnote{Actually, we can show that $R_i^0,\dots, R_i^{i-1}\in \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{i-2}]$} \[ Q_{i}^{i+k} = \sum_{j=0}^{i-1} k^j R_i^j, \qquad \forall k \geqslantslant 0. \] Furthermore, ${|\I|}splaystyle R_{i}^{i-1}=\frac{BP_{i-1}}{(i-1)!}$. \mathrm end{lemma} \begin{proof} We prove the first part of the statement by induction on $i$. For $i=1$, one easily checks that \begin{equation}\label{Eq:initialisation:i=1} Q_1^{1+k} = B, \qquad \forall k \in \mathbb{N}. \mathrm end{equation} Assuming the desired property for $i$, we have to prove that there exist $R_{i+1}^0, \dots, R_{i+1}^{i} \in \mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{i}]$ such that \[ Q_{i+1}^{i+1+k} = \sum_{j=0}^{i} k^j R_{i+1}^{j}, \qquad \forall k \geqslantslant 0. \] Using the definition of $Q_{i+1}^{i+1 +\mathrm ell}$ and the recurrence relation \mathrm eqref{Eq:recurrence_relation} yields \begin{equation}\label{Eq:increments} Q_{i+1}^{i+1 +\mathrm ell} = \Psi(Q_{i}^{i+\mathrm ell}) + Q_{i+1}^{i+\mathrm ell}, \qquad \forall \mathrm ell \geqslantslant 1. \mathrm end{equation} Consequently, for all $k \geqslantslant 0$, \begin{align} Q_{i+1}^{i+1+ k} & = \sum_{\mathrm ell = 1}^{k} \leqslantslantft( Q_{i+1}^{i+1+\mathrm ell} - Q_{i+1}^{i+\mathrm ell} \right) + Q_{i+1}^{i+1} \nonumber\\ & = \sum_{\mathrm ell = 1}^{k} \leqslantslantft( \Psi(Q_{i}^{i+\mathrm ell})\right) + Q_{i+1}^{i+1} \tag{by \mathrm eqref{Eq:increments}}\\ & = \sum_{\mathrm ell = 1}^{k} \leqslantslantft( \sum_{j=0}^{i-1} \mathrm ell^j \Psi(R_{i}^{j})\right) + Q_{i+1}^{i+1} \tag{by induction hypothesis}\\ & = \sum_{j=0}^{i-1}\leqslantslantft( \sum_{\mathrm ell = 1}^{k} \mathrm ell^j \right) \Psi(R_{i}^{j}) + Q_{i+1}^{i+1} \nonumber\\ & = \sum_{j=0}^{i-1} S^j(k) \Psi(R_{i}^{j}) + Q_{i+1}^{i+1}, \qquad \text{with~}S^j(k)=\sum_{\mathrm ell = 1}^{k} \mathrm ell^j. \nonumber \mathrm end{align} Note that $Q_{i+1}^{i+1}, \Psi(R_{i}^{j}) \in \mathrm{End}(\mathbb{R}^n)[X_0, \dots, X_i]$ for all $j \in \{0, \dots, i-1\}$ ($Q_{i+1}^{i+1} = \partial{P_{i+1}}/\partial{X_0}$). Moreover, according to Faulhaber's formula, we have \[ S^j(k) = \frac{k^{j+1}}{j+1} + T^j(k), \qquad \forall j, k\in \mathbb{N}, \] where $T^j(k)$ is a polynomial in the variable $k$ of degree $j$ with no constant term. Consequently, \begin{align} Q_{i+1}^{i+1+ k} &= \frac{k^{i}}{i}\Psi(R_{i}^{i-1}) + \leqslantslantft( T^{i-1}(k)\Psi(R_{i}^{i-1}) + \sum_{j=0}^{i-2}S^j(k) \Psi(R_{i}^{j}) \right) + Q_{i+1}^{i+1} \nonumber\\ &= k^{i} R_{i+1}^{i} + \sum_{j=1}^{i-1} k^j R_{i+1}^j + R_{i+1}^0 \nonumber\\ &= \sum_{j=0}^{i} k^j R_{i+1}^j, \nonumber \mathrm end{align} with $R_{i+1}^i = \Psi(R_{i}^{i-1})/i$, $R_{i+1}^0 = Q_{i+1}^{i+1}$ and $R_{i+1}^{j} \in \mathrm{End}(\mathbb{R}^n)[X_0, \dots, X_i]$ for all $j \in \{0, \dots, i\}$. The second part of the statement easily follows by induction. Indeed, \begin{align*} BP_0 = Q_1^1 = \sum_{j=0}^{0} 0^j R_1^j = R_1^0, \mathrm end{align*} and \[ R_{i+1}^i = \frac{\Psi(R_{i}^{i-1})}{i} = \frac{1}{i} \Psi \leqslantslantft( \frac{1}{(i-1)!}BP_{i-1} \right) = \frac{1}{i!} B\Psi(P_{i-1}) = \frac{1}{i!} BP_i. \] The statement follows. \mathrm end{proof} \begin{corollary}\label{C:CBPk} Let $i, m\in \mathbb{N}$, $i \geqslantslant 1$. Let $v\in\mathbb{R}^i$ and $\omega_0\in \mathbb{R}^n$. Either there exists $k_0\geqslantslantq i$ such that $CB^m Q_i^k(v) \omega_0\neq 0$ for all $k\geqslantslantq k_0$ or $C B^mQ_i^k(v)\omega_0=0$ for all $k\geqslantslantq i$. \mathrm end{corollary} \begin{proof} By Lemma \ref{lemma:polynomial}, we have $Q_i^k=\sum_{j=0}^{i-1} (k-i)^j R_i^j$ for all integer $k\geqslantslantq i$. If $CB^m R_i^j(v) \omega_0= 0$ for all $j \in \{0, \dots, i-1\}$, then $C B^mQ_i^k(v)\omega_0=0$ for all $k\geqslantslantq i$. Otherwise, there exists $j \in \{0, \dots, i-1\}$ such that $CB^m R_i^j(v) \omega_0\neq 0$. Let $(k_0,\dots k_{i-1}) \in\mathbb{N}^i$ with $k_0<\dots<k_{i-1}$. We have \begin{align*} C B^m \begin{pmatrix} Q_i^{i+k_0}(v)\\ \vdots\\ Q_i^{i+k_{i-1}}(v) \mathrm end{pmatrix} \omega_0 = \begin{pmatrix} 1&k_0&\dots&k_0^i\\ \vdots&\vdots&&\vdots\\ 1&k_{i-1}&\dots&k_{i-1}^i\\ \mathrm end{pmatrix} C B^m \begin{pmatrix} R_i^{0}(v)\\ \vdots\\ R_i^{{i-1}}(v) \mathrm end{pmatrix} \omega_0. \mathrm end{align*} Since $k_0,\dots k_{i-1}$ are pairwise different, the Vandermonde matrix is invertible. Consequently, there exits $j\in\{0, \dots, i-1\}$ such that $C B^mQ_i^{i+k_j}(v)\omega_0\neq0$. Hence, there exists at most $i-1$ positive integers $k_j$ such that $C B^mQ_i^{i+k_j}(v)\omega_0=0$. Thus, there exists $k_0 \geqslantslantq i$ such that $CB^m Q_i^k(v) \omega_0\neq 0$ for all $k\geqslantslantq k_0$. \mathrm end{proof} For all $P\in\mathrm{End}(\mathbb{R}^n)[X_0,\dots X_{k-1}]$ and all $v\in\mathbb{R}^\mathbb{N}$, we set $P(v) = P(v_0,\dots,v_{k-1})$. \begin{corollary}\label{C:rank_N} Let $v\in\mathbb{R}^\mathbb{N}$, $\omega_0\in \mathbb{R}^n$ and $m\in \mathbb{N}$. If there exists $i \in \mathbb{N}\setminus\{0\}$ such that $CB^{m+1}P_{i-1}(v)\omega_0\neq 0$, then there exists $k_0\in \mathbb{N}$ such that, for all $N\in \mathbb{N} \setminus \{0\}$, \begingroup the mapping \footnote{Note that $\varphi(\cdot)=F_{\{k_0,\dots, k_0+N-1\}}^m(\cdot,\omega_0)$, with $F_{\{k_0,\dots, k_0+N-1\}}^m$ defined as in Section~\ref{Subsec:part2}} $\varphi:J^{k_0+N-1}_0(\mathbb{R}, \mathbb{R})=\mathbb{R}^{k_0+N}\to\mathbb{R}^N$ defined by $$\varphi(\cdot) = (CB^m P_{k_0}(\cdot)\omega_0,\dots,CB^mP_{k_0+N-1}(\cdot)\omega_0)$$ \endgroup has a rank $N$ differential at $(v_0,\dots,v_{k_0+N-1})$. \mathrm end{corollary} \begin{proof} Assume that there exists $i \geqslantslant 1$ such that $CB^{m+1}P_{i-1}(v)\omega_0\neq 0$. Since, according to Lemma \ref{lemma:polynomial}, $R_{i}^{i-1}=BP_{i-1}/(i-1)!$, this is equivalent to $CB^{m} R_{i}^{i-1}(v) \omega_0 \neq 0$. Thus, reasoning as in the proof of Corollary~\ref{C:CBPk}, the sequence $\big(CB^{m}Q_i^k(v)\omega_0\big)_{k\geqslantslantq i}$ is not constant equal to zero. Set \begin{equation}\label{E:first-non-zero-diagonal} i_0 = \min \leqslantslantft\{ i \in \mathbb{N}\setminus\{0\} : \big(CB^{m}Q_i^k(v)\omega_0\big)_{k\geqslantslantq i} \not\mathrm equiv 0 \right\}. \mathrm end{equation} As a consequence of Corollary~\ref{C:CBPk}, there exists $k_0\in \mathbb{N}$ such that $C B^mQ_{i_0}^k(v) \omega_0\neq 0$ for all $k\geqslantslantq k_0$, \mathrm emph{i.e.~} \[ \dfrac{\partial \leqslantslantft(CB^m P_{k}\omega_0\right)}{\partial X_{k-i_0}}(v_0,\dots,v_{k_0+N-1}) = \dfrac{\partial \leqslantslantft(CB^m P_{k}\omega_0\right)}{\partial X_{k-i_0}}(v) \neq 0, \qquad \forall k\geqslantslantq k_0, \] and (by construction of $i_0$) \[ \dfrac{\partial \leqslantslantft(CB^m P_{k}\omega_0\right)}{\partial X_{\mathrm ell}}(v_0,\dots,v_{k_0+N-1}) = \dfrac{\partial \leqslantslantft(CB^m P_{k}\omega_0\right)}{\partial X_{\mathrm ell}}(v) = 0, \qquad \forall \mathrm ell>k-i_0. \] In other words, \begin{equation}\label{E:differential-CB^mPk} D\varphi(v_0,\dots,v_{k_0+N-1}) = \begin{pmatrix} * & \hdots & * & a_0(v) & 0 & & \hdots & & 0 \\ \vdots & & & \ddots & \ddots & \ddots & & & \vdots\\ * & & \hdots & & * & a_{N-1}(v) & 0 & \hdots & 0\\ \mathrm end{pmatrix}, \mathrm end{equation} with $a_i(v) = C B^mQ_{i_0}^{k_0+i}(v) \omega_0$. The statement follows. \mathrm end{proof} \subsection{Observability away from the target and proof of Theorem~\ref{Thm:main}} \label{Subsec:part2} Using the results of the previous section, we are now able to prove our main Theorem~\ref{Thm:main}. In this section, we assume that $0\notin\mathcal{K}x$. From now on $t \mapsto \leqslantslantft(\hat{x}(t), \varepsilon(t), \xi(t), \omega(t) \right)$, or simply $(\hat{x}, \varepsilon, \xi, \omega)$, denotes the solution to \mathrm eqref{E:kalman_coupled} with initial condition $(\hat{x}_0, \varepsilon_0, \xi_0, \omega_0)$. Let us introduce some new notation. For any $k\in\mathbb{N}$, define the map $G^k$ by: \fonction{G^k}{J^k(\mathbb{R}^n, \mathbb{R}) \times \mathcal{K}eps\times \mathcal{K}P}{J^k_0(\mathbb{R}, \mathbb{R})}{ \Big( j^k\delta(\hat{x}_0), \varepsilonilon_0, \xi_0 \Big) }{j^k \big((\lambda+\delta) \circ \hat{x} \big)(0).} For any finite subset $\mathcal{I}\subset\mathbb{N}$ and any $m\in\mathbb{N}$, set ${k_{\I}} = \max \mathcal{I}$ and define the maps, $F_{\mathcal{I}}^m$ and $H^{m}_{\mathcal{I}}$ as follows: \fonction{F_{\mathcal{I}}^m}{J^{k_{\I}}_0(\mathbb{R}, \mathbb{R})\times \mathbb S^{n-1}}{\mathbb{R}^{{|\I|}}}{(v, \omega_0)}{ \big( CB^m P_{k}(v)\omega_0 \big)_{k\in\mathcal{I}},} \[ H^{m}_{\mathcal{I}} = F_{\mathcal{I}}^m \circ \leqslantslantft( G^{{k_{\I}}} \times \mathbbm{I}_{\mathbb S^{n-1}}\right). \] \begin{remark}\label{R:on-the-def-of-F} Notice that for any $m, k_0 \in \mathbb{N}$ and any $N \in \mathbb{N}\setminus\{0\}$ such that $\mathcal{I} \subset \{k_0, \dots, k_0+N-1\}$, the map $F_{\mathcal{I}}^m$ satisfies \[ F_{\mathcal{I}}^m = \pi_{\mathcal{I}} \circ F_{\{k_0, \dots, k_0+N-1\}}^m, \] where $\pi_{\mathcal{I}}: J^{k_0+N-1}_0(\mathbb{R}, \mathbb{R}) = \mathbb{R}^{k_0+N} \to \mathbb{R}^{|\mathcal{I}|}$ denotes the canonical projection onto the factors that correspond to indices in $\mathcal{I}$. \mathrm end{remark} Now we state the following proposition, which leads directly to Theorem~\ref{Thm:main}. \begin{proposition}\label{P:main} \begingroup For all $m \in \{0, \dots, n-1\}$, define \begin{equation*} E_m = \begin{cases} \mathbb S^{n-1} &\mbox{if } m=0 \\ \leqslantslantft\{\omega_0\in \mathbb S^{n-1} \;:\; CB^i\omega_0 = 0,\quad ~\forall i \in\intset{0}{m-1}\right\} &\mbox{otherwise}. \mathrm end{cases} \mathrm end{equation*} \endgroup Suppose $(C, A)$ and $(C, B)$ are observable pairs. Then for every $m \in \intset{0}{n-1}$, there exist $k\in\mathbb{N}$, a positive real number $\mathrm eta$ and a dense open subset $\mathcal{O}O_m\subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ such that for all $(\delta, \hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0) \in \mathcal{O}O_m \times \mathcal{K}\times E_m$ \[ H^m_{\intset{0}{k}}(j^{k}\delta(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0) \neq 0. \] \mathrm end{proposition} \begin{proof} The proof strongly relies on the results of Section~\ref{sec:prel} and on the Goresky-MacPherson transversality theorem (see \cite[Part I, Chapter 1]{GMP}). We prove the proposition by finite descending induction on $m$. Note that since the pair $(C, B)$ is observable, we have \( \mathrm emptyset = E_n \subset E_{n-1} \subset \cdots \subset E_1 \varsubsetneq E_0 = \mathbb S^{n-1}. \) For $m=n-1$, the result is immediate because, by observability of the pair $(C, B)$, $CB^{n-1}\omega_0 \neq 0$ for all $\omega_0 \in E_{n-1}$. Hence, for $k=0$ and any positive real number $\mathrm eta$, we have for all $(\delta, \hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0) \in \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta) \times \mathcal{K}\times E_{n-1}$, \[ H^{n-1}_{\{0\}}(j^{0}\delta(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0) = CB^{n-1}\omega_0 \neq 0. \] Now suppose $1 \leqslantslant m \leqslantslant n-1$. Note that, by definition of $E_{m-1} \setminus E_{m}$, \begin{equation}\label{E:on_E_m-1-E_m} CB^{m-1}\omega_0 \neq 0, \quad \forall \omega_0 \in E_{m-1} \setminus E_{m}. \mathrm end{equation} Assume that we are given a $k\in\mathbb{N}$, a positive real number $\mathrm eta$ and a dense open subset $\mathcal{O}O_m \subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ such that \begin{equation}\label{E:induction-hypothesis} H^m_{\{0, \dots, k\}}(j^{k}\delta(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0) \neq 0, \qquad \forall (\delta, \hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0) \in \mathcal{O}O_m \times \mathcal{K}\times E_m. \mathrm end{equation} Choose $(\delta, \hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0) \in \mathcal{O}O_m \times \mathcal{K}\times E_m$ and put $u(t) = (\lambda+\delta)\big( \hat{x}(t) \big)$. Equation~\mathrm eqref{E:induction-hypothesis} implies that $CB^m P_{i}(u^{(0)}, \dots, u^{(k)})\omega_0\neq0$ for an integer $i \in \{0, \dots, k\}$, so, by Corollary~\ref{C:rank_N} there exists $k_0\in\mathbb{N}$ such that, for any positive integer $k_1$, the map $F_{\{k_0, \dots, k_0+k_1-1\}}^{m-1}$ has a rank $k_1$ differential at $(u^{(0)}, \dots, u^{(k_0+k_1-1)})$. Let $i_0 \in \mathbb{N}$ be defined as in the proof of Corollary~\ref{C:rank_N}. Let $p\in\mathbb{N}\setminus\{0\}$ be such that $\hat{x}^{(p)}\neq 0$ and $\hat{x}^{(q)} = 0$ for all $q<p$ (which exists by hypothesis~\ref{NFOT} and $0\notin\mathcal{K}x$), and choose $\mathrm ell \in \{1, \dots n\}$ so that $\hat{x}_\mathrm ell^{(p)}\neq 0$. Put \[ {j_0} = \min\big\{ j\geqslantslantq k_0 \;:\; j - i_0 \mathrm equiv 0 \pmod p \big\} \footnote{Index $j_0$ corresponds to the smallest index $j \geqslantslant k_0$ such that $\hat{x}^{(p)}_\mathrm ell$ appears in $u^{(j-i_0)}$.} \quad\text{and}\quad \mathcal{I} = \big\{ {j_0} + rp \;:\; r\in\intset{0}{N-1} \big\}, \] where $N$ is a positive integer. The (partial) differential of $G^{m}_{\mathcal{I}}$ with respect to \[ w = \leqslantslantft.\leqslantslantft( \delta, \parfrac{}{x_\mathrm ell}\delta, \dots, \leqslantslantft(\parfrac{}{x_\mathrm ell}\right)^{{k_{\I}}}\delta \right)\right|_{x=\hat{x}_0} \] at $X_0 = (j^{{k_{\I}}}\delta(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0)$ is the submatrix $D_wG^{m}_{\mathcal{I}}(X_0)$ obtained from $DG^{m}_{\mathcal{I}}(X_0)$ by deleting all columns that do not correspond to partial derivatives with respect to $w$. In other words, \[ D_wG^{m}_{\mathcal{I}}(X_0) =\begin{pmatrix} \mathtt{col}({0}) & \cdots & \mathtt{col}({{k_{\I}}-1}) \mathrm end{pmatrix}. \] Each column $\mathtt{col}({i})$, $i \in \{0, \dots, {k_{\I}}-1\}$ of $D_wG^{m}_{\mathcal{I}}(X_0)$ satisfies \begin{align*} \mathtt{col}({i})^* = \begin{pmatrix} 0 &\cdots &0 &b_i(X_0) &* &\cdots * \mathrm end{pmatrix}^*,\qquad b_i(X_0) \neq 0, \mathrm end{align*} where the non zero coefficient $b_i(X_0)$ appears at the $ip\,$th row. According to Fa{\`a} di Bruno formula, we have \[ b_i(X_0) = n_i\leqslantslantft(\hat{x}^{(p)}_\mathrm ell \right)^i, \] $n_i$ being a positive integer for each $i \in \{0, \dots, {k_{\I}}-1\}$. It is clear from the definition of $F_{\mathcal{I}}^m$ and Remark \ref{R:on-the-def-of-F} thereafter that $DF_{\mathcal{I}}^m$ is the submatrix of $DF_{\{k_0, \dots, {k_{\I}}\}}^m$ (see equation~\mathrm eqref{E:differential-CB^mPk}) obtained by keeping the $i\,$th rows for $i \in \mathcal{I}$. Therefore, \begin{align} \rank \leqslantslantft( DH^{m}_{\mathcal{I}}(X_0) \right) &\geqslantslant \rank \leqslantslantft( D_vF_{\mathcal{I}}^m\leqslantslantft(G^{{k_{\I}}}(X_0), \omega_0\right) \circ D_w G^{{k_{\I}}}(X_0) \right) \nonumber\\ &= \rank \begin{pmatrix} * & \cdots & * & c_0(X_0) & 0 & & \cdots & & 0 \\ \vdots & & & \ddots & \ddots & \ddots & & & \vdots \\ * & & \cdots & & * & c_{N-1}(X_0) & 0 & \cdots & 0 \\ \mathrm end{pmatrix}, \nonumber \mathrm end{align} where \( c_r(X_0) = a_{j_0+rp}\leqslantslantft(G^{{k_{\I}}}(X_0), \omega_0\right) b_{j_0+rp}(X_0), \) $r \in \{0, \dots, N-1\}$. Hence $H_\mathcal{I}^{m-1}$ has a rank $N$ differential at $X_0$. For any $k \in \mathbb{N}$, any compact subset $K \subset \mathbb{R}^n$ and any $\mathrm eta>0$, $k \in \mathbb{N}$, define \[ \mathcal{M}(k, K, \mathrm eta) = \leqslantslantft\{ \alpha \in J^k(\mathbb{R}^n, \mathbb{R}) : \mathrm exists f \in \mathbb{N}NN(k, K, \mathrm eta), \quad \mathrm exists a \in K, \quad \alpha = j^k f(a) \right\}. \] Clearly, $\mathcal{M}(k, K, \mathrm eta)$ is an open submanifold of $J^k(\mathbb{R}^n, \mathbb{R})$. Since the rank is a semi-continuous map, there exists a neighborhood \( V \subset \mathcal{M}({k_{\I}}, \mathcal{K}x, \mathrm eta) \times \mathcal{K}eps \times \mathcal{K}P \times E_m \) of $(j^{{k_{\I}}}_0(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0)$ such that $H_\mathcal{I}^{m-1}$ has a rank $N$ on $V$. Let $\rho \in (0, \mathrm eta)$ and $\mathcal{C}(\rho) = \mathcal{C}x \times \mathcal{C}eps \times \mathcal{C}P \times \Omega_m$ be a semi-algebraic compact subset of $\mathcal{K}\times E_m$ such that \[ W := \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho) \times \mathcal{C}eps \times \mathcal{C}P \times \Omega_m \subset V. \] Let $B = \big(H_\mathcal{I}^{m-1}|_W \big)^{-1}(0)$ and $Z = \pi(B)$, where $\pi$ is the projection that is parallel to $\mathcal{C}eps \times \mathcal{C}P \times \Omega_m$. Then, and because $\mathcal{C}eps \times \mathcal{C}P \times \Omega_m$ is compact, $Z\subset \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)$ is a closed semi-algebraic subset. Hence, according to the Goresky-McPherson transversality theorem (\cite[Part I, Chapter 1, page 38, Proposition]{GMP}), the set \[ \tilde\OOO(\rho) = \leqslantslantft\{ f \in C^\infty\big(\mathbb{R}^n, \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)\big) \;:\; f|_{\mathcal{C}x} \text{~is transversal to~} Z \right\} \] is open and dense (in the Whitney $C^\infty$ topology) in $C^\infty\big(\mathbb{R}^n, \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)\big)$. Moreover, since $H_\mathcal{I}^{m-1}|_W$ is a submersion, we have $\codim_{\mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)} Z \geqslantslantq \codim_{\mathbb{R}^N}\{0\} - {|\I|}m (\mathcal{C}(\rho)\times E_m) = N - {|\I|}m (\mathcal{C}(\rho)\times E_m)$. Picking $N$ sufficiently large, we have \[ \codim_{\mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)} Z > n \] in which case, transversal necessarily means to avoid. It follows that \begin{align*} \tilde\OOO(\rho) &= \leqslantslantft\{ f \in C^\infty\big(\mathbb{R}^n, \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)\big) \;:\; \forall \hat{x}\in\mathcal{C}x, f(\hat{x}) \notin Z \right\} \\ &= \leqslantslantft\{ f \in C^\infty\big(\mathbb{R}^n, \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)\big) \;:\; \forall (\hat{x}, \varepsilon, \xi, \omega)\in \mathcal{C}(\rho), \big( f(\hat{x}), \varepsilon, \xi, \omega \big) \notin B \right\} \\ &= \leqslantslantft\{ f \in C^\infty\big(\mathbb{R}^n, \mathcal{M}({k_{\I}}, \mathcal{K}x, \rho)\big) \;:\; \forall (\hat{x}, \varepsilon, \xi, \omega)\in \mathcal{C}(\rho), H_\mathcal{I}^{m-1}\big( f(\hat{x}), \varepsilon, \xi, \omega \big) \neq 0 \right\}. \mathrm end{align*} By compactness of $\mathcal{K}\times E_m$, there exists $q \in \mathbb{N}$ such that \begin{equation}\label{E:compacite} \mathcal{K}\times E_m = \bigcup_{i=1}^{q} \mathcal{C}(\rho_i). \mathrm end{equation} Set \( \mathrm eta = \min \{\rho_i \;:\; i=1, \dots, q\}>0, \) \( k = \max \{{k_{\I}}(\rho_i) \;:\; i=1, \dots, q\} \) and define \( \tilde\OOO = \bigcap_{i=1}^q \tilde\OOO(\rho_i). \) According to \mathrm eqref{E:compacite}, \begin{multline*} \tilde\OOO = \Big\{ f \in C^\infty \big(\mathbb{R}^n, \mathcal{M}(k, \mathcal{K}x, \mathrm eta) \big) \;:\; \forall (\hat{x}, \varepsilon, \xi, \omega)\in \mathcal{K}\times E_m,\\ H_{\{0, \dots, k\}}^{m-1} \big(f(\hat{x}), \varepsilon, \xi, \omega \big) \neq 0 \Big\}. \mathrm end{multline*} Also, by definition of $E_{m-1}$ and $E_m$, $H_{\{0\}}^{m-1}(\omega) = CB^{m-1}\omega \neq 0$ for all $\omega\in E_{m-1} \setminus E_m$. Thus, \begin{multline*} \tilde\OOO = \Big\{ f \in C^\infty \big(\mathbb{R}^n, \mathcal{M}(k, \mathcal{K}x, \mathrm eta) \big) \;:\; \forall (\hat{x}, \varepsilon, \xi, \omega)\in \mathcal{K}\times E_{m-1},\\ H_{\{0, \dots, k\}}^{m-1} \big(f(\hat{x}), \varepsilon, \xi, \omega \big) \neq 0 \Big\} \mathrm end{multline*} is an open dense subset of $C^\infty(\mathbb{R}^n, \mathcal{M}(k, \mathcal{K}x, \mathrm eta))$. Then $\mathcal{O}O_{m-1} := \{ \tau\circ f\;:\; f\in \tilde\OOO\}$ where $\tau$ is the target map is an open dense subset of $ \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ and \begin{multline*} \mathcal{O}O_{m-1} = \Big\{ \delta \in \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta) \;:\; \forall (\hat{x}_0, \varepsilon_0, \xi_0, \omega_0)\in \mathcal{K}\times E_{m-1},\\ H_{\{0, \dots, k\}}^{m-1}(j^{k}\delta(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0) \neq 0 \Big\}. \mathrm end{multline*} It concludes the induction and the proof. \mathrm end{proof} \begin{proof}[Proof of Theorem~\ref{Thm:main}] Applying Proposition~\ref{P:main} to $m=0$ and recalling the definition of $H^0_{\{0,\dots,k\}}$, we immediately get the main Theorem~\ref{Thm:main}. \mathrm end{proof} A straightforward consequence of Theorem~\ref{Thm:main} is the following corollary, that deals with the observability of \mathrm eqref{E:observation_system}, as announced in Remark \ref{rk:obs1}. \begin{corollary}\label{obs} Assume that $(C, A)$ and $(C, B)$ are observable pairs. \begingroup Assume that $0\notin \mathcal{K}x$. \endgroup Then there exist $\mathrm eta>0$, $k\in\mathbb{N}$ and an open dense subset $\mathcal{O}O \subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ such that for all $(\delta, \hat{x}_0, \varepsilon_0, \xi_0)\in \mathcal{O}O \times \mathcal{K}$, system~\mathrm eqref{E:observation_system} is observable in any time $T>0$ for the control $u = (\lambda + \delta) \circ \hat{x}$, where $\hat{x}$ follows \mathrm eqref{E:kalman_coupled} with initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$ and feedback perturbation $\delta$. \mathrm end{corollary} \begin{proof} Applying Proposition~\ref{P:main} to $m = 0$, we find that there exist $\mathrm eta>0$, $k\in\mathbb{N}$ and an open dense subset $\mathcal{O}O \subset \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)$ such that for all $(\delta, \hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0) \in \mathcal{O}O\times \mathcal{K}\times E_0$, $H^0_{\{0, \dots, k\}}(j^{k}\delta(\hat{x}_0), \varepsilonilon_0, \xi_0, \omega_0) \neq 0$. Let $(\delta, \hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0) \in \mathcal{O}O\times \mathcal{K} \times \mathbb S^{n-1}$, and let $(\hat{x}, \varepsilonilon, \xi, \omega)$ denote the solution of \mathrm eqref{E:kalman_coupled} with initial conditions $(\hat{x}_0, \varepsilonilon_0, \xi_0, \omega_0)$. From the definition of $H^0_{\{0, \dots, k\}}$ it follows that there exists $i\in\mathbb{N}$ such that $C\omega^{(i)}(0) \neq 0$. Consequently, $C\omega|_{[0, T]} \not\mathrm equiv 0$, which was to be proved. \mathrm end{proof} As stated in Remark \ref{rk:obs1}, we now want to complete the compact $\mathcal{K}x$ with a neighborhood of zero as in Corollary~\ref{Cor:main}. We do so in the following section. \subsection{Observability near the target and proof of Corollary~\ref{Cor:main}} \label{sec:target} We use Theorem~\ref{Thm:main} to prove Corollary~\ref{Cor:main}. In order to do so, we need the following notations and lemmas. For any control $u\in C^\infty(\mathbb{R}_+,\mathbb{R})$, let $\Phi_u:\mathbb{R}_+\to \mathrm{End} (\mathbb{R}^n)$ be the flow of the time-varying linear ordinary differential equation~\mathrm eqref{E:omega}. So $\Phi_u(t)\omega_0$ is the solution of \mathrm eqref{E:omega} at time $t\in\mathbb{R}_+$ with initial condition $\omega_0\in\mathbb{R}^n$. Notice for instance that $\Phi_0(t)=\mathrm e^{At}$. Recall that an input $u\in C^\infty(\mathbb{R}_+, \mathbb{R})$ is said to make system~\mathrm eqref{E:observation_system} observable in time $T>0$ if for all $\omega_0\in \mathbb S^{n-1}$ there exists $t\in[0, T]$ such that $C\Phi_u(t)\omega_0\neq 0$. \begin{lemma}\label{L:lemma1} Let $T>0$, $\mathrm eta_0 = \max\{\leqslantslantft| C\Phi_0(t)\omega_0 \right|\;:\; t\in [0,T],~\omega_0\in \mathbb S^{n-1} \}$ and $u\in C^\infty(\mathbb{R}_+, \mathbb{R})$. If \begin{equation}\label{E:hypo-lem} \forall t\in [0,T], \forall \omega_0\in \mathbb S^{n-1},\quad \leqslantslantft| C\Phi_u(t) \omega_0-C\Phi_0(t)\omega_0 \right| < \mathrm eta_0, \mathrm end{equation} then $u$ makes system~\mathrm eqref{E:observation_system} observable in time $T$. \mathrm end{lemma} \begin{proof} Let $t\in[0, T]$ and $\omega_0\in \mathbb S^{n-1}$ be such that $|C\Phi_0(t)\omega_0| = \mathrm eta_0$. Using \mathrm eqref{E:hypo-lem}, we get \begin{align*} \leqslantslantft| C\Phi_u(t) \omega_0 \right| &\geqslantslantq \leqslantslantft| C\Phi_0(t) \omega_0 \right| - \leqslantslantft| C\Phi_u(t) \omega_0-C\Phi_0(t) \omega_0 \right| > 0, \mathrm end{align*} which shows that $u$ makes system~\mathrm eqref{E:observation_system} observable in time $T$. \mathrm end{proof} \begin{lemma}\label{L:lemma2} Let $T>0$. Let $M=\sup\{\|\Phi_0(t)\|\;:\; t\in [0,T]\}$. Let $u\in C^\infty(\mathbb{R}_+, \mathbb{R})$ and let $u_M=\sup\{|u(t)|\;:\; t\in [0,T]\}$. Then there exists a constant $K>0$ such that for all $t\in [0,T]$ and all $\omega_0\in \mathbb S^{n-1}$, \begin{equation}\label{E:estimation} \leqslantslantft| \Phi_u(t)\omega_0 -\Phi_0(t)\omega_0 \right| < MKu_M\mathrm e^{K u_M}. \mathrm end{equation} \mathrm end{lemma} \begin{proof} By the variation of constants formula, for all $t\in [0,T]$ and all $\omega_0\in\mathbb S^{n-1}$, $$ \Phi_u(t)\omega_0 -\Phi_0(t)\omega_0 = \int_0^t\Phi_0(t-s)Bu(s)\Phi_u(s){|\I|}f s \, \omega_0. $$ Iterating integrals, we get a (formal) series expansion \begin{equation}\label{E:interated_int} \int_0^{s_0}\Phi_0(s_0-s_1)Bu(s)\Phi_u(s){|\I|}f s_1 = \sum_{k=0}^{+\infty} J_k \mathrm end{equation} where $$ J_k = \int_{0}^{s_0} \dotsi \int_{0}^{s_k} \Psi_k(s_0,\dots,s_{k+1}) \Phi_0(s_{k+1}) u(s_0) \dotsi u(s_{k+1}) {|\I|}f s_1 \dotsm {|\I|}f s_{k+1} $$ with $\Psi_k(s_0,\dots,s_{k+1})=\Phi_0(s_0-s_1) B \dotsm \Phi_0(s_{k}-s_{k+1}) B$. Then $ \|\Psi_k(s_0,\dots,s_{k+1})\|\leqslantslantq M^{k+1} \|B\|^{k+1} $ and $$ \|J_k\|\leqslantslantq M^{k+2} \|B\|^{k+1} u_M^{k+1}\int_{0}^{s_0} \dotsi \int_{0}^{s_k} {|\I|}f s_1 \dotsm {|\I|}f s_{k+1} \leqslantslantq M^{k+2} \|B\|^{k+1} u_M^{k+1} \frac{T^{k+1}}{(k+1)!}. $$ Thus $$ \begin{aligned} \sum_{k=0}^{+\infty} \|J_k\| &\leqslantslantq \sum_{k=0}^{+\infty} M^{k+2} \|B\|^{k+1} u_M^{k+1} \frac{T^{k+1}}{(k+1)!} \\ &\leqslantslantq M^{2} \|B\| u_M T \sum_{k=0}^{+\infty} M^{k} \|B\|^{k} u_M^{k} \frac{T^{k}}{k!} \mathrm end{aligned} $$ which proves the convergence of the series expansion~\mathrm eqref{E:interated_int} and inequality \mathrm eqref{E:estimation} with \( K = M\|B\|T. \) \mathrm end{proof} \begin{proposition}\label{Prop_zero} Assume that the pair $(C,A)$ is observable. Assume that $0$ is in the interior of $\mathcal{K}x$. Let $T>0$. Then there exists $\mathbb{R}RR>0$ such that $B(0,\mathbb{R}RR)\subset\mathcal{K}x$ and $\eta_1>0$ such that the following property holds: Let $(\hat{x},\varepsilon,\xi,\omega)$ be the solution of \mathrm eqref{E:kalman_coupled} with initial condition $(\hat{x}_0,\varepsilon_0,\xi_0,\omega_0)\in B(0,\mathbb{R}RR)\times \mathbb{R}^n\times\Prics\times \mathbb S^{n-1}$. Let $\delta\in{C^\infty(\R^n, \R)}$ such that $\delta(0)=0$ and $\sup\{|\delta(x)|\;:\; x\in\mathcal{K}x\}<\eta_1$. If $\hat{x}(t)\in B(0,\mathbb{R}RR)$ for all $t\in [0,T]$, then the control $u:t\mapsto (\lambda+\delta)(\hat{x}(t))$ makes system~\mathrm eqref{E:observation_system} observable in time $T$. \mathrm end{proposition} \begin{proof} Let $T>0$ and $\mathrm eta_0$ be as in the statement of Lemma~\ref{L:lemma1}. The observability of the pair $(C,A)$ yields $\mathrm eta_0>0$. Let $\eta_1>0$ be such that $MK\eta_1\mathrm e^{K \eta_1} < \mathrm eta_0$. For all $R>0$ and all $\delta\in{C^\infty(\R^n, \R)}$ satisfying $\delta(0)=0$ and $\sup\{|\delta(x)|\;:\; x\in\mathcal{K}x\}<\eta_1$, let $u_M(\mathbb{R}RR, \delta)=\sup\{|(\lambda+\delta)(x)|\;:\; x\in B(0,\mathbb{R}RR)\}$. Since $\lambda+\delta$ is continuous and $\lambda(0)=\delta(0)=0$, $u_M(\cdot, \delta)$ is a continuous non decreasing function on $\mathbb{R}_+$ such that $u_M(0, 0)=0$ and $u_M(\mathbb{R}RR, \delta) \leqslantslantq u_M(\mathbb{R}RR, 0) + \eta_1$. Then, we can choose $\mathbb{R}RR>0$ such that $MK(u_M(\mathbb{R}RR, 0) + \eta_1)\mathrm e^{K (u_M(\mathbb{R}RR, 0) + \eta_1)}< \mathrm eta_0$. Since $u_M(\cdot, 0)$ is non decreasing, it is possible to choose $\mathbb{R}RR$ such that $B(0,\mathbb{R}RR)\subset\mathcal{K}x$. Now, fix $\delta\in{C^\infty(\R^n, \R)}$ satisfying $\delta(0)=0$ and $\sup\{|\delta(x)|\;:\; x\in\mathcal{K}x\}<\eta_1$. Let $(\hat{x},\varepsilon,\xi,\omega)$ be the solution of \mathrm eqref{E:kalman_coupled} with initial condition $(\hat{x}_0,\varepsilon_0,\xi_0,\omega_0)\in B(0,\mathbb{R}RR)\times \mathbb{R}^n\times\Prics\times \mathbb S^{n-1}$. Then $MKu_M(\mathbb{R}RR, \delta)\mathrm e^{K u_M(\mathbb{R}RR, \delta)}< \mathrm eta_0$. Hence, from Lemmas \ref{L:lemma1} and \ref{L:lemma2}, if $\hat{x}(t)\in B(0,\mathbb{R}RR)$ for all $t\in [0,T]$, then the control $u:t\mapsto (\lambda+\delta)(\hat{x}(t))$ makes system~\mathrm eqref{E:observation_system} observable in time $T$. \mathrm end{proof} \begingroup \begin{proof}[Proof of Corollary~\ref{Cor:main}] Let $\mathbb{R}RR>0$ and $\eta_1$ be as in Proposition~\ref{Prop_zero}. Let $\mathbb{R}R\in (0, \mathbb{R}RR)$ and $\rho\in(0, \mathbb{R}R)$. We apply Corollary~\ref{obs} to the compact $\mathcal{K}x \setminus B(0, \mathbb{R}R)$. Since the statement holds for some $\mathrm eta$ small enough, we assume without loss of generality that $\mathrm eta<\eta_1$: there exist $\mathrm eta\in(0, \mathrm eta_1)$, $k\in\mathbb{N}$ and an open dense subset $\mathcal{O}O \subset \mathbb{N}NN(k, \mathcal{K}x\setminus B(0, \mathbb{R}R), \mathrm eta)$ such that for all $(\delta, \hat{x}_0, \varepsilon_0, \xi_0)\in \mathcal{O}O \times \leqslantslantft(\mathcal{K}x \setminus B(0, \mathbb{R}R)\right) \times \mathcal{K}eps \times \mathcal{K}P$, system~\mathrm eqref{E:observation_system} is observable in any time $T>0$ for the control $u = (\lambda + \delta) \circ \hat{x}$, where $\hat{x}$ follows \mathrm eqref{E:kalman_coupled} with initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$ and feedback perturbation $\delta$. Let $$\mathcal{O}O' = \leqslantslantft\{{\tilde\delta}\in \mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)\cap VV_\rho\;:\; \mathrm exists \delta\in\mathcal{O}O, \forall x\in \mathcal{K}x\setminus B(0, \mathbb{R}R), {\tilde\delta}(x)=\delta(x)\right\}.$$ Then $\mathcal{O}O'$ is open and dense in $\mathbb{N}NN(k, \mathcal{K}x, \mathrm eta)\cap VV_\rho$ (in the Whitney $C^\infty$ induced topology) since $\mathcal{O}O$ is open and dense in $\mathbb{N}NN(k, \mathcal{K}x\setminus B(0, \mathbb{R}R), \mathrm eta)$. Moreover, if ${\tilde\delta}\in\mathcal{O}O'$, then system~\mathrm eqref{E:observation_system} is still observable in any time $T>0$ for the control $u = (\lambda + {\tilde\delta}) \circ \hat{x}$ with initial conditions $(\hat{x}_0, \varepsilon_0, \xi_0)$ in $\leqslantslantft(\mathcal{K}x \setminus B(0, \mathbb{R}R)\right) \times \mathcal{K}eps \times \mathcal{K}P$. Let $({\tilde\delta}, \hat{x}_0, \varepsilon_0, \xi_0)\in \mathcal{O}O' \times \mathcal{K}$. If $\hat{x}_0\notin B(0, \mathbb{R}R)$, then the result holds from above. On the other hand, assume that $\hat{x}_0\in B(0, \mathbb{R}R)$. If $\hat{x}(t)\in B(0, \mathbb{R}RR)$ for all $t\in[0, T]$, then according to Proposition~\ref{Prop_zero}, \mathrm eqref{E:observation_system} is observable in time $T$ for the control $u = (\lambda + {\tilde\delta}) \circ \hat{x}$. Otherwise, there exists $t_0\in(0, T)$ such that $\hat{x}(t_0)\notin B(0, \mathbb{R}R)$. Apply Corollary~\ref{obs} with the new initial condition $(\hat{x}(t_0), \varepsilon(t_0), \xi(t_0))$ and with the same perturbation ${\tilde\delta}$. Then \mathrm eqref{E:observation_system} is observable in time $T>t_0$ for the control $u = (\lambda + {\tilde\delta}) \circ \hat{x}$. \mathrm end{proof} \begin{proof}[Proof of Corollary~\ref{cor:feedback}] Let $T>0$ and $\lambda\in\Lambda$. Let $\mathbb{R}RR$, $\mathrm eta$, $k$ and $\mathcal{O}O$ be as in Corollary~\ref{Cor:main}. Since $\mathcal{O}O$ is dense (in the Whitney $C^\infty$ topology) in $\mathbb{N}NN(k, \mathcal{K}x, \mathrm eta) \cap VV_\mathbb{R}RR$, for all neighborhood $\mathscr U$ of $\lambda\in \Lambda$, there exists $\delta \in \mathcal{O}O$ such that $\lambda+\delta\in \mathscr U\capFF_T$. Hence, $FF_T$ is a dense subset of $\Lambda$. Moreover, \begin{align*} FF_T &= \leqslantslantft\{\lambda\in\Lambda\;:\; \forall (\hat{x}_0, \varepsilon_0, \xi_0, \omega_0)\in\mathcal{K}\times\mathbb S^{n-1}, \mathrm exists t\in[0, T], C\omega(t)\neq0 \right\}\\ &= \bigcap_{(\hat{x}_0, \varepsilon_0, \xi_0, \omega_0)\in\mathcal{K}\times\mathbb S^{n-1}} h_{\hat{x}_0, \varepsilon_0, \xi_0, \omega_0}^{-1}(\mathcal{C}^\infty([0, T], \mathbb{R})\setminus\{0\}) \mathrm end{align*} where $h_{\hat{x}_0, \varepsilon_0, \xi_0, \omega_0}:\Lambda\to C^\infty([0, T], \mathbb{R})$ is given by $h_{\hat{x}_0, \varepsilon_0, \xi_0, \omega_0}(\lambda) = C\omega|_{[0, T]}$ where $\omega$ is the solution of \mathrm eqref{E:kalman_coupled} with initial condition $(\hat{x}_0, \varepsilon_0, \xi_0, \omega_0)$ and $\delta\mathrm equiv0$. The map $h$ is continuous, the set $\mathcal{C}^\infty([0, T], \mathbb{R})\setminus\{0\}$ is open and the set $\mathcal{K}\times\mathbb S^{n-1}$ is compact. Thus $FF_T$ is open in $\Lambda$. \mathrm end{proof} \endgroup \section{Application to classical observers} \label{sec:appli} In this section, we show that there exist observers such that the key hypotheses~\ref{FC} and \ref{NFOT} are satisfied. In particular, we show that both the Luenberger observer and the Kalman observer satisfy these hypotheses, as stated in Theorem~\ref{thm:appli}. Hence, the main Theorem~\ref{Thm:main} and its Corollary~\ref{Cor:main} apply to these observers. While \ref{FC} has already been studied for such observers (see \mathrm emph{e.g.~} \cite{Besancon, Gauthier_book}), \ref{NFOT} is more difficult to check, and relies on the fact that the observer dynamics is somehow compatible with the Kalman observability decomposition. For the sake of generality, we state the results of this section for an arbitrary output dimension $m$ (\mathrm emph{i.e.~} $C\in\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$). Let $\Prics_n \subset \mathrm{End}(\mathbb{R}^n)$ denote the subset of real positive-definite symmetric endomorphism on $\mathbb{R}^n$. Regarding hypothesis~\ref{FC}, the following result is well-known. \begin{proposition}\label{prop:fc} Assume that $\lambda$ is bounded over $D(\lambda)$. Let $Q\in\Prics_n$. For all $\xi\in\Prics_n$ and all $u\in\mathbb{R}$, consider the following well-known observers: \begin{align} &f^\mathrm{Luenberger}(\xi, u) = 0\tag{Luenberger observer}\\ &f^\mathrm{Kalman}_{Q}(\xi, u) = \xi {C^\infty(\R^n, \R)}u{u}^* + {C^\infty(\R^n, \R)}u{u}\xi + Q - \xi C^*C\xi \tag{Kalman observer} \mathrm end{align} and $\mathcal{L}L(\xi) = \xi C^*$. Then the coupled system~\mathrm eqref{E:kalman_coupled} given by $(f, \mathcal{L}L)$ satisfies the hypothesis~\ref{FC} for any $f\in\{f^\mathrm{Luenberger}, f^\mathrm{Kalman}_{Q}\}$. \mathrm end{proposition} Let us investigate hypothesis~\ref{NFOT}. First, we state sufficient conditions for it to hold, and then show that they are satisfied by both the Kalman and Luenberger observers. For all $A_0\in C^{\infty}\leqslantslantft(\mathbb{R}_+,\mathrm{End}(\mathbb{R}^n)\right)$ and for all $C_0 \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$, let $f(\cdot, A_0, C_0)$ be a forward complete time-varying vector field over $\Prics_n$. Let $\mathcal{L}L:\xis_n\times\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m) \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$. For all $T\in\mathrm{GL}(\mathbb{R}^n)$, for all $(\bar{A}, \bar{C})\in\mathrm{End}(\mathbb{R}^n)\times\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ and for all $\xi\in \Prics_n$, let $(\bar{f}, \bar{\mathcal{L}L})$ be defined by \begin{align} \begin{cases} \bar{f}(T\xi T^*, T\bar{A}T^{-1}, \bar{C}T^{-1}) = Tf(\xi, \bar{A}, \bar{C})T^*\\ \bar{\mathcal{L}L}(T\xi T^*, \bar{C}T^{-1}) = T\mathcal{L}L(\xi, \bar{C}). \mathrm end{cases}\label{Hyp:T} \mathrm end{align} For all $(\bar{A}, \bar{C}, \bar{b}) \in \mathrm{End}(\mathbb{R}^n) \times \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)\times\mathbb{R}^n$, we consider the following dynamical observer system \begin{equation}\label{E:Observer_system_const} \leqslantslantft\{ \begin{aligned} &\dot{\hat{x}}=\bar{A} \hat{x} + \bar{b} - \bar{\mathcal{L}L}(\xi, \bar{C})\bar{C}\varepsilon \\ &\dot{\varepsilon}=\leqslantslantft( \bar{A} -\bar{\mathcal{L}L}(\xi, \bar{C})\bar{C} \right) \varepsilon \\ &\dot{\xi}= \bar{f}(\xi,\bar{A}, \bar{C}). \mathrm end{aligned} \right. \mathrm end{equation} For all $k\in\{1,\dots,n\}$, let $(\bar{A}, \bar{C}) \in \mathrm{End}(\mathbb{R}^n) \times \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ having the following structure: \begin{equation}\label{eq:struct} \begin{aligned} \bar{A} = \begin{pmatrix} A_{11} & 0\\ A_{21} & A_{22} \mathrm end{pmatrix},\qquad \bar{C} = \begin{pmatrix} C_1 & 0 \mathrm end{pmatrix}, \mathrm end{aligned} \mathrm end{equation} with suitable matrices $A_{11} \in \mathrm{End}(\mathbb{R}^k)$, $A_{21} \in \mathcal{L}(\mathbb{R}^k, \mathbb{R}^{n-k})$, $A_{22} \in \mathrm{End}(\mathbb{R}^{n-k})$ and $C_1 \in \mathcal{L}(\mathbb{R}^k, \mathbb{R}^m)$. For any solution of \mathrm eqref{E:Observer_system_const}, set similarly \begin{equation*} \begin{aligned} \hat{x} = \begin{pmatrix} \hat{x}_{1} \\ \hat{x}_{2} \mathrm end{pmatrix},\quad \varepsilon = \begin{pmatrix} \varepsilon_{1} \\ \varepsilon_{2} \mathrm end{pmatrix},\quad \bar{b} = \begin{pmatrix} b_{1} \\ b_{2} \mathrm end{pmatrix},\quad \xi = \begin{pmatrix} \xi_{11} & \xi_{12}\\ \xi^*_{12} & \xi_{22} \mathrm end{pmatrix}. \mathrm end{aligned} \mathrm end{equation*} \begin{proposition}\label{prop:appli} Assume that the pair $(C, A)$ is observable. Assume that for all $T\in\mathrm{GL}(\mathbb{R}^n)$, for all $(\bar{f}, \bar{\mathcal{L}L})$ as in \mathrm eqref{Hyp:T}, for all $k\in\{1,\dots,n\}$ and for all $(\bar{A}, \bar{C}) \in \mathrm{End}(\mathbb{R}^n) \times \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ as in \mathrm eqref{eq:struct}, the following hypotheses hold. \begin{enumerate}[label={\textnormal{H\arabic*.}}, ref={(\textnormal{H\arabic*})}] \item There exists $(f_{11}, \mathcal{L}L_1)$ such that \begin{equation}\label{E:kalman_coupled_obs} \leqslantslantft\{ \begin{aligned} &\dot{\hat{x}}_1= A_{11}\hat{x}_1 + b_1 - \mathcal{L}L_1(\xi_{11}, C_1)C_1 \varepsilon_1 \\ &\dot{\varepsilon}_1=\leqslantslantft(A_{11} - \mathcal{L}L_1(\xi_{11}, C_1)C_1\right) \varepsilon_1 \\ &\dot{\xi}_{11}= f_{11}(\xi_{11}, A_{11}, C_1) \mathrm end{aligned} \right. \mathrm end{equation} where $(f_{11},\mathcal{L}L_1)$ is such that \begin{align*} \bar{f}(\xi,\bar{A}, \bar{C}) = \begin{pmatrix} f_{11}(\xi_{11}, A_{11}, C_1) && *\\ * && * \mathrm end{pmatrix},\qquad \bar{\mathcal{L}L}(\xi, \bar{C}) = \begin{pmatrix} \mathcal{L}L_1(\xi_{11}, C_1)\\ * \mathrm end{pmatrix}. \mathrm end{align*} \label{Hyp:11} \item If $(C_1, A_{11})\in\mathcal{L}(\mathbb{R}^k,\mathbb{R}^m)\times\mathrm{End}(\mathbb{R}^k)$ is an observable pair, then the solutions of \mathrm eqref{E:Observer_system_const} are such that for any initial conditions, ${\mathcal{L}L}_{11}(\xi_{11}(t), {C}_{1}){C}_{1}\varepsilon_{1}(t)\to 0$ as $t\to+\infty$. \label{Hyp:conv} \item For all $\xi_{11}\in\Prics_k$ and all ${C}_{1}\in\mathcal{L}(\mathbb{R}^k,\mathbb{R}^m),\ \ker {\mathcal{L}L}_{1}(\xi_{11}, {C}_{1}) \cap \mathcal{I}mage {C}_1 = \{0\}$. \label{Hyp:ker} \mathrm end{enumerate} Then the coupled system~\mathrm eqref{E:kalman_coupled} given by $(f(\cdot, {C^\infty(\R^n, \R)}u{u}, C),\mathcal{L}L(\cdot, C))$ satisfies the hypothesis~\ref{NFOT}. \mathrm end{proposition} \begin{remark} In the case where $T$ is the identity matrix and $k = n$, \ref{Hyp:11} is clearly satisfied, \ref{Hyp:conv} means that the correction term $\mathcal{L}L(\xi, \bar{C})\bar{C}\varepsilon$ converges to zero for any observable pair $(\bar{A}, \bar{C})$, and \ref{Hyp:ker} means that the correction term is null if and only if $\bar{C}\varepsilon = 0$. We will see in Theorem~\ref{thm:appli} that these hypotheses are clearly satisfied for the Luenberger and Kalman observers. \mathrm end{remark} \begin{remark} Hypothesis~\ref{Hyp:11} can be seen as a compatibility condition between the observer dynamics and the Kalman observability decomposition: when $\bar{A}$ is of the standard form \mathrm eqref{eq:struct}, the observer acts autonomously on the upper left matrix block, which will correspond to the observable part of the system. \mathrm end{remark} This proposition is a consequence of the series of lemmas that follows. Until the end of the proof of Proposition~\ref{prop:appli}, assume that its hypotheses are satisfied. For any $\mu:\mathbb{R}^n\to \mathbb{R}$, $F_\mu$ denotes the vector field over $\mathbb{R}^n$ given by \( F_{\mu}(x)= {C^\infty(\R^n, \R)}u{\mu(x)} x+ b \mu(x). \) \begin{lemma} \label{lem1} For all $R>0$, there exists $\mathrm eta>0$ such that for all $\delta \inVV_R$ satisfying $\sup\{\abs{\delta(x)}\;:\; x\in \mathcal{K}x\} < \mathrm eta$, $0$ is the unique equilibrium point of $F_{\lambda+\delta}$ lying in $\mathcal{K}x$. \mathrm end{lemma} \begin{proof} Let $R>0$ and $\delta\inVV_R$. Let $x\in\mathcal{K}x$ be such that $F_{\lambda+\delta}(x) = 0$. Then, \begin{align*} 0=F_{\lambda+\delta}(x) = F_{\lambda}(x) + \delta(x)(Bx + b). \mathrm end{align*} Then $|F_{\lambda}(x)| = \abs{\delta(x)}|Bx + b|$. Set $C_1 = \inf \{|F_{\lambda}(x)|\;:\; x\in \mathcal{K}x \backslash B(0, R)\} $. Since $0$ is not in the closure of $\mathcal{K}x \backslash B(0, R)$, we get by uniqueness of the equilibrium point of $F_\lambda$ that $C_1>0$. Set also $C_2 = \sup \{|Bx+b|\;:\; x\in \mathcal{K}x\}$. Since $\mathcal{K}x$ is compact, $C_2<+\infty$. Set $\mathrm eta = \frac{C_1}{C_2}$. Assume that $\sup \{\abs{\delta(x)}: x\in \mathcal{K}x\} < \mathrm eta$. Then, \begin{align*} F_{\lambda}(x) & \leqslantslantq \mathrm eta \abs{Bx + b} \leqslantslantq C_1. \mathrm end{align*} Hence $x\in B(0, R)$ by definition of $C_1$. Then $\delta(x) = 0$. Hence $F_{\lambda}(x) = 0$. Thus, $x=0$ since $0$ is the unique equilibrium point of $F_{\lambda}$. Moreover, by definition of $VV_R$, $F_{\lambda+\delta}(0) = 0$. \mathrm end{proof} \begin{lemma} \label{lem2} Assume that the pair $(C, A)$ is observable. Let $(u_0, \hat{x}_0, \varepsilon_0, \xi_0) \in \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^n \times\Prics$. Let $(\hat{x}, \varepsilon, \xi)$ be the solution of \mathrm eqref{E:Observer_system} given by the initial condition $(\hat{x}_0, \varepsilon_0, \xi_0)$ and the constant input $u\mathrm equivu_0$. If $\hat{x}$ is constant, then for all $t\in\mathbb{R}_+$, $\mathcal{L}L(\xi(t), C)C\varepsilon(t) = 0$. \mathrm end{lemma} \begin{proof} Let $(u_0, \hat{x}_0, \varepsilon_0, \xi_0) \in \mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n\times\Prics $. Let $(\hat{x}, \varepsilon, \xi)$ be the solution of \mathrm eqref{E:Observer_system} given by the initial condition $(\hat{x}_0, \varepsilon_0, \xi_0)$ and the constant input $u\mathrm equivu_0$. Assume that $\hat{x}$ is constant, \mathrm emph{i.e.~} $\hat{x} \mathrm equiv \hat{x}_0$. Set $A_0=A+u_0 B$ and $b_0=b u_0 $. Then $\dot{\hat{x}} \mathrm equiv 0$ yields \begin{align*} A_0 \hat{x} + b_0 - \mathcal{L}L(\xi, C)C \varepsilon \mathrm equiv 0. \mathrm end{align*} Since $\hat{x}$ is constant, so is $\mathcal{L}L(\xi)C \varepsilon$. Then, set $K = \mathcal{L}L(\xi, C)C\varepsilon$. It remains to show that $K=0$. Let $k = \rank \mathcal{O}(C, A_0)$ where $\mathcal{O}(C, A_0)$ is defined by \mathrm eqref{E:Kalman-matrix} Since $C\neq 0$ (since $(C, A_0)$ is observable), $k\geqslantslantq1$. According to the Kalman observability decomposition, there exists an invertible endomorphism $T\in\mathrm{GL}(\mathbb{R}^n)$ such that $\bar{A} = TA_0T^{-1}$ and $\bar{C} = CT^{-1}$ have the following structure: \begin{equation} \begin{aligned} \bar{A} = \begin{pmatrix} A_{11} & 0\\ A_{21} & A_{22} \mathrm end{pmatrix},\qquad \bar{C} = \begin{pmatrix} C_1 & 0 \mathrm end{pmatrix}, \mathrm end{aligned} \mathrm end{equation} with suitable matrices $A_{11} \in \mathrm{End}\big(\mathbb{R}^k\big)$, $A_{21} \in \mathcal{L}\big(\mathbb{R}^k, \mathbb{R}^{n-k}\big)$, $A_{22} \in \mathrm{End}\big(\mathbb{R}^{n-k}\big)$ and $C_1 \in \mathcal{L}\big(\mathbb{R}^k, \mathbb{R}^m\big)$. Moreover, the pair $(C_1, A_{11})$ is observable. For the sake of readability, we omit the horizontal bars over the submatrices (for instance, $A_{11}$ is a submatrix of $\bar{A}$ and not of $A$). Similarly, set \begin{equation*} \begin{array}{lllll} \bar{x} = Tx = \begin{pmatrix} x_{1} \\ x_{2} \mathrm end{pmatrix},\qquad && \bar{\hat{x}} = T\hat{x} = \begin{pmatrix} \hat{x}_{1} \\ \hat{x}_{2} \mathrm end{pmatrix},\qquad && \bar{\varepsilon} = T\varepsilon = \begin{pmatrix} \varepsilon_{1} \\ \varepsilon_{2} \mathrm end{pmatrix},\\ \bar{b}_0 = Tb_0 = \begin{pmatrix} b_{1} \\ b_{2} \mathrm end{pmatrix},\qquad && \bar{K} = TK = \begin{pmatrix} K_{1} \\ K_{2} \mathrm end{pmatrix},\qquad && \bar{\xi} = T\xi T^* = \begin{pmatrix} \xi_{11} & \xi_{12}\\ \xi^*_{12} & \xi_{22} \mathrm end{pmatrix}. \mathrm end{array} \mathrm end{equation*} Then, according to \mathrm eqref{Hyp:T}, we have the following observed control system on $\bar{x}$, and the corresponding observer: \begin{equation}\label{E:observation_system_normal} \leqslantslantft\{ \begin{aligned} &\dot{\bar{x}}= \bar{A} \bar{x}+ \bar{b}_0 \\ &y= \bar{C} \bar{x}\\ &\dot{\bar{\hat{x}}}=\bar{A} \bar{\hat{x}} + \bar{b}_0 - \bar{\mathcal{L}L}(\xi, \bar{C})\bar{C} \bar{\varepsilon} \\ &\dot{\bar{\varepsilon}}=\leqslantslantft(\bar{A}- \bar{\mathcal{L}L}(\xi, \bar{C})\bar{C} \right) \bar{\varepsilon} \\ &\dot{\bar{\xi}}= \bar{f}(\bar{\xi}, \bar{A}, \bar{C}). \mathrm end{aligned} \right. \mathrm end{equation} Then, according to hypothesis~\ref{Hyp:11}, we can write \begin{equation} \leqslantslantft\{ \begin{aligned} &\dot{\xi}_{11}= f_{11}(\xi_{11}, A_{11}) \\ &\dot{\hat{x}}_1= A_{11}\hat{x}_1 + b_1 - \mathcal{L}L_1(\xi_{11}, C_1)C_1 \varepsilon_1 \\ &\dot{\varepsilon}_1=\leqslantslantft(A_{11} - \mathcal{L}L_1(\xi_{11}, C_1)C_1\right) \varepsilon_1. \mathrm end{aligned} \right. \mathrm end{equation} Since the pair $(C_1, A_{11})$ is observable, \ref{Hyp:11} and \ref{Hyp:conv} yield $\mathcal{L}L_1(\xi_{11}(t), C_1)C_1\varepsilon_1(t) \to 0$ as $t\to +\infty$. The equality $K_1 = \mathcal{L}L_1(\xi_{11}(t), C_1)C_1\varepsilon_1(t)$ thus yields $K_1 = 0$. Then, by hypotheses~\ref{Hyp:11} and \ref{Hyp:ker}, $\bar{C}\varepsilon\mathrm equiv C_1\varepsilon_1\mathrm equiv0$. Hence $K=0$. Finally, we have $K = T^{-1}\bar{K} = 0$. \mathrm end{proof} \begin{lemma} Let $(\delta, \hat{x}_0, \varepsilon_0, \xi_0) \in {C^\infty(\R^n, \R)}\times \mathcal{K}$. Let $(\hat{x}, \varepsilon, \xi)$ be the solution of \mathrm eqref{E:kalman_coupled} given by $(\delta, \hat{x}_0, \varepsilon_0, \xi_0)$. Set $u_0 = (\lambda+\delta)(\hat{x}_0)$. Let $(\hat{x}an, \varepsilonan, {\Pric}_{\omega})$ be the solution of \mathrm eqref{E:Observer_system} given by the initial condition $(\hat{x}_0, \varepsilon_0, \xi_0)$ and the constant input $u\mathrm equivu_0$. If $\hat{x}^{(i)}(0) = 0$ for all $i\in\mathbb{N}\setminus\{0\}$, then $\hat{x}an$ is constant and \begin{equation} (\varepsilonan^{(k)}(0), {\Pric}_{\omega}^{(k)}(0)) = (\varepsilon^{(k)}(0), \xi^{(k)}(0)) \mathrm end{equation} for all $k\in\mathbb{N}$. \label{lem4} \mathrm end{lemma} \begin{proof} Assume that $\hat{x}^{(i)}(0) = 0$ for all $i\in\mathbb{N}\setminus\{0\}$. Then, for all $i\in\mathbb{N}\setminus\{0\}$, \begin{align} {C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x})}^{(i)}(0) = 0. \label{eqA} \mathrm end{align} According to the ODE version of the Cauchy-Kovalevskaya theorem, $(\hat{x}an, \varepsilonan, {\Pric}_{\omega})$ is analytic in a neighborhood of $0$. Hence, it is sufficient to show that \begin{equation} (\hat{x}an^{(k)}(0), \varepsilonan^{(k)}(0), {\Pric}_{\omega}^{(k)}(0)) = (\hat{x}^{(k)}(0), \varepsilon^{(k)}(0), \xi^{(k)}(0)) \label{eqan} \mathrm end{equation} for all $k\in\mathbb{N}$. By definition of $(\hat{x}, \varepsilon, \xi)$ and $(\hat{x}an, \varepsilonan, {\Pric}_{\omega})$, we have \begin{align*} (\hat{x}an(0), \varepsilonan(0), {\Pric}_{\omega}(0)) = (\hat{x}_0, \varepsilon_0, \xi_0) = (\hat{x}(0), \varepsilon(0), \xi(0)). \mathrm end{align*} Let $k\in\mathbb{N}$. Assume that for all $i\in\intset{0}{k}$, \mathrm eqref{eqan} is satisfied. Then we prove that \mathrm eqref{eqan} is also satisfied for $i=k+1$. Using Faà di Bruno's formula and \mathrm eqref{eqA}, we get \begin{align} \xi^{(k+1)}(0) &= f\leqslantslantft( \xi, {C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x})}, C \right)^{(k)}(0)\nonumber\\ &= f\leqslantslantft( \xi, {C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x}(0))}, C \right)^{(k)}(0)\tag{by \mathrm eqref{eqA}}\\ &= f\leqslantslantft( {\Pric}_{\omega}, {C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x}(0))}, C \right)^{(k)}(0)\tag{by induction hypothesis}\\ &= {\Pric}_{\omega}^{(k+1)}(0).\nonumber \mathrm end{align} Likewise, we obtain \(\varepsilon^{(k+1)}(0) = \varepsilonan^{(k+1)}(0)\) and \(\hat{x}^{(k+1)}(0) = \hat{x}an^{(k+1)}(0)\). \mathrm end{proof} \begin{lemma}\label{l:appli} Assume that the pair $(C, A)$ is observable. Let $(\hat{x}_0, \varepsilon_0, \xi_0) \in \mathcal{K}$. Let $R>0$, $\mathrm eta>0$ as in Lemma \ref{lem1} and $\delta\in VV_R$ satisfying $\sup\{\abs{\delta(x)} \;:\; x\in \mathcal{K}x\} < \mathrm eta$. Let $(\hat{x}, \varepsilon, \xi)$ be the solution of \mathrm eqref{E:kalman_coupled} given by $(\delta, \hat{x}_0, \varepsilon_0, \xi_0)$. If for all $i\in\mathbb{N}\setminus\{0\}$, $\hat{x}^{(i)}(0) = 0$, then $\hat{x} \mathrm equiv \varepsilon \mathrm equiv 0$. \mathrm end{lemma} \begin{proof} Assume that for all $i\in\mathbb{N}\setminus\{0\}$, $\hat{x}^{(i)}(0) = 0$. Set $u_0 = (\lambda+\delta)(\hat{x}_0)$. Let $(\hat{x}an, \varepsilonan, {\Pric}_{\omega})$ be the solution of \mathrm eqref{E:kalman_coupled} given by the initial condition $(\hat{x}_0, \varepsilon_0, \xi_0)$ and the constant input $u\mathrm equivu_0$. According to Lemma \ref{lem4}, $\hat{x}an \mathrm equiv \hat{x}_0$ and for all $k\in \mathbb{N}$, $(\varepsilonan^{(k)}(0), {\Pric}_{\omega}^{(k)}(0)) = (\varepsilon^{(k)}(0), \xi^{(k)}(0))$. Then, by Lemma \ref{lem2}, we get that $\mathcal{L}L({\Pric}_{\omega}, C)C\varepsilonan \mathrm equiv 0$. Hence, ${C^\infty(\R^n, \R)}u{u_0}\hat{x}an + b u_0 \mathrm equiv 0$ \mathrm emph{i.e.~} ${C^\infty(\R^n, \R)}u{(\lambda+\delta)(\hat{x}_0)}\hat{x}an(t) + b (\lambda+\delta)(\hat{x}_0) = 0$ for all $t\in\mathbb{R}_+$. In particular, at $t=0$ we have that $F_{\lambda+\delta}(\hat{x}_0) = 0$. Hence, from Lemma \ref{lem1}, $\hat{x}_0 = 0$. By uniqueness of the solution of \mathrm eqref{E:kalman_coupled} for a given initial condition, it remains to prove that $\varepsilon_0 = 0$ in order to get that $\hat{x} \mathrm equiv \varepsilon \mathrm equiv 0$. Since the pair $(C, A)$ is observable, it is sufficient to prove that $CA^k\varepsilon_0 = 0$ for all $k\in\mathbb{N}$. We proceed by induction. From Lemma \ref{lem2}, $\mathcal{L}L({\Pric}_{\omega}(0), C)C\varepsilonan(0) = 0$. Then, according to hypothesis~\ref{Hyp:ker}, $C\varepsilon_0 = C\varepsilonan(0) = 0$. Let $k\in \mathbb{N}$. Assume that $CA^i\varepsilon_0 = 0$ for all $i\in\intset{0}{k-1}$. We prove in the following that $CA^k\varepsilon_0 = 0$. From Lemma \ref{lem2}, $(\mathcal{L}L({\Pric}_{\omega}, C)C\varepsilonan)^{(i)}(0) = 0$ for all $i\in\mathbb{N}$. Hence, by Lemma \ref{lem4}, we get for all $i\in\mathbb{N}$, $(\mathcal{L}L(\xi, C)C\varepsilon)^{(i)}(0) = (\mathcal{L}L({\Pric}_{\omega}, C)C\varepsilonan)^{(i)}(0) = 0$ and then $C\varepsilon^{(i)}(0) = C{C^\infty(\R^n, \R)}u{u_0}^i\varepsilon_0 = CA^i\varepsilon_0$ since $u_0 = (\lambda+\delta)(\hat{x}_0) =(\lambda+\delta)(0) = 0 $. Then, \begin{align} 0 &= (\mathcal{L}L({\Pric}_{\omega}, C)C\varepsilonan)^{(k)}(0) \tag{by Lemma \ref{lem2}}\\ &= (\mathcal{L}L(\xi, C)C\varepsilon)^{(k)}(0) \tag{by Lemma \ref{lem4}}\\ &= \sum_{i=0}^k \binom{k}{i} \mathcal{L}L(\xi, C)^{(k-i)}(0) C \varepsilon^{(i)}(0) \tag{by Leibniz rule}\\ &= \sum_{i=0}^k \binom{k}{i} \mathcal{L}L(\xi, C)^{(k-i)}(0) C A^i\varepsilon_0 \nonumber\\ &=\mathcal{L}L(\xi_0, C)C A^k\varepsilon_0. \tag{by induction hypothesis} \mathrm end{align} Thus, by hypothesis~\ref{Hyp:ker}, $C A^k\varepsilon_0 = 0$, which concludes the induction and the proof. \mathrm end{proof} This concludes the series of lemmas necessary to prove Proposition~\ref{prop:appli} and Theorem~\ref{thm:appli}. \begin{proof}[Proof of Proposition~\ref{prop:appli}] The statement follows directly from the contrapositive of Lemma~\ref{l:appli}. \mathrm end{proof} \begin{proof}[Proof of Theorem~\ref{thm:appli}] Recall that, according to Proposition~\ref{prop:fc}, the Luenberger observer and the Kalman observer satisfy \ref{FC}. It remains to show that the sufficient conditions stated in the Proposition~\ref{prop:appli} are satisfied by these observers to conclude the proof of Theorem~\ref{thm:appli}. Let $Q\in\Prics_n$. For all $(\bar{A}, \bar{C})\in\mathrm{End}(\mathbb{R}^n)\times\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ and all $\xi\in \Prics_n$, let \begin{align} &f^\mathrm{Luenberger}(\xi, \bar{A}, \bar{C}) = 0\tag{Luenberger observer}\\ &f^\mathrm{Kalman}_{Q}(\xi, \bar{A}, \bar{C}) = \xi \bar{A}^* + \bar{A} \xi + Q - \xi\bar{C}^*\bar{C}\xi \tag{Kalman observer} \mathrm end{align} and $\mathcal{L}L(\xi, C) = \xi \bar{C}^*$. Let $f\in\{f^\mathrm{Luenberger}, f^\mathrm{Kalman}_{Q}\}$. According to Proposition \ref{prop:fc}, the time-varying vector field $f$ is forward complete. For all $T\in\mathrm{GL}(\mathbb{R}^n)$, for all $(\bar{A}, \bar{C})\in\mathrm{End}(\mathbb{R}^n)\times\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ and for all $\xi\in \Prics_n$, let $(\bar{f}, \bar{\mathcal{L}L})$ be defined by \begin{align} \begin{cases} \bar{f}(T\xi T^*, T\bar{A}T^{-1}, \bar{C}T^{-1}) = Tf(\xi, \bar{A}, \bar{C})T^*\\ \bar{\mathcal{L}L}(T\xi T^*, \bar{C}T^{-1}) = T\mathcal{L}L(\xi, \bar{C}). \mathrm end{cases} \mathrm end{align} Then \begin{align*} \bar{\mathcal{L}L}(T\xi T^*, \bar{C}T^{-1}) = T\mathcal{L}L(\xi, \bar{C}) = T \xi \bar{C}^* = T \xi T^* (\bar{C} T^{-1})^* = \mathcal{L}L(T\xi T^*, \bar{C}T^{-1}). \mathrm end{align*} Hence $\bar{\mathcal{L}L} = \mathcal{L}L$. Moreover, if $f = f^\mathrm{Luenberger}$, then $\bar{f} = f = 0$. Otherwise, if $f = f^\mathrm{Kalman}_{Q}$ and then \begin{align*} \bar{f}(T\xi T^*, T\bar{A}T^{-1}, \bar{C}T^{-1}) &= Tf(\xi, \bar{A}, \bar{C})T^*\\ &= T\xi \bar{A}^* + \bar{A} \xi + Q - \xi\bar{C}^*\bar{C}\xi T^*\\ &= T\xi T^* (T\bar{A}T^{-1})^* + (T\bar{A}T^{-1}) T\xi T^* \\ & \quad + TQT^* - T\xi T^* (\bar{C}T^{-1})^*\bar{C}T^{-1} T\xi T^*\\ &= f^\mathrm{Kalman}_{TQT^*}(T\xi T^*, T\bar{A}T^{-1}, \bar{C}T^{-1}), \mathrm end{align*} Hence it is sufficient to prove that, for all $(\bar{A}, \bar{C}) \in \mathrm{End}(\mathbb{R}^n) \times \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ satisfying \mathrm eqref{eq:struct}, $(f, \mathcal{L}L)$ satisfies hypotheses~\ref{Hyp:11}, \ref{Hyp:conv} and \ref{Hyp:ker}. Hypothesis~\ref{Hyp:11} requires some computations to check that if $(\bar{A}, \bar{C})$ is of the form \mathrm eqref{eq:struct}, then \mathrm eqref{E:kalman_coupled_obs} is satisfied with \begin{align} f_{11}(\xi_{11}, \bar{A}_{11}, \bar{C}_{1}) = \begin{cases} 0 & \text{ if } f = f^\mathrm{Luenberger}\\ \xi_{11} \bar{A}_{11}^* + \bar{A}_{11} \xi_{11} + Q_{11} - \xi_{11}\bar{C}_{1}^*\bar{C}_{1}\xi_{11} & \text{ if } f = f^\mathrm{Kalman}_{Q} \mathrm end{cases} \mathrm end{align} and $\mathcal{L}L_1(\xi_{11}, \bar{C}_{1}) = \xi_{11}\bar{C}_{1}^*$. Hence, for any $f\in\{f^\mathrm{Luenberger}, f^\mathrm{Kalman}_{Q}\}$, $f_{11}$ is an observer of the same form than $f$ acting on $\mathbb{R}^k$. Hypothesis~\ref{Hyp:conv} follows from the fact that these well-known observers guaranty that the correction term $\mathcal{L}L_1(\xi_{11}, \bar{C}_1)\bar{C}_1\varepsilon_{1}$ goes to 0 as soon as the pair $(\bar{C}_1, \bar{A}_{11})$ is observable (see \mathrm emph{e.g.~} \cite[Chapter 1, Theorems 3 and 4]{Besancon}). Hypothesis~\ref{Hyp:ker} is clear: for all $\xi_{11}\in\Prics_k$ and all $\bar{C}_1\in\mathcal{L}(\mathbb{R}^k,\mathbb{R}^m)$, if $\varepsilon_1\in\mathbb{R}^k$ is such that $\xi_{11} \bar{C}_1^* \bar{C}_1 \varepsilon_1 = 0$, then $\bar{C}_1\varepsilon_1 = 0$ since $\xi_{11}$ is invertible. Thus the conclusion of Proposition~\ref{prop:appli} holds. \mathrm end{proof} \section*{Acknowledgments} The authors would like to thank Vincent Andrieu and Daniele Astolfi for many fruitful discussions. \begingroup They would also like to thank the anonymous reviewer, for suggesting the addition of Corollary~\ref{cor:feedback} to the paper. \endgroup \mathrm end{document}

\betagin{document} \title{Quadratic points on modular curves with infinite Mordell--Weil group} \author{Josha Box} \betagin{abstract} Bruin and Najman \cite{bruin} and Ozman and Siksek \cite{ozman} have recently determined the quadratic points on each modular curve $X_0(N)$ of genus 2, 3, 4, or 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell--Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on $X_0(N)^+$, and we specify the exceptional quadratic points on $X_0(N)$ not coming from $X_0(N)^+$. In particular we determine the $j$-invariants of the corresponding elliptic curves and whether they are $\mathbb{Q}$-curves or have complex multiplication. \end{abstract} \date{\today} \thanks{2010 \emph{Mathematics Subject Classification.} 11G05, 14G05, 11G18.\\ \mathbf{1}ent \emph{Key words and phrases}. Modular Curves, Quadratic Points, Mordell--Weil, Jacobian, Chabauty. \\ \mathbf{1}ent During the work on this article, the author was supported by an EPSRC DTP studentship.} \maketitle \tableofcontents \section{Introduction} Let $N$ be a positive integer. In a celebrated paper \cite{mazur1}, Mazur determined exactly which modular curves $X_1(N)$ admit non-cuspidal rational points and, for prime values of $N$, he repeated this for $X_0(N)$ \cite{mazur0}. These results for $X_0(N)$ were later extended to composite levels by Kenku \cite{kenku}. Since this work of Mazur, people have been interested in obtaining similar results for points of low degree $d$. For $d=2$ (Kamienny \cite{kamienny}), $d=3$ (Parent \cite{parent}), and $d\in \{4,5,6\}$ (Derickx, Kamienny, Stein and Stoll \cite{dkss}), the prime values of $N$ such that $X_1(N)$ has a non-cuspidal degree $d$ point have been determined explicitly. Moreover, Merel's uniform boundedness theorem \cite{merel} proves the existence of an upper bound $B_d$ such that $X_1(N)$ has no non-cuspidal points for primes $N\geq B_d$. The low degree points on $X_0(N)$, however, are naturally more abundant than on $X_1(N)$, which complicates their study. There are two obvious potential sources of infinitely many quadratic points on $X_0(N)$: a degree 2 map $X_0(N)\to \mathbb{P}^1$ over $\mathbb{Q}$ (which exists if and only if $X_0(N)$ is hyperelliptic) and a degree 2 map over $\mathbb{Q}$ to an elliptic curve with infinite Mordell--Weil group (the existence of which implies that $X_0(N)$ is bielliptic). In both of these cases the rational points on the image give rise to infinitely many quadratic points on $X_0(N)$. Using Faltings' theorem \cite{faltings} on abelian varieties, Abramovich and Harris \cite{abramovich} show that the set of quadratic points on $X_0(N)$ is infinite in these two cases \emph{only}. Building on the work of Harris and Silverman \cite{harris}, Bars \cite{bars} then determined all bielliptic $X_0(N)$ and decided that exactly 10 of these have a quotient of positive Mordell--Weil rank. Moreover, Ogg \cite{ogg} decided for which 19 values of $N$ the curve $X_0(N)$ is hyperelliptic. Consequently, the values of $N$ where $X_0(N)$ has genus at least 2 and admits infinitely many quadratic points are 22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 43, 46, 47, 48, 50, 53, 59, 61, 65, 71, 79, 83, 89, 101 and 131. (Of course there are also infinitely many quadratic points on each of the genus 0 and genus 1 curves.) The careful reader will have noticed there must be one curve that is both hyperelliptic and bielliptic with quotient of positive rank: this is the infamous $X_0(37)$. Even when infinite, the quadratic points on $X_0(N)$ can be described. Recently, Bruin and Najman \cite{bruin} determined the finitely many exceptional quadratic points (those not coming from $\mathbb{P}^1(\mathbb{Q})$) on all hyperelliptic $X_0(N)$ except for $N=37$, and they moreover proved in those cases that the quadratic points coming from $\mathbb{P}^1(\mathbb{Q})$ correspond to $\mathbb{Q}$-curves. The remaining case $N=37$ is also the only hyperelliptic one where $J_0(N)(\mathbb{Q})$ is infinite. Subsequently, Ozman and Siksek \cite{ozman} determined all finitely many quadratic points on $X_0(N)$ when it is non-hyperelliptic of genus 2, 3, 4 or 5 and the Mordell--Weil group $J_0(N)(\mathbb{Q})$ is finite.\\ In this paper, we complement the work of Ozman and Siksek by describing all quadratic points on the modular curves $X_0(N)$ of genus 2, 3, 4 and 5 whose Mordell--Weil group is infinite. The corresponding values of $N$, together with the genus $g$ of $X_0(N)$ and the rank $r$ of $J_0(N)(\mathbb{Q})$ are listed in the table below. \betagin{table}[h!] \centering \betagin{tabular}{c c c c c c c c c } $N$ & 37 & 43 & 53 & 61 & 57 & 65 & 67 & 73 \\ \hline $g$ & 2 & 3 & 4 & 4 & 5 & 5 & 5 & 5\\[-1.2ex] $r$ &1 & 1 & 1 & 1 & 1 & 1 & 2& 2\\ \end{tabular} \label{Nlist} \end{table} The values of $N$ such that $X_0(N)$ has genus 2, 3, 4 or 5 can be determined via the genus formula, and the rank can be found using the Modular Arithmetic Geometry package in \texttt{Magma} that uses Stein's algorithms \cite{stein}, \cite{stein2}; this was done in Section 2 of \cite{ozman}. We denote by $X_0(N)^+$ the quotient of $X_0(N)$ by the Atkin--Lehner involution $w_N$ corresponding to $N$. As mentioned before, the case $N=37$ is special because $X_0(N)$ is hyperelliptic. The Atkin--Lehner involution and the hyperelliptic involution do not coincide, causing both $\mathbb{P}^1$ and the rank 1 elliptic curve $X_0(37)^+$ to contribute infinitely many quadratic points. Despite this, a description of all quadratic points is still possible, albeit slightly less satisfying than in the other cases. We give this description in Proposition \ref{prop37} in the final Section \ref{hypersection}. Until then, we assume $N$ to be in the above list but unequal to 37. The main trick to studying quadratic points on a curve $X$ is to instead study the rational points on its symmetric square $X^{(2)}$. This set $X^{(2)}(\mathbb{Q})$ consist of pairs $\{P,\overline{P}\}$ of a genuinely quadratic point and its conjugate, as well as pairs $\{P,Q\}$ of rational points $P,Q\in X(\mathbb{Q})$. Following the suggestion of Ozman and Siksek in \cite{ozman}, we have studied the rational points on $X_0(N)^{(2)}$ using the (relative) symmetric Chabauty method developed by Siksek in \cite{siksek}, in combination with a Mordell--Weil sieve. We describe the Chabauty method in more detail in Sections \ref{coleman}--\ref{relativechab}, while the sieve is explained in Section \ref{sieve}. The values of $N$ in our list for which $X_0(N)$ has finitely many quadratic points are 57, 67 and 73. The symmetric Chabauty method was successful in determining all twelve quadratic points on $X_0(57)$. For the remaining values of $N$ we used the relative symmetric Chabauty method also developed by Siksek in the same paper \cite{siksek}. In those cases we found that $\mathrm{rk}\,J_0^+(N)(\mathbb{Q})=\mathrm{rk}\,J_0(N)(\mathbb{Q})$, allowing us to use Chabauty relative to the degree 2 map $X_0(N)\to X_0(N)^+$ to determine all finitely many quadratic points on $X_0(N)$ not coming from $X_0(N)^+(\mathbb{Q})$. For $N\in \{67,73\}$, it then remains to determine the rational points on $X_0(N)^+$, which is hyperelliptic of genus 2 and rank 2 in both cases. These curves $X_0(67)^+$ and $X_0(73)^+$ turn out to be beautiful test cases for the explicit quadratic Chabauty method developed by Balakrishnan and Dogra \cite{dogra} and Balakrishnan, Dogra, M\"uller, Tuitman and Vonk \cite{tuitman} following Kim's work \cite{kim1}, \cite{kim2} on non-abelian Chabauty. Their rational points were determined recently in joint work by Balakrishnan, Best, Bianchi, Lawrence, M\"uller, Triantafillou and Vonk \cite{bbbmtv}, thus giving us a complete list of all quadratic points on $X_0(67)$ and $X_0(73)$. For $N\in \{43,53,61,65\}$, $X_0(N)^+$ is a rank 1 elliptic curve, and we have computed explicitly the map $X_0(N)\to X_0(N)^+$ as well as generators for $X_0(N)^+(\mathbb{Q})$. The \texttt{Magma} code to verify all these computations can be found at \[ \texttt{\href{https://github.com/joshabox/quadraticpoints/}{https://github.com/joshabox/quadraticpoints/}} \;. \] \betagin{theorem}[Main theorem] All finitely many quadratic points on the modular curves $X_0(N)$ for $N\in \{57,67,73\}$ are as described in the tables in Section \ref{results}. For $N\in \{43,53,61,65\}$, the set of quadratic points on $X_0(N)$ is infinite. The tables in Section \ref{results} give all the finitely many quadratic points not coming from $X_0(N)^+(\mathbb{Q})$ as well as generators for the Mordell--Weil groups $X_0(N)^+(\mathbb{Q})$ of the elliptic curves $X_0(N)^+$. \end{theorem} We call points on $X_0(N)$ not coming from $X_0(N)^+(\mathbb{Q})$ \emph{exceptional}. If a quadratic point is non-exceptional, the Atkin--Lehner involution $w_N$ defines an $N$-isogeny from the corresponding elliptic curve to its conjugate. This implies that this elliptic curve is a $\mathbb{Q}$-curve: it is $\overline{\mathbb{Q}}$-isogenous to all its Galois conjugates. In attempts to prove modularity results over quadratic fields one is often led to the study of quadratic points on modular curves; see for example the proof of Freitas, Le Hung and Siksek \cite{freitas} for elliptic curves over totally real quadratic fields. Since $\mathbb{Q}$-curves are automatically modular \cite{ribet}, it is of interest to know the remaining quadratic points.\\ The author would like to express his sincere gratitude to Samir Siksek for various invaluable suggestions and enlightening conversations, which have without doubt improved this work greatly. We also thank the anonymous referees for their detailed feedback including several useful suggestions and corrections that improved the quality of this paper. \section{Relative Symmetric Chabauty and the Mordell--Weil sieve}\label{chabauty} In this section we describe the relative symmetric Chabauty method developed by Siksek in \cite{siksek}. Thoughout this Section, let $X/\mathbb{Q}$ be a (smooth projective) non-hyperelliptic curve of genus $g\geq 3$ with Jacobian $J$. Let $p$ be a prime of good reduction for $X$. \subsection{Chabauty--Coleman}\label{coleman} When the rank $r$ of $J(\mathbb{Q})$ and the genus $g$ of $X$ satisfy the \emph{Chabauty assumption} \[ r<g, \] the method of Chabauty \cite{chabauty} and Coleman \cite{coleman} can be used to find a finite set of points containing $X(\mathbb{Q}_p)\cap \overline{J(\mathbb{Q})}$, where the closure is inside the $p$-adic topology on $J(\mathbb{Q}_p)$. Here we use a rational point $\infty$ to embed $X\to J$ via $P\mapsto [P-\infty]$. In order to bound $X(\mathbb{Q}_p)\cap \overline{J(\mathbb{Q})}$, one needs to find at least one global differential $\omegaega \in \mathcal{O}mega_{X/\mathbb{Q}_p}(X)$ whose Coleman integrals $\int_0^D\omegaega$ vanish on all $D\in X(\mathbb{Q}_p)\cap J(\mathbb{Q})$. We call such a differential a \emph{vanishing differential}. Such integrals can locally be written as a $p$-adic power series in a local parameter and one can then bound its number of zeros using the theory of Newton polygons. Combining this information for several primes $p$, one hopes to determine all the rational points on $X$. A great source for more information about Chabauty--Coleman is the survey article by McCallum and Poonen \cite{poonen}. \subsection{Symmetric Chabauty} \label{symmchab} \subsubsection{Introduction}To study all quadratic points on $X$ at once, one often instead studies the rational points on the symmetric square $X^{(2)}$. We denote points on $X^{(2)}$ as a 2-set $\{P,Q\}$ of points $P,Q\in X$. Recall that the set $X^{(2)}(\mathbb{Q})$ consists of pairs $\{P,\overline{P}\}$ of a genuinely quadratic point $P$ on $X$ and its Galois conjugate, as well as pairs $\{P,Q\}$ of rational points $P,Q$ on $X$. As $X$ is non-hyperelliptic of genus $g\geq 3$, also $X^{(2)}$ can be embedded in $J$ via $\{P,Q\}\mapsto [P+Q-2\cdot \infty]$. Now one can use the same method to potentially determine a set containing \[ X^{(2)}(\mathbb{Q}_p)\cap \overline{J(\mathbb{Q})}, \] where $p$ is a prime. However, as $X^{(2)}$ is 2-dimensional, we now need at least two linearly independent differentials vanishing on $X^{(2)}(\mathbb{Q}_p)\cap \overline{J(\mathbb{Q})}$. Their existence is guaranteed when the analogous Chabauty assumption \betagin{align} \label{chabautycondition} r<g-1 \end{align} is satisfied. (In general, for $d$-fold symmetric powers this is $r<g-(d-1)$.) However, unlike in the classical case for $X^{(1)}$, finding two linearly independent vanishing differentials alone need not yield an effective upper bound for the number of rational points on $X^{(2)}$. In fact, even when (\ref{chabautycondition}) is satisfied, $X^{(2)}(\mathbb{Q})$ may still be infinite due to the existence of a degree 2 map $X\to C$ to a curve $C$ with infinitely many rational points. This happens for $X=X_0(N)$ with $N\in \{43,53,61,65\}$. Moreover, even if $C(\mathbb{Q})$ and $X^{(2)}(\mathbb{Q})$ are both finite, Chabauty's method still fails to find an upper bound for $X^{(2)}(\mathbb{Q})$ when no vanishing differential exists on $C$. This happens for $X=X_0(N)$ with $N\in \{67,73\}$. For $N=57$, this symmetric Chabauty method does succeed in determining all quadratic points. \subsubsection{Precise formulation} \label{precisesymm} Let $\mathcal{X}$ be a proper $\mathbb{Z}_p$-scheme with generic fibre $X$, which is smooth over $\mathrm{Spec} (\mathbb{Z}_p$). Smoothness here simply means that $\widetilde{X}:=\mathcal{X}_{\mathbb{F}_p}$ is also non-singular, c.f. Proposition 2.9 in \cite{advancedsilverman}. This is in fact the minimal proper regular model of $X$; see the section on schemes in \cite{hindry}. Note that in particular we assume that $X$ has good reduction at $p$. Inside the global differential forms $\mathcal{O}mega_{X/\mathbb{Q}_p}(X)$, we have the sub-$\mathbb{Z}_p$-module $\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})$. Coleman integration defines a bilinear pairing \betagin{align} \label{pairing} \mathcal{O}mega_{X/\mathbb{Q}_p}(X)\times J(\mathbb{Q}_p) \to \mathbb{Q}_p, \;\; (\omegaega,\left[\sum_i P_i-Q_i\right])\mapsto \sum_i\int_{Q_i}^{P_i}\omegaega, \end{align} with kernel equal to $J(\mathbb{Q}_p)_{\mathrm{tors}}$ on the right and 0 on the left. Define $V$ to be the annihilator of $J(\mathbb{Q})$ with respect to the pairing (\ref{pairing}), and $\mathcal{V}:=V\cap \mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})$. Let $\widetilde{V}$ be the image of $\mathcal{V}$ under the reduction map $\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})\to \mathcal{O}mega_{\widetilde{X}/\mathbb{F}_p}(\widetilde{X})$ (see the proof of Lemma \ref{surjectivelemma} for a definition of this reduction map). A priori $\widetilde{V}$ is, despite being a space of differentials on the reduction $\widetilde{X}$, defined in terms of the differentials on $\mathcal{X}$. In Section \ref{vanishingdiffs} we show that, when $X=X_0(N)$ for $N$ in our list, we can give an alternative description of $\widetilde{V}$ in terms of data on the reduction $\widetilde{X}$ only. This saves us from having to compute expansions of the vanishing differentials on $\mathcal{X}$, which involves the computationally non-trivial problem of lifting uniformisers around the point of expansion (see Section 2.5 in \cite{siksek}). Consider $\mathcal{Q}=\{Q_1,Q_2\}\in X^{(2)}(\mathbb{Q})$. If $Q_1=Q_2$, we say that $Q_1$ has multiplicity $m=2$; else $Q_1$ has multiplicity $m=1$. Now choose a basis $\omegaega_1,\ldots, \omegaega_k$ for $\widetilde{V}$. Also choose a prime $v$ of $\mathbb{Q}(Q_1)$ above $p$, reductions respect to which we also denote by a tilde. Choose uniformisers $t_{\widetilde{Q}_j}$ at $\widetilde{Q}_j$ for each $j$. Now for each $i$ and $j$, we can expand $\omegaega_i$ around $\widetilde{Q}_j$ as a formal power series: \[ \omegaega_i=(a_0(\omegaega_i,t_{\widetilde{Q}_j})+a_1(\omegaega_i,t_{\widetilde{Q}_j})t_{\widetilde{Q}_j}^1+\ldots)\mathrm{d} t_{\widetilde{Q}_j}. \] When $m=1$, define \[ \widetilde{\mathcal{A}}:=\betagin{pmatrix} a_0(\omegaega_1,t_{\widetilde{Q_1}}) & a_0(\omegaega_1,t_{\widetilde{Q_2}}) \\ \vdots & \vdots \\ a_0(\omegaega_k,t_{\widetilde{Q_1}}) & a_0(\omegaega_k,t_{\widetilde{Q_2}}) \end{pmatrix} \] and when $m=2$ and $p>2$, set \[ \widetilde{\mathcal{A}}:=\betagin{pmatrix} a_0(\omegaega_1,t_{\widetilde{Q_1}}) & a_1(\omegaega_1,t_{\widetilde{Q_1}})/2 \\ \vdots & \vdots \\ a_0(\omegaega_k,t_{\widetilde{Q_1}}) & a_1(\omegaega_k,t_{\widetilde{Q_1}})/2 \end{pmatrix}. \] \betagin{theorem}[Symmetric Chabauty, Siksek] Suppose that $p>2$ and that $p\neq 3$ when $[\mathbb{F}_p(\widetilde{Q_1}):\mathbb{F}_p]=1$. If $\mathrm{rank} (\widetilde{\mathcal{A}})=2$ then $\mathcal{Q}$ is the unique point of $X^{(2)}(\mathbb{Q})$ in its residue class modulo $p$. \label{theorem1} \end{theorem} \betagin{proof} This is a special case of Theorem 1 in \cite{siksek} except for one difference: we work with expansions of differentials on the reduction $\widetilde{X}$, rather than reducing the coefficients of the expansion of differentials on $\mathcal{X}$. To justify this, note that expansions of the vanishing differentials on $\mathcal{X}$ in \cite{siksek} are taken with respect to uniformisers $t_{Q_j}$ at the points $Q_j$ that also reduce to uniformisers $t_{\widetilde{Q}_j}$ at the reduced points. So if $\eta \in \mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})$ has expansion \[ \eta=(a_0+a_1t_{Q_j}+\ldots)\mathrm{d} t_{Q_j} \] then the expansion of $\widetilde{\eta}$ with respect to $t_{\widetilde{Q}_j}$ has the reductions $\widetilde{a}_i$ as coefficients. Therefore, the reduction of the matrix $\mathcal{A}$ in \cite{siksek} equals our $\widetilde{\mathcal{A}}$, at least up to a change of uniformiser and possibly the removal of redundant rows (in case some vanishing differentials agree modulo $p$, c.f. Remark \ref{p=2remark}). These changes leave the rank unaffected. \end{proof} Maarten Derickx \cite{dkss} has written down a similar criterion in terms of the differentials on $\widetilde{X}$ to show points are alone in their residue classes. His approach using the theory of formal immersions was already implicit in the works of Mazur \cite{mazur5} and Kamienny \cite{kamienny2}. It has the advantage of not introducing denominators in the matrix $\widetilde{\mathcal{A}}$ and can thus also be applied to $p=2$ when $m=2$. On the other hand, Siksek's method allows slightly more flexibility in the choice of the vanishing differentials. \betagin{remark} It would be interesting to determine if the combination of sufficiently many vanishing differentials for $X^{(2)}$ as well as for the curves $C$ admitting a degree 2 map $X\to C$ suffices for the Chabauty method to succeed on $X^{(2)}$, and whether a more intrinsic condition on $X$ exists; but we do not pursue this here. \end{remark} \subsection{The relative case}\label{relativechab} \subsubsection{Introduction} When $X=X_0(N)$, we found that, in the cases where Chabauty fails, it is indeed due to a degree 2 map $X\to C$ for some curve $C$. Moreover, except for $N=37$ (a case treated separately in Section \ref{hypersection}), there is a single such curve $C$. We thus slightly change perspective and use a relative version of Chabauty's method to instead determine all \emph{exceptional points} of $X^{(2)}(\mathbb{Q})$, i.e. those that do not come from $C(\mathbb{Q})$ via the degree 2 map $X\to C$. In each case $C(\mathbb{Q})$ can be described separately, either because it is an elliptic curve and we can find generators (this happens for $X_0(N)$ with $N\in \{43,53,61,65\})$ or because it is a hyperelliptic curve and its rational points have already been determined using quadratic Chabauty in \cite{bbbmtv} (this happens for $X_0(N)$ with $N\in \{67,73\}$). The main idea behind the relative Chabauty method is the following. Suppose that we have a degree 2 map $X\to C$, where $C$ is a curve with good reduction at the prime $p$. Then $J$ is isogenous to $J(C)\times A$ for some Abelian variety $A$, and we have a trace map $\mathrm{Tr}: \mathcal{O}mega_{X/\mathbb{Q}_p}(X)\to \mathcal{O}mega_{C/\mathbb{Q}_p}(C)$. Denote the map $J\to A$ by $\pi$. As long as we can find sufficiently many vanishing differentials $\omegaega$ whose image (also denoted by $\omegaega$) in $\mathcal{O}mega_{J/\mathbb{Q}_p}(J)$ is in the subspace $\pi^*\mathcal{O}mega_{A/\mathbb{Q}_p}(A)$, Chabauty's method is expected to succeed in determining the exceptional points using only these differentials. In terms of $X$, the condition $\omegaega \in \pi^*\mathcal{O}mega_{A/\mathbb{Q}_p}$ means that $\omegaega$ is in the kernel of the trace map. In practice $A(\mathbb{Q})$ will have rank zero and all differentials in $\pi^*\mathcal{O}mega_{A/\mathbb{Q}_p}(A)$ are vanishing differentials, c.f. Lemma \ref{vanishingdiffs}. \betagin{remark}\label{rationalrk} Suppose that we can find one rational point $P$ on $X$. Then each other rational point $Q\in X(\mathbb{Q})$ will occur in $X^{(2)}(\mathbb{Q})$ at least twice: as $\{Q,Q\}$ and as $\{P,Q\}$. At most one of these pairs is the pullback of the image of $Q$ under $X\to C$, so at least one pair will occur as an exceptional point of $X^{(2)}(\mathbb{Q})$. This means that, as a byproduct, we also determine the rational points when we find these exceptional points. Indeed each of the modular curves we consider contains a rational point. \end{remark} \subsubsection{Precise formulation} We continue with the notation from Section \ref{precisesymm}. Assume that $X$ admits a degree 2 map $\rho: X\to C$ to a curve $C$ and consider a proper smooth $\mathbb{Z}_p$-scheme $\mathcal{C}$ with $C$ as generic fibre. We assume that $\rho$ extends to a morphism $\mathcal{X}\to \mathcal{C}$, thus obtaining a corresponding reduced degree 2 map $\widetilde{X}=\mathcal{X}_{\mathbb{F}_p}\to \mathcal{C}_{\mathbb{F}_p}$ between non-singular curves. Note that in particular we assume that $C$ has good reduction at $p$. We denote the points of $X^{(2)}(\mathbb{Q})$ coming from rational points of $C$ by $\rho^*C(\mathbb{Q})$. Define $\mathcal{V}_0:=\mathcal{V}\cap \mathrm{ker}(\mathrm{Tr}: \mathcal{O}mega_{X/\mathbb{Q}_p}(X)\to \mathcal{O}mega_{C/\mathbb{Q}_p}(C)$ and let $\widetilde{V}_0$ be its image under the reduction map. As in the non-relative case, we show in Section \ref{vanishingdiffs} how $\widetilde{V}_0$ can be computed in terms of $\widetilde{X}$ only. We now consider a point $\mathcal{Q}=\{Q_1,Q_2\}\in \rho^*C(\mathbb{Q})$ and choose a basis $\omegaega_1,\ldots,\omegaega_k$ for $\widetilde{V}_0$. Define $N$ as in Section \ref{precisesymm} and again let $t_{\widetilde{Q}_1}$ be a uniformiser at the reduction $\widetilde{Q}_1$ of $Q_1$ at the chosen prime $v$ above $p$. \betagin{theorem}[Relative symmetric Chabauty, Siksek] \label{theorem2} Suppose that $p>2$ when $[\mathbb{F}_p(\widetilde{Q_1}):\mathbb{F}_p]=1$. If there exists $i\in \{1,\ldots,k\}$ such that \[ \frac{\omegaega_i}{\mathrm{d} t_{\widetilde{Q}_1}}|_{t_{\widetilde{Q}_1}=0} \neq 0 \] then every point in $X^{(2)}(\mathbb{Q})$ belonging to the residue class of $\mathcal{Q}$ comes from a point in $C(\mathbb{Q})$. \end{theorem} \betagin{proof} This is a special case of Theorem 2 in \cite{siksek}, with the same difference as in Theorem \ref{theorem1}; see its proof for the justification. \end{proof} Note that $\frac{\omegaega_i}{\mathrm{d} t_{\widetilde{Q}_1}}|_{t_{\widetilde{Q}_1}=0}$ is simply the constant coefficient of the expansion of $\omegaega_i$ at $t_{\widetilde{Q}_1}$, and the non-zero condition says that the corresponding $k\times 1$ matrix has rank 1. It is the combination of Theorem \ref{theorem1} for the points in $X^{(2)}(\mathbb{Q})\setminus \rho^*C(\mathbb{Q})$ and Theorem \ref{theorem2} for the points in $\rho^*C(\mathbb{Q})$ that provably determines $X^{(2)}(\mathbb{Q})$ for all curves $X$ we consider. \subsection{The Mordell--Weil sieve}\label{sieve} Again consider a (smooth) non-singular curve $X/\mathbb{Q}$ with Jacobian $J$. If $X$ admits a degree 2 map to another curve over $\mathbb{Q}$, we denote it by $\rho: X\to C$. In Chabauty's method the choice of a prime $p$ (of good reduction for $X$ and, if appropriate, also for $C$) is free to choose. The happy outcome is a number of $p$-adic discs in $X^{(2)}$ where no unknown rational points can lie. The Mordell--Weil sieve is a way of combining this $p$-adic information for several primes $p$ to make sure all of $X^{(2)}$ is covered. We have implemented the sieve described in Section 5 of \cite{siksek}, with one minor difference to be pointed out in Remark \ref{differenceremark}. Choose primes $p_1,\ldots,p_r$ (see the next Section on how to choose wisely). We assume given the following input: \betagin{itemize} \item[(i)] a number of divisors $D_1,\ldots, D_n$ generating a subgroup $G$ of $J(\mathbb{Q})$ of finite index, \item[(ii)] a positive integer $I$ such that $I\cdot J(\mathbb{Q})\subset G$, and \item[(ii)] a (finite) list $\mathcal{L'}$ of known rational points for $X^{(2)}$, which, in the case of a degree 2 map $X\to C$, may also include points of $\rho^*C(\mathbb{Q})$. \end{itemize} In case of a degree 2 map $X\to C$, define $\mathcal{L}:=\mathcal{L}'\cup \rho^*C(\mathbb{Q})$. Else, let $\mathcal{L}:=\mathcal{L}'$. Suppose that $\mathcal{P}$ is a hypothetical point in $X^{(2)}(\mathbb{Q})\setminus \mathcal{L}$. Our objective is to show that such $\mathcal{P}$ cannot exist. Let $\phi: \mathbb{Z}^n\to J(\mathbb{Q})$ be the map given by $\phi(a_1,\ldots,a_n)=\sum_i a_i D_i$. It has image $G$. We choose a rational degree 2 divisor $\infty$ and define $\iota: X^{(2)}(\mathbb{Q})\to G$ by $\mathcal{Q}\mapsto I\cdot [\mathcal{Q}- \infty]$. Let $p$ be one of the primes $p_i$. We denote reduction modulo $p$ by a tilde. Also define $\iota_p: X^{(2)}(\mathbb{F}_p)\to J(\mathbb{F}_p)$ by $\mathcal{R}\mapsto I\cdot [\mathcal{R}-\widetilde{\infty}]$. Finally define $\phi_p$ to make the diagram \[ \xymatrixcolsep{6pc}\xymatrix{ \mathcal{L} \ar[r]^{i} \ar[rd] & X^{(2)}(\mathbb{Q}) \ar[r]^{\iota} \ar[d]^{\mathrm{red}} & G \ar[d]^{\mathrm{red}} & \ar[l]_{\phi} \ar[ld]^{\phi_p} \mathbb{Z}^n \\ & \widetilde{X}^{(2)}(\mathbb{F}_p) \ar[r]_{\iota_p}& J(\mathbb{F}_p) & } \] commute. Note that $\iota_p(\widetilde{\mathcal{P}})=\mathrm{red}(\iota(\mathcal{P}))$, hence $\iota_p(\widetilde{\mathcal{P}})\in \mathrm{Im}(\phi_p)$. Since $\mathcal{P}\notin \mathcal{L}$, our Chabauty method limits the possible values of $\widetilde{\mathcal{P}}$, which in turn reduces the union $W_p$ of $\ker(\phi_p)$-cosets of $\mathbb{Z}^n$ that could possibly be mapped to $\iota_p(\widetilde{\mathcal{P}})$ under $\phi_p$. The key observation here is that $\mathbb{Z}^n$ is independent of $p$, and that these $W_p$ can thus be compared for different values of $p$. In particular, if they have empty common intersection, then our hypothetical point $\mathcal{P}$ cannot exist. \betagin{definition} \label{sievedef} Define $\mathcal{M}_p\subset \widetilde{X}^{(2)}(\mathbb{F}_p)$ to be the set of points $\mathcal{R}\in \iota_p^{-1}(\mathrm{Im}(\phi_p))$ satisfying one of the following: \betagin{itemize} \item[(i)] $\mathcal{R}\notin \mathrm{red}(\mathcal{L'})$, \item[(ii)] $\mathcal{R}=\widetilde{\mathcal{Q}}$ for some $\mathcal{Q}\in \mathcal{L'}$ \emph{not} satisfying the conditions of Theorem \ref{theorem1} or \item[(iii)] there is a degree 2 map $X\to C$ and $\mathcal{R}=\widetilde{\mathcal{Q}}$ for some $\mathcal{Q}\in \mathcal{L'}\cap \rho^*C(\mathbb{Q})$ \emph{not} satisfying the conditions of Theorem \ref{theorem2}. \end{itemize} \end{definition} Then by construction, the reduction $\widetilde{\mathcal{P}}$ of our hypothetical point is in $\mathcal{M}_p$. We conclude the following. \betagin{theorem}[Mordell--Weil sieve] If \[ \bigcap_{i=1}^r\phi_{p_i}^{-1}(\iota_{p_i}(\mathcal{M}_{p_i}))=\emptyset \] then $X^{(2)}(\mathbb{Q})=\mathcal{L}$. \end{theorem} \betagin{remark} For each prime $p$, we note that $\widetilde{X}^{(2)}(\mathbb{F}_p)$, $J(\widetilde{X})(\mathbb{F}_p)$, $\iota_p$, $\phi_p$ and $\mathrm{red}|_{\mathcal{L}}$ can all be computed explicitly. Therefore, each coset $\phi_p^{-1}(\iota_p(\mathcal{M}_{p_i}))$ can be determined explicitly. The computations we perform with cosets of subgroups of $\mathbb{Z}^n$ require only linear algebra over $\mathbb{Z}$ and can be done almost instantaneously by computer algebra systems. \end{remark} \betagin{remark} \label{differenceremark} The only difference between this sieve and the one described in Section 5 of \cite{siksek} is that we work with cosets in $\mathbb{Z}^n$ rather than cosets of increasingly large finite quotients of $\mathbb{Z}^n$. This saves us from an explosion of the size of such finite quotients caused by the Chinese Remainder Theorem. \end{remark} \subsubsection{Prime-choosing heuristics} An interesting aspect of the Mordell--Weil sieve is the question of which primes to choose in which order. We have found the naive choice of a small number of small primes to be sufficient when $N\notin \{67,73\}$. For the remaining two cases, however, we used the prime-choosing heuristics as described in this section. These were inspired by those in Section 11 of \cite{bugeaud}. Note that, even though this strategy works for us, it may not be optimal. For example, we attempt to choose the primes one by one in a near-optimal sense, rather than attempting to find a near-optimal \emph{set} of primes. A more detailed discussion about choosing primes in the Mordell--Weil sieve can be found in \cite{bruinstoll}. Suppose that we have already chosen the primes $p_1,\ldots,p_k$ and we would like to choose the next prime $p_{k+1}$. Let us write $W_k:=\bigcap_{i=1}^k\phi_{p_i}^{-1}(\iota_{p_i}(\mathcal{M}_{p_i}))$. This is a union of $A_k$-cosets, where $A_k\subset \mathbb{Z}^n$ is the subgroup $\bigcap_{i=1}^k \ker \phi_{p_i}\subset \mathbb{Z}^n$. \betagin{choice} We first establish a bounded range of primes $\mathbb{P}$ in which computations are reasonably fast, e.g. all primes up to 100. Then we choose $p_{k+1}$ to be the prime of good reduction in $\mathbb{P}\setminus \{p_1,\ldots,p_k\}$ minimising \[ \frac{[A_k: \ker (\phi_{p_{k+1}}|_{A_k})]}{p_{k+1}^{g-2}}. \] \end{choice} Let us explain the ideas behind this choice. Let $w_k$ be the number of $A_k$-cosets in $W_k$. We aim to choose the next prime $p_{k+1}$ in such a way that $w_{k+1}$ is as small as possible. Note that, after choosing $p_{k+1}$, $W_{k+1}=W_k\cap \phi_{p_{k+1}}^{-1}(\iota_{p_{k+1}}(\mathcal{M}_{p_{k+1}}))$ consisting of $A_{k+1}=\ker (\phi_{p_{k+1}}|_{A_k})$-cosets. So a priori, $w_k$ gets multiplied by a factor \[ [A_k: \ker (\phi_{p_{k+1}}|_{A_k})]. \] Next, to form $W_{k+1}$ we remove the $A_{k+1}$-cosets not mapping to $\iota_{p_{k+1}}(\widetilde{X}^{(2)}(\mathbb{F}_p))$. Since $\widetilde{X}^{(2)}(\mathbb{F}_p)\to J(\widetilde{X})(\mathbb{F}_p),\; \mathcal{Q}\mapsto [\mathcal{Q}-\infty]$ is injective, one would expect (assuming the image of $\widetilde{X}^{(2)}(\mathbb{F}_p)$ in $J(\widetilde{X})(\mathbb{F}_p)$ is randomly behaved) a proportion \[ \frac{\#\widetilde{X}^{(2)}(\mathbb{F}_p)}{\#J(\widetilde{X})(\mathbb{F}_p)} \] of the cosets to remain. Finally, we also remove the $A_{k+1}$-cosets mapping to elements $D\in J(\widetilde{X})(\mathbb{F}_p)$ such that each point in $\iota_{p_{k+1}}^{-1}(D)$ is excluded by Chabauty. This is a contribution that we decide to neglect when choosing our prime, both because it is hard to estimate and because we work with a fixed finite set $\mathcal{L}'\subset \mathcal{L}$. As the size of $p$ increases, $\mathrm{red}(\mathcal{L'})$ thus forms an increasingly small proportion of $\widetilde{X}^{(2)}(\mathbb{F}_p)$. Next, we note that $\#J(\mathbb{F}_p)=p^g+o(p^g)$ as $p\to \infty$ and $\widetilde{X}^{(2)}(\mathbb{F}_p)=p^2+o(p^2)$ as $p\to \infty$ by classical bounds. Hence we approximate $\#\widetilde{X}^{(2)}(\mathbb{F}_p)/\#J(\mathbb{F}_p)$ by $p^{2-g}$. Therefore, we approximate $w_{k+1}$ by \[ \frac{[A_k: \ker (\phi_{p_{k+1}}|_{A_k})]}{p_{k+1}^{g-2}}w_k, \] which explains our choice. Note that we do need to make a little precomputation before choosing each new prime, but this precomputation is relatively fast compared to checking the Chabauty method at each point. Moreover, the relatively expensive part of this precomputation is computing $J(\mathbb{F}_p)$ and $\phi_p$ for each $p\in \mathbb{P}$, which only needs to be done once. \section{The input} We continue with the notation from the previous Section. When $X=X_0(N)$ for $N \in \{43, 53, 61, 57, 65, 67,73\}$, we always have a degree 2 map $X_0(N)\to X_0(N)^+$, where $X_0(N)^+$ is the quotient of $X_0(N)$ by the Atkin--Lehner involution $w_N$. Except for $N=57$, we will use relative symmetric Chabauty with respect to $C=X_0(N)^+$. In this Section we describe how we computed the necessary input to make this work. This consists of the following four parts: \betagin{enumerate} \item[(i)] explicit defining equations for $X_0(N)$ as well as the corresponding Atkin--Lehner involutions, \item[(ii)] a list $\mathcal{L}'$ of known quadratic points on $X$, including points coming from $C(\mathbb{Q})$ in the case of a degree 2 map $X\to C$, \item[(iii)] a list of generators of a subgroup $G\subset J(\mathbb{Q})$ and an integer $I$ such that $I\cdot J(\mathbb{Q})\subset G$, \item[(iv)] at least 2 linearly independent vanishing differentials for each prime $p$, which are in the kernel of $\mathrm{Tr}: \mathcal{O}mega_{X/\mathbb{Q}_p}(X)\to \mathcal{O}mega_{C/\mathbb{Q}_p}(C)$ in case of a degree 2 map $X\to C$. \end{enumerate} In this Section we describe how we obtain this input. We note that knowing more than two vanishing differentials will improve the speed and the chance of success of the sieve. \subsection{A model for $X_0(N)$} The Small Modular Curves package in \texttt{Magma} is a great tool for computing models of modular curves $X_0(N)$. As, however, $X_0(65),X_0(67)$ and $X_0(73)$ are not in this database, we instead compute models for all our modular curves using the code written by Ozman and Siksek; see Section 3 of \cite{ozman}. This, too, uses the canonical embedding to compute the models. Moreover, Ozman and Siksek also explicitly compute the action of the Atkin--Lehner operators on their models, as well as the equations for the $j$-invariant map $X_0(N)\to X(1)$. For primes $p$ where the chosen model has good reduction, we define $\mathcal{X}_0(N)$ to be the $\mathbb{Z}_p$-scheme defined by these equations. In these cases, $\mathcal{X}_0(N)$ is our minimal proper regular model, c.f. Section 2.2.2. \subsection{Searching for quadratic points} We assume henceforth that we have found a model for $X$ in $\mathbb{P}^k$, with coordinates $x_0,\ldots,x_k$. We search for quadratic points on $X$ by intersecting $X\subset \mathbb{P}^k$ with hyperplanes of the form \[ b_0x_0+\ldots + b_kx_k=0, \] where $b_0,\ldots,b_k\in \mathbb{Z}$ are chosen coprime and up to a certain bound. When the decomposition of the divisor (over $\mathbb{Q}$) corresponding to this intersection contains effective degree 2 divisors, we have found quadratic points. In practice it sufficed for us to consider only the hyperplanes defined by $|b_i|\leq 10$ for $i\in \{1,\ldots,k\}$. Unlike searching for rational points, this can be a time consuming exercise: for the genus 5 curve $X_0(67)\subset \mathbb{P}^4$, for example, the search took multiple hours. This is largely due to the time taken to decompose these hyperplane intersections into linear combinations of irreducible effective divisors. \subsection{Determining subgroups of the Mordell--Weil groups} We first consider the general case, where $X/\mathbb{Q}$ is again any (projective non-singular) curve, $\Gamma$ is a finite subgroup of $\mathrm{Aut}_{\mathbb{Q}}(X)$ and $C=X/\Gamma$. Denote the quotient map by $\rho: X\to C$. This map has degree $\deg\rho = \#\Gamma$. We still assume that $X$ has a rational point. We now denote the Jacobian of $X$ by $J(X)$ to emphasize the difference with the Jacobian $J(C)$ of $C$. After choosing compatible base points for the maps $\iota_X: X\to J(X)$ and $\iota_C: C\to J(C)$, we then obtain a commuting diagram \sqcommdiag{X}{J(X)}{C}{J(C).}{\rho}{\iota_X}{\rho_*}{\iota_C} We now make the extra assumption that $\mathrm{rk} J(X)(\mathbb{Q})=\mathrm{rk} J(C)(\mathbb{Q})$; denote this common rank by $r$. Let $G$ be the subgroup of $J(X)(\mathbb{Q})$ generated by $J(X)(\mathbb{Q})_{\mathrm{tors}}$ and $\rho^*J(C)(\mathbb{Q})$. \betagin{proposition} We have that $(\#\Gamma)\cdot J(X)(\mathbb{Q})\subset G$. \label{mwgroup} \end{proposition} \betagin{proof} Let $P_1,\ldots,P_r \in J(C)(\mathbb{Q})$ be linearly independent generators of $J(C)(\mathbb{Q})/J(C)(\mathbb{Q})_{\mathrm{tors}}$. Set $D_i=\rho^*P_i$ for each $i$. Then $\rho_*D_i=\deg(\rho)P_i=(\#\Gamma)P_i$ and hence $D_1,\ldots, D_r$ are also linearly independent. We consider each $D_i$ as an element of $J(X)(\mathbb{Q})/J(X)(\mathbb{Q})_{\mathrm{tors}}$ and each $P_i$ as an element of $J(C)(\mathbb{Q})/J(C)(\mathbb{Q})_{\mathrm{tors}}$, while still denoting the maps between these quotients by $\rho_*$ and $\rho^*$. As $\rho_*D_i=(\#\Gamma)P_i$ for each $i$, we see that $\rho_*G=(\#\Gamma)\cdot J(C)(\mathbb{Q})/J(C)(\mathbb{Q})_{\mathrm{tors}}$. In particular, as we assume $\rho_*$ to be injective modulo torsion (equal rank), we find that $(\#\Gamma)\cdot J(X)(\mathbb{Q})/J(X)(\mathbb{Q})_{\mathrm{tors}}\subset G/J(X)(\mathbb{Q})_{\mathrm{tors}}$. \end{proof} We now consider the special case $X=X_0(N)$ for $N$ in our list. Let us first compute the torsion subgroup of the Jacobian. For each $N$, let $C_0(N)$ be the subgroup of $J_0(N)(\overline{\mathbb{Q}})$ generated by the classes of differences of cusps, and $C_0(N)(\mathbb{Q})$ its subgroup fixed by $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. This is called the \emph{rational cuspidal subgroup}. By the Manin-Drinfeld theorem \cite{manin}, \cite{drinfeld}, $C_0(N)(\mathbb{Q})\subset J_0(N)(\mathbb{Q})_{\mathrm{tors}}$ for each $N$, and a conjecture of Ogg proved by Mazur \cite{mazur1} tells us that $J_0(N)(\mathbb{Q})_{\mathrm{tors}}=C_0(N)(\mathbb{Q})$ for prime values of $N$. Moreover, Mazur also showed in that case that the order of $J_0(N)(\mathbb{Q})_{\mathrm{tors}}$ is the numerator of $(p-1)/12$. Recent results (\cite{oggconj1},\cite{oggconj2},\cite{oggconj3},\cite{oggconj4},\cite{ozman}) have verified the equality $C_0(N)(\mathbb{Q})=J_0(N)(\mathbb{Q})_{\mathrm{tors}}$ for various non-prime values of $N$, to which we can now add 57 and 65. \betagin{lemma} For $N\in \{43,53,61,67,73\}$ the torsion subgroup of $J_0(N)(\mathbb{Q})$ is generated by the difference of the two cusps, the orders of which are, respectively, 7, 13, 5, 11 and 6. The torsion subgroups of $J_0(57)(\mathbb{Q})$ and $J_0(65)(\mathbb{Q})$ also equal their rational cuspidal subgroups, which are $\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/30\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/84\mathbb{Z}$ respectively as abstract groups. \label{torsionlemma} \end{lemma} \betagin{proof} For the prime values of $N$, this is the aforementioned theorem of Mazur. For $N=57$, we use \texttt{Magma} code of Ozman and Siksek \cite{ozman} to compute $C_0(57)(\mathbb{Q})$ as a group, which gives $\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/30\mathbb{Z}$. As 5 is a prime of good reduction for $X_0(N)$, we find that $J_0(57)(\mathbb{Q})_{\mathrm{tors}}$ injects into $J_0(57)(\mathbb{F}_5)$. We compute that \[ J_0(57)(\mathbb{F}_5)\simeq \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/180\mathbb{Z} \] so that the index $[J_0(57)(\mathbb{Q})_{\mathrm{tors}}:C_0(57)(\mathbb{Q})]$ divides 18. Looking at $J_0(57)(\mathbb{F}_{23})$, we find that the index is coprime to 3, and from $J_0(57)(\mathbb{F}_{11})$ we learn that the index is odd, leaving index 1 as only option. A similar computation at the primes 3 and 11 allows us to determine $J_0(65)(\mathbb{Q})_{\mathrm{tors}}$. \end{proof} For $N\in \{43,53,65,61,67,73\}$, let $\Gamma$ be the subgroup of $\mathrm{Aut}(X_0(N))$ generated by $w_N$, and for $N=57$ let $\Gamma=\langle w_{19},w_{3}\rangle$. Define $C(N):=X_0(N)/\Gamma$. For each $N\neq 57$, this curve will be the degree two quotient of $X_0(N)$ denoted before in greater generality by $C$. Note that $C(N)=X_0(N)^+$ for $N\neq 57$ and $C(N)$ is the curve often denoted by $X_0(N)^*$ for $N=57$. We verify that, with our choice of $X=X_0(N)$ and $\Gamma$, the hypothesis for Proposition \ref{mwgroup} holds true. \betagin{lemma} \label{ranklemma} For each $N$ in our list, we have $ \mathrm{rk}J_0(N)(\mathbb{Q}) = \mathrm{rk} J(C(N))(\mathbb{Q}). $ \end{lemma} \betagin{proof} We checked this using the algorithm of Stein \cite{stein} as implemented in the modular arithmetic geometry package in \texttt{Magma}. For each $N$, $J_0(N)\sim \prod_f A_f$ is isogenous to a product of simple modular abelian varieties $A_f$ corresponding to Galois orbits $f$ of newforms of weight 2 and level $N$. Now $J(C(N))$ is isogenous to the product of those $A_f$ where $f$ is invariant under $\Gamma$. For each $N$ we found $J(C(N))$ to be simple, hence $J(C(N))\sim A_f$ for some orbit $f$. We checked that the values of the $L$-series $L(g,s)$ are non-zero at $s=1$ for the eigenforms $g$ not conjugate to $f$. This means that the corresponding $A_g$ have analytic rank 0, and hence algebraic rank zero by a theorem of Kolyvagin and Logachev \cite{kolyvagin}. \end{proof} Write $r(N):=\mathrm{rk} J_0(N)(\mathbb{Q})$. We note that $r(N)=2$ for $N\in \{67,73\}$ and $r(N)=1$ for the other values of $N$. Also $C(N)$ has genus 2 for $N\in \{67,73\}$ and genus 1 otherwise, making $C(N)$ either an elliptic curve or a hyperelliptic curve. When $C(N)$ is an elliptic curve, we can compute a basis for its Mordell--Weil group using Cremona's algorithm \cite{cremona} implemented in \texttt{Magma}. When it is hyperelliptic of genus 2, we manage to do the same using height bounds and an algorithm of Stoll \cite{stoll}. We have written down the explicit generators in Section \ref{results}. Together with Lemma \ref{torsionlemma}, this gives us a complete description of a suitable subgroup of $J_0(N)(\mathbb{Q})$ for each of the values of $N$ we consider. \subsection{The vanishing differentials}\label{vanishingdiffs} For $N\neq 57$, let $V$ be the image of $1-w_N^*: \mathcal{O}mega_{X_0(N)/\mathbb{Q}}(X_0(N))\to \mathcal{O}mega_{X_0(N)/\mathbb{Q}}(X_0(N))$. For $N=57$, let $V\subset \mathcal{O}mega_{X_0(57)/\mathbb{Q}}(X_0(57))$ be the sum of the images of $1-w_3^*$ and $1-w_{19}^*$. \betagin{lemma} All differentials $\omegaega\in V$ annihilate $J_0(N)(\mathbb{Q})$ via the integration pairing, i.e. \[ \int_{0}^{D}\omegaega =0 \text{ for all }D\in J_0(N)(\mathbb{Q}). \] Moreover, $V$ is contained in the kernel of $\mathrm{Tr}: \mathcal{O}mega_{X_0(N)/\mathbb{Q}}(X_0(N))\to \mathcal{O}mega_{C(N)/\mathbb{Q}}(C(N))$. \end{lemma} \betagin{proof} Note that, as the Atkin--Lehner maps are involutions, the image of $1-w_N^*$ is the kernel of $1+w_N^*$, which is the trace map for $N\neq 57$. For $N=57$, both the images of $1-w_3^*$ and $1-w_{19}^*$ are in the kernel of the trace map $1+w_3^*+w_{19}^*+w_{57}^*$. If we identify $V$ with a subspace of $\mathcal{O}mega_{J_0(N)/\mathbb{Q}}(J_0(N))$, this means that $V\subset \pi^*\mathcal{O}mega_{A/\mathbb{Q}}$, where $J_0(N)\sim J(C)\times A$ as in the proof of Lemma \ref{ranklemma} and $\pi$ is the map $J_0(N)\to A$. It can also be seen directly using the algorithm of Stein \cite{stein} implemented in the modular abelian varieties package in \texttt{Magma} that the image of $1-w_N^*: J_0(N)\to J_0(N)$ is $A$ for $N\neq 57$, and that the union of the images of $1-w_3^*$ and $1-w_{19}^*$ is $A$ for $N=57$. Let $\omegaega\in V$ and consider $\eta\in \mathcal{O}mega_{A/\mathbb{Q}}$ such that $\omegaega=\pi^*\eta$. By Lemma \ref{ranklemma}, we find that $A(\mathbb{Q})$ is torsion. Consider $D\in J_0(N)(\mathbb{Q})$ and let $n$ be the order of $\pi(D)$ in $A(\mathbb{Q})$. By the additive property of Coleman integration, we find \[ \int_0^{D}\omegaega=\int_0^{\pi(D)}\eta =\frac{1}{n}\int_0^{n\pi(D)}\eta =0, \] as desired. \end{proof} As we have computed equations for $X_0(N)$ as well as the Atkin--Lehner involutions, we can compute $V$ explicitly for each $N$. Note that $\dim(V)=\mathrm{genus}(X)-\mathrm{genus}(C)\geq 2$ for every value of $N$ we consider. This indicates that the relative symmetric Chabauty method may succeed. However, in order to apply Theorems \ref{theorem1} and \ref{theorem2}, we need to compute the image $\widetilde{V}$ of $\mathcal{V}=V\cap \mathcal{O}mega_{\mathcal{X}_0(N)/\mathbb{Z}_p}(\mathcal{X}_0(N))$ under the reduction map. To this end, consider a prime $p$ of good reduction for $X=X_0(N)$ and, when appropriate, for $C(N)$. Note that each Atkin--Lehner involution $w_M$ for $M\mid N$ is integral and thus extends to a map on the proper minimal regular model $\mathcal{X}_0(N)\to \mathrm{Spec} \mathbb{Z}_p$. In particular, we obtain reduced maps $\widetilde{w}_M: \widetilde{X}_0(N)\to \widetilde{X}_0(N)$. The following proposition allows us to easily compute $\widetilde{V}$ in \texttt{Magma}. \betagin{proposition} \label{vanishingprop} For $N\neq 57$, $\widetilde{V}$ is the image of $1-\widetilde{w}_N^*: \mathcal{O}mega_{\widetilde{X}_0(N)/\mathbb{F}_p}(\widetilde{X}_0(N))\to \mathcal{O}mega_{\widetilde{X}_0(N)/\mathbb{F}_p}(\widetilde{X}_0(N))$. For $N=57$, $\widetilde{V}$ is the sum of the images of $1-\widetilde{w}_3^*$ and $1-\widetilde{w}_{19}^*$. \end{proposition} The proof of this proposition relies on the following lemma, which in turn depends heavily on the fact that $X$ has good reduction at $p$. This lemma is commonly known by experts, but we could not find a reference for it. \betagin{lemma} \label{surjectivelemma} Let $X/\mathbb{Q}$ be a (smooth projective) curve with minimal proper regular model $\mathcal{X}/\mathbb{Z}_p$. Then the reduction map $\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})\to \mathcal{O}mega_{\widetilde{X}/\mathbb{F}_p}(\widetilde{X})$ is surjective. \end{lemma} \betagin{proof} On the level of sheaves, the Cartesian diagram defining the fibre $\widetilde{X}$ yields the isomorphism \[ \mathcal{O}mega_{\widetilde{X}/\mathbb{F}_p}\simeq \mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}|_{\widetilde{X}}\otimes_{\mathbb{Z}_p}\mathbb{F}_p, \] where $\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}|_{\widetilde{X}}$ is the sheaf associated to \[ U\mapsto \mathrm{Li}m_{\substack{\longrightarrow\\V\supset U}}\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(V) \] (see Proposition 8.10 in \cite{hartshorne}). Moreover, this isomorphism defines the reduction map $\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_{p}}(\mathcal{X})\to \mathcal{O}mega_{\widetilde{X}/\mathbb{F}_p}(\widetilde{X})$ on global sections. We first note that every Zariski-open subset of $\mathcal{X}$ containing $\widetilde{X}$ equals $\mathcal{X}$: if $V\subset \mathcal{X}$ is a Zariski-open set containing $\widetilde{X}$, then its complement is a closed subset (in $\mathcal{X}$) of the generic fibre $X$. However, the closure (in $\mathcal{X}$) of any point $P\in X$ contains its reduction $\widetilde{P}\in \mathcal{X}$, so $\mathcal{X}\setminus V=\emptyset$. Now for simplicity, we write $A=\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_{p}}(\mathcal{X})$ and $B=\mathcal{O}mega_{\widetilde{X}/\mathbb{F}_p}(\widetilde{X})$. We will show that $A/pA\simeq B$. Note that $pA$ is in the kernel of $A\to B$. Next, by flatness of $\mathcal{X}\to \mathrm{Spec} \mathbb{Z}_p$, we have that $\mathrm{rank}_{\mathbb{Z}_p} A=g =\dim_{\mathbb{F}_p}B$. Therefore, the induced map $A/pA\to B$ is a linear map of $\mathbb{F}_p$-vector spaces of equal dimension. It thus suffices to show that each $\omegaega\in \ker (A\to B)$ is a $p$-fold. We first show this to be true locally. Consider $\omegaega\in \ker(A\to B)$. By the above isomorphism of sheaves, there exists an open cover $\{U\}$ of $\widetilde{X}$ such that \[ \mathcal{O}mega_{\widetilde{X}/\mathbb{F}_p}(U)\simeq\mathrm{Li}m_{\substack{\longrightarrow\\V\supset U}}\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(V)\otimes_{\mathbb{Z}_p}\mathbb{F}_p \;\;\; \text{ for each } U. \] Note that it does not matter whether we tensor with $\mathbb{F}_p$ inside or outside the limit. By this isomorphism, there must be open $V_U\supset U$ and $\eta_U\in \mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(V_U)$ such that $\omegaega|_{V_U}=p\eta_U$. Since $\bigcup_UV_U$ is an open subset of $\mathcal{X}$ containing $\widetilde{X}$, the $V_U$ must cover $\mathcal{X}$ entirely. As $(\omegaega|_{V_U})_{V_U}$ lifts to the global section $\omegaega$ and we are working in characteristic zero, also the $\eta_U$ must agree on overlaps and lift to $\eta\in A$ such that $\omegaega=p\eta$. \end{proof} \betagin{proofofprop} LLet $\widetilde{W}$ be the image of $1-\widetilde{w}_N^*$ for $N\neq 57$ and the sum of the images of $1-\widetilde{w}_3^*$ and $1-\widetilde{w}_{19}^*$ for $N=57$. We want to show that $\widetilde{W}=\widetilde{V}$. Note first that $\widetilde{V}\subset \widetilde{W}$ because if $\omegaega\in \ker(1-w_M^*)$ then $\widetilde{\omegaega}\in \ker (1-\widetilde{w}_M^*)$, so we need to prove the opposite inclusion. To this end, consider $\widetilde{\omegaega}\in \widetilde{W}$ and $\widetilde{\eta}$ such that $\widetilde{\omegaega}=(1-w_M^*)(\widetilde{\eta}$). Here $M=N$ for $N\neq 57$ and for $N=57$ we may assume $\widetilde{\omegaega}$ to be of this form for $M\in \{3,19\}$. By the previous lemma, $\widetilde{\eta}$ lifts to some $\eta \in \mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})$. Then $(1-w_M^*)(\eta) \in \mathcal{V}$ reduces to $\widetilde{\omegaega}$, as desired. \end{proofofprop} \betagin{remark} \label{p=2remark} When $p\neq 2$, the rank of $\mathcal{V}$ equals the dimension of $\widetilde{V}$: for such $p$, $\mathcal{V}$ is the set of those $\omegaega\in \mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})$ fixed by $\frac12(1-w_N^*)$, and if $\frac12(1-w_N^*)$ fixes $p\omegaega$ then it fixes $\omegaega$ too. For $p=2$, however, $\frac12(1-w_N^*)$ may not map $\mathcal{O}mega_{\mathcal{X}/\mathbb{Z}_p}(\mathcal{X})$ into itself. In fact we found for $X=X_0(53)$ that $\mathrm{rank} (\mathcal{V})=3$ but $\dim \widetilde{V}=1$. This means that the matrix $\widetilde{\mathcal{A}}$ of Theorem \ref{theorem1} or the corresponding $1\times k$ matrix of Theorem \ref{theorem2} is reduced in size compared to the matrix $\mathcal{A}$ defined in Theorems 1 and 2 of \cite{siksek}, but note that this does not affect the rank. \end{remark} \section{Results for non-hyperelliptic $X_0(N)$}\label{results} In this section we list all the (exceptional) quadratic points found for $X_0(N)$ with \[ N\in \{43,53,61,57,65,67,73\} \] as well as the elliptic curves they correspond to. We also list any non-cuspidal rational points, c.f. Remark \ref{rationalrk}. Models for $X_0(N)$ are always given in projective space. With $C$ we always mean the quotient of $X_0(N)$ by one or more Atkin--Lehner involutions such that $\mathrm{rk}J(C)(\mathbb{Q}) = \mathrm{rk}J_0(N)(\mathbb{Q})$, and we denote the map $X\to C$ by $\rho$. For elliptic curves, $O$ always denotes the zero element, which equals $(0:1:0)$ in every case. The column denoted by CM lists the discriminant of the order by which the elliptic curve has complex multiplication if it does, and NO if it does not have CM. The column $\mathbb{Q}$-curve denotes the Atkin--Lehner operators defining an isogeny between the elliptic curve and its conjugate if there is one, and NO if it is not a $\mathbb{Q}$-curve. \subsection{$X_0(43)$} Model for $X_0(43)$: \betagin{align*} x_0^3x_2 - 2x_0^2x_1^2 &+ 2x_0^2x_1x_2 - 2x_0^2x_2^2 + x_0x_1^3 + 3x_0x_1^2x_2 - 5x_0x_1x_2^2 \\&+ 3x_0x_2^3 - 9x_1^4 + 24x_1^3x_2 - 28x_1^2x_2^2 + 16x_1x_2^3 - 4x_2^4=0. \end{align*} Genus $X_0(43)$: 3.\\ Cusps: $(1:0:0)$, $(1:1:1)$.\\ $C=X_0(43)^+$: elliptic curve $y^2 + y = x^3 + x^2$ of conductor 43.\\ Group structure of $J(C)(\mathbb{Q})$: $J(C)(\mathbb{Q})= \mathbb{Z}\cdot [Q_C-O]$, where $Q_C:=(0 : -1 : 1)$.\\ Group structure of $G\subset J_0(43)(\mathbb{Q})$: $G= \mathbb{Z}\cdot D_1\oplus \mathbb{Z}/7\mathbb{Z}\cdot D_{\mathrm{tor}}$, where $D_{\mathrm{tor}}=[(1:1:1)-(1:0:0)]$ and $D_1=[P+\overline{P}-(1:1:1)-(1:0:0)]=\rho^*[Q_C-O]$ for \[ P:=\left(\frac18(\sqrt{-7} + 3) : \frac18(-\sqrt{-7} + 5) : 1\right) \in X_0(43)(\mathbb{Q}(\sqrt{-7})) \] satisfying $\rho(P)=Q_C$.\\ Primes used in sieve: 5,7,11.\\ The following table contains a list of all quadratic points (up to Galois conjugacy) not coming from $X_0^+(43)(\mathbb{Q})$ via $\rho: X_0(43)\to X_0^+(43)$ and all non-cuspidal rational points. \betagin{table}[h!] \centering \betagin{tabular}{c c c c c c} Name & $\theta^2$ & Coordinates & $j$-invariant & CM & $\mathbb{Q}$-curve \\ [0.5ex] \hline \hline $P_0$ & - & $\left(0: \frac45 : \frac25 : 1\right)$ & -884736000 & -43 & -\\ $P_1$ & -131 & $\left(\frac{1}{72}(\theta + 65) : \frac{1}{24}(\theta + 17) : 1\right)$ & \tiny{$\substack{(2646379314349235349704820159442238\theta \\+ 70713216722735823130811605070229181)\\ /3410605131648480892181396484375}$} & NO & NO \\ $P_2$ & -131 & $\left(\frac{1}{25}(-2\theta + 9) : 2/5 : 1\right)$ & \tiny{$\substack{(245508467396487686583118\theta \\- 5588464515419225929913951)\\/1641284836972685388135}$} &NO & NO \\ $P_3$ & -71 & $\left(\frac14(\theta + 1) : 1 : 0\right)$ & $\frac{-49\theta-977}{4}$ & NO & NO \\ $P_4$ & -71 & $ \left(\frac{1}{15}(-\theta + 8) : \frac{1}{30}(-\theta + 23) : 1\right)$ & \tiny{$\substack{(5111948195521623101849\theta\\ - 22519853936617719563123)\\/17592186044416}$} & NO & NO \end{tabular} \label{table43} \end{table} \subsection{$X_0(53)$} Model for $X_0(53)$: \betagin{align*} &9x _0^2 x_3 - 5 x_0 x_3^2 - 27 x_1^3 - 18 x_1^2 x_2 + 78 x_1^2 x_3 - 18 x_1 x_2^2 + 30 x_1 x_2 x_3 \\ &\quad\quad\quad\quad\quad\quad\;\;\;\;\;- 64 x_1 x_3^2 - 57 x_2^3 + 136 x_2^2 x_3 - 104 x_2 x_3^2 + 49 x_3^3=0,\\ &x_0 x_2 - 2 x_0 x_3 - 3x_1^2 + 5 x_1 x_3 - 2 x_2^2 + x_2 x_3 - 2 x_3^2=0. \end{align*} Genus $X_0(53)$: 4.\\ Cusps: $(1:0:0:0)$, $(1:1:1:1)$.\\ $C=X_0(53)^+$: elliptic curve $y^2 + xy + y = x^3 - x^2$ of conductor 53.\\\ Group structure of $J(C)(\mathbb{Q})$: $J(C)(\mathbb{Q})=\mathbb{Z}\cdot [Q_C-O]$, where $Q_C:=(0:-1:1)$.\\ Group structure of $G\subset J(X)(\mathbb{Q})$: $G= \mathbb{Z}\cdot D_1\oplus \mathbb{Z}/13\mathbb{Z}\cdot D_{\mathrm{tor}}$, where $D_{\mathrm{tor}}=[(1:1:1:1)-(1:0:0:0)]$ and $D_1=[P+\overline{P}-(1:1:1:1)-(1:0:0:0)]=\rho^*[Q_C-O]$ for \[ P:=\left( 0, \frac16(-\sqrt{-11} + 5): 1: 1 \right)\in X_0(53)(\mathbb{Q}(\sqrt{-11})) \] satisfying $\rho(P)=Q_C$.\\ Primes used in sieve: 11,7. \\ There are {\bf no} quadratic points on $X_0(53)$ not coming from $X_0^+(53)(\mathbb{Q})$ via $\rho: X_0(53)\to X_0^+(53)$ and {\bf no} non-cuspidal rational points. In particular, all quadratic points correspond to $\mathbb{Q}$-curves. \subsection{$X_0(61)$} Model for $X_0(61)$: \betagin{align*} & x_0^2 x_3 + x_0 x_1 x_3 - 2 x_0 x_3^2 - 2 x_1^3 - 6 x_1^2 x_2 + 5 x_1^2 x_3 - 5 x_1 x_2^2 + 4 x_1 x_2 x_3 - 6 x_2^3 + 14 x_2^2 x_3 - 11 x_2 x_3^2 + 4 x_3^3=0,\\ & x_0 x_2 - x_1^2 - x_1 x_2 - 2 x_2^2 + 2 x_2 x_3 - x_3^2=0 \end{align*} Genus of $X_0(61)$: 4.\\ Cusps: $(1:0:0:0)$, $(1:0:1:1)$.\\ $C=X_0(61)^+$: elliptic curve $y^2 + xy = x^3 + 6x^2 + 11x + 6$ of conductor 61.\\ Group structure of $J(C)$: $J(C)(\mathbb{Q})= \mathbb{Z}\cdot [Q_C-O]$, where $Q_C=(-1:1:1)$.\\ Group structure of $G\subset J_0(61)(\mathbb{Q})$: $G= \mathbb{Z}\cdot D_1\oplus \mathbb{Z}/5\mathbb{Z}\cdot D_{\mathrm{tor}}$, where $D_{\mathrm{tor}}=[(1:0:1:1)-(1:0:0:0)]$ and $D_1=[P+\overline{P}-(1:1:1:1)-(1:0:1:1)]=\rho^*[Q_C-O]$ for \[ P:=\left(0: \frac12(\sqrt{-3} - 1): 1: 1\right)\in X_0(61)(\mathbb{Q}(\sqrt{-3})) \] satisfying $\rho(P)=Q_C$.\\ Primes used in sieve: 7.\\ There are {\bf no} quadratic points on $X_0(61)$ not coming from $X_0^+(61)(\mathbb{Q})$ via $\rho: X_0(61)\to X_0^+(61)$ and {\bf no} non-cuspidal rational points. In particular, all quadratic points correspond to $\mathbb{Q}$-curves. \subsection{$X_0(57)$} Model for $X_0(57)$: \betagin{align*} &x_0x_2 - x_1^2 + 2x_1x_3 + 2x_1x_4 - 2x_2^2 - 2x_2x_3 + 3x_2x_4 - x_3^2 - 2x_3x_4 - x_4^2=0,\\ &x_0x_3 - x_1x_2 - 2x_1x_4 + 4x_2x_3 - 6x_2x_4 - x_3^2 + 5x_3x_4 - 5x_4^2=0,\\ &x_0x_4 - x_2^2 + x_2x_3 - 2x_2x_4 - 2x_4^2=0. \end{align*} Genus of $X_0(57)$: 5.\\ Cusps: $(1:0:0:0:0),(1:1:0:1:0),(3:3:1:2:1),(3:9/2:-1/2:7/2:1)$.\\ $C=X_0(57)^*=X_0(57)/\langle w_3,w_{19}\rangle$: elliptic curve $y^2 + y = x^3 - x^2 - 2x + 2$.\\ Group structure of $J(C)$: $J(C)(\mathbb{Q})\simeq \mathbb{Z}$, $[Q_C-O]\mapsto 1$, where $Q_C=(2:-2:1)$.\\ Group structure of $G\subset J_0(57)(\mathbb{Q})$: $G= \mathbb{Z}\cdot D_1\oplus \mathbb{Z}/6\mathbb{Z}\cdot D_{\mathrm{tor},1}\oplus \mathbb{Z}/30\mathbb{Z}\cdot D_{\mathrm{tor},2}$, where $D_{\mathrm{tor},1}=[(1:1:0:1:0)-(1:0:0:0:0)]$, $D_{\mathrm{tor},2}=[(3:3:1:2:1)-(1:0:0:0:0)]$ and $D_1=[\mathrm{Trace}(P)-\sum_{c\in \mathrm{cusps}}c]$ for \betagin{align*} P:=&(\frac{1}{13}(-9\alpha^3 - 2\alpha^2 - 3\alpha + 34): \frac{1}{13}(15\alpha^3 + 12\alpha^2 - 21\alpha + 30): \frac{1}{13}(-5\alpha^3 - 4\alpha^2 + 7\alpha + 3):\\ &\frac{1}{13}(8\alpha^3 + 9\alpha^2 - 19\alpha + 29):1),\;\; \alpha^2=\frac{1+\sqrt{-3}}{2} \end{align*} satisfying $\rho(P)=Q_C$.\\ The following are {\bf all} quadratic points on $X_0(57)$ up to Galois conjugacy. There are {\bf no} non-cuspidal rational points. \\ \betagin{table}[h!] \centering \betagin{tabular}{c c c c c c} Name & $\theta^2$ & Coordinates & $j$-invariant & CM & $\mathbb{Q}$-curve \\[0.5ex] \hline \hline $P_1$ & -23 & \tiny{$\left(\frac{1}{32}(-11 \theta + 47): \frac{1}{16}(-7 \theta + 43): \frac18( \theta - 5): \frac14(- \theta + 9): 1\right)$} & $\frac{343\theta + 4021}{4}$ & NO & NO \\ $P_2$ & -23 & $\left(\frac12( \theta - 3): \frac12( \theta + 3): 1: 1: 0\right)$ &$\frac{402878\theta + 2212325}{32}$&NO & NO \\ $P_3$ & -23 & $\left(\frac12(- \theta + 3): - \theta + 2: -1: \frac12(- \theta + 1): 1\right)$ & \tiny{$\substack{(152503825515346075337\theta \\ - 1518681643605456439979)\\/288230376151711744}$}& NO & NO \\ $P_4$ & -23 & $ \left(\frac18( \theta - 1): \frac18( \theta + 11): \frac14(- \theta - 1): \frac18( \theta + 15): 1 \right)$ & \tiny{$\substack{ (-56278625425021601\theta\\ - 102516814328210867)\\/1048576}$} & NO & NO\\ $P_5$ & -3 & $\left(2:\frac12(- \theta + 7): 0: \frac12(- \theta + 5): 1\right)$&-12288000& -27&$w_{19}$ \\ $P_6$ & -3 &$\left(- \theta: - \theta + 3: \frac12( \theta - 1): - \theta + 2: 1\right)$ &0&-3 & $w_{19}$ \\ $P_7$ & -3 &$\left(\frac12(- \theta + 3): \frac12(- \theta + 3): \frac12( \theta - 1): 2: 1\right)$ &54000&-12 &$w_{19},w_{57}$ \\ $P_8$ & -3 & $\left(\frac12(-5 \theta + 1): \frac12(-5 \theta+ 1): \frac12( \theta- 3): - \theta + 1: 1\right)$ &0& -3&$w_{19},w_{57}$ \\ $P_9$ & -2 &$\left(\frac13( \theta + 4): \frac13(4 \theta + 7): \frac13(- \theta - 1): \frac13(2 \theta + 5): 1 \right)$ &8000&-8 & $w_{57}$ \\ $P_{10}$ & -2 &$\left(\frac12(- \theta+ 2): 3:\frac12 \theta: \frac12(- \theta + 6): 1\right)$ &8000& -8&$w_{57}$ \\ $P_{11}$ & 13 & $\left(\frac13(2 \theta + 14): \frac13(2 \theta + 14): \frac16(- \theta - 1): \frac16(5 \theta + 29): 1\right)$&\tiny{$\substack{(3387888351672962316333\theta \\ - 12215205167504087643323)/2}$}&NO &$w_{3}$ \\ $P_{12}$ & 13 &$\left(\frac12(5 \theta + 23): \frac12(-3 \theta - 3): \frac12( \theta + 3): \frac12(- \theta + 3): 1\right)$ & $\frac{-17787\theta - 61763}{2}$&NO&$w_3$ \end{tabular} \label{table57} \end{table} \noindent Primes used in sieve: 11,13. \subsection{$X_0(65)$} Model for $X_0(65)$: \betagin{align*} &x_0 x_2 - x_1^2 + x_1 x_4 - 2 x_2^2 - x_2 x_3 + 3 x_2 x_4 - x_3^2 + 2 x_3 x_4 - 2 x_4^2=0,\\ &x_0 x_3 - x_1 x_2 - 2 x_2^2 - x_2 x_3 + 4 x_2 x_4 - x_3^2 + 2 x_3 x_4 - 2 x_4^2=0,\\ &x_0 x_4 - x_1 x_3 - 2 x_2^2 - 3 x_2 x_3 + 5 x_2 x_4 + 3 x_3 x_4 - 3 x_4^2=0. \end{align*} Genus of $X_0(65)$: 5.\\ Cusps: $(1:0:0:0:0),(1:1:1:1:1),(1/3:2/3:2/3:2/3:1),(1/2:1/2:1/2:1/2:1)$.\\ $C=X_0(65)^+$: elliptic curve $y^2 + xy = x^3 - x$ of conductor 65.\\ Group structure of $J(C)$: $J(C)(\mathbb{Q})= \mathbb{Z}\cdot [Q_C-O]\oplus \mathbb{Z}/2\mathbb{Z}\cdot [(0:0:1)-O]$, where $Q_C=(1:0:1)$.\\ Group structure of $G\subset J_0(65)(\mathbb{Q})$: $G= \mathbb{Z}\cdot D_1 \oplus \mathbb{Z}/2\mathbb{Z}\cdot (-9D_{\mathrm{tor},1}+2D_{\mathrm{tor},2})\oplus \mathbb{Z}/84\mathbb{Z}\cdot (17D_{\mathrm{tor},1}+13D_{\mathrm{tor},2})$, where $D_{\mathrm{tor},1}=[(1:1:1:1:1)-(1:0:0:0:0)]$, $D_{\mathrm{tor},2}=[(1/3:2/3:2/3:2/3:1)-(1:0:0:0:0)]$ and $D_1=[P+\overline{P}-(1:0:0:0:)-(1:1:1:1:1)]=\rho^*([Q_C-O])$ for \[ P=\left(0:1:\frac12(1+i):1:1\right)\in X_0(65)(\mathbb{Q}(i)) \] satisfying $\rho(P)=Q_C$. \\ There are {\bf no} quadratic points on $X_0(65)$ that do not come from $X_0(65)^+(\mathbb{Q})$ via $\rho: X_0(65)\to X_0(65)^+$ and {\bf no} non-cuspidal rational points. In particular, all quadratic points correspond to $\mathbb{Q}$-curves.\\ Primes used in sieve: 17, 23. \subsection{$X_0(67)$} Model for $X_0(67)$: \betagin{align*} &x_0 x_2 - x_1^2 + 2 x_1 x_3 + 2 x_1 x_4 - 2 x_2^2 - 2 x_2 x_3 + 3 x_2 x_4 - x_3^2 - 2 x_3 x_4 - x_4^2=0,\\ &x_0 x_3 - x_1 x_2 - 2 x_1 x_4 + 4 x_2 x_3 - 6 x_2 x_4 - x_3^2 + 5 x_3 x_4 - 5 x_4^2=0,\\ &x_0 x_4 - x_2^2 + x_2 x_3 - 2 x_2 x_4 - 2 x_4^2=0. \end{align*} Genus of $X_0(67)$: 5.\\ Cusps: $(1:0:0:0:0),(1/2 : 1 : 1/2 : 1/2 : 1)$.\\ $C=X_0(67)^+$: genus 2 hyperelliptic curve $y^2 = x^6 - 2x^5 + x^4 + 2x^3 + 2x^2 + 4x + 1$.\\ Group Structure of $C$: $J(C)(\mathbb{Q})= \mathbb{Z}\cdot [Q_1-(1:-1:0)]\oplus \mathbb{Z}[Q_2-(1:-1:0)]$, where $Q_1=(1:1:0)$ and $Q_2=(0:1:1)$. \\ Group Structure of $G\subset J_0(67)(\mathbb{Q})$: $G = \mathbb{Z}\cdot D_1\oplus \mathbb{Z} D_2\oplus \mathbb{Z}/11\mathbb{Z}\cdot D_{\mathrm{tor}}$, where $D_{\mathrm{tor}}:=[(1/2 : 1 : 1/2 : 1/2 :1)-(1:0:0:0:0)]$, $D_1:=[P_7+\overline{P}_7-(1:0:0:0:0)-(1/2 : 1 : 1/2 : 1/2 :1)]$ and $D_2:=[P_1+\overline{P}_1-(1:0:0:0:0)-(1/2 : 1 : 1/2 : 1/2 :1)]$ for $P_1,P_7$ defined in the table below satisfying $\rho(P_1)=Q_1$ and $\rho(P_7)=Q_2$.\\ There are {\bf no} quadratic points on $X_0(67)^+$ that do not come from $X_0(67)^+(\mathbb{Q})$ via $\rho: X_0(67)\to X_0(67)^+$ and there is one non-cuspidal rational point. \\ Moreover, we deduce that the following table gives a complete list of {\bf all} quadratic points and non-cuspidal rational points on $X_0(67)$ up to Galois conjugacy. \betagin{table}[h!] \centering \betagin{tabular}{c c c c c c} Name & $\theta^2$ & Coordinates & $j$-invariant & CM & $\mathbb{Q}$-curve \\ [0.5ex] \hline \hline $P_0$ & - & $\left(\frac34 : \frac{7}{12} : \frac{7}{12} : \frac13 : 1\right)$ & \small{-147197952000} & -67 & - \\ $P_1$ & -2 & \tiny{$\left( \frac{1}{18}(- \theta + 4): \frac{1}{18}( \theta + 14): \frac{1}{18} (- \theta + 4): \frac19 (- \theta + 4): 1 \right)$} & 8000 & -8 & $w_{67}$ \\ $P_2$ & -3 & \tiny{$\left( 0: \frac16 ( \theta + 3): \frac16 ( \theta + 3): \frac16 ( \theta + 3): 1 \right)$} &54000&-12 & $w_{67}$ \\ $P_3$ & -3 & \tiny{$\left( \frac{1}{26} (3 \theta + 5): 1: \frac{1}{26} (- \theta + 7): \frac{1}{13} ( \theta + 6): 1 \right)$} & -12288000& -27 & $w_{67}$ \\ $P_4$ & -3 & \tiny{$\left( \frac{1}{91} (18 \theta + 22): \frac{1}{182} (-27 \theta + 149): \frac{1}{182} (15 \theta + 79): \frac{1}{182} (-3 \theta + 57): 1\right)$} & 0 & -3 & $w_{67}$\\ $P_5$ & -7 & \tiny{$\left( \frac{1}{16} ( \theta + 5): \frac{1}{16} (- \theta + 11): \frac{1}{16} ( \theta + 5): \frac{1}{16} ( \theta + 5): 1 \right)$}&-3375& -7&$w_{67}$ \\ $P_6$ & -7 & \tiny{$\left( \frac{1}{20} (- \theta + 1): \frac{1}{20} (- \theta + 21): \frac{1}{20} ( \theta + 7): \frac{1}{20} (- \theta+ 9): 1 \right)$} &16581375&-28 & $w_{67}$ \\ $P_7$ & -11 & \tiny{$\left( 0: \frac{1}{11} ( \theta + 11): \frac{1}{22} (- \theta + 11): \frac{1}{22} ( \theta + 11): 1 \right)$} &-32768&-11 &$w_{67}$ \\ $P_8$ & -43 & \tiny{$\left( \frac{1}{106} (- \theta - 13): \frac{1}{53} (-2 \theta + 27): \frac{1}{106} (-5 \theta + 41): \frac{1}{53} (-2 \theta + 27): 1 \right)$} &-884736000& -43&$w_{67}$ \end{tabular} \label{table67} \end{table} \noindent Primes used in sieve: 73, 59, 53, 31, 19, 5. \subsection{$X_0(73)$} Model for $X_0(73)$: \betagin{align*} &x_0 x_2 - 2 x_1^2 + 2 x_1 x_2 - 2 x_1 x_4 - x_2^2 + 3 x_2 x_3 + 3 x_3^2 - x_4^2=0,\\ &x_0 x_3 - 1/2 x_1 x_2 - x_1 x_3 + 1/2 x_2^2 - 1/2 x_2 x_3 + x_2 x_4 - 4 x_3^2 + 9/2 x_3 x_4 - 1/2 x_4^2=0,\\ &x_0 x_4 - x_1 x_3 + x_1 x_4 - x_2 x_3 - 5 x_3^2 + 4 x_3 x_4=0. \end{align*} Genus of $X_0(73)$: 5.\\ Cusps: $(1:0:0:0:0),(1:1:1:0:0)$.\\ $C=X_0(73)^+$: genus 2 hyperelliptic curve $y^2 = x^6 + 2x^5 + x^4 + 6x^3 + 2x^2 - 4x + 1$.\\ Group Structure of $J(C)$: $J(C)(\mathbb{Q}) = \mathbb{Z}\cdot [Q_1-(1:1:0)]\oplus \mathbb{Z}\cdot [Q_2-(1:1:0)]$, where $Q_1:=(0:-1:1)$ and $Q_2:=(0:1:1)$. \\ Group Structure of $G\subset J_0(73)(\mathbb{Q})$: $G= \mathbb{Z}\cdot D_1\oplus \mathbb{Z}\cdot D_2 \oplus \mathbb{Z}/6\mathbb{Z}\cdot D_{\mathrm{tor}}$, where $D_{\mathrm{tor}}=[(1:0:0:0:0)-(1:1:1:0:0)]$, $D_1=[P_3+\overline{P}_3-(1:0:0:0:0)-(1:1:1:0:0)]$, $D_2=[P_6+\overline{P}_6-(1:0:0:0:0)-(1:1:1:0:0)]$ for $P_3$, $P_6$ defined in the table below and satisfying $\rho(P_3)=Q_1$ and $\rho(P_6)=Q_2$. \\ The only quadratic points on $X_0(73)$ that do not come from $X_0(73)^+(\mathbb{Q})$ via $\rho: X_0(73)\to X_0(73)^+$ are $P_1,P_2$ as defined in the table below. There are {\bf no} non-cuspidal rational points. \\ Moreover, we deduce that the following table gives a complete list of {\bf all} quadratic points on $X_0(73)$ up to Galois conjugacy. \betagin{table}[h!] \centering \betagin{tabular}{c c c c c c} Name & $\theta^2$ & Coordinates & $j$-invariant & CM & $\mathbb{Q}$-curve \\ [0.5ex] \hline \hline $P_1$ & -31 & \tiny{$\left( \frac{1}{32} (-\theta - 33): \frac{1}{16} (-\theta - 9): \frac{1}{32} (-3 \theta - 35): \frac{1}{32} (\theta + 17): 1 \right)$} & \tiny{$\substack{(-218623729131479023842537441\theta\\- 75276530483988147885303471)\\/18889465931478580854784}$} & NO & NO \\ $P_2$ & -31 & \tiny{$\left( \frac{1}{32} (-\theta - 31): \frac{1}{16} (\theta - 17): -\frac{3}{2}: \frac{1}{2}: 1 \right)$} &$\frac{-6561\theta + 1809}{4}$&NO & NO \\ $P_3$ & -19 & \tiny{$\left( 1/7 (\theta - 10): \frac{1}{14} (\theta - 17): \frac{1}{14} (\theta - 17): \frac{1}{14} (\theta + 11): 1 \right)$ } & -884736& -19 & $w_{73}$ \\ $P_4$ & -1 & \tiny{$\left( \frac{1}{5} (\theta - 7): \frac{1}{5} (2 \theta - 4): \frac{1}{5} (3 \theta - 6): \frac{1}{5} (\theta + 3): 1 \right)$} & 287496& -16 & $w_{73}$\\ $P_5$ & -1 & \tiny{$\left( \frac{1}{13} (2 \theta - 16): \frac{1}{13} (-8 \theta - 14): \frac{1}{13} (-2 \theta - 23): \frac{1}{13} (2 \theta + 10): 1 \right)$}&1728& -4&$w_{73}$ \\ $P_6$ & -2 &\tiny{$\left( \frac{1}{6} (-\theta - 8): -1: \frac{1}{6} (-\theta - 8): \frac{1}{6} (-\theta + 4): 1 \right)$} & 8000&-8 & $w_{73}$\\ $P_7$ & -127 & \tiny{$\left( \frac{1}{32} (-\theta - 47): \frac{1}{176} (-5 \theta - 163): \frac{1}{22} (-\theta - 26): \frac{1}{44} (-\theta + 29): 1 \right)$} &\tiny{$\substack{(14758692270140155157349165\theta\\ + 81450017206599109708140525)\\/18889465931478580854784}$}&NO & $w_{73}$ \\ $P_8$ & -3 & \tiny{$\left( \frac{1}{26} (-9 \theta - 41): \frac{1}{55} (-15 \theta - 51): \frac{1}{26} (-9 \theta - 28): \frac{1}{52} (-9 \theta + 37): 1 \right)$} &0&-3 &$w_{73}$ \\ $P_9$ & -3 & \tiny{$\left( -1: \frac{1}{4} (\theta - 3): \frac{1}{4} (\theta - 5): \frac{1}{2}: 1 \right)$} &54000& -12&$w_{73}$ \\ $P_{10}$ &-1&\tiny{$\left( -1: \frac12(\theta - 3): -2: 1: 1 \right)$} &-12288000&-27&$w_{73}$\\ $P_{11}$ &-67&\tiny{$\left( \frac{1}{23} (-4 \theta - 43): \frac{1}{46} (-7 \theta - 35): \frac{1}{23} (-3 \theta - 15): \frac{1}{23} (-\theta + 18): 1 \right)$}&-147197952000&-67&$w_{73}$ \end{tabular} \label{table73} \end{table} \noindent Primes used in sieve: 43, 67, 41, 17, 37, 13. \section{The hyperelliptic curve $X_0(37)$} \label{hypersection} This curve deserves a special section due to its peculiar nature. A model for $X_0(37)$, computed using the Small Modular Curves package in \texttt{Magma}, is given by \[ X_0(37):\;y^2 = x^6 + 8x^5 - 20x^4 + 28x^3 - 24x^2 + 12x - 4. \] It is hyperelliptic of genus 2. We note that Mazur and Swinnerton--Dyer \cite[Section 5]{mazursd} already computed a model for this curve, as well as for $X_0(37)^+$, in 1974 without the aid of computers. They noticed that 37 is the smallest value of $N$ such that $X_0(N)^+$ has positive genus. Being hyperelliptic, $X_0(37)$ admits a hyperelliptic involution, mapping $(x,y)\mapsto (x,-y)$. Naturally, one is led to wonder: what is its moduli interpretation? Is it perhaps the Atkin--Lehner involution? There are exactly nineteen values of $N$ such that $X_0(N)$ is hyperelliptic, as was discovered by Ogg \cite{ogg}. Only in three of those nineteen cases, the hyperelliptic involution is not an Atkin--Lehner involution. For two of these, $N=40$ and $N=48$, there are different matrices in $\mathrm{SL}_2(\mathbb{R})$ inducing the hyperelliptic involution. Not for $N=37$, however. Lehner and Newman computed in 1964 in a page of corrections to their paper \cite{lehner} that the hyperelliptic involution of $X_0(37)$ is not induced by any automorphism of the complex upper half plane. Instead, the two cusps $(1:1:0)$ and $(1:1:1)$ (in the closure in $\mathbb{P}(1,3,1)$ of the above model) are mapped to the non-cuspidal rational points $(1:-1:0)$ and $(1:-1:1)$ respectively. We now know by \cite{mazur0} that in fact these are the only non-cuspidal rational points. This observation of Lehner and Newman means that any moduli interpretation of the hyperelliptic involution must include an interpretation of the cusps. Such an interpretation in terms of ``generalized elliptic curves" was given by Deligne and Rapoport \cite{deligne}, but we do not attempt to describe the hyperelliptic involution in those terms here. Define $\infty_+:=(1:1:0)$ and $\infty_-:=(1:-1:0)$ and let $i$ be the hyperelliptic involution on $X_0(37)$. We then have three non-trivial involutions $i,w_{37}$ and $i\circ w_{37}=w_{37}\circ i$. In fact, we compute in \texttt{Magma} that $\mathrm{Aut}_{\mathbb{Q}}(X_0(37))=\langle i,w_{37}\rangle\simeq \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. We compute $w_{37}$ to be \[ w:=w_{37}: X_0(37)\to X_0(37), \; (x:y:z)\mapsto (x:y:x-z). \] We refer to the four mentioned rational points as $\infty_+$, $\infty_-$, $w(\infty_+)$ and $w(\infty_-)$. The fact that $i\neq w_{37}$ has dramatic consequences for the abundance of quadratic points on $X_0(37)$. The hyperelliptic covering map $x: X_0(37)\to \mathbb{P}^1$ is one source of infinitely many quadratic points. Unlike in the cases $N=40$ and $N=48$ (see \cite{bruin}), here the quotient $\rho: X_0(37)\to X_0(37)^+:=X_0(37)/\langle w_{37}\rangle$ is an elliptic curve of rank 1. A model is given by \[ X_0(37)^+: \;y^2 + y = x^3 - x \] and its Mordell-Weil group is the free abelian group generated by the point $Q_1:=(0:-1:1)$ (choosing $O=\rho(\infty_+)=(0:1:0)$). Hence this bielliptic quotient $X_0(37)^+$ is a second source of infinitely many quadratic points. Moreover, we also have the quotient map $\pi: X_0(37)\to E:=X_0(37)/\langle i\circ w_{37}\rangle$. A model for $E$ is \[ E: \;y^2 + y = x^3 + x^2 - 23x - 50. \] This elliptic curve has Mordell-Weil group $\mathbb{Z}/3\mathbb{Z}$, generated by the point $Q_E:=(8:18:1)$ (choosing $O=\pi(\infty_+)=(0:1:0)$). Another source of points, albeit just three. In this Section we describe how these different sources fit together to make up all points on $X_0(37)^{(2)}(\mathbb{Q})$. Consider the map \[ \kappa: X_0(37)^{(2)}(\mathbb{Q})\to J_0(37)(\mathbb{Q}), \; \{P,Q\}\mapsto [P+Q - \infty_+-\infty_-]. \] \betagin{lemma} Let $X$ be any hyperelliptic curve of genus 2 with involution $i$. Then an effective degree 2 divisor $D$ is linearly equivalent to the canonical divisor $K$ if and only if $D$ is of the form $P+i(P)$ for some point $P$ on $X$. \end{lemma} \betagin{proof} This is well-known. Riemann--Roch implies that $\ell(K-P-i(P))=\ell(P+i(P))-1$, and $\deg (K-P-i(P))=0$. Post-composed with an automorphism of $\mathbb{P}^1$, the hyperelliptic covering map $X\to \mathbb{P}^1$ defines a non-constant function in the Riemann--Roch space $L(P+i(P))$, so that $K=[P+i(P)]$. Conversely, suppose that $P+Q\sim R + i(R)$ for points $P,Q,R$ on $X$. Note that $\ell(R+i(R))=2$ by Riemann--Roch, hence $L(R+i(R))=\{\alpha+\betata f : \alpha,\betata \in \overline{\mathbb{Q}}\}$, where $f$ is the hyperelliptic covering map with poles in $R,i(R)$. This shows that $R+i(R)$ is only linearly equivalent to degree 2 divisors of the form $P+i(P)$. \end{proof} \betagin{lemma} Let $X$ be a hyperelliptic genus 2 curve with Jacobian $J$, hyperelliptic involution $i$ and $\kappa: X^{(2)}\to J, \;P\mapsto [P-\infty-i(\infty)]$ for some point $\infty$ on $X$. The fibre $\kappa^{-1}(\{0\})$ consists of the effective divisors linearly equivalent to the canonical divisor $K$. Moreover, $\kappa$ is a bijection away from this fibre. \end{lemma} \betagin{proof} This is again a well-known consequence of Riemann--Roch. \end{proof} \betagin{proposition} Let $P_1=(1/2(\sqrt{-3} + 1): -1: 1)\in X_0(37)(\overline{\mathbb{Q}})$. The Jacobian $J_0(37)(\mathbb{Q})= \mathbb{Z}\cdot D\oplus\mathbb{Z}/3\mathbb{Z} D_{\mathrm{tor}}$, where $D=[P_1+\overline{P_1}-\infty_+-\infty_-]=\rho^*([Q_1-O])$ and $D_{\mathrm{tor}}=[\infty_++w(\infty_-)-\infty-\infty_-]=[\infty_+-w(\infty_+)]=\pi^*[Q_E-O]$. \end{proposition} \betagin{proof} As $X_0(37)$ is hyperelliptic of genus 2, its Mordell--Weil group can be determined using height bounds and an algorithm of Stoll \cite{stoll}. As 37 is prime, we know by Mazur's \cite{mazur1} proof of Ogg's conjecture that the torsion subgroup equals the rational cuspidal subgroup, and we check that the difference of the cusps has order 3, equals $\pi^*([Q_E-O])$ and is of the form $\kappa(\{\infty_+,i(w(\infty_+)\})$. \end{proof} Note that $\rho(P_1)\neq Q_1$ because the basedivisor of $\kappa$ is not $w_{37}$-invariant. This gives us a description of all points in $X_0(37)^{(2)}(\mathbb{Q})$. First, there is the set $\mathcal{Q}_i$ of quadratic points $Q_x$ with rational $x$-coordinate $x$. Though easy to describe coordinatewise, we lack a moduli interpretation of elliptic curves corresponding to these points. Then we have the points $\mathcal{P}_{1,0}:=\{P_1,\overline{P_1}\}$ and $\mathcal{P}_{0,1}:=\{\infty_+,w(\infty_-)\}$. For any $(a,b)\in \mathbb{Z}\times \mathbb{Z}/3\mathbb{Z} \setminus \{(0,0)\}$, there is another point $P_{a,b}\in X_0(37)^{(2)}(\mathbb{Q})$ defined as the unique effective degree 2 divisor $P$ such that $P-\infty_+ -\infty_- \sim a\mathcal{P}_{1,0}+b\mathcal{P}_{0,1}-(a+b)(\infty_++\infty_-)$ for any lift of $b$ to $\mathbb{Z}$. To compute $\mathcal{P}_{a,b}$, one simply computes the 1-dimensional Riemann--Roch space $L(a\mathcal{P}_{1,0}+b\mathcal{P}_{0,1}-(a+b-1)(\infty_++\infty_-)).$ We conclude the following. \betagin{proposition}\label{prop37} The map \[ \mathbb{P}^1(\mathbb{Q})\sqcup (\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z} \setminus \{(0,0)\})\to X^{(2)}(\mathbb{Q}),\; x\mapsto Q_x, \; (a,b)\mapsto P_{a,b} \] is a bijection. Moreover, $w_{37}$ interchanges the two points in $\mathcal{P}\in X_0(37)^{(2)}(\mathbb{Q})$ if and only if $\mathcal{P}=P_{a,0}$ for some $a\in \mathbb{Z}$, and in that case $\mathcal{P}$ corresponds to a $\mathbb{Q}$-curve. \end{proposition} \betagin{remark}If $P$ is a quadratic point such that $\rho(P)\in X_0(37)^+(\mathbb{Q})$, then $[P+\overline{P}-\infty_+ -\infty_-]$ is not equal to $\rho^*[\rho(P)-O]=[P+\overline{P}-\infty_+ -w(\infty_+)]$. As $[w(\infty_+)-\infty_-]=-3[P_1-\infty_+-\infty_-]$, this amounts to a shift: if $\rho(P)=n[Q_1-O]$ then $\{P,\overline{P}\}=P_{n-3,0}$. This justifies the claim that $P_{a,b}$ is fixed by $w_{37}$ if and only if $b=0$, despite the incompatiblity of the basedivisors. \end{remark} The three points $\{P,Q\}$ in $X_0(37)^{(2)}(\mathbb{Q})$ such that $Q=i\circ w_{37}(P)$ are $\{\infty_+,w(\infty_-)\}$, $\{w(\infty_+),\infty_-\}$ and $\{R,\overline{R}\}$, where $R=(2:2\sqrt{37}:1)$, so $E(\mathbb{Q})$ only contributes one pair of genuinely quadratic points. Yet, this is an interesting pair. We see that $i$ also interchanges the two points, so $\{R,\overline{R}\}=Q_2$. Again, this is no problem, because $[R+\overline{R}-\infty_+-\infty_-]\neq \pi^*([\pi(R)-O])$ due to $\infty_-$ not mapping to $O$ under $\pi$. As both $i$ and $i\circ w$ interchange $R$ and $\overline{R}$, we find that $w_{37}$ fixes $R$ and $\overline{R}$. The corresponding elliptic curve thus has a degree 37 endomorphism and must have CM by the maximal order in $\mathbb{Q}(\sqrt{-37})$. It is no coincidence that the point is defined over $\mathbb{Q}(\sqrt{37})$: this follows from Theorem 4.1 in \cite{advancedsilverman} as the Hilbert class field of $\mathbb{Q}(\sqrt{-37})$ is $\mathbb{Q}(\sqrt{-37},\sqrt{37})$. The $j$-invariant of the elliptic curve corresponding to $R$ is $3260047059360000\sqrt{37} + 19830091900536000$. \textsc{Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom} \emph{E-mail address}: \texttt{j.box@warwick.ac.uk} \end{document}

\begin{document} \begin{center} \textbf{A characterization of compact operators via the non-connectedness of the attractors of a family of IFSs} by ALEXANDRU\ MIHAIL (Bucharest) and RADU\ MICULESCU (Bucharest) \end{center} {\small Abstract}. {\small In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space }$X${\small , }$S$ {\small \ and }$T${\small \ bounded linear operators from }$X${\small \ to }$ X${\small \ such that }$\left\Vert S\right\Vert ,\left\Vert T\right\Vert <1$ {\small \ and }$w\in X${\small , let us consider the IFS }$ S_{w}=(X,f_{1},f_{2})${\small , where }$f_{1},f_{2}:X\rightarrow X${\small \ are given by }$f_{1}(x)=S(x)${\small \ and }$f_{2}(x)=T(x)+w${\small , for all }$x\in X${\small . On one hand we prove that if the operator }$S${\small \ is compact, then there exists a family }$(K_{n})_{n\in \mathbb{N}}${\small \ of compact subsets of }$X${\small \ such that }$A_{\mathcal{S}_{w}}$ {\small \ is\ not connected, for all }$w\in H-\underset{n\in \mathbb{N}}{ \cup }K_{n}${\small . One the other hand we prove that if }$H$ {\small is an infinite dimensional Hilbert space, then a bounded linear operator }$ S:H\rightarrow H${\small \ having the property that }$\left\Vert S\right\Vert <1${\small \ is compact provided that for every bounded linear operator }$T:H\rightarrow H${\small \ such that }$\left\Vert T\right\Vert <1$ {\small \ there exists a sequence }$(K_{T,n})_{n}${\small \ of compact subsets of }$H${\small \ such that }$A_{\mathcal{S}_{w}}${\small \ is\ not connected for all }$w\in H-\underset{n}{\cup }K_{T,n}${\small . Consequently, given an infinite dimensional Hilbert space }$H$,{\small \ there exists a complete characterization of the compactness of an operator }$ S:H\rightarrow H${\small \ by means of the non-connectedness of the attractors of a family of IFSs related to the given operator.} \textbf{1. Introduction. }IFSs were introduced in their present form by John Hutchinson (see [9]) and popularized by Michael Barnsley (see [2]). They are one of the most common and most general ways to generate fractals. Although the fractals sets are defined by means of measure theory concepts (see [7]), they have very interesting topological properties. The connectivity of the attractor of an iterated function system has been studied, for example, in [14] (for the case of an iterated multifunction system) and in [6] (for the case of an infinite iterated function system). It is well known the role of the compact operators theory in functional --------------------------- \textit{2000 Mathematical\ Subject Classification}{\small : }Primary 28A80, 47B07; Secondary 54D05 \textit{Key words and phrases}: iterated function systems, attractors, connectivity, compact operators analysis and, in particular, in the theory of the integral equations. In this frame, a natural question is to provide equivalent characterizations for compact operators. Let us mention some results on this direction. A bounded operator $T$ on a separable Hilbert space $H$ is compact if and only if $\underset{n\rightarrow \infty }{\lim } <Te_{n},e_{n}>=0$ (or equivalently $\underset{n\rightarrow \infty }{\lim } Te_{n}=0$), for each orthonormal basis $\{e_{n}\}$ for $H$ (see [1], [8], [16] and [17]) if and only if every orthonormal basis $\{e_{n}\}$ for $H$ has a rearrangement $\{e_{\sigma (n)}\}$ such that $\sum \frac{1}{n} \left\Vert Te_{\sigma (n)}\right\Vert <\infty $ (see [18]). In a more general framework, in [10] a characterization of the compact operators on a fixed Banach space in terms of a construction due to J.J.M. Chadwick and A.W. Wickstead (see [3]) is presented and in [11] a purely structural characterization of compact elements in a $C^{\ast }$ algebra is given. In contrast to the above mentioned characterizations of the compact operators which are confined to the framework of the functional analysis, in this paper we present such a characterization by means of the non-connectedness of the attractors of a family of IFSs related to the considered operator. In this way we establish an unexpected connection between the theory of compact operators and the theory of iterated function systems. \textbf{2.} \textbf{Preliminary results. }In this paper, for a function $f$\ and $n\in \mathbb{N}$, by $f^{[n]}$ we mean the composition of $f$ by itself $n$ times. DEFINITION\ 2.1. Let $(X,d)$\ be a metric space. A function $f:X\rightarrow X $\ is called a \textit{contraction} in case there exists $\lambda \in (0,1) $ such that \begin{equation*} d(f(x),f(y))\leq \lambda d(x,y)\text{,} \end{equation*} for all $x,y\in X$. THEOREM\ 2.2 (The Banach-Cacciopoli-Picard contraction principle). \textit{ If }$X$\textit{\ is a complete metric space, then for each contraction }$ f:X\rightarrow X$\textit{\ there exists a unique fixed point }$x^{\ast }$ \textit{\ of }$f$\textit{.} \textit{Moreover } \begin{equation*} x^{\ast }=\underset{n\rightarrow \infty }{\lim }f^{[n]}(x_{0})\text{\textit{, }} \end{equation*} \textit{\ for each }$x_{0}\in X$\textit{.} NOTATION. Given a metric space $(X,d)$, by $K(X)$\ we denote the set of non-empty compact subsets of $X$. DEFINITION\ 2.3. For a metric space $(X,d)$, the function $h:K(X)\times K(X)\rightarrow \lbrack 0,+\infty )$ defined by \begin{equation*} h(A,B)=\max (d(A,B),d(B,A))= \end{equation*} \begin{equation*} =\inf \{r\in \lbrack 0,\infty ):A\subseteq B(B,r)\text{ and }B\subseteq B(A,r)\}\text{,} \end{equation*} where \begin{equation*} B(A,r)=\{x\in X:d(x,A)<r\} \end{equation*} and \begin{equation*} d(A,B)=\underset{x\in A}{\sup }d(x,B)=\underset{x\in A}{\sup }(\underset{ y\in B}{\inf }d(x,y))\text{,} \end{equation*} turns out to be a metric which is called the \textit{Hausdorff-Pompeiu metric }. REMARK 2.4. The metric space $(K(X),h)$ is complete, provided that $(X,d)$ is a complete metric space. DEFINITION\ 2.5. Let $(X,d)$\ be a complete metric space. An \textit{ iterated function system} (for short an IFS) on $X$, denoted by $ S=(X,(f_{k})_{k\in \{1,2,...,n\}})$, consists of a finite family of contractions $(f_{k})_{k\in \{1,2,...,n\}}$, $f_{k}:X\rightarrow X$. THEOREM\ 2.6. \textit{Given} $\mathcal{S}=(X,(f_{k})_{k\in \{1,2,...,n\}})$ \textit{an iterated function system on }$X$, \textit{the function} $F_{ \mathcal{S}}:K(X)\rightarrow K(X)$ \textit{defined by} \begin{equation*} F_{\mathcal{S}}(C)=\underset{k=1}{\overset{n}{\cup }}f_{k}(C)\text{,} \end{equation*} \textit{for all} $C\in K(X)$, \textit{which is called the set function associated to }$\mathcal{S}$\textit{, turns out to be a contraction and its unique fixed point, denoted by }$A_{\mathcal{S}}$\textit{, is called the attractor of the IFS }$\mathcal{S}$. REMARK 2.7. For each $i\in \{1,2,...,n\}$, the fixed point of $f_{i}$\ is an element of $A_{\mathcal{S}}$. REMARK 2.8. If $A\in K(X)$ has the property that $F_{\mathcal{S} }(A)\subseteq A$, then $A_{\mathcal{S}}\subseteq A$. $\mathtt{Proof}$. The proof is similar to the one of Lemma 3.6 from [13]. $ \square $ DEFINITION\ 2.9. Let $(X,d)$\ be a metric space and $(A_{i})_{i\in I}$\ a family of nonempty subsets of $X$. The \textit{family} $(A_{i})_{i\in I}$\ is said to be \textit{connected} if for every $i,j\in I$, there exist $n\in \mathbb{N}$\ and $\{i_{1},i_{2},...,i_{n}\}\subseteq I$\ such that $i_{1}=i$ , $i_{n}=j$\ and $A_{i_{k}}\cap A_{i_{k+1}}\neq \emptyset $\ for every $k\in \{1,2,..,n-1\}$. THEOREM\ 2.10\textit{\ }(see [12], Theorem 1.6.2, page 33)\textit{. Given an IFS }$\mathcal{S}=(X,(f_{k})_{k\in \{1,2,...,n\}})$\textit{, where }$(X,d)$ \textit{\ is a complete metric space, the following statements are equivalent:} \textit{1)\ the family }$(f_{i}(A_{\mathcal{S}}))_{i\in \{1,2,...,n\}}$ \textit{\ is connected;} \textit{2) }$A_{\mathcal{S}}$\textit{\ is arcwise connected.} \textit{3) }$A_{\mathcal{S}}$\textit{\ is connected.} PROPOSITION 2.11. \textit{For a given complete metric space }$(X,d)$\textit{ , let us consider the IFSs }$\mathcal{S}=(X,f_{1},f_{2})$\textit{\ and }$ \mathcal{S}^{^{\prime }}=(X,f_{1}^{[m]},f_{2})$\textit{, where }$m\in \mathbb{N}$\textit{.} \textit{If }$A_{\mathcal{S}^{^{\prime }}}$\textit{\ is connected, then }$A_{ \mathcal{S}}$\textit{\ is connected.} $\mathtt{Proof}$. Since $F_{\mathcal{S}^{^{\prime }}}(A_{\mathcal{S} })=f_{1}^{[m]}(A_{\mathcal{S}})\cup f_{2}(A_{\mathcal{S}})\subseteq A_{ \mathcal{S}}$, we get (using Remark 2.8) $A_{\mathcal{S}^{^{\prime }}}\subseteq A_{\mathcal{S}}$ and hence $f_{2}(A_{\mathcal{S}^{^{\prime }}})\subseteq f_{2}(A_{\mathcal{S}})$. Because $f_{1}^{[m]}(A_{\mathcal{S} ^{^{\prime }}})\subseteq f_{1}(A_{\mathcal{S}})$, it follows that $ f_{1}^{[m]}(A_{\mathcal{S}^{^{\prime }}})\cap f_{2}(A_{\mathcal{S}^{^{\prime }}})\subseteq f_{1}(A_{\mathcal{S}})\cap f_{2}(A_{\mathcal{S}})$ (*). Since $ A_{\mathcal{S}^{^{\prime }}}$\textit{\ }is connected, taking into account Theorem 2.10, we deduce that $f_{1}^{[m]}(A_{\mathcal{S}^{^{\prime }}})\cap f_{2}(A_{\mathcal{S}^{^{\prime }}})\neq \varnothing $, which, using $(\ast )$ , implies that $f_{1}(A_{\mathcal{S}})\cap f_{2}(A_{\mathcal{S}})\neq \varnothing $. Then, using again Theorem 2.10, we infer that $A_{\mathcal{S} } $\textit{\ }is connected. $\square $ PROPOSITION 2.12\textbf{\ }(see [5], page 238, lines 11-12). \textit{Assume that }$H$\textit{\ is a Hilbert space. Let us consider a self-adjoint operator }$N:H\rightarrow H$\textit{\ and }$E$ \textit{its spectral decomposition. Then for each }$\lambda \in \mathbb{R}$\textit{\ we have} \begin{equation*} NE((-\infty ,\lambda ))\leq \lambda E((-\infty ,\lambda )) \end{equation*} \textit{and} \begin{equation*} \lambda E((\lambda ,\infty ))\leq NE((\lambda ,\infty ))\text{,} \end{equation*} \textit{for all }$\lambda \in \mathbb{R}$\textit{.} PROPOSITION 2.13\textbf{\ }(see [5], page 226, Observation 7). \textit{ Assume that }$H$\textit{\ is a Hilbert space. Let us consider two self-adjoint operators }$N_{1},N_{2}:H\rightarrow H$\textit{.} \textit{If} \begin{equation*} 0\leq N_{1}\leq N_{2}\text{,} \end{equation*} \textit{then} \begin{equation*} \left\Vert N_{1}\right\Vert \leq \left\Vert N_{2}\right\Vert \text{.} \end{equation*} PROPOSITION 2.14\textbf{\ }(see [19], ex. 25, page 344). \textit{Assume that }$H$\textit{\ is a Hilbert space. Let us consider a normal operator }$ N:H\rightarrow H$\textit{, }$g$ \textit{a bounded Borel function on }$\sigma (N)$\textit{\ and }$S=g(T)$\textit{. If }$E_{N}$\textit{\ and }$E_{S}$ \textit{\ are the spectral decomposition of }$N$\textit{\ and }$S$\textit{, then } \begin{equation*} E_{S}(\omega )=E_{N}(g^{-1}(\omega ))\text{,} \end{equation*} \textit{for every Borel set }$\omega \subseteq \sigma (S)$\textit{.} PROPOSITION 2.15 (see [4], Proposition 4.1, page 278). \textit{Assume that }$ H$\textit{\ is a Hilbert space. Let us consider a normal operator} $ N:H\rightarrow H$ \textit{and} $E$ \textit{its spectral decomposition.\ Then }$N$\textit{\ is compact if and only if }$E(\{z\mid \left\vert z\right\vert >\varepsilon \})$\textit{\ has finite rank, for every }$\varepsilon >0$ \textit{.} PROPOSITION 2.16. \textit{Assume that }$H$\textit{\ is a Hilbert space. Let us consider a bounded linear operator }$A:H\rightarrow H$ \textit{which is invertible. Then }$Id_{H}-A^{\ast }A$\textit{\ is compact if and only if }$ Id_{H}-AA^{\ast }$\textit{\ is compact.} $\mathtt{Proof}$. According to the well known polar decomposition theorem there exists an unitary operator $U:H\rightarrow H$ and a positive operator $ P:H\rightarrow H$ such that $P^{2}=A^{\ast }A$ and $A=UP$. Then \begin{equation*} Id_{H}-AA^{\ast }=Id_{H}-UP(UP)^{\ast }=Id_{H}-UPP^{\ast }U^{\ast }=Id_{H}-UP^{2}U^{\ast }= \end{equation*} \begin{equation*} =UU^{\ast }-UP^{2}U^{\ast }=U(Id_{H}-P^{2})U^{\ast }=U(Id_{H}-A^{\ast }A)U^{\ast }\text{.} \end{equation*} Hence $Id_{H}-AA^{\ast }=U(Id_{H}-A^{\ast }A)U^{\ast }$ and $Id_{H}-A^{\ast }A=U^{\ast }(Id_{H}-AA^{\ast })U$. From the last two relations we obtain the conclusion. $\square $ COROLLARY 2.17. \textit{Assume that }$H$\textit{\ is a Hilbert space. Let us consider a bounded linear operator }$S:H\rightarrow H$ \textit{such that }$ \left\Vert S\right\Vert <1$\textit{. Then }$S+S^{\ast }-SS^{\ast }$\textit{\ is compact if and only if }$S+S^{\ast }-S^{\ast }S$\textit{\ is compact.} $\mathtt{Proof}$. The operator $A=Id_{H}-S$ is invertible since $\left\Vert S\right\Vert <1$. According to Proposition 2.16 $Id_{H}-A^{\ast }A$\textit{\ }is compact if and only if $Id_{H}-AA^{\ast }$ is compact i.e. $S+S^{\ast }-SS^{\ast }$ is compact if and only if $S+S^{\ast }-S^{\ast }S$ is compact. $\square $ PROPOSITION 2.18\textbf{\ }(see [19], ex. 14, page 324). \textit{Assume that }$H$\textit{\ is a Hilbert space and let us consider a bounded linear operator }$S:H\rightarrow H$\textit{. If }$S^{\ast }S$\textit{\ is a compact operator, then }$S$\textit{\ is compact.} \textbf{3.}\ \textbf{A sufficient condition for the compactness of an operator. }In this section, $H$ is an infinite-dimensional Hilbert space. We shall use the notation $Id_{H}$ for the function $Id_{H}:H\rightarrow H$, given by $Id_{H}(x)=x$, for all $x\in H$ . If $S$ and $T$ are bounded linear operators from $H$ to $H$ such that $\left\Vert S\right\Vert ,\left\Vert T\right\Vert <1$, then $S$ and $T$ are contractions. For $w\in X$, we consider the IFS $S_{w}=(X,f_{1},f_{2})$, where $f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $f_{2}(x)=T(x)+w$, for all $x\in X$. THEOREM\ 3.1. \textit{In the preceding framework, let us consider a bounded linear operator }$S:H\rightarrow H$ \textit{satisfying the condition} $ \left\Vert S\right\Vert <1$\textit{. If for every bounded linear operator }$ T:H\rightarrow H$\textit{\ such that }$\left\Vert T\right\Vert <1$\textit{\ there exists a sequence }$(K_{T,n})_{n}$\textit{\ of compact subsets of }$H$ \textit{\ having the property that }$A_{\mathcal{S}_{w}}$\textit{\ is\ not connected for all }$w\in H-\underset{n}{\cup }K_{T,n}$\textit{, then the operator }$S$\textit{\ is compact.} $\mathtt{Proof}$. For each $m\in \mathbb{N}$ let us consider the bounded linear operator $U=S^{[m]}$. Obviously $\left\Vert U\right\Vert <1$. Let us consider $P_{\varepsilon }=E((-\infty ,1-\varepsilon ))$ and $\overset{\sim } {P_{\varepsilon }}=E((1+\varepsilon ,\infty ))$, where $E$ is the spectral decomposition of the positive (so self-adjoint, so normal) bounded linear operator \begin{equation*} N=(Id_{H}-U)^{\ast }(Id_{H}-U)=Id_{H}-U-U^{\ast }+U^{\ast }U\text{.} \end{equation*} We claim\textit{\ that }$P_{\varepsilon }$\textit{\ has finite rank for every }$\varepsilon >0$. Indeed, if there is to be an $\varepsilon _{0}>0$\textit{\ }such that $ P_{\varepsilon _{0}}$ has infinite rank, then\textit{\ }let us consider the operator $T=(Id_{H}-U)P_{\varepsilon _{0}}$ and remark that \begin{equation*} NP_{\varepsilon _{0}}=NP_{\varepsilon _{0}}^{2}=NP_{\varepsilon _{0}}^{\ast }P_{\varepsilon _{0}}=P_{\varepsilon _{0}}^{\ast }NP_{\varepsilon _{0}}=P_{\varepsilon _{0}}^{\ast }((Id_{H}-U)^{\ast }(Id_{H}-U))P_{\varepsilon _{0}}= \end{equation*} \begin{equation*} =((Id_{H}-U)P_{\varepsilon _{0}})^{\ast }((Id_{H}-U)P_{\varepsilon _{0}})\geq 0\text{.} \end{equation*} Hence, according to Proposition 2.12, we have $0\leq NP_{\varepsilon _{0}}\leq (1-\varepsilon _{0})P_{\varepsilon _{0}}$ and therefore, using Proposition 2.13, it follows that $\left\Vert NP_{\varepsilon _{0}}\right\Vert \leq 1-\varepsilon _{0}$. Consequently we obtain \begin{equation*} \left\Vert T\right\Vert ^{2}=\left\Vert T^{\ast }T\right\Vert =\left\Vert (Id_{H}-U)P_{\varepsilon _{0}})^{\ast }(Id_{H}-U)P_{\varepsilon _{0}}\right\Vert = \end{equation*} \begin{equation*} =\left\Vert P_{\varepsilon _{0}}^{\ast }(Id_{H}-U)^{\ast }(Id_{H}-U)P_{\varepsilon _{0}}\right\Vert =\left\Vert P_{\varepsilon _{0}}NP_{\varepsilon _{0}}\right\Vert \leq \end{equation*} \begin{equation*} \leq \left\Vert P_{\varepsilon _{0}}\right\Vert \left\Vert NP_{\varepsilon _{0}}\right\Vert =\left\Vert NP_{\varepsilon _{0}}\right\Vert \leq 1-\varepsilon _{0} \end{equation*} and thus \begin{equation*} \left\Vert T\right\Vert \leq \sqrt{1-\varepsilon _{0}}<1\text{.} \end{equation*} For $w\in H$, let us consider, besides $\mathcal{S}_{w}$, the IFS $\mathcal{S }_{w}^{^{\prime }}=(H,f,f_{2})$, where $f:H\rightarrow H$ is given by $ f(x)=U(x)$, for all $x\in H$. Now let us choose an arbitrary $w\in (Id_{H}-T)P_{\varepsilon _{0}}(H)$. On one hand, since $0$ is the fixed point of $f$, using Remark 2.7, we infer that $0\in A_{\mathcal{S}_{w}}$. On the other hand, using the same argument, we get that $e$, the fixed point of $f_{2}$, belongs to $A_{\mathcal{S}_{w}}$ , that is $e=U^{-1}(w)=(Id_{H}-T)^{-1}(w)\in A_{\mathcal{S}_{w}^{^{\prime }}} $. Since $f(e)=f_{2}(0)=w$, we obtain $w\in f(A_{\mathcal{S} _{w}^{^{\prime }}})\cap f_{2}(A_{\mathcal{S}_{w}^{^{\prime }}})$, which implies $f(A_{\mathcal{S}_{w}^{^{\prime }}})\cap f_{2}(A_{\mathcal{S} _{w}^{^{\prime }}})\neq \emptyset $, and therefore, according to Theorem 2.10, $A_{\mathcal{S}_{w}^{^{\prime }}}$is connected. We conclude (using Proposition 2.11) that $A_{\mathcal{S}_{w}}$ is connected. Consequently there exists a bounded linear operator $T:H\rightarrow H$ having $\left\Vert T\right\Vert <1$ such that $A_{\mathcal{S}_{w}}$ is connected for every $w\in (Id_{H}-T)P_{\varepsilon _{0}}(H)$. According to the hypothesis there exists a sequence $(K_{T,n})_{n}$\textit{\ }of compact subsets of $H$\ having the property that $A_{\mathcal{S}_{w}}$\ is\ not connected, for all $w\in H-\underset{n}{\cup }K_{T,n}m.$ Therefore we obtain $(Id_{H}-T)P_{\varepsilon _{0}}(H)\subseteq \underset{n}{ \cup }K_{T,n}$ which (taking into account the fact that $(Id_{H}-T)P_{ \varepsilon _{0}}(H)$ is infinite dimensional, that the closed unit ball in a normed linear space $X$ is compact if and only if $X$ is infinite dimensional and Baire's theorem) generates a contradiction. We assert \textit{that }$\overset{\sim }{P_{\varepsilon }}$\textit{\ has finite rank for every }$\varepsilon >0$. Indeed, if by contrary we suppose that there exists $\varepsilon _{0}>0$ \textit{\ }such that $\overset{\sim }{P_{\varepsilon _{0}}}$ has infinite rank, let $R_{\varepsilon _{0}}$ designates the orthogonal projection of $H$ onto $(Id_{H}-U)\overset{\sim }{P_{\varepsilon _{0}}}(H)$ and let us consider the bounded linear operator $T=(Id_{H}-U)^{-1}R_{\varepsilon _{0}}$ . Based upon Proposition 2.12, we have \begin{equation*} N\overset{\sim }{P_{\varepsilon _{0}}}=(Id_{H}-U)^{\ast }(Id_{H}-U)\overset{ \sim }{P_{\varepsilon _{0}}}\geq (1+\varepsilon _{0})\overset{\sim }{ P_{\varepsilon _{0}}}\text{,} \end{equation*} which implies that \begin{equation*} \left\Vert (Id_{H}-U)\overset{\sim }{P_{\varepsilon _{0}}}(x)\right\Vert ^{2}=<N\overset{\sim }{P_{\varepsilon _{0}}}(x),\overset{\sim }{ P_{\varepsilon _{0}}}(x)>\geq (1+\varepsilon _{0})\left\Vert \overset{\sim }{ P_{\varepsilon _{0}}}(x)\right\Vert ^{2}\text{,} \end{equation*} i.e. \begin{equation} \sqrt{1+\varepsilon _{0}}\left\Vert \overset{\sim }{P_{\varepsilon _{0}}} (x)\right\Vert \leq \left\Vert (Id_{H}-U)\overset{\sim }{P_{\varepsilon _{0}} }(x)\right\Vert \text{,} \tag{0} \end{equation} for each $x\in H$. So, as for each $u\in H$ there exists $x_{u}\in H$ such that $R_{\varepsilon _{0}}(u)=(Id_{H}-U)\overset{\sim }{P_{\varepsilon _{0}}} (x_{u})$, we infer that \begin{equation*} \left\Vert T(u)\right\Vert =\left\Vert (Id_{H}-U)^{-1}R_{\varepsilon _{0}}(u)\right\Vert =\left\Vert (Id_{H}-U)^{-1}(Id_{H}-U)\overset{\sim }{ P_{\varepsilon _{0}}}(x_{u})\right\Vert = \end{equation*} \begin{equation*} =\left\Vert \overset{\sim }{P_{\varepsilon _{0}}}(x_{u})\right\Vert \overset{ (0)}{\leq }\frac{1}{\sqrt{1+\varepsilon _{0}}}\left\Vert (Id_{H}-U)\overset{ \sim }{P_{\varepsilon _{0}}}(x)\right\Vert = \end{equation*} \begin{equation*} =\frac{1}{\sqrt{1+\varepsilon _{0}}}\left\Vert R_{\varepsilon _{0}}(u)\right\Vert \leq \frac{1}{\sqrt{1+\varepsilon _{0}}}\left\Vert R_{\varepsilon _{0}}\right\Vert \left\Vert u\right\Vert =\frac{1}{\sqrt{ 1+\varepsilon _{0}}}\left\Vert u\right\Vert \end{equation*} i.e. $\left\Vert T(u)\right\Vert \leq \frac{1}{\sqrt{1+\varepsilon _{0}}} \left\Vert u\right\Vert $, for each $u\in H$, which takes on the form \begin{equation*} \left\Vert T\right\Vert \leq \frac{1}{\sqrt{1+\varepsilon _{0}}}<1\text{.} \end{equation*} For $w\in H$, let us consider, besides $\mathcal{S}_{w}$, the IFS $\mathcal{S }_{w}^{^{\prime }}=(H,f,f_{2})$, where $f:H\rightarrow H$ is given by $ f(x)=U(x)$, for all $x\in H$. Now let us choose an arbitrary $w\in (Id_{H}-T)\overset{\sim }{ P_{\varepsilon _{0}}}(H)$. Then there exists $u\in H$ such that $w=(Id_{H}-T) \overset{\sim }{P_{\varepsilon _{0}}}(u)$. On one hand, since $0$ is the fixed point of $f$, using Remark 2.7, we infer that $0\in A_{\mathcal{S} _{w}^{^{\prime }}}$. On the other hand, using the same argument, we get that $e$ (the fixed point of $f_{2}$) belongs to $A_{\mathcal{S}_{w}^{^{\prime }}} $, that is $e=U^{-1}(w)=(Id_{H}-T)^{-1}(w)\in A_{\mathcal{S} _{w}^{^{\prime }}}$, and therefore $f(e)\in A_{\mathcal{S}_{w}^{^{\prime }}}$ . Since $f(0)=0 $, on one hand we infer that \begin{equation} 0\in f(A_{\mathcal{S}_{w}^{^{\prime }}})\text{.} \tag{1} \end{equation} On the other hand we have \begin{equation*} f_{2}(f(e))=TU(e)+w=TU(Id_{H}-T)^{-1}(w)+(Id_{H}-T)(Id_{H}-T)^{-1}(w)= \end{equation*} \begin{equation*} =(Id_{H}-T(Id_{H}-U))(Id_{H}-T)^{-1}(w)= \end{equation*} \begin{equation*} =(Id_{H}-T(Id_{H}-U))(Id_{H}-T)^{-1}(Id_{H}-T)\overset{\sim }{P_{\varepsilon _{0}}}(u)= \end{equation*} \begin{equation*} =(Id_{H}-T(Id_{H}-U))\overset{\sim }{P_{\varepsilon _{0}}}(u)=\overset{\sim } {P_{\varepsilon _{0}}}(u)-(Id_{H}-U)^{-1}R_{\varepsilon _{0}}(Id_{H}-U)) \overset{\sim }{P_{\varepsilon _{0}}}(u)= \end{equation*} \begin{equation*} =\overset{\sim }{P_{\varepsilon _{0}}}(u)-(Id_{H}-U)^{-1}(Id_{H}-U))\overset{ \sim }{P_{\varepsilon _{0}}}(u)=0\text{,} \end{equation*} so \begin{equation} 0\in f_{2}(A_{\mathcal{S}_{w}^{^{\prime }}})\text{.} \tag{2} \end{equation} From $(1)$ and $(2)$ we obtain $0\in f(A_{\mathcal{S}_{w}^{^{\prime }}})\cap f_{2}(A_{\mathcal{S}_{w}^{^{\prime }}})$, i.e. $f(A_{\mathcal{S} _{w}^{^{\prime }}})\cap f_{2}(A_{\mathcal{S}_{w}^{^{\prime }}})\neq \varnothing $, so, relying on Theorem 2.10, $A_{S_{w}^{^{\prime }}}$ is connected. We appeal to Proposition 2.11 to deduce that $A_{\mathcal{S}_{w}}$ is connected. Consequently there exists a bounded linear operator $T:H\rightarrow H$ having $\left\Vert T\right\Vert <1$ such that $A_{\mathcal{S}_{w}}$is connected for every $w\in (Id_{H}-T)\overset{\sim }{P_{\varepsilon _{0}}}(H)$ . Taking into account the hypothesis there exists a sequence $(K_{T,n})_{n}$ \textit{\ }of compact subsets of $H$\ having the property that $A_{S_{w}}$\ is\ not connected for all $w\in H-\underset{n}{\cup }K_{T,n}m.$ Thus we obtain the inclusion $(Id_{H}-T)\overset{\sim }{P_{\varepsilon _{0}}} (H)\subseteq \underset{n}{\cup }K_{T,n}$ which generates a contradiction by invoking the same arguments that we used in the final part of the previous claim's proof. Now\textit{\ we state that }$Id_{H}-(Id_{H}-U)^{\ast }(Id_{H}-U)$ \textit{is compact.} If $\mathcal{E}$ is the spectral decomposition of $Id_{H}-N$, using Proposition 2.14, we obtain $E((-\infty ,1-\varepsilon )\cup (1+\varepsilon ,\infty ))=E(g^{-1}((-\infty ,-\varepsilon )\cup (\varepsilon ,\infty )))= \mathcal{E(}(-\infty ,-\varepsilon )\cup (\varepsilon ,\infty )\mathcal{)}= \mathcal{E(}(-\infty ,-\varepsilon )\cup (\varepsilon ,\infty )\mathcal{)}$, where $g(x)=1-x$. Since from the above two claims we infer that the operator $E(((-\infty ,1-\varepsilon )\cup (1+\varepsilon ,\infty )))=E((-\infty ,1-\varepsilon ))+E((1+\varepsilon ,\infty ))$ has finite rank, we get that $ \mathcal{E(}(-\infty ,-\varepsilon )\cup (\varepsilon ,\infty )\mathcal{)}$ has finite rank, for every $\varepsilon >0$. Proposition 2.15\ assures us that $Id_{H}-N$ is compact, i.e. $Id_{H}-(Id_{H}-U)^{\ast }(Id_{H}-U)=U+U^{\ast }-U^{\ast }U$ is compact. \textit{Hence} \begin{equation*} S^{[m]}+(S^{[m]})^{\ast }-S^{[m]}(S^{[m]})^{\ast } \end{equation*} \textit{is compact}, \textit{for every }$m\in \mathbb{N}$. For $m=1$, we get that $S+S^{\ast }-S^{\ast }S$ is compact. Note that, by Corollary 2.17, $S+S^{\ast }-SS^{\ast }$ is compact and hence $SS^{\ast }-S^{\ast }S$ is compact (3). Consequently $S^{\ast }(S^{\ast }S-SS^{\ast })S=(S^{\ast })^{[2]}S^{[2]}-S^{\ast }SS^{\ast }S$ is compact. (4) Moreover, for $m=2$, we obtain that $S^{[2]}+(S^{\ast })^{[2]}-(S^{\ast })^{[2]}S^{[2]}$ is compact. (5) But \begin{equation*} (S+S^{\ast }-S^{\ast }S)(S+S^{\ast }-S^{\ast }S)= \end{equation*} \begin{equation*} =(S+S^{\ast })^{[2]}-(S+S^{\ast }-S^{\ast }S)S^{\ast }S-S^{\ast }S(S+S^{\ast }-S^{\ast }S)-S^{\ast }SS^{\ast }S \end{equation*} is compact. Since $S+S^{\ast }-S^{\ast }S$ is compact, we infer that \begin{equation*} (S+S^{\ast })^{[2]}-S^{\ast }SS^{\ast }S= \end{equation*} \begin{equation*} =S^{[2]}+(S^{\ast })^{[2]}+SS^{\ast }+S^{\ast }S-S^{\ast }SS^{\ast }S= \end{equation*} \begin{equation*} =S^{[2]}+(S^{\ast })^{[2]}-(S^{\ast })^{[2]}S^{[2]}+SS^{\ast }+S^{\ast }S+(S^{\ast })^{[2]}S^{[2]}-S^{\ast }SS^{\ast }S \end{equation*} is compact. (6) Then, from (4), (5) and (6), we get that $SS^{\ast }+S^{\ast }S$ is compact. (7) From (3) and (7) we deduce that $S^{\ast }S$ is a compact operator and, using Proposition 2.18, we conclude that $S$ is compact. $\square $ \textbf{4. A necessary condition for the compactness of an operator. }In this section $X$ is a Banach space. We shall designate by $Id_{X}$ the function $Id_{X}:X\rightarrow X$, given by $Id_{X}(x)=x$, for all $x\in X$. If $S$ and $T$ be bounded linear operator from $X$ to $X$ such that $ \left\Vert S\right\Vert ,\left\Vert T\right\Vert <1$, then $S$ and $T$ are contractions and $T^{[n]}-Id_{X}$ is invertible, for each $n\in \mathbb{N}$. For $w\in X$, we consider the IFS $S_{w}=(X,f_{1},f_{2})$, where $ f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $ f_{2}(x)=T(x)+w $, for all $x\in X$. THEOREM\ 4.1. \textit{In the above mentioned setting, if the operator }$S$ \textit{is compact, then} \textit{there exists a family }$(K_{n})_{n\in \mathbb{N}}$ \textit{of compact subsets of }$X$\textit{\ such that }$A_{ \mathcal{S}_{w}}$\textit{\ is\ not connected, for all }$w\in H-\underset{ n\in \mathbb{N}}{\cup }K_{n}$\textit{.} $\mathtt{Proof}$. The proof given in Theorem 5, from [15], applies with little change. More precisely let $C_{0}$ be the compact set $\overline{ S(B(0,1))}$. Let $X^{^{\prime }},X_{1},X_{2},...,X_{n},...$ be given by \begin{equation*} X^{^{\prime }}=S(X)=\underset{k\in \mathbb{N}}{\cup }kC_{0} \end{equation*} and \begin{equation*} X_{n}=(T-Id_{X})(T^{[n]}-Id_{X})^{-1}(X^{^{\prime }}-T^{[n]}(X^{^{\prime }})) \text{,} \end{equation*} for each $n\in \mathbb{N}$. We have \begin{equation*} X_{n}=(T-Id_{X})(T^{[n]}-Id_{X})^{-1}(\underset{k\in \mathbb{N}}{\cup } kC_{0}-T^{[n]}(\underset{l\in \mathbb{N}}{\cup }lC_{0}))= \end{equation*} \begin{equation*} =(T-Id_{X})(T^{[n]}-Id_{X})^{-1}(\underset{k\in \mathbb{N}}{\cup }kC_{0}- \underset{l\in \mathbb{N}}{\cup }lT^{[n]}(C_{0}))= \end{equation*} \begin{equation*} =(T-Id_{X})(T^{[n]}-Id_{X})^{-1}(\underset{k,l\in \mathbb{N}}{\cup } (kC_{0}-lT^{[n]}(C_{0}))\text{,} \end{equation*} for each $n\in \mathbb{N}$ and since $kC_{0}-lT^{[n]}(C_{0})$ is compact for all $k,l\in \mathbb{N}$, we infer that $X_{n}$ is a countable union of compact subsets of $X$. Therefore there exists a family\textit{\ }$ (K_{n})_{n\in \mathbb{N}}$ of compact subsets of $X$ such that $\underset{ n\in \mathbb{N}}{\cup }X_{n}=\underset{n\in \mathbb{N}}{\cup }K_{n}$. The rest of the proof of the Theorem mentioned above does not require any modification. Hence $A_{\mathcal{S}_{w}}$\textit{\ }is disconnected, for each $w\in X\smallsetminus \underset{n\in \mathbb{N}}{\cup }X_{n}=X\smallsetminus \underset{n\in \mathbb{N}}{\cup }K_{n}$. $\square $ REMARK 4.2. If $X$ is infinite dimensional, then $W\overset{not}{=} X\smallsetminus \underset{n\in \mathbb{N}}{\cup }X_{n}=X\smallsetminus \underset{n\in \mathbb{N}}{\cup }K_{n}$ is dense in $X$. $\mathtt{Proof}$. Indeed, let us note that $K_{n}$ is a closed set. Moreover $\overset{\circ }{K_{n}}=\varnothing $ since if this is not the case, then the closure of the unit ball of the infinite-dimensional space $X$ is compact which is a contradiction. Consequently $X_{n}$ is nowhere dense, for each $n\in \mathbb{N}$, and therefore $W$ is dense in $X$. \begin{center} \textbf{References} \end{center} [1] D. Baki\'{c} and B. Gulja\v{s}, \textit{Which operators approximately annihilate orthonormal bases?}, Acta Sci. Math. (Szeged) 64 (1998), No.3-4, 601-607. [2] M.F. Barnsley, \textit{Fractals everywhere}, Academic Press Professional, Boston, 1993. [3] J. J. M. Chadwick and A.W. Wickstead, \textit{A quotient of ultrapowers of Banach spaces and semi-Fredholm operators}, Bull. London Math. Soc. 9 (1977), 321-325. [4] J. B. Conway, \textit{A course in functional analysis}, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985. [5] R. Cristescu, \textit{No\c{t}iuni de Analiz\u{a} Func\c{t}ional\u{a} Liniar\u{a}}, Editura Academiei Rom\^{a}ne, Bucure\c{s}ti, 1998. [6] D. Dumitru and A. Mihail, \textit{A sufficient condition for the connectedness of the attractors of an infinite iterated function systems}, An. \c{S}tiin\c{t}. Univ. Al. I. Cuza Ia\c{s}i. Mat. (N.S.), LV (2009), f.1, 87-94. [7] K.J. Falconer, \textit{Fractal geometry, Mathematical foundations and applications}, John Wiley \& Sons, Ltd., Chichester, 1990. [8] P. A. Fillmore and J.P. Williams, \textit{On operator ranges}, Adv. Math. 7 (1971), 254-281. [9] J.E. Hutchinson, \textit{Fractals and self-similarity}, Indiana Univ. Math. J. 30 (1981), 713-747. [10] K. Imazeki, \textit{Characterizations of compact operators and semi-Fredholm operators}, TRU Math. 16 (1980), No.2, 1-8. [11] J. M. Isidro, \textit{Structural characterization of compact operators} , in Current topics in operator algebras, (Nara, 1990), 114-129, World Sci. Publ., River Edge, NJ, 1991. [12] J. Kigami, \textit{Analysis on Fractals,} Cambridge University Press, 2001. [13] R. Miculescu and A. Mihail, \textit{Lipscomb's space }$\omega ^{A}$ \textit{\ is the attractor of an infinite IFS containing affine transformations of }$l^{2}(A)$, Proc. Amer. Math. Soc. 136\ (2008), No. 2, 587-592. [14] A. Mihail, \textit{On the connectivity of the attractors of iterated multifunction systems}, Real Anal. Exchange, 34 (2009), No. 1, 195-206. [15] A. Mihail and R. Miculescu, \textit{On a family of IFSs whose attractors are not connected}, to appear in J. Math. Anal. Appl., DOI information: 10.1016/j.jmaa.2010.10.039. [16] K. Muroi and K. Tamaki, \textit{On Ringrose's characterization of compact operators}, Math. Japon. 19 (1974), 259-261. [17] J. R. Ringrose, \textit{Compact non-self-adjoint operators}, Van Nostrand Reinhold Company, London, 1971. [18] \'{A}. Rod\'{e}s-Us\'{a}n, \textit{A characterization of compact operators}, in Proceedings of the tenth Spanish-Portuguese on mathematics, III (Murcia, 1985), 178-182, Univ. Murcia, Murcia, 1985. [19] W. Rudin, \textit{Functional analysis}, 2nd ed., International Series in Pure and Applied Mathematics. New York, NY: McGraw-Hill, 1991. Department of Mathematics Faculty of Mathematics and Informatics University of Bucharest Academiei Street, no. 14 010014 Bucharest, Romania E-mail: mihail\_alex@yahoo.com \qquad\ \ \ \ \ miculesc@yahoo.com \end{document}

\begin{equation}gin{document} \title{Numerical Method for Solving Electromagnetic Wave Scattering by One and Many Small Perfectly Conducting Bodies} \begin{equation}gin{abstract} In this paper, we investigate the problem of electromagnetic (EM) wave scattering by one and many small perfectly conducting bodies and present a numerical method for solving it. For the case of one body, the problem is solved for a body of arbitrary shape, using the corresponding boundary integral equation. For the case of many bodies, the problem is solved asymptotically under the physical assumptions $a\ll d \ll {\langle}mbda$, where $a$ is the characteristic size of the bodies, $d$ is the minimal distance between neighboring bodies, ${\langle}mbda=2\pi/k$ is the wave length and $k$ is the wave number. Numerical results for the cases of one and many small bodies are presented. Error analysis for the numerical method are also provided. \end{abstract} \noindent\textbf{Key words:} electromagnetic scattering; many bodies; perfectly conducting body; integral equation; EM waves. \\ \noindent\textbf{MSC:} 35J05; 35J57; 78A45; 78A25; 70F10. \section{Introduction} {\langle}bel{sec0} Many real-world electromagnetic (EM) problems like EM wave scattering, EM radiation, etc \cite{Tsang2004}, cannot be solved analytically and exactly to get a solution in a closed form. Thus, numerical methods have been developed to tackle these problems approximately. Computational Electromagnetics (CEM) has evolved enormously in the past decades to a point that its methods can solve EM problems with extreme accuracy. These methods can be classified into two categories: Integral Equation (IE) method and Differential Equation (DE) method. Typical IE methods include: Method of Moment (MoM) developed by Roger F. Harrington (1968) \cite{harrington1996field}, Fast Multipole Method (FMM) first introduced by Greengard and Rokhlin (1987) \cite{GR1987} and then applied to EM by Engheta et al (1992) \cite{engheta1985fast}, Partial Element Equivalent Circuit (PEEC) method \cite{ruehli1974equivalent}, and Discrete Dipole Approximation \cite{devoe1964optical}. Typical DE methods are: Finite Difference Time Domain (FDTD) developed by Kane Yee (1966) \cite{yee1966numerical}, Finite Element Method (FEM) \cite{zienkiewicz1977finite}, Finite Integration Technique (FIT) proposed by Thomas Weiland (1977) \cite{weiland2001discrete}, Pseudospectral Time Domain (PSTD) \cite{liu1997pstd}, Pseudospectral Spatial Domain (PSSD) \cite{tyrrell2005pseudospectral}, and Transmission Line Matrix (TLM) \cite{hoefer1985transmission}. Among these methods, FDTD has emerged as one of the most popular techniques for solving EM problems due to its simplicity and ability to provide animated display of the EM field. However, FDTD requires the entire computational domain be gridded \cite{stavroulakis2013biological}, that results in very long solution times. Furthermore, as a DE method, it does not take into account the radiation condition in exact sense \cite{kunz1993finite, buchanan1993simulation}, which leads to certain error in the solution. On the other hand, spurious solutions might exist in DE methods \cite{mur2002causes, lu2004elimination, zhao2007spurious}. Most importantly, most of DE methods are not suitable if the number of bodies is very large. In \cite{R620,R635,R652,andriychuk2012electromagnetic,ramm2014application,ramm2014calculation}, A. G. Ramm has developed a theory of EM wave scattering by many small perfectly conducting and impedance bodies. In this theory, the EM wave scattering problem is solved asymptotically under the physical assumptions: $a\ll d \ll {\langle}mbda$, where $a$ is the characteristic size of the bodies, $d$ is the minimal distance between neighboring bodies, ${\langle}mbda=2\pi/k$ is the wave length and $k$ is the wave number. In \cite{TEMImpedance}, a numerical method is developed for solving EM wave scattering by many small impedance bodies. In this paper, the problem of EM wave scattering by one and many small perfectly conducting bodies is considered. A numerical method for solving this problem asymptotically based on the above theory is presented. For the case of one body, the problem is solved for a body of arbitrary shape, using the corresponding boundary integral equation. For the case of many small bodies, the problem is solved under the basic assumptions $a\ll d \ll {\langle}mbda$ and the assumption about the distribution of the small bodies \begin{equation} \mathcal{N}(\Delta)=\frac{1}{a^3} \int_\Delta N(x)dx[1+o(1)], \quad a\to 0, \end{equation} in which $\Delta$ is an arbitrary open subset of the domain $\Omega$ that contains all the small bodies, $\mathcal{N}(\Delta)$ is the number of the small bodies in $\Delta$, and $N(x)$ is the distribution function of the bodies \begin{equation} N(x)\ge 0, \quad N(x) \in C(\Omega). \end{equation} In Sections \ref{sec1} and \ref{sec2}, the theory of EM wave scattering by one and many small perfectly conducting bodies is presented. The numerical methods for solving these problems are also described in details. Furthermore, error analysis for the numerical methods of solving the EM scattering problem are also provided. In Section \ref{sec3} these methods are tested and numerical results are discussed. \section{EM wave scattering by one perfectly conducting body} {\langle}bel{sec1} Let $D$ be a bounded perfectly conducting body, $a=\frac{1}{2}$diam$D$, $S$ be its $C^2$-smooth boundary, and $D':={\mathbb R}RR \setminus D$. Let $\epsilon$ and $\mu$ be the dielectric permittivity and magnetic permeability constants of the medium in $D'$. Let $E$ and $H$ denote the electric and magnetic fields, respectively, $E_0$ be the incident field and $v_E$ be the scattered field. The problem of electromagnetic wave scattering by one perfectly conducting body can be stated as follows \begin{equation}gin{align} &\nabla \times E=i \omega \mu H, \quad \text{in }D':={\mathbb R}RR \setminus D, {\langle}bel{eq1.1.1} \\ &\nabla \times H=-i \omega \epsilon E, \quad \text{in }D', {\langle}bel{eq1.1.2} \\ &[N,[E, N]]=0, \quad \text{on } S:=\partial D, {\langle}bel{eq1.1.3} \\ &E=E_0+v_E, {\langle}bel{eq1.1.4} \\ &E_0=\mathcal{E} e^{ik\alpha\cdot x}, \quad \mathcal{E} \cdot \alpha=0, \quad \alpha \in S^2, {\langle}bel{eq1.1.5} \\ &\frac{\partial v_E}{\partial r}-ikv_E=o\left(\frac{1}{r}\right), \quad r:=|x|\to \infty, {\langle}bel{eq1.1.6} \end{align} where $\omega>0$ is the frequency, $k=2\pi/{\langle}mbda=\omega \sqrt{\epsilon \mu}$ is the wave number, $ka \ll 1$, ${\langle}mbda$ is the wave length, $\mathcal{E}$ is a constant vector, and $\alpha$ is a unit vector that indicates the direction of the incident wave $E_0$. This incident wave satisfies the relation $\nabla\cdot E_0=0$. The scattered field $v_E$ satisfies the radiation condition \eqref{eq1.1.6}. Here, $N$ is the unit normal vector to the surface $S$, pointing out of $D$. By $[\cdot,\cdot]$ the vector product is denoted and $\alpha \cdot x$ is the scalar product of two vectors. The solution to problem \eqref{eq1.1.1}-\eqref{eq1.1.6} can be found in the form \begin{equation} {\langle}bel{eq1.3.2} E(x)=E_0(x)+\nabla \times \int_S g(x,t) J(t)dt, \quad g(x,t):=\frac{e^{ik|x-t|}}{4\pi |x-t|}, \end{equation} see \cite{R620}. Here, $E$ is a vector in ${\mathbb R}RR$ and $\nabla \times E$ is a pseudo-vector, that is a vector-like object which changes sign under reflection of its coordinate axes. $E_0$ is the incident plane wave defined in \eqref{eq1.1.5} and $J$ is an unknown pseudo-vector that is to be found. $J$ is assumed to be tangential to $S$ and continuous. $J$ can be found by applying the boundary condition \eqref{eq1.1.3}, or equivalently $[N,E]=0$, to \eqref{eq1.3.2} and solving the resulting boundary integral equation \begin{equation} {\langle}bel{eq1.3.2b} \frac{J}{2}+AJ:=\frac{J(s)}{2}+\int_S [N_s,[\nabla_s g(s,t),J(t)]]dt=-[N_s,E_0], \end{equation} or, equivalently \begin{equation} {\langle}bel{eq1.3.2c} (I+2A)J=F, \end{equation} where $F:=-2[N_s,E_0]$. Equation \eqref{eq1.3.2c} is of Fredholm type since $A$ is compact, see \cite{R652}. Once we have $J$, $E$ can be computed by formula \eqref{eq1.3.2} and $H$ can be found by the formula \begin{equation} {\langle}bel{eq1.3.3} H=\frac{\nabla\times E}{i\omega \mu}. \end{equation} If $D$ is sufficiently small, then equation \eqref{eq1.3.2c} is uniquely solvable in $C(S)$ and its solution $J$ is tangential to $S$, see \cite{R620}. The asymptotic formula for $E$ when the radius $a$ of the body $D$ tends to zero can be derived as follows, see \cite{R620}. Rewrite equation \eqref{eq1.3.2} as \begin{equation} {\langle}bel{eq1.3.4} E(x)=E_0(x)+[\nabla g(x,x_1), Q]+ \nabla\times \int_{S} [g(x,t)-g(x,x_1)] J(t)dt, \end{equation} where $x_1 \in D$, an arbitrary point inside the small body $D$, and \begin{equation} {\langle}bel{eq1.3.5} Q:=\int_S J(t)dt. \end{equation} Since \begin{equation}gin{align} &|\nabla g(x,x_1)]| = O\left(\frac{k}{d}+\frac{1}{d^2}\right), \quad d=|x-x_1|, \\ &|g(x,t)-g(x,x_1)| = O\left(\left(\frac{k}{d}+\frac{1}{d^2}\right)a\right), \quad a=|t-x_1|, \quad \text{ and} \\ &|\nabla[g(x,t)-g(x,x_1)]| = O\left(\frac{ak^2}{d}+\frac{ak}{d^2}+\frac{a}{d^3}\right), \end{align} the second term in \eqref{eq1.3.4} is much greater than the last term \begin{equation} {\langle}bel{eq1.3.6} \left|[\nabla g(x,x_1), Q]\right| \gg \left|\nabla\times \int_{S} [g(x,t)-g(x,x_1)] J(t)dt\right|, \quad a \to 0. \end{equation} Then, the asymptotic formula for $E$ when $a$ tends to zero is \begin{equation} {\langle}bel{eq1.3.8} E(x)=E_0(x)+[\nabla_x g(x,x_1),Q], \quad a \to 0, \end{equation} where $|x-x_1|\gg a$, $x_1 \in D$. Thus, when $D$ is sufficiently small, instead of finding $J$, we can just find one pseudovector $Q$. The analytical formula for $Q$ is derived as follows, see \cite{R652}. By integrating both sides of \eqref{eq1.3.2b} over $S$, one gets \begin{equation} {\langle}bel{eq1.3.9} \int_S \frac{J(s)}{2}ds +\int_S ds \int_S dt [N_s,[\nabla_s g(s,t),J(t)]]=-\int_S [N_s,E_0]ds. \end{equation} This is equivalent to \begin{equation} {\langle}bel{eq1.3.10} \frac{Q}{2}+\int_S dt \int_S ds \nabla_s g(s,t)N_s\cdot J(t)-\int_S dt J(t) \int_S ds \frac{\partial g(s,t)}{\partial N_s}=-\int_D \nabla\times E_0 dx. \end{equation} When $a\to 0$, this equation becomes \begin{equation} {\langle}bel{eq1.3.11} \frac{Q}{2}+e_p\int_S dt \int_S ds \frac{\partial g(s,t)}{\partial s_p}N_q(s) J_q(t)+\frac{1}{2}\int_S dt J(t) =-|D| \nabla\times E_0, \quad 1\le p,q\le 3, \end{equation} where in the second term, summations over the repeated indices are understood, $e_p$, $1\le p\le 3$, are the orthogonal unit vectors in ${\mathbb R}RR$, $|D|$ is the volume of $D$, $|D|=c_D a^3$, and in the third term we use this estimate \begin{equation} {\langle}bel{eq1.3.12} \int_S ds \frac{\partial g(s,t)}{\partial N_s}\simeq \int_S ds \frac{\partial g_0(s,t)}{\partial N_s}=-\frac{1}{2}, \quad g_0(s,t):=\frac{1}{4\pi|s-t|}. \end{equation} Let \begin{equation} {\langle}bel{eq1.3.13} \Gamma_{pq}(t):=\int_S ds \frac{\partial g(s,t)}{\partial s_p}N_q(s), \end{equation} then equation \eqref{eq1.3.11} can be rewritten as follows \begin{equation} {\langle}bel{eq1.3.14} \frac{Q}{2}+e_p\int_S dt \Gamma_{pq}(t) J_q(t)+\frac{Q}{2}=-|D| \nabla\times E_0, \end{equation} or \begin{equation} {\langle}bel{eq1.3.15} Q+\Gamma Q=-|D| \nabla\times E_0, \end{equation} where $\Gamma$ is a $3\times 3$ constant matrix and it is defined by \begin{equation} {\langle}bel{eq1.3.16a} \Gamma Q=e_p\int_S dt \Gamma_{pq}(t) J_q(t), \end{equation} in which summations are understood over the repeated indices. Thus, $Q$ can be written as \begin{equation} {\langle}bel{eq1.3.16} Q=-|D|(I+\Gamma)^{-1} \nabla\times E_0, \quad a \to 0, \end{equation} where $I:=I_3$, the $3\times 3$ identity matrix. This formula is asymptotically exact as $a \to 0$. \subsection{Numerical method for solving EM wave scattering by one perfectly conducting spherical body}{\langle}bel{sec3.1} In this section, we consider the EM wave scattering problem by a small perfectly conducting spherical body. Instead of solving the problem \eqref{eq1.1.1}-\eqref{eq1.1.6} directly, we will solve its corresponding boundary integral equation \eqref{eq1.3.2b} for the unknown vector $J$ \begin{equation} {\langle}bel{eq3.0.1} \frac{J(s)}{2}+\int_S [N_s,[\nabla_s g(s,t),J(t)]]dt=-[N_s,E_0]. \end{equation} Then the solution $E$ to the EM wave scattering problem by one perfectly conducting body can be computed by either the exact formula \eqref{eq1.3.2} or the asymptotic formula \eqref{eq1.3.8}. Scattering by a sphere has been discussed in many papers, for example \cite{Mie1908} in which Mie solves the EM wave scattering problem by separation of variables. The EM field, scattered by a small body, is proportional to $O(a^3)$. Suppose $S$ is a smooth surface of a spherical body. Let $S$ be partitioned into $P$ non-intersecting subdomains $S_{ij}, 1 \le i \le m_{\theta}, 1 \le j \le m_{\phi}$, using spherical coordinates, where $m_{\theta}$ is the number of intervals of $\theta$ between $0$ and $2\pi$ and $m_{\phi}$ defines the number of intervals of $\phi$ between $0$ and $\pi$. Then $P=m_{\theta}m_{\phi}+2$, which includes the two poles of the sphere. $m_{\theta}$ is defined in this way: $m_{\theta}=m_{\phi}+|\phi-\frac{\pi}{2}|6m_{\phi}$. This means the closer it is to the poles of the sphere, the more intervals for $\theta$ are used. Then the point $(\theta_i,\phi_j)$ in $S_{ij}$ is chosen as follows \begin{equation}gin{align} &\theta_i = i\frac{2\pi}{m_{\theta}}, \quad 1 \le i \le m_{\theta}, {\langle}bel{eq3.0.2} \\ &\phi_j = j\frac{\pi}{m_{\phi}+1}, \quad 1 \le j \le m_{\phi}. {\langle}bel{eq3.0.3} \end{align} Note that there are many different ways to distribute collocation points. However, the one that we describe here will guarantee convergence to the solution to \eqref{eq3.0.1} with fewer collocation points used from our experiment. Furthermore, one should be careful when choosing the distribution of collocation points on a sphere. If one chooses $\phi_j = j\frac{\pi}{m_{\phi}}, 1 \le j \le m_{\phi}$, then when $j= m_{\phi}$, $\phi_j=\pi$ and thus there is only one point for this $\phi$ regardless of the value of $\theta$ as shown in \eqref{eq3.0.4}. The position of a point in each $S_{ij}$ can be computed by \begin{equation} {\langle}bel{eq3.0.4} (x,y,z)_{ij} = a(\cos\theta_i \sin\phi_j,\sin\theta_i \sin\phi_j,\cos\phi_j), \end{equation} and the outward-pointing unit normal vector $N$ to $S$ at this point is \begin{equation} {\langle}bel{eq3.0.5} N_{ij}=N(\theta_i,\phi_j)=(\cos\theta_i \sin\phi_j,\sin\theta_i \sin\phi_j,\cos\phi_j). \end{equation} For a star-shaped body with a different shape, only the normal vector $N$ needs to be recomputed. Rewrite the integral equation \eqref{eq3.0.1} as \begin{equation} {\langle}bel{eq3.0.6a} \frac{J(s)}{2}+\int_S \nabla_s g(s,t)N_s\cdot J(t)dt-\int_S \frac{\partial g(s,t)}{\partial N_s} J(t)dt=-[N_s,E_0]. \end{equation} This integral equation can be discretized as follows \begin{equation} {\langle}bel{eq3.0.6} J(i)+2\sum_{j \neq i}^P [\nabla_s g(i,j)N_s(i)\cdot J(j)-J(j)\nabla_s g(i,j)\cdot N_s(i)]\Delta_{j} = F(i), \quad 1 \le i \le P, \end{equation} in which by $i$ the point $(x_i,y_i,z_i)$ is denoted, $F(i):=-2[N_s,E_0](i)$, and $\Delta_j$ is the surface area of the subdomain $j$. This is a linear system with unknowns $J(i):=(X_i,Y_i,Z_i), 1 \le i \le P$. This linear system can be rewritten as follows \begin{equation}gin{align} &X_i+\sum_{j\neq i}^P a_{ij}X_j+ b_{ij}Y_j+ c_{ij}Z_j = F_x(i), {\langle}bel{eq3.0.7}\\ &Y_i+\sum_{j\neq i}^P a'_{ij}X_j+ b'_{ij}Y_j+ c'_{ij}Z_j = F_y(i), {\langle}bel{eq3.0.8}\\ &Z_i+\sum_{j\neq i}^P a''_{ij}X_j+ b''_{ij}Y_j+ c''_{ij}Z_j = F_z(i), {\langle}bel{eq3.0.9} \end{align} where by the subscripts $x,y,z$ the corresponding coordinates are denoted, e.g. $F(i)=(F_x,F_y,F_z)(i)$, and \begin{equation}gin{align} &a_{ij}:=2[\nabla g(i,j)_x N_x(i)-\nabla g(i,j)\cdot N(i)]\Delta_j,\\ &b_{ij}:=2\nabla g(i,j)_x N_y(i)\Delta_j,\\ &c_{ij}:=2\nabla g(i,j)_x N_z(i)\Delta_j, \end{align} for $i\neq j$; when $i=j$: $a_{ii}=1, b_{ii}=0$, and $c_{ii}=0$, \begin{equation}gin{align} &a'_{ij}:=2\nabla g(i,j)_y N_x(i)\Delta_j,\\ &b'_{ij}:=2[\nabla g(i,j)_y N_y(i)-\nabla g(i,j)\cdot N(i)]\Delta_j,\\ &c'_{ij}:=2\nabla g(i,j)_y N_z(i)\Delta_j, \end{align} for $i\neq j$; when $i=j$: $a'_{ii}=0, b'_{ii}=1$, and $c'_{ii}=0$, \begin{equation}gin{align} &a''_{ij}:=2\nabla g(i,j)_z N_x(i)\Delta_j,\\ &b''_{ij}:=2\nabla g(i,j)_z N_y(i)\Delta_j,\\ &c''_{ij}:=2[\nabla g(i,j)_z N_z(i)-\nabla g(i,j)\cdot N(i)]\Delta_j, \end{align} for $i\neq j$; when $i=j$: $a''_{ii}=0, b''_{ii}=0$, and $c''_{ii}=1$. \subsection{Error analysis}{\langle}bel{sec3.1.1} Recall the boundary integral equation \eqref{eq1.3.2b} \begin{equation} {\langle}bel{eq3.0.10} \frac{J(s)}{2}+\int_S [N_s,[\nabla_s g(s,t),J(t)]]dt=-[N_s,E_0]. \end{equation} Integrate both sides of this equation over $S$ and get \begin{equation} {\langle}bel{eq3.0.11} Q+\Gamma Q=-|D| \nabla\times E_0, \end{equation} see Section \ref{sec1}. Once $J$ is found from solving \eqref{eq3.0.10}, $Q$ can be computed by $Q=\int_S J(t)dt$. Then one can validate the values of $J$ and $Q$ by checking the following things \begin{equation}gin{itemize} \item Is $J$ tangential to $S$ as shown in Section \ref{sec1}? One needs to check $J(s)\cdot N_s$. \item Is $Q=\int_S J(t)dt$ correct? The relative error of $Q$ can be computed as follows \begin{equation} {\langle}bel{eq3.0.12} \text{Error} = \frac{|Q+\Gamma Q-RHS|}{|RHS|}, \end{equation} where $RHS:=-|D| \nabla\times E_0$. This will give the error of the numerical method for the case of one body. \end{itemize} Furthermore, one can also compare the value of the asymptotic $Q_a$ in formula \eqref{eq1.3.16} with the exact $Q_e$ defined in \eqref{eq1.3.5} by \begin{equation} {\langle}bel{eq3.0.13} \text{Error} = \frac{|Q_e-Q_a|}{|Q_e|}, \end{equation} and check the difference between the asymptotic $E_a$ in \eqref{eq1.3.8} and the exact $E_e$ defined in \eqref{eq1.3.2} by computing this relative error \begin{equation} {\langle}bel{eq3.0.14} \text{Error} = \frac{|E_e-E_a|}{|E_e|}. \end{equation} \subsection{General method for solving EM wave scattering by one perfectly conducting body}{\langle}bel{sec3.4} In this section, we present a general method for solving the EM wave scattering problem by one perfectly conducting body, whose surface is parametrized by $f(u,v) = (x(u,v), y(u,v),$ $ z(u,v))$. \begin{equation}gin{itemize} \item Step 1: One needs to partition the surface of the body into $P$ non-intersecting subdomains. In each subdomain, choose a collocation point. The position of the collocation points can be computed using $f(u,v)=(x(u,v),y(u,v),z(u,v))$, see for example \eqref{eq3.0.2}-\eqref{eq3.0.4}. \item Step 2: Find the unit normal vector $N$ of the surface from the function $f$. \item Step 3: Solve the linear system \eqref{eq3.0.7}-\eqref{eq3.0.9} for $X_i,Y_i$, and $Z_i, 1 \le i \le P$. Then vector $J$ in the boundary integral equation \eqref{eq1.3.2b} is computed by $J(i):=(X_i,Y_i,Z_i)$ at the point $i$ on the surface. \item Step 4: Compute the electric field $E$ using \eqref{eq1.3.2}. \end{itemize} \section{EM wave scattering by many small perfectly conducting bodies} {\langle}bel{sec2} Consider a bounded domain $\Omega$ containing $M$ small bodies $D_m$, $1\le m \le M$, and $S_m$ are their corresponding smooth boundaries. Let $D:=\bigcup_{m=1}^M D_m \subset \Omega$ and $D'$ be the complement of $D$ in ${\mathbb R}RR$. We assume that $S=\bigcup_{m=1}^M S_m$ is C$^2$-smooth. $\epsilon$ is the dielectric permittivity constant and $\mu$ is the magnetic permeability constant of the medium. Let $E$ and $H$ denote the electric and magnetic fields, respectively. $E_0$ is the incident field and $v$ is the scattered field. The problem of electromagnetic wave scattering by many small perfectly conducting bodies involves solving the following system \begin{equation}gin{align} &\nabla \times E=i \omega \mu H, \quad \text{in }D':={\mathbb R}RR \setminus D, \quad D:=\bigcup_{m=1}^M D_m, {\langle}bel{eq2.1.1} \\ &\nabla \times H=-i \omega \epsilon E, \quad \text{in }D', {\langle}bel{eq2.1.2} \\ &[N,[E, N]]=0, \quad \text{on } S, {\langle}bel{eq2.1.3} \\ &E=E_0+v, {\langle}bel{eq2.1.4} \\ &E_0=\mathcal{E} e^{ik\alpha\cdot x}, \quad \mathcal{E} \cdot \alpha=0, \quad \alpha \in S^2. {\langle}bel{eq2.1.5} \end{align} where $v$ satisfies the radiation condition \eqref{eq1.1.6}, $\omega>0$ is the frequency, $k=2\pi/{\langle}mbda$ is the wave number, $ka \ll 1$, $a:=\frac{1}{2}\max_m \text{diam}D_m$, and $\alpha$ is a unit vector that indicates the direction of the incident wave $E_0$. Furthermore, \begin{equation} {\langle}bel{eq2.1.7} \epsilon=\epsilon_0, \quad \mu=\mu_0 \quad \text{ in } \Omega':={\mathbb R}RR \setminus \Omega. \end{equation} Assume that the distribution of small bodies $D_m$, $1 \le m \le M$, in $\Omega$ satisfies the following formula \begin{equation} {\langle}bel{eq2.1.8} \mathcal{N}(\Delta)=\frac{1}{a^3} \int_\Delta N(x)dx[1+o(1)], \quad a\to 0, \end{equation} where $\mathcal{N}(\Delta)$ is the number of small bodies in $\Delta$, $\Delta$ is an arbitrary open subset of $\Omega$, and $N(x)$ is the distribution function \begin{equation} {\langle}bel{eq2.1.9} N(x)\ge 0, \quad N(x) \in C(\Omega). \end{equation} Note that $E$ solves this equation \begin{equation} {\langle}bel{eq2.1.11} \nabla\times\nabla\times E=k^2 E, \quad k^2=\omega^2\epsilon \mu, \end{equation} if $\mu=$const. Once we have $E$, then $H$ can be found from this relation \begin{equation} {\langle}bel{eq2.1.10} H=\frac{\nabla\times E}{i\omega\mu}. \end{equation} From \eqref{eq2.1.10} and \eqref{eq2.1.11}, one can get \eqref{eq2.1.2}. Thus, we need to find only $E$ which satisfies the boundary condition \eqref{eq2.1.3}. It was proved in \cite{R620} that under the radiation condition and the assumptions $a \ll d \ll {\langle}mbda$, the problem \eqref{eq2.1.1}-\eqref{eq2.1.5} has a unique solution and its solution is of the form \begin{equation} {\langle}bel{eq2.1.13} E(x)=E_0(x)+\sum_{m=1}^M \nabla\times \int_{S_m} g(x,t) J_m(t)dt, \end{equation} where $J_m$ are unknown continuous functions that can be found from the boundary condition. Let \begin{equation} {\langle}bel{eq2.1.15} Q_m:=\int_{S_m}J_m(t)dt. \end{equation} When $a \to 0$, the asymptotic solution for the electric field is given by \begin{equation} {\langle}bel{eq2.1.22} E(x)=E_0(x)+\sum_{m=1}^M [\nabla g(x,x_m), Q_m], \quad a\to 0. \end{equation} Therefore, instead of finding $J_m(t), \forall t \in S, 1 \le m \le M$, to get the solution $E$, one can just find $Q_m$. This allows one to solve the EM scattering problem with a very large number of small bodies which is impossible to do before. The analytic formula for $Q_m$ can be derived by using formula \eqref{eq1.3.16} and replacing $E_0$ in this formula by the effective field $E_{e}(x_m)$ acting on the m-th body \begin{equation} {\langle}bel{eq2.1.24} Q_m=-|D_m|(I+\Gamma)^{-1} \nabla\times E_{e}(x_m), \quad 1 \le m \le M, \quad x_m \in D_m, \end{equation} where the effective field acting on the m-th body is defined as \begin{equation} {\langle}bel{eq2.1.23} E_e(x_m)=E_0(x_m)+\sum_{j\ne m}^M [\nabla g(x_m,x_j), Q_j]\quad 1 \le m \le M. \end{equation} When $a \to 0$, the effective field $E_e(x)$ is asymptotically equal to the field $E(x)$ in \eqref{eq2.1.22} as proved in \cite{R620} and \cite{R652}. Let $E_{em}:=E_{e}(x_m)$, where $x_m$ is a point in $D_m$. From \eqref{eq2.1.24}, and \eqref{eq2.1.23}, one gets \begin{equation} {\langle}bel{eq2.1.28} E_{em}=E_{0m}-\sum_{j\ne m}^M [\nabla g(x_m,x_j), (I+\Gamma)^{-1}\nabla\times E_{ej}]|D_j|,\quad 1 \le m \le M. \end{equation} \subsection{Numerical method for solving EM wave scattering by many small perfectly conducting bodies}{\langle}bel{sec3.5} For finding the solution to EM wave scattering in the case of many small perfectly conducting bodies, we need to find $E_{em}$ in \eqref{eq2.1.28}. Apply the operator $(I+\Gamma)^{-1}\nabla \times$ to both sides of \eqref{eq2.1.28} and let $A_m:=(I+\Gamma)^{-1}\nabla \times E_{em}$. Then \begin{equation}gin{align} A_m=A_{0m}-(I+\Gamma)^{-1}\sum_{j\ne m}^M |D_j|\left(\nabla_x\times [\nabla g(x,x_j),A_j] \right) |_{x=x_m}, \quad 1 \le m \le M, {\langle}bel{eq3.5.1} \end{align} Solving this system yields $A_m$, for $1 \le m \le M$. Then $E$ can be computed by \begin{equation} {\langle}bel{eq3.5.2} E(x)=E_0(x)+\sum_{m=1}^M [\nabla g(x,x_m), Q_m], \end{equation} where \begin{equation} {\langle}bel{eq3.5.2a} Q_m=-|D_m|A_m, \quad 1 \le m \le M. \end{equation} Equation \eqref{eq3.5.1} can be rewritten as follows \begin{equation} {\langle}bel{eq3.5.3} A_m=A_{0m}-\sum_{j\ne m}^M \tau[k^2 g(x_m,x_j)A_j+(A_j\cdot \nabla_x)\nabla g(x,x_j)|_{x=x_m}]|D_j|, \end{equation} where $1 \le m \le M$, $\tau:=(I+\Gamma)^{-1}$, and $A_m$ are vectors in ${\mathbb R}RR$. Let $A_i:=(X_i,Y_i,Z_i)$ then one can rewrite the system \eqref{eq3.5.3} as \begin{equation}gin{align} &X_i+\sum_{j\neq i}^M a_{ij}X_j + b_{ij}Y_j + c_{ij}Z_j = F_x(i), {\langle}bel{eq3.5.7}\\ &Y_i+\sum_{j\neq i}^M a'_{ij}X_j + b'_{ij}Y_j + c'_{ij}Z_j = F_y(i), {\langle}bel{eq3.5.8}\\ &Z_i+\sum_{j\neq i}^M a''_{ij}X_j + b''_{ij}Y_j + c''_{ij}Z_j = F_z(i), {\langle}bel{eq3.5.9} \end{align} in which by the subscripts $x,y,z$ the corresponding coordinates are denoted, e.g. $F(i)=(F_x,F_y,F_z)(i)$, where $F(i):=A_{0i}$ and \begin{equation}gin{align} &a_{ij}:=[k^2 g(i,j) + \partial_{x} \nabla g(i,j)_x]|D_j|\tau(1,1),\\ &b_{ij}:=\partial_{y} \nabla g(i,j)_x|D_j|\tau(1,1),\\ &c_{ij}:=\partial_{z} \nabla g(i,j)_x|D_j|\tau(1,1), \end{align} for $i\neq j$, here $\tau(1,1)$ is the entry (1,1) of matrix $\tau$ in \eqref{eq3.5.3}; when $i=j$: $a_{ii}=1, b_{ii}=0$, and $c_{ii}=0$; \begin{equation}gin{align} &a'_{ij}:=\partial_{x} \nabla g(i,j)_y|D_j|\tau(2,2),\\ &b'_{ij}:=[k^2 g(i,j) + \partial_{y} \nabla g(i,j)_y]|D_j|\tau(2,2),\\ &c'_{ij}:=\partial_{z} \nabla g(i,j)_y|D_j|\tau(2,2), \end{align} for $i\neq j$; when $i=j$: $a'_{ii}=0, b'_{ii}=1$, and $c'_{ii}=0$; \begin{equation}gin{align} &a''_{ij}:=\partial_{x} \nabla g(i,j)_z|D_j|\tau(3,3),\\ &b''_{ij}:=\partial_{y} \nabla g(i,j)_z|D_j|\tau(3,3),\\ &c''_{ij}:=[k^2 g(i,j) + \partial_{z} \nabla g(i,j)_z]|D_j|\tau(3,3), \end{align} for $i\neq j$; when $i=j$: $a'_{ii}=0, b'_{ii}=0$, and $c'_{ii}=1$. \subsection{Error analysis} The error of the solution to the EM wave scattering problem by many small perfectly conducting bodies can be estimated as follows. From the solution $E$ of the electromagnetic scattering problem by many small bodies given in \eqref{eq2.1.13} \begin{equation} {\langle}bel{eq3.5.48} E(x)=E_0(x)+\sum_{m=1}^M \nabla\times \int_{S_m} g(x,t) J_m(t)dt, \end{equation} we can rewrite it as \begin{equation} {\langle}bel{eq3.5.49} E(x)=E_0(x)+\sum_{m=1}^M [\nabla g(x,x_m), Q_m]+\sum_{m=1}^M \nabla\times \int_{S_m} [g(x,t)-g(x,x_m)] J_m(t)dt. \end{equation} Comparing this with the asymptotic formula for $E$ when $a \to 0$ given in \eqref{eq2.1.22} \begin{equation} {\langle}bel{eq3.5.50} E(x)=E_0(x)+\sum_{m=1}^M [\nabla g(x,x_m), Q_m], \end{equation} we have the error of this asymptotic formula is \begin{equation} {\langle}bel{eq3.5.51} \text{Error}=\left|\sum_{m=1}^M \nabla\times \int_{S_m} [g(x,t)-g(x,x_m)] J_m(t)dt\right| \sim \frac{1}{4\pi}\left(\frac{ak^2}{d}+\frac{ak}{d^2}+\frac{a}{d^3}\right)\sum_{m=1}^M|Q_m|, \end{equation} where $d=\min_m|x-x_m|$ and \begin{equation} {\langle}bel{eq3.5.52} Q_m=-|D_m|(I+\Gamma)^{-1} \nabla\times E_{e}(x_m), \quad 1 \le m \le M, \quad x_m \in D_m, \quad a \to 0, \end{equation} because \begin{equation} {\langle}bel{eq3.5.53} |\nabla[g(x,t)-g(x,x_m)]| = O\left(\frac{ak^2}{d}+\frac{ak}{d^2}+\frac{a}{d^3}\right), \quad a=\max_m|t-x_m|. \end{equation} \section{Experiments} {\langle}bel{sec3} \subsection{EM wave scattering by one perfectly conducting spherical body}{\langle}bel{sec3.1.2} To illustrate the idea of the numerical method, we use the following physical parameters to solve the EM wave scattering problem by one small perfectly conducting sphere, i.e solving the linear system \eqref{eq3.0.7}-\eqref{eq3.0.9} \begin{equation}gin{itemize} \item Speed of wave, $c=(3.0E+10)$ cm/sec. \item Frequency, $\omega=(5.0E+14)$ Hz. \item Wave number, $k = (1.05E+05)$ cm$^{-1}$. \item Wave length, ${\langle}mbda= (6.00E-05)$ cm. \item Direction of incident plane wave, $\alpha = (0, 1, 0)$. \item Magnetic permeability, $\mu = 1$. \item Vector $\mathcal{E} = (1, 0, 0)$. \item Incident field vector, $E_0$: $E_{0}(x)=\mathcal{E} e^{ik\alpha\cdot x}$. \item The body is a sphere of radius $a$, centered at the origin. \end{itemize} We use GMRES iterative method, see \cite{GMRES}, to solve the linear system \eqref{eq3.0.7}-\eqref{eq3.0.9}. For a spherical body, matrix $\Gamma$ in \eqref{eq1.3.15} can be computed analytically as follows. Recall that \begin{equation} \Gamma_{pq}(t):=\int_S \frac{\partial g(s,t)}{\partial s_p}N_q(s) ds, \quad 1\le p,q\le 3, \end{equation} where \begin{equation} N=(\cos\theta \sin\phi,\sin\theta \sin\phi,\cos\phi) \end{equation} and \begin{equation} \frac{\partial g(s,t)}{\partial s_p} \simeq \frac{\partial g_0(s,t)}{\partial s_p} = -\frac{s_p-t_p}{4\pi|s-t|^3}, \quad g_0(s,t):=\frac{1}{4\pi|s-t|}. \end{equation} We choose a coordinate system centered at the center of the sphere such that $t=(0,0,a)$ and $s=aN$. Then \begin{equation} \Gamma_{pq}(t):=-\frac{a^2}{4\pi}\int_0^{2\pi} d\theta \int_0^\pi d\phi \sin\phi \frac{(s_p-t_p)N_q}{a^3 8\sin^3\frac{\phi}{2}}, \quad 1\le p,q\le 3. \end{equation} When \begin{equation}gin{align} &p=q=1: \quad \Gamma_{11}(t) = -1/3, \\ &p=q=2: \quad \Gamma_{22}(t) = -1/3, \\ &p=q=3: \quad \Gamma_{33}(t) = 1/6, \\ &p \neq q: \qquad\quad \Gamma_{pq}(t) = 0. \end{align} Therefore, matrix $\Gamma$ is \begin{equation} \Gamma\simeq\left[ \begin{equation}gin{array}{ccc} -1/3 & 0 & 0 \\ 0 & -1/3 & 0 \\ 0 & 0 & 1/6 \end{array} \right] \end{equation} For example, Table \ref{tab3.1.2.0} shows the exact and asymptotic vector $Q$ when the radius of the body is $a=(1.0E-09)$ cm and the number of collocation points used to solve the integral equation \eqref{eq3.0.1} is $P=766$. Note that $a=(1.0E-09)$ cm satisfies $ka \ll 1$. The point $x_1$ in \eqref{eq1.3.8} is taken at the center of the body, the origin. Table \ref{tab3.1.2.1} and \ref{tab3.1.2.2} show the exact and asymptotic vector $E=(E_x,E_y,E_z)$, the electric field, at the point $x$ outside of the body, respectively. The distance $|x-x_1|$ is measured in cm in these tables. \begin{equation}gin{table}[htbp] \centering \caption{Vector $Q_e$ and $Q_a$ when $P=766$ collocation points and $a=(1.0E-09)$ cm.} \begin{equation}gin{tabular}{rccc} \toprule \multicolumn{4}{c}{P=766, a=1.0E-09} \\ \midrule \multicolumn{1}{c}{} & \multicolumn{3}{c}{ 1.0E-21 *} \\ $Q_e$ & 0.0000 + 0.0000i & 0.0000 + 0.0000i & 0.0000 + 0.3925i \\ $Q_a$ & 0.0000 + 0.0000i & 0.0000 + 0.0000i & 0.0000 + 0.3760i \\ \bottomrule \end{tabular} {\langle}bel{tab3.1.2.0} \end{table} \begin{equation}gin{table}[htbp] \centering \caption{Vector $E_e$ for one perfectly conducting body with $a=(1.0E-09)$ cm and $P=766$ collocation points.} \begin{equation}gin{tabular}{crrr} \toprule $|x-x_1|$ & \multicolumn{3}{c}{$E_{e}(x)$} \\ \midrule 1.73E-08 & 1.0000 + 0.0010i & 0.0001 + 0.0000i & 0.0004 + 0.0000i \\ 1.73E-07 & 0.9999 + 0.0105i & 0.0000 + 0.0000i & 0.0000 + 0.0000i \\ 1.73E-06 & 0.9945 + 0.1045i & 0.0000 + 0.0000i & 0.0000 + 0.0000i \\ \bottomrule \end{tabular} {\langle}bel{tab3.1.2.1} \end{table} \begin{equation}gin{table}[htbp] \centering \caption{Vector $E_a$ for one perfectly conducting body with $a=(1.0E-09)$ cm and $P=766$ collocation points.} \begin{equation}gin{tabular}{crrr} \toprule $|x-x_1|$ & \multicolumn{3}{c}{$E_{a}(x)$} \\ \midrule 1.73E-08 & 1.0000 + 0.0010i & 0.0000 + 0.0000i & 0.0000 + 0.0000i \\ 1.73E-07 & 0.9999 + 0.0105i & 0.0000 + 0.0000i & 0.0000 + 0.0000i \\ 1.73E-06 & 0.9945 + 0.1045i & 0.0000 + 0.0000i & 0.0000 + 0.0000i \\ \bottomrule \end{tabular} {\langle}bel{tab3.1.2.2} \end{table} \begin{equation}gin{table}[htbp] \centering \caption{Relative errors between the asymptotic and exact formulas for $E$ when $P=766$ collocation points and $a=(1.0E-09)$ cm.} \begin{equation}gin{tabular}{cc} \toprule $|x-x_1|$ & $E_{e}$ vs $E_{a}$ \\ \midrule 1.73E-08 & 4.67E-04 \\ 1.73E-07 & 4.67E-07 \\ 1.73E-06 & 4.70E-10 \\ \bottomrule \end{tabular} {\langle}bel{tab3.1.2.3} \end{table} In this case, we also verify the following things:\\ a) Is $J$ tangential to $S$? \\ In fact, this vector $J$ is tangential to the surface $S$ of the body, $J\cdot N_s= O(10^{-14})$. \\ b) How accurate is the asymptotic formula \eqref{eq1.3.16} for $Q$?\\ We check the accuracy of the asymptotic formula for $Q$ in \eqref{eq1.3.16} by comparing it with the exact formula \eqref{eq1.3.5}, see Section \ref{sec3.1.1}, and the relative error is $4.21E-02$. The more collocation points used, the little this relative error is. \\ c) How accurate is the asymptotic formula \eqref{eq1.3.8} for $E$?\\ The accuracy of the asymptotic formula for $E$ in \eqref{eq1.3.8} can be checked by comparing it with the exact formula \eqref{eq1.3.2} at several points $x$ outside of the body, $|x-x_1|\gg a$ where $x_1$ is the center of the body, see the error analysis in Section \ref{sec3.1.1}. The relative errors are given in Table \ref{tab3.1.2.3}. \begin{equation}gin{table}[htbp] \centering \caption{Relative errors of the asymptotic $E$ and $Q$ when $P=1386$ collocation points.} \begin{equation}gin{tabular}{ccccc} \toprule \multicolumn{5}{c}{$P=1386, |x-x_1|=1.73E-05$} \\ \midrule $a$ & 1.00E-07 & 1.00E-08 & 1.00E-09 & 1.00E-10 \\ $E_{e}$ vs $E_{a}$ & 1.08E-06 & 1.08E-09 & 1.08E-12 & 1.12E-15 \\ $Q_{e}$ vs $Q_{a}$ & 1.96E-02 & 1.96E-02 & 1.96E-02 & 1.89E-02 \\ \bottomrule \end{tabular} {\langle}bel{tab3.1.2.4} \end{table} Table \ref{tab3.1.2.4} compares the asymptotic $Q_a$ versus exact $Q_e$ and asymptotic $E_a$ versus exact $E_e$, when $P=1386$ collocation points, $|x-x_1|=1.73E-05$ cm, and with various $a$. The errors shown in this table are relative errors, see the error analysis in Section \ref{sec3.1.1}. As one can see from this table, the smaller the radius $a$ is, compared to the distance from the point of interest to the center of the body, the more precise the asymptotic formulas of $E$ and $Q$ are. Furthermore, the numerical results also depend on the number of collocation points used. The more collocation points used, the more accurate the results is. \subsection{EM wave scattering by one perfectly conducting ellipsoid body}{\langle}bel{sec3.2} In this section, we consider the EM wave scattering problem by a small perfectly conducting ellipsoid body. The method for solving the problem in this setting is the same as that of Section \ref{sec3.1} except that one needs to recompute the unit normal vector $N$. To get the solution of this problem, one can follow the steps in Section \ref{sec3.1}. In particular, one needs to solve the linear system \eqref{eq3.0.7}-\eqref{eq3.0.9}. To illustrate the idea, we use the same physical parameters as described in Section \ref{sec3.1.2}, except that the body now is an ellipsoid. Let $S$ be its smooth surface. The way we partition $S$ into many subdomains $S_{ij}$ is the same as the way we partition a spherical body as described in section \ref{sec3.1}. Then, the position of the collocation point in each subdomain $S_{ij}$ is defined by \begin{equation} {\langle}bel{eq3.2.1} (x,y,z)_{ij} = (a\cos\theta_i \sin\phi_j,b\sin\theta_i \sin\phi_j,c\cos\phi_j), \end{equation} where $a, b,$ and $c$ are the lengths of the semi-principal axes of the ellipsoid. The outward-pointing normal vector $n$ to $S$ at this point is \begin{equation} {\langle}bel{eq3.2.2} n_{ij}=n(\theta_i,\phi_j)=2\left(\frac{\cos\theta_i \sin\phi_j}{a},\frac{\sin\theta_i \sin\phi_j}{b},\frac{\cos\phi_j}{c}\right), \end{equation} and the corresponding unit normal vector $N$ is \begin{equation} {\langle}bel{eq3.2.3} N_{ij}=N(\theta_i,\phi_j)=n_{ij}/|n_{ij}|. \end{equation} \begin{equation}gin{table}[htb] \centering \caption{Vector $E_e$ for one perfectly conducting ellipsoid body with $a=(1.0E-08)$ cm, $b=(1.0E-09)$ cm, $c=(1.0E-09)$ cm, and $P=1052$ collocation points.} \begin{equation}gin{tabular}{crrr} \toprule $|x-x_1|$ & \multicolumn{3}{c}{$E_{e}(x)$} \\ \midrule 1.01E-07 & 0.9998 + 0.0010i & -0.0000 + 0.0000i & -0.0000 - 0.0000i \\ 1.01E-06 & 0.9999 + 0.0105i & -0.0000 + 0.0000i & -0.0000 - 0.0000i \\ 1.01E-05 & 0.9945 + 0.1045i & -0.0000 + 0.0000i & -0.0000 - 0.0000i \\ \bottomrule \end{tabular} {\langle}bel{tab3.2.1} \end{table} \begin{equation}gin{table}[htb] \centering \caption{Vector $E_a$ for one perfectly conducting ellipsoid body with $a=(1.0E-08)$ cm, $b=(1.0E-09)$ cm, $c=(1.0E-09)$ cm, and $P=1052$ collocation points.} \begin{equation}gin{tabular}{crrr} \toprule $|x-x_1|$ & \multicolumn{3}{c}{$E_{a}(x)$} \\ \midrule 1.01E-07 & 1.0000 + 0.0010i & -0.0000 + 0.0000i & 0.0000 - 0.0000i \\ 1.01E-06 & 0.9999 + 0.0105i & -0.0000 + 0.0000i & 0.0000 - 0.0000i \\ 1.01E-05 & 0.9945 + 0.1045i & -0.0000 + 0.0000i & 0.0000 - 0.0000i \\ \bottomrule \end{tabular} {\langle}bel{tab3.2.2} \end{table} For example, Tables \ref{tab3.2.1} and \ref{tab3.2.2} show the exact and asymptotic vector $E=(E_x,E_y,E_z)$, the electric field, got from solving this EM wave scattering problem with one perfectly conducting ellipsoid body, when the semi-principle axes of the body are $a=(1.0E-08)$ cm, $b=(1.0E-09)$ cm, $c=(1.0E-09)$ cm, and the number of collocation points is $P=1052$. Note that $a$, $b$, and $c$ satisfy $k\max(a,b,c) \ll 1$. The point $x_1$ in \eqref{eq1.3.8} is taken at the center of the ellipsoid body. Each row in Tables \ref{tab3.2.1} and \ref{tab3.2.2} shows the exact and asymptotic $E=(E_x,E_y,E_z)$, respectively, at the point $x$ outside of the body. The distance $|x-x_1|$ is measured in cm in these tables. As for the case of one body, we need to verify the following things:\\ a) Is $J$ tangential to $S$? \\ In fact, this vector $J$ is tangential to the surface $S$ of the body, $J\cdot N_s= O(10^{-13})$. \\ b) Are $Q$ and $J$ correct? We check the relative error described in Section \ref{sec3.1.1}, $\text{Error} = \frac{|Q+\Gamma Q-RHS|}{|RHS|}=14\%$. The more collocation points used, the smaller this error is, for example, with $P=1762$ collocation points, this error is only 3.6\%.\\ c) How accurate is the asymptotic formula \eqref{eq1.3.8} for $E$?\\ The accuracy of the asymptotic formula for $E$ in \eqref{eq1.3.8} can be checked by comparing it with the exact formula \eqref{eq1.3.2} at several points $x$ outside of the body, $|x-x_1|\gg \max(a,b,c)$ where $x_1$ is the center of the body. The relative errors are given in Table \ref{tab3.2.3}. \begin{equation}gin{table}[htb] \centering \caption{Relative errors between the asymptotic and exact formulas for $E$ when $P=1052$ collocation points, $a=(1.0E-08)$ cm, $b=(1.0E-09)$ cm, and $c=(1.0E-09)$ cm.} \begin{equation}gin{tabular}{cc} \toprule $|x-x_1|$ & $E_{e}$ vs $E_{a}$ \\ \midrule 1.01E-07 & 1.73E-04 \\ 1.01E-06 & 1.73E-07 \\ 1.01E-05 & 1.73E-10 \\ \bottomrule \end{tabular} {\langle}bel{tab3.2.3} \end{table} \begin{equation}gin{table}[htb] \centering \caption{Relative errors of the asymptotic $E$ when $P=1052$ collocation points.} \begin{equation}gin{tabular}{ccccc} \toprule \multicolumn{5}{c}{$P=1052, |x-x_1|=1.73E-07$} \\ \midrule $a$ & 1.00E-07 & 1.00E-08 & 1.00E-09 & 1.00E-10 \\ $b$ & 1.00E-08 & 1.00E-09 & 1.00E-10 & 1.00E-11 \\ $c$ & 1.00E-08 & 1.00E-09 & 1.00E-10 & 1.00E-11 \\ $E_{e}$ vs $E_{a}$ & 2.65E-02 & 2.76E-05 & 2.76E-08 & 2.76E-11 \\ \bottomrule \end{tabular} {\langle}bel{tab3.2.4} \end{table} Table \ref{tab3.2.4} shows the relative errors between the asymptotic $E$ versus exact $E$, when $P=1052$ collocation points, $|x-x_1|=1.73E-07$ cm, and with various semi-principle axes $a$, $b$, and $c$. As one can see from this table, the smaller the semi-principle axes are, compared to the distance from the point of interest to the center of the body, the more accurate the asymptotic formulas of $E$ is. \subsection{EM wave scattering by one perfectly conducting cubic body}{\langle}bel{sec3.3} In this section, we consider the EM wave scattering problem by a small perfectly conducting cubic body. Again, the method for solving the problem in this setting is the same as that of Section \ref{sec3.1} except that one needs to recompute the unit normal vector $N$. One can follow the steps outlined in Section \ref{sec3.1} to solve this problem. That means, one needs to solve the linear system \eqref{eq3.0.7}-\eqref{eq3.0.9}. For illustration purpose, we use the same physical parameters as described in Section \ref{sec3.1.2}, except that the body now is a cube. Suppose the cube is placed in the first octant where the origin is one of its vertices, one can use the standard unit vectors in ${\mathbb R}RR$ as the unit normal vectors to the surfaces of the cube. \begin{equation}gin{table}[htb] \centering \caption{Vector $E_e$ for one perfectly conducting body with $a=(1.0E-07)$ cm and $M=600$ collocation points.} \begin{equation}gin{tabular}{crrr} \toprule $|x-x_1|$ & \multicolumn{3}{c}{$E_e(x)$} \\ \midrule 1.73E-04 & -0.5000 - 0.8660i & -0.0000 + 0.0000i & -0.0000 + 0.0000i \\ 1.73E-05 & 0.5000 + 0.8660i & 0.0000 + 0.0000i & 0.0000 + 0.0000i \\ 1.73E-06 & 0.9945 + 0.1045i & 0.0006 + 0.0000i & 0.0006 + 0.0000i \\ \bottomrule \end{tabular} {\langle}bel{tab3.3.1} \end{table} \begin{equation}gin{table}[htb] \centering \caption{Vector $E_a$ for one perfectly conducting body with $a=(1.0E-07)$ cm and $M=600$ collocation points.} \begin{equation}gin{tabular}{crrr} \toprule $|x-x_1|$ & \multicolumn{3}{c}{$E_a(x)$} \\ \midrule 1.73E-04 & -0.5000 - 0.8660i & 0.0000 - 0.0000i & 0.0000 + 0.0000i \\ 1.73E-05 & 0.5000 + 0.8660i & -0.0000 + 0.0000i & -0.0000 - 0.0000i \\ 1.73E-06 & 0.9945 + 0.1045i & -0.0000 + 0.0000i & -0.0000 - 0.0000i \\ \bottomrule \end{tabular} {\langle}bel{tab3.3.2} \end{table} For example, Tables \ref{tab3.3.1} and \ref{tab3.3.2} show the exact and asymptotic vector $E=(E_x,E_y,E_z)$, the electric field, got from solving this EM wave scattering problem with one perfectly conducting cubic body, when the half side of the body is $a=(1.0E-07)$ cm and the number of collocation points is $M=600$. Note that $a=1.0E-07$ cm satisfies $ka \ll 1$. The point $x_1$ in \eqref{eq1.3.8} is taken at the center of the cubic body. Each row in Tables \ref{tab3.3.1} and \ref{tab3.3.2} shows the exact and asymptotic $E=(E_x,E_y,E_z)$, respectively, at the point $x$ outside of the body. The distance $|x-x_1|$ is measured in cm in these tables. As before, for the case of one body, we need to verify the following things:\\ a) Is $J$ tangential to $S$? \\ In fact, this vector $J$ is tangential to the surface $S$ of the body, $J\cdot N_s= O(10^{-13})$. \\ b) How accurate is the asymptotic formula \eqref{eq1.3.16} for $Q$?\\ We check the relative error described in Section \ref{sec3.1.1}, $\text{Error} = \frac{|Q+\Gamma Q-RHS|}{|RHS|}=1.13\%$. \\ c) How accurate is the asymptotic formula \eqref{eq1.3.8} for $E$?\\ The accuracy of the asymptotic formula for $E$ in \eqref{eq1.3.8} can be checked by comparing it with the exact formula \eqref{eq1.3.2} at several points $x$ outside of the body, $|x-x_1|\gg a$ where $x_1$ is the center of the body. The relative errors are given in Table \ref{tab3.3.3}. \begin{equation}gin{table}[htb] \centering \caption{Relative errors between the asymptotic and exact formulas for $E$ when $M=600$ collocation points and $a=(1.0E-07)$ cm.} \begin{equation}gin{tabular}{cc} \toprule $|x-x_1|$ & $E_e$ vs $E_a$ \\ \midrule 1.73E-03 & 1.19E-08 \\ 1.73E-04 & 1.19E-07 \\ 1.73E-05 & 1.52E-06 \\ 1.73E-06 & 8.64E-04 \\ \bottomrule \end{tabular} {\langle}bel{tab3.3.3} \end{table} \begin{equation}gin{table}[htb] \centering \caption{Relative errors of the asymptotic $E$ when $M=600$ collocation points.} \begin{equation}gin{tabular}{cccc} \toprule \multicolumn{4}{c}{$M=600, |x-x_1|=1.73E-06$} \\ \midrule $a$ & 1.00E-07 & 1.00E-08 & 1.00E-09 \\ $E_e$ vs $E_a$ & 8.64E-04 & 6.49E-07 & 6.32E-10 \\ \bottomrule \end{tabular} {\langle}bel{tab3.3.4} \end{table} Table \ref{tab3.3.4} shows the relative errors between the asymptotic $E$ versus exact $E$, when $M=600$ collocation points, $|x-x_1|=(1.73E-06)$ cm, and with various $a$. From this table, we can see that the smaller the side of the cube is, compared to the distance from the point of interest to the center of the body, the more accurate the asymptotic formula of $E$ is. \subsection{EM wave scattering by many small perfectly conducting bodies} To illustrate the idea, consider a domain $\Omega$ as a unit cube placed in the first octant such that the origin is one of its vertices. This domain $\Omega$ contains $M$ small bodies. Suppose these small bodies are particles. We use GMRES iterative method, see \cite{GMRES}, to solve the linear system \eqref{eq3.5.7}-\eqref{eq3.5.9}. The following physical parameters are used to solve the EM wave scattering problem \begin{equation}gin{itemize} \item Speed of wave, $c=(3.0E+10)$ cm/sec. \item Frequency, $\omega=(5.0E+14)$ Hz. \item Wave number, $k =(1.05E+05)$ cm$^{-1}$. \item Wave length, ${\langle}mbda= (6.00E-05)$ cm. \item Direction of incident plane wave, $\alpha = (0, 1, 0)$. \item Magnetic permeability, $\mu = 1$. \item Volume of the domain $\Omega$ that contains all the particles, $|\Omega| = 1$ cm$^3$. \item The distance between two neighboring particles, $d = (1.00E-07)$ cm. \item Vector $\mathcal{E} = (1, 0, 0)$. \item Vector $A_0$: $A_{0m}:=(I+\Gamma)^{-1}\nabla \times E_0(x)|_{x=x_m}=(I+\Gamma)^{-1}\nabla \times \mathcal{E} e^{ik\alpha\cdot x}|_{x=x_m}$. \end{itemize} Note that the distance $d$ satisfies the assumption $d \ll {\langle}mbda$. The radius $a$ of the particles is chosen variously so that it satisfies the assumption $ka \ll 1$. For illustration purpose, the problem of EM wave scattering by many small perfectly conducting bodies is solved with $M=27$ and $1000$ particles. \begin{equation}gin{table}[htbp] \centering \caption{Vector $E$ when $M=27$ particles, $d=(1.0E-07)$ cm and $a=(1.0E-09)$ cm.} \begin{equation}gin{tabular}{rrr} \toprule \multicolumn{3}{c}{$M=27$, $d=1.0E-07$, $a=1.0E-09$}\\ \midrule 1.00E+00+1.01E-14i & 5.69E-17-1.01E-14i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.19E-14i & 0.00E+00+0.00E+00i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.01E-14i & -5.69E-17+1.01E-14i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.05E-02i & 1.24E-16-1.19E-14i & -1.36E-29-5.20E-36i \\ 1.00E+00+1.05E-02i & 0.00E+00+0.00E+00i & -4.80E-30-5.20E-36i \\ 1.00E+00+1.05E-02i & -1.24E-16+1.19E-14i & -1.22E-30-5.20E-36i \\ 1.00E+00+2.09E-02i & 1.54E-16-1.01E-14i & -3.40E-30-1.04E-35i \\ 1.00E+00+2.09E-02i & 0.00E+00+0.00E+00i & -2.43E-30-1.04E-35i \\ 1.00E+00+2.09E-02i & -1.54E-16+1.01E-14i & -1.20E-30-1.04E-35i \\ 1.00E+00+1.19E-14i & 6.63E-17-1.19E-14i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.40E-14i & 4.80E-30+0.00E+00i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.19E-14i & -6.63E-17+1.19E-14i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.05E-02i & 1.47E-16-1.40E-14i & -4.80E-30-5.20E-36i \\ 1.00E+00+1.05E-02i & 2.61E-30+0.00E+00i & -2.61E-30-5.20E-36i \\ 1.00E+00+1.05E-02i & -1.47E-16+1.40E-14i & -9.24E-31-5.20E-36i \\ 1.00E+00+2.09E-02i & 1.82E-16-1.19E-14i & -2.43E-30-1.04E-35i \\ 1.00E+00+2.09E-02i & 9.24E-31+0.00E+00i & -1.85E-30-1.04E-35i \\ 1.00E+00+2.09E-02i & -1.82E-16+1.19E-14i & -1.01E-30-1.04E-35i \\ 1.00E+00+1.01E-14i & 5.69E-17-1.01E-14i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.19E-14i & 2.43E-30+0.00E+00i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.01E-14i & -5.69E-17+1.01E-14i & 0.00E+00+0.00E+00i \\ 1.00E+00+1.05E-02i & 1.24E-16-1.19E-14i & -1.22E-30-5.20E-36i \\ 1.00E+00+1.05E-02i & 1.85E-30+0.00E+00i & -9.24E-31-5.20E-36i \\ 1.00E+00+1.05E-02i & -1.24E-16+1.19E-14i & -5.03E-31-5.20E-36i \\ 1.00E+00+2.09E-02i & 1.54E-16-1.01E-14i & -1.20E-30-1.04E-35i \\ 1.00E+00+2.09E-02i & 9.98E-31+0.00E+00i & -1.01E-30-1.04E-35i \\ 1.00E+00+2.09E-02i & -1.54E-16+1.01E-14i & -6.54E-31-1.04E-35i \\ \bottomrule \end{tabular} {\langle}bel{tab1.0} \end{table} For example, Tables \ref{tab1.0} show the result of solving the EM wave scattering problem with $M=27$ particles in the unit cube in which the distance between neighboring particles is $d=(1.0E-07)$ cm and the radius of the particles is $a=(1.0E-09)$ cm. Each row in Tables \ref{tab1.0} is a vector $E(i)=(E_x,E_y,E_z)(i)$ at the point $i$ in the cube. The norm of this asymptotic solution $E$ is $5.20E+00$ and the error of the solution is $8.16E-10$. This error is computed using \eqref{eq3.5.51}. Table \ref{tab1} shows the relative errors of $E$ when there are $M=27$ particles in the cube, the distance between neighboring particles is $d=(1.0E-07)$ cm, and with various radius $a$. Figure \ref{fig1} shows the relative error of the asymptotic $E$. From this figure, one can see that when the ratio $a/d$ decreases from $1.0E-01$ to $1.0E-04$, the error of the asymptotic solution decreases linearly and rapidly from $8.16E-06$ to about $8.16E-18$. The smaller the ratio $a/d$ is, the better the asymptotic formula \eqref{eq2.1.22} approximates $E$. \begin{equation}gin{table}[htbp] \centering \caption{Error of the asymptotic solution $E$ when $M=27$ and $d=(1.0E-07)$ cm.} \begin{equation}gin{tabular}{ccccc} \toprule \multicolumn{5}{c}{M=27, d=1.0E-07} \\ \midrule a & 1.00E-08 & 1.00E-09 & 1.00E-10 & 1.00E-11 \\ a/d & 1.00E-01 & 1.00E-02 & 1.00E-03 & 1.00E-04 \\ Norm of E & 5.20E+00 & 5.20E+00 & 5.20E+00 & 5.20E+00 \\ Error of E & 8.16E-06 & 8.16E-10 & 8.16E-14 & 8.16E-18 \\ \bottomrule \end{tabular} {\langle}bel{tab1} \end{table} \begin{equation}gin{figure}[htbp] \centering \includegraphics[scale=0.95]{M27} \caption{Error of the asymptotic solution $E$ when $M=27$ and $d=(1.0E-07)$ cm.} {\langle}bel{fig1} \end{figure} Table \ref{tab2} and Figure \ref{fig2} show the results of solving the problem with $M=1000$ particles, when the distance between neighboring particles is $d=(1.0E-07)$ cm, and with different radius $a$. From these table and figure, one can see that the relative error of the asymptotic solution in this case is also very small, less than $3.02E-04$, when the ratio $a/d < 1.0E-01$. In this case, the error of the asymptotic $E$ is greater than that of the previous case when $M=27$. However, this time, the error is also decreasing quickly and linearly when the ratio $a/d$ decreases from $1.0E-01$ to $1.0E-04$. Therefore, the asymptotic formula \eqref{eq2.1.22} for the solution $E$ is applicable when $a \ll d$. \begin{equation}gin{table}[htbp] \centering \caption{Error of the asymptotic solution $E$ when $M=1000$ and $d=(1.0E-07)$ cm.} \begin{equation}gin{tabular}{ccccc} \toprule \multicolumn{5}{c}{M=1000, d=1.0E-07} \\ \midrule a & 1.00E-08 & 1.00E-09 & 1.00E-10 & 1.00E-11 \\ a/d & 1.00E-01 & 1.00E-02 & 1.00E-03 & 1.00E-04 \\ Norm of E & 3.16E+01 & 3.16E+01 & 3.16E+01 & 3.16E+01 \\ Error of E & 3.02E-04 & 3.02E-08 & 3.02E-12 & 3.02E-16 \\ \bottomrule \end{tabular} {\langle}bel{tab2} \end{table} \begin{equation}gin{figure}[htbp] \centering \includegraphics[scale=0.95]{M1000} \caption{Error of the asymptotic solution $E$ when $M=1000$ and $d=(1.0E-07)$ cm.} {\langle}bel{fig2} \end{figure} \section{Conclusions} {\langle}bel{sec4} In this paper, we present a numerical method for solving the EM wave scattering by one and many small perfectly conducting bodies. One of the advantages of this method is that it is relatively easy to implement. Furthermore, one can get an asymptotically exact solution to the problem when the characteristic size of the bodies tends to zero. To illustrate the applicability and efficiency of the method, we use it to solve the EM wave scattering problem by one and many small perfectly conducting bodies. Numerical results of these experiments are presented and error analysis of the asymptotic solutions for the case of one and many bodies are also discussed. For the case of one small body, one can always find the exact solution using the described method. For the case of many small bodies, the accuracy of our method is high if $a \ll d \ll {\langle}mbda$. The problem of EM wave scattering is much harder to treat, compared to scalar wave scattering \cite{Nakayama1981, Ito1985, TMaterial, TFastScalar}. 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\begin{document} \bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}is{\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}isplaystyle} \begin{center} { {\sc Stability, NIP, and NSOP; Model Theoretic Properties of Formulas \\ via Topological Properties of Function Spaces }} { {\sc Karim Khanaki}} {\footnotesize Department of science, Arak University of Technology, \\ P.O. Box 38135-1177, Arak, Iran; e-mail: khanaki@arakut.ac.ir \\ School of Mathematics, Institute for Research in Fundamental Sciences (IPM), \\ P.O. Box 19395-5746, Tehran, Iran; e-mail: khanaki@ipm.ir} \end{center} {\sc Abstract.} {\small We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the `almost definability' and `Baire~1 definability' of coheirs assuming NIP. We show that a formula $\phi(x,y)$ has the strict order property if and only if there is a convergent sequence of continuous functions on the space of $\phi$-types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-\v{S}mulian theorem.} {\small{\sc Keywords}: Talagrand's stability, independence property, coheir, strict order property, continuous~logic, relative weak compactness, angelic space.} AMS subject classification: 03C45, 03C90, 46E15, 46A50 \noindent\hrulefill {\small \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ableofcontents \noindent\hrulefill \section{Introduction} \label{1} In \cite{Ros} Rosenthal introduced the independence property for families of real-valued functions and used this property for proving his celebrated $l^1$ theorem: a Banach space is either `good' (every bounded sequence has a weak-Cauchy subsequence) or `bad' (contains an isomorphic copy of $l^1$). After this and another work of Rosenthal \cite{Ros2}, Bourgain, Fremlin and Talagrand \cite{BFT} found some topological and measure theoretical criteria for the independence property and proved that the space of functions of the first Baire class on a Polish space is angelic; a topological notion for which the terminology was introduced by Fremlin. This theorem asserts that a set of continuous functions on a Polish space is either `good' (its closure is precisely the set of limits of its sequences) or `bad' (its closure contains non-measurable functions). In fact these dichotomies correspond to the NIP/IP dichotomy in continuous logic; see Fact~\ref{BFT} below. In this paper we propose a generalization of Shelah's dividing lines for classification of first order theories which deals with real-valued formulas instead of 0-1~valued formulas. The principal aim of this paper is to study and characterize some model theoretic properties of formulas, such as OP, IP and SOP, in terms of topological and measure theoretical properties of function spaces. This study enables us to obtain new results and to reach a better understanding of the known results. Let us give the background and our own point of view. In Shelah's stability theory, the set-theoretic criteria lead to ranks or combinatorial properties of a particular formula. There are known interactions between some of these combinatorial properties and some topological properties of function spaces. As an example, a formula $\phi(x,y)$ has the order property (OP) if there exist $a_ib_i,i<\omegaega$ such that $\phi(a_i,b_j)$ holds if and only if $i<j$. One can assume that $\phi$ is a 0-1 valued function, such that $\phi(a,b)=1$ iff $\phi(a,b)$ holds. Then $\phi$ has the order property iff there exist $a_i,b_j$ such that $\lim_i\lim_j\phi(a_i,b_j)=1\neq 0=\lim_j\lim_i\phi(a_i,b_j)$. Thus failure of the order property, or stability, is equivalent to the requirement that the double limits $\lim_i\lim_j\phi$ and $\lim_j\lim_i\phi$ be the same. Using a crucial result due to Eberlein and Grothendieck, the latter is a topological property of a family of functions; see Fact~\ref{Fact2} below. Similarly, using the result of Bourgain, Fremlin and Talagrand mentioned above, one can obtain some topological and measure theoretical characterizations of NIP formulas. Therefore, it seems reasonable that one studies real-valued formulas and hopes to obtain new classes of functions (formulas) and develop a sharper stability theory by making use of topological properties of function spaces instead of only combinatorial properties of formulas. In this paper (except in Section 4) we work in continuous logic which is an extension of classical first order logic; thus our results hold in the latter case. The following is a summary of the main results of this paper: Propositions \ref{SCP->NSOP}, \ref{NSOP=SCP}, \ref{Shelah=Eberlein}, Theorems \ref{NIP-compactness} and \ref{almost-dfn} are new results. Also, Definitions \ref{NIP-formula} and \ref{universal dfn} are new. Propositions \ref{NIP-almost}, \ref{NIP-dfn}, \ref{Keisler-NIP} and Theorem \ref{Baire-dfn} have not previously been published but are essentially just translations from functional analysis. Section~3 focuses on NIP in the framework of continuous logic and Section~4 focuses on SOP in classical model theory. This is not the end of the story if one defines a notion of non-forking extension in NIP theories such that it satisfies symmetry and transitivity. Moreover, one can study sensitive families of functions, dynamical systems and chaotic maps and their connections with stability theory. We will study them in a future work. It is worth recalling another line of research. After the preparation of the first version of this paper, we came to know that simultaneously in \cite{Iba14} and \cite{S2} the relationship between NIP and Rosenthal's dichotomy was noticed in the contexts of $\bar {a}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\b{\bar {b}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\c{\bar {c}lphaeph_0$-categorical structures in continuous logic and classical first order setting, respectively. Independently, the relationship between NIP in integral logic and Talagrand's stability was studied in \cite{K}. This paper is organized as follows: In the second section, we briefly review continuous logic and stability. In the third section, we study Talagrand's stability and its relationship with NIP in logic, and give some characterizations of NIP in terms of measure and topology. The result of Bourgain, Fremlin and Talagrand is used in this section for proving of definability of coheirs in NIP theories. In the fourth section, we study the SOP and point out the correspondence between Shelah's theorem and the Eberlein-\v{S}mulian theorem. \noindent {\bf Acknowledgements.} I am very much indebted to Professor David H. Fremlin for his kindness and his helpful comments. I am grateful to M\'{a}rton Elekes for valuable comments and observations, particularly Example~\ref{exa} below. I thank the anonymous referees for their detailed suggestions and corrections; they helped to improve significantly the exposition of this paper. I would like to thank the Institute for Basic Sciences (IPM), Tehran, Iran. Research partially supported by IPM grant 93030032.} \noindent\hrulefill \section{Continuous Logic} \label{2} In this section we give a brief review of continuous logic from \cite{BU} and \cite{BBHU}. Results stated without proof can be found there. The reader who is familiar with continuous logic can skip this section. \subsection{Syntax and semantics} \label{syntax} A {\em language} is a set $L$ consisting of constant symbols and function/relation symbols of various arities. To each relation symbol $R$ is assigned a bound $\flat_R\in[0,\infty)$ and we assume that its interpretations is bounded by $\flat_R$. It is always assumed that $L$ contains the metric symbol $d$ and $\flat_d=1$. We use $\mathbb{R}$ as value space and its common operations $+,\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes$ and scalar products as connectives. Moreover to each relation symbol $R$ (function symbol $F$) is assigned a modulus of uniform continuity $\Delta_R$ ($\Delta_F$). We also use the symbols `sup' and `inf' as quantifiers. Let $L$ be a language. {\em $L$-terms} and their bound are inductively define as follows: \begin{itemize} \item Constant symbols and variables are terms. \item If $F$ is a $n$-ary function symbol and $t_1, \ldots,t_n$ are terms, then $F(t_1,\ldots, t_n)$ is a term. All $L$-terms are constructed in this way. \end{itemize} \begin{dfn} $L$-formulas and their bounds are inductively defined as follows: \begin{itemize} \item Every $r\in\mathbb{R}$ is an atomic formula with bound $|r|$. \item If $R$ is a $n$-ary relation symbol and $t_1,\ldots,t_n$ are terms, $R(t_1,\ldots,t_n)$ is an atomic formula with bound $\flat_R$. \item If $\phi,\psi$ are formula and $r\in\mathbb{R}$ then $\phi+\psi,\phi\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes\psi$ and $r\phi$ are formulas with bound resp $\flat_\phi+\flat_\psi, \flat_\phi\flat_\psi, |r|\flat_\phi$. \item If $\phi$ is a formula and $x$ is a variable, $\sup_x\phi$ and $\inf_x\phi$ are formulas with the same bound as $\phi$. \end{itemize} \end{dfn} \begin{dfn} A {\em prestructure} in $L$ is pseudo-metric space $(M, d)$ equipped with: \begin{itemize} \item for each constant symbol $c\in L$, an element $c^M\in M$ \item for each $n$-ary function symbol $F$ a function $F^M : M^n\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o M$ such that $$d_n^M(\bar x,\bar y)\leqslant\Delta_F(\epsilonilon) \Longrightarrow d^M(F^M(\bar x),F^M(\bar y))\leqslant\epsilonilon$$ \item for each $n$-ary relation symbol $R$ a function $R^M : M^n\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o [-\flat_R,\flat_R]$ such that $$d_n^M(\bar x,\bar y)\leqslant\Delta_R(\epsilonilon) \Longrightarrow |R^M(\bar x)-R^M(\bar y)|\leqslant\epsilonilon.$$ \end{itemize} \end{dfn} If $M$ is a prestructure, for each formula $\phi(\bar x)$ and $\bar a\in M$, $\phi^M(\bar a)$ is defined inductively starting from atomic formulas. In particular, $(\sup_y \phi)^M(\bar a)=\sup_{b\in M}\phi^M(\bar a, b)$. Similarly for $\inf_y\phi$. \begin{proposition} Let $M$ be an $L$-prestructure and $\phi(\bar x)$ a formula with $|\bar x|=n$. Then $\phi^M(\bar x)$ is a real-valued function on $M^n$ with a modulus of uniform continuity $\Delta_\phi$ and $|\phi^M(\bar a)|\leqslant\flat_\phi$ for every $\bar a$. \end{proposition} Interesting prestructures are those which are {\em complete} metric spaces. They are called {\em $L$-structures}. Every prestructure can be easily transformed to a complete $L$-structure by first taking the quotient metric and then completing the resulting metric space. By uniform continuity, interpretations of function and relation symbols induce well-defined function and relations on the resulting metric space. \subsection{Compactness, types, stability} \label{compactness} Let $L$ be a language. An expression of the form $\phi\leqslant\psi$, where $\phi,\psi$ are formulas, is called a {\em condition}. The equality $\phi=\psi$ is called a condition again. These conditions are called closed if $\phi,\psi$ are sentences. A {\em theory} is a set of closed conditions. The notion $M\models T$ is defined in the obvious way. $M$ is then called a model of $T$. A theory is {\em satisfiable} if has a model. An ultraproduct construction can be defined. The most important application of this construction in logic is to prove the \L o\'{s} theorem and to deduce the compactness theorem. \begin{thm}[Compactness Theorem] Let $T$ be an $L$-theory and $\mathcal{C}$ a class of $L$-structures. Suppose that $T$ is finitely satisfiable in $\mathcal{C}$. Then there exists an ultraproduct of structures from $\mathcal{C}$ that is a model of $T$. \end{thm} There are intrinsic connections between some concepts from functional analysis and continuous logic. For example, types are well known mathematical objects, {\em Riesz homomorphisms}. To illustrate this, there are two options; Gelfand representation of $C^*$-algebras, and Kakutani representation of $M$-spaces. We work in a real-valued logic, so we use the latter. Suppose that $L$ is an arbitrary language. Let $M$ be an $L$-structure, $A\subseteq M$ and $T_A=Th({M}, a)_{a\in A}$. Let $p(x)$ be a set of $L(A)$-conditions in free variable $x$. We shall say that $p(x)$ is a {\em type over} $A$ if $p(x)\cup T_A$ is satisfiable. A {\em complete type over} $A$ is a maximal type over $A$. The collection of all such types over $A$ is denoted by $S^{M}(A)$, or simply by $S(A)$ if the context makes the theory $T_A$ clear. The {\em type of $a$ in $M$ over $A$}, denoted by $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{tp}^{M}(a/A)$, is the set of all $L(A)$-conditions satisfied in $M$ by $a$. If $\phi(x,y)$ is a formula, a {\em $\phi$-type} over $M$ is a maximal consistent set of formulas of the form $\phi(x,a)\geqslant r$, for $a\in M$ and $r\in\mathbb{R}$. The set of $\phi$-types over $M$ is denoted by $S_\phi(M)$. The definition of a $\phi$-type over a set $A$ which is not a model needs a few more steps (see Definition~6.6 in \cite{BU}). We now give a characterization of complete types in terms of functional analysis. Let $\mathcal{L}_A$ be the family of all interpretations $\phi^{M}$ in $M$ where $\phi$ is an $L(A)$-formula with a free variable $x$. Then $\mathcal{L}_A$ is an Archimedean Riesz space of measurable functions on $M$ (see \cite{Fremlin3}). Let $\sigmama_A({M})$ be the set of Riesz homomorphisms $I: {\mathcal L}_A\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \mathbb{R}$ such that $I(\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{1}) = 1$, where $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{1}$ is the constant $1$ function on $M$. The set $\sigmama_A({M})$ is called the {\em spectrum} of $T_A$. Note that $\sigmama_A({M})$ is a weak* compact subset of the dual space $\mathcal{L}_A^*$ of $\mathcal{L}_A$. The next proposition shows that a complete type can be coded by a Riesz homomorphism and gives a characterization of complete types. In fact, by the Kakutani representation theorem, the map $S^{M}(A)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\sigmama_A({M})$, defined by $p\mapsto I_p$ where $I_p(\phi^M)=r$ if $\phi(x) = r$ is in $p$, is a bijection. By adapting the proof of Proposition~5.6 of \cite{K}, one can show that: \begin{proposition} \label{key} Suppose that $M$, $A$ and $T_A$ are as above. \begin{itemize} \item [{\em (i)}] The map $S^{M}(A)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\sigmama_A({M})$ defined by $p\mapsto I_p$ is bijective. \item [{\em (ii)}] A set $p$ of $L(A)$-conditions is an element of $S^{M}(A)$ if and only if there is an elementary extension $N$ of $M$ and $a\in N$ such that $p=\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{tp}^{N}(a/A)$. \end{itemize} \end{proposition} We equip $S^{M}(A)=\sigmama_A({M})$ with the related topology induced from $\mathcal{L}_A^*$. Therefore, $S^{M}(A)$ is a compact and Hausdorff space. For any complete type $p$ and formula $\phi$, we let $\phi(p)=I_p(\phi^{M})$. It is easy to verify that the topology on $S^{M}(A)$ is the weakest topology in which all the functions $p\mapsto \phi(p)$ are continuous. This topology is sometimes called the {\em logic topology}. The same things are true for $S_\phi(M)$. \begin{dfn} \label{stab2} A formula $\phi(x,y)$ is called {\em stable in a structure} $M$ if there are no $\epsilonilon>0$ and infinite sequences $a_n, b_n \in M$ such that for all $i<j$: $|\phi(a_i,b_j) - \phi(a_j,b_i)| \geqslant \epsilonilon$. A formula $\phi$ is {\em stable in a theory} $T$ if it is stable in every model of $T$. If $\phi$ is not stable in $M$ we say that it has the {\em order property} (or short the OP). Similarly, $\phi$ has the OP in $T$ if it is not stable in some model of $T$. \end{dfn} It is easy to verify that $\phi(x,y)$ is stable in $M$ if whenever $a_n,b_m\in M$ form two sequences we have $$\lim_n\lim_m\phi(a_n,b_m)=\lim_m\lim_n\phi(a_n,b_m),$$ provided both limits exist. \begin{lem} \label{stable formula} Let $\phi(x,y)$ be a formula. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] The formula $\phi$ is stable. \item [{\em (ii)}] There are no distinct real numbers $r,s$ and infinite sequence $(a_ib_i\colon i < \omegaega)$ such that $\phi(a_i,b_j)=r$ for $i<j$ and $\phi(a_i,b_j)=s$ for $i\geq j$. \end{itemize} \end{lem} By the following result, stability of a formula $\phi(x,y)$ is equivalent to the family of functions being relatively weakly compact. In everything that follows, if $X$ is a topological space then $C_b(X)$ denotes the Banach space of bounded real-valued functions on $X$, equipped with the supremum norm. A subset $A\subseteq C_b(X)$ is relatively weakly compact if it has compact closure in the weak topology on $C_b(X)$. If $X$ is a compact space, then we write $C(X)$ instead of $C_b(X)$. \begin{fct}[\cite{Fremlin4}, Proposition~462E] \label{Grothendieck-lemma} Let $X$ be a compact topological space, and $A$ a subset of $C(X)$. Then $A$ is weakly compact in $C(X)$ iff it is norm-bounded and pointwise compact. \end{fct} In \cite{Gro}, Grothendieck says that the following is based on an idea of Eberlein. (In \cite{Pillay-Grothendieck}, Pillay correctly pointed out this.) \begin{fct}[Eberlein-Grothendieck criterion, \cite{Gro}, Th\'{e}or\`{e}me~6] \label{Criterion} Let $X$ be an arbitrary topological space, $X_0\subseteq X$ a dense subset. Then the following are equivalent for a subset $A\subseteq C_b(X)$: \begin{itemize} \item [{\em (i)}] The set $A$ is relatively weakly compact in $C_b(X)$. \item [{\em (ii)}] The set $A$ is bounded, and for any sequences $\{f_n\}_{1}^\infty\subseteq A$ and $\{x_n\}_{1}^\infty\subseteq X_0$, we have $$\lim_n \lim_m f_n(x_m) =\lim_m \lim_n f_n(x_m),$$ whenever both limits exist. \end{itemize} \end{fct} The following is a model-theoretic version of the Eberlein-Grothendieck criterion, as pointed out by Ben~Yaacov in \cite{Ben-Gro} (see Fact 2, the discussion before Theorem 3 and Theorem 5 therein). \begin{cor} \label{Fact2} Let $M$ be a structure and $\phi(x,y)$ a formula. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $\phi(x,y)$ is stable in $M$. \item [{\em (ii)}] The set $A=\{\phi(x,b):S_x(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \mathbb{R}~|b\in M\}$ is relatively weakly compact in $C(S_x(M))$. \end{itemize} \end{cor} \noindent\hrulefill \section{NIP} \label{4} In this section we study Talagrand's stability and its relationship to NIP in continuous logic. Then, we give some characterizations of NIP in terms of topology and measure, and deduce various forms of definability of coheirs for NIP models. \subsection{Independent family of functions} In \cite{Ros} Rosenthal introduced the independence property for families of real-valued functions and used it for proving his dichotomy. As we will see shortly, this notion corresponds to a generalization of the IP for real-valued formulas. \begin{dfn}[\cite{GM}, Definition~2.8] \label{NIP-family} A family $F$ of real-valued functions on a set $X$ is said to be {\em independent} (or has the {\em independence property}, short IP) if there exist real numbers $s<r$ and a sequence $f_n\in F$ such that for each $k\geqslant 1$ and for each $I\subseteq\{1,\ldots,k\}$, there is $x\in X$ with $f_i(x)\leqslant s$ for $i\in I$ and $f_i(x)\geqslant r$ for $i\notin I$. In this case, sometimes we say that every finite subset of the sequence $f_n$ is shattered by $X$. If $F$ has not the independence property then we say that it has the {\em dependent property} (or the NIP). \end{dfn} We have the following remarkable topological characterizations of this property. More details and several equivalent presentations can be found in \cite{GM}. \begin{fct}[\cite{GM}, Theorem~2.11] \label{NIP-convergence} Let $X$ be a compact space and $F\subseteq C(X)$ a bounded subset. The following conditions are equivalent: \begin{itemize} \item [{\em (i)}] $F$ does not contain an independent sequence. \item [{\em (ii)}] Each sequence in $F$ has a pointwise convergent subsequence in $\mathbb{R}^X$. \end{itemize} \end{fct} \begin{dfn} \label{RSC} We say that a (bounded) family $F$ of real-valued function on a set $X$ has the {\em relative sequential compactness in ${\mathbb{R}}^X$} (short RSC) if every sequence in $F$ has a pointwise convergent subsequence in ${\mathbb{R}}^X$. \end{dfn} As we will see shortly, the following statement is a generalization of a model theoretic fact, i.e. IP implies OP. \begin{fct} \label{IP->OP} Let $X$ be a compact space and $F\subseteq C(X)$ a bounded subset. If $F$ is relatively weakly compact in $C(X)$, then $F$ has the RSC. \end{fct} \begin{proof} Suppose that $F$ is relatively weakly compact in $C(X)$. (Not that, by Fact~\ref{Grothendieck-lemma} above, the weak topology and pointwise topology are the same.) By the Eberlein-\v{S}mulian theorem, each sequence in $F$ has a subsequence converging to an element of $C(X)$. So, in particular, $F$ has the RSC. \end{proof} \subsection{Talagrand's stability and almost NIP} Historically, Talagrand's stability (see Definition~\ref{Talagrand-stable} below), which we call the almost dependence property, arose naturally when Talagrand and Fremlin were studying pointwise compact sets of measurable functions; they found that in many cases a set of functions was relatively pointwise compact because it was almost dependent (see Fact~\ref{almost-NIP} below). Later it appeared that the concept was connected with Glivenko-Cantelli classes in the theory of empirical measures, as explained in \cite{Talagrand}. In this subsection we study this property and show that it is the `correct' counterpart of NIP in integral logic (see \cite{K}). Then, we point out the connection between NIP in continuous logic and this property. \begin{dfn}[Talagrand's stability, \cite{Fremlin4}, 465B] \label{Talagrand-stable} Let $A\subseteq C(X)$ be a pointwise bounded family of real-valued continuous functions on $X$. Suppose that $\mu$ is a measure on $X$. We say that $A$ is {\em $\mu$-stable}, if $A$ is a stable set of functions in the sense of Definition~465B in \cite{Fremlin4}, that is, whenever $E\subseteq M$ is measurable, $\mu(E)>0$ and $s<r$ in $\mathbb{R}$, there is some $k\geqslant 1$ such that $(\mu^{2k})^*D_k(A, E,s,r)<(\mu E)^{2k}$ where \begin{align*} D_k(A, E,s,r) = \bigcup_{f\in A}\big\{w\in & E^{2k}:f(w_{2i})\leqslant s, ~f(w_{2i+1})\geqslant r \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{ for } i<k\big\}. \end{align*} \end{dfn} Now we invoke the first result connecting this notion. First, we need a notion and a notation. If $X$ is any set and $A$ a subset of ${\Bbb R}^X$, then the topology of {\em pointwise convergence} on $A$ is that inherited from the usual product topology of ${\Bbb R}^X$; that is, the coarsest topology on $A$ for which the map $f\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o f(x) : A\mapsto {\Bbb R}$ is continuous for every $x\in X$. We will denote the pointwise closure of $A$ in ${\Bbb R}^X$ by $cl_p(A)$. \begin{fct}[{\cite[465D]{Fremlin4}}] \label{almost-NIP} Let $X$ be a compact Housdorff space and $A\subseteq C(X)$ be a pointwise bounded family of real-valued continuous functions from $X$. Suppose that $\mu$ is a Radon measure on $X$. If $A$ is $\mu$-stable, then $cl_p(A)$ is $\mu$-stable and every element in $cl_p(A)$ is $\mu$-measurable. \end{fct} In \cite{Fremlin75} Fremlin obtained a remarkable result, which has become known as Fremlin's dichotomy: a set of measurable functions on a perfect measure space is either `good' (relatively countably compact for the pointwise topology and relatively compact for the topology of convergence in measure) or `bad' (with neither property). We recall that a subset $A$ of a topological space $X$ is {\em relatively countably compact} if every sequence of $A$ has a cluster point in $X$. \begin{fct}[Fremlin's dichotomy, \cite{Fremlin4}, 463J] Let $(X,\Sigma,\mu)$ be a perfect $\sigmama$-finite measure space, and $\{f_n\}$ a sequence of real-valued measurable functions on $X$. Then \begin{enumerate} \item[] either $\{f_n\}$ has a subsequence which is convergent almost everywhere \item[] or $\{f_n\}$ has a subsequence with no measurable cluster point in $\mathbb{R}^X$. \end{enumerate} \end{fct} We now define the notion of $\mu$-almost NIP and we will see shortly the connection between this notion and NIP. \begin{dfn}[$\mu$-almost NIP] Let $A\subseteq C(X)$ be a pointwise bounded family of real-valued continuous functions on $X$. Suppose that $\mu$ is a measure on $X$. We say that $A$ has the {\em $\mu$-almost NIP}, if every sequence in $A$ has a subsequence which is convergent $\mu$-almost everywhere. \end{dfn} Let $(X,\Sigma,\mu)$ be a finite Radon measure on a compact space $X$, and $\mathcal{L}^0$ the set of all real-valued measurable functions on $X$. Let $A\subseteq \mathcal{L}^0$ be a bounded family. Then we say that $A$ satisfies condition (M), if for all $s<r$ and all $k$, the set $D_k(A,X,r,s)$ is measurable (this applies, in particular, if $A$ is countable). \begin{proposition} \label{NIP-Fremlin} Let $(X,\Sigma,\mu)$ be a finite Radon measure on a compact space $X$, and $A\subseteq \mathcal{L}^0$ a bounded family of real-valued measurable functions on $X$. Consider the following statements. \begin{itemize} \item [{\em (i)}] $A$ is $\mu$-stable. \item [{\em (ii)}] There do not exist measurable set $E$ with $\mu(E)>0$ and $s<r$ in $\mathbb{R}$, such that for each $n$, and almost all $w\in E^n$, for each subset $I$ of $\{1,\ldots,n\}$, there is $f\in A$ with $$f(w_i)<s \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{ if } i\in I \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{ and } f(w_i)>r \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{ if } i\notin I.$$ \item [{\em (iii)}] $A$ has the $\mu$-almost NIP. \end{itemize} Then (i)~$\Rightarrow$~(ii). If $A$ satisfies condition (M), then (ii)~$\Rightarrow$~(i). (i)~$\Rightarrow$~(iii), but (iii)~$\nRightarrow$~(i) and (iii)~$\nRightarrow$~(ii). \end{proposition} \begin{proof} (i)~$\Rightarrow$~(ii) is evident. (M)$\wedge$(ii)~$\Rightarrow$~(i) is Proposition~4 in \cite{Talagrand}. (i)~$\Rightarrow$~(iii): Let $\{f_n\}$ be any sequence in $A$, and take an arbitrary subsequence of it (still denoted by $\{f_n\}$). Let $\mathcal{D}$ be a non-principal ultrafilter on $\mathbb{N}$, and then define $f(x)=\lim_{\mathcal{D}} f_i(x)$ for all $x\in X$. (By the assumption, there is a real number $r$ such that $|h|\leqslant r$ for each $h\in A$, and therefore $f$ is well defined.) Since $A$ is $\mu$-stable and $f\in \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{cl}_p(\{f_n\})$, the function $f$ is measurable (see Fact~\ref{almost-NIP}). So every subsequence of $\{f_n\}$ has a measurable cluster point. Fremlin's dichotomy now tells us that $\{f_n\}$ has a subsequence which is convergent almost everywhere. (iii)~$\nRightarrow$~(i)$\vee$(ii): In \cite{SF} Shelah and Fremlin found that in a model of set theory there is a separable pointwise compact set $A$ of real-valued Lebesgue measurable functions on the unit interval which it is not $\mu$-stable. Thus we see that (iii)~$\nRightarrow$~(i). Since the set $A$ is separable, it satisfies condition (M) and therefore (ii) fails. \end{proof} Professor Fremlin kindly pointed out to us that Shelah's model, described in their paper \cite{SF}, in fact deals with the point that there is a countable set of \emph{continuous} functions which is relatively pointwise compact in ${\mathcal L}^0(\mu)$ for a Radon measure $\mu$, but that it is not $\mu$-stable. Of course, in some cases, there are still some things to say (see Theorem~\ref{NIP-compactness} below). For a Hausdorff space $X$, $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{M}_r(X)$ will be the space of universally measurable functions, i.e. a function $f$ is an element of $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{M}_r(X)$ iff $f$ is $\mu$-measurable for every Radon measure $\mu$ on $X$. \begin{fct}[BFT Criterion, \cite{BFT}, Theorem~2F] \label{BFT} Let $X$ be a compact Hausdorff space, and $F\subseteq C(X)$ be bounded. Then the following are equivalent. \begin{itemize} \item [{\em (i)}] $F$ has the NIP (see Definition~\ref{NIP-family} above). \item [{\em (ii)}] $F$ is relatively compact in $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{M}_r(X)$ for the topology of pointwise convergence. \item [{\em (iii)}] $F$ has the RSC (see Definition~\ref{RSC} above). \item [{\em (iv)}] Each sequence in $F$ has a subsequence which is convergent $\mu$-almost everywhere for every Radon measure $\mu$ on $X$. \item [{\em (v)}] For each Radon measure $\mu$ on $X$, each sequence in $F$ has a subsequence which is convergent $\mu$-almost everywhere. \end{itemize} \end{fct} \begin{proof} The equivalence (i)--(iii) is the equivalence (ii)~$\Leftrightarrow$~(vi)~$\Leftrightarrow$~(iv) of Theorem~2F of \cite{BFT}. (See also Fact~\ref{NIP-convergence} above.) Fremlin's dichotomy and the equivalence (v)~$\Leftrightarrow$~(vi)~$\Leftrightarrow$~(iv) of Theorem~2F of \cite{BFT} imply (v)~$\Leftrightarrow$~(i)~$\Leftrightarrow$~(iv). \end{proof} We will see that the BFT criterion in NIP theories plays a role similar to the role played by the Eberlein-Grothendieck criterion in stable theories. \subsection{NIP in a model} In \cite{Sh} Shelah introduced the independence property (IP) for 0-1 valued formulas; a formula $\phi(x,y)$ has the IP if for each $n$ there exist $b_1,\ldots,b_n$ in the monster model such that each nontrivial Boolean combination of $\phi(x,b_1),\ldots,\phi(x,b_n)$ is satisfiable. By some set-theoretic considerations, a formula $\phi(x,y)$ has IP if and only if $\sup\{|S_\phi(A)|:A\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{ of size }\kappa\}=2^\kappa$ for some infinite cardinal $\kappa$. Although this property was introduced for counting types, its negation (NIP) is a successful extension of local stability and also an active domain of research in classical first order logic and other areas of mathematics. The following generalization of NIP (in the framework of continuous logic) also has a natural topological presentation. \begin{dfn} \label{NIP-formula} Let $M$ be a structure, and $\phi(x,y)$ a formula. We say that $\phi(x,y)$ is NIP on $M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes M$ (or on $M$) if for each sequence $(a_n)\subseteq M$, and $r>s$, there are some {\em finite} disjoint subsets $E,F$ of ${\Bbb N}$ such that $$\Big\{b\in M:\big( \bigwedge_{n\in E}\phi^{\sf M}(a_n,b)\leqslant s\big)\wedge\big(\bigwedge_{n\in F}\phi^{\sf M}(a_n,b)\geqslant r\big)\Big\}=\emptyset.$$ \end{dfn} \begin{lem} \label{equivalence} Let $M$ be a structure, and $\phi(x,y)$ a formula. Then the following are equivalent. \begin{itemize} \item [(i)] $\phi(x,y)$ is NIP on $M$. \item [(ii)] For each sequence $(a_n)\subseteq M$, each saturated elementary extension ${N}\succeq{M}$, and $r>s$, there are disjoint subsets $E,F$ of ${\Bbb N}$ such that $$\Big\{b\in N: \big( \bigwedge_{n\in E}\phi(a_n,b)\leqslant s\big)\wedge\big(\bigwedge_{n\in F}\phi(a_n,b)\geqslant r\big)\Big\}=\emptyset.$$ \item [(iii)] For each sequence $\phi(a_n,y)$ in the set $A=\{\phi(a,y):S_y(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o{\Bbb R}~|~a\in M\}$, where $S_y(M)$ is the space of all complete types on $M$ in the variable $y$, and $r>s$ there are {\em finite} disjoint subsets $E,F$ of ${\Bbb N}$ such that $$\Big\{y\in S_y(M):\big( \bigwedge_{n\in E}\phi(a_n,y)\leqslant s\big)\wedge\big(\bigwedge_{n\in F}\phi(a_n,y)\geqslant r\big)\Big\}=\emptyset.$$ \item [(iv)] The condition (iii) holds for {\em arbitrary} disjoint subsets $E,F$ of ${\Bbb N}$. \item [(v)] Every sequence $\phi(a_n,y)$ in $A$ has a convergent subsequence in ${\Bbb R}^X$, equivalently $A$ has the RSC. \end{itemize} \end{lem} \begin{proof} (ii)~$\Rightarrow$~(i) follows from the compactness theorem and saturation. (Indeed, suppose that (i) fails, and then consider a suitable type and get a contradiction.) (i)~$\Rightarrow$~(iii) is just a restatement of the notion of type. (iv)~$\Rightarrow$~(iii): Suppose that (iii) fails for the sequence $(a_n)\subseteq M$ and $s<r$. Let $E=\{i_n:n\in \Bbb N\}$ and $F=\{j_n:n\in \Bbb N\}$ be arbitrary disjoint subsets of ${\Bbb N}$. Let $E_m=\{i_n:n\leq m\}$ and $F_m=\{j_n:n\leq m\}$ for each $m\in\Bbb N$. So, for each $m$, there is some $y_m\in S_y(M)$ such that $ \bigwedge_{n\in E_m}\phi(a_n,y_m)\leqslant s$ and $\bigwedge_{n\in F_m}\phi(a_n,y_m)\geqslant r$. Let $z$ be a cluster point of the sequence $(y_m)$. (Note that $z\in S_y(M)$ because the type space is compact.) Since the $\phi(a_n,y)$ are continuous, it is easy to verify that $ \bigwedge_{n\in E}\phi(a_n,z)\leqslant s$ and $\bigwedge_{n\in F}\phi(a_n,z)\geqslant r$. As $E,F$ are arbitrary, (iv) fails. (iii)~$\Rightarrow$~(iv) is evident. (iii)~$\Leftrightarrow$~(v): It is easy to verify that for the set $A$, the dependence property in Definition~\ref{NIP-family} is equivalent to the condition (iii). Now, by Fact~\ref{NIP-convergence} the proof is completed. (ii)~$\Leftrightarrow$~(iv) follows from saturation (and the notion of type). \end{proof} Some similar notions are studied in \cite{Iba14}. Note that the notion NIP on a model is `double local', i.e. $\phi$ can be NIP on a model, but not in a theory. If $\phi(x,y)$ is a formula, we let $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\phi}(y, x)= \phi(x,y)$. Hence $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\phi}$ is the same formula as $\phi$, but we have exchanged the role of variables and parameters. \begin{rmk} Let $M$ be a structure and $\phi(x,y)$ a formula. The space $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$ of all $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi$-types on $M$ is the quotient of $S_y(M)$ given by the family of functions $\{\phi(a,y):a\in M\}$ (see \cite{BU}, Fact~4.7). So in Definition~\ref{NIP-formula} above, $S_y(M)$ can be replace by $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. \end{rmk} \begin{thm}[NIP and $\mu$-stability] \label{NIP-compactness} Let $M$ be an $\bar {a}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\b{\bar {b}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\c{\bar {c}lphaeph_0$-saturated $L$-structure, $\phi(x;y)$ a formula, $A=\{\phi(a,y):a\in M\}$ and $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{A}=\{\phi(x,b):b\in M\}$. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $\phi$ is NIP on $M$. \item [{\em (ii)}] $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde A$ is $\mu$-stable for all Radon measures $\mu$ on $S_\phi(M)$. \end{itemize} \end{thm} \begin{proof} (i)~$\Rightarrow$~(ii): By the compactness theorem of continuous logic, since $M$ is $\bar {a}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\b{\bar {b}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\c{\bar {c}lphaeph_0$-saturated and $\phi(x,y)$ is NIP on $M$, there is some integer $n$ such that no subset (of $M$) of size $n$ is shattered by $\phi(x,y)$. We note that by Proposition~465T of \cite{Fremlin4}, the conditions (i) and (ii) of Proposition~\ref{NIP-Fremlin} are equivalent. So if $E\subseteq M$, $\mu(E)>0$, $r>s$, then for each $(a_1,\ldots,a_n)\in E^n$ there is a set $I\subseteq \{1,\ldots,n\}$ such that $$\Big\{y\in S_y(M):\big( \bigwedge_{i\in I}\phi(a_i,y)\leqslant s\big)\wedge\big(\bigwedge_{i\notin I}\phi(a_i,y)\geqslant r\big)\Big\}=\emptyset,$$ where $S_y(M)$ is the space of all complete types on $M$ in the variable $y$. Since $M\subseteq S_y(M)$, the set ${\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde A}$ is $\mu$-stable for every Radon measure $\mu$ on $S_\phi(M)$. (ii)~$\Rightarrow$~(i): Suppose that $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde A$ is $\mu$-stable for every Radon measure $\mu$ on $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{X}=S_\phi(M)$. Thus, by Fact~\ref{almost-NIP}, $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde A$ is relatively compact in $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{M}_r({\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde X})$ (the space of all $\mu$-measurable functions on $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde X$ for each Radon measure $\mu$ on ${\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde X}$). By the BFT criterion, for each sequence $\phi(x,a_n)$ in $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde A$, and $r<s$, there is some $I\subseteq {\mathbb{N}}$ such that $$\Big\{x\in S_x(M):\big( \bigwedge_{i\in I}{\phi}(x,a_i)\leqslant s\big)\wedge\big(\bigwedge_{i\notin I}{\phi}(x,a_i)\geqslant r\big)\Big\}=\emptyset.$$ Thus the dual formula ${\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde \phi}(y,x)$ is NIP on $M$. So, by applying the direction (i)~$\Rightarrow$~(ii) to the formula $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde \phi$, we see that ${\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde A}}=A$ is $\mu$-stable for every Radon measure $\mu$ on $X=S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\phi}}(M)$. Thus, again by the BFT criterion and Proposition~\ref{NIP-Fremlin}, we conclude that $\phi(x,y)$ is NIP on $M$. \end{proof} In fact the proof of the previous result says more: if $M$ is $\bar {a}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\b{\bar {b}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\c{\bar {c}lphaeph_0$-saturated, then $\phi$ is NIP on $M$ if and only if $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde \phi$ is NIP on $M$. \begin{cor} Under the assumptions in Theorem~\ref{NIP-compactness}, $\phi$ is NIP on $M$ if and only if $A$ is $\mu$-stable for every Radon measure $\mu$ on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. \end{cor} The previous results also show why the $\mu$-stability is the `correct' notion of NIP in integral logic (see \cite{K}). The following is a translation of the BFT criterion into (continuous) model theory. Note that here we do not need any saturation conditions on the model. \begin{proposition}[NIP and $\mu$-almost NIP] \label{NIP-almost} Let $M$ be an $L$-structure, $\phi(x;y)$ a formula and $A=\{\phi(a,y):a\in M\}$. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $\phi$ is NIP on $M$. \item [{\em (ii)}] $A$ has the $\mu$-almost NIP for all Radon measures $\mu$ on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. \end{itemize} \end{proposition} \begin{proof} This is the equivalence (i)~$\Leftrightarrow$~(v) of Fact~\ref{BFT} with $X=S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$ and $F=A$. \end{proof} \begin{rmk} One can not expect the notion `NIP on a model' to be symmetrical. It is easy to make examples such that $\phi$ is NIP on $M$ but $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi$ is not NIP on $M$. Of course, if $M$ is $\bar {a}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\b{\bar {b}}\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}ef\c{\bar {c}lphaeph_0$-saturated, they are the same. \end{rmk} \subsection{Almost definable coheirs} \label{3} It is well known that every type on a stable model is definable (see \cite{Ben-Gro}). Here we want to give a counterpart of this fact for NIP theories. In \cite{K} it is shown that if a formula $\phi$ (in integral logic) is $\mu$-stable on a model $M$, then every type in $S_\phi(M)$ is $\mu$-almost definable. We present the notion of `coheir' here. Let $M^*$ be a saturated elementary extension of $M$. A type $p(x)\in S_\phi(M^*)$ is called a {\em coheir (of a type) over $M$} if for every condition $\varphi=0$ in $p(x)$ and every $\epsilonilon>0$, the condition $|\varphi|\leq \epsilonilon$ is satisfiable in $M$. (In classical ($\{0,1\}$-valued) model theory, this means that every formula in $p(x)$ is realized in $M$.) In this case we say that $p$ is $M$-finitely satisfiable. It is easy to verify that a type $p(x)\in S_\phi(M^*)$ is a coheir over $M$ if and only if there are $(a_i\in M: i\in I)$ and an ultrafilter $\mathcal D$ on $I$ such that $\lim_{i,\mathcal D}tp(a_i/M)=p$, where the $\mathcal D$-limit is taken in the logic topology. (For the definition of $\mathcal D$-limit, see Section~5 of \cite{BBHU}.) By Proposition~\ref{key}, $p$ is a coheir over $M$ if and only if there are $(a_i\in M: i\in I)$ and an ultrafilter $\mathcal D$ on $I$ such that $\lim_{i,\mathcal D}I_{p_i}=I_p$, where $p_i=tp(a_i/M)$ and the $\mathcal D$-limit is taken in the weak* topology on $\sigmama_{M^*}(M^*)$. Here we say a function $\psi:X\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \mathbb{R}$ on a topological space $X$ is {\em universally measurable}, if it is $\mu$-measurable for every probability Radon measure $\mu$ on $X$. \begin{dfn} \label{universal dfn} Let $M^*$ be a saturated elementary extension of $M$ and $p(x)\in S_\phi(M^*)$ be a coheir of a type over $M$. We say that a universally measurable function $\psi:S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\mathbb{R}$ {\em defines} $p$ if $\phi(p,a)=\psi(tp_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(a/M))$ for all $a\in M^*$, and in this case we say that $p$ is {\em universally definable}. \end{dfn} The above notion is well defined since every coheir is $M$-finitely satisfiable and so $M$-invariant. \begin{rmk}[\cite{Pillay-Grothendieck}, Remark~2.1] There is a correspondence between the set of all coheirs of types over $M$ and the closure of the set $A=\{\phi^a(y):S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o{\mathbb{R}}~|a\in M\}$, where $\phi^a(q)=\phi(a,q)$ for all $q\in S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Indeed, let $M^*$ be a saturated elemetary extension of $M$. Then for any global $M$-finitely satisfiable $\phi$-type $p(x)\in S_\phi(M^*)$ there is a function $\psi_p$ in the closure $A$ such that $\psi_p(tp_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(a/M))=\phi(p,a)$ for all $a\in M^*$. Indeed, suppose that $tp_\phi(a_i/M^*)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o p$ in the logic topology, where $a_i\in M$. Define $\psi_p(y)=\lim_i\phi(a_i,y)$ for all $y\in S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Now, it is easy to verify that $\psi_p(tp_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(a/M))=\phi(p,a)$ for all $a\in M^*$. To summarize, let $S_\phi^{M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{-fs}}(M^*)$ be the set of all global $M$-finitely satisfiable $\phi$-types (over the monster model $M^*$). Then the map $S_\phi^{M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{-fs}}(M^*)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \overline{A}$, defined by $p\mapsto\psi_p$, is a homeomorphic embedding of $S_\phi^{M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{-fs}}(M^*)$ in the pointwise convergence topology on $\overline{A}\subseteq {\Bbb R}^{S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)}$. \end{rmk} For simplicity, we will write $\phi(p,a)=\phi(p,b)$ where $b=tp_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(a/M)$ and $a\in M^*$. The following is a translation of the BFT criterion: \begin{proposition} \label{NIP-dfn} Let $M$ be a structure and $\phi(x,y)$ a formula. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $\phi$ is NIP on $M$. \item [{\em (ii)}] Every coheir of a $\phi$-type over $M$ is definable by a universally measurable relation $\psi(y)$ over $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. \end{itemize} \end{proposition} \begin{proof} (i)~$\Rightarrow$~(ii): Let $A=\{\phi^a(y):S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o{\mathbb{R}}~|a\in M\}$. By NIP, $A$ is relatively compact in $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extbf{M}_r(S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M))$ (see the BFT criterion). Suppose that $p_{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o p\in S_{\phi}(M^*)$ where $p_{a_i}$ is realized by $a_i\in M$ and $M^*$ is a saturated elementary extension of $M$. (We note that the set of all types realized in $M$ is dense in the set of all coheirs.) Thus $\phi^{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\psi$ pointwise where $\psi$ is universally measurable, and $\psi$ defines $p$. (ii)~$\Rightarrow$~(i): Suppose that $\phi^{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\psi$ pointwise. We can assume that $p_{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o p\in S_{\phi}(M^*)$. Suppose that $p$ is definable by a universally measurable relation $\varphi$, so we have $\psi=\varphi$ on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. So, $\psi$ is measurable for all Radon measures on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Again by the BFT criterion, $\phi$ is NIP. \end{proof} \begin{rmk} \label{invariant-type} In \cite{HP}, the authors showed that, in 0-1 valued logic, every global invariant type admits a Borel definition assuming NIP. This implies, in particular, that every global $M$-invariant type admits a universally measurable definition. Note that Proposition~\ref{NIP-dfn} is an extension of their result to continuous logic for global finitely satisfiable types. Moreover, one can show that $\phi(x,y)$ is NIP on a separable model $M$ if and only if every global $M$-finitely satisfiable $\phi$-type is Baire 1 definable (see also Fact~\ref{Polish-compact}, Theorem~\ref{Baire-dfn} and Remark~\ref{NIP=Baire 1} below). \end{rmk} Here we mention a characterization of NIP in terms of measure algebra. For this, a definition is needed. Let $\phi(x,y)$ be a formula, $r\in\mathbb{R}$ and $a\in M$. By $\{\phi(x,a)\geqslant r\}$ we denote the set $\{p\in S_\phi(M):\phi(p,a)\geqslant r \}$. The set $\{ \phi(x,a)\leqslant r\}$ has the obvious meaning. The measure algebra generated by $\phi$ on $S_\phi(M)$ is the measure algebra generated by all sets of the forms $\{\phi(x,a)\geqslant r\}$ and $\{\phi(x,b)\leqslant s\}$ where $a,b\in M$ and $r,s\in \mathbb{R}$. One can assume that all $r,s$ are rational numbers. Now, a straightforward translation of the proof for classical first order theories, as can be found in \cite[Theorem~3.14]{Keisler}, implies that: \begin{proposition} \label{Keisler-NIP} Let $T$ be a theory and $\phi(x,y)$ a formula. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $\phi$ is NIP. \item [{\em (ii)}] For every sufficiently saturated model $M$, each Radon measure on $S_\phi(M)$ has a countably generated measure algebra (which is the measure algebra generated by $\phi$). \end{itemize} \end{proposition} Now we are going to give another characterization of NIP. First we need some definitions. Let $\psi$ be a measurable function on $(S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M),\mu)$ where $\mu$ is a probability Radon measure on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Then $\psi$ is called an {\em almost ${\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}$-definable relation over $M$} if there is a sequence $g_n:S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \mathbb{R}$, $|g_n|\leqslant |{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}|$, of continuous functions such that $\lim_n g_n(p)=\psi(p)$ for almost all $p\in S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. (We note that by the Stone-Weierstrass theorem every continuous function $g_n:S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \mathbb{R}$ can be expressed as a uniform limit of algebraic combinations of (at most countably many) functions of the form $p\mapsto {\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(p, b)$, $b\in M$.) An almost $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi$-definable relation $\psi(y)$ over $M$ defines a coheir $p(x) \in S_\phi(M^*)$ (of a $\phi$-type over $M$) if the set $A_0\subseteq S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$ is measurable and $\mu(A_0)=1$, where $A_0=\{b\in S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M):\phi(p,b) =\psi(b)\}$. In this case we say that $p$ is ($\mu$-)\emph{almost definable}. It is easy to check that almost definability is well defined. Suppose that every coheir $p$ is almost definable by a measurable function $\psi^p$. Then, we say that $p$ is {\em almost equal to $q$}, denoted by $p\equiv_\mu q$, if $\psi^p=\psi^q$ $\mu$-almost everywhere. For a coheir $p(x)$, define $[p]_\mu=\{q\in S_\phi(M^*): p\equiv_\mu q \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{ and $q$ is a coheir}\}$ and $[S_\phi]_\mu(M)=\{[p]_\mu:p\in S_\phi(M^*) \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ext{ is a coheir}\}$. Then $[S_\phi]_\mu(M)$ has a natural topology which is defined by metric $d([p]_\mu,[q]_\mu)=\int|\psi^p-\psi^q|d\mu$ for coheirs $p,q\in S_\phi(M^*)$. Recall that the density character of a topological space $X$, is the least infinite cardinal number of a dense subset of $X$. When measuring the size of a structure we will use its density character (as a metric space), denoted $\|M\|$, rather than its cardinality. Similarly, since $[S_\phi]_\mu(M)$ is a metric space, we measure the size $[S_\phi]_\mu(M)$ by its density character $\|[S_\phi]_\mu(M)\|$. \begin{thm}[Almost definability of coheirs] \label{almost-dfn} Let $T$ be a theory and $\phi(x,y)$ a formula. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $\phi$ is NIP. \item [{\em (ii)}] For every model $M$ and measure $\mu$ on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$, every coheir of a type over $M$ is $\mu$-almost definable, and $\|[S_{\phi}]_\mu(M)\|\leqslant\|M\|$. \end{itemize} \end{thm} \begin{proof} (i)~$\Rightarrow$~(ii): Suppose that the coheir $p\in S_\phi(M^*)$ is definable by a universally measurable relation $\psi$ on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Let $\mu$ be a Radon measure on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Then there is a sequence $g_n$ of continuous functions on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$ such that $g_n\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \psi$ in $L^1(\mu)$ (see \cite[7.9]{Folland}), and hence a subsequence (still denoted by $g_n$) that converges to $\psi$ $\mu$-almost everywhere. So $p$ is $\mu$-almost definable. Moreover, by the Stone-Weierstrass theorem, $\|C(S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M))\|\leqslant\|M\|$. Now, since $C(S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M))$ is dense in $L^1(\mu)$ (again see \cite[7.9]{Folland}), $\|L^1(\mu)\|\leqslant\|M\|$. By definition, $\|[S_{\phi}]_\mu(M)\|\leqslant\|M\|$ and the proof is completed. (ii)~$\Rightarrow$~(i): Let $p\in S_\phi(M^*)$ be a coheir of a type over $M$. Suppose that $p_{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o p$ where $p_{a_i}$ is realized by $a_i\in M$. Then the function $\psi(y)=\lim_i\phi(a_i,y)$ is measurable for all Radon measures on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. Indeed, by definition, for each Radon measure $\mu$, there is a measurable function $\psi_\mu$ such that $\psi_\mu(b)=\phi(p,b)$ $\mu$-almost everywhere. Since $\mu$ is Radon (and so is complete), and $\psi=\psi_\mu$ almost everywhere, $\psi$ is $\mu$-measurable (see \cite[2.11]{Folland}). Then, by Proposition~\ref{NIP-dfn}, the proof is completed. \end{proof} \subsection{Baire 1 definable types} More results can be reached, if one works in a separable model. Let $X$ be a Polish space. A function $f:X\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o{\mathbb{R}}$ is of Baire class 1 if it can be written as the pointwise limit of a sequence of continuous functions. The set of Baire class 1 functions on $X$ is denoted by $B_1(X)$. \begin{fct}[BFT Criterion for Polish spaces, \cite{BFT}, Corollary~4G] \label{Polish-compact} Let $X$ be a Polish space, and $A\subseteq C(X)$ pointwise bounded set. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $A$ is relatively compact in $B_1(X)$. \item [{\em (ii)}] $A$ is relatively sequentially compact in ${\mathbb{R}}^X$, or $A$ has the RSC. \end{itemize} \end{fct} Fremlin's notion of an angelic topological space is as follows: a regular Hausdorff space X is {\em angelic} if (i) every relatively countably compact set in $X$ is relatively compact, (ii) the closure of a relatively compact set is precisely the set of limits of its sequences. The following is the principal result of \cite{BFT}. \begin{fct}[\cite{BFT}, Theorem~3F] \label{Polish-angelic} If $X$ is a Polish space, then $B_1(X)$ is angelic under the topology of pointwise convergence. \end{fct} Let $M$ be a structure and $\phi(x,y)$ a formula. A Baire class 1 function $\psi:S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o{\mathbb{R}}$ defines $p\in S_\phi(M)$ if $\phi(p,b)=\psi(b)$ for all $b\in M$. We say $p$ is Baire 1 definable if some Baire class 1 function $\psi$ defines it. The following is another criterion for NIP. \begin{thm}[Baire 1 definability of types] \label{Baire-dfn} Let $\phi(x,y)$ be a NIP formula on a separable model $M$. Then every $p\in S_{\phi}(M)$ is definable by a Baire 1 function $\psi(y)$ on $S_{\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi}(M)$. \end{thm} \begin{proof} The proof is an easy consequence of Fact~\ref{Polish-compact}. Suppose that $p_{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o p\in S_{\phi}(M)$ where $a_i\in M$. (Recall that the set of all types realized in $M$ is dense in $S_{\phi}(M)$.) For each $a\in M$, define $\phi^a:M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o {\Bbb R}$ by $\phi^a(b)=\phi(a,b)$. Since $\phi$ is NIP, the set $\hat{A}=\{\phi(a,y):S_y(M)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o{ \mathbb{R}}:a\in M\}$ is relatively sequentially compact in ${\Bbb R}^{S_y(M)}$, and in particular the set $A=\{\phi^a:a\in M\}$ is relatively sequentially compact in ${\Bbb R}^M$. Now by Fact~\ref{Polish-compact}, since $M$ is Polish, also $A$ is relatively compact in $B_1(M)$. Thus, there is a $\psi\in B_1(M)$ such that $\phi^{a_i}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o \psi$, so $p$ is definable by a Baire class 1 function. Moreover, since $B_1(M)$ is angelic, there is some sequence $\phi^{a_n},a_n\in M$ such that $\phi^{a_n}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\psi$. \end{proof} \begin{rmk} \label{NIP=Baire 1} Note that one can say more: $\phi$ is NIP on $M$ if and only if every coheir is Baire 1 definable. This is discussed in detail in \cite{KP}. \end{rmk} \noindent\hrulefill \section{SOP} \label{5} In this section we work in the classical logic. One reason for restricting our attention to the classical case is to make this section more accessible to model-theorists and other interested readers. In \cite{Sh} Shelah introduced the strict order property as complementary to the independence property: a theory has OP iff it has IP or SOP. In functional analysis, the Eberlein-\v{S}mulian theorem states that a subset of a Banach space is not relatively weakly compact iff it has a sequence without any weak Cauchy subsequence or it has a weak Cauchy sequence with no weak limit. In fact there is a correspondence between the Eberlein-\v{S}mulian theorem and Shelah's result above. To determine this correspondence, we first give a topological description of the strict order property, and then study the above dividing line. In classical ($\{0,1\}$-valued) model theory a formula $\phi(x,y)$ has the {\em strict order property} (or short SOP) if there exists a sequence $(a_i:i<\omegaega)$ in the monster model $\mathcal{U}$ such that for all $i<\omegaega$, $$\phi({\mathcal{U}},a_i)\subsetneqq\phi({\mathcal U},a_{i+1}).$$ The acronym SOP stands for the strict order property and NSOP is its negation. We can assume that $\phi(x,y)$ is a 0-1 valued function on $\mathcal U$ such that $\phi(a,b)=1$ iff $\models\phi(a,b)$. Then $\phi(x,y)$ has the strict order property if and only if there are sequences $(a_i,b_j:i,j<\omegaega)$ in $\mathcal U$ such that for each $b\in{\mathcal U}$, the sequence $\{\phi(b,a_i)\}_i$ is increasing -- therefore the pointwise limit $\psi(x):=\lim_i\phi(x,a_i)$ is well-defined -- and $\phi(b_j,a_j)<\phi(b_j,a_{j+1})$ for all $j<\omegaega$. Now, suppose that the $\phi(x,a_i)$ are continuous functions on $S_\phi({\mathcal U})$, the space of all complete $\phi$-types. Suppose that $\phi$ has not the SOP, and $\phi(x,a_i)\nearrow\psi(x)$. Then $\psi:S_\phi({\mathcal U})\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\{0,1\}$ is continuous, because there is a $k$ such that $\phi(x,a_k)=\phi(x,a_{k+1})=\cdots$. Conversely, suppose that $\phi(x,a_i)\nearrow\psi(x)$ and $\psi$ is continuous. It is a standard fact that an increasing sequence of continuous functions on a compact space which converges to a continuous function converges uniformly (Dini's Theorem). Therefore, our sequence is eventually constant, because the logic is 0-1 valued. Therefore, it seems right to say that the SOP in classical logic is equivalent to the existence of a pointwise convergent sequence (not necessary increasing) of continuous functions such that its limit is not continuous. Our next goal is to convince the reader that by a technical consideration this is indeed the case. In functional analysis, a Banach space $X$ is called {\em weakly sequentially complete} if every weak Cauchy sequence has a weak limit. Similarly we define the following notion and will observe that this notion corresponds to NSOP on the model-theoretic side. \begin{dfn} Let $X$ be a topological space and $F\subseteq C(X)$. We say that $F$ has the {\em weak sequential completeness property} (or short SCP) if the limit of each pointwise convergent sequence $\{f_n\}\subseteq F$ is continuous. \end{dfn} As we will see shortly, the following statement is a generalization of a well known model theoretic fact, i.e. SOP implies OP. \begin{fct} \label{SOP->OP} Let $X$ be a compact space and $F\subseteq C(X)$ a bounded subset. If $F$ is relatively weakly compact in $C(X)$, then $F$ has the SCP. \end{fct} \begin{proof} Suppose that $F$ is relatively weakly compact in $C(X)$, and $\{f_n\}$ is a sequence in $F$ which pointwise converges to $f$. Since the pointwise topology and weak topology are the same (see Fact~\ref{Grothendieck-lemma} above), so $f$ as a cluster point of $\{f_n\}$ is continuous. \end{proof} The next result is another application of the Eberlein-Grothendieck criterion: \begin{thm} \label{nip+scp=stable} Let $X$ be a compact space and $A\subseteq C(X)$ be bounded. Then $A$ is relatively weakly compact in $C(X)$ iff it has RSC and SCP. \end{thm} \begin{proof} First we show that $cl_p(A)\subseteq C(X)$ if every sequence of $A$ has a convergent subsequence in ${ \mathbb{R}}^X$ and the limit of every convergent sequence of $A$ is continuous. Suppose that $A$ has RSC and SCP. Let $\{f_n\}_n\subseteq A$ and $\{a_m\}_m\subseteq X$, and suppose that the double limits $\lim_m\lim_n f_n(a_m)$ and $\lim_n\lim_m f_n(a_m)$ exist. Let $a$ be a cluster point of $\{a_m\}_m$. By RSC, there is a convergent subsequence $f_{n_k}$ such that $f_{n_k}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o f$. Therefore $\lim_m\lim_{n_k} f_{n_k}(a_m)=\lim_mf(a_m)$ and $\lim_{n_k}\lim_m f_{n_k}(a_m)=\lim_{n_k}f_{n_k}(a)=f(a)$. By SCP, $\lim_mf(a_m)=f(a)$. Since the double limits exist, it is easy to verify that $\lim_m\lim_n f_{n}(a_m)=\lim_m\lim_{n_k} f_{n_k}(a_m)$ and $\lim_n\lim_m f_{n}(a_m)=\lim_{n_k}\lim_m f_{n_k}(a_m)$. So $A$ has the double limit property and thus it is relatively weakly compact in $C(X)$. The converse follows from Facts~\ref{IP->OP} and \ref{SOP->OP}. \end{proof} \begin{proposition} \label{SCP->NSOP} If the set $\{\phi(x,a):a\in\mathcal{U}\}$ has the SCP, then $\phi(x,y)$ is NSOP. \end{proposition} \begin{proof} Suppose, for a contradiction, that $\{\phi(x,a):a\in\mathcal{U}\}$ has the SCP and $\phi$ is SOP. By SOP, there are $(a_ib_i:i<\omegaega)$ in the monster model $\mathcal U$ such that $\phi({\mathcal U},a_i)\leqslant\phi({\mathcal U},a_{i+1})$ and $\phi(b_j,a_i)<\phi(b_i,a_j)$ for all $i<j$. Let $b$ be a cluster point of $\{b_i\}_{i<\omegaega}$. By SCP, $\phi(S_\phi({\mathcal U}),a_i)\nearrow\psi$ and $\psi$ is continuous. But $\lim_i\lim_j\phi(b_j,a_i)=0<1=\lim_i\lim_j\phi(b_i,a_j)$ and by continuity $\psi(b)<\psi(b)$, a contradiction. \end{proof} The following example shows that the converse does not hold in analysis. It was suggested to us by M\'{a}rton Elekes. \begin{exa} \label{exa} Let $X$ be the Cantor set. Let $H=\{0\}\cup(X\cap(2/3,1))$. (We note that $H$ is $\Delta_2^0$, i.e. it is $F_\sigmama$ and $G_\bar {d}}\def\x{\bar {x}}\def\y{\bar {y}elta$ at the same time, but neither open nor closed.) Then it is easy to see that there exists a sequence $H_n$ of clopen subsets of $X$ such that if $f_n$ is the characteristic function of $H_n$ and $f$ is the characteristic function of $H$ then $f_n\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o f$ pointwise. Let $A =\{f_n :n<\omegaega\}$. Then all $f_n$ are continuous, uniformly bounded (even 0-1 valued), the pointwise closure is $A\cup\{f\}$ (which are all Baire class 1 functions), and all monotone sequences in $A$ are eventually constant: indeed, if there were a true monotone subsequence then its limit would be the characteristic function of an open or a closed set, but $H$ is neither open nor closed. Also, we note that $A$ has the RSC but it is not relatively weakly compact in $C(X)$. \end{exa} Again we give a topological presentation of a model theoretic property. For this, we need some definitions. Let $M$ be a saturated enough structure and $\phi:M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes M\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\{0,1\}$ a formula. For subsets $B,D\subseteq M$, we say that $\phi(x,y)$ has the {\em order property on } $B\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes D$ (short OP on $B\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes D$) if there are sequences $(a_i)\subseteq B$, $(b_i)\subseteq D$ such that $\phi(a_i,b_j)$ holds if and only if $i<j<\omegaega$. We will say that $\phi(x,y)$ has the {\em NIP on $B\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes D$}, if for the set $A=\{\phi(a,y):S_y(D)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\{0,1\}~|a\in B\}$, any of the cases in Lemma~\ref{equivalence} holds. \begin{proposition} \label{NSOP=SCP} Suppose that $T$ is a theory. Then the following are equivalent: \begin{itemize} \item [{\em (i)}] $T$ is NSOP. \item [{\em (ii)}] For each indiscernible sequence $(a_n)_{n<\omegaega}$ and formula $\phi(x,y)$, if the sequence $(\phi(x,a_n))_{n<\omegaega}$ pointwise converges on $S_\phi(\mathcal{U})$, then its limit is continuous. \end{itemize} \end{proposition} \begin{proof} (i) $\Rightarrow$ (ii): Suppose that there are an indiscernible sequence $(a_n)_{n<\omegaega}$ and a formula $\phi(x,y)$ such that the sequence $(\phi(x,a_n))_{n<\omegaega}$ pointwise converges but its limit is not continuous. Since the limit is not continuous, $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\phi}(y,x)=\phi(x,y)$ has OP on $\{a_n\}_{n<\omegaega}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes S_\phi({\mathcal U})$. Since every sequence in $\{\phi(x,a_n)\}_{n<\omegaega}$ has a pointwise convergent subsequence, $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\phi}(y,x)$ is NIP on $\{a_n\}_{n<\omegaega}\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}imes S_\phi({\mathcal U})$. The following argument is classic (see \cite{Poi} and \cite{S}). Since $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde{\phi}(y,x)$ has OP, there is a sequence $\{b_N\}\subseteq S_\phi({\mathcal U})$ such that $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,b_N)$ holds if and only if $i<N$. By NIP, there is some integer $n$ and $\eta : n \rightarrow \{0,1\}$ such that $\bigwedge_{i<n} \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x)^{\eta(i)}$ is inconsistent. (Recall that for a formula $\varphi$, we use the notation $\varphi^0$ to mean $\neg\varphi$ and $\varphi^1$ to mean $\varphi$.) Starting with that formula, we change one by one instances of $\neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x) \wedge \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i+1},x)$ to $\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x) \wedge \neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i+1},x)$. Finally, we arrive at a formula of the form $\bigwedge_{i<N} \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x) \wedge \bigwedge_{N\leq i<n} \neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x)$. The tuple $b_N$ satisfies that formula. Therefore, there is some $i_0<n$, $\eta_0 : n \rightarrow \{0,1\}$ such that $$\bigwedge_{i\neq i_0, i_0+1} \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x)^{\eta_0(i)} \wedge \neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i_0},x) \wedge \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i_0+1},x)$$ is inconsistent, but $$\bigwedge_{i\neq i_0, i_0+1} \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x)^{\eta_0(i)} \wedge \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i_0},x) \wedge \neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i_0+1},x)$$ is consistent. Let us define $\varphi(\bar a,x)=\bigwedge_{i\neq i_0,i_0+1} \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x)^{\eta_0(i)}$. Increase the sequence $(a_i : i<\omegaega)$ to an indiscernible sequence $(a_i:i\in \mathbb Q)$. Then for $i_0 \leq i<i' \leq i_0+1$, the formula $\varphi(\bar a,x) \wedge \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x) \wedge \neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i'},x)$ is consistent, but $\varphi(\bar a,x) \wedge \neg\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_i,x) \wedge \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(a_{i'},x)$ is inconsistent. Thus the formula $\psi(x,y) = \varphi(\bar a,x) \wedge \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}ilde\phi(y,x)$ has the strict order property. (ii) $\Rightarrow$ (i): Suppose that the formula $\phi(x,y)$ has SOP as witnessed by a sequence $(a_nb_n:n<\omegaega)$. Then the formula $\psi(y_1,y_2)=\forall x(\phi(x,y_1)\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\phi(x,y_2))$ defines a continuous pre-order for which the sequence $(a_n:n<\omegaega)$ forms an infinite chain. Replace $(a_n)_{n<\omegaega}$ by an indiscernible sequence $(c_n)_{n<\omegaega}$, and return to $\phi(x,y)$. Therefore, $\phi(x,y)$ has SOP as witnessed by the sequence $(c_nb_n:n<\omegaega)$. Now, $\phi(S_\phi({\mathcal U}),c_n)\nearrow\varphi$ but $\varphi$ is not continuous. \end{proof} We now provide a proof of Shelah's theorem (\cite{Sh}, Theorem~4.1). \begin{cor} \label{Shelah-continuous} Suppose that $T$ is NIP and NSOP. Then $T$ is stable. \end{cor} \begin{proof} Let $\phi(x,y)$ be a formula, $(a_n)_{n<\omegaega}$ an indiscernible sequence, and $(b_n)_{n<\omegaega}$ an arbitrary sequence. Suppose that the double limits $\lim_m\lim_n\phi(b_n,a_m)$ and $\lim_n\lim_m\phi(b_n,a_m)$ exist. By NIP, there is a convergent subsequence $\phi(x,a_{m_k})$ such that $\phi(x,a_{m_k})\bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}o\psi(x)$ on $S_\phi(\mathcal{U})$. Therefore, $\lim_n\lim_k\phi(b_n,a_{m_k})=\lim_n\psi(b_n)$ and $\lim_k\lim_n\phi(b_n,a_{m_k})=\lim_k\phi(b,a_{m_k})=\psi(b)$ where $b$ is a cluster point of $\{b_n\}$. By NSOP, $\lim_n\psi(b_n)=\psi(b)$. So the double limits are the same and thus $T$ is stable. (Compare Theorem~\ref{nip+scp=stable}.) \end{proof} \subsection*{Theorems of Eberlein-\v{S}mulian and Shelah} The well known compactness theorem of Eberlein and \v{S}mulian says that relative compactness, relative sequential compactness and relative countable compactness are equivalent for the weak topology of a Banach space. Now, we show the correspondence between Shelah's theorem and the Eberlein-\v{S}mulian theorem. \begin{proposition} \label{Shelah=Eberlein} Suppose that $X$ is a space of the form $S_\phi(M)$ and $A=\{\phi(a,y):a\in M\}$ where $M$ is a sufficiently saturated model of a theory $T$ and $\phi(x,y)$ a formula. Then the following are equivalent. \begin{itemize} \item [{\em (i)}] {\bf The Eberlein-\v{S}mulian theorem:} For every $A\subseteq C(X)$, the following statements are equivalent: \begin{itemize} \item [{\em (a)}] The weak closure of $A$ is weakly compact in $C(X)$. \item [{\em (b)}] Each sequence of elements of $A$ has a subsequence that is weakly convergent in $C(X)$. \end{itemize} \item [{\em (ii)}] {\bf Shelah's theorem:} The following statements are equivalent: \begin{itemize} \item [{\em (a$'$)}] $T$ is stable. \item [{\em (b$'$)}] $T$ has the NIP and the NSOP. \end{itemize} \end{itemize} \end{proposition} \begin{proof} First, we note that by the Eberlein-Grothendieck criterion, (a)~$\Leftrightarrow$~(a$'$). It suffices to show that (b)~$\Leftrightarrow$~(b$'$). Suppose that $(f_n)$ is a sequence of the form $(\phi(a_n,y))$ where $(a_n)$ is an indiscernible sequence. By (b), there is a subsequence $(f_{n_k})$ that is convergent. Therefore, $T$ has NIP. Again by (b), its limit is continuous, so $T$ has NSOP, and (b$'$) holds. Conversely, suppose that $T$ has NIP and NSOP. Let $(f_n)$ be a sequence of the form $(\phi(a_n,y))$ where $(a_n)$ is an arbitrary sequence. By NIP, $(f_n)$ has a convergent subsequence $(f_{n_k})$. Replace $(a_n)$ by an indiscernible sequence $(c_n)$. Then, by NSOP, $f=\lim_kf_{n_k}$ is continuous. So, (b) holds. \end{proof} To summarize: $$\begin{array}{cccccc} \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{Logic:~~~~~~~~~} & \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{Stable} & \Longleftrightarrow & \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{NIP} & + & \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{ NSOP} \\ \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{ } & & & & & \\ \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{Analysis:~~~~~} & \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{Weakly Compact} & \Longleftrightarrow & \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{RSC} & + & \bar {t}}\def\u{\bar {u}}\def\e{\mathbf {e}extrm{SCP} \end{array}$$ Of course, the Eberlein-\v{S}mulian theorem is proved for arbitrary Banach spaces (even normed spaces), but it follows easily from the case $C(X)$ (see \cite{Fremlin4}, Theorem~462D). On the other hand, the above argument implicitly shows that countable compactness implies compactness. Earlier we defined angelic topological spaces. Roughly an angelic space is one for which the conclusions of the Eberlein-\v{S}mulian theorem hold. By the previous observations one can say that `first order logic is angelic.' \noindent\hrulefill \end{document}

\begin{document} \title{Brownian Bridge Asymptotics for the Subcritical Bernoulli Bond Percolation.} \author{Yevgeniy Kovchegov\\ \small{Email: yevgeniy@math.stanford.edu}\\ \small{Fax: 1-650-725-4066}} \maketitle \begin{abstract} For the $d$-dimensional model of a subcritical bond percolation ($p<p_c$) and a point $\mathbf{\vec{a}}$ in $\mathbb{Z}^d$, we prove that a cluster conditioned on connecting points $(0,...,0)$ and $n \mathbf{\vec{a}}$ if scaled by $\frac{1}{n \| \mathbf{\vec{a}} \|}$ along $\mathbf{\vec{a}}$ and by $\frac{1}{\sqrt{n}}$ in the orthogonal direction converges asymptotically to Time $\times$ ($d-1$)-dimensional Brownian Bridge. \end{abstract} \section{Introduction.} \subsection{Percolation and Brownian Bridge.} We begin by briefly stating the notion of a bond percolation based on the material rigorously presented in \cite{grimmett} and \cite{kesten}, and the notion of the Brownian Bridge as well as the word description of the result connecting the two that we have obtained and made the primary objective of this paper. \textbf{Percolation:} For each edge of the $d$-dimensional square lattice $\mathbb{Z}^d$ in turn, we declare the edge $open$ with probability $p$ and $closed$ with probability $1-p$, independently of all other edges. If we delete the closed edges, we are left with a random subgraph of $\mathbb{Z}^d$. A connected component of the subgraph is called a ``cluster", and the number of edges in a cluster is the ``size" of the cluster. The probability $\theta (p)$ that the point $(0,0)$ belongs to a cluster of an infinite size is zero if $p=0$, and one if $p=1$. However, there exists a critical probability $0< p_c <1$ such that $\theta (p) = 0$ if $p < p_c$ and $\theta (p) >0$ if $p > p_c$. In the first case, we say that we are dealing with a $subcritical$ percolation model, and in the second case, we say that we are dealing with a $supercritical$ percolation model. \textbf{Brownian Bridge:} defined as a sample-continuous Gaussian process $B^0$ on $[0,1]$ with mean $0$ and $\bold{E} B^0_s B^0_t = s(1-t)$ for $0 \leq s \leq t \leq 1$. So, $B^0_0 = B^0_1 = 0$ a.s. Also, if $B$ is a Brownian motion, then the process $B_t - tB_1$ ($0 \leq t \leq 1$) is a Brownian Bridge. For more details see \cite{bill}, \cite{dudley} and \cite{durrett}. The process $B^{0, \mathbf{\vec{a}}}_t \equiv B^0_t +t \mathbf{\vec{a}}$ is a Brownian Bridge connecting points zero and $\vec{\mathbf{a}}$. \textbf{History of the problem:} Below, we consider the $d$-dimensional model of a subcritical bond percolation ($p<p_c$) and a point $\mathbf{\vec{a}}$ in $\mathbb{Z}^d$, conditioned on the event of zero being connected to $n \mathbf{\vec{a}}$. We first show that a specifically chosen path connecting points zero and $n \mathbf{\vec{a}}$ and going through some appropriately defined points on the cluster (regeneration points), if scaled $\frac{1}{n \| \mathbf{\vec{a}}\|}$ times along $\mathbf{\vec{a}}$ and $\frac{1}{\sqrt{n}}$ times in the direction orthogonal to $\mathbf{\vec{a}}$, converges to Time $\times$ ($d-1$)-dimensional Brownian Bridge as $n \rightarrow +\infty$, where the scaled interval connecting points zero and $n \mathbf{\vec{a}}$ serves as a $[0,1]$ time interval. In other words, we prove that a scaled ``skeleton" going through the regeneration points of the cluster converges to Time $\times$ ($d-1$)-dimensional Brownian Bridge. In a subsequent step, we show that if scaled, then the hitting area of the orthogonal hyper-planes shrinks, implying that for $n$ large enough, all the points of the scaled cluster are within an $\varepsilon$-neighborhood of the points in the ``skeleton". One of the major tools used in this research was the renewal technique developed in \cite{acc}, \cite{ccc}, \cite{ioffe}, \cite{cc} and \cite{ioffe1} as part of the derivation of the Ornstein-Zernike estimate for the subcritical bond percolation model and ``similar" processes. A major result related to the study is that for $\mathbf{\vec{a}}=(1,0,...,0)$, the hitting distribution of the cluster in the intermediate planes, $x_1 =tn\mathbf{\vec{a}}$, $0<t<1$ obeys a multidimensional local limit theorem (see \cite{ccc}). Dealing with all other $\mathbf{\vec{a}} \not= (k,0,...,0)$ became possible only after the corresponding technique further mastering the regeneration structures and equi-decay profiles was developed in \cite{ioffe} and \cite{ioffe1}. This technique played a central role in obtaining the research results. \subsection{Asymptotic Convergence.}\label{intro:2} Here we state a version of a local CLT and a technical result that we later prove.\\ \textbf{Local Limit Theorem:} In this paper we are going to use the version of the local CLT borrowed from \cite{durrett}: Let $X_1, X_2,... \in \mathbb{R}$ be i.i.d. with $\bold{E} X_i =0$, $\bold{E} X_i^2 = \sigma ^2 \in (0, \infty)$, and having a common lattice distribution with span $h$. If $S_n = X_1 + ... + X_n$ and $P[X_i \in b + h \mathbb{Z}] = 1$ then $P[S_n \in nb + h \mathbb{Z}] = 1$. We put $$p_n(x) = P[S_n / \sqrt{n} =x] \mbox{ for } x \in \Lambda_n = \{ (nb + hz) / \sqrt{n} \mbox{ : } z \in \mathbb{Z} \} $$ and $$n(x)= (2 \pi \sigma^2)^{-1/2} \exp(-x^2/2\sigma^2) \mbox{ for } x \in (-\infty, \infty)$$ \begin{CLT} Under the above hypotheses, $\sup_{x \in \Lambda_n} |{\frac{\sqrt{n}}{h} p_n(x) - n(x)}| \rightarrow 0$ as $n \rightarrow \infty$. \end{CLT} \textbf{Technical Result Concerning Convergence to the Brownian Bridge} (to be used in Chapter 2.4, and is proved in Chapter 3.2): The following technical result is going to be proved in the section \ref{bb} of this paper, however since we are to use it in the section \ref{percol}, we will state the result below as part of the introduction. Let $X_1, X_2,...$ be i.i.d. random variables on $\mathbb{Z}^d$ with the span of the lattice distribution equal to one (see \cite{durrett}, section 2.5), and let there be a $\bar{\lambda} > 0$ such that the moment-generating function $$\bold{E}(e^{\theta \cdot X_1}) <\infty$$ for all $\theta \in B_{\bar{\lambda}}$. \\ Now, for a given vector $\mathbf{\vec{a}} \in \mathbb{Z}^d$, let $X_1+...+X_i =[t_i, Y_i]_f \in \mathbb{Z}^d$ when written in the new orthonormal basis such that $\mathbf{\vec{a}} = [\| \mathbf{\vec{a}} \|, 0]_f$ (in the new basis $[\cdot , \cdot ]_f \in \mathbb{R} \times \mathbb{R}^{d-1}$). Also let $P[\mathbf{\vec{a}} \cdot X_i]>0]=1$. We define the process $[t, Y_{n,k}^*(t)]_f$ to be the interpolation of $0$ and $[\frac{1}{n \| \mathbf{\vec{a}} \|}t_i, \frac{1}{\sqrt{n}}Y_i]_f^{i=0,1,...,k}$, in Section 2.2 we will show that \begin{TECHthm} The process $$\{ Y^*_{n,k}\mbox{ for some } k \mbox{ such that } [t_k, Y_k]_f =n \mathbf{\vec{a}} \}$$ conditioned on the existence of such $k$ converges weakly to the Brownian Bridge (of variance that depends only on the law of $X_1$). \end{TECHthm} \section{The Main Result in Subcritical Percolation.} \label{percol} In this section we work only with subcritical percolation probabilities $p<p_c$. \subsection{Preliminaries.} Here we briefly go over the definitions that one can find in Section 4 of \cite{ioffe}. \\ We start with the inverse correlation length $\xi_p(\vec{x})$: $$\xi_p(\vec{x}) \equiv -\lim_{n \rightarrow \infty} \frac{1}{n} P_p (0 \leftrightarrow [n\vec{x}]),$$ where the limit is always defined due to the FKG property of the Bernoulli bond percolation (see \cite{grimmett}). Now, $\xi_p(\vec{x})$ is the support function of the compact convex set $$ \bold{K}^p \equiv \bigcap_{\vec{n} \in \mathbb{S}^{d-1}} \lbrace \vec{r} \in \mathbb{R}^d \mbox{ : } \vec{r} \cdot \vec{n} \leq \xi_p(\vec{n}) \rbrace ,$$ with non-empty interior int\{$\bold{K}^p$\} containing point zero. \\ Let $\mathbf{\vec{r}} \in \partial \bold{K}^p$, and let $\vec{e}$ be a basis vector such that $\vec{e} \cdot \mathbf{\vec{r}}$ is maximal. For $\vec{x},\vec{y} \in \mathbb{Z}^d$ define $$S^r_{\vec{x},\vec{y}} \equiv \lbrace \vec{z} \in \mathbb{R}^d \vert \mathbf{\vec{r}} \cdot \vec{x} \leq \mathbf{\vec{r}} \cdot \vec{z} \leq \mathbf{\vec{r}} \cdot \vec{y} \rbrace .$$ Note that $S^r_{\vec{x},\vec{y}} = \emptyset$ if $ \mathbf{\vec{r}} \cdot \vec{y} < \mathbf{\vec{r}} \cdot \vec{x} $. \\ Let $\bold{C}^r_{\vec{x},\vec{y}}$ denote the corresponding common open cluster of $x$ and $y$ when we run the percolation process on $S^r_{\vec{x},\vec{y}}$. \begin{Def} For $\vec{x},\vec{y} \in \mathbb{Z}^d$ lets define $h_r$-connectivity $\lbrace \vec{x} \leftarrow^{h_r} \rightarrow \vec{y} \rbrace$ of $\vec{x}$ and $\vec{y}$ to be the event that\\ 1. $\vec{x}$ and $\vec{y}$ are connected in the restriction of the percolation configuration to the slab $S^r_{\vec{x},\vec{y}}$. \\ 2. If $\vec{x} \not= \vec{y}$, then $\bold{C}^r_{\vec{x},\vec{y}} \bigcap S^r_{\vec{x},\vec{x}+\vec{e}} = \lbrace \vec{x},\vec{x}+\vec{e} \rbrace$ and $\bold{C}^r_{\vec{x},\vec{y}} \bigcap S^r_{\vec{y}-\vec{e},\vec{y}} = \lbrace \vec{y}-\vec{e},\vec{y} \rbrace$. \\ 3. If $\vec{x}=\vec{y}$ and all the edges adjoined to $\vec{x}$ and perpendicular to $\vec{e}$ are closed. \end{Def} Set $$h_r(\vec{x}) \equiv P_p \lbrack 0 \leftarrow^{h_r} \rightarrow \vec{x} \rbrack .$$ Notice that $h_r(0) = (1-p)^{2(d-1)}$. \begin{Def} For $\vec{x},\vec{y} \in \mathbb{Z}^d$ lets define $f_r$-connectivity $\lbrace \vec{x} \leftarrow^{f_r} \rightarrow \vec{y} \rbrace$ of $\vec{x}$ and $\vec{y}$ to be the event that\\ 1. $\vec{x} \not= \vec{y}$\\ 2. $\vec{x} \leftarrow^{h_r} \rightarrow \vec{y}$ .\\ 3. For no $\vec{z} \in \mathbb{Z}^d \setminus \lbrace \vec{x},\vec{y} \rbrace $ both $ \lbrace \vec{x} \leftarrow^{h_r} \rightarrow \vec{z} \rbrace \mbox{ and } \lbrace \vec{z} \leftarrow^{h_r} \rightarrow \vec{y} \rbrace$ take place. \end{Def} Set $$f_r(\vec{x}) \equiv P_p \lbrack 0 \leftarrow^{f_r} \rightarrow \vec{x} \rbrack .$$ Notice that $f_r(0)=0$. \begin{Def} Suppose $0 \leftarrow^{h_r} \rightarrow \vec{x}$, we say that $\vec{z} \in \mathbb{Z}^d$ is \textbf{a regeneration point} of $\bold{C}_{0,\vec{x}}^r$ if \\ 1. $\mathbf{\vec{r}} \cdot \vec{e} \leq \mathbf{\vec{r}} \cdot \vec{z} \leq \mathbf{\vec{r}} \cdot (\vec{y}-\vec{e}) $ \\ 2. $S_{\vec{z}-\vec{e},\vec{z}+\vec{e}}^r \bigcap \bold{C}_{0,\vec{x}}^r$ contains exactly three points: $\vec{z}-\vec{e}$, $\vec{z}$ and $\vec{z}+\vec{e}$, where $\vec{e}$ is defined as before.\\ Let also $\vec{x}$ itself be a regeneration point. \end{Def} The following Ornstein-Zernike equality is due to be used soon: \begin{thm*} $\exists$ $A(\cdot, \cdot)$ on $(0,p_c) \times \mathbf{S}^{d-1}$ s. t. \begin{eqnarray} \label{oz} P_p[0 \leftrightarrow \vec{x}] = \frac{A(p, n(\vec{x}))} {\|\vec{x}\|^{\frac{d-1}{2}} } e^{-\xi_p(\vec{x})} (1+o(1)) \end{eqnarray} uniformly in $\vec{x} \in \mathbb{Z}^d$, where $n(\vec{x}) \equiv \frac{\vec{x}}{\|\vec{x}\|}$. \end{thm*} We refer to \cite{ioffe} for the proof of the theorem. \subsection{Measure $Q_{r_0}^r(x)$.} It had been proved in section 4 of \cite{ioffe} that for a given $\mathbf{\vec{r}}_0 \in \partial \bold{K}^p$ there exists $\bar{\lambda} >0$ such that $$F_{r_0}(\mathbf{\vec{r}})= \frac{1}{(1-p)^{2(d-1)}} \sum_{x \in \mathbb{Z}^d} f_{\mathbf{\vec{r}}_0}(x) e^{\mathbf{\vec{r}} \cdot \vec{x}} =1 \mbox{ whenever } \mathbf{\vec{r}} \in B_{\bar{\lambda}}(\mathbf{\vec{r}}_0) \bigcap \partial \bold{K}^p$$ and therefore $$Q_{r_0}^r(\vec{x}) \equiv \frac{1}{(1-p)^{2(d-1)}}f_{r_0}(\vec{x}) e^{\mathbf{\vec{r}} \cdot \vec{x}} \mbox{ is a measure on } \mathbb{Z}^d .$$ Also, it was shown that $$\mu=\mu_{r_0}(\mathbf{\vec{r}}) \equiv \bold{E}_{r_0}^rX = \sum_{\vec{x} \in \mathbb{Z}^d}\vec{x}Q_{r_0}^r(\vec{x}) = \nabla_r logF_{r_0}(\mathbf{\vec{r}}) \not= 0$$ and $$F_{r_0}(\mathbf{\vec{r}}) < \infty \mbox{ for all } \mathbf{\vec{r}} \mbox{ in } B_{\bar{\lambda}}(\mathbf{\vec{r}}_0).$$ The later implies $$F_{r_0}(\mathbf{\vec{r}}) =\sum_{\vec{x} \in \mathbb{Z}^d}f_{r_0}(\vec{x}) e^{\mathbf{\vec{r}} \cdot \vec{x}} = \sum_{\vec{x} \in \mathbb{Z}^d}Q_{r_0}^{r_0}(\vec{x}) e^{\theta \cdot \vec{x}} < \infty$$ for $\theta = \mathbf{\vec{r}}-\mathbf{\vec{r}}_0 \in B_{\bar{\lambda}}(0)$,\\ i.e. the moment generating function $\bold{E}_{r_0}^{r_0} (e^{\theta \cdot X_1})$ of the law $Q_{r_0}^{r_0}$ is finite for all $\theta \in B_{\bar{\lambda}}(0)$.\\ Now, there is a renewal relation (see section 1 and section 4 of \cite{ioffe}), $$ h_{r_0}(\vec{x})=\frac{1}{(1-p)^{2(d-1)}} \sum_{\vec{z} \in \mathbb{Z}^d} f_{r_0}(\vec{z})h_{r_0}(\vec{x}-\vec{z}) \mbox{ with } h_{r_0}(0)=(1-p)^{2(d-1)}$$ and therefore $$h_{r_0}([N \mu])=(1-p)^{2(d-1)}e^{-r \cdot [N \mu]} \sum_{k} \bigotimes_1^k Q_{r_0}^r(X_1+...+X_k=[N \mu]) \mbox{ for } N>0 ,$$ where $X_1,X_2,...$ is a sequence of i.i.d. random variables distributed according to $Q_{r_0}^r$, as $h_{r_0}$-connection is a chain of $f_{r_0}$-connections with junctions at the regeneration points of $\bold{C}_{0,x}^{r_0}$. \\ \subsection{Important Observation.} The probability that $0 \leftarrow^{h_{r_0}} \rightarrow x$ with exactly $k$ regeneration points $x_1, x_1+x_2, ... , \sum_{i=1}^k x_i =x$ \begin{eqnarray} \label{imp} P_X & \equiv & P[0 \leftarrow^{h_{r_0}} \rightarrow x \mbox{ ; regeneration points: } x_1, x_1+x_2, ... , \sum_{i=1}^k x_i =x] \nonumber \\ & = & \frac{1}{(1-p)^{2(d-1)(k-1)}} P[0 \leftarrow^{f_{r_0}} \rightarrow x_1] P[x_1 \leftarrow^{f_{r_0}} \rightarrow x_1+x_2]... P[\sum_{i=1}^{k-1} x_i \leftarrow^{f_{r_0}} \rightarrow \sum_{i=1}^k x_i =x] \nonumber \\ & = & \frac{1}{(1-p)^{2(d-1)(k-1)}} f_{r_0}(x_1)f_{r_0}(x_2)...f_{r_0}(x_k). \end{eqnarray} \subsection{The Result.} In this section we fix $\bold{\vec{a}} \in \mathbb{Z}^d$, and let $\bold{r}=\bold{r}_0= \bold{\vec{a}} \mathbb{R}^+\bigcap \partial \bold{K}^p$. Then we recall that $$\bold{E}_{r_0}^r (e^{\theta \cdot X_1}) < \infty$$ for all $\theta \in B_{\bar{\lambda}}(0)$. We also denote $h(x) \equiv h_{r_0}(x)$ and $f(x) \equiv f_{r_0}(x)$. \\ First, we introduce a new basis $\{ \vec{f_1},\vec{f_2},..., \vec{f_d} \}$, where $\vec{f_1} = \frac{\bold{\vec{a}}}{\| \bold{\vec{a}} \|}$. We use $[\cdot,\cdot]_f \in \mathbb{R} \times \mathbb{R}^{d-1}$ to denote the coordinates of a vector with respect to the new basis. Obviously $\mathbf{\vec{a}}=[\|\mathbf{\vec{a}} \|, 0]_f$. We want to prove that the process corresponding to the last $d-1$ coordinates in the new basis of the scaled ($\frac{1}{n \| \bold{\vec{a}} \|}$ times along $\bold{\vec{a}}$ and $\frac{1}{\sqrt{n}}$ times in the orthogonal d-1 dimensions) interpolation of regeneration points of $\bold{C}_{0, n \bold{\vec{a}}}^{r_0}$ conditioned on ${0 \leftarrow^{h}\rightarrow n \bold{\vec{a}}}$ converges weakly to the Brownian Bridge $B^o(t)$ (with variance that depends only on measure $Q_{r_0}^r$) where $t$ represents the scaled first coordinate in the new basis.\\ Let $X_1, X_2,...$ be i.i.d. random variables distributed according to $Q_{r_0}^r$ law. We interpolate $0,X_1,(X_1+X_2),...,(X_1+...+X_k)$ and scale by $\frac{1}{n\|\mathbf{\vec{a}} \|} \times \frac{1}{\sqrt{n}}$ along $<\mathbf{\vec{a}}> \times <\mathbf{\vec{a}}>^{\bot}$ to get the process $[t, Y^*_{n,k}(t)]_f$. The technical theorem (see Chapters (\ref{intro:2}) and (\ref{bb:gen})) implies the following \begin{thm} The process $$\{ Y^*_{n,k}\mbox{ for some } k \mbox{ such that } X_1+...+X_k= n \bold{\vec{a}} \}$$ conditioned on the existence of such $k$ converges weakly to the Brownian Bridge (with variance that depends only on measure $Q_{r_0}^r$). \end{thm} Now, let for $y_1,...,y_k \in \mathbb{Z}^d$ with positive increasing first coordinates $\gamma (y_1,...,y_k)$ be the last $(d-1)$ coordinates in the new basis of the scaled ($\frac{1}{n \|\mathbf{\vec{a}}\|} \times \frac{1}{\sqrt{n}}$) interpolation of points $0,y_1,...,y_k$ (where the first coordinate is time). Notice that $\gamma (y_1,...,y_k) \in C_o[0,1]^{d-1}$ as a function of scaled first coordinate whenever $y_k= n \bold{\vec{a}}$. \\ By the important observation (\ref{imp}) we've made before, for any function $F(\cdot )$ on $C[0,1]^{d-1}$,\\ \\ $\sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} F(\gamma (x_1, x_1+x_2, ... , \sum_{i=1}^k x_i))$ $$ \times P[0 \leftarrow^{h_{r_0}} \rightarrow x \mbox{ ; regeneration points: } x_1, x_1+x_2, ... , \sum_{i=1}^k x_i =x]$$ $$=\sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} F(\gamma (x_1, x_1+x_2, ... , \sum_{i=1}^k x_i)) \frac{1}{(1-p)^{2(d-1)(k-1)}} f(x_1)...f(x_k)$$ $$= (1-p)^{2(d-1)} e^{-r \cdot n \bold{\vec{a}}} \sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} F(\gamma (x_1, x_1+x_2, ... , \sum_{i=1}^k x_i)) Q_{r_0}^r(x_1)...Q_{r_0}^r(x_k).$$ \\ Therefore, for any $A \subset C[0,1]^{d-1}$\\ \\ $P_p[ \gamma (\mbox{regeneration points}) \in A \mbox{ } | \mbox{ } 0 \leftarrow^{h}\rightarrow n \bold{\vec{a}} ]$ $$ =\frac{ \sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} I_A (\gamma (x_1, x_1+x_2, ... , \sum_{i=1}^k x_i)) \frac{1}{(1-p)^{2(d-1)(k-1)}} f(x_1)...f(x_k) } { \sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} \frac{1}{(1-p)^{2(d-1)(k-1)}} f(x_1)...f(x_k) }$$ $$ =\frac{\sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} I_A (\gamma (x_1, x_1+x_2, ... , \sum_{i=1}^k x_i)) Q_{r_0}^r(x_1)...Q_{r_0}^r(x_k)} {\sum_k \sum_{x_1+...+x_k= n \bold{\vec{a}}} Q_{r_0}^r(x_1)...Q_{r_0}^r(x_k)}$$ $$= P[Y^*_{n,k} \in A \mbox{ for the } k \mbox{ such that } X_1+...+X_k= n \bold{\vec{a}} \mbox{ } | \mbox{ } \exists k \mbox{ such that } X_1+...+X_k= n \bold{\vec{a}}] .$$ Hence, we have proved the following \begin{cor} The process corresponding to the last $d-1$ coordinates (in the new basis $\{ \vec{f_1},\vec{f_2},...,\vec{f_d} \}$) of the scaled $({\frac{1}{n \|\mathbf{\vec{a}} \|} \times \frac{1}{\sqrt{n}} })$ interpolation of regeneration points of $\bold{C}_{0,n \bold{\vec{a}}}^{r_0}$ (where the first coordinate is time) conditioned on ${0 \leftarrow^{h}\rightarrow n \bold{\vec{a}}}$ converges weakly to the Brownian Bridge (with variance that depends only on measure $Q_{r_0}^r$). \end{cor} \subsection{Shrinking of the Cluster. Main Theorem.} Here for $\bold{\vec{a}} \in \mathbb{Z}^d$ we let $\bold{r}_0= \bold{\vec{a}} \mathbb{R}^+\bigcap \partial \bold{K}^p$ again. Before we proceed with the proof that the scaled percolation cluster $\bold{C}_{0,n \bold{\vec{a}}}^{r_0}$ shrinks to the scaled interpolation skeleton of regeneration points, we need to prove the following \begin{prop} If $\mathbf{\vec{r}} = \nabla \xi_p(\mathbf{\vec{r}}_0)$ then $Q_{r_0}^r$ is a probability measure. \end{prop} \begin{proof} First we notice that $\mathbf{\vec{r}}_0 \cdot \mathbf{\vec{r}} = \mathbf{\vec{r}}_0 \cdot \nabla \xi_p(\mathbf{\vec{r}}_0) = D_{\mathbf{\vec{r}}_0}(\xi_p(\mathbf{\vec{r}}_0)) = \xi_p(\mathbf{\vec{r}}_0)$, and thus $$H_{r_0}(\mathbf{\vec{r}}) \equiv \frac{1}{(1-p)^{2(d-1)}} \sum_{\vec{x} \in \mathbb{Z}^d} h_{r_0}(x)e^{\mathbf{\vec{r}} \cdot \vec{x}} \geq \sum_{\vec{x} \in <\mathbf{\vec{a}}> \cap \mathbb{Z}^d} h_{r_0}(x)e^{\mathbf{\vec{r}} \cdot \vec{x}} = \sum_{\vec{x} \in <\mathbf{\vec{a}}> \cap \mathbb{Z}^d} h_{r_0}(x)e^{\xi_p(\vec{x})} = +\infty$$ for $d \leq 3$ by Ornstein-Zernike equation (\ref{oz}). For all other $d$ we sum over all $\vec{x}$ inside a small enough cone around $\mathbf{\vec{a}}$ to get $H_{r_0}(\mathbf{\vec{r}}) = +\infty$.\\ Now, for all $\vec{n} \in \mathbb{S}^{d-1}$, $\vec{n} \cdot \nabla \xi_p(\mathbf{\vec{r}}_0) = D_{\vec{n}} \xi_p(\mathbf{\vec{r}}_0) \leq \xi_p(\vec{n})$ by convexity of $\xi_p$, and therefore $\mathbf{\vec{r}} = \nabla \xi_p(\mathbf{\vec{r}}_0) \in \partial \mathbf{K}^p$. Notice that due to the strict convexity of $\xi_p$ and the way $\mathbf{K}^p$ was defined, $\mathbf{\vec{r}} = \nabla \xi_p(\mathbf{\vec{r}}_0)$ is the only point on $\partial \mathbf{K}^p$ such that $\mathbf{\vec{r}}_0 \cdot \mathbf{\vec{r}} = \xi_p(\mathbf{\vec{r}}_0)$.\\ Now, Ornstein-Zernike equation (\ref{oz}) also implies that the sums $H_{r_0}(\tilde{r})$ and $F_{r_0}(\tilde{r})$ are finite whenever $\tilde{r} \in \alpha \bold{K}^p = \bigcap_{\vec{n} \in \mathbb{S}^{d-1}} \lbrace \vec{r} \in \mathbb{R}^d \mbox{ : } \vec{r} \cdot \vec{n} \leq \alpha \xi_p(\vec{n}) \rbrace $ with $\alpha \in (0,1)$, and due to the recurrence relation of $f_{r_0}$ and $h_{r_0}$ connectivity functions, $H_{r_0}(\tilde{r}) = \frac{1}{1 -F_{r_0}(\tilde{r})}$ (see \cite{ioffe}). Therefore $F_{r_0}(\mathbf{\vec{r}}) \equiv \frac{1}{(1-p)^{2(d-1)}} \sum_{\vec{x} \in \mathbb{Z}^d} f_{r_0}(x)e^{\mathbf{\vec{r}} \cdot \vec{x}} = 1$, where the probability measure $Q_{r_0}^r$ has an exponentially decaying tail due to the same reasoning as in chapter 4 of \cite{ioffe} ("mass-gap" property). \end{proof} With the help of the proposition above we shell show that the consequent regeneration points are situated relatively close to each other: \begin{lem*} $$P_p[\max_{i} |x_i - x_{i-1}|>n^{1/3}, \mbox{ } x_i \mbox{- reg. points } | \mbox{ } 0 \leftarrow^h \rightarrow n \bold{\vec{a}} ] <\frac{1}{n}$$ for $n$ large enough. \end{lem*} \begin{proof} Let $\mathbf{\vec{r}} \equiv \nabla \xi_p(\mathbf{\vec{r}}_0) = \nabla \xi_p(\mathbf{\vec{a}})$. Since $\xi_p(x)$ is strictly convex (see section 4 in \cite{ioffe}), $$ \frac{ \xi_p(\bold{\vec{a}}) - \xi_p(\bold{\vec{a}} -\frac{\vec{x}}{n})} {(\frac{\|\vec{x}\|}{n})} < \frac{\vec{x}}{\|\vec{x}\|} \cdot \nabla \xi_p(\bold{\vec{a}}) $$ for $\vec{x} \in \mathbb{Z}^d$ ($\vec{x} \not= 0$), and therefore $$\xi_p(n \bold{\vec{a}})- \xi_p(n \bold{\vec{a}}-\vec{x}) = \|\vec{x}\| \frac{ \xi_p(\bold{\vec{a}}) - \xi_p(\bold{\vec{a}} -\frac{\vec{x}}{n})}{(\frac{\|\vec{x}\|}{n})} < \vec{x} \cdot \nabla \xi_p(\bold{\vec{a}}) = \mathbf{\vec{r}} \cdot \vec{x}.$$ Thus, since $Q_{r_0}^r(x)$ decays exponentially and therefore $$\frac{f(x)}{(1-p)^{2(d-1)}} e^{\xi_p(n \bold{\vec{a}})- \xi_p(n \bold{\vec{a}}-x)} < Q_{r_0}^r(x)$$ and also decays exponentially. Hence by Ornstein-Zernike result (\ref{oz}), $$P_p[ n^{1/3} < |x| , \mbox{ } x \mbox{-first reg. point } | 0 \leftarrow^h \rightarrow n \bold{\vec{a}} ] =\sum_{ n^{1/3} < |x| } \frac{f(x)}{(1-p)^{2(d-1)}} \frac{h(n \bold{\vec{a}}-x)} {h(n \bold{\vec{a}})} < \frac{1}{n^2} $$ for $n$ large enough. So, since the number of the regeneration points is no greater than $n$, $$P_p[\max_{i} |x_i - x_{i-1}|>n^{1/3}, \mbox{ } x_i \mbox{- reg. points } | \mbox{ } 0 \leftarrow^h \rightarrow n \bold{\vec{a}} ] <\frac{1}{n}$$ for $n$ large enough. \end{proof} Now, it is really easy to check that there is a constant $\lambda_f >0$ such that $$f(\vec{x}) > e^{-\lambda_f \|\vec{x}\|}$$ for all $\vec{x}$ such that $f(\vec{x}) \not= 0$ (here we only need to connect points $\vec{e}$ and $\vec{x} - \vec{e}$ with two non-intersecting open paths surrounded by the closed edges), and there exists a constant $\lambda_u >0$ such that $$P_p[\mbox{ percolation cluster } \bold{C}(0) \not\subset [\mathbb{R}; B_{R}^{d-1}(0)]_f ] < e^{-\lambda_u R} $$ for $R$ large enough due to the exponential decay of the radius distribution for subcritical probabilities (see \cite{grimmett}). Hence, for a given $\epsilon >0$ $$P_p[\mbox{ cluster } \bold{C}_{0,\vec{x}}^{r_0} \not\subset [\mathbb{R}, B_{\epsilon \sqrt{n}}^{d-1}(0)]_f \mbox{ } | \mbox{ } 0 \leftarrow^f \rightarrow x] < e^{ \lambda_f \|\vec{x}\| -\lambda_u \epsilon \sqrt{n}}, $$ and therefore, summing over the regeneration points, we get $$P_p[\mbox{ scaled cluster } \bold{C}_{0, n \bold{\vec{a}} }^{r_0} \not\subset \epsilon \mbox{-neighbd. of } [0,1] \times \gamma (\mbox{ reg. points }) \mbox{ } | \mbox{ } 0 \leftarrow^g \rightarrow n \bold{\vec{a}} ]$$ $$ < \frac{1}{n} + n e^{ \lambda_f n^{1/3} -\lambda_u \epsilon \sqrt{n}} $$ for $n$ large enough.\\ We can now state the main result of this paper: \begin{Mthm} The process corresponding to the last $d-1$ coordinates (in the new basis $\{ \vec{f_1},\vec{f_2},...,\vec{f_d} \}$) of the scaled $({\frac{1}{n \|\mathbf{\vec{a}} \|} \times \frac{1}{\sqrt{n}} })$ interpolation of regeneration points of $\bold{C}_{0,n \bold{\vec{a}}}^{r_0}$ (where the first coordinate is time) conditioned on ${0 \leftarrow^{h}\rightarrow n \bold{\vec{a}}}$ converges weakly to the Brownian Bridge (with variance that depends only on measure $Q_{r_0}^r$). \\ Also for a given $\epsilon >0$ $$P_p[\mbox{ scaled cluster } \bold{C}_{0, n \bold{\vec{a}} }^{r_0} \not\subset \epsilon \mbox{-neighbd. of } [0,1] \times \gamma (\mbox{ reg. points }) \mbox{ } | \mbox{ } 0 \leftarrow^h \rightarrow n \bold{\vec{a}} ] \rightarrow 0$$ as $n \rightarrow \infty$. \end{Mthm} \section{Convergence to Brownian Bridge.} \label{bb} As it was mentioned in the introduction, this chapter is entirely dedicated to proving the Technical Theorem that we have already used in the proof of the main result. \subsection{Simple Case.} Let $Z_1, Z_2,...$ be i.i.d. random variables on $\mathbb{Z}$ with the span of the lattice distribution equal to one (see \cite{durrett}, section 2.5) and mean $\mu = \bold{E}Z_1 <\infty$, $\sigma^2=Var(Z_1)<\infty$. Also let point zero be inside of the closed convex hull of $\{ z \mbox{ : } P[Z_1 = z]>0 \}$. \\ Consider a one dimensional plane and a walk $X_j$ that starts with $X_0=0$ and for a given $X_j$, the (j+1)-st step to be $X_{j+1}=X_j + Z_{j+1}$. After interpolation we get $$X(t)=X_{[t]}+(t-[t])(X_{[t]+1}-X_{[t]})$$ for $0\leq{t}<\infty$.\\ And define $\bar{X}(t) = (t,X(t))$ to be a two dimensional walk.\\ Now, if for a given integer $n>0$ we define $X_n(t)\equiv{\frac{X(nt)}{\sqrt{n}}}$ for $0\leq{t}\leq{1}$, then $X_n(t)$ would belong to $C[0,1]$ and $X_n(0)=0$. \begin{thm}\label{simpleT} $X_n(t)$ conditioned on $X_n(1)=0$ converges weakly to the Brownian Bridge. \end{thm} First we need to prove the theorem when $\mu =0$. For this we need to prove that \begin{lem} For $A_0 \subseteq {C[0,1]}$, let $P_n(A_0)=P[X_n\in{A_0}|X_n(1)=0]$ to be the law of $X_n$ conditioned on $X_n(1)=0$. Then \\ (a) For $\mu=0$, the finite-dimensional distributions of $P_n$ converge weakly to a Gaussian distributions.\\ (b) There are positive $\{C_n\}_{n=1,2,...}\rightarrow{C}$ ($C =\sigma^2$ when $\mu=0$) such that $0<C<\infty$ and $$Cov_{P_n}(X_n(s),X_n(t)) = C_ns(1-t) + O(\frac{1}{n})$$ for all $0\leq{s}\leq{t}\leq{1}$. More precisely: $Cov_{P_n}(X_n(s),X_n(t))=C_ns(1-t)$ if $[ns]<[nt]$ and \\ $Cov_{P_n}(X_n(s),X_n(t))=C_ns(1-t) - C_n\frac{\epsilon_1 (1- \epsilon_2 )}{n}$ if $[ns]=[nt]$, where $\epsilon_1 = \frac{ns-[ns]}{n} \in [0,1)$ and $\epsilon_2 = \frac{nt-[nt]}{n} \in [0,1)$. \end{lem} and we need \begin{lem} For $\mu=0$, the probability measures $P_n$ induced on the subspace of $X_n(t)$ trajectories in $C[0,1]$ are tight. \end{lem} \begin{proof}[Proof of Lemma 1:] (a) Though it is not difficult to show that a finite-dimensional distribution of $P_n$ converges weakly to a gaussian distribution, here we only show the convergence for one and two points on the interval (in case of one point $t \in [0,1]$, we show that the limit variance has to be equal to $t(1-t) \sigma^2$). Take $t \in \frac{1}{n}\mathbb{Z} \cap (0,1)$ and let $\alpha =\frac{k}{\sqrt{n}}$, then by the Local CLT, \begin{eqnarray} \label{phi} P[X(tn)=k] = \frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{t}}(\alpha) + o(\frac{1}{\sqrt{n}}), \mbox{ where } \Phi_{v}(x) \equiv \frac{1}{v \sqrt{2\pi}}e^{-\frac{x^2}{2 v^2}} \end{eqnarray} is the normal density function, and the error term is uniformly bounded by a $o(\frac{1}{\sqrt{n}})$ function independent of $k$. \\ Therefore, substituting (\ref{phi}), $$P_n[X_n(t)=\alpha] = \frac{(\frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{t}}(\alpha) + o(\frac{1}{\sqrt{n}})) (\frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{1-t}}(\alpha)+ o(\frac{1}{\sqrt{n}}))} {\frac{1}{\sqrt{n}} \Phi_{\sigma}(0) + o(\frac{1}{\sqrt{n}}) } = \frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{t(1-t)}}(\alpha) + o(\frac{1}{\sqrt{n}}) .$$ Thus for a set $A$ in $\mathbb{R}$, $$P_n[X_n(t)\in{A}] = \sum_{k\in\sqrt{n}A}[\frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{t(1-t)}}(\alpha)+ o(\frac{1}{\sqrt{n}})] = N[0, t(1-t)\sigma^2](A) + o(1)$$ -here the limit variance is equal to $t(1-t)\sigma^2$. Given that the variance $\sigma^2 <0$, the convergence follows. \\ The same method works for more than one point, here we do it for two: Let $\alpha_1 = \frac{k_1}{\sqrt{n}}$ and $\alpha_2 = \frac{k_2}{\sqrt{n}}$, then as before, for $t_1<t_2$ in $\frac{1}{n}\mathbb{Z} \cap (0,1)$, writing the conditional probability as a ratio of two probabilities, and representing the probabilities according to (\ref{phi}), we get $$P_n[X_n(s)=\alpha_1 , X_n(t)=\alpha_2] = \frac{\sqrt{|\mathcal{A}|}}{2\pi \sigma^2 } \exp \left \{-\frac{(\alpha_1, \alpha_2)\mathcal{A} (\alpha_1, \alpha_2)^T} {2\sigma^2} \right \} +o(\frac{1}{n}) . $$ \\ \\ where $$\mathcal{A}={\left( \begin{array}{cc} \frac{t_2}{(t_2-t_1)t_1} & -\frac{1}{t_2-t_1} \\ -\frac{1}{t_2-t_1} & \frac{1-t_1}{(t_2-t_1)(1-t_2)} \end{array} \right)}.$$ \\ Thus for sets $A_1$ and $A_2$ in $\mathbb{R}$, \begin{eqnarray*} P_n[X_n(t_1) \in A_1 , X_n(t_2) \in A_2] & = & \sum_{k_1 \in \sqrt{n}A_1, k_2 \in \sqrt{n}A_2} [\frac{\sqrt{|\mathcal{A}|}}{2\pi \sigma^2 } \exp \left \{ -\frac{(\alpha_1, \alpha_2) \mathcal{A} (\alpha_1, \alpha_2)^T} {2\sigma^2} \right \}+o(\frac{1}{n})] \\ \\ & = & N[0, \mathcal{A}^{-1}](A_1 \times A_2) + o(1) \end{eqnarray*} Observe that $(\sigma^2 \mathcal{A}^{-1})={\left( \begin{array}{cc} t_1(1-t_1)\sigma^2 & t_1(1-t_2)\sigma^2 \\ t_1(1-t_2)\sigma^2 & t_2(1-t_2)\sigma^2 \end{array} \right)}$ is the covariance matrix, and the part (b) of the lemma follows in case $\mu =0$.\\ (b) Though the estimate above produces the needed variance in case when the mean $\mu =0$ , in general, we need to apply the following approach: We first consider the case when $s<t$ and both $s,t \in \frac{1}{n}\mathbb{Z} \cap (0,1)$ where $$\mathbf{E}[X_n(s) \mbox{ }|\mbox{ } X_n(t) =y] = \mathbf{E}[Z_1 +...+ Z_{sn} |Z_1 +...+ Z_{tn} = y] = \frac{s}{t} y ,$$ and therefore \begin{eqnarray*} Cov_{P_n}(X_n(s),X_n(t)) & = & \frac{s}{t} \mathbf{E}[X^2_n(t)|X_n(1)=0] \end{eqnarray*} as $\{ -X_n(1-t) \mbox{ }|\mbox{ } X_n(1)=0 \}$ and $\{ X_n(t) \mbox{ }|\mbox{ } X_n(1)=0 \}$ are identically distributed.\\ Now, by symmetry (time reversal), $$Cov_{P_n}(X_n(s),X_n(t)) =Cov_{P_n}(X_n(1-t),X_n(1-s)) =\frac{1-t}{1-s} \mathbf{E}[X^2_n(s)|X_n(1)=0],$$ and therefore $$\frac{\mathbf{E}[X^2_n(s)|X_n(1)=0]} {\mathbf{E}[X^2_n(t)|X_n(1)=0]}=\frac{s(1-s)}{t(1-t)}.$$ Hence, there exists a constant $C_n$ such that for all $t \in \frac{1}{n}\mathbb{Z} \cap (0,1)$ $$\frac{\mathbf{E}[X^2_n(t)|X_n(1)=0]}{t(1-t)} \equiv C_n.$$ Thus we have shown that for $s \leq t$ in $\frac{1}{n}\mathbb{Z} \cap [0,1]$, $$Cov_{P_n}(X_n(s),X_n(t)) = \frac{s}{t} \mathbf{E}[X^2_n(t)|X_n(1)=0] = \frac{s}{t}C_n t(1-t) = C_n s(1-t).$$ Now, consider the general case: $s=s_0 + \frac{\epsilon_1}{n} \leq t = t_0 + \frac{\epsilon_2}{n}$, where $ns_0, nt_0 \in \mathbb{Z}$ and $\epsilon_1, \epsilon_2 \in [0,1)$. Then the covariance \begin{eqnarray*} Cov_{P_n}(X_n(s),X_n(t)) & = & (1-\epsilon_1)(1-\epsilon_2)Cov_{P_n}(X_n(s_0),X_n(t_0))\\ & + & (1-\epsilon_1)\epsilon_2Cov_{P_n}(X_n(s_0),{X_n(t_0+\frac{1}{n})}) \\ & + &\epsilon_1(1-\epsilon_2)Cov_{P_n}(X_n(s_0+\frac{1}{n}),X_n(t_0))\\ & + &\epsilon_1\epsilon_2Cov_{P_n}(X_n(s_0+\frac{1}{n}),X_n(t_0+\frac{1}{n})) \end{eqnarray*} Therefore \begin{eqnarray*} Cov_{P_n}(X_n(s),X_n(t)) & = & C_ns(1-t) \mbox{ when } s_0<t_0 \mbox{ (}[ns]<[nt]\mbox{),} \end{eqnarray*} and \begin{eqnarray*} Cov_{P_n}(X_n(s),X_n(t)) & = & C_ns(1-t) - C_n\frac{\epsilon_1 (1- \epsilon_2 )}{n} \mbox{ when } s_0=t_0 \mbox{ (}[ns]=[nt]\mbox{).} \end{eqnarray*} Now, plugging in $s=t=\frac{1}{2}$ we get $$C_n =4\mathbf{E}[X^2_n(\frac{1}{2})|X_n(1)=0] \mbox{ when }n\mbox{ is even,}$$ and $$C_n =4\mathbf{E}[X^2_n(\frac{1}{2})|X_n(1)=0] (\frac{n}{n-1}) \mbox{ when }n\mbox{ is odd.}$$ Therefore $$C_n =4\mathbf{E}[X^2_n(\frac{1}{2})|X_n(1)=0] (1+O(\frac{1}{n})) \rightarrow{C} =\sigma^2$$ as $\{ X_n(\frac{1}{2}), P_n \}$ converges in distribution as $n \rightarrow +\infty$. \end{proof} \begin{proof}[Proof of Lemma 2:] Before we begin the proof of tightness, we notice that the only real obstacle we face is that the process is conditioned on $X_n=0$. The tightness for the case without the conditioning has been proved years ago as part of the Donsker's Theorem (see Chapter 10 in \cite{bill}). With the help of the local CLT we are essentially removing the difference between the two cases. Given a $\lambda >0$ and let $m=[n \delta]$ for a given $0< \delta \leq 1$, then for any $\mu >0$, \begin{eqnarray*} P_{\lambda} & \equiv & P[ \max_{0\leq i \leq m} X_i \geq \lambda \sqrt{n} > X_m> -\lambda \sqrt{n}| X_n = 0] \\ & = & \sum_{a=-[\lambda \sqrt{n}]}^{[\lambda \sqrt{n}]} \frac{P[\max_{0 \leq i \leq m}X_i > [\lambda \sqrt{n}] \mbox{ ; } X_m =a \mbox{ ; } X_n=0]} {P[X_n = 0]}\\ & = & \sum_{a=-[\lambda \sqrt{n}]}^{[\lambda \sqrt{n}]} \frac{P[\max_{0 \leq i \leq m}X_i > [\lambda \sqrt{n}] \mbox{ ; } X_m =a] P[X_{n-m}=-a]} {P[X_n = 0]}\\ & \leq & \max_{-[\lambda \sqrt{n}] \leq a \leq [\lambda \sqrt{n}]} (\frac{P[X_{n-m}=-a]} {P[X_n = 0]}) \times \sum_{a=-[\lambda \sqrt{n}]}^{[\lambda \sqrt{n}]} P[\max_{0 \leq i \leq m}X_i > [\lambda \sqrt{n}] \mbox{ ; } X_m =a] \\ & \leq & 2P[\max_{0\leq i \leq m}X_i \geq \lambda \sqrt{n} \geq X_m \geq -\lambda \sqrt{n} ] \end{eqnarray*} \\ for $n$ large enough, where by the local CLT, $$ \max_{-[\lambda \sqrt{n}] \leq a \leq [\lambda \sqrt{n}]} (\frac{P[X_{n-m}=-a]} {P[X_n = 0]}) \leq 2 $$ for $n$ large enough as $n-m$ linearly depends on $n$. \\ Therefore, the probability $$P[\max_{0\leq i \leq m} |X_i| \geq \lambda \sqrt{n} |X_n=0] \leq 2P_{\lambda} + P[|X_m| \geq \lambda \sqrt{n} |X_n=0],$$ where $$P_{\lambda} \leq 2P[\max_{0\leq i \leq m} X_i \geq \lambda \sqrt{n}].$$ Now, due to the point-wise convergence, we can proceed as in Chapter 10 of \cite{bill} by bounding the two remaining probabilities: $$P[\max_{0\leq i \leq m} |X_i| \geq \lambda \sqrt{n}] \leq 2 P[|X_m| \geq \frac{1}{2} \lambda \sqrt{n} ] \rightarrow 2P[|\sqrt{\delta}N| \geq \frac{\lambda}{2 \sigma}] \leq \frac{16 \delta^{3/2} \sigma^3}{\lambda^3} \mathbf{E}[|N|^3]$$ and similarly $$P[|X_m| \geq \lambda \sqrt{n} | X_n=0] \rightarrow P[|\sqrt{\delta(1-\delta)}N| \geq \frac{\lambda}{\sigma}] \leq \frac{\delta^{3/2} \sigma^3}{\lambda^3} \mathbf{E}[|N|^3].$$ Thus, for all integer $k \in [0, n-m]$, $$P[\max_{0\leq i \leq m} |X_{k+i} - X_k| \geq \lambda \sqrt{n} |X_n=0] = P[\max_{0\leq i \leq m} |X_i| \geq \lambda \sqrt{n} |X_n=0] \leq 70\frac{\delta^{3/2} \sigma^3}{\lambda^3} \mathbf{E}[|N|^3]$$ for $n$ large enough, (see Chapter 10 in \cite{bill}). Therefore $\{ P_n \}$ are tight (see Chapter 8 of \cite{bill}). \end{proof} \begin{proof}[Proof of Theorem \ref{simpleT}:] The lemmas above imply the convergence when the mean $\mu =0$. Now, for $\mu \not= 0$, there exists a $\rho \in \mathbb{R}$ such that $\sum_{z \in \mathbb{Z}} z e^{\rho z} P[Z_1 =z] = 0$. Then we let $\hat{Z}_1,\hat{Z}_2,...$ be i.i.d. random variables with their distribution defined in the following fashion: $$P[\hat{Z}_j =z] \equiv \frac{e^{\rho z}}{C_{\rho}} P[Z_j =z]$$ for all $j$ and $z \in \mathbb{R}$, where $C_{\rho} \equiv \sum_{z \in \mathbb{Z}} P[\hat{Z}_1 =z]= \sum_{z \in \mathbb{Z}} e^{\rho z} P[Z_1 =z]$. Then the law of $Z_1,...,Z_n$ conditioned on $Z_1+...+Z_n=0$ is the same as that of $\hat{Z}_1,...,\hat{Z}_n$ conditioned on $\hat{Z}_1+...+\hat{Z}_n=0$, and the case is reduced to that of $\mu =0$ as $\bold{E} \hat{Z}_j =0$. We also estimate the covariance equal to $\hat{C} s(1-t)$ for all $0 \leq s \leq t \leq 1$, where as before $$\hat{C} = \lim_{n \rightarrow +\infty} \bold{E}[Z_1^2 \mbox{ }| \mbox{ } Z_1 +...+ Z_n =0].$$ \end{proof} Observe that the result can be modified for $X_1, X_2,...$ defined on a multidimensional lattice $\mathbb{L} \subset \mathbb{R}^d$,$d>1$, if we condition on $X_n(1)=\bold{a}(n) = \bold{a+o(1)} \in \{ z\sqrt{n} \mbox{ : } z \in \bigoplus^n_1 \mathbb{L} \}$. We again let point zero be inside the closed convex hull of $\{ z \mbox{ : } P[Z_1 = z]>0 \}$. In this case the process $\tilde{X}_n(t) = X_n(t) + (\bold{a} - \bold{a}(n))t$ converges to the Brownian Bridge $B^{0, \bold{a}}$, and convergence is uniform whenever $\bold{a}(n)$ uniformly converges to zero thanks to the Local CLT. \begin{thm} $\tilde{X}_n(t)$ conditioned on $X_n(1)=\bold{a}(n) = \bold{a}+o(1)$ converges weakly to the Brownian Bridge. \end{thm} Here, as before, if we take $t \in \frac{1}{n} \mathbb{Z} \cap [0,1]$ and let $\alpha =\frac{k}{\sqrt{n}}$, then \begin{eqnarray*} P[X_n(t)=\alpha \mbox{ }|\mbox{ } X_n(1)=\bold{a}(n)] & = & \frac{(\frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{t}}(\alpha) + o(\frac{1}{\sqrt{n}})) (\frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{1-t}}(\bold{a}(n) -\alpha) + o(\frac{1}{\sqrt{n}}))} {\frac{1}{\sqrt{n}} \Phi_{\sigma}(\bold{a}(n)) + o(\frac{1}{\sqrt{n}})} \\ & = & \frac{1}{\sqrt{n}} \Phi_{\sigma \sqrt{t(1-t)}} (\alpha-\bold{a}(n)t) + o(\frac{1}{\sqrt{n}}) . \end{eqnarray*} \subsection{General Case.}\label{bb:gen} As before, for a given non-zero vector $\bold{\vec{a}} \in \mathbb{Z}^d$, we let $X_1, X_2,...$ be i.i.d. random variables on $\mathbb{Z}^d$ with the span of the lattice distribution equal to one (see \cite{durrett}) such that the probability $P[\bold{\vec{a}} \cdot X_1 >0] =1$, the mean $\mu = \bold{E}X_1 <\infty$ and there is a constant $\bar{\lambda} >0$ such that the moment-generating function $$\bold{E}(e^{\theta \cdot X_1}) <\infty$$ for all $\theta \in B_{\bar{\lambda}}$. Also we let $\bold{P_{\vec{a}}}$ denote the projection map on $<\bold{\vec{a}}>$ and $\bold{P^{\bot}_{\vec{a}}}$ denote the orthogonal projection on $<\bold{\vec{a}}>^{\bot}$. Now we can decompose the mean $\mu = \mu_{a} \times \mu_{or}$, where $\mu_{a} \equiv \bold{P_{\vec{a}}}\mu$ and $\mu_{or} \equiv \bold{P^{\bot}_{\vec{a}}}\mu$.\\ As before we introduce a new basis $\{ \vec{f_1},\vec{f_2},..., \vec{f_d} \}$, where $\vec{f_1} = \frac{\bold{\vec{a}}}{\| \bold{\vec{a}} \|}$. We again use $[\cdot,\cdot]_f \in \mathbb{R} \times \mathbb{R}^{d-1}$ to denote the coordinates of a vector with respect to the new basis. We denote $X_i=[T_i,Z_i]_f \in \mathbb{Z} \times \mathbb{Z}^{d-1}$, where $[T_i, 0]_f = \bold{P_{\vec{a}}}X_i$ and $[0, Z_i]_f = \bold{P^{\bot}_{\vec{a}}}X_i$, and we let $X_1+...+X_i =[t_i, Y_i]_f \in \mathbb{Z} \times \mathbb{Z}^{d-1}$. Note: $T_i$ and $Z_i$ don't have to be independent. Interpolating $Y_i$, we get $$Y(t)=Y_{[t]}+(t-[t])(Y_{[t]+1} - Y_{[t]})$$ for $0 \leq t \leq \infty$ and if we now define $Y_n(t)\equiv{\frac{Y(nt)}{\sqrt{n}}}$ for $0\leq{t}\leq{1}$, then the following theorem easily follows from the previous result: \begin{cor} $Y_n(t)$ conditioned on $Y_n(1)=0$ converges weakly to the Brownian Bridge. \end{cor} Since the first coordinate $T_i$ is positive with probability one, the next step will be to interpolate $[t_i,Y_i]_f$, and prove that if scaled and conditioned on $[t_n,Y_n]_f= X_1+...+X_n= [n \| \bold{\vec{a}} \|,0]_f = n\bold{\vec{a}}$ it will converge weakly to the Brownian Bridge (with the first coordinate being the time axis). Now, the last theorem implies the result for $P[[T_i,0]_f=\mu_a]=1$, we want the same result for $\bold{E}T_i=\| \mu_a \|$ and $VarT_i<\infty$. \\ We first let $\bar{X}_i \equiv X_i - \mu_a$, then $\bold{E}\bar{X}_i= \mu_{or}$ and $Var \bar{X}_i <\infty$. We again interpolate: $$\bar{X}(t)=\bar{X}_{[t]}+(t-[t])(\bar{X}_{[t]+1} - \bar{X}_{[t]})$$ for $0 \leq t \leq \infty$, and scale $\bar{X}_k(t)\equiv{\frac{\bar{X}(kt)}{\sqrt{k}}}$. Note: the last $d-1$ coordinates of $\bar{X}_k(t)$ w.r.t. the new basis are $Y_k(t)$ (e.g. $\bold{P^{\bot}_{\vec{a}}}\bar{X}_k(t) = [0, Y_k(t)]_f$). \\ From here on we denote $S_j \equiv [t_j, Y_j]_f =X_1 +...+X_j$ and $\bar{S}_j \equiv \bar{X}_1 +...+\bar{X}_j =S_j- j\mu_a$ for any positive integer $j$. As a first important step, we state another important \begin{cor} For $k=k(n)= [\frac{n \| \bold{\vec{a}} \|}{\| \mu_a \|} + k_0 \sqrt{n}]$, $\{ \bar{X}_k(t) -(k_0\sqrt{\frac{\| \mu_a \|}{\| \bold{\vec{a}} \|}} \mu_a +\frac{n\bold{\vec{a}} - k\mu_a}{\sqrt{k}})t \}$ conditioned on $\bar{X}_k(1)= n \bold{\vec{a}} -k\mu_a$ ${(e.g. [t_k,Y_k]_f= n \bold{\vec{a}} )}$ converges weakly to the Brownian Bridge $B^{0, - k_0 \sqrt{\frac{\| \mu_a \|}{\| \bold{\vec{a}} \|}} \mu_a}$. \end{cor} Observe that $n \bold{\vec{a}} -k\mu_a = -k_0 \sqrt{n}\mu_a +o(\sqrt{n})$ and that the convergence is uniform for all $k_0$ in a compact set . Now, looking only at the last $d-1$ coordinates of $\bar{X}_k(t)$, w.r.t. the new basis the last Corollary implies: \begin{lem} For $k=k(n)= [\frac{n \| \bold{\vec{a}} \|}{\| \mu_a \|} + k_0 \sqrt{n}]$, $Y_k(t)$ conditioned on $t_k=n \| \bold{\vec{a}} \|$ and $Y_k(1)=0$ converges weakly to the Brownian Bridge. \end{lem} Note that convergence is uniform for $k_0$ in a compact set. \\ What the Lemma above says is the following: the interpolation of $[\frac{i}{k}, \frac{1}{\sqrt{k}}Y_i]_f$ conditioned on $[t_k, Y_k]_f=n \bold{\vec{a}}$ converges to Time$\times$Brownian Bridge. Now, define the process $[t, Y_{n,k}^*(t)]_f$ to be the interpolation of $[\frac{1}{n \| \bold{\vec{a}} \|}t_i, \frac{1}{\sqrt{n}}Y_i]^{i=0,1,...,k}_f$, then \begin{thm} For $k=k(n)= [\frac{n \| \bold{\vec{a}} \|}{\| \mu_a \|} + k_0 \sqrt{n}]$, $\sqrt{\frac{n}{k}} Y^*_{n,k}(t)$ conditioned on $t_k=n \| \bold{\vec{a}} \|$ and $Y_k(1)=0$ converges weakly to the Brownian Bridge. \end{thm} \begin{proof}[Proof:] Here we observe that the mean $\bold{E} [\frac{t_i}{n \| \bold{\vec{a}} \|} - \frac{t_{i-1}}{n\| \bold{\vec{a}} \|}]$ is actually equal to $\frac{\| \mu_a \|}{n \| \bold{\vec{a}} \|} = \frac{1}{k -k_0 \sqrt{n}}+o(\frac{1}{n})$, and that for a given $\epsilon >0$, the probability of the $\|[\frac{1}{n \| \bold{\vec{a}} \|}t_i, \frac{1}{\sqrt{n}}Y_i]_f - [\frac{i}{k}, \frac{1}{\sqrt{k}}Y_i]_f \| = |\frac{t_j}{n \| \bold{\vec{a}} \|} - \frac{j}{k}|$ exceeding $\epsilon$ for some $j\leq k$, \begin{eqnarray*} P[ \max_{0\leq j \leq k} |t_j- \frac{n \| \bold{\vec{a}} \|}{k} j|\geq n\epsilon \mbox{ }| \mbox{ } S_n = n \bold{\vec{a}}] & \leq & P[ \max_{0\leq j \leq k} \| S_j -\frac{n \| \bold{\vec{a}} \| j}{k}\mu_a \| \geq n \epsilon \mbox{ } | \mbox{ } S_k = n \bold{\vec{a}}]\\ & \leq & P[ \max_{0\leq j \leq k} |\bar{S}_j| \geq n \frac{\epsilon}{2} \mbox{ } | \mbox{ } \bar{S}_k = [n \| \bold{\vec{a}} \| -k\| \mu_a \|, 0]_f ]\\ & \rightarrow & 0 \end{eqnarray*} as $n \rightarrow +\infty$ since $n \| \bold{\vec{a}} \| -k\| \mu_a \| = -\| \mu_a \| k_0 \sqrt{n} + o(\sqrt{n})$. \end{proof} Now, the next step is to prove that the process $$\{Y^*_{n,k} \mbox{ for some } k \mbox{ such that } [t_k, Y_k]_f= n \bold{\vec{a}} \}$$ conditioned on the existence of such $k$ converges weakly to the Brownian Bridge.\\ First of all the last theorem implies \begin{lem} For given $k=k(n)= [\frac{n \| \bold{\vec{a}} \|}{\| \mu_a \|} + k_0 \sqrt{n}]$, $Y^*_{n,k}(t)$ conditioned on $t_k=n \| \bold{\vec{a}} \|$ and $Y_k(1)=0$ converges weakly to the Brownian Bridge. \end{lem} For a fixed $M>0$, convergence is also uniform on $k \in [\frac{n \| \bold{\vec{a}} \|}{\| \mu_a \|} -M\sqrt{n}, \frac{n \| \bold{\vec{a}} \|}{\| \mu_a \|} +M\sqrt{n}]$. For the future purposes we denote $\kappa \equiv \frac{\| \mu_a \|}{\| \bold{\vec{a}} \|}$ and $I_M \equiv [\frac{n}{\kappa} -M\sqrt{n}, \frac{n}{\kappa}+M\sqrt{n}] \bigcap \mathbb{Z}$.\\ Finally, we want to prove the following technical result, in which we use the uniformity of convergence for all $k=k(n) \in I_M$ and the truncation techniques to show the convergence of $Y^*_{n,k}$ to the Brownian Bridge in case when we condition only on the existence of such $k$. \begin{TECHthm} The process $$\{ Y^*_{n,k}\mbox{ for some } k \mbox{ such that } [t_k, Y_k]_f = n \bold{\vec{a}} \}$$ conditioned on the existence of such $k$ converges weakly to the Brownian Bridge. \end{TECHthm} \begin{proof}[Proof:] Take $M$ large, notice that for $A \subset C^{d-1}[0,1]$, $$\max_{k \in I_M} |P[Y^*_k \in A \mid [t_k, Y_k]_f = n \bold{\vec{a}}] -P[ B^o \in A] | = o(1),$$ where the Brownian Bridge $B^o$ is scaled up to the same constant for all those $k$.\\ Hence, $$lim_{n \rightarrow +\infty} \frac{\sum_{k \in I_M} P[S_k= n \bold{\vec{a}}]P[Y^*_{n,k} \in A | S_k= n \bold{\vec{a}} ]} {\sum_{k \in I_M} P[S_k= n \bold{\vec{a}} ]} = P[ B^o \in A].$$ Therefore we are only left to prove the truncation argument as $M \rightarrow +\infty$. Now, for any $\epsilon >0$ there exists $M>0$ such that $$(1+\epsilon) \sum_{k \in I_M} P[S_k= n \bold{\vec{a}} ] \leq \sum_k P[S_k= n \bold{\vec{a}} ] \leq (1+2\epsilon) \sum_{k \in I_M} P[S_k= n \bold{\vec{a}} ]$$ for $n$ large enough, as by the large deviation upper bound, there is a constant $\bar{C}_{LD} >0$ such that $$P[S_k= n \bold{\vec{a}} ] \leq e^{- \bar{C}_{LD} \frac{(n-k\kappa)^2}{k} \wedge |n-k\kappa|} ,$$ and therefore $\exists C_{LD}>0$ such that $$\sum_{|n-k \kappa| >n^{2/3} } P[S_k= n \bold{\vec{a}}] < e^{-C_{LD} n^{1/3} }.$$ Also, by the local CLT, $$P[S_k= n \bold{\vec{a}}]=P[\bar{S}_k= (n-k \kappa) \bold{\vec{a}}] =\frac{1}{k^{d/2} \sqrt{Var\bar{X}_1 (2 \pi)^d }} e^{-\frac{1}{2Var\bar{X}_1} \frac{(n-k \kappa)^2}{k} } +o(\frac{1}{k^{d/2}})$$ implying $$\sum_{|n-k \kappa| \leq n^{2/3} } P[S_k= n \bold{\vec{a}} ] = \frac{1}{ n^{\frac{d-1}{2}} } [\int_{-\infty}^{+\infty} \frac{1}{\sqrt{Var\bar{X}_1 (2 \pi)^d }} e^{-\frac{x^2}{2Var\bar{X}_1} } dx +o(1)]$$ where $$\sum_{k \in I_M} P[S_k= n \bold{\vec{a}} ] = \frac{1}{ n^{\frac{d-1}{2}} } [\int_{-M}^M \frac{1}{\sqrt{Var\bar{X}_1 (2 \pi)^d }} e^{-\frac{x^2}{2Var\bar{X}_1} } dx +o(1)].$$ Therefore $$\frac{1}{1+2\epsilon} \frac{\sum_{k \in I_M} P[S_k= n \bold{\vec{a}}]P[Y^*_{n,k} \in A | S_k= n \bold{\vec{a}}]} {\sum_{k \in I_M} P[S_k=n \bold{\vec{a}}]} \leq \frac{\sum_{k} P[S_k= n \bold{\vec{a}} ]P[Y^*_{n,k} \in A | S_k= n \bold{\vec{a}} ]} {\sum_{k} P[S_k= n \bold{\vec{a}} ]}$$ $$ \leq \frac{1}{1+\epsilon} \frac{\sum_{k \in I_M} P[S_k= n \bold{\vec{a}} ]P[Y^*_{n,k} \in A | S_k= n \bold{\vec{a}} ]} {\sum_{k \in I_M} P[S_k= n \bold{\vec{a}} ]}$$ for all $A \subset C^{d-1}[0,1]$. Taking the $\liminf$ and $\limsup$ of the fraction in the middle completes the proof. \end{proof} \end{document}

\begin{document} \begin{center} \large \bf Birationally rigid Fano-Mori fibre spaces \end{center} \centerline{A.V.Pukhlikov} \parshape=1 3cm 10cm \noindent {\small \quad\quad\quad \quad\quad\quad\quad \quad\quad\quad {\bf }\newline In this paper we prove the birational rigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a Fano complete intersection of index 1 and codimension $k\geqslant 3$ in the projective space ${\mathbb P}^{M+k}$ for $M$ sufficiently high, satisfying certain natural conditions of general position, in the assumption that the fibre space $V\slash S$ is sufficiently twisted over the base. The dimension of the base $S$ is bounded from above by a constant, depending only on the dimension $M$ of the fibre (as the dimension of the fibre $M$ grows, this constant grows as $\frac12 M^2$). Bibliography: 28 items.} AMS classification: 14E05, 14E07 Key words: Fano variety, Mori fibre space, birational map, birational rigidity, linear system, maximal singularity, multi-quadratic singularity. \section*{Introduction} {\bf 0.1. Fano complete intersections.} In the present paper we study the birational geometry of algebraic varieties, fibred into Fano complete intersections of codimension $k\geqslant 3$ (fibrations into Fano hypersurfaces were studied in \cite{Pukh15a}, into Fano complete intersections of codimension 2 in \cite{Pukh2022a}). We start with a description of fibres of these fibre spaces. Let us fix an integer $k\geqslant 3$ and set $$ \varepsilon(k)=\mathop{\rm min} \left\{a\in {\mathbb Z}\,\left|\, a\geqslant 1, \left(1+\frac{1}{k}\right.\right)^a\geqslant 2\right\}. $$ Now let us fix $M\in{\mathbb Z}$, satisfying the inequality \begin{equation}\label{14.11.22.1} M\geqslant 10 k^2+8k+2\varepsilon(k)+3. \end{equation} The right hand side of that inequality denote by the symbol $\rho(k)$. Let $$ \underline{d}=(d_1,\dots, d_k) $$ be an ordered tuple of integers, $$ 2\leqslant d_1\leqslant d_2\leqslant\dots\leqslant d_k, $$ satisfying the equality $$ d_1+\dots +d_k=M+k. $$ Fano varieties, considered in this paper, are complete intersections of type $\underline{d}$ in the complex projective space ${\mathbb P}^{M+k}$. More precisely, let the symbol ${\cal P}_{a,N}$ stand for the space of homogeneous polynomials of degree $a\in {\mathbb Z}_+$ in $N\geqslant 1$ variables. Set $$ {\cal P}=\prod^k_{i=1} {\cal P}_{d_i,M+k+1} $$ to be the space of all tuples $$ \underline{f}=(f_1,\dots, f_k) $$ of homogeneous polynomials of degree $d_1$, \dots, $d_k$ on ${\mathbb P}^{M+k}$. If for $\underline{f}\in {\cal P}$ the scheme of common zeros of the polynomials $f_1,\dots, f_k$ is an irreducible reduced factorial variety $F=F(\underline{f})$ of dimension $M$ with terminal singularities, then $F$ is a primitive Fano variety: $$ \mathop{\rm Pic} F={\mathbb Z} H_F,\quad K_F=-H_F, $$ where $H_F$ is the class of a hyperplane section of the variety $F$ (the Lefschetz theorem). Assuming that this is the case, let us give the following definition. {\bf Definition 0.1.} The variety $F$ is {\it divisorially canonical}, if for every effective divisor $D\sim nH_F$ the pair $(F,\frac{1}{n}D)$ is canonical, that is, for every exceptional divisor $E$ over $F$ the inequality $$ \mathop{\rm ord}\nolimits_E D\leqslant n\cdot a(E) $$ holds, where $a(E)$ is the discrepancy of $E$ with respect to $F$. Below is the first main result of the present paper. {\bf Theorem 0.1.} {\it There exist a Zariski open subset ${\cal F}\subset{\cal P}$, such that for every tuple $\underline{f}\in{\cal F}$ the scheme of common zeros of the tuple $\underline{f}$ is an irreducible reduced factorial divisorially canonical variety $F(\underline{f})$ of dimension $M$ with terminal singularities, and the codimension of the complement ${\cal P}\setminus{\cal F}$ satisfies the inequality} $$ \mathop{\rm codim}(({\cal P}\setminus{\cal F})\subset{\cal P}) \geqslant M-k+5+{M-\rho(k)+2 \choose 2}. $$ (Thus for a fixed $k$ and growing $M$ the codimension of the complement ${\cal P}\setminus{\cal F}$ grows as $\frac12 M^2$.) It is convenient to express the property of divisorial canonicity in terms of the {\it global canonical threshold} of the variety $F$. Recall that for a Fano variety $X$ with the Picard number 1 and terminal ${\mathbb Q}$-factorial singularities its global canonical threshold $\mathop{\rm ct}(X)$ is the supremum of $\lambda\in{\mathbb Q}_+$ such that for every effective divisor $D\sim -nK_X$ (here $n\in{\mathbb Q}_+$) the pair $\left(X,\frac{\lambda}{n}D\right)$ is canonical. Therefore, Theorem 0.1 claims that for every $\underline{f}\in{\cal F}$ the inequality $\mathop{\rm ct} (F(\underline{f}))\geqslant 1$ holds. If in the definition of the global canonical threshold instead of ``for every effective divisor $D\sim -nK_X$'' we put ``for a general divisor $D$ in any linear system $\Sigma\subset|-nK_X|$ with no fixed components'', we get the definition of the {\it mobile canonical threshold} $\mathop{\rm mct}(X)$; obviously, $\mathop{\rm mct}(X)\geqslant\mathop{\rm ct}(X)$. The inequality $\mathop{\rm mct}(X)\geqslant 1$ is equivalent to the birational superrigidity of the Fano variety $X$, see \cite{Ch05c}. If in the definition of the global canonical threshold the property of the pair $(X,\frac{\lambda}{n}D)$ to be canonical we replace by the log canonicity of that pair, we get the definition of the {\it global log canonical threshold} $\mathop{\rm lct}(X)$; again, $\mathop{\rm lct}(X)\geqslant\mathop{\rm ct}(X)$. For simplicity we write $F\in{\cal F}$ instead of $F=F(\underline{f})$ for $\underline{f}\in{\cal F}$. {\bf 0.2. Fano-Mori fibre spaces.} By a {\it Fano-Mori fibre space} we mean a surjective morphism of projective varieties $$ \pi\colon V\to S, $$ where $\dim V\geqslant 3 + \dim S$, the base $S$ is non-singular and rationally connected, and the following conditions are satisfied: (FM1) every scheme fibre $F_s=\pi^{-1}(s)$, $s\in S$, is an irreducible reduced factorial Fano variety with terminal singularities and the Picard group $\mathop{\rm Pic} F_s\cong {\mathbb Z}$, (FM2) the variety $V$ itself is factorial and has at most terminal singularities, (FM3) the equality $$ \mathop{\rm Pic} V={\mathbb Z} K_V\oplus\pi^* \mathop{\rm Pic}S $$ holds. So Fano-Mori fibre spaces are Mori fibre spaces with additional very good properties. {\bf Definition 0.2.} A Fano-Mori fibre space $\pi\colon V\to S$ is {\it stable with respect to fibre-wise birational modifications}, if for every birational morphism $\sigma_S\colon S^+\to S$, where $S^+$ is a non-singular projective variety, the morphism $$ \pi_+\colon V^+=V\mathop{\times}\nolimits_S S^+\to S^+ $$ is a Fano-Mori fibre space. We will consider birational maps $\chi\colon V\dashrightarrow V'$, where $V$ is the total space of a Fano-Mori fibre space and $V'$ is the total space of a fibre space $\pi'\colon V'\to S'$ which belongs to one of the two classes: (1) {\it rationally connected fibre spaces}, that is, $V'$ and $S'$ are non-singular and the base $S'$ and a fibre of general position $(\pi')^{-1}(s')$ are rationally connected, (2) {\it Mori fibre spaces}, where $V'$ and $S'$ are projective and the variety $V'$ has ${\mathbb Q}$-factorial terminal singularities. For a birational map $\chi\colon V\dashrightarrow V'$, where $V'/S'$ is a rationally connected fibre space, we want to answer the question: is it fibre-wise, that is, is there a rational dominant map $\beta\colon S\dashrightarrow S'$, making the diagram \begin{equation}\label{15.11.22.1} \begin{array}{rcccl} & V & \stackrel{\chi}{\dashrightarrow} & V' & \\ \pi\!\!\!\!\! & \downarrow & & \downarrow & \!\!\!\!\!\pi' \\ & S & \stackrel{\beta}{\dashrightarrow} & S' \end{array} \end{equation} a commutative one, that is, $\pi'\circ\chi=\beta\circ\pi$? For a birational map $\chi\colon V\dashrightarrow V'$, where $V'/S'$ is a Mori fibre space with the additional properties (2) (only such Mori fibre spaces are considered in this paper), we want to answer the question: is there a {\it birational} map $\beta\colon S\dashrightarrow S'$, for which the diagram (\ref{15.11.22.1}) is commutative? If the answer to this question is always affirmative (that is, it is affirmative for every fibre space from the class (2)), then the fibre space $V/S$ is {\it birationally rigid}. Now let us state the second main result of the present paper. {\bf Theorem 0.2.} {\it Assume that a Fano-Mori fibre space $\pi\colon V\to S$ is stable with respect to fibre-wise birational modifications, and moreover, {\rm (i)} for every point $s\in S$ the fibre $F_s$ satisfies the inequalities $\mathop{\rm lct} (F_s)\geqslant 1$ and $\mathop{\rm mct} (F_s)\geqslant 1$, {\rm (ii)} (the $K$-condition) every mobile (that is, with no fixed components) linear system on $V$ is a subsystem of a complete linear system $|-nK_V+\pi^* Y|$, where $Y$ is a pseudoeffective class on $S$, {\rm (iii)} for every family ${\overline{\cal C}}$ of irreducible curves on $S$, sweeping out a dense subset of the base $S$, and $\overline{C}\in{\overline{\cal C}}$, no positive multiple of the class $$ -(K_V\cdot \pi^{-1}(\overline{C}))-F\in A^{\dim S} V, $$ where $A^iV$ is the numerical Chow group of classes of cycles of codimension $i$ on $V$ and $F$ --- the class of a fibre of the projection $\pi$, is represented by an effective cycle on $V$. Then for every rationally connected fibre space $V'\slash S'$ every birational map $\chi\colon V\dashrightarrow V'$ (if such maps exist) is fibre-wise, and the fibre space $V\slash S$ itself is birationally rigid.} By what was said in Subsection 0.1, the assumption (i) can be replaced by the single inequality $\mathop{\rm ct}(F_s)\geqslant 1$ for every $s\in S$, that is, it is sufficient to assume that every fibre of the fibre space $V\slash S$ is a divisorially canonical variety. As we will see from the proof of Theorem 0.2, instead of the conditions (ii) and (iii) it is sufficient to require that for every family $\overline{\cal C}$ of irreducible curves on $S$, sweeping out a dense subset, and $\overline{C}\in \overline{\cal C}$ the class $$ -N(K_V\cdot \pi^{-1}(\overline{C}))-F\in A^{\dim S} V $$ is not represented by an effective cycle on $V$ for any $N\geqslant 1$. The last condition is especially easy to verify: it is enough to have a numerically effective $\pi$-ample class $H_V$ on $V$, satisfying the inequality \begin{equation}\label{18.11.22.1} \left(K_V\cdot \pi^{-1}(\overline{C})\cdot H_V^{\dim V-\dim S}\right)\geqslant 0 \end{equation} for every dense family $\overline{\cal C}\ni\overline{C}$. {\bf 0.3. An explicit construction of a fibre space.} Now let us construct a large class of Fano-Mori fibre spaces, satisfying the conditions of Theorem 0.2. Let $S$ be a non-singular projective rationally connected positive-dimensional variety and $\pi_X\colon X\to S$ a locally trivial fibration with the fibre ${\mathbb P}^{M+k}$, where $k$ and $M$ are the same as in Subsection 0.1. We say that the subvariety $V\subset X$ of codimension $k$ is a {\it fibration into complete intersections of type} $\underline{d}$, if the base $S$ can be covered by Zariski open subsets $U$, over which the fibration $\pi_X$ is trivial, $\pi^{-1}_X(U)\cong U\times{\mathbb P}^{M+k}$, and for every $U$ there is a regular map $$ \Phi_{U}\colon U\to{\cal P}, $$ such that $V\cap\,\pi^{-1}_X(U)$ in the sense of the above-mentioned trivialization is the scheme of common zeros of a tuple $$ \underline{f}(s)=\Phi_{U}(s)=(f_1(x_*,s),\dots,f_k(x_*,s)), $$ where $x_*$ are homogeneous coordinates on ${\mathbb P}^{M+k}$ and $s$ runs through $U$. Below (in \S 1) it will be clear, that the open subset ${\cal F}$ from Theorem 0.1 is invariant under the action of the group $\mathop{\rm Aut}{\mathbb P}^{M+k}$. For that reason, the following definition makes sense. {\bf Definition 0.3.} A fibration $V\subset X$ into complete intersections of type $\underline{d}$ is a ${\cal F}$-{\it fibration}, if for any trivialization of the bundle $\pi_X$ over an open set $U\subset S$ we have $\Phi_{U}(U)\subset{\cal F}$. Obviously, if the inequality \begin{equation}\label{18.11.22.2} \mathop{\rm dim} S\leqslant M-k+4+{M-\rho(k)+2 \choose 2} \end{equation} holds, then we may assume that $V$ is a ${\cal F}$-fibration. Set $\pi=\pi_X|_V$. Now from Theorems 0.1 and 0.2 it is easy to obtain the third main result of the present paper. {\bf Theorem 0.3.} {\it Any ${\cal F}$-fibration $\pi\colon V\to S$ constructed above is a Fano-Mori fibre space. If the conditions (ii) and (iii) of Theorem 0.2 hold, then for every rationally connected fibre space $V'/S'$ every birational map $\chi\colon V\dashrightarrow V'$ is fibre-wise, and the fibre space $V\slash S$ itself is birationally rigid.} {\bf Example 0.1.} Let $H_X$ be a numerically effective divisorial class on $X$, the restriction of which onto the fibre $\pi_X^{-1}(s)\cong {\mathbb P}^{M+k}$ is the class of a hyperplane. Let $\Delta_1$,\dots, $\Delta_k$ be very ample classes on the base $S$. Let us construct a ${\cal F}$-fibration $V/S$ as a complete intersection of $k$ general divisors $$ V=G_1\cap \dots \cap G_k, $$ where $G_i\in|d_i H_X+\pi_X^*\Delta_i|$. Let us find out, when $V/S$ satisfies the conditions (ii) and (iii) of Theorem 0.2. Write $$ K_X=-(M+k+1) H_X+\pi^*_X \Delta_X, $$ then we get $$ K_V=\left.\left(-H_X+\pi^*_X\left(\Delta_X+\sum^k_{i=1}\Delta_i\right) \right)\right|_V. $$ It is easy to check that the inequality (\ref{18.11.22.1}) in this case takes the form of the estimate $$ \left(\left(\Delta_X+\sum^k_{i=1}\left(1-\frac{1}{d_i}\right)\Delta_i\right)\cdot \overline{C}\right)\geqslant \left(H_X^{M+k+1}\cdot \pi_X^{-1}(\overline{C})\right), $$ where for the class $H_V$ we took $H_X|_V$. This inequality must be satisfied for every dense family $\overline{\cal C}\ni\overline{C}$. Let us consider a very particular case, when $X={\mathbb P}^m\times{\mathbb P}^{M+k}$ and $G_i$ are divisors of bi-degree $(m_i,d_i)$, $i=1,\dots,k$. Taking for $H_X$ the pull back on $X$ of the class of a hyperplane in ${\mathbb P}^{M+k}$, we get that the last inequality is equivalent to the numerical inequality \begin{equation}\label{18.11.22.3} \sum^k_{i=1}\left(1-\frac{1}{d_i}\right)m_i\geqslant m+1. \end{equation} If it is satisfied and the dimension $m=\mathop{\rm dim} S$ satisfies the inequality (\ref{18.11.22.2}), then the intersection $V=G_1\cap\dots\cap G_k$ of general (in the sense of Zariski topology) divisors of bi-degree $(m_1,d_1),\dots,(m_k,d_k)$, fibred over $S={\mathbb P}^m$, is a birationally rigid Fano-Mori fibre space and every birational map of $V$ onto the total space of a rationally connected fibre space is fibre-wise. The inequality (\ref{18.11.22.3}) shows that this claim holds for almost all tuples $(m_1,\dots,m_k)\in{\mathbb Z}^k_+$ (except for finitely many of them), that is, for almost all families of Fano-Mori fibre spaces, obtained by means of this construction. Note that the condition (\ref{18.11.22.3}) is lose to a criterial one: if $$ m_1+\dots+m_k\leqslant m, $$ then the projection of $V$ onto ${\mathbb P}^{M+k}$ defines on $V$ a structure of a Fano-Mori fibre space (and a rationally connected fibre space), which is ``transversal'' to the original structure $\pi\colon V\to S$ (and is not fibre-wise), so that in this case $V/S$ is not birationally rigid. {\bf 0.4. The structure of the paper.} The paper is organized in the following way. In \S 1 we produce the explicit local conditions defining the open subset ${\cal F}\subset{\cal P}$. The proof of divisorial canonicity of a variety $F\in{\cal F}$ (that is, of the inequality $\mathop{\rm ct}(F)\geqslant 1$) is reduced in \S 1 to a number of technical facts that will be shown in the subsequent sections (\S\S 3-7). In \S 2 we show Theorem 0.2. The proof of Theorem 0.1 consists of several pieces. The fact that the local conditions for the singularities that a variety $F\in{\cal F}$ can have ({\it multi-quadratic singularities}, see Subsection 1.2), guarantee that the variety $F$ is factorial and its singularities are terminal, is proven in \S 4, where we give a general definition of multi-quadratic singularities and study their properties. The estimate for the codimension of the complement ${\cal P}\setminus {\cal F}$ (which is very important for constructing families of Fano-Mori fibre spaces, satisfying the assumptions of Theorem 0.2) is shown in \S 8. However, the main (and the hardest) part of the proof of Theorem 0.1 is to show that a variety $F\in{\cal F}$ is divisorially canonical. We assume that for some effective divisor $D\sim nH_F$ the pair $(F,\frac{1}{n}D)$ is not canonical, that is, for some exceptional divisor $E$ over $F$ the inequality $$ \mathop{\rm ord}\nolimits_E D>n\cdot a(E) $$ holds. Now we have to show that this assumption leads to a contradiction. In Subsections 1.3-1.6 it is shown how (using the inequalities for the multiplicity of subvarieties of the variety $F$ at a given point, proven in \S 7) to obtain a contradiction in the case when a point of general position $o\in B$, where $B$ is the centre of $E$ on $F$, either is non-singular on $F$, or is a quadratic singularity. The hardest task is to obtain a contradiction when the point $o$ is a multi-quadratic singularity of the variety $F$. A plan of solving this problem is given in Subsection 1.7, where we introduce the concept of a {\it working triple} and describe the procedure of construction a sequence of subvarieties of the variety $F$, in which each subvariety is a hyperplane section of the previous one and the last subvariety delivers the desired contradiction. This program is realized in \S 3, where we study the properties of working triples, however a number of key technical facts is only stated there --- their proof is put off for a greater clarity of exposition. These key facts are shown in \S\S 5,6 (and the proof makes use of the facts on linear subspaces on complete intersections of quadrics, proven in Subsection 4.5). Finally, in \S 7 we prove the estimates for the multiplicities of certain subvarieties of the variety $F$ at given points in terms of the degrees of these subvarieties in ${\mathbb P}^{M+k}$. Here we use the well known technique of hypertangent divisors. For the purposes of our proof of Theorem 0.1 we have to somewhat modify this technique. {\bf 0.5. General remarks.} The birational rigidity of Fano-Mori fibre spaces over a positive-dimensional base was one of the most important topics in birational geometry in the past 40 years. For its history and place in the context of the modern birational geometry of rationally connected varieties, see \cite[Subsection 0.4]{Pukh2022a}. Here we just mention a few recent papers in the areas that are close to the direction, to which the present paper belongs. These areas are: the birational rigidity, explicit birational geometry of Mori fibre spaces (including the studies of their groups of birational automorphisms and, wider, Sarkisov links), the rationality problem, computing and estimating the global canonical thresholds and, related to these problems, the theory of $K$-stability. In the papers \cite{KrylovOkadaetal22,AbbanKrylov22,Krylov18} important results on the birational rigidity and rigidity-type results for fibrations over ${\mathbb P}^1$ were obtained. The paper \cite{Stibitz21} links the Sarkisov program with the problem of estimating the canonical threshold of certain divisors on Fano varieties. The papers \cite{AbbanOkada18,KrylovOkada20} prove the stable non-rationality of very general conic bundles and fibrations into del Pezzo surfaces, respectively, over a higher-dimensional base. The problem of stable rationality for hypersurfaces of various bi-degrees in the products of projective spaces (see Example 0.1 above) is considered in \cite{NicaiseOttem22}. The theory of $K$-stability, which is on the border of birational geometry, is investigated in many papers (especially in the recent past), in particular, see \cite{StibitzZhuang19,Zhuang21,CheltsovPark22,CheltsovDenisovaetal22}; we mentioned the papers that are the closest to the birational rigidity-type problems. Finally, there was a lot of development recently in the direction of applying the theory of Sarkisov links and relations between them to the study of the groups of birational automorphisms of such varieties that have a very large this group, see, for instance, \cite{BlancYasinsky20,BlancLamyZ21}. Getting back to the topic of this paper, we note that its immediate predecessor is \cite{Pukh2022a}, however, that paper investigates the non canonical singularities, the centre of which is contained in the set of bi-quadratic points of the variety (from the technical viewpoint, this is the hardest part of the proof of divisorial canonicity), using the secant varieties of subvarieties of codimension 2 on an intersection of two quadrics. It is not possible to apply this approach to subvarieties of higher codimension on an intersection of $k\geqslant 3$ quadrics, and the present paper is based on a completely different construction (which applies to the bi-quadratic singularities, considered in \cite{Pukh2022a}, as well). The author is grateful to the members of Divisions of Algebraic Geometry and Algebra at Steklov Institute of Mathematics for the interest to his work, and also to the colleagues in Algebraic Geometry research group at the University of Liverpool for general support. \section{Fano complete intersections} In this section we describe the local conditions defining the open subset ${\cal F}\subset{\cal P}$ (Subsections 1.2 and 1.4). For a complete intersection $F\in{\cal F}$ the proof of its divisorial canonicity is reduced to a number of technical claims, which will be shown later. A more detailed plan of the proof of Theorem 0.1 is given in Subsection 1.1. {\bf 1.1. A plan of the proof of Theorem 0.1.} In order to prove Theorem 0.1, one has to give an explicit definition of the open set ${\cal F}\subset{\cal P}$. This definition consists of two groups of conditions, which should be satisfied by the polynomials $f_1,\dots, f_k$ at every point $o\in {\mathbb P}^{M+k}$ at which they all vanish. The first group of conditions is about the singularities of the complete intersection $F(\underline{f})$: they can be quadratic or multi-quadratic of a rank bounded from below. The corresponding definitions and facts are given in Subsection 1.2. Assuming that the conditions of the first group are satisfied, we get that the scheme of common zeros of the polynomials $f_1,\dots, f_k$ is an irreducible reduced factorial variety $F=F(\underline{f})\subset {\mathbb P}^{M+k}$ with terminal singularities, and so $\mathop{\rm Pic} F={\mathbb Z} H_F$ and $K_F=-H_F$, so that the question, is it divisorially canonical, makes sense. Assuming that $F$ is not divisorially canonical, let us fix an effective divisor $D_F\sim n(D_F)H_F$, where $n(D_F)\geqslant 1$, such that the pair $$ \left(F,\frac{1}{n(D_F)}D_F\right) $$ is not canonical, that is, there is an exceptional divisor $E$ over $F$, satisfying the Noether-Fano inequality: $$ \mathop{\rm ord}\nolimits_E D_F>n(D_F)\, a(E). $$ We have to show that the existence of such a divisor leads to a contradiction. Let $B\subset F$ be the centre of the exceptional divisor $E$ on $F$. The information about the singularities of the varieties $F$ makes it possible to easily exclude the option when $\mathop{\rm codim}(B\subset F)=2$. This is done in Subsection 1.3. After that in Subsection 1.4 we produce the second group of local conditions for the tuple of polynomials $\underline{f}\in{\cal F}$: now they are the regularity conditions. Assuming that they are satisfied at every point $o\in F$, we exclude the option $B\not\subset\mathop{\rm Sing}F$ in Subsection 1.5, and in Subsection 1.6 the option that the point $o\in B$ of general position is a quadratic singularity of $F$. In Subsection 1.7 we describe the procedure of excluding the multi-quadratic case, when the point $o\in B$ of general position is a multi-quadratic singularity of the type $2^l$, $l\in\{2,\dots,k\}$. This is the hardest part of the work, which is completed in the subsequent sections. {\bf 1.2. Multi-quadratic singularities.} Let $o\in{\mathbb P}^{M+k}$ be a point, at which $f_1,\dots,f_k$ all vanish. Let us consider a system of affine coordinates $z_*=(z_1,\dots,z_{M+k})$ with the origin at the point $o$ on an affine chart ${\mathbb A}^{M+k}\subset{\mathbb P}^{M+k}$, containing that point. Write down $$ \begin{array}{ccccc} f_1=f_{1,1}+f_{1,2}+ & \dots & + f_{1,d_1}, & &\\ f_2=f_{2,1}+f_{2,2}+ & \dots & & + f_{2,d_2}, & \\ & \dots & & &\\ f_k=f_{k,1}+f_{k,2}+ & \dots & & & +f_{k,d_k}, \end{array} $$ where we use the same symbols $f_i$ for the non-homogeneous polynomials in $z_*$, corresponding to the original polynomials $f_i$, and $f_{i,a}$ is a homogeneous polynomial of degree $a$ in $z_*$. Obviously, if the linear forms $f_{1,1},\dots,f_{k,1}$ are linearly independent, then in a neighborhood of the point $o$ the scheme of common zeros of the polynomials $f_1,\dots,f_k$ is a non-singular complete intersection of codimension $k$. In order to give the definition of a multi-quadratic singularity, we will need the concept of the rank of a tuple of quadratic forms. {\bf Definition 1.1.} (\cite{Pukh2022a}) {\it The rank of the tuple of quadratic forms} $q_1,\dots,q_l$ in $N$ variables is the number $$ \mathop{\rm rk}(q_1,\dots,q_l)=\mathop{\rm min}\{\mathop{\rm rk}(\lambda_1 q_1+\dots+\lambda_l q_l)\,|\,(\lambda_1,\dots,\lambda_l)\neq (0,\dots,0)\}. $$ Obviously, $\mathop{\rm rk}(q_1,\dots,q_l)\leqslant N$. For that reason, in the sequel the inequality $\mathop{\rm rk}(q_*)\geqslant a$ means implicitly that the forms $q_i$ depend on a sufficient $(\geqslant a)$ number of variables. Take $l\in\{1,2,\dots,k\}$. {\bf Definition 1.2.} The tuple $\underline{f}$ has at the point $o$ a {\it multi-quadratic singularity of type} $2^l$ of rank $a$, if the following conditions are satisfied: \begin{itemize} \item $\mathop{\rm dim}\langle f_{1,1},\dots,f_{1,k}\rangle=k-l$ (and in order to simplify the notations we assume that the forms $$ f_{l+1},\dots, f_k $$ are linearly independent), \item the rank of the tuple of quadratic forms $$ f^*_{i,2}=f_{i,2}-\sum^k_{j=l+1}\lambda_{i,j}f_{j,2}, $$ $i=1,\dots,l$, where $\lambda_{i,j}\in{\mathbb C}$ are defined by the equalities $$ f_{i,1}=\sum^k_{j=l+1}\lambda_{i,j}f_{j,1}, $$ is equal to the number $a$. \end{itemize} Now the first condition, defining the subset ${\cal F}\subset{\cal P}$, is stated in the following way. (MQ1) For every point $o\in{\mathbb P}^{M+k}$, such that $$ f_1(o)=\dots=f_k(o)=0 $$ either the linear forms $f_{1,1},\dots,f_{k,1}$ are linearly independent, or $\underline{f}$ has at the point $o$ a multi-quadratic singularity of type $2^l$, where $l\in\{1,2,\dots,k\}$, of rank $$ \geqslant 2l+4k+2\varepsilon(k)-1. $$ {\bf Theorem 1.1.} {\it Assume that $\underline{f}$ satisfies the condition (MQ1). Then the scheme of common zeros of the polynomials $f_1,\dots,f_k$ is an irreducible reduced factorial variety F=F(\underline{f}) --- a complete intersection of codimension $k$ with terminal singularities, and, moreover,} $$ \mathop{\rm codim}(\mathop{\rm Sing}F\subset F)\geqslant 4k+2\varepsilon(k). $$ {\bf Proof} is given in \S 4 (Subsections 4.1-4.3). Assume that $\underline{f}$ satisfies the condition (MQ1). For a point $o\in F=F(\underline{f})$ the symbol $T_oF$ stands for the subspace $\{f_{1,1}=\dots=f_{k,1}=0\}\subset{\mathbb C}^{M+k}$. For the proof of Theorem 0.1 we will need one more property of the tuple $\underline{f}$, which we include in the definition of the subset ${\cal F}$. (MQ2) For any point $o\in F$, which is a multi-quadratic of type $2^l$, where $l\geqslant 2$, the rank of the tuple of quadratic forms $$ f_{1,2}|_{T_oF},\dots,f_{k,2}|_{T_oF} $$ is at least $10k^2+8k+2\varepsilon(k)+5$. The condition (MQ2) for multi-quadratic points of type $2^l$ with $l\geqslant 2$ implies the condition (MQ1), because the rank of a quadratic form, restricted to a hyperplane, drops at most by 2, however for the convenience of references we state the conditions (MQ1) and (MQ2) independently of each other. These conditions are used in the proof of Theorem 0.1 in different ways. So every tuple $\underline{f}\in{\cal F}$ satisfies (MQ1) and (MQ2). {\bf 1.3. Subvarieties of codimension 2.} Following the plan, given in Subsection 1.1, let us fix an effective divisor $D_F\sim n(D_F)H_F$, $n(D_F)\geqslant 1$, such that the pair $(F,\frac{1}{n(D_F)}D_F)$ is not canonical. By the symbol $$ \mathop{\rm CS}\left(F,\frac{1}{n(D_F)}D_F\right) $$ we denote the union of the centres on $F$ of all exceptional divisors over $F$, satisfying the Noether-Fano inequality (that is to say, of all non-canonical singularities of that pair). This is a closed subset of $F$. Let $B$ be an irreducible component of maximal dimension of that set. {\bf Proposition 1.1.} {\it The following inequality holds:} $\mathop{\rm codim}(B\subset F)\geqslant 3$. {\bf Proof.} Assume the converse: $\mathop{\rm codim}(B\subset F)=2$. Then $B\not\subset\mathop{\rm Sing}F$. Moreover, let $P\subset{\mathbb P}^{M+k}$ be a general linear subspace of codimension $2k+2$. Theorem 1.1 implies that $P\cap\mathop{\rm Sing}F=\emptyset$, so that $F\cap P$ is a non-singular complete intersection of type $\underline{d}$ in $P\cong{\mathbb P}^{2k+2}$. Furthermore, the pair $$ \left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) $$ is not canonical, and the irreducible subvariety $B\cap P$ is an irreducible component of maximal dimension of the set $$ \mathop{\rm CS}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right), $$ so that (as $F\cap P$ is non-singular) $$ \mathop{\rm mult}\nolimits_{B\cap P}D_F|_{F\cap P}>n(D_F). $$ However, $D_F|_{F\cap P}\sim n(D_F)H_{F\cap P}$ (where $H_{F\cap P}$ is the class of a hyperplane section of $F\cap P$), so that by \cite[Proposition 3.6]{Pukh06b} or \cite{Suzuki15} we get a contradiction, proving the proposition. Q.E.D. {\bf 1.4. Regularity conditions.} In order to continue the proof of Theorem 0.1, we need a second group of conditions defining the set ${\cal F}$. Let $o\in F$ be a point. We use the notations of Subsection 1.2. By the symbol $T_oF$ we denote the {\it linear} tangent space $$ \{f_{1,1}=\dots=f_{k,1}=0\}\subset{\mathbb C}^{M+k}, $$ and by the symbol ${\mathbb P}(T_oF)$ its projectivization. Let ${\cal S}=(h_1,\dots,h_M)$ be the sequence of homogeneous polynomials $$ f_{i,j}|_{{\mathbb P}(T_oF)}, $$ where $j\geqslant 2$, placed in the lexicographic order: $(i_1,j_1)$ precedes $(i_2,j_2)$, if $j_1<j_2$ or $j_1=j_2$, but $i_1<i_2$. By the symbol ${\cal S}[-m]$ denote the sequence ${\cal S}$ with the last $m$ members removed. Finally, the symbol ${\cal S}[-m]|_{\Pi}$ stands for the restriction of that sequence (that is, the restriction of each its member) onto a linear subspace $\Pi\subset{\mathbb P}(T_oF)$. The regularity conditions depend on the type of the singularity $o\in F$. First, let the point $o\in F$ be non-singular, so that ${\mathbb P}(T_oF)\cong{\mathbb P}^{M-1}$. In that case the regularity condition is stated in the following way. (R1) The sequence $$ {\cal S}[-(k+\varepsilon(k)+3)]|_{\Pi} $$ is regular for every subspace $\Pi\subset{\mathbb P}(T_oF)$ of codimension $k+\varepsilon(k)-1$. The condition (R1) is assumed for every non-singular point $o\in F$. It implies the following key fact. {\bf Theorem 1.2.} {\it Let $P\subset{\mathbb P}^{M+k}$ be an arbitrary linear subspace of codimension $\leqslant k+ \varepsilon(k)-1$. Then for every non-singular point $o\in F\cap P$ and every prime divisor $Y\sim n(Y)H_{F\cap P}$ on $F\cap P$ the inequality} $$ \mathop{\rm mult}\nolimits_oY\leqslant2n(Y) $$ {\it holds.} {\bf Proof} is given in \S 7 (Subsections 7.1, 7.2). Now let $o\in F$ be a quadratic singularity (this case corresponds to the value $l=1$ in Definition 1.2). Here ${\mathbb P}(T_oF)\cong{\mathbb P}^M$. In this case the regularity condition is stated as follows. (R2) The sequence $$ {\cal S}[-4]|_{\Pi} $$ is regular for every hyperplane $\Pi\subset{\mathbb P}(T_oF)$. The condition (R2) is assumed for every quadratic singular point $o\in F$ and implies the following key fact. {\bf Theorem 1.3.} {\it Let $o\in F$ be a quadratic singularity and $W\ni o$ the section of $F$ by a hyperplane that is not tangent to $F$ at the point $o$, and $Y\sim n(Y)H_W$ a prime divisor on $W$. Then the following inequality holds:} $$ \mathop{\rm mult}\nolimits_oY\leqslant 4n(Y). $$ {\bf Proof} is given in \S 7 (Subsection 7.3). (The symbol $H_W$ stands for the class of a hyperplane section of the variety $W$; the linear form, defining the hyperplane that cuts out $W$, is not a linear combination of the forms $f_{1,1},\dots,f_{k,1}$.) Now let $o\in F$ be a multi-quadratic point of type $2^l$, where $l\in\{2,\dots,k\}$. Here we will need two regularity conditions. In the first of them the symbol $T_oF$ means the projective closure of the linear subspace $$ \{f_{1,1}=\dots=f_{k,1}=0\}\subset{\mathbb C}^{M+k} $$ in ${\mathbb P}^{M+k}$. (R3.1) For every subspace $P\subset T_oF$ of codimension $\varepsilon(k)$, containing the point $o$, the scheme of common zeros of the polynomials $$ f_1|_P,\dots,f_k|_P,\quad f_{i,2}|_P\quad\mbox{for all}\quad i: d_i\geqslant 3, $$ is an irreducible reduced subvariety of codimension $k+k_{\geqslant 3}$ in $P$, where $$ k_{\geqslant 3}=\sharp\{i=1,\dots,k\,|\, d_i\geqslant 3\}. $$ Note that in the condition (R3.1) the homogeneous polynomials $f_{i,2}$ in the {\it affine} coordinates $z_*$ are considered as quadratic forms in {\it homogeneous} coordinates on ${\mathbb P}^{M+k}$. In the second regularity condition for multi-quadratic points the symbol $T_oF$ means a linear subspace in ${\mathbb C}^{M+k}$. (R3.2) For every linear subspace $\Pi\subset{\mathbb P}(T_oF)$ of codimension $\varepsilon(k)$ the sequence $$ {\cal S}[-m^*]|_{\Pi} $$ is regular, where $m^*=\mathop{\rm max}\{\varepsilon(k)+4-l,0\}$. The conditions (R3.1) and (R3.2) are assumed for every multi-quadratic singular point $o\in F$. They imply the following key inequality. In Theorem 1.4, stated below, the symbol $T_oF$ stands for the projective closure of the embedded tangent space, that is, a linear subspace in ${\mathbb P}^{M+k}$, containing the point $o$. {\bf Theorem 1.4.} {\it Let $P\subset T_oF$ be an arbitrary linear subspace of codimension $\leqslant\varepsilon(k)$ and $Y\ni o$ a prime divisor on $F\cap P$, $Y\sim n(Y)H_{F\cap P}$. Then the following inequality holds:} $$ \mathop{\rm mult}\nolimits_o Y\leqslant\frac32\cdot 2^kn(Y). $$ {\bf Proof} is given in \S 7 (Subsections 7.4, 7.5). (The symbol $H_{F\cap P}$ stands for the class of a hyperplane section of the variety $F\cap P$; we will show below, see \S 4, that $F\cap P$ is an irreducible factorial complete intersection.) Summing up, let us give a complete definition of the subset ${\cal F}\subset{\cal P}$: it consists of the tuples $\underline{f}$, satisfying the conditions (MQ1,2), the condition (R1) at every non-singular point $o\in F(\underline{f})$, the condition (R2) at every quadratic point $o\in F(\underline{f})$ and the conditions (R3.1,2) at every multi-quadratic point $o\in F(\underline{f})$. The inequality for the codimension of the complement ${\cal P}\setminus{\cal F}$, given in Theorem 0.1, is shown in \S 8. {\bf 1.5. Exclusion of the non-singular case.} We carry on with the proof of divisorial canonicity of the variety $F\in{\cal F}$. In the notations of Subsection 1.3 assume that the point of general position $o\in B$ is a non-singular point of $F$. We know (Proposition 1.1), that $\mathop{\rm codim}(B\subset F)\geqslant 3$. Consider a general subspace $P\ni o$ of dimension $k+3$. Then $F\cap P$ is a non-singular three-dimensional variety and the point $o$ is a connected component of the set $$ \mathop{\rm CS}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) $$ (if $\mathop{\rm codim}(B\subset F)\geqslant 4$, then $\mathop{\rm CS}$ can be replaced, by inversion of adjunction, by $\mathop{\rm LCS}$), that is, outside the point $o$ in a neighborhood of that point the pair \begin{equation}\label{23.11.22.1} \left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) \end{equation} is canonical. It is well known (see \cite[Proposition 3]{Pukh05} or \cite[Chapter 7, Proposition 2.3]{Pukh13a}), it follows from here that either the inequality $$ \mathop{\rm mult}\nolimits_oD_F>2n(D_F) $$ holds, or on the exceptional divisor $E\cong{\mathbb P}^{M-1}$ of the blow up $F^+\to F$ of the point $o$ there is a hyperplane $\Theta\subset E$ (uniquely determined by the pair (\ref{23.11.22.1})), such that the inequality $$ \mathop{\rm mult}\nolimits_oD_F+\mathop{\rm mult}\nolimits_{\Theta}D^+_F> 2n(D_F) $$ holds, where $D^+_F$ is the strict transform of $D_F$ on $F^+$. The first option is impossible as it contradicts Theorem 1.2. In the second case denote by the symbol $|H-\Theta|$ the projectively $k$-dimensional linear system of hyperplane sections of $F$, a general element of which $W\ni o$ is non-singular at the point $o$ and satisfies the equality $$ W^+\cap E=\Theta. $$ The restriction $D_W=(D_F\circ W)$ is an effective divisor on $W$, and $n(D_W)=n(D_F)$ and the inequality $$ \mathop{\rm mult}\nolimits_oD_W\geqslant\mathop{\rm mult}\nolimits_oD_F+\mathop{\rm mult}\nolimits_{\Theta}D^+_F>2n(D_W) $$ holds, which again contradicts Theorem 1.2. We have shown the following fact. {\bf Proposition 1.2.} {\it The subvariety $B$ is contained in the singular locus of $F$:} $B\subset\mathop{\rm Sing} F$. {\bf 1.6. Exclusion of the quadratic case.} Again let $o\in B$ be a point of general position. {\bf Proposition 1.3.} {\it The point $o$ is a multi-quadratic singularity of type} $2^l$, $l\geqslant 2$. {\bf Proof.} Assume the converse: the point $o$ is a quadratic singularity of $F$. Let $P\ni o$ be a general $(k+3)$-dimensional linear subspace in ${\mathbb P}^{M+k}$. By the condition (MQ1) and Theorem 1.1 the intersection $F\cap P$ is a three-dimensional variety with the unique singular point $o$, which is a non-degenerate quadratic singularity. This intersection can be constructed in two steps: first, we consider the inetrsection $F\cap P'$ with a general linear subspace $P'\subset{\mathbb P}^{M+k}$, $P'\ni o$, of dimension $$ k+\mathop{\rm codim}(\mathop{\rm Sing}F\subset F) $$ and after that the intersection with a general subspace $P\subset P'$, $P\ni o$, of dimension $(k+3)$. Now we get: the pair $$ \left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) $$ is not log canonical, but canonical out side the point $o$. Let us consider the blow up $$ \varphi_P\colon P^+\to P $$ of the point $o$ with the exceptional divisor ${\mathbb E}_P\cong{\mathbb P}^{k+2}$ and let $(F\cap P)^+\subset P^+$ be the strict transform of $F\cap P$ on $P^+$, so that $(F\cap P)^+\to F\cap P$ is the blow up of the quadratic singularity $o$ with the exceptional divisor $E_P=(F\cap P)^+\cap{\mathbb E}_P$, which is a non-singular two-dimensional quadric in the three-dimensional subspace $\langle E_P\rangle\subset {\mathbb E}_P$. Obviously, $a(E_P, F\cap P)=1$, so that, writing down $$ D_P=D_F|_{F\cap P}\sim n(D_F)H_{F\cap P} $$ and $D_P^+\sim n(D_F)H_{F\cap P}-\nu E_P$ (the strict transform of $D_P$ on $(F\cap P)^+$), we obtain two options: \begin{itemize} \item either $\nu>2n(D_F)$, so that $E_P$ is a non log canonical singularity of the pair $(F\cap P,\frac{1}{n(D_F)} D_P)$, \item or $n(D_F)<\nu\leqslant 2n(D_F)$, and then the closed set $$ \mathop{\rm LCS}\left(\left(F\cap P,\frac{1}{n(D_F)} D_P\right), (F\cap P)^+\right) $$ --- the union of the centres of all non log canonical singularities of the original pair $(F\cap P,\frac{1}{n(D_F)} D_P)$ on $(F\cap P)^+$ is a connected closed subset of the non-singular quadric $E_P$, which can be either a (possibly reducible) connected curve $C_P\subset E_P$, or a point $x_P\in E_P$. \end{itemize} (It is well known, see, for instance, \cite[Chapter 2, Proposition 3.7]{Pukh13a}, that the inequality $\nu\leqslant n(D_F$) is impossible.) In the case $\nu>2n(D_F)$ we get $$ \mathop{\rm mult}\nolimits_o D_P=\mathop{\rm mult}\nolimits_o D_F> 4n(D_F), $$ which contradicts Theorem 1.3, so that this case is impossible. Coming back to the original variety $F$, let us consider the blow ups $\varphi_{\mathbb P}\colon({\mathbb P}^{M+k})^+\to{\mathbb P}^{M+k}$ and $\varphi\colon F^+\to F$ of the point $o$, where $F^+$ is identified with the strict transform of $F$ on $({\mathbb P}^{M+k})^+$, with the exceptional divisors ${\mathbb E}$ and $E$, respectively, so that $E=F^+\cap{\mathbb E}$ is a quadratic hypersurface $E$ in the subspace $\langle E\rangle\subset{\mathbb E}$ of codimension $(k+1)$. By the condition for the rank (MQ1) the case of a point $x_P\in E_P$ is impossible: in that case the quadric $E$ would contain a linear subspace of codimension 2 (with respect to $E$), which can not happen. Now, arguing word for word as in \cite[Subsection 3.2]{Pukh2022a} and using \cite[Theorem 3.1]{Pukh2022a}, we get that on the quadric $E$ there is a hyperplane section $\Lambda\subset E$, such that $$ \nu+\mathop{\rm mult}\nolimits_{\Lambda}D^+_F>2n(D_F). $$ Taking the linear system $|H_F-\Lambda|$ (of the projective dimension $(k-1)$) of hyperplane sections of the variety $F$, a general divisor $W\in|H_F-\Lambda|$ in which contains the point $o$ and its strict transform $W^+$ cuts out $\Lambda$ on $E$ (that is, $W^+\cap E=\Lambda$), we set $D_W=(D_F\circ W)$ and obtain the inequality $$ \mathop{\rm mult}\nolimits_oD_W=2(\nu+\mathop{\rm mult}\nolimits_{\Lambda}D^+_F)>4n(D_F)=4n(D_W), $$ which contradicts Theorem 1.3. This completes the proof of Proposition 1.3. {\bf 1.7. Exclusion of the multi-quadratic case.} This is the hardest and the longest part of our work. Fix a point $o\in B$ of general position, which by what was proven is a multi-quadratic singularity of type $2^l$, satisfying the conditions (MQ1,2). The pair $(F,\frac{1}{n(D_F)}D_F)$ has a non-canonical singularity, the centre $B$ of which is a component of the maximal dimension of the set $\mathop{\rm CS}(F,\frac{1}{n(D_F)}D_F)$, so that in a neighborhood of the point $o$ this pair is canonical outside $B$. We will show that this is impossible. This will be done in a few steps, and now we describe the scheme of the proof and state the key intermediate claims. {\bf Definition 1.3.} A pair $[X,o]$, where $$ X\subset{\mathbb P}(X)={\mathbb P}^{N(X)} $$ is an irreducible reduced factorial complete intersection of type $\underline{d}$ in the projective space ${\mathbb P}(X)$, $\mathop{\rm dim}X=N(X)-k\geqslant 3$, and $o\in X$ is a point, is called a {\it complete intersection with a marked point} or, for brevity, a {\it marked complete intersection of level} $(l_X,c_X)$), where $l_X$, $c_X$ are positive integers, satisfying the inequalities $$ 2\leqslant l_X\leqslant k\quad\mbox{and} \quad c_X\geqslant l_X+4, $$ if the following conditions are satisfied: (MC1) the inequality $$ \mathop{\rm codim}(\mathop{\rm Sing}X\subset X)\geqslant c_X $$ holds, (MC2) the point $o\in X$ is a multi-quadratic singularity of type $2^{l_X}$, the rank of which satisfies the inequality $$ \mathop{\rm rk}(o\in X)\geqslant 2l_X+c_X-1, $$ (MC3) the non-singular part $X\setminus\mathop{\rm Sing}X$ of the variety $X$ satisfies the condition of divisorial canonicity, $$ \mathop{\rm ct}(X\setminus\mathop{\rm Sing}X)\geqslant 1, $$ that is, for every effective divisor $A\sim aH_X$ we have $\mathop{\rm CS}(X,\frac{1}{a}A)\subset \mathop{\rm Sing}X$. The non-singular set of integers $$ I_X=[k+l_X+3,k+c_X-1]\cap{\mathbb Z} $$ is called the {\it admissible set} of the marked complete intersection $[X,o]$. {\bf Remark 1.1.} (i) Since $X\subset{\mathbb P}(X)$ is a complete intersection, the factoriality of the variety $X$ follows from Grothendieck's theorem \cite{CL} by the condition (MC1). For that reason $\mathop{\rm Pic}X={\mathbb Z}H_X$, where $H_X$ is the class of a hyperplane section. (ii) By (MC1) for every $m\leqslant k+c_X-1$ and a general subspace $P\ni o$ of dimension $m$ in ${\mathbb P}(X)$ the point $o$ is the only singularity of the variety $X\cap P$. (iii) Let ${\mathbb P}(X)^+\to{\mathbb P}(X)$ be the blow up of the point $o$ with the exceptional divisor ${\mathbb E}_X\cong{\mathbb P}^{N(X)-1}$. The strict transform $X^+\subset{\mathbb P}(X)^+$ is the result of blowing up the point $o$ on $X$ with the exceptional divisor $E_X=X^+\cap{\mathbb E}_X$. Obviously, $E_X$ is an irreducible reduced non-degenerate complete intersection of $l_X$ quadrics in a linear subspace of codimension $(k-l_X)$ in ${\mathbb E}_X$ (this follows from (MC2), see Proposition 1.4). {\bf Proposition 1.4.} {\it The following inequality holds:} $$ \mathop{\rm codim}(\mathop{\rm Sing}E_X\subset E_X)\geqslant c_X. $$ {\bf Proof} see in \S 4 (Subsection 4.2; by the condition (MQ2) the claim of the proposition follows from Proposition 4.2, (ii)). {\bf Remark 1.2.} Proposition 1.4 implies the estimate $$ \mathop{\rm codim}(\mathop{\rm Sing}E_X\subset{\mathbb E}_X)\geqslant k+c_X. $$ Therefore for every $m\leqslant k+c_X$ and a general subspace $P\ni o$ of dimension $m$ in ${\mathbb P}(X)$ the strict transform $P^+\subset{\mathbb P}(X)^+$ does not meet the set $\mathop{\rm Sing}E_X$, since $P^+\cap{\mathbb E}_X$ is a general linear subspace of dimension $m-1\leqslant k+c_X-1$ in ${\mathbb E}_X$. Therefore, for $m=k+c_X$ an isolated, and for $m\leqslant k+c_X-1$ the unique singularity $o$ of the variety $X\cap P$ is resolved by the blow up of that point, and moreover the exceptional divisor $$ E_{X\cap P}=P^+\cap E_X $$ of that blow up is a non-singular complete intersection of $l_X$ quadrics in the linear subspace of codimension $(k-l_X)$ in ${\mathbb E}_{X\cap P}=P^+\cap{\mathbb E}_X$. The discrepancy of that exceptional divisor is $$ a(E_{X\cap P})=a(E_{X\cap P}, X\cap P)=m-1-k-l_X, $$ so that for $m=k+l_X+3$ we have: $a(E_{X\cap P})=2$. The meaning of the lower end of the admissible set is in that equality. In the following definition we use the notations of Remarks 1.1 and 1.2. We continue to consider a marked complete intersection $[X,o]$ of level $(l_X,c_X)$. {\bf Definition 1.4.} A triple $(X,D,o)$, where $D\sim n(D)H_X$ is an effective divisor on $X$, $n(D)\geqslant 1$, is called a {\it working triple}, if for a general subspace $P\ni o$ of dimension $k+c_X-1$ in ${\mathbb P}(X)$ the pair \begin{equation}\label{28.11.22.1} \left(X\cap P,\frac{1}{n(D)}D|_{X\cap P}\right) \end{equation} is not log canonical at the point $o$. {\bf Remark 1.3.} Since the point $o$ is the unique singularity of the variety $X\cap P$, and by (MC3) the pair (\ref{28.11.22.1}) is canonical outside the point $o$, there is a non log canonical singularity of that pair, the centre of which on $X\cap P$ is precisely the point $o$. By inversion of adjunction, the same is true for a general subspace $P\ni o$ of dimension $m\leqslant k+c_X-2$. Let us introduce one more notation. For the strict transform $D^+$ of the divisor $D$ on $X^+$ write $$ D^+\sim n(D)H_X-\nu(D)E_X $$ (in order to simplify the notations, the pull back of the divisorial class $H_X$ on $X^+$ is denoted by the same symbol $H_X$). Respectively, for a general subspace $P\ni o$ in ${\mathbb P}(X)$ of dimension $m\leqslant k+c_X-1$ we have $$ D_P=D|_{X\cap P}\sim n(D)H_{X\cap P} $$ and $$ D^+_P\sim n(D)H_{X\cap P}-\nu(D)E_{X\cap P}, $$ where $H_{X\cap P}=H_X|_{X\cap P}$ is the class of a hyperplane section of the variety $X\cap P\subset P\cong{\mathbb P}^m$. {\bf Proposition 1.5.} {\it Assume that $c_X\geqslant 2l_X+4$. Then the inequality} $\nu(D)>n(D)$ {\it holds.} {\bf Proof} is given in \S 3 (Subsection 3.2). Let us come back to the task of excluding the multi-quadratic case. Recall that $F\in{\cal F}$, so that we can use the conditions (MQ1,2) and the statement of Theorem 1.4. We fix a point of general position $o\in B$, where $B$ is an irreducible component of the maximal dimension of the closed set $\mathop{\rm CS}(F,\frac{1}{n(D_F)}D_F)$. {\bf Proposition 1.6.} {\it The pair $[F,o]$ is a marked complete intersection of level $(l,c_F)$, where $c_F=4k+2\varepsilon(k)$, and $(F,D_F,o)$ is a working triple.} {\bf Proof} is given in \S 3 (Subsection 3.1). Assume now that $l\leqslant k-1$. The symbol $T_oF$ stands again for a subspace of codimension $(k-l)$ of the projective space ${\mathbb P}^{M+k}$. Set $$ T=F\cap T_oF. $$ This is subvariety of codimension $(k-l)$ in $F$ and a complete intersection of type $\underline{d}$ in ${\mathbb P}(T)=T_oF$. {\bf Remark 1.4.} Let us state here two well known facts which we will use many times in the sequel: when a quadratic form is restricted to a hyperplane, its rank either remains the same or drops by 1 or 2; when a complete intersection in the projective space is intersected with a hyperplane, the codimension of its singular locus either remains the same or drops by 1 or 2 (for a proof of the second claim, see \cite{IP} or \cite{Pukh00a}). If $l=k$, then for uniformity of notations we set $T=F$. {\bf Proposition 1.7.} {\it The pair $[T,o]$ is a marked complete intersection of level $(k,c_T)$, where $c_T=2k+2\varepsilon(k)+4$. There is an effective divisor $D_T\sim n(D_T)H_T$ on $T$, such that $(T,D_T,o)$ is a working triple.} {\bf Proof} is given in \S 3 (Subsection 3.4). Proposition 1.5 (taking into account Remark 1.4) implies that $\nu(D_T)>n(D_T)$. Now the main stage in the exclusion of the multi-quadratic case (and thus in the proof of Theorem 0.1) is given by the following claim. {\bf Proposition 1.8.} {\it There is a sequence of marked complete intersections $$ [R_0=T,o],\quad [R_1,o],\quad\dots,\quad [R_a,o], $$ where $a\leqslant\varepsilon(k)$ and ${\mathbb P}(R_{i+1})$ is a hyperplane in ${\mathbb P}(R_{i})$, containing the point $o$, and of effective divisors $D_i\sim n(D_i)H_{R_i}$ on $R_i$, $n(D_i)\geqslant 1$, such that $D_0=D_T$ and $$ (R_0,D_0,o),\quad (R_1,D_1,o),\quad\dots,\quad (R_a,D_a,o) $$ are working triples, and moreover for every $i=0,\dots,a-1$ the inequality $$ 2-\frac{\nu(D_{i+1})}{n(D_{i+1})}<\frac{1}{1+\frac{1}{k}}\left( 2-\frac{\nu(D_i)}{n(D_i)}\right) $$ holds and} $\nu(D_a)>\frac32 n(D_a)$. {\bf Proof} is given in \S 3 (Subsection 3.5) and \S 5. Now let us complete the exclusion of the multi-quadratic case. The variety $R_a$ is a section of $T=F\cap T_oF$ by a subspace of codimension $\leqslant\varepsilon(k)$, containing the point $o$, and $D_a$ is an effective divisor on $R_a$, satisfying the inequality $$ \mathop{\rm mult}\nolimits_o D_a=2^k\nu(D_a)>\frac32\cdot 2^kn(D_a). $$ This contradicts Theorem 1.4. The contradiction completes the proof of divisorial canonicity of the variety $F\in{\cal F}$. \section{Fano-Mori fibre spaces} In this section we prove Theorem 0.2. In Subsection 2.1 we associate with a birational map $\chi\colon V\dashrightarrow V'$ a mobile linear system $\Sigma$ on $V$ and state the key Theorem 2.1 about this system. In Subsection 2.2 we construct a fibre-wise birational modification of the fibre space $V/S$ for the system $\Sigma$. In Subsection 2.3 we consider a mobile algebraic family of irreducible curves ${\cal C}$ on $V$, and use it to prove (in Subsection 2.4) Theorem 2.1, which implies the first claim of Theorem 0.2 (that $\chi$ is fibre-wise). In Subsection 2.5 we prove the birational rigidity of the fibre space $V/S$. {\bf 2.1. The mobile linear system $\Sigma$.} Assume that the Fano-Mori fibre space $\pi\colon V\to S$ satisfies all conditions of Theorem 0.2. Fix a fibre space $\pi'\colon V'\to S'$ that belongs to one of the two classes: either the class of rationally connected fibre spaces (and then we say that the rationally connected case is being considered), or the class of Mori fibre spaces in the sense of Subsection 0.2 (and then we say that the case of a Mori fibre space is being considered). We will study both cases simultaneously. In the rationally connected case let $Y'\ni\mathop{\rm Pic}S'$ be a very ample class. Set $$ \Sigma'=|(\pi')^*Y'|=|-mK_V'+(\pi')^*Y'|, $$ where $m=0$. This is a mobile complete linear system on $V'$ (it defines the morphism $\pi'$). In the case of a Mori fibre space let $$ \Sigma'=|-m K_V'+(\pi')^*Y'| $$ be a complete linear system on $V'$, where $m\geqslant 0$ and $Y'$ is a very ample divisorial class on $S'$, and moreover, for $m\geqslant 1$ the system $\Sigma'$ is very ample. In both cases set $$ \Sigma=(\chi^{-1})_*\Sigma'\subset|-n K_V+{\pi}^*Y| $$ to be the strict transform of $\Sigma'$ on $V$ with respect to the birational map $\chi\colon V\dashrightarrow V'$. Note that if $m=0$ and $n=0$, then by construction of these linear systems the map $\chi$ is fibre-wise. {\bf Theorem 2.1.} {\it The following inequality holds:} $n\leqslant m$. {\bf Proof.} Assume the converse: $n>m$. In particular, if $m=0$, then $\chi$ is not fibre-wise. Let us show that this assumption leads to a contradiction. {\bf 2.2. A fibre-wise birational modification of the fibre space $V/S$.} Let $\sigma_S\colon S^+\to S$ be a composition of blow ups with non-singular centres, $$ S^+=S_N\stackrel{\sigma_{S,N}}{\to}S_{N-1} \to\dots\stackrel{\sigma_{S,1}}{\to}S_0=S, $$ where $\sigma_{S,i+1}\colon S_{i+1}\to S_i$ blows up a non-singular subvariety $Z_{S,i}\subset S_i$. Set $V_i=V\times_SS_i$ and $\pi_i\colon V_i\to S_i$; by the assumption on the stability with respect to birational modifications of the base $V_i/S_i$ is a Fano-Mori fibre space. Obviously, $$ V_{i+1}=V_i\times_{S_i}S_{i+1} $$ is the result of the blow up $\sigma_{i+1}\colon V_{i+1}\to V_i$ of the subvariety $Z_i=\pi^{-1}_i(Z_{S,i})\subset V_i$. Therefore, we get the commutative diagram $$ \begin{array}{ccccccccccccccc} V^+ & = & V_N & \stackrel{\sigma_N}{\to} & \dots & \to & V_{i+1} & \stackrel{\sigma_{i+1}}{\to} & V_i & \to & \dots & \stackrel{\sigma_1}{\to} & V_0 & = & V \\ & & \downarrow & & \dots & & \downarrow & & \downarrow & & \dots & & \downarrow & & \\ S^+ & = & S_N & \stackrel{\sigma_{S,N}}{\to} & \dots & \to & S_{i+1} & \stackrel{\sigma_{S,i+1}}{\to} & S_i & \to & \dots & \stackrel{\sigma_{S,1}}{\to} & S_0 & = & S, \end{array} $$ where the vertical arrows $\pi\colon V_i\to S_i$ are Fano-Mori fibre spaces. The symbol $\Sigma^i$ stands for the strict transform of the system $\Sigma$ on $V_i$, $\Sigma^+=\Sigma^N$. In these notations, let us consider a sequence of blow ups $\sigma_{S,*}$ such that for every $i=0,1,\dots,N-1$ $$ Z_i\subset\mathop{\rm Bs}\Sigma^i, $$ and the base set of the system $\Sigma^+$ contains entirely no fibre $\pi^{-1}_+(s_+)$, where $s_+\in S^+$ and $\pi_+=\pi_N$. (If this is true already for the original system $\Sigma$, then we set $\sigma_S=\mathop{\rm id}_S$, $S^+=S$ and $V^+=V$, there is no need to make any blow ups; but we will soon see that this case is impossible.) By the assumptions on the fibre space $V/S$ the fibre $\pi^{-1}_+(s_+)$ is isomorphic to the fibre $F_s=\pi^{-1}(s)$ of the original fibre space, where $s=\sigma_S(s_+)$. Let ${\cal T}$ be the set of all prime $\sigma_S$-exceptional divisors on $S^+$. We get: $$ \Sigma^+\subset\left|-n\sigma^*K_V+\pi^*_+\left(\sigma^*_SY-\sum_{T\in{\cal T}}b_TT\right)\right|= $$ $$ =\left|-nK^++\pi^*_+\left(\sigma^*_SY+\sum_{T\in{\cal T}} (na_T-b_T)T\right)\right|, $$ where $\sigma\colon V^+\to V$ is the composition of the morphisms $\sigma_i$, $K^+=K_{V^+}$, $b_T\geqslant 1$ and $a_T\geqslant 1$ for all $T\in{\cal T}$, $a_T=a(T,S)$ is the discrepancy of $T$ with respect to $S$. Let $\varphi\colon\widetilde{V}\to V^+$ be the resolution of singularities of the composite map $\chi_+=\chi\circ\sigma\colon V^+\dashrightarrow V'$, ${\cal E}$ the set of prime $\varphi$-exceptional divisors on $\widetilde{V}$ and $\psi=\chi\circ\sigma\circ\varphi\colon\widetilde{V}\to V'$ is a birational morphism. {\bf Proposition 2.1.} {\it For a general divisor $D^+\in\Sigma^+$ the pair $(V^+,\frac{1}{n}D^+)$ is canonical.} {\bf Proof.} Assume that this is not the case. Then there is an exceptional divisor $E\in{\cal E}$, satisfying the Noether-Fano inequality $$ \mathop{\rm ord}\nolimits_ED^+=\mathop{\rm ord}\nolimits_E\Sigma^+>na(E,V^+) $$ (we write $D^+,\Sigma^+$ instead of $\varphi^*D^+,\varphi^*\Sigma^+$ for simplicity). Set $B=\varphi(E)\subset V^+$. There are two options: (1) $\pi_+(B)=S^+$, (2) $\pi_+(B)$ is a proper irreducible closed subset $S^+$. If (1) is the case, then the fibre $F=F_s$ of general position intersects $B$. The restriction $$ \Sigma^+_F=\Sigma^+|_F\subset|-nK_F| $$ is a mobile linear system, and moreover, the pair $(F,\frac{1}{n}D^+_F)$ is not canonical for $D^+_F=D^+|_F$. This contradicts the condition $\mathop{\rm mct}(F)\geqslant 1$. Therefore, (2) is the case. Let $p\in B$ be a point of general position and $F=\pi^{-1}_+(\pi_+(p))$, so that $p\in F$. Since $F\not\subset\mathop{\rm Bs}\Sigma^+$, the restriction $D^+_F=D^+|_F$ is well defined (although the linear system $\Sigma^+_F$ may have fixed components). By inversion of adjunction the pair $(F,\frac{1}{n}D^+_F)$ is not log canonical. This contradicts the condition $\mathop{\rm lct}(F)\geqslant 1$. Q.E.D. for the proposition. Denote by the symbol $\widetilde{\Sigma}$ the strict transform of the system $\Sigma^+$ on $\widetilde{V}$. Obviously, \begin{equation}\label{05.11.22.1} \widetilde{\Sigma}=\psi^*\Sigma'=|-m\psi^*K'+\psi^*(\pi')^*Y'|, \end{equation} where $K'=K_{V'}$, that is, $\widetilde{\Sigma}$ is a complete linear system. We have another presentation for this linear system: $$ \widetilde{\Sigma}=\left|\varphi^*D^+-\sum_{E\in{\cal E}}b_EE\right|= $$ \begin{equation}\label{05.11.22.2} =\left|-n\widetilde{K}+\varphi^*\pi^*_+\left(\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T\right)+\sum_{E\in{\cal E}}(na_E-b_E)E\right|, \end{equation} where $\widetilde{K}=K_{\widetilde{V}}$, $D^+\in\Sigma^+$ is a general divisor and $a_E=a(E,V^+)$ is the discrepancy. {\bf 2.3. The mobile system of curves.} Take a family of irreducible curves ${\cal C'}$ on $V'$, contracted by the projection $\pi'$, sweeping out a Zariski dense subset of the variety $V'$ and not meeting the set where the birational map $\psi^{-1}$ is not determined. Assume that for a general pair of points $p,q$ in a fibre of general position of the projection $\pi'$ there is a curve $C'\in {\cal C'}$, containing the both points. In the rationally connected case the curves of the family ${\cal C'}$ are rational (the existence of such family is shown in \cite[Chapter II]{Kol96}), in the case of a Mori fibre space we do not require this. For a curve $C'\in{\cal C'}$ set $\widetilde{C}=\psi^{-1}(C')$ (at every point of the curve $C'$ the map $\psi^{-1}$ is an isomorphism), thus we get a family $\widetilde{{\cal C}}$ of irreducible curves on $\widetilde{V}$. Both in the rationally connected case and the case of a Mori fibre space the inequality $$ (C'\cdot K')<0 $$ holds, so that $(\widetilde{C}\cdot\widetilde{K})=(C'\cdot K')<0$. Furthermore, $$ (\widetilde{C}\cdot\widetilde{D})=(C'\cdot D')=-m(C'\cdot K')\geqslant 0, $$ and $(\widetilde{C}\cdot\widetilde{D})=0$ if and only if $m=0$ (since obviously $(C'\cdot(\pi')^*Y')=0$). Let ${\cal C}^+=\varphi_*\widetilde{{\cal C}}$ be the image of the family $\widetilde{{\cal C}}$ on $V^+$ and ${\cal C} =\sigma_*{\cal C}^+$ its image on $V$. {\bf Proposition 2.2.} {\it The curves $C\in{\cal C}$ are not contracted by the projection} $\pi$. {\bf Proof.} Assume the converse: $\pi(C)$ is a point on $S$. By the construction of the family ${\cal C}'$ this means that the map $\chi^{-1}$ is fibre-wise: there is a rational dominant map $\beta'\colon S'\dashrightarrow S$, such that the diagram $$ \begin{array}{ccc} V & \stackrel{\phantom{xxx}\chi^{-1}}{\dashleftarrow} & V'\\ \downarrow & & \downarrow \\ S & \stackrel{\phantom{xx}\beta'}{\dashleftarrow} & S' \end{array} $$ is commutative, and moreover, $\dim S' > \dim S$ (otherwise $\beta'$ is birational and then $\chi$ is fibre-wise, contrary to our assumption). In that case for a point $s\in S$ of general position the fibre $F_s=\pi^{-1}(s)$ is birational to $(\pi')^{-1}(\beta')^{-1}(s)$. Here $\dim (\beta')^{-1}(s)\geqslant 1$ and either the fibre $(\pi')^{-1}(s')$ for a point $s'\in (\beta')^{-1}(s)$ of general position is rationally connected, or the anti-canonical class of the variety $(\pi')^{-1}(\beta')^{-1}(s)$ is $\pi'$-ample, and we get a contradiction with the condition $\mathop{\rm mct} (F_s)\geqslant 1$ (the fibre $F_s$ is a birationally superrigid Fano variety). Q.E.D. for the proposition. For a general curve $C\in{\cal C}$ set $$ \pi_*C=d_C\overline{C}, $$ where $d_C\geqslant 1$. Replacing, if necessary, the family $\cal {C}'$ by some open subfamily, we may assume that the integer $d_C$ does not depend on $C$. For the corresponding curve $C^+\in\cal{C}^+$ we have $(\pi_+)_*C^+=d_C\overline{C}^+$, where $\overline{C}^+$ is the strict transform of the curve $\overline{C}$ on $S^+$. {\bf 2.4. Proof of Theorem 2.1.} Recall that we assume that $n>m$. Using the two presentations (\ref{05.11.22.1}) and (\ref{05.11.22.2}) for the class of a divisor $\widetilde{D}\in\widetilde{\Sigma}$, we get $$ d_C\left(\overline{C}^+\cdot\left(\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T\right)\right)+\sum_{E\in{\cal E}}(na_E-b_E)(\overline{C}\cdot E)= (n-m)(\widetilde{C}\cdot\widetilde{K})<0, $$ whence, taking into account the inequalities $b_E\leqslant na_E$ for all $E\in{\cal E}$ (Proposition 2.1), it follows that $$ \left(\overline{C}^+\cdot \left(\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T\right)\right)<0. $$ However, the class $Y$ is pseudo-effective, so that $$ (\overline{C}^+\cdot\sigma^*_S Y)=(\overline{C}\cdot Y)\geqslant 0, $$ and $(\overline{C}^+\cdot T)\geqslant 0$ for all $T\in{\cal T}$, so that ${\cal T}\neq\emptyset$ and for some $T\in{\cal T}$, such that $(\overline{C}^+\cdot T)>0$, the inequality $b_T>na_T$ holds. Since $a_T\geqslant 1$ for all $T\in{\cal T}$, we conclude that $$ \left(\overline{C}^+\cdot\left(\sigma^*_SY-\sum_{T\in{\cal T}} b_T T \right)\right)< -n\left(\overline{C}^+\cdot \sum_{T\in{\cal T}} a_T T \right)\leqslant -n. $$ For a general curve $\overline{C}^+$ consider the algebraic cycle of the scheme-theoretic intersection $$ (D^+\circ \pi_+^{-1}(\overline{C}^+))=\left(\left(\sigma^* D-\pi_+^*\left(\sum_{T\in{\cal T}}b_T T\right)\right)\circ\pi_+^{-1}(\overline{C}^+)\right). $$ The numerical class of that effective cycle is $$ n(\sigma^*(-K_V)\cdot\pi^{-1}_+(\overline{C}^+))+\left(\overline{C}^+ \cdot\left(\sigma^*_S Y-\sum_{T\in{\cal T}}b_TT\right)\right)F $$ (where $F$ is the class of a fibre of the projection $\pi_+$), and the class of the effective cycle $\sigma_*(D^+\circ\pi^{-1}_+(\overline{C}^+ ))$ in the numerical Chow group is $$ -n(K_V\cdot\pi^{-1}(\overline{C}))+bF, $$ where $b<-n$. This contradicts the condition (iii) of Theorem 0.2. the proof of Theorem 2.1 is complete. Therefore, in both cases (that of a rationally connected fibre space and of a Mori fibre space) the map $\chi$ is fibre-wise. The first claim of Theorem 0.2 (in the rationally connected case) is shown. It remains to prove the birational rigidity. {\bf 2.5. Proof of birational rigidity.} Starting from this moment, we assume that $V'/S'$ is a Mori fibre space and the birational map $\chi\colon V\dashrightarrow V'$ is fibre-wise, however, the corresponding map of the bases $\beta\colon S\dashrightarrow S'$ is not birational: $\mathop{\rm dim}S>\mathop{\rm dim}S'$ and the fibres $\beta^{-1}(s')$ for $s'\in S'$ are of positive dimension. We have to obtain a contradiction, showing that this case is impossible. First of all, let us consider the fibre-wise modification of the fibre space $V/S$ (Subsection 2.2). Now we will need a composition of blow ups $\sigma_S\colon S^+\to S$ with non-singular centres such that as in Subsection 2.2, none of the fibres of the Fano-Mori fibre space $V^+/S^+$ is contained in the base set $\mathop{\rm Bs}\Sigma^+$ and, in addition, $\sigma_S$ resolves the singularities of the rational dominant map $\beta\colon S\dashrightarrow S'$, that is, $$ \beta_+=\beta\circ\sigma_S\colon S^+\to S' $$ is a morphism. (So that the inclusion $Z_i=\pi^{-1}_i(Z_{S,i})\subset\mathop{\rm Bs}\Sigma^i$, see Subsection 2.2, no longer takes place for all $i=0,\dots,N-1$.) The fibre $\beta^{-1}_+(s')$ over a point $s'\in S'$ of general position is an irreducible non-singular subvariety of positive dimension. Set $G(s')=(\pi')^{-1}(s')$ and let $G^+(s')$ be the strict transform of $G(s')$ on $V^+$. Obviously, $$ G^+(s')=\pi^{-1}_+(\beta^{-1}_+(s')) $$ is a union of fibres of the projection $\pi_+$ over the points of the variety $\beta^{-1}_+(s')$. Since $\pi'\colon V'\to S'$ is a Mori fibre space, we have the equality $\rho(V')=\rho(S')+1$. Let ${\cal E}'$ be the set of all $\psi$-exceptional divisors $E'\in{\cal E}'$, satisfying the equality $\pi'(\psi'(E'))=S'$. Furthermore, let ${\cal Z}\subset\mathop{\rm Pic}\widetilde{V}\otimes{\mathbb Q}$ be the subspace, generated by the subspace $\psi^*(\pi')^*\mathop{\rm Pic}S'\otimes{\mathbb Q}$ and the classes of all $\psi$-exceptional divisors on $\widetilde{V}$, the images of which on $V'$ do not cover $S'$. Then the equality $$ \mathop{\rm Pic}\widetilde{V}\otimes{\mathbb Q}={}\mathbb Q\widetilde{K}\oplus\left(\bigoplus_{E'\in{\cal E}'}{\mathbb Q}E'\oplus {\cal Z}\right) $$ holds, in particular, the subspace in brackets is a hyperplane in $\mathop{\rm Pic}\widetilde{V}\otimes{\mathbb Q}$. Writing down the class $\widetilde{K}$ with respect to the morphisms $\varphi$ and $\psi$, we get the equality \begin{equation}\label{08.11.22.1} \varphi^*K^++\sum_{E\in{\cal E}}a^+_EE=\psi^* K'+\sum_{E'\in{\cal E}'}a'(E')E'+Z_1, \end{equation} where $Z_1\in{\cal Z}$ is some effective class, $a^+_E=a(E,V^+)$ for $\varphi$-exceptional divisors $E\in{\cal E}$ and $a'(E')=a(E',V')$ for $\psi$-exceptional divisors $E'\in{\cal E}'$, covering $S'$. Here all $a^+_E\geqslant 1$ and $a'(E')>0$. Using the $\psi$-presentation (\ref{05.11.22.1}) and the $\varphi$-presentation (\ref{08.11.22.1}) of the divisorial class $\widetilde{D}$ and expressing $K'$ from the formula (\ref{08.11.22.1}), we get the following equality in $\mathbb{\rm Pic}\widetilde{V}\otimes{\mathbb Q}$: \begin{equation}\label{05.12.22.1} (m-n)\varphi^*K^++\varphi^*\pi^*_+Y_++\sum_{E\in{\cal E}}(ma^+_E-b_E)E=m\sum_{E'\in{{\cal E}}'}a'(E')E'+Z_2, \end{equation} where $Y_+=\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T$ and $Z_2=mZ_1+\psi^*(\pi')^*Y'\in{\cal Z}$ is an effective class. Applying to both sides of (\ref{05.12.22.1}) $\varphi_*$ and restricting onto a fibre of general position of the projection $\pi_+$, we get that $$ (m-n)K^+|_{\pi_+^{-1}(s_+)} $$ is an effective class. Since $m\geqslant n$ and the fibre $\pi_+^{-1}(s_+)$ is a Fano variety, we conclude that $m=n$ and (\ref{05.12.22.1}) turns into \begin{equation}\label{08.11.22.2} \varphi^*\pi^*_+Y_++\sum_{E\in{\cal E}}(na^+_E-b_E)E=n\sum_{E'\in{{\cal E}}'}a'(E')E'+Z_2. \end{equation} By Proposition 2.1, we have $b_E\leqslant na^+_E$ for all $E\in{\cal E}$. Again we apply $\varphi_*$ and get that the class $Y_+$ is effective on $S^+$. Now let us consider the defined above fibre $G=G(s')$ of general position of the morphism $\pi'$ and its strict transforms $\widetilde{G}$ on $\widetilde{V}$ and $G^+$ on $V^+$ (the symbol $s'$ for simplicity of notations is omitted). Obviously, for every $Z\in{\cal Z}$ we have $Z|_{\widetilde{G}}=0$. Furthermore, for any linear combination with non-negative coefficients $$ \left.\left(\sum_{E'\in{\cal E'}}b'_{E'}E'\right)\right|_{\widetilde{G}} $$ is a fixed divisor on $\widetilde{G}$. Now let $\Delta$ be a very ample divisor on $S^+$. Then the restriction $\varphi^*\pi^*_+\Delta|_{\widetilde{G}}$ is mobile (recall that $\beta^{-1}_+(s')$ is a variety of positive dimension, so that $\Delta|_{\beta^{-1}_+(s')}$ is a mobile class). Therefore, $$ \varphi^*\pi^*_+\Delta\not\in\bigoplus_{E'\in{\cal E'}}{\mathbb Q}E'\oplus {\cal Z}, $$ whence we conclude that $$ \mathop{\rm Pic}\widetilde{V}\otimes{\mathbb Q}={\mathbb Q}[\varphi^*\pi^*_+\Delta]\oplus\left(\bigoplus_{E'\in{\cal E'}}{\mathbb Q}E'\oplus{\cal Z}\right). $$ However, this can not be the case. Let $F^+\subset G^+$ be a fibre of general position of the morphism $\pi_+$ and $\widetilde{F}\subset\widetilde{G}$ its strict transform on $\widetilde{V}$. Restricting (\ref{08.11.22.2}) onto $\widetilde{F}$, we obtain the equality $$ \sum_{E\in{\cal E}}(na_E^+-b_E)E|_{\widetilde{F}}= n\sum_{E'\in{\cal E'}}a'(E')E'|_{\widetilde{F}}, $$ where on the right hand side it is a linear combination of all divisors $E'|_{\widetilde{F}}$, $E'\in{\cal E'}$, with {\it positive} coefficients (it is here that we use the assumption that the singularities of the variety $V'$ are terminal, see Subsection 0.2), and on the left hand side it is a linear combination of $\varphi$-exceptional divisors $E|_{\widetilde{F}}$, $E\in{\cal E}$, with non-negative coefficients. Since by construction $\pi^*_+\Delta|_{F^+}=0$, we have $\varphi^*\pi^*_+\Delta|_{\widetilde{F}}=0$, whence it follows that the restriction of {\it every} divisorial class in $\mathop{\rm Pic}\widetilde{V}\otimes{\mathbb Q}$ onto $\widetilde{F}$ is fixed (is a linear combination of $\varphi$-exceptional divisors $E|_F,E\in{\cal E}$), which is impossible. This contradiction completes the proof of Theorem 0.2. \section{Hyperplane sections} This section is an immediate follow up of \S 1: we develop the technique of working triples and consider its first applications. {\bf 3.1. The working triple $(F,D_F,o)$.} Let us prove Proposition 1.6. Proposition 1.2, shown in Subsection 1.5, implies the condition (MC3). Theorem 1.1 gives the condition (MC1) for $c_F=4k+2\varepsilon(k)$ (the inequality $c_F\geqslant l+4$ is satisfied in the obvious way, since $l\leqslant k$). Finally, the condition (MQ1) gives precisely (MC2). Therefore, $[F,o]$ is indeed a marked complete intersection of level $(l,c_F)$. Consider a general subspace $P^{\sharp}\ni o$ of dimension $k+c_F$ in ${\mathbb P}^{M+k}$. The pair $$ \left(F\cap P^{\sharp},\frac{1}{n(D_F)}D_F|_{F\cap P^{\sharp}}\right) $$ is not canonical. By (MC1) the singularities of the variety ${F\cap P^{\sharp}}$ are zero-dimensional, and moreover, $o\in\mathop{\rm Sing}{F\cap P^{\sharp}}$ and $$ \mathop{\rm CS}\left(F\cap P^{\sharp},\frac{1}{n(D_F)}D_F|_{F\cap P^{\sharp}}\right)\subset\mathop{\rm Sing} (F\cap P^{\sharp}), $$ and the point $o$ is an (isolated) centre of some non canonical singularity of that pair. For a general subspace $P\ni o$ of dimension $k+c_F-1$ take a general hyperplane in $P^{\sharp}$, containing the point $o$. By inversion of adjunction we have the equalities $$ \{o\}=\mathop{\rm LCS}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right)= \mathop{\rm CS}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right), $$ and this is precisely (\ref{28.11.22.1}). Q.E.D. for Proposition 1.6. As we explained in Subsection 1.7, from now our work is constructing a certain special sequence of working triples. This sequence starts with the working triple $(F,D_F,o)$. In order to construct the sequence, we will need certain facts about working triples. {\bf 3.2. Multiplicity at the marked point.} Let us prove Proposition 1.5. We use the notations of Subsection 1.7, work with a working triple $(X,D,o)$, where $[X,o]$ is a marked complete intersection. Assume that $\nu(D)\leqslant 2n(D)$ (otherwise, there is nothing to prove). Since for a general subspace $P\ni o$ of dimension $m\in I_X$ the inequality $a(E_{X\cap P})\geqslant 2$ holds (see Remark 1.2), the pair $$ \left((X\cap P)^+,\frac{1}{n(D)}D^+_P\right) $$ is not log canonical, and moreover, $$ \mathop{\rm LCS}\left((X\cap P)^+,\frac{1}{n(D)}D^+_P\right)\subset E_{X\cap P}. $$ Let $B(P)\subset E_{X\cap P}$ be the centre of some non log canonical singularity of that pair. Then the inequality $$ \mathop{\rm mult}\nolimits_{B(P)}D^+_P>n(D) $$ holds, and the more so $$\mathop{\rm mult}\nolimits_{B(P)}D^+_P|_{E_{X\cap P}}>n(D). $$ Considering a general subspace $P^*\ni o$ of the minimal admissible dimension $k+l_X+3$ in $I_X$ as a general subspace of codimension $\geqslant l_X$ in a general subspace $P\ni o$ of the maximal admissible dimension $k+c_X-1$ in $I_X$ (recall that by assumption $c_X\geqslant 2l_X+4$), we see that the centre $B(P)$ of some non log canonical singularity is of dimension $\geqslant l_X$. However, $E_{X\cap P}$ is a non-singular complete intersection of $l_X$ quadrics in the projective space of dimension $l_X+c_X-2$, and the divisor $D^+_P|_{E_{X\cap P}}$ is cut out on $E_{X\cap P}$ by a hypersurface of degree $\nu(D)$ in that projective space. Therefore (for example, by \cite[Proposition 3.6]{Pukh06b}), the inequality $$ \nu(D)\geqslant\mathop{\rm mult}\nolimits_{B(P)} D^+_P|_{E_{X\cap P}} $$ holds. Therefore, $\nu(D)>n(D)$. Q.E.D. for Proposition 1.5. {\bf 3.3. Transversal hyperplane sections.} We still work with an arbitrary working triple $(X,D,o)$, where $[X,o]$ is a marked complete intersection of level $(l_X, c_X)$. {\bf Proposition 3.1.} {\it Let $R\ni o$ be the section of the variety $X$ by a hyperplane ${\mathbb P}(R)\subset{\mathbb P}(X)$, which is not tangent to $X$ at the point $o$. Then $D\neq bR$ for $b\geqslant 1$. Moreover, if $D$ contains $R$ as a component, that is, $$ D=D^*+bR, $$ where $b\geqslant 1$, then $(X,D^*,o)$ is a working triple.} {\bf Proof.} If $c_X\geqslant 2l_X+4$, then the first claim (that $D$ is not a multiple of $R$) follows immediately from Proposition 1.5: indeed, the hyperplane ${\mathbb P}(R)$ is not tangent to $X$ at the point $o$, that is, for the strict transform $R^+$ on the blow up of that point we have $$ R^+\sim H_X-E_X, $$ so that the equality $D=bR$ implies that $n(D)=b=\nu(D)$, which contradicts Proposition 1.5. However we will show now that the additional assumptions for the parameters $l_X$ and $c_X$ are not needed. By Remark 1.4 the condition (MC1) for $X$ implies the inequality \begin{equation}\label{13.09.22.1} \mathop{\rm codim}(\mathop{\rm Sing}R\subset R)\geqslant c_X-2. \end{equation} Since the hyperplane ${\mathbb P}(R)$ is not tangent to $X$ at the point $o$, this point is a multi-quadratic singularity of the variety $R$ of type $2^{l_X}$, the rank of which (by Remark 1.4 and the condition (MC2)) satisfies the inequality $$ \mathop{\rm rk}(o\in R)\geqslant 2l_X+c_X-3. $$ Consider a general linear subspace $P\ni o$ in ${\mathbb P}(X)$ of dimension $k+c_X-2$. That dimension, generally speaking, does not belong to $I_X$ and only the inequality $$ a(E_{X\cap P})\geqslant 1 $$ holds. The variety $X\cap P$ has a unique singularity, the point $o$, and its strict transform $(X\cap P)^+$ and the exceptional divisor $E_{X\cap P}$ are non-singular. The intersection $P\cap{\mathbb P}(R)$ is a general linear subspace of dimension $k+c_X-3$ in ${\mathbb P}(R)$, containing the point $o$. For that reason $R\cap P$ has a unique singularity, the point $o$, and moreover, the exceptional divisor $$ E_{R\cap P}=(R\cap P)^+\cap{\mathbb E}_X=R^+\cap E_{X\cap P} $$ is non-singular, and the map $(R\cap P)^+\to R\cap P$ is the blow up of the point $o$ on $R\cap P$, which resolves the singularities of that variety. From here, taking into account that $$ \nu(R)=1\leqslant a(E_{X\cap P}), $$ it follows that the pair $(X\cap P,R\cap P)$ is canonical. By inversion of adjunction we get that for every $m\in I_X$ and a general subspace $P^{\sharp}\ni o$ of dimension $m$ the pair $(X\cap P^{\sharp}, R\cap P^{\sharp})$ is canonical. Therefore, $D\neq bR$, $b\geqslant 1$, and the first claim of the proposition is shown. Assume now that $D=D^*+bR$, where $b\geqslant 1$. Then for a general subspace $P\ni o$ of dimension $k+c_X-1$ in ${\mathbb P}(X)$ the pair $(X\cap P, \frac{1}{n(D)}D_P)$ is not log canonical at the point $o$. As we saw above, the pair $(X\cap P, R_P)$ is log canonical (and even canonical). The condition of being log canonical is linear, so we conclude that the pair $$ \left(X\cap P, \frac{1}{n(D^*)} D^*|_{X\cap P}\right) $$ is not log canonical at the point $o$. Therefore, $(X,D^*,o)$ is a working triple. Q.E.D. for the proposition. {\bf Theorem 3.1 (on the transversal hyperplane section).} {\it Let $[X,o]$ be a marked complete intersection of level $(l_X=k,c_X)$, where $c_X\geqslant k+6$, and $(X,D,o)$ a working triple. Let $R\ni o$ be a hyperplane section, which is not a component of the divisor $D$. Assume that the inequality $\mathop{\rm ct}(R\backslash\mathop{\rm Sing}R)\geqslant 1$ holds. Then $(R,(D\circ R),o)$ is a working triple on the marked complete intersection $[R,o]$ of level $(l_R=k,c_R)$, where} $c_R=c_X-2$. {\bf Proof.} First of all, let us check that $[R,o]$ is a marked complete intersection of level $(k,c_R)$. The inequality $c_R\geqslant k+4$ holds by assumption. Furthermore, $$ \mathop{\rm codim} (\mathop{\rm Sing}R\subset R) \geqslant c_X-2=c_R, $$ so that the condition (MC1) is satisfied. Furthermore, the point $o\in R$ is a multi-quadratic singularity, the rank of which satisfies the inequality $$ \mathop{\rm rk}(o\in R)\geqslant\mathop{\rm rk}(o\in X)-2\geqslant 2k+c_R-1, $$ so that the condition (MC2) holds. The condition (MC3) holds by assumption. The bound for the codimension of the singular set $\mathop{\rm Sing}R$ guarantees that the complete intersection $R\subset{\mathbb P}(R)$ is irreducible, reduced and factorial. Therefore, $[R,o]$ is a marked complete intersection of level $(k,c_R)$. Set $$ I_R=[2k+3,k+c_R-1]. $$ Obviously, $(D\circ R)\sim n(D\circ R)H_R=n(D)H_R$, where $H_R$ is the class of a hyperplane section of $R$. It remains to check that for a general subspace $P\ni o$ of dimension $k+c_R-1$ in ${\mathbb P}(R)$ the pair \begin{equation}\label{14.09.22.1} \left(R\cap P\,\frac{1}{n(D\circ R)}(D\circ R)|_{R\cap P}\right) \end{equation} is not log canonical at the point $o$. In order to do this, we present $P$ as the intersection $$ P=P^{\sharp}\cap{\mathbb P}(R), $$ where $P^{\sharp}\ni o$ is a general subspace of dimension $$ k+c_R=k+c_X-2 $$ in ${\mathbb P}(X)$. As $k+c_R\in I_X$, the point $o$ is the only singularity of the variety $X\cap P^{\sharp}$ (and the only singularity of the variety $R\cap P$), and $$ \{o\}=\mathop{\rm LCS}\left(X\cap P^{\sharp},\frac{1}{n(D)}D|_{X\cap P^{\sharp}}\right). $$ The variety $R\cap P$ is the section of the variety $X\cap P^{\sharp}$ by the hyperplane $P= P^{\sharp}\cap{\mathbb P}(R)$, containing the point $o$, so that by inversion of adjunction the pair (\ref{14.09.22.1}) is not log canonical. At the same time, it is canonical outside the point $o$ since the subspace $P\subset{\mathbb P}(R)$ is generic, the non-singular part $R\backslash\mathop{\rm Sing}R$ is divisorially canonical and the equality $\{o\}=\mathop{\rm Sing}(R\cap P)$ holds. Therefore, the pair (\ref{14.09.22.1}) is not log canonical precisely at the point $o$, which completes the proof of Theorem 3.1. {\bf 3.4. Tangent hyperplane sections.} Now let us consider a marked complete intersection $[X,o]$ of level $(l_X,c_X)$, where $l_X\leqslant k-1$. Let $R$ be the section of the variety $X$ by a hyperplane ${\mathbb P}(R)\subset{\mathbb P}(X)$, which is tangent to $X$ at the point $o$. By the symbol ${\mathbb P}(R)^+$ denote the strict transform of the hyperplane ${\mathbb P} (R)$ on ${\mathbb P}(X)^+$ and set $$ {\mathbb E}_R={\mathbb P}(R)^+\cap {\mathbb E}_X. $$ Obviously, ${\mathbb E}_R\cong{\mathbb P}^{N(X)-2}$ is the exceptional divisor of the blow up of the point $o$ on the hyperplane ${\mathbb P}(R)$. Set also $E_R=R^+\cap{\mathbb E}_X$. Obviously, $E_R\subset{\mathbb E}_R$, and $$ \mathop{\rm codim}(E_R\subset{\mathbb E}_R)=k. $$ {\bf Proposition 3.2.} {\it Assume that $c_X\geqslant l_X+5$ and the point $o\in R$ is a multi-quadratic singularity of type $2^{l_X+1}$, and moreover, the inequality $$ \mathop{\rm rk} (o\in R)\geqslant 2l_X+c_X-2 $$ holds. Then $D\neq bR$ for $b\geqslant 1$. Moreover, if the divisor $D$ contains $R$ as a component, that is, $$ D=D^*+bR, $$ where $b\geqslant 1$, then $(X,D^*,o)$ is a working triple.} {\bf Proof} is completely similar to the proof Proposition 3.1, but we give it in full details, because there some small points where the two arguments are different. The inequality (\ref{13.09.22.1}) holds in this case again. Let us use the additional assumption about the singularity $o\in R$. Consider a general linear subspace $P\ni o$ in ${\mathbb P}(X)$ of dimension $k+l_X+3$ (it is the minimal admissible dimension). We get the equality $a(E_{X\cap P})=2$. Obviously, $$ R^+\sim H_X-2E_X $$ and, respectively, on $(X\cap P)^+$ we have $$ (R\cap P)^+\sim H_{X\cap P}-2E_{X\cap P}. $$ Arguing as in the transversal case, we note that the intersection $P\cap{\mathbb P}(R)$ is a general subspace of dimension $k+l_X+2$ in ${\mathbb P}(R)$. Taking into account that by the inequality (\ref{13.09.22.1}) the inequality $$ \mathop{\rm codim}(\mathop{\rm Sing}R\subset{\mathbb P}(R))\geqslant k+c_X-2 $$ holds, and that by assumption $c_X\geqslant l_X+5$, we see that $R\cap P$ has a unique singularity, the point $o$. Furthermore, by the assumption about the rank of the singular point $o\in R$ we get the inequality $$ \mathop{\rm codim}(\mathop{\rm Sing}E_R\subset E_R)\geqslant c_X-3\geqslant l_X+2, $$ so that $$ \mathop{\rm codim}(\mathop{\rm Sing}E_R\subset{\mathbb E}_R)\geqslant k+l_X+2. $$ The exceptional divisor $E_{R\cap P}$ is the section of the subvariety $E_R\subset{\mathbb E}_R$ by a general linear subspace of dimension $k+l_X+1$, whence we conclude that the variety $E_{R\cap P}$ is non-singular. Thus we have shown that the singularity $o\in R\cap P$ is resolved by one blow up. Therefore, the pair $$ ((X\cap P)^+,(R\cap P)^+) $$ is canonical, so that the pair $$ (X\cap P),(R\cap P) $$ is canonical, too. We have shown that $D\neq bR$ for $b\geqslant 1$. By inversion of adjunction for every $m\in I_X$ and a general subspace $P^{\sharp}\ni o$ of dimension $m$ the pair $(X\cap P^{\sharp}, R\cap P^{\sharp})$ is canonical (recall that $n(R)=1$). Repeating the arguments given in the transversal case (the proof of Proposition 3.1) word for word, we complete the proof of Proposition 3.2. {\bf Remark 3.1.} If for $l_X\leqslant k-1$ the intersection $X\cap T_oX$ has the point $o$ as a multi-quadratic singularity of type $2^k$, the rank of which satisfies the inequality $$ \mathop{\rm rk}(o\in X\cap T_oX)\geqslant 2l_X+c_X-2, $$ the the assumption about the rank $\mathop{\rm rk}(o\in R)$ in the statement of Proposition 3.2 holds automatically for every tangent hyperplane at the point $o$. {\bf Theorem 3.2 (on the tangent hyperplane section).} {\it Let $[X,o]$ be a marked complete intersection of level $(l_X,c_X)$, where $2\leqslant l_X\leqslant k-1$ and $c_X\geqslant l_X+7$, and $(X,D,o)$ a working triple. Let $R$ be the section of $X$ by a hyperplane which is tangent to $X$ at the point $o$, and assume that $R$ is not a component of the divisor $D$. Assume that the point $o\in R$ is a multi-quadratic singularity of type $2^{l_R}$, where $l_R=l_X+1$, the rank of which satisfies the inequality $$ \mathop{\rm rk}(o\in R)\geqslant 2l_R+c_R-1=2l_X+c_X-1, $$ where $c_R=c_X-2$, and also that the inequality $\mathop{\rm ct}(R\backslash\mathop{\rm Sing}R)\geqslant 1$ holds. Then $(R,(D\circ R),o)$ is a working triple on the marked complete intersection $[R,o]$ of level} $(l_R,c_R)$. {\bf Proof} is similar to the transversal case (Theorem 3.1), and we just emphasize the necessary modifications. The fact that $[R,o]$ is a marked complete intersection of level $(l_R,c_R)$ is checked in the tangent case even easier than in the transversal one, because the assumption about the singularity $o\in R$ is among the assumptions of the theorem. A general subspace $P\ni o$ of dimension $k+c_R-1=k+c_X-3$ in ${\mathbb P}(R)$ is again presented as the intersection $P=P^{\sharp}\cap{\mathbb P}(R)$, where $P^{\sharp}\ni o$ is a general subspace of dimension $k+c_R\in I_X$ in ${\mathbb P}(X)$, and now, repeating the arguments in the transversal case and using inversion of adjunction, we get that $(R,(D\circ R),o)$ is a working triple. Q.E.D. for the theorem. {\bf Proof of Proposition 1.7.} We assume that $l\leqslant k-1$. recall that the symbol $T$ stands for the intersection $F\cap T_oF$; this is a subvariety of codimension $(k-l)$ in $F$. Let us construct a sequence of subvarieties $$ T_0=F\supset T_1\supset\dots\supset T_{k-l}=T, $$ where $T_{i+1}$ is the section of $T_i\ni o$ by some hyperplane ${\mathbb P}(T_{i+1})=\langle T_{i+1}\rangle\ni o$, which is tangent to $T_i$ at the point $o$. Theorem 1.1 implies that the inequality $$ c_F\geqslant l+3(k-l)+4 $$ holds (the inequality of Theorem 1.1 for the codimension $c_F$ is much stronger, but for the clarity of exposition we give the weakest estimate that is sufficient for the proof of Proposition 1.7; this remark also applies to the estimate of the rank of the multi-quadratic singularity $o\in T$ below). Furthermore, the condition (MQ2) implies that $o\in T$ is a multi-quadratic singularity of type $2^k$, and moreover, the inequality \begin{equation}\label{19.09.22.1} \mathop{\rm rk}(o\in T)\geqslant 2k+c_F-1 \end{equation} holds. Finally, by Theorem 1.2 for every hyperplane section $W$ of every subvariety $T_i$, $i=0,1,\dots,k-l$, every non-singular point $p\in W$ and every prime divisor $Y$ on $W$ the inequality \begin{equation}\label{19.09.22.2} \frac{\mathop{\rm mult}_p}{\mathop{\rm deg}}Y\leqslant\frac{2}{\mathop{\rm deg} F} \end{equation} holds. Then for all $i=0,1,\dots,k-l$ the pair $[T_i,o]$ is a marked complete intersection of level $$ (l_i=l+i,c_i=c_F-2i). $$ Indeed, the inequality $c_i\geqslant l_i+4$ is true by the definition of the numbers $l_i$, $c_i$, the condition (MC1) follows from Remark 1.4, the point $o\in T_i$ by construction is a multi-quadratic singularity of type $2^{l+1}$, and moreover, by (\ref{19.09.22.1}) we have $$ \mathop{\rm rk}(o\in T_i)\geqslant 2l+c_F-1=2l_i+c_i-1, $$ and, finally, repeating the proof of Proposition 1.1 and the arguments of Subsection 1.5 word for word, we get that by (\ref{19.09.22.2}) the condition (MC3) holds. Therefore, $[T,o]$ is a marked complete intersection of level $(k,c_{k-l})$, where $c_{k-l}=c_F-2(k-l)\geqslant k+4$. Recall (Proposition 1.6), that $c_F=4k+2\varepsilon(k)$. Since $l\geqslant 2$, the inequality $$ c_{k-l}\geqslant c_T=2k+2\varepsilon(k)+4 $$ holds, so that $[T,o]$ is a marked complete intersection of level $(k,c_T)$, as we claimed. It remains to construct the working triple $(T,D_T,o)$. We will construct a sequence of working triples $(T_i,D_i,o)$, where $i=0,1,\dots,k-l$ and $D_0=D_F$. Assume that $(T_i,D_i,o)$ is already constructed and $i\leqslant k-l-1$. Let us check that all assumptions that allow us to apply Proposition 3.2 are satisfied. Indeed, the fact that $i\leqslant k-l-1$ implies the inequality $c_i\geqslant l_i+7$. The point $o\in T_{i+1}$ is a multi-quadratic singularity of type $2^{l_i+1}$, the rank of which satisfies the inequality $$ \mathop{\rm rk}(o\in T_{i+1})\geqslant 2l_{i+1}+c_{i+1}-1=2l_i+c_i-1 $$ (see above). Applying Proposition 3.2, we remove $T_{i+1}$ from the effective divisor $D_i$ (if it is necessary) and obtain the working triple $(T_i,D^*_i,o)$, where the effective divisor $D^*_i$ does not contain $T_{i+1}$ as a component. it ie easy to see that we have all assumptions of Theorem 3.2. Set $$ D_{i+1}=(D^*_i\circ T_{i+1}). $$ Now $(T_{i+1}, D_{i+1}, o)$ is a working triple. Proof of Proposition 1.7 is complete. {\bf 3.5. Plan of the proof of Proposition 1.8.} Recall that by the condition (MQ2) the inequality $$ \mathop{\rm rk} (o\in T)\geqslant 10k^2+8k+2\varepsilon(k)+5 $$ holds. The pair $[T,o]$ is a marked complete intersection of level $(k,c_T)$, where $c_T=2k+2\varepsilon(k)+4$. Let $$ R_0=T,R_1,\dots,R_a, $$ where $a\leqslant\varepsilon(k)$, be an {\it arbitrary} sequence of subvarieties in $T$, where $R_{i+1}$ is the section of $R_i$ by the hyperplane ${\mathbb P}(R_{i+1})$ in ${\mathbb P}(R_i)$, containing the point $o$. Set $c_i=c_T-2i$, where $i=0,1,\dots,a$. {\bf Proposition 3.3.} {\it The pair $[R_i,o]$ is a marked complete intersection of level} $(k,c_i)$. {\bf Proof.} Since $a\leqslant\varepsilon(k)$, the inequality $c_i\geqslant k+4$ holds in an obvious way (in fact, $c_i\geqslant 2k+4)$. The condition (MC1) holds by Remark 1.4. The condition (MC3) is obtained by repeating the proof of Proposition 1.1 and the arguments of Subsection 1.5 word for word, taking into account Theorem 1.2. Finally, again by Remark 1.4 the inequality $$ 10k^2+8k+2\varepsilon(k)+4\geqslant 2k+c_i+2i-1 $$ implies the condition (MC2). Q.E.D. for the proposition. Now let us construct for every $i=0,1,\dots,a$ an effective divisor $D_i$ on $R_i$ in the same way as we did it in Subsection 3.4 in the proof of Proposition 1.7, applying instead of Proposition 3.2 its ``transversal'' analog, Proposition 3.1, and Theorem 3.1 instead of Theorem 3.2. More precisely, if the effective divisor $D_i$, where $i\leqslant a-1$, is already constructed, we remove from this divisor all components that are hyperplane sections (if there are such components), and obtain an effective divisor $D^*_i$ that does not contain hyperplane sections as components, and such that $(R_i,D^*_i,o)$ is a working triple (Proposition 3.1). {\bf Proposition 3.4.} {\it The following inequality holds:} $$ \frac{\nu(D^*_i)}{n(D^*_i)}\geqslant\frac{\nu(D_i)}{n(D_i)}. $$ {\bf Proof.} It is sufficient to consider the case when $D^*_i$ is obtained from $D_i$ by removing one hyperplane section $Z\ni o$. Write down $$ D_i=D^*_i+bZ, $$ where $b\geqslant 1$. Since $c_i\geqslant 2k+4$, we can apply Proposition 1.5: $\nu(D_i)>n(D_i)$. On the other hand, $\nu(Z)=n(Z)=1$. Set $\nu(D_i)=\alpha n(D_i)$, where $\alpha>1$. We get $$ \frac{\nu(D^*_i)}{n(D^*_i)}= \frac{\alpha n(D^*_i)+(\alpha-1)b}{n(D^*_i)}>\alpha, $$ which proves the proposition. Q.E.D. (If we remove from $D_i$ a hyperplane section that does not contain the point $o$, the claim of Proposition 3.4 is trivial.) Now we apply Theorem 3.1, setting $D_{i+1}=(D^*_i\circ R_{i+1})$, this cycle of the scheme-theoretic intersection is well defined as an effective divisor on $R_{i+1}$, and moreover, $(R_{i+1}, D_{i+1}, o)$ is a working triple and $$ \frac{\nu(D_{i+1})}{n(D_{i+1})}\geqslant\frac{\nu(D^*_i)}{n(D^*_i)} \geqslant\frac{\nu(D_i)}{n(D_i)}. $$ We emphasize that $R_1,\dots, R_a$ is an arbitrary sequence of consecutive hyperplane sections. By Remark 1.4, for all $i=0,1,\dots,a$ the inequality $c_i\geqslant 2k+4$ holds, and the rank of the multi-quadratic singularity $o\in R_i$ of type $2^k$ is at least $10k^2+8k+5$. By Theorem 1.4 and Proposition 1.5 w have the inequalities $$ n(D_i)<\nu(D_i)\leqslant\frac32n(D_i). $$ Therefore, at every step of our construction the assumptions of the following claim are satisfied. {\bf Theorem 3.3 (on the special hyperplane section).} {\it Let $[X,o]$ be a marked complete intersection of level $(k,c_X)$, where $c_X\geqslant 2k+4$ and the inequality $$ \mathop{\rm rk}(o\in X)\geqslant 10k^2+8k+5 $$ holds. Let $(X,D,o)$ be a working triple, where the effective divisor $D$ does not contain hyperplane sections and satisfies the inequalities $$ n(D)<\nu(D)\leqslant\frac32n(D). $$ Then there is a section $R\ni o$ of the variety $X$ by a hyperplane ${\mathbb P}(R)=\langle R\rangle\subset {\mathbb P}(X)= {\mathbb P}^{N(X)}$, such that the effective divisor $D_R=(R\circ D)$ on $R$ satisfies the inequality} $$ 2-\frac{\nu(D_R)}{n(D_R)}<\frac{1}{1+\frac{1}{k}}\left(2- \frac{\nu(D)}{n(D)}\right). $$ Now by the definition of the integer $\varepsilon(k)$ and what was said above, Theorem 3.3 immediately implies Proposition 1.8. {\bf Proof} of Theorem 3.3 is given in \S 5. \section{Multi-quadratic singularities} In this section we consider the properties of multi-quadratic singularities, the rank of which is bounded from below: they are factorial, stable with respect to blow ups and terminal. In Subsection 4.5 we study linear subspaces on complete intersections of quadrics and the properties of projections from these subspaces. {\bf 4.1. The definition and the first properties.} Let ${\cal X}$ be an (irreducible) algebraic variety, $o\in{\cal X}$ a point. {\bf Definition 4.1.} The point $o$ s a {\it multi-quadratic singularity} of the variety ${\cal X}$ of type $2^l$ and rank $r\geqslant 1$, if in some neighborhood of this point ${\cal X}$ can be realized as a subvariety of a non-singular $N=(\mathop{\rm dim} {\cal X}+ l)$-dimensional variety ${\cal Y}\ni o$, and for some system $(u_1, \dots, u_N)$ of local parameters on ${\cal Y}$ at the point $o$ the subvariety ${\cal X}$ is the scheme of common zeros of regular functions $$ \alpha_1,\dots,\alpha_l\in{\cal O}_{o,{\cal Y}}\subset{\mathbb C}[[u_1,\dots,u_N]], $$ which are represented by the formal power series $$ \alpha_i=\alpha_{i,2}+\alpha_{i,3}+\dots, $$ where $\alpha_{i,j}(u_1,\dots,u_N)$ are homogeneous polynomials of degree $j$ and $$ \mathop{\rm rk}(\alpha_{1,2},\dots,\alpha_{l,2})=r. $$ (Obviously, the order of the formal power series, representing $\alpha_i$, and the rank of the tuple of quadratic forms $\alpha_{i,2}$ do not depend on the choice of the local parameters on ${\cal Y}$ at the point $o$.) It is convenient to work in a more general context. Assume that in a neighborhood of the point $o$ the variety ${\cal X}$ is realized as a subvariety ${\cal X}\subset{\cal Z}$, where $\mathop{\rm dim}{\cal Z}=\mathop{\rm dim}{\cal X}+e=N({\cal Z})$, and for a certain system of local parameters $(v_1,\dots,v_{N({\cal Z})})$ on ${\cal Z}$ at the point $o$ the subvariety ${\cal X}$ is the scheme of common zeros of regular functions $$ \beta_1,\dots,\beta_e\in{\cal O}_{o,{{\cal Z}}}\subset{\mathbb C}[[v_*]], $$ which are represented by the formal power series $$ \beta_i=\beta_{i,1}+\beta_{i,2}+\dots, $$ where $\beta_{i,j}(v_*)$ are homogeneous polynomials of degree $j$. Assume that for some $l\in\{0,1,\dots,e\}$ $$ \mathop{\rm dim}\langle\beta_{1,1},\dots,\beta_{e,1}\rangle=e-l, $$ where we assume (for the convenience of notations), that the linear forms $\beta_{j,1}$ for $l+1\leqslant j\leqslant e$ are linearly independent, so that for $1\leqslant i\leqslant l$ and $l+1\leqslant j\leqslant e$ there are uniquely determined numbers $a_{i,j}$, such that $$ \beta_{i,1}=\sum^e_{j=l+1}a_{i,j}\beta_{j,1}. $$ Set ${\cal Y}=\{\beta_j=0\,|\, l+1\leqslant j\leqslant e\}$ and $$ \beta_i^*=\beta_i-\sum_{j=l+1}^e a_{i,j}\beta_j. $$ Then (in a neighborhood of the point $o$) the variety ${\cal Y}$ is non-singular, and ${\cal X}\subset {\cal Y}$ is realized as the scheme of common zeros of the regular functions $\beta^*_i$, $1\leqslant i\leqslant l$. Set $$ T_o{\cal Y}=T_o{\cal X}=\{\beta_{j,1}=0\,|\, l+1\leqslant j\leqslant e\}. $$ If $$ \mathop{\rm rk}\left(\beta^*_{i,2}|_{T_o{\cal X}}\, |\, 1\leqslant i\leqslant l\right)=r, $$ then obviously $o\in{\cal X}$ is a multi-quadratic singularity of rank $r$. The rank of the multi-quadratic point $o\in{\cal X}$ is denoted by the symbol $\mathop{\rm rk} (o\in{\cal X})$ or just $\mathop{\rm rk}(o)$, if it is clear which variety is meant. For uniformity of notations we treat a non-singular point as a multi-quadratic one of type $2^0$. {\bf Proposition 4.1.} {\it Assume that $o\in{\cal X}$ is a multi-quadratic singularity of type $2^l$, where $l\geqslant 1$, and of rank $r\geqslant 2l$. Then in a neighborhood of the point $o$ every point $p\in{\cal X}$ is either non-singular, or a multi-quadratic of type $2^b$, where $b\in\{1,\dots,l\}$, of rank} $\geqslant r-2(l-b)$. {\bf Proof.} Using the notations for the embedding ${\cal X}\subset{\cal Z}$, introduced above, with $e=l$ (so that $\beta_{i,1}=0$ for all $i=1,\dots,l$) and setting $N({\cal Z})=N$, consider an open set $U\subset{\cal Z}$, $U\ni o$, such that for every point $p\in U$ the ``shifted'' functions $$ v^{(p)}_i=v_i-v_i(p),\quad i=1,\dots,N, $$ form a system of local parameters at the point $p$, and in the formal expansion $$ \beta_i=\beta^{(p)}_{i,0}+\beta^{(p)}_{i,1}+\beta^{(p)}_{i,2}+\dots $$ with respect to the system of parameters $v^{(p)}_*$ the quadratic components satisfy the inequality $$ \mathop{\rm rk}(\beta^{(p)}_{i,2}\,|\, 1\leqslant i\leqslant l)\geqslant r. $$ If the point $p$ is a common zero of $\beta_1,\dots,\beta_l$, then $\beta^{(p)}_{i,0}=0$ for $1\leqslant i\leqslant l$. Set $$ T_p{\cal X}=\{\beta^{(p)}_{i,1}=0\,|\, 1\leqslant i\leqslant l\} $$ and assume (for the convenience of notations) that the forms $\beta^{(p)}_{i,1}$ for $b+1\leqslant i\leqslant l$ are linearly independent, where $$ \mathop{\rm dim}\langle\beta^{(p)}_{i,1}\,|\,\,1\leqslant i\leqslant l\rangle=l-b. $$ Since $\mathop{\rm codim}(T_p{\cal X}\subset T_p{\cal Z})=l-b$, by Remark 1.4 the inequality $$ \mathop{\rm rk}(\beta^{(p)}_{i,2}|_{T_p{\cal X}}\,|\, 1\leqslant i\leqslant l) \geqslant r - 2(l-b) $$ holds. It is easy to see from the construction of the quadratic forms $\beta^{(p)*}_{i,2}$, $1\leqslant i\leqslant b$, that every linear combination of these forms with coefficients $(\lambda_1,\dots,\lambda_b)\neq(0,\dots,0)$ is a linear combination of the original forms $\beta^{(p)}_{i,2}$, $1\leqslant i\leqslant l$, not all coefficients in which are equal to zero. Therefore, the point $p$ is a multi-quadratic singularity of rank $\geqslant r-2(l-b)$, as we claimed. Q.E.D. for the proposition. {\bf 4.2. Complete intersections of quadrics.} In the notations of Definition 4.1 let ${\cal Y}^+\to{\cal Y}$ be the blow up of the point $o$ with the exceptional divisor $E_{\cal Y}\cong{\mathbb P}^{N-1}$ and ${\cal X}^+\subset{\cal Y}^+$ the strict transform of ${\cal X}$ on ${\cal Y}^+$, so that ${\cal X}^+\to{\cal X}$ is the blow up of the point $o$ on ${\cal X}$ with the exceptional divisor $E_{\cal Y}|_{{\cal X}^+}=E_{\cal X}$. Therefore, $E_{\cal X}$ is the scheme of common zeros of the quadratic forms $\alpha_{i,2}$, $i=1,\dots,l$, on $E_{\cal Y}\cong{\mathbb P}^{N-1}$. Let $q_1,\dots,q_l$ be quadratic forms on ${\mathbb P}^{N-1}$, where $N\geqslant l+4$. By the symbol $q_{[1,l]}$ we denote the tuple $(q_1,\dots,q_l)$. {\bf Proposition 4.2.} (i) {\it Assume that the inequality $$ \mathop{\rm rk} q_{[1,l]}\geqslant 2l+3 $$ holds. Then the scheme of common zeros of the forms $q_1,\dots,q_l$ is an irreducible non-degenerate factorial variety $Q\subset{\mathbb P}^{N-1}$ of codimension $l$, that is, a complete intersection of type $2^l$.} (ii) {\it Assume that for some $e\geqslant 4$ the inequality $$ \mathop{\rm rk} q_{[1,l]}\geqslant 2l+e-1 $$ holds. Then the following inequality is true:} $$ \mathop{\rm codim}(\mathop{\rm Sing}Q\subset Q)\geqslant e. $$ {\bf Proof} is given below in Subsection 4.4. {\bf Corollary 4.1.} (i) {\it Assume that the rank of the tuple $\alpha_{*,2}=(\alpha_{1,2},\dots,\alpha_{l,2})$ of quadratic forms satisfies the inequality $$ \mathop{\rm rk} (\alpha_{*,2})\geqslant 2l+3. $$ Then in a neighborhood of the point $o$ the scheme of common zeros of the regular functions $\alpha_1,\dots,\alpha_l$ is an irreducible reduced factorial subvariety ${\cal X}$ of codimension $l$ in ${\cal Y}$.} (ii) {\it Assume that for some $e\geqslant 4$ the inequality $$ \mathop{\rm rk} (\alpha_{*,2})\geqslant 2l+e-1 $$ holds. Then in a neighborhood of the point $o$ the following inequality is true} $$ \mathop{\rm codim}(\mathop{\rm Sing}{\cal X}\subset{\cal X})\geqslant e. $$ {\bf Proof.} Both claims obviously follow from Proposition 4.2, taking into account Grothendieck's theorem \cite{CL} on the factoriality of a complete intersection, the singular set of which is of codimension $\geqslant 4$. Therefore, for $r\geqslant 2l+3$ the assumption in Definition 4.1 that ${\cal X}$ is an irreducible variety, is unnecessary: in a neighborhood of the point $o$ the scheme of common zeros of the functions $\alpha_*$ is automatically irreducible and reduced, and moreover, it is a factorial variety. This proves all claims of Theorem 1.1, except for that the singularities of the variety $F$ are terminal. {\bf 4.3. Stability with respect to blow ups.} Let $\underline{r}=(r_1,r_2,\dots,r_k)$ be a tuple of integers, satisfying the inequalities $r_{i+1}\geqslant r_i+2$ for $i=1,\dots,k-1$, where $r_1\geqslant 5$. Again, let ${\cal Y}$ be a non-singular $N$-dimensional variety, where $N\geqslant k+3$, and ${\cal X}\subset{\cal Y}$ an (irreducible) subvariety of codimension $k$, every point $o\in{\cal X}$ of which is either non-singular, or a multi-quadratic singularity of type $2^l$, where $l\in\{1,\dots,k\}$, of rank $\geqslant r_l$. Somewhat abusing the terminology, we say in this case that ${\cal X}$ has {\it multi-quadratic singularities of type} $\underline{r}$. {\bf Theorem 4.1.} {\it In the assumptions above let $B\subset{\cal X}$ be an irreducible subvariety of codimension $\geqslant 2$. Then there is an open subset $U\subset{\cal X}$, such that $U\cap B\neq\emptyset$, $U\cap B$ is non-singular and the blow up $$ \sigma_B\colon U_B\to U $$ along $B$ gives a quasi-projective variety $U_B$ with multi-quadratic singularities of type} $\underline{r}$. {\bf Proof.} If a point of general position $o\in B$ is non-singular on ${\cal X}$, there is nothing to prove. If $o\in{\cal X}$ is a multi-quadratic singularity of type $2^l$, where $l>k$, then a certain Zariski open subset $U\subset{\cal X}$, $U\ni o$, has multi-quadratic singularities of type $(r_1,\dots,r_l)$ (see Subsection 4.1), so that it is sufficient to consider the case when a point of general position $o\in B$ is a multi-quadratic point of type $2^k$ on ${\cal X}$. Passing over to an open subset, we may assume that the subvariety $B$ is non-singular. Let $(u_1,\dots,u_N)$ be a system of local parameters at the point $o$, such that $B=\{u_1=\dots=u_m=0\}$. Since $B\subset{\cal X}$, the subvariety ${\cal X}\subset{\cal Y}$ is the scheme of common zeros of regular functions $$ \beta_1,\dots,\beta_k\in{\cal O}_{o,{\cal Y}}\subset{\cal O}_{o,B}[[u_1,\dots,u_m]], $$ where for all $i=1,\dots,k$ $$ \beta_i=\beta_{i,2}+\beta_{i,3}+\dots, $$ where $\beta_{i,j}$ are homogeneous polynomials of degree $j$ in $u_1,\dots,u_m$ with coefficients from ${\cal O}_{o,B}$. Again replacing ${\cal Y}$, if necessary, by an open subset, containing the point $o$, we have $$ \beta_i\in{\cal O}({\cal Y})\subset{\cal O}(B)[[u_1,\dots,u_m]], $$ so that all coefficients of the forms $\beta_{i,j}$ are regular functions on $B$; in particular, $$ \beta_{i,2}=\sum_{1\leqslant j_1\leqslant j_2\leqslant m}A_{j_1,j_2}u_{j_1}u_{j_2}, $$ where $A_{j_1,j_2}\in{\cal O}(B)$. In terms of the embedding ${\cal O}_{o,{\cal Y}}\subset{\mathbb C}[[u_1,\dots,u_N]]$ we get the presentation $$ \beta_i=\overline{\beta}_{i,2}+\overline{\beta}_{i,3}+\dots, $$ where $\overline{\beta}_{i,j}$ is a homogeneous polynomial of degree $j$ in $u_*$, and moreover, in the right hand side there are no monomials that do not contain the variables $u_1,\dots,u_m$, or that contain precisely one of them (in the power 1): every monomial in the right hand side is divisible by some quadratic monomial in $u_1,\dots,u_m$. Let ${\cal Y}_B\to{\cal Y}$ be the blow up of the subvariety $B$ and ${\cal X}_B\subset{\cal Y}_B$ the strict transform of ${\cal X}$. Obviously, the morphism ${\cal X}_B\to{\cal X}$ in the blow up of $B$ on ${\cal X}$. The symbol $E_B$ denotes the exceptional divisors of the blow up of $B$ on ${\cal Y}$. Since outside $E_B$ the varieties ${\cal X}_B$ and ${\cal X}$ are isomorphic, it is sufficient to show that every point $p\in{\cal X}_B\cap E_B$ is either non-singular, or a multi-quadratic singularity of the variety $U_B$ of type $2^l$, where $l\geqslant 1$, and of rank $\geqslant r_l$. We assume that the point $p$ lies over the point $o\in U$ and is a singularity of the variety $U_B$. By a linear change of local parameters $u_1,\dots,u_m$ we may ensure that at the point $p\in{\cal Y}_B$ there is a system of local parameters $$ (v_1,\dots,v_m,u_{m+1},\dots,u_N), $$ linked to the original system of parameters $u_*$ by the standard relations $$ u_1=v_1,\,\,u_2=v_1v_2,\dots,\,\,u_m=v_1v_m. $$ The local equation of the exceptional divisor $E_B$ at the point $p$ is $v_1=0$, and the subvariety ${\cal X}_B\subset{\cal Y}_B$ at that point is defined by the equations $$ \widetilde{\beta}_1,\dots,\widetilde{\beta}_k\in{\cal O}_{p,{\cal Y}_B}\subset{\mathbb C}[[v_1,\dots,v_m,u_{m+1},\dots,u_N]]. $$ Write down $\widetilde{\beta}_i=\widetilde{\beta}_{i,1} +\widetilde{\beta}_{1,2}+\dots$ and assume that for some $l\in\{1,\dots,k\}$ the linear forms $\widetilde{\beta}_{j,1}$, $l+1\leqslant j\leqslant k$, are linearly independent, and moreover, $$ \mathop{\rm dim}\langle\widetilde{\beta}_{i,1}\,|\,1\leqslant i\leqslant k\rangle=k-l, $$ so that there are relations $$ \widetilde{\beta}_{i,1}=\sum^k_{j=l+1}a_{i,j}\widetilde{\beta}_{i,1}, $$ $i=1,\dots,l$. Replacing the original system of local equations $\beta_1,\dots,\beta_k$ by $$ \beta_i-\sum^k_{j=l+1}a_{i,j}\beta_j,\,\,i=1,\dots,l,\quad \beta_{l+1},\dots,\beta_k, $$ we may assume that the linear forms $\widetilde{\beta}_{i,1}$, $i=1,\dots,l$, are identically zero. In that case the following claim is true. {\bf Lemma 4.1.} {\it For $i=1,\dots,l$ the quadratic forms $\overline{\beta}_{i,2}$ depend only on $u_2,\dots,u_m$ and $$ \widetilde{\beta}_{i,2}=\overline{\beta}_{i,2}(v_2,\dots,v_m)+\beta^{\sharp}_{i,2}, $$ where every monomial in the quadratic form $\beta^{\sharp}_{i,2}$ is divisible either by $v_1$, or by} $u_i$, $m+1\leqslant i\leqslant N$. {\bf Proof.} This is obvious because every monomial in $\overline{\beta}_{i,j}$ is divisible by some quadratic monomial in $u_1,\dots,u_m$, and $\widetilde{\beta}_{i,1}\equiv 0$ for $i=1,\dots,l$, and by the standard formulas, transforming regular functions under a blow up. Q.E.D. for the lemma. The lemma gives us the inequality $$ \mathop{\rm rk}(\widetilde{\beta}_{i,2},\, 1\leqslant i\leqslant l)\geqslant\mathop{\rm rk}(\overline{\beta}_{i,2},\, 1\leqslant i\leqslant l)\geqslant r_k. $$ Setting $T_p{\cal X}_B=\{\widetilde{\beta}_{i,1}=0\,|\, l+1\leqslant j\leqslant k\}$ and using Remark 1.4, we get $$ \mathop{\rm rk}(\widetilde{\beta}_{i,2}|_{T_p{\cal X}_B},\, 1\leqslant i\leqslant l)\geqslant r_k-2(k-l)\geqslant r_l. $$ Therefore, $p\in{\cal X}_B$ is a multi-quadratic singularity of type $2^l$ and rank $\geqslant r_l$. Q.E.D. for Theorem 4.1. {\bf Corollary 4.2.} {\it Assume that $\cal X$ has multi-quadratic singularities of type $\underline{r}$, where $r_l\geqslant 3l+1$ for all $l=1,\dots,k$. Then the singularities of ${\cal X}$ are terminal.} {\bf Proof.} In the notations of the proof of Theorem 4.1 it is sufficient to show the inequality $$ a({\cal X}_B\cap E_B,{\cal X})\geqslant 1. $$ Assume that a point $o\in B$ of general position is a multi-quadratic singularity of type $2^l$. From the claim (ii) of Corollary 4.1 we get the inequality $$ \mathop{\rm codim}(B\subset{\cal X})\geqslant l+2, $$ so that $\mathop{\rm codim}(B\subset{\cal Y})\geqslant k+l+2$ and for that reason $$ a(E_B,{\cal Y})\geqslant k+l+1. $$ By the adjunction formula $$ a({\cal X}_B\cap E_B,{\cal X})=a(E_B,{\cal Y})-(k-l)-2l, $$ which implies the required inequality. Q.E.D. for the corollary. This completes the proof of Theorem 1.1. {\bf 4.4. Singularities of complete intersections.} Let us show Proposition 4.2. We will prove the claims (i) and (ii) simultaneously: by Grothendieck's theorem on parafactoriality \cite{CL,Call1994} the claim (ii) for $e=4$ implies the factoriality of the variety $Q$. We argue by induction on $l\geqslant 1$. For one quadric ($l=1$) the claims (i) and (ii) are obvious. Since $$ \mathop{\rm rk} q_{[1,l-1]}\geqslant \mathop{\rm rk} q_{[1,l]}, $$ we may assume that the claims (i) and (ii) are true for the tuple of quadratic forms $q_1$, \dots, $q_{l-1}$. In particular, the scheme of their common zeros $Q_{l-1}$ is an irreducible reduced factorial complete intersection of type $2^{l-1}$ in ${\mathbb P}^{N-1}$, so that $\mathop{\rm Pic} Q_{l-1}={\mathbb Z} H_{l-1}$, where $H_{l-1}$ is the class of a hyperplane section: every effective divisor on $Q_{l-1}$ is cut out on $Q_{l-1}$ by a hypersurface in ${\mathbb P}^{N-1}$. The scheme of common zeros of the quadratic forms $q_1$, \dots, $q_l$ is the divisor of zeros of the form $q_l$ on the variety $Q_{l-1}$. This divisor is reducible or non-reduced if and only if there is a form $q^*_l$ of rank $\leqslant 2$ such that $$ q_l-q^*_l\in\langle q_1,\dots,q_{l-1}\rangle, $$ and in that case $\mathop{\rm rk}q_{[1,l]}\leqslant 2$, which contradicts the assumption. Therefore, $Q$ is an irreducible reduced complete intersection. It is easy to see that $Q\subset{\mathbb P}^{N-1}$ is non-degenerate. Since $$ \mathop{\rm rk}q_{[1,l-1]}\geqslant 2(l-1)+(e+2)-1 $$ (for the claim (i) we set $e=4$), we have $$ \mathop{\rm codim}(\mathop{\rm Sing}Q_{l-1}\subset Q_{l-1})\geqslant e+2, $$ so that $$ \mathop{\rm codim}((Q\cap\mathop{\rm Sing}Q_{l-1})\subset Q)\geqslant e+1. $$ It is easy to see that a point $p\in Q$, which is non-singular on $Q_{l-1}$, is singular on $Q$ if and only if for some $\lambda_1,\dots,\lambda_{l-1}$ the quadric $$ q_l-\lambda_1q_1-\dots-\lambda_{l-1}q_{l-1}=0 $$ is singular at that point. Since the singular set of a quadric of rank $r$ in ${\mathbb P}^{N-1}$ has dimension $N-1-r$, we conclude that the dimension of the set $$ \mathop{\rm Sing}Q\cap(Q_{l-1}\setminus\mathop{\rm Sing}Q_{l-1}) $$ does not exceed $N-1-\mathop{\rm rk}q_{[1,l]}+(l-1)$, whence it follows that the codimension of that set with respect to $Q$ is at least $\mathop{\rm rk}q_{[1,l]}-2l+1\geqslant e$. Q.E.D. for Proposition 4.2. {\bf 4.5. Linear subspaces and projections.} Now let us consider the questions that are naturally close to Proposition 4.2 and its proof. These questions are of key importance in the proof of Theorem 3.3 (which will be given in \S 5). Since in Theorem 3.3 the multi-quadratic singularity is of type $2^k$, starting from this moment we consider $k$ quadratic forms $q_1$, \dots, $q_k$ in $N$ variables (that is, on ${\mathbb P}^{N-1}$), and the tuple of them is denoted by the symbol $q_{[1,k]}$. The symbol $Q$, as above, stands for the complete intersection of these $k$ quadrics $\{q_i=0\}$ in ${\mathbb P}^{N-1}$. {\bf Proposition 4.3.} {\it Assume that for some $b\geqslant 0$ the inequality $$ \mathop{\rm rk} q_{[1,k]}\geqslant 2(1+b)k+3 $$ holds. Then for every point $p\in Q\setminus \mathop{\rm Sing} Q$ there is a linear space $\Pi\subset {\mathbb P}^{N-1}$ of dimension $b$, such that $p\in\Pi\subset Q$, and moreover,} $\Pi\cap\mathop{\rm Sing} Q=\emptyset$. {\bf Proof} contains the (obvious) construction of such linear subspaces. We argue by induction on $b$. If $b=0$, then $\Pi$ is the point $p$ itself and there is nothing to prove. Assume that $b\geqslant 1$ and for $b-1$ the claim of the Proposition is true. Consider the linear subspace $T=T_pQ$ of codimension $k$ in ${\mathbb P}^{N-1}$. Obviously, every linear space in ${\mathbb P}^{N-1}$ that contains the point $p$ and is contained in $Q$, is contained in $T$, too. Furthermore, $Q\cap T$ is defined by the quadratic forms $q_1|_T,\dots,q_k|_T$. Since $\mathop{\rm rk} q_{[1,k]}|_T\geqslant\mathop{\rm rk} q_{[1,k]}-2k$, the inequality $$ \mathop{\rm rk} q_{[1,k]}|_T\geqslant 2bk+3 $$ holds, where every quadric $\{q_i|_T=0\}$, $i=1,\dots,k$, is by construction a cone with the vertex at $p$. Therefore, $Q\cap T$ is a cone with the vertex at the point $p$. Let $P\subset T$ be a hyperplane in $T$ that does not contain the point $p$. Then the cone $Q\cap T$ is a cone is the cone with the base $Q\cap P$, where $Q\cap P$ is a complete intersection of the quadrics $\{q_i|_P=0\}$, where, obviously, $$ \mathop{\rm rk} q_{[1,k]}|_P=\mathop{\rm rk} q_{[1,k]}|_T\geqslant 2(1+(b-1))k+3. $$ By the induction hypothesis, there is a linear subspace $\Pi^{\sharp}\subset P$ of dimension $(b-1)$, such that $\Pi^{\sharp}\subset Q\cap P$ and $\Pi^{\sharp}\cap\mathop{\rm Sing}(Q\cap P)=\emptyset$. Furthermore, the set of singular points $\mathop{\rm Sing}(Q\cap T)$ is a cone with the vertex $p$, the base of which is $\mathop{\rm Sing}(Q\cap P)$, so that for the subspace $\Pi=\langle p,\Pi^{\sharp}\rangle$, which is a cone with the vertex $p$ and the base $\Pi^{\sharp}$, we have $\Pi\cap\mathop{\rm Sing}(Q\cap T)=\{p\}$. Since $T\cap\mathop{\rm Sing}Q\subset\mathop{\rm Sing}(Q\cap T)$ and $p\not\in\mathop{\rm Sing}Q$, we get $\Pi\cap\mathop{\rm Sing}Q=\emptyset$, which completes the proof of the proposition. {\bf Proposition 4.4.} {\it Let $b\geqslant \beta\geqslant 0$ be some integers. Assume that the inequality $$ \mathop{\rm rk} q_{[1,k]}\geqslant 2k(b+\beta+1)+2\beta+3 $$ holds. Then for every linear subspace $P\subset{\mathbb P}^{N-1}$ of codimension $\beta$ and a general linear subspace $\Pi\subset Q$, $\Pi\cap\mathop{\rm Sing}=\emptyset$, of dimension $b$ the intersection $P\cap\Pi$ has codimension $\beta$ in $\Pi$.} {\bf Proof.} Again we argue by induction on $b,\beta$; the case $\beta=0$ is trivial, only the equality $\Pi\cap\mathop{\rm Sing} Q=\emptyset$ for a general subspace $\Pi\subset Q$ of dimension $\beta\geqslant 0$ is needed, and it is true by Proposition 4.3. Let us show our claim in the assumption that it is true for $\beta-1$. First of all, note that $$ \mathop{\rm rk} q_{[1,k]}|_P\geqslant 2k(b+\beta+1)+3>2k+3, $$ so that by Proposition 4.2 the intersection $Q\cap P$ is an irreducible reduced complete intersection of type $2^k$ in $P$; in particular, a point of general position $p\in Q\cap P$ is non-singular. This means that $$ T_p(Q\cap P)=T_pQ\cap P $$ is of codimension $k$ in $P$, so that $T_pQ$ and $P$ are in general position. The property to be in general position is an open property, therefore for a point of general position $p\in Q$ (in particular, $p\not\in P$) the linear subspaces $T_pQ$ and $P$ are in general position and their intersection $T_pQ\cap P$ is of codimension $k$ in $P$ and of codimension $\beta$ in $T_pQ$. Consider a general hypersurface $Z$ in $T_pQ$, containing the subspace $T_pQ\cap P$ and not containing the point $p$. We have $$ \mathop{\rm rk} q_{[1,k]}|_Z\geqslant 2k(b+\beta)+2\beta+1=2k(b+(\beta-1)+1)+2(\beta-1))+3, $$ so that by the induction hypothesis for a general linear subspace $\Pi^{\sharp}\subset Q\cap Z$ of dimension $(b-1)$ that does not meet the set $\mathop{\rm Sing}(Q\cap Z)$ the intersection $$ (P\cap T_pQ)\cap\Pi^{\sharp}=P\cap\Pi^{\sharp} $$ is of codimension $\beta-1=\mathop{\rm codim}((P\cap T_pQ)\subset Z)$ with respect to $\Pi^{\sharp}$. Then the linear space $$ \Pi=\langle p,\Pi^{\sharp}\rangle\subset T_pQ $$ of dimension $b$ is contained in $Q$, does not meet the set $\mathop{\rm Sing}Q$ (see the proof of Proposition 4.3) and, finally, the subspace $$ P\cap\Pi=P\cap T_pQ\cap\Pi=P\cap T_pQ\cap Z\cap\Pi=P\cap\Pi^{\sharp} $$ is of codimension $\beta$ with respect to $\Pi$. Q.E.D. for the proposition. {\bf Corollary 4.3.} {\it In the assumptions of Proposition 4.4, where $\beta\geqslant k$, let $Y\subset Q$ be an irreducible subvariety of codimension $\beta-k$. Then the restriction onto $Y$ of the projection $$ \mathop{\rm pr}\nolimits_{\Pi}\colon {\mathbb P}^{N-1}\dashrightarrow {\mathbb P}^{N-b-2} $$ from a general subspace $\Pi\subset Q$ of dimension $b$ is dominant.} {\bf Proof.} Let $p\in Y$ be a non-singular point. We apply Proposition 4.4 to the subspace $P=T_pY\subset{\mathbb P^{N-1}}$ of codimension $\beta$. A general subspace $\Pi\subset Q$ of dimension $b\geqslant\beta$ does not contain the point $p$ and is in general position with $P$, so that $\mathop{\rm pr}_{\Pi}|_P$ is regular in a neighborhood of the point $p$ and its differential at the point $p$ is an epimorphism. Therefore, $\mathop{\rm pr}_{\Pi}|_Y$ is regular at the point $p$ and its differential at that point is an epimorphism. Q.E.D. for the corollary. Note an important particular case. {\bf Corollary 4.4.} {\it Assume that $b\geqslant k$ and the inequality $$ \mathop{\rm rk}\nolimits_{[1,k]}\geqslant 2k(b+k+2)+3 $$ holds. Then the restriction of the projection $\mathop{\rm pr}_{\Pi}$ from a general subspace $\Pi\subset Q$ of dimension $b$ onto $Q$ is dominant and its general fibre is a linear subspace of dimension} $b+1-k$. {\bf Proof.} That it is dominant, follows from the previous corollary, so that the dimension of a general fibre is $b+1-k$. Furthermore, $\mathop{\rm pr}_{\Pi}$ fibres ${\mathbb P}^{N-1}$ (more precisely, ${\mathbb P}^{N-1}\setminus\Pi$) into linear subspaces $\Pi^{\sharp}\supset\Pi$ of dimension $b+1$. The centre $\Pi$ of the projection is a hyperplane in $\Pi^{\sharp}$. Since $\Pi\subset Q$, the quadric $\{q_i|_{\Pi^{\sharp}}=0\}$ is the union of two hyperplanes, one of which is $\Pi$. Now the claim of the corollary is obvious. Let $\Pi\subset Q$ be a linear subspace of dimension $b\geqslant k$, not meeting the set $\mathop{\rm Sing}Q$, and $\sigma\colon\widetilde{Q}\to Q$ and $\sigma_{\mathbb P}\colon\widetilde{{\mathbb P}^{N-1}}\to{\mathbb P}^{N-1}$ the blow ups of $\Pi$ on $Q$ and ${\mathbb P}^{N-1}$, respectively, so that we can identify $\widetilde{Q}$ with the strict transform of $Q$ on $\widetilde{{\mathbb P}^{N-1}}$. By the symbols $E_Q$ and $E_{\mathbb P}$ we denote the exceptional divisors of these blow ups; we consider $E_Q$ as a subvariety in $E_{\mathbb P}$. Let $\varphi\colon\widetilde{Q}\to{\mathbb P}^{N-b-2}$ and $\varphi_{\mathbb P}\colon\widetilde{{\mathbb P}^{N-1}} \to{\mathbb P}^{N-b-2}$ be the regularizations of the rational maps $\mathop{\rm pr}_{\Pi}|_Q$ and $\mathop{\rm pr}_{\Pi}$, respectively. We have the natural identification $E_{\mathbb P}=\Pi\times{\mathbb P}^{N-b-2}$, where the map $$ \varphi_{\mathbb P}|_{E_{\mathbb P}}\colon E_{\mathbb P}\to{\mathbb P}^{N-b-2} $$ is the projection onto the second factor. In the assumptions of Corollary 4.4 the morphism $\varphi$ is surjective and for a point of general position $p\in{\mathbb P}^{N-b-2}$ the fibre $\varphi^{-1}(p)$ is a linear subspace of dimension $b+1-k$ in $\varphi^{-1}_{\mathbb P}(p)\cong{\mathbb P}^{b+1}$, which is not contained entirely in the hyperplane $$ \varphi^{-1}_{\mathbb P}(p)\cap E_{\mathbb P}=\left(\varphi_{\mathbb P}|_{E_{\mathbb P}}\right)^{-1}(p), $$ which identifies naturally with $\Pi$, and for that reason $\varphi^{-1}(p)\cap E_{\mathbb P}$ identifies naturally with a subspace of dimension $b-k$ in $\Pi$ (and a hyperplane in $\varphi^{-1}(p)$). However, $$ \varphi^{-1}(p)\cap E_{\mathbb P}=\varphi^{-1}(p)\cap E_Q=(\varphi|_{E_Q})^{-1}(p), $$ so that arguing by dimensions, we conclude that the restriction $\varphi|_{E_Q}$ is surjective. {\bf Proposition 4.5.} {\it In the assumptions of Corollary 4.4, let $Y\subset\Pi$ be an irreducible closed subset, and assume that $$ b\geqslant k+\mathop{\rm codim} (Y\subset\Pi). $$ Then the restriction $\varphi|_{\sigma^{-1}(Y)}$ is surjective, so that for a point of general position $p\in{\mathbb P}^{N-b-2}$ the intersection $\varphi^{-1}(p)\cap \sigma^{-1}(Y)$ is non-empty and each of its components is of codimension $\mathop{\rm codim} (Y\subset\Pi)$ in the projective space} $\varphi^{-1}(p)\cap E_{\mathbb P}$. {\bf Proof.} Obviously, $$ \sigma^{-1}(Y)=\sigma^{-1}_{\mathbb P}(Y)\cap\widetilde{Q}= \sigma^{-1}_{\mathbb P}(Y)\cap E_Q. $$ Since $\varphi^{-1}(p)\subset\widetilde{Q}$, the equality $$ \varphi^{-1}(p)\cap\sigma^{-1}(Y)=\varphi^{-1}(p)\cap\sigma^{-1}_{\mathbb P}(Y) $$ holds, but $\sigma^{-1}(Y)=Y\times{\mathbb P}^{N-b-2}$ in terms of the direct decomposition of the exceptional divisor $E_{\mathbb P}$. Therefore, identifying the fibre of the projection $\varphi_{\mathbb P}|_{E_{\mathbb P}}$ with the projective space $\Pi$, we get that the intersection $\varphi^{-1}(p)\cap\sigma^{-1}(Y)$ identifies naturally with the intersection of $Y$ and the linear subspace $\varphi^{-1}(p)\cap E_{\mathbb P}$ of dimension $b-k$ in $\Pi$. By our assumption this intersection is non-empty, so that the morphism $\varphi|_{\sigma^{-1}(Y)}$ is surjective. Q.E.D. for the proposition. \section{The special hyperplane section} In this section we prove Theorem 3.3. {\bf 5.1. Start of the proof.} We use the notations of Subsection 1.7 and the assumptions of Theorem 3.3. Recall that $$ I_X=[2k+3,k+c_X-1]\cap {\mathbb Z} $$ is the set of admissible dimensions for the working triple $(X,D,o)$. Consider a general subspace $P\ni o$ in ${\mathbb P}(X)$ of the minimal admissible dimension $2k+3$. Since $a(E_{X\cap P})=2$ and $\nu(D)\leqslant\frac32n(D)<2n(D)$, we conclude that the pair $$ \left((X\cap P)^+,\frac{1}{n(D)} D|^+_{X\cap P}\right) $$ is not log canonical, but canonical outside the exceptional divisor $E_{X\cap P}$. By the inequality $\nu(D)<2n(D)$ we can apply the connectedness principle to this pair: \begin{equation}\label{16.01.23.1} \mathop{\rm LCS}\left((X\cap P)^+,\frac{1}{n(D)}D|^+_{X\cap P}\right) \end{equation} is a proper connected closed subset of the exceptional divisor $E_{X\cap P}$. There are the following options: $(1)_P$ this subset contains a divisor, $(2)_P$ some irreducible component of maximal dimension $B(P)\subset E_{X\cap P}$ in this set has a positive dimension and codimension $\geqslant 2$ in $E_{X\cap P}$, $(3)_P$ this subset is a point. {\bf Remark 5.1.} In the case $(1)_P$ the divisor in the subset (\ref{16.01.23.1}) is unique and is a hyperplane section of the variety $E_{X\cap P}\subset{\mathbb E}_{X\cap P}$, since $D|^+_{X\cap P}$ has along this subvariety the multiplicity $>n(D)$ (since it is the centre of some non log canonical singularity), whereas the restriction $D^+|_{E_{X\cap P}}$ is cut out on $E_{X\cap P}$ by a hypersurface of degree $\nu(D)<2n(D)$. Since $P\ni o$ is a subspace of general position, we go back to the original variety $X$ and get that the pair $(X^+,\frac{1}{n(D)}D|^+)$ is not log canonical, and moreover, for the centre $B\subset E_X$ of some non log canonical singularity of that pair one of the three option takes place: (1) $B$ is a hyperplane section of $E_X\subset{\mathbb E}_X$, (2) $\mathop{\rm codim}(B\subset E_X)\in\{2,\dots,k+1\}$, (3) $B$ is a linear subspace of codimension $2k+2$ in ${\mathbb E}_X$, which is contained in $E_X$. {\bf Proposition 5.1.} {\it The option (1) does not take place.} {\bf Proof.} Assume the converse: $B$ is a hyperplane section of $E_X$. Let $R\subset X$, $R\ni o$ be the uniquely determined hyperplane section, such that $R^+\cap E_X=B$ (in other words, ${\mathbb P}(R)^+\cap{\mathbb E}_X$ is the hyperplane in ${\mathbb E}_X$ that cuts out $B$ on $E_X$). Since $\mathop{\rm mult}_BD^+>n(D)$, we get that for the effective divisor $D_R=(D\circ R)$ on $R$ the inequality $$ \nu(D_R)\geqslant\nu(D)+\mathop{\rm mult}\nolimits_BD^+>2n(D)=2n(D_R) $$ holds, which is impossible by Theorem 1.4. Q.E.D. for the proposition. {\bf Proposition 5.2.} {\it The option (3) does not take place.} {\bf Proof.} Since $\mathop{\rm codim}(B\subset E_X)=k+2$, this is impossible by the Lefschetz theorem (in order to apply the Lefschetz theorem, it is sufficient to have the inequality $\mathop{\rm codim}(\mathop{\rm Sing}E_X\subset E_X)\geqslant 2k+6$, for which by Proposition 4.2 it is sufficient to have the inequality $\mathop{\rm rk}(o\in X)\geqslant 4k+5$; we have a much stronger condition for the rank of the singularity). Q.E.D. for the proposition. Therefore, the option (2) takes place. By construction (or arguing by dimension), $B\not\subset\mathop{\rm Sing}E_X$. Recall that there is a non log canonical singularity of the pair $(X^+,\frac{1}{n(D)}D^+)$, the centre of which is $B$. Let $p\in B$ be a point of general position; in particular, $p\not\in\mathop{\rm Sing}E_X$ and the more so $p\not\in\mathop{\rm Sing} X^+$. Applying inversion of adjunction in the word for word the same way as in \cite[Chapter 7, Proposition 2.3]{Pukh13a} (that is, restricting $D^+$ onto a general non-singular surface, containing the point $p$), we get the alternative: either $\mathop{\rm mult}_BD^+>2n(D)$, or on the blow up $$ \varphi_p\colon X^{(p)}\to X^+ $$ of the point $p$ with the exceptional divisor $E(p)\subset X^{(p)}$, $E(p)\cong{\mathbb P}^{N(X)-1}$, there is a hyperplane $\Theta(p)\subset E(p)$ in $E(p)$, satisfying the inequality \begin{equation}\label{22.09.22.1} \mathop{\rm mult}\nolimits_BD^++\mathop{\rm mult}\nolimits_{\Theta(p)}D^{(p)}>2n(D), \end{equation} where $D^{(p)}$ is the strict transform of the divisor $D^+$ on $X^{(p)}$, and moreover, the hyperplane $\Theta(p)$ is uniquely determined by the pair $(X^+,\frac{1}{n(D)}D^+)$ and varies algebraically with the point $p\in B$. The case when the inequality $\mathop{\rm mult}\nolimits_BD^+>2n(D)$ holds, is excluded (with simplifications) by the arguments, excluding the option $(2)_{\Theta}$, given below, see Subsection 5.3, Remark 5.2. There are two options for the hyperplane $\Theta(p)$: $(1)_{\Theta}$ $\Theta(p)\neq {\mathbb P}(T_pE_X)$ (where we identify $E(p)$ with the projectivization of the tangent space $T_pX^+$), so that $\Theta(p)$ intersects ${\mathbb P}(T_pE_X)$ by some hyperplane $\Theta_E(p)$, $(2)_{\Theta}$ the hyperplanes $\Theta(p)$ and ${\mathbb P}(T_pE_X)$ in $E(p)$ are equal. Below (see Subsection 5.3, Remark 5.2) we show that the option $(2)_{\Theta}$ does not take place: it implies that $E_X\subset D^+$, which is impossible; the same arguments exclude the inequality $\mathop{\rm mult}\nolimits_BD^+>2n(D)$, too. Therefore, we may assume that the option $(1)_{\Theta}$ takes place. {\bf 5.2. The existence of the special hyperplane section.} Adding the upper index $(p)$ means the strict transform on $X^{(p)}$: we used this principle for the divisor $D$ above and will use it for other subvarieties on $X^+$. Our aim is to prove the following claim. {\bf Theorem 5.1.} {\it There is a hyperplane section $\Lambda$ of the exceptional divisor $E_X\subset{\mathbb E}_X$, containing $B$, satisfying the inequality $$ \mathop{\rm mult}\nolimits_{\Lambda}D^+>\frac{2n(D)-\nu(D)}{k+1}. $$ Moreover, for a point of general position $p\in B$ the following equality holds:} $$ \Lambda^{(p)}\cap E(p)=\Theta_E(p). $$ {\bf Proof.} Let $L\subset E_X,L\ni p$ be a line in the projective space ${\mathbb E}_X$, such that $L\cap\mathop{\rm Sing}E_X=\emptyset$ and $$ L^{(p)}\cap E_(p)\in\Theta_E(p). $$ {\bf Lemma 5.1.}{\it The line $L$ is contained in $D^+$.} {\bf Proof.} Assume the converse. Then $D^+|_L$ is an effective divisor on $L$ of degree $\nu(D)\leqslant\frac32n(D)<2n(D)$. At the same time, the divisor $D^+|_L$ contains the point $p$ with multiplicity $>2n(D)$ due to the inequality (\ref{22.09.22.1}). The contradiction proves the lemma. Q.E.D. {\bf Proposition 5.3.} {\it The following inequality holds:} $$ \mathop{\rm mult}\nolimits_LD^+>\frac{2n(D)-\nu(D)}{k+1}. $$ {\bf Proof} is given in \S 6. Let us go back to the proof of Theorem 5.1. We will construct the set $\Lambda\subset E_X$ explicitly, and then prove that it is a hyperplane section. The exceptional divisor $E_X$ is a complete intersection of $k$ quadrics in ${\mathbb E}_X$: $$ E_X=\{q_1=\dots q_k=0\}, $$ using the notations of Subsection 4.5. Let $U_B\subset B$ be a non-empty Zariski open subset, where $$ U_B\cap\mathop{\rm Sing} E_X=\emptyset $$ and for every point $p\in B$ the option $(1)_{\Theta}$ takes place. By the assumption on the rank of the multi-quadratic point $o\in X$ for $p\in U_{B}$ the set $E_X\cap T_pE_X$ (where $T_pE_X\subset{\mathbb E}_X$ is the embedded tangent space, that is, a linear subspace of codimension $k$ in ${\mathbb E}_X$) is irreducible and reduced, and moreover, every hyperplane section of that set is also irreducible and reduced. Indeed, by Proposition 4.2, in order to have these properties it is sufficient to have the inequality $\mathop{\rm rk}q_{[1,k]}\geqslant 4k+5$, because by Remark 1.4 it implies the inequality $$ \mathop{\rm rk}q_{[1,k]}|_{T_pE_X}\geqslant 2k+5 $$ and we can apply Proposition 4.2. Obviously, $E_X\cap T_pE_X$ is a cone with the vertex $p$, consisting of all lines $L\subset E_X$, $L\ni p$. The singular set of that cone is of codimension $\geqslant 6$ (Proposition 4.2), and so for a general line $L\ni p$ $$ L\cap\mathop{\rm Sing}(E_X\cap T_p E_X)=\{p\}, $$ so that $L\cap\mathop{\rm Sing}E_X=\emptyset$ and the same is true for every hyperplane section of the cone $E_X\cap T_pE_X$, containing the point $p$, since its singular set is of codimension $\geqslant 4$ (Remark 1.4). Let ${\cal L}(p)$ be the union of all lines $L\subset E_X,L\ni p$, such that $$ L^{(p)}\cap E(p)\in\Theta_E(p). $$ Obviously, ${\cal L}(p)$ is the section of the cone $E_X\cap T_pE_X$ by some hyperplane, containing the point $p$ (this hyperplane corresponds to the hyperplane $\Theta_E(p$)). As we have shown above, ${\cal L}(p)$ is an irreducible closed subset of codimension $k+1$ in $E_X$, and $$ \mathop{\rm mult}\nolimits_{{\cal L}(p)} D^+>\frac{2n(D)-\nu(D)}{k+1}. $$ Set $$ \Lambda=\overline{\mathop{\bigcup}\limits_{p\in U_B}{\cal L}(p)} $$ (the overline means the closure). By what was said above, the inequality $$ \mathop{\rm mult}\nolimits_{\Lambda}D^+>\frac{2n(D)-\nu(D)}{k+1} $$ holds. {\bf Theorem 5.2.} {\it The subset $\Lambda\subset E_X$ is a hyperplane section of the variety} $E_X\subset{\mathbb E}_X$. We will prove Theorem 5.2 in two steps: first, we will show that $\Lambda$ is a prime divisor on $E_X$, and then, that this divisor is a hyperplane section. By construction, the set $\Lambda$ is irreducible. {\bf 5.3. The set $\Lambda$ is a divisor.} By our assumption about the rank of the point $o\in X$ for $b=k+1$ the inequality \begin{equation}\label{01.10.22.1} \mathop{\rm rk} q_{[1,k]}\geqslant 2k(b+2k+2)+2(2k+1)+3 \end{equation} holds. By Corollary 4.3, for a general subspace $\Pi\subset E_X$ of dimension $b$ the restriction onto $B$ of the projection $$ \mathop{\rm pr}\nolimits_{\Pi}\colon {\mathbb P}^{N(X)-1}\dashrightarrow {\mathbb P}^{N(X)-b-2} $$ from the subspace $\Pi$ is dominant. Let $s\in {\mathbb P}^{N(X)-b-2}$ be a point of general position. By the symbol $\langle\Pi,s\rangle$ denote the closure $$ \overline{\mathop{\rm pr}\nolimits_{\Pi}^{-1}(s)}\subset {\mathbb P}^{N(X)-1} $$ (this is a $(\mathop{\rm dim}\Pi+1)$-dimensional subspace) and set $$ E_X(\Pi,s)=E_X\cap \langle\Pi,s\rangle. $$ For the blow ups $\sigma\colon\widetilde{E}_X\to E_X$ and $\sigma_{\mathbb P}\colon\widetilde{{\mathbb P}^{N(X)-1}}\to{\mathbb P}^{N(X)-1}$ of the subspace $\Pi$ on $E_X$ and ${\mathbb P}^{N(X)-1}$, respectively, let $\varphi\colon\widetilde{E}_X\to{\mathbb P}^{N(X)-b-2}$ and $\varphi_{\mathbb P}\colon\widetilde{{\mathbb P}^{N(X)-1}}\to{\mathbb P}^{N(X)-b-2}$ be the regularizations of the projections $\mathop{\rm pr}_{\Pi}|_{E_X}$ and $\mathop{\rm pr}_{\Pi}$, respectively. Obviously, the fibre $\varphi^{-1}_{\mathbb P}(s)$ identifies naturally with $\langle\Pi,s\rangle$, and the fibre $\varphi^{-1}(s)$ with $E_X(\Pi,s)$. The fibre of the surjective morphism $\varphi|_{\sigma^{-1}(B)}$ over the point $s$ we denote by the symbol $B(s)$; this is a possibly reducible closed subset in $\varphi^{-1}_{\mathbb P}(s)$, each irreducible component of which is of codimension $c_B=\mathop{\rm codim} (B\subset E_X)$ and is not contained entirely in the hyperplane $\Pi$ (with respect to the identification $\varphi^{-1}_{\mathbb P}(s)=\langle\Pi,s\rangle$). Write down $B(s)$ as a union of irreducible components: $$ B(s)=\mathop{\bigcup}\limits_{i\in I} B_i(s), $$ and let $p\in B_i(s)$ be a point of general position on one of them; in particular, $p\not\in\Pi$, so that the projection $\mathop{\rm pr}\nolimits_{\Pi}$ is regular at that point and $p\not\in B_j(s)$ for $j\neq i$. We will consider the point $p$ as a point of general position on $B$, which was introduced in Subsection 5.1, and use the notations for the blow up $\varphi_p$ of this point and for objects linked to this blow up. Note that for $b=k+1$ we have the inequality $$ \mathop{\rm dim}B(s)=\mathop{\rm dim}B_i(s) \geqslant 1. $$ The set of lines $L\subset E_X(\Pi,s)$, $L\ni p$, such that $L^{(p)}\cap E(p)\in\Theta(p)$, forms a hyperplane in $E_X(\Pi,s)$, which we denote by the symbol $\Lambda(\Pi,s,p)$. By construction, $\Lambda(\Pi,s,p)\subset\Lambda$. Since any non-trivial algebraic family of hyperplanes in a projective space sweeps out that space and for a general point $s$ we have $E_X(\Pi,s)\not\subset\Lambda$ (otherwise $\Lambda=E_X$, which is impossible), we conclude that the hyperplane $\Lambda(\Pi,s,p)$ does not depend on the choice of a point of general position $p\in B_i(s)$, so that $$ \Lambda(\Pi,s,p)=\Lambda(\Pi,s,B_i(s)) $$ is a hyperplane in $\varphi^{-1}(s)=E_X(\Pi,s)$, containing the component $B_i(s)$. Therefore, for a general point $s$ the intersection $\Lambda\cap E_X(\Pi,s)$ contains a divisor in $E_X(\Pi,s)$, whence we get that $\Lambda\subset E_X$ is a (prime) divisor on $E_X$, as we claimed. This divisor is cut out on $E_X$ by a hypersurface of degree $d_{\Lambda}$ in ${\mathbb E}_X$. It remains to show that $d_{\Lambda}=1$. {\bf Remark 5.2.} We promised above that the option $(2)_{\Theta}$ does not take place. Indeed, if it does, then every line $L\ni p$ in $E_X(\Pi,s)$ is contained in $\Lambda$, so that $E_X(\Pi,s)\subset\Lambda$ and for that reason $E_X\subset\Lambda$, which is absurd. In a similar way, if $\mathop{\rm mult}_BD^+>\nu(D)$, then every line in $E_X(\Pi,s)$, meeting $B$, is contained in $\Lambda$, so that $E_X\subset\Lambda$, which is impossible. Therefore, the inequality $$ \mathop{\rm mult}\nolimits_BD^+\leqslant\nu(D) $$ holds. {\bf 5.4. The divisor $\Lambda$ is a hyperplane section.} Let us consider the intersection $\Lambda\cap E_X(\Pi,s)$ for a general point $s$ in more details. This is a possibly reducible divisor, each component of which has multiplicity 1, containing at least one hyperplane. If in this divisor there are components of degree $\geqslant 2$, then the union of hyperplanes in $\Lambda(\Pi,s)$ gives a proper closed subset of $\Lambda_1(\Pi,s)$, which is also a divisor. Then $$ \overline{\bigcup_s\Lambda_1(\Pi,s)} $$ (the union is taken over a non-empty open subset in ${\mathbb P}^{N(X)-b-2}$) is a proper closed subset in $\Lambda$, which is of codimension 1 in $E_X$, which is impossible as $\Lambda$ is a prime divisor. We conclude that $\Lambda(\Pi,s)$ is a union of precisely $d_{\Lambda}$ distinct hyperplanes in $E_X(\Pi,s)$. Assume that $d_{\Lambda}\geqslant 2$. By our assumptions about the rank $\mathop{\rm rk}(o\in X)$ the inequality (\ref{01.10.22.1}) holds for $b=3k$: $$ \mathop{\rm rk}q_{[1,k]}\geqslant 10k^2+8k+5. $$ Again we apply Corollary 4.3, now to a general subspace $\Pi^*=E_X(\Pi,s)$ of dimension $b^*=b+1-k\geqslant 2k+1$. The subspace $\Pi^*$ does not meet the set $\mathop{\rm Sing} E_X$ and the restriction of the projection from $\Pi^*$ $$ \mathop{\rm pr}\nolimits_{\Pi^*}\colon{\mathbb P}^{N(X)-1}\dashrightarrow {\mathbb P}^{N(X)-b^*-2} $$ onto $B$ is dominant. Let $s^*\in{\mathbb P}^{N(X)-b^*-2}$ be a point of general position. We use the notations introduced above and write $E_X(\Pi^*,s^*)$. For the blow ups of the subspace $\Pi^*$ we use the symbols $\sigma_{\Pi^*}$ and $\sigma_{{\mathbb P},\Pi^*}$, respectively, and for the regularized projections the symbols $\varphi_{\Pi^*}$ and $\varphi_{{\mathbb P},\Pi^*}$. The symbol $\langle\Pi^*,s^*\rangle$ has the same meaning as above. Set $$ E^*=\sigma_{\Pi^*}^{-1}(\Pi^*)\quad\mbox{and}\quad E^*_{\mathbb P}= \sigma_{{\mathbb P},\Pi^*}^{-1}(\Pi^*) $$ to be the exceptional divisors of the blow up of $\Pi^*$ on $E_X$ and ${\mathbb P}^{N(X)-1}$. By the arguments immediately before the statement of Proposition 4.5, the map $\varphi_{\Pi^*}|_{E^*}$ is surjective, and by Proposition 4.5 (which applies since $b^*\geqslant k+1$) the intersection $$ \varphi_{\Pi^*}^{-1}(s^*)\cap \sigma_{\Pi^*}^{-1}(\Lambda\cap \Pi^*) $$ is non-empty and each of its irreducible components is of codimension 1 in the projective space $\varphi_{\Pi^*}^{-1}(s^*)\cap E^*_{\mathbb P}$. By what was shown above, $\Lambda\cap{\Pi^*}$ is a union of $d_{\Lambda}$ distinct hyperplanes $\Lambda^*_i$, $i\in I$. In a similar way, $$ \Lambda\cap E_X(\Pi^*,s^*)=\sigma^{-1}_{\Pi^*}(\Lambda)\cap\varphi^{-1}_{\Pi^*}(s^*) $$ is the union of $d_{\Lambda}$ distinct hyperplanes in $\varphi^{-1}_{\Pi^*}(s^*)$, none of which coincides with the hyperplane $\varphi^{-1}_{\Pi^*}(s^*)\cap E_{\mathbb P}^*$. Note that the strict transform of the divisor $\Lambda$ with respect to the blow up $\sigma_{\Pi^*}$ is just its full inverse image $\sigma^{-1}_{\Pi^*}(\Lambda)$, since $\Lambda\not\subset\Pi^*$. Furthermore, $$ \sigma^{-1}_{\Pi^*}(\Lambda)\cap E^*=\sigma^{-1}_{\Pi^*}(\Lambda\cap \Pi^*)=\bigcup_{i\in I}\sigma^{-1}_{\Pi^*}(\Lambda^*_i), $$ and every intersection $\varphi^{-1}_{\Pi^*}(s^*)\cap\sigma^{-1}_{\Pi^*}(\Lambda^*_i)$ is a hyperplane in $\varphi^{-1}_{\Pi^*}(s^*)\cap E^*_{\mathbb P}$. It follows that each irreducible component of set $\Lambda\cap E_X(\Pi^*,s^*)$ intersects the hyperplane $\varphi^{-1}_{\Pi^*}(s^*)\cap E^*_{\mathbb P}$ by one of the hyperplanes $\sigma^{-1}_{\Pi^*}(\Lambda^*_i)\cap\varphi^{-1}_{\Pi^*}(s^*)$, $i\in I$. Thus one can write down $$ \Lambda\cap E_X(\Pi^*,s^*)=\bigcup_{i\in I}\Lambda_i(\Pi^*,s^*), $$ where $\Lambda_i(\Pi^*,s^*)$ is a hyperplane in $E_X(\Pi^*,s^*)$, satisfying the equality $$ \Lambda_i(\Pi^*,s^*)\cap E^*=\sigma^{-1}_{\Pi^*}(\Lambda^*_i)\cap\varphi^{-1}_{\Pi^*}(s^*). $$ In other words, the choice of a component of the intersection $\Lambda\cap\Pi^*$ determines uniquely the component of the intersection of $\Lambda$ with $\langle\Pi^*,s^*\rangle=\varphi^{-1}_{\Pi^*}(s^*)=E_X(\Pi^*,s^*)$ for a general point $s^*$. Now set $$ \Lambda_i=\sigma_{\Pi^*}\left(\overline{\mathop{\bigcup}\limits_{s^*} (\Pi^*,s^*)}\right), $$ where the union is taken over a non-empty Zariski open subset of the projective space ${\mathbb P}^{N(X)-b^*-2}$. This is a prime divisor on $E_X$, and moreover, $\Lambda_i\subset\Lambda$ and for that reason $\Lambda_i=\Lambda$, whence we conclude that all hyperplanes $\Lambda_i(\Pi^*,s^*)$ are the same, which is a contradiction with the assumption that $d_{\Lambda}\geqslant 2$. Thus $d_{\Lambda}=1$ and $\Lambda$ is a hyperplane section of $E_X\subset {\mathbb E}_X$. Q.E.D. for Theorem 5.2 and therefore, for Theorem 5.1. {\bf 5.5. The construction of a new working triple.} Now we can complete the proof Theorem 3.3 and construct the new working triple $(R,D_R,o)$. Let $R\ni o$ the section of $X$ by the hyperplane ${\mathbb P}(R)=\langle R\rangle$, such that $$ R^+\cap {\mathbb E}_X=R^+\cap E_X=\Lambda $$ (in other words, the hyperplane ${\mathbb P}(R)^+ \cap {\mathbb E}_X$ cuts out $\Lambda$ on $E_X$). Since $R$ is not a component of the effective divisor $D_X$, the scheme-theoretic intersection $(R\circ D_X)$ is well defined, and we treat this intersection as an effective divisor on $R$. Set $D_R=(R\circ D_X)$ in that sense. On $R^+\subset X^+$ with the exceptional divisor $$ E_R=(R^+\cap E_X)=\Lambda\subset {\mathbb E}_R= {\mathbb P}(R)^+\cap{\mathbb E}_X $$ we have the equivalence $$ D^+_R\sim n(D_R) H_R-\nu(D_R) E_R, $$ where $H_R$ is the class of a hyperplane section of $R$ and $$ \nu(D_R)\geqslant\nu(D_X)+\mathop{\rm mult}\nolimits_\Lambda D^+_X>\nu(D_X)+\frac{2n(D_X)-\nu(D_X)}{k+1}. $$ Again $[R,o]$ is a marked complete intersection, of level $(k,c_R)$, where $c_R=c_X-2$, $(R,D_R,o)$ is a working triple, and the inequality $$ 2n(D_R)-\nu(D_R)<\left(1-\frac{1}{k+1}\right)(2n(D_X)-\nu(D_X)) $$ holds (since $n(D_R)=n(D_X)$). The procedure of constructing the special hyperplane section is complete. Q.E.D. for Theorem 3.3. \section{Multiplicity of a line} In this section we prove Proposition 5.3. {\bf 6.1. Blowing up a point and a curve.} Since we completed our study of working triples, the symbol $X$ is now free and will mean an arbitrary non-singular quasi-projective variety of dimension $\geqslant 3$. Let $C\subset X$ be a non-singular projective curve, $p\in C$ a point. Furthermore, let $$ \sigma_C\colon X(C)\to X $$ be the blow up of the curve $C$ with the exceptional divisor $E_C$ and $\sigma^{-1}_C(p)\cong {\mathbb P}^{\dim X-2}$ the fibre over the point $p$. Let $$ \sigma\colon X(C,\sigma^{-1}_C(p))\to X(C) $$ be the blow up of that fibre with the exceptional divisor $E$ and $E_C^{(p)}$ the strict transform of $E_C$ on that blow up. On the other hand, consider the blow up $$ \varphi_p\colon X(p)\to X $$ of the point $p$ with the exceptional divisor $E_p$ and denote by the symbol $C(p)$ the strict transform of the curve $C$ on $X(p)$. Finally, let $$ \varphi\colon X(p,C(p))\to X(p) $$ be the blow up of the curve $C(p)$, $E_{C(p)}$ the exceptional divisor of that blow up and $E^C_p$ the strict transform of $E_p$. {\bf Proposition 6.1.} {\it The identity map $\mathop{\rm id}_X$ extends to an isomorphism $$ X(C,\sigma^{-1}_C(p))\cong X(p,C(p)), $$ identifying the subvarieties $E$ and $E^C_p$ and the subvarieties $E^p_C$ and $E_{C(p)}$.} {\bf Proof.} This is a well known fact, which can be checked by elementary computations in local parameters. Q.E.D. for the proposition. Taking into account the identifications above, we will use the notations $E^C_p$ and $E_{C(p)}$, and forget about $E$ and $E^p_C$. The variety $X(C,\sigma^{-1}_C(p))$ will be denoted by the symbol $\widetilde{X}$. Let $D$ be an effective divisor on $X$. The symbols $D^C$ and $D^p$ stand for its strict transforms on $X(C)$ and $X(p)$, respectively, and the symbol $\widetilde{D}$ for its strict transform on $\widetilde{X}$. Set $$ \mu=\mathop{\rm mult}\nolimits_CD\quad\mbox{and}\quad\mu_p=\mathop{\rm mult}\nolimits_pD, $$ where, of course, $\mu_p\geqslant\mu$. {\bf Lemma 6.1.} {\it The following equality holds:} $$ \mathop{\rm mult}\nolimits_{\sigma^{-1}_C(p)}D^C=\mu_p-\mu. $$ {\bf Proof.} (This is a well known fact, and we give a proof for the convenience of the reader, and also because a similar argument is used below.) We have the sequence of obvious equalities: $$ \sigma^*_CD=D^C+\mu E_C, $$ so that $$ \sigma^*\sigma^*_CD=\widetilde{D}+\mu E_{C(p)}+(\mu+\mathop{\rm mult}\nolimits_{\sigma^{-1}_{C(p)}}D^C)E^C_p. $$ Considering the second sequence of blow ups, we get $$ \varphi^*_pD=D^p+\mu_pE_p $$ and, respectively, $$ \varphi^*\varphi^*_pD=\widetilde{D}+\mu E_{C(p)}+\mu_pE^C_p. $$ Comparing the two presentations of the same effective divisor, we get the claim of the lemma. {\bf 6.2. Blowing up two points and a curve.} In the notations of the previous subsection, let us consider the point $$ q=C(p)\cap E_p. $$ Set $\mu_q=\mathop{\rm mult}_qD^p$. Obviously, $$ \mu_q\geqslant\mathop{\rm mult}\nolimits_{C(p)}D^p=\mathop{\rm mult}\nolimits_CD=\mu. $$ Let $$ \varphi_q\colon X(p,q)\to X(p) $$ the blow up of the point $q$ with the exceptional divisor $E_q$ and $C(p,q)$ the strict transform of the curve $C(p)$. Finally, let $$ \varphi_{\sharp}\colon X^{\sharp}\to X(p,q) $$ be the blow up of the curve $C(p,q)$ with the exceptional divisor $E_{C(p,q)}$ and $E^{\sharp}_q$ the strict transform of $E_q$. Note that the curve $C(p)$ intersects $E_p$ transversally and therefore $C(p,q)$ does not meet the strict transform $E^q_p$ of the divisor $E_p\subset X(p)$ on $X(p,q)$. {\bf Proposition 6.2.} {\it The restriction of the divisor $D^C$ onto the exceptional divisor $E_C$ contains the fibre $\sigma^{-1}_C(p)$ with multiplicity at least} $\mu_p+\mu_q-2\mu$. {\bf Proof.} Obviously, on $X^{\sharp}$ we have the equality $$ \varphi^*_{\sharp}\varphi^*_q\varphi^*_p D=D^{\sharp}+ (\mu_q+\mu_p)E^{\sharp}_q+\mu_p E^q_p+\mu E_{C(p,q)}, $$ where $D^{\sharp}$ is the strict transform of $D$ on $X^{\sharp}$. On the other hand, using the constructions of Subsection 6.1, we see that $X^{\sharp}$ can be obtained as the blow up of the curve $C(p)$ on $X(p)$ with the subsequent blowing up of the fibre of the exceptional divisor $E_{C(p)}$ over the point $q$ or, applying the construction of Subsection 6.1 twice, as the blow up of the curve $C$ on $X$ with the subsequent blowing up of the fibre $\sigma^{-1}_C(p)$ and then the blowing up of the subvariety $$ E^C_p\cap E^p_C. $$ In the last presentation the three prime exceptional divisors are $$ E^{\sharp}_C=E_{C(p,q)},\quad E^{\sharp}_p\quad\mbox{and}\quad E^{\sharp}_q. $$ We denote the blow up $X^{\sharp}\to\widetilde{X}$ of the subvariety $E^C_p\cap E^p_C$, mentioned above, by the symbol $\sigma_{\sharp}$. Thus we obtain the following commutative diagram of birational morphisms: $$ \begin{array}{ccccc} X(C) & \stackrel{\sigma}{\leftarrow} & \widetilde{X} & \stackrel{\sigma_{\sharp}}{\leftarrow} & X^{\sharp}\\ \downarrow & & \downarrow & & \downarrow\\ X &\stackrel{\varphi_p}{\leftarrow} & X(p) &\stackrel{\varphi_q}{\leftarrow} & X(p,q),\\ \end{array} $$ where the vertical arrows (from the left to the right) are $\sigma_C$, $\varphi$ and $\varphi_{\sharp}$, respectively. We have the equality $$ \sigma^*_{\sharp}\sigma^*\sigma^*_CD=\varphi^*_{ \sharp}\varphi^*_q\varphi^*_pD. $$ This pull back can be written down as $$ D^{\sharp}+\mu E^{\sharp}_C+\mu_pE^{\sharp}_p+(\mu_p+\mu_q)E^{\sharp}_q, $$ since $E_{C(p,q)}=E^{\sharp}_C$ and $E^{\sharp}_q$ is the exceptional divisor of the blow up $\sigma_{\sharp}$. On the other hand, $D^C=\sigma^{*}_C D-\mu E_C$ and, besides, $$ \sigma^*_{\sharp}\sigma^*E_C=E^{\sharp}_C+E^{\sharp}_p+2E^{\sharp}_q, $$ so that the exceptional divisor $E^{\sharp}_q$ comes into the pull back of the divisor $D^C$ on $X^{\sharp}$ with multiplicity $(\mu_p+\mu_q-2\mu)$. However, the blow ups $\sigma$ and $\sigma_{\sharp}$ do not change the divisor $E_C$, as they blow up subvarieties of codimension 1 on the variety: $$ \sigma\circ\sigma_{\sharp}|_{E^{\sharp}_C}\colon E^{\sharp}_C\to E_C $$ is an isomorphism. Since the restriction $D^{\sharp}$ onto $E^{\sharp}_C$ is an effective divisor, it follows from here that the restriction of the divisor $D^C$ onto $E_C$ contains the fibre $\sigma^{-1}(p)$ (which is precisely the restriction of $E^{\sharp}_q$ onto $E^{\sharp}_C$ in terms of the isomorphism between $E^{\sharp}_C$ and $E_C$, discussed above) with multiplicity $\geqslant\mu_p+\mu_q-2\mu$. Proof of Proposition 6.2 is complete. {\bf 6.3. The multiplicity of an infinitely near line.} Let us come back to the proof of Proposition 5.3. We will obtain its claim from a more general fact. Let $o\in{\cal X}$ be a germ of a multi-quadratic singularity of type $2^k$, where ${\cal X}\subset{\cal Y}$, ${\cal Y}$ is non-singular, $\mathop{\rm codim}({\cal X}\subset{\cal Y})=k$ and the inequality $$ \mathop{\rm rk}(o\in{\cal X})\geqslant 2k+3 $$ holds, so that $\mathop{\rm codim}(\mathop{\rm Sing{\cal X}\subset{\cal X}})\geqslant 4$ and ${\cal X}$ is factorial. Let $\sigma_{\cal Y}\colon{\cal Y}^+\to{\cal Y}$ be the blow up of the point $o$ with the exceptional divisor $E_{\cal Y}$, ${\cal X}^+\subset{\cal Y}^+$ the strict transform, so that $$ \sigma=\sigma_{\cal Y}|_{{\cal X}^+}\colon{\cal X}^+\to{\cal X} $$ is the blow up of the point $o$ on ${\cal X}$ with the exceptional divisor $E_{\cal X}$, which is a complete intersection of $k$ quadrics in $E_{\cal Y}\cong{\mathbb P}^{\rm dim{\cal Y}-1}$. By Proposition 4.2 $$ \mathop{\rm codim}(\mathop{\rm Sing} E_{\cal X}\subset E_{\cal X})\geqslant 4. $$ Let $L\subset E_{\cal X}$ be a line, where $L\cap \mathop{\rm Sing} E_{\cal X}=\emptyset$ and $p\in L$ a point. Let us blow up this point on ${\cal Y}^+$ and ${\cal X}^+$, respectively: $$ \sigma_{p,{\cal Y}}\colon {\cal Y}_p\to {\cal Y}^+\quad\mbox{and}\quad \sigma_p\colon {\cal X}_p\to {\cal X}^+ $$ are these blow ups with the exceptional divisors $E_{p,{\cal Y}}$ and $E_p$. Set $$ q=L^{(p)}\cap E_p, $$ where $L^{(p)}\subset {\cal X}_p$ is the strict transform. Let $D_{\cal X}$ be an effective divisor on ${\cal X}$. For its strict transform on ${\cal X}^+$ we have the equality $$ D^+_{\cal X}=\sigma^* D_{\cal X}-\nu E_{\cal X} $$ for some $\nu\in{\mathbb Z}_+$. Furthermore, we denote the strict transform of $D^+_{\cal X}$ on ${\cal X}_p$ by the symbol $D^{(p)}_{\cal X}$ and set $$ \mu_p=\mathop{\rm mult}\nolimits_pD^+_{\cal X}\quad\mbox{and} \quad\mu_q=\mathop{\rm mult}\nolimits_qD^{(p)}_{\cal X}. $$ Set also $\mu=\mathop{\rm mult}_LD^+_{\cal X}$; obviously, $\mu\leqslant\mu_p$. {\bf Theorem 6.1.} {\it The following inequality holds:} $$ \mu\geqslant\frac{1}{k+1}(\mu_p+\mu_q-\nu). $$ {\bf Proof.} Let $P\subset E_{\cal Y}$ be a general linear subspace of dimension $(k+2)$, containing the line $L$. {\bf Lemma 6.2.} {\it The surface $S=P\cap{\cal X}^+=P\cap E_{\cal X}$ is non-singular.} {\bf Proof.} We argue by induction on $\mathop{\rm dim}E_{\cal X}\geqslant 2$. If $\mathop{\rm dim}E_{\cal X}=2$, then there is nothing to prove. Let $\mathop{\rm dim}E_{\cal X}\geqslant 3$. The hyperplanes in $E_{\cal Y}$, tangent to $E_{\cal X}$ at at least one point of the line $L$, form a $k$-dimensional family. The hyperplanes, containing the line $L$, form a family (a linear subspace) of codimension 2 in the dual projective space for $E_{\cal Y}$, that is, of dimension $k+\mathop{\rm dim}E_{\cal X}-2\geqslant k+1$, so that for a general hyperplane $R_{\cal Y}\supset L$ in $E_{\cal Y}$ we have: $E_{\cal X}\cap R_{\cal Y}$ is non-singular along $L$ (and, of course, for the codimension of the singular set we have the equality $\mathop{\rm codim}(\mathop{\rm Sing}(E_{\cal X}\cap R_{\cal Y})\subset(E_{\cal X}\cap R_{\cal Y}))=\mathop{\rm codim}(\mathop{\rm Sing}E_{\cal X}\subset E_{\cal X}$)). Applying the induction hypothesis, we complete the proof of the lemma. Q.E.D. Let ${\cal Z}\subset{\cal Y}$, ${\cal Z}\ni o$, be a general subvariety of dimension $(k+3)$, non-singular at the point $o$, such that ${\cal Z}^+\cap E_{\cal Y}=P$, and $$ {\cal X}_P={\cal X}\cap{\cal Z} $$ (the notation ${\cal X}_P$ is chosen for convenience: ${\cal X}_P$ is determined by ${\cal Z}$, not by $P$). Then ${\cal X}_P$ is a three-dimensional variety with the isolated multi-quadratic singularity $o\in{\cal X}_P$, and the blow up of the point $o$ resolves this singularity: the exceptional divisor ${\cal X}^+_P\cap E_{\cal Y}$ is the non-singular surface $S$. The restriction of the divisor $D_{\cal X}$ onto ${\cal X}_P$ is denoted by the symbol $D_P$, and its strict transform on ${\cal X}^+_P$ by the symbol $D^+_P$. {\bf Lemma 6.3.} {\it The normal sheaf ${\cal N}_{L/{\cal X}^+_P}\cong{\cal O}_L(-\alpha)\oplus{\cal O}_L(-\beta)$, where $\alpha+\beta=k$ and $\alpha\geqslant\beta\geqslant 1$.} {\bf Proof.} Since $P\cong{\mathbb P}^{k+2}$, by the adjunction formula $K_S=(k-3)H_{S}$, where $H_S$ is the class of a hyperplane section of $S\subset P$, whence it follows that $(L^2)_{S}=1-k$. Furthermore, the surface $S$ is the exceptional divisor of the blow up of the point $o$ on ${\cal X}_P$ ($S={\cal X}^+_P\cap E_{\cal Y}$), so that ${\cal O}_{{\cal X}^+_P}(S)|_L={\cal O}_L(-1)$ and we have the exact sequence $$ 0\to{\cal N}_{L/S}\to{\cal N}_{L/{\cal X}^+_P}\to{\cal N}_{S/{\cal X}^+_P}|_L\to 0 $$ or $$ 0\to{\cal O}_L(1-k)\to{\cal N}_{L/{\cal X}^+_P}\to{\cal O}(-1)\to 0. $$ From this the claim of the lemma follows at once. Q.E.D. Let $\sigma_L\colon{\cal X}_{P,L}\to{\cal X}^+_P$ be the blow up of the line $L$, and $E_{P,L}\subset{\cal X}_{P,L}$ the exceptional divisor. The lemma implies that $E_{P,L}$ is a ruled surface of type ${\mathbb F}_{\alpha-\beta}$ and its Picard group is ${\mathbb Z}s\oplus{\mathbb Z}f$, where $f$ is the class of a fibre, $s$ the class of the exceptional section, $s^2=-(\alpha-\beta)$. Again from the lemma shown above it follows that $$ (E^3_{P,L})_{{\cal X}_{P,L}}=(E_{P,L}|^2_{E_{P,L}})= -\mathop{\rm deg}{\cal N}_{L/{\cal X}^+_P}=k, $$ so that $$ -E_{P,L}|_{P,L}=s+\frac12(k+\alpha-\beta)f. $$ Obviously (since the subspace $P$ is general), $$ \mathop{\rm mult}\nolimits_pD^+_P=\mu_p. $$ Let ${\cal X}^{(p)}_P\subset{\cal X}_p$ be the strict transform of ${\cal X}^+_P$ on ${\cal X}_p$. By construction, $q\in{\cal X}^{(p)}_P$. Setting $D^{(p)}_P=D^{(p)}_{\cal X}|_{{\cal X}^{(p)}_P}$, we obtain $$ \mu_q=\mathop{\rm mult}\nolimits_qD^{(p)}_P. $$ Finally, let $D^{(L)}_P$ be the strict transform of the divisor $D^+_P$ on ${\cal X}_{P,L}$. Obviously, $$ D^{(L)}_P=\sigma^*_LD^+_P-\mu E_{P,L}, $$ so that, writing the pull back on ${\cal X}_{P,L}$ of the restriction $E_{\cal X}|_{{\cal X}^+_P}$ for simplicity as the restriction $E_{\cal X}|_{{\cal X}_{P,L}}$, we have $$ (-\nu E_{\cal X}|_{{\cal X}_{P,L}}-\mu E_{P,L})|_{E_{P,L}}\sim $$ $$ \sim\nu f+\mu(s+\frac12(k+\alpha-\beta)f)=\mu s+(\nu+\frac12\mu (k+\alpha-\beta))f. $$ By Proposition 6.2, this effective divisor contains the fibre $\sigma^{-1}_L(p)$ with multiplicity at least $\mu_p+\mu_q-2\mu$, whence we get the inequality $$ \nu+\frac12\mu(k+\alpha-\beta)\geqslant\mu_p+\mu_q-2\mu, $$ which after easy transformations gives us that $$ \mu>\frac{2(\mu_p+\mu_q)-2\nu}{k+(\alpha-\beta)+4}. $$ The denominator of the right hand side is maximal when $\alpha=k-1$ and $\beta=1$ and so $$ \mu>\frac{2(\mu_p+\mu_q)-2\nu}{2k+2}=\frac{(\mu_p+\mu_q)-\nu}{k+1}. $$ Q.E.D. for the theorem. Proposition 5.3 follows immediately from the theorem that we have just shown, taking into account the construction of the line $L$ and the inequality (\ref{22.09.22.1}). \section{Hypertangent divisors} In this section we prove Theorems 1.2, 1.3 and 1.4. {\bf 7.1. Non-singular points. Tangent divisors.} Let us start the proof of Theorem 1.2. Obviously, it is sufficient to consider the case when the subspace $P$ is of maximal admissible codimension $k+\varepsilon(k)-1$ in ${\mathbb P}^{M+k}$. Theorem 1.1 and Remark 1.4 imply that the inequality $$ \mathop{\rm codim}(\mathop{\rm Sing}(F\cap P)\subset(F\cap P))\geqslant 2k+2 $$ holds. In particular, $F\cap P$ is a factorial complete intersection of codimension $k$ in ${\mathbb P}\cong{\mathbb P}^{M-\varepsilon(k)+1}$. Moreover, by the lefschetz theorem, applied to the section of the variety $F\cap P$ by a general linear subspace of dimension $3k+1$ in $P$ (this section is a non-singular complete intersection of codimension $k$ in ${\mathbb P}^{3k+1}$), we get that the section of $F\cap P$ by an arbitrary linear subspace of codimension $a\leqslant k$ is irreducible and reduced, since for the numerical Chow group we have $$ A^aF\cap P={\mathbb Z}H^a_{F\cap P}, $$ where $H_{F\cap P}$ is the class of a hyperplane section. Assume that Theorem 1.2 is not true and $\mathop{\rm mult_o}Y>2n(Y)$. We will argue precisely as in \cite[\S 2]{Pukh01}, see also \cite[Chapter 3, Section 2.1]{Pukh13a}. Let $T_1,\dots,T_k$ be the tangent hyperplane sections of $F\cap P$ at the point $o$ (in the notations of Subsection 1.4 they are defined by the linear forms $f_{i,1}|_P$, $i=1,\dots,k$). By what was said above, for each $i=1,\dots,k$ the intersection $T_1\cap\dots\cap T_i$ is of codimension $i$ in $F\cap P$, coincides with the scheme-theoretic intersection $(T_1\circ\dots\circ T_i)$ and its multiplicity at the point $o$ equals precisely $2^i$, since the quadratic forms $$ f_{1,2}|_{T_o(F\cap P)},\dots,f_{k,2}|_{T_o(F\cap P)} $$ satisfy the regularity condition. Now we argue as in \cite[\S 2]{Pukh01}. We set $Y_1=Y$ and see that $Y_1\neq T_1$, because $\mathop{\rm mult}_oT_1=2n(T_1)=2$. We consider the cycle $(Y_1\circ T_1)$ of the scheme-theoretic intersection and take for $Y_2$ the component of that cycle that has the maximal value of the ratio $\mathop{\rm mult}_o/\mathop{\rm deg}$. Assume that the subvariety $Y_i$ of codimension $i$ in $F\cap P$, satisfying the inequality $$ \mathop{\rm mult}\nolimits_oY_i>\frac{2^i}{\mathop{\rm deg}F}\mathop{\rm deg}Y_i, $$ is already constructed, and $i\leqslant k-1$. Then $$ Y_i\neq T_1\cap\dots\cap T_i, $$ however by construction $Y_i$ is contained in the divisors $T_1,\dots,T_{i-1}$, so that $Y_i\not\subset T_i$ and the cycle of scheme-theoretic intersection $(Y_i\circ T_i)$ of codimension $i+1$ is well defined. For $Y_{i+1}$ we take the component of this cycle with the maximal value of the ratio $\mathop{\rm mult}_o/\mathop{\rm deg}$. Completing this process, we obtain an irreducible subvariety $Y_{k+1}\subset(F\cap P)$ of codimension $k+1$, satisfying the inequality $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}} Y_{k+1}>\frac{2^{k+1}}{\mathop{\rm deg}F}. $$ {\bf 7.2. Non-singular points. Hypertangent divisors.} We continue the proof of Theorem 1.2. In the notations of Subsection 1.4 for each $j=2,\dots, d_k-1$ construct the hypertangent linear systems $$ \Lambda_j=\left|\sum^k_{i=1}\sum^{\min\{j,d_i-1\}}_{\alpha=1} f_{i,[1,\alpha]}s_{i,j-\alpha}\right|_{F\cap P}, $$ where $f_{i,[1,\alpha]}=f_{i,1}+\dots +f_{i,\alpha}$ is the left segment of the polynomial $f_i$ of length $\alpha$, the polynomials $s_{i,j-\alpha}$ are homogeneous polynomials of degree $j-\alpha$, running through the spaces ${\cal P}_{j-\alpha,M+k}$ independently of each other and the restriction onto $F\cap P$ means the restriction onto the affine part of that variety in ${\mathbb A}^{M+k}_{z_*}$ followed by the closure. Let $h_a$, where $a\geqslant k+1$, be the $a$-th polynomial in the sequence ${\cal S}$. Then $h_a=f_{i,j}|_{{\mathbb P}(T_oF)}$ for some $i$ and $j\geqslant 3$. Set ${\cal H}_a=\Lambda_{j-1}$. In this way we obtain a sequence of linear systems ${\cal H}_{k+1}$, ${\cal H}_{k+2}$,\dots, ${\cal H}_{M}$, where the system $\Lambda_j$ occurs, in the notations of \cite[\S 2]{Pukh01}, $$ w^+_j=\sharp\{i, 1\leqslant i\leqslant k\,|\, j\leqslant d_i-1\} $$ times. By the symbol ${\cal H}[-m]$ we denote the space $$ \prod^{M-m}_{a=k+1}{\cal H}_a $$ of all tuples $(D_{k+1},\dots,D_{M-m})$ of divisors, where $D_a\in{\cal H}_a$. For $a\in\{k+1,\dots,M\}$ set $$ \beta_a=\frac{j+1}{j}, $$ if ${\cal H}_a=\Lambda_j$. The number $\beta_a$ is called the {\it slope} of the divisor $D_a$. It is easy to see that \begin{equation}\label{24.10.22.1} \prod^M_{a=k+1}\beta_a=\frac{d_1\dots d_k}{2^k}=\frac{\mathop{\rm deg} F}{2^k}. \end{equation} Set $m_*=k+\varepsilon(k)+3$. Let $$ (D_{k+1},\dots,D_{M-m_*})\in{\cal H}[-m_*] $$ be a general tuple. The technique of hypertangent divisors, applied in precisely the same way as in \cite[\S 2]{Pukh01} or \cite[Chapter 3, Subsection 2.2]{Pukh13a}, see also \cite[Proposition 2.1]{Pukh2022a}, gives the following claim. {\bf Proposition 7.1.} {\it There is a sequence of irreducible subvarieties $$ Y_{k+1},Y_{k+2},\dots,Y_{M-m_*}, $$ where $Y_{k+1}$ has been constructed above, such that $\mathop{\rm codim}(Y_i\subset(F\cap P))=i$, the subvariety $Y_i$ is not contained in the support of the divisor $D_{i+1}$ for $i\leqslant M-m_*-1$, the subvariety $Y_{i+1}$ is an irreducible component of the effective cycle $(Y_i\circ D_{i+1})$ and the following inequality holds:} $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_{i+1}\geqslant \beta_{i+1}\frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_i. $$ There is no need to give a {\bf proof} of that claim because it is identical to the arguments mentioned above. Note only that the key point in the construction of the sequence of subvarieties $Y_i$ is the fact that $Y_i$ is not contained in the support of a general divisor $D_{i+1}\in{\cal H}_{i+1}$, and this fact follows from the regularity condition (R1). Since $\mathop{\rm dim}(F\cap P)=M+1-k-\varepsilon(k)$, the subvariety $Y^*=Y_{M-m_*}$ is of dimension 4 and satisfies the inequality $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y^*> \frac{\displaystyle\frac{2^{k+1}}{\mathop{\rm deg}F}\cdot\frac{\mathop{\rm deg} F}{2^k}}{\displaystyle\frac32\cdot\prod\limits^M_{a=M-m_*+1}\beta_a}= \frac43\frac{1}{\prod\limits^M_{a=M-m_*+1}\beta_a}. $$ (The number $\frac32$ appears in the denominator because the hypertangent divisor $D_{k+1}$ is skipped in the procedure of intersection, in the same way and for the same reason as in \cite[\S 2]{Pukh01}, and its slope is $\frac32$.) Now the inequality \begin{equation}\label{14.10.22.1} \frac43\geqslant\prod^M_{a=M-m_*+1}\beta_a, \end{equation} shown below in Proposition 7.2, completes the proof of Theorem 1.2. {\bf Proposition 7.2.} {\it Assume that for $k=3,4,5$ the dimension $M$ is, respectively, at least $96$, $160$, $215$, and for $k\geqslant 6$ the inequality $M\geqslant 8k^2+2k$ holds. then the inequality (\ref{14.10.22.1}) is true.} {\bf Proof.} Using the obvious fact that the function $\frac{t+1}{t}$ is decreasing, it is easy to see that the right hand side of the inequality (\ref{14.10.22.1}) with $k$ and $M$ fixed attains the maximum when the degrees $d_1,\dots,d_k$ are equal or ``almost equal'' in the following sense: let $M\equiv e\mathop{\rm mod}k$ with $e\in\{0,1,\dots,k-1\}$, then the ``almost equality'' means that $$ d_1=\dots=d_{k-e}=\frac{M-e}{k}+1,\quad d_{k-e+1}=\dots=d_k=\frac{M-k}{k}+2. $$ For $k\in\{3,\dots,9\}$ the claim of the proposition can be checked for each case of almost equal degrees, that is, for each possible value of $e$, manually, computing $\varepsilon(k)$ explicitly. For $k\geqslant 10$ it is easy to see that $\varepsilon(k)\leqslant k-3$, so that $m_*\leqslant 2k$. Therefore (again considering the case of almost equal degrees) the right hand side of (\ref{14.10.22.1}) does not exceed the number $$ \left(\frac{\frac{M}{k}}{\frac{M}{k}-2}\right)^k= \left(\frac{M}{M-2k}\right)^k, $$ from which we get that (\ref{14.10.22.1}) is true if $$ M\geqslant 2k\frac{(1+\frac13)^\frac{1}{k}}{(1+\frac13)^\frac{1}{k}-1}. $$ If in the numerator and denominator we replace $(1+\frac{1}{3})^{\frac{1}{k}}$ by the smaller number $1+\frac{1}{4k}$, the right hand side of the last inequality gets higher. This proves the proposition. Q.E.D. Note that for $M\geqslant \rho(k)$ (see the inequality (\ref{14.11.22.1}) in Subsection 0.1) the assumptions of the previous proposition are satisfied. This completes the proof of Theorem 1.2. {\bf 7.3. Quadratic points.} Let us show Theorem 1.3. Note first of all that if $Y$ is a section of the variety $W$ by a hyperplane that is tangent to $W$ at the point $o$ (that is, the equation of the hyperplane is a linear combination of the forms $f_{1,1},\dots,f_{k,1}$, restricted onto the hyperplane ${\mathbb P}(W)\cong{\mathbb P}^{M+k-1}$), then $$ \mathop{\rm mult}\nolimits_oY=4n(Y)=4, $$ so that the claim of the theorem is optimal. Thus we assume the converse: the inequality $$ \mathop{\rm mult}\nolimits_oY>2n(Y) $$ holds. We argue as in the non-singular case (Subsection 7.1): let $T_1,\dots,T_{k-1}$ be the tangent hyperplane sections, given by $(k-1)$ independent forms taken from the set $\{f_{1,1},\dots,f_{k,1}\}$. Since $\mathop{\rm codim}(\mathop{\rm Sing}F\subset F)\geqslant2k+2$, all scheme-theoretic intersections $(T_1\circ\dots\circ T_i)$, $1\leqslant i\leqslant k-1$, are irreducible, reduced and coincide with the set-theoretic intersection $T_1\cap\dots\cap T_i$, and moreover, by the condition (R2) the equality $$ \mathop{\rm mult}\nolimits_oT_1\cap\dots\cap T_i=2^{i+1} $$ holds. In particular, $\mathop{\rm mult}_oT_1=4n(T_1)=4$, so that $Y\neq T_1$. Arguing as in Subsection 7.1, we construct a sequence of irreducible subvarieties $Y_1=Y,Y_2,\dots,Y_k$, where $\mathop{\rm codim}(Y_i\subset W)=i$ and the inequality $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_k>\frac{2^{k+1}}{\mathop{\rm deg}F} $$ holds. Now the proof of Theorem 1.3 repeats the arguments of Subsection 7.2, where $m_*$ is replaced by 4. Since $4<m_*$, the inequality (\ref{14.10.22.1}) guarantees the inequality which is obtained from (\ref{14.10.22.1}) when $m_*$ is replaced by 4. This completes the proof of Theorem 1.3. {\bf 7.4. Multi-quadratic points. Tangent divisors.} We start the proof of Theorem 1.4, the structure of which is similar to the structure of the proof of Theorem 1.2. At first we argue as in Subsection 7.1: it is sufficient to consider a linear subspace $P$ in $T_oF$ of maximal admissible codimension $\varepsilon(k)$. Assume that the prime divisor $Y$ on $F\cap P$ satisfies the inequality $$ \mathop{\rm mult}\nolimits_oY>\frac32\cdot2^kn(Y), $$ or the equivalent inequality $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y>\frac32\cdot\frac{2^k}{\mathop{\rm deg}F}, $$ and consider the second hypertangent linear system (which in this case plays the role of the tangent linear system) $$ \Lambda_2=\left|\sum_{d_i\geqslant 3}s_{i,0}f_{i,2}\right|_{F\cap P}, $$ where $s_{i,0}\in{\mathbb C}$ are constants, independent of each other. Instead of the Lefschetz theorem we use the condition (R3.1): the system of equations $f_{i,2}|_{F\cap P}=0$, where $d_i\geqslant 3$, defines an irreducible reduced subvariety of codimension $k+k_{\geqslant 3}$ in $P$, and by (R3.2) the multiplicity of that subvariety at the point $o$ is precisely $2^k\cdot(\frac32)^{k_{\geqslant 3}}$. More precisely, for a general tuple $(D_{2,1},\dots,D_{2,k_{\geqslant 3}})$ of divisors in the system $\Lambda_2$ the following claim is true: for each $i=1,\dots,k_{\geqslant 3}$ the cycle $(D_{2,1}\circ\dots\circ D_{2,i}$) of the scheme-theoretic intersection of the divisors $D_{2,1},\dots,D_{2,i}$ is an irreducible reduced subvariety of codimension $i$ in $F\cap P$, the multiplicity of which at the point $o$ is $2^k\cdot(\frac32)^i$. Arguing as in Subsection 7.1, we construct a sequence $Y_1=Y,Y_2,\dots,Y_{k_{\geqslant 3}}$ of irreducible subvarieties of codimension $\mathop{\rm codim}(Y_i\subset(F\cap P))=i$, where $Y_{i+1}$ is an irreducible component of the cycle $(Y_i\circ D_{2,i})$ with the maximal value of the ratio $\mathop{\rm mult}_o/\mathop{\rm deg}$. Therefore, $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_{k_{\geqslant 3}}> \left(\frac32\right)^{k_{\geqslant 3}}\cdot\frac{2^k}{\mathop{\rm deg}F}. $$ It follows from here that $Y_{k_{\geqslant 3}}\neq D_{2,1}\cap\dots\cap D_{2,k_{\geqslant 3}}$, but since by construction $$ Y_{k_{\geqslant 3}}\subset D_{2,1}\cap\dots\cap D_{2,k_{\geqslant 3}-1}, $$ we conclude that $Y_{k_{\geqslant 3}}\not\subset D_{2,k_{\geqslant 3}}$, so that the effective cycle $(Y_{k_{\geqslant 3}}\circ D_{2,k_{\geqslant 3}})$ of the scheme-theoretic intersection of these varieties is well defined and one of its components $Y_{k_{\geqslant 3}+1}$ satisfies the inequality $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_{k_{\geqslant 3}+1}> \left(\frac32\right)^{k_{\geqslant 3}+1}\cdot\frac{2^k}{\mathop{\rm deg}F}. $$ {\bf 7.5. Multi-quadratic points. Hypertangent divisors.} Now we argue almost word for word as in Subsection 7.2: construct the hypertangent systems $$ \Lambda_j=\left|\sum^j_{\alpha=2}\sum_{d_i\geqslant \alpha+1}f_{i,[2,\alpha]}s_{i,j-\alpha}\right|_{F\cap P}, $$ where $j=3,\dots,d_k-1$ and all symbols have the same meaning as in Subsection 7.2. If $h_a$, where $a\geqslant k+k_{\geqslant 3}+1$, is the $a$-th polynomial in the sequence ${\cal S},h_a=f_{i,j}|_{{\mathbb P}(T_oF)}$, for some $i$ and $j\geqslant 4$, then we set ${\cal H}_a=\Lambda_{j-1}$ and obtain the sequence of linear systems $$ {\cal H}_{k+k_{\geqslant 3}+1},\quad {\cal H}_{k+k_{\geqslant 3}+2},\quad\dots,\quad {\cal H}_M. $$ By the symbol ${\cal H}[-m]$ we denote the space $$ \prod^{M-m}_{a=k+k_{\geqslant 3}+1}{\cal H}_a. $$ Instead of the equality (\ref{24.10.22.1}) we get the equality $$ \prod^M_{a=k+k_{\geqslant 3}+1}\beta_a=\frac{\mathop{\rm deg}F}{2^k\left(\frac32\right)^{k_{\geqslant 3}}}. $$ Let $(D_{k+k_{\geqslant 3}+1},\dots,D_{M-m^*})\in{\cal H}[-m^*]$ be a general tuple. Now the technique of hypertangent divisors, applied in the word for word the same way as in Subsection 7.2, gives the following claim. {\bf Proposition 7.3.} {\it There is a sequence of irreducible subvarieties $$ Y_{k_{\geqslant 3}+1},Y_{k_{\geqslant 3}+2},\dots,Y_{M-k-m^*}, $$ where $Y_{k_{\geqslant 3}+1}$ is constructed above, such that $\mathop{\rm codim}(Y_i\subset(F\cap P))=i$, the subvariety $Y_i$ is not contained in the support of the divisor $D_{k+i+1}$ for $i\leqslant M-m^*-1$, the subvariety $Y_{i+1}$ is an irreducible component of the effective cycle $(Y_i\circ D_{k+i+1})$ and the following inequality holds:} $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_{i+1}\geqslant\beta_{k+i+1}\frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y_i. $$ Now since $\mathop{\rm dim}F\cap P=M-(k-l)-\varepsilon(k)$, by the definition of the number $m^*$ the last subvariety $Y^*=Y_{M-k-m^*}$ in that sequence is of dimension $\geqslant 4$ and satisfies the inequality $$ \frac{\mathop{\rm mult}_o}{\mathop{\rm deg}}Y^*> \frac{ \displaystyle \left(\frac32\right)^{k_{\geqslant 3}+1}\cdot\frac{2^k}{\mathop{\rm deg}F}\cdot \frac{\mathop{\rm deg}F}{2^k \left(\frac32\right)^{k_{\geqslant 3}}}}{\displaystyle \frac43\prod\limits^M_{a=M-m_*+1}\beta_{k+a}}= \frac98\frac{1}{\displaystyle\prod\limits^M_{a=M-m_*+1}\beta_{k+a}}. $$ (The number $\frac43$ appears in the denominator of the right hand side, because the hypertangent divisor $D_{k+k_{\geqslant 3}+1}$ is skipped in the process of constructing the sequence $Y_*$, see the similar remark above, before the inequality (\ref{14.10.22.1}).) If $m^*=0$, then the product in the denominator is assumed to be equal to 1. Now the inequality \begin{equation}\label{28.10.22.1} \frac98\geqslant\prod^M_{a=M-m^*+1}\beta_{k+a}, \end{equation} shown below in Proposition 7.4, completes the proof of Theorem 1.4. {\bf Proposition 7.4.} {\it Assume that for $k\in\{3,\dots,7\}$ the number $M$ is at least the number shown in the corresponding column of the table \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline $k$ & $3$ & $4$ & $5$ & $6$ & $7$\\ \hline $M\geqslant$ & $128$ & $204$ & $255$ & $357$ & $477$\\ \hline \end{tabular}, \end{center} and for $k\geqslant 8$ the inequality $M\geqslant 9k^2+k$ holds. Then the inequality (\ref{28.10.22.1}) holds.} {\bf Proof.} As in the non-singular case (the proof of Proposition 7.2), we see that the right hand side of the inequality (\ref{28.10.22.1}) for $k$ and $M$ fixed, attains the maximum when the degrees $d_i$ are equal or ``almost equal''. For $k\in\{3,\dots,7\}$ the claim of the proposition is checked manually. For $k\geqslant 8$ we have $\varepsilon(k)\leqslant k-2$, so that $m^*\leqslant k$. Therefore (considering the case of equal or almost equal degrees) the right hand side of the inequality (\ref{28.10.22.1}) does not exceed the number $$ \left(\frac{M}{M-k}\right)^k, $$ which, in its turn, does not exceed $\frac98$ for $M\geqslant 9k^2+k$, which is easy to check by elementary computations, similar to the proof Proposition 7.2. Q.E.D. \section{The codimension of the complement} In this section we show the estimate for the codimension of the complement ${\cal P}\setminus{\cal F}$, given in Theorem 0.1. {\bf 8.1. Preliminary constructions.} Set $$ \gamma=M-k+5+{M-\rho(k)+2\choose 2}, $$ see Subsection 0.1. We consider $\gamma$ as a function of $M$ with $k\geqslant 3$ fixed, where $M\geqslant\rho(k)$. Let $o\in{\mathbb P}^{M+k}$ be an arbitrary point. The symbol ${\cal P}(o)$ stands for the linear subspace of the space ${\cal P}$, consisting of all tuples $\underline{f}$, vanishing at the point $o$: $\underline{f}(o)=(0,\dots,0)$. Obviously, $\mathop{\rm codim}({\cal P}(o)\subset{\cal P})=k$. Fixing the point $o$, we use the notations of Subsections 1.2-1.4, considering the polynomials $f_i$ as non-homogeneous polynomials in the affine coordinates $z_*$. By the symbols $$ {\cal B}_{MQ1},\,{\cal B}_{MQ2},\,{\cal B}_{R1},\,{\cal B}_{R2},\,{\cal B}_{R3.1},\,{\cal B}_{R3.2} $$ we denote the subsets of the subspace ${\cal P}(o)$, consisting of the tuples $\underline{f}$ that do not satisfy the conditions $$ (MQ1),\,(MQ2),\,(R1),\,(R2),\,(R3.1),\,(R3.2) $$ at the point $o$, respectively. Since the point $o$ varies in ${\mathbb P}^{M+k}$, it is sufficient to show that the codimension of each of the six sets ${\cal B}_*$ in ${\cal P}(o)$ is at least $\gamma+M$. Furthermore, for an arbitrary tuple $$ \underline{\xi}=(\xi_1,\dots,\xi_k) $$ of linear forms in $z_*$ the symbol ${\cal P}(o,\underline{\xi})$ denotes the affine subspace, consisting of the tuples $\underline{f}$, such that $$ f_{1,1}=\xi_1,\quad\dots,\quad f_{k,1}=\xi_k. $$ By the symbol $\mathop{\rm dim}\underline{\xi}$ denote the dimension $$ \mathop{\rm dim}\langle\xi_1,\dots,\xi_k\rangle, $$ so that ${\cal P}(o)$ is fibred into disjoint subsets $$ {\cal P}^{(i)}(o)=\bigcup_{\mathop{\rm dim}\underline{\xi}=i}{\cal P}(o,\underline{\xi}), $$ where $i=0,1,\dots,k$. Obviously, the equality $$ \mathop{\rm codim}({\cal P}^{(i)}(o)\subset{\cal P}(o))=(k-i)(M+k-i) $$ holds. In particular, ${\cal P}^{(k)}(o)$ consists of the tuples $\underline{f}$, such that the scheme of their common zeros is a non-singular subvariety of codimension $k$ in a neighborhood of the point $o$. Set $$ {\cal B}_{R1}(\underline{\xi})={\cal B}_{R1}\cap{\cal P}(o,\underline{\xi}). $$ For the case of a non-singular point it is sufficient to prove the inequality $$ \mathop{\rm codim}({\cal B}_{R1}(\underline{\xi})\subset{\cal P}(o,\underline{\xi}))\geqslant \gamma+M, $$ where $\mathop{\rm dim}\underline{\xi}=k$. Furthermore, let ${\cal B}_{MQ1}(\underline{\xi})={\cal B}_{MQ1}\cap{\cal P} (o,{\underline{\xi}})$, where $\mathop{\rm dim}\underline{\xi}=i\leqslant k-1$, be the set of the tuples $\underline{f}$, such that the condition (MQ1) for $l=k-i$ is not satisfied, and ${\cal B}_{MQ2}(\underline{\xi})={\cal B}_{MQ2}\cap{\cal P }(o,\underline{\xi})$, where $\mathop{\rm dim}\underline{\xi}=i\leqslant k-2$, be the set of the tuples $\underline{f}$, such that the condition (MQ2) for $l=k-i$ is not satisfied. In a similar way, we define the sets ${\cal B}_{R2}(\underline{\xi})$ for $\mathop{\rm dim}\underline{\xi}=k-1$ and ${\cal B}_{R3.1}(\underline{\xi})$, ${\cal B}_{R3.2}(\underline{\xi})$ for $\mathop{\rm dim}\underline{\xi}\leqslant k-2$. Clearly, it is sufficient to prove that for $\mathop{\rm dim}\underline{\xi}=i$ the codimension of the set ${\cal B}_*(\underline{\xi})$ in ${\cal P}(o,\underline{\xi})$ is at least $$ \gamma+M-(k-i)(M+k-i). $$ In the conditions (R1),(R2) and (R3.2) we have also an arbitrary subspace $\Pi\subset{\mathbb P}(T_oF)$ of the corresponding codimension, and in the condition (R3.1) an arbitrary subspace $P$ in the embedded tangent space $T_oF\subset{\mathbb P}^{M+k}$ of codimension $\varepsilon(k)$, containing the point $o$. For an arbitrary subspace $\Pi\subset{\mathbb P}(T_oF)$ of the corresponding codimension let $$ {\cal B}_{R1}(\underline{\xi},\Pi),\quad {\cal B}_{R2}(\underline{\xi},\Pi),\quad {\cal B}_{R3.2}(\underline{\xi},\Pi) $$ be the set of tuples $\underline{f}\in{\cal P}(o,\underline{\xi})$, such that the respective condition (R1),(R2) and (R3.2) is violated precisely for that subspace $\Pi$. In a similar way we define the subset ${\cal B}_{R3.1}(\underline{\xi},P)$. These definitions are meaningful because the tangent space $T_oF$ is given by the fixed linear forms $\xi_i$ and for that reason is fixed. Since the subspace $\Pi$ varies in a $(\mathop{\rm dim}\Pi+1)\mathop{\rm codim}(\Pi\subset{\mathbb P}(T_oF))$-dimensional Grassmanian, the estimate for the codimension of the set ${\cal B}_*(\underline{\xi},\Pi)$ in ${\cal P}(o,\underline{\xi})$ should be stronger than the estimate for the codimension of the set ${\cal B}_*(\underline{\xi})$ by that number. Similarly, $P$ varies in a $$ \varepsilon(k)(\mathop{\rm dim}T_oF-\varepsilon(k)) $$ -dimensional family, so that the estimate for the codimension of the set ${\cal B}_{R3.1}(\underline{\xi},P)$ should be stronger than the estimate for ${\cal B}_{R3.1}(\underline{\xi})$ by that number. Now everything is ready to consider each of the six subsets ${\cal B}_*$. {\bf 8.2. The conditions (MQ1) and (MQ2).} For a non-singular point $o\in F$ these conditions contain no restrictions, so we assume that $\dim\underline{\xi}\leqslant k-1$. It is well known that the closed subset of quadratic forms of rank $\leqslant r\leqslant N-1$ in the space ${\cal P}_{2,N}$ has the codimension $$ {N-r+1\choose 2}. $$ From here it is easy to see that the closed subset of tuples $(q_1,\dots,q_e)=q_{[1,e]}$ of quadratic forms in $N$ variables, defined by the condition $$ \mathop{\rm rk}q_{[1,e]}\leqslant r, $$ is of codimension $$ \geqslant{N-r+1\choose 2}-(e-1) $$ in the space ${\cal P}^{\times e}_{2,N}$. As we noted in Subsection 1.2 (after stating the condition (MQ2)), for $l\geqslant 2$ the condition (MQ2) is stronger than (MQ1), therefore it is sufficient to estimate the codimension of the set ${\cal B}_{MQ2}$ (in the case of quadratic points, when $l=1$, it is easy to check that the codimension of the set ${\cal B}_{MQ1}$ is higher than required). So we assume that $\mathop{\rm dim}\underline{\xi}=k-l\leqslant k-2$. The condition (MQ2) requires the rank of the tuple of quadratic forms $q_{[1,k]}$, where $q_i=f_{i,2}|_{T_oF}$, to be at least $\rho(k)+2$, see (\ref{14.11.22.1}) in Subsection 0.1. Taking into account the variation of the tuple $\underline{\xi}$, from what was said above it is easy to obtain that the codimension of the set ${\cal B}_{MQ2}\cap{\cal P}^{k-l}(o)\geqslant$ $$ -k+1+l(M+l)+{M+l-\rho(k)\choose 2}. $$ The minimum of this expression is attained for $l=2$ and it is easy to check that this minimum is precisely $\gamma+M$. Therefore, the codimension of the set ${\cal B}_{MQ2}$ is at least $\gamma$, the codimension of the set ${\cal B}_{MQ1}$ for $l\geqslant 2$ is higher. For $l=1$ the last codimension is also higher. This completes our consideration of the conditions (MQ1) and (MQ2). {\bf 8.3. Regularity at the non-singular and quadratic points.} Let us estimate the codimension of the set ${\cal B}_{R1}(\underline{\xi},\Pi)$ in the space ${\cal P}(o,\underline{\xi})$. Here $\mathop{\rm dim}\underline{\xi}=k$ and $\Pi\subset{\mathbb P}(T_oF)$ is a subspace of codimension $k+\varepsilon(k)-1=m_*-4$. Let $$ {\cal G}(\underline{d})=\prod^{M-m_*}_{i=1}{\cal P}_{\mathop{\rm deg}h_i,\mathop{\rm dim}\Pi+1} $$ be the space, parameterizing all sequences ${\cal S}[-m_*]|_{\Pi}$. Since the polynomials $h_i$ are distinct homogeneous components of the polynomials of the tuple $\underline{f}$, restricted onto the subspace $\Pi$, the codimension of the subset ${\cal B}_{R1}(\underline{\xi},\Pi)$ in ${\cal P}(o,\underline{\xi})$ is equal to the codimension of the subset ${\cal B}\subset{\cal G}(\underline{d})$, which consists of the sequences that do not satisfy the condition (R1). Using the approach that was applied in \cite{Pukh98b,Pukh13a,EvansPukh2} and many other papers, let us present ${\cal B}$ as a disjoint union $$ {\cal B}=\bigsqcup^{M-m_*}_{i=1}{\cal B}_i, $$ where ${\cal B}_i$ consists of sequences $$ (h_1|_{\Pi},\dots,h_{M-m_*}|_{\Pi}), $$ such that the first $i-1$ polynomials form a regular sequence but $h_i$ vanishes on one of the components of the set of their common zeros. The ``projection method'' estimates the codimension of ${\cal B}_i$ in ${\cal G}(\underline{d})$ from below by the integer \begin{equation}\label{17.10.22.1} {\mathop{\rm dim}\Pi-i+1+\mathop{\rm deg}h_i\choose \mathop{\rm deg}h_i} ={\mathop{\rm dim}\Pi-i+1+\mathop{\rm deg}h_i\choose \mathop{\rm dim}\Pi-i+1} \end{equation} (we will use both presentations). It follows easily from here (see \cite[\S3]{EvansPukh2}), that the worst estimate corresponds to the case of equal or ``almost equal'' degrees $d_i$, described above. We will consider this case. Thus we need to estimate from below the minimum of $M-m_*$ integers (\ref{17.10.22.1}). Here are the first $(k+1)$ of them: $$ {\mathop{\rm dim}\Pi+2\choose 2},\,{\mathop{\rm dim}\Pi+1\choose 2},\,\dots,\,{\mathop{\rm dim}\Pi+3-k\choose 2},\,{\mathop{\rm dim}\Pi+3-k\choose 3}. $$ We call the left hand side of the equality (\ref{17.10.22.1}) the presentation of type (I), the right hand side is the presentation of type (II). Let us write down each of the numbers (\ref{17.10.22.1}) in the form $$ {A(i)\choose B(i)}, $$ where $A(i)\geqslant 2B(i)$, using the presentation of type (I) or of type (II). At first (for the starting segment of the sequence) we use the presentation of type (I). It is easy to see that when we change $i$ by $i+1$, we have one of the two options: \begin{itemize} \item either $\mathop{\rm deg}h_{i+1}=\mathop{\rm deg}h_i$, and then $A(i+1)=A(i)-1$ and $B(i+1)=B(i)$, so that $$ {A(i+1)\choose B(i+1)}<{A(i)\choose B(i)}, $$ and moreover, $C(i)=A(i)-2B(i)$ decreases: $C(i+1)=C(i)-1$, \item or $\mathop{\rm deg}h_{i+1}=\mathop{\rm deg}h_i+1$, and then $A(i+1)=A(i)$ and $B(i+1)=B(i)+1$, so that $C(i+1)=C(i)-2$ and if $C(i+1)\geqslant 0$, then $$ {A(i+1)\choose B(i+1)}>{A(i)\choose B(i)}. $$ \end{itemize} This is how it goes on until the ``equilibrium'': $C(i_*)\geqslant 0$, but $C(i_*+1)<0$, and after that we use the presentation of type (II). Now when we change $i$ by $(i+1)$, we have one of the two options: \begin{itemize} \item either $\mathop{\rm deg}h_{i+1}=\mathop{\rm deg}h_i$, and then $A(i+1)=A(i)-1$ and $B(i+1)=B(i)-1$, so that $C(i+1)=C(i)+1$ and $$ {A(i+1)\choose B(i+1)}<{A(i)\choose B(i)}, $$ \item or $\mathop{\rm deg}h_{i+1}=\mathop{\rm deg}h_i+1$, and then $A(i+1)=A(i)$ and $B(i+1)=B(i)-1$, so that $C(i+1)=C(i)+2$ and $$ {A(i+1)\choose B(i+1)}<{A(i)\choose B(i)}. $$ \end{itemize} Therefore, after the ``equilibrium'' our sequence is strictly decreasing. Moreover, if $$ {A(i_1)\choose B(i_1)}\quad\mbox{and}\quad {A(i_2)\choose B(i_2)} $$ are two numbers in our sequence, where $i_1\leqslant i_*$ and $i_2>i_*$ and $B(i_1)\geqslant B(i_2)$, then, obviously, $$ {A(i_1)\choose B(i_1)}>{A(i_2)\choose B(i_2)}. $$ Recall that the degrees $d_i$ are equal or ``almost equal''. {\bf Lemma 8.1.} {\it For $M\geqslant 3k^2$ the following inequality holds:} $i_*<M-m_*$. {\bf Proof.} Elementary computations, using the equality $C(i+k)=C(i)-(k+1)$ if $C(i+k)\geqslant 0$. Q.E.D. for the lemma. Therefore, the ``equilibrium'' is reached earlier than the sequence $h_i,\dots,h_{M-m_*}$ comes to an end, so that there is a non-empty segment after the ``equilibrium''. By construction, $B(M-m_*)=4$. By what was said above, the minimum of the numbers ${A(i)\choose B(i)}$ for $i=1,\dots,M-m_*$ is the minimum of the following three numbers: $$ {\mathop{\rm dim}\Pi+3-k\choose 2},\quad {\mathop{\rm dim}\Pi+4-2k\choose 3},\quad {\mathop{\rm deg}h_{M-m_*}+4\choose 4}. $$ {\bf Lemma 8.2.} {\it For $\mathop{\rm dim}\Pi\geqslant 3k+1$ the following inequality holds:} $$ {\mathop{\rm dim}\Pi+4-2k\choose 3}>{\mathop{\rm dim}\Pi+3-k\choose 2}. $$ {\bf Proof.} Elementary computations. Q.E.D. {\bf Lemma 8.3.} {\it For $M\geqslant 2\sqrt{3}k^2$ the following inequality holds:} $$ {\mathop{\rm deg}h_{M-m_*}+4\choose 4}>{\mathop{\rm dim}\Pi+3-k\choose 2}. $$ {\bf Proof.} It is easy to check the inequalities $$ \frac{(M-2k)^2}{2}\geqslant{\mathop{\rm dim}\Pi+3-k\choose 2} $$ and $$ {\mathop{\rm deg}h_{M-m_*}+4\choose 4}\geqslant\frac{1}{24} \left(\frac{M}{k}+1\right)\left(\frac{M}{k}\right)\left(\frac{M}{k}-1\right) \left(\frac{M}{k}-2\right), $$ so that it is sufficient to show that for $M\geqslant 2\sqrt{3}k^2$ the inequality $$ \left(\frac{M^2}{k^2}-1\right)\left(\frac{M^2}{k^2}-2\frac{M}{k}\right) >12(M-2k)^2 $$ holds or, equivalently, $M(M^2-k^2)>12k^4(M-2k)$. It is easy to check the last inequality, considering the cubic polynomial $$ t^3-(12k^4+k^2)t+24k^5 $$ in the real variable $t$. Q.E.D. for the lemma. The work that was carried out above gives the inequality $$ \mathop{\rm codim}({\cal B}\subset{\cal G}(\underline{d}))\geqslant{M+3-2k-\varepsilon(k)\choose 2}. $$ From here by elementary computations (taking into account the variation of the subspace $\Pi$, see Subsection 8.1) it is easy to obtain the required the inequality $\mathop{\rm codim}({\cal B}_{R1}\subset{\cal P}(o))\geqslant\gamma+M$. This completes the proof in the case of smooth points. It is easy to see that the methods used above give a stronger estimate for the codimension of the set ${\cal B}_{R2}$, because the dimension of the subspace $\Pi$ is higher. The computations are completely similar to the computations given above for the case of a non-singular point, for that reason we do not consider the case of a quadratic point and move on to estimating the codimension of the sets ${\cal B}_{R3.1}$ and ${\cal B}_{R3.2}$. {\bf 8.4. Regularity at the multi-quadratic points.} Let us estimate the codimension of the set ${\cal B}_{R3.2}(\underline{\xi},\Pi)$, where $\Pi\subset{\mathbb P}(T_oF)$ is an arbitrary subspace of codimension $\varepsilon(k)$. Our arguments are completely similar to the arguments of Subsection 8.3 for a non-singular point and give a stronger estimate for the codimension. We just point out the necessary changes in the constructions of Subsection 8.3. Set $$ {\cal G}(\underline{d})=\prod^{M-m^*}_{i=1}{\cal P}_{\mathop{\rm deg} h_i,\mathop{\rm dim}\Pi+1}. $$ Denote by the symbol ${\cal B}$ the subset in ${\cal G}(\underline{d})$, consisting of the sequences that do not satisfy the condition (R3.2). Again we break ${\cal B}$ into subsets: $$ {\cal B}=\bigsqcup^{M-m^*}_{i=1}{\cal B}_i, $$ where ${\cal B}_i$ has the same meaning as in Subsection 8.3 (but for the multi-quadratic point $o$). Again the codimension of ${\cal B}_i$ in ${\cal G}(\underline{d})$ is bounded from below by the number (\ref{17.10.22.1}), and for $k$ and $M$ fixed the worst estimate corresponds to the case of equal or ``almost equal'' degrees $d_i$. Arguing precisely in the same way as in the non-singular case (Subsection 8.3), we see, since the dimension of the subspace $\Pi$ is higher than in the non-singular case, that the claim of Lemma 8.1 is true. Note that if $m^*=0$, then in the notations of Subsection 8.3 we have $B(M-m^*)\geqslant 4$. Thus, replacing $B(M-m^*)$ by 4, we get that $\mathop{\rm codim}({\cal B}\subset{\cal G}(\underline{d}))$ is bounded from below by the least of the three numbers $$ {\mathop{\rm dim}\Pi+3-k\choose 2},\quad {\mathop{\rm dim}\Pi+4-2k\choose 3},\quad {\mathop{\rm deg}h_{M-m^*}+4\choose 4}. $$ The claim of Lemma 8.2 is true since, as we noted above, $\mathop{\rm dim} \Pi$ in the multi-quadratic case is higher than in the non-singular case. Obviously, $m^*<m_*$, so that $\mathop{\rm deg} h_{M-m^*}\geqslant\mathop{\rm deg}h_{M-m_*}$ and the claim of Lemma 8.3 is also true. As a result, we get the inequality $$ \mathop{\rm codim}({\cal B}\subset{\cal G}(\underline{d}))\geqslant{M+2+l-k-\varepsilon(k)\choose 2}, $$ where $\mathop{\rm dim}\underline{\xi}=k-l$. The minimum of the right hand side is attained for $l=2$ and it is easy to see that this minimum is significantly higher than in the non-singular case. It is easy to check, taking into account the variation of the subspace $\Pi$, that $$ \mathop{\rm codim}(B_{R3.2}\subset{\cal P}(o))>\gamma+M. $$ This completes our consideration of the condition (R3.2) in the multi-quadratic case. {\bf 8.5. The condition (R3.1).} In order to estimate the codimension of the set ${\cal B}_{R3.1}(\underline{\xi},P)$, we need the following known general fact. Take $e\geqslant 1$ and let $\underline{w}=(w_1,\dots,w_e)\in{\mathbb Z}^e$ be a tuple of integers, where $2\leqslant w_1\leqslant\dots\leqslant w_e$. Set $$ {\cal P}(\underline{w})=\prod^e_{i=1}{\cal P}_{w_i,N+1} $$ to be the space of tuples $\underline{g}=(g_1,\dots,g_e)$ of homogeneous polynomials in $N+1$ variables, $\mathop{\rm deg}g_i=w_i$, which we consider as homogeneous polynomials on ${\mathbb P}^N$. Let $$ {\cal B}^*(\underline{w})\subset{\cal P}(\underline{w}) $$ be the set of tuples $\underline{g}$, such that the scheme of their common zeros is not an irreducible reduced subvariety of codimension $e$ in ${\mathbb P}^N$. {\bf Theorem 8.1.} {\it The following inequality holds:} $$ \mathop{\rm codim}({\cal B}^*(\underline{w})\subset{\cal P}(\underline{w}))\geqslant\frac12(N-e-1)(N-e-4)+2. $$ {\bf Proof:} this is Theorem 2.1 in \cite{Pukh2022a}. Let us estimate the codimension of the set ${\cal B}_{R3.1}(\underline{\xi},\Pi)$. In order to do this, consider in the projective space $P$ a hypersurface $P^{\sharp}$ that does not contain the point $o$, for instance, the intersection of the hyperplane ``at infinity'' with respect to the system of affine coordinates $(z_1,\dots,z_{M+k})$ with the subspace $P$. If the scheme of common zeros of the tuple of polynomials, consisting of $$ f_1|_P,\dots,f_k|_P $$ and the polynomials $f_{i,2}|_P$ for $i$ such that $d_i\geqslant 3$, is not an irreducible reduced subvariety of codimension $k+k_{\geqslant 3}$ in $P$ (that is, the condition (R3.1) is violated, see Subsection 1.4), then the scheme of common zeros of the set of polynomials \begin{equation}\label{27.02.23.1} f_1|_{P^{\sharp}},\dots,f_k|_{P^{\sharp}},f_{i,2}|_{P^{\sharp}}\quad \mbox{for} \quad d_i\geqslant 3, \end{equation} respectively, is reducible, non-reduced or is of codimension $<k+k_{\geqslant 3}$ in $P^{\sharp}$. However, for each $i$, such that $d_i\geqslant 3$, the homogeneous polynomials $$ f_i|_{P^{\sharp}}=f_{i,d_i}|_{P^{\sharp}}\quad\mbox{and}\quad f_{i,2}|_{P^{\sharp}} $$ on the projective space $P^{\sharp}$ are linear combinations of disjoint sets of monomials in $f_i$, so that the coefficients of those polynomials belong to disjoint subsets of coefficients of the polynomial $f_i$. Therefore (re-ordering the polynomials of the tuple (\ref{27.02.23.1}) so that their degrees do not decrease), applying Theorem 8.1 to the tuple (\ref{27.02.23.1}), we get that the codimension of the set ${\cal B}_{R3.1}(\underline{\xi},P)$ is at least $$ \frac12(\mathop{\rm dim}P^{\sharp}-k-k_{\geqslant 3}-1) (\mathop{\rm dim}P^{\sharp}-k-k_{\geqslant 3}-4)+2, $$ where $\mathop{\rm dim}P^{\sharp}=M+l-\varepsilon(k)-1$, $\mathop{\rm dim}\underline{\xi}=k-l$. It is easy to check by elementary computations that this estimate (with the correction due to the variation of the subspace $P$ and the set of linear forms $\underline{\xi}$) is stronger than we need. This completes the proof of the estimate for the codimension of the complement ${\cal P}\backslash{\cal F}$ in Theorem 0.1. Note that (for the technique of estimating the codimension that we used) the estimate of Theorem 0.1 is optimal for the condition (MQ2), that requirement turns out to be the strongest. \begin{flushleft} Department of Mathematical Sciences,\\ The University of Liverpool \end{flushleft} \noindent{\it pukh@liverpool.ac.uk} \end{document}

\begin{document} \begin{abstract} When $S$ is a discrete subsemigroup of a discrete group $G$ such that $G = S^{-1} S$, it is possible to extend circle-valued multipliers {}from $S$ to $G$; to dilate (projective) isometric representations of $S$ to (projective) unitary representations of $G$; and to dilate/extend actions of $S$ by injective endomorphisms of a C*-algebra to actions of $G$ by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of $S$ is isomorphic to a full corner in the (twisted) crossed product by the dilated action of $G$. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost--Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product $C_0(\mathbb A_f) \rtimes \Q^*_+$, corresponding to the multiplicative action of the positive rationals on the additive group $\mathbb A_f $ of finite adeles. \end{abstract} \maketitle \section*{Introduction} In recent years there has been renewed interest in crossed products by semigroups of endomorphisms, viewed now as universal algebras in contrast to their original presentation as corners in crossed products by groups. This new approach, initiated by Stacey \cite{sta} following a strategy pioneered by Raeburn for crossed products by group actions \cite{rae}, is based on the explicit formulation of a semigroup crossed product as the universal C*-algebra of a covariance relation. As such, it motivated the development of specific techniques and brought about new insights and applications, e.g. \cite{sta,alnr,murnew,sri,twi-units,quasilat,bc-alg, hecke5,diri}. Nevertheless, the implicit view of semigroup crossed products as corners continues to have a very important role: it is often invoked to prove the existence of nontrivial universal objects and it allows one to import results from the well-developed theory of crossed products by groups. When the endomorphisms are injective and the semigroup is abelian the two approaches are equivalent, and the proof involves using a direct limit to transform the endomorphisms into automorphisms and the isometries into unitaries. This has been done when the abelian semigroup is $\mathbb N$ \cite{cun,sta}, when it is totally ordered \cite{sri}, and, in general, when it is cancellative \cite{murnew}. As crossed products by more general (nonabelian) semigroups are being considered from the universal property point of view, the need arises to determine whether a realization as corners in crossed products by groups is true and useful in those cases too. This is the main task undertaken in the present work. A step away from commutativity of the acting semigroup was taken in \cite{semico} where isometric representations and multipliers of {\em normal} cancellative semigroups were extended using the same direct limits (the semigroup $S$ is normal if $xS = Sx$ for every $x \in S$, in which case the natural notions of right and left orders on $S$ coincide). Here we will go further and consider discrete semigroups that can be embedded in a discrete group and for which the right order is cofinal; since cofinality is a key ingredient of a directed system, this class is, arguably, the most general one for which the usual direct limit construction would work without a major modification. Based on the results presented below one may argue that the relevant object is the action of an ordered group, and that there are two ways of looking at it; the first is as an automorphic action on a C*-algebra {\em taken together with a distinguished subalgebra which is invariant under the action of the positive cone}, and the second is simply as the endomorphic action of this positive cone on the invariant subalgebra. We show that these two points of view are equivalent: to go from the former to the latter one just cuts down the automorphisms to endomorphisms of the invariant subalgebra and restricts to the positive cone, and the process is reversed by way of a dilation-extension construction, \thmref{dil-ext}, which constitutes our first main result. We also explicitly state and prove two additional features of this equivalence that, in our opinion, have not previously received enough attention. The first one is that the minimal automorphic dilation is canonically {\em unique}, which for instance allows one to test a good candidate, as done in Subsection \ref{diladeles} below. The second one is that the crossed product by the semigroup action is realized as a {\em full} corner in the crossed product by a group action, so the equivalence of the two approaches technically translates into the strong Morita equivalence of the crossed products. This is done in \thmref{fulcor}, which is our second main result. A modicum of extra work shows that these results are also valid for twisted crossed products and projective isometric representations with circle-valued multipliers. This requires the easy generalization, to Ore semigroups, of results known for semigroups that are abelian \cite{arv,din,che,kle} or normal \cite{semico,murpro}, which is done in the preliminary subsections \ref{multipls} and \ref{twi.cross.prod}. The arguments given are for projective isometric representations and twisted crossed products, but setting all multipliers to be identically $1$ will lighten the burden slightly for those interested in the dilation-extension itself and not in projective representations, twisted crossed products, and extensions of multipliers. In the final section we give an application to the semigroup dynamical system from number theory \cite{bc-alg} which has the Bost-Connes Hecke C*-algebra \cite{bos-con} as its crossed product. Starting with the $p$-adic version of the system \cite[Section 5.4]{diri} we show how one is quite naturally led to consider the ring of finite adeles with the multiplicative action of the positive rationals. This establishes a natural heuristic link between the Bost-Connes Hecke C*-algebra and the space $\mathcal A/\mathbb Q^*$, which lies at the heart of Connes's recent formulation of the Riemann Hypothesis as a trace formula \cite{con-cr,con-rzf}. \section{Preliminaries} In this first section we gather the basic definitions and results concerning the semigroups on which we will be interested. We also generalize other results about isometries and crossed products that are valid, with more or less the same proofs, in the present setting, although they were originally stated for particular cases. \subsection{Ore semigroups.} \begin{definition} An {\em Ore semigroup} $S$ is a cancellative semigroup such that $Ss \cap St \neq \emptyset $ for every pair $s, t \in S$. Ore semigroups are also known as {\em right--reversible} semigroups. (We leave the obvious symmetric consideration of left--reversibility to the reader.) \end{definition} \begin{theorem}[Ore, Dubreil] A semigroup $S$ can be embedded in a group $G$ with $S^{-1} S = G$ if and only if it is an Ore semigroup. In this case, the group $G$ is determined up to canonical isomorphism and every semigroup homomorphism $\phi$ from $S$ into a group $\mathcal G$ extends uniquely to a group homomorphism $\varphi : G \to \mathcal G$. \end{theorem} \begin{proof} See e.g. theorems 1.23, 1.24 and 1.25 in \cite{cli-pre} for the first part. We only need to prove the assertion about extending $\phi$. Since $G = S^{-1} S$, given $x,y \in S$ there exist $u,v \in S$ such that $v^{-1} u = y x^{-1}$, and hence the element $ux = vy $ is in $S x \cap S y$, proving that $S$ is directed by the relation defined by $s \preceq_r t$ if $t\in Ss$. An easy argument shows that $\varphi(x^{-1} y) = \phi(x)^{-1} \phi(y) $ defines a group homomorphism from $G = S^{-1} S $ to $\mathcal G$ that extends $\phi$. \end{proof} \begin{remark} The last assertion of the theorem generalizes \cite[Lemma 1.1]{semico}. Here we have found it more convenient, for compatibility with the rest of \cite{semico}, to work with the {\em right order} $\preceq_r$ determined by $S$ on $G$ via $x \preceq_r y$ if $y \in S x$. \end{remark} To illustrate the class of semigroups being considered, we list a few examples which have appeared recently in the context of semigroup actions: \begin{itemize} \item Abelian semigroups, (notably the multiplicative nonzero integers in an algebraic number field \cite{hecke5}); \item Semigroups obtained by pulling back the positive cone from a totally ordered quotient \cite{phi-rae}; \item Normal semigroups, in particular semidirect products \cite{semico,murpro}; \item Groups of matrices over the integers having positive determinant \cite[Example 4.3]{bre}; \end{itemize} \subsection{Extending multipliers and dilating isometries.}\label{multipls} Let $\lambda$ be a circle--valued multiplier on $S$, that is, a function $\lambda : S\times S \to \mathbb T$ such that $$ \lambda(r,s) \lambda(rs,t) = \lambda(r,st) \lambda(s,t), \quad r,s,t \in S. $$ A {\em projective isometric representation} of $S$ with multiplier $\lambda$ on a Hilbert space $H$ (an isometric $\lambda$--representation of $S$ on $H$) is a family $\{V_s: s\in S\}$ of isometries on $H$ such that $V_s V_t = \lambda(s,t) V_{st}$. A twisted version of Ito's dilation theorem \cite{ito} was obtained in \cite[Theorem 2.1]{semico}, where projective isometric representations of normal semigroups were dilated to projective unitary representations. Essentially the same proof, inspired on Douglas's \cite{dou}, works for Ore semigroups and gives the following. \begin{theorem}\label{dilation} Suppose $S$ is an Ore semigroup and let $\{V_s: s \in S\}$ be an isometric $\lambda$--representation of $S$ on a Hilbert space $H$, where $\lambda$ is a multiplier on $S$. Then there exists a unitary $\lambda$--representation of $S$ on a Hilbert space $\mathcal H$ containing a copy of $H$ such that \begin{enumerate} \item[(i)] $U_s$ leaves $H$ invariant and $U_s |_H = V_s$; and \item[(ii)] $\bigcup_{s\in S} U_s^*H$ is dense in $\mathcal H$. \end{enumerate} \end{theorem} \begin{proof} Verbatim from the proof of \cite[Theorem 2.1]{semico}, except for the following minor modification of the part of the argument where normality is used to obtain an admissible value for the function $f_t $. The value $st$ used there has to be substituted by any (fixed) $z \in Ss \cap St$, and thus the fourth paragraph there should be replaced by the following one. Suppose now that $f \in H_0$ and $t \in S$, and consider the function $f_t $ defined by $f_t(x) = \lambda(x,t) f(xt)$ for $x \in S$. If $s \in S$ is admissible for $f$, let $z \in Ss \cap S t$. We will show that $s_0 := zt^{-1}$ is admissible for $f_t$. For every $x \in Ss_0$, $xt \in Sz$, and since $z$ is admissible for $f$ \begin{eqnarray*} \lambda(x,t)f(xt) &= &\lambda(x,t) \overline{\lambda(xtz^{-1}, z)}V_{xtz^{-1}} f(z)\\ & = & \overline{\lambda(xtz^{-1}, zt^{-1})}V_{xtz^{-1}}\lambda(zt^{-1},t)f(zt^{-1}t)\\ & = & \overline{\lambda(xs_0^{-1}, s_0)}V_{xs_0^{-1}} f_t(s_0) \end{eqnarray*} where the second equality holds by the multiplier property applied to the elements $xtz^{-1}$, $zt^{-1}$, and $t$ in $S$. This proves that $s_0$ is admissible for $f_t$, so $f_t \in H_0$. \end{proof} Since the results of \cite{semico} concerning discrete normal semigroups depend only on this dilation theorem and on the unique extension of group--valued homomorphisms, they too are valid for Ore semigroups and we list them here for reference. \begin{theorem}\label{semimult} Suppose $S$ is an Ore semigroup and let $G = S^{-1} S$. Then \begin{enumerate} \item Every multiplier on $S$ extends to a multiplier on $G$. \item Restriction of multipliers on $G$ to multipliers on $S$ gives an isomorphism of $H^2(G,\mathbb T)$ onto $H^2(S,\mathbb T)$. \item Suppose $\lambda$ is a multiplier on $S$ and let $V$ be a $\lambda$--representation of $S$ by isometries on $H$. Assume $\mu$ is a multiplier on $G$ extending $\lambda$. Then there exists a unitary $\mu$--representation $U$ of $G$ on a Hilbert space $\mathcal H$ containing a copy of $H$ such that $U_s|H = V_s$ for $s \in S$, and $\bigcup_{s\in S}U_s^* H$ dense in $\mathcal H$. Moreover, $U$ and $\mathcal H$ are unique up to canonical isomorphism. \end{enumerate} \end{theorem} \begin{proof} The proofs of all but the last statement about uniqueness are as in Theorem 2.2, Corollary 2.3 and Corollary 2.4 of \cite{semico}, provided one considers the left-quotients $x = t^{-1} s$ instead of the right-quotients used there. In order to prove the uniqueness statement suppose $(U', \mathcal H ')$ is another unitary $\mu$-representation such that ${U'}_s|H = V_s$ and $\bigcup_{s\in S}{U'}_s^* H$ is dense in $\mathcal H'$. It is easy to see that the map $$ W: U_s^* h \mapsto {U'}_s^* h , \qquad s\in S, h \in H $$ is isometric, and that it extends to an isomorphism of $\mathcal H $ to $ \mathcal H'$ because of the density condition. It only remains to show that $W$ intertwines $U$ and $U'$. Since $S$ is an Ore semigroup, for every $x$ and $s$ in $S$ there exist $z $ and $t$ in $S$ such that $x s^{-1} = t ^{-1} z$. Then $tx = zs$, so \begin{eqnarray*} W U_x (U_s^* h) &=& W U_x U_{tx}^* U_{zs} U_s^* h = \mu(t,x) \overline{\mu(z,s)} W U_t^* U_{z} h = \mu(t,x) \overline{\mu(z,s)} W U_t^* (V_z h)\\ &=& \mu(t,x) \overline{\mu(z,s)} {U'}_t^* (V_z h) = \mu(t,x) \overline{\mu(z,s)}{U'}_t^* {U'}_z h = {U'}_x ({U'}_s^* h ) \\ &=& {U'}_x W (U_s^* h) \end{eqnarray*} This shows that $WU_x = U'_x W$ for every $x\in S$, hence for every $x \in G$. \end{proof} \subsection{Twisted semigroup crossed products}\label{twi.cross.prod} Suppose $A$ is a unital C*-algebra and let $\alpha$ be an action of the discrete semigroup $S$ by not necessarily unital endomorphisms of $A$. Let $\lambda$ be a circle-valued multiplier on $S$. A {\em twisted covariant representation} of the semigroup dynamical system $(A, S, \alpha)$ with multiplier $\lambda$ is a pair $(\pi, V)$ in which \begin{enumerate} \item $\pi$ is a unital representation of $A$ on $H$, \item $V: S \to Isom( H)$ is a projective isometric representation of $S$ with multiplier $\lambda$, i.e., $V_s V_t = \lambda(s,t) V_{st}$, and \item the covariance condition $\pi(\alpha_t(a)) = V_t \pi(a) V_t^*$ holds for every $a\in A$ and $ t\in S$. \end{enumerate} When dealing with twisted covariant representations with a specific multiplier $\lambda$, we will refer to the dynamical system as a twisted dynamical system and denote it by $(A, S, \alpha, \lambda)$. The (twisted) crossed product associated to $(A, S, \alpha, \lambda)$ is a C*-algebra $A\rtimes_{\alpha, \lambda}S$ together with a unital homomorphism $i_A :A \rightarrow A\rtimes_{\alpha, \lambda}S$ and a projective $\lambda$-representation of $S$ as isometries $i_{S}: S \rightarrow A\rtimes_{\alpha, \lambda}S $ such that \begin{enumerate} \item $(i_A, i_{S})$ is a twisted covariant representation for $(A, S, \alpha, \lambda)$, \item for any other covariant representation $(\pi, V)$ there is a representation $\pi \times V$ of $A\rtimes_{\alpha, \lambda}S$ such that $\pi = (\pi\times V )\circ i_A$ and $V = (\pi\times V) \circ i_{S}$, and \item $A\rtimes_{\alpha, \lambda}S$ is generated by $i_A(A)$ and $i_{S} (S)$ as a C*-algebra. \end{enumerate} The existence of a nontrivial universal object associated to $(A, S, \alpha, \lambda)$ depends on the existence of a nontrivial twisted covariant representation with multiplier $\lambda$. For general endomorphisms such representations need not exist, even in the untwisted case. For instance, the action of $\mathbb N$ by surjective shift-endomorphisms of $c_0$ described in Example 2.1(a) of \cite{sta} does not admit any nontrivial covariant representations. We will assume that our endomorphisms are injective, hence nontriviality of the semigroup crossed product will follow from its realization as a corner in a nontrivial classical crossed product. See \cite{sta,murnew} for abelian semigroups, and Remark \ref{nontriv} below. There are other possible covariance conditions which yield nontrivial crossed products even if the endomorphisms fail to be injective, see e.g. \cite{murpac} and \cite{pet}. We will not deal with them here, but we refer to \cite{lam} for an interesting comparative discussion of the different constructions. \begin{remark} It is immediate from the definition that the crossed product $A \rtimes S$ is generated, as a C*-algebra, by the monomials $v_x^* a v_y $ with $a \in A$, and $ x,y \in S$, but more is true for Ore semigroups: the products of such monomials can be simplified using covariance to obtain another monomial of the same type. Specifically, in order to simplify the product $v_x^* a v_y v_r^* b v_s$ we begin by finding elements $t$ and $z$ in $S$ such that $y r^{-1}= t^{-1} z$, so that $ty = zr$, (such elements do exist because $S$ is an Ore semigroup). It follows that \begin{eqnarray*} v_x^* a v_y v_r^* b v_s &=& \lambda(y,t) \overline{\lambda(z,r)} v_x^* a v_y v_{ty}^* v_{zr} v_r^* b v_s\\ &=& \lambda(y,t) \overline{\lambda(z,r)} v_x^* a v_yv_y^* v_t^* v_z v_rv_r^* b v_s\\ &=& \lambda(y,t) \overline{\lambda(z,r)} v_x^* v_t^* \alpha_t(a\alpha_y(1)) \alpha_z(\alpha_r(1) b ) v_z v_s\\ &=& \lambda(y,t) \overline{\lambda(z,r)} \overline{\lambda(t,x)} \lambda(z,s) v_{tx}^* \alpha_t(a\alpha_y(1)) \alpha_z(\alpha_r(1) b ) v_{zs}, \end{eqnarray*} hence the linear span of such monomials is dense in the crossed product. \end{remark} \section{The minimal automorphic dilation.} \label{min-aut-ext} There are two steps in realizing a semigroup crossed product as a corner in a crossed product by a group action. The first one is the dilation-extension of a semigroup action by injective endomorphisms to a group action by automorphisms, and the second one is the corresponding dilation-extension of covariant representations of the semigroup dynamical system to covariant representations of the dilated system. \subsection{A dilation-extension theorem.} \begin{theorem}\label{dil-ext} Assume $S$ is an Ore semigroup with enveloping group $G = S^{-1} S$ and let $\alpha$ be an action of $S$ by injective endomorphisms of a unital C*-algebra $A$. Then there exists a C*-dynamical system $(B, G, \beta)$, unique up to isomorphism, consisting of an action $\beta$ of $G$ by automorphisms of a C*-algebra $B$ and an embedding $i: A \to B$ such that \begin{enumerate} \item $\beta$ dilates $\alpha$, that is, $\beta_s\circ i = i \circ \alpha_s$ for $s \in S$, and \item $(B, G, \beta)$ is minimal, that is, $\bigcup_{s\in S}\beta_s^{-1}(i(A))$ is dense in $B$. \end{enumerate} \end{theorem} \begin{proof} By right--reversibility, $S$ is directed by $\preceq_r$ so one may follow the argument of \cite[Section 2]{murnew}. However, extra work is needed here: since $G$ need not be abelian, the choice of embeddings in the directed system must be carefully matched to the choice of right-order $\preceq_r$ on $S$. Consider the directed system of C*-algebras determined by the maps $\alpha_y^x = \alpha_{yx^{-1}}$ from $A_x := A $ into $A_y := A$, for $x \in S$ and $y \in Sx$, i.e. for $x \preceq_r y$ in $S$. By \cite[Proposition 11.4.1(i)]{kad-rin} there exists an inductive limit C*-algebra $A_\infty$ together with embeddings $\alpha^x : A_x \to A_\infty$ such that $\alpha^x = \alpha^y \circ \alpha^x_y$ whenever $x\preceq_r y$, and such that $\bigcup_{x\in S} \alpha^x(A_x)$ is dense in $A_\infty$. The next step is to extend the endomorphism $\alpha_s$ to an automorphism of $A_\infty$. For any fixed $s \in S$ the subset $Ss$ of $S$ is cofinal, so $A_\infty$ is also the inductive limit of the directed subsystem $(A_x, x\in Ss)$, and, for this subsystem, we may consider new embeddings $\psi^x : A_x \to A_\infty$ defined by $\psi^x (a) = \alpha^{xs^{-1}}(a)$ for $x \in Ss$ and $a \in A_x$. By \cite[Proposition 11.4.1(ii)]{kad-rin} there is an automorphism $\tilde{\alpha}_s$ of $A_\infty$ such that $\tilde{\alpha}_s \circ \alpha^x = \psi^x$ for every $x \in Ss$. Since $ \alpha^1 = \alpha^s \circ \alpha^1_s$ and $\psi^x = \alpha^{xs^{-1}}$, the choice $x = s$ gives $$ \tilde{\alpha}_s \circ \alpha^1 = \tilde{\alpha}_s \circ \alpha^s \circ \alpha^1_s = \alpha^1 \circ \alpha_s $$ so that (1) holds with $\beta = \tilde{\alpha}$ and $i = \alpha^1 : A_1 \to A_\infty$. Since $\tilde{\alpha}_s^{-1}(i(A)) = \alpha^s (A_s)$, (2) also holds. Uniqueness of the dilated system follows from \cite[Proposition 11.4.1(ii)]{kad-rin}: $A_\infty$ is the closure of the union of the subalgebras $\tilde{\alpha}_s^{-1}(i(A))$ with $s \in S$, if $(B, G, \beta)$ is another minimal dilation with embedding $j : A \to B$ then there is an isomorphism $\theta: A_\infty \rightarrow B$ given by $\theta \circ \tilde{\alpha}_{s^{-1}}( i(a)) = \beta_{s^{-1}} (j(a))$ for $a \in A$ and hence which intertwines $\tilde{\alpha}$ and $\beta$. \end{proof} \begin{definition} A system $(B, G, \beta)$ satisfying the conditions (1) and (2) of \thmref{dil-ext} is called the {\em minimal automorphic dilation} of $(A,S,\alpha)$. If $\lambda$ is a multiplier on $S$ with extension $\mu$ to $G$, we say that the twisted system $(B, G, \beta, \mu)$ is the minimal automorphic dilation of the twisted system $(A,S,\alpha,\lambda)$. (By \thmref{semimult} the extended multiplier exists and is unique up to a coboundary.) \end{definition} \begin{lemma}\label{dil-cov-rep} Let $(\pi,V)$ be a covariant representation for the twisted system $(A,S,\alpha,\lambda)$ on the Hilbert space $H$, and let $\tilde{V}$ be the minimal projective unitary dilation of $V$ on $\mathcal H$ given by \thmref{dilation}. Then there exists a representation $\tilde{\pi}$ of $B$ on $\mathcal H$ such that $(\tilde{\pi},\tilde{V})$ is covariant for the minimal automorphic dilation $(B, G, \beta, \mu)$ and $\tilde{\pi} \circ i = \pi$ on $H$. \end{lemma} \begin{proof} We work with the dense subspace $\mathcal H_0 = \bigcup_{t\in S} U_t^* H$ of $\mathcal H$ and the dense subalgebra $B_0 = \bigcup_{s\in S} \beta_s^{-1}(i(A))$. If $\xi \in \mathcal H_0$ there exists $t \in S$ such that $U_t \xi \in H$; assume $b = \beta_t^{-1}(i(a))$, since we want $(\tilde{\pi}, \tilde{V})$ to be covariant, the only choice is to define $\tilde{\pi}$ by $$ \tilde{\pi}(b) \xi = \tilde{\pi}(\beta_t^{-1}(i(a))) \xi = \tilde{V}_t^* \tilde{\pi}(i(a)) \tilde{V}_t \xi = \tilde{V}_t^* \pi(a) \tilde{V}_t \xi $$ because $\tilde{\pi}$ restricted to $i(A)$ and cut down to $H$ has to be equal to $\pi$. Of course we have to show that this actually defines an operator $\tilde{\pi} (b)$ on $\mathcal H$ for each $b \in B_0$, that $\tilde{\pi}$ extends to a homomorphism from all of $B$ to $B(\mathcal H)$, and that $(\tilde{\pi},\tilde{V})$ is covariant. The first step is to define $\tilde{\pi}(b)$ on $\mathcal H_0$ for a fixed $b \in B_0$. We begin by fixing $b \in B_0$, $a \in A$ and $s\in S$ such that $b = \beta_s^{-1}(i(a))$. For $\xi \in \tilde{V}^*_{t_0} H$ with $t_0$ in the cofinal set $Ss$, we let \begin{equation} \label{dil-rep} \varphi(b) \xi = \tilde{V}_{t_0}^* \pi (\alpha_{{t_0}s^{-1}}(a)) \tilde{V}_{t_0} \xi. \end{equation} If $t \in St_0$ then $\xi \in \tilde{V}^*_t H$, and \begin{eqnarray*} \tilde{V}_{t}^* \pi (\alpha_{{t}s^{-1}}(a)) \tilde{V}_{t} \xi_0 &=& \tilde{V}_{t_0}^* \tilde{V}_{tt_0^{-1}}^* \pi(\alpha_{t t_0^{-1}} \circ \alpha_{{t_0}s^{-1}}(a)) \tilde{V}_{tt_0^{-1}} \tilde{V}_{t_0} \xi \\ & = & \tilde{V}_{t_0}^* \tilde{V}_{tt_0^{-1}}^* V_{t t_0^{-1}} \pi( \alpha_{{t_0}s^{-1}}(a)) V_{t t_0^{-1}}^* \tilde{V}_{tt_0^{-1}} \tilde{V}_{t_0} \xi \\ & = & \tilde{V}_{t_0}^* \pi( \alpha_{{t_0}s^{-1}}(a)) \tilde{V}_{t_0} \xi. \end{eqnarray*} So the definition of $\phi (b) \xi $ could have been given using any $t \in St_0$ in place of $t_0$. Next we show that $\phi (b) \xi $ is also independent of $s$ and $a$, in the sense that if $b$ is also equal to $ \beta_{s'}^{-1}(i(a')) $ then $\alpha_{{t}{s'}^{-1}}(a')$ is equal to $\alpha_{{t} s^{-1}}(a)$ for $t$ in a cofinal set. To see this let $t \in Ss \cap Ss'$. Then $\alpha^t \circ \alpha^{s'}_t (a') = \alpha^{s'}(a') = \beta_{s'}^{-1}(i(a')) = \beta_{s}^{-1}(i(a)) = \alpha^{s}(a) = \alpha^t \circ \alpha^{s}_t (a)$, and since the embedding $\alpha^t$ is injective, it follows that $\alpha_{t{s'}^{-1}}(a') = \alpha_{ts^{-1}}(a)$. The map $\varphi(b) : \mathcal H_0 \to \mathcal H_0$ is clearly linear, and since the endomorphisms are injective, $\| \varphi (b) \xi\| \leq \|b\| \| \xi \|$. Thus $\phi(b)$ can be uniquely extended to a bounded linear operator (also denoted $\varphi(b)$) on all of $\mathcal H$ such that $\| \varphi(b)\| \leq \|b\|$. For any $s$ the map $\operatorname{Ad}_{\tilde{V}_{t_0}^*} \circ \pi \circ \alpha_{{t_0}s^{-1}}$ is a *-homomorphism on $A$, and by cofinality of $\preceq_r$, for any $b_1$ and $ b_2 $ in $B_0$ there exist $s \in S$ and $a_1$ and $a_2$ in $A$ such that $b_1 = \beta_s^{-1}(i(a_1))$ and $b_2 = \beta_s^{-1}(i(a_2))$. It follows easily from (\ref{dil-rep}) that $\varphi : B_0 \to B(\mathcal H)$ is a *-homomorphism which can be extended to a representation $\tilde{\pi}$ of $B$ on $\mathcal H$. Putting $a = 1$ in (\ref{dil-rep}) shows that $\tilde{\pi}$ is nondegenerate and there only remains to check that $(\tilde{\pi},\tilde{V})$ is a covariant pair for $(B,G, \beta, \mu)$. Suppose first $x \in S$ and $b \in B_0$; we can assume that $b = \beta^{-1}_s (i(a))$ for some $a \in A$ and $s \in Sx$. Let $\xi \in \tilde{V}^*_t H$; we can assume $t \in Ss \subset Sx$, and we observe that $\tilde{V}_x \xi \in \tilde{V}^*_{tx^{-1}} H$. Then \begin{eqnarray*} \tilde{\pi}(\beta_x(b)) \tilde{V}_x \xi & = & \tilde{\pi}(\beta_{x s^{-1}}(i(a))) \tilde{V}_x \xi\\ & = & \tilde{\pi}(\beta^{-1}_{sx^{-1}}(i(a))) \tilde{V}_x \xi\\ & = & \tilde{V}^*_{tx^{-1}} \pi(\alpha_{tx^{-1} xs^{-1}}(i(a))) \tilde{V}_{tx^{-1}}\tilde{V}_x \xi\\ & = & \tilde{V}^*_{tx^{-1}} \pi(\alpha_{ts^{-1}}(i(a))) \tilde{V}_{tx^{-1}}\tilde{V}_x \xi\\ & = & \tilde{V}^*_{x^{-1}}\tilde{V}^*_{t} \pi(\alpha_{ts^{-1}}(i(a))) \tilde{V}_t \xi\\ & = & \tilde{V}_{x} \tilde{\pi}(\beta^{-1}_{s}(i(a))) \xi, \end{eqnarray*} and since $\mathcal H_0$ is dense in $\mathcal H$ and $B_0$ is dense in $B$, the pair $(\tilde{\pi}, \tilde{V})$ satisfies the covariance relation. \end{proof} \subsection{Full corners.} Once we know how to dilate covariant representations from the semigroup action to the group action we can establish the relation between the respective crossed products. Before proving our main result we recall that if $p$ is a projection in the C*-algebra $A$ then the algebra $p A p$ is a {\em corner} in $A$, which is said to be {\em full} if the linear span of $ApA$ is dense in $A$. The most relevant feature of full corners is that if $pAp$ is a full corner in $A$, then $pA$ is a full Hilbert bimodule implementing the Morita equivalence, in the sense of Rieffel, of $pAp$ to $A$. \begin{theorem}\label{fulcor} Suppose $(A,S,\alpha, \lambda)$ is a twisted semigroup dynamical system in which $S$ is an Ore semigroup acting by injective endomorphisms and $\lambda $ is a multiplier on $S$. Let $(B,G,\beta, \mu)$ be the minimal automorphic dilation, with embedding $i: A \to B$. Then $A\rtimes_{\alpha,\lambda} S$ is canonically isomorphic to $i(1) (B\rtimes_{\beta, \mu} G) i(1)$, which is a full corner. As a consequence, the crossed product $A\rtimes_{\alpha,\lambda} S$ is Morita equivalent to $B \rtimes_{\beta, \mu} G$. \end{theorem} \begin{proof} Let $U$ be the projective unitary representation of $G$ in the multiplier algebra of $B\rtimes_{\beta, \mu} G$, and notice that $$i(1) U_s i(1) = U_s i(1), \qquad s \in S, $$ because $i(A)$ is invariant under $\beta_s$. Define $v_s = U_s i(1) $. Then $v_s^* v_s = i(1) U_s ^* U_s i(1) = i(1)$ and $v_s v_t = U_s i(1) U_t i(1) = U_s U_t i(1) = \mu(s,t) U_{st} i(1) = \lambda(s,t) v_{st}$, so $v$ is a projective isometric representation of $S$ with multiplier $\lambda$. Since $i(1) (B\rtimes_{\beta, \mu} G) i(1)$ is generated by the elements $i(1)U_x^* i(a) U_y i(1) = v_x^* i(a) v_y$, the isomorphism will be established by uniqueness of the crossed product once we show that the pair $(i,v)$ is universal. Suppose $(\pi,V)$ is a covariant representation for the twisted system $(A,S,\alpha,\lambda)$, and let $(\tilde{\pi},\tilde{V})$ be the corresponding dilated covariant representation of $(B,G,\beta, \mu)$ given by \lemref{dil-cov-rep}. By the universal property of $ B\rtimes_{\beta, \mu} G $ there is a homomorphism $$ (\tilde{\pi} \times \tilde{V}) : B\rtimes_{\beta, \mu} G \to C^*(\tilde{\pi}, \tilde{V}) $$ such that $\tilde{\pi}(b) \tilde{V}_s = (\tilde{\pi} \times \tilde{V}) (i_B(b) U_s)$ . Let $\rho $ be the restriction of $ (\tilde{\pi} \times \tilde{V}) $ to $i(1) (B\rtimes_{\beta, \mu} G) i(1)$, cut down to the invariant subspace $H$. By \lemref{dil-cov-rep} $$ \rho(i(a)) = (\tilde{\pi} \times \tilde{V})(i(a)) = \tilde{\pi} \circ i(a) = \pi(a), \qquad a \in A, $$ while $$ \rho(v_s) = (\tilde{\pi} \times \tilde{V}) (U_s i(1)) = \tilde{V}_s \pi(1) = V_s, \qquad s \in S $$ Thus $\rho \circ i = \pi$ and $\rho \circ v = V$, so $(i,v)$ is universal for $(A,S,\alpha, \lambda)$. Finally we prove that the corner is full, i.e., that the linear span of the elements of the form $X i(1) Y$ with $X, Y \in B \rtimes_{\beta, \mu} G $ is a dense subset of $ B \rtimes_{\beta, \mu} G$. It is easy to see that the elements of the form $U_s^* b U_t$ span a dense subset of $ B \rtimes_{\beta, \mu} G$ because $G = S^{-1} S$, where $b$ may be replaced with $U_r^* i(a) U_r$ by minimality of the dilation. Thus the elements $U_y^* i(\alpha_z(a)) U_x$ with $x,y,z \in S$ and $a \in A$ span a dense subset of $B \rtimes_{\beta, \mu} G$, and since $i(\alpha_z(a)) = i(1) i(\alpha_z(a))$, the proof is finished. \end{proof} \begin{remark} \label{nontriv} If one drops the assumption of injectivity of the endomorphisms, it is still possible to carry out the constructions and the arguments in the proofs of the preceding theorems. However, the resulting homomorphism $i: A \to B$ may not be an embedding any more. Indeed, Example 2.1(a) of \cite{sta} shows that the limit algebra $B$ may turn out to be the $0$ C*-algebra, yielding a trivial dilated system. We notice that the dilated system $(B,G,\beta,\mu)$ has nontrivial covariant representations if and only if $B \neq 0$, and these representations, when cut down to $i(A)$, give nontrivial covariant representations of the original semigroup system $(A,S,\alpha,\lambda)$. Thus, following \cite[Proposition 2.2]{sta} which deals with the case $S = \mathbb N$, we conclude that the crossed product $A\rtimes_{\alpha,\lambda} S$ is nontrivial if and only if the limit algebra $B$ is not $0$. Clearly, this is the case when, for instance, the endomorphisms are injective. \end{remark} \section{An example from number theory.} As an application of the preceding theory we consider the semigroup dynamical system from \cite{bc-alg} whose crossed product is the Bost-Connes Hecke C*-algebra \cite{bos-con}. Since Morita equivalence implies that the representation theory of the semigroup dynamical system is equivalent to that of the dilated system, it is quite useful to have an explicit formulation of the dilation. We point out that since the semigroup in question is abelian, this application is somewhat independent from the rest of the material on nonabelian semigroups. In fact, the example could be dealt with by enhancing \cite[Section 2]{murnew} with the uniqueness and fullness properties discussed above, which are easier to prove for abelian semigroups. \subsection{Finite Adeles.} The natural setting for identifying the ingredients of the minimal automorphic dilation of the semigroup dynamical system introduced in \cite{bc-alg} will be the (dual) $p$-adic picture described in \cite[Proposition 32]{diri}, in which the algebra is $C(\prod_p \mathbb Zp)$ and the endomorphisms $\alpha_n$ consist of `division by $n$' in $\prod_p \mathbb Zp$: $$\alpha_n (f) (x) = \left\{\begin{array}{ll} f(x/n)&\mbox{if $n | x$}\\ 0&\mbox{ otherwise}. \end{array} \right. $$ By \cite[Corollary 2.10]{bc-alg} the crossed product associated to this system is canonically isomorphic to the Bost-Connes Hecke C*-algebra $\mathcal C_{\mathbb Q}$. The ring ${\mathcal Z} := \prod_p \mathbb Zp$ has lots of zero divisors and hence no fraction field. However, the diagonally embedded copy of $\N^{\times}$ is a multiplicative set with no zero divisors, and we may enlarge ${\mathcal Z}$ to a ring in which division by an element of $\N^{\times}$ is always possible. Our motivation is to extend the endomorphisms $\alpha_n$ defined above to automorphisms. The algebraic part is easy: we consider the ring $(\N^{\times})^{-1} {\mathcal Z}$ of formal fractions ${z}/{n}$ with $z\in {\mathcal Z}$ and $n \in \N^{\times}$, with the obvious rules of addition and multiplication (and simplification!), \cite[II.\S3]{lan-alg}. This ring has a universal property with respect to homomorphisms of ${\mathcal Z}$ that send $\N^{\times}$ into units. Since $\N^{\times} $ has no zero divisors, the canonical map $z \rightarrow {z}/{1}$ is an embedding of ${\mathcal Z}$ in $(\N^{\times})^{-1} {\mathcal Z}$. The topological aspect requires a moments thought, after which we declare that the subring ${\mathcal Z}$ must retain its compact topology and be relatively open. Since we want division by $n\in \N^{\times}$ to be an automorphism, this determines a topology on the compact open sets $(1/n) {\mathcal Z}$ and hence on their union, $(\N^{\times})^{-1} {\mathcal Z}$, which becomes a locally compact ring containing ${\mathcal Z}$ as a compact open subring. The ring we have just defined is (isomorphic to) the locally compact ring $\mathbb A_f$ of finite adeles, which is usually defined as the restricted product, over the primes $p \in \mathcal P$ of the $p$-adic numbers $\mathbb Q_p$ with respect to the $p$-adic integers $\mathbb Zp$: $$ \mathbb A_f : = \{ (a_p) : a_p \in\mathbb Q_p \text{ and } a_p \in \mathbb Zp \text{ for all but finitely many }p \in \mathcal P \}, $$ with $ \prod_p \mathbb Zp$ as its maximal compact open subring. The isomorphism is implemented by the map from $(\N^{\times})^{-1} {\mathcal Z}$ into $\mathbb A_f$ given by the universal property; this map is clearly injective and, since every finite adele can be written as $z/n$ with $z\in {\mathcal Z}$ and $n \in \N^{\times}$, it is also surjective. Specifically, for each $a_p \in \mathbb Q_p$ there exists $k_p$ such that $p^{k_p} a_p = z_p \in \mathbb Zp$ and a sequence $a = (a_p)_{p\in \mathcal P}$ is an adele if and only if $k_p$ can be taken to be $0$ for all but finitely many $p$'s, in which case $ n = \prod_p p^{-k_p} \in \N^{\times}$ and $a = (na)/n$, with $na = (na_p)_{p\in \mathcal P} \in \prod_p \mathbb Zp$. \subsection{The minimal automorphic dilation of $(C(\mathcal Z), \N^{\times}, \alpha)$.}\label{diladeles} The rational numbers are embedded in $\mathbb A_f$, and division by a nonzero rational is clearly a homeomorphism so $$ \beta_r(f)(a) = f(r^{-1} a), \quad a \in\mathbb A_f, r \in \Q^*_+ $$ defines an action of $\Q^*_+ = (\N^{\times})^{-1} \N^{\times} $ by automorphisms of $C_0(\mathbb A_f)$. Since ${\mathcal Z}$ is compact and open, its characteristic function $ 1_{{\mathcal Z}}$ is a projection in $C_0(\mathbb A_f)$ and there is an obvious embedding $i$ of $C({\mathcal Z})$ as the corresponding ideal of $ C_0(\mathbb A_f)$, given by $$ i(f) (a) = \left\{ \begin{array}{ll} f(a) & \text{ if } a \in {\mathcal Z}\\ 0 & \text{ if } a \notin {\mathcal Z}. \end{array} \right. $$ \begin{proposition}\label{BC-min-aut-ext} The C*-dynamical system $(C_0(\mathbb A_f), \Q^*_+, \beta)$ is the minimal automorphic dilation of the semigroup dynamical system $(C({\mathcal Z}), \N^{\times}, \alpha)$, and hence $\mathcal C_{\mathbb Q}$ is the full corner of $C_0(\mathbb A_f)\rtimes_\beta \Q^*_+$ determined by the projection $1_{\mathcal Z}$. \end{proposition} \begin{proof} The embedding clearly intertwines $\alpha_n$ and $\beta_n$, in the sense that $\beta_n (i(f)) = i(\alpha_n(f))$, and the union of the compact subgroups $(1/n) {\mathcal Z}$ is dense in $\mathbb A_f$, so the union of the subalgebras $\beta_{1/n}(i( C({\mathcal Z})) )$ is dense in $C_0(\mathbb A_f) $, and the result follows from \thmref{dil-ext} and \thmref{fulcor}. \end{proof} Since the discrete multiplicative group $\Q^*_+$ acts by homotheties on the locally compact additive group $\mathbb A_f$, and since $\mathbb A_f$ is self-dual, we obtain another characterization of $\mathcal C_{\mathbb Q}$ as a full corner in the group C*-algebra of the semidirect product $\mathbb A_f \rtimes \Q^*_+$. One should bear in mind, however, that the self duality of $\mathbb A_f$ is not canonical. \begin{corollary} Let $e_{{\mathcal Z}} \in C^*(\mathbb A_f)$ be the Fourier transform of ${1_{\mathcal Z}} \in C_0(\mathbb A_f)$. Then $$ \mathcal C_{\mathbb Q} \cong e_{{\mathcal Z}} C^*(\mathbb A_f \rtimes \Q^*_+)e_{{\mathcal Z}}.$$ \end{corollary} \begin{proof} The action of $\Q^*_+$ on $\mathbb A_f$ is by homotheties, which are group automorphisms, so $C^*(\mathbb A_f \rtimes \Q^*_+)$ is isomorphic to the crossed product $C^*(\mathbb A_f) \rtimes_\beta \Q^*_+$. Moreover, the self-duality of the additive group of $\mathbb A_f$ satisfies $\langle rx,y\rangle = \langle x,ry\rangle$ for $r \in \Q^*_+$, thus $C^*(\mathbb A_f)$ is covariantly isomorphic to $C_0(\mathbb A_f)$, so $C^*(\mathbb A_f) \rtimes_\beta \Q^*_+$ is isomorphic to $C_0(\mathbb A_f) \rtimes_\beta \Q^*_+$, and the claim follows from Proposition \ref{BC-min-aut-ext}. \end{proof} \begin{remark} One of the principles of noncommutative geometry advocates that if $G$ is a group acting on a space $X$, then the quotient space $X/G$ has a noncommutative version in the associated crossed product $C_0(X) \rtimes G$, which is often more tractable. Accordingly, if we allow back in the all-important place at infinity which is left out from $\mathcal A_f$ and if we substitute $\Q^*_+$ by $\mathbb Q^*$, cf. \cite[Remarks 33]{bos-con}, then our \proref{BC-min-aut-ext} gives an explicit path leading from the Bost-Connes Hecke C*-algebra to the space $\mathcal A/\mathbb Q^*$, on which the construction of \cite{con-cr, con-rzf} is based. \end{remark} \end{document}

\begin{document} \title{APX-Hardness of the Minimum Vision Points Problem} \begin{abstract} Placing a minimum number of guards on a given watchman route in a polygonal domain is called the {\em minimum vision points problem}. We prove that finding the minimum number of vision points on a shortest watchman route in a simple polygon is APX-Hard. We then extend the proof to the class of rectilinear polygons having at most three dent orientations. \end{abstract} \graphicspath{{Figures/}} \section{Introduction}\label{intro} The problem of guarding polygonal domains is known as the {\em Art Gallery Problem}. A {\em guard\/} is a point in the domain and the visibility of the guard is defined to be those points that can be reached from the guard by line segments that do not intersect the exterior of the domain. When the domain is a simple polygon, Aggarwal~\cite{Aggarwal} and Lee and Lin~\cite{Lee} independently prove that finding the minimum number of guards is NP-hard; see also O'Rourke~\cite{ORourke}, this is later strengthened to APX-hardness and $\exists\mathbb{R}$-hardness~\cite{Abrahamsen,Broden-APX,Eidenbenz}. Computing the watchman route is another way to solve the guarding problem. A {\em watchman route\/} is a closed tour traced by a moving guard who sees the complete polygon while tracing the tour. There exist polynomial time algorithms that compute the shortest watchman route for simple polygons~\cite{Dror-fixed,Tan-floating,Tan-fixed}. Surveillance devices that trace the watchman route to guard a polygonal domain may be unable to accurately engage their vision systems continuously and could potentially only do so at discrete points along the tour. Such points are called {\em vision points}. The optimization problem of finding a minimum number of vision points on a shortest watchman route is denoted the {\em minimum vision points problem}~({\sc mvpp}) and is the focus of our current results. The {\sc mvpp}\ has the potential of being computationally easier than the original art gallery problem and indeed {\sc mvpp}\ admits a polynomial time exact solution for straight-walkable polygons and street polygons~\cite{Carlsson-Nilsson} whereas the art gallery problem has been shown NP-hard already for monotone polygons~\cite{Krohn-Nilsson}, monotone polygons being a subset of straight-walkable polygons; see also Ghosh and Burdick~\cite{Ghosh} for results for polygons with holes. On the negative side, Carlsson~{\em et~al}.~\cite{Carlsson-NPHard} prove the NP-hardness of finding a minimum number of vision points on a shortest watchman route in a simple polygon and the claim by Abrahamsen~{\em et~al}.~\cite{Abrahamsen} that the art gallery problem is $\exists\mathbb{R}$-hard also for guards restricted to the boundary of a polygon further strengthens the complexity of {\sc mvpp}\ since it shows that the {\sc mvpp}\ is also $\exists\mathbb{R}$-hard in simple polygons. Given a simple polygon \mmathp{\bf P}\!, we can add small notches (consisting of a constant number of edges) at each of the convex vertices of \mmathp{\bf P}\!, thus creating a new polygon \mmathp{\bf P}'\ such that the shortest watchman route follows each edge of the original polygon~\mmathp{\bf P}\!, whereby the minimum set of vision points for \mmathp{\bf P}'\ is a superset of the minimum guard set of \mmathp{\bf P}\ restricted to the boundary. The hardness results exhibited for these problems increases the importance of obtaining good approximation methods. We show that there is a limit to how well such approximation methods can be by showing that {\sc mvpp}\ is APX-hard for simple polygons and we also extend the result to also hold for rectilinear polygons having three dent orientations~\cite{Culberson,Motwani,Motwani-perfect}, thus likely excluding the existence of polynomial time approximation schemes for~{\sc mvpp}. \section{A Reduction for Simple Polygons}\label{simple} We make a gap preserving reduction~\cite{Ausiello} from {\sc max2sat{\tt(}3{\tt)}}\ to {\sc mvpp}\ in simple polygons, where {\sc max2sat{\tt(}3{\tt)}}\ is the following problem. \begin{quote} {\sc max2sat{\tt(}3{\tt)}}\\ {\sc Instance}: a set of $n$ boolean variables $u_1,\ldots,u_n$ and a set of $m$ clauses $c_1,\ldots,c_m$, each consisting of a disjunction of exactly two distinct literals formed from the $n$ variables such that each variable occurs at most three times in the clauses.\\ {\sc Solution:} an assignment to the variables that satisfies the largest number of clauses. \end{quote} Berman and Karpinski~\cite{Berman} show that it is NP-hard to approximate {\sc max2sat{\tt(}3{\tt)}}\ by a factor $2012/2011-\epsilon$, for any $\epsilon>0$. Given an instance of {\sc max2sat{\tt(}3{\tt)}}, we can assume that no variable occurs only non-negated or negated in the clauses, otherwise we simply assign it the appropriate truth value to satisfy those clauses it is contained in. Since a variable then occurs two or three times in the clauses, there is one version, non-negated or negated, that occurs exactly once. We call this the {\em lone literal\/} of the variable. We also assume that at least one variable occurs three times in the clauses and without loss of generality that $u_1$ is such a variable with $\bar{u}_1$ as the lone literal in clause $c_1$. This will be used later to argue the structure of a canonical set of vision points. \begin{lemma}\label{lem:maxsat} For an instance of {\sc max2sat{\tt(}3{\tt)}}\ having $n$ variables and $m$ clauses, such that $M$ of them are satisfiable, it holds that: \begin{enumerate} \item $n\leq m$, \item $M\geq3m/4$, \item there exists an optimal solution such that any unsatisfied clause consists only of lone literals. \end{enumerate} \end{lemma} \begin{proof} Claim~1.\@~holds since each variable appears as at least two literals in the clauses. For Claim~2., a random assignment will satisfy each clause with probability at least $3/4$ so in total $3m/4$ clauses are satisfied in expectation. Therefore, at least one assignment must exist having at least $3m/4$ clauses satisfied. For Claim~3., any unsatisfied clause that contains a non-lone literal can be satisfied by reversing the assignment of that variable. Only the clause containing the lone literal can become unsatisfied by this operation so the number of satisfied clauses does not decrease. \end{proof} We call a solution to a {\sc max2sat{\tt(}3{\tt)}}\ instance that obeys Claims~1.,~2.,~and~3.~in Lemma~\ref{lem:maxsat}, {\em locally maximal}. To prove APX-Hardness of {\sc mvpp}\ in simple polygons, we reduce {\sc max2sat{\tt(}3{\tt)}}\ to {\sc mvpp}. We modify the NP-hardness proof by Aggarwal~\cite{Aggarwal} and Lee and Lin~\cite{Lee} in the same way as Carlsson~{\em et al}.~\cite{Carlsson-NPHard}; see Figure~\ref{vppredfig}. \begin{figure} \caption{\label{vppredfig} \label{vppredfig} \end{figure} The original NP-hardness proof constructs a {\em reduction polygon\/} for a {\sc max2sat{\tt(}3{\tt)}}\ instance consisting of a base rectangle with clause gadgets along the upper segment and variable gadgets along the lower segment. Each clause gadget $c_k$ is a structure with two {\em chimneys\/} corresponding to the literals in the clause having a designated point $q_k$ that is seen using two guards if and only if the clause is satisfied. Each variable gadget consists of two {\em wells}, visible from the point $x$, one corresponding to the literal $u_i$ and the other corresponding to the literal $\bar{u}_i$. Each variable gadget has a spike $t$, the only vertices seeing $t$ being its adjacent vertices along the polygon boundary and the two vertices $v_i$ and $v'_i$ marked red in Figure~\ref{setupfig}. Each literal chimney in a clause $c_k$ has two red vertices $r_{ik}$ and $r'_{ik}$ that see it and they are connected to the corresponding variable gadget of $u_i$. For the clause $c_k=(l_i\vee l_{j})$ the literal chimney of $l_i$ is connected to variable gadget $u_i$ by adding spikes $s_{ik}$ and $s'_{ik}$ depending on whether $l_i$ is $u_i$ or $\bar{u}_i$ as illustrated in Figures~\ref{setupfig}(a) and~(b). At least one guard in the clause gadget $c_k$ must be placed at the lower vertex $r'_{ik}$ or $r'_{jk}$ to see both the chimney and the point $q_k$, the rightmost point of the clause gadget. The chimney is made thin enough so that no point sees the top $y_{ik}$ of more than one chimney; see Figures~\ref{vppredfig} and~\ref{setupfig}. \begin{figure} \caption{\label{setupfig} \label{setupfig} \end{figure} To adapt the construction for {\sc mvpp}, we extend it by adding two {\em caves\/} (thin corridors each with a $90^{\circ}$ bend) to each clause structure, one at the top of each chimney, one cave is added at the bottom of each well structure, and three extra caves, on the top right sides of the well structures and one at~$x$, are also added. This gives us $2n+2m+3$ caves. To guard the polygon using vision points, the shortest watchman route must enter each of the caves and thus has a vision point in each cave, these are marked green in Figures~\ref{vppredfig} and~\ref{setupfig}. The caves tie down the shortest watchman route to ensure that the route passes the {\em critical guard points\/} that are used in the constructions of Aggarwal~\cite{Aggarwal} and Lee and Lin~\cite{Lee}. These critical guard points are marked red in Figures~\ref{vppredfig} and~\ref{setupfig}. The polygon and shortest watchman route have polynomial sized descriptions in the size of the {\sc max2sat{\tt(}3{\tt)}}\ instance. Disregarding the vision points in the caves, each clause gadget requires at least two more vision points (and at most three) and each variable gadget requires at least one vision point given a vision point at $x$. Thus, if an assignment to the {\sc max2sat{\tt(}3{\tt)}}\ instance satisfies $M$ clauses, we can guard the polygon using $ V = 2n+2m+3+n+2M+3(m-M)+1=3n+5m-M+4 $ vision points by placing one in each cave, one at $x$, one on the critical guard point corresponding to the assigned truth value in the variable gadget, one on the matching critical guard point of each literal chimney and, if the clause $c_k$ is not satisfied, one extra vision point at the rightmost of the critical guard points $r'_{ik}$ in the clause structure to see~$q_k$. We call such a placement a {\em canonical vision point set\/} and prove that we can assume that any vision point set is canonical. (A canonical vision point set is given in Figure~\ref{vppredfig} consisting of the cave guards in green, the point $x$ and the subset of red points that have white centers.) Given a set of vision points, we modify it to be canonical without increasing its size as follows. Clearly each green point must be a vision point otherwise not all caves are seen. We can also assume that point $x$ is a vision point, otherwise each variable gadget must have two further vision points and, since each clause gadget must also have two further vision points, we obtain at least $4n+4m+3$ vision points. Without loss of generality, these are the critical guard points $v_i$ and $v'_i$ in the variable gadgets and $r'_{ik}$ and $r'_{jk}$ in each clause gadget $c_k=(l_i\vee l_{j})$, giving exactly $4n+4m+3$ vision points guarding the polygon. Since $u_1$ occurs in three clauses and has $\bar{u}_1$ as lone literal in clause $c_1$, we remove the vision point at $v'_1$, place it at $x$, and move the vision point at $r'_{1,1}$ to $r_{1,1}$ if necessary, thus neither decreasing coverage nor increasing the size of the vision point set. The top point $y_{ik}$ of a clause gadget chimney of $l_i$ in clause $c_k$ sees two connected components of the watchman route that we denote $w$ and $w'$, $w$ containing $y_{ik}$. The component $w$ contains the chimney's two critical guard points $r_{ik}$ and $r'_{ik}$, $r'_{ik}$ seeing $q_{k}$. If $w'$ contains vision points, we move them to $r'_{ik}$, if the path from $y_{ik}$ to $r_{ik}$ of $w$ contains vision points, we move them to $r_{ik}$, and if the path from $y_{ik}$ to $r'_{ik}$ of $w$ contains vision points, we move them to $r'_{ik}$. If $r'_{ik}$ has a vision point after these moves, we remove all other vision points that see $y_{ik}$ (except the green cave one), otherwise we keep one at $r_{ik}$. Together with $x$, this vision point will guard at least as much as the original vision points on $w$ and~$w'$ (except for a disregardable portion of the other literal chimney in the clause gadget). The apex of the spike $t$ in a variable gadget $u_i$ sees three connected components of the watchman route. We denote these by $w_1$, $w_2$, and $w_3$ in increasing order of distance to $t$ and note that $w_2$ contains critical guard point $v_i$ and $w_3$ contains $v'_i$. If $w_1$ or $w_2$ have vision points, we move them to $v_i$ and if $w_3$ has vision points, we move them to $v'_i$ and remove any duplicates from $v_i$ and $v'_i$. Together with $x$, these points will guard at least as much as the original vision points on $w_1$, $w_2$, and $w_3$. If both $v_i$ and $v'_i$ have vision points, we remove the one that corresponds to the lone literal in the clause gadget of some clause $c_k$ and place one at $r'_{ik}$ unless point $q_k$ is already seen by the other vision points in the clause gadget. The process described above never adds vision points so the size of a canonical vision point set is no larger than the original set. We state this as a lemma. \begin{lemma}\label{lem:canonical} Any vision point set on a shortest watchman route in a reduction polygon can be transformed to a canonical vision point set of no larger size than the original set. \end{lemma} Berman and Karpinski~\cite{Berman} show that it is NP-hard to approximate {\sc max2sat{\tt(}3{\tt)}}\ by a factor $2012/2011-\epsilon$, for any $\epsilon>0$. Assume from the discussion above that we have a polynomial time approximation algorithm for {\sc mvpp}\ that produces $V=3n+5m-M+4$ canonical vision points for some value $M$. We can assume that $M$ corresponds to some locally maximal solution of the {\sc max2sat{\tt(}3{\tt)}}\ instance for which the optimum is \ensuremath{M_{\rm opt}}. Given an optimal solution to the {\sc max2sat{\tt(}3{\tt)}}\ instance, we construct a canonical vision point set in the reduction polygon by assigning vision points according to the truth values in the {\sc max2sat{\tt(}3{\tt)}}\ solution. Let $V'=3n+5m-\ensuremath{M_{\rm opt}}+4$ be the number of vision points placed in this way in the reduction polygon and let \ensuremath{V_{\rm opt}}\ be the minimum number of vision points in the reduction polygon. Since $V'\geq\ensuremath{V_{\rm opt}}$, $\ensuremath{M_{\rm opt}}/M\geq2012/2011-\epsilon$, and $m\geq4$, we have by Lemmata~\ref{lem:maxsat} and~\ref{lem:canonical} the ratio \begin{align} \frac{V}{\ensuremath{V_{\rm opt}}} & \geq \frac{V}{V'} = \frac{ 3n+5m-M+4 }{ 3n+5m-\ensuremath{M_{\rm opt}}+4 } \geq \frac{ 9m-M }{ 9m-M(2012/2011-\epsilon) } \geq \frac{ 22121 }{ 22120 }-\delta, \end{align} for any $\delta>0$ dependent on~$\epsilon$, which proves the APX-hardness of {\sc mvpp}\ in simple polygons. We have proved the following theorem. \begin{theorem} For every $\delta>0$, it is NP-hard to approximate {\sc mvpp}\ in a simple polygon to within a factor of\/~$22121/22120-\delta$. \end{theorem} \section{The {\sc mvpp}\ in Rectilinear Polygons with Three Dents}\label{threedent} The concept of {\em dents\/} in rectilinear polygons was introduced by Culberson and Reckhow~\cite{Culberson} and Motwani~{\em et~al}.~\cite{Motwani,Motwani-perfect} and they develop algorithms for orthogonal covering problems in rectilinear polygons with restricted number of dent orientations. A {\em dent\/} in a rectilinear polygon is simply a boundary edge where both endpoints are reflex. Thus, we identify dents with four different orientations, {\em north}, {\em south}, {\em east}, and {\em west\/}; see Figure~\ref{dentfig}(a). \begin{figure} \caption{\label{dentfig} \label{dentfig} \end{figure} Monotone rectilinear polygons have dents of one or two (opposite) orientations and for these, optimal linear time algorithms for {\sc mvpp}\ exist~\cite{Carlsson-Nilsson,Carlsson-NPHard}. We settle the complexity status for polygons with three dent orientations here but for rectilinear polygons having two (non-opposite) dent orientations the complexity status remains unknown. This should be contrasted with the classical art gallery problem, where linear time algorithms for computing the minimum number of point guards exist only for rectilinear polygons having one dent orientation (histograms)~\cite{Carlsson-NPHard}, for rectilinear polygons having two non-opposite dent orientations, the art gallery problem can be shown to be APX-hard by modifying the proof by Brodén~{\em et~al}.~\cite{Broden-APX} slightly. For the classical art gallery problem, the complexity is unknown for rectilinear monotone polygons having two opposite dent orientations but we suspect that the problem is indeed NP-hard given the recent NP-hardness proof for vertex guarding rectilinear staircase polygons~\cite{Gibson-staircase}. We modify the reduction introduced in the previous section to be rectilinear and furthermore to only contain dents of three different orientations. To this end, we introduce the {\em rectilinear spike emulator}, also used by Katz and Roisman~\cite{Katz}. A spike as used in Section~\ref{simple} is a thin corridor that can only be seen along a thin visibility cone. We can emulate the effect with a rectilinear gadget as shown in Figure~\ref{dentfig}(b) using one extra guard (green in the figure) and, as long as the original spike has reflex vertices with larger $x$-coordinates than its convex vertices, the rectilinear spike gadget never introduces an east dent. \begin{figure} \caption{\label{rectilineargadgetfig} \label{rectilineargadgetfig} \end{figure} As in Section~\ref{simple}, each variable gadget consists of two rectilinear wells, corresponding to the literals $u_i$ and $\bar{u}_i$. The point $x$ is not necessary, since each well is covered by a green guard at the bottom. Each variable gadget has a rectilinear spike $t$ seen by the two critical guard points $v_i$ and $v'_i$ marked red in Figure~\ref{rectilineargadgetfig}. As before, each clause gadget has two rectilinear chimneys corresponding to the literals in the clause and each chimney in a clause $c_k$ has two critical guard points $r_{ik}$ and $r'_{ik}$ that see it and they are connected to the corresponding variable gadget of $u_i$ by adding rectilinear spike emulators $s_{ik}$ and $s'_{ik}$ as illustrated in Figure~\ref{rectilineargadgetfig}. Again, we note that at least one guard in a clause gadget $c_k=(l_i\vee l_{j})$ must be placed at the lower vertex $r'_{ik}$ or $r'_{jk}$ to see both the chimney and the point $q_k$, placed in rectilinear spike emulator at the top edge of the clause gadget; see Figure~\ref{rectilineargadgetfig}. We add caves at the top of the chimneys, at the bottom of the variable gadgets, on the right side of the base rectangle and two caves, ensuring that these do not introduce east dents. These tie down the shortest watchman route to make it pass all the critical guard points. The convex vertices of the shortest watchman route each require a vision point, giving us $2n+8m+2$ such green vision points. In the same way as in Section~\ref{simple}, we can argue that any algorithm produces a canonical vision point set consisting of $ V = 2n+8m+2+n+2M+3(m-M) = 3n+11m-M+2 $ vision points, choosing the remaining ones from the set of critical guard points; see Figure~\ref{rvppredfig} for a full example of a canonical vision point set consisting of the green points and the subset of the red points that have white centers in a rectilinear polygon with three dent orientations. \begin{figure} \caption{\label{rvppredfig} \label{rvppredfig} \end{figure} Using the result by Berman and Karpinski~\cite{Berman}, that it is NP-hard to approximate {\sc max2sat{\tt(}3{\tt)}}\ by a factor $2012/2011-\epsilon$, for any $\epsilon>0$, we obtain as before the ratio \begin{align} \frac{V}{\ensuremath{V_{\rm opt}}} & \geq \frac{ 3n+11m-M+2 }{ 3n+11m-\ensuremath{M_{\rm opt}}+2 } \geq \frac{ 15m-M }{ 15m-M(2012/2011-\epsilon) } \geq \frac{ 38209 }{ 38208 }-\delta, \end{align} for any $\delta>0$ dependent on $\epsilon$, proving the APX-hardness of {\sc mvpp}\ in rectilinear polygons having three dent orientations. We have proved the following theorem. \begin{theorem} For every $\delta>0$, it is NP-hard to approximate {\sc mvpp}\ in a simple rectilinear polygon with three dent orientations to within a factor of\/~$38209/38208-\delta$. \end{theorem} \end{document}

\begin{document} \begin{center} {\Large \textbf{Using MM principles to deal with incomplete data in K-means clustering}}\\ {\large Mini Project: MM Optimization Algorithms}\\ {\large Ali Beikmohammadi}\\ \textit{Department of Computer and Systems Sciences\\ Stockholm University\\ SE-164 07 Kista, Sweden \\ \texttt{beikmohammadi@dsv.su.se} \\} \end{center} \begin{center} \rule{\textwidth}{0.2mm} \end{center} \begin{abstract} Among many clustering algorithms, the K-means clustering algorithm is widely used because of its simple algorithm and fast convergence. However, this algorithm suffers from incomplete data, where some samples have missed some of their attributes. To solve this problem, we mainly apply MM principles to restore the symmetry of the data, so that K-means could work well. We give the pseudo-code of the algorithm and use the standard datasets for experimental verification. The source code for the experiments is publicly available in the following link: \url{https://github.com/AliBeikmohammadi/MM-Optimization/blob/main/mini-project/MM \end{abstract} \begin{center} \rule{\textwidth}{0.2mm} \end{center} \section{Background and Introduction} Clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense or another) to each other than to those in other groups (clusters). It is the main task of exploratory data mining, and a common technique for statistical data analysis used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics \cite{boyd2018introduction, jain1999data, bijuraj2013clustering}. From a theoretical point of view, all proposed methods of clustering could be appointed to five major classes: Partitioning methods, Hierarchical methods, Density-based methods, Grid-based methods, and Model-based methods. Particularly, the simplest form of clustering is partitional clustering which aims at subdividing the dataset into a set of K groups so that specific clustering criteria are optimized, where K is the number of groups pre-specified by the analyst \cite{likas2003global}. The most widely used criterion is the clustering error criterion which for each point computes its squared distance from the corresponding cluster center and then takes the sum of these distances for all points in the data set \cite{likas2003global, bishop2006pattern, webb2011statistical}. There are different types of partitioning clustering methods. A popular clustering method that minimizes the clustering error is the K-means algorithm \cite{macqueen1967some}, in which, each cluster is represented by the center or means of the data points belonging to the cluster. K-means algorithm has many advantages such as simple mathematical ideas, fast convergence, easy implementation, and easily adaptation to new examples. \cite{li2017parallel}. Therefore, the application fields are very broad, including different types of document classification, topic discovery, patient clustering, customer market segmentation, student clustering, and weather zones, the construction of recommendation systems based on user interests, and so on \cite{boyd2018introduction, li2017parallel}. However, the K-means algorithm is a local search procedure and it is well known that it suffers from the serious drawback such as, choosing K manually, being dependent on initial values, difficulty in clustering data of varying sizes and density, sensitivity to outliers, scaling with the number of dimensions, and the need for access to full data \cite{pena1999empirical}. Fortunately, the researchers have proposed some solutions to address many of these challenges \cite{yuan2019research, kanungoy2000cient, huang1998extensions}. In this report, we will focus on the last weakness, where we have an incomplete data. The assumption of not having access to all the attributes of a sample is completely realistic. For example, in many cases where these components of each sample are collected from the environment through various sensors, there is always the possibility that some of the sensors will fail. Under this condition, we would not have access to those features for that particular sample. The first idea when dealing with such incomplete samples is to withdraw them from the data set, so that it is possible to use a standard K-means clustering algorithm, known as Lloyd's algorithm \cite{lloyd1982least}. But, in this project, with the help of MM optimization principles \cite{Kenneth-2016}, the symmetry of the dataset is reconstructed. By doing this, on the one hand, we make the most of the available measurements, and on the other hand, we are able to use the standard K-means clustering algorithm. The rest of this work is organized as follows. In Section \ref{section2}, we describe the formulation associated with the original problem which is K-means clustering algorithm. We apply MM principles to design our solution in Section \ref{section3} to deal with incomplete data. Experimental results and analyses are reported in Sections \ref{section4}. \section{Problem Formulation: K-means Clustering Algorithm} \label{section2} The K-means clustering algorithm was first proposed in 1957 by Stuart Lloyd, and independently by Hugo Steinhaus \cite{lloyd1982least, boyd2018introduction}. Sometimes, it is called the Lloyd algorithm. The name `K-means' has been used from the 1960s. The K-means algorithm is based on alternating two procedures. The first is one of assignment of objects to groups. An object is usually assigned to the group to whose mean it is closest in the Euclidean sense. The second procedure is the calculation of new group means based on the assignments. The process terminates when no movement of an object to another group will reduce the within-group sum of squares. To be more specific, given a set of observations $(x_1, x_2, ..., x_m)$, where each observation is a $d$-dimensional real vector, K-means clustering aims to partition the $m$ observations into $K (\le m)$ sets $C = \{C_1, C_2, ..., C_K\}$ so as to minimize the within-cluster sum of squares. Formally, the objective is to: \begin{equation} \label{eq:1} \text{minimize} f(\mu) = \sum_{k=1}^{K} \sum_{x_i \in C_k} \|x_i-\mu_k\|^2, \end{equation} where $\mu_k$ is the center of cluster $k$. The solution to the centroid $\mu_k$ is as follows: \begin{equation} \label{eq:2} \begin{split} \frac {\partial f(\mu)} { \partial\mu_k} &= \frac {\partial} {\partial\mu_k} \sum_{k=1}^{K} \sum_{x_i \in C_k} (x_i-\mu_k)^2\\ &= \sum_{k=1}^{K} \sum_{x_i \in C_k}\frac {\partial} {\partial\mu_k} (x_i-\mu_k)^2\\ &= \sum_{x_i \in C_k} 2 (x_i-\mu_k). \end{split} \end{equation} Let Equation \ref{eq:2} be zero; then $\mu^\star_k=\frac {1} { |C_k|}\sum_{x_i \in C_k}x_i$. In terms of iterative solving this problem, the Algorithm \ref{code1} shows all the needed steps, where the central idea is to randomly extract K sample points from the sample set as the center of the initial cluster; Divide each sample point into the cluster represented by the nearest center point; then the center point of all sample points in each cluster is the center point of the cluster. Repeat the above steps until the center point of the cluster is almost unchanged ($\le \epsilon$) or reaches the set number of iterations. \begin{minipage}{0.5\textwidth} \centering \begin{algorithm}[H] \label{code1} \DontPrintSemicolon \KwInput{$K, \epsilon$} \KwOutput{$\mu_1, \mu_2, ... , \mu_K$} \KwData{$X$} $n=0$ \; Randomly pick K samples of X as K centroids: $\mu_1^n, \mu_2^n, ... , \mu_K^n \in \mathbb{R}^d$ \; \Repeat{$\sum_{k=1}^{K}\|\mu_k^n-\mu_k^{n-1}\|^2 \le \epsilon$}{ $n \leftarrow n+1$\; $C_k \leftarrow \emptyset $ for all $k=1, ... , K$\; \tcc{Cluster Assignment Step} \ForEach{$x_i \in X$}{ $k^\star\leftarrow \text{arg min}_k \|x_i-\mu_k^n\|^2$ \tcc{Assign $x_i$ to closest centroid} $C_{k^\star}\leftarrow C_{k^\star} \cup x_i$ } \tcc{Centroid Update Step} \ForEach{$k=1 $ to $ K$}{ $\mu_k^n=\frac {1} { |C_k|}\sum_{x_i \in C_k}x_i$} } \caption{K-means} \end{algorithm} \end{minipage} \begin{minipage}{0.5\textwidth} \centering \begin{algorithm}[H] \label{code2} \DontPrintSemicolon \KwInput{$K, \epsilon$} \KwOutput{$\mu_1, \mu_2, ... , \mu_K$} \KwData{$X, O_X$} $n=0$ \; Randomly pick K samples among those X that are fully observed as K centroids: $\mu_1^n, \mu_2^n, ... , \mu_K^n \in \mathbb{R}^d$ \; Randomly replace unobserved data with corresponding element of one of $\mu_1^n, \mu_2^n, ... , \mu_K^n$ \; \Repeat{$\sum_{k=1}^{K}\|\mu_k^n-\mu_k^{n-1}\|^2 \le \epsilon$}{ $n \leftarrow n+1$\; $C_k \leftarrow \emptyset $ for all $k=1, ... , K$\; \tcc{Cluster Assignment Step} \ForEach{$x_i \in X$}{ $k^\star\leftarrow \text{arg min}_k \|x_i-\mu_k^n\|^2$ \tcc{Assign $x_i$ to closest centroid} $C_{k^\star}\leftarrow C_{k^\star} \cup x_i$ } \tcc{Centroid Update Step} \ForEach{$k=1 $ to $ K$}{ $\mu_k^n=\frac {1} { |C_k|}\sum_{x_i \in C_k}x_i$} \tcc{Unobserved Data Replacement Step} \ForEach{$x_i \in X$ that $j \notin O_{x_i}$}{ ${x_i}_j \leftarrow {\mu^n_k}_j$\; } } \caption{MM K-means} \end{algorithm} \end{minipage} However, as it turns out, this algorithm can only be used when all d features of each observation are available. In the next section, we overcome this limitation by applying MM principles. \section{Restoring Data Symmetry by Applying MM Principles} \label{section3} Let $O_{x_i}$ denote the set of indexes observed in sample $x_i$. Then, the objective function mentioned in Equation \ref{eq:1} should change to \begin{equation} \label{eq:3} \text{minimize} f(\mu) = \sum_{k=1}^{K} \sum_{x_i \in C_k} \Big[\sum_{j \in O_{x_i}}({x_i}_j-{\mu_k}_j)^2\Big], \end{equation} due to incomplete $x_i$s. Note that, ${\mu_k}_j$ is $j$th index of $\mu_k$. Therefore, incomplete data prevents the use of the standard K-means algorithm and following Algorithm \ref{code1}. However, since $f(\mu)$ is a convex function, we can apply an interesting majorization to $f(\mu)$ by following the MM principles so that: \begin{equation} \label{eq:4} f(\mu) \le \sum_{k=1}^{K} \sum_{x_i \in C_k} \Big[\sum_{j \in O_{x_i}}({x_i}_j-{\mu_k}_j)^2 + \sum_{j \notin O_{x_i}}({\mu^n_k}_j-{\mu_k}_j)^2\Big] =g(\mu|\mu^n). \end{equation} As we know, majorization combines two conditions: the tangency condition $g(\mu^n|\mu^n)= f(\mu^n)$ and the domination condition $g(\mu|\mu^n)\ge f(\mu)$ for all $\mu$ \cite{Kenneth-2016}. Here, since $\sum_{j \notin O_{x_i}}({\mu^n_k}_j-{\mu^n_k}_j)^2 =0$ and $\sum_{j \notin O_{x_i}}({\mu^n_k}_j-{\mu_k}_j)^2 \ge 0$, both conditions are satisfied. Therefore, by simply substituting the corresponding ${\mu^n_k}_j$ with the unobserved data ${x_i}_j$ in each iteration, the data symmetry is restored, and one can apply standard K-means algorithm. Particularly, Algorithm \ref{code2} shows a step-by-step process of our proposed method. In the next section, we examine its performance in detail. \section{Results and Discussion} \label{section4} \begin{figure} \caption{Comparison of the K-means algorithm with the proposed MM K-means algorithm when 10\%, 30\% and 50\% of the data are unobserved randomly. The first column represents noisy circles, noisy moons, blobs with varied variances, anisotropicly distributed, and blobs dataset, respectively. Note that the black dots in each plot represent the cluster centers reached by each algorithm after 100 iterations.} \label{fig:fig1} \end{figure} \begin{figure} \caption{(Left) Blobs dataset with three clusters of green, blue, and orange, where samples that have lost at least one of their elements are marked with the black circle. Note that, here, 50\% of all elements, i.e., 50\% of 500*2, are assumed unobserved. (Right) The result of applying the proposed MM K-means Algorithm to the Blobs dataset (left plot) in more detail, where the centers of the clusters are marked with black dots.} \label{fig:fig2} \end{figure} For comparison, we have created five different datasets shown in the first column of Figure \ref{fig:fig1} using the scikit-learn library \cite{pedregosa2011scikit}. To make good visualization, each sample is assumed to consist of only two elements, i.e., $x_i \in \mathbb{R}^2$ \$. We have also collected 500 samples in each dataset. Also, the first two datasets, namely noisy circles and noisy moons, consist of two clusters. Note that, when making them, the amount of noise is set to 0.05. The rest of the datasets, blobs with varied variances, anisotropicly distributed, and blobs dataset,comprised of three clusters. Details on how to create datasets can be found in the source code in the following link: \url{https://github.com/AliBeikmohammadi/MM-Optimization/blob/main/mini-project/MM \begin{table*} \centering \begin{tabular}{ccccccccc} \hline \hline \textbf{Dataset} & \textbf{Algorithm} & \textbf{Time} & \textbf{Homogeneity}& \textbf{Completeness}& \textbf{V-measure}& \textbf{ARI}& \textbf{AMI}& \textbf{Silhouette Coefficient} \\ \hline \hline noisy circles & original dataset & 0.00 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 0.111 \\ & K-means on Complete Data & 0.09 & 0.000 & 0.000& 0.000& -0.002 &-0.001 &0.352\\ & MM K-means on 90\% Data & 0.13& 0.000& 0.000& 0.000& -0.002& -0.001& 0.355\\ & MM K-means on 70\% Data & 0.12& 0.000& 0.000& 0.000& -0.002& -0.001& 0.355\\ & MM K-means on 50\% Data & 0.16& 0.000& 0.000& 0.000& -0.002& -0.001& 0.347\\ \hline noisy moons& original dataset &0.00& 1.000& 1.000& 1.000& 1.000& 1.000& 0.387\\ & K-means on Complete Data& 0.10& 0.385& 0.385& 0.385& 0.483& 0.384& 0.497\\ & MM K-means on 90\% Data & 0.08& 0.385& 0.385& 0.385& 0.483& 0.384& 0.497\\ & MM K-means on 70\% Data & 0.17& 0.386& 0.386& 0.386& 0.483& 0.385& 0.495\\ & MM K-means on 50\% Data & 0.21& 0.387& 0.394& 0.391& 0.467& 0.390& 0.484\\ \hline varied & original dataset & 0.00& 1.000& 1.000& 1.000& 1.000& 1.000& 0.569\\ & K-means on Complete Data& 0.10& 0.726& 0.743& 0.734& 0.731& 0.733& 0.639\\ & MM K-means on 90\% Data & 0.17& 0.723& 0.740& 0.731& 0.727& 0.730& 0.639\\ & MM K-means on 70\% Data & 0.27& 0.702& 0.723& 0.712& 0.701& 0.711& 0.631\\ & MM K-means on 50\% Data & 0.29& 0.737& 0.752& 0.745& 0.745& 0.744& 0.635\\ \hline aniso & original dataset & 0.00& 1.000& 1.000& 1.000& 1.000& 1.000& 0.459\\ & K-means on Complete Data& 0.11& 0.600& 0.600& 0.600& 0.578& 0.599& 0.503\\ & MM K-means on 90\% Data & 0.27& 0.613& 0.615& 0.614& 0.585& 0.613& 0.503\\ & MM K-means on 70\% Data & 0.31& 0.642& 0.647& 0.645& 0.618& 0.643& 0.504\\ & MM K-means on 50\% Data & 0.27& 0.657& 0.680& 0.668& 0.619& 0.667& 0.490\\ \hline blobs & original dataset & 0.00& 1.000& 1.000& 1.000& 1.000& 1.000& 0.829\\ & K-means on Complete Data& 0.05& 1.000& 1.000& 1.000& 1.000& 1.000& 0.829\\ & MM K-means on 90\% Data & 0.09& 1.000& 1.000& 1.000& 1.000& 1.000& 0.829\\ & MM K-means on 70\% Data & 0.17& 1.000& 1.000& 1.000& 1.000& 1.000& 0.829\\ & MM K-means on 50\% Data & 0.16& 1.000& 1.000& 1.000& 1.000& 1.000& 0.829\\ \hline \hline \end{tabular} \caption{Numerical comparison of the K-means algorithm with the proposed MM K-means algorithm when 10\%, 30\% and 50\% of the data are unobserved randomly. Note that, ARI and AMI stand for Adjusted Rand Index and Adjusted Mutual Information, respectively. Also, anisotropicly distributed dataset and blobs with varied variances dataset are mentioned with aniso and varied, respectively.} \label{tbl:table1} \end{table*} Four experiments were performed on each dataset. First, the K-means algorithm is trained on the whole dataset as the baseline. Second, after removing 10\% of the total constituent elements of the samples, the proposed MM K-means algorithm is tested. The third and fourth are done exactly as same as the second experiment, with the difference that 30\% and 50\% of the data are assumed not to be observed, respectively. In all experiments, the number of iterations is set to 100. Following the algorithms \ref{code1} and \ref{code2}, the results shown in Figure \ref{fig:fig1} are obtained. These results visually confirm that the proposed MM K-means algorithm is fully consistent with the K-means algorithm and was able to restore the data symmetry. Specifically, the black dots shown in each plot in Figure \ref{fig:fig1}, which represent the centers of each cluster, indicate that both algorithms have converged to the same point. But, as expected, these found centers can not always guarantee the finding of perfectly correct clusters. In fact, we were able to use MM principle to help reconstruct the data, not to improve the K-means algorithm. However, it can be argued that the same principle can be used as an umbrella over other more efficient clustering methods that suffer from the inability to work with missing data. Interestingly, the proposed method is robust against increasing the amount of missing data, where by increasing the amount of unobserved data from 10\% to 50\%, it has a negligible effect on the results. To prove this claim more, Table \ref{tbl:table1} provides a numerical comparison with full details. In this comparison, known criteria in the field of clustering have been used, which are: Homogeneity, Completeness, V-measure, Adjusted Rand Index, Adjusted Mutual Information, and Silhouette Coefficient. Regardless of the criteria chosen, all the results confirm that, firstly, the data reconstruction is very well done and, secondly, the increase in the amount of lost data has little effect on the performance. Figure \ref{fig:fig2} also shows the proper scattering of incomplete data (assuming 50\% of incomplete data) in the dataset. However, it is clear that the algorithm has been able to find cluster centers similar to the ones in which we have access to all data, by properly reconstructing missing data. Finally, as shown in Figure \ref{fig:fig1} and Table \ref{tbl:table1}, the execution time of the proposed MM K-means algorithm is longer because it requires updating the missing elements in each iteration. However, in conclusion, applying MM principles to reconstruct data symmetry can play an important role in maximizing data usage along with the possibility of using standard algorithms. It is hoped that in the future, this technique can be used to improve the performance of deep learning-based methods for various applications such as plant identification \cite{beikmohammadi2020swp, BEIKMOHAMMADI2022117470}, handwritten digit recognition \cite{beikmohammadi2021hierarchical}, and human action detection \cite{beikmohammadi2019mixture}. \end{document}

\begin{document} \begin{abstract} Let $Y$ be the variety of (skew) symmetric $n\times n$-matrices of rank $\leftarrowe r$. In paper we construct a full faithful embedding between the derived category of a non-commutative resolution of $Y$, constructed earlier by the authors, and the derived category of the classical Springer resolution of $Y$. \end{abstract} \maketitle \section{Introduction} \leftarrowabel{ref-1-0} Throughout $k$ is an algebraically closed field of characteristic zero. If $\Lambda$ is a right noetherian ring then we write ${\cal D}(\Lambda)$ for $D^b_f(\Lambda)$, the bounded derived category of right $\Lambda$-modules with finitely generated cohomology. Similarly for a noetherian scheme/stack $X$ we write ${\cal D}(X):=D^b_{\mathop{\text{\uparrowpshape{coh}}}}(X)$. If $Y$ is the determinantal variety of $n\times n$-matrices of rank $\leftarrowe r$ then in \cite{VdB100} (and independently in \cite{SegalDonovan}) a ``non-commutative crepant resolution'' \cite{Leuschke,VdB32} $\Lambda$ for $k[Y]$ was constructed. Such an NCCR is a $k[Y]$-algebra which has in particular the property that ${\cal D}(\Lambda)$ is a ``strongly crepant categorical resolution'' of~$\operatorname{Perf}(Y)$ (the derived category of perfect complexes on $Y$) in the sense of~\cite[Def.\ 3.5]{Kuznetsov}. This NCCR was constructed starting from a tilting bundle on the standard Springer type resolution of singularities $Z\rightarrow Y$ where $Z$ is a vector bundle over a Grassmannian. Indeed the main properties of $\Lambda$ were derived from the existence of a derived equivalence between ${\cal D}(\Lambda)$ and ${\cal D}(Z)$. In this paper we discuss suitably adapted versions of these results for determinantal varieties of symmetric matrices and skew symmetric matrices. It turns out that both settings are very similar but notationally cumbersome to treat together. So we present our main results and arguments in the skew symmetric case. The modifications needed for the symmetric case will be discussed briefly in Section~\rightarrowef{symsec}. Let $n>r>0$ with $2|r$ and now let $Y$ be the variety of skew symmetric $n\times n$-matrices of rank $\leftarrowe r$. If $n$ is odd then in \cite{SpenkoVdB} we constructed an NCCR $\Lambda$ for $k[Y]$ (the existence of the resulting strongly crepant categorical resolution of $Y$ was conjectured in \cite[Conj.\ 4.9]{Kuznetsov4}). The construction of $\Lambda$ also works when $n$ is even but then $\Lambda$ is not an NCCR, albeit very close to one. In particular one may show that ${\cal D}(\Lambda)$ is a ``weakly crepant categorical resolution'' of $\operatorname{Perf}(Y)$, again in the sense of \cite{Kuznetsov} (see \cite{Abuaf} for an entirely different construction of such resolutions). In contrast to \cite{VdB100,SegalDonovan} the construction of the NCCR $\Lambda$ is based on invariant theory and does not use geometry. Nonetheless it is well known that also in this case $Y$ has a canonical (commutative) Springer type resolution of singularities $Z\rightarrow Y$ and our main concern below will be the relationship between the resolutions $\Lambda$ and $Z$. In particular we will construct a $k[Y]$-linear embedding \begin{equation} \leftarrowabel{ref-1.1-1} {\cal D}(\Lambda)\hookrightarrow {\cal D}(Z). \end{equation} For $n$ odd such an inclusion is expected by the fact that NCCRs are conjectured to yield minimal categorical resolutions. Note that the embedding \eqref{ref-1.1-1} turns out to be somewhat non-trivial. The image of $\Lambda$ is a coherent sheaf of ${\cal O}_Z$-modules, but it is not a vector bundle. As already mentioned, the construction of $\Lambda$ uses invariant theory. We explain this next. Let $H$, $V$ be vector spaces of dimension $n$, $r$ with $V$ being in addition equipped with a symplectic bilinear form $\leftarrowangle-,-\rightarrowangle$. The corresponding symplectic group is denoted by $\Sp(V)$. If ${{\mathop{\text{Ch}}i}}$ is a partition with $l({{\mathop{\text{Ch}}i}})\leftarrowe r/2$ then we let $S^{\leftarrowangle{{\mathop{\text{Ch}}i}}\rightarrowangle}V$ be the irreducible representation of $\Sp(V)$ with highest weight~${{\mathop{\text{Ch}}i}}$. If ${{\mathop{\text{Ch}}i}}=({{\mathop{\text{Ch}}i}}_1,\leftarrowdots,{{\mathop{\text{Ch}}i}}_{r})\in\ZZ^r$ is a dominant $\operatorname {GL}(V)$-weight then we let $S^{{\mathop{\text{Ch}}i}} V$ be the irreducible $\operatorname {GL}(V)$-representation with highest weight ${{\mathop{\text{Ch}}i}}$. Put $ X=\operatorname {Hom}(H,V) $ and let $T$ be the coordinate ring of $X$: \[ T=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(H\otimes_k V^\vee). \] Put \begin{equation} \leftarrowabel{ref-1.2-2} M({{\mathop{\text{Ch}}i}}):= (S^{\leftarrowangle{{\mathop{\text{Ch}}i}}\rightarrowangle}V\otimes_k T)^{\Sp(V)}\,. \end{equation} Thus $M(\mathop{\text{Ch}}i)$ is a ``module of covariants'' in the sense of \cite{VdB9}. Let $B_{m,n}$ be the set of partitions contained in a box with $m$ rows and $n$ columns. Put \begin{equation} \leftarrowabel{ref-1.3-3} M=\bigoplus_{{{\mathop{\text{Ch}}i}}\in B_{r/2,\leftarrowfloor n/2\rightarrowfloor-r/2}} M({{\mathop{\text{Ch}}i}}) \end{equation} and $ \Lambda=\operatorname {End}_{R}(M) $. In \cite{SpenkoVdB} the following result (which improves on \cite{WeymanZhao}) was proved: \begin{theorem} \leftarrowabel{ref-1.1-4} One has $\operatorname {gl\,dim}\Lambda<\infty$. Moreover if $n$ is odd then $\Lambda$ is a Cohen-Macaulay $R:=T^{\Sp(V)}$-module. In other words, in the terminology of \cite{Leuschke,VdB32}, when $n$ is odd $\Lambda$ is a non-commutative crepant resolution (NCCR) of $R$. \end{theorem} By the first fundamental theorem for the symplectic group $R$ is a quotient of $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 H)$ so that dually $\operatorname {Spec} R\hookrightarrow \wedge^2 H^\vee\subset \operatorname {Hom}_k(H,H^\vee)$. The second fundamental theorem for the symplectic group yields \[ \operatorname {Spec} R=\{\psi\mid \psi\in \operatorname {Hom}_k(H,H^\vee),\psi+\psi^\vee=0,\operatorname {rk} \psi\leftarrowe r\}\,. \] so that $\operatorname {Spec} R\cong Y$ with $Y$ as introduced above. Below we identify $R$ with $k[Y]$. We now discuss the Springer resolution $p:Z\rightarrow Y$ as well as the inclusion ${\cal D}(\Lambda)\hookrightarrow {\cal D}(Z)$ announced in \eqref{ref-1.1-1}. Let $ F=\operatorname{Gr}(r,H)$ be the Grassmannian of $r$-dimensional quotients $H\twoheadrightarrow Q$ of $H$ and put \[ Z=\{(\phi,Q)\mid Q\in F,\phi\in \operatorname {Hom}_k(Q,Q^\vee),\phi+\phi^\vee=0\}\,. \] The Springer resolution $p:Z\rightarrow Y\hookrightarrow \operatorname {Hom}_k(H,H^\vee)$ of $Y$ sends $(\phi,Q)$ to the composition \[ [H\twoheadrightarrow Q\xrightarrow{\phi} Q^\vee \hookrightarrow H^\vee]\in \operatorname {Hom}_k(H,H^\vee)\,. \] Using again the fundamental theorems for the symplectic group we have \begin{equation} \leftarrowabel{ref-1.4-5} \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes_k V^\vee)^{\Sp(V)}\cong\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q) \end{equation} (since $\downarrowim Q=\downarrowim V$, there are no relations on the righthand side). For a partition~${{\mathop{\text{Ch}}i}}$ with $l({{\mathop{\text{Ch}}i}})\leftarrowe r/2$ we put \begin{equation} \leftarrowabel{eq:mq} M_Q({{\mathop{\text{Ch}}i}})=(\downarrowet Q)^{\otimes r-n}\otimes_k (S^{\leftarrowangle {{\mathop{\text{Ch}}i}}\rightarrowangle} V\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes_k V^\vee))^{\Sp(V)} \end{equation} where we consider $M_Q({{\mathop{\text{Ch}}i}})$ as a $\operatorname {GL}(Q)$-equivariant $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q)$-module via \eqref{ref-1.4-5}. Choose a specific $(H\twoheadrightarrow Q)\in F$. One has $F=\operatorname {GL}(H)/P_Q$ where $P_Q$ is the parabolic subgroup of $\operatorname {GL}(H)$ that stabilizes the kernel of $H\twoheadrightarrow Q$. We regard $\operatorname {GL}(Q)$-equivariant objects tacitly as $P_Q$-equivariant objects through the canonical morphism $P_Q\twoheadrightarrow \operatorname {GL}(Q)$. Taking the fiber in $Q$ defines an equivalence between $\mathop{\text{\uparrowpshape{coh}}}(\operatorname {GL}(H),Z)$ and $\operatorname{mod}(P_Q,{\cal Z}_Q)$ where ${\cal Z}_Q:=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q)$, whose inverse will be denoted by $\widetilde{?}$. Put \[ {\cal M}_{Z}({{\mathop{\text{Ch}}i}})=\widetilde{M_Q({{\mathop{\text{Ch}}i}})}\in \mathop{\text{\uparrowpshape{coh}}}(\operatorname {GL}(H),Z)\,. \] \begin{theorem} (see \S\rightarrowef{ref-5.2-29}) \leftarrowabel{ref-1.2-6} Let $\mu,\leftarrowambda\in B_{r/2,n-r}$. \begin{enumerate} \item \leftarrowabel{ref-1-7} We have for $i>0$. \[ \operatorname {Ext}^i_Z({\cal M}_Z(\leftarrowambda),{\cal M}_Z(\mu))=0. \] \item There are isomorphisms as $R$-modules \begin{equation} \leftarrowabel{ref-1.5-8} R\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(Z,{\cal M}_Z(\leftarrowambda))\cong M(\leftarrowambda)\,. \end{equation} \item \leftarrowabel{ref-3-9} Applying $p_\ast$ induces an isomorphism \begin{align} \operatorname {Hom}_Z({\cal M}_Z(\leftarrowambda),{\cal M}_Z(\mu))&\overset{p_\ast}{\cong} \operatorname {Hom}_Y(p_\ast{\cal M}_Z(\leftarrowambda),p_\ast{\cal M}_Z(\mu))\leftarrowabel{ref-1.6-10}\\ &\cong \operatorname {Hom}_R(\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(Z,{\cal M}_Z(\leftarrowambda)),\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(Z,{\cal M}_Z(\mu)))&&\text{($Y$ is affine)}\nonumber\\ &\cong \operatorname {Hom}_R(M(\leftarrowambda),M(\mu))&& \text{(by \eqref{ref-1.5-8})}\nonumber \end{align} \end{enumerate} \end{theorem} From this theorem it follows in particular that \[ {\cal M}_Z:=\bigoplus_{{{\mathop{\text{Ch}}i}}\in B_{r/2,\leftarrowfloor n/2\rightarrowfloor-r/2}} {\cal M}_Z({{\mathop{\text{Ch}}i}}) \] satisfies \[ \operatorname {Ext}^i_Z({\cal M}_Z,{\cal M}_Z)= \begin{cases} \Lambda&\text{if $i=0$}\\ 0&\text{if $i>0$} \end{cases} \] and we obtain the following more precise version of \eqref{ref-1.1-1}: \begin{corollary} \leftarrowabel{ref-1.3-11} There is a full exact embedding \[ -\overset{L}{\otimes}_{\Lambda} {\cal M}_Z: {\cal D}(\Lambda)\hookrightarrow {\cal D}(Z)\,. \] \end{corollary} \begin{remark} Put \[ M':=\bigoplus_{\mathop{\text{Ch}}i\in B_{r/2,n-r}} M(\mathop{\text{Ch}}i) \] and $\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma=\operatorname {End}_R(M')$. It follows from \cite[Thm 1.5.1]{SpenkoVdB} (applied with $\Delta=\epsilon \bar{\Sigma}$ for a sufficiently small $\epsilon>0$) that $\operatorname {gl\,dim} \mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma<\infty$. See the computation in \S6 in loc.\ cit.. We have $\Lambda=e\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma e$ for a suitable idempotent $e$. The fact that $\operatorname {gl\,dim} \Lambda<\infty$ implies that $\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma$ cannot be an NCCR by \cite[Ex.\ 4.34]{Wemyss1} (see also \cite[Remark 3.6]{SpenkoVdB}). In the terminology of \cite{SpenkoVdB} $\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma$ is a (non-crepant) non-commutative resolution of $R$. As in Corollary \rightarrowef{ref-1.3-11} we still have an embedding ${\cal D}(\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma)\subset {\cal D}(Z)$. \end{remark} \begin{comment} \begin{remark} The methods in this paper apply almost verbatim to determinantal varieties for symmetric matrices but, as is often the case, the statements and arguments become a bit more involved (for starters there is a distinction between $r$ even and odd). This is why we restrict ourselves to skew symmetric matrices in this paper. The details for the symmetric case will be discussed elsewhere. \end{remark} \end{comment} \section{A $\operatorname {GL}(Q)$-equivariant free resolution of $M_Q(\leftarrowambda)$} \leftarrowabel{ref-3-12} In this section we discuss some of the properties of the $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q)$-modules ${M_{Q}}(\leftarrowambda)$ introduced in the introduction. We basically restate some results from \cite{WeymanSam} in our current language. To do this it will be convenient to consider \[ N_Q(\mathop{\text{Ch}}i):= (S^{\leftarrowangle {{\mathop{\text{Ch}}i}}\rightarrowangle} V\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes_k V^\vee))^{\Sp(V)} \] so that $M_Q(\mathop{\text{Ch}}i)=(\downarrowet Q)^{\otimes r-n}\otimes_k N_Q(\mathop{\text{Ch}}i)$. Since $\downarrowet Q$ is one-dimensional, $M_Q(\leftarrowambda)$ and $N_Q(\leftarrowambda)$ have identical properties. The following fact will not be used although it seems interesting to know \begin{lemma} ${N_Q}(\leftarrowambda)$ is a reflexive $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2Q)$-module. \end{lemma} \begin{proof} This follows for example from the fact that $\operatorname {Spec}\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes_k V^\vee)\rightarrow \operatorname {Spec}\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q)$ contracts no divisor. \end{proof} \leftarrowabel{ref-3-13} \begin{comment} Choose a basis for $Q$. Using this basis we will identify the Weyl group of $\operatorname {GL}(Q)$ with $S_r$ and we will write the $\operatorname {GL}(Q)$-weights as $r$-tuples: $\mu=(\mu_1,\leftarrowdots,\mu_r)$ with the dominant weights being characterized by $\mu_1\ge \operatorname {cd}ots\ge\operatorname {cd}ots\ge \mu_r$. For a dominant weight $\mu$ we write $S^\mu Q$ for the corresponding irreducible $\operatorname {GL}(Q)$-representation. The twisted Weyl group action on weights will be denoted by ``$\ast$''. The $i$'th simple reflection $s_i$ acts by $s_i{\ast} (\leftarrowdots,\mu_i,\mu_{i+1},\leftarrowdots):= (\leftarrowdots,\mu_{i+1}-1,\mu_{i}+1,\leftarrowdots)$. If there is a $w\in S_r$ such that $w{\ast}\mu$ is dominant then we say that~$\mu$ is regular, otherwise that it is singular. If~$\mu$ is regular then we write $\mu^+$ for the (necessarily unique) dominant weight in the twisted Weyl group orbit of $\mu$. We also write $i(\mu)$ for the minimal number of (twisted) simple reflections required to make $\mu$ dominant. \end{comment} Recall that a border strip is a connected skew Young diagram not containing any $2\times 2$ square. The size of a border strip is the number of boxes it contains. We follow \cite{WeymanSam} and associate to some partitions $\leftarrowambda$ a partition $\tau_{r}(\leftarrowambda)$ and a number $i_{r}(\leftarrowambda)$. The definition of $(\tau_r(\leftarrowambda),i_r(\leftarrowambda))$ is inductive. If $l(\leftarrowambda)\leftarroweq r/2$ then $\tau_{r}(\leftarrowambda)=\leftarrowambda$, $i_r(\leftarrowambda)=0$. Suppose now that $l(\leftarrowambda)>r/2$. If there exists a non empty border strip $R_\leftarrowambda$ of size $2 l(\leftarrowambda)-r-2$ starting at the first box in the bottom row of $\leftarrowambda$ such that $\leftarrowambda\setminus R_\leftarrowambda$ is a partition then $\tau_r(\leftarrowambda):=\tau_r(\leftarrowambda\setminus R_\leftarrowambda)$, and $i_r(\leftarrowambda):=c(R_\leftarrowambda)+i_r(\leftarrowambda\setminus R_\leftarrowambda)$, where $c(R_\leftarrowambda)$ is the number of columns of $R_\leftarrowambda$. Otherwise $\tau_r(\leftarrowambda)$ is undefined and $i_r(\leftarrowambda)=\infty$. \begin{comment} Recall that a partition has Frobenius coordinates $(a_1,\leftarrowdots,a_u;b_1,\leftarrowdots,b_u)$, $a_1>\operatorname {cd}ots>a_u\ge 1$, $b_1>\operatorname {cd}ots>b_u\ge 1$ if for all $i$ the box $(i,i)$ has arm length $a_i-1$ and leg length $b_i-1$. Let ${Q_{-1}}(m)$ be the set of partitions $\mathop{\text{Ch}}i$ with $|\mathop{\text{Ch}}i|=m$ whose Frobenius coordinates are of the form $(a_1,\leftarrowdots,a_u{{;}} a_1{+}1,\leftarrowdots,a_u{+}1)$. For partitions $\downarrowelta,\mathop{\text{Ch}}i$ such that $l(\downarrowelta)$, $l(\mathop{\text{Ch}}i)\leftarrowe r/2$ put $(\downarrowelta|\mathop{\text{Ch}}i):=(\downarrowelta_1,\leftarrowdots,\downarrowelta_{r/2},\mathop{\text{Ch}}i_1,\leftarrowdots,\mathop{\text{Ch}}i_{r/2})$ with the latter being viewed as a weight for $\operatorname {GL}(Q)$. \end{comment} From \cite[Corollary 3.16]{WeymanSam} we extract the following result (the role of $\operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)$ is played by the ring $A$ in loc.\ cit. and our $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes_k V^\vee)$ is denoted by $B$). \begin{proposition} \leftarrowabel{ref-3.2-14} Assume $\mathop{\text{Ch}}i$ is a partition with $l(\mathop{\text{Ch}}i)\leftarrowe r/2$. Then ${N_Q}(\mathop{\text{Ch}}i)$ has a $\operatorname {GL}(Q)$-equivariant free resolution as a $\operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)$-module which in homological degree $t\ge 0$ is the direct sum of $S^{\leftarrowambda} Q\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q)$ for $\leftarrowambda$ satisfying $(\tau_r(\leftarrowambda),i_r(\leftarrowambda))=(\mathop{\text{Ch}}i,t)$. \end{proposition} \begin{example} \leftarrowabel{ref-3.3-15} Write $[\mu_1,\mu_2,\leftarrowdots]$ for $S^{\mu} Q\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q)$. Assume $r=4$. Then the above resolution of ${N_Q}(a,b)$ has the form \[ 0\rightarrow [a,b,1,1]\rightarrow [a,b] \] if $b\ge 1$. If $b=0$ then the resolution has only one term given by $[a]$. \end{example} \begin{example} \leftarrowabel{ref-3.4-16} Assume $r=6$. Now the resolution of ${N_Q}(a,b,c)$ is \[ 0\rightarrow [a,b,c,2,2,2] \rightarrow [a,b,c,2,1,1] \rightarrow [a,b,c,1,1]\rightarrow [a,b,c] \] if $c\ge 2$. If $c=1$ then we have \[ 0\rightarrow [a,b,1,1,1]\rightarrow [a,b,1] \] If $c=0,b\ge 1$ we get \[ 0 \rightarrow [a,b,1,1,1,1]\rightarrow [a,b] \] Finally for $c=b=0$ the resolution has again only a single term given by $[a]$. \end{example} \begin{remark} \leftarrowabel{sam2} For $\mathop{\text{Ch}}i_{r/2}\ge r/2-1$ we give an explict description the resolution of $N_Q(\mathop{\text{Ch}}i)$ (including the differentials) in Appendix \rightarrowef{ref-A-50}. Steven Sam informed us of an alternative (and more general) approach as follows. There is an action of $\mathfrak{so}(Q+Q^*)$ on $\operatorname{Sym}} \def\Sp{\operatorname{Sp}(Q\otimes_k V^*)$, which commutes with the $\Sp(V)$-action. Therefore $\mathfrak{so}(Q+Q^*)$ acts on $N_Q(\mathop{\text{Ch}}i)$. The resolution of $N_Q(\mathop{\text{Ch}}i)$ in Proposition \rightarrowef{ref-3.2-14} can be upgraded to an $\mathfrak{so}(Q+Q^*)$-equivariant resolution, which is a BGG-resolution by parabolic Verma modules of the irreducible highest weight representation $N_Q(\mathop{\text{Ch}}i)$. This follows by \cite[Lemma 5.14, Theorem 5.15, Corollary 6.8]{EHP} since $N_Q(\mathop{\text{Ch}}i)$ is unitary \cite[Proposition 4.1]{ChengZhang}. In this way, using \cite[Section 5.3]{EHP} and \cite[Proposition 3.7]{Lepowsky}, one may in fact give an explicit description of the resolution of $N_Q(\mathop{\text{Ch}}i)$ also for general $\mathop{\text{Ch}}i$. However an analogue of the uniqueness claim of Proposition \rightarrowef{prop:uniqueness} is apparently not yet available in the literature. \end{remark} \begin{comment} is computed using Weyman's ``geometric method'' (see \cite[Lemma 3.11, Lemma 3.12, Prop.\ 3.13]{WeymanSam}). We do not need the definition of $M_\mathop{\text{Ch}}i$, just the shape of its resolution which we now describe. Let $G$ be the Grassmannian of $r/2$ dimensional quotients of $Q$ and let ${\cal P}$, ${\cal S}$ be respectively the universal quotient and subbundle on $G$. The resolution of $M_\mathop{\text{Ch}}i$ given by the geometric method is in homological degree $t\ge 0$ equal to \begin{equation} \leftarrowabel{ref-3.1-17} \bigoplus_{j\in \NN} H^j(G,\wedge^{t+j}(\wedge^2{\cal S})\otimes_G S^\mathop{\text{Ch}}i {\cal P}) \otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)\,. \end{equation} We have \begin{equation} \leftarrowabel{ref-3.2-18} \wedge^{k}(\wedge^2{\cal S})\cong \bigoplus_{\mu\in {Q_{-1}}(2k)} S^\mu{\cal S}\,. \end{equation} It now suffices to invoke Bott's theorem to see that the resolution \eqref{ref-3.1-17} has the same shape as the resolution introduced in the statement of Proposition \rightarrowef{ref-3.2-14}. \end{comment} Looking at the Examples \rightarrowef{ref-3.3-15}, \rightarrowef{ref-3.4-16} suggests the following easy consequence of Proposition \rightarrowef{ref-3.2-14} which is crucial for what follows: \begin{corollary} \leftarrowabel{ref-3.6-19} The summands of the resolution of ${N_Q}(\mathop{\text{Ch}}i)$ given in Proposition \rightarrowef{ref-3.2-14} are all of the form $S^\downarrowelta Q\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)$ with $\downarrowelta_1=\mathop{\text{Ch}}i_1$. \end{corollary} \begin{proof} Note first of all that $l(\downarrowelta)\leftarrowe r$ (otherwise $S^\downarrowelta Q=0$). A border strip $R$ of size $\leftarroweq 2 l(\leftarrowambda)-r-2$ starting at the first box in the bottom row of a partition $\leftarrowambda$ with $r\ge l(\leftarrowambda)>r/2$ has at most $2 l(\leftarrowambda)-r-2$ rows. So if we remove $R$ then the first $l(\leftarrowambda)-(2 l(\leftarrowambda)-r-2)=-l(\leftarrowambda)+r+2\ge 2$ rows of $\leftarrowambda$ are unaffected. If $S^\downarrowelta Q\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)$, $\downarrowelta\neq \mathop{\text{Ch}}i$, appears in the resolution of ${N_Q}(\mathop{\text{Ch}}i)$ then $\mathop{\text{Ch}}i$ is by Proposition \rightarrowef{ref-3.2-14} obtained from $\downarrowelta$ by a sequence of border strip removals as in the previous paragraph. Thus $\downarrowelta_1=\mathop{\text{Ch}}i_1$ (and also $\downarrowelta_2=\mathop{\text{Ch}}i_2$). \begin{comment} We assume $r\ge 4$ since otherwise the statement is trivial. Put $\rightarrowho=(r,r-1,\leftarrowdots,2,1)$. The twisted Weyl group action can be described as $w{\ast}\theta=w(\theta+\rightarrowho)-\rightarrowho$. In fact to compute $\theta^+$ one proceeds as follows: first compute $\theta'=\theta+\rightarrowho$. Order the components of $\theta'$ in descending order to obtain $\theta''$. If $\theta''$ contains repeated entries then $\theta$ is singular. Otherwise $\theta^+:=\theta''-\rightarrowho$. We apply this with $\theta=(\mathop{\text{Ch}}i{|}\mu)$ as in Proposition \rightarrowef{ref-3.2-14}. The fact $l(\mu)\leftarrowe r/2$ and $\mu\in \cup_k{Q_{-1}}(2k)$ implies $\mu_1\leftarrowe r/2-1$ (the arm length of $(1,1)$ in $\mu$ must be one less than the leg length). From this one computes $\theta'_1,\theta'_2\ge \theta'_{u}$ for $u\ge 2$. Indeed if $u\leftarrowe r/2$ then this follows the fact that $\mathop{\text{Ch}}i$ is a partition and if $u\ge r/2+1$ then it follows from the fact that $\theta'_1,\theta'_2\ge r-1$ and $\theta'_u=\theta_u+r-u+1=\mu_{u-r}+r-u+1\leftarrowe \mu_1+r-(r/2+1)+1\leftarrowe r-1$. Hence $\theta''_1=\theta'_1$, $\theta''_2=\theta_2'$ and thus if $\theta$ is regular $\theta^+_1=\theta'_1-r=\theta_1$ and similarly $\theta^+_2=\theta'_2-(r-1)=\theta_2$. \end{comment} \end{proof} \section{The Springer resolution} \leftarrowabel{ref-4-20} Let $\sigma:V\rightarrow V^\vee$, $\sigma+\sigma^\vee=0$ be the isomorphism corresponding to the symplectic form on $V$. Consider the following diagram. \begin{equation} \leftarrowabel{ref-4.1-21} \xymatrix{ {{E}}\ar[r]^{\tilde{p}}\ar[d]_{\tilde{q}}&X\ar[d]^q\\ Z\ar[r]_p\ar[d]_\pi&Y\ar@{^(->}[r]&\wedge^2 H^\vee\\ F } \end{equation} where $X=\operatorname {Hom}(H,V)$ is as above and \begin{equation} \leftarrowabel{ref-4.2-22} {{Y}}=\{\psi\in \operatorname {Hom}(H,H^\vee)\mid \psi+\psi^\vee=0,\operatorname {rk} \psi\leftarrowe r\}\subset \wedge^2 H^\vee\,, \end{equation} \[ F=\operatorname{Gr}(r,H):=\{\text{$r$-dimensional quotients of $H$}\}\,, \] \begin{equation} \leftarrowabel{ref-4.3-23} Z=\{(\phi,Q)\mid Q\in F,\phi\in \operatorname {Hom}(Q,Q^\vee),\phi+\phi^\vee=0\}\,, \end{equation} \begin{equation} \leftarrowabel{ref-4.4-24} {{E}}=\{(\epsilon,Q)\mid Q\in F, \epsilon\in \operatorname {Hom}(Q,V)\}\,. \end{equation} If $\theta:H\rightarrow V\in X$ then $q(\theta)\in {{Y}}$ is the composition \[ q(\theta)=[H\xrightarrow{\theta}V\xrightarrow{\sigma}V^\vee \xrightarrow{\theta^\vee} H^\vee]\,. \] If $(\phi,Q)\in Z$ then $p(\phi,Q)\in {{Y}}$ is the composition \[ p(\phi,Q)=[H\twoheadrightarrow Q\xrightarrow{\phi} Q^\vee \hookrightarrow H^\vee]\,. \] The map $\pi:Z\rightarrow F$ is the projection $(\phi,Q)\mapsto Q$. If $(\epsilon,Q)\in {{E}}$ then $\tilde{p}(\epsilon,Q)$ is the composition \[ [H\twoheadrightarrow Q\xrightarrow{\epsilon} V] \] and $\tilde{q}(\epsilon,Q)$ is $(\phi,Q)$ where $\phi$ is the composition \[ [Q\xrightarrow{\epsilon} V\xrightarrow{\sigma} V^\vee \xrightarrow{\epsilon^\vee} Q^\vee]\,. \] In the diagram \eqref{ref-4.2-22}, $X$, $Z$, $E$ are smooth, $p$ is a resolution of singularities and $\pi$ and $\pi\tilde{q}$ are vector bundles. The coordinate ring of $X$ is $T=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(H\otimes_k V^\vee)$. For the other schemes in \eqref{ref-4.1-21} we have \begin{align*} Y&=\operatorname {Spec} T^{\Sp(V)}\\ Z&=\uparrownderline{\operatorname {Spec}}_F {\cal Z}\\ {{E}}&=\uparrownderline{\operatorname {Spec}}_F {\cal E} \end{align*} with ${\cal Z}$, ${\cal E}$ being the sheaves of ${\cal O}_F$-algebras given by \begin{equation} \leftarrowabel{ref-4.5-25} \begin{aligned} {\cal Z}&=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_F(\wedge^2{\cal Q})\\ {\cal E}&=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_F({\cal Q}\otimes_k V^\vee) \end{aligned} \end{equation} where ${\cal Q}$ is the tautological quotient bundle on $F$. From \eqref{ref-4.5-25} obtain in particular \begin{lemma} \leftarrowabel{ref-4.1-26} If $U\subset F$ is an affine open then $\pi^{-1}(U)$ and $(\pi\tilde{q})^{-1}(U)$ are affine and moreover $k[\pi^{-1}(U)]= k[(\pi\tilde{q})^{-1}(U)]^{\Sp(V)}$. \end{lemma} Now let $Y_0$ be the open subscheme of $Y$ of those $\psi\in Y$ (see \eqref{ref-4.2-22}) which have rank exactly $r$ and put $X_0=q^{-1}(Y_0)$, $Z_0=p^{-1}(Y_0)$, ${{E}}_0=\tilde{p}^{-1}(X_0)=\tilde{q}^{-1}(Z_0)$. Then it is easy to see that $X_0\subset X=\operatorname {Hom}(H,V)$ is the open subscheme of those $\theta:H\rightarrow V$ which are surjective and that $Z_0\subset Z$ is the open subscheme of those $(\phi,Q)\in Z$ (see \eqref{ref-4.3-23}) where $\phi$ is an isomorphism. Finally ${{E}}_0$ is the open subscheme of ${{E}}$ of those $(\epsilon,Q)$ where $\epsilon$ is an isomorphism. The restricted morphisms $\tilde{q}_0:E_0\rightarrow Z_0$, $q_0:X_0\rightarrow Y_0$ are $\Sp(V)$-torsors and the restricted morphisms $\tilde{p}_0:E_0\rightarrow X_0$, $p_0:Z_0\rightarrow Y_0$ are isomorphisms. \section{A splitting functor} \leftarrowabel{ref-5-27} \subsection{Preliminaries} The idea of using splitting functors was suggested to us by Sasha Kuznetsov. Recall that a (full) triangulated subcategory of a triangulated category is right admissible if the inclusion functor has a right adjoint. Following \cite[Def.\ 3.1]{Kuznetsov3} we say that a functor $\Phi:{\cal B}\rightarrow {\cal A}$ is right splitting if $\operatorname {ker} \Phi$ is right admissible in~${\cal B}$,~$\Phi$ restricted to $(\operatorname {ker} \Phi)^\perp$ is fully faithful and finally $\operatorname {im}\Phi=\Phi(\operatorname {ker}\Phi^\perp)$ is right admissible in ${\cal A}$. Left splitting functors are defined in a similar way. We see that splitting functors are categorical versions of partial isometries between Hilbert space. According to \cite[Lem.\ 3.2, Cor.\ 3.4]{Kuznetsov3} a right splitting functor $\Phi$ has a right adjoint~$\Phi^!$ which is a left splitting functor. According to \cite[Thm 3.3(3r)]{Kuznetsov3} if $\Phi$ is right splitting then~$\Phi$ and $\Phi^!$ induce inverse equivalences between $\operatorname {im} \Phi\subset {\cal A}$ and $\operatorname {im} \Phi^!\subset {\cal B}$. Below we will use the following criterion to verify that a certain functor is splitting. \begin{lemmas} \leftarrowabel{ref-5.1.1-28} Assume that $\Phi:{\cal B}\rightarrow {\cal A}$ is an exact functor between triangulated categories. Assume that $\Phi$ has a right adjoint $\Phi^!$ such that the composition of the counit map $\Phi \Phi^!\rightarrow \operatorname{id}_{{\cal A}}$ with $\Phi$ yields a natural isomorphism $\Phi \Phi^!\Phi\rightarrow \Phi$. Then $\Phi$ is a right splitting functor. \end{lemmas} \begin{proof} This is equivalent to the criterion \cite[Thm 3.3(4r)]{Kuznetsov3}. In the latter case we start from the unit map $\operatorname{id}_{{\cal A}} \rightarrow \Phi\Phi^! $ and we require that the resulting $\Phi\rightarrow \Phi \Phi^!\Phi$ is an isomorphism. As the composition $\Phi\rightarrow \Phi \Phi^!\Phi \rightarrow \Phi$ is the identity, it follows that if one of these maps is an isomorphism then so is the other. \end{proof} \subsection{The functor} \leftarrowabel{ref-5.2-29} The diagram \eqref{ref-4.1-21} may be transformed into a diagram of quotient stacks \[ \xymatrix{ {{E/\Sp(V)}}\ar[r]^{\tilde{p}_s}\ar[d]_{\tilde{q}_s}&X/\Sp(V)\ar[d]^{q_s}\\ Z\ar[r]_p\ar[d]_\pi&Y\ar@{^(->}[r]&\wedge^2 H^\vee\\ F } \] which is compatible with the natural maps $E\rightarrow E/\Sp(V)$, $X\rightarrow X/\Sp(V)$. This means in particular that $L\tilde{q}^\ast_s$, $Lq^\ast_s$, $R\tilde{p}_{s,\ast}$, $L\tilde{p}_{s}^\ast$, $\tilde{p}^!_{s,\ast}$ may be computed like their non-stacky counterparts. We will use this without further comment. We define the functor $\Phi$ as the composition \[ \Phi:{\cal D}(Z)\xrightarrow{L\tilde{q}^\ast_s} {\cal D}(E/\Sp(V)) \xrightarrow{R\tilde{p}_{s,\ast}} {\cal D}(X/\Sp(V)) \] The functor $\Phi$ has a right adjoint $\Phi^!$ given by the composition \[ \Phi^!:{\cal D}(X/\Sp(V))\xrightarrow{\tilde{p}^!_s} {\cal D}(E/\Sp(V))\xrightarrow{R\tilde{q}_{s\ast}} {\cal D}(Z) \] where $ \tilde{p}_s^!=\omega_{E/X}\otimes_E L\tilde{p}_s^\ast(-) $ and $\tilde{q}_{s\ast}$ is given by taking $\Sp(V)$-invariants. From Lemma \rightarrowef{ref-4.1-26} and the fact that $\Sp(V)$ is reductive it follows that $\tilde{q}_{s\ast}$ is an exact functor. \begin{theorems} \leftarrowabel{ref-5.2.1-30} \begin{enumerate} \item $\Phi$ is a right splitting functor. \leftarrowabel{ref-1-31} \item $\operatorname {im} \Phi$ is the smallest triangulated subcategory of ${\cal D}(X/\Sp(V))$ containing $S^{\leftarrowangle \leftarrowambda\rightarrowangle} V\otimes_k {\cal O}_X$ for $\leftarrowambda\in B_{r/2,n-r}$. \leftarrowabel{ref-2-32} \item $\operatorname {im} \Phi^!$ is the smallest triangulated subcategory of ${\cal D}(Z)$ containing ${\cal M}_Z(\leftarrowambda)$ for $\leftarrowambda\in B_{r/2,n-r}$. \leftarrowabel{ref-3-33} \item For $\leftarrowambda\in B_{r,n-r}$ we have \leftarrowabel{ref-4-34} \[ \Phi(\pi^\ast((\downarrowet {\cal Q})^{\otimes r-n} \otimes_F S^\leftarrowambda {\cal Q}))\cong S^\leftarrowambda V\otimes_k {\cal O}_X\,. \] \item For $\leftarrowambda\in B_{r/2,n-r}$ we have \leftarrowabel{ref-5-35} \[ \Phi({\cal M}_Z(\leftarrowambda))\cong S^{\leftarrowangle \leftarrowambda\rightarrowangle} V\otimes_k{\cal O}_X\,. \] \item For $\leftarrowambda\in B_{r/2,n-r}$ we have \leftarrowabel{ref-6-36} \[ \Phi^!(S^{\leftarrowangle \leftarrowambda\rightarrowangle} V\otimes_k{\cal O}_X)\cong {\cal M}_Z(\leftarrowambda). \] \end{enumerate} \end{theorems} The proof is based on a series of lemmas. Most arguments are quite standard. See \cite{VdB100, WeymanBook}. \begin{lemmas}\leftarrowabel{ref-5.2.2} \begin{enumerate} \item We have \begin{equation} \leftarrowabel{ref-5.1-37} \omega_{E/X}= (\pi\tilde{q})^\ast (\downarrowet {\cal Q})^{\otimes r-n} \end{equation} as $\operatorname {GL}(H)\times\Sp(V)$-equivariant coherent sheaves. \item Moreover \begin{equation} \leftarrowabel{ref-5.2-38} R\tilde{p}_{s,\ast} \omega_{E/X}={\cal O}_X. \end{equation} \end{enumerate} \end{lemmas} \begin{proof} \begin{enumerate} \item For clarity we will work $\operatorname {GL}(H)\times \operatorname {GL}(V)$-equivariantly. Using the identification $E=\uparrownderline{\operatorname {Spec}} {\cal E}$ with ${\cal E}=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_F({\cal Q}\otimes_k V^\vee)$ (see \eqref{ref-4.5-25}) we find that~$\omega_E$ corresponds to the sheaf of graded ${\cal E}$-modules given by \[ \omega_{{\cal E}}=\omega_{F}\otimes_F\downarrowet ({\cal Q}\otimes_k V^\vee)\otimes_F{\cal E}\,. \] From the fact that $\Omega_F=\operatorname {\mathcal{H}\mathit{om}}_F({\cal Q},{\cal R})$ where ${\cal R}=\operatorname {ker}(H\otimes_k{\cal O}_F\rightarrow {\cal Q})$ one computes \[ \omega_F=(\downarrowet H)^{\otimes r}\otimes_F(\downarrowet {\cal Q})^{\otimes -n}\,. \] We also have \[ \downarrowet({\cal Q}\otimes_k V^\vee)=(\downarrowet Q)^{\otimes r}\otimes_k (\downarrowet V)^{\otimes -r} \] so that ultimately we get \[ \omega_{\cal E}=(\downarrowet H)^{\otimes r}\otimes_k (\downarrowet V)^{\otimes -r}\otimes_k (\downarrowet {\cal Q})^{\otimes r-n}\otimes_F {\cal E}\, \] and hence \[ \omega_E=(\downarrowet H)^{\otimes r}\otimes_k (\downarrowet V)^{\otimes -r}\otimes_k(\pi\tilde{q})^\ast (\downarrowet {\cal Q})^{\otimes r-n}. \] One also has $\omega_X=(\downarrowet H)^{\otimes r}\otimes_k (\downarrowet V)^{\otimes -n}\otimes_k {\cal O}_X$ which yields \[ \omega_{E/X}=(\downarrowet V)^{\otimes n-r}\otimes_k(\pi\tilde{q})^\ast (\downarrowet {\cal Q})^{\otimes r-n}. \] It now suffices to note that $\downarrowet V$ is a trivial $\Sp(V)$-representation. \item It is easy to show this directly from \eqref{ref-5.1-37} but one may also argue that~$X$, being smooth, has rational singularities and hence $R\tilde{p}_{s,\ast}(\omega_E)=\omega_X$. Tensoring with $\omega_X^{-1}$ yields the desired result.}\end{proof \end{enumerate} \downarrowef}\end{proof{}\end{proof} On $E$ there is a tautological map \[ \epsilon:(\pi\tilde{q})^\ast({\cal Q})\rightarrow V\otimes_k {\cal O}_E \] whose fiber in a point $(\epsilon,Q)\in E$ is simply $\epsilon:Q\rightarrow V$. From this description it is clear that $\epsilon{|}E_0$ is an isomorphism. \begin{lemmas} \leftarrowabel{ref-5.2.3-39} Assume $\leftarrowambda\in B_{r,n-r}$. The map $S^\leftarrowambda \epsilon$ becomes an isomorphism after applying the functor $R\tilde{p}_\ast (\omega_{E/X}\otimes_E-)$. \end{lemmas} \begin{proof} By \eqref{ref-5.2-38} we have \begin{equation} \leftarrowabel{ref-5.3-40} R\tilde{p}_\ast(\omega_{E/X}\otimes_E(S^\leftarrowambda V\otimes_k {\cal O}_E))= S^\leftarrowambda V\otimes_k R\tilde{p}_\ast(\omega_{E/X})= S^\leftarrowambda V\otimes_k {\cal O}_X. \end{equation} When viewed as $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(H\otimes_k V^\vee)$-module $R^i\tilde{p}_\ast(\omega_{E/X}\otimes_E S^\leftarrowambda((\pi\tilde{q}_s)^\ast({\cal Q})))$ is given by \begin{equation} \leftarrowabel{ref-5.4-41} H^i(F,S^\leftarrowambda {\cal Q} \otimes_F (\downarrowet {\cal Q})^{\otimes r-n}\otimes_F {\cal E}) \end{equation} (using \eqref{ref-5.1-37}). It follows from \cite[Prop.\ 1.4]{BLV1000} that \eqref{ref-5.4-41} is zero for $i>0$. So $R^i\tilde{p}_\ast(\omega_{E/X}\otimes_E S^\leftarrowambda((\pi\tilde{q}_s)^\ast({\cal Q})))=0$ for $i>0$. We now consider $i=0$. We claim that $\tilde{p}_\ast(\omega_{E/X}\otimes_E S^\leftarrowambda((\pi\tilde{q}_s)^\ast({\cal Q}))$ is maximal Cohen-Macaulay. To this end we have to show that $\operatorname {RHom}_X(\tilde{p}_\ast(\omega_{E/X}\otimes_E S^\leftarrowambda((\pi\tilde{q}_s)^\ast({\cal Q})),{\cal O}_X)$ has no higher cohomology or equivalently $\operatorname {Ext}^i_E( S^\leftarrowambda((\pi\tilde{q})^\ast({\cal Q})),{\cal O}_E)=0$ for $i>0$. In other words we should have \[ H^i(F,(S^\leftarrowambda {\cal Q})^\vee \otimes_F {\cal E})=0 \] for $i>0$. This follows again from \cite[Prop.\ 1.4]{BLV1000}. Combining this with \eqref{ref-5.3-40} we see that $R\tilde{p}_\ast(\omega_{E/X}\otimes_X S^\leftarrowambda\epsilon)$ is a map between maximal Cohen-Macaulay ${\cal O}_X$-modules. Since this map is an isomorphism on $X_0$ and $\codim (X-X_0)\ge 2$ we conclude that $R\tilde{p}_\ast(\omega_{E/X}\otimes_X S^\leftarrowambda\epsilon)$ is indeed an isomorphism. \end{proof} Put ${\cal N}_Z(\leftarrowambda):=\widetilde{N_Q(\leftarrowambda)}$ where the notation $\tilde{?}$ was introduced in the introduction and $N_Q(\leftarrowambda)$ was introduced in \S\rightarrowef{ref-3-12}. From Lemma \rightarrowef{ref-4.1-26} we deduce \begin{equation} \leftarrowabel{ref-5.5-42} {\cal N}_Z(\leftarrowambda)\cong R\tilde{q}_{s,\ast}(S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_E) \end{equation} so that by adjunction we get a map \begin{equation} \leftarrowabel{ref-5.6-43} L\tilde{q}^\ast_s{\cal N}_Z(\leftarrowambda)\rightarrow S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_E. \end{equation} \begin{lemmas} \leftarrowabel{ref-5.2.4-44} Assume $\leftarrowambda\in B_{r/2,n-r}$. The map \eqref{ref-5.6-43} becomes an isomorphism after applying the functor $R\tilde{p}_{s,\ast}(\omega_{E/X}\otimes_E-)$. \end{lemmas} \begin{proof} Note that \eqref{ref-5.6-43} is an isomorphism on $E_0$ since $E_0\rightarrow Z_0$ is an $\Sp(V)$-torsor and so $L\tilde{q}^\ast_s$ and $R\tilde{q}_{s,\ast}$ define inverse equivalences between ${\cal D}(E_0/\Sp(V))$ and ${\cal D}(Z_0)$. By Corollary \rightarrowef{ref-3.6-19} we have a $\operatorname {GL}(H)$-equivariant resolution \[ \operatorname {cd}ots \rightarrow P_1(\pi^\ast{\cal Q})\rightarrow P_0(\pi^\ast{\cal Q})\rightarrow {\cal N}_Z(\leftarrowambda)\rightarrow 0 \] where the $P_i$ are polynomial functors which are finite sums of Schur functors $S^\mathop{\text{Ch}}i$ with $\mathop{\text{Ch}}i\in B_{r,n-r}$. It follows that the cone of \eqref{ref-5.6-43} is described by a $\operatorname {GL}(H)\times \Sp(V)$-equivariant complex of the form \begin{equation} \leftarrowabel{ref-5.7-45} \operatorname {cd}ots \rightarrow P_1((\pi\tilde{q})^\ast{\cal Q})\rightarrow P_0((\pi\tilde{q})^\ast{\cal Q})\rightarrow S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_E\rightarrow 0 \end{equation} and moreover this complex is exact when restricted to $E_0$. Using Lemma \rightarrowef{ref-5.2.3-39} and \eqref{ref-5.2-38} applying $R\tilde{p}_{\ast}(\omega_{E/X}\otimes_X-)$ to \eqref{ref-5.7-45} yields a $\operatorname {GL}(H)\times \Sp(V)$-equivariant complex on $X$ \begin{equation} \leftarrowabel{ref-5.8-46} \operatorname {cd}ots \rightarrow P_1(V)\otimes_k {\cal O}_X \rightarrow P_0(V)\otimes_k {\cal O}_X\rightarrow S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_X\rightarrow 0 \end{equation} This complex is exact on $X_0$ (since $X_0\cong E_0$) but we must prove it is exact on $X$. The morphisms in \eqref{ref-5.8-46} are determined by $\operatorname {GL}(H)\times\Sp(V)$-equivariant maps \begin{align*} P_{i+1}(V)&\rightarrow P_i(V)\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(H\otimes_k V^\vee)\\ P_{0}(V)&\rightarrow S^{\leftarrowangle\leftarrowambda\rightarrowangle}(V)\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(H\otimes_k V^\vee) \end{align*} which by $\operatorname {GL}(H)$-equivariance must necessarily be obtained from $\Sp(V)$-equivariant maps \begin{align*} P_{i+1}(V)&\rightarrow P_i(V)\\ P_{0}(V)&\rightarrow S^{\leftarrowangle\leftarrowambda\rightarrowangle}(V) \end{align*} We conclude that \eqref{ref-5.8-46} is of the form \begin{equation} \leftarrowabel{ref-5.9-47} (\operatorname {cd}ots \rightarrow P_2(V)\rightarrow P_1(V)\rightarrow P_0(V)\rightarrow S^{\leftarrowangle\leftarrowambda\rightarrowangle} V\rightarrow 0)\otimes_k {\cal O}_X \end{equation} in a way which is compatible with $\operatorname {GL}(H)\times \Sp(V)$-actions. Restricting to $X_0$ we see that \[ \operatorname {cd}ots \rightarrow P_2(V)\rightarrow P_1(V)\rightarrow P_0(V)\rightarrow S^{\leftarrowangle\leftarrowambda\rightarrowangle} V\rightarrow 0 \] must be exact. But then \eqref{ref-5.9-47} is also exact and hence so is \eqref{ref-5.8-46}. \end{proof} \begin{lemmas} \leftarrowabel{ref-5.2.5-48} Let $\leftarrowambda\in B_{r/2,n-r}$. The counit map \[ \Phi\Phi^!(S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_X) \rightarrow S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_X \] is an isomorphism. \end{lemmas} \begin{proof} We have \[ \tilde{p}_{s}^!(S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_X) =S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k \omega_{E/X}. \] Hence we have to show that the counit map \[ L\tilde{q}_s^\ast R{\tilde{q}}_{s,\ast} (S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k \omega_{E/X}) \rightarrow S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k \omega_{E/X} \] becomes an isomorphism after applying $R\tilde{p}_{s,\ast}$. Using \eqref{ref-5.1-37} we see that it is sufficient to prove that \[ L\tilde{q}_s^\ast R{\tilde{q}}_{s,\ast} (S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_E) \rightarrow S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_E \] becomes an isomorphism after applying $R\tilde{p}_{s,\ast}(\omega_{E/X}\otimes_X-)$. This is precisely Lemma \rightarrowef{ref-5.2.4-44}. \end{proof} \begin{proof}[Proof of Theorem \rightarrowef{ref-5.2.1-30}] \begin{enumerate} \item[\eqref{ref-6-36}] We have by \eqref{ref-5.1-37} and \eqref{ref-5.5-42} \begin{align*} \Phi^!(S^{\leftarrowangle \leftarrowambda\rightarrowangle} V\otimes_k{\cal O}_X)&=R\tilde{q}_{s,\ast}(\omega_{E/X}\otimes_E (S^{\leftarrowangle \leftarrowambda\rightarrowangle} V\otimes_k{\cal O}_E))\\ &=\pi^\ast(\downarrowet {\cal Q})^{\otimes r-n} \otimes_Z {\cal N}_Z(\leftarrowambda)\\ &={\cal M}_Z(\leftarrowambda). \end{align*} \item[\eqref{ref-5-35}] Using \eqref{ref-5.1-37}\eqref{ref-5.2-38} and Lemma \rightarrowef{ref-5.2.4-44} we have \begin{align*} \Phi({\cal M}_Z(\leftarrowambda))&=R\tilde{p}_{s,\ast}(\omega_{E/X} \otimes_E L\tilde{q}_{s}^\ast {\cal N}_Z(\leftarrowambda))\\ &=S^{\leftarrowangle\leftarrowambda\rightarrowangle} V\otimes_k R\tilde{p}_{s,\ast}\omega_{E/X}\\ &=S^{\leftarrowangle\leftarrowambda\rightarrowangle} V\otimes_k {\cal O}_X. \end{align*} \item[\eqref{ref-4-34}] Using \eqref{ref-5.1-37}\eqref{ref-5.2-38} and Lemma \rightarrowef{ref-5.2.3-39} we have \begin{align*} \Phi(\pi^\ast((\downarrowet {\cal Q})^{\otimes r-n} \otimes_F S^\leftarrowambda {\cal Q}))&=R\tilde{p}_{s,\ast}(\omega_{E/X}\otimes_E L(\pi\tilde{q}_s)^\ast(S^\leftarrowambda {\cal Q}))\\ &=S^\leftarrowambda V\otimes_k R\tilde{p}_{s,\ast}(\omega_{E/X})\\ &=S^\leftarrowambda V\otimes_k {\cal O}_X. \end{align*} \item[\eqref{ref-1-31}] We use Lemma \rightarrowef{ref-5.1.1-28}. So we have to prove that the counit map $\Phi\Phi^{!}(A)\rightarrow A$ is an isomorphism for every object of the form $A=\Phi(B)$ with $B\in {\cal D}(Z)$. It is clearly sufficient to check this for $B$ running through a set of generators of ${\cal D}(Z)$. The sheaves $(\downarrowet {\cal Q})^{\otimes r-n}\otimes_F S^\leftarrowambda{\cal Q}$ for $\leftarrowambda\in B_{r,n-r}$ generate ${\cal D}(F)$ \cite{Kapranov3}. Hence since $Z\rightarrow F$ is affine it follows that the sheaves $\pi^\ast((\downarrowet {\cal Q})^{\otimes r-n}\otimes_F S^\leftarrowambda{\cal Q})$ generate ${\cal D}(Z)$. By \eqref{ref-4-34} we have $\Phi(\pi^\ast((\downarrowet {\cal Q})^{\otimes r-n} \otimes_F S^\leftarrowambda {\cal Q})) \cong S^\leftarrowambda V\otimes_k {\cal O}_X$ and $S^\leftarrowambda V$ is a sum of $S^{\leftarrowangle\mu\rightarrowangle}V$ with $\mu_1\leftarrowe\leftarrowambda_1$, for example by careful inspection of the formula \cite[\S2.4.2]{HTW}. It now suffices to invoke Lemma \rightarrowef{ref-5.2.5-48} (or, with a bit of handwaving, \eqref{ref-5-35}\eqref{ref-6-36}). \item[\eqref{ref-2-32}] This has been proved as part of \eqref{ref-1-31}. \item[\eqref{ref-3-33}] By \cite[Thm 3.3(3r)]{Kuznetsov3} it follows that $\operatorname {im} \Phi^!=\Phi^!(\operatorname {im} \Phi)$. It now suffices to invoke \eqref{ref-2-32}\eqref{ref-6-36}.}\end{proof \end{enumerate} \downarrowef}\end{proof{}\end{proof} \begin{proof}[Proof of Theorem \rightarrowef{ref-1.2-6}] \begin{enumerate} \item Since by Theorem \rightarrowef{ref-5.2.1-30}(6) ${\cal M}_Z(\leftarrowambda),{\cal M}_Z(\mu)\in \operatorname {im} \Phi^!$ we have by Theorem \rightarrowef{ref-5.2.1-30}(5) \begin{align*} \operatorname {Ext}^i_Z({\cal M}_Z(\leftarrowambda),{\cal M}_Z(\mu))&=\operatorname {Ext}^i_{X/\Sp(V)}(\Phi({\cal M}_Z(\leftarrowambda)),\Phi({\cal M}_Z(\mu)))\\ &=\operatorname {Ext}^i_{X/\Sp(V)}(S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_X,S^{\leftarrowangle \mu\rightarrowangle}V\otimes_k {\cal O}_X) \end{align*} which is zero for $i>0$ (since $\Sp(V)$ is reductive). Note that we also find \begin{equation} \leftarrowabel{ref-5.10-49} \begin{aligned} \operatorname {Hom}_Z({\cal M}_Z(\leftarrowambda),{\cal M}_Z(\mu))&= \operatorname {Hom}_X(S^{\leftarrowangle \leftarrowambda\rightarrowangle}V\otimes_k {\cal O}_X,S^{\leftarrowangle \mu\rightarrowangle}V\otimes_k {\cal O}_X)^{\Sp(V)}\\ &\cong\operatorname {Hom}_R(M(\leftarrowambda),M(\mu)) \end{aligned} \end{equation} by \cite[Lemma 4.1.3]{SpenkoVdB}. \item We have by \eqref{ref-5.1-37}\eqref{ref-5.2-38} \begin{align*} Rp_\ast {\cal M}_Z(\leftarrowambda)&=Rp_\ast R\tilde{q}_{s,\ast}(\omega_{E/X}\otimes_k S^{\leftarrowangle \leftarrowambda\rightarrowangle }V)\\ &=Rq_{s,\ast} R\tilde{p}_{s,\ast}(\omega_{E/X}\otimes_k S^{\leftarrowangle \leftarrowambda\rightarrowangle }V)\\ &=Rq_{s,\ast}( S^{\leftarrowangle \leftarrowambda\rightarrowangle }V\otimes_k R\tilde{p}_{s,\ast}(\omega_{E/X}))\\ &=Rq_{s,\ast} (S^{\leftarrowangle \leftarrowambda\rightarrowangle }V\otimes_k {\cal O}_X)\\ &=(S^{\leftarrowangle \leftarrowambda\rightarrowangle }V\otimes_k {\cal O}_X)^{\Sp(V)} \end{align*} Taking global sections yields what we want. \item By \eqref{ref-5.10-49} and \eqref{ref-1.5-8} both sides of \eqref{ref-1.6-10} are reflexive $R$-modules. Since~$p_\ast$ induces an isomorphism on $Y_0$ between both sides of \eqref{ref-1.6-10} (viewed as sheaves on $Y$) and $\codim(Y-Y_0)\ge 2$ \eqref{ref-1.6-10} must be an isomorphism. }\end{proof \end{enumerate} \downarrowef}\end{proof{}\end{proof} \section{Symmetric matrices}\leftarrowabel{symsec} In this section we present modification needed to treat determinantal varieties of symmetric matrices. We keep the same notation as in the introduction, but now we equip $V$ with a symmetric bilinear form so that $r=\downarrowim V$ does not need to be even, $Y$ is the variety of $n\times n$ symmetric matrices of rank $\leftarroweq r$, $G=O(V)$, while $X=\operatorname {Hom}(H,V)$, $T=\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(H\otimes V^\vee)$ remain the same, put $R=T^{O(V)}$. By the fundamental theorems for the orthogonal group we have $Y\cong \operatorname {Spec} R$. If $\mathop{\text{Ch}}i$ is a partition with $\mathop{\text{Ch}}i^t_1+\mathop{\text{Ch}}i^t_2\leftarroweq r$, where $\mathop{\text{Ch}}i^t$ denotes the transpose partition, we write $S^{[\mathop{\text{Ch}}i]}V$ for the corresponding irreducible representation of $O(V)$ (see \cite[\S 19.5]{FH}), and call such a partition admissible. By $\mathop{\text{Ch}}i^\sigma$ we denote the conjugate partition of $\mathop{\text{Ch}}i$; i.e., $(\mathop{\text{Ch}}i^\sigma)^t_1=r-\mathop{\text{Ch}}i^t_1$, $(\mathop{\text{Ch}}i^\sigma)^t_k=\mathop{\text{Ch}}i^t_k$ for $k>1$. Note that either $l(\mathop{\text{Ch}}i)\leftarrowe r/2$ or $l(\mathop{\text{Ch}}i^\sigma)\leftarrowe r/2$. We have $S^{[\leftarrowambda^\sigma]}V=\downarrowet V\otimes_k S^{[\leftarrowambda]} V$ \cite[\S6.6, Lemma 2]{Procesi3}. In \cite{SpenkoVdB} a non-commutative resolution of $R$ has been constructed, which is crepant in case $n$ and $r$ have opposite parity. Let $B_{k,l}^{a}$ denote the set admissible partitions in $B_{k,l}$. We put \begin{equation} \leftarrowabel{ref-6-1} M=\bigoplus_{{{\mathop{\text{Ch}}i}}\in B_{r,\leftarrowfloor (n-r)/2\rightarrowfloor +1 }^{a}} M({{\mathop{\text{Ch}}i}}), \end{equation} where $M(\mathop{\text{Ch}}i)=(S^{[\mathop{\text{Ch}}i]} V\otimes_k T)^{O(V)}$ and write $\Lambda=\operatorname {End}_R(M)$. \begin{theorem} One has $\operatorname {gl\,dim}\Lambda<\infty$. $\Lambda$ is a non-commutative crepant resolution of $R$ if $n$ and $r$ have opposite parity.\footnote{In case $n$, $r$ have the same parity then there is a \emph{twisted} non-commutative crepant resolution. We do not consider such resolutions in this paper.} \end{theorem} In the symmetric case we also have an analogous Springer resolution where we adapt the definitions in the obvious way. The fundamental theorems for the orthogonal group yield $\operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes V)^{O(V)}\cong \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\operatorname{Sym}} \def\Sp{\operatorname{Sp}^2(Q))$. We only slightly change the definition of $M_Q(\mathop{\text{Ch}}i)$, now \[ M_Q({{\mathop{\text{Ch}}i}})=\downarrowet(V)^{\gamma_{r,n}}\otimes(\downarrowet Q)^{\otimes r-n}\otimes_k (S^{[ {{\mathop{\text{Ch}}i}}]} V\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(Q\otimes_k V^\vee))^{O(V)}, \] where $\gamma_{r,n}=0$ (resp. $\gamma_{r,n}=1$) if $r$ and $n$ have the same (resp. opposite) parity. As in the skew symmetric case ${\cal M}_{Z}({{\mathop{\text{Ch}}i}})=\widetilde{M_Q({{\mathop{\text{Ch}}i}})}\in \mathop{\text{\uparrowpshape{coh}}}(\operatorname {GL}(H),Z)$. To give an analogue of Proposition \rightarrowef{ref-3.2-14} we need to adapt the definitions of $\tau_r(\leftarrowambda)$, $i_r(\leftarrowambda)$ following \cite[\S 4.4]{WeymanSam}. The differences (denoted by D1, D2, D3 in loc.\ cit.) are that we remove border strips $R_\leftarrowambda$ of size $2 l(\leftarrowambda)-r$ instead of $2l(\leftarrowambda)-r-2$ and in the definition of $i_r(\leftarrowambda)$ we use $c(R_\leftarrowambda)-1$ instead of $c(R_\leftarrowambda)$. Finally if the total number of border strips removed is odd, then we replace the end result $\mu$ with $\mu^\sigma$. With these modifications and replacing $B_{r/2,n-r}$ by $B_{r,n-r}^{a}$ Proposition \rightarrowef{ref-3.2-14} remains true also in the symmetric case by \cite[Corollary 4.23]{WeymanSam} in the case $r$ is odd, and by \cite[(4.2), Theorem 4.4]{WeymanSam} in the case $r$ is even. Also Corollary \rightarrowef{ref-3.6-19} remains valid. In its proof we only need to additionally note that one can also remove a border strip of size $l(\leftarrowambda)$ (which affects the first row) but this can only happen in the case $\leftarrowambda=(1^r)$ and in this case, since the number of borders strips removed is odd, $\tau_r(\leftarrowambda)=(0)^\sigma=\leftarrowambda$. In particular, $\tau_r(\leftarrowambda)_1=\leftarrowambda_1$ still holds. We now present modifications needed in statements of other results. \begin{itemize} \item In Theorem \rightarrowef{ref-1.2-6} we replace $B_{r/2,n-r}$ by $B_{r,n-r}^{a}$. \item In Theorem \rightarrowef{ref-5.2.1-30} we replace $S^{\leftarrowangle \leftarrowambda\rightarrowangle}V$ by $S^{[ {{\leftarrowambda}}]} V$, and $B_{r/2,n-r}$ by $B_{r,n-r}^{a}$. Item (4) needs to be modified as \[ \Phi(\pi^\ast((\downarrowet {\cal Q})^{\otimes r-n} \otimes_F S^\leftarrowambda {\cal Q}))\cong S^\leftarrowambda V\otimes_k (\downarrowet V)^{\gamma_{r,n}}\otimes_k {\cal O}_X\,. \] \item In Lemma \rightarrowef{ref-5.2.2} we have \[ \omega_{E/X}=(\downarrowet V)^{\gamma_{r,n}}\otimes (\pi\tilde{q})^\ast (\downarrowet {\cal Q})^{\otimes r-n} \] as $\operatorname {GL}(H)\times O(V)$-equivariant coherent sheaves. \end{itemize} One can easily check that the proofs obtained in the skew symmetric case also apply almost verbatim in the symmetric case. \appendix \section{More on the resolution of ${N_Q}(\mathop{\text{Ch}}i)$ in the symplectic case} \leftarrowabel{ref-A-50} We refer to Remark \rightarrowef{sam2} for an alternative approach, suggested to us by Steven Sam, towards the results in this Appendix. We believe that our elementary arguments are still of independent interest. Recall that a partition has Frobenius coordinates $(a_1,\leftarrowdots,a_u;b_1,\leftarrowdots,b_u)$, $a_1>\operatorname {cd}ots>a_u\ge 1$, $b_1>\operatorname {cd}ots>b_u\ge 1$ if for all $i$ the box $(i,i)$ has arm length $a_i-1$ and leg length $b_i-1$. Let ${Q_{-1}}(m)$ be the set of partitions $\mathop{\text{Ch}}i$ with $|\mathop{\text{Ch}}i|=m$ whose Frobenius coordinates are of the form $(a_1,\leftarrowdots,a_u{{;}} a_1{+}1,\leftarrowdots,a_u{+}1)$. For partitions $\downarrowelta,\mathop{\text{Ch}}i$ such that $l(\downarrowelta)$, $l(\mathop{\text{Ch}}i)\leftarrowe r/2$ put $(\downarrowelta|\mathop{\text{Ch}}i):=(\downarrowelta_1,\leftarrowdots,\downarrowelta_{r/2},\mathop{\text{Ch}}i_1,\leftarrowdots,\mathop{\text{Ch}}i_{r/2})$ with the latter being viewed as a weight for $\operatorname {GL}(Q)$. For $\alpha\in {Q_{-1}}(2k)$, $\beta\in {Q_{-1}}(2(k-1))$, $l(\alpha),l(\beta)\leftarrowe r/2$ we put $\beta\subset_2 \alpha$ if $\beta\subset \alpha$ and $\alpha/\beta$ does not consist of two boxes next to each other. For $\mathop{\text{Ch}}i$ a partition with $l(\mathop{\text{Ch}}i)\leftarrowe r/2$ and $\mathop{\text{Ch}}i_{r/2}\ge r/2-1$ put \[ S_{\mathop{\text{Ch}}i,k}=\{(\mathop{\text{Ch}}i|\mu)\mid \mu\in {Q_{-1}}(2k), l(\mu)\leftarrowe r/2\}\,. \] Note that if $\mu\in {Q_{-1}}(2k)$ and $l(\mu)\leftarrowe r/2$ then $\mu_1\leftarrowe r/2-1$. Hence all elements of $S_{\mathop{\text{Ch}}i,k}$ are dominant. For $\pi=(\mathop{\text{Ch}}i|\alpha)\in S_{\mathop{\text{Ch}}i,k}$, $\tau=(\mathop{\text{Ch}}i|\beta)\in S_{\mathop{\text{Ch}}i,k-1}$ put $\tau\subset_2\pi$ if $\beta\subset_2\alpha$. If $\tau\subset_2\pi$ then by the Pieri rule $S^\tau Q$ is a summand with multiplicity one of $\wedge^2 Q\otimes_k S^\pi Q$. We call any non-zero $\operatorname {GL}(Q)$-equivariant map \[ \phi_{\pi,\tau}:S^\pi Q\rightarrow \wedge^2 Q\otimes_k S^\tau Q \] a Pieri map. Needless to say that a Pieri map is only determined up to a non-zero scalar. By analogy of \cite[\S7]{VdB100} we call a collection of Pieri-maps $\phi_{\pi,\tau}$ such that $\tau\subset_2 \pi$ a Pieri system. We say that two Pieri systems $\phi_{\pi,\tau}$, $\phi'_{\pi,\tau}$ are equivalent if there exist non-zero scalars $(c_\sigma)_\sigma$ such that \[ \phi'_{\pi,\tau}=\frac{c_\tau}{c_\pi}\phi_{\pi,\tau}\, \] for all $\pi,\tau$. We will now make Proposition \rightarrowef{ref-3.2-14} more explicit for partitions with $\mathop{\text{Ch}}i_{r/2}\ge r/2-1$. \begin{proposition} \leftarrowabel{prop:uniqueness} Assume $\mathop{\text{Ch}}i$ is a partition with $l(\mathop{\text{Ch}}i)\leftarrowe r/2$ and $\mathop{\text{Ch}}i_{r/2}\ge r/2-1$. Then $N_{Q}(\mathop{\text{Ch}}i)$ has a $\operatorname {GL}(Q)$-equivariant resolution $P_\bullet$ as a $\operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)$-module such that \[ P_k=\bigoplus_{\pi\in S_{\mathop{\text{Ch}}i,k}}S^{\pi} Q\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}_k(\wedge^2 Q) \] and such that the differential $P_k\rightarrow P_{k-1}$ is the sum of maps for $\tau\subset_2\pi$: \begin{equation} \leftarrowabel{ref-A.1-51} S^\pi Q\otimes_k \operatorname{Sym}(\wedge^2 Q)\xrightarrow{\phi_{\pi,\tau}\otimes 1} S^\tau Q\otimes_k \wedge^2 Q\otimes_k \operatorname{Sym}(\wedge^2 Q) \rightarrow S^\tau Q\otimes_k \operatorname{Sym}(\wedge^2 Q) \end{equation} where the $(\phi_{\pi,\tau})_{\pi,\tau}$ are Pieri maps and the last map is obtained from the multiplication $\wedge^2 Q\otimes_k \operatorname{Sym}(\wedge^2 Q)\rightarrow\operatorname{Sym}(\wedge^2 Q)$. Moreover every choice of Pieri maps such that the compositions $P_k\rightarrow P_{k-1}\rightarrow P_{k-2}$ are zero yields isomorphic resolutions, and the isomorphism is given by scalar multiplication. \end{proposition} \begin{proof} We will first discuss uniqueness up to scalar multiplication of maps in the resolutions. The condition that \eqref{ref-A.1-51} forms a complex may be expressed as follows. For $\pi\in S_{\mathop{\text{Ch}}i,k}$, $\sigma\in S_{\mathop{\text{Ch}}i,k-2}$ put \begin{equation} \leftarrowabel{ref-A.2-52} \{(\tau_i)_i\in I\}:=\{\tau\in S_{\mathop{\text{Ch}}i,k-1}\mid \sigma\subset_2 \tau\subset_2 \pi\}\,. \end{equation} Then \eqref{ref-A.1-51} forms a complex if and only if the compositions \begin{equation} \leftarrowabel{ref-A.3-53} S^\pi Q\xrightarrow{(\phi_{\pi,\tau_i})_i} \bigoplus_i \wedge^2 Q\otimes S^{\tau_i} Q \xrightarrow{(1\otimes\phi_{\tau_i,\sigma})_i} \wedge^2 Q\otimes \wedge^2 Q\otimes S^{\sigma} Q\rightarrow S^2(\wedge^2 Q )\otimes S^{\sigma} Q \end{equation} are zero. We must show that any two Pieri-systems satisfying \eqref{ref-A.3-53} are equivalent. Let $\alpha\in {Q_{-1}}(2k)$, $\beta\in {Q_{-1}}(2(k-1))$. We may express the relation $\beta\subset_2 \alpha$ in terms of Frobenius coordinates. If $\alpha=(a_1,\leftarrowdots,a_u{;}a_1+1,\leftarrowdots,a_u+1)$ and $\beta=(b_1,\leftarrowdots,b_v{;}b_1+1,\leftarrowdots,b_v+1)$ then $\beta\subset_2\alpha$ if and only if $u=v$ and $(a_1,\leftarrowdots,a_u)=(b_1,\leftarrowdots,b_t+1,\leftarrowdots,b_v)$ for some $t$, or else $u=v+1$ and $(a_1,\leftarrowdots,a_{u})=(b_1,\leftarrowdots,b_v,1)$. From this it follows in particular that \eqref{ref-A.2-52} contains at most two elements. Like in the proof of \cite[Prop.\ 7.1(iv)]{VdB100} we can now build a contractible cubical complex $\PP$ with vertices $\cup_k S_{\mathop{\text{Ch}}i,k}$ and edges the pairs $\tau\subset_2\pi$ such that if $\phi_{\pi,\tau}$, $\phi'_{\pi,\tau}$ are two Pieri-systems satisfying \eqref{ref-A.1-51} then $\phi'_{\pi,\tau}/\phi_{\pi,\tau}$ is a 1-cocycle for $\PP$. Since $\PP$ is contractible this 1-cocycle is a coboundary which turns out to express exactly that $\phi'_{\pi,\tau}$ and $\phi_{\pi,\tau}$ are equivalent. We now discuss the existence of $P_\bullet$. To this end we introduce some notation. Let $G$ be the Grassmannian of $r/2$ dimensional quotients of $Q$ and let ${\cal P}$, ${\cal S}$ be respectively the universal quotient and subbundle on~$G$. The resolution of ${N_Q}(\mathop{\text{Ch}}i)$ constructed in \cite[Lemma 3.11, Lemma 3.12, Prop.\ 3.13]{WeymanSam} (denoted by $M_\mathop{\text{Ch}}i$ in loc.\ cit.) using the ``geometric method'' is now obtained by applying $\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,-\otimes_G S^\mathop{\text{Ch}}i{\cal P})$ to the Koszul complex \[ \wedge^\bullet (\wedge^2 {\cal S}) \otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q) \] obtained from the inclusion $\wedge^2{\cal S}\subset {\cal O}_G\otimes_k \wedge^2 Q$. So the resulting complex is \begin{equation} \leftarrowabel{ref-A.4-54} \mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,\wedge^\bullet(\wedge^2{\cal S})\otimes_G S^\mathop{\text{Ch}}i {\cal P})\otimes_k \operatorname{Sym}} \def\Sp{\operatorname{Sp}(\wedge^2 Q)\,. \end{equation} Using the decomposition \begin{equation} \leftarrowabel{ref-3.2-18} \wedge^{k}(\wedge^2{\cal S})\cong \bigoplus_{\mu\in {Q_{-1}}(2k)} S^\mu{\cal S}\, \end{equation} we obtain from Lemma \rightarrowef{ref-A.2-56} below that the differential in \eqref{ref-A.4-54} is given by the composition \begin{multline} \leftarrowabel{ref-A.5-55} \mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,S^\mathop{\text{Ch}}i {\cal P}\otimes_G S^\alpha {\cal S}) \xrightarrow{\phi_{\alpha,\beta,{\cal S}}} \mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,S^\mathop{\text{Ch}}i {\cal P}\otimes_G \wedge^2{\cal S} \otimes_G S^\beta {\cal S}) \hookrightarrow\\ \mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,S^\mathop{\text{Ch}}i {\cal P}\otimes_G (\wedge^2 Q\otimes_k S^\beta {\cal S})) =\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,S^\mathop{\text{Ch}}i {\cal P}\otimes_G S^\beta {\cal S}) \otimes_k \wedge^2Q \end{multline} where $\phi_{\alpha,\beta,{\cal S}}$ is a Pieri map. Now for each pair $(\mathop{\text{Ch}}i,\alpha)\in S_{\mathop{\text{Ch}}i,k}$ choose an isomorphism $\mathop{\uparrownderline{\uparrownderline{{\Gamma}}}}\nolimitsamma(G,S^\mathop{\text{Ch}}i {\cal P}\otimes_G S^\alpha {\cal S})\cong S^{(\mathop{\text{Ch}}i|\alpha)} Q$. Then \eqref{ref-A.5-55} becomes a $\operatorname {GL}(Q)$-equivariant morphism \[ \phi_{\mathop{\text{Ch}}i,\alpha,\beta}:S^{(\mathop{\text{Ch}}i|\alpha)}Q \rightarrow S^{(\mathop{\text{Ch}}i|\beta)}Q \otimes_k \wedge^2 Q\,. \] \begin{sublemma} If $\beta\subset_2\alpha$ then $\phi_{\mathop{\text{Ch}}i,\alpha,\beta}$ is not zero and hence it is a Pieri map. \end{sublemma} \begin{proof} In \eqref{ref-A.5-55} $\phi_{\alpha,\beta,{\cal S}}$ is a monomorphism. So it induces a monomorphism on global sections. The compositions of two monomorphisms is again monomorphism. This can only be zero if its source is zero, which is not the case since $(\mathop{\text{Ch}}i|\alpha)\in S_{\mathop{\text{Ch}}i,k}$ is dominant. \end{proof} It follows that \eqref{ref-A.4-54} becomes a complex of the shape asserted in the statement of the proposition, finishing the proof. \end{proof} A version for vector bundles of the following lemma was used. \begin{lemma} \leftarrowabel{ref-A.2-56} Let $R$ be a vector space of dimension $n$. Let $\alpha\in {Q_{-1}}(2k)$, $\beta\in {Q_{-1}}(2(k-1))$ with $\beta\subset_2\alpha$ and $l(\alpha)\leftarrowe n$. Then following composition is non-zero \[ \phi_{\alpha,\beta}:S^\alpha R\hookrightarrow \wedge^k(\wedge^2 R)\xrightarrow{\phi} \wedge^2R \otimes_k \wedge^{k-1} (\wedge^2 R) \twoheadrightarrow \wedge^2 R\otimes_l S^\beta R \] where the first and last map are obtained from the $\operatorname {GL}(R)$-equivariant decomposition $\wedge^k(\wedge^2 R)\cong\bigoplus_{\alpha\in {Q_{-1}}(2k)} S^\alpha R$, $\wedge^{k-1}(\wedge^2 R)\cong\bigoplus_{\beta\in {Q_{-1}}(2(k-1))} S^\beta R$ and the middle map is the canonical one. \end{lemma} \begin{proof} Choose a basis $\{e_1,\leftarrowdots,e_n\}$ for $R$ and let $U$ be the unipotent subgroup of $\operatorname {GL}(R)$ given by upper triangular matrices with 1's on the diagonal, written in the basis $\{e_1,\leftarrowdots,e_n\}$. In other words $u\in U$ if and only if $u\operatorname {cd}ot e_i=e_i+\sum_{j<i} \leftarrowambda_j e_j$ for $i=1,\leftarrowdots,r$. The $U$-invariant vectors in $\wedge^k(\wedge^2 R)$ corresponding to the decomposition \begin{equation} \leftarrowabel{ref-A.6-57} \wedge^k(\wedge^2 R)\cong\bigoplus_{\alpha\in {Q_{-1}}(2k)} S^\alpha R \end{equation} were explicitly written down in \cite[Prop. 2.3.9]{WeymanBook}. To explain this let $\alpha\in {Q_{-1}}(2k)$ and write it in Frobenius coordinates as $(a_1,\leftarrowdots,a_u{;}a_1+1,\leftarrowdots,a_u+1)$. Then the highest weight vector of the $S^\alpha R$-component in \eqref{ref-A.6-57} is given by $u_\alpha:= \bigwedge_{i< j\leftarrowe i+a_i} v_{ij}$ for $v_{ij}=e_i\wedge e_j$ (we do not care about the sign of $u_\alpha$ so the ordering of the product is unimportant). If we represent $\alpha$ by a Young diagram then the index set of the exterior product corresponds to the boxes strictly below the diagonal which makes it easy to visualize why $u_\alpha$ is $U$-invariant and why it has weight $\alpha$ for the maximal torus corresponding of the diagonal matrices in $\operatorname {GL}(R)$. We have $\phi(u_\alpha)=\sum_{ij}\pm v_{ij}\otimes \hat{u}_{\alpha,ij}$ where $\hat{u}_{\alpha,ij}$ is obtained from $u_{\alpha}$ by removing the factor $v_{ij}$. Thus $\phi_{\alpha,\beta}(u_\alpha) =\sum_{ij}\pm v_{ij}\otimes \mathop{\text{pr}}\nolimits_\beta(\hat{u}_{\alpha,ij})$ where $\mathop{\text{pr}}\nolimits_\beta:\wedge^{k-1}(\wedge^2 Q) \rightarrow S^\beta R$ is the projection. Since the $v_{ij}$ are linearly independent in $\wedge^2 R$ it follows that $\phi_{\alpha,\beta}(u_\alpha)$ can only be zero if $\mathop{\text{pr}}\nolimits_\beta(\hat{u}_{\alpha,ij})$ is zero for all $i,j$. Now if $\beta\subset_2\alpha$ then there exist $i,j$ such that $\hat{u}_{\alpha,ij}=\pm u_{\beta}$. Since by definition $\mathop{\text{pr}}\nolimits_\beta(u_\beta)=u_\beta\neq 0$ we obtain $\mathop{\text{pr}}\nolimits_\beta(\hat{u}_{\alpha,ij})\neq 0$ and thus also $\phi_{\alpha,\beta}(u_\alpha)\neq 0$. \end{proof} \downarrowef$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \downarrowef$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \downarrowef$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \mathop{\text{pr}}\nolimitsovidecommand{\bysame}{\leftarroweavevmode\hbox to3em{\hrulefill}\thinspace} \mathop{\text{pr}}\nolimitsovidecommand{\MR}{\rightarrowelax\ifhmode\uparrownskip\space\fi MR } \mathop{\text{pr}}\nolimitsovidecommand{\MRhref}[2]{ \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \mathop{\text{pr}}\nolimitsovidecommand{\href}[2]{#2} \end{document}

\begin{document} \title{A Systematic Study of Online Class Imbalance Learning with Concept Drift} \author{Shuo~Wang,~\IEEEmembership{Member,~IEEE,} Leandro L.~Minku,~\IEEEmembership{Member,~IEEE,} and Xin~Yao,~\IEEEmembership{Fellow,~IEEE} \thanks{S. Wang and X. Yao are with the Centre of Excellence for Research in Computational Intelligence and Applications (CERCIA), School of Computer Science, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. E-mail: \{S.Wang, X.Yao\}@cs.bham.ac.uk.} \thanks{L. L. Minku is with Department of Informatics, University of Leicester, Leicester LE1 7RH, UK. E-mail: leandro.minku@leicester.ac.uk.}} \markboth{IEEE Transactions on Neural Networks and Learning Systems,~Vol.~xx, No.~x, February~2017} {Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for IEEE Journals} \maketitle \begin{abstract} As an emerging research topic, online class imbalance learning often combines the challenges of both class imbalance and concept drift. It deals with data streams having very skewed class distributions, where concept drift may occur. It has recently received increased research attention; however, very little work addresses the combined problem where both class imbalance and concept drift coexist. As the first systematic study of handling concept drift in class-imbalanced data streams, this paper first provides a comprehensive review of current research progress in this field, including current research focuses and open challenges. Then, an in-depth experimental study is performed, with the goal of understanding how to best overcome concept drift in online learning with class imbalance. Based on the analysis, a general guideline is proposed for the development of an effective algorithm. \end{abstract} \begin{IEEEkeywords} Online learning, class imbalance, concept drift, resampling. \end{IEEEkeywords} \IEEEpeerreviewmaketitle \section{Introduction} \label{sec:intro} With the wide application of machine learning algorithms to the real world, class imbalance and concept drift have become crucial learning issues. Applications in various domains such as risk management~\cite{Sousa2016pe}, anomaly detection~\cite{Meseguer2010bo}, software engineering~\cite{Wang2013hi}, and social media mining~\cite{Sun2016ba} are affected by both class imbalance and concept drift. Class imbalance happens when the data categories are not equally represented, i.e., at least one category is minority compared to other categories~\cite{He2009kx}. It can cause learning bias towards the majority class and poor generalization. Concept drift is a change in the underlying distribution of the problem, and is a significant issue specially when learning from data streams~\cite{Minku:2010uq}. It requires learners to be adaptive to dynamic changes. Class imbalance and concept drift can significantly hinder predictive performance, and the problem becomes particularly challenging when they occur simultaneously. This challenge arises from the fact that one problem can affect the treatment of the other. For example, drift detection algorithms based on the traditional classification error may be sensitive to the imbalanced degree and become less effective; and class imbalance techniques need to be adaptive to changing imbalance rates, otherwise the class receiving the preferential treatment may not be the correct minority class at the current moment. Although there have been papers studying data streams with an imbalanced distribution and data streams with concept drift respectively, very little work discusses the cases when both class imbalance and concept drift exist. This paper aims to provide a systematic study of handling concept drift in class-imbalanced data streams. We focus on online (i.e. one-by-one) learning, which is a more difficult case than chunk-based learning, because only a single instance is available at a time. We first give a comprehensive review of current research progress in this field, including problem definitions, problem and approach categorization, performance evaluation and up-to-date approaches. It reveals new challenges and research gaps. Most existing work focuses on the concept drift in posterior probabilities (i.e. real concept drift~\cite{Gama2014re}, changes in $P\left(y \mid \mathbf{x} \right)$). The challenges in other types of concept drift have not been fully discussed and addressed. Especially, the change in prior probabilities $P\left( y \right)$ is closely related to class imbalance and has been overlooked by most existing work. Most proposed concept drift detection approaches are designed for and tested on balanced data streams. Very few approaches aim to tackle class imbalance and concept drift simultaneously. Among limited solutions, it is still unclear which approach is better and when. It is also unknown whether and how applying class imbalance techniques (e.g. resampling methods) affects concept drift detection and online prediction. To fill in the research gaps, we then provide an experimental insight into how to best overcome concept drift in online learning with class imbalance, by focusing on three research questions: 1) what are the challenges in detecting each type of concept drift when the data stream is imbalanced? 2) Among the proposed methods designed for online class imbalance learning with concept drift, which one performs better for which type of concept drift? 3) Would applying class imbalance techniques (e.g. resampling methods) facilitate concept drift detection and online prediction? Six recent approaches, DDM-OCI~\cite{Wang2013bp}, LFR~\cite{Wang2015zo}, PAUC-PH~\cite{Brzezinski2015ib}~\cite{Brzezinski2017el}, OOB~\cite{Wang2014vb}, RLSACP~\cite{Ghazikhani2013wm} and ESOS-ELM~\cite{Mirza2015nt}, are compared and analyzed in depth under each of the three fundamental types of concept drift (i.e. changes in prior probability $P\left( y \right)$, class-conditional probability density function (pdf) $p\left(\mathbf{x}\mid y\right)$ and posterior probability $P\left(y \mid \mathbf{x} \right)$) in artificial data streams, as well as real-world data sets. To the best of our knowledge, they are the very few methods that are explicitly designed for online learning problems with class imbalance and concept drift so far. Finally, based on the review and experimental results, we provide some guidelines for developing an effective algorithm for learning from imbalanced data streams with concept drift. We stress the importance of studying the mutual effect of class imbalance and concept drift. The contributions of this paper include: this is the first comprehensive study that looks into concept drift detection in class-imbalanced data streams; data problems are categorized in different types of concept drift and class imbalance with illustrative applications; existing approaches are compared and analysed systematically in each type; pros and cons of each approach are investigated; the results provide guidance for choosing the appropriate technique and developing better algorithms for future learning tasks; this is also the first work exploring the role of class imbalance techniques in concept drift detection, which sheds light on whether and how to tackle class imbalance and concept drift simultaneously. The rest of this paper is organized as follows. Section~\ref{sec:framework} formulate the learning problem, including a learning framework and detailed problem descriptions and introduction of class imbalance and concept drift individually. Section~\ref{sec:overcomeboth} reviews the combined issue of class imbalance and concept drift, including example applications and existing solutions. Section~\ref{sec:exp} carries out the experimental study, aiming to find out the answers to the three research questions. Section~\ref{sec:con} draws the conclusions and points out potential future directions. \section{Online Learning Framework with Class Imbalance and Concept Drift} \label{sec:framework} In data stream applications, data arrives over time in streams of examples or batches of examples. The information up to a specific time step $t$ is used to build/update predictive models, which then predict the new example(s) arriving at time step $t+1$. Learning under such conditions needs chunk-based learning or online learning algorithms, depending on the number of training examples available at each time step. According to the most agreed definitions~\cite{Minku:2010uq}~\cite{Ditzler2015hq}, chunk-based learning algorithms process a batch of data examples at each time step, such as the case of daily internet usage from a set of users; online learning algorithms process examples one by one and the predictive model is updated after receiving each example~\cite{Oza2001fk}, such as the case of sensor readings at every second in engineering systems. The term ``incremental learning" is also frequently used under this scenario. It is usually referred to as any algorithm that can process data streams with certain criteria met~\cite{Polikar2001oq}. On one hand, online learning can be viewed as a special case of chunk-based learning. Online learning algorithms can be used to deal with data coming in batches. They both build and continuously update a learning model to accommodate newly available data, and simultaneously maintain its performance on old data, giving rise to the stability-plasticity dilemma~\cite{Grossberg1988as}. On the other hand, the way of designing online and chunk-based learning algorithms can be very different~\cite{Minku:2010uq}. Most chunk-based learning algorithms are not suitable for online learning tasks, because batch learners process a chunk of data each time, possibly using an offline learning algorithm for each chunk. Online learning requires the model being adapted immediately upon seeing the new example, and the example is then immediately discarded, which allows to process high-speed data streams. From this point of view, designing online learning algorithm can be more challenging but so far has received much less attention than the other. First, the online learner needs to learn from a single data example, so it needs a more sophisticated training mechanism. Second, data streams are often non-stationary (concept drift). The limited availability of training examples at the current moment in online learning hinders the detection of such changes and the application of techniques to overcome the change. Third, it is often seen that data is class imbalanced in many classification tasks, such as the fault detection task in an engineering system, where the fault is always the minority. Class imbalance aggravates the learning difficulty~\cite{He2009kx} and complicates the data status~\cite{Wang2013po}. However, there is a severe lack of research addressing the combined issue of class imbalance and concept drift in online learning. To fill in this research gap, this paper aims at a comprehensive review of the work done to overcome class imbalance and concept drift, a systematic study of learning challenges, and an in-depth analysis of the performance of current approaches. We begin by formalizing the learning problem in this section. \subsection{Learning Procedure} \label{subsec:procedure} In supervised online classification, suppose a data generating process provides a sequence of examples $\left( \mathbf{x_t}, y_t \right)$ arriving one at a time from an unknown probability distribution $p_t\left(x,y \right)$. $\mathbf{x_t}$ is the input vector belonging to an input space $X$, and $y_t$ is the corresponding class label belonging to the label set $Y = \left\{ c_1, \ldots, c_N \right\}$. We build an online classifier $F$ that receives the new input $\mathbf{x_t}$ at time step $t$ and then makes a prediction. The predicted class label is denoted by $\hat{y}_t$. After some time, the classifier receives the true label $y_t$, used to evaluate the predictive performance and further train the classifier. This whole process will be repeated at following time steps. It is worth pointing out that we do not assume new training examples always arrive at regular and pre-defined intervals here. In other words, the actual time interval between time step $t$ and $t+1$ may be different from the actual time interval between $t+1$ and $t+2$. One challenge arises when data is class imbalanced. Class imbalance is an important data feature, commonly seen in applications such as spam filtering~\cite{Nishida2008fk} and fault diagnosis~\cite{Meseguer2010bo}~\cite{Wang2013hi}. It is the phenomenon when some classes of data are highly under-represented (i.e. minority) compared to other classes (i.e. majority). For example, if $P\left ( c_i \right )\ll P\left ( c_j \right )$, then $c_j$ is a majority class and $c_i$ is a minority class. The difficulty in learning from imbalanced data is that the relatively or absolutely underrepresented class cannot draw equal attention to the learning algorithm, which often leads to very specific classification rules or missing rules for this class without much generalization ability for future prediction. It has been well-studied in offline learning~\cite{Japkowicz:2002ul}, and has attracted growing attention in data stream learning in recent years~\cite{Hoens2012uq}. In many applications, such as energy forecasting and climate data analysis~\cite{Monteiro2009nw}, the data generator operates in nonstationary environments. It gives rise to another challenge, called ``concept drift". It means that the probability density function (pdf) of the data generating process is changing over time. For such cases, the fundamental assumption of traditional data mining -- the training and testing data are sampled from the same static and unknown distribution -- does not hold anymore. Therefore, it is crucial to monitor the underlying changes, and adapt the model to accommodate the changes accordingly. When both issues exist, the online learner needs to be carefully designed for effectiveness, efficiency and adaptivity. An online class imbalance learning framework was proposed in~\cite{Wang2013po} as a guide for algorithm design. The framework breaks down the learning procedure into three modules -- a class imbalance detector, a concept drift detector and an adaptive online learner, as illustrated in Fig.~\ref{fig:framework}. \begin{figure} \caption{Learning framework for online class imbalance learning~\cite{Wang2013po} \label{fig:framework} \end{figure} The class imbalance detector reports the current class imbalance status of data streams. The concept drift detector captures concept drifts involving changes in classification boundaries. Based on the information provided by the first two modules, the adaptive online learner determines when and how to respond to the detected class imbalance and concept drift, in order to maintain its performance. The learning objective of an online class imbalance algorithm can be described as ``recognizing minority-class data effectively, adaptively and timely without sacrificing the performance on the majority class"~\cite{Wang2013po}. \subsection{Problem Descriptions} \label{subsec:probdesp} A more detailed introduction about class imbalance and concept drift is given here individually, including the terminology, research focuses and state-of-the-art approaches. The purpose of this section is to understand the fundamental issues that we need to take extra care of in online class imbalance learning. We also aim at understanding whether and how the current research in class imbalance learning and concept drift detection are individually related to their combined issue elaborated later in Section~\ref{sec:overcomeboth}, rather than to provide an exhaustive list of approaches in the literature. Among others, we will answer the following questions: \textit{can existing class imbalance techniques process data streams? Would existing concept drift detectors be able to handle imbalanced data streams?} \subsubsection{\textbf{Class imbalance}} In class imbalance problems, the minority class is usually much more difficult or expensive to be collected than the majority class, such as the spam class in spam filtering and the fraud class in credit card application. Thus, misclassifying a minority-class example is more costly. Unfortunately, the performance of most conventional machine learning algorithms is significantly compromised by class imbalance, because they assume or expect balanced class distributions or equal misclassification costs. Their training procedure with the aim of maximizing overall accuracy often leads to a high probability of the induced classifier predicting an example as the majority class, and a low recognition rate on the minority class. In reality, it is common to see that the majority class has accuracy close to 100\% and the minority class has very low accuracy between 0\%-10\%~\cite{Kubat:1998bs}. The negative effect of class imbalance on classifiers, such as decision trees~\cite{Japkowicz:2002ul}, neural networks~\cite{Visa:2005ai}, k-Nearest Neighbour (kNN)~\cite{Kubat:1997yg}~\cite{Batista:2004oq}~\cite{Zhang:2003ve} and SVM~\cite{Yan:2003vn}~\cite{Wu:2003qe}, has been studied. A classifier that provides a balanced degree of predictive performance for all classes is required. The major research questions in this area are summarized and answered as follows:\\ \noindent (a) \textit{How do we define the imbalanced degree of data?} It seems to be a trivial question. However, there is no consensus on the definition in the literature. To describe how imbalanced the data is, researchers choose to use the percentage of the minority class in the data set~\cite{Hulse:2007eu}, the size ratio between classes~\cite{Lopez2013ls}, or simply a list of the number of examples in each class~\cite{Chawla:2002yq}. The coefficient of variance is used in~\cite{Hoens2012lq}, which is less straightforward. The description of imbalance status may not be a crucial issue in offline learning, but becomes more important in online learning, because there is no static data set in online scenarios. It is necessary to have some measurement automatically describing the up-to-date imbalanced degree and techniques monitoring the changes in class imbalance status. This will help the online learner to decide when and how to tackle class imbalance. The issue of changes in class imbalance status is relevant to concept drift, which will be further discussed in the next subsection. To define the imbalanced degree suitable for online learning, a real-time indicator was proposed -- time-decayed class size~\cite{Wang2013po}, expressing the size percentage of each class in the data stream. It is updated incrementally at each time step by using a time decay (forgetting) factor, which emphasizes the current status of data and weakens the effect of old data. Based on this, a class imbalance detector was proposed to determine which classes should be regarded as the minority/majority and how imbalanced the current data stream is, and then used for designing better online classifiers~\cite{Wang2014vb}~\cite{Wang2013hi}. The merit of this indicator is that it is suitable for data with arbitrary number of classes. \noindent (b) \textit{When does class imbalance matter?} It has been shown that class imbalance is not the only problem responsible for the performance reduction of classifiers. Classifiers' sensitivity to class imbalance also depends on the complexity and overall size of the data set. Data complexity comprises issues such as overlapping~\cite{Batista:2005uq}~\cite{Prati:2004kx} and small disjuncts~\cite{Jo2004bh}. The degree of overlapping between classes and how the minority class examples distribute in data space aggravate the negative effect of class imbalance. The small disjunct problem is associated with the within-class imbalance~\cite{Japkowicz:2003pd}. Regarding the size of the training data, a very large domain has a good chance that the minority class is represented by a reasonable number of examples, and thus may be less affected by imbalance than a small domain containing very few minority class examples. In other words, the rarity of the minority class can be in a relative or absolute sense in terms of the number of available examples~\cite{He2009kx}. In particular, authors in~\cite{Napierala2012bo}~\cite{Napierala2016gr} distinguished and analysed four types of data distributions in the minority class -- safe, borderline, outliers and rare examples. Safe examples are located in the homogenous regions populated by the examples from one class only; borderline examples are scattered in the boundary regions between classes, where the examples from both classes overlap; rare examples and outliers are singular examples located deeper in the regions dominated by the majority class. Borderline, rare and outlier data sets were found to be the real source of difficulties in learning imbalanced data sets offline, which have also been shown to be the harder cases in online applications~\cite{Wang2014vb}. Therefore, for any developed algorithms dealing with imbalanced data online, it is worth discussing their performance on data with different types of distributions. \noindent (c) \textit{How can we tackle class imbalance effectively (state-of-the-art solutions)?} A number of algorithms have been proposed to tackle class imbalance at the data and algorithm levels. Data-level algorithms include a variety of resampling techniques, manipulating training data to rectify the skewed class distributions. They oversample minority-class examples (i.e. expanding the minority class), undersample majority-class examples (i.e. shrinking the majority class), or combine both, until the data set is relatively balanced. Random oversampling and random undersampling are the simplest and most popular resampling techniques, where examples are randomly chosen to be added or removed. There are also smart resampling techniques (a.k.a guided resampling). For example, SMOTE~\cite{Chawla:2002yq} is a widely used oversampling method, which generates new minority-class data points based on the similarities between original minority-class examples in the feature space. Other smart oversampling techniques include Borderline-SMOTE~\cite{Han:2005sf}, ADASYN~\cite{He:2008cr}, MWMOTE~\cite{Barua2014mq}, to name but a few. Smart undersampling techniques include Tomek links~\cite{TOMEK:1976la}, One-sided selection~\cite{Kubat:1997kx}, Neighbourhood cleaning rule~\cite{Jorma:2001kx}, etc. The effectiveness of resampling techniques have been proved in real-world applications~\cite{Hao2014po}. They work independently of classifiers, and are thus more versatile than algorithm-level methods. The key is to choose an appropriate sampling rate~\cite{Estabrooks:2004ve}, which is relatively easy for two-class data sets, but becomes more complicated for multi-class data sets~\cite{Saez2016yq}. Empirical studies have been carried out to compare different resampling methods~\cite{Hulse:2007eu}. Particularly, it is shown that smart resampling techniques are not necessarily superior to random oversampling and undersampling; besides, they cannot be applied to online scenarios directly, because they work on a static data set for the relation among the training examples. Some initial effort has been made recently, to extend smart resampling techniques to online learning~\cite{Mao2015xp}. Algorithm-level methods address class imbalance by modifying their training mechanism with the direct goal of better accuracy on the minority class, including one-class learning~\cite{Japkowicz:1995qf}, cost-sensitive learning~\cite{Liu:2006yq} and threshold methods~\cite{Weiss:2003eu}. They require different treatments for specific kinds of learning algorithms. In other words, they are algorithm-dependent, so they are not as widely used as data-level methods. Some online cost-sensitive methods have been proposed, such as CSOGD~\cite{Wang2014fn} and RLSACP~\cite{Ghazikhani2013wm}. They are restricted to the perceptron-based classifiers, and require pre-defined misclassification costs of classes that may or may not be updated during the online learning. Finally, ensemble learning (also known as multiple classifier systems)~\cite{Polikar:2006xy} has become a major category of approaches to handling class imbalance~\cite{Galar:2011uq}. It combines multiple classifiers as base learners and aims to outperform every one of them. It can be easily adapted for emphasizing the minority class by integrating different resampling techniques~\cite{Li:2007hc}~\cite{Liu:2009kx}~\cite{Chawla:2003yq}~\cite{Blaszczynski2015ge} or by making base classifiers cost-sensitive~\cite{Joshi:2001kx}~\cite{Chawla:2007kx}~\cite{Guo:2004qe}~\cite{Fan:1999rm}. A few ensemble methods are available for online class imbalance learning, such as OOB and UOB~\cite{Wang2014vb} applying random oversampling and undersampling in Online Bagging~\cite{Oza:2005ve}, and WOS-ELM~\cite{Mirza2013la} training a set of cost-sensitive online extreme learning machines. It is worth pointing out that, the aforementioned online learning algorithms designed for imbalanced data are not suitable for non-stationary data streams. They do not involve any mechanism handling drifts that affect classification boundaries, although OOB and UOB can detect and react to class imbalance changes. \noindent (d) \textit{How do we evaluate the performance of class imbalance learning algorithms?} Traditionally, overall accuracy and error rate are the most frequently used metrics of performance evaluation. However, they are strongly biased towards the majority class when data is imbalanced. Therefore, other performance measures have been adopted. Most studies concentrate on two-class problems. By convention, the minority class is treated to be the positive, and the majority class is treated to be the negative. Table~\ref{tab:confusion} illustrates the confusion matrix of a two-class problem, producing four numbers on testing data. \begin{table}[htp] \caption{Confusion matrix for a two-class problem.} \label{tab:confusion} \centering \begin{tabular}{|c|c|c|} \hline & Predicted as positive & Predicted as negative\\ \hline Actual positive & True positive (TP) & False negative (FN) \\ Actual negative & False positive (FP) & True negative (TN) \\ \hline \end{tabular} \end{table} From the confusion matrix, we can derive the expressions for \textit{recall} and \textit{precision}: \begin{equation} \label{eq:recall} recall = \frac{TP}{TP+FN}, \end{equation} \begin{equation} \label{eq:precision} precision = \frac{TP}{TP+FP}. \end{equation} Recall (i.e. TP rate) is a measure of completeness -- the proportion of positive class examples that are classified correctly to all positive class examples. Precision is a measure of exactness -- the proportion of positive class examples that are classified correctly to the examples predicted as positive by the classifier. The learning objective of class imbalance learning is to improve recall without hurting precision. However, improving recall and precision can be conflicting. Thus, F-measure is defined to show the trade-off between them. \begin{equation} \label{eq:F} Fm = \frac{\left ( 1+\beta^2 \right )\cdot recall\cdot precision}{\beta^2\cdot precision+recall}, \end{equation} where $\beta$ corresponds to the relative importance of recall and precision. It is usually set to 1. Kubat et al.~\cite{Kubat:1997kx} proposed to use G-mean to replace overall accuracy: \begin{equation} \label{eq:G} Gm = \sqrt{\frac{TP}{TP+FN}\times \frac{TN}{TN+FP}}. \end{equation} It is the geometric mean of positive accuracy (i.e. TP rate) and negative accuracy (i.e. TN rate). A good classifier should have high accuracies on both classes, and thus a high G-mean. According to~\cite{He2009kx}, any metric that uses values from both rows of the confusion matrix for addition (or subtraction) will be inherently sensitive to class imbalance. In other words, the performance measure will change as class distribution changes, even though the underlying performance of the classifier does not. This performance inconsistency can cause problems when we compare different algorithms over different data sets. Precision and F-measure, unfortunately, are sensitive to the class distribution. Therefore, recall and G-mean are better options. To compare classifiers over a range of sample distributions, AUC (abbr. of the Area Under the ROC curve) is the best choice. A ROC curve depicts all possible trade-offs between TP rate and FP rate, where FP rate = $FP/\left( TN+FP \right)$. TP rate and FP rate can be understood as the benefits and costs of classification with respect to data distributions. Each point on the curve corresponds to a single trade-off. A better classifier should produce a ROC curve closer to the top left corner. AUC represents a ROC curve as a single scalar value by estimating the area under the curve, varying in [0, 1]. It is insensitive to the class distribution, because both TP rate and FP rate use values from only one row of the confusion matrix. AUC is usually generated by varying the classification decision threshold for separating positive and negative classes in the testing data set~\cite{Maloof:2003dq}~\cite{Fawcett:2006uq}. In other words, calculating AUC requires a set of confusion matrices. Therefore, unlike other measures based on a single confusion matrix, AUC cannot be used as an evaluation metric in online learning without memorizing data. Although a recent study has modified AUC for evaluating online classifiers~\cite{Brzezinski2015ib}, it still needs to collect recently received examples. The properties of the above measures are summarized in Table~\ref{tab:imbalance-metric}. They are defined under the two-class context. They cannot be used to evaluate multi-class data directly, except for recall. Their multi-class versions have been developed~\cite{Sokolova2009dl}~\cite{Sun:2006pd}~\cite{Hand:2001kx}. The ``multi-class" and ``online" columns in the table show whether the corresponding measure can be used directly without modification in multi-class and online data scenarios. \begin{table}[htp] \caption{Performance evaluation measures for class imbalance problems.} \label{tab:imbalance-metric} \centering \begin{tabular}{|c|c|c|c|} \hline Measures& Multi-class & Online & Sensitive to\\ &&&Imbalance\\ \hline recall & yes & yes & no\\ \hline precision & no~\cite{Sokolova2009dl} & yes & yes\\ \hline Fm & no~\cite{Sokolova2009dl} & yes & yes\\ \hline Gm & yes~\cite{Sun:2006pd} & yes & no\\ \hline AUC & no (See MAUC~\cite{Hand:2001kx}) & no (See PAUC~\cite{Brzezinski2015ib}) & no\\ \hline \end{tabular} \end{table} \subsubsection{\textbf{Concept drift}} Concept drift is said to occur when the joint probability $P\left( \mathbf{x},y \right)$ changes~\cite{Gama2014re}~\cite{Minku2012vn}~\cite{Minku2010uq2}. The key research topics in this area include:\\ \noindent (a) \textit{How many types of concept drift are there? Which type is more challenging?} Concept drift can manifest three fundamental forms of changes corresponding to the three major variables in the Bayes' theorem~\cite{Kelly1999fs}: 1) a change in prior probability $P\left( y \right)$; 2) a change in class-conditional pdf $p\left(\mathbf{x}\mid y \right)$; 3) a change in posterior probability $P\left( y\mid \mathbf{x} \right)$. The three types of concept drift are illustrated in Figure~\ref{fig:drift}. Comparing to the original data distribution shown in Figure~\ref{fig:drift}(a), \begin{figure} \caption{Illustration of 3 concept drift types.} \label{fig:drift} \end{figure} Fig.~\ref{fig:drift}(b) shows the $P\left( y \right)$ type of concept drift without affecting $p\left( \mathbf{x}\mid y \right)$ and $P\left( y\mid \mathbf{x} \right)$. The decision boundary remains unaffected. The prior probability of the circle class is reduced in this example. Such change can lead to class imbalance. A well-learnt discrimination function may drift away from the true decision boundary, due to the imbalanced class distribution. Fig.~\ref{fig:drift}(c) shows the $p\left( \mathbf{x}\mid y \right)$ type of concept drift without affecting $P\left( y \right)$ and $P\left( y\mid \mathbf{x} \right)$. The true decision boundary remains unaffected. Elwell and Polikar claimed that this type of drift is the result of an incomplete representation of the true distribution in current data, which simply requires providing supplemental data information to the learning model~\cite{Elwell2011dq}. Fig.~\ref{fig:drift}(d) shows the $P\left( y\mid \mathbf{x} \right)$ type of concept drift. The true boundary between classes changes after the drift, so that the previously learnt discrimination function does not apply any more. In other words, the old function becomes unsuitable or partially unsuitable, and the learning model needs to be adapted to the new knowledge. The posterior distribution change clearly indicates the most fundamental change in the data generating function. This is classified as \textit{real concept drift}. The other two types belong to \textit{virtual concept drift}~\cite{Hoens2012uq}, which does not change the decision (class) boundaries. In practice, one type of concept drift may appear in combination with other types. Existing studies primarily focus on the development of drift detection methods and techniques to overcome the real drift. There is a significant lack of research on virtual drift, which can also deteriorate classification performance. As illustrated in Fig.~\ref{fig:drift}(b), even though these types of drift do not affect the true decision boundaries, they can cause a well-learnt decision boundary to become unsuitable. Unfortunately, the current techniques for handling real drift may not be suitable for virtual drift, because they present very different learning difficulties and require different solutions. For instance, the methods for handling real drift often choose to reset and retrain the classifier, in order to forget the old concept and better learn the new concept. This is not an appropriate strategy for data with virtual drift, because the examples from previous time steps may still remain valid and help the current classification in virtual drift cases. It would be more effective and efficient to calibrate the existing classifier than retraining it. Besides, techniques for handling real drift typically rely on feedback about the performance of the classifier, while techniques for handling virtual drift can operate without such feedback~\cite{Gama2014re}. From our point of view, all three types are equally important. Particularly, the two virtual types require more research effort than currently dedicated work by our community. A systematic study of the challenges in each type will be given in Section~\ref{sec:exp}. \begin{table*}[htp] \caption{Categorization of concept drift techniques. See~\cite{Ditzler2015hq} for the full list of techniques under each category.} \label{tab:driftdetector} \centering \begin{tabular}{|c|c|l|} \hline \multirow{14}{*}{\textbf{Active}} & \multirow{6}{*}{\textbf{Step1. Change}} & \textbf{Hypothesis tests}: assess the validity of a hypothesis by comparing the distributions of two sets of fix-length \\ && data sequences. \\ \cline{3-3} && \textbf{Change-point methods}: identify the change point by analyzing all possible partitions of a fixed data sequence. \\ \cline{3-3} &\multirow{2}{*}{\textbf{detection}}& \textbf{Sequential hypothesis tests}: provide a one-off detection of change or no change, by inspecting incoming \\ && examples one by one (sequentially). \\ \cline{3-3} && \textbf{Change detection tests}: analyze the statistical behavior of streams of data in a fully sequential manner, such \\ && as a feature value or classification error. They are either based on a pre-defined threshold or some statistical \\ && features representing current data. \\ \cline{2-3} & \multirow{5}{*}{\textbf{Step2. Classifier}} & \textbf{Windowing}: the classifier is retrained based on a window with up-to-date examples. The window length can \\ && be either fixed or adaptive. \\ \cline{3-3} && \textbf{Weighting}: all received examples are weighted according to time or classification error, which are then used to \\ &\multirow{1}{*}{\textbf{adaptation}}& update the classifier. \\ \cline{3-3} && \textbf{Random Sampling}: the examples used to retrain the classifier are randomly chosen based on certain rules. \\ \cline{3-3} && \textbf{Ensemble}: build a new model in the classifier for the new concept. \\ \hline \multirow{2}{*}{\textbf{Passive}} & \multicolumn{2}{|l|}{\textbf{Single classifier}: update a single classifier, such as decision trees, online information network, and extreme learning machine.} \\ \cline{2-3} & \multicolumn{2}{|l|}{\textbf{Ensemble}: add, remove or modify the models in an ensemble classifier.} \\ \hline \end{tabular} \end{table*} Concept drift has further been characterized by its speed, severity, cyclical nature, etc. A detailed and mutually exclusive categorization can be found in~\cite{Minku2010uq2}. For example, according to speed, concept drift can be either abrupt, when the generating function is changed suddenly (usually within one time step), or gradual, when the distribution evolves slowly over time. They are the most commonly discussed types in the literature, because the effectiveness of drift detection methods can vary with the drifting speed. While most methods are quite successful in detecting abrupt drifts, as future data is no longer related to old data~\cite{Ditzler2013nh}, gradual drifts are often more difficult, because the slow change can delay or hide the hint left by the drift. We can see some drift detection methods specifically designed for gradual concept drift, such as Early Drift Detection method (EDDM)~\cite{Baena-Garca2006tg}. \noindent (b) \textit{How can we tackle concept drift effectively (state-of-the-art solutions)?} There is a wide range of algorithms for learning in non-stationary environments. Most of them assume and specialize in some specific types of concept drift, although real-world data often contains multiple types. They are commonly categorized into two major groups: active vs. passive approaches, depending on whether an explicit drift detection mechanism is employed. Active approaches (also known as trigger-based approaches) determine whether and when a drift has occurred before taking any actions. They operate based on two mechanisms -- a change detector aiming to sense the drift accurately and timely, and an adaptation mechanism aiming to maintain the performance of the classifier by reacting to the detected drift. Passive approaches (also known as adaptive classifiers) evolve the classifier continuously without an explicit trigger reporting the drift. A comprehensive review of up-to-date techniques tackling concept drift is given by Ditzler et al.~\cite{Ditzler2015hq}. They further organise these techniques based on their core mechanisms, summarized in Table~\ref{tab:driftdetector}. This table will help us to understand how online class imbalance algorithms are designed, which will be introduced in details in Section~\ref{sec:overcomeboth}. There exist other ways to classify the proposed algorithms, such as Gama et al.'s taxonomy based on the four modules of an adaptive learning system~\cite{Gama2014re}, and Webb et al.'s quantitative characterization~\cite{Webb2016jq}. This paper adopts the one proposed by Ditzler et al.~\cite{Ditzler2015hq} for its simplicity. The best algorithm varies with the intended applications. A general observation is that, while active approaches are quite effective in detecting abrupt drift, passive approaches are very good at overcoming gradual drift~\cite{Elwell2011dq}~\cite{Ditzler2015hq}. It is worth noting that most algorithms do not consider class imbalance. It is unclear whether they will remain effective if data becomes imbalanced. For example, some algorithms determine concept drift based on the change in the classification error, including OLIN~\cite{Cohen2008vn}, DDM~\cite{Gama2004kl} and PERM~\cite{Harel2014oa}. As we have explained in Section~\ref{subsec:probdesp} 1), the classification error is sensitive to the imbalance degree of data, and does not reflect the performance of the classifier very well when there is class imbalance. Therefore, these algorithms may not perform well when concept drift and class imbalance occur simultaneously. Some other algorithms are specifically designed for data streams coming in batches, such as AUE~\cite{Brzezinski2014fo} and the Learn++ family~\cite{Elwell2011dq}. These algorithms cannot be applied to online cases directly. \noindent (c) \textit{How do we evaluate the performance of concept drift detectors and online classifiers?} To fully test the performance of drift detection approaches (especially an active detector), it is necessary to discuss both data with artificial concept drifts and real-world data with unknown drifts. Using data with artificial concept drifts allows us to easily manipulate the type and timing of concept drifts, so as to obtain an in-depth understanding of the performance of approaches under various conditions. Testing on data from real-world problems helps us to understand their effectiveness from the practical point of view, but the information about when and how concept drift occurs is unknown in most cases. The following aspects are usually considered to assess the accuracy of active drift detectors. Their measurement is based on data with artificial concept drifts where drifts are known. \begin{itemize} \item True detection rate: the possibility of detecting the true concept drift. It shows the accuracy of the detection approach. \item False alarm rate: the possibility of reporting a concept drift that does not exist (false-positive rate). It characterizes the costs and reliability of the detection approach. \item Delay of detection: an estimate of how many time steps are required on average to detect a drift after the actual occurrence. It reflects how much time would be taken before the drift is detected. \end{itemize} Wang and Abraham~\cite{Wang2015zo} use a histogram to visualize the distribution of detection points from the drift detection approach over multiple runs. It reflects all the three aspects above in one plot. It is worth nothing that there are trade-offs between these measures. For example, an approach with a high true detection rate may produce a high false alarm rate. A very recent algorithm, Hierarchical Change-Detection Tests (HCDTs), was proposed to explicitly deal with the trade-off~\cite{Alippi2017hw}. After the performance of drift detection approaches is better understood, we need to quantify the effect of those detections on the performance of predictive models. All the performance metrics introduced in the previous section of ``class imbalance" can be used. The key question here is how to calculate them in the streaming settings with evolving data. The performance of the classifier may get better or worse every now and then. There are two common ways to depict such performance over time -- holdout and prequential evaluation~\cite{Gama2014re}. Holdout evaluation is mostly used when the testing data set (holdout set) is available in advance. At each time step or every few time steps, the performance measures are calculated based on the valid testing set, which must represent the same data concept as the training data at that moment. However, this is a very rigorous requirement for data from real-world applications. In prequential evaluation, data received at each time step is used for testing before it is use for training. From this, the performance measures can be incrementally updated for evaluation and comparison. This strategy does not require a holdout set, and the model is always tested on unseen data. When the data stream is stationary, the prequential performance measures can be computed based on the accumulated sum of a loss function from the beginning of the training. However, if the data stream is evolving, the accumulated measure can mask the fluctuation in performance and the adaptation ability of the classifier. For example, consider that an online classifier correctly predicts 90 out of 100 examples received so far (90\% accuracy on data with the original concept). Then, an abrupt concept drift occurs at time step 101, which makes the classifier only correctly predict 3 out of 10 examples from the new concept (30\% accuracy on data with the new concept). If we use the accumulated measure based on all the historical data, the overall accuracy will be 93/110, which seems to be high but does not reflect the true performance on the new data concept. This problem can be solved by using a sliding window or a time-based fading factor that weigh observations~\cite{Gama2013qp}. \section{Overcoming Class Imbalance and Concept Drift Simultaneously} \label{sec:overcomeboth} Following the review of class imbalance and concept drift in Section~\ref{sec:framework}, this section reviews the combined issue, including example applications and existing solutions. When both exist, one problem affects the treatment of the other. For example, the drift detection algorithms based on the traditional classification error may be sensitive to imbalanced degree and become less effective; the class imbalance techniques need to be adaptive to changing $P\left( y \right)$, otherwise the class receiving the preferential treatment may not be the correct minority class at the current moment. Therefore, their mutual effect should be considered during the algorithm design. \subsection{Illustrative Applications} \label{subsec:application} The combined problems of concept drift and class imbalance have been found in many real-world applications. Three examples are given here, to help us understand each type of concept drift. \subsubsection{Environment monitoring with $P\left( y \right)$ drift} Environment monitoring systems usually consist of various sensors generating streaming data in high speed. Real-time prediction is required. For example, a smart building has sensors deployed to monitor hazardous events. Any sensor fault can cause catastrophic failures. Machine learning algorithms can be used to build models based on the sensor information, aiming to predict faults in sensors accurately and timely~\cite{Wang2013hi}. First, the data is characterized by class imbalance, because obtaining a fault in such systems can be very expensive. Examples representing faults are the minority. Second, the number of faults varies with the faulty condition. If the damage gets worse over time, the faults will occur more and more frequently. It implies a prior probability change, a type of virtual concept drift. \subsubsection{Spam filtering with $p\left( \mathbf{x}\mid y \right)$ drift} Spam filtering is a typical classification problem involving class imbalance and concept drift~\cite{Lindstrom2010qg}. First of all, the spam class is the minority and suffers from a higher misclassification cost. Second, the spammers are actively working on how to break through the filter. It means that the adversary actions are adaptive. For example, one of the spamming behaviours is to change email content and presentation in disguise, implying a possible class-conditional pdf ($p\left( \mathbf{x}\mid y \right)$) change~\cite{Gama2014re}. \subsubsection{Social media analysis with $P\left( y\mid \mathbf{x} \right)$ drift} Social media (e.g. twitter, facebook) is becoming a valuable source of timely information on the internet. It attracts a growing number of people, sharing, communicating, connecting and creating user-generated data. Consider the example where a company would like to make relevant product recommendations to people who have shown some type of interest in their tweets. Machine learning algorithms can be used to discover who is interested in the product from the large amount of tweets~\cite{Li2012la}. The number of users who have shown the interest is always very small. Their information tends to be overwhelmed by other unrelated messages. Thus, it is utterly important to overcome the imbalanced distribution and discover the hidden information. Another challenge is users' interest changing from time to time. Users may lose their interest in the current trendy product very quickly, causing posterior probability ($P\left( y\mid \mathbf{x} \right)$) changes. Although the above examples are associated with only one type of concept drift, different types often coexist in real-world problems, which are hard to know in advance. For the example of spam filtering, which email belongs to spam also depends on users' interpretation. Users may re-label a particular category of normal emails as spam, which indicates a posterior probability change. \subsection{Approaches to Tackling Both Class Imbalance and Concept Drift} \label{subsec:approach} Some research efforts have been made to address the joint problem of concept drift and class imbalance, due to the rising need from practical problems~\cite{Pan2015wp}~\cite{Sousa2016pe}. Uncorrelated Bagging is one of the earliest algorithms, which builds an ensemble of classifiers trained on a more balanced set of data through resampling and overcomes concept drift passively by weighing the base classifier based on their discriminative power~\cite{Gao2008uq}~\cite{Gao2007fk}~\cite{Wu2014nq}. Selectively recursive approaches SERA~\cite{Chen2009dz} and REA~\cite{Chen2011tg} use similar ideas to Uncorrelated Bagging of building an ensemble of weighted classifiers, but with a ``smarter" oversampling technique. Learn++.CDS and Learn++.NIE are more recent algorithms, which tackle class imbalance through the oversampling technique SMOTE~\cite{Chawla:2002yq} or a sub-ensemble technique, and overcome concept drift through a dynamic weighting strategy~\cite{Ditzler2013mk}. HUWRS.IP~\cite{Hoens2013gh} improves HUWRS~\cite{Hoens:2011ys} to deal with imbalanced data streams by introducing an instance propagation scheme based on a Na\"{i}ve Bayes classifier, and uses Hellinger distance as a weighting measure for concept drift detection. This method relies on finding examples that are similar to the current minority-class concept, which however may not exist. So, Hellinger Distance Decision Tree (HDDT) was proposed to use Hellinger distance as the decision tree splitting criteria that is imbalance-insensitive~\cite{Pozzolo2014wl}. All these approaches belong to chunk-based learning algorithms. Their core techniques work when a batch of data is received at each time step, i.e. they are not suitable for online processing. Developing a true online algorithm for concept drift is very challenging because of the difficulties in measuring minority-class statistics using only one example at a time~\cite{Ditzler2015hq}. To handle class imbalance and concept drift in an online fashion, a few methods have been proposed recently. Drift Detection Method for Online Class Imbalance (DDM-OCI)~\cite{Wang2013bp} is one of the very first algorithms detecting concept drift actively in imbalanced data streams online. It monitors the reduction in minority-class recall (i.e. true positive rate). If there is a significant drop, a drift will be reported. It was shown to be effective in cases when minority-class recall is affected by the concept drift, but not when the majority class is mainly affected. A Linear Four Rates (LFR) approach was then proposed to improve DDM-OCI, which monitors four rates from the confusion matrix -- minority-class recall and precision and majority-class recall and precision, with statistically-supported bounds for drift detection~\cite{Wang2015zo}. If any of the four rates exceeds the bound, a drift will be confirmed. Instead of tracking several performance rates for each class, prequential AUC (PAUC)~\cite{Brzezinski2015ib}~\cite{Brzezinski2017el} was proposed as an overall performance measure for online scenarios, and was used as the concept drift indicator in Page-Hinkley (PH) test~\cite{Page1954qg}. However, it needs access to historical data. DDM-OCI, LFR and PAUC-based PH test are active drift detectors designed for imbalanced data streams, and are independent of classification algorithms. They aim at concept drift with classification boundary changes by default. Therefore, if a concept drift is reported, they will reset and retrain the online model. Although these drift detectors are designed for imbalanced data, they themselves do not handle class imbalance. It is still unclear how they perform when working with class imbalance techniques. \begin{table*}[htp] \caption{Online approaches to tackling concept drift and class imbalance, and their properties.} \label{tab:onlinemethod} \centering \begin{tabular}{|c|c|c|c|c|c|c|} \hline Approaches & Category? & Class & Access to & Additional data? & Multi-class? & $P\left( y \right)$ drift? \\ & & imbalance? & old data? & & & \\ \hline DDM-OCI~\cite{Wang2013bp} & Active (change detection test + windowing) & No & No & No & No & No \\ \hline LFR~\cite{Wang2015zo} & Active (change detection test + windowing) & No & No & No & No & No \\ \hline PAUC-PH~\cite{Brzezinski2015ib} & Active (change detection test + windowing) & No & Yes & No & No & No \\ \hline RLSACP~\cite{Ghazikhani2013wm}/ONN~\cite{Ghazikhani2014qb} & Passive (single classifier) & Yes & Yes & No & No & Yes \\ \hline ESOS-ELM~\cite{Mirza2015nt} & Passive+Active (ensemble) & Yes & No & Yes & No & No \\ \hline OOB/UOB using CID~\cite{Wang2014vb} & Active (weighting) & Yes & No & No & No & Yes \\ \hline \end{tabular} \end{table*} Besides the above active approaches, the perceptron-based algorithms RLSACP~\cite{Ghazikhani2013wm}, ONN~\cite{Ghazikhani2014qb} and ESOS-ELM~\cite{Mirza2015nt} adapt the classification model to non-stationary environments passively, and involve mechanisms to overcome class imbalance. RLSACP and ONN are single-model approaches with the same general idea. Their error function for updating the perceptron weights is modified, including a forgetting function for model adaptation and an error weighting strategy as the class imbalance treatment. The forgetting function has a pre-defined form, allowing the old data concept to be forgotten gradually. The error weights in RLSACP are incrementally updated based either on the classification performance or the imbalance rate from recently received data. It was shown that weight updating based on the imbalance rate leads to better performance. ESOS-ELM is an ensemble approach, maintaining a set of online sequential extreme learning machines (OS-ELM)~\cite{Liang2006bp}. For tackling class imbalance, resampling is applied in a way that each OS-ELM is trained with approximately equal number of minority- and majority-class examples. For tackling concept drift, voting weights of base classifiers are updated according to their performance G-mean on a separate validation data set from the same environment as the current training data. In addition to the passive drift detection technique, ESOS-ELM includes an independent module -- ELM-store, to handle recurring concept drift. ELM-store maintains a pool of weighted extreme learning machines (WELM)~\cite{Mirza2013la} to retain old information. It adopts a threshold-based technique and hypothesis testing to detect abrupt and gradual concept drift actively. If a concept drift is reported, a new WELM will be built and kept in ELM-store. If any stored model performs better than the current OS-ELM ensemble, indicating a possible recurring concept, it will be introduced in the ensemble. ESOS-ELM assumes the imbalance rate is known in advance and fixed. It needs a separate data set for initializing OS-ELMs and WELMs, which must include examples from all classes. It is also necessary to have validation data sets reflecting every data concept for concept drift detection, which can be a quite restrictive requirement for real-world data. With a different goal of concept drift detection from the above, a class imbalance detection (CID) approach was proposed, aiming at $P\left( y \right)$ changes~\cite{Wang2013po}. It reports the current imbalance status and provides information of which classes belong to the minority and which classes belong to the majority. Particularly, a key indicator is the real-time class size $w_k^{(t)}$, the percentage of class $c_k$ at time step t. When a new example $\mathbf{x_t}$ arrives, $w_k^{(t)}$ is incrementally updated by the following equation~\cite{Wang2013po}: \begin{equation} w_k^{(t)}=\theta w_k^{(t-1)}+\left ( 1-\theta \right ) \left[ \left (\mathbf{x_t},c_k \right ) \right], (k=1,\ldots,N) \label{eq:wk} \end{equation} where $\left[ \left( \mathbf{x_t}, c_k \right) \right] = 1$ if the true class label of $\mathbf{x_t}$ is $c_k$, and 0 otherwise. $\theta$ $\left(0<\theta<1\right)$ is a pre-defined time decay (forgetting) factor, which reduces the contribution of older data to the calculation of class sizes along with time. It is independent of learning algorithms, so it can be used with any type of online classifiers. For example, it has been used in OOB and UOB~\cite{Wang2014vb} for deciding the resampling rate adaptively and overcoming class imbalance effectively over time. OOB and UOB integrate oversampling and undersampling respectively into ensemble algorithm Online Bagging (OB)~\cite{Oza:2005ve}. Oversampling and undersampling are one of the simplest and most effective techniques of tackling class imbalance~\cite{Hulse:2007eu}. The properties of the above online approaches are summarized in Table~\ref{tab:onlinemethod}, answering the following six questions in order: \begin{itemize} \item How do they handle concept drift (the type based on the categorization in Table~\ref{tab:driftdetector})? \item Do they involve any class imbalance technique to improve the predictive performance of online models, in addition to concept drift detection? \item Do they need access to previously received data? \item Do they need additional data sets for initialisation or validation? \item Can they handle data streams with more than two classes (multi-class data)? \item Do they involve any mechanism handling $P\left( y \right)$ drift? \end{itemize} \section{Performance Analysis} \label{sec:exp} With a complete review of online class imbalance learning, we aim at a deep understanding of concept drift detection in imbalanced data streams and the performance of existing approaches introduced in Section~\ref{subsec:approach}. Three research questions will be looked into through experimental analysis: \textit{1) what are the difficulties in detecting each type of concept drift?} Little work has given separate discussions on the three fundamental types of concept drift, especially the $P\left( y \right)$ drift. It is important to understand their differences, so that the most suitable approaches can be used for the best performance. \textit{2) Among existing approaches designed for imbalanced data streams with concept drift, which approach is better and when?} Although a few approaches have been proposed for the purpose of overcoming concept drift and class imbalance, it is still unclear how well they perform for each type of concept drift. \textit{3) Whether and how do class imbalance techniques affect concept drift detection and online prediction?} No study has looked into the mutual effect of applying class imbalance techniques and concept drift detection methods. Understanding the role of class imbalance techniques will help us to develop more effective concept drift detection methods for imbalanced data. \subsection{Data Sets} \label{subsec:data} For an accurate analysis and comparable results, we choose two most commonly used artificial data generators, SINE1~\cite{Gama2004kl} and SEA~\cite{Street2001bh}, to produce imbalanced data streams containing three simulated types of concept drift. This is one of the very few studies that individually discuss $P\left( y \right)$, $p\left( \mathbf{x}\mid y \right)$ and $P\left(y \mid \mathbf{x} \right)$ types of concept drift in depth. In addition, each generator produces two data streams with a different drifting speed -- abrupt and gradual drifts. The drifting speed is defined as the inverse of the time taken for a new concept to completely replace the old one~\cite{Minku2010uq2}. According to speed, drifts can be either abrupt, when the generating function is changed completely in only one time step, or gradual, otherwise. The data streams with a gradual concept drift are denoted by `g' in the following experiment, i.e. SINE1g~\cite{Baena-Garca2006tg} and SEAg. Every data stream has 3000 time steps, with one concept drift starting at time step 1501. The new concept in SINE1 and SEA fully takes over the data stream from time step 1501; the concept drift in SINE1g and SEAg takes 500 time steps to complete, which means that the new concept fully replaces the old one from time step 2001. The detailed settings for generating each type of concept drift are included in the individual subsections. After the detailed analysis of the three types of concept drift, three real-world data sets are included in our experiment with unknown concept drift, which are PAKDD 2009 credit card data (PAKDD)~\cite{Linhart2010ln}, Weather data~\cite{Ditzler2013nh} and UDI TweeterCrawl data~\cite{LiWDWC12}. Data in PAKDD are collected from the private label credit card operation of a Brazilian retail chain. The task of this problem is to identify whether the client has a good or bad credit. The ``bad" credit is the minority class, taking 19.75\% of the provided modelling data. Because the data have been collected from a time interval in the past, gradual market change occurs. The Weather data set aims to predict whether rain precipitation was observed on each day, with inherent seasonal changes. The class of ``rain" is the minority at IR of 31\%. The original Tweet data include 50 million tweets posted mainly from 2008 to 2011. The task is to predict the tweet topic. We choose a time interval, containing 8774 examples and covering seven tweet topics~\cite{Wang2016wl}. Then, we further reduce it to 2-class data by using only two out of seven topics for our experiment. These real-world data will help us to understand the effectiveness of existing concept drift and class imbalance approaches in practical scenarios, which usually have more complex data distributions and concept drift. \subsection{Experimental and Evaluation Settings} \label{subsec:setting} \begin{table*}[htp] \caption{Artificial data streams with $P\left( y \right)$ concept drift.} \label{tab:pydata} \centering \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline ID & Data& Speed & \multicolumn{3}{|c|}{Class +1} & \multicolumn{3}{|c|}{Class -1}\\ \cline{4-9} &&& Concept & Old $P\left( y \right)$ & New $P\left( y \right)$ & Concept & Old $P\left( y \right)$ & New $P\left( y \right)$ \\ \hline 1&SINE1 & Abrupt & \multirow{2}{*}{Points below $y=\sin \left ( x \right )$} & \multirow{2}{*}{0.1} & \multirow{2}{*}{0.9} & \multirow{2}{*}{Points above or on $y=\sin \left ( x \right )$} & \multirow{2}{*}{0.9} & \multirow{2}{*}{0.1} \\ \cline{1-3} 2&SINE1g & Gradual &&&&&& \\ \hline 3&SEA & Abrupt & \multirow{2}{*}{$x_1+x_2 \leq 7$} & \multirow{2}{*}{0.5} & \multirow{2}{*}{0.1} & \multirow{2}{*}{$x_1+x_2 > 7$} & \multirow{2}{*}{0.5} & \multirow{2}{*}{0.9} \\ \cline{1-3} 4&SEAg & Gradual &&&&&& \\ \hline \end{tabular} \end{table*} The approaches listed in Table~\ref{tab:onlinemethod}, which are explicitly designed for the combined problem of class imbalance and concept drift, are discussed in our experiment. For the three active drift detection methods -- DDM-OCI, LFR and PAUC-PH, they are used with the traditional Online Bagging (abbr. OB)~\cite{Oza:2005ve} and OOB with CID~\cite{Wang2014vb} respectively for classification. Because OOB applies oversampling to overcome class imbalance and OB does not, it can help us to observe the role of class imbalance techniques (oversampling in our experiment) in concept drift detection. UOB is not chosen, for the consideration that undersampling may cause unstable performance which may indirectly affect our observation~\cite{Wang2014vb}. Between RLSACP and ONN, due to their similarity and the more theoretical support in RLSACP, only RLSACP is included in our experiment. Considering RLSACP and ESOS-ELM are perceptron-based methods, we use the Multilayer Perceptron (MLP) classifier as the base learner of OB and OOB. The number of neurons in the hidden layer of MLPs is set to the average of the number of attributes and classes in data, which is also the number of perceptrons in RLSACP and ESOS-ELM. All ensemble methods maintain 15 base learners. For ESOS-ELM, we disable the ``ELM-Store", which is designed for recurring concept drift; we allow that its ensemble size can grow to 20. In addition, ESOS-ELM requires an initialisation data set to initialize ELMs, and validation data sets to adjust misclassification costs. When dealing with artificial data, we use the first 100 examples to initialize ESOS-ELM, and generate a separate validation data set for each concept stage. We track the performance of all the methods from time step 101. In summary, ten algorithms join the comparison from Table~\ref{tab:onlinemethod}: OB, OOB, DDM-OCI+OB/OOB, PAUC-PH+OB/OOB, LFR+OB/OOB, RLSACP and ESOS-ELM. OB is the baseline without involving any class imbalance and concept drift techniques. To evaluate the effectiveness of concept drift detection methods and online learners, we adopt prequential test (as described in Section~\ref{sec:framework}) for its simplicity and popularity. Prequential recall of each class (defined in Eq.~\ref{eq:recall}) and prequential G-mean (defined in Eq.~\ref{eq:G}) are tracked over time for comparison, because they are insensitive to imbalance rates. When discussing the generated artificial data sets with ground truth known, we also compare the true detection rate (abbr. TDR), total number of false alarms (abbr. FA) and delay of detection (abbr. DoD) (as defined in Section~\ref{sec:framework}) among methods using any of the three active drift detectors (i.e. DDM-OCI, LFR and PAUC-PH). The calculation of TDR, FA and DoD is based on the following understanding: before a real concept drift occurs, all the reported alarms are considered as false alarms; after a real concept drift occurs, the first detection is seen as the true alarm; after that and before the next new real concept drift, the consequent detections are considered as false alarms. Furthermore, because we are particularly interested in how the learner performs on the new data concept in the artificial data sets, we calculate the average recall and G-mean over all the time steps before the concept drift starts and after the concept drift completely ends. It is worth noting that the recall and G-mean values are reset to 0 when the drift starts and ends for an accurate analysis. We use the Wilcoxon Sign Rank test at the confidence level of 95\% as our significance test in this paper. \subsection{Comparative Study on Artificial Data} \label{subsec:artificial_analysis} \noindent C.1. $\mathbf P\left( y \right)$ \textbf{Concept Drift} This section focuses on the $P\left( y \right)$ type of concept drift, without $p\left( \mathbf{x} \mid y \right)$ and $P\left( y \mid \mathbf{x} \right)$ changes. Data streams SINE1 and SINE1g have a severe class imbalance change, in which the minority (majority) class during the first half of data streams becomes the majority (minority) during the latter half. SEA and SEAg have a less severe change, in which the data stream presented to be balanced during the first half becomes imbalanced during the latter half. The concrete setting for each data stream is summarized in Table~\ref{tab:pydata}. Table~\ref{tab:pyDetectors} compares the detection performance of the three active concept drift detectors, in terms of TDR, FA and DoD. The first column is the data ID number, as denoted in Table~\ref{tab:pydata}. We can see that DDM-OCI and LFR are sensitive to class imbalance changes in data. They present very high true detection rate; especially, LFR has 100\% TDR in all cases regardless of whether resampling is used to tackle class imbalance. PAUC-PH does not report any concept drift, showing 0\% TDR in all cases. This is because DDM-OCI and LFR use time-decayed metrics as the indicator of concept drift, which have higher sensitivity to performance change in general than the prequential AUC used by PAUC-PH. LFR shows even higher TDR than DDM-OCI, because it tracks four rates in the confusion matrix instead of one. For the same reason, DDM-OCI and LFR have a higher chance of issuing false alarms than PAUC-PH. For DDM-OCI, oversampling in OOB increases the probability of reporting a concept drift by observing TDR in SEA and SEAg, compared to OB. This is because more examples are used for training in OOB, which improves the performance on the minority class for concept drift detection. \begin{table}[htp] \caption{Performance of the 3 active concept drift detectors on artificial data with $P\left( y \right)$ changes: TDR, FA and DoD. The `-' symbol indicates that no concept drift is detected.} \label{tab:pyDetectors} \centering \begin{tabular}{|c|c|c|c|c|} \hline & Method & TDR & FA & DoD \\ \hline \multirow{6}{*}{\begin{turn}{90}SINE1\end{turn}} & DDM-OCI+OB & 100\% & 0 & 94 \\ \cline{2-5} &DDM-OCI+OOB & 100\% & 2.22 & 45 \\ \cline{2-5} &LFR+OB & 100\% & 24 & 91\\ \cline{2-5} &LFR+OOB & 100\% & 26.16 & 63\\ \cline{2-5} &PAUC-PH+OB & 0\% & 1.03 & -\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 1.28 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SINE1g\end{turn}} & DDM-OCI+OB & 100\% & 1.09 & 281 \\ \cline{2-5} &DDM-OCI+OOB & 100\% & 4.38 & 118 \\ \cline{2-5} &LFR+OB & 100\% & 18.01 & 383 \\ \cline{2-5} &LFR+OOB & 100\% & 21.15 & 153 \\ \cline{2-5} &PAUC-PH+OB & 0\% & 1 & - \\ \cline{2-5} &PAUC-PH+OOB & 0\% & 1 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SEA\end{turn}} &DDM-OCI+OB & 45\% & 11.9 & 255\\ \cline{2-5} &DDM-OCI+OOB & 94\% & 14.1 & 301\\ \cline{2-5} &LFR+OB & 100\% & 0.73 & 35\\ \cline{2-5} &LFR+OOB & 100\% & 6.51 & 45\\ \cline{2-5} &PAUC-PH+OB & 0\% & 1 & -\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 1 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SEAg\end{turn}} & DDM-OCI+OB & 92\% & 15.1 & 80\\ \cline{2-5} & DDM-OCI+OOB & 100\% & 16.56 & 93\\ \cline{2-5} & LFR+OB & 100\% & 2.27 & 121\\ \cline{2-5} & LFR+OOB & 100\% & 6.3 & 324\\ \cline{2-5} & PAUC-PH+OB & 0\% & 1 & -\\ \cline{2-5} & PAUC-PH+OOB & 0\% & 1.01 & - \\ \hline \end{tabular} \end{table} Table~\ref{tab:pyLearners} compares recall and G-mean of all models over the new data concept, i.e. performance over time steps 1501-3000 for data streams with an abrupt change and performance over time steps 2001-3000 for data streams with a gradual change, showing whether and how well the drift detector can help with learning after concept drift is completed. The first column is the data ID number, as denoted in Table~\ref{tab:pydata}. In SINE1 and SINE1g, the negative class presents to be the minority after the change; in SEA and SEAg, the positive class presents to be the minority after the change. \begin{table}[htp] \caption{Performance of online learners on artificial data with $P\left( y \right)$ changes: means and standard deviations of average recall of each class and average G-mean over the new data concept. The significantly best values among all methods are shown in bold italics.} \label{tab:pyLearners} \centering \resizebox{\columnwidth}{!}{ \begin{tabular}{|c|c|c|c|c|} \hline & Method & Class+1 Recall & Class-1 Recall & G-mean \\ \hline \multirow{10}{*}{\begin{turn}{90}SINE1\end{turn}} & DDM-OCI+OB & 0.887$\pm$0.004 & 0.170$\pm$0.009 & 0.317$\pm$0.009 \\ \cline{2-5} &DDM-OCI+OOB & 0.979$\pm$0.007 & 0.049$\pm$0.016 & 0.188$\pm$0.033 \\ \cline{2-5} &LFR+OB & 0.870$\pm$0.004 & 0.183$\pm$0.019 & 0.334$\pm$0.022\\ \cline{2-5} &LFR+OOB & 0.952$\pm$0.011 & 0.061$\pm$0.023 & 0.221$\pm$0.042\\ \cline{2-5} &PAUC-PH+OB & 0.889$\pm$0.004 & 0.168$\pm$0.008 & 0.316$\pm$0.007\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.992$\pm$0.002} & 0.692$\pm$0.013 & 0.828$\pm$0.008 \\ \cline{2-5} &RLSACP & 0.962$\pm$0.004 & 0.072$\pm$0.014 & 0.217$\pm$0.026 \\ \cline{2-5} &ESOS-ELM & 0.176$\pm$0.136 & \textbf{0.999$\pm$0.001} & 0.358$\pm$0.192 \\ \cline{2-5} &OB & 0.889$\pm$0.004 & 0.170$\pm$0.009 & 0.318$\pm$0.009 \\ \cline{2-5} &OOB & \textbf{0.992$\pm$0.002} & 0.699$\pm$0.014 & \textbf{0.832$\pm$0.008}\\ \hline \multirow{10}{*}{\begin{turn}{90}SINE1g\end{turn}} & DDM-OCI+OB & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 & 0.000$\pm$0.000\\ \cline{2-5} &DDM-OCI+OOB & 0.997$\pm$0.004 & 0.008$\pm$0.005 & 0.050$\pm$0.016\\ \cline{2-5} &LFR+OB & 0.972$\pm$0.006 & 0.031$\pm$0.027 & 0.138$\pm$0.079\\ \cline{2-5} &LFR+OOB & 0.956$\pm$0.011 & 0.036$\pm$0.026 & 0.150$\pm$0.076\\ \cline{2-5} &PAUC-PH+OB & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 & 0.000$\pm$0.000\\ \cline{2-5} &PAUC-PH+OOB & 0.989$\pm$0.001 & 0.708$\pm$0.002 & \textbf{0.835$\pm$0.002}\\ \cline{2-5} &RLSACP & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.001 & 0.002$\pm$0.013\\ \cline{2-5} &ESOS-ELM & 0.109$\pm$0.102 & \textbf{0.997$\pm$0.000} & 0.273$\pm$0.165\\ \cline{2-5} &OB & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 & 0.000$\pm$0.000\\ \cline{2-5} &OOB & 0.989$\pm$0.002 & 0.709$\pm$0.002 & \textbf{0.835$\pm$0.001} \\ \hline \multirow{10}{*}{\begin{turn}{90}SEA\end{turn}} &DDM-OCI+OB & 0.003$\pm$0.031 & \textbf{0.999$\pm$0.000} & 0.007$\pm$0.055\\ \cline{2-5} &DDM-OCI+OOB & 0.146$\pm$0.072 & 0.965$\pm$0.013 & 0.344$\pm$0.086\\ \cline{2-5} &LFR+OB & 0.020$\pm$0.009 & 0.996$\pm$0.001 & 0.113$\pm$0.053\\ \cline{2-5} &LFR+OOB & 0.059$\pm$0.031 & 0.981$\pm$0.007 & 0.221$\pm$0.054\\ \cline{2-5} &PAUC-PH+OB & 0.323$\pm$0.010 & 0.995$\pm$0.001 & 0.559$\pm$0.009\\ \cline{2-5} &PAUC-PH+OOB & 0.514$\pm$0.015 & 0.943$\pm$0.007 & \textbf{0.688$\pm$0.010}\\ \cline{2-5} &RLSACP & 0.021$\pm$0.023 & 0.993$\pm$0.007 & 0.070$\pm$0.077\\ \cline{2-5} &ESOS-ELM & \textbf{0.608$\pm$0.214} & 0.829$\pm$0.140 & \textbf{0.681$\pm$0.142}\\ \cline{2-5} &OB & 0.324$\pm$0.009 & 0.996$\pm$0.001 & 0.561$\pm$0.008\\ \cline{2-5} &OOB & 0.515$\pm$0.016 & 0.945$\pm$0.006 & \textbf{0.689$\pm$0.010} \\ \hline \multirow{10}{*}{\begin{turn}{90}SEAg\end{turn}} & DDM-OCI+OB & 0.040$\pm$0.073 & 0.998$\pm$0.001 & 0.124$\pm$0.136\\ \cline{2-5} &DDM-OCI+OOB & 0.142$\pm$0.071 & 0.973$\pm$0.014 & 0.334$\pm$0.096\\ \cline{2-5} &LFR+OB & 0.003$\pm$0.006 & \textbf{0.999$\pm$0.000} & 0.019$\pm$0.035\\ \cline{2-5} &LFR+OOB & 0.076$\pm$0.084 & 0.976$\pm$0.018 & 0.217$\pm$0.123\\ \cline{2-5} &PAUC-PH+OB & 0.365$\pm$0.029 & 0.997$\pm$0.000 & 0.600$\pm$0.023\\ \cline{2-5} &PAUC-PH+OOB & 0.489$\pm$0.024 & 0.951$\pm$0.011 & \textbf{0.679$\pm$0.017}\\ \cline{2-5} &RLSACP & 0.002$\pm$0.006 & \textbf{0.999$\pm$0.001} & 0.011$\pm$0.035\\ \cline{2-5} &ESOS-ELM & \textbf{0.562$\pm$0.208} & 0.809$\pm$0.143 & 0.646$\pm$0.130\\ \cline{2-5} &OB & 0.371$\pm$0.029 & 0.997$\pm$0.001 & 0.605$\pm$0.023\\ \cline{2-5} &OOB & 0.484$\pm$0.032 & 0.951$\pm$0.012 & \textbf{0.675$\pm$0.022} \\ \hline \end{tabular} } \end{table} \begin{table*}[htp] \caption{Artificial data streams with $p\left( \mathbf{x} \mid y \right)$ concept drift.} \label{tab:pxydata} \centering \begin{tabular}{|c|c|c|c|c|c|c|} \hline ID & Data& Speed & \multicolumn{2}{|c|}{Class +1} & \multicolumn{2}{|c|}{Class -1}\\ \cline{4-7} &&& Old concept & New concept & Old concept & New concept \\ \hline 1&SINE1 & Abrupt & \multirow{2}{*}{Points below $y=\sin \left ( x \right )$} & \multirow{2}{*}{Points below $y=\sin \left ( x \right )$} & Points above or on $y=\sin \left ( x \right )$ & Points above or on $y=\sin \left ( x \right )$ \\ \cline{1-3} 2&SINE1g & Gradual &&&and $P\left ( x<0.5 \right ) = 0.9$&and $P\left ( x<0.5 \right ) = 0.1$ \\ \hline 3&SEA & Abrupt & \multirow{2}{*}{$x_1+x_2 \leq 7$} & \multirow{2}{*}{$x_1+x_2 \leq 7$} & $x_1+x_2 > 7$ & $x_1+x_2 > 7$ \\ \cline{1-3} 4&SEAg & Gradual &&&and $P\left ( x_1<5 \right ) = 0.9$&and $P\left ( x_1<5 \right ) = 0.1$ \\ \hline \end{tabular} \end{table*} In terms of minority-class recall, we can see that ESOS-ELM performs the significantly best, but ESOS-ELM sacrifices majority-class recall, especially in SINE1 and SINE1g. In terms of G-mean, OOB and OOB using PAUC-PH perform the significantly best, which shows they can best balance the performance between classes. It is worth noting that PAUC-PH is the drift detection method with 0\% TDR based on Table~\ref{tab:pyDetectors}. It means that OOB plays the main role in learning. It also explains that OOB and OOB using PAUC-PH have very close performance. All the OB and OOB models using the other active drift detectors do not show competitive recall and G-mean. Especially for those using DDM-OCI and LFR, the high number of false alarms causes too much resetting and performance loss; OOB can increase the chance of producing a false alarm, because more minority-class examples join the training. Therefore, we conclude that, for $P\left( y \right)$ type of concept drift, it is not necessary to apply any drift detection techniques that are not specifically designed for class imbalance changes; the use of these drift detectors could be even detrimental to the predictive performance due to false alarms and performance resetting; the adaptive resampling in OOB is sufficient to deal with the change and maintain the predictive performance; when using OOB with other active concept drift detectors, the number of false alarms and performance resetting need to be carefully considered. \noindent C.2. $\mathbf p\left(\mathbf{x} \mid y\right)$ \textbf{Concept Drift} The data streams in this section only involve $p\left( \mathbf{x} \mid y \right)$ type of concept drift, without $P\left( y \right)$ and $P\left( y \mid \mathbf{x} \right)$ changes. The class imbalance ratio is fixed to 1:9 and we let the positive class be the minority, so that the data stream is constantly imbalanced. The concept drift in each data stream is controlled by $p\left ( \mathbf{x} \right )$ of the negative class, as shown in Table~\ref{tab:pxydata}. Table~\ref{tab:pxyDetectors} compares the detection performance of the three active concept drift detectors. Similar to our previous results, DDM-OCI and LFR are more sensitive to $P\left( x \mid y \right)$ changes than PAUC-PH. When DDM-OCI and LFR work with OOB, their TDR shows 100\%; and LFR has higher FA and shorter DOD than DDM-OCI, due to more indicators it monitors. PAUC-PH shows 0\% TDR in most cases of working with both OB and OOB. Different from $P\left( y \right)$ changes, when DDM-OCI and LFR work with OB, their TDR is rather low, which suggests that their sensitivity is dependent on the class imbalance techniques. Unlike the cases with class imbalance changes, where it is possible for the minority-class examples to become more frequent, the data streams generated in this section have a fixed minority class with a constantly small prior probability. In other words, it would be more difficult to recognize examples from this minority class, which indirectly affects the detection sensitivity of DDM-OCI and LFR. When oversampling is applied, which introduces more training examples for the minority class, the performance metrics (G-mean, recall and precision) monitored by DDM-OCI and LFR can be substantially improved. It also increases the possibility of reporting a concept drift. This explains the low detection rate of DDM-OCI and LFR when working with OB and their high detection rate when working with OOB. \begin{table}[htp] \caption{Performance of the 3 active concept drift detectors on artificial data with $p\left(\mathbf{x} \mid y\right)$ changes: TDR, FA and DoD. The `-' symbol indicates that no concept drift is detected.} \label{tab:pxyDetectors} \centering \begin{tabular}{|c|c|c|c|c|} \hline & Method & TDR & FA & DoD \\ \hline \multirow{6}{*}{\begin{turn}{90}SINE1\end{turn}} & DDM-OCI+OB & 0\% & 0 & - \\ \cline{2-5} &DDM-OCI+OOB & 100\% & 1.25 & 594 \\ \cline{2-5} &LFR+OB & 0\% & 0.05 & -\\ \cline{2-5} &LFR+OOB & 100\% & 3.99 & 528\\ \cline{2-5} &PAUC-PH+OB & 4\% & 0.45 & 232\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 0.45 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SINE1g\end{turn}} & DDM-OCI+OB & 0\% & 0 & - \\ \cline{2-5} &DDM-OCI+OOB & 100\% & 1.37 & 387 \\ \cline{2-5} &LFR+OB & 0\% & 0 & -\\ \cline{2-5} &LFR+OOB & 100\% & 5.45 & 258 \\ \cline{2-5} &PAUC-PH+OB & 0\% & 1.04 & -\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 1 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SEA\end{turn}} &DDM-OCI+OB & 16\% & 1 & 1394\\ \cline{2-5} &DDM-OCI+OOB & 100\% & 4.03 & 473\\ \cline{2-5} &LFR+OB & 100\% & 0.31 & 52\\ \cline{2-5} &LFR+OOB & 100\% & 13.48 & 59\\ \cline{2-5} &PAUC-PH+OB & 0\% & 0 & -\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 0.85 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SEAg\end{turn}} & DDM-OCI+OB & 90\% & 0.15 & 238\\ \cline{2-5} & DDM-OCI+OOB & 100\% & 4.03 & 279\\ \cline{2-5} & LFR+OB & 29\% & 0 & 1154\\ \cline{2-5} & LFR+OOB & 100\% & 12.75 & 196\\ \cline{2-5} & PAUC-PH+OB & 0\% & 1 & -\\ \cline{2-5} & PAUC-PH+OOB & 0\% & 1 & - \\ \hline \end{tabular} \end{table} Table~\ref{tab:pxyLearners} compares recall and G-mean of all models over the new data concept. As we expected, almost all OB models show significantly worse minority-class recall and G-mean. On SINE1 and SINE1g data, minority-class recall of OB models is as low as 0, which may hinder the detection of any concept drift. Among the OOB models, those using DDM-OCI and LFR perform significantly worse than OOB using PAUC-PH and OOB itself, and the latter two show very close performance. This is because DDM-OCI and LFR trigger concept drift with false alarms, and cause model resetting multiple times. Along with the resetting, the useful and valid information learnt in the past is forgotten at the same time. For the two passive models, RLSACP and ESOS-ELM do not perform very well compared to OOB. Generally speaking, for imbalanced data streams with $p\left( \mathbf{x} \mid y \right)$ changes, class imbalance seems to be a more important issue than concept drift, considering that the learning model without triggering any concept drift detection achieves the best performance. Besides, while the adopted class imbalance technique can improve the final prediction, it can also improve the performance of active concept drift detection methods, depending on their working mechanism. \begin{table}[htp] \caption{Performance of online learners on artificial data with $p\left(\mathbf{x} \mid y\right)$ changes: means and standard deviations of average recall of each class and average G-mean over the new data concept. The significantly best values among all methods are shown in bold italics.} \label{tab:pxyLearners} \centering \resizebox{\columnwidth}{!}{ \begin{tabular}{|c|c|c|c|c|} \hline & Method & Class+1 Recall & Class-1 Recall & G-mean \\ \hline \multirow{10}{*}{\begin{turn}{90}SINE1\end{turn}} & DDM-OCI+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 \\ \cline{2-5} &DDM-OCI+OOB & 0.036$\pm$0.025 & 0.997$\pm$0.002 & 0.145$\pm$0.052 \\ \cline{2-5} &LFR+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &LFR+OOB & 0.061$\pm$0.036 & 0.994$\pm$0.005 & 0.200$\pm$0.066\\ \cline{2-5} &PAUC-PH+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.689$\pm$0.038} & 0.985$\pm$0.004 & \textbf{0.811$\pm$0.027} \\ \cline{2-5} &RLSACP & 0.090$\pm$0.028 & 0.939$\pm$0.012 & 0.251$\pm$0.045 \\ \cline{2-5} &ESOS-ELM & 0.058$\pm$0.122 & \textbf{1.000$\pm$0.000} & 0.113$\pm$0.208 \\ \cline{2-5} &OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 \\ \cline{2-5} &OOB & \textbf{0.696$\pm$0.020} & 0.985$\pm$0.004 & \textbf{0.817$\pm$0.013}\\ \hline \multirow{10}{*}{\begin{turn}{90}SINE1g\end{turn}} & DDM-OCI+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &DDM-OCI+OOB & 0.035$\pm$0.064 & 0.993$\pm$0.006 & 0.096$\pm$0.135\\ \cline{2-5} &LFR+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &LFR+OOB & 0.038$\pm$0.062 & 0.992$\pm$0.008 & 0.111$\pm$0.132\\ \cline{2-5} &PAUC-PH+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.801$\pm$0.032} & 0.988$\pm$0.003 & \textbf{0.884$\pm$0.019}\\ \cline{2-5} &RLSACP & 0.072$\pm$0.049 & 0.952$\pm$0.009 & 0.173$\pm$0.102\\ \cline{2-5} &ESOS-ELM & 0.077$\pm$0.112 & 0.991$\pm$0.035 & 0.162$\pm$0.215\\ \cline{2-5} &OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &OOB & \textbf{0.802$\pm$0.034} & 0.988$\pm$0.003 & \textbf{0.884$\pm$0.021} \\ \hline \multirow{10}{*}{\begin{turn}{90}SEA\end{turn}} &DDM-OCI+OB & 0.001$\pm$0.000 & \textbf{0.999$\pm$0.000} & 0.002$\pm$0.006\\ \cline{2-5} &DDM-OCI+OOB & 0.144$\pm$0.027 & 0.973$\pm$0.007 & 0.332$\pm$0.040\\ \cline{2-5} &LFR+OB & 0.036$\pm$0.012 & 0.984$\pm$0.005 & 0.144$\pm$0.048\\ \cline{2-5} &LFR+OOB & 0.085$\pm$0.039 & 0.971$\pm$0.015 & 0.243$\pm$0.069\\ \cline{2-5} &PAUC-PH+OB & 0.130$\pm$0.027 & 0.983$\pm$0.004 & 0.341$\pm$0.042\\ \cline{2-5} &PAUC-PH+OOB & 0.459$\pm$0.044 & 0.923$\pm$0.010 & 0.645$\pm$0.030\\ \cline{2-5} &RLSACP & 0.000$\pm$0.001 & \textbf{0.999$\pm$0.001} & 0.001$\pm$0.006\\ \cline{2-5} &ESOS-ELM & 0.202$\pm$0.158 & 0.967$\pm$0.071 & 0.394$\pm$0.167\\ \cline{2-5} &OB & 0.130$\pm$0.027 & 0.983$\pm$0.004 & 0.341$\pm$0.042\\ \cline{2-5} &OOB & \textbf{0.477$\pm$0.031} & 0.919$\pm$0.010 & \textbf{0.657$\pm$0.021} \\ \hline \multirow{10}{*}{\begin{turn}{90}SEAg\end{turn}} & DDM-OCI+OB & 0.002$\pm$0.007 & \textbf{1.000$\pm$0.000} & 0.010$\pm$0.035\\ \cline{2-5} &DDM-OCI+OOB & 0.100$\pm$0.040 & 0.978$\pm$0.008 & 0.257$\pm$0.066\\ \cline{2-5} &LFR+OB & 0.101$\pm$0.027 & 0.999$\pm$0.000 & 0.269$\pm$0.058\\ \cline{2-5} &LFR+OOB & 0.050$\pm$0.029 & 0.980$\pm$0.011 & 0.182$\pm$0.065\\ \cline{2-5} &PAUC-PH+OB & 0.107$\pm$0.025 & 0.999$\pm$0.000 & 0.278$\pm$0.046\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.348$\pm$0.023} & 0.939$\pm$0.017 & \textbf{0.553$\pm$0.019}\\ \cline{2-5} &RLSACP & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.002\\ \cline{2-5} &ESOS-ELM & 0.183$\pm$0.137 & 0.964$\pm$0.090 & 0.368$\pm$0.161\\ \cline{2-5} &OB & 0.106$\pm$0.021 & 0.999$\pm$0.000 & 0.279$\pm$0.040\\ \cline{2-5} &OOB & \textbf{0.345$\pm$0.027} & 0.943$\pm$0.018 & \textbf{0.552$\pm$0.022} \\ \hline \end{tabular} } \end{table} \noindent C.3. $\mathbf P\left(y \mid \mathbf{x}\right)$ \textbf{Concept Drift} \begin{table*}[htp] \caption{Artificial data streams with $P\left( y \mid \mathbf{x} \right)$ concept drift.} \label{tab:pyxdata} \centering \begin{tabular}{|c|c|c|c|c|c|c|} \hline ID & Data& Speed & \multicolumn{2}{|c|}{Class +1} & \multicolumn{2}{|c|}{Class -1}\\ \cline{4-7} &&& Old concept & New concept & Old concept & New concept \\ \hline 1&SINE1 & Abrupt & \multirow{2}{*}{Points below $y=\sin \left ( x \right )$} & \multirow{2}{*}{Points above/on $y=\sin \left ( x \right )$} & \multirow{2}{*}{Points above/on $y=\sin \left ( x \right )$} & \multirow{2}{*}{Points below $y=\sin \left ( x \right )$} \\ \cline{1-3} 2&SINE1g & Gradual &&&& \\ \hline 3&SEA & Abrupt & \multirow{2}{*}{$x_1+x_2 \leq 7$} & \multirow{2}{*}{$x_1+x_2 \leq 13$} & \multirow{2}{*}{$x_1+x_2 > 7$} & \multirow{2}{*}{$x_1+x_2 > 13$} \\ \cline{1-3} 4&SEAg & Gradual &&&&\\ \hline \end{tabular} \end{table*} The data streams in this section only involve $P\left( y \mid \mathbf{x} \right)$ type of concept drift, without $P\left( y \right)$ and $p\left( \mathbf{x} \mid y \right)$ changes. Following the settings in Section~\ref{subsec:artificial_analysis}.2, we fix the class imbalance ratio to 1:9 and let the positive class be the minority, so that the data stream is constantly imbalanced. As shown in Table~\ref{tab:pyxdata}, the data distribution in SINE1 and SINE1g involves a concept swap, and this change occurs probabilistically in SINE1g; the data distribution in SEA and SEAg has a concept threshold moving, and this change occurs continuously in SEAg. The change in SEA and SEAg is less severe than the change in SINE1 and SINE1g, because some of the examples from the old concept are still valid under the new concept after the threshold moves completely. The concept drift discussed in this section belongs to the real concept drift category, which affects the classification boundary and is expected to be captured by all concept drift detectors. According to Table~\ref{tab:pyxDetectors}, we can see that DDM-OCI and LFR have difficulty in detecting the concept drift when working with OB, because of the poor recall and G-mean produced by OB, which is also observed and explained in Section~\ref{subsec:artificial_analysis}.2. When DDM-OCI and LFR work with OOB, their detection rate TDR is greatly improved (above 90\% in most cases). This is because the improved performance metrics facilitate the detection. LFR is more sensitive to the change, which produces higher FA and shorter DoD. Different from previous observations in terms of concept drift detection performance, PAUC-PH working with OB produces 100\% TDR and low FA on data streams SINE1 and SINE1g, but PAUC-PH does not work well with OOB on the same data. It is interesting to see that oversampling does not always play a positive role in drift detection. One possible reason is that class imbalance techniques may sometimes hide the performance drop caused by the real concept drift, while it tries to maintain the overall predictive performance, especially for AUC type of metrics in our case. On data streams SEA and SEAg, PAUC-PH does not report any concept drift, probably due to the less severe concept drift. \begin{table}[htp] \caption{Performance of the 3 active concept drift detectors on artificial data with $P\left(y \mid \mathbf{x}\right)$ changes: TDR, FA and DoD. The `-’ symbol indicates that no concept drift is detected.} \label{tab:pyxDetectors} \centering \begin{tabular}{|c|c|c|c|c|} \hline & Method & TDR & FA & DoD \\ \hline \multirow{6}{*}{\begin{turn}{90}SINE1\end{turn}} & DDM-OCI+OB & 0\% & 0 & - \\ \cline{2-5} &DDM-OCI+OOB & 97\% & 1.02 & 1166 \\ \cline{2-5} &LFR+OB & 0\% & 0 & -\\ \cline{2-5} &LFR+OOB & 91\% & 3.92 & 783\\ \cline{2-5} &PAUC-PH+OB & 100\% & 1.03 & 884\\ \cline{2-5} &PAUC-PH+OOB & 2\% & 1.28 & 1180 \\ \hline \multirow{6}{*}{\begin{turn}{90}SINE1g\end{turn}} & DDM-OCI+OB & 0\% & 0 & - \\ \cline{2-5} &DDM-OCI+OOB & 69\% & 2.16 & 165 \\ \cline{2-5} &LFR+OB & 0\% & 1 & -\\ \cline{2-5} &LFR+OOB & 85\% & 6.21 & 306 \\ \cline{2-5} &PAUC-PH+OB & 100\% & 1.03 & 1119\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 1 & - \\ \hline \multirow{6}{*}{\begin{turn}{90}SEA\end{turn}} &DDM-OCI+OB & 61\% & 0.39 & 23\\ \cline{2-5} &DDM-OCI+OOB & 100\% & 3.87 & 151\\ \cline{2-5} &LFR+OB & 10\% & 0.02 & 865\\ \cline{2-5} &LFR+OOB & 100\% & 13.73 & 65\\ \cline{2-5} &PAUC-PH+OB & 0\% & 1 & -\\ \cline{2-5} &PAUC-PH+OOB & 0\% & 1 & -\\ \hline \multirow{6}{*}{\begin{turn}{90}SEAg\end{turn}} & DDM-OCI+OB & 100\% & 0 & 71\\ \cline{2-5} & DDM-OCI+OOB & 100\% & 3.9 & 342\\ \cline{2-5} & LFR+OB & 3\% & 0.02 & 1036\\ \cline{2-5} & LFR+OOB & 100\% & 13.59 & 123\\ \cline{2-5} & PAUC-PH+OB & 0\% & 1 & -\\ \cline{2-5} & PAUC-PH+OOB & 0\% & 1 & - \\ \hline \end{tabular} \end{table} \begin{table}[htp] \caption{Performance of online learners on artificial data with $P\left(y \mid \mathbf{x}\right)$ changes: means and standard deviations of average recall of each class and average G-mean over the new data concept. The significantly best values among all methods are shown in bold italics.} \label{tab:pyxLearners} \centering \resizebox{\columnwidth}{!}{ \begin{tabular}{|c|c|c|c|c|} \hline & Method & Class+1 Recall & Class-1 Recall & G-mean \\ \hline \multirow{10}{*}{\begin{turn}{90}SINE1\end{turn}} & DDM-OCI+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 \\ \cline{2-5} &DDM-OCI+OOB & 0.004$\pm$0.003 & 0.998$\pm$0.002 & 0.030$\pm$0.016 \\ \cline{2-5} &LFR+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &LFR+OOB & 0.013$\pm$0.010 & 0.996$\pm$0.006 & 0.062$\pm$0.036\\ \cline{2-5} &PAUC-PH+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.031$\pm$0.013} & 0.941$\pm$0.009 & \textbf{0.098$\pm$0.026} \\ \cline{2-5} &RLSACP & 0.000$\pm$0.001 & \textbf{0.999$\pm$0.001} & 0.003$\pm$0.010 \\ \cline{2-5} &ESOS-ELM & 0.000$\pm$0.000 & 0.997$\pm$0.003 & 0.000$\pm$0.000 \\ \cline{2-5} &OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000 \\ \cline{2-5} &OOB & \textbf{0.033$\pm$0.012} & 0.942$\pm$0.009 & \textbf{0.102$\pm$0.022}\\ \hline \multirow{10}{*}{\begin{turn}{90}SINE1g\end{turn}} & DDM-OCI+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &DDM-OCI+OOB & 0.014$\pm$0.017 & 0.993$\pm$0.006 & 0.069$\pm$0.074\\ \cline{2-5} &LFR+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &LFR+OOB & 0.019$\pm$0.018 & 0.993$\pm$0.006 & 0.086$\pm$0.077\\ \cline{2-5} &PAUC-PH+OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.031$\pm$0.011} & 0.993$\pm$0.002 & \textbf{0.103$\pm$0.026}\\ \cline{2-5} &RLSACP & 0.000$\pm$0.001 & \textbf{1.000$\pm$0.000} & 0.001$\pm$0.008\\ \cline{2-5} &ESOS-ELM & 0.000$\pm$0.000 & 0.907$\pm$0.140 & 0.000$\pm$0.000\\ \cline{2-5} &OB & 0.000$\pm$0.000 & \textbf{1.000$\pm$0.000} & 0.000$\pm$0.000\\ \cline{2-5} &OOB & 0.027$\pm$0.010 & 0.995$\pm$0.002 & 0.093$\pm$0.028 \\ \hline \multirow{10}{*}{\begin{turn}{90}SEA\end{turn}} &DDM-OCI+OB & 0.013$\pm$0.022 & \textbf{0.999$\pm$0.001} & 0.050$\pm$0.085\\ \cline{2-5} &DDM-OCI+OOB & 0.110$\pm$0.031 & 0.968$\pm$0.008 & 0.311$\pm$0.057\\ \cline{2-5} &LFR+OB & 0.149$\pm$0.025 & \textbf{0.999$\pm$0.000} & 0.378$\pm$0.036\\ \cline{2-5} &LFR+OOB & 0.031$\pm$0.022 & 0.964$\pm$0.016 & 0.144$\pm$0.071\\ \cline{2-5} &PAUC-PH+OB & 0.153$\pm$0.023 & \textbf{0.999$\pm$0.000} & 0.384$\pm$0.031\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.292$\pm$0.017} & 0.967$\pm$0.008 & \textbf{0.530$\pm$0.015}\\ \cline{2-5} &RLSACP & 0.013$\pm$0.013 & 0.995$\pm$0.001 & 0.072$\pm$0.063\\ \cline{2-5} &ESOS-ELM & 0.065$\pm$0.068 & 0.997$\pm$0.022 & 0.222$\pm$0.106\\ \cline{2-5} &OB & 0.152$\pm$0.023 & \textbf{0.999$\pm$0.000} & 0.383$\pm$0.032\\ \cline{2-5} &OOB & 0.287$\pm$0.014 & 0.966$\pm$0.008 & 0.525$\pm$0.012 \\ \hline \multirow{10}{*}{\begin{turn}{90}SEAg\end{turn}} & DDM-OCI+OB & 0.000$\pm$0.002 & \textbf{1.000$\pm$0.000} & 0.001$\pm$0.013\\ \cline{2-5} &DDM-OCI+OOB & 0.042$\pm$0.022 & 0.988$\pm$0.006 & 0.163$\pm$0.059\\ \cline{2-5} &LFR+OB & 0.145$\pm$0.032 & 0.999$\pm$0.000 & 0.356$\pm$0.066\\ \cline{2-5} &LFR+OOB & 0.024$\pm$0.018 & 0.985$\pm$0.006 & 0.112$\pm$0.065\\ \cline{2-5} &PAUC-PH+OB & 0.152$\pm$0.019 & 0.999$\pm$0.000 & 0.370$\pm$0.027\\ \cline{2-5} &PAUC-PH+OOB & \textbf{0.288$\pm$0.034} & 0.974$\pm$0.010 & \textbf{0.512$\pm$0.036}\\ \cline{2-5} &RLSACP & 0.009$\pm$0.018 & \textbf{1.000$\pm$0.000} & 0.043$\pm$0.077\\ \cline{2-5} &ESOS-ELM & 0.138$\pm$0.088 & 0.993$\pm$0.057 & 0.336$\pm$0.106\\ \cline{2-5} &OB & 0.149$\pm$0.025 & 0.999$\pm$0.000 & 0.364$\pm$0.042\\ \cline{2-5} &OOB & \textbf{0.282$\pm$0.032} & 0.974$\pm$0.008 & \textbf{0.506$\pm$0.034} \\ \hline \end{tabular} } \end{table} \begin{figure*} \caption{Time-decayed G-mean curves (decay factor = 0.995) from OOB, PAUC-PH+OOB and ESOS-ELM on real-world data.} \label{fig:realdata} \end{figure*} The recall and G-mean over the new data concept in Table~\ref{tab:pyxLearners} further confirms the above analysis. The OB models produce very low minority-class recall and thus low G-mean. RLSACP and ESOS-ELM do not perform well on the new data concept either. By comparing the models that captures concept drifts (DDM-OCI+OOB, LFR+OOB, PAUC-PH+OB) and the models without reporting any concept drift (PAUC-PH+OOB, OOB), it seems that class imbalance causes a more difficult learning issue than the real concept drift in our cases. The models solely tackling class imbalance produce the significantly best recall and G-mean. The rather low imbalance ratio (i.e. 1:9) could be a reason. It would be worth discussing various imbalance levels in data with concept drift in our future work, in order to find out when it is worthwhile considering concept drift in imbalanced data streams. By comparing the results in Table~\ref{tab:pyxLearners}, Table~\ref{tab:pxyLearners} and Table~\ref{tab:pyLearners}, the $P\left( y \mid \mathbf{x} \right)$ type of concept drift indeed leads to the most performance reduction. It is consistent with our understanding that the real concept drift is the most radical type of change in data. However, existing approaches do not seem to tackle it well when data streams are very imbalanced. To develop better concept drift detection methods, the key issues here include how to best have them and class imbalance techniques work together and how to tackle the performance loss brought by false alarms. \subsection{Comparative Study on Real-World Data} \label{subsec:real_analysis} After the detailed analysis of the three types of concept drift, we now look into the performance of the above learning models on the three real-world data sets (PAKDD~\cite{Linhart2010ln}, Weather~\cite{Ditzler2013nh} and Tweet~\cite{LiWDWC12}) described in Section~\ref{subsec:data}. Based on the experimental results on the artificial data, we focus on the best active (PAUC+OOB) and the best passive concept drift detection methods (ESOS-ELM) here for a clear observation, in comparison with OOB. The three methods use the same parameter settings as before. The initialisation and validation data required by ESOS-ELM is the first 2\% examples of each data set. Without knowing the true concept drifts in real-world data, we calculate and track the time-decayed G-mean by setting the decay factor to 0.995, which means that the old performance is forgotten at the rate of 0.5\%. All the compared metrics are the average of 100 runs in the following figures. Fig.~\ref{fig:realdata} presents the time-decayed G-mean curves from OOB, PAUC-PH+OOB and ESOS-ELM on the three real-world data sets. The average number of reported drift by PAUC-PH is 1, 3 and 1 on Weather, PAKDD and Tweet data respectively. Compared to the artificial cases, we obtain some similar results: the passive approach ESOS-ELM does not perform as well as the other two methods; OOB and PAUC-PH show very close G-mean over time on Weather and PAKDD data, which suggests the importance of tackling class imbalance adaptively. In the PAKDD plot, we can see that the G-mean level is relatively stable without significant drop; differently, G-mean in the Tweet plot is reducing. It may suggest that the concept drift in PAKDD is less significant or influential than that in Tweet. Compared to the gradual market and environment change in PAKDD, the tweet topic change can be much faster and more noticeable. Therefore, although PAUC-PH detects 3 concept drifts in PAKDD, the two methods, OOB and PAUC-PH+OOB, does not show much difference. In tweet, PAUC-PH+OOB presents better G-mean than using OOB alone, showing the positive effect of the active concept drift detector in fast changing data streams. \subsection{Further Discussions} \label{subsec:discussion} In this section, we summarize and further discuss the results in the above comparative study on the artificial and real-world data. We also answer the research questions proposed at the beginning of this paper. When dealing with imbalanced data streams with concept drift, we have obtained the following: \begin{itemize} \item When both class imbalance and concept drift exist, class imbalance status and class imbalance changes are shown to be more crucial issues than the traditional concept drift (i.e. $p\left( \mathbf{x} \mid y \right)$ and $P\left( y \mid \mathbf{x} \right)$ changes) in terms of the online prediction performance. It is necessary to adopt adaptive class imbalance techniques (e.g. OOB discussed in our experiment), in addition to using concept drift detection methods alone (e.g. DDM-OCI, LFR). Most existing papers that proposed new concept drift detection methods for imbalanced data so far did not consider the effect of class imbalance techniques on final prediction and concept drift detection. \item $P\left( y \mid \mathbf{x} \right)$ concept drift (i.e. real concept drift) is the most severe type of change in data, compared to $p\left( \mathbf{x} \mid y \right)$ and $P\left( y \right)$ concept drift. This is based on the observation on the final prediction performance. For all three types of concept drift, existing concept drift approaches do not show much benefit in performance improvement. Concept drift is hard to be detected when no class imbalance technique is applied. Their drift detection performance is affected by the class imbalance technique, depending on their detection mechanism. \item For $P\left( y \right)$ concept drift, it is not necessary to apply any concept drift detection methods that are not designed for class imbalance changes, due to their false alarms and model resetting. It is crucial to detect and handle the class imbalance change in time. \item From the results on real-world data, we see that the effectiveness of traditional concept drift detectors (e.g. PAUC-PH) depends on the type of concept drift. For fast and significant concept drift, applying PAUC-PH seems to be more beneficial to the prediction performance. \item Among existing methods designed for imbalanced data with concept drift (4 active methods and 2 passive methods), the passive methods (i.e. ESOS-ELM and RLSACP) do not perform well in general. Although they contain both class imbalance and concept drift techniques, firstly, their class imbalance technique is not effectively adaptive to class imbalance changes, so that wrong imbalance status might be used during learning; secondly, they are restricted to the use of certain perceptron-based classifiers, so that the disadvantages of the classifiers are also inherited by the online model. For example, the training of OS-ELM in ESOS-ELM requires initialisation and validation data sets reflecting the correct data concepts, and the weighted OS-ELM was found to over-emphasize the minority class and present large performance variance sometimes in earlier studies~\cite{Wang2014vb}. \item Among the three active methods discussed in this work, which are DDM-OCI, LFR and PAUC-PH, DDM-OCI and LFR are more sensitive to concept drift than PAUC-PH, with a higher detection rate but also higher false alarms. In addition, the detection performance of DDM-OCI and LFR can be greatly improved by OOB. The explanation can be found in the previous analysis. \end{itemize} Overall, all these results suggest us that class imbalance and concept drift need to be studied simultaneously, when we design an algorithm to deal with imbalanced data with concept drift. Their mutual effect must be taken into consideration. Hence, we propose the following key issues to be considered for an effective algorithm: \begin{itemize} \item Is the class imbalance technique effective in predicting minority-class examples? \item Is the class imbalance technique adaptive to class imbalance changes? \item Is the concept drift technique effective in detecting different types of concept drift, in terms of detection rate, false alarms and detection promptness? Which type of concept drift is it designed for? Which type of concept drift does it perform better? \item Is the detection performance of the concept drift technique affected by the class imbalance technique? And how? \item How can we have the class imbalance technique and concept drift technique work together, to achieve better detection rate, fewer false alarms, less detection delay or better online prediction? \end{itemize} \section{Conclusion} \label{sec:con} This paper gives the first systematic study of handling concept drift in class-imbalanced data streams. In the context of online learning, we provide a thorough review and an experimental insight into this problem. First, a comprehensive review is given, including the problem description and definitions, the individual learning issues and solutions in class imbalance and concept drift respectively, the combined challenges and existing solutions in online class imbalance learning with concept drift, and example applications. The review reveals research gaps in the field of online class imbalance learning with concept drift. Specifically, little work has looked into the concept drift issue in imbalanced data streams systematically, although a few methods have been proposed for this purpose; $P\left( y \right)$ type of concept drift is closely related to the class imbalance issue, but it has not been investigated properly so far; most existing concept drift detection methods are only designed for or tested on balanced data streams. Second, to fill in these research gaps, we carry out a thorough empirical study by looking into the following research questions: 1) what are the challenges in detecting each type of concept drift when the data stream is imbalanced (i.e. changes in $P\left( y \right)$, $p\left( \mathbf{x} \mid y \right)$, and $P\left( y \mid \mathbf{x} \right)$)? 2) Among the proposed methods designed for online class imbalance learning with concept drift, i.e. DDM-OCI~\cite{Wang2013bp}, LFR~\cite{Wang2015zo}, PAUC-PH~\cite{Brzezinski2015ib}, OOB~\cite{Wang2014vb}, RLSACP~\cite{Ghazikhani2013wm} and ESOS-ELM~\cite{Mirza2015nt}, which one performs better for which type of concept drift? 3) Would applying class imbalance techniques (e.g. resampling methods) facilitate the concept drift detection and online prediction? By generating artificial data streams with different types of class imbalance and concept drift and experimenting on real-world data, we make the following conclusions. For the first research question, a $P\left( y \right)$ change can be easily tackled by an adaptive class imbalance technique (e.g. OOB used in this work). The traditional concept drift detectors, such as LFR, DDM-OCI and PAUC-PH, do not perform well in detecting a $p\left( \mathbf{x} \mid y \right)$ change. The prediction performance on an imbalanced data stream with $p\left( \mathbf{x} \mid y \right)$ changes can be effectively improved by solely using an adaptive class imbalance technique. A $P\left( y \mid \mathbf{x} \right)$ change is the most challenging case for learning, where the traditional active and passive concept drift detection methods do not bring much performance improvement. Class imbalance is shown to be a more crucial issue in terms of final prediction performance. For the second research question, the two passive methods, RLSACP and ESOS-ELM, do not perform well in general. DDM-OCI and LFR are sensitive to different types of concept drift, with a high detection rate but also high false alarms. PAUC-PH is more conservative in terms of drift detection. Based on the observation on minority-class recall and G-mean, the combination PAUC-PH and OOB was shown to be the best approach among all. For the third research question, it is necessary to apply adaptive class imbalance techniques when learning from imbalanced data streams with concept drift -- they bring the most prediction performance improvement. In our experiment, our class imbalance technique OOB facilitates the concept drift detection of DDM-OCI and LFR. This paper also provides guidelines for future algorithm design. Several important issues are pointed out for consideration. There are still many challenges and learning issues in this field that are worth of ongoing research, such as more effective concept drift detection methods for imbalanced data streams, studying the mutual effect of class imbalance and concept drift, and more real-world applications with different types of class imbalance and concept drift. \end{document}

\begin{document} \title{Dynamics of Entanglement and Uncertainty Relation in Coupled Harmonic Oscillator System : Exact Results} \author{DaeKil Park$^{1,2}$} \affiliation{$^1$Department of Electronic Engineering, Kyungnam University, Changwon 631-701, Korea \\ $^2$Department of Physics, Kyungnam University, Changwon 631-701, Korea } \begin{abstract} The dynamics of entanglement and uncertainty relation is explored by solving the time-dependent Schr\"{o}dinger equation for coupled harmonic oscillator system analytically when the angular frequencies and coupling constant are arbitrarily time-dependent. We derive the spectral and Schmidt decompositions for vacuum solution. Using the decompositions we derive the analytical expressions for von Neumann and R\'{e}nyi entropies. Making use of Wigner distribution function defined in phase space, we derive the time-dependence of position-momentum uncertainty relations. In order to show the dynamics of entanglement and uncertainty relation graphically we introduce two toy models and one realistic quenched model. While the dynamics can be conjectured by simple consideration in the toy models, the dynamics in the realistic quenched model is somewhat different from that in the toy models. In particular, the dynamics of entanglement exhibits similar pattern to dynamics of uncertainty parameter in the realistic quenched model. \end{abstract} \maketitle \section{Introduction} Nickname of quantum entanglement\cite{schrodinger-35,text,horodecki09} is `spooky action at a distance' due to EPR paradox\cite{epr-35}. Although the debate related to EPR paradox does seem to be far from complete conclusion, many theorists use the entanglement as a physical resource to develop the various quantum information processing such as quantum teleportation\cite{teleportation}, superdense coding\cite{superdense}, quantum cloning\cite{clon}, and quantum cryptography\cite{cryptography,cryptography2}. It is also quantum entanglement, which makes the quantum computer outperform the classical one\cite{qcreview,computer}. Furthermore, many experimentalists have tried to realize such quantum information processing in the laboratory for last decade. In particular, quantum cryptography seems to approaching to the commercial level\cite{white}. In addition to quantum technology quantum entanglement is important notion in various branches of physics. The von Neumann\cite{woot-98} and R\'{e}nyi entropies\cite{renyi96}, which are frequently used to measure the bipartite entanglement, enable us to understand the Hawking-Bekenstein entropy\cite{bekenstein73,hawking76,hooft85,luca86,mark93,solo11} of black holes more deeply. They are also important to study on the quantum criticality\cite{eisert10,vidal03} and topological matters\cite{levin06,jiang12}. Another important cornerstone in quantum mechanics is a uncertainty relation\cite{cohen}, which arises due to wave-particle dual property in the isolated systems. In this paper we examine the dynamics of the entanglement and uncertainty relation in coupled harmonic oscillator system, where the angular frequencies and coupling constant are arbitrarily time-dependent. The harmonic oscillator system is used in many branches of physics due to the fact that its mathematical simplicity provides a clear illustration of abstract ideas. For example, this system was used in Ref. \cite{han97} to discuss on the effect of the rest of universe\cite{feyman72}. It was shown that ignoring the rest of universe appears as an increase of uncertainty and entropy in the system in which we are interested. The analytical expression of von Neumann entropy was derived for a general real Gaussian density matrix in Ref. \cite{luca86} and it was generalized to massless scalar field in Ref. \cite{mark93}. Putting the scalar field system in the spherical box, the author in Ref. \cite{mark93} has shown that the total entropy of the system is proportional to surface area. This result gives some insight into a question why the Hawking-Bekenstein entropy is proportional to the area of the event horizon. Recently, the entanglement is computed in the coupled harmonic oscillator system using a Schmidt decomposition\cite{maka17}. The von Neumann and R\'{e}nyi entropies are also explicitly computed in the similar system, called two site Bose-Hubbard model\cite{ghosh17}. The coupled harmonic oscillator system is also used in other branches such as molecular chemistry\cite{ikeda99,fillaux05} and biophysics\cite{bio1,bio2}. This paper is organized as follows. In next section the diagonalization of Hamiltonian is discussed briefly. In Sec. III we derive the solutions for time-dependent Schr\"{o}dinger equation (TDSE) explicitly in the coupled harmonic oscillator system. In Sec. IV we derive the spectral and Schmidt decompositions for the vacuum solution. Using the decompositions we derive von Neumann and R\'{e}nyi entropies analytically if the oscillators are in the ground states initially. In Sec. V we discuss on the dynamics of position-momentum uncertainty relation by making use of Wigner distribution function. In Sec. VI we introduce two toy models and one realistic quenched model, and derive the time-dependence of entanglement and uncertainty relation explicitly. It is shown that in the quenched model the pattern of uncertainty is similar to that of entanglement. In Sec. VII a brief conclusion is given. In appendix A the dynamics in the excited states is discussed briefly by assuming that the two oscillators are in ground and first-excited states initially. \section{Diagonalization of Hamiltonian} Let us consider the following Hamiltonian of coupled harmonic oscillator system \begin{equation} \langlebel{hamil-1} H = \frac{1}{2} (p_1^2 + p_2^2) + \frac{1}{2} \left( \otimesmega_1^2 (t) x_1^2 + \otimesmega_2^2 (t) x_2^2 \right) - J (t) x_1 x_2 \end{equation} where $\{x_i, p_i \} \hspace{.1cm} (i=1, 2)$ are the canonical coordinates and momenta, and frequencies $\otimesmega_j \hspace{.1cm} (j=1, 2)$ and coupling parameter $J$ are arbitrarily dependent on time. For simplicity, we assume that the oscillators have unit masses. Now, we define a rotation angle $\alpha$ as \begin{eqnarray} \langlebel{define-1} \left( \begin{array}{c} y_1 \\ y_2 \end{array} \right) = \left( \begin{array}{cc} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right). \end{eqnarray} If we choose $\alpha$ as \begin{equation} \langlebel{angle} \alpha = \frac{1}{2} \tan^{-1} \left( \frac{2 J}{\otimesmega_1^2 - \otimesmega_2^2} \right) \end{equation} with $-\pi / 4 \leq \alpha \leq \pi / 4$, the Hamiltonian is diagonalized as a form: \begin{equation} \langlebel{hamil-2} H = \frac{1}{2} \left( \tilde{p}_1^2 + \tilde{p}_2^2 \right) + \frac{1}{2} \left( \tilde{\otimesmega}_1^2 (t) y_1^2 + \tilde{\otimesmega}_2^2 (t) y_2^2 \right) \end{equation} where \begin{eqnarray} \langlebel{freq-1} &&\tilde{\otimesmega}_1^2 = \otimesmega_1^2 + J \tan \alpha = \frac{1}{2} \left[ \left( \otimesmega_1^2 + \otimesmega_2^2 \right) + \epsilon (\otimesmega_1^2 - \otimesmega_2^2) \sqrt{(\otimesmega_1^2 - \otimesmega_2^2)^2 + 4 J^2} \right] \\ \nonumber &&\tilde{\otimesmega}_2^2 = \otimesmega_2^2 - J \tan \alpha = \frac{1}{2} \left[ \left( \otimesmega_1^2 + \otimesmega_2^2 \right) - \epsilon (\otimesmega_1^2 - \otimesmega_2^2) \sqrt{(\otimesmega_1^2 - \otimesmega_2^2)^2 + 4 J^2} \right] \end{eqnarray} with $\epsilon(x) = x / |x|$. Of course, $\tilde{p}_j = -i \partial / \partial y_j \hspace{.1cm} (j=1,2)$ are canonical momenta of $y_j$. In next section we will use the diagonalized Hamiltonian (\ref{hamil-2}) to solve the TDSE of the original Hamiltonian (\ref{hamil-1}). \section{solutions of TDSE} Consider a Hamiltonian of single harmonic oscillator with time-dependent frequency \begin{equation} \langlebel{hamil-3} H = \frac{p^2}{2} + \frac{1} {2} \otimesmega^2 (t) x^2. \end{equation} The TDSE of this system was exactly solved in Ref. \cite{lewis68,lohe09}. The linearly independent solutions $\psi_n (x, t) \hspace{.1cm} (n=0, 1, \cdots)$ are expressed in a form \begin{equation} \langlebel{TDSE-1} \psi_n (x, t) = e^{-i E_n \tau(t)} e^{\frac{i}{2} \left( \frac{\dot{b}}{b} \right) x^2} \phi_n \left( \frac{x}{b} \right) \end{equation} where \begin{eqnarray} \langlebel{TDSE-2} && E_n = \left( n + \frac{1}{2} \right) \otimesmega(0) \hspace{1.0cm} \tau (t) = \int_0^t \frac{d s}{b^2 (s)} \\ \nonumber &&\phi_n (x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{ \otimesmega (0)} {\pi b^2} \right)^{1/4} H_n \left(\sqrt{\otimesmega (0)} x \right) e^{-\frac{\otimesmega (0)}{2} x^2 }. \end{eqnarray} In Eq. (\ref{TDSE-2}) $H_n (z)$ is $n^{th}$-order Hermite polynomial and $b(t)$ satisfies the Ermakov equation \begin{equation} \langlebel{ermakov-1} \ddot{b} + \otimesmega^2 (t) b = \frac{\otimesmega^2 (0)}{b^3} \end{equation} with $b(0) = 1$ and $\dot{b} (0) = 0$. Solution of the Ermakov equation was discussed in Ref. \cite{pinney50}. If $\otimesmega(t)$ is time-independent, $b(t)$ is simply one. If $\otimesmega (t)$ is instantly changed as \begin{eqnarray} \langlebel{instant-1} \otimesmega (t) = \left\{ \begin{array}{cc} \otimesmega_i & \hspace{1.0cm} t = 0 \\ \otimesmega_f & \hspace{1.0cm} t > 0, \end{array} \right. \end{eqnarray} then $b(t)$ becomes \begin{equation} \langlebel{scale-1} b(t) = \sqrt{ \frac{\otimesmega_f^2 - \otimesmega_i^2}{2 \otimesmega_f^2} \cos (2 \otimesmega_f t) + \frac{\otimesmega_f^2 + \otimesmega_i^2}{2 \otimesmega_f^2}}. \end{equation} Recently. the solution (\ref{scale-1}) is extensively used to discuss the entanglement dynamics for the sudden quenched states of two site Bose-Hubbard model in Ref. \cite{ghosh17}. Since TDSE is a linear differential equation, the general solution of TDSE is $\Psi (x, t) = \sum_{n=0}^{\infty} c_n \psi_n (x, t)$ with $\sum_{n=0}^{\infty} |c_n|^2 = 1$. The coefficient $c_n$ is determined by making use of the initial condition. Using Eqs. (\ref{hamil-2}) and (\ref{TDSE-1}) the general solution for TDSE of the coupled harmonic oscillators is $\Psi (x_1, x_2 : t) = \sum_n \sum_m c_{n,m} \psi_{n,m} (x_1, x_2 : t)$, where $\sum_n \sum_m |c_{n, m}|^2 = 1$ and \begin{eqnarray} \langlebel{solu-1} &&\psi_{n,m} (x_1, x_2 : t) = \frac{1}{\sqrt{2^{n+m} n! m!}} \left( \frac{\otimesmega'_1 \otimesmega'_2} {\pi^2} \right)^{1/4} \mbox{Exp} \Bigg[-i (E_n \tau_1 + E_m \tau_2) \\ \nonumber &&\hspace{2.0cm} - \frac{\otimesmega'_1}{2} (x_1 \cos \alpha - x_2 \sin \alpha )^2 - \frac{\otimesmega'_2}{2} (x_1 \sin \alpha + x_2 \cos \alpha )^2 \\ \nonumber &&\hspace{2.0cm}+ \frac{i}{2} \left\{ \left( \frac{\dot{b_1}}{b_1} \right) (x_1 \cos \alpha - x_2 \sin \alpha )^2 + \left( \frac{\dot{b_2}}{b_2} \right) (x_1 \sin \alpha + x_2 \cos \alpha )^2 \right\} \Bigg] \\ \nonumber &&\hspace{2.0cm} \times H_n \left[\sqrt{\otimesmega'_1} (x_1 \cos \alpha - x_2 \sin \alpha) \right] H_n \left[\sqrt{\otimesmega'_2} (x_1 \sin \alpha + x_2 \cos \alpha) \right]. \end{eqnarray} In Eq. (\ref{solu-1}) $b_j \hspace{.1cm} (j=1,2)$ satisfy the Ermakov equations $\ddot{b_j} + \tilde{\otimesmega}_j^2 (t) b_j = \frac{\tilde{\otimesmega}_j^2 (0)}{b_j^3}$ respectively, and \begin{equation} \langlebel{boso-1} \tau_j = \int_0^t \frac{d s}{b_j^2 (s)} \hspace{1.0cm} \otimesmega'_j = \frac{\tilde{\otimesmega}_j (0)}{ b_j^2}. \end{equation} The corresponding density matrix is defined as \begin{equation} \langlebel{density-1} \rho (x_1, x_2: x'_1, x'_2 : t) = \Psi (x_1, x_2 : t) \Psi^* (x'_1, x'_2 : t). \end{equation} If two oscillators are $n^{th}-$ and $m^{th}-$states initially, the density matrix becomes \begin{eqnarray} \langlebel{density-2} && \rho_{n,m} (x_1, x_2: x'_1, x'_2 : t) = \psi_{n,m} (x_1, x_2 : t) \psi^*_{n, m} (x'_1, x'_2 : t) \\ \nonumber && = \frac{\sqrt{\otimesmega'_1 \otimesmega'_2}}{2^{n+m} n! m! \pi} H_n \left[\sqrt{\otimesmega'_1} (x_1 \cos \alpha - x_2 \sin \alpha) \right] H_n \left[\sqrt{\otimesmega'_1} (x'_1 \cos \alpha - x'_2 \sin \alpha) \right] \\ \nonumber &&\hspace{2.0cm} \times H_m \left[\sqrt{\otimesmega'_2} (x_1 \sin \alpha + x_2 \cos \alpha) \right] H_m \left[\sqrt{\otimesmega'_2} (x'_1 \sin \alpha + x'_2 \cos \alpha) \right] \\ \nonumber &&\times \mbox{Exp} \left[-\frac{x_1^2}{2} (v_1 \cos^2 \alpha + v_2 \sin^2 \alpha ) - \frac{x_2^2}{2} (v_1 \sin^2 \alpha + v_2 \cos^2 \alpha) + x_1 x_2 \sin \alpha \cos \alpha (v_1 - v_2) \right] \\ \nonumber &&\times \mbox{Exp} \left[-\frac{{x'_1}^{2}}{2} (v_1^* \cos^2 \alpha + v_2^* \sin^2 \alpha ) - \frac{{x'_2}^2}{2} (v_1^* \sin^2 \alpha + v_2^* \cos^2 \alpha) + x'_1 x'_2 \sin \alpha \cos \alpha (v_1^* - v_2^*) \right] \end{eqnarray} where $v_j = \otimesmega'_j - i \frac{\dot{b_j}}{b_j}$. In next section we will discuss on the entanglement of the vacuum state $\rho_{0,0} (x_1, x_2: x'_1, x'_2 : t)$. \section{Entanglement} In order to explore the entanglement of the vacuum states we will derive the Schmidt decomposition of $\psi_{0,0} (x_1, x_2: t)$ and the spectral decomposition of the reduced density matrix explicitly. The reduced density matrix of the first oscillator is given by \begin{equation} \langlebel{density-3} \rho_{(0,0)}^A (x_1, x'_1: t) \equiv \int d x_2 \rho_{0, 0} (x_1, x_2 : x'_1, x_2: t). \end{equation} The explicit expression of the reduced density matrix is \begin{equation} \langlebel{density-4} \rho_{(0,0)}^A (x_1, x'_1: t) = \sqrt{\frac{2 a_1}{\pi}} \mbox{Exp} \left[ - \left\{ (a_1 + a_3) - i a_2 \right\} x_1^2 - \left\{ (a_1 + a_3) + i a_2 \right\} {x'_1}^2 + 2 a_3 x_1 x'_1 \right] \end{equation} where \begin{equation} \langlebel{boso-2} a_1 = \frac{\otimesmega'_1 \otimesmega'_2}{2 D} \hspace{.5cm} a_2 = \frac{\otimesmega'_1 \frac{\dot{b}_2}{b_2} \sin^2 \alpha + \otimesmega'_2 \frac{\dot{b}_1}{b_1} \cos^2 \alpha}{2 D} \hspace{.5cm} a_3 = \frac{\sin^2 \alpha \cos^2 \alpha \left[ (\otimesmega'_1 - \otimesmega'_2)^2 + \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2 \right]}{4 D} \end{equation} with $D = \otimesmega'_1 \sin^2 \alpha + \otimesmega'_2 \cos^2 \alpha$. One can show easily \begin{eqnarray} \langlebel{trace-1} &&\mbox{Tr} \left[\rho_{(0,0)}^A \right] \equiv \int dx \rho_{(0,0)}^A (x, x: t) = 1 \\ \nonumber &&\mbox{Tr} \left[\left(\rho_{(0,0)}^A \right)^2\right] \equiv \int dx dx' \rho_{(0,0)}^A (x, x': t) \rho_{(0,0)}^A (x', x: t) = \sqrt{\frac{a_1}{a_1 + 2 a_3}} = \sqrt{\frac{\otimesmega'_1 \otimesmega'_2}{\bar{\eta}}} \end{eqnarray} where $\bar{\eta} = D \tilde{D} + \sin^2 \alpha \cos^2 \alpha \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2$ with $\tilde{D} = \otimesmega'_1 \cos^2 \alpha + \otimesmega'_2 \sin^2 \alpha$. First equation of Eq. (\ref{trace-1}) guarantees the probability conservation and second one denotes the mixedness of $ \rho_{(0,0)}^A (x, x': t)$. If it is one, this means that $\rho_{(0,0)}^A (x, x': t)$ is pure state. If, on the contrary, it is zero, this means that $\rho_{(0,0)}^A (x, x': t)$ is completely mixed state. If $\tilde{\otimesmega}_j$ are independent of time, $\otimesmega'_j = \tilde{\otimesmega}_j$ and $\mbox{Tr} \left[\left(\rho_{(0,0)}^A \right)^2\right]$ becomes $\sqrt{\otimesmega'_1 \otimesmega'_2 / (D \tilde{D})}$. Thus, if $\alpha = 0$, $ \rho_{(0,0)}^A (x, x': t)$ becomes pure state. The most strong mixed states occur at $\alpha = \pm \pi / 4$, In this case $\mbox{Tr} \left[\left(\rho_{(0,0)}^A \right)^2\right]$ becomes $2 \sqrt{\otimesmega'_1 \otimesmega'_2} / (\otimesmega'_1 + \otimesmega'_2)$. In order to derive the spectral decomposition of $\rho_{(0,0)}^A (x, x': t)$ we should solve the following eigenvalue equation; \begin{equation} \langlebel{eigen-1} \int_{-\infty}^{\infty} dx' \rho_{(0,0)}^A (x, x': t) f_n (x', t) = p_n (t) f_n (x', t). \end{equation} Similar problem was discussed in Ref. \cite{ghosh17,luca86,mark93}. It is not difficult to show that the eigenvalue and normalized eigenfunction in this case are \begin{equation} \langlebel{eigen-2} f_n (x, t) = \frac{1}{\sqrt{2^n n!}} \left(\frac{\epsilon}{\pi}\right)^{1/4} H_n (\sqrt{\epsilon} x) \mbox{Exp} \left[-\frac{\epsilon}{2} x^2 + i a_2 x^2 \right] \hspace{.5cm} p_n(t) =(1 - \xi) \xi^n \end{equation} where \begin{equation} \langlebel{eigen-3} \epsilon = 2 \sqrt{(a_1 + a_3)^2 - a_3^2} \hspace{1.0cm} \xi = \frac{a_3}{(a_1 + a_3) + \frac{\epsilon}{2}}. \end{equation} Thus, the spectral decomposition of $ \rho_{(0,0)}^A (x, x': t)$ can be written as \begin{equation} \langlebel{spectral-1} \rho_{(0,0)}^A (x, x': t) = \sum_{n=0}^{\infty} p_n (t) f_n (x, t) f_n^* (x', t). \end{equation} This can be proved explicitly by making use of a mathematical formula $$\sum_{n=0}^{\infty} \frac{t^n}{n!} H_n (x) H_n (y) = (1 - 4 t^2)^{-1/2} \mbox{Exp}\left[\frac{4 t x y - 4 t^2 (x^2 + y^2)}{1 - 4 t^2} \right].$$ Thus R\'{e}nyi and von Neumann entropies are given by \begin{eqnarray} \langlebel{entropy-1} && S_{n} \equiv \frac{1}{1 - n} \ln \mbox{Tr} \left(\left( \rho_{(0,0)}^A (x, x': t) \right)^n \right) = \frac{1}{1 - n} \ln \frac{(1 - \xi)^n}{1 - \xi^n} \\ \nonumber && S_{von} \equiv \lim_{n \rightarrow 1} S_n = - \ln (1 - \xi) - \frac{\xi}{1 - \xi} \ln \xi \end{eqnarray} where $n$ is any positive integer. It is worthwhile noting that when $\alpha = 0$, $\xi$ becomes zero which results in vanishing R\'{e}nyi and von Neumann entropies. It is obvious because $\alpha = 0$ corresponds to $J = 0$ and, two oscillators are completely decoupled. In order to derive the Schmidt decomposition of $\psi_{0,0} (x_1, x_2: t)$ we should solve the eigenvalue equation of other party. The reduced density matrix of other party is given by \begin{eqnarray} \langlebel{density-B-1} && \rho_{(0,0)}^B (x_2, x'_2: t) \equiv \int d x_1 \rho_{0, 0} (x_1, x_2 : x_1, x'_2: t) \\ \nonumber &&\hspace{1.0cm}= \sqrt{\frac{2 \tilde{a}_1}{\pi}} \mbox{Exp} \left[ - \left\{ (\tilde{a}_1 + \tilde{a}_3) - i \tilde{a}_2 \right\} x_2^2 - \left\{ (\tilde{a}_1 + \tilde{a}_3) + i \tilde{a}_2 \right\} {x'_2}^2 + 2 \tilde{a}_3 x_2 x'_2 \right] \end{eqnarray} where \begin{equation} \langlebel{bosoB-2} \tilde{a}_1 = \frac{\otimesmega'_1 \otimesmega'_2}{2 \tilde{D}} \hspace{.5cm} \tilde{a}_2 = \frac{\otimesmega'_1 \frac{\dot{b}_2}{b_2} \cos^2 \alpha + \otimesmega'_2 \frac{\dot{b}_1}{b_1} \sin^2 \alpha}{2 \tilde{D}} \hspace{.5cm} \tilde{a}_3 = \frac{\sin^2 \alpha \cos^2 \alpha \left[ (\otimesmega'_1 - \otimesmega'_2)^2 + \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2 \right]}{4 \tilde{D}}. \end{equation} The eigenvalues of $\rho_{(0,0)}^B$ are exactly the same with those of $\rho_{(0,0)}^A$ and the eigenfunction becomes \begin{equation} \langlebel{eigen-B-1} \tilde{f}_n (x, t) = \frac{1}{\sqrt{2^n n!}} \left(\frac{\tilde{\epsilon}}{\pi}\right)^{1/4} H_n (\sqrt{\tilde{\epsilon}} x) \mbox{Exp} \left[-\frac{\tilde{\epsilon}}{2} x^2 + i \tilde{a}_2 x^2 \right] \end{equation} where $\tilde{\epsilon} = 2 \sqrt{(\tilde{a}_1 + \tilde{a}_3)^2 - \tilde{a}_3^2}$. Then one can find the Schmidt decomposition of $\psi_{0,0} (x_1, x_2: t)$, which is \begin{equation} \langlebel{schmidt-1} \psi_{0,0} (x_1, x_2: t) = \sum_n \sqrt{p_n} \left[ f_n (x_1, t) e^{-i n \theta / 2} e^{-i (E_0 \tau_1 - \varphi / 4)} \right] \left[ \tilde{f}_n (x_2, t) e^{-i n \theta / 2} e^{-i (E_0 \tau_2 - \varphi / 4)} \right] \end{equation} where \begin{equation} \langlebel{schmidt-2} \theta = \tan^{-1} \left( \frac{Z_2}{Z_1} \right) \hspace{1.0cm} \varphi = \tan^{-1} \left( \frac{(\kappa - 1) Z_1 Z_2}{Z_1^2 + \kappa Z_2^2} \right). \end{equation} In Eq. (\ref{schmidt-2}) $Z_1$, $Z_2$, and $\kappa$ are given by \begin{eqnarray} \langlebel{schmidt-3} &&\kappa = \left[ 1 + \frac{\sin^2 \alpha \cos^2 \alpha}{\otimesmega'_1 \otimesmega'_2} \left\{ (\otimesmega'_1 - \otimesmega'_2)^2 + \left(\frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2 \right\} \right]^{1/2} \\ \nonumber && \hspace{2.0cm}Z_1 = \otimesmega'_1 - \otimesmega'_2 \hspace{1.0cm} Z_2 = \frac{1}{\kappa} \left(\frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right). \end{eqnarray} If $\tilde{\otimesmega}_j$ are independent of time, $Z_2$ becomes zero, which results in $\theta = \varphi = 0$. From the Schmidt decomposition one can construct other bipartite entanglement measures such as St\"{u}ckelberg entropy. Furthermore, Schmidt basis is very useful to discuss on the entanglement in quantum optics, atom-field interaction, and electron-electron correlation\cite{grobe94,ekert95}. In this paper, however, we consider only R\'{e}nyi and von Neumann entropies as bipartite entanglement measures. \section{Uncertainty Relations} In order to discuss on the time-dependence of the uncertainty it is convenient to compute the Wigner distribution function defined \begin{equation} \langlebel{wigner-1} W (x_1, x_2: p_1, p_2: t) \equiv \frac{1}{\pi^2} \int dy_1 dy_2 e^{-2 i (p_1 y_1 + p_2 y_2)} \Psi^* (x_1 + y_1, x_2 + y_2: t) \Psi (x_1 - y_1, x_2 - y_2: t). \end{equation} Many interesting properties of the Wigner function are discussed in Ref.\cite{feyman72,noz91}. In particular, it is convenient to introduce the Wigner distribution function in the density matrix formalism when we want to study the uncertainty relations in detail. If we choose the wave function as $\psi_{0,0} (x_1, x_2: t)$, the Wigner function becomes \begin{eqnarray} \langlebel{wigner-2} &&W_{(0,0)} (x_1, x_2: p_1, p_2: t) = \frac{1}{\pi^2} \mbox{Exp} \bigg[ -A_1 x_1^2 - A_2 x_2^2 - B_1 p_1^2 - B_2 p_2^2 \\ \nonumber &&\hspace{1.0cm}+ 2 A_3 x_1 x_2 + 2 B_3 p_1 p_2 + 2 F (x_1 p_2 + x_2 p_1) + 2 D_{11} x_1 p_1 + 2 D_{22} x_2 p_2 \bigg] \\ \nonumber \end{eqnarray} where \begin{eqnarray} \langlebel{wigner-3} &&A_1 = \frac{1}{\otimesmega'_1 \otimesmega'_2} \left[ \otimesmega'_1 \otimesmega'_2 \tilde{D} + \otimesmega'_2 \left(\frac{\dot{b}_1}{b_1}\right)^2 \cos^2 \alpha + \otimesmega'_1 \left( \frac{\dot{b}_2}{b_2} \right)^2 \sin^2 \alpha \right] \\ \nonumber &&A_2 = \frac{1}{\otimesmega'_1 \otimesmega'_2} \left[ \otimesmega'_1 \otimesmega'_2 D + \otimesmega'_2 \left(\frac{\dot{b}_1}{b_1}\right)^2 \sin^2 \alpha + \otimesmega'_1 \left( \frac{\dot{b}_2}{b_2} \right)^2 \cos^2 \alpha \right] \\ \nonumber &&A_3 = \frac{\sin \alpha \cos \alpha}{\otimesmega'_1 \otimesmega'_2} \left[ \otimesmega'_1 \otimesmega'_2 (\otimesmega'_1 - \otimesmega'_2) + \otimesmega'_2 \left(\frac{\dot{b}_1}{b_1}\right)^2 - \otimesmega'_1 \left( \frac{\dot{b}_2}{b_2} \right)^2\right] \\ \nonumber && B_1 = \frac{D}{\otimesmega'_1 \otimesmega'_2} \hspace{1.0cm} B_2 = \frac{\tilde{D}}{\otimesmega'_1 \otimesmega'_2} \hspace{1.0cm} B_3 = - \frac{\sin \alpha \cos \alpha}{\otimesmega'_1 \otimesmega'_2} (\otimesmega'_1 - \otimesmega'_2) \\ \nonumber && F = \frac{\sin \alpha \cos \alpha}{\otimesmega'_1 \otimesmega'_2} \left(\otimesmega'_2 \frac{\dot{b}_1}{b_1} - \otimesmega'_1 \frac{\dot{b}_2}{b_2} \right) \hspace{1.0cm} D_{11} = - \frac{1}{\otimesmega'_1 \otimesmega'_2} \left(\otimesmega'_2 \frac{\dot{b}_1}{b_1} \cos^2 \alpha + \otimesmega'_1 \frac{\dot{b}_2}{b_2} \sin^2 \alpha \right) \\ \nonumber &&\hspace{2.0cm} D_{22} = - \frac{1}{\otimesmega'_1 \otimesmega'_2} \left(\otimesmega'_2 \frac{\dot{b}_1}{b_1} \sin^2 \alpha + \otimesmega'_1 \frac{\dot{b}_2}{b_2} \cos^2 \alpha \right). \end{eqnarray} The Wigner function $W_{(0,0)} (x_1, p_1: t)$ is defined from $W_{(0,0)} (x_1, x_2: p_1, p_2: t)$ as \begin{equation} \langlebel{wigner-4} W_{(0,0)} (x_1, p_1: t) = \int dx_2 dp_2 W_{(0,0)} (x_1, x_2: p_1, p_2: t). \end{equation} Using Eq. (\ref{wigner-2}) one can show \begin{equation} \langlebel{wigner-5} W_{(0,0)} (x_1, p_1: t) = \frac{1}{\pi} \sqrt{\frac{\otimesmega'_1 \otimesmega'_2}{\bar{\eta}}} e^{-\alpha_1 x_1^2 - \alpha_2 p_1^2 + 2 \alpha_3 x_1 p_1} \end{equation} where \begin{eqnarray} \langlebel{wigner-6} &&\alpha_1 = \frac{1}{\bar{\eta}} \left[ \tilde{D} \otimesmega'_1 \otimesmega'_2 + \otimesmega'_2 \left(\frac{\dot{b}_1}{b_1}\right)^2 \cos^2 \alpha + \otimesmega'_1 \left( \frac{\dot{b}_2}{b_2} \right)^2 \sin^2 \alpha \right] \\ \nonumber && \alpha_2 = \frac{D}{\bar{\eta}} \hspace{1.0cm} \alpha_3 = - \frac{1}{\bar{\eta}} \left( \otimesmega'_2 \frac{\dot{b}_1}{b_1} \cos^2 \alpha + \otimesmega'_1 \frac{\dot{b}_2}{b_2} \sin^2 \alpha \right). \end{eqnarray} It is worthwhile noting that $\alpha_j \hspace{.1cm} (j=1,2,3)$ satisfy $$ \alpha_1 \alpha_2 - \alpha_3^2 = \frac{\otimesmega'_1 \otimesmega'_2}{\bar{\eta}}.$$ One can show straightforwardly \begin{eqnarray} \langlebel{wigner-7} &&\int dx_1 dp_1 W_{(0,0)} (x_1, p_1: t) = 1 = \mbox{Tr} \left[\rho_{(0,0)}^A \right] \\ \nonumber &&2 \pi \int dx_1 dp_1 W_{(0,0)}^2 (x_1, p_1: t) = \sqrt{\frac{\otimesmega'_1 \otimesmega'_2}{\bar{\eta}}} = \mbox{Tr} \left[\left(\rho_{(0,0)}^A \right)^2\right]. \end{eqnarray} In terms of the Wigner distribution function the average of a quantity ${\cal O} (x_1, p_1)$ is defined as \begin{equation} \langlebel{average-1} <{\cal O}> (x_1, p_1) \equiv \int {\cal O} (x_1, p_1) W_{(0,0)} (x_1, p_1) dx_1 dp_1. \end{equation} Then it is straightforward to show that $<x_1> = <p_1> = 0$ and \begin{equation} \langlebel{average-2} <x_1^2> = \frac{D}{2 \otimesmega'_1 \otimesmega'_2} \hspace{1.0cm} <p_1^2> = \frac{1}{2} \left[ \tilde{D} + \frac{1}{\otimesmega'_1} \left(\frac{\dot{b}_1}{b_1}\right)^2 \cos^2 \alpha + \frac{1}{\otimesmega'_2} \left( \frac{\dot{b}_2}{b_2} \right)^2 \sin^2 \alpha \right]. \end{equation} Thus, the position-momentum uncertainty for the vacuum state becomes \begin{equation} \langlebel{uncertainty-1} \left(\Delta x_1 \Delta p_1 \right)^2 = \frac{1}{4} \Omega (t) \end{equation} where \begin{equation} \langlebel{uncertainty-2} \Omega (t) = \left( \frac{1}{\otimesmega'_1} \cos^2 \alpha + \frac{1}{\otimesmega'_2} \sin^2 \alpha \right) \left[ \left\{ \otimesmega'_1 + \frac{1}{\otimesmega'_1} \left(\frac{\dot{b}_1}{b_1}\right)^2 \right\} \cos^2 \alpha + \left\{ \otimesmega'_2 + \frac{1}{\otimesmega'_2} \left( \frac{\dot{b}_2}{b_2} \right)^2 \right\} \sin^2 \alpha \right]. \end{equation} Using Eq. (\ref{density-B-1}) one can compute also the uncertainty between $x_2$ and $p_2$. In this case the uncertainty becomes $(\Delta x_2 \Delta p_2)^2 = \tilde{\Omega} (t) / 4$, where \begin{equation} \langlebel{uncertainty-3} \tilde{\Omega} (t) = \left( \frac{1}{\otimesmega'_1} \sin^2 \alpha + \frac{1}{\otimesmega'_2} \cos^2 \alpha \right) \left[ \left\{ \otimesmega'_1 + \frac{1}{\otimesmega'_1} \left(\frac{\dot{b}_1}{b_1}\right)^2 \right\} \sin^2 \alpha + \left\{ \otimesmega'_2 + \frac{1}{\otimesmega'_2} \left( \frac{\dot{b}_2}{b_2} \right)^2 \right\} \cos^2 \alpha \right]. \end{equation} If $\tilde{\otimesmega}_j$ are time-independent, both $\Omega$ and $\tilde{\Omega}$ reduces to $1 + \frac{ (\otimesmega'_1 - \otimesmega'_2)^2}{\otimesmega'_1 \otimesmega'_2} \sin^2 \alpha \cos^2 \alpha $. Thus minimum uncertainty occurs at $\alpha = 0$ while maximum uncertainty occurs at $\alpha = \pm \pi / 4$. \section{Dynamics of Entanglement and Uncertainty} In this section we examine the dynamics of entanglement and uncertainty in three models. First two models are toy models, which are introduced to examine the effect of time-dependence of the angular frequencies and rotation angle $\alpha$ in the dynamics. As third model we introduce more realistic quenched model, where the dynamics can be solved analytically. Although we can consider more general case by solving the Ermakov equation (\ref{ermakov-1}) numerically, this fully general model is not explored in this paper because we would like to confine ourselves to analytic cases. The first model we consider is a simple case that one of the angular frequencies $\tilde{\otimesmega}_j$ is zero at late time. We choose \begin{eqnarray} \langlebel{model1-1} \tilde{\otimesmega}_1 (t) = \left\{ \begin{array}{cc} \otimesmega_{1, i} & \hspace{0.25cm} t = 0 \\ \otimesmega_{1, f} = 0 & \hspace{0.5cm} t > 0 \end{array} \right. \hspace{1.0cm} \tilde{\otimesmega}_2 (t) = \left\{ \begin{array}{cc} \otimesmega_{2, i} & \hspace{0.25cm} t = 0 \\ \otimesmega_{2, f} & \hspace{0.25cm} t > 0. \end{array} \right. \end{eqnarray} From Eq. (\ref{freq-1}) this is achieved by $\otimesmega_1 \otimesmega_2 = \pm J$ with $\otimesmega_2^2 > \otimesmega_1^2$ at $t > 0$. In this case $b_1 (t)$ and $b_2(t)$ become \begin{equation} \langlebel{model1-2} b_1 (t) = \sqrt{1 + \otimesmega_{1, i}^2 t^2} \hspace{1.0cm} b_2 (t) = \sqrt{ \left(\frac{\otimesmega_{2, f}^2 - \otimesmega_{2, i}^2}{2 \otimesmega_{2, f}^2}\right) \cos (2 \otimesmega_{2, f} t) + \left( \frac{\otimesmega_{2, f}^2 + \otimesmega_{2, i}^2}{2 \otimesmega_{2, f}^2}\right)}. \end{equation} The time-dependence of the von Neumann entropy for $\alpha = \pi / 4$ (red solid line), $\alpha = \pi / 12$ (blue dashed line), and $\alpha = \pi / 24$ (black dotted line) is plotted in Fig. 1(a) when $\otimesmega_{1, i} = 1$, $\otimesmega_{1, f} = 0$, $\otimesmega_{2, i} = 2$, and $\otimesmega_{2, f} = 0.5$. It exhibits an increasing behavior with oscillation. This oscillation is mainly due to $b_2 (t)$. The figure shows that the coupled harmonic oscillator is more entangled with increasing $|\alpha|$. This can be expected from the fact that the oscillators become separable when $\alpha = 0$. The time-dependence of uncertainty $\Omega (t) = (2 \Delta x_1 \Delta p_1)^2$ is plotted in Fig. 1(b) for $\alpha = \pi / 4$ (red solid line), $\alpha = \pi / 8$ (blue dashed line) and $\alpha = 0$ (black dotted line). The uncertainty is maximized at the separable oscillator system and is minimized at $|\alpha| = \pi /4$ at most domain of time. However, this order is reversed at the small $t$ region (for our case $0 < t < 0.773$). In this region the uncertainty is maximized at $\alpha = \pi / 4$ and is minimized at $\alpha = 0$. The oscillatory behavior is also due to $b_2(t)$. Second simple model we consider is a case that one of the angular frequencies $\tilde{\otimesmega}_j$ is imaginary at late time. We choose \begin{eqnarray} \langlebel{model2-1} \tilde{\otimesmega}_1 (t) = \left\{ \begin{array}{cc} \otimesmega_{1, i} & \hspace{0.25cm} t = 0 \\ i \otimesmega_{1, f} & \hspace{0.5cm} t > 0 \end{array} \right. \hspace{1.0cm} \tilde{\otimesmega}_2 (t) = \left\{ \begin{array}{cc} \otimesmega_{2, i} & \hspace{0.25cm} t = 0 \\ \otimesmega_{2, f} & \hspace{0.25cm} t > 0. \end{array} \right. \end{eqnarray} From Eq. (\ref{freq-1}) this is achieved by $J^2 > \otimesmega_1^2 \otimesmega_2^2$ with $\otimesmega_2^2 > \otimesmega_1^2$ at $t > 0$. In this case $b_2(t)$ is not changed and $b_1(t)$ becomes \begin{equation} \langlebel{model2-2} b_1 (t) = \sqrt{ \left(\frac{\otimesmega_{1, f}^2 + \otimesmega_{1, i}^2}{2 \otimesmega_{1, f}^2}\right) \cosh (2 \otimesmega_{1, f} t) + \left( \frac{\otimesmega_{1, f}^2 - \otimesmega_{1, i}^2}{2 \otimesmega_{1, f}^2}\right)}. \end{equation} The time-dependence of the von Neumann entropy for $\alpha = \pi / 4$ (red solid line), $\alpha = \pi / 8$ (blue dashed line), and $\alpha = \pi / 24$ (black dotted line) is plotted in Fig. 2(a) when $\otimesmega_{1, i} = 1$, $\otimesmega_{1, f} = 0.7 $, $\otimesmega_{2, i} = 2$, and $\otimesmega_{2, f} = 0.5$. Like a previous case it exhibits an increasing behavior with oscillation. The difference is the fact that the von Neumann entropy in the present case is rapidly increasing in time compared to the previous case. This seems to be mainly due to exponential behavior of $b_1(t)$ in time. The time-dependence of uncertainty $\Omega (t) = (2 \Delta x_1 \Delta p_1)^2$ is plotted in Fig. 2(b) for $\alpha = \pi / 4$ (red solid line), $\alpha = \pi / 8$ (blue dashed line) and $\alpha = 0$ (black dotted line). Although whole behavior is similar to the previous case, the oscillatory behavior disappears in this case. This is due to the rapid increasing behavior of $\Omega (t)$, thus the amplitude of oscillation is negligible. In this case also the order of uncertainty is reversed at the small $t$ region (for this case $0 \leq t \leq 0.713$). The final and more realistic model we consider is a quenched model. In this model we choose the original angular frequencies $\otimesmega_j$ as \begin{eqnarray} \langlebel{model3-1} \otimesmega_1 (t) = \left\{ \begin{array}{cc} \otimesmega_{1, i} & \hspace{0.25cm} t = 0 \\ \otimesmega_{1, f} & \hspace{0.5cm} t > 0 \end{array} \right. \hspace{1.0cm} \otimesmega_2 (t) = \left\{ \begin{array}{cc} \otimesmega_{2, i} & \hspace{0.25cm} t = 0 \\ \otimesmega_{2, f} & \hspace{0.25cm} t > 0. \end{array} \right. \end{eqnarray} In this case the rotation angle $\alpha$ is completely determined by Eq. (\ref{angle}) if $J$ is given. Also $\tilde{\otimesmega}_{1,i}$, $\tilde{\otimesmega}_{1,f}$, $\tilde{\otimesmega}_{2,i}$, and $\tilde{\otimesmega}_{2,f}$ are completely determined by Eq. (\ref{freq-1}). The scale factor $b_j (t)$ become \begin{eqnarray} \langlebel{model3-2} b_j (t) = \sqrt{ \left(\frac{\tilde{\otimesmega}_{j, f}^2 - \tilde{\otimesmega}_{j, i}^2}{2 \tilde{\otimesmega}_{j, f}^2}\right) \cos (2 \tilde{\otimesmega}_{j, f} t) + \left( \frac{\tilde{\otimesmega}_{j, f}^2 + \tilde{\otimesmega}_{j, i}^2}{2 \tilde{\otimesmega}_{j, f}^2}\right)} \hspace{1.0cm} (j=1, 2). \end{eqnarray} The time-dependence of the von Neumann entropy $S_{von}$, R\'{e}nyi entropy $S_n$, and uncertainty $\Omega(t)$ is plotted in Fig. 3 when $\otimesmega_{1, i} = 1$, $\otimesmega_{1, f} = 1.3$, $\otimesmega_{2, i} = 1.5$, and $\otimesmega_{2, f} = 1.8$ with varying $J$ (Fig. 3(a), Fig. 3(c)) or $n$ (Fig. 3(b)). In Fig. 3(a) the von Neumann entropy is plotted for $J=1.1$ (red solid line), $J = 0.9$ (blue dashed line), and $J = 0.6$ (black dotted line). Unlike the previous toy models the large $\alpha$ (or large $J$) does not guarantees higher entanglement in the full range of time in this realistic model. Another difference is a fact that the time-dependence of the von Neumann entropy exhibits a double oscillatory behavior. This is due to the fact that the trigonometric functions are involved in both $b_1 (t)$ and $b_2 (t)$. The time-dependence of the R\'{e}nyi entropy is plotted for $n=2$ (red solid line), $n=4$ (blue dashed line), and $n = 100$ (black dotted line). In this figure $J$ is fixed as $1.1$. It also exhibits a double oscillatory behavior. With increasing $n$ the R\'{e}nyi entropy decreases, and eventually approaches to $S_{\infty} = - \ln (1 - \xi)$. Most striking difference arises in the dynamics of the uncertainty $\Omega (t) = (2 \Delta x_1 \Delta p_1)^2$. This is plotted on Fig. 3(c) for $J = 1.1$ (red solid line), $J = 0.9$ (blue dashed line), and $J = 0.6$ (black dotted line). In the previous toy models large $\alpha$ yields small $\Omega (t)$ at large time region. However this does not hold in this realistic model. In this model large $J$ yields large $\Omega (t)$ in most region of time domain. The surprising fact is that $S_{von}$ and $\Omega$ exhibit similar pattern. We do not know whether or not this is universal property. If so, one can use the uncertainty as a candidate of entanglement measure after rescaling it appropriately. It also exhibits a double oscillatory behavior due to the scale factors $b_j (t)$. \section{Conclusions} The dynamics of the entanglement and uncertainty relation is examined by solving the TDSE of the coupled harmonic oscillator system when the angular frequencies $\otimesmega_j$ and coupling constant $J$ are arbitrarily time-dependent and two oscillators are in ground states initially. To show the dynamics pictorially we introduce two toy models and one realistic quenched model. While the dynamics can be conjectured by simple consideration in the toy models, the dynamics in the realistic quenched model is somewhat different from that in the toy models. In particular, the dynamics of entanglement exhibits similar behavior to dynamics of uncertainty parameter in the realistic quenched model. We do not know whether or not this is general feature. It is natural to ask how the dynamics of entanglement and uncertainty relation is changed in the excited states. This issue is examined in appendix A, where the two oscillators are in ground and first-excited states initially. In this case we fail to compute the entanglement analytically because we do not know how to derive the eigenfunctions and the corresponding eigenvalues explicitly. However, the uncertainty relation is derived exactly in the appendix. Another interesting issue related to the entanglement of the coupled harmonic oscillators is multipartite entanglement. Consider the three coupled harmonic oscillator system, whose Hamiltonian is \begin{equation} \langlebel{con-8} H = \frac{1}{2} (p_1^2 + p_2^2 + p_3^2) + \frac{1}{2} \left( \otimesmega_1^2 (t) x_1^2 + \otimesmega_2^2 (t) x_2^2 + \otimesmega_3^2 (t) x_3^2 \right) - ( J_{12} (t) x_1 x_2 + J_{13} (t) x_1 x_3 + J_{23} (t) x_2 x_3). \end{equation} We conjecture that the TDSE of this system can be solved analytically. However, computation of the tripartite entanglement seems to be formidable task. First of all we do not know what kind entanglement measure can be computed. In qubit system we usually use the three tangle\cite{ckw} or $\pi$ tangle\cite{ou07-1} to measure the tripartite entanglement. However, it is not clear whether these tangles can be computed analytically in the coupled harmonic oscillator system or not. We hope to visit this issue in the future. {\bf Acknowledgement}: This work was supported by the Kyungnam University Foundation Grant, 2016. \begin{appendix}{\centerline{\bf Appendix A}} \setcounter{equation}{0} \renewcommand{A.\arabic{equation}}{A.\arabic{equation}} In this appendix we examine how to extend the main results of this paper to the excite states. If, for example, two oscillators are in ground and first-excited states initially, the reduced density matrix becomes \begin{equation} \langlebel{con-1} \rho_{(0, 1)}^A (x_1, x'_1 : t) = 2 \otimesmega'_2 \rho_{(0, 0)}^A (x_1, x'_1 : t) \left[ \frac{\cos^2 \alpha}{2 D} + F_1 x_1^2 + F_1^* {x'_1}^2 + F_2 x_1 x'_1 \right] \end{equation} where $ \rho_{(0, 0)}^A (x_1, x'_1 : t)$ is given in Eq. (\ref{density-4}) and \begin{eqnarray} \langlebel{con-2} &&F_1 = \frac{\sin^2 \alpha \cos^2 \alpha}{4 D^2} \bigg[ \left(\otimesmega'_1 - \otimesmega'_2 \right) \left\{\otimesmega'_1 (1 + \sin^2 \alpha) + \otimesmega'_2 \cos^2 \alpha \right\} \\ \nonumber && \hspace{3.0cm}- \cos^2 \alpha \left( \frac{\dot{b_1}}{b_1} - \frac{\dot{b_2}}{b_2} \right)^2 - 2 i \otimesmega'_1 \left( \frac{\dot{b_1}}{b_1} - \frac{\dot{b_2}}{b_2} \right) \bigg] \\ \nonumber && F_2 = \frac{1}{D} (2 a_3 \cos^2 \alpha + \otimesmega'_1 \sin^2 \alpha ). \end{eqnarray} The explicit expression of $a_3$ is given in Eq. (\ref{boso-2}). Then one can show \begin{eqnarray} \langlebel{con-3} &&\mbox{Tr} \left[\rho_{(0,1)}^A \right] \equiv \int dx \rho_{(0,1)}^A (x, x: t) = 1 \\ \nonumber &&\mbox{Tr} \left[\left(\rho_{(0,1)}^A \right)^2\right] \equiv \int dx dx' \rho_{(0,1)}^A (x, x': t) \rho_{(0,1)}^A (x', x: t) = \mbox{Tr} \left[\left(\rho_{(0,0)}^A \right)^2\right] r(t) \end{eqnarray} where $r(t)$ is ratio of mixedness between $\rho_{(0,0)}^A$ and $\rho_{(0,1)}^A$ and its explicit expression is \begin{eqnarray} \langlebel{con-4} &&r(t) = 4 {\otimesmega'_2}^2 \Bigg[ \frac{\cos^4 \alpha}{4 D^2} + \frac{\cos^2 \alpha}{4 D a_1 (a_1 + 2 a_3)} \left\{ (F_1 + F_1^*) a_1 + (F_1 + F_1^* + F_2) a_3 \right\} \\ \nonumber && + \frac{1}{16 a_1^2 (a_1 + 2 a_3)^2} \bigg[ a_1^2 \left\{ (F_1 + F_1^*)^2 + 4 |F_1|^2 + F_2^2 \right\} + a_3^2 \left\{ 3 (F_1 + F_1^*)^2 + 3 F_2 (2 F_1 + 2 F_1^* + F_2) \right\} \\ \nonumber && \hspace{4.0cm} + 2 a_1 a_3 \left\{ (F_1 + F_1^*)^2 + 4 |F_1|^2 + F_2 (3 F_1 + 3 F_1^* + F_2) \right\} \bigg] \Bigg]. \end{eqnarray} We expect that the entanglement between ground and first-excited harmonic oscillators is very small compared to that between two ground state harmonic oscillators. However, it is difficult to show this explicitly because the analytic derivation of eigenvalues and eigenfunctions for $ \rho_{(0,1)}^A (x, x': t)$ does not seem to be simple matter, at least for us. We hope to discuss the dynamics of entanglement for general excited $(m,n)$ state in the future. The time-dependence of the uncertainty $\Delta x_1 \Delta p_1$ for $\rho_{(0,1)}^A$ can be computed analytically. The Wigner function $W_{(0,1)} (x_1, p_1, t)$ for this state becomes \begin{equation} \langlebel{con-5} W_{(0,1)} (x_1, p_1, t) = W_{(0,0)} (x_1, p_1, t) \left[h_0 (t) + h_1 (t) x_1^2 + h_2 (t) p_1^2 + 2 h_3 (t) x_1 p_1 \right] \end{equation} where $W_{(0,0)} (x_1, p_1, t)$ is the Wigner function for $\psi_{0,0} (x_1, x_2, t)$ given in Eq. (\ref{wigner-5}) and \begin{eqnarray} \langlebel{con-6} &&h_0 (t) = \frac{\otimesmega'_1 \otimesmega'_2} {\bar{\eta}} \cos 2 \alpha \\ \nonumber &&h_1 (t) = \frac{2 \otimesmega'_2 \sin^2 \alpha}{\bar{\eta}^2} \left\{ \left[ \otimesmega'_1 \tilde{D} + \cos^2 \alpha \frac{\dot{b}_1}{b_1} \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right) \right]^2 + \left[ \otimesmega'_1 \frac{\dot{b}_2}{b_2} \sin^2 \alpha + \otimesmega'_2 \frac{\dot{b}_1}{b_1} \cos^2 \alpha \right]^2 \right\} \\ \nonumber &&h_2 (t) = \frac{2 \otimesmega'_2 \sin^2 \alpha}{\bar{\eta}^2} \left[ D^2 + \cos^4 \alpha \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2 \right] \\ \nonumber &&h_3 (t) = \frac{2 \otimesmega'_2 \sin^2 \alpha}{\bar{\eta}^2} \Bigg\{ \cos^2 \alpha \left(\frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right) \left[ \otimesmega'_1 \tilde{D} + \cos^2 \alpha \frac{\dot{b}_1}{b_1} \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right) \right] \\ \nonumber && \hspace{9.0cm}+ D \left[ \otimesmega'_1 \frac{\dot{b}_2}{b_2} \sin^2 \alpha + \otimesmega'_2 \frac{\dot{b}_1}{b_1} \cos^2 \alpha \right] \Bigg\}. \end{eqnarray} Then, it is straightforward to show that the uncertainty relation for $\rho_{(0,1)}^A$ becomes $(\Delta x_1 \Delta p_1)^2 = \Gamma (t) / 4$, where \begin{eqnarray} \langlebel{con-7} &&\Gamma (t) = \left( \frac{\bar{\eta}}{\otimesmega'_1 \otimesmega'_2} \right)^2 \left[ \left(h_0 \alpha_2 + \frac{h_2}{2} \right) + \frac{3 \bar{\eta}}{2 \otimesmega'_1 \otimesmega'_2} \left(h_1 \alpha_2^2 + h_2 \alpha_3^2 + 2 h_3 \alpha_2 \alpha_3 \right) \right] \\ \nonumber && \hspace{4.0cm} \times \left[ \left(h_0 \alpha_1 + \frac{h_1}{2} \right) + \frac{3 \bar{\eta}}{2 \otimesmega'_1 \otimesmega'_2} \left(h_2 \alpha_1^2 + h_1 \alpha_3^2 + 2 h_3 \alpha_1 \alpha_3 \right) \right] \end{eqnarray} where $\alpha_j$ are defined in Eq. (\ref{wigner-6}). The time-dependence of the ratios $r (t)$ and $\Gamma (t) / \Omega (t)$ for the realistic quenched model is plotted in Fig. 4, where $\otimesmega_{1, i} = 1$, $\otimesmega_{1, f} = 1.3$, $\otimesmega_{2, i} = 1.5$, and $\otimesmega_{2, f} = 1.8$ are chosen. The red solid, blue dashed, and black dotted lines correspond to $J=1.1$, $J=0.9$, and $J=0.6$ respectively. The fact $r(t) < 1$ in the full range of time indicates that $\rho_{(0,1)}^A$ is more mixed than $\rho_{(0,0)}^A$. It is of interest to note that $\rho_{(0,1)}^A$ becomes more mixed compared to $\rho_{(0,0)}^A$ with increasing the coupling constant $J$. Fig. 4(b) indicates that the uncertainty $\Delta x_1 \Delta p_1$ increases in $\rho_{(0,1)}^A$ compared to that of $\rho_{(0,0)}^A$. The increasing rate becomes larger with increasing the coupling constant $J$. \end{appendix} \end{document}

\begin{document} \textup{$\mathfrak{m}$}aketitle \begin{abstract} We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $\theta$ and the infinite-volume Brownian disk of perimeter $\sigma$. We also obtain various coupling and limit results clarifying the relation between these objects. \end{abstract} \textup{$\mathfrak{f}$}ootnote{{\it Acknowledgment of support.} The research of EB was supported by the Swiss National Science Foundation grant P300P2\_161011, and performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universit\'e de Lyon, within the program ``Investissements d'Avenir'' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). GM acknowledges support of the grants ANR-14-CE25-0014 (GRAAL) and ANR-15-CE40-0013 (Liouville), and of Fondation Simone et Cino Del Duca. } \setcounter{tocdepth}{2} \begin{figure} \caption{Schematic representation of the non-trivial scaling limits in the regimes $1\ll \sigma_n\ll\sqrt{n} \end{figure} \tableofcontents \section{Introduction} In this work, we obtain a complete classification of possible scaling limits of finite random planar quadrangulations with a boundary when their size tends to infinity. Recall that a planar map is a proper embedding of a finite connected graph in the two-dimensional sphere. The graph may have loops and multiple edges. The faces of a map are the connected components of the complement of its edges. A planar quadrangulation with a boundary is a particular planar map where its faces have degree four, i.e., are incident to four oriented edges (an edge is counted twice if it lies entirely in the face), except possibly one face which may have an arbitrary (even) degree. This face is referred to as the external face, where the other faces that form quadrangles are called internal faces. The boundary of the map is given by the oriented edges that are incident to the external face, and the number of such edges is called the size of the boundary, or the {\it perimeter} of the map. The size of the map is given by the number of internal faces. We do not ask for the boundary to be a simple curve. We always consider rooted maps with a boundary, which means that we distinguish one oriented edge of the boundary such that the root face lies to the left of that edge. This edge will be called the root edge, and its origin the root vertex. As usual, two (rooted) maps are considered equivalent if they differ by an orientation- and root-preserving homeomorphism of the sphere. We are interested in scaling limits of planar maps picked uniformly at random among all quadrangulations with a boundary when the size and (possibly) the perimeter of the map tend to infinity. This means that we view the vertex set of the quadrangulation as a metric space for the graph distance and consider (under a suitable rescaling of the distance) distributional limits of such metric spaces, either in the global or local Gromov-Hausdorff topology. In~\cite{LG3} and independently in~\cite{Mi2}, it was shown that uniformly chosen quadrangulations of size $n$, equipped with the graph distance $d_{\textup{gr}}$ rescaled by a factor $n^{-1/4}$, converge to a random compact metric space called the Brownian map. The latter turns out to be a universal object which appears as the distributional limit of many classes of random maps. We refer to the recent overview~\cite{Mi3} for various aspect of the Brownian map and for more references. Here we shall deal with quadrangulations of size $n$ having a boundary of size $2\sigma_n$, and we will distinguish three boundary regimes as $n$ tends to infinity: $${\bf a})\,\,\sigma_n/\sqrt{n}\rightarrow 0;\textup{$\mathfrak{q}$}uad{\bf b})\,\,\sigma_n/\sqrt{n}\rightarrow \sqrt{2}\sigma\;\hbox{ for some }\sigma\in(0,\infty);\textup{$\mathfrak{q}$}uad{\bf c})\,\, \sigma_n/\sqrt{n}\rightarrow \infty.$$ Bettinelli~\cite{Be3} showed that in regime {\bf a}), the boundary becomes negligible in the scale $n^{-1/4}$, and the Brownian map appears in the limit when $n$ tends to infinity. In regime {\bf b}), he obtained under the same rescaling convergence along appropriate infinite subsequences to a random metric space called the Brownian disk $\mathcal BD_\sigma$. Uniqueness of this limit was later established by Bettinelli and Miermont in~\cite{BeMi}. For the third regime {\bf c}), it is shown in~\cite{Be3} that a rescaling by $\sigma_n^{-1/2}$ leads in the limit to Aldous's continuum random tree $\textup{\textsf{CRT}}$~\cite{Al1,Al2}. The scaling factors considered by Bettinelli~\cite{Be3} ensure that the diameter of the rescaled planar map stays bounded in probability. Consequently, the limits he obtains are random compact metric spaces, and the right notion of convergence is the Gromov-Hausdorff convergence in the space of (isometry classes of) compact metric spaces. We will study all possible scalings $a_n\rightarrow\infty$ in all the above boundary regimes. When $a_n$ grows slower then the diameter of the map as $n$ tends to infinity, the right notion of convergence is the {\it local} Gromov-Hausdorff convergence. Depending on the ratio of perimeter and scaling parameter, the boundary will in the limit be either invisible, or of a size comparable to the full map, or dominate the map. In the process we obtain two new one-parameter families of limit spaces: the Brownian half-plane $\mathcal BHP_\theta$ with parameter $\theta\in[0,\infty)$ and the infinite-volume Brownian disk $\textup{\textsf{IBD}}_\sigma$ with boundary length $\sigma\in(0,\infty)$. The Brownian disk $\mathcal BD_\sigma$ and the Brownian half-plane $\mathcal BHP=\mathcal BHP_0$ play a central role in this work. The latter can be seen as the Gromov-Hausdorff tangent cone in distribution of $\mathcal BD_\sigma$ at its root, and also as the scaling limit of the so-called uniform infinite half-planar quadrangulation $\textup{\textsf{UIHPQ}}$, which in turn arises as the local limit in the sense of Benjamini and Schramm of uniform quadrangulations with $n$ faces and a boundary growing slower than $n$. The space $\mathcal BHP_\theta$ for $\theta>0$ can be understood as an interpolation between $\mathcal BHP$ (when $\theta\to 0$) and the so-called infinite continuum random tree $\textup{\textsf{ICRT}}$ introduced by Aldous~\cite{Al1} (when $\theta\to \infty$). The $\textup{\textsf{IBD}}_\sigma$ in turn interpolates between $\mathcal BHP$ (when $\sigma\rightarrow \infty$) and the Brownian plane $\mathcal BP$ introduced by Curien and Le Gall~\cite{CuLG,CuLG2} (when $\sigma\rightarrow 0$). See also the recent work of Budzinski~\cite{Bu} for a hyperbolic version of the Brownian plane. These interpretations are easy consequences of our results. We refer to Remark~\ref{rem:exercises} and the exercises there for the exact statements. For a better overview, we begin with a rough list of our main results on scaling limits of finite-size quadrangulations with a boundary (including results of~\cite{Be3} and~\cite{BeMi}). We then mention further results that will be obtained below, including limit statements on $\mathcal BD_{\sigma}$. The precise formulations can be found in Section~\ref{sec:mainresults}, after a proper definition of the limit spaces and a reminder on the notion of convergence in Section~\ref{sec:def}. As in many works in this context, our approach is based on the Bouttier-Di Francesco-Guitter bijection~\cite{BoGu,BoDFGu}, which establishes a one-to-one correspondence between (finite-size) quadrangulations with a boundary on the one hand and discrete labeled forests and bridges on the other hand. The bijection is recalled in Section~\ref{sec:encoding}. Section~\ref{sec:auxiliaryresults} contains some more auxiliary results, mostly convergence results on forests and bridges when their size tends to infinity. The statements proved there are of some independent interest, but can also be skipped at first reading. Section~\ref{sec:proofs} contains all the proofs of our main statements. \subsection{Overview over the main results} \textup{$\mathfrak{l}$}bel{sec:overview} For any $\sigma_n\in \textup{$\mathfrak{m}$}athbb N=\{1,2,\ldots\}$, we write $Q_n^{\sigma_n}$ for a uniformly distributed rooted quadrangulation with $n$ inner faces and a boundary of size $2\sigma_n$. The vertex set of $Q_n^{\sigma_n}$ is denoted by $V(Q_n^{\sigma_n})$, $\rho_n$ represents the root vertex and $d_{\textup{gr}}$ stands for the graph distance on $V(Q_n^{\sigma_n})$. For any two sequences $(a_n,n\in\textup{$\mathfrak{m}$}athbb N),(b_n,n\in\textup{$\mathfrak{m}$}athbb N)$ of reals, we write $a_n \ll b_n$ or $b_n \gg a_n$ if and only if $a_n/b_n \to 0$ as $n\to\infty$, and we write $a_n\sim b_n$ if $a_n/b_n\to 1$. We denote by $\circ$ the trivial one-point metric space and write $\textup{$\textsf{s-Lim}$}$ ($\textup{$\textsf{s-Lim}_{\textup{loc}}$}$) for the distributional scaling limit of $(V(Q_n^{\sigma_n}),a_n^{-1}d_{\textup{gr}},\rho_n)$ in the Gromov-Hausdorff topology (in the local Gromov-Hausdorff topology) as $n$ tends to infinity. \paragraph{The regime $\sigma_n \ll \sqrt{n}$.} \begin{mdframed} \begin{itemize} \item If $1\ll a_n \ll \sqrt{\sigma_n}$, then $\textup{$\textsf{s-Lim}_{\textup{loc}}$}=\mathcal BHP.$ \item If $1\ll a_n \sim (1/9)^{1/4}\sqrt{2\sigma_n/\sigma}$, $\sigma \in (0,\infty)$, then $\textup{$\textsf{s-Lim}_{\textup{loc}}$}=\textup{\textsf{IBD}}_\sigma.$ \item If $\sqrt{\sigma_n} \ll a_n \ll n^{1/4}$, then $\textup{$\textsf{s-Lim}_{\textup{loc}}$}=\mathcal BP.$ \item If $a_n \sim (8/9)^{1/4}n^{1/4}$, then (see~\cite{Be3}) $\textup{$\textsf{s-Lim}$}=\mathcal BM.$ \item If $a_n \gg n^{1/4}$, then $\textup{$\textsf{s-Lim}$}=\circ\,.$ \end{itemize} \end{mdframed} \paragraph{The regime $\sigma_n \sim \sigma \sqrt{2n}$, $\sigma\in(0,\infty)$.} \begin{mdframed} \begin{itemize} \item If $1 \ll a_n \ll n^{1/4}$, then $\textup{$\textsf{s-Lim}_{\textup{loc}}$}=\mathcal BHP.$ \item If $a_n \sim (8/9)^{1/4}n^{1/4}$, then (see~\cite{Be3} and~\cite{BeMi}) $\textup{$\textsf{s-Lim}$}=\mathcal BD_\sigma.$ \item If $a_n \gg n^{1/4}$, then $\textup{$\textsf{s-Lim}$}=\circ\,.$ \end{itemize} \end{mdframed} \paragraph{The regime $\sigma_n\gg \sqrt{n}$.} \begin{mdframed} \begin{itemize} \item If $\sigma_n\ll n$ and $\lim_{n\rightarrow\infty} (9/4)^{1/4}a_n/\sqrt{2n/\sigma_n}=\sqrt{\theta}\in[0,\infty)$, then $\textup{$\textsf{s-Lim}_{\textup{loc}}$}=\mathcal BHP_\theta.$ \item If $\textup{$\mathfrak{m}$}ax\{1,\sqrt{n/\sigma_n}\} \ll a_n \ll \sqrt{\sigma_n}$, then $\textup{$\textsf{s-Lim}_{\textup{loc}}$}=\textup{\textsf{ICRT}}.$ \item If $a_n \sim \sqrt{2\sigma_n}$ (see~\cite{Be3}), then $\textup{$\textsf{s-Lim}$}=\textup{\textsf{CRT}}.$ \item If $a_n \gg \sqrt{\sigma_n} $, then $\textup{$\textsf{s-Lim}$}=\circ\,.$ \end{itemize} \end{mdframed} The new results in these listings are covered by Theorems~\ref{thm:BP},~\ref{thm:IBD},~\ref{thm:BHP1},~\ref{thm:BHP3} and~\ref{thm:ICRT} below. In the regime $\sigma_n\ll\sqrt{n}$ in the first list, the last three convergences include the case of bounded $\sigma_n$. In the last regime $\sigma_n\gg \sqrt{n}$, we allow $\sigma_n$ to grow faster than $n$. The scaling constants are chosen in such a way that the description of the limiting objects is the most natural. See also Figure~\ref{fig:regimes-schema} for a schematic representation. Figure~\ref{fig:usersmanual} shows all possible regimes in one diagram, in which the $x$-axis denotes the limiting possible values for the logarithm of the boundary length $\log(\sigma_n)/\log(n)$ in units of $\log(n)$, and the $y$-axis corresponds to the limit of the logarithm of the scaling factor $\log(a_n)/\log(n)$ in units of $\log(n)$. For the specific value $y=0$, it will be assumed that $a_n=1$, so that we are really in the regime of local limits, without any rescaling. Similarly, for some specific values of $(x,y)$, that are shown on the colored lines, we will require some particular scaling behaviors that are detailed in the list above. For instance, for $x=1/2$ and $y=1/4$, we really ask that $\sigma_n\sim \sigma\sqrt{2n}$ for some $\sigma>0$ and $a_n\sim (8/9)^{1/4}n^{1/4}$. As it is shown in Theorem~\ref{thm:UIHPQ-BHP}, the $\mathcal BHP$ can also be obtained from the $\textup{\textsf{UIHPQ}}$ by zooming-out around the root: $\textup{$\mathfrak{l}$}mbda\cdot\textup{\textsf{UIHPQ}}\rightarrow\mathcal BHP$ in distribution in the local Gromov-Hausdorff sense as $\textup{$\mathfrak{l}$}mbda \to 0$. Here, $\textup{$\mathfrak{l}$}mbda\cdot\textup{\textsf{UIHPQ}}$ is obtained from $\textup{\textsf{UIHPQ}}$ by keeping the same set of points, but rescaling the metric by a factor $\textup{$\mathfrak{l}$}mbda$, see Section~\ref{sec:locGH} below. Many of our results, for example those involving the Brownian half-planes $\mathcal BHP_\theta$, $\theta\geq 0$, are based on coupling methods, which yield in fact stronger statements than those mentioned above. In particular, couplings will allow us to deduce that the topology of $\mathcal BHP_\theta$ is that of a closed half-plane, whereas $\textup{\textsf{IBD}}_\sigma$ is homeomorphic to the pointed closed disk (Corollaries~\ref{cor:topology-BHP} and~\ref{cor:topology-IBD}). The above results will moreover enable us to determine the limiting behavior of the Brownian disk $\mathcal BD_{T,\sigma}$ of volume $T$ and perimeter $\sigma$ when zooming-in around its root vertex, or, equivalently by scaling, by blowing up its volume and perimeter. Depending on the behavior of the ``perimeter'' function $\sigma(\cdot):(0,\infty)\rightarrow (0,\infty)$ for large volumes $T$, we observe $\mathcal BP$, $\textup{\textsf{IBD}}_\varsigma$, $\mathcal BHP_\theta$ or the $\textup{\textsf{ICRT}}$ as the distributional limit in the local Gromov-Hausdorff sense of $\mathcal BD_{T,\sigma(T)}$ when $T\rightarrow\infty$. See Figure~\ref{fig:diagram2} below and Corollaries~\ref{cor:BD1},~\ref{cor:BD4},~\ref{cor:BD2},~\ref{cor:BD3}. \begin{figure} \caption{The user's manual to this paper, displaying all possible regimes and limits for the rescaled pointed space $(V(Q_n^{\sigma_n} \end{figure} \section{Definitions} \textup{$\mathfrak{l}$}bel{sec:def} In this section, we define our limit objects and recall some facts about the (local) Gromov-Hausdorff convergence and the local limits of maps. All our limit metric spaces will be defined in terms of certain random processes. To make the presentation unified, we will denote by $(X,W)$ the canonical continuous process in $\textup{$\mathfrak{m}$}athcal{C}(I,\textup{$\mathfrak{m}$}athbb R)^2$, where $I$ will be an interval of the form $I=[0,T]$ for some $T>0$, or $I=\textup{$\mathfrak{m}$}athbb R$. The set $\textup{$\mathfrak{m}$}athcal{C}(I,\textup{$\mathfrak{m}$}athbb R)$ of continuous functions on $I$ is equipped with the compact-open topology (topology of uniform convergence over compact subsets of $I$). For $t\in I\cap [0,\infty)$, we write $\underline{X}_t=\inf_{[0,t]}X,$ and in case $I=\textup{$\mathfrak{m}$}athbb R$, we put for $t<0$ $\underline{X}_t=\inf_{(-\infty,t]}X.$ If $Y=(Y_t:t\geq 0)$ is a real-valued process indexed by the positive real half-line, we write $\mathbb Pi(Y)$ for its {\it Pitman transform} defined as $\mathbb Pi(Y)_t=Y_t-2\underline{Y}_t$, $t\geq 0$. We will often use the fact that if $B=(B_t,t\geq 0)$ is a standard Brownian motion, then its Pitman transform $\mathbb Pi(B)$ has the law of a three-dimensional Bessel process, and $\inf_{[t,\infty)}\mathbb Pi(B)= -\inf_{[0,t]}B$ for every $t\geq 0$. See~\cite[Theorem 0.1 (ii)]{Pi}. \subsection{Metric spaces coded by real functions} \textup{$\mathfrak{l}$}bel{sec:continuum-tree-coded} \paragraph{Real trees. } If $f$ is an element of $\textup{$\mathfrak{m}$}athcal{C}(I,\textup{$\mathfrak{m}$}athbb R)$, and $s,t\in I$, we denote by $\underline{f}(s,t)$ the quantity $$ \underline{f}(s,t)=\left\{ \begin{array}{lcl} \inf_{[s,t]}f & \textup{$\mathfrak{m}$}box{ if }& s\leq t\\ \inf_{I\setminus[t,s]}f & \textup{$\mathfrak{m}$}box{ if }&s>t \end{array}\right., $$ and for $s,t\in I$ we let $$d_f(s,t)=f(s)+f(t)-2\textup{$\mathfrak{m}$}ax(\underline{f}(s,t),\underline{f}(t,s))\, .$$ This defines a pseudo-metric on $I$, which is a class function for the equivalence relation $\{d_f=0\}$. Therefore, we can define the quotient space $\textup{$\mathfrak{m}$}athcal{T}_f=I/\{d_f=0\}$, on which $d_f$ induces a true distance, still denoted by $d_f$ for simplicity. Since we assumed that $I$ contains $0$, it is natural to ``root'' the space $(\textup{$\mathfrak{m}$}athcal{T}_f,d_f)$ at the point $\rho$ given by the equivalence class $[0]=\{s\in I:d_f(0,s)=0\}$ of $0$. The metric space $(\textup{$\mathfrak{m}$}athcal{T}_f,d_f,\rho)$ is called the {\it continuum tree coded by $f$}. In more precise terms, it is a rooted $\textup{$\mathfrak{m}$}athbb R$-tree, which is also compact if $I$ is compact. This fact is well-known in the ``classical case'' where $f$ is a non-negative function on an interval $[0,T]$, and $f(0)=f(T)=0$, see, e.g.~\cite[Section 3]{LGMi}, but it remains true in this more general context. This fact will not be used in this paper, so we do not prove it here. Note that the space $(\textup{$\mathfrak{m}$}athcal{T}_f,d_f)$ comes with a natural Borel $\sigma$-finite measure, $\textup{$\mathfrak{m}$}u_f$, which is defined as the push-forward of the Lebesgue measure on $I$ by the canonical projection $p_f:I\to \textup{$\mathfrak{m}$}athcal{T}_f$. \paragraph{Metric gluing of a real tree on another. } Let $f,g$ be two elements of $\textup{$\mathfrak{m}$}athcal{C}(I,\textup{$\mathfrak{m}$}athbb R)$. These functions code two $\textup{$\mathfrak{m}$}athbb R$-trees $\textup{$\mathfrak{m}$}athcal{T}_f,\textup{$\mathfrak{m}$}athcal{T}_g$ in the preceding sense. We now define a new metric space $(M_{f,g},D_{f,g})$ by informally quotienting the space $(\textup{$\mathfrak{m}$}athcal{T}_g,d_g)$ by the equivalence relation $\{d_f=0\}$. Formally, for $s,t\in I$, we let \begin{equation} \textup{$\mathfrak{l}$}bel{eq:Dfg} D_{f,g}(s,t)=\inf\left\{\sum_{i=1}^kd_g(s_i,t_i):\begin{array}{l} k\geq 1, \, s_1,\ldots,s_k,t_1,\ldots,t_k\in I,s_1=s,t_k=t,\\ d_f(t_i,s_{i+1})=0\textup{$\mathfrak{m}$}box{ for every }i\in \{1,2,\ldots,k-1\} \end{array} \right\}\, . \end{equation} This defines a pseudo-metric on $I$, and we let $M_{f,g}$ be the quotient space $I/\{D_{f,g}=0\}$, endowed with the true metric inherited from $D_{f,g}$ (and again, still denoted by $D_{f,g}$). Again, this space is naturally pointed at the equivalence class of $0$ for $\{D_{f,g}=0\}$, which we still denote by $\rho$. Again, the space $(M_{f,g},D_{f,g})$ is naturally endowed with the measure $\textup{$\mathfrak{m}$}u_{f,g}$, defined as the push-forward of the Lebesgue measure on $I$ by the canonical projection $p_{f,g}:I\to M_{f,g}$. \subsection{Random snakes}\textup{$\mathfrak{l}$}bel{sec:random-snakes} The definition for most of our limiting random spaces depend on the notion of a random snake, which we now introduce. Let $f\in \textup{$\mathfrak{m}$}athcal{C}(I,\textup{$\mathfrak{m}$}athbb R)$ be a continuous path on an interval $I$. The random snake driven by $f$ is a random Gaussian process $(Z^f_s,s\in I)$ satisfying $Z^f_0=0$ a.s.\ and $$\textup{$\mathfrak{m}$}athbb E[|Z^f_s-Z^f_t|^2]=d_f(s,t)\, .$$ These specifications characterize the law of $Z^f$: roughly speaking, it can be seen as Brownian motion indexed by the tree $\textup{$\mathfrak{m}$}athcal{T}_f$, see, e.g., Section 4 of~\cite{LGMi}. It is easy to see and well-known that the process $Z^f$ admits a continuous modification as soon as $f$ is a locally H\"older-continuous function on $I$. In this case, we always work with this modification. The snake driven by a random function $Y$ is then defined as the random Gaussian process $Z^Y$ conditionally given $Y$. In all our applications, $Y$ will be considered under probability distributions that make it a H\"older-continuous function with probability one. More specifically, except for the case of the infinite-volume Brownian disk, see below, we will either let $Y=X$ for $X$ the canonical process on $\textup{$\mathfrak{m}$}athcal{C}(I,\textup{$\mathfrak{m}$}athbb R)$ (namely for the Brownian map and the Brownian plane), or $Y=X-\underline{X}$ (for the Brownian disk and the Brownian half-planes). \subsection{Limit random metric spaces} We apply the preceding constructions to a variety of random versions of the functions $f,g$. \subsubsection{Compact spaces}\textup{$\mathfrak{l}$}bel{sec:compact-spaces} In this section the processes considered all take values in $\textup{$\mathfrak{m}$}athcal{C}([0,T],\textup{$\mathfrak{m}$}athbb R)$ for some $T>0$. \paragraph{Continuum random tree {\normalfont $\textup{\textsf{CRT}}_T$, $T>0$.}} The continuum random tree was introduced by Aldous~\cite{Al1, Al2} and is defined as follows. \begin{defn} Let $T>0$. The continuum random tree $\textup{\textsf{CRT}}_T$ with volume $T$ is the random rooted real tree $(\textup{$\mathfrak{m}$}athcal{T}_X,d_x,\rho)$ for the probability distribution that makes the canonical process $X$ of $\textup{$\mathfrak{m}$}athcal{C}([0,T],\textup{$\mathfrak{m}$}athbb R)$ the standard Brownian excursion with duration $T$. \end{defn} The term ``CRT'' usually denotes $\textup{\textsf{CRT}}_1$ with volume $T=1$, in which case $X$ is taken under the law of the normalized Brownian excursion. We simply write $\textup{\textsf{CRT}}$ instead of $\textup{\textsf{CRT}}_1$. Note the scaling relation, for $\textup{$\mathfrak{l}$}mbda,T>0$: $$\textup{$\mathfrak{l}$}mbda\cdot\textup{\textsf{CRT}}_T =_d \textup{\textsf{CRT}}_{\textup{$\mathfrak{l}$}mbda^2 T} \, .$$ This comes from the fact that, if $\textup{$\mathfrak{m}$}athbbm{e}^T$ is a Brownian excursion with duration $T$, then $\textup{$\mathfrak{l}$}mbda\textup{$\mathfrak{m}$}athbbm{e}^{T}(\cdot/\textup{$\mathfrak{l}$}mbda^2)$ has same distribution as $\textup{$\mathfrak{m}$}athbbm{e}^{\textup{$\mathfrak{l}$}mbda^2 T}$. We should also discuss the role of $\rho$ in the above definition. The re-rooting property of $\textup{\textsf{CRT}}_T$~\cite[(20)]{Al2} states, roughly speaking, that if $\rho'$ is a random variable with distribution $\textup{$\mathfrak{m}$}u_X/\textup{$\mathfrak{m}$}u_X(1)$ (the normalized version of the measure defined above), then $(\textup{$\mathfrak{m}$}athcal{T}_X,d_X,\rho')$ has same distribution as $(\textup{$\mathfrak{m}$}athcal{T}_X,d_X,\rho)$. In this sense, the point $\rho$ plays no distinguished role in the construction of $\textup{\textsf{CRT}}_T$. \paragraph{Brownian map {\normalfont $\mathcal BM_T$, $T>0$.}} The Brownian map is roughly speaking the metric gluing of the tree coded by a snake driven by a normalized Brownian excursion, on the tree coded by the excursion itself. \begin{defn} The Brownian map $\mathcal BM_T$ with volume $T$ is the metric space $(M_{X,W},D_{X,W},\rho)$ for the probability law that makes $X$ a Brownian excursion of duration $T$, and $W$ is the snake driven by $X$. \end{defn} We write $\mathcal BM$ instead of $\mathcal BM_1$. The scaling properties of Gaussian processes imply easily that for $\textup{$\mathfrak{l}$}mbda>0$, $$\textup{$\mathfrak{l}$}mbda\cdot \mathcal BM_T=_d \mathcal BM_{\textup{$\mathfrak{l}$}mbda^4 T}\, .$$ Just as for $\textup{\textsf{CRT}}_T$, the point $\rho$ in $\mathcal BM_T$ should be seen as a random choice according to the normalized measure $\textup{$\mathfrak{m}$}u_{X,W}/\textup{$\mathfrak{m}$}u_{X,W}(1)$, which is known as the re-rooting property of the Brownian map (Theorem 8.1 of~\cite{LG4}). The latter is a crucial property for characterizing the Brownian map, see, e.g., the recent work~\cite{MiSh}. \paragraph{Brownian disk {\normalfont $\mathcal BD_{T,\sigma}$, $\sigma\in(0,\infty)$, $T>0$.}} The Brownian disk first appears in~\cite{Be3} as limiting metric space along suitable infinite subsequences. Uniqueness of the limit and a concrete description of the metric were obtained in~\cite{BeMi}. The description is slightly more elaborate than that of the Brownian map. For $t\geq 0$, we let $\underline{X}_t=\inf_{[0,t]}X$. \begin{defn} \textup{$\mathfrak{l}$}bel{def:BD} The Brownian disk $\mathcal BD_{T,\sigma}$ with volume $T$ and boundary length $\sigma$ is the metric space $(M_{X,W},D_{X,W},\rho)$ under the probability measures that makes $X$ a first-passage Brownian bridge from $0$ to $-\sigma$ of duration $T$, and conditionally given $X$, $(W_t,0\leq t\leq T)$ has same distribution as $(\sqrt{3}\, \gamma_{-\underline{X}_t}+Z_t,0\leq t\leq T)$, where \begin{itemize} \item $(Z_t,0\leq t\leq T)=Z^{X-\underline{X}}$ is the random snake driven by the reflected process $(X_t-\underline{X}_t,0\leq t\leq T)$, i.e., (a continuous modification of) the centered Gaussian process with covariances given by $$ \textup{$\mathfrak{m}$}athbb E\left[Z_sZ_t\right]=\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}(X-\underline{X}). $$ \item $(\gamma_x,0\leq x\leq \sigma)$ is a standard Brownian bridge with duration $\sigma$, independent of $Z^{X-\underline{X}}$. \end{itemize} \end{defn} The Brownian disks are homeomorphic to the closed unit disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}$, where $\textup{$\mathfrak{m}$}athbb{D}=\{z\in \textup{$\mathfrak{m}$}athbb{C}:|z|<1\}$, see~\cite[Proposition 21]{Be3} (cited as Lemma~\ref{lem:proof-prop-refpr} below). They enjoy the following scaling property: For $\textup{$\mathfrak{l}$}mbda>0$, $$\textup{$\mathfrak{l}$}mbda\cdot \mathcal BD_{T,\sigma}=_d \mathcal BD_{\textup{$\mathfrak{l}$}mbda^4 T,\textup{$\mathfrak{l}$}mbda^2\sigma}\, .$$ If $T=1$, we will simply write $\mathcal BD_\sigma$ instead of $\mathcal BD_{1,\sigma}$. Contrary to the Brownian tree or the Brownian map, $\rho$ does not play the role of a random point distributed according to $\textup{$\mathfrak{m}$}u_{X,W}/\textup{$\mathfrak{m}$}u_{X,W}(1)$. The reason is that $\rho$ is a.s.\ a point of the boundary of the disk, which is of zero measure, see~\cite{BeMi} for more details. \subsubsection{Non-compact spaces}\textup{$\mathfrak{l}$}bel{sec:non-compact-spaces} In this subsection, all processes take values in $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$. \paragraph{Infinite continuum random tree {\normalfont $\textup{\textsf{ICRT}}$.}} The $\textup{\textsf{ICRT}}$ is process $2$ in~\cite{Al1} and can be defined as follows. \begin{defn} \textup{$\mathfrak{l}$}bel{def:ICRT} The infinite continuum random tree $\textup{\textsf{ICRT}}$ is the random rooted real tree $(\textup{$\mathfrak{m}$}athcal{T}_X,d_X,\rho)$, for the probability distribution under which the canonical process $X$ in $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$ is such that $(X_t,t\geq 0)$ and $(X_{-t},t\geq 0)$ are two independent standard three-dimensional Bessel processes started at $0$. \end{defn} This results in an a.s.\ non-compact real tree, which enjoys the remarkable self-similarity property that $\textup{$\mathfrak{l}$}mbda\cdot \textup{\textsf{ICRT}}=_d\textup{\textsf{ICRT}}$ for every $\textup{$\mathfrak{l}$}mbda>0$. Note that if we let $Y$ be the canonical process in $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb{R},\textup{$\mathfrak{m}$}athbb{R})$ such that $(Y_t,t\geq 0)$ and $(Y_{-t},t\geq 0)$ are two independent standard Brownian motions, then the random rooted real tree $(\textup{$\mathfrak{m}$}athcal{T}_Y,d_Y,[0])$ has same distribution as $\textup{\textsf{ICRT}}$. This follows readily from the fact that $\mathbb Pi((Y_t,t\geq 0))$ has the law of a three-dimensional Bessel process. \paragraph{Brownian plane {\normalfont $\mathcal BP$.}} The Brownian plane was introduced in~\cite{CuLG}. \begin{defn} \textup{$\mathfrak{l}$}bel{def:BP} The Brownian plane $\mathcal BP$ is the pointed space $(M_{X,W},D_{X,W},\rho)$ under the probability distribution such that \begin{itemize} \item $(X_t,t\geq 0)$ and $(X_{-t},t\geq 0)$ are two independent three-dimensional Bessel processes. \item Given $X=(X_t, t\in\textup{$\mathfrak{m}$}athbb{R})$, $W$ has same distribution as the random snake $Z^X$ driven by $X$. \end{itemize} \end{defn} The Brownian plane is a.s.\ homeomorphic to $\textup{$\mathfrak{m}$}athbb R^2$, and is invariant under scaling: for $\textup{$\mathfrak{l}$}mbda>0$, $$\textup{$\mathfrak{l}$}mbda\cdot \mathcal BP=_d\mathcal BP\, .$$ \paragraph{Brownian half-planes {\normalfont $\mathcal BHP_{\theta}$, $\theta\in[0,\infty)$.}} The Brownian half-planes are the first truly new limiting metric spaces that we encounter in this study. Recall that $\underline{X}_t=\inf_{[0,t]}X$ for $t\geq 0$ and $\underline{X}_{t}=\inf_{(-\infty,t]}X$ for $t<0$. \begin{defn} \textup{$\mathfrak{l}$}bel{def:BHP} Let $\theta\geq 0$ be fixed. The Brownian half-plane $\mathcal BHP_\theta$ with skewness parameter $\theta$ is the pointed space $(M_{X,W},D_{X,W},\rho)$ under the probability distribution such that \begin{itemize} \item $(X_t,t\geq 0)$ is a standard Brownian motion with linear drift $-\theta$, and $(X_{-t},t\geq 0)$ is the Pitman transform $\mathbb Pi(X')$ of an independent copy $X'$ of $(X_t,t\geq 0)$. \item Given $X$, $W$ has same distribution as $(\sqrt{3}\, \gamma_{-\underline{X}_t}+Z_t,t\in \textup{$\mathfrak{m}$}athbb R)$, where \begin{itemize} \item $(Z_t,t\in\textup{$\mathfrak{m}$}athbb R)=Z^{X-\underline{X}}$ is the snake driven by the process $(X_t-\underline{X}_t,t\in \textup{$\mathfrak{m}$}athbb R)$, i.e., the centered Gaussian process with covariances given by $$ \textup{$\mathfrak{m}$}athbb E\left[Z_sZ_t\right]=\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}(X-\underline{X}). $$ \item $(\gamma_x,x\in \textup{$\mathfrak{m}$}athbb R)$ is a two-sided standard Brownian motion with $\gamma_0=0$, independent of $Z^{X-\underline{X}}$. \end{itemize} \end{itemize} \end{defn} The scaling property enjoyed by $\mathcal BHP_\theta$ is that for $\textup{$\mathfrak{l}$}mbda>0$, $$\textup{$\mathfrak{l}$}mbda\cdot \mathcal BHP_\theta=_d\mathcal BHP_{\theta/\textup{$\mathfrak{l}$}mbda^2}\, .$$ This makes the value $\theta=0$ special in the sense that the space is self-similar in distribution in this case (just as $\textup{\textsf{ICRT}}$ or $\mathcal BP$). Keep in mind that we often write $\mathcal BHP$ instead of $\mathcal BHP_0$. We will see in Corollary~\ref{cor:topology-BHP} that for every $\theta\geq 0$, $\mathcal BHP_\theta$ is a.s.\ homeomorphic to the closed half-plane $\overline{\textup{$\mathfrak{m}$}athbb{H}}=\textup{$\mathfrak{m}$}athbb R\times\textup{$\mathfrak{m}$}athbb R_+$. \begin{remark} Note that a random metric space also called the Brownian half-plane first appeared in the recent work~\cite{CaCu}, where it is conjectured that it arises as the scaling limit of the uniform infinite half-planar quadrangulation $\textup{\textsf{UIHPQ}}$, the definition of which is recalled in Section~\ref{sec:constr-UIHPQ}. Theorem~\ref{thm:UIHPQ-BHP} below states indeed that the scaling limit of $\textup{\textsf{UIHPQ}}$ is the space $\mathcal BHP_0$. However, an important caveat is that the definition of the Brownian half-plane from~\cite{CaCu} is different from ours: it is still of the form $(M_{X,W},D_{X,W},\rho)$, but for processes $(X,W)$ having a very different law from the one presented in Definition~\ref{def:BHP} (with $\theta=0$). We do not actually prove that the two definitions coincide, since we believe that this would require some specific work. Nonetheless, we prefer to stick to the name ``Brownian half-plane'' since we feel that this should be the proper denomination for the scaling limit of the $\textup{\textsf{UIHPQ}}$. See also Remark~\ref{rem:BHP-char} below. \end{remark} \paragraph{Infinite-volume Brownian disk {\normalfont $\textup{\textsf{IBD}}_\sigma$, $\sigma\in(0,\infty)$.}} The infinite-volume Brownian disk $\textup{\textsf{IBD}}_\sigma$ should be thought of as a Brownian disk $\mathcal BD_\sigma$ filled in with a Brownian plane $\mathcal BP$. The definition is a bit elaborate; we give some explanation in Remark~\ref{rem:def-IBD} below. Let $(B_t,t\geq 0)$ be a standard Brownian motion with $B_0=0$, and $T_{x}=\inf\{t\geq 0:B_t<-x\}$ the first hitting time of $(-\infty,-x)$. Let $R,R'$ be two independent three-dimensional Bessel processes independent of $B$, and $U_0$ be a uniform random variable in $[0,\sigma]$, independent of $B,R,R'$. We set $$Y^\sigma_t=\left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }\,}l} R'_{-t+T_\sigma-T_{U_0}}+\sigma -U_0 & t \leq T_{U_0}-T_\sigma\\ B_{T_\sigma+t}+\sigma & T_{U_0}-T_\sigma\leq t \leq 0\\ B_t & 0\leq t\leq T_{U_0}\\ -U_0+R_{t-T_{U_0}} & t \geq T_{U_0} \end{array}\right. . $$ \begin{defn} \textup{$\mathfrak{l}$}bel{def:IBD} Let $\sigma>0$ be fixed. The infinite-volume Brownian disk $\textup{\textsf{IBD}}_\sigma$ with boundary length $\sigma$ is the pointed space $(M_{X,W},D_{X,W},\rho)$ under the probability distribution such that \begin{itemize} \item $(X_t,t\in \textup{$\mathfrak{m}$}athbb R)$ is given by the process $Y^\sigma$ described above. \item Given $X$, $W$ has same distribution as $(\sqrt{3}\, \gamma_{-\underline{\underline{X}}^\sigma_t}+Z_t,t\in \textup{$\mathfrak{m}$}athbb R)$, where \begin{itemize} \item $$\underline{\underline{X}}_t=\left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }\,}l} \textup{$\mathfrak{m}$}in\left\{\inf_{(-\infty,t]}X,\, \inf_{[0,\infty)} X+\sigma\right\} &t\leq 0\\ \textup{$\mathfrak{m}$}in_{[0,t]}X & t\geq 0 \end{array}\right.,$$ and $\underline{\underline{X}}^\sigma=\underline{\underline{X}}-\sigma$ on $(-\infty,0)$, $\underline{\underline{X}}^\sigma=\underline{\underline{X}}$ on $[0,\infty)$. \item $(Z_t,t\in \textup{$\mathfrak{m}$}athbb R)=Z^{X-\underline{\underline{X}}}$ is the random snake driven by the process $X-\underline{\underline{X}}$. \item $(\gamma_x,0\leq x\leq \sigma)$ is a standard Brownian bridge with duration $\sigma$, independent of $Z^{X-\underline{\underline{X}}}$. \end{itemize} \end{itemize} \end{defn} The infinite-volume Brownian disks enjoy the scaling property $$\textup{$\mathfrak{l}$}mbda\cdot \textup{\textsf{IBD}}_{\sigma}=_d\textup{\textsf{IBD}}_{\textup{$\mathfrak{l}$}mbda^2\sigma}\, ,$$ for $\textup{$\mathfrak{l}$}mbda,\sigma>0$. We will prove in Corollary~\ref{cor:topology-IBD} below that for every $\sigma>0$, $\textup{\textsf{IBD}}_\sigma$ is a.s.\ homeomorphic to the pointed closed disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$. \begin{remark} \textup{$\mathfrak{l}$}bel{rem:def-IBD} We give some intuition behind the above definition. Recall that in the case of the Brownian disk $\mathcal BD_{T,\sigma}$, the contour process $X$ is given by a first-passage Brownian bridge from $0$ to $-\sigma$ of duration $T$. Here, in the case of $\textup{\textsf{IBD}}_\sigma$, we consider a Brownian motion $B$ stopped upon first hitting $-\sigma$. When for the first time level $-U_0$ is hit, where $U_0$ is uniform on $[0,\sigma]$, the encoding of a Brownian plane ``inside'' the disk starts. The encoding of the latter is given by two independent three-dimensional Bessel processes $R$ and $R'$, as in the definition of $\mathcal BP$. The part of the Brownian plane encoded by $R'$ as well as the trees encoded by $B$ along $(U_0,\sigma]$ appear in the definition of $Y^\sigma$ to the left of zero. \end{remark} \begin{figure} \caption{The contour process $X=_dY^\sigma$ of the infinite Brownian disk $\textup{\textsf{IBD} \end{figure} \paragraph{Uniform infinite half-planar quadrangulation {\normalfont $\textup{\textsf{UIHPQ}}$}.} The (non-compact) random metric space $\textup{\textsf{UIHPQ}}$ $Q_{\infty}^{\infty}=(V(Q_{\infty}^{\infty}),d_{\textup{gr}},\rho)$ is an infinite rooted random quadrangulation with an infinite boundary. It arises as the distributional limit of $Q_n^{\sigma_n}$, $1\ll\sigma_n\ll n$, for the so-called local metric $d_{\textup{map}}$, see Proposition~\ref{prop:Qn-UIHPQ}. We defer to Section~\ref{sec:locallimit} for a definition of the metric and to Section~\ref{sec:constr-UIHPQ} for a precise construction of the $\textup{\textsf{UIHPQ}}$. \subsection{Notion of convergence} \textup{$\mathfrak{l}$}bel{S-notionconvergence} \subsubsection{Gromov-Hausdorff convergence} Given two pointed compact metric spaces $\textup{$\mathfrak{m}$}athbf{E}=(E,d,\rho)$ and $\textup{$\mathfrak{m}$}athbf{E}' =(E',d',\rho')$, the Gromov-Hausdorff distance between $\textup{$\mathfrak{m}$}athbf{E}$ and $\textup{$\mathfrak{m}$}athbf{E}'$ is given by $$ d_{\textup{GH}}(\textup{$\mathfrak{m}$}athbf{E},\textup{$\mathfrak{m}$}athbf{E}') = \inf\left\{d_{\textup{H}}(\varphi(E),\varphi'(E))\vee\mathrm{d}elta(\varphi(\rho),\varphi'(\rho'))\right\}, $$ where the infimum is taken over all isometric embeddings $\varphi : E\rightarrow F$ and $\varphi' : E'\rightarrow F$ of $E$ and $E'$ into the same metric space $(F,\mathrm{d}elta)$, and $d_{\textup{H}}$ denotes the Hausdorff distance between compact subsets of $F$. The space of all isometry classes of pointed compact metric spaces $(\textup{$\mathfrak{m}$}athbb{K},d_{\textup{GH}})$ forms a Polish space. We will use a well-known alternative characterization of the Gromov-Hausdorff distance {\it via} correspondences, which we recall here for the reader's convenience. A {\it correspondence} between two pointed metric spaces $\textup{$\mathfrak{m}$}athbf{E}=(E,d,\rho)$, $\textup{$\mathfrak{m}$}athbf{E}'=(E',d',\rho')$ is a subset $\mathcal{R}\subset E\times E'$ such that $(\rho,\rho')\in \mathcal{R}$, and for every $x\in E$ there exists at least one $x'\in E'$ such that $(x,x')\in E\times E'$ as well as for every $y'\in E'$, there exists at least one $y\in E$ such that $(y,y')\in\mathcal{R}$. The distortion of $\mathcal{R}$ with respect to $d$ and $d'$ is given by $$ \textup{dis}(\mathcal{R}) = \sup\left\{|d(x,y)-d'(x',y')| : (x,x'), (y,y')\in\mathcal{R}\right\}. $$ Then it holds that (see, for example,~\cite{BuBuIv}) $$ d_{\textup{GH}}(\textup{$\mathfrak{m}$}athbf{E},\textup{$\mathfrak{m}$}athbf{E}') = \textup{$\mathfrak{f}$}rac{1}{2}\inf_\mathcal{R}\textup{dis}(\mathcal{R}), $$ where the infimum is taken over all correspondences between $\textup{$\mathfrak{m}$}athbf{E}$ and $\textup{$\mathfrak{m}$}athbf{E}'$. The convergences listed in the overview above which involve compact limiting spaces, i.e., $\mathcal BM$, $\mathcal BD_\sigma$, $\textup{\textsf{CRT}}$ and the trivial one-point space, hold in distribution in $(\textup{$\mathfrak{m}$}athbb{K},d_{\textup{GH}})$. \subsubsection{Local Gromov-Hausdorff convergence} \textup{$\mathfrak{l}$}bel{sec:locGH} For non-compact spaces like $\mathcal BHP_\theta$, $\textup{\textsf{IBD}}_\sigma$ or $\textup{\textsf{ICRT}}$, the Gromov-Hausdorff convergence is too restrictive. Instead, the right notion is convergence in the so-called local Gromov-Hausdorff sense, which, roughly speaking, requires only convergence of balls of a fixed radius seen as compact metric spaces. We give here a quick reminder of this form of convergence; for more details, we refer to Chapter $8$ of~\cite{BuBuIv}. As in~\cite{CuLG}, we can restrict ourselves to the case of (pointed) complete and locally compact length spaces (see our discussion below). More precisely, a metric space $(E,d)$ is a length space if for every pair $(x,y)$ of points in $E$, the distance $d(x,y)$ agrees with the infimum over the lengths of continuous paths from $x$ to $y$. Here, a continuous path from $x$ to $y$ is a continuous function $\gamma : [0,T]\rightarrow E$ with $\gamma(0)=x$ and $\gamma(T)=y$ for some $T\geq 0$, and the length of $\gamma$ is given by $$ L(\gamma)=\sup_{\tau}\sum_{k=1}^{n-1} d(\gamma(t_k),\gamma(t_{k+1})), $$ where the supremum is taken over all subdivisions $\tau$ of $[0,T]$ of the form $0=t_1<t_2<\mathrm{d}ots<t_n=T$ for some $n\in\textup{$\mathfrak{m}$}athbb N$. Note that in a complete and locally compact length space $(E,d)$, there exists between any two points $x,y\in E$ with $d(x,y)<\infty$ a continuous path of minimal length, see~\cite[Theorem 2.5.23]{BuBuIv}. Now let $\textup{$\mathfrak{m}$}athbf{E}=(E,d,\rho)$ be a pointed metric space, that is a metric space with a distinguished point $\rho\in E$. We denote by $B_r(\textup{$\mathfrak{m}$}athbf{E})$ the closed ball of radius $r$ around $\rho$ in $\textup{$\mathfrak{m}$}athbf{E}$. Equipped with the restriction of $d$, we view $B_r(\textup{$\mathfrak{m}$}athbf{E})$ as a pointed compact metric space, with distinguished point given by $\rho$. By a small abuse of notation, we shall also view $B_r(\textup{$\mathfrak{m}$}athbf{E})$ as a set and write $x\in B_r(\textup{$\mathfrak{m}$}athbf{E})$ if $x\in E$ is at distance at most $r$ from $\rho$. Given pointed complete and locally compact length spaces $(\textup{$\mathfrak{m}$}athbf{E}_n)_n$ and $\textup{$\mathfrak{m}$}athbf{E}$, the sequence $(\textup{$\mathfrak{m}$}athbf{E}_n)_n$ converges to $\textup{$\mathfrak{m}$}athbf{E}$ in the local Gromov-Hausdorff sense if for every $r\geq 0$, $$ d_{\textup{GH}}(B_r(\textup{$\mathfrak{m}$}athbf{E}_n),B_r(\textup{$\mathfrak{m}$}athbf{E}))\rightarrow 0\textup{$\mathfrak{q}$}uad\textup{ as }n\rightarrow\infty. $$ This notion of convergence is metrizable (see~\cite[Section 2.1]{CuLG} for a definition of the metric) and turns the space $\textup{$\mathfrak{m}$}athbb{K}_{bcl}$ of isometry classes of pointed boundedly compact length spaces into a Polish space. We are interested in limits of quadrangulations; however, as discrete planar maps the latter are clearly not length spaces. Following~\cite{CuLG}, we may nonetheless interpret a (finite or infinite) quadrangulation $Q$ as a pointed complete and locally finite length space ${\textup{$\mathfrak{m}$}athbf Q}$. Namely, we replace each edge of $Q$ by an Euclidean segment of length one such that two segments can intersect only at their endpoints, and they do so if and only if the corresponding edges in $E$ share one or two vertices. The resulting metric space ${\textup{$\mathfrak{m}$}athbf Q}$ is then a union of copies of the interval $[0,1]$, one for each edge of $Q$. The distance between two points is simply given by the length of a shortest path between them. With the root vertex of $Q$ as distinguished point, this new metric space ${\textup{$\mathfrak{m}$}athbf Q}$ is a (pointed) complete and locally compact length space. Moreover, it is easy to see that $d_{\textup{GH}}(B_r(Q),B_r({\bf Q}))\leq 1$ for every $r\geq 0$. \\ \newline {\noindent\bf Notation:} Given a pointed metric space $\textup{$\mathfrak{m}$}athbf{E}=(E,d,\rho)$ and $\textup{$\mathfrak{l}$}mbda >0$, we write $\textup{$\mathfrak{l}$}mbda\cdot\textup{$\mathfrak{m}$}athbf{E}$ for the dilated (or rescaled) space $(E,\textup{$\mathfrak{l}$}mbda\cdot d,\rho)$. In particular, if $\textup{$\mathfrak{l}$}mbda,\mathrm{d}elta>0$, $\textup{$\mathfrak{l}$}mbda\cdot B_\mathrm{d}elta(\textup{$\mathfrak{m}$}athbf{E})=B_{\textup{$\mathfrak{l}$}mbda \mathrm{d}elta}(\textup{$\mathfrak{l}$}mbda\cdot\textup{$\mathfrak{m}$}athbf{E})$. \begin{remark} \textup{$\mathfrak{l}$}bel{rem:localGH} From our observation above, we deduce that our limit results for quadrangulations $Q_n^{\sigma_n}$ in the local Gromov-Hausdorff sense will follow if we show that for each $r\geq 0$, $B_r(a_n^{-1}\cdot Q_n^{\sigma_n})$ converges in distribution in $\textup{$\mathfrak{m}$}athbb{K}$ towards the ball of radius $r$ in the corresponding limit space (all our limit spaces are already locally compact length spaces). We therefore do not have to deal with the more complicated notion of local Gromov-Hausdorff convergence for general (pointed) metric spaces, see~\cite[Definition 8.1.1]{BuBuIv}. \end{remark} \subsubsection{Local limits of maps} \textup{$\mathfrak{l}$}bel{sec:locallimit} Local limits of maps in the sense of Benjamini and Schramm~\cite{BeSc} concern the convergence of combinatorial balls. More specifically, given a rooted planar map $\textup{$\mathfrak{m}$}$ and $r\geq 0$, write $\textup{Ball}_r(\textup{$\mathfrak{m}$})$ for the combinatorial of radius $r$, that is the submap of $\textup{$\mathfrak{m}$}$ formed by all the vertices $v$ of $\textup{$\mathfrak{m}$}$ with $d_{\textup{gr}}(\varrho,v)\leq r$, together with the edges of $\textup{$\mathfrak{m}$}$ in between such vertices. For two rooted maps $\textup{$\mathfrak{m}$}$ and $\textup{$\mathfrak{m}$}'$, the local distance between $\textup{$\mathfrak{m}$}$ and $\textup{$\mathfrak{m}$}'$ is defined as $$ d_{\textup{map}}(\textup{$\mathfrak{m}$},\textup{$\mathfrak{m}$}') = \left(1+\sup\{r\geq 0:\textup{Ball}_r(\textup{$\mathfrak{m}$})=\textup{Ball}_r(\textup{$\mathfrak{m}$}')\}\right)^{-1}. $$ The metric $d_{\textup{map}}$ induces a topology on the space of all finite quadrangulations (with or without boundary). {\it Infinite quadrangulations} are the elements in the completion of this space with respect to $d_{\textup{map}}$ that are not finite quadrangulations (the $\textup{\textsf{UIHPQ}}$ is a random infinite quadrangulation with an infinite boundary). See~\cite{CuMeMi} for more on this. \section{Main results} \textup{$\mathfrak{l}$}bel{sec:mainresults} We formulate now in a proper way our main results, which cover together with the results of~\cite{Be3,BeMi} all the convergences listed in the introduction. The proofs will be given in Section~\ref{sec:proofs}. \subsection{Scaling limits of quadrangulations with a boundary} \textup{$\mathfrak{l}$}bel{sec:results-scalinglimits} We let $Q_n^{\sigma_n}$ be uniformly distributed over the set of all rooted planar quadrangulations with $n$ inner faces and a boundary of perimeter $2\sigma_n$, $\sigma_n\in\textup{$\mathfrak{m}$}athbb N$. Recall that we write $V(Q_n^{\sigma_n})$ for the vertex set of $Q_n^{\sigma_n}$, $\rho_n$ for its root vertex and $d_{\textup{gr}}$ for the graph distance on $V(Q_n^{\sigma_n})$. Always, $(a_n,n\in\textup{$\mathfrak{m}$}athbb N)$ a sequence of (strictly) positive reals. All convergences in this section are in law, with respect to the local Gromov-Hausdorff topology. We always let $n\rightarrow\infty$. \begin{thm} \textup{$\mathfrak{l}$}bel{thm:BP} Assume $\sigma_n\ll\sqrt{n}$. If $\sqrt{\sigma_n} \ll a_n \ll n^{1/4}$, then $$ (V(Q_n^{\sigma_n}),a_n^{-1} d_{\textup{gr}},\rho_n) \longrightarrow \mathcal BP. $$ \end{thm} \begin{thm} \textup{$\mathfrak{l}$}bel{thm:IBD} Assume $1\ll\sigma_n\ll\sqrt{n}$ and $a_n \sim (4/9)^{1/4}\sqrt{\sigma_n/\sigma}$ for some $\sigma \in (0,\infty)$. Then $$ (V(Q_n^{\sigma_n}),a_n^{-1} d_{\textup{gr}},\rho_n) \longrightarrow\textup{\textsf{IBD}}_{\sigma}. $$ \end{thm} \begin{thm} \textup{$\mathfrak{l}$}bel{thm:BHP1} Assume $1\ll \sigma_n\ll n$ and $1\ll a_n \ll \textup{$\mathfrak{m}$}in\{\sqrt{\sigma_n},\,\sqrt{n/\sigma_n}\}$. Then $$ (V(Q_n^{\sigma_n}),a_n^{-1} d_{\textup{gr}},\rho_n) \longrightarrow \mathcal BHP. $$ \end{thm} \begin{thm} \textup{$\mathfrak{l}$}bel{thm:BHP3} Assume $\sqrt{n}\ll\sigma_n\ll n$ and $a_n\sim 2\sqrt{\theta n/3\sigma_n}$ for some $\theta\in (0,\infty)$. Then $$ (V(Q_n^{\sigma_n}),a_n^{-1} d_{\textup{gr}},\rho_n) \longrightarrow\mathcal BHP_\theta. $$ \end{thm} \begin{thm} \textup{$\mathfrak{l}$}bel{thm:ICRT} Assume $\sigma_n\gg \sqrt{n}$ and $\textup{$\mathfrak{m}$}ax\{1,\,\sqrt{n/\sigma_n}\} \ll a_n \ll \sqrt{\sigma_n}$. Then $$ (V(Q_n^{\sigma_n}),a_n^{-1} d_{\textup{gr}},\rho_n) \longrightarrow\textup{\textsf{ICRT}}. $$ \end{thm} When the scaling sequence $(a_n,n\in\textup{$\mathfrak{m}$}athbb N)$ satisfies $a_n\gg\textup{$\mathfrak{m}$}ax\{\sqrt{\sigma_n},n^{1/4}\}$, then the limiting space is the trivial one-point metric space. This is a direct consequence of the results in~\cite{Be3}, for example. The Brownian half-plane $\mathcal BHP$ does also arise as the weak scaling limit of the $\textup{\textsf{UIHPQ}}$ (similarly, the Brownian plane $\mathcal BP$ is the scaling limit of the so-called uniform infinite planar quadrangulation $\textup{\textsf{UIPQ}}$, see the first part of~\cite[Theorem 2]{CuLG}). \begin{thm} \textup{$\mathfrak{l}$}bel{thm:UIHPQ-BHP} $$ \textup{$\mathfrak{l}$}mbda\cdot\textup{\textsf{UIHPQ}} \xrightarrow[]{\textup{$\mathfrak{l}$}mbda\rightarrow 0}\mathcal BHP. $$ \end{thm} \subsection{Couplings and topology} For proving Theorem~\ref{thm:BHP1}, we follow a strategy similar to that in Curien and Le Gall~\cite{CuLG}. As an intermediate step, we establish a coupling between the Brownian disk $\mathcal BD_\sigma$ and the Brownian half-plane $\mathcal BHP_\theta$, which we also apply to determine the topology of $\mathcal BHP_\theta$. \begin{thm} \textup{$\mathfrak{l}$}bel{thm:coupling-BD-BHP} Let $\varepsilon>0$, $r\geq 0$. Let $\sigma(\cdot):(0,\infty)\rightarrow(0,\infty)$ be a function satisfying $\lim_{T\rightarrow\infty}\sigma(T)/T=\theta\in[0,\infty)$ and $\liminf_{T\rightarrow\infty}\sigma(T)/\sqrt{T}>0$. Then there exists $T_0=T_0(\varepsilon,r,\sigma)$ such that for all $T\geq T_0$, one can construct copies of $\mathcal BD_{T,\sigma(T)}$ and $\mathcal BHP_\theta$ on the same probability space such that with probability at least $1-\varepsilon$, there exist two isometric open subsets $O_\mathcal BD$, $O_{\mathcal BHP}$ in these spaces which are both homeomorphic to the closed half-plane $\overline{\textup{$\mathfrak{m}$}athbb{H}}$ and contain the balls $B_r(\mathcal BD_{T,\sigma(T)})$ and $B_r(\mathcal BHP_\theta)$, respectively. \end{thm} We remark that for the proof of Theorem~\ref{thm:BHP1}, it would be sufficient to show that the balls of radius $r$ around the root in the corresponding spaces are isometric. From the stronger version of the coupling stated above, we can however additionally deduce \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:topology-BHP} For every $\theta \geq 0$, the space $\mathcal BHP_\theta$ is a.s. homeomorphic to the closed half-plane $\overline{\textup{$\mathfrak{m}$}athbb{H}}=\textup{$\mathfrak{m}$}athbb R\times\textup{$\mathfrak{m}$}athbb R_+$. \end{corollary} Since the Brownian half-plane $\mathcal BHP=\mathcal BHP_0$ is scale-invariant, i.e., $\textup{$\mathfrak{l}$}mbda\cdot\mathcal BHP =_d\mathcal BHP$ for every $\textup{$\mathfrak{l}$}mbda>0$, Theorem~\ref{thm:coupling-BD-BHP} moreover implies that $\mathcal BHP$ is locally isometric to the disk $\mathcal BD_\sigma(=\mathcal BD_{1,\sigma})$. \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:isometry-BD-BHP} Fix $\sigma\in(0,\infty)$, and let $\varepsilon>0$. Then one can find $\mathrm{d}elta>0$ and construct on the same probability space copies of $\mathcal BD_\sigma$ and $\mathcal BHP$ such that with probability at least $1-\varepsilon$, $B_\mathrm{d}elta(\mathcal BHP)$ and $B_\mathrm{d}elta(\mathcal BD_\sigma)$ are isometric. \end{corollary} The proof of Corollary~\ref{cor:isometry-BD-BHP} is immediate from the scaling properties of $\mathcal BD_{T,\sigma}$ and $\mathcal BHP$, whereas Corollary~\ref{cor:topology-BHP} needs an extra argument, which we give in Section~\ref{sec:proof-coupling-BD-BHP}. \begin{remark} \textup{$\mathfrak{l}$}bel{rem:BHP-char} The local isometry between $\mathcal BHP$ and $\mathcal BD_\sigma$ together with the fact that $\mathcal BHP$ is scale-invariant uniquely characterizes the law of $\mathcal BHP$ in the set of all probability measures on $\textup{$\mathfrak{m}$}athbb{K}_{bcl}$. This follows from the argument in the proof of~\cite[Proposition 3.2]{CuLG2}, where a similar characterization of the Brownian plane is given. \end{remark} For establishing Theorem~\ref{thm:BHP1}, we shall also need a coupling between the $\textup{\textsf{UIHPQ}}$ and $Q_n^{\sigma_n}$ when $\sigma_n$ grows slower than $n$. \begin{prop} \textup{$\mathfrak{l}$}bel{prop:Qn-UIHPQ} Assume $1\ll \sigma_n\ll n$, and put $\vartheta_n=\textup{$\mathfrak{m}$}in\left\{\sigma_n,\, n/\sigma_n\right\}$. Given any $\varepsilon>0$, there exist $\mathrm{d}elta>0$ and $n_0\in\textup{$\mathfrak{m}$}athbb N$ such that for every $n\geq n_0$, one can construct copies of $Q_n^{\sigma_n}$ and $\textup{\textsf{UIHPQ}}$ on the same probability space such that with probability at least $1-\varepsilon$, the balls $B_{\mathrm{d}elta \sqrt{\vartheta_n}}(Q_n^{\sigma_n})$ and $B_{\mathrm{d}elta \sqrt{\vartheta_n}}(\textup{\textsf{UIHPQ}})$ are isometric. Moreover, we have the local convergence $$ (V(Q_n^{\sigma_n}),d_{\textup{gr}},\rho_n) \longrightarrow \textup{\textsf{UIHPQ}} $$ in distribution for the metric $d_{\textup{map}}$, as $n\rightarrow\infty$. \end{prop} Note that the above mentioned $\textup{\textsf{UIPQ}}$ is in turn the weak limit in the sense of $d_{\textup{map}}$ for uniform quadrangulations {\it without} a boundary, see Krikun~\cite{Kr}. For proving Theorem~\ref{thm:IBD} and showing that the topology of $\textup{\textsf{IBD}}_\sigma$ is that of a pointed closed disk, we couple the Brownian disk $\mathcal BD_{T,\sigma}$ for large volumes $T$ with the infinite-volume Brownian disk $\textup{\textsf{IBD}}_\sigma$. \begin{thm} \textup{$\mathfrak{l}$}bel{thm:coupling-BD-IBD} Fix $\sigma\in(0,\infty)$, and let $\varepsilon>0$, $r\geq 0$. There exists $T_0=T_0(\varepsilon,r,\sigma)$ such that for all $T\geq T_0$, we can construct copies of $\mathcal BD_{T,\sigma}$ and $\textup{\textsf{IBD}}_\sigma$ on the same probability space such that with probability at least $1-\varepsilon$, there exist two isometric open subsets $A_{\mathcal BD}$, $A_{\textup{\textsf{IBD}}}$ in these spaces which are both homeomorphic to the pointed closed disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$ and contain the balls $B_r(\mathcal BD_{T,\sigma})$ and $B_r(\textup{\textsf{IBD}}_\sigma)$, respectively. \end{thm} It will be straightforward to deduce \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:topology-IBD} For each $\sigma\in(0,\infty)$, the space $\textup{\textsf{IBD}}_\sigma$ is a.s. homeomorphic to the pointed closed disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$, where $\textup{$\mathfrak{m}$}athbb{D}=\{z\in \textup{$\mathfrak{m}$}athbb{C}:|z|<1\}$. \end{corollary} In order to prove Theorem~\ref{thm:IBD}, we finally need a coupling of balls in the quadrangulations $Q_n^{\sigma_n}$ and $Q_{R\sigma_n^2}^{\sigma_n}$ of a radius of order $\sqrt{\sigma_n}$, when $1\ll \sigma_n\ll \sqrt{n}$ and $R$ is large. \begin{prop} \textup{$\mathfrak{l}$}bel{prop:coupling-Qn-largevol} Assume $1\ll \sigma_n\ll \sqrt{n}$. Given any $\varepsilon>0$ and $r>0$, there exist $R_0>0$ and $n_0\in\textup{$\mathfrak{m}$}athbb N$ such that for every integer $R\geq R_0$ and every $n\geq n_0$, on can construct copies of $Q_n^{\sigma_n}$ and $Q_{R\sigma_n^2}^{\sigma_n}$ on the same probability space such that with probability at least $1-\varepsilon$, the balls $B_{r\sqrt{\sigma_n}}(Q_n^{\sigma_n})$ and $B_{r\sqrt{\sigma_n}}(Q_{R\sigma_n^2}^{\sigma_n})$ are isometric. \end{prop} Some of our results involving $\textup{\textsf{UIHPQ}}$, $\mathcal BHP$ and $\mathcal BD_\sigma$ are depicted in Figure~\ref{fig:diagram1}, which should be compared with~\cite[Figure 1]{CuLG}. \begin{figure} \caption{Illustration of~\cite[Theorem 1]{BeMi} \end{figure} \subsection{Limits of the Brownian disk} \textup{$\mathfrak{l}$}bel{sec:results-BDlimits} Our statements from the last two sections imply various limit results for the Brownian disk $\mathcal BD_{T,\sigma(T)}$ when zooming-in around its root. We let $\sigma(\cdot):(0,\infty)\rightarrow(0,\infty)$ be a function of the volume of the Brownian disk that specifies its perimeter. All of the following convergences hold in distribution with respect to the local Gromov-Hausdorff topology when the volume $T$ of the disk tends to infinity. \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:BD1} Assume $\lim_{T\rightarrow\infty}\sigma(T)= 0$. Then $$ \mathcal BD_{T,\sigma(T)}\longrightarrow\mathcal BP. $$ \end{corollary} \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:BD4} Assume $\lim_{T\rightarrow\infty}\sigma(T)= \varsigma\in(0,\infty)$. Then $$ \mathcal BD_{T,\sigma(T)}\longrightarrow\textup{\textsf{IBD}}_\varsigma. $$ \end{corollary} \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:BD2} Assume $\sigma(T)\rightarrow\infty$ and $\sigma(T)/T\rightarrow\theta\in[0,\infty)$ as $T\rightarrow\infty$. Then $$ \mathcal BD_{T,\sigma(T)}\longrightarrow\mathcal BHP_{\theta}. $$ \end{corollary} \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:BD3} Assume $\sigma(T)/T\rightarrow\infty$ as $T\rightarrow\infty$. Then $$ \mathcal BD_{T,\sigma(T)}\longrightarrow\textup{\textsf{ICRT}}. $$ \end{corollary} Note that Corollary~\ref{cor:BD2} includes the case where $\sigma(T)=\sqrt{T}$. Then $\theta=0$, and since by scaling, $T^{1/4}\cdot \mathcal BD_1=_d \mathcal BD_{T,\sqrt{T}}$, it follows that $\mathcal BHP$ is the tangent cone in distribution of any disk $\mathcal BD_{A,L}$ for fixed $A,L>0$. See~\cite[Section 8.2]{BuBuIv} for an explanation of this terminology in the context of boundedly compact length spaces, and compare with~\cite[Theorem 1]{CuLG}, where it is shown that the Brownian plane is the tangent cone of the Brownian map at its root. For completeness, but without going into details, let us mention that identically to the proof of Corollary~\ref{cor:BD1} (or Corollary~\ref{cor:BD3}), a combination of~\cite[Theorem 1]{BeMi} and~\cite[Theorem 4]{Be3} (or ~\cite[Theorem 4]{Be3}) leads to the convergences $$ \mathcal BD_{T,\sigma}\xrightarrow[]{\sigma\to 0}\mathcal BM_T,\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \mathcal BD_{T,\sigma}\xrightarrow[]{T\to 0}\textup{\textsf{CRT}}_{3\sigma} $$ in law in the sense of the {\it global} Gromov-Hausdorff topology. The factor $3$ in $\textup{\textsf{CRT}}_{3\sigma}$ stems from the particular normalization of the Brownian disk. \begin{remex} \textup{$\mathfrak{l}$}bel{rem:exercises} We leave it as an exercise to the reader to find the right combination of our (or Bettinelli's, cf.~\cite{Be3}) foregoing results to deduce the following additional results on tangent cones (in distribution, with respect to the local Gromov-Hausdorff topology): $$\textup{\textsf{CRT}}_T\xrightarrow[]{T\rightarrow\infty} \textup{\textsf{ICRT}},\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \mathcal BHP_\theta\xrightarrow[]{\theta \to 0} \mathcal BHP,\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \textup{\textsf{IBD}}_\sigma\xrightarrow[]{\sigma\rightarrow\infty} \mathcal BHP.$$ Combining results from the regime $\sigma_n\ll \sqrt{n}$ in the first and from $\sigma_n\gg \sqrt{n}$ in the second case, one may also prove the following scaling results in law: $$ \mathcal BHP_\theta\xrightarrow[]{\theta\rightarrow\infty} \textup{\textsf{ICRT}}\textup{$\mathfrak{q}$}uad,\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \textup{\textsf{IBD}}_\sigma\xrightarrow[]{\sigma\rightarrow 0} \mathcal BP.$$ \end{remex} \begin{figure} \caption{Zooming-in around the root of the Brownian disk $\mathcal BD_{T,\sigma(T)} \end{figure} \section{Encoding of quadrangulations with a boundary} \textup{$\mathfrak{l}$}bel{sec:encoding} We will use a variant of the Cori-Vauquelin-Schaeffer~\cite{CoVa,Sc} bijection developed by Bouttier, Di Francesco and Guitter~\cite{BoDFGu} to encode quadrangulations with a boundary. More specifically, we will encode planar quadrangulation of size $n$ with a boundary of size $2\sigma$ in terms of $\sigma$ trees with $n$ edges in total, which are attached to a discrete bridge of length $\sigma$. We first introduce the encoding objects. Our notation is inspired by~\cite{Be2,Be3}. \subsection{Encoding in the finite case} \textup{$\mathfrak{l}$}bel{sec:encoding-finite} \subsubsection{Well-labeled tree, forest and bridge} \textup{$\mathfrak{l}$}bel{sec:welllabeledforest} A {\it well-labeled tree} $(\tau,(\ell(u))_{u\in\tau})$ of size $|\tau|=n$ consists of a rooted plane tree $\tau$ with $n$ edges together with integer labels $(\ell(u))_{u\in V(\tau)}$ attached to the vertices of $\tau$, such that the root has label $0$, and $|\ell(u)-\ell(v)| \leq 1$ whenever $u$ and $v$ are neighbors. A {\it well-labeled forest} with $\sigma$ trees and $n$ tree edges is a collection $\textup{$\mathfrak{f}$}=(\tau_0,\ldots,\tau_{\sigma-1})$ of $\sigma$ trees with $n$ edges in total, together with a labeling of vertices $\textup{$\mathfrak{l}$} :\cup_{i=0}^{\sigma-1}V(\tau_i)\rightarrow\textup{$\mathfrak{m}$}athbb{Z}$, which has the property that for each $i=0,\ldots,\sigma-1$, the tree $\tau_i$ together with the restriction $\textup{$\mathfrak{l}$}\restriction V(\tau_i)$ forms a well-labeled tree. The vertex set of $\textup{$\mathfrak{f}$}$ is $V(\textup{$\mathfrak{f}$})=\cup_{i=0}^{\sigma-1}V(\tau_i)$. Note that $|V(\textup{$\mathfrak{f}$})|=n+\sigma$. The size of $\textup{$\mathfrak{f}$}$ is given by $|\textup{$\mathfrak{f}$}|= n$, i.e., its number of edges. We write $(0),\ldots,(\sigma-1)$ for the root vertices of $\tau_0,\ldots,\tau_{\sigma-1}$. If $u$ is a vertex of a tree of $\textup{$\mathfrak{f}$}$, $\textup{$\mathfrak{r}$}(u)$ denotes the root of this tree. In particular, the vertex set of the $j$th tree of $\textup{$\mathfrak{f}$}$ is the set $\{u\in V(\textup{$\mathfrak{f}$}):\textup{$\mathfrak{r}$}(u)=(j-1)\}$, $j=1,\ldots,\sigma$. We write $t(\textup{$\mathfrak{f}$})=\sigma$ for the number of trees of $\textup{$\mathfrak{f}$}$. We will often identify the root vertices with the integers $0,\ldots,\sigma-1$ and consequently regard $\textup{$\mathfrak{r}$}(u)$ as a number. We call the pair $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})$ a {\it well-labeled forest} and denote by $$\mathbb Fo_\sigma^n = \{(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}): t(\textup{$\mathfrak{f}$}) = \sigma, |\textup{$\mathfrak{f}$}| = n\}$$ the set of all well-labeled forests of size $n$ with $\sigma$ trees. A {\it bridge} of length $\sigma\geq 1$ is a sequence of numbers $(\textup{\textsf{b}}(0),\textup{\textsf{b}}(1), \ldots ,\textup{\textsf{b}}(\sigma))$ with $\textup{\textsf{b}}(0)=0$ and such that $\textup{\textsf{b}}(i+1) - \textup{\textsf{b}}(i) \in \textup{$\mathfrak{m}$}athbb N_0\cup\{-1\}$, and $\textup{\textsf{b}}(\sigma) \le 0$. By linear interpolation between integer values, we will view $\textup{\textsf{b}}: [0,\sigma]\to\textup{$\mathfrak{m}$}athbb R$ as a continuous function and write $\mathcal Br_\sigma\subset\textup{$\mathfrak{m}$}athcal{C}([0,\sigma],\textup{$\mathfrak{m}$}athbb R)$ for the set of all possible bridges of length $\sigma$. The terminal value $\textup{\textsf{b}}(\sigma)$ of a bridge has a special interpretation: It keeps the information where to find the root in the quadrangulation associated to a triplet $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})\in\mathbb Fo_\sigma^n\times\mathcal Br_\sigma$, see Section~\ref{sec:BDG-bijection} below. \subsubsection{Contour pair and label function} \textup{$\mathfrak{l}$}bel{sec:contourlabel-finite} Consider a well-labeled forest $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})$ of size $n$ with $\sigma$ trees. In order to define its contour pair and label function, it is convenient to associate to $\textup{$\mathfrak{f}$}$ a representation in the plane, as depicted in Figure~\ref{fig:finiteforest}: We add $\sigma-1$ edges which link the root vertices $(0),\ldots,(\sigma-1)$, such that vertex $(i-1)$ gets connected to $(i)$ for $i=1,\ldots,\sigma-1$, plus an extra vertex $(\sigma)$ and an extra edge linking $(\sigma-1)$ to $(\sigma)$. We extend $\textup{$\mathfrak{l}$}$ to $(\sigma)$ by setting $\textup{$\mathfrak{l}$}((\sigma))=0$. We refer to the segment connecting the roots of $\textup{$\mathfrak{f}$}$ and the extra vertex $(\sigma)$ as the {\it floor} of $\textup{$\mathfrak{f}$}$. \begin{figure} \caption{On the left: A proper representation of a finite forest $\textup{$\mathfrak{f} \end{figure} The {\it facial sequence} $\textup{$\mathfrak{f}$}(0),\ldots,\textup{$\mathfrak{f}$}(2n+\sigma)$ of $\textup{$\mathfrak{f}$}$ is the sequence of vertices obtained from exploring (the embedding of) $\textup{$\mathfrak{f}$}$ in the contour order, starting from vertex $(0)$. In other words, $\textup{$\mathfrak{f}$}(0),\ldots,\textup{$\mathfrak{f}$}(2n+\sigma-1)$ is given by the sequence of vertices of the discrete contour paths of the trees $\tau_0,\ldots,\tau_{\sigma-1}$, and the sequence terminates with value $\textup{$\mathfrak{f}$}(2n+\sigma)=(\sigma)$. See, e.g.,~\cite[Section 2]{LGMi} for more on contour paths. Given a well-labeled forest $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})$, we define its {\it contour pair} $(C_\textup{$\mathfrak{f}$},L_\textup{$\mathfrak{f}$})$ by $$ C_\textup{$\mathfrak{f}$}(j) =d_{\textup{$\mathfrak{f}$}}(\textup{$\mathfrak{f}$}(j),(\sigma))-\sigma, \textup{$\mathfrak{q}$}uad L_\textup{$\mathfrak{f}$}(j) = \textup{$\mathfrak{l}$}(\textup{$\mathfrak{f}$}(j)),\textup{$\mathfrak{q}$}uad j=0,\ldots,2n+\sigma. $$ Here, $d_{\textup{$\mathfrak{f}$}}$ denotes the graph distance on the representation of $\textup{$\mathfrak{f}$}$ in the plane. We call $C_\textup{$\mathfrak{f}$}$ the {\it contour function} of $\textup{$\mathfrak{f}$}$, since it is obtained from concatenating the contour paths of the trees $\tau_0,\ldots,\tau_{\sigma-1}$, with an additional $-1$ step after a tree has been visited. Note that $L_\textup{$\mathfrak{f}$}(\textup{$\mathfrak{f}$}(j))=0$ if $\textup{$\mathfrak{f}$}(j)$ lies on the floor of $\textup{$\mathfrak{f}$}$. See again Figure~\ref{fig:finiteforest} for an illustration. Now consider additionally a bridge $\textup{\textsf{b}}\in\mathcal Br_\sigma$. Put $\underline{C}_{\textup{$\mathfrak{f}$}}(j)=\textup{$\mathfrak{m}$}in_{[0,j]}C_{\textup{$\mathfrak{f}$}}$. The function $$ \textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}}(j) = L_{\textup{$\mathfrak{f}$}}(j) + \textup{\textsf{b}}(-\underline{C}_{\textup{$\mathfrak{f}$}}(j)),\textup{$\mathfrak{q}$}uad j=0,\mathrm{d}ots,2n+\sigma, $$ is called the {\it label function} associated to $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})$. The label function plays an important role in measuring distances in the quadrangulation associated through the Bouttier-Di Francesco-Guitter bijection, see Section~\ref{sec:distances}. By linear interpolation between integers, we extend all three functions $C_\textup{$\mathfrak{f}$}$, $L_\textup{$\mathfrak{f}$}$ and $\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}$ to continuous real-valued functions on $[0,2n+\sigma]$. \subsection{Encoding in the infinite case} \textup{$\mathfrak{l}$}bel{sec:encoding-infinite} We next introduce the infinite analogs of the objects from the previous section. They will encode certain infinite quadrangulations with an infinite boundary. \subsubsection{Well-labeled infinite forest and infinite bridge} A {\it well-labeled infinite forest} is an infinite collection $\textup{$\mathfrak{f}$}=(\tau_{i},i\in\textup{$\mathfrak{m}$}athbb Z)$ of finite rooted plane trees, together with a labeling of vertices $\textup{$\mathfrak{l}$} :\cup_{i\in\textup{$\mathfrak{m}$}athbb Z}V(\tau_i)\rightarrow\textup{$\mathfrak{m}$}athbb{Z}$ such that for each $i\in\textup{$\mathfrak{m}$}athbb Z$, $\tau_i$ together with the restriction of $\textup{$\mathfrak{l}$}$ to $V(\tau_i)$ forms a well-labeled tree. We write again $(k)$ for the root vertex of $\tau_k$ and often identify $(k)$ with $k\in\textup{$\mathfrak{m}$}athbb Z$. We call the pair $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})$ an {\it well-labeled infinite forest} and denote by $\mathbb Fo_\infty$ the set of all well-labeled infinite forests. An {\it infinite bridge} is a sequence of numbers $\textup{\textsf{b}}=(\textup{\textsf{b}}(i),i\in\textup{$\mathfrak{m}$}athbb Z\cup\{\partial\})$ with $\textup{\textsf{b}}(0)=0$, $\textup{\textsf{b}}(i+1)- \textup{\textsf{b}}(i) \in \textup{$\mathfrak{m}$}athbb N_0\cup\{-1\}$ for all $i\in\textup{$\mathfrak{m}$}athbb Z$ and $\textup{\textsf{b}}(\partial)\in\{\textup{\textsf{b}}(-1)-1,\ldots,0\}$. The extra value $\textup{\textsf{b}}(\partial)$ will keep track of the position of the root in the quadrangulation. Often, we consider only the values $\textup{\textsf{b}}(i)$, $i\in\textup{$\mathfrak{m}$}athbb Z$, and then view $\textup{\textsf{b}}$ as a continuous function from $\textup{$\mathfrak{m}$}athbb R$ to $\textup{$\mathfrak{m}$}athbb R$, by linear interpolation between integer values. We write $\mathcal Br_\infty$ for the set of all infinite bridges $\textup{\textsf{b}}$ which have the property that $\inf_{i\in \textup{$\mathfrak{m}$}athbb N}\textup{\textsf{b}}(i)=-\infty$, and $\inf_{i\in \textup{$\mathfrak{m}$}athbb N}\textup{\textsf{b}}(-i)=-\infty$. \subsubsection{Contour pair and label function in the infinite case} \textup{$\mathfrak{l}$}bel{sec:contourlabel-infinite} We consider a well-labeled infinite forest $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})\in\mathbb Fo_\infty$. Again, we view $\textup{$\mathfrak{f}$}$ as a graph properly embedded in the plane (Figure~\ref{fig:infiniteforest}): We identify the set of roots of the trees of $\textup{$\mathfrak{f}$}$ with $\textup{$\mathfrak{m}$}athbb Z$ and connect neighboring roots by an edge. We obtain what we call the {\it floor} of $\textup{$\mathfrak{f}$}$. The trees $\tau_i$ of $\textup{$\mathfrak{f}$}$ are drawn in the upper half-plane and attached to the floor. The {\it facial sequence} $(\textup{$\mathfrak{f}$}(i),i\in\textup{$\mathfrak{m}$}athbb Z)$ of $\textup{$\mathfrak{f}$}$ is defined as follows: $(\textup{$\mathfrak{f}$}(0),\textup{$\mathfrak{f}$}(1),\ldots)$ is the sequence of vertices of the contour paths of the trees $\tau_i, i\in\textup{$\mathfrak{m}$}athbb N_0$, in the contour order, starting from the root of the tree $\tau_0$, and $(\textup{$\mathfrak{f}$}(-1),\textup{$\mathfrak{f}$}(-2),\ldots)$ is given by the sequence of vertices of the contour paths $\tau_{-1},\tau_{-2},\ldots$, in the {\it counterclockwise} order, starting from the root of the tree $\tau_{-1}$. \begin{figure} \caption{On the left: A proper representation of an infinite forest $\textup{$\mathfrak{f} \end{figure} In analogy to the finite case, given a well-labeled infinite tree $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})$, its {\it contour pair} $(C_{\textup{$\mathfrak{f}$}},L_{\textup{$\mathfrak{f}$}})$ is a tuple functions defined {\it via} $$ C_{\textup{$\mathfrak{f}$}}(j) =d_{\textup{$\mathfrak{f}$}}(\textup{$\mathfrak{f}$}(j),\textup{$\mathfrak{r}$}(\textup{$\mathfrak{f}$}(j)))-\textup{$\mathfrak{r}$}(\textup{$\mathfrak{f}$}(j)), \textup{$\mathfrak{q}$}uad L_{\textup{$\mathfrak{f}$}}(j) = \textup{$\mathfrak{l}$}(\textup{$\mathfrak{f}$}(j)),\textup{$\mathfrak{q}$}uad j\in\textup{$\mathfrak{m}$}athbb Z, $$ where $d_{\textup{$\mathfrak{f}$}}$ is the graph distance on the embedding of $\textup{$\mathfrak{f}$}$, and $\textup{$\mathfrak{r}$}(\textup{$\mathfrak{f}$}(j))$ denotes the root of the tree $\textup{$\mathfrak{f}$}(j)$ belongs to. Be aware of the small abuse of notation: In the expression for $C_{\textup{$\mathfrak{f}$}}$, $\textup{$\mathfrak{r}$}(\textup{$\mathfrak{f}$}(j))$ is first viewed as a vertex and then as an integer. Note that $\lim_{j\rightarrow\infty}C_\textup{$\mathfrak{f}$}(j)\rightarrow -\infty$ and $\lim_{j\rightarrow -\infty}C_\textup{$\mathfrak{f}$}(j)\rightarrow +\infty$ for every infinite forest. As for a finite forest, we call $C_{\textup{$\mathfrak{f}$}}$ the {\it contour function} of $\textup{$\mathfrak{f}$}$. If additionally $\textup{\textsf{b}}\in \mathcal Br_\infty$, we define the {\it label function} associated to $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})$ by $$\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}(j)= L_\textup{$\mathfrak{f}$}(j) + \textup{\textsf{b}}(\underline{C}_\textup{$\mathfrak{f}$}(j)),\textup{$\mathfrak{q}$}uad j\in\textup{$\mathfrak{m}$}athbb Z,\textup{$\mathfrak{q}$}uad \textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}(\partial)=\textup{\textsf{b}}(\partial),$$ where $\underline{C}_\textup{$\mathfrak{f}$}(j)=\inf_{(-\infty,j]}C_\textup{$\mathfrak{f}$}$ for $j<0$ and $\underline{C}_\textup{$\mathfrak{f}$}(j)=\textup{$\mathfrak{m}$}in_{[0,j]}C_\textup{$\mathfrak{f}$}$ for $j\geq 0$, as above. Again by linear interpolation between integers, we view $C_\textup{$\mathfrak{f}$},L_\textup{$\mathfrak{f}$}$ and $\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}$ as continuous functions on $\textup{$\mathfrak{m}$}athbb R$. \subsection{Bouttier-Di Francesco-Guitter bijection} \textup{$\mathfrak{l}$}bel{sec:BDG-bijection} Recall that a rooted quadrangulation with a boundary comes with a distinguished edge along the boundary, the root edge, whose origin is the root vertex. We write $\mathcal{Q}_n^{\sigma}$ for the set of all rooted quadrangulations with $n$ inner faces and a boundary of size $2\sigma$. A {\it pointed quadrangulation with a boundary} is a pair $(\textup{$\mathfrak{q}$},v^{\bullet})$, where $\textup{$\mathfrak{q}$}$ is a rooted quadrangulation with a boundary and $v^{\bullet}\in V(\textup{$\mathfrak{q}$})$ is a distinguished vertex. The set of all rooted pointed quadrangulations with $n$ internal faces and $2\sigma$ boundary edges is denoted by $$\mathcal{Q}_{n,\sigma}^\bullet=\left\{(\textup{$\mathfrak{q}$},v^{\bullet}) : \textup{$\mathfrak{q}$}\in\mathcal{Q}_n^{\sigma}, v^{\bullet}\in V(\textup{$\mathfrak{q}$})\right\}.$$ \subsubsection{The finite case} The Bouttier-Di Francesco-Guitter bijection~\cite{BoDFGu} provides us with a bijection $$\mathbb Phi_n:\mathbb Fo_\sigma^n \times \mathcal Br_\sigma\longrightarrow \mathcal{Q}_{n,\sigma}^\bullet.$$ We shall here content ourselves with the description of the mapping from the encoding objects to the quadrangulations. We follow largely the presentation in~\cite{Be3}, where also a description of the reverse direction can be found. In this regard, let $((\textup{$\mathfrak{f}$},\textup{\textsf{b}}),\textup{$\mathfrak{l}$})\in\mathbb Fo_\sigma^n\times\mathcal Br_{\sigma}$. Out of this triplet, we will now construct a rooted pointed quadrangulation $(\textup{$\mathfrak{q}$},v^{\bullet})\in\mathcal{Q}_{n,\sigma}^\bullet$. Recall the facial sequence $\textup{$\mathfrak{f}$}(0),\ldots,\textup{$\mathfrak{f}$}(2n+\sigma)$ of $\textup{$\mathfrak{f}$}$ obtained from exploring the trees of $\textup{$\mathfrak{f}$}$ in the contour order, as well as the associated label function $\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}$. We view $\textup{$\mathfrak{f}$}$ as embedded in the plane (as explained above) and add an additional vertex $v^{\bullet}$ inside the only face of $\textup{$\mathfrak{f}$}$, with label $\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}(v^{\bullet})=-\infty$. The vertex set of $\textup{$\mathfrak{q}$}$ is given by $V(\textup{$\mathfrak{f}$})\cup\{v^{\bullet}\}$. Note that by definition, the additional vertex $(\sigma)$ which forms part of the embedding of $\textup{$\mathfrak{f}$}$ is not an element of $V(\textup{$\mathfrak{f}$})$. In order to specify the edges between the vertices of $\textup{$\mathfrak{q}$}$, we define for $i=0,\ldots,2n+\sigma-1$ the {\it successor} $\textup{succ}(i)\in\{0,\ldots,2n+\sigma-1\}\cup\{\infty\}$ of $i$ to be the first number $k$ in the list $(i+1,\ldots,2n+\sigma-1,0,\ldots,i-1)$ with the property that $\textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}}(k)=\textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}}(i)-1$, with $\textup{succ}(i)=\infty$ if there is no such number. Letting $\textup{$\mathfrak{f}$}(\infty)=v^{\bullet}$, we now follow the facial sequence of $\textup{$\mathfrak{f}$}$ and draw for every $i=0,\ldots,2n+\sigma-1$ an arc between $\textup{$\mathfrak{f}$}(i)$ and $\textup{$\mathfrak{f}$}(\textup{succ}(i))$, in such a way that it neither crosses arcs that were previously drawn, nor edges of the embedding of $\textup{$\mathfrak{f}$}$. Since any vertex of $\textup{$\mathfrak{f}$}$ which is not a leaf is visited at least twice in the contour exploration, there can be several arcs connecting $\textup{$\mathfrak{f}$}(i)$ and $\textup{$\mathfrak{f}$}(\textup{succ}(i))$. By a small abuse of language, we therefore speak of the arc connecting $i$ to $\textup{succ}(i)$ and write $$ i\curvearrowright\textup{succ}(i)\textup{$\mathfrak{q}$}uad\hbox{or}\textup{$\mathfrak{q}$}uad i\curvearrowleft\textup{succ}(i) $$ for the oriented arc from $i$ towards $\textup{succ}(i)$ or from $\textup{succ}(i)$ towards $i$, respectively. \begin{figure} \caption{The Bouttier-Di Francesco-Guitter bijection $\mathbb Phi_n$ applied to an element $((\textup{$\mathfrak{f} \end{figure} The arcs between the vertices $V(\textup{$\mathfrak{f}$})\cup\{v^{\bullet}\}$ form the edges of $\textup{$\mathfrak{q}$}$, and it remains to specify the root edge of $\textup{$\mathfrak{q}$}$: The root vertex is given by $\textup{$\mathfrak{f}$}(\textup{succ}^{-\textup{\textsf{b}}(\sigma)}(0))$, and the root edge is in case $\textup{\textsf{b}}(\sigma)>\textup{\textsf{b}}(\sigma-1)-1$ given by $\textup{succ}^{-\textup{\textsf{b}}(\sigma)}(0)\curvearrowright\textup{succ}^{-\textup{\textsf{b}}(\sigma)+1}(0)$, and in case $\textup{\textsf{b}}(\sigma)=\textup{\textsf{b}}(\sigma-1)-1$ by $2n+\sigma-1\curvearrowleft\textup{succ}(2n+\sigma-1)$. Note that in the second case, we have indeed $\textup{$\mathfrak{f}$}(\textup{succ}(2n+\sigma-1))=\textup{$\mathfrak{f}$}(\textup{succ}^{-\textup{\textsf{b}}(\sigma)}(0))$, i.e., $\textup{succ}(2n+\sigma-1)$ is the root vertex. \subsubsection{The infinite case} Let $\mathcal{Q}$ denote the completion of the space of all rooted finite quadrangulations with a boundary with respect to $d_{\textup{map}}$. We extend $\mathbb Phi_n$ to a mapping $$\mathbb Phi:\left({\cup}_{n,\sigma\in\textup{$\mathfrak{m}$}athbb N}\mathbb Fo_{\sigma}^n\times\mathcal Br_\sigma\right)\cup\left(\mathbb Fo_\infty\times\mathcal Br_\infty\right)\longrightarrow \mathcal{Q}$$ as follows. For elements $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})\in \mathbb Fo_{\sigma}^n\times\mathcal Br_\sigma$, we let $\mathbb Phi(((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}}))=\mathbb Phi_n((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})$, where we view the latter as an element in $\mathcal{Q}_n^{\sigma_n}$, by simply forgetting its distinguished vertex. Now let $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})\in\mathbb Fo_\infty\times\mathcal Br_\infty$. For $i\in\textup{$\mathfrak{m}$}athbb Z$, we define the {\it successor} $\textup{succ}_\infty(i)$ to be the smallest number $k$ greater than $i$ such that $\textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}}(k)=\textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}}(i)-1$. Note that since $\inf_{i\in\textup{$\mathfrak{m}$}athbb N}\textup{\textsf{b}}(i)=-\infty$, the definition make sense. We consider a proper embedding of $\textup{$\mathfrak{f}$}$ in the plane as described above and draw an arc between $\textup{$\mathfrak{f}$}(i)$ and $\textup{$\mathfrak{f}$}(\textup{succ}_\infty(i))$, for any $i\in\textup{$\mathfrak{m}$}athbb Z$, as indicated by Figure~\ref{fig:uihpq}. Again we can do this in a way such that arcs do not cross. The vertex set of $\mathbb Phi(((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}}))$ is given by $V(\textup{$\mathfrak{f}$})$, and the edges are the arcs we constructed. Finally, we follow a rooting convention which is analogous to the finite case (we adapt the notion $i\curvearrowright\textup{succ}_\infty(i)$ in the obvious way): The root vertex is given by $\textup{$\mathfrak{f}$}(\textup{succ}_\infty^{-\textup{\textsf{b}}(\partial)}(0))$, and the root edge is in case $\textup{\textsf{b}}(\partial)>\textup{\textsf{b}}(-1)-1$ given by $\textup{succ}_\infty^{-\textup{\textsf{b}}(\partial)}(0)\curvearrowright\textup{succ}^{-\textup{\textsf{b}}(\partial)+1}(0)$, and in case $\textup{\textsf{b}}(\partial)=\textup{\textsf{b}}(-1)-1$ by $-1\curvearrowleft\textup{succ}_\infty(-1)$. \begin{figure} \caption{The Bouttier-Di Francesco-Guitter mapping applied to an element $((\textup{$\mathfrak{f} \end{figure} \begin{remark} Notice that a triplet $((\textup{$\mathfrak{f}$},\textup{\textsf{b}}),\textup{$\mathfrak{l}$})$ in $\mathbb Fo_\sigma^n\times\mathcal Br_{\sigma}$ or in $\mathbb Fo_\infty\times\mathcal Br_{\infty}$ is uniquely determined by its associated contour and label functions $(C_\textup{$\mathfrak{f}$},\textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}})$. In particular, it makes sense to speak of the quadrangulation associated to $(C_\textup{$\mathfrak{f}$},\textup{$\mathfrak{L}$}_{\textup{$\mathfrak{f}$}})$. The distinguished vertex $v^{\bullet}$ in the finite case will play no particular role in our statements, since we view quadrangulations as metric spaces pointed at their root vertices. \end{remark} \subsection{Construction of the \normalfont{\textup{\textsf{UIHPQ}}}} \textup{$\mathfrak{l}$}bel{sec:constr-UIHPQ} We first introduce a $\mathbb Fo_\infty$-valued random element $(\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty)$ together with a $\mathcal Br_\infty$-valued random element $\textup{\textsf{b}}_\infty$, which will encode the $\textup{\textsf{UIHPQ}}$. \subsubsection{Uniformly labeled critical infinite forest} Let $\tau$ be a finite random plane tree. Conditionally on $\tau$, we assign a sequence of i.i.d random variables with the uniform distribution on $\{-1,0,1\}$ to the edges of $\tau$. The label $\ell(u)$ of a vertex $u$ of $\tau$ is defined to be the sum of the random variables along the edges of the (unique) path from the root to $u$. Such a random labeling $\ell:V(\tau)\rightarrow\textup{$\mathfrak{m}$}athbb{Z}$ is referred to as a {\it uniform labeling}. If the tree $\tau$ is a Galton-Watson tree with a geometric offspring distribution of parameter $1/2$, we say that $\tau$ is a {\it critical geometric Galton-Watson tree}. If $\ell$ is a uniform labeling of $\tau$, we refer to the pair $(\tau,(\ell(u))_{u\in\tau})$ as a {\it uniformly labeled critical geometric Galton-Watson tree}. A {\it uniformly labeled critical infinite forest} is a random element $(\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty)$ taking values in $\mathbb Fo_\infty$ such that the pairs $(\tau_i,\textup{$\mathfrak{l}$}_\infty\restriction V(\tau_i))$, $i\in\textup{$\mathfrak{m}$}athbb Z$, are independent uniformly labeled critical geometric Galton-Watson trees. \subsubsection{Uniform infinite bridge} \textup{$\mathfrak{l}$}bel{sec:infinitebridge} Let $\textup{\textsf{b}}_\infty=(\textup{\textsf{b}}_\infty(i),i\in\textup{$\mathfrak{m}$}athbb{Z})$ be a two-sided random walk starting from $0$ at time $0$, i.e., $\textup{\textsf{b}}_\infty(0)=0$, which has independent increments given by $$ \mathbb P(\textup{\textsf{b}}_\infty(i)-\textup{\textsf{b}}_\infty(i-1)=k)=2^{-k-2},\textup{$\mathfrak{q}$}uad k\in\textup{$\mathfrak{m}$}athbb{N}_0\cup\{-1\},\textup{$\mathfrak{q}$}uad\hbox{for }i\in\textup{$\mathfrak{m}$}athbb{Z}\setminus\{0\}, $$ and $$ \mathbb P(-\textup{\textsf{b}}_\infty(-1)=k)=(k+2)2^{-(k+3)},\textup{$\mathfrak{q}$}uad k\in\textup{$\mathfrak{m}$}athbb{N}_0\cup\{-1\}. $$ Note that $-\textup{\textsf{b}}_\infty(-1)$ has same law as $G+G'-1$ for $G$ and $G'$ two independent geometric random variables of parameter $1/2$. This follows from the well-known fact that $G+G'+1$ is distributed as a size-biased geometric random variable. We refer to Section~\ref{sec:bridges} for more explanations. Next, given $\textup{\textsf{b}}_\infty(-1)$, we let $\textup{\textsf{b}}_\infty(\partial)$ be a uniformly distributed random variable in $\{\textup{\textsf{b}}_\infty(-1)-1,\ldots,0\}$, independent of everything else. We call the random element $\textup{\textsf{b}}_\infty=(\textup{\textsf{b}}_\infty(i), i\in\textup{$\mathfrak{m}$}athbb Z\cup\{\partial\})$ with values in $\mathcal Br_\infty$ the {\it uniform infinite bridge}. We review now the construction of the $\textup{\textsf{UIHPQ}}$ given in~\cite{CuMi}. Note that there, the encoding is defined in a slightly different (but equivalent) manner, and the root edge is oriented in the opposite direction. The following definition is justified by Proposition~\ref{prop:Qn-UIHPQ}. \begin{defn} Let $(\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty)$ be a uniformly labeled critical infinite forest, and let $\textup{\textsf{b}}_\infty$ be a uniform infinite bridge independent of $(\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty)$. The uniform infinite half-planar quadrangulation $\textup{\textsf{UIHPQ}}$ is the (rooted) random infinite quadrangulation $Q_{\infty}^{\infty}=(V(Q_{\infty}^{\infty}),d_{\textup{gr}},\rho)$ with an infinite boundary obtained from applying the Bouttier-Di Francesco-Guitter mapping $\mathbb Phi$ to $((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_\infty)$. \end{defn} In~\cite{CuMi}, it was shown that in the sense of $d_{\textup{map}}$, there are the weak convergences $$ Q_n^{\sigma_n}\xrightarrow[]{n \to \infty}Q_{\infty}^{\sigma},\textup{$\mathfrak{q}$}uad Q_{\infty}^{\sigma}\xrightarrow[]{\sigma \to \infty}Q_{\infty}^{\infty},$$ where $Q_{\infty}^{\sigma}$ is the so-called (rooted) uniform infinite planar quadrangulation with a boundary of perimeter $2\sigma$. We also point at the recent work~\cite{CaCu}, where a construction of the $\textup{\textsf{UIHPQ}}$ with a positivity constraint on labels is given, similarly to the Chassaing-Durhuus construction~\cite{ChDu} of the $\textup{\textsf{UIPQ}}$. \begin{remark} We stress that while we use the notation $(\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$})$ for both a finite or infinite (deterministic) well-labeled forest, and similarly, $\textup{\textsf{b}}$ represents a finite or infinite bridge, $(\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty)\in\mathbb Fo_\infty$ and $\textup{\textsf{b}}_\infty\in\mathcal Br_\infty$ will always stand for {\it random} elements with the particular law just described. We will implicitly assume that $\textup{\textsf{b}}_\infty$ is independent of $(\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty)$. Similarly, for given $\sigma_n$, $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ will denote a random element with the uniform distribution on $\mathbb Fo_n^{\sigma_n}\times\mathcal Br_{\sigma_n}$, see Section~\ref{sec:usualsetting}. \end{remark} \subsection{Some ramifications} We gather here some consequences and remarks which we will tacitly use in the following. We begin with some observations concerning the Bouttier-Di Francesco-Guitter bijection. \subsubsection{Distances} \textup{$\mathfrak{l}$}bel{sec:distances} Let $(\textup{$\mathfrak{q}$},v^{\bullet})\in\textup{$\mathfrak{m}$}athcal{Q}_{n,\sigma}^\bullet$ be a (rooted) pointed quadrangulations of size $n$ with a boundary of size $2\sigma$. Then $(\textup{$\mathfrak{q}$},v^{\bullet})$ corresponds to a pair $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})\in\mathbb Fo_\sigma^n \times\mathcal Br_\sigma$ {\it via} the Bouttier-Di Francesco-Guitter bijection, and the sets $V(\textup{$\mathfrak{q}$})\setminus\{v^{\bullet}\}$ and $V(\textup{$\mathfrak{f}$})$ are identified through this bijection. Recall that the label function $\textup{$\mathfrak{L}$}=\textup{$\mathfrak{L}$}_f$ represents the labels in the forest shifted tree by tree according to the values of the bridge $\textup{\textsf{b}}$. By a slight abuse of notation, we will view $\textup{$\mathfrak{L}$}$ also as a function on $V(\textup{$\mathfrak{q}$})\setminus\{v^{\bullet}\}$ (or $V(\textup{$\mathfrak{f}$})$): If $v\in V(\textup{$\mathfrak{q}$})\setminus\{v^{\bullet}\}$, there is at least one $i\in\{0,\ldots,2n+\sigma-1\}$ such that $v$ is visited in the $i$th step of the contour exploration, and we let $\textup{$\mathfrak{L}$}(v)=\textup{$\mathfrak{L}$}(i)$. Note that this definition makes sense, since $\textup{$\mathfrak{L}$}(i) = \textup{$\mathfrak{L}$}(j)$ if $\textup{$\mathfrak{f}$}(i)=\textup{$\mathfrak{f}$}(j)$. Write $d_{{\bf q}}$ for the graph distance on ${\bf q}$. From the description of the bijection above, we deduce that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:distance-vdot} d_{{\bf q}}(u,v^{\bullet})=\textup{$\mathfrak{L}$}(u)-\textup{$\mathfrak{m}$}in\textup{$\mathfrak{L}$} +1. \end{equation} Moreover, if $v_0$ is the root vertex of $\textup{$\mathfrak{q}$}$, we know that its distance to vertex $\textup{$\mathfrak{f}$}(0)=(0)$ is \begin{equation} \textup{$\mathfrak{l}$}bel{eq:distance-root-0} d_{{\bf q}}(v_0,(0))= -\textup{\textsf{b}}(\sigma). \end{equation} In general, there is no simple formula for distances in $\textup{$\mathfrak{q}$}$. However, as we explain next, there exist lower and upper bounds in terms of $\textup{$\mathfrak{L}$}$. We first discuss a lower bound. If $u,v\in V(\textup{$\mathfrak{f}$})$ are vertices of the same tree $\tau$ of $\textup{$\mathfrak{f}$}$, i.e., $\textup{$\mathfrak{r}$}(u)=\textup{$\mathfrak{r}$}(v)$, we let $[[u,v]]$ be the vertex set of the unique injective path in $\tau$ connecting $u$ to $v$. If $(i)$, $(j)$ are two tree roots of $\textup{$\mathfrak{f}$}$ with $i<j$, we let $[[(i),(j)]]$ denote the sequence of root vertices $(i),(i+1),\ldots,(j)$. For the remaining cases, if $\textup{$\mathfrak{r}$}(u)< \textup{$\mathfrak{r}$}(v)$, we put $$[[u,v]] = [[u,\textup{$\mathfrak{r}$}(u)]]\cup [[ \textup{$\mathfrak{r}$}(u),\textup{$\mathfrak{r}$}(v)]] \cup[[v,\textup{$\mathfrak{r}$}(v)]],$$ whereas if $\textup{$\mathfrak{r}$}(v)<\textup{$\mathfrak{r}$}(u)$, we let $$[[u,v]] = [[u,\textup{$\mathfrak{r}$}(u)]]\cup [[ \textup{$\mathfrak{r}$}(u),(\sigma-1)]]\cup [[(0),\textup{$\mathfrak{r}$}(v)]] \cup [[v,\textup{$\mathfrak{r}$}(v)]].$$ Now let $u,v\in V(\textup{$\mathfrak{q}$})\setminus\{v^{\bullet}\}$. The so-called {\it cactus bound} states that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:cactus1} d_{{\bf q}}(u,v) \geq \textup{$\mathfrak{L}$}(u)+\textup{$\mathfrak{L}$}(v)-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[[ u,v]]}\textup{$\mathfrak{L}$}, \textup{$\mathfrak{m}$}in_{[[ v,u]]}\textup{$\mathfrak{L}$}\right\}. \end{equation} See~\cite[Proposition 2.3.8]{Mi3} for a proof in a slightly different context, which is readily adapted to our setting. Since vertex $(0)$ has label $\textup{$\mathfrak{L}$}(0)=0$ and $\textup{$\mathfrak{L}$}$ coincides with the values of the bridge along the floor of $\textup{$\mathfrak{f}$}$, the distance $d_{{\bf q}}((0),u)$ for $u\in V(\textup{$\mathfrak{q}$})\setminus\{v^{\bullet}\}$ is lower bounded by \begin{equation} \textup{$\mathfrak{l}$}bel{eq:cactus2} d_{{\bf q}}((0),u) \geq -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,\textup{$\mathfrak{r}$}(u)]}\textup{\textsf{b}}, \textup{$\mathfrak{m}$}in_{[\textup{$\mathfrak{r}$}(u),\sigma-1]}\textup{\textsf{b}}\right\}. \end{equation} For an upper bound of $d_{{\bf q}}(u,v)$ when $u,v\in V(\textup{$\mathfrak{q}$})\setminus\{v^{\bullet}\}$, choose $i,j\in\{0,\ldots,2n+\sigma-1\}$ such that $\textup{$\mathfrak{f}$}(i)= u$ and $\textup{$\mathfrak{f}$}(j)=v$. Define $$ \overrightarrow{[i,j]} = \left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }}l} \{i,\ldots, j\} & i\leq j\\ \{i,\ldots,2n+\sigma-1\} \cup \{ 0,\ldots,j\} & i>j \end{array}\right.. $$ Then there is the upper bound (see~\cite[Lemma 3]{Mi1} for a proof) \begin{equation} \textup{$\mathfrak{l}$}bel{eq:dist-upperbound} d_{{\bf q}}(u,v) \leq \textup{$\mathfrak{L}$}(u)+\textup{$\mathfrak{L}$}(v)-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{\overrightarrow{[ i,j]}}\textup{$\mathfrak{L}$}(\textup{$\mathfrak{f}$}),\,\textup{$\mathfrak{m}$}in_{\overrightarrow{[ j,i]}}\textup{$\mathfrak{L}$}(\textup{$\mathfrak{f}$})\right\}+2. \end{equation} Bounds similar to~\eqref{eq:cactus1},~\eqref{eq:cactus2} and~\eqref{eq:dist-upperbound} can be formulated for infinite quadrangulations $\textup{$\mathfrak{q}$}_\infty$ constructed from triplets $((\textup{$\mathfrak{f}$},\textup{\textsf{b}}),\textup{$\mathfrak{l}$})\in\mathbb Fo_\infty\times \mathcal Br_\infty$. For example, if $u,v\in V(\textup{$\mathfrak{f}$})$ with $\textup{$\mathfrak{r}$}(u)\leq \textup{$\mathfrak{r}$}(v)$, the cactus bound~\eqref{eq:cactus1} reads \begin{equation} \textup{$\mathfrak{l}$}bel{eq:cactus3} d_{{\bf q}_\infty}(u,v) \geq \textup{$\mathfrak{L}$}(u)+\textup{$\mathfrak{L}$}(v)-2\textup{$\mathfrak{m}$}in_{[[ u,v]]}\textup{$\mathfrak{L}$}. \end{equation} \subsubsection{Bridges} \textup{$\mathfrak{l}$}bel{sec:bridges} We will need some properties of elements in $\mathcal Br_\sigma$. Firstly, as it is shown in~\cite[Lemma 6]{Be3}, by identifying a bridge $(\textup{\textsf{b}}(i),0\leq i\leq \sigma)\in\mathcal Br_\sigma$ with the sequence \begin{equation} \textup{$\mathfrak{l}$}bel{eq:correspondence-bridge} \big(\underbrace{+1,+1,\mathrm{d}ots,+1}_{\textup{\textsf{b}}(0)-\textup{\textsf{b}}(\sigma) \textup{ times}},-1,\underbrace{+1,+1,\mathrm{d}ots,+1}_{\textup{\textsf{b}}(1)-\textup{\textsf{b}}(0)+1 \textup{ times}},-1,\underbrace{+1,+1,\mathrm{d}ots,+1}_{\textup{\textsf{b}}(2)-\textup{\textsf{b}}(1)+1 \textup{ times}},\mathrm{d}ots, -1, \hspace*{-2mm}\underbrace{+1,+1,\mathrm{d}ots,+1}_{\textup{\textsf{b}}(\sigma)-\textup{\textsf{b}}(\sigma-1)+1 \textup{ times}}\big), \end{equation} one obtains a one-to-one correspondence between $\mathcal Br_\sigma$ and the set of sequences in $\{-1,+1\}^{2\sigma}$ counting exactly~$\sigma$ times the number $-1$. As a consequence, $|\mathcal Br_\sigma|={2\sigma \choose \sigma}$. It is helpful to adopt the following point of view. Imagine that we mark $\sigma$ points on the discrete circle $\textup{$\mathfrak{m}$}athbb{Z}/\textup{mod}\, 2\sigma$ uniformly at random. Marked points obtain label $-1$, unmarked points label $+1$. Now choose uniformly at random one of the $2\sigma$ circle points as the origin. By walking around the circle in the clockwise order starting from the chosen origin, one observes a sequence of consecutive $+1$ and $-1$, which is distributed as~\eqref{eq:correspondence-bridge} when $\textup{\textsf{b}}$ is chosen uniformly at random in $\mathcal Br_\sigma$. In particular, $(\textup{\textsf{b}}(\sigma)-\textup{\textsf{b}}(\sigma-1)+1)+(\textup{\textsf{b}}(0)-\textup{\textsf{b}}(\sigma)+1)=- \textup{\textsf{b}}(\sigma-1)+2$ has the law of a size-biased pick among all $\sigma$ consecutive segments of the form $(+1,+1,\ldots,+1,-1)$. When $\sigma$ tends to infinity, it is readily seen that $-\textup{\textsf{b}}(\sigma-1)$ converges in distribution to $G+G'-1$, where $G$ and $G'$ are two independent geometric random variables of parameter $1/2$. This explains the particular law of the increment $-\textup{\textsf{b}}_\infty(-1)$ of a uniform infinite bridge $\textup{\textsf{b}}_\infty$ that forms part of the encoding of the $\textup{\textsf{UIHPQ}}$. Next, let $(X_i,i\in\textup{$\mathfrak{m}$}athbb N)$ be a sequence of i.i.d. random variables with distribution $$ \mathbb P(X_1=k)=2^{-k-2},\textup{$\mathfrak{q}$}uad k\geq -1. $$ Put $\Sigma_j = \sum_{i=1}^jX_i$, with $\Sigma_0=0$. Fix $0\leq k\leq \sigma$, and denote by $S^{(k)}=(S^{(k)}(j),j=0,\ldots,\sigma)$ the discrete bridge distributed as $(\Sigma_j,j=0,\ldots,\sigma)$ conditioned on $\{\Sigma_\sigma =-k\}$. Then the above considerations imply that $S^{(k)}$ is uniformly distributed over the set $\{\textup{\textsf{b}}\in\mathcal Br_\sigma: \textup{\textsf{b}}(\sigma)=-k\}$. Secondly, we can compute \begin{equation} \textup{$\mathfrak{l}$}bel{eq:law-bsigma}\mathbb P\left(\textup{\textsf{b}}(\sigma)=-k\right)=\textup{$\mathfrak{f}$}rac{1}{2}\textup{$\mathfrak{f}$}rac{(2\sigma-k-1)!}{(2\sigma-1)!}\textup{$\mathfrak{f}$}rac{\sigma!}{(\sigma-k)!}\leq 2^{-k}, \end{equation} and $\mathbb P\left(\textup{\textsf{b}}(\sigma)=-k\right)\rightarrow 2^{-k-1}$ as $\sigma\rightarrow\infty$. See~\cite[Proof of Proposition 7]{Be3} for a complete argument. \subsubsection{Forests} \textup{$\mathfrak{l}$}bel{sec:forests} In the rest of this paper, we will often use the following well-known fact (see, e.g.,~\cite[Section 2]{LGMi}): If $\textup{$\mathfrak{f}$}=(\tau_0,\ldots,\tau_{\sigma-1})$ is chosen uniformly at random among all forests with $\sigma$ trees and $n$ edges, then the corresponding discrete contour path $(C_{\textup{$\mathfrak{f}$}}(j),j=0,\ldots,\sigma)$, is distributed as a simple random walk path starting at $0$ and conditioned to end at $-\sigma$ at time $2n+\sigma$. As a consequence, we have for $j\in\{0,\ldots,\sigma\}$ and positive integers $k_i$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:forest-hittingtime} \mathbb P\left(|\tau_0|=k_0,\ldots, |\tau_{j-1}|=k_{j-1}\right)=\mathbb P\left(T_{-j}=2(k_0+\ldots+k_{j-1})+j\,|\,T_{-\sigma}=2n+\sigma\right), \end{equation} where $T_{-i}$ denotes the first hitting time of $-i$ of a simple random walk started at $0$. Also note that the joint law of the trees $(\tau_0,\ldots,\tau_{\sigma-1})$ is invariant under permutation of its components. Moreover, the sequence of trees $(\tau_0,\ldots,\tau_{\sigma-1})$ has the law of $\sigma$ independent critical geometric Galton-Watson trees conditioned to have total size $n$. In this context, we recall (see, e.g.,~\cite[Section 2.2]{LGMi}) that if $\mathbb P_{\textup{GW}}$ is the law of critical geometric Galton-Watson tree and $\tau$ a given finite tree, then \begin{equation} \textup{$\mathfrak{l}$}bel{eq:criticalGW} \mathbb P_{\textup{GW}}\left(\tau\right)=(1/2)\,4^{-|\tau|}. \end{equation} Probabilities as in~\eqref{eq:forest-hittingtime} can be computed using Kemperman's formula (see, e.g.,~\cite[Chapter 6]{Pi}). It tells us that if $(S_i,i\in \textup{$\mathfrak{m}$}athbb N_0)$ is a simple random walk started at $0$, then \begin{equation} \textup{$\mathfrak{l}$}bel{eq:Kemperman} \mathbb P\left(T_{j}=k\right)=\textup{$\mathfrak{f}$}rac{|j|}{k}\mathbb P(S_k=j),\textup{$\mathfrak{q}$}uad j\in\textup{$\mathfrak{m}$}athbb{Z},\,k\in\textup{$\mathfrak{m}$}athbb N. \end{equation} By applying Kemperman's formula to $\mathbb P(T_{-\sigma}=2n+\sigma)$ and counting paths, we obtain $$ |\mathbb Fo_\sigma^n| = 3^n\textup{$\mathfrak{f}$}rac{\sigma}{2n+\sigma}{2n+\sigma\choose n}. $$ Note that the factor $3^n$ accounts for the $3^n$ possible labelings of a forest with $n$ tree edges. For estimating $\mathbb P(S_k=j)$ when $k$ and $j$ are large, one typically applies a local central limit theorem. Setting $$ \overline{p}(k,j)=\textup{$\mathfrak{f}$}rac{2}{\sqrt{2\pi k}}\exp\left(-\textup{$\mathfrak{f}$}rac{j^2}{2k}\right),\textup{$\mathfrak{q}$}uad j\in\textup{$\mathfrak{m}$}athbb{Z},\,k\in\textup{$\mathfrak{m}$}athbb N, $$ and $\overline{p}(0,j)=\mathrm{d}elta_0(j)$, one has (see, e.g.,~\cite[Theorem 1.2.1]{La}) \begin{equation} \textup{$\mathfrak{l}$}bel{eq:localCLT} \mathbb P(S_k=j)=\overline{p}(k,j) + O\left(1/k^{3/2}\right) \end{equation} if $k+j$ is even, and $\mathbb P(S_k=j)=0$ otherwise. For us, it will mostly be sufficient to record that $\mathbb P(S_k=j)\leq Ck^{-1/2}$ for some $C>0$ uniformly in $j$ and $k$. However, in the boundary regime $\sigma_n\gg \sqrt{n}$, we will sometimes find ourselves in an atypical regime for simple random walk, where the control provided by~\eqref{eq:localCLT} is not good enough. In this case, we use the following asymptotic expression due to Bene\v{s}~\cite[Theorem 1.3, first case]{Be}. For $x\ll m$ such that $x+m$ is even, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:localCLT2} \mathbb P(S_m = x) = \sqrt{\textup{$\mathfrak{f}$}rac{2}{\pi m}}\exp\left(-\sum_{\ell = 1}^\infty \textup{$\mathfrak{f}$}rac{1}{2\ell (2 \ell -1)} \textup{$\mathfrak{f}$}rac{x^{2 \ell}}{ m^{2 \ell -1}}\right) \left(1 + O\left(\textup{$\mathfrak{f}$}rac{x^2}{m^2} + \textup{$\mathfrak{f}$}rac{1}{m}\right) \right). \end{equation} Note that as it is remarked in~\cite{Be}, this expression can also be obtained from~\cite[Theorem 6.1.6]{BoBo} by an explicit calculation of the rate function. \subsubsection{Remarks on notation} \textup{$\mathfrak{l}$}bel{sec:usualsetting} We always let $\textup{$\mathfrak{m}$}athbb N=\{1,2,\ldots\}$, $\textup{$\mathfrak{m}$}athbb N_0=\textup{$\mathfrak{m}$}athbb N\cup\{0\}$. Recall that for real sequences $(a_n,n\in\textup{$\mathfrak{m}$}athbb N),(b_n,n\in\textup{$\mathfrak{m}$}athbb N)$, $a_n \ll b_n$ or $b_n \gg a_n$ means that $a_n/b_n \to 0$ as $n\to\infty$, and $a_n\sim b_n$ means $a_n/b_n\to 1$. Moreover, we write $a_n\lesssim b_n$ if $a_n\leq C b_n$ for some constant $C>0$ independent of $n$. Sometimes, we also use the Landau Big-O and Little-o notation, in a way that will be clear from the context. Given a random variable (or sequence) $U$ and an event $\textup{$\mathfrak{m}$}athcal{E}$, we write $\textup{$\mathfrak{m}$}athcal{L}(U)$ and $\textup{$\mathfrak{m}$}athcal{L}(U|\textup{$\mathfrak{m}$}athcal{E})$ for the law of $U$ and the conditional law of $U$ given $\textup{$\mathfrak{m}$}athcal{E}$, respectively. The total variation norm of a probability measure is denoted by $\|\cdot\|_{\textup{TV}}$. We now specify a (notational) framework in which we will often work. \begin{mdframed} {\bf The usual setting.} For each $n\in\textup{$\mathfrak{m}$}athbb N$, we let $Q_n^{\sigma_n}=(V(Q_n^{\sigma_n}),\rho_n,d_{\textup{gr}})$ be uniformly distributed over the set $\textup{$\mathfrak{m}$}athcal Q_n^{\sigma_n}$ of rooted quadrangulations with $n$ internal faces and $2\sigma_n$ boundary edges. Given $Q_n^{\sigma_n}$, we choose $v^{\bullet}_n$ uniformly at random among the elements of $V(Q_n^{\sigma_n})$, and then $(Q_n^{\sigma_n},v^{\bullet}_n)$ is uniformly distributed over $Q_{n,\sigma_n}^\bullet$ and corresponds through the Bouttier-Di Francesco-Guitter bijection to a triplet $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ uniformly distributed over the set $\mathbb Fo_{\sigma_n}^n \times \mathcal Br_{\sigma_n}$. We let $(C_n,L_n)$ be the contour pair corresponding to $(\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n)$ and write $$ \textup{$\mathfrak{L}$}_n = \left(L_n(t) + \textup{\textsf{b}}_n(-\underline{C}_n(t)), 0\leq t\leq 2n+\sigma_n\right) $$ for the label function associated to $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$. The random triplet $((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_\infty)$ represents a uniformly labeled critical infinite forest and an independent uniform infinite bridge and encodes the $\textup{\textsf{UIHPQ}}$ $Q_\infty^\infty=(V(Q_\infty^{\infty}),\rho,d_{\textup{gr}})$. We write $(C_\infty,L_\infty)$ for the corresponding contour pair and $\textup{$\mathfrak{L}$}_\infty$ for the label function. While $B_r(Q_n^{\sigma_n})$ denotes the closed ball of radius $r$ around the root $\rho_n$ in $Q_n^{\sigma_n}$, we will also consider the ball $B_r^{(0)}(Q_n^{\sigma_n})$ around the vertex $\textup{$\mathfrak{f}$}_n(0)=(0)$, and similarly for the $\textup{\textsf{UIHPQ}}$. \end{mdframed} \section{Auxiliary results} \textup{$\mathfrak{l}$}bel{sec:auxiliaryresults} In this part we collect general results and observations which will be useful later on. Our statements on Galton-Watson trees might be of some interest on its own. \subsection{Convergence of forests} The first two lemmas in this section provide the necessary control over the trees of a forest $\textup{$\mathfrak{f}$}_n$ chosen uniformly at random in $\mathbb Fo^{\sigma_n}_n$ in the regime $\sigma_n\ll\sqrt{n}$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:GW1} Assume $\sigma_n\ll\sqrt{n}$. Denote by $(\tau_i)_{1\leq i\leq \sigma_n}$ a family of $\sigma_n$ independent critical geometric Galton-Watson trees. Then $$ \liminf_{\mathrm{d}elta\mathrm{d}ownarrow 0}\liminf_{n\rightarrow\infty}\mathbb P\left(\exists !\, i\in\{1,\ldots,\sigma_n\}\hbox{ with }|\tau_i|\geq (1-\mathrm{d}elta) n \mathcal Big| \sum_{i=1}^{\sigma_n}|\tau_i|=n\right) = 1. $$ \end{lemma} \begin{proof} We use the contour function representation of the forest $\textup{$\mathfrak{f}$}_n$ as a simple random walk, conditioned on first hitting $-\sigma_n$ at time $2n+\sigma_n$ (and interpolated linearly between integer times). We let $(S_i,0\leq i\leq 2n+\sigma_n)$ denote such a conditioned random walk. Under our assumptions, it holds that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:1} \left(\textup{$\mathfrak{f}$}rac{S_{(2n+\sigma_n) t}}{\sqrt{2n}}\right)_{0\leq t\leq 1}\xrightarrow[n\to\infty]{(d)}\textup{$\mathfrak{m}$}athbbm{e}\, , \end{equation} in distribution in $\textup{$\mathfrak{m}$}athcal{C}([0,1],\textup{$\mathfrak{m}$}athbb R)$, where $\textup{$\mathfrak{m}$}athbbm{e}$ is the normalized Brownian excursion. This ``folklore'' result is implicit in \cite{Be2}, so we recall quickly how to obtain it, omitting some details. First, by~\cite{BeChPi}, one can represent the conditioned random walk as a cyclic shift of a simple random walk that is conditioned to hit $-\sigma_n$ at time $2n+\sigma_n$, but not necessarily for the first time. More precisely, calling $S'$ this new random walk, we let $\nu_n$ be a uniform random variable in $\{0,1,\ldots,\sigma_n-1\}$, and we let $$A_n=\inf\left\{i\geq 0: S'_i=\textup{$\mathfrak{m}$}in\{ S'_j: 0\leq j \leq 2n+\sigma_n\}+\nu_n\right\}\, .$$ Then, the sequence $(S''_i,0\leq i\leq 2n+\sigma_n)$ defined by $$S''_i=\left\{\begin{array}{lll}S'_{A_n+i}-S'_{A_n} & \textup{$\mathfrak{m}$}box{ if }& 0\leq i\leq 2n+\sigma_n-A_n\\ -\sigma_n+S'_{i-2n-\sigma_n+A_n} -S'_{A_n}& \textup{$\mathfrak{m}$}box{ if }&2n+\sigma_n-A_n<i\leq 2n+\sigma_n \end{array}\right. $$ has same distribution as $(S_i,0\leq i\leq 2n+\sigma_n)$. Now it is classical that under the assumption that $\sigma_n=o(\sqrt{n})$, $$\left(\textup{$\mathfrak{f}$}rac{S'_{(2n+\sigma_n)t}}{\sqrt{2n}}\right)_{0\leq t\leq 1}\xrightarrow[n\to\infty]{(d)}\textup{$\mathfrak{m}$}athbbm{b}\, ,$$ where $\textup{$\mathfrak{m}$}athbbm{b}$ is a standard Brownian bridge. From this, one deduces that $$\left(\textup{$\mathfrak{f}$}rac{S_{(2n+\sigma_n)t}}{\sqrt{2n}}\right)_{0\leq t\leq 1}=_d\left(\textup{$\mathfrak{f}$}rac{S''_{(2n+\sigma_n)t}}{\sqrt{2n}}\right)_{0\leq t\leq 1}\xrightarrow[n\to\infty]{(d)}V\textup{$\mathfrak{m}$}athbbm{b}\, ,$$ where $V\textup{$\mathfrak{m}$}athbbm{b}=(\textup{$\mathfrak{m}$}athbbm{b}_{s+s_*}-\textup{$\mathfrak{m}$}athbbm{b}_{s_*},0\leq s\leq 1)$ is the Vervaat transform of $\textup{$\mathfrak{m}$}athbbm{b}$, that is the cyclic shift of $\textup{$\mathfrak{m}$}athbbm{b}$ at the a.s.\ unique time $s_*$ where it attains its overall minimum. Here, the bridge $\textup{$\mathfrak{m}$}athbbm{b}$ is extended periodically by $\textup{$\mathfrak{m}$}athbbm{b}_{s+1}=\textup{$\mathfrak{m}$}athbbm{b}_{s}$ for $s\in [0,1]$. Finally, we use the well-known fact that $\textup{$\mathfrak{m}$}athbbm{e}$ and $V\textup{$\mathfrak{m}$}athbbm{b}$ have the same distribution. Now notice that the quantities $\tau_1,\ldots,\tau_{\sigma_n}$ are equal to (half) the lengths of the excursions of $S$ above its infimum process, in the order in which they appear. Hence, the convergence \eqref{eq:1} clearly implies that the largest of these quantities satisfies $$\textup{$\mathfrak{f}$}rac{\textup{$\mathfrak{m}$}ax\{\tau_i: 1\leq i\leq\sigma_n\}}{n}\longrightarrow 1$$ in probability as $n\to\infty$, while all the other quantities are negligible compared to $n$ in probability. \end{proof} \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:GW2} Assume $\sigma_n\ll\sqrt{n}$. Denote by $(\tau_i)_{1\leq i\leq \sigma_n}$ a family of $\sigma_n$ independent critical geometric Galton-Watson trees. Write $i_\ast$ for the smallest index such that $|\tau_{i_\ast}|\geq \textup{$\mathfrak{m}$}ax_{1\leq i\leq \sigma_n, i\neq i_\ast}|\tau_j|$. Then $$ \lim_{n\rightarrow\infty}\left\|\textup{$\mathfrak{m}$}athcal{L}\mathcal Big((\tau_i)_{1\leq i\leq \sigma_n, i\neq i_\ast}\,\mathcal Big|\,\sum_{i=1}^{\sigma_n}|\tau_i|=n\mathcal Big) -\textup{$\mathfrak{m}$}athcal{L}\left((\tau_i)_{1\leq i\leq \sigma_n-1}\right)\right\|_{\textup{TV}}=0. $$ \end{lemma} \begin{proof} For $\mathrm{d}elta>0$, and $F$ a bounded and measurable function, \begin{align*} \lefteqn{\textup{$\mathfrak{m}$}athbb E\left[F((\tau_i)_{i\neq i_\ast})\,\mathcal Big|\,\sum_{i=1}^{\sigma_n}|\tau_i|=n\right]}\\ &=\textup{$\mathfrak{m}$}athbb E\left[F((\tau_i)_{i\neq i_\ast})1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{|\tau_{i_\ast}|\geq \mathrm{d}elta n\}\cap\{|\tau_{i_\ast}|>|\tau_i|\textup{$\mathfrak{m}$}box{\small{ for all }}i\neq i_\ast\}}\,\mathcal Big|\,\sum_{i=1}^{\sigma_n}|\tau_i|=n\right] + \|F\|_\infty R_n^{(\mathrm{d}elta)}, \end{align*} where by Lemma~\ref{lem:GW1}, the error term $R_n^{(\mathrm{d}elta)}$ satisfies $\limsup_{\mathrm{d}elta\mathrm{d}ownarrow 0}\limsup_{n\rightarrow\infty}R_n^{(\mathrm{d}elta)}=0$. Therefore it remains to consider the expectation in the last display for small but fixed $\mathrm{d}elta$. Put $p(k,m)= \mathbb P(\sum_{i=1}^{k}|\tau_i|=m)$, and write $\tau$ instead of $\tau_\sigma$. Using exchangeability of the trees, the expectation becomes \begin{align*} \lefteqn{\textup{$\mathfrak{f}$}rac{\sigma_n}{p(\sigma_n,n)}\textup{$\mathfrak{m}$}athbb E\left[F((\tau_i)_{1\leq i\leq \sigma_n-1})1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{|\tau_{\sigma_n}|>\textup{$\mathfrak{m}$}ax_{1\leq i\leq \sigma_n-1}|\tau_i|\}\cap\{|\tau_{\sigma_n}|\geq \mathrm{d}elta n\}\cap\{\sum_{i=1}^{\sigma_n}|\tau_i|=n\}}\right] = } \\ &\textup{$\mathfrak{m}$}athbb E\mathcal Bigg[F((\tau_i)_{1\leq i\leq \sigma_n-1})\underbrace{\textup{$\mathfrak{f}$}rac{\sigma_n}{p(\sigma_n,n)}\mathbb P\mathcal Big(|\tau_{\sigma_n}|>\textup{$\mathfrak{m}$}ax_{1\leq i\leq \sigma_n-1}|\tau_i|; |\tau_{\sigma_n}|\geq \mathrm{d}elta n; \sum_{i=1}^{\sigma_n}|\tau_i|=n\,\mathcal Big|\,(\tau_i)_{1\leq i\leq \sigma_n-1}\mathcal Big)}_{=Z_n}\mathcal Bigg]. \end{align*} In order to conclude, it suffices to show that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:GW2-1} \limsup_{n\rightarrow\infty}\textup{$\mathfrak{m}$}athbb E\left[\left|Z_n-1\right|\right]=0. \end{equation} Let $K\in\textup{$\mathfrak{m}$}athbb N$. We split into \begin{equation} \textup{$\mathfrak{l}$}bel{eq:GW2-2} \textup{$\mathfrak{m}$}athbb E[|Z_n-1|]=\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|\leq K\sigma_n^2\}}\right] +\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2\}}\right]. \end{equation} We first show that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:GW2-3} \limsup_{K\rightarrow\infty}\limsup_{n\rightarrow\infty}\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2\}}\right]=0. \end{equation} We estimate $$ \textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2\}}\right]\leq \textup{$\mathfrak{m}$}athbb E\left[Z_n1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2\}}\right]+\mathbb P\left(\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2\right). $$ Recall that the last term is equal to $\mathbb P(T_{-\sigma_n+1}>2K\sigma_n^2+\sigma_n-1)$, where $T_{k}$ is as above the first hitting time of $k$ of a simple random walk $(S_i,i\in\textup{$\mathfrak{m}$}athbb N_0)$ started at zero. Standard random walk estimates (e.g., Kemperman's formula~\eqref{eq:Kemperman} together with~\eqref{eq:localCLT}) entail that $$ \limsup_{K\rightarrow\infty}\limsup_{n\rightarrow\infty}\mathbb P\left(T_{-\sigma_n+1}> 2K\sigma_n^2+\sigma_n-1\right)=0. $$ The first term on the right hand side in the next to last display is estimated by \begin{align*} \textup{$\mathfrak{m}$}athbb E\left[Z_n1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2\}}\right]&\leq\textup{$\mathfrak{f}$}rac{\sigma_n}{p(\sigma_n,n)} \mathbb P\left(\sum_{i=1}^{\sigma_n-1}|\tau_i|> K\sigma_n^2;\,\sum_{i=1}^{\sigma_n}|\tau_i|=n;\,|\tau_{\sigma_n}|\geq \mathrm{d}elta n\right)\\ &\leq \textup{$\mathfrak{f}$}rac{\sigma_n}{p(\sigma_n,n)}\sum_{m= K\sigma_n^2}^{n-\lceil\mathrm{d}elta n\rceil}p(\sigma_n-1,m)p(1,n-m)\\ &\lesssim n^{3/2}\sum_{m= K\sigma_n^2}^{n-\lceil\mathrm{d}elta n\rceil}\textup{$\mathfrak{f}$}rac{\sigma_n}{(2m+\sigma_n-1)^{3/2}}\textup{$\mathfrak{f}$}rac{1}{(2(n-m)+1)^{3/2}}\\ &\lesssim \sigma_n\sum_{m= K\sigma_n^2}^{\lceil n/2\rceil}\textup{$\mathfrak{f}$}rac{1}{m^{3/2}} + \sigma_n \sum_{m=\lceil n/2\rceil+1}^{n-\lceil\mathrm{d}elta n\rceil}\textup{$\mathfrak{f}$}rac{1}{(n-m)^{3/2}}\,\lesssim\, \textup{$\mathfrak{f}$}rac{1}{K^{1/2}}+ \textup{$\mathfrak{f}$}rac{\sigma_n}{\sqrt{\mathrm{d}elta n}}. \end{align*} Recalling that $\sigma_n\ll \sqrt{n}$, this finishes the proof of~\eqref{eq:GW2-3}. We turn to the first term on the right hand side of~\eqref{eq:GW2-2}. First note that on the event $$ \left\{\sum_{i=1}^{\sigma_n-1}|\tau_i|\leq K\sigma_n^2;\,\sum_{i=1}^{\sigma_n}|\tau_i|=n\right\}, $$ we have $|\tau_{\sigma_n}|\geq \mathrm{d}elta n$ and $|\tau_{\sigma_n}|>\textup{$\mathfrak{m}$}ax_{1\leq i\leq \sigma_n-1}|\tau_i|$ almost surely, provided $n$ is large enough. Therefore, \begin{align*} \lefteqn{\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|\leq K\sigma_n^2\}}\right]}\\ &= \textup{$\mathfrak{m}$}athbb E\left[\left|\textup{$\mathfrak{f}$}rac{\sigma_n}{p(\sigma_n,n)}\mathbb P\mathcal Big(|\tau_{\sigma_n}|=n-\sum_{i=1}^{\sigma_n-1}|\tau_i|\,\mathcal Big|\,(\tau_i)_{1\leq i\leq \sigma_n-1}\mathcal Big)-1\right|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|\leq K\sigma_n^2\}}\right]\\ &=\sum_{m=0}^{K\sigma_n^2}\left|\textup{$\mathfrak{f}$}rac{\sigma_np(1,n-m)}{p(\sigma_n,n)}-1\right|p(\sigma_n-1,m). \end{align*} We now show that the terms inside the absolute value in the last display are of order $o(1)$ as $n$ tends to infinity, uniformly in $m$ with $m\leq K\sigma_n^2$. First, by Kemperman's formula~\eqref{eq:Kemperman}, $$ \textup{$\mathfrak{f}$}rac{\sigma_np(1,n-m)}{p(\sigma_n,n)}= \textup{$\mathfrak{f}$}rac{2n+\sigma_n}{(2(n-m)+1)}\textup{$\mathfrak{f}$}rac{\mathbb P\left(S_{2(n-m)+1}=1\right)}{\mathbb P\left(S_{2n+\sigma_n}=\sigma_n\right)}. $$ Since $\sigma_n^2\ll n$, we have $2n+\sigma_n/(2(n-m)+1)\sim 1$. For the fraction of the two probabilities involving simple random walk, we apply the local central limit theorem~\eqref{eq:localCLT} and obtain $$ \limsup_{n\rightarrow\infty}\sup_{m\leq K\sigma_n^2}\left|\textup{$\mathfrak{f}$}rac{\mathbb P\left(S_{2(n-m)+1}=1\right)}{\mathbb P\left(S_{2n+\sigma_n}=\sigma_n\right)}-1\right|=0. $$ This shows $$ \limsup_{n\rightarrow\infty}\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{\sigma_n-1}|\tau_i|\leq K\sigma_n^2\}}\right]=0. $$ With~\eqref{eq:GW2-3}, we have proved that~\eqref{eq:GW2-1} holds, completing thereby the proof of the lemma. \end{proof} The next statement will prove useful for the regimes $1\ll\sigma_n\ll\sqrt{n}$ and $\sigma_n\sim \sigma\sqrt{2n}$, $\sigma\in(0,\infty)$, as well as for the local convergence of $Q_n^{\sigma_n}$ towards the $\textup{\textsf{UIHPQ}}$ when $1\ll\sigma_n\ll n$. We stress that if $\sigma_n\ll \sqrt{n}$, the following lemma is already a corollary of Lemmas~\ref{lem:GW1} and~\ref{lem:GW2}. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:GW3} Assume $1\ll\sigma_n\ll n$. Denote by $(\tau_i)_{1\leq i\leq \sigma_n}$ a family of $\sigma_n$ independent critical geometric Galton-Watson trees. If $k_n$ is a sequence of positive integers with $k_n\leq \sigma_n$ and $k_n=o(\sigma_n\wedge (n/\sigma_n))$ as $n\rightarrow\infty$, then $$ \limsup_{n\rightarrow\infty}\left\|\textup{$\mathfrak{m}$}athcal{L}\mathcal Big((\tau_i)_{1\leq i\leq k_n}\,\mathcal Big|\,\sum_{i=1}^{\sigma_n}|\tau_i|=n\mathcal Big) -\textup{$\mathfrak{m}$}athcal{L}\left((\tau_i)_{1\leq i\leq k_n}\right)\right\|_{\textup{TV}}=0. $$ \end{lemma} \begin{proof} The arguments are similar to those in the proof of Lemma~\ref{lem:GW2}. We set again $p(k,m)= \mathbb P(\sum_{i=1}^{k}|\tau_i|=m)$. We have for bounded and measurable $F$ \begin{align*} &\textup{$\mathfrak{m}$}athbb E\left[F((\tau_i)_{1\leq i\leq k_n})\,\mathcal Big|\,\sum_{i=1}^{\sigma_n}|\tau_i|=n\right]\\ &= \textup{$\mathfrak{m}$}athbb E\mathcal Bigg[F((\tau_i)_{1\leq i\leq k_n})\underbrace{\textup{$\mathfrak{f}$}rac{1}{p(\sigma_n,n)}\mathbb P\left(\sum_{i=1}^{\sigma_n}|\tau_i|=n\, \,\mathcal Big|\,(\tau_i)_{1\leq i\leq k_n}\right)}_{=Z_n}\mathcal Bigg], \end{align*} and the claim follows if we show that $$\limsup_{n\rightarrow\infty}\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|\right]=0.$$ We argue now similarly to Lemma~\ref{lem:GW2}. With $K\in\textup{$\mathfrak{m}$}athbb N$, we split into $$ \textup{$\mathfrak{m}$}athbb E[|Z_n-1|]=\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|\leq Kk_n^2\}}\right] +\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|> Kk_n^2\}}\right] $$ and bound the second term by $$ \textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|> Kk_n^2\}}\right]\leq \textup{$\mathfrak{m}$}athbb E\left[Z_n1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|> Kk_n^2\}}\right]+\mathbb P\left(\sum_{i=1}^{k_n}|\tau_i|> Kk_n^2\right). $$ The last term in the above display is estimated in the same way as the analogous term in Lemma~\ref{lem:GW2}. For the first term, we have \begin{align*} \lefteqn{\textup{$\mathfrak{m}$}athbb E\left[Z_n1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|> Kk_n^2\}}\right] \leq \textup{$\mathfrak{f}$}rac{1}{p(\sigma_n,n)}\sum_{m= Kk_n^2}^{n}p(k_n,m)p(\sigma_n-k_n,n-m)}\\ &\lesssim \textup{$\mathfrak{f}$}rac{n^{3/2}}{\sigma_n}\sum_{m= Kk_n^2}^{\lceil n/2\rceil}\textup{$\mathfrak{f}$}rac{k_n}{m^{3/2}}\textup{$\mathfrak{f}$}rac{\sigma_n-k_n}{(n-m)^{3/2}} + \textup{$\mathfrak{f}$}rac{k_n}{\sigma_n}\sum_{m=\lceil n/2\rceil+1}^{n}p(\sigma_n-k_n,n-m)\lesssim \textup{$\mathfrak{f}$}rac{1}{\sqrt{K}}+ \textup{$\mathfrak{f}$}rac{k_n}{\sigma_n}. \end{align*} Recalling that $k_n\ll\sigma_n$, we obtain $$ \limsup_{K\rightarrow\infty}\limsup_{n\rightarrow\infty}\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|> Kk_n^2\}}\right]=0. $$ It remains to show that for fixed $K$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:GW3-1} \limsup_{n\rightarrow\infty}\textup{$\mathfrak{m}$}athbb E\left[|Z_n-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|\leq K k_n^2\}}\right]=0. \end{equation} We write \begin{align*} \lefteqn{\textup{$\mathfrak{m}$}athbb E\left[|Z_{n}-1|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|\leq Kk_n^2\}}\right]}\\ &= \textup{$\mathfrak{m}$}athbb E\left[\left|\textup{$\mathfrak{f}$}rac{1}{p(\sigma_n,n)}\mathbb P\mathcal Big(\sum_{i= k_n+1}^{\sigma_n}|\tau_i|=n-\sum_{i=1}^{k_n}|\tau_i|\,\mathcal Big|\,(\tau_i)_{1\leq i\leq k_n}\mathcal Big)-1\right|1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{\sum_{i=1}^{k_n}|\tau_i|\leq K k_n^2\}}\right]\\ &=\sum_{m=0}^{K k_n^2}\left|\textup{$\mathfrak{f}$}rac{p(\sigma_n-k_n,n-m)}{p(\sigma_n,n)}-1\right|p(k_n,m). \end{align*} Again, our proof will be complete if we show that the terms inside the absolute value are of order $o(1)$, uniformly in $m$ with $m\leq K k_n^2$. For such $m$, let $$x_n=\sigma_n - k_n, \textup{$\mathfrak{q}$}uad y_n = 2(n-m)+ \sigma_n -k_n.$$ Since the case where $\sigma_n$ is much larger than $\sqrt{n}$ is also included in our statement, we apply the refined version~\eqref{eq:localCLT2} (and first Kemperman's formula), which gives \begin{align} \textup{$\mathfrak{f}$}rac{p(\sigma_n-k_n,n-m)}{p(\sigma_n,n)}&\sim\textup{$\mathfrak{f}$}rac{\mathbb P\left(S_{y_n}=x_n\right)}{\mathbb P\left(S_{2n+\sigma_n}=\sigma_n\right)}\nonumber\\ &\sim \exp\left(-\sum_{\ell = 1}^\infty \textup{$\mathfrak{f}$}rac{1}{2\ell (2 \ell -1)} \left(\textup{$\mathfrak{f}$}rac{x_n^{2 \ell}}{ {y_n}^{2 \ell -1}} -\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{ (2n+\sigma_n)^{2\ell-1}}\right)\right), \textup{$\mathfrak{l}$}bel{eq:GW3-2} \end{align} everything uniformly in $m$ with $m\leq K k_n^2$. By Taylor's expansion, we obtain \begin{multline*} \textup{$\mathfrak{f}$}rac{x_n^{2 \ell}}{ {y_n}^{2 \ell -1}} -\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{ (2n+\sigma_n)^{2\ell-1}} = \\ \textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}}\left[ -2 \ell \textup{$\mathfrak{f}$}rac{k_n}{\sigma_n} + (2 \ell -1) \textup{$\mathfrak{f}$}rac{2(m+k_n)}{2n+\sigma_n} + O\left(\left(\textup{$\mathfrak{f}$}rac{k_n}{\sigma_n}\right)^2 \right ) + O\left(\left(\textup{$\mathfrak{f}$}rac{m+k_n}{2n+\sigma_n}\right)^2\right) \right]. \end{multline*} Since $k_n=o(\sigma_n\wedge (n/\sigma_n))$, we have for $m\leq K k_n^2$ \begin{align*} -2\ell\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}} \textup{$\mathfrak{f}$}rac{ k_n}{\sigma_n} & = \textup{$\mathfrak{f}$}rac{\sigma_n^{2(\ell-1)}}{(2n+\sigma_n)^{2 (\ell -1)}}\,o(1) ,\textup{$\mathfrak{q}$}uad\hbox{ and}\\ (2\ell-1) \textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}} \textup{$\mathfrak{f}$}rac{2(m+k_n)}{2n+\sigma_n} & = \textup{$\mathfrak{f}$}rac{\sigma_n^{2(\ell-1)}}{(2n+\sigma_n)^{2 (\ell -1)}}\,o(1). \end{align*} In particular, all terms of the sum inside the exponential in~\eqref{eq:GW3-2} tend to zero as $n\rightarrow\infty$. Moreover, for $n$ large, each term is bounded by $2^{-2(\ell-1)}$, which is summable. We finish the proof of the lemma by an application of dominated convergence, giving $$ \limsup_{n\rightarrow\infty}\sup_{m\leq Kk_n^2}\left|\textup{$\mathfrak{f}$}rac{p(\sigma_n-k_n,n-m)}{p(\sigma_n,n)}-1\right|=0. $$ \end{proof} \subsection{Convergence of bridges} Here, we collect two convergence results of a bridge $\textup{\textsf{b}}_n$ uniformly distributed in $\mathcal Br_{\sigma_n}$ which are valid in all regimes $\sigma_n\gg 1$. The first lemma follows from~\cite[Lemma 10]{Be1} (recall the remarks above on the distribution of $\textup{\textsf{b}}_n$). \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:bridge0} Assume $\sigma_n\rightarrow\infty$, and let $\textup{\textsf{b}}_n$ be a bridge of length $\sigma_n$ uniformly distributed in $\mathcal Br_{\sigma_n}$. Then $(\textup{\textsf{b}}_n(\sigma_n s)/\sqrt{2\sigma_n},0\leq s\leq 1)$ converges as $n\rightarrow\infty$ to a standard Brownian bridge $\textup{$\mathfrak{m}$}athbbm{b}$, and the convergence holds in distribution in the space $\textup{$\mathfrak{m}$}athcal{C}([0,1],\textup{$\mathfrak{m}$}athbb{R})$. \end{lemma} The next lemma provides a finer convergence without normalization for the bridge restricted to the first and last $k_n$ values when $k_n=o(\sigma_n)$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:bridge1} Assume $\sigma_n\rightarrow\infty$. Let $\textup{\textsf{b}}_n$ be uniformly distributed in $\mathcal Br_{\sigma_n}$, and let $\textup{\textsf{b}}_\infty$ be a uniform infinite bridge as defined under Section~\ref{sec:infinitebridge}. Then, if $k_n$ is a sequence of positive integers with $k_n\leq \sigma_n$ and $k_n=o(\sigma_n)$ as $n\rightarrow\infty$, \begin{equation*} \begin{split}\limsup_{n\rightarrow\infty}&\left\|\textup{$\mathfrak{m}$}athcal{L}((\textup{\textsf{b}}_n(\sigma_n-k_n),\ldots,\textup{\textsf{b}}_n(\sigma_n-1),\textup{\textsf{b}}_n(0),\textup{\textsf{b}}_n(1),\ldots,\textup{\textsf{b}}_n(k_n)))\right.\\ &\left. -\,\textup{$\mathfrak{m}$}athcal{L}((\textup{\textsf{b}}_\infty(-k_n),\ldots,\textup{\textsf{b}}_\infty(-1),\textup{\textsf{b}}_\infty(0),\textup{\textsf{b}}_\infty(1),\ldots,\textup{\textsf{b}}_\infty(k_n)))\right\|_{\textup{TV}}=0.\end{split} \end{equation*} \end{lemma} \begin{proof} Let $(b(i) : -k_n\leq i\leq k_n)$ be a $\textup{$\mathfrak{m}$}athbb{Z}$-valued sequence with $b(0)=0$ and $b(i+1)-b(i)\in\textup{$\mathfrak{m}$}athbb{N}_0\cup\{-1\}$. Note that by definition, both $(\textup{\textsf{b}}_n(\sigma_n-k_n),\ldots,\textup{\textsf{b}}_n(0),\ldots,\textup{\textsf{b}}_n(k_n))$ and $(\textup{\textsf{b}}_\infty (-k_n),\ldots,\textup{\textsf{b}}_\infty (0),\ldots,\textup{\textsf{b}}_\infty (k_n))$ are only supported on such sequences. By definition of $\textup{\textsf{b}}_\infty$, we obtain $$ \mathbb P\left(\textup{\textsf{b}}_\infty(i)=b(i),\, -k_n\leq i\leq k_n\right)=(-b(-1)+2)2^{-(b(k_n)-b(-k_n))-4k_n-1}. $$ Next recall the interpretation of the increments of $\textup{\textsf{b}}_n$ explained in Section~\ref{sec:bridges}. We get \begin{align*} \lefteqn{\mathbb P\left(\textup{\textsf{b}}_n(i)=b(i),\,\textup{\textsf{b}}_n(\sigma_n-i)=b(-i),\,1\leq i\leq k_n\right)}\\ &=\sum_{j=b(-1)-1}^0\mathbb P\left(\textup{\textsf{b}}_n(i)=b(i),\,\textup{\textsf{b}}_n(\sigma_n-i)=b(-i),\,1\leq i\leq k_n,\,\textup{\textsf{b}}_n(\sigma_n)=j\right)\\ &=(-b(-1)+2)\mathbb P\left(\textup{\textsf{b}}_n(i)=b(i),\,\textup{\textsf{b}}_n(\sigma_n-i)=b(-i),\,1\leq i\leq k_n,\,\textup{\textsf{b}}_n(\sigma_n)=0\right)\\ &=(-b(-1)+2){2\sigma_n-(b(k_n)-b(-k_n))-4k_n-1\choose \sigma_n-2k_n-1}\mathcal Big/{2\sigma_n\choose\sigma_n}. \end{align*} Here, the next to last line follows from the fact that $\textup{\textsf{b}}_n$ is uniformly distributed in $\mathcal Br_{\sigma_n}$, and the last line follows from counting the possibilities to put $(\sigma_n-2k_n-1)$ times the number $-1$ in the remaining $(2\sigma_n-(b(k_n)-b(-k_n))-4k_n-1)$ spots. We now concentrate on $b$ such that $|b(k_n)-b(-k_n)|\leq K\sqrt{k_n}$ for some fixed constant $K>0$. We put $B_n=b(k_n)-b(-k_n)+1$. An application of Stirling's formula shows that $$ \textup{$\mathfrak{f}$}rac{{2\sigma_n-4k_n-B_n\choose \sigma_n-2k_n-1}}{{2\sigma_n\choose\sigma_n}}\sim 2^{-4k_n-B_n}\left(\textup{$\mathfrak{f}$}rac{\sigma_n-2k_n-B_n/2}{\sigma_n-2k_n-B_n}\right)^{\sigma_n-2k_n-B_n}\left(\textup{$\mathfrak{f}$}rac{\sigma_n-2k_n-B_n/2}{\sigma_n-2k_n}\right)^{\sigma_n-2k_n} $$ as $n\rightarrow\infty$, uniformly in $b$ with $|b(k_n)-b(-k_n)|\leq K\sqrt{k_n}$. Next, observe that $$ \left(\textup{$\mathfrak{f}$}rac{\sigma_n-2k_n-B_n/2}{\sigma_n-2k_n}\right)^{\sigma_n-2k_n}=\left(1-\textup{$\mathfrak{f}$}rac{B_n/2}{\sigma_n-k_n}\right)^{\sigma_n-k_n} \sim\exp\left(-\textup{$\mathfrak{f}$}rac{B_n}{2}(1+O\left(B_n/\sigma_n)\right)\right), $$ and similarly $$ \left(\textup{$\mathfrak{f}$}rac{\sigma_n-2k_n-B_n/2}{\sigma_n-2k_n-B_n}\right)^{\sigma_n-2k_n-B_n}\sim\exp\left(\textup{$\mathfrak{f}$}rac{B_n}{2}\left(1-O(B_n/\sigma_n)\right)\right). $$ Note that $B^2_n/\sigma_n=o(1)$ as $n\rightarrow\infty$ uniformly in the sequences $b$ under consideration. Now let $0<\varepsilon<1$. Putting the above estimates together, we deduce that there exist $n'=n'(K,\varepsilon)$ sufficiently large such that for all $n\geq n'$, \begin{align*} (1-\varepsilon)(2^{-(b(k_n)-b(-k_n))-4k_n-1})&\leq{2\sigma_n-(b(k_n)-b(-k_n))-4k_n-1\choose \sigma_n-2k_n-1}\mathcal Big/{2\sigma_n\choose\sigma_n}\nonumber\\ &\leq (1+\varepsilon)(2^{-(b(k_n)-b(-k_n))-4k_n-1}). \end{align*} Using the aforementioned interpretation of $\textup{\textsf{b}}_n$ (or Lemma~\ref{lem:bridge0}), it is immediate to check that both $\textup{\textsf{b}}_n(-k_n)$ and $\textup{\textsf{b}}_n(k_n)$ are of order $\sqrt{k_n}$ for large $n$, i.e., we find $K>0$ such that $|\textup{\textsf{b}}_n(k_n)-\textup{\textsf{b}}_n(-k_n)|\leq K\sqrt{k_n}$ with probability at least $1-\varepsilon$, provided $n$ is sufficiently large. By Donsker's invariance principle, we see that a similar bound holds for $\textup{\textsf{b}}_\infty$. For any set $\textup{$\mathfrak{m}$}athcal{E}_n$ of $\textup{$\mathfrak{m}$}athbb{Z}$-valued sequences of length $2k_n+1$, we thus obtain \begin{align*} \begin{split}&|\,\mathbb P\left(\left(\textup{\textsf{b}}_n(\sigma_n-k_n),\ldots,\textup{\textsf{b}}_n(0),\ldots,\textup{\textsf{b}}_n(k_n)\right)\in \textup{$\mathfrak{m}$}athcal{E}_n\right)\\ &\,-\,\mathbb P\left(\left(\textup{\textsf{b}}_\infty (-k_n),\ldots,\textup{\textsf{b}}_\infty (0),\ldots,\textup{\textsf{b}}_\infty(k_n)\right)\in \textup{$\mathfrak{m}$}athcal{E}_n\right)|\end{split}\\ \textup{$\mathfrak{q}$}uad &\leq \mathbb P\left(|\textup{\textsf{b}}_n(k_n)-\textup{\textsf{b}}_n(-k_n)|\geq K\sqrt{k_n}\right) + \mathbb P\left(|\textup{\textsf{b}}_\infty(k_n)-\textup{\textsf{b}}_\infty(-k_n)|\geq K\sqrt{k_n}\right)\\ &\textup{$\mathfrak{q}$}uad + \sum_{b\in\textup{$\mathfrak{m}$}athcal{E}_n:\atop |b(k_n)-b(-k_n)|\leq K\sqrt{k_n}}\mathcal Bigg[\mathcal Big|\mathbb P\left(\textup{\textsf{b}}_\infty(i)=b(i),\, -k_n\leq i\leq k_n\right)\cdot\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \textup{$\mathfrak{f}$}rac{\mathbb P\left(\textup{\textsf{b}}_n(i)=b(i),\,\textup{\textsf{b}}_n(\sigma_n-i)=b(-i),\,1\leq i\leq k_n\right)}{\mathbb P\left(\textup{\textsf{b}}_\infty(i)=b(i),\, -k_n\leq i\leq k_n\right)}-1\mathcal Big|\mathcal Bigg]\\ &\textup{$\mathfrak{q}$}uad \leq 2\varepsilon + (1+\varepsilon)-1\leq 3\varepsilon. \end{align*} This finishes the proof. \end{proof} \subsection{Root issues} We work in the usual setting introduced in Section~\ref{sec:usualsetting}. As the next lemma shows, instead of showing distributional convergence of balls in $Q_n^{\sigma_n}$ or $Q_\infty^\infty$ around the roots, we can as well consider the corresponding balls around $(0)$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:ball0} Let $(a_n)_{n\geq 1}$ be a sequence of reals with $a_n\rightarrow\infty$ as $n\rightarrow\infty$. Let $r\geq 0$. Then, in the notation from above, we have the following convergences in probability as $n\rightarrow\infty$. \begin{enumerate} \item $d_{\textup{GH}}\left(B_r\left((a_n^{-1}\cdot Q_n^{\sigma_n}\right),B_r^{(0)}\left((a_n^{-1}\cdot Q_n^{\sigma_n}\right)\right)\rightarrow 0, $ \item $d_{\textup{GH}}\left(B_r\left((a_n^{-1}\cdot Q_\infty^{\infty})\right),B_r^{(0)}\left((a_n^{-1}\cdot Q_\infty^{\infty}\right)\right)\rightarrow 0. $ \end{enumerate} \end{lemma} The proof will be a consequence of the following general lemma. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:localGHconv} Let $r\geq 0$, and let $\textup{$\mathfrak{m}$}athbf{E}=(E,d,\rho)$ and $\textup{$\mathfrak{m}$}athbf{E}'=(E',d',\rho')$ be two pointed complete and locally compact length spaces. Let $\mathcal{R}\subset E\times E'$ be a subset with the following properties: \begin{itemize} \item $(\rho,\rho')\in\mathcal{R}$, \item for all $x \in B_r(\textup{$\mathfrak{m}$}athbf{E})$, there exists $x'\in E'$ such that $(x,x')\in\mathcal{R}$, \item for all $y' \in B_r(\textup{$\mathfrak{m}$}athbf{E}')$, there exists $y\in E$ such that $(y,y')\in\mathcal{R}$. \end{itemize} Then, $d_{\textup{GH}}(B_r(\textup{$\mathfrak{m}$}athbf{E}),B_r(\textup{$\mathfrak{m}$}athbf{E}'))\leq (3/2)\textup{dis}(\mathcal{R})$. \end{lemma} \begin{remark} Note that $\mathcal{R}$ is not necessarily a correspondence; nonetheless, the definition of the distortion $\textup{dis}(\mathcal{R})$ from Section~\ref{S-notionconvergence} makes sense (we allow it to take the value $+\infty$). \end{remark} \begin{proof}[Proof of Lemma~\ref{lem:localGHconv}] We construct a correspondence $\tilde{\mathcal{R}}$ between $B_r(\textup{$\mathfrak{m}$}athbf{E})$ and $B_r(\textup{$\mathfrak{m}$}athbf{E}')$. For each $x\in B_r(\textup{$\mathfrak{m}$}athbf{E})$, there exists by assumption $x'=x'_{x}\in E'$ such that $(x,x')\in\mathcal{R}$. Since $d'(x',\rho')\leq d(x,\rho) + \textup{dis}(\mathcal{R})$, we see that in fact $x'\in B_{r+\textup{dis}(\mathcal{R})}(\textup{$\mathfrak{m}$}athbf{E}')$. We choose $z'=z'(x)\in B_r(\textup{$\mathfrak{m}$}athbf{E}')$ that minimizes $d'(x',z')$. Note that such a $z'$ exists in a complete and locally compact length space. Then $d'(x',z')\leq \textup{dis}(\mathcal{R})$. In an entirely similar way, using the third property of $\mathcal{R}$ instead of the second, we assign to each $y'\in B_r(\textup{$\mathfrak{m}$}athbf{E}')$ an element $z=z(y')\in B_r(\textup{$\mathfrak{m}$}athbf{E})$. In this notation, we now define $$ \tilde{\mathcal{R}}=\left\{(x,z'(x)) : x\in B_r(\textup{$\mathfrak{m}$}athbf{E})\right\}\cup \left\{(z(y'),y') : y'\in B_r(\textup{$\mathfrak{m}$}athbf{E}')\right\}. $$ Clearly, $\tilde{\mathcal{R}}$ is a correspondence between $B_r(\textup{$\mathfrak{m}$}athbf{E})$ and $B_r(\textup{$\mathfrak{m}$}athbf{E}')$, and a straightforward application of the triangle inequality shows that in fact $\textup{dis}(\tilde{\mathcal{R}}) \leq 3\textup{dis}(\mathcal{R})$. This proves our claim and hence the lemma. \end{proof} \begin{proof}[Proof of Lemma~\ref{lem:ball0}] We show only (a), the proof of (b) is similar. We apply Lemma~\ref{lem:localGHconv} as follows. Instead of considering the pointed quadrangulation $(V(Q_n^{\sigma_n}),d_{\textup{gr}},\rho_n)$, we may work with the corresponding pointed length space $\textup{$\mathfrak{m}$}athbf{E}_n=(E_n,d,(0))$ obtained from replacing edges by Euclidean segments of length one, as explained in Section~\ref{sec:locGH} (the distance $d$ between two points is given by the length of a shortest path between them). Similarly, we replace $(V(Q_n^{\sigma_n}),d_{\textup{gr}},(0))$ by $\textup{$\mathfrak{m}$}athbf{E}'_n=(E_n,d,\rho_n)$. Define $$ \mathcal{R}_n=\{(\rho_n,(0))\}\cup\{(x,x) : x\in E_n\}. $$ Then $\mathcal{R}_n$ fulfills trivially the properties of Lemma~\ref{lem:localGHconv}, and we have dis$(\mathcal{R}_n)\leq d(\rho_n,(0)) =-\textup{\textsf{b}}_n(\sigma_n)$ by~\eqref{eq:distance-root-0}. Since $\textup{\textsf{b}}_n(\sigma_n)$ is stochastically bounded, see~\eqref{eq:law-bsigma}, the claim follows. \end{proof} \section{Main proofs} \textup{$\mathfrak{l}$}bel{sec:proofs} We start now with the proofs of the main results. To facilitate the reading, we will sometimes include a paragraph ``Idea of the proof'', where we informally explain the basic strategy. \subsection{Brownian plane (Theorem~\ref{thm:BP})} \textup{$\mathfrak{l}$}bel{sec:proof-thmBP} Recall that Theorem~\ref{thm:BP} deals with the regime $\sigma_n\ll\sqrt{n}$ and $\sqrt{\sigma_n} \ll a_n$. \begin{mdframed} {\bf Idea of the proof.} Let $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ be uniformly distributed over the set $\mathbb Fo_{\sigma_n}^n \times \mathcal Br_{\sigma_n}$, and let $\textup{$\mathfrak{L}$}_n$ be the associated label function. Thanks to Lemmas~\ref{lem:GW1} and~\ref{lem:GW2}, we know that for large $n$, $\textup{$\mathfrak{f}$}_n$ has a unique largest tree $\tau$ of a size of order $n$, and all the other $\sigma_n-1$ trees behave as independent critical geometric Galton-Watson trees. As a consequence, both the maximal and minimal values of the label function $\textup{$\mathfrak{L}$}_n$ restricted to these $\sigma_n-1$ non-largest trees are of order $\sqrt{\sigma_n}$, see the proof of Lemma~\ref{lem:BP2}. Under a rescaling of distances by the factor $a_n^{-1}$, this implies by a result of Bettinelli~\cite[Lemma 23]{Be3} that the part of the quadrangulation encoded by the forest without its largest tree $\tau$ is negligible in the limit $n\rightarrow\infty$ for the local Gromov-Hausdorff topology. Conditionally on its size, $\tau$ is uniformly distributed among all plane trees, and (up to the removal of a single edge) so is the associated quadrangulation among all quadrangulations with $|\tau|$ faces and no boundary. This allows us to apply the second part of~\cite[Theorem 3]{CuLG}, which states that the Brownian plane appears as the scaling limit $m\rightarrow\infty$ of uniform quadrangulations with $m$ faces when the scaling factor approaches zero slower than $m^{-1/4}$. \end{mdframed} To make things precise, we recall \begin{lemma}[Lemma 23 of~\cite{Be3}] \textup{$\mathfrak{l}$}bel{lem:BP1} Let $\sigma\in\textup{$\mathfrak{m}$}athbb N$. Let $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})\in \mathbb Fo_{\sigma}^n \times \textup{$\mathfrak{m}$}athcal B_{\sigma}$. Fix any tree $\tau$ of $\textup{$\mathfrak{f}$}$. Let $b\in\{-1,0\}$. We view $(\tau,\textup{$\mathfrak{l}$}_{|\tau})$ as an element of $\mathbb Fo_1^{|\tau|}$ and denote by $\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$}\in\textup{$\mathfrak{m}$}athcal{Q}_n^\sigma$ and $\textup{$\mathfrak{q}$}_{\tau}\in\textup{$\mathfrak{m}$}athcal{Q}_{|\tau|}^1$ the quadrangulations associated to $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})$ and $((\tau,\textup{$\mathfrak{l}$}_{|\tau}),(0,b))$, respectively, through the Bouttier-Di Francesco-Guitter bijection (the distinguished vertices are omitted). Then $$ d_{\textup{GH}}\left(V(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$}),V(\textup{$\mathfrak{q}$}_\tau)\right)\leq 2\left (\textup{$\mathfrak{m}$}ax_{\textup{$\mathfrak{f}$}\setminus \textup{$\mathfrak{m}$}athring{\tau}}\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}-\textup{$\mathfrak{m}$}in_{\textup{$\mathfrak{f}$}\setminus \textup{$\mathfrak{m}$}athring{\tau}}\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$} +1\right), $$ where $\textup{$\mathfrak{m}$}athring{\tau}$ stand for the tree $\tau$ without its root vertex, and $\textup{$\mathfrak{L}$}_\textup{$\mathfrak{f}$}$ is the label function associated to $((\textup{$\mathfrak{f}$},\textup{$\mathfrak{l}$}),\textup{\textsf{b}})$ as defined in Section~\ref{sec:contourlabel-finite}. \end{lemma} \begin{remark} As always, we interpret $V(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$})$ and $V(\textup{$\mathfrak{q}$}_\tau)$ as pointed metric spaces (pointed at their root vertices and endowed with the graph distance). Note that~\cite[Lemma 23]{Be3} is formulated in terms of the unpointed Gromov-Hausdorff distance, but the proof carries over to the pointed version used here. \end{remark} Let $r\geq 0$. For the balls $B_r(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$})$ and $B_r(\textup{$\mathfrak{q}$}_\tau)$ around the root vertices, we claim that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BP-1} d_{\textup{GH}}\left(B_r(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$},B_r(\textup{$\mathfrak{q}$}_\tau\right) \leq 3 d_{\textup{GH}}\left(V(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$}),V(\textup{$\mathfrak{q}$}_\tau)\right) +8. \end{equation} Indeed, we may first replace both $V(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$})$ and $V(\textup{$\mathfrak{q}$}_\tau)$ by the corresponding length spaces ${\textup{$\mathfrak{m}$}athbf Q}_\textup{$\mathfrak{f}$}$ and ${\textup{$\mathfrak{m}$}athbf Q}_\tau$ as explained in Section~\ref{sec:locGH}. We obtain $$ \left|d_{\textup{GH}}\left(B_r(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$}),B_r(\textup{$\mathfrak{q}$}_\tau)\right)- d_{\textup{GH}}\left(B_r({\textup{$\mathfrak{m}$}athbf Q}_\textup{$\mathfrak{f}$}),B_r({\textup{$\mathfrak{m}$}athbf Q}_\tau)\right)\right|\leq 2. $$ For estimating the Gromov-Hausdorff distance on the right, we note that every correspondence between ${\textup{$\mathfrak{m}$}athbf Q_\textup{$\mathfrak{f}$}}$ and ${\textup{$\mathfrak{m}$}athbf Q}_\tau$ satisfies the requirements of Lemma~\ref{lem:localGHconv}, so that by this lemma $$ d_{\textup{GH}}\left(B_r({\textup{$\mathfrak{m}$}athbf Q}_\textup{$\mathfrak{f}$}),B_r({\textup{$\mathfrak{m}$}athbf Q}_\tau)\right)\leq (3/2)\inf_\mathcal{R}\textup{dis}(\mathcal{R}) = 3d_{\textup{GH}}\left({\textup{$\mathfrak{m}$}athbf Q_\textup{$\mathfrak{f}$}},{\textup{$\mathfrak{m}$}athbf Q}_\tau\right)\leq 3d_{\textup{GH}}\left(V(\textup{$\mathfrak{q}$}_\textup{$\mathfrak{f}$}),V(\textup{$\mathfrak{q}$}_\tau)\right)+ 6, $$ where the infimum is taken over all correspondences between ${\textup{$\mathfrak{m}$}athbf Q_\textup{$\mathfrak{f}$}}$ and ${\textup{$\mathfrak{m}$}athbf Q}_\tau$, and the equality follows from the alternative description of the Gromov-Hausdorff distance in terms of correspondences. We are now in position to prove Theorem~\ref{thm:BP}. \begin{proof}[Proof of Theorem~\ref{thm:BP}] Recall from\cite[Theorem 4]{Be3} that $$ (V(Q_n^{\sigma_n}),(8/9)^{-1/4}n^{-1/4} d_{\textup{gr}},\rho_n) \xrightarrow[n \to \infty]{(d)} \mathcal BM $$ in the Gromov-Hausdorff topology, where $\mathcal BM$ is the Brownian map. This result immediately implies that when $a_n\gg n^{1/4}$, then $(V(Q_n^{\sigma_n}),a_n^{-1}d_{\textup{gr}},\rho_n)$ converges to the trivial metric space consisting of a single point, which proves the second part of the theorem. For the first part and the rest of this proof, we assume $\sqrt{\sigma_n} \ll a_n \ll n^{1/4}$. We have to show that for each $r\geq 0$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BP-2} B_r\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\xrightarrow[n \to \infty]{(d)}B_r(\mathcal BP) \end{equation} in distribution in $\textup{$\mathfrak{m}$}athbb{K}$. Let $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ be uniformly distributed in $\mathbb Fo_{\sigma_n}^n \times \mathcal Br_{\sigma_n}$, and write $Q_n^{\sigma_n}$ for the (rooted and pointed) quadrangulation associated through the Bouttier-Di Francesco-Guitter bijection, as usual. We denote by $\tau_\ast^{(n)}$ the largest tree of $\textup{$\mathfrak{f}$}_n$ (we take that with the smallest index if several trees attain the largest size). We let $b_n\in\{-1,0\}$ be uniformly distributed and independent of everything else and denote by $\hat{Q}_n$ the quadrangulation encoded by $((\tau_\ast^{(n)},{\textup{$\mathfrak{l}$}_n}_{|\tau_\ast^{(n)}}),(0,b_n))$, in the same way as in Lemma~\ref{lem:BP1}. We obtain from~\eqref{eq:BP-1} together with Lemma~\ref{lem:BP1} that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BP-3} d_{\textup{GH}}\left(B_r(a_n^{-1}\cdot Q_n^{\sigma_n}),B_r(a_n^{-1}\cdot \hat{Q}_n)\right)\leq \textup{$\mathfrak{f}$}rac{6}{a_n}\left(\textup{$\mathfrak{m}$}ax_{\textup{$\mathfrak{f}$}_n\setminus \textup{$\mathfrak{m}$}athring{\tau}_{\ast}^{(n)}}\textup{$\mathfrak{L}$}_{n}-\textup{$\mathfrak{m}$}in_{\textup{$\mathfrak{f}$}_n\setminus \textup{$\mathfrak{m}$}athring{\tau}_{\ast}^{(n)}}\textup{$\mathfrak{L}$}_n\right)+o(1) \end{equation} as $n\rightarrow\infty$, where in the notation of Lemma~\ref{lem:BP1}, $\textup{$\mathfrak{m}$}athring{\tau}_{\ast}^{(n)}$ stands for the tree ${\tau}_{\ast}^{(n)}$ without its root, and $\textup{$\mathfrak{L}$}_n$ is the label function of $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$. We claim that the right hand side in the last display converges to zero in probability. In this regard, recall that $$\textup{$\mathfrak{L}$}_n = \left(L_n(t) + \textup{\textsf{b}}_n(-\underline{C}_n(t)),\,0\leq t\leq 2n+\sigma_n\right).$$ By Lemma~\ref{lem:bridge0}, the values of $\textup{\textsf{b}}_n$ are of order $\sqrt{\sigma_n}\ll a_n$, so that we may replace $\textup{$\mathfrak{L}$}_n$ by $L_n$ in~\eqref{eq:BP-3}. Denote by $\textup{$\mathfrak{f}$}'_n=\textup{$\mathfrak{f}$}_n\setminus \tau_{\ast}^{(n)}$ the forest obtained from $\textup{$\mathfrak{f}$}_n$ by removing $\tau_{\ast}^{(n)}$, i.e., if $\tau_{\ast}^{(n)}$ is the tree of $\textup{$\mathfrak{f}$}_n$ with index $i$, then $\textup{$\mathfrak{f}$}'_n=(\tau_0^{(n)},\ldots,\tau_{i-1}^{(n)},\tau_{i+1}^{(n)},\ldots,\tau_{\sigma_n-1}^{(n)})$. We let $\textup{$\mathfrak{l}$}'$ be the labeling of $\textup{$\mathfrak{f}$}_n$ restricted to $\textup{$\mathfrak{f}$}'_n$, and write $(C'_n,L'_n)$ for the contour pair corresponding to $(\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n)$. We view both $C'_n$ and $L'_n$ as continuous functions on $[0,\infty)$ by letting $C'_n(s)=C'_n(s\wedge (2(n-|\tau_{\ast}^{(n)}|)+\sigma_n-1))$, and similarly with $L'_n$. The convergence to zero of the right hand side in~\eqref{eq:BP-3} is now a consequence of the following lemma. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:BP2} In the notation from above, we have for sequences $a_n$ satisfying $a_n\gg \sqrt{\sigma_n}$, $$ \left(\textup{$\mathfrak{f}$}rac{1}{a^2_n}C'_n,\textup{$\mathfrak{f}$}rac{1}{a_n}L'_n\right)\xrightarrow[n\to \infty]{(p)} (0,0)\textup{$\mathfrak{q}$}uad\textup{in }\textup{$\mathfrak{m}$}athcal{C}([0,\infty),\textup{$\mathfrak{m}$}athbb{R})^2. $$ \end{lemma} \begin{proof} Let $(\tilde{\tau}_i,(\tilde{\ell}_i(u))_{u\in\tau_i})$, $i=0,\ldots,\sigma_n-2$, be a sequence of $\sigma_n-1$ uniformly labeled critical geometric Galton-Watson trees. Consider the forest $\tilde{\textup{$\mathfrak{f}$}}_n=(\tilde{\tau}_0,\ldots,\tilde{\tau}_{\sigma_n-2})$ together with the labeling $\tilde{\textup{$\mathfrak{l}$}}_n$ given by $\tilde{\textup{$\mathfrak{l}$}}_n\restriction V(\tilde{\tau_i})=\tilde{\ell}_i$, for all $i$. Let $(\tilde{C}_n,\tilde{L}_n)$ denote the contour pair associated to $(\tilde{\textup{$\mathfrak{f}$}}_n,\tilde{\textup{$\mathfrak{l}$}}_n)$, continuously extended to $[0,\infty)$ outside $[0,2\sum_{i=0}^{\sigma_n-2}|\tilde{\tau}_i|+\sigma_n-1]$ as described above. By Lemma~\ref{lem:GW2}, we can for each $\varepsilon>0$ couple the pairs $(C_n',L_n')$ and $(\tilde{C}_n,\tilde{L}_n)$ on the same probability space such that with probability at least $1-\varepsilon$, we have the equality $$ (C'_n,L'_n)=(\tilde{C}_n,\tilde{L}_n) $$ as elements of $\textup{$\mathfrak{m}$}athcal{C}([0,\infty),\textup{$\mathfrak{m}$}athbb{R})^2$, provided $n$ is sufficiently large. Our claim therefore follows if \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BP-3a} \left(\textup{$\mathfrak{f}$}rac{1}{a^2_n}\tilde{C}_n,\textup{$\mathfrak{f}$}rac{1}{a_n}\tilde{L}_n\right)\xrightarrow[n\to \infty]{(p)} (0,0). \end{equation} From Section~\ref{sec:forests}, we know that the law of $\tilde{C}_n$ agrees with that of a simple random walk started from $0$ and stopped upon hitting $-(\sigma_n-1)$, with linear interpolation between integer values. Donsker's invariance principle thus shows that $((1/\sigma_n)\tilde{C}_n(\sigma_n^2t),t\geq 0)$ converges in distribution to a standard Brownian motion $(B_{t\wedge T_{-1}}, t\geq 0)$ stopped upon hitting $-1$. Arguments like in~\cite[Proof of Theorem 4.3]{LGMi} then imply convergence of the finite-dimensional laws on $\textup{$\mathfrak{m}$}athcal{C}([0,\infty),\textup{$\mathfrak{m}$}athbb R)^2$ of the tuple $((1/\sigma_n)\tilde{C}_n(\sigma_n^2\cdot),(1/\sqrt{\sigma_n})\tilde{L}_n(\sigma_n^2\cdot))$, and tightness of the second component follows {\it via} Kolmogorov's criterion from moment bounds on $\tilde{C}_n$ as in~\cite[Lemma 2.3.1]{LGMi} (in our case, these bounds are in fact easier to establish, since we consider an unconditioned random walk). We do not repeat the arguments here, but refer the reader to~\cite{LGMi} or~\cite[Section 5]{Be1} for more details. We obtain the convergence in distribution $$ \left(\textup{$\mathfrak{f}$}rac{1}{\sigma_n}\tilde{C}_n(\sigma^2_n\cdot),\textup{$\mathfrak{f}$}rac{1}{\sqrt{\sigma_n}}\tilde{L}_n(\sigma^2_n\cdot)\right) \xrightarrow[n \to \infty]{(d)} \left(B_{\cdot\wedge T_{-1}}, Z\right)\textup{$\mathfrak{q}$}uad\textup{in }\textup{$\mathfrak{m}$}athcal{C}([0,\infty),\textup{$\mathfrak{m}$}athbb{R})^2, $$ where $Z=(Z_t,t\geq 0)$ is the Brownian snake driven by $(B_{t\wedge T_{-1}}, t\geq 0)$. Since $a^2_n\gg \sigma_n$, this last result implies clearly~\eqref{eq:BP-3a} and hence the assertion of the lemma. \end{proof} Going back to~\eqref{eq:BP-3}, it remains to show that for $\varepsilon>0$, $F:\textup{$\mathfrak{m}$}athbb{K}\rightarrow\textup{$\mathfrak{m}$}athbb{R}$ continuous and bounded and $n\geq n_0$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BP-4} \left|\textup{$\mathfrak{m}$}athbb E\left[F\left(B_r\left(a_n^{-1}\cdot \hat{Q}_n\right)\right)\right]-\textup{$\mathfrak{m}$}athbb E\left[F(B_r(\mathcal BP))\right]\right|\leq \varepsilon. \end{equation} Let $\mathrm{d}elta>0$. We estimate \begin{align*} \lefteqn{\left|\textup{$\mathfrak{m}$}athbb E\left[F\left(B_r\left(a_n^{-1}\cdot \hat{Q}_n\right)\right)\right]-\textup{$\mathfrak{m}$}athbb E\left[F(B_r(\mathcal BP))\right]\right| \leq 2\sup|F|\,\mathbb P\left(|\tau_{\ast}^{(n)}|\leq \mathrm{d}elta n\right)}\\ &\textup{$\mathfrak{q}$}uad + \sum_{k=\lceil \mathrm{d}elta n\rceil}^n\mathbb P\left(|\tau_{\ast}^{(n)}|=k\right)\left|\textup{$\mathfrak{m}$}athbb E\left[F\left(B_r\left(a_n^{-1}\cdot \hat{Q}_n\right)\right)\,\mathcal Big|\,|\tau_{\ast}^{(n)}|=k\right]-\textup{$\mathfrak{m}$}athbb E\left[F(B_r(\mathcal BP))\right]\right|. \end{align*} For $\mathrm{d}elta=\mathrm{d}elta(F,\varepsilon)>0$ small and $n=n(\mathrm{d}elta,\varepsilon)\in\textup{$\mathfrak{m}$}athbb N$ sufficiently large, we have by Lemma~\ref{lem:GW1} $$ 2\sup|F|\,\mathbb P\left(|\tau_{\ast}^{(n)}|\leq \mathrm{d}elta n\right) \leq \varepsilon/2. $$ Concerning the summands in the second term, we note that conditionally on $|\tau_{\ast}^{(n)}|=k$, the quadrangulation $\hat{Q}_n^{\sigma_n}$ is uniformly distributed among all quadrangulations in $\mathcal{Q}_k^{1}$, i.e., those with $k$ inner faces and a boundary of size $2$. Removing the only edge of the boundary which is not the root edge, we obtain a quadrangulation uniformly distributed among all quadrangulations with $k$ faces and no boundary. Clearly, the removal of this edge does not change the underlying metric space. By~\cite[Theorem 2]{CuLG}, we therefore get for $k\geq \lceil \mathrm{d}elta n\rceil$ and $n$ sufficiently large, recalling that $a_n \ll n^{1/4}$, $$\left|\textup{$\mathfrak{m}$}athbb E\left[F\left(B_r\left(a_n^{-1}\cdot \hat{Q}_n)\right)\right)\,\mathcal Big|\,|\tau_{\ast}^{(n)}|=k\right]-\textup{$\mathfrak{m}$}athbb E\left[F(B_r(\mathcal BP))\right]\right|\leq \varepsilon/2. $$ This shows~\eqref{eq:BP-4} and hence~\eqref{eq:BP-2}. \end{proof} \subsection{Coupling Brownian disk \& half-planes (Theorem~\ref{thm:coupling-BD-BHP} and Corollary~\ref{cor:topology-BHP})} \textup{$\mathfrak{l}$}bel{sec:proof-coupling-BD-BHP} Let us first show how Corollary~\ref{cor:topology-BHP} follows from Theorem~\ref{thm:coupling-BD-BHP}. \begin{proof}[Proof of Corollary~\ref{cor:topology-BHP}] Theorem~\ref{thm:coupling-BD-BHP} implies that with probability $1$, for every $r>0$, the ball $B_r(\mathcal BHP_\theta)$ is included in an open set of $\mathcal BHP_\theta$ homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{H}}$. This shows that $\mathcal BHP_\theta$ is a simply connected topological surface with a boundary, and that this boundary is connected and non-compact: it must therefore be homeomorphic to $\textup{$\mathfrak{m}$}athbb R$. We construct a surface $S$ without boundary by gluing a copy $H$ of the closed half-plane $\overline{\textup{$\mathfrak{m}$}athbb{H}}$ to $\mathcal BHP_\theta$ along the boundary. This non-compact surface is still simply connected by van Kampens' Theorem, and in particular, it is one-ended. Therefore, it must be homeomorphic to $\textup{$\mathfrak{m}$}athbb R^2$, see~\cite{Ri}. Now if $\phi$ is a homeomorphism from the boundary of $\mathcal BHP_\theta$ to $\textup{$\mathfrak{m}$}athbb R$, then the Jordan-Schoenflies Theorem (in fact a simple variation of the latter) implies that $\phi$ can be extended to a homeomorphism $\overline{\phi}$ from $S$ to $\textup{$\mathfrak{m}$}athbb R^2$, and the two halves $\mathcal BHP_\theta$ and $H$ of $S$ must be sent {\it via} $\overline{\phi}$ to the two half-spaces $\overline{\textup{$\mathfrak{m}$}athbb{H}}$ and $-\overline{\textup{$\mathfrak{m}$}athbb{H}}$. In particular, $\overline{\phi}$ induces a homeomorphism from $\mathcal BHP_\theta$ to a closed half-plane, as wanted. \end{proof} We turn to Theorem~\ref{thm:coupling-BD-BHP}, and in this regard, we begin with proving the following weaker statement (compare with Proposition 4 of~\cite{CuLG} for the Brownian map and plane). \begin{prop} \textup{$\mathfrak{l}$}bel{prop:isometry-BD-BHP} Let $\varepsilon>0$, $r\geq 0$. Let $\sigma(\cdot):(0,\infty)\rightarrow(0,\infty)$ be a function satisfying $\lim_{T\rightarrow\infty}\sigma(T)/T=\theta\in[0,\infty)$ and $\liminf_{T\rightarrow\infty}\sigma(T)/\sqrt{T}>0$. Then there exists $T_0=T_0(\varepsilon,r,\sigma)$ such that for all $T\geq T_0$, one can construct copies of $\mathcal BD_{T,\sigma(T)}$ and $\mathcal BHP_\theta$ on the same probability space such that with probability at least $1-\varepsilon$, the balls $B_r(\mathcal BD_{T,\sigma(T)})$ and $B_r(\mathcal BHP_\theta)$ of radius $r$ around the respective roots are isometric. \end{prop} Before proving Proposition~\ref{prop:isometry-BD-BHP}, we recapitulate for the reader's convenience in the next section the definitions of $\mathcal BD_{T,\sigma}$ and $\mathcal BHP_\theta$. \subsubsection{Brownian half-plane and disk} \textup{$\mathfrak{l}$}bel{sec:recapBHPBD} In order to ease the reading of the proofs which follow, we use a notation which differs slightly from that in Section~\ref{sec:def}. Let $\sigma(\cdot):(0,\infty)\rightarrow(0,\infty)$ be a perimeter function as given in the statement of Proposition~\ref{prop:isometry-BD-BHP}. Recall that $\sigma(\cdot)$ is a function of the volume $T$ of the disk. For defining the {\bf Brownian disk }$\mathcal BD_{T,\sigma(T)}$ of volume $T$ and boundary length $\sigma(T)$, we consider a contour function $F=(F_t,0\leq t\leq T)$ and a label function $W=(W_t,0\leq t\leq T)$ given as follows. \begin{itemize} \item $F$ has the law of a first-passage Brownian bridge on $[0,T]$ from $0$ to $-\sigma(T)$. \item Given $F$, the function $W$ has same distribution as $(b_{-\underline{F}_t}+Z_t,0\leq t\leq T)$, where \begin{itemize} \item $(Z_t,t\in\textup{$\mathfrak{m}$}athbb R)=Z^{F-\underline{F}}$ is a continuous modification of the centered Gaussian process with covariances given by $$ \textup{$\mathfrak{m}$}athbb E\left[Z_sZ_t\right]=\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}F-\underline{F}, $$ with $\underline{F}_t=\inf_{[0,t]}F$. \item $(b_x,0\leq x\leq \sigma(T))$ is a standard Brownian bridge with duration $\sigma(T)$ and scaled by the factor $\sqrt{3}$, independent of $Z^{F-\underline{F}}$. \end{itemize} \end{itemize} The pseudo-metrics $d_{F}$ and $d_{W}$ on $[0,T]$ are given by $$ d_{F}(s,t)=F_s+F_t-2\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}F, $$ and $$ d_{W}(s,t)=W_s+W_t-2\textup{$\mathfrak{m}$}ax\left(\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]} W,\,\textup{$\mathfrak{m}$}in_{[0,s\wedge t]\cup[s\vee t,T]}W\right). $$ We shall write $D$ instead of $D_{F,W}$, i.e., $$ D(s,t)=\inf\left\{\sum_{i=1}^kd_{W}(s_i,t_i):\begin{array}{l} k\geq 1, \, s_1,\ldots,s_k,t_1,\ldots,t_k\in I,s_1=s,t_k=t,\\ d_{F}(t_i,s_{i+1})=0\textup{$\mathfrak{m}$}box{ for every }i\in \{1,\ldots,k-1\} \end{array} \right\}\, . $$ The Brownian disk $\mathcal BD_{T,\sigma(T)}$ has the law of the pointed metric space $([0,T]/\{D=0\},D,\rho)$, with $\rho$ being the equivalence class of $0$. \paragraph{} The {\bf Brownian half-plane }$\mathcal BHP_{\theta}$, $\theta\in[0,\infty)$ is given in terms of contour and label processes $X^{\theta}=(X^{\theta}_t,t\in\textup{$\mathfrak{m}$}athbb R)$ and $W^{\theta}=(W^{\theta}_t,t\in\textup{$\mathfrak{m}$}athbb R)$ specified as follows: \begin{itemize} \item $(X^{\theta}_t,t\geq 0)$ has the law of a one-dimensional Brownian motion $B=(B_t,t\geq 0)$ with drift $-\theta$ and $B_0=0$, and $(X^{\theta}_{-t},t\geq 0)$ has the law of the Pitman transform of an independent copy of $B$. \item Given $X^{\theta}$, the (label) function $W^{\theta}$ has same distribution as $(\gamma_{-\underline{X}_t^{\theta}}+\textup{$\mathfrak{m}$}athbb Zha_t,t\in \textup{$\mathfrak{m}$}athbb R)$, where \begin{itemize} \item given $X^{\theta}$, $\textup{$\mathfrak{m}$}athbb Zha=(\textup{$\mathfrak{m}$}athbb Zha_t,t\in\textup{$\mathfrak{m}$}athbb R)=Z^{X^{\theta}-\underline{X}^{\theta}}$ is a continuous modification of the centered Gaussian process with covariances given by $$ \textup{$\mathfrak{m}$}athbb E\left[\textup{$\mathfrak{m}$}athbb Zha_s\textup{$\mathfrak{m}$}athbb Zha_t\right]=\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}X^{\theta}-\underline{X}^{\theta}, $$ with $\underline{X}_t^{\theta}=\inf_{[0,t]}X^{\theta}$ for $t\geq 0$, and $\underline{X}_t^{\theta}=\inf_{(-\infty,t]}X^{\theta}$ for $t<0$. \item $(\gamma_x,x\in \textup{$\mathfrak{m}$}athbb R)$ is a two-sided Brownian motion with $\gamma_0=0$ and scaled by the factor $\sqrt{3}$, independent of $\textup{$\mathfrak{m}$}athbb Zha$. \end{itemize} \end{itemize} For notational simplicity, we include here the scaling factor $\sqrt{3}$ already in the definition of $\gamma$. The pseudo-metrics $d_{X^{\theta}}$ and $d_{W^{\theta}}$ on $\textup{$\mathfrak{m}$}athbb{R}$ are given by $$ d_{X^{\theta}}(s,t)=X^{\theta}_s+X^{\theta}_t-2\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}X^{\theta}\textup{$\mathfrak{q}$}uad d_{W^{\theta}}(s,t)=W^{\theta}_s+W^{\theta}_t-2\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}W^{\theta}, $$ and we write $D_{\theta}$ instead of $D_{X^{\theta},W^{\theta}}$, cf.~\eqref{eq:Dfg}, i.e., $$ D_{\theta}(s,t)=\inf\left\{\sum_{i=1}^kd_{W^{\theta}}(s_i,t_i):\begin{array}{l} k\geq 1, \, s_1,\ldots,s_k,t_1,\ldots,t_k\in I,s_1=s,t_k=t,\\ d_{X^{\theta}}(t_i,s_{i+1})=0\textup{$\mathfrak{m}$}box{ for every }i\in \{1,\ldots,k-1\} \end{array} \right\}\, . $$ Then the Brownian half-plane $\mathcal BHP_\theta$ has the law of the pointed metric space $(\textup{$\mathfrak{m}$}athbb R/\{D_{\theta}=0\},D_{\theta},\rho_{\theta})$, with $\rho_{\theta}$ being the equivalence class of $0$. \begin{remark} Be aware that all the quantities in the definition of $\mathcal BD_{T,\sigma(T)}$ depend on $T$ or $\sigma(T)$ (like $F,b,W,Z$ or the pseudo-metric $D$). The real $T$ measuring the volume will be chosen sufficiently large later on, but for the ease of reading, we mostly suppress $T$ from the notation. \end{remark} \subsubsection{Absolute continuity relation between contour functions} A key step in proving Proposition~\ref{prop:isometry-BD-BHP} is to relate the contour function $X^{\theta}$ for $\mathcal BHP_\theta$ to the contour function $F$ for $\mathcal BD_{T,\sigma(T)}$, in spirit of~\cite[Proposition 3]{CuLG}. We fix once for all a perimeter function $\sigma(\cdot):(0,\infty)\rightarrow(0,\infty)$ as given in the statement of Proposition~\ref{prop:isometry-BD-BHP}, and let $\theta=\lim_{T\rightarrow\infty}\sigma(T)/T\in[0,\infty)$. For given $T>0$, which we will choose large enough later on, we let $F$ be a first-passage Brownian bridge on $[0,T]$ from $0$ to $-\sigma(T)$, $B$ a one-dimensional Brownian motion on $[0,\infty)$ with drift $-\theta$ and $B_0=0$, and, by a small abuse of notation, $\mathbb Pi$ the Pitman transform of an independent copy of $B$. Now assume $\alpha,\beta>0$ with $\alpha+\beta<T$. We consider the pair $((F_t)_{0\leq t\leq \alpha},(F_{T-t})_{0\leq t\leq \beta})$ as an element of the space $\textup{$\mathfrak{m}$}athcal{C}([0,\alpha],\textup{$\mathfrak{m}$}athbb{R})\times \textup{$\mathfrak{m}$}athcal{C}([0,\beta],\textup{$\mathfrak{m}$}athbb{R})$. We write $(\omega,\omega')$ for the generic element of this space. We next introduce some probability kernels. Let $t>0$. For $x\in\textup{$\mathfrak{m}$}athbb{R}$, the heat kernel is denoted $$ p_t(x)=\textup{$\mathfrak{f}$}rac{1}{\sqrt{2\pi t}}\exp\left(-\textup{$\mathfrak{f}$}rac{x^2}{2t}\right). $$ For $x,y>0$, the transition density of Brownian motion killed upon hitting $0$ is given by $$ p_t^\ast(x,y) = p_t(y-x)-p_t(y+x). $$ The density of the first hitting time of level $x>0$ of Brownian motion started at $0$ is $$ g_t(x) = \textup{$\mathfrak{f}$}rac{x}{t}p_t(x). $$ The transition density of a three-dimensional Bessel process takes the form \begin{equation} \textup{$\mathfrak{l}$}bel{eq:kernel-bessel} r_t(x,y)=\left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }}l} 2yg_t(y)& x=0\\ x^{-1}p_t^{\ast}(x,y)y& x,y>0 \end{array}\right.. \end{equation} In~\cite[Theorem 1]{PiRo}, Pitman and Rogers show that the Pitman transform of a one-dimensional Brownian motion with drift $-\theta$ has the law of the radial part of a three-dimensional Brownian motion with a drift of magnitude $\theta$. In particular, if $\theta=0$, it has the law of a three-dimensional Bessel process, and for all $\theta \geq 0$, it is a transient process. In~\cite[Theorem 3]{PiRo}, it is moreover shown that its transition density is given by \begin{equation} \textup{$\mathfrak{l}$}bel{eq:kernel-qttheta} q_t^{(\theta)}(x,y)= \exp\left(-(t/2)\theta^2\right)h^{-1}(x\theta)r_t(x,y)h(y\theta), \end{equation} where $$ h(x)=\left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }}l} x^{-1}\sinh x& x>0\\ 1& x=0 \end{array}\right.. $$ Recall that $\sigma(\cdot)$ satisfies $\lim_{T\rightarrow\infty}\sigma(T)/T=\theta\in[0,\infty)$, $\liminf_{T\rightarrow\infty}\sigma(T)/\sqrt{T}>0$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:abs-cont-F} In the notation from above, the law of $$ \left((F_t)_{0\leq t\leq \alpha},(F_{T-t})_{0\leq t\leq \beta}\right) $$ is absolutely continuous with respect to the law of $$ \left((B_t)_{0\leq t\leq \alpha},(\mathbb Pi_t-\sigma(T))_{0\leq t\leq \beta}\right), $$ with density given by the function $$\varphi_{T,\alpha,\beta}(\omega,\omega') = \textup{$\mathfrak{m}$}athbbm{1}_{\{\omega_s > -\sigma(T)\textup{ for }s\in[0,\alpha]\}}(\omega)\,\,\textup{$\mathfrak{f}$}rac{p^\ast_{T-(\alpha+\beta)}(\omega_\alpha +\sigma(T), \omega'_\beta+\sigma(T))}{2(\omega'_\beta+\sigma(T))g_T(\sigma(T))}\,\textup{$\mathfrak{f}$}rac{\exp\left(\omega_\alpha\theta+\textup{$\mathfrak{f}$}rac{\alpha+\beta}{2}\theta^2\right)}{h((\omega'_\beta+\sigma(T))\theta)}. $$ Moreover, with $\mathbb P_{\alpha,\beta}$ denoting the joint (product) law of $((B_t)_{0\leq t\leq \alpha},(\mathbb Pi_t)_{0\leq t\leq \beta})$, the following holds true: For each $\varepsilon >0$, there exists $T_0>0$ and a measurable set $E=E(\varepsilon,T_0)\subset \textup{$\mathfrak{m}$}athcal{C}([0,\alpha],\textup{$\mathfrak{m}$}athbb{R})\times \textup{$\mathfrak{m}$}athcal{C}([0,\beta],\textup{$\mathfrak{m}$}athbb{R})$ with $\mathbb P_{\alpha,\beta}(E)\geq 1-\varepsilon$ such that for $T\geq T_0$, $$ \sup_{(\omega,\omega'+\sigma(T))\in E}\left|\varphi_{T,\alpha,\beta}(\omega,\omega')-1\right| \leq \varepsilon. $$ \end{lemma} Note that $\varphi_{T,\alpha,\beta}$ depends on the second coordinate $\omega'$ only through its endpoint $\omega'_\beta$. \begin{proof} We show that the finite-dimensional distributions of $F$ agree with those of $$\varphi_{T,\alpha,\beta}(\omega,\omega')\mathbb P_{\alpha,\beta}(\textup{d}\omega,\textup{d}\omega').$$ Note that the law of the first-passage Brownian bridge $F$ is specified by $F_T=-\sigma(T)$ and \begin{equation} \textup{$\mathfrak{l}$}bel{eq:fpb} \textup{$\mathfrak{m}$}athbb E\left[f\left((F_t)_{0\leq t\leq T'}\right)\right] = \textup{$\mathfrak{m}$}athbb E\left[f\left((\tilde{\gamma})_{0\leq t\leq T'}\right)\textup{$\mathfrak{m}$}athbbm{1}_{\{\underline{\tilde{\gamma}}_{T'}>-\sigma(T)\}}\textup{$\mathfrak{f}$}rac{g_{T-T'}(\tilde{\gamma}_{T'}+\sigma(T))}{g_{T}(\sigma(T))}\right] \end{equation} for all $0\leq T'<T$ and all functions $f\in \textup{$\mathfrak{m}$}athcal{C}([0,T'],\textup{$\mathfrak{m}$}athbb{R})$, where $\tilde{\gamma}$ is a standard one-dimensional Brownian motion started from zero (without drift). Let us next simplify notation. For $x\in\textup{$\mathfrak{m}$}athbb{R}$, write $\tilde{x}=x+\sigma(T)$. For $0<t_1<t_2<\mathrm{d}ots<t_p$ and $x_1,\mathrm{d}ots,x_p>-\sigma(T)$, let $$ G_{t_1,\mathrm{d}ots,t_p}(x_1,\mathrm{d}ots,x_p)=p_{t_1}^\ast(\sigma(T),\tilde{x}_1)p_{t_2-t_1}^\ast(\tilde{x}_1,\tilde{x}_2)\mathrm{d}ots p_{t_p-t_{p-1}}^\ast(\tilde{x}_{p-1},\tilde{x}_p). $$ For $0<t'_1<t'_2<\mathrm{d}ots<t'_q$ and $x_{p+1},\mathrm{d}ots,x_{p+q}>-\sigma(T)$, let $$ H_{t'_1,\mathrm{d}ots,t'_q}(x_{p+q},\mathrm{d}ots,x_{p+1})=g_{t'_1}(\tilde{x}_{p+q})p_{t'_2-t'_1}^\ast(\tilde{x}_{p+q},\tilde{x}_{p+q-1})\mathrm{d}ots p_{t'_q-t'_{q-1}}^\ast(\tilde{x}_{p+2},\tilde{x}_{p+1}). $$ Now fix $0<t_1<t_2<\mathrm{d}ots<t_p=\alpha$ and $0<t'_1<t'_2<\mathrm{d}ots<t'_q=\beta$. We infer from~\eqref{eq:fpb} that the density of the $(p+q)$-tuple $(F_{t_1},\mathrm{d}ots,F_{t_p},F_{T-t'_q},\mathrm{d}ots,F_{T-t'_1})$ is given by the function \begin{equation} \textup{$\mathfrak{l}$}bel{eq:abs-cont-F-density} f_{t_1,\mathrm{d}ots,t_p,t'_1,\mathrm{d}ots,t'_q}(x_1,\mathrm{d}ots,x_{p+q})= G_{t_1,\mathrm{d}ots,t_p}(x_1,\mathrm{d}ots,x_p)H_{t'_1,\mathrm{d}ots,t'_q}(x_{p+q},\mathrm{d}ots,x_{p+1})\cdot \textup{$\mathfrak{f}$}rac{p^\ast_{T-(\alpha+\beta)}(\tilde{x}_p,\tilde{x}_{p+1})}{g_T(\sigma(T))}. \end{equation} From Girsanov's theorem, we know that the finite-dimensional laws $(B_{t_1},\mathrm{d}ots,B_{t_p})$ of a one-dimensional Brownian motion $B$ with drift $-\theta$ are absolutely continuous with respect to those of a standard Brownian motion $\gamma$ without drift, with a density given by $$ \exp\left(-\theta \gamma_{t_p}-\alpha\theta^2/2\right). $$ Next, we see from~\eqref{eq:kernel-qttheta} that the finite-dimensional laws of $(\mathbb Pi_{t_1'}-\sigma(T),\mathrm{d}ots, \mathbb Pi_{t_q'}-\sigma(T))$ have density $$ \pi_{t'_1,\mathrm{d}ots,t'_q}(x_{p+q},\mathrm{d}ots,x_{p+1})= 2\tilde{x}_{p+1}\exp\left(-(\beta/2)\theta^2\right)h(\tilde{x}_{p+1}\theta)H_{t'_1,\mathrm{d}ots,t'_q}(x_{p+q},\mathrm{d}ots,x_{p+1}), $$ for $x_{p+q},\mathrm{d}ots,x_{p+1}>-\sigma(T)$. By~\eqref{eq:abs-cont-F-density} and the last two observations, the first claim of the statement follows. For the second, for every $\mathrm{d}elta>0$, by continuity of $B$ and $\mathbb Pi$, we can find a constant $K=K(\mathrm{d}elta,\alpha,\beta)>0$ such that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:Aalphabeta} \mathbb P_{\alpha,\beta}\left(\textup{$\mathfrak{m}$}in_{[0,\alpha]}B>-K,\,\textup{$\mathfrak{m}$}ax_{[0,\beta]}\mathbb Pi<K\right)\geq 1-\mathrm{d}elta. \end{equation} The second claim now follows from~\eqref{eq:Aalphabeta} and the fact that for every $\mathrm{d}elta'>0$, if $T$ is large enough, we have \begin{equation} \textup{$\mathfrak{l}$}bel{eq:density1} \left|\textup{$\mathfrak{f}$}rac{p^\ast_{T-(\alpha+\beta)}(x+\sigma(T), y+\sigma(T))}{2(y+\sigma(T))g_T(\sigma(T))}\,\textup{$\mathfrak{f}$}rac{\exp\left(x\theta+\textup{$\mathfrak{f}$}rac{\alpha+\beta}{2}\theta^2\right)}{h((y+\sigma(T))\theta)} -1\right|\leq \mathrm{d}elta' \end{equation} uniformly in $x\in\textup{$\mathfrak{m}$}athbb{R}$ with $|x| \leq K$ and $y\geq -\sigma(T)$ with $|y+\sigma(T)|\leq K$. The last display in turn follows from a straightforward but somewhat tedious calculation; we give some indication for the case $\lim_{T\rightarrow\infty}\sigma(T)/T=\theta>0$. First, as $T\rightarrow\infty$, \begin{align*} \lefteqn{\textup{$\mathfrak{f}$}rac{p^\ast_{T-(\alpha+\beta)}(x+\sigma(T), y+\sigma(T))}{2(y+\sigma(T))g_T(\sigma(T))}\,\textup{$\mathfrak{f}$}rac{\exp\left(x\theta+\textup{$\mathfrak{f}$}rac{\alpha+\beta}{2}\theta^2\right)}{h((y+\sigma(T))\theta)}}\\ &\sim\left(\textup{$\mathfrak{f}$}rac{\exp\left(-\textup{$\mathfrak{f}$}rac{(y-x)^2}{2(T-(\alpha+\beta))}\right)-\exp\left(-\textup{$\mathfrak{f}$}rac{(x+y+2\sigma(T))^2}{2(T-(\alpha+\beta))}\right)}{\exp\left(-\theta^2T/2\right)}\right)\textup{$\mathfrak{f}$}rac{\exp\left(x\theta+\textup{$\mathfrak{f}$}rac{\alpha+\beta}{2}\theta^2\right)}{2\sinh\left(\theta(y+\sigma(T))\right)}. \end{align*} Then, uniformly in $x$ and $y$ as specified above, we find $$ \exp\left(-\textup{$\mathfrak{f}$}rac{(y-x)^2}{2(T-(\alpha+\beta))}+\theta^2T/2\right)\sim \exp\left((-x+y+\sigma(T))\theta-(\alpha+\beta)\theta^2/2\right), $$ and $$ \exp\left(-\textup{$\mathfrak{f}$}rac{(x+y+2\sigma(T))^2}{2(T-(\alpha+\beta))}+\theta^2T/2\right) \sim \exp\left((-x-y-\sigma(T))\theta-(\alpha+\beta)\theta^2/2\right). $$ Putting these three estimates together,~\eqref{eq:density1} follows. The case $\lim_{T\rightarrow\infty}\sigma(T)/T=0$ with $\liminf_{T\rightarrow\infty}\sigma(T)/\sqrt{T}>0$ is similar but easier (note that the expression for $\varphi_{T,\alpha,\beta}$ simplifies when $\theta=0$). \end{proof} We need a similar absolute continuity property for the Brownian bridge $b$ on $[0,\sigma(T)]$ from $0$ to $0$ with respect to two independent linear Brownian motions $\gamma$ and $\gamma'$ scaled by $\sqrt{3}$. Let $\alpha,\beta>0$ such that $\alpha+\beta<\sigma(T)$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:abs-cont-b} The law of $$ \left((b_t)_{0\leq t\leq \alpha},(b_{L-t})_{0\leq t\leq \beta}\right) $$ is absolutely continuous with respect to the law of $$ \left((\gamma_t)_{0\leq t\leq \alpha},(\gamma'_{t})_{0\leq t\leq \beta}\right), $$ with density given by the function $$\tilde{\varphi}_{T,\alpha,\beta}(\omega,\omega') = \textup{$\mathfrak{f}$}rac{p_{\sigma(T)-(\alpha+\beta)}(\omega'_\beta-\omega_\alpha)}{p_{\sigma(T)}(0)}. $$ Moreover, with $\mathbb P_{\alpha,\beta}$ denoting the joint law of $((\gamma_t)_{0\leq t\leq \alpha},(\gamma'_{t})_{0\leq t\leq \beta})$, the following holds true: For each $\varepsilon >0$, there is $T_0>0$ and a measurable set $E=E(\varepsilon,T_0)\subset \textup{$\mathfrak{m}$}athcal{C}([0,\alpha],\textup{$\mathfrak{m}$}athbb{R})\times \textup{$\mathfrak{m}$}athcal{C}([0,\beta],\textup{$\mathfrak{m}$}athbb{R})$ with $\mathbb P_{\alpha,\beta}(E)\geq 1-\varepsilon$ such that for $T\geq T_0$, $$ \sup_{(\omega,\omega')\in E}\left|\tilde{\varphi}_{T,\alpha,\beta}(\omega,\omega')-1\right| \leq \varepsilon. $$ \end{lemma} \begin{proof} The first part is immediate from the fact that the law of the Brownian bridge $b$ is specified by $b_{\sigma(T)}=0$ and \begin{equation} \textup{$\mathfrak{m}$}athbb E\left[f\left((b_t)_{0\leq t\leq T'}\right)\right] = \textup{$\mathfrak{m}$}athbb E\left[f\left((\gamma_t)_{0\leq t\leq T'}\right)\textup{$\mathfrak{f}$}rac{p_{\sigma(T)-T'}(\gamma_{T'})}{p_{\sigma(T)}(0)}\right] \end{equation} for all $0\leq T'<\sigma(T)$ and all $f\in \textup{$\mathfrak{m}$}athcal{C}([0,T'],\textup{$\mathfrak{m}$}athbb{R})$. The proof of the second part is very similar to that of Lemma~\ref{lem:abs-cont-F}. We omit the details. \end{proof} \subsubsection{Isometry of balls in {\normalfont$\ \mathcal BD_{T,\sigma(T)}$} and {\normalfont $\mathcal BHP_\theta$}} Recall the definition of $\sigma(T)$ and $\theta$ from the statement of the proposition. In the proof that follows, we will choose $T>0$ sufficiently large. We work with the following processes: \begin{itemize} \item $F$ a first-passage Brownian bridge on $[0,T]$ from $0$ to $-\sigma(T)$; \item $b$ a Brownian bridge on $[0,\sigma(T)]$ from $0$ to $0$, multiplied by $\sqrt{3}$, independent of $F$; \item $B$ a Brownian motion on $[0,\infty)$ with drift $-\theta$, started from $B_0=0$; \item $\mathbb Pi$ the Pitman transform of an independent copy of $B$; \item $\gamma$ a two-sided Brownian motion on $\textup{$\mathfrak{m}$}athbb{R}$ with $\gamma_0=0$, scaled by the factor $\sqrt{3}$, independent of $(B,\mathbb Pi)$; \item $Z,W$ and $\textup{$\mathfrak{m}$}athbb Zha,W^{\theta}$ the random processes associated with $F,b$ and $B,\mathbb Pi,\gamma$ as described in Section~\ref{sec:recapBHPBD}. \end{itemize} \begin{proof}[Proof of Proposition~\ref{prop:isometry-BD-BHP}] For $x\in\textup{$\mathfrak{m}$}athbb{R}$, let $$\eta_{\textup{l}}(x)=\inf\{t\geq 0:B_t\leq -x\},\textup{$\mathfrak{q}$}uad\eta_{\textup{r}}(x)=\sup\{t\geq 0:\mathbb Pi_t=x\}.$$ We fix $\varepsilon>0$ and $r\geq 0$ and first introduce some auxiliary events. For $A>0$, define \begin{equation*} \textup{$\mathfrak{m}$}athcal{E}^1(A)=\left\{\begin{split}&\textup{$\mathfrak{m}$}in_{[0,A]}\gamma < -6r,\,\textup{$\mathfrak{m}$}in_{[A,A^2]}\gamma < -6r,\,\textup{$\mathfrak{m}$}in_{[A^2,A^3]}\gamma < -6r,\\ &\textup{$\mathfrak{m}$}in_{[-A,0]}\gamma < -6r,\,\textup{$\mathfrak{m}$}in_{[-A^2,-A]}\gamma < -6r,\,\textup{$\mathfrak{m}$}in_{[-A^3,-A^2]}\gamma < -6r\end{split}\right\}. \end{equation*} Next, for $u_0>0$, $A>0$, let $$\textup{$\mathfrak{m}$}athcal{E}^2(A,u_0)=\left\{\eta_{\textup{l}}(A^3)\leq u_0\right\},\textup{$\mathfrak{q}$}uad \textup{$\mathfrak{m}$}athcal{E}^3(A,u_0)=\left\{\eta_{\textup{r}}(A^3)\leq u_0\right\}. $$ For $u_2\geq u_1>0$, let $$ \textup{$\mathfrak{m}$}athcal{E}^4(u_1,u_2)=\left\{\inf_{[u_2,\infty)}\mathbb Pi>\textup{$\mathfrak{m}$}in_{[u_1,u_2]}\mathbb Pi\right\}. $$ For $u_4\geq u_3>0$ and $T\geq u_4$, let $$ \textup{$\mathfrak{m}$}athcal{E}^5(u_3,u_4,T)=\left\{\textup{$\mathfrak{m}$}in_{[0,T-u_4]}F>\textup{$\mathfrak{m}$}in_{[T-u_4,T-u_3]}F\right\}. $$ Standard properties of Brownian motion imply that there exist $A>0$ such that $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^1)\geq 1-\varepsilon/10$, and we fix $A$ accordingly. Then, we can find $u_0>0$ such that $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^2)\geq 1-\varepsilon/10$ and $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^3)\geq 1-\varepsilon/10$, due to the fact that $\mathbb Pi$ is transient. Then, we can find $u_1$ and $u_2$ with $u_2\geq u_1\geq u_0$ such that $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^4)\geq 1-\varepsilon/10$. At last,we claim that we can find $u_4$, $u_3$ satisfying $u_4\geq u_3\geq u_2$ and $T'_0$ with $T'_0\geq 2u_4$ such that for $T\geq T'_0$, $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^5)\geq 1-\varepsilon/10$. To see this, one can use the fact that $F+\sigma(T)$ is a bridge of a three-dimensional Bessel process from $\sigma(T)$ to $0$ with duration $T$, see, e.g.,~\cite{BeChPi}. Its time-reversal $\tilde{F}=F(T-\cdot)+\sigma(T)$ is then a Bessel bridge from $0$ to $\sigma(T)$ with duration $T$, see~\cite[Chapter XI $\S 3$]{ReYo}. Write $\tilde{F}^1$ for a Bessel bridge from $0$ to $\sigma(T)/\sqrt{T}$ with duration $1$. Using this representation we have \begin{align} \textup{$\mathfrak{l}$}bel{eq:E5} 1-\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^5(u_3,u_4,T))&=\mathbb P\left(\textup{$\mathfrak{m}$}in_{[u_3,u_4]}\tilde{F}\geq\textup{$\mathfrak{m}$}in_{[u_4,T]}\tilde{F}\right) = \mathbb P\left(\textup{$\mathfrak{m}$}in_{[u_3/T,u_4/T]}\tilde{F}^1\geq\textup{$\mathfrak{m}$}in_{[u_4/T,1]}\tilde{F}^1\right)\nonumber\\ &=\mathbb P\left(\textup{$\mathfrak{m}$}in_{[u_3/T,u_4/T]}\tilde{F}^1\geq\textup{$\mathfrak{m}$}in_{[u_4/T,1/2]}\tilde{F}^1\right)+o(1)\end{align} as $T\rightarrow\infty$, where we used scaling at the second step and for the last equality that $u_4/T\rightarrow 0$ for a fixed $u_4$ as $T\to\infty$. Note that the law of $(\tilde{F}^1_{t_1},\ldots,\tilde{F}^1_{t_k})$ for $0<t_1<\mathrm{d}ots<t_k<1$ has density $$r_{t_1}(0,x_1)r_{t_2-t_1}(x_1,x_2)\mathrm{d}ots r_{t-t_k}\left(x_k,\sigma(T)/\sqrt{T}\right)\mathcal Big/r_1\left(0,\sigma(T)/\sqrt{T}\right),$$ see again~\cite[Chapter XI $\S 3$]{ReYo}. For a moment, let us denote by $\tilde{\mathbb Pi}^1$ the Pitman transform of a Brownian motion with drift $-\sigma(T)/\sqrt{T}$. Its transition kernel is given by $q_t^{(\sigma(T)/\sqrt{T})}$, see~\eqref{eq:kernel-qttheta} above. A small calculation involving the last display and the explicit form of $q_t^{(\sigma(T)/\sqrt{T})}$ shows that the Bessel bridge $\tilde{F}^1$ restricted to $[0,1/2]$ is absolutely continuous with respect to $\tilde{\mathbb Pi}^1$, with a density that can be written as $\mathbb Xi(\tilde{\mathbb Pi}_{1/2}^1)$ for some measurable and bounded function $\mathbb Xi$. The probability on the right hand side in~\eqref{eq:E5} is therefore bounded from above by \begin{align*} \|\mathbb Xi\|_\infty\mathbb P\left(\textup{$\mathfrak{m}$}in_{[u_3/T,u_4/T]}\tilde{\mathbb Pi}^1\geq\textup{$\mathfrak{m}$}in_{[u_4/T,1/2]}\tilde{\mathbb Pi}^1\right)+o(1) &=\|\mathbb Xi\|_\infty\mathbb P\left(\textup{$\mathfrak{m}$}in_{[u_3,u_4]}\mathbb Pi\geq\textup{$\mathfrak{m}$}in_{[u_4,T/2]}\mathbb Pi\right)+o(1)\\ &=\|\mathbb Xi\|_\infty\mathbb P\left(\textup{$\mathfrak{m}$}in_{[u_3,u_4]}\mathbb Pi\geq\inf_{[u_4,\infty)}\mathbb Pi\right)+o(1), \end{align*} where for the first equality, we used scaling again and the fact that $\sigma(T)/T\rightarrow\theta$, and for the second equality transience of $\mathbb Pi$. Identically to the event $\textup{$\mathfrak{m}$}athcal{E}^4$, transience of $\mathbb Pi$ implies that the last probability on the right can be made as small as we wish if we choose $u_4$ large enough. This shows that for each choice of $u_3\geq u_2$, we find $u_4\geq u_3$ and $T'_0$ such that $\textup{$\mathfrak{m}$}athcal{E}^5(u_3,u_4,T)\geq 1-\varepsilon$ for all $T\geq T'_0$. We now fix numbers $A,u_4\geq u_3\geq\mathrm{d}ots\geq u_0$ and $T'_0$ as discussed above. By Lemmas~\ref{lem:abs-cont-F} and~\ref{lem:abs-cont-b}, we deduce that we can find $T_0>T'_0$ such that every for $T\geq T_0$, the processes $F,b,B,\mathbb Pi,\gamma$ can be coupled on the same probability space such that the event $$ \textup{$\mathfrak{m}$}athcal{E}^6(T)=\left\{ \begin{split}& F_t=B_t,\,F_{T-t}=\mathbb Pi_t-\sigma(T) &\hbox{ for }t\in[0,u_4]\\ &b_x = \gamma_x,\,b_{L-x}=\gamma_{-x} &\hbox{ for }x\in [0,A^3] \end{split}\right\}. $$ has probability at least $1-\varepsilon/2$, $F$ is independent of $b$, $B$ is independent of $\mathbb Pi$, and $\gamma$ is independent of $(B,\mathbb Pi)$. Now fix $T\geq T_0$. We assume that $F,b,B,\mathbb Pi,\gamma$ have been coupled as above. Recall that the snake $(Z_t,0\leq t\leq T)$ and the label function $(W_t,0\leq t\leq T)$ of the Brownian disk $\mathcal BD_{T,\sigma(T)}$ are defined in terms of $F$ and $b$, see Section~\ref{sec:recapBHPBD}. We put $$ \textup{$\mathfrak{m}$}athbb{W}_t=\left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }}l} W_t& t\in[0,u_1]\\ W_{T+t} & t\in[-u_1,0] \end{array}\right.. $$ Given $F$, the process $(\textup{$\mathfrak{m}$}athbb{W}_t)_{t\in[-u_1,u_1]}$ is Gaussian; moreover, if we restrict ourselves to the event $\textup{$\mathfrak{m}$}athcal{E}^5$, we have $${\underline F}_{T-t}=\textup{$\mathfrak{m}$}in_{[T-u_4,T-t]}F\textup{$\mathfrak{q}$}uad\hbox{for }t\in [0,u_1].$$ Hence the covariance of $(\textup{$\mathfrak{m}$}athbb{W}_t)_{t\in[-u_1,u_1]}$ is on $\textup{$\mathfrak{m}$}athcal{E}^5$ a function of the tuple \begin{equation} \textup{$\mathfrak{l}$}bel{eq:tuple1} \left((F_t)_{0\leq t\leq u_4},(F_{T-t})_{0\leq t\leq u_4}\right). \end{equation} We turn to the Brownian half-plane and its label function $W^{\theta}=(W^{\theta}_t,t\in\textup{$\mathfrak{m}$}athbb{R})$, which we define in terms of $B$, $\mathbb Pi$ and $\gamma$, see again Section~\ref{sec:recapBHPBD}. Conditionally on $(B, \mathbb Pi)$, $W^{\theta}$ is a Gaussian process. Moreover, since on the event $\textup{$\mathfrak{m}$}athcal{E}^4$, $$ \inf_{[t,\infty)}\mathbb Pi = \textup{$\mathfrak{m}$}in_{[t,u_4]}\mathbb Pi\textup{$\mathfrak{q}$}uad\hbox{for }t\in[0,u_1], $$ the covariance of the restriction of $W^{\theta}$ to $[-u_1,u_1]$ is on $\textup{$\mathfrak{m}$}athcal{E}^4$ given by exactly the same function of the tuple $((B_t)_{0\leq t\leq u_4},(\mathbb Pi_t)_{0\leq t\leq u_4})$ as the covariance of $(\textup{$\mathfrak{m}$}athbb{W}_t)_{t\in[-u_1,u_1]}$. Since a shift of $\mathbb Pi$ does not affect the covariance, we can instead consider the tuple \begin{equation} \textup{$\mathfrak{l}$}bel{eq:tuple2} \left((B_t)_{0\leq t\leq u_4},(\mathbb Pi_t-\sigma(T))_{0\leq t\leq u_4}\right). \end{equation} On the event $\textup{$\mathfrak{m}$}athcal{E}^5$, both tuples of processes~\eqref{eq:tuple1} and~\eqref{eq:tuple2} coincide. On the event $\textup{$\mathfrak{m}$}athcal{E}^4\cap\textup{$\mathfrak{m}$}athcal{E}^5\cap\textup{$\mathfrak{m}$}athcal{E}^6$, we can therefore construct $W$ and $W^{\theta}$ such that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:idWWla} W_t = W^{\theta}_t,\textup{$\mathfrak{q}$}uad W_{T-t} = W^{\theta}_{-t}\textup{$\mathfrak{q}$}uad\hbox{ for all }t\in[0,u_1],\textup{$\mathfrak{q}$}uad \end{equation} We shall now work on the event $\textup{$\mathfrak{m}$}athcal{F}=\textup{$\mathfrak{m}$}athcal{E}^1\cap\textup{$\mathfrak{m}$}athcal{E}^2\cap\textup{$\mathfrak{m}$}athcal{E}^3\cap\textup{$\mathfrak{m}$}athcal{E}^4\cap\textup{$\mathfrak{m}$}athcal{E}^5\cap\textup{$\mathfrak{m}$}athcal{E}^6$, which has probability at least $1-\varepsilon$, and assume that the identity~\eqref{eq:idWWla} holds true. We follow a strategy similar to~\cite{CuLG}. Let $s,t\in [0,T]$. If either $s,t\in[0,T/2]$, or $s,t\in[T/2,T]$, we let $$ \tilde{d}_{W}(s,t) = W_s +W_t -2\textup{$\mathfrak{m}$}in_{[s\wedge t, s\vee t]}W. $$ Otherwise, we set $$ \tilde{d}_{W}(s,t) = W_s +W_t -2\textup{$\mathfrak{m}$}in_{[0,s\wedge t]\cup[s\vee t,T]}W. $$ We shall need the continuous analog of the cactus bound~\eqref{eq:cactus1} for the pseudo-metric $D$ belonging to the Brownian disk $\mathcal BD_{T,\sigma(T)}$. It reads \begin{equation} \textup{$\mathfrak{l}$}bel{eq:cactusDBD} D(s,t) \geq W_s+W_t-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[s\wedge t,s\vee t]}W,\textup{$\mathfrak{m}$}in_{[0,s\wedge t]\cup[s\vee t,T]}W\right\},\textup{$\mathfrak{q}$}uad{s,t\in[0,T]}. \end{equation} See, for example,~\cite{CuLG} for a proof of the corresponding bound in the context of the Brownian map, which can easily be adapted to the Brownian disk. In the notation from above, we have the following \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:DBD} Assume $\textup{$\mathfrak{m}$}athcal{F}$ holds. \begin{enumerate} \item For every $t\in [\eta_{\textup{l}}(A),T-\eta_{\textup{r}}(A)]$, $D(0,t) > r$. \item For every $s,t\in [0,\eta_{\textup{l}}(A)]\cup[0,T-\eta_{\textup{r}}(A)]$ with $\textup{$\mathfrak{m}$}ax\{D(0,s), D(0,t)\}\leq r$, it holds that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:DBD-rhs} D(s,t) =\inf_{s_1,t_1,\mathrm{d}ots,s_k,t_k}\sum_{i=1}^k \tilde{d}_{W}(s_i,t_i), \end{equation} where the infimum is over all possible choices of $k\in\textup{$\mathfrak{m}$}athbb N$ and reals $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in [0,\eta_{\textup{l}}(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$ such that $s_1=s,t_k=t$, and $d_{F}(t_i,s_{i+1})=0$ for $1\leq i\leq k-1$. \end{enumerate} \end{lemma} \begin{proof} (a) If $t\in [\eta_{\textup{l}}(A),T-\eta_{\textup{r}}(A)]$, then by the cactus bound~\eqref{eq:cactusDBD} in the first inequality, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:bound-DBD1} D(0,t)\geq W_t-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,t]}W,\textup{$\mathfrak{m}$}in_{[t,T]}W\right\}\geq -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,\eta_{\textup{l}}(A)]}W,\textup{$\mathfrak{m}$}in_{[T-\eta_{\textup{r}}(A),T]}W\right\}. \end{equation} Let us show how to bound the first minimum on the right hand side. On the event $\textup{$\mathfrak{m}$}athcal{E}^6$, $b_x=\gamma_x$ for $x\in[0,A^3]$ and $F_t=B_t$ for $t\in[0,u_4]$. On the event $\textup{$\mathfrak{m}$}athcal{E}^2$, we know that $\eta_{\textup{l}}(A)$, the first instant when $B$ attains the value $-A$, is bounded from above by $u_0$, which satisfies $u_0 \leq u_1\leq u_4$. Moreover, on $\textup{$\mathfrak{m}$}athcal{E}^1$, $\textup{$\mathfrak{m}$}in_{[0,A]}\gamma < -6r$. Going back to the definition of $W$ (and using the fact that $Z_t$ equals zero if $\underline{F}$ attains a new minimum at $t$), we obtain that the first minimum on the right hand side is bounded from above by $-6r$. For the second minimum on the right of~\eqref{eq:bound-DBD1}, we first observe that on the event $\textup{$\mathfrak{m}$}athcal{E}^6$, we have also $b_{L-x}=\gamma_{-x}$ for $x\in[0,A^3]$ and $F_{T-t}=\mathbb Pi_t-\sigma(T)$ for $t\in [0,u_4]$. Now on $\textup{$\mathfrak{m}$}athcal{E}^5\cap\textup{$\mathfrak{m}$}athcal{E}^6$, we have $$\underline{F}_{T-t}=\textup{$\mathfrak{m}$}in_{[T-u_4,T-t]}F=\textup{$\mathfrak{m}$}in_{[t,u_4]}(\mathbb Pi-\sigma(T))\textup{$\mathfrak{q}$}uad\hbox{ for } t\in[0,u_1].$$ But on $\textup{$\mathfrak{m}$}athcal{E}^3$, $\eta_{\textup{r}}(A)\leq u_0\leq u_1$, so that in particular $$ \underline{F}_{T-\eta_{\textup{r}}(A)} =\textup{$\mathfrak{m}$}in_{[\eta_{\textup{r}}(A),u_4]}(\mathbb Pi-\sigma(T))\geq A-\sigma(T),$$ where for the last inequality we used the fact that $\mathbb Pi_t\geq A$ for $t\geq \eta_{\textup{r}}(A)$. Since on $\textup{$\mathfrak{m}$}athcal{E}^1$, also $\textup{$\mathfrak{m}$}in_{[-A,0]}\gamma < -6r$, the second minimum is bounded above again by $-6r$. This proves $D(0,t)\geq 6r$ whenever $t\in [\eta_{\textup{l}}(A),T-\eta_{\textup{r}}(A)]$, which is more than we claimed.\\ (b) Recall that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:dist-DBD} D(s,t)=\inf\left\{\sum_{i=1}^k d_{W}(s_i,t_i) :\begin{split}& k\geq 1, s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in[0,T],s_1=s, t_k=t,\\ & d_{F}(t_i,s_{i+1})=0\hbox{ for every }i\in\{1,\ldots,k-1\}\end{split}\right\}. \end{equation} Since $D(s,t)\leq D(0,s)+D(0,t)\leq 2r$ for $s,t$ as in the statement, it suffices to look at $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in[0,T]$ with \begin{equation} \textup{$\mathfrak{l}$}bel{eq:bound-Dcirc} \sum_{i=1}^k d_W(s_i,t_i) \leq 3r. \end{equation} We now argue that on the right hand side of~\eqref{eq:dist-DBD}, we can restrict ourselves to reals $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in[0,\eta_\ell(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T].$ Suppose that there is $i\in\{1,\ldots,k\}$ such that $t_i$ is not included in $[0,\eta_\ell(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$. Note that from the cactus bound and the fact that $W_0=W_T=0$, we have $|W_s|\leq r$ whenever $D(0,s)\leq r$. Therefore, the cactus bound gives $$ D(s,t_i) \geq -r -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[s\wedge {t_i},s\vee t_i]}W,\textup{$\mathfrak{m}$}in_{[0,s\wedge t_i]\cup[s\vee t_i,T]}W\right\}. $$ Recall that by assumption $s \in [0,\eta_\ell(A)]\cup[T-\eta_{\textup{r}}(A),T]$. If $t_i\in[0,T]\setminus([0,\eta_\ell(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T])$, then both minima on the right hand side are taken over subsets which include either $[\eta_\ell(A),\eta_\ell(A^2)]$ or $[T-\eta_{\textup{r}}(A^2),T-\eta_{\textup{r}}(A)]$. Therefore, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:bound-DBD2} D(s,t_i) \geq -r - \textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[\eta_\ell(A),\eta_\ell(A^2)]}W,\textup{$\mathfrak{m}$}in_{[T-\eta_{\textup{r}}(A^2),T-\eta_{\textup{r}}(A)]}W\right\}. \end{equation} We can now argue similarly to (a) to show that both minima are bounded from above by $-6r$. Concerning the first minimum, we know on $\textup{$\mathfrak{m}$}athcal{E}^6$ that $b_x=\gamma_x$ for $x\in[0,A^3]$ and $F_t=B_t$ for $t\in[0,u_4]$. On $\textup{$\mathfrak{m}$}athcal{E}^2$, we have $\eta_{\textup{l}}(A^2)\leq u_0$. Since $[-A^2,-A]\subset B([\eta_{\textup{l}}(A),\eta_{\textup{l}}(A^2)]$, the bound on the first minimum follows from the fact that on $\textup{$\mathfrak{m}$}athcal{E}^1$, $\textup{$\mathfrak{m}$}in_{[A,A^2]}\gamma < -6r$. The second minimum is treated similarly and left to the reader. With these bounds, we obtain $D(s,t_i) \geq 5r$. On the other hand, we know from~\eqref{eq:bound-Dcirc} that $D(s,t_i) \leq 3r$, a contradiction. The case where $s_i$ is not included in $[0,\eta_\ell(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$ for some $i\in\{1,\ldots,k\}$ is analogous. Therefore, we can restrict ourselves in~\eqref{eq:dist-DBD} to reals $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in[0,\eta_\ell(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$. We still have to show that we can replace $d_{W}$ in~\eqref{eq:dist-DBD} by $\tilde{d}_{W}$. Let $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in[0,\eta_\ell(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$ with $s_1=s$, $t_k=t$ and such that~\eqref{eq:bound-Dcirc} holds. Assume first that there is $i\in\{1,\ldots,k\}$ such that $s_i\in[0,\eta_\ell(A^2)]$ and $t_i\in[T-\eta_{\textup{r}}(A^2),T]$, and let us show that then $d_{W}(s_i,t_i)=\tilde{d}_{W}(s_i,t_i)$. First, by~\eqref{eq:bound-Dcirc} in the first inequality, $$ 3r\geq d_{W}(s,s_i)\geq W_s-W_{s_i}. $$ Since $W_s\geq -r$, this shows $W_{s_i}\geq -4r$, and identically one obtains $W_{t_i}\geq -4r$. Using again~\eqref{eq:bound-Dcirc}, \begin{align*} 3r\geq d_{W}(s_i,t_i)&=W_{s_i} +W_{t_i} - 2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[s_i,t_i]} W,\,\textup{$\mathfrak{m}$}in_{[0,s_i]\cup[t_i,T]}W\right\}\\ &\geq -8r-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[s_i,t_i]} W,\,\textup{$\mathfrak{m}$}in_{[0,s_i]\cup[t_i,T]}W\right\} \end{align*} We claim that this last inequality can only hold if the maximum is attained at the second minimum (which means precisely $d_{W}(s_i,t_i)=\tilde{d}_{W}(s_i,t_i)$). Indeed, if $s_i\in[0,\eta_\ell(A^2)]$ and $t_i\in[T-\eta_{\textup{r}}(A^2),T]$, then $[s_i,t_i]$ contains the interval $[\eta_\ell(A^2),\eta_\ell(A^3)]$. Arguing in the same way as for the first minimum in~\eqref{eq:bound-DBD2}, we deduce that $\textup{$\mathfrak{m}$}in_{[s_i,t_i]}W\leq -6r$, which proves our claim. The case where $t_i\in[0,\eta_\ell(A^2)]$ and $s_i\in[T-\eta_{\textup{r}}(A^2),T]$ is treated by symmetry. Assume now both $s_i,t_i$ lie in $[0,\eta_\ell(A^2)]$. Then the interval $[s_i\vee t_i,T]$ contains the interval $[\eta_\ell(A^2),\eta_\ell(A^3)]$, so that $\textup{$\mathfrak{m}$}in_{[s_i\vee t_i,T]}W\leq -6r$ by the same reasoning, which gives again $d_{W}(s_i,t_i)=\tilde{d}_{W}(s_i,t_i)$. If both $s_i,t_i$ lie in $[T-\eta_{\textup{r}}(A^2),T]$, then $[0,s_i\wedge t_i]$ contains $[T-\eta_{\textup{r}}(A^3),T-\eta_{\textup{r}}(A^2)]$, and the minimum of $W$ over this interval is again bounded from above by $-6r$, using arguments as for the second minimum in~\eqref{eq:bound-DBD1} (or~\eqref{eq:bound-DBD2}). This leads to $d_{W}(s_i,t_i)=\tilde{d}_{W}(s_i,t_i)$ also in this case, which completes the proof of (b). \end{proof} We turn to the analogous statement for the pseudo-distance function $D_{\theta}$ of the Brownian half-plane. Recall the definition of $(X^{\theta},W^{\theta})$ for the case of the Brownian half-plane $\mathcal BHP_\theta$, cf. Section~\ref{sec:recapBHPBD}. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:DH} Assume $\textup{$\mathfrak{m}$}athcal{F}$ holds. \begin{enumerate} \item For every $t'\in (-\infty,-\eta_{\textup{r}}(A)]\cup[\eta_{\textup{l}}(A),\infty)$, $D_{\theta}(0,t') > r$. \item For every $s',t'\in [-\eta_{\textup{r}}(A),\eta_{\textup{l}}(A)]$ with $\textup{$\mathfrak{m}$}ax\{D_{\theta}(0,s'), D_{\theta}(0,t')\}\leq r$, it holds that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:DH-rhs} D_{\theta}(s',t') =\inf_{s'_1,t'_1,\mathrm{d}ots,s'_k,t'_k}\sum_{i=1}^k d_{W^{\theta}}(s'_i,t'_i), \end{equation} where the infimum is over all possible choices of $k\in\textup{$\mathfrak{m}$}athbb N$ and reals $s'_1,\mathrm{d}ots,s'_k,t'_1,\mathrm{d}ots,t'_k\in [-\eta_{\textup{r}}(A^2),\eta_{\textup{l}}(A^2)]$ such that $s'_1=s',t'_k=t'$, and $d_{X^{\theta}}(t'_i,s'_{i+1})=0$ for $1\leq i\leq k-1$. \end{enumerate} \end{lemma} \begin{proof} Essentially, one can rely on the identity~\eqref{eq:idWWla} and then follow the proof of Lemma~\ref{lem:DBD}. The starting point is the cactus bound for $D_{\theta}$, which reads $$ D_{\theta}(s',t')\geq W^{\theta}_{s'}+W^{\theta}_{t'}-2\textup{$\mathfrak{m}$}in_{[s'\wedge t',s'\vee t']}W^{\theta},\textup{$\mathfrak{q}$}uad s',t'\in\textup{$\mathfrak{m}$}athbb{R}. $$ We refer again to the proof in~\cite{CuLG}, which can be transferred to this setting. Let us now sketch how to prove (a); the proof of (b) is left to the reader. If $t'\in (-\infty,-\eta_{\textup{r}}(A)]$, the cactus bound gives $$ D_{\theta}(0,t')\geq -\textup{$\mathfrak{m}$}in_{[t',0]}W^{\theta}. $$ The very definitions of $W^{\theta}$ and $\eta_{\textup{r}}(A)$ together with the fact that on $\textup{$\mathfrak{m}$}athcal{E}^1$, $\textup{$\mathfrak{m}$}in_{[-A,0]}\gamma<-6r$, entail that the minimum is bounded from above by $-6r$. The same bound holds if $t'\in [\eta_{\textup{l}}(A),\infty)$, which proves (a). \end{proof} Combining Lemmas~\ref{lem:DBD} and~\ref{lem:DH}, we obtain the following corollary. For the rest of the proof of Proposition~\ref{prop:isometry-BD-BHP}, we put for a point $u\in[0,T]$ $$ I(u)= \left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }}l} u& u\in[0,T/2]\\ u-T& u\in[T/2,T] \end{array}\right.. $$ \begin{corollary} \textup{$\mathfrak{l}$}bel{cor:DBD-DH} Assume $\textup{$\mathfrak{m}$}athcal{F}$ holds. Let $s,t\in [0,\eta_\ell(A)]\cup[T-\eta_{\textup{r}}(A),T]$. Then it holds that $\textup{$\mathfrak{m}$}ax\{D(0,s),D(0,t)\}\leq r$ if and only if $\textup{$\mathfrak{m}$}ax\{D_{\theta}(0,I(s)),D_{\theta}(0,I(t))\}\leq r$. Under these conditions, $$ D(s,t)=D_{\theta}(I(s),I(t)). $$ \end{corollary} \begin{proof} Take $s,t\in [0,\eta_\ell(A)]\cup[T-\eta_{\textup{r}}(A),T]$ with $\textup{$\mathfrak{m}$}ax\{D(0,s),D(0,t)\}\leq r$. We claim that the right hand side of the expression~\eqref{eq:DBD-rhs} for $D(s,t)$ agrees with the right hand side of~\eqref{eq:DH-rhs} for $s'_1=I(s), t'_k=I(t)$. First note that we have $\textup{$\mathfrak{m}$}ax\{\eta_{\textup{l}}(A^2),\eta_{\textup{r}}(A^2)\}\leq T/2$ and thus $u\in[0,\eta_{\textup{l}}(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$ if and only if $I(u)\in[-\eta_{\textup{r}}(A^2),\eta_{\textup{l}}(A^2)]$. Now let $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in[0,\eta_{\textup{l}}(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$ such that $s_1=s$ and $t_k=t$. On $\textup{$\mathfrak{m}$}athcal{F}$, we have $d_{F}(t_i,s_{i+1}) = 0$ if and only if $d_{X^{\theta}}(I(t_i),I(s_{i+1}))=0$, and $D(s_i,t_i)= D_{\theta}(I(s_i),I(t_{i+1}))$ for each $i\in\{1,\ldots,k\}$, which proves our claim. In order to see that the right hand side of~\eqref{eq:DH-rhs} for $s'_1=I(s), t'_k=I(t)$ agrees with $D_{\theta}(I(s),I(t))$, we still have to show that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:condI} I(s),I(t)\in[-\eta_{\textup{r}}(A),\eta_{\textup{l}}(A)]\textup{$\mathfrak{q}$}uad\hbox{and}\textup{$\mathfrak{q}$}uad \textup{$\mathfrak{m}$}ax\{D_{\theta}(0,I(s)),D_{\theta}(0,I(t))\}\leq r. \end{equation} The first statement is clear since $s,t\in [0,\eta_\ell(A)]\cup[T-\eta_{\textup{r}}(A),T]$. For the second statement, the right hand side of~\eqref{eq:DH-rhs} specialized to $s'_1=I(s), t'_k=0$ yields an upper bound on $D_{\theta}(0,I(s))$, and then the equality of the right hand sides~\eqref{eq:DBD-rhs} and~\eqref{eq:DH-rhs} shows $D_{\theta}(0,I(s))\leq D(0,s)\leq r$. Entirely similar, one sees $D_{\theta}(0,I(t))\leq r$. A symmetry argument shows that~\eqref{eq:condI} implies $\textup{$\mathfrak{m}$}ax\{D(0,s),D(0,t)\}\leq r$. Finally, invoking again the equality of the right hand sides~\eqref{eq:DBD-rhs} and~\eqref{eq:DH-rhs} shows $D(s,t)=D_{\theta}(I(s),I(t))$. \end{proof} We finish now the proof of Proposition~\ref{prop:isometry-BD-BHP} by showing that the balls $B_r(\mathcal BD_{T,\sigma(T)})$ and $B_r(\mathcal BHP_\theta)$ are isometric on the event~$\textup{$\mathfrak{m}$}athcal{F}$. Write $\textup{\textsf{Y}}$ for the pointed metric space $([0,T]/\{D=0\},D,\rho)$, so that $\textup{\textsf{Y}}$ has the law of $\mathcal BD_{T,\sigma(T)}$. By Lemma~\ref{lem:DBD} (a), points in $B_r(\textup{\textsf{Y}})$ are on $\textup{$\mathfrak{m}$}athcal{F}$ equivalence classes of the form $[s]$ for $s\in[-\eta_{\textup{r}}(A),\eta_{\textup{l}}(A)]$. By the last statement of Corollary~\ref{cor:DBD-DH}, we deduce that the map $I$ from above can be viewed as an isometric map from $B_r(\textup{\textsf{Y}})$ to the quotient $\textup{$\mathfrak{m}$}athbb Zsf=(\textup{$\mathfrak{m}$}athbb R/\{D_\theta=0\},D_\theta,\rho_\theta)$, which has the law of $\mathcal BHP_\theta$. Moreover, from Lemma~\ref{lem:DH} (a) we see that $I$ maps $B_r(\textup{\textsf{Y}})$ onto $B_r(\textup{$\mathfrak{m}$}athbb Zsf)$, and it sends $\rho$, the equivalence class of $0$ in $\textup{\textsf{Y}}$, to $\rho_{\theta}$, the equivalence class of $0$ in $\textup{$\mathfrak{m}$}athbb Zsf$. This completes the proof of the proposition. \end{proof} We end this section by improving Proposition~\ref{prop:isometry-BD-BHP} to the statement of Theorem~\ref{thm:coupling-BD-BHP}. \subsubsection{Proof of Theorem~\ref{thm:coupling-BD-BHP}} We will need some known facts about the Brownian disks of finite volume, mostly from Bettinelli~\cite{Be3,Be4}. With $\textup{\textsf{Y}}=([0,T]/\{D=0\},D,\rho)$ as in the previous section, we let $p_\textup{\textsf{Y}}:[0,T]\to \textup{\textsf{Y}}$ be the canonical projection. \begin{lemma}[Proposition 17 in~\cite{Be4}] \textup{$\mathfrak{l}$}bel{lem:proof-prop-refpr-1} Let $s,t\in [0,T]$ with $s\neq t$ be such that $p_\textup{\textsf{Y}}(s)=p_\textup{\textsf{Y}}(t)$ (equivalently $D(s,t)=0$). Then either $d_{F}(s,t)=0$ or $d_{W}(s,t)=0$. Moreover, the topology of $\textup{\textsf{Y}}$ is equal to the quotient topology of $[0,T]/\{D=0\}$. \end{lemma} \begin{lemma}[Theorem 2 and Proposition 21 in~\cite{Be3}] \textup{$\mathfrak{l}$}bel{lem:proof-prop-refpr}Almost surely, the space $\textup{\textsf{Y}}$ is homeomorphic to the closed unit disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}$, and the boundary of $\textup{\textsf{Y}}$ as a topological surface is determined by $$p_\textup{\textsf{Y}}^{-1}(\partial\textup{\textsf{Y}})=\{s\in [0,T]:F_s=\underline{F}_s\}\, .$$ \end{lemma} Let a real-valued function $f$ be defined on an interval $J$, and $t\in J$. We say that $t$ is a right-increase point of $f$ if there exists $\varepsilon>0$ such that $[t,t+\varepsilon]\subset J$ and $f(s)\geq f(t)$ for every $s\in [t,t+\varepsilon]$. Left-increase points are defined similarly, and a unilateral increase point is a time $t$ which is either a left-increase point or a right increase point. Note for instance that the preceding lemma implies that a point of $\partial \textup{\textsf{Y}}$ is necessarily of the form $p_\textup{\textsf{Y}}(s)$ where $s$ is a unilateral increase point of $F$. \begin{lemma}[Lemma 12 in~\cite{Be3}] \textup{$\mathfrak{l}$}bel{lem:proof-prop-refpr-3} Almost surely, the sets of unilateral increase points of $F$ and $W$ are disjoint. \end{lemma} The following lemma is not strictly needed but useful. See also~\cite{LGWe}. \begin{lemma}[Lemma 11 in~\cite{Be3}] \textup{$\mathfrak{l}$}bel{lem:proof-prop-refpr-2} Almost surely, there exists a unique $s_\ast\in (0,T)$ such that $W_{s_\ast}=\textup{$\mathfrak{m}$}in_{[0,T]}W$. \end{lemma} For $t\in[0,T)$, define $$\mathbb Phi_t(r)=\inf\{s\geq_\circ t:W_s=W_t-r\}\, ,\textup{$\mathfrak{q}$}quad 0\leq r\leq W_t-W_{s_\ast}\, ,$$ where in the notation $\geq _\circ$, it should be understood that we consider the cyclic order in $[0,T]$ when $T$ and $0$ are identified. More precisely, identifying $[0,T)$ with $\textup{$\mathfrak{m}$}athbb R/T\textup{$\mathfrak{m}$}athbb Z$, for $s,t\in [0,T)$, let $[s,t]_\circ$ be the cyclic interval from $s$ to $t$, namely, $[s,t]_\circ=[s,t]$ if $s\leq t$ and $[s,t]_\circ=[s,T)\cup[0,t]$ if $t<s$. Then $\mathbb Phi_t(r)=s$ if and only if $W_u>W_t-r$ for every $u\in [t,s]_\circ\setminus\{s\}$, and $W_s=W_t-r$. The properties that we will need are summarized in the following statement. For the rest of the proof, the time $s_\ast\in (0,T)$ is specified as in Lemma~\ref{lem:proof-prop-refpr-2}. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:proof-prop-refpr-4} The following properties hold almost surely. \begin{enumerate} \item For every $t\in [0,T]$, the path $\Gamma_t=p_\textup{\textsf{Y}}\circ \mathbb Phi_t$ is a geodesic path from $p_\textup{\textsf{Y}}(t)$ to $x_\ast=p_\textup{\textsf{Y}}(s_\ast)$. \item For every geodesic path $\Gamma$ to $x_\ast$ in $\textup{\textsf{Y}}$, there exists a unique $t\in [0,T)$ such that $\Gamma_t=\Gamma$. \item For every $t\in [0,T]$, the path $\Gamma_t$ intersects $\partial \textup{\textsf{Y}}$, if at all, only at its origin $\Gamma_t(0)$. \item Let $s,t\in [0,T]$. Then the intersection of the images $\{\Gamma_s(r):0<r<D(s_\ast,s)\}$ and $\{\Gamma_t(r):0<r\leq D(s_\ast,t)$ (excluding the starting point) is the set $$\left\{\Gamma_s(D(s_\ast,s)-r):0\leq r\leq \textup{$\mathfrak{m}$}ax\left\{\inf_{[s,t]_\circ} W,\inf_{[t,s]_\circ}W\right\}-W_{s_\ast}\right\}.$$ In particular, as soon as $s,t\in [0,T)$ and $s \neq t$, there exists $\varepsilon>0$ such that $\{\Gamma_s(r):0<r\leq \varepsilon\}$ and $\{\Gamma_t(r):0<r\leq \varepsilon\}$ are disjoint. \end{enumerate} \end{lemma} \begin{proof} (a) and (b) are proved in~\cite[Proposition 23]{Be4}. To prove (c), we notice that from the definition of $\mathbb Phi_t$, every point in the set $\{\Gamma_t(r):0<r\leq D(s_\ast,t)\}$ must be of the form $p_\textup{\textsf{Y}}(s)$, where $s$ is a left-increase point of $W$. By Lemma~\ref{lem:proof-prop-refpr-3} it cannot be a unilateral increase point of $F$, and thus $p_\textup{\textsf{Y}}(s)$ is not in $\partial \textup{\textsf{Y}}$ by Lemma~\ref{lem:proof-prop-refpr}. To prove (d) we first note that whenever $a<\textup{$\mathfrak{m}$}ax\left\{\inf_{[s,t]_\circ} W,\,\inf_{[t,s]_\circ}W\right\}$, it must hold that $\inf\{u\geq_\circ s:W_u=a\}=\inf\{u\geq_\circ t:W_u=a\}$, and the fact that $\Gamma_s(D(s_\ast,s)-r)=\Gamma_t(D(s_\ast,t)-r)$ for $r$ in the range given in the statement is a simple rewriting of this property and of the fact that $D(s_\ast,s)=W_s-W_{s_\ast}$, which is the length of the geodesic $\Gamma_s$. On the other hand, if $a>\textup{$\mathfrak{m}$}ax\left\{\inf_{[s,t]_\circ} W,\,\inf_{[t,s]_\circ}W\right\}$, then it is simple to see that $s_a=\inf\{u\geq_\circ s:W_u=a\}$ and $t_a=\inf\{u\geq_\circ t:W_u=a\}$ are such that $d_{W}(s_a,t_a)>0$, and since both points are left increase points for $W$, this implies that $p_\textup{\textsf{Y}}(s_a)\neq p_\textup{\textsf{Y}}(t_a)$ by Lemmas~\ref{lem:proof-prop-refpr-3} and~\ref{lem:proof-prop-refpr-1}. We leave to the reader to check that this implies (d). \end{proof} Let $a_0>0$, which will be fixed later on, and let $O^0_{\mathcal BD}=[0,\eta_{\textup{l}}(a_0)]\cup[T-\eta_{\textup{r}}(a_0),T]$, where $\eta_{\textup{l}}$ and $\eta_{\textup{r}}$ are defined as in the proof of Proposition~\ref{prop:isometry-BD-BHP}. We reason on the event that $s_\ast\notin O^0_\mathcal BD$, which will later be granted (with high probability) by the fact that $T$ is bound to go to $\infty$. For now we only assume that $\sigma(T)>2a_0$ so that by Lemma \ref{lem:proof-prop-refpr}, the points $x_{\textup{l}}=p_\textup{\textsf{Y}}(\eta_{\textup{l}}(a_0))$ and $x_{\textup{r}}=p_\textup{\textsf{Y}}(T-\eta_{\textup{r}}(a_0))$ are distinct elements of $\partial\textup{\textsf{Y}}$ (outside an event of zero probability). Let $t_\ast\in O^0_\mathcal BD$ be such that $W_{t_\ast}=\textup{$\mathfrak{m}$}in_{O^0_\mathcal BD}W$ (this defines $t_\ast$ uniquely a.s.\,, but we are not going to need this fact explicitly.) By (d) in Lemma~\ref{lem:proof-prop-refpr-4}, together with the fact that $s_\ast\notin O^0_\mathcal BD$, the paths $\Gamma_{\eta_{\textup{l}}(a_0)}$ and $\Gamma_{T-\eta_{\textup{r}}(a_0)}$ are disjoint until they first meet at the point $y_\ast=p_\textup{\textsf{Y}}(t_\ast)$. We let $P$ be the union of the segments of $\Gamma_{\eta_{\textup{l}}(a_0)}$ and $\Gamma_{T-\eta_{\textup{r}}(a_0)}$ between $x_{\textup{l}},x_{\textup{r}}$ and $y_\ast$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:proof-prop-refpr-5} The set $P$ is a simple curve in $\textup{\textsf{Y}}$ from $x_{\textup{l}}$ to $x_{\textup{r}}$, that intersects $\partial\textup{\textsf{Y}}$ only at $x_{\textup{l}}$ and $x_{\textup{r}}$. Letting $O_\mathcal BD$ be the connected component of $\textup{\textsf{Y}}\setminus P$ that contains $p_\textup{\textsf{Y}}(0)$, then $O_\mathcal BD$ is a.s. homeomorphic to the closed half-plane $\overline{\textup{$\mathfrak{m}$}athbb{H}}$, and is the interior of the set $p_\textup{\textsf{Y}}(O^0_\mathcal BD)$ in $\textup{\textsf{Y}}$. \end{lemma} \begin{proof} The fact that $P$ is a simple path follows from the discussion around its definition, and the fact that it intersects the boundary only at its extremities follows at once from (c) in Lemma \ref{lem:proof-prop-refpr-4}. The fact that $O_\mathcal BD$ is a.s. homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{H}}$ follows at once from this and the fact that $\textup{\textsf{Y}}$ is homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{D}}$. It remains to show that $O_\mathcal BD$ is the interior of the set $p_\textup{\textsf{Y}}(O^0_\mathcal BD)$. Note that the curve $\beta:x\textup{$\mathfrak{m}$}apsto p_\textup{\textsf{Y}}(\inf\{s\in [0,T]:F_s=-x\})$ is a continuous curve from $[0,\sigma(T)]$ to $\partial \textup{\textsf{Y}}$ with same starting and ending point, and taking distinct values otherwise. If we view $\beta$ as defined on a circle $\textup{$\mathfrak{m}$}athbb R/\sigma(T)\textup{$\mathfrak{m}$}athbb Z$, then it realizes a homeomorphism onto $\partial \textup{\textsf{Y}}$. In particular, $p_\textup{\textsf{Y}}(O^0_\mathcal BD)$ contains the segment $S$ of $\partial \textup{\textsf{Y}}$ between $x_{\textup{l}}$ and $x_{\textup{r}}$ that contains $\rho=p_\textup{\textsf{Y}}(0)$ (including $x_{\textup{l}}$, $x_{\textup{r}}$), while $p_\textup{\textsf{Y}}([0,T]\setminus O^0_\mathcal BD)$ contains the other segment which is equal to $S'=\partial \textup{\textsf{Y}}\setminus S$. For every $s\in [0,T]$, let $$\mathbb Xi_s(r)=\sup\{t\leq s:F_t=F_s-r\}\, ,\textup{$\mathfrak{q}$}quad 0\leq r\leq F_s-\underline{F}_s\, .$$ Then $p_\textup{\textsf{Y}}\circ\mathbb Xi_s$ defines a continuous path in $\textup{\textsf{Y}}$ from $p_\textup{\textsf{Y}}(s)$ to the point $\pi(s)=p_\textup{\textsf{Y}}(\sup\{t\leq s:F_t=\underline{F}_t\})$ which is in $\partial \textup{\textsf{Y}}$. Moreover, for every $r\in (0,F_s-\underline{F}_s]$, the point $\mathbb Xi_s(r)$ is a right-increase point of $F$, so by Lemma \ref{lem:proof-prop-refpr-3} it does not belong to $P\setminus\{x_{\textup{l}},x_{\textup{r}}\}$, since the latter set contains only points of the form $p_\textup{\textsf{Y}}(t)$ where $t$ is a unilateral increase point of $W$. Clearly, $\pi(s)\in S$ if $s\in O^0_\mathcal BD$, while $\pi(s)\in S'$ otherwise. We have proved that there exists a continuous path from $x$ to $p_\textup{\textsf{Y}}(0)$ not intersecting $P$ for every $x\in O_\mathcal BD$, while there exists a continuous path from $x$ to $S'$ not intersecting $P$ for every $x\in \textup{\textsf{Y}}\setminus p_\textup{\textsf{Y}}(O^0_\mathcal BD)$, and this shows that $O_\mathcal BD$ and $\textup{\textsf{Y}}\setminus p_\textup{\textsf{Y}}(O^0_\mathcal BD)$ are the two connected components of $\textup{\textsf{Y}}\setminus P$. \end{proof} \begin{figure} \caption{Proof of Theorem~\ref{thm:coupling-BD-BHP} \end{figure} To finish the proof of Theorem~\ref{thm:coupling-BD-BHP}, fix $r>0$, and let $a_0$ be large enough so that $$\mathbb P\left(\textup{$\mathfrak{m}$}in_{[0,a_0]}\gamma<-2r,\textup{$\mathfrak{m}$}in_{[-a_0,0]}\gamma<-2r\right)>1-\varepsilon/4\, .$$ Then, we choose $r_0>r$ such that, with $W^{\theta}$ the label function of $\mathcal BHP_\theta$, $$\mathbb P(\omega(W^{\theta},[-\eta_{\textup{r}}(a_0),\eta_{\textup{l}}(a_0)])\leq r_0/2)\geq 1-\varepsilon/4\, ,$$ where $\omega(f,I)=\sup_I f-\inf_I f$ is the modulus of continuity of $f$ over the set $I$. We use this value of $r_0$ to apply Proposition \ref{prop:isometry-BD-BHP}. Fix $\varepsilon>0$. Then for every $T\geq T_0(\varepsilon/2)$ large enough, we can construct copies of $\textup{\textsf{Y}}=\mathcal BD_{T,\sigma(T)}$ and $\mathcal BHP_\theta$ such that $B_{r_0}(\textup{\textsf{Y}})$ and $B_{r_0}(\mathcal BHP_\theta)$ are isometric with probability at least $1-\varepsilon/2$. More precisely, we are going to use the event $\textup{$\mathfrak{m}$}athcal{F}$ specified in the proof of Proposition~\ref{prop:isometry-BD-BHP} on which this property holds (in the definition of $\textup{$\mathfrak{m}$}athcal{F}$ we have to make sure that $A$ is chosen so that $A>a_0$), and which implies the property that $s_\ast\notin O^0_\mathcal BD$, on which our analysis so far is based. The probability of $$\textup{$\mathfrak{m}$}athcal{F}'=\textup{$\mathfrak{m}$}athcal{F}\cap \left\{\textup{$\mathfrak{m}$}in_{[0,a_0]}\gamma<-2r,\textup{$\mathfrak{m}$}in_{[-a_0,0]}\gamma<-2r\right\} \cap \{\omega(W^{\theta},[-\eta_{\textup{r}}(a_0),\eta_{\textup{l}}(a_0)])\leq r_0/2\}$$ is then at least $1-\varepsilon$. On this event we claim that $$B_{r}(\textup{\textsf{Y}})\subset O_\mathcal BD\subset B_{r_0}(\textup{\textsf{Y}})\, .$$ The second inclusion comes from the fact that $D(0,s)\leq d_{W}(0,s)\leq 2\omega(W^{\theta},[-\eta_{\textup{r}}(a_0),\eta_{\textup{l}}(a_0)])$ for every $s\in [0,\eta_{\textup{l}}(a_0)]\cup [T-\eta_{\textup{r}}(a_0),T]$ (recall that $\textup{$\mathfrak{m}$}athbb{W}=W^{\theta}$ on the set $[-\eta_{\textup{r}}(A^3),\eta_{\textup{l}}(A^3)]$ on $\textup{$\mathfrak{m}$}athcal{F}$). The first inclusion comes from the cactus bound, with the fact that $\textup{$\mathfrak{m}$}in_{[0,a_0]}\gamma<-2r$ and $\textup{$\mathfrak{m}$}in_{[-a_0,0]}\gamma<-2r$, just as in the proof of (a) in Lemma \ref{lem:DBD}. To be more precise, this shows that $B_{2r}(\textup{\textsf{Y}})\subset p_\textup{\textsf{Y}}([0,\eta_{\textup{l}}(a_0)]\cup [T-\eta_{\textup{r}}(a_0),T])$, and since $O_\mathcal BD$ is equal to the interior of the latter set, the wanted inclusion follows. We refer to Figure~\ref{fig:topo-BHP} for an illustration. Finally, we prove that $O_\mathcal BHP=I(O_\mathcal BD)$ is an open subset of $\mathcal BHP_\theta$, where $I$ is the isometry defined before Corollary \ref{cor:DBD-DH}, or more precisely the induced map on $\textup{\textsf{Y}}$. Let $x\in O_\mathcal BHP$, so that $x=I(y)$ for some $y\in O_\mathcal BD\subset B_{r_0}(\textup{\textsf{Y}})$. Let $\mathrm{d}elta>0$ be such that $y+B_\mathrm{d}elta(\textup{\textsf{Y}})\subset O_\mathcal BD$, and $x'\in \mathcal BHP_\theta$ be such that $D_{\theta}(x,x')<\mathrm{d}elta$. Then $D_{\theta}(0,x')\leq D_{\theta}(0,x)+\mathrm{d}elta<r_0$, so that $x'\in B_{r_0}(\mathcal BHP_\theta)$. Therefore, there exists $y'\in B_{r_0}(\textup{\textsf{Y}})$ with $x'=I(y')$, and one has $D(y,y')=D_{\theta}(x,x')<\mathrm{d}elta$, so that $y'\in O_\mathcal BD$, and thus $x'\in O_\mathcal BHP$. This proves that $O_\mathcal BHP$ is open and concludes the proof of Theorem~\ref{thm:coupling-BD-BHP}. \subsection{Coupling quadrangulations of large volumes (Propositions~\ref{prop:Qn-UIHPQ} and~\ref{prop:coupling-Qn-largevol})} We begin with the proof of Proposition~\ref{prop:Qn-UIHPQ}, which is in spirit of~\cite[Lemma 8 and Proposition 9]{CuLG}. \begin{proof}[Proof of Proposition~\ref{prop:Qn-UIHPQ}] Assume $1\ll \sigma_n\ll n$. Set $\vartheta_n=\textup{$\mathfrak{m}$}in\{\sigma_n,\, n/\sigma_n\}$, and let $\varepsilon>0$. Let $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ be uniformly distributed over the set $\mathbb Fo_{\sigma_n}^n \times \mathcal Br_{\sigma_n}$, and consider a triplet $((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_{\infty})$ of a uniformly labeled critical infinite forest and a uniform infinite bridge. We first argue that we can find $\mathrm{d}elta>0$ and $n_0$ such that for all $n\geq n_0$, we can construct $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ and $((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_{\infty})$ on the same probability space such that on an event of probability at least $1-\varepsilon$, the corresponding balls of radius $2\mathrm{d}elta \sqrt{\vartheta_n}$ around the vertices $\textup{$\mathfrak{f}$}_n(0)$ and $\textup{$\mathfrak{f}$}_\infty(0)$ in the associated quadrangulations are isometric. For $0\leq k\leq\sigma_n-1$, write $\tau(\textup{$\mathfrak{f}$}_n,k)$ for the tree of $\textup{$\mathfrak{f}$}_n$ rooted at $(k)$, and put $\tau(\textup{$\mathfrak{f}$}_n,\sigma_n)=\tau(\textup{$\mathfrak{f}$}_n,0)$. Similarly, define $\tau(\textup{$\mathfrak{f}$}_\infty,k)$ to be the tree of $\textup{$\mathfrak{f}$}_\infty$ rooted at $(k)$, where now $k\in\textup{$\mathfrak{m}$}athbb{Z}$. As a consequence of Lemmas~\ref{lem:GW3} and~\ref{lem:bridge1}, there exist $\mathrm{d}elta'>0$ and $n'_0\in\textup{$\mathfrak{m}$}athbb N$ such that for $n\geq n'_0$, with $A_n=\lfloorloor \mathrm{d}elta'\vartheta_n\rfloorloor$, we can construct $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ and $((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_{\infty})$ on the same probability space such that if we let \begin{align*} \textup{$\mathfrak{m}$}athcal{E}^1(n,\mathrm{d}elta') &= \textup{$\mathfrak{q}$}uad\left\{\tau(\textup{$\mathfrak{f}$}_n,i)=\tau(\textup{$\mathfrak{f}$}_\infty,i),\,\tau(\textup{$\mathfrak{f}$}_n,\sigma_n-i)=\tau(\textup{$\mathfrak{f}$}_\infty,-i),\,0\leq i\leq A_n\right\}\\ &\,\textup{$\mathfrak{q}$}uad \cap\left\{\textup{\textsf{b}}_n(i)=\textup{\textsf{b}}_\infty(i),\,\textup{\textsf{b}}_n(\sigma_n-i)=\textup{\textsf{b}}_\infty(-i),\,1\leq i\leq A_n\right\}\\ &\,\textup{$\mathfrak{q}$}uad\cap\left\{\textup{$\mathfrak{l}$}_n|_{\tau(\textup{$\mathfrak{f}$}_n,i)}= \textup{$\mathfrak{l}$}_\infty|_{\tau(\textup{$\mathfrak{f}$}_\infty,i)},\,\textup{$\mathfrak{l}$}_n|_{\tau(\textup{$\mathfrak{f}$}_n,\sigma_n-i)}=\textup{$\mathfrak{l}$}_\infty|_{\tau(\textup{$\mathfrak{f}$}_\infty,-i)},\,0\leq i \leq A_n\right\}, \end{align*} then $\textup{$\mathfrak{m}$}athcal{E}^1(n,\mathrm{d}elta')$ has probability at least $1-\varepsilon/3$. We fix such a $\mathrm{d}elta'$. For $\mathrm{d}elta>0$ and $n\in\textup{$\mathfrak{m}$}athbb N$, put $$\textup{$\mathfrak{m}$}athcal{E}^2(n,\mathrm{d}elta)=\left\{\textup{$\mathfrak{m}$}in_{[0,\,A_n]}\textup{\textsf{b}}_\infty <-5\mathrm{d}elta \sqrt{\vartheta_n},\, \textup{$\mathfrak{m}$}in_{[-A_n,\,0]}\textup{\textsf{b}}_\infty<-5\mathrm{d}elta\sqrt{\vartheta_n}\right\}\cap\left\{-\textup{\textsf{b}}_\infty(-1)<\mathrm{d}elta^{-1}\right\},$$ and let $$\textup{$\mathfrak{m}$}athcal{E}^3(n,\mathrm{d}elta)=\left\{\textup{$\mathfrak{m}$}in_{[A_n+1,\,\sigma_n-(A_n+1)]}\textup{\textsf{b}}_n <-5\mathrm{d}elta\sqrt{\vartheta_n}\right\}.$$ Donsker's invariance principle applied to $(\textup{\textsf{b}}_{\infty}(i),i\in\textup{$\mathfrak{m}$}athbb{Z})$ guarantees that we can find $\mathrm{d}elta>0$ such that for all sufficiently large $n$, $$ \mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}^2(n,\mathrm{d}elta)\right)\geq 1-\varepsilon/3. $$ Moreover, provided $n$ is large enough and $\mathrm{d}elta>0$ is sufficiently small, Lemma~\ref{lem:bridge0} ensures that $$ \mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}^3(n,\mathrm{d}elta)\right)\geq 1-\varepsilon/3. $$ We fix $n_0\geq n'_0$ and $\mathrm{d}elta>0$ such that for all $n\geq n_0$, the bounds in the last two displays hold. From now on, we work on the event $\textup{$\mathfrak{m}$}athcal{E}^1(n,\mathrm{d}elta')\cap\textup{$\mathfrak{m}$}athcal{E}^2(n,\mathrm{d}elta)\cap\textup{$\mathfrak{m}$}athcal{E}^3(n,\mathrm{d}elta)$. Let $(Q_n^{\sigma_n},v^{\bullet})=\mathbb Phi_n(((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n))$ and $Q_\infty^\infty=\mathbb Phi_{\infty}(((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_{\infty}))$ be the quadrangulations constructed from the triplets $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ and $((\textup{$\mathfrak{f}$}_\infty,\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_{\infty})$ {\it via} the Bouttier-Di Francesco-Guitter mapping. We denote by $d_n$ and $d_{\infty}$ the graph distances on $V(Q_n^{\sigma_n})$ and $V(Q_\infty^\infty)$. We write $$\textup{$\mathfrak{f}$}'_n=\left(\tau(\textup{$\mathfrak{f}$}_n,\sigma_n-A_n),\ldots,\tau(\textup{$\mathfrak{f}$}_n,\sigma_n-1),\tau(\textup{$\mathfrak{f}$}_n,0),\ldots,\tau(\textup{$\mathfrak{f}$}_n,A_n)\right)$$ for the forest obtained from restricting $\textup{$\mathfrak{f}$}_n$ to the last $A_n$ and the first $A_n+1$ trees, and identically $$\textup{$\mathfrak{f}$}'_\infty=\left(\tau(\textup{$\mathfrak{f}$}_{\infty},-A_n),\ldots,\tau(\textup{$\mathfrak{f}$}_{\infty},-1),\tau(\textup{$\mathfrak{f}$}_\infty,0),\ldots,\tau(\textup{$\mathfrak{f}$}_\infty,A_n)\right).$$ Recall the cactus bounds~\eqref{eq:cactus1} and~\eqref{eq:cactus3} for $Q_n^{\sigma_n}$ and $Q_\infty^\infty$, respectively. For vertices $v\in V(\textup{$\mathfrak{f}$}_n)\setminus V(\textup{$\mathfrak{f}$}'_n)$, we obtain, with $(0)=\textup{$\mathfrak{f}$}_n(0)$, $$ d_n((0),v) \geq -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,A_n]}\textup{\textsf{b}}_n, \textup{$\mathfrak{m}$}in_{[-A_n,0]}\textup{\textsf{b}}_n\right\} \geq 5\mathrm{d}elta\sqrt{\vartheta_n}, $$ and identically, for vertices $v\in V(\textup{$\mathfrak{f}$}_\infty)\setminus V(\textup{$\mathfrak{f}$}'_\infty)$, writing now $(0)$ for $\textup{$\mathfrak{f}$}_\infty(0)$, $ d_{\infty}((0),v) \geq 5\mathrm{d}elta\sqrt{\vartheta_n}. $ We argue now similarly to the second part in the proof of~\cite[Lemma 8]{CuLG}. Firstly, if $u\in V(\textup{$\mathfrak{f}$}_n)$ is any vertex with $d_n((0),u)\leq 5\mathrm{d}elta\sqrt{\vartheta_n}-1$, then any vertex on a geodesic path from $(0)$ to $u$ in $Q_n^{\sigma_n}$ satisfies the same bound and must therefore belong to one of the trees in $\textup{$\mathfrak{f}$}'_n$. From the construction of edges in the Bouttier-Di Francesco-Guitter mapping, we deduce that any edge of $Q_n^{\sigma_n}$ on such a geodesic path corresponds to an edge of $Q_\infty^{\infty}$ with the same endpoints, provided none of these edges in $Q_n^{\sigma_n}$ connect two vertices $w$ and $w'$ such that the set of vertices between $w$ and $w'$ in the cyclic contour order around the forest $\textup{$\mathfrak{f}$}_n$ contains the vertices of $\textup{$\mathfrak{f}$}_n\setminus \textup{$\mathfrak{f}$}_n'$. But on the event $\textup{$\mathfrak{m}$}athcal{E}^3(n,\mathrm{d}elta)$, the set of vertices between $w$ and $w'$ would in particular contain a (root) vertex of $\textup{$\mathfrak{f}$}_n\setminus\textup{$\mathfrak{f}$}_n'$ with label less than $-5\mathrm{d}elta\sqrt{\vartheta_n}$. This would imply that both vertices $w$ and $w'$ of such an edge have a label which is smaller than $-5\mathrm{d}elta\sqrt{\vartheta_n}$, too, in contradiction to the fact that $d_n((0),v)\leq 5\mathrm{d}elta\sqrt{\vartheta_n}-1$ for all vertices $v$ on a geodesic between $(0)$ and $u$. We deduce that if $u\in V(\textup{$\mathfrak{f}$}_n)$ satisfies $d_n((0),u)\leq 5\mathrm{d}elta\sqrt{\vartheta_n}-1$, then $d_\infty((0),u)\leq d_n((0),u)$. Since in turn any edge of $Q_\infty^{\infty}$ on a geodesic between $(0)$ and a vertex $u\in V(\textup{$\mathfrak{f}$}_\infty)$ with $d_\infty((0),u)\leq 5\mathrm{d}elta\sqrt{\vartheta_n}-1$ does correspond to an edge of $Q_n^{\sigma_n}$ with the same endpoints, we obtain also $d_n((0),u)\leq d_\infty((0),u)$. Therefore, we have that vertices with distance at most $5\mathrm{d}elta\sqrt{\vartheta_n}-1$ from $(0)$ are the same in $Q_n^{\sigma_n}$ and $Q_\infty^{\infty}$. We claim that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:prop-BHP1-eq1} d_n(u,v)=d_{\infty}(u,v)\textup{$\mathfrak{q}$}uad\hbox{whenever }u,v\in B^{(0)}_{2\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n}). \end{equation} Indeed, if $u,v$ are vertices in $B^{(0)}_{2\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n})$, then any geodesic connecting $u$ to $v$ in $Q_n^{\sigma_n}$ must lie entirely in $B^{(0)}_{4\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n})$, and any edge on such a geodesic corresponds to an edge in $Q_\infty^\infty$. Since the converse is also true, we obtain~\eqref{eq:prop-BHP1-eq1}, and with the correspondence of edges between $Q_n^{\sigma_n}$ and $Q_\infty^\infty$ alluded to above we deduce that the balls $B^{(0)}_{2\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n})$ and $B^{(0)}_{2\mathrm{d}elta\sqrt{\vartheta_n}}(Q_\infty^{\infty})$ are isometric on an event of probability at least $1-\varepsilon$. Finally, recall from the Bouttier-Di Francesco-Guitter bijection that the root vertex $\rho_n$ of $Q_n^{\sigma_n}$ is given by $\textup{$\mathfrak{f}$}_n(\textup{succ}^{-\textup{\textsf{b}}_n(\sigma_n)}(0))$, where conditionally on $\textup{\textsf{b}}_n(\sigma_n-1)$, $\textup{\textsf{b}}_n(\sigma_n)$ is uniformly distributed on $\{\textup{\textsf{b}}_n(\sigma_n-1)-1,\ldots,0\}$. Similarly, the root vertex $\rho$ of $Q_\infty^\infty$ is given by $\textup{$\mathfrak{f}$}_\infty(\textup{succ}^{-\textup{\textsf{b}}_\infty(\partial)}(0))$, where conditionally on $\textup{\textsf{b}}_\infty(-1)$, $\textup{\textsf{b}}_\infty(\partial)$ is uniformly distributed on $\{\textup{\textsf{b}}_\infty(-1)-1,\ldots,0\}$. On the event $\textup{$\mathfrak{m}$}athcal{E}^{1}(n,\mathrm{d}elta')\cap \textup{$\mathfrak{m}$}athcal{E}^{2}(n,\mathrm{d}elta)$, we can couple $\textup{\textsf{b}}_n(\sigma_n)$ and $\textup{\textsf{b}}_\infty(\partial)$ such that $\textup{\textsf{b}}_n(\sigma_n)=\textup{\textsf{b}}_\infty(\partial)$. Moreover, for $n$ large enough, we have on this event $B_{\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n})\subset B^{(0)}_{2\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n})$ and $B_{\mathrm{d}elta\sqrt{\vartheta_n}}(Q_{\infty}^{\infty})\subset B_{2\mathrm{d}elta\sqrt{\vartheta_n}}^{(0)}(Q_{\infty}^{\infty})$. Therefore, we have equality of $B_{\mathrm{d}elta\sqrt{\vartheta_n}}(Q_n^{\sigma_n})$ and $B_{\mathrm{d}elta\sqrt{\vartheta_n}}(Q_{\infty}^{\infty})$ on the event $\textup{$\mathfrak{m}$}athcal{E}^{1}(n,\mathrm{d}elta')\cap \textup{$\mathfrak{m}$}athcal{E}^{2}(n,\mathrm{d}elta)\cap\textup{$\mathfrak{m}$}athcal{E}^{3}(n,\mathrm{d}elta)$. Local convergence of $Q_n^{\sigma_n}$ towards $\textup{\textsf{UIHPQ}}$ in the sense of $d_{\textup{map}}$ is a direct consequence of this, and the proposition is proved. \end{proof} We now turn to the proof of Proposition~\ref{prop:coupling-Qn-largevol}. We will adopt the notion of~\cite[Section 4.3.1]{CuLG} concerning pruned (pointed) trees. More precisely, a (finite) pointed tree consists of a pair $\boldsymbol{\tau}=(\tau,\xi)$, where $\tau$ is a tree of finite size and $\xi$ is a distinguished vertex of $\tau$. Given such a pointed tree $\boldsymbol{\tau}=(\tau,\xi)$ and $h$ an integer with $0\leq h<|\xi|$, $\textup{$\mathfrak{m}$}athscr{P}(\boldsymbol{\tau},h)$ represents the subtree of $\tau$ containing all the vertices $u\in V(\tau)$ such that the height of the most recent common ancestor of $u$ and $\xi$ is strictly less than $h$, together with the ancestor $\left[\xi\right]_h$ of $\xi$ at height exactly $h$. By pointing $\textup{$\mathfrak{m}$}athscr{P}(\boldsymbol{\tau},h)$ at $\left[\xi\right]_h$, this subtree is itself considered as a pointed tree. If $h\geq |\xi|$, we agree that $\textup{$\mathfrak{m}$}athscr{P}(\boldsymbol{\tau},h)=(\{\emptyset\},\emptyset)$, where $\emptyset$ represents the root vertex of $\tau$. It is straightforward to see that if $\boldsymbol{\tau}=(\tau,\xi)$ is a pointed tree and $h$ and $h'$ are two integers with $h'\geq h\geq 0$, then \begin{equation} \textup{$\mathfrak{l}$}bel{eq:pruningconsistent} \textup{$\mathfrak{m}$}athscr{P}\left((\boldsymbol{\tau},h'),h\right)=\textup{$\mathfrak{m}$}athscr{P}\left(\boldsymbol{\tau},h\right). \end{equation} \begin{proof}[Proof of Proposition~\ref{prop:coupling-Qn-largevol}] We assume $1\ll \sigma_n\ll \sqrt{n}$ and fix $\varepsilon>0$ and $r>0$ for the rest of this proof. We let $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ be uniformly distributed over the set $\mathbb Fo_{\sigma_n}^n \times \mathcal Br_{\sigma_n}$, and for given $R\in\textup{$\mathfrak{m}$}athbb N$, we let $((\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n),\textup{\textsf{b}}'_n)$ be uniformly distributed over $\mathbb Fo_{\sigma_n}^{R\sigma_n^2} \times \mathcal Br_{\sigma_n}$. Identically to the proof of Proposition~\ref{prop:Qn-UIHPQ}, it suffices to show that we can find $R_0>0$ and $n_0\in\textup{$\mathfrak{m}$}athbb N$ such that for all integers $R\geq R_0$ and all $n\geq n_0$, we can construct $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ and $((\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n),\textup{\textsf{b}}'_{n})$ on the same probability space such that on an event of probability at least $1-\varepsilon$, the corresponding balls of radius $2r\sqrt{\sigma_n}$ around the vertices $\textup{$\mathfrak{f}$}_n(0)$ and $\textup{$\mathfrak{f}$}'_n(0)$ in the associated quadrangulations are isometric. For $0\leq k\leq\sigma_n-1$, we let $\tau(\textup{$\mathfrak{f}$}_n,k)$ be the tree of $\textup{$\mathfrak{f}$}_n$ rooted at $(k)$ and denote by $i_\ast$ the smallest index such that $|\tau(\textup{$\mathfrak{f}$}_n,i_\ast)|\geq |\tau(\textup{$\mathfrak{f}$}_n,k)|$ for all $0\leq k\leq \sigma_n-1$. We shall point the tree $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$, by choosing conditionally on $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$ a vertex $\xi_n\in V(\tau(\textup{$\mathfrak{f}$}_n,i_\ast))$ uniformly at random. We write $(\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n)$ for the pointed tree obtained in this way, and for $h\in\textup{$\mathfrak{m}$}athbb N$, we write $\textup{$\mathfrak{l}$}_n|_{\textup{$\mathfrak{m}$}athscr{P}((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),h)}$ for the restriction of the labels $\textup{$\mathfrak{l}$}_n$ of $\textup{$\mathfrak{f}$}_n$ to the subtree $\textup{$\mathfrak{m}$}athscr{P}((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),h)$ of $(\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n)$ pruned at height $h$, see the notation above. Finally, we let $(\tau_i,\ell_i)$, $0\leq i\leq \sigma_n-1$, be a sequence of independent uniformly labeled critical geometric Galton-Watson trees. For $H\in\textup{$\mathfrak{m}$}athbb N$, set $H_n=H\sigma_n$. Recall that the law of $((\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n),\textup{\textsf{b}}'_n)$ depends on $R\in\textup{$\mathfrak{m}$}athbb N$. We claim that for each fixed integer $H\in\textup{$\mathfrak{m}$}athbb N$, provided $n$ and $R$ are sufficiently large, we can construct $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$, $((\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n),\textup{\textsf{b}}'_n)$, $\xi_n$, $\xi'_n$ and $(\tau_i,\ell_i)$ for $0\leq i\leq \sigma_n-1$ on the same probability space such that the event \begin{align*} \textup{$\mathfrak{m}$}athcal{E}_1(n,R,H) &= \left\{i_\ast=i'_\ast\right\}\cap\left\{\tau(\textup{$\mathfrak{f}$}_n,i)=\tau(\textup{$\mathfrak{f}$}'_n,i)=\tau_i,\,0\leq i\leq \sigma_n-1,\, i\neq i_\ast\right\}\\ &\,\textup{$\mathfrak{q}$}uad\cap \left\{\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),H_n\right)=\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast),\xi'_n),H_n\right)\neq (\{\emptyset\},\emptyset)\right\}\\ &\,\textup{$\mathfrak{q}$}uad\cap\left\{\textup{\textsf{b}}_n(i)=\textup{\textsf{b}}'_n(i),\,0\leq i\leq \sigma_n\right\}\\ &\,\textup{$\mathfrak{q}$}uad\cap \left\{\textup{$\mathfrak{l}$}_n|_{\tau(\textup{$\mathfrak{f}$}_n,i)}= \textup{$\mathfrak{l}$}'_n|_{\tau(\textup{$\mathfrak{f}$}'_n,i)}=\ell_i,\,0\leq i \leq \sigma_n-1, i\neq i_\ast\right\}\\&\,\textup{$\mathfrak{q}$}uad \cap\left\{\textup{$\mathfrak{l}$}_n|_{\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),H_n\right)}=\textup{$\mathfrak{l}$}'_n|_{\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast),\xi'_n),H_n\right)}\right\} \end{align*} has probability at least $1-\varepsilon/2$. Let us look separately at the different sets on the right hand side. Firstly, from Lemma~\ref{lem:GW1} we know that $\textup{$\mathfrak{f}$}_n$ has with high probability a unique largest tree of order $\sigma_n^2$, and its index is uniform in $\{0,\ldots,\sigma_n-1\}$. Moreover, Lemma~\ref{lem:GW2} asserts that the other trees of $\textup{$\mathfrak{f}$}_n$ are close in total variation to $\sigma_n-1$ critical geometric Galton-Watson trees. The same holds for $\textup{$\mathfrak{f}$}'_n$, from which we deduce that $\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{f}$}'_n$ and $\tau_i$, $0\leq i\leq \sigma_n-1$, can be coupled such that the intersection of the first two events on the right hand side has probability at least $1-\varepsilon/3$, say. For the event on the second line concerning the pruned trees, we use that fact that conditionally on $|\tau(\textup{$\mathfrak{f}$}_n,i_\ast)|=m_n$, $(\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n)$ is uniformly distributed over the set of all pointed trees of size $m_n$. A similar statement holds for $\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast)$. Now by Lemma~\ref{lem:GW1}, for any $K>0$, the probability that $|\tau(\textup{$\mathfrak{f}$}_n,i_\ast)|\geq K\sigma^2_n$ tends to one with increasing $n$, since $n\gg \sigma_n^2$. Similarly, for any given $K>0$, by choosing $R$ large enough, we can ensure that $|\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast)|\geq K\sigma^2_n$ holds with a probability as close to one as we wish for large $n$. An application of Proposition $7$ of~\cite{CuLG} therefore shows that both $\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),H_n\right)$ and $\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast),\xi'_n),H_n\right)$ are for large $R$ and $n$ close in total variation to the so-called uniform infinite tree (or Kesten's tree) pruned at height $H_n$. Applying the triangle inequality, we see that the total variation distance between $\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),H_n\right)$ and $\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast),\xi'_n),H_n\right)$ can be made as small as we wish, provided $R$ and $n$ are taken sufficiently large. Combining the above coupling with this last observation, we infer that we can in fact couple $\textup{$\mathfrak{f}$}_n$, $\textup{$\mathfrak{f}$}'_n$, $\xi_n$, $\xi_n'$ and $\tau_i$ for $0\leq i\leq \sigma_n-1$ such that the intersection of the first three events on the right hand side has probability at least $1-\varepsilon/2$ for large $R$ and $n$. Since the bridges $\textup{\textsf{b}}_n$ and $\textup{\textsf{b}}'_n$ have both the same law and are independent of the trees, we can additionally assume that the probability space carries realizations of $\textup{\textsf{b}}_n$ and $\textup{\textsf{b}}'_n$ such that $\textup{\textsf{b}}_n\equiv \textup{\textsf{b}}'_n$. A similar arguments allows us to couple the labelings $\textup{$\mathfrak{l}$}_n$, $\textup{$\mathfrak{l}$}'_n$, and $\ell_i$ such that the last two events on the right hand side in the definition of $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)$ hold true. This proves the claim about $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)$. We will now work on the event $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)$ and let $(Q_n^{\sigma_n},v^{\bullet})=\mathbb Phi_n(((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n))$ and $(Q_{R\sigma_n^2}^{\sigma_n},w^{\bullet})=\mathbb Phi_{R\sigma_n^2}(((\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n),\textup{\textsf{b}}'_n))$ be the quadrangulations constructed from the triplets $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)$ and $((\textup{$\mathfrak{f}$}'_n,\textup{$\mathfrak{l}$}'_n),\textup{\textsf{b}}'_n)$, respectively. Recall that $[\xi_n]_{H_n}$ denotes the ancestor of $\xi_n$ in $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$ at height $H_n$. Let $$M_n=-\textup{$\mathfrak{m}$}in_{[[\emptyset,[\xi_n]_{H_n}]]}\textup{$\mathfrak{l}$}_n,$$ where $[[\emptyset,[\xi_n]_{H_n}]]$ is the vertex set of the unique injective path in $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$ connecting the (tree) root $\emptyset$ to $[\xi_n]_{H_n}$. By definition of the labeling $\textup{$\mathfrak{l}$}_n$, conditionally on the tree, $M_n$ has the law of the maximum attained by a random walk started at zero and stopped after $H_n$ many steps, with increments uniformly distributed in $\{-1,0,1\}$. Setting $$\textup{$\mathfrak{m}$}athcal{E}_2(n,H)=\left\{M_n\geq 5r\sqrt{\sigma_n}\right\}, $$ we can ensure by an application of Donsker's invariance principle that for $H\in\textup{$\mathfrak{m}$}athbb N$ sufficiently large (recall that $r$ was fixed at the beginning, and $H_n=H\sigma_n$), the event $\textup{$\mathfrak{m}$}athcal{E}_2(n,H)$ has probability at least $1-\varepsilon/2$. In particular, by choosing $H\in\textup{$\mathfrak{m}$}athbb N$ large enough, we obtain that $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)\cap\textup{$\mathfrak{m}$}athcal{E}_2(n,H)$ has probability at least $1-\varepsilon$ for all $R$, $n\in\textup{$\mathfrak{m}$}athbb N$ sufficiently large. It remains to convince ourselves that on the event $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)\cap \textup{$\mathfrak{m}$}athcal{E}_2(n,H)$, the balls $B^{(0)}_{2r\sqrt{\sigma_n}}(Q_n^{\sigma_n})$ and $B^{(0)}_{2r\sqrt{\sigma_n}}(Q_{R\sigma_n^2}^{\sigma_n})$ are isometric. Since the arguments are very close to those provided in the proofs of Proposition~\ref{prop:Qn-UIHPQ} above and~\cite[Lemma 8]{CuLG}, we only sketch them in order to avoid too much repetition. Write $\emptyset=u_0,u_1,\ldots,u_{H_n}=[\xi_n]_{H_n}$ for the vertices of the non-backtracking path connecting $\emptyset$ to $[\xi_n]_{H_n}$ in $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$. Let $k_n\in\{0,\ldots,H_n\}$ such that $$\textup{$\mathfrak{l}$}_{n}(u_{k_n})=-\textup{$\mathfrak{m}$}in_{[[\emptyset,[\xi_n]_{H_n}]]}\textup{$\mathfrak{l}$}_n.$$ Recall the identification of $V(Q_n^{\sigma_n})\setminus\{v^{\bullet}\}$ with $V(\textup{$\mathfrak{f}$}_n)$. Denote by $d_n$ the graph distance on $V(Q_n^{\sigma_n})$. If $v$ is a vertex of $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$ that does not belong to the subtree $\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),k_n\right)$, then the ancestral lines of $v$ and $\xi_n$ coincide at least up to level $k_n$. In particular, they both contain the vertex $u_{k_n}$. For such vertices $v$, we obtain from the cactus bound~\eqref{eq:cactus1} on the event $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)\cap \textup{$\mathfrak{m}$}athcal{E}_2(n,H)$ the bound $$ d_n((0),v)\geq 5r\sqrt{\sigma_n}, $$ with $(0)=\textup{$\mathfrak{f}$}_n(0)$. See~\cite[Proof of Lemma 8]{CuLG} for the complete argument (note however that $(0)$ might be the root of a tree different from $\tau(\textup{$\mathfrak{f}$}_n,i_\ast)$). On $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)$, using additionally~\eqref{eq:pruningconsistent}, $$ \textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}_n,i_\ast),\xi_n),k_n\right)= \textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast),\xi'_n),k_n\right), $$ and the labelings $\textup{$\mathfrak{l}$}_n$ and $\textup{$\mathfrak{l}$}'_n$ restricted to the subtrees on the left and right, respectively, agree. Therefore, a similar inequality holds for $Q_{R\sigma^2_n}^{\sigma_n}$, for vertices $v'$ of $\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast))$ which do not belong to the subtree $\textup{$\mathfrak{m}$}athscr{P}\left((\tau(\textup{$\mathfrak{f}$}'_n,i'_\ast),\xi'_n),k_n\right)$. Adapting now the reasoning of~\cite[Proof of Lemma 8]{CuLG} to our situation (see also the proof of Proposition~\ref{prop:Qn-UIHPQ} above), we obtain that vertices with distance at most $5r\sqrt{\sigma_n}-1$ from $(0)$ are the same in $Q_n^{\sigma_n}$ and $Q_{R\sigma^2_n}^{\sigma_n}$ on the event $\textup{$\mathfrak{m}$}athcal{E}_1(n,R,H)\cap \textup{$\mathfrak{m}$}athcal{E}_2(n,H)$, and, with $d_n'$ being the graph distance in $Q_{R\sigma^2_n}^{\sigma_n}$, $$ d_n(u,v)=d'_n(u,v)\textup{$\mathfrak{q}$}uad\hbox{whenever }u,v\in B^{(0)}_{2r\sqrt{\sigma_n}}(Q_n^{\sigma_n}). $$ This finishes our proof. \end{proof} \subsection{Brownian half-plane with zero skewness (Theorems~\ref{thm:BHP1} and~\ref{thm:UIHPQ-BHP})} We work in the usual setting from Section~\ref{sec:usualsetting}. In particular, $Q_n^{\sigma_n}$ denotes a random variable with the uniform distribution over the set $\textup{$\mathfrak{m}$}athcal Q_n^{\sigma_n}$ of rooted quadrangulations with $n$ internal faces and $2\sigma_n$ boundary edges. Our proofs of Theorems~\ref{thm:BHP1} and~\ref{thm:UIHPQ-BHP} are essentially consequences of the coupling of balls between the Brownian disk $\mathcal BD_{T,\sqrt{T}}$ and the Brownian half-plane $\mathcal BHP$ (Proposition~\ref{prop:isometry-BD-BHP}), of the fundamental convergence \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BeMi} \left(V(Q_n^{\sigma_n}),(8/9)^{-1/4}n^{-1/4}d_{\textup{gr}},\rho_n\right) \xrightarrow[n \to \infty]{(d)} \mathcal BD_\sigma=\mathcal BD_{1,\sigma} \end{equation} proved in~\cite[Theorem 1]{BeMi} for the regime $\sigma_n\sim\sigma\sqrt{2n}$ when $\sigma\in(0,\infty)$ is a fixed real and of the coupling between $Q_n^{\sigma_n}$ and the $\textup{\textsf{UIHPQ}}$ $Q_\infty^\infty$ (Proposition~\ref{prop:Qn-UIHPQ}). \begin{proof}[Proof of Theorem~\ref{thm:UIHPQ-BHP}] In view of Remark~\ref{rem:localGH}, the result follows if we show that for every $r\geq 0$ and every sequence of positive reals $a_n\rightarrow\infty$, $$ B_r\left(a_n^{-1}\cdot Q_\infty^{\infty}\right)\xrightarrow[n\to \infty]{(d)}B_{r}(\mathcal BHP)$$ in distribution in $\textup{$\mathfrak{m}$}athbb{K}$. For notational simplicity, we restrict ourselves to the case $r=1$. Fix $\varepsilon>0$. By Proposition~\ref{prop:isometry-BD-BHP}, we find $T_0=T_0(\varepsilon)>0$ such that for all $T\geq T_0$, we can construct copies of $\mathcal BD_{T,\sqrt{T}}$ and $\mathcal BHP$ on the same probability space such that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BHP-coupling1} B_1\left(\mathcal BD_{T,\sqrt{T}}\right)=B_1(\mathcal BHP) \end{equation} with probability at least $1-\varepsilon$. Let $\sigma_n=\lceil \sqrt{2n}\rceil$. By Proposition~\ref{prop:Qn-UIHPQ}, there exists $\mathrm{d}elta>0$ such that for $n$ large enough, we can couple $Q_{n}^{\sigma_{n}}$ and $Q_{\infty}^{\infty}$ on the same probability space such that with probability at least $1-\varepsilon$, $ B_{\mathrm{d}elta\sqrt{\sigma_n}}\left(Q_{n}^{\sigma_{n}}\right) =B_{\mathrm{d}elta\sqrt{\sigma_n}}\left(Q_{\infty}^{\infty}\right). $ We can and will assume that $\mathrm{d}elta< 2T_0^{-1/4}$. We put $m_n=\lceil \mathrm{d}elta^{-4}a_n^4\rceil$. Then $a_n\leq \mathrm{d}elta\sqrt{\sigma_{m_n}}$. With $m_n$ taking the role of $n$, the last observation enables us to find a coupling between $Q_{m_n}^{\sigma_{m_n}}$ and $Q_{\infty}^{\infty}$ on the same probability space such that for large $n$, we have with probability at least $1-\varepsilon$ \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BHP-coupling2} B_{a_n}\left(Q_{m_n}^{\sigma_{m_n}}\right) =B_{a_n}\left(Q_{\infty}^{\infty}\right). \end{equation} Let $F:\textup{$\mathfrak{m}$}athbb{K}\rightarrow\textup{$\mathfrak{m}$}athbb{R}$ be bounded and continuous, and put $T=\mathrm{d}elta^{-4}(8/9)$. Note that $T\geq T_0$. We work with a coupling of $Q_{m_n}^{\sigma_{m_n}}$ and $Q_{\infty}^{\infty}$ as well as with a coupling of $\mathcal BD_{T,\sqrt{T}}$ and $\mathcal BHP$ such that the properties just mentioned hold. Then \begin{align*} \lefteqn{\left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_1\left(a_n^{-1}\cdot Q_\infty^{\infty}\right)\right)\right]-\textup{$\mathfrak{m}$}athbb E\left[F\left(B_1(\mathcal BHP)\right)\right]\right|}\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \leq \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( a_n^{-1}\cdot B_{a_n}\left(Q_\infty^{\infty}\right)\right)-F\left( a_n^{-1}\cdot B_{a_n}\left(Q_{m_n}^{\sigma_{m_n}}\right)\right)\right]\right|\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad + \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_{1}\left(a_n^{-1}\cdot Q_{m_n}^{\sigma_{m_n}}\right)\right)\right]-\textup{$\mathfrak{m}$}athbb E\left[F\left(B_{1}\left(\mathcal BD_{T,\sqrt{T}}\right)\right)\right]\right|\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad + \left|\textup{$\mathfrak{m}$}athbb E\left[F\left(B_{1}\left(\mathcal BD_{T,\sqrt{T}}\right)\right)-F\left(B_1\left(\mathcal BHP\right)\right)\right]\right|. \end{align*} Using the coupling~\eqref{eq:BHP-coupling2} for the first and the coupling~\eqref{eq:BHP-coupling1} for the third summand on the right hand side, we see that both of them are bounded from above by $2\varepsilon\sup F$. The second summand converges to zero as $n\rightarrow\infty$, using~\eqref{eq:BeMi} and the scaling relation $\mathcal BD_{T,\sqrt{T}}=_d T^{1/4}\mathcal BD_{1}$. This concludes the proof of Theorem~\ref{thm:UIHPQ-BHP}. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:BHP1}] We have to show that when $1\ll\sigma_n\ll n$, we have for every $r\geq 0$ and any sequence $1\ll a_n\ll\textup{$\mathfrak{m}$}in\{\sqrt{\sigma_n},\,\sqrt{n/\sigma_n}\}$, $ B_r(a_n^{-1}\cdot Q_n^{\sigma_n})\longrightarrow B_{r}(\mathcal BHP)$ in distribution in $\textup{$\mathfrak{m}$}athbb{K}$ as $n\rightarrow\infty$. Let $\varepsilon>0$ and $r\geq 0$. By Proposition~\ref{prop:Qn-UIHPQ}, we can couple $Q_{n}^{\sigma_{n}}$ and $Q_{\infty}^{\infty}$ on the same probability space such that with probability at least $1-\varepsilon$, for $n\geq n_0$, $ B_{ra_n}\left(Q_{n}^{\sigma_{n}}\right) =B_{ra_n}\left(Q_{\infty}^{\infty}\right). $ Then, for $F:\textup{$\mathfrak{m}$}athbb{K}\rightarrow\textup{$\mathfrak{m}$}athbb{R}$ bounded and continuous, \begin{align*} \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_r\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\right)-F\left(B_r(\mathcal BHP)\right)\right]\right| &\leq \left|\textup{$\mathfrak{m}$}athbb E\left[F\left(a_n^{-1}\cdot B_{ra_n}\left(Q_n^{\sigma_n}\right)\right)-F\left(a_n^{-1}\cdot B_{ra_n}\left(Q_\infty^\infty\right)\right)\right]\right|\\ &\textup{$\mathfrak{q}$}uad + \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_r\left(\mathcal BHP\right)\right)-F\left(a_n^{-1}\cdot B_{ra_n}\left(Q_\infty^\infty\right)\right)\right]\right|. \end{align*} Under our coupling, the first summand behind the inequality is bounded by $2\varepsilon\sup|F|$ provided $n\geq n_0$. By Theorem~\ref{thm:UIHPQ-BHP}, the second summand converges to zero as $n\rightarrow\infty.$ \end{proof} \begin{remark} \textup{$\mathfrak{l}$}bel{rem:jointconv-CLBr} Notice that in our proofs of the couplings Proposition~\ref{prop:isometry-BD-BHP} (between $\mathcal BD_\sigma$ and $\mathcal BHP$) and Proposition~\ref{prop:Qn-UIHPQ} (between $Q_n^{\sigma_n}$ and $\textup{\textsf{UIHPQ}}$), we construct in fact joint couplings of contour functions, label functions and balls in the corresponding metric spaces. As a consequence, the theorems proved in this section can be strengthened in a way we now exemplify based on Theorem~\ref{thm:UIHPQ-BHP}. Recall that we view the contour and label functions $C_\infty$ and $\textup{$\mathfrak{L}$}_\infty$ which specify the $\textup{\textsf{UIHPQ}}$ $Q_\infty^\infty$ as (random) continuous functions on $\textup{$\mathfrak{m}$}athbb R$. The Brownian half-plane $\mathcal BHP$ is constructed from contour and label functions $X^0=(X^0_t,t\in\textup{$\mathfrak{m}$}athbb R)$ and $W^0=(W^0_t,t\in\textup{$\mathfrak{m}$}athbb R)$ as specified in Section~\ref{sec:recapBHPBD}. We now claim that for each $r\geq 0$ and any positive sequence $a_n\rightarrow\infty$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:jointconv-CLBr} \left(\textup{$\mathfrak{f}$}rac{C_\infty((9/4)a_n^4\cdot)}{(3/2)a_n^2},\textup{$\mathfrak{f}$}rac{\textup{$\mathfrak{L}$}_\infty\left((9/4)a_n^4\cdot\right)}{a_n}, B_r\left(a_n^{-1}\cdot Q_\infty^\infty\right)\right) \xrightarrow[n\to\infty]{(d)}\left(X^0,W^0,B_{r}(\mathcal BHP)\right) \end{equation} jointly in the space $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)^2\times\textup{$\mathfrak{m}$}athbb{K}$. The convergence does also hold with $B_r(a_n^{-1}\cdot Q_\infty^\infty)$ replaced by $B_r^{(0)}(a_n^{-1}\cdot Q_\infty^\infty)$. To see why~\eqref{eq:jointconv-CLBr} holds, one has to slightly enhance the proof of Theorem~\ref{thm:UIHPQ-BHP}. Since all the necessary arguments were already given, we restrict ourselves to a sketch proof and leave it to the reader to fill in the details. We assume $r=1$ for simplicity. Let $T>0$, denote by $(F,W)$ the contour and label function of $\mathcal BD_{T,\sqrt{T}}$, and set $F(-t)=F(T-t)+\sqrt{T}$ and $W(-t)=W(T-t)$ for $t\in[-T,0]$. Now fix $K>0$. Firstly, the arguments in the proof of Proposition~\ref{prop:isometry-BD-BHP} show that for $T>0$ large, one can construct a coupling such that with high probability, Equality~\eqref{eq:BHP-coupling1} holds jointly with an equality of $(F,W)$ and $(X^{\theta},W^{\theta})$ on $[-K,K]^2\subset [-T,T]^2$. Secondly, let $m_n=\lceil\mathrm{d}elta^{-4}a_n^4\rceil$ and $\sigma_{m_n}=\lceil\sqrt{2m_n}\rceil$ as in the proof of Theorem~\ref{thm:UIHPQ-BHP}. We extend the contour function $C_{m_n}$ of $Q_{m_n}^{\sigma_{m_n}}$ to $t\in [-(2m_n+\sigma_{m_n}),0]$ by setting $C_{m_n}(t)=C_{m_n}(2m_n+\sigma_{m_n}+t)+\sigma_{m_n}$. Similarly, we extend the label function $\textup{$\mathfrak{L}$}_{m_n}$ by letting $\textup{$\mathfrak{L}$}_{m_n}(t)=\textup{$\mathfrak{L}$}_{m_n}(2m_n+\sigma_{m_n}+t)$ for $t\in[-(2m_n+\sigma_{m_n}),-1]$, and then by linear interpolation on $[-1,0]$ between $\textup{$\mathfrak{L}$}_{m_n}(-1)$ and $\textup{$\mathfrak{L}$}_{m_n}(0)=0$. From the proof of Proposition~\ref{prop:Qn-UIHPQ} we see that for $\mathrm{d}elta$ small and $n$ large, one can construct a coupling such that with high probability, Equality~\eqref{eq:BHP-coupling2} holds jointly with an equality of $(C_{m_n},\textup{$\mathfrak{L}$}_{m_n})$ and $(C_\infty,\textup{$\mathfrak{L}$}_\infty)$ on $[-Ka_n^4,Ka_n^4]^2$. Thanks to~\cite{Be3,BeMi}, we already know that the convergence of $a_n^{-1}\cdot Q_{m_n}^{\sigma_{m_n}}$ to $\mathcal BD_{T,\sqrt{T}}$ (with $T=\mathrm{d}elta^{-4}(8/9)$) holds jointly with the convergence $$\left(\textup{$\mathfrak{f}$}rac{C_{m_n}((9/4)a_n^4\cdot)}{(3/2)a_n^{2}}, \textup{$\mathfrak{f}$}rac{\textup{$\mathfrak{L}$}_{m_n}((9/4)a_n^4\cdot)}{a_n}\right) \xrightarrow[n\to\infty]{(d)}\left(F,W\right) $$ on $\textup{$\mathfrak{m}$}athcal{C}([-T, T]^2,\textup{$\mathfrak{m}$}athbb R)^2$. Putting these observations together,~\eqref{eq:jointconv-CLBr} follows. We come back to Display~\eqref{eq:jointconv-CLBr} in the proof of Theorem~\ref{thm:BHP3} below. \end{remark} \subsection{Brownian half-plane with non-zero skewness (Theorem~\ref{thm:BHP3})} \textup{$\mathfrak{l}$}bel{sec:brownian-half-plane-1} Theorem~\ref{thm:BHP3} covers the regime $\sqrt{n}\ll \sigma_n\ll n$ when $a_n\sim 2\sqrt{\theta n/3\sigma_n}$ for some $\theta\in (0,\infty).$ The parameter $\theta$ measures the skewness of the limiting Brownian half-plane. Note that the regimes where $\mathcal BHP$ corresponding to the choice $\theta=0$ appears is already treated in Theorem~\ref{thm:BHP1}. We work in the usual setting introduced Section~\ref{sec:usualsetting}; in particular, the pair $(Q_n^{\sigma_n},v^{\bullet})$ consisting of a quadrangulation and a distinguished vertex is uniformly distributed over $\mathcal{Q}_{n,\sigma_n}^\bullet$ and encoded by a triplet $((\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n),\textup{\textsf{b}}_n)\in\mathbb Fo_{\sigma_n}^n \times \textup{$\mathfrak{m}$}athcal B_{\sigma_n}$. The associated contour pair is denoted $(C_n,L_n)$, and the corresponding label function takes the form $\textup{$\mathfrak{L}$}_n(t) = L_n(t) + \textup{\textsf{b}}_n(-\underline{C}_n(t))$, $0\leq t\leq 2n+\sigma_n$. It will be convenient to view both $C_n$ and $\textup{$\mathfrak{L}$}_n$ as continuous functions on $\textup{$\mathfrak{m}$}athbb R$. Let $N=2n+\sigma_n$. We extend $C_n$ first to $[-N,N]$ by $C_n(t)=C_n(N+t)+\sigma_n$ for $t\in[-N,0]$, and then to all reals $t$ by setting $C_n(t)=C_n(t\vee (-N)\wedge N)$. Similarly, we let $\textup{$\mathfrak{L}$}_n(t)=\textup{$\mathfrak{L}$}_n(N+t)$ for $t\in[-N,-1]$, with linear interpolation on $[-1,0]$ between $\textup{$\mathfrak{L}$}_n(-1)$ and $0$. Outside $[-N,N]$, we set $\textup{$\mathfrak{L}$}_n(t)=\textup{$\mathfrak{L}$}_n(t\vee (-N)\wedge N)$. In this way, we interpret $C_n$ and $\textup{$\mathfrak{L}$}_n$ as functions in $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$. Recall that they completely determine $(Q_n^{\sigma_n},v^{\bullet})$. \begin{mdframed} {\bf Idea of the proof.} Fix $r\geq 0$. The ball $B_{ra_n}(Q_n^{\sigma_n})$ of radius $ra_n$ around the root in $Q_n^{\sigma_n}$ is with high probability encoded by the union of the first $ca_n^2$ and last $ca_n^2$ trees of $\textup{$\mathfrak{f}$}_n$ for some $c>0$, together with their labels and the corresponding bridge values along the floor of $\textup{$\mathfrak{f}$}_n$. In Lemma~\ref{lem:RN-Deriv}, we calculate the Radon-Nikodym derivative of the law of these $2ca_n^2$ trees with respect to the law of $2ca_n^2$ independent critical geometric Galton-Watson trees. In this way, we explicitly relate the laws of $B_{ra_n}(Q_n^{\sigma_n})$ and $B_{ra_n}(Q_\infty^{\infty})$ to each other. Since we already know that $a_n^{-1}\cdot B_{ra_n}(Q_\infty^\infty)$ converges to $B_r(\mathcal BHP_0)$ jointly with its properly rescaled contour and label functions, see Remark~\ref{rem:jointconv-CLBr}, it remains to identify the limiting Radon-Nikodym derivative, which we find to be the Radon-Nikodym derivative of a (two-sided) Brownian motion with drift $-\theta$ with respect to standard Brownian motion. An application of the Pitman transform then concludes the proof. \end{mdframed} Let us now give the details and first introduce some supplementary notation. Given a continuous function $f:\textup{$\mathfrak{m}$}athbb{R}\rightarrow\textup{$\mathfrak{m}$}athbb{R}$ and $x\in\textup{$\mathfrak{m}$}athbb{R}$, let $$ U_x(f)=\inf\{t\leq 0: f(t)=x\}\in [-\infty,0],\textup{$\mathfrak{q}$}uad T_x(f)=\inf\{t\geq 0: f(t)=x\}\in [0,\infty]. $$ In words, $U_x(f)$ is the time of the last visit to $x$ to the left of $0$, with $U_x(f)=-\infty$ if there is no such time, and $T_x(f)$ is the first time $f$ visits $x$ to the right of $0$, with $T_x(f)=\infty$ if there is no such time. Of course, we can also apply $T_x$ to functions in $\textup{$\mathfrak{m}$}athcal{C}([0,\infty),\textup{$\mathfrak{m}$}athbb R)$, and $U_x$ to functions in $\textup{$\mathfrak{m}$}athcal{C}((-\infty,0],\textup{$\mathfrak{m}$}athbb R)$. For $f\in\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$ and $x>0$, set $$ v(f,x) = \textup{$\mathfrak{f}$}rac{1}{2}\left(T_{-x}(f)-U_{x}(f) -2x\right) $$ whenever all terms on the right hand side are finite, and $v(f,x)=\infty$ otherwise. Note that if $x$ is an integer and $f$ is the contour path of an infinite forest, then $v(f,x)$ is the total number of edges of the $2x$ trees that are encoded by $f$ along the interval $[U_{x}(f),T_{-x}(f)]$. Let $s>0$ be given. For the rest of this section, we will always set $s_n=\lfloorloor (3/2)sa_n^2\rfloorloor$. Since $a_n^2\ll\sigma_n\ll n$, we will implicitly assume that $n$ is so large such that $s_n<\sigma_n<n$. We first prove an absolute-continuity relation on the interval $[U_{s_n},T_{-s_n}]$ between $C_n$ and the contour function $C_\infty$ of a critical infinite forest. For that purpose, we define two probability laws $\mathbb P_{n,r}$, $\mathbb Q_{n,r}$ on $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$ as follows: \begin{align*} \mathbb P_{n,s}&=\textup{$\mathfrak{m}$}athcal{L}\left(\left(C_n(t\vee U_{s_n}(C_n)\wedge T_{-s_n}(C_n)),t\in\textup{$\mathfrak{m}$}athbb R\right)\right),\\ \mathbb Q_{n,s}&=\textup{$\mathfrak{m}$}athcal{L}\left(\left(C_\infty(t\vee U_{s_n}(C_\infty)\wedge T_{-s_n}(C_\infty)),t\in\textup{$\mathfrak{m}$}athbb R\right)\right). \end{align*} \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:RN-Deriv} Let $s>0$. The laws $\mathbb P_{n,s}$ and $\mathbb Q_{n,s}$ are absolutely continuous with respect to each other. Moreover, given $\varepsilon>0$, there exists $n_0\in\textup{$\mathfrak{m}$}athbb N$ such that for all $n\geq n_0$, with $s_n=\lfloorloor (3/2)sa_n^2\rfloorloor$, $$ \sum_{f\in\textup{supp}(\mathbb P_{n,s})}\left|\mathbb P_{n,s}(f)-\e^{2s\theta-\textup{$\mathfrak{f}$}rac{v(f,s_n)}{(9/4)a_n^4}\theta^2}\mathbb Q_{n,s}(f)\right|\leq \varepsilon, $$ where \textup{supp}$(\mathbb P_{n,s})\subset \textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$ denotes the support of $\mathbb P_{n,s}$ (which is equal to \textup{supp}$(\mathbb Q_{n,s})$). \end{lemma} \begin{proof} From the constructions of $C_n$ and $C_\infty$, it is clear that each realization of $\mathbb P_{n,s}$ is a realization of $\mathbb Q_{n,s}$, and vice versa. Now let $s>0$ and $\varepsilon>0$. We first show that there exists $c_v>0$ such that for $n$ sufficiently large, \begin{equation}\textup{$\mathfrak{l}$}bel{eq:RN-Deriv-toshow1} \sum_{f\in\textup{supp}(\mathbb P_{n,s}):\atop v(f,s_n)>c_v a_n^4}\left|\mathbb P_{n,s}(f)-\e^{2s\theta-\textup{$\mathfrak{f}$}rac{v(f,s_n)}{(9/4)a_n^4}\theta^2}\mathbb Q_{n,s}(f)\right|\leq \varepsilon/2. \end{equation} Since $\theta$ and $s$ are fixed, the last display follows if we show that for some $c_v>0$, $$ \mathbb P_{n,s}\left(f\in \textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R): v(f,s_n)>c_v a_n^4\right)\leq \varepsilon/4,\textup{$\mathfrak{q}$}uad\mathbb Q_{n,s}\left(f\in \textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R): v(f,s_n)>c_v a_n^4\right)\leq \varepsilon/4. $$ Write $T_k$ for the first hitting time of $k$ of a simple random walk started at zero. By construction of $C_\infty$, we have $$ \mathbb Q_{n,s}\left(\{v(f,s_n)>c_v a_n^4\}\right)= \mathbb P\left(T_{-2s_n}>2c_v a_n^4+2s_n\right), $$ and standard random walk estimates give the existence of $n_0\in \textup{$\mathfrak{m}$}athbb N$ and $c_v>0$ (depending on $s$, but $s$ is fixed) such that for $n\geq n_0$, $\mathbb Q_{n,s}(\{v(f,s_n)>c_v a_n^4\})\leq \varepsilon/4.$ Similarly, $$ \mathbb P_{n,s}\left(\{v(f,s_n)>c_v a_n^4\}\right) = \mathbb P\left(T_{-2s_n}>2c_va_n^4+2s_n\,|\,T_{-\sigma_n}=2n+\sigma_n\right), $$ and since $\sigma_n\gg \sqrt{n}$, it is easy to check that the probability on the right is bounded by the unconditioned probability $\mathbb P\left(T_{-2s_n}>2c_v a_n^4+2s_n\right)\leq \varepsilon/4$. This shows~\eqref{eq:RN-Deriv-toshow1}. It remains to argue that for fixed $c_v$ and all $n$ large enough, we have also \begin{equation}\textup{$\mathfrak{l}$}bel{eq:RN-Deriv-toshow2} \sum_{f\in\textup{supp}(\mathbb P_{n,s}):\atop v(f,s_n)\leq c_v a_n^4}\left|\mathbb P_{n,s}(f)-\e^{2s\theta-\textup{$\mathfrak{f}$}rac{v(f,s_n)}{(9/4)a_n^4}\theta^2}\mathbb Q_{n,s}(f)\right|\leq \varepsilon/2. \end{equation} In this regard, consider a sequence $f_n\in\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$ of functions in the support of $\mathbb P_{n,s}$ such that $v_n=v(f_n,s_n)\leq c_v a_n^4$. Let $$x_n=\sigma_n - 2s_n, \textup{$\mathfrak{q}$}uad y_n = 2(n-v_n)+ \sigma_n -2s_n.$$ We can assume that both $x_n$ and $y_n$ are positive numbers. Let $(S(i), i \in \textup{$\mathfrak{m}$}athbb N_0)$ denote a simple random walk started at $S(0)=0$. The probability $\mathbb P_{n,s}(f_n)$ is given by the probability to observe $2s_n$ particular trees of total size $v_n$ in a forest of size $n$ with $\sigma_n$ trees. By Kemperman's formula~\eqref{eq:Kemperman}, we obtain \begin{align} \mathbb P_{n,s}(f_n) &= \textup{$\mathfrak{f}$}rac{\textup{$\mathfrak{f}$}rac{x_n}{y_n }2^{y_n} \mathbb P(S(y_n) = x_n)}{\textup{$\mathfrak{f}$}rac{\sigma_n}{2n+\sigma_n} 2^{2n+\sigma_n} \mathbb P(S(2n+\sigma_n)= \sigma_n)}\nonumber\\ &= \textup{$\mathfrak{f}$}rac{x_n}{y_n}\textup{$\mathfrak{f}$}rac{2n+\sigma_n}{\sigma_n}2^{-2(v_n+s_n)} \textup{$\mathfrak{f}$}rac{\mathbb P(S(y_n) = x_n)}{\mathbb P(S(2n+\sigma_n)= \sigma_n)}\,. \textup{$\mathfrak{l}$}bel{eq:RN-Deriv-eq1} \end{align} By definition of $C_\infty$, $\mathbb Q_{n,s}(f_n)$ is given by a particular realization of $2s_n$ independent critical geometric Galton-Watson trees with $v_n$ edges in total. Therefore, by~\eqref{eq:criticalGW}, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:RN-Deriv-eq2} \mathbb Q_{n,s}(f_n)=2^{-2(v_n + s_n)}. \end{equation} Moreover, by assumption on $\sigma_n$ and $a_n$, we have uniformly in all possible choices of $f_n$ that satisfy $v_n\leq c_va_n^4$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:RN-Deriv-eq3} \left|\textup{$\mathfrak{f}$}rac{x_n}{y_n}\textup{$\mathfrak{f}$}rac{2n+\sigma_n}{\sigma_n}-1\right| = o(1). \end{equation} Since $\sigma_n \gg \sqrt n$, the fraction of random walk probabilities in~\eqref{eq:RN-Deriv-eq1} is not controlled well-enough by a standard local central limit theorem as formulated in~\eqref{eq:localCLT}. Instead, we use~\eqref{eq:localCLT2} and obtain \begin{equation} \textup{$\mathfrak{l}$}bel{eq:RN-Deriv-eq4} \textup{$\mathfrak{f}$}rac{\mathbb P(S(y_n) = x_n)}{\mathbb P(S(2n+\sigma_n)= \sigma_n )} = \exp\left(-\sum_{\ell = 1}^\infty \textup{$\mathfrak{f}$}rac{1}{2\ell (2 \ell -1)} \left(\textup{$\mathfrak{f}$}rac{x_n^{2 \ell}}{ {y_n}^{2 \ell -1}} -\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{ (2n+\sigma_n)^{2\ell-1}}\right)\right)\left(1+o(1)\right). \end{equation} We now analyze the terms in the sum inside the exponential in the last display, similarly to the proof of Lemma~\ref{lem:GW3}. Firstly, \begin{multline} \textup{$\mathfrak{f}$}rac{x_n^{2 \ell}}{ {y_n}^{2 \ell -1}} -\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{ (2n+\sigma_n)^{2\ell-1}} = \\ \textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}}\left[ -2 \ell \textup{$\mathfrak{f}$}rac{2s_n}{\sigma_n} + (2 \ell -1) \textup{$\mathfrak{f}$}rac{2(v_n + s_n)}{2n+\sigma_n} + O\left(\left(\textup{$\mathfrak{f}$}rac{s_n}{\sigma_n}\right)^2 \right ) + O\left(\left(\textup{$\mathfrak{f}$}rac{v_n+ s_n}{2n+\sigma_n}\right)^2\right) \right]. \textup{$\mathfrak{l}$}bel{eq:RN-Deriv-eq5} \end{multline} We now observe that \begin{align*} -2\ell\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}} \textup{$\mathfrak{f}$}rac{ 2s_n}{\sigma_n} & = (-4\ell s\theta+o(1))\textup{$\mathfrak{f}$}rac{\sigma_n^{2(\ell-1)}}{(2n+\sigma_n)^{2 (\ell -1)}},\textup{$\mathfrak{q}$}uad\hbox{ and}\\ (2\ell-1) \textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}} \textup{$\mathfrak{f}$}rac{2(v_n + s_n)}{2n+\sigma_n} & = (2\ell-1)\textup{$\mathfrak{f}$}rac{2v_n}{(9/4)a_n^4}\left(\theta^2+o(1)\right)\textup{$\mathfrak{f}$}rac{\sigma_n^{2(\ell-1)}}{(2n+\sigma_n)^{2 (\ell -1)}}. \end{align*} Since $\sigma_n\ll n$, we deduce from the last display that if $\ell\geq 2$, all the terms in~\eqref{eq:RN-Deriv-eq5} converge to $0$ as $n\rightarrow\infty$. If $\ell =1$, \begin{align*} -2\ell\textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}} \textup{$\mathfrak{f}$}rac{ 2s_n}{\sigma_n} & = -4s\theta+o(1),\textup{$\mathfrak{q}$}uad\hbox{ and}\\ (2\ell-1) \textup{$\mathfrak{f}$}rac{\sigma_n^{2\ell}}{(2n+\sigma_n)^{2 \ell -1}} \textup{$\mathfrak{f}$}rac{2(v_n + s_n)}{2n+\sigma_n} & = \textup{$\mathfrak{f}$}rac{2v_n}{(9/4)a_n^4}\theta^2 +o(1). \end{align*} For $n$ large enough, $\sigma_n/(2n+\sigma_n) <1/2$, so that each term in the sum in~\eqref{eq:RN-Deriv-eq4} is bounded by $C(1/2)^{2(\ell-1)}$ for some universal constant $C>0$, which is summable. Therefore, by dominated convergence $$ \textup{$\mathfrak{f}$}rac{\mathbb P(S(y_n) = x_n)}{\mathbb P(S(2n+\sigma_n)= \sigma_n )} = \exp\left( 2s \theta - \textup{$\mathfrak{f}$}rac{v_n}{(9/4)a_n^4}\theta^2\right) +o(1). $$ Note that all the error terms above do depend on $f_n$ only through the constant $c_v$. Combining the last display with~\eqref{eq:RN-Deriv-eq2} and~\eqref{eq:RN-Deriv-eq3},~\eqref{eq:RN-Deriv-toshow2} and hence the claim of the lemma follow. \end{proof} \begin{remark} Note that $C_\infty$ is a discrete analog of the contour function of the Brownian half-plane $\mathcal BHP$: The process $(C_\infty(i),i\in \textup{$\mathfrak{m}$}athbb N_0)$ is a simple random walk, and if $S=(S(i),i\in\textup{$\mathfrak{m}$}athbb N_0)$ denotes another (independent) simple random walk, then it is straightforward to check that $$ (C_\infty(-i),i\in\textup{$\mathfrak{m}$}athbb N)=_d \left(S(i+1)-2\textup{$\mathfrak{m}$}in_{0\leq \ell\leq i+1}S(\ell) +1,\,i\in\textup{$\mathfrak{m}$}athbb N\right), $$ i.e., $(C_\infty(-i),i\in\textup{$\mathfrak{m}$}athbb N)$ is a discrete Pitman-type transform of a simple random walk. In particular, $-U_k(C_\infty)=_d T_{-k}(S)$. \end{remark} For proving Theorem~\ref{thm:BHP3}, it is convenient to introduce some more notation. Let us first define rescaled versions of the contour and label functions $C_n$ and $\textup{$\mathfrak{L}$}_n$ that capture the information encoded by the first $s_n=\lfloorloor (3/2)sa_n^2\rfloorloor$ trees $(\tau_0,\ldots,\tau_{s_n-1})$ and the last $s_n$ trees $(\tau_{\sigma_n-s_n},\ldots,\tau_{\sigma_n-1})$ of $\textup{$\mathfrak{f}$}_n$, \begin{align*} C_{n,s}=\left(C_{n,s}(t),t\in\textup{$\mathfrak{m}$}athbb R\right)&=\left(\textup{$\mathfrak{f}$}rac{1}{(3/2)a_n^2}C_{n}\left((9/4)a_n^4t\vee U_{s_n}(C_n)\wedge T_{-s_n}(C_n)\right),t\in \textup{$\mathfrak{m}$}athbb R\right),\\ \textup{$\mathfrak{L}$}_{n,s}=\left(\textup{$\mathfrak{L}$}_{n,s}(t),t\in\textup{$\mathfrak{m}$}athbb R\right)&=\left(\textup{$\mathfrak{f}$}rac{1}{a_n}\textup{$\mathfrak{L}$}_{n}\left((9/4)a_n^4t\vee U_{s_n}(C_n)\wedge T_{-s_n}(C_n)\right),t\in \textup{$\mathfrak{m}$}athbb R\right). \end{align*} Let $((\textup{$\mathfrak{f}$}_{\infty},\textup{$\mathfrak{l}$}_\infty),\textup{\textsf{b}}_\infty)$ encode the $\textup{\textsf{UIHPQ}}$, with $C_\infty$ and $\textup{$\mathfrak{L}$}_\infty$ denoting the associated contour and label functions. In analogy to the last display, we set \begin{align*} C^{\infty}_{n,s}=\left(C^{\infty}_{n,s}(t),t\in\textup{$\mathfrak{m}$}athbb R\right)&=\left(\textup{$\mathfrak{f}$}rac{1}{(3/2)a_n^2}C_{\infty}\left((9/4)a_n^4t\vee U_{s_n}(C_{\infty})\wedge T_{-s_n}(C_{\infty})\right),t\in \textup{$\mathfrak{m}$}athbb R\right),\\ \textup{$\mathfrak{L}$}_{n,s}^\infty=\left(\textup{$\mathfrak{L}$}_{n,s}^\infty(t),t\in\textup{$\mathfrak{m}$}athbb R\right)&=\left(\textup{$\mathfrak{f}$}rac{1}{a_n}\textup{$\mathfrak{L}$}_{\infty}\left((9/4)a_n^4t\vee U_{s_n}(C_{\infty})\wedge T_{-s_n}(C_{\infty})\right),t\in \textup{$\mathfrak{m}$}athbb R\right). \end{align*} We recapitulate the definition of the contour and label functions $X^\theta=(X^\theta(t),t\in\textup{$\mathfrak{m}$}athbb R)$ and $W^\theta=(W^\theta(t),t\in\textup{$\mathfrak{m}$}athbb R)$ which encode the Brownian half-plane $\mathcal BHP_\theta$: $(X^\theta(t),t\geq 0)$ is given by a Brownian motion with linear drift $-\theta$, and $(X^\theta(-t),t\geq 0)$ is the Pitman transform of an (independent) copy of $(X^\theta(t),t\geq 0)$. Moreover, conditionally on $X^\theta$, the label function $W^\theta=(W^\theta(t),t\in\textup{$\mathfrak{m}$}athbb R)$ is given by $W^\theta(t)=\gamma(-\underline{X}^\theta(t))+Z^\theta(t)$, $t\in \textup{$\mathfrak{m}$}athbb R$, where $Z^\theta=(Z^\theta(t),t\in\textup{$\mathfrak{m}$}athbb R)$ is the random snake driven by $X^\theta-\underline{X}^\theta$, and $\gamma=(\gamma(t),t\in\textup{$\mathfrak{m}$}athbb R)$ is a two-sided Brownian motion with $\gamma(0)=0$ and scaled by the factor $\sqrt{3}$, independent of $Z^{X^\theta-\underline{X}^\theta}$. We set \begin{align*} X^{\theta,s}=\left(X^{\theta,s}(t),t\in\textup{$\mathfrak{m}$}athbb R\right)&=\left(X^\theta\left(t\vee U_{s}(X^\theta)\wedge T_{-s}(X^\theta)\right),t\in \textup{$\mathfrak{m}$}athbb R\right),\\ W^{\theta,s}=\left(W^{\theta,s}(t),t\in\textup{$\mathfrak{m}$}athbb R\right)&=\left(W^\theta\left(t\vee U_{s}(X^\theta)\wedge T_{-s}(X^\theta)\right),t\in \textup{$\mathfrak{m}$}athbb R\right). \end{align*} Finally, for $f\in\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)$, put $$ \textup{$\mathfrak{l}$}mbda_{n,s}(f)=\exp\left(2s\theta-\textup{$\mathfrak{f}$}rac{v(f,s_n)}{(9/4)a_n^4}\theta^2\right). $$ \begin{proof}[Proof of Theorem~\ref{thm:BHP3}] Let $r\geq 0$. By Lemma~\ref{lem:ball0}, our claim follows if we show that $$ B_r^{(0)}\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\xrightarrow[n \to \infty]{(d)}B_r(\mathcal BHP_{\theta})$$ in distribution in $\textup{$\mathfrak{m}$}athbb{K}$, where we recall that $\theta=\lim_{n\rightarrow\infty}(3/2)a_n^2\sigma_n/2n.$ For $n\in \textup{$\mathfrak{m}$}athbb N$ and $s>0$, define the events $$ \textup{$\mathfrak{m}$}athcal{E}^1(n,s)=\left\{\textup{$\mathfrak{m}$}in_{[0,\,s_n]}\textup{\textsf{b}}_n <-3ra_n,\, \textup{$\mathfrak{m}$}in_{[\sigma_n-s_n,\,\sigma_n-1]}\textup{\textsf{b}}_n<-3ra_n\right\}\cap\left\{\textup{$\mathfrak{m}$}in_{[s_n+1,\sigma_n-(s_n+1)]}\textup{\textsf{b}}_n<-3ra_n\right\}$$ and similarly \begin{align*}\textup{$\mathfrak{m}$}athcal{E}^2(n,s)&=\left\{\textup{$\mathfrak{m}$}in_{[0,\,s_n]}\textup{\textsf{b}}_\infty <-\,3ra_n,\, \textup{$\mathfrak{m}$}in_{[-s_n,\,0]}\textup{\textsf{b}}_\infty<-3ra_n\right\},\\ \textup{$\mathfrak{m}$}athcal{E}^3(s)&=\left\{\textup{$\mathfrak{m}$}in_{[0,s]}\gamma <-3r,\, \textup{$\mathfrak{m}$}in_{[-s,0]}\gamma<-3r\right\}. \end{align*} Let $\varepsilon>0$ be given. Applying Lemma~\ref{lem:bridge0}, we find $n_0\in \textup{$\mathfrak{m}$}athbb N$ and $s>0$ sufficiently large such that for $n\geq n_0$, $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^1(n,s))\geq 1-\varepsilon$. For possibly larger values of $n$ and $s$, Donsker's invariance principle shows that also $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^2(n,s))\geq 1-\varepsilon$, and standard properties of Brownian motion give $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^3(s))\geq 1-\varepsilon$ for $s$ large enough. We now fix $s>0$ and $n_0\in\textup{$\mathfrak{m}$}athbb N$ such that for all $n\geq n_0$, each of the events $\textup{$\mathfrak{m}$}athcal{E}^1,\textup{$\mathfrak{m}$}athcal{E}^2,\textup{$\mathfrak{m}$}athcal{E}^3$ has probability at least $1-\varepsilon$. As in the proof of Proposition~\ref{prop:Qn-UIHPQ}, we write $\tau(\textup{$\mathfrak{f}$}_\infty,k)$ for the tree of $\textup{$\mathfrak{f}$}_\infty$ which is attached to $(k)$, $k\in\textup{$\mathfrak{m}$}athbb Z$. We identify $V(\textup{$\mathfrak{f}$}_\infty)$ with $V(Q_\infty^\infty)$, as usual. Recall that the root $\rho$ of $\textup{\textsf{UIHPQ}}$ is at distance at most $-\textup{\textsf{b}}_\infty(-1)+1$ away from $(0)$. On the event $\textup{$\mathfrak{m}$}athcal{E}^2(n,s)$, the cactus bound~\eqref{eq:cactus3} thus gives for vertices $v\in V(Q_\infty^\infty)$ which do not belong to any of the trees $\tau(\textup{$\mathfrak{f}$}_\infty,k)$, $k=-s_n,\ldots,s_n$, $$ d_{\infty}(0,v) \geq -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,s_n]}\textup{\textsf{b}}_\infty, \textup{$\mathfrak{m}$}in_{[-s_n,0]}\textup{\textsf{b}}_\infty\right\} \geq 3ra_n $$ for large $n$. Since for vertices $u,v$ in $B_{ra_n}^{(0)}(Q_\infty^\infty)$, any geodesic between $u$ and $v$ in $Q_\infty^\infty$ lies entirely in $B_{2ra_n}^{(0)}(Q_\infty^\infty)$, we obtain from the construction of edges in the Bouttier-Di Francesco-Guitter mapping that the submap $B_{ra_n}^{(0)}(Q_\infty^\infty)$ is a measurable function of $(C_{n,s}^\infty,\textup{$\mathfrak{L}$}_{n,s}^\infty)$. A similar argument which we leave to the reader (see also the first part of the proof of Proposition~\ref{prop:Qn-UIHPQ}) shows that on $\textup{$\mathfrak{m}$}athcal{E}^1(n,s)$, the submap $B_{ra_n}^{(0)}(Q_n^{\sigma_n})$ is given by the {\it same} function of $(C_{n,s},\textup{$\mathfrak{L}$}_{n,s})$. Moreover, on $\textup{$\mathfrak{m}$}athcal{E}^3(s)$, $B_r(\mathcal BHP)$ is determined by $(X_{0,s},W_{0,s})$. By Lemma~\ref{lem:bridge1}, recalling that $a^2_n\ll\sigma_n$, we have for large $n$ \begin{equation*} \begin{split}&\left\|\textup{$\mathfrak{m}$}athcal{L}((\textup{\textsf{b}}_n(\sigma_n-s_n),\ldots,\textup{\textsf{b}}_n(\sigma_n-1),\textup{\textsf{b}}_n(0),\textup{\textsf{b}}_n(1),\ldots,\textup{\textsf{b}}_n(s_n)))\right.\\ &\textup{$\mathfrak{q}$}uad\left. -\,\textup{$\mathfrak{m}$}athcal{L}((\textup{\textsf{b}}_\infty(-s_n),\ldots,\textup{\textsf{b}}_\infty(-1),\textup{\textsf{b}}_\infty(0),\textup{\textsf{b}}_\infty(1),\ldots,\textup{\textsf{b}}_\infty(s_n)))\right\|_{\textup{TV}}\leq \varepsilon.\end{split} \end{equation*} Combining this bound with Lemma~\ref{lem:RN-Deriv}, the above observations entail that for any measurable and bounded $F: \textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)^2\times\textup{$\mathfrak{m}$}athbb{K}\rightarrow\textup{$\mathfrak{m}$}athbb R$ and $n$ large enough \begin{multline} \textup{$\mathfrak{l}$}bel{eq:BHP3-mainproof-eq1} \left|\textup{$\mathfrak{m}$}athbb E\left[F\left(C_{n,s},\textup{$\mathfrak{L}$}_{n,s},B_r^{(0)}\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\right)1 \textup{$\mathfrak{m}$}kern -6mu 1_{\textup{$\mathfrak{m}$}athcal{E}^1(n,s)}\right] -\right.\\ \left.\textup{$\mathfrak{m}$}athbb E\left[\textup{$\mathfrak{l}$}mbda_{n,s}(C_\infty)F\left(C^\infty_{n,s}, \textup{$\mathfrak{L}$}_{n,s}^\infty,B_r^{(0)}\left(a_n^{-1}\cdot Q_\infty^\infty\right)\right)1 \textup{$\mathfrak{m}$}kern -6mu 1_{\textup{$\mathfrak{m}$}athcal{E}^2(n,s)}\right]\right|\leq C\varepsilon, \end{multline} where $C>0$ is a constant that depends only on $F$ and $\theta,s$, which are fixed. Recall from the proof of Lemma~\ref{lem:RN-Deriv} that for each $\mathrm{d}elta>0$, we find $c_\mathrm{d}elta>0$ such that $\mathbb P(v(C_\infty,s_n)>c_\mathrm{d}elta a_n^4)\leq \mathrm{d}elta$. The joint convergence~\eqref{eq:jointconv-CLBr} thus implies $$ \left(C^\infty_{n,s}, \textup{$\mathfrak{L}$}_{n,s}^\infty,B_r^{(0)}\left(a_n^{-1}\cdot Q_\infty^\infty\right)\right)\xrightarrow[n\to\infty]{(d)} \left(X^{0,s},W^{0,s},B_r(\mathcal BHP)\right) $$ in $\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)^2\times \textup{$\mathfrak{m}$}athbb{K}$, and $$\textup{$\mathfrak{f}$}rac{v(C_\infty,s_n)}{(9/4)a_n^4}\xrightarrow[n \to \infty]{(d)}\textup{$\mathfrak{f}$}rac{1}{2}\left(T_{-s}-U_s\right)(X^{0}), $$ where, in hopefully obvious notation, $X^0$ stands for the contour function of the Brownian half-plane with zero skewness, and $X^{0,s}$,$W^{0,s}$ were defined above in terms of $\mathcal BHP$. For large $n$, we can therefore ensure that \begin{multline} \textup{$\mathfrak{l}$}bel{eq:BHP3-mainproof-eq2} \mathcal Big|\textup{$\mathfrak{m}$}athbb E\left[\textup{$\mathfrak{l}$}mbda_{n,s}(C_\infty)F\left(C^\infty_{n,s}, \textup{$\mathfrak{L}$}_{n,s}^\infty,B_r^{(0)}\left(a_n^{-1}\cdot Q_\infty^\infty\right)\right)\right] -\\ \textup{$\mathfrak{m}$}athbb E\left[\exp\left(2s\theta-(T_{-s}-U_s)(X^0)\theta^2/2\right)F\left(X^{0,s},W^{0,s},B_r\left(\mathcal BHP\right)\right)\right]\mathcal Big|\leq \varepsilon. \end{multline} We will now rewrite the second expectation in the last display using Girsanov's (and implicitly Pitman's) transform. More specifically, an application of Girsanov's theorem for Brownian motion with drift $-\theta$ (see, e.g.,~\cite[Chapter 3.5]{KaSc}) shows that for $G:\textup{$\mathfrak{m}$}athcal{C}(\textup{$\mathfrak{m}$}athbb R,\textup{$\mathfrak{m}$}athbb R)\rightarrow\textup{$\mathfrak{m}$}athbb R$ continuous and bounded, $$ \textup{$\mathfrak{m}$}athbb E\left[\exp\left(2s\theta-(T_{-s}-U_s)(X^0)\theta^2/2\right)G\left(X^{0,s}\right)\right]=\textup{$\mathfrak{m}$}athbb E\left[G\left(X^{\theta,s}\right)\right]. $$ Since on the event $\textup{$\mathfrak{m}$}athcal{E}^3(s)$, $B_r(\mathcal BHP)$ is a measurable function of $(X^{0,s},W^{0,s})$ (and $B_r(\mathcal BHP_\theta)$ is given by the {\it same} measurable function of $(X^{\theta,s},W^{\theta,s})$), we obtain \begin{multline} \textup{$\mathfrak{l}$}bel{eq:BHP3-mainproof-eq3} \textup{$\mathfrak{m}$}athbb E\left[\exp\left(2s\theta-(T_{-s}-U_s)(X^0)\theta^2/2\right)F\left(X^{0,s},W^{0,s},B_r\left(\mathcal BHP\right)\right)1 \textup{$\mathfrak{m}$}kern -6mu 1_{\textup{$\mathfrak{m}$}athcal{E}^3(s)}\right]\\ =\textup{$\mathfrak{m}$}athbb E\left[F\left(X^{\theta,s},W^{\theta,s},B_r\left(\mathcal BHP_\theta\right)\right)1 \textup{$\mathfrak{m}$}kern -6mu 1_{\textup{$\mathfrak{m}$}athcal{E}^3(s)}\right]. \end{multline} Using that the three events $\textup{$\mathfrak{m}$}athcal{E}^1(n,s)$, $\textup{$\mathfrak{m}$}athcal{E}^2(n,s)$ and $\textup{$\mathfrak{m}$}athcal{E}^3(s)$ have all probability at least $1-\varepsilon$, a combination of~\eqref{eq:BHP3-mainproof-eq1},~\eqref{eq:BHP3-mainproof-eq2} and~\eqref{eq:BHP3-mainproof-eq3} shows that for large $n$ $$ \left|\textup{$\mathfrak{m}$}athbb E\left[F\left(C_{n,s},\textup{$\mathfrak{L}$}_{n,s},B_r^{(0)}\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\right)\right] - \textup{$\mathfrak{m}$}athbb E\left[F\left(X^{\theta,s},W^{\theta,s},B_r\left(\mathcal BHP_\theta\right)\right)\right]\right|\leq C'\varepsilon $$ for some $C'$ depending only on $F$ and $s,\theta$. Clearly, this implies our claim. \end{proof} \subsection{Coupling Brownian disks (Theorem~\ref{thm:coupling-BD-IBD} and Corollary~\ref{cor:topology-IBD})} \textup{$\mathfrak{l}$}bel{sec:proof-coupling-BD-IBD} The main ideas are similar to those of Section~\ref{sec:proof-coupling-BD-BHP}, but closer in spirit to those of~\cite{CuLG}. We begin with showing how Theorem~\ref{thm:coupling-BD-IBD} implies that $\textup{\textsf{IBD}}_\sigma$ is homeomorphic to the pointed closed disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$. \begin{proof}[Proof of Corollary~\ref{cor:topology-IBD}] The arguments are similar to the proof of Corollary~\ref{cor:topology-BHP}. First, Theorem~\ref{thm:coupling-BD-IBD} shows that with probability $1$, for every $r>0$, the ball $B_r(\textup{\textsf{IBD}}_\sigma)$ is contained in a set homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$. In particular, $\textup{\textsf{IBD}}_\sigma$ is a non-compact surface with a boundary homeomorphic to the circle $\textup{$\mathfrak{m}$}athbb{S}^1$, and it has only one end. Let us glue a copy $D$ of $\overline{\textup{$\mathfrak{m}$}athbb{D}}$ along the boundary of $\textup{\textsf{IBD}}_\sigma$, hence obtaining a non-compact surface $S$ without boundary, which is now simply connected. This surface is thus homeomorphic to $\textup{$\mathfrak{m}$}athbb R^2$. Again, the Jordan-Schoenflies theorem shows that any homeomorphism from the boundary of $\textup{\textsf{IBD}}_\sigma$ to $\textup{$\mathfrak{m}$}athbb{S}^1$ can be extended to a homeomorphism from $S$ to $\textup{$\mathfrak{m}$}athbb R^2$, and this homeomorphism must send $\textup{\textsf{IBD}}_\sigma$ to the unbounded region $\{z:|z|\geq 1\}$, which in turn is homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus\{0\}$, as wanted. \end{proof} As in Section~\ref{sec:proof-coupling-BD-BHP}, we first prove the following simplification of Theorem~\ref{thm:coupling-BD-IBD}. \begin{prop}\textup{$\mathfrak{l}$}bel{prop:isometry-BD-IBD} Fix $\sigma \in (0,\infty)$, and let $\varepsilon > 0$, $r \geq 0$. There exists $T_0 = T_0(\varepsilon, r, \sigma)$ such that for all $T \geq T_0$, we can construct copies of $\mathcal BD_{T,\sigma}$ and $\textup{\textsf{IBD}}_\sigma$ on the same probability space such that with probability at least $1-\varepsilon$, the balls $B_r(\mathcal BD_{T,\sigma})$ and $B_r(\textup{\textsf{IBD}}_\sigma)$ of radius $r$ around the respective roots are isometric. \end{prop} The crucial step in the proof of the proposition is to show how one can couple the processes encoding $\mathcal BD_{T,\sigma}$ and $\textup{\textsf{IBD}}_\sigma$. The rest of the proof then uses arguments very close to those given in~\cite[Section 3.2]{CuLG} and in Section~\ref{sec:proof-coupling-BD-BHP} above. \subsubsection{Coupling of contour and label functions} Throughout this section, $\sigma\in(0,\infty)$ is fixed, and $T$ denotes always a strictly positive real. We recall that the main building block of the Brownian disk $\mathcal BD_{T,\sigma}$ is a first-passage Brownian bridge from $0$ to $-\sigma$ and duration $T$. Let $B=(B_t,t\geq 0)$ be a standard Brownian motion. In this section, it will be convenient to write $T_x=\inf\{t\geq 0: B_t<-x\}$ for the first hitting time of $(-\infty,-x)$ of the process $B$, so that $(T_x,0\leq x\leq \sigma)$ is a stable subordinator of index $1/2$ and Laplace exponent $-\log\textup{$\mathfrak{m}$}athbb E[\exp(-\textup{$\mathfrak{l}$}mbda T_1)]=\sqrt{2\textup{$\mathfrak{l}$}mbda}$. Let us write the jump sizes of $(T_x,0\leq x\leq \sigma)$, together with the times in $[0,\sigma]$ at which they occur, as a point measure $$\textup{$\mathfrak{m}$}athcal{M}=\sum_{i\geq 1}\mathrm{d}elta_{(\Delta_i,U_i)}\, ,$$ so that $T_{U_i}-T_{U_i-}=\Delta_i$. By well-known properties of subordinators, this measure is Poisson with intensity measure $(2\pi y^3)^{-1/2}\mathrm{d} y\otimes \mathrm{d} u1 \textup{$\mathfrak{m}$}kern -6mu 1_{[0,\sigma]}(u)$. As a consequence, the random variables $U_i,i\geq 1$, are i.i.d.\ uniform in $[0,\sigma]$ and independent of $(\Delta_1,\Delta_2,\ldots)$. This property will remain true when we condition the measure $\textup{$\mathfrak{m}$}athcal{M}$ on events that involve only the sequence $(\Delta_1,\Delta_2,\ldots)$. The first-passage bridge consists in the process $(B_t,0\leq t\leq T)$ conditioned on the event $\{T_{\sigma}=T\}=\{\sum_i\Delta_i=T\}$. In order to describe the conditional law of $\textup{$\mathfrak{m}$}athcal{M}$, we follow Pitman~\cite[Chapter 4]{Pi} and fix the ordering $\Delta_1,\Delta_2,\ldots$ as the {\em size-biased ordering} of the jumps, so that conditionally given $(\Delta_1,\ldots,\Delta_i)$, $\Delta_{i+1}$ is chosen from all the remaining jumps with probability that is proportional to its size. \begin{lemma}[\cite{Pi}] \textup{$\mathfrak{l}$}bel{lem:coupl-brown-disks} Conditionally given $\{T_{\sigma}=T\}$, the law of $\Delta_1$ is $$\mathbb P(\Delta_1\in \mathrm{d} y| T_{\sigma}=T)=\textup{$\mathfrak{f}$}rac{\sigma \mathrm{d} y}{T\, (2\pi y)^{1/2}}\textup{$\mathfrak{f}$}rac{q_\sigma(T-y)}{q_\sigma(T)}=\e^{\sigma^2/2T}\sqrt{\textup{$\mathfrak{f}$}rac{T}{y}}q_\sigma(T-y)\mathrm{d} y\, ,$$ and conditionally given $\{T_{\sigma}=T\, ,\, \Delta_1=y\}$, the remaining jumps $(\Delta_2,\Delta_3,\ldots)$ have the same distribution as $(\Delta_1,\Delta_2,\ldots)$ conditionally given $\{T_{\sigma}=T-y\}$. \end{lemma} This allows us to obtain the main technical lemma of this section, which one should see as the continuum version of Lemmas~\ref{lem:GW1} and~\ref{lem:GW2}: it says that given, $T_{\sigma}=T$, the jumps behave as those of the unconditioned subordinator $(T_{x},0\leq x\leq \sigma)$, with the exception of the largest jump of size approximately $T$. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:coupl-brown-disks-1} \begin{enumerate} \item For every $\mathrm{d}elta\in (0,1)$, one has $$\liminf_{T\to\infty}\mathbb P\left(\Delta_1>(1-\mathrm{d}elta)T\, \bigg|\, \sum_i\Delta_i=T\right)=1\, .$$ \item One has $$\lim_{T\rightarrow \infty}\left\|\textup{$\mathfrak{m}$}athcal{L}\left(\Delta_2,\Delta_3,\ldots\, \bigg|\, \sum_i\Delta_i=T\right)-\textup{$\mathfrak{m}$}athcal{L}(\Delta_1,\Delta_2,\ldots)\right\|_{\textup{$\mathfrak{m}$}athrm{TV}}=0.$$ \end{enumerate} \end{lemma} \begin{proof} From the description of the conditional law of $\Delta_1$ given in Lemma~\ref{lem:coupl-brown-disks}, we obtain $$\mathbb P(\Delta_1>(1-\mathrm{d}elta)T\, |\, \sum_i\Delta_i=T)=\e^{\sigma^2/2T}\int_0^{\mathrm{d}elta T}\mathrm{d} x\sqrt{\textup{$\mathfrak{f}$}rac{T}{T-x}}q_\sigma(x)\mathrm{d} x\underset{T\to\infty}{\longrightarrow} \int_0^\infty q_\sigma(x)\mathrm{d} x=1\, ,$$ by dominated convergence since $\sqrt{T/(T-x)}\leq (1-\mathrm{d}elta)^{-1/2}$. This proves (a). For (b), one can use the second part of Lemma~\ref{lem:coupl-brown-disks} to obtain the disintegration $$\textup{$\mathfrak{m}$}athcal{L}\left(\Delta_2,\Delta_3,\ldots\, \bigg|\, \sum_i\Delta_i=T\right) =\int_0^T \mathrm{d} x\, \e^{\sigma^2/2T}\sqrt{\textup{$\mathfrak{f}$}rac{T}{T-x}}\, q_\sigma(x)\textup{$\mathfrak{m}$}athcal{L}\left(\Delta_1,\Delta_2,\ldots\, \bigg|\, \sum_i\Delta_i=x\right)\, .$$ Since $q_\sigma$ is the density function of $T_{\sigma}=\sum_i\Delta_i$, we also have the disintegration $$\textup{$\mathfrak{m}$}athcal{L}\left(\Delta_1,\Delta_2,\ldots\right) =\int_0^\infty \mathrm{d} x \, q_\sigma(x)\textup{$\mathfrak{m}$}athcal{L}\left(\Delta_1,\Delta_2,\ldots\, \bigg|\, \sum_i\Delta_i=x\right)\, ,$$ which entails that \begin{align*} \lefteqn{\left\|\textup{$\mathfrak{m}$}athcal{L}\left(\Delta_2,\Delta_3,\ldots\, \bigg|\, \sum_i\Delta_i=T\right)-\textup{$\mathfrak{m}$}athcal{L}(\Delta_1,\Delta_2,\ldots)\right\|_{\textup{$\mathfrak{m}$}athrm{TV}}}\\ &\leq \int_T^\infty q_\sigma(x)\mathrm{d} x +\int_0^T\left| \e^{\sigma^2/2T}\sqrt{\textup{$\mathfrak{f}$}rac{T}{T-x}}-1\right|q_\sigma(x)\, \mathrm{d} x\, . \end{align*} The first integral obviously converges to $0$, and we can split that second integral at $T/2$ and rewrite it, after simple manipulations, as $$\int_0^{T/2}\left|\e^{\sigma^2/2T}\sqrt{\textup{$\mathfrak{f}$}rac{T}{T-x}}-1\right|q_\sigma(x)\, \mathrm{d} x + T\int_0^{1/2}\left|\e^{\sigma^2/2T}\sqrt{\textup{$\mathfrak{f}$}rac{1}{x}}-1\right|q_\sigma(T(1-x))\mathrm{d} x\, .$$ The first term converges to $0$ by dominated convergence, and the second vanishes as well since $q_\sigma(T(1-x))\leq 2\sigma/\sqrt{\pi T^3}$ for every $x\in [0,1/2]$. \end{proof} In the next proposition, we let $(F_t,0\leq t\leq T)$ be a first-passage bridge of Brownian motion hitting $-\sigma$ for the first time at $T$. We will let $T^F(x)=\inf\{t\geq 0:F_t<-x\}\wedge T$ for $0\leq x\leq \sigma$. Similarly, we let $\Delta_0^F,\Delta_1^F,\Delta_2^F,\ldots$ be the jump sizes of $T^F$ ranked in size-biased order, and $U^F_0,U^F_1,\ldots$ be the corresponding levels. For $i\geq 0$, we let $$e^F_i(t)=U_i^F+F(T^F(U_i^F-)+t)\, ,\textup{$\mathfrak{q}$}quad 0\leq t\leq \Delta_i^F,$$ be the excursion of $F$ above level $-U_i$ --- note that $\Delta_i^F=T^F(U_i)-T^F(U_i-)$. We also let $B$ be a (unconditioned) standard Brownian motion, and let $\Delta_1,\Delta_2,\ldots$ be the jump sizes of the first-hitting time subordinator $(T_{x},0\leq x\leq \sigma)$. We let $R,R'$ be two independent three-dimensional Bessel processes, independent of $B$. Finally, we let $U_0$ be a uniform random variable in $[0,\sigma]$, independent of $B,R,R'$. Figure~\ref{fig:coupling-contour-IBD} illustrates the following proposition. \begin{figure} \caption{The coupling of contour functions stated as Proposition~\ref{prop:coupl-brown-disks-2} \end{figure} \begin{prop} \textup{$\mathfrak{l}$}bel{prop:coupl-brown-disks-2} For every $\varepsilon\in (0,1)$ and $\alpha,\beta>0$, there exists $T^0>0$ such that for every $T>T^0$, it is possible to couple $F,B,R,R',U_0$ on the same probability space in such a way that with probability at least $1-\varepsilon$, one has $U_0=U_1^F$ and $$F_t=B_t,\,0\leq t\leq T^F(U_0-)=T_{U_0}\, ,\textup{$\mathfrak{q}$}quad F_{T-t}=B_{T_{\sigma}-t},\,0\leq t\leq T-T^F(U_0)=T_\sigma-T_{U_0}\, ,$$ and $$e^F_0(t)=R_t,\textup{$\mathfrak{q}$}uad 0\leq t\leq \alpha\, ,\textup{$\mathfrak{q}$}quad e^F_0(\Delta_0^F-t)=R'_t,\textup{$\mathfrak{q}$}uad 0\leq t\leq \beta\, ,$$ and finally $$\inf_{[\alpha,\infty)}R\wedge \inf_{[\beta,\infty)}R'=\textup{$\mathfrak{m}$}in_{[\alpha,\Delta^F_0-\beta]}e^F_0\, .$$ \end{prop} \begin{proof} By Lemma~\ref{lem:coupl-brown-disks-1}, for $T$ large enough, say $T>T^1$, it is possible to couple two sequences $\Delta_1,\Delta_2,\ldots$ and $\Delta'_0,\Delta'_1,\Delta'_2,\ldots$ on the same probability space such that \begin{itemize} \item $(\Delta_1,\Delta_2,\ldots)$ has the law of the jump sizes of $(T_{x},0\leq x\leq \sigma)$ ranked in size-biased order, and \item $(\Delta'_0,\Delta'_1,\Delta'_2,\ldots)$ has the law of $(\Delta_1,\Delta_2,\ldots)$ conditionally given $\sum_{i\geq 1}\Delta_i=T$, \end{itemize} in such a way that on an event $\textup{$\mathfrak{m}$}athcal{E}_1$ of probability at least $1-\varepsilon/2$, one has $$\Delta_i=\Delta'_{i}\, ,\textup{$\mathfrak{q}$}quad i\geq 1\, ,\textup{$\mathfrak{q}$}quad \textup{$\mathfrak{m}$}box{ and }\textup{$\mathfrak{q}$}quad \Delta'_0>T/2\, .$$ Extending the probability space if necessary, we can assume that it also supports an independent family of random variables $e_0,e_1,e_2,\ldots$ that are independent normalized Brownian excursions, and $U_0,U_1,U_2,\ldots$ that are independent uniform random variables in $[0,\sigma]$, independent of all the rest. By It{\^o}'s synthesis of Brownian motion from its excursions, if we set, for $i\geq 1$, $$B_t=-U_i+\textup{$\mathfrak{f}$}rac{e_i(\Delta_i(t-\sum_{j:U_j<U_i}\Delta_j))}{\sqrt{\Delta_i}}\, ,$$ whenever $\sum_{j\geq 1:U_j<U_i}\Delta_i< t\leq \sum_{j\geq 1:U_j\leq U_i}\Delta_j$, then this a.s.\ extends to a continuous path $(B_t,0\leq t\leq \sum_{i\geq 1}\Delta_i)$ which is a trajectory of Brownian motion stopped when first hitting $-\sigma$, which occurs at time $T_{\sigma}=\sum_{i\geq 1}\Delta_i$. Similarly, setting, this time for $i\geq 0$, $$F(t)=-U_i+\textup{$\mathfrak{f}$}rac{e_i(\Delta_i'(t-\sum_{j:U_j<U_i}\Delta'_j))}{\sqrt{\Delta'_i}}\, ,$$ whenever $\sum_{j\geq 0:U_j<U_i}\Delta'_i< t\leq \sum_{j\geq 0:U_j\leq U_i}\Delta'_j$, this extends to a trajectory of a first-passage bridge $(F(t),0\leq t\leq T)$ from $0$ to $-\sigma$, as the notation suggests, and if we set $\Delta^F_i=\Delta'_i$ for $i\geq 0$ then $(\Delta^F_i,i\geq 0)$ is indeed a size-biased ordering of the jumps of the first hitting time process of negative values of $F$. On the event $\textup{$\mathfrak{m}$}athcal{E}_1$, the two processes $B$ and $F$ coincide on the interval $[0,\sum_{j\geq 1:U_j<U_0}\Delta_j]$, and likewise, $B_{T_{\sigma}-\cdot}$ and $F(T-\cdot)$ coincide on $[0,\sum_{j\geq 1:U_j>U_0}\Delta_j]$. This yields the first displayed identity in the statement, since by construction $$\sum_{j\geq 1:U_j<U_0}\Delta'_j =T^F(U_0-)\, ,\textup{$\mathfrak{q}$}quad \sum_{j\geq 1:U_j<U_0}\Delta_j=\sum_{j\geq 1:U_j\leq U_0}\Delta_j=T_{U_0}\, , $$ while we have $$\sum_{j\geq 1:U_j>U_0}\Delta'_j =T-T^F(U_0)\, ,\textup{$\mathfrak{q}$}quad \sum_{j\geq 1:U_j>U_0}\Delta_j=\sum_{j\geq 1:U_j\geq U_0}\Delta_j=T_{\sigma}-T_{U_0}\, . $$ Finally, in this construction, and still in restriction to $\textup{$\mathfrak{m}$}athcal{E}_1$, $e_0^F=e_0(\Delta_0'\cdot)/\sqrt{\Delta_0'}$ is an excursion of Brownian motion with duration $\Delta'_0>T/2$. At this point, we can apply Proposition 3 in~\cite{CuLG}, in the same way as in the proof of Proposition 4 therein. Up to a further extension of the probability space, as soon as $T$ is chosen large enough, say $T>T^2$, we can couple this ``long'' excursion with two independent Bessel processes $R,R'$ (and independent of all previously defined random variables) in such a way that the three last identities of the statement are satisfied on an event $\textup{$\mathfrak{m}$}athcal{E}_2$ with probability at least $1-\varepsilon/2$. This yields the wanted result with $T^0=T^1\vee T^2$, since the intersection $\textup{$\mathfrak{m}$}athcal{E}_1\cap\textup{$\mathfrak{m}$}athcal{E}_2$ has probability at least $1-\varepsilon$. \end{proof} \subsubsection{Isometry of balls in {\normalfont $\mathcal BD_{T,\sigma}$} and {\normalfont $\textup{\textsf{IBD}}_\sigma$}} We fix $\sigma\in(0,\infty)$, $\varepsilon>0$ and let $r\geq 0$. With the coupling from the preceding section at hand, the proof the proposition is a minor modification of~\cite[Proof of Proposition 4]{CuLG} (see also Proposition~\ref{prop:isometry-BD-BHP} and its proof). We will point at the necessary modifications and then leave it to reader to fill in the remaining details. We work in the notation and with the processes of Proposition~\ref{prop:coupl-brown-disks-2} and denote additionally by $\gamma= (\gamma_u,0\leq u\leq \sigma)$ a standard Brownian bridge with duration $\sigma$, multiplied by $\sqrt{3}$. \begin{proof}[Proof of Proposition~\ref{prop:isometry-BD-IBD}] Let us first introduce a few events. For $K>0$, put $$ \textup{$\mathfrak{m}$}athcal{E}^1(K)=\left\{\textup{$\mathfrak{m}$}ax_{[0,\sigma]}\gamma < K\right\}. $$ Then, given $A>0$, with $\zeta=(\zeta_t,t\geq 0)$ denoting a Brownian motion started at $0$, let $$ \textup{$\mathfrak{m}$}athcal{E}^2(A,K)=\left\{\textup{$\mathfrak{m}$}in_{[0,A]}\zeta<-10r-K,\,\textup{$\mathfrak{m}$}in_{[A,A^2]}\zeta<-10r-K,\,\textup{$\mathfrak{m}$}in_{[A^2,A^4]}\zeta<-10r-K\right\}, $$ and for $A>0$ and $\alpha>0$, set $$ \textup{$\mathfrak{m}$}athcal{E}^3(A,\alpha)=\left\{\inf_{[\alpha,\infty)}R\wedge \inf_{[\alpha,\infty)}R'>A^4\right\}. $$ We first choose $K$ sufficiently large such that $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^1)\geq 1-\varepsilon/6.$ Then, standard properties of Brownian motion allow us to find $A>0$ such that $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^2)\geq 1-\varepsilon/6$ as well, and with such a fixed $A$, we find by transience of the Bessel process an $\alpha>0$ large enough such that $\mathbb P(\textup{$\mathfrak{m}$}athcal{E}^3)>1-\varepsilon/3$. Let us next recall the contour process $Y^\sigma$ of $\textup{\textsf{IBD}}_\sigma$ specified just before Definition~\ref{def:IBD} in terms of the Bessel processes $R$ and $R'$ and the Brownian motion $B$ stopped at times $T_{U_0}$ and $T_\sigma$. We obtain that on the coupling event $\textup{$\mathfrak{m}$}athcal{E}^4=\textup{$\mathfrak{m}$}athcal{E}^4(\alpha,T)$ described in the statement of Proposition~\ref{prop:coupl-brown-disks-2} (with $\beta=\alpha$), in the notation from there, $$ F_t=Y^\sigma_t\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{for } t\in[0,T_{U_0}+\alpha]\,,\textup{$\mathfrak{q}$}uad \textup{$\mathfrak{m}$}box{ and }\textup{$\mathfrak{q}$}uad F_{T-t}+\sigma=Y^\sigma_{-t}\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{for } t\in[0,T_\sigma-T_{U_0}+\alpha]\,.$$ Concerning $\mathcal BD_{T,\sigma}$, we use the notation from Section~\ref{sec:recapBHPBD} (note however that $\sigma$ is now a constant not depending on the volume $T$). We build the label process of $\mathcal BD_{T,\sigma}$ in the following way: Consider a Brownian bridge $\gamma$ as specified above, independent of $(F,B,R,R',U_0)$. Then let $Z=Z^{F-\underline{F}}$ be the random snake driven by $F-\underline{F}$, and set $$W_t=\gamma_{-\underline{F}_t}+Z_t,\textup{$\mathfrak{q}$}uad 0\leq t\leq T.$$ Concerning $\textup{\textsf{IBD}}^\sigma$, we write $\textup{$\mathfrak{m}$}athbb Zi=Z^{Y^\sigma-\underline{\underline{Y}}^\sigma}$ for the random snake driven by $Y^\sigma-\underline{\underline{Y}}^\sigma$, see Definition~\ref{def:IBD}, and $W^{\textup{I}}_t=\gamma_{-\underline{\underline{Y}}^\sigma_t}+\textup{$\mathfrak{m}$}athbb Zi$ for the label process associated with $\textup{\textsf{IBD}}_\sigma$. Of course, we choose to use the same bridge $\gamma$ to construct $W$ and $W^{\textup{I}}$. We now work always conditionally on $(F,B,R,R',U_0)$. Similarly to the considerations around~\eqref{eq:tuple1} and~\eqref{eq:tuple2} in the proof of Proposition~\ref{prop:isometry-BD-BHP}, one checks that on the event $\textup{$\mathfrak{m}$}athcal{E}^4$, the covariance function of $$(W_t,0\leq t\leq T_{U_0}+\alpha),(W_{T-t},0\leq t\leq T_\sigma-T_{U_0}+\alpha)$$ on the one hand, and $$(W^{\textup{I}}_t,0\leq t\leq T_{U_0}+\alpha),(W^{\textup{I}}_{-t},0\leq t\leq T_\sigma-T_{U_0}+\alpha)$$ on the other hand, are the same. Consequently, we may assume that $W$ and $W^{\textup{I}}$ are coupled in such a way that, on the event $\textup{$\mathfrak{m}$}athcal{E}^4$, $$ W_t = W^{\textup{I}}_t\textup{$\mathfrak{q}$}uad\hbox{ for all }t\in[0,T_{U_0}+\alpha],\textup{$\mathfrak{q}$}uad W_{T-t} = W^{\textup{I}}_{-t}\textup{$\mathfrak{q}$}uad\hbox{ for all }t\in[0,T_\sigma-T_{U_0}+\alpha]. $$ From Proposition~\ref{prop:coupl-brown-disks-2}, we derive that for the choice of $\alpha$ from above, the coupling event $\textup{$\mathfrak{m}$}athcal{E}^4(\alpha,T)$ has probability at least $1-\varepsilon/3$ provided $T$ is sufficiently large, and we shall work with such a $T$. The reminder of the proof is now close to~\cite[Proof of Proposition 4]{CuLG}. For every $x\geq 0$, let \begin{align*} \eta_{\textup{l}}(x)&=\sup\{0\leq t\leq \Delta_0^F/2:e_0^F(t)=x\}+T_{U_0},\\ \eta_{\textup{r}}(x)&=\Delta_0^F-\inf\{\Delta_0^F/2\leq t\leq \Delta_0^F:e_0^F(t)=x\}+T_\sigma-T_{U_0}, \end{align*} and \begin{align} \textup{$\mathfrak{l}$}bel{eq:etaIBD} \eta_{\textup{l}}^{\textup{I}}(x)&=\sup\{t\geq 0: R_t=x\}+T_{U_0},\nonumber\\ \eta_{\textup{r}}^{\textup{I}}(x)&=\sup\{t\geq 0: R_t'=x\} +T_\sigma-T_{U_0}. \end{align} Then the process $(Z^\textup{I}_{\eta_{\textup{l}}^\textup{I}(x)}, x\geq 0)$ has the law of a standard Brownian motion started at $Z^\textup{I}_{T_{U_0}}=0$. Choosing this Brownian motion in the definition of the event $\textup{$\mathfrak{m}$}athcal{E}^2$ from above, so that on $\textup{$\mathfrak{m}$}athcal{E}^2$, we have \begin{equation} \textup{$\mathfrak{l}$}bel{eq:boundZi} \textup{$\mathfrak{m}$}in_{x\in[0,A]}Z^\textup{I}_{\eta_{\textup{l}}^{\textup{I}}(x)}<-6r-K,\,\textup{$\mathfrak{m}$}in_{x\in[A,A^2]}Z^{\textup{I}}_{\eta_{\textup{l}}^{\textup{I}}(x)}<-6r-K,\,\textup{$\mathfrak{m}$}in_{x\in[A^2,A^4]}Z^{\textup{I}}_{\eta_{\textup{l}}^{\textup{I}}(x)}<-6r-K, \end{equation} we shall from now on work on the intersection of events \begin{equation} \textup{$\mathfrak{l}$}bel{eq:couplingIBD-BD-defF} \textup{$\mathfrak{m}$}athcal{F}=\textup{$\mathfrak{m}$}athcal{E}^1\cap \textup{$\mathfrak{m}$}athcal{E}^2\cap\textup{$\mathfrak{m}$}athcal{E}^3\cap\textup{$\mathfrak{m}$}athcal{E}^4, \end{equation} which has probability at least $1-\varepsilon$. On $\textup{$\mathfrak{m}$}athcal{E}^3\cap\textup{$\mathfrak{m}$}athcal{E}^4$, we note that $\textup{$\mathfrak{m}$}in_{[\alpha,\Delta_0^F-\alpha]}e^F_0=\inf_{[\alpha,\infty)}R \wedge \inf_{[\alpha,\infty)} R' >A^4$, whence for $x\in[0,A^4]$, $\eta_{\textup{l}}(x)=\eta_{\textup{l}}^{\textup{I}}(x)< T_{U_0}+\alpha$ and $\eta_{\textup{r}}(x)=\eta_{\textup{r}}^{\textup{I}}(x)< T_{\sigma}-T_{U_0}+\alpha$. It follows that for any $x\in[0,A^4]$, $$Z_{\eta_{\textup{l}}(x)}= Z_{\eta_{\textup{l}}^\textup{I}(x)}^{\textup{I}}= Z_{-\eta_{\textup{r}}^\textup{I}(x)}^{\textup{I}} = Z_{T-\eta_{\textup{r}}(x)}.$$ We are now almost in a setting where we can appeal to the reasoning in~\cite[Section 3.2]{CuLG}. We should still adapt the definition of $\tilde{d}_W(s,t)$ given just before Lemma~\ref{lem:DBD} to the setting considered here. Let $s,t\in [0,T]$. If $s,t$ lie both in either $[0,T_{U_0}+\Delta_0^F/2]$ or in $[T_{U_0}+\Delta_0^F/2,T]$, we let $$ d'_{W}(s,t) = W_s +W_t -2\textup{$\mathfrak{m}$}in_{[s\wedge t, s\vee t]}W. $$ Otherwise, we set $$ d'_{W}(s,t) = W_s +W_t -2\textup{$\mathfrak{m}$}in_{[0,s\wedge t]\cup[s\vee t,T]}W. $$ Recall the definition of the pseudo-metric $D(s,t)$ associated to the Brownian disk $\mathcal BD_{T,\sigma}$. The following statement replaces Lemma~\ref{lem:DBD} and is close to~\cite[Lemma 5(i)]{CuLG}. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:DBD2} Assume $\textup{$\mathfrak{m}$}athcal{F}$ holds. \begin{enumerate} \item For every $t\in [\eta_{\textup{l}}(A),T-\eta_{\textup{r}}(A)]$, $D(0,t) > r$. \item For every $s,t\in [0,\eta_{\textup{l}}(A)]\cup[0,T-\eta_{\textup{r}}(A)]$ with $\textup{$\mathfrak{m}$}ax\{D(0,s), D(0,t)\}\leq r$, it holds that $$ D(s,t) =\inf_{s_1,t_1,\mathrm{d}ots,s_k,t_k}\sum_{i=1}^k d'_{W}(s_i,t_i), $$ where the infimum is over all possible choices of $k\in\textup{$\mathfrak{m}$}athbb N$ and reals $s_1,\mathrm{d}ots,s_k,t_1,\mathrm{d}ots,t_k\in [0,\eta_{\textup{l}}(A^2)]\cup[T-\eta_{\textup{r}}(A^2),T]$ such that $s_1=s,t_k=t$, and $d_{F}(t_i,s_{i+1})=0$ for $1\leq i\leq k-1$. \end{enumerate} \end{lemma} \begin{proof} One can follow the same line of reasoning as in~\cite[Proof of Lemma 5(i)]{CuLG}, with one small modification, which is apparent from the proof of (a), so let us prove this part. If $t\in [\eta_{\textup{l}}(A),T-\eta_{\textup{r}}(A)]$, then by the cactus bound~\eqref{eq:cactusDBD}, $$ D(0,t)\geq W_t-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,t]}W,\textup{$\mathfrak{m}$}in_{[t,T]}W\right\}\geq -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,\eta_{\textup{l}}(A)]}W,\textup{$\mathfrak{m}$}in_{[T-\eta_{\textup{r}}(A),T]}W\right\}. $$ Recalling that $W_t=\gamma_{-\underline{F}_t}+Z_t$, we first remark that on the event $\textup{$\mathfrak{m}$}athcal{E}^3\cap\textup{$\mathfrak{m}$}athcal{E}^4$, since $\eta_{\textup{l}}(A)< T_{U_0}+\alpha$, we have $Z=\textup{$\mathfrak{m}$}athbb Zi$ on $[0,\eta_{\textup{l}}(A)]$. Since $\eta_{\textup{l}}([0,A])\subset [0,\eta_{\textup{l}}(A)]$, it follows now from~\eqref{eq:boundZi} that the minimum of $Z$ on $[0,\eta_{\textup{l}}(A)]$ is bounded from above by $-6r-K$. But on $\textup{$\mathfrak{m}$}athcal{E}^1$, $\textup{$\mathfrak{m}$}ax \gamma < K$, so that $\textup{$\mathfrak{m}$}in_{[0,\eta_{\textup{l}}(A)]}W\leq -6r$. A similar argument holds for the second minimum, so that in fact $D(0,t)\geq 6r$ whenever $t\in[\eta_{\textup{l}}(A),T-\eta_{\textup{r}}(A)]$. For (b), one can follow~\cite[Proof of Lemma 5(i)]{CuLG}, or modify the proof of (b) in Lemma~\ref{lem:DBD}. \end{proof} Entirely similar, one finds the corresponding statement for the pseudo-metric $D^{\textup{I}}$ of $\textup{\textsf{IBD}}_\sigma$ that replaces Lemma~\ref{lem:DH}: In the statement there, $\eta_{\textup{r}}$ and $\eta_{\textup{l}}$ have to be replaced by $\eta_{\textup{r}}^{\textup{I}}$ and $\eta_{\textup{l}}^{\textup{I}}$ as defined under~\eqref{eq:etaIBD}, and $D_\theta$, $d_{W^\theta}$ by $D^{\textup{I}}$ and $d_{W^{\textup{I}}}$. Following again~\cite{CuLG}, or adapting the second part of the proof of Proposition~\ref{prop:isometry-BD-BHP}, these two lemmas lead to the stated isometry between $B_r(\mathcal BD_{T,\sigma})$ and $B_r(\textup{\textsf{IBD}}_\sigma)$ on the event $\textup{$\mathfrak{m}$}athcal{F}$ of probability at least $1-\varepsilon$, finishing thereby the proof of Proposition~\ref{prop:isometry-BD-IBD}. \end{proof} It remains to show how Proposition~\ref{prop:isometry-BD-IBD} can be improved to the coupling stated in Theorem~\ref{thm:coupling-BD-IBD}. \subsubsection{Proof of Theorem~\ref{thm:coupling-BD-IBD}} The proof is close in spirit to that of Theorem~\ref{thm:coupling-BD-BHP}: for a fixed $r>0$, we must find some $r_0>r$ large enough so that for all $T$ sufficiently large, the ball $B_{r_0}(\mathcal BD_{T,\sigma})$ contains with high probability an open set $A_{\mathcal BD}$ that is homeomorphic to the pointed closed disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$ and that, in turn, contains the ball $B_r(\mathcal BD_{T,\sigma})$ with high probability. Then we will apply the coupling Proposition~\ref{prop:isometry-BD-IBD} in order to couple the balls $B_{r_0}(\mathcal BD_{T,\sigma})$ and $B_{r_0}(\textup{\textsf{IBD}}_{\sigma})$. The set $A_{\mathcal BD}$ will be defined as a region bounded by certain geodesic paths. We use a similar notation to that of the proof of Theorem~\ref{thm:coupling-BD-BHP}, setting $\textup{\textsf{Y}}=([0,T]/\{D=0\},D,\rho)$ and letting $p_\textup{\textsf{Y}}$ be the associated canonical projection, so that $Y$ has the law of the Brownian disk $\mathcal BD_{T,\sigma}$ with root $\rho=p_\textup{\textsf{Y}}(0)$. We will also use the geodesic paths $\Gamma_s,s\in [0,T]$, in $\textup{\textsf{Y}}$ respectively from $p_\textup{\textsf{Y}}(s)$ to $x_*=p_\textup{\textsf{Y}}(s_*)$ defined around Lemma~\ref{lem:proof-prop-refpr-4}, together with the properties stated there. Let $a_0>0$ be a large number to be specified later on. We let $A_\mathcal BD^0=[0,\eta_{\textup{l}}(a_0)]\cup [T-\eta_{\textup{r}}(a_0),T]$, where $\eta_{\textup{l}},\eta_{\textup{r}}$ are defined in the proof of Proposition~\ref{prop:isometry-BD-IBD}. We will work on the event that $s_*\notin A_\mathcal BD^0$, a fact that will be granted later with high probability by the fact that $T$ is going to infinity. With our notation, the definition of $A_\mathcal BD^0$ is the exact same as $O_\mathcal BD^0$ in the proof of Theorem \ref{thm:coupling-BD-BHP}, but note that by contrast, the points $p_\textup{\textsf{Y}}(\eta_{\textup l}(a_0))$ and $p_\textup{\textsf{Y}}(T-\eta_{\textup r}(a_0))$ are now equal, and we denote it by $x_0$. Note in passing that $x_0\notin \partial \textup{\textsf{Y}}$ by Lemma \ref{lem:proof-prop-refpr}. We let $t_*\in A^0_\mathcal BD$ be such that $W_{t_*}=\textup{$\mathfrak{m}$}in_{A^0_\mathcal BD}W$. The geodesic paths $\Gamma_{\eta_l(a_0)}$ and $\Gamma_{T-\eta_r(a_0)}$ both start from $x_0$, but by (d) in Lemma \ref{lem:proof-prop-refpr-4}, they become disjoint until they meet again for the first time at the point $p_\textup{\textsf{Y}}(t_*)$. Therefore, the segments of these geodesics between $x_0$ and $p_\textup{\textsf{Y}}(t_*)$ form a simple loop $P$, which is disjoint from the boundary $\partial \textup{\textsf{Y}}$ by (c) in Lemma \ref{lem:proof-prop-refpr}. We point at Figure~\ref{fig:topo-IBD} for an illustration. The analog of Lemma \ref{lem:proof-prop-refpr-5} is the following. \begin{lemma} \textup{$\mathfrak{l}$}bel{sec:proof-theor-refthm:c} The set $P$ is a simple loop in $\textup{\textsf{Y}}$ containing $x_0$, and that does not intersect $\partial \textup{\textsf{Y}}$. Letting $A_\mathcal BD$ be the connected component of $\textup{\textsf{Y}}\setminus P$ that contains $p_\textup{\textsf{Y}}(0)$, then $A_\mathcal BD$ is almost surely homeomorphic to the pointed closed disk $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus \{0\}$, and is the interior of the set $p_\textup{\textsf{Y}}(A^0_\mathcal BD)$ in $\textup{\textsf{Y}}$. \end{lemma} \begin{proof} The proof is very similar to that of Lemma \ref{lem:proof-prop-refpr-5}. The fact that $A_\mathcal BD$ is a.s.\ homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{D}}\setminus\{0\}$ is a direct consequence of the fact that $\textup{\textsf{Y}}$ is homeomorphic to $\overline{\textup{$\mathfrak{m}$}athbb{D}}$ and that $P$ is a simple loop not intersecting $\partial \textup{\textsf{Y}}$. The only thing that remains to be proved given our discussion so far is that $A_\mathcal BD$ is indeed the interior of $p_\textup{\textsf{Y}}(A^0_\mathcal BD)$. However, using the exact same definition of the paths $\mathbb Xi_s$, it is simple to see that a point in $p_\textup{\textsf{Y}}(A^0_\mathcal BD)$ is linked to $\partial \textup{\textsf{Y}}$, and hence to $p_\textup{\textsf{Y}}(0)$, by a simple path that intersects $P$, if at all, only at its starting point. It remains to verify that for every $x\in \textup{\textsf{Y}}\setminus p_\textup{\textsf{Y}}(A^0_\mathcal BD)$, we can find a simple path from $x$ to $p_\textup{\textsf{Y}}(s_*)$ that possibly intersects $P$ only at its origin. Writing $x=p_\textup{\textsf{Y}}(s)$, we leave it to the reader that such a path can be obtained by concatenating segments of the paths $p_\textup{\textsf{Y}}\circ \mathbb Xi_s$ and $p_\textup{\textsf{Y}}\circ \mathbb Xi_{s_*}$. The conclusion follows. \end{proof} Now for a fixed $r>0$ and $\varepsilon>0$, we choose $a_0>0$ large enough so that $$\mathbb P\left(\textup{$\mathfrak{m}$}in_{[0,a_0]}\textup{$\mathfrak{m}$}athbb Zi_{\eta^\textup{I}_{\textup l}(\cdot)}<-2r\right)\geq 1-\varepsilon/4,$$ and then let $r_0>r$ be large enough so that $$\mathbb P\left(\omega(W^{\textup{I}},[-\eta_r^{\textup{I}}(a_0),\eta_l^{\textup{I}}(a_0)])\leq r_0/2\right)\geq 1-\varepsilon/4\, .$$ We use this value of $r_0$ to apply the coupling of $\textup{\textsf{Y}}=\mathcal BD_{T,\sigma}$ and $\textup{\textsf{IBD}}_\sigma$ of Proposition \ref{prop:isometry-BD-IBD}, guaranteeing that the balls of radius $r_0$ in these pointed spaces are isometric with probability at least $1-\varepsilon/2$. From there, we conclude exactly as in the end of the proof of Theorem \ref{thm:coupling-BD-BHP}, replacing $O_\mathcal BD$ and $O_\mathcal BHP=I(O_\mathcal BD)$ by $A_\mathcal BD$ and $A_\mathcal BHP=I(A_\mathcal BD)$, where $I$ is defined as before Corollary \ref{cor:DBD-DH} and defines an isometry between $B_{r_0}(\textup{\textsf{Y}})$ and $B_{r_0}(\textup{\textsf{IBD}}_{\sigma})$ on the coupling event $\textup{$\mathfrak{m}$}athcal{F}$ given by~\eqref{eq:couplingIBD-BD-defF} (note that on this event, one has in particular $\eta_{\textup l}(x)=\eta^{\textup{I}}_{\textup l}(x)$ and $\eta_{\textup r}(x)=\eta_{\textup r}^{\textup{I}}(x)$ for every $x\leq A^4$, and without loss of generality, we can choose $A$ so that $A^4>a_0$.) This completes the proof of Theorem~\ref{thm:coupling-BD-IBD}. \begin{figure} \caption{Illustration of the proof of Theorem~\ref{thm:coupling-BD-IBD} \end{figure} \subsection{Infinite-volume Brownian disk (Theorem~\ref{thm:IBD})} \textup{$\mathfrak{l}$}bel{sec:proof-thmIBD} For proving Theorem~\ref{thm:IBD}, we will combine the convergence towards the Brownian disk $\mathcal BD_{T,\sigma}$ proved in~\cite[Theorem 1]{BeMi} (see Display~\eqref{eq:BeMi}) with the couplings Theorem~\ref{thm:coupling-BD-IBD} and Proposition~\ref{prop:coupling-Qn-largevol}. We work in the usual setting specified in Section~\ref{sec:usualsetting}. \begin{proof}[Proof of Theorem~\ref{thm:IBD}] Assume $1\ll \sigma_n\ll\sqrt{n}$ and $a_n\sim (4/9)^{1/4}\sqrt{\sigma_n/\sigma}$ for some $\sigma\in(0,\infty)$. We have to show that for each $r\geq 0$, $$ B_r\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\xrightarrow[n\to \infty]{(d)}B_{r}(\textup{\textsf{IBD}}_\sigma)$$ in distribution in $\textup{$\mathfrak{m}$}athbb{K}$. We fix $\varepsilon>0$ and $r\geq 0$. By Theorem~\ref{thm:coupling-BD-IBD}, we find $T_0$ such that for all $T\geq T_0$, we can construct on the same probability space copies of $\mathcal BD_{T,\sigma}$ and $\textup{\textsf{IBD}}_\sigma$ such that with probability at least $1-\varepsilon$, we have an isometry of balls \begin{equation} \textup{$\mathfrak{l}$}bel{eq:IBD-coupling1} B_r(\mathcal BD_{T,\sigma})=B_r(\textup{\textsf{IBD}}_\sigma). \end{equation} By Proposition~\ref{prop:coupling-Qn-largevol}, we find $R_0\geq T_0/(2\sigma^2)$ such that for $R\geq R_0$ and $n$ sufficiently large, we can construct on the same probability space copies of $Q_n^{\sigma_n}$ and $Q_{R\sigma_n^2}^{\sigma_n}$ such that with probability at least $1-\varepsilon$, there is the isometry \begin{equation} \textup{$\mathfrak{l}$}bel{eq:IBD-coupling2} B_{ra_n}(Q_n^{\sigma_n})=B_{ra_n}\left(Q_{R\sigma_n^2}^{\sigma_n}\right). \end{equation} Now let $F:\textup{$\mathfrak{m}$}athbb{K}\rightarrow\textup{$\mathfrak{m}$}athbb R$ be a bounded and continuous function and $R\geq R_0$. We assume that $Q_{n}^{\sigma_n}$ and $Q_{R\sigma_n^2}^{\sigma_n}$ are constructed on the same probability space such that~\eqref{eq:IBD-coupling2} holds, and similarly $\mathcal BD_{2R\sigma^2,\sigma}$ and $\textup{\textsf{IBD}}_\sigma$ so that~\eqref{eq:IBD-coupling1} is satisfied. We write \begin{align*} \lefteqn{\left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_r\left(a_n^{-1}\cdot Q_n^{\sigma_n}\right)\right)\right]-\textup{$\mathfrak{m}$}athbb E\left[F\left(B_r\left(\textup{\textsf{IBD}}_\sigma\right)\right)\right]\right|}\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\leq \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( a_n^{-1}\cdot B_{ra_n}\left(Q_n^{\sigma_n}\right)\right)-F\left( a_n^{-1}\cdot B_{ra_n}\left(Q_{R\sigma_n^2}^{\sigma_n}\right)\right)\right]\right|\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad + \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( a_n^{-1}\cdot B_{ra_n}\left(Q_{R\sigma_n^2}^{\sigma_n}\right)\right)\right]-\textup{$\mathfrak{m}$}athbb E\left[F\left(B_r\left(\mathcal BD_{2\sigma^2R,\sigma}\right)\right)\right]\right|\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad + \left| \textup{$\mathfrak{m}$}athbb E\left[F\left(B_r(\mathcal BD_{2\sigma^2R,\sigma})\right)-F\left(B_r\left(\textup{\textsf{IBD}}_\sigma\right)\right)\right]\right|. \end{align*} Using~\eqref{eq:IBD-coupling2} and~\eqref{eq:IBD-coupling1} (note that $2\sigma^2R\geq T_0$), the first and third summand on the right hand side are bounded from above by $2\varepsilon\sup F$. The scaling property $\textup{$\mathfrak{l}$}mbda\cdot \mathcal BD_{1,\sigma}=_d \mathcal BD_{\textup{$\mathfrak{l}$}mbda^4 ,\textup{$\mathfrak{l}$}mbda^2\sigma}$ for $\textup{$\mathfrak{l}$}mbda>0$ combined with the convergence~\eqref{eq:BeMi} implies that the second summand converges to zero as $n\rightarrow\infty$. This finishes the proof of Theorem~\ref{thm:IBD}. \end{proof} \subsection{Brownian disk limits (Corollaries~\ref{cor:BD1},~\ref{cor:BD4},~\ref{cor:BD2} and~\ref{cor:BD3}).} \textup{$\mathfrak{l}$}bel{sec:proofs-BDlimits} \begin{proof}[Proof of Corollaries~\ref{cor:BD1},~\ref{cor:BD4},~\ref{cor:BD2} and~\ref{cor:BD3}.] We have to show that for each $r\geq 0$, when $T$ tends to infinity, $B_r(\mathcal BD_{T,\sigma(T)})$ converges in law to the ball of radius $r$ around the root in the limit space $\textup{$\mathfrak{m}$}athcal{X}$ that appears in the corresponding corollary. As usual, we consider only the case $r=1$. Let $F:\textup{$\mathfrak{m}$}athbb{K}\rightarrow\textup{$\mathfrak{m}$}athbb{R}$ be bounded and continuous. For $T\in\textup{$\mathfrak{m}$}athbb N$ and $n\in\textup{$\mathfrak{m}$}athbb N$, we set $$ m_n(T)=Tn,\textup{$\mathfrak{q}$}uad \sigma_n(T)=\lfloorloor\sigma(T)\sqrt{2n}\rfloorloor,\textup{$\mathfrak{q}$}uad a_n=(8/9)^{1/4}n^{1/4}. $$ We write \begin{align*} \lefteqn{\left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_1\left(\mathcal BD_{T,\sigma(T)}\right)\right)\right] - \textup{$\mathfrak{m}$}athbb E\left[F\left( B_1\left(\textup{$\mathfrak{m}$}athcal{X}\right)\right)\right]\right|}\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad \leq \left|\textup{$\mathfrak{m}$}athbb E\left[F\left( B_1\left(\mathcal BD_{T,\sigma(T)}\right)\right)\right] -\textup{$\mathfrak{m}$}athbb E\left[F\left(a_n^{-1}\cdot B_{a_n}\left(Q_{m_n(T)}^{\sigma_n(T)}\right)\right)\right)\right|\\ &\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{q}$}uad +\left|\textup{$\mathfrak{m}$}athbb E\left[F\left(a_n^{-1}\cdot B_{a_n}\left(Q_{m_n(T)}^{\sigma_n(T)}\right)\right)\right] -\textup{$\mathfrak{m}$}athbb E\left[F\left(B_1\left(\textup{$\mathfrak{m}$}athcal{X}\right)\right)\right]\right|. \end{align*} For each fixed $T\in\textup{$\mathfrak{m}$}athbb N$, the convergence~\cite[Theorem 1]{BeMi} towards the Brownian disk with volume $T$ and perimeter $\sigma(T)$ (see~\eqref{eq:BeMi} above) implies that the first summand on the right hand side is bounded by $\varepsilon$, provided $n\geq n_0(T)$. Concerning the second summand, we argue by contradiction that for large enough $T$, there exists $n_0=n_0=(T,\varepsilon)$ such that for any $n\geq n_0$, the second summand is bounded by $\varepsilon$ as well. Indeed, assuming this is not the case, we find a sequence of integers $(T_k,k\in\textup{$\mathfrak{m}$}athbb N)$ with $T_k\rightarrow\infty$ and a sequence of integers $(n_k,k\in \textup{$\mathfrak{m}$}athbb N)$ with $n_k$ depending on $T_k$ and $n_k\rightarrow\infty$, such that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:BDlimits-eq1} \left|\textup{$\mathfrak{m}$}athbb E\left[F\left(a_{n_k}^{-1}\cdot B_{a_{n_k}}\left(Q_{m_{n_k}(T_k)}^{\sigma_{n_k}(T_k)}\right)\right)\right] -\textup{$\mathfrak{m}$}athbb E\left[F\left(B_1\left(\textup{$\mathfrak{m}$}athcal{X}\right)\right)\right]\right|>\varepsilon. \end{equation} We put $T_0=n_0=0$ and for $n\in\textup{$\mathfrak{m}$}athbb N$, $$ \tilde{a}_n= (8/9)^{1/4}\left(n/T_k\right)^{1/4},\textup{$\mathfrak{q}$}uad \tilde{\sigma}_n=\lfloorloor \sigma(T_k)\sqrt{2n/T_k}\rfloorloor,\textup{$\mathfrak{q}$}uad \textup{ if } T_{k}n_{k-1}< n\leq T_k n_k. $$ For concreteness, we now consider the framework of Corollary~\ref{cor:BD1}, where $\textup{$\mathfrak{m}$}athcal{X}=\mathcal BP$ and $\sigma(T)\rightarrow 0$ as $T$ tends to infinity. Then $\tilde{\sigma}_n\ll\sqrt{n}$, and $\sqrt{\tilde{\sigma}_n}\ll \tilde{a}_n\ll n^{1/4}$, so that we can apply the convergence towards $\mathcal BP$ proved in Theorem~\ref{thm:BP} (with $\sigma_n$ and $a_n$ there replaced by $\tilde{\sigma}_n$ and $\tilde{a}_n$). However, observing the quadrangulations at sizes $m_k=T_kn_k$,~\eqref{eq:BDlimits-eq1} contradicts the convergence towards $\mathcal BP$. In the setting of Corollary~\ref{cor:BD4}, we use Theorem~\ref{thm:IBD} instead of Theorem~\ref{thm:BP} and the fact that $\sigma(T)\rightarrow\varsigma\in(0,\infty)$ as $T\rightarrow\infty$. An identical argument allows us to finish the proof in this case, with $\textup{$\mathfrak{m}$}athcal{X}$ given by $\textup{\textsf{IBD}}_\varsigma$. In the framework of Corollary~\ref{cor:BD3} where $\sigma(T)/T\rightarrow\infty$ as $T$ tends to infinity, we apply Theorem~\ref{thm:ICRT} instead. Let us finally look at Corollary~\ref{cor:BD2}. There, $\sigma(T)\rightarrow\infty$ and $\sigma(T)/T\rightarrow\theta\in[0,\infty)$. If $\theta=0$, then, along sequences $(T_m,m\in\textup{$\mathfrak{m}$}athbb N)$ tending to infinity for which $\sigma(T_m)/\sqrt{T_m}\rightarrow 0$ as $m\rightarrow\infty$, we follow the same argumentation by contradiction and use Theorem~\ref{thm:BHP1}, whereas if $\liminf_{m\rightarrow\infty}\sigma(T_m)/\sqrt{T_m}>0$, the corollary is a direct consequence of Theorem~\ref{thm:coupling-BD-BHP}, and so is it in the case $\theta>0$. \end{proof} \subsection{Infinite continuum random tree (Theorem~\ref{thm:ICRT})} We use the second construction of $\textup{\textsf{ICRT}}=(\mathcal{T}_Y,d_Y,[0])$ from Section~\ref{sec:def} in terms of a standard two-sided Brownian motion $(Y_t,t\in\textup{$\mathfrak{m}$}athbb{R})$ with $Y_0=0$. Recall that Theorem~\ref{thm:ICRT} treats the regime $\sigma_n \gg \sqrt{n}$, and we explicitly allow $\sigma_n$ to grow faster than $n$. \begin{mdframed}{\bf Idea of the proof.} Let $(\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n)$ be a random uniform element of $\mathbb Fo_{\sigma_n}^n$. We show that under the rescaling by a factor $\textup{$\mathfrak{m}$}ax\{1,\sqrt{n/\sigma_n}\} \ll a_n \ll \sqrt{\sigma_n}$, the labels of the first and last $ca_n^2$ trees of $\textup{$\mathfrak{f}$}_n$ converge to zero. For the rescaled submap $a_n^{-1}B_{ra_n}(Q_n^{\sigma_n})$, this means that for large $n$, the boundary dominates and folds the map into a tree-shaped object, which we identify with the $\textup{\textsf{ICRT}}$ in the limit $n\rightarrow\infty$. \end{mdframed} \begin{proof}[Proof of Theorem~\ref{thm:ICRT}] We work in the usual setting, cf.~\ref{sec:usualsetting}. We will construct a set $\textup{$\mathfrak{m}$}athcal{R}_n$ which shares the properties of Lemma~\ref{lem:localGHconv}, and for that reason, we shall consider both $V(Q_n^{\sigma_n})$ and the corresponding length space ${\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}$, cf. Section~\ref{sec:locGH}. Let $r\geq 0$. Still the vertex set $V(\textup{$\mathfrak{f}$}_n)$ is naturally identified with points of ${\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}$, and we may consider the ball $B_r^{(0)}({\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n})$ in ${\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}$ of radius $r$ around $\textup{$\mathfrak{f}$}_n(0)=(0)$ (with respect to the shortest-path metric $d$). Since the Gromov-Hausdorff distance between $B_r^{(0)}(Q_n^{\sigma_n})$ and $B_r^{(0)}({\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n})$ is bounded by one, the theorem follows from Lemma~\ref{lem:ball0} if we show that $$ B_r^{(0)}\left(a_n^{-1}\cdot {\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}\right)\xrightarrow[n \to \infty]{(d)}B_r(\mathcal{T}_Y),$$ for any scaling sequence $a_n$ with $\textup{$\mathfrak{m}$}ax\{1,\sqrt{n/\sigma_n}\} \ll a_n \ll \sqrt{\sigma_n}$. For ease of reading, we restrict ourselves to the case $r=1$. Define $$ Y_n(t)= \left\{\begin{array}{l@{\textup{$\mathfrak{q}$}uad\textup{$\mathfrak{m}$}box{if }}l} \textup{\textsf{b}}_n(\sigma_n -1 + (a_n^2/2)t)& -(\sigma_n-2)/a_n^2\leq t<0\\ \textup{\textsf{b}}_n((a_n^2/2)t)& 0\leq t\leq \sigma_n/a_n^2 \end{array}\right.. $$ We extend the definition of $Y_n$ to all reals by letting $Y_n\equiv\textup{\textsf{b}}_n(\sigma_n/2)$ on $\{|t|> \sigma_n/a_n^2\}$. Note that $Y_n$ is c\`adl\`ag, with only one possible jump at $t=0$ of height $-\textup{\textsf{b}}_n(\sigma_n-1)$, which is stochastically bounded in probability as $n\rightarrow \infty$. From Lemma~\ref{lem:bridge0} we know that $ (1/\sqrt{2\sigma_n})(\textup{\textsf{b}}_n(\sigma_nt),0\leq t\leq 1)$ converges to a standard Brownian bridge $\textup{$\mathfrak{m}$}athbbm{b}$ in $\textup{$\mathfrak{m}$}athcal{C}([0,1],\textup{$\mathfrak{m}$}athbbm{R})$ as $n\rightarrow\infty$. From this and the fact that $a_n\ll \sqrt{\sigma_n}$, we obtain by standard reasoning (see, e.g., the proof of~\cite[Lemma 10]{Be1}) \begin{equation} \textup{$\mathfrak{l}$}bel{eq:convYn-Y} \left(\textup{$\mathfrak{f}$}rac{Y_n(t)}{a_n},t\in\textup{$\mathfrak{m}$}athbb{R}\right)\xrightarrow[n \to \infty]{(d)}\left(Y_t,t\in\textup{$\mathfrak{m}$}athbb{R}\right), \end{equation} where $Y=(Y_t,t\in\textup{$\mathfrak{m}$}athbb{R})$ is a two-sided Brownian motion with $Y_0=0$, and the convergence holds in the space of c\`adl\`ag functions on $\textup{$\mathfrak{m}$}athbb{R}$ equipped with the compact-open topology. By the Skorokhod representation theorem, we may assume that \eqref{eq:convYn-Y} holds almost surely uniformly over compacts. Let $\varepsilon>0$. By standard properties of Brownian motion, we find $\alpha>0$ and $n_0\in \textup{$\mathfrak{m}$}athbb N$ such that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:YYn-prop} \textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,\alpha]}Y_,\textup{$\mathfrak{m}$}in_{[-\alpha,0]}Y\right\}<-1\textup{$\mathfrak{q}$}uad\textup{and}\textup{$\mathfrak{q}$}uad \textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[0,\alpha]}Y_n,\textup{$\mathfrak{m}$}in_{[-\alpha,0]}Y_n\right\} < -a_n \end{equation} with probability at least $1-\varepsilon$ for $n\geq n_0$. We fix such an $\alpha$ and work on the event where~\eqref{eq:YYn-prop} holds true. In the following, we make no difference between the root vertices $(0),\ldots,(\sigma_n-1)$ and the integers $0,\ldots,\sigma_n-1$. Moreover, recall that if $v$ is a vertex of a tree of $\textup{$\mathfrak{f}$}_n$, $\textup{$\mathfrak{r}$}(v)$ denotes the root vertex of that tree. For $v\in V(\textup{$\mathfrak{f}$}_n)$, the cactus bound~\eqref{eq:cactus2} gives \begin{equation} \textup{$\mathfrak{l}$}bel{dgr1v} d_{\textup{gr}}((0),v) \geq -\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[ 0,\textup{$\mathfrak{r}$}(v)]}\textup{\textsf{b}}_n, \textup{$\mathfrak{m}$}in_{[\textup{$\mathfrak{r}$}(v),\sigma_n-1]}\textup{\textsf{b}}_n\right\}. \end{equation} Every point $v$ in the length space ${\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}$ lies on a segment $e$ of length one connecting two vertices in $V(Q_n^{\sigma_n})$. We associate to each $v\in{\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}$ that endpoint $v'\in V(Q_n^{\sigma_n})$ of $e$ which satisfies $d(0,v)\geq d_{\textup{gr}}(0,v')$. We then extend the definition of $\textup{$\mathfrak{r}$}$ to all points $v$ in ${\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}$ by letting $\textup{$\mathfrak{r}$}(v)=\textup{$\mathfrak{r}$}(v')$, where we agree that $\textup{$\mathfrak{r}$}(v^{\bullet})=\infty$. Next define the subset of vertices $$ A_n = \left\{v\in {\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}: \textup{$\mathfrak{r}$}(v) \in [0,\alpha a_n^2/2] \cup [\sigma_n-1-\alpha a_n^2/2,\sigma_n-1]\right\}. $$ On the event where~\eqref{eq:YYn-prop} holds, we have for vertices $v\in {\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}\setminus A_n$ by~\eqref{dgr1v} (and by~\eqref{eq:distance-vdot} for $v$ with $v'=v^{\bullet}$) for $n$ sufficiently large $$d(0,v)\geq d_{\textup{gr}}(0,v') >a_n.$$ In words, $A_n$ contains the ball of radius $a_n$ around $(0)$ in $({\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n},d)$. We now consider the $\textup{\textsf{ICRT}}$ $(\mathcal{T}_Y, d_Y,[0])$ defined in terms of $Y$ and write $p_Y:\textup{$\mathfrak{m}$}athbb{R}\rightarrow \mathcal{T}_Y$ for the canonical projection. Define a subset $\textup{$\mathfrak{m}$}athcal{R}_n\subset {\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}\times\mathcal{T}_Y$ by putting \begin{equation*} \textup{$\mathfrak{m}$}athcal{R}_n=\left\{(v,p_Y(t)) :\begin{split}& v\in A_n, t\in[0,\alpha]\hbox{ with }\textup{$\mathfrak{r}$}(v) = \lfloorloor (a_n^2/2)t\rfloorloor,\hbox{ or }\\ &v\in A_n, t\in[-\alpha,0]\hbox{ with }\textup{$\mathfrak{r}$}(v) = \lfloorloor\sigma_n-1-(a_n^2/2)t\rfloorloor\end{split}\right\}. \end{equation*} On the event where~\eqref{eq:YYn-prop} is true, the set $\textup{$\mathfrak{m}$}athcal{R}_n$ fulfills the conditions of Lemma~\ref{lem:localGHconv} (with $\rho=(0)$, $\rho'=p_Y(0)$, $r=a_n$). Thus it remains to show that $\lim_{n\rightarrow\infty}a_n^{-1}\textup{dis}(\textup{$\mathfrak{m}$}athcal{R}_n)= 0$ in probability. Note that in the notation from above, we have $|d(v,w)- d_{\textup{gr}}(v',w')| \leq 2$. Recall that the label function $\textup{$\mathfrak{L}$}_n$ is given by the labels $\textup{$\mathfrak{l}$}_n$ of the tree vertices, shifted according to the label of the corresponding root vertex which is carried by the bridge $\textup{\textsf{b}}_n$. By the distance bounds~\eqref{eq:cactus1} and~\eqref{eq:dist-upperbound}, we obtain for $v,w\in A_n$ with $\textup{$\mathfrak{r}$}(v)\leq \textup{$\mathfrak{r}$}(w)$, \begin{equation*} \begin{split} \left|d(v,w)-\left(\textup{\textsf{b}}_n(\textup{$\mathfrak{r}$}(v))+\textup{\textsf{b}}_n(\textup{$\mathfrak{r}$}(w))-2\textup{$\mathfrak{m}$}ax\left\{\textup{$\mathfrak{m}$}in_{[\textup{$\mathfrak{r}$}(v),\textup{$\mathfrak{r}$}(w)]} \textup{\textsf{b}}_n,\textup{$\mathfrak{m}$}in_{[0,\textup{$\mathfrak{r}$}(v)]\cup[\textup{$\mathfrak{r}$}(w),\sigma_n-1]}\textup{\textsf{b}}_n\right\}\right)\right|\\ \leq 3\left(\sup_{A_n} \textup{$\mathfrak{l}$}_n-\inf_{A_n} \textup{$\mathfrak{l}$}_n\right)+4. \end{split} \end{equation*} Since $Y_n$ converges to $Y$, cf.~\eqref{eq:convYn-Y}, the last display entails that \begin{equation} \textup{$\mathfrak{l}$}bel{eq:thmICRT-distortion} \limsup_{n\rightarrow\infty}\textup{$\mathfrak{f}$}rac{1}{a_n}\textup{dis}(\textup{$\mathfrak{m}$}athcal{R}_n) \leq \limsup_{n\rightarrow\infty}\textup{$\mathfrak{f}$}rac{3\left(\sup_{A_n} \textup{$\mathfrak{l}$}_n-\inf_{A_n} \textup{$\mathfrak{l}$}_n\right)}{a_n}, \end{equation} and we are reduced to show that the right hand side equals zero. For that purpose, recall that $(C_n,L_n)$ denotes the contour pair associated to $(\textup{$\mathfrak{f}$}_n,\textup{$\mathfrak{l}$}_n)$, where $C_n=(C_n(t),0\leq t\leq 2n+\sigma_n)$ is distributed as a (linearly interpolated) simple random walk conditioned to first hit $-\sigma_n$ at time $N=2n+\sigma_n$. For $t\geq 0$, put $$ \tilde{C}_n(t) = \textup{$\mathfrak{f}$}rac{1}{a_n^2}C_n\left(\left(N/\sigma_n\right)a_n^2t\wedge N\right),\textup{$\mathfrak{q}$}uad\tilde{L}_n(t)=\textup{$\mathfrak{f}$}rac{1}{a_n}L_n\left(\left(N/\sigma_n\right)a_n^2t\wedge N\right). $$ The following lemma will complete our proof of Theorem~\ref{thm:ICRT}. \begin{lemma} \textup{$\mathfrak{l}$}bel{lem:ICRT} In the regime $\sigma_n\gg \sqrt{n}$, we have for sequences $a_n$ of positive reals that satisfy $\textup{$\mathfrak{m}$}ax\{1,\sqrt{n/\sigma_n}\}\ll a_n\ll\sqrt{\sigma_n}$ $$ \left((\tilde{C}_n(t),\tilde{L}_n(t)),t\geq 0\right)\xrightarrow[n\to \infty]{(p)} \left((-t,0),t\geq 0\right)\textup{$\mathfrak{q}$}uad\textup{in }\textup{$\mathfrak{m}$}athcal{C}([0,\infty),\textup{$\mathfrak{m}$}athbb{R})^2. $$ \end{lemma} Splitting the set $A_n$ from above into the disjoint sets $A_n=A_n^+\cup A_n^{-}$, where $$ A_n^+=\{v\in {\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}: \textup{$\mathfrak{r}$}(v) \in [0,\alpha a_n^2/2]\},\textup{$\mathfrak{q}$}uad A_n^-=\{v\in {\textup{$\mathfrak{m}$}athbf Q}_n^{\sigma_n}: \textup{$\mathfrak{r}$}(v) \in [\sigma_n-1-\alpha a_n^2/2,\sigma_n-1]\}, $$ Lemma~\ref{lem:ICRT} shows that $\sup_{A_n^+}\textup{$\mathfrak{l}$}_n-\inf_{A_n^+}\textup{$\mathfrak{l}$}_n = o(a_n)$ in probability as $n\rightarrow\infty$. By exchangeability of the trees of $\textup{$\mathfrak{f}$}_n$, we obtain however also that $\sup_{A_n^-}\textup{$\mathfrak{l}$}_n-\inf_{A_n^-}\textup{$\mathfrak{l}$}_n = o(a_n)$. Thus the right hand side of~\eqref{eq:thmICRT-distortion} is equal to zero, and the proof of the theorem follows. \end{proof} It remains to prove Lemma~\ref{lem:ICRT}. \begin{proof}[Proof of Lemma~\ref{lem:ICRT}] We first prove convergence of the first component. Let $0<\varepsilon<1$. We have to show that for each $K>0$, as $n\rightarrow\infty$, \begin{equation} \mathbb P\left(\sup_{t\in[0,K]}\left|\tilde{C}_n(t)+t\right|\geq \varepsilon\right)\longrightarrow 0. \end{equation} Set $N=2n+\sigma_n$, and denote by $(S_i,i\in\textup{$\mathfrak{m}$}athbb N_0)$ a simple random walk started from $S_0=0$. Write $T_{-\sigma_n}$ for its first hitting time of $-\sigma_n$. Fix $K\geq 1$, and set $K_n=\lceil(N/\sigma_n)a_n^2K\rceil$. Note that $K_n\ll N$. By definition of $\tilde{C}_n$, we obtain \begin{equation} \textup{$\mathfrak{l}$}bel{eq:lemICRT-eq1} \mathbb P\left(\sup_{t\in[0,K]}\left|\tilde{C}_n(t)+t\right|\geq \varepsilon\right) \leq \mathbb P\left(\sup_{0\leq i \leq K_n }\left|S_i+\textup{$\mathfrak{f}$}rac{\sigma_n}{N}i\right|\geq \varepsilon a_n^2\,\big |\,T_{-\sigma_n}=N\right). \end{equation} With the abbreviation $$\textup{$\mathfrak{m}$}athcal{E}_n=\left\{\sup_{0\leq i \leq K_n}\left|S_i+\textup{$\mathfrak{f}$}rac{\sigma_n}{N}i\right|\geq \varepsilon a_n^2\right\}, $$ we claim that as $n\rightarrow\infty$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:lemICRT-eq2} \mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\,\big |\,T_{-\sigma_n}=N\right)\leq 3 \mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\,\big |\,S_N=-\sigma_n\right) + o(1). \end{equation} Indeed, on the one hand, we have by the Markov property at time $K_n$, \begin{align*} \lefteqn{\mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\cap\{S_{K_n}< \sigma_n\} \big |\,T_{-\sigma_n}=N\right)}\\ &=\textup{$\mathfrak{m}$}athbb E\left[1 \textup{$\mathfrak{m}$}kern -6mu 1_{\textup{$\mathfrak{m}$}athcal{E}_n\cap\{S_{K_n}< \sigma_n\}\cap\{S_i>-\sigma_n \hbox{\small{ for all }}i\leq K_n\}} \textup{$\mathfrak{f}$}rac{\mathbb P\left(T_{-(\sigma_n+S_{K_n})}=N-K_n\,|\,S_{K_n}\right)}{\mathbb P\left(T_{-\sigma_n}=N\right)}\right]. \end{align*} By Kemperman's formula, we obtain on the event $\{|S_{K_n}| <\sigma_n\}$ for $n$ sufficiently large \begin{align*} \textup{$\mathfrak{f}$}rac{\mathbb P\left(T_{-(\sigma_n+S_{K_n})}=N-K_n\,|\,S_{K_n}\right)}{\mathbb P\left(T_{-\sigma_n}=N\right)} &=\textup{$\mathfrak{f}$}rac{|\sigma_n+S_{K_n}|}{N-K_n}\textup{$\mathfrak{f}$}rac{N}{\sigma_n}\textup{$\mathfrak{f}$}rac{\mathbb P\left(S_{N-K_n}=-(\sigma_n+S_{K_n})\,|\,S_{K_n}\right)}{\mathbb P\left(S_N=-\sigma_n\right)}\\ &\leq 3\textup{$\mathfrak{f}$}rac{\mathbb P\left(S_{N-K_n}=-(\sigma_n+S_{K_n})\,|\,S_{K_n}\right)}{\mathbb P\left(S_N=-\sigma_n\right)}. \end{align*} Plugging this into the expression above, we arrive at $$\mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\cap\{S_{K_n}<\sigma_n\} \big |\,T_{-\sigma_n}=N\right)\leq 3\mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\cap\{S_{K_n}< \sigma_n\} \big |\,S_N=-\sigma_n\right). $$ On the other hand, arguing as above, we obtain \begin{align*} \lefteqn{\mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\cap\{S_{K_n}\geq \sigma_n\} \big |\,T_{-\sigma_n}=N\right)=\mathbb P\left(S_{K_n}\geq \sigma_n \big |\,T_{-\sigma_n}=N\right)}\\ &\textup{$\mathfrak{q}$}uad= \textup{$\mathfrak{f}$}rac{N}{(N-K_n)\sigma_n}\textup{$\mathfrak{m}$}athbb E\left[(\sigma_n+S_{K_n})\textup{$\mathfrak{f}$}rac{\mathbb P\left(S_{N-K_n}=-(\sigma_n+S_{K_n})\,|\,S_{K_n}\right)}{\mathbb P\left(S_N=-\sigma_n\right)}1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{S_{K_n}\geq \sigma_n\}}\right]. \end{align*} Clearly, on the event $\{S_{K_n}\geq \sigma_n\}$, the fraction of the two probabilities inside the expectation is bounded by $1$. Moreover, keeping in mind that $K_n\ll\sigma_n^2$, we have by the local central limit theorem~\eqref{eq:localCLT} $$ \textup{$\mathfrak{m}$}athbb E\left[(\sigma_n+S_{K_n})1 \textup{$\mathfrak{m}$}kern -6mu 1_{\{S_{K_n}\geq \sigma_n\}}\right]\leq 2\sum_{\ell=\sigma_n}^{K_n}\ell\,\mathbb P\left(S_{K_n}=\ell\right)\lesssim K_n^{1/2}. $$ We obtain $\mathbb P\left(\textup{$\mathfrak{m}$}athcal{E}_n\cap\{S_{K_n}\geq \sigma_n\} \big |\,T_{-\sigma_n}=N\right)=o(1),$ and~\eqref{eq:lemICRT-eq2} follows. In order to conclude the convergence of the first component, we now show that the probability on the right hand side of~\eqref{eq:lemICRT-eq2} tends to zero as $n\rightarrow\infty$. Our proof is based on a change-of-measure argument in spirit of~\cite[Proof of Lemma 24]{Be3}. We consider the random walk $\tilde{S}=(\tilde{S}_i,i\in\textup{$\mathfrak{m}$}athbb N_0)$ started at zero with step distribution $$ \mathbb P\left(\tilde{S}_{i+1}-\tilde{S}_i=1+\textup{$\mathfrak{f}$}rac{\sigma_n}{N}\right)=\textup{$\mathfrak{f}$}rac{1-\sigma_n/N}{2},\textup{$\mathfrak{q}$}uad\mathbb P\left(\tilde{S}_{i+1}-\tilde{S}_i=-1+\textup{$\mathfrak{f}$}rac{\sigma_n}{N}\right)=\textup{$\mathfrak{f}$}rac{1+\sigma_n/N}{2}. $$ The walk $\tilde{S}$ is a martingale, and a direct computation shows that for every measurable function $f$ and any $\ell\in\textup{$\mathfrak{m}$}athbb N_0$, $$ \textup{$\mathfrak{m}$}athbb E\left[f\left(\left(S_i+\textup{$\mathfrak{f}$}rac{\sigma_n}{N}i\right)_{0\leq i\leq \ell}\right)\right]=\textup{$\mathfrak{m}$}athbb E\left[\left(1-\textup{$\mathfrak{f}$}rac{\sigma_n^2}{N^2}\right)^{-\ell/2}\left(\textup{$\mathfrak{f}$}rac{1-\sigma_n/N}{1+\sigma_n/N}\right)^{-\textup{$\mathfrak{f}$}rac{1}{2}(\tilde{S}_\ell-\ell\sigma_n/N)}f\left((\tilde{S}_i)_{0\leq i\leq \ell}\right)\right]. $$ With the last display, we can rewrite the probability on the right hand side of~\eqref{eq:lemICRT-eq2} as $$ \mathbb P\left(\sup_{0\leq i \leq K_n}\left|S_i+\textup{$\mathfrak{f}$}rac{\sigma_n}{N}i\right|\geq \varepsilon a_n^2\,\mathcal Big |\,S_{N}=-\sigma_n\right)=\mathbb P\left(\sup_{0\leq i \leq K_n}\left|\tilde{S}_i\right|\geq \varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=0\right). $$ We estimate \begin{align}\textup{$\mathfrak{l}$}bel{eq:lemICRT-eq3} \lefteqn{\mathbb P\left(\sup_{0\leq i \leq K_n}\left|\tilde{S}_i\right|\geq \varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=0\right)}\nonumber\\ &\leq \mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=0\right) +\mathbb P\left(\sup_{0\leq i \leq K_n}\tilde{S}_i\geq \varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=0\right), \end{align} and we show next that the first summand on the right tends to zero as $n\rightarrow\infty$. First, note that for $k\in\textup{$\mathfrak{m}$}athbb Z$, $(\tilde{S}_i,0\leq i\leq N)$ is under $\mathbb P(\cdot\,|\,\tilde{S}_N=2k)$ uniformly distributed among all paths starting at $0$ at time $0$, ending at $2k$ at time $N$ and making upward steps of size $ 1+\sigma_n/N$ and downward steps of size $-1+\sigma_n/N$. Switching an upward step chosen uniformly at random into a downward step gives a path with law $\mathbb P(\cdot\,|\,\tilde{S}_N=2(k-1))$, which lies below the original one. Therefore, for all $k\leq 0$ such that $\mathbb P(\tilde{S}_{N}=2k) >0$, $$ \mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=0\right)\leq\mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=2k\right). $$ From this inequality, we deduce that \begin{align*} \lefteqn{\mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\,\mathcal Big |\,\tilde{S}_{N}=0\right)}\\ &\leq {\mathbb P\left(\tilde{S}_N\leq 0\right)}^{-1}\sum_{k=0}^\infty\mathbb P\left(\tilde{S}_N=-2k\right)\mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\,\mathcal Big|\,\tilde{S}_{N}=2k\right)\\ &\leq {\mathbb P\left(\tilde{S}_N\leq 0\right)}^{-1}\mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\right). \end{align*} The central limit theorem bounds the probability $\mathbb P(\tilde{S}_N\leq 0)$ from below by $1/3$ for $n$ large enough. From Doob's inequality, we get $$ \mathbb P\left(\inf_{0\leq i \leq K_n}\tilde{S}_i\leq -\varepsilon a_n^2\right)\leq \mathbb P\left(\sup_{0\leq i\leq K_n}|\tilde{S}_i|\geq \varepsilon a_n^2\right)\leq \textup{$\mathfrak{f}$}rac{1}{\varepsilon^2a_n^4}\textup{$\mathfrak{m}$}athbb E\left[\tilde{S}_{K_n}^2\right] \leq \textup{$\mathfrak{f}$}rac{4K_n}{\varepsilon^2a_n^4} = o(1), $$ where we have used that $a_n^2\gg \textup{$\mathfrak{m}$}ax\{1,n/\sigma_n\}$. The second term of~\eqref{eq:lemICRT-eq3} involving the supremum of $\tilde{S}$ up to time $K_n$ is treated similarly, by switching downward into upward steps. We conclude that the probability on the right hand side of~\eqref{eq:lemICRT-eq1} converges to zero as $n$ tends to infinity. It remains to prove the joint convergence of $(\tilde{C}_n,\tilde{L}_n)$ stated in the lemma. Let again $K>0$. By what we just have proved and Skorokhod's theorem, we can assume that $(\tilde{C}_n(t),t\geq 0)$ converges almost surely to $(t, t\geq 0)$ in $\textup{$\mathfrak{m}$}athcal{C}([0,K])$. Finite-dimensional convergence of $(\tilde{C}_n,\tilde{L}_n)$ then follows from standard arguments as for example given in~\cite[Proof of Theorem 4.3]{LGMi}. We are left with showing tightness of the laws of $\tilde{L}_n$ on $\textup{$\mathfrak{m}$}athcal{C}([0,K])$. By the theorem of Arzel\`a-Ascoli, we have to show that for every $\varepsilon>0$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:lemICRT-eq4} \lim_{\mathrm{d}elta\mathrm{d}ownarrow 0}\limsup_{n\rightarrow\infty}\mathbb P\left(\sup_{s,t\in[0,K], |s-t|\leq \mathrm{d}elta}\left|\tilde{L}_n(s)-\tilde{L}_n(t)\right|\geq \varepsilon\right) =0. \end{equation} Recall that $K_n=\lceil(N/\sigma_n)a_n^2K\rceil$. Since by the first part of the lemma, $$ \mathbb P\left(\sup_{i\leq K_n}|C_n(i)| > 2Ka_n^2\right)\longrightarrow 0$$ as $n$ tends to infinity, we may instead show tightness of the laws of $\tilde{L}_n$ given the event $\textup{$\mathfrak{m}$}athcal{E}_n'=\{\sup_{i\leq K_n}|C_n(i)| \leq 2Ka_n^2\}$. By Kolmogorov's criterion, tightness follows if we show that there exists a constant $M<\infty$ such that for all $n$ and all $s,t \in [0,K]$, \begin{equation} \textup{$\mathfrak{l}$}bel{eq:lemICRT-tightness} \textup{$\mathfrak{m}$}athbb E\left[|\tilde{L}_n(s)-\tilde{L}_n(t)|^{4}\,\mathcal Big |\,\textup{$\mathfrak{m}$}athcal{E}_n'\right]\leq M|s-t|^2. \end{equation} Since $L_n$ is Lipschitz, we can restrict ourselves to the case where $(N/\sigma_n)a_n^2s$ and $(N/\sigma_n)a_n^2t$ are integers. Let $$\Delta \tilde{C}_n(s,t) =\tilde{C}_n(s)+\tilde{C}_n(t)-2\textup{$\mathfrak{m}$}in_{[s,t]}\tilde{C}_n.$$ By definition of the contour pair $(C_n,L_n)$, conditionally given $C_n$, the difference $a_n(\tilde{L}_n(s)-\tilde{L}_n(t))$ is distributed as a sum of i.i.d. variables $\eta_i$ with the uniform law on $\{-1,0,1\}$. By construction, the sum involves at most $a_n^2\Delta\tilde{C}_n(s,t)$ summands. We thus obtain for some $M'>0$ $$ \textup{$\mathfrak{m}$}athbb E\left[|\tilde{L}_n(s)-\tilde{L}_n(t)|^{4}\,\mathcal Big |\,\textup{$\mathfrak{m}$}athcal{E}_n'\right]\leq \textup{$\mathfrak{f}$}rac{1}{a_n^{4}}\textup{$\mathfrak{m}$}athbb E\left[\left(\sum_{i=1}^{a_n^2\Delta\tilde{C}_n(s,t)}\eta_i\right)^{4}\,\mathcal Big |\,\textup{$\mathfrak{m}$}athcal{E}_n'\right] \leq M'\textup{$\mathfrak{m}$}athbb E\left[|\Delta\tilde{C}_n(s,t)|^2\,\mathcal Big |\,\textup{$\mathfrak{m}$}athcal{E}_n'\right]. $$ By the first part of the lemma, $|\Delta\tilde{C}_n(s,t)|$ converges to $|s-t|$ in probability, and $|\Delta\tilde{C}_n(s,t)|$ is uniformly bounded on $\textup{$\mathfrak{m}$}athcal{E}_n'$ by $8K$. Therefore, the expectation on the right converges to $|s-t|^2$, and~\eqref{eq:lemICRT-tightness} follows. This completes the proof of the lemma. \end{proof} \noindent {\bf Acknowledgments.} We would like to thank Lo\"ic Richier for helpful discussions, and for introducing EB to IPE. \end{document}

\begin{document} \begin{bibunit} \title{The mixed problem for the Laplacian in Lipschitz domains} \author{ Katharine A. Ott \footnote{Research supported, in part, by the National Science Foundation.} \\ Department of Mathematics \\University of Kentucky \\ Lexington, Kentucky \and Russell M. Brown \\ Department of Mathematics \\University of Kentucky \\ Lexington, Kentucky } \date{} \maketitle \abstract{ We consider the mixed boundary value problem, or Zaremba's problem, for the Laplacian in a bounded Lipschitz domain $\Omega$ in $ {\bf R} ^ n$, $n\geq 2$. We decompose the boundary $ \partial \Omega= D\cup N$ with $D$ and $N$ disjoint. The boundary between $D$ and $N$ is assumed to be a Lipschitz surface in $\partial \Omega$. We find an exponent $q_0>1$ so that for $ p $ between $ 1 $ and $q_0$ we may solve the mixed problem for $L^p$. Thus, if we specify Dirichlet data on $D$ in the Sobolev space $\sobolev 1 p (D)$ and Neumann data on $N$ in $ L^ p (N)$, the mixed problem with data $f_N$ and $f_D$ has a unique solution and the non-tangential maximal function of the gradient lies in $L^p( \partial \Omega)$. We also obtain results for $p=1$ when the data comes from Hardy spaces. {\em Keywords: }Mixed boundary value problem, Laplacian {\em Mathematics subject classification: }35J25 } \section{Introduction} Over the past thirty years, there has been a great deal of interest in studying boundary value problems for the Laplacian in Lipschitz domains. A fundamental paper of Dahlberg \cite{BD:1977} treated the Dirichlet problem. Jerison and Kenig \cite{JK:1982c} treated the Neumann problem and provided a regularity result for the Dirichlet problem. Another boundary value problem of interest is the mixed problem or Zaremba's problem where we specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. To state this boundary value problem, we let $ \Omega$ be a bounded open set in $ {\bf R} ^n$ and suppose that we have written $ \partial \Omega = D\cup N$ where $D$ is an open subset of the boundary and $ N =\partial \Omega \setminus D$. We consider the {\em $L^p$-mixed problem } by which we mean the boundary value problem \begin{equation} \label{MP} \left\{ \begin{array}{ll} \Delta u = 0, \qquad & \mbox{in } \Omega\\ u = f_D, \qquad & \mbox{on } D\\ \bigfrac { \partial u }{ \partial \nu} = f_N,\qquad & \mbox{on } N \\ \nontan{(\nabla u ) } \in L^ p ( \partial \Omega). \end{array} \right. \end{equation} Here, we are using $ \nontan {(\nabla u)}$ to denote the non-tangential maximal function of $\nabla u$ and the restriction to the boundary of $u$ and $\nabla u$ are defined using non-tangential limits. See section \ref{Definitions} for details. The normal derivative at the boundary $ \partial u /\partial \nu$ is defined as $ \nabla u \cdot \nu$ where $\nu $ is the outer unit normal defined a.e.~on the boundary. Our goals are to find conditions on $\Omega$, $N$ and $D$ which allow us to show that (\ref{MP}) has at most one solution and to find conditions on $ \Omega$, $N$, $D$, $f_N$ and $f_D$ which guarantee the existence of solutions. The study of the mixed problem in Lipschitz domains is listed as an open problem in Kenig's CBMS lecture notes \cite[Problem 3.2.15]{CK:1994}. Recall that simple examples show that we cannot expect to find solutions whose gradient lies in $L^2$ of the boundary. For example, the function $ \mathop{\rm Re}\nolimits \sqrt z$ on the upper half-plane has zero Neumann data on the positive real axis and zero Dirichlet data on the negative real axis but the gradient is not locally square integrable on the boundary of the upper half-space. This appears to present a technical problem as the standard technique for studying boundary value problems has been the Rellich identity which produces estimates in $L^2$. In 1994, one of the authors observed that the Rellich identity could be used to study the mixed problem in a restricted class of Lipschitz domains \cite{RB:1994b}. Roughly speaking, this work requires that the sets $N$ and $D$ meet at an angle less than $\pi$. Based on this work and the methods used by Dahlberg and Kenig to study the Neumann problem \cite{DK:1987}, J. Sykes \cite{JS:1999,SB:2001} established results for the mixed problem in a restricted class of Lipschitz graph domains. I.~Mitrea and M.~Mitrea \cite{MM:2007} have studied the mixed problem for the Laplacian with data taken from a large family of function spaces, but with a restriction on the class of domains. Brown and I.~Mitrea have studied the mixed problem for the Lam\'e system \cite{MR2503013} and Brown, I.~Mitrea, M.~Mitrea and Wright have considered a mixed problem for the Stokes system \cite{MR2563727}. More recently, Lanzani, Capogna and Brown \cite{LCB:2008} used a variant of the Rellich identity to establish an estimate for the mixed problem in two-dimensional graph domains when the data comes from weighted $L^2$ spaces and the Lipschitz constant is less than one. The present work also relies on weighted estimates, but uses a simpler, more flexible approach that applies to all Lipschitz domains. Several other authors have treated the mixed problem in various settings. Verchota and Venouziou \cite{MR2500502} treat a large class of three dimensional polyhedral domains under the condition that the Neumann and Dirichlet faces meet at an angle of less than $\pi$. Maz'ya and Rossman \cite{MR:2005,MR:2007,MR:2006} have studied the Stokes system in polyhedral domains. Finally, we note that Savar\'e \cite{GS:1997} has shown that on smooth domains, we may find solutions in the Besov space $ B^ {2,\infty}_{ 3/2}$. This result seems to be very close to optimal. The example $ \mathop{\rm Re}\nolimits \sqrt z$ described above shows that we cannot hope to obtain an estimate in the Besov space $ B^ {2,2}_{3/2}$. We outline the rest of the paper and describe the main tools of the proof. Our first main result is an existence result for the mixed problem when the Neumann data is an atom for a Hardy space. We begin with the weak solution of the mixed problem and use Jerison and Kenig's results for the Dirichlet problem and Neumann problem \cite{JK:1982c} to obtain estimates for the gradient of the solution on the interior of $D$ or $N$. This leads to a weighted estimate where the weight is a power of the distance to the common boundary between $D$ and $N$. The estimate involves a term in the interior of the domain $\Omega$. We handle this term by showing that the gradient of a weak solution lies in $L^ p (\Omega)$ for some $p>2$. The $L^p(\Omega)$ estimates for the gradient of a weak solution are proved in section \ref{Reverse} using the reverse H\"older technique of Gehring \cite{FG:1973} and Giaquinta and Modica \cite{MR549962}. Using this weighted estimate for solutions of the mixed problem, we obtain existence for solutions with Hardy space data by extending the methods of Dahlberg and Kenig \cite{DK:1987}. Uniqueness of solutions is proven in section \ref{Unique}. With the Hardy space results in hand, we establish the existence of solutions to the mixed problem when the Neumann data is in $L^p ( N)$ and the Dirichlet data is in the Sobolev space $ \sobolev 1 p (D)$. This is done in sections \ref{BoundaryReverse} and \ref{LpSection} by adapting the reverse H\"older technique used by Shen to study boundary value problems for elliptic systems \cite{ZS:2007}. The novel feature in our work is that we are able to use the estimates in Hardy spaces proven in section \ref{Atoms}, whereas Shen's work begins with existence in $L^2$. \section{Definitions and preliminaries} \label{Definitions} We say that a bounded, connected open set $\Omega$ is a Lipschitz domain if the boundary is locally the graph of a Lipschitz function. To make this precise, for $M>0$, $ x\in \partial \Omega $ and $ r>0$, we define a {\em coordinate cylinder} $\cyl x r $ to be $\cyl x r = \{ y : |y'-x'|< r , \ |y_n -x_n | < ( 1+M)r \}$. We use coordinates $(x', x_n ) = ( x_1, x'', x_n ) \in {\bf R} \times {\bf R} ^ { n- 2 } \times {\bf R}$ and assume that this coordinate system is a translation and rotation of the standard coordinates. We say that $ \Omega$ is a {\em Lipschitz domain } if for each $x$ in $\partial \Omega$, we may find a coordinate cylinder and a Lipschitz function $\phi : {\bf R} ^ { n-1} \rightarrow {\bf R}$ with Lipschitz constant $M$ so that \begin{eqnarray*} \Omega \cap \cyl x r & =& \{ (y', y_n ) : y_ n > \phi (y') \} \cap \cyl x r \\ \partial \Omega \cap \cyl x r & = & \{ (y', y_n ) : y_ n = \phi (y') \}\cap \cyl x r . \end{eqnarray*} For a Lipschitz domain $ \Omega$, we define a {\em decomposition of the boundary for the mixed problem}, $\partial \Omega = D \cup N$, as follows. We assume that $D$ is a relatively open subset of $ \partial \Omega$, $N= \partial \Omega \setminus D$ and let $\Lambda $ be the boundary (relative to $\partial \Omega$) of $D$. For each $x$ in $ \Lambda$, we require that a coordinate cylinder centered at $x$ have some additional properties. We ask that there be a coordinate system $(x_1, x'', x_n)$, a coordinate cylinder $\cyl x r $, a function $\phi$ as above and also a Lipschitz function $\psi: {\bf R}^{ n-2} \rightarrow {\bf R}$ with Lipschitz constant $M$ so that \begin{eqnarray*} \cyl x r \cap D & = & \{ (y_1, y'', y_n ) : y_ 1 > \psi (y''), \ y_n = \phi(y') \} \cap \cyl x r \\ \cyl x r \cap N & = & \{ (y_1, y'', y_n ) : y_ 1 \leq \psi (y''), \ y_n = \phi(y') \} \cap \cyl x r . \end{eqnarray*} We fix a covering of the boundary by coordinate cylinders $\{ \cyl {x_i } { r _i } \}_{i=1} ^ L$ so that each $\cyl {x_i } { 100r _i } $ is also a coordinate cylinder. We assume that for each $i$, the cylinder $\cyl {x_i} { 100r_i} \cap \partial \Omega \subset D$, $\cyl {x_i }{ 100r_i} \cap \partial \Omega \subset N $ or $\cyl {x_i} {100r_i}$ is one the coordinate cylinders from the definition of the boundary decomposition for the mixed problem. We let $r_0 = \min \{ r_i : i =1, \dots ,L\}$ be the smallest radius in the collection. We will call a Lipschitz domain $ \Omega$ and a decomposition of the boundary $ \partial \Omega = N \cup D$ satisfying the above properties a {\em standard domain for the mixed problem}. We will use $ \delta (y) = \mathop{\rm dist}\nolimits (y, \Lambda)$ to denote the distance from a point $y $ to $\Lambda$. We will let $ \ball x r = \{ y : |y-x| < r \}$ denote the standard ball in $ {\bf R} ^n$ and then $ \sball x r = \ball x r \cap \partial \Omega$ will denote a {\em surface ball}. Throughout this paper we will need to be careful of several points. The surface balls may not be connected and we will use the notation $ \sball x r $ where $ x$ may not be on the boundary. We use $ \dball x r $ to stand for $ \ball x r \cap \Omega$. Since $\Lambda$ is a Lipschitz graph, we may find a constant $c = c(n,M) >0$ so that we have the property \begin{equation} \label{SurfProp} \mbox{If $ x\in \Lambda$ and $ 0<r< r_0$, then $\sigma ( \sball x {r} \cap D ) > c r^ { n-1}$. } \end{equation} Here and throughout this paper, we use $ \sigma$ for surface measure. Our main tool for estimating solutions will be the non-tangential maximal function. We fix $ \alpha > 0$ and for $ x\in \partial \Omega$ we define a {\em non-tangential approach region } by $$ \ntar x = \{ y \in \Omega : |x-y | \leq ( 1+ \alpha) \mathop{\rm dist}\nolimits (y, \partial \Omega) \}. $$ Given a function $u$ defined on $ \Omega$, we define the {\em non-tangential maximal function } by $$ \nontan{u} (x) = \sup _{ y \in \ntar x } |u (y) |, \qquad x \in \partial \Omega. $$ It is well-known that for different values of $ \alpha$, the non-tangential maximal functions have comparable $L^p$-norms. Thus, the dependence on $\alpha $ is not important for our purposes and we suppress the value of $\alpha $ in our notation. In (\ref{MP}), we define the restriction of $u$ and $ \nabla u$ to the boundary using non-tangential limits. Thus, for a function $v$ defined in $ \Omega$ and $ x \in \partial \Omega$, we define $$ v(x) = \lim _{ \Gamma (x) \ni y \rightarrow x } v(y) $$ provided the limit exists. It is well-known that if $v$ is harmonic in a Lipschitz domain, then the non-tangential limits exist at almost every point where the non-tangential maximal function is finite. In addition, if the non-tangential maximal function of $ \nabla u$ lies in $L^p( \partial \Omega)$, then according to the argument in \cite[Lemma 2.2]{RB:1995a}, as corrected in Wright \cite{MR2713677}, the non-tangential maximal function of $u$ lies in an $L^p$-space and hence has non-tangential limits a.e.. Many of our estimates will be of a local, scale invariant form and hold on a scale $r$ that is less than $r_0$. The constants in these local estimates will depend on the constant $M$, the dimension $n$, and any $L^p$-indices that appear in the estimate. If a constant depends on $M$, $n$, any $L^p$-indices and also depends on the collection of coordinate cylinders which cover $ \partial \Omega$ and the constant in the coercivity condition (\ref{coerce}), then we say that the constant depends on the global character of $ \Omega$, $N$ and $D$. We will use $L^p(E)$ to denote $L^p$-spaces. If $ E\subset \partial \Omega$, then we use the $(n-1)$-dimensional measure on the boundary to define the $L^p$-space. Otherwise, the $L^p$-norm is taken with respect to $n$-dimensional Lebesgue measure. For $ \Omega$ an open subset of $ {\bf R} ^n$, $k=1,2,\dots$ and $ 1\leq p \leq \infty$, we use $ \sobolev kp(\Omega)$ to denote the Sobolev space of functions having $k$ derivatives in $L^p( \Omega) $. We introduce notation for the tangential gradient of a function defined on the boundary, $ \tangrad u$. If $u$ is a smooth function defined in a neighborhood of $ \partial \Omega$, then we have that $\tangrad u = \nabla u - (\nabla u \cdot \nu )\nu$. See \cite[p.~580]{GV:1984} for more details. For $D$ an open subset of $\partial \Omega$, we use $ \sobolev 1 p(D)$ to denote the Sobolev space of functions defined on $D$ and having one derivative in $L^p(D)$. The norm in this space is given by $ \|f\|_{ \sobolev 1 p (D)}= \| f\|_{ L^ p (D)}+\|\tangrad f \|_ { L^ p (D)}$. Before stating the main theorem, we recall the definitions of atoms and atomic Hardy spaces. We say that $a$ is an {\em atom for the boundary }$ \partial \Omega$ if $a$ is supported in a surface ball $ \Delta _r (x)$ for some $x$ in $ \partial \Omega$, $\|a\|_{L^\infty(\partial \Omega)} \leq 1/\sigma(\Delta _r(x))$ and $ \int _{ \partial \Omega } a \, d\sigma = 0. $ When we consider the mixed problem, we will want to consider atoms for the subset $N$. We say that {\em $ a$ is an atom for $N$} if $a$ is the restriction to $N$ of a function $\tilde a $ which is an atom for $ \partial \Omega$. For $ N $ a subset of $ \partial \Omega$, the Hardy space $ H^1( N)$ is the collection of functions $f$ which can be represented as $ \sum \lambda_j a_j$ where each $a_j$ is an atom for $N$ and the coefficients satisfy $\sum | \lambda _j|< \infty $. This includes, of course, the case where $N = \partial \Omega$ and then we obtain the standard definition. It is easy to see that the Hardy space $H^1(N)$ is the restriction to $N$ of elements of the Hardy space $ H^ 1 ( \partial \Omega)$. We give a similar definition for the Hardy-Sobolev space $H^ { 1,1}$. We say that $ A$ is an {\em atom for $H^ { 1,1 } ( \partial \Omega)$ } if $A$ is supported in a surface ball $ \sball x r $ for some $x \in \partial \Omega$ and $\| \nabla _t A \| _ { L^ \infty ( \partial \Omega ) } \leq 1/\sigma ( \sball x r )$. We say that $A$ is an {\em atom for $H^ { 1 ,1 } (D) $ } if $A$ is the restriction to $D$ of an atom $ \tilde A$ for $ \partial \Omega$. Again, the space $H^ { 1,1 }( D )$ is the collection generated by taking sums of $H^ {1,1}(D) $ atoms with coefficients in $ \ell ^1$. See the article of Coifman and Weiss \cite{CW:1976} for more information about Hardy spaces. We are now ready to state our main theorem. \begin{theorem} \label{main} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem. a) For $ p \geq 1$, the $L^p$-mixed problem has at most one solution. b) If $f_N$ lies in $H^ 1(N)$ and $f _D $ lies in $H^{1,1}(D)$, the $L^1$-mixed problem has a solution which satisfies the estimate $$ \| (\nabla u )^*\| _ { L^ 1 (\partial \Omega )} \leq C ( \| f_N\|_ { H ^ 1( N) } + \|f_D\| _ { H ^ { 1,1 } (D)}) . $$ c) There exists $q_0 > 1 $, depending only on $M$ and $n$ so that for $p$ satisfying $1< p< q_0$, we have: If $f_N \in L^ p (N)$ and $ f_D \in \sobolev 1 p(D)$, then the $L^p$-mixed problem has a solution $u$ which satisfies $$ \| (\nabla u ) ^ * \| _ { L^ p ( \partial \Omega ) } \leq C ( \| f_N \| _{L^ p(N) } + \|f_D \| _ { \sobolev 1 p (D)} ) . $$ The constants in the estimates depend on the global character of the domain and the index $p$. \end{theorem} The rest of the paper is devoted to the proof of this theorem. We outline the main steps of the proof. \begin{proof}[Outline of the proof] We begin by recalling that for the Dirichlet problem with data from a Sobolev space, we obtain non-tangential maximal function estimates for the gradient of the solution. This is treated for $p=2 $ by Jerison and Kenig \cite{JK:1982c} and for $ 1< p < 2$ by Verchota \cite{GV:1982,GV:1984}. The Hardy space problem was studied by Dahlberg and Kenig \cite{DK:1987} and by D.~Mitrea in two dimensions \cite[Theorem 3.6]{MR1883390}. Using these results, it suffices to prove Theorem \ref{main} in the case when the Dirichlet data is zero. The existence when the Neumann data is taken from the atomic Hardy space and the Dirichlet data is zero is given in Theorem \ref{HardyTheorem}. The existence for $L^p$ data appears in section \ref{LpSection}. It suffices to establish uniqueness when $p=1$ and this is treated in Theorem \ref{uRuniq}. \end{proof} \section{Higher integrability of the gradient of a weak solution} \label{Reverse} It is well-known that one can obtain higher integrability of the gradient of weak solutions of an elliptic equation. An early result of this type is due to Meyers \cite{NM:1963}. Meyers's result has been extended to the mixed problem by Gr\"oger \cite{MR990595}. However, we choose to obtain our estimates using the reverse H\"older technique introduced by Gehring \cite{FG:1973} and Giaquinta and Modica \cite{MR549962} (we use the formulation from Giaquinta \cite[p.~122]{MG:1983}). This approach allows us to include non-zero boundary data and obtain local, scale-invariant results. At a few points of the proof it will be simpler if we are working in a coordinate cylinder $Z$ where we have that $ \partial \Omega\cap Z $ lies in a hyperplane. Thus, we will establish results for divergence form elliptic operators with bounded measurable coefficients as this class is preserved by a change of variable that will flatten part of the boundary. We will consider several formulations of the mixed problem. Our goal is to obtain solutions whose gradient lies in $L^p( \partial \Omega)$ for $p$ near 1. Our argument begins with a weak solution whose gradient lies in $L^2 ( \Omega)$. We will show that under appropriate assumptions on the data, this solution will have a gradient in $L^p ( \partial \Omega)$. We describe a weak formulation of the mixed boundary value problem. Some of the results of this section will hold for solutions of divergence form operators. Thus, we define weak solutions in this more general setting. For $D$ a subset of the boundary, we let $W^{1,2}_D(\Omega)$ be the closure in $W^{1,2}(\Omega)$ of functions in $ C_0^ \infty ( {\bf R} ^ n )$ for which $ \mathop{\rm supp}\nolimits u \cap \bar D = \emptyset$. We let $ W_D^{1/2,2}(\partial \Omega) $ be the restrictions to $ \partial \Omega$ of the space $ W^ {1,2}_D( \Omega)$. We define $W^{-1/2,2}_D( \partial \Omega)$ to be the dual of $W^ { 1/2,2} _D( \partial \Omega)$. The Neumann data $f_N$ will be taken from the space $W^{-1/2,2}_D(\partial \Omega)$. If $A(x) $ is a symmetric matrix with bounded, measurable entries and satisfies the ellipticity condition $\lambda |\xi |^2 \geq A(x) \xi \cdot \xi \geq \lambda^{-1} |\xi |^ 2 $ for some $ \lambda >0$ and all $ \xi \in {\bf R} ^ n$, we consider the problem \begin{equation}\label{WeakMix} \left \{ \begin{array}{ll} {\mathop{\rm div}\nolimits} A \nabla u = 0 , \qquad & \mbox{in } \Omega \\ u = 0 , \qquad & \mbox{on } D \\ A\nabla u \cdot \nu = f _N, & \mbox{on }N. \end{array} \right. \end{equation} We say that $u$ is a {\em weak solution }of this problem if $u \in W^ { 1,2 } _D ( \Omega) $ and we have $$ \int _ \Omega A \nabla u \cdot \nabla v \, dy = \langle f _N , v \rangle _ { \partial \Omega} , \qquad \mbox{for all } v \in W^ { 1,2 }_D ( \Omega). $$ Here, we are using $ \langle \cdot , \cdot \rangle_{ \partial \Omega}$ to denote the pairing between $ W^ { 1/2,2 } _D( \partial \Omega)$ and the dual $ W^ {- 1/2,2 } _D( \partial \Omega)$. To establish existence of weak solutions of the mixed problem, we assume the coercivity condition \begin{equation} \label{coerce} \|u\|_{ L^2 ( \Omega) } \leq c \|\nabla u \|_{ L^2( \Omega)}, \qquad u \in W^ { 1 , 2 } _D ( \Omega) . \end{equation} Under this assumption, the existence and uniqueness of weak solutions to (\ref{WeakMix}) is a consequence of the Lax-Milgram theorem. It is easy to see that (\ref{coerce}) holds when $ \Omega$, $N$ and $D$ is a standard domain for the mixed problem. If $f_N$ is a function on $N$, then we may identify $f_N$ with an element of the space $W^{-1/2,2}_D( \partial \Omega)$ by $$ \langle f_N , \phi \rangle_{ \partial \Omega} = \int_{N } f_N \phi \, d\sigma, \qquad \mbox{for all } \phi \in W^ { 1/2,2} _D( \partial \Omega). $$ From Sobolev embedding we have $ W^ { 1/2, 2}_D( \partial \Omega) \subset L^p ( \partial \Omega)$, where $p = 2 ( n-1)/(n-2)$ if $n\geq 3$ or $ p < \infty$ when $n=2$. Thus the integral on the right-hand side will be well-defined if we have $f_N$ in $L^ { 2(n-1)/n}( N )$ when $n \geq 3 $ or $ L^ p ( N )$ for any $ p > 1$ when $n =2$. \note { Outline of the proof of existence of weak solution of the mixed problem (\ref{MP}). 1. We assume that the Neumann data, $f_N$ lies in the dual of $W^ { 1/2,2}_D( \partial \Omega)$ and the Dirichlet data $f_D$ lies in $W^ {1/2, 2} ( \partial \Omega)$ and thus is the restriction to $ \partial \Omega$ of a function $ W^ { 1,2}( \Omega)$. 2. We solve the Dirichlet problem $$\left\{ \begin{array}{ll} \Delta u = 0 , \qquad & \mbox{in } \Omega\\ u = f_D, \qquad & \mbox{on } \partial \Omega. \end{array} \right. $$ The solution satisfies $u - f_D$ lies in $ W^ { 1,2}_0 ( \Omega)$ and $$ \int _ \Omega \nabla u\cdot \nabla \phi \, dx = 0 , \qquad \phi \in W^{ 1,2}_0 (\Omega ) . $$ By Dirichlet's principle, we have $ \|\nabla u \|_{L^2 ( \Omega)}\leq \|\nabla f_D\|_ {L^ 2 ( \Omega)}$. 3. If $f$ is in $ W^ {1,2} ( \Omega)$ and $ \Delta f$ is in the dual of $ W^ {1,2}( \Omega)$, then we may define the normal derivative of $f$ at the boundary as an element of the dual of $ W^ { 1/2,2}( \partial \Omega)$ by $$ \langle \partial f/\partial \nu, \phi \rangle_{\partial \Omega} = \int _ {\Omega} \nabla f \cdot \nabla \phi \, dx + \langle \Delta f , \phi\rangle_\Omega, \qquad \phi \in W^ { 1,2}( \Omega). $$ In particular, if $f$ is weakly harmonic, then we may define the normal derivative. 4. By point 2, we may assume that the Dirichlet data in (\ref{MP}) is given as the boundary values of a harmonic function. We write the solution of (\ref{MP}) $ u = f_D+v$ where $f_D$ is a harmonic representative of the boundary data in (\ref{MP}). We let $v$ be a solution of the mixed problem $$ \left\{ \begin{array}{ll} \Delta v = 0 , \qquad & \mbox{in } \Omega\\ \partial v /\partial \nu = f_N - \partial f_D /\partial \nu, \qquad & \mbox{on } N\\ v = 0 , \qquad & \mbox{on } D. \end{array} \right. $$ The weak formulation of this problem is $$ \int _ \Omega \nabla v \cdot \nabla \phi\,dx = \langle f_N, \phi\rangle_{\partial \Omega} - \int_ \Omega \nabla \phi \cdot \nabla f_D\,dx, \qquad \phi \in W^ { 1,2}_D ( \Omega) $$ where we have substituted $ \langle \partial f_D/\partial \nu , \phi \rangle _{\partial \Omega} = \int _ \Omega \nabla f_D \cdot \nabla \phi \, dx. $ The existence of $v$ in $W^ { 1,2}_D( \Omega) $ satisfying this weak formulation follows from Lax-Milgram. It is clear that $u= f_D+v$ satisfies $$ \int_\Omega \nabla u \cdot \nabla \phi \, dx = \langle f_N, \phi \rangle_ {\partial \Omega}, \qquad \phi \in W^ { 1,2}_D ( \Omega) $$ and that we have $u-f_D\in W^ { 1,2}_D ( \Omega)$. 5. Uniqueness is easy. If $u$ is a weak solution of (\ref{MP}) with zero data, then we may use $u\in W^ {1,2}_D( \Omega)$ as the test function in the weak formulation to obtain $$ \int_ \Omega |\nabla u |^2 \, dx = 0. $$ Since we assume (\ref{coerce}) for functions in $W^ { 1,2}_D( \Omega)$, it follows that $u$ is zero. } \note { We recall a result from Giaquinta \cite[p.~122]{MG:1983}. In this result and throughout this paper, we use the notation $ -\!\!\!\!\!\!\int _A f\, dx$ to denote the average of a function $f$ on a set $A$. Let $Q$ be a cube in $ {\bf R} ^n$ and suppose that whenever $Q_{2r}(x)\subset Q$, we have $$ -\!\!\!\!\!\!\int _{Q_r(x) } g^q \, dy \ \leq A \left ( -\!\!\!\!\!\!\int _ {Q_ {2r}( x)} g \, dy \right ) ^ { q} + -\!\!\!\!\!\!\int _ {Q_{2r}(x) } f^ q \, dy . $$ Then there is an $\epsilon >0$ which depends on $A$, $q$ and $n$ so that for $ p \in [q,q+\epsilon)$, we have $$ \left ( -\!\!\!\!\!\!\int_ {Q/2} g^p\, dy \right ) ^ { 1/p} \leq C \left ( -\!\!\!\!\!\!\int _ Q g^q \, dy \right ) ^ { 1/q} + \left( -\!\!\!\!\!\!\int_ Q f^ p \, dy \right )^ {1/p}. $$ In particular, if $f$ is in $L^ p(Q)$, then $g$ is in $L^ p_{loc}( Q)$. } We define a sub-linear operator $P$ which takes functions on $\partial \Omega$ to functions in $ \Omega$ by $$ Pf(x) = \sup _{ s > 0 } \frac 1 {s^ { n-1} } \int _{ \sball x s } |f |\, d\sigma, \qquad x \in \Omega $$ and a local version of $P$ by $$ P_r f(x) = \sup _{ r> s > 0 } \frac 1 {s^ { n-1} } \int _{ \sball x s } |f |\, d\sigma , \qquad \qquad x \in \Omega . $$ On the boundary, we have that $Pf$ is the Hardy-Littlewood maximal function $$ Mf(x)= Pf(x) = \sup _{ s> 0} \frac 1 {s ^ {n-1}} \int _ { \sball x s} |f |\, d\sigma, \qquad x\in \partial \Omega . $$ The following result is probably well-known, but we could not find a reference. \begin{lemma} \label{PEstimate} For $1< p < \infty$, $ 1 \leq q \leq pn/( n -1) $, $ x \in \partial \Omega $ and $r < r_0$, we have \begin{equation}\label{Plocal} \left ( -\!\!\!\!\!\!\int _{\dball x r } |P_rf|^ q \, dy \right ) ^ { 1/q } \leq C \left (\frac 1 { r^ { n-1} } \int _{ \sball x { 2r} } |f |^ p \, d\sigma \right ) ^ { 1/p} . \end{equation} The constant in this estimate depends only on the Lipschitz constant $M$ and the dimension. \end{lemma} \begin{proof} We begin by considering the case where $ \Omega = \{ (y', y_n ) : y _ n > 0\}$ is a half-space. We use coordinates $ y = (y',y _n )$ and we claim that \begin{eqnarray} \label{P1} Pf(y', y_n ) & \leq & Mf(y',0) \\ \label{P2} Pf(y) & \leq & C\|f\|_{ L^ p ( \partial \Omega) } y _n ^ {( 1-n ) /p}, \qquad y _ n > 0 . \end{eqnarray} The estimate (\ref{P1}) follows easily since $ \sball {(y', y_n )} s \subset \sball {(y', 0)} s $. To establish the second estimate, we observe that if $ s < y _n $, then $\sball y s = \emptyset$ and hence $$ Pf(y) = \sup _{ s \geq y _n } \frac 1 { s^ { n-1}} \int _ { \sball y s } |f| \, d\sigma \leq C y _n ^ { (1-n)/p} \| f\|_{ L ^ p ( \partial \Omega) } . $$ We claim that we have the following weak-type estimate for $Pf$, \begin{equation} \label{Pclaim} |\{ x\in \Omega : Pf(x) > \lambda \}| \leq C \| f\|_ { L^ p (\partial \Omega ) } ^ p \lambda ^ { -pn /(n-1) } , \qquad \lambda > 0 . \end{equation} To prove (\ref{Pclaim}), we may assume $ \| f \| _{ L^ p ( \partial \Omega )} = 1$. With this normalization, the observation (\ref{P2}) implies that $\{ y ' : Pf(y', y _n) > \lambda \} = \emptyset $ if $ y _n > c \lambda ^ { -p /( n-1)} $. Thus, we may use Fubini's theorem to write \begin{eqnarray*} |\{ x \in \Omega : Pf(x) > \lambda \} | & = & \int _ 0 ^ { c \lambda ^ { -p/( n-1)} } \sigma ( \{ y ' :Pf(y', y _n ) > \lambda \}) \, dy _n \\ &\leq & C\int _0 ^ {c\lambda ^ { -p/ ( n-1 ) }} \sigma ( \{ y ' : Mf ( y',0) > c \lambda \}) \, dy _n \\ & = & C \lambda ^ { -p n / ( n-1)} \end{eqnarray*} where we used (\ref{P1}), the weak-type $(p,p)$ inequality for the maximal operator on ${\bf R} ^ { n-1}$ and our normalization of the $L^p$-norm of $f$. From the weak-type estimate (\ref{Pclaim}) and the Marcinkiewicz interpolation theorem we obtain that there is a constant $C$ depending on $p$ and $n$ so that for $p>1$, \begin{equation}\label{Pglobal} \| Pf\|_{ L^ {pn/(n-1)} ( \Omega)} \leq C \| f \|_{L^p ( {\bf R} ^ { n -1} )} . \end{equation} To obtain the estimate (\ref{Plocal}), we observe that if $ y \in \ball x r $ then $ \ball y r \subset \ball x {2r}$ and hence $$ P_r f( y) \leq P _r ( \chi _ { \sball x { 2r} } f ) (y) ,\qquad y \in \ball x r . $$ Thus in the case where $ \Omega$ is a half-space, the result (\ref{Plocal}) follows from (\ref{Pglobal}) and H\"older's inequality. Finally, to obtain the local result on a general Lipschitz domain, one may change variables so that the boundary is flat near $ \sball x r$. This introduces the dependence on the constant $M$. \end{proof} \note { In applying the change of variables, it is helpful to note that for a bi-Lipschitz transformation $ \Phi$, we have $$ B_ { cr} ( \Phi(x) ) \subset \Phi( B_r ( x )) \subset B_ { Cr} ( \Phi(x) ). $$ Do we need to multiply the radius by a constant in the statement? Check changes to Lemma \ref{RHEstimate} now that we use Lemma \ref{MSIRHI}. Does Lemma \ref{MSIRHI} below duplicate one of the estimates used in the $H^1$ part of the paper. } We recall several versions of the Poincar\'e and Sobolev inequalities. \begin{lemma} \label{YAPI} Let $ \Omega $ be a convex domain of diameter $d$. Suppose that $ S $ is a subset of $ \bar \Omega $ that satisfies: a) for some $ r $ with $ 0 < r< d$ we have $ \sigma ( S \cap \ball x r ) = r ^ { n-1} $ and b) there is a constant $A$ so that $ \sigma ( S \cap \ball x t ) \leq A t^ { n-1}$ for $ t >0$. Let $u$ be a function in $ W^ { 1,p } ( \Omega)$ and suppose that $u$ vanishes on $S$. Then for $ 1< p < n $, we have a constant $C$ $$ \left ( \int _ \Omega |u|^ p \, dy \right ) ^ { 1/p } \leq \frac { C d^ n } { |\Omega | ^ { 1/p'} } { r ^ {1 - n / p } } \left ( \int _ { \Omega } |\nabla u | ^ p \, dy \right) ^ { 1/p } . $$ The constant $C$ depends on $p$, the dimension $n$ and $A$. \end{lemma} \begin{proof} It suffices to consider functions $u$ which are smooth in $ \bar \Omega$ and vanish on $S$. We follow the proof of Corollary 8.2.2 in the book of Adams and Hedberg \cite{MR1411441}, except that we substitute the Riesz capacity for the Bessel capacity in order to obtain a scale-invariant estimate. Following their arguments, we obtain that if $u$ vanishes on $S$, then \begin{equation} \label{Fractional} |u(x) | \leq \frac { d^ n }{ |\Omega| } ( I_ 1 ( |\nabla u | ) (x) + \| \nabla u \|_{ L^ p ( \Omega) } \| I _ 1 ( \mu ) \| _ { L^ { p ' } ( \Omega ) } ) . \end{equation} Here $ I _ 1( f ) ( x) = \int _ { \Omega } f(y) | x-y |^ { 1-n } \, dy$ is the first-order fractional integral and $\mu $ is any non-negative measure on $ S$ normalized so that $\mu(S) =1$. To estimate $ \| I _1( \mu)\|_ {L^ { p'} ( \Omega)} $ we use Theorem 4.5.4 of Adams and Hedberg \cite{MR1411441} which gives that $$ \int _ { {\bf R} ^ n } ( I _ 1( \mu ) ) ^ { p ' } \, dy \leq C \int_{ {\bf R} ^ n } \dot W ^ \mu _ { 1, p } \, d\mu $$ where $\dot W ^ \mu _ { 1, p } (x) $ is the Wolff potential of $\mu$ defined by $$ \dot W ^ \mu _ { 1,p}(x) = \int _ 0 ^ \infty ( \mu ( \ball x t ) t ^ { p -n } ) ^ { 1/ ( p-1)} \, dt/t. $$ Our assumptions imply that with $ \mu = r ^ { 1-n } \sigma$ denoting normalized surface measure on $S$, we have $ I _ 1 ( \mu) (x) \leq C r^ { ( p-n ) /( p-1)}$ where $C$ depends only on $A$. Using this estimate for the Wolff potential and Young's convolution inequality to estimate $I_1(|\nabla u|)$, the Lemma follows from (\ref{Fractional}). \end{proof} The next inequality is also taken from Adams and Hedberg \cite[Corollary 8.1.4]{MR1411441}. Let $1/q + 1/n < 1 $ and assume that $ \Omega$ is a convex domain of diameter $d$. We let $ \bar u = -\!\!\!\!\!\!\int _\Omega u \, dy$ and then we may find a constant $C = C_{q,n}$ depending only on $q$ and $n$ so that \begin{equation}\label{SoPo1} \int _ { \Omega } |u -\bar u | ^ q \, dy \leq C\frac { d^ n } { |\Omega |} \left ( \int _ { \Omega } |\nabla u |^ {nq/(n+q)}\, dy \right) ^ { ( n+q)/n} . \end{equation} Finally, we suppose that $\Omega$ is a domain and $ \dball x r$ lies in a coordinate cylinder $Z$ so that $ \partial \Omega \cap Z$ lies in a hyperplane and let $\bar u = -\!\!\!\!\!\!\int _{\dball x r} u\,dy$. Provided $ \dball x r \subset Z$, we have \begin{equation}\label{SoPo2} \left( \int _{ \sball x {r} } | u -\bar u | ^ { q} \, d\sigma \right) ^ { 1/q} \leq C \left( \int_{ \dball x { r}}|\nabla u |^ p \, dy \right ) ^ { 1/p }. \end{equation} In the inequality (\ref{SoPo2}), $p$ and $q$ are related by $ 1/q = 1/p - ( 1- 1/p) / (n-1) $ and $ p > 1$. \begin{lemma} \label{MSIRHI} Let $ \Omega $, $N$ and $D$ be a standard domain for the mixed problem. Suppose that (\ref{SurfProp}) holds, let $ x \in \Omega$ and $ 0 < r < r_0$. Let $u$ be a weak solution of the mixed problem for a divergence form elliptic operator with zero Dirichlet data and Neumann data $f_N$. We have the estimate $$ \left ( -\!\!\!\!\!\!\int _{ \dball x { r} } |\nabla u | ^ 2 \, dy \right ) ^ { 1/2 } \leq C \left [ -\!\!\!\!\!\!\int _ {\dball x {2r} } |\nabla u | \, dy +\left ( \frac 1 { r^ { n-1}} \int _ {N \cap \sball x {2r} } |f_N|^ { p} \, d\sigma \right) ^ { 1/p} \right ] . $$ Here, $p=2$ if $n = 2$ and $ p = 2( n-1)/(n-2)$ for $n\geq 3$. The constant $C$ depends only on $M$ and the dimension $n$. \end{lemma} \begin{proof} Changing variables to flatten the boundary of a Lipschitz domain preserves the class of elliptic operators with bounded measurable coefficients, thus it suffices to consider the case where the ball $\sball x r $ lies in a hyperplane. We may rescale to set $ r = 1$. We claim that we can find an exponent $a$ so that for $s$ and $t$ which satisfy $ 1/2\leq s < t \leq 1$, we have \begin{eqnarray} \lefteqn{ \left ( \int _ { \dball x s } |\nabla u |^ 2 \, dy \right ) ^ { 1/2} } \nonumber \\ & \leq & \frac C { ( t-s ) ^ a } \left ( \int _ { \dball x t } | \nabla u | ^ q \, dy \right ) ^ { 1/ q} + \left ( \int _ {N \cap \sball x 1 } | f_N|^ { p} \, d\sigma \right) ^ { 1/p} \label{pqClaim} \end{eqnarray} where we may choose the exponents $ p = 2 ( n-1)/( n-2) $ and $q = 2n/(2n+2)$ if $ n \geq 3$ or $ p =2$ and $ q = 4/3$ if $n = 2$. We give the details when $ n \geq 3$. In the argument that follows, let $ \epsilon = (t-s)/2$ and choose $ \eta$ to be a cut-off function which is one on $\ball x s$, supported in $ \ball x { s+ \epsilon}$ and satisfies $ |\nabla \eta | \leq C /\epsilon $. We let $ v = \eta ^ 2 ( u -E) $ where $E$ is a constant. If we choose $E$ so that $ v \in W^ { 1,2 } _ D ( \Omega) $, the weak formulation of the mixed problem and H\"older's inequality gives for $ 1 < p < \infty $ \begin{eqnarray} \int _ \Omega | \nabla u |^ 2 \eta ^ 2 \, dy & \leq & C \left [ \int _ \Omega | u - E |^ 2 |\nabla \eta | ^ 2 \, dy + \left ( \int _{ N\cap \sball x { s+ \epsilon }} | u - E| ^ { p'}\, d\sigma \right ) ^ { 2/ p ' } \right . \nonumber \\ & & \qquad \left. + \left( \int _{N \cap \sball x { s+ \epsilon } } |f_N|^ p \, d\sigma \right ) ^ { 2/ p }\right] . \label{John} \end{eqnarray} We consider two cases: a) $ \ball x { s+ \epsilon } \cap D = \emptyset $ and b) $\ball x { s+ \epsilon } \cap D \neq \emptyset$. In case a) we may chooose $ E = \bar u = -\!\!\!\!\!\!\int _ { \dball x { s+ \epsilon } } u \, dy $. We use the Poincar\'e-Sobolev inequality (\ref{SoPo1}) and the inequality (\ref{SoPo2}) to estimate the first two terms on the right of (\ref{John}) and conclude that \begin{eqnarray*} \lefteqn{ \int _ { \dball x s } |\nabla u |^ 2 \, dy }\\ & \leq & C \left [ \frac 1 { ( t-s) ^ 2 } \left ( \int _ { \dball x { s+ \epsilon } } |\nabla u | ^ { \frac {2n} {n+2} } \, dy \right ) ^ { \frac { n+2 } n } \right . \\ & & \quad + \left . \left ( \int _ { \dball x { s+ \epsilon } } |\nabla u | ^ { \frac { np} { np-n + 1 } } \, dy \right ) ^ { \frac { 2 ( np -n + 1)} { pn } } + \left ( \int _ { N \cap \sball x 1 } |f_N|^ { p} d\sigma \right) ^ {\frac 1 p } \right ]. \end{eqnarray*} If $n \geq 3$, we may choose $p = 2 ( n-1) / ( n-2) $ and then we have that $ np/( np -n + 1) = 2n /(n+2)$ to obtain the claim. We now turn to case b). Since $ \ball x { s+ \epsilon}$ meets the set $D$, we cannot subtract a constant from $u$ and remain in the space of test functions, $ W^ { 1,2 } _D ( \Omega)$. Thus, we let $E=0 $ in (\ref{John}). We let $ \bar u $ be the average value of $u$ on $ \dball x {s+ 2\epsilon} $ and obtain $$ \int _ { \dball x { s+ \epsilon } } u ^ 2 |\nabla \eta |^ 2 \, dy \leq \frac C { \epsilon ^ 2} \left [ \int _ { \dball x { s+ 2\epsilon } } | u - \bar u | ^ 2 \, dy + \bar u ^ 2 \right ] . $$ Since $ \ball x { s+ \epsilon } \cap D \neq \emptyset $, our assumption (\ref{SurfProp}) on the set $D$ implies that we may find a point $ \tilde x \in \Lambda $ so that $\ball { \tilde x } \epsilon \subset \ball x t $ and so that $ \sigma ( \ball { \tilde x } \epsilon \cap D ) \geq c \epsilon ^ { n -1} $. As $c$ depends on $M$ our final constant may be taken to depend on $M$. Using (\ref{SoPo1}) and the Poincar\'e inequality of Lemma \ref{YAPI} we conclude that \begin{eqnarray} \int _ { \dball x { s+ \epsilon }} u^2 |\nabla \eta | ^ 2 \, dy &\leq & C \left [ \frac 1 { \epsilon ^ 2 } \left ( \int _ { \dball x {s+2\epsilon } } | \nabla u | ^ { 2n / ( n+2) } \, dy \right ) ^ { ( n + 2) /n } \right. \nonumber \\ \label{Paul} & & \left. \qquad + \frac 1 {\epsilon^{ 2n/q}} \left ( \int _ { \dball x { s + 2\epsilon } } |\nabla u | ^ q \, dy \right ) ^ { 2/ q} \right] \end{eqnarray} for $ 1 < q < n $. A similar argument using (\ref{SoPo2}) and Lemma \ref{YAPI} gives us \begin{eqnarray} \nonumber \left( \int _ { \sball x { s+ 2\epsilon } } | u |^ { p'} \, d\sigma \right ) ^ { 1/ p' } & \leq & \left ( \int _ { \sball x { s + 2\epsilon }} | u -\bar u | ^ { p ' } \, d\sigma \right ) ^ { 1/ p' } + | \bar u | \\ \nonumber & \leq & C \left [ \left ( \int _ { \dball x { s+ 2\epsilon } } |\nabla u | ^ { np / ( np - n + 1) } \, dy \right ) ^ { ( np -n + 1 ) / ( np) } \right. \\ \label{George} & &\left. \quad + \epsilon ^ { 1- n/q} \left ( \int _ { \dball x { s+ 2\epsilon } } |\nabla u | ^ q \, dy \right ) ^ { 1/ q} \right ] \end{eqnarray} where the use of Lemma \ref{YAPI} requires that we have $ 1< q< n$. We use (\ref{Paul}) and (\ref{George}) in (\ref{John}) and choose $ q = 2n/( n+2) $ and $ p = 2 ( n -2 ) /( n-1) $ if $ n \geq 3$. Once we recall that $ t -s = 2 \epsilon $, we obtain (\ref{pqClaim}). Finally, we may use the techniques given in \cite[pp.~80-82]{MR1239172} or \cite[pp.~1004--1005]{FS:1984} to see that the claim (\ref{pqClaim}) implies the estimate \begin{equation} \left ( \int _ { \dball x {1/2} } |\nabla u |^ 2 \, dy \right ) ^ { 1/2} \leq C \left[ \int _ { \dball x 1 } | \nabla u | \, dy + \left ( \int _ {N \cap \sball x 1 } |f_N|^ { p} d\sigma \right) ^ { 1/p} \right] \end{equation} with $p$ as in (\ref{pqClaim}). When the dimension $n=2$, the exponent $2n/(n+2)$ is 1 and it is not clear that we have (\ref{SoPo1}) as used to obtain (\ref{Paul}). However, from (\ref{SoPo1}) and H\"older's inequality we can show $$ \left( \int_{ \dball x { s+ 2\epsilon } } |u - \bar u | ^ 2 \, dy \right ) ^ { 1/2} \leq C \left( \int_{ \dball x { s+ 2\epsilon } } |\nabla u | ^ {4/3} \, dy \right ) ^ { 3/4} . $$ This may be substituted for (\ref{SoPo1}) in the above argument to obtain (\ref{pqClaim}) when $ n =2$. \end{proof} \note { The argument of Fabes and Stroock will give any $p$, not just $p=1$ on the right-hand side. We give proofs of (\ref{SoPo1}) and (\ref{SoPo2}). Oops, this is a proof of a version of (\ref{SoPo2}) that we are no longer using. If $u$ is in $W^ {1,p} ( \Omega)$ and $ u =0$ on a subset $S\subset \sball x r $ with $\sigma(S) > cr^ { n-1}$ and $ r< r_0$, then we have that $$ \int _{ \sball x r } u^p \, d\sigma \leq \frac {Cr^ { n+p}} {\sigma(S)} \int _ {\Omega \cap \ball x {Cr}} |\nabla u | ^ p \, dy. $$ \begin{proof} 1. We first consider the case where $ \Omega = \{ x: x_n > 0\}$ and suppose that our ball, $B= B_1(0) $, is centered at the origin. We let $ \bar u = -\!\!\!\!\!\!\int _{ B_r^ +} u dy$, extend $u-\bar u$ to a function $E(u-\bar u) $ on $ {\bf R} ^ n _+$ by reflecting in the ball $ |x|=1$ and multiplying by a cut-off function $ \eta$ which is one on $B$ and supported in $2B$. Let $ v=\eta E(u-\bar u)$ denote the resulting function. 2. According to Runst and Sickel \cite{MR98a:47071}, we have the trace theorem $$ \|v\|_{ B^ {p,p}_{ 1-1/p}} \leq \|v\|_{ W^ { 1,p}( {\bf R} ^ n _ +)}. $$ 3. Using Poincar\'e inequalities and properties of the extension operator, we can show $$ \|v\|_ {W^ { 1,p}({\bf R}^n_+)} \leq \|\nabla u \|_ { L^p( B_+)}. $$ 4. Recalling the definition of the Besov norm and that $u=0$ on $S$, we have \begin{eqnarray*} \frac { \sigma ( S) } { 2^{n-2+p}} \int _\Delta |u|^p \, d\sigma & \leq & \int_ \Delta \int _ \Delta \frac { |u(x',0) - u(y',0) | ^ p }{ |x'-y'| ^ { n-2 +p} } \, d\sigma d\sigma \\ & \leq & \int_ {{\bf R} ^ { n-1} } \int _ {{\bf R} ^ { n-1} } \frac { |v(x',0) - v(y',0) | ^ p }{ |x'-y'| ^ { n-2 +p } } \, d\sigma d\sigma \\ & \leq & \|v\|^p_ {W^ { 1,p}({\bf R}^n_+)} \end{eqnarray*} where we use $ \Delta $ to denote the ball $ \{ x': |x'|<1\}$. This uses that $v(x',0) - v(y',0)= u(x',0)-u(y',0)$ for $x', y' \in \Delta$. The inequalities in 3 and 4 give the result when $ \Omega $ is a half-space and $r=1$. Rescaling, gives the result for general $r$. For a general Lipschitz domain, we may change variables to reduce to the problem in a half space. The image of $ \sball x r$ will be contained in a ball of radius $\sqrt{1+M^2}r$ where $M$ depends on the Lipschitz constant. \end{proof} If $ \bar u = -\!\!\!\!\!\!\int _{ \Omega \cap \ball x r } u \, dy $ or if $u$ vanishes on $ S\subset \partial \Omega \cap \sball x r$ and $ \sigma (S) > cr^ { n-1}$, then we have $$ \left( \int _ { \partial \Omega \cap \sball x r } |u- \bar u | ^ {(n-1)p/(n-p) } \, d\sigma \right) ^ {\frac {n-p} {(n-1)p} } \leq C\left( \int _ { \Omega \cap \ball x {Cr} } |\nabla u |^ p \, dy\right) ^ { 1/p} . $$ \begin{proof}[Proof of SoPo2] 1. We change variables to reduce to the case of a half-space and then rescale to obtain a radius of 1. Note that the change of variables, $ (x',x_n) \rightarrow ( x', \phi(x') + x_n)$ has Jacobian 1, so that this preserves mean value zero. 2. In the case where $ \bar u$ is the average, we may extend and cut off as in the previous result and let $ v = \eta E( u-\bar u)$. We apply the trace theorem to obtain that $v$ is in a Besov space $B^ { p,p}_{ 1-1/p}$ (\cite{MR98a:47071}[p.~75]. Next, we can use the embedding for Besov spaces on the boundary to conclude that $$\| v\|_ {L^ {(n-1)p/(n-p)} ( \partial \Omega)} \leq \|v\|_ {B^ { p,p} _{ 1- 1 / p} (\Omega)}.$$ As $v =u - \bar u$ on $ \sball 0 1$, we have the desired result. 3. In the case where $ \sball 0 1 $ intersects the boundary and $u$ vanishes on a set $S$, we use the previous result to conclude that we have the estimate $$ \int_{ \Delta } u ^ p \leq C\int _ { \ball 0 1 \cap {\bf R} ^ n _+} |\nabla u | ^ p \, dy. $$ 4. From this inequality, we then can show that $$ \int _{B_+} u^p \,dx \leq C \int _ \Delta u^p + \int _{B_+} |\nabla u | ^ p \, dy. $$ 5. Since $u$ is in $W^ { 1,p}$, we may then extend and multiply by a cut-off function and obtain a function $v$ which is in $W^ { 1,p}( {\bf R}^n _+)$. Applying the trace theorem, we conclude that $v$ is in the Besov space $ B^ {p,p}_{ 1-1/1p}$ of the boundary and this space embeds into $ L^ { (n-1)p/(n-p)}$. 6. Given the result in a half space, the result stated in a Lipschitz domain follows by a change of variables. \end{proof} } \begin{lemma} \label{RHEstimate} Let $ \Omega$, $D$ and $N$ be a standard domain for the mixed problem. Let $ x \in \Omega$ and suppose that $r$ satisfies $ 0< r < r_0$. Let $u$ be a weak solution of the mixed problem (\ref{WeakMix}) with zero Dirichlet data and Neumann data $f$ in $L^p(N)$ which is supported in $ N \cap \sball x r $. There exists $ p_0=p_0(n,M) > 2 $ so that for $t $ in $[2,p_0) $ if $n \geq 3$ or $t$ in $(2,p_0)$ if $n =2 $, we have the estimate \begin{eqnarray*} \lefteqn{ \left ( -\!\!\!\!\!\!\int _{ \dball x r } |\nabla u |^t \, dy \right )^ { 1/t}} \\ & \leq & C\left[ -\!\!\!\!\!\!\int _ {\dball x {2r} } |\nabla u |\,dy +\left( \frac 1 { r^ { n-1} } \int _{ \sball x {2r} \cap N } |f|^{t(n-1)/n}\, d\sigma\right) ^ { n/(t(n-1))}\right] . \end{eqnarray*} The constant in this estimate depends on $t$, $M$ and $n$. \end{lemma} \note{ In applications, we seem to only need $f$ bounded. Is it worth the trouble to keep track of the exponents? } \begin{proof} According to Lemma \ref{MSIRHI}, $ \nabla u$ satisfies a reverse H\"older inequality and thus we may apply a result of Giaquinta \cite[p.~122]{MG:1983} to conclude that there exists $p_0 > 2$ so that we have $$ \left( -\!\!\!\!\!\!\int _ {\dball x r } | \nabla u|^ t \, dy \right) ^ { 1/t} \leq C\left [ -\!\!\!\!\!\!\int_ {\dball x {2r} } |\nabla u | \, dy + \left( -\!\!\!\!\!\!\int _ { \dball x { 2r} } (P_ {2r} |f|^p)^{ t/p} \, dy \right) ^ { 1/ t }\right] $$ for $t $ in $[2,p_0)$ and $p$ as in Lemma \ref{MSIRHI}. From this, we may use Lemma \ref{PEstimate} to obtain $$ \left( -\!\!\!\!\!\!\int _ {\dball x r } | \nabla u|^ t \, dy \right) ^ { 1/t} \leq C\left [ -\!\!\!\!\!\!\int_ {\dball x {2r} } |\nabla u | \, dy + \left( -\!\!\!\!\!\!\int _ { \sball x { 4r} } |f|^{ t(n-1)/n} \, d\sigma \right) ^ { n/ t(n-1) }\right] $$ when $ n \geq 3$ and $ t $ is in $[2,p_0)$. If $n=2$ we need $ t >2$ so that $f$ is raised to a power larger than 1. Now a simple argument that involves covering $ \sball x r$ by surface balls of radius $r/4$ allows us to conclude the estimate of the Lemma. \end{proof} \section{Estimates for solutions with atomic data} \label{Atoms} We establish an estimate for the solution of the mixed problem when the Neumann data is an atom for $ H^1$ and the Dirichlet data is zero. The key step is to establish decay of the solution as we move away from the support of the atom. We will measure the decay by taking $L^q$-norms in dyadic rings around the support of the atom. Thus, given a surface ball $ \sball x r$, $ x \in \partial \Omega$, we define $\Sigma_k = \sball x { 2^ k r} \setminus \sball x {2^ { k-1} r}$ and define $ S_k = \dball x {2^k r } \setminus \dball x { 2^ { k-1} r } $. \begin{theorem} \label{AtomicTheorem} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem. Let $u$ be a weak solution of the mixed problem with Neumann data $a$ which is an atom which is supported in $N \cap\sball x r $ and zero Dirichlet data. If $p_0$ is as in Lemma \ref{RHEstimate} and $ 1< q < p_0/2$, then we have $ \nabla u \in L^ q ( \partial \Omega)$, \begin{equation} \label{LocalPart} \left( \int _{\sball x {8r} } |\nabla u |^q \, d \sigma \right)^ { 1/q} \leq C \sigma (\sball x {8r} )^ {-1/q'} \end{equation} and for $ k \geq 4$, \begin{equation} \label{Decay} \left ( \int _{ \sring k} |\nabla u |^q \,d\sigma \right) ^ { 1/q} \leq C 2^ {-\beta k} \sigma( \sring k ) ^ {- 1/q'} . \end{equation} Here, $ \beta $ is as in Lemma \ref{Green} and the constant $C$ in the estimates (\ref{LocalPart}) and (\ref{Decay}) depends on $q$ and the global character of the domain. \end{theorem} If $r < r_0 $ and $x$ is in $\partial \Omega$, then we may construct a star-shaped Lipschitz domain $ \locdom x r = Z_r(x) \cap \Omega$ where $\cyl x r $ is the coordinate cylinder defined above. Given a function $v$ defined in $ \Omega$, $x \in \partial \Omega$, and $r>0$, we define a truncated non-tangential maximal function $ \nontan{v_r} $ by $$ \nontan v_r (x) = \sup _{ y \in \ntar x \cap \ball x r } |v(y)|. $$ \begin{lemma} \label{NeumannRegularity} Let $\Omega$ be a Lipschitz domain. Suppose that $ x \in \partial \Omega $ and $0< r < r_0$. Let $u$ be a harmonic function in $ \locdom x {4r} $. If $\nabla u \in L^2 ( \locdom x {4r} )$ and $ \partial u /\partial \nu $ is in $L^ 2 (\partial \Omega \cap \partial \locdom x {4r} ) $, then we have $ \nabla u \in L^2 ( \sball x r)$ and $$ \int _ { \sball x {r}} (\nontan {( \nabla u )}_r)^2 \, d\sigma \leq C \left ( \int _ { \partial \Omega \cap \partial \locdom x {4r} } \left |\frac { \partial u }{ \partial \nu } \right | ^ 2 \, d\sigma + \frac 1 r \int _ {\locdom x {4r} } |\nabla u |^2 \, dy\right). $$ The constant $C$ depends only on the dimension $n$ and $M$. \end{lemma} \begin{proof} Since the estimate only involves $\nabla u$, we may subtract a constant from $u$ so that $ \int _ {\locdom x r } u \, dy = 0$. We pick a smooth cut-off function $ \eta$ which is one on $ Z_{3r}(x)$ and zero outside $Z_{4r}(x)$. Since we assume that $\nabla u$ is in $L^2( \locdom x {4r } )$, it follows that $\Delta ( \eta u) = u \Delta \eta + 2 \nabla u \cdot \nabla \eta $ is in $L^2( \locdom x {4r} )$. Thus, with $ \Xi$ the usual fundamental solution of the Laplacian, $w = \Xi*(\Delta( \eta u))$ will be in the Sobolev space $W^ {2,2}( {\bf R} ^n)$. We have defined $ \Delta (\eta u )$ to be zero outside $ \locdom x {4r}$ in order to make sense of the convolution in the definition of $w$. Next, we let $v$ be the solution of the Neumann problem $$ \left\{ \begin{array} {ll} \Delta v = 0, \qquad & \mbox {in } \locdom x {4r } \\ \bigfrac { \partial v } { \partial \nu } = \bigfrac { \partial( \eta u) } { \partial \nu }- \bigfrac {\partial w }{ \partial \nu}, \qquad & \mbox{on }\partial \locdom x {4r } . \end{array} \right. $$ According to Jerison and Kenig \cite{JK:1982c}, the solution $v$ will have non-tangential maximal function in $L^2 ( \partial \locdom x {4r} )$. By uniqueness of weak solutions to the Neumann problem, we may add a constant to $v$ so that we have $ \eta u = v+w$. As $w$ and all its derivatives are bounded in $ \locdom x {2r}$ and the non-tangential maximal function of $\nabla v$ is in $L^2( \partial \locdom x {4r})$, we obtain the Lemma. \end{proof} The proof of the following Lemma for the regularity problem is identical to the proof of Lemma \ref{NeumannRegularity}. \begin{lemma} \label{DirichletRegularity} Let $\Omega$ be a Lipschitz domain. Suppose that $ x \in \partial \Omega $ and $0< r < r_0$. Let $u$ be a harmonic function in $ \locdom x { 4r} $. If $\nabla u \in L^2 (\locdom x {4r} )$ and $ \tangrad u $ is in $L^ 2 (\partial \Omega \cap \partial \locdom x { 4r} )$, then we have $ \nabla u \in L^2 ( \sball x r)$ and $$ \int _ { \sball x {r}} (\nontan {( \nabla u )}_r)^2 \, d\sigma \leq C \left( \int _ { \partial \Omega \cap \partial \locdom x { 4r} }| \tangrad u |^2 \, d\sigma + \frac 1 r \int _ {\locdom x { 4r} } |\nabla u |^2 \, dy\right). $$ The constant $C$ depends only on the dimension $n$ and $M$. \end{lemma} The following weighted estimate will be an intermediate step towards our estimates for solutions with atomic data. In the next lemma, $\Omega$ is a bounded Lipschitz domain and the boundary is written $ \partial \Omega = D \cup N$. Recall that $ \delta(x)$ denotes the distance from $x$ to the set $\Lambda$. \begin{lemma} \label{Whitney} Let $\Omega$, $D$ and $N$ be a standard domain for the mixed problem. Let $u$ be a weak solution of the mixed problem (\ref{WeakMix}) with Neumann data $f_N \in L^2 (N)$ and zero Dirichlet data. Let $\epsilon \in {\bf R}$, $x \in \partial \Omega$ and $0< r < r_0$ and assume that for some $A>0$, $ \delta (x) \leq Ar$. Then we have $$ \int _ {\sball x r } (\nontan{( \nabla u)}_{c\delta}) ^2 \, \delta ^{1-\epsilon} d\sigma \leq C \left ( \int _ { \sball x {2r}} |f_N|^2 \delta ^ { 1- \epsilon } \, d\sigma + \int _ { \dball x { 2r} } |\nabla u |^2 \, \delta ^ { -\epsilon } \, dy \right ) . $$ The constant in this estimate depends on $M$, $n$, $\epsilon$ and $A$. \end{lemma} \note { Using a Hardy inequality, we can probably show that $u$ in $L^2( N ; \delta \, d\sigma )$ implies that $u$ is in the dual of $W^ { 1/2, 2}_D ( \partial \Omega)$. } The proof below uses a Whitney decomposition and thus it is simpler if we use surface cubes, rather than the surface balls used elsewhere. A {\em surface cube }is the image of a cube in $ {\bf R} ^ { n-1}$ under the mapping $ x' \rightarrow ( x' , \phi(x'))$. Obviously, each cube will lie in a coordinate cylinder. \begin{proof} We may assume that $ \dball x {2r}$ is contained in a coordinate cylinder $ Z_{ 2r_0}$. If $Z_ { 100r_0} \cap \partial \Omega \subset N$ or $Z_ { 100r_0} \cap \partial \Omega \subset D$, then the estimate of the Lemma follows easily from Lemma \ref{NeumannRegularity} or Lemma \ref{DirichletRegularity} since we have that $ \delta (y) $ is equivalent to $r$ for $y \in \dball x {2r}$. This equivalency follows from our assumption that $ \delta (x) < Ar$ and that $Z_{ 100r_0}$ does not intersect $\Lambda$. If $ Z_ { 100r_0}$ meets both $D$ and $N$, we begin by finding a decomposition of $ ( \partial \Omega \cap Z_{ 4r_0}) \setminus \Lambda $ into non-overlapping surface cubes $ \{ Q_j \}$ which satisfy: 1) For each cube $ Q_j$, we have constants $c''$ and $c'$ so that $c''\delta (y) \leq \mathop{\rm diam}\nolimits (Q_j) \leq c ' \delta (y)$ for $ y \in Q_j$. The constant $c'$ may be chosen as small as we like. 2) We let $T(Q) = \{ y \in \Omega : \mathop{\rm dist}\nolimits (y, Q ) < \mathop{\rm diam}\nolimits Q\}$. Then the family $ \{ T(2Q_j)\}$ has bounded overlaps and thus $$ \sum \chi _{ T(2Q_j)}\leq C (n, M, c''). $$ To construct the family of surface cubes, begin with a Whitney decomposition of $ {\bf R} ^ { n-1}\setminus \{ ( \psi (x''), x'' ): x''\in {\bf R} ^ { n -2}\}$ and then map the cubes onto the boundary with the map $x' \rightarrow ( x', \phi(x'))$. Here, $ \phi$ and $\psi $ are the functions used to describe $\partial \Omega$ and $\Lambda$ in the coordinate cylinder $Z_{ r_0}$. As the surface cubes $ Q_j$ are connected and $ \delta $ never vanishes on $Q_j$, we have that either $ Q_j \subset N$ or that $ Q_j \subset D$. We choose the constant $ c'$ small so that $ Q _ j \cap \sball x r \neq \emptyset$ implies that $T(2Q_j)\subset \dball x { 2r} $. Let $r_j$ be the diameter of the cube $r_j$. Applying Lemma \ref{NeumannRegularity} or Lemma \ref{DirichletRegularity}, we conclude that \begin{equation} \label{Whit1} \int _ { Q_j } |\nabla u |^ 2 \, d\sigma \leq C \left ( \int _ { 2Q_j \cap N} \left| \frac { \partial u } { \partial \nu } \right | ^ 2 \, d\sigma + \frac 1 { r_j } \int _ { T( 2Q_j)} |\nabla u |^ 2 \, dy \right) . \end{equation} We multiply equation (\ref{Whit1}) by $ r_j ^ { 1-\epsilon } $, choose $c'$ small so that $ r _j $ is equivalent to $ \delta (y)$ in $T(2Q_j)$ and obtain \begin{equation} \label{Whit2} \int _ { Q_j } |\nabla u |^ 2 \delta ^ { 1- \epsilon} \, d\sigma \leq C \left ( \int _ { 2Q_j \cap N} \left | \frac { \partial u } { \partial \nu } \right | ^ 2 \delta ^ { 1- \epsilon } \, d\sigma + \int _ { T( 2Q_j)} |\nabla u |^ 2 \delta ^ { -\epsilon }\, dy \right) . \end{equation} We sum over $j$ such that $ Q_j \cap \sball x r \neq \emptyset$ and use that the family $ \{ T(2Q_j)\}$ has bounded overlaps to obtain the Lemma. \end{proof} An important part of the proof of our estimate for the mixed problem is to show that a solution with Neumann data an atom will decay as we move away from the support of the atom. This decay is encoded in estimates for the Green function for the mixed problem. These estimates rely in large part on the work of de Giorgi \cite{EG:1957}, Moser \cite{JM:1961} and Nash \cite{JN:1958}, on H\"older continuity of weak solutions of elliptic equations with bounded and measurable coefficients, and the work of Littman, Stampacchia and Weinberger \cite{LSW:1963} who constructed the fundamental solution of such operators. Also, see Kenig and Ni \cite{MR87f:35065} for the construction of a global fundamental solution in two dimensions. Given the free space fundamental solution, the Green function may be constructed by reflection in a manner similar to the construction given for graph domains in \cite{LCB:2008}. A similar argument was used by Dahlberg and Kenig \cite{DK:1987} and by Kenig and Pipher \cite{KP:1993} in their studies of the Neumann problem. Once we have a Green function which satisfies the correct boundary conditions in a coordinate cylinder, we may solve a weak version of the mixed problem to obtain a Green function in all of $ \Omega$. \begin{lemma} \label{Green} Let $\Omega$, $N$ and $D$ be a standard domain for the mixed problem. There exists a Green function $G(x,y)$ for the mixed problem which satisfies: 1) If $G_x(y) = G(x,y)$, then $ G_x$ is in $W^ { 1,2}_D ( \Omega \setminus \ball x r )$ for all $r>0$, 2) $\Delta G_x = \delta _x$, the Dirac $\delta$-measure at $x$, 3) If $f_N$ lies in $ W^ {-1/2, 2} _D ( \partial \Omega)$, then the solution of the mixed problem with $f_D=0$ can be represented by $$ u ( x) = - \langle f_N , G_x\rangle _{\partial \Omega} , $$ 4) The Green function is H\"older continuous away from the pole and satisfies the estimates $$ |G(x,y) - G(x,y')| \leq \frac { C|y-y'|^ \beta } { |x-y |^ { n-2+\beta }} , \qquad |x-y| > 2 |y-y'|, $$ $$| G(x,y) | \leq \frac C { |x-y|^ { n-2} }, \qquad n\geq 3, $$ and with $ d = \mathop{\rm diam}\nolimits( \Omega)$, $$| G(x,y) | \leq C( 1+ \log (d/ |x-y|)) , \qquad n = 2. $$ \end{lemma} \note { Construction of a Green function. 1. We begin by recalling that an elliptic operator with bounded measurable coefficients has a Green function in all of $ {\bf R}^n$. This is proven by Littman, Stampacchia and Weinberger \cite{LSW:1963} for dimensions $n \geq 3$. The details when $n=2$ may be found in Kenig and Ni \cite{MR87f:35065}. 2. (Moser, \cite{JM:1961}) If $u$, defined in $\ball x {2r}$, is a solution of an elliptic operator with bounded measurable coefficients, then $u$ is H\"older continuous and satisfies the estimates below. \begin{eqnarray*} |u(x) | & \leq & \frac 1 { r^n} \int _ {\ball x {2r}} |u(y) | \, dy \\ |u(y) -u(z) | &\leq & C ( |y-z|/r) ^ \beta \sup _{\ball x {2r}} |u(y)|, \qquad y, z \in \ball x r. \end{eqnarray*} 3. We cover $ \partial \Omega$ by a collection of coordinate cylinders $\{ Z_i\}_{ i =1, \dots, N}$, with $ Z_ i = \cyl {x_i} { r_i}$ and we assume that for each $i$, $4Z_i = \cyl { x_i} {4r_i}$ is also a coordinate cylinder. We also assume that each coordinate cylinder satisfies one of the following case a) $ 4Z_i \cap \partial \Omega\subset D$, b) $ 4Z_i \cap \partial \Omega\subset N$, c) $ \Lambda \cap 4Z_i$ is given as a graph as in the definition for cylinders centered at a point in $ \Lambda $. 4. Fix $x$ and suppose that $x$ lies in one of the cylinders $Z_i$. Using the reflection argument as discussed Dahlberg and Kenig (for the pure Neumann or Dirichlet case) or in Lanzani, Capogna and Brown \cite{LCB:2008}, we can construct a first approximation to the Green function $G_0(x,y)$ which satsfies $ \Delta_y G_0(x,y) = \delta_x$n for $ y \in 4Z_i$ (and with $\delta_x$ denoting the $\delta$-function), $G_0(x, y ) = 0$ for $y \in D \cap 4Z_i$, and $\partial G_0(x,y) /\partial \nu_y =0$ for $y \in N \cap 4Z_i$. Since $G_0$ is not defined in all of $ \Omega$, we need to introduce a cut-off function $ \eta$ which is one on $ 2Z_i$ and zero outside $4Z_i$. We note that $ G_0$ vanishes on $D\cap 4Z_i$. Thus, we have $$ \left\{ \begin{array}{ll} \Delta_y h(x,y) = \Delta_y \eta(y) G_0(x,y) , \qquad & \mbox{in } \Omega\\ h(x,y) = 0, \qquad &y \in D \\ \partial h(x,y) / \partial \nu_y = \partial\eta (y) G_0 (x,y) /\partial \nu, \qquad & y \in N . \end{array} \right. $$ 5. We can estimate $$ \| \partial \eta G_0(x,\cdot )/ \partial \nu \|_ { W_D^ {-1/2,2} (\partial \Omega) } + \|\Delta \eta G_0(x,\cdot)\|_{ H^ { -1}( \Omega)} \leq C $$ where the constant is independent of $x$. This is because the data for this mixed problem is zero. Thus, the solution $ h(x,\cdot)$ to the boundary value problem in 4. satisfies $$ \|h(x, \cdot ) \| _ {L^2 ( \partial \Omega) } + \| \nabla h(x,\cdot) \|_{ L^2 ( \partial \Omega)} \leq C. $$ (Check details.) 6. We may define the Green function $G(x,y) = h(x,y) + G_0(x,y)$. The pointwise estimates of the Lemma follow from the estimates for $G_0$, the estimate for $h$ in 5. and the boundedness of solutions in 2. The H\"older continuity follows from the upper bounds for the fundamental solution and the estimate in 2. This construction give $G$ for $x$ in a coordinate cylinder. For $x$ in the interior, we may let $G_0$ be $ \eta \Xi$ where $\eta $ is smooth function which is one in a neighborhood of $x$ and zero near the boundary. 7. We now turn to the representation formula. We write $ G(x,y) = \Xi (x-y) - g(x,y)$ where $\Xi$ is the free space fundamental solution and $g$ is defined by this equation. We let $u$ be a weak solution with Neumann data $f_N$ and $f_D=0$. We fix $x$ and let $ \eta$ be a cut-off function which is one in a neighborhood of the boundary and 0 in neighborhood of $x$. As $u$ lies in $W^ {1,2}_D ( \Omega)$ and $ \eta G(x,\cdot) $ also lies in $W^ { 1,2}_D( \Omega)$ we may apply the weak formulation of the mixed problem to obtain that \begin{eqnarray*} \langle f _N , G(x, \cdot)\rangle_ {\partial \Omega} & =& \int _ \Omega \nabla u \cdot \nabla ( \eta G(x, \cdot))\, dy\\ 0 = \langle u , \partial G(x,\cdot ) / \partial \nu \rangle _{ \partial \Omega } & = & \int_{ \Omega } \nabla u \cdot \nabla ( \eta G(x, \cdot) )+ u \Delta ( \eta G( x, \cdot) )\,d y . \end{eqnarray*} Subtracting these expressions gives that \begin{eqnarray*} \langle f _N , G(x, \cdot)\rangle & = & - \int_ { \Omega} 2 \nabla \eta \cdot \nabla G(x,\cdot) + \Delta \eta G(x,\cdot)\, dy \\ & = & \int_ { \Omega} 2u \nabla (1-\eta) \cdot \nabla G(x,\cdot) +u \Delta( 1- \eta) G(x,\cdot)\, dy. \end{eqnarray*} Since the function $\nabla( 1- \eta)$ is supported away from $x$ and the boundary, we may integrate by parts in the first term in the integral below to obtain, \begin{eqnarray*} \int_ { \Omega} 2u \nabla (1-\eta) \cdot \nabla G(x,\cdot) +u \Delta( 1- \eta) G(x,\cdot)\, dy & = & - \int_ \Omega 2\nabla u \cdot \nabla ( 1-\eta) G(x, \cdot) \\ & & \qquad + u \Delta (1-\eta) G(x, \cdot) \, dy \\ & = & - \int _ \Omega G(x, \cdot) \Delta ( u ( 1-\eta))\, dy. \end{eqnarray*} From the standard properties of a fundamental solution, we obtain that $$u(x) = - \langle f_N, G(x, \cdot) \rangle_{\partial \Omega} .$$ } \begin{lemma} \label{Energy} Let $u$ be a weak solution of the mixed problem (\ref{WeakMix}) with Neumann data $f$ in $L^ {p}(N)$ where $p = (2n-2)/n $ for $ n\geq 3$. Then we have the estimate $$ \int _{\Omega } |\nabla u | ^2 \, dy \leq C\| f\|^2_ {L^ p ( N)} . $$ If $n =2$, we have $$ \int _{\Omega } |\nabla u | ^2 \, dy \leq C\| f\|^2_ {H^1 (N)} . $$ In each case, the constant $C$ depends on $\Omega$ and the constant in (\ref{coerce}). \end{lemma} \begin{proof} When $ n \geq 3$, we use that $ W^ { 1/2, 2}_D( \partial \Omega ) \subset L^ { 2(n-1) /( n-2)} ( \partial \Omega)$. By duality, we see that $ L^ { 2(n-1)/n}( \partial \Omega) \subset W^ { -1/2, 2}_D( \partial \Omega ) $ and since the weak solution of the mixed problem satisfies $$ \int_\Omega |\nabla u |^2 \, dy \leq C \| f\|^2 _ { W^ { -1/2,2}_D( \partial \Omega)} $$ the Lemma follows. When $ n=2$, the proof above fails since we do not have $ W^ { 1/2, 2}_D(\partial \Omega ) \subset L^ \infty( \partial \Omega)$. However, we do have the embedding $W^ { 1/2, 2}_D(\partial \Omega ) \subset BMO(\partial \Omega) $. Since $ \phi \in W^ { 1/2, 2}_D( \partial \Omega)$ vanishes on $D$ and $ f \in H^1(N)$ has an extension $\tilde f$ which lies in $H^1( \partial \Omega)$, we obtain the result for $n=2$. \end{proof} Finally, we give a technical lemma that will be used below. \begin{lemma} \label{DorN} Let $ \Omega $, $N$ and $D$ be a standard domain for the mixed problem and suppose that $0 < r < r_0$, $ x \in \partial \Omega$ and $ \delta (x) > r \sqrt { 1+M^2}$. Then we have $ \sball x r \subset N$ or $ \sball x r \subset D$. \end{lemma} \begin{proof} We fix $ y \in \sball x r $. Since $ r< r_0$, we may find a coordinate cylinder $Z$ which contains $ \sball x r$. We let $ \phi $ be the function whose graph gives $ \partial \Omega$ near $ Z $. Since $ y \in\sball x r$, we have $|x'-y'| < r$. We let $ x'(t) = ( 1-t) x' + t y '$ and then $ \gamma (t) = ( x' (t) , \phi (x' (t)))$ gives a path in $ \partial \Omega$ joining $x$ to $y$ and of length at most $ r\sqrt { 1+ M^ 2}$. Since $ \delta (x )> r\sqrt { 1+M^2}$ and $ \delta$ is Lipschitz with constant one, we have that $ \delta( \gamma(t) ) >0$ for $ 0 \leq t \leq 1$. Since $ \gamma(t)$ does not pass through $ \Lambda $ we must have $x$ and $y$ both lie in $D$ or both lie in $N$. As $y$ is an arbitrary point in $\sball x r$, it follows that $ \sball x r $ lies entirely in $D$ or entirely in $N$. \end{proof} \begin{proof}[Proof of \ref{AtomicTheorem}] It suffices to restrict attention to atoms which are supported in a surface ball $\sball x r$, with $ x\in\partial \Omega$ and $0< r< r_0$ since an atom which is supported in a larger surface ball can be sub-divided into a finite number of atoms which are supported in balls of the form $ \sball x { r_0}$. The increase in the constant due to this step will depend on the global character of the domain. Thus, we fix an atom $a$ that is supported in the set $ \sball x r \cap N$ and begin the proof of (\ref{LocalPart}). We consider two cases: a) $ \delta (x) \leq 16r \sqrt { 1+M^2}$, and b) $ \delta (x) > 16r \sqrt { 1+M^2}$. In case a) we fix $ q$ between 1 and 2 and use H\"older's inequality with exponents $ 2/q$ and $2 /(2-q)$ to find \begin{eqnarray*} \left( \int _{ \sball x { 8r}} |\nabla u | ^ q\, d\sigma \right) ^ { 1/q} & \leq& \left( \int_ { \sball x { 8r}} |\nabla u |^ 2\delta^ { 1-\epsilon} \, d\sigma \right) ^ { 1/2} \left ( \int _ { \sball x { 8r}} \delta ^ { \frac { q ( \epsilon -1)} { 2-q}}\,d \sigma \right) ^ { \frac {2-q}{ 2q}} \\ & \leq & C r^ { (n-1) ( \frac 1 q -\frac 1 2 ) + \frac { \epsilon -1}2} \left( \int_{ \sball x { 8r}}|\nabla u | ^2 \delta ^ { 1- \epsilon} \, d\sigma \right)^ { 1/2} . \end{eqnarray*} The second inequality requires that $q$ and $ \epsilon$ satisfy $ q ( \epsilon -1) /( 2- q) > -1$ or $ q < 1 /( 1-\epsilon /2)$. Next, we use Lemma \ref{Whitney} and our assumption that $ \delta(x) \leq 16r \sqrt { 1+M^2}$ to bound the weighted $ L^ 2 ( \delta ^ { 1-\epsilon } d\sigma )$ norm of $ \nabla u $. This gives us \begin{eqnarray*} \left( \int _ { \sball x {8r} } |\nabla u | ^ q \, d\sigma \right) ^ { 1/q} & \leq & C \left [ \left ( \int _ { \sball x { r} \cap N } |a |^ 2 \delta ^ { 1-\epsilon} \, d\sigma \right ) ^ { 1/2} \right. \\ & & \quad + \left . \left ( \int _ { \dball x { 16r}} | \nabla u | ^ 2 \delta ^ { -\epsilon } \, dy \right ) ^ { 1/2} \right ] r ^ { ( n-1)( \frac 1 q - \frac 1 2 ) + \frac { \epsilon -1 } 2} . \end{eqnarray*} We estimate the integral over $ \dball x {16r}$ in this last expression with H\"older's inequality and obtain \begin{eqnarray*} \lefteqn{ \left ( \int _ { \dball x { 16r}} | \nabla u | ^ 2 \delta ^ { -\epsilon } \, dy \right ) ^ {1/ 2 } } \qquad \\ & \leq & C \left ( \int _ { \dball x {16r}} |\nabla u | ^ p \, dy \right) ^ { 1/p} \left ( \int _ { \dball x { 16r} } \delta ^ { - \epsilon p/( p-2)}\, dy \right ) ^ { 1/ 2 - 1/ p } \\ & \leq & C r ^ { n ( \frac 1 2 - \frac 1 p) - \epsilon / 2 }\left ( \int _ { \dball x { 16r}}|\nabla u | ^ p \, dy \right ) ^ { 1/p} . \end{eqnarray*} The second inequality depends on our assumption on $ \Lambda $ and holds when $ \epsilon p /( p-2) < 2 $ or $p > 2 / ( 1- \epsilon /2)$. Now we may use the three previous displayed equations and Lemma \ref{RHEstimate} to obtain $$ \left ( \frac 1 { r^ { n-1}}\int _ { \sball x { 8r}}|\nabla u | ^ q \, d\sigma \right ) ^ { 1/q} \leq C \left [ \left ( \frac 1 { r^ n } \int _ {\dball x { 32r}} |\nabla u |^ 2\, dy \right ) ^ { 1/2} + r ^ { 1-n } \right ]. $$ In this last step, we have used the normalization of $a$, $ \| a \|_{ L^ \infty } \leq 1/ \sigma ( \sball x r )$ to estimate the term involving the Neumann data from Lemma \ref{RHEstimate}. Finally, we may use the Lemma \ref{Energy} and the normalization of the atom to obtain that $( r^ { -n} \int _ \Omega |\nabla u |^ 2 \, dy ) ^ { 1/2} \leq C r^ { 1-n}$ which gives the estimate (\ref{LocalPart}). In case b), we use Lemma \ref{DorN} to conclude that $ \sball x { 16r} \subset N$. Next, we use H\"older's inequality, that $a$ is supported in $ \sball x r$ and Lemma \ref{NeumannRegularity} to obtain \begin{eqnarray} \lefteqn{ \left( \frac 1 { r ^ { n-1}} \int _{ \sball x { 8r} } |\nabla u | ^ q \, d\sigma \right ) ^ { 1/q} } \nonumber \\ & \leq &C \left (\frac 1 { r ^ { n-1}} \int _ { \sball x { 8r} } |\nabla u | ^ 2 \, dy \right ) ^ { 1/2} \nonumber \\ & \leq & C \left [ \left ( \frac 1 { r^ { n-1}} \int _ { \sball x r \cap N } |a|^2\, d\sigma \right ) ^ { 1/2} + \left(\frac 1 { r ^ n } \int _ { \dball x { 16r}}|\nabla u | ^ 2 \, dy \right ) ^ { 1/2} \right]. \label{EasyCase} \end{eqnarray} Using the normalization of the atom $a$ and Lemma \ref{Energy}, the right-hand side of (\ref{EasyCase}) may be estimated by $ \sigma(\sball x { 8r} ) ^ { -1} $ and we obtain (\ref{LocalPart}) in this case. Now we turn our attention to the proof of the estimate (\ref{Decay}). Our first step is to observe that the solution $u$ satisfies the estimate \begin{equation} \label{AtomDecay} |u(y) | \leq \frac { Cr ^ \beta } { |x-y |^ { n-2+ \beta } }, \qquad |y-x| > 2r. \end{equation} To establish (\ref{AtomDecay}), we begin with the representation formula in part 3) of Lemma \ref{Green} and claim that we may find $\bar x $ in $\sball x r $ so that $$ u ( y ) = - \int _ { \sball x r \cap N } a(z) ( G(y,z) - G( y, \bar x) ) \, d\sigma . $$ If $ \sball x r \subset N$, then we may let $ \bar x = x$ and use that $a$ has mean value zero to obtain the above representation. If $\sball x r \cap D \neq \emptyset $, then we choose $\bar x \in D \cap \sball x r$ and use that $ G( y, \cdot ) $ vanishes on $D$. Now the estimate (\ref{AtomDecay}) follows easily from the normalization of the atom and the estimates for the Green function in part 4) of Lemma \ref{Green}. We will consider three cases in the proof of (\ref{Decay}): a) $2^k r < r_0$ and $ \delta (x) \leq 2\cdot 2^k r\sqrt{ 1+M^2}$, b) $2^k r < r_0$ and $ \delta (x) > 2\cdot 2^k r\sqrt { 1+M^2}$, c) $2^k r \geq r_0$. The details are similar to the proof of (\ref{LocalPart}), thus we will be brief. We begin with case a) and use H\"older's inequality with exponents $2/q$ and $ 2/(2-q)$ to obtain $$ \left( \int _ { \Sigma _k } |\nabla u | ^ q \, d\sigma \right ) ^ { 1/q} \leq C \left( \int _ { \Sigma _k } |\nabla u |^2 \delta ^ { 1- \epsilon } \, d\sigma \right ) ^ {1/2} ( 2^ k r ) ^ { (n-1) ( \frac 1 q - \frac 1 2 ) + \frac { \epsilon -1} 2 }. $$ As in the proof of the estimate (\ref{LocalPart}), this requires that $ 1 < q < 1/ ( 1 - \epsilon /2)$. From Lemma \ref{Whitney} we have $$ \left( \int _ { \Sigma_k } | \nabla u | ^ 2 \delta ^ { 1-\epsilon } \, d\sigma \right) ^ { 1/2} \leq C \left ( \sum _ { j = k -1} ^ { k+1} \int _ { S_j } |\nabla u | ^ 2\delta ^ { -\epsilon } \, dy \right) ^ { 1/2} $$ This estimate requires $ k \geq 2$ so that $ \sring {k-1} \cap \sball x {r} = \emptyset$. Then H\"older and the reverse H\"older estimate in Lemma \ref{RHEstimate} gives $$ \left( \int _ {S_k} |\nabla u | ^ 2 \delta ^ { -\epsilon } \, dy \right) ^ {\frac 1 2 } \leq C \left ( \int _ { S_k } |\nabla u | ^ p \, dy \right) ^ { \frac 1 p } \leq C ( 2^ k r) ^ {- \frac { \epsilon} 2} \left ( \sum _ { j = k -1} ^ { k+1} \int _{S_j} |\nabla u |^2 \, dy \right) ^ { \frac 1 2 } . $$ Here we need $ k \geq 2$ so that $ \sball x r \cap \Sigma _ { k-1} = \emptyset $ and the term in involving the Neumann data in Lemma \ref{RHEstimate} vanishes. Finally, from Caccioppoli and our estimate (\ref{AtomDecay}) for $u$, we obtain that $$ \left( \int _{S_k} |\nabla u | ^ 2 \, dy \right) ^ { 1/2} \leq \frac C { 2^ k r} \left( \sum _ { j = k -1} ^ {k+1} \int_{S_j} |u|^2 \, dy \right) ^ { 1/2} \leq C 2 ^ { -k\beta} (2^k r ) ^ {1 -n/2}. $$ Again, we need $ k\geq 2$ so that the data for the mixed problem is zero when we apply Caccioppoli's inequality. Combining the four previous estimates gives $$ \left( \int _ { \Sigma _k } |\nabla u | ^ q \, d\sigma \right) ^ { 1/q} \leq C 2 ^ { -k\beta} \sigma ( \Sigma _k ) ^ { -1/q'} $$ for $k \geq 4 $. We need $k \geq 4$ in order to fatten up the set $ \sring k $ three times: once to apply Lemma \ref{Whitney}, once to apply Lemma \ref{RHEstimate} and once to apply Caccioppoli's inequality. Now we consider case b). Since $ \delta (x) \geq 2( 2^ k r) \sqrt { 1+M^ 2} $, we have $ \sball x {2 \cdot 2^ k r } \subset N$ by Lemma \ref{DorN}. Hence, we may use Lemma \ref{NeumannRegularity}, Caccioppoli's inequality and (\ref{AtomDecay}) to obtain (\ref{Decay}). Finally, we consider case c) where $ 2^k r > r_0$. We recall that we have a covering of $ \partial \Omega$ by coordinate cylinders. In each coordinate cylinder, we may use Lemma \ref{NeumannRegularity}, Lemma \ref{DirichletRegularity} or Lemma \ref{Whitney} and the techniques given above to obtain $$ \left( \int _ { Z_ { r_0} \cap \partial \Omega } |\nabla u |^ q \, d\sigma\right)^ { 1/q} \leq C r_0 ^ { (1-n)/q'}. $$ Adding these estimates gives (\ref{Decay}) with a constant that depends on the global character of the domain. \end{proof} We now show that the non-tangential maximal function of our weak solutions lies in $L^1$ when the Neumann data is an atom. \begin{theorem} \label{HardyTheorem} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem. If $ f_N$ is in $ H^1 (N)$, then there exists $u$ a solution of the $L^1$-mixed problem (\ref{MP}) with Neumann data $f_N$ and zero Dirichlet data and this solution satisfies $$ \|\nontan{ (\nabla u) } \|_{L^1( \partial \Omega)} \leq C \|f_N\|_ { H^1 (N)}. $$ The constant $C$ in this estimate depends on the global character of $ \Omega$, $N$ and $D$. \end{theorem} \note { Theorem restated to give existence in the Hardy space, rather than for an atom. Check to make sure the proof proves the theorem. } \begin{proof} We begin by considering the case when $f_N$ is an atom and we let $u$ be the weak solution of the mixed problem with Neumann data an atom $a$ and zero Dirichlet data. The result for data in $H^1(N)$ follows easily from the result for an atom. We establish a representation for the gradient of $u$ in terms of the boundary values of $u$. Let $x \in \Omega$ and $j$ be an index ranging from $1$ to $n$. We claim \begin{eqnarray} \label{RepFormula} \frac {\partial u } { \partial x_j} (x) & = & \int _{ \partial \Omega } \sum _{ i=1 } ^ n \frac { \partial \Xi }{ \partial y_i } (x-\cdot)( \nu _i \frac { \partial u }{ \partial y_j } - \frac { \partial u }{ \partial y _i } \nu _j ) \nonumber \\ & & \qquad + \frac {\partial \Xi }{\partial y_j } ( x-\cdot) \frac { \partial u }{ \partial \nu } \, d\sigma . \end{eqnarray} If $u$ is smooth up to the boundary, the proof of (\ref{RepFormula}) is a straightforward application of the divergence theorem. However, it takes a bit more work to establish this result when we only have that $u$ is a weak solution. Thus, we suppose that $\eta $ is a smooth function which is zero in a neighborhood of $ \Lambda $ and supported in a coordinate cylinder. Using the coordinate system for our coordinate cylinder, we set $ u_ \tau (y) = u(y+\tau e_n)$ where $e_n$ is the unit vector the $x_n$ direction and $ \tau >0$. Applying the divergence theorem gives \begin{eqnarray} \lefteqn{ \int _{ \partial \Omega }\eta ( \frac { \partial \Xi }{ \partial \nu } (x- \cdot ) \frac { \partial u_\tau }{ \partial y_j } - \nabla \Xi (x-\cdot ) \cdot \nabla u_\tau \nu _j + \frac {\partial \Xi }{\partial y_j } ( x-\cdot ) \frac { \partial u_\tau }{ \partial \nu } ) \, d\sigma } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \nonumber\\ & = & \eta (x) \frac {\partial u_\tau }{\partial x_j } (x) +\int _\Omega \nabla \eta \cdot \nabla \Xi(x- \cdot ) \frac { \partial u_\tau }{ \partial y_j} \nonumber \\ & & \qquad- \nabla _y \Xi (x-\cdot ) \cdot \nabla u_\tau \frac { \partial \eta } {\partial y_j} \nonumber\\ & & \qquad + \frac { \partial \Xi }{ \partial y_j } (x- \cdot ) \nabla u_\tau \cdot \nabla \eta \, dy . \label{dadgumidentity} \end{eqnarray} Thanks to the truncated maximal function estimate in Lemma \ref{Whitney}, we may let $ \tau $ tend to zero from above and conclude that the same identity holds with $u_\tau$ replaced by $u$. Next, we suppose that $ \eta$ is of the form $ \eta \phi_ \epsilon$ where $ \phi_ \epsilon =0$ on $\{ x: \delta (x) < \epsilon\}$, $ \phi_ \epsilon =1$ on $\{ x: \delta (x) > 2\epsilon \}$ and we have the estimate $ | \nabla \phi_ \epsilon (x) | \leq C/\epsilon $. Since we assume the boundary between $D$ and $N$ is a Lipschitz surface, we have the following estimate for $ \epsilon $ sufficiently small \begin{equation} \label{creasecollar} |\{ x: \delta (x) \leq 2 \epsilon \}| \leq C \epsilon ^2. \end{equation} Using our estimate for $ \nabla \phi_\epsilon$ and the inequality (\ref{creasecollar}), we have $$ | \int _\Omega \eta \nabla \phi_\epsilon \cdot \nabla \Xi(x-\cdot) \frac { \partial u }{ \partial y_j} \,d y | \leq C \left ( \int_ { \{ y: \delta (y) < 2\epsilon \} } |\nabla u | ^ 2\, dy \right ) ^ { 1/2} $$ and the last term tends to zero with $\epsilon$ since the gradient of a weak solution lies in $L^2 ( \Omega)$. Using this and similar estimates for the other terms in (\ref{dadgumidentity}), gives \begin{eqnarray*} \lefteqn{ \lim _{ \epsilon \rightarrow 0^+} \int _\Omega \nabla( \phi_\epsilon \eta ) \cdot \nabla_y \Xi(x- \cdot ) \frac { \partial u }{ \partial y_j} - \nabla _y \Xi (x- \cdot) \cdot \nabla u \frac { \partial (\phi_ \epsilon \eta ) } {\partial y_j} } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ & & \\ + \frac { \partial \Xi }{ \partial y_j } (x-\cdot ) \nabla u \cdot \nabla(\phi_\epsilon \eta ) \, dy &= & \int _\Omega \nabla \eta \cdot \nabla_y \Xi(x-\cdot ) \frac { \partial u }{ \partial y_j} \\ & & \qquad - \nabla_y \Xi (x- \cdot ) \cdot \nabla u \frac { \partial \eta } {\partial y_j} \\ & &\qquad + \frac {\partial \Xi }{ \partial y_j } (x-\cdot ) \nabla u \cdot \nabla \eta \, dy . \end{eqnarray*} Thus we obtain the identity (\ref{dadgumidentity}) with $ u_\tau$ replaced by $u$ and without the support restriction on $\eta$. Finally, we choose a partition of unity which consists of functions that are either supported in a coordinate cylinder, or whose support does not intersect the boundary of $\Omega$. Summing as $\eta $ runs over this partition gives us the representation formula (\ref{RepFormula}) for $u$. As we have $ \nabla u \in L^ q ( \partial \Omega)$ for some $q>1$, it follows from the theorem of Coifman, McIntosh and Meyer \cite{CMM:1982} that $ \nontan { ( \nabla u )} $ lies in $L^ q ( \partial \Omega)$. However, a bit more work is needed to obtain the correct $L^1 $ estimate for $ \nontan { ( \nabla u )} $. We claim \begin{eqnarray*} \int_{ \partial \Omega } \frac { \partial u}{\partial \nu } \, d\sigma &=&0 \\ \int_{ \partial \Omega } \nu _j \frac{ \partial u }{ \partial y_i } - \nu _i \frac{ \partial u }{ \partial y_j } \, d\sigma & = & 0. \end{eqnarray*} Since $ \nontan { ( \nabla u ) } $ lies in $L^q( \partial \Omega)$, the proof of these two identities is a standard application of the divergence theorem. Using these results and the estimates for $\nabla u$ in Theorem \ref{AtomicTheorem}, we can show that $\partial u /\partial \nu$ and $ \nu_j \partial u /\partial y_i-\nu _i \partial u/\partial y_j$ are molecules on the boundary (see \cite{CW:1976}) and hence it follows from the representation formula (\ref{RepFormula}) that $\nontan{(\nabla u )}$ lies in $L^1 ( \partial \Omega)$ and satisfies the estimate $$ \| \nontan {(\nabla u) }\|_{L^1 ( \partial \Omega)}\leq C. $$ Finally, the existence of non-tangential limits at the boundary follows from the estimate for the non-tangential maximal function. Once we know the limits exist it is easy to see that the boundary data for the $L^1$-mixed problem must agree with the boundary data for the weak formulation. \end{proof} \section{Uniqueness of solutions} \label{Unique} In this section we establish uniqueness of solutions to the $L^1$-mixed problem (\ref{MP}). We use the existence result established in section \ref{Atoms} and argue by duality that if $u$ is a solution of the mixed problem with zero Dirichlet and Neumann data, then $u$ is also a solution of the regularity problem with zero data and hence is zero. \begin{theorem} \label{uRuniq} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem. Suppose that $u$ solves the $L^1$-mixed problem (\ref{MP}) with data $ f_N = 0$ and $ f_D=0$. If $ ( \nabla u ) ^ * \in L^ 1 ( \partial \Omega)$, then $u = 0$. \end{theorem} Given a Lipschitz domain $ \Omega$, we may construct a sequence of smooth approximating domains. A careful exposition of this construction may be found in the dissertation of Verchota (\cite[Appendix A]{GV:1982}, \cite[Theorem 1.12]{GV:1984}). We will need this approximation scheme and a few extensions. Given a Lipschitz domain $ \Omega$, Verchota constructs a family of smooth domains $ \{ \Omega_k \}$ with $ \bar \Omega _k \subset \Omega$. In addition, he finds bi-Lipschitz homeomorphisms $ \Lambda _k : \partial \Omega \rightarrow \partial \Omega _k$ which are constructed as follows. We choose a smooth vector field $ V$ so that for some $ \tau = \tau (M) >0$, $ V\cdot \nu \leq - \tau $ a.e.~on $\partial \Omega$ and define a flow $ f( \cdot ,\cdot ) : {\bf R} ^ n \times {\bf R} \rightarrow {\bf R} ^ n$ by $ \frac d { dt} f(x,t) = V(f(x,t))$, $f(x,0) = x$. One may find $ \xi > 0$ so that \begin{equation}\label{Gdef} {\cal O} = \{ f(x, t) : x \in \partial \Omega , - \xi < t < \xi \} \end{equation} is an open set and the map $ (x,t) \rightarrow f(x,t)$ from $ \partial \Omega \times ( -\xi, \xi) \rightarrow {\cal O}$ is bi-Lipschitz. Since the vector field $V$ is smooth, we have \begin{equation} \label{DThing} Df(x,t) = I _n + O(t) \end{equation} where $ I_n$ is the $n\times n$ identity matrix and $DF$ denotes the derivative of a map $F$. In addition, we have a Lipschitz function $t_k (x)$ defined on $ \partial \Omega$ so that $ \Lambda_k (x) = f(x, t_k ( x)) $ is a bi-Lipschitz homeomorphism, $ \Lambda _k : \partial \Omega \rightarrow \partial \Omega _k$. We may find a collection of coordinate cylinders $\{ Z _i \}$ so that each $ Z_i$ serves as a coordinate cylinder for $ \partial \Omega$ and for each of the approximating domains $ \partial \Omega_k$. If we fix a coordinate cylinder $Z$, we have functions $ \phi$ and $\phi_k$ so that $\partial \Omega \cap Z = \{ ( x', \phi ( x' )) : x' \in {\bf R} ^ { n -1} \} \cap Z$ and $\partial \Omega_ k \cap Z = \{ ( x' , \phi_k( x' )) : x' \in {\bf R} ^ { n -1} \} \cap Z$. The functions $ \phi _k $ are $ C^ \infty $ and $ \|\nabla' \phi _k \| _ { L ^ \infty ( {\bf R} ^ { n -1} )} $ is bounded in $k$, $ \lim _{ k \rightarrow \infty } \nabla' \phi _k ( x' ) = \nabla' \phi (x' )$ a.e. and $ \phi_k$ converges to $\phi$ uniformly. Here we are using $ \nabla'$ to denote the gradient on ${\bf R} ^ {n-1}$. We let $ \pi: {\bf R} ^ n \rightarrow {\bf R} ^ { n-1}$ be the projection $ \pi (x', x_n ) = x' $ and define $ S_k (x' ) = \pi(\Lambda_k (x' ,\phi (x')))$. According to Verchota, the map $S_k $ is bi-Lipschitz and has a Jacobian which is bounded away from 0 and $ \infty$. We let $T_k$ denote $S_k^ { -1} $ and assume that both are defined in a neighborhood of $ \pi(Z)$. We claim that \begin{equation} \label{Dclaim} \lim _ { k \rightarrow \infty } DT_k (S_k(x') ) = I _ { n-1}, \qquad \mbox{a.e.~in $\pi(Z)$}, \end{equation} and the sequence $\| DT_k\|_ { L ^ \infty ( \pi (Z) )}$ is bounded in $k$. To establish (\ref{Dclaim}), it suffices to show that $ DS_k $ converges to $ I_{ n-1}$ and that the Jacobian determinant of $ DS_k$ is bounded away from zero and infinity. The bound on the Jacobian is part of Verchota's construction (see \cite[p.~119]{GV:1982}). As a first step, we compute the derivatives of $t_k (x', \phi (x'))$. We first observe that \begin{eqnarray*} \lefteqn{\frac { \partial } { \partial x_i } f((x', \phi(x') ) , t_k (x', \phi (x' ))) } ~~~~~~ \\ & = & \frac { \partial f}{ \partial x_i } ( (x', \phi (x' )), t_k (x' , \phi (x'))) +\frac { \partial \phi }{ \partial x_i }( x' ) \frac { \partial f}{ \partial x_n } (( x', \phi (x' )), t_k (x' , \phi (x'))) \\ & & \qquad + V(f((x', \phi(x') ) , t_k (x', \phi (x' )))) \frac \partial { \partial x_i } t _k (x' , \phi (x' )). \end{eqnarray*} Since $ f((x', \phi ( x' )), t_k (x', \phi (x')))$ lies in $ \partial \Omega_k$, the derivative is tangent to $ \partial \Omega_k$ and we have \begin{equation}\label{Tangential} \frac { \partial }{ \partial x_i } f((x', \phi ( x' )), t_k (x', \phi (x'))) \cdot \nu _ k (y) = 0, \qquad \mbox{a.e.~in }\pi (Z), \end{equation} where $ y = ( S_k (x'), \phi_k ( S_k (x' )))$ and $ \nu _k $ is the normal to $ \partial \Omega_k$. Solving equation (\ref{Tangential}) for $ \frac \partial { \partial x_i } t_k $ gives \begin{eqnarray*} \frac \partial { \partial x_i } t _k (x' , \phi (x' )) & = & - ( V(y) \cdot \nu _k (y) ) ^ { -1} \left ( \frac { \partial f }{ \partial x_i }( ( x' , \phi ( x' )), t_k (x', \phi (x' ))) \right . \\ & & \qquad + \left. \frac { \partial \phi } { \partial x_i } (x' ) \frac { \partial f }{ \partial x_n }( ( x' , \phi ( x' )), t_k (x', \phi (x' ))) \right)\cdot \nu_k (y) . \end{eqnarray*} Since $\lim _{ k \rightarrow \infty } t_k (x', \phi (x')) = 0$ uniformly for $x' \in \pi (Z)$, (\ref{DThing}) holds, and $ \nu _ k ( y) $ converges pointwise a.e.~and boundedly to $ \nu (x)$, we obtain that \begin{equation} \label{tderiv} \lim _ { k \rightarrow \infty } \frac \partial { \partial x_i } t _k (x' , \phi (x' )) = 0, \qquad \mbox{a.e.~in $ \pi (Z)$}. \end{equation} Given (\ref{DThing}), (\ref{tderiv}), and recalling that $ S_k( x') = \pi(f( (x', \phi(x')), t_k (x', \phi (x')))$, (\ref{Dclaim}) follows. \note { The proof of this Lemma is in RMB notebook 19, page 49. } \begin{lemma} \label{Verchota} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem. If $u$ is in $\sobolev 1 1 (\partial \Omega _k)$ and $w$ is the weak solution of the mixed problem with Neumann data an atom for $N$ and zero Dirichlet data, then we have $$ \int _ { \partial \Omega _k } u \frac { \partial w }{ \partial \nu } \,d\sigma \leq C_w \| u \|_{ W^ { 1,1 }( \partial \Omega _k )}. $$ \end{lemma} \begin{proof} This may be proven using generalized Riesz transforms as in \cite[Section 5]{GV:1984}. Also, see more recent treatments by Sykes and Brown \cite[section 3]{SB:2001} and Kilty and Shen \cite[section 7]{KS:2010}. Verchota's argument uses square function estimates to show that the generalized Riesz transforms are bounded operators on $L^p ( \partial \Omega)$. In the proof of this Lemma, we need that the Riesz transforms of $w$ are bounded functions. From the estimate for the Green function in Lemma \ref{Green} and the representation of $ w= - \langle G, a\rangle_{\partial \Omega} $, we conclude that $ w$ is H\"older continuous. The H\"older continuity, and hence boundedness, of the Riesz transforms of $w$ follow from the following characterization of H\"older continuous harmonic functions. A harmonic function $u$ in a Lipschitz domain $\Omega$ is H\"older continuous of exponent $\alpha$, $0< \alpha < 1$, if and only if $ \sup _{ x\in \Omega } \mathop{\rm dist}\nolimits(x, \partial \Omega )^ {1- \alpha } |\nabla u (x) | $ is finite. \end{proof} We will need the following technical lemma on approximation of functions with $ \nontan{(\nabla u )} $ in $L^ 1 ( \partial \Omega)$. The proof relies on the approximation scheme of Verchota outlined above. In our application, we are interested in studying functions in Sobolev spaces on the family of approximating domains. Working with derivatives makes the argument fairly intricate. \begin{lemma} \label{TechnicalMonstrosity} Let $\Omega$, $N$ and $D$ be a standard domain for the mixed problem. If $ u $ satisfies $ \nontan { ( \nabla u ) } \in L^ 1 ( \partial \Omega)$ and $ \nabla u$ has non-tangential limits a.e.~on $ \partial \Omega$, then we may find a sequence of Lipschitz functions $ U_j$ so that $$ \lim _{ k\rightarrow \infty } \| u - U _j \| _ { \sobolev 1 1 ( \partial \Omega _k )} \leq C /j. $$ If the non-tangential limits of $u$ are zero a.e. on $D$, then we may arrange that $U_j|_{\partial \Omega}$ is zero on $ D$. The constant $C$ may depend on $ \Omega$ and $u$. \end{lemma} \begin{proof} To prove the Lemma, it suffices to consider a function $u$ which is zero outside one of the coordinate cylinders $Z$ as given in Verchota's approximation scheme. We have $ u( x' , \phi (x')) \in \sobolev 1 1 ( {\bf R} ^ { n -1})$, where we have set this function to be zero outside $ \pi (Z)$. Hence, there exists a sequence of Lipschitz functions $ u _j $ so that $\int _{ {\bf R} ^ { n-1} } | \nabla' u (x', \phi (x' )) - \nabla' u _ j ( x', \phi (x' )) | \, dx' \leq 1/j$ where $ \nabla' $ denotes the gradient in $ {\bf R} ^ { n -1}$. We extend $u_j$ to a neighborhood of $ \partial \Omega$ by $$U _ j ( f(x,t) ) = \eta ( f(x, t) ) u _j (x) , \qquad x \in \partial \Omega$$ where $ \eta$ is a smooth cutoff function which is one on a neighborhood of $ \partial \Omega$ and supported in the set $ {\cal O} $ defined in (\ref{Gdef}). If we have that $u$ is zero on $D$, then standard approximation results for Sobolev spaces allow us to choose $u_j$ to be zero in a neighborhood of $\bar D$. This relies on our assumption on $ \Lambda $. We consider \begin{eqnarray*} \lefteqn{ \int _ {\pi ( Z)} | \nabla' u (x' , \phi _k (x' )) - \nabla' U_j ( x', \phi_k (x' ))| \, dx' } ~~~~~~~~~~~~~~~~~~~~ & & \\ & \leq & \int _ {\pi ( Z)} | \nabla' u (x' , \phi _k (x' )) - \nabla' u ( x', \phi (x' ))| \, dx' \\ & & \qquad + \int _ {\pi ( Z)} | \nabla' u (x' , \phi (x' )) - \nabla' u_j ( x', \phi (x' ))| \, dx'\\ & & \qquad + \int _{ \pi ( Z)} | \nabla' u_j (x' , \phi (x' )) - \nabla' U_j ( x', \phi_k (x' ))| \, dx'\\ & = & A_k +B +C_k. \end{eqnarray*} We have that $ \lim _{ k \rightarrow \infty } A_k =0 $ since we assume that $ \nontan {(\nabla u )} \in L ^ 1 ( \partial \Omega )$, $ \nabla u $ has non-tangential limits a.e., and $ \nabla' \phi _k $ converges pointwise a.e.~and boundedly to $ \nabla' \phi$. By our choice of $ u_j$, we have $ B\leq C/j$. Finally, our construction of $U_j$ and our definition of $T_k$ (before (\ref{Dclaim})) imply that $ U_j (x', \phi _k ( x' )) = u _j ( T_k (x') , \phi ( T_ k (x')))$ and hence we have \begin{eqnarray*} C_k & \leq & \int _ { \pi (Z)} | ( I _ { n-1} - DT_ k ( x' )) \nabla' u _ j ( x', \phi (x')) | \, dx' \\ & & \qquad + \int _ { \pi (Z)} | DT_k (x' ) ( \nabla' u _ j ( x' , \phi(x' ))) - \nabla' u _ j ( T_k (x' ), \phi ( T_k (x' )))| \, d x' \\ & = & C_ { k, 1 } + C_ {k,2 } . \end{eqnarray*} We have that $ \lim _ { k\rightarrow \infty } C_ { k,1 } = 0$ since $ \nabla' u _ j $ is bounded and (\ref{Dclaim}) holds. Since $ T_k ( x' )$ converges uniformly to $x' $, $DT_k $ is bounded and the Jacobian of $S_k$ is bounded, we have that $ \lim _ { k \rightarrow \infty } C_{ k,2} = 0$. \end{proof} \begin{proof}[Proof of Theorem \ref{uRuniq}] We let $ u$ be a solution of the $L^1$-mixed problem, (\ref{MP}), with $f_N = 0$ and $f_D=0$ and we wish to show that $u$ is zero. We fix $a$ an atom for $N$ and let $ w$ be a solution of the mixed problem with Neumann data $a$ and zero Dirichlet data. Our goal is to show that \begin{equation}\label{AtomClaim} \int _ { N } a u \, d\sigma =0. \end{equation} This implies that $u$ is zero on $ \partial \Omega$ and then Dahlberg and Kenig's result for uniqueness of solutions of the regularity problem \cite{DK:1987} implies that $u=0$ in $ \Omega$. We turn to the proof of (\ref{AtomClaim}). Applying Green's second identity in one of the approximating domains $ \Omega _k$ gives us \begin{equation}\label{uniq42} \int _ { \partial \Omega _k } w \frac { \partial u }{ \partial \nu} \, d\sigma = \int _ { \partial \Omega _k } u \frac { \partial w }{ \partial \nu} \, d\sigma, \qquad k =1,2\dots . \end{equation} We have $ \nontan { ( \nabla u )} $ is in $L^1 ( \partial \Omega )$ while $ w$ H\"older continuous and hence bounded. Recalling that $w$ is zero on $D$ and $ \partial u /\partial \nu$ is zero on $N$, we may use the dominated convergence theorem to obtain \begin{equation}\label{uniq43} \lim _ { k \rightarrow \infty } \int _ { \partial \Omega _k } w \frac { \partial u }{ \partial \nu } \, d\sigma =0 . \end{equation} Thus, our claim will follow if we can show that \begin{equation} \label{ClaimFollow} \lim _{ k \rightarrow \infty } \int _ { \partial \Omega _k } u \frac { \partial w }{ \partial \nu } \, d\sigma = \int _{\partial \Omega } ua \, d\sigma . \end{equation} Note that the existence of the limit in (\ref{ClaimFollow}) follows from (\ref{uniq42}) and (\ref{uniq43}). We let $U_j$ be the sequence of functions from Lemma \ref{TechnicalMonstrosity} and consider \begin{eqnarray} \lefteqn{ \left | \int _{ \partial \Omega } ua \, d\sigma -\lim _{ k \rightarrow \infty} \int _{ \partial \Omega _k } u \frac { \partial w }{ \partial \nu } \, d\sigma \right | } \label{U1}\\ & \leq & \left| \int_{ \partial \Omega } ua \, d\sigma - \lim _{ k \rightarrow \infty } \int _{ \partial \Omega _k } U_j \frac { \partial w }{ \partial \nu } \, d\sigma \right | + \limsup _{ k \rightarrow \infty } \left | \int _ { \partial \Omega _k } ( u - U_j ) \frac { \partial w}{ \partial \nu } \, d\sigma \right| . \nonumber \end{eqnarray} Because we have that $ \nontan {( \nabla w ) } $ is in $L^ 1( \partial \Omega)$ and $U_j$ is bounded, we may take the limit of the first term on the right of (\ref{U1}) and obtain $$ \left | \int_{ \partial \Omega } ua \, d\sigma - \lim _{ k \rightarrow \infty } \int _{ \partial \Omega _k } U_j \frac { \partial w }{ \partial \nu } \, d\sigma \right | = \left | \int _{ N } ( u -U _j ) a \, d\sigma\right | \leq C/j . $$ Here we use that $U_j|_{ D } =0$. According to Lemmata \ref{Verchota} and \ref{TechnicalMonstrosity}, the second term on the right of (\ref{U1}) is bounded by $ C_w/j$. As $j$ is arbitrary, we obtain (\ref{ClaimFollow}) and hence the Theorem. \end{proof} \section{A Reverse H\"older inequality at the boundary} \label{BoundaryReverse} In this section we establish an estimate in $L^p( \partial \Omega)$ for the gradient of a solution to the mixed problem. This is the key estimate that is used in section \ref{LpSection} to establish $L^p$-estimates for the mixed problem. \begin{lemma} \label{newLocal} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem. Let $u$ be a weak solution of the mixed problem with Neumann data $f_N$ in $L^ \infty ( N)$ and zero Dirichlet data. Let $ p_0 >2$ be as in Lemma \ref{RHEstimate} and fix $ q$ satisfying $ 1< q < p_0/2$. For $ x \in \partial \Omega $ and $ r$ with $ 0 < r < r_0$ we have $$ \left ( -\!\!\!\!\!\!\int _ { \sball x r } { \nontan {( \nabla u ) _{cr} } }^ q \, d\sigma \right) ^ { 1/q} \leq C \left [ -\!\!\!\!\!\!\int _{ \dball x { 2r} } | \nabla u | \, dy + \|f_N\|_{ L^ \infty ( \sball x { 2r} \cap N)} \right]. $$ The constant $c=1/16$ and $C$ depends on $M$, $n$ and $q$. \end{lemma} \begin{proof} We fix $ x\in \partial \Omega$ and $r$ with $0< r < r_0$. We claim that we have \begin{equation}\label{LocalClaim} \left ( -\!\!\!\!\!\!\int _ { \sball x {4r}} |\nabla u |^ q \, d\sigma \right ) ^ { 1/q} \leq C \left ( -\!\!\!\!\!\!\int _ { \dball x {16r}} |\nabla u | \, dy + \| f_N\| _ { L^ \infty ( \sball x { 16r} \cap N)} \right). \end{equation} We will consider two cases: a) $ \delta (x) \leq 8r \sqrt { 1+M^2}$ , b) $ \delta (x) > 8r \sqrt { 1+M^2} $. We give the proof in case a). Since we assume $ 1 < q < p_0/2$, we may choose $ \epsilon $ satisfying $ 2-2/q < \epsilon < 2 - 4 /p_0$. We apply H\"older's inequality with exponents $ 2/q$ and $2/(2-q)$ to obtain \begin{eqnarray*} \lefteqn{ \left ( \int _ { \sball x { 4r}} |\nabla u | ^ q \, d\sigma \right ) ^ { \frac 1 q} } \qquad \\ & \leq & \left ( \int _ { \sball x { 4r} } |\nabla u |^ 2 \delta ^ { 1-\epsilon } \, d\sigma \right ) ^ { \frac 1 2 } \left ( \int _ {\sball x { 4r} } \delta ^ { ( \epsilon -1)q /( 2-q)}\, d\sigma \right ) ^ { \frac 1 q - \frac 1 2 } \\ & \leq & C r ^ { ( n-1) ( \frac 1 q - \frac 1 2 ) + \frac { \epsilon -1 } 2 }\left ( \int _ { \sball x { 4r} } |\nabla u | ^ 2 \delta ^ { 1-\epsilon } \, d\sigma \right ) ^ {1/2} \end{eqnarray*} where we use that $ q (\epsilon -1) / ( 2-q) > -1$ or $ 2 - \frac 2 q < \epsilon $ which implies that the integral of $ \delta ^ { ( \epsilon -1) q /( 2-q)}$ is finite. Next, we use Lemma \ref{Whitney} and our hypothesis that $ \delta (x) \leq 8r \sqrt { 1+M^2}$ to obtain \begin{eqnarray*} \lefteqn { \left ( \int _ { \sball x { 4r} } |\nabla u | ^ 2 \delta ^ { 1- \epsilon } \, d\sigma \right ) ^ { 1/2 } } \\ & \leq & C \left [ \left ( \int _ {\dball x {8r} } |\nabla u |^ 2 \, \delta ^ { - \epsilon } \, dy \right ) ^ { 1/2} + \left ( \int _ { \sball x { 8r}\cap N} |f_N |^ 2 \delta ^ { 1- \epsilon } \, d\sigma \right ) ^ { 1/2} \right ] \\ & \leq & C \left [ \left ( \int _ { \dball x { 8r} } |\nabla u |^ 2\delta ^ { - \epsilon } \, dy \right ) ^ { 1/2 } + r ^ { \frac { n- \epsilon } 2 } \| f_N \| _ { L^ \infty ( \sball x { 8r } \cap N ) } \right ] . \end{eqnarray*} To estimate $(\int _ {\dball x {8r}}|\nabla u | ^ 2 \delta ^ { -\epsilon } \, dy ) ^ {1/2}$, we choose $p >2$, use H\"older's inequality with exponents $ p/2 $ and $ p / (p-2)$, and Lemma \ref{RHEstimate} to find \begin{eqnarray*} \lefteqn{ \left( \int _ { \dball x { 8r}}|\nabla u | ^ 2 \delta ^ { -\epsilon } \, dy \right ) ^ { 1/2} } \\ & \leq & \left( \int _ { \dball x { 8r}}\delta ^ {- \epsilon p / ( p-2)} \, dy \right ) ^ { \frac 1 2 - \frac 1 p }\left ( \int _ { \dball x { 8r } }|\nabla u | ^ p \, dy \right ) ^ { 1/p } \\ & \leq & C r ^ { - \frac \epsilon 2 + \frac n 2 } \left [ -\!\!\!\!\!\!\int _ { \dball x { 16r } }|\nabla u | \, dy + \left ( -\!\!\!\!\!\!\int _ { \sball x { 16r} \cap N}|f_N | ^ { \frac { p _0 ( n-1)} n} \, d\sigma \right ) ^ { \frac n { p_0 ( n-1) } } \right ]. \end{eqnarray*} Combining the two previous displayed inequalities gives the estimate $$ \left( \int _ { \sball x { 4r} } |\nabla u | ^ q \, d\sigma \right ) ^ { 1/q} \leq C r ^ { ( n-1)/q} \left ( -\!\!\!\!\!\!\int_ { \dball x { 16r}} |\nabla u | \, dy + \| f _N \| _ { L^ \infty ( \sball x { 16r} \cap N )}\right ), $$ which gives the claim (\ref{LocalClaim}). Now we consider the proof of (\ref{LocalClaim}) in case b). Here, we use $ \delta (x) >8r \sqrt{ 1+M^2}$ and Lemma \ref{DorN} to conclude that $ \sball x {8r}\subset N$ or that $ \sball x {8r} \subset D$. Then we may use Lemma \ref{NeumannRegularity} or Lemma \ref{DirichletRegularity} to conclude that $$ \int _ { \sball x { 4r}} |\nabla u | ^ 2 \, d\sigma \leq C \left( \int _ { \sball x { 8r} \cap N} |f_N |^ 2 \, d\sigma + \frac 1 r \int_{\dball x { 8r} } |\nabla u | ^ 2 \, dy \right ). $$ Next, Lemma \ref{RHEstimate} gives $$ \left ( -\!\!\!\!\!\!\int _ { \dball x { 8r}} |\nabla u | ^ 2 \, dy \right ) ^ { 1/2} \leq C \left ( -\!\!\!\!\!\!\int _ { \dball x { 16r} } |\nabla u | \, dy + \| f\| _ { L^ \infty ( \sball x { 16r} \cap N) } \right). $$ Using the two previous estimates and H\"older's inequality, we obtain the claim (\ref{LocalClaim}) in case b). To obtain the estimate for the non-tangential maximal function, we choose a cutoff function $ \eta $ which is one on $ \ball x { 3r}$ and supported in $ \ball x { 4r}$. By repeating the arguments in the proof of Theorem \ref{HardyTheorem}, we may show that for $z $ in $ \Omega$ and $j =1,\dots,n$, we have the following representation for the derivatives of $u$: \begin{eqnarray*} (\eta\frac {\partial u}{\partial z_j}) (z) & = & \int _ { \partial \Omega } \eta ( \frac {\partial \Xi }{ \partial \nu } ( z-\cdot ) \frac { \partial u }{\partial y _j } - \nu _j \nabla_y \Xi (z- \cdot ) \cdot \nabla u + \frac { \partial \Xi }{\partial y _j }(z-\cdot) \frac { \partial u }{ \partial \nu } ) \, d\sigma \\ & & \qquad - \int _ { \Omega} \nabla \eta \cdot \nabla_y \Xi ( z-\cdot )\frac { \partial u }{ \partial y _j } - \frac { \partial \eta }{ \partial y _j } \nabla_y \Xi ( z-\cdot)\cdot \nabla u \\ & & \qquad \qquad \qquad + \nabla \eta \cdot \nabla u \frac { \partial \Xi }{ \partial y _j } ( z- \cdot ) \, dy . \end{eqnarray*} From this representation and the theorem of Coifman, McIntosh and Meyer \cite{CMM:1982}, we obtain $$ \left ( -\!\!\!\!\!\!\int _ { \sball x r } { \nontan {( \nabla u ) _{r} } }^ q \, d\sigma \right) ^ { 1/q} \leq C \left [ -\!\!\!\!\!\!\int_{ \dball x { 4r}} |\nabla u | \, dy + \left( -\!\!\!\!\!\!\int _ { \sball x { 4r}}|\nabla u | ^ q \, d \sigma \right) ^ { 1/q}\right]. $$ From this estimate, the claim (\ref{LocalClaim}) and a covering argument, we obtain the Theorem. \end{proof} \section{Estimates for solutions with data from $L^p$, $p> 1$} \label{LpSection} In this section, we use the following variant of an argument developed by Shen \cite{ZS:2007} to establish $L^p$-estimates for elliptic problems in Lipschitz domains. Shen's argument is based on earlier in work of Caffarelli and Peral \cite{MR1486629}. As the argument depends on a Calder\'on-Zygmund decomposition into dyadic cubes, it will be stated using surface cubes rather than the surface balls $ \sball xr $ used elsewhere in this paper. Let $Q _0$ be a cube in the boundary and let $F$ be defined on $4 Q_0$. Let the exponents $ p $ and $q$ satisfy $ 1< p < q$. Assume that for each $Q \subset Q_0$, we may find two functions $F_Q$ and $R_Q$ defined in $2Q$ such that \begin{eqnarray} \label{Shen1} |F| & \leq & |F_Q| + |R_Q | , \\ -\!\!\!\!\!\!\int_{ 2Q} |F_Q| \, d\sigma &\leq & C \left( -\!\!\!\!\!\!\int_{ 4Q} |f|^ p \, d\sigma \right ) ^ {1/p}, \label{Shen2} \\ \left ( -\!\!\!\!\!\!\int_{2Q} |R_Q|^ q \, d\sigma \right) ^ {1/q} &\leq & C \left [ -\!\!\!\!\!\!\int _{ 4Q} |F|\,d \sigma + \left( -\!\!\!\!\!\!\int_{4Q} |f|^ p \, d\sigma \right) ^ {1/p} \right]. \label{Shen3} \end{eqnarray} Under these assumptions, for $r$ in the interval $ ( p, q)$, we have $$ \left( -\!\!\!\!\!\!\int _{Q_0} |F|^ r \, d\sigma \right)^ { 1/r } \leq C \left[ -\!\!\!\!\!\!\int _{4Q_0} |F|\, d\sigma + \left( -\!\!\!\!\!\!\int_{4Q_0} |f|^ r \, d\sigma \right ) ^ { 1/r} \right ] . $$ The constant in this estimate will depend on the Lipschitz constant of the domain, the $L^p$ indices involved and the constants in the estimates in the conditions (\ref{Shen2}--\ref{Shen3}). The argument to obtain this conclusion is more or less the same as in Shen \cite[Theorem 3.2]{ZS:2007}. The main differences arise because the last term in (\ref{Shen3}) require us to substitute the maximal function $M(|f|^p)^ {1/p}$ for $M(f)$. We omit a detailed proof. Our hypotheses hypotheses differ from Shen's in that Shen has $p=1$ in (\ref{Shen2}) and (\ref{Shen3}) while we have $p>1$. We need to change Shen's formulation because we begin with results in Hardy spaces, rather than $L^p$-spaces. In our application, we will let $ 4Q_0$ be a cube with sidelength comparable to $r_0$. We let $u$ be a solution of the mixed problem with Neumann data $f$ in $L^p(N)$ and Dirichlet data zero. We define $f$ to be zero in $D$. Since $L^p(N)$ is contained in the Hardy space $H^1(N)$, we may use Theorem \ref{AtomicTheorem} to obtain a solution of the mixed problem with Neumann data $f$ on $N$ and zero Dirichlet data on $D$. Let $F = \nontan{(\nabla u )}$ and given a cube $Q\subset Q_0$ and with diameter $r$, define $ F_Q$ and $R_Q$ as follows. We let $\bar f _{ 4Q} =0$ if $ 4Q \cap D \neq \emptyset $ and $ \bar f _{ 4Q} = -\!\!\!\!\!\!\int _{ 4Q} f \, d\sigma $ if $ 4Q \subset N$. Set $g = \chi _{ 4Q} ( f-\bar f _{ 4Q})$ and $h = f-g$. As both $g$ and $h$ are elements of the Hardy space $ H^ 1(N)$, we may use Theorem \ref{HardyTheorem} to find solutions of the $L^1$-mixed problem with Neumann data $g$ or $h$. We let $v$ be the solution with Neumann data $g$ and $w$ be the solution with Neumann data $h$. According to the uniqueness result Theorem \ref{uRuniq} we have $ u = v+w$. We let $R_Q = \nontan { ( \nabla w )}$ and $ F_Q = \nontan {( \nabla v ) } $ so that (\ref{Shen1}) holds. We turn our attention to establishing (\ref{Shen2}) and (\ref{Shen3}). To establish (\ref{Shen2}), observe that the $H^1$-norm of $g$ satisfies the bound $$ \| g \|_{ H^1 (N)} \leq C \| f \| _ {L^p( 4Q)} \sigma (Q) ^ { 1/p'}. $$ With this, the estimate (\ref{Shen2}) follows from Theorem \ref{AtomicTheorem}. Now we turn to the estimate (\ref{Shen3}) for $ F_Q = \nontan {( \nabla w ) } $. We note that the Neumann data $h$ is constant on $ 4Q \cap N$. We define a maximal operator by taking the supremum over that part of the cone that is far from the boundary, $$ \nontan { ( \nabla w )_+}(x) = \sup _{ y \in \ntar x \setminus \ball x {Ar} } |\nabla w (y)| $$ where $A$ is to be chosen. A simple geometric argument gives that \begin{equation} \label{far} \nontan { ( \nabla w )_+}(x) \leq C -\!\!\!\!\!\!\int _{ 4Q} \nontan { ( \nabla w ) } \, d\sigma , \qquad x \in 2Q. \end{equation} The estimate for $ \nontan { ( \nabla w )_{Ar} } $ uses the local estimate for the mixed problem in Lemma \ref{newLocal} to conclude that \begin{eqnarray} \nonumber \left ( -\!\!\!\!\!\!\int _ { 2Q} { \nontan { ( \nabla w ) _ { Ar} }} ^ q \, d\sigma \right) ^ { 1/q} & \leq & C \left [ \|h\|_{L^ \infty ( 4Q)} + -\!\!\!\!\!\!\int _ { T( 3Q) } |\nabla w | \, d\sigma \right ] \\ & \leq & C \left [ \left( -\!\!\!\!\!\!\int_{4Q} |f|\, d\sigma \right) + -\!\!\!\!\!\!\int _ {4Q} \nontan { ( \nabla w ) } \, d \sigma \right ]. \label{near} \end{eqnarray} provided that the constant $A$ in the definition of $ \nontan{ (\nabla w )_+} $ is chosen sufficiently small. Recall that $ T(Q)$ was defined at the beginning of the proof of Lemma \ref{Whitney}. From the estimates (\ref{far}) and (\ref{near}), we conclude that \begin{equation} \label{New3} \left( -\!\!\!\!\!\!\int _{ 2Q} ( R_Q ) ^ q \, d\sigma \right) ^ { 1/q} \leq C \left[ -\!\!\!\!\!\!\int _ {4Q} |f| \, d\sigma + \left( -\!\!\!\!\!\!\int _{ 4Q} \nontan { ( \nabla w ) }\, d\sigma \right) ^ { 1/p } \right ]. \end{equation} We have $ \nontan {( \nabla w )} \leq \nontan { ( \nabla v ) } + \nontan {( \nabla u ) } $ and hence we may estimate the term involving $ \nontan { ( \nabla w )}$ by $$ -\!\!\!\!\!\!\int _ { 4Q} \nontan {( \nabla w ) } \, d\sigma \leq -\!\!\!\!\!\!\int _ { 4Q} \nontan {( \nabla u ) } \, d\sigma + -\!\!\!\!\!\!\int _ { 4Q} \nontan {( \nabla v ) } \, d\sigma \leq -\!\!\!\!\!\!\int _ { 4Q} \nontan {( \nabla u ) } \, d\sigma + C \left ( -\!\!\!\!\!\!\int _ { 4Q} |f| ^ p \, d\sigma \right ) ^ { 1/p} $$ where we have used Theorem \ref{HardyTheorem} to estimate the term involving $\nontan{(\nabla v)}$. Combining this with (\ref{New3}) gives (\ref{Shen3}). Applying the technique of Shen outlined above gives the $L^p$-estimate and thus we obtain the following theorem. \begin{theorem} Let $ \Omega$, $N$ and $D$ be a standard domain for the mixed problem and let $p$ satisfy $ 1 < p < p_0/2$ where $p_0$ is from Lemma \ref{RHEstimate}. Given data $f_N$ in $L^p(N)$, we may solve the $L^p$-mixed problem with Neumann data $f_N$ and Dirichlet data 0 and this solution satisfies the estimate $$ \| \nontan {( \nabla u )} \|_{ L^ p ( \partial \Omega )} \leq C \| f _N \| _{ L^ p (\partial \Omega )} . $$ The constant $C$ depends on the global character of the domain and the index $p$. \end{theorem} \section{Further questions} This work adds to our understanding of the mixed problem in Lipschitz domains. However, there are several avenues which are not yet explored. \begin{enumerate} \item Can we study the inhomogeneous mixed problem and obtain results similar to those of Fabes, Mendez and M. Mitrea \cite{FMM:1998} and I.~Mitrea and M.~Mitrea \cite{MM:2007}? \item Is there an extension to $p < 1$ as the work of Brown \cite{RB:1995a}? \item Can we study the mixed problem for more general decompositions of the boundary, $ \partial \Omega = D \cup N$? To what extent is the condition that the boundary between $D$ and $N$ be a Lipschitz graph needed? \item Can we extend these techniques to elliptic systems and higher order elliptic equations? \end{enumerate} \note { Index of notation. \begin{tabular}{rl} \sc Symbol & Meaning \rm \\ $\delta(x) $ & the distance from $x$ to the crease $\Lambda$ \\ $D$ & region where we specify Dirichlet data\\ $N$ & region where we specify Neumann data \\ $P$& $P$ operator \\ $\Xi$ & fundamental solution \\ $\ntar x $ & non-tangential approach region\\ $\Lambda $ & boundary between $N$ and $D$ \\ $\Omega $ & domain \\ $ \locdom x r $ & domain of size $r$ near a boundary point $x$. \\ $\dball x r $ & $\Omega \cap \ball x r $ \\ $T(Q) $ & in proof of Lemma \ref{Whitney} $T(Q) = \{ x \in \bar \Omega : \mathop{\rm dist}\nolimits(x, Q) < \mathop{\rm diam}\nolimits(Q)\} $ \end{tabular} } \def$'${$'$} \def$'${$'$} \def$'${$'$} \end{bibunit} \small \noindent \today \end{document}

\mathfrak{b}egin{document} \title{Higher indescribability and derived topologies} \author[Brent Cody]{Brent Cody} \address[Brent Cody]{ Virginia Commonwealth University, Department of Mathematics and Applied Mathematics, 1015 Floyd Avenue, PO Box 842014, Richmond, Virginia 23284, United States } \email[B. ~Cody]{bmcody@vcu.edu} \urladdr{http://www.people.vcu.edu/~bmcody/} \mathfrak{b}egin{abstract} We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{\kappa^+,\kappa^+}$-indescribability and ${\mathbb P}i^1_\xi$-indescribability of a cardinal $\kappa$ for all $\xi<\kappa^+$. In this context, universal ${\mathbb P}i^1_\xi$ formulas exist, there is a normal ideal associated to ${\mathbb P}i^1_\xi$-indescribability and the notions of ${\mathbb P}i^1_\xi$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given a regular cardinal $\mu$, we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's \cite{MR3894041} sequence $\langlengle\tau_\xi:\xi<\mu\mathop{\rm ran}glengle$ of derived topologies on $\mu$ to $\langlengle\tau_\xi:\xi<\mu^+\mathop{\rm ran}glengle$. Finally, we prove that for all $\xi<\mu^+$, if there is a stationary set of $\alpha<\mu$ that have a high enough degree of indescribability, then there are stationarily-many $\alpha<\mu$ that are nonisolated points in the space $(\mu,\tau_{\xi+1})$. \end{abstract} {\mathop{\rm sub}}jclass[2010]{Primary 03E55, 54A35; Secondary 03E05} \keywords{Derived topology, diagonal Cantor derivative, indescribable cardinals, stationary reflection} \maketitle \tableofcontents $\hat{\text{s}}$ection{Introduction}\langlebel{section_introduction} When working with certain large cardinals, set theorists often use reflection arguments. For example, if $\kappa$ is a measurable cardinal then it is inaccessible, and furthermore, there are normal measure one many $\alpha<\kappa$ which are inaccessible; we say that the inaccessibility of a measurable cardinal $\kappa$ \emph{reflects} below $\kappa$. In this article we consider generalizations of this kind of reflection so that we may reflect attributes of large cardinals that are expressible by formulas whose lengths can be strictly longer than the large cardinal under consideration. We will see that in many cases, if $\kappa$ is a measurable cardinal and $\kappa$ has some property, which is expressible by a formula $\varphi$ whose length is less than $\kappa^+$, then the set of $\alpha<\kappa$ such that a canonically defined \emph{restricted} version of this formula $\varphi\mathrm{|}^\kappa_\alpha$ is true of $\alpha$, is normal measure one. We use this kind of generalized reflection to define the $L_{\kappa^+,\kappa^+}$-indescribability and ${\mathbb P}i^1_\xi$-indescribability of a cardinal $\kappa$ for all $\xi<\kappa^+$, thus generalizing the notions of indescribability previously considered in \cite{MR3894041}. Let us note that a precursor to this type of reflection principle was studied by Sharpe and Welch (see \cite[Definition 3.21]{MR2817562}). We then use our notion of ${\mathbb P}i^1_\xi$-indescribability to establish the nondiscreteness of certain topological spaces which are generalizations of the derived topologies considered in \cite{MR3894041}, and which are defined by using a diagonal version of the Cantor derivative operator (see the definition of $\tau_\xi$ and $d_\xi$ in Section \ref{section_higher_derived_topologies} and see Remark \ref{remark_example} for a simple case). We believe the results presented below will open up new avenues for future work in many directions. For example, in order to define the restriction of formulas (Definition \ref{definition_restriction} and Definition \ref{definition_restriction_2}) and then to establish basic properties of ${\mathbb P}i^1_\xi$-indescribability, we introduce the \emph{canonical reflection functions} (see Definition \ref{definition_canonical_reflection_functions} and Section \ref{section_canonical_reflection_functions}), which are interesting in their own right and will likely have applications in areas far removed from this paper. We also expect that the notion of restriction of formulas defined below will have applications in the study of infinitary logics and model theory. Note that \cite{MR457191} and \cite{MR360274} both contain results involving a notion of restriction of $L_{\infty,\omega}$ formulas to countable sets; we suspect that these results, as well as other results in this area \cite{MR457191}, will have analogues involving our notion of restriction. Furthermore, let us note that the notion of higher ${\mathbb P}i^1_\xi$-indescribability also allows for a finer analysis of the large cardinal hierarchy as in \cite{MR4206111} and \cite{cody_holy_2022}. Finally, the notions and results contained herein, particularly those on higher $\xi$-stationarity and higher derived topologies (see Section \ref{section_higher_derived_topologies}), should also allow for generalizations of many results concerning iterated stationary reflection properties and characterizations of indescribability in G\"{o}del's constructible universe (see \cite{MR1029909}, \cite{MR3416912}, \cite{MR3894041} and \cite{MR4094556}). Before we discuss the restriction of formulas in general, let us give some examples. For cardinals $\kappa$ and $\mu$, recall that $L_{\kappa,\mu}$ denotes the infinitary logic which allows for conjunctions of $<\kappa$-many formulas that together contain $<\mu$-many free variables and quantification (universal and existential) over $<\mu$-many variables at once. If $\kappa$ is a measurable cardinal and $\varphi$ is any sentence in the $L_{\kappa,\kappa}$ language of set theory such that $V_\kappa\models\varphi$, then the set of $\alpha<\kappa$ such that $V_\alpha\models\varphi$ is normal measure one in $\kappa$. On the other hand, for any cardinal $\kappa$ there are $L_{\kappa^+,\kappa^+}$ sentences which are true in $V_\kappa$ and false in $V_\alpha$ for all $\alpha<\kappa$. For example, for each $\eta<\kappa$ there is a natural $L_{\kappa^+,\kappa^+}$ formula $\chi_\eta(x)$ such that for all $\alpha\leq\kappa$ and all $a\in V_\alpha$ we have $V_\alpha\models\chi_\eta(a)$ if and only if $a$ is an ordinal and $a$ has order type at least $\eta$. Now $\chi=\mathfrak{b}igwedge_{\eta<\kappa}\exists x\chi_\eta(x)$ is an $L_{\kappa^+,\kappa^+}$ sentence such that $V_\kappa\models\chi$, and yet there is no $\alpha<\kappa$ such that $V_\alpha\models\chi$. However, the \emph{restriction} $\chi\mathrm{|}^\kappa_\alpha:=\mathfrak{b}igwedge_{\eta<\alpha}\exists x\chi_\eta(x)$ of $\chi$ to $\alpha$ holds in $V_\alpha$ for all $\alpha<\kappa$. In what follows we will define the restriction of $L_{\kappa^+,\kappa^+}$ formulas in generality, which will allow for similar reflection results. However, the main focus of this article is on a different kind of infinitary formula. Generalizing the notions of ${\mathbb P}i^1_n$ and $\mathbb{S}igma^1_n$ formulas (see \cite{MR0281606} or \cite[Section 0]{MR1994835}), Bagaria \cite{MR3894041} defined the classes of ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas for all ordinals $\xi$. For example, if $\xi$ is a limit ordinal, a formula is ${\mathbb P}i^1_\xi$ if it is of the form $\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta$ where each $\varphi_\zeta$ is ${\mathbb P}i^1_\zeta$. A formula is $\mathbb{S}igma^1_{\xi+1}$ if it is of the form $\exists X\psi$ where $\psi$ is ${\mathbb P}i^1_\xi$. Throughout this article, first-order variables will be written as lower case letters and second-order variables will be written as upper case. For more on the definition of ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas, see Section \ref{section_definition_pi1xi}. Given a cardinal $\kappa$, Bagaria defined a set $S{\mathop{\rm sub}}seteq\kappa$ to be ${\mathbb P}i^1_\xi$-indescribable in $\kappa$ if and only if for all $A{\mathop{\rm sub}}seteq V_\kappa$ and all ${\mathbb P}i^1_\xi$ sentences $\varphi$, if $(V_\kappa,\in,A)\models\varphi$ then there is an $\alpha\in S$ such that $(V_\alpha,\in,A\cap V_\alpha)\models\varphi$. Bagaria pointed out that, using his definition, no cardinal $\kappa$ can be ${\mathbb P}i^1_\kappa$-indescribable because the ${\mathbb P}i^1_\kappa$ sentence $\chi$ defined above is true in $V_\kappa$ but false in $V_\alpha$ for all $\alpha<\kappa$. We introduce a modification of Bagaria's notion of ${\mathbb P}i^1_\xi$-indescribability which allows for a cardinal $\kappa$ to be ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\kappa^+$. Given a cardinal $\kappa$ and an ordinal $\xi<\kappa^+$, we say that a set $S{\mathop{\rm sub}}seteq\kappa$ is \emph{${\mathbb P}i^1_\xi$-indescribable in $\kappa$} if and only if for all ${\mathbb P}i^1_\xi$ sentences $\varphi$ (with first and second-order parameters from $V_\kappa$), if $V_\kappa\models\varphi$\footnote{Note that $\varphi$ may involve finitely-many second-order parameters $A_1,\ldots,A_n{\mathop{\rm sub}}seteq V_\kappa$, and when we write $V_\kappa\models\varphi$ we mean $(V_\kappa,\in,A_1,\ldots,A_n)\models\varphi$. Since this abbreviated notion will not cause confusion and greatly simplifies notation, we will use it throughout the paper without further comment.} then there is some $\alpha\in S$ such that a canonically defined restriction of $\varphi$ is true in $V_\alpha$, which we express by writing $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$ (see Definition \ref{definition_indescribability} for details). In order to define the notions of restriction of $L_{\kappa^+,\kappa^+}$ formulas and restriction of ${\mathbb P}i^1_\xi$ formulas, we use a sequence of functions $\langleF^\kappa_\xi:\xi<\kappa^+\rangle$ we call the \emph{sequence of canonical reflection functions at $\kappa$}, which is part of the set theoretic folklore and which is closely related to the sequence $\langlef^\kappa_\xi:\xi<\kappa^+\rangle$ of canonical functions at $\kappa$. Before defining the canonical reflection functions, let us recall some basic properties of canonical functions. Given a regular cardinal $\kappa$, the ordering defined on $^\kappa\mathop{{\rm ORD}}$ by letting $f<g$ if and only if $\{\alpha<\kappa: f(\alpha)<g(\alpha)\}$ contains a club, is a well-founded partial ordering. The Galvin-Hajnal \cite{MR376359} norm $\|f\|$ of such a function is defined to be the rank $f$ in the relation $<$. For each $\xi<\kappa^+$, there is a \emph{canonical} function $f^\kappa_\xi:\kappa\to\kappa$ of norm $\xi$, in the sense that $\|f^\kappa_\xi\|=\xi$ and whenever $\|h\|=\xi$ the set $\{\alpha<\kappa: f^\kappa_\xi(\alpha)\leq h(\alpha)\}$ contains a club (see \cite[Page 99]{MR2768680}). For concreteness, we will use the following definition of $f^\kappa_\xi$ for $\xi<\kappa^+$. If $\xi<\kappa$ we let $f^\kappa_\xi:\kappa\to\kappa$ be the function with constant value $\xi$. If $\kappa\leq\xi<\kappa^+$ we fix a bijection $b_{\kappa,\xi}:\kappa\to\xi$ and define $f^\kappa_\xi$ by letting $f^\kappa_\xi(\alpha)=\mathop{\rm ot}\nolimits(b_{\kappa,\xi}[\alpha])$ for all $\alpha<\kappa$. For convenience, we take $b_{\kappa,\kappa}$ to be the identity function $\mathop{\rm id}_\kappa:\kappa\to\kappa$, which implies that $f^\kappa_\kappa=\mathop{\rm id}_\kappa$. It is easy to see that for all $\zeta<\xi<\kappa^+$ we have $f^\kappa_\zeta<f^\kappa_\xi$ and that $f^\kappa_\xi$ is a canonical function of norm $\xi$. The sequence $\vec{f}=\langlef^\kappa_\xi:\xi<\kappa^+\rangle$ is sometimes referred to as the sequence of canonical functions at $\kappa$. Although, this terminology is slightly misleading as the canonical functions are only well-defined modulo the nonstationary ideal. \mathfrak{b}egin{definition}\langlebel{definition_canonical_reflection_functions} Suppose $\kappa$ is a regular cardinal. For each $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ let $b_{\kappa,\xi}:\kappa\to\xi$ be a bijection. We define the corresponding sequence of \emph{canonical reflection functions $\vec{F}=\langleF^\kappa_\xi:\xi<\kappa^+\rangle$ at $\kappa$} where $F^\kappa_\xi:\kappa\to P_\kappa\kappa^+$ for each $\xi<\kappa^+$ as follows. \mathfrak{b}egin{enumerate} \item For $\xi<\kappa$ we let $F^\kappa_\xi(\alpha)=\xi$ for all $\alpha<\kappa$. \item For $\kappa\leq\xi<\kappa^+$ we let $F^\kappa_\xi(\alpha)=b_{\kappa,\xi}[\alpha]$ for all $\alpha<\kappa$. \end{enumerate} For each $\xi<\kappa^+$ and $\alpha<\kappa$ we let $\pi^\kappa_{\xi,\alpha}:F^\kappa_\xi(\alpha)\to f^\kappa_\xi(\alpha)$ be the transitive collapse of $F^\kappa_\xi(\alpha)$. \end{definition} Notice that for all $\xi<\kappa^+$ we have $f^\kappa_\xi(\alpha)=\mathop{\rm ot}\nolimits(F^\kappa_\xi(\alpha))$ by definition. It is not difficult to see that for $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$, the $\xi^{th}$ canonical reflection function $F^\kappa_\xi$ is independent, modulo the nonstationary ideal, of which bijection $b_{\kappa,\xi}:\kappa\to\xi$ is used in its definition. That is, if $b_{\kappa,\xi}^1:\kappa\to\xi$ and $b_{\kappa,\xi}^2:\kappa\to\xi$ are two bijections then the set $\{\alpha<\kappa: b_{\kappa,\xi}^1[\alpha]=b_{\kappa,\xi}^2[\alpha]\}$ contains a club. In Section \ref{section_canonical_reflection_functions}, we establish many basic structural properties of the canonical reflection functions which will be used later in the paper. A particularly useful application of canonical functions \cite[Proposition 2.34]{MR2768692} is that that the $\xi^{th}$ canonical function at a regular cardinal $\kappa$ represents the ordinal $\xi$ in any generic ultrapower by any normal ideal on $\kappa$. An easy result below (see Proposition \ref{proposition_useful_object}) shows that whenever $I$ is a normal ideal on $\kappa$, $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic and $j:V\to V^\kappa/G{\mathop{\rm sub}}seteq V[G]$ is the corresponding generic ultrapower embedding, the $\xi^{th}$ canonical reflection function $F^\kappa_\xi$ represents $j"\xi$ in the generic ultrapower, that is, $j(F^\kappa_\xi)(\kappa)=j"\xi$. In Section \ref{section_definition_pi1xi}, given a regular cardinal $\kappa$, we review the definitions of ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas over $V_\kappa$; when we say that $\varphi$ is ${\mathbb P}i^1_\xi$ \emph{over} $V_\kappa$ we mean that $\varphi$ is ${\mathbb P}i^1_\xi$ in Bagaria's sense, but $\varphi$ is also allowed to have any number of first-order parameters from $V_\kappa$ and finitely-many second-order parameters from $V_\kappa$ (see Definition \ref{definition_over}). In Definition \ref{definition_restriction}, we use canonical reflection functions to define the notion of restriction of ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas by transfinite induction on $\xi<\kappa^+$. For example, if $\varphi=\varphi(X_1,\ldots,X_m,A_1,\ldots, A_n)$ is a ${\mathbb P}i^1_\xi$ formula over $V_\kappa$ and $\xi<\kappa$, then we define \[\varphi\mathrm{|}^\kappa_\alpha=\varphi(X_1,\ldots,X_m,A_1\cap V_\alpha,\ldots,A_n\cap V_\alpha).\] As another example, suppose $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ and $\xi$ is a limit ordinal. If \[\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta\] is a ${\mathbb P}i^1_\xi$ formula and $\alpha<\kappa$, then we define \[\varphi\mathrm{|}^\kappa_\alpha=\mathfrak{b}igwedge_{\zeta<f^\kappa_\xi(\alpha)}\varphi_{(\pi^\kappa_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^\kappa_\alpha\] provided that this formula is a ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ formula over $V_\alpha$. As a consequence of this definition, it follows that there is a club $C$ in $\kappa$ such that for all regular $\alpha\in C$, $\varphi$ is a ${\mathbb P}i^1_\kappa$ formula over $V_\kappa$ and $\varphi\mathrm{|}^\kappa_\alpha=\mathfrak{b}igwedge_{\zeta<\alpha}\varphi_\alpha$. One nice feature of our definition of restriction is that it leads to a convenient way to represent ${\mathbb P}i^1_\xi$ formulas in normal generic ultrapowers. Suppose $\kappa$ is weakly Mahlo, $I$ is a normal ideal on $\kappa$ and $\varphi$ is a ${\mathbb P}i^1_\xi$ formula over $V_\kappa$ for some $\xi<\kappa^+$. Then, a result of \cite{cody_holy_2022} (see Lemma \ref{lemma_represent} below) shows that whenever $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic over $V$ and $j:V\to V^\kappa/G$ is the corresponding generic ultrapower embedding, we have $j({\mathbb P}hi)(\kappa)=\varphi$ where ${\mathbb P}hi$ is the function with domain $\kappa$ defined by ${\mathbb P}hi(\alpha)=\varphi\mathrm{|}^\kappa_\alpha$. For a given cardinal $\kappa$ and ordinal $\xi<\kappa^+$, in Definition \ref{definition_indescribability} we say that $S{\mathop{\rm sub}}seteq\kappa$ is \emph{${\mathbb P}i^1_\xi$-indescribable} if and only if for all ${\mathbb P}i^1_\xi$ sentences $\varphi$ over $V_\kappa$, whenever $V_\kappa\models\varphi$ there must be an $\alpha\in S$ such that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$.\footnote{Sharpe and Welch \cite[Definition 3.21]{MR2817562} extended the notion of ${\mathbb P}i^1_n$-indescribability of a cardinal $\kappa$ where $n<\omega$ to that of ${\mathbb P}i^1_\xi$-indescribability where $\xi<\kappa^+$ by demanding that the existence of a winning strategy for a particular player in a certain finite game played at $\kappa$ implies that the same player has a winning strategy in the analogous game played at some cardinal less than $\kappa$. The relationship between their notion and the one defined here is not known.} Our last result in Section \ref{section_definition_pi1xi} states that if $\kappa$ is a measurable cardinal then $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\kappa^+$, and furthermore, the set of $\alpha<\kappa$ such that $\alpha$ is ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta<\alpha^+$ is normal measure one in $\kappa$. In Section \ref{section_other}, given a regular cardinal $\kappa$ and an $L_{\kappa^+,\kappa^+}$ formula $\varphi$ in the language of set theory, we use the canonical reflection functions at $\kappa$ to define a notion of restriction $\varphi\mathrm{|}^\kappa_\alpha$ by induction on subformulas, for all $\alpha\leq\kappa$. For a regular cardinal $\kappa$, we say that a set $S{\mathop{\rm sub}}seteq\kappa$ is \emph{$L_{\kappa^+,\kappa^+}$-indescribable} if and only if for all $L_{\kappa^+,\kappa^+}$ sentences in the language of set theory with $V_\kappa\models\varphi$ there is an $\alpha<\kappa$ such that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$. Proposition \ref{proposition_Lkappa+} states that if $\kappa$ is a measurable cardinal, then $\kappa$ is $L_{\kappa^+,\kappa^+}$-indescribable and furthermore, the set of regular cardinals $\alpha<\kappa$ that are $L_{\alpha^+,\alpha^+}$-indescribable is normal measure one in $\kappa$. Generalizing the results of L\'evy \cite{MR0281606} and Bagaria \cite{MR3894041} on universal formulas, in Section \ref{section_universal}, we establish the existence of universal ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas at a regular cardinal $\kappa$ for all $\xi<\kappa^+$ in an appropriate sense. Using universal formulas, we prove Theorem \ref{theorem_normal_ideal}, which states that if $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable where $\xi<\kappa^+$, then the collection \[{\mathbb P}i^1_\xi(\kappa)=\{X{\mathop{\rm sub}}seteq\kappa:\text{$X$ is not ${\mathbb P}i^1_\xi$-indescribable}\}\] is a nontrivial normal ideal on $\kappa$. In Section \ref{section_hierarchy}, again using the existence of universal ${\mathbb P}i^1_\xi$ formulas discussed above, we prove Theorem \ref{theorem_expressing_indescribability}, which states that given a regular cardinal $\kappa$ and $\xi<\kappa^+$, the ${\mathbb P}i^1_\xi$-indescribability of a set $S{\mathop{\rm sub}}seteq\kappa$ is, in an appropriate sense, expressible by a ${\mathbb P}i^1_{\xi+1}$ formula. We then prove two hierarchy results for ${\mathbb P}i^1_\xi$-indescribability. For example, as a consequence of these results, if $\kappa$ is $\kappa+n+1$-indescribable, where $n<\omega$, then the set of $\alpha<\kappa$ which are $\alpha+n$-indescribable is in the filter ${\mathbb P}i^1_{\kappa+n+1}(\kappa)^*$. More generally, our first hierarchy result, Corollary \ref{corollary_hierarchy}, states that if $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable where $\xi<\kappa^+$ and $\zeta<\xi$, then the set of $\alpha<\kappa$ which are ${\mathbb P}i^1_{f^\kappa_\zeta(\alpha)}$-indescribable is in the filter ${\mathbb P}i^1_\xi(\kappa)^*$. Our second hierarchy result, Corollary \ref{corollary_proper}, states that if $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable where $\xi<\kappa^+$, then for all $\zeta<\xi$ we have ${\mathbb P}i^1_\zeta(\kappa){\mathop{\rm sub}}setneq{\mathbb P}i^1_\xi(\kappa)$. The proofs of these two hierarchy results require several lemmas which are interesting in their own right. For example, Proposition \ref{proposition_double_restriction} state that for any weakly Mahlo cardinal $\kappa$ and ordinal $\xi<\kappa^+$, if $\varphi$ is any ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula then there is a club $C{\mathop{\rm sub}}seteq\kappa$ such that for all regular $\alpha\in C$, the set of $\mathfrak{b}eta<\alpha$ for which \[(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta=\varphi\mathrm{|}^\kappa_\mathfrak{b}eta\] is club in $\alpha$. Recall that, by results of Sun \cite{MR1245524} and Hellsten \cite{MR2026390}, one can characterize ${\mathbb P}i^1_n$-indescribable subsets of a ${\mathbb P}i^1_n$-indescribable cardinal $\kappa$ by using a natural base for the filter ${\mathbb P}i^1_n(\kappa)^*$ dual to ${\mathbb P}i^1_n(\kappa)$. For a regular cardinal $\kappa$, a set $C{\mathop{\rm sub}}seteq\kappa$ is a \emph{${\mathbb P}i^1_0$-club in $\kappa$} if it is club in $\kappa$. We say that $C{\mathop{\rm sub}}seteq\kappa$ is \emph{${\mathbb P}i^1_{n+1}$-club} in $\kappa$, where $n<\omega$, if it is ${\mathbb P}i^1_n$-indescribable in $\kappa$ and whenever $C\cap\alpha$ is ${\mathbb P}i^1_n$-indescribable in $\alpha$ we have $\alpha\in C$. Then, if $\kappa$ is ${\mathbb P}i^1_n$-indescribable, a set $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_n$-indescribable if and only if $S\cap C\neq\varnothing$ for all ${\mathbb P}i^1_n$-clubs $C{\mathop{\rm sub}}seteq\kappa$. This result is due to Sun \cite{MR1245524} for $n=1$ and to Hellsten \cite{MR2026390} for $n<\omega$. In Section \ref{section_higher_xi_clubs}, we generalize this to ${\mathbb P}i^1_\xi$-indescribable subsets of ${\mathbb P}i^1_\xi$-indescribable cardinals for all $\xi<\kappa^+$. That is, for all $\xi<\kappa^+$, we introduce a notion of ${\mathbb P}i^1_\xi$-club subset of $\kappa$ such that if $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable then a set $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable if and only if $S\cap C\neq\varnothing$ for all ${\mathbb P}i^1_\xi$-clubs $C{\mathop{\rm sub}}seteq\kappa$. For more results involving ${\mathbb P}i^1_\xi$-clubs, one should consult \cite{MR3985624}, \cite{MR4050036}, \cite{MR4230485} and \cite{MR4082998}. Finally, in Section \ref{section_higher_derived_topologies}, we generalize some of the results of Bagaria \cite{MR3894041} on derived topologies on ordinals. Given a nonzero ordinal $\mathfrak{d}elta$, Bagaria defined a transfinite sequence of topologies $\langle\tau_\xi:\xi\in\mathop{{\rm ORD}}\rangle$ on $\mathfrak{d}elta$, called the \emph{derived topologies on $\mathfrak{d}elta$}, and proved---using the definitions of \cite{MR3894041}---that if there is an $\alpha<\mathfrak{d}elta$ which is ${\mathbb P}i^1_\xi$-indescribable then the $\tau_{\xi+1}$ topology on $\mathfrak{d}elta$ is non-discrete. However, using the definitions of \cite{MR3894041}, $\alpha$ can be ${\mathbb P}i^1_\xi$-indescribable only if $\xi<\alpha$. Thus, Bagaria obtained the non-discreteness of the $\tau_\xi$ topologies on $\mathfrak{d}elta$ only for $\xi<\mathfrak{d}elta$. Given a regular cardinal $\mu$, in Section \ref{section_higher_derived_topologies}, using \emph{diagonal Cantor derivatives}, we present a natural extension of Bagaria's notion of derived topologies on $\mu$ by defining a transfinite sequence of topologies $\langle\tau_\xi:\xi<\mu^+\rangle$ on $\mu$ such that for $\xi<\mu$ our $\tau_\xi$ is the same as that of \cite{MR3894041} and Bagaria's conditions for the nondiscreteness of the topologies $\tau_{\xi+1}$ for $\xi<\mu$ can be generalized to all $\xi<\mu^+$ (see Theorem \ref{theorem_xi_s_nonisolated} and Corollary \ref{corollary_nondiscreteness_from_indescribability}). \mathfrak{b}egin{remark}\langlebel{remark_example} Let us describe the simplest of the new topologies introduced in this article. If $\langle\tau_\xi:\xi<\mu\rangle$ is Bagaria's sequence of derived topologies on a regular $\mu$, we define $d_\mu:P(\mu)\to P(\mu)$ by letting \[d_\mu(A)=\{\alpha<\mu:\text{$\alpha$ is a limit point of $A$ in the $\tau_\alpha$ topology on $\mu$}\}.\] We then define a new topology $\tau_\mu$ declaring $C{\mathop{\rm sub}}seteq\mu$ to be closed in the space $(\mu,\tau_\mu)$ if and only if $d_\mu(C){\mathop{\rm sub}}seteq C$. That is, we let $U\in\tau_\mu$ if and only if $d_\mu(\mu$\hat{\text{s}}$etminus U){\mathop{\rm sub}}seteq\mu$\hat{\text{s}}$etminus U$ for $U{\mathop{\rm sub}}seteq\mu$. \end{remark} \mathfrak{b}egin{comment} \cite{MR4206111} \cite{MR3416912} \cite{MR4050036} \cite{MR2817562} \cite{BrickhillWelch} \cite{Brickhill:Thesis} \cite{MR4081067} \cite{MR1245524} \cite{MR2252250} \cite{MR2653962} \cite{MR2026390} \cite{MR4082998} \cite{MR0539973} \cite{MR457191} \cite{MR4094556} \end{comment} $\hat{\text{s}}$ection{Canonical reflection functions}\langlebel{section_canonical_reflection_functions} In this section we establish the basic properties of the canonical reflection functions at a regular cardinal. Although some of these results are folklore, we include proofs for the reader's convenience. Many of the proofs in the current section will establish that certain sets defined using canonical reflection functions are in the club filter on a given regular cardinal $\kappa$. These results will be established by using generic ultrapower embeddings\footnote{The author would like to thank Peter Holy for suggesting the use of generic ultrapowers in the arguments of the current section.}; some background material on generic ultrapowers may be found in \cite{MR2768692}, but we will only that which is summarized here. Recall that if $\kappa$ is a regular cardinal, $I$ is a normal ideal on $\kappa$ and $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic over $V$, then, working in the forcing extension $V[G]$ there is a canonical $V$-normal $V$-ultrafilter $U_G{\mathop{\rm sub}}seteq P(\kappa)$ obtained from $G$ such that $U_G$ extends the filter $I^*$ dual to $U$ and we may form the corresponding generic ultrapower $j:V\to V^\kappa/U_G{\mathop{\rm sub}}seteq V[G]$. Further recall that the critical point of $j$ is $\kappa$ and equals the equivalence class of the identity function $\mathop{\rm id}:\kappa\to\kappa$. Thus, for all $X\in P(\kappa)^V$ we have $X\in U$ if and only if $\kappa\in j(X)$. Furthermore, the ultrapower $V^\kappa/U_G$ is wellfounded up to $(\kappa^+)^V$, $H(\kappa^+){\mathop{\rm sub}}seteq V^\kappa/U_G$ and when $\kappa$ is inaccessible we have $H(\kappa)=H(\kappa)^{V^\kappa/U_G}$. As is standard practice, in what follows we will often write $V^\kappa/G$ to mean $V^\kappa/U_G$. The following two propositions will be used throughout the article. \mathfrak{b}egin{proposition}\langlebel{proposition_framework} Suppose $\kappa$ is a regular uncountable cardinal and $S{\mathop{\rm sub}}seteq\kappa$. Then $S$ contains a club subset of $\kappa$ if and only if whenever $G$ is generic for $P(\kappa)/{\mathop{\rm NS}}_\kappa$ it follows that $\kappa\in j(S)$ where $j:V\to V^\kappa/G$ is the generic ultrapower embedding obtained from $G$. \end{proposition} \mathfrak{b}egin{proposition}\langlebel{proposition_framework2} The following are equivalent when $\kappa$ is a regular uncountable cardinal\footnote{Notice that when the set of regular cardinals less than $\kappa$ is not stationary in $\kappa$, i.e. when $\kappa$ is not weakly Mahlo, then both (1) and (2) hold trivially. In later sections, when we apply Proposision \ref{proposition_framework2}, and the results derived from it in the current section, $\kappa$ will in fact be weakly Mahlo.} and $E{\mathop{\rm sub}}seteq\kappa$. \mathfrak{b}egin{enumerate} \item There is a club $C{\mathop{\rm sub}}seteq\kappa$ such that for all regular uncountable $\alpha\in C$ we have $\alpha\in E$. \item Whenever $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ is generic over $V$ such that $\kappa$ is regular in $V^\kappa/G$ and $j:V\to V^\kappa/G$ is the corresponding generic ultrapower embedding, we have $\kappa\in j(E)$. \end{enumerate} \end{proposition} \mathfrak{b}egin{proof} It is trivial to see that (1) implies (2). If (1) is false then the set $S$ of regular cardinals in $\kappa$\hat{\text{s}}$etminus E$ is stationary in $\kappa$. Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic over $V$ with $S\in G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. Then $\kappa\in j(S)$, which implies $\kappa$ is regular in $V^\kappa/G$ and $\kappa\notin j(E)$ contradicting (2). \end{proof} The next result shows that, for regular $\kappa$, the $\xi^{th}$ canonical reflection function $F^\kappa_\xi$ (see Definition \ref{definition_canonical_reflection_functions}) represents a useful object in any generic ultrapower obtained from a normal ideal on $\kappa$. \mathfrak{b}egin{proposition}\langlebel{proposition_useful_object} Suppose $\kappa$ is a regular cardinal and $I$ is a normal ideal on $\kappa$. Let $G$ be generic for $P(\kappa)/I$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. Then, for all $\xi<\kappa^+$, the $\xi^{th}$ canonical reflection function $F^\kappa_\xi$ represents $j"\xi$ in the generic ultrapower, that is, $j(F^\kappa_\xi)(\kappa)=j"\xi$. \end{proposition} \mathfrak{b}egin{proof} Let $j:V\to V^\kappa/G$ be the generic ultrapower obtained from a generic filter $G{\mathop{\rm sub}}seteq P(\kappa)/I$ over $V$. Since $\mathop{\rm crit}(j)=\kappa$, it is easy to see that for $\xi\leq\kappa$ we have $j(F^\kappa_\xi)(\kappa)=j"\xi$. Now suppose $\kappa<\xi<\kappa^+$ and let $b_{\kappa,\xi}:\kappa\to\xi$ be the bijection such that $F^\kappa_\xi(\alpha)=b_{\kappa,\xi}[\alpha]$ for all $\alpha<\kappa$. By elementarity, $j(b_{\kappa,\xi}):j(\kappa)\to j(\xi)$ is a bijection in $M$ and $j(b_{\kappa,\xi})(\alpha)=j(b_{\kappa,\xi}(\alpha))$ for all $\alpha<\kappa$. Thus, $j(F^\kappa_\xi)(\kappa)=j(b_{\kappa,\xi})[\kappa]=j"\xi$. \end{proof} \mathfrak{b}egin{corollary} Suppose $U$ is a normal measure on $\kappa$ and $j:V\to M$ is the corresponding ultrapower embedding. For all $\xi<\kappa^+$, the $\xi^{th}$ canonical reflection function $F^\kappa_\xi$ represents $j"\xi$ in the ultrapower, that is, $j(F^\kappa_\xi)(\kappa)=j"\xi$. \end{corollary} Next we show that at least some of the canonical reflection functions at a regular $\kappa$ are, in fact, canonical; in Remark \ref{remark_not_canonical}, we show that this partial canonicity result is the best possible. \mathfrak{b}egin{lemma}\langlebel{lemma_canonicity} Suppose $\kappa$ is regular.\mathfrak{b}egin{enumerate} \item For all $\xi<\kappa^+$ the set $\{\alpha<\kappa: F^\kappa_\zeta(\alpha){\mathop{\rm sub}}setneq F^\kappa_\xi(\alpha)\}$ contains a club subset of $\kappa$ for all $\zeta<\xi$. \item If $\xi<\kappa^+$ is a limit ordinal then the set \[\{\alpha<\kappa: F^\kappa_\xi(\alpha)=\mathfrak{b}igcup_{\zeta\in F^\kappa_\xi(\alpha)} F^\kappa_\zeta(\alpha)\}\] contains a club subset of $\kappa$. \item If $\xi<\kappa^+$ is a limit ordinal the function $F^\kappa_\xi$ is canonical in the sense that whenever $F:\kappa\to P_\kappa\kappa^+$ is a function such that for all $\zeta<\xi$ the set $\{\alpha<\kappa: F^\kappa_\zeta(\alpha){\mathop{\rm sub}}seteq F(\alpha)\}$ contains a club, then the set $\{\alpha<\kappa: F^\kappa_\xi(\alpha){\mathop{\rm sub}}seteq F(\alpha)\}$ contains a club subset of $\kappa$. \end{enumerate} \end{lemma} \mathfrak{b}egin{proof} For (1), suppose $\zeta<\xi<\kappa^+$ and let $C=\{\alpha<\kappa: F^\kappa_\zeta(\alpha){\mathop{\rm sub}}setneq F^\kappa_\xi(\alpha)\}$. Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic over $V$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower. By Proposition \ref{proposition_useful_object}, $\kappa\in j(C)$ and thus by Proposition \ref{proposition_framework} we see that $C$ contains a club subset of $\kappa$. Similarly, for (2), suppose $\xi$ is a limit ordinal, let $C=\{\alpha<\kappa: F^\kappa_\xi(\alpha)=\mathfrak{b}igcup_{\zeta\in F^\kappa_\xi(\alpha)}F^\kappa_\zeta(\alpha)\}$ and let $j:V\to V^\kappa/G$ be the generic ultrapower obtained by forcing with $P(\kappa)/{\mathop{\rm NS}}_\kappa$. Working in $V^\kappa/G$, if $\zeta$ is an ordinal less than $j(\kappa)^+$, we let $\overline{F}^{j(\kappa)}_\zeta$ denote the $\zeta$-th cannonical reflection function at $j(\kappa)$. For each $\zeta<\xi$ we let $j(\langleF^\kappa_\zeta(\alpha):\alpha<\kappa\rangle)=\langle\overline F^{j(\kappa)}_{j(\zeta)}(\alpha):\alpha<j(\kappa)\rangle$. Notice that \mathfrak{b}egin{align*}j(C)&=\{\alpha<\kappa: j(F^\kappa_\xi)(\alpha)=\mathfrak{b}igcup_{\zeta\in j(F^\kappa_\xi)(\alpha)}\overline F^{j(\kappa)}_\zeta(\alpha)\}\\ &=\{\alpha<\kappa: j"\xi=\mathfrak{b}igcup_{\zeta\in j"\xi}\overline{F}^{j(\kappa)}_\zeta(\alpha)\} \end{align*} For each $\zeta\in j"\xi$ we have $\overline F^{j(\kappa)}_\zeta(\kappa)=\overline F^{j(\kappa)}_{j(j^{-1}(\zeta))}(\kappa)=j(F^\kappa_{j^{-1}(\zeta)})(\kappa)=j"(j^{-1}(\zeta))=(j"\xi)\cap\zeta$, it follows that $\kappa\in j(C)$. For (3), suppose $\xi<\kappa^+$ is a limit and let $F$ be as in the statement of the lemma. By assumption, if $j:V\to V^\kappa/G$ is any generic ultrapower obtained by forcing with $P(\kappa)/{\mathop{\rm NS}}_\kappa$, then $j(F^\kappa_\zeta)(\kappa)=j"\zeta{\mathop{\rm sub}}seteq j(F)(\kappa)$ for all $\zeta<\xi$. By (2), we know that $j(F^\kappa_\xi)(\kappa)=j"\xi=\mathfrak{b}igcup_{\zeta<\xi}j"\zeta$ and hence $j(F^\kappa_\xi)(\kappa){\mathop{\rm sub}}seteq j(F)(\kappa)$. \end{proof} \mathfrak{b}egin{remark}\langlebel{remark_not_canonical} Let us point out that Lemma \ref{lemma_canonicity}(3) does not hold if $\xi<\kappa^+$ is a successor ordinal. For example, supose $\xi=\kappa+1$ and $F:\kappa\to P_\kappa\kappa^+$ is defined by $F(\alpha)=\alpha$. Let $j:V\to V^\kappa/G$ be any generic ultrapower obtained by forcing with $P(\kappa)/{\mathop{\rm NS}}_\kappa$. Since $j(F)(\kappa)=\kappa$ and $j"(\kappa+1)=\kappa\cup\{j(\kappa)\}$ we see that $\{\alpha<\kappa: F^\kappa_\kappa(\alpha){\mathop{\rm sub}}seteq F(\alpha)\}$ contains a club in $\kappa$ and $\{\alpha<\kappa: F^\kappa_{\kappa+1}(\alpha){\mathop{\rm sub}}seteq F(\alpha)\}$ is nonstationary in $\kappa$. \end{remark} The following lemma shows that the canonical reflection functions at a regular cardinal satisfy a natural kind of coherence property. \mathfrak{b}egin{lemma}\langlebel{lemma_coherence} Suppose $\kappa$ is a regular cardinal and $\xi<\kappa^+$ is a limit ordinal. Let $\pi^\kappa_{\xi,\alpha}:F^\kappa_\xi(\alpha)\to f^\kappa_\xi(\alpha)$ be the transitive collapse of $F^\kappa_\xi(\alpha)$ for each $\alpha<\kappa$. Then the set \[C=\{\alpha<\kappa:(\forall\zeta\in F^\kappa_\xi(\alpha))\ F^\kappa_\xi(\alpha)\cap\zeta=F^\kappa_\zeta(\alpha)\}\] contains a club subset of $\kappa$. \end{lemma} \mathfrak{b}egin{proof} Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic over $V$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. Let $\vec{F}=\langleF^\kappa_\zeta:\zeta<\kappa^+\rangle$ and notice that $j(\vec{F})=\langle\overline F^{j(\kappa)}_\zeta:\zeta<j(\kappa^+)\rangle$ where $\overline F^{j(\kappa)}_\zeta$ is the $\zeta$-th canonical reflection function at $j(\kappa)$ in $V^\kappa/G$. We have \[j(C)=\{\alpha<j(\kappa):(\forall\zeta\in j(F^\kappa_\xi)(\alpha))\ j(F^\kappa_\xi)(\alpha)\cap\zeta = \overline F^{j(\kappa)}_\zeta(\alpha)\}.\] Since $j(F^\kappa_\xi)(\kappa)=j"\xi$ and for each $\zeta\in j"\xi$ we have $\overline F^{j(\kappa)}_\zeta(\kappa)=\overline F^{j(\kappa)}_{j(j^{-1}(\zeta))}(\kappa)=j(F^\kappa_{j^{-1}(\zeta)})(\kappa)=j"\zeta$, it follows that $\kappa\in j(C)$. \end{proof} Next we will show that for all limit ordinals $\xi<\kappa^+$, for club many $\alpha<\kappa$, the value of $f^\kappa_\xi(\alpha)$ is determined by the values of $f^\kappa_\zeta(\alpha)$ for $\zeta\in F^\kappa_\xi(\alpha)$. \mathfrak{b}egin{lemma}\langlebel{lemma_canonical_functions_at_limits} Suppose $\kappa$ is regular and $\xi<\kappa^+$ is a limit ordinal. Then the set \[D=\{\alpha<\kappa: f^\kappa_\xi(\alpha)=\mathfrak{b}igcup_{\zeta\in F^\kappa_\xi(\alpha)}f^\kappa_\zeta(\alpha)\}\] contains a club subset of $\kappa$. \end{lemma} \mathfrak{b}egin{proof} Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic over $V$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. Let $j(\langlef^\kappa_\zeta:\zeta<\kappa^+\rangle)=\langle\overline f^{j(\kappa)}_\zeta:\zeta<j(\kappa^+)\rangle$. We have \[j(D)=\{\alpha<j(\kappa): j(f^\kappa_\xi)(\alpha)=\mathfrak{b}igcup_{\zeta\in j(F^\kappa_\xi)(\alpha)} \overline f^{j(\kappa)}_\zeta(\alpha)\}.\] Since $\xi=\mathfrak{b}igcup_{\zeta\in j"\xi} j^{-1}(\zeta)$, it follows that $\kappa\in j(D)$. \end{proof} The next two lemmas follow easily from Proposition \ref{proposition_framework} and confirm our intuition that for a regular cardinal $\kappa$ and ordinal $\xi<\kappa^+$, for club-many $\alpha<\kappa$ the value $f^\kappa_\xi(\alpha)$ behaves like $\alpha$'s version of $\xi$. \mathfrak{b}egin{lemma}\langlebel{lemma_limits} Suppose $\kappa$ is regular and $\xi<\kappa^+$ is a limit ordinal. Then the set \[D=\{\alpha<\kappa: \text{$f^\kappa_\xi(\alpha)$ is a limit ordinal}\}\] contains a club subset of $\kappa$. \end{lemma} \mathfrak{b}egin{lemma}\langlebel{lemma_successor} Suppose $\kappa$ is regular. For all $\zeta<\kappa^+$ the following sets are closed unbounded in $\kappa$. \mathfrak{b}egin{align*} D_0&=\{\alpha<\kappa: F^\kappa_{\zeta+1}(\alpha)\cap\zeta=F^\kappa_\zeta(\alpha)\}\\ D_1&=\{\alpha<\kappa: F^\kappa_{\zeta+1}(\alpha)=F^\kappa_\zeta(\alpha)\cup\{\zeta\}\}\\ D_2&=\{\alpha<\kappa: f^\kappa_{\zeta+1}(\alpha)=f^\kappa_\zeta(\alpha)+1\} \end{align*} \end{lemma} Next we prove a proposition which generalizes a folklore result concerning canonical functions (see Corollary \ref{corollary_crazy}) to canonical reflection functions, and which draws a connection between the canonical reflection functions at a regular cardinal $\kappa$ and the canonical reflection functions at regular $\alpha<\kappa$. The following proposition was originally established in a previous version of this article using a more complicated proof; the proof below is due to Cody and Holy and appears in \cite{cody_holy_2022}. \mathfrak{b}egin{proposition}\langlebel{proposition_crazy} Suppose $\kappa$ is regular and $\xi<\kappa^+$. For each $\alpha<\kappa$ let \[\pi^\kappa_{\xi,\alpha}:F^\kappa_\xi(\alpha)\to f^\kappa_\xi(\alpha)\] be the transitive collapse of $F^\kappa_\xi(\alpha)$. Then there is a club $C^\kappa_\xi{\mathop{\rm sub}}seteq\kappa$ such that for all regular uncountable $\alpha\in C^\kappa_\xi$ the set \[D^\alpha_\xi=\{\mathfrak{b}eta<\alpha : \pi^\kappa_{\xi,\alpha}[F^\kappa_\xi(\mathfrak{b}eta)]=F^\alpha_{f^\kappa_\xi(\alpha)}(\mathfrak{b}eta)\}\] is in the club filter on $\alpha$. \end{proposition} \mathfrak{b}egin{proof} In order to prove the existence of such a club, we will use Proposition \ref{proposition_framework2}. Suppose $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ is generic over $V$ such that $\kappa$ is regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower. For each regular uncountable $\alpha<\kappa$ let $D^\alpha_\xi=\{\mathfrak{b}eta<\alpha : \pi^\kappa_{\xi,\alpha}[F^\kappa_\xi(\mathfrak{b}eta)]=F^\alpha_{f^\kappa_\xi(\alpha)}(\mathfrak{b}eta)\}$. We must show that \[\kappa\in j(\{\alpha\in{\rm REG}\cap\kappa: D^\alpha_\xi\text{ contains a club subset of $\kappa$}\}).\] Let $\vec{D}=\langleD^\alpha_\xi:\alpha\in{\rm REG}\cap\kappa\rangle$, $\vec{\pi}=\langle\pi^\kappa_{\xi,\alpha}:\alpha<\kappa\rangle$ and $\vec{F}=\langleF^\alpha_{f^\kappa_\xi(\alpha)}:\alpha\in{\rm REG}\cap\kappa\rangle$. By elementarity it follows that in $V^\kappa/G$, $j(\vec{\pi})_\kappa$ is a bijection from $j(F^\kappa_\xi)(\kappa)=j"\xi$ to $j(f^\kappa_\xi)(\kappa)=\xi$. Thus the set $\{j(\vec{\pi})_\kappa[j(F^\kappa_\xi)(\mathfrak{b}eta)]:\mathfrak{b}eta<\kappa\}$ is cofinal in $[\xi]^{<\kappa}$. Also by elementarity, we see that $j(\vec{F})_\kappa$ is the $\xi$-th canonical reflection function at $\kappa$ in $V^\kappa/G$ and hence the set $\{j(\vec{F})_\kappa(\mathfrak{b}eta):\mathfrak{b}eta<\kappa\}$ is cofinal in $[\xi]^{<\kappa}$. By the usual catching up argument, in $V^\kappa/G$ the set $j(\vec{D})_\kappa$ contains a club subset of $\kappa$. \end{proof} The following folklore result (see \cite[Section 5]{MR1077260}) easily follows from Proposition \ref{proposition_crazy}, or can be established directly using an argument which is easier than that of Proposition \ref{proposition_crazy}. \mathfrak{b}egin{corollary}\langlebel{corollary_crazy} Suppose $\kappa$ is regular and $\xi<\kappa^+$. Then there is a club $C^\kappa_\xi{\mathop{\rm sub}}seteq\kappa$ such that for all regular uncountable $\alpha\in C^\kappa_\xi$ the set \[D^\alpha_\xi=\{\mathfrak{b}eta<\alpha : f^\kappa_\xi(\mathfrak{b}eta)=f^\alpha_{f^\kappa_\xi(\alpha)}(\mathfrak{b}eta)\}\] is in the club filter on $\alpha$. \end{corollary} $\hat{\text{s}}$ection{Restricting ${\mathbb P}i^1_\xi$ formulas and consistency of higher ${\mathbb P}i^1_\xi$-indescribability} \langlebel{section_definition_pi1xi} We begin this section with a precise definition of ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas \emph{over $V_\kappa$}, where $\kappa$ is a regular cardinal and $\xi$ is an ordinal. The following definition is similar to \cite[Definition 4.1]{MR3894041}, the only difference being that we allow for first and second order parameters from $V_\kappa$. Recall that throughout the article we use capital letters to denote second-order variables and lower case letters to denote first-order variables. \mathfrak{b}egin{definition}\langlebel{definition_over} Suppose $\kappa$ is a regular cardinal. We define the notions of ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$, for all ordinals $\xi$ as follows. \mathfrak{b}egin{enumerate} \item A formula $\varphi$ is ${\mathbb P}i^1_0$, or equivalently $\mathbb{S}igma^1_0$, over $V_\kappa$ if it is a first order formula in the language of set theory, however we allow for free variables and parameters from $V_\kappa$ of two types, namely of first and of second order. \item A formula $\varphi$ is ${\mathbb P}i^1_{\xi+1}$ over $V_\kappa$ if it is of the form $\forall X_{k_1}\cdots\forall X_{k_m}\psi$ where $\psi$ is $\mathbb{S}igma^1_\xi$ over $V_\kappa$ and $m\in\omega$. Similarly, $\varphi$ is $\mathbb{S}igma^1_{\xi+1}$ over $V_\kappa$ if it is of the form $\exists X_{k_1}\cdots\exists X_{k_m}\psi$ where $\psi$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$ and $m\in\omega$.\footnote{We follow the convention that uppercase letters represent second order variables, while lower case letters represent first order variables. Thus, in the above, all quantifiers displayed are understood to be second order quantifiers, i.e., quantifiers over subsets of $V_\kappa$.} \item When $\xi$ is a limit ordinal, a formula $\varphi$, with finitely many second-order free variables and finitely many second-order parameters, is ${\mathbb P}i^1_\xi$ over $V_\kappa$ if it is of the form \[\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta\] where $\varphi_\zeta$ is ${\mathbb P}i^1_\zeta$ over $V_\kappa$ for all $\zeta<\xi$. Similarly, $\varphi$ is $\mathbb{S}igma^1_\xi$ if it is of the form \[\mathfrak{b}igvee_{\zeta<\xi}\varphi_\zeta\] where $\varphi_\zeta$ is $\mathbb{S}igma^1_\zeta$ over $V_\kappa$ for all $\zeta<\xi$. \end{enumerate} \end{definition} \mathfrak{b}egin{definition}\langlebel{definition_restriction} By induction on $\xi<\kappa^+$, we define $\varphi\mathrm{|}^\kappa_\alpha$ for all ${\mathbb P}i^1_\xi$ formulas $\varphi$ over $V_\kappa$ and all regular $\alpha<\kappa$ as follows. First assume that $\xi<\kappa$. If \[\varphi=\varphi(X_1,\ldots,X_m,A_1,\ldots,A_n),\] with free second order variables $X_1,\ldots,X_m$ and second order parameters $A_1,\ldots,A_n$, then we define \[\varphi\mathrm{|}^\kappa_\alpha=\varphi(X_1,\ldots,X_m,A_1\cap V_\alpha,\ldots,A_n\cap V_\alpha).\] If $\xi=\zeta+1$ is a successor ordinal and $\varphi=\forall X_{k_1}\ldots\forall X_{k_m}\psi$ is ${\mathbb P}i^1_{\zeta+1}$ over $V_\kappa$, then we define \[\varphi\mathrm{|}^\kappa_\alpha=\forall X_{k_1}\ldots\forall X_{k_m}(\psi\mathrm{|}^\kappa_\alpha).\] We define $\varphi\mathrm{|}^\kappa_\alpha$ analogously when $\varphi$ is $\mathbb{S}igma^1_{\zeta+1}$. If $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ is a limit ordinal, and \mathfrak{b}egin{align}\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\psi_\zeta\langlebel{equation_defn_restriction}\end{align} is ${\mathbb P}i^1_\xi$ over $V_\kappa$, then we define \[\varphi\mathrm{|}^\kappa_\alpha=\mathfrak{b}igwedge_{\zeta\in f^\kappa_\xi(\alpha)}\psi_{(\pi^\kappa_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^\kappa_\alpha\] in case $\psi_{(\pi^\kappa_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^\kappa_\alpha$ is a ${\mathbb P}i^1_\zeta$ formula over $V_\alpha$ for every $\zeta<f^\kappa_\xi(\alpha)$. We leave $\varphi\mathrm{|}^\kappa_\alpha$ undefined otherwise. We define $\varphi\mathrm{|}^\kappa_\alpha$ similarly when $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ is a limit ordinal and $\varphi$ is $\mathbb{S}igma^1_\xi$. \end{definition} \mathfrak{b}egin{remark}\langlebel{remark_definition_of_restriction} A few remarks about Definition \ref{definition_restriction} are in order. \mathfrak{b}egin{enumerate} \item An easy inductive argument on $\xi<\kappa^+$ shows that if $\varphi$ is a ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$, and $\alpha<\kappa$ is regular, then whenever $\varphi\mathrm{|}^\kappa_\alpha$ is defined, it is a ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ or $\mathbb{S}igma^1_{f^\kappa_\xi(\alpha)}$ formula over $V_\alpha$ respectively. \item Recall that we defined the sequence of canonical reflection functions $\langleF^\kappa_\xi:\xi<\kappa^+\rangle$, the sequence of canonical functions $\langlef^\kappa_\xi:\xi<\kappa^+\rangle$ and the transitive collapses $\pi^\kappa_{\xi,\alpha}:F^\kappa_\xi(\alpha)\to f^\kappa_\xi(\alpha)$ in a particular way making use of fixed sequence of bijections $\langleb_{\kappa,\xi}:\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa\rangle$. Thus, the definition of $\varphi\mathrm{|}^\kappa_\alpha$ given above clearly depends on our choice of bijections $\langleb_{\kappa,\xi}:\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa\rangle$. Below we will see that our definition of $\varphi\mathrm{|}^\kappa_\alpha$ is independent of this choice of bijections modulo the nonstationary ideal. See the paragraph after Definition \ref{definition_indescribability} for details. \end{enumerate} \end{remark} In order to establish some basic properties of the restriction operation from Definition \ref{definition_restriction}, let us consider how it behaves with respect to generic ultrapowers. We will want to apply elementary embeddings to ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas, which will be viewed as set theoretic objects. \mathfrak{b}egin{remark}\langlebel{remark_coding} Assume that $\varphi$ is either a ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$ for some $\xi<\kappa^+$. Let $j:V\to V^\kappa/G$ be the generic ultrapower embedding obtained by forcing with $P(\kappa)/I$ where $I$ is some normal ideal on $\kappa$. We will leave it to the reader to check that any reasonable coding of formulas has the following properties. \mathfrak{b}egin{enumerate} \item If $\xi<\kappa$, and $A_1,\ldots,A_n$ are all second order parameters appearing in $\varphi$, then \[j(\varphi(A_1,\ldots,A_n))=\varphi(j(A_1),\ldots,j(A_n)).\] \item $j(\forall X\,\varphi)=\forall X\,j(\varphi)$. \item If $\xi\ge\kappa$ is a limit ordinal, and $\varphi$ is either of the form $\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\psi_\zeta$, or of the form $\mathfrak{b}igvee_{\zeta<\xi}\psi_\zeta$, let $\vec\psi=\langlengle\psi_\zeta\mid\zeta<\xi\mathop{\rm ran}glengle$. Then, \[j(\varphi)=\mathfrak{b}igwedge_{\zeta<j(\xi)}j(\vec\psi)_\zeta\quad\textrm{or}\quad j(\varphi)=\mathfrak{b}igvee_{\zeta<j(\xi)}j(\vec\psi)_\zeta\] respectively. \end{enumerate} \end{remark} Regarding the assumption of the next lemma, and also of some later results, note that $\kappa$ will be regular in a generic ultrapower $V^\kappa/G$ obtained by forcing with a normal ideal on $\kappa$ if and only if $G$ contains the set of regular cardinals below $\kappa$. This is of course only possible if that latter set is a stationary subset of $\kappa$, i.e., if $\kappa$ is weakly Mahlo. Let us note that the assumption that $\kappa$ is regular in the generic ultrapower is needed to ensure that $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa$ is defined. \mathfrak{b}egin{lemma}[{Cody-Holy \cite{cody_holy_2022}}]\langlebel{lemma_j_of} Suppose $\kappa$ is a regular cardinal and $\varphi$ is a ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$ for some $\xi<\kappa^+$. Whenever $I$ is a normal ideal on $\kappa$, $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic over $V$ and $j:V\to V^\kappa/G$ is the corresponding generic ultrapower such that $\kappa$ is regular in $V^\kappa/G$, it follows that $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa$ is ${\mathbb P}i^1_\xi$ in $V^\kappa/G$ and furthermore, \[j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi.\] \end{lemma} \mathfrak{b}egin{proof} We proceed by induction on $\xi<\kappa^+$. By Remark \ref{remark_coding}(1) and the definition of the restriction operation, the case when $\xi<\kappa$ is easy since \[j(\varphi(A_1,\ldots,A_n))\mathrm{|}^{j(\kappa)}_\kappa=\varphi(j(A_1),\ldots,j(A_n))\mathrm{|}^{j(\kappa)}_\kappa=\varphi(A_1,\ldots,A_n).\] Suppose $\xi=\zeta+1$ is a successor ordinal above $\kappa$ and $\varphi=\forall X\psi(X)$ where $\psi(X)$ is a $\mathbb{S}igma^1_\zeta$ formula over $V_\kappa$. By Remark \ref{remark_coding}(2), \[j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=j(\forall X\psi(X))\mathrm{|}^{j(\kappa)}_\kappa=\forall X j(\psi(X))\mathrm{|}^{j(\kappa)}_\kappa=\forall X\psi(X).\] Essentially the same argument works when $\varphi=\exists X\psi(X)$ and $\psi(X)$ is a ${\mathbb P}i^1_\zeta$ formula over $V_\kappa$. Suppose $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ is a limit and $\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\psi_\zeta$ is a ${\mathbb P}i^1_\xi$ formula over $V_\kappa$. Let $\vec\psi=\langlengle\psi_\zeta\mid\zeta<\xi\mathop{\rm ran}glengle$, and let $\vec\pi=\langlengle\pi^\kappa_{\xi,\alpha}\mid\alpha<\kappa\mathop{\rm ran}glengle$. By elementarity $j(\vec{\pi})_\kappa$ is the transitive collapse of $j(F^\kappa_\xi)(\kappa)=j"\xi$ to $j(f^\kappa_\xi)(\kappa)=\xi$ and hence $j(\vec{\pi})_\kappa\mathrm{|}trict j"\xi=j^{-1}\mathrm{|}trict j"\xi$. Furthermore, For each $\zeta<\xi$ we have $j(\vec{\psi})_{j(\vec{\pi})_\kappa^{-1}(\zeta)}\mathrm{|}^{j(\kappa)}_\kappa=j(\vec{\psi})_{j(\zeta)}\mathrm{|}^{j(\kappa)}_\kappa=j(\psi_\zeta)\mathrm{|}^{j(\kappa)}_\kappa$, which is ${\mathbb P}i^1_\zeta$ in $V^\kappa/G$ by our inductive hypothesis. Thus we have \mathfrak{b}egin{align*} j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa&=\mathfrak{b}igwedge_{\zeta<f^{j(\kappa)}_{j(\xi)}(\kappa)}j(\vec{\psi})_{j(\vec{\pi})_\kappa^{-1}(\zeta)}\mathrm{|}^{j(\kappa)}_\kappa\\ &=\mathfrak{b}igwedge_{\zeta<\xi}j(\psi_\zeta)\mathrm{|}^{j(\kappa)}_\kappa\\ &=\varphi. \end{align*} The case when $\varphi$ is a $\mathbb{S}igma^1_\xi$ formula is treated in exactly the same way. \end{proof} A nice feature of our definition of restriction is that it provides a convenient way to represent ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas in generic ultrapowers. \mathfrak{b}egin{lemma}[{Cody-Holy \cite{cody_holy_2022}}]\langlebel{lemma_represent} Suppose $I$ is a normal ideal on $\kappa$, $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic over $V$, $j:V\to V^\kappa/G$ is the corresponding generic ultrapower and $\kappa$ is regular in $V^\kappa/G$. Suppose $\varphi$ is a ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$ for some $\xi<\kappa^+$ and let ${\mathbb P}hi:\kappa\to V_\kappa$ be such that ${\mathbb P}hi(\alpha)=\varphi\mathrm{|}^\kappa_\alpha$ for every regular $\alpha<\kappa$. Then, ${\mathbb P}hi$ represents $\varphi$ in $V^\kappa/G$. That is, $j({\mathbb P}hi)(\kappa)=\varphi$. \end{lemma} \mathfrak{b}egin{proof} This is an easy consequence of Lemma \ref{lemma_j_of} since \[j({\mathbb P}hi)(\kappa)=j(\langle\varphi\mathrm{|}^\kappa_\alpha:\alpha\in\kappa\cap{\rm REG}\rangle)_\kappa=j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi.\] \end{proof} As an easy consequence of Lemma \ref{lemma_j_of}, we see that for each $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$, the definition of $\varphi\mathrm{|}^\kappa_\alpha$ where $\varphi$ is ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ over $V_\kappa$ is independent of which bijection $b_{\kappa,\xi}$ is used in its computation, modulo the nonstationary ideal on $\kappa$. \mathfrak{b}egin{corollary}\langlebel{corollary_restriction_is_well_defined} Suppose $\kappa$ is a regular cardinal, $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ and $\varphi$ is ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$. Let $b_{\kappa,\xi}$ and $\mathfrak{b}ar{b}_{\kappa,\xi}$ be bijections from $\kappa$ to $\xi$, and let $\varphi\mathrm{|}^\kappa_\alpha$ and $\varphi\mathfrak{b}ar{\mathrm{|}}^\kappa_\alpha$ denote the restriction of $\varphi$ to a regular $\alpha<\kappa$ defined using $b_{\kappa,\xi}$ and $\mathfrak{b}ar{b}_{\kappa,\xi}$ respectively. Then there is a club $C{\mathop{\rm sub}}seteq\kappa$ such that for all regular $\alpha\in C$ we have $\varphi\mathrm{|}^\kappa_\alpha=\varphi\mathfrak{b}ar\mathrm{|}^\kappa_\alpha$ \end{corollary} \mathfrak{b}egin{proof} To prove the existence of such a club we use Proposition \ref{proposition_framework2}. Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic over $V$ such that $\kappa$ is regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower. Lemma \ref{lemma_j_of} implies that $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi=j(\varphi)\mathfrak{b}ar{\mathrm{|}}^{j(\kappa)}_\kappa$. \end{proof} The following lemma was essentially established in an earlier version of the current article using a more complicated proof. The following simplified proof is due to Cody-Holy and appears in \cite{cody_holy_2022}. \mathfrak{b}egin{lemma}\langlebel{lemma_restriction_is_nice} Suppose $\kappa$ is weakly Mahlo. For any $\xi<\kappa^+$, if $\varphi$ is a ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$, then there is a club subset $C_\varphi$ of $\kappa$ such that for any regular $\alpha\in C_\varphi$, $\varphi\mathrm{|}^\kappa_\alpha$ is defined, and therefore a ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ or $\mathbb{S}igma^1_{f^\kappa_\xi(\alpha)}$ formula over $V_\alpha$ respectively by Remark \ref{remark_definition_of_restriction}. \end{lemma} \mathfrak{b}egin{proof} Suppose $\varphi$ is a ${\mathbb P}i^1_\xi$ formula over $V_\kappa$. To prove the existence of $C_\varphi$ we use Proposition \ref{proposition_framework2}. Suppose $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ is generic over $V$ such that $\kappa$ is regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower. By Lemma~\ref{lemma_j_of}, $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi$ is clearly defined and is a ${\mathbb P}i^1_\xi$ formula in $V^\kappa/G$. \end{proof} \mathfrak{b}egin{definition}\langlebel{definition_indescribability} Suppose $\kappa$ is a cardinal and $\xi<\kappa^+$. A set $S{\mathop{\rm sub}}seteq\kappa$ is \emph{${\mathbb P}i^1_\xi$-indescribable} if for every ${\mathbb P}i^1_\xi$ sentence $\varphi$ over $V_\kappa$, if $V_\kappa\models\varphi$ then there is some $\alpha\in S$ such that $V_\alpha\models\varphi|^\kappa_\alpha.$ \end{definition} It easily follows from Corollary \ref{corollary_restriction_is_well_defined} that the above notion of indescribability does not depend on which sequence $\langleb_{\kappa,\xi}:\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa\rangle$ is used to compute restrictions of formulas. In Proposition \ref{proposition_measurable} below, we establish that the notion of indescribability given in Definition \ref{definition_indescribability} is relatively consistent by showing that every measurable cardinal $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\kappa^+$ and, in terms of consistency strength, the existence of a cardinal $\kappa$ which is ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\kappa^+$ is strictly weaker than the existence of a measurable cardinal. \mathfrak{b}egin{proposition}\langlebel{proposition_measurable} Suppose $U$ is a normal measure on a measurable cardinal $\kappa$. Then $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\kappa^+$ and the set \[X=\{\alpha<\kappa:\text{$\alpha$ is ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\alpha^+$}\}\] is in $U$. \end{proposition} \mathfrak{b}egin{proof} Let $j:V\to M$ be the usual ultrapower embedding obtained from $U$ where $M$ is transitive and $j$ has critical point $\kappa$. Let us show that the set $X$ is in $U$; the fact that $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable for all $\xi<\kappa^+$ follows by a similar argument. Notice that it follows directly from Lemma \ref{lemma_restriction_is_nice} that for any $\xi<\kappa^+$ if $\varphi$ is any ${\mathbb P}i^1_\xi$ formula over $V_\kappa$ then, in $M$, the formula $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$ (because $\kappa\in j(C_\varphi)$). Furthermore, by Lemma \ref{lemma_j_of} we have \mathfrak{b}egin{align} j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi.\langlebel{equation_measurable} \end{align} It will suffice to show that, in $M$, $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable for all limit ordinals $\xi<\kappa^+$. Fix a limit ordinal $\xi<\kappa^+$ and suppose \[(V_\kappa\models\varphi)^M\] where $\varphi$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$ in $M$. Since $H(\kappa^+)^V= H(\kappa^+)^M$, we have $\varphi\in V$ and $\varphi$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$ in $V$. It follows by (\ref{equation_measurable}), that \[\left((\exists\alpha<j(\kappa))\ V_\alpha\models j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha\right)^M\] and thus, by elementarity, there is some $\alpha<\kappa=\mathop{\rm crit}(j)$ such that \[V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha.\] Thus, $(V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha)^M$ and hence $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable in $M$. \end{proof} $\hat{\text{s}}$ection{Restricting $L_{\kappa^+,\kappa^+}$ formulas and consistency of $L_{\kappa^+,\kappa^+}$-indescribability}\langlebel{section_other} We will need to apply elementary embeddings to formulas of $L_{\kappa^+,\kappa^+}$, so let us consider some assumptions regarding the set-theoretic nature of these formulas. For example, we assume that if $j:V\to M$ is an elementary embedding with critical point $\kappa$ then $j(\lnot\varphi)=\lnot j(\varphi)$ and $j(\exists x\psi)=\exists j(\vec{x})\psi$; we also make additional assumption as in Remark \ref{remark_coding}(3) above, but we will not discuss this further. Typically, when defining $L_{\kappa^+,\kappa^+}$ formulas, one begins by fixing a supply of $\kappa^+$-many variables that can be used to form $L_{\kappa^+,\kappa^+}$ sentences. However, without loss of generality, we will assume that we begin with a supply of $\kappa$-many variables $\{x_\eta:\eta<\kappa\}$. This assumption does not weaken the expressive power of $L_{\kappa^+,\kappa^+}$ sentences or theories (without parameters) in the language of set theory, because any particular sentence defined using a supply of $\kappa^+$-many variables only actually mentions $\kappa$-many. This assumption allows us to write $L_{\kappa^+,\kappa^+}$ formulas beginning with existential quantifiers in the form $\exists\langlex_{\alpha_\eta}:\eta<\gamma\rangle\psi$, where the domain of the sequence of variables being quantified over is simply some $\gamma\leq\kappa$, rather than some $\xi<\kappa^+$ and $\langle\alpha_\eta:\eta<\gamma\rangle$ is some increasing sequence of ordinals less than $\kappa$. Another consequence of this assumption is that we may assume that all variables are elements of $V_\kappa$ when $\kappa=|V_\kappa|$, and hence if $j:V\to M$ is an elementary embedding with critical point $\kappa$, we will have $j(x)=x$ for all variables $x$. We will also assume that for all cardinals $\alpha<\kappa$ the set $\{x_\eta:\eta<\alpha\}$ constitutes the supply of $\alpha$-many variables used to form all $L_{\alpha^+,\alpha^+}$ sentences. For a regular cardinal $\kappa$ and an ordinal $\alpha<\kappa$, we define $\varphi\mathrm{|}^\kappa_\alpha$ for all $L_{\kappa^+,\kappa^+}$ formulas $\varphi$ by induction of subformulas. For more on such induction principles, see \cite[Page 64]{MR0539973}. \mathfrak{b}egin{definition}\langlebel{definition_restriction_2} Suppose $\kappa$ is a regular cardinal and $\alpha<\kappa$ is a cardinal. We define $\varphi\mathrm{|}^\kappa_\alpha$ for all formulas $\varphi$ of $L_{\kappa^+,\kappa^+}$ in a given signature by induction on complexity of $\varphi$. \mathfrak{b}egin{enumerate} \item If $\varphi$ is a term equation $t_1=t_2$ or a relational formula of the form $R(t_1,\ldots,t_k)$ we define $\varphi\mathrm{|}^\kappa_\alpha$ to be $\varphi$. \item If $\varphi$ is of the form $\lnot\psi$ where $\psi\mathrm{|}^\kappa_\alpha$ has already been defined, we let $\varphi\mathrm{|}^\kappa_\alpha$ be the formula $\lnot(\psi\mathrm{|}^\kappa_\alpha)$. \item If $\varphi$ is of the form $\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta$ where $\xi<\kappa^+$ and $\varphi_\zeta\mathrm{|}^\kappa_\alpha$ has been defined for all $\zeta<\xi$, then we define \[\varphi\mathrm{|}^\kappa_\alpha=\mathfrak{b}igwedge_{\zeta < f^\kappa_\xi(\alpha)}\varphi_{(\pi^\kappa_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^\kappa_\alpha,\] provided that this definition of $\varphi\mathrm{|}^\kappa_\alpha$ is a formula of $L_{\alpha^+,\alpha^+}$; otherwise we leave $\varphi\mathrm{|}^\kappa_\alpha$ undefined. \item If $\varphi$ is of the form $\exists\langlex_{\alpha_\eta}:\eta<\gamma\rangle\psi$ where $\gamma\leq\kappa$, $\langle\alpha_\eta:\eta<\gamma\rangle$ is an increasing sequence of ordinals less than $\kappa$ and $\psi\mathrm{|}^\kappa_\alpha$ has already been defined, we let \[\varphi\mathrm{|}^\kappa_\alpha=\exists\langlex_{\alpha_\eta}:\eta<\alpha\cap\gamma\rangle\ \psi\mathrm{|}^\kappa_\alpha,\] provided that this definition of $\varphi^\kappa_\alpha$ is a formula of $L_{\alpha^+,\alpha^+}$; otherwise we leave $\varphi\mathrm{|}^\kappa_\alpha$ undefined. \end{enumerate} \end{definition} As for the notion of restriction of ${\mathbb P}i^1_\xi$ formulas considered in Section \ref{section_definition_pi1xi} above, one can easily show that for $L_{\kappa^+,\kappa^+}$ formulas, the definition of $\varphi\mathrm{|}^\kappa_\alpha$ is independent of our choice of bijections $\langleb_{\kappa,\xi}:\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa\rangle$ modulo the nonstationary ideal on $\kappa$. Notice that, in Definition \ref{definition_restriction_2}(4), it is at least conceivable that some of the bound variables of $\varphi$ could become free variables of $\varphi\mathrm{|}^\kappa_\alpha$. However, it easily follows from the next lemma that this can happen only for a nonstationary set of $\alpha$, and hence this aspect of the definition can be ignored in all of the cases that we care about. \mathfrak{b}egin{lemma}\langlebel{lemma_alternative_indescribability} Suppose $\kappa$ is a regular cardinal and $\varphi$ is an $L_{\kappa^+,\kappa^+}$ formula in the language of set theory. If $I$ is a normal ideal on $\kappa$, $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic over $V$ and $j:V\to V^\kappa/G$ is the corresponding generic ultrapower such that $\kappa$ is regular in $V^\kappa/G$, then it follows that $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi$ is a formula of $L_{\kappa^+,\kappa^+}$ in $V^\kappa/G$. \end{lemma} \mathfrak{b}egin{proof} When $\varphi$ is a relational formula, it follows by our assumptions on the set-theoretic nature of such formulas that $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi\mathrm{|}^{j(\kappa)}_\kappa=\varphi$. If the result holds for $\psi$ and $\varphi$ is of the form $\lnot \varphi$, then it clearly holds for $\varphi$ too. Now suppose $\varphi$ is of the form \[\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta,\] where $\xi<\kappa^+$. Define sequences $\vec{\varphi}=\langle\varphi_\zeta:\zeta<\xi\rangle$ and $\vec{\pi}=\langle\pi^\kappa_{\xi,\alpha}:\alpha<\kappa\rangle$. Recall that $j(\vec{\pi})_\kappa \mathrm{|}trict j"\xi=j^{-1}\mathrm{|}trict j"\xi$. We have \mathfrak{b}egin{align*} j(\varphi)=\mathfrak{b}igwedge_{\zeta<j(\xi)} j(\vec{\varphi})_\zeta \end{align*} and thus \mathfrak{b}egin{align*} j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa&=\mathfrak{b}igwedge_{\zeta<j(f^\kappa_\xi)(\kappa)} j(\vec{\varphi})_{j(\vec{\pi})_\kappa^{-1}(\zeta)}\mathrm{|}^{j(\kappa)}_\kappa\\ &=\mathfrak{b}igwedge_{\zeta<\xi} j(\vec{\varphi})_{j(\zeta)}\mathrm{|}^{j(\kappa)}_\kappa\\ &=\mathfrak{b}igwedge_{\zeta<\xi} j(\varphi_\zeta)\mathrm{|}^{j(\kappa)}_\kappa\\ &=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta\\ &=\varphi. \end{align*} Now suppose $\varphi$ is of the form $\exists\langlex_{\alpha_\eta}:\eta<\gamma\rangle\psi$ where $\gamma\leq\kappa$ and $\langle\alpha_\eta:\eta<\gamma\rangle$ is an increasing sequence of ordinals less than $\kappa$. Let $\vec{x}=\langlex_{\alpha_\eta}:\eta<\gamma\rangle$. We have \[j(\varphi)=\exists j(\vec{x}) j(\psi)\] and thus \mathfrak{b}egin{align*} j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa&=\exists j(\vec{x})\mathrm{|}trict(\kappa\cap j(\gamma)) \ j(\psi)\mathrm{|}^{j(\kappa)}_\kappa\\ &=\exists\vec{x}\psi\\ &=\varphi. \end{align*} \end{proof} One can easily show that the following definition of $L_{\kappa^+,\kappa^+}$-indescribability is not dependent on which sequence of bijections is used to compute restrictions of $L_{\kappa^+,\kappa^+}$ formulas. \mathfrak{b}egin{definition} Suppose $\kappa$ is a regular cardinal. A set $S{\mathop{\rm sub}}seteq\kappa$ is \emph{$L_{\kappa^+,\kappa^+}$-indescribable} if for all sentences $\varphi$ of $L_{\kappa^+,\kappa^+}$ in the language of set theory, if $V_\kappa\models\varphi$ then there is some $\alpha<\kappa$ such that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$. \end{definition} From Lemma \ref{lemma_alternative_indescribability} and an argument similar to that given above for Proposition \ref{proposition_measurable} we obtain the following, which shows that the existence of a cardinal $\kappa$ which is $L_{\kappa^+,\kappa^+}$-indescribable is strictly weaker than the existence of a measurable cardinal. \mathfrak{b}egin{proposition}\langlebel{proposition_Lkappa+} Suppose $U$ is a normal measure on a measurable cardinal $\kappa$. Then $\kappa$ is $L_{\kappa^+,\kappa^+}$-indescribable and the set \[\{\alpha<\kappa:\text{$\alpha$ is $L_{\alpha^+,\alpha^+}$-indescribable}\}\] is in $U$. \end{proposition} $\hat{\text{s}}$ection{Higher ${\mathbb P}i^1_\xi$-indescribability ideals} In this section, given a regular cardinal $\kappa$, we prove the existence of universal ${\mathbb P}i^1_\xi$ formulas for all $\xi<\kappa^+$ and use such formulas to show that the natural ideal on $\kappa$ associated to ${\mathbb P}i^1_\xi$-indescribability is normal. We then use universal formulas to show that ${\mathbb P}i^1_\xi$-indescribability is, in a sense, expressible by a ${\mathbb P}i^1_{\xi+1}$ formula. This leads to several hierarchy results and a characterization of ${\mathbb P}i^1_\xi$-indescribability in terms of the natural filter base consisting of the ${\mathbb P}i^1_\xi$-club subsets of $\kappa$. \mathfrak{b}egin{remark}\langlebel{remark_diagonal} Let us make a brief remark about normal ideals and a notion of diagonal intersection we will use in several places below. Recall that an ideal $I$ on a regular cardinal $\kappa$ is \emph{normal} if and only if for any positive set $S\in I^+=\{X{\mathop{\rm sub}}seteq\kappa: X\notin I\}$ and every function $f:S\to\kappa$ with $f(\alpha)<\alpha$ for all $\alpha\in S$, there is a positive set $T\in P(S)\cap I^+$ such that $f$ is constant on $T$. Equivalently, $I$ is normal if and only if the filter $I^*$ dual to $I$ is closed under diagonal intersection; that is, whenever $\vec{C}=\langleC_\alpha:\alpha<\kappa\rangle$ is a sequences of sets in $I^*$ then $\mathop{\text{\Large$\mathfrak{b}igtriangleup$}}\vec{C}=\mathop{\text{\Large$\mathfrak{b}igtriangleup$}}_{\alpha<\kappa}C_\alpha=\{\alpha<\kappa:\alpha\in\mathfrak{b}igcap_{\mathfrak{b}eta<\alpha}C_\mathfrak{b}eta\}$ is in $I^*$. Since diagonal intersections are independent, modulo the nonstationary ideal, of the particular enumeration of the sets involved, it follows that an ideal $I$ on $\kappa$ is normal if and only if for all $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$ whenever $\vec{C}=\langleC_\zeta:\zeta<\xi\rangle$ is a sequence of sets in $I^*$, the set \[\mathop{\text{\Large$\mathfrak{b}igtriangleup$}}\vec{C}=\mathop{\text{\Large$\mathfrak{b}igtriangleup$}}_{\zeta<\xi}C_\zeta=\{\alpha<\kappa: \alpha\in \mathfrak{b}igcap_{\zeta\in b_{\kappa,\xi}[\alpha]}C_\zeta\}\] is in $I^*$, where $b_{\kappa,\xi}:\kappa\to\xi$ is a bijection. In what follows, we will often make use of the fact that the club filter on a regular $\kappa$ is closed under such diagonal intersections. \end{remark} {\mathop{\rm sub}}section{Universal ${\mathbb P}i^1_\xi$ formulas and normal ideals}\langlebel{section_universal} For a regular cardinal $\kappa>\omega$, we define the notion of universal ${\mathbb P}i^1_\xi$ formula, where $\xi<\kappa^+$, as follows. If $\xi<\kappa$ then we adopt a definition of universal ${\mathbb P}i^1_\xi$ formula over $V_\kappa$, which is similar to that of \cite{MR3894041}, but we need a different notion for $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$. \mathfrak{b}egin{definition}\langlebel{definition_universal} Suppose $\kappa$ is a regular cardinal and $\xi<\kappa$. We say that a ${\mathbb P}i^1_\xi$ formula ${\mathbb P}si(X_1,\ldots,X_n,Y_\xi)$ over $V_\kappa$, where $X_1,\ldots,X_n,Y_\xi$ are second-order variables, is a \emph{universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with $n$ free variables} if for all ${\mathbb P}i^1_\xi$ formulas $\varphi(X_1,\ldots,X_n)$ over $V_\kappa$, with all free variables displayed, there is a $K_\varphi\in V_\kappa$, referred to as a \emph{code for $\varphi$} such that for all $A_1,\ldots,A_n{\mathop{\rm sub}}seteq V_\kappa$ and all regular $\alpha\in \kappa$\hat{\text{s}}$etminus\xi$ we have \[V_\alpha\models\varphi(A_1,\ldots,A_n)\text{ if and only if }V_\alpha\models{\mathbb P}si(A_1,\ldots,A_n,K_\varphi).\] On the other hand, suppose $\xi\in\kappa^+$\hat{\text{s}}$etminus\kappa$. We say that a ${\mathbb P}i^1_\xi$ formula ${\mathbb P}si(X_1,\ldots,X_n,Y_\xi)$ over $V_\kappa$, where $X_1,\ldots,X_n,Y_\xi$ are second-order variables, is a \emph{universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with $n$ free second order variables} if for all ${\mathbb P}i^1_\xi$ formulas $\varphi(X_1,\ldots,X_n)$ over $V_\kappa$, with all free variables displayed, there is a $K_\varphi{\mathop{\rm sub}}seteq\kappa$ and there is a club $C_\varphi{\mathop{\rm sub}}seteq\kappa$ such that for all $A_1,\ldots,A_n{\mathop{\rm sub}}seteq V_\kappa$ and all regular $\alpha\in C_\varphi\cup\{\kappa\}$ we have \[V_\alpha\models\varphi(A_1,\ldots,A_n)\mathrm{|}^\kappa_\alpha\text{ if and only if }V_\alpha\models{\mathbb P}si(A_1,\ldots,A_n,K_\varphi)\mathrm{|}^\kappa_\alpha.\] When $n=0$, the intended meaning is that $\varphi$ is a ${\mathbb P}i^1_\xi$ sentence over $V_\kappa$ and ${\mathbb P}si_{\xi,0}(Y)$ has one free-variable. The notion of \emph{universal $\mathbb{S}igma^1_\xi$ formula at $\kappa$ for formulas with $n$ free second-order variables} is defined similarly. \end{definition} We will use the following lemma to prove that universal ${\mathbb P}i^1_\xi$ formulas exist at regular $\kappa$ where $\kappa\leq\xi<\kappa^+$. \mathfrak{b}egin{lemma}\langlebel{lemma_no_increase} Suppose $\kappa$ is regular and $1\leq\zeta<\kappa^+$. Suppose $\psi_\zeta(W_1,\ldots,W_n,Y,Z)$ is a ${\mathbb P}i^1_\zeta$ formula over $V_\kappa$ and $\varphi(X,Y)$ is a ${\mathbb P}i^1_0$ formula over $V_\kappa$ where all free second-order variables are displayed. Then there is a ${\mathbb P}i^1_\zeta$ formula $\varphi_\zeta(X,Z)$ over $V_\kappa$ and a club $C_\zeta$ in $\kappa$ such that for all $A,B{\mathop{\rm sub}}seteq V_\kappa$ and for all regular $\alpha\in C_\zeta\cup\{\kappa\}$ we have \[V_\alpha\models\forall Y\forall W_1\cdots\forall W_n(\varphi(A\cap V_\alpha,Y)\lor\psi_\zeta(W_1,\ldots, W_n,Y,B)\mathrm{|}^\kappa_\alpha)\] if and only if \[V_\alpha\models \varphi_\zeta(A,B)\mathrm{|}^\kappa_\alpha.\] Furthermore, a similar statement holds for $\mathbb{S}igma^1_\zeta$ formulas $\psi_\zeta'$ over $V_\kappa$. \end{lemma} \mathfrak{b}egin{proof} We provide a proof for the cases in which $\psi_\zeta$ is a ${\mathbb P}i^1_\zeta$ formula over $V_\kappa$. The other case in which the formulas are $\mathbb{S}igma^1_\zeta$ is similar. We proceed by induction on $\zeta$. If $\zeta=1$ then $\psi_1(W_1,\ldots, W_n,Y,Z)$ is of the form $\forall W\psi_0(W_1,\ldots, W_n,W,Y,Z)$ where $\psi_0(W_1,\ldots,W_n,W,Y,Z)$ is ${\mathbb P}i^1_0$ over $V_\kappa$ and we see that \[\forall Y\forall W_1\cdots W_n(\varphi(X,Z)\lor\psi_1(W_1,\ldots,W_n,Y,Z))\] is equivalent over $V_\kappa$ to the ${\mathbb P}i^1_1$ formula \[\varphi_1(X,Z)=\forall Y\forall W_1\cdots\forall W_n\forall W(\varphi(X,Y)\lor\psi_0(W_1,\ldots,W_n,W,Y,Z)).\] Since restrictions of ${\mathbb P}i^1_1$ formulas are trivial, this establishes the base case taking $C_1=\kappa$. If $\zeta=\eta+1<\kappa^+$ is a successor ordinal, then $\psi_{\eta+1}(W_1,\ldots,W_n,Y,Z)$ is of the form $\forall W \psi_\eta'(W_1,\ldots,W_n,W,Y,Z)$ where $\psi_\eta'$ is $\mathbb{S}igma^1_\eta$ over $V_\kappa$. Clearly the formula \[\varphi_{\eta+1}:=\forall Y\forall W_1\cdots\forall W_n\forall W(\varphi(X,Y)\lor \psi_\eta'(W_1,\ldots,W_n,W,Y,Z))\] is ${\mathbb P}i^1_{\eta+1}$ over $V_\kappa$ and satisfies the desired property together with the club $C_{\eta+1}=C_\eta$ obtained from the inductive hypothesis. If $\zeta<\kappa^+$ is a limit ordinal, then \[\psi_\zeta(W_1,\ldots,W_n,Y,Z)=\mathfrak{b}igwedge_{\eta<\zeta}\psi_\eta(W_1,\ldots,W_n,Y,Z)\] where $\psi_\eta$ is ${\mathbb P}i^1_\eta$ over $V_\kappa$ for all $\eta<\zeta$. In this case, the formula \[\forall Y\forall W_1\cdots\forall W_n(\varphi(X,Y)\lor\psi_\zeta(W_1,\ldots,W_n,Y,Z))\] is equivalent over $V_\kappa$ to \[\mathfrak{b}igwedge_{\eta<\zeta}\forall Y\forall W_1\cdots\forall W_n(\varphi(X,Y)\lor\psi_\eta(W_1,\ldots,W_n,Y,Z)),\] and by our inductive hypothesis, for each $\eta<\zeta$, there is a ${\mathbb P}i^1_\eta$ formula $\varphi_\eta(X,Z)$ over $V_\kappa$ and a club $C_\eta$ in $\kappa$ such that for all $A,B{\mathop{\rm sub}}seteq V_\kappa$ and all regular $\alpha\in C_\eta\cup\{\kappa\}$ we have \[V_\alpha\models\forall Y\forall W_1\cdots\forall W_n(\varphi(A,Y)\lor\psi_\eta(W_1,\ldots, W_n,Y,B)\mathrm{|}^\kappa_\alpha)\] if and only if \[V_\alpha\models\varphi_\eta(A,B)\mathrm{|}^\kappa_\alpha.\] It is easy to verify that the formula \[\varphi_\zeta(X,Y)=\mathfrak{b}igwedge_{\eta<\zeta}\varphi_\eta(X,Y)\] and a club subset $C_\zeta$ of $\mathop{\text{\Large$\mathfrak{b}igtriangleup$}}_{\eta<\zeta}C_\eta=\{\alpha<\kappa:\alpha\in\mathfrak{b}igcap_{\eta\in F^\kappa_\zeta(\alpha)}C_\eta\}$ are as desired.\footnote{Recall that $C_\zeta$ is in fact in the club filter on $\mu$ by Remark \ref{remark_diagonal}.} \end{proof} The following proposition generalizes results of L\'{e}vy \cite{MR0281606} and Bagaria \cite{MR3894041}; Levy proved the case in which $\xi<\omega$ and Bagaria proved the case in which $\xi<\kappa$. \mathfrak{b}egin{theorem}\langlebel{theorem_universal} Suppose $\kappa>\omega$ is a regular cardinal and $\xi$ is an ordinal with $\xi<\kappa^+$. For each $n<\omega$ there is a universal ${\mathbb P}i^1_\xi$ formula ${\mathbb P}si^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)$ and a universal $\mathbb{S}igma^1_\xi$ formula $\mathfrak{b}ar{{\mathbb P}si}^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)$ at $\kappa$ for formulas with $n$ free second-order variables. \end{theorem} \mathfrak{b}egin{proof} The case in which $\xi<\kappa$ follows directly from the proof of \cite[Proposition 4.4]{MR3894041}. Suppose $\xi=\zeta+1$ is a successor ordinal with $\kappa<\zeta+1<\kappa^+$ and the result holds for all $\eta\leq\zeta$. Let us show that there is a universal ${\mathbb P}i^1_{\zeta+1}$ formula at $\kappa$ for formulas with $n$ free second-order variables; a similar argument works for $\mathbb{S}igma^1_{\zeta+1}$ formulas, which we leave to the reader. Let $\mathfrak{b}ar{{\mathbb P}si}_{\zeta,n+1}^\kappa(X_1,\ldots,X_n,X_{n+1},Y_\xi)$ be a universal $\mathbb{S}igma^1_\zeta$ formula at $\kappa$ for formulas with $n+1$ free second-order variables obtained from the induction hypothesis. We will show that \[{\mathbb P}si_{\zeta+1,n}^\kappa(X_1,\ldots,X_n,Y_\xi)=\forall W\mathfrak{b}ar{{\mathbb P}si}_{\zeta,n+1}^\kappa(X_1,\ldots,X_n,W,Y_\xi)\] is the desired formula. Suppose $\varphi(X_1,\ldots,X_n)=\forall W\varphi_\zeta(X_1,\ldots,X_n,W)$ is any ${\mathbb P}i^1_{\zeta+1}$ formula with $n$ free second-order variables, where $\varphi_\zeta$ is $\mathbb{S}igma^1_\zeta$ with $n+1$ free second-order variables.\footnote{Note that if we had here a block of quantifiers $\forall W_1\cdots\forall W_k$, they could be collapsed to a single one by modifying $\varphi_\zeta$ without changing the fact that $\varphi_\zeta$ is $\mathbb{S}igma^1_\zeta$.} Let $C_{\varphi_\zeta}$ and $K_{\varphi_\zeta}$ be as obtained from the inductive hypothesis. Fix $A_1,\ldots,A_n{\mathop{\rm sub}}seteq V_\kappa$. Then for all regular $\alpha\in C_{\varphi_\zeta}\cup\{\kappa\}$ we have \mathfrak{b}egin{align*} V_\alpha\models\varphi(A_1,\ldots,A_n)\mathrm{|}^\kappa_\alpha&\iff (\forall W{\mathop{\rm sub}}seteq V_\alpha) V_\alpha\models \varphi_\zeta(A_1,\ldots,A_n,W)\mathrm{|}^\kappa_\alpha\\ &\iff (\forall W{\mathop{\rm sub}}seteq V_\alpha) V_\alpha\models \mathfrak{b}ar{{\mathbb P}si}_{\zeta,n+1}^\kappa(A_1,\ldots,A_n,W,K_{\varphi_\zeta})\mathrm{|}^\kappa_\alpha\\ &\iff V_\alpha\models \forall W\mathfrak{b}ar{{\mathbb P}si}_{\zeta,n+1}^\kappa(A_1,\ldots,A_n,W,K_{\varphi_\zeta})\mathrm{|}^\kappa_\alpha\\ &\iff V_\alpha\models{\mathbb P}si_{\zeta+1,n}^\kappa(A_1,\ldots,A_n, K_{\varphi_\zeta})\mathrm{|}^\kappa_\alpha, \end{align*} which establishes the successor case of the induction. Suppose $\xi$ is a limit ordinal with $\kappa\leq\xi<\kappa^+$ and the result holds for all $\zeta<\xi$. We will show that there is a universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with $1$ free second-order variable; the proof for $n$ free second-order variables is essentially the same but one must replace the single variable $X$ with a tuple $X_1,\ldots,X_n$ in the appropriate places. We let $\Gamma:\kappa\times\kappa\to\kappa$ be the usual definable pairing function and for $A{\mathop{\rm sub}}seteq\kappa$ and $\eta<\kappa$ we let \[(A)_\eta=\{\mathfrak{b}eta<\kappa:\Gamma(\eta,\mathfrak{b}eta)\in A\}\] be the ``$\eta^{th}$ slice'' of $A$. Suppose $\zeta<\xi$. Using the inductive hypothesis, we let ${\mathbb P}si^\kappa_{\zeta,1}$ be a universal ${\mathbb P}i^1_\zeta$ formula at $\kappa$ for formulas with $1$ free variable. We will define the desired universal formula ${\mathbb P}si_{\xi,1}^\kappa(X,Y_\xi)$ by simply taking the conjunction of the ${\mathbb P}si^\kappa_{\zeta,1}$'s for $\zeta<\xi$, with the proviso that we must take care to use the right slice of the code $K_\varphi{\mathop{\rm sub}}seteq\kappa$ we will define for an arbitrary ${\mathbb P}i^1_\xi$ formula $\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta$, where $(K_\varphi)_{b_{\kappa,\xi}^{-1}(\zeta)}=K_{\varphi_\zeta}$. With this in mind, note that we will define ${\mathbb P}si_{\xi,1}^\kappa(X,Y_\xi)$ in such a way that it is equivalent to \[\mathfrak{b}igwedge_{\zeta<\xi}{\mathbb P}si^\kappa_{\zeta,1}(X,(Y_\xi)_{b_{\kappa,\xi}^{-1}(\zeta)})\] over $V_\kappa$. In order to verify that the following definition of ${\mathbb P}si^\kappa_{\xi,1}$ produces a ${\mathbb P}i^1_\xi$ formula over $V_\kappa$, we must check that ${\mathbb P}si^\kappa_{\zeta,1}(X,(Y_\xi)_{b_{\kappa,\xi}^{-1}(\zeta)})$ is expressible by a ${\mathbb P}i^1_\zeta$ formula over $V_\kappa$. Suppose $\zeta<\xi$. Notice that for $A{\mathop{\rm sub}}seteq V_\kappa$ and $B{\mathop{\rm sub}}seteq\kappa$ the sentence ${\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)})$ is equivalent to \[\forall Y(Y=(B)_{b_{\kappa,\xi}^{-1}(\zeta)}\rightarrow {\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)})),\] over $V_\kappa$ where $Y=(B)_{b_{\kappa,\xi}^{-1}(\zeta)}$ is expressible as a ${\mathbb P}i^1_0$ formula over $V_\kappa$ using $b_{\kappa,\xi}^{-1}(\zeta)$ as a parameter and using a first order quantifier over ordinals. Thus, by Lemma \ref{lemma_no_increase}, we may let ${\mathcal T}heta^\kappa_\zeta(X,Y_\xi)$ be a ${\mathbb P}i^1_\xi$ formula over $V_\kappa$ such that for all $A{\mathop{\rm sub}}seteq V_\kappa$ and $B{\mathop{\rm sub}}seteq\kappa$, $V_\kappa\models{\mathcal T}heta_\zeta^\kappa(A,B)$ if and only if $V_\kappa\models {\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)})$, and furthermore, we can assume ${\mathcal T}heta_\zeta^\kappa$ has the property that there is a club $D_\zeta$ in $\kappa$ such that for all regular $\alpha\in D_\zeta$ and all $A,B{\mathop{\rm sub}}seteq V_\kappa$ we have \[V_\alpha\models {\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)})\mathrm{|}^\kappa_\alpha\text{ if and only if }V_\kappa\models{\mathcal T}heta_\zeta^\kappa(A,B)\mathrm{|}^\kappa_\alpha.\] Now let us check that \[{\mathbb P}si_{\xi,1}^\kappa(X,Y_\xi)=\mathfrak{b}igwedge_{\zeta<\xi}{\mathcal T}heta_\zeta^\kappa(X,Y_\xi)\] is a universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with $1$ free variable. Suppose $\varphi(X)=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta(X)$ is any ${\mathbb P}i^1_\xi$ formula over $V_\kappa$ with one free second-order variable, where each $\varphi_\zeta$ is ${\mathbb P}i^1_\zeta$ over $V_\kappa$. Let $K_\varphi=\{\Gamma(b_{\kappa,\xi}^{-1}(\zeta),\mathfrak{b}eta):\zeta<\xi\langlend\mathfrak{b}eta\in K_{\varphi_\zeta}\}$ code the sequence $\langleK_{\varphi_\zeta}:\zeta<\xi\rangle$, where the codes $K_{\varphi_\zeta}$ are obtained by the inductive hypothesis. Notice that for each $\zeta<\xi$ we have $(K_\varphi)_{b_{\kappa,\xi}^{-1}(\zeta)}=K_{\varphi_\zeta}$. Fix $A{\mathop{\rm sub}}seteq V_\kappa$. It follows easily from the definitions of $K_\varphi$ and ${\mathbb P}si_{\xi,1}^\kappa(X,Y_\xi)$ that \mathfrak{b}egin{align*} V_\kappa\models{\mathbb P}si_{\xi,1}^\kappa(A,K_\varphi)&\iff V_\kappa\models\mathfrak{b}igwedge_{\zeta<\xi}{\mathbb P}si^\kappa_{\zeta,1}(A,(K_{\varphi})_{b_{\kappa,\xi}^{-1}(\zeta)})\\ &\iff V_\kappa\models\mathfrak{b}igwedge_{\zeta<\xi}{\mathbb P}si^\kappa_{\zeta,1}(A,(K_{\varphi_\zeta}))\\ &\iff V_\kappa\models\varphi(A). \end{align*} Next let us show that there is a club $C$ in $\kappa$ such that for all regular $\alpha\in C$ we have $V_\alpha\models\varphi(A)\mathrm{|}^\kappa_\alpha$ if and only if $V_\alpha\models{\mathbb P}si_{\xi,1}^\kappa(A,K_\varphi)\mathrm{|}^\kappa_\alpha$. To prove that such a club exists we use Proposition \ref{proposition_framework2}. Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic over $V$ such that $\kappa$ is regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. We must show that $\kappa\in j(T)$ where \mathfrak{b}egin{align} T=\{\alpha\in{\rm REG}: V_\alpha\models\varphi(A)\mathrm{|}^\kappa_\alpha\iff V_\alpha\models{\mathbb P}si_{\xi,1}^\kappa(A,K_\varphi)\mathrm{|}^\kappa_\alpha\}. \end{align} By Lemma \ref{lemma_j_of} we have \[j(\varphi(A))\mathrm{|}^{j(\kappa)}_\kappa=\varphi(A)=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta(A)\] and \[j({\mathbb P}si_{\xi,1}^\kappa(A,K_\varphi))\mathrm{|}^{j(\kappa)}_\kappa={\mathbb P}si_{\xi,1}^\kappa(A,K_\varphi)=\mathfrak{b}igwedge_{\zeta<\xi}{\mathcal T}heta_\zeta^\kappa(A,K_\varphi).\] By our inductive hypothesis, for each $\zeta<\xi$ there is a club $C_{\varphi_\zeta}$ in $\kappa$ such that for all regular $\alpha\in C_{\varphi_\zeta}$ we have $V_\alpha\models\varphi_\zeta(A)\mathrm{|}^\kappa_\alpha$ if and only if $V_\alpha\models{\mathbb P}si_{\zeta,1}^\kappa(A,K_{\varphi_\zeta})\mathrm{|}^\kappa_\alpha$. Since for all $\zeta<\xi$ we have $\kappa\in j(C_{\varphi_\zeta}\cap D_\zeta)$ and $(K_\varphi)_{b_{\kappa,\xi}^{-1}(\zeta)}=K_{\varphi_\zeta}$, it follows that in $V^\kappa/G$ we have \mathfrak{b}egin{align*} V_\kappa\models\varphi_\zeta(A)&\iff V_\kappa\models{\mathbb P}si_{\zeta,1}^\kappa(A,(K_\varphi)_{b_{\kappa,\xi}^{-1}(\zeta)})\\ &\iff V_\kappa\models{\mathcal T}heta_\zeta^\kappa(A,K_\varphi). \end{align*} Hence $\kappa\in j(T)$. \end{proof} \mathfrak{b}egin{remark}\langlebel{remark_universal_formula_depends_on_bijections} Notice that contained within the proof of Theorem \ref{theorem_universal} is a construction via transfinite recursion on $\xi$ of a universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with $n$ free second-order variables. Furthermore, when $\xi$ is a limit, let us emphasize that the definition of ${\mathbb P}si^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)$ depends not only on the chosen bijection $b_{\kappa,\xi}:\kappa\to\xi$, but on the entire history of bijections $b_{\kappa,\zeta}:\kappa\to\zeta$ chosen at previous limit steps $\zeta<\xi$ in the construction. \end{remark} Generalizing work of Bagaria \cite{MR3894041}, as our first application of the existence of universal formulas, we show that there are natural normal ideals on $\kappa$ associated to ${\mathbb P}i^1_\xi$-indescribability for all $\xi<\kappa^+$. \mathfrak{b}egin{theorem}\langlebel{theorem_normal_ideal} If a cardinal $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable where $\xi<\kappa^+$, then the collection \[{\mathbb P}i^1_\xi(\kappa)=\{X{\mathop{\rm sub}}seteq\kappa:\text{$X$ is not ${\mathbb P}i^1_\xi$-indescribable}\}\] is a nontrivial normal ideal on $\kappa$. \end{theorem} \mathfrak{b}egin{proof} Suppose $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable where $\xi<\kappa^+$. It is easy to see that \[{\mathbb P}i^1_\xi(\kappa)=\{X{\mathop{\rm sub}}seteq\kappa:\text{$X$ is not ${\mathbb P}i^1_\xi$-indescribable}\}\] is a nontrivial ideal on $\kappa$, so we just need to prove it is normal. Suppose $S\in{\mathbb P}i^1_\xi(\kappa)^+$ and fix a regressive function $f:S\to\kappa$. For the sake of contradiction, assume that for all $\eta<\kappa$ the set $f^{-1}(\{\eta\})=\{\alpha\in S: f(\alpha)=\eta\}$ is not in ${\mathbb P}i^1_\xi(\kappa)^+$. Then, for each $\eta<\kappa$ there is some ${\mathbb P}i^1_\xi$ formula $\varphi_\eta(X)$ over $V_\kappa$ and some $A_\eta{\mathop{\rm sub}}seteq V_\kappa$ such that $V_\kappa\models\varphi_\eta(A_\eta)$ but \mathfrak{b}egin{align}V_\alpha\models\lnot\varphi_\eta(A_\eta)\mathrm{|}^\kappa_\alpha\text{ for all }\alpha\in S \text{ such that }f(\alpha)=\eta.\langlebel{equation_will_contradict} \end{align} Let ${\mathbb P}si^\kappa_{\xi,1}(X,Y_\xi)$ be the universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with one free second-order variable, let $K_{\varphi_\eta}{\mathop{\rm sub}}seteq\kappa$ be the code for $\varphi_\eta$ and let $C_{\varphi_\eta}$ be the club subset of $\kappa$ as in Definition \ref{definition_universal}. Then for all $\eta<\kappa$ we have \[V_\kappa\models{\mathbb P}si^\kappa_{\xi,1}(A_\eta,K_{\varphi_\eta}).\] We would like to show that the formula $\mathfrak{b}igwedge_{\eta<\kappa}{\mathbb P}si^\kappa_{\xi,1}(A_\eta,K_{\varphi_\eta})$ is equivalent to a single ${\mathbb P}i^1_\xi$ formula over $V_\kappa$. Let $A=\{\Gamma(\eta,\mathfrak{b}eta): \eta<\kappa\langlend\mathfrak{b}eta\in A_\eta\}{\mathop{\rm sub}}seteq\kappa$ and $K=\{\Gamma(\eta,\mathfrak{b}eta):\eta<\kappa\langlend\mathfrak{b}eta\in K_{\varphi_\eta}\}{\mathop{\rm sub}}seteq\kappa$ code the sequences $\langleA_\eta:\eta<\kappa\rangle$ and $\langleK_{\varphi_\eta}:\eta<\kappa\rangle$ respectively. Let \[C=\mathop{\text{\Large$\mathfrak{b}igtriangleup$}}_{\eta<\kappa}C_{\varphi_\eta}=\{\zeta<\kappa:\zeta\in\mathfrak{b}igcap_{\eta<\zeta}C_{\varphi_\eta}\}\] and notice that $C$ is in the club filter on $\kappa$. By a straightforward application of Lemma \ref{lemma_no_increase}, there is a ${\mathbb P}i^1_\xi$ sentence $\varphi(A,K,C)$ such that \[V_\kappa\models\varphi(A,K,C) \text{ if and only if } V_\kappa\models\mathfrak{b}igwedge_{\eta<\kappa}{\mathbb P}si^\kappa_{\xi,1}(A_\eta,K_{\varphi_\eta}),\] and furthermore, there is a club $D{\mathop{\rm sub}}seteq\kappa$ such that for all regular $\alpha\in D$ we have \[V_\alpha\models\varphi(A,K,C)\mathrm{|}^\kappa_\alpha\text{ if and only if }V_\alpha\models\mathfrak{b}igwedge_{\eta<\alpha}{\mathbb P}si^\kappa_{\xi,1}(A_\eta,K_{\varphi_\eta})\mathrm{|}^\kappa_\alpha.\] Since $S$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$, there is some regular $\alpha\in S\cap C\cap D$ such that $V_\alpha\models\varphi(A,K,C)\mathrm{|}^\kappa_\alpha$. Since $\alpha\in D$ we have $V_\alpha\models \mathfrak{b}igwedge_{\eta<\alpha}{\mathbb P}si_{\xi,1}^\kappa(A_\eta,K_{\varphi_\eta})\mathrm{|}^\kappa_\alpha$ and since $\alpha\in C$ we have $V_\alpha\models\mathfrak{b}igwedge_{\eta<\alpha} \varphi_\eta(A_\eta)$, which contradicts (\ref{equation_will_contradict}) since $f(\alpha)<\alpha$. \end{proof} As an easy consequence of Theorem \ref{theorem_normal_ideal} we obtain the following a characterization of ${\mathbb P}i^1_\xi$-indescribable subsets of a cardinal in terms of generic elementary embeddings, which we will use below to characterize ${\mathbb P}i^1_\xi$-indescribable sets in terms of a natural filter base (see Theorem \ref{theorem_xi_clubs}(1)). \mathfrak{b}egin{proposition}\langlebel{proposition_generic_characterization} Suppose $\kappa$ is a regular cardinal, $\xi<\kappa^+$ and $S{\mathop{\rm sub}}seteq\kappa$. The following are equivalent. \mathfrak{b}egin{enumerate} \item The set $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. \item There is some poset ${\mathbb P}$ such that whenever $G{\mathop{\rm sub}}seteq {\mathbb P}$ is generic over $V$, there is an elementary embedding $j:V\to M{\mathop{\rm sub}}seteq V[G]$ in $V[G]$ with critical point $\kappa$ such that \mathfrak{b}egin{enumerate} \item $\kappa\in j(S)$ and \item for all ${\mathbb P}i^1_\xi$ sentences $\varphi$ over $V_\kappa$ in $V$ we have $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi$ and \[(V_\kappa\models\varphi)^V \implies (V_\kappa\models\varphi)^M.\] \end{enumerate} \end{enumerate} \end{proposition} \mathfrak{b}egin{proof} Suppose $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathbb P}i^1_\xi(\kappa)$ be generic over $V$ with $S\in G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. Note that the normality of the ideal ${\mathbb P}i^1_\xi(\kappa)$ implies that the critical point of $j$ is $\kappa$ and $\kappa\in j(S)$. Suppose $\varphi$ is a ${\mathbb P}i^1_\xi$ sentence over $V_\kappa$ and $V_\kappa\models\varphi$. Then the set \[C=\{\alpha<\kappa:\text{$\varphi\mathrm{|}^\kappa_\alpha$ is defined and $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$}\}\] is in the filter dual to ${\mathbb P}i^1_\xi(\kappa)$. Thus $\kappa\in j(C)$ and since $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi$ by Lemma \ref{lemma_j_of}, we see that (2b) holds. Conversely, let $j$ be as in (2). Fix a ${\mathbb P}i^1_\xi$ sentence $\varphi$ over $V_\kappa$ with $V_\kappa\models\varphi$. Then it follows by (2) that, in $M$, there is some $\alpha\in j(S)$ such that $V_\alpha\models j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha$. Hence by elementarity, there is an $\alpha\in S$ such that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$. \end{proof} {\mathop{\rm sub}}section{A hierarchy result}\langlebel{section_hierarchy} In order to prove the hierarchy results below (Corollary \ref{corollary_hierarchy} and Corollary \ref{corollary_proper}), we first need to establish a connection between universal formulas at $\kappa$ and universal formulas at regular $\alpha<\kappa$. \mathfrak{b}egin{comment} \mathfrak{b}egin{lemma}\langlebel{lemma_elementary} Suppose $\kappa$ is regular and $\xi<\kappa^+$. The set of all $M\prec H_{\kappa^+}$ such that $|M|<\kappa$, $M\cap\kappa<\kappa$ and $M\cap \kappa^+=F^\kappa_\xi(M\cap\kappa)$ is club in $[H_{\kappa^+}]^{<\kappa}$. \end{lemma} \mathfrak{b}egin{proof} Let $C$ be the collection of all $M\prec H_{\kappa^+}$ such that $|M|<\kappa$ and $M\cap\kappa^+=F^\kappa_\xi(\kappa)$. Suppose $\langleM_\zeta:\zeta<\gamma\rangle$ is an increasing chain of elements of $C$ where $\gamma<\kappa$ and set $M=\mathfrak{b}igcup_{\zeta<\gamma}M_\zeta$. Clearly we have $M\prec H_{\kappa^+}$, $|M|<\kappa$ and $M\cap\kappa=\mathfrak{b}igcup_{\zeta<\gamma}M_\zeta\cap\kappa<\kappa$. Furthermore, \[M\cap\kappa^+=\mathfrak{b}igcup_{\zeta<\gamma}M_\zeta\cap\kappa^+=\mathfrak{b}igcup_{\zeta<\gamma}F^\kappa_\xi(M_\zeta\cap\kappa)=F^\kappa_\xi(M\cap\kappa),\] and hence $C$ is a closed subset of $[H_{\kappa^+}]^{<\kappa}$. To see that $C$ is unbounded in $[H_{\kappa^+}]^{<\kappa}$, fix $a\in H_{\kappa^+}$. Let $M_0\prec H_{\kappa^+}$ be such that $a\in M_0$ and $M_0\cap\kappa<\kappa$. Choose $\alpha_0<\kappa$ large enough so that $M_0\cap\kappa^+{\mathop{\rm sub}}seteq F^\kappa_\xi(\alpha_0)$. Given that $M_n$ and $\alpha_n$ have been defined where $n<\omega$, let $M_{n+1}\prec H_{\kappa^+}$ be such that $M_n\cup F^\kappa_\xi(\alpha_n){\mathop{\rm sub}}seteq M_{n+1}$ and let $\alpha_{n+1}\in\kappa$\hat{\text{s}}$etminus(\alpha_n+1)$ be such that $M_{n+1}\cap\kappa^+{\mathop{\rm sub}}seteq F^\kappa_\xi(\alpha_{n+1})$. Now notice that if we let $M_\omega=\mathfrak{b}igcup_{n<\omega}M_n$ we have $a\in M\prec H_{\kappa^+}$, $M\cap\kappa=\mathfrak{b}igcup_{n<\omega}\alpha_n<\kappa$ and $M\cap\kappa^+=F^\kappa_\xi(M\cap\kappa)$. \end{proof} \mathfrak{b}egin{corollary} Suppose $\kappa$ is a regular cardinal, $\xi<\kappa^+$ is a limit ordinal and $b_{\kappa,\xi}:\kappa\to\xi$ is a bijection. There is a club $C$ in $\kappa$ such that for all $\alpha\in C$ there is an $M\prec H_{\kappa^+}$ such that $M$ contains $\kappa$, $\xi$ and $b_{\kappa,\xi}$ as elements, and the following hold. \mathfrak{b}egin{enumerate} \item $M\cap\kappa=\alpha$ \item $M\cap\kappa^+=F^\kappa_\xi(\alpha)$ \item If $\pi_M:M\to N$ is the transitive collapse of $M$ then $\pi_M\mathrm{|}trict F^\kappa_\xi(\alpha)$ is the transitive collapse of $F^\kappa_\xi(\alpha)$ and \[\pi_M(b_{\kappa,\xi}):\alpha\to f^\kappa_\xi(\alpha)\] is a bijection. \end{enumerate} \end{corollary} \mathfrak{b}egin{proof} Suppose $b_{\kappa,\xi}:\kappa\to\xi$ is a bijection and let $D$ be the club subset of $[H_{\kappa^+}]^{<\kappa}$ from the statement of Lemma \ref{lemma_elementary}. We can build an increasing continuous chain $\langleM_\zeta:\zeta<\kappa\rangle$ of elementary substructures of $H_{\kappa^+}$ such that $\kappa,\xi,b_{\kappa,\xi}\in M_0$. It is easy to see that $C=\{M_\zeta\cap\kappa:\zeta<\kappa\}$ is the desired club subset of $\kappa$. \end{proof} \end{comment} \mathfrak{b}egin{lemma}\langlebel{lemma_restricting_universal_formulas} Suppose $\kappa>\omega$ is regular. Fix any $\xi<\kappa^+$ and $n<\omega$, let ${\mathbb P}si^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)$ and $\mathfrak{b}ar{{\mathbb P}si}^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)$ be, respectively, universal ${\mathbb P}i^1_\xi$ and $\mathbb{S}igma^1_\xi$ formulas at $\kappa$ for formulas with $n$ free second-order variables, which were defined by transfinite recursion in the proof of Theorem \ref{theorem_universal}. There are clubs $C_{\xi,n}$ and $D_{\xi,n}$ in $\kappa$ such that the following hold. \mathfrak{b}egin{enumerate} \item For all regular $\alpha\in C_{\xi,n}$ the formula ${\mathbb P}si^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)\mathrm{|}^\kappa_\alpha$ is a universal ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ formula at $\alpha$ for formulas with $n$ free second-order variables. \item For all regular $\alpha\in D_{\xi,n}$ the formula $\mathfrak{b}ar{{\mathbb P}si}^\kappa_{\xi,n}(X_1,\ldots,X_n,Y_\xi)\mathrm{|}^\kappa_\alpha$ is a universal $\mathbb{S}igma^1_{f^\kappa_\xi(\alpha)}$ formula at $\alpha$ for formulas with $n$ free second-order variables. \end{enumerate} \end{lemma} \mathfrak{b}egin{proof} We proceed by induction on $\xi$. Suppose $\xi<\kappa^+$ is a limit ordinal. The case in which $\xi$ is a successor ordinal is easier and is left to the reader. We will now prove (1); the proof of (2) is similar. Recall that \[{\mathbb P}si^\kappa_{\xi,1}(X,Y_\xi)=\mathfrak{b}igwedge_{\zeta<\xi}{\mathcal T}heta^\kappa_{\zeta,1}(X,Y_\xi)\] where each ${\mathcal T}heta^\kappa_{\zeta,1}(X,Y_\xi)$ is a ${\mathbb P}i^1_\zeta$ formula over $V_\kappa$ equivalent to ${\mathbb P}si^\kappa_{\zeta,1}(X,(Y_\xi)_{b_{\kappa,\xi}^{-1}(\zeta)})$ in the sense that there is a club $D_\zeta$ in $\kappa$ such that for all regular $\alpha\in D_\zeta\cup\{\kappa\}$ we have $V_\alpha\models{\mathcal T}heta^\kappa_{\zeta,1}(A,B)\mathrm{|}^\kappa_\alpha$ if and only if $V_\alpha\models {\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)})\mathrm{|}^\kappa_\alpha$ for all $A{\mathop{\rm sub}}seteq V_\kappa$ and $B{\mathop{\rm sub}}seteq\kappa$. To prove (1), we will use Proposition \ref{proposition_framework2}. Suppose $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ is generic such that $\kappa$ is regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. To show that the desired club $C_{\xi,1}$ exists, we must show that $\kappa\in j(T)$ where $T$ is the set of regular cardinals $\alpha<\kappa$ such that ${\mathbb P}si^\kappa_{\xi,1}(X,Y_\xi)\mathrm{|}^\kappa_\alpha$ is a universal ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ formula at $\alpha$ for formulas with $1$ free variable. By Lemma \ref{lemma_j_of} we have $j({\mathbb P}si^\kappa_{\xi,1}(X,Y_\xi))\mathrm{|}^{j(\kappa)}_\kappa={\mathbb P}si^\kappa_{\xi,1}(X,Y_\xi)$ and $\kappa\in j(D_\zeta)$ for all $\zeta<\xi$. Thus, working in $V^\kappa/G$, if $A{\mathop{\rm sub}}seteq V_\kappa$ and $B{\mathop{\rm sub}}seteq\kappa$ then \mathfrak{b}egin{align*} V_\kappa\models{\mathbb P}si^\kappa_{\xi,1}(A,B)&\iff V_\kappa\models \mathfrak{b}igwedge_{\zeta<\xi}j({\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)}))\mathrm{|}^{j(\kappa)}_\kappa\\ &\iff V_\kappa\models \mathfrak{b}igwedge_{\zeta<\xi}{\mathbb P}si^\kappa_{\zeta,1}(A,(B)_{b_{\kappa,\xi}^{-1}(\zeta)}).\\ \end{align*} Now it is straightforward to verify $\kappa\in j(T)$, that is, ${\mathbb P}si^\kappa_{\xi,1}(X,Y_\xi)$ is a universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ for formulas with $1$ free variable in $V^\kappa/G$; we give a brief outline of how to do this here. Still working in $V^\kappa/G$, fix a ${\mathbb P}i^1_\xi$ formula $\varphi_\xi(X)=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta(X)$. From our inductive assumption, working in $V^\kappa/G$, we may fix codes $K_{\varphi_\zeta}{\mathop{\rm sub}}seteq\kappa$ such that $V_\kappa\models\varphi_\zeta(A)$ if and only if $V_\kappa\models{\mathbb P}si^\kappa_{\zeta,1}(A,K_{\varphi_\zeta})$. Then we let $K_\varphi=\{\Gamma(b_{\kappa,\xi}^{-1}(\zeta),\mathfrak{b}eta):\zeta<\xi\langlend\mathfrak{b}eta\in K_{\varphi_\zeta}\}$ and proceed exactly as in the proof of Theorem \ref{theorem_universal}, except that here we work in $V^\kappa/G$. Thus we conclude $\kappa\in j(T)$. \end{proof} Next we show that for $\xi<\kappa^+$, the ${\mathbb P}i^1_\xi$-indescribability of a set $S{\mathop{\rm sub}}seteq\kappa$, is expressible by a ${\mathbb P}i^1_{\xi+1}$ formula over $V_\kappa$ in the following sense. \mathfrak{b}egin{theorem}\langlebel{theorem_expressing_indescribability} Suppose $\kappa>\omega$ is inaccessible and $\xi<\kappa^+$. There is a ${\mathbb P}i^1_{\xi+1}$ formula ${\mathbb P}hi^\kappa_\xi(Z)$ over $V_\kappa$ and a club $C{\mathop{\rm sub}}seteq\kappa$ such that for all $S{\mathop{\rm sub}}seteq\kappa$ we have \[\text{$S$ is a ${\mathbb P}i^1_\xi$-indescribable subset of $\kappa$ if and only if $V_\kappa\models{\mathbb P}hi^\kappa_\xi(S)$}\] and for all regular $\alpha\in C$ we have \[\text{$S\cap\alpha$ is a ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable subset of $\alpha$ if and only if $V_\alpha\models{\mathbb P}hi^\kappa_\xi(S)\mathrm{|}^\kappa_\alpha$}.\] \end{theorem} \mathfrak{b}egin{proof} We let $R{\mathop{\rm sub}}seteq\kappa$ be a set, defined as follows, coding information about which $\alpha<\kappa$ and which $a{\mathop{\rm sub}}seteq\alpha$ satisfy $V_\alpha\models{\mathbb P}si^\kappa_{\xi,0}(a)\mathrm{|}^\kappa_\alpha$. For each regular $\alpha<\kappa$ let $\langlea^\alpha_\mathfrak{b}eta:\mathfrak{b}eta<\mathfrak{d}elta_\alpha\rangle$ be a sequence of subsets of $\alpha$ such that for all $a{\mathop{\rm sub}}seteq\alpha$ we have $V_\alpha\models{\mathbb P}si^\kappa_{\xi,0}(a)\mathrm{|}^\kappa_\alpha$ if and only if $a=a^\alpha_\mathfrak{b}eta$ for some $\mathfrak{b}eta<\mathfrak{d}elta_\alpha$. Let $\Gamma:\kappa\times\kappa\times\kappa\to\kappa$ be the usual definable bijection. We let \[R=\{\Gamma(\alpha,\mathfrak{b}eta,\gamma): \text{($\alpha$ is regular)}\langlend \mathfrak{b}eta<\mathfrak{d}elta_\alpha\langlend \gamma\in a^\alpha_\mathfrak{b}eta\}.\] For $\alpha,\mathfrak{b}eta<\kappa$ we define \[R_{(\alpha,\mathfrak{b}eta)}=\{\gamma:\Gamma(\alpha,\mathfrak{b}eta,\gamma)\in R\}\] to be the $(\alpha,\mathfrak{b}eta)^{th}$ slice of $R$ so that when $\alpha$ is regular and $\mathfrak{b}eta<\mathfrak{d}elta_\alpha$ we have $R_{(\alpha,\mathfrak{b}eta)}=a^\alpha_\mathfrak{b}eta$. Now we let \[{\mathbb P}hi^\kappa_\xi(Z)=\forall X[{\mathbb P}si^\kappa_{\xi,0}(X)\rightarrow(\exists Y{\mathop{\rm sub}}seteq Z\cap{\rm REG})(Y\in{\mathop{\rm NS}}_\kappa^+)\langlend(\forall\eta\in Y)(\exists\mathfrak{b}eta)(X\cap\eta=R_{(\eta,\mathfrak{b}eta)})].\] Since the part of ${\mathbb P}hi^\kappa_\xi$ to the right of the $\rightarrow$ is $\mathbb{S}igma^1_2$ over $V_\kappa$, and since ${\mathbb P}si^\kappa_{\xi,0}(X)$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$ and appears to the left of the $\rightarrow$ in ${\mathbb P}hi^\kappa_\xi$, it follows that ${\mathbb P}hi^\kappa_\xi$ is expressible by a ${\mathbb P}i^1_{\xi+1}$ formula over $V_\kappa$. In what follows, we will identify ${\mathbb P}hi^\kappa_\xi$ with this ${\mathbb P}i^1_{\xi+1}$ formula. First let us show that $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$ if and only if $V_\kappa\models{\mathbb P}hi^\kappa_\xi(S)$. Suppose $S$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. To see that $V_\kappa\models{\mathbb P}hi^\kappa_\xi(S)$, fix $K{\mathop{\rm sub}}seteq\kappa$ such that $V_\kappa\models{\mathbb P}si^\kappa_{\xi,0}(K)$. Then $D_0=\{\alpha<\kappa: V_\alpha\models{\mathbb P}si^\kappa_{\xi,0}(K)\mathrm{|}^\kappa_\alpha\}$ is in the filter ${\mathbb P}i^1_\xi(\kappa)^*$ and thus $Y=S\cap D_0\cap{\rm REG}$ is, in particular, stationary in $\kappa$. If $\alpha\in Y$ then we have $V_\alpha\models{\mathbb P}si^\kappa_{\xi,0}(K)\mathrm{|}^\kappa_\alpha$ and hence $V_\alpha\models{\mathbb P}si^\kappa_{\xi,0}(K\cap\alpha)\mathrm{|}^\kappa_\alpha$, which implies that $K\cap\alpha=R_{(\alpha,\mathfrak{b}eta)}$ for some $\mathfrak{b}eta<\mathfrak{d}elta_\alpha$. Conversely, suppose $V_\kappa\models {\mathbb P}hi^\kappa_\xi(S)$ and let us show that $S$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. Fix a ${\mathbb P}i^1_\xi$ sentence $\varphi$ such that $V_\kappa\models\varphi$. Then, by Theorem \ref{theorem_universal}, $V_\kappa\models{\mathbb P}si^\kappa_{\xi,0}(K_\varphi)$ and thus there is a $Y{\mathop{\rm sub}}seteq S\cap{\rm REG}$ stationary in $\kappa$ such that for all $\alpha\in Y$ we have $V_\alpha\models{\mathbb P}si^\kappa_{\xi,0}(K_\varphi)\mathrm{|}^\kappa_\alpha$. By Theorem \ref{theorem_universal} there is a club $D_\varphi{\mathop{\rm sub}}seteq\kappa$ such that for all regular $\alpha\in D_\varphi$ we have $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$ if and only if $V_\alpha\models{\mathbb P}si_{\xi,0}(K_\varphi)\mathrm{|}^\kappa_\alpha$. Thus we may choose a regular $\alpha\in Y\cap D_\varphi\cap{\rm REG}{\mathop{\rm sub}}seteq S$ and observe that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$. Hence $S$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. To prove the second part of the statement we will use Proposition \ref{proposition_framework2}. Fix $S{\mathop{\rm sub}}seteq\kappa$. Suppose $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ is generic, $\kappa$ is regular in $V^\kappa/G$ and $j:V\to V^\kappa/G$ is the corresponding generic ultrapower. Let $E$ be the set of ordinals $\alpha<\kappa$ such that $S\cap\alpha$ is a ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable subset of $\alpha$ if and only if $V_\alpha\models{\mathbb P}hi^\kappa_{\xi,0}(S)\mathrm{|}^\kappa_\alpha$. We must show that $\kappa\in j(E)$. By Lemma \ref{lemma_j_of} we have $j({\mathbb P}hi^\kappa_{\xi,0}(S))\mathrm{|}^{j(\kappa)}_\kappa={\mathbb P}hi^\kappa_{\xi,0}(S)$, and thus we must show that in $V^\kappa/G$, $S$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$ if and only if $V_\kappa\models {\mathbb P}hi^\kappa_{\xi,0}(S)$. By Lemma \ref{lemma_restricting_universal_formulas}, it follows that in $V^\kappa/G$, ${\mathbb P}si^\kappa_{\xi,0}(X)$ is a universal ${\mathbb P}i^1_\xi$ formula at $\kappa$ and therefore we can proceed to verify $\kappa\in j(E)$ by using the argument in the previous paragraph, but working in $V^\kappa/G$. \end{proof} We obtain our first hierarchy result as an easy corollary of Theorem \ref{theorem_expressing_indescribability}. \mathfrak{b}egin{corollary}\langlebel{corollary_hierarchy} Suppose $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$ where $\xi<\kappa^+$ and let $\zeta<\xi$. Then the set \[C=\{\alpha<\kappa:\text{$S\cap\alpha$ is ${\mathbb P}i^1_{f^\kappa_\zeta(\alpha)}$-indescribable}\}\] is in the filter ${\mathbb P}i^1_\xi(\kappa)^*$. \end{corollary} \mathfrak{b}egin{proof} Since $\zeta<\xi$, it follows that $S$ is ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$, and thus $V_\kappa\models{\mathbb P}hi^\kappa_\zeta(S)$, where ${\mathbb P}hi^\kappa_\zeta(Z)$ is the ${\mathbb P}i^1_{\zeta+1}$ formula over $V_\kappa$ obtained from Theorem \ref{theorem_expressing_indescribability}. By Theorem \ref{theorem_expressing_indescribability}, there is a club $D$ in $\kappa$ such that for every regular $\alpha\in D$, \[\text{$S\cap\alpha$ is ${\mathbb P}i^1_{f^\kappa_\zeta(\alpha)}$-indescribable if and only if $V_\alpha\models{\mathbb P}hi^\kappa_\zeta(S)\mathrm{|}^\kappa_\alpha$}.\] Since the set \[D\cap\{\alpha<\kappa: V_\alpha\models{\mathbb P}hi^\kappa_\zeta(S)\mathrm{|}^\kappa_\alpha\}\] is in the filter ${\mathbb P}i^1_\zeta(\kappa)^*$, we see that \[\{\alpha<\kappa:\text{$S\cap\alpha$ is ${\mathbb P}i^1_{f^\kappa_\zeta(\alpha)}$-indescribable}\}\in{\mathbb P}i^1_\zeta(\kappa)^*{\mathop{\rm sub}}seteq {\mathbb P}i^1_\xi(\kappa)^*.\] \end{proof} Next, in order to show that when $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable, we have a proper containment ${\mathbb P}i^1_\zeta(\kappa){\mathop{\rm sub}}setneq{\mathbb P}i^1_\xi(\kappa)$ for all $\zeta<\xi$ (see Corollary \ref{corollary_proper}), we need several preliminary results. Before we show that the restriction of a restriction of a given ${\mathbb P}i^1_\xi$ formula $\varphi$, is often equal to a single restriction of $\varphi$, we need a lemma, which is established using an argument similar to that of Lemma \ref{lemma_j_of}. \mathfrak{b}egin{lemma}[{Cody-Holy \cite{cody_holy_2022}}]\langlebel{lemma_double_restriction} Suppose $I$ is a normal ideal on $\kappa$ and $G{\mathop{\rm sub}}seteq P(\kappa)/I$ is generic such that $\kappa$ is regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower. If $\varphi$ is either a ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ formula over $V_\kappa$ for some $\xi<\kappa^+$, and $\alpha<\kappa$ is regular such that $\varphi\mathrm{|}^\kappa_\alpha$ is defined, then \[j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha=\varphi\mathrm{|}^\kappa_\alpha,\] with the former being calculated in $V^\kappa/G$, and the latter being calculated in $V$. \end{lemma} \mathfrak{b}egin{proof} By induction on $\xi<\kappa^+$. This is immediate in case $\xi<\kappa$, for then by Remark \ref{remark_coding}(1), $j(\varphi(A_1,\ldots,A_n))=\varphi(j(A_1),\ldots,j(A_n))$, and thus $j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha=\varphi\mathrm{|}^\kappa_\alpha$ by the definition of the restriction operation in this case. It is also immediate for successor steps above $\kappa$, for then by Remark \ref{remark_coding}(2), $j(\forall\vec X\psi)=\forall\vec X j(\psi)$. At limit steps $\xi\ge\kappa$, if $\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\psi_\zeta$ is a ${\mathbb P}i^1_\xi$ formula, let $\vec\psi=\langlengle\psi_\zeta\mid\zeta<\xi\mathop{\rm ran}glengle$, and let $\vec\pi=\langlengle\pi^\kappa_{\xi,\alpha}\mid\alpha<\kappa\mathop{\rm ran}glengle$. Then, by Remark \ref{remark_coding}(3), $j(\varphi)=\mathfrak{b}igwedge_{\zeta<j(\xi)}j(\vec\psi)_\zeta$, and therefore, assuming for now that $j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha$ is defined, \[j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha=\mathfrak{b}igwedge_{\zeta\in j(f^\kappa_\xi)(\alpha)}j(\vec\psi)_{j(\vec\pi)_\alpha^{-1}(\zeta)}\mathrm{|}^{j(\kappa)}_\alpha=\mathfrak{b}igwedge_{\zeta\in j(f^\kappa_\xi)(\alpha)}j(\psi_{j^{-1}(j(\vec\pi)_\alpha^{-1}(\zeta))})\mathrm{|}^{j(\kappa)}_\alpha,\] using that $j(\vec\pi)_\alpha^{-1}[j(f^\kappa_\xi)(\alpha)]=j(F^\kappa_\xi)(\alpha){\mathop{\rm sub}}seteq j(F^\kappa_\xi)(\kappa)=j"\xi$. By our inductive hypothesis, for each $\gamma\in\xi$ and every regular $\alpha<\kappa$, $j(\psi_\gamma)\mathrm{|}^{j(\kappa)}_\alpha=\psi_\gamma\mathrm{|}^\kappa_\alpha$. Thus, \[j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha=\mathfrak{b}igwedge_{\zeta\in j(f^\kappa_\xi)(\alpha)}\psi_{j^{-1}(j(\vec\pi)_\alpha^{-1}(\zeta))}\mathrm{|}^\kappa_\alpha.\] Now, \[\varphi\mathrm{|}^\kappa_\alpha=\mathfrak{b}igwedge_{\zeta\in f^\kappa_\xi(\alpha)}\psi_{(\pi^\kappa_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^\kappa_\alpha.\] Since $\alpha<\kappa$ we have $j(f^\kappa_\xi)(\alpha)=f^\kappa_\xi(\alpha)$, and furthermore \[(\pi^\kappa_{\xi,\alpha})^{-1}[f^\kappa_\xi(\alpha)]=F^\kappa_\xi(\alpha)=(j^{-1}\circ j(\vec\pi)_\alpha^{-1})[j(f^\kappa_\xi)(\alpha)],\] showing the above restrictions of $\varphi$ and of $j(\varphi)$ to be equal,\footnote{Being somewhat more careful here, this in fact also uses that the maps $\pi^\kappa_{\xi,\alpha}$, $j$, and $j(\vec\pi)_\alpha$ are order-preserving, so that both of the above conjunctions are taken of the same formulas \emph{in the same order}.} and thus in particular also showing that $j(\varphi)\mathrm{|}^{j(\kappa)}_\alpha$ is defined, as desired. The case when $\varphi$ is a $\mathbb{S}igma^1_\xi$ formula is treated in exactly the same way. \end{proof} We can now easily deduce the following, which was originally established in an earlier version of this article using a different proof. The proof included below is due to the author and Peter Holy. \mathfrak{b}egin{proposition}\langlebel{proposition_double_restriction} Suppose $\kappa$ is weakly Mahlo, and $\xi<\kappa^+$. For any formula $\varphi$ which is either ${\mathbb P}i^1_\xi$ or $\mathbb{S}igma^1_\xi$ over $V_\kappa$, there is a club $D{\mathop{\rm sub}}seteq\kappa$ such that for all regular uncountable $\alpha\in D$, $\varphi\mathrm{|}^\kappa_\alpha$ is defined, and the set $D_\alpha$ of all ordinals $\mathfrak{b}eta<\alpha$ such that $(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta$ is defined and $(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta=\varphi\mathrm{|}^\kappa_\mathfrak{b}eta$, is in the club filter on $\alpha$. \end{proposition} \mathfrak{b}egin{proof} Assume for a contradiction that the conclusion of the proposition fails. By Lemma \ref{lemma_restriction_is_nice}, this means that there is a stationary set $T$ consisting of regular and uncountable cardinals $\alpha$ such that the set $D_\alpha$ has stationary complement $E_\alpha{\mathop{\rm sub}}seteq\alpha$. Using Lemma \ref{lemma_restriction_is_nice} once again, we may assume that $(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta$ is defined for every $\alpha\in T$ and every $\mathfrak{b}eta\in E_\alpha$. Let $\vec E$ denote the sequence $\langlengle E_\alpha\mid\alpha\in T\mathop{\rm ran}glengle$. Assume that $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ is generic over $V$ with $T\in G$ and $j:V\to V^\kappa/G$ is the corresponding generic ultrapower. Then, $\kappa\in j(T)$, and thus $j(\vec E)_\kappa$ is stationary in $V^\kappa/G$. But, \[j(\vec E)_\kappa=\{\mathfrak{b}eta<\kappa\mid(j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa)\mathrm{|}^\kappa_\mathfrak{b}eta\ne j(\varphi)\mathrm{|}^{j(\kappa)}_\mathfrak{b}eta\}.\] Note that by Lemma \ref{lemma_double_restriction}, $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi$. But then, by Lemma \ref{lemma_restriction_is_nice} and Lemma \ref{lemma_double_restriction}, $j(\vec E)_\kappa$ is nonstationary in $V^\kappa/G$, which gives our desired contradiction. \end{proof} Recall that for an uncountable regular cardinal $\kappa$, if $S{\mathop{\rm sub}}seteq\kappa$ is stationary in $\kappa$ and for each $\alpha\in S$ we have a set $S_\alpha{\mathop{\rm sub}}seteq\alpha$ which is stationary in $\alpha$, then it follows that $\mathfrak{b}igcup_{\alpha\in S}S_\alpha$ is stationary in $\kappa$. We generalize this to ${\mathbb P}i^1_\xi$-indescribability for all $\xi<\kappa^+$ as follows (this result was previously known \cite[Lemma 3.1]{MR4206111} for $\xi<\kappa$). \mathfrak{b}egin{lemma}\langlebel{lemma_union} Suppose $S$ is a ${\mathbb P}i^1_\xi$-indescribable subset of $\kappa$ where $\xi<\kappa^+$. Further suppose that $S_\alpha$ is a ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable subset of $\alpha$ for each $\alpha\in S$. Then $\mathfrak{b}igcup_{\alpha\in S}S_\alpha$ is a ${\mathbb P}i^1_\xi$-indescribable subset of $\kappa$. \end{lemma} \mathfrak{b}egin{proof} Suppose $\xi<\kappa^+$ and $\varphi$ is some ${\mathbb P}i^1_\xi$ sentence over $V_\kappa$ such that $V_\kappa\models\varphi$. By Lemma \ref{lemma_restriction_is_nice}, \[C_\varphi=\{\alpha<\kappa:\text{$\varphi\mathrm{|}^\kappa_\alpha$ is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ over $V_\kappa$}\}\] is in the club filter on $\kappa$. By Proposition \ref{proposition_double_restriction}, there is a club $D_\varphi{\mathop{\rm sub}}seteq\kappa$ such that for all regular $\alpha\in D_\varphi$ the set of $\mathfrak{b}eta<\alpha$ such that $(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta=\varphi\mathrm{|}^\kappa_\mathfrak{b}eta$ is in the club filter on $\alpha$. Thus, $S\cap C_\varphi\cap D_\varphi$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. Hence there is a regular uncountable $\alpha\in S\cap C_\varphi\cap D_\varphi$ such that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$. Let $E$ be a club subset of $\alpha$ such that for all $\mathfrak{b}eta\in E$ we have $(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta=\varphi\mathrm{|}^\kappa_\mathfrak{b}eta$. Since $S_\alpha\cap E$ is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable in $\alpha$ and $\varphi\mathrm{|}^\kappa_\alpha$ is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$ over $V_\alpha$, there is some $\mathfrak{b}eta\in S_\alpha\cap E$ such that $V_\mathfrak{b}eta\models(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta$. Since $(\varphi\mathrm{|}^\kappa_\alpha)\mathrm{|}^\alpha_\mathfrak{b}eta=\varphi\mathrm{|}^\kappa_\mathfrak{b}eta$, it follows that $\mathfrak{b}igcup_{\alpha\in S}S_\alpha$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. \end{proof} \mathfrak{b}egin{lemma}\langlebel{lemma_set_of_non} For all ordinals $\xi$, if $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$ where $\xi<\kappa^+$, then the set \[T=\{\alpha<\kappa:\text{$S\cap\alpha$ is not ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable in $\alpha$}\}\] is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. \end{lemma} \mathfrak{b}egin{proof} We proceed by induction on $\xi$. For $\xi<\omega$ this is a well-known result, which follows directly from \cite[Lemma 3.2]{MR4206111}. Suppose $\xi\in\kappa^+$\hat{\text{s}}$etminus\omega$ and, for the sake of contradiction, suppose $S$ is ${\mathbb P}i^1_\xi$-indescribable and $T$ is not ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. Then $\kappa$\hat{\text{s}}$etminus T$ is in the filter ${\mathbb P}i^1_\xi(\kappa)^*$ and is thus ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. By Corollary \ref{corollary_crazy}, there is a club $C{\mathop{\rm sub}}seteq\kappa$ such that for all regular uncountable $\alpha\in C$, the set \[D_\alpha=\{\mathfrak{b}eta<\alpha: f^\kappa_\xi(\mathfrak{b}eta)=f^\alpha_{f^\kappa_\xi(\alpha)}(\mathfrak{b}eta)\}\] is in the club filter on $\alpha$. Let $D$ be the set of regular uncountable cardinals less than $\kappa$, and note that $D\in {\mathbb P}i^1_1(\kappa)^*{\mathop{\rm sub}}seteq{\mathbb P}i^1_\xi(\kappa)^*$. Notice that $(\kappa$\hat{\text{s}}$etminus T)\cap C\cap D$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. For each $\alpha\in (\kappa$\hat{\text{s}}$etminus T)\cap C\cap D$, it follows by induction that the set \[T_\alpha=\{\mathfrak{b}eta<\alpha:\text{$S\cap\mathfrak{b}eta$ is not ${\mathbb P}i^1_{f^\alpha_{f^\kappa_\xi(\alpha)}(\mathfrak{b}eta)}$-indescribable}\}\] is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable in $\alpha$. Thus, for each $\alpha\in (\kappa$\hat{\text{s}}$etminus T)\cap C\cap D$ the set $T_\alpha\cap D_\alpha$ is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-indescribable in $\alpha$. Now it follows by Lemma \ref{lemma_union} that the set \[\mathfrak{b}igcup_{\alpha\in (\kappa$\hat{\text{s}}$etminus T)\cap C\cap D}(T_\alpha\cap D_\alpha){\mathop{\rm sub}}seteq T\] is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$, a contradiction. \end{proof} Now we show that for regular $\kappa$, whenever $\zeta<\xi<\kappa^+$ and the ideals under consideration are nontrivial, we have ${\mathbb P}i^1_\zeta(\kappa){\mathop{\rm sub}}setneq{\mathbb P}i^1_\xi(\kappa)$. \mathfrak{b}egin{corollary}\langlebel{corollary_proper} Suppose $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable where $\xi<\kappa^+$. Then for all $\zeta<\xi$ we have ${\mathbb P}i^1_\zeta(\kappa){\mathop{\rm sub}}setneq{\mathbb P}i^1_\xi(\kappa)$. \end{corollary} \mathfrak{b}egin{proof} The fact that ${\mathbb P}i^1_\zeta(\kappa){\mathop{\rm sub}}seteq{\mathbb P}i^1_\xi(\kappa)$ follows easily from the fact that the class of ${\mathbb P}i^1_\xi$ formulas includes the ${\mathbb P}i^1_\zeta$ formulas. To see that the proper containment holds, consider the set \[C=\{\alpha<\kappa:\text{$\alpha$ is ${\mathbb P}i^1_{f^\kappa_\zeta(\alpha)}$-indescribable}\}.\] By Corollary \ref{corollary_hierarchy} and Proposition \ref{lemma_set_of_non}, we have $\kappa$\hat{\text{s}}$etminus C\in {\mathbb P}i^1_\xi(\kappa)$\hat{\text{s}}$etminus{\mathbb P}i^1_\zeta(\kappa)$. \end{proof} {\mathop{\rm sub}}section{Higher ${\mathbb P}i^1_\xi$-clubs}\langlebel{section_higher_xi_clubs} Now we present a characterization of the ${\mathbb P}i^1_\xi$-indescribability of sets $S{\mathop{\rm sub}}seteq\kappa$ in terms of a natural base for the filter ${\mathbb P}i^1_\xi(\kappa)^*$. \mathfrak{b}egin{definition}\langlebel{definition_Pi1xi_club} Suppose $\kappa$ is a regular cardinal. We define the notion of ${\mathbb P}i^1_\xi$-club subset of $\kappa$ for all $\xi<\kappa^+$ by induction.\mathfrak{b}egin{enumerate} \item A set $C{\mathop{\rm sub}}seteq\kappa$ is \emph{${\mathbb P}i^1_0$-club} if it is closed and unbounded in $\kappa$. \item We say that $C$ is \emph{${\mathbb P}i^1_{\zeta+1}$-club in $\kappa$} if $C$ is ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$ and $C$ is \emph{${\mathbb P}i^1_\zeta$-closed}, in the sense that there is a club $C^*$ in $\kappa$ such that for all $\alpha\in C^*$, whenever $C\cap\alpha$ is ${\mathbb P}i^1_{f_\zeta^\kappa(\alpha)}$-indescribable in $\alpha$ we must have $\alpha\in C$. \item If $\xi$ is a limit, we say that $C{\mathop{\rm sub}}seteq\kappa$ is \emph{${\mathbb P}i^1_\xi$-club in $\kappa$} if $C$ is ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta<\xi$ and $C$ is \emph{${\mathbb P}i^1_\xi$-closed}, in the sense that there is a club $C^*$ in $\kappa$ such that for all $\alpha\in C^*$, whenever $C\cap\alpha$ is ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta < f^\kappa_\xi(\alpha)$, we must have $\alpha\in C$. \end{enumerate} \end{definition} Let us show that, when the ${\mathbb P}i^1_\xi$-indescribability ideal ${\mathbb P}i^1_\xi(\kappa)$ is nontrivial, the ${\mathbb P}i^1_\xi$-club subsets of $\kappa$ form a filter base for the dual filter ${\mathbb P}i^1_\xi(\kappa)^*$ and a set being ${\mathbb P}i^1_\xi$-club in $\kappa$ is expressible by a ${\mathbb P}i^1_\xi$ sentence. \mathfrak{b}egin{theorem}\langlebel{theorem_xi_clubs} Suppose $\kappa$ is a regular cardinal. For all $\xi<\kappa^+$, if $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable then the following hold. \mathfrak{b}egin{enumerate} \item A set $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable if and only if $S\cap C\neq\varnothing$ for all ${\mathbb P}i^1_\xi$-clubs $C{\mathop{\rm sub}}seteq\kappa$. \item There is a ${\mathbb P}i^1_\xi$ formula $\chi^\kappa_\xi(X)$ over $V_\kappa$ such that for all $C{\mathop{\rm sub}}seteq \kappa$ we have \[\text{$C$ is ${\mathbb P}i^1_\xi$-club in $\kappa$ if and only if } V_\kappa\models \chi^\kappa_\xi(C)\] and there is a club $D_\xi$ in $\kappa$ such that for all regular $\alpha\in D_\xi$ and all $C{\mathop{\rm sub}}seteq\kappa$ we have \[\text{$C\cap\alpha$ is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-club in $\alpha$ if and only if } V_\alpha\models\chi^\kappa_\xi(C)\mathrm{|}^\kappa_\alpha.\] \end{enumerate} \end{theorem} \mathfrak{b}egin{proof} Sun \cite[Theorem 1.17]{MR1245524} proved that the theorem holds for $\xi=1$, and Hellsten \cite[Theorem 2.4.2]{MR2026390} generalized this to the case in which $\xi<\omega$. We provide a proof of the case in which $\xi<\kappa^+$ is a limit ordinal; the case in which $\xi<\kappa^+$ is a successor is similar, but easier. Suppose $\xi<\kappa^+$ is a limit ordinal and that both (1) and (2) hold for all ordinals $\zeta<\xi$. For the forward direction of (1), suppose $S{\mathop{\rm sub}}seteq\kappa$ is ${\mathbb P}i^1_\xi$-indescribable and fix $C{\mathop{\rm sub}}seteq\kappa$ a ${\mathbb P}i^1_\xi$-club subset of $\kappa$. Then, in particular, for each $\zeta<\xi$, $C$ is ${\mathbb P}i^1_\zeta$-indescribable and, by Theorem \ref{theorem_expressing_indescribability}, we see that \[V_\kappa\models\mathfrak{b}igwedge_{\zeta<\xi}{\mathbb P}hi^\kappa_\zeta(C).\] Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathbb P}i^1_\xi(\kappa)$ be generic over $V$ with $S\in G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower. Then $\kappa\in j(S)$ and by the proof of Proposition \ref{proposition_generic_characterization}, we have $\left(V_\kappa\models\mathfrak{b}igwedge_{\zeta<\xi}{\mathbb P}hi^\kappa_\zeta(C)\right)^{V^\kappa/G}.$ For each $\zeta<\xi$, let $C_\zeta$ be the club subset of $\kappa$ obtained from Theorem \ref{theorem_expressing_indescribability} and notice that $\kappa\in j(C_\zeta)$ and hence in $V^\kappa/G$ the set $C$ is ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$. Since $C$ is a ${\mathbb P}i^1_\xi$-club subset of $\kappa$ there is a club $C^*{\mathop{\rm sub}}seteq\kappa$ as in Definition \ref{definition_Pi1xi_club}. Since $\kappa\in j(C^*)$ and $j(C)\cap\kappa$ is ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta<\xi=j(f^\kappa_\xi)(\kappa)$, it follows that $\kappa\in j(C)$. Therefore by elementarity $S\cap C\neq\varnothing$. For the reverse direction of (1), suppose $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable and $S{\mathop{\rm sub}}seteq\kappa$ intersects every ${\mathbb P}i^1_\xi$-club. It suffices to show that if $\varphi=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta$ is any ${\mathbb P}i^1_\xi$ sentence over $V_\kappa$ such that $V_\kappa\models\varphi$, then the set \[C=\{\alpha\in D: V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha\}\] contains a ${\mathbb P}i^1_\xi$-club, where $D{\mathop{\rm sub}}seteq\kappa$ is a club subset of $\kappa$ such that for all regular $\alpha\in D$, $\varphi\mathrm{|}^\kappa_\alpha$ is defined. First, let us argue that $C$ is ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta<\xi$. Suppose not. Then for some fixed $\zeta<\xi$, $C$ is not ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$ and hence $\kappa$\hat{\text{s}}$etminus C$ is in the filter ${\mathbb P}i^1_\zeta(\kappa)^*$. Since $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable by assumption, and since ${\mathbb P}i^1_\zeta(\kappa)^*{\mathop{\rm sub}}seteq{\mathbb P}i^1_\xi(\kappa)^*{\mathop{\rm sub}}seteq{\mathbb P}i^1_\xi(\kappa)^+$, we see that $\kappa$\hat{\text{s}}$etminus C$ is ${\mathbb P}i^1_\xi$-indescribable in $\kappa$. Since $(\kappa$\hat{\text{s}}$etminus C)\cap D$ is ${\mathbb P}i^1_\xi$-indescribable and $V_\kappa\models\varphi$ there is an $\alpha\in (\kappa$\hat{\text{s}}$etminus C)\cap D$ such that $V_\alpha\models\varphi\mathrm{|}^\kappa_\alpha$, a contradiction. Next we must argue that $C$ is ${\mathbb P}i^1_\xi$-closed. We must show that there is a club $C^*$ in $\kappa$ such that for all regular $\alpha\in C^*$, if $C\cap\alpha$ is ${\mathbb P}i^1_\zeta$-indescribable in $\alpha$ for all $\zeta<f^\kappa_\xi(\alpha)$ then $\alpha\in C$. We will use Proposition \ref{proposition_framework2}. Let $G{\mathop{\rm sub}}seteq P(\kappa)/{\mathop{\rm NS}}_\kappa$ be generic with $\kappa$ regular in $V^\kappa/G$ and let $j:V\to V^\kappa/G$ be the corresponding generic ultrapower embedding. It suffices to show that in $V^\kappa/G$, if $C$ is ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$ for all $\zeta<\xi$ then $\alpha\in j(C)$. Assume that in $V^\kappa/G$, $C$ is ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$ for all $\zeta<\xi$ but $\kappa\notin j(C)$. Since $j(\varphi)\mathrm{|}^{j(\kappa)}_\kappa=\varphi$, it follows from the definition of $C$ that for some $\zeta<\xi$, $(V_\kappa\models\lnot\varphi_\zeta)^{V^\kappa/G}$. But in $V^\kappa/G$, $C$ is ${\mathbb P}i^1_\zeta$-indescribable in $\kappa$ and so there is some $\alpha\in C$ such that $(V_\alpha\models\lnot\varphi_\zeta\mathrm{|}^\kappa_\alpha)^{V^\kappa/G}$, which contradicts the definition of $C$. Now, let us show that (2) holds for the limit ordinal $\xi$. The definition of ``$X$ is ${\mathbb P}i^1_\xi$-club'' is equivalent over $V_\kappa$ to \[\left(\mathfrak{b}igwedge_{\eta<\xi}{\mathbb P}hi^\kappa_\eta(X)\right)\langlend(\exists C^*)\left[(\text{$C^*$ is club})\langlend (\forall\mathfrak{b}eta\in C^*)\left(\mathfrak{b}igwedge_{\zeta< f^\kappa_\xi(\alpha)}(X\cap\mathfrak{b}eta\in{\mathbb P}i^1_\zeta(\mathfrak{b}eta)^+)\rightarrow\mathfrak{b}eta\in X\right)\right].\] We define a set $R_\xi{\mathop{\rm sub}}seteq \kappa$ that codes all relevant information about which subsets of $\alpha$, for $\alpha<\kappa$, are ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta<f^\kappa_\xi(\alpha)$ as follows. We let $R_\xi{\mathop{\rm sub}}seteq\kappa$ be such that for each regular $\alpha<\kappa$, if $\alpha$ is ${\mathbb P}i^1_\zeta$-indescribable for all $\zeta<f^\kappa_\xi(\alpha)$, then the sequence \[\langle(R_\xi)_\eta:\alpha\leq\eta<2^\alpha\rangle\] is an enumeration of the subsets of $\alpha$ that are ${\mathbb P}i^1_\zeta$-indescribable in $\alpha$ for all $\zeta<f^\kappa_\xi(\alpha)$. Otherwise, we define $(R_\xi)_\eta=\varnothing$. Now we let \[\mathfrak{b}ar\chi^\kappa_\xi(X)=(\exists C^*)\left[(\text{$C^*$ is club})\langlend(\forall\mathfrak{b}eta\in C^*)(\exists\eta (X\cap\mathfrak{b}eta=(R_\xi)_\eta)\rightarrow \mathfrak{b}eta\in X)\right]\] and \[\chi^\kappa_\xi(X)=\left(\mathfrak{b}igwedge_{\eta<\xi}{\mathbb P}hi^\kappa_\eta(X)\right)\langlend\mathfrak{b}ar\chi^\kappa_\xi(X).\] Since the second part $\mathfrak{b}ar\chi^\kappa_\xi(X)$ of the definition of $\chi^\kappa_\xi(X)$ is $\mathbb{S}igma^1_1$, it is also trivially ${\mathbb P}i^1_2$, and thus we see that $\chi^\kappa_\xi(X)$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$. Clearly, for all $C{\mathop{\rm sub}}seteq\kappa$ we have \[\text{$C$ is ${\mathbb P}i^1_\xi$-club in $\kappa$}\iff V_\kappa\models\chi_\xi(C).\] To complete the proof of (2), one may use Proposition \ref{proposition_framework2}, along with Theorem \ref{theorem_expressing_indescribability} to show that there is a club $D_\xi$ in $\kappa$ such that for all regular $\alpha\in D_\xi$ we have that for all $C{\mathop{\rm sub}}seteq\kappa$, \[\text{$C\cap\alpha$ is ${\mathbb P}i^1_{f^\kappa_\xi(\alpha)}$-club in $\alpha$}\iff V_\alpha\models\chi_\xi(C)\mathrm{|}^\kappa_\alpha.\] Let us note that the remaining details are similar to the proof of Theorem \ref{theorem_expressing_indescribability}(2), and are therefore left to the reader. \end{proof} $\hat{\text{s}}$ection{Higher $\xi$-stationarity, $\xi$-s-stationarity and derived topologies}\langlebel{section_higher_derived_topologies} In this section we define natural generalizations of Bagaria's notions of $\xi$-stationarity, $\xi$-s-stationarity and derived topologies. Given a regular cardinal $\mu$, we will define a sequence of topologies $\langle\tau_\xi:\xi<\mu^+\rangle$ on $\mu$ such that the sequence $\langle\tau_\xi:\xi<\mu\rangle$ is Bagaria's sequence of derived topologies, and Bagaria's characterization of nonisolated points in the spaces $(\mu,\tau_\xi)$ for $\xi<\mu$ has a natural generalization to $\tau_\xi$ for $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ (see Theorem \ref{theorem_xi_s_nonisolated}). We also show that Bagaria's result, in which he obtains the nondiscreteness of the topologies $\tau_\xi$ for $\xi<\mu$ from an indescribability hypothesis, can be generalized to $\tau_\xi$ for all $\xi<\mu^+$ using higher indescribability (see Corollary \ref{corollary_nondiscreteness_from_indescribability}). Let us now discuss a generalization of Bagaria's derived topologies. Recall that, under certain conditions, one can specify a topology on a set $X$ by stating what the limit point operation must be. If $d:P(X)\to P(X)$ is a function satisfying properties (1) and (2) in Definition \ref{definition_cantor_derivative}, then one can define a topology $\tau_d$ on $X$ by demanding that a set $C{\mathop{\rm sub}}seteq X$ be closed if and only if $d(C){\mathop{\rm sub}}seteq C$. Furthermore, if $d$ also satisfies property (3) in Definition \ref{definition_cantor_derivative}, then $d$ equals the limit point operator in the space $(X,\tau_d)$. \mathfrak{b}egin{definition}\langlebel{definition_cantor_derivative} Given a set $X$, we say that a function $d:P(X)\to P(X)$ is a \emph{Cantor derivative} on $X$ provided that the following conditions hold. \mathfrak{b}egin{enumerate} \item $d(\varnothing)=\varnothing$. \item For all $A,B\in P(X)$ we have \mathfrak{b}egin{enumerate} \item $A{\mathop{\rm sub}}seteq B$ implies $d(A){\mathop{\rm sub}}seteq d(B)$, \item $d(A\cup B){\mathop{\rm sub}}seteq d(A)\cup d(B)$ and \item for all $x\in X$, $x\in d(A)$ implies $x\in d(A$\hat{\text{s}}$etminus\{x\})$. \end{enumerate} \item $d(d(A)){\mathop{\rm sub}}seteq d(A)\cup A$. \end{enumerate} If $d$ satisfies only (1) and (2) then we say that $d$ is a \emph{pre-Cantor derivative}. \end{definition} \mathfrak{b}egin{fact}\langlebel{fact_cantor_derivatives} If $d:P(X)\to P(X)$ is a pre-Cantor derivative on $X$ then the collection \[\tau=\{U{\mathop{\rm sub}}seteq X: d(X$\hat{\text{s}}$etminus U){\mathop{\rm sub}}seteq X$\hat{\text{s}}$etminus U\}\] is a topology on $X$. Furthermore, if $d$ is a Cantor derivative on $X$ then \[d(A)=\{x\in X:\text{$x$ is a limit point of $A$ in $(X,\tau)$}\}\] for all $A{\mathop{\rm sub}}seteq X$. \end{fact} \mathfrak{b}egin{proof} Clearly $\varnothing\in\tau$ since $d(X){\mathop{\rm sub}}seteq X$. Furthermore, $X\in\tau$ because $d(\varnothing)=\varnothing$ by assumption. Suppose $I$ is some index set and for each $i\in I$ we have a set $C_i{\mathop{\rm sub}}seteq X$ with $d(C_i){\mathop{\rm sub}}seteq C_i$. By (2a), it follows that for every $j\in I$ we have $d\left(\mathfrak{b}igcap_{i\in I}C_i\right){\mathop{\rm sub}}seteq d(C_j)$ and hence $d\left(\mathfrak{b}igcap_{i\in I}C_i\right){\mathop{\rm sub}}seteq \mathfrak{b}igcap_{i\in I}C_i.$ Furthermore, if $I=\{0,1\}$ we have $d(C_0\cup C_1){\mathop{\rm sub}}seteq d(C_0)\cup d(C_1){\mathop{\rm sub}}seteq C_0\cup C_1$. Thus, $\tau$ is a topology on $X$. For $A{\mathop{\rm sub}}seteq X$, let $A'$ denote the set of limit points of $A$ in $(X,\tau)$. Let us show that $d(A)=A'$. Suppose $x\in d(A)$ and fix $U\in \tau$ with $x\in U$. For the sake of contradiction, suppose that $(U\cap A)$\hat{\text{s}}$etminus\{x\}=\varnothing$, and notice that $A$\hat{\text{s}}$etminus\{x\}{\mathop{\rm sub}}seteq A$\hat{\text{s}}$etminus U{\mathop{\rm sub}}seteq X$\hat{\text{s}}$etminus U$. Thus $d(A$\hat{\text{s}}$etminus \{x\}){\mathop{\rm sub}}seteq d(X$\hat{\text{s}}$etminus U){\mathop{\rm sub}}seteq X$\hat{\text{s}}$etminus U$, and since $x\in U$, this implies $x\notin d(A$\hat{\text{s}}$etminus \{x\})$. But this implies $x\notin d(A)$ by (2c), a contradiction. Thus for any $A{\mathop{\rm sub}}seteq X$ we have $d(A){\mathop{\rm sub}}seteq A'$. For any set $A{\mathop{\rm sub}}seteq X$, since the closure $\overline{A}=A\cup A'$ is the smallest closed set containing $A$, since $A\cup d(A)$ is closed (by (2b) and (3)) and since $d(A){\mathop{\rm sub}}seteq A'$, it follows that $\overline{A}=A\cup A'=A\cup d(A)$. Now fix $A{\mathop{\rm sub}}seteq X$. We have \mathfrak{b}egin{align*} x\in A' &\iff x\in \overline{A$\hat{\text{s}}$etminus\{x\}}\\ &\iff x\in (A$\hat{\text{s}}$etminus \{x\})\cup d(A$\hat{\text{s}}$etminus\{x\})\\ &\iff x\in d(A$\hat{\text{s}}$etminus\{x\})\\ &\iff x\in d(A). \end{align*} \end{proof} Given an ordinal $\mathfrak{d}elta$, Bagaria defined the sequence of derived topologies $\langle\tau_\xi:\xi<\mathfrak{d}elta\rangle$ on $\mathfrak{d}elta$ as follows. \mathfrak{b}egin{definition}[Bagaria \cite{MR3894041}]\langlebel{definition_bagaria} Let $\tau_0$ be the interval topology on $\mathfrak{d}elta$. That is, $\tau_0$ is the topology on $\mathfrak{d}elta$ generated\footnote{Recall that, given a set $X$ and a collection ${\mathcal B}{\mathop{\rm sub}}seteq P(X)$, the \emph{topology generated by ${\mathcal B}$} is the smallest topology on $X$ which contains ${\mathcal B}$. That is, the topology generated by ${\mathcal B}$ is the collection of all unions of finite intersections of members of ${\mathcal B}$ together with the set $X$.} by the collection ${\mathcal B}_0$ consisting of $\{0\}$ and all open intervals of the form $(\alpha,\mathfrak{b}eta)$ where $\alpha<\mathfrak{b}eta\leq\mathfrak{d}elta$. We let $d_0:P(\mathfrak{d}elta)\to P(\mathfrak{d}elta)$ be the limit point operator of the space $(\mathfrak{d}elta,\tau_0)$. If $\xi<\mathfrak{d}elta$ is an ordinal and the sequences $\langleB_\zeta:\zeta\leq\xi\rangle$, $\langle\tau_\zeta:\zeta\leq\xi\rangle$ and $\langled_\zeta:\zeta\leq\xi\rangle$ have been defined, we let $\tau_{\xi+1}$ be the topology generated by the collection \[{\mathcal B}_{\xi+1}={\mathcal B}_\xi\cup\{d_\xi(A): A{\mathop{\rm sub}}seteq\mathfrak{d}elta\}\] and we let \[d_{\xi+1}(A)=\{\alpha<\mathfrak{d}elta:\text{$\alpha$ is a limit point of $A$ in the $\tau_{\xi+1}$ topology}\}.\] When $\xi<\mathfrak{d}elta$ is a limit ordinal, we define ${\mathcal B}_\xi=\mathfrak{b}igcup_{\zeta<\xi}{\mathcal B}_\zeta$, let $\tau_\xi$ be the topology generated by ${\mathcal B}_\xi$ and define $d_\xi$ to be the limit point operator of the space $(\mathfrak{d}elta,\tau_\xi)$. \end{definition} Bagaria proved that a point $\alpha<\mathfrak{d}elta$ is not isolated in $(\mathfrak{d}elta,\tau_\xi)$ if and only if it is $\xi$-s-reflecting (see \cite[Definition 2.8]{MR3894041} or Definition \ref{definition_xi_s_stationarity}). Since no ordinal $\alpha<\mathfrak{d}elta$ can be $\mathfrak{d}elta$-s-reflecting (see Remark \ref{remark_nontrivial}), it follows that the topology $\tau_\mathfrak{d}elta$ generated by $\mathfrak{b}igcup_{\zeta<\mathfrak{d}elta}{\mathcal B}_\zeta$ is discrete. In what follows, by using diagonal Cantor derivatives, we extend Bagaria's definition of derived topologies to allow for more nontrivial cases. One may want to review Remark \ref{remark_example} before reading the following. \mathfrak{b}egin{definition}\langlebel{definition_tau_xi} Suppose $\mu$ is a regular cardinal. We define three sequences of functions $\langle{\mathcal B}_\xi:\xi<\mu^+\rangle$, $\langle{\mathcal T}_\xi:\xi<\mu^+\rangle$ and $\langled_\xi:\xi<\mu^+\rangle$, and one sequence $\langle\tau_\xi:\xi<\mu^+\rangle$ of topologies on $\mu$ by transfinite induction as follows. For $\xi<\mu$ we let $\tau_\xi$ and $d_\xi$ be defined as Definition \ref{definition_bagaria}, and we let ${\mathcal B}_\xi$ and ${\mathcal T}_\xi$ be functions with domain $\mu$ such that for all $\alpha<\mu$, we have ${\mathcal T}_\xi(\alpha)=\tau_\xi$ and ${\mathcal B}_\xi(\alpha)={\mathcal B}_\xi$ is the subbasis for $\tau_\xi$ as in Definition \ref{definition_bagaria}. Suppose $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ and we have already defined $\langle{\mathcal B}_\zeta:\zeta<\xi\rangle$, $\langle{\mathcal T}_\zeta:\zeta<\xi\rangle$, $\langled_\zeta:\zeta<\xi\rangle$ and $\langle\tau_\zeta:\zeta<\xi\rangle$. We let ${\mathcal B}_\xi$ and ${\mathcal T}_\xi$ be the functions with domain $\mu$ such that for each $\alpha\in \mu$ we have \[{\mathcal B}_\xi(\alpha)={\mathcal B}_0\cup\{d_\zeta(A):\zeta\in F^\mu_\xi(\alpha)\langlend A{\mathop{\rm sub}}seteq\mu\}\] and ${\mathcal T}_\xi(\alpha)$ is the topology on $\mu$ generated by ${\mathcal B}_\xi(\alpha)$. We define $d_\xi:P(\mu)\to P(\mu)$ by letting \[d_\xi(A)=\{\alpha<\mu:\text{$\alpha$ is a limit point of $A$ in the ${\mathcal T}_\xi(\alpha)$ topology}\}\] for $A{\mathop{\rm sub}}seteq\mu$. Then we let $\tau_\xi$ be the topology\footnote{It is easily seen that this $d_\xi$ is a pre-Cantor derivative as in Definition \ref{definition_cantor_derivative}, and thus $\tau_\xi$ is in fact a topology on $\mu$.} \[\tau_\xi=\{U{\mathop{\rm sub}}seteq\mu: d_\xi(\mu$\hat{\text{s}}$etminus U){\mathop{\rm sub}}seteq\mu$\hat{\text{s}}$etminus U\}.\] \end{definition} For all $\xi<\mu^+$ and $\alpha<\mu$, since ${\mathcal T}_\xi(\alpha)$ is the topology generated by ${\mathcal B}_\xi(\alpha)$, it follows that the collection of finite intersections of members of ${\mathcal B}_\xi(\alpha)$ is a basis for ${\mathcal T}_\xi(\alpha)$. That is, the collection of sets of the form \[I\cap d_{\xi_0}(A_0)\cap\cdots\cap d_{\xi_{n-1}}(A_{n-1})\] where $n<\omega$, $I\in{\mathcal B}_0$ is an interval in $\mu$, the ordinals $\xi_0\leq\cdots\leq\xi_{n-1}$ are in $F^\mu_\xi(\alpha)$ and $A_i{\mathop{\rm sub}}seteq\mu$ for $i<n$, is a basis for the ${\mathcal T}_\xi(\alpha)$ topology on $\mu$. Next let us show that the diagonal Cantor derivatives $d_\xi$ in Definition \ref{definition_tau_xi} are in fact Cantor derivatives as in Definition \ref{definition_cantor_derivative}, and thus each $d_\xi$ is the Cantor derivative of the space $(\mu,\tau_\xi)$ for all $\xi<\mu^+$. \mathfrak{b}egin{lemma}\langlebel{lemma_d_xi_is_cantor} Suppose $\mu$ is regular. For all $\xi<\mu^+$ and all $A{\mathop{\rm sub}}seteq\mu$ we have \[d_\xi(d_\xi(A)){\mathop{\rm sub}}seteq d_\xi(A).\] \end{lemma} \mathfrak{b}egin{proof} For $\xi<\mu$ this follows easily from the fact that $d_\xi$ is defined to be the Cantor derivative of the space $(\mu,\tau_\xi)$. Suppose $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ and $\alpha\in d_\xi(d_\xi(A))$. Then $\alpha$ is a limit point of the set \[d_\xi(A)=\{\mathfrak{b}eta<\alpha:\text{$\mathfrak{b}eta$ is a limit point of $A$ in ${\mathcal T}_\xi(\mathfrak{b}eta)$}\}\] in the topology ${\mathcal T}_\xi(\alpha)$ on $\mu$ generated by ${\mathcal B}_\xi(\alpha)$. To show $\alpha\in d_\xi(A)$, we must show that $\alpha$ is a limit point of $A$ in the topology ${\mathcal T}_\xi(\alpha)$. Fix a basic open neighborhood $U$ of $\alpha$ in the ${\mathcal T}_\xi(\alpha)$ topology. Then $U$ is of the form \[I\cap d_{\xi_0}(A_0)\cap\cdots\cap d_{\xi_{n-1}}(A_{n-1})\] for some $\xi_i\in F^\mu_\xi(\alpha)$ and some $A_i{\mathop{\rm sub}}seteq\mu$ where $i<n$. Since $\alpha$ is a limit point of $d_\xi(A)$ in the ${\mathcal T}_\xi(\alpha)$ topology on $\mu$ and since ${\mathcal B}_0{\mathop{\rm sub}}seteq{\mathcal T}_\xi(\alpha)$, it follows that for all $\eta<\alpha$, $\alpha$ is a limit point of the set $d_\xi(A)$\hat{\text{s}}$etminus\eta$ in the ${\mathcal T}_\xi(\alpha)$ topology. Since $\xi_i\in F^\mu_\xi(\alpha)$ for $i<n$ and since $\alpha$ is a limit ordinal, we can choose a $\mathfrak{b}eta<\alpha$ such that $\xi_i\in F^\mu_\xi(\mathfrak{b}eta)$ for all $i<n$. Since $\alpha$ is a limit point of $d_\xi(A)$\hat{\text{s}}$etminus\mathfrak{b}eta$ in the ${\mathcal T}_\xi(\alpha)$ topology, we may choose an $\eta\in (d_\xi(A)$\hat{\text{s}}$etminus\mathfrak{b}eta)\cap U\cap\alpha$. Since $\eta\geq\mathfrak{b}eta$ we have $\xi_i\in F^\mu_\xi(\eta)$ for all $i<n$ and thus $U\in {\mathcal B}_\xi(\eta){\mathop{\rm sub}}seteq {\mathcal T}_\xi(\eta)$. But since $\eta$ is a limit point of $A$ in the ${\mathcal T}_\xi(\eta)$ topology we have $A\cap U\cap\eta\neq\varnothing$. Thus $\alpha\in d_\xi(A)$. \end{proof} The following result is an easy consequence of Fact \ref{fact_cantor_derivatives} and Lemma \ref{lemma_d_xi_is_cantor}. \mathfrak{b}egin{corollary} Suppose $\mu$ is a regular cardinal. For each $\xi<\mu^+$, the function $d_\xi$ is the Cantor derivative of the space $(\mu,\tau_\xi)$. \end{corollary} Let us present the following generalizations of Bagaria's notions of $\xi$-stationarity and $\xi$-s-stationarity, which will allow us to characterize the nondiscreteness of points in the spaces $(\mu,\tau_\xi)$ for $\xi<\mu^+$. \mathfrak{b}egin{definition}\langlebel{definition_xi_stationary} Suppose $\mu$ is a regular cardinal. A set $A{\mathop{\rm sub}}seteq\mu$ is $0$-stationary in $\alpha<\mu$ if and only if $A$ is unbounded in $\alpha$. For $0<\xi<\alpha^+$, where $\alpha$ is regular, we say that $A$ is \emph{$\xi$-stationary in $\alpha$} if and only if for every $\zeta<\xi$, every set $S$ that is $\zeta$-stationary in $\alpha$ \emph{$\zeta$-reflects} to some $\mathfrak{b}eta\in A$, i.e., $S$ is $f^\alpha_\zeta(\mathfrak{b}eta)$-stationary in $\mathfrak{b}eta$. We say that an ordinal $\alpha<\mu$ is \emph{$\xi$-reflecting} if it is $\xi$-stationary in $\alpha$ as a subset of $\mu$. \end{definition} \mathfrak{b}egin{definition}\langlebel{definition_xi_s_stationarity} Suppose $\mu$ is a regular cardinal. $A$ set $A{\mathop{\rm sub}}seteq\mu$ is \emph{$0$-simultaneously stationary in $\alpha$} (\emph{$0$-s-stationary in $\alpha$} for short) if and only if $A$ is unbounded in $\alpha$. For $0<\xi<\alpha^+$, where $\alpha$ is regular, we say that $A$ is \emph{$\xi$-simultaneously stationary in $\alpha$} (\emph{$\xi$-s-stationary in $\alpha$} for short) if and only if for every $\zeta<\xi$, every pair of subsets $S$ and $T$ that are $\zeta$-s-stationary in $\alpha$ \emph{simultaneously $\zeta$-reflect} to some $\mathfrak{b}eta\in A$, i.e., $S$ and $T$ are both $f^\alpha_\zeta(\mathfrak{b}eta)$-s-stationary in $\mathfrak{b}eta$. We say that $\alpha$ is $\xi$-s-reflecting if it is $\xi$-s-stationary in $\alpha$. \end{definition} \mathfrak{b}egin{remark}\langlebel{remark_nontrivial} Bagaria defined a set $A{\mathop{\rm sub}}seteq\mu$ to be $\xi$-stationary in $\alpha<\mu$ if and only if for every $\zeta<\xi$, for every $S{\mathop{\rm sub}}seteq\mu$ that is $\zeta$-stationary in $\alpha$ there is a $\mathfrak{b}eta\in A\cap\alpha$ such that $S$ is $\zeta$-stationary in $\mathfrak{b}eta$. Since $f^\alpha_\zeta$ equals the constant function $\zeta$ when $\zeta<\alpha$, it follows that Bagaria's notion of $A$ being $\xi$-stationary in $\alpha$ is equivalent to ours when $\xi<\alpha$. Bagaria comments in the paragraphs following \cite[Definition 2.6]{MR3894041} that, under his definition, no ordinal $\alpha$ can be $(\alpha+1)$-reflecting, because if $\alpha$ is the least such ordinal there is a $\mathfrak{b}eta<\alpha$ such that $\alpha\cap\mathfrak{b}eta=\mathfrak{b}eta$ is $\alpha$-stationary and thus $(\mathfrak{b}eta+1)$-stationary in $\mathfrak{b}eta$. Let us show that such an argument does \emph{not} work to rule out the existence of ordinals $\alpha$ which are $\alpha+1$-reflecting under our definition. Suppose $\alpha$ is $(\alpha+1)$-reflecting, as in Definition \ref{definition_xi_stationary}. Then there is some $\mathfrak{b}eta<\alpha$ that is $f^\alpha_\alpha(\mathfrak{b}eta)$-reflecting, but $f^\alpha_\alpha(\mathfrak{b}eta)=\mathfrak{b}eta$ and thus the conclusion is that $\mathfrak{b}eta$ is $\mathfrak{b}eta$-reflecting, and Bagaria shows that some ordinals (namely some large cardinals) $\mathfrak{b}eta$ can be $\mathfrak{b}eta$-reflecting. \end{remark} In order to streamline the proof of the characterization of the nonisolated points in $(\mu,\tau_\xi)$ for $\xi<\mu^+$, we will use the following auxiliary notion of $\xi$-$\hat{\text{s}}$-stationarity, which is often equivalent to $f^\mu_\xi(\alpha)$-s-stationarity as shown in Lemma \ref{lemma_s_hat} below. \mathfrak{b}egin{definition}\langlebel{definition_xi_s_hat_stationary} Suppose $\mu$ is a regular cardinal. A set $A{\mathop{\rm sub}}seteq\mu$ is \emph{$0$-simultaneously hat stationary in $\alpha$} ($0$-$\hat{\text{s}}$-stationary for short) if and only if $A$ is unbounded in $\alpha$. For $0<\xi<\mu^+$, we say that $A$ is \emph{$\xi$-simultaneously hat stationary in $\alpha$} (\emph{$\xi$-$\hat{\text{s}}$-stationary in $\alpha$} for short) if and only if for every $\zeta\in F^\mu_\xi(\alpha)$, every pair of subsets $S$ and $T$ of $\mu$ that are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ \emph{simultaneously $\zeta$-$\hat{\text{s}}$-reflect} to some $\mathfrak{b}eta\in A$, i.e., $S$ and $T$ are both $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. We say that $\alpha$ is \emph{$\xi$-$\hat{\text{s}}$-reflecting} if it is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$. \end{definition} \mathfrak{b}egin{remark} Notice that when $\xi<\mu$, a set $A{\mathop{\rm sub}}seteq\mu$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ if and only if for all $\zeta\in F^\mu_\xi(\alpha)=\xi$, every pair of subsets $S$ and $T$ of $\mu$ that are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ simultaneously $\zeta$-$\hat{\text{s}}$-reflect to some $\mathfrak{b}eta\in A$. Thus, for $\xi<\mu$, Definition \ref{definition_xi_s_hat_stationary} agrees with \cite[Definition 2.8]{MR3894041}. Furthermore, $A$ is $\mu$-$\hat{\text{s}}$-stationary in $\alpha$ if and only if it is $\alpha$-$\hat{\text{s}}$-stationary in $\alpha$. Also notice that for $\zeta<\xi<\mu$, there is a club $C_{\zeta,\xi}$ in $\mu$ such that for all $\alpha\in C_{\zeta,\xi}$ we have $F^\mu_\zeta(\alpha){\mathop{\rm sub}}seteq F^\mu_\xi(\alpha)$ and hence $\alpha$ being $\xi$-$\hat{\text{s}}$-stationary implies $\alpha$ is $\zeta$-$\hat{\text{s}}$-stationary. \end{remark} We will need the following lemma, which generalizes \cite[Proposition 2.9]{MR3894041}. \mathfrak{b}egin{lemma}\langlebel{lemma_intersect_with_club} Suppose $\mu$ is a regular cardinal and $\xi<\mu^+$. There is a club $B_\xi{\mathop{\rm sub}}seteq\mu$ such that for all regular $\alpha\in B_\xi$ if $A$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ ($f^\mu_\xi(\alpha)$-s-stationary in $\alpha$) and $C$ is a club subset of $\alpha$, then $A\cap C$ is also $\xi$-$\hat{\text{s}}$-stationary ($f^\mu_\xi(\alpha)$-s-stationary in $\alpha$) in $\alpha$.\end{lemma} \mathfrak{b}egin{proof} We will prove the lemma for $\xi$-$\hat{\text{s}}$-stationarity; the proof for $\xi$-s-stationarity is similar. When $\xi<\mu$ the lemma follows directly from \cite[Proposition 2.9]{MR3894041}. Suppose $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ is a limit ordinal and the result holds for $\zeta<\xi$. For each $\zeta<\xi$ let $B_\zeta$ be the club subset of $\mu$ obtained from the inductive hypothesis. Let $B_\xi$ be a club subset of $\mu$ such that for all $\alpha\in B_\xi$ we have \mathfrak{b}egin{enumerate} \item[(i)] $\alpha\in\mathfrak{b}igcap_{\zeta\in F^\mu_\xi(\alpha)}B_\zeta$, \item[(ii)] $\mathop{\rm ot}\nolimits(F^\mu_\xi(\alpha))$ is a limit ordinal and \item[(iii)] $(\forall\zeta\in F^\mu_\xi(\alpha))$ $F^\mu_\xi(\alpha)\cap\zeta=F^\mu_\zeta(\alpha)$. \end{enumerate} Suppose $\alpha\in B_\xi$, let $A{\mathop{\rm sub}}seteq\mu$ be $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ and let $C$ be a club subset of $\alpha$. Since $\mathop{\rm ot}\nolimits(F^\mu_\xi(\alpha))$ is a limit ordinal, to show that $A\cap C$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$, it suffices to show that $A\cap C$ is $\eta$-$\hat{\text{s}}$-stationary in $\alpha$ for all $\eta\in F^\mu_\xi(\alpha)$. Since $F^\mu_\eta(\alpha){\mathop{\rm sub}}seteq F^\mu_\xi(\alpha)$ it follows that $A$ is $\eta$-$\hat{\text{s}}$-stationary in $\alpha$. Then, because $\alpha\in B_\eta$, it follows by the inductive hypothesis that $A\cap C$ is $\eta$-$\hat{\text{s}}$-stationary in $\alpha$. Now suppose $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ and the result holds for $\zeta\leq\xi$. We will show that it holds for $\xi+1$. For each $\zeta\leq\xi$, let $B_\zeta$ be the club subset of $\mu$ obtained by the inductive hypothesis. Let $B_{\xi+1}$ be a club subset of $\mu$ such that for all $\alpha\in B_{\xi+1}$ we have $\alpha\in \mathfrak{b}igcap_{\zeta\in F^\mu_{\xi+1}(\alpha)} B_\zeta$. Suppose $\alpha\in B_{\xi+1}$, let $A$ be $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$ and suppose $C$ is a club subset of $\alpha$. To show that $A\cap C$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, fix any sets $S,T{\mathop{\rm sub}}seteq\alpha$ that are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta\in F^\mu_{\xi+1}(\alpha)$. Since $\alpha\in B_\zeta$, it follows by the inductive hypothesis that both $S\cap C$ and $T\cap C$ are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, there is some $\mathfrak{b}eta\in A$ such that both $S\cap C$ and $T\cap C$ are $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Thus, $\mathfrak{b}eta\in A\cap C$ and both $S$ and $T$ are $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$, which establishes that $A\cap C$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. \end{proof} \mathfrak{b}egin{lemma}\langlebel{lemma_s_hat} Suppose $\mu$ is a regular cardinal. For all $\xi<\mu^+$ there is a club $C_\xi{\mathop{\rm sub}}seteq\mu$ such that for all regular $\alpha\in C_\xi$ a set $X{\mathop{\rm sub}}seteq\alpha$ is $f^\mu_\xi(\alpha)$-s-stationary in $\alpha$ if and only if it is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$. \end{lemma} \mathfrak{b}egin{proof} Suppose $\xi<\mu$. Let $C_\xi=\mu$\hat{\text{s}}$etminus\xi$ and suppose $\alpha\in C_\xi$ is regular and $X{\mathop{\rm sub}}seteq\alpha$. Since $F^\mu_\xi(\alpha)=f^\mu_\xi(\alpha)=\xi$ it is easy to see that from the definitions that $X$ is $\xi$-s-stationarity in $\alpha$ if and only if it is $\xi$-$\hat{\text{s}}$-stationarity in $\alpha$. Now suppose that $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ is a limit ordinal and the result holds for $\zeta<\xi$; the case in which $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ is a successor is easier and is therefore left to the reader. For each $\zeta<\xi$ let $C_\zeta$ be the club obtained by the inductive hypothesis, and let $B_\xi$ be the club obtained from Lemma \ref{lemma_intersect_with_club}. Let $C_\xi$ be a club subset of $\mu$ such that for all $\alpha\in C_\xi$ we have \mathfrak{b}egin{enumerate} \item[(i)] $\alpha\in B_\xi\cap \mathfrak{b}igcap_{\zeta\in F^\mu_\xi(\alpha)}d_0(C_\zeta)$, \item[(ii)] $\mathop{\rm ot}\nolimits(F^\mu_\xi(\alpha))$ is a limit ordinal, \item[(iii)] $f^\mu_\xi(\alpha)=\mathfrak{b}igcup_{\zeta\in F^\mu_\xi(\alpha)}f^\mu_\zeta(\alpha)$, \item[(iv)] for all $\zeta\in F^\mu_\xi(\alpha)$ there is a club $D^\alpha_{\xi,\zeta}$ in $\alpha$ such that for all $\mathfrak{b}eta\in D^\alpha_{\xi,\zeta}$ we have $f^\alpha_{f^\mu_\xi(\alpha)}(\mathfrak{b}eta)=f^\mu_\xi(\mathfrak{b}eta)$, \item[(v)] $(\forall\zeta\in F^\mu_\xi(\alpha))$ $F^\mu_\xi(\alpha)\cap \zeta=F^\mu_\zeta(\alpha)$. \end{enumerate} Suppose $X$ is $f^\mu_\xi(\alpha)$-s-stationary in $\alpha$ and fix sets $S,T{\mathop{\rm sub}}seteq\alpha$ that are $\eta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\eta\in F^\mu_\xi(\alpha)$. Since $\alpha\in d_0(C_\eta){\mathop{\rm sub}}seteq C_\eta$, it follows by the inductive hypothesis that both $S$ and $T$ are $f^\mu_\eta(\alpha)$-s-stationary in $\alpha$. By (iii) we have $f^\mu_\eta(\alpha)<f^\mu_\xi(\alpha)$, and since $\alpha\in B_\xi$ it follows by Lemma \ref{lemma_intersect_with_club} that $X\cap C_\eta\cap D^\alpha_{\xi,\zeta}$ is $f^\mu_\xi(\alpha)$-s-stationarity in $\alpha$. Hence there is a $\mathfrak{b}eta\in X\cap C_\eta\cap D^\alpha_{\xi,\zeta}$ such that both $S$ and $T$ are $f^\alpha_{f^\mu_\eta(\alpha)}(\mathfrak{b}eta)$-s-stationary in $\mathfrak{b}eta$. Since it follows from (v) that $f^\alpha_{f^\mu_\eta(\alpha)}(\mathfrak{b}eta)=f^\mu_\eta(\mathfrak{b}eta)$, both $S$ and $T$ are $f^\mu_\eta(\mathfrak{b}eta)$-s-stationary in $\mathfrak{b}eta$. Since $\mathfrak{b}eta\in C_\eta$ it follows that $S$ and $T$ are both $\eta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Thus $S$ is $\xi$-$\hat{\text{s}}$-stationar in $\alpha$. Conversely, suppose $X$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ and fix sets $S,T{\mathop{\rm sub}}seteq\alpha$ that are $\eta$-s-stationary in $\alpha$ for some $\eta<f^\mu_\xi(\alpha)$. Let $\pi^\mu_{\xi,\alpha}:F^\mu_\xi(\alpha)\to f^\mu_\xi(\alpha)$ be the transitive collapse of $F^\mu_\xi(\alpha)$ and let $\hat\eta=(\pi^\mu_{\xi,\alpha})^{-1}(\eta)$. Since $\alpha\in C_{\hat\eta}$, it follows that $S$ and $T$ are $\hat\eta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $X\cap D^\alpha_{\xi,\hat\eta}$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$, there is a $\mathfrak{b}eta\in X\cap D^\alpha_{\xi,\hat\eta}\cap\alpha$ such that $S$ and $T$ are both $f^\mu_{\hat\eta}(\mathfrak{b}eta)$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Since $\mathfrak{b}eta\in D^\alpha_{\xi,\hat\eta}$, the sets $S$ and $T$ are $f^\alpha_{f^\mu_{\hat\eta}(\alpha)}(\mathfrak{b}eta)$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Since $\eta=f^\mu_{\hat\eta}(\alpha)$ we see that both $S$ and $T$ are $f^\alpha_\eta(\mathfrak{b}eta)$-s-stationary in $\mathfrak{b}eta$. Thus $X$ is $\xi$-s-stationary. \end{proof} In order to characterize the nonisolated points of the spaces $(\mu,\tau_\xi)$, for $\xi<\mu^+$, in terms of $\eta$-s-reflecting cardinals, we will need the following proposition, which generalizes \cite[Proposition 2.10]{MR3894041}. \mathfrak{b}egin{proposition}\langlebel{proposition_meat} Suppose $\mu$ is a regular cardinal. \mathfrak{b}egin{enumerate} \item For all $\xi<\mu^+$ there is a club $C_\xi{\mathop{\rm sub}}seteq\mu$ such that for all $A{\mathop{\rm sub}}seteq\mu$ we have \[d_\xi(A)\cap C_\xi=\{\alpha<\mu:\text{$A$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$}\}\cap C_\xi.\] \item For all $\xi<\mu^+$ there is a club $D_\xi{\mathop{\rm sub}}seteq\mu$ such that for all $\alpha\in D_\xi$ and all $A{\mathop{\rm sub}}seteq\mu$ we have that $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$ if and only if $A\cap d_\zeta(S)\cap d_\zeta(T)\neq\varnothing$ (equivalently, if and only if $A\cap d_\zeta(S)\cap d_\zeta(T)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$) for every $\zeta\in F^\mu_{\xi+1}(\alpha)$ and every pair $S,T$ of subsets of $\alpha$ that are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. \item For all $\xi<\mu^+$ there is a club $E_\xi{\mathop{\rm sub}}seteq\mu$ such that for all $\alpha\in E_\xi$ and all $A{\mathop{\rm sub}}seteq\mu$, if $A$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ and $A_i$ is $\zeta_i$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta_i\in F^\mu_\xi(\alpha)$, for all $i<n$ where $n<\omega$, then $A\cap d_{\zeta_0}(A_0)\cap \cdots\cap d_{\zeta_{n-1}}(A_{n-1})$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$. \end{enumerate} \end{proposition} \mathfrak{b}egin{proof} We will prove (1) -- (3) by simultaneous induction on $\xi$, for $\xi$-$\hat{\text{s}}$-stationarity. For $\xi<\mu$, (1) -- (3) follow directly from \cite[Proposition 2.10]{MR3894041}, taking $C_\xi=D_\xi=E_\xi=\mu$. Let us first show that if $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ is a limit ordinal and (1) -- (3) hold for all $\zeta<\xi$, then (1) -- (3) hold for $\xi$. First we will show that for $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ a limit ordinal, if (1) holds for $\zeta<\xi$ then (1) holds for $\xi$. For each $\zeta<\xi$, let $C_\zeta$ be the club subset of $\mu$ obtained from (1). Let $C_\xi$ be a club subset of $\mu$ such that for all $\alpha\in C_\xi$ we have \mathfrak{b}egin{enumerate} \item[(i)] $\alpha\in\mathfrak{b}igcap_{\zeta\in F^\mu_\xi(\alpha)}C_\zeta$, \item[(ii)] $\mathop{\rm ot}\nolimits(F^\mu_\xi(\alpha))$ is a limit ordinal and \item[(iii)] $(\forall\zeta\in F^\mu_\xi(\alpha))$ $F^\mu_\xi(\alpha)\cap\zeta=F^\mu_\zeta(\alpha)$. \end{enumerate} Now fix $A{\mathop{\rm sub}}seteq\mu$ and suppose $\alpha\in d_\xi(A)\cap C_\xi$. Then $\alpha$ is a limit point of $A$ in the ${\mathcal T}_\xi(\alpha)$ topology on $\mu$. For each $\zeta\in F^\mu_\xi(\alpha)$ we have $F^\mu_\zeta(\alpha){\mathop{\rm sub}}seteq F^\mu_\xi(\alpha)$, which implies ${\mathcal T}_\zeta(\alpha){\mathop{\rm sub}}seteq{\mathcal T}_\xi(\alpha)$, and hence $\alpha$ is a limit point of $A$ in the ${\mathcal T}_\zeta(\alpha)$ topology on $\mu$. Thus $\alpha\in \mathfrak{b}igcap_{\zeta\in F^\mu_\xi(\alpha)}d_\zeta(A)$. Since $\alpha\in \mathfrak{b}igcap_{\zeta\in F^\mu_\xi(\alpha)}C_\zeta$, it follows that $A$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ for all $\zeta\in F^\mu_\xi(\alpha)$. By (ii) and (iii), this implies that $A$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$. Conversely, suppose $A$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ and $\alpha\in C_\xi$. To show that $\alpha\in d_\xi(A)$ we must show that $\alpha$ is a limit point of $A$ in the ${\mathcal T}_\xi(\alpha)$ topology on $\mu$ generated by ${\mathcal B}_\xi(\alpha)$. Fix a basic open neighborhood $U$ of $\alpha$ in ${\mathcal T}_\xi(\alpha)$. Then $U$ is of the form \[I\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})\] where $I$ is an interval in $\mu$, $n<\omega$, and for all $i<n$ we have $\zeta_i\in F^\mu_\xi(\alpha)$ and $A_i{\mathop{\rm sub}}seteq\mu$. By (ii) we can choose some $\eta\in F^\mu_\xi(\alpha)$ with $\eta>\max\{\zeta_i: i<n\}$. By (iii), for each $i<n$ we have $\zeta_i\in F^\mu_\xi(\alpha)\cap\eta=F^\mu_\eta(\alpha)$ and hence $U$ is an open neighborhood of $\alpha$ in the ${\mathcal T}_\eta(\alpha)$ topology. Since $F^\mu_\eta(\alpha){\mathop{\rm sub}}seteq F^\mu_\xi(\alpha)$, it follows that $A$ is $\eta$-$\hat{\text{s}}$-stationary in $\alpha$, and since $\alpha\in C_\eta$ we have that $\alpha\in d_\eta(A)$. Thus $\alpha$ is a limit point of $A$ in the ${\mathcal T}_\eta(\alpha)$ topology, so $A\cap U$\hat{\text{s}}$etminus\{\alpha\}\neq\varnothing$. This shows that $\alpha$ is a limit point of $A$ in the ${\mathcal T}_\xi(\alpha)$ topology. Let us show that for $\xi\in \mu^+$\hat{\text{s}}$etminus\mu$ a limit ordinal, if (3) holds for $\zeta<\xi$, then (3) holds for $\xi$. Let $E_\xi$ be a club subset of $\mu$ such that for all $\alpha\in E_\xi$ we have \mathfrak{b}egin{enumerate} \item[(i)] $\alpha\in \mathfrak{b}igcap_{\zeta\in F^\mu_\xi(\alpha)}E_\zeta$, \item[(ii)] $\mathop{\rm ot}\nolimits(F^\mu_\xi(\alpha))$ is a limit ordinal and \item[(iii)] $(\forall\zeta\in F^\mu_\xi(\alpha))$ $F^\mu_\xi(\alpha)\cap\zeta=F^\mu_\zeta(\alpha)$. \end{enumerate} Suppose $\alpha\in E_\xi$. Let $A{\mathop{\rm sub}}seteq\mu$ be $\xi$-$\hat{\text{s}}$-stationary in $\alpha$ and, for $i<n$, suppose $A_i$ is $\zeta_i$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta_i\in F^\mu_\xi(\alpha)$. We must show that $A\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$. Fix a pair of sets $S,T{\mathop{\rm sub}}seteq\mu$ that are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta\in F^\mu_\xi(\alpha)$. Using (ii), choose $\eta\in F^\mu_\xi(\alpha)$ with $\eta>\max(\{\zeta_i: i< n\}\cup\{\zeta\})$. Since $F^\mu_\xi(\alpha)\cap\eta=F^\mu_\eta(\alpha)$, it follows that $A$ is $\eta$-$\hat{\text{s}}$-stationary in $\alpha$, and by our assumption that (3) holds for $\eta<\xi$ and the fact that $\alpha\in E_\eta$, it follows that $A\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})$ is $\eta$-$\hat{\text{s}}$-stationary in $\alpha$. Thus, there is a $\mathfrak{b}eta\in A\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})$ such that both $S$ and $T$ are $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Now we will show that for a limit ordinal $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$, if (1) and (3) hold for $\zeta\leq\xi$, then (2) holds for $\xi$. For each $\zeta\leq\xi$, let $B_\zeta$ be the club subset of $\mu$ obtained from Lemma \ref{lemma_intersect_with_club}. Let $D_\xi$ be a club subset of $\mu$ such that for all $\alpha\in D_\xi$ we have \mathfrak{b}egin{enumerate}[(i)] \item $\mathop{\rm ot}\nolimits(F^\mu_\xi(\alpha))$ is a limit ordinal, \item $(\forall\zeta\in F^\mu_\xi(\alpha))$ $F^\mu_\xi(\alpha)\cap\zeta=F^\mu_\zeta(\alpha)$ and \item $\alpha\in\mathfrak{b}igcap_{\zeta\in F^\mu_{\xi+1}(\alpha)}(B_\zeta\cap d_0(C_\zeta)\cap d_0(E_\zeta))$ where the $C_\zeta$'s and $E_\zeta$'s are obtained by the inductive hypothesis from (1) and (3) respectively. \end{enumerate} Suppose $\alpha\in D_\xi$. For the forward direction of (2), let $A$ be $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$ and fix a pair $S,T$ of subsets of $\alpha$ that are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta\in F^\mu_{\xi+1}(\alpha)$. Since $\alpha\in d_0(C_\zeta)$, it follows that $C_\zeta$ is closed and unbounded in $\alpha$ and hence the set $A\cap C_\zeta$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. Hence there exists a $\mathfrak{b}eta\in A\cap C_\zeta$ such that $S$ and $T$ are both $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$, and since $\mathfrak{b}eta\in C_\zeta$ we have $\mathfrak{b}eta\in A\cap d_\zeta(S)\cap d_\zeta(T)$ by (1). To see that $A\cap d_\zeta(S)\cap d_\zeta(T)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$, fix sets $X,Y{\mathop{\rm sub}}seteq\alpha$ that are $\eta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\eta\in F^\mu_\zeta(\alpha)$. Since $\alpha\in E_\zeta$, it follows by (3) that $S\cap d_\eta(X)$ and $T\cap d_\eta(Y)$ are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $A\cap C_\zeta$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$ there is some $\mathfrak{b}eta\in A\cap C_\zeta$ such that $S\cap d_\eta(X)$ and $T\cap d_\eta(Y)$ are both $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Since $\mathfrak{b}eta\in C_\zeta$, it follows that $\mathfrak{b}eta\in A\cap C_\zeta\cap d_\zeta(S\cap d_\eta(X))\cap d_\zeta(T\cap d_\eta(Y))\neq\varnothing$. Now we have \[\varnothing\neq A\cap C_\zeta\cap d_\zeta(S\cap d_\eta(X))\cap d_\zeta(T\cap d_\eta(Y)){\mathop{\rm sub}}seteq A\cap C_\zeta\cap d_\zeta(S)\cap d_\zeta(T)\cap d_\eta(X)\cap d_\eta(Y),\] and hence $A\cap d_\zeta(S)\cap d_\zeta(T)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. For the reverse direction of (2), suppose that $\alpha\in D_\xi$ and for all $\zeta\in F^\mu_{\xi+1}(\alpha)$, if $S,T{\mathop{\rm sub}}seteq\alpha$ are both $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ then $A\cap d_\zeta(S)\cap d_\zeta(T)\neq\varnothing$. To show that $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, fix $\zeta\in F^\mu_{\xi+1}(\alpha)$ and suppose $S,T{\mathop{\rm sub}}seteq\alpha$ are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. By Lemma \ref{lemma_intersect_with_club} and the fact that $\alpha\in B_\zeta\cap d_0(C_\zeta)$, it follows that $S\cap C_\zeta$ and $T$ are both $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Thus, by (1), there is a $\mathfrak{b}eta\in A\cap d_\zeta(S\cap C_\zeta)\cap d_\zeta(T)$. Now since $\mathfrak{b}eta\in C_\zeta\cap d_\zeta(S)\cap d_\zeta(T)$, it follows by (1) that $S$ and $T$ are both $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Hence $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. It remains to show that if $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ is an ordinal and (1), (2) and (3) hold for $\zeta\leq\xi$, then (1), (2) and (3) also hold for $\xi+1$. Given that (1), (2) and (3) hold for $\zeta\leq\xi$, let us show that (3) holds for $\xi+1$. For $\zeta\leq\xi$, let $C_\zeta$, $D_\zeta$ and $E_\zeta$ be the club subsets of $\mu$ obtained from (1), (2) and (3) respectively. For each $\zeta\leq\xi$, let $B_\zeta$ be the club subset of $\mu$ obtained from Lemma \ref{lemma_intersect_with_club}. Let $E_{\xi+1}$ be a club subset of $\mu$ such that for all $\alpha\in E_{\xi+1}$ we have \mathfrak{b}egin{enumerate} \item[(i)] $\alpha\in \mathfrak{b}igcap_{\zeta\in F^\mu_{\xi+1}(\alpha)} (B_\zeta\cap d_0(C_\zeta)\cap D_\zeta\cap E_\zeta)$, \item[(ii)] $\alpha\in C_\xi\cap D_\xi\cap E_\xi$ and \item[(iii)] $(\forall\zeta\in F^\mu_{\xi+1}(\alpha))$ $F^\mu_{\xi+1}(\alpha)\cap\zeta=F^\mu_\zeta(\alpha)$. \end{enumerate} Suppose $\alpha\in E_{\xi+1}$ and $A{\mathop{\rm sub}}seteq\mu$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. Let $n<\omega$ and for each $i<n$ suppose $\zeta_i\in F^\mu_{\xi+1}(\alpha)$ and $A_i$ is $\zeta_i$-$\hat{\text{s}}$-stationary in $\alpha$. We must show that $A\cap d_{\zeta_0}(A_0)\cap \cdots d_{\zeta_{n-1}}(A_{n-1})$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. Fix sets $S,T{\mathop{\rm sub}}seteq\alpha$ which are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta\in F^\mu_{\xi+1}(\alpha)$. We must show that there is a $\mathfrak{b}eta\in A\cap d_{\zeta_0}(A_0)\cap \cdots d_{\zeta_{n-1}}(A_{n-1})$ such that $S\cap\mathfrak{b}eta$ and $T\cap\mathfrak{b}eta$ are $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Since $\alpha\in C_\zeta$, it follows by an inductive application of (1) that in order to prove (3) holds for $\xi+1$, it suffices to show that \mathfrak{b}egin{align} A\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})\cap d_\zeta(S)\cap d_\zeta(T)\cap C_\zeta&\neq\varnothing.\langlebel{eqn_for_3} \end{align} Let us proceed to prove \ref{eqn_for_3} by induction on $n$. First, let us consider the case in which $\zeta_0=\zeta$. Since $\alpha\in D_\zeta$ and $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, it follows inductively from (2) that the set $d_\zeta(S)\cap d_\zeta(T)$\hat{\text{s}}$upseteq A\cap d_\zeta(S)\cap d_\zeta(T)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Now since $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, the sets $A_0$ and $d_\zeta(S)\cap d_\zeta(T)$ are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ and since $\alpha\in D_\zeta$, it follows from (2) that $A\cap d_{\zeta_0}(A_0)\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. By Lemma \ref{lemma_intersect_with_club}, since $\alpha\in B_\zeta\cap d_0(C_\zeta)$ we have that $A\cap d_{\zeta_0}(A_0)\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))\cap C_\zeta$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $A\cap d_{\zeta_0}(A_0)\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))\cap C_\zeta{\mathop{\rm sub}}seteq A\cap d_{\zeta_0}(A_0)\cap d_\zeta(S)\cap d_\zeta(T)\cap C_\zeta$, this establishes (\ref{eqn_for_3}) in case $n=1$ and $\zeta_0=\zeta$. Second, let us consider the case in which $n=1$ and $\zeta_0<\zeta$. Since $F^\mu_\zeta(\alpha){\mathop{\rm sub}}seteq F^\mu_{\xi+1}(\alpha)$, it follows that $A$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $\alpha\in E_\zeta$ and $\zeta\in F^\mu_{\xi+1}(\alpha)$, we may inductively apply (3) to see that $A\cap d_{\zeta_0}(A_0)$ and thus also $d_{\zeta_0}(A_0)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Hence because $\alpha\in D_\zeta$, it follows by an inductive application of (2) that $A\cap d_\zeta(d_{\zeta_0}(A_0))\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ and by Lemma \ref{lemma_intersect_with_club} and the fact that $\alpha\in B_\zeta\cap d_0(C_\zeta)$, we see that the set $A\cap d_\zeta(d_{\zeta_0}(A_0))\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))\cap C_\zeta$ is also $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $A\cap d_\zeta(d_{\zeta_0}(A_0))\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))\cap C_\zeta{\mathop{\rm sub}}seteq A\cap d_{\zeta_0}(A_0)\cap d_\zeta(S)\cap d_\zeta(T)\cap C_\zeta$, this establishes (\ref{eqn_for_3}) in the second case where $n=1$ and $\zeta_0<\zeta$. Thirdly, suppose $n=1$ and $\zeta_0>\zeta$. Then by an inductive application of (2), the set $A\cap d_{\zeta_0}(A_0)$ is $\zeta_0$-$\hat{\text{s}}$-stationary in $\alpha$. Since $\alpha \in B_\zeta\cap d_0(C_{\zeta_0})$, it follows from by Lemma \ref{lemma_intersect_with_club} that $A\cap d_{\zeta_0}(A_0)\cap C_\zeta$ is also $\zeta_0$-$\hat{\text{s}}$-stationary in $\alpha$. Since $\zeta\in F^\mu_{\zeta_0}(\alpha)$, we see that there is some $\mathfrak{b}eta\in A\cap d_{\zeta_0}(A_0)\cap C_\zeta$ such that both $S$ and $T$ are $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$, and thus $\mathfrak{b}eta\in A\cap d_{\zeta_0}(A_0)\cap d_\zeta(S)\cap d_\zeta(T)\cap C_\zeta$. This establishes that (6) holds for $n=1$. Now suppose $n>1$. Since $\alpha\in D_\zeta$, it follows inductively by (2) that $d_\zeta(S)\cap d_\zeta(T)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $\alpha\in E_\zeta$, it follows by an inductive application of (3) that $d_\zeta(d_\zeta(S)\cap d_\zeta(T))$ is $\mu$-$\hat{\text{s}}$-stationary in $\alpha$. Furthermore, since $\alpha\in C_{\zeta_{n-1}}$ we have that $A_{n-1}$ is $\zeta_{n-1}$-$\hat{\text{s}}$-stationary in $\alpha$ and thus we see that, again by an inductive application of (3) using the fact that $\alpha\in E_{\zeta_{n-1}}$ the set $d_{\zeta_{n-1}}(A_{n-1})$ is $\zeta_{n-1}$-$\hat{\text{s}}$-stationary in $\alpha$. Also, by the inductive hypothesis on $n$, the set $A\cap d_{\zeta_0}(A_0)\cap \cdots\cap d_{\zeta_{n-1}}(A_{n-2})$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. Therefore, by an inductive application of (2), the set \[A\cap d_{\zeta_0}(A_0)\cap \cdots\cap d_{\zeta_{n-2}}(A_{n-2})\cap d_{\zeta_{n-1}}(d_{\zeta_{n-1}}(A_{n-1}))\cap d_\zeta(d_\zeta(S)\cap d_\zeta(T))\cap C_\zeta\] which is contained in \[A\cap d_{\zeta_0}(A_0)\cap \cdots\cap d_{\zeta_{n-2}}(A_{n-2})\cap d_{\zeta_{n-1}}(A_{n-1})\cap d_\zeta(S)\cap d_\zeta(T)\cap C_\zeta\] is $\mu$-$\hat{\text{s}}$-stationary in $\alpha$. This establishes (\ref{eqn_for_3}) and hence (3) holds for $\xi+1$. Next, given that (1) and (2) hold for $\zeta\leq\xi$ and (3) holds for $\zeta\leq\xi+1$, let us show that (1) holds for $\xi+1$. For each $\zeta\leq\xi$, let $C_\zeta$ be the club subset of $\mu$ obtained from (1). Also let $E_{\xi+1}$ be the club obtained from (3). For each $\zeta\leq\xi+1$, let $B_\zeta$ be the club subset of $\mu$ obtained from Lemma \ref{lemma_intersect_with_club}. Now we let $C_{\xi+1}$ be a club subset of $\mu$ such that for all $\alpha\in C_{\xi+1}$ we have \mathfrak{b}egin{enumerate} \item[(i)] $\alpha\in \mathfrak{b}igcap_{\zeta\in F^\mu_{\xi+1}(\alpha)} (B_\zeta\cap d_0(C_\zeta))$ and \item[(ii)] $\alpha\in B_{\xi+1}\cap E_{\xi+1}$. \end{enumerate} Suppose $\alpha\in d_{\xi+1}(A)\cap C_{\xi+1}$. Then $\alpha$ is a limit point of $A$ in the ${\mathcal T}_{\xi+1}(\alpha)$ topology on $\mu$. To show that $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, fix $\zeta\in F^\mu_{\xi+1}(\alpha)$ and suppose $S,T{\mathop{\rm sub}}seteq\alpha$ are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Since $\alpha\in C_{\xi+1}$ we have $\alpha\in d_0(C_\zeta)$ and thus $\alpha\in d_\zeta(S)\cap d_\zeta(T)$. Since $d_0(C_\zeta)\cap d_\zeta(S)\cap d_\zeta(T)\in {\mathcal B}_{\xi+1}(\alpha)$ is a basic open neighborhood of $\alpha$ in the ${\mathcal T}_{\xi+1}(\alpha)$ topology, and since $\alpha\in d_{\xi+1}(A)$, it follows that there is some $\mathfrak{b}eta\in A\cap d_0(C_\zeta)\cap d_\zeta(S)\cap d_\zeta(T)$\hat{\text{s}}$etminus\{\alpha\}$. Since $\mathfrak{b}eta\in C_\zeta$, it follows by the inductive hypothesis on (1) that both $S$ and $T$ are $\zeta$-$\hat{\text{s}}$-stationary in $\mathfrak{b}eta$. Thus, $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. Now suppose $\alpha\in C_{\xi+1}$ and $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. To show that $\alpha\in d_{\xi+1}(A)$, fix a basic open set $U\in{\mathcal B}_{\xi+1}(\alpha)$ in the ${\mathcal T}_{\xi+1}(\alpha)$ topology with $\alpha\in U$. Then $U$ is of the form $I\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})$, where $I\in{\mathcal B}_0(\alpha)$ is an interval in $\mu$, $n<\omega$ and for all $i<n$ we have $\zeta_i\in F^\mu_{\xi+1}(\alpha)$ and $A_i{\mathop{\rm sub}}seteq\mu$. Since $\alpha\in C_{\xi+1}$ and $\alpha\in I\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})$, it follows by the inductive hypothesis that $A_i$ is $\zeta_i$-$\hat{\text{s}}$-stationary in $\alpha$ for all $i<n$. Then since $A$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$, it follows from the fact that (3) holds for $\xi+1$ and $\alpha\in E_{\xi+1}$, that $A\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$. Furthermore, since $I\cap\alpha$ is a club subset of $\alpha$ and $\alpha\in B_{\xi+1}$, Lemma \ref{lemma_intersect_with_club} implies that $A\cap I\cap d_{\zeta_0}(A_0)\cap\cdots\cap d_{\zeta_{n-1}}(A_{n-1})\cap\alpha$ is $(\xi+1)$-$\hat{\text{s}}$-stationary in $\alpha$ and is thus nonempty. This establishes that $\alpha$ is a limit point of $A$ in the ${\mathcal T}_{\xi+1}(\alpha)$ topology, that is, $\alpha\in d_{\xi+1}(A)$. Finally, given that (1) and (3) hold for $\zeta\leq\xi+1$, let us show that (2) holds for $\xi+1$. Let $D_{\xi+1}$ be a club subset of $\mu$ such that for all $\alpha\in D_{\xi+1}$ we have \mathfrak{b}egin{enumerate} \item[(i)] $(\forall\zeta\in F^\mu_{\xi+2}(\alpha))$ $F^\mu_{\xi+2}(\alpha)\cap\zeta=F^\mu_\zeta(\alpha)$; \item[(ii)] $\alpha\in\mathfrak{b}igcap_{\zeta\in F^\mu_{\xi+2}(\alpha)}C_\zeta\cap E_\zeta$; \item[(iii)] $\alpha\in B_{\xi+2}\cap C_{\xi+1}$ where $B_{\xi+2}$ is the club subset of $\mu$ obtained from Lemma \ref{lemma_intersect_with_club} and $C_{\xi+1}$ is obtained from our inductive assumption on (1); and \item[(iv)] for all $\zeta\in F^\mu_{\xi+2}(\alpha)$ and all $\eta\in F^\mu_\zeta(\alpha)$ we have \[\alpha\in d_0(\{\mathfrak{b}eta<\mu: F^\mu_\eta(\mathfrak{b}eta){\mathop{\rm sub}}seteq F^\mu_\zeta(\mathfrak{b}eta)\}).\] \end{enumerate} Suppose $A$ is $(\xi+2)$-$\hat{\text{s}}$-stationary in $\alpha$ and $\alpha\in D_{\xi+1}$. Let $S,T{\mathop{\rm sub}}seteq\alpha$ be $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$ for some $\zeta\in F^\mu_{\xi+2}(\alpha)$. We must show that $A\cap d_\zeta(S)\cap d_\zeta(T)$ is $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. Fix $\eta\in F^\mu_\zeta(\alpha)$ and suppose $X$ and $Y$ are $\eta$-$\hat{\text{s}}$-stationary subsets of $\alpha$. By an inductive application of (1), it will suffice to show that \[A\cap d_\zeta(S)\cap d_\zeta(T)\cap d_\eta(X)\cap d_\eta(Y)\cap C_\eta\neq\varnothing.\] Since $\alpha\in E_\zeta$ it follows inductively by (3) that the sets $S\cap d_\eta(X)$ and $T\cap d_\eta(Y)$ are $\zeta$-$\hat{\text{s}}$-stationary in $\alpha$. By (iv) the set $\{\mathfrak{b}eta<\alpha: F^\mu_\eta(\mathfrak{b}eta){\mathop{\rm sub}}seteq F^\mu_\zeta(\mathfrak{b}eta)\}$ is club in $\alpha$ and therefore, by Lemma \ref{lemma_intersect_with_club} since $\alpha\in B_{\xi+2}$, the set \[A\cap\{\mathfrak{b}eta<\alpha: F^\mu_\eta(\mathfrak{b}eta){\mathop{\rm sub}}seteq F^\mu_\zeta(\mathfrak{b}eta)\}\] is $(\xi+2)$-$\hat{\text{s}}$-stationary in $\alpha$. Since $\zeta\in F^\mu_{\xi+2}(\alpha)$ and since $\alpha\in E_\zeta$, it follows by (3) for $\zeta$ that there is some $\mathfrak{b}eta\in A$ such that \[\mathfrak{b}eta\in d_\zeta(S\cap d_\eta(X))\cap d_\zeta(T\cap d_\eta(Y)).\] Thus we have \[\mathfrak{b}eta\in d_\zeta(S)\cap d_\zeta(d_\eta(X))\cap d_\zeta(T)\cap d_\zeta(d_\eta(Y)).\] Since $F^\mu_\eta(\mathfrak{b}eta){\mathop{\rm sub}}seteq F^\mu_\zeta(\mathfrak{b}eta)$, it follows that ${\mathcal T}_\eta(\mathfrak{b}eta){\mathop{\rm sub}}seteq {\mathcal T}_\zeta(\mathfrak{b}eta)$, and therefore we obtain \[\mathfrak{b}eta\in d_\zeta(S)\cap d_\eta(d_\eta(X))\cap d_\zeta(T)\cap d_\eta(d_\eta(Y)).\] By Lemma \ref{lemma_d_xi_is_cantor}, we have \[\mathfrak{b}eta\in d_\zeta(S)\cap d_\zeta(T)\cap d_\eta(X) \cap d_\eta(Y)\] as desired. \end{proof} Now we are ready to characterize the nonisolated points of the spaces $(\mu_\xi,\tau_\xi)$ (on a club) in terms of $\eta$-$\hat{\text{s}}$-reflecting cardinals. The following is a generalization of \cite[Theorem 2.11]{MR3894041}. \mathfrak{b}egin{theorem}\langlebel{theorem_xi_s_hat_nonisolated} Suppose $\mu$ is a regular cardinal. For all $\xi<\mu^+$ if $C_\xi$ is the club subset of $\mu$ obtained from Proposition \ref{proposition_meat}(1), then for all $\alpha\in C_\xi$, the ordinal $\alpha$ is not isolated in the $\tau_\xi$ topology on $\mu$ if and only if $\alpha$ is $\xi$-$\hat{\text{s}}$-reflecting. \end{theorem} \mathfrak{b}egin{proof} For $\xi<\mu$ the result follows directly from \cite[Theorem 2.11]{MR3894041}. Suppose $\xi\in\mu^+$\hat{\text{s}}$etminus\mu$ and $\alpha\in C_\xi$. By Definition \ref{definition_tau_xi}, we have \mathfrak{b}egin{align*} \text{$\alpha$ is not isolated in the $\tau_\xi$ topology} &\iff \{\alpha\}\notin\tau_\xi\\ &\iff d_\xi(\mu$\hat{\text{s}}$etminus\{\alpha\})\not{\mathop{\rm sub}}seteq\mu$\hat{\text{s}}$etminus\{\alpha\}\\ &\iff \alpha\in d_\xi(\mu$\hat{\text{s}}$etminus\{\alpha\})\\ &\iff \text{$\alpha$ is $\xi$-$\hat{\text{s}}$-stationary in $\alpha$}\\ &\iff \text{$\alpha$ is $\xi$-$\hat{\text{s}}$-reflecting.} \end{align*} \end{proof} By applying Lemma \ref{lemma_s_hat} and Theorem \ref{theorem_xi_s_hat_nonisolated} we easily obtain the following. \mathfrak{b}egin{theorem}\langlebel{theorem_xi_s_nonisolated} Suppose $\mu$ is a regular cardinal. For all $\xi<\mu^+$ there is a club $C_\xi{\mathop{\rm sub}}seteq\mu$ such that for all $\alpha \in C_\xi$ we have that $\alpha$ is not isolated in the $\tau_\xi$ topology on $\mu$ if and only if $\alpha$ is $f^\mu_\xi(\alpha)$-s-reflecting. \end{theorem} In order to show that ${\mathbb P}i^1_\xi$-indescribability can be used to obtain the nondiscreteness of the topologies $\tau_\xi$ on $\mu$ for $\xi<\mu^+$, we need the following expressibility result. \mathfrak{b}egin{lemma}\langlebel{lemma_expressing_s_stationarity} Suppose $\kappa$ is a regular cardinal. For all $\xi<\kappa^+$ there is a formula ${\mathbb P}i^1_\xi$ formula $\varphi_\xi(X)$ over $V_\kappa$ and a club $C_\xi$ subset of $\kappa$ such that for all $A{\mathop{\rm sub}}seteq\kappa$ we have \[\text{$A$ is $\xi$-s-stationary in $\kappa$ if and only if $V_\kappa\models\varphi_\xi(A)$}\] and for all $\alpha\in C_\xi$ we have \[\text{$A$ is $f^\kappa_\xi(\alpha)$-s-stationary in $\alpha$ if and only if $V_\alpha\models\varphi_\xi(A)\mathrm{|}^\kappa_\alpha$}\] \end{lemma} \mathfrak{b}egin{proof} We follow the proof of \cite[Proposition 4.3]{MR3894041} and proceed by induction on $\xi$. We let $\varphi_0(X)$ be the natural ${\mathbb P}i^1_0$ formula asserting that $X$ is $0$-s-stationary (i.e. unbounded) in $\kappa$. Suppose $\xi<\kappa^+$ is a limit ordinal and the result holds for $\zeta<\xi$. We let \[\varphi_\xi(X)=\mathfrak{b}igwedge_{\zeta<\xi}\varphi_\zeta(X).\] Clearly $\varphi_\xi(X)$ is ${\mathbb P}i^1_\xi$ over $V_\kappa$. Using our inductive assumption about the $\varphi_\zeta$'s, it is easy to verify that $A$ is $\xi$-s-stationary in $\kappa$ if and only if $V_\kappa\models\varphi_\xi(A)$. Furthermore, using an argument involving generic ultrapowers similar to those of Theorem \ref{theorem_expressing_indescribability} and Theorem \ref{theorem_xi_clubs}(2), the existence of the desired club $C_\xi$ is straightforward, and is therefore left to the reader. Suppose $\xi=\zeta+1<\kappa^+$ is a successor. We let $\varphi_{\zeta+1}(X)$ be the natural ${\mathbb P}i^1_{\zeta+1}$ formula equivalent to \mathfrak{b}egin{align*} \left(\mathfrak{b}igwedge_{\eta<\zeta}\varphi_\eta(X)\right)&\langlend \forall S\forall T(\varphi_\zeta(S)\langlend\varphi_\zeta(T)\rightarrow\\ &(\exists\mathfrak{b}eta\in A)(\text{$S$ and $T$ are $f^\kappa_\zeta(\mathfrak{b}eta)$-s-stationary in $\mathfrak{b}eta$})). \end{align*} Note that, by an argument similar to that for Theorem \ref{theorem_xi_clubs}(2), we can code information about which subsets of $\mathfrak{b}eta$ are $f^\kappa_\zeta(\mathfrak{b}eta)$-s-stationary in $\mathfrak{b}eta$ into a subset of $\kappa$ and verify that the above formula is in fact equivalent to a ${\mathbb P}i^1_{\zeta+1}$ formula over $V_\kappa$. The verification that the desired club $C_{\zeta+1}$ exists and that $\varphi_{\zeta+1}$ satisfies the requirements of the lemma is similar to the proof of Theorem \ref{theorem_expressing_indescribability} and Theorem \ref{theorem_xi_clubs}(2), and is thus left to the reader. \end{proof} The next proposition, which is a generalization of \cite[Proposition 4.3]{MR3894041}, will allow us to obtain the nondiscreteness of the topologies $\tau_\xi$ from an indescribability hypothesis. \mathfrak{b}egin{proposition}\langlebel{proposition_indescribability_implies_reflecting} If a cardinal $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable for some $\xi<\kappa^+$, then it is $(\xi+1)$-s-reflecting. \end{proposition} \mathfrak{b}egin{proof} Suppose $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable and suppose that $S$ and $T$ are $\zeta$-s-stationary in $\kappa$ where $\zeta\leq\xi$. Then we have \[V_\kappa\models\varphi_\zeta(S)\langlend\varphi_\zeta(T)\] where $\varphi_\zeta(X)$ is the ${\mathbb P}i^1_\zeta$ formula obtained in Lemma \ref{lemma_expressing_s_stationarity}. Let $C_\zeta$ be the club subset of $\kappa$ from the statement of Lemma \ref{lemma_expressing_s_stationarity}. Since $\kappa$ is ${\mathbb P}i^1_\xi$-indescribable, there is an $\alpha\in C_\zeta$ such that \[V_\alpha\models\varphi_\zeta(S)\mathrm{|}^\kappa_\alpha\langlend\varphi_\zeta(T)\mathrm{|}^\kappa_\alpha,\] which implies that $S$ and $T$ are both $f^\kappa_\zeta(\alpha)$-s-stationary in $\alpha$. Hence $\kappa$ is $(\xi+1)$-s-stationary. \end{proof} Finally, we conclude that from an indescribability hypothesis, one can prove that the $\tau_{\xi+1}$ topology is not discrete. \mathfrak{b}egin{corollary}\langlebel{corollary_nondiscreteness_from_indescribability} Suppose $\mu$ is a regular cardinal and $\xi<\mu^+$. If the set \[S=\{\alpha<\mu:\text{$\alpha$ is $f^\mu_\xi(\alpha)$-indescribable}\}\] is stationary in $\mu$ (for example, this will occur if $\mu$ is ${\mathbb P}i^1_{\xi+1}$-indescribable), then there is an $\alpha<\mu$ which is nonisolated in the space $(\mu,\tau_{\xi+1})$. \end{corollary} \mathfrak{b}egin{proof} Fix $\xi<\mu^+$. Let $C$ be the club subset of $\mu$ obtained from Theorem \ref{theorem_xi_s_nonisolated}; that is, $C{\mathop{\rm sub}}seteq\mu$ is club such that for all $\alpha\in C$ we have $\alpha$ is $f^\mu_{\xi+1}(\alpha)$-s-reflecting if and only if $\alpha$ is not isolated in the $\tau_{\xi+1}$ topology. Let $D=\{\alpha<\mu: f^\mu_{\xi+1}(\alpha)=f^\mu_\xi(\alpha)+1\}$ be the club subset of $\mu$ obtained from Lemma \ref{lemma_successor}. 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\begin{document} \begin{abstract} We introduce two partial compactifications of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The subcategory of massless objects is thick and there is an induced classical stability on the quotient category. We study deformations of lax stability conditions. Second we consider the space arising by identifying lax stability conditions which are deformation-equivalent with fixed charge. This second space is stratified by stability spaces of Verdier quotients of the triangulated category by thick subcategories of massless objects. We illustrate our results through examples in which the Grothendieck group has rank $2$. For these, our partial compactification can be explicitly described and related to the wall-and-chamber structure of the stability space. \end{abstract} \keywords{Lax stability condition, triangulated category, massless semistable object} \subjclass[2020]{18G80, 16E35, 14F08} \title{Partial Compactification of Stability Manifolds \ by Massless Semistable Objects} {\small \tableofcontents } \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} {\centering \begin{tabular}{ccc} \scalebox{0.80}{ \definecolor{qqwwzz}{rgb}{0,0.4,0.6} \definecolor{qqwuqq}{rgb}{0,0.39215686274509803,0} \definecolor{uuuuuu}{rgb}{0.26666666666666666,0.26666666666666666,0.26666666666666666} \definecolor{ffvvqq}{rgb}{1,0.3333333333333333,0} \definecolor{ffzztt}{rgb}{1,0.6,0.2} \definecolor{ccqqqq}{rgb}{0.8,0,0} \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm, scale=0.4] \draw [line width=0.8pt,dash pattern=on 1pt off 1pt,color=ffvvqq,fill=ffvvqq,fill opacity=0.2] (8.46,-1.2) circle (6.885007262160295cm); 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\draw [fill=qqwwzz, color=qqwwzz] (1.931011111205963,-0.25164144761592944) circle (1.5pt); \draw [fill=ccqqqq, color=ccqqqq] (13.56782222177876,-0.21207066848262546) circle (2.5pt); \draw [fill=qqwwzz, color=qqwwzz] (12.522120083885705,2.9155435194013815) circle (2.5pt); \draw [fill=qqwwzz, color=qqwwzz] (11.524053904569175,3.4053087759508673) circle (2.5pt); \draw [fill=qqwwzz, color=qqwwzz] (5.394582956471153,3.404401546476862) circle (2.5pt); \draw [fill=qqwwzz, color=qqwwzz] (4.396661802227889,2.914340861732336) circle (2.5pt); \draw [fill=ccqqqq, color=ccqqqq] (5.053494543849391,2.742641721126842) circle (2.5pt); \end{tikzpicture}}\\ Ginzburg algebra $\cat{C} = \cat{D}^b(\Gamma_{\!2} A_2)$ && Bound path algebra $\cat{C} = \cat{D}^b(\Lambda_{1,2,0})$ \end{tabular} Examples of quotient stability spaces $\qstab{\cat{C}}^*$ up to $\mathbb{C}$ action; see Figure~\ref{a2 figure}. \par } \section*{Glossary} \subsection*{Slicings} \begin{compacthang} \item $\slice{\cat{C}}$, the set of locally finite slicings on $\cat{C}$; page~\pageref{sub:slicings} \item $P \in \slice{\cat{C}}$ is \defn{adapted} to thick $\cat{N} \subset \cat{C}$ if $P$ restricts to $\cat{N}$ and if $P(I) \cap \cat{N} \subset P(I)$ are Serre subcategories for all $I=[\varphi,\varphi+1)$ and $I=(\varphi,\varphi+1]$; page~\pageref{adapted slicing} \item $P \in \slice{\cat{C}}$ is \defn{well-adapted} to thick $\cat{N} \subset \cat{C}$ if it is adapted and the quotient slicing $P_{\cat{C}/\cat{N}}$ is locally finite; page~\pageref{well-adapted slicing} \end{compacthang} \subsection*{Charges} \begin{compacthang} \item $v\colon K(\cat{C}) \to \Lambda$, a surjective homomorphism onto a lattice with fixed inner product; page~\pageref{charges} \item $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}} \subset \mor{\Lambda}{\mathbb{C}}$, the set of charge maps $\Lambda\to\mathbb{C}$ vanishing on $\Lambda_\cat{N}$ \item $\mor{\Lambda_\cat{N}}{\mathbb{C}} \subset \mor{\Lambda}{\mathbb{C}}$, charges on $\Lambda_\cat{N}$, subset via the inner product; page~\pageref{charges} \end{compacthang} \subsection*{The spaces} \begin{compacthang} \item $\stab{\cat{C}} \subseteq \slice{\cat{C}} \times \mor{\Lambda}{\cat{C}}$, the set of stability conditions (i.e.\ supported pre-stability conditions) on $\cat{C}$ whose charge map factors as $Z \colon K(\cat{C}) \xrightarrow{v} \Lambda \to \mathbb{C}$; page~\pageref{stability conditions} \item $\mathcal{Z}\colon \stab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$, the charge map (also for larger spaces); pages~\pageref{def:charge map} and \pageref{def:space of degenerate stability conditions} \item $\legstab{\cat{C}} \subseteq \slice{\cat{C}} \times \mor{\Lambda}{\mathbb{C}}$, the set of lax stability conditions (semistable objects can have mass 0, with the support condition); page~\pageref{spaces of stability conditions} \item $\lstab{\cat{C}} = \legstab{\cat{C}} \cap \overline{\stab{\cat{C}}}$, adding the closure condition; page~\pageref{spaces of stability conditions} \item $\qstab{\cat{C}} = \lstab{\cat{C}}/{\sim}$, the space of quotient stability conditions, where two lax stability conditions are equivalent if they have the same charge and lie in the same connected component of the corresponding fibre of $\lstab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$; page~\pageref{space qstab} \end{compacthang} \[ \begin{tikzcd} \stab{\cat{C}} \ar[hook]{r} & \lstab{\cat{C}} \ar[hook]{r} \ar[twoheadrightarrow]{d} & \legstab{\cat{C}} \ar[hook]{r} & \slice{\cat{C}} \times \mor{\Lambda}{\cat{C}} \ar{d}{\mathcal{Z}} \\ & \qstab{\cat{C}} & & \mor{\Lambda}{\mathbb{C}} \end{tikzcd} \] \begin{compacthang} \item $\lnstab{\cat{N}}{\cat{C}} = \{ \sigma \in \lstab{\cat{C}} \mid \cat{N}_\sigma =\cat{N} \}$, the subset with massless subcategory $\cat{N}$; page~\pageref{def:space of degenerate stability conditions} \item $\wnstab{\cat{N}}{\cat{C}} = \{ \sigma \in \overline{\stab{\cat{C}}} \mid \cat{N}_\sigma = \cat{N}, \mu_\cat{N}(\sigma) \in \stab{\cat{C}/\cat{N}}\}$; page \pageref{def:space of degenerate stability conditions} \end{compacthang} \subsection*{Open subsets} \begin{compacthang} \item $B_\varepsilon(\sigma) = \{ (Q,W) : d(P,Q)<\varepsilon \ \text{and}\ ||W-Z||_\sigma < \sin(\pi \varepsilon) \}$; page~\pageref{semi-norm neighbourhoods} \item $V_\varepsilon(\sigma) = \{ (Q,W) \in B_\varepsilon(\sigma) : ||W_\cat{N}||_\sigma <\sin(\pi \varepsilon)\}$ for $\sigma\in\lstab{\cat{N}}{\cat{C}}$; page~\pageref{sub:neighbourhoods of strata} \item Open neighbourhoods: $\lnstab{\cat{N}}{\cat{C}} \subseteq \lustab{\cat{N}}{\cat{C}} \subseteq \lvstab{\cat{N}}{\cat{C}} \subseteq \lbstab{\cat{N}}{\cat{C}} \subseteq \lstab{\cat{C}}$: \item $\lbstab{\cat{N}}{\cat{C}} = \bigcup_{\sigma \in \lnstab{\cat{N}}{\cat{C}}} B_\varepsilon(\sigma) \cap \lstab{\cat{C}}$; page~\pageref{massless part} \item $\lvstab{\cat{N}}{\cat{C}} = \bigcup_{\sigma\in \lnstab{\cat{N}}{\cat{C}}} V_\varepsilon(\sigma) \cap \lstab{\cat{C}}$; page~\pageref{sub:neighbourhoods of strata} \item $\lustab{\cat{N}}{\cat{C}} = \{ \tau \in \lvstab{\cat{N}}{\cat{C}} : \tau \in B_\varepsilon(\Phi_\cat{N}(\tau))\}$; page~\pageref{smaller neighbourhoods} \end{compacthang} \subsection*{The maps} \begin{compacthang} \item $\mu_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$, the map sending a lax stability condition $(P,Z)$ with massless subcategory $\cat{N}$ to the massive stability condition $\mu_\cat{N}(P,Z) = (P_{\cat{C}/\cat{N}},Z)$ on the quotient. It extends to a continuous map $\mu_\cat{N} \colon \overline{\lnstab{\cat{N}}{\cat{C}}} \to \lstab{\cat{C}/\cat{N}}$; page~\pageref{massive part} \item $\rho_\cat{N} \colon \lbstab{\cat{N}}{\cat{C}} \to \lstab{\cat{N}}$, the restriction map $\rho_\cat{N}(P,Z) = (P_\cat{N}, Z_\cat{N}) = (P\cap\cat{N},Z|_{\Lambda_\cat{N}})$; page~\pageref{cor:restriction map} \item $\Phi_\cat{N} \colon \lvstab{\cat{N}}{\cat{C}} \to \lnstab{\cat{N}}{\cat{C}}$, a deformation retraction; page~\pageref{deformation retraction} \end{compacthang} \subsection*{Support propagation} \defn{Support propagates} from a component $\Sigma \subset \lnstab{\cat{N}}{\cat{C}}$ (\text{page~\pageref{def:global support propagation}}) if \[ \underset{\varepsilon>0}{\mbox{\Large $\mathsurround0pt\exists$}} \: \: \underset{\sigma\in\Sigma}{\mbox{\Large $\mathsurround0pt\forall$}} \left\{ \tau = (Q,W) \in B_\varepsilon(\sigma) \mathrel{\bigg|} \begin{array}{l} \rho_\cat{N}(\tau)\in\lstab{\cat{N}}\} \text{ and} \\ ||Z-(W-W_\cat{N})||_\sigma < \sin(\pi\varepsilon) \end{array} \right\} \subset \lnstab{\cat{N}}{\cat{C}}. \] \addtocontents{toc}{\protect\setcounter{tocdepth}{1}} \section{Introduction} \noindent The space of stability conditions on a non-zero triangulated category is always non-compact when non-empty: the mass of an object may tend to zero or infinity, and the phases of objects may tend to infinity. We construct a partial compactification in which we add boundary strata where the masses of objects in certain thick subcategories vanish. The points of these boundary strata can be interpreted as stability conditions on quotient categories. We have in mind applications to the study of the topology of stability spaces and of their wall-and-chamber structure, and also to the construction of new stability conditions. Firstly, the partial compactification is always contractible, and so maybe a useful stepping stone in establishing the conjectured contractibility of stability spaces. Secondly, under a suitable technical condition, a neighbourhood of each boundary stratum has a simple product structure. This provides new information about the boundary of the stability space. Thirdly, there is a close connection between boundary strata and walls which we hope will prove useful in understanding the wall-and-chamber structure. Roughly, the stratum where a stable object's mass vanishes is the end point of the walls where that object is a destabilising subobject or quotient. And fourthly, the key technical ingredient in the local model is a deformation result for stability conditions with massless objects. Under suitable conditions, this allows one to construct stability conditions from ones on a thick subcategory and on the quotient by it. This is reminiscent of the tilting process by which stability conditions are constructed on complex algebraic surfaces and $3$-folds by perturbing the charge of a `very-weak stability condition'. The key extra data required to perform this tilting is a Bogomolov--Gieseker type inequality. We require a stronger condition because we impose the extra requirement that the deformation is continuous in the slicing metric. \subsubsection*{Description of results} Let $\cat{C}$ be a triangulated category and $v\colon K(\cat{C}) \to \Lambda$ a surjective homomorphism from its Grothendieck group onto a finite rank lattice. Write $\slice{\cat{C}}$ for the space of locally finite slicings of $\cat{C}$. The space of stability conditions $\stab{\cat{C}}$ is the subspace of $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ consisting of pairs $(P,Z)$ with $Z(c) \in \mathbb{R}_{>0}e^{i\pi \varphi}$ whenever $0\neq c\in P(\varphi)$ is semistable of phase $\varphi$, and satisfying the \defn{support property} $\inf M_{(P,Z)}>0$ where \[ M_{(P,Z)} = \left\{ \frac{|Z(c)|}{||v(c)||} : 0\neq c \in \cat{C}\ \text{stable} \right\} \] is the \defn{(normalised) mass distribution}. The support property is usually stated in terms of \defn{semistable} objects, but it is equivalent to consider only stable objects and this turns out to be crucial when one allows semistable objects with zero mass. For simplicity in the introduction we assume that all stability spaces are connected. A pair $(P,Z)$ in the boundary of $\stab{\cat{C}}$ in $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ satisfies the conditions \begin{enumerate} \item $Z(c) \in \mathbb{R}_{\geq 0}e^{i\pi \varphi}$ whenever $0\neq c\in P(\varphi)$ and \item $\inf M_{(P,Z)} =0$. \end{enumerate} We call such a pair $(P,Z)$ a \defn{lax pre-stability condition} and refer to $c\in P(\varphi)$ with $Z(c) =0$ as a \defn{massless} semistable object. We say the pair is a \defn{lax stability condition} if it satisfies the modified support property $\inf ( M_{(P,Z)}-\{0\} ) > 0$, i.e.\ if zero is an isolated point of the normalised mass distribution. Intuitively, this forces a separation between massive and massless objects which leads to a `de-coupling' of the massive and massless parts of the theory. This manifests geometrically in local product descriptions near such points in the boundary. The \defn{space of lax stability conditions} $\lstab{\cat{C}}$ is the subset of points in the closure of $\stab{\cat{C}}$ satisfying this modified support property. An example may help to understand the restrictions we impose on boundary points --- see also \S\ref{dense phase case}. The stability space $\stab{X}$ of a strictly positive genus smooth complex projective curve $X$ is isomorphic to $\mathbb{H}\times \mathbb{C}$, where $\mathbb{H}$ is the strict upper half-plane in $\mathbb{C}$. The boundary points $\mathbb{Q}\times \mathbb{C}$ where the masses of line bundles vanish do not appear in $\lstab{X}$ because the slicings do not converge as we approach them. Nor do the points $(\mathbb{R}-\mathbb{Q})\times \mathbb{C}$ appear because no objects become massless at these points, and therefore since $\inf M_{(P,Z)} =0$ we must also have $\inf ( M_{(P,Z)}-\{0\} ) = 0$. This example shows that any partial compactification of the stability space including points where our support property fails is likely to have rather complicated local geometry. It would be pleasant to have a partial compactification including the points $\mathbb{Q}\times \mathbb{C}$, but the topology on it would have to allow for the slicings to vary discontinuously. The techniques we use rely on the convergence of slicings so prohibit consideration of such boundary points. In contrast, in the genus zero case $X=\mathbb{P}^1$ there is one boundary stratum in $\lstab{X}$ corresponding to the vanishing mass of each line bundle $\mathcal{O}(n)$ for $n\in \mathbb{Z}$. Returning to the general situation, the full subcategory $\cat{N}$ of massless objects in a lax stability condition $(P,Z)$ is thick, and there is an induced stability condition on the quotient $\cat{C}/\cat{N}$ with the same charge $Z$ and for which the semistable objects of phase $\varphi$ are those in the isomorphism closure of $P(\varphi)$ in $\cat{C}/\cat{N}$. More precisely, the induced stability condition has charge $Z$ considered as an element of the subspace $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}} \subset \mor{\Lambda}{\mathbb{C}}$ where $\Lambda_\cat{N}$ is the saturation of the subgroup $\{ v(c) : c\in \cat{N}\}$ of $\Lambda$. It satisfies the support property with respect to the homomorphism $K(\cat{C}/\cat{N}) \to \Lambda/\Lambda_\cat{N}$ induced from $v$. When $\Lambda_\cat{N}$ has rank one, $\lstab{\cat{C}}$ contains a real-codimension one boundary stratum homeomorphic to $\stab{\cat{C}/\cat{N}}\times \mathbb{R}$. Up to shift, the massless objects in $\cat{N}$ are all semistable with common phase, and this phase is recorded by the factor $\mathbb{R}$. There is a local product description at each point of this boundary stratum. The normal factor is homeomorphic to \[ \lstab{\cat{N}} \cong \stab{\cat{N}} \cup \lnstab{\cat{N}}{\cat{N}}\cong \mathbb{C}\cup (-\infty+i\mathbb{R}) \] where $\lnstab{\cat{N}}{\cat{N}}$ is the unique boundary stratum in $\lstab{\cat{N}}$ at which all objects in $\cat{N}$ are massless. In summary, $\stab{\cat{C}}$ is the interior of a manifold with boundary each of whose boundary components is homeomorphic to $\stab{\cat{C}/\cat{N}}\times \mathbb{R}$ for a some thick subcategory $\cat{N}$ of $\cat{C}$. Under additional assumptions this picture generalises to higher rank $\Lambda_\cat{N}$, allowing us to describe higher codimension strata in the boundary of $\stab{\cat{C}}$. Namely if `support propagates' along a boundary stratum where objects in $\cat{N}$ are massless then a neighbourhood of that stratum is homeomorphic to a neighbourhood of \[ \stab{\cat{C}/\cat{N}} \times \lnstab{\cat{N}}{\cat{N}}\subset \stab{\cat{C}/\cat{N}} \times \lstab{\cat{N}}. \] This `support propagation' condition is satisfied by points in the boundary of any `finite type' component, but we do not know whether it holds more generally. The fibres of the charge map $\lstab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}} \colon (P,Z) \mapsto Z$ are not discrete, because the phases of massless objects may vary whilst the charge remains constant. The massless subcategory $\cat{N}$ and the induced stability condition on $\stab{\cat{C}/\cat{N}}$ are locally constant on the fibres. We define the \defn{space of quotient stability conditions} $\qstab{\cat{C}}$ to be the topological quotient space of $\lstab{\cat{C}}$ by the equivalence relation identifying points in the same component of a fibre. Each point of $\qstab{\cat{C}}$ can be interpreted as a stability condition on the quotient $\cat{C}/\cat{N}$ by the massless subcategory $\cat{N}$. The space of quotient stability conditions contains $\stab{\cat{C}}$ as a subspace, and has a complex codimension one boundary stratum $\stab{\cat{C}/\cat{N}}$ for each boundary component $\stab{\cat{C}/\cat{N}}\times \mathbb{R}$ in $\lstab{\cat{C}}$ arising from a rank one massless subcategory $\cat{N}$. When support propagates from all boundary strata in $\lstab{\cat{C}}$ we can say more. The space of quotient stability conditions decomposes as a union \[ \qstab{\cat{C}} = \bigcup_{i\in I} \stab{\cat{C}/\cat{N}_i} \] of stability spaces of quotients of $\cat{C}$. The mass of each object $c\in \cat{C}$ extends to a continuous function $m_\bullet(c) \colon \qstab{\cat{C}} \to \mathbb{R}_{\geq0}$ and $\stab{\cat{C}/\cat{N}}$ is the subset where this mass vanishes if and only if $c\in \cat{N}$. It follows that each stratum is locally closed with closure \[ \overline{\stab{\cat{C}/\cat{N}_i}} = \bigcup_{i\leq j} \stab{\cat{C}/\cat{N}_j} \] where the indexing set $I$ is partially-ordered by the inclusion of thick subcategories. (We do not have a general characterisation of which thick subcategories $\cat{N}_i$ appear.) Moreover, under the strong propagation assumption, the space of quotient stability conditions has a local product structure: an open neighbourhood of $\stab{\cat{C}/\cat{N}}$ in $\qstab{\cat{C}}$ is homeomorphic to a neighbourhood of the `central' fibre of the second projection \[ \stab{\cat{C}/\cat{N}} \times \qstab{\cat{N}} \to \qstab{\cat{N}}, \] that is the fibre over the point stratum of $\qstab{\cat{N}}$ where all objects of $\cat{N}$ are massless. Intuitively, the massive and massless parts of the theory `de-couple' and can be treated independently of each other. The charge extends to a continuous map $\qstab{\cat{C}} \to \mor{\Lambda}{\cat{C}}$ given on a stratum by \[ \stab{\cat{C}/\cat{N}} \to \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}} \hookrightarrow \mor{\Lambda}{\mathbb{C}}, \] i.e.\ the composite of the charge map for the stability space of the quotient and the natural inclusion. The fibres are discrete and the restriction to each stratum is a local homeomorphism. This allows us to view $\qstab{\cat{C}}$ as a `stratified branched cover' of $\mor{\Lambda}{\mathbb{C}}$. However, care is required since some fibres may be empty, corresponding to the existence of `forbidden' massless subcategories. Moreover, some points may be infinitely ramified. For example, when $\Lambda_\cat{N}$ has rank one $\qstab{\cat{N}} \cong \mathbb{C} \cup \{-\infty\}$ with charge map the extension of the exponential $\mathbb{C} \to \mathbb{C}^*$ by $\exp(-\infty)=0$ to create a branched cover infinitely ramified over the origin. Indeed, the local product description shows that this is the typical situation along a codimension one stratum. There are several other recent constructions of (partial) compactifications of stability spaces. Bolognese \cite{bolognese2020local} constructs a metric completion, at least when the charge map is a covering map, which seems to be closely related to $\qstab{\cat{C}}$. Its points can also be interpreted as stability conditions on quotients of $\cat{C}$ by thick subcategories of massless objects. The difference is that she uses a notion of `limiting support' for Cauchy sequences of stability conditions, and it is not immediately obvious how this relates to the notion of support we use to define lax stability conditions --- see \S\ref{bolognese comparison}. Bapat, Deopurkar and Licata \cite{bapat2020thurston} take a very different approach. They consider, by analogy with the Thurston compactification of Teichm\"uller space, the closure of the image of \[ \eqstab{\cat{C}} \to \mathbb{R}\mathbb{P}^{\, \cat{C}} \colon \sigma \mapsto [m_\sigma(c) : c\in \cat{C} ] \] and conjecture that under `mild conditions' on $\cat{C}$ this is a (real) manifold with boundary whose interior is $\stab{\cat{C}}/\mathbb{C}$. The above map extends continuously to $\pcQstab{\cat{C}}$ allowing us to compare the two spaces, which we do in two simple examples in \S\ref{bapat et al comparison}. Under appropriate conditions it seems reasonable to hope that this extension is a homeomorphism between the interiors, and has dense image in the boundary. \subsubsection*{Discrepancies between classical and lax stability conditions} In many ways lax stability conditions behave much as classical stability conditions do. However, in some respects lax stability conditions have weaker categorical and analytical properties than classical stability conditions. We highlight the differences. Let $\sigma = (P,Z)$ be a pair consisting of a slicing $P$ on $\cat{C}$ and a charge $Z\in\mor{\Lambda}{\mathbb{C}}$. \begin{enumerate} \item The definitional distinction is that the mass of a non-zero semistable object $c\in P(\varphi)$ is classically required to be positive but can be zero if $\sigma$ is lax. \item The support property for a stability condition $\sigma$ implies that the slicing $P$ is locally finite. By contrast, the support condition for a lax stability condition $\sigma$ does not imply local finiteness of the slicing (which we therefore impose as a separate condition). \item The slices $P(\varphi)$ are always abelian length categories if $\sigma$ is a classical stability condition. If $\sigma$ is a lax stability condition then we only know that $P(\varphi)$ is a quasi-abelian length category; in particular, we don't know whether the Jordan--H\"older property holds. \item A classical stability condition $\sigma$ induces a norm $||-||_\sigma$ on $\mor{\Lambda}{\mathbb{C}}$ whereas a lax stability condition induces a semi-norm. \item If $\sigma$ is a classical stability condition then any sufficiently close element of $\slice{\cat{C}}\times\mor{\Lambda}{\mathbb{C}}$ is again a stability condition. By contrast, we do not know if elements near a lax stability condition have the support property; in general they may only be lax \defn{pre-}stability conditions. \item Finally, the space of classical stability conditions $\stab{\cat{C}}$ forms a complex manifold modelled on $\mor{\Lambda}{\mathbb{C}}$. The space of lax stability conditions $\lstab{\cat{C}}$ is a stratified space under good conditions. \end{enumerate} \begin{table} \begin{center} \begin{tabular}{llcc} \toprule & & Stability conditions & lax stability conditions \\ \midrule (1) & Masses are & positive; & non-negative. \\ (2) & Slicings are locally-finite & automatically; & by additional condition. \\ (3) & Slices $P(\varphi)$ are & abelian categories; & quasi-abelian categories. \\ (4) & $||-||_\sigma$ is a & norm; & semi-norm. \\ (5) & Small deformations are & stability conditions; & lax pre-stability conditions. \\ (6) & Geometric structure: & complex manifold; & stratified space.\\ \bottomrule \end{tabular} \end{center} \caption{Summary of discrepancies between ordinary and lax stability conditions} \label{table:discrepancies} \end{table} \subsubsection*{Structure of the article} Section~\ref{notation} fixes notation. Section~\ref{slicings and thick subcategories} discusses the relationship between slicings on $\cat{C}$ and on a thick subcategory $\cat{N}$ and the quotient $\cat{C}/\cat{N}$. This is a key ingredient of the deformation results in Section~\ref{deformations}. In Section~\ref{degenerate and quotient stability conditions} we define lax stability conditions and prove that the massless subcategory of a lax stability condition is thick and that there is a naturally induced classical stability condition on the quotient. Section~\ref{spaces of stability conditions} contains the first results on the space of lax stability conditions, including the continuity of masses and phases and the `semi-continuity' of massless subcategories. Section~\ref{deformations} is the technical heart of the paper. In Section~\ref{tangential deformation} we extend Bridgeland's result on lifting charge deformations to the context of classical stability conditions. In Section~\ref{propagation} we discuss the extent to which the support property `propagates' when we deform a lax stability condition. In Section~\ref{topology of dstab} the deformation results are applied to describe the local topology of the space of lax stability conditions. In Section~\ref{space qstab} we finally define the space $\qstab{\cat{C}}$ of quotient stability conditions. The results about the local structure of $\lstab{\cat{C}}$ descend to corresponding results about the local structure of $\qstab{\cat{C}}$. Section~\ref{rank one} examines support propagation in the simplest case in which the saturation $\Lambda_\cat{N}$ of the image of $K(\cat{N}) \to K(\cat{C})$ has rank one. In this case the massless subcategory $\cat{N}$ is generated by a set of stable objects with common phase. Support propagates from the corresponding boundary stratum in $\lstab{\cat{C}}$ and the stratum is homeomorphic to $\stab{\cat{C}/\cat{N}} \times \mathbb{R}$ and has real codimension one. The corresponding stratum in $\qstab{\cat{C}}$ has complex codimension one. In Section~\ref{finite type components} we consider the partial compactification of a `finite type component' $\stab{\cat{C}}$. This case is much simpler than the general one because $\lstab{\cat{C}} = \overline{\stab{\cat{C}}}$. We show that the massless subcategories occurring in a finite type component of $\stab{\cat{C}}$ are precisely those generated by subsets of simple objects in the heart of a stability condition in the component. We also show that support propagates from each boundary stratum so that the strong forms of our local structure results apply, in particular each stratum in $\lstab{\cat{C}}$ and in $\qstab{\cat{C}}$ has a product neighbourhood. The universal cover $G$ of $\mathrm{GL}_2^+(\mathbb{R})$ acts on $\stab{\cat{C}}$. In Section~\ref{Orbit closures} we describe the closure of the $G$-orbit of a stability condition $\sigma$ in $\qstab{\cat{C}}$ in terms of the phase diagram of $\sigma$, i.e.\ the set of `occupied' phases for which there is a non-zero semistable object. This is a key ingredient of Section~\ref{2d case} in which we illustrate our results in some simple two-dimensional examples. In each case we are able to identify $\eqstab{\cat{C}}$ holomorphically as either $\mathbb{C}$ or the the Poincar\'e disk, and also identify its wall-and-chamber structure. The walls are given by smooth analytic curves with endpoints on the boundary where the destabilising subobject and quotient respectively become massless. Finally, in Section~\ref{comparisons} we compare our approach to those of Bolognese \cite{bolognese2020local} and Bapat, Deopurkar and Licata \cite{bapat2020thurston}. \subsection*{Acknowledgments} We would like to thank Lasse Rempe for very helpful discussions about Riemann surface theory, and in particular explaining approaches to the `type problem' for two-dimensional stability spaces --- any errors are of course entirely ours. We are grateful to the London Mathematical Society and the Mathematisches Forschungsinstitut Oberwolfach for financial support through their `Research in Pairs' schemes, grants no.\ 41434 and 1815p. The second named author was supported by EPRSC grant no.\ EP/V050524/1. \section{Notation and preliminaries} \label{notation} \subsection{Quasi-abelian categories} \label{sub:quasi-abelian} It has been known since Bridgeland's original article \cite{MR2376815} that quasi-abelian categories are important in the theory of stability conditions. In this text they appear even more prominently because slices of lax stability conditions are in general not abelian categories (as with stability conditions) but only quasi-abelian; see slicing $P_t$ in Example~\ref{ex:slicings}. Recall, e.g.\ from \cite{MR1779315}, that a \defn{quasi-abelian category} is an additive category with kernels and cokernels and such that the pullback of a strict epimorphism is a strict epimorphism, and the pushout of a strict monomorphism is a strict monomorphism. Here a \defn{strict morphism} is one for which the canonical morphism from its coimage to its image is an isomorphism. A \defn{length} quasi-abelian category is one which is both artinian and noetherian. \begin{example}[{\cite[Examples 3.5 and 6.9]{MR4021926}}] Let $\mathcal{E}$ be the full additive subcategory of finite-dimensional ${\mathbf{k}}$-vector spaces generated by ${\mathbf{k}}^2$ and ${\mathbf{k}}^3$. In this example, any non-zero map ${\mathbf{k}}^2\to{\mathbf{k}}^3$ has kernel and cokernel $0$. Therefore the coimage ${\mathbf{k}}^2$ and image ${\mathbf{k}}^3$ are not isomorphic and the morphism is not strict. In particular, ${\mathbf{k}}^2$ and ${\mathbf{k}}^3$ are simple objects of $\mathcal{E}$, and $\mathcal{E}$ is a length quasi-abelian category. However, the Jordan--H\"older property fails: ${\mathbf{k}}^6 = {\mathbf{k}}^2\oplus{\mathbf{k}}^2\oplus{\mathbf{k}}^2 = {\mathbf{k}}^3\oplus{\mathbf{k}}^3$ has two Jordan--H\"older filtrations with non-isomorphic factors and of different lengths. \end{example} \subsection{Slicings} \label{sub:slicings} Let $\cat{C}$ be a triangulated category with shift functor $c\mapsto c[1]$. A \defn{slicing} $P$ on $\cat{C}$ is a collection of full additive subcategories $P(\varphi)$ for each $\varphi \in \mathbb{R}$ such that \begin{enumerate} \item $P(\varphi+1) = P(\varphi)[1]$ for all $\varphi\in \mathbb{R}$; \item $\Mor{\cat{C}}{c}{c'}=0$ whenever $c\in P(\varphi)$ and $c'\in P(\varphi')$ with $\varphi > \varphi'$; \item each $0\neq c\in \cat{C}$ admits a finite filtration i.e.\ a finite sequence of morphisms \[ \begin{tikzcd} 0=c_0 \ar{r} & c_1 \ar{r} \ar{d} & c_2 \ar{r} \ar{d} & \cdots \ar{r} & c_{n-1} \ar{r} & c_n = c \ar{d}\\ & a_1 \ar[dashed]{ul} & a_2 \ar[dashed]{ul} & & & a_n \ar[dashed]{ul} \end{tikzcd} \] with cones $a_i \in P(\varphi_i)$ where $\varphi_1 > \varphi_2 > \cdots > \varphi_n$. \end{enumerate} The objects in $P(\varphi)$ are called \defn{semistable of phase $\varphi$}, the filtration is the \defn{Harder--Narasimhan filtration} (henceforth abbreviated to HN filtration) of $c$, and the objects $a_i$ are called the semistable factors of $c$. The filtration, in particular the semistable factors, are determined uniquely up to isomorphism when they exist. The \defn{maximal and minimal phases} of $0\neq c \in \cat{C}$ are $\varphi^+(c)=\varphi_1$ and $\varphi^-(c)=\varphi_n$, respectively. For any slicing $P$ and interval $I\subset \mathbb{R}$ let $P(I)$ denote the full subcategory of $\cat{C}$ on those objects whose semistable factors with respect to the slicing have phases in $I$. When $I=(a,b]$ we omit the outer brackets and simply write write $P(a,b]$ and so on. The category $P(I)$ is quasi-abelian when $I$ has length strictly less than one. A \defn{stable} object is a simple semistable object, that is a semistable object of some phase $\varphi$ with no proper strict subobjects in the quasi-abelian category $P(\varphi)$. The slicing $P$ is \defn{locally finite} if there is some $\varepsilon>0$ such that $P(\varphi-\varepsilon,\varphi+\varepsilon) $ is a length quasi-abelian category for each $\varphi\in \mathbb{R}$. In particular, for a locally finite slicing each slice $P(\varphi)$ is a quasi-abelian length category. It follows that each semistable object has a finite composition series whose factors are stable objects. However, we do not know in general that the set of these stable factors, nor the multiplicity with which each occurs, are well-defined --- see \cite{enomoto2021jordanholder} for a discussion of when a quasi-abelian category satisfies the Jordan--H\"older Theorem. Let $\slice{\cat{C}}$ denote the space of locally finite slicings on $\cat{C}$. This has a metric \[ d(P,Q) = \sup_{0\neq c\in\cat{C}} \max \left\{ |\varphi_P^-(c)-\varphi_Q^-(c)|, |\varphi_P^+(c)-\varphi_Q^+(c)| \right\}. \] For any slicing $P$ and $\varphi\in \mathbb{R}$ the inclusions of $P(-\infty, \varphi)$ and $P(-\infty,\varphi]$ into $\cat{C}$ have respective left adjoints $H_P^{<\varphi}$ and $H_P^{\leq \varphi}$. Dually, the inclusions of $P(\varphi,\infty)$ and $P[\varphi,\infty)$ into $\cat{C}$ have respective right adjoints $H_P^{>\varphi}$ and $H_P^{\geq \varphi}$. We use the notation \[ H_P^{(\varphi,\psi)} = H_P^{>\varphi} \circ H_P^{<\psi} = H_P^{<\psi} \circ H_P^{>\varphi}, \] and similarly for semi-closed and closed intervals. We also use the shorthand $H_P^\varphi = H_P^{[\varphi,\varphi]}$. When $I$ is an interval of strict length one, that is $I$ is either $(\varphi,\varphi+1]$ or $[\varphi,\varphi+1)$ for some $\varphi\in \mathbb{R}$, the subcategory $P(I)$ is the heart of a bounded t-structure on $\cat{C}$ and $H^I_P \colon \cat{C} \to P(I)$ is the associated cohomological functor taking triangles in $\cat{C}$ to long exact sequences in $P(I)$. \begin{remark} The right adjoint to the inclusion $P(0,\infty) \hookrightarrow \cat{C}$ is the truncation \emph{below} associated to the bounded t-structure on $\cat{C}$ with heart $P(0,1]$, i.e.\ it is the functor classically denoted $\tau^{\leq 0}$. This unfortunate clash of notation arises because the factors in a HN filtration are ordered by decreasing phase. To avoid confusion we use the notation $H^{>0}_P$ instead. \end{remark} For $c\in \cat{C}$ the $P$-semistable factors of $H_P^{(\varphi,\psi)}(c)$ are precisely the $P$-semistable factors of $c$ with phases in the interval $(\varphi,\psi)$, and $H_P^\varphi (c)$ is the $P$-semistable factor with phase $\varphi$, or zero if no such factor exists. If $Q$ is another slicing with $Q(-\infty,\varphi)\subset P(-\infty,\psi)$ then \[ H_Q^{<\varphi} = H_Q^{<\varphi} H_P^{<\psi} \] and similarly for closed intervals, and the dual cases. We omit the subscript $P$ from the notation when the slicing is understood from the context. \subsection{Charges} \label{charges} Fix a finite rank lattice $\Lambda$ and a surjective homomorphism $v\colon K(\cat{C}) \to \Lambda$. For $Z\in \mor{\Lambda}{\mathbb{C}}$ and $c\in \cat{C}$ we abuse notation by writing $Z(c)$ for $Z(v([c]))$. Suppose $\cat{N}$ is a thick subcategory of $\cat{C}$. The Verdier quotient $\cat{C}/\cat{N}$ is a triangulated category with the same objects as $\cat{C}$ and a morphism $c' \to c$ in $\cat{C}/\cat{N}$ given by a roof $c' \leftarrow c'' \to c$ where the cone of the first morphism $c' \leftarrow c''$ is in $\cat{N}$; see, for example, \cite{MR1438306, MR0222093}. The homomorphism $K(\cat{N}) \to K(\cat{C})$ induced from the inclusion $\cat{N} \to \cat{C}$ need not be injective, but its cokernel is $K(\cat{C}/\cat{N})$. Let $\Lambda_\cat{N} \subset \Lambda$ be the minimal primitive sublattice containing the image of $K(\cat{N}) \to K(\cat{C}) \to \Lambda$, so that $\Lambda/\Lambda_\cat{N}$ is again a lattice. Let $v_\cat{N} \colon K(\cat{N}) \to \Lambda_\cat{N}$ and $v_{\cat{C}/\cat{N}} \colon K(\cat{C}/\cat{N}) \to \Lambda/\Lambda_\cat{N}$ denote the induced homomorphisms. The map $v_\cat{N}$ may not be surjective but it always has finite index. We also fix an inner product $\langle \cdot ,\cdot \rangle$ on $\Lambda \otimes \mathbb{R}$ and denote the associated norm by $||\cdot ||$. This norm restricts to a norm on $\Lambda_\cat{N}\otimes \mathbb{R}$ and also induces a norm on the quotient \[ \Lambda/\Lambda_\cat{N} \otimes \mathbb{R} \cong (\Lambda\otimes \mathbb{R}) / (\Lambda_\cat{N}\otimes \mathbb{R}) \] defined by $|| \lambda + \Lambda_\cat{N}\otimes \mathbb{R} || = \inf\{ || \lambda+\alpha || : \alpha \in \Lambda_\cat{N}\otimes \mathbb{R}\}$. Alternatively, this is given by identifying $\Lambda/\Lambda_\cat{N} \otimes \mathbb{R}$ with the orthogonal complement of $\Lambda_\cat{N}\otimes \mathbb{R}$ and taking the restriction of $||\cdot||$. The orthogonal projection $\Lambda\otimes \mathbb{R} \to \Lambda_\cat{N}\otimes \mathbb{R}$ induces a splitting $\mor{\Lambda_\cat{N}}{\mathbb{C}} \hookrightarrow \mor{\Lambda}{\mathbb{C}}$ and we use this to identify $\mor{\Lambda_\cat{N}}{\mathbb{C}}$ with its image in $\mor{\Lambda}{\mathbb{C}}$. \subsection{Spaces of stability conditions} We work with stability conditions on $\cat{C}$ whose charges factor through $v \colon K(\cat{C}) \to \Lambda$ and satisfy the support condition. (See \S\ref{stability conditions} for the definition.) We denote the space of these by $\stab{\cat{C}}$, leaving the lattice $\Lambda$ implicit. Stability conditions on thick subcategories $\cat{N}$ of $\cat{C}$, and on the quotients $\cat{C}/\cat{N}$ by these play a prominent role. The charges of these are always understood to factor through $v_\cat{N}$ and $v_{\cat{C}/\cat{N}}$ respectively. We denote the respective spaces of stability conditions by $\stab{\cat{N}}$ and $\stab{\cat{C}/\cat{N}}$, again omitting the lattices from the notation. \section{Restriction, descent and glueing of slicings} \label{slicings and thick subcategories} \noindent Let $\cat{N}\subset \cat{C}$ be a thick subcategory, and $\cat{C}/\cat{N}$ the quotient triangulated category. We investigate the relationship between slicings of $\cat{C}$ and slicings of $\cat{N}$ and $\cat{C}/\cat{N}$. In this section we do not assume that slicings are locally finite unless stated otherwise. \begin{definition} A slicing $P$ of $\cat{C}$ is \defn{compatible} with a pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ of slicings of $\cat{N}$ and $\cat{C}/\cat{N}$ if there are inclusions of objects $P_\cat{N}(\varphi) \subset P(\varphi) \subset P_{\cat{C}/\cat{N}}(\varphi)$ for each $\varphi \in \mathbb{R}$. \end{definition} \begin{proposition} \label{uniqueness of compatibility} There is at most one pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ compatible with each slicing $P$. When it exists, $P_\cat{N}(\varphi)=P(\varphi)\cap\cat{N}$ and $P_{\cat{C}/\cat{N}}(\varphi)$ is the isomorphism closure of $P(\varphi)$ in $\cat{C}/\cat{N}$. Conversely, there is at most one slicing $P$ compatible with each pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$. When it exists $c\in P(\varphi)$ if and only if $c\in P_{\cat{C}/\cat{N}}(\varphi)$ and \begin{equation} \label{glued slicing condition} \Mor{\cat{C}}{b}{c}=0=\Mor{\cat{C}}{c}{d} \end{equation} for all $b\in P_\cat{N}(\psi)$ with $\psi>\varphi$ and all $d\in P_\cat{N}(\psi')$ with $\psi'<\varphi$. \end{proposition} \begin{proof} Suppose the pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ is compatible with $P$. Then $P_\cat{N}(\varphi)\subset P(\varphi)$ so that the HN filtration of any $c\in \cat{N}$ with $P_\cat{N}$-semistable factors is also the HN filtration with $P$-semistable factors. The uniqueness of HN filtrations implies that $P(\varphi)\cap \cat{N} \subset P_\cat{N}(\varphi)$, hence that $P_\cat{N}(\varphi)=P(\varphi)\cap \cat{N}$ for each $\varphi\in \mathbb{R}$. Denote by $P(\varphi)_{\cat{C}/\cat{N}}$ the closure of $P(\varphi)$ in $\cat{C}/\cat{N}$ under isomorphisms. We claim $P_{\cat{C}/\cat{N}}(\varphi) = P(\varphi)_{\cat{C}/\cat{N}}$. Since $P(\varphi)\subset P_{\cat{C}/\cat{N}}(\varphi)$ it is clear that $P(\varphi)_{\cat{C}/\cat{N}} \subset P_{\cat{C}/\cat{N}}(\varphi)$. Moreover, again by uniqueness, the HN filtration of any $c\in \cat{C}$ with $P$-semistable factors descends to the HN filtration of $c$ with $P_{\cat{C}/\cat{N}}$-semistable factors if we simply ignore any factors in $\cat{N}$. Thus if $c\in P_{\cat{C}/\cat{N}}(\varphi)$ it has a HN filtration in $\cat{C}$ with all factors in $\cat{N}$ except for one factor, say $c'$, in $P(\varphi)$. Thus $c\cong c'$ in $\cat{C}/\cat{N}$ and $P_{\cat{C}/\cat{N}}(\varphi) \subset P(\varphi)_{\cat{C}/\cat{N}}$ establishing the claim. In the other direction, if $P$ is compatible with $(P_\cat{N},P_{\cat{C}/\cat{N}})$ then by definition $P(\varphi)\subset P_{\cat{C}/\cat{N}}(\varphi)$. We saw above that any $c\in P_{\cat{C}/\cat{N}}(\varphi)$ has a HN filtration in $\cat{C}$ with all factors in $\cat{N}$ apart from a single factor in $P(\varphi)$. Therefore $H_P^{>\varphi}(c)$ and $H_P^{<\varphi}(c)$ are in $\cat{N}$ and so vanish precisely when (\ref{glued slicing condition}) holds. Thus $c\in P(\varphi)$ if and only if $c\in P_{\cat{C}/\cat{N}}(\varphi)$ and (\ref{glued slicing condition}) holds. This shows that $P$, when it exists, is uniquely determined by the pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$. \end{proof} \begin{lemma} \label{local-finiteness of compatible slicing} If $P$ is compatible with a pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ of locally finite slicings then $P$ is also locally finite. \end{lemma} \begin{proof} Let $I \subset \mathbb{R}$ be an interval such that both $P_\cat{N}(I)$ and $P_{\cat{C}/\cat{N}}(I)$ are quasi-abelian length categories. Let $a_0 \hookrightarrow a_1 \hookrightarrow \cdots \hookrightarrow a$ be an increasing sequence of strict subobjects of $a$ in $P(I)$. This can be considered as an increasing sequence of strict subobjects in $P_{\cat{C}/\cat{N}}(I)$ via the quotient functor $\cat{C} \to \cat{C}/\cat{N}$ since the strict exact structures on $P_\cat{N}(I)$ and $P_{\cat{C}/\cat{N}}(I)$ are induced by the triangulated structures on $\cat{C}$ and $\cat{C}/\cat{N}$, respectively, and the quotient functor is exact. Since $P_{\cat{C}/\cat{N}}(I)$ is length this chain stabilises, i.e.\ there is some $n\in \mathbb{N}$ such that $a_n\cong a_{n+1}\cong \cdots \cong a$ in $P_{\cat{C}/\cat{N}}(I)$. Pushing the sequence of strict monomorphisms out along $a_{n+1}\to a_{n+1}/a_n$ we obtain a diagram \[ \begin{tikzcd} a_{n+1} \ar[hookrightarrow]{r} \ar[->>]{d} & a_{n+2}\ar[hookrightarrow]{r} \ar[->>]{d} & \cdots \ar[hookrightarrow]{r} & a \ar[->>]{d}\\ a_{n+1}/a_n \ar[hookrightarrow]{r} & a_{n+2}/a_n\ar[hookrightarrow]{r} & \cdots \ar[hookrightarrow]{r} & a/a_n \end{tikzcd} \] whose bottom row is an increasing sequence of strict subobjects of $a/a_n$ in $P_\cat{N}(I)$. Since the latter is length this bottom row also stabilises. It follows that the original sequence stabilises so that $P(I)$ is noetherian. The proof that it is artinian is dual. \end{proof} In the next sections we discuss the more subtle question of when compatible slicings exist. \subsection{Restriction and descent} Let $\cat{N}\subset \cat{C}$ be a thick subcategory. We say that a slicing $P$ of $\cat{C}$ \defn{restricts} to $\cat{N}$ if the HN factors of each $c\in \cat{N}$ lie in $\cat{N}$. In that case the full subcategories $P(\varphi)\cap \cat{N}$ define a slicing $P_\cat{N}(\varphi)$ of $\cat{N}$. The question of when $P$ \defn{descends} to a compatible slicing of $\cat{C}/\cat{N}$ is more involved. \begin{definition} \label{adapted slicing} A slicing $P$ of $\cat{C}$ is \defn{adapted} to a thick subcategory $\cat{N}$ if it restricts to $\cat{N}$ and $P(I) \cap \cat{N}$ is a Serre subcategory of $P(I)$ for each strict length one interval $I \subset \mathbb{R}$. \end{definition} \begin{lemma} \label{quotient semistables} Let $P$ be a slicing of $\cat{C}$ adapted to the thick subcategory $\cat{N}$. Fix $c\in P(\varphi)$ and $b\in \cat{C}$. Then the following two conditions are equivalent: \begin{enumerate} \item $b \cong c$ in $\cat{C}/\cat{N}$ \item the only semistable factor of $b$ not in $\cat{N}$ is a factor $b_0\in P(\varphi)$ with $b_0\cong c$ in $\cat{C}/\cat{N}$. \end{enumerate} \end{lemma} \begin{proof} Clearly if $b$ has a unique semistable factor $b_0 \in P(\varphi)$ not in $\cat{N}$ then $b\cong b_0$ in $\cat{C}/\cat{N}$, and thus if $b_0\cong c$ in $\cat{C}/\cat{N}$ then also $b\cong c$ in $\cat{C}/\cat{N}$. For the other direction, it is enough to show that, given $c\in \cat{C}$ with $H^{<\varphi}(c)$ and $H^{>\varphi}(c)$ in $\cat{N}$ and a morphism in $\Mor{\cat{C}}{c}{b}$ or $\Mor{\cat{C}}{b}{c}$ with cone in $\cat{N}$, then $H^{<\varphi}(b)$ and $H^{>\varphi}(b)$ are also in $\cat{N}$. The cases are similar, and we only consider the first in which there is an exact triangle $d[-1]\to c\to b \to d$ with $d\in \cat{N}$. Applying the cohomological functor $H^{(\varphi,\varphi+1]}$ yields a long exact sequence \[ \cdots \to H^{(\varphi,\varphi+1]}(c) \to H^{(\varphi,\varphi+1]}(b) \to H^{(\varphi,\varphi+1]}(d) \to \cdots. \] The assumptions on $c$ and $d$ imply that the first and third terms are in $P(\varphi,\varphi+1]\cap \cat{N}$. Since this is a Serre subcategory of $P(\varphi,\varphi+1]$ so too is the middle term. For the same reason $H^{(\varphi+n,\varphi+n+1]}(b) \in \cat{N}$ for all $n\in \mathbb{N}$, which implies $H^{>\varphi}(b)\in \cat{N}$. To show that $H^{<\varphi}(b)\in \cat{N}$ one proceeds similarly using the cohomological functor $H^{[\varphi-1,\varphi)}$. \end{proof} \begin{proposition} \label{quotient slicing} Let $P$ be a slicing of $\cat{C}$. Then the following conditions are equivalent: \begin{enumerate} \item $P$ is adapted to $\cat{N}$. \item There is a pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ of slicings of $\cat{N}$ and $\cat{C}/\cat{N}$ compatible with $P$. \end{enumerate} \end{proposition} \begin{proof} If there is a compatible pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ then we saw above that $P_\cat{N}(\varphi)= P(\varphi)\cap \cat{N}$ so that $P$ restricts to $\cat{N}$. Moreover, the quotient functor $\cat{C}\to \cat{C}/\cat{N}$ restricts to an exact functor $P(\varphi,\varphi+1] \to P_{\cat{C}/\cat{N}}(\varphi,\varphi+1]$ between abelian categories with $P(\varphi,\varphi+1]\cap \cat{N}$ as kernel. Hence the latter is a Serre subcategory for each $\varphi\in \mathbb{R}$. The argument showing $P[\varphi,\varphi+1) \cap \cat{N}$ is a Serre subcategory is similar. Thus $P$ is adapted to $\cat{N}$. Now suppose that $P$ is adapted to $\cat{N}$. The subcategories $P_\cat{N}(\varphi) \coloneqq P(\varphi)\cap\cat{N}$ define a slicing of $\cat{N}$. We must show that $P$ also descends to a slicing of $\cat{C}/\cat{N}$. Proposition~\ref{uniqueness of compatibility} shows that we must define $P_{\cat{C}/\cat{N}}(\varphi)$ to be the closure of $P(\varphi)$ under isomorphisms in $\cat{C}/\cat{N}$. By construction $P_{\cat{C}/\cat{N}}(\varphi)$ is a full additive subcategory of $\cat{C}/\cat{N}$ for each $\varphi\in \mathbb{R}$, satisfying $P_{\cat{C}/\cat{N}}(\varphi+1) = P_{\cat{C}/\cat{N}}(\varphi)[1]$. Moreover, ignoring any factors in $\cat{N}$, the image in $\cat{C}/\cat{N}$ of the HN filtration of $0\neq c \in \cat{C}$ with respect to the slicing $P$ provides a finite filtration with factors in these subcategories and with strictly decreasing phases. Therefore to show that $P_{\cat{C}/\cat{N}}$ is a slicing we must show that there are no non-zero morphisms in $\cat{C}/\cat{N}$ from $c\in P_{\cat{C}/\cat{N}}(\varphi)$ to $c'\in P_{\cat{C}/\cat{N}}(\varphi')$ when $\varphi > \varphi'$. It is enough to show that $\Mor{\cat{C}/\cat{N}}{c}{c'}=0$ for $c\in P(\varphi)$ and $c'\in P(\varphi')$. A morphism in $\Mor{\cat{C}/\cat{N}}{c}{c'}$ is represented by a `roof' $c \leftarrow b \to c'$ in $\cat{C}$ where the cone on the left hand morphism is in $\cat{N}$. By Lemma~\ref{quotient semistables}, $H^{>\varphi}(b)$ and $H^{<\varphi} (b)$ are in $\cat{N}$ and moreover, $\Mor{\cat{C}}{H^{>\varphi}(b)}{c}=0= \Mor{\cat{C}}{H^{>\varphi}(b)}{c'}$ because $c\in P(\varphi)$ and $c'\in P(\varphi')$ for $\varphi'<\varphi$. Thus we can construct dashed arrows to form a commutative diagram \[ \begin{tikzcd} {} & b\ar{dl} \ar{d} \ar{dr} & {}\\ c & H^{\leq \varphi}(b) \ar[dashed]{l} \ar[dashed]{r} & c' \\ {} & H^\varphi(b) \ar{u} \ar[dashed]{ul} \ar[dashed]{ur} \ar{u} & {} \end{tikzcd} \] in which the morphisms $b\to H^{\leq \varphi}(b)$ and $H^\varphi(b)\to H^{\leq \varphi}(b)$ are the canonical ones. These are isomorphisms in $\cat{C}/\cat{N}$. Therefore the bottom `roof' $c \leftarrow H^\varphi(b) \to c'$ represents the same morphism on $\cat{C}/\cat{N}$. Since $\Mor{\cat{C}}{H^\varphi(b)}{c'}=0$ we deduce $\Mor{\cat{C}/\cat{N}}{c}{c'}=0$ as required. \end{proof} \begin{corollary} \label{cor:slicing inequalities} Let $P$ and $Q$ be slicings on $\cat{C}$ that are adapted to $\cat{N}$, and let $P_\cat{N}, Q_\cat{N} \in \slice{\cat{N}}$, $P_{\cat{C}/\cat{N}}, Q_{\cat{C}/\cat{N}} \in \slice{\cat{C}/\cat{N}}$ be the induced slicings. Then \[ d(P_\cat{N},Q_\cat{N}) \leq d(P,Q) \qquad\text{and}\qquad d(P_{\cat{C}/\cat{N}},Q_{\cat{C}/\cat{N}}) \leq d(P,Q) . \] \end{corollary} \begin{proof} The HN filtrations of $c\in\cat{N}$ with respect to $P$ and to $P_\cat{N}$ coincide. The first statement follows since the supremum defining $d(P_\cat{N},Q_\cat{N})$ is taken over a subset of that defining $d(P,Q)$. The second statement follows because \begin{align*} d(P, Q) < \varepsilon & \iff P(\varphi) \subset Q(\varphi-\varepsilon,\varphi+\varepsilon) & \forall \varphi \in \mathbb{R} \\ & \mathrm{im}plies P(\varphi) \subset Q_{\cat{C}/\cat{N}}(\varphi-\varepsilon,\varphi+\varepsilon) & \forall \varphi \in \mathbb{R} \\ & \iff P_{\cat{C}/\cat{N}}(\varphi) \subset Q_{\cat{C}/\cat{N}}(\varphi-\varepsilon,\varphi+\varepsilon) & \forall \varphi \in \mathbb{R}\\ & \iff d(P_{\cat{C}/\cat{N}}, Q_{\cat{C}/\cat{N}}) < \varepsilon \end{align*} where we have used the fact that $P_{\cat{C}/\cat{N}}(\varphi)$ is the isomorphism closure of $P(\varphi)$ in $\cat{C}/\cat{N}$, and similarly for $Q_{\cat{C}/\cat{N}}(\varphi)$. \end{proof} If $P$ is a locally finite slicing adapted to a thick subcategory $\cat{N}$ then the restriction $P_\cat{N}$ is clearly locally finite. However, we do not know whether the slicing on the quotient $P_{\cat{C}/\cat{N}}$ is locally finite when $P$ is so. Therefore we introduce the following enhancement of Definition~\ref{adapted slicing}. \begin{definition} \label{well-adapted slicing} A locally finite slicing $P$ of $\cat{C}$ is \defn{well-adapted} to a thick subcategory $\cat{N}$ if it is adapted to it and the quotient slicing $P_{\cat{C}/\cat{N}}$ is locally finite. \end{definition} \begin{corollary} Let $P$ be a slicing of $\cat{C}$. Then the following conditions are equivalent: \begin{enumerate} \item $P$ is locally finite and well-adapted to $\cat{N}$. \item There is a pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ of locally finite slicings of $\cat{N}$ and $\cat{C}/\cat{N}$ compatible with $P$. \end{enumerate} \end{corollary} \begin{proof} If $P$ is locally finite and well-adapted to $\cat{N}$ then the restricted slicing $P_\cat{N}$ and quotient slicing $P_{\cat{C}/\cat{N}}$ exist and are locally finite. The slicing $P$ is compatible with them. Conversely, if $P$ is compatible with the pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ of locally finite slicings then $P$ is locally finite by Lemma~\ref{local-finiteness of compatible slicing}, it is adapted to $\cat{N}$ by Proposition~\ref{quotient slicing} and indeed is well-adapted since $P_{\cat{C}/\cat{N}}$ is locally finite by assumption. \end{proof} \subsection{Glueing} Let $\cat{N}\subset \cat{C}$ be a thick subcategory. In this subsection we establish a criterion for when slicings of $\cat{N}$ and $\cat{C}/\cat{N}$ can be glued to a compatible slicing on $\cat{C}$. An important consequence is that the set of pairs of slicings which glue to a locally finite slicing is open. \begin{proposition} \label{glueing slicings} Let $(Q_\cat{N},Q_{\cat{C}/\cat{N}})$ be a pair of slicings of $\cat{N}$ and $\cat{C}/\cat{N}$. Then the following conditions are equivalent: \begin{enumerate} \item There is a compatible locally finite slicing $Q$ of $\cat{C}$. \item There is a locally finite slicing $P$ of $\cat{C}$ compatible with a pair $(P_\cat{N},P_{\cat{C}/\cat{N}})$ such that $d(P_\cat{N},Q_\cat{N})<\varepsilon$ and $d(P_{\cat{C}/\cat{N}},Q_{\cat{C}/\cat{N}})<\varepsilon$ where $\varepsilon>0$ is sufficiently small that the categories $P(\varphi-2\varepsilon,\varphi+2\varepsilon)$ are length for each $\varphi\in \mathbb{R}$. \end{enumerate} \end{proposition} One direction is trivial, if there is a compatible locally finite slicing $Q$ then we set $P=Q$ and are done. The proof of the other direction is rather long, so we break it down into a number of results. We retain the notation and assumptions of the statement throughout this section. By Proposition~\ref{uniqueness of compatibility} there is at most one choice for the slicing: for $\varphi\in \mathbb{R}$ the full subcategory $Q(\varphi)$ must be defined by $c\in Q(\varphi) \iff c\in Q_{\cat{C}/\cat{N}}(\varphi)$ and, for any $b\in Q_\cat{N}(\psi)$, \begin{enumerate}[(i)] \item \label{ss1} $\psi > \varphi$ implies $\Mor{\cat{C}}{b}{c}=0$ and \item \label{ss2} $\psi < \varphi$ implies $\Mor{\cat{C}}{c}{b}=0$. \end{enumerate} That is, $Q(\varphi) = Q_{\cat{C}/\cat{N}}(\varphi) \cap Q_\cat{N}(\mathit{>}\varphi)^\perp \cap {}^\perp Q_\cat{N}(\mathit{<}\varphi)$. Clearly $Q(\varphi+1) = Q(\varphi)[1]$ for all $\varphi \in \mathbb{R}$. Moreover, $Q(\varphi)\cap \cat{N} = Q_\cat{N}(\varphi)$ and $Q(\varphi)\subset Q_{\cat{C}/\cat{N}}(\varphi)$. As usual, we extend the notation to intervals $I\subset \mathbb{R}$ by defining $Q(I)$ to be the extension-closure of the $Q(\varphi)$ for $\varphi\in I$. By definition $Q(I) \subset Q_{\cat{C}/\cat{N}}(I)$ for any interval $I$. \begin{lemma} \label{slicing distance} For each $\varphi \in \mathbb{R}$ we have $Q(\varphi) \subset P(\varphi-\varepsilon,\varphi+\varepsilon)$. \end{lemma} \begin{proof} Suppose $c\in Q(\varphi)$. Then as $Q(\varphi) \subset Q_{\cat{C}/\cat{N}}(\varphi)\subset P_{\cat{C}/\cat{N}}(\varphi-\varepsilon, \varphi+\varepsilon)$ we know that $H_P^{\geq \varphi+\varepsilon}(c) , H_P^{\leq \varphi-\varepsilon}(c) \in \cat{N}$. Thus $H_P^{\geq \varphi+\varepsilon}(c) \in Q_\cat{N}(\mathit{>}\varphi)$ and $H_P^{\leq \varphi-\varepsilon}(c) \in Q_\cat{N}(\mathit{<}\varphi)$ and so by the definition of $Q(\varphi)$ we have \[ \Mor{\cat{C}}{H_P^{\geq \varphi+\varepsilon}(c)}{c} = 0 = \Mor{\cat{C}}{c}{H_P^{\leq \varphi-\varepsilon}(c)}. \] It follows that $H_P^{\geq \varphi+\varepsilon}(c) = 0 = H_P^{\leq \varphi-\varepsilon}(c)$ so that $c\in P(\varphi-\varepsilon,\varphi+\varepsilon)$ as claimed. \end{proof} \begin{lemma} \label{reconstructed slicing} If $c\in Q(\varphi)$ and $c'\in Q(\varphi')$ with $\varphi > \varphi'$ then $\Mor{\cat{C}}{c}{c'}=0$. \end{lemma} \begin{proof} Suppose $\gamma \in \Mor{\cat{C}}{c}{c'}$. Since $c \in Q_{\cat{C}/\cat{N}}(\varphi)$ and $c' \in Q_{\cat{C}/\cat{N}}(\varphi')$ the morphism $\gamma$ vanishes in $\cat{C}/\cat{N}$. Hence it must factor through some $b\in \cat{N}$. We therefore have a diagram \[ \begin{tikzcd} c \ar{rr}{\gamma} \ar{dr} \ar[dashed]{dd} && c' \\ & b \ar{ur} \ar{dr} &\\ H_{Q_\cat{N}}^{\geq \varphi}(b) \ar{ur} && H_{Q_\cat{N}}^{<\varphi}(b) \ar{ll}[swap]{[1]} \end{tikzcd} \] in which the upper triangle commutes and the lower triangle is exact. Condition~\ref{ss2} implies that $\Mor{\cat{C}}{c}{H^{<\varphi}(b)}=0$, and hence that there is a dashed morphism making the left hand triangle commute. Condition~\ref{ss1} and the condition $\varphi'<\varphi$ imply that $\Mor{\cat{C}}{H^{\geq\varphi}(b)}{c'}=0$. Hence $\gamma=0$ as claimed. \end{proof} It remains to check that each $c\in \cat{C}$ has a HN filtration with respect to $Q$. We do so by induction on the length of the HN filtration of $c$ in $\cat{C}/\cat{N}$ with respect to $Q_{\cat{C}/\cat{N}}$. The next result provides the base case. \begin{lemma} \label{HN base case} Suppose $c\in Q_{\cat{C}/\cat{N}}(\varphi)$. Then $c$, considered as an object of $\cat{C}$, has a HN filtration with respect to $Q$ with all factors in $\cat{N}$ except for a single factor in $Q(\varphi)$. \end{lemma} \begin{proof} Let $0 < \varepsilon < \frac{1}{2}$ and $\cat{A}= P(\varphi-\varepsilon, \varphi+\varepsilon)$ and to begin with, assume $c\in \cat{A}$. We first show that $c\in Q(\varphi)$ if and only if $\Mor{\cat{C}}{b}{c} = 0 = \Mor{\cat{C}}{c}{d}$ for all $b\in \cat{A} \cap Q_\cat{N}(\mathit{>}\varphi)$ and $d\in \cat{A} \cap Q_\cat{N}(\mathit{<}\varphi)$. One direction is clear: when $c\in Q(\varphi)$ the vanishing conditions follow immediately from the definition of $Q$. For the other direction suppose $b\in Q_\cat{N}(\mathit{>}\varphi)$. There is an exact triangle \[ H_P^{\geq \varphi+\varepsilon}(b) \to b \to H_P^{< \varphi+\varepsilon}(b) \to H_P^{\geq\varphi+\varepsilon}(b)[1]. \] Note that $\Mor{\cat{C}}{H^{\geq \varphi+\varepsilon}(b)}{c} = 0$ because $c \in \cat{A} \subset P(\mathit{<}\varphi+\varepsilon)$. Therefore any morphism from $b$ to $c$ factors through $h \coloneqq H_P^{< \varphi+\varepsilon}(b)$. Since $d(P_\cat{N},Q_\cat{N})<\varepsilon$ we have \[ H_P^{\geq\varphi+\varepsilon}(b)[1] \in P_\cat{N}(\geq \varphi+\varepsilon+1) \subset Q_\cat{N}(\geq \varphi+1) \subset Q_\cat{N}(>\varphi). \] Since $Q_\cat{N}(\mathit{>}\varphi)$ is extension-closed the above triangle shows that $h \in Q_\cat{N}(\mathit{>}\varphi)$. Moreover, $h \in \cat{A}$ because $b\in Q_\cat{N}(\mathit{>}\varphi) \subset P_\cat{N}(\mathit{>}\varphi-\varepsilon)$. Therefore $\Mor{\cat{C}}{h}{c}=0$ by assumption, and hence $\Mor{\cat{C}}{b}{c}=0$ too. A dual argument shows that $\Mor{\cat{C}}{c}{d}=0$ for all $d\in Q_\cat{N}(\mathit{<}\varphi)$. Hence $c\in Q(\varphi)$ as claimed. We now use this criterion to construct a HN filtration for $c \in \cat{A}$ with all factors in $\cat{N}$ except for a single factor in $Q(\varphi)$ isomorphic to $c$ in $\cat{C}/\cat{N}$. Let $b$ be a maximal strict subobject of $c$ in the subcategory $\cat{A}\cap Q_\cat{N}(\mathit{>}\varphi)$ of $\cat{A}$. We can always find such a $b$ (possibly zero) because $\cat{A}$ is a quasi-abelian length category. Then $c'=c/b$ has no non-zero strict subobjects in $\cat{A}\cap Q_\cat{N}(\mathit{>}\varphi)$ because if $b' \hookrightarrow c'$ is such a strict subobject we can pullback to obtain a commutative diagram \[ \begin{tikzcd} b \ar[equals]{d} \ar[hookrightarrow]{r} & b'' \ar[hookrightarrow]{d}\ar[->>]{r} & b'\ar[hookrightarrow]{d} \\ b \ar[hookrightarrow]{r} & c\ar[->>]{r} & c' \end{tikzcd} \] whose rows are strict short exact sequences and whose vertical morphisms are strict monomorphisms. In particular $b''$ is a strict subobject of $c$ in $\cat{A}\cap Q_\cat{N}(\mathit{>}\varphi)$ and by maximality of $b$ we deduce that $b\cong b''$ and therefore that $b'=0$. By assumption $b$ has a HN filtration with respect to $Q_\cat{N}$. Using this we can construct a finite filtration of $c$ in $\cat{A}$ whose quotients are a sequence of $Q_\cat{N}$-semistable objects of strictly decreasing phase in $\cat{A}\cap Q_\cat{N}(\mathit{>}\varphi)$, except for the final quotient $c'$ which has no non-zero strict subobjects in $\cat{A} \cap Q_\cat{N}(\mathit{>}\varphi)$. A dual argument constructs a finite filtration of this final quotient $c'$ whose first term $c''$ has no non-zero strict quotients in $\cat{A} \cap Q_\cat{N}(\mathit{<}\varphi)$ and whose other quotients form a sequence of $Q_\cat{N}$-semistable objects of strictly decreasing phase in $\cat{A}\cap Q_\cat{N}(\mathit{<}\varphi)$. It follows that $c''$ cannot have any non-zero strict subobjects in $\cat{A} \cap Q_\cat{N}(\mathit{>}\varphi)$ either, for any such would lift to a subobject of $c'$. To summarise we have constructed a strict subquotient $c''$ of $c$ in $\cat{A}$ such that \begin{enumerate} \item $c''\cong c$ in $\cat{C}/\cat{N}$, in particular $c''\in Q_{\cat{C}/\cat{N}}(\varphi)$; \item $c''$ has no non-zero strict subobjects in $\cat{A} \cap Q_\cat{N}(\mathit{>}\varphi)$; \item $c''$ has no non-zero strict quotients in $\cat{A} \cap Q_\cat{N}(\mathit{<}\varphi)$. \end{enumerate} Recalling that the image of any morphism in the quasi-abelian category $\cat{A}$ is a strict subobject of the target, and dually that the coimage is a strict quotient of the source, we conclude that $\Mor{\cat{C}}{b}{c''} = 0 = \Mor{\cat{C}}{c''}{d}$ for all $b\in \cat{A} \cap Q_\cat{N}(\mathit{>}\varphi)$ and $d\in \cat{A} \cap Q_\cat{N}(\mathit{<}\varphi)$. Hence $c''\in Q(\varphi)$ and concatenating the filtrations of $b$ and of $c'$ using iterated applications of the octahedral axiom yields the desired HN filtration. Now consider the general case, i.e.\ remove the assumption that $c\in \cat{A}$. By the first part $c_0=H_P^{(\varphi-\varepsilon,\varphi+\varepsilon)}(c)$ has a HN filtration with respect to $Q$. Noting $H_P^{\geq \varphi + \varepsilon} H_P^{> \varphi - \varepsilon}(c) = H_P^{\geq \varphi + \varepsilon}(c)$ and applying the octahedral axiom, there is a commutative diagram \[ \begin{tikzcd} H_P^{\geq \varphi+\varepsilon}(c) \ar{r} \ar[equals]{d}& c_1 \ar{r} \ar{d}& H_Q^{>\varphi}(c_0) \ar{d} \\ H_P^{\geq \varphi+\varepsilon}(c) \ar{r} \ar{d}& H_P^{>\varphi-\varepsilon}(c) \ar{r} \ar{d}& c_0 \ar[]{d}\\ 0 \ar{r} & H_Q^{\leq \varphi}(c_0) \ar[equals]{r}& H_Q^{\leq \varphi}(c_0) \end{tikzcd} \] whose rows and columns extend to exact triangles. Moreover $H_P^{\geq \varphi+\varepsilon}(c) \in \cat{N}$ because \[ c\in Q_{\cat{C}/\cat{N}}(\varphi) \subset P_{\cat{C}/\cat{N}}(\varphi-\varepsilon, \varphi+\varepsilon). \] Indeed $H_P^{\geq \varphi+\varepsilon}(c) \in Q_\cat{N}(\mathit{>}\varphi)$ because $d(P_\cat{N},Q_\cat{N})<\varepsilon$. Considering the top row, and recalling that $H_Q^{>\varphi}(c_0)\in \cat{N}$ too, shows that $c_1 \in Q_\cat{N}(\mathit{>}\varphi)$. Therefore, by considering the middle column, we can construct a HN filtration for $c$ with $Q$-semistable factors by concatenating the filtrations of $c_1$ and of $H_Q^{\leq \varphi}(c_0)$. \end{proof} \begin{proof}[Proof of Proposition~\ref{glueing slicings}] Suppose $c\in \cat{C}$ has a HN filtration of length $k\in \mathbb{N}$ in $\cat{C}/\cat{N}$ with $Q_{\cat{C}/\cat{N}}$-semistable factors. We show that $c$ has a HN filtration in $\cat{C}$ with $Q$-semistable factors. If $k=0$ then $c\in \cat{N}$ and we simply take the HN filtration with respect to $Q_\cat{N}$. If $k=1$ then the result holds by Lemma~\ref{HN base case}. Therefore we assume that $k>1$ and that the result holds for any object with a strictly shorter HN filtration in $\cat{C}/\cat{N}$. Choose a representative $b\in Q_{\cat{C}/\cat{N}}(\psi)$ for the highest phase factor of $c$ so that there is an exact triangle $b\to c \to d \to b[1]$ in $\cat{C}$. By induction we may assume both $b$ and $d$ have HN filtrations with $Q$-semistable factors. In particular there is an exact triangle \[ H^{\geq \psi}_Q(b) \to b \to H^{<\psi}_Q(b) \to H^{\geq \psi}_Q(b)[1] \] in $\cat{C}$. Since $H^{\geq \psi}_Q(b) \in Q(\mathit{\geq}\psi) \subset Q_{\cat{C}/\cat{N}}(\mathit{\geq}\psi)$ and $H^{<\psi}_Q(b)\in Q(\mathit{<}\psi) \subset Q_{\cat{C}/\cat{N}}(\mathit{<}\psi)$ we deduce that $H^{<\psi}_Q(b)\in \cat{N}$ and that $H^{\geq \psi}_Q(b) \to b$ is an isomorphism in $\cat{C}/\cat{N}$. Therefore we may assume $b\in Q(\mathit{\geq}\psi)$. Having done so we argue similarly with $d$. There is an exact triangle \[ H^{\geq \psi}_Q(d) \to d \to H^{<\psi}_Q(d) \to H^{\geq \psi}_Q(d)[1] \] where now $H^{\geq \psi}_Q(d)\in \cat{N}$ and $d \to H^{<\psi}_Q(d)$ is an isomorphism in $\cat{C}/\cat{N}$. Hence there is a commutative diagram \[ \begin{tikzcd} b \ar{r} \ar[equals]{d}& b' \ar{r} \ar{d}& H_Q^{\geq\psi}(d) \ar{d} \\ b\ar{r} \ar{d}& c\ar{r} \ar{d}& d \ar[]{d}\\ 0 \ar{r} & H_Q^{<\psi}(d) \ar[equals]{r}& H_Q^{<\psi}(d) \end{tikzcd} \] whose rows and columns extend to exact triangles. By considering the top row we see that $b'\in Q(\mathit{\geq} \psi)$. Thus the middle column shows that we may assume, by judicious choice of representatives, that $b\in Q(\mathit{\geq} \psi)$ and $d\in Q(\mathit{<} \psi)$. Having done so, we obtain a $Q$ HN filtration for $c$ by concatenating those of $b$ and $d$. Clearly $d(P,Q)<\varepsilon$ because $Q(\varphi)\subset P(\varphi-\varepsilon,\varphi+\varepsilon)$ for all $\varphi\in \mathbb{R}$ by Lemma~\ref{slicing distance}. Finally $Q$ is locally finite because $Q(\varphi-\varepsilon,\varphi+\varepsilon) \subset P(\varphi-2\varepsilon,\varphi+2\varepsilon)$ and the latter is length. \end{proof} \begin{example} \label{ex:slicings} Let $\cat{C} = \cat{D}^b(\mathbb{P}^1)$ be the bounded derived category of coherent sheaves on the projective line $\mathbb{P}^1$. All complexes in $\cat{C}$ decompose into direct sums of their cohomologies (this holds for any smooth curve) and, moreover, all coherent sheaves decompose into direct sums of line bundles $\mathcal{O}(n)$ and torsion sheaves; the latter have the skyscraper sheaves $\mathcal{O}_x$ for $x\in\mathbb{P}^1$ as their minimal non-zero subsheaves. For a slicing in $\cat{C}$, up to shift and direct sums, various $\mathcal{O}(n)$ and $\mathcal{O}_x$ occur as cones of HN filtrations and, conversely, the decomposition properties of $\cat{C}$ imply that HN filtrations exist trivially for any family of subcategories $P(\varphi)$ with $P(\varphi+1) = P(\varphi)[1]$ and Hom-vanishing $\mor{P(\mathit{>}\varphi)}{P(\varphi)} = 0$ and such that all $\mathcal{O}_x$ and $\mathcal{O}(n)$ are in the heart $P(0,1]$, again up to shift. Thus the following assignments for $\varphi\in(0,1]$ give slicings on $\cat{C}$: \[ P_t(\varphi) = \begin{cases} \mathbb{C}lext{ \mathcal{O}(n) : n\in \mathbb{Z} } & \varphi=\frac{1}{2}\\ \mathbb{C}lext{ \mathcal{O}_x : x\in \mathbb{P}^1 } & \varphi=1\\ 0 & \textrm{else}; \end{cases} , \qquad P_g(\varphi) = \begin{cases} \mathbb{C}lext{ \mathcal{O}(n) : n\in \mathbb{Z} } & \varphi = \frac{1}{\pi} \arg(-n+i) \in (0,1)\\ \mathbb{C}lext{ \mathcal{O}_x : x\in \mathbb{P}^1 } & \varphi = 1\\ 0 & \textrm{else}; \end{cases} \] \[ P_b(\varphi) = \begin{cases} \mathbb{C}lext{ \mathcal{O} } & \varphi=\frac{1}{2}\\ \mathbb{C}lext{ \mathcal{O}_x, \mathcal{O}(n), \mathcal{O}(-n)[1] : x\in\mathbb{P}^1, n\in\mathbb{N}_{>0} } & \varphi=1\\ 0 & \textrm{else}; \end{cases} , \qquad P_c(\varphi) = \begin{cases} \coh(\mathbb{P}^1) & \varphi=1\\ 0 & \textrm{else}. \end{cases} \] $P_t$ separates torsion sheaves and line bundles into two slices. $P_g$ is the geometric slicing induced by the classical slope of coherent sheaves $\mu(A) = \deg(A)/\rk(A)$; see Example~\ref{ex:seminorm balls asymmetric}. The slicing $P_b$ occurs in the boundary of the stability space; see Example~\ref{ex:norm example} and Subsection~\ref{p1}. These three slicings are locally finite. Note that the slice $P_t(\frac{1}{2})$ contains the infinite chain $\cdots\to\mathcal{O}(-1)\to\mathcal{O}\to\mathcal{O}(1)\to\cdots$. Nonetheless $P_t$ is locally finite because any non-zero morphism $\mathcal{O}(n)\to\mathcal{O}(m)$ for $n<m$ is not strict in $P_t(\frac{1}{2})$ since it has image $\mathcal{O}(n)$ and coimage $\mathcal{O}(m)$. In particular, each $\mathcal{O}(n)$ is simple in the quasi-abelian (but not abelian) category $P_t(\frac{1}{2})$. In $P_c$, a single slice contains the whole heart; it is not locally finite and used in Example~\ref{support and local-finiteness}. See \cite{MR2084563} for the classification of stability conditions and bounded t-structures on $\cat{C}$. \end{example} \begin{example} \label{ex:thick subcategories} We continue the above example by considering some thick subcategories of $\cat{C} = \cat{D}^b(\mathbb{P}^1)$: first the subcategory $\cat{N} = \cat{N}_\mathcal{O} = \thick{}{\mathcal{O}}$ generated by the trivial line bundle. Because every line bundle is an exceptional object, $\cat{N} \cong \cat{D}^b({\mathbf{k}})$ and $\cat{N}$ is an admissible subcategory, i.e.\ the inclusion $\cat{N} \hookrightarrow \cat{C}$ has both adjoints. In particular, there is a canonical equivalence $\cat{C}/\cat{N} \cong \cat{N}^\perp \cong \thick{}{\mathcal{O}(-1)}$. Next, for any point $x\in\mathbb{P}^1$, let $\cat{N}_x$ be the subcategory generated by the skyscraper sheaf $\mathcal{O}_x$. Moreover, let $\cat{T}$ be the subcategory of all torsion objects in $\cat{C}$. Both subcategories are thick and neither is admissible. We have $\cat{N}_x \subsetneq \cat{T}$ and $\cat{C}/\cat{T} \cong \cat{N}_\mathcal{O}$. The quotient $\cat{C}/\cat{N}_x$ is not Hom-finite: the objects $\mathcal{O}$ and $\mathcal{O}(-1)$ are isomorphic in the quotient but the morphisms $y\colon \mathcal{O}(-1) \to \mathcal{O}$ for $y\neq x$ induce non-zero elements of $\End{\cat{C}/\cat{N}_x}{\mathcal{O}}$. \end{example} \section{Lax stability conditions and quotient categories} \label{degenerate and quotient stability conditions} \subsection{Stability conditions} \label{stability conditions} \noindent Let $\cat{C}$ be a triangulated category and $v\colon K(\cat{C}) \to \Lambda$ a surjective homomorphism from the Grothendieck group to a finite rank lattice. Let $\slice{\cat{C}}$ be the space of locally finite slicings. A \defn{pre-stability condition} on $\cat{C}$ is a pair $(P, Z) \in \slice{\cat{C}}\times \mor{ \Lambda}{ \mathbb{C} }$ such that $0\neq c \in P(\varphi)$ implies $Z(c) = m(c) \exp(i\pi \varphi)$ for some $m(c) \in \mathbb{R}_{>0}$. An object $c\in P(\varphi)$ is said to be \defn{semistable of phase $\varphi$} and $m(c)$ is its \defn{mass}. The \defn{mass} of any $c \in \cat{C}$ is defined to be \[ m(c) \coloneqq \sum_{i=1}^k m(c_i) \] where $c_i \in P(\varphi_i)$ for $i=1,\ldots,k$ are the semistable factors of $c$ with respect to the slicing $P$. The maximal and minimal phases of $0\neq c \in \cat{C}$ are $\varphi^+(c)=\varphi_1$ and $\varphi^-(c)=\varphi_k$ respectively. A \defn{stability condition} is a pre-stability condition $\sigma=(P,Z)$ for which there exists $K>0$ such that \[ m(c) = |Z(v(c))| \geq \frac{1}{K}||v(c)|| \] for all semistable $c\in \cat{C}$. This latter condition is referred to as the \defn{support property} \cite[\S 2.1]{MR2681792}; it is independent of the choice of norm because $\dim (\Lambda\otimes \mathbb{R}) <\infty$. The support property has three important consequences. First, and most obviously, it implies that the \defn{infimal mass} \[ \mu_\sigma = \inf \{ m(c) : 0\neq c\in \cat{C}\} \geq \frac{1}{K} \inf\{ ||\lambda|| : 0\neq \lambda \in \Lambda\} \] is strictly positive. Second, it implies that the slicing $P$ is locally finite, in fact that $P(I)$ is a length category for any interval $I\subset \mathbb{R}$ of length $|I|<1$. This is because if $c\in P(I)$ has a composition series in $P(I)$ with $n$ non-zero factors then elementary trigonometry shows that \[ |Z(c)| \geq n \mu_\sigma \cos\Big(\frac{\pi}{2} |I|\Big) \] so that the length $n$ of any composition series of $c$ in $P(I)$ is bounded above. Third, it implies that the generalised norm \[ U \mapsto ||U||_\sigma = \sup\left \{ \frac{|U(c)|}{|Z(c)|} : 0\neq c\in P(\varphi), \varphi\in \mathbb{R} \right\} \] defined in \cite{MR2373143} is actually a norm on $\mor{\Lambda}{\mathbb{C}}$ because $||U||_\sigma \leq K||U||$ where \[ ||U|| = \sup \{ |U(\lambda)| : \lambda\in \Lambda\otimes\mathbb{R},\ ||\lambda||=1\} \] denotes the operator norm. In fact the support property is equivalent to $||\cdot||_\sigma$ being a norm, see \cite[Appendix B]{MR2852118}. The central result in the theory of stability conditions is the following deformation theorem. \begin{theorem}[{\cite[Theorem 7.1 and Lemma 6.2]{MR2373143}}] \label{deformation thm} Let $\sigma = (P,Z)$ be a pre-stability condition. Then for any $0<\varepsilon<1/8$ and $W \in \mor{\Lambda}{\mathbb{C}}$ with $|| W-Z||_\sigma< \sin(\pi\varepsilon)$ there is a unique pre-stability condition $\tau = (Q,W)$ with $d(P,Q)<\varepsilon$. Moreover, if $\sigma$ is a stability condition then so is $\tau$. \end{theorem} \noindent Let $\stab{\cat{C}}$ be the set of stability conditions and $\mathcal{Z} \colon \stab{\cat{C}} \to \mor{\Lambda}{\cat{C}}$ the second projection. We refer to this as the \defn{charge map}. \label{def:charge map} The above deformation theorem shows that the charge map is a local homeomorphism, and therefore that $\stab{\cat{C}}$ can be given the structure of a, possibly empty, complex manifold of dimension $\rk(\Lambda)$. \subsection{Lax stability conditions} \label{sub:degenerate stability conditions} We start with a modified version of Bridgeland's notion of stability condition without the condition that the masses of non-zero objects have to be positive, and with a concomitantly modified support condition. \begin{definition}[Lax pre-stability condition] A \defn{lax pre-stability condition} on $\cat{C}$ is a pair $(P, Z) \in \slice{\cat{C}}\times \mor{ \Lambda}{ \mathbb{C} }$ such that $0\neq c \in P(\varphi)$ implies $Z(c) = m(c) \exp(i\pi \varphi)$ for some $m(c) \in \mathbb{R}_{\geq 0}$. \end{definition} \noindent As in the classical case we refer to $m(c)$ as the \defn{mass} of a semistable object $c\in P(\varphi)$. The \defn{mass} of any $c \in \cat{C}$ is again defined as $m(c) \coloneqq m(c_1)+\cdots+m(c_k)$ where $c_1 \in P(\varphi_1),\ldots,c_k \in P(\varphi_k)$ are the semistable factors of $c$ with respect to the slicing $P$. We define the maximal and minimal phases of $0\neq c \in \cat{C}$ to be $\varphi^+(c)=\varphi_1$ and $\varphi^-(c)=\varphi_k$ as before. \begin{definition} An object $c\in \cat{C}$ is called \defn{massive} if $m(c)>0$, and \defn{massless} if $m(c)=0$. Note that $0\in\cat{C}$ has no semistable factors, so that $m(0)=0$, i.e.\ $0$ is always a massless object. The \defn{massless subcategory} $\cat{N}$ of a lax pre-stability condition $\sigma$ is the full subcategory on the massless objects. When $\cat{N}=0$, i.e.\ the mass of every non-zero semistable object is strictly positive, $\sigma$ is a pre-stability condition; sometimes for emphasis we say it is \defn{classical}. \end{definition} \begin{proposition} \label{massless thick} The massless subcategory $\cat{N}$ of a lax pre-stability condition $\sigma=(P,Z)$ is a thick subcategory of $\cat{C}$ to which the slicing $P$ is adapted. \end{proposition} \begin{proof} We start with the second claim. Clearly every semistable factor of a massless object is massless since the mass of an object is the sum of the masses of its semistable factors. Moreover, for any interval $I$ of the form $(\varphi, \varphi+1]$ or $[\varphi, \varphi+1)$ the full subcategory $P(I)$ is the heart of a t-structure, and hence abelian. The intersection $P(I) \cap \cat{N}$ is a Serre subcategory because $m(c)=0 \iff Z(c)=0$ for $c\in P(I)$. Therefore $P$ is adapted to $\cat{N}$. It is clear that $\cat{N}$ is closed under shifts, so we need only show it is closed under extensions and direct summands. Suppose that $b \in \cat{C}$ sits in a triangle $a \to b \to c \to a[1]$ with $a, c \in \cat{N}$. Taking cohomology with respect to the t-structure with heart $P(0,1]$ we obtain a long exact sequence $\cdots \to H^ia \to H^ib \to H^ic \to \cdots$ of objects of $P(0,1]$. By assumption $m(a)=0=m(c)$, so that $m( H^ia ) = 0 = m( H^ic)$ for each $i\in \mathbb{Z}$ since the HN filtration of an object $x$ of $\cat{C}$ is a refinement of the decomposition of $x$ into its cohomology with respect to the heart $P(0,1]$. It follows from the fact that $\cat{N} \cap P(0,1]$ is a Serre subcategory that $m(H^ib)=0$ too. Hence $m(b) = \sum_{i\in \mathbb{Z}} m(H^ib) = 0$, and $\cat{N}$ is extension-closed. Since the set of semistable factors of $a\oplus b$ is the union of the sets of semistable factors of $a$ and $b$, we obtain that $\cat{N}$ is thick from the following chain of equivalences: \[ a\oplus b \in \cat{N} \iff m(a\oplus b)=0 \iff m(a)+m(b)=0 \iff m(a)=0=m(b) \iff a, b \in \cat{N}. \qedhere \] \end{proof} \begin{lemma} \label{massless simples} Suppose $\sigma = (P,Z)$ is a lax pre-stability condition. Then $\cat{N}=\triang{}{S}$ is the triangulated closure of the set $S$ of stable massless objects in the heart $P(0,1]$. \end{lemma} \begin{proof} Evidently $\triang{}{S} \subset \cat{N}$. To see the other inclusion consider the HN filtration of a massless object $c\in \cat{N}$. There are finitely many semistable factors. Each factor is massless, and has a finite composition series with stable massless objects. Up to shift each of these stable objects has phase in $(0,1]$. Therefore $c\in \triang{}{S}$ and $\cat{N}\subset \triang{}{S}$. \end{proof} \begin{remark} When $\sigma=(P,Z)$ is a \emph{classical} pre-stability condition the slices $P(\varphi)$ are abelian categories \cite[Lemma 5.2]{MR2373143}. However, this is not necessarily the case when $\sigma$ is lax since the argument relies on the charge of each semistable object being non-zero --- see Example~\ref{degenerate supported versus weak}. Nevertheless, each slice $P(\varphi)$ is a quasi-abelian length category, and so each semistable object $c$ has at least one finite composition series with stable factors. Indeed, every composition series of $c$ must have finite length, but the lengths need not be the same, and the multi-sets of stable factors need not be unique up to isomorphism. \end{remark} \begin{definition}[Lax stability condition] \label{degenerate stability condition} A \defn{lax stability condition} is a lax pre-stablity condition $\sigma=(P,Z)$ for which there exists $K>0$ such that \[ m(s) = |Z(v(s))| \geq \frac{1}{K}||v(s)|| \] for all \emph{massive stable} objects $s\in \cat{C}$. We refer to this as the \defn{support condition} for a lax pre-stability condition. \end{definition} When all non-zero objects are massive this support condition coincides with the usual one in \S\ref{stability conditions}. Since every massive stable object is semistable it is clear that the usual support condition implies the above one. In the other direction, for a semistable object $c$ the inequality $m(c) \geq ||v(c)|| / K$ follows by applying the above condition to the stable factors of $c$, each of which is massive. It is clear that massless objects must be excluded from any analogue of the support property for lax pre-stability conditions. The reason it is important to consider only \emph{stable} massive objects is to avoid support failing simply because there is a massive semistable object $b$ and a non-zero massless semistable object $c$ of the same phase. In that situation the sums $b\oplus c^n$ for $n\in \mathbb{N}$ are semistable with fixed mass $m(b\oplus c^n)=m(b)$ but with $|| v( b\oplus c^n) || \to \infty$ as $n\to \infty$. The support condition guarantees a `mass gap' between the massive and massless objects of a lax stability condition. \begin{lemma} There is a uniform lower bound on the mass of massive objects of a lax stability condition. \end{lemma} \begin{proof} Suppose $c\in \cat{C}$ is massive. Then \begin{align*} m(c) &\geq \inf \{m(s) : \text{massive stable}\ s \in P(\varphi),\ \varphi\in \mathbb{R}\} \\ & \geq \inf \left\{\frac{||v(s)||}{K} : \text{massive stable}\ s \in P(\varphi),\ \varphi\in \mathbb{R}\right\} \\ & \geq \frac{\inf \{||\lambda|| : 0\neq \lambda\in \Lambda\}}{K}. \end{align*} where the last term is strictly positive because $\Lambda$ has finite rank. \end{proof} However, the presence of massless objects means that, unlike in the classical setting, the support condition does not imply that the slicing is locally finite. \begin{example} \label{support and local-finiteness} Let $\cat{C} = \cat{D}^b(\mathbb{P}^1)$ and $\sigma=(P,Z)$ with charge $Z=0$ and the slicing $P=P_c$ of Example~\ref{ex:slicings}, i.e.\ $P(1)=\coh(\mathbb{P}^1)$. Then $\sigma$ satisfies the support property since every non-zero object is massless. However, it is not locally finite since $P(1-\varepsilon,1+\varepsilon)=P(1)$ is not length for any $0<\varepsilon<1/2$. \end{example} \subsection{Semi-norms and support} A lax pre-stability condition $\sigma = (P,Z)$ defines a (generalised) semi-norm \[ W \mapsto || W ||_\sigma = \sup \left \{ \frac{|W(s)|}{|Z(s)|} \ : \ \text{massive stable}\ s \in P(\varphi),\ \varphi\in \mathbb{R} \right\} \] on $\mor{\Lambda}{\mathbb{C}}$. By convention we set $\sup(\varnothing)=0$ so that $||W||_\sigma=0$ for all $W\in \mor{\Lambda}{\mathbb{C}}$ when $\sigma$ has no massive objects. The adjective `generalised' refers to the fact that we allow $||W||_\sigma=\infty$ if the supremum does not exist, see Example~\ref{ex:non-supported} below. This is only a \emph{semi-}norm because it is possible for a non-zero charge $W$ to vanish on all \emph{massive} stable objects, so that $||W||_\sigma=0$. When $\sigma$ is classical this is the usual (generalised) norm because if $c$ is semistable then \[ |W(c)| \leq \sum_{s\in S} |W(s)| \leq ||W||_\sigma \sum_{s\in S} |Z(s)| = ||W||_\sigma |Z(c)| \] where $S$ is the multi-set of stable factors of $c$, all of which are necessarily massive since $\sigma$ is classical. \begin{definition}[Full stability condition] A lax pre-stability condition $\sigma$ is \defn{full} if the semi-norm $||\cdot||_\sigma$ is bounded on the unit ball in $\mor{\Lambda}{\mathbb{C}}$, i.e.\ if there exists $K>0$ such that \[ ||W||_\sigma \leq K||W|| \] for all $W\in \mor{\Lambda}{\mathbb{C}}$, where $||W|| = \sup \{ |W(\lambda)| : \lambda\in \Lambda\otimes \mathbb{R},\ ||\lambda||=1\}$ is the operator norm. This is independent of the norm on $\Lambda\otimes \mathbb{R}$, and reduces to the usual notion \cite{MR2376815} when $\sigma$ is classical. \end{definition} The next result is a simple extension of \cite[Proposition B.4]{MR2852118} and \cite[Lemma 11.4]{MR3573975}, following \cite[\S 2.1]{MR2681792}, to the case of lax stability conditions. \begin{proposition} \label{equiv support conditions} For a lax pre-stability condition $\sigma = (P,Z)$ with charge factoring through $v \colon K(\cat{C}) \to \Lambda$ the following are equivalent: \begin{enumerate} \item $\sigma$ is a lax stability condition; \item $\sigma$ is full; \item there exists a quadratic form $\Delta$ on $\Lambda\otimes \mathbb{R}$ such that \begin{enumerate} \item $\Delta(v(s))\geq 0$ for each massive stable $s\in \cat{C}$; \item $\Delta$ is negative definite on $\ker Z \subset \Lambda\otimes \mathbb{R}$. \end{enumerate} \end{enumerate} \end{proposition} \begin{proof} $(1) \mathrm{im}plies (2), (3)$. Suppose $\sigma$ satisfies the support property. Then for $W\in \mor{\Lambda}{\mathbb{C}}$ we have \begin{align*} || W ||_\sigma &= \sup\left\{ \frac{|W(s)|}{|Z(s)|} : \text{massive stable}\ s\in P(\varphi),\ \varphi\in \mathbb{R} \right\} \\ & \leq K \sup \left\{ \frac{|W(s)|}{||v(s)||} : \text{massive stable}\ s\in P(\varphi),\ \varphi\in \mathbb{R} \right\}\\ &\leq K\sup \left\{ \frac{|W(\lambda)|}{||\lambda||} : 0\neq \lambda\in \Lambda \right\} = K||W|| \end{align*} so that $\sigma$ is full. Moreover, the quadratic form $\Delta(\lambda) = K^2|Z(\lambda)|^2 - ||\lambda||^2$ on $\Lambda\otimes \mathbb{R}$ satisfies the properties of the third condition in the statement. $(2) \mathrm{im}plies (1)$. Now suppose $\sigma$ is full. Assume, for a contradiction, that $\sigma$ does not satisfy the support property. Then there is a sequence $(s_n)$ of massive stable objects for which \[ m_\sigma(s_n) = |Z(s_n)| < \frac{||v(s_n)|| }{n}. \] One can choose $W_n\in \mor{\Lambda}{\mathbb{C}}$ with $||W_n||=1$ and $|W_n(s_n)|=||v(s_n)||$. But then \[ ||W_n||_\sigma \geq \frac{|W_n(s_n)|}{|Z(s_n)|} > n\frac{|W_n(s_n)|}{||v(s_n)||} = n \] so that $||\cdot||_\sigma$ is not bounded on the (compact) unit ball. This contradicts the fact that $\sigma$ is full, so $\sigma$ does not satisfy support after all. $(3) \mathrm{im}plies (1)$. Finally, suppose that $\Delta$ is a quadratic form with $\Delta(v(s))\geq 0$ for every massive stable object $s\in\cat{C}$ and whose restriction to $\ker Z$ is negative definite. In particular if $\Delta(\lambda) >0$ then $\lambda \not \in \ker Z$ so $|Z(\lambda)|^2>0$. Therefore, because the unit ball is compact, there exists $K>0$ with \[ \lambda \mapsto K^2|Z(\lambda)|^2 - \Delta(\lambda) \] a positive definite form on $\Lambda\otimes\mathbb{R}$. If $||\cdot||$ is the induced norm then \[ K^2|Z(s)|^2= ||v(s)||^2 +\Delta(v(s)) \geq ||v(s)||^2 \] for each massive stable object $s\in \cat{C}$. Therefore $\sigma$ satisfies support. \end{proof} \begin{example} \label{not well-supported} Let $\cat{C} = \cat{D}^b(\mathbb{P}^1)$. Its Grothendieck group is $\Lambda = K(\cat{C}) = K(\mathbb{P}^1) \cong \mathbb{Z}^2$ using the basis $[\mathcal{O}]$ (structure sheaf) and $[\mathcal{O}_x]$ (skyscraper sheaves). The inner product is chosen so that this basis is orthonormal. Let $\sigma=(P,Z)$ be the lax pre-stability condition defined by the charge $Z=0$ and the slicing $P=P_t$ from Example~\ref{ex:slicings}, i.e.\ $P(1) = \mathbb{C}lext{\mathcal{O}_x : x\in\mathbb{P}^1}$ and $P(1/2)=\mathbb{C}lext{\mathcal{O}(n) : n\in\mathbb{Z}}$. Every object is massless so $\sigma$ trivially satisfies the support property and so is a lax stability condition. Let $\tau=(P,W)$ be the lax pre-stability condition with the same slicing as $\sigma$ but charge $W(\mathcal{O}_x)=0$, $W(\mathcal{O})=i$. Since there are no massive $\sigma$-stable objects, $||W||_\sigma=0$. The massless subcategory of $\tau$ is $\thick{}{\mathcal{O}_x : x\in \mathbb{P}^1}$ and the massive stable objects are, up to shifts, the line bundles $\mathcal{O}(n)$ for $n\in \mathbb{Z}$. Therefore \[ || U ||_\tau = \sup\left\{ \frac{|U(b)|}{|W(b)|} : b\ \text{massive $\tau$-stable} \right\} = \sup\left\{| U (\mathcal{O}(n)) | : n\in \mathbb{Z}\right\}, \] which is infinite for example when $U(\mathcal{O}_x) = 1$ and $U(\mathcal{O})=0$. Thus $\tau$ does not satisfy support and is not a lax stability condition. \end{example} \subsection{Stability conditions on quotients} A lax pre-stability condition $\sigma$ on $\cat{C}$ with massless subcategory $\cat{N}$ induces a pre-stability condition $\mu_\cat{N}(\sigma)$ on the quotient $\cat{C}/\cat{N}$. We think of this as the `massive part' of $\sigma$, and refer to it as the \defn{associated pre-stability condition on the quotient}. We say `stability condition on the quotient' to distinguish these from the `quotient stability conditions' of Section~\ref{space qstab}. \begin{proposition} \label{prop:massive stability condition} Let $\sigma = (P,Z)$ be a lax pre-stability condition with massless subcategory $\cat{N}$. Put $\mu_\cat{N}(\sigma)=(P_{\cat{C}/\cat{N}}, Z) \in \slice{\cat{C}/\cat{N}}\times \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. \begin{enumerate} \item If the slicing $P$ is well-adapted to $\cat{N}$ then $\mu_\cat{N}(\sigma)$ is a pre-stability condition on $\cat{C}/\cat{N}$. \item If $\sigma$ is a lax stability condition then $\mu_\cat{N}(\sigma)$ is a stability condition on $\cat{C}/\cat{N}$. \end{enumerate} \end{proposition} \begin{proof} (1) By Proposition~\ref{massless thick} and Proposition~\ref{quotient slicing}, $P$ is adapted to $\cat{N}$ and so is compatible with a pair of slicings $(P_\cat{N},P_{\cat{C}/\cat{N}})$ on the massless subcategory $\cat{N}$ and the quotient $\cat{C}/\cat{N}$. If it is well-adapted then $P_{\cat{C}/\cat{N}}$ is locally finite. The charge $Z$ lies in the subspace $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ and is compatible with the slicing $P_{\cat{C}/\cat{N}}$. There are no massless objects in $\cat{C}/\cat{N}$. Therefore, $\mu_\cat{N}(\sigma)=(P_{\cat{C}/\cat{N}},Z)$ is a pre-stability condition on $\cat{C}/\cat{N}$. (2) Let $\sigma$ be a lax stability condition. The support property implies that the slicing $P_{\cat{C}/\cat{N}}$ is locally finite because there are no massless objects in $\cat{C}/\cat{N}$. Hence $P$ is well-adapted to $\cat{N}$ and $\mu_\cat{N}(\sigma)$ is a lax pre-stability condition by (1). We claim that $\mu_\cat{N}(\sigma)$ satisfies the support property. By Lemma~\ref{quotient semistables}, the HN filtration of a $\mu_\cat{N}(\sigma)$-semistable object $b$ has a unique massive $\sigma$-semistable factor $c$. Let $S$ be a multi-set of massive stable factors of $c$. Then there is some constant $K>0$ with \[ m_{\mu_\cat{N}(\sigma)}(b) = m_\sigma(c) = \sum_{s\in S} m_\sigma(s) \geq \frac{1}{K} \sum_{s\in S} || v(s)|| \geq \frac{1}{K} || \sum_{s\in S} v(s) || =\frac{1}{K}||v_{\cat{C}/\cat{N}}(b)|| \] using the restricted norm on $(\Lambda/\Lambda_\cat{N})\otimes \mathbb{R}$ for last term. Hence $\mu_\cat{N}(\sigma)$ satisfies the support property. \end{proof} \begin{remark} \label{degenerate quotient} We can generalise Proposition~\ref{prop:massive stability condition} slightly. The same argument shows that if $\cat{M}\subset \cat{N}$ is a thick subcategory of the massless subcategory of $\sigma = (P,Z)$ to which the slicing is well-adapted, then \begin{enumerate} \item there is a lax pre-stability condition $\mu_\cat{M}(\sigma) = (P_{\cat{C}/\cat{M}}, Z)$ on $\cat{C}/\cat{M}$; \item $\mu_\cat{M}(\sigma)$ is a lax stability condition when $\sigma$ is a lax stability condition. \end{enumerate} The last point follows from the same calculation as before (or the lemma below), but note that we now need the extra local-finiteness assumption as this no longer follows from the support property when there are massless objects. \end{remark} Given a lax pre-stability condition $\sigma$ with massless subcategory $\cat{N}$ then by the proposition there is an induced pre-stability condition $\mu_\cat{N}(\sigma)$ on the quotient. Moreover, $\sigma$ restricts to a fully lax pre-stability condition $\rho_\cat{N}(\sigma)$ on $\cat{N}$ by restricting the slicing of $\sigma$ to $\cat{N}$ and assigning the zero charge map. This latter construction is taken up again in Corollary~\ref{cor:restriction map}. \begin{lemma} \label{semi-norm restricts to quotient norm} Suppose $\sigma=(P,Z)$ is a lax pre-stability condition with massless subcategory $\cat{N}$ and that the slicing is well-adapted to a thick subcategory $\cat{M} \subset \cat{N}$. Then $||W||_{\mu_\cat{M}(\sigma)}\leq ||W||_\sigma$ for all $W\in \mor{\Lambda/\Lambda_\cat{M}}{\mathbb{C}}$ with equality when $\cat{M}=\cat{N}$. \end{lemma} \begin{proof} Suppose $c$ is a massive $\mu_\cat{M}(\sigma)$-stable object of phase $\varphi$. Then by Lemma~\ref{HN base case} its HN filtration with respect to $\sigma$ has a single massive semistable factor $b$ also of phase $\varphi$, with all other factors in $\cat{M}$. Furthermore, any composition series of $b$ in $P(\varphi)$ must have exactly one massive stable factor, $a$ say, with all other factors in $\cat{M}$ since otherwise $c$ would fail to be $\mu_\cat{M}(\sigma)$-stable. Therefore $a\cong c$ in $\cat{C}/\cat{M}$ so that $W(a)=W(c)$. It follows that \begin{align*} || W ||_{\mu_\cat{M}(\sigma)} &= \sup \left\{ \frac{|W(c)|}{|Z(c)|} : c\ \text {massive $\mu_\cat{M}(\sigma)$-stable} \right\}\\ &\leq \sup \left\{ \frac{|W(a)|}{|Z(a)|} : a\ \text {massive $\sigma$-stable} \right\} = ||W||_\sigma. \end{align*} Now suppose that $\cat{M}=\cat{N}$ and $b\in P(\varphi)$ is a massive stable object. Let $S$ be a multi-set of $\mu_\cat{N}(\sigma)$-stable factors of $b$ considered as an object of $P_{\cat{C}/\cat{N}}(\varphi)$. Then \[ |W(b)| \leq \sum_{s\in S}|W(s)| \leq ||W||_{\mu_\cat{N}(\sigma)} \sum_{s\in S} |Z(s)| = ||W||_{\mu_\cat{N}(\sigma)} |Z(b)| \] by the triangle inequality, the definition of the semi-norm $||\cdot ||_{\mu_\cat{N}(\sigma)}$, the fact that $W(s)=0$ when $s\in \cat{N}$, and the fact that all $s\in S$ have phase $\varphi$. Rearranging, $||W||_\sigma \leq ||W||_{\mu_\cat{N}(\sigma)}$. \end{proof} The support property satisfied by a lax stability condition is stronger than that for the induced stability condition on the quotient. The following technical lemma will be useful later in Lemma~\ref{support failure criterion}. The case $\cat{M}=\cat{N}$ provides a criterion for distinguishing a lax pre-stability condition which satisfies support on the quotient $\cat{C}/\cat{N}$ from a genuine lax stability condition. \begin{lemma} \label{support criterion} Suppose $\sigma$ is a lax pre-stability condition with massless subcategory $\cat{N}$. Let $\cat{M}$ be a thick subcategory of $\cat{N}$ such that each object of $\cat{M}$ has HN filtration in $\cat{M}$, and assume that $\mu_\cat{M}(\sigma)$ is a lax stability condition. Then any sequence $(b_n)$ of massive $\sigma$-stable objects with $m_\sigma(b_n) / || v(b_n)|| \to 0$ as $n\to \infty$ contains a subsequence $(c_n)$ with \[ \lim_{n\to \infty} \left( \frac{v(c_n)}{||v(c_n)||} \right) = \lambda \in \Lambda_\cat{M}\otimes \mathbb{R}. \] Moreover, $\sigma$ is a lax stability condition if and only if there is no such sequence $(c_n)$. \end{lemma} \begin{proof} Suppose there is such a sequence $(c_n)$. Then \[ \frac{m_\sigma(c_n)}{||v(c_n)||} = \left| Z\left( \frac{v(c_n)}{||v(c_n)||} \right)\right| \to |Z(\lambda)| = 0 \] since $\Lambda_\cat{M} \otimes \mathbb{R} \subset \Lambda_\cat{N} \otimes \mathbb{R} \subset \ker(Z)$. Therefore $\sigma$ does not satisfy the support property. Conversely, suppose that $\sigma$ does not satisfy the support property. Then there is a sequence $(b_n)$ of massive stable objects with $m_\sigma(b_n) / ||v(b_n)|| \to 0$. Let $v_{\cat{C}/\cat{M}} \colon K(\cat{C}) \to \Lambda / \Lambda_\cat{M}$ be the composite of $v$ and the quotient by the primitive sublattice $\Lambda_\cat{M}$. Note that $Z(v_{\cat{C}/\cat{M}}(b_n)) \neq 0$ because $b_n$ is massive and that $m_\sigma(b_n) / || v_{\cat{C}/\cat{M}}(b_n)||$ is bounded below because $\mu_\cat{M}(\sigma)$ satisfies the support property by assumption. Therefore writing \[ \frac{m_\sigma(b_n)}{||v(b_n)||} = \frac{m_\sigma(b_n)}{||v_{\cat{C}/\cat{M}}(b_n)||} \cdot \frac{||v_{\cat{C}/\cat{M}}(b_n)||}{||v(b_n)||} \] we deduce that $ v_{\cat{C}/\cat{M}}(b_n) / ||v(b_n)|| \to 0$. Passing to a subsequence $(c_n)$ such that the unit vectors $v(c_n) / ||v(c_n)||$ converge we have \[ \lim_{n\to \infty} \left( \frac{ v(c_n) } { || v(c_n)||}\right) = \lim_{n\to \infty} \left( \frac{ v(c_n) - v_{\cat{C}/\cat{M}}(c_n) } { || v(c_n)||}\right) \in \Lambda_\cat{M}\otimes \mathbb{R} \] where we consider $v_{\cat{C}/\cat{M}}(c_n) \in \Lambda\otimes \mathbb{R}$ via the orthogonal splitting. \end{proof} \begin{example} \label{degenerate supported versus weak} Let $\tau$ be the lax pre-stability condition on $\cat{C} = \cat{D}^b(\mathbb{P}^1)$ with massless subcategory $\cat{N}=\thick{}{\mathcal{O}_x : x\in \mathbb{P}^1}$ defined in Example~\ref{not well-supported}. The quotient $\cat{C}/\cat{N} \simeq \thick{\cat{C}/\cat{N}}{\mathcal{O}}$ is generated by a single object with charge $i$ and phase $1/2$ in the stability condition on the quotient $\mu_\cat{N}(\tau)$. It follows that $\mu_\cat{N}(\tau)$ satisfies the support property so that $\tau$ is a lax pre-stability condition which satisfies support on the quotient $\cat{C}/\cat{N}$. However, the sequence $\mathcal{O}(n)$ of massive stable objects shows that $\tau$ does not satisfy the support property so that $\tau$ is not a lax stability condition. \end{example} Recall that pre-stability conditions $\sigma=(P,Z)$ and $\tau=(Q,Z)$ with the same charge and with $d(P,Q)<1$ are equal \cite[Lemma 6.4]{MR2373143}. There is an analogue for lax pre-stability conditions; the only difference is that the slicing on the massless objects is not determined by the charge so we must fix this too. \begin{corollary} \label{degenerate uniqueness} If $\sigma=(P,Z)$ and $\tau=(Q,Z)$ are lax pre-stability conditions with the same charge $Z$, the same massless subcategory $\cat{N}$, the same massless slicing $P_\cat{N}=Q_\cat{N}$ and $d(P,Q)<1$, then $\sigma=\tau$. \end{corollary} \begin{proof} The induced pre-stability conditions $\mu_\cat{N}(\sigma)$ and $\mu_\cat{N}(\tau)$ have the same charge and the distance between their slicings is $d(P_{\cat{C}/\cat{N}},Q_{\cat{C}/\cat{N}}) \leq d(P,Q) <1$. Hence $P_{\cat{C}/\cat{N}}=Q_{\cat{C}/\cat{N}}$ by \cite[Lemma 6.4]{MR2373143}. Since $P_\cat{N}=Q_\cat{N}$, and the glued slicing is unique, we deduce that $P=Q$. \end{proof} \section{The space of lax stability conditions} \label{spaces of stability conditions} \noindent Fix a finite rank lattice $\Lambda$ with a surjective homomorphism $v\colon K(\cat{C}) \to \Lambda$ and a norm $||\cdot||$ on $\Lambda \otimes \mathbb{R}$. Let $\stab{\cat{C}}$ be the space of stability conditions whose charges factor through $v$. Recall \cite[\S 6]{MR2373143} that this has the subspace topology from the inclusion $\stab{\cat{C}} \subset \slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ where the right hand side has the topology from the metric \[ d( (Q,W) , (P,Z) ) = \max\{ d(P,Q) , ||W-Z|| \} \] arising from the metric on $\slice{\cat{C}}$ and the operator norm on $\mor{\Lambda}{\mathbb{C}}$. \begin{definition} \label{def:space of degenerate stability conditions} Let $\legstab{\cat{C}}$ be the subset of lax stability conditions in $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ equipped with the subspace topology. Let \[ \lstab{\cat{C}} = \legstab{\cat{C}} \cap \overline{\stab{\cat{C}}} \] be the subspace of lax stability conditions in the boundary of $\stab{\cat{C}}$. Since this will be the principal object we study we refer to it as the \defn{space of lax stability conditions}. We also introduce the larger space $\wstab{\cat{C}}$ of lax pre-stability conditions which satisfy support on the quotient of $\cat{C}$ by the massless subcategory and which lie in the closure of $\stab{\cat{C}}$. For a thick subcategory $\cat{N}$ of $\cat{C}$, let $\lnstab{\cat{N}}{\cat{C}} \subset \lstab{\cat{C}}$ and $\wnstab{\cat{N}}{\cat{C}} \subset \wstab{\cat{C}}$ denote the subsets where the massless subcategory is $\cat{N}$, respectively. For a thick subcategory $\cat{N}$ of $\cat{C}$, the subset $\lnstab{\cat{N}}{\cat{C}} \subset \lstab{\cat{C}}$ will be called a \defn{stratum} of $\lstab{\cat{C}}$. For each space, the \defn{charge map} $\mathcal{Z}$ is the second projection, e.g.\ $\mathcal{Z} \colon \lstab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$. \end{definition} The subspace $\lnstab{\cat{N}}{\cat{C}}$ for $\cat{N}=0$ is just the space of (classical) stability conditions: $\lnstab{0}{\cat{C}} = \stab{\cat{C}}$. In the other extreme case $\cat{N}=\cat{C}$ of lax stability conditions with zero charge map, the subspace $\lnstab{\cat{C}}{\cat{C}}$ is homeomorphic to the closure in $\slice{\cat{C}}$ of the set of slicings of stability conditions in $\stab{\cat{C}}$. By Proposition~\ref{prop:massive stability condition} (2), there is a map $\mu_{\cat{N}} \colon \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$. Note that $\lstab{\cat{C}} \subset \wstab{\cat{C}}$ and $\lnstab{\cat{N}}{\cat{C}} \subset \wnstab{\cat{N}}{\cat{C}}$ because the support property for $\sigma$ implies the support property for $\mu_\cat{N}(\sigma)$, but not {\it vice versa} in general. \begin{remark} For a lax pre-stability condition $\sigma = (P,Z)$ to be in $\lstab{\cat{C}}$, it needs to have two properties: the support condition and the closure condition, i.e.\ $\sigma \in \overline{\stab{\cat{C}}}$. Of these two, the support condition is the main issue. Indeed, Corollary~\ref{inductive dstab criterion} and Lemma~\ref{dstab criterion} imply that if $\sigma$ has the support property and the restricted slicing on the massless subcategory $\cat{N}$ is in the closure of the set of slicings occuring in $\stab{\cat{N}}$ then $\sigma \in \overline{\stab{\cat{C}}}$. Theorem~\ref{all deformations remain in bdy} refines this result by showing that under the above assumptions all sufficiently small deformations of $\sigma$ are also in $\overline{\stab{\cat{C}}}$. One way to find an example of a lax stability condition not in $\overline{\stab{\cat{C}}}$ would be to find one whose massless subcategory $\cat{N}$ had empty stability space. However, since the thick subcategory $\cat{N}$ always comes with the slicing restricted from $P$, so in particular carries bounded t-structures, the obstruction to $\stab{\cat{N}} \neq \varnothing$ could only be the absence of compatible stability functions on the heart. We do not know if such examples exist. \end{remark} \begin{lemma} \label{length lemma} Suppose $\sigma = (P,Z)$ is a lax pre-stability condition in $\overline{\stab{\cat{C}}}$. Then $P(\varphi-\varepsilon,\varphi+\varepsilon)$ is length for any $0<\varepsilon<1/2$. \end{lemma} \begin{proof} Fix $0<\varepsilon<1/2$ and $n>0$ with $\varepsilon+1/n< 1/2$. Since $\sigma\in \overline{\stab{\cat{C}}}$ there is $\tau = (Q,W) \in \stab{\cat{C}}$ with $d(P,Q)<1/n$. Then, for all $\varphi\in \mathbb{R}$, \[ P(\varphi-\varepsilon,\varphi+\varepsilon) \subset Q(\varphi-\varepsilon-\tfrac{1}{n},\varphi+\varepsilon+\tfrac{1}{n}). \] Since the latter is length, so is the former because the strict exact structure is inherited from the triangulated structure on $\cat{C}$. \end{proof} \subsection{Semi-norm neighbourhoods} \label{semi-norm neighbourhoods} For $\varepsilon>0$ and a lax pre-stability condition $\sigma =(P,Z)$ define a subset of $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ by \[ B_\varepsilon(\sigma) = \{ (Q,W) : d(P,Q)<\varepsilon \ \text{and}\ ||W-Z||_\sigma < \sin(\pi \varepsilon) \}. \] Proposition~\ref{equiv support conditions} implies that $B_\varepsilon(\sigma)$ is open precisely when $\sigma$ is a lax stability condition. In this case it contains all sufficiently small metric balls about $\sigma$, but it need not be contained within any such metric ball because $||\cdot ||_\sigma$ is only a \emph{semi}-norm. If $\sigma$ does not satisfy the support property then $B_\varepsilon(\sigma)$ need not even contain any metric ball about $\sigma$. It is also important to note that the condition $||W-Z||_\sigma < \sin(\pi \varepsilon)$ is asymmetric in $W$ and $Z$ because $Z$ is the charge of $\sigma$. This asymmetry is illustrated in the example below. \begin{example} \label{ex:seminorm balls asymmetric} Let $\sigma_g = (P_g, Z_g)$ the classical geometric stability condition on $\cat{C} = \cat{D}^b(\mathbb{P}^1)$ with charge $Z_g = -\deg + i \cdot \mathrm{rk}$, and slicing $P_g$ from Example~\ref{ex:slicings}, i.e.\ $P_g(1) = \langle \mathcal{O}_x : x\in\mathbb{P}^1 \rangle$ and $P_g(\varphi) = \langle \mathcal{O}(n) \rangle$ for $\varphi = \frac{1}{\pi}\arg(-n+i) \in (0,1)$. And let $\sigma_d = (P_d,Z_d) \coloneqq (P_g,0)$ be the lax stability condition with the same slicing but zero charge so that all objects are massless. Now $d(P_g,P_d)=0$ and $|| Z_g-Z_d ||_{\sigma_g} = || Z_g ||_{\sigma_g} = 1$ and $|| Z_d-Z_g ||_{\sigma_d} = || Z_g ||_{\sigma_d} = 0$ because there are no massive $\sigma_d$-stable objects. Thus $\sigma_g \in B_\varepsilon(\sigma_d)$ but $\sigma_d \not \in B_\varepsilon(\sigma_g)$ for any $\varepsilon>0$. \end{example} For classical $\sigma$ the intersections of the $B_\varepsilon(\sigma)$ with $\stab{\cat{C}}$ form a basis for the topology, see \cite[\S 6]{MR2373143}. The semi-norm neighbourhoods $B_\varepsilon(\sigma)$ are similarly useful for studying the topology of $\lstab{\cat{C}}$. Clearly, if $\sigma\in \lstab{\cat{C}}$ then $B_\varepsilon(\sigma) \cap \stab{\cat{C}} \neq \varnothing$ for any $\varepsilon>0$. The next result is a partial converse. \begin{lemma} Suppose $\sigma$ is a lax pre-stability condition such that $B_\varepsilon(\sigma) \cap \stab{\cat{C}} \neq \varnothing$. Then $\sigma$ is a lax stability condition. \end{lemma} \begin{proof} Let $\sigma=(P,Z)$ and suppose $\tau=(Q,W) \in B_\varepsilon(\sigma)\cap \stab{\cat{C}}$. Suppose $c$ is a massive $\sigma$-stable object, and let $S$ be the set of its $\tau$-semistable factors. Then \[ \begin{array}{rcccc} m_\sigma(c) = |Z(c)| &>& \dfrac{1}{1+\sin(\pi\varepsilon)}|W(c)| &\geq& \dfrac{\cos(2\pi\varepsilon)}{(1+\sin(\pi\varepsilon))} \sum_{s\in S}|W(s)| \\[2ex] &\geq& \dfrac{\cos(2\pi\varepsilon)}{K(1+\sin(\pi\varepsilon))} \sum_{s\in S}||v(s)|| &\geq& \dfrac{\cos(2\pi\varepsilon)}{K(1+\sin(\pi\varepsilon))} || v(c)||, \end{array}\] where we have used successively the norm bound $||W-Z||_\sigma < \sin(\pi \varepsilon)$ and the triangle inequality, the fact that $d(P,Q)<\varepsilon$, the support property for $\tau$, and the triangle inequality for the norm on $\Lambda\otimes \mathbb{R}$. \end{proof} The HN factors of a massless object are, by definition, massless. In fact, this property persists in an open neighbourhood in the following sense. \begin{lemma} \label{local persistence of massless factors} Let $\sigma =(P,Z)$ be a lax pre-stability condition with massless subcategory $\cat{N}$. If $d(P,Q)<1/8$ then $Q$ restricts to a slicing $Q_\cat{N}$ on $\cat{N}$. In particular this applies to the slicing of any lax pre-stability condition in $B_\varepsilon(\sigma)$ for $0<\varepsilon<1/8$. \end{lemma} \begin{proof} Let $d(P,Q) < \varepsilon < 1/8$. We must show that the $Q$-semistable factors of any $c\in \cat{N}$ lie in $\cat{N}$. Suppose $b\in Q(\varphi)$ is a $Q$-semistable factor of $c\in \cat{N}$. Then $b$ is also a $Q$-semistable factor of $c' = H_P^{(\varphi-\varepsilon,\varphi+\varepsilon)}(c)$ because $Q(\varphi) \subset P(\varphi-\varepsilon,\varphi+\varepsilon)$ so that \[ H_Q^\varphi (c') = H_Q^\varphi H_P^{(\varphi-\varepsilon,\varphi+\varepsilon)}(c) = H_Q^\varphi (c) = b. \] Moreover, $c'\in \cat{N}$ too, since each of its $P$-semistable factors lies in $\cat{N}$. In particular, $Z(c')=0$. Let $b_1,\ldots,b_m$ be the $Q$-semistable factors of $c'$, so that $b=b_i$ for some $1\leq i \leq m$. Since $c'\in P(\varphi-\varepsilon,\varphi+\varepsilon) \subset Q(\varphi-2\varepsilon,\varphi+2\varepsilon)$ we have \[ b_1, \ldots,b_m \in Q(\varphi-2\varepsilon,\varphi+2\varepsilon) \subset P(\varphi-3\varepsilon,\varphi+3\varepsilon). \] Now let $b_{ij}$ for $j=1, \ldots, n_i$ be the $P$-semistable factors of $b_i$ for $1\leq i \leq m$. Since $6\varepsilon <1$ the equation \[ \sum_{i,j}Z(b_{ij}) = \sum_i Z(b_i) = Z(c') = 0 \] implies that $Z(b_{ij})=0$, and hence that $b_{ij}\in\cat{N}$, for each $1\leq i \leq m$ and $1\leq j\leq n_i$. In particular, all $P$-semistable factors of $b$ lie in $\cat{N}$, so that $b\in \cat{N}$ as claimed. \end{proof} Recall the constructions of restriction and quotient stability conditions: \[ \begin{array}{rl @{\qquad} l} \rho_\cat{N} \colon& \lnstab{\cat{N}}{\cat{C}} \to \lstab{\cat{N}}, & \sigma = (P,Z) \mapsto \rho_\cat{N}(\sigma) = (P_\cat{N},0); \\ \mu_\cat{N} \colon& \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}, & \sigma = (P,Z) \mapsto \mu_\cat{N}(\sigma) = (P_{\cat{C}/\cat{N}},Z); \\ \end{array} \] for the latter, see Proposition~\ref{prop:massive stability condition}. The next lemma says that these maps are contractions and hence, in particular, continuous. \begin{lemma} \label{lemma:continuous-restriction} Let $\sigma$ and $\tau$ be lax pre-stability conditions on $\cat{C}$ such that their slicings are adapted to a thick subcategory $\cat{N}$ of $\cat{C}$. Then \[ d(\mu_\cat{N}(\sigma),\mu_\cat{N}(\tau)) \leq d(\sigma,\tau) \qquad\text{and}\qquad d(\rho_\cat{N}(\sigma),\rho_\cat{N}(\tau)) \leq d(\sigma,\tau). \] \end{lemma} \begin{proof} Writing $\sigma = (P,Z)$ and $\tau = (Q,W)$, Corollary~\ref{cor:slicing inequalities} gives $d(P_\cat{N},Q_\cat{N}) \leq d(P,Q)$ and $d(P_{\cat{C}/\cat{N}},Q_{\cat{C}/\cat{N}}) \leq d(P,Q)$. For any $U \in \mor{\Lambda}{\mathbb{C}}$ let $U_\cat{N} = U|_{\Lambda_\cat{N}} \in \mor{\Lambda_\cat{N}}{\mathbb{C}}$ be the restriction. Using the embedding $\mor{\Lambda_\cat{N}}{\mathbb{C}} \hookrightarrow \mor{\Lambda}{\mathbb{C}}$ arising from the inner product on $\Lambda\otimes \mathbb{R}$, see \S\ref{charges}, we may also consider $U_\cat{N} \in \mor{\Lambda}{\mathbb{C}}$. Write (in this proof only) $U'_\cat{N} = U-U_\cat{N}$ for the component in the orthogonal complement $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ to $\mor{\Lambda_\cat{N}}{\mathbb{C}}$. Because $U = U_\cat{N} + U'_\cat{N}$ is an orthogonal decomposition: $||U||^2 = ||U_\cat{N}||^2 + ||U'_\cat{N}||^2$. Applied to $U \coloneqq Z-W$, this shows that $||Z_\cat{N}-W_\cat{N}|| \leq ||Z-W||$ and $||Z'_\cat{N}-W'_\cat{N}|| \leq ||Z-W||$. The claimed inequalities follow because $\rho_\cat{N}(\sigma) = (P_\cat{N},Z_\cat{N})$ and $\mu_\cat{N}(\sigma) = (P_{\cat{C}/\cat{N}}, Z'_\cat{N})$, and likewise for $\tau$. \end{proof} If $\sigma=(P,Z) \in \lnstab{\cat{N}}{\cat{C}}$ then its restriction to the massless subcategory produces a lax stability condition $\rho_\cat{N}(\sigma) = (P\cap\cat{N}, 0)$ in which all objects are massless. By Lemma~\ref{local persistence of massless factors}, this construction extends to nearby lax stability conditions. \begin{corollary} \label{cor:restriction map} Given $\sigma=(P,Z)\in\lnstab{\cat{N}}{\cat{C}}$ and $0 < \varepsilon < 1/16$ there is a continuous map \begin{align*} \rho_\cat{N} \colon \{ \tau=(Q,W) \in \lstab{\cat{C}} : d(P,Q)<\varepsilon \} &\to \lstab{\cat{N}} \\ \tau = (Q,W) &\mapsto \rho_\cat{N}(\tau) = (Q\cap\cat{N}, W_\cat{N}), \end{align*} in particular this restricts to a map $\rho_\cat{N} \colon B_\varepsilon(\sigma) \cap \lstab{\cat{C}} \to \lstab{\cat{N}}$. \end{corollary} \begin{proof} The restriction, $\rho_\cat{N}(\tau)$, is a lax pre-stability condition by Lemma~\ref{local persistence of massless factors}. It also satisfies the support property because $Q_\cat{N}(\varphi) \subset Q(\varphi)$ and so a $\rho_\cat{N}(\tau)$-semistable object $s$ is also $\tau$-semistable, hence $v(s) = v_\cat{N}(s) \in \Lambda_\cat{N}$. In particular, the restricted charge map $W_\cat{N}$ satisfies \[ m_{\rho_\cat{N}(\tau)}(s) = |W_\cat{N}(v_\cat{N}(s))| = |W(v(s))| \geq \frac{1}{K} || v(s) || = \frac{1}{K} || v_\cat{N}(s) || . \] Next, we check that $\rho_\cat{N}(\tau) \in \overline{\stab{\cat{N}}}$. Since $\tau \in \lstab{\cat{C}} \subseteq \overline{\stab{\cat{C}}}$, for each $\varepsilon > 0$ there exists $\sigma_\varepsilon = (P_\varepsilon, Z_\varepsilon) \in \stab{\cat{C}}$ with $d(\tau,\sigma_\varepsilon) < \varepsilon$ in the metric on $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$. As $d(P,P_\varepsilon) \leq d(P,Q) + d(Q,P_\varepsilon) < \varepsilon + \varepsilon = 2\varepsilon$, the slicing $P_\varepsilon$ restricts to $\cat{N}$ by Lemma~\ref{local persistence of massless factors}. Hence, $d(\rho_\cat{N}(\tau), \rho_\cat{N}(\sigma_\varepsilon))<\varepsilon$ in the metric on $\slice{\cat{N}}\times \mor{\Lambda_\cat{N}}{\mathbb{C}}$. It follows that $\rho_\cat{N}(\tau) \in \overline{\stab{\cat{N}}}$. Finally, the map $\rho_\cat{N}$ is continuous by Lemma~\ref{lemma:continuous-restriction}. \end{proof} The next result shows that the massless subcategory of a lax pre-stability condition varies semi-continuously. \begin{lemma} \label{thick subcats in nbhd} Suppose $\sigma$ is a lax pre-stability condition with massless subcategory $\cat{N}$. If $\tau = (Q,W) \in B_\varepsilon(\sigma)$ for $0<\varepsilon<1/8$, then the massless subcategories $\cat{N}_\tau \subset \cat{N}$ are nested. Moreover, the inclusion is equality if and only if $W\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}} \subset \mor{\Lambda}{\mathbb{C}}$. \end{lemma} \begin{proof} Let $\sigma=(P,Z)$. For $\sigma$-stable $c\in \cat{C}$ there is an inequality \begin{equation} \label{mass inequality 1} (1-\sin(\pi\varepsilon))|Z(c)| \leq |W(c)|. \end{equation} This is evident if $c$ is massless, and follows from the definition of the norm and the (reverse) triangle inequality if it is massive. Now suppose $b \in Q(\varphi)$ is $\tau$-semistable. Let $S$ be a (finite) multi-set of $\sigma$-stable factors of $b$. Since $\tau\in B_\varepsilon(\sigma)$ we know that $S\subset P(\varphi-\varepsilon,\varphi+\varepsilon) \subset Q(\varphi-2\varepsilon,\varphi+2\varepsilon)$. Hence, using elementary trigonometry and the inequality \eqref{mass inequality 1} we have \begin{align*} |W(b)| \geq \cos(2\pi\varepsilon) \sum_{s\in S} |W(s)| \geq (1-\sin(\pi\varepsilon)) \cos(2\pi\varepsilon)\sum_{s\in S} |Z(s)|. \end{align*} Therefore $m_\tau(b) \geq (1-\sin(\pi\varepsilon)) \cos(2\pi\varepsilon) m_\sigma(b)$. In particular, if $b\in \cat{N}_\tau$ then $b\in \cat{N}$. For the equality statement, clearly, $W\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ if and only if $W(c)=0$ for all $c\in \cat{N}$. In particular, if $\cat{N}_\tau=\cat{N}$ then $W\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. Conversely, if $W\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ then, by Lemma~\ref{local persistence of massless factors}, the $\tau$-semistable factors of any $c\in \cat{N}$ are also in $\cat{N}$ which implies that $m_\tau(c)=0$ because $|W(c)| \leq \sum_{s\in S} |W(s)| \leq ||W||_\sigma \sum_{s \in S} |Z(s)| = 0$, where $S$ is the multi-set of $\tau$-semistable factors of $c$. Hence $\cat{N}_\tau\supset \cat{N}$ and so we have equality as claimed. \end{proof} We next show that the quotient stability map extends to the closure of $\lnstab{\cat{N}}{\cat{C}}$. \begin{proposition} \label{massive part} The map $\mu_\cat{N}\colon \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$ extends to a continuous map \[ \mu_\cat{N} \colon \overline{\lnstab{\cat{N}}{\cat{C}}} \to \lstab{\cat{C}/\cat{N}} . \] Moreover, $\mu_\cat{N}(\sigma) \in \stab{\cat{C}/\cat{N}} \iff \sigma \in \lnstab{\cat{N}}{\cat{C}}$. \end{proposition} \begin{proof} By Proposition~\ref{prop:massive stability condition}, the assignment $\sigma =(P,Z) \mapsto (P_{\cat{C}/\cat{N}},Z) = \mu_\cat{N}(\sigma)$ defines a map $\lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$. It is continuous by Lemma~\ref{lemma:continuous-restriction}. If $\sigma$ is in the boundary of $\lnstab{\cat{N}}{\cat{C}}$ in $\lstab{\cat{C}}$ then it has massless subcategory $\cat{N}_\sigma\supset \cat{N}$ by Lemma~\ref{thick subcats in nbhd}. We claim that the slicing $P$ is well-adapted to $\cat{N}$. Suppose $c\in \cat{N}$ has a $\sigma$-semistable factor $b \not \in \cat{N}$. For sufficiently close $\tau=(Q,W)\in \lnstab{\cat{N}}{\cat{C}}$ the HN filtration of $c$ with respect to $\tau$ will be the concatenation of the filtrations of its $\sigma$-semistable factors. Since $b \not \in \cat{N}$ its filtration must contain at least one factor not in $\cat{N}$. This contradicts Proposition~\ref{massless thick} which says that $Q$ is adapted to $\cat{N}$. Therefore $P$ restricts to $\cat{N}$ after all. Now suppose that $b\in \cat{N}\cap P(I)$ and that $0\to a \to b\to c\to 0$ is a short exact sequence in the abelian category $P(I)$. Then for sufficiently close $\tau = (Q,W)$ in $\lnstab{\cat{N}}{\cat{C}}$ and suitable strict length one interval $J$ it is also a short exact sequence in $Q(J)$. Therefore $a, c\in \cat{N}$ because $Q$ is adapted to $\cat{N}$ by Proposition~\ref{massless thick}. Since $\cat{N}\cap P(I)$ is clearly extension-closed it is therefore a Serre subcategory of $P(I)$. Thus $P$ is adapted to $\cat{N}$. By Proposition~\ref{quotient slicing} there is a slicing $P_{\cat{C}/\cat{N}}$ on the quotient. Since $d(P_{\cat{C}/\cat{N}}, Q_{\cat{C}/\cat{N}})\leq d(P,Q)$ and $\sigma \in \overline {\lnstab{\cat{N}}{\cat{C}}}$ this slicing is the limit of slicings appearing in $\stab{\cat{C}/\cat{N}}$. Thus $P_{\cat{C}/\cat{N}}$ is locally finite by Lemma~\ref{length lemma}. Therefore $P$ is well-adapted to $\cat{N}$ as claimed. It follows that $\mu_\cat{N}(\sigma)$ is well-defined by Remark~\ref{degenerate quotient}, is in the boundary of $\stab{\cat{C}/\cat{N}}$, inherits the support property from $\sigma$ and has massless subcategory $\cat{N}_\sigma/\cat{N}$. Hence it is in $\lstab{\cat{C}/\cat{N}}$ as claimed. \end{proof} Finally we relate the semi-norms associated to nearby lax pre-stability conditions. This is the (weaker) analogue of the fact that the norms associated to stability conditions in the same component of $\stab{\cat{C}}$ are equivalent, cf.\ \cite[Lemma 6.2]{MR2373143}. \begin{lemma} \label{semi-norm bound} Let $\sigma$ be a lax pre-stability condition with massless subcategory $\cat{N}$ and let $\tau \in B_\varepsilon(\sigma)$. Then for any $U \in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ \[ ||U||_\tau \leq \frac{||U||_\sigma}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)} . \] \end{lemma} \begin{proof} Let $\sigma=(P,Z)$ and $\tau=(Q,W)$. Suppose $b\in Q(\varphi)$ is a massive $\tau$-stable object and $S$ is a multi-set of its $\sigma$-stable factors. Note that $|U(s)| \leq ||U||_\sigma |Z(s)|$ for each $s\in S$ by definition of $||\cdot||_\sigma$ when $s$ is massive, and trivially when $s$ is massless since then $U(s)=0$. Therefore, using (\ref{mass inequality 1}) and the fact that $S \subset P(\varphi-\varepsilon,\varphi+\varepsilon) \subset Q(\varphi-2\varepsilon,\varphi+2\varepsilon)$, we have \[ |U(b)| \leq \sum_{s\in S}|U(s)| \leq ||U||_\sigma \sum_{s\in S}|Z(s)| \leq \frac{||U||_\sigma}{1-\sin(\pi\varepsilon)} \sum_{s\in S}|W(s)|\leq \frac{||U||_\sigma }{1-\sin(\pi\varepsilon)} \cdot \frac{|W(b)|}{\cos(2\pi\varepsilon)} . \] Dividing by $|W(b)|$ and taking the supremum over all massive $\tau$-stable $b\in \cat{C}$ gives the result. \end{proof} \subsection{Continuity of masses and phases} \begin{proposition} \label{mass and phase continuity} For each $0\neq c\in \cat{C}$ the functions $\sigma \mapsto m_\sigma(c)$ and $\sigma \mapsto \varphi^\pm_\sigma(c)$ are continuous on $\lstab{\cat{C}}$. \end{proposition} \begin{proof} Fix $0 \neq c \in \cat{C}$. The result is immediate for the minimal and maximal phases $\varphi^\pm_\sigma(c)$. To show that the mass is continuous consider $\sigma=(P,Z)\in \lstab{\cat{C}}$. For sufficiently small $\varepsilon >0$ and $\tau =(Q,W)$ with $d(P,Q)<\varepsilon$ the HN filtration of $c$ with respect to $\tau$ is the concatenation of the filtrations of the $\sigma$-semistable factors $\{c_i\}$ of $c$. Hence \[ m_\tau(c) - m_\sigma(c) = \sum_i ( m_\tau(c_i) - m_\sigma(c_i) ). \] Therefore it suffices to consider the case in which $c$ is $\sigma$-semistable. Assume $c\in P(\varphi)$ and let $S$ be a multi-set of $\tau$-stable factors of $c$. Since $d(P,Q)<\varepsilon$ and $c\in P(\varphi)$ we have $S\subset Q(\varphi-\varepsilon,\varphi+\varepsilon)$. By the triangle inequality and elementary trigonometry \[ |W(c)| \leq \sum_{s\in S} |W(s)| \leq \frac{|W(c)|}{\cos(2\pi \varepsilon)} \] and therefore \[ | m_\tau(c) - m_\sigma(c) | = \left| \sum_{s\in S}|W(s)| - |Z(c)| \right| \leq \max\left\{ |Z(c)|-|W(c)| , \frac{|W(c)|}{\cos(2\pi \varepsilon)} -|Z(c)|\right\}. \] Applying the triangle inequality to each term on the right-hand side and the operator norm bounds $|W(c)-Z(c)| \leq ||W-Z|| \cdot ||v(c)||$ and $|Z(c)| \leq ||Z|| \cdot ||v(c)||$ we obtain \[ | m_\tau(c) - m_\sigma(c)| \leq \max \left\{ ||W-Z|| , \frac{||W-Z|| + (1-\cos(2\pi\varepsilon))||Z||}{\cos(2\pi\varepsilon)}\right\} || v(c)||. \] Requiring $||W-Z||<\varepsilon$, in addition to $d(P,Q)<\varepsilon$, we see the bound can be made arbitrarily small by reducing $\varepsilon$. The result follows. \end{proof} \begin{proposition} \label{local constancy 1} For any $c\in \cat{C}$ the function $\sigma \mapsto m_\sigma(c)$ is locally constant on the fibres of the charge map $\mathcal{Z} \colon\lstab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$. Moreover, the set of phases of the massive semistable factors of $c$ is also locally constant on the fibres of $\mathcal{Z}$. \end{proposition} \begin{proof} Fix $c\in \cat{C}$ and $\sigma \in \lstab{\cat{C}}$. Then for $\tau$ sufficiently close to $\sigma$ the HN filtration of $c$ with respect to $\tau$ is the concatenation of the filtrations of the $\sigma$-semistable factors of $c$. Suppose $c_i$ is one of these $\sigma$-semistable factors. The charges of the $\tau$-semistable factors of $c_i$ lie in a cone of angle $2\pi\varepsilon$ in $\mathbb{C}$, centred on the phase of $c_i$. If $\mathcal{Z}(\tau) = \mathcal{Z}(\sigma)$ then all but one of the $\tau$-semistable factors of $c_i$ must be massless because otherwise the massive semistable factors would already destabilise $c_i$ with respect to $\sigma$. The unique massive factor must have the same charge, in particular the same mass, as $c_i$. It follows that $m_\tau(c) = m_\sigma(c)$, and also that the sets of phases of the massive factors of $c$ with respect to $\sigma$ and $\tau$ are the same. \end{proof} \begin{corollary} \label{local constancy 2} The subcategory $\cat{N}$ of massless objects, and the stability condition on the quotient $\mu_\cat{N}(\sigma)$ are locally constant on the fibres of the charge map $\mathcal{Z} \colon \lstab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$. \end{corollary} \begin{proof} The massless subcategory $\cat{N}$ and the semistable objects of the stability condition on the quotient $\mu_\cat{N}(\sigma)$ are locally constant on the fibres of the projection by Proposition~\ref{local constancy 1}. By construction the charge of $\mu_\cat{N}(\sigma)$ is constant. \end{proof} \subsection{Group actions} \label{group actions} Let $\aaut{\Lambda}{\cat{C}}$ be the subgroup of auto-equivalences $\alpha \colon \cat{C} \to \cat{C}$ which descend (necessarily uniquely) to an isomorphism $[\alpha] \colon \Lambda \to \Lambda$ with $v\circ \alpha = [\alpha]\circ v$. Then $\aaut{\Lambda}{\cat{C}}$ acts smoothly on the left of $\stab{\cat{C}}$ via \[ (P,Z) \mapsto \left(\alpha \circ P, Z \circ [\alpha]^{-1} \right). \] There is also a smooth right action of the universal cover $G$ of the orientation-preserving component $\mathrm{GL}_2^+(\mathbb{R})$. An element $g\in G$ corresponds to a pair $(T_g,\theta_g)$ where $T_g$ is the projection of $g$ to $\mathrm{GL}_2^+(\mathbb{R})$ under the covering map and $\theta_g\colon\mathbb{R} \to \mathbb{R}$ is an increasing map with $\theta_g(t+1)=\theta_g(t)+1$ which induces the same map as $T_g$ on the circle $\mathbb{R}/2\mathbb{Z} = (\mathbb{R}^2-\{0\}) / \mathbb{R}_{>0}$. The element acts by \begin{equation} \label{G action} (P,Z) \mapsto \left( P\circ \theta_g, T_g^{-1} \circ Z\right) \end{equation} where we think of the central charge as taking values in $\mathbb{R}^2$. This action preserves the semistable and stable objects and the HN filtrations of all objects. The subgroup consisting of pairs with $T$ conformal is isomorphic to $\mathbb{C}$ with $w\in \mathbb{C}$ acting via \[ (P,Z) \mapsto \big(P( \varphi + \mathrm{Re}\, w), \exp(-i\pi w)Z )\big) \] i.e.\ by rotating the phases and rescaling the masses of semistable objects. The $\mathbb{C}$ action is free provided $\cat{C}\neq 0$. Clearly the charge map $\mathcal{Z} \colon \stab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$ is equivariant with respect to these actions and the evident actions on $\mor{\Lambda}{\mathbb{C}}$. The group actions preserve the semi-norms $||\cdot||_\sigma$ for $\sigma \in\lstab{\cat{C}}$ in the sense that \begin{equation} \label{semi-norm invariance} || \alpha\cdot U\cdot w||_{\alpha\cdot \sigma\cdot w} = ||U||_\sigma \end{equation} for any automorphism $\alpha\in\aaut{\Lambda}{\cat{C}}$, element $w\in \mathbb{C}$ and charge $U\in \mor{\Lambda}{\mathbb{C}}$. They also preserve the semi-norm neighbourhoods: $\alpha \cdot B_\varepsilon(\sigma) \cdot w = B_\varepsilon(\alpha \cdot \sigma\cdot w)$ for any $\alpha \in \aaut{\Lambda}{\cat{C}}$ and $w\in \mathbb{C}$. \begin{lemma} \label{actions on dstab} The actions of $\aaut{\Lambda}{\cat{C}}$ and of $G$ on $\stab{\cat{C}}$ extend uniquely to continuous actions on $\lstab{\cat{C}}$ so that the charge map is equivariant. Elements of $G$ preserve $\lnstab{\cat{N}}{\cat{C}}$ and each $\alpha \in \aaut{\Lambda}{\cat{C}}$ maps $\lnstab{\cat{N}}{\cat{C}}$ to $\lnstab{\alpha(\cat{N})}{\cat{C}}$. The map $\mu_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$ is $G$-equivariant and such that \[ \begin{tikzcd} \lnstab{\cat{N}}{\cat{C}} \ar{r}{\alpha}\ar{d}{\mu_{\cat{N}}} & \lnstab{\alpha(\cat{N})}{\cat{C}} \ar{d}{\mu_{\alpha(\cat{N})}}\\ \stab{\cat{C}/\cat{N}} \ar{r}{\alpha} & \stab{\cat{C}/\alpha(\cat{N})} \end{tikzcd} \] commutes for each $\alpha \in \aaut{\Lambda}{\cat{C}}$. In particular $\mu_\cat{N}$ is equivariant for the subgroup of $\aaut{\Lambda}{\cat{C}}$ preserving $\cat{N}$. \end{lemma} \begin{proof} The actions of $\aaut{\Lambda}{\cat{C}}$ and $G$ extend to continuous actions on $\slice{\cat{C}} \times \mor{\Lambda}{\mathbb{C}}$ which preserve the subsets of lax and classical stability conditions. Hence they preserve $\lstab{\cat{C}}$. The equivariance of the charge map and the properties of the $\mu_\cat{N}$ are easy to verify. \end{proof} \subsection{Neighbourhoods of strata} \label{massless part} Fix $0<\varepsilon<1/8$ and consider the open neighbourhood \[ \lbstab{\cat{N}}{\cat{C}}= \mathbb{H}nionOfOpens{B_\varepsilon(\sigma)} \] of $\lnstab{\cat{N}}{\cat{C}}$ in $\lstab{\cat{C}}$. Intuitively this is the subset where objects of $\cat{N}$, and only those, are close to massless. These neighbourhoods are $\mathbb{C}$-invariant and compatible with the action of automorphisms in the sense that $\alpha\cdot \lbstab{\cat{N}}{\cat{C}} = \lbstab{\alpha(\cat{N})}{\cat{C}}$. Note that $\lbstab{0}{\cat{C}}=\stab{\cat{C}}$, and that $\lbstab{\cat{C}}{\cat{C}}=\lstab{\cat{C}}$ because the closure of the $\mathbb{C}$-orbit of any $\sigma \in \lstab{\cat{C}}$ contains a lax stability condition with massless subcategory $\cat{C}$. For $\sigma\in\lnstab{\cat{N}}{\cat{C}}$, Corollary~\ref{cor:restriction map} shows that restriction $(Q,W) \mapsto (Q\cap \cat{N} , W|_{\Lambda_\cat{N}})$ gives a continuous map $\rho_\cat{N}\colon B_\varepsilon(\sigma) \cap \lstab{\cat{C}} \to \lstab{\cat{N}}$. It clearly extends to $\lbstab{\cat{N}}{\cat{C}}$ so that there is a commutative diagram \[ \label{diag:restriction map} \begin{tikzcd} \lbstab{\cat{N}}{\cat{C}} \ar{r}{\rho_\cat{N}} \ar{d}[swap]{\mathcal{Z}} & \lstab{\cat{N}} \ar{d}{\mathcal{Z}} \\ \mor{\Lambda}{\mathbb{C}} \ar[->>]{r} & \mor{\Lambda_\cat{N}}{\mathbb{C}} \end{tikzcd} \] of continuous maps which are equivariant for the right action of $\mathbb{C}$ and for the left action of the subgroup of automorphisms preserving $\cat{N}$. More generally, $\alpha\cdot \rho_{\cat{N}}(\sigma) = \rho_{\alpha(\cat{N})}(\alpha \cdot \sigma)$ as elements of $\lstab{\alpha(\cat{N})}$ for any $\alpha\in \aaut{\Lambda}{\cat{C}}$. \section{Deforming lax stability conditions} \label{deformations} \noindent The technical heart of the theory of stability conditions is Theorem~\ref{deformation thm} which governs their deformation. We cannot expect such a simple result for lax stability conditions, but it turns out that it is still possible to deform them in a reasonable way. The heuristic is that the massive and massless parts of a lax stability condition deform independently. \subsection{Tangential, normal and fibrewise deformations} \label{normal deformation} \label{tangential deformation} For a lax stability condition $\sigma = (P,Z) \in \lnstab{\cat{N}}{\cat{C}}$ with massless subcategory $\cat{N}$, the base of the charge map $\mathcal{Z}\colon\lstab{\cat{C}}\to\mor{\Lambda}{\mathbb{C}}$ decomposes as \[ \mor{\Lambda_\cat{N}}{\mathbb{C}} \oplus \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}},\] with $Z\in\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. Here, as elsewhere, we consider $\mor{\Lambda_\cat{N}}{\mathbb{C}}$ as a subspace of $\mor{\Lambda}{\mathbb{C}}$ using the splitting arising from the inner product on $\Lambda\otimes \mathbb{R}$ --- see \S\ref{charges}. It is geometrically appealing to distinguish three (not mutually exclusive) cases of deformation: \begin{enumerate} \item A \defn{tangential deformation} of $\sigma$ is given by varying the charge in $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. Such a deformation fixes the massless subcategory and hence stays inside $\lnstab{\cat{N}}{\cat{C}}$. \item A \defn{normal deformation} of $\sigma$ is given by varying the charge in $\mor{\Lambda_\cat{N}}{\mathbb{C}}$. Such a deformation moves out of $\lnstab{\cat{N}}{\cat{C}}$ into $\lnstab{\cat{M}}{\cat{C}}$ for some thick subcategory $\cat{M}$ of $\cat{N}$. We think of this as deforming in a normal slice to the stratum. \item A \defn{fibrewise deformation} of $\sigma$ takes place when the charge function is fixed, i.e.\ only the slicing is deformed. In the classical setting, the charge map $\stab{\cat{C}}\to\mor{\Lambda}{\mathbb{C}}$ has discrete fibres, so there are no non-trivial fibrewise deformations. However for the lax stability condition $\sigma$ we can potentially vary the slicing on $\cat{N}$ in a continuous way, as this is not controlled by the charge function. \end{enumerate} We begin with a variant of Theorem~\ref{deformation thm} which shows that a lax stability condition with massless subcategory $\cat{N}$ can be freely deformed in the normal direction with respect to a nearby \emph{stability} condition on $\cat{N}$. \begin{theorem} \label{thm:deformation to non-degenerate} Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\cat{C}$ with massless subcategory $\cat{N}$. Then there is some $0<\varepsilon_0<1/8$ such that for any $0<\varepsilon <\varepsilon_0$ and pre-stability condition $\tau_\cat{N}=(Q_\cat{N},W_\cat{N})$ on $\cat{N}$ with \begin{itemize} \item $||W_\cat{N}||_\sigma<\sin(\pi\varepsilon)$ \item and $d(P_\cat{N},Q_\cat{N})<\varepsilon$ \end{itemize} there is a unique pre-stability condition $\tau=(Q,W)$ on $\cat{C}$ with \begin{itemize} \item charge $W=Z+W_\cat{N}$, \item restricted slicing $Q\cap \cat{N}=Q_\cat{N}$ \item and $d(P,Q)<\varepsilon$. \end{itemize} If $\sigma \in \legstab{\cat{C}}$ and $\tau_\cat{N} \in \stab{\cat{N}}$ then $\tau \in \stab{\cat{C}}$. \end{theorem} \begin{remark} \label{continuity of deformation} This is a normal deformation at least whenever $\tau_\cat{N}$ is a stability condition --- the charge $Z$ is changed by $W_\cat{N} \in \mor{\Lambda_\cat{N}}{\mathbb{C}}$, and since the deformation $\tau$ is in $\stab{\cat{C}}$ which has discrete fibres over $\mor{\Lambda}{\cat{C}}$, the slicing of $\tau$ cannot be deformed in the fibre. The construction defines a continuous map \begin{equation} \delta_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \stab{\cat{N}} \dashrightarrow \stab{\cat{C}} \end{equation} where the fibre product denotes the set of pairs $(\sigma, \tau_\cat{N}) \in \lnstab{\cat{N}}{\cat{C}} \times \stab{\cat{N}} $ whose slicings agree on $\cat{N}$ and the dashed arrow indicates that the map is only defined on the open subset where the charge $W_\cat{N}$ of $\tau_\cat{N}$ satisfies $|| W_\cat{N}||_\sigma <\sin(\pi\varepsilon)$ for some suitably small $\varepsilon>0$. The continuity of the charge $Z+W_\cat{N}$ of $\delta_\cat{N}(\sigma,\tau_\cat{N})$ is evident; the continuity of the slicing follows from the fact that $\stab{\cat{C}}$ is locally homeomorphic to $\mor{\Lambda}{\mathbb{C}}$ and Corollary~\ref{degenerate uniqueness}. \end{remark} \begin{proof} Choose $0<\varepsilon_0<1/8$ sufficiently small so that $P(\varphi-4\varepsilon_0,\varphi+4\varepsilon_0)$ is a length category for all $\varphi\in \mathbb{R}$ and fix $0<\varepsilon<\varepsilon_0$. Recall that $P(s,t)$ is a \defn{thin subcategory}, see \cite[Definition 7.2]{MR2373143}, if $0<t-s<1-2\varepsilon$, and that this implies that it is quasi-abelian. The charge $W=Z+W_\cat{N}$ defines a skewed stability function, see \cite[Definition 4.4]{MR2373143}, on any thin subcategory $P(s,t)$. That is, $W \colon K(P(s,t)) \to \mathbb{C}$ is a group homomorphism taking every non-zero object into a rotated copy of the strict half-plane $\mathbb{H} \cup \mathbb{R}_{<0}$. To see why, suppose that $c\in P(\varphi)$ for some $\varphi\in (s,t)$. Let $A$ be a finite multi-set of stable factors of $c$ in the quasi-abelian length category $P(\varphi)$. If, on the one hand, $a\in A$ is a massless stable object in $\cat{N}$ then $a\in Q_\cat{N}(\varphi-\varepsilon,\varphi+\varepsilon)$ and so $W(a) = (Z+W_\cat{N})(a) = W_\cat{N}(a)$ is non-zero and therefore one can assign the phase $\frac{1}{\pi}\arg W_{\cat{N}}(a) \in (s-\varepsilon,t+\varepsilon)$ to $a$. On the other hand, if $a\not \in \cat{N}$ is a massive stable factor then \[ |W(a)-Z(a)| < \sin(\pi\varepsilon) |Z(a)| \] because $||W-Z||_\sigma = ||W_\cat{N}||_\sigma <\sin(\pi\varepsilon)$. Therefore $W(a)\neq 0$ and differs in phase from $Z(a)$ by less than $\varepsilon$ and again the phase of $a$ with respect to $W$ lies in $(s - \varepsilon, t +\varepsilon)$. Since $W(c) = \sum_{a\in A} W(a)$ we conclude that $W(c)\neq0$ too and the phase of $c$, $\frac{1}{\pi} \arg W(c) \in (s-\varepsilon,t+\varepsilon)$. The remainder of the proof follows that of Theorem~\ref{deformation thm} in \cite[\S7]{MR2373143} verbatim. This is possible because, after the above initial step of showing that $W$ defines a skewed stability function on each thin subcategory, the charge $Z$ and the masses of objects with respect to $\sigma$ play no role in the proof, one only uses the locally finite slicing $P$. Therefore the same argument goes through even though $\sigma$ is lax, and we can construct a unique pre-stability condition $\tau=(Q,W)$ with $d(P,Q)<\varepsilon$. By Lemma~\ref{local persistence of massless factors} the slicing $Q$ restricts to a slicing $Q\cap\cat{N}$ on $\cat{N}$ with $d(P_\cat{N},Q\cap\cat{N})<\varepsilon$. Since $\sigma$ has massless subcategory $\cat{N}$ we know $Z\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ so that $W|_{\Lambda_\cat{N}}=W_\cat{N}$. Therefore $\tau=(Q,W)$ restricts to a pre-stability condition on $\cat{N}$ with charge $W_\cat{N}$ and slicing within distance $2\varepsilon$ of $Q_\cat{N}$. By Corollary~\ref{degenerate uniqueness} it follows that the restriction is $\tau_\cat{N}$, i.e.\ that $Q\cap \cat{N} = Q_\cat{N}$ as claimed. Finally, we must verify that $\tau$ satisfies the support property when $\sigma$ and $\tau_\cat{N}$ do. We may assume that $\sigma$ and $\tau_\cat{N}$ satisfy $K$-support for the same constant $K$. Suppose $b \in Q(\varphi)$ and let $S$ be a multi-set of its $\sigma$-stable factors. Since $d(P,Q)<\varepsilon$ these factors lie in $P(\varphi-\varepsilon,\varphi+\varepsilon) \subset Q(\varphi-2\varepsilon,\varphi+2\varepsilon)$. Therefore \[ m_\tau(b) = |W(b)| \geq \cos(2\pi\varepsilon) \sum_{s\in S} |W(s)|. \] We consider massive factors in $S - \cat{N}$ and massless ones in $S\cap \cat{N}$ separately. For a massive $\sigma$-stable factor $s$ the triangle inequality in $\mathbb{C}$, the norm estimate $||W-Z||_\sigma < \sin(\pi \varepsilon)$, and support for $\sigma$ imply that \[ |W(s)| \geq |Z(s)| - | Z(s)-W(s) | > (1-\sin(\pi\varepsilon)) |Z(s)| \geq \frac{1-\sin(\pi\varepsilon)}{K}||v(s)||. \] For a massless factor $s$ in $S\cap\cat{N}$ consider the set $T$ of its $\tau$-semistable factors. Since $d(P,Q)<\varepsilon$ these lie in $Q(\psi-\varepsilon,\psi+\varepsilon)$ where $s\in P(\psi)$. Moreover, since the slicing $Q$ restricts to the slicing $Q_\cat{N}$ on $\cat{N}$ we know that each $t\in T$ is a $\tau_\cat{N}$-semistable object in $\cat{N}$, and in particular that $W(t) = Z(t)+W_\cat{N}(t) = W_\cat{N}(t)$. Together with support for $\tau_\cat{N}$ and the triangle inequality for the norm $||\cdot||$ these observations yield the inequality \[ |W(s)| \geq \cos(2\pi\varepsilon) \sum_{t\in T} |W(t)| = \cos(2\pi\varepsilon) \sum_{t\in T} |W_\cat{N}(t)| \geq \frac{\cos(2\pi\varepsilon)}{K} \sum_{t\in T} || v(t) || \geq \frac{\cos(2\pi\varepsilon)}{K} ||v(s)||. \] Combining these estimates for the massive and massless factors of $b$, and using the triangle inequality for the norm again, gives \[ m_\tau(b) \geq \frac{1}{L} \sum_{s\in S} ||v(s)|| \geq \frac{1}{L} ||v(b)|| \] where $L = K \max \{ 1/(1-\sin(\pi\varepsilon)), 1/\cos(\pi\varepsilon) \}$. Hence $\tau$ satisfies the support property and so is a stability condition in $\stab{\cat{C}}$. \end{proof} \begin{example} \label{ex:norm example} Let $\cat{C} = \cat{D}^b(\mathbb{P}^1)$ and $\Lambda=K(\mathbb{P}^1) \cong \mathbb{Z}^2$ with basis $[\mathcal{O}], [\mathcal{O}_x]$. The inner product is chosen so that this basis is orthonormal. Let $\sigma=(P,Z)$ be the lax stability condition with charge $Z(\mathcal{O})=0$, $Z(\mathcal{O}_x)=-1$ and slicing $P=P_b$ from Example~\ref{ex:slicings}, i.e.\ $P(1) = \mathbb{C}lext{ \mathcal{O}_x, \mathcal{O}(n), \mathcal{O}(-n)[1] : x\in\mathbb{P}^1, n\in\mathbb{N}_{>0}}$ and $P(1/2) = \mathbb{C}lext{\mathcal{O}}$. The massless subcategory $\cat{N}=\thick{}{\mathcal{O}}$ and the massive stable objects are, up to shifts, the skyscrapers $\mathcal{O}_x$ for $x\in \mathbb{P}^1$, the line bundle $\mathcal{O}(1)$ and the shifted line bundle $\mathcal{O}(-1)[1]$. This lax stability condition can be deformed to a classical one using the previous result. Let $\tau_\cat{N} = (Q_\cat{N},W_\cat{N})$ where $Q_\cat{N}=P_\cat{N}$ is the restricted slicing with $Q_\cat{N}(1/2)=\clext{\mathcal{O}}$ and $W_\cat{N}(\mathcal{O}) = ri$ for some $r>0$. Considered as a charge in $\mor{\Lambda}{\mathbb{C}}$ via the orthogonal splitting we also have $W_\cat{N}(\mathcal{O}_x)=0$. Therefore \[ || W_\cat{N} ||_\sigma = \sup\left\{ \frac{|W_\cat{N}(c)|}{|Z(c)|} : c\ \text{massive $\sigma$-stable} \right\} = \sup\left\{ \frac{r}{n} : n\neq 0\right\} = r \] and the conditions of Theorem~\ref{thm:deformation to non-degenerate} are satisfied. The deformed stability condition $\tau=(Q,W)$ has charge $W(\mathcal{O}_x)=-1$, $W(\mathcal{O}(n))=-n+ri$ and heart $Q(0,1]= \coh(\mathbb{P}^1)$. Note that $d(P,Q)=\arctan(r)$ so that the slicing converges to $P$ as $r\to 0$. \end{example} \begin{example} \label{ex:non-supported} In contrast there are lax pre-stability conditions which cannot be deformed to classical ones. Again on $\cat{C} = \cat{D}^b(\mathbb{P}^1)$, let $\tau=(Q,W)$ be defined by the charge $W(\mathcal{O}_x)=0$, $W(\mathcal{O})=i$ and slicing $Q=P_t$ from Example~\ref{ex:slicings}, i.e.\ $Q(1/2) = \mathbb{C}lext{ \mathcal{O}(n) : n\in\mathbb{Z} }$ and $Q(1) = \mathbb{C}lext{ \mathcal{O}_x : x\in\mathbb{P}^1 }$. The massless subcategory $\cat{N}=\thick{}{\mathcal{O}_x : x\in \mathbb{P}^1}$ and the massive stable objects are, up to shifts, the line bundles $\mathcal{O}(n)$ for $n\in \mathbb{Z}$. Let $\tau_\cat{N} = (Q_\cat{N},W_\cat{N})$ where $W_\cat{N}(\mathcal{O}_x) = w$ for some $0\neq w\in \mathbb{C}$ and $Q_\cat{N}$ is a compatible slicing. Considering $W_\cat{N}$ as a charge in $\mor{\Lambda}{\mathbb{C}}$ via the orthogonal splitting we also have $W_\cat{N}(\mathcal{O})=0$. Therefore \[ || W_\cat{N} ||_\tau = \sup\left\{ \frac{|W_\cat{N}(c)|}{|W(c)|} : c \text{ massive $\tau$-stable} \right\} = \sup\{|n||w| : n\in \mathbb{Z}\} = \infty. \] Thus the conditions of Theorem~\ref{thm:deformation to non-degenerate} are not satisfied. Indeed we have already seen in Examples~\ref{not well-supported} and~\ref{degenerate supported versus weak} that $\tau$ is not a lax stability condition, so not in $\lstab{\cat{C}}$, because it does not satisfy the support property as well as not being in the closure of $\stab{\cat{C}}$. \end{example} Theorem~\ref{thm:deformation to non-degenerate} leads to the following inductive criterion for recognising when a lax stability condition is in $\lstab{\cat{C}}$. \begin{corollary} \label{inductive dstab criterion} Suppose $\sigma \in \legstab{\cat{C}}$ is a lax stability condition with massless subcategory $\cat{N}$. Then $\sigma\in \lstab{\cat{C}} \iff \rho_\cat{N}(\sigma) \in \lstab{\cat{N}}$. \end{corollary} \begin{proof} If $\sigma =(P,Z) \in \lnstab{\cat{N}}{\cat{C}}$ then $\rho_\cat{N}(\sigma) \in \lstab{\cat{N}}$ by Lemma~\ref{local persistence of massless factors}. Conversely, if $\rho_\cat{N}(\sigma)=(P_\cat{N},0) \in \lstab{\cat{N}}$ then we can choose a sequence of stability conditions $(Q_n,W_n) \in \stab{\cat{N}}$ converging to $(P_\cat{N},0)$ in the sense that $d(P_\cat{N},Q_n) \to 0$ and $W_n\to 0$ in the operator norm on $\mor{\Lambda_\cat{N}}{\mathbb{C}}$. It follows that $W_n\to 0$ in the operator norm on $\mor{\Lambda}{\mathbb{C}}$, where as usual we consider $W_n\in \mor{\Lambda}{\mathbb{C}}$ via the fixed splitting $\mor{\Lambda_\cat{N}}{\mathbb{C}} \hookrightarrow \mor{\Lambda}{\mathbb{C}}$. Since $\sigma$ satisfies the support property this implies $||W_n||_\sigma\to 0$. Therefore we can apply Theorem~\ref{thm:deformation to non-degenerate} to lift this sequence uniquely to a sequence of stability conditions $(P_n, Z+W_n) \in \stab{\cat{C}}$ converging to $(P,Z)=\sigma$. Hence $\sigma\in \lstab{\cat{C}}$ as claimed. \end{proof} The above criterion is tautological, and useless, when every object is massless i.e.\ when $\cat{N}=\cat{C}$. In that case we have the following result. \begin{lemma} \label{dstab criterion} Suppose $P$ is a slicing. Then \[ (P,0)\in \lstab{\cat{C}} \iff P\in \overline{\{Q : (Q,W)\in \stab{\cat{C}} \}}. \] \end{lemma} \begin{proof} Suppose $(P,0) \in \lstab{\cat{C}}$. Then there is a sequence $(P_n,Z_n)$ of stability conditions in $\stab{\cat{C}}$ converging to it, in particular with $P_n\to P$. Conversely, suppose $P_n\to P$ and there exists a charge $Z_n$ such that $(P_n,Z_n)\in \stab{\cat{C}}$ for each $n\in \mathbb{N}$. Then $P$ is locally finite by Lemma~\ref{length lemma}. Moreover, $(P_n, Z_n / n||Z_n||) \in \stab{\cat{C}}$ and converges to $(P,0)$ in $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ as $n\to \infty$. So $(P,0) \in \overline{\stab{\cat{C}}}$ and since the support property is automatic when all objects are massless in fact $(P,0)\in \lstab{\cat{C}}$. \end{proof} \begin{example} Suppose $\cat{C} = \cat{D}^b({\mathbf{k}} A_2)$, where $A_2$ is the quiver $1 \longrightarrow 2$. Define a sequence of stability conditions $\tau_n = (Q_n, W_n)$ via $W_n(S_1) = -1/n$ and $W_n(S_2) = i/e^n$, where $S_1$ and $S_2$ are the simple modules at $1$ and $2$ in the standard heart, respectively. Note that the indecomposable projective module $P_1$ in $\mod{{\mathbf{k}} A_2}$ is also $\tau_n$-semistable. The limit slicing $Q = \lim_{n\to \infty} Q_n$ is given by $Q(1/2) = \add{S_2}$ and $Q(1) = \add{S_1 \oplus P_1}$. Note that $Q$ is not a slicing for any (pre-)stability condition on $\cat{D}^b({\mathbf{k}} A_2)$ because the slice $Q(1)$ is not abelian. \end{example} We now refine Theorem~\ref{thm:deformation to non-degenerate} by considering deformations of a lax pre-stability condition with massless subcategory $\cat{N}$ to one with massless subcategory $\cat{M}\subset \cat{N}$. We saw in the proof of Proposition~\ref{massive part} that if $\sigma=(P,Z)$ is in the closure of $\lnstab{\cat{M}}{\cat{C}}$ then $P$ is well-adapted to $\cat{M}$. Therefore it is natural to impose this condition. The result is weaker than Theorem~\ref{thm:deformation to non-degenerate} in that even if we start with lax stability conditions on $\cat{C}$ and $\cat{N}$, we only show that the resulting deformation is a lax pre-stability condition which satisfies support on the quotient $\cat{C}/\cat{M}$ but not necessarily on $\cat{C}$; the set of such is denoted $\weakstab{\cat{C},\cat{M}}$ below. In the notation of Remark~\ref{continuity of deformation}, we get a continuous map \begin{equation} \delta_{\cat{N}, \cat{M}} \colon \legstab{\cat{C},\cat{N}} \times_{\slice{\cat{N}}} \legstab{\cat{N},\cat{M}} \dashrightarrow \weakstab{\cat{C},\cat{M}} \end{equation} where the domain is the set of pairs $(\sigma,\tau_\cat{N})$ whose slicings agree on $\cat{N}$, and are well-adapted to the subcategory $\cat{M}$. It is not clear that this is an open subset of the fibre product. It is a natural question when the image of this map is within lax stability conditions; this is discussed in Subsection~\ref{propagation}. \begin{proposition} \label{deformation off stratum} Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\cat{C}$ with massless subcategory $\cat{N}$ such that $P$ is well-adapted to the thick subcategory $\cat{M}$ of $\cat{N}$. Then there is some $0< \varepsilon_0<1/8$ such that for any $0<\varepsilon<\varepsilon_0$ and lax pre-stability condition $\tau_\cat{N}=(Q_\cat{N},W_\cat{N})$ on $\cat{N}$ with \begin{itemize} \item massless subcategory $\cat{M}$, \item $||W_\cat{N}||_\sigma <\sin(\pi\varepsilon)$ \item and $d(P_\cat{N},Q_\cat{N})<\varepsilon$ \end{itemize} there is a unique lax pre-stability condition $\tau=(Q,W)$ on $\cat{C}$ with \begin{itemize} \item massless subcategory $\cat{M}$, \item $W=Z+W_\cat{N}$, \item restricted slicing $Q \cap \cat{N} = Q_\cat{N}$ \item and $d(P,Q)<\varepsilon$. \end{itemize} If in addition $\sigma\in\legstab{\cat{C}}$ and $\tau_\cat{N}\in\legstab{\cat{N}}$ then $\tau \in \weakstab{\cat{C}}$. \end{proposition} \begin{remark} We do not know if a version of the last statement holds that includes the closure property, i.e.\ whether $\sigma\in\lstab{\cat{C}}$ and $\tau_\cat{N}\in\lstab{\cat{N}}$ implies $\tau\in\wstab{\cat{C}}$. \end{remark} \begin{proof} For $\cat{M}=0$ this is Theorem~\ref{thm:deformation to non-degenerate}. When $\cat{M}\neq 0$ the strategy is to reduce to this case by taking the quotient by $\cat{M}$ and then lifting back up to $\cat{C}$ using Proposition~\ref{glueing slicings}. We first observe that $Q_{\cat{N}}$ is well-adapted to $\cat{M}$. It is adapted to $\cat{M}$ by Proposition~\ref{massless thick}, so we just need to see that $Q_{\cat{N}/\cat{M}}$ is locally finite. The restricted slicing $P_{\cat{C}/\cat{M}}(\varphi) \cap \cat{N}/\cat{M} = P_{\cat{N}/\cat{M}}(\varphi)$ for each $\varphi \in \mathbb{R}$ because the proof of Proposition~\ref{uniqueness of compatibility} shows that each is the full subcategory of $\cat{N}/\cat{M}$ consisting of those objects having an HN filtration with one factor in $P(\varphi)\cap \cat{N}$ and all others in $\cat{M}$. Hence, since $d(P_{\cat{N}/\cat{M}},Q_{\cat{N}/\cat{M}}) \leq d(P_\cat{N},Q_\cat{N}) < \varepsilon$ by Corollary~\ref{cor:slicing inequalities}, and $P_{\cat{N}/\cat{M}}$ is a locally finite slicing, so is $Q_{\cat{N}/\cat{M}}$. Now, by Proposition~\ref{prop:massive stability condition}, $\tau_\cat{N}$ induces a pre-stability condition $\mu_\cat{M}(\tau_{\cat{N}}) = (Q_{\cat{N}/\cat{M}},W_\cat{N})$ on $\cat{N}/\cat{M}$. Moreover, since $P$ is well-adapted to $\cat{M}$, Remark~\ref{degenerate quotient} shows there is a lax pre-stability condition $\mu_\cat{M}(\sigma) = (P_{\cat{C}/\cat{M}},Z)$ on $\cat{C}/\cat{M}$ with massless subcategory $\cat{N}/\cat{M}$. We verify that these satisfy the conditions of Theorem~\ref{thm:deformation to non-degenerate}. The charge $W_\cat{N}$ is in $\mor{\Lambda_\cat{N}/\Lambda_\cat{M}}{\mathbb{C}} $ and the splitting $\mor{\Lambda_\cat{N}}{\mathbb{C}} \hookrightarrow \mor{\Lambda}{\mathbb{C}}$ restricts to one $\mor{\Lambda_\cat{N}/\Lambda_\cat{M}}{\mathbb{C}} \hookrightarrow \mor{\Lambda/\Lambda_\cat{M}}{\mathbb{C}}$. We use this to consider $W_\cat{N}$ as an element of $\mor{\Lambda/\Lambda_\cat{M}}{\mathbb{C}}$. With this identification $||W_\cat{N}||_{\mu_\cat{M}(\sigma)} \leq ||W_\cat{N}||_\sigma < \sin(\pi\varepsilon)$ by Lemma~\ref{semi-norm restricts to quotient norm}. Since the restricted slicing $P_{\cat{C}/\cat{M}}(\varphi) \cap \cat{N}/\cat{M} = P_{\cat{N}/\cat{M}}(\varphi)$ for each $\varphi \in \mathbb{R}$ and $d(P_{\cat{N}/\cat{M}},Q_{\cat{N}/\cat{M}}) \leq d(P_\cat{N},Q_\cat{N}) < \varepsilon$ the conditions of Theorem~\ref{thm:deformation to non-degenerate} are satisfied. Applying that result we construct a pre-stability condition $(Q_{\cat{C}/\cat{M}}, W_{\cat{C}/\cat{M}})$ on $\cat{C}/\cat{M}$ where $W_{\cat{C}/\cat{M}} = Z + W_\cat{N}$ and $d(P_{\cat{C}/\cat{M}},Q_{\cat{C}/\cat{M}})<\varepsilon$. Since $d(P_\cat{M},Q_\cat{M}) \leq d(P_\cat{N},Q_\cat{N}) < \varepsilon$ we can use Proposition~\ref{glueing slicings} to glue $Q_\cat{M}$ and $Q_{\cat{C}/\cat{M}}$ to a locally finite slicing $Q$ with $d(P,Q)<\varepsilon$ by Lemma~\ref{slicing distance}. By Lemma~\ref{local persistence of massless factors} this slicing $Q$ restricts to $\cat{N}$. It follows from the construction that the restriction is the slicing glued from $Q_\cat{M}$ and $Q_{\cat{N}/\cat{M}}$, which by uniqueness is $Q_\cat{N}$. Thus we have constructed a lax pre-stability condition $\tau = (Q,W)$ with $d(P,Q)<\varepsilon$, massless subcategory $\cat{M}$, charge $W=Z+W_\cat{N}$ and slicing $Q$ restricting to $Q_\cat{N}$ on $\cat{N}$. Corollary~\ref{degenerate uniqueness} implies that $\tau$ is unique with these properties. Finally, when $\sigma$ and $\tau_\cat{N}$ are lax stability conditions then $\mu_\cat{M}(\sigma)$ is a lax stability condition and $\mu_\cat{M}(\tau_\cat{N})$ a stability condition. Therefore by the last part of Theorem~\ref{thm:deformation to non-degenerate} the pre-stability condition on the quotient $\mu_\cat{M}(\tau)=(Q_{\cat{C}/\cat{M}}, Z+W_\cat{N})$ is in $\stab{\cat{C}/\cat{M}}$. Hence, $\tau \in \weakstab{\cat{C}}$ as claimed. \end{proof} The deformations described in this result are in general neither purely normal nor fibrewise, but a mixture of the two: the charge $W$ is changed by $W_\cat{N} \in \mor{\Lambda_\cat{N}}{\mathbb{C}}$, so for $\cat{M}\neq\cat{N}$, a component of the deformation occurs in the normal direction, but for $\cat{M}\neq 0$ the slicing on $\cat{N}$ is not fully determined by $W_\cat{N}$, and the choice of $Q_\cat{N}$ determines the fibrewise deformation. When $\cat{M}=\cat{N}$ the deformation is purely fibrewise since the charge, indeed the associated pre-stability condition on the quotient, is fixed and only the massless slicing is deformed. \begin{corollary} \label{cor:deformation in fibres} Suppose $\sigma=(P,Z)$ is a lax pre-stability condition on $\cat{C}$ with massless subcategory $\cat{N}$. Then there is some $\varepsilon_0>0$ such that for any $0<\varepsilon<\varepsilon_0$ and $Q_\cat{N}\in \slice{\cat{N}}$ with $d(P_\cat{N},Q_\cat{N})<\varepsilon$ there is a unique lax pre-stability condition $\tau=(Q,Z)$ on $\cat{C}$ with \begin{itemize} \item massless subcategory $\cat{N}$ \item massless slicing $Q_\cat{N}$ \item and $d(P,Q)<\varepsilon$. \end{itemize} If $\sigma$ is a lax stability condition then we may choose $\varepsilon_0=1/4$. \end{corollary} \begin{proof} This is the special case $\cat{M}=\cat{N}$ of Proposition~\ref{deformation off stratum}, i.e.\ $\tau_\cat{N} = (Q_\cat{N},0)$ except for the last statement. If $\sigma$ is a lax stability condition then $P(I)$ is length for any interval $I$ of length strictly less than one by Lemma~\ref{length lemma}. Therefore we may choose $\varepsilon_0=1/4$. \end{proof} Finally, we consider tangential deformations where the charge is varied in $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ and the massless slicing remains fixed. \begin{proposition} \label{def in bdy} Suppose $\sigma = (P,Z)$ is a lax pre-stability condition on $\cat{C}$ with massless subcategory $\cat{N}$. Then for any $0<\varepsilon<1/8$ and $W\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ satisfying $||W-Z||_\sigma<\sin(\pi\varepsilon)$ there is a unique lax pre-stability condition $\tau=(Q,W)$ with $d(P,Q)<\varepsilon$ and massless slicing $Q_\cat{N}=P_\cat{N}$. If in addition $\sigma \in \lnstab{\cat{N}}{\cat{C}}$ and $\tau \in \legstab{\cat{C}}$ then $\tau \in \lnstab{\cat{N}}{\cat{C}}$. \end{proposition} \begin{proof} By Lemma~\ref{semi-norm restricts to quotient norm}, $||W-Z||_{\mu_\cat{N}(\sigma)}= ||W-Z||_\sigma<\sin(\pi\varepsilon)$ where $\mu_\cat{N}(\sigma) = (P_{\cat{C}/\cat{N}},Z)$ is the induced stability condition in $\stab{\cat{C}/\cat{N}}$. Therefore, Theorem~\ref{deformation thm} allows us to construct a stability condition $(Q_{\cat{C}/\cat{N}},W)$ in $\stab{\cat{C}/\cat{N}}$ with $d(P_{\cat{C}/\cat{N}}, Q_{\cat{C}/\cat{N}}) < \varepsilon$. By Proposition~\ref{glueing slicings} the slicings $P_\cat{N}$ and $Q_{\cat{C}/\cat{N}}$ can be glued to a locally finite slicing $Q$ on $\cat{C}$. Since $Q(\varphi) \subseteq Q_{\cat{C}/\cat{N}}(\varphi)$ for all $\varphi\in \mathbb{R}$, we have thus constructed a lax pre-stability condition $\tau=(Q,W)$ with massless subcategory $\cat{N}$, restricted slicing $Q_\cat{N}=P_\cat{N}$, and such that $d(P,Q)<\varepsilon$. Uniqueness follows from Corollary~\ref{degenerate uniqueness}. Since $P_\cat{N}=Q_\cat{N}$ the last statement follows from Corollary~\ref{inductive dstab criterion}. \end{proof} \subsection{Support propagation} \label{propagation} In the classical case support propagates in components of the stability space: all nearby deformations of a stability condition are also stability conditions, and not just pre-stability conditions. We now discuss the extent to which this remains true for lax stability conditions. \begin{definition} We say that \defn{support propagates} from \begin{itemize} \item \label{def:local support propagation} a lax stability condition $\sigma \in \lnstab{\cat{N}}{\cat{C}}$ if there is $\varepsilon>0$ such that any $\tau =(Q,W) \in B_\varepsilon(\sigma)$ with $|| Z-(W-W_\cat{N})||_\sigma < \sin(\pi\varepsilon)$ and $\rho_\cat{N}(\tau)\in\lstab{\cat{N}}$ is in $\lstab{\cat{C}}$. See Figure~\ref{propagation conditions} for a schematic illustration of the charge conditions. \item \label{def:global support propagation} a component $\Sigma$ of $\lnstab{\cat{N}}{\cat{C}}$ if there is an $\varepsilon>0$ such that support propagates from all $\sigma\in\Sigma$ with respect to $\varepsilon$. \item $\lnstab{\cat{N}}{\cat{C}}$ if it propagates from all components. \end{itemize} \end{definition} \begin{figure} \caption{The definition of support propagation from $\sigma=(P,Z)\in \lnstab{\cat{N} \label{propagation conditions} \end{figure} This condition on the lax stability condition $\sigma$ means that nearby lax pre-stability conditions $\tau$ have the support property provided $|| Z-(W-W_\cat{N})||_\sigma < \sin(\pi\varepsilon)$ and $\rho_\cat{N}(\tau)\in\lstab{\cat{N}}$. The last condition is necessary because it is implied by $\tau\in B_\varepsilon(\sigma) \cap \lstab{\cat{C}}$. We illustrate this in the degenerate example below. \begin{example} Consider the classical geometric stability condition $\sigma_g = (P_g, Z_g)$ on $\cat{C} = \cat{D}^b(\mathbb{P}^1)$ from Examples~\ref{ex:slicings} and~\ref{ex:seminorm balls asymmetric}. By sending all charges to zero uniformly, we obtain a lax stability condition $\sigma = (P_g,0)$ with massless subcategory $\cat{N} = \cat{C}$. Consider the lax stability condition $\tau = (Q,0)$ given by $Q(\varphi) = P_g(\varphi)$ for $\varphi \notin \frac{1}{2}\mathbb{Z}$ and $Q(\frac{1}{2} + \varepsilon) = P_g(\frac{1}{2}) = \clext{\mathcal{O}}$ for sufficiently small $\varepsilon > 0$. Since we have not changed the phase of any other slices, $Q$ cannot be compatible with any linear charge. Hence, $\tau \in B_\varepsilon(\sigma)$ is a lax stability condition with the same charge as $\sigma$ but $\tau \notin \lstab{\cat{N}} = \lstab{\cat{C}}$. \end{example} \begin{example} Support propagates in the following cases: \begin{enumerate} \item For $\stab{\cat{C}}$, i.e.\ classical stability conditions, by Theorem~\ref{deformation thm}. \item For $\lnstab{\cat{C}}{\cat{C}}$, i.e.\ for lax stability conditions $(P,0)$ with massless subcategory $\cat{N}=\cat{C}$. The propagation condition is tautological in this case because $||\cdot||_\sigma=0$ and $\rho_\cat{N}=\id$. \item For $\lnstab{\cat{N}}{\cat{C}}$ where $\rk(\Lambda_\cat{N})=1$. This result is a combination of Theorem~\ref{thm:deformation to non-degenerate} (normal deformations) and Theorem~\ref{codim 1 support propagation} (tangential deformations). The condition that $\rho_\cat{N}(\tau) \in \lstab{\cat{N}}$ is necessary in this case. \item For any component of $\lnstab{\cat{N}}{\cat{C}}$ in the boundary of a finite type component $\stab{\cat{C}}$, i.e.\ a component in which every heart is a length abelian category, by Corollary~\ref{finite type support propagation}. In this case we take $\Lambda = K(\cat{C})$. \end{enumerate} \end{example} \label{q:support propagation} We do not know of any examples in which support does not propagate. However, it is entirely possible that such examples exist. The next result is a refinement of Corollary~\ref{inductive dstab criterion}. \begin{corollary} Suppose support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Let $\sigma$ be a lax stability condition with massless subcategory $\cat{N}$ and $\cat{M}\subset \cat{N}$ be a thick subcategory. Then \[ \sigma\in \overline{\lnstab{\cat{M}}{\cat{C}}} \iff \rho_\cat{N}(\sigma)\in \overline{ \lnstab{\cat{M}}{\cat{N}}}. \] \end{corollary} \begin{proof} If $ \sigma = (P,Z) \in \overline{\lnstab{\cat{M}}{\cat{C}}}$ then the continuity of $\rho_\cat{N}$ implies $\rho_\cat{N}(\sigma) \in \overline{\lnstab{\cat{M}}{\cat{N}}}$. Now suppose that $\rho_\cat{N}(\sigma)=(P_\cat{N},0)$ is in the closure of $\lnstab{\cat{M}}{\cat{N}}$. We always have $\overline{\lnstab{\cat{M}}{\cat{N}}} \subseteq \legstab{\cat{N}} \cap \overline{\stab{\cat{N}}} = \lstab{\cat{N}}$. Thus $\sigma \in \lnstab{\cat{N}}{\cat{C}}$ by Corollary~\ref{inductive dstab criterion}. Next, $P_\cat{N}$ is well-adapted to $\cat{M}$ by the proof of Proposition~\ref{massive part}, i.e.\ $P_{\cat{N}/\cat{M}}$ is locally finite and $P_\cat{N}$ is adapted to $\cat{M}$. Thus $P$, being well-adapted to $\cat{N}$ by Proposition~\ref{prop:massive stability condition}, is also adapted to $\cat{M}$ and moreover $P_{\cat{C}/\cat{N}}$ is locally finite. As $P_{\cat{C}/\cat{M}}$ is compatible with the pair $(P_{\cat{N}/\cat{M}}, P_{\cat{C}/\cat{N}})$, we conclude from Lemma~\ref{local-finiteness of compatible slicing} that $P_{\cat{C}/\cat{M}}$ is locally finite. Hence $P$ is actually well-adapted to $\cat{M}$. Thus we can apply Proposition~\ref{deformation off stratum} to $\sigma$ and a sequence of lax stability conditions in $\lnstab{\cat{M}}{\cat{N}}$ converging to $\rho_\cat{N}(\sigma)$ to construct a sequence of lax pre-stability conditions with massless subcategory $\cat{M}$ which satisfy support on the quotient $\cat{C}/\cat{M}$ and which converge to $\sigma$. We conclude by noting that support propagation implies this is eventually a sequence of lax stability conditions in $\lnstab{\cat{M}}{\cat{C}}$. \end{proof} \begin{remark} If we do not assume support propagates then we can only conclude that $\sigma$ is in $\lnstab{\cat{N}}{\cat{C}}$ and is in the limit in $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ of a sequence of lax pre-stability conditions with massless subcategory $\cat{M}$ which satisfy support on the quotient $\cat{C}/\cat{M}$. \end{remark} Definition~\ref{def:local support propagation} asks for two properties of lax pre-stability conditions occurring as small deformations of $\sigma\in\lnstab{\cat{N}}{\cat{C}}$: firstly the support property and secondly the property of being in the closure of $\stab{\cat{C}}$. The next result shows that the latter is automatic. Thus the terminology `support propagation' is justified. \begin{theorem} \label{all deformations remain in bdy} Suppose $\sigma =(P,Z) \in \lnstab{\cat{N}}{\cat{C}}$. Then for any sufficiently small $\varepsilon >0$ the set of lax pre-stability conditions $\tau = (Q,W)\in B_\varepsilon(\sigma)$ with $|| Z-(W-W_\cat{N})||_\sigma < \sin(\pi\varepsilon)$ and $\rho_\cat{N}(\tau) \in \lstab{\cat{N}}$ is contained in $\overline{\stab{\cat{C}}}$. \end{theorem} \begin{proof} Let $\sigma = (P,Z) \in \lnstab{\cat{N}}{\cat{C}}$ so that $\rho_\cat{N}(\sigma)=(P_\cat{N},Z_\cat{N}=0)$. Since $\sigma$ satisfies the support property there is some $K>0$ for which $|| v(c) || \leq K|Z(c)|$ whenever $c$ is a massive stable object. Recall from Proposition~\ref{equiv support conditions} that this implies that $|| \cdot ||_\sigma \leq K||\cdot ||$ on $\mor{\Lambda}{\mathbb{C}}$. Let $\tau = (Q,W) \in B_\varepsilon(\sigma)$ be a lax pre-stability condition with $\rho_\cat{N}(\tau) \in \lstab{\cat{N}}$ and $||Z-(W-W_\cat{N})||_\sigma< \sin(\pi\varepsilon)$. Thus $d(P_\cat{N},Q_\cat{N}) \leq d(P,Q)<\varepsilon$ by Corollary~\ref{cor:slicing inequalities} and \[ ||W_\cat{N}||_\sigma \leq ||Z-W||_\sigma + ||Z-(W-W_\cat{N})||_\sigma < 2\sin(\pi\varepsilon) \] by the triangle inequality. To construct $\tau' \in \stab{\cat{C}}$ suitably close to $\tau$ we proceed in a sequence of steps. We first deform $\sigma$ to $\sigma' \in \stab{\cat{C}}$ using Theorem~\ref{thm:deformation to non-degenerate}. We then apply Bridgeland's deformation theorem, Theorem~\ref{deformation thm}, to deform the charge of $\sigma'$ to obtain a stability condition $\tau'$ whose charge is closer to that of $\tau$. Finally, we check that $\tau'$ is indeed suitably close to $\tau$. \noindent {\bf Step 1:} {\it Deforming $\sigma$ to $\sigma'\in\stab{\cat{C}}$.} \noindent Since $\rho_\cat{N}(\tau) \in \lstab{\cat{N}} \subset \overline{\stab{\cat{N}}}$, for any $\varepsilon>0$ there is $\tau'_\cat{N} = (Q'_\cat{N},W_\cat{N}+W'_\cat{N}) \in \stab{\cat{N}}$ such that $d(Q_\cat{N},Q'_\cat{N}) < \varepsilon$ and $||(W_\cat{N}+W'_\cat{N})-W_\cat{N}|| = ||W'_\cat{N}|| < \sin(\pi \varepsilon)/K$. The latter inequality implies $||W_\cat{N}'||_\sigma \leq \sin(\pi\varepsilon)$ and then the triangle inequality gives \[ d(P_\cat{N},Q'_\cat{N}) < 2\varepsilon \quad \text{and} \quad ||W_\cat{N} + W'_\cat{N}||_\sigma < 3\sin(\pi\varepsilon). \] Set $\delta \coloneqq \max \{ 2\varepsilon, \arcsin(3 \sin (\pi\varepsilon)) / \pi \}$ and note that $\delta\to 0$ as $\varepsilon\to 0$. In particular we can choose $\varepsilon>0$ sufficiently small that $\delta<1/8$. By Theorem~\ref{thm:deformation to non-degenerate} there is a unique stability condition $\sigma' = (P',Z') \in B_\delta(\sigma)$ with charge $Z' = Z + W_\cat{N} + W'_\cat{N}$ and restricted slicing $P' \cap \cat{N} = Q'_\cat{N}$. \noindent {\bf Step 2:} {\it Deform the charge of $\sigma'$ to get $\tau' \in \stab{\cat{C}}$.} \noindent Set $U \coloneqq W-Z-W_\cat{N} \in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. Then by Lemma~\ref{semi-norm bound} \[ ||U||_{\sigma'} \leq \frac{||U||_\sigma}{C} \leq \frac{||W-Z||_\sigma + ||W'_\cat{N}||_\sigma}{C} \leq \frac{2\sin(\pi\varepsilon)}{C} < \sin(3\pi\varepsilon) \] where $C \coloneqq (1-\sin(\pi\delta))\cos(2\pi\delta) \to 1$ as $\varepsilon \to 0$ so that the final inequality holds for sufficiently small $\varepsilon>0$. Possibly further shrinking $\varepsilon$ we can apply Theorem~\ref{deformation thm} to obtain a stability condition $\tau' = (Q',W') \in B_{3\varepsilon}(\sigma')$ with $W' = Z' + U = W+W'_\cat{N}$. \noindent {\bf Step 3:} {\it $\tau'$ is close to $\tau$.} \noindent By construction $||W'-W|| = || W'_\cat{N}|| < \sin(\pi \varepsilon)$. Therefore it suffices to show that $d(Q',Q) < \varepsilon$. Consider a $\tau'$-semistable object $b\in Q'(\varphi)$. Using $d(P, Q)<\varepsilon$, $d(P, P')<\delta$ and $d(P', Q')<3\varepsilon$ gives $d(Q, Q')<\delta+4\varepsilon$ and hence \[ Q'(\varphi) \subset Q( \varphi-\delta-4\varepsilon, \varphi+\delta+4\varepsilon ) \subset P( \varphi-\delta-5\varepsilon, \varphi+\delta+5\varepsilon ) \eqqcolon \cat{A}. \] For sufficiently small $\varepsilon>0$ Lemma~\ref{length lemma} ensures that $\cat{A}$ is a quasi-abelian length category. The simple objects in $\cat{A}$ are the $\sigma$-stable objects. The HN filtration of $b$ with respect to $\tau$ is obtained by grouping the simple factors in some composition series of $b$ in $\cat{A}$ into a sequence of $\tau$-semistable objects of strictly decreasing phase. Therefore it suffices to show that the phases of these simple factors with respect to $\tau$ and $\tau'$ differ by less than $\varepsilon$. If $a\in \cat{A}$ is a massless simple object then there is nothing to prove because $d(Q'_\cat{N},Q_\cat{N})<\varepsilon$. If $a\in \cat{A}$ is massive then it suffices to show that $|W_\cat{N}'(a)| / |W(a)|$ can be made arbitrarily small, uniformly for all such $a$, by choosing $||W_\cat{N}'||$ sufficiently small. We do so by making a series of estimates. The operator norm estimate and the support property for $\sigma$ provide a bound \[ |W'_\cat{N}(a)| \leq ||W'_\cat{N}|| \cdot ||v(a)|| \leq K||W'_\cat{N}||\cdot |Z(a)| \] for some constant $K>0$. Moreover, since $Z\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ and $d(P,Q)<\varepsilon$ we have \[ | Z(a)| \leq \sum_{t\in T} |Z(t)| \leq \sum_{t\in T} ||Z||_\tau |W(t)| \leq \frac{||Z||_\tau}{\cos(2\pi\varepsilon)}|W(a)| \] where $T$ is a set of $\tau$-stable factors of the object $a$. Finally, \[ ||Z||_\tau \leq \frac{||Z||_\sigma}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)} = \frac{1}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)} \] by Lemma~\ref{semi-norm bound} because $Z\in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. Combining these estimates, we have the bound \[ \frac{|W_\cat{N}'(a)| }{ |W(a)| } \leq \frac{K||W'_\cat{N}||}{(1-\sin(\pi\varepsilon))\cos(2\pi\varepsilon)^2}, \] uniformly in $a$. The result follows because we can make the right hand side smaller than any given $\varepsilon>0$ by rescaling $W_\cat{N}'$. \end{proof} \begin{proposition} \label{continuity of deformation 2} Suppose support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Then the map $\delta_\cat{N}$ defined in Remark~\ref{continuity of deformation} extends to a continuous map \[ \delta_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \lstab{\cat{N}} \dashrightarrow \lstab{\cat{C}} \] defined on the open subset of pairs $(\sigma,\tau_\cat{N}) = \big((P,Z), (P\cap\cat{N},W_\cat{N})\big)$ with $||W_\cat{N}||_\sigma < \sin (\pi\varepsilon)$ for some suitably small $\varepsilon>0$. \end{proposition} \begin{proof} Let $\cat{M} \subset \cat{N}$ be the massless subcategory of $\tau_\cat{N}$. Then $P_\cat{N} = P\cap\cat{N}$ is adapted to $\cat{M}$ by Proposition~\ref{massless thick}. It follows that the $P$-semistable factors of an object $c\in \cat{M}$ are the $P_\cat{N}$-semistable factors, thus are in $\cat{M}$. Moreover, for any strict length one interval $I$ the intersection $P(I)\cap \cat{M} = P_\cat{N}(I) \cap \cat{M}$ is a Serre subcategory of $P_\cat{N}(I)$, which is in turn a Serre subcategory of $P(I)$. Hence, $P$ is adapted to $\cat{M}$. Therefore, by Proposition~\ref{quotient slicing}, there is an induced slicing $P_{\cat{C}/\cat{M}}$ on $\cat{C}/\cat{M}$. This induced slicing is compatible with $P_{\cat{C}/\cat{N}}$ and $P_{\cat{N}/\cat{M}}$, both of which are locally finite because they are respectively the slicings of the classical stability conditions $\mu_\cat{N}(\sigma)$ and $\mu_\cat{M}(\tau_\cat{N})$. Hence $P_{\cat{C}/\cat{M}}$ is locally finite by Lemma~\ref{local-finiteness of compatible slicing} so that $P$ is well-adapted to $\cat{M}$. Thus Proposition~\ref{deformation off stratum} lets us define $\delta_\cat{N}(\sigma,\tau_\cat{N})$ as the unique lax pre-stability condition $(Q,W=Z+W_\cat{N})$ with $d(P,Q)<\varepsilon$ and restricted slicing $Q\cap\cat{N}=P_\cat{N}$. The assumption that support propagates implies $\delta_\cat{N}(\sigma,\tau_\cat{N})\in \lstab{\cat{C}}$ so that $\delta_\cat{N}$ is well-defined as a map of sets. The restriction of $\delta_\cat{N}$ to (an open subset of) $\lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \stab{\cat{N}}$ is continuous by Remark~\ref{continuity of deformation}. The proof of Theorem~\ref{all deformations remain in bdy} shows that the extension is continuous on each normal slice $\{\sigma\} \times_{\slice{\cat{N}}} \lstab{\cat{N}}$. Together these facts imply the extension is continuous. \end{proof} \section{The topology of the space of lax stability conditions} \label{topology of dstab} \noindent We apply the deformation results of \S\ref{deformations} to study the topology of the space of lax stability conditions. Our starting point is the following consequence of support propagation. \begin{proposition} \label{support propagation homeo} Suppose that support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Then the following maps are homeomorphisms onto unions of components: \begin{enumerate}[itemsep=1ex] \item $\mu_\cat{N} \times \rho_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}} \times \lnstab{\cat{N}}{\cat{N}}$; \item $\mu_\cat{N}\times \id \colon \lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \lstab{\cat{N}} \to \stab{\cat{C}/\cat{N}} \times \lstab{\cat{N}}$. \end{enumerate} Here $\lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \lstab{\cat{N}}$ denotes the set of pairs $(P,Z)$ in $\lnstab{\cat{N}}{\cat{C}}$ and $(Q_\cat{N},W_\cat{N})$ in $\lstab{\cat{N}}$ such that the restricted slicing $P_\cat{N}=Q_\cat{N}$. \end{proposition} \begin{proof} (1) The quotient map $\mu_\cat{N}\colon\lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$ is well-defined by Proposition~\ref{prop:massive stability condition} and continuous by Proposition~\ref{massive part}. The restriction map $\rho_\cat{N}\colon\lnstab{\cat{N}}{\cat{C}} \to \lnstab{\cat{N}}{\cat{N}}$ is continuous by Corollary~\ref{cor:restriction map}. The product $\mu_\cat{N}\times\rho_\cat{N}$ is injective: given $\sigma = (P,Z) \in \lnstab{\cat{N}}{\cat{C}}$, the slicings $P_\cat{N}$ and $P_{\cat{C}/\cat{N}}$ determine $P$ uniquely by Propositions~\ref{uniqueness of compatibility},~\ref{quotient slicing} and~\ref{massless thick}; the charge $Z$ is determined by its factorisation through $\Lambda/\Lambda_\cat{N}$. By Corollary~\ref{cor:deformation in fibres}, $\sigma$ can be deformed along nearby slicings in $\lnstab{\cat{N}}{\cat{N}} \subseteq \slice{\cat{N}}$ to a unique lax pre-stability condition which, by the assumption that support propagates from $\lnstab{\cat{N}}{\cat{C}}$, is in fact in $\lnstab{\cat{N}}{\cat{C}}$. Proposition~\ref{def in bdy} is the analogous statement about tangential deformations, i.e.\ deforming charges in $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ or, equivalently, deforming the stability condition on the quotient $\cat{C}/\cat{N}$. Therefore, there is an open metric ball around $(\mu_\cat{N}(\sigma),\rho_\cat{N}(\sigma))$ in the image of $\mu_\cat{N} \times \rho_\cat{N}$. Thus the map in (1) is open. The statement about surjectivity on connected components follows. (2) As in the first part, the map $\mu_\cat{N}\times \id$ is injective because the slicing of $\sigma$ in $\lnstab{\cat{N}}{\cat{C}}$ can be reconstructed uniquely from the slicing of $\mu_\cat{N}(\sigma)$ and the slicing on $\cat{N}$. It is surjective because given $\left( \sigma_{\cat{C}/\cat{N}}, \sigma_\cat{N}\right)$ in a component meeting the image of $\mu_\cat{N}\times \id$ we can apply the first part to construct $\sigma \in \lnstab{\cat{N}}{\cat{C}}$ with $\mu_\cat{N}(\sigma) = \sigma_{\cat{C}/\cat{N}}$ and restricted slicing that of $\sigma_\cat{N}$. Then $(\sigma,\sigma_\cat{N})$ is the required lift of $\left( \sigma_{\cat{C}/\cat{N}}, \sigma_\cat{N}\right)$. The inverse is continuous because the reconstruction of the slicing on $\cat{C}$ from those on $\cat{C}/\cat{N}$ and $\cat{N}$ is continuous in the slicing metric; see Lemma~\ref{slicing distance}. \end{proof} We show in Corollary~\ref{local structure} that the product description of $\lnstab{\cat{N}}{\cat{C}}$ extends to a deformation retract neighbourhood, and we describe how pairs of strata fit together in Proposition~\ref{everything commutes}. \subsection{Neighbourhoods of strata} \label{sub:neighbourhoods of strata} Our aim is to construct a deformation retract neighbourhood of $\lnstab{\cat{N}}{\cat{C}}$. The construction depends on the splitting $\mor{\Lambda_\cat{N}}{\mathbb{C}} \hookrightarrow \mor{\Lambda}{\mathbb{C}}$ induced from the inner product on $\Lambda\otimes\mathbb{R}$, and therefore we need a neighbourhood which is adapted to this. We define one as follows. Fix $0<\varepsilon<1/8$ and for $\sigma\in \lnstab{\cat{N}}{\cat{C}}$ set \[ V_\varepsilon(\sigma) = \{ \tau =(Q,W) \in B_\varepsilon(\sigma) : ||W_\cat{N}||_\sigma <\sin(\pi \varepsilon)\} \] where $W_\cat{N}$ is the restriction of $W$ to $\Lambda_\cat{N}$ considered as an element of $\mor{\Lambda}{\mathbb{C}}$ via the splitting. Thus $V_\varepsilon(\sigma)$ is open in $\slice{\cat{C}}\times\mor{\Lambda}{\mathbb{C}}$. Then define an open subset of $\lstab{\cat{C}}$ by \[ \lvstab{\cat{N}}{\cat{C}} = \mathbb{H}nionOfOpens{V_\varepsilon(\sigma)} \] These smaller neighbourhoods have the same good properties as the $\lbstab{\cat{N}}{\cat{C}}$, namely they are $\mathbb{C}$-invariant and satisfy $\alpha\cdot \lvstab{\cat{N}}{\cat{C}} = \lvstab{\alpha(\cat{N})}{\cat{C}}$ for automorphisms $\alpha\in \aaut{\Lambda}{\cat{C}}$. Furthermore, $\lvstab{0}{\cat{C}}=\stab{\cat{C}}$ and $\lvstab{\cat{C}}{\cat{C}}=\lstab{\cat{C}}$ because the closure of the $\mathbb{C}$-orbit of any $\sigma \in \lstab{\cat{C}}$ contains a lax stability condition with massless subcategory $\cat{C}$. \begin{lemma} \label{deformation retraction} Assume that support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Then, for sufficiently small $\varepsilon>0$, the identity on $\lnstab{\cat{N}}{\cat{C}}$ extends to a continuous map \[ \Phi_{\cat{N}} \colon \lvstab{\cat{N}}{\cat{C}} \longrightarrow \lnstab{\cat{N}}{\cat{C}} \] mapping $(Q,W)$ to the unique element of $\lnstab{\cat{N}}{\cat{C}}$ with charge $W-W_\cat{N}$, massless slicing $Q_\cat{N}$ and slicing within distance $5\varepsilon$ of $Q$. Moreover, $\Phi_{\cat{N}}$ is $\mathbb{C}$-equivariant and satisfies $\alpha\cdot \Phi_{\cat{N}}(\tau) = \Phi_{\alpha(\cat{N})}(\alpha\cdot \tau)$ for each automorphism $\alpha\in \aaut{\Lambda}{\cat{C}}$ and $\tau \in \lvstab{\cat{N}}{\cat{C}}$. \end{lemma} \begin{remark} The map $\Phi_\cat{N}$ is induced by the projection $\pi_\cat{N}$ of charges onto their $\Lambda/\Lambda_\cat{N}$-component, $\pi_\cat{N}(W) = W - W_\cat{N}$: \[\begin{tikzcd} \lvstab{\cat{N}}{\cat{C}} \ar{rr}{\Phi_{\cat{N}}} \ar{d}{\mathcal{Z}} & & \lnstab{\cat{N}}{\cat{C}} \ar{d}{\mathcal{Z}}\\ \mor{\Lambda}{\mathbb{C}} \ar{r}{\pi_\cat{N}} & \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}} \ar[r, hook] & \mor{\Lambda}{\mathbb{C}} \end{tikzcd}\] \end{remark} \begin{proof} Given $\tau =(Q,W) \in \lvstab{\cat{N}}{\cat{C}}$ choose $\sigma =(P,Z) \in \lnstab{\cat{N}}{\cat{C}}$ such that $\tau \in B_\varepsilon(\sigma)$ and $||W_\cat{N}||_\sigma <\sin(\pi\varepsilon)$ where $\rho_\cat{N}(\tau) = (Q_\cat{N},W_\cat{N})$ and the slicing restricts by Lemma~\ref{local persistence of massless factors}. Then $W-W_\cat{N} \in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$ and \[ || Z - (W-W_\cat{N})||_\sigma \leq ||Z-W||_\sigma + ||W_\cat{N}||_\sigma < 2\sin(\pi\varepsilon) \leq\sin(3\pi\varepsilon) \] where we assume $\varepsilon\leq1/6$ which enables the final trigonometric inequality. Hence, for sufficiently small $\varepsilon>0$, we can apply Proposition~\ref{def in bdy} to construct a lax pre-stability condition $\sigma'=(P', W-W_\cat{N})$ with restricted slicing $P_\cat{N}'=P_\cat{N}$ and $d(P,P') < 3 \varepsilon$. Therefore $\sigma'\in B_{3\varepsilon}(\sigma)$, the restriction $\rho_\cat{N}(\sigma')=\rho_\cat{N}(\sigma)$, and $|| Z-(W-W_\cat{N})||_\sigma <\sin(3\pi\varepsilon)$. Thus we can apply the support propagation assumption to conclude that $\sigma'$ is in $\lnstab{\cat{N}}{\cat{C}}$ when $\varepsilon$ is sufficiently small. Since $\tau \in B_\varepsilon(\sigma)$ we know that $d(P,Q)<\varepsilon$. Hence $d(P_\cat{N},Q_\cat{N})<\varepsilon$ too. Therefore we can deform the massless slicing of $\sigma'$ using Corollary~\ref{cor:deformation in fibres} to obtain a lax pre-stability condition $\Phi_\cat{N}(\tau)=(Q',W-W_\cat{N})$ with massless slicing $Q_\cat{N}'=Q_\cat{N}$ and $d(P',Q') < \varepsilon$, from which it follows that $d(Q,Q') < 5\varepsilon$. It follows from Corollary~\ref{degenerate uniqueness} that $\Phi_\cat{N}(\tau)$ is well-defined and independent of the particular choice of $\sigma$ used to construct it. For sufficiently small $\varepsilon>0$ the support propagation assumption implies that $\Phi_\cat{N}(\tau)$ is in $\lnstab{\cat{N}}{\cat{C}}$. By construction $\Phi_\cat{N}$ is continuous in the charge and in the restricted slicing on $\cat{N}$. Since $\rho_\cat{N} \Phi_\cat{N}(\tau) = (Q_\cat{N},0)$, in order to see that $\Phi_\cat{N} \colon \lvstab{\cat{N}}{\cat{C}} \to \lnstab{\cat{N}}{\cat{C}}$ is continuous, it is sufficient to show that $\mu_\cat{N} \Phi_\cat{N}(\tau)$ varies continuously in $\tau$. If $\tau'$ is sufficiently close to $\tau$ then, since the slicing distance between $\tau$ and $\Phi_\cat{N}(\tau)$ is less than $5\varepsilon$, the slicing distance between $\Phi_\cat{N}(\tau)$ and $\Phi_\cat{N}(\tau')$ can be made strictly less than $1$. Therefore the slicing distance between $\mu_\cat{N}\Phi_\cat{N}(\tau)$ and $\mu_\cat{N}\Phi_\cat{N}(\tau')$ is also strictly less than $1$. Deformations of classical stability conditions with slicing within distance $1$ are uniquely determined by the deformation of the charge, i.e.\ if the charge deforms continuously then so does the stability condition. Since $\mu_\cat{N}\Phi_\cat{N}(\tau)$ and $\mu_\cat{N}\Phi_\cat{N}(\tau')$ are classical stability conditions whose slicings are within distance $1$ of each other whose charges are also close, it follows that $\mu_\cat{N}\Phi_\cat{N}(\tau)$ varies continuously with $\tau$. Moreover $\Phi_\cat{N}$ restricts to the identity on $\lnstab{\cat{N}}{\cat{C}}$. The characterisation of $\Phi_\cat{N}(\tau)$ in the statement follows immediately from the above construction, as does the compatibility with the actions of $\mathbb{C}$ and of $\aaut{\Lambda}{\cat{C}}$. \end{proof} Using the map $\Phi_\cat{N}$ we can, for sufficiently small $\varepsilon>0$, define a smaller, and even better behaved, neighbourhood of $\lnstab{\cat{N}}{\cat{C}}$ namely \[ \label{smaller neighbourhoods} \lustab{\cat{N}}{\cat{C}} = \left\{ \tau \in \lvstab{\cat{N}}{\cat{C}} \colon \tau\in B_\varepsilon(\Phi_\cat{N}(\tau)) \right\}. \] The compatibility of $\Phi_\cat{N}$ with the actions of $\mathbb{C}$ and $\aaut{\Lambda}{\cat{C}}$ ensures that these neighbourhoods are $\mathbb{C}$-invariant and that $\lustab{\cat{N}}{\cat{C}}$ is mapped to $\lustab{\alpha(\cat{N})}{\cat{C}}$ by $\alpha \in \aaut{\Lambda}{\cat{C}}$. In the two extreme cases of $\cat{N}=0$ and $\cat{C}$ the extra condition is vacuous, and we have equalities \begin{align*} \lustab{0}{\cat{C}} &= \lvstab{0}{\cat{C}} = \lbstab{0}{\cat{C}} = \stab{\cat{C}}, \\ \lustab{\cat{C}}{\cat{C}} &= \lvstab{\cat{C}}{\cat{C}} = \lbstab{\cat{C}}{\cat{C}} = \lstab{\cat{C}} . \end{align*} \begin{corollary} \label{local structure} Suppose support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Then, for sufficiently small $\varepsilon>0$ the map \[ \Phi_\cat{N}\times \rho_\cat{N} \colon \lustab{\cat{N}}{\cat{C}} \longrightarrow \lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \lstab{\cat{N}} \] is a homeomorphism onto the set of pairs $(\sigma,\tau_\cat{N})$ with $||W_\cat{N}||_\sigma < \sin(\pi\varepsilon)$ where $\tau_\cat{N}=(Q_\cat{N}, W_\cat{N})$. \end{corollary} \begin{proof} The map $\Phi_\cat{N}\times \rho_\cat{N}$ is continuous and, by definition of $\lustab{\cat{N}}{\cat{C}}$, has image in the subset \[ \left\{ \left(\sigma,\tau_\cat{N}=(Q_\cat{N},W_\cat{N}) \right) : ||W_\cat{N}||_\sigma < \sin(\pi\varepsilon) \right\}. \] The assumption that support propagates means that the map $\delta_\cat{N}$ defined in Proposition~\ref{continuity of deformation 2} exists. We claim it is inverse to $\Phi_\cat{N}\times \rho_\cat{N}$. Let $\tau=\delta_\cat{N}(\sigma,\tau_\cat{N})$, where $\sigma = (P,Z)$. By construction $\rho_\cat{N}(\tau)=\tau_\cat{N}$. Moreover $\tau\in \lvstab{\cat{N}}{\cat{C}}$ so that $\Phi_\cat{N}(\tau)$ is well-defined. Then $\Phi_\cat{N}(\tau)$ has charge $Z$, massless subcategory $\cat{N}$ and massless slicing $Q_\cat{N}$. The slicing of $\Phi_\cat{N}(\tau)$ is within distance $5\varepsilon$ of $P$ so that $\Phi_\cat{N}(\tau)=\sigma$ by Corollary~\ref{degenerate uniqueness} provided that $5\varepsilon <1$. We conclude that $\tau \in B_\varepsilon(\Phi_\cat{N}(\tau))$ so that $\tau \in \lustab{\cat{N}}{\cat{C}}$ and $(\Phi_\cat{N}\times \rho_\cat{N})\circ \delta_\cat{N}$ is the identity. Now set $\tau'= \delta_\cat{N}(\Phi_\cat{N}(\tau) , \rho_\cat{N}(\tau))$. This is a lax stability condition with the same charge, massless subcategory and massless slicing as $\tau$. Moreover, its slicing is within distance $5\varepsilon$ of that of $\tau$. Thus $\tau'=\tau$ by Corollary~\ref{degenerate uniqueness} if again $5\varepsilon <1$ and $\delta_\cat{N} \circ (\Phi_\cat{N}\times \rho_\cat{N})$ is the identity too. \end{proof} Clearly $\{ (\sigma,(Q_\cat{N}, W_\cat{N})) : ||W_\cat{N}||_\sigma < \sin(\pi\varepsilon)\}$ is an open neighbourhood of $\lnstab{\cat{N}}{\cat{C}}$ in $\lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \lstab{\cat{N}}$, and therefore $\lustab{\cat{N}}{\cat{C}}$ is an open neighbourhood of $\lnstab{\cat{N}}{\cat{C}}$. Indeed it is a deformation retract neighbourhood. \begin{definition} \label{def retract} Suppose support propagates from $\lnstab{\cat{N}}{\cat{C}}$ so that $\Phi_\cat{N} \times \rho_\cat{N}$ is a homeomorphism on $\lustab{\cat{N}}{\cat{C}}$ for some sufficiently small $\varepsilon>0$. For $t\in [0,1]$ let \[ \Phi_{\cat{N},t} \colon \lustab{\cat{N}}{\cat{C}} \to \lustab{\cat{N}}{\cat{C}} \] be the map corresponding to $(\sigma, (Q_\cat{N},W_\cat{N})) \mapsto (\sigma, (Q_\cat{N},tW_\cat{N}))$ under $\Phi_\cat{N}\times \rho_\cat{N}$. Note that $\Phi_{\cat{N},0}=\Phi_\cat{N}$ and $\Phi_{\cat{N},1}=\id$. \end{definition} When $\cat{N}=\cat{C}$ the retraction $\Phi_{\cat{N},t}$ is defined on the entire space of lax stability conditions because $\lustab{\cat{C}}{\cat{C}}=\lstab{\cat{C}}$. In this case $\Phi_{\cat{N},t}(Q,W) = (Q,tW) = (Q,W) \cdot i\log(t)/\pi$ is obtained by dilating the masses of objects using the right action of $i\mathbb{R} \subset \mathbb{C}$ defined in \S\ref{group actions}. More generally \begin{equation} \label{pi and rho commute} \begin{tikzcd} \lustab{\cat{N}}{\cat{C}} \ar{r}{\Phi_{\cat{N},t}} \ar{d}{\rho_\cat{N}} & \lustab{\cat{N}}{\cat{C}} \ar{d}{\rho_\cat{N}}\\ \lstab{\cat{N}} \ar{r}{\Phi_{\cat{N},t}} & \lstab{\cat{N}} \end{tikzcd} \end{equation} commutes for each $t\in [0,1]$ where the bottom map is given by mass dilation in $\lstab{\cat{N}}$. It is also clear that the maps $\Phi_{\cat{N},t}$ preserve the intersections $\lustab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}$ for thick $\cat{M}\subset \cat{N}$. \subsection{Pairs of strata} We now consider how the subsets $\lnstab{\cat{M}}{\cat{C}}$ and $\lnstab{\cat{N}}{\cat{C}}$ fit together when $\cat{M}\subset \cat{N}$ are thick subcategories. The first observation is that the \defn{frontier condition} fails: that is in general \[ \lnstab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}} \neq \varnothing \centernot \mathrm{im}plies \lnstab{\cat{N}}{\cat{C}} \subset \overline{\lnstab{\cat{M}}{\cat{C}}}. \] \begin{example} Let $\cat{C}=\cat{D}^b(\mathbb{P}^1)$ and consider the pair $\cat{M} = \thick{}{\mathcal{O}} \subset \cat{D}^b(\mathbb{P}^1) = \cat{N}$. The above intersection contains the slicings $P$ occurring in $\stab{\cat{C}}$ for which $\mathcal{O}$ is semistable, so is non-empty. However, it is not the whole of $\lnstab{\cat{C}}{\cat{C}}$ because it does not contain any slicings for which $\mathcal{O}$ is unstable, such as those in which the only two semistable objects are $\mathcal{O}(1)$ and $\mathcal{O}(2)$ and their shifts. \end{example} Nevertheless the situation is quite well-behaved. When we restrict to $\overline{\lnstab{\cat{M}}{\cat{C}}}$ the maps $\Phi_\cat{N}$ and $\rho_\cat{N}$ are compatible with passing to the quotient via $\mu_\cat{M}$. Effectively then we can reduce to the case $\cat{M}=0$. \begin{lemma} \label{mass maps preserve nbhds} Suppose $\cat{M}\subset \cat{N}$ are thick subcategories such that support propagates from both $\lnstab{\cat{N}}{\cat{C}}$ and $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$. Then for sufficiently small $\varepsilon>0$ the map $\mu_\cat{M}$ restricts to a map \[ \lustab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}\longrightarrow \lustab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}. \] \end{lemma} \begin{proof} The support propagation assumptions ensure that the maps $\Phi_{\cat{N},t}$ for $t\in [0,1]$ and $\Phi_{\cat{N}/\cat{M}}$ are defined. Let $\tau =(Q,W) \in \lustab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}$ and $\rho_\cat{N}(\tau) = (Q_\cat{N},W_\cat{N})$. Then $\Phi_\cat{N}(\tau)$ is also in $\overline{\lnstab{\cat{M}}{\cat{C}}}$ because $\Phi_{\cat{N},t}$ preserves the intersection $\lustab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}$, and $\mu_\cat{M}(\Phi_\cat{N}(\tau))$ is well-defined and lies in $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$; see Remark~\ref{degenerate quotient}. Then \[ || W_\cat{N}||_{\mu_\cat{M}(\Phi_\cat{M}(\tau))} \leq || W_\cat{N}||_{\Phi_\cat{M}(\tau)} <\sin(\pi\varepsilon) \] by Lemma~\ref{semi-norm restricts to quotient norm} and, by Lemma~\ref{lemma:continuous-restriction}, the distance between the slicings of $\mu_\cat{M}(\Phi_\cat{N}(\tau))$ and $\mu_\cat{M}(\tau)$ is less than that between those of $\Phi_\cat{N}(\tau)$ and $\tau$, which in turn is less than $\varepsilon$ because $\tau\in B_\varepsilon(\Phi_\cat{N}(\tau))$. Therefore, we also have $\mu_\cat{M}(\tau) \in B_\varepsilon(\mu_\cat{M}(\Phi_\cat{M}(\tau)))$. This shows that $\mu_\cat{M}(\tau)$ is in $\lvstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$ so that $\Phi_{\cat{N}/\cat{M}}(\mu_\cat{M}(\tau))$ is well-defined. Applying Corollary~\ref{degenerate uniqueness} then shows that \[ \Phi_{\cat{N}/\cat{M}}(\mu_\cat{M}(\tau)) = \mu_\cat{M}(\Phi_\cat{N}(\tau)) \] because they have the same charge $W-W_\cat{N}$, the same massless subcategory $\cat{N}/\cat{M}$ and massless slicing, and their slicings are within distance one of each other for sufficiently small $\varepsilon>0$. Therefore, $\mu_\cat{M}(\tau) \in B_\varepsilon(\Phi_{\cat{N}/\cat{M}}(\mu_\cat{M}(\tau)))$ and so is in $\lustab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$ as claimed. \end{proof} \begin{proposition} \label{everything commutes} Suppose $\cat{M}\subset \cat{N}$ are thick subcategories such that support propagates from both $\lnstab{\cat{N}}{\cat{C}}$ and $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$. Then for sufficiently small $\varepsilon>0$ there is a commutative diagram \[ \begin{tikzcd}[scale cd=0.95] \stab{\cat{C}/\cat{N}} \ar[equals]{d} & \lnstab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}} \ar{d}{\mu_\cat{M}} \ar{l}[swap]{\mu_\cat{N}} & \lustab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}\ar{l}[swap]{\Phi_\cat{N}} \ar{r}{\rho_\cat{N}} \ar{d}{\mu_\cat{M}}& \overline{\lnstab{\cat{M}}{\cat{N}} \ar{d}{\mu_\cat{M}}}\\ \stab{\cat{C}/\cat{N}} & \lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}} \ar{l}{\mu_{\cat{N}/\cat{M}}} & \lustab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}} \ar{l}{\Phi_{\cat{N}/\cat{M}}} \ar{r}[swap]{\rho_{\cat{N}/\cat{M}}}& \lstab{\cat{N}/\cat{M}}. \end{tikzcd} \] \end{proposition} \begin{proof} The support propagation assumptions and the previous lemma ensure the maps in the central square are well-defined, and that the square commutes. Consider the left hand square and choose \[ \sigma = (P,Z)\in \lnstab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}. \] Recall $\mu_\cat{N}(\sigma) = (P_{\cat{C}/\cat{N}},Z)$ from Proposition~\ref{prop:massive stability condition} where $P_{\cat{C}/\cat{N}}(\varphi)$ is the isomorphism closure of $P(\varphi)$ in $\cat{C}/\cat{N}$. Similarly, $\mu_\cat{M}(\sigma) = (P_{\cat{C}/\cat{M}},Z)$ where $P_{\cat{C}/\cat{M}}(\varphi)$ is the isomorphism closure of $P(\varphi)$ in $\cat{C}/\cat{M}$. Applying $\mu_{\cat{N}/\cat{M}}$ the charge remains the same, namely $Z$, and the category of semistable objects of phase $\varphi$ becomes the isomorphism closure of $P_{\cat{C}/\cat{M}}(\varphi)$ in $(\cat{C}/\cat{M})/(\cat{N}/\cat{M}) \simeq \cat{C}/\cat{N}$ which is, as before, the isomorphism closure of $P(\varphi)$. Hence $\mu_\cat{N} = \mu_{\cat{N}/\cat{M}} \circ \mu_{\cat{M}}$. Now consider the right hand square. Choose \[ \sigma = (P,Z)\in \lustab{\cat{N}}{\cat{C}} \cap \overline{\lnstab{\cat{M}}{\cat{C}}}. \] Both $\mu_\cat{M}\circ \rho_\cat{N}(\sigma)$ and $\rho_\cat{\cat{N}/\cat{M}}\circ \mu_\cat{M}(\sigma)$ have charge $Z|_{\Lambda_\cat{N}}$. On the one hand, the category of semistable objects of $\mu_\cat{M}\circ \rho_\cat{N}(\sigma)$ with phase $\varphi$ is the isomorphism closure of $P(\varphi)\cap\cat{N}$ in $\cat{N}/\cat{M}$. On the other hand, that of $\rho_\cat{\cat{N}/\cat{M}}\circ \mu_\cat{M}(\sigma)$ is the isomorphism closure of $P(\varphi)$ in $\cat{C}/\cat{M}$ intersected with $\cat{N}/\cat{M}$. This clearly contains the former and hence is the same as the former since nested slicings are equal. Therefore $\mu_\cat{M}\circ \rho_\cat{N}=\rho_\cat{\cat{N}/\cat{M}}\circ \mu_\cat{M}$. \end{proof} \section{The space of quotient stability conditions} \label{space qstab} \subsection{Definition} Define an equivalence relation on the points of $\lstab{\cat{C}}$ by $\sigma \sim \tau$ if they have the same charge and lie in the same connected component of the corresponding fibre of $\lstab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$. We refer to an equivalence class as a \defn{quotient stability condition on $\cat{C}$} and denote the class of $\sigma$ by $[\sigma]$. If $\sigma \sim \tau$ then by Corollary~\ref{local constancy 2} they have the same massless subcategory, $\cat{N}$ say, and induce the same stability condition $\mu_\cat{N}(\sigma)=\mu_\cat{N}(\tau)$ in $\stab{\cat{C}/\cat{N}}$. \begin{remark} \label{rmk: equiv rel} If support propagates from $\lnstab{\cat{N}}{\cat{C}}$ then \[ \mu_\cat{N} \times \rho_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \to \lstab{\cat{C}/\cat{N}} \times \lnstab{\cat{N}}{\cat{N}} \] is a homeomorphism onto a union of components by Proposition~\ref{support propagation homeo}. In this case $\sigma$ and $\tau$ are in the same component of the fibre of the charge map precisely when $\rho_\cat{N}(\sigma)$ and $\rho_\cat{N}(\tau)$ are in the same component of $\lnstab{\cat{N}}{\cat{N}}$. Conjecturally, non-empty stability spaces are contractible, in particular connected. If this is the case, and support propagates from all strata $\lnstab{\cat{N}}{\cat{C}}$, then a quotient stability condition is specified by a choice of massless subcategory $\cat{N}$ and stability condition in $\stab{\cat{C}/\cat{N}}$. Which thick subcategories $\cat{N}$ arise as massless categories remains a subtle question. \end{remark} The \defn{space of quotient stability conditions} is defined to be $\qstab{\cat{C}} = \lstab{\cat{C}} / {\sim}$ equipped with the quotient topology. By definition the charge map factors through the quotient, and we denote this factorisation also by $\mathcal{Z}$. The theme of this section is that $\qstab{\cat{C}}$ is a stratified space in a reasonable way. The first step is to specify a decomposition. Let $\qnstab{\cat{N}}{\cat{C}} = \lnstab{\cat{N}}{\cat{C}}/{\sim}$ be the subspace of quotient stability conditions with massless subcategory $\cat{N}$. Evidently \[ \qstab{\cat{C}} = \bigsqcup_{\mathrm{thick}~ \cat{N} \subset \cat{C}}\qnstab{\cat{N}}{\cat{C}} \] is a disjoint union of these subsets. The \defn{strata} are the connected components of these pieces. The maximal and minimal dimensional strata are easy to identify. The fibres of the charge map on $\stab{\cat{C}}$ are discrete so there are homeomorphisms \[ \stab{\cat{C}}\cong \lnstab{0}{\cat{C}} \cong \qnstab{0}{\cat{C}}. \] Thus the usual space of stability conditions embeds continuously in $\qnstab{0}{\cat{C}}$ as a union of strata. At the other extreme $\qnstab{\cat{C}}{\cat{C}} = \pi_0\left(\mathcal{Z}^{-1}(0)\right)$ is the set of components of the fibre $\mathcal{Z}^{-1}(0) = \lnstab{\cat{C}}{\cat{C}}$, with the discrete topology. These are the $0$-dimensional strata. If $\stab{\cat{C}}$ is connected there is a unique such stratum by Lemma~\ref{dstab criterion}. \subsection{The stratification of the space of quotient stability conditions} We show that, under suitable technical assumptions, $\qstab{\cat{C}}$ is a \emph{stratified space}. By this we mean that the strata $\qnstab{\cat{N}}{\cat{C}}$ are locally closed subspaces satisfying the frontier condition, i.e.\ the closure of each such stratum is a union of strata. For $\sigma\in \qnstab{\cat{N}}{\cat{C}}$ and $c\in \cat{C}$ the mass $m_\sigma(c)$ is the mass of $c$ in the corresponding stability condition on $\cat{C}/\cat{N}$. For massive $c\in \cat{C}$ we define $\tilde{\varphi}^\pm_\sigma(c)$ to be the minimal and maximal phases of the HN factors of $c$ in $\cat{C}/\cat{N}$. \begin{lemma} For each $c\in \cat{C}$ the mass $m_\bullet(c) \colon \qstab{\cat{C}} \to \mathbb{R}_{\geq 0}$ is continuous. The phases $\tilde{\varphi}_\bullet^\pm(c) \colon m_\bullet(c)^{-1}(\mathbb{R}_{>0}) \to \mathbb{R}$ are well-defined and $\tilde{\varphi}_\bullet^-(c)$ and $\tilde{\varphi}_\bullet^+(c)$ are respectively upper and lower semi-continuous. \end{lemma} \begin{proof} Recall that the mass of $c$ is continuous on $\lstab{\cat{C}}$ and is constant on equivalence classes, since it depends only on the massive HN factors. The continuity of masses on $\qstab{\cat{C}}$ follows immediately. The statements for phases follow similarly using the fact that the minimal and maximal phases are respectively bounded above and below by the minimal and maximal phases of the massive factors (which are constant on equivalence classes). \end{proof} It follows immediately from the continuity of masses that the subsets $\qnstab{\cat{N}}{\cat{C}}$ and hence also their connected components, i.e.\ the strata, are locally closed. The following consequence of Proposition~\ref{support propagation homeo} identifies the strata as components of spaces of stability conditions on various quotient categories. \begin{corollary} \label{stratum structure 2} Suppose that support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Then the factorisation of $\mu_\cat{N} \colon \lnstab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}$ through the quotient map induces a homeomorphism between the component of $\sigma \in \qnstab{\cat{N}}{\cat{C}}$ and the component of $\mu_\cat{N}(\sigma) \in \stab{\cat{C}/\cat{N}}$. Thus each stratum of $\qstab{\cat{C}}$ can be given the structure of a complex manifold in such a way that the restriction of the charge map is a local homeomorphism to $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. \end{corollary} We begin by describing the equivalence relation in a neighbourhood of $\lnstab{\cat{N}}{\cat{C}}$. \begin{lemma} \label{equiv rel lemma} Suppose support propagates from both $\lnstab{\cat{N}}{\cat{C}}$ and $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$ for each thick $\cat{M}\subset \cat{N}$. Let $\sigma,\tau \in\lustab{\cat{N}}{\cat{C}}$ for some sufficiently small $\varepsilon>0$. Then $\sigma\sim \tau$ if and only if $\mu_\cat{N}(\Phi_\cat{N}(\sigma)) = \mu_\cat{N}(\Phi_\cat{N}(\tau))$ and $\rho_\cat{N}(\sigma)\sim \rho_\cat{N}(\tau)$ as elements of $\lstab{\cat{N}}$. \end{lemma} \begin{proof} Suppose $\sigma\sim\tau$. Then $\rho_\cat{N}(\sigma)\sim \rho_\cat{N}(\tau)$ as elements of $\lstab{\cat{N}}$. Moreover, $\sigma$ and $\tau$ have the same massless subcategory, $\cat{M}$ say, so that $\sigma,\tau \in \lustab{\cat{N}}{\cat{C}} \cap \lnstab{\cat{M}}{\cat{C}}$. Since we also have $\mu_\cat{M}(\sigma)=\mu_\cat{M}(\tau)$ Proposition~\ref{everything commutes} implies that $\mu_\cat{N}(\Phi_\cat{N}(\sigma)) = \mu_\cat{N}(\Phi_\cat{N}(\tau))$. Conversely, suppose $\mu_\cat{N}(\Phi_\cat{N}(\sigma)) = \mu_\cat{N}(\Phi_\cat{N}(\tau))$ and $\rho_\cat{N}(\sigma)\sim \rho_\cat{N}(\tau)$. Then $\sigma$ and $\tau$ have the same massless subcategory, namely the common massless subcategory $\cat{M}$ of $\rho_\cat{N}(\sigma)$ and $\rho_\cat{N}(\tau)$. Because $\Phi_\cat{N}(\sigma)$ and $\Phi_\cat{N}(\tau)$ have the same charge, as do $\rho_\cat{N}(\sigma)$ and $\rho_\cat{N}(\tau)$, we get that $\sigma$ and $\tau$ have the same charge. Next, $\rho_\cat{M}(\rho_\cat{N}(\sigma))$ and $\rho_\cat{M}(\rho_\cat{N}(\tau))$ are in the same component of $\lnstab{\cat{M}}{\cat{M}}$, since $\rho_\cat{N}(\sigma)\sim\rho_\cat{N}(\tau)$. Then $\sigma\sim\tau$ by Remark~\ref{rmk: equiv rel}, as $\rho_\cat{M}\circ \rho_\cat{N} = \rho_\cat{M}$. \end{proof} \begin{lemma} Suppose support propagates from both $\lnstab{\cat{N}}{\cat{C}}$ and $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$ for each thick $\cat{M}\subset \cat{N}$. Then, for sufficiently small $\varepsilon >0$, the open neighbourhood $\lustab{\cat{N}}{\cat{C}}$ is a union of equivalence classes. \end{lemma} \begin{proof} Suppose $\tau \in \lustab{\cat{N}}{\cat{C}}$. By definition, the class $[\tau]$ is connected in the subspace topology from $\lstab{\cat{C}}$. Its intersection with $\lustab{\cat{N}}{\cat{C}}$ is an open subset of $[\tau]$, so it suffices to show this intersection is also closed in $[\tau]$. By Corollary~\ref{local structure}, $\lustab{\cat{N}}{\cat{C}}$ is homeomorphic via $\Phi_\cat{N}\times \rho_\cat{N}$ to \[ \{ \left(\sigma,\tau_\cat{N}=(Q_\cat{N}, W_\cat{N})\right) : ||W_\cat{N}||_\sigma < \sin(\pi\varepsilon)\} \subset \lnstab{\cat{N}}{\cat{C}} \times_{\slice{\cat{N}}} \lstab{\cat{N}}. \] And by the previous result $\tau'\in [\tau] \cap \lustab{\cat{N}}{\cat{C}}$ if and only if $\mu_\cat{N}(\Phi_\cat{N}(\tau')) = \mu_\cat{N}(\Phi_\cat{N}(\tau))$ and $\rho_\cat{N}(\tau') \sim \rho_\cat{N}(\tau)$. The set of such $\tau'$ is closed in the above fibre product, therefore also in $\lustab{\cat{N}}{\cat{C}}$ and in $[\tau]$. \end{proof} These lemmas show that $\qustab{\cat{N}}{\cat{C}} = \lustab{\cat{N}}{\cat{C}}/{\sim}$ is an open neighbourhood of $\qnstab{\cat{N}}{\cat{C}}$ in $\qstab{\cat{C}}$ and allow us to describe the stratification within this neighbourhood. \begin{corollary} \label{local structure 2} Suppose that support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Then for sufficiently small $\varepsilon >0$ the map \[ \mu_\cat{N}\circ\Phi_\cat{N} \times [\rho_\cat{N}] \colon \qustab{\cat{N}}{\cat{C}} \to \stab{\cat{C}/\cat{N}}\times \qstab{\cat{N}}, \quad [\sigma] \mapsto (\mu_\cat{N}\left( \Phi_\cat{N}(\sigma)\right) , [\rho_\cat{N}(\sigma)]) \] is a homeomorphism onto an open subset. Moreover, if for each thick $\cat{M}\subset \cat{N}$ support also propagates from $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$ then the homeomorphism above restricts to a biholomorphism from $\qustab{\cat{N}}{\cat{C}} \cap \qnstab{\cat{M}}{\cat{C}}$ onto (an open subset of) a union of components of $\stab{\cat{C}/\cat{N}} \times \qnstab{\cat{M}}{\cat{N}}$. In particular, locally we have \[ \overline{\qnstab{\cat{M}}{\cat{C}}} \cong \overline{\stab{\cat{C}/\cat{N}}\times \qnstab{\cat{M}}{\cat{N}}} \cong \stab{\cat{C}/\cat{N}}\times \overline{\qnstab{\cat{M}}{\cat{N}}}. \] \end{corollary} \begin{proof} The existence of the homeomorphism, and the fact that it is stratum-preserving, follow from Corollary~\ref{local structure} and Lemma~\ref{equiv rel lemma}. It restricts to a holomorphic isomorphism because it is compatible with the charge maps. \end{proof} \begin{theorem} \label{qstab stratified} Suppose that support propagates from both $\lnstab{\cat{N}}{\cat{C}}$ and $\lnstab{\cat{N}/\cat{M}}{\cat{C}/\cat{M}}$ for each thick $\cat{M}\subset \cat{N}$. Then the space of quotient stability conditions is stratified by complex manifolds. More precisely, \[ \qstab{\cat{C}} = \bigsqcup_{\mathrm{thick}~ \cat{N} \subset \cat{C}}\qnstab{\cat{N}}{\cat{C}} \] decomposes into a disjoint union of locally closed subsets. Each $\qnstab{\cat{N}}{\cat{C}}$ is a complex manifold of dimension $\mathrm{rk}(\Lambda / \Lambda_\cat{N})$; we refer to its connected components as strata. This decomposition satisfies the frontier condition: the closure of each stratum is a union of strata. \end{theorem} \begin{proof} We have already seen that there is such a decomposition into locally closed subsets $\qnstab{\cat{N}}{\cat{C}}$ and, by Corollary~\ref{stratum structure 2}, $\qnstab{\cat{N}}{\cat{C}}$ is a complex manifold locally homeomorphic to $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. For the last part, consider $[\sigma] \in \qnstab{\cat{N}}{\cat{C}}$. Note that $[\rho_\cat{N}(\sigma)] \in \qnstab{\cat{N}}{\cat{N}}$ which is a discrete set of points. If $[\rho_\cat{N}(\sigma)] \in \overline{\qnstab{\cat{M}}{\cat{N}}}$ then $[\sigma] \in \overline{\qnstab{\cat{M}}{\cat{C}}}$ by the last part of Corollary~\ref{local structure 2}. Since $\qnstab{\cat{N}}{\cat{N}}$ is discrete and we have a local product description, an open neighbourhood of $[\sigma]$ in $\qnstab{\cat{N}}{\cat{C}}$ is contained in \[ \stab{\cat{C}/\cat{N}} \times \{ \, [\rho_\cat{N}(\sigma)] \, \} \subset \stab{\cat{C}/\cat{N}}\times \overline{\qnstab{\cat{M}}{\cat{N}}}. \] Therefore by the formula at the end of Corollary ~\ref{local structure 2} the connected component of $[\sigma]$ in $\qnstab{\cat{N}}{\cat{C}}$ is in the closure of $\qnstab{\cat{M}}{\cat{C}}$. \end{proof} \begin{corollary} \label{deformation retraction 2} Suppose support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Let $\Phi_{\cat{N},t}$ be the deformation retraction of Definition~\ref{def retract}. Then \[ [\Phi_{\cat{N},t}]\colon \qustab{\cat{N}}{\cat{C}} \to \qustab{\cat{N}}{\cat{C}}, \quad [\sigma] \mapsto [\Phi_{\cat{N},t}(\sigma)] \] is an almost-stratum-preserving deformation retraction of $\qustab{\cat{N}}{\cat{C}}$ onto $\qnstab{\cat{N}}{\cat{C}}$. By this we mean that the track of each point under the deformation retraction remains in the same stratum until the last point, at which it may enter a higher codimension stratum. \end{corollary} \begin{proof} Suppose $\sigma, \tau \in \lustab{\cat{N}}{\cat{C}}$ and $\sigma \sim\tau$. Then $\Phi_{\cat{N},t}(\sigma)\sim\Phi_{\cat{N},t}(\tau)$ by Lemma~\ref{equiv rel lemma}, because $\Phi_\cat{N}(\Phi_{\cat{N},t}(\sigma)) = \Phi_\cat{N}(\sigma)$ and $\rho_\cat{N} (\Phi_{\cat{N},t}(\sigma)) = \rho_\cat{N}(\sigma) \cdot i\log(t)/\pi$. Thus $\Phi_{\cat{N},t}$ descends to the quotient. Since $\Phi_{\cat{N},t}(\sigma)$ has the same massless subcategory as $\sigma$ for $t\in (0,1]$ the resulting deformation retraction is almost-stratum-preserving, i.e.\ its tracks remain in the same stratum until the last moment, when they move into the deeper stratum $\qnstab{\cat{N}}{\cat{C}}$. \end{proof} \subsection{Group actions} To complete our description of the space of quotient stability conditions, we note that the actions of $\aaut{\Lambda}{\cat{C}}$ and $G$ descend to it, and that they respect the stratification and local models. \begin{corollary} \label{actions on qstab} The actions of $\aaut{}{\cat{C}}$ and of $G$ on $\stab{\cat{C}}$ extend uniquely to continuous actions on $\qstab{\cat{C}}$. The maps in the diagram \[ \begin{tikzcd} \stab{\cat{C}} \ar[hookrightarrow]{r} \ar{dr}[swap]{\mathcal{Z}}& \lstab{\cat{C}} \ar[->>]{r} \ar{d}{\mathcal{Z}} & \qstab{\cat{C}} \ar{dl}{\mathcal{Z}} \\ & \mor{\Lambda}{\mathbb{C}} \end{tikzcd} \] are equivariant with respect to these actions. The actions respect the stratification: elements of $G$ preserve $\qnstab{\cat{N}}{\cat{C}}$ whereas the action of an automorphism $\alpha \in \aaut{\Lambda}{\cat{C}}$ takes $\qnstab{\cat{N}}{\cat{C}}$ to $\qnstab{\alpha(\cat{N})}{\cat{C}}$. Moreover, the neighbourhood $\qustab{\cat{N}}{\cat{C}}$ is $\mathbb{C}$-invariant and $\alpha\in \aaut{\Lambda}{\cat{C}}$ maps it to $\qustab{\alpha(\cat{N})}{\cat{C}}$. When support propagates from $\lnstab{\cat{N}}{\cat{C}}$ the actions are compatible with the local model in Corollary~\ref{local structure 2} in that \[ \begin{tikzcd} \qustab{\cat{N}}{\cat{C}} \ar{r} \ar{d}{\alpha} &\stab{\cat{C}/\cat{N}} \times \qstab{\cat{N}}\ar{d}{\alpha \times \alpha}\\ \qustab{\alpha(\cat{N})}{\cat{C}} \ar{r} &\stab{\cat{C}/\alpha(\cat{N})} \times \qstab{\alpha(\cat{N})} \end{tikzcd} \] commutes, and that the horizontal maps are $\mathbb{C}$-equivariant with respect to the evident diagonal action of $\mathbb{C}$ on the right hand terms. \end{corollary} \begin{proof} It is easy to check that the actions preserve the equivalence relation and so descend to the quotient. Lemma~\ref{actions on dstab} implies they respect the stratification as stated. The final part follows from the properties of the maps $\mu_\cat{N}$, $\rho_\cat{N}$ and $\Phi_\cat{N}$ --- see Lemma~\ref{actions on dstab}, Section~\ref{massless part}, and Lemma~\ref{deformation retraction} respectively. \end{proof} \section{Codimension one strata} \label{rank one} \noindent We investigate the case in which the massless subcategory is non-zero but `as small as possible'. More precisely, throughout this section we fix a thick subcategory $\cat{N}$ for which the saturation $\Lambda_\cat{N}$ of the image of $K(\cat{N}) \to K(\cat{C}) \to \Lambda$ is a rank one lattice. We abuse terminology by saying that the massless subcategory $\cat{N}$ has rank one. We assume that $\lnstab{\cat{N}}{\cat{C}}$ is non-empty. For simplicity of notation we also assume that it is connected; otherwise we consider each component separately. The main result of this section, Theorem~\ref{codim 1 support propagation}, is that $\lnstab{\cat{N}}{\cat{C}}$ is a component of $\stab{\cat{C}/\cat{N}} \times \mathbb{R}$. In particular it is a real codimension one boundary stratum in $\lstab{\cat{C}}$. \subsection{Objects and phases} When $\mathrm{rk}(\Lambda_\cat{N})=1$ the massless subcategory $\cat{N}$ has a simple form. In particular, up to shift, all the massless objects must have the same phase. This remains true even for nearby stability conditions. \begin{lemma} \label{common massless phase} Suppose $\sigma \in \overline{\stab{\cat{C}}}$ is a lax pre-stability condition with rank one massless subcategory $\cat{N}$. Then $\cat{N}=\triang{}{S}$ is the triangulated closure of a set $S$ of stable objects of the same phase. \end{lemma} \begin{proof} Write $\sigma = (P,Z)$. Suppose $0\neq c\in \cat{N}$ is a $\sigma$-semistable object with phase $0< \varphi(c)\leq 1$. Given $0<\varepsilon <1/8$ we can choose $\tau = (Q,W) \in \stab{\cat{C}}$ with $d(P,Q)<\varepsilon$. By Lemma~\ref{local persistence of massless factors} all $\tau$-semistable factors of $c$ are in $\cat{N}$. Since they have phases in $(\varphi(c)-\varepsilon, \varphi(c)+\varepsilon)$ and $m_\tau(c)>0$ we deduce that $W|_{\Lambda_\cat{N}}\neq 0$. Therefore we can choose a generator $\lambda \in \Lambda_\cat{N}$ with $W(\lambda)\in \mathbb{H} \cup \mathbb{R}_{<0}$. Let $\psi \in (0,1]$ be the phase of $W(\lambda)$. Then $|\varphi(c)-\psi|<\varepsilon$ and if $c'\in \cat{N}$ is another $\sigma$-semistable object with phase $0< \varphi(c')\leq 1$ we have \[ |\varphi(c)-\varphi(c')| \leq |\varphi(c)-\psi|+|\psi-\varphi(c')| < 2\varepsilon. \] It follows that $\varphi(c) = \varphi(c') = \varphi$, say, for all $\sigma$-semistable objects in $\cat{N}$, i.e.\ $\cat{N} \cap P(0,1] \subset P(\varphi)$. Since $\sigma\in \overline{\stab{\cat{C}}}$ its slicing $P$ is locally finite so that $P(\psi)$ is a quasi-abelian length category. The full subcategory $\cat{N} \cap P(0,1]$ is closed under extensions, strict subobjects and strict quotients. Therefore $\cat{N} \cap P(0,1] = \clext{S}$ is the extension-closure of a subset $S$ of simple objects of $P(\varphi)$. Hence $\cat{N}=\triang{}{S}$. \end{proof} \begin{corollary} \label{common phase nearby} Fix $0<\varepsilon<1/8$. There is a set $S$ of objects with $\cat{N}=\triang{}{S}$ and such that for each $\sigma$ in the open neighbourhood $\lbstab{\cat{N}}{\cat{C}}$ of $\lnstab{\cat{N}}{\cat{C}}$ all objects in $S$ are stable of the same phase. \end{corollary} \begin{proof} Let $\sigma =(P,Z) \in \lnstab{\cat{N}}{\cat{C}}$. By the previous lemma there is $0<\varphi\leq 1$ and a subset $S$ of simple objects in $P(\varphi)$ with $\cat{N}=\triang{}{S}$. Suppose $\tau=(Q,W) \in B_\varepsilon(\sigma)\cap \lstab{\cat{C}}$. Lemma~\ref{local persistence of massless factors} implies that \[ \clext{S} \subset Q(\varphi-\varepsilon,\varphi+\varepsilon) \cap \cat{N} \subset P(\varphi-2\varepsilon,\varphi+2\varepsilon) \cap \cat{N}. \] Since $P(0,1] \cap \cat{N} = \clext{S}$ and $0<\varepsilon<1/8$ the right-hand side above is $\clext{S}[k]$ for $k=0$ or $\pm 1$. Since it contains $\clext{S}$ we must have $k=0$ so that the above containments are equalities. Therefore all $\tau$-semistable factors of each $s\in S$ are also in $\clext{S}$, and since each object $s\in S$ is simple in $\clext{S}$ this implies that each $s\in S$ is $\tau$-semistable. It follows as in the proof of the previous lemma that all $s\in S$ have the same phase, so that in fact $\clext{S} = Q(\psi) \cap \cat{N}$ for some $\psi\in (\varphi-\varepsilon,\varphi+\varepsilon)$. Since each $s \in S$ is simple, we deduce that each $s \in S$ is actually $\tau$-stable. The result now follows from the definition \[ \lbstab{\cat{N}}{\cat{C}} = \bigcup_{\sigma \in \lnstab{\cat{N}}{\cat{C}}} B_\varepsilon(\sigma) \cap \lstab{\cat{C}} \] of the neighbourhood and the fact that we have assumed $\lnstab{\cat{N}}{\cat{C}}$ is connected. \end{proof} \subsection{The neighbourhood of a stratum} The neighbourhood of the stratum $\lnstab{\cat{N}}{\cat{C}}$ has a simple form when $\mathrm{rk}(\Lambda_\cat{N})=1$: it is a fibration over $\mathbb{C}$. Recall that $\stab{\cat{N}}$ is the space of stability conditions on $\cat{N}$ whose charges factor through $\Lambda_\cat{N} \cong \mathbb{Z}$. The space of non-zero charges is $\mor{\Lambda_\cat{N}}{\mathbb{C}} - \{0\} \cong \mathbb{C}^*$. As $\mathbb{C}$ acts freely on $\stab{\cat{N}}$ we deduce that each component is homeo\-mor\-phic to the universal cover $\mathbb{C}$, with $\mathbb{C}$ acting freely and transitively. The image of the continuous restriction map \[ \rho_\cat{N} \colon \lbstab{\cat{N}}{\cat{C}} \to \lstab{\cat{N}}, \quad (P,Z) \mapsto (P \cap \cat{N}, Z|_{\Lambda_\cat{N}}) \] consists of lax stability conditions in which the simple objects of the generating set $S$ are stable and have a common phase. \begin{lemma} Fix $0<\varepsilon<1/8$. The map $\rho_\cat{N}$ restricts to a holomorphic fibration from $\lbstab{\cat{N}}{\cat{C}} \cap \stab{\cat{C}}$ onto the component of $\stab{\cat{N}}$ consisting of stability conditions with heart a shift of $\clext{S}$. \end{lemma} \begin{proof} If $\sigma = (P,Z) \in \lbstab{\cat{N}}{\cat{C}} \cap \stab{\cat{C}}$ then clearly $\rho_\cat{N}(\sigma) \in \stab{\cat{N}}$. Moreover the objects of $S$ are stable with respect to $\rho_\cat{N}(\sigma)$ and by Corollary~\ref{common phase nearby} have a common phase. It follows that the heart $P(0,1] \cap \cat{N}$ is a shift of the abelian length category $\clext{S}$. The $\mathbb{C}$-equivariance of $\rho_\cat{N}$ implies that the restriction is surjective onto this component. Indeed it is a holomorphic submersion whose fibres are biholomorphic to one another, and therefore it is a holomorphic fibration. \end{proof} The only massless subcategory appearing in the boundary of this component is $\cat{N}$ itself. A lax stability condition is determined by a common phase $\varphi\in\mathbb{R}$ for the objects in $S$. Thus we have a commuting diagram \[ \begin{tikzcd} \lstab{\cat{N}}\ar{d}[swap]{\mathcal{Z}} \ar{r}{\simeq} & \mathbb{C} \cup (-\infty+i\mathbb{R}) \ar{d}{\exp} \\ \mor{\Lambda_\cat{N}}{\mathbb{C}} \ar{r}{\simeq} & \mathbb{C} \end{tikzcd} \] where we define $\exp(-\infty + ir)= 0$ for $r\in \mathbb{R}$. In the next section we show that the restriction of $\rho_\cat{N}$ to $\lnstab{\cat{N}}{\cat{C}}$ is a fibration over $-\infty+i\mathbb{R}$ with fibre $\stab{\cat{C}/\cat{N}}$. \subsection{The boundary stratum} In this section we show that the boundary stratum $\lnstab{\cat{N}}{\cat{C}}$ is (a component of) the product $\stab{\cat{C}/\cat{N}} \times \mathbb{R}$ of the space of stability conditions on the quotient category $\cat{C}/\cat{N}$ and a factor $\mathbb{R}$ recording the common phase of the massless objects. Recall that, for simplicity of notation only, we assume that $\lnstab{\cat{N}}{\cat{C}}$ is connected. We first observe that there are inclusions \[ \lnstab{\cat{N}}{\cat{C}} \subset \wnstab{\cat{N}}{\cat{C}} \subset \stab{\cat{C}/\cat{N}} \times \mathbb{R}, \] where we abuse notation by identifying $\sigma \in \wnstab{\cat{N}}{\cat{C}}$ with its image under the continuous inclusion $\mu_\cat{N}\times \rho_\cat{N}$, and where $\wnstab{\cat{N}}{\cat{C}}$ denotes the set of lax pre-stability conditions $\sigma$ in the closure of $\stab{\cat{C}}$ with massless subcategory $\cat{N}$ and such that $\mu_\cat{N}(\sigma)$ is a stability condition on $\cat{C}/\cat{N}$; see Definition~\ref{def:space of degenerate stability conditions}. \begin{theorem} \label{codim 1 support propagation} Support propagates from $\lnstab{\cat{N}}{\cat{C}}$. Therefore, $\lnstab{\cat{N}}{\cat{C}}$ is a component of $\stab{\cat{C}/\cat{N}} \times \mathbb{R}$ and $\qnstab{\cat{N}}{\cat{C}}$ is a component of $\stab{\cat{C}/\cat{N}}$. \end{theorem} \begin{proof} Let $\sigma = (P,Z) \in \lnstab{\cat{N}}{\cat{C}}$. We show that support propagates from $\sigma$ in two steps: \begin{enumerate} \item for $\tau$ sufficiently close to $\sigma$, we have $\tau \in \wnstab{\cat{N}}{\cat{C}}$; and, \item any $\tau \in \weakstab{\cat{C}}$ with massless subcategory $\cat{N}$ satisfies the support property. \end{enumerate} \noindent {\bf Step 1:} Choose $\varepsilon>0$ sufficiently small that we can apply Theorem~\ref{thm:deformation to non-degenerate} and Theorem~\ref{all deformations remain in bdy}. Let $\tau = (Q,W) \in B_\varepsilon(\sigma)$. If $W|_{\Lambda_\cat{N}} \neq 0$ then $\tau$ is a (classical) pre-stability condition in $B_\varepsilon(\sigma)$ and is therefore actually a stability condition by the argument at the end of the proof of Theorem~\ref{thm:deformation to non-degenerate}. Therefore we may assume that $W \in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. Then by Theorem~\ref{all deformations remain in bdy} \begin{equation} \{ \tau =(Q,W) \in B_\varepsilon(\sigma) : W \in \mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}} \} \subset \wnstab{\cat{N}}{\cat{C}} \end{equation} because $\mu_\cat{N}(\sigma)$ is a stability condition and so therefore is any sufficiently small deformation of it. (The condition that $\rho_\cat{N}(\tau)$ is in $\lstab{\cat{N}}$ is automatic because $\lnstab{\cat{N}}{\cat{N}}\cong \mathbb{R}$.) \noindent {\bf Step 2:} It remains to show that any $\tau\in \wnstab{\cat{N}}{\cat{C}}$ satisfies the support property, for which we need the following lemma, whose proof we defer. \begin{lemma} \label{support failure criterion} Suppose $\sigma =(P,Z) \in \wnstab{\cat{N}}{\cat{C}}$. Let $S \subset P(\varphi)$ be a set of stable objects such that $\cat{N}=\triang{}{S}$. Then, up to shifts, any sequence $(b_n)$ of massive stable objects with $m(b_n) / ||v(b_n)|| \to 0$ contains a subsequence $(c_n)$ whose phases converge and such that \[ \lim_{n\to \infty} \left( \frac{v(c_n)}{||v(c_n)|| }\right) = \frac{v(s)}{||v(s)||} \] for any $s\in S$. Moreover, $\lim_{n\to \infty} \varphi(c_n) \in \varphi + 2\mathbb{Z}$ for any such subsequence. \end{lemma} Let $S \subset Q(\varphi)$ be a set of $\tau$-stable objects such that $\cat{N}=\triang{}{S}$. By Lemma~\ref{support failure criterion} any sequence $(b_n)$ of massive stable objects with $m(b_n) / || v(b_n)|| \to 0$ has, up to shifts, a subsequence $(c_n)$ with $\varphi(c_n) \to \varphi$. Without loss of generality we may assume $\varphi(c_n) \leq \varphi$ for all $n\in \mathbb{N}$; the other case is similar. Since $c_n$ is stable and not in $S$ this implies that $\mor{s}{c_n}=0$ for all $s\in S$. Perturbing the common phase of the massless objects slightly using Corollary~\ref{cor:deformation in fibres}, we can find $\tau' = (Q',W)$ also in $\wnstab{\cat{N}}{\cat{C}}$ with $\mu_\cat{N}(\tau')=\mu_\cat{N}(\tau)$ and $S\subset Q'(\varphi')$ where $\varphi'>\varphi$. The $c_n$ remain massive and stable for $\tau'$ by Proposition~\ref{uniqueness of compatibility} because \[ c_n \in Q_{\cat{C}/\cat{N}}(\varphi(c_n)) =Q'_{\cat{C}/\cat{N}}(\varphi(c_n)) \] where $\varphi(c_n)<\varphi'$ and $\mor{s}{c_n}=0$ for all $s\in S$. However now $\varphi'(c_n) = \varphi(c_n) \to \varphi < \varphi'$ contradicting Lemma~\ref{support failure criterion}. We conclude that there is no sequence $(b_n)$ of massive stable objects with $m(b_n) / || v(b_n)|| \to 0$. Therefore $\tau$ satisfies the support property and so is in $\lnstab{\cat{N}}{\cat{C}}$. The final statement on the space of quotient stability conditions follows from Proposition~\ref{support propagation homeo} and the definition of $\qnstab{\cat{N}}{\cat{C}}$ because $\lnstab{\cat{N}}{\cat{N}} \cong \mathbb{R}$. \end{proof} \begin{proof}[Proof of Lemma~\ref{support failure criterion}] If $\sigma$ satisfies the support property then there is no sequence $(b_n)$ with $m(b_n) / ||v(b_n)|| \to 0$ and the result is vacuously true. Therefore we assume $\sigma$ does not satisfy the support property i.e.\ there is a sequence $(b_n)$ of massive stable objects with $m(b_n) / ||v(b_n)|| \to 0$. Since $\Lambda_\cat{N}$ has rank one and by Corollary~\ref{common phase nearby} the objects in $S$ are stable with a common phase in nearby classical stability conditions, there is $\lambda\in \Lambda_\cat{N} \otimes \mathbb{R}$ with $||\lambda||=1$ such that $v(s) \in \mathbb{R}_{>0}\cdot \lambda$ for all $s\in S$. Lemma~\ref{support criterion} tells us that there is a subsequence $(c_n)$ such that \[ \lim_{n\to \infty} \left( \frac{v(c_n)}{||v(c_n)|| }\right) \in \Lambda_\cat{N}\otimes \mathbb{R}. \] By replacing $c_n$ with $c_n[1]$ if necessary we can ensure that the limit is $\lambda = v(s) / ||v(s)||$. Further shifting by an even integer if necessary we may assume that $\varphi(c_n) \in [\varphi-1,\varphi+1]$ for each $n\in \mathbb{N}$. Therefore, we can pass to a subsequence $(c_n)$ whose phases converge. It remains to show that the limit is the common phase $\varphi$ of the objects $s\in S$. Suppose that $(\sigma_m)$ is a sequence in $\stab{\cat{C}}$ tending to $\sigma$. By the argument in the proof of Corollary~\ref{common phase nearby}, for sufficiently large $m\in \mathbb{N}$, the objects in $S$ are $\sigma_m$-stable with common phase $\varphi_m(s) $ for all $s\in S$. For similar reasons, $c_n$ is massive for each $\sigma_m$ and its charge has a well-defined phase $\varphi_m(c_n)$ in the interval $(\varphi(c_n)-1/8,\varphi(c_n)+1/8)$. Since $d(P_m,P) \to 0$ and $\varphi(c_n)\to \varphi'$, say, for any $\varepsilon>0$ we can find $M,N\in \mathbb{N}$ such that \[ | \varphi_m(c_n) - \varphi' | \leq |\varphi_m(c_n)-\varphi(c_n) | + |\varphi(c_n)-\varphi'| < \varepsilon \] whenever $m\geq M$ and $n\geq N$. Therefore we can switch the limits to compute \[ \varphi' = \lim_{n\to \infty} \varphi(c_n) = \lim_{n\to \infty} \lim_{m\to \infty} \varphi_m(c_n) = \lim_{m\to \infty} \lim_{n\to \infty} \varphi_m(c_n) = \lim_{m\to \infty} \varphi_m(s) =\varphi \] where the penultimate step follows because the phase $\varphi_m$ only depends on the normalised vector in $\Lambda \otimes \mathbb{R}$ and $v(c_n) / ||v(c_n)|| \to v(s) / ||v(s)||$. \end{proof} \begin{remark} In Step 2 of the proof of Theorem~\ref{codim 1 support propagation} we established the support property for all elements of $\wnstab{\cat{N}}{\cat{C}}$. This special situation in the codimension one case is not obvious in general: Example~\ref{ex:non-supported} gives an example of a lax pre-stability condition which satisfies support on the quotient of $\cat{C}/\cat{N}$ but does not satisfy the stronger support property required to be a lax stability condition. However, this is not quite a counter-example because it is not in the closure of $\stab{\cat{C}}$ since the slicings do not converge as we approach it. \end{remark} \section{Finite type components} \label{finite type components} \noindent Let $\cat{C}$ be a triangulated category with $K(\cat{C}) \cong \mathbb{Z}^n$ and set $v=\id \colon K(\cat{C}) \to \Lambda$. For simplicity of notation we assume that $\stab{\cat{C}}$ is connected; otherwise consider a component. Say that a stability condition $\sigma=(P,Z)$ has \defn{discrete phase distribution} if $\{\varphi \in \mathbb{R} : P(\varphi) \neq 0\}$ is a discrete subset of $\mathbb{R}$, and that it has \defn{algebraic heart} if $P(0,1]$ is a finite length abelian category with finitely many simple objects. Finally, we say that $\stab{\cat{C}}$ has \defn{finite type} if each stability condition has an algebraic heart admitting only finitely many torsion pairs \cite{MR3858773}. \begin{lemma} \label{finite type cpt} Assume that $K(\cat{C})\cong \mathbb{Z}^n$ is free abelian. The following are equivalent: \begin{enumerate} \item Each $\sigma\in \stab{\cat{C}}$ has algebraic heart. \item $\stab{\cat{C}}$ has finite type. \item $\stab{\cat{C}}$ has dimension $n$ and each stability condition has discrete phase distribution. \end{enumerate} \end{lemma} \begin{proof} $(1) \mathrm{im}plies (2)$: Suppose each $\sigma\in \stab{\cat{C}}$ has algebraic heart. Then the set of hearts of stability conditions in $\stab{\cat{C}}$ is closed under HRS-tilting at simple objects of these hearts (see, for example, \cite{MR1327209,MR2739061} for the definition), because a stability condition with such a heart is given by freely assigning a charge in the upper half plane to each simple object of the heart. Moreover, by \cite[Corollary 3.27]{MR3858773} the poset of strata is pure of length $n=\rk K(\cat{C})$, i.e.\ each maximal chain in the poset has length $n$. This implies that each heart has only finitely many torsion pairs because the poset of torsion pairs in the heart has a uniform bound on the valency of each element and on the length of each chain. Therefore $\stab{\cat{C}}$ has finite type. $(2) \mathrm{im}plies (3)$: Now suppose $\stab{\cat{C}}$ has finite type. The phase distribution of each $\sigma \in \stab{\cat{C}}$ must be discrete, for otherwise rotating phases would yield an infinite sequence of tilts, and thus infinitely many torsion pairs in the heart of $\sigma$. $(3) \mathrm{im}plies (1)$: Finally, suppose each $\sigma \in \stab{\cat{C}}$ has discrete phase distribution. Then for each $\sigma$ there exists $\varepsilon>0$ such that $P_\sigma(0,\varepsilon)=0$. Hence by \cite[Lemma 3.1]{MR3858773} the heart of $\sigma$ is algebraic. \end{proof} In general, we do not know which thick subcategories $\cat{N}$ of $\cat{C}$ can appear as massless subcategories. If $\cat{N}$ occurs as a massless subcategory then, by Corollary~\ref{local structure}, both $\stab{\cat{C}/\cat{N}}$ and $\stab{\cat{N}}$ are non-empty. However this is not sufficient, as can be already seen for $\cat{D}^b(\mathbb{P}^1)$. \begin{example} Let $\cat{C}=\cat{D}^b(\mathbb{P}^1)$ with $\Lambda = K(\cat{C}) \cong \mathbb{Z}^2$ and let $\cat{N}=\thick{}{\mathcal{O}_x : x\in \mathbb{P}^1}$. Note $\Lambda_\cat{N} = \mathrm{im}(K(\cat{N}) \to K(\cat{C}) \to \Lambda) \cong \mathbb{Z}$ even though $K(\cat{N}) \cong \mathbb{Z}^{\mathbb{P}^1}$. With these lattices, $\stab{\cat{N}} \cong \mathbb{C}$ and $\stab{\cat{C}/\cat{N}} \cong \mathbb{C}$ are non-empty. However, it follows from Proposition \ref{prop: massless stables = algebraic simples} that $\cat{N}$ does not occur as a massless subcategory because the skyscrapers $\mathcal{O}_x$ are not simple objects in any algebraic heart. \end{example} For a finite type component of a stability space the situation is much simpler: we can classify the massless subcategories completely, and show that support propagates from each boundary stratum. \begin{corollary} \label{boundary algebraic} Suppose $\stab{\cat{C}}$ has finite type and $\sigma = (P,Z) \in \overline{\stab{\cat{C}}}$. Then for any $\varphi \in \mathbb{R}$ the full subcategory $P(\varphi,\varphi+1]$ is the heart of some classical stability condition in $\stab{\cat{C}}$, and so, in particular, is algebraic. Moreover, $\sigma$ has discrete phase distribution. \end{corollary} \begin{proof} Rotating phases by the $\mathbb{C}$ action it suffices to prove the first part for $\varphi=0$. Choose $\tau=(Q,W)$ in $\stab{\cat{C}}$ with $d(P,Q)<1/2$ so that the hearts $P(0,1]$ and $Q(0,1]$ are both contained in $Q(-1/2,3/2]$. Then $P(0,1]$ is obtained by first tilting from $Q(0,1]$ to $Q(-1/2,1/2]$ and then tilting from that to $P(0,1]$. Since the set of hearts of stability conditions in the finite-type component $\stab{\cat{C}}$ is closed under tilting, $P(0,1]$ is the heart of some stability condition in $\stab{\cat{C}}$. Therefore $P(0,1]$ is algebraic. Since $P(\varphi,\varphi+1]$ is algebraic there exists $\varepsilon>0$, depending on $\varphi$, for which $P(\varphi, \varphi+\varepsilon)=0$. It follows that $\sigma$ has discrete phase distribution. \end{proof} \begin{corollary} \label{finite type massless} Suppose $\stab{\cat{C}}$ has finite type. Then $\cat{N}$ is the massless subcategory of some $\sigma \in \lstab{\cat{C}}$ if and only if $\cat{N}$ is the triangulated closure of a finite subset of simple objects in the heart of some classical stability condition $\tau \in \stab{\cat{C}}$. \end{corollary} \begin{proof} Suppose $\cat{N}$ is generated by a finite subset $I$ of simple objects in the heart $Q(0,1]$ of some classical $\tau = (Q,W)\in \stab{\cat{C}}$. By deforming the charge we may assume that \[ W(s) = \begin{cases} -r & \iff s \in I, \\ -1 & \iff s\not \in I, \end{cases} \] for some $r\in \mathbb{R}_{>0}$. Clearly the charges converge as $r\to 0$. Moreover, since $Q(0,1]= Q(1)$ for all $r$ the slicings also converge as $r\to 0$. In the limit as $r\to 0$ we obtain a lax stability condition $\sigma$ with massless subcategory $\cat{N}$ and the same slicing $Q$. Therefore the only stable objects of $\sigma$ are the simple objects of $Q(0,1]$. Since there are finitely many of these, $\sigma$ satisfies the support property and therefore lies in $\lstab{\cat{C}}$. Now suppose $\sigma = (P,Z) \in \lstab{\cat{C}}$. By Lemma~\ref{massless simples} the massless subcategory $\cat{N}$ is the triangulated closure of the set of stable massless objects in $P(0,1]$. Since $P(0,1]$ is algebraic each stable massless object has a composition series whose factors are massless simple objects, each of which is of course also stable. Therefore $\cat{N}=\triang{}{S}$ is the triangulated closure of the set $S$ of massless simple objects in $P(0,1]$. The result follows because $P(0,1]$ is the heart of some classical stability condition in $\stab{\cat{C}}$ by Corollary~\ref{boundary algebraic}. \end{proof} \begin{remark} \label{non-admissible} Corollary~\ref{finite type massless} tells us that massless subcategories need not be admissible. An example is the derived-discrete algebra $A = \Lambda(2,1,0)$; see \cite{MR2041666} for notation and \cite[\S 8]{MR3178243} for a detailed description of $\cat{D}^b(A)$. One of the two simple modules $S$ is a $2$-spherical object. Therefore, its thick hull $\cat{N} \coloneqq \thick{}{S}$ occurs as a massless subcategory. On the other hand, the duality property $\mor{S}{-} = \mor{-}{S[2]}^*$ implies that an adjoint of the inclusion $\cat{N} \hookrightarrow \cat{D}^b(A)$ would lead to a splitting $\cat{D}^b(A) \cong \cat{N} \oplus \cat{M}$ but $\cat{D}^b(A)$ is indecomposable. \end{remark} \begin{proposition} \label{boundary support} If $\stab{\cat{C}}$ has finite type, then $\lstab{\cat{C}} = \overline{\stab{\cat{C}}}$. \end{proposition} \begin{proof} Suppose $\sigma = (P,Z) \in \overline{\stab{\cat{C}}}$ does not satisfy the support property. Then we can find a sequence $(c_n)$ of massive stable objects with $m_\sigma(c_n) / ||v(c_n)|| \to 0$. Shifting the objects if necessary we may assume they all have phase in $(0,1]$ and hence, by passing to a subsequence, that the phases $\varphi(c_n) \to \varphi$ as $n\to \infty$. Corollary~\ref{boundary algebraic} states that $\sigma$ has a discrete phase distribution so, again passing to a subsequence if required, we may assume that $\varphi(c_n)=\varphi$ for all $n\in \mathbb{N}$. Thus the $c_n$ are simple objects in $P(\varphi)$, and therefore also simple in $P(\varphi-1,\varphi]$. There are only finitely many iso-classes of such simple objects because $P(\varphi-1,\varphi]$ is algebraic by Corollary~\ref{boundary algebraic}. So, passing to a subsequence for a final time, we may assume that $(c_n)$ is a constant sequence. Since the $c_n$ are massive this contradicts the fact that $m_\sigma(c_n) / ||v(c_n)|| \to 0$. We conclude that no such sequence exists, i.e.\ that $\sigma$ satisfies support after all. \end{proof} \begin{corollary} \label{finite type support propagation} If $\stab{\cat{C}}$ has finite type, then support propagates from $\lnstab{\cat{N}}{\cat{C}}$. \end{corollary} \begin{proof} This follows immediately from the fact that $\lstab{\cat{C}}=\overline{\stab{\cat{C}}}$ and Theorem~\ref{all deformations remain in bdy} which show that deformations of an element of $\lnstab{\cat{N}}{\cat{C}}$ remain in the closure of $\stab{\cat{C}}$. \end{proof} \begin{corollary} \label{quotients finite type} Suppose $\stab{\cat{C}}$ has finite type and that $\sigma\in \lnstab{\cat{N}}{\cat{C}}$. Then $\mu_\cat{N}(\sigma)$ is in a finite type component of $\stab{\cat{C}/\cat{N}}$. \end{corollary} \begin{proof} By Corollary~\ref{finite type support propagation}, the entire component of $\mu_\cat{N}(\sigma)$ in $\stab{\cat{C}/\cat{N}}$ is in the image of $\lnstab{\cat{N}}{\cat{C}}$ under $\mu_\cat{N}$. The slicing of each lax stability condition $\sigma=(P,Z)$ in $\lnstab{\cat{N}}{\cat{C}}$ has discrete phase distribution by Corollary~\ref{boundary algebraic}. Since $P_{\cat{C}/\cat{N}}(\varphi)=0$ when $P(\varphi)=0$ the slicing $P_{\cat{C}/\cat{N}}$ of $\mu_\cat{N}(\sigma)$ also has discrete phase distribution. Thus the component of $\mu_\cat{N}(\sigma)$ has finite type by Lemma~\ref{finite type cpt}. \end{proof} \begin{remark} We have thus shown that the conditions for the structure results of \S\ref{topology of dstab} and \S\ref{space qstab} are satisfied. If $\stab{\cat{C}}$ is a finite type component then the space $\lstab{\cat{C}}$ of lax stability conditions is a union of components of $\stab{\cat{C}/\cat{N}} \times \lnstab{\cat{N}}{\cat{N}}$, where $\cat{N}$ is the triangulated closure of a finite set of simple objects in the heart of a stability condition in $\stab{\cat{C}}$. Each component of $\stab{\cat{C}/\cat{N}}$ which appears has finite type. The space $\qstab{\cat{C}}$ of quotient stability conditions is therefore stratified by these finite type components. \end{remark} \section{Closures of $G$-orbits} \label{Orbit closures} \noindent Recall that the the universal cover $G$ of $\mathrm{GL}_2^+(\mathbb{R})$ acts on stability spaces and our partial compactifications. In this section we describe the closures of $G$-orbits. The phase diagrams $\Phi_\sigma = \{ \varphi + \mathbb{Z} : P(\varphi) \neq 0 \} \subset \mathbb{R}/\mathbb{Z}$ of stability conditions $\sigma = (P,Z)$ in the same orbit are related by orientation-preserving diffeomorphisms of the circle $\mathbb{R}/\mathbb{Z}$. The structure of the closure of the orbit in both $\stab{\cat{C}}$ and in $\qstab{\cat{C}}^*$ can be described in terms of the phase diagram. Here $\qstab{\cat{C}}^*$ is the space obtained by removing the strata where all objects are massless from $\qstab{\cat{C}}$. This is more convenient to consider because the action of $\mathbb{C}$ on $\qstab{\cat{C}}^*$ is free and in fact we will describe the closure of $(\sigma\cdot G)/\mathbb{C}$ in $\eqstab{\cat{C}}$ and in $\pcQstab{\cat{C}}$. Fix a stability condition $\sigma=(P,Z)$. If there is only one point in the phase diagram $\Phi_\sigma$ then there is a non-trivial stabiliser and $(\sigma\cdot G)/\mathbb{C}$ is a point. In this case the orbit is closed in $\eqstab{\cat{C}}$ and also in $\pcQstab{\cat{C}}$. Henceforth, we assume that $\Phi_\sigma$ consists of at least two points. In particular the image of the charge is the whole of $\mathbb{C}$ so that the $G$-orbit through $\sigma$ is free and $(\sigma\cdot G)/\mathbb{C}$ is biholomorphic to the Poincar\'e disk $\mathbb{D}$ because $G \cong \mathbb{C} \times \mathbb{H}$. We show that the closures of the orbit in $\eqstab{\cat{C}}$ and in $\pcQstab{\cat{C}}$ are homeomorphic to subsets of the closed disk, with appropriate topologies. Our constructions require an explicit identification of the Poincar\'e disk and the strict upper half-plane; we choose $f \colon \mathbb{D} \to \mathbb{H}$ given by \[ f(w)= i\frac{1+w}{1-w}. \] Note that $f$ extends to a homeomorphism from the closure of the disk to $\mathbb{H}\cup\mathbb{R}\cup\{\infty\}$. For each $w\in \overline{\mathbb{D}}$ we choose a charge $Z_w = M_w \circ Z$ where $M_w \in \End{\mathbb{R}}{\mathbb{C}}$ satisfies \[ M_w(1)=1\quad \text{and} \quad M_w(i) = f(w) \qquad (w\neq 1) \] and $M_1(1)=0$, $M_1(i) =-1$. Note that $M_w(i) \in \mathbb{H}$ when $w\in \mathbb{D}$ so that there is a unique compatible slicing $P_w$ with $P_w(0,1]= P(0,1]$. Mapping $w$ to the $\mathbb{C}$-orbit of $\sigma_w = (P_w,Z_w)$ gives an explicit identification $\mathbb{D} \cong (\sigma\cdot G)/\mathbb{C}$. The reason for this particular choice is that \begin{align*} P(\varphi) \subset \ker Z_w & \iff M_w(e^{i\pi \varphi})=0\\ &\iff M_w(1) \cos(\pi\varphi) + f(w) \sin(\pi\varphi)=0\\ &\iff \left( w=1 \ \text{and}\ \varphi=0 \right)\ \text{or}\ \left( w=e^{2\pi i \varphi}\ \text{and}\ \varphi\neq 0 \right)\\ &\iff w=e^{2\pi i \varphi} \end{align*} so that points on $\partial \mathbb{D}$ correspond to charges for which semistable objects of a certain phase have vanishing mass in a natural way. For $w =e^{2\pi i \varphi}\in \partial \mathbb{D}$ and $c \in P(\varphi')$ the sine rule yields \begin{equation} \label{sine rule charge equation} Z_w(c) = \begin{cases} |Z(c)| \dfrac{\sin\pi(\varphi-\varphi')}{\sin\pi\varphi} & \text{ if } w \neq 1; \\ -|Z(c)| \sin \pi \varphi' & \text{ if } w = 1. \end{cases} \end{equation} When $P(\varphi)=0$ there is a unique choice of slicing $P_w$ for $w=e^{2\pi i \varphi}$ which is compatible with $Z_w$ and with $P(0,1] \subset P_w[0,1]$, namely \[ P_w(1)=P(\varphi,\varphi+1] = P[\varphi,\varphi+1) \] and all slices with phase in $(0,1)$ are zero. One can verify that $\sigma_w=(P_w,Z_w)$ is a pre-stability condition. \begin{lemma} \label{lem: non-degenerate support} Suppose $w=e^{2\pi i \varphi} \in \partial \mathbb{D}$ and $P(\varphi)=0$. Then $\sigma_w=(P_w,Z_w)$ is a stability condition if and only if $\varphi$ is not an accumulation point of the phase diagram $\Phi_\sigma$. \end{lemma} \begin{proof} Suppose $\varphi\neq 0$ so that $M_w(1)=1$. If $\varphi$ is an accumulation point of $\Phi_\sigma$ and $\varepsilon>0$ then using the first equation in \eqref{sine rule charge equation} shows that one can choose $\varphi' \in \Phi_\sigma$ sufficiently close to $\varphi$ that \[ |Z_w(c)| \leq \frac{\varepsilon}{||Z||} |Z(c)| \leq \varepsilon ||v(c)|| \] for any $c\in P(\varphi')$. Hence the support property fails since $c$ remains semistable for $\sigma_w$. Conversely, if $\varphi$ is not an accumulation point then there is $L>0$ such that \[ |Z_w(c)| \geq L |Z(c)| \geq KL ||v(c)|| \] for all $c\in P(\varphi')$ where $\varphi'\in (0,1]$, and $K$ is a support constant for $\sigma$. Therefore the same inequality holds for all $c\in P_w(1)=P(\varphi,\varphi+1]$ so that $\sigma_w$ satisfies the support property as claimed. The case $\varphi=0$ is similar but uses the second equation in \eqref{sine rule charge equation}. \end{proof} When $P(\varphi)\neq 0$ there is a one-parameter family of compatible slicings $P_w^\psi$ for $w = e^{2\pi i \varphi}$ compatible with $Z_w$ and with $P(0,1] \subset P_w[0,1]$. They differ by the choice of phase for the massless objects in $P(\varphi)$. Namely, for each $\psi\in [0,1]$ there is a unique such slicing $P_w^\psi$ with \[ P_w^\psi(1) \supset P(\varphi,\varphi+1) \quad \text{and}\quad P_w^\psi(\psi)\supset P(\varphi) \] and all other slices with phase in $(0,1]$ zero. One can verify that $\sigma^\psi_w=(P^\psi_w,Z_w)$ is a lax pre-stability condition. We now give criteria for when it is in the closure of the orbit, and when it satisfies the support property. \begin{lemma} \label{lem: degenerate closure} Suppose $w=e^{2\pi i \varphi} \in \partial \mathbb{D}$ and $P(\varphi)\neq 0$. The lax pre-stability condition $\sigma_w^\psi$ is in the closure of the orbit $\sigma\cdot G$ in $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ if and only if one of the following conditions holds \begin{enumerate} \item \label{degenerate closure 1} $\varphi$ is an isolated point of $\Phi_\sigma$ and $\psi\in[0,1]$; \item \label{degenerate closure 2}$\varphi$ is an accumulation point of $\Phi_\sigma$ such that $P(\varphi-\varepsilon,\varphi)=0$ for some $\varepsilon>0$ and $\psi=1$; \item \label{degenerate closure 3}$\varphi$ is an accumulation point of $\Phi_\sigma$ such that $P(\varphi,\varphi+\varepsilon)=0$ for some $\varepsilon>0$ and $\psi=0$. \end{enumerate} In particular, the slicing $P_w^\psi$ is locally-finite in each of the above cases. \end{lemma} \begin{proof} If there is a sequence in $\Phi_\sigma$ tending to $\varphi$ from below then the slicings of a sequence in $\sigma\cdot G$ can only converge to $P_w^\psi$ if $\psi=0$. Similarly, if there is a sequence in $\Phi_\sigma$ tending to $\varphi$ from above then the slicings of a sequence in $\sigma\cdot G$ can only converge to $P_w^\psi$ if $\psi=1$. It follows that if $\sigma_w^\psi$ is in the closure then we are in one of the three cases in the statement. In each of those cases one can construct a sequence $(w_n)$ in $\mathbb{D}$ converging to $w\in \partial \mathbb{D}$ and with $\sigma_{w_n} \to \sigma_w^\psi$ by controlling the limiting phase of objects in $P(\varphi)$ appropriately. \end{proof} \begin{lemma} \label{lem: degenerate support} Suppose $w=e^{2\pi i \varphi} \in \partial \mathbb{D}$ and $P(\varphi) \neq 0$ and moreover that the lax pre-stability condition $\sigma_w^\psi$ is in the closure of the orbit $\sigma\cdot G$. Then it satisfies the support property if, and only if, there is some $\varepsilon>0$ such that either \begin{enumerate} \item \label{accumulation above} $P(\varphi-\varepsilon,\varphi)=0$ and no simple object in $P[\varphi,\varphi+1)$ lies in $P(\varphi,\varphi+\varepsilon)$; or, \item \label{accumulation below} $P(\varphi,\varphi+\varepsilon)=0$ and no simple object in $P(\varphi,\varphi+1]$ lies in $P(\varphi+1-\varepsilon,\varphi+1)$. \end{enumerate} In particular, if $\varphi$ is isolated both conditions are satisfied and $\sigma_w^\psi$ satisfies the support property. \end{lemma} \begin{proof} Under the assumption $P(\varphi-\varepsilon,\varphi)=0$ the heart $P[\varphi,\varphi+1)$ is length so that it makes sense to speak of simple objects, and similarly for the second case. We only need to check the support property for massive stable objects. Each such is also $\sigma$ stable. If $\varphi$ is an isolated point of $\Phi_\sigma$ then any massive $\sigma_w^\psi$ stable object lies in a slice $P(\varphi')$ whose phase $\varphi'$ is bounded away from $\varphi$. Again the support property follows as in the proof of Lemma~\ref{lem: non-degenerate support}. If $\varphi$ is an accumulation point of $\Phi_\sigma$ with $P(\varphi-\varepsilon,\varphi)=0$ then $\psi=1$ by Lemma~\ref{lem: degenerate closure}. It is enough to consider massive stable objects in $P_w^\psi(1) = P[\varphi,\varphi+1)$. These are the simple objects of this heart which are not in $P(\varphi)$. Therefore each such lies in $P(\varphi')$ for some $\varphi'\in (\varphi,\varphi+1)$. The result then follows as in the proof of Lemma~\ref{lem: non-degenerate support}. The other case is similar. \end{proof} \begin{remark} \label{degenerate support remark} If $\cat{C}$ has an algebraic heart then any length heart is algebraic, i.e.\ has finitely many iso-classes of simple objects. In this situation the conditions simplify to $P(\varphi-\varepsilon,\varphi)=0$ or $P(\varphi,\varphi+\varepsilon)=0$, i.e.\ to $\varphi$ being isolated or at worst a one-sided accumulation point. \end{remark} \begin{definition} We define subsets of the closure of the Poincar\'e disk by \begin{align*} \mathbb{D}_\sigma &= \mathbb{D} \cup \{e^{2\pi i\varphi} : \varphi \not \in \overline{\Phi_\sigma}\}\\ \text{and}\ \mathbb{D}^Q_\sigma &= \mathbb{D}\cup\{ e^{2\pi i \varphi} : \varphi\ \text{satisfies either (\ref{accumulation above}) or (\ref{accumulation below}) of Lemma~\ref{lem: degenerate support} for some}\ \varepsilon >0 \}. \end{align*} Note that $\mathbb{D}_\sigma \subset \mathbb{D}^Q_\sigma$ because (\ref{accumulation above}) and (\ref{accumulation below}) are satisfied when $\varphi \not \in \overline{\Phi_\sigma}$. For each $w \in \mathbb{D}^Q_\sigma$ we have defined a quotient stability condition $\overline{\sigma}_w \in \qstab{\cat{C}}^*$, namely $\overline{\sigma}_w=[\sigma_w]$ for $w\in \mathbb{D}_\sigma$ and $\overline{\sigma}_w=[\sigma^\psi_w]$ for $w\in \mathbb{D}^Q_\sigma-\mathbb{D}_\sigma$. \end{definition} \begin{proposition} \label{orbit closure bijection} Let $\overline{\sigma\cdot G}/\mathbb{C}$ denote the closure of the orbit of $\sigma$ in $\pcQstab{\cat{C}}$. Then \[ \mathbb{D}^Q_\sigma \to \overline{\sigma\cdot G}/\mathbb{C} \quad \text{given by} \quad w \mapsto \overline{\sigma}_w\cdot \mathbb{C} \] is a bijection, and restricts to a bijection between $\mathbb{D}_\sigma$ and the closure of the orbit of $\sigma$ in $\eqstab{\cat{C}}$. \end{proposition} \begin{proof} By construction, and Lemma~\ref{lem: degenerate closure}, $\overline{\sigma}_w$ is in $\overline{\sigma\cdot G}$ so the map is well-defined. Since the image of the charge $Z$ is the whole of $\mathbb{C}$ we can find $\lambda,\mu\in \Lambda\otimes \mathbb{R}$ with $Z(\lambda)=i$ and $Z(\mu)=1$. Then the assignment \[ (Q,W) \mapsto f^{-1}\left( \frac{W(\lambda)}{W(\mu)} \right) \] descends to a map $\overline{\sigma\cdot G}/\mathbb{C} \to \overline{\mathbb{D}}$ taking $\overline{\sigma}_w$ to $w$. We claim this is the inverse. Certainly, if $w$ is the image of $(Q,W)$ then $W$ is in the same $\mathbb{C}$ orbit as $Z_w$. It follows that the image of this map is precisely $\mathbb{D}^Q_\sigma$, because we have shown that there are no quotient stability conditions in $\overline{\sigma\cdot G}$ with charge $Z_w$ for $w\not \in \mathbb{D}^Q_\sigma$. Moreover, there is a unique quotient stability condition in $\overline{\sigma\cdot G}$ with charge $Z_w$, which establishes the claim. \end{proof} \begin{corollary} \label{orbit closed} The orbit $\sigma\cdot G$ is closed in $\stab{\cat{C}}$ and in $\qstab{\cat{C}}^*$ if and only if the phase diagram $\Phi_\sigma$ is dense. \end{corollary} \begin{proof} This follows immediately from the above result since $\mathbb{D}_\sigma^Q = \mathbb{D}$ when $\Phi_\sigma$ is dense. Alternatively, note that the Bridgeland metric induces the standard hyperbolic metric on the quotient $(\sigma\cdot G)/\mathbb{C} \cong \mathbb{D}$ up to a factor \cite[Proposition 4.1]{MR3007660}. Since the hyperbolic metric is complete, the orbit is closed. \end{proof} Finally we describe the topology on $\mathbb{D}^Q_\sigma$ for which the bijection in Proposition~\ref{orbit closure bijection} is a homeomorphism. This is the topology in which a sequence $(w_n)$ converges to $w$ if and only if the charges $Z_{w_n}$ converge in $\mor{\Lambda}{\mathbb{C}}/\mathbb{R}_{>0}$ and the slicings $P_{w_n}$ converge in $\slice{\cat{C}}$. Since $d(P_w,P_{w'})\leq 1$ for $w,w'\in \mathbb{D}^Q_\sigma$ the convergence of charges implies uniform convergence of the phases of massive objects. This occurs whenever $w_n\to w$ in the subspace topology from $\mathbb{C}$. To ensure that the phase of the massless objects in $P(\varphi)$ converges we have to refine this topology in a neighbourhood of each $w\in \mathbb{D}_\sigma^Q-\mathbb{D}_\sigma$ so that convergence also implies that \[ \lim_{n\to \infty} \frac{w_n-w}{|w_n-w|} \] is a well-defined unit tangent vector in $T_w\mathbb{C}$. In other words, the required topology arises from the real blow-up $\beta \colon \widetilde{\mathbb{C}} \to \mathbb{C}$ at the points in $\mathbb{D}^Q_\sigma - \mathbb{D}_\sigma$. More precisely, let $\widetilde{\mathbb{D}}^Q_\sigma$ be the subspace of $\beta^{-1}(\mathbb{D}^Q_\sigma)$ consisting of $\beta^{-1}(\mathbb{D}_\sigma)$ and those points $(w=e^{2\pi i \varphi},v)$ in the exceptional divisors where the unit tangent vector $v\in T_w\mathbb{C}$ is such that $\varphi$ and \[ \psi= \frac{1}{\pi} \arg d_wf(v) \] satisfy one of the conditions (\ref{degenerate closure 1}), (\ref{degenerate closure 2}) or (\ref{degenerate closure 3}) in Lemma~\ref{lem: degenerate closure}. The above discussion leads to the following description of orbit closures. \begin{corollary} \label{orbit closure} Equip $\mathbb{D}^Q_\sigma$ with the quotient topology from $\beta \colon \widetilde{\mathbb{D}}^Q_\sigma \to \mathbb{D}^Q_\sigma$. Then \[ \mathbb{D}^Q_\sigma \to \overline{\sigma\cdot G}/\mathbb{C} \colon w \mapsto \overline{\sigma}_w\cdot \mathbb{C} \] is a homeomorphism, and restricts to a homeomorphism between $\mathbb{D}_\sigma$ and the closure of the orbit of $\sigma$ in $\eqstab{\cat{C}}$. \end{corollary} \begin{remark} In fact, although we have not filled in all the details, the subspace $\widetilde{\mathbb{D}}^Q_\sigma$ is homeomorphic to the closure of $(\sigma\cdot G)/\mathbb{C}$ in $\lstab{\cat{C}}^*\!/\mathbb{C}$. \end{remark} \section{Two-dimensional stability spaces} \label{2d case} \noindent We illustrate our results in the simplest non-trivial case in which the stability space is a $2$-dimensional complex manifold. In this context there is a very close relationship between the boundary strata we add and the wall-and-chamber structure of the stability space. \subsection{Walls and chambers} Suppose that $\Lambda$ is a rank two lattice, and moreover that if $\cat{C}$ contains a length heart then that heart has two iso-classes of simple objects and $\Lambda=K(\cat{C}) \cong \mathbb{Z}^2$. This assumption is reasonable because otherwise the stability space of $\cat{C}$ is naturally higher-dimensional. Since $\rk(\Lambda)=2$ the quotient $\eqstab{\cat{C}}$ is a non-compact Riemann surface and \[ \stab{\cat{C}} \cong \eqstab{\cat{C}} \times \mathbb{C} \] as a complex manifold because all holomorphic bundles on a non-compact Riemann surface are holomorphically trivial \cite[Theorem 30.4]{MR1185074}. Therefore it suffices to describe $\eqstab{\cat{C}}$. The action of $\mathbb{C}$ preserves the set of semistable objects, so the wall-and-chamber structure of the stability space descends to a partition of $\eqstab{\cat{C}}$ into open chambers and real codimension one walls between them. The charge map descends to a holomorphic map $\eqstab{\cat{C}} \to \mathbb{P}( \mor{\Lambda}{\mathbb{C}} ) \cong \mathbb{C}\mathbb{P}^1$ which by abuse of notation we also denote $\mathcal{Z}$. The equatorial copy of $\mathbb{R}\mathbb{P}^1$ consists of the charges with rank one image. The action of the universal cover $G$ of $\mathrm{GL}_2^+(\mathbb{R})$ also preserves the sets of semistable objects, so each chamber and each wall is a union of orbits. The image of a free orbit in $\eqstab{\cat{C}}$ is a copy of $G/\mathbb{C}\cong \mathbb{H}$ biholomorphic to its image under the charge map. This image is either the Southern or Northern hemisphere in $\mathbb{C}\mathbb{P}^1$. We refer to these free orbits as \defn{cells}. The image of a non-free orbit in $\eqstab{\cat{C}}$ is a point. The heart of a stability condition in such an orbit is length, and so by our assumption has two isoclasses of simple objects, say $s$ and $t$. There is a one-parameter family consisting of images of orbits through stability conditions for which $s$ and $t$ are semistable of the same phase and where the ratio $m(s)/m(t)$ varies in $\mathbb{R}_{>0}$. This describes a real analytic curve in $\eqstab{\cat{C}}$ which we refer to as a \defn{cell-wall}. Each cell-wall is isomorphic to its image in $\mathbb{P}\mor{\Lambda}{\cat{C}}$ which is an arc in the equator from the point $Z(s)=0$ to $Z(t)=0$. The following lemma is immediate. \begin{lemma} Suppose $\cat{C}$ admits an algebraic heart with two simple objects. Then there is a bijection between the set of cell-walls and the set of algebraic hearts (up to shift). \end{lemma} If there are non-split extensions between the simple objects $s$ and $t$ then a cell-wall is a genuine wall in the stability space along which these extensions are strictly semistable. All walls are of this form; in particular no two walls intersect. If there are no non-split extensions, for instance if the heart is semisimple, then the cell-wall lies in the same chamber as the two neighbouring cells. \begin{lemma} Each chamber of $\eqstab{\cat{C}}$ is a linear chain of cells, separated by cell-walls. If it is a doubly-infinite chain then it is biholomorphic to $\mathbb{C}$, otherwise it is biholomorphic to $\mathbb{H}$. \end{lemma} \begin{proof} Let $C$ be a cell. If it has no cell-walls in its closure then it is a chamber and we are done. Let $W$ be a cell-wall in the closure, and let $s$ and $t$ be simple objects in a corresponding algebraic heart. Their phases agree on $W$ and we may assume $\varphi(t) - 1 < \varphi(s)<\varphi(t)$ in $C$. Suppose $W'$ is another cell-wall in the closure of $C$ which is not an actual wall. Let $s'$ and $t'$ be simple objects in a corresponding algebraic heart. Since $s$ and $t$ are semistable on $W'$, some shifts $s[m]$ and $t[n]$ lie in this heart. Indeed, since we assume $\varphi(s)<\varphi(t)$ in $C$, we may apply a shift so that the heart contains $s[1]$ and $t$. Since these are indecomposable and there are no non-split extensions between $s'$ and $t'$ we deduce, after swapping $s'$ and $t'$ if necessary, that $s[1]$ and $t$ are respectively self-extensions of $s'$ and $t'$. However, the classes of $s[1]$ and $t$ are primitive in $\Lambda=K(\cat{C})$ so in fact $s'=s[1]$ and $t'=t$. It follows that the closure of $C$ in $\eqstab{\cat{C}}$ is precisely $W\cup C \cup W'$. Arguing inductively we conclude that the chamber is either a linear chain of cells as claimed or a cycle. The possibility that it is a cycle is easily excluded since the phase difference between $s$ and $t$ is well-defined and monotonic as we move along a chain of cells. For the final part note that the chamber is an open subset of the universal cover $\mathbb{C}$ of $\mathbb{P}\mor{\Lambda}{\cat{C}} - \{ Z(s)=0,Z(t)=0\}$. If the chain of cells is doubly-infinite then it is the entirety of $\mathbb{C}$. Otherwise it is a proper simply-connected open subset of $\mathbb{C}$ and is therefore biholomorphic to $\mathbb{H}$ by the Riemann Mapping Theorem. \end{proof} \subsection{The Speiser graph} The topology and geometry of $\eqstab{\cat{C}}$ is encoded combinatorially in what we call the \defn{Speiser graph}. This is the dual graph $\Sp{\cat{C}}$ to the cell and cell-wall decomposition, i.e.\ it has a vertex for each cell and an edge between two vertices when they have a common cell-wall in their closures. (More properly the Speiser graph is defined when $\eqstab{\cat{C}} \to \mathbb{P}\mor{\Lambda}{\cat{C}}$ has a finite set $S$ of singular values. It is the preimage in $\eqstab{\cat{C}}$ of the dual graph in $\mathbb{P}\mor{\Lambda}{\cat{C}}-S$ to a Jordan curve passing through the singular values in $S$ in cyclic order, see for example \cite[\S 2]{MR752802}. However, this definition is too restrictive for our setting where there may be infinitely many singular values.) Since the cells and cell-walls are contractible the Speiser graph encodes the homotopy type of $\eqstab{\cat{C}}$. Indeed we can embed the Speiser graph in $\eqstab{\cat{C}}$ so that each vertex is in the corresponding cell and each edge is a smooth curve crossing the corresponding cell-wall once transversely (and not crossing any other cell-wall). Then $\eqstab{\cat{C}}$ deformation retracts onto the embedded Speiser graph. \begin{conjecture} \label{Speiser conjecture 1} Each component of the Speiser graph $\Sp{\cat{C}}$ is a tree and therefore each component of $\eqstab{\cat{C}}$ is contractible. \end{conjecture} When the Speiser graph is a tree the Uniformisation Theorem, see for example \cite[Theorem 27.9]{MR1185074}, tells us that $\eqstab{\cat{C}}$ is biholomorphic to either $\mathbb{C}$ (parabolic type) or to the Poincar\'e disk $\mathbb{D}$ (hyperbolic type). Either case may occur, for instance if $\cat{C}$ is the bounded derived category of representations of the quiver with two vertices and no arrows then $\eqstab{\cat{C}} \cong \mathbb{C}$ whereas if $\cat{C}=\cat{D}^b(X)$ is the bounded derived category of coherent sheaves on a smooth complex projective curve $X$ of strictly positive genus then $\eqstab{\cat{C}} \cong \mathbb{D}$ \cite{MR2373143, MR2335991}. The type is parabolic if the Brownian motion on $\eqstab{\cat{C}}$ is recurrent and hyperbolic if it is transient \cite{MR32123,MR0056100}. The idea behind the following conjecture is that the Brownian motion can be combinatorially modelled by the random walk on the Speiser graph with suitable transition probabilities on each edge (provided at least that we are not in the trivial case in which the Speiser graph has a single vertex and no edges). See \cite{MR752802} for a nice discussion. \begin{conjecture} \label{Speiser conjecture 2} Assume the Speiser graph contains at least one edge. Equip the charge space $\mathbb{P}\mor{\Lambda}{\mathbb{C}}\cong \mathbb{C}\mathbb{P}^1$ with the constant curvature Riemannian metric in which the equator (consisting of charges with rank one image) has length one. Assign each edge in the Speiser graph a transition probability given by the length of the image in charge space of the corresponding cell-wall. Then $\eqstab{\cat{C}}$ is parabolic if the resulting random walk is recurrent and hyperbolic if it is transient. \end{conjecture} The following criterion, proved by comparing with the random walk on the standard $2$-dimensional lattice, is useful for establishing recurrence and shows that it does not depend delicately on the transition probabilities. \begin{theorem}[{\cite[\S 2.4]{MR920811}}] \label{recurrence criterion} Suppose $\Gamma$ is an infinite connected graph, with a uniform upper bound on the valency of its vertices. Assign transition probabilities in $[\varepsilon, 1-\varepsilon]$, where $\varepsilon>0$, to each edge. Then the resulting random walk is recurrent if the vertices of $\Gamma$ can be embedded in $\mathbb{R}^2$ with a uniform lower bound on the distance between any two vertices, and a uniform upper bound on the length of each edge. \end{theorem} \subsection{The space of quotient stability conditions} We remove the boundary strata of $\lstab{\cat{C}}$ where all objects are massless, and denote the resulting space by $\lstab{\cat{C}}^*$. Therefore $\mathrm{rk}(\Lambda_\cat{N})= 1$ when the massless subcategory $\cat{N}\neq 0$. In this case $\lnstab{\cat{N}}{\cat{C}}$ is a union of components each homeomorphic to $\mathbb{C} \times \mathbb{R}$ by Theorem~\ref{codim 1 support propagation}. The action of $\mathbb{C}$ on $\lstab{\cat{C}}^*$ is free and the quotient is a union of the open subset $\eqstab{\cat{C}}$ and a copy of $\mathbb{R}$ for each boundary component in $\lstab{\cat{C}}^*$. Passing to $\qstab{\cat{C}}^*$ replaces each of these copies of $\mathbb{R}$ by a point. Recall, we abuse notation by using $\mathcal{Z}$ to denote the map $\pcLstab{\cat{C}} \to \mathbb{P}( \mor{\Lambda}{\mathbb{C}} ) \cong \mathbb{C}\mathbb{P}^1$ induced from the charge map on the space of lax stability conditions. This maps points of $\qnstab{\cat{N}}{\cat{C}}\!/\mathbb{C}$ where $\cat{N}$ is massless to the point $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. By Corollary~\ref{local structure 2} such a point has a punctured neighbourhood in $\pcQstab{\cat{C}}$ isomorphic as a complex manifold to the universal cover of a punctured disk centred at $\mor{\Lambda/\Lambda_\cat{N}}{\mathbb{C}}$. The partial compactification $\pcQstab{\cat{C}}$ is obtained locally by adding a point to the universal cover over the puncture. Such a point is called a \defn{logarithmic singularity} of $\mathcal{Z}$. It follows that $\pcQstab{\cat{C}}$ embeds in the \defn{Mazurkiewicz completion} which is the union of $\eqstab{\cat{C}}$ with all logarithmic singularities of $\mathcal{Z}$, see for example \cite{eremenko2021singularities}. We do not claim that this embedding is a homeomorphism, i.e.\ that every logarithmic singularity occurs as a boundary point in $\pcQstab{\cat{C}}$. The space $\pcLstab{\cat{C}}$ is recovered by performing a real blow-up at each of the boundary points, resolving them to copies of $\mathbb{R}$. Note that $\pcLstab{\cat{C}}$ is \emph{not} a Riemann surface with boundary because each boundary stratum has a holomorphic function which vanishes along it, namely the charge of (any) object which becomes massless on that stratum. The next result classifies the stable massless objects. \begin{proposition} \label{prop: massless stables = algebraic simples} An object is massless and stable at some point in $\qstab{\cat{C}}^*$ precisely if it is simple in some algebraic heart of $\cat{C}$. \end{proposition} \begin{proof} Suppose $s$ is simple in an algebraic heart $\cat{H}$. Let $t$ be the other simple object. Define stability conditions $(P,Z_n)$ for $n\in \mathbb{N}$ with slicing $P(1)=P(0,1] =\cat{H}$ and charge $Z_n(s)=-1/n$ and $Z_n(t)=-1$. The limit $\sigma=\lim_{n\to \infty}(P,Z_n)$ is a lax stability condition in $\lstab{\cat{C}}^*$ for which $s$ is massless and stable. (By construction the limit is in the closure of $\stab{\cat{C}}$ with locally-finite slicing $P$, and the support property follows immediately because, up to shift, $t$ is the only massive stable object.) Now suppose $s$ is a massless stable object at some point in $\qstab{\cat{C}}^*$. Since $\mathrm{rk}(\Lambda)=2$ each free $G$-orbit in $\stab{\cat{C}}$ is open, and $\qstab{\cat{C}}^*$ is the union of their closures. Hence $s$ is massless and stable at some point in the closure of a free orbit $\sigma\cdot G$ where $\sigma=(P,Z)\in \stab{\cat{C}}$. Then by Remark~\ref{degenerate support remark} and Proposition~\ref{orbit closure bijection} the object $s$ is stable in some slice $P(\varphi)$ such that $P(\varphi-\varepsilon, \varphi)=0$ or $P(\varphi,\varphi+\varepsilon)=0$ for sufficiently small $\varepsilon>0$ (or both). In the first case $P[\varphi,\varphi+1)$ is a length heart in which $s$ is simple and in the second $P(\varphi-1,\varphi]$. \end{proof} \begin{corollary} \label{rank 2 boundary points} The boundary points of $\pcQstab{\cat{C}}$ are in bijection with iso-classes of massless stable objects up to shift. \end{corollary} \begin{proof} We must show that there is a unique boundary point of $\pcQstab{\cat{C}}$ at which a given stable object is massless. Suppose $\sigma \in \lnstab{\cat{N}}{\cat{C}}$ where $\cat{N} = \triang{\cat{C}}{s}$ is generated by the stable massless object $s$. By Proposition~\ref{prop: massless stables = algebraic simples}, $s$ is simple in some algebraic heart and our standing assumption $K(\cat{C})\cong \mathbb{Z}^2$ implies $K(\cat{C}/\cat{N})\cong \mathbb{Z}$ and $\stab{\cat{C}/\cat{N}}\cong \mathbb{C}$. Thus the induced stability condition $\sigma_{\cat{C}/\cat{N}}=(P_{\cat{C}/\cat{N}},Z_{\cat{C}/\cat{N}}=Z)$ on the quotient is fixed up to the action of $\mathbb{C}$. By Proposition~\ref{uniqueness of compatibility} the slicing $P$ is uniquely determined by $P_{\cat{C}/\cat{N}}$ and $P_\cat{N}$, i.e.\ by the slicing of the quotient and the phase of $s$. However, by Theorem~\ref{codim 1 support propagation} the component of $\sigma$ in $\lnstab{\cat{N}}{\cat{C}}$ is isomorphic to $\stab{\cat{C}/\cat{N}}\times \mathbb{R} \cong \mathbb{C}\times \mathbb{R}$ so that all possible such $\sigma$ occur in the same component of $\lnstab{\cat{N}}{\cat{C}}$. Hence they all determine the same point of $\pcQstab{\cat{C}}$ as claimed. \end{proof} Recall that cell-walls in $\eqstab{\cat{C}}$ correspond to algebraic hearts (up to shift) and that the masses of the two simple objects vanish at the respective ends of the cell-wall. Therefore the cell-wall starts and ends at the boundary points where these simple objects become massless. \subsection{The exchange graph} The \defn{exchange graph} $\EG{\cat{C}}$ has one vertex for each algebraic heart of a stability condition in $\stab{\cat{C}}$ and an edge whenever two hearts are related by a simple HRS tilt; see \cite{MR1327209,MR2739061}. Each vertex of $\EG{\cat{C}}$ is $4$-valent because we can tilt left or right at each of the two simple objects of the corresponding heart. The shift acts freely on $\EG{\cat{C}}$ and we denote the quotient, the \defn{projective exchange graph}, by $\EG{\cat{C}}/\mathbb{Z}$. This quotient has one vertex for each algebraic heart up to shift, i.e.\ one vertex for each cell-wall. If there is an edge between two vertices then they share a common end point in the boundary of $\pcQstab{\cat{C}}$. The massless stable object at this boundary point is (up to shift) the common simple object of the hearts corresponding to the two vertices. The exchange graph can be embedded into $\pcQstab{\cat{C}}$ by placing a vertex on each cell-wall and embedding edges as smooth curves in the unique cell containing the two cell-walls corresponding to its vertices in its closure. \subsection{Dense phase case} \label{dense phase case} Suppose that $\stab{\cat{C}}$ contains a stability condition $\sigma$ whose phase diagram $\Phi_\sigma$ is dense. Then the $G$-orbit of $\sigma$ is free and by Corollary~\ref{orbit closed} closed. Since the orbit is also open it is an entire connected component, consisting of a single chamber with no walls. This is the case in which the Speiser graph is trivial. Every stability condition in this component has dense phases and therefore no stability condition in this component has an algebraic heart. Our theory adds no boundary points to this component. This situation occurs for the space $\stab{X}$ of numerical stability conditions on a smooth complex projective curve $X$ of genus $g>0$, see \cite{MR2373143,MR2335991} respectively for the elliptic curve and higher genus cases. It also occurs for the space $\stab{Q}$ of stability conditions on the bounded derived category of finite-dimensional representations of a $2$-vertex quiver $Q$ with an oriented cycle, see \cite[Remark 3.33]{MR3289326} for the existence of a dense-phase stability condition. \subsection{Non-dense phase case} \label{non-dense phase case} Now suppose that there is at least one $\sigma$ in $\stab{\cat{C}}$ with non-dense phases. By the above, every stability condition in the component of $\sigma$ has non-dense phases, and therefore lies in the $\mathbb{C}$ orbit of a stability condition with length heart. In this situation we assume $\Lambda=K(\cat{C})\cong\mathbb{Z}^2$. Using the description of the closure of the orbit $(\sigma\cdot G)/\mathbb{C}$ in Proposition~\ref{orbit closure bijection} and Corollary~\ref{orbit closure}, we can describe the component of $\sigma$ in $\pcQstab{\cat{C}}$. First we determine the chamber containing $\sigma$ and its walls. The latter correspond to non-trivial algebraic hearts whose simple objects are stable in the chamber. Then we find the stable objects in the adjacent chambers and repeat. In the examples we consider there are only finitely many chambers up to the action of $\aaut{\Lambda}{\cat{C}}$ so this is an effective strategy. It is easy to find the cell decompositions of chambers (since each chamber is a linear chain of cells) and thence construct the Speiser graph. In our examples this is always a tree, so $\stab{\cat{C}}$ is contractible. The type of $\eqstab{\cat{C}}$, either parabolic or hyperbolic, is known from previous results and we confirm Conjecture~\ref{Speiser conjecture 2} in these cases by showing that the random walk on the Speiser graph is respectively recurrent or transient. In the former case $\aaut{\Lambda}{\cat{C}}$ acts by rotations and translations of $\mathbb{C}$, and in the latter by rotations, translations and ideal rotations of $\mathbb{D}$. \subsection{The $A_2$ quiver} \label{a2} Let $Q$ be the $A_2$ quiver with two vertices and one arrow. In this section we consider several stability spaces associated with algebras associated with the $A_2$ quiver, namely the classical $A_2$ path algebra, and the the associated $2$-Calabi--Yau Ginzburg algebra. \subsubsection{Classic $A_2$} First we consider the bounded derived category $\cat{D}^b(A_2)$ of finite-dimensional representations of the classic $A_2$. Its stability space $\stab{A_2}\cong \mathbb{C}^2$ was first described by King \cite{elephant}, see \cite{MR4064773} for further references. The standard heart of $\cat{D}^b(A_2)$ has two exceptional simple objects $s$ and $t$ with one non-split extension $0\to s\to e\to t \to 0$ between them. It is easy to construct a stability condition in which $s$, $e$ and $t$ are the only stable objects up to shift. Its phase diagram has three isolated phases. By Corollary~\ref{orbit closure} the cell containing this stability condition has three cell-walls, and three boundary points where respectively $s$, $e$ and $t$ become massless. The object $e$ destabilises as we cross the cell-wall along which $s$ and $t$ have the same phase, and we enter a chamber in which only $s$ and $t$ are stable. Since $e$ is the unique indecomposable extension between any shifts of $s$ and $t$, this chamber is a chain of cells indexed by $\mathbb{N}$. Similar considerations apply to the other two walls of the initial cell. Thus the stability space has four chambers, one in which $s$, $e$ and $t$ are all stable and three in which pairs of them are stable. The Speiser graph has one central vertex with three infinite linear graphs attached. The random walk on this is recurrent by Theorem~\ref{recurrence criterion} in agreement with the fact that $\eqstab{\cat{C}} \cong \mathbb{C}$ is parabolic. The category $\cat{D}^b(A_2)$ is fractional Calabi--Yau; the Serre functor $S$ satisfies $S^3=[1]$. This acts by rotation on $\pcQstab{\cat{C}}$ preserving the central chamber and cyclically permuting the other three chambers, and also the three boundary points. See Figure~\ref{a2 figure} for an illustration. \subsubsection{$2$-Calabi--Yau $A_2$} Now consider the $2$-Calabi--Yau category $\cat{D}^b(\Gamma_{\!2} A_2)$, where $\Gamma_{\!2} A_2$ is the Ginzburg dg algebra of the $A_2$ quiver; see \cite[\S 7.2]{MR3050703}, for example, for details of the construction. The stability space $\eqstab{\Gamma_{\!2} A_2} \cong \mathbb{D}$ is the universal cover of the thrice punctured Riemann sphere and was first described in \cite{MR2230573}. See \cite{MR4064773} for a detailed discussion and further references, and also \cite{MR3406522,MR3858773} for more general discussions of the stability spaces of the Ginzburg dg-algebras associated to Dynkin quivers. The standard heart of $\cat{D}^b(\Gamma_{\!2} A_2)$ has two $2$-spherical simple objects $s$ and $t$, with one non-split extension $0\to s \to e\to t \to 0$ and $0\to t\to f \to s \to 0$ in each direction. Each $2$-spherical object in $\cat{D}^b(\Gamma_{\!2} A_2)$ generates a twist automorphism. For example, applying the twist $\textsc{tw}_{s}$ about $s$ to the triangle $s\to e\to t \to s[1]$ yields the triangle $s[-1] \to t \to f \to s$, and then applying $\textsc{tw}_{t}$ yields the rotation $e[-1] \to t[-1] \to s \to e$ of the original triangle. In particular $e$ and $f$ are also $2$-spherical. The subgroup of $\aut{\cat{D}^b(\Gamma_{\!2} A_2)}$ generated by $\textsc{tw}_{s}$ and $\textsc{tw}_{t}$ is isomorphic to the Artin--Tits braid group $B_3$ of the $A_2$ quiver, i.e.\ the braid group on three strands. The centre $Z(B_3)$ is generated by a single automorphism which acts as the Serre functor $S = [2]$. Let $\mathcal{S}$ be the set of equivalence classes of spherical objects in $\cat{D}^b(\Gamma_{\!2} A_2)$ up to isomorphism and shift. We abuse notation by using the same notation for spherical objects and their classes in $\mathcal{S}$; this is harmless since the twists $\textsc{tw}_{s}=\textsc{tw}_{s[1]}$ agree. The quotient $B_3/Z(B_3)\cong \mathrm{PSL}_2(\mathbb{Z})$ acts on $\mathcal{S}$ and the stabiliser of $s$ is the infinite cyclic subgroup generated by $\textsc{tw}_{s}$. From the above examples $t$ is in the orbit of $s$ (indeed the action on $\mathcal{S}$ is transitive although we do not need this). Now consider the stability space $\stab{\Gamma_{\!2} A_2}$. As in the $\cat{D}^b(A_2)$ case one can easily construct a stability condition in which, up to shift, the stable objects are the two simple objects $s$ and $t$ of the standard heart and one, $e$ say, of the two extensions between them. The phase diagram has three isolated phases so the corresponding cell in $\eqstab{\Gamma_{\!2} A_2}$ has three cell-walls. As before $e$ destabilises as we cross the wall where $s$ and $t$ have the same phase, but now the other extension $f$ becomes stable on the far side of the wall. Thus we enter the chamber obtained by applying $\textsc{tw}_{s}$ to the initial one. Similar considerations apply to the other walls of the initial chamber. Therefore $\mathrm{PSL}_2(\mathbb{Z})$ acts transitively on the chambers in $\eqstab{\Gamma_{\!2} A_2}$, each of which is a single cell bounded by three walls. There are three stable $2$-spherical objects in each chamber whose respective masses vanish at the three boundary points of the chamber. The action of $\mathrm{PSL}_2(\mathbb{Z})$ on chambers is free because no pair of distinct spherical objects, {\it a fortiori} no triple, is fixed. The action on walls is also free and it quickly follows from the examples of twist actions that it is transitive. In conformity with Corollary~\ref{finite type massless} and Corollary~\ref{rank 2 boundary points}, the only massless stable objects are the simple objects of the hearts of stability conditions (all of which are algebraic; see e.g.\ \cite{MR3858773}) and there is one boundary point in $\pcQstab{\Gamma_{\!2} A_2}$ for each (up to shift and isomorphism). The Speiser graph is the Cayley graph of $\mathrm{PSL}_2(\mathbb{Z})$ with respect to the generating set consisting of the images of $\textsc{tw}_{s}$, $\textsc{tw}_{e}$ and $\textsc{tw}_{t}$. It is an infinite trivalent tree as expected from Conjecture~\ref{Speiser conjecture 1} and the random walk on it is transient as expected from Conjecture~\ref{Speiser conjecture 2}. The twist $\textsc{tw}_{s}$ acts by a hyperbolic isometry fixing the boundary point at which $s$ is massless. Therefore, $\textsc{tw}_{s}$ acts either by an ideal rotation about that point or a translation, since, locally at the fixed point the action universally covers the action on $\mathbb{P}\mor{\Lambda}{\mathbb{C}}$. This is given by the matrix \[ \begin{pmatrix} -1 & 0 \\ 1 & 1 \end{pmatrix} \] with respect to the basis $\{ [s], [t]\}$ of $\Lambda=K(\cat{D}^b(\Gamma_{\!2} A_2))$. Since the eigenvalues are $\pm 1$ the twist acts by an ideal rotation. Up to isometry, the three boundary points where $s$, $e$ and $t$ are massless can be chosen arbitrarily on $\partial \mathbb{D}$ and this fixes the remaining boundary points of $\pcQstab{{\Gamma_{\!2} A_2}}$ uniquely. They form a dense subset of $\partial \mathbb{D}$. See Figure~\ref{a2 figure} for an illustration. \begin{figure} \caption{Illustrations of $\pcQstab{\cat{C} \label{a2 figure} \label{p1 figure} \label{lambda figure} \end{figure} \subsection{A discrete derived category} \label{discrete} Let $Q = \Lambda_{2,1,0}$ be the bound quiver with two vertices, one arrow in each direction, and the zero relation given by the composite of these arrows. The (principal component of the) stability space $\stab{\Lambda_{2,1,0}}\cong \mathbb{C}^2$ was first described in \cite{MR2739061}, see also \cite{MR3483114,MR3858773} for proofs that the stability space is connected and generalisations to other discrete derived categories. Let $s$ be the simple representation at the vertex with no relation, and $t=t_0$ the other simple representation. The object $s$ is $2$-spherical and $t_0$ is exceptional. Since $\textsc{tw}_{s}(s)=s[-1]$ the twist $\textsc{tw}_{s}$ generates an infinite cyclic subgroup of automorphisms. Set $t_n =\textsc{tw}_{s}^n(t_0)$. There are unique non-split extensions $0\to s \to t_{-1} \to t_0 \to 0$ and $0\to t_0 \to t_1 \to s \to 0$. In particular there is a stability condition in which $s$, $t_{-1}$ and $t_0$ are the only stable objects up to shift. This lives in a chamber with three walls, and three boundary points at which these objects are respectively massless. Crossing the wall where $t_{-1}$ destabilises we enter a chamber in which $t_1$ is stable. As in the previous example this chamber is the image of the initial one under the action of the twist $\textsc{tw}_{s}$, and similarly crossing the wall where $t_0$ destabilises we enter the image of the initial chamber under $\textsc{tw}_{s}^{-1}$. However, if we cross the wall where the spherical object $s$ destabilises then we enter a chamber in which only $t_0$ and $t_1$ are stable. Unlike the previous chambers which consist of a single cell, this is the union of a sequence of cells, and cell-walls upon which the phases of $t_0$ and $t_1[n]$ agree for $n\in \mathbb{N}$. In summary, there is one free orbit of chambers with three stable objects (one spherical and two exceptional) and one free orbit of chambers with two stable objects (both exceptional) under the action generated by $\textsc{tw}_{s}$. As expected, there is one boundary point in $\pcQstab{\Lambda_{2,1,0}}$ at which each of $s$ and $\{ t_n : n\in \mathbb{Z}\}$ is massless. Clearly $\textsc{tw}_{s}$ fixes the former and acts freely and transitively on the latter. Its square $\textsc{tw}_{s}^2$ acts trivially on the Grothendieck group so the images of the boundary points labelled by the $t_n$ map to one of two points in charge space according to whether $n$ is even or odd. The Speiser graph has vertices $\mathbb{Z} \times \mathbb{N}$ with edges $(m,0)$ to $(m+1,0)$ and $(m,n)$ to $(m,n+1)$ for $m\in\mathbb{Z}$ and $n\in \mathbb{N}$. This is a tree as predicted by Conjecture~\ref{Speiser conjecture 1}. Moreover, the random walk on the Speiser graph is recurrent by Theorem~\ref{recurrence criterion}, as predicted by Conjecture~\ref{Speiser conjecture 2}. The twist $\textsc{tw}_{s}$ acts by an isometry without fixed points on $\eqstab{\Lambda_{2,1,0}} \cong \mathbb{C}$ and therefore acts by a translation of $\mathbb{C}$. This is illustrated in Figure~\ref{lambda figure}. \subsection{The projective line} \label{p1} The stability space $\eqstab{\mathbb{P}^1}\cong \mathbb{C}$ was first described in \cite{MR2219846}, see also \cite{MR2335991}. The `classical' stability condition has heart the coherent sheaves, with stable objects the skyscrapers $\mathcal{O}_x$ for $x\in \mathbb{P}^1$ and the line bundles $\mathcal{O}(k)$ for $k\in \mathbb{Z}$. The cell containing it has a sequence of cell-walls indexed by $\mathbb{Z}$ separated by boundary points at which $\mathcal{O}(k)$ is massless for $k\in \mathbb{Z}$. An example for a lax stability condition at the boundary is given by the slicing $P_b$ of Example~\ref{ex:slicings} with charge map $Z(\mathcal{O})=0$ and $Z(\mathcal{O}(1)) = i$. These boundary points accumulate on the boundary at a point where the charge of the skyscrapers vanishes. However, Proposition~\ref{prop: massless stables = algebraic simples} shows this point is not in $\pcQstab{\mathbb{P}^1}$ because the skyscrapers are not simple in any algebraic heart and so cannot become massless (see also Example~\ref{degenerate supported versus weak}). We indicate this omitted point by a white dot in Figure~\ref{p1 figure} and label it by $\mathcal{O}_x$ to indicate that the skyscrapers are stable in the adjacent chamber. Crossing the wall spanning the points where $\mathcal{O}(k)$ and $\mathcal{O}(k+1)$ are massless the skyscrapers and all other line bundles destabilise and we enter a chamber in which the only stable objects are $\mathcal{O}(k)$ and $\mathcal{O}(k+1)$. This chamber is the union of a sequence of cells separated by walls on which the phases of $\mathcal{O}(k+1)$ and $\mathcal{O}(k)[n]$ for $n>0$ agree. The Speiser graph is the union of $\mathbb{Z}$ copies of the graph with vertices $\mathbb{N}$ and edges from $n$ to $n+1$ joined at the $0$ vertices. This is a tree as expected from Conjecture~\ref{Speiser conjecture 1}. The central vertex has infinite valence so Theorem~\ref{recurrence criterion} does not apply. Nevertheless the random walk is recurrent as expected from Conjecture~\ref{Speiser conjecture 2}. The infinite cyclic group generated by the automorphism $-\otimes \mathcal{O}(1)$ preserves the chamber containing the `classical' stability condition (but does not fix any stability condition in this chamber) and acts freely and transitively on the chambers in which only two objects are stable. It also acts freely and transitively on the boundary points. It follows that it acts by a translation on $\eqstab{\mathbb{P}^1}\cong \mathbb{C}$. See Figure~\ref{p1 figure} for an illustration. Superficially, this closely resembles the previous example. However, there are several important (and inter-related) differences. In this case there is a chamber bounded by a countably infinite family of walls; there are stable objects whose mass does not vanish; the images of the boundary points accumulate in charge space. \section{Comparisons with other constructions} \label{comparisons} \noindent We compare our partial compactification of the stability space with two alternative approaches, namely Bolognese's `local compactification' \cite{bolognese2020local} and Bapat, Deopurkar and Licata's `Thurston compactification' \cite{bapat2020thurston}. \subsection{Bolognese's `local compactification'} \label{bolognese comparison} In \cite{bolognese2020local} Bolognese constructs an alternative `local compactification' of $\stab{\cat{C}}$ using a metric completion of $\stab{\cat{C}}$. In order to do so she assumes that $\mathcal{Z} \colon \stab{\cat{C}} \to \mor{\Lambda}{\mathbb{C}}$ is a cover of the complement of a locally finite union $\Delta \subset \mor{\Lambda}{\mathbb{C}}$ of submanifolds. She fixes an inner product on the underlying real space of $\mor{\Lambda}{\mathbb{C}}$ and gives $\stab{\cat{C}}$ the geodesic metric $d_B$ induced from the pullback of the associated metric. Since $\stab{\cat{C}}$ is locally homeomorphic to $\mor{\Lambda}{\mathbb{C}}$ with its norm topology this metric induces the usual topology on $\stab{\cat{C}}$. Her local compactification $\hatstab{\cat{C}}$ is the subspace of the metric completion consisting of equivalence classes of Cauchy sequences satisfying the limiting support property below. As a topological space $\hatstab{\cat{C}}$ is independent of the choice of inner product \cite[Lemma 3.6]{bolognese2020local}. \begin{definition}[{Limiting support property \cite[Definition 4.3]{bolognese2020local}}] A Cauchy sequence $(\sigma_n)$ in the metric $d_B$ on $\stab{\cat{C}}$ has the \defn{limiting support property} if $\liminf_{n\to \infty} C_n = C >0$ where for each $n\in \mathbb{N}$ the constant $C_n$ is the infimum of those $K>0$ such that $|Z_n(c)| > K||v(c)||$ for every $c\in P_n(\varphi)$ with $\lim_{n\to \infty} Z_n(c) \neq 0$. (The set of such constants $K$ is non-empty because each $\sigma_n$ is in $\stab{\cat{C}}$ and so satisfies the support property.) This property is well-defined on equivalence classes of Cauchy sequences by \cite[Lemma 4.4]{bolognese2020local}. \end{definition} In fact Bolognese shows that one can construct $\hatstab{\cat{C}}$ using only $\mathcal{Z}$-local Cauchy sequences, that is Cauchy sequences $(\sigma_n)$ for $d_B$ which eventually lie in an open subset $U\subset \stab{\cat{C}}$ homeomorphic to its image via $\mathcal{Z}$. More precisely, she shows that any Cauchy sequence is equivalent to a $\mathcal{Z}$-local one, and that if two $\mathcal{Z}$-local Cauchy sequences are equivalent then they are $\mathcal{Z}$-local with respect to the same open $U$ \cite[Theorem 3.7]{bolognese2020local}. Moreover, each $\mathcal{Z}$-local Cauchy sequence determines a thick subcategory of objects which become massless in the limit, and a well-defined stability condition on the quotient category \cite[Propositions 4.2 and Theorem 6.1]{bolognese2020local}. Finally, $\mathcal{Z}$-local Cauchy sequences are equivalent precisely when they determine the same massless subcategory and stability condition on the quotient \cite[Theorem 6.2]{bolognese2020local}. \begin{lemma} Suppose $(\sigma_n)$ is a $\mathcal{Z}$-local Cauchy sequence for $d_B$ on $\stab{\cat{C}}$. Then $(\sigma_n)$ converges to a lax pre-stability condition $\sigma$ in the product metric on $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$. \end{lemma} \begin{proof} By construction the sequence $(Z_n)$ of charges converges in norm in $\mor{\Lambda}{\mathbb{C}}$, say $Z_n \to Z$ as $n\to \infty$. The full subcategories $P(\varphi) \coloneqq \{ c\in \cat{C} : \varphi_{\sigma_n}^\pm(c) \to \varphi \}$ define a slicing \cite[Proposition 5.3]{bolognese2020local}. Bolognese notes in the proof of \cite[Proposition 5.2]{bolognese2020local} that the sequence $(P_n)$ of slicings is Cauchy in the slicing metric on $\slice{\cat{C}}$. It follows that $P_n \to P$ in $\slice{\cat{C}}$. For, if this were not the case, then, after passing to a subsequence, we may assume there is some $\varepsilon >0$ with $d(P,P_n)\geq \varepsilon$ for all $n\in\mathbb{N}$. In other words there is a sequence $(\varphi_n)$ of phases and objects $c_n \in P(\varphi_n)$ such that $c_n \not \in P_n(\varphi_n-\varepsilon, \varphi_n+\varepsilon)$. However, we know that for each $n\in \mathbb{N}$ there is some $N$ with $c_n \in P_m(\varphi_n-\varepsilon/2, \varphi_n+\varepsilon/2)$ whenever $m\geq N$. Choosing $N$ sufficiently large that we also have $d(P_n,P_m)<\varepsilon/2$ for all $m,n\geq N$ leads to a contradiction. Therefore $(\sigma_n)$ converges to $(P,Z)$ in the product metric on $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$. It is easy to confirm that $\sigma=(P,Z)$ is a lax pre-stability condition, i.e.\ that $Z(c) \in \mathbb{R}_{\geq 0} e^{\pi i \varphi}$ whenever $c\in P(\varphi)$. \end{proof} Let $\Bstab{\cat{C}}$ be the subset of the closure of $\stab{\cat{C}}$ in $\slice{\cat{C}}\times \mor{\Lambda}{\mathbb{C}}$ consisting of the limits of $\mathcal{Z}$-local Cauchy sequences for $d_B$ which satisfy the limiting support property. In order to compare Bolognese's local compactification with our constructions, we need to be able to compare this with $\lstab{\cat{C}}$. Unfortunately the relationship is not obvious since our support property is phrased in terms of massive stable objects in some $P(\varphi)$ and Bolognese's limiting support property is phrased in terms of semistable objects in some $P_n(\varphi)$ which remain massive in the limit. Since the HN filtration of a $\sigma_n$-semistable object with respect to $\sigma$ may contain massless objects, and similarly the other way round, there is no direct argument relating the two notions of support. If $\Bstab{\cat{C}}=\lstab{\cat{C}}$ then there should be a homeomorphism $\hatstab{\cat{C}} \cong \qstab{\cat{C}}$. For instance, this is so for the example of the $A_1$ quiver computed in Bolognese's paper. More generally an inclusion in either direction should extend to a map between $\hatstab{\cat{C}}$ and $\qstab{\cat{C}}$ in the corresponding direction. \subsection{Bapat, Deopurkar and Licata's `Thurston compactification'} \label{bapat et al comparison} In \cite{bapat2020thurston}, the authors take a different approach to compactifying the stability space, more precisely the quotient $\eqstab{\cat{C}}$, of a ${\mathbf{k}}$-linear triangulated category $\cat{C}$. By analogy with Thurston's compactification of Teichm\"uller space, they consider the map \[ m\colon \eqstab{\cat{C}} \to \mathbb{P}(\mathbb{R}^\cat{C}) \colon \sigma\cdot \mathbb{C} \mapsto [m_\sigma(c) : 0\neq c\in \cat{C} ]. \] When $\mathbb{P}(\mathbb{R}^\cat{C})$ has the topology induced from the product topology on $\mathbb{R}^\cat{C}$ this map is continuous, for instance by Proposition~\ref{mass and phase continuity}. Automorphisms of $\cat{C}$ act on $\mathbb{P}(\mathbb{R}^\cat{C})$ by pre-composing a real-valued function on the objects of $\cat{C}$ with the inverse automorphism. The map $m$ is equivariant for $\aaut{\Lambda}{\cat{C}}$ because $m_{\alpha \cdot \sigma(c)} = m_\sigma(\alpha^{-1}c)$. Under appropriate conditions on $\cat{C}$, Bapat, Deopurkar and Licata conjecture that $m$ is a homeomorphism onto its image $M(\cat{C})$, and that the closure $\overline{M(\cat{C})}$ is a manifold with boundary and interior $M(\cat{C})$. Moreover, motivated by the description of boundary points of Thurston's compactification as functionals given by unsigned intersections with closed curves, they conjecture that there is a suitable class $\mathcal{S}$ of objects such that the functionals \[ \overline{\mathrm{hom}}(s) \coloneqq \bigg[ \sum_{n\in \mathbb{Z}} \dim_{\mathbf{k}} \Mor{\cat{C}}{s}{c[n]} \bigm| 0\neq c\in \cat{C} \bigg] \qquad (s\in \mathcal{S}) \] form a dense subset of the boundary $\partial M(\cat{C})$. These conjectures hold for the $2$-Calabi--Yau category associated to the $A_2$ quiver \cite[\S5]{bapat2020thurston}. More generally, when $\cat{C}$ is the $2$-Calabi--Yau category associated to any connected quiver, they show $m$ is injective, that the closure of the image is compact, and that the functionals of all $2$-spherical objects lie in the boundary. Let $\lstab{\cat{C}}^* = \lstab{\cat{C}} - \lnstab{\cat{C}}{\cat{C}}$ be the space of lax stability conditions with the stratum where all objects are massless deleted, and define $\qstab{\cat{C}}^*$ similarly. We can extend $m$ to a map from $\pcLstab{\cat{C}}$ defined in the same way; the extension is also continuous by Proposition~\ref{mass and phase continuity}. This extension factorises continuously through the quotient \[ \begin{tikzcd} \pcLstab{\cat{C}} \ar{d}[swap]{q} \ar{r}{m} & \mathbb{P}(\mathbb{R}^\cat{C}) \\ \pcQstab{\cat{C}} \ar[dashed]{ur}[swap]{m} \end{tikzcd} \] because the masses depend only on the associated quotient stability condition. (We abuse notation by denoting the extension and the factorisation by $m$.) By construction the images of both $\pcLstab{\cat{C}}$ and $\pcQstab{\cat{C}}$ are contained within $\overline{M(\cat{C})}$. At least when $\cat{C}$ is the $2$-Calabi--Yau category associated to a connected quiver, Bapat, Deopurkar and Licata expect the boundary points in the image of $\pcQstab{\cat{C}}$ to be dense in the boundary \cite[Remark 4.9]{bapat2020thurston}. We show that this is true for the $A_2$ quiver. Before discussing that example we consider the non Calabi--Yau case. We slightly modify the construction by considering the map \[ \pcQstab{\cat{C}} \to \mathbb{P}(\mathbb{R}^\mathcal{S}) \colon \sigma \mapsto [ m_\sigma(s) : s\in \mathcal{S} ] \] where $\mathcal{S}$ is a suitable class of objects with the property that the masses of objects in $\mathcal{S}$ uniquely determine the masses of all objects. \begin{example} Recall the description of $\eqstab{A_2}$ in \S\ref{a2}. In each stability condition in $\stab{A_2}$ the semistable objects are, up to shift, a subset of at least two of $\{s,e,t\}$ where $s$ and $t$ are the two simple representations and $e$ the extension between them. It follows that the masses of objects in $\mathcal{S}=\{s,e,t\}$ determine the masses of all objects. This remains true for lax stability conditions in $\lstab{A_2}$. Therefore it suffices to consider the map \[ \eqstab{A_2} \to \mathbb{P}(\mathbb{R}^3) \colon \sigma \mapsto [m_\sigma(s):m_\sigma(e):m_\sigma(t)]. \] The image is cut out by the inequalities $x_0,x_1,x_2>0$ (the masses are strictly positive) together with the cyclic permutations of the inequality $x_0-x_1+x_2\leq 0$ (the mass of an extension is bounded by the sum of the masses of its factors). If we normalise so that $x_0+x_1+x_2=1$ then the image can be viewed as the shaded triangle, with vertices omitted, in the $2$-simplex in Figure~\ref{a2 Thurston compactification}. In particular we see that the map is not injective because when, for example, $e$ is unstable the masses of $s$ and $t$ do not suffice to determine their phases. The three boundary points in $\pcQstab{A_2}$ where the masses of $s$, $e$ and $t$ respectively vanish are mapped to the three omitted vertices. So in this example, $\pcQstab{A_2}$ surjects onto Bapat, Deopurkar and Licata's compactification, and the boundaries coincide. The boundary points correspond precisely to the functionals $\overline{\mathrm{hom}}(c)$ for $c\in \mathcal{S}$. \begin{figure} \caption{The image of $\eqstab{A_2} \label{a2 Thurston compactification} \end{figure} \end{example} \begin{example} Let $\mathcal{S}$ be the set of equivalence classes of spherical objects in $\cat{D}^b(\Gamma_{\!2} A_2)$ up to isomorphism and shift. Then by \cite[Proposition 5.7]{bapat2020thurston} the map \[ m \colon \eqstab{\Gamma_{\!2} A_2} \to \mathbb{P}(\mathbb{R}^\mathcal{S}) \colon \sigma \mapsto [m_\sigma(s) : s\in \mathcal{S}] \] is a homeomorphism onto its image which we denote $M(\Gamma_{\!2} A_2)$. Moreover, $M(\Gamma_{\!2} A_2)$ is homeomorphic to an open disk by \cite[Proposition 5.20]{bapat2020thurston}. After choosing an element $s\in \mathcal{S}$ to map to $[1:0]$, the action of the Artin--Tits braids group induces a bijection $\mathcal{S} \cong \mathbb{P}(\mathbb{Z}^2)$. Using this identification the map \[ \mathcal{S} \to \mathbb{P}(\mathbb{R}^\mathcal{S}) \colon s \mapsto [ \mathrm{hom}(s)(s') : s'\in \mathcal{S} ] \] extends uniquely to a homeomorphism from $\mathbb{P}(\mathbb{R}^2)$ onto the boundary of $M(\Gamma_{\!2} A_2)$ in $\mathbb{P}(\mathbb{R}^S)$ by \cite[Propositions 5.13 and 5.18]{bapat2020thurston}. The closure $\overline{M(\Gamma_{\!2} A_2)}$ is homeomorphic to a closed disk. The functional $\overline{\mathrm{hom}}(s)$ is the unique fixed point of the spherical twist $\textsc{tw}_{s}$. Now consider the extension $m\colon \pcQstab{\Gamma_{\!2} A_2} \to \overline{M(\Gamma_{\!2} A_2)}$. Recall from \S\ref{a2} and Figure~\ref{a2 figure} that the partial compactification $\pcQstab{\Gamma_{\!2} A_2}$ contains one boundary point for each $s\in\mathcal{S}$ at which the objects in the class $s$ become massless. This boundary point is fixed by $\textsc{tw}_{s}$ because $\textsc{tw}_{s}$ acts by a shift on $s$. The equivariance of $m$ implies that this point is mapped to $\overline{\mathrm{hom}}(s)$. (At first sight this looks odd because $\sum_{n\in \mathbb{Z}}\mor{s}{s[n]}=2$ is non-zero. However we are working in an infinite-dimensional projective space and \cite[Proposition 4.5]{bapat2020thurston} states that \[ \lim_{n\to \infty} \frac{ m_\sigma\left(\textsc{tw}^n_{s}(s')\right)}{n} = m_\sigma(s) \, \overline{\mathrm{hom}}(s)(s') \] for any $s'\in \mathcal{S}$ and stability condition $\sigma$ in which $s$ is stable. Since $\textsc{tw}_{s}$ fixes $s$ up to a shift, this implies that $\overline{\mathrm{hom}}(s)(s)=0$ as expected.) We conclude that \[ m \colon \pcQstab{\Gamma_{\!2} A_2} \to \overline{M(\Gamma_{\!2} A_2)} \] is a continuous embedding, restricting to a homeomorphism between the interiors, and whose image is dense in the boundary. This accords with the expectations of \cite[Remark 4.9]{bapat2020thurston}, and provides a modular interpretation of the boundary points $\overline{\mathrm{hom}}(s)$ as quotient stability conditions. \end{example} \section{Open questions} \noindent We end with four open questions. (1) and (2) relate to the properties of composition series in quasi-abelian categories. This technical issue plays a role because the slices $P(\varphi)$ of a lax stability condition are quasi-abelian but, in contrast to the situation for classical stability conditions, need not be abelian. (3) and (4) relate to support properties for lax stability conditions. There are various different notions in the literature, and it would be good to understand which are equivalent, and to what extent each has the crucial propagation property enjoyed by support for classical stability conditions. \begin{enumerate} \item Are there examples where the Jordan-H\"older property fails for the slices $P(\varphi)$ of a lax stability condition? If there are then, whilst the HN filtrations of objects are unique, their refinements to filtrations with stable factors would not be. Presumably this would have implications for the wall-and-chamber structure. \item Let $P$ be a locally finite slicing on $\cat{C}$ and $\cat{N} \subset \cat{C}$ a thick subcategory such that $P$ descends to $\cat{C}/\cat{N}$. Is the slicing $P_{\cat{C}/\cat{N}}$ on the quotient also locally finite? What if $P$ is the slicing of a lax pre-stability condition with massless subcategory $\cat{N}$? (If $P$ is the slicing of a lax stability condition then the support property guarantees that $P_{\cat{C}/\cat{N}}$ is locally finite, see Proposition~\ref{prop:massive stability condition}.) \item Are there examples where support propagation fails? See Definition~\ref{def:local support propagation} and the discussion on page~\pageref{q:support propagation}. This is crucial for the range of applicability of our results. \item What is the relationship between the support property for a lax pre-stability condition $\sigma$ we use and Bolognese's notion of limiting support for a sequence of stability conditions converging to $\sigma$? Understanding this is key to clarifying the relationship between our partial compactification and Bolognese's, see \S\ref{bolognese comparison}. \end{enumerate} {\small\texttt{ \noindent \begin{tabular}{lll} \textit{\textrm{Contact:}} & nathan.broomhead@plymouth.ac.uk, & d.pauksztello@lancaster.ac.uk, \\ & david.ploog@uis.no, & jonathan.woolf@liverpool.ac.uk \end{tabular}}} \end{document}

\begin{document} \pagestyle{plain} \title{\LARGE \bf Verification of the Incremental Merkle Tree Algorithm with Dafny} \thispagestyle{empty} \begin{abstract} The Deposit Smart Contract (DSC) is an instrumental component of the Ethereum 2.0 Phase 0 infrastructure. We have developed the first machine-checkable version of the incremental Merkle tree algorithm used in the DSC. We present our new and original correctness proof of the algorithm along with the Dafny machine-checkable version. The main results are: 1) a new proof of total correctness; 2) a software artefact with the proof in the form of the complete Dafny code base and 3) new provably correct optimisations of the algorithm. \end{abstract} \iffalse \section*{CAV 2021 industrial and case-study papers.} Industrial Experience Reports and Case Studies should not exceed 10 pages, not counting references. These papers are expected to describe the use of formal methods techniques in industrial settings or in new application domains. Papers in this category do not necessarily need to present original research results but are expected to contain novel applications of formal methods techniques as well as an \textbf{evaluation of these techniques} in the chosen application domain. Such papers are encouraged to \textbf{discuss the unique challenges of transferring research ideas to a real-world setting} and \textbf{reflect on any lessons learned from this technology transfer experience}. FM 2021: FM in practice Industrial applications of formal methods, experience with formal methods in industry, tool usage reports, experiments with challenge problems. The authors are encouraged to explain how formal methods overcame problems, led to improved designs, or provided new insights. \fi \section{Introduction}\label{sec-intro} Blockchain-based decentralised platforms process transactions between parties and record them in an immutable distributed ledger. Those platforms were once limited to handle simple transactions but the next generation of platforms will routinely run \emph{decentralised applications} (DApps) that enable users to make complex transactions (sell a car, a house or more broadly, swap assets) without the need for an institutional or governmental trusted third-party. \paragraph{\bf \itshape Smart Contracts.} More precisely, the transactions are \emph{programmatically} performed by \emph{programs} called \emph{smart contracts}. If there are real advantages having smart contracts act as third-parties to process transactions, there are also lots of risks that are inherent to computer programs: they can contain \emph{bugs}. Bugs can trigger runtime errors like \emph{division by zero} or \emph{array-out-of-bounds}. In a networked environment these types of vulnerabilities can be exploited by malicious attackers over the network to disrupt or take control of the computer system. Other types of bugs can also compromise the business logic of a system, {e.g.},~ an implementation may contain subtle errors ({e.g.},~ using a \verb|+=| operator in C instead of \verb|=+|) that make them deviate from the initial intended specifications. Unfortunately it is extremely hard to guarantee that programs and henceforth smart contracts implement the correct business logics, that they are free of common runtime errors, or that they never run into a non-terminating computation.\footnote{In the Ethereum ecosystem, programs can only use a limited amount of resources, determined by the \emph{gas limit}. So one could argue that non-terminating computations are not problematic as they cannot arise: when the gas limit is reached a computation is aborted and has no side effects. It follows that a non-terminating computation (say an infinite loop due to a programming error) combined with a finite gas limit will abort and will result in the system being unable to successfully process some or all \emph{valid} transactions and this is a serious issue.} There are notorious examples of smart contract vulnerabilities that have been exploited and publicly reported: in 2016, a \emph{reentrance} vulnerability in the Decentralised Autonomous Organisation (DAO) smart contract was exploited to steal more than USD50 Million. There may be several non officially reported similar attacks that have resulted in the loss of assets. \paragraph{\bf \itshape The Deposit Smart Contract in Ethereum 2.0.} The next generation of Ethe\-reum-based networks, Ethereum 2.0, features a new \emph{proof-of-stake} consen\-sus protocol. Instead of miners used in Ethereum 1.x, the new protocol relies on \emph{va\-lidators} to create and \emph{validate} blocks of transactions that are added to the led\-ger. The protocol is designed to be fault-tolerant to up to $1/3$ of Byzantine ({i.e.},~ malicious or dishonest) validators. To discourage validators to deviate from an honest behaviour, they have to \emph{stake} some assets in Ether (a crypto-currency), and if they are dishonest they can be \emph{slashed} and lose (part of) their stake. The process of staking is handled by the \emph{Deposit Smart Contract (DSC)}: a validator sends a transaction (``stake some Ether'') by \emph{calling} the DSC. The DSC has a \emph{state} and can update/record the history of deposits that have occurred so far. As a result the DSC is a mission-critical component of Ethereum 2.0, and any errors/crashes could result in inaccurate tracking of the deposits or downtime which in turn may compromise the integrity/availability of the whole system. This could be mitigated if the DSC was a simple piece of code, but, for performance reasons, it relies on sophisticated data structures and algorithms to maintain the list of deposits so that they can be communicated over the network efficiently: the history of deposits is summarised as a unique number, a \emph{hash}, computed using a \emph{Merkle} (or \emph{Hash}) tree. The tree is built incrementally using the \emph{incremental Merkle tree algorithm}, and as stated in~\cite{deposit-cav-2020}: \begin{quote} \em ``The efficient incremental algorithm leads to the DSC implementation being unintuitive, and makes it non-trivial to ensure its correctness.'' \end{quote} \paragraph{\bf \itshape Related Work.} In this context, it is not surprising that substantial efforts, au\-di\-ting, review~\cite{suhabe-dsc}, testing and formal verification~\cite{formal-inc-merkle-rv,deposit-cav-2020} has been invested to guarantee the reliability and integrity ({e.g.},~ resilience to potential attacks) of the DSC. The DSC has been the focus of an end-to-end analysis~\cite{deposit-cav-2020}, including the bytecode\footnote{A limitation is that the bytecode is proved using a non-trusted manual specification.} that is executed on the Ethereum Virtual Machine (EVM). However, the \emph{incremental Merkle tree algorithm} has not been \emph{mechanically verified} yet, even though a pen and paper proof has been proposed~\cite{formal-inc-merkle-rv} and \emph{partially} mechanised using the K-framework~\cite{k-2019}. An example of the limitations of the mechanised part of the proof in~\cite{formal-inc-merkle-rv} is that it does not contain a formal (K-)de\-finition of Merkle trees. The mechanised sections (lemmas 7 and 9) pertain to some invariants of the algorithm but not to a proper correctness specification based on Merkle trees. The K-framework and KEVM, the formalisation of the EVM in K, has been used to analyse a number of other smart contracts~\cite{rv-sc}. There are several techniques and tools\footnote{\url{https://github.com/leonardoalt/ethereum_formal_verification_overview}.} {e.g.},~ ~\cite{solc-vstte-19,fmbc-20,DBLP:conf/cpp/AmaniBBS18,csi-mythx,harvey}, for auditing and analysing smart contracts writ\-ten in Solidity (a popular language to write Ethereum smart contracts) or EVM bytecode, but they offer limited capabilities to verify complex functional requirements. Interesting properties of incremental Merkle trees were established in~\cite{ogawa} using the MONA prover. This work does not prove the algorithms in the DSC which are designed to minimise gas consumption and hence split into parts: insert a value in a tree, and compute the root hash. Moreover, some key lemmas in the proofs could not be discharged by MONA. The gold standard in program correctness is a complete logical proof that can be \emph{mechanically checked} by a prover. This is the problem we address in this paper: to design a \emph{machine-checkable proof} for the DSC algorithms (not the bytecode) using the Dafny language and verifier. The DSC has been deployed in Nov\-ember 2020. To the best of our knowledge, our analysis, completed in October 2020, provided the first fully mechanised proof that the code logic was correct, and free of runtime errors. There seem to be few comparable case-studies of Dafny-verified (or other verification-aware programming languages like Whiley~\cite{whiley-setss-2018}) code bases. The most notorious and complex one is probably the IronFleet/IronClad~\cite{ironfleet-2015} dis\-tri\-bu\-ted system, along with some non-trivial algorithms like DPPL~\cite{dpll-dafny-19} or Red-Black trees~\cite{DBLP:journals/jar/Pena20}, or operating systems, FreeRTOS scheduler~\cite{matias_program_2014}, and ExpressOS~\cite{DBLP:conf/asplos/MaiPXKM13}. Other proof assistants like Coq~\cite{Paulin-Mohring2012}, Isabelle/HOL~\cite{Nipkow-Paulson-Wenzel:2002} or Lean~\cite{DBLP:conf/cade/MouraKADR15} have also been extensively used to write machine-checkable proofs of algorithms~\cite{lammich-ijcar-2020,lammich-mc-ta,timsort-jdk,ghc-mergesort} and software systems~\cite{DBLP:conf/sosp/KleinEHACDEEKNSTW09,DBLP:journals/jar/Leroy09}. \paragraph{\bf \itshape Our Contribution.} We present a thorough analysis of the incremental Merkle tree algorithm used in the DSC. Our results are available as software artefacts, written using the CAV-awarded Dafny\footnote{\url{https://github.com/dafny-lang/dafny}} verifi\-ca\-tion-aware programming language~\cite{dafny-ieee-2017}. This provides a self-contained machine checkable and reproducible proof of the DSC algorithms. Our contribution is many-fold and includes: \begin{itemize} \item a \emph{new original simple proof} of the incremental Merkle tree algorithm. In contrast to the previous non-mechanised proof in~\cite{formal-inc-merkle-rv} we do not attempt to directly prove the existing algorithm, but rather to \emph{design} and refine it. Our proof is \emph{parametric} in the height of the tree, and \emph{hash} functions; \item a \emph{logical specification} using a formal definition of Merkle trees, and a \emph{new functional version} of the algorithm that is proved correct against this specification; the functional version is used to specify the invariants for the proof of the imperative original version~\cite{vitalik-merkle} of the algorithm; \item a repository\footnote{\url{https://github.com/ConsenSys/deposit-sc-dafny}} with the complete Dafny source code of the specification, the algorithms and the proofs, and comprehensive documentation; \item some new provably correct simplifications/optimisations; \item some reflections on the practicality of using a verification-aware programming language like Dafny and some lessons learned from this experience. \end{itemize} \section{Incremental Merkle Trees}\label{sec-example} \paragraph{\bf \itshape Merkle Trees.} A \emph{complete (or perfect) binary tree} is such that each non-leaf node has exactly two children, and the two children have the same \emph{height}. An example of a complete binary tree is given in \figref{fig-compute-root-hash}. A \emph{Merkle (or hash) tree} is a complete binary tree the nodes of which are decorated with \emph{hashes} (fixed-size bit-vectors). The hash values of the leaves are given and the hash values of the internal (non-leaf) nodes are computed by \emph{combining} the values of their children with a binary function $\mathbf{hash}$. It follows that a Merkle tree is a complete binary tree decorated with a \emph{synthesised attribute} defined by a binary function. Merkle trees are often used in distributed ledger systems to define a \emph{property} of a collection of elements {e.g.},~ a list $L$ of values. This property can then be used \emph{instead of the collection itself} to verify,\footnote{More precisely the verification result holds with high probability as the chosen hashing functions may (rarely) generate collisions.} using a mechanism called \emph{Merkle proofs}, that data received from a node in the distributed system is not corrupted. This is a crucial optimisation as the size of the collection is usually large (typically up to $2^{32}$) and using a compact representation is instrumental to obtain time and space efficient communication and a reasonable transactions' processing throughput. \begin{quote} \it In this work, we are not concerned with Merkle proofs but rather with the (efficient) computation of the $\mathbf{hash}$ attribute on a Merkle tree. \end{quote} The actual function used to compute the values of the internal nodes is not re\-le\-vant in the incremental Merkle tree algorithms' functional logics and without loss of generality we may treat it as a parameter {i.e.},~ a given binary function.\footnote{In the code base, the $\mathbf{hash}$ function is uninterpreted and its type is generic.} In the sequel we assume that the decorations of the nodes are integers, and we use in the examples a simple function $\mathbf{hash} : \text{Int} \times \text{Int} \longrightarrow \text{Int}$ defined by $\mathbf{hash}(x, y) = x - y - 1$ instead of an actual ({e.g.},~ \texttt{sha256}-based) hash function. \iffalse \begin{figure} \caption{A Merkle tree of height $3$ for a list $L = [3, 6, 2, -2, 4]$ and $\mathbf{hash} \label{fig-ex1} \end{figure} \fi \paragraph{\bf \itshape Properties of Lists with Merkle Trees.} A complete binary tree of height\footnote{The height is the length of the longest path from the root to any leaf.} $h$ has $2^h$ leaves and $2^{h + 1} - 1$ nodes. Given a list $L$ of integers (type \text{Int}) of size $|L| = 2^h$, we let $T(L)$ be the Merkle tree for $L$: the values of the leaves of $T(L)$, from left to right, are the elements of $L$ and $T(L)$ is attributed with the $\mathbf{hash}$ function. The value of the attribute at the root of $T(L)$, the \emph{root hash}, defines a property of the list $L$. It is straightforward to extend this de\-fi\-ni\-tion to lists $L$ of size $|L| \leq 2^h$ by right-padding the list with \emph{zeroes} (or any other default values.) Given a list $L$ of size $|L| \leq 2^h$, let $\overline{L}$ denote $L$ right-padded with $2^h - |L|$ default values. The Merkle tree associated with $L$ is $T(\overline{L})$, and the root hash of $L$ is the root hash of $T(\overline{L})$. Computing the root hash of a tree $T(\overline{L})$ requires to traverse all the nodes of the tree and thus is \emph{exponential} in the height of the tree. \paragraph{\bf \itshape The Incremental Merkle Tree Problem.} A typical use case of a Merkle tree in the context of Ethereum 2.0 is to represent properties of lists that \emph{grow monotonically}. In the DSC, a Merkle tree is used to record the list of validators and their stakes or deposits. A compact representation of this list, as the root hash of a Merkle tree, is com\-mu\-nicated to the nodes in the network rather than the tree (or list) itself. However, as mentioned before, each time a new deposit is appended to the list, computing the new root hash using a standard synthesised-attribute computation algorithm requires exponential time in $h$. This is clearly impractical in a distributed system like Ethereum in which the height of the tree is $32$ and the number of nodes is $2^{33} - 1$. Given (a tree height) $h > 0$, $L$ a list with $|L| < 2^h$, and $e$ a new element to add to $L$, the incremental Merkle tree problem (IMTP) is defined as follows:\footnote{Polynomial in the height of the tree $h$. The operator $+$ is list concatenation.} \begin{quote} \it Can we find $\alpha(L)$ a \textbf{polynomial-space abstraction} of $T(L)$ such that we can compute in \textbf{polynomial-time}: 1) the root hash of $T(L)$ from $\alpha(L)$, and 2) the abstraction $\alpha(L + [e])$ from $\alpha(L)$ and $e$? \end{quote} Linear-time/space algorithms to solve the IMTP were originally pro\-po\-sed by V. Buterin in~\cite{vitalik-merkle}. However, the correctness of these algorithms is not obvious. In the next section, we analyse the IMTP, and we present the main properties that en\-able us to \emph{design} polynomial-time recursive algorithms and to \emph{verify} them. \section{Recursive Incremental Merkle Tree Algorithm}\label{sec-3} In this section we present the main ideas of the \emph{recursive} algorithms to insert a new value in a Merkle tree and to compute the new root hash (after a new value in inserted) by re-using (\emph{dynamic programming}) previously computed results. \begin{figure} \caption{A Merkle tree of height $3$ for list $L_2 = [3, 6, 2, -2, 4]$ and $\mathbf{hash} \label{fig-compute-root-hash} \end{figure} \paragraph{\bf \itshape Notations.} A \emph{path} $\pi$ from the root of a tree to a node can be defined as a se\-quence of bits (left or right) in $\{0, 1\}^*$. In a Merkle tree of height $h$, the \emph{length}, $|\pi|$, of $\pi$ is at most $h$. $\nu(\pi)$ is the \emph{node} at the end of $\pi$. If $|\pi| = h$ then $\nu(\pi)$ is a leaf. For instance $\nu(\varepsilon)$ is the root of the tree, $\nu(0)$ in \figref{fig-compute-root-hash} is the node carrying the value $-8$ and $\nu(1.0.0)$ is a leaf. The \emph{right sibling of a left node} of the form $\nu(\pi.0)$ is the node $\nu(\pi.1)$. Left siblings are defined symmetrically. A node in a Merkle tree is associated with a \emph{level} which is the distance from the node to a leaf in the tree. Leaves are at level $0$ and the root is at level $h$. In a Merkle tree, level $0$ has $2^h$ leaves that can be indexed left to right from $0$ to $2^h -1$. The \emph{$n$-th leaf} of the tree for $0 \leq n < 2^h$ is the leaf at index $n$. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=Recursive Algorithm to Compute the Root Hash., captionpos=t, label={algo-compute-root}] computeRootUp(p:seq<bit>,left:seq<int>,right:seq<int>,seed:int):int requires |p| == |left| == |right| // vectors have the same sizes decreases p { if |p| == 0 then seed else if last(p) == 0 then // node at end of p is a left node computeRootUp(init(p),init(left),init(right),hash(seed,last(right))) else // node at end of p is a right node computeRootUp(init(p),init(left),init(right),hash(last(left),seed)) } \end{lstlisting} \end{minipage} \paragraph{\bf \itshape Computation of the Root Hash on a Path.} We first show that the root hash can be computed if we know the values of the \emph{siblings} of the nodes on \emph{any} path, and the value at the end of the path. For instance, If we know the values of the left and right siblings (shaded nodes) of the nodes on $\pi_1$ (green path in \figref{fig-compute-root-hash}), and the value at the end of $\pi_1$, we can compute the root hash of the tree by propagating upwards the attribute $\mathbf{hash}$. The value of the $\mathbf{hash}$ attribute at $\nu(1.0)$ is $\mathbf{hash}(4,\nu(1.0.1)) = 3$, at $\nu(1)$ it is $\mathbf{hash}(3, \nu(1.1)) = 3$ and at the root $\mathbf{hash}(\nu(0), \nu(1)) = \mathbf{hash}(-8, 3) = -12$. Algorithm \texttt{computeRootUp} (Listing~\ref{algo-compute-root}) computes\footnote{For $l = l' + x$, $\mathtt{last}(l) = x$, $\mathtt{init}(l) = l'$, and for $l = x + l'$, $\mathtt{first}(l) = x$, $\mathtt{tail}(l) = l'$.} bottom-up in time linear in $|\mathtt{p}|$ the root hash with \texttt{left} the list of values of the left siblings (top-down) on a path \texttt{p} (top-down), \texttt{right} the values of the right siblings (top-down) and \texttt{seed} the value at $\nu(\mathtt{p})$. The generic version (uninterpreted hash) of the algorithm is provided in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/synthattribute/ComputeRootPath.dfy#L57}{\tt ComputeRootPath.dfy} file. For the green path $\mathtt{pi}_1 = [1,0,0]$ in~\figref{fig-compute-root-hash}, $\mathtt{left} = [-8, i_1, i_0]$, $\mathtt{right} = [-1, -1, 0]$ and the seed is $4$. The evaluation of \texttt{computeRootUp} returns $-12$. Given a path $\pi$, if the leaves on the right of $\nu(\pi)$ all have the default value $0$, the values of the right siblings on the path $\pi$ only depend on the \emph{level} of the sibling in the tree. For example, the leaves on the right of $\pi_1$ (orange in \figref{fig-compute-root-hash}) all have the default value $0$. The root hash of a tree in which all the leaves have the same default value only depends on the level of the root: $0$ at level $0$, $\mathbf{hash}(0, 0)$ at level $1$, $\mathbf{hash}(\mathbf{hash}(0, 0), \mathbf{hash}(0, 0))$ at level $2$ and so on. Let $\mathbf{zero}^l$ be defined by: $\mathbf{zero}^l = 0 \text{ if $l = 0$ else $\mathbf{hash}(\mathbf{zero}_0^{l -1}, \mathbf{zero}_0^{l -1})$}$. \vspace*{-0.5em} \begin{quote}\em Given a path $\pi$, if all the leaves on the right of $\nu(\pi)$ have the default value, any right sibling value at level $l$ on $\pi$ is equal to $\mathbf{zero}^l$. \end{quote} As an example in \figref{fig-compute-root-hash}, the right siblings on $\pi_1 = 1.0.0$ have values $0$ at level $0$, node $\nu(1.0.1)$, and $\mathbf{hash}(0,0) = \mathbf{zero}^1 = -1$ at level $1$, node $\nu(1.1)$. \vspace*{-0.2em} If a path \texttt{p} leads to a node with the default value $0$ and all the leaves right of $\nu(\mathtt{p})$ have the default value $0$, the root hash depends only on the values of the \texttt{left} and default \texttt{right} siblings. Hence the root hash can be obtained by \texttt{computeRootUp(p, left, right, 0)}. For the path $\mathtt{pi2} = [1, 0, 1]$ (\figref{fig-compute-root-hash}), $\mathtt{left} = [-8, i_1, 4]$, $\mathtt{right} = [-1, -1, 0]$, \texttt{computeRootUp(pi2, left, right, 0)} returns $-12$. As a result, to compute the root hash of a tree $T(\overline{L})$, we can use a compact abstraction $\alpha(L)$ of $T(\overline{L})$ composed of the left siblings vector $b$ and the right siblings default values $z$ (\figref{fig-compute-root-hash}) of the path to the $|L|$-th leaf in $T(\overline{L})$. \paragraph{\bf \itshape Insertion: Update the Left Siblings.} Assume $\pi_1$ is a path to the $n$-th leaf and $ n < 2^h - 1$ (not the last leaf), where the next value $v$ is to be inserted. As we have shown before, if we have $b_1$ holding the values of left siblings of $\pi_1$, $z$ and $v$, we can compute the new attribute values of the nodes on $\pi_1$ and the new root hash after $v$ is inserted. Let $\pi_2$ be the path to the $n+1$-th leaf. If we can compute the values $b_2$ of the left siblings of $\pi_2$ as a function of $b_1$, $z$ and $v$, we have an efficient algorithm to \emph{incrementally} compute the root hash of a Merkle tree: we keep track of the values of the left siblings $b$ on the path to the next available leaf, and iterate this process each time a new value is inserted. As $\nu(\pi_1)$ is not the last leaf, $\pi_1$ must contain at least one $0$, and has the form\footnote{$x^k, x \in \{0,1\}$ denotes the sequence of $k$ $x$'s.} $\pi_1 = w.0.1^k$ with $w \in \{0, 1\}^*, k \geq 0$. Hence, the path $\pi_2$ to the $n+1$-th leaf is $w.1.0^k$, arithmetically $\pi_2 = \pi_1 + 1$. An example of two consecutive paths is given in \figref{fig-compute-root-hash} with $\pi_1$ (green) and $\pi_2$ (blue) to the leaves at indices $4$ and $5$. The related forms of $\pi_1$ (a path) and $\pi_2$ (the successor path) are useful to figure out how to incrementally compute the left siblings vector $b_2$ for $\pi_2$: \begin{itemize} \item as the initial prefix $w$ is the same in $\pi_1$ and $\pi_2$, the values of the left siblings on the nodes of $w$ are the same in $b_1$ and $b_2$; \item all the nodes in the suffix $0^k$ of $\pi_2$ are left nodes and have right siblings. It follows that the corresponding $k$ values in $b_2$ are irrelevant as they correspond to right siblings, and we can re-use the corresponding $b_1$ values; \item hence $b_2$ is equal to $b_1$ except possibly for the level of the node at $\nu(w.0)$. \end{itemize} We now illustrate how to compute the new value in the vector $b_2$ on the example of \figref{fig-compute-root-hash}. Let $\pi_1 = w.0$ and $\pi_2 = w.1$ with $w = 1.0$ and $|w| = 2$. For the top levels $2$ and $1$, $b_2$ is the same as $b_1$: $b_2[2] = b_1[2] = -8$ and $b_2[1] = b_1[1] = i_1$. For level $0$, the level of the node $\nu(w.0)$, the value at $\nu(w.0) = \nu(1.0.0)$ becomes the left sibling of the node $\nu(1.0.1)$ on $\pi_2$ at this level. So the new value of the left sibling on $\pi_2$ is exactly the new value, $4$, of the node $\nu(1.0.0)$ after $4$ is inserted. More generally, when computing the new root hash bottom-up on $\pi_1$, the first time we encounter a left node, at level $d$, we update the corresponding value of $b$ with the computed value of the attribute on $\pi_1$ at level $d$. Algorithm\footnote{$+$ stands for list concatenation.} \texttt{insertValue} in Listing~\ref{algo-compute-sib} computes, in linear-time, the list of values of the left siblings (top-down) of the path $\mathtt{p} + 1$ using as input the list (top-down) of values left (resp. right) siblings \texttt{left} (resp. \texttt{right}) of \texttt{p} and \texttt{seed} the new value inserted at $\nu(\mathtt{p})$. The generic (non-interpreted hash) algorithm is provided in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/paths/NextPathInCompleteTreesLemmas.dfy#L101}{\tt NextPathInCompleteTreesLemmas.dfy} file. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=Recursive Algorithm to Compute the New Left Siblings., captionpos=t, label={algo-compute-sib}] insertValue(p:seq<bit>,left:seq<int>,right:seq<int>,seed:int):seq<int> requires |left| == |right| == |p| >= 1 decreases p { if |p| == 1 then // note that first(p) == last(p) in this case if first(p) == 0 then [seed] else left else if last(p) == 0 then // we encounter a left node. Stop recursion. init(left) + [seed] else // right node,move up on the path. insertValue(init(p),init(left),init(right),hash(last(left),seed)) + [last(left)] } \end{lstlisting} \end{minipage} We illustrate how the algorithm \texttt{insertValue} works with the example of \figref{fig-compute-root-hash}. Assume we insert the seed $4$ at the end of the (green) path \texttt{pi1 = [1,0,0]}. The left (resp. right) siblings' values are given by $\mathtt{left} = [-8, i_1, i_0]$ (resp. $\mathtt{right} = [-1, -1, 0]$). \texttt{insertValue} computes the values of the left siblings on the (blue) path \texttt{pi2 = [1,0,1]} after $4$ is inserted at the end of $\pi_1$: the first call terminates the algorithm and returns $[-8, i_1, 4]$ which is the list of left siblings that are needed on $\pi_2$. In the next section we describe how to verify the recursive algorithms and the versions implemented in the DSC. \section{Verification of the Algorithms} In order to verify the implemented (imperative style/Solidity) versions of the algorithms of the DSC, we first prove total correctness of the recursive versions (Section~\ref{sec-3}) and them use them to prove the code implemented in the DSC. \begin{quote}\em In this section, the Dafny code has been simplified and sometimes even altered while retaining the main features, for the sake of clarity. The code in this section may not compile. We provide the links to the files with the full code in the text and refer the reader to those files. \end{quote} \paragraph{\bf \itshape Correctness Specification.} The (partial) correctness of our algorithms reduces to checking that they compute the same values as the ones obtained with a synthesised attribute on a Merkle tree. We have specified the data types \texttt{Tree}, \texttt{MerkleTree} and \texttt{CompleteTrees} and the relation between Merkle trees and lists of values (see \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/paths/NextPathInCompleteTreesLemmas.dfy#L101https://github.com/ConsenSys/deposit-sc-dafny/tree/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/trees}{\tt trees} folder.) The root hash of a \texttt{MerkleTree} \texttt{t} is \texttt{t.rootv}. The (specification) function \texttt{buildMerkle(h, L, $\mathbf{hash}$)} returns a \texttt{MerkleTree} of height \texttt{h}, the leaves of which are given by the values (right-padded) $\overline{\mathtt{L}}$, and the values on the internal nodes agree with the definition of the synthesised attribute $\mathbf{hash}$, {i.e.},~ what we previously defined in Section~\ref{sec-example} as $T(\overline{\texttt{L}})$. It follows that \texttt{buildMerkle(h, L, $\mathbf{hash}$).rootv} is the root hash of a Merkle tree with leaves $\overline{\mathtt{L}}$. \paragraph{\bf \itshape Total Correctness.} The total correctness proof for the \texttt{computeRootUp} function amounts to showing that 1) the algorithm always terminates and 2) the result of the computation is the same as the hash of the root of the tree. In Dafny, to prove termination, we need to provide a ranking function (strictly decreasing and bounded from below.) The length of the path \texttt{p} is a suitable ranking function (see the \texttt{decreases} clause in Listing~\ref{algo-compute-root}) and is enough for Dafny to prove termination of \texttt{computeRootUp}. We establish property 2) by proving a \emph{lemma} (Listing~\ref{proof-computeRoot}): the pre-conditions (\texttt{requires}) of the lemma are the assumptions, and the post-conditions (\texttt{ensures}) the intended property. The body of the lemma (with a non-interpreted hash function) which provides the machine-checkable proof is available in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/synthattribute/ComputeRootPath.dfy#L85}{computeRootPath.dfy} file. This lemma requires that the tree \texttt{r} is a Merkle tree, and that the lists \texttt{left} (resp. \texttt{right}) store the values of left (resp. right) siblings of the nodes on a path \texttt{p}. Moreover, the value at the end of \texttt{p} should be \texttt{seed}. Under these assumptions the conclusion (\texttt{ensures}) is that \texttt{computeRootUp} returns the value of the root hash of \texttt{r}. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=Correctness Proof Specification for \texttt{ComputeRootUp}., captionpos=t, label={proof-computeRoot}] lemma computeRootUpIsCorrectForTree( p:seq<bit>,r:Tree<int>,left:seq<int>,right:seq<int>,seed:int) // size of p is the height of the tree r requires |p| == height(r) // r is a Merkle tree for attribute hash requires isCompleteTree(r) requires isDecoratedWith(hash,r) // the value at the end of the path p in r is seed requires seed == nodeAt(p,r).v // vectors of same sizes requires |right| == |left| == |p| // Left and right contain values of left and right siblings of p in r. requires forall i :: 0 <= i < |p| ==> // the value of the sibling of the node at p[..i] in r siblingValueAt(p,r,i + 1) == // are stored in left and right if p[i] == 0 then right[i] else left[i] // Main property: computeRootUp computes the hash of the root of r ensures r.rootv == computeRootUp(p,left,right,seed) \end{lstlisting} \end{minipage} The proof of lemma \texttt{computeRootUpIsCorrectForTree} requires a few intermediate sub-lem\-mas of moderate difficulty. The main step in the proof is to establish an equivalence between a bottom-up computation \texttt{computeRootUp} and the top-down definition of (attributed) Merkle trees. All the proofs are by induction on the tree or the path. The complete Dafny code for algorithm is available in \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/synthattribute/ComputeRootPath.dfy}{computeRootPath.dfy} file. Termination for \texttt{insertValue} is proved by using a ranking function (decreases clause in Listing~]\ref{algo-compute-sib}). The functional correctness of \texttt{insertValue} reduces to proving that, assuming \texttt{left} (resp. \texttt{right}) contains the values of the left (resp. right) siblings of the nodes on \texttt{p}, then \texttt{insertValue(p, left, right, seed)} returns the values of the nodes that are left siblings on the successor path. The specification of the corresponding lemma is given in Listing~\ref{proof-sib}. The code for this lemma is in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/paths/NextPathInCompleteTreesLemmas.dfy}{NextPathInCompleteTreesLemmas.dfy} file. The main proof is based on several sub-lemmas that are not hard conceptually but cannot be easily discharged using solely the built-in Dafny induction strategies. They require some intermediate proof hints (verified calculations) to deal with all the nodes on the path \texttt{p}. Note that for this lemma, we require that the leaves are indexed (from left to right) to be able to uniquely identify each leaf of \texttt{r}. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=Correctness Proof Specification for \texttt{ComputeRootUp}., captionpos=t, label={proof-sib}] lemma insertValuetIsCorrectInATree( p: seq<bit>,r:Tree<int>,left:seq<int>,right:seq<int>,seed:T,k :nat) // r is a Merkle tree requires isCompleteTree(r) requires isDecoratedWith(f, r) // leaves are uniquely indexed from to right requires hasLeavesIndexedFrom(r, 0) // k is an index which is not the index of the last leaf requires k < |leavesIn(r)| - 1 requires 1 <= |p| == height(r) // The leaf at index k is the leaf at the end of p requires nodeAt(p, r) == leavesIn(r)[k] // The value of the leaf at the end of p is seed requires seed == nodeAt(p,r).v requires |p| == |left| == |right| // Left and right contain the values of the siblings on p requires forall i :: 0 <= i < |p| ==> siblingAt(take(p,i + 1), r).v == if p[i] == 0 then right[i] else left[i] // A path to a leaf that is not the rightmost one has a zero ensures exists i :: 0 <= i < |p| && p[i] == 0 // insertValue computes the values of the left siblings // of the successor path of `p`. ensures forall i :: 0 <= i < |p| && nextPath(p)[i] == 1 ==> computeLeftSiblingOnNextPathFromLeftRight(p,left,right,f,seed)[i] == siblingAt(take(nextPath(p),i + 1),r).v \end{lstlisting} \end{minipage} \paragraph{\bf \itshape Index Based Algorithms.} The algorithms that implement the DSC do not use a bitvector to encode a path, but rather, a \emph{counter} that records the number of values inserted so far and the height of the tree. In order to prove the algorithms actually implemented in the DSC, we first recast the \texttt{computeRootUp} and \texttt{insertValue} algorithms to use a counter and the height $h$ of a tree. In this step, we use a parameter \texttt{k} that is the index of the next available leaf where a new value can be inserted. The leaves are indexed left to right from $0$ to $2^{h} - 1$ and hence $k$ is the number of values that have been inserted so far. It follows that the leaves with indices $k \leq i \leq 2^{h} - 1$ have the default value. The correspondence between the bitvector encoding of the path to the leaf at index $k$ and the value $k$ is straightforward: the encoding of the path $p$ is the value of $k$ in binary over $h$ bits. We can rewrite left \texttt{computeRootUp} to use use $k$ and $h$ (\texttt{computeRootUpWithIndex}, Listing~\ref{algo-comp-root-indexed}) and prove it computes the same value as \texttt{computeRootUp}. A similar proof can be established for the \texttt{insertValue} algorithm. The index based algorithms and the proofs that they are equivalent (compute the same values as) to \texttt{computeRootUp} and \texttt{insertValue} are available in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/algorithms/IndexBasedAlgorithm.dfy}{IndexBasedAlgorithm.dfy} file. Dafny can discharge the equivalence proofs with minimal proof hints using the builtin induction strategies. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=\texttt{ComputeRootUpWithIndex}., captionpos=t, label={algo-comp-root-indexed}] computeRootUpWithIndex( h:nat,k:nat,left:seq<int>,right:seq<int>,seed:int):int requires |left| == |right| == h // the index is in the range of indices for a tree of height h requires k < power2(h) // Indexed algorithm computes the same value as computeRootUp ensures computeRootUpWithIndex(h,k,left,right,f,seed) == // natToBitList(k,h) is the binary encoding of k over h bits computeRootUp(natToBitList(k,h),left,right,f,seed) // ranking function decreases h { if h == 0 then seed else if k computeRootUpWithIndex(h-1,k/2,init(left),init(right),hash(seed,last(right))) else // right node computeRootUpWithIndex(h-1,k/2,init(left),init(right),hash(last(left),seed)) } \end{lstlisting} \end{minipage} \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=Implemented Version of \texttt{computeRootUp}., captionpos=t, label={algo-get-root}] method get_deposit_root() returns (r:int) // The result of get_deposit_root_() is the root value of the Merkle tree. ensures r == buildMerkle(values,TREE_HEIGHT,hash).rootv { // Store the expected result in a ghost variable. // values is a ghost variable of ther DSC and record all the inserted values ghost var e := computeRootUpWithIndex(TREE_HEIGHT,count,branch,zero_hashes,0); // Start with default value for r. r := 0; var h := 0; var size := count; while h < TREE_HEIGHT // Main invariant: invariant e == computeRootUpWithIndex( TREE_HEIGHT - h,size, take(branch,TREE_HEIGHT - h),take(zero_hashes,TREE_HEIGHT - h),r) { if size r := hash(branch[h],r); } else { r := hash(r,zero_hashes[h]); } size := size / 2; h := h + 1; } } \end{lstlisting} \end{minipage} \paragraph{\bf \itshape Total Correctness of the Algorithms Implemented in the DSC.} In this section we present the final proof of (total) correctness for the algorithms implemented in the DSC (Solidity-like version.) Our proof establishes that the imperative versions, with while loops and dynamic memory allocation (for arrays) are correct, terminate and are memory safe. The DSC is an object and has a state defined by a few variables: \texttt{count} is the number of inserted values (initially zero), \texttt{branch} is a vector that stores that value of the left siblings of the path leading to the leaf at index \texttt{count}, and \texttt{zero\_hashes} is what we previously defined as $z$. The algorithm that computes the root hash of the Merkle tree in the DSC is \texttt{get\_deposit\_root()}. \texttt{get\_deposit\_root()} does not have any \emph{seed} parameter as it computes the root hash using the default value ($0$). The correctness proof of \texttt{get\_deposit\_root()} uses the functional (proved correct) algorithm \texttt{computeRootUpWithIndex} as an invariant. Listing~\ref{algo-get-root} is a simplified version (for clarity) of the full code available in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/DepositSmart.dfy}{DepositSmart.dfy} file. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=The \texttt{deposit} method., captionpos=t, label={algo-deposit}] method deposit(v:int) // The tree cannot be full. requires count < power2(TREE_HEIGHT) - 1 // branch and zero_hashes hold the values of the siblings requires areSiblingsAtIndex(|values|, buildMerkle(values,TREE_HEIGHT,hash),branch, zero_hashes) // Correctness property ensures areSiblingsAtIndex(|values|, buildMerkle(values,TREE_HEIGHT,hash),branch,zero_hashes) { var value := v; var size : nat := count; var i : nat := 0; // Store the expected result in e. ghost var e := computeLeftSiblingsOnNextpathWithIndex( TREE_HEIGHT,old(size),old(branch),zero_hashes,v); while size // Main invariant: invariant e == computeLeftSiblingsOnNextpathWithIndex( TREE_HEIGHT - i,size, take(branch,TREE_HEIGHT - i), take(zero_h,TREE_HEIGHT - i),value) + drop(branch,TREE_HEIGHT - i) decreases size { value := f(branch[i],value); size := size / 2; i := i + 1; } // 0 <= i < |branch| and no there is no index-out-of-bounds error branch[i] := value; count := count + 1; values := values + [v]; } \end{lstlisting} \end{minipage} The algorithm that inserts a value \texttt{v} in the tree is \texttt{deposit(v)} in the implemented version of the DSC. Listing~\ref{algo-deposit} is an optimised version of the original algorithm. The simplification is explained in Section~\ref{sec-findings}. The correctness of the algorithm is defined by ensuring that, if at the beginning of the computation the vectors \texttt{branch} (resp, \texttt{zero\_hashes}) contain values of the left (resp. right) siblings of the path leading to the leaf at index \texttt{count}, then at the end of the computation, after \texttt{v} is inserted, this property still holds. The proof of this invariant requires a number of proof hints for Dafny to verify it. We use the functional version of the algorithm to specify a loop invariant (not provided in Listing~\ref{algo-deposit}). The termination proof is easy using \texttt{size} as the decreasing ranking function. However, a difficulty in this proof is memory safety, i.e. to guarantee that the index \texttt{i} used to access \texttt{branch[i]} is within the range of the indices of \texttt{branch}. We have also proved the initialisation functions \texttt{init\_zero\_hashes()} and \texttt{constructor}. The full code of the imperative version of the DSC is available in the \href{https://github.com/ConsenSys/deposit-sc-dafny/blob/50b16f96021368a839f932b5d666729405a305b0/src/dafny/smart/DepositSmart.dfy}{DepositSmart.dfy} file. \section{Findings and Lessons Learned}\label{sec-findings} \paragraph{\bf \itshape Methodology.} In contrast to the previous attempts to analyse the DSC, we have adopted a textbook approach and used standard algorithms' design techniques ({e.g.},~ dynamic programming, refinement, recursion.) This has several advantages over a direct proof ({e.g.},~ \cite{formal-inc-merkle-rv}) of the imperative code including: \begin{itemize} \item the design of simple algorithms and proofs; \item recursive and language-agnostic recursive versions of the algorithms; \item new and provably correct simplifications/optimisations. \end{itemize} \paragraph{\bf \itshape Algorithmic Considerations.} Our implementations and formal proofs have resulted in the identification of two previously unknown/unconfirmed optimisations. First, it is not necessary to initialise the vector of left siblings, \texttt{b}, and the algorithms are correct for any initial value of this vector. Second, the original version of the \texttt{deposit} algorithm (which we have proved correct too) has the form\footnote{The complete Solidity source code is freely available on GitHub at \url{https://github.com/ethereum/eth2.0-specs/blob/dev/solidity_deposit_contract/deposit_contract.sol}} given in Listing~\ref{algo-solidity-deposit}. Our formal machine-checkable proof revealed\footnote{This finding was not uncovered in any of the previous audits/analyses.} that indeed the condition \texttt{C1} is always true and the loop always terminates because \texttt{C2} eventually becomes true. As witnessed by the comment after the loop in the Solidity code of the DSC, this property was expected but not confirmed, and the Solidity contract authors did not take the risk to simplify the code. Our result shows that the algorithm can be simplified to \texttt{while not(C2) do ... od}. \noindent\begin{minipage}{\linewidth} \begin{lstlisting}[language=dafny,caption=Solidity Version of the DSC Deposit Function., captionpos=t, label={algo-solidity-deposit}] deposit( ... ) { while C1 do if C2 return; ... od // As the loop should always end prematurely with the `return` statement, // this code should be unreachable. We assert `false` just to be safe. assert(false); } \end{lstlisting} \end{minipage} This is interesting not only from a safety and algorithmic perspectives, but also because it reduces the computation cost (in gas/Ether) of executing the \texttt{deposit} method. This simplification proposal is currently being discussed with the DSC developer, however the currently deployed version still uses the non-optimised code. \paragraph{\bf \itshape Verification Effort.} The verification effort for this project is 12 person-weeks resulting in $3500$ lines of code and $1000$ lines of documentation. This assumes familiarity with program verification, Hoare logic and Dafny. Table~\ref{tab-stats}, page~\pageref{tab-stats} provides some insights into the code base. The filenames in \textcolor{green!50!black}{green} are the ones that require the less number of hints for Dafny to check a proof. In this set of files the hints mostly consist of simple \emph{verified calculations} ({e.g.},~ empty sequence is a neutral element for lists \texttt{[] + l == l + [] == l}.) Most of the results on sequences (\texttt{helpers} package) and simplifications of sequences of bits (\texttt{seqofbits} package) are in this category and require very few hints. This also applies for the proofs\footnote{The file \texttt{CommuteProof.dfy} in this package is not needed for the main proof but was originally used and provides an interesting result, so it is still in code base.} of the \texttt{algorithms} package, {e.g.},~ proving that the versions using the index of a leaf instead of the binary encoding of a path are equivalent. The filenames in \textcolor{orange}{orange} require some non-trivial proof hints beyond the implicit induction strategies built in Dafny. For instance in \textcolor{orange}{NextPathInCompleteTrees.dfy} and \textcolor{orange}{PathInCompleteTrees.dfy}, we had to provide several annotations and structure for the proofs. This is due to the fact that the proofs involve properties on a Merkle tree $t_1$ and its \emph{successor} $t_2$ (after a value is inserted) which is a new tree, and on a path $\pi_1$ in $t_1$ and its successor $\pi_2$ in $t_2$. The filenames in \textcolor{red}{red} require a lot of hints. For the files in the \texttt{synthattribute} package it is mostly calculation steps. Some steps are not absolutely necessary but adding them reduces the verification time by on order of magnitude (on our system configuration, MacBookPro 16GBRAM). The hardest proof is probably the correctness of the \texttt{deposit} method in \textcolor{red}{DepositSmart.dfy}. The proof requires non trivial lemmas and invariants. The difficulty stems from a combination of factors: first the while loop of the algorithm (Listing~\ref{algo-deposit}) maintains a constraint between \texttt{size} and \texttt{i}, the latter being used to access the array elements in \texttt{branch}. Proving that there is no array-of-bounds error ({i.e.},~ $i$ is within the size of \texttt{branch}) requires to prove some arithmetic properties. Second, the proof of the main invariant (Listing~\ref{algo-deposit}) using the functional specification \texttt{computeLeftSiblingsOnNextpathWithIndex} is complex and had to be structured around additional lemmas. \input{table.tex} Overall, almost $90\%$ of the lines of code are (non-executable) proofs, and function definitions used in the proofs. The verified algorithms implemented in the DSC functional are provided in \texttt{DepositSmart.dfy} and account for less than $10\%$ of the code. Considering the criticality of the DSC, 12 person-weeks can be considered a moderate effort well worth the investment: the result is an unparalleled level of trustworthiness that can inspire confidence in the Ethereum platform. According to our experts (ConsenSys Diligence) in the verification of Smart Contracts, the size of such an effort is realistic and practical considering the level of guarantees provided. The only downside is the level of verification expertise required to design the proofs. The trust base in our work is composed of the Dafny verification engine (verification conditions generator) and the SMT-solver Z3. \paragraph{\bf \itshape Dafny Experience.} Dafny is rather has excellent documentation, support for data structures, functional (side-effect free) and object-oriented programming. The automated verification engine has a lot of built-in strategies ({e.g.},~ induction, calculations) and a good number of specifications are proved fully automatically without providing any hints. The Dafny proof strategies and constructs that we mostly used are \emph{verified calculations} and \emph{induction}. The side-effect free proofs seem to be handled much more efficiently (time-wise) than the proofs using mutable data structures. In the current version we have used the \texttt{autocontracts} attribute for the DSC object which is a convenient way of proving memory safety using a specific invariant (given by the \texttt{Valid} predicate). This could probably be optimised as Dafny has some support to specify precisely the side-effects using \emph{frames} (based on \emph{dynamic framing}.) Overall, Dafny is a practical option for the verification of mission-critical smart contracts, and a possible avenue for adoption could be to extend the Dafny code generator engine to support Solidity, a popular language for wri\-ting smart contracts for the Ethereum network, or to automatically translate Solidity into Dafny. We are currently evaluating these options with our colleagues at ConsenSys Diligence, as well as the benefits of our technique to the analysis of other critical smart contracts. The software artefacts including the implementations, proofs, documentation and a Docker container to reproduce the results are freely available as a GitHub repository at \url{https://github.com/ConsenSys/deposit-sc-dafny}. \paragraph{\bf \itshape Acknowledgements.} I wish to thank Suhabe Bugrara, ConsenSys Mesh, for helpful dis\-cus\-sions on the Deposit Smart Contract previous work and the anonymous reviewers of a preliminary version of this paper. \end{document}

\begin{document} \title{Enhanced output entanglement with reservoir engineering} \author{\ Xiao-Bo Yan} \email{yxb@itp.ac.cn} \affiliation{Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China} \date{\today} \pacs{42.50.Ex, 42.50.Wk, 07.10.Cm} \begin{abstract} We study the output entanglement in a three-mode optomechanical system via reservoir engineering by shifting the center frequency of filter function away from resonant frequency. We find the bandwidth of the filter function can suppress the entanglement in the vicinity of resonant frequency of the system, while the entanglement will become prosperous if the center frequency departs from the resonant frequency. We obtain the approximate analytical expressions of the output entanglement, and from which we give the optimal center frequency at which the entanglement takes the maximum. Furthermore, we study the effects of time delay between the two output fields on the output entanglement, and obtain the optimal time delay for the case of large filter bandwidth. \end{abstract} \maketitle \section{Introduction} Cavity optomechanics \cite{Aspelmeyer2014} exploring the interaction between macroscopic mechanical resonators and light fields, has received increasing attention for the potential to detect of tiny mass, force and displacement \cite{Kippenberg2008,Marquardt2009,Verlot2010,Mahajan2013}. The common optomechanical cavity contains one end mirror being a macroscopic mechanical oscillator or a vibrating membrane \cite{Gigan2006,Kleckner2006,Thompson2008,Groblacher2009,wenzhijia,xiaoboyan}. In these optomechanical systems, the motion of mechanical oscillator can be effected by the radiation pressure of cavity field, and this interaction can generate various quantum phenomena. Such as ground-state cooling of mechanical modes \cite{Marquardt2007,Wilson-Rae2007,Bhattacharya2007,Chan2011,Teufel2011,BingHe2017}, electromagnetically induced transparency and normal mode splitting \cite{Huang2009,Weis2010,Safavi-Naeini2011,LiuYX2013,Kronwald2013}, nonlinear interaction effects \cite{Komar2013,Lemonde2013,Borkje2013,LuXY2013} and quantum state transfer between photons with vastly differing wavelengths \cite{Tian2010,Stannigel2010,WangYD2012a,WangYD2012b}. Entanglement is the characteristic element of quantum theory because it is responsible for nonlocal correlations between observables and an essential ingredient in most applications in quantum information. For these reasons, there are a number of theoretical and experimental works on entanglement between macroscopic objects such as, between atomic ensembles \cite{Julsgaard2011,Krauter2011}, and between superconducting qubits \cite{Berkley2003,Neeley2010,DiCarlo2010,Flurin2012}. Recently, quantum entanglement in cavity optomechanics has received increasing attention for the potential to use the interaction to generate various entanglement between subsystems. For example, quantum entanglement between mechanical resonators \cite{Bhattacharya2008,Chen2014,Liao2014,Yang2015}, between different optical modes \cite{Paternostro2007,Wipf2008,Genes2008,Barzanjeh2011,Barzanjeh2012,Barzanjeh2013,Wang2013,Tian2013,Kuzyk2013,Wang2015,Deng2015,Deng2016} , and between mechanical resonators and light modes \cite{Vitali2007,Hofer2011,Akram2012,Sinha2015,BingHe1704} have been studied theoretically and the entanglement between mechanical motion and microwave fields has been demonstrated in a recent experiment \cite{Palomaki2013}. Here, we consider a three-mode optomechanical system in which two cavities are coupled to a common mechanical resonator (see Fig. 1). This setup has been realized in several recent experiments \cite{Dong2012,Hill2012,Andrews2014}. Because in such a system the parametric-amplifier interaction and the beam-splitter interaction can entangle the two intracavity modes, the output cavity ones are also entangle with each other. In previous works \cite{Wang2015,Deng2016}, the entanglement of two output optical fields with their center frequencies same as the resonant frequencies of the cavities has been studied. In Ref. \cite{Wang2015}, the entanglement between the two output fields is enhanced obviously via reservoir engineering \cite{Poyatos1996,Muschik2011}: cooling the Bogoliubov mode through enhancing mechanical decay results in large entanglement between the two target output fields. But these output entanglement in Ref. \cite{Wang2015,Deng2016} will be largely limited by the bandwidth of filter function, and the optimal time delay in Ref. \cite{Wang2015} between the two output fields only suitable for the case of little bandwidth of filter function. In this paper, we first study the effect of filter bandwidth on the output entanglement between the two optical fields without time delay. We find the bandwidth will strongly suppress the output entanglement, specifically as the center frequency of the output fields in the vicinity of resonant frequency. While the output entanglement will become prosperous if the center frequency of output fields departs from the resonant frequency. We will see that the physics behind this phenomenon is the reservoir engineering mechanism because shifting the center frequency can cool the temperature of the system. We obtain all the approximate analytical expressions of the output entanglement in various case, and from which we give the corresponding optimal center frequencies making the entanglement maximum. Finally, we study the effect of the time delay between the two output fields on the output entanglement according to the reservoir engineering mechanism, from which we obtain the approximate analytical expression of the optimal time delay for the case of large filter bandwidth. We think the results of this paper may be used for reference to experimental and theoretical physicists who work on entanglement or quantum information processing. The rest of this paper is organized as follows. In Section II, we introduce the three-mode optomechanical model with a corresponding equivalent model, and the definition of canonical mode operators of the two output optical fields. In Section III, we study the entanglement between the two output optical fields by shifting the center frequency of filter function from resonant frequency. And we study the effects of time delay on the output entanglement. Finally, the conclusions are given in the Section IV. \section{system and an equivalent model} \begin{figure} \caption{(Color online) A three-mode optomechanical system with a mechanical resonator (mode $\hat{b} \label{Fig1} \end{figure} We consider a three-mode optomechanical system in which two cavities are coupled a common mechanical resonator (see Fig. 1). The standard optomechanical Hamiltonian \begin{align} H & =\omega_{m}\hat{b}^{\dag}\hat{b}+\sum_{i=1,2}[\omega_{i}\hat{a} _{i}^{\dag}\hat{a}_{i}+g_{i}(\hat{b}^{\dag}+\hat{b})\hat{a}_{i}^{\dag}\hat {a}_{i}] \label{Eq1} \end{align} governs the system's dynamics, where $\hat{a}_{i}$ is the annihilation operator for cavity $i$ with frequency $\omega_{i}$ and damping rate $\kappa_{i}$, $\hat{b}$ is the annihilation operator for mechanics resonator with frequency $\omega_{m}$ and damping rate $\gamma$, and $g_{i}$ is the optomechanical coupling strength. In order to generate the steady entanglement between the two output fields, we drive cavity 1 (2) at the red (blue) sideband with respect to mechanical resonator: $\omega_{d1}=\omega_{1} -\omega_{m}$ and $\omega_{d2}=\omega_{2}+\omega_{m}$. If we work in a rotating frame with respect to the free Hamiltonian, following the standard linearization procedure, and make the rotating-wave approximation (in this paper, we focus on the resolved-sideband regime $\omega_{m}\gg\kappa _{1},\kappa_{2}$), hence, the Hamiltonian of the system can be written as \begin{align} \hat{H}_{int} & =G_{1}\hat{b}^{\dag}\hat{d}_{1}+G_{2}\hat{b}\hat{d}_{2}+H.c. \end{align} Here, $\hat{d}_{i}=\hat{a}_{i}-\bar{a}_{i}$, $\bar{a}_{i}$ being the classical cavity amplitude. $G_{i}$ is the effective coupling strength. The combined swapping and entangling interactions in $\hat{H}_{int}$ lead to a net entangling interaction between the two intracavity modes as discussed in \cite{Wang2013}. Based on Eq. (2), the dynamics of the system is described by the following quantum Langevin equations for relevant annihilation operators of mechanical and optical modes \begin{align} \frac{d}{dt}\hat{b} & =-\frac{\gamma}{2}\hat{b}-i(G_{1}\hat{d}_{1}+G_{2} \hat{d}_{2}^{\dag})-\sqrt{\gamma}\hat{b}^{in},\nonumber\\ \frac{d}{dt}\hat{d}_{1} & =-\frac{\kappa_{1}}{2}\hat{d}_{1}-iG_{1}\hat {b}-\sqrt{\kappa_{1}}\hat{d}_{1}^{in},\label{Eq3}\\ \frac{d}{dt}\hat{d}_{2}^{\dag} & =-\frac{\kappa_{2}}{2}\hat{d}_{2}^{\dag }+iG_{2}\hat{b}-\sqrt{\kappa_{2}}\hat{d}_{2}^{in,\dag},\nonumber \end{align} In Eq. (3), $\hat{b}^{in},\hat{d}_{i}^{in}$ are the input noise operators of mechanical resonator and cavity $i (i=1,2)$, whose correlation functions are $\langle\hat{b}^{in}(t)\hat{b}^{in,\dag}(t^{\prime})\rangle=N_{m} \delta(t-t^{\prime})$ and $\langle\hat{d}_{i}^{in}(t)\hat{d}_{i}^{in,\dag }(t^{\prime})\rangle=N_{i}\delta(t-t^{\prime})$ respectively. Here, $N_{m}$ and $N_{i}$ are the average thermal populations of mechanical mode and cavity $i$, respectively. In the following discussion, we mainly concentrate on how the effects of the center frequency departing from the resonance, the bandwidth of filter function on the entanglement, so we assume these average thermal populations are zero (zero temperature). According to the Routh-Hurwitz stability conditions \cite{DeJesus1987} and we focus on the regime of strong cooperativities $C_{i}\equiv4G_{i}^{2}/(\gamma\kappa_{i} )\gg1$ and $\kappa_{i}\gg\gamma$ in this paper, the stability condition of our system can be obtained as $G_{1}^{2}/G_{2}^{2}>\max(\kappa_{1}/\kappa _{2},\kappa_{2}/\kappa_{1})$ for $\kappa_{1}\neq\kappa_{2}$, and the system is always stable if $\kappa_{1}=\kappa_{2}$ and $G_{2}\leq G_{1}$ \cite{Wang2013,Wang2015}. For simplicity, we adopt a rectangle filter with a bandwidth $\sigma$ centered about the frequency $\omega$ to generate the output temporal modes. Then, the canonical mode operators of the two output fields can be described as \begin{align} \hat{D}_{i}^{out}[\omega,\sigma,\tau_{i}]=\frac{1}{\sqrt{\sigma}}\int _{\omega_{-}}^{\omega_{+}}d\omega^{\prime}e^{-i\omega^{\prime}\tau_{i}}\hat {d}_{i}^{out}[\omega^{\prime}]. \end{align} Here, $\omega_{\pm}=\omega\pm\frac{\sigma}{2}$, and $\tau_{i}$ is the absolute time at which the wavepacket of interest is emitted from cavity $i$. The frequency-resolved output modes $\hat{d}_{i}^{out} [\omega]\equiv\int d\omega e^{i\omega t}\hat{d}_{i}^{out}[t]/\sqrt{2\pi}$ which can be obtained straightforwardly from the system Langevin equations and input-output relations \cite{Gardiner2004}. And we use the logarithmic negativity \cite{Vidal2002,Plenio2005} to quantify the entanglement between the two output cavity modes $\hat{D}_{1}^{out}[\omega,\sigma,\tau_{1}]$ and $\hat{D}_{2}^{out}[-\omega,\sigma,\tau_{2}]$. Without loss of generality, we set $\tau_{2}=0$, and we write $\hat{D}_{i} ^{out}[\omega,\sigma,\tau_{i}]$ as $\hat{D}_{i}$ for simplicity in the following. It can be proofed that our system can be mapped to a two-mode squeezed thermal state \cite{Wang2015} \begin{align} \hat{\rho}_{12}=\hat{S}_{12}(R_{12})[\hat{\rho}_{1}^{th}(\bar{n}_{1} )\otimes\hat{\rho}_{2}^{th}(\bar{n}_{2})]\hat{S}_{12}^{\dag}(R_{12}) \end{align} Here, \begin{align} \hat{S}_{12}(R_{12})=\exp[R_{12}\hat{D}_{1}\hat{D}_{2}-H.c.] \end{align} is the two-mode squeeze operator, with $R_{12}$ being the squeezing parameter, and $\rho_{i}^{th}(\bar{n}_{i})$ describes a single-mode thermal state with average population $\bar{n}_{i}$. Hence, the output fields are thus completely characterized by just three parameters: $\bar{n}_{1}$, $\bar{n}_{2}$, $R_{12} $. The relationship between the two-mode squeezed thermal state and our system can be obtained as follows \begin{align} \bar{n}_{1} & =\frac{\langle\hat{D}_{1}^{\dag}\hat{D}_{1}\rangle-\langle \hat{D}_{2}^{\dag}\hat{D}_{2}\rangle-1+\sqrt{A^{2}-4|\langle\hat{D}_{1}\hat {D}_{2}\rangle|^{2}}}{2},\nonumber\\ \bar{n}_{2} & =\frac{\langle\hat{D}_{2}^{\dag}\hat{D}_{2}\rangle-\langle \hat{D}_{1}^{\dag}\hat{D}_{1}\rangle-1+\sqrt{A^{2}-4|\langle\hat{D}_{1}\hat {D}_{2}\rangle|^{2}}}{2},\nonumber\\ R_{12} & =\frac{1}{2}\mathtt{arctanh}(\frac{2|\langle\hat{D}_{1}\hat{D} _{2}\rangle|}{A}), \end{align} here, $\langle\hat{D}_{1}^{\dag}\hat{D}_{1}\rangle$, $\langle\hat{D}_{2} ^{\dag}\hat{D}_{2}\rangle$, $\langle\hat{D}_{1}\hat{D}_{2}\rangle$ are the correlators of the output cavity modes, which can be obtained by Langevin equations Eq. (3) and input-output relation, and $A=\langle\hat{D}_{1}^{\dag }\hat{D}_{1}\rangle+\langle\hat{D}_{2}^{\dag}\hat{D}_{2}\rangle+1$. According to Eq. (5) and Eq. (6), the output entanglement $E_{n}$ of this two-mode squeezed thermal state (if $E_{n}\geq0$) can be simply given by \begin{align} E_{n}=-\ln(n_{R}-\sqrt{n_{R}^{2}-(1+2\bar{n}_{1})(1+2\bar{n}_{2})}) \end{align} with $n_{R}=(\bar{n}_{1}+\bar{n}_{2}+1)\cosh2R_{12}$. It can be seen from Eq. (8) that the entanglement will increase with the increase of the squeezing parameter $R_{12}$, while decrease with the increase of the average populations $\bar{n}_{1}, \bar{n}_{2}$. In the following, it can be seen that shifting the center frequency of filter function from the resonance can evidently cool the temperature of the system (decrease the average populations $\bar{n}_{1}, \bar{n}_{2}$). \section{cavity output entanglement} For simplicity, we set equal cavity damping rate $\kappa_{1}=\kappa_{2} =\kappa$, equal coupling $G_{1}=G_{2}=G$, and $\gamma\ll\sigma,\kappa,G$ in the following. We discuss the output entanglement on two cases: shifting the filter center frequency $\omega$ from the resonant frequency (the resonant frequency is zero in the rotating frame) under the condition of small bandwidth ($\sigma\ll\kappa$), and large bandwidth ($\sigma=\kappa$) respectively. \begin{figure} \caption{(a) The entanglement vs the normalized center frequency $\omega/\kappa$. The black-solid line is numerical result, the red-dashed line is plotted according to analytical expression Eq. (10). (b) The squeezing parameter $R_{12} \label{Fig2} \end{figure} \subsection{Small bandwidth} In this section we discuss the effects of small bandwidth $\sigma$ ($\sigma\ll\kappa$) on the entanglement between the two output fields. If we shift the filter center frequency $\omega$ to satisfy $0\leq\omega\leq\frac{\sigma}{2}$ (in the rotating frame), the approximate analytical expression of the output entanglement can be written as \begin{equation} E_{n}\approx\frac{\pi\gamma}{2\sigma}. \end{equation} It can be seen from Eq. (9) that the entanglement between output fields is not related to the filter center frequency $\omega$ and the coupling strength $G$. And increasing the mechanical decay rate $\gamma$ can enhance the output entanglement in the vicinity of resonant frequency $\omega=0$ just as what the author did in Ref. \cite{Wang2015}, which is the reservoir engineering mechanism because increasing mechanical decay rate $\gamma$ can cool the Bogoliubov mode \cite{Wang2015}. If the mechanical damping rate $\gamma$ satisfies $\gamma\ll\sigma$, the entanglement will almost equal to zero. It can also be seen from Eq. (9) that the output entanglement can be largely suppressed by increasing the filter bandwidth $\sigma$. If the center frequency $\omega$ satisfies $\frac{\sigma}{2}<\omega <\frac{\kappa}{2}$, and the coupling strength $G$ is weak coupling ($G<\kappa$), the analytical expression of the entanglement can be simplified to \begin{equation} E_{n}\approx-\ln\frac{20G^{4}\sigma^{2}+3\kappa^{2}\omega^{4}} {3\omega^{2}(64G^{4}+\sqrt{2}\kappa^{2}\omega^{2})}. \end{equation} \begin{figure} \caption{(a) The entanglement vs the normalized center frequency $\omega/\kappa$. The black-solid line is numerical result, the red-dashed line is plotted according to analytical expression Eq. (11). (b) The squeezing parameter $R_{12} \label{Fig3} \end{figure} The entanglement is plotted in Fig. 2(a) with parameters $\gamma=1, \sigma=10, \kappa=10^{5}, G=\kappa/10$. The black-solid line is numerical result according to logarithmic negativity, while the red-dashed line is plotted according to simplified analytical expression Eq. (10). The entanglement is not monotonic with the change of center frequency $\omega$, and will reach a maximum as the optimal center frequency satisfy $\omega_{opt}\approx6^{1/4}G(\sigma /\kappa)^{1/2}$. The entanglement will appear a peak value at resonant frequency ($\omega=0$) for the case $\sigma=0$ \cite{Wang2015}, but the peak will emerge at some a center frequency $\omega$ for the case $\sigma\neq0$. We can give a clear reason for this phenomenon from Fig. 2(b) in which the squeezing parameter $R_{12}$ (red-dashed line), the thermal populations $\bar{n}_{1}$ (blue-dotted line), $\bar{n}_{2}$ (black-solid line) vs the normalized center frequency $\omega/\sigma$ are plotted. It can be seen from Fig. 2(b) the two thermal populations $\bar{n}_{1}$, $\bar{n}_{2}$ are very large (the temperature of the equivalent two-mode squeezing thermal state is very high) for $\omega<\sigma/2$, then the entanglement is almost zero. But if the center frequency $\omega$ become larger ($\omega>\sigma/2$), the two thermal populations $\bar{n}_{1}$, $\bar{n}_{2}$ will decrease rapidly while the squeezing parameter $R_{12}$ decrease very slowly. Hence, the entanglement become larger with the increase of center frequency $\omega$ until the highest point. As a result, the optimal center frequency $\omega_{opt}$ at which the entanglement reaches a maximum must be greater than $\sigma/2$. If the coupling strength $G$ is strong coupling ($G>\kappa$), and the filter center frequency $\omega$ still satisfies $\frac{\sigma}{2}<\omega<\frac{\kappa}{2}$, the analytical expression of the entanglement can be simplified to \begin{equation} E_{n}\approx-\frac{1}{2}\ln[\frac{G^{8}\sigma^{4}+G^{4}\sigma^{2} \omega^{4}\kappa^{2}+2\omega^{10}\kappa^{2}}{144G^{8}\omega^{4}}], \end{equation} which reaches a maximum as the optimal center frequency satisfy $\omega_{opt}\approx(G^{8}\sigma^{4}/3\kappa^{2})^{1/5}$. The entanglement is plotted in Fig. 3(a) with parameters $\gamma=1, \sigma=10, \kappa=10^{5}, G=10\kappa$. The black-solid line is numerical result according to logarithmic negativity, while the red-dashed line is plotted according to simplified analytical expression Eq. (11). It can be seen from Fig. 2, Fig. 3 that the curves of entanglement plotted by simplified analytical expressions fits the numerical results very well, the squeezing parameter $R_{12}$ of strong coupling is larger than the case of weak coupling, and the two thermal populations $\bar{n}_{1}$, $\bar{n}_{2}$ of strong coupling will also decrease rapidly as the center frequency $\omega>\sigma/2$ just as the case of weak coupling. That is the reason why the entanglement of strong coupling will be larger than the one of weak coupling. According to the above analysis that the optimal center frequency $\omega_{opt}$ must be greater than $\sigma/2$, hence $\omega_{opt}$ will be far away from the resonant frequency $\omega$ ($\omega=0$) if $\sigma$ is very large. We will discuss the case $\sigma=\kappa$ in the following. \subsection{Large bandwidth} \begin{figure} \caption{(a) The entanglement $En$ vs the normalized center frequency $\omega/\kappa$: The red-solid line is the entanglement plotted with the optimal time delay Eq. (14), the blue-dashed line is the entanglement plotted with the numerical optimal time delay making the entanglement $En$ maximum, the black-solid line is the entanglement plotted according to analytical expression Eq. (12) without time delay, and the green-dashed-dotted line is the entanglement plotted by numerical result according to the logarithmic negativity without time delay. (b) The optimal time delay $\tau_{opt} \label{Fig4} \end{figure} For $G<\kappa$ and large $\sigma$, such as $G=\kappa/10$ and $\sigma=\kappa$, the entanglement will be very small. Hence, in this section, we just discuss the entanglement of strong coupling $G>\kappa$ with the bandwidth $\sigma=\kappa$. Because of $\sigma=\kappa\gg\gamma$, the entanglement almost be zero when $0\leq\omega\leq\frac{\kappa}{2}$ according to Eq. (9). The analytical expression of the entanglement can be simplified to \begin{equation} E_{n}\approx\ln[\sqrt{2}(\frac{3G^{4}\kappa^{2}(\omega^{2}+\frac {3\kappa^{2}}{4})+G^{2}\kappa^{2}\omega^{4}+\omega^{8}} {3G^{4}\kappa ^{4}+2G^{2}\omega^{2}\kappa^{4}+\omega^{8}})] \end{equation} for $\frac{\kappa}{2}\lesssim\omega\lesssim7\kappa,$ and the optimal center frequency $\omega_{opt}\approx\sqrt{G\kappa}$. In Fig. 4(a), we plot the entanglement vs center frequency $\omega/\kappa$ according to the analytical expression Eq. (9), Eq. (12) (black-solid line) and the numerical result according to the logarithmic negativity (green-dashed-dotted line) under the parameters: $\gamma=1,\sigma=\kappa=10^{5},G=10\kappa$. It can be seen from Fig. 4(a) that there still is large entanglement even with large bandwidth ($\sigma=\kappa$). This because shifting center frequency can effectively cool the two thermal populations $\bar{n}_{1}$, $\bar{n}_{2}$ via reservoir engineering as above. And the tendencies of the two thermal populations $\bar{n}_{1}$, $\bar{n}_{2}$ and the squeezing parameter $R_{12}$ are almost the same as the previous cases in Fig. 2(b), Fig. 3(b), we don't discuss them any more. As the above analysis, large bandwidth $\sigma$ must strongly influence the entanglement of the two output fields. According to the definition of the canonical mode operators $\hat{D}_{i}$ (see Eq. (4)), the correlator of the output cavity modes $\langle\hat{D}_{1}\hat{D}_{2}\rangle$ is connected with time delay $\tau$, while the other two correlators $\langle\hat{D}^{\dag}_{1}\hat{D}_{1}\rangle$, $\langle\hat{D}^{\dag}_{2}\hat{D}_{2}\rangle$ are not. The expression $\langle\hat{D}_{1}\hat{D}_{2}\rangle$ can be written explicitly as \begin{equation} \langle\hat{D}_{1}\hat{D}_{2}\rangle=\int_{\omega_{-}}^{\omega_{+}}\frac{e^{-i\tau\Omega}(8G^{2}\kappa+(\gamma+2i\Omega)(\kappa ^{2}+4\Omega^{2}))}{-(\gamma^{2}+4\Omega^{2})(\kappa^{2}+4\Omega^{2})^{2}/(8G^{2}\kappa) }d\Omega. \end{equation} The effect of time delay $\tau$ on entanglement $E_{n}$ can be seen easily form the equivalent two-mode squeezing thermal state. From Eq. (7), we can see that the two-mode squeezing parameters $\bar{n}_{1}$, $\bar{n}_{2}$, and $R_{12}$ are affected by time delay $\tau$ just through the correlator $\langle\hat{D}_{1}\hat{D}_{2}\rangle$. More specifically, $\bar{n}_{1}$, $\bar{n}_{2}$ will decrease and $R_{12}$ will increase if the modulus $|\langle\hat{D}_{1}\hat{D}_{2}\rangle|$ becomes large as other parameters fixed except for time delay $\tau$. Hence, we can assert categorically that the output entanglement $E_{n}$ will increase with the increasing of the modulus of the correlator $\langle\hat{D}_{1}\hat{D}_{2}\rangle$. The optimal time delay $\tau_{opt}$ is the delay which makes the $|\langle\hat{D}_{1}\hat{D}_{2}\rangle|$ reach a maximum. After obtaining the approximate analytical expression about $|\langle\hat{D}_{1}\hat{D}_{2}\rangle|$ and making some corrections, we find the optimal time delay is \begin{equation} \tau_{opt}\approx \begin{cases} \frac{3G^{2}\kappa(\omega^{2}-\frac{\kappa^{2}}{8})}{G^{4}\kappa^{2} +\omega^{6}}, & \omega\geq\frac{\kappa}{2}.\\ \frac{\pi\kappa}{2(2+\pi)G^{2}}, & 0\leq\omega<\frac{\kappa}{2}. \end{cases} \end{equation} We plot the output entanglement $E_{n}$ with optimal time delay $\tau_{opt}$ (red-solid line) based on Eq. (14), and that with numerical optimal time delay which makes the entanglement $E_{n}$ reach a maximum (blue-dashed line) in Fig. 4(a) and the corresponding time delays are plotted in Fig. 4 (b) with the parameters: $\gamma=1,\sigma=\kappa=10^{5},G=10\kappa$, and they all fit very well. It can be seen from Fig. 4(a) that the time delay $\tau$ strongly affects the entanglement $E_{n}$ as long as the center frequency $\omega$ is not big enough compared with bandwidth $\sigma$, while has no effect on the entanglement $E_{n}$ as $\omega\gg\sigma$. The reason is that the effect of fixing $\sigma$ and increasing $\omega$ is equivalent to that of fixing $\omega$ and decreasing $\sigma$. And the time delay $\tau$ has no effect on entanglement for the case of $\sigma\rightarrow0$, which can be seen according to Eq. (13) that the factor $e^{-i\tau\Omega}$ can be extracted out of the integration for small bandwidth $\sigma$ with the result that the modulus $|\langle\hat{D}_{1}\hat{D}_{2}\rangle|$ will be not related to $\tau$. The steep entanglement in the vicinity $\omega=\sigma/2$ is because of the special rectangle filter and reaches a local minimum ($En_{min}\approx1.68$) at $\omega=\sigma/2$ according to the numerical result. \section{Conclusions} In summary, we have studied theoretically the output entanglement between two output cavity fields via reservoir engineering by shifting the center frequency of the causal filter function away from the resonance ($\omega=0$ in the rotating frame) in a three-mode cavity optomechanical system. We find that the nonzero bandwidth $\sigma$ can largely suppress the entanglement $En$, specifically in the vicinity of resonant frequency $En\sim1/\sigma$. While the output entanglement will become prosperous, if we shift the center frequency of output fields away from the resonant frequency. This is because shifting center frequency can effectively cool the two-mode squeezing thermal state which is equivalent to our model. We obtain all the approximate analytical expressions of the output entanglement, and from which we give the corresponding optimal center frequencies $\omega_{opt}$. In addition, we find the time delay $\tau$ between the two output optical fields can evidently effect the output entanglement. And we obtain the analytical expression of the optimal time delay $\tau_{opt}$ in the case of large filter bandwidth ($\sigma=\kappa$). Our results can also be applied to other parametrically coupled three-mode bosonic systems, and may be useful to experimentalists to obtain large entanglement. \end{document}

\begin{document} \title[The Swing Lemma and $\E C_1$-diagrams] {Using the Swing Lemma and $\E C_1$-diagrams for congruences of planar semimodular lattices} \author[G.\ Gr\"atzer]{George Gr\"atzer} \email{gratzer@me.com} \urladdr{http://server.maths.umanitoba.ca/homepages/gratzer/} \address{University of Manitoba} \date{June 6, 2021} \begin{abstract} A planar semimodular lattice $K$ is \emph{slim} if $\SM{3}$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No Child Property}: \emph{Let~$\mathcal{P}$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in \mathcal{P}$ and let $z$ be a~maximal element of $\mathcal{P}$. If $x \neq y$ and $x, y \prec z$ in $\mathcal{P}$, then there is no element $u$ of $\mathcal{P}$ such that $u \prec x, y$ in $\mathcal{P}$.} We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Cz\'edli's, to verify his four properties. \end{abstract} \subjclass[2000]{06C10} \keywords{Rectangular lattice, slim planar semimodular lattice, congruence lattice} \maketitle \section{Introduction}\label{S:Introduction} Let $K$ be a planar semimodular lattice. We call the lattice $K$ \emph{slim} if $\SM{3}$ is not a~sublattice of~$K$. In the paper \cite[Theorem 1.5]{gG14a}, I found a property of congruences of slim, planar, semimodular lattices. In the same paper (see also Problem 24.1 in G. Gr\"atzer~\cite{CFL2}), I~proposed the following: \tbf{Problem.} Characterize the congruence lattices of slim planar semimodular lattices. G. Cz\'edli ~\cite[Corollaries 3.4, 3.5, Theorem 4.3]{gCa} found four new properties of congruence lattices of slim, planar, semimodular lattices. \begin{theoremn}\label{T:main} Let $K$ be a slim, planar, semimodular lattice with at least three elements and let~$\E P$ be the ordered set of join-irreducible congruences of $K$. \begin{enumeratei} \item \emph{Partition Property:} The set of maximal elements of $\E P$ can be divided into the disjoint union of two nonempty subsets such that no two distinct elements in the same subset have a common lower cover.\label{E:LC} \item \emph{Maximal Cover Property:} If $v \in \E P$ is covered by a maximal element $u$ of $\E P$, then $u$ is not the only cover of $v$. \item \emph{No Child Property:} Let $x \neq y \in \E P$ and let $u$ be a maximal element of $\E P$. If $x,y \prec u$ in $\E P$, then there is no element $z \in \E P$ such that $z \prec x, y$ in $\E P$. \item \emph{Four-Crown Two-pendant Property:} There is no cover-preserving embedding of the ordered set $\E R$ in Figure~\ref{F:notation} into $\E P$ satisfying the property\tup{:} any maximal element of~$\E R$ is a maximal element of $\E P$. \end{enumeratei} \end{theoremn} In this paper, we will provide a short and direct proof of this theorem using only the Swing Lemma and $\E C_1$-diagrams, see Section~\ref{S:Tools}. \begin{figure} \caption{The Four-crown Two-pendant ordered set $\E R$ with notation; the covering $\SfS 7$ sublattice with edge and element notation} \label{F:notation} \end{figure} \subsection*{Outline} Section~\ref{S:Motivation} provides the motivation for Cz\'edli's Theorem. Section~\ref{S:Tools} provides the tools we need: the Swing Lemma, $\E C_1$-diagrams, and forks. Section~\ref{S:partition} proves the Partition Property, Section~\ref{S:Maximal} does the Maximal Cover Property, while Section~\ref{S:Child} verifies the No Child Property. Finally, The Four-Crown Two-pendant Property is proved in Section~\ref{S:Crown}. \section{Motivation}\label{S:Motivation} In my paper \cite{GLS98a} with H. Lakser and E.\,T. Schmidt, we proved that every finite distributive lattice $D$ can be represented as the congruence lattice of a semimodular lattice $L$. To our surprise, the semimodular lattice $K$ we constructed was \emph{planar}. G.~Gr\"atzer and E.~Knapp~\cite{GKn07}--\cite{GK10} started the study of planar semimodular lattices. I continued it with my ``Notes on planar semimodular lattices'' series (started with Knapp): \cite{gG13}, \cite{GW10} (with T. Wares), \cite{CG12} (with G. Cz\'edli), \cite{gG19}, \cite{gG21b}. See also G. Cz\'edli and E.\,T. Schmidt \cite{CS13} and G. Cz\'edli \cite{gC14}--\cite{gCb}. A major subchapter of the theory of planar semimodular lattices started with the observation that in the construction of the lattice $K$, as in the first paragraph of this section, $\SM{3}$ sublattices play a crucial role. It was natural to raise the question what can be said about congruence lattices of slim, planar, semimodular (SPS) lattices (see [CFL2, Problem~24.1], originally raised in G. Gr\"atzer~\cite{gG14a}). In~\cite{gG14a}, I~found the first necessary condition and G. Cz\'edli \cite{gC14a} proved that this condition is not sufficient (see also my related papers \cite{gG15a} and \cite{gG19}). A number of papers developed tools to tackle this problem: the Swing Lemma (G. Gr\"atzer~\cite{gG15}), trajectory coloring (G. Cz\'edli \cite{gC14}), special diagrams (G. Cz\'edli \cite{gC17}), lamps (G. Cz\'edli \cite{gCa}). Some of these results require long proofs. The proof of the trajectory coloring theorem is just shy of 20 pages, while the basic theory of lamps and its application to Theorem~\ref{T:main} is 23 pages. There are a number of surveys of this field, see the book chapters G.~Cz\'edli and G.~Gr\"atzer~\cite{CG14} and G.~Gr\"atzer~\cite{gG13b} in G.~Gr\"atzer and F.~Wehrung, eds.\,~\cite{LTS1}. My~presentation \cite{gG21a} provides a gentle review for the background of this topic. \section{The tools we need}\label{S:Tools} Most basic concepts and notation not defined in this paper are available in Part~I of the book \cite{CFL2}, see \verb+https://www.researchgate.net/publication/299594715+\\ \indent {\tt arXiv:2104.06539} \noindent It is available to the reader. We will reference it, for instance, as [CFL2, page 52]. In particular, we use the notation $C \persp D$, $C \perspup D$, and $C \perspdn D$ for perspectivity, up-perspectivity, and down-perspectivity, respectively. As usual, for planar lattices, a prime interval (or covering interval) is called an \emph{edge}. For a finite lattice $K$ and a~finite ordered set $R$, a \emph{cover-preserving} embedding $\ge \colon R \to K$ is an embedding~$\ge$ mapping edges of $R$ to edges of $K$. We define a \emph{cover-preserving} sublattice similarly. For the lattice $\SfS 7$ of Figure~\ref{F:notation}, we need a variant: an $\SfS 7$ sublattice $\SfS{}$ (a sublattice isomorphic to $\SfS 7$) is a \emph{peak sublattice} if the three top edges ($L$, $M$, and $R$ in Figure~\ref{F:notation}) are edges in $K$. By G. Gr\"atzer and E. Knapp \cite{GKn09}, every slim, planar, semimodular lattice $K$ has a congruence-preserving extension (see [CFL2, page 43]) $\hat K$ to a slim rectangular lattice. Any of the properties (i)--(iv) holds for $K$ if{}f it holds for $\hat K$. Therefore, in~the rest of this paper, we can assume that $K$ is a slim rectangular lattice, simplifying the discussion. \subsection{Swing Lemma}\label{S:Swing} For an edge $E$ of an SPS lattice $K$, let $E = [0_E, 1_E]$ and define $\Col{E}$, the \emph{color of}~$E$, as $\con E$, the (join-irreducible) congruence generated by collapsing $E$ (see [CFL2, Section 3.2]). We write $\E P$ for $\Ji {\Con K}$, the ordered set of join-irreducible congruences of $K$. As in my paper~\cite{gG15}, for the edges $U, V$ of an SPS lattice $K$, we define a binary relation: $U$~\emph{swings} to $V$, in formula, $U \swing V$, if $1_U = 1_V$, the element $1_U = 1_V$ of~$K$ covers at least three elements, and $0_V$ is neither the left-most nor the right-most element covered by $1_U = 1_V$; if also $0_U$ is such, then the swing is \emph{interior}, otherwise, it is \emph{exterior}, denoted by $U \inswing V$ and $U \exswing V$, respectively. \begin{named}{Swing Lemma [G. Gr\"atzer~\cite{gG15}]} Let $K$ be an SPS lattice and let $U$ and $V$ be edges in $K$. Then $\Col V \leq\Col U$ if{}f there exists an edge $R$ such that $U$ is up-perspective to $R$ and there exists a sequence of edges and a~sequence of binary relations \begin{equation*}\label{E:sequence} R = R_0 \mathbin{\gr}_1 R_1 \mathbin{\gr}_2 \dots \mathbin{\gr}_n R_n = V, \end{equation*} where each relation $\mathbin{\gr}_i$ is $\perspdn$ \pr{down-perspective} or $\swing$ \pr{swing}. In~addition, this sequence also satisfies \begin{equation*}\label{E:geq} 1_{R_0} \geq 1_{R_1} \geq \dots \geq 1_{R_n}. \end{equation*} \end{named} The following statements are immediate consequences of the Swing Lemma, see my papers~\cite{gG15} and \cite{gG14e}. \begin{corollary}\label{C:equal} We use the assumptions of the Swing Lemma. \begin{enumeratei} \item The equality $\Col U = \Col V$ holds in $\E P$ if{}f there exist edges $S$ and $T$ in $K$, such that \begin{equation*}\label{E:xx} U \perspup S,\ S \inswing T,\ T \perspdn V. \end{equation*} \item Let us further assume that the element $0_U$ is meet-irreducible. Then the equality $\Col U = \Col V$ holds in $\E P$ if{}f there exists an edge $T$ such that $U \inswing T \perspdn V$. \item If the lattice $K$ is rectangular and $U$ is on the upper boundary of $K$, then the equality $\Col U = \Col V$ holds in $\E P$ if{}f $U \perspdn V$. \end{enumeratei} \end{corollary} Note that in (i) the edges $S, T, U, V$ need not be distinct, so we have as special cases $U = V$, $U \persp V$, $S = T$, and others. \begin{corollary}\label{C:cov} We use the assumptions of the Swing Lemma. \begin{enumeratei} \item The covering $\Col V \prec \Col U$ holds in $\E P$ if{}f there exist edges $R_1, \dots, R_4$ in~$K$, such that \begin{equation*} U \perspup R_1,\ R_1 \inswing R_2,\ R_2 \perspdn R_3,\ R_3 \exswing R_4,\ R_4 \perspdn V. \end{equation*} \item If the element $0_U$ is meet-irreducible, then the covering $\Col V \prec \Col U$ holds in $\E P$ if{}f there exist edges $S, T$ in $K$, so that \begin{equation*} U \perspdn S \exswing T \perspdn V. \end{equation*} \end{enumeratei} \end{corollary} \begin{corollary}\label{C:covnew} Let $K$ be a slim rectangular lattice, let $U$ and $V$ be edges in $K$, and let $U$ be in the upper-left boundary of $K$. \begin{enumeratei} \item The covering $\Col V \prec \Col U$ holds in $\E P$ if{}f there exist edges $S, T$ in $K$, such that \begin{equation}\label{E:seq5} U \perspdn S \exswing T \perspdn V. \end{equation} \item Define the element $t = 1_S = 1_T \in K$ and let $S = E_1, E_2, \dots, E_n = W$ enumerate, from left to right, all the edges $E$ of $K$ with $1_E = t$. Then \begin{align} \col{S} &\neq \col{W},\label{E:1}\\%\eqref{E:1} \col{E_2} = \cdots &= \col{E_{n-1}} = \col{T},\label{E:2}\\%\eqref{E:2} \col{T} &\prec \col{S}, \col{W}.\label{E:3} \end{align} \end{enumeratei} \end{corollary} \begin{corollary}\label{C:max} Let the edge $U$ be on the upper edge of the rectangular lattice $K$. Then $\Col U$ is a maximal element of $\E P$. \end{corollary} The converse of this statement is stated in Corollary~\ref{C:max1}. \subsection{$\E C_1$-diagrams}\label{S:diagrams} In the diagram of a planar lattice $K$, a \emph{normal edge} (\emph{line}) has a~slope of $45\degree$ or $135\degree$. If it is the first, we call it a~\emph{normal-up edge} (\emph{line}), otherwise, a \emph{normal-down edge} (\emph{line}). Any edge of slope strictly between $45\degree$ and $135\degree$ is \emph{steep}. \begin{definition}\label{D:well} A diagram of an rectangular lattice $K$ is a \emph{$\E C_1$-diagram} if the middle edge of any covering $\SfS 7$ is steep and all other edges are normal. \end{definition} This concept was introduced in G.~Cz\'edli~\cite[Definition 5.3(B)] {gC17}, see also G.~Cz\'edli \cite[Definition 2.1]{gCa} and G. Cz\'edli and G.~Gr\"atzer~\cite[Definition 3.1]{CG21}. The following is the existence theorem of $\E C_1$-diagrams in G. Cz\'edli \cite[Theorem 5.5]{gC17}. \begin{theorem}\label{T:well} Every rectangular lattice lattice $K$ has a $\E C_1$-diagram. \end{theorem} See the illustrations in this paper for examples of $\E C_1$-diagrams. For a short and direct proof for the existence of $\E C_1$-diagrams, see my paper~\cite{gG21b}. \emph{In this paper, $K$ denotes a slim rectangular lattice with a fixed $\E C_1$-diagram and~$\E P$ is the ordered set of join-irreducible congruences of $K$}. Let $C$ and $D$ be maximal chains in an interval $[a,b]$ of $K$ such that $C \ii D = \set{a,b}$. If there is no element of~$K$ between $C$ and $D$, then we call $C \uu D$ a~\emph{cell}. A~four-element cell is a \text{\emph{$4$-cell}}. Opposite edges of a $4$-cell are called \emph{adjacent}. Planar semimodular lattices are $4$-cell lattices, that is, all of its cells are $4$-cells, see G.~Gr\"atzer and E. Knapp \cite[Lemmas 4, 5]{GKn07} and [CFL2,~Section 4.1] for more detail. The following statement illustrates the use of $\E C_1$-diagrams. \begin{lemma}\label{L:application} Let $K$ be a slim rectangular lattice $K$ with a fixed $\E C_1$-diagram and let~$X$ be a normal-up edge of $K$. Then $X$ is up-perspective either to an edge in the upper-left boundary of $K$ or to a steep edge. \end{lemma} \begin{proof} If $X$ is not steep nor it is in the upper-left boundary of $K$, then there is a~$4$-cell $C$ whose lower-right edge is $X$. If the upper-left edge is steep or it is in the upper-left boundary, then we are done. Otherwise, we proceed the same way until we reach a~steep edge or an edge the upper-left boundary. \end{proof} \begin{corollary}\label{C:max1} Let the edge $U$ be on the upper edge of $K$. Then $\Col U$ is a maximal element of $\E P$. Conversely, if $u$ is a maximal element of $\E P$, then there is an edge $U$ on the upper edge of $K$ so that $\Col U = u$. \end{corollary} \subsection{Trajectories}\label{S:Trajectories} G. Cz\'edli and E.\,T. Schmidt \cite{CS11} introduced a \emph{trajectory} in $K$ as a maximal sequence of consecutive edges, see also [CFL2, Section~4.1]. The \emph{top edge}~$T$ of a trajectory is either in the upper boundary of $K$ or it is steep by Lemma~\ref{L:application}. For such an edge~$T$, we denote by $\traj T$ the trajectory with top edge~$T$. By G.~Gr\"atzer and E. Knapp \cite[Lemma 8]{GKn07}, an element $a$ in an SPS lattice~$K$ has at most two covers. Therefore, a trajectory has at most one top edge and at most one steep edge. So we conclude the following statement. \begin{lemma}\label{L:disj} Let $K$ be a slim rectangular lattice $K$ with a fixed $\E C_1$-diagram. Let $X$ and $Y$ be distinct steep edges of $K$. Then $\traj X$ and $\traj Y$ are disjoint. \end{lemma} \section{The Partition Property}\label{S:partition} First, we verify the Partition Property for the slim rectangular lattice $K$ and with a fixed $\E C_1$-diagram. We start with a lemma. \begin{lemma}\label{L:disjoint} Let $X $ and $Y$ be distinct edges on the upper-left boundary of $K$. Then there is no edge $Z$ of $K$ such that $\col Z \prec \col X, \col Y$. \end{lemma} \begin{proof} By way of contradiction, let $Z$ be an edge such that $\col Z \prec \col X, \col Y$. Since $X$ and $Y$ are on the upper-left boundary, Corollary~\ref{C:covnew}(i) applies. Therefore, there exist normal-up edges $S_X, S_Y$ and steep edges $T_X, T_Y$ such that \[ X \perspdn S_X \exswing T_X,\q Y \perspdn S_Y \exswing T_Y,\q Z \in \traj {T_X} \ii \traj {T_Y}. \] By Lemma~\ref{L:disj}, the third formula implies that $T_X = T_Y$ and xo $X = Y$, contrary to the assumption. \end{proof} By Corollary~\ref{C:max1}, the set of maximal elements of $\E P$ is the same as the set of colors of edges in the upper boundaries. We can partition the set of edges in the upper boundaries into the set of edges~$\E L$ in the upper-left boundary and the set of edges~$\E R$ in the upper-right boundary. If $X $ and $Y$ are distinct edges in $\E L$, then there is no edge $Z$ of $K$ such that $\col Z \prec \col X, \col Y$ by Lemma~\ref{L:disjoint}. By symmetry, this verifies the Partition Property. \section{The Maximal Cover Property}\label{S:Maximal} Next, we verify the Maximal Cover Property for the slim rectangular lattice~$K$ and with a fixed $\E C_1$-diagram. Let $x \in \E P$ be covered by a maximal element $u$ of $\E P$ in $K$. By Corollary~\ref{C:max1}, we can choose an edge $U$ of color $u$ on the upper boundary of $K$, by symmetry, on the upper-left boundary of $K$. By Corollary~\ref{C:covnew}(ii), we can choose the edges $S, T$ in $K$ so that $U \perspdn S \exswing T$, $\col S = u$, and $\col T = x$. By Corollary~\ref{C:covnew}(ii), specifically, by equations \eqref{E:1} and \eqref{E:3}, we have $x \prec u, \col{W}$ and $u \neq \col{W}$, verifying the Maximal Cover Property. \section{The No Child Property}\label{S:Child} In this section, we verify the No Child Property for the slim rectangular lattice~$K$ and with a fixed $\E C_1$-diagram. Let $x,y,z,u \in \E P$ with $x \neq y \in \E P$, let $u$ be a maximal element of $\E P$, and let $x, y \prec u$ in $\E P$. By way of contradiction, let us assume that there is an element $z \in \E P$ such that $z \prec x,y$ in $\E P$. By Corollary~\ref{C:max1}, the element~$u$ colors an edge~$U$ on the upper boundary of~$K$, say, in the upper-left boundary. By Corollary~\ref{C:cov}(i), for $z \prec x \in \E P$, we get a peak sublattice $\SfS 7$ in which the middle edge $Z$ is colored by $z$ and upper-left edge $X$ is colored by $x$, or symmetrically. The upper-right edge $Y$ must have color~$y$. Now we apply Corollary~\ref{C:covnew}(ii) to the edge $U$ and middle edge $Z$ of the peak sublattice $\SfS 7$, obtaining that $U \perspdn Y \swing Z$, in particular, $U \perspdn Y$. This is a contradiction, since $U$ is normal-up and $Y$ is normal-down. \section{The Four-Crown Two-pendant Property}\label{S:Crown} Finally, we verify the Four-Crown Two-pendant Property for the slim rectangular lattice $K$ and with a fixed $\E C_1$-diagram. By way of contradiction, assume that the ordered set $\E R$ of Figure~\ref{F:notation} is a cover-preserving ordered subset of $\E P$, where $a,b,c,d$ are maximal elements of $\E P$. By~Corollary~\ref{C:max1}, there are edges $A,B,C,D$ on the upper boundary of $K$, so that $\col A = a$, $\col B = b$, $\col C=c$, $\col D = d$. By left-right symmetry, we can assume that the edge $A$ is on the upper-left boundary of $K$. Since $p \prec a, b$ in $\E P$, it follows from Lemma~\ref{L:disjoint} that the edge $B$ is on the upper-right boundary of $K$, and so is $D$. Similarly, $C$ is on the upper-left boundary of $K$. There are four cases, (i) $C$ is below $A$ and $B$ is below $D$; (ii)~$C$~is below $A$ and $D$ is below $B$; and so on. The first two are illustrated in Figure~\ref{F:CABDx}. \begin{figure} \caption{Illustrating the proof of The Four-Crown Two-pendant Property} \label{F:CABDx} \end{figure} We consider the first case. By Corollary~\ref{C:cov}(ii), there is a peak sublattice $\SfS 7$ with middle edge $P$ (as in the first diagram of Figure~\ref{F:CABDx}) so that $A$ and $B$ are down-perspective to the upper-left edge and the upper-right edge of this peak sublattice, respectively. We define, similarly, the edge $Q$ for $C$ and $B$, the edge $S$ for $A$ and~$D$, the edge $R$ for $C$ and $D$, and the edge $U$ for $R$ and $P$. The ordered set $\E R$ is a cover-preserving subset of $\E P$, so we get, similarly, the peak sublattice~$\SfS 7$ with middle edge $U$. Finally, $v \prec q, s$ in $\E R$, therefore, there is a peak sublattice~$\SfS 7$ with middle edge $V$ with upper-left edge $V_l$ and the upper-right edge~$V_r$ so that $S \perspdn V_l$ and $S \perspdn V_r$, or symmetrically. This concludes the proof of the Four-Crown Two-pendant Property and of Cz\'edli's Theorem. Of course, the diagrams in Figure~\ref{F:CABDx} are only illustrations. The grid could be much larger, the edges $A, C$ and $B, D$ may not be adjacent, and there maybe lots of other elements in $K$. However, our argument does not utilize the special circumstances in the diagrams. The second case is similar, except that we get the edge $V$ and cannot get the edge $U$. The third and fourth cases follow the same way. \appendix \section{Two more illustrations for Section~\ref{S:Crown}}\label{S:appendix} \begin{figure} \caption{Two more illustrations for Section~\ref{S:Crown} \label{F:CABDx2} \end{figure} \end{document}

\begin{document} \title{Training variational quantum algorithms is NP-hard} \author{Lennart Bittel} \ensuremath\mathrm{e}mail{lennart.bittel@uni-duesseldorf.de} \author{Martin Kliesch} \ensuremath\mathrm{e}mail{mail@mkliesch.eu} \affiliation{\hhu} \begin{abstract} \Aclp{VQA} are proposed to solve relevant computational problems on near term quantum devices. Popular versions are \aclp{VQE} and \aclp{QAOA} that solve ground state problems from quantum chemistry and binary optimization problems, respectively. They are based on the idea of using a classical computer to train a parameterized quantum circuit. We show that the corresponding classical optimization problems are \class{NP}-hard. Moreover, the hardness is robust in the sense that, for every polynomial time algorithm, there are instances for which the relative error resulting from the classical optimization problem can be arbitrarily large assuming $\class{P} \neq \class{NP}$. Even for classically tractable systems composed of only logarithmically many qubits or free fermions, we show the optimization to be \class{NP}-hard. This elucidates that the classical optimization is intrinsically hard and does not merely inherit the hardness from the ground state problem. Our analysis shows that the training landscape can have many far from optimal persistent local minima. This means that gradient and higher order descent algorithms will generally converge to far from optimal solutions. \ensuremath\mathrm{e}nd{abstract} \maketitle \acresetall \section{Introduction} Recent years have seen enormous progress toward large-scale quantum computation. A central goal of this effort is the implementation of a type of quantum computation that solves computational problems of practical relevance faster than any classical computer. However, the noisy nature of quantum gates and the high overhead cost of noise reduction and error correction limit near term devices to shallow circuits~\cite{Pre18}. \Acp{VQA} have been proposed to bring us a step closer to this goal. Here, an optimization problem is captured by a loss function given by expectation values of observables w.r.t.\ states generated from a parametrized quantum circuit. Then a classical computer trains the quantum circuit by optimizing the expectation value over the circuit's parameters. Figure~\ref{fig:sketch} illustrates a possible \ac{VQA} routine. Popular candidates to be used on near term devices are \acp{QAOA} \cite{FarGolGut14} and \acp{VQE} \cite{PerMcCSha14}; see Ref.~\cite{Cerezo20VariationalQuantumAlgorithms} for a review. \acp{VQE} are proposed, for instance, to solve electronic structure problems, which are central to quantum chemistry and material science. Proposals of \acp{QAOA} include improved algorithms for quadratic optimization problems over binary variables such as the problem of finding the maximum cut of a graph (\class{MaxCut}). For hybrid classical-quantum computation to be successful, two challenges need to be overcome. First, one needs to find parameterized quantum circuits that have the expressive power to yield a sufficiently good approximation to the optimal solution of relevant optimization problems (i.e., the model mismatch is small). Second, the classical optimization over the parameters of the quantum circuit needs to be solved quickly enough and with sufficient accuracy. We will focus on this second challenge. \begin{figure} \ensuremath\mathrm{i}ncludegraphics[width=1\linewidth]{sketch} \caption{Sketch of a \ac{VQA} optimization routine. This work addresses the complexity of the classical optimization part (red). } \label{fig:sketch} \ensuremath\mathrm{e}nd{figure} For the classical optimization several heuristic approaches are known, most of which are based on gradient descent ideas and higher order methods. This is convenient, as with the parameter shift rule \cite{Schuld19EvaluatingAnalyticGradients}- the gradient can be calculated efficiently. Methods include standard BFGS optimization and extensions \cite{Byrd1995LimitedMemoryAlgorithm} and natural gradient descent \cite{Stokes2020QuantumNaturalGradient}, which has a favorable performance for at least certain easy instances \cite{Wierichs2020AvoidingLocalMinima}. Second order methods require significant overhead in the number of measurements but can yield better accuracy \cite{Mari2020EstimatingTheGradient}. Quantum analytic descent \cite{Koczor2020QuantumAnalyticDescent} uses certain classical approximations of the objective function in order to reduce the number of quantum circuit evaluations at the cost of a higher classical computation effort. However, it has also been shown recently that there are certain obstacles that need to be overcome to render the classical optimization successful. The training landscape can have so-called \ensuremath\mathrm{e}mph{barren plateaus} where the loss function is effectively constant and hence yields a vanishing gradient, which prevents efficient training. This phenomenon can be caused, for example by random initializations \cite{McClean2018BarrenPlateausIn} and nonlocality of the observable defining the loss function \cite{Cerezo20CostFunctionDependent}. Also, sources of randomness given by noise in the gate implementations can cause similar effects \cite{Wang2020Noise-InducedBarren}. Moreover, the problem of barren plateaus cannot be fully resolved by higher order methods \cite{Cerezo20ImpactOfBarren}. In this work, we show that the existence of persistent local minima can also render the training of \acp{VQA} infeasible. For this purpose, we encode the \class{NP}-hard \class{MaxCut}\ problem into the corresponding classical optimization task for several versions of \acp{VQA}, which have many far from optimal local minima. Specifically, we obtain hardness results concerning the optimization in four different settings: (i) We use an oracle description of a quantum computer and show that the classical optimization of \acs{VQA} is an \class{NP}-hard problem, even if it needs to be solved only within constant relative precision. Next, we remove the oracle from the problem formulation by focusing on classically tractable systems where the underlying ground state problem is efficiently solvable. Here, we consider quantum systems where (ii) the Hilbert space dimension scales polynomially in the number of parameters (i.e., logarithmically many qubits) or (iii) is composed of free fermions. (iv) If the setup is restricted to the \ac{QAOA} type, we show that our hardness results also hold. \subsection{Connection to complexity theory} The decision version of \ac{VQA} optimization is in the complexity class \class{QCMA}, problems that can be verified with a classical proof on a quantum computer. The class \class{QMA}, which allows for the proof to be a quantum state, contains \class{QCMA}. Much about the relationship between classical \class{MA}, \class{QCMA}, and \class{QMA}\ is still unknown. Notably, finding the ground state energy of a local Hamiltonian is \class{QMA}-hard \cite{KitSheVya02,Kempe04TheComplexityOf}. This means that if $\class{QCMA}\neq \class{QMA}$, then \ac{VQA} algorithms will not be able to solve the local Hamiltonian problem, but only problems contained in \class{QCMA}. Our results imply that even if the relevant energy eigenstates are contained in the \ac{VQA} ansatz, the classical optimization may still be at least as difficult as solving \class{NP}\ problems (Section~\ref{sec:VQAopt_q}). \subsection{Notation} We use the notation $[n]\coloneqq \{1,\dots,n\}$. The Pauli matrices are denoted by $\sigma_x$, $\sigma_y$, and $\sigma_z$. An operator $X$ acting on subsystem $j$ of a larger quantum system is denoted by $X^{(j)}$ - e.g., $\sigma_x^{(1)}$ is the Pauli-$x$ matrix acting on subsystem $1$. By $\|X\|$ we refer to the operator norm of operator $X$. The number of edges of the graph with the adjacency matrix $A$ is denoted by $\left|E(A)\right|$. By $\MaxCut(A)$ we denote the solution of \class{MaxCut}\ for an adjacency matrix $A$; see Problem~\ref{p:maxcut}. Throughout, we consider only adjacency matrices $A$ of undirected, unweighted graphs with at least one edge; i.e., \daggermat\ is a nonzero symmetric binary matrix with a vanishing diagonal. \section{A continuous \texorpdfstring{\NoCaseChange{\class{MaxCut}}}{MaxCut} optimization} We introduce a continuous, trigonometric problem which we show to be \class{NP}-hard to optimize and approximate. This is related to earlier work on the optimization of trigonometric functions~\cite{Pfister2018BoundingMultivariateTrigonometric} for which \class{NP}-hardness is known. For the specific class of functions, we also show the existence of an approximation ratio explicitly. Below, we use this problem to obtain hardness results for various \ac{VQA} versions. \begin{problem}[\class{MaxCut}]\label{p:maxcut} \\ \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem [Instance] The adjacency matrix \daggermat\ of an unweighted undirected graph. \ensuremath\mathrm{i}tem [Task] Find $S\subset [n]$ that maximizes $\sum_{i\ensuremath\mathrm{i}n S,j\ensuremath\mathrm{i}n [n]\setminus S} A_{i,j}$. \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} {\class{MaxCut}} is famously known to be \class{NP}-hard. Additionally \class{MaxCut}\ is \class{APX} -hard, meaning that every polynomial time approximation algorithm there exist some instances, where the approximation ratio $\alpha$, the ratio between the algorithmic solution and the optimal solution, is bounded by $\alpha\leq \alpha_{\max}<1$, assuming that $\class{P}\neq \class{NP}$. It was shown that if the unique games conjecture is true, then the best approximation ratio of a polynomial algorithm is $\alpha_{\max}=\min_{0<\theta<\pi}\frac{\theta/\pi}{(1-\cos(\theta)/2)}\approx 0.8786$~\cite{khot_optimal_2007}, which is also what the best known algorithms can guarantee~\cite{goemans_improved_1995}. Without the use this conjecture, it has been proven that $\alpha_{\max}\leq\frac{16}{17}\approx0.941$~\cite{hastad_optimal_2001}. For our purposes we define a continuous, trigonometric version of \class{MaxCut}. Minima of real valued functions are given by real numbers that may not have an efficient numerical representation. However, it is commonly said that a minimization problem is solved if it is solved to exponential precision, which is the convention we will also be using throughout this paper. The intuitive notion is that the hardness does not come from the difficulty of representing the minimum. \begin{problem}[Continuous \class{MaxCut}]\label{p:shift_maxcut} \\ \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem [Instance] The adjacency matrix \daggermat\ of an unweighted graph. \ensuremath\mathrm{i}tem [Task] Find $\vec\phi\ensuremath\mathrm{i}n[0,2\pi)^d$ that minimizes \begin{equation} \mu (\vec \phi)\coloneqq \frac{1}{4}\sum_{i,j=1}^dA_{i,j}[\cos(\phi_i)\cos(\phi_j)-1]. \ensuremath\mathrm{e}nd{equation} \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} \begin{lemma}\label{lem:continuous_MaxCut} Problem~\ref{p:shift_maxcut} is \class{NP}-hard. Moreover, if $\class{P}\neq\class{NP}$, for every polynomial time algorithm there exists an approximation ratio, which is at most that of \class{MaxCut}. \ensuremath\mathrm{e}nd{lemma} \begin{proof} We will show that it suffices to look at $\vec\phi$ from the discrete subset $\{0,\pi\}^d$. For this purpose, we analyze the dependence of $\mu$ on one coordinate $\phi_i$ of $\vec \phi$. Denoting the vector obtained from $\vec\phi$ by replacing $\phi_i$ with $x$ by $\left.\vec\phi\right|_{\phi_i=x}$, we write $\mu$ as \begin{align} \mu(\left. \vec \phi\right|_{\phi_i=x}) = \cos(x) \left[\frac{1}{2}\sum_{j:\; j\neq i} A_{i,j}\cos(\phi_j)\right]+C\,, \ensuremath\mathrm{e}nd{align} where we have used that $A_{i,i} = 0$ for all $i$, $A=A^T$ and $C$ is independent of $x$. Since the only dependence on $x$ is given by the cosine, it follows that for any $\vec \phi\ensuremath\mathrm{i}n [0,2\pi)^d$ \begin{equation} \mu(\vec\phi)\geq\min\{\mu(\left.\vec\phi\right|_{\phi_i=0}),\mu(\left.\vec\phi\right|_{\phi_i=\pi})\}\,. \ensuremath\mathrm{e}nd{equation} This observation implies that we can iteratively choose each $\phi_i$ to be in $\{0,\pi\}$ for $i\ensuremath\mathrm{i}n[d]$, without increasing $\mu$. Therefore, an algorithm returning continuous values of $\phi_i$ can be turned discrete in polynomial time without reducing the approximating power. The discrete problem can be written as a quadratic unconstrained binary optimization \begin{equation} \begin{aligned} \min_{\vec\phi\ensuremath\mathrm{i}n \{0,\pi\}^d}\mu(\vec \phi) &= \frac{1}{4}\min_{\vec v \ensuremath\mathrm{i}n\{-1,1\}^d} \sum_{i, j=1}^dA_{i,j}(v_iv_j-1)\\ &=\frac{1}{4} \min_{\vec v \ensuremath\mathrm{i}n\{-1,1\}^d} \sum_{i,j=1}^d A_{i,j}\left(-2\delta_{v_i\neq v_j}\right)\\ &=-\max_{S \subset [d]} \sum_{i\ensuremath\mathrm{i}n S,j\ensuremath\mathrm{i}n [d]\setminus S} A_{i,j} \, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} where $\delta_{v_i\neq v_j}=1-\delta_{v_i,v_j}$ and we used again that $A$ is symmetric. Therefore, we obtain a (many-one) reduction of {\class{MaxCut}} to Problem~\ref{p:shift_maxcut}, implying Problem~\ref{p:shift_maxcut} is \class{NP}-hard, which finishes the proof. \ensuremath\mathrm{e}nd{proof} We note that the derivative and the Hessian of $\mu$ is given by \begin{equation} \begin{aligned} \frac{\partial\mu(\vec\phi)}{\partial \phi_i} &=-\frac 12 \sin(\phi_i)\sum_{j:\; j\neq i} A_{i,j}\cos(\phi_j)\\ \frac{\partial^2\mu(\vec\phi)}{\partial \phi_i\partial \phi_k} &=-\frac 12\delta_{i,k} \cos(\phi_i)\sum_{j:\; j\neq i} A_{i,j}\cos(\phi_j)\\ &+\frac{A_{i,j}}{2} \sin(\phi_i)\sin(\phi_k) \, . \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} At the relevant discrete set $\vec\phi\ensuremath\mathrm{i}n\{0,\pi\}^d$, the derivative vanishes and the Hessian is diagonal, meaning a point in the set describes a local minimum whenever changing any single $\phi_i$ increases the objective function. The same minima (in the discrete \class{MaxCut}\ formulation) are also achieved by a greedy algorithm: start with a random bipartition of the vertex set, then repeatably change a single vertex assignment if it increases the cut until the cut cannot be increased any further by this update rule. This algorithm has an approximation ratio of $\alpha=\frac{1}{2}$, meaning it only guarantees to approximate \class{MaxCut}\ to half its optimal solution. If Problem~\ref{p:shift_maxcut} is solved with gradient based methods, any local minimum can be the final result, therefore gradient based algorithms also only have an approximation ratio of $\alpha=\frac 1 2$, which is significantly worse than what modern \class{MaxCut}\ solvers can achieve~\cite{goemans_improved_1995}. \section{\texorpdfstring{\Aclp{VQA}}{VQAs}} \Ac{VQA} is a general framework of hybrid quantum computers, where classically tunable parameters $\vec \phi$ of a unitary circuit are used to minimize the expectation value of an observable. First, in Section~\ref{sec:VQAopt_q}, we consider such computing schemes, where the quantum part is composed of qubits. In order to show that the classical simulation is hard, we assume oracle access to an idealized quantum device. Next, in Section~\ref{sec:VQEopt_poly}, we show that the problem is also hard for \ac{VQA} settings with small Hilbert space dimensions (or logarithmically many qubits) so that the oracle can be replaced by efficient classical simulation. In Section~\ref{sec:VQAopt_QAOA} we use the same setting, but consider \acp{QAOA} instances instead. Last, in Section~\ref{sec:VQAopt_ff}, we analyze \acp{VQA} in free Fermionic systems, where the oracle can be replaced by efficient free fermionic calculations. \subsection{\texorpdfstring{\Ac{VQA}}{VQA} optimization with quantum computer access} \label{sec:VQAopt_q} The common application of \acp{VQA} is within quantum computing, where a quantum computer is used to estimate the expectation value and a classical algorithm chooses the circuit parameters of the quantum computer. For the classical optimization, we describe the information obtained from the quantum computer with oracle calls made by the classical algorithm. \begin{problem}[\Ac{VQA} minimization, oracular formulation]\label{p:VQA_O} \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem[Instance] A set of generators $\{H_i \}_{i \ensuremath\mathrm{i}n \{1,\dots, L\}}$ and an observable $O$ acting on $\mcH=(\mathbb{C}^2)^{\otimes N}$, given in terms of their Pauli basis representation. \ensuremath\mathrm{i}tem[Oracle access] We set $\ket{\class{P}si(\vec \phi)}\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\ket{\vec 0}$ with $U_i(\phi)=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_i \phi}$. The oracle $\mc O$ returns $\operatorname{O}Ex{\vec \phi} \coloneqq \sandwich{\class{P}si(\vec \phi)}{O}{\class{P}si(\vec \phi)}$, for a given $\vec \phi$, up to any desired polynomial additive error. \ensuremath\mathrm{i}tem[Task] Find $\vec \phi \ensuremath\mathrm{i}n \mathbb{R}^L$ that minimizes $\operatorname{O}Ex{\vec \phi}$ provided access to $\mc O$. \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} We use the oracle to outsource difficult computations, which is similar to how a quantum computer would in a physical implementation. The motivation of our oracle is that Problem~\ref{p:VQA_O} captures the complexity of only the classical optimization effort in hybrid quantum computations. The oracle can be seen as postselecting on the successful runs only, therefore making the return deterministic. \begin{proposition}[Hardness of \ac{VQA} optimization, oracular formulation] \label{thm:VQA1} Assuming $\class{P}\neq \class{NP}$ there is no deterministic classical algorithm that solves Problem~\ref{p:VQA_O} in polynomial time. \ensuremath\mathrm{e}nd{proposition} It is straightforward to show that Problem~\ref{p:VQA_O} is \class{NP}-hard to solve. Essentially, we use a diagonal observable for which the ground state problem is \class{NP}-hard and use unitaries to reach every computational basis state. \begin{proof} We prove the proposition via a reduction of Problem~\ref{p:shift_maxcut} to Problem~\ref{p:VQA_O}. For this, let $N=d$ and let $O$ the usual Ising Hamiltonian encoding of \class{MaxCut} \begin{align} O&\coloneqq\frac{1}{4}\sum_{i,j=1}^dA_{i,j}(\sigma_z^{(i)}\sigma^{(j)}_z-1)\, . \ensuremath\mathrm{e}nd{align} We use $L=d$ layers with \begin{align} H_i&\coloneqq \frac{\sigma_y^{(i)}}{2} \quad,\quad i\ensuremath\mathrm{i}n [d] \,, \ensuremath\mathrm{e}nd{align} as generators. By direct calculation we find that \begin{equation} \begin{aligned} \operatorname{O}Ex{\vec \phi}&=\bra{\class{P}si(\vec\phi)}O\ket{\class{P}si(\vec \phi)}\\ \label{eq:oracle_replacement} &=\frac{1}{4}\sum_{i,j=1}^dA_{i,j}\left[\cos(\phi_i)\cos(\phi_j)-1\right]\\ &=\mu(\vec\phi) \, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} which is the objective function of Problem~\ref{p:shift_maxcut}. \ensuremath\mathrm{e}nd{proof} To analyze the overall approximation power of an algorithm we define the \ensuremath\mathrm{e}mph{approximation error} for an instance as \begin{align} \Deltai\coloneqq \frac{\braket{O}_a-\lambda_{\min}(O)}{\lambda_{\max}(O)-\lambda_{\min}(O)}\, , \ensuremath\mathrm{e}nd{align} where $\lambda_{\min}(O)$ is the smallest eigenvalue of the observable $O$ and $\lambda_{\max}(O)$ is the largest; the expectation value of the final output of the algorithm is $\braket{O}_a\geq \lambda_{\min}(O)$. We normalize by the \ensuremath\mathrm{e}mph{spectral width} \begin{align} w(O)\coloneqq\lambda_{\max}(O)-\lambda_{\min}(O)\,, \ensuremath\mathrm{e}nd{align} as this ensures that $\Deltai\ensuremath\mathrm{i}n [0,1]$. There are two error contributions: (i) the \ensuremath\mathrm{e}mph{model mismatch} $\Deltai_m$ is the approximation error resulting from the ansatz class being unable to represent the ground state and (ii) the \ensuremath\mathrm{e}mph{optimization error} $\Deltai_o$ is the error due to the classical algorithm not converging to the optimal solution within the class. That is, \begin{align} \Deltai&= \frac{\braket{O}_{\min}-\lambda_{\min}(O)}{w(O)}+\frac{\braket{O}_a-\braket{O}_{\min}}{w(O)}\\ &=\qquad\Deltai_{\mathrm{m}}\qquad+\qquad\Deltai_o \, , \ensuremath\mathrm{e}nd{align} where $\braket{O}_{\min}$ refers to the smallest expectation value over the ansatz class, i.e., the global minimum over the circuit parameters. Since we are interested in classical algorithms, we define an optimization error, in a similar manner to how approximation ratios are defined for \class{NP}\ optimization problems (the complexity class \class{APX}), over all considered instances. \begin{definition}[Optimization error]\label{def:approx_val} The \ensuremath\mathrm{e}mph{optimization error} of an optimization algorithm $\Delta\ensuremath\mathrm{i}n [0,1]$ is the smallest number such that \begin{align} \Delta \geq \frac{\braket{O}_a-\braket{O}_{\min}}{w(O)} \ensuremath\mathrm{e}nd{align} for all considered \ac{VQA} instances. \ensuremath\mathrm{e}nd{definition} \begin{corollary}\label{cor:apvalexp} If $\class{P}\neq\class{NP}$, then there exists no polynomial time algorithm which can guarantee any optimization error $\Delta<1$ for all \acp{VQA} defined by Problem~\ref{p:VQA_O}. \ensuremath\mathrm{e}nd{corollary} \begin{proof} We prove this statement by relating the optimization error of a \ac{VQA} to the approximation ratio of \class{MaxCut}\ and by introducing a boosting technique to amplify errors in the setting of Problem~\ref{p:VQA_O}. From the proof of Proposition~\ref{thm:VQA1} we obtain $w(O)=\MaxCut(A)$ and the optimal solution is also $\left|\braket{O}_{\min} \right|=\MaxCut(A)$, as there is no model mismatch. From the algorithm we get $\braket{O}_{a}=\MaxCut(A)a$, where $\MaxCut(A)a$ is the approximation of the continuous \class{MaxCut}\ problem (Problem~\ref{p:shift_maxcut}) and, therefore, an approximation to \class{MaxCut}\ itself. With this argument it follows that \begin{equation} \Delta\geq1-\alpha\, , \ensuremath\mathrm{e}nd{equation} where $\alpha$ is the approximation ratio related to Problem~\ref{p:shift_maxcut} of the algorithm. To boost this result we introduce a variable $k$ and choose operators for $k\times d$ qubits \begin{align} \tilde O&=(-1)^{k-1}O^{\otimes k} \, ,\\ \tilde U(\vec \phi)&=U(\vec \phi)^{\otimes k} \, . \ensuremath\mathrm{e}nd{align} We can verify that the generators and $\tilde O$ only have $\func{poly}(d)$ many terms for constant $k$. For the expectation value this gives \begin{equation} \begin{aligned} \braket{\tilde O(\vec \phi)}&=(-1)^{k-1}\bra{\class{P}si(\vec \phi)}^{\otimes k}O^{\otimes k}\ket{\class{P}si(\vec \phi)}^{\otimes k}\\ &=-\left|\operatorname{O}Ex{\vec\phi}\right|^k=-|\mu(\vec\phi)|^k \, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} where the introduced sign ensures that the problem remains a minimization for all $k$. We obtain $w(\tilde O)=\bigl|\braket{\tilde O}_{\min}\bigr|=\MaxCut(A)^k$. This yields \begin{equation} \begin{aligned} \Delta&\geq \sup_A \frac{\left|\MaxCut(A)^k-\left|\braket{\tilde O}_a\right|\right|}{\MaxCut(A)^k}\\ &=\sup_A \left(1-\frac{|\MaxCut(A)a|^k}{\MaxCut(A)^k}\right)\\ &= (1-\alpha^k)\,, \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} Therefore, no optimization error strictly smaller than $\Delta<1$ can exist for all instances (if $\class{P}\neq \class{NP}$), as this would mean in return, the algorithm could solve Problem~\ref{p:shift_maxcut} to arbitrary precision. \ensuremath\mathrm{e}nd{proof} \subsection{Logarithmic number of qubits --- polynomial Hilbert space dimension} \label{sec:VQEopt_poly} We can improve on the previous result by allowing only $N\ensuremath\mathrm{i}n O(\log(d))$ many qubits, where $d$ is the input length of the \class{MaxCut}\ instance. This drastically reduces the system's size and complexity. Notably, since the Hilbert space is now only of polynomial dimension, both the calculation of expectation values and the ground state problem can be computed efficiently. Yet we show that \ac{VQA} optimization is still \class{NP}-hard. This means that the classical optimization does not merely inherit the hardness of the ground state problem but rather is intrinsically difficult. Since the expectation value is efficiently numerically simulatable, we do not require oracle access to a quantum computer to analyze the problem. Also, for convenience, instead of the Pauli-basis we use the computational basis of the Hilbert space $\mcH$ of dimension $\dim(\mcH)=2^N\ensuremath\mathrm{e}qqcolon n$. This gives the following problem description. \begin{problem}[\Ac{VQA} minimization problem]\label{p:VQA} \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem [Instance] An initial state $\ket{\class{P}si_0}\ensuremath\mathrm{i}n \mathbb{C}^n$, a set of generators $\{H_i \}_{i \ensuremath\mathrm{i}n \{1,\dots, L\}}\subset \mcHerm(\mathbb{C}^n)$, where $L$ is the number of layers and an observable $O\subset \mcHerm(\mathbb{C}^n)$. \ensuremath\mathrm{i}tem[Task] For $\ket{\class{P}si(\vec \phi)}\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\ket{\class{P}si_0}$ with $U_i(\phi)=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_i \phi}$, find a $\vec \phi \ensuremath\mathrm{i}n \mathbb{R}^L$ that minimizes $\operatorname{O}Ex{\vec\phi}\coloneqq~\sandwich{\class{P}si(\vec \phi)}{O}{\class{P}si(\vec \phi)}$. \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} \begin{theorem}[Hardness of \ac{VQA} optimization]\label{thm:VQA2} \Ac{VQA} optimization (Problem~\ref{p:VQA}) is \class{NP}-hard. \ensuremath\mathrm{e}nd{theorem} \begin{proof} We prove the theorem via a many-one reduction from Problem~\ref{p:shift_maxcut} Let \daggermat\ be the adjacency matrix of an unweighted graph. On the Hilbert space $\mcH=\mathbb{C}^{2d}$ we first define an observable in the standard basis as \begin{align}\label{eq:defOprime} O'&\coloneqq \frac{d}{8}\cdot A\otimes \begin{pmatrix} 1&1\\mathds{1}&1 \ensuremath\mathrm{e}nd{pmatrix}\,, \ensuremath\mathrm{e}nd{align} where $\otimes$ denotes the Kronecker product. For the actual observable we modify the diagonal as \begin{align}\label{eq:Observable_loc_VQA} O_{i,j}=\left\{\begin{matrix} O'_{i,j}& i\neq j\\ -\sum_{\alpha=1}^{2d} O'_{\alpha,j} & i=j\\ \ensuremath\mathrm{e}nd{matrix}\right.\,. \ensuremath\mathrm{e}nd{align} The initial state and generators are chosen as \begin{align} \ket{\class{P}si_0}&\coloneqq \frac{1}{\sqrt{2d}} \sum_{j=1}^{2d} \ket{j} \, ,\\\label{eq:VQA_U_small} H_i&\coloneqq \ketbra{2i-1}{2i-1}- \ketbra{2i}{2i}\,, \ensuremath\mathrm{e}nd{align} where we take $L=d$ layers. As the parametrized state we obtain \begin{equation} \begin{aligned} \ket{\class{P}si(\vec\phi)} &\coloneqq U_{d}(\phi_d)\dots U_1(\phi_1) \ket{\class{P}si_0} \\ &= \frac{1}{\sqrt{2d}}\sum_{j=1}^{2d}\left( \ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} \phi_j}\ket{2j-1}+\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} \phi_j}\ket{2j} \right) \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} and \begin{equation}\label{eq:VQA} \begin{aligned} \operatorname{O}Ex{\vec \phi}&=\bra{\class{P}si(\vec \phi)}O\ket{\class{P}si(\vec \phi)} \\ &=\frac{1}{16} \sum_{s,p\ensuremath\mathrm{i}n\{+,-\}}\sum_{i,j=1}^d\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} s \phi_i }A_{i,j} \ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} p \phi_j }-\frac{1}{4}\sum_{i,j=1}^d A_{i,j} \\ &=\frac{1}{8}\sum_{i,j=1}^d A_{i,j} \left(\cos(\phi_i-\phi_j)+\cos(\phi_i+\phi_j)-2\right)\\ &=\frac{1}{4}\sum_{i,j=1}^d A_{i,j}\left[\cos(\phi_i)\cos(\phi_j)-1\right]\\ &=\mu(\vec\phi) \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} as corresponding expectation value. This completes the reduction of Problem~\ref{p:shift_maxcut} to Problem~\ref{p:VQA}. \ensuremath\mathrm{e}nd{proof} From this result, \class{NP}-completeness follows for the decision version. \begin{problem}[\Ac{VQA} minimization, decision version]\label{p:VQA_d} \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem[Instance] An initial state $\ket{\class{P}si_0}\ensuremath\mathrm{i}n \mathbb{C}^n$, a set of generators $\{H_i \}_{i \ensuremath\mathrm{i}n \{1,\dots, L\}}\subset \mcHerm(\mathbb{C}^n)$, where $L$ are the number of layers, an observable $O\ensuremath\mathrm{i}n\mcHerm(\mathbb{C}^n)$ and a threshold $a\ensuremath\mathrm{i}n \mathbb{R}$. \ensuremath\mathrm{i}tem[Task] For $\ket{\class{P}si_{\vec \phi}}\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\ket{\class{P}si_0}$ with $U_i(\phi)~=~\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_i \phi}$, determine wheter there exists $\vec \phi \ensuremath\mathrm{i}n \mathbb{R}^d$ for which $\sandwich{\class{P}si(\vec \phi)}{O}{\class{P}si(\vec \phi)}\leq a$. \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} \begin{corollary} Problem~\ref{p:VQA_d} is \class{NP}-complete. \ensuremath\mathrm{e}nd{corollary} \begin{proof} As calculating the expectation value of observable on polynomial dimensional Hilbert spaces is in \class{P}, $\vec \phi$ is a valid proof for the \ensuremath\mathrm{e}mph{yes} instances, which can be verified in polynomial time and is therefore in \class{NP}. Together with hardness of problem~\ref{p:VQA}, this means Problem~\ref{p:VQA_d} is \class{NP}-complete. \ensuremath\mathrm{e}nd{proof} We now show that $L=1$ layer is sufficient to show hardness. For this purpose we will use certain properties of Hamiltonian spectra. \begin{definition}[Approximate ergodic energy spectrum] Let $\ensuremath\mathrm{e}psilon>0$. We call a set $\{E_i\}_{i\ensuremath\mathrm{i}n n}\subset \mathbb{R}$ an \ensuremath\mathrm{e}mph{$\ensuremath\mathrm{e}psilon$-approximate ergodic energy spectrum} if for all $\vec \phi \ensuremath\mathrm{i}n [0,2\pi)^n$ there exists $t\ensuremath\mathrm{i}n \mathbb{R}_0^+$ such that \begin{equation} \modnorm{\phi_i - E_i t} \leq \ensuremath\mathrm{e}psilon \ensuremath\mathrm{e}nd{equation} for all $i\ensuremath\mathrm{i}n [n]$, where $\modnorm{x} \coloneqq \ensuremath\mathrm{i}nf_{k\ensuremath\mathrm{i}n \mathbb{Z}} |x-2\pi k| \ensuremath\mathrm{i}n [0,\pi] $. \ensuremath\mathrm{e}nd{definition} Generic energy spectra are exactly ($\ensuremath\mathrm{e}psilon=0$) ergodic. For our purpose we want to show that there are also efficiently expressible approximate ergodic energy spectra. \begin{lemma}[Approximate ergodic energy spectra]\label{l:ph_to_en} \\ Let $m\ensuremath\mathrm{i}n \mathbb{N}$. Then \begin{equation} E_i\coloneqq \frac{2\pi}{m^i} \ensuremath\mathrm{e}nd{equation} with $i \ensuremath\mathrm{i}n [n]$ defines an $\ensuremath\mathrm{e}psilon$-approximate ergodic energy spectrum with \begin{equation} \ensuremath\mathrm{e}psilon = \frac {4\pi} m\, . \ensuremath\mathrm{e}nd{equation} \ensuremath\mathrm{e}nd{lemma} We provide a proof in Appendix~\ref{ap:erd_en}. The chosen energies can be expressed with $n\times \lceil\log_2(m)\rceil$ bits of precision. With this property we can show the following theorem. \begin{theorem}\label{t:unitary} \Ac{VQA} optimization (Problem~\ref{p:VQA}) is \class{NP}-hard for $L=1$ layer. \ensuremath\mathrm{e}nd{theorem} \begin{proof} For the single layer, we choose the generators as a linear combination of terms from Eq.~\ensuremath\mathrm{e}qref{eq:VQA_U_small} \begin{align}\label{eq:Ham_ev} H=\sum_{j=1}^d E_j \left( \ket{2j-1}\bra{2j-1}- \ket{2j}\bra{2j}\right)\, \ensuremath\mathrm{e}nd{align} meaning $U(\phi)=\ensuremath\mathrm{e}xp(-\ensuremath\mathrm{i} \phi H)=U(\vec E \phi)$. The initial state and $O$ remain identical. This leads to the expectation value \begin{align} \operatorname{O}Ex{\phi}=\sum_{i,j=1}^d A_{i,j}\left[\cos(E_i \phi)\cos(E_j \phi)-1\right]=\mu(\vec E\phi)\,. \ensuremath\mathrm{e}nd{align} If $\{E_i\}_{i}$ are chosen as in Lemma~\ref{l:ph_to_en} then $\operatorname{O}Ex{\phi}$ approximates $\mu(\vec\phi)$ with $\vec \phi = \vec E \phi$ to arbitrary precision, which we have shown to be \class{NP}-hard to optimize in Lemma~\ref{lem:continuous_MaxCut}. \ensuremath\mathrm{e}nd{proof} By viewing the \ac{VQA} in Theorem~\ref{t:unitary} as a continuous time evolution for logarithmically many qubits, we obtain the following result (we are unaware of this statement being explicitly proven before). \begin{corollary} For a system with logarithmically many qubits, we consider the expectation value of a (unitarily) time evolved observable $\langle O(t)\rangle$, starting from some initial state. Minimizing the expectation value over $t\ensuremath\mathrm{i}n \mathbb{R}_0^+$ is then \class{NP}-hard. \ensuremath\mathrm{e}nd{corollary} \subsection{\texorpdfstring{\Aclp{QAOA}}{QAOA} for a logarithmic number of qubits} \label{sec:VQAopt_QAOA} \Acp{QAOA} can be seen as certain types of \acp{VQA}, which are inspired by adiabatic computation, where a slow enough transition between two Hamiltonians $H_b$ and $H_c$ guarantees remaining in the ground state as long as the Hamiltonians are gapped and level crossings are avoided~\cite{Albash2018AdiabaticQuantumComputation}. \Acp{QAOA} capture a time-discrete version of this approach by alternatingly applying the time evolutions of the Hamiltonians. Accordingly, parameter vectors $\vec \beta,\vec \gamma \ensuremath\mathrm{i}n \mathbb{R}^{L}$ need to be chosen, which define how long each Hamiltonian is applied. We demonstrate that the hardness of \ac{VQA} optimization for logarithmically many qubits also translates to \ac{QAOA} problems. Formally, the problem is as follows. \begin{problem}[\Ac{QAOA} minimization problem]\label{p:QAOA} \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem[Instance] Two Hamiltonians $H_b,H_c\ensuremath\mathrm{i}n \mcHerm(\mathbb{C}^n)$ and the number of layers $L$ in unary notation\footnote{This means that the length of the input scales linearly with $L$.}. \ensuremath\mathrm{i}tem[Task] For a tunable state $\ket{\class{P}si(\vec \beta, \vec \gamma)}\coloneqq U_b(\beta_L)U_c(\gamma_L)\cdots U_b(\beta_1)U_c(\gamma_1) \ket{\class{P}si_0}$, where $\ket{\class{P}si_0}$ is the ground state of $H_b$, $U_b(\beta)=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_b \beta}$ and $U_c(\gamma)=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_c \gamma}$, find $\vec \beta,\vec \gamma \ensuremath\mathrm{i}n \mathbb{R}^d$ which minimize $\operatorname{O}Ex{\vec \beta, \vec \gamma}\coloneqq\sandwich{\class{P}si(\vec \beta, \vec \gamma)}{H_c}{\class{P}si(\vec \beta, \vec \gamma)}$. \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} \begin{theorem}[Hardness of optimization in \acp{QAOA}]\label{thm:QAOA} Problem~\ref{p:QAOA} is \class{NP}-hard for $L=1$ layer. \ensuremath\mathrm{e}nd{theorem} \begin{proof} We will perform a reduction from single layer \ac{VQA} to \ac{QAOA}, which implies that Problem~\ref{p:QAOA} is \class{NP}-hard. We consider the Hilbert space $\mcH = \mathbb{C}^{2d+1}$. For $H_b$ we take \begin{align} H_b&=\mathrm{diag}(E_1,-E_1,E_2,-E_2,\dots ,E_d,-E_d,-1)\, , \ensuremath\mathrm{e}nd{align} where $|E_i|<1$ for all $i\ensuremath\mathrm{i}n[d]$. For $H_c$ \begin{align} H_c&=O\oplus 0 +\tau\left(\ket{+_{2d}}\bra{2d+1}+\ket{2d+1}\bra{+_{2d}}\right)\, , \ensuremath\mathrm{e}nd{align} where $\ket{+_{2d}}=\sum_{j=1}^{2d}\ket{j}/\sqrt{2d}$, $\tau$ is some real constant that we adjust later and the observable $O$ is as defined in Eq.~\ensuremath\mathrm{e}qref{eq:Observable_loc_VQA}. $O\oplus 0$ refers to $O$ being embedded in the first $2d$ computational states in the Hilbert space. By design, $\lambda_{\min}(H_b)=-1$ is the ground state energy with ground state $\ket{2d+1}$. \\ For the state we obtain \begin{equation} \begin{aligned} \ket{\class{P}si(\gamma)}&\coloneqq U_c(\gamma)\ket{2d+1}\\ &=\cos(\tau\gamma)\ket{2d+1}+\ensuremath\mathrm{i}\sin(\tau\gamma)\ket{+_{2d}}\\ \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} after applying the first Hamiltonian, where we used that $\ket{+_{2d}}$ is an eigenstate of $O\oplus 0$. This gives the final state \begin{equation*} \begin{aligned} \ket{\class{P}si(\beta,\gamma)}&=U_b(\beta)U_c(\gamma)\ket{2d+1} \\ &= \cos(\tau\gamma)\ensuremath\mathrm{e}^{\ensuremath\mathrm{i}\beta}\ket{2d+1}\\&+\ensuremath\mathrm{i}\sin(\tau\gamma)\frac{1}{\sqrt{2d}}\sum_{j=1}^d\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} E_j\beta}\ket{2j-1}+\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} E_j\beta}\ket{2j}\, . \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation*} From this variational state we derive the expectation value \begin{equation}\nonumber \begin{aligned} \operatorname{O}Ex{\beta,\gamma}&=\bra{\class{P}si(\beta,\gamma)} H_c\ket{\class{P}si(\beta,\gamma)}\\ &=\sin^2(\tau\gamma)f(\beta)+2\tau\cos(\tau\gamma)\sin(\tau\gamma)g(\beta) \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} with \begin{align}\label{eq:O_QAOA1} f(\beta)&=\frac{1}{4}\sum_{i,j}(A_{i,j}\left[\cos(E_i\beta)\cos(E_j\beta)-1\right]\, ,\\ g(\beta)&=-\frac{\sin(\beta)}{d}\sum_{i=1}^d\cos(E_j\beta)\,. \ensuremath\mathrm{e}nd{align} For $\tau\ll1$ the contribution of $g$ becomes insignificant and $\gamma=\frac{\pi}{2\tau}$ minimizes the objective function as $f(\beta)\leq0$, meaning the problem is equivalent to minimizing $f(\beta)=\mu(\vec E \beta )$ which approximates $\mu(\vec\phi)$ to arbitrary precision if $\vec E$ is chosen as in Lemma~\ref{l:ph_to_en} and therefore gives a reduction from Problem~\ref{p:shift_maxcut}. \ensuremath\mathrm{e}nd{proof} In the proof of Theorem~\ref{thm:QAOA}, the energies of $H_b$ span many orders of magnitude. This means a potential quantum computer needs to be incredibly precise to implement such a \ac{QAOA}. We will show, that this is not required, but that also for very simple spectra, hardness results can be obtained. \begin{theorem}\label{thm:QAOA2} QAOA optimization (Problem~\ref{p:QAOA}) is \class{NP}-hard for periodic optimization $\vec \beta,\vec\gamma\ensuremath\mathrm{i}n [0,2\pi)^L$, even if we restrict $\|H_b\|\leq3$ and $\|H_c\|\leq3$ (i.e $E_i\ensuremath\mathrm{i}n \{-3,-2,\dots,3\}$). \ensuremath\mathrm{e}nd{theorem} \begin{proof}[Proof outline] In Appendix~\ref{ap:multiLQAOA}, we construct explicit Hamiltonians $H_b$ and $H_c$ from an adjacency matrix $A$, where the solution is $\left<H_c\right>_{\min}=1-\frac{2\MaxCut(A)}{\left|E(A)\right|}$ and $\|H_c\|=1$. We do this by embedding a modified version of Problem~\ref{p:shift_maxcut} into the \ac{QAOA} circuit in such a way, that deviations from the intended structure are penalized by increasing the expectation value. \ensuremath\mathrm{e}nd{proof} From this we can derive bounds on the optimization errors for \ac{VQA} in this restricted setting. Here, we are unable to use the same boosting technique to increase the hardness result further. \begin{corollary} All polynomial time algorithms for \Ac{QAOA} and therefore \ac{VQA} optimization (Problems~\ref{p:QAOA} and \ref{p:VQA}) have an optimization error $\Delta\geq \frac{1-\alpha_{\max}}{2}$, where $\alpha_{\max}$ is the approximation ratio of \class{MaxCut}. \ensuremath\mathrm{e}nd{corollary} \begin{proof} For the Hamiltonians in proof of Theorem~\ref{thm:QAOA2}, we have $w(H_c)=2$ and the lowest achievable expectation value is $\left<H_c\right>_{\min}=1-\frac{2\MaxCut(A)}{\left|E(A)\right|}$, where $\left|E(A)\right|$ is the number of edges of the the graph. From this we can calculate an upper limit on the possible guaranteed precision of an optimization algorithm for all instances \begin{equation} \begin{aligned} \Delta\geq&\sup_A\left(\frac{\left|\left<H_c\right>_{\min}-\left<H_c\right>_{\mathrm{a}}\right|}{w(H_c)}\right)\\ &\geq\frac{1}{2}\sup_A\left|1-\frac{2\MaxCut(A)}{\left|E(A)\right|}-\left(1-\frac{2|\MaxCut(A)a|}{\left|E(A)\right|}\right)\right|\\ &=\frac{1}{2}\sup_A\left(\frac{2\MaxCut(A)}{\left|E(A)\right|}-\frac{2|\MaxCut(A)a|}{\left|E(A)\right|}\right)\\ &=\frac{1}{2}(1-\alpha)\frac{2\MaxCut(A)}{\left|E(A)\right|}\\ &\geq \frac{1-\alpha}{2}\, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} where the supremum goes over all adjacency matrices and $|\MaxCut(A)a|$ is the approximation of \class{MaxCut}\ from the algorithm and $\alpha$ is the approximation ratio of the algorithm; in the last step we used that $\MaxCut(A)\geq \frac{\left|E(A)\right|}{2}$. This means that if $\class{P}\neq\class{NP}$, any polynomial time algorithm is only able to guarantee \ac{QAOA} and therefore general \ac{VQA} minimization to an optimization error $\Delta\geq\frac{1-\alpha_{\max}}{2}$. \ensuremath\mathrm{e}nd{proof} For gradient based methods this means $\Delta\geq 1/4$ for logarithmically many qubits, as $\alpha = 1/2$ was shown. \subsection{Free fermionic models} \label{sec:VQAopt_ff} Free fermionic models are a certain class of fermionic many-body systems that are without actual particle-particle interactions. They are especially interesting for us, as they can be simulated efficiently for so-called Gaussian input states and observables. Fermionic creation and annihilation operators are denoted by $c_j^\dagger$ and $c_j$. They satisfy the anticommutation relations $\{c_i^\dagger,c_j\}=\delta_{i,j}$ and $\{c_i,c_j\}=0$ for all $i,j$. We call an operator \ensuremath\mathrm{e}mph{quadratic} or \ensuremath\mathrm{e}mph{Gaussian} if it is a quadratic polynomial in the creation and annihilation operators. We will consider \ensuremath\mathrm{e}mph{(balanced) quadratic observables} of the form \begin{equation}\label{eq:quadraticO} H = \sum_{i,j} h_{i,j}\, c^{\dagger}_{i}c_{j} \ensuremath\mathrm{e}nd{equation} and will call $h$ the \ensuremath\mathrm{e}mph{coefficient matrix} of $H$, which is Hermitian. Also, in the following, we denote operators by capital and their respective coefficient matrices by lowercase letters. A quantum state is \ensuremath\mathrm{e}mph{Gaussian} if it can be arbitrarily well approximated by a thermal state of a quadratic Hamiltonian. For a Hamiltonian $H$ we denote its ground state by \begin{equation}\label{eq:gs} \rho[H] = \lim_{\beta\rightarrow \ensuremath\mathrm{i}nfty }\frac{\ensuremath\mathrm{e}^{-\beta H}}{\ensuremath\mathrm{i}ntercalr[\ensuremath\mathrm{e}^{-\beta H}]}\,. \ensuremath\mathrm{e}nd{equation} From this we can define the \ac{VQA} problem in the free fermionic setting. \begin{problem}[\Ac{VQA} minimization problem, free fermions]\label{p:VQA_fermionic} \begin{description}[noitemsep,leftmargin=0.5cm,font=\normalfont] \ensuremath\mathrm{i}tem [Instance] Coefficient matrices $h_{0}, h_1, \dots, h_L, o \ensuremath\mathrm{i}n \mcHerm(\mathbb{C}^n)$. \ensuremath\mathrm{i}tem[Task] The coefficient matrices define quadratic observables $H_0, H_1, \dots, H_L$ and $O$ via \ensuremath\mathrm{e}qref{eq:quadraticO} and $\rho_0= \rho[H_0]$. For the evolved state \[\rho(\vec \phi)\coloneqq U_L(\phi_L)\cdots U_1(\phi_1)\rho_{0}U^\dagger_1(\phi_1)\cdots U^\dagger_L(\phi_L)\,,\] with $U_i(\phi)=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_i \phi}$, find a $\vec \phi \ensuremath\mathrm{i}n \mathbb{R}^L$ that minimizes $\operatorname{O}Ex{\vec\phi}\coloneqq \ensuremath\mathrm{i}ntercalr[O\rho(\vec\phi)]$. \ensuremath\mathrm{e}nd{description} \ensuremath\mathrm{e}nd{problem} \begin{theorem} Problem~\ref{p:VQA_fermionic} is \class{NP}-hard, even if the initial state $\rho_{0}$ is pure. \ensuremath\mathrm{e}nd{theorem} \begin{proof} We prove the theorem via a reduction of Problem~\ref{p:shift_maxcut} to Problem~\ref{p:VQA_fermionic}. Therefore, we consider a Hermitian adjacency matrix $A \ensuremath\mathrm{i}n \{0,1\}^{d\times d}$. For the \ac{VQA} setup, we use $n=d\times2 $ fermionic modes $c_{i}$ with $i \ensuremath\mathrm{i}n [2d]$ and $L=d$ layers. To encode Problem~\ref{p:shift_maxcut} we define $h_{0},\{h_i\}_{i\ensuremath\mathrm{i}n [L]},o\ensuremath\mathrm{i}n \mathrm{Herm}(\mathbb{C}^{2d\times 2d})$ as follows: \begin{align} h_0&=\left(\mathds{1}-\frac{\vec 1}{n}\right), \\ h_i&=\vec E_i\otimes \left(\begin{matrix} 1&0\\0&-1\ensuremath\mathrm{e}nd{matrix}\right)\ , \quad i\ensuremath\mathrm{i}n[d]\, , \ensuremath\mathrm{e}nd{align} where $\vec 1_{a,b}=1$ and $E_{i;a,b}=\delta_{i,a,b}$ (Kronecker delta) for all $i,a,b$. The coefficient matrix $o$ is given by the matrix $O$ defined in Eqs.~\ensuremath\mathrm{e}qref{eq:defOprime} and \ensuremath\mathrm{e}qref{eq:Observable_loc_VQA}, which is used for the encoding of the adjacency matrix $A$. We define $\Gamma_{i,j}\coloneqq\ensuremath\mathrm{i}ntercalr(c_j^\dagger c_i\rho_0)$ to be the correlation matrix of $\rho_0$, which can be evaluated to $\Gamma=\vec 1/(2d)$ using the identity \ensuremath\mathrm{e}qref{eq:ap_cov}. As the eigenvalues of $h_0$ are $\lambda=(-1,1,\cdots,1)$, $\rho_{0}$ describes a pure state, cp.\ Appendix~\ref{ap:therm_st}. From Eq.~\ensuremath\mathrm{e}qref{eq:ap_heisev} we obtain the coefficient matrix of $O(\vec\phi)$ in the Heisenberg picture as \begin{align} o(\vec \phi)=\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} h_d\phi_d}\cdots\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} h_1\phi_1}o\,\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} h_1\phi_1}\cdots\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} h_d\phi_d}\,. \ensuremath\mathrm{e}nd{align} With these prerequisites we can derive the following expectation value: \begin{equation} \begin{aligned} \operatorname{O}Ex{\vec\phi}&=\ensuremath\mathrm{i}ntercalr(O(\vec\phi)\rho_{0})\\ &=\ensuremath\mathrm{i}ntercalr\left(\sum_{i,j=1}^{2d}o(\vec\phi)_{i,j} c_i^\dagger c_j \rho_{0}\right)\\ &=\sum_{i,j}o(\vec\phi)_{i,j} \Gamma_{j,i}\\ &=\frac{1}{2d}\sum_{i,j}o(\vec\phi)_{i,j}\\ &=\frac{1}{4}A_{i,j}\left(\cos(\phi_i)\cos(\phi_j)-1\right)=\mu(\vec\phi)\, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} where the last step analogously follows Eq.~\ensuremath\mathrm{e}qref{eq:VQA}. As this gives the objective function from Problem~\ref{p:shift_maxcut}, this completes the desired reduction. \ensuremath\mathrm{e}nd{proof} \section{Conclusion and outlook} Our results show that classical training poses challenge in \ac{VQA} based hybrid quantum computations. Not only is optimizing \ac{VQA} algorithms \class{NP}-hard, but also that no polynomial time algorithm can have an optimization error $\Delta<1$ in all instances (assuming $\class{P}\neq \class{NP}$). Additionally, for significantly simpler systems, such as those composed of logarithmically many qubits or free fermions, the hardness results already hold. This also shows that hardness does not merely derive from the ground state problem. We have extended these results further to optimization on a single layer of gates, to continuous unitary time evolution and to \ac{QAOA} problems. We encoded \class{NP}-hard problems into local extrema of the optimization landscape of \ac{VQA} problems. Gradient descent type optimization and also higher order methods can converge to any local minimum, determined mostly by the initialization. From this we could explicitly show, that even for logarithmically many qubits, these methods have an approximation error of $\Delta\geq \frac{1}{4}$. For our particular \ac{VQA}, this is significantly worse than what modern efficient \class{MaxCut}\ solvers can guarantee. This emphasizes the need for effective initialization procedures for \ac{VQA} algorithms and poses the challenge of finding non-local heuristics for \ac{VQA} optimization to overcome the problem of these persistent local minima to reach smaller optimization errors. In order to put our results into perspective, we briefly compare them to other hardness results for relevant optimization problems. For instance, optimization within the \ac{DMRG} method is \class{NP}-hard \cite{Eisert06ComputationalDifficultyOf}. However, hardness holds only for errors scaling as the inverse of the bond dimension and there are variants where convergence can be rigorously guaranteed \cite{LanVazVid15}. \ac{VQA} optimization is arguably more similar to the optimization in the Hartree-Fock method. Despite being \class{NP}-hard \cite{Schuch09ComputationalComplexityOf} it is widely used in many practical calculations. It is our hope that this work helps of identify and overcome optimization challenges also for practically relevant \ac{VQA} problems. \section*{Acknowledgments} We thank David Wierichs, Sevag Gharibian, Raphael Brieger and Thomas Wagner for helpful comments on our manuscript and Jens Watty, Christian Gogolin and David Gross for fruitful discussions on the nature of \acp{VQE} and \acp{QAOA}. We also thank the anonymous Referee B for valuable comments, which have helped us to improve this paper. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Emmy Noether program (Grant No. 441423094) and by the German Federal Ministry of Education and Research (BMBF) within the funding program ``Quantum technologies—From basic research to market'' in the joint project MANIQU (Grant No. 13N15578). \section*{Appendices} \appendix \section{Proof of Lemma~\ref{l:ph_to_en} on ergodic energy spectra} \label{ap:erd_en} Starting from the definition of $\vec E$ \begin{equation} E_i\coloneqq \frac{2\pi}{m^i}\, , \ensuremath\mathrm{e}nd{equation} let $\vec \phi \ensuremath\mathrm{i}n [0,2\pi)^n$ be the desired phase vector. For this we define \begin{equation} s_i\coloneqq \left\lfloor\frac{\phi_i m}{2\pi}\right\rfloor \ensuremath\mathrm{i}n \{0, \dots, m-1\}\, \ensuremath\mathrm{e}nd{equation} and \begin{equation} t(\vec s)\coloneqq \sum_{j=1}^{n} s_{j} m^{j-1} \ensuremath\mathrm{i}n\{0,\dots,m^n-1\}\, . \ensuremath\mathrm{e}nd{equation} Then \begin{equation} \begin{aligned} & \phi_i - E_it \\ &= \left(\phi_i - \frac{2\pi s_i}{m}\right) + \left(\frac{2\pi s_i}{m} - 2\pi\sum_{j=1}^n s_{j}m^{j-1-i} \right) \\ &= \left(\phi_i - \frac{2\pi s_i}{m}\right) - \underbrace{\left(2\pi\sum_{j=1}^{i-1} s_{j}m^{j-1-i} \right)}_{\left|{\,\cdot\,}\right|\leq 2\pi/m} \\ & \qquad \qquad \qquad\qquad - \underbrace{\left( 2\pi\sum_{j=i+1}^n s_{j}m^{j-1-i} \right)}_{\ensuremath\mathrm{i}n 2\pi \mathbb{Z}}\, . \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} Hence, \begin{equation} \modnorm{\phi_i - E_it} \leq \frac{4\pi}{m}\, , \ensuremath\mathrm{e}nd{equation} which is what we wanted to show. \qed \section{Proof of Theorem~\ref{thm:QAOA2} on multilayer QAOAs} \label{ap:multiLQAOA} Now we construct a many-one reduction from Problem~\ref{p:shift_maxcut} to a multilayer \ac{QAOA} optimization. For this purpose, we first define some useful objects. Let $\mc K \coloneqq \mathbb{C}^d \otimes \mathbb{C}^d \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, where $d$ will be the size of an adjacency matrix to encode {\class{MaxCut}} and the number of layers ($L=d$). We define a larger Hilbert space $\mcH$ as a direct sum $\mcH=\mcH_1\oplus\dots\oplus \mcH_{2d+1}$ with $\mcH_\ensuremath\mathrm{e}ll\cong\mc K$ for $\ensuremath\mathrm{e}ll\ensuremath\mathrm{i}n [2d+1]$. We canonically identify each $\mc H_\ensuremath\mathrm{e}ll$ with the corresponding subspace $\mc H_\ensuremath\mathrm{e}ll\subset \mcH$ and denote the canonical basis states by $\{\ket{i,j,a,b}_\ensuremath\mathrm{e}ll\}$, where $i,j\ensuremath\mathrm{i}n[d]$, $a,b\ensuremath\mathrm{i}n \{0,1\}$ and $\ensuremath\mathrm{e}ll$ indicates the subspace $\mcH_\ensuremath\mathrm{e}ll$. Next, we define four two-level unitary evolutions by \begin{align*} \ket{\psi_0(\phi)}&=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\phi/2}\cos(\phi/2)\ket 1+\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\phi/2}\sin(\phi/2)\ket 2 , \\ \ket{\psi_1(\phi)}&=\cos(\phi)\ket 1+\sin(\phi)\ket 2 , \\ \ket{\psi_2(\phi)}&=\ensuremath\mathrm{e}^{\ensuremath\mathrm{i}\phi}\cos(\phi)\ket 1+\ensuremath\mathrm{i}\,\ensuremath\mathrm{e}^{\ensuremath\mathrm{i}\phi}\sin(\phi)\ket 2 , \\ \ket{\psi_3(\phi)}&=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\phi}\cos(\phi)\ket 1-\ensuremath\mathrm{i}\,\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\phi}\sin(\phi)\ket 2 , \ensuremath\mathrm{e}nd{align*} which are generated as $\ket{\psi_i(\phi)}=\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H_i\phi}\ket 1$ by \begin{align*} H_{0}&=\frac{1}{2}\begin{pmatrix}1 &-i \\ensuremath\mathrm{i} &1\ensuremath\mathrm{e}nd{pmatrix}, &H_1&=\begin{pmatrix}0 &-i \\ensuremath\mathrm{i} &0\ensuremath\mathrm{e}nd{pmatrix}, \\ H_2&=\begin{pmatrix}-1 &-1 \\-1 &-1\ensuremath\mathrm{e}nd{pmatrix},& H_3&=\begin{pmatrix}1 &1 \\mathds{1} &1\ensuremath\mathrm{e}nd{pmatrix}, \ensuremath\mathrm{e}nd{align*} with eigenvalues $\{0,1\}$, $\{-1,1\}$, $\{-2,0\}$ and $\{0,2\}$, respectively. Based on these evolutions, we define \ensuremath\mathrm{e}mph{transfer Hamiltonians} $H_T,H_T^{(\kappa)}\ensuremath\mathrm{i}n \mcHerm(\mc K\oplus\mc K)$ as \begin{equation} \begin{split} H_T^{(\kappa)} \coloneqq \sum_{i,j,a,b,x,y} & \begin{cases} H_{1\,x,y}& \text{ if } i=j\quad \mathrm{or}\quad a=0,\\ H_{2\,x,y}& \text{ if } i=\kappa\quad \mathrm{or}\quad j=\kappa ,b=0,\\ H_{3\,x,y}& \text{ if } j=\kappa ,b=1,\\ H_{1\,x,y}& \text{ otherwise } \ensuremath\mathrm{e}nd{cases} \\ &\times\ket{i,j,a,b}_x\bra{i,j,a,b}_y \\ \phantom . \\ \phantom . \ensuremath\mathrm{e}nd{split} \ensuremath\mathrm{e}nd{equation} and \begin{equation} H_T\coloneqq \sum_{i,j,a,b,x,y}H_{0\, x,y}\ket{i,j,a,b}_x\bra{i,j,a,b}_y \ensuremath\mathrm{e}nd{equation} with $x,y\ensuremath\mathrm{i}n \{1,2\}$. Let \daggermat\ be the adjacency matrix of an unweighted graph with at least one edge. We will construct $H_b$ such that it has the ground state \begin{equation} \ket{\mathrm{gs}_b}\coloneqq \frac{1}{2\sqrt{\sum_{i,j} A_{i,j}}}\sum_{i\neq j,a,b}A_{i,j}\ket{i,j,a,b} \ensuremath\mathrm{i}n \mc K. \ensuremath\mathrm{e}nd{equation} For this construction it will be helpful to denote \begin{equation} H_{\mathrm{gs}}\coloneqq -3\ketbra{\mathrm{gs}_b}{\mathrm{gs}_b} . \ensuremath\mathrm{e}nd{equation} The solution of \class{MaxCut}\ will be captured by the last subspace $\mcH_{2d+1}\subset \mcH$. For this we define $H_p\ensuremath\mathrm{i}n \mcHerm(\mc K)$ as \begin{equation} H_p=\frac{1}{2}\sum_{i,j,a,b,\tilde a,\tilde b} \delta_{a\neq \tilde a}\ket{i,j,a,b}\bra{i,j,\tilde a,\tilde b}, \ensuremath\mathrm{e}nd{equation} where $\delta_{a \neq \tilde a} \coloneqq 1-\delta_{a,\tilde a}$. Finally, we define $H_b,H_c\ensuremath\mathrm{i}n \mcHerm(\mcH)$ as \begin{equation}\label{eq:HbHc} \begin{aligned} H_b&= H_{\mathrm{gs}}\oplus H_T^{(1)}\oplus\cdots\oplus H_T^{(d)} , \\ H_c&=H_T\oplus\cdots\oplus H_T\oplus H_p\, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} where the ground state of $H_b$ is $\ket{\mathrm{gs}_b}_1$ given as the embedded state $\ket{\mathrm{gs}_b}_1 = \ket{\mathrm{gs}_b}\oplus 0 \ensuremath\mathrm{i}n \mcH$. Similarly, $\ket{\mathrm{gs}_b}_\ensuremath\mathrm{e}ll\ensuremath\mathrm{i}n \mcH_\ensuremath\mathrm{e}ll\subset \mcH$ is defined. For the first layer, this gives the state \begin{widetext} \begin{equation} \begin{aligned} \ket{\class{P}si_0} & = \ket{\mathrm{gs}_b}_1=\frac{1}{2\sqrt{\sum A_{i,j}}}\sum_{i\neq j,a,b}A_{i,j}\ket{i,j,a,b}_{1} , \\ U_{c}(\gamma_1)\ket{\class{P}si_0} &=\sin(\gamma_1/2)\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\gamma_1/2}\ket{\mathrm{gs}_b}_{2}+\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\gamma_1/2}\cos(\gamma_1/2)\ket{\mathrm{gs}_b}_{1}, \\ U_b(\beta_1)U_c(\gamma_1)\ket{\class{P}si_0} &=\frac{\sin(\beta_1)\sin(\gamma_1/2)\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\gamma_1/2}}{2\sqrt{\sum A_{i,j}}} \sum_{i\neq j,a,b}A_{i,j}\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} a(\delta_{i,1}(\beta_1+\frac{\pi}{2})+(-1)^{b}\delta_{j,1}(\beta_1+\frac{\pi}{2}))}\ket{i,j,a,b}_{3}+\ldots; \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} only the highest $\mcH_i$ subspace is shown, as this is the relevant one. Applying all $d$ layers of the \ac{QAOA} gives \begin{equation} \begin{aligned} \ket{\class{P}si(\vec \beta\vec\gamma)} &= U_b(\beta_d)U_c(\gamma_d) \dots U_b(\beta_1)U_c(\gamma_1)\ket{\class{P}si_0} \\ &=\frac{\prod_k\sin(\beta_k)\sin(\gamma_k/2)\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i}\gamma_k/2}}{2\sqrt{\sum A_{i,j}}} \sum_{i\neq j,a,b}A_{i,j}\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} a(\beta_i+\frac{\pi}{2}+(-1)^{b}(\beta_j+\frac{\pi}{2}))}\ket{i,j,a,b}_{2d+1}+\ldots \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} And thus the expectation value of $H_c$ becomes \begin{equation} \begin{aligned} \bra{\class{P}si(\vec \beta\vec\gamma)}H_c\ket{\class{P}si(\vec \beta\vec\gamma)} &=\frac{1}{2}\frac{\prod_k\sin^2(\beta_k)\sin^2(\gamma_k/2)}{4\sum A_{i,j}}\\&\times \sum_{i,j=1}^d2A_{i,j}\left(\cos(\beta_i+\beta_j+\pi)+\cos(\beta_i-\beta_j)+\cos(-\beta_i-\beta_j-\pi)+\cos(-\beta_i+\beta_j)\right)+\braket{O_{\mathrm{rest}}} \\ &=\frac{1}{2}\frac{\prod_k\sin^2(\beta_k)\sin^2(\gamma_k/2)}{4\sum A_{i,j}}\sum_{i,j=1}^d8A_{i,j}\sin(\beta_i)\sin(\beta_j) +\braket{O_{\mathrm{rest}}} \\ &\geq \frac{\prod_k\sin^2(\beta_k)\sin^2(\gamma_k/2)}{\sum A_{i,j}} \sum_{i,j=1}^dA_{i,j}\sin(\beta_i)\sin(\beta_j)\, , \ensuremath\mathrm{e}nd{aligned} \ensuremath\mathrm{e}nd{equation} \ensuremath\mathrm{e}nd{widetext} where we used that $A_{i,j}^2=A_{i,j}$ and $\braket{O_{\mathrm{rest}}}\geq 0$ denotes the expectation valueof $H_c$ within $\mcH_1\oplus\cdots\oplus \mcH_{2d}$. The expression \begin{equation*} f(\vec\beta)\coloneqq \sum_{i,j}A_{i,j}\sin(\beta_i)\sin(\beta_j)=4\mu(\vec \beta-\pi/2)+2\left|E(A)\right| \ensuremath\mathrm{e}nd{equation*} is minimized for a shifted solution of Problem~\ref{p:shift_maxcut} with local extrema $\beta_i \ensuremath\mathrm{i}n \{\pi/2,3\pi/2\}$. Its minimum value is non-positive, as $\MaxCut(A)\geq \left|E(A)\right|/2$. This means that \begin{equation*} g(\vec \beta,\vec \gamma)\coloneqq \prod_k\sin^2(\beta_k)\sin^2(\gamma_k/2) \ensuremath\mathrm{e}nd{equation*} needs to be maximized. This can be achieved trivially by setting $\gamma_i = \pi$ for all $i$ and choosing $\vec \beta$ to be a local extremum, where the function evaluates to $1$. This also minimizes $\braket{O_{\mathrm{rest}}}=0$. This means the problem is equivalent to minimizing $\mu(\vec \beta-\pi/2)$, which completes the reduction from Problem~\ref{p:shift_maxcut}. Similarly, it follows that an algorithm approximating this \ac{QAOA} also returns a lower bound to $\MaxCut(A)$. Finally, we show the claimed norm bounds on $H_b$ and $H_c$ from Eq.~\ensuremath\mathrm{e}qref{eq:HbHc}. Direct calculations reveal that $\norm{H_{\mathrm{gs}}}=3$, $\norm{H_T^{(\kappa)}} = 2$, $\norm{H_T} = 1$ and $\norm{H_p} = 1$. Hence, $\norm{H_c} = 1$ and $\norm{H_b} = 3$. \qed \section{Free fermions} \label{ap:therm_st} In this section, we provide some basics on free fermions for the special case of particle number preserving Hamiltonians. Throughout, we consider $n$ fermionic modes with annihilation operators $c_1, \dots, c_n$. First, we explain how time evolution can be simulated efficiently. With the commutation relation $[c_i^\dagger c_j,c_k^{\dagger}c_l]=\delta_{j,k}c_i^\dagger c_l-\delta_{i,l}c_k^\dagger c_j$ the time evolution in the Heisenberg picture becomes \begin{align} \dot{O}&=\ensuremath\mathrm{i}[H,O]=\ensuremath\mathrm{i}\sum_{i,j=1}^n[h,o]_{i,j}c_i^\dagger c_j \, , \ensuremath\mathrm{e}nd{align} where $o$ and $h$ are again the coefficient matrices of $O$ and $H$, as in \ensuremath\mathrm{e}qref{eq:quadraticO}. With $\dot{O}=\sum_{i,j=1}^n \dot o_{i,j} c_i^\dagger c_j$ we obtain \begin{align} \dot{o}&=\ensuremath\mathrm{i}[h,o]\, . \ensuremath\mathrm{e}nd{align} For $O(t) = \ensuremath\mathrm{e}^{\ensuremath\mathrm{i} H t} O \, \ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} H t}$ this gives \begin{equation}\label{eq:ap_heisev} o(t) = \ensuremath\mathrm{e}^{\ensuremath\mathrm{i} h t} o \,\ensuremath\mathrm{e}^{-\ensuremath\mathrm{i} h t}\, , \ensuremath\mathrm{e}nd{equation} meaning that the Hilbert space unitary $\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} H t}$ is represented by the unitary $n\times n$ matrix $\ensuremath\mathrm{e}^{\ensuremath\mathrm{i} ht}$ on the level of second moments. Secondly we derive an expression for the covariance matrix $\Gamma$ for thermal states. Quadratic observables \ensuremath\mathrm{e}qref{eq:quadraticO} can be written in a normal form. This form can be obtained by observing that unitary mode transformations leave the commutation relations invariant: For \begin{equation} \tilde c_i=\sum_{j=1}^n u_{i,j}c_j \ensuremath\mathrm{e}nd{equation} with $u\ensuremath\mathrm{i}n \U(n)$ being unitary matrix, \begin{align} \{\tilde c_i^\dagger,\tilde c_j\} =\sum_{k,l=1}^n u_{i,k}u_{j,l}^*\{c_k ,c_l^\dagger\} =\delta_{i,j} \, . \ensuremath\mathrm{e}nd{align} Hence, with basic linear algebra one can find a transformation $u\ensuremath\mathrm{i}n \U(n)$ such that \begin{equation}\label{eq:nf} H=\sum_{i,j=1}^n h_{i,j}c_i^\dagger c_j = \sum_{i,j=1}^n \tilde h_{i,j}\tilde c_i^\dagger \tilde c_j = \sum_{i=1}^n \lambda_i \tilde c_i^\dagger \tilde c_i \, , \ensuremath\mathrm{e}nd{equation} where $h=u^\dagger \tilde h u$ and $\tilde h = \diag(\lambda)$. This describe $n$ decoupled modes each with eigenenergies $E_i\ensuremath\mathrm{i}n\{0,\lambda_i\}$. The total energy of an eigenstate is therefore $E=\sum_{i=1}^n E_i$. We note that the ground state energy is non-degenerate if $\lambda_i\neq 0$ for all $i\ensuremath\mathrm{i}n[n]$. The normal form \ensuremath\mathrm{e}qref{eq:nf} allows us to write the partition function of a thermal state at inverse temperature $\beta$ as \begin{align}\label{eq:Z} Z=\ensuremath\mathrm{i}ntercalr[\ensuremath\mathrm{e}xp(-\beta H)]=\prod_{i}(\ensuremath\mathrm{e}^{-\beta\lambda_i}+1) \, . \ensuremath\mathrm{e}nd{align} Hence, the covariance matrix of the corresponding thermal state w.r.t.\ $\{\tilde c_i\}$ is given by \begin{align} \tilde\Gamma(\beta)_{i,j} &= \bigl< \tilde c_j^\dagger \tilde c_i\bigr>_\beta \\ &= -\delta_{i,j}\frac{\partial}{\partial (\beta\lambda_i)}\ln(Z) \\ &= \frac{\delta_{i,j}}{\ensuremath\mathrm{e}^{-\beta\lambda_i}+1} \, , \ensuremath\mathrm{e}nd{align} where we have denoted the expectation value of the thermal state by $\langle {\,\cdot\,} \rangle_\beta$, used the normal form \ensuremath\mathrm{e}qref{eq:nf} in the second step and \ensuremath\mathrm{e}qref{eq:Z} in the last step. In compact notation, \begin{equation} \tilde \Gamma(\beta) = (\ensuremath\mathrm{e}^{-\beta \tilde h} + 1)^{-1} \, . \ensuremath\mathrm{e}nd{equation} Therefore, the covariance matrix w.r.t.\ $\{c_i\}$ is \begin{align} \Gamma(\beta)_{i,j}&=\left< c_j^\dagger c_i\right>_\beta = \sum_{k,l=1}^n u_{k,i}^*u_{l,j} \left<\tilde c_l^\dagger \tilde c_k\right>_\beta\, , \ensuremath\mathrm{e}nd{align} that is \begin{equation}\label{eq:ap_cov} \Gamma(\beta) = u^\dagger\tilde \Gamma u=\left(\ensuremath\mathrm{e}^{-\beta h}+\mathds{1}\right)^{-1}\, . \ensuremath\mathrm{e}nd{equation} \begin{acronym}[POVM]\ensuremath\mathrm{i}temsep.5\baselineskip \acro{NISQ}{noisy and intermediate scale quantum} \acro{VQE}{variational quantum eigensolver} \acro{VQA}{variational quantum algorithm} \acro{QAOA}{quantum approximate optimization algorithm} \acro{DMRG}{density matrix renormalization group} \acro{POVM}{positive operator valued measure} \acro{PVM}{projector-valued measure} \acro{CP}{completely positive} \acro{CPT}{completely positive and trace preserving} \acro{DFE}{direct fidelity estimation} \acro{MUBs}{mutually unbiased bases} \acro{SIC}{symmetric, informationally complete} \acro{SFE}{shadow fidelity estimation} \acro{RB}{randomized benchmarking} \acro{AGF}{average gate fidelity} \acro{XEB}{cross-entropy benchmarking} \acro{SPAM}{state preparation and measurement} \acro{TV}{total variation} \acro{HOG}{heavy outcome generation} \acro{BOG}{binned outcome generation} \acro{QPT}{quantum process tomography} \acro{GST}{gate set tomography} \acro{MW}{micro wave} \acro{rf}{radio frequency} \ensuremath\mathrm{e}nd{acronym} \ensuremath\mathrm{e}nd{document}

\begin{document} \baselineskip=1.6pc \begin{center} {\bf High order maximum principle preserving finite volume method for convection dominated problems } \end{center} \centerline{ Pei Yang \fracootnote{Department of Mathematics, University of Houston, Houston, 77204. E-mail: peiyang@math.uh.edu.} Tao Xiong \fracootnote{Department of Mathematics, University of Houston, Houston, 77204. E-mail: txiong@math.uh.edu.} Jing-Mei Qiu \fracootnote{Department of Mathematics, University of Houston, Houston, 77204. E-mail: jingqiu@math.uh.edu. The first, second and the third authors are supported by Air Force Office of Scientific Computing YIP grant FA9550-12-0318, NSF grant DMS-1217008.} Zhengfu Xu \fracootnote{Department of Mathematical Science, Michigan Technological University, Houghton, 49931. E-mail: zhengfux@mtu.edu. Supported by NSF grant DMS-1316662.} } \centerline{\bf Abstract} In this paper, we investigate the application of the maximum principle preserving (MPP) parametrized flux limiters to the high order finite volume scheme with Runge-Kutta time discretization for solving convection dominated problems. Such flux limiter was originally proposed in {\em [Xu, Math. Comp., 2013]} and further developed in {\em[Xiong et. al., J. Comp. Phys., 2013]} for finite difference WENO schemes with Runge-Kutta time discretization for convection equations. The main idea is to limit the temporal integrated high order numerical flux toward a first order MPP monotone flux. In this paper, we generalize such flux limiter to high order finite volume methods solving convection-dominated problems, which is easy to implement and introduces little computational overhead. More importantly, for the first time in the finite volume setting, we provide a general proof that the proposed flux limiter maintains high order accuracy of the original WENO scheme for linear advection problems without any additional time step restriction. For general nonlinear convection-dominated problems, we prove that the proposed flux limiter introduces up to $\mathcal{O}({\Delta x^3} + \Delta t^3)$ modification to the high order temporal integrated flux in the original WENO scheme without extra time step constraint. We also numerically investigate the preservation of up to ninth order accuracy of the proposed flux limiter in a general setting. The advantage of the proposed method is demonstrated through various numerical experiments. \noindent {\bf Keywords:} Convection diffusion equation; High order WENO scheme; Finite volume method; Maximum principle preserving; Flux limiters \section{Introduction} \label{sec1} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} Recently, there is a growing interest in designing high order maximum principle preserving (MPP) schemes for solving scalar convection-dominated problems \cite{zhang2010maximum, zhang2012maximumcd, mpp_xu, jiang2013parametrized, mpp_xuMD, mpp_xqx}, positivity preserving schemes for compressible Euler and Navier-Stokes equations \cite{hu2013positivity, pp_euler, perthame1996positivity, zhang2010positivity}. The motivation of this family of work arises from the observation that many existing high order conservative methods break down when simulating fluid dynamics in extreme cases such as near-vacuum state. To illustrate the purpose of the family of the MPP methods, we shall consider the solution to the following problem \begin{equation} u_t+f(u)_x=a(u)_{xx}, \qquad u(x,0)=u_0(x), \label{ad1} \end{equation} with $a'(u)>0$. The solution to (\ref{ad1}) satisfies the maximum principle, i.e. \begin{equation}\label{maxmin} \text{if } u_M=\max_x u_0(x), \text{} u_m=\min_x u_0(x), \text{ then } u(x,t)\in [u_m, u_M]. \end{equation} Within the high order finite volume (FV) Runge-Kutta (RK) weighted essentially non-oscillatory (WENO) framework, we would like to maintain a discrete form of (\ref{maxmin}): \begin{equation}\label{maxminD} \text{if } u_M=\max_x u_0(x), \text{} u_m=\min_x u_0(x), \text{ then } \bar u^n_j \in [u_m, u_M] \text{ for any } n, \text{ } j, \end{equation} where $\bar u^n_j$ approximates the cell average of the exact solution with high order accuracy on a given $j$th spatial interval at time $t^n$. Efforts for designing MPP high order schemes to solve (\ref{ad1}) can be found in recent work by Zhang et al. \cite{zhang2012maximumcd, yzhang2012maximum}, as a continuous research effort to design high order FV and discontinuous Galerkin (DG) MPP schemes based on a polynomial rescaling limiter on the reconstructed (for FV) or representing (for DG) polynomials \cite{zhang2010maximum}. This approach requires the updated {\em cell average} to be written as a convex combination of some local quantities within the range $[u_m, u_M]$. For convection-diffusion problems which do not have a finite speed of propagation, it is difficult to generalize such approach to design MPP schemes that are higher than third order accurate. In \cite{jiang2013parametrized}, an alternative approach via a parametrized flux limiter, developed earlier by Xu et al. \cite{mpp_xu,mpp_xqx}, is proposed for the finite difference (FD) RK WENO method in solving convection diffusion equations. The flux limiter is applied to convection and diffusion fluxes together to achieve (\ref{maxminD}) for the approximated point values in the finite difference framework. In this paper, we continue our effort in applying the MPP flux limiters to high order FV RK WENO methods to maintain (\ref{maxminD}) with efficiency. Furthermore, we provide some theoretical analysis on the preservation of high order accuracy for the proposed flux limiter in FV framework. Finally, we remark that our current focus is on convection-dominated diffusion problems for which explicit temporal integration proves to be efficient. For the regime of medium to large diffusion, where implicit temporal integration is needed for simulation efficiency, we refer to earlier work in \cite{fujii1973some, farago2005discrete, farago2006discrete, farago2012discrete} and references therein for the construction of the MPP schemes with finite element framework. The generalization of the current flux limiter is not yet available and is subject to future investigation. The MPP methods in \cite{zhang2010maximum, mpp_xu, mpp_xqx} are designed base on the observation that first order monotone schemes in general satisfy MPP property (\ref{maxminD}) with proper Courant-Friedrichs-Lewy (CFL) numbers, while regular high order conservative schemes often fail to maintain (\ref{maxminD}). The MPP flux limiting approach is to seek a linear combination of the first order monotone flux with the high order flux, in the hope of that such combination can achieve both MPP property and high order accuracy under certain conditions, e.g. some mild time step constraint. This line of approach is proven to be successful in \cite{mpp_xqx, jiang2013parametrized} for the FD RK WENO schemes and it is later generalized to the high order semi-Lagrangian WENO method for solving the Vlasov-Poisson system \cite{mpp_vp}. A positivity preserving flux limiting approach is developed in \cite{pp_euler} to ensure positivity of the computed density and pressure for compressible Euler simulations. Technically, the generalization of such MPP flux limiters from FD WENO \cite{jiang2013parametrized} to FV WENO method is rather straightforward. Taking into the consideration that FV method offers a more natural framework for mass conservation and flexibility in handling irregular computational domain, we propose to apply the MPP flux limiters to the high order FV RK WENO method to solve (\ref{ad1}). The proposed flux limiting procedure is rather easy to implement even with the complexity of the flux forms in multi-dimensional FV computation. Moreover, a general theoretical proof on preserving both MPP and high order accuracy without additional time step constraint can be done for FV methods when solving a linear advection equation; such result does not hold for high order FD schemes \cite{mpp_xqx}. In this paper, for the first time, we establish a general proof that, there is no further time step restriction, besides the CFL condition under the linear stability requirement, to preserve high order accuracy when the high order flux is limited toward an upwind first order flux for solving linear advection problem, when the parametrized flux limiters are applied to FV RK WENO method. In other words, both the MPP property and high order accuracy of the original scheme can be maintained without additional time step constraint. For a general nonlinear convection problem, we prove that the flux limiter preserves up to third order accuracy and the discrete maximum principle with no further CFL restriction. {This proof relies on tedious Taylor expansions, and it is difficult to generalize it to results with higher order accuracy (fourth order or higher). On the other hand, such analysis can be extended to a convection-dominated diffusion problem as done in \cite{jiang2013parametrized}.} Furthermore, numerical results indicate that mild CFL restriction is needed for the MPP flux limiting finite volume scheme without sacrificing accuracy. For more discussions, see Section 3. The paper is organized as follows. In Section 2, we provide the numerical algorithm of the high order FV RK WENO schemes with MPP flux limiters. In Section 3, theoretical analysis is given for a linear advection problem and general nonlinear problems. Numerical experiments are demonstrated in Section 4. We give a brief conclusion in Section 5. \section{A MPP FV method} \label{sec2} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} In this section, we propose a high order FV scheme for the convection-diffusion equation. In the proposed scheme, the high order WENO reconstruction of flux is used for the convection term, while a high order compact reconstruction of flux is proposed for the diffusion term. For simplicity, we first consider a one dimensional (1D) case. The following uniform spatial discretization is used for a 1D bounded domain $[a, b]$, \begin{equation} a=x_{\fracrac{1}{2}}<x_{\fracrac{3}{2}}<\cdots<x_{N-\fracrac{1}{2}}<x_{N+\fracrac{1}{2}}=b, \ \Delta x =\fracrac{b-a}{N}. \end{equation} with the computational cell and cell center defined as \begin{equation} I_j=[x_{j-\fracrac{1}{2}}, x_{j+\fracrac{1}{2}}], \ x_j=\fracrac{1}{2}(x_{j-\fracrac{1}{2}}+x_{j+\fracrac{1}{2}}), \ j=1,2,\cdots,N. \end{equation} Let $\bar{u}_j$ denote approximation to the cell average of $u$ over cell $I_j$. The FV scheme is designed by integrating equation (\ref{ad1}) over each computational cell $I_j$ and then dividing it by $\Delta x$, \begin{equation}\label{1dscheme} \fracrac{d\bar{u}_j}{dt} = -\fracrac{1}{\Delta x} (\hat{H}_{j+\fracrac{1}{2}}^C - \hat{H}_{j-\fracrac{1}{2}}^C) + \fracrac{1}{\Delta x} (\hat{H}_{j+\fracrac{1}{2}}^D - \hat{H}_{j-\fracrac{1}{2}}^D), \end{equation} where $\hat{H}_{j+\fracrac{1}{2}}^C$ and $\hat{H}_{j+\fracrac{1}{2}}^D$ are the numerical fluxes for convection and diffusion terms respectively. For the convection term, one can adopt any monotone flux. For example, in our simulations, we use the Lax-Friedrichs flux \begin{equation} \hat{H}_{j+\fracrac{1}{2}}^C (u_{j+\fracrac{1}{2}}^-, u_{j+\fracrac{1}{2}}^+)= \fracrac{1}{2}\big(f(u_{j+\fracrac{1}{2}}^-) + \alpha u_{j+\fracrac{1}{2}}^-\big) + \fracrac{1}{2}\big(f(u_{j+\fracrac{1}{2}}^+) - \alpha u_{j+\fracrac{1}{2}}^+\big), \ \alpha ={\max}_{u_{m}\le u \le u_{M}} |f'(u)|. \end{equation} Here $u_{j+\fracrac{1}{2}}^- \doteq P(x_{j+\fracrac{1}{2}})$, where $P(x)$ is obtained by reconstructing a $(2k+1)^{th}$ order polynomial whose averages agree with those in a left-biased stencil $\{\bar{u}_{j-k}, \cdots, \bar{u}_{j+k}\}$, \[ \fracrac{1}{\Delta x}\int_{I_{l}} P(x)dx = \bar{u}_{l},\ l=j-k,\cdots,j+k. \] The reconstruction procedure for $u_{j+\fracrac{1}{2}}^+$ can be done similarly from a right-biased stencil. To suppress oscillation around discontinuities and maintain high order accuracy around smooth regions of the solution, the WENO mechanism can be incorporated in the reconstruction. Details of such procedure can be found in $\cite{Shu_book}$. For the diffusion term, we propose the following {\em compact} reconstruction strategy for approximating fluxes at cell boundaries $a(u)_x|_{x_{j+\fracrac12}}$. Without loss of generality, we consider a fourth order reconstruction, while similar strategies can be extended to schemes with arbitrary high order. Below we let $u_j$ denote approximation to the point values of $u$ at $x_j$. \begin{enumerate} \item Reconstruct $\{ u_{l}\}_{l=j-1}^{j+2}$ from the cell averages $\{ \bar u_{l}\}_{l=j-1}^{j+2}$ by constructing a cubic polynomial $P(x)$, such that \[ \fracrac{1}{\Delta x}\int_{I_{l}} P(x)dx = \bar{u}_{l},\ l=j-1,\cdots,j+2. \] Then $u_l = P(x_l)$, $l=j-1, \cdots j+2$. We use $\mathcal{R}_1$ to denote such reconstruction procedure, $$( u_{j-1}, u_{j}, u_{j+1}, u_{j+2}) = \mathcal{R}_1(\bar u_{j-1}, \bar u_{j}, \bar u_{j+1}, \bar u_{j+2}).$$ As a reference, the reconstruction formulas for $\mathcal{R}_1$ are provided below, \begin{align} & u_{j-1}=\fracrac{11}{12}\bar{u}_{j-1} + \fracrac{5}{24}\bar{u}_{j} -\fracrac{1}{6}\bar{u}_{j+1} + \fracrac{1}{24}\bar{u}_{j+2}, \nonumber \\ & u_j=-\fracrac{1}{24}\bar{u}_{j-1} + \fracrac{13}{12}\bar{u}_{j} -\fracrac{1}{24}\bar{u}_{j+1}, \nonumber\\ & u_{j+1}=-\fracrac{1}{24}\bar{u}_{j} + \fracrac{13}{12}\bar{u}_{j+1} - \fracrac{1}{24}\bar{u}_{j+2}, \nonumber\\ & u_{j+2}=\fracrac{1}{24}\bar{u}_{j-1} - \fracrac{1}{6}\bar{u}_{j} + \fracrac{5}{24}\bar{u}_{j+1} + \fracrac{11}{12}\bar{u}_{j+2}. \nonumber \end{align} \item Construct an interplant $Q(x)$ such that $$Q(x_l)=a(u_l),\ l=j-1,\cdots, j+2.$$ Then let $$\hat{H}_{j+\fracrac{1}{2}}^D=Q'(x)|_{x_{j+\fracrac{1}{2}}}.$$ Such procedure is denoted as \[\hat{H}_{j+\fracrac{1}{2}}^D = \mathcal{R}_2(a(u_{j-1}), a(u_{j}), a(u_{j+1}), a(u_{j+2})). \] As a reference, we provide the formula for $\mathcal{R}_2$ below \begin{align} & \hat{H}_{j+\fracrac{1}{2}}^D = \fracrac{1}{24} a(u_{j-1}) - \fracrac{9}{8} a(u_j) +\fracrac{9}{8} a(u_{j+1}) - \fracrac{1}{24} a(u_{j+2}). \nonumber \end{align} \end{enumerate} \begin{rem} The reconstruction processes for $\mathcal{R}_1$ and $\mathcal{R}_2$ operators are designed such that $\hat{H}_{j+\fracrac{1}{2}}^D$ is reconstructed from a compact stencil with a given order of accuracy. Because of such design, for the linear diffusion term $a(u)=u$, $\mathcal{R}_1$ and $\mathcal{R}_2$ can be combined and the strategy above turns out to be a classical fourth order central difference from a five-cell stencil with $$\hat{H}_{j+\fracrac{1}{2}}^D = \fracrac{1}{\Delta x}(\fracrac{1}{2} \bar{u}_{j-1} -\fracrac{15}{12} \bar{u}_j + \fracrac{15}{12}\bar{u}_{j+1} -\fracrac{1}{12}\bar{u}_{j+2}).$$ \end{rem} If each of $u_l$ ($l=j-1, \cdots j+2$) in Step 1 is reconstructed from symmetrical stencils (having the same number of cells from left and from right), the reconstruction of $\hat{H}_{j+\fracrac{1}{2}}^D$ will depend on a much wider stencil $\{u_{j-3}, \cdots u_{j+4}\}$. Such non-compact way of reconstructing numerical fluxes for diffusion terms will introduce some numerical instabilities when approximating nonlinear diffusion terms in our numerical tests, whereas the proposed compact strategy does not encounter such difficulty. We use the following third order total variation diminishing (TVD) RK method \cite{gottlieb2009high} for the time discretization of (\ref{1dscheme}), which reads \begin{align}\label{rk} &u^{(1)}=\bar u^n+\Delta t L(\bar u^n), \nonumber \\ &u^{(2)}=\bar u^n+\Delta t (\fracrac{1}{4}L(\bar u^n) + \fracrac{1}{4}L(u^{(1)})), \\ &\bar u^{n+1}=\bar u^n+\Delta t (\fracrac{1}{6} L(\bar u^n) + \fracrac{1}{6} L(u^{(1)}) + \fracrac{2}{3} L(u^{(2)})), \nonumber \end{align} where $L(\bar u^n)$ denotes the right hand side of equation (\ref{1dscheme}). Here $\bar u^n$ and $u^{(s)}$, $s = 1, 2$ denote the numerical solution of $u$ at time $t^n$ and corresponding RK stages. The fully discretized scheme \eqref{rk} can be rewritten as \begin{equation}\label{1dschemefully} \bar u_j^{n+1}=\bar u_j^{n}-\lambda (\hat{H}_{j+\fracrac{1}{2}}^{rk} - \hat{H}_{j-\fracrac{1}{2}}^{rk}) \end{equation} with $\lambda = \fracrac{\Delta t}{\Delta x}$ and \[ \hat{H}_{j+\fracrac{1}{2}}^{rk}=\fracrac{1}{6}(\hat{H}_{j+\fracrac{1}{2}}^{C,n} - \hat{H}_{j+\fracrac{1}{2}}^{D,n}) + \fracrac{1}{6}(\hat{H}_{j+\fracrac{1}{2}}^{C,(1)} - \hat{H}_{j+\fracrac{1}{2}}^{D,(1)}) + \fracrac{2}{3}(\hat{H}_{j+\fracrac{1}{2}}^{C,(2)} - \hat{H}_{j+\fracrac{1}{2}}^{D,(2)}). \] Here $\hat{H}_{j+\fracrac{1}{2}}^{C,(s)},\ \hat{H}_{j+\fracrac{1}{2}}^{D,(s)}\ (s=1,2)$ are the numerical fluxes at the intermediate stages in the RK scheme (\ref{rk}). It has been known that the numerical solutions from schemes with a first order monotone flux for the convection term together with a first order flux for the diffusion term satisfy the maximum principle, if the time step is small enough \cite{yzhang2012maximum}. However, if the numerical fluxes are of high order such as the one from the reconstruction process proposed above, the MPP property for the numerical solutions does not necessarily hold under the same time step constraint. Next we apply the parametrized flux limiters proposed in \cite{mpp_xqx} to the scheme (\ref{1dschemefully}) to preserve the discrete maximum principle (\ref{maxminD}). We modify the numerical flux $\hat{H}_{j+\fracrac{1}{2}}^{rk}$ in equation (\ref{1dschemefully}) with \begin{equation}\label{modified flux} \tilde{H}_{j+\fracrac{1}{2}}^{rk}=\theta_{j+\fracrac{1}{2}} \hat{H}_{j+\fracrac{1}{2}}^{rk} + (1-\theta_{j+\fracrac{1}{2}}) \hat{h}_{j+\fracrac{1}{2}}, \end{equation} by carefully seeking local parameters $\theta_{j+\fracrac{1}{2}}$, such that the numerical solutions enjoy the MPP property yet $\theta_{j+\fracrac12}$ is as close to $1$ as possible. In other words, $\tilde{H}_{j+\fracrac{1}{2}}^{rk}$ is as close to the original high order flux $\hat{H}_{j+\fracrac{1}{2}}^{rk}$ as possible. Here $\hat{h}_{j+\fracrac{1}{2}}$ denotes the first order flux for convection and diffusion terms, using which in the scheme \eqref{1dscheme} with a forward Euler time discretization guarantees the maximum principle of numerical solutions. For example, we can take \[ \hat{h}_{j+\fracrac{1}{2}}=\hat{h}^C_{j+\fracrac{1}{2}} - \hat{h}^D_{j+\fracrac{1}{2}} = \fracrac{1}{2}\big(f(\bar{u}_{j}) + \alpha \bar{u}_{j}\big) + \fracrac{1}{2}\big(f(\bar{u}_{j+1}) - \alpha \bar{u}_{j+1}\big)- \fracrac{a(\bar{u}_{j+1})-a(\bar{u}_j)}{\Delta x} \] with $\alpha = {\max}_{u_{m}\le u \le u_{M}} |f'(u)|$. The goal of the procedures outlined below is to adjust $\theta_{j+\fracrac12}$, so that with the modified flux $\tilde{H}_{j+\fracrac{1}{2}}^{rk}$, the numerical solutions satisfy the maximum principle, \begin{equation} u_m \le \bar u_j^n - \lambda (\tilde{H}_{j+\fracrac{1}{2}}^{rk} - \tilde{H}_{j-\fracrac{1}{2}}^{rk}) \le u_M, \quad \fracorall j. \end{equation} Detailed procedures in decoupling the above inequalities have been intensively discussed in our previous work, e.g. \cite{mpp_xqx}. Below we only briefly describe the computational algorithm for the proposed limiter. Let $F_{j+\fracrac{1}{2}} \doteq \hat{H}_{j+\fracrac{1}{2}}^{rk}-\hat{h}_{j+\fracrac{1}{2}}$ and \[ \Gamma _j^M \doteq u_M - (\bar u_j^n - \lambda (\hat{h}_{j+\fracrac{1}{2}} - \hat{h}_{j-\fracrac{1}{2}})), \ \Gamma _j^m \doteq u_m - (\bar u_j^n - \lambda (\hat{h}_{j+\fracrac{1}{2}} - \hat{h}_{j-\fracrac{1}{2}})). \] The MPP property is satisfied with the modified flux (\ref{modified flux}) when the following inequalities are hold, \begin{equation}a && \lambda \theta_{j-\fracrac{1}{2}} F_{j-\fracrac{1}{2}} - \lambda \theta_{j+\fracrac{1}{2}} F_{j+\fracrac{1}{2}} - \Gamma_j^M \le 0, \label{max} \\ && \lambda \theta_{j-\fracrac{1}{2}} F_{j-\fracrac{1}{2}} - \lambda \theta_{j+\fracrac{1}{2}}F_{j+\fracrac{1}{2}} - \Gamma_j^m \ge 0. \label{min} \end{equation}a We first consider the inequality \eqref{max}. We seek a local pair of numbers $(\Lambda _{-\fracrac{1}{2},I_j} ^M, \Lambda _{+\fracrac{1}{2},I_j} ^M)$ such that (1) $\Lambda _{\pm\fracrac{1}{2},I_j} ^M\in[0, 1]$ and is as close to $1$ as possible, (2) for any $\theta_{j-\fracrac{1}{2}} \in [0,\Lambda _{-\fracrac{1}{2},I_j} ^M], \ \theta_{j+\fracrac{1}{2}} \in [0,\Lambda _{+\fracrac{1}{2},I_j} ^M]$, the inequality (\ref{max}) holds. The inequality \eqref{max} can be decoupled based on the following four different cases: \begin{enumerate}[(a)] \item If $F_{j-\fracrac{1}{2}}\le 0$ and $F_{j+\fracrac{1}{2}} \ge 0$, then $(\Lambda _{-\fracrac{1}{2},I_j} ^M, \Lambda _{+\fracrac{1}{2},I_j} ^M)=(1,1). $ \item If $F_{j-\fracrac{1}{2}}\le 0$ and $F_{j+\fracrac{1}{2}} < 0$, then $(\Lambda _{-\fracrac{1}{2},I_j} ^M, \Lambda _{+\fracrac{1}{2},I_j} ^M)=(1, \min(1, \fracrac{\Gamma_j^M}{-\lambda F_{j+\fracrac{1}{2}}})). $ \item If $F_{j-\fracrac{1}{2}}\ > 0$ and $F_{j+\fracrac{1}{2}} \ge 0$, then $(\Lambda _{-\fracrac{1}{2},I_j} ^M, \Lambda _{+\fracrac{1}{2},I_j} ^M)=(\min(1, \fracrac{\Gamma_j^M}{\lambda F_{j-\fracrac{1}{2}}}), 1). $ \item If $F_{j-\fracrac{1}{2}}\ > 0$ and $F_{j+\fracrac{1}{2}} < 0$, then $$(\Lambda _{-\fracrac{1}{2},I_j} ^M, \Lambda _{+\fracrac{1}{2},I_j} ^M)=(\min(1,\fracrac{\Gamma_j^M}{\lambda F_{j-\fracrac{1}{2}}-\lambda F_{j+\fracrac{1}{2}}}), \min(1,\fracrac{\Gamma_j^M}{\lambda F_{j-\fracrac{1}{2}}-\lambda F_{j+\fracrac{1}{2}}})). $$ \end{enumerate} Similarly, we can find a local pair of numbers $(\Lambda _{-\fracrac{1}{2},I_j} ^m, \Lambda _{+\fracrac{1}{2},I_j} ^m)$ such that for any $$\theta_{j-\fracrac{1}{2}} \in [0,\Lambda _{-\fracrac{1}{2},I_j} ^m], \ \theta_{j+\fracrac{1}{2}} \in [0,\Lambda _{+\fracrac{1}{2},I_j} ^m]$$ (\ref{min}) holds. There are also four different cases: \begin{enumerate}[(a)] \item If $F_{j-\fracrac{1}{2}} \ge 0$ and $F_{j+\fracrac{1}{2}} < 0$, then $(\Lambda _{-\fracrac{1}{2},I_j} ^m, \Lambda _{+\fracrac{1}{2},I_j} ^m)=(1,1).$ \item If $F_{j-\fracrac{1}{2}} \ge 0$ and $F_{j+\fracrac{1}{2}} > 0$, then $(\Lambda _{-\fracrac{1}{2},I_j} ^m, \Lambda _{+\fracrac{1}{2},I_j} ^m)=(1,\min(1,\fracrac{\Gamma_j^m}{-\lambda F_{j+\fracrac{1}{2}}})).$ \item If $F_{j-\fracrac{1}{2}} < 0$ and $F_{j+\fracrac{1}{2}} < 0$, then $(\Lambda _{-\fracrac{1}{2},I_j} ^m, \Lambda _{+\fracrac{1}{2},I_j} ^m)=(\min(1,\fracrac{\Gamma_j^m}{\lambda F_{j-\fracrac{1}{2}}}),1).$ \item If $F_{j-\fracrac{1}{2}} < 0$ and $F_{j+\fracrac{1}{2}} \ge 0$, then \[(\Lambda _{-\fracrac{1}{2},I_j} ^m, \Lambda _{+\fracrac{1}{2},I_j} ^m)=(\min(1,\fracrac{\Gamma_j^m}{\lambda F_{j-\fracrac{1}{2}}-\lambda F_{j+\fracrac{1}{2}}}), \min(1,\fracrac{\Gamma_j^m}{\lambda F_{j-\fracrac{1}{2}}-\lambda F_{j+\fracrac{1}{2}}})).\] \end{enumerate} \noindent Finally, the local limiter parameter $\theta_{j+\fracrac{1}{2}}$ at the cell boundary $x_{j+\fracrac{1}{2}}$ is defined as \begin{equation} \theta_{j+\fracrac{1}{2}} = min(\Lambda_{+\fracrac{1}{2},I_j}^M, \Lambda_{+\fracrac{1}{2},I_j}^m, \Lambda_{-\fracrac{1}{2},I_{j+1}}^M, \Lambda_{-\fracrac{1}{2},I_{j+1}}^m), \end{equation} so that the numerical solutions $\bar u_j^{n+1}$, $\fracorall j, n$ satisfy the maximum principle. The extension of the FV RK scheme and the MPP flux limiter from 1D case to two dimensional (2D) convection-diffusion problems is straightforward. For example, we consider a 2D problem on a rectangular domain $[a, b] \times [c, d]$, \begin{equation} \label{eq: 2d} u_t + f(u)_x + g(u)_y = a(u)_{xx} + b(u)_{yy}. \end{equation} Without loss of generality, we consider a set of uniform mesh $$a=x_{\fracrac{1}{2}}<x_{\fracrac{3}{2}}<\cdots<x_{N-\fracrac{1}{2}}<x_{N_x+\fracrac{1}{2}}=b, \ \Delta x =\fracrac{b-a}{N_x},$$ $$c=y_{\fracrac{1}{2}}<y_{\fracrac{3}{2}}<\cdots<y_{N-\fracrac{1}{2}}<y_{N_y+\fracrac{1}{2}}=d, \ \Delta y =\fracrac{d-c}{N_y},$$ with $I_{i,j}=[x_{i-\fracrac{1}{2}},x_{i+\fracrac{1}{2}}]\times [y_{j-\fracrac{1}{2}},y_{j+\fracrac{1}{2}}]$. A semi-discrete FV discretization of \eqref{eq: 2d} gives \begin{align}\label{2dscheme} \fracrac{d}{dt} \bar{u}_{i,j} & + \fracrac{1}{\Delta x} (\hat{f}_{i+\fracrac{1}{2},j}-\hat{f}_{i-\fracrac{1}{2},j})+ \fracrac{1}{\Delta y} (\hat{g}_{i,j+\fracrac{1}{2}}-\hat{g}_{i,j-\fracrac{1}{2}}) \nonumber \\ &=\fracrac{1}{\Delta x} (\widehat{(a_x)}_{i+\fracrac{1}{2},j}-\widehat{(a_x)}_{i-\fracrac{1}{2},j}) + \fracrac{1}{\Delta y} (\widehat{(b_y)}_{i,j+\fracrac{1}{2}}-\widehat{(b_y)}_{i,j-\fracrac{1}{2}}), \end{align} where $\bar{u}_{i,j}=\fracrac{1}{\Delta x \Delta y} \int \int_{I_{i,j}} u dxdy$ and $\hat{f}_{i+\fracrac{1}{2},j}=\fracrac{1}{\Delta y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} f(x_{i+\fracrac{1}{2}},y)dy$ is the average of the flux over the right boundary of cell $I_{i,j}$. $\hat{g}_{i,j+\fracrac{1}{2}}$, $\widehat{(a_x)}_{i+\fracrac{1}{2},j}$, $\widehat{(b_y)}_{i,j+\fracrac{1}{2}}$ can be defined similarly. The flux $\hat{f}_{i+\fracrac{1}{2},j}$ is evaluated by applying the Gaussian quadrature rule for integration, \begin{align} & \hat{f}_{i+\fracrac{1}{2},j} = \fracrac{1}{2} \underset{i_g}{\Sigma} \omega_{i_g} f(u_{i+\fracrac{1}{2},i_g}). \end{align} Here $\underset{i_g}{\Sigma}$ represents the summation over the Gaussian quadratures with $\omega_{i_g}$ being quadrature weights and $u_{i+\fracrac{1}{2},i_g}$ is the approximated value to $u(x_{i+\fracrac{1}{2}},y_{i_g})$ with $y_{i_g}$ being the Gaussian quadrature points over $[y_{j-\fracrac{1}{2}},y_{j+\fracrac{1}{2}}]$. $u_{i+\fracrac{1}{2},i_g}$ can be reconstructed from $\{\bar{u}_{i,j}\}$ in the following two steps. Firstly, we reconstruct $\fracrac{1}{\Delta x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u(x,y_{i_g})dx$ from $\{\bar{u}_{i,j}\}$. To do this, we construct a polynomial $Q(y)$ such that \begin{equation} \fracrac{1}{\Delta y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} Q(y)dy =\fracrac{1}{\Delta x\Delta y} \int_{I_{i,j}} u(x,y)dx dy =\bar{u}_{i,j}, \end{equation} with $j$ belongs to a reconstruction stencil in the $y$-direction as in the one-dimensional case. Then $Q(y_{i_g})$ is a high order approximation to $\fracrac{1}{\Delta x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u(x,y_{i_g})dx$. We let $\mathcal{R}_{y}$ to denote such reconstruction process in $y$-direction. Secondly, we construct a polynomial $P(x)$ such that \begin{equation} \fracrac{1}{\Delta x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} P(x)dx = \fracrac{1}{\Delta x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u(x,y_{i_g})dx, \end{equation} with $i$ belongs to a reconstruction stencil in the $x$-direction as in the one-dimensional case. Then $u_{i+\fracrac{1}{2},i_g}=P(x_{i+\fracrac{1}{2}})$. Such 1D reconstruction process is denoted as $\mathcal{R}_{x}$. The 2D reconstructing procedure can be summarized as the following flowchart \begin{equation} \centering \{ \bar{u}_{i,j}\} \overset{\mathcal{R}_y}{\longrightarrow} \{\fracrac{1}{\Delta x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u(x,y_{i_g})dx \} \overset{\mathcal{R}_x}{\longrightarrow} \{u_{i+\fracrac{1}{2},i_g}\}. \end{equation} Detailed information on the 2D reconstruction procedure is similar to those described in \cite{Shu_book}. The 2D MPP flux limiter is applied in a similar fashion as those in \cite{mpp_xuMD, jiang2013parametrized, mpp_xqx}. Thus details are omitted for brevity. \begin{rem} The proposed generalization of the parametrized flux limiter to convection-diffusion problems is rather straightforward. In comparison, it is much more difficult to generalize the polynomial rescaling approach in \cite{zhang2010maximum} to schemes with higher than third order accuracy for convection diffusion problems. The approach there relies on rewriting the updated cell average as a convex combination of some local quantities within the range $[u_m, u_M]$; this is more difficult to achieve with the diffusion terms \cite{zhang2012maximumcd, yzhang2012maximum}. Moreover, the proposed flux limiter introduces very mild time step constraint to preserve both MPP and high order accuracy of the original FV RK scheme, see the next section for more discussions. \end{rem} \section{Theoretical properties} \label{sec3} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} In this section, we provide accuracy analysis for the MPP flux limiter applied to the high order FV RK scheme solving pure convection problems. Specifically, we will prove that the proposed parametrized flux limiter as in equation \eqref{modified flux} introduces a high order modification in space and time to the temporal integrated flux of the original scheme, assuming that the solution is smooth enough. A general proof on preservation of {\em arbitrary} high order accuracy will be provided for linear problems. Then by performing Taylor expansions around extrema, we prove that the modification from the proposed flux limiter is of at least third order, for FV RK schemes that are third order or higher in solving general nonlinear problems. The entropy solution $u(x, t)$ to a scalar convection problem \begin{equation} u_t+f(u)_x=0, \quad u(x,0)=u_0(x). \label{eq: adv} \end{equation} satisfies \begin{eqnarray} \label{eq: weak} \fracrac{d}{dt} \int_{x_{j-\fracrac12}}^{x_{j+\fracrac12}} u(x,t) dx =f(u(x_{j+\fracrac12}, t))-f(u(x_{j-\fracrac12}, t)). \end{eqnarray} Integrating (\ref{eq: weak}) over the time period $[t^n, t^{n+1}]$, we have \begin{eqnarray} \label{eq: int} \bar u_j(t^{n+1}) = \bar u_j(t^n)-\lambda (\check f_{j+\fracrac12} -\check f_{j-\fracrac12}), \end{eqnarray} where $\lambda=\Delta t/\Delta x$ and \begin{equation} \label{cellaverage} \bar u_j(t)=\fracrac{1}{\Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}} u(x, t) dx, \quad \check f_{j-1/2} =\fracrac{1}{\Delta t} \int_{t^n}^{t^{n+1}} f(u(x_{j-1/2}, t)) dt. \end{equation} The entropy solution satisfies the maximum principle in the form of \begin{equation} \label{IQL} u_m\le\bar u_j(t^n)-\lambda (\check f_{j+\fracrac12} -\check f_{j-\fracrac12}) \le u_M. \end{equation} For schemes with $(2k+1)^{th}$ order finite volume spatial discretization \eqref{1dschemefully} and $p^{th}$ order RK time discretization, we assume \begin{equation} |\check f_{j+\fracrac12}-\hat H^{rk}_{j+\fracrac12}|=\mathcal{O}(\Delta x^{2k+1} + \Delta t^p), \quad \fracorall j. \label{assumption} \end{equation} Our analysis is in the sense of local truncation analysis assuming the difference between $\bar u_j(t^n)$ and $\bar u^n_j$ is of high order ($\mathcal{O}(\Delta x^{2k+1} + \Delta t^p)$). Under a corresponding $(2k+1)^{th}$ order reconstruction, the difference between the point values $u(x_j, t^n)$ and $u^n_j$ is also of high order. In the following, we use them interchangeably when such high order difference allows. For the MPP flux limiter, we only consider the maximum value part as in equation \eqref{max}. The proof of equation \eqref{min} for the minimum value would be similar. We would like to prove that the difference between $\hat H^{rk}_{j+\frac12}$ and $\tilde H^{rk}_{j+\frac12}$ in \eqref{modified flux} is of high order in both space and time, that is \begin{equation} \label{trc} |\hat H^{rk}_{j+\fracrac12}-\tilde H^{rk}_{j+\fracrac12}|=\mathcal{O}(\Delta x^{2k+1} + \Delta t^p), \quad \fracorall j. \end{equation} There are four cases of the maximum value part \eqref{max} outlined in the previous section. The estimate \eqref{trc} can be easily checked for case (a) and (d) under the assumption \eqref{assumption} and the fact (\ref{IQL}), see arguments in \cite{mpp_xqx}. Below we will only discuss case (b), as the argument for case (c) would be similar. First we give the following lemma: \begin{lem} \label{lem1} Consider applying the MPP flux limiter \eqref{modified flux} for the maximum value part \eqref{max} with case (b), to prove \eqref{trc}, it suffices to have \begin{equation} |u_M-(\bar u_j-\lambda(\check f_{j+\fracrac{1}{2}}-\hat h_{j-\fracrac12}))| = \mathcal{O}(\Delta x^{2k+1} + \Delta t^p), \label{3rd} \end{equation} if $u_M-(\bar u_j-\lambda(\hat H^{rk}_{j+\fracrac{1}{2}}-\hat h_{j-\fracrac12})) < 0$. \end{lem} \begin{proof} For case (b), we are considering the case when \begin{equation*} \Lambda_{+\fracrac12, {I_j}}=\fracrac{\Gamma^M_j}{-\lambda F_{j+\fracrac12}} < 1, \end{equation*} which is equivalent to $u_M-(\bar u_j-\lambda(\hat{H}^{rk}_{j+\fracrac{1}{2}}-\hat h_{j-\fracrac12})) < 0$, and \begin{equation*} \qquad \tilde H^{rk}_{j+\fracrac12} -\hat H^{rk}_{j+\fracrac12} = \fracrac {\Gamma^M_j+\lambda F_{j+\fracrac12}}{-\lambda}=\fracrac{u_M-(\bar u_j -\lambda (\hat H^{rk}_{j+\fracrac12}-\hat h_{j-\fracrac12}))}{-\lambda}, \end{equation*} which indicates that it suffices to have \eqref{3rd} to obtain (\ref{trc}) with the assumption (\ref{assumption}). \end{proof} \begin{thm} \label{general_proof} Assuming $f'(u) >0$ and $\lambda \max_u |f'(u)|\le 1$, we have \begin{equation} \label{UL} \bar u_j(t^n)-\lambda (\check f_{j+\fracrac12} - f(\bar u_{j-1}(t^n))) \le u_M \end{equation} if $u(x, t)$ is the entropy solution to (\ref{eq: adv}) subject to initial data $u_0(x)$. \end{thm} \begin{proof} Consider the problem (\ref{eq: adv}) with a different initial condition at time level $t^n$, \begin{eqnarray} \tilde u(x,t^n) = \begin{cases} u(x,t^n) \quad & x\ge x_{j-\fracrac12},\\ \bar u_{j-1}(t^n) \quad & x< x_{j-\fracrac12}, \end{cases} \end{eqnarray} here $u(x,t^n)$ is the exact solution of (\ref{eq: adv}) at time level $t^n$. Assuming $\tilde u(x, t)$ is its entropy solution corresponding to the initial data $\tilde u(x, t^n)$, instantly we have \begin{eqnarray} \label{eq: eql1} \bar {\tilde u}_j(t^n)=\bar u_j(t^n). \end{eqnarray} Since $f'(u) >0$, we have \begin{eqnarray} \label{eq: eql2} f(\tilde u(x_{j-\fracrac12}, t)) =f(\bar u_{j-1}(t^n)), \end{eqnarray} for $t\in [t^n, t^{n+1}]$. Since $\lambda \max_u |f'(u)|\le 1$, the characteristic starting from $x_{j-\fracrac12}$ would not hit the side $x_{j+\fracrac12}$, therefore \begin{eqnarray} \label{eq: eql3} \tilde u(x_{j+\fracrac12}, t)= u(x_{j+\fracrac12}, t) \end{eqnarray} for $t\in [t^n, t^{n+1}]$. Also since $\tilde u$ satisfies the maximum principle $\tilde u \le u_M$, we have \begin{eqnarray*} \bar {\tilde u}^{n+1}_j = \bar {\tilde u}^n_j -\lambda (\check {\tilde f}_{j+\fracrac12} - \check {\tilde f}_{j-\fracrac12}) \le u_M, \end{eqnarray*} where \begin{equation} \label{cellaverage1} \check {\tilde f}_{j-1/2} =\fracrac{1}{\Delta t} \int_{t^n}^{t^{n+1}} f(\tilde u(x_{j-1/2}, t)) dt. \end{equation} Substituting (\ref{eq: eql1}), (\ref{eq: eql2}) and (\ref{eq: eql3}) into the above inequality, it follows that \begin{eqnarray*} \bar {u}_j(t^n)-\lambda (\check {f}_{j+\fracrac12} - f(\bar u_{j-1}(t^n)) \le u_M. \end{eqnarray*} \end{proof} For the case $f'(u)< 0$, we have the following \begin{thm} \label{general_proof2} Assuming $f'(u) < 0$ and $\lambda \max_u |f'(u)|\le 1$, we have \begin{equation} \label{DL} \bar u_j (t^n)-\lambda (\check f_{j+\fracrac12} - f(\bar u_{j} (t^n))) \le u_M, \end{equation} if $u(x, t)$ is the entropy solution to problem (\ref{eq: adv}) subject to initial data $u_0(x)$. \end{thm} \begin{proof} The proof is similar. The only difference is that in this case, we shall consider an auxiliary problem (\ref{eq: adv}) with initial data \begin{eqnarray} \tilde {\tilde u}(x, t^n) = \begin{cases} u(x, t^n) \quad & x\ge x_{j+\fracrac12},\\ \bar u_{j} (t^n) \quad & x< x_{j+\fracrac12}. \end{cases} \end{eqnarray} \end{proof} Theorem \ref{general_proof} and \ref{general_proof2} implies the first {\bf main result} \begin{thm} \label{UW} For the cases stated in Theorem \ref{general_proof} and \ref{general_proof2}: $f'(u)>0$ or $f'(u)<0$, with $\lambda \max_u |f'(u)|\le 1$, the estimate $$|\hat H^{rk}_{j+\fracrac12}-\tilde H^{rk}_{j+\fracrac12}|=\mathcal{O}(\Delta x^{2k+1} + \Delta t^p), \quad \fracorall j$$ holds if equation $$|\check f_{j+\fracrac12}-\hat H^{rk}_{j+\fracrac12}|=\mathcal{O}(\Delta x^{2k+1} + \Delta t^p), \quad \fracorall j$$ holds, when $\hat h_{j-\fracrac12}$ is the first order Godunov flux for the modification in (\ref{modified flux}). \end{thm} \begin{proof} The theorem can be proved by combining earlier arguments in this section, observing that $\hat h_{j-\fracrac12} = f(\bar u^n_{j-1})$ if $f'(u)> 0$, otherwise $\hat h_{j-\fracrac12} = f(\bar u^n_{j})$. \end{proof} The conclusion from Theorem \ref{UW} is that the MPP flux limiters for high order FV RK scheme does not introduce extra CFL constraint to preserve the high order accuracy of the original scheme. In the linear advection case, Theorem \ref{UW} simply indicates that \begin{rem} \label{DON} The MPP flux limiters preserve high order accuracy under the CFL requirement $\lambda \max_u |f'(u)|\le 1$ for linear advection problems when high order numerical fluxes are limited to the first order upwind flux. Without much difficulty, we can generalize the results in Theorem \ref{general_proof}, \ref{general_proof2} to two dimensional linear advection problems. \end{rem} It is difficult to generalize the above approach to general convection-dominated diffusion problems. However, we believe this is one important step toward a complete proof. Below, by performing Taylor expansions around extrema, we provide a proof of \eqref{trc} with third order spatial and temporal accuracy ($k=1, p=3$) for a general nonlinear problem. We consider a first order monotone flux $\hat h_{j-\fracrac12}=\hat h(\bar u_{j-1}, \bar u_j)$ in the proposed parametrized flux limiting procedure \eqref{modified flux}. And we define \begin{equation} L_{1,j} = \frac{\hat h(\bar u_{j-1}, \bar u_j )-f(\bar u_{j-1})}{\bar u_{j}-\bar u_{j-1}}, \quad L_{2,j} = - \frac{f(\bar u_{j})-\hat h(\bar u_{j-1}, \bar u_j )}{\bar u_{j}-\bar u_{j-1}}, \label{lipschitz} \end{equation} where $L_{1,j}$ and $L_{2,j}$ are two coefficients related to the monotonicity condition \cite{harten1983high}. Let $L=\max_j |L_{1,j}+L_{2,j}|$, we have \begin{thm} \label{thm: accuracy} Consider a third order (or higher) finite volume RK discretization for a pure convection problem \eqref{eq: adv}, with a first order monotone flux $\hat h_{j-\fracrac12}=\hat h(\bar u_{j-1}, \bar u_j)$ in \eqref{modified flux}. The estimate \eqref{trc} holds with $k=1, p=3$ under the CFL condition $1-\lambda L \ge 0$. \end{thm} \begin{proof} Using the earlier argument, we will only prove \eqref{3rd}, assuming $u_M-(\bar u_j-\lambda(\hat H^{rk}_{j+\fracrac{1}{2}}-\hat h_{j-\fracrac12})) < 0$. We mimic the proof for the finite difference scheme in \cite{mpp_xqx}. First we use the 3-point Gauss Lobatto quadrature to approximate $\check f_{j+\frac12}$, \begin{align} \check f_{j+\frac12}=\fracrac16 f(u(x_{j+\fracrac12},t^n+\Delta t))+\fracrac23 f((x_{j+\fracrac12},t^n+\fracrac{\Delta t}{2}))+\fracrac16 f((x_{j+\fracrac12},t^n))+\mathcal{O}(\Delta t^3). \label{glrule} \end{align} Following the characteristics, we get \begin{align} \check f_{j+\frac12}=\fracrac16 f(u(x_{j+\fracrac{1}{2}}-\lambda_{1}\Delta x,t^n))+ \fracrac23 f(u(x_{j+\fracrac{1}{2}}-\lambda_{2}\Delta x,t^n )) + \fracrac16 f(u(x_{j+\fracrac{1}{2}},t^n))+\mathcal{O}(\Delta t^3),\label{glrule3} \end{align} where $\lambda_{1}$ and $\lambda_{2}$ can be determined from \begin{eqnarray} \lambda_{1}=\lambda f'(u(x_{j+\fracrac12}-\lambda_{1}\Delta x,t^n)), \quad \lambda_{2}=\fracrac{\lambda}{2} f'(u(x_{j+\fracrac12}-\lambda_{2}\Delta x,t^n)). \label{lamb} \end{eqnarray} For the finite volume method, $u(x^*,t^n)$ in (\ref{glrule3}) can be approximated by a second order polynomial reconstruction from $\bar u_{j-1}$, $\bar u_j$ and $\bar u_{j+1}$. Denoting $u_1=u(x_{j+\fracrac{1}{2}}-\lambda_{1}\Delta x,t^n)$, $u_2=u(x_{j+\fracrac{1}{2}}-\lambda_{2}\Delta x,t^n)$ and $u_3=u(x_{j+\fracrac{1}{2}},t^n)$, we have \begin{subequations} \label{3rdr} \begin{align} &u_1=\frac16\left((5+6\lambda_1-6\lambda_1^2)\bar u_j+(-1+3\lambda_1^2)\bar u_{j-1}+(2-6\lambda_1+3\lambda_1^2)\bar u_{j+1}\right) +O(\Delta x^3), \\ &u_2=\frac16\left((5+6\lambda_2-6\lambda_2^2)\bar u_j+(-1+3\lambda_2^2)\bar u_{j-1}+(2-6\lambda_2+3\lambda_1^2)\bar u_{j+1}\right) +O(\Delta x^3), \\ &u_3=\frac16\left(5\bar u_j-\bar u_{j-1}+2\bar u_{j+1}\right) +O(\Delta x^3). \end{align} \end{subequations} We prove (\ref{3rd}) case by case. We first consider the case $x_M \in I_j$, with $u_M=u(x_M)$, $u'_M=0$ and $u''_M\le 0$. We perform Taylor expansions around $x_M$, \begin{subequations} \label{tayloru} \begin{align} &\bar u_j=u_M+u'_M(x_j-x_M)+u''_M \left(\fracrac{(x_j-x_M)^2}{2}+\fracrac{\Delta x^2}{24}\right)+O(\Delta x^3), \label{tayloru0} \\ &\bar u_{j+1}=u_M+u'_M(x_j-x_M+\Delta x)+u''_M \left(\fracrac{(x_j-x_M+\Delta x)^2}{2}+\fracrac{\Delta x^2}{24}\right)+O(\Delta x^3), \label{taylorup} \\ &\bar u_{j-1}=u_M+u'_M(x_j-x_M)+u''_M \left(\fracrac{(x_j-x_M-\Delta x)^2}{2}+\fracrac{\Delta x^2}{24}\right)+O(\Delta x^3). \label{taylorum} \end{align} \end{subequations} Denoting $z=(x_j-x_M)/\Delta x$, the approximation in \eqref{3rdr} can be rewritten as \begin{subequations} \label{3rdr2} \begin{align} &u_1=u_M+u'_M\Delta x(\frac12-\lambda_1+z)+u''_M\frac{\Delta x^2}{2}(\frac14-\lambda_1+\lambda_1^2+z-2\lambda_1z+z^2)+O(\Delta x^3), \\ &u_2=u_M+u'_M\Delta x(\frac12-\lambda_2+z)+u''_M\frac{\Delta x^2}{2}(\frac14-\lambda_2+\lambda_2^2+z-2\lambda_2z+z^2)+O(\Delta x^3), \\ &u_3=u_M+u'_M\Delta x(\frac12+z)+u''_M\frac{\Delta x^2}{2}(\frac14+z+z^2)+O(\Delta x^3). \end{align} \end{subequations} Based on similar Taylor expansions of \eqref{tayloru}, for the flux function $f$, from \eqref{tayloru} and \eqref{3rdr2}, we would have \begin{subequations} \label{taylorf} \begin{align} f(\bar u_j)&= f(u_M)+ f'(u_M) \Big(u'_M \Delta x z +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{12}+z^2)\Big) \nonumber \\ &+\frac12 f''(u_M)\Big(u'_M z \Delta x+u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{12}+z^2)\Big)^2+O(\Delta x^3), \label{taylorf0} \\ f(\bar u_{j-1})&= f(u_M)+ f'(u_M) \Big(u'_M \Delta x (z-1) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{13}{12}-2z+z^2)\Big) \nonumber \\ &+\frac12 f''(u_M)\Big(u'_M \Delta x (z-1) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{13}{12}-2z+z^2)\Big)^2+O(\Delta x^3), \label{taylorfm} \\ f(u_1)&= f(u_M)+ f'(u_M) \Big(u'_M \Delta x (1/2-\lambda_1+z) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{4}-\lambda_1+\lambda_1^2+z-2 \lambda_1 z+z^2)\Big) \nonumber \\ &+\frac12 f''(u_M)\Big(u'_M \Delta x (1/2-\lambda_1+z) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{4}-\lambda_1+\lambda_1^2+z-2 \lambda_1 z+z^2)\Big)^2+O(\Delta x^3), \label{taylorf1} \\ f(u_2)&= f(u_M)+ f'(u_M) \Big(u'_M \Delta x (1/2-\lambda_2+z) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{4}-\lambda_2+\lambda_2^2+z-2 \lambda_2 z+z^2)\Big) \nonumber \\ &+\frac12 f''(u_M)\Big(u'_M \Delta x (1/2-\lambda_2+z) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{4}-\lambda_2+\lambda_2^2+z-2 \lambda_2 z+z^2)\Big)^2+O(\Delta x^3), \label{taylorf2} \\ f(u_3)&= f(u_M)+ f'(u_M) \Big(u'_M \Delta x (1/2+z) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{4}+z+z^2)\Big)\nonumber \\ &+\frac12 f''(u_M)\Big(u'_M \Delta x (1/2+z) +u''_M\fracrac{\Delta x^2}{2}(\fracrac{1}{4}+z+z^2)\Big)^2+O(\Delta x^3). \label{taylorf3} \end{align} \end{subequations} Now denoting $\lambda_1 =\lambda_0+\eta_1 \Delta x+\mathcal{O}(\Delta x^2)$ and $\lambda_2=\fracrac{\lambda_0}{2}+\eta_2\Delta x+\mathcal{O}(\Delta x^2)$, where $\lambda_0=\lambda f'(u_M)$, we can determine $\eta_1$ and $\eta_2$ by substituting them into (\ref{lamb}) and we have \begin{align*} &\lambda_1=\lambda_0+f''(u_M)u'_M \lambda (z+\fracrac12-\lambda_0)\Delta x+\mathcal{O}(\Delta x^2), \nonumber \\ &\lambda_2=\fracrac{\lambda_0}{2}+f''(u_M)u'_M \fracrac{\lambda}{2} (z+\fracrac12-\fracrac{\lambda_0}{2})\Delta x+\mathcal{O}(\Delta x^2). \end{align*} For the first order monotone flux $\hat h_{j-\frac12}=\hat h(\bar u_{j-1}, \bar u_j )$, it can be written as \begin{equation} \hat h_{j-\frac12}=f(\bar u_{j-1})+L_{1,j} (\bar u_{j}-\bar u_{j-1}), \quad L_{1,j} = \frac{\hat h(\bar u_{j-1}, \bar u_j )-f(\bar u_{j-1})}{\bar u_{j}-\bar u_{j-1}}, \label{hupwind} \end{equation} where $f(\bar u_{j-1})=\hat h(\bar u_{j-1},\bar u_{j-1})$ due to consistence. $L_{1,j}$ is negative and bounded due to the monotonicity and Lipschitz continuous conditions. On the other hand, $\hat h_{j-\frac12}$ can also be written as \begin{equation} \hat h_{j-\frac12}=f(\bar u_{j})+L_{2,j} (\bar u_{j}-\bar u_{j-1}), \quad L_{2,j} = - \frac{f(\bar u_{j})-\hat h(\bar u_{j-1}, \bar u_j )}{\bar u_{j}-\bar u_{j-1}}, \label{hdownwind} \end{equation} where $f(\bar u_{j})=\hat h(\bar u_{j},\bar u_{j})$, and $L_{2,j}$ is negative and bounded. With above notations and $u'_M=0$, we now discuss the following two cases: \begin{itemize} \item If $f'(u_M)\ge 0$, we have $\lambda_0=\lambda f'(u_M)\in[0,1]$ since $\lambda \max_u |f'(u)| \le 1$. We take $\hat h_{j-\frac12}$ as in \eqref{hupwind} and we have \begin{equation} \bar u_j-\lambda\left(\check f_{j+\frac12}-\hat h_{j-\fracrac12}\right) =u_M+\fracrac{u''_M}{12}\Delta x^2 g(z,\lambda_0)+\mathcal{O}(\Delta x^3+\Delta t^3), \label{star3n} \end{equation} where \begin{equation} g(z,\lambda_0)=g_1(z,\lambda_0)- 6\lambda L_{1,j} (1-2z), \end{equation} with \begin{equation} g_1(z,\lambda_0)=\frac12+(5\lambda_0+3\lambda_0^2-2\lambda_0^3)+6(-3\lambda_0+\lambda_0^2)z+6z^2. \end{equation} $\lambda L_{1,j}(1-2z)\le 0$ for $z\in[-\frac12, \frac12]$ and $L_{1,j}\le 0$. The minimum value of function $g_1$ with respect to $z$ is \begin{equation} (g_1)_{min}=g_1(z,\lambda_0)\Big|_{z=-\fracrac12\lambda_0(\lambda_0-3)}=\frac12+\fracrac{\lambda_0}{2}(\lambda_0-2)(\lambda_0-1)(5-3\lambda_0)\ge 0, \end{equation} so that $g(z,\lambda_0)\ge 0$. Since $u''_M\le 0$, from (\ref{star3n}) we obtain (\ref{3rd}). \item If $f'(u_M)<0$, we have $\lambda_0\in[-1,0]$. We take $\hat h_{j-\frac12}$ in \eqref{hdownwind}, similarly we have (\ref{star3n}) and \begin{equation} g(z,\lambda_0)=g_2(z,\lambda_0)-6\lambda L_{2,j} (1-2z), \end{equation} with \begin{equation} g_2(z,\lambda_0)=\frac12+(-\lambda_0+3\lambda_0^2-2\lambda_0^3)+6(-\lambda_0+\lambda_0^2)z+6z^2. \end{equation} $\lambda L_{2,j}(1-2z)\le 0$ for $z\in[-\frac12, \frac12]$ and $L_{2,j}\le 0$. The minimum value of $g_2$ with respect to $z$ is \begin{equation} (g_2)_{min}=g_2(z,\lambda_0)\Big|_{z=-\fracrac12\lambda_0(\lambda_0-1)} =\frac12+\fracrac{\lambda_0}{2}(\lambda_0+1)(\lambda_0-1)(2-3\lambda_0)\ge 0, \end{equation} that is $g(z,\lambda_0) \ge 0$. Since $u''_M\le 0$, from (\ref{star3n}) we also obtain (\ref{3rd}). \end{itemize} Now if $x_M \notin I_j$, however there is a local maximum point $x^{loc}_M$ inside the cell of $I_j$, the above analysis still holds. We then consider that $u(x)$ reaches its local maximum $u^{loc}_M$ over $I_j$ at $x^{loc}_M=x_{j-\fracrac12}$, we have $u'_{j-\fracrac12}<0$. We take $\hat h_{j-\frac12}$ as an average of \eqref{hupwind} and \eqref{hdownwind}. From the Taylor expansions in (\ref{taylorf}), following the same procedure as above, with $z=(x_j-x^{loc}_M)/\Delta x=(x_j-x_{j-\fracrac12})/\Delta x=1/2$, we have \begin{align} \bar u_j-\lambda\left(\check f_{j+\frac12}-\hat h_{j-\fracrac12}\right) =u_{j-\fracrac12}+u'_{j-\fracrac12} \Delta x s_1 + (u'_{j-\fracrac12})^2 \Delta x^2 s_2 +u''_{j-\fracrac12}\fracrac{\Delta x^2}{2}s_3+\mathcal{O}(\Delta x^3+\Delta t^3), \label{star3} \end{align} where \begin{align*} & s_1=\fracrac12(-2\lambda_0+\lambda_0^2)+\fracrac{1}{2}(1+\lambda(L_{1,j}+L_{2,j})), \\ & s_2=-f''(u_{j-\fracrac12})\fracrac{\lambda }{8}(3-4\lambda_0+4\lambda_0^2),\qquad s_3=\fracrac{1}{3}(1-2\lambda_0+3\lambda_0^2-\lambda_0^3). \end{align*} (\ref{star3}) can be rewritten as \begin{align} \bar u_j-\lambda\left(\check f_{j+\frac12}-\hat h_{j-\fracrac12}\right) =& u(x_{j-\fracrac12}-\sqrt{s_3}\Delta x)+u'_{j-\fracrac12}\Delta x \big(\fracrac12(-2\lambda_0+\lambda_0^2)+\sqrt{s_3} \nonumber \\ +&\frac12(1+\lambda(L_{1,j}+L_{2,j}))\big)+(u'_{j-\fracrac12})^2\Delta x^2 s_2 +\mathcal{O}(\Delta x^3+\Delta t^3). \label{star31} \end{align} It is easy to check that $s_3 > 0$ and $\fracrac12(-2\lambda_0+\lambda_0^2)+\sqrt{s_3}>0$ for $\lambda_0=\lambda f'(u_M)\in[-1,1]$. From the CFL condition $1+\lambda (L_{1,j}+L_{2,j}) \ge 1-\lambda L \ge 0$, we obtain $u'_{j-\fracrac12}\Delta x \big(\fracrac12(-2\lambda_0+\lambda_0^2)+\sqrt{s_3}+\frac12(1+\lambda(L_{1,j}+L_{2,j}))\big)\le 0$ since $u'_{j-\fracrac12}<0$. Now to prove \eqref{3rd}, it is sufficient to show $u(x_{j-\fracrac12}-\sqrt{s_3}\Delta x)+\Delta x^2 (u'_{j-\fracrac12})^2 s_2\le u_M$ or $u'_{j-\fracrac12} = \mathcal{O}(\Delta x)$. If $[x_{j-\fracrac12}-\sqrt{s_3}\Delta x-\Delta x, x_{j-\fracrac12}-\sqrt{s_3}\Delta x]$ is not a monotone region, there is a point $x^{\#,1}$ in this region, such that $u'(x^{\#,1})=0$. Similarly, if $[x_{j-\fracrac12}-\sqrt{s_3}\Delta x-\Delta x, x_{j-\fracrac12}-\sqrt{s_3}\Delta x]$ is a monotone increasing region, since $u'_{j-\fracrac12}<0$, there is one point $x^{\#,2}$ in $[x_{j-\fracrac12}-\sqrt{s_3}\Delta x, x_{j-\fracrac12}]$, such that $u'(x^{\#,2})=0$. For these two cases, $u'_{j-\fracrac12}=\mathcal{O}(\Delta x)$. We then focus on the case when $[x_{j-\fracrac12}-\sqrt{s_3}\Delta x-\Delta x, x_{j-\fracrac12}-\sqrt{s_3}\Delta x]$ is a monotone decreasing region. We assume \[ u(x_{j-\fracrac12}-\sqrt{s_3}\Delta x)+ c \Delta x^2 >u_M \] where $c=|(u'_{j-\fracrac12})^2 s_2|$. Since \[ u(x_{j-\fracrac12}-\sqrt{s_3}\Delta x)=u(x_{j-\fracrac12}-\sqrt{s_3}\Delta x-\Delta x)+u'(x^{\#,3}) \Delta x, \] where $u'(x^{\#,3})<0$, we have \[ u'(x^{\#,3}) \Delta x +c\Delta x^2>0, \] which implies $|u'(x^{\#,3})|\le c \Delta x$, therefore, $u'_{j-\fracrac12} =\mathcal{O}(\Delta x)$. $x^{loc}_M=x_{j+\fracrac{1}{2}}$ with $u'_{j+\frac12} \ge 0$ can be proved similarly. Combining the above discussion, (\ref{3rd}) is proved. \end{proof} Therefore, for the general nonlinear convection problem, the MPP flux limiters preserve the third order accuracy of the original FV RK scheme without extra CFL constraint. \begin{rem} The above proof relies on characteristic tracing. It is difficult to directly generalize such approach to the convection-diffusion problem. On the other hand, similar strategy as that used in \cite{jiang2013parametrized} by using a Lax-Wendroff strategy, i.e. transforming temporal derivatives into spatial derivatives by repeating using PDEs and its differentiation versions, can be directly applied here. A similar conclusion can be obtained that {the MPP flux limiters preserve the third order accuracy of the original FV RK scheme for the convection dominated diffusion equation without extra CFL constraint}. To save some space, we will not repeat the algebraically tedious details here. \end{rem} \begin{rem} It is technically difficult to generalize the proof in Theorem~\ref{thm: accuracy} to higher than third order, especially with the use of general monotone fluxes, for example, global Lax-Friedrich flux \begin{equation} \label{eq: gLxf} \hat h_{j-\fracrac12}=\hat h(\bar u_{j-1}, \bar u_j)=\fracrac12\big(f(\bar u_j)+f(\bar u_{j-1})-\alpha(\bar u_j-\bar u_{j-1})\big), \quad \alpha=\max_{u}|f'(u)|. \end{equation} On the other hand, the use of the global Lax-Friedrich flux with an extra large $\alpha$ is not unusual; yet it is quite involved to theoretically or numerically investigate such issue in a nonlinear system. Instead, we use a monotone but over-diffusive flux with \begin{equation} \label{eq: over_diff} \hat{h}_{j+\fracrac12} = \frac12 \big((1+\alpha) \bar{u}_j + (1- \alpha) \bar{u}_{j+1}\big), \quad \alpha>\max_{u}|f'(u)|=1, \end{equation} for a linear advection equation $u_t + u_x = 0$ with a set of carefully chosen initial conditions. Such scenario is set up to mimic the use of global Lax-Friedrich flux with an extra large $\alpha$ for general nonlinear systems. In Table~\ref{tab301}-\ref{tab303} below, we present the accuracy test for using the parametrized flux limiter with an over-diffusive first order monotone flux \eqref{eq: over_diff} with $\alpha=1.2$ on a linear 5th, 7th and 9th order FV RK schemes, which denoted to be ``FVRK5'', ``FVRK7'', ``FVRK9'' respectively. A mild CFL constraint around $0.7$ with time step $\Delta t=CFL \Delta x/\alpha$ is observed to be sufficient to maintain the high order accuracy of the underlying scheme with the MPP flux limiter. \end{rem} \begin{table}[ht]\fracootnotesize \centering \begin{tabular}{|c||c||c|c|c|c|c||c|c|} \hline $CFL$& & mesh & $L^1$ error & order & $L^\infty$ error & order & Umin & Umax \\ \hline \multirow{14}{*}{$0.9$} &\multirow{7}{*}{Non-} & 20 & 1.29E-02 & --& 2.00E-02 & -- & -0.013805229 & 0.960012218 \\ \cline{3-9} &\multirow{7}{*}{MPP} & 40 & 5.62E-04 & 4.52& 9.27E-04 & 4.43 & -0.000670411 & 0.988524452 \\ \cline{3-9} & & 80 & 1.87E-05 & 4.91& 3.13E-05 & 4.89 & -0.000025527 & 0.998060523 \\ \cline{3-9} & & 160 & 5.96E-07 & 4.97& 9.94E-07 & 4.98 & -0.000000471 & 0.999076363 \\ \cline{3-9} & & 320 & 1.87E-08 & 4.99& 3.12E-08 & 4.99 & -0.000000025 & 0.999931894 \\ \cline{3-9} & & 640 & 5.85E-10 & 5.00& 9.76E-10 & 5.00 & -0.000000001 & 0.999980112 \\ \cline{3-9} & & 1280 & 1.83E-11 & 5.00& 3.05E-11 & 5.00 & 0.000000000 & 0.999992161 \\ \cline{2-9} &\multirow{7}{*}{MPP} & 20 & 9.97E-03 & --& 1.82E-02 & -- & 0.000000000 & 0.960132209 \\ \cline{3-9} & & 40 & 5.52E-04 & 4.18& 1.31E-03 & 3.80 & 0.000000000 & 0.988525623 \\ \cline{3-9} & & 80 & 1.89E-05 & 4.87& 4.62E-05 & 4.83 & 0.000000000 & 0.998060523 \\ \cline{3-9} & & 160 & 6.04E-07 & 4.96& 2.01E-06 & 4.52 & 0.000000325 & 0.999076363 \\ \cline{3-9} & & 320 & 1.91E-08 & 4.98& 7.25E-08 & 4.79 & 0.000000010 & 0.999931894 \\ \cline{3-9} & & 640 & 6.04E-10 & 4.99& 2.95E-09 & 4.62 & 0.000000001 & 0.999980112 \\ \cline{3-9} & & 1280 & 1.90E-11 & 4.99& 1.33E-10 & 4.47 & 0.000000000 & 0.999992161 \\ \hline \multirow{14}{*}{$0.7$} &\multirow{7}{*}{Non-} & 20 & 1.30E-02 & --& 2.01E-02 & -- & -0.014015296 & 0.959761206 \\ \cline{3-9} &\multirow{7}{*}{MPP} & 40 & 5.66E-04 & 4.52& 9.35E-04 & 4.43 & -0.000680048 & 0.988513480 \\ \cline{3-9} & & 80 & 1.89E-05 & 4.90& 3.17E-05 & 4.88 & -0.000025848 & 0.998060157 \\ \cline{3-9} & & 160 & 6.03E-07 & 4.97& 1.01E-06 & 4.98 & -0.000000482 & 0.999076351 \\ \cline{3-9} & & 320 & 1.89E-08 & 4.99& 3.16E-08 & 4.99 & -0.000000026 & 0.999931893 \\ \cline{3-9} & & 640 & 5.92E-10 & 5.00& 9.87E-10 & 5.00 & -0.000000001 & 0.999980112 \\ \cline{3-9} & & 1280 & 1.85E-11 & 5.00& 3.09E-11 & 5.00 & 0.000000000 & 0.999992161 \\ \cline{2-9} &\multirow{7}{*}{MPP} & 20 & 9.95E-03 & --& 1.81E-02 & -- & 0.000000000 & 0.959688278 \\ \cline{3-9} & & 40 & 5.55E-04 & 4.16& 1.40E-03 & 3.70 & 0.000000000 & 0.988514505 \\ \cline{3-9} & & 80 & 1.91E-05 & 4.86& 4.90E-05 & 4.84 & 0.000000000 & 0.998060157 \\ \cline{3-9} & & 160 & 6.09E-07 & 4.97& 1.86E-06 & 4.72 & 0.000000000 & 0.999076351 \\ \cline{3-9} & & 320 & 1.91E-08 & 5.00& 6.03E-08 & 4.94 & 0.000000002 & 0.999931893 \\ \cline{3-9} & & 640 & 5.95E-10 & 5.00& 1.91E-09 & 4.98 & 0.000000000 & 0.999980112 \\ \cline{3-9} & & 1280 & 1.85E-11 & 5.00& 5.61E-11 & 5.09 & 0.000000000 & 0.999992161 \\ \hline \end{tabular} \caption{$L^1$ and $L^\infty$ errors and orders for $u_t+u_x=0$ with initial condition $u(x,0)=\sin^4(x)$. $T=1$. The over-diffusive global Lax-Friedrichs flux \eqref{eq: over_diff} is used with $\alpha=1.2$. FVRK5. } \label{tab301} \end{table} \begin{table}[ht]\fracootnotesize \centering \begin{tabular}{|c||c||c|c|c|c|c||c|c|} \hline $CFL$& & mesh & $L^1$ error & order & $L^\infty$ error & order & Umin & Umax \\ \hline \multirow{14}{*}{$0.9$} &\multirow{6}{*}{Non-} & 20 & 4.13E-03 & --& 6.38E-03 & -- & -0.004489835 & 0.972363581 \\ \cline{3-9} &\multirow{6}{*}{MPP} & 40 & 4.69E-05 & 6.46& 7.37E-05 & 6.44 & -0.000005603 & 0.989301523 \\ \cline{3-9} & & 80 & 3.99E-07 & 6.88& 6.38E-07 & 6.85 & 0.000001412 & 0.998091183 \\ \cline{3-9} & & 160 & 3.20E-09 & 6.96& 5.10E-09 & 6.97 & 0.000000392 & 0.999077344 \\ \cline{3-9} & & 320 & 2.51E-11 & 6.99& 4.01E-11 & 6.99 & 0.000000002 & 0.999931925 \\ \cline{3-9} & & 640 & 1.97E-13 & 7.00& 3.14E-13 & 6.99 & 0.000000000 & 0.999980113 \\ \cline{2-9} &\multirow{6}{*}{MPP} & 20 & 3.60E-03 & --& 6.39E-03 & -- & 0.000517069 & 0.972406897 \\ \cline{3-9} & & 40 & 4.78E-05 & 6.23& 1.04E-04 & 5.94 & 0.000064524 & 0.989302277 \\ \cline{3-9} & & 80 & 6.29E-07 & 6.25& 2.95E-06 & 5.15 & 0.000003451 & 0.998091182 \\ \cline{3-9} & & 160 & 1.42E-08 & 5.47& 2.09E-07 & 3.82 & 0.000000602 & 0.999077344 \\ \cline{3-9} & & 320 & 4.87E-10 & 4.87& 1.44E-08 & 3.86 & 0.000000012 & 0.999931925 \\ \cline{3-9} & & 640 & 1.78E-11 & 4.78& 1.01E-09 & 3.83 & 0.000000001 & 0.999980113 \\ \hline \multirow{14}{*}{$0.7$} &\multirow{6}{*}{Non-} & 20 & 4.12E-03 & --& 6.38E-03 & -- & -0.004485289 & 0.972368315 \\ \cline{3-9} &\multirow{6}{*}{MPP} & 40 & 4.69E-05 & 6.46& 7.37E-05 & 6.44 & -0.000005556 & 0.989301572 \\ \cline{3-9} & & 80 & 3.98E-07 & 6.88& 6.38E-07 & 6.85 & 0.000001412 & 0.998091183 \\ \cline{3-9} & & 160 & 3.19E-09 & 6.96& 5.10E-09 & 6.97 & 0.000000392 & 0.999077344 \\ \cline{3-9} & & 320 & 2.51E-11 & 6.99& 4.00E-11 & 6.99 & 0.000000002 & 0.999931925 \\ \cline{3-9} & & 640 & 1.96E-13 & 7.00& 3.14E-13 & 7.00 & 0.000000000 & 0.999980113 \\ \cline{2-9} &\multirow{6}{*}{MPP} & 20 & 3.62E-03 & --& 6.59E-03 & -- & 0.000515735 & 0.972263646 \\ \cline{3-9} & & 40 & 4.65E-05 & 6.28& 8.94E-05 & 6.20 & 0.000054894 & 0.989301394 \\ \cline{3-9} & & 80 & 3.98E-07 & 6.87& 6.38E-07 & 7.13 & 0.000001412 & 0.998091183 \\ \cline{3-9} & & 160 & 3.19E-09 & 6.96& 5.10E-09 & 6.97 & 0.000000392 & 0.999077344 \\ \cline{3-9} & & 320 & 2.51E-11 & 6.99& 4.00E-11 & 6.99 & 0.000000002 & 0.999931925 \\ \cline{3-9} & & 640 & 1.96E-13 & 7.00& 3.14E-13 & 7.00 & 0.000000000 & 0.999980113 \\ \hline \end{tabular} \caption{$L^1$ and $L^\infty$ errors and orders for $u_t+u_x=0$ with initial condition $u(x,0)=\sin^4(x)$. $T=1$. The over-diffusive global Lax-Friedrichs flux \eqref{eq: over_diff} is used with $\alpha=1.2$. FVRK7. } \label{tab302} \end{table} \begin{table}[ht]\fracootnotesize \centering \begin{tabular}{|c||c||c|c|c|c|c||c|c|} \hline $CFL$& & mesh & $L^1$ error & order & $L^\infty$ error & order & Umin & Umax \\ \hline \multirow{14}{*}{$0.9$} &\multirow{5}{*}{Non-} & 20 & 1.29E-03 & --& 2.00E-03 & -- & -0.001216056 & 0.975890071 \\ \cline{3-9} &\multirow{5}{*}{MPP} & 40 & 3.99E-06 & 8.34& 6.19E-06 & 8.34 & 0.000053321 & 0.989362841 \\ \cline{3-9} & & 80 & 8.67E-09 & 8.85& 1.37E-08 & 8.82 & 0.000002016 & 0.998091807 \\ \cline{3-9} & & 160 & 1.75E-11 & 8.95& 2.76E-11 & 8.96 & 0.000000397 & 0.999077349 \\ \cline{3-9} & & 320 & 3.44E-14 & 8.99& 5.51E-14 & 8.97 & 0.000000002 & 0.999931925 \\ \cline{2-9} &\multirow{5}{*}{MPP} & 20 & 1.20E-03 & --& 2.37E-03 & -- & 0.000393260 & 0.975868904 \\ \cline{3-9} & & 40 & 8.91E-06 & 7.08& 3.54E-05 & 6.06 & 0.000092174 & 0.989363425 \\ \cline{3-9} & & 80 & 2.90E-07 & 4.94& 2.72E-06 & 3.70 & 0.000003586 & 0.998091812 \\ \cline{3-9} & & 160 & 1.15E-08 & 4.65& 2.02E-07 & 3.75 & 0.000000600 & 0.999077349 \\ \cline{3-9} & & 320 & 4.32E-10 & 4.74& 1.30E-08 & 3.96 & 0.000000013 & 0.999931925 \\ \hline \multirow{14}{*}{$0.7$} &\multirow{5}{*}{Non-} & 20 & 1.29E-03 & --& 2.00E-03 & -- & -0.001216106 & 0.975890020 \\ \cline{3-9} &\multirow{5}{*}{MPP} & 40 & 3.99E-06 & 8.34& 6.19E-06 & 8.34 & 0.000053321 & 0.989362841 \\ \cline{3-9} & & 80 & 8.67E-09 & 8.85& 1.37E-08 & 8.82 & 0.000002016 & 0.998091807 \\ \cline{3-9} & & 160 & 1.75E-11 & 8.95& 2.76E-11 & 8.96 & 0.000000397 & 0.999077349 \\ \cline{3-9} & & 320 & 3.44E-14 & 8.99& 5.60E-14 & 8.94 & 0.000000002 & 0.999931925 \\ \cline{2-9} &\multirow{5}{*}{MPP} & 20 & 1.20E-03 & --& 2.47E-03 & -- & 0.000419926 & 0.975868183 \\ \cline{3-9} & & 40 & 3.99E-06 & 8.23& 6.19E-06 & 8.64 & 0.000053321 & 0.989362841 \\ \cline{3-9} & & 80 & 8.67E-09 & 8.85& 1.37E-08 & 8.82 & 0.000002016 & 0.998091807 \\ \cline{3-9} & & 160 & 1.75E-11 & 8.95& 2.76E-11 & 8.96 & 0.000000397 & 0.999077349 \\ \cline{3-9} & & 320 & 3.44E-14 & 8.99& 5.59E-14 & 8.95 & 0.000000002 & 0.999931925 \\ \hline \end{tabular} \caption{$L^1$ and $L^\infty$ errors and orders for $u_t+u_x=0$ with initial condition $u(x,0)=\sin^4(x)$. $T=1$. The over-diffusive global Lax-Friedrichs flux \eqref{eq: over_diff} is used with $\alpha=1.2$. FVRK9. } \label{tab303} \end{table} \section{Numerical simulations} \label{sec4} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} In this section, we present numerical tests of the proposed MPP high order FV RK WENO method for convection diffusion problems. Schemes with and without MPP limiters are compared. In these tests, the time step size for the RK method is chosen such that \begin{equation} \Delta t = \min \Big( \fracrac{CFLC}{\max|f'(u)|}\Delta x , \fracrac{CFLD}{\max|a'(u)|} \Delta x^2 \Big), \end{equation} for one dimensional problems and \begin{equation} \Delta t = \min \Big( \fracrac{CFLC}{\max|f'(u)|/\Delta x + \max|g'(u)|/\Delta y} , \fracrac{CFLD}{\max|a'(u)|/\Delta x^2 + \max|b'(u)|/\Delta y^2} \Big), \end{equation} for two dimensional problems. Here CFLC (CFLD resp.) represents the CFL number for the convection (diffusion resp.) term. In our tests, we take $CFLC = 0.6$ for convection-dominated problems and $CFLD = 0.8$ for pure diffusion problems. Herein we let ``MPP" and ``NonMPP" denote the scheme with and without the MPP limiter, and $U_{\max}\ (U_{\min} \ \mbox{resp.})$ denote the maximum (minimum resp.) value among the numerical cell averages $\bar{u}_j$. To better illustrate the effectiveness of the MPP limiters, we use linear weights instead of WENO weights in the reconstruction procedure for the convection term. \subsection{Basic Tests} \begin{exa}(1D Linear Problem) \begin{equation}\label{1dlinear} u_t+u_x=\epsilon u_{xx}, \ x\in [0, 2\pi], \ \epsilon = 0.00001. \end{equation} We test the proposed scheme on the problem (\ref{1dlinear}) with initial condition $u(x,0)=\sin^4(x)$ and periodic boundary condition. The exact solution is \begin{equation} \label{1daccuracy} u(x,t)=\fracrac{3}{8} - \fracrac{1}{2}\exp(-4\epsilon t)\cos(2(x-t)) + \fracrac{1}{8}\exp(-16\epsilon t)\cos(4(x-t)). \end{equation} The $L_1$ and $L_\infty$ errors and orders of convergence for the scheme with and without MPP limiters are shown in Table \ref{tableexample1}. It is observed that the MPP limiter avoids overshooting and undershooting of the numerical solution while preserve high order accuracy. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|l ||c ||c |c ||c |c ||c |c |} \hline & mesh &$L_1$ error &order &$L_{\infty}$ error &order &Umax &Umin \\ \hline \hline \multirow{6}{*}{Non-} & \multicolumn{1}{l ||}{50} & \multicolumn{1}{l |}{1.68E-04 } & \multicolumn{1}{l ||}{---} & \multicolumn{1}{l |}{2.76E-04 } & \multicolumn{1}{l ||}{---} & \multicolumn{1}{l |}{0.996998594480 } & \multicolumn{1}{l |}{-0.000182938402}\\ \cline{2-8} \multirow{6}{*}{MPP} & \multicolumn{1}{l ||}{100} & \multicolumn{1}{l |}{5.47E-06 } & \multicolumn{1}{l ||}{4.94 } & \multicolumn{1}{l |}{9.11E-06 } & \multicolumn{1}{l ||}{4.92 } & \multicolumn{1}{l |}{0.997933416789 } & \multicolumn{1}{l |}{-0.000005718342 }\\ \cline{2-8} & \multicolumn{1}{l ||}{200} & \multicolumn{1}{l |}{1.72E-07 } & \multicolumn{1}{l ||}{4.99 } & \multicolumn{1}{l |}{2.87E-07 } & \multicolumn{1}{l ||}{4.99 } & \multicolumn{1}{l |}{0.999579130130 } & \multicolumn{1}{l |}{-0.000000153518 }\\ \cline{2-8} & \multicolumn{1}{l ||}{400} & \multicolumn{1}{l |}{5.38E-09 } & \multicolumn{1}{l ||}{5.00 } & \multicolumn{1}{l |}{9.00E-09 } & \multicolumn{1}{l ||}{5.00} & \multicolumn{1}{l |}{0.999905929907 } & \multicolumn{1}{l |}{-0.000000002134 }\\ \cline{2-8} & \multicolumn{1}{l ||}{800} & \multicolumn{1}{l |}{1.68E-10 } & \multicolumn{1}{l ||}{5.00} & \multicolumn{1}{l |}{2.81E-10 } & \multicolumn{1}{l ||}{5.00} & \multicolumn{1}{l |}{0.999945898951 } & \multicolumn{1}{l |}{0.000000001890 }\\ \cline{2-8} \hline \hline \multirow{6}{*}{MPP} & \multicolumn{1}{l ||}{50} & \multicolumn{1}{l |}{1.71E-04 } & \multicolumn{1}{l ||}{---} & \multicolumn{1}{l |}{2.87E-04 } & \multicolumn{1}{l ||}{---} & \multicolumn{1}{l |}{0.996998296191 } & \multicolumn{1}{l |}{0.000000000000 }\\ \cline{2-8} & \multicolumn{1}{l ||}{100} & \multicolumn{1}{l |}{5.46E-06 } & \multicolumn{1}{l ||}{4.93 } & \multicolumn{1}{l |}{1.34E-05 } & \multicolumn{1}{l ||}{4.42 } & \multicolumn{1}{l |}{ 0.997933416819 } & \multicolumn{1}{l |}{ 0.000000016274 }\\ \cline{2-8} & \multicolumn{1}{l ||}{200} & \multicolumn{1}{l |}{1.72E-07 } & \multicolumn{1}{l ||}{ 5.00 } & \multicolumn{1}{l |}{4.91E-07 } & \multicolumn{1}{l ||}{4.77 } & \multicolumn{1}{l |}{0.999579130130 } & \multicolumn{1}{l |}{0.000000013987 }\\ \cline{2-8} & \multicolumn{1}{l ||}{400} & \multicolumn{1}{l |}{5.38E-09 } & \multicolumn{1}{l ||}{ 5.03 } & \multicolumn{1}{l |}{1.25E-08 } & \multicolumn{1}{l ||}{5.29 } & \multicolumn{1}{l |}{0.999905929907 } & \multicolumn{1}{l |}{0.000000001048 }\\ \cline{2-8} & \multicolumn{1}{l ||}{800} & \multicolumn{1}{l |}{1.68E-10 } & \multicolumn{1}{l ||}{5.01 } & \multicolumn{1}{l |}{ 2.81E-10 } & \multicolumn{1}{l ||}{5.48 } & \multicolumn{1}{l |}{ 0.999945898951 } & \multicolumn{1}{l |}{ 0.000000001890 }\\ \cline{2-8} \hline \end{tabular} \caption{Accuracy tests for 1D linear equation \eqref{1dlinear} with exact solution \eqref{1daccuracy} at time $T=1.0$.} \label{tableexample1} \end{table} We then test problem (\ref{1dlinear}) with the initial condition having rich solution structures \begin{equation} \label{1ddiscontinuous} u_0(x)= \begin{cases} \fracrac{1}{6}(G(x,\beta,z-\delta)+G(x,\beta,z+\delta)+4G(x,\beta,z)), \ \ \ &-0.8\le x \le -0.6; \\ 1, \ \ \ &-0.4\le x \le -0.2; \\ 1-|10(x-0.1)|, \ \ \ &0 \le x \le 0.2; \\ \fracrac{1}{6}(F(x,\gamma,a-\delta)+F(x,\gamma,a+\delta)+4F(x,\gamma,a)), \ \ \ &0.4\le x \le 0.6; \\ 0, \ \ \ &\mbox{ otherwise}. \end{cases} \end{equation} where $G(x,\beta, z)=e^{-\beta(x-z)^2}$ and $F(x,\gamma,a)=\sqrt{\max(1-\gamma^2(x-a)^2,0)}$. The constants involved are $a=0.5, z=-0.7, \delta=0.005, \gamma=10$ and $\beta=\log 2/(36\delta^2)$ and the boundary condition is periodic. The maximum and minimum cell averages are listed in Table \ref{table1_1}. In Figure \ref{figure1}, the effectiveness of the MPP limiters in controlling the numerical solution within theoretical bounds can be clearly observed. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|l ||c |c ||c |c |} \hline & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline 50 & 1.106238399422 &-0.114766938420 & 1.000000000000 & 0.000000000000 \\ \hline 100 & 1.056114534445 &-0.067351423479 & 1.000000000000 & 0.000000000000 \\ \hline 200 & 1.054864483784 &-0.054928012204 & 1.000000000000 & 0.000000000000 \\ \hline 400 & 1.048250067722 &-0.048250171364 & 1.000000000000 & 0.000000000000 \\ \hline 800 & 1.031246517796 &-0.031246517794 & 1.000000000000 & 0.000000000000 \\ \hline \end{tabular} \caption{The maximum and minimum values of the numerical cell averages for problem \eqref{1dlinear} with initial conditions \eqref{1ddiscontinuous} at time $T=1.0$.} \label{table1_1} \end{table} \begin{figure} \caption{Left: Comparison of the FV RK scheme with and without MPP limiters for 1d linear problem \eqref{1dlinear} \label{figure1} \end{figure} \end{exa} \begin{exa} (1D Nonlinear Equation) We test the FV RK scheme with and without MPP limiters on Burgers' equation \begin{equation}\label{burgersequation} u_t+(\fracrac{u^2}{2})_x=\epsilon u_{xx}, \ x\in [-1,1], \ \epsilon = 0.0001, \end{equation} with initial condition \begin{equation}\nonumber u(x,0)=\begin{cases} 2, \ \ \ &|x|<0.5; \\ 0, \ \ \ &\mbox{otherwise}, \end{cases} \end{equation} and periodic boundary conditions. The results in Table \ref{tableexample2} shows that the numerical solution goes beyond the theoretical bounds if no limiters are applied and stays within the theoretical range if MPP limiters are applied. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|l ||c |c ||c |c |} \hline & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline 50 & 2.349929038912 &-0.063536142936 & 1.818784698878 & 0.000000000000 \\ \hline 100 & 2.438970633433 &-0.135799476071 & 1.879377697365 & 0.000000000000 \\ \hline 200 & 2.217068598684 &-0.095548979222 & 1.913720603302 & 0.000000000000 \\ \hline 400 & 2.216719764740 &-0.095114086983 & 1.938439146468 & 0.000000000000 \\ \hline 800 & 2.210614277385 &-0.092745597929 & 1.959770865698 & 0.000000000000 \\ \hline \end{tabular} \caption{The maximum and minimum values of the numerical cell averages for Burgers' equation \eqref{burgersequation} at time $T=0.05$.} \label{tableexample2} \end{table} \end{exa} \begin{exa} (2D Linear Problem) \begin{equation}\label{2dlinear} u_t+u_x+u_y=\epsilon (u_{xx}+u_{yy}), \ \ \ (x,y)\in [0,2\pi]^2, \ \ \ \epsilon =0.001. \end{equation} We first consider the problem with initial condition $u(x,y,0)=\sin^4(x+y)$ and periodic boundary condition. The exact solution to the problem is \begin{equation} \label{2daccuracy} u(x,y,t)=\fracrac{3}{8}-\fracrac{1}{2}\exp(-8\epsilon t)\cos(2(x+y-2t))+\fracrac{1}{8}\exp(-32\epsilon t)\cos(4(x+y-2t)). \end{equation} The $L_1$ and $L_\infty$ errors and orders of convergence for the FV RK scheme with and without MPP limiters are shown in Table \ref{tableexample3}. High order accuracy is preserved when the MPP limiters are applied to control the numerical solution within the theoretical bounds. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|l ||c ||c |c ||c |c ||c |c |} \hline & mesh &$L_1$ error &order &$L_{\infty}$ error &order &Umax &Umin \\ \hline \hline \multirow{5}{*}{NonMPP} & \multicolumn{1}{c ||}{$16\times 16$} & \multicolumn{1}{l |}{4.86E-03 } & \multicolumn{1}{l ||}{--- } & \multicolumn{1}{l |}{9.30E-03 } & \multicolumn{1}{l ||}{--- } & \multicolumn{1}{l |}{0.919696089900 } & \multicolumn{1}{l |}{ 0.000159282060 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$32\times 32$} & \multicolumn{1}{l |}{2.85E-04 } & \multicolumn{1}{l ||}{4.29 } & \multicolumn{1}{l |}{4.49E-04 } & \multicolumn{1}{l ||}{4.37 } & \multicolumn{1}{l |}{0.986054820018 } & \multicolumn{1}{l |}{-0.000283832731 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$64\times 64$} & \multicolumn{1}{l |}{9.82E-06 } & \multicolumn{1}{l ||}{4.84 } & \multicolumn{1}{l |}{1.62E-05 } & \multicolumn{1}{l ||}{4.79 } & \multicolumn{1}{l |}{0.995960434630 } & \multicolumn{1}{l |}{-0.000004482350 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$128\times 128$} & \multicolumn{1}{l |}{3.12E-07 } & \multicolumn{1}{l ||}{4.96 } & \multicolumn{1}{l |}{5.22E-07 } & \multicolumn{1}{l ||}{4.95 } & \multicolumn{1}{l |}{0.998407179488 } & \multicolumn{1}{l |}{ 0.000001288422 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$256\times 256$} & \multicolumn{1}{l |}{9.73E-09 } & \multicolumn{1}{l ||}{5.00 } & \multicolumn{1}{l |}{1.63E-08 } & \multicolumn{1}{l ||}{5.01 } & \multicolumn{1}{l |}{0.998990497491 } & \multicolumn{1}{l |}{0.000000740680 }\\ \hline \hline \multirow{5}{*}{MPP} & \multicolumn{1}{c ||}{$16\times 16$} & \multicolumn{1}{l |}{4.86E-03 } & \multicolumn{1}{l ||}{ --- } & \multicolumn{1}{l |}{ 9.30E-03 } & \multicolumn{1}{l ||}{--- } & \multicolumn{1}{l |}{ 0.919696089900 } & \multicolumn{1}{l |}{ 0.000159282060 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$32\times 32$} & \multicolumn{1}{l |}{2.87E-04 } & \multicolumn{1}{l ||}{ 4.27 } & \multicolumn{1}{l |}{4.49E-04 } & \multicolumn{1}{l ||}{4.37 } & \multicolumn{1}{l |}{0.986054818813 } & \multicolumn{1}{l |}{ 0.000000000000 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$64\times 64$} & \multicolumn{1}{l |}{9.82E-06 } & \multicolumn{1}{l ||}{ 4.85 } & \multicolumn{1}{l |}{1.64E-05 } & \multicolumn{1}{l ||}{4.77 } & \multicolumn{1}{l |}{0.995960434630 } & \multicolumn{1}{l |}{0.000000000000 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$128\times 128$} & \multicolumn{1}{l |}{3.12E-07 } & \multicolumn{1}{l ||}{4.97 } & \multicolumn{1}{l |}{ 5.22E-07 } & \multicolumn{1}{l ||}{4.97 } & \multicolumn{1}{l |}{ 0.998407179488 } & \multicolumn{1}{l |}{ 0.000001288422 }\\ \cline{2-8} & \multicolumn{1}{c ||}{$256\times 256$} & \multicolumn{1}{l |}{9.73E-09 } & \multicolumn{1}{l ||}{5.00 } & \multicolumn{1}{l |}{1.63E-08 } & \multicolumn{1}{l ||}{5.01 } & \multicolumn{1}{l |}{0.998990497491 } & \multicolumn{1}{l |}{ 0.000000740680 }\\ \hline \end{tabular} \caption{Accuracy tests for 2D linear equation \eqref{2dlinear} with exact solution \eqref{2daccuracy} at time $T=1.0$.} \label{tableexample3} \end{table} We then consider problem (\ref{2dlinear}) with initial condition \begin{equation} \label{2ddiscontinuous} u(x,0)=\begin{cases} 1, \ \ \ \ &(x,y)\in [\fracrac{\pi}{2},\fracrac{3\pi}{2}]\times[\fracrac{\pi}{2},\fracrac{3\pi}{2}]; \\ 0, \ \ & \mbox{otherwise on }[0,2\pi]\times [0,2\pi], \end{cases} \end{equation} and periodic boundary condition. The results are shown in Table \ref{tableexample32}, which indicates the effectiveness of the MPP limiter. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times16$ & 1.196476571354 &-0.102486638966 & 1.000000000000 & 0.000000000000 \\ \hline $32\times32$ & 1.317444117818 &-0.169214623680 & 1.000000000000 & 0.000000000000 \\ \hline $64\times64$ & 1.341696522446 &-0.182902057169 & 1.000000000000 & 0.000000000000 \\ \hline $128\times128$ & 1.225931525834 &-0.116989442889 & 1.000000000000 & 0.000000000000 \\ \hline $256\times256$ & 1.108731559448 &-0.055808238605 & 1.000000000000 & 0.000000000000 \\ \hline \end{tabular} \caption{Maximum and minimum cell averages in the 2D linear problem \eqref{2dlinear} with initial condition \eqref{2ddiscontinuous} at time $T=0.1$.} \label{tableexample32} \end{table} \end{exa} \begin{exa} (1D Buckley-Leverett Equation) Consider the problem \begin{equation}\label{1dbuckley-leverett} u_t+f(u)_x=\epsilon(\nu(u)u_x)_x, \ \ \ \epsilon = 0.01, \end{equation} where $$\nu(u)=\begin{cases} 4u(1-u), \ \ \ \ &0\le u \le 1; \\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &\mbox{otherwise}, \end{cases} \quad \mbox{and} \quad f(u)=\fracrac{u^2}{u^2+(1-u)^2}. $$ The initial condition is $$u(x,0)=\begin{cases} 1-3x, \ \ \ \ 0\le x < \fracrac{1}{3}; \\ 0, \ \ \ \ \ \ \ \ \ \ \ \fracrac{1}{3} \le x \le 1, \end{cases} $$ and the boundary conditions are $u(0,t)=1$ and $u(1,t)=0$. The numerical results are shown in Table \ref{tableexample4}. The numerical solution goes below $0$ if MPP limiters are not applied, and stays within the theoretical bounds $[0,1]$ when MPP limiters are applied. Figure \ref{figure4} illustrates the effectiveness of MPP limiters near the undershooting of the numerical solution. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|l ||c |c ||c |c |} \hline & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline 50 & 1.000000000000000 &-0.002643266424381 & 1.000000000000000 & 0.000000000000000 \\ \hline 100 & 1.000000000000000 &-0.001813338703220 & 1.000000000000000 & 0.000000000000000 \\ \hline 200 & 1.000000000000000 &-0.000942402907667 & 1.000000000000000 & 0.000000000000000 \\ \hline 400 & 1.000000000000000 &-0.000491323673758 & 1.000000000000000 & 0.000000000000000 \\ \hline 800 & 1.000000000000000 &-0.000247268741213 & 1.000000000000000 & 0.000000000000000 \\ \hline \end{tabular} \caption{The maximum and minimum values for 1D Buckley-Leverett problem \eqref{1dbuckley-leverett} at time $T=0.2$.} \label{tableexample4} \end{table} \begin{figure} \caption{Left: Solutions for 1D Buckley-Leverett equation \eqref{1dbuckley-leverett} \label{figure4} \end{figure} \end{exa} \begin{exa}(2D Buckley-Leverett Equation) Consider \begin{equation}\label{2dbuckley-leverett} u_t+f(u)_x+g(u)_y=\epsilon(u_{xx}+u_{yy}), \ \ \ (x,y)\in [-1.5,1.5]^2, \ \ \ \epsilon =0.01 \end{equation} where $$f(u)=\fracrac{u^2}{u^2+(1-u)^2},\ \ g(u)=f(u)(1-5(1-u)^2),$$ with initial condition $$u(x,y,0)=\begin{cases} 1, \ \ \ \ &x^2+y^2<0.5; \\ 0, \ \ \ \ &\mbox{otherwise on }[-1.5,1.5]^2, \end{cases} $$ and periodic boundary conditions. The numerical results in Table \ref{tableexample5} show that the MPP limiters effectively control the numerical solution within the theoretical range $[0,1]$. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 1.190542402917 &-0.142603740886 & 1.000000000000 & 0.000000000000 \\ \hline $32\times 32$ & 1.183357844800 &-0.174592560044 & 1.000000000000 & 0.000000000000 \\ \hline $64\times 64$ & 1.148424330885 &-0.167227853261 & 1.000000000000 & 0.000000000000 \\ \hline $128\times 128$ & 1.084563025034 &-0.083883559766 & 1.000000000000 & 0.000000000000 \\ \hline $256\times 256$ & 0.998736899089 &-0.018463025969 & 0.998566263416 & 0.000000000000 \\ \hline \end{tabular} \caption{Maximum and minimum cell averages for 2D Buckley-Leverett problem \eqref{2dbuckley-leverett} at time $T=0.5$.} \label{tableexample5} \end{table} \end{exa} \begin{exa} (1D Porous Medium Equation) Consider \begin{equation}\label{1dporous} u_t=(u^m)_{xx}, \ m>1, \ \ \ x\in [-2\pi,2\pi] \end{equation} whose solution is the Barenblatt solution in the following form \begin{equation} B_m(x,t)=t^{-k}\Big[(1-\fracrac{k(m-1)}{2m}\fracrac{|x|^2}{t^{2k}})_{+}\Big]^{\fracrac{1}{m+1}}, \end{equation} with $k=\fracrac{1}{m+1}$ and $u_+=\max(u,0)$. The boundary conditions are assumed to be zero at both ends. Starting from time $T_0=1$, we compute the numerical solution of the problem up to time $T=2$ by the FV RK scheme and the results are shown in Table \ref{tableexample6}. Obviously, there are undershoots when regular FV RK scheme are applied. And the MPP limiters can effectively eliminate the overshoots in the numerical solution. Also the plot in Figure \ref{figure6} shows the effectiveness of the MPP limiters. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline $N=100$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline m & Umax & Umin & Umax & Umin \\ \hline 2 & 0.793283780606 &-0.000338472445 & 0.793283375962 & 0.000000000000 \\ \hline 3 & 0.840666629482 &-0.001792679096 & 0.840663542409 & 0.000000000000 \\ \hline 5 & 0.890829374423 &-0.005693908465 & 0.890821177490 & 0.000000000000 \\ \hline 8 & 0.925837535365 &-0.003841778007 & 0.925826127818 & 0.000000000000 \\ \hline \end{tabular} \caption{Maximum and minimum cell average values for 1D porous medium problem \eqref{1dporous} with $m=2,3,5,8$ at time $T=2$.} \label{tableexample6} \end{table} \begin{figure} \caption{Left: Plot for 1D porous medium problem \eqref{1dporous} \label{figure6} \end{figure} \end{exa} \begin{exa}(2D Porous Medium Equation) Consider \begin{equation}\label{2dporous} u_t=(u^m)_{xx}+(u^m)_{yy}, \ \ m=2, \ \ \ (x,y)\in [-1,1]^2 \end{equation} with initial condition $$u(x,y,0)=\begin{cases} 1, \ \ \ \ &(x,y)\in [-\fracrac{1}{2},\fracrac{1}{2}]^2; \\ 0, \ \ \ \ & \mbox{otherwise on } [-\fracrac{1}{2},\fracrac{1}{2}]^2, \end{cases}$$ and periodic boundary conditions. We produce the numerical results at time $T=0.005$, as shown in Table \ref{tableexample7}. The results show that the MPP limiters perform effectively at avoiding overshooting and undershooting of the numerical solution. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16$ & 1.000485743751 &-0.000349298087 & 0.999827816078 & 0.000000000000 \\ \hline $32\times 32$ & 0.999625786453 &-0.001200636807 & 0.999573139639 & 0.000000000000 \\ \hline $64\times 64$ & 0.999537081790 &-0.000855830629 & 0.999533087178 & 0.000000000000 \\ \hline $128\times 128$ & 0.999527411822 &-0.000474775257 & 0.999526635569 & 0.000000000000 \\ \hline $256\times 256$ & 0.999525567240 &-0.000261471521 & 0.999525309113 & 0.000000000000 \\ \hline \end{tabular} \caption{Maximum and minimum cell average values for 2D porous medium problem \eqref{2dporous} at time $T=0.005$.} \label{tableexample7} \end{table} \end{exa} \subsection{Incompressible Flow Problems} In this subsection, we test the proposed scheme on incompressible flow problems in the form \begin{align} &\omega_t+(u\omega)_x+(v\omega)_y=\fracrac{1}{Re}(\omega_{xx}+\omega_{yy}), \end{align} where $\langle u,v \rightarrowngle$ is the divergence-free velocity field and Re is the Reynold number. The theoretical solution satisfies the maximum principle due to the divergence-free property of the velocity field. For the numerical solution to satisfy the maximum principle, discretized divergence-free condition needs to be considered, hence special treatment needs to be taken when low order flux for the convection term is designed. For details, see \cite{mpp_xqx}, according to which we design the low order monotone flux for the following incompressible problems. \noindent \begin{exa}(Rotation with Viscosity) \begin{equation}\label{rbr} u_t+(-yu)_x+(xu)_y=\fracrac{1}{Re}(u_{xx}+u_{yy}), \ \ (x,y)\in[-\pi,\pi]^2. \end{equation} The initial condition is shown in Figure \ref{figure8_1} and the boundary condition is assumed to be periodic. The numerical solution at time $T=0.1$ is shown in Table \ref{tableexample8}, which indicates that there are overshooting and undershooting in the numerical solution by regular FV RK scheme and they can be avoided by applying the MPP limiter. The solutions with and without MPP limiter are also compared in Figure \ref{figure8_2}. From Table \ref{tableexample8} and Figure \ref{figure8_2}, the effectiveness of the MPP limiter can be better illustrated when Renold number is larger. This is because the overshooting and undershooting are more apparent when Reynold number is larger, which corresponds to less diffusion. \begin{figure} \caption{Initial condition for Example 3.8 and Example 3.9.} \label{figure8_1} \end{figure} \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline Re=$100$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 0.947915608973 &-0.041388485669 & 0.947719795318 & 0.000000000000 \\ \hline $32\times 32 $ & 0.999789765557 &-0.048836983632 & 0.996173203589 & 0.000000000000 \\ \hline $64\times 64 $ & 1.008171330748 &-0.039241271474 & 0.999999999928 & 0.000000000000 \\ \hline $128\times 128 $ & 1.002125190412 &-0.027962451582 & 0.999999999920 & 0.000000000000 \\ \hline $256\times 256 $ & 1.000099518450 &-0.012262487330 & 0.999999999983 & 0.000000000000 \\ \hline \hline Re=$10000$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 0.949247968412 &-0.042285048496 & 0.949049295419 & 0.000000000000 \\ \hline $32\times 32 $ & 1.002247494119 &-0.053653247391 & 0.996943318800 & 0.000000000000 \\ \hline $64\times 64 $ & 1.012845607701 &-0.049914946698 & 0.999999462216 & 0.000000000000 \\ \hline $128\times 128 $ & 1.009050027036 &-0.050526262050 & 0.999999999977 & 0.000000000000 \\ \hline $256\times 256 $ & 1.007608558521 &-0.058482843302 & 0.999999999995 & 0.000000000000 \\ \hline \end{tabular} \caption{The maximum and minimum cell averages for rotation problem (\ref{rbr}) with two different Reynold numbers at $T=0.1$.} \label{tableexample8} \end{table} \begin{figure} \caption{Left: Cutting plots for rotation problem (\ref{rbr} \label{figure8_2} \end{figure} \end{exa} \begin{exa}(Swirling Deformation with Viscosity) \begin{align}\label{sd} &u_t+(-\cos^2(\fracrac{x}{2})\sin(y)g(t)u)_x+(\sin(x)\cos^2(\fracrac{y}{2})t(t))u)_y=\fracrac{1}{Re}(u_{xx}+u_{yy}), \end{align} where $(x,y)\in[-\pi,\pi]^2$ and $g(t)=\cos(\pi t /T)\pi$. The initial condition is the same as in Example 4.8 and the boundary conditions are also periodic. Similarly, we also compare the results for different Reynold numbers Re=$100$ and Re=$10000$. As shown in Table \ref{tableexample9}, the MPP limiter plays the role of eliminating overshooting and undershooting in the numerical solution, especially for problems with larger Reynold number. This can also be observed in Figure \ref{figure9_1}. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline Re=$100$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 0.873440241699 &-0.010737472197 & 0.842184825192 & 0.000000000000 \\ \hline $32\times 32 $ & 0.971822334038 &-0.011947680561 & 0.942384582101 & 0.000000000000 \\ \hline $64\times 64 $ & 0.997563271155 &-0.005935366467 & 0.986960253479 & 0.000000000000 \\ \hline $128\times 128 $ & 1.000886437426 &-0.001258903421 & 0.998925498573 & 0.000000000000 \\ \hline $256\times 256 $ & 1.000040508119 &-0.000036182185 & 0.999992956155 & 0.000000000000 \\ \hline \hline Re=$10000$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 0.874953790056 &-0.011212471543 & 0.846813512747 & 0.000000000000 \\ \hline $32\times 32 $ & 0.973964125865 &-0.014299538733 & 0.942368749644 & 0.000000000000 \\ \hline $64\times 64 $ & 1.000873875979 &-0.006640227946 & 0.988604733672 & 0.000000000000 \\ \hline $128\times 128 $ & 1.002350640870 &-0.002755842119 & 0.999375840770 & 0.000000000000 \\ \hline $256\times 256 $ & 1.000734372263 &-0.000563730690 & 0.999998986667 & 0.000000000000 \\ \hline \end{tabular} \caption{The maximum and minimum cell averages for swirling deformation problem (\ref{sd}) with two different Reynold numbers at T=$0.1$.} \label{tableexample9} \end{table} \begin{figure} \caption{Left: Cutting plots for swirling deformation problem (\ref{sd} \label{figure9_1} \end{figure} \end{exa} \begin{exa}(Vortex Patch) Consider the problem \begin{align}\label{vp} &\omega_t+(u\omega)_x+(v\omega)_y=\fracrac{1}{Re}(\omega_{xx}+\omega_{yy}), \\ & \Delta \psi = \omega,\ \langle u,v \rightarrowngle = \langle -\psi_y, \psi_x \rightarrowngle, \end{align} with the following initial condition \begin{equation} \omega(x,y,0)=\begin{cases} -1, \ &\fracrac{\pi}{2} \le x \le \fracrac{3\pi}{2}, \ \fracrac{\pi}{4} \le \fracrac{3\pi}{4}, \\ 1, \ \ &\fracrac{\pi}{2} \le x \le \fracrac{3\pi}{2}, \ \fracrac{5\pi}{4} \le \fracrac{7\pi}{4}, \\ 0, \ \ &\mbox{otherwise}, \end{cases} \end{equation} and periodic boundary condition. The maximum and minimum cell averages of the numerical solution with two Reynold numbers Re=$100$ and Re=$10000$, obtained by regular FV RK scheme and the scheme with the MPP limiter are compared in Table \ref{tableexample10}, from which we can observe the effectiveness of the MPP limiter in controlling overshooting and undershooting in the numerical solution. The contour plot of the solution is presented in Figure \ref{figure10}, which shows that the solution obtained by FV RK scheme with the MPP limiter is comparable to that obtained by regular FV RK scheme. \begin{table}[h]\fracootnotesize \centering \begin{tabular}{|c ||c |c ||c |c |} \hline Re=$100$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 1.035853749815 &-1.035699868274 & 1.000000000000 &-1.000000000000 \\ \hline $32\times 32 $ & 1.054573231517 &-1.054663726026 & 1.000000000000 &-1.000000000000 \\ \hline $64\times 64 $ & 1.044017351861 &-1.044000125346 & 1.000000000000 &-1.000000000000 \\ \hline $128\times 128 $ & 1.010637311054 &-1.010641150928 & 1.000000000000 &-1.000000000000 \\ \hline $256\times 256 $ & 1.000000232315 &-1.000000231632 & 1.000000000000 &-1.000000000000 \\ \hline \hline Re=$10000$ & \multicolumn{2}{c ||}{NonMPP} & \multicolumn{2}{c|}{MPP} \\ \hline mesh & Umax & Umin & Umax & Umin \\ \hline $16\times 16 $ & 1.036117022938 &-1.035951331163 & 1.000000000000 &-1.000000000000 \\ \hline $32\times 32 $ & 1.060652217270 &-1.060764279809 & 1.000000000000 &-1.000000000000 \\ \hline $64\times 64 $ & 1.086490500643 &-1.086296444198 & 1.000000000000 &-1.000000000000 \\ \hline $128\times 128 $ & 1.127323843780 &-1.127407543973 & 1.000000000000 &-1.000000000000 \\ \hline $256\times 256 $ & 1.129384376147 &-1.129395445889 & 1.000000000000 &-1.000000000000 \\ \hline \end{tabular} \caption{The maximum and minimum cell averages for vortex patch problem (\ref{vp}) at time T=$0.1$ with Re=$100$ and Re=$10000$. } \label{tableexample10} \end{table} \begin{figure} \caption{Contours of the numerical solution for vortex patch problem (\ref{vp} \label{figure10} \end{figure} \end{exa} \section{Conclusion} \label{sec6} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} In this paper, we have successfully generalized the MPP flux limiters to the high order FV RK WENO schemes solving convection-dominated problems. For a special case, $f'(u)>0$ or $f'(u)<0$, we provide a complete analysis that the original high order FV RK WENO scheme coupled with the MPP flux limiters maintains high order accuracy and MPP property when Godunov type flux is used as the first order flux, toward which the high order numerical flux is limited. For a general setting, we rely on the Taylor expansion around extrema to prove that the FV RK schemes with MPP flux limiters preserve up to third order accuracy without addition CFL constraint. Establishing analysis for accuracy preservation under suitable constraints in a general setting will be part of our future work. \label{appendix A} In this section we derive the MPP flux limiters for two-dimensional problem. The basic idea is the same as for one-dimensional problem, i.e., necessary conditions for the numerical solutions to satisfy maximum principle will be derived, based on similar inequalities as (\ref{max}) and (\ref{min}). Before further discussion, we briefly review the finite volume scheme for two dimensional problem. Similarly as in one dimensional case, we integrate Equation (\ref{ad2}) over cell $I_{i,j}$ and divide it by $\triangle x \triangle y$, then we have \begin{align}\label{2dint} \fracrac{1}{\triangle x \triangle y} \fracrac{d}{dt} \int \int_{I_{i,j}} u dxdy + & \fracrac{1}{\triangle x \triangle y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} f(u)|_{x_{i+\fracrac{1}{2}}}dy - \fracrac{1}{\triangle x \triangle y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} f(u)|_{x_{i-\fracrac{1}{2}}}dy \nonumber \\ + & \fracrac{1}{\triangle x \triangle y} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} g(u)|_{y_{j-\fracrac{1}{2}}}dx - \fracrac{1}{\triangle x \triangle y} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} g(u)|_{y_{j-\fracrac{1}{2}}}dx \nonumber \\ =& \fracrac{1}{\triangle x \triangle y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} a(u)_x|_{x_{i+\fracrac{1}{2}}}dy - \fracrac{1}{\triangle x \triangle y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} a(u)_x|_{x_{i-\fracrac{1}{2}}}dy \nonumber \\ + & \fracrac{1}{\triangle x \triangle y} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} b(u)_y|_{y_{j-\fracrac{1}{2}}}dx - \fracrac{1}{\triangle x \triangle y} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} b(u)_y|_{y_{j-\fracrac{1}{2}}}dx. \end{align} Denote $\fracrac{1}{\triangle x \triangle y} \int \int_{I_{i,j}} u dxdy$ with $\bar{u}_{i,j}$, then the corresponding numerical scheme is \begin{align}\label{2dscheme} \fracrac{d}{dt} \bar{u}_{i,j} + & \fracrac{1}{\triangle x} (\hat{f}_{i+\fracrac{1}{2},j}-\hat{f}_{i-\fracrac{1}{2},j})+ \fracrac{1}{\triangle y} (\hat{g}_{i,j+\fracrac{1}{2}}-\hat{g}_{i,j-\fracrac{1}{2}}) \nonumber \\ &=\fracrac{1}{\triangle x} (\widehat{(a_x)}_{i+\fracrac{1}{2},j}-\widehat{(a_x)}_{i-\fracrac{1}{2},j}) + \fracrac{1}{\triangle y} (\widehat{(b_y)}_{i,j+\fracrac{1}{2}}-\widehat{(b_y)}_{i,j-\fracrac{1}{2}}). \end{align} where $\hat{f}_{i+\fracrac{1}{2},j}$ can be understood as the average of the flux function over the right boundary of cell $I_{i,j}$, and $\hat{g}_{i,j+\fracrac{1}{2}}, \widehat{(a_x)}_{i+\fracrac{1}{2},j}, \widehat{(b_y)}_{i,j+\fracrac{1}{2}}$ have similar meanings. Applying Gaussian quadrature integration rules to the integrals involved in (\ref{2dint}), we have \begin{align} & \hat{f}_{i+\fracrac{1}{2},j} = \fracrac{1}{2} \underset{i_g}{\Sigma} \omega_{i_g} f(u_{i+\fracrac{1}{2},i_g}),\\ & \hat{g}_{i,j+\fracrac{1}{2}} = \fracrac{1}{2} \underset{i_g}{\Sigma} \omega_{i_g} g(u_{i_g,j+\fracrac{1}{2}}),\\ & \widehat{(a_x)}_{i+\fracrac{1}{2},j} = \fracrac{1}{2} \underset{i_g}{\Sigma} \omega_{i_g} a_x( u_{i+\fracrac{1}{2},i_g} ),\\ & \widehat{(b_y)}_{i,j+\fracrac{1}{2}} = \fracrac{1}{2} \underset{i_g}{\Sigma} \omega_{i_g} b_y( u_{i_g,j+\fracrac{1}{2}} ), \end{align} where $\underset{i_g}{\Sigma}$ represents the summation over the Gaussian quadratures and $u_{i+\fracrac{1}{2},i_g},u_{i_g,j+\fracrac{1}{2}}$ are the approximate values to $u(x_{i+\fracrac{1}{2}},y_{i_g}),u(x_{i_g},y_{j+\fracrac{1}{2}})$, and can be reconstructed from $\{\bar{u}_{i,j}\}$. Details of the reconstruction procedure can be found in Appendix A. Hence after discretized temporally with TVD Runge-Kutta method in the way similar to the case for the one dimensional problem, the scheme (\ref{2dscheme}) becomes \begin{equation} u_{i,j}^{n+1}=u_{i,j}^n - \lambda_x(\hat{H}^{rk}_{i+\fracrac{1}{2},j}-\hat{H}^{rk}_{i-\fracrac{1}{2},j}) -\lambda_y(\hat{G}^{rk}_{i,j+\fracrac{1}{2}}-\hat{G}^{rk}_{i,j-\fracrac{1}{2}}), \end{equation} where $\lambda_x=\fracrac{\triangle t}{\triangle x}$ and $\lambda_y=\fracrac{\triangle t}{\triangle y}$, and \begin{align} & \hat{H}^{rk}_{i+\fracrac{1}{2},j}=\fracrac{1}{6}(\hat{f}_{i+\fracrac{1}{2},j}^n - \widehat{(a_x)}_{i+\fracrac{1}{2},j}^n) + \fracrac{1}{6}(\hat{f}_{i+\fracrac{1}{2},j}^1 - \widehat{(a_x)}_{i+\fracrac{1}{2},j}^1) + \fracrac{2}{3} (\hat{f}_{i+\fracrac{1}{2},j}^2 - \widehat{(a_x)}_{i+\fracrac{1}{2},j}^2),\\ & \hat{G}^{rk}_{i,j+\fracrac{1}{2}}=\fracrac{1}{6}(\hat{g}_{i+\fracrac{1}{2},j}^n - \widehat{(b_x)}_{i+\fracrac{1}{2},j}^n) + \fracrac{1}{6}(\hat{g}_{i+\fracrac{1}{2},j}^1 - \widehat{(b_x)}_{i+\fracrac{1}{2},j}^1) + \fracrac{2}{3} (\hat{g}_{i+\fracrac{1}{2},j}^2 - \widehat{(b_x)}_{i+\fracrac{1}{2},j}^2). \end{align} $\hat{H}^{rk}_{i+\fracrac{1}{2},j}$ and $\hat{G}^{rk}_{i,j+\fracrac{1}{2}}$ can be understood as the average integral of the numerical fluxes in the temporal direction. Similarly as for the one dimensional case, we modify the fluxes as follows, \begin{align} & \tilde{H}^{rk}_{i+\fracrac{1}{2},j}=\theta_{i+\fracrac{1}{2},j} \hat{H}^{rk}_{i+\fracrac{1}{2},j} + (1-\theta_{i+\fracrac{1}{2},j}) \hat{h}_{i+\fracrac{1}{2},j}, \label{2dmax}\\ & \tilde{G}^{rk}_{i,j+\fracrac{1}{2}}=\theta_{i,j+\fracrac{1}{2}} \hat{G}^{rk}_{i,j+\fracrac{1}{2}}+ (1-\theta_{i,j+\fracrac{1}{2}}) \hat{g}_{i,j+\fracrac{1}{2}}, \label{2dmin} \end{align} where $\hat{h}_{i+\fracrac{1}{2},j}$ and $\hat{g}_{i,j+\fracrac{1}{2}}$ are low order monotone flux that satisfy maximum principle, so that \begin{equation}\label{constraint} u_m \le u_{i,j}^n - \lambda_x(\tilde{H}^{rk}_{i+\fracrac{1}{2},j}-\tilde{H}^{rk}_{i-\fracrac{1}{2},j}) -\lambda_y(\tilde{G}^{rk}_{i,j+\fracrac{1}{2}}-\tilde{G}^{rk}_{i,j-\fracrac{1}{2}}) \le u_M, \end{equation} with $u_m=\underset{x,y}{min} \ u_0(x,y)$ and $u_M=\underset{x,y}{max} \ u_0(x,y)$. Introducing the notations \begin{align} &F_{i-\fracrac{1}{2},j}=\lambda_x (\hat{H}^{rk}_{i-\fracrac{1}{2},j}-\hat{h}_{i-\fracrac{1}{2},j}), \nonumber \\ &F_{i+\fracrac{1}{2},j}=-\lambda_x (\hat{H}^{rk}_{i+\fracrac{1}{2},j}-\hat{h}_{i+\fracrac{1}{2},j}), \nonumber \\ &F_{i,j-\fracrac{1}{2}}=\lambda_y (\hat{G}^{rk}_{i,j-\fracrac{1}{2}}-\hat{g}_{i,j-\fracrac{1}{2}}), \nonumber \\ &F_{i,j+\fracrac{1}{2}}=-\lambda_y (\hat{G}^{rk}_{i,j+\fracrac{1}{2}}-\hat{g}_{i,j+\fracrac{1}{2}}), \nonumber \end{align} and plugging the modified fluxes (\ref{2dmax}) and (\ref{2dmin}) into (\ref{constraint}), we have \begin{align} &\theta_{i+\fracrac{1}{2},j} F_{i+\fracrac{1}{2},j} + \theta_{i-\fracrac{1}{2},j} F_{i-\fracrac{1}{2},j} + \theta_{i,j+\fracrac{1}{2}} F_{i,j+\fracrac{1}{2}} + \theta_{i,j-\fracrac{1}{2}} F_{i,j-\fracrac{1}{2}} \le \Gamma_{i,j}^M, \label{2dmax1}\\ &\theta_{i+\fracrac{1}{2},j} F_{i+\fracrac{1}{2},j} + \theta_{i-\fracrac{1}{2},j} F_{i-\fracrac{1}{2},j} + \theta_{i,j+\fracrac{1}{2}} F_{i,j+\fracrac{1}{2}} + \theta_{i,j-\fracrac{1}{2}} F_{i,j-\fracrac{1}{2}} \ge \Gamma_{i,j}^m, \label{2dmin1} \end{align} where \begin{align} & \Gamma_{i,j}^M = u_M - (u_{i,j} - \lambda_x(\hat{h}_{i+\fracrac{1}{2},j}-\hat{h}_{i-\fracrac{1}{2},j}) - \lambda_y(\hat{g}_{i,j+\fracrac{1}{2}} - \hat{g}_{i,j-\fracrac{1}{2}}) ) \ge 0, \\ & \Gamma_{i,j}^m = u_m - (u_{i,j} - \lambda_x(\hat{h}_{i+\fracrac{1}{2},j}-\hat{h}_{i-\fracrac{1}{2},j}) - \lambda_y(\hat{g}_{i,j+\fracrac{1}{2}} - \hat{g}_{i,j-\fracrac{1}{2}}) ) \le 0. \end{align} Similarly as in the one dimensional case, we need to find numbers $\Lambda_{L,i,j}, \Lambda_{R,i,j}, \Lambda_{D,i,j}, \Lambda_{U,i,j}$ such that if \begin{equation} (\theta_{i-\fracrac{1}{2},j}, \theta_{i+\fracrac{1}{2},j}, \theta_{i,j-\fracrac{1}{2}}, \theta_{i,j+\fracrac{1}{2}}) \in [0,\Lambda_{L,i,j}]\times [0,\Lambda_{R,i,j}]\times [0,\Lambda_{D,i,j}]\times [0,\Lambda_{U,i,j}], \end{equation} then (\ref{2dmax1}) and (\ref{2dmin1}) hold. We do similar analysis as for the one dimensional case, and the results are listed in Table \ref{Lambdasmax} and Table \ref{Lambdasmin}. (In the table, $'+'$ means that the corresponding variable is positive ($>0$) and $'-'$ means that it's non-positive ($\le 0$). And $\mathbf{F}=(F_{i-\fracrac{1}{2},j},F_{i+\fracrac{1}{2},j},F_{i,j-\fracrac{1}{2}},F_{i,j+\fracrac{1}{2}})$.) Both the cases for maximum value and minimum value should be considered, so the numbers $\Lambda_{L,i,j}, \Lambda_{R,i,j}, \Lambda_{D,i,j}, \Lambda_{U,i,j}$ are \begin{equation} \begin{cases} \Lambda_{L,i,j}=min(\Lambda_{L,i,j}^M,\Lambda_{L,i,j}^m),\\ \Lambda_{R,i,j}=min(\Lambda_{R,i,j}^M,\Lambda_{R,i,j}^m),\\ \Lambda_{D,i,j}=min(\Lambda_{D,i,j}^M,\Lambda_{D,i,j}^m),\\ \Lambda_{U,i,j}=min(\Lambda_{U,i,j}^M,\Lambda_{U,i,j}^m). \end{cases} \end{equation} Finally we define the local limiter parameters as \begin{equation} \begin{cases} \theta_{i+\fracrac{1}{2},j}=min(\Lambda_{R,i,j},\Lambda_{L,i+1,j}), \\ \theta_{i,j+\fracrac{1}{2}}=min(\Lambda_{U,i,j},\Lambda_{D,i,j+1}). \end{cases} \end{equation} With these limiters, the numerical solution at each time step will satisfy the maximum principle. We will demonstrate the results in Section \ref{sec4}. Given cell averages $\{ \bar{u}_{i,j} \}$, we need to reconstruct $u_{i+\fracrac{1}{2},i_g}$, the value of $u$ at $(x_{i+\fracrac{1}{2}},y_{i_g})$ where $y_{i_g}$ is the $i_g$-th Gaussian quadrature point on the interval $[y_{j-\fracrac{1}{2}},y_{j+\fracrac{1}{2}}]$. To reconstruct $u_{i+\fracrac{1}{2},i_g}$, we consider $P(x)=u_(x,y_{i_g})$, then $u_{i+\fracrac{1}{2},i_g}=P(x_{i+\fracrac{1}{2}})$ and \begin{equation} \fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} P(x)dx = \fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y_{i_g})dx. \end{equation} So as long as we have $\fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y_{i_g})dx$, the averages of $u(x,y)$ over the segments $\{ x\in [x_{i-\fracrac{1}{2}}, x_{i+\fracrac{1}{2}}], y=y_{i_g}\}$, then we can reconstruct $u_{i+\fracrac{1}{2},i_g}$ in the same way as in [ref]. So it suffices to reconstruct $\fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y_{i_g})dx$. To do this, we consider $Q(y)=\fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y))dx$, then $\fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y_{i_g})dx=Q(y_{i_g})$ and \begin{equation} \fracrac{1}{\triangle y} \int_{y_{j-\fracrac{1}{2}}}^{y_{j+\fracrac{1}{2}}} Q(y)dy =\fracrac{1}{\triangle y} \fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y))dx dy =\bar{u}_{i,j}. \end{equation} So $\fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y_{i_g})dx$ can be reconstructed from cell averages $\{ \bar{u}_{i,j}\}$ with the method in [ref]. If we denote the standard reconstructing procedure in [ref] by $\mathcal{R}$, then the whole two dimensional finite volume reconstructing procedure can be illustrated as \begin{equation} \centering \{ \bar{u}_{i,j}\} \overset{\mathcal{R}}{\longrightarrow} \{\fracrac{1}{\triangle x} \int_{x_{i-\fracrac{1}{2}}}^{x_{i+\fracrac{1}{2}}} u_(x,y_{i_g})dx \} \overset{\mathcal{R}}{\longrightarrow} \{u_{i+\fracrac{1}{2},i_g}\}. \end{equation} \end{document}

\begin{document} \title{Quasi-Clifford algebras, Quadratic forms over $\mathbb{F}_2$, and Lie Algebras} \author{Hans Cuypers} \begin{abstract} Let $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ be a graph, whose vertices $v\in \mathcal{V}$ are colored black and white and labeled with invertible elements $\lambda_v$ from a commutative and associative ring $R$ containing $\pm 1$. Then we consider the associative algebra $\mathfrak{C}(\mathcal{G}amma)$ with identity element $\mathbf{1}$ generated by the elements of $\mathcal{V}$ such that for all $v,w\in \mathcal{V}$ we have $$\begin{array}{lll} v^2 &=\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is white},\\ v^2 &=-\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is black},\\ vw+wv&=0&\textrm{if } \{v,w\}\in \mathcal{E},\\ vw-wv&=0&\textrm{if } \{v,w\}\not\in \mathcal{E}.\\ \end{array}$$ If $\mathcal{G}amma$ is the complete graph, $\mathfrak{C}(\mathcal{G}amma)$ is a Clifford algebra, otherwise it is a so-called quasi-Clifford algebra. We describe this algebra as a twisted group algebra with the help of a quadratic space $(V,Q)$ over the field $\mathbb{F}_2$. Using this description, we determine the isomorphism type of $\mathfrak{C}(\mathcal{G}amma)$ in several interesting examples. As the algebra $\mathfrak{C}(\mathcal{G}amma)$ is associative, we can also consider the corresponding Lie algebra and some of its subalgebras. In case $\lambda_v=1$ for all $v\in \mathcal{V}$, and all vertices are black, we find that the elements $v,w\in \mathcal{V}$ satisfy the following relations $$\begin{array}{lll} [v,w]&=0&\textrm{if } \{v,w\}\not\in \mathcal{E},\\ {[v,[v,w]]}&=-w&\textrm{if } \{v,w\}\in \mathcal{E}.\\ \end{array}$$ In case $R$ is a field of characteristic $0$, we identify these algebras as quotients of the compact subalgebras of Kac-Moody Lie algebras and prove that they admit a so-called generalized spin representation. \end{abstract} \maketitle \section{Introduction} Let $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ be a graph, whose vertices $v$ are colored black or white and labeled with invertible elements $\lambda_v$ from a commutative and associative ring $R$ containing $\pm 1$. (By default, an arbitrary graph is considered to have black vertices and all labels equal to $1$.) Then we consider the associative algebra $\mathfrak{C}(\mathcal{G}amma)$ with identity element $\mathbf{1}$ generated by the elements of $\mathcal{V}$ such that for all $v,w\in \mathcal{V}$ we have $$\begin{array}{lll} v^2 &=\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is white},\\ v^2 &=-\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is black},\\ vw+wv&=0&\textrm{if } v\sim w,\\ vw-wv&=0&\textrm{if } v\not \sim w.\\ \end{array}$$ Here $v\sim w$ denotes that $\{v,w\}$ is an edge in $\mathcal{E}$. If $\mathcal{G}amma$ contains no edges, all vertices are white and $\lambda_v=1$ for all $v\in \mathcal{V}$, then the algebra $\mathfrak{C}(\mathcal{G}amma)$ is a Grassmann algebra. On the other hand, if $\mathcal{G}amma$ is the complete graph on $n$ vertices, $R=\mathbb{R}$ and $\lambda_v=1$ for all $v\in \mathcal{V}$, then the algebra $\mathfrak{C}(\mathcal{G}amma)$ is a Clifford algebra $\mathrm{Cl}(p,q)$, where $n=p+q$ and $p$ vertices are colored white, while $q$ vertices have the color black. This construction also appears in \cite{Tanya1,gintz}, where ordinary finite graphs with all vertices black and $R$ the field of complex numbers are considered. For an arbitrary finite graph $\mathcal{G}amma$ and field $R$, we obtain a so-called quasi-Clifford algebra as studied by Gastineau-Hills in \cite{quasi} in connection with orthogonal designs (see also \cite{leopardi1,Leopardi2,Seberry1} and the recent book \cite{Seberry}). In this paper we first describe for arbitrary black and white colored graphs $\mathcal{G}amma$ the algebra $\mathfrak{C}(\mathcal{G}amma)$ as a twisted group algebra with the help of an $\mathbb{F}_2$-space $V$ and a bilinear form $g$ on $V$. Their isomorphism type turns out to depend only on the quadratic form $Q$ obtained by $Q(v)=g(v,v)$ for $v\in V$. This is shown in the Sections \ref{sect:twisted} and \ref{sect:generators}. Given such a quadratic form we determine the structure of the algebra, focusing on the case where $\lambda_v=1$ for all $v\in \mathcal{V}$. The algebras obtained are called special by Gastineau-Hills \cite{quasi}, and are up to a center isomorphic to (sums of) Clifford algebras. Using the description as twisted group algebras, we determine the isomorphism type of $\mathfrak{C}(\mathcal{G}amma)$ for several interesting graphs $\mathcal{G}amma$. This is done in Section \ref{sect:examples}. We apply our results to complete graphs and obtain quickly the classification of Clifford algebras. But we also consider graphs of type $A_n, D_n$ and $E_n$. As the algebra $\mathfrak{C}(\mathcal{G}amma)$ is associative, we can also consider the corresponding Lie algebra and some of its subalgebras. In particular, we determine the isomorphism type of the Lie algebras generated by the generators in $\mathcal{V}$. See Section \ref{sect:lie}. In case $\lambda_v=1$ for all $v\in\mathcal{V}$, and all vertices are black, we find that the elements $v,w\in \mathcal{V}$ satisfy the following relations, where $[\cdot,\cdot]$ denotes the Lie product: $$\begin{array}{lll} [v,w]&=0&\textrm{if } v\not\sim w,\\ {[v,[v,w]]}&=-w&\textrm{if } v \sim w.\\ \end{array}$$ In case $R$ is a field of characteristic $0$, we identify these Lie algebras with quotients of compact subalgebras of Kac-Moody Lie algebras and prove that they admit a so-called generalized spin representation. In particular, using the computations of Section \ref{sect:examples} and \ref{sect:lie}, we are able to identify various quotients of these compact Lie subalgebras of Kac-Moody algebras and construct spin representations of such algebras extending the results of \cite{ Damour,Buyl,kohlspin}. This is the topic of Section \ref{sect:kac}. \section{A class of algebras obtained from bilinear forms over $\mathbb{F}_2$} \label{sect:twisted} In this section we provide a description of a class of algebras as twisted group algebras. The finite dimensional algebras we describe turn out to be quasi-Clifford algebras as introduced by Gastineau-Hills \cite{quasi}. Our description as twisted group algebra is closely related to the description of Clifford algebras as twisted group algebras, see \cite{Twisted_clifford}, and relates our algebras to quadratic spaces over the field with two elements as in \cite{Elduque_clifford}. (See also the work of Shaw \cite{shaw1,shaw2,shaw3,shaw4}.) Let $V$ be an $\mathbb{F}_2$ vector space (with addition $\pluscirc$) equipped with a bilinear form $g:V\times V\rightarrow \mathbb{F}_2$. Let $\mathcal{B}$ be a basis for $V$ and $\mathcal{B}^*$ a dual basis, where $b^*$ denotes the dual of $b\in \mathcal{B}$. Now assume $R$ is a commutative and associative ring, and $R^*$ its set of invertible elements including the distinct elements $1$ and $-1$. Then let $\Lambda:\mathcal{B}\rightarrow R^*$ be a map which we extend to $V\times V$ by $$\Lambda(v,w):=\prod_{b\in \mathcal{B}}\ \Lambda(b)^{b^*(v)b^*(w)}$$ for all $v,w\in V$. Notice that this is well defined, also for infinite dimensional spaces $V$, since almost all values of $b^*(v)b^*(w)$ are $0$, in which case $\Lambda(b)^{b^*(v)b^*(w)}$ equals $1$. The algebra $\mathfrak{C}{(V,g,\Lambda)}$ is then the $R$-algebra with basis $\{e_v\mid v\in V\}$, unit element $e_0=\mathbf{1}$, and multiplication defined by $$ \begin{array}{ll} e_ve_w&=(-1)^{g(v,w)}\cdot\Lambda(v,w) e_{v\pluscirc w}\\ \end{array}$$ for all $v,w\in V$. If $\Lambda(v)=1$ for all $v\in V$, we write $\mathfrak{C}(V,g)$ instead of $\mathfrak{C}{(V,g,\Lambda)}$. Notice that elements $e_v$ and $e_w$, where $v\neq w\in V$ satisfy the relations $$\begin{array}{lll} e_ve_w-e_we_v=0& \textrm{if } f(v,w)=0\\ e_ve_w+e_we_v=0& \textrm{if } f(v,w)=1.\\ \end{array}$$ \begin{proposition} The algebra $\mathfrak{C}{(V,g,\Lambda)}$ is associative. \end{proposition} \begin{proof} Let $u,v,w\in V$, then $$\begin{array}{ll} e_u(e_ve_w)&=e_u(-1)^{g(v,w)}\Lambda(v,w)e_{v\pluscirc w}\\ &=(-1)^{g(v,w)+g(u,v\pluscirc w))}\Lambda(u,v\pluscirc w)\Lambda(v,w)e_{u\pluscirc v\pluscirc w}\\ &=(-1)^{g(v,w)+g(u,v)+g(u,w)}\Lambda(u,v\pluscirc w)\Lambda(v,w)e_{u\pluscirc v\pluscirc w}, \end{array}$$ while $$\begin{array}{ll} (e_ue_v)e_w&=(-1)^{g(u,v)}\Lambda(u,v)e_{u\pluscirc v}e_w\\ &=(-1)^{g(u,v)+g(u\pluscirc v,w)}\Lambda(u,v)\Lambda(u\pluscirc v,w) e_{u\pluscirc v\pluscirc w}\\ & =(-1)^{g(u,v)+g(u,w)+g(v,w)}\Lambda(u,v)\Lambda(u\pluscirc v,w)e_{u\pluscirc v\pluscirc w}.\end{array}$$ So, we find the algebra to be associative, if and only if the function $\Lambda$ satisfies $$\Lambda(u,v)\cdot\Lambda (u\pluscirc v,w)=\Lambda(v,w)\cdot \Lambda(u,v\pluscirc w).$$ This identity follows from the observation that for all $u,v,w$ and $b^*\in B^*$ we have $$\begin{array}{ll} b^*(u)b^*(v)+b^*(u\pluscirc v)b^*(w)&=b^*(u)b^*(v)+ b^*(u)b^*(w)+ b^*(v)b^*(w)\\ & =b^*(v)b^*(w)+(b^*(v)+b^*(w))b^*(u)\\ &=b^*(v)b^*(w)+b^*(v\pluscirc w)b^*(u).\\ \end{array}$$ \end{proof} \section{From relations to algebra} \label{sect:generators} Let $R$ be a commutative and associative ring with distinct elements $1,-1$. Suppose $V$ is an $\mathbb{F}_2$-space equipped with a bilinear form $g$ and for some basis $\mathcal{B}$ of $V$ a map $\Lambda:\mathcal{B}\rightarrow R^*$ which we extend to a map $\Lambda:V\times V\rightarrow R^*$ defined by $\Lambda(v,w):=\prod_{b\in \mathcal{B}}\ \Lambda(b)^{b^*(v)b^*(w)}$ for all $v,w\in V$. Then we can consider the algebra $\mathfrak{C}(V,g,\Lambda)$ as defined in \cref{sect:twisted}. We identify the elements $v\in V$ with the basis vectors $e_v$ of $\mathfrak{C}(V,g,\Lambda)$. The algebra $\mathfrak{C}(V,g,\Lambda)$ is defined with the help of the basis $\mathcal{B}$. For any other basis $\mathcal{V}$ of $V$ we find that $\mathcal{V}$ also generates the algebra. The elements $v\neq w\in \mathcal{V}$ then satisfy the following relations: $$ \begin{array}{ll} v^2&=(-1)^{Q(v)}\Lambda(v,v)\mathbf{1}\\ vw&=(-1)^{f(v,w)}wv,\\ \end{array}$$ where $Q$ is the quadratic form on $V$ defined by $Q(v)=g(v,v)$ and $f$ is the symmetric bilinear form associated to $Q$ and given by $f(v,w)=g(v,w)+g(w,v)$ for all $v,w\in V$. We can capture this information in a black and white colored graph. This graph has vertex set $\mathcal{V}$. Two vertices $v\neq w$ are adjacent if and only if $vw=-wv$. A vertex $v\in\mathcal{V}$ is labeled by $\Lambda(v,v)$ and is colored black or white. Its color is black if and only if $v^2=-\Lambda(v,v) v$. In this section we reverse this process by showing that each such graph determines the generators and relations of an associative algebra isomorphic to an algebra $\mathfrak{C}(V,g,\Lambda)$. So, let $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ be a black and white colored graph with vertex set $\mathcal{V}$ and edge set $\mathcal{E}$, and the vertices $v\in \mathcal{V}$ labeled by nonzero invertible elements $\lambda_v$ from a commutative and associative ring $R$ containing the distinct elements $1$ and $-1$. Then consider ${V}_\mathcal{G}amma$, the vector space of finite subsets of $\mathcal{V}$, where for two finite subsets $v,w$ of $\mathcal{V}$ the sum $v\pluscirc w$ is defined to be the symmetric difference of $v$ and $w$. Put a total ordering $<$ on the vertex set of $\mathcal{G}amma$. Let $u$ and $w$ be two finite subsets of $\mathcal{V}$ and let $g_\mathcal{G}amma(u,w)$ denote the number of ordered pairs $(x,y)\in u\times w$, where $x<y$ and $\{x,y\}$ is an edge, or $x=y$ is a black vertex, modulo $2$. Then $g_\mathcal{G}amma(u,v\pluscirc w)=g_\mathcal{G}amma(u,v)+g_\mathcal{G}amma(u,w)$, for any finite subsets $u,v,w$ of $\mathcal{V}$, as the ordered edges $(x,z)$ with $x<z$ and $z\in v\cap w$, are counted twice at the right hand site of the equation, just as black vertices in the intersection of $u$ and $v\cap w$. Similarly we find $g_\mathcal{G}amma(v\pluscirc w,u)=g_\mathcal{G}amma(v,u)+g_\mathcal{G}amma(w,u)$. So, $g_\mathcal{G}amma:{V}_\mathcal{G}amma\times {V}_\mathcal{G}amma\rightarrow \mathbb{F}_2$ is bilinear. The map $Q_\mathcal{G}amma:V_\mathcal{G}amma\rightarrow \mathbb{F}_2$ given by $Q_\mathcal{G}amma(v)=g_\mathcal{G}amma(v,v)$ for all $v\in V_\mathcal{G}amma$ is a quadratic form with associated symmetric (and also alternating) form $f_\mathcal{G}amma$ given by $f_\mathcal{G}amma(u,w)=g_\mathcal{G}amma(u,w)+g_\mathcal{G}amma(w,u)$. Now we define an associative algebra $\mathfrak{C}(\mathcal{G}amma)$ over $R$ with basis the set of element of $ {V}_\mathcal{G}amma$, in which the elements $v\neq w\in \mathcal{V}$ (after being identified with the subset $\{v\}$) satisfy the following relations: $$\begin{array}{lll} v^2 &=\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is white},\\ v^2 &=-\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is black},\\ vw+wv&=0&\textrm{if } v\sim w,\\ vw-wv&=0&\textrm{if } v\not \sim w.\\ \end{array}$$ The product is defined as follows. The element $\emptyset$ is the unit element of $\mathfrak{C}(\mathcal{G}amma)$ and will be denoted by $\mathbf{1}$. For $v,w$ being finite subsets of $\mathcal{V}$, we define the product of $v$ and $w$ by $$vw=(-1)^{g_\mathcal{G}amma(v,w)}(\prod_{x\in v\cap w}\lambda_x)\, v\pluscirc w.$$ Clearly this definition of the product is forced upon us by the relations and associativity of the product. But then it is straightforward to check that with $\Lambda_\mathcal{G}amma(v,w)=\prod_{x\in v\cap w}\lambda_v$ we have the following. \begin{theorem} The algebra $\mathfrak{C}(\mathcal{G}amma)$ is isomorphic to $\mathfrak{C}(V_\mathcal{G}amma,g_\mathcal{G}amma,\Lambda_\mathcal{G}amma)$. \end{theorem} By construction, the algebra $\mathfrak{C}(\mathcal{G}amma)$ is the universal associative algebra satisfying the relations prescribed by the graph $\mathcal{G}amma$. So, we have: \begin{theorem} An associative algebra $\mathfrak{C}$ with unit element $\mathbf{1}$ generated by a set of elements $\mathcal{V}$ satisfying the relations $$\begin{array}{lll} v^2 &=\pm\lambda_v\mathbf{1},&\lambda_v\in R^*\\ vw+wv&=0&\textrm{or}\\ vw-wv&=0&\\ \end{array}$$ for $v\neq w\in \mathcal{V}$, is isomorphic to a quotient of $\mathfrak{C}(\mathcal{G}amma)$, where $\mathcal{G}amma$ is the black and white colored graph with vertex set $\mathcal{V}$, two vertices being adjacent if and only if they do not commute and each vertex $v$ is labeled with $\lambda_v$ and $v$ is black if and only if $v^2=-\lambda_v\mathbf{1}$. \end{theorem} Let $V$ be an $\mathbb{F}_2$-space equipped with a bilinear form $g$. Let $Q$ be the quadratic form given by $Q(v)=g(v,v)$ for all $v$ and denote by $f$ the associated alternating form given by $f(u,v)=g(u,v)+g(v,u)=Q(u+v)+Q(u)+Q(v)$ for all $v,w\in V$. Then the above results imply that, up to isomorphism, the algebra $\mathfrak{C}(V,g,\Lambda)$ is determined by the quadratic form $Q$. For this reason we will also write $\mathfrak{C}(V,Q,\Lambda)$ to denote (the isomorphism class of) an algebra $\mathfrak{C}(V,g,\Lambda)$. Moreover, two algebras $\mathfrak{C}(V,Q,\Lambda)$ and $\mathfrak{C}(V,Q',\Lambda)$, with $Q$ and $Q'$ quadratic forms, are isomorphic when the two forms $Q$ and $Q'$ are equivalent, i.e., when there is a $\gamma\in \mathrm{GL}(V)$ with $Q(v)=Q'(v^\gamma)$ for all $v\in V$. We collect this information in the following theorem. \begin{theorem} Let $(V,Q)$ be quadratic $\mathbb{F}_2$-space with basis $\mathcal{V}$ and $\Lambda:\mathcal{V}\rightarrow R^*$ a map. Suppose $f$ is the symmetric form associated to $Q$. Suppose $g$ is a bilinear form on $V$ with $Q(v)=g(v,v)$ for all $v\in V$. Then the algebra $\mathfrak{C}(V,g,\Lambda)$ is isomorphic to $\mathfrak{C}(\mathcal{G}amma)$ where $\mathcal{G}amma$ is the graph with vertex set $\mathcal{V}$, in which two vertices $v,w$ are adjacent if and only if $f(v,w)=1$, a vertex $v$ is labeled by $\Lambda(v)$ and colored black or white, according to $v^2=-\Lambda(v)\mathbf{1}$ or $+\Lambda(v)\mathbf{1}$, respectively. \end{theorem} \section{Algebras and quadratic forms} \label{sect:quadric} As we have seen in the previous section, the algebras $\mathfrak{C}(\mathcal{G}amma)$, where $\mathcal{G}amma$ is a black and white colored graph whose vertices are labeled by invertible elements from an associative ring $R$ are, up to isomorphism, algebras $\mathfrak{C}(V,Q,\Lambda)$ for some quadratic space $(V,Q)$ over the field $\mathbb{F}_2$ and a map $\Lambda:V\rightarrow R^*$. The classification of quadratic forms on vector spaces of finite dimension over the field of $2$ elements is well known. We discuss this briefly. The radical of $f$, defined as ${\rm Rad}(f)=\{v\in V\mid f(v,w)=0$ for all $w\in V\}$, is a subspace of $V$. It contains the radical of $Q$, defined as ${\rm Rad}(Q)=\{v\in{\rm Rad}(f)\mid Q(v)=0\}$, as a subspace of codimension at most $1$. We call the form $Q$ \emph{nondegenerate} if and only if ${\rm Rad}(f)=\{0\}$ and \emph{almost nondegenerate} if ${\rm Rad}(Q)=\{0\}$, but ${\rm Rad}(f)\neq \{0\}$. In dimension one there is, up to isomorphism, a unique nontrivial quadratic form $Q(x)=x^2$, which is almost nondegenerate. It is called of $0$-type. In dimension 2 we have, up to isomorphism, exactly two nondegenerate forms, $Q(x_1,x_2)=x_1x_2$, called of $+$-type, and $Q(x_1,x_2)=x_1^2+x_1x_2+x_2^2$, called of $-$-type. In dimension $n>2$ we can distinguish, up to isomorphism, the following forms: \hspace{0.5cm}\begin{minipage}{11cm} \begin{enumerate} \item[$+$-type:] $V$ is an orthogonal sum $V_1\perp\cdots\perp V_k\perp {\rm Rad}(Q)$, where all $V_i$ are $2$-spaces of $+$-type. \item[$-$-type:] $V$ is an orthogonal sum $V_1\perp\cdots\perp V_k\perp {\rm Rad}(Q)$, where all $V_i$ are $2$-spaces of $+$-type, except for one, which is of $-$-type. \item[$0$-type:] $V$ is an orthogonal sum $V_1\perp\cdots\perp V_k\perp {\rm Rad}(Q)$, where all $V_i$ are $2$-spaces of $+$-type, except for one, which is one dimensional and of $0$-type. \\ Notice, in this case we find the radical of $f$ to be larger than the radical of $Q$. \end{enumerate} \end{minipage} One of the key observations in the proof of this classification is that the type of a direct orthogonal sum of two spaces is determined by the type of the summants. The orthogonal direct sum of spaces of type $x$ and type $y$, where $x,y=\pm$ or $0$, gives us a space of type $x\cdot y$. We will frequently use these observations in the sequel. We note that the number of isomorphism classes of quadratic spaces $(V,Q)$ over $\mathbb{F}_2$ of infinite dimension is much larger, see \cite{Hall_extra}. The decomposition of $(V,Q)$ into pairwise orthogonal subspaces provides a decomposition of the algebra $\mathfrak{C}(V,Q,\Lambda)$ into tensor products. Indeed, if we suppose $R$ is a field, then the following proposition yields this decomposition. \begin{proposition} \label{decompose} Let $R$ be a field. Suppose $(V,Q)$ is finite dimensional and can be decomposed as a direct orthogonal sum $(V_1,Q_1)\perp (V_2,Q_2)$. Then $\mathfrak{C}(V,Q,\Lambda)$ is isomorphic to $\mathfrak{C}(V_1,Q_1,\Lambda_1)\otimes \mathfrak{C}(V_2,Q_2,\Lambda_2)$, where $\Lambda_i$ is the restriction of $\Lambda$ to $V_i\times V_i$. \end{proposition} \begin{proof} The map $\phi$ that sends each tensor $e_{v_1}\otimes e_{v_2}\in \mathfrak{C}(V_1,Q_1,\Lambda_1)\otimes \mathfrak{C}(V_2,Q_2,\Lambda_2)$, with $v_1\in V_1, v_2\in V_2$ to $e_{v_1\pluscirc v_2}$ extends uniquely to a linear map $$\phi:\mathfrak{C}(V_1,Q_1,\Lambda_1)\otimes \mathfrak{C}(V_2,Q_2,\Lambda_2)\rightarrow \mathfrak{C}(V,Q,\Lambda).$$ Moreover, as the elements $e_{v_1}$ and $e_{v_2}$ commute in $\mathfrak{C}(V,Q,\Lambda)$, it is straightforward to check that $\phi$ is a surjective homomorphism of algebras. As the dimensions of $\mathfrak{C}(V,Q,\Lambda)$ and $\mathfrak{C}(V_1,Q_1,\Lambda_1)\otimes \mathfrak{C}(V_2,Q_2,\Lambda_2)$ coincide, we find an isomorphism. \end{proof} The structure of the algebra $\mathfrak{C}(V,Q,\Lambda)$ not only depends on the quadratic space $(V,Q)$, but also on the ring $R$ and of course the values $\Lambda$ takes in $R$. In case $R=\mathbb{F}$ is a field, we can use the above Proposition \ref{decompose} and only have to consider small dimensional cases for $V$ to find the structure of the algebra $\mathfrak{C}(V,Q,\Lambda)$. These small dimensional cases are worked out in \cite{quasi}. For later use we describe the situation in the case where $R=\mathbb{F}$ is a field and $\Lambda$ is $1$. In this situation we consider three types of fields, type I, II and III, defined by: \hspace{0.5cm}\begin{minipage}{11cm} \begin{enumerate} \item[type I:] There is an element $i\in \mathbb{F}$ with $i^2=-1$. \item[type II:] There is no $i\in \mathbb{F}$ with $i^2=-1$, but there are $x,y\in\mathbb{F}$ with $x^2+y^2=-1$. \item[type III:] There are no $x,y\in \mathbb{F}$ with $x^2+y^2=-1$. \end{enumerate} \end{minipage} If $V$ is $1$-dimensional, then $\mathfrak{C}(V,Q)$ is isomorphic to $\mathbb{F}\times \mathbb{F}$ in case $Q$ is trivial on $V$ or $\mathbb{F}$ is a field of type I. If $Q$ is non-trivial on $V$ and $\mathbb{F}$ is of type II or III, then $\mathfrak{C}(V,Q)$ is isomorphic to $\mathbb{F}[i]$, where $i^2=-1$. Now assume that $V=\langle e_1,e_2\rangle$ is $2$-dimensional and suppose $Q$ is of $+$-type, $Q(e_1)=Q(e_2)=0$ and $f(e_1,e_2)=1$. Then we can identify $\mathfrak{C}(V,Q)$ with $M(2,\mathbb{F})$, the algebra of $2\times 2$-matrices via the map $$e_1\mapsto \begin{pmatrix}0 & 1 \\ 1&0\end{pmatrix}$$ and $$e_2\mapsto \begin{pmatrix}1 & 0 \\ 0&-1\end{pmatrix}.$$ If $Q$ is of $-$-type, then we may assume that $Q(e_1)=Q(e_2)=f(e_1,e_2)=1$ and we can identify $\mathfrak{C}(V,Q)$ with $M(2,\mathbb{F})$ via the map $$e_1\mapsto \begin{pmatrix}0&1\\-1&0\end{pmatrix}\textrm{ and }e_2\mapsto \begin{pmatrix}i&0\\0&i\end{pmatrix}$$ if $\mathbb{F}$ is of type I, and $$e_1\mapsto \begin{pmatrix}0&1\\-1&0\end{pmatrix}\textrm{ and }e_2\mapsto \begin{pmatrix}x&-y\\-y&-x\end{pmatrix}$$ if $\mathbb{F}$ is of type II and $x,y\in \mathbb{F}$ with $x^2+y^2=-1$. If $\mathbb{F}$ is of type III, we can identify $\mathfrak{C}(V,Q)$ with the matrix algebra $$\left\langle \begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\\ \end{pmatrix}, \begin{pmatrix}0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\\ \end{pmatrix} \right\rangle.$$ This algebra can be identified with the algebra $\mathbb{H}$ of quaternions over $\mathbb{F}$. This implies that for finite dimensional spaces $(V,Q)$ the algebra $\mathfrak{C}(V,Q)$ is determined, up to isomorphism, by the following parameters: \begin{enumerate}[\rm(a)] \item Dimension $n$ of $\overline{V}:=V/\mathrm{Rad}(Q)$; \item Dimension $r$ of $\mathrm{Rad}(f)$; \item Type of $\overline{Q}$, the form induced by $Q$ on $\overline{V}$; \item Type of $\mathbb{F}$. \end{enumerate} We can now describe the various isomorphism classes of the algebras $\mathfrak{C}(V,Q)$ in terms of these parameters. \begin{proposition} Let $(V,Q)$ be a nontrivial, finite dimensional quadratic space over the field $\mathbb{F}_2$. Then the isomorphism type of the algebra $\mathfrak{C}(V,Q)$ over a field $\mathbb{F}$ of characteristic $\neq 2$ is given in Table \ref{isotable}. \end{proposition} \begin{table}[h] \begin{tabular}{|l|l|l|l|} \hline $\mathrm{dim}(\overline{V})$ & Type($\overline{Q}$) &Type of $\mathbb{F}$& Algebra\\ \hline\hline $n=0 \pmod{2}$& $+$ & I& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$ \\ $n=0 \pmod{2}$& $-$ & I& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$\\ $n=1 \pmod{2}$& $0$ & I& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}})^{2^{r}}$\\ $n=0 \pmod{2}$& $+$ & II& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$ \\ $n=0 \pmod{2}$& $-$ & II& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$\\ $n=1 \pmod{2}$& $0$ & II& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}}\otimes\mathbb{F}[i])^{2^{r-1}}$\\ $n=0 \pmod{2}$& $+$ & III& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$ \\ $n=0 \pmod{2}$& $-$ & III& $(M(2,\mathbb{F})^{\otimes\frac{n-2}{2}}\otimes \mathbb{H})^{2^{r}}$\\ $n=1 \pmod{2}$& $0$ & III& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}}\otimes\mathbb{F}[i])^{2^{r-1}}$\\ \hline \end{tabular} \caption{The isomorphism types of the algebras $\mathfrak{C}(V,Q)$.}\label{isotable} \end{table} We end this section with describing two involutions, related to the grading, reversion and conjugation involutions of Clifford algebras. Let $H$ be a hyperplane of $V$ and define $\tau_H:\mathfrak{C}(V,Q)\rightarrow \mathfrak{C}(V,Q)$ by linear expansion of $$\tau(v)=\begin{cases} v\ \mathrm{if}\ v\in H\\ -v\ \mathrm{if}\ v\not \in H.\\ \end{cases}$$ The second involution $\tau_Q:\mathfrak{C}(V,Q)\rightarrow \mathfrak{C}(V,Q)$ is defined as the linear expansion of $$\tau(v)=\begin{cases} v\ \mathrm{if}\ Q(v)=0\\ -v\ \mathrm{if}\ Q(v)=1.\\ \end{cases}$$ \begin{proposition}\label{auto} The involution $\tau_H$ is an automorphism of $\mathfrak{C}(V,Q)$. The involution $\tau_Q$ is an anti-automorphism of $\mathfrak{C}(V,Q)$. \end{proposition} \begin{proof} First consider $\tau_H$, where $H$ is a hyperplane of $V$. It suffices to check for $u,v\in V\setminus \{0\}$ that $\tau_H(uv)=\tau_H(u)\tau_H(v)$. As $H$ is a hyperplane, $\tau_H$ fixes either all three vectors $u,v,u\pluscirc v$ or negates two of them and, indeed, we find $\tau_H(uv)=\tau_H(u)\tau_H(v)$. To check that $\tau_Q$ is an anti-automorphism, we have to check $\tau_Q(uv)=\tau_Q(v)\tau_Q(u)$. If $Q(u)=Q(v)=0$, then $Q(u\pluscirc v)=0$ and $uv=vu$, or $Q(u\pluscirc v)=1$ and $uv=-vu$. In both cases $\tau_Q(uv)=\tau_Q(v)\tau_Q(u)$. If $Q(u)=Q(v)=1$, then $Q(u\pluscirc v)=0$ and $uv=vu$ or $Q(u\pluscirc v)=1$ and $uv=-vu$. Again, in both cases $\tau_Q(uv)=\tau_Q(v)\tau_Q(u)$. Finally, if $Q(u)=0$ and $Q(v)=1$ (or $Q(u)=1$ and $Q(v)=0$), then $Q(u\pluscirc v)=0$ and $uv=-vu$ or $Q(u\pluscirc v)=1$ and $uv=vu$. Also now we can check $\tau_Q(uv)=\tau_Q(v)\tau_Q(u)$. \end{proof} \begin{proposition} Let $\tau$ be a nontrivial linear map of $\mathfrak{C}(V,Q)$ mapping any $v\in V$ to $\pm v$. If $\tau$ is an automorphism of $\mathfrak{C}(V,Q)$, then $\tau=\tau_H$ for some hyperplane $H$ of $V$. If $\tau$ is an anti-automorphism of $\mathfrak{C}(V,Q)$, then $\tau=\tau_Q$ or $\tau_Q\tau_H$ for some hyperplane $H$ of $V$. \end{proposition} \begin{proof} First assume that $\tau$ is an automorphism. If $\tau$ negates two vectors $v,w\in V\setminus \{0\}$, then $v\pluscirc w$, should be fixed. So, the vectors in $V$ fixed by $\tau$ form a hyperplane $H$ of $V$ and $\tau=\tau_H$. Next, assume that $\tau$ is an anti-automorphism. The $\tau_Q\tau$ is an automorphism, and by the above, we either have $\tau=\tau_Q$ or $\tau_Q\tau_H$ for some hyperplane $H$ of $V$. \end{proof} \begin{remark}\label{transpose} The anti-automorphism $\tau_Q$ acts on the matrix algebras of Table \ref{isotable} by transposition followed by complex or quaternion conjugation (if applicable) on $\mathbb{F}[i]$ or $\mathbb{H}$, respectively. This can easily be checked in small dimensional cases, as described above, and hence on the tensor products. See also \cite{transpoI}. \end{remark} \section{Examples}\label{sect:examples} In this section we consider a few examples of algebras given by some black and white colored graph $\mathcal{G}amma$. We only consider cases where the ring $R=\mathbb{F}$ is a field and where the values of the vertices are $\pm 1$. Up to changing the colors of the vertices, we can assume the map $\Lambda$ to be the constant map $1$. When drawing a graph $\mathcal{G}amma$ we use the color gray for a vertex to indicate that we have not yet determined whether its color should be black or white. \begin{example}[Clifford algebras and graphs of type $A$] Let $\mathcal{G}amma$ be the complete graph on $n$ vertices with $p$ white vertices and $q$ black vertices. Then of course $\mathfrak{C}(\mathcal{G}amma)$ is isomorphic to the Clifford algebra $\mathrm{Cl}(p,q)$. Consider the corresponding quadratic space $(V,Q)=(V_\mathcal{G}amma,Q_\mathcal{G}amma)$ obtained from $\mathcal{G}amma$. Suppose the vectors $e_1,\dots, e_n\in V$ correspond to the vertices of $\mathcal{G}amma$, where $Q(e_i)=1$ for all $i$ with $1\leq i\leq q$. Then with $f_1=e_1,f_2=e_1\pluscirc e_2, f_3=e_2\pluscirc e_3,\dots, f_n=e_{n-1}\pluscirc e_n$ we find a spanning set for $V$ with corresponding graph of type $A_n$ as in Figure \ref{newgraphAa}. \begin{figure} \caption{Graph of type $A_n$ obtained by changing the generators.} \label{newgraphAa} \end{figure} All vertices are black, except for $f_{q+1}$, which is white. (If $q=n$, then all vertices are black, if $q=0$, only $f_1$ is white.) This implies that for $q\geq 1$ we find $\mathrm{Cl}(p,q)$ to be isomorphic to $\mathrm{Cl}(q-1,p+1)$. Just read the diagram from right to left. Now let $g_1=f_1$, and for $i$ with $2\leq 2i\leq n$ let $g_i=f_{2i}$ and $g_{i-1}=f_1 \pluscirc f_3 \pluscirc \dots \pluscirc f_{2i-1}.$ If $n$ is odd, then let $g_n=f_1\pluscirc f_n$. Then the graph on these vertices is given in Figure \ref{Angraph}. \begin{figure} \caption{The graphs for $n$ even (left) or odd (right).} \label{Angraph} \end{figure} First assume $n$ is even. Notice that $Q(g_{2i})=1$ for all $i$, except when $q\leq n$ is odd. Then $Q(g_{q+1})=0$. Moreover, $Q(g_{1})=1$ and $Q(g_{2i+1})$ is $i\pmod{2}$ if $q$ is odd. For even $q$ we find that $Q(g_{2i+1})$ is $i+1\pmod{2}$ if $2i+1\leq q$ and $i\pmod{2}$ for $2i+1>q$. For odd $n$, we find $Q(g_n)=Q(g_1)+Q(g_{n-1})$. From this information we can deduce the type of $Q$. In particular, we see that the type of $Q$ is multiplied with $-1$ if we raise $p$ or $q$ with $4$, and hence stays the same if we add $8$ to $q$ or $p$ (Bott-periodicity). Indeed, adding $4$ to $p$ or $q$ adds \noindent to the graph and multiplies the type of $Q$ with $-1$. For small values of $p$ and $q$ we have collected this information in Table \ref{Cliffordtable}. \begin{table} \begin{tabular}{|l||l|l|l|l|l|} \hline $p\setminus q$ & 0 & 1 & 2 & 3 & 4 \\ \hline\hline 0 & $+$ & 0 & $-$ &$-$ & $-$ \\ 1 & $+$ & $+$ & 0 &$-$ & $-$ \\ 2 & $+$ & $+$ & $+$ &$0$ & $-$ \\ 3 & $0$ & $+$ & $+$ &$+$ & $0$ \\ 4 & $-$ & $0$ & $+$ &$+$ & $+$ \\ \hline \end{tabular} \caption{Type of $Q$ for small values of $p+q$.} \label{Cliffordtable} \end{table} Using the results of Table \ref{Cliffordtable} and the above information, we find in Table \ref{IsoCliffordTable} the isomorphism type of the Clifford algebras over fields $\mathbb{F}$ of type III. \begin{table} \begin{tabular}{|l|l|l|} \hline $p-q\pmod{8}$ & Type $Q$ & $\mathrm{Cl}(p,q)$\\ \hline\hline 0,2 & $+$ & $M(2^{\frac{p+q}{2}},\mathbb{F})$\\ 4,6 & $-$ & $M(2^{\frac{p+q-2}{2}},\mathbb{H})$\\ 1 & $+$ & $M(2^{\frac{p+q-1}{2}},\mathbb{F})^2$\\ 3,7 & $0$ & $M(2^{\frac{p+q-1}{2}},\mathbb{F}[i])$\\ 5 & $-$ & $M(2^{\frac{p+q-3}{2}},\mathbb{H})^2$\\ \hline \end{tabular} \caption{Isomorphism type of the Clifford algebras.} \label{IsoCliffordTable} \end{table} We notice that the above also classifies the algebras $\mathfrak{C}(\mathcal{G}amma)$ where $\mathcal{G}amma$ is a graph of type $A_n$ as in Figure \ref{Graph_An}, since we can replace the vertices $f_i$ by $e_i$, i.e., by reversing the above described process, and end up with a complete graph. In particular, we find that we only have to consider those graphs of type $A_n$ in which at most one vertex is white. \begin{figure} \caption{Graph of type $A_n$.} \label{Graph_An} \end{figure} \end{example} \begin{example}[Graphs of type $D$] Next we consider graphs of type $D_n$, where $n\geq 4$. See Figure \ref{Graph_Dn}. \begin{figure} \caption{Graphs of type $D_n$.} \label{Graph_Dn} \end{figure} To classify the corresponding algebras we only have to consider the cases where at most one of the vertices $2,\dots, n$ is white. Moreover, we notice that $e_1\pluscirc e_2$ is an element which is in the radical of the form $f$ induced on $V=\langle e_1,\dots, e_n\rangle$. If both the vertices $1$ and $2$ are black or both are white, we find $e_1\pluscirc e_2$ to be in the radical of $Q_\mathcal{G}amma$ and $\mathfrak{C}(\mathcal{G}amma)$ is the direct product $\mathfrak{C}(\mathcal{G}amma_1)\times \mathfrak{C}(\mathcal{G}amma_1)$, where $\mathcal{G}amma_1$ is obtained from $\mathcal{G}amma$ by deleting vertex $1$. If only one of the two vertices $1$ and $2$ is black, then $Q_\mathcal{G}amma(e_1+e_2)=1$ and we find $\mathfrak{C}(\mathcal{G}amma)$ to be isomorphic to $\mathfrak{C}(\mathcal{G}amma_1)\otimes \mathbb{F}[i]$. \end{example} \begin{example}[Graphs of type $E$] Let $\mathcal{G}amma$ be a graph of type $E_n$, where $n\geq 1$ as in Figure \ref{En}. \begin{figure} \caption{Graphs of type $E_n$.} \label{En} \end{figure} Assume that all vertices are colored black. Consider the quadratic form on $V=\mathbb{F}^{n}$ given by $$Q(x_1,\dots,x_n)=(\sum_{i=1}^n\ x_i^2)+x_2x_4+x_1x_3+x_3x_4+\dots +x_{n-1}x_n.$$ Then $Q(e_i)=1$ and $f(e_i,e_j)=1$ if and only if $i$ is adjacent to $j$. So, $\mathfrak{C}(\mathcal{G}amma)$ is isomorphic to $\mathfrak{C}(V,Q)$. For $n\geq 4$ even, we find that we can split $V$ into the orthogonal sum of the spaces $$\langle e_1,e_3\rangle\perp \langle e_2, e_2 \pluscirc e_4 \pluscirc \dots \pluscirc e_{n}\rangle \perp \langle e_5,e_6\rangle\perp \langle e_8, e_5\pluscirc e_7\rangle\perp\dots\perp \langle e_n, e_5 \pluscirc e_7 \pluscirc\dots \pluscirc e_{n-1}\rangle.$$ Such a $2$-dimensional space is of $+$ type if the second generator is of even weight, and of $-$ type if the second generator is of odd weight. So we find $Q$ to be of $+$-type if $n=0,2\pmod{8}$ and of $-$-type for $n=4,6\pmod{8}$. For $n\geq 5$ odd we find the vector $e_2\pluscirc e_5$ (for $n=5$) or $e_2 \pluscirc e_5 \pluscirc e_7\pluscirc e_9 \pluscirc\cdots \pluscirc e_n$ (for $n\geq 9$) to span the radical of $f$, the bilinear form associated to $Q$. This vector is isotropic if and only if $n=1\pmod{4}$. It remains to find the type of the form induced on $V/\mathrm{Rad}(f)$ in case $n=1\pmod{4}$. As modulo $e_2 \pluscirc e_5 \pluscirc e_7 \pluscirc e_9\pluscirc \cdots \pluscirc e_n$, we find that $e_2$ is in the subspace spanned by $e_1,e_3,e_4, \dots, e_n$, the type of $Q$ is determined by the type of $Q$ restricted to this subspace. As above we find that this is of $+$-type if $n-1=0,2\pmod{8}$ and of $-$-type if $n-1=4,6\pmod{8}$. So, also for graphs $E_n$ we find Bott-periodicity. The information is summarized in \cref{EnCliffordTable}. \begin{table} \begin{tabular}{|l|l|l|l|l|} \hline $\dim(V)$ & Type $Q$&Type of $\mathbb{F}$&$\mathfrak{C}(\mathcal{G}amma)$\\ \hline\hline $n=0\pmod{8}$ &$+$&I, II, III& $M(2,\mathbb{F})^{\otimes\frac{n}{2}}$\\ $n=1\pmod{8}$ &$+$ &I, II, III& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}})^2$\\ $n=2\pmod{8}$ &$+$&I, II, III& $M(2,\mathbb{F})^{\otimes\frac{n}{2}}$\\ $n=3\pmod{8}$ &$0$ &I& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}})^2$\\ $n=3\pmod{8}$ &$0$ &II,III& $M(2,\mathbb{F})^{\otimes\frac{n-1}{2}}\otimes\mathbb{F}[i]$\\ $n=4\pmod{8}$ &$-$ &I,II& $M(2,\mathbb{F})^{\otimes\frac{n}{2}}$\\ $n=4\pmod{8}$ &$-$ &III&$M(2,\mathbb{F})^{\otimes\frac{n-2}{2}}\otimes \mathbb{H}$ \\ $n=5\pmod{8}$ &$-$ &I, II& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^2$\\ $n=5\pmod{8}$ &$-$ &III&$(M(2,\mathbb{F})^{\otimes\frac{n-3}{2}}\otimes \mathbb{H})^2$ \\ $n=6\pmod{8}$ &$-$ &I, II& $M(2,\mathbb{F})^{\otimes\frac{n}{2}}$\\ $n=6\pmod{8}$ &$-$ &III& $M(2,\mathbb{F})^{\otimes\frac{n-2}{2}}\otimes \mathbb{H}$\\ $n=7\pmod{8}$ &$0 $ &I& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}})^2$\\ $n=7\pmod{8}$ &$0 $ &II,III& $M(2,\mathbb{F})^{\otimes\frac{n-1}{2}}\otimes\mathbb{F}[i]$\\ \hline \end{tabular} \caption{Algebras $\mathfrak{C}(E_n)$.} \label{EnCliffordTable} \end{table} \end{example} \begin{algorithm}\label{cliffordalgorithm} In this section we have seen three examples on how to identify the algebra $\mathfrak{C}(\mathcal{G}amma)$ from the graph $\mathcal{G}amma$. The described method can be turned into an algorithm, which consists of the following steps: \begin{enumerate}[\rm(a)] \item Apply a (modified) Gram-Schmidt procedure to decompose $V_\mathcal{G}amma$ into an orthogonal sum of nondegenerate $2$-dimensional spaces and $1$-dimensional spaces. \item Determine the type of $Q_\mathcal{G}amma$ by taking the product of the types of the nondegenerate $2$-dimensional spaces and $1$-dimensional spaces from step (a) on which $Q_\mathcal{G}amma$ is nontrivial. \item Determine the isomorphism type of $\mathfrak{C}(\mathcal{G}amma)$ using the type of $Q_\mathcal{G}amma$ as computed in step (b) and \cref{isotable}. \end{enumerate} If $n$ denotes the number of vertices of $\mathcal{G}amma$, then this algorithm has complexity of order $n^3$, as the Gram-Schmidt procedure has complexity of order $n^3$. \end{algorithm} \section{Lie algebras} \label{sect:lie} We continue with the notation of the previous sections. Consider the algebra $\mathfrak{C}(V,Q,\Lambda)$ as in Section \ref{sect:twisted}, where $(V,Q)$ is a quadratic space over the field $\mathbb{F}_2$ and $\Lambda:V\times V\rightarrow R^*$ is defined as in Section \ref{sect:twisted}. Then $\mathfrak{C}(V,Q,\Lambda)$ is an associative algebra and we can consider the associated Lie algebra, where the Lie bracket is defined by the linear expansion of $$\begin{array}{ll} [u,v]&=\frac{1}{2}(uv-vu)\\ &=\frac{1}{2}((-1)^{g(u,v)}-(-1)^{g(v,u)}))\Lambda(u,v) \cdot\ u\pluscirc v \\ &=-f(u,v)\displaystyle\Lambda(u,v) (u\pluscirc v),\\ \end{array}$$ for all $u,v\in{V}$. Here $g$ is a bilinear form with $Q(v)=g(v,v)$ for $v\in V$, and $f(u,v)=g(u,v)+g(v,u)$ the corresponding alternating form defined by $Q$. Notice that we identify the values of $f(u,v)\in \mathbb{F}_2$ with $0$ and $1$ in $R$. This Lie algebra does depend only on the symplectic space $(V,f)$ and the map $\Lambda$, and can actually be defined for any symplectic space $(V,f)$, even if $2$ is not invertible in $R^*$. We denote this Lie algebra by $\mathfrak{g}(V,f,\Lambda)$. As the elements of $V$ form a basis for $\mathfrak{C}(V,Q,\Lambda)$, they also form a basis for $\mathfrak{g}(V,f,\Lambda)$. Elements $u,w\in {V}$ satisfy the following relations in $\mathfrak{g}(V,f,\Lambda)$: $$[u,[u,w]] =-f(u,w)\Lambda(u,w)\Lambda(u,u\pluscirc w) w=-f(u,w)\Lambda(u,u) w.$$ Clearly, the element $\mathbf{1}$ is in the center of this Lie algebra, but so are all elements $u\in {V}$ that are in the radical of $f$. We now concentrate on the case where $R$ is a field $\mathbb{F}$ of characteristic $\neq 2$, and $\Lambda(u,v)=1$ for all $u,v\in V$. In this case we write $\mathfrak{g}(V,f)$ for $\mathfrak{g}(V,f,\Lambda)$. If $u,v\in V$ with $r_0:=u\pluscirc v$ in the radical of $f$, we find $$\begin{array}{ll} [u+v,w]&=[u,w]+[v,w]\\ &=-f(u,w) (u\pluscirc w) -f(v,w)(v\pluscirc w)\\ &=-f(u,w)((u\pluscirc w) + (v\pluscirc w)).\\ \end{array}$$ As $(u\pluscirc w) \pluscirc (v\pluscirc w)=u\pluscirc v=r_0$, we find that the linear span of the elements $u+v$, where $v=u\pluscirc r_0$, is an ideal of $\mathfrak{g}(V,f,\Lambda)$, which we denote by $\mathfrak{I}_{r_0}^+$. Similarly we find $$[u-v,w]=-f(u,w)(u\pluscirc w - v\pluscirc w)$$ so that $\mathfrak{I}_{r_0}^-$, the linear span of the elements $u-v$, where $v=u\pluscirc r_0$ is also an ideal. This implies the following. \begin{proposition}\label{liedecomposition} Let $0\neq r_0\in {\rm Rad}(f)$, then $\mathfrak{g}(V,f)=\mathfrak{I}_{r_0}^+\oplus \mathfrak{I}_{r_0}^-$. Moreover, $\mathfrak{g}/\mathfrak{I}_r^+$ is isomorphic to $\mathfrak{g}(\overline{V},\overline{f})$, where $(\overline{V},\overline{f})$ is the quotient space of $(V,f)$ modulo $\langle r_0\rangle$. \end{proposition} Using the above proposition and the information in \cref{isotable}, we can deduce the isomorphism types of the Lie algebras $\mathfrak{g}(V,f)$ obtained from the various algebras $\mathfrak{C}(V,Q)$. This information can be found in \cref{isoLietable}. Here $r$ denotes the dimension of the radical and $(\overline{V},\overline{Q})$ is obtained from $(V,Q)$ by taking the quotient modulo the radical of $Q$. \begin{table}[h] {\small \begin{tabular}{|l|l|l|l|l|l|} \hline $\mathrm{dim}(\overline{V})$ & $\overline{Q}$ &$\mathbb{F}$&$\mathfrak{C}(V,Q)$ &$\mathfrak{g}(V,f)$&$\mathfrak{g}(V,Q)$\\ \hline\hline $n=0 \pmod{2}$& $+$ & I& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$& $\mathfrak{gl}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$ &$\mathfrak{so}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$\\ $n=0 \pmod{2}$& $-$ & I& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$&$\mathfrak{gl}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$ &$\mathfrak{sp}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$\\ $n=1 \pmod{2}$& $0$ & I& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}})^{2^{r}}$&$\mathfrak{gl}(2^{\frac{n-1}{2}},\mathbb{F})^{2^r}$&$\mathfrak{sl}(2^{\frac{n-1}{2}},\mathbb{F})^{2^r}$ \\ $n=0 \pmod{2}$& $+$ & II& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$&$\mathfrak{gl}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$&$\mathfrak{so}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$ \\ $n=0 \pmod{2}$& $-$ & II& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$&$\mathfrak{gl}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$&$\mathfrak{so}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$ \\ $n=1 \pmod{2}$& $0$ & II& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}}\otimes\mathbb{F}[i])^{2^{r-1}}$&$\mathfrak{gl}(2^{\frac{n-1}{2}},\mathbb{F}[i])^{2^{r-1}}$ &$\mathfrak{su}(2^{\frac{n-1}{2}},\mathbb{F}[i])^{2^{r-1}}$\\ $n=0 \pmod{2}$& $+$ & III& $(M(2,\mathbb{F})^{\otimes\frac{n}{2}})^{2^{r}}$& $\mathfrak{gl}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$&$\mathfrak{so}(2^{\frac{n}{2}},\mathbb{F})^{2^r}$\\ $n=0 \pmod{2}$& $-$ & III& $(M(2,\mathbb{F})^{\otimes\frac{n-2}{2}}\otimes \mathbb{H})^{2^{r}}$&$\mathfrak{gl}(2^{\frac{n-2}{2}},\mathbb{H})^{2^r}$&$\mathfrak{sp}(2^{\frac{n-2}{2}},\mathbb{H})^{2^r}$\\ $n=1 \pmod{2}$& $0$ & III& $(M(2,\mathbb{F})^{\otimes\frac{n-1}{2}}\otimes\mathbb{F}[i])^{2^{r-1}}$&$\mathfrak{gl}(2^{\frac{n-1}{2}},\mathbb{F}[i])^{2^{r-1}}$&$\mathfrak{su}(2^{\frac{n}{2}},\mathbb{F}[i])^{2^{r-1}}$\\ \hline \end{tabular} } \caption{The isomorphism types of the Lie algebras $\mathfrak{g}(V,f)$ and $\mathfrak{g}(V,Q)$. }\label{isoLietable} \end{table} Although the Lie algebra $\mathfrak{g}$ of the algebra $\mathfrak{C}(V,Q)$ only depends on the symplectic form $f$ but not on $Q$, it does contain a Lie subalgebra that is related to $Q$, and in fact is the centralizer of $-\tau_Q$. \begin{proposition}\label{automorphism} Let $H$ be a hyperplane of $V$. Then $\tau_H$ and $-\tau_Q$ are automorphisms of $\mathfrak{g}$. \end{proposition} \begin{proof} By \ref{auto} we find $\tau_H$ to be an automorphism. So, we consider $-\tau_Q$. Let $u,v\in V$, then $$\begin{array}{ll} -\tau_Q([u,v])&=-\tau_Q(uv-vu)\\ &=-(\tau_Q(v)\tau_Q(u)-\tau_Q(u)\tau_Q(v))\\ &=-[\tau_Q(v),\tau_Q(u)]\\ &=[\tau_Q(u),\tau_Q(v)]\\ &=[-\tau_Q(u),-\tau_Q(v)]. \end{array}$$ \end{proof} The centralizer in $\mathfrak{g}(V,f)$ of an automorphism $\sigma$ is a Lie subalgebra, which we denote by $\mathfrak{g}_\sigma(V,f)$. Clearly $\mathfrak{g}_{\tau_H}(V,f)$ is isomorphic to $\mathfrak{g}(H,f_{\mid H})$. The subalgebra $\mathfrak{g}_{-\tau_Q}(V,f)$ depends on $Q$ and therefore is also denoted by $\mathfrak{g}(V,Q)$. It is the linear span of the set $\{ v\in V\mid Q(v)=1\}$ of non-isotropic vectors in $V$ inside $\mathfrak{g}(V,f)$. The isomorphism types of these subalgebras can also be found in Table \ref{isoLietable}. They can be deduced using the description of the matrix algebras as given in Section \ref{sect:quadric} and Remark \ref{transpose}. \begin{remark}\label{Shirokov} If we fix a hyperplane $H$ of $V$, then the group $\langle -\tau_Q,\tau_H\rangle$ is elementary abelian of order $2^2$. The Lie algebra $\mathfrak{g}(V,f)$ can be decomposed as $$\mathfrak{g}(V,f)=\mathfrak{g}_{1,1}\oplus \mathfrak{g}_{1,-1}\oplus\mathfrak{g}_{-1,1}\oplus\mathfrak{g}_{-1,-1},$$ where $\mathfrak{g}_{i,j}$ for $i,j=\pm 1$ denotes the intersection of the $i$-eigenspace of $-\tau_Q$ and $j$-eigenspace of $\tau_H$. Notice that for $i,j,k,l=\pm 1$ we have $$[\mathfrak{g}_{i,j},\mathfrak{g}_{k,l}]\subseteq \mathfrak{g}_{ik,jl}.$$ So, we find in $\mathfrak{g}(V,f)$ Lie subalgebras $\mathfrak{g}_{1,1}\oplus \mathfrak{g}_{1,-1}$, $\mathfrak{g}_{1,1}\oplus \mathfrak{g}_{-1,1}$ and $\mathfrak{g}_{1,1}\oplus \mathfrak{g}_{-1,-1}$, which are just the centralizers of the involutions $-\tau_Q,\tau_H$ and $-\tau_Q\tau_H$ in $\langle -\tau_Q,\tau_H\rangle$. Notice that $\tau_Q\tau_H=\tau_{Q'}$ where $Q'$ is the quadratic form defined by $Q'(v)=Q(v)+\phi_H(v)$ for all $v\in V$, with $\phi_H$ being the linear form on $V$ with kernel equal to $H$. The form $Q'$ has also $f$ as its associated symplectic form. These decompositions and the corresponding Lie subalgebras are investigated by Shirokov in \cite{Shirokov15,Shirokov16,Shirokov18} in case we are dealing with a real Clifford algebra. Actually, several results of \cite{Shirokov15,Shirokov16,Shirokov18} follow directly from the above considerations and \cref{isoLietable}. When $\mathbb{F}$ is a field of type III, one can also consider the $\mathbb{F}$-Lie subalgebras $$\mathfrak{g}_{1,1}\oplus i\mathfrak{g}_{k,l}$$ (where $k,l=\pm 1$) of the Lie algebra $\mathfrak{g}(V,Q)$ defined over $\mathbb{F}[i]$ with $i^2=-1$. See also \cite{Shirokov15,Shirokov16,Shirokov18}. \end{remark} \section{Lie algebras obtained from graphs} Let $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ be a black and white colored graph with all labels equal to $1$. Then let $\mathfrak{g}(\mathcal{G}amma)$ be the Lie algebra of $\mathfrak{C}(\mathcal{G}amma)$. The vertices in $\mathcal{V}$ do generate $\mathfrak{C}(\mathcal{G}amma)$, but need not generate the Lie algebra $\mathfrak{g}(\mathcal{G}amma)$. In this section we provide a characterization of the Lie algebras $\mathfrak{g}(\mathcal{G}amma)$ and its subalgebra generated by the vertices of $\mathcal{G}amma$. So, consider a connected black and white colored graph $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ and consider the Lie algebra $\mathfrak{g}(\mathcal{G}amma)$ over a field $\mathbb{F}$, with characteristic different from $2$. As in the previous sections we identify $\mathfrak{C}(\mathcal{G}amma)$ with the algebra $\mathfrak{C}(V_\mathcal{G}amma,Q_\mathcal{G}amma)$. By $f_\mathcal{G}amma$ we denote the bilinear form associated to $Q_\mathcal{G}amma$. The Lie subalgebra of $\mathfrak{g}(\mathcal{G}amma)$ generated by the vertices of $\mathcal{G}amma$ will be studied with the help of the geometry of $(V_\mathcal{G}amma,Q_\mathcal{G}amma)$ and $(V_\mathcal{G}amma,f_\mathcal{G}amma)$. We denote this subalgebra by $\mathfrak{K}(\mathcal{G}amma)$. Notice that the coloring of the vertices of $\mathcal{G}amma$ has no effect on the isomorphism type of this Lie algebra. So, from now on we assume that all vertices are black. Let $(V,Q)$ be a quadratic space over $\mathbb{F}_2$ with addition $\pluscirc$. If $v\neq w\in V$ are nonzero vectors with $Q(v)=Q(w)=f(v,w)=1$, then we call the $2$-dimensional subspace $\langle v,w\rangle$ an \emph{elliptic line} of $(V,Q)$. We identify this $2$-space with the set of three nonzero vectors $\{v,w,v\pluscirc w\}$ contained in it. By $\Pi(V,Q)$ we denote the partial linear space $(P,L)$ where $P$ consists of all the vectors $v$ of $V\setminus {\rm Rad}(f)$ with $Q(v)=1$ and whose lines in $L$ are the elliptic lines. (Notice that a vector $v$ with $Q(v)=1$ but $v\in {\rm Rad}(f)$ is not in $P$.) It is a so-called \emph{cotriangular space}, having the property that for each point $p$ and line $\ell$ not on $p$, the point $p$ is collinear to $0$ or all but one of the points of $\ell$. A \emph{subspace} of $\Pi(V,Q)$ is a subset $S$ of the point set of $\Pi$ such that each line meeting $S$ in two points is contained in $S$. A subspace $S$ is often identified with the partial linear space $(S, \{\ell\in L\mid \ell\subseteq S\})$. As the intersection of subspaces is again a subspace, we can define the subspace generated by a subset $X$ of $P$ to be the intersection of all subspaces containing $X$. Cotriangular spaces (and their subspaces) have been studied by several authors, see for example \cite{Hall,Shult, Seidel}. Their connection with Lie algebras has been considered in \cite{cotriangleLie,Erik}. Notice that $\mathcal V$ is a basis for $V_\mathcal{G}amma$ and $\mathcal{G}amma$ is connected. Then the subspace of $\Pi(V_\mathcal{G}amma,Q_\mathcal{G}amma)$ generated by $\mathcal V$ is denote by $\Pi(\mathcal{G}amma)$. \begin{proposition} Let $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ be a connected graph. The subspace ${\Pi}(\mathcal{G}amma)$ of $\Pi(V_\mathcal{G}amma,Q_\mathcal{G}amma)$ is a basis for $\mathfrak{K}(\mathcal{G}amma)$. \end{proposition} \begin{proof} This follows immediately from the following observation: if $v,w\in V_\mathcal{G}amma$ are collinear points in $\Pi(\mathcal{G}amma)$, then $Q_\mathcal{G}amma(v)=Q_\mathcal{G}amma(w)=1=f_\mathcal{G}amma(v,w)$. So $Q(v\pluscirc w)=1$ and $v\pluscirc w$ is a point of $\Pi(\mathcal{G}amma)$ and $[v,w]=\pm v\pluscirc w$. If $v,w$ are not collinear, then $[v,w]=0$. \end{proof} Let $\Pi=(P,L)$ be an arbitrary cotriangular space with point set $P$ and set of lines $L$. Then on $P$ we can define an equivalence relation $\sim$, where two points $p,q\in P$ are equivalent if and only if the set of points collinear with but different from $p$ coincides with the set of points collinear with, but different from $q$. Notice that two points that are collinear, are never equivalent. Now for each line $\ell\in L$ we can consider $\overline{\ell}$ to be the set of three equivalence classes of the points on $\ell$. If $\overline{P}$ denotes the equivalence classes of $P$ and $\overline{L}$ the set $\{\overline{\ell}\mid \ell\in L\}$, then $\overline{\Pi}=(\overline{P},\overline{L})$ is also a cotriangular space. Moreover, it is \emph{reduced}, meaning that no two distinct points are $\sim$-equivalent. If $\mathcal{V}$ is a subset of $P$ and $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ the graph with vertex set $\mathcal V$ and two vertices $v,w\in \mathcal V$ adjacent if and only if $f(v,w)=1$, then $\overline\mathcal{G}amma$ denotes the graph with vertices the $\sim$-equivalence classes of the vertices in $\mathcal V$ and two classes adjacent if and only if there are vertices adjacent vertices in these classes. Besides the cotriangular spaces obtained from the elliptic lines of a quadratic space over the field $\mathbb{F}_2$, there is a second class of examples. Let $\Omega$ be a finite set and $P$ be the set of unordered pairs of elements from $\Omega$. As lines we take the triples of points contained in any subset of $\Omega$ of size $3$. This space will be denoted as $\mathcal{T}(\Omega)$. As follows from the work of Hall \cite{Hall}, cotriangular spaces come only in these two types: \begin{theorem}\cite{Hall} Let $\Pi$ be a connected and reduced cotriangular space. Then up to isomorphism $\Pi$ is one of the following. \begin{enumerate}[(a)] \item The geometry $\Pi(V,Q)$ of elliptic lines in an orthogonal space $(V,Q)$ over $\mathbb{F}_2$, where the radical of $Q$ is $\{0\}$. \item The geometry $\mathcal{T}(\Omega)$ for some set $\Omega$. \end{enumerate} \end{theorem} Hall also determined how these spaces can embed in each other. In particular, he has proven the following result. \begin{theorem}\cite{Hall}\label{subspacethm} Let $(V,Q)$ be an orthogonal space over $\mathbb{F}_2$, where ${\rm Rad}(Q)=\{0\}$ Let $\Pi$ be a proper connected subspace of $\Pi(V,Q)$, where ${\rm Rad}(Q)=\{0\}$. Then either there is a proper subspace $U$ of $V$ such that the points of $\Pi$ are in $P\cap U$, or $\Pi$ is isomorphic to $\mathcal{T}(\Omega)$ for some set $\Omega$. Moreover, in the latter case, $V$ can be identified with the vector subspace of $\mathbb{F}_2\Omega$ of even weight vectors, and $Q$ takes the value $1$ on all weight $2$ vectors. \end{theorem} \begin{corollary}\label{subspacecor} Let $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ be a connected graph. Then either $\Pi(\overline\mathcal{G}amma)=\overline{\Pi(V_\mathcal{G}amma,Q_\mathcal{G}amma)}$, or $\overline\mathcal{G}amma$ is a line graph and $\Pi(\overline\mathcal{G}amma)$ isomorphic to $\mathcal{T}(\Omega)$ for some set $\Omega$. \end{corollary} \begin{proof} Suppose $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ is a connected graph. As we can identify $\overline{\Pi(V_\mathcal{G}amma,Q_{\mathcal{G}amma})}$ with $\Pi(V_{\overline\mathcal{G}amma},Q_{\overline\mathcal{G}amma})$, we can assume $\mathcal{G}amma=\overline{\mathcal{G}amma}$. If $\Pi(\mathcal{G}amma)\neq\Pi(V_{\mathcal{G}amma},Q_{\mathcal{G}amma})$, then, as the vertices in ${\mathcal{G}amma}$ linearly span $V_\mathcal{G}amma$, the above \cref{subspacethm} can be applied to find $\Pi(\mathcal{G}amma)$ to be isomorphic to $\mathcal{T}(\Omega)$ for some set $\Omega$. But then $\mathcal{G}amma$ is a line graph of a graph with vertex set $\Omega$. \end{proof} We use the above theorem and its corollary to determine when $\mathfrak{K}(\mathcal{G}amma)$ and $\mathfrak{g}(\mathcal{G}amma)$ do or do not coincide. In order to describe the Lie algebras thus obtained we need to introduce one more class of Lie algebras connected to the cotriangular spaces $\mathcal{T}(\Omega)$. So, let $\Omega$ be a set and $\mathcal{T}(\Omega)$ the corresponding cotriangular space. Then the points of $\mathcal{T}(\Omega)$ can be identified with the vectors of weight $2$ in the $\mathbb{F}_2$ vector space $\mathbb{F}_2\Omega$ with the elements of $\Omega$ as basis and addition $\pluscirc$. On the space $\mathbb{F}_2\Omega$ we can define a quadratic form $Q$ by $Q(\omega)=0$ and $Q(\omega\pluscirc \omega')=1$ for all distinct $\omega,\omega'\in \Omega$. Then, consider $\mathfrak{C}(\mathbb{F}_2\Omega,Q)$ and in its Lie algebra $\mathfrak{g}(\mathbb{F}_2\Omega,Q)$ the subalgebra $\mathfrak{g}(\Omega)$ spanned by the weight two vectors. For two weight two vectors $\omega_1\pluscirc \omega_2$ and $\omega_3\pluscirc \omega_4$ we have $$ [\omega_1\pluscirc \omega_2,\omega_3\pluscirc \omega_4] =-f(\omega_1\pluscirc \omega_2,\omega_3\pluscirc \omega_4)\omega_1\pluscirc \omega_2\pluscirc\omega_3\pluscirc \omega_4,$$ where $f$ is the bilinear form associated to $Q$. This is equal to $0$ if $\omega_1\pluscirc \omega_2=\omega_3\pluscirc\omega_4$ or $\omega_1,\dots,\omega_4$ are all distinct, and $- \omega_2\pluscirc \omega_3$ if $\omega_1,\omega_2,\omega_3$ are distinct, and $\omega_4=\omega_2$. So indeed, $\mathfrak{g}(\Omega)$ is a Lie subalgebra. We can identify the Lie algebra $\mathfrak{g}(\Omega)$ with a Lie subalgebra of $\mathfrak{gl}(\mathbb{F}\Omega)$. Indeed, an element $\omega_1\pluscirc\omega_2$, where $\omega_1,\omega_2$ are distinct element from $\Omega$ acts linearly on $\mathbb{F}\Omega$ as $\epsilon_{\omega_1\pluscirc\omega_2}$, which is defined by $$\mathrm{\epsilon}_{\omega_1\pluscirc\omega_2}(\omega)=f(\omega_1\pluscirc\omega_2,\omega)(-1)^{g(\omega_1\pluscirc\omega_2,\omega)} \omega_1\pluscirc\omega_2\pluscirc\omega_3\\ $$ for all $\omega_3\in \Omega$. Here $g$ is a bilinear form with $g(v,v)=Q(v)$ for all $v\in \mathbb{F}_2\Omega$. So, $\mathrm{\epsilon}_{\omega_1\pluscirc\omega_2}(\omega_1)=\pm\omega_2$ and $\mathrm{\epsilon}_{\omega_1\pluscirc\omega_2}(\omega_2)=\mp\omega_1$, while $\mathrm{\epsilon}_{\omega_1\pluscirc\omega_2}(\omega)=0$ for $\omega\in \Omega$ different from $\omega_1,\omega_2$. One easily checks that $\epsilon$ maps $\mathfrak{g}(\Omega)$ to the Lie algebra of finitary anti-symmetric linear maps in $\mathfrak{gl}(\mathbb{F}\Omega)$. In particular, if $|\Omega|=n$ is finite, then $\mathfrak{g}(\Omega)$ is isomorphic to $\mathfrak{so}(n,\mathbb{F})$. \begin{theorem}\label{graphliealgebrathm} Suppose $\mathcal{G}amma$ is a connected graph and all its vertices are black. If ${\overline\mathcal{G}amma}$ is not a line graph, then $\mathfrak{K}(\mathcal{G}amma)$ admits a quotient isomorphic to $\mathfrak{g}(\overline\mathcal{G}amma)$. If $\overline\mathcal{G}amma$ is a line graph, then $\Pi(\overline\mathcal{G}amma)\simeq \mathcal{T}(\Omega)$ for some set $\Omega$ and $\mathfrak{K}(\mathcal{G}amma)$ admits a quotient isomorphic to $\mathfrak{g}(\Omega)$. \end{theorem} \begin{proof} The elements of $\mathcal{G}amma$ generate a subalgebra $\mathfrak{K}(\mathcal{G}amma)$ of $\mathfrak{g}(\mathcal{G}amma)$. Clearly if, $u,v\in V_\mathcal{G}amma$ are in $\mathfrak{K}(\mathcal{G}amma)$, then so is $[u,v]$. This implies that the elements of $V_\mathcal{G}amma$ that are contained in $\mathfrak{K}(\mathcal{G}amma)$ form a subspace $S$ of the the geometry $\Pi:=\Pi(V_\mathcal{G}amma,Q_\mathcal{G}amma)$. Now let $R$ be the radical of $Q_\mathcal{G}amma$ on $V_\mathcal{G}amma$. For points $p,q$ of $\Pi$ we have $p\equiv q$ if and only if $p+q\in R$. As factoring out the radical of $Q_\mathcal{G}amma$ also implies taking a quotient of $\mathfrak{K}(\mathcal{G}amma)$, we find that $\mathfrak{K}(\mathcal{G}amma)$ admits a quotient isomorphic to $\mathfrak{K}(\overline\mathcal{G}amma)$. Moreover, $S$ is mapped to a subspace $\overline{S}$ of $\overline\Pi$. If $\overline{\mathcal{G}amma}$ is not a line graph, then, by \cref{subspacecor}, this subspace $\overline{S}$ is the full cotriangular space $\Pi$, and $\mathfrak{K}(\overline\mathcal{G}amma)= \mathfrak{g}(\overline\mathcal{G}amma)$. If $\overline\mathcal{G}amma$ is a line graph of a graph with vertex set $\Omega$, then its vertices can be identified with pairs from $\Omega$, and we find $\overline S$ to be isomorphic to $\mathcal{T}(\Omega)$. But then $\mathfrak{K}(\mathcal{G}amma)$ admits a quotient isomorphic to $\mathfrak{g}(\Omega)$. \end{proof} We notice that due to Beineke's characterization of line graphs, see \cite{Beineke}, we can conclude that $\overline{\mathcal{G}amma}$ is not a line graph if it contains an induced subgraph $\Delta$ which is one of the nine graphs from Figure \ref{forbidden}. The three graphs on the first row of Figure \ref{forbidden} are not reduced, while the others are. So, if $\Delta$ is one of these three graphs contained as an induced subgraph in some reduced graph $\mathcal{G}amma$, then $\mathcal{G}amma$ contains a vertex distinguishing the vertices that have in $\Delta$ the same set of neighbors. So, in $\mathcal{G}amma$ we find two vertices if $\Delta$ is the first graph and one vertex in case $\Delta$ is the second or third graph, such that adding these vertices to $\Delta$ we obtain a reduced graph. This implies that $\mathcal{G}amma$ contains a reduced connected subgraph $\mathcal{G}amma_0$ on $6$ vertices which is not a line graph. In particular, if we determine the quadratic space $(V_{\mathcal{G}amma_0},Q_{\mathcal{G}amma_0})$ for this subgraph, then this is a nondegenerate orthogonal $\mathbb{F}_2$-space of $+$-or $-$-type. But, if $(V_{\mathcal{G}amma_0},Q_{\mathcal{G}amma_0})$ is of $+$-type, then its cotriangular space $\Pi(V_{\mathcal{G}amma_0},Q_{\mathcal{G}amma_0})$ is isomorphic to $\mathcal{T}(\Omega)$, where $\Omega$ is of size $8$, contradicting that $\mathcal{G}amma_0$ is not a line graph. We have proven the following. \begin{proposition}\label{6graph} Suppose $\mathcal{G}amma$ is a connected graph such that $\overline{\mathcal{G}amma}$ is not a line graph. Then $\mathcal{G}amma$ contains a subgraph $\mathcal{G}amma_0$ on $6$ vertices spanning a nondegenerate $6$-dimensional orthogonal $\mathbb{F}_2$ space $(V_{\mathcal{G}amma_0},Q_{\mathcal{G}amma_0})$ of $-$-type. \end{proposition} \begin{figure} \caption{The nine forbidden subgraphs for a line graph.} \label{forbidden} \end{figure} \begin{corollary}\label{6graphcor} Suppose $\mathcal{G}amma$ is a connected graph such that $\overline{\mathcal{G}amma}$ is not a line graph. Then $\mathfrak{K}(\mathcal{G}amma)$, defined over a field $\mathbb{F}$ of odd characteristic, contains a subalgebra isomorphic to $\mathfrak{sp}(4,\mathbb{H})$, where $\mathbb{H}$ is a quaternion algebra over $\mathbb{F}$. \end{corollary} \begin{proof} Let $\mathcal{G}amma_0$ be the subgraph on $6$ vertices guaranteed by Proposition \ref{6graph}. Then $\mathfrak{K}(\mathcal{G}amma_0)$ is the subalgebra we are looking for. \end{proof} \begin{remark} The Lie algebra $\mathfrak{sp}(4,\mathbb{H})$, where $\mathbb{H}$ are the real quaternions, is the maximal compact Lie subalgebra of a split real Lie algebra of type $\mathfrak{e}_6$. See \cref{compact}. \end{remark} \begin{remark} \cref{6graph} and \cref{6graphcor} are closely related to some results of Seven \cite{seven}. See in particular \cite[Theorem 2.7]{seven}. Seven shows, among other things, the following: Let $(V,Q)$ be an orthogonal space over the field with two elements with corresponding bilinear form $f$. To each vector $v$ with $Q(v)=1$ we can assign a transvection $\tau_v:V\rightarrow V$ in the orthogonal group of $\mathrm{O}(V,Q)$, such that for all $w\in V$ we have $$\tau_v(w)=w+f(v,w)v.$$ Let $\mathcal{V}$ be a basis of anisotropic vectors of $V$, and denote by $\mathcal{G}amma$ the graph where two elements $v,w\in\mathcal{V}$ are adjacent if and only if $f(v,w)=1$. If $\mathcal{G}amma$ is connected, but $\overline{\mathcal{G}amma}$ is not a line graph, then $\mathcal{G}amma$ contains an induced subgraph $\mathcal{G}amma_0$ on six points that generate a nondegenerate $6$-dimensional orthogonal $\mathbb{F}_2$ space $(V_{\mathcal{G}amma_0},Q_{\mathcal{G}amma_0})$ of $-$-type on which the corresponding transvections induce the orthogonal group $\mathrm{O}(V_{\mathcal{G}amma_0},Q_{\mathcal{G}amma_0})$ which is isomorphic to the Weyl group of type $E_6$. \end{remark} \begin{remark} We notice that we can consider the various algebras of this and the previous section over a ring $R$. In particular, we can consider the Lie algebras $\mathfrak{g}(V,f)$ and $\mathfrak{g}(V,Q)$, as well as $\mathfrak{g}(\Omega)$ for some quadratic $\mathbb{F}_2$-space $(V,Q)$ with associated bilinear form $f$ and set $\Omega$ over the integers $\mathbb{Z}$. If we reduce scalars modulo an odd prime $p$, we obtain the Kaplansky Lie algebras as considered in \cite{Erik}, and if we reduce scalars modulo $2$ we find the Lie algebras considered by Kaplansky in \cite{kaplansky}. See also \cite{cotriangleLie}. \end{remark} \begin{algorithm} Let $\mathcal{G}amma$ be a finite connected black colored graph. The above considerations also provide an algorithm to determine $\mathfrak{K}(\overline\mathcal{G}amma)$ as in \cref{graphliealgebrathm} from the input $\mathcal{G}amma$. \begin{enumerate}[{\rm (a)}] \item Find the decomposition into the $\sim$-equivalence of $\mathcal{V}$. This can be done using a standar partition algorithm. See for example Algorithm 2 in \cite{partition}. \item Take a single vertex from each $\sim$-class and determine the induced subgraph of $\mathcal{G}amma$. This graph is isomorphic to $\overline{\mathcal{G}amma}$. \item Determine, if possible, a graph $\Delta$ such that $\overline\mathcal{G}amma$ is the line graph of $\Delta$ (several algorithms exist, see for example \cite{alglinegraph}). In case $\overline{\mathcal{G}amma}$ is $K_3$, the complete graph on $3$ points, there are two graphs $\Delta$, namely $K_3$ and $K_{1,3}=D_4$ having $\overline{\mathcal{G}amma}$ as line graph. In this case, take $\Delta$ to be $K_3$. \item If $\overline\mathcal{G}amma$ is the line graph of the graph $\Delta$, then $\mathfrak{K}(\overline\mathcal{G}amma)$ is isomorphic to $\mathfrak{g}(\Omega)$, where $\Omega$ is the vertex set of $\Delta$. \item If $\overline\mathcal{G}amma$ is not a line graph, then $\mathfrak{K}(\overline\mathcal{G}amma)$ equals $\mathfrak{g}(\overline\mathcal{G}amma)$, the Lie algebra of $\mathcal{C}(\overline\mathcal{G}amma)$. The isomorphism type of the latter can be determined using \cref{isoLietable} and \cref{cliffordalgorithm}. \end{enumerate} If the input of the algorithm is a graph on $n$ vertices, then the complexity of the algorithm is of order at most $n^3$, as for each step there exist algorithms of order at most $n^3$. \end{algorithm} \section{Spin representations and compact subalgebras of Kac-Moody algebras} \label{sect:kac} Suppose $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ is a graph with all vertices colored black. Then the generators $x\neq y\in \mathcal{V}$ of the Lie algebra $\mathfrak{K}(\mathcal{G}amma)$ do satisfy the relations $$\begin{array}{rll} [x,y]&=0 & \textrm{ if }(x,y)\not\in \mathcal{E}\\ {[x,[x,y]]}&=-y & \textrm{ if }(x,y)\in \mathcal{E}.\\ \end{array}$$ So, the free Lie algebra $\mathfrak{g}_\mathcal{G}amma$ with generators in $\mathcal{V}$ subject to the above relations has then $\mathfrak{K}(\mathcal{G}amma)$ as a quotient. The next result is motivated by, and a generalisation of the results of \cite{kohlspin}. We consider linear representations $\phi$ of the free Lie algebra $\mathfrak{g}_\mathcal{G}amma$ into $\mathfrak{gl}(W)$, the general linear Lie algebra on a vector space $W$ over a field of characteristic not $2$. If $x,y$ are two linear maps on $W$, then by $xy$ we denote the composition, and we consider the Lie product of $\mathfrak{gl}(W)$ to be defined as $$[x,y]=\frac{1}{2}(xy-yx).$$ Such a representation $\phi$ is called a \emph{generalized spin representation} of $\mathfrak{g}_\mathcal{G}amma$, if and only if $$\phi(x)^2=-\mathbf{1}_W$$ for all generators $x\in \mathcal{V}$. Our first observation is that $\mathfrak{g}_\mathcal{G}amma$ always admits such a representation. \begin{proposition}\label{spin_exists} The Lie algebras $\mathfrak{g}_\mathcal{G}amma$ and $\mathfrak{K}(\mathcal{G}amma)$ admit a generalized spin representation. \end{proposition} \begin{proof} As $\mathfrak{K}(\mathcal{G}amma)$ is a quotient of $\mathfrak{g}_\mathcal{G}amma$, we only have to show that $\mathfrak{K}(\mathcal{G}amma)$ admits a generalized spin representation. As the elements of $\mathfrak{K}(\mathcal{G}amma)$ act by left multiplication on $\mathfrak{C}(\mathcal{G}amma)$, and $x^2=-\mathbf{1}$ for all $x\in \mathcal{V}$, we have found a generalized spin representation. \end{proof} For finitely generated $\mathfrak{g}_\mathcal{G}amma$ generalized spin representations have been constructed by \cite{kohlspin}, generalizing \cite{Buyl,Damour} in which such representations have been constructed for graphs of type $E_{9}$ and $E_{10}$. The following characterization of the generalized spin representation is also obtained in \cite{kohlspin}. \begin{theorem}\label{spinthm} Suppose $\phi:\mathfrak{g}_\mathcal{G}amma\rightarrow \mathfrak{gl}(W)$ for some vector space $W$ over a field of characteristic $\neq 2$ is a linear representation of $\mathfrak{g}_\mathcal{G}amma$. If $\phi$ is a generalized spin representation, then $\phi(\mathfrak{g}_\mathcal{G}amma)$ is isomorphic to a quotient of $\mathfrak{K}(\mathcal{G}amma)$. \end{theorem} \begin{proof} We identify the elements $x\in X$ with their images under $\phi$ and compute in $\mathrm{End}(W)$. Then, as $x^2=- \mathbf{1}_W$ , we find $x$ to be invertible invertible with inverse $- \mathbf{1}_W$. Now for $x,y\in X$ we have $ xy-yx=0$ or $\frac{1}{4}(x(xy-yx)-(xy-yx)x)=- y$. Suppose we are in the latter case. Then $x^2y-2xyx+yx^2=-4y$ and hence $2y-2xyx=0$. Now multiplying with $x$ yields $2(xy+ yx)=0$ from which we deduce $xy+yx=0$. So, the (images under $\phi$ of the) elements $x\in \mathcal{V}$ satisfy, as linear maps from $W$ to itself, the relations of the generators of $\mathfrak{C}(\mathcal{G}amma)$, where all vertices of $\mathcal{G}amma$ are considered to be black. But then the subalgebra of $\mathrm{End}(W)$ generated by $\phi(\mathcal{V})$ is isomorphic to a quotient of $\mathfrak{C}(\mathcal{G}amma)$. In particular, $\mathfrak{g}(\mathcal{G}amma)$ maps onto a subalgebra of $\mathfrak{gl}(W)$ containing $\phi(\mathfrak{g}_\mathcal{G}amma)$ as the subalgebra generated by the elements of $\phi(x)$ with $x\in \mathcal{V}$. This implies that $\phi(\mathfrak{g}_\mathcal{G}amma)$ is isomorphic to a quotient of $\mathfrak{K}(\mathcal{G}amma)$. \end{proof} A result of Berman \cite{Berman} relates the free Lie algebra $\mathfrak{g}_\mathcal{G}amma$ to the so-called compact subalgebras of Kac-Moody algebras over fields $\mathbb{F}$ of characteristic $0$. Let us explain this connection, restricting ourselves to the simply laced case. Let $A=(a_{ij})$ be a generalized Cartan matrix indexed by the set $\mathcal{V}$, which is simply laced. That means $$a_{ii}=2$$ and $$a_{ij}=a_{ji}=0 \textrm{ or } -1$$ for $i\neq j \in \mathcal{V}$. Then the Kac-Moody Lie algebra $\mathfrak{KM}(A)$ is the free Lie algebra over $\mathbb{F}$ with generators $$e_i,f_i,h_i,\textrm{ where }i\in \mathcal{V}$$ subject to the relations $$[h_i,h_j]=0,\ [h_i,e_j]=a_{ij}e_j,\ [h_i,f_j]=-a_{ij}f_j \ \textrm{ for all } i,j\in \mathcal{V}$$ $$[e_i,f_j]=0,\ [e_i,f_i]=h_i,\ [e_i,[e_i,e_j]]=0=[f_i,[f_i,f_j]] \textrm{ for all } i\neq j \in \mathcal{V}.$$ The so-called compact subalgebra $\mathfrak{K}(A)$ of $\mathfrak{KM}(A)$ is the Lie subalgebra generated by the elements $$e_i+f_i, \ i\in \mathcal{V}.$$ If for each $x\in \mathcal{V}$ we denote by $x$ the element $e_x+f_x$, and consider the associated graph $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$ with vertex set $\mathcal{V}$ and two distinct vertices $x,y$ adjacent if and only if $a_{xy}\neq 0$, then we obtain the following. \begin{lemma} Let $x\neq y\in \mathcal{V}$. Then $$\begin{array}{rll} [x,y]&=0 & \textrm{ if }(x,y)\not\in \mathcal{E}\\ {[x,[x,y]]}&=-y & \textrm{ if }(x,y)\in \mathcal{E}.\\ \end{array}$$ \end{lemma} \begin{proof} If $x$ and $y$ are non-adjacent then clearly $[x,y]=0$. So, assume $x$ and $y$ are adjacent. Then $$\begin{array}{ll} [x,[x,y]]&=[e_x+f_x,[e_x+f_x,e_y+f_y]]]\\ &=[e_x+f_x,[e_x,e_y]+[f_x,f_y]]\\ &=[e_x,[e_x,e_y]]+ [e_x,[f_x,f_y]]+[f_x,[e_x,e_y]]+ [f_x,[f_x,f_y]]\\ &=[e_x,[f_x,f_y]]+[f_x,[e_x,e_y]]\\ &=-[f_y,[e_x,f_x]]-[e_y,[f_x,e_x]]\\ &=-[f_y,h_x]+[e_y,h_x]\\ &=-f_y-e_y\\ &=-y. \end{array}$$ \end{proof} \begin{theorem}(Berman \cite{Berman})\label{theorem_Berman} Let $\mathbb{F}$ be a field of characteristic $0$ and $A=(a_{ij})$ a simply laced generalized Cartan matrix with associated graph $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$. Let $\mathfrak{KM}(A)$ the Kac-Moody Lie algebra over $\mathbb{F}$. Then the compact Lie subalgebra $\mathfrak{K}(A)$ of $\mathfrak{KM}(A)$ is isomorphic to the free Lie algebra $\mathfrak{g}_\mathcal{G}amma$ over $\mathbb{F}$ generated by $\mathcal{V}$ subject to the relations $$\begin{array}{rl} [x,y]=0 & \textrm{ if }(x,y)\not\in \mathcal{E}\\ {[x,[x,y]]}=-y & \textrm{ if }(x,y)\in \mathcal{E}\\ \end{array}$$ for $x\neq y\in \mathcal{V}$. \end{theorem} Combining the above Theorem \ref{theorem_Berman} with Proposition \ref{spin_exists}, we obtain the following. \begin{corollary}\label{cor_Berman} Let $\mathbb{F}$ be a field of characteristic $0$ and $A=(a_{ij})$ a simply laced generalized Cartan matrix with associated graph $\mathcal{G}amma=(\mathcal{V},\mathcal{E})$. Let $\mathfrak{KM}(A)$ the Kac-Moody Lie algebra over $\mathbb{F}$. The compact Lie subalgebra $\mathfrak{K}(A)$ of $\mathfrak{KM}(A)$ admits a quotient isomorphic to $\mathfrak{K}(\mathcal{G}amma)$, and in particular, admits a spin representation. \end{corollary} \begin{example} If $\mathcal{G}amma$ is the graph $E_{10}$, with all vertices black, and $\mathbb{F}$ is a field of type III, for example $\mathbb{R}$, then $\mathfrak{C}(\mathcal{G}amma)$ is isomorphic to $\mathfrak{C}(V,Q)$, where $(V,Q)$ is a nondegenerate form of $+$-type. But then $\mathfrak{g}(\mathcal{G}amma)=\mathfrak{K}(\mathcal{G}amma)$ is isomorphic to $\mathfrak{so}(32,\mathbb{F})$. So, if $\mathbb{F}=\mathbb{R}$, we find that the compact Lie subalgebra $\mathfrak{K}(E_{10})$ of $\mathfrak{KM}(E_{10})$ admits a quotient isomorphic to $\mathfrak{g}(\mathcal{G}amma)$. Using \cref{EnCliffordTable}, we obtain similar results for graphs of type $E_n$ for all $n$. See also \cite{Buyl,Damour,kohlspin}. \end{example} \begin{example}\label{compact} As in \cite{kohlspin} we can use the above result to determine the maximal compact Lie subalgebra $\mathfrak{K}$ of the semi-simple split real Lie algebras of type $A_n$, $D_n$, where $n\geq 1$ and $E_n$, where $6\leq n\leq 8$. Indeed, using \ref{cor_Berman}, we find that the maximal compact Lie subalgebra $\mathfrak{K}$ of a semi-simple split real Lie algebras $\mathfrak{g}$ of type $A_n$, $D_n$, where $n\geq 1$ and $E_n$, where $6\leq n\leq 8$, admits a quotient isomorphic to $\mathfrak{K}(\mathcal{G}amma)$, where $\mathcal{G}amma$ is the corresponding graph of type $A_n$, $D_n$, or $E_n$. Using the results of the previous sections, we find these quotients to be as in Table \ref{maxcompact}. This provides a lowerbound for the dimension of $\mathfrak{K}$ which coincides with the upperbound of the dimension of $\mathfrak{K}$ that one can obtain from the Iwasawa decomposition of $\mathfrak{g}$. These results can also be found in \cite{Tanya2}, where $\mathfrak{K}$ is embedded in the Lie algebra of a (generalized) Clifford algebra. \begin{table} \begin{tabular}{|l|l|l|}\hline Type of $\mathfrak{g}$ & Maximal compact subalgebra $\mathfrak{K}$& dimension\\ \hline\hline $A_n$ & $\mathfrak{so}(n+1,\mathbb{R})$& $\binom{n+1}{2}$\\ $D_n \ (n>3)$ & $\mathfrak{so}(n,\mathbb{R})\oplus \mathfrak{so}(n,\mathbb{R})$&$n(n-1)$\\ $E_6$ & $\mathfrak{sp}(4,\mathbb{H})$& $36$\\ $E_7$ & $\mathfrak{su}(8,\mathbb{C})$ & $63$\\ $E_8$ & $\mathfrak{so}(16,\mathbb{R})$ & $120$\\ \hline \end{tabular} \caption{Maximal compact subalgebras of the split real Lie algebras.}\label{maxcompact} \end{table} \end{example} \parindent=0pt Hans Cuypers\\ Department of Mathematics and Computer Science \\ Eindhoven University of Technology\\ P.O. Box 513 5600 MB, Eindhoven\\ The Netherlands\\ email: f.g.m.t.cuypers@tue.nl \end{document}

\begin{document} \mainmatter \title{Hybrid Graph Embedding Techniques in Estimated Time of Arrival Task } \titlerunning{Hybrid Graph Embedding Techniques in Estimated Time of Arrival Task} \author{Vadim Porvatov\inst{1,2}\printfnsymbol{1} \and Natalia Semenova\inst{1}\thanks{equal contribution} \and Andrey Chertok\inst{1,3}} \authorrunning{V. Porvatov et al.} \institute{Sberbank, Moscow 117997, Russia, \and National University of Science and Technology ``MISIS'', Moscow 119991, Russia, \and Artificial Intelligence Research Institute (AIRI),\\ \email{eighonet@gmail.com}\\ \email{semenova.bnl@gmail.com}\\ \email{achertok@sberbank.ru}\\ \url{}} \maketitle \begin{abstract} Recently, deep learning has achieved promising results in the calculation of Estimated Time of Arrival (ETA), which is considered as predicting the travel time from the start point to a certain place along a given path. ETA plays an essential role in intelligent taxi services or automotive navigation systems. A common practice is to use embedding vectors to represent the elements of a road network, such as road segments and crossroads. Road elements have their own attributes like length, presence of crosswalks, lanes number, etc. However, many links in the road network are traversed by too few floating cars even in large ride-hailing platforms and affected by the wide range of temporal events. As the primary goal of the research, we explore the generalization ability of different spatial embedding strategies and propose a two-stage approach to deal with such problems. \keywords{Graph Embedding, Machine Learning, ETA, Geospatial \\ Linked Data.} \end{abstract} \section{Introduction} The modern state of traffic induces a remarkable number of forecasting challenges in a variety of related areas. According to the industrial needs, a relevant computation of the estimated time of vehicle arrival can be considered as one of the most actual problems in the logistics domain. In particular, intelligent traffic management systems \cite{6531823} require significant accuracy in case of arrival time estimation. Besides such an application, computation of ETA also appears as a common issue in the commercial areas which are strongly dependent on optimal routing. The explicit examples of such services are taxi \cite{distribution_cat}, railway \cite{eta_trains}, vessels \cite{PARK2021100012} and aircraft transportation \cite{eta_planes}. Accurate prediction of ETA for cars is a complex task requiring the relevant processing of heterogeneous data. It is frequently represented as time series and graph structure with feature vectors associated with its nodes and/or edges. In comparison with other vehicles, computation of ETA for cars is considerably influenced by the road network topology, nonlinear traffic dynamics, unexpected temporal events, and unstable weather conditions, Figure \ref{usage_freqs}. The stochastic nature of the introduced problem requires an implementation of a powerful domain-specific regression model with a high generalization ability. \begin{figure} \caption{Demonstration of temporal traffic dynamics: cumulative frequencies of car activity and distribution of trips duration for Abakan and Omsk in the two hours interval.} \label{usage_freqs} \end{figure} Machine learning proved its outstanding efficiency in a wide range of regression tasks. However, not every model can be efficiently applied to the ETA forecasting due to the mentioned constraints of available data. Previously performed attempts of a simple model implementation (e. g., linear regressions and gradient boosting) were reported as inefficient \cite{lr, WDR}, while the more sophisticated approaches allowed to achieve more optimistic results \cite{sun_2020_metric_learning_eta}. Thus, in order to obtain a better performance, we assume the necessity of applying graph neural networks \cite{gnn} as a part of the presented pipeline. According to the extensive growth of graph machine learning in recent years, many promising architectures \cite{kipf2017semisupervised, Perozzi_2014} emerged and soon were applied in a wide range of graph-related studies \cite{sweet_physics, protein_chem}. These models quickly became useful in terms of feature extraction in downstream tasks. Applied to the underlying graph structure of a city road network, such algorithms have the potential to dramatically increase the expressiveness of regression models and therefore should be explored. In the present paper, we propose and compare different architectures of the hybrid graph neural network for ETA prediction. Our main contributions are the following: \begin{itemize} \item We introduce and publish the first to our best knowledge dataset\footnote[1]{to receive an access to data you need to send a request to semenova.bnl@gmail.com} with intermediate trip points. This dataset is relevant for consistent ETA prediction task and future usage as a benchmark. We provide common information about trips and city road network as well as road structural properties, marking, and weather conditions (other features are described in Section 3 in detail). Additionally, the route data includes auxiliary information which can be used both for evaluation of the ETA and independent prediction of real traveled distance as a separate problem. \item Absence of methodological review of subgraph embeddings in the domain of interest encourages us to overwhelm such a limitation. Instead of focusing on more general approaches which include both spatial and recurrent temporal aspects, we prefer to precisely explore the domain of spatial embeddings as an underdeveloped one at the present moment. \item We conduct a comprehensive evaluation of our method on two real-world datasets which correspond to tangibly different cities. Obtained results of computational experiments motivate us to further develop our research in accordance with achieved significant performance improvements. \end{itemize} \section{Related work} As it has been mentioned above, the ETA-related tasks are a fundamental part of logistic services. In overwhelming number of cases, they demand two properties from the predictive algorithms: computational efficiency and relevant accuracy. The first part of this challenge was unequivocally solved by simple learning models like gradient tree boosting, multi-layer perceptron, and linear regression. However, the quality of these models cannot be reported as sufficient even beyond the commercial logistics. Along with the simple learning models, deterministic algorithms were also developed in huge amount \cite{10.1145/2820783.2820836, 10.1145/2623330.2623656}. In the majority of cases they cannot be compared with learning models in terms of quality. However, some of them were inspiring enough to influence the future development of their concepts in a more sophisticated way. \begin{figure} \caption{Edges usage frequencies projected as a heatmap on the road networks of Abakan (a) and Omsk (b). The patterns of edges demand are clearly distinguishable as the topology of networks remains significantly different.} \label{projection} \end{figure} Limitations of mentioned approaches were partially overwhelmed in DeepTTE \cite{deeptte} and MURAT \cite{murat}. The first approach includes a recurrent neural network (RNN) which subsequently predicts the travel time along the trip. As many other recent methods, this algorithm is dependent on intermediate GPS coordinates. At the same time, the second method is closely related to the proposed architecture in the sense of graph embedding usage. In spite of the deep development of the temporal forecasting part, no more than one spatial embedding method was observed in any of this papers. The most recent studies introduce new solutions with the potential to significantly increase the quality of ETA prediction. WDR \cite{WDR} is a wide-deep architecture that outperformed a lot of previously established approaches. Its further improvement and computational experiments led the same authors to the design of RNML-ETA architecture \cite{sun_2020_metric_learning_eta} which allows to achieve even better results. Simultaneously, another intriguing paper \cite{hstgcn} emerged as a prospective modification of ST-GCN methods family\cite{stgcn, astgcn, stsgcn}. All of these methods use datasets with intermediate points in contrary to the overwhelming majority of early papers. Following this positive trend, we continue studies in the same direction. \section{Data} In the present work, we use two datasets related to the city networks of Abakan and Omsk. The cities have significantly different scales. Hence, their infrastructure pattern cannot be compared directly. Such a diversity allows us to check the generalization ability of the proposed architectures in a more explicit way. General properties of the dataset are established in Table 1 when the frequencies of road network segments usage are represented in Figure \ref{projection}(a, b). \begin{table}[b] \caption{Description of the datasets in terms of common networks characteristics} \begin{center} \begin{tabular}{l@{\quad}@{\hspace{6em}}l@{\hspace{6em}}l} \hline \multicolumn{1}{l}{\rule{0pt}{12pt}Property}& \multicolumn{1}{l}{Abakan}& \multicolumn{1}{l}{Omsk}\\[2pt] \hline\rule{-3pt}{12pt} Nodes & 65524 & 231688\\ Edges & 340012 & 1149492\\ Total trips number & 119986 & 120000\\ Trips coverage & 0.535 & 0.392\\ Edges usage median & 12 & 8\\[2pt] \hline \end{tabular} \end{center} \end{table} Each dataset consists of both road networks and the routes associated with their edges. City networks contain an abundant number of meaningful features that can be translated to the predictive model in different ways. The route sample includes information about the start and destination point and a set of visited nodes during the ride. The trip data was collected in the period from December 1, 2020 up to December 31, 2020 by subsidiary companies of Sberbank. A comprehensive description of the proposed data is given in Table 2 for the city network and in Table 3 for car routes. \begin{table} \begin{center} \caption{Edge features of city network} \begin{tabular}{l@{\quad}l@{\hspace{1.0em}}l}\hline \multicolumn{1}{l}{\rule{0pt}{12pt}Feature}& \multicolumn{1}{l}{Values}& \multicolumn{1}{l}{Description}\\[2pt] \hline\\[-8pt]\rule{-3pt}{12pt} Road class & \begin{tabular}[c]{@{}l@{}}fake road, intra-quarter driveway, \\ dirt road, other city street, main \\ city street, highway, intercity \\ road, federal highway, cycle path, \\ walkway\end{tabular} & \begin{tabular}[c]{@{}l@{}}General road segments \\ categories\end{tabular} \\ Length & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Length of a road \\ segment in meters\end{tabular} \\ Width & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Width of a road \\ segment in meters\end{tabular} \\ Def speed & \{3, 15, 20, 60, 90\} & \begin{tabular}[c]{@{}l@{}}Speed limit on a \\ road section in km/h\end{tabular} \\ Lanes & \{0, 1, 2, 3, 4, 5\} & \begin{tabular}[c]{@{}l@{}}Number of lanes in \\ a road segment\end{tabular} \\ Barrier & \{0, 1\} & \begin{tabular}[c]{@{}l@{}}Defines the presence \\ of road barriers\end{tabular} \\ Payment flag & \{0, 1\} & \begin{tabular}[c]{@{}l@{}}Defines a road segment \\ as toll\end{tabular} \\ Turn restrictions & \{0, 1\} & \begin{tabular}[c]{@{}l@{}}Defines an ability to \\ turn on a road section\end{tabular} \\ Pedo offset & \{0, 1\} & \begin{tabular}[c]{@{}l@{}}Defines the presence of \\ crosswalk offsets\end{tabular} \\ Bad road & \{0, 1\} & \begin{tabular}[c]{@{}l@{}}Defines the condition \\ of a road segment\end{tabular} \\ Style & \begin{tabular}[c]{@{}l@{}}undefined, archway, crosswalk,\\ stairway, bridge, overground way,\\ invisible, normal, park path,\\ park footpath, subway, pedestrian \\ bridge, underground way, tunnel, \\ living zone, ford\end{tabular} & \begin{tabular}[c]{@{}l@{}}Additional road segments \\ categories\end{tabular} \\[29pt] \hline \end{tabular} \end{center} \end{table} \begin{table} \begin{center} \caption{Features of trip dataset} \begin{tabular}{l@{\quad}@{\hspace{2.5em}}l@{\hspace{2.8em}}l}\hline \multicolumn{1}{l}{\rule{0pt}{12pt}Feature}& \multicolumn{1}{l}{Values}& \multicolumn{1}{l}{Description}\\[2pt] \hline\rule{-3pt}{12pt} Nodes &$\{\hat{V} \subset V\}$ & \begin{tabular}[c]{@{}l@{}}Subset of nodes\end{tabular} \\ Dist to a & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Length of a segment between actual start \\ point and its projection on the first edge\end{tabular} \\ Dist to b & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Length of a segment between actual end \\ point and its projection on the last edge\end{tabular} \\ Start point part & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Part of the first edge where the trip \\ starts in meters\end{tabular} \\ Finish point part & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Part of the last edge where the \\ trip ends in meters\end{tabular} \\ Start UTC & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}} Start time of the trip in UTC format\end{tabular} \\ Real time of arrival & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Trip duration in seconds\end{tabular} \\[2pt] Real dist* & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}} Actual traveled distance in meters \end{tabular} \\ Rebuild count* & $\mathbf{Z_{+}}$ & \begin{tabular}[c]{@{}l@{}}Number of route rebuilds that corresponds \\ to the destination change \end{tabular} \\[2pt] \hline \end{tabular} \end{center} \end{table} According to the complexity of input data, it cannot be directly translated to a predictive model as an input. In order to correctly solve the desired task, it is recommended to filter the established dataset and perform feature engineering. Trips that have a rebuild count more than 1 should be optionally separated from the main volume of routes as well as anomaly short and long routes. Values of start (finish) point parts and dist to a(b) can be also added or subtracted from the total estimated length of the route in order to obtain a better spatial resolution of subgraph embeddings. \section{Methods} The task can be mathematically formulated as a regression problem that extended by a special procedure of an automatic feature engineering. In order to handle this challenge, we generate vector representations of the road segments via GNNs, aggregate them to the trips embeddings and then apply a regression model which predicts ETA. Given a graph $G = (V, A, X)$ of the city road network, where $V = \{v_1, v_2, ... , v_n\}$ denotes the set of graph vertexes (road segments), $A$: $n\times n$ $\xrightarrow{} \{0, 1\}$ denotes the adjacency matrix (each edge encodes connectivity of the road segments), and $X$: $n \times m$ $\xrightarrow{}$ $\mathbf{R}$ is a matrix of node features. The goal is to compute such a representation of each node $v_i \in V$ that can be effectively aggregated in accordance with structural properties of the route $s_j:= \{v_{j_1}, ..., v_{j_t}\}$, $s_j \in S$. There are two main aggregation strategies that potentially allow to construct a meaningful route subgraph embedding. The first one based on basic summation of all representations of the nodes that are included to the exact route \begin{equation} z_{s_j} = \sum_{i=1}^{\#s_j} Z(v_{j_i}), \end{equation} where Z($\cdot$) is the node embedding function. Another approach related to initial graph extension by virtual nodes. This procedure induces a new graph $\hat{G}$($V'$, $A'$, $ X'$), where $V'$ = $\{v_1, ..., v_n, v_{n+1}, ...,$ $v_{n+\#S}\}$, $A'$:($n + \#S$)$\times$($n + \#S$) $\xrightarrow{}$ $\{0, 1\}$, $\forall v_i, v_n \in V$ adjacency matrix defined as $A'(v_i$, $v_j$) = $A$($v_i$, $v_j$). For the other edges, we propose the bijective function $f:V'$\textbackslash $V$ $\xrightarrow{}$ S that defines $\forall v'_{k} \in V'$\textbackslash $V$ and $\forall v_l \in$ $f(v'_k)$ values in remaining part of the extended adjacency matrix as $A'(v_l$, $v'_k$) = 1. In agreement with this method, \begin{equation} z_{s_j} = Z(f^{-1}(s_j)). \end{equation} For both strategies it is crucial to find the appropriate node embedding function $Z(\cdot)$ which has a significant impact on the relevance of the final route subgraph representations. We propose graph convolutional networks \cite{kipf2017semisupervised}, GAT \cite{gat}, and GraphSAGE \cite{graphsage} as the main candidates for nodes representation learning. The ideas behind these methods are quite similar as they all encode nodes to vectors of a fixed size via a repeated aggregation over a local neighborhood. However, while the GCN is based on mean aggregation, GraphSAGE pretends to be a more flexible and representative instrument due to its different aggregators and embedding concatenation stage. On the other hand, GAT adopts the mechanism of attention \cite{attention_first} firstly proposed in Natural Language Processing (NLP) to the needs of graph machine learning. To explicitly reveal the relevance of the mentioned approaches, in the following we briefly introduce the main aspects of each method. \textbf{Graph Convolutional Network (GCN)}. For a given graph $G(V, A, X)$ this method defines an effective approach to network information aggregation. Single graph convolution layer is its atomic unit that can be represented as \begin{equation} H^{(l+1)}=\sigma\left(\tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}} H^{(l)} W^{(l)}\right), \end{equation} where $l+1$ is the current convolution layer number, $\sigma$ is an arbitrary nonlinear function (e. g., ReLU), $H^{(0)}$ = $X$, $\tilde{A}=A+I_{N}$, $\tilde{D}_{i i}=\sum_{j} \tilde{A}_{i j}$ and $W^l$ is the matrix of learning parameters. \textbf{GraphSAGE}. This algorithm mostly inherits the notation of convolutions from the GCN architecture, but instead of using full graph it directly computes convolution for each node $v$ in the iterative manner \begin{equation} h_{v}^{l+1} = \sigma\left(W^{l} \cdot \operatorname{CONCAT}\left(h_{v}^{l}, h_{N(v)}^{l+1}\right)\right), \end{equation} where $h_{N(v)}^{l+1}$ can be extracted by a few different aggregate functions for the set of neighbour nodes $N(v)$. \textbf{Graph Attention Network}. The last considered method is based on the attention mechanism which also avoids transductive GCN constraints and apply the iterative aggregation procedure \begin{equation} h^{l+1}_i =\operatorname{CONCAT}_{k=1}^{K} \sigma\left( \; \sum_{j \in N(i)} \alpha_{i j}^{k} W^{k} h^{l}_{j}\right). \end{equation} The attention coefficient is computed as follows: \begin{equation} \alpha_{i j}=\frac{\exp \left(\sigma\left(a^{T}\cdot\operatorname{CONCAT}(W h^{l}_{i}, W h^{l}_{j})\right)\right)}{\sum_{k \in N(i)} \exp \left(\sigma\left(a^{T}\cdot\operatorname{CONCAT}(W h^{l}_{i}, W h^{l}_{k})\right)\right)}, \end{equation} where $a^T$ is a transposed vector of attention trainable parameters. In order to boost the expressiveness of these methods and convert supervised setups to unsupervised, we propose to embed them as a part of the Deep Graph InfoMax pipeline \cite{dgi}. This approach is based on minimizing of a two-component loss function \begin{equation} L=\frac{1}{N+M}\sum_{i=1}^{N} E_{G}\left[\log D(d_{i}, T)\right]+\sum_{j=1}^{M} E_{C}\left[\log \left(1-D(\tilde{d}_{j}, T)\right)\right] \end{equation} which aims to learn how to distinguish initial nodes representations $d$ and corrupted ones $\tilde{d}$, Figure \ref{dgi}. \begin{figure} \caption{Deep Graph Infomax corrupts feature vectors of the input graph G by function S (in the used realisation it shuffles features), constructs regular and corrupted node embeddings by applying $Z(\cdot)$, and finally estimates their similarity to the ground-truth vector T by the discriminator function D.} \label{dgi} \end{figure} Once embeddings of routes $z_{s_j}$ are computed, each vector can be extended by additional information about the weather conditions and corresponding temporal categorical features. After these manipulations with route vectors $z_{s_j}$ they can be finally fed to the regression model. \section{Results} In order to perform the training and evaluation of proposed architectures, we need to split the datasets into three samples. We trained our model on the first 100 000 trips, while the test and validation steps were performed on equal parts of the remaining datasets. Following the evaluation standards, we use a common set of metrics for the ETA prediction task: Mean Average Error (Eq. 8), Mean Average Percentage Error (Eq. 9), and Rooted Mean Square Error (Eq. 10). \begin{equation} \begin{aligned} \mathrm{MAE} &=\frac{1}{N} \sum_{i=1}^{N}\left|y_{i}-y_{i}^{\prime}\right|, \end{aligned} \end{equation} \begin{equation} \begin{aligned} \mathrm{MAPE} &=\frac{100}{N} \sum_{i=1}^{N}\left|\frac{y_{i}-y_{i}^{\prime}}{y_{i}}\right|, \end{aligned} \end{equation} \begin{equation} \begin{aligned} \mathrm{RMSE} &=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(y_{i}-y_{i}^{\prime}\right)^{2}}. \end{aligned} \end{equation} \subsection{Implementation details} Computational experiments were provided with the use of StellarGraph\cite{StellarGraph} library. All models were trained on 2 GPU Tesla V100, the total training time of the pipeline for the best models is 9 hours. During the embedding construction process, we used three types of each observed architecture with the number of layers from 1 to 3 and the fixed output of size 128. Neural networks weights were trained by Adam optimizer\cite{Adam} due to its good convergence and stability. We use the static learning rate parameters $L_1$ = 0.001 for node embedding generation and $L_2$ = 0.0001 for regression. \subsection{Experiments} We performed series of computational experiments varying the strategy of subgraph embedding generation and the method of node representation extraction. As the final regression model, we leverage a multi-layer perceptron (MLP). For the purpose of Deep Graph InfoMax tests extension, we also compute the values of the metrics for regular unsupervised GraphSAGE and regression baseline to illustrate the general capabilities of different approaches. The final values of metrics for each configuration are shown in Table 4. \begin{table}[] \begin{center} \label{results_table} \caption{Evaluation results on test sample} \begin{tabular}{l@{\quad}cccccc} \hline \rule{-4pt}{12pt} & \multicolumn{3}{c}{Abakan} & \multicolumn{3}{c}{Omsk}\\[2pt] \hline \rule{-4pt}{12pt} & MAE & RMSE & MAPE & MAE & RMSE & MAPE\\ Baseline(MLP only) & 111.05 & 316.39 & 27.129 & 145.819 & 296.86 & 25.019\\ GraphSAGE + VN & 111.23 & 316.82 & 27.213 & 146.003 & 297.028 & 25.108 \\ GraphSAGE + Sum & 96.575 & 310.114 & 22.881 & \textbf{129.831} &\textbf{ 279.773} &\textbf{ 22.416} \\ DGI(GCN) + Sum & 97.927 & 310.628 & 23.506 & 141.017 & 289.32 & 24.335 \\ DGI(GAT) + Sum & 101.808 & 313.01 & 25.737 & 133.262 & 283.22 & 23.175 \\ DGI(GS) + Sum & \textbf{95.819} & \textbf{309.627} & \textbf{22.622} & 130.296 & 280.058 & 22.593 \\ \hline \end{tabular} \end{center} \end{table} As it seen from the table, the best performance was achieved by the GraphSAGE setup with Deep Graph InfoMax in the case of Abakan. Meanwhile, common GraphSAGE also demonstrates promising embeddings quality (especially for Omsk) which is slightly different from its DGI modification. The error distributions of the best models for each dataset are shown in Figure 4. \begin{figure} \caption{Error distribution for the regression models trained on Abakan (a) and Omsk (b) datasets.} \label{error} \label{confusion} \end{figure} Unfortunately, the test series of virtual nodes route embeddings turned down our pursuit to report any significant results. We conclude that the expressiveness of this method is limited in the area of interest, despite previous positive attempts of implementation in other tasks \cite{drugs}. However, such a result was partially foreordained by the studies which also explored subgraph embeddings \cite{subgnn}. \section{Conclusion and Outlook} In this work, we implemented and explored a pipeline that includes state-of-the-art algorithms of graph machine learning that emerged in recent years. We trained and tested our model on two consistent datasets which correspond to cities with different road topology types. Our results allow us to conclude that GraphSAGE-based models capture spatial patterns of city networks more substantially. Our own perspectives include future development and modification of more specific methods based on obtained results. As the primary goal of this research was to find the most efficient methods of subgraph embedding construction in the context of ETA problem, we intend to use this knowledge to construct a more complex spatial approach in the upcoming papers. In the spotlight of our research, we also have an idea to design an powerful generalizing approach to various kinds of road networks with the potential of applying it to a bunch of cities. \end{document}

\begin{document} \title{ Sufficient stochastic maximum principle for the optimal control of semi-Markov modulated jump-diffusion with application to Financial optimization.} \baselineskip20pt \parskip10pt \parindent.4in \begin{abstract} \nonumberi \textcolor{red}{ Paper forthcoming in Stochastic Analysis and Applications}\\ The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem and to a quadratic loss minimization problem. \end{abstract} \nonumberindent\\ {\bf Keywords}: semi-Markov modulated jump diffusions, sufficient stochastic maximum principle, dynamic programming, risk-sensitive control, quadratic loss-minimization.\\ {\bf AMS subject classification} 93E20; 60H30;46N10. \section{Introduction} \indent The stochastic maximum principle is a stochastic version of the Pontryagin maximum principle which states that the any optimal control must satisfy a system of forward-backward stochastic differential {equations,} called the optimality system, and should maximize a functional, called the Hamiltonian. The converse indeed is true and gives the sufficient stochastic maximum principle. In this article we will derive sufficient stochastic maximum principle for a class of process called as the semi-Markov modulated jump-diffusion process. In this process the drift, the diffusion and the jump kernel term is modulated by an semi-Markov process. \\ \indent An early investigation of stochastic maximum principle and its application to finance has been credited to Cadenillas and Karatzas \cite{CK}. Framstadt et al. \cite{Fr} formulated the stochastic maximum principle for jump-diffusion process and applied it to a quadratic portfolio optimization problem. Their work has been partly generalized by Donnelly \cite{Do} who considered a Markov chain modulated diffusion process in which the drift and the diffusion term is modulated by a Markov chain. Zhang et al. \cite{Zh} studied sufficient maximum principle of a process similar to that studied by Donnelly additionally with a jump term whose kernel is also modulated by a Markov chain. It can be noted that the Markov modulated process has been quite popular with its recent applications to finance for example Options pricing (Deshpande and Ghosh \cite{DG}) and references therein and to portfolio optimization refer Xhou and Yin \cite{XY}. However application of semi-Markov modulated process to portfolio optimization in which the portfolio wealth process is a semi-Markov modulated diffusion are not many, see for example Ghosh and Goswami \cite{GG}. Even so it appears that the sufficient maximum principle has not been formulated for the case of a semi-Markov modulated diffusion process with jumps and studied further in the context of quadratic portfolio optimization. Moreover, application of the sufficient stochastic maximum principle in the context of risk-sensitive control portfolio optimization with the portfolio wealth process following a semi-Markov modulated diffusion process has not been studied. This article aims to provide answers to these missing dots and connect them together. For the same reasons, alongwith providing a popular application of the sufficient stochastic maximum principle to a quadratic loss minimization problem when the portfolio wealth process follows a semi-Markov modulated jump-diffusion, we also provide an example of risk-sensitive portfolio optimization for the diffusion part of the said dynamics. \\ \indent The article is organized as follows. In the next section we formally describe basic terminologies used in the article. In section 3 we detail the control problem that we are going to study. The sufficient maximum principle is proven in Section 4. This is followed by establishing its connection with the dynamic programming. We conclude the article by illustrating its applications to risk-sensitive control optimization and to a quadratic loss minimization problem. \section{Mathematical Preliminaries} We adopt the following notations that are valid for the whole paper:\\ $\mathbb{R}$: the set of real numbers\\ $r,M$: any positive integer greater than 1.\\ $ {{\mathcal{X}=\{1,...,M\}}}.$\\ $\mathcal{C}^{1,2,1}([0,T] \times \mathbb{R}^{r} \times \mathcal{X} \times \mathbb{R}_{+})$: denote the family of all functions on $[0,T] \times \mathbb{R}^{r} \times \mathcal{X} \times \mathbb{R}_{+}$ which are twice continuously differentiable in $x$ and continuously differentiable in $t$ and $y$.\\ $v^{'}$, $A^{'}$: the transpose of the vector (say )$v$ and matrix say $A$ respectively.\\ $||v||$: Euclidean norm of a vector $v$.\\ $|A|$: norm of a matrix $A$.\\ $tr(A)$: trace of a square matrix $A$.\\ $C^{m}_{b}(\mathbb{R}^{r})$: Set of real $m$-times continuously differentiable functions which are bounded together with their derivatives upto the $m^{th}$ order. \\ \indent We assume that the probability space ($\Omega,\mathcal{F},\{\mathcal{F}({t})\},\mathbb{P}$) is complete with filtration $\{\mathcal{F}({t})\}_{t \geq 0}$ and is right-continuous and $\mathcal{F}({0})$ contains all $\mathbb{P}$ null sets. Let $\{{\theta}({t})\}_{t\geq 0}$ be a semi-Markov process taking values in $ {\mathcal{X}}$ with transition probability $ {p_{ij}}$ and conditional holding time distribution $F^{h}(t|i)$. Thus if $0 \leq t_{0}\leq t_{1}\leq ...$ are times when jumps occur, then \begin{eqnarray}\label{2.1} P(\theta({t_{n+1}})=j,t_{n+1}-t_{n} \leq t|\theta({t_{n}})=i)=p_{ij}F^{h}(t|i). \end{eqnarray} Matrix $[p_{ij}]_{\{i,j=1,...,M\}}$ is irreducible and for each $i$, $F^{h}(\cdot|i)$ has continuously differentiable and bounded density $f^{h}(\cdot|i)$. For a fixed $t$, let $n(t) \triangleq \max\{n: t_n \leq t\}$ and $Y(t) \triangleq t- t_{n(t)}$. Thus $Y(t)$ represents the amount of time the proess $\theta(t)$ is at the current state after the last jump. The process ($\theta{(t)},Y{(t)}$)defined on ($\Omega,\mathcal{F},\mathbb{P}$) is jointly Markov and the differential generator $\mathcal{L}$ given as follows (Chap.2, \cite{GS}) \begin{eqnarray}\label{2.3} \mathcal{L}\phi(i,y)=\frac{d}{dy}\phi(i,y)+\frac{f^{h}(y|i)}{1-F^{h}(y|i)}\sum_{j \neq i,j \in \mathcal{X}}{p_{ij}[\phi(j,0)-\phi(i,y)]}. \end{eqnarray} for $\phi:\mathcal{X} \times \mathbb{R_{+}}\rightarrow \mathbb{R}$ is { $C^{1}$} function.\\ \indent We first represent semi-Markov process $\theta(t)$ as a stochastic integral with respect to a Poisson random measure. With that perspective in mind, embed $\mathcal{X}$ in $\mathbb{R}^{M}$ by identifying $i$ with $e_{i} \in \mathbb{R}^{M}$. For $y \in [0,\infty)$ $i, j \in \mathcal{X}$, define \begin{eqnarray*} \lambda_{ij}(y)&=&p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)} \geq 0 ~~\mbox{and}~~ \forall~~ i \neq j, \\ \lambda_{ii}(y)&=&-\sum_{j\in \mathcal{X},j \neq i}^{M}{\lambda_{ij}(y)}~~ \forall~~ i~~ \in \mathcal{X}. \end{eqnarray*} \indent For $i \neq j \in \mathcal{X}$ , $y \in \mathbb{R}_{+}$ let $\Lambda_{ij}(y)$ be consecutive (with respect to lexicographic ordering on $\mathcal{X}\times \mathcal{X}$) left-closed, right-open intervals of the real line, each having length $\lambda_{ij}(y)$. Define the functions $\bar{h}:\mathcal{X}\times \mathbb{R}_{+}\times\mathbb{R}\rightarrow \mathbb{R}^{r}$ and $\bar{g}:\mathcal{X}\times \mathbb{R}_{+}\times \mathbb{R} \rightarrow \mathbb{R}_{+}$ by $$ \bar{h}(i,y,z) = \left\{ \begin{array}{rl} j-i &\mbox{ if $z \in \Lambda_{ij}(y)$} \\ 0 &\mbox{ otherwise} \end{array} \right. $$ $$ \bar{g}(i,y,z) = \left\{ \begin{array}{rl} y &\mbox{ if $z \in \Lambda_{ij}(y), j \neq i$} \\ 0 &\mbox{ otherwise} \end{array} \right. $$ \\ \indent Let $\mathcal{M}(\mathbb{R}_{+} \times \mathbb{R})$ be the set of all nonnegative integer-valued $\sigma$-finite measures on { Borel } $\sigma$-field of ($\mathbb{R}_{+}\times \mathbb{R}$). {The process $\{\tilde{\theta}{(t)},Y{(t)}\}$ is defined} by the following stochastic integral equations: \begin{eqnarray}\label{2.2} \begin{split} \tilde{\theta}{(t)}=\tilde{\theta}{(0)}+\int_{0}^{t}\int_{\mathbb{R}}{\bar{h}(\tilde{\theta}{(u-)},Y{(u-)},z)N_{1}(du,dz)},\\ Y{(t)}=t-\int_{0}^{t}\int_{\mathbb{R}}{\bar{g}(\tilde{\theta}{(u-)},Y{(u-)},z)N_{1}(du,dz)}, \end{split} \end{eqnarray} where $N_{1}(dt,dz)$ is an $\mathcal{M}$($\mathbb{R}_{+}\times \mathbb{R}$)-valued Poisson random measure with intensity $dt m(dz)$ independent of the $\mathcal{X}$-valued random variable $\tilde{\theta}{(0)}$, where $m(\cdot)$ is a Lebesgue measure on $\mathbb{R}$. As usual by definition $Y(t)$ represents the amount of time, process $\tilde{\theta}(t)$ is at the current state after the last jump. We define the corresponding compensated or centered one dimensional Poisson measure as $\tilde{N}_{1}(ds,dz)=N_{1}(ds,dz)-dsm(dz)$. It was shown in Theorem 2.1 of Ghosh and Goswami \cite{GG} that $\tilde{\theta}{(t)}$ is a semi-Markov process with transition probability matrix $[p_{ij}]_{\{i,j=1,...,M\}}$ with conditional holding time distributions $F^{h}(y|i)$. {Since by definition $\theta(t)$ is also a semi-Markov process with transition probability matrix $[p_{ij}]_{\{i,j=1,...,M\}}$ with conditional holding time distributions $F^{h}(y|i)$ defined on the same underlying probability space, by equivalence, $\tilde{\theta}{(t)}=\theta{(t)}$ for $t \geq 0$}.\\ { {\bf Remark 2.1}~~The semi-Markov process with conditional density $f^{h}(y|i)=\tilde{\lambda}_{i}e^{-\tilde{\lambda}_{i}y}$ for some $\tilde{\lambda}_{i}>0$, $i =1,2...,M$, is infact a Markov chain.} \section{The control problem} Let $\mathcal{U} \subset \mathbb{R}^{r}$ be a closed subset. { Let $\mathbb{B}_{0}$ be the family of Borel sets $\Gamma \subset \mathbb{R}^{r}$ whose closure $\bar{\Gamma}$ does not contain {0}. For and Borel set $B \subset \Gamma$, one dimensional poisson random measure $ N(t,B)$ counts the number of jumps on $[0,t]$ with values in $B$.} { For a predictable process $u:[0,T] \times \Omega \rightarrow \mathcal{U}$ with right continuous left limit paths, consider the controlled process $X$ with given initial condition $X(0)=x \in \mathbb{R}^{r}$ given by} \begin{eqnarray}\label{3.1} dX({t})=b(t,X({t}),u({t}),\theta({t}))dt+\sigma(t,X({t}),u({t}),\theta({t}))dW({t})+\int_{\Gamma}g(t,X({t}),u({t}),\theta({t})),\gamma){N}(dt,d\gamma),\nonumber\\ \end{eqnarray} where $X(t) \in \mathbb{R}^{r}$ and $W(t)=(W_{1}(t),...,W_{r}(t))$ is $r$-dimensional standard Brownian motion. The coefficients $b(\cdot,\cdot,\cdot,\cdot):[0,T] \times \mathbb{R}^{r}\times \mathcal{U} \times \mathcal{X} \rightarrow \mathbb{R}^{r}$,$\sigma(\cdot,\cdot,\cdot,\cdot):[0,T] \times \mathbb{R}^{r}\times \mathcal{U}\times \mathcal{X} \rightarrow \mathbb{R}^{r} \times \mathbb{R}^{r}$ and $g(\cdot,\cdot,\cdot,\cdot,\cdot):[0,T] \times \mathbb{R}^{r}\times \mathcal{U} \times \mathcal{X} \times \Gamma \rightarrow \mathbb{R}^{r}$ { and satisfy the following conditions,\\ {\bf Assumption (A1)}\\ \textit{(At most linear growth)~~ There exists a constant $ C_{1}< \infty $ for any $ i \in \mathcal{X} $ such that}\\ ${|\sigma(t,x,u,i)|}^{2} +{||b(t,x,u,i)||}^{2}+\int_{\mathbb{R}}{{||g(t,x,u,i, \gamma)||}^{2}}\lambda(d\gamma) \leq C_{1}(1+||x||^{2})$\\ \textit{(Lipschitz continuity)~~ There exists a constant $C_{2}< \infty$ for any $ i \in \mathcal{X} $ such that}\\ ${|\sigma(t,x,u,i)-\sigma(t,y,u,i)|}^{2} +{||b(t,x,u,i)-b(t,y,u,i)||}^{2}+\int_{\Gamma}{||g(t,x,u,i,\gamma)-g(t,y,u,i,\gamma)||^{2}}\lambda(d\gamma) \leq C_{2}||x-y||^{2}$ $\forall x,y \in \mathbb{R}^{r}$.\\ Then $X(t)$ is a unique cadlag adapted solution given by (\ref{3.1}) refer Theorem 1.19 of \cite{Oks}.}\\ \indent Define $a(t,x,u,i)=\sigma(t,x,u,i)\sigma'(t,x,u,i)$ is a $\mathbb{R}^{r\times r}$ matrix and $a_{kl}(t,x,u,i)$ is the $(k,l)^{th}$ element of the matrix $a$ while $b_{k}(t,x,u,i)$ is the $k^{th}$ element of the vector $b(t,x,u,i)$. We assume that $N(\cdot,\cdot), N_{1}(\cdot,\cdot)$ and $\theta_{0},W_{t},X_{0}$ defined on ($\Omega,\mathcal{F},\mathbb{P}$) are independent. For future use we define the compensated Poisson measure $\tilde{N}(dt,d\gamma)=N(dt,d\gamma)-{\lambda} \pi(d\gamma)dt$, where $\pi(\cdot)$ is the jump distribution { (is a probability measure) and $0<{\lambda}<\infty$ is the jump rate} { such that $\int_{\Gamma}{\min({||\gamma||}^{2},1)}\lambda{(d\gamma)}<\infty$.}\\ \indent Consider the performance criterion \begin{eqnarray}\label{3.2} J^{u}(x,i,y)=E^{x,i,y}[\int_{0}^{T}{f_{1}(t,X({t}),u(t),\theta({t}),Y(t))dt+f_{2}(X(T),\theta(T),Y(T))}], \end{eqnarray} where $f_{1}:[0,T] \times \mathbb{R}^{r}\times \mathcal{U} \times \mathcal{X} \times \mathbb{R}_{+} \rightarrow \mathbb{R}$ is continuous and $f_{2}: \mathbb{R}^{r} \times \mathcal{X} \times \mathbb{R}_{+}\rightarrow \mathbb{R}$ is concave. We say that the admissible class of controls $u \in \mathcal{A}(T)$ if \begin{eqnarray*} E^{x,i,y}\bigg[\int_{0}^{T}|f_{1}(t,X(t),u(t),\theta(t),Y(t))|dt+f_{2}(X(T),\theta(T),Y(T))]\bigg]<\infty. \end{eqnarray*} The problem is to maximize $J^{u}$ over all $u \in \mathcal{A}(T)$ i.e. we seek $\hat{u} \in \mathcal{A}(T)$ such that \begin{eqnarray}\label{3.3} J^{\hat{u}}(x,i,y)=\sup_{u \in \mathcal{A}(T)}J^{u}(x,i,y), \end{eqnarray} where $\hat{u}$ is an optimal control.\\ Define a Hamiltonian $\mathcal{H}: [0,T] \times \mathbb{R}^{r}\times \mathcal{U} \times \mathcal{X} \times \mathbb{R}_{+} \times \mathbb{R}^{r} \times \mathbb{R}^{r \times r} \times \mathbb{R}^{r} \rightarrow \mathbb{R}$ by, \begin{eqnarray}\label{3.4} \mathcal{H}(t,x,u,i,y,p,q,\eta)&:=& f_{1}(t,x,u,i,y)+\bigg(b^{'}(t,x,u,i)-\int_{\Gamma}{g^{'}(t,x,u,i,\gamma)}\pi(d\gamma)\bigg)p+tr(\sigma^{'}(t,x,u,i)q)\nonumber\\ &+&\bigg(\int_{{\Gamma}}{g^{'}(t,x,u,i,\gamma)}\pi(d\gamma)\bigg)\eta. \end{eqnarray} We assume that the Hamiltonian $\mathcal{H}$ is differentiable with respect to $x$. {The adjoint equation corresponding to $u$ and $X^{u}$ in the unknown adapted processes $p(t) \in \mathbb{R}^{r}$,$ q(t) \in \mathbb{R}^{r \times r}$, $\eta:\mathbb{R}_{+} \times \mathbb{R}^{r}-\{0\}\rightarrow \mathbb{R}^{r }$ and $\tilde{\eta}(t,z)=(\eta^{(1)}(t,z),...,\eta^{(r)}(t,z))^{'}$, where $\tilde{\eta}^{(n)}(t,z) \in \mathbb{R}^{r \times r}$ for each $n=1,2,...,r$, is the backward stochastic differential equation (BSDE)}, \begin{eqnarray}\label{3.5} dp(t)&=& -\nabla_{x}\mathcal{H}(t,X(t),u(t),\theta(t),p(t),q(t),\eta(t,\gamma))dt+q^{'}(t)dW(t)+\int_{\Gamma}{\eta(t,\gamma) \tilde{N}(dt,d\gamma)}\nonumber \\&+&\int_{\mathbb{R}}\tilde{\eta}(t,z) \tilde{N}_{1}(dt,dz), \nonumber \\ p(T)&=&\nabla_{x}f_{2}(X(T),\theta(T),Y(T)).~~a.s. \end{eqnarray} We have assumed that $\mathcal{H}$ is differentiable with respect to $x=X(t)$ and is denoted as \\$\nabla_{x}\mathcal{H}(t,X(t),u(t),\theta(t),p(t),q(t),\eta(t,\gamma))$. { As per Remark 2.1, for the special case where the semi-Markov process has exponential holding time distribution, we would have (\ref{3.5}) to be a BSDE with Markov chain switching. For this special case, Cohen and Elliott \cite{CE} have provided conditions for uniqueness of the solution. However, corresponding uniqueness result for the semi-Markov modulated BSDE as in (3.5) seems not available in the literature. Since this paper concerns sufficient conditions, we will assume ad hoc that a solution to this BSDE exists and is unique. }\\ {\bf Remark 3.1}~~ Notice that there are jumps in the adjoint equation (3.5) attributed to jumps in the semi-Markov process $\theta({t})$. This is because the drift, the diffusion and the jump kernel of the process $X({t})$ is modulated by a semi-Markov process. Also note that the unknown process $\tilde{\eta}(t,z)$ in the adjoint equations (\ref{3.5}) does not appear in the Hamiltonian (\ref{3.4}). \section{Sufficient Stochastic Maximum principle} In this section we state and prove the sufficient stochastic maximum principle.\\ {\bf Theorem 4.1}(Sufficient Maximum principle) Let $\hat{u} \in \mathcal{A}(T)$ with corresponding solution $\hat{X} \triangleq X^{\hat{u}}$. Suppose there exists a solution ($\hat{p}(t),\hat{q}(t),\hat{\eta}(t,\gamma),\hat{\tilde{\eta}}(t,z)$)of the adjoint equation (\ref{3.5}) satisfying \\ \begin{eqnarray}\label{4.1} &&E \int_{0}^{T}{||\bigg(\sigma(t,\hat{X}(t),\theta(t))-\sigma(t,X^{u}(t),\theta(t))\bigg)^{'}\hat{p}(t)||^{2}}dt< \infty \\ &&E \int_{0}^{T}{||\hat{q}^{'}(t)\bigg(\hat{X}(t)-X^{u}(t)\bigg)||^{2}}dt< \infty \\ &&E \int_{0}^{T}{||(\hat{X}(t)-X^{u}(t))^{'}\hat{\eta}(t,\gamma)||^{2}\pi(d\gamma)}dt< \infty \\ &&E \int_{0}^{T}{|\bigg(\hat{X}(t)-X^{u}(t)\bigg)^{'}\hat{\tilde{\eta}}(t,z)|^{2}m(dz)}dt< \infty. \end{eqnarray} for all admissible controls $u \in \mathcal{A}(T)$. If we further suppose that \\ 1. \begin{eqnarray}\label{4.5} \mathcal{H}(t,\hat{X}({t}),\hat{u}({t}),\theta(t),Y(t),\hat{p}({t}),\hat{q}({t}),\hat{\eta}{(t,\cdot)})=\sup_{u \in\mathcal{A}(T)} \mathcal{H}(t,\hat{X}({t}),{u}({t}),\theta(t),Y(t),\hat{p}({t}),\hat{q}({t}),\hat{\eta}{(t,\cdot)}). \end{eqnarray} 2. for each fixed pair $(t,i,y) \in ([0,T] \times \mathcal{X} \times \mathbb{R}_{+})$,~~$\hat{\mathcal{H}}(x):= \sup_{ u \in \mathcal{A}(T)}\mathcal{H}(t,x,u,i,y,\hat{p}(t),\hat{q}(t),\hat{\eta}(t,\cdot))$ exists and is a concave function of $x$. Then $\hat{u}$ is an optimal control.\\ {\textit{Proof}}~~Fix $u \in \mathcal{A}(T)$ with corresponding solution $X=X^{u}$. For sake of brevity we would henceforth represent ($t,\hat{X}(t-),\hat{u}(t-),\theta(t-),Y(t-)$) by ($t,\hat{X}(t-)$) and ($t,{X}(t-),{u}(t-),\theta(t-),Y(t-)$) by ($t,{X}(t-)$). Then, \begin{eqnarray*} J(\hat{u})-J(u)=E\bigg(\int_{0}^{T}\bigg({f_{1}(t,\hat{X}(t))-f_{1}(t,X(t))}\bigg)dt+f_{2}(\hat{X}(T),\theta(T),Y(T))-f_{2}(X(T),\theta(T),Y(T))\bigg). \end{eqnarray*} By use of concavity of $f_{2}(\cdot,i,y)$ we have for each $i \in \mathcal{X},~ y \in \mathbb{R}_{+}$ and (\ref{3.5}) to obtain the inequalities, \begin{eqnarray*} E\bigg(f_{2}(\hat{X}(T),\theta(T),Y(T))-f_{2}({X}(T),\theta(T),Y(T))\bigg) & \geq & E\bigg((\hat{X}(T)-X(T))^{'}\nabla_{x}f_{2}(\hat{X}(T),\theta(T),Y(T))\bigg) \nonumber \\ &\geq & E \bigg((\hat{X}(T)-X(T))^{'}\hat{p}(T)\bigg). \end{eqnarray*} which gives \begin{eqnarray}\label{4.6} J(\hat{u})-J(u) \geq E {\int_{0}^{T}{\bigg(f_{1}(t,\hat{X}(t))-f_{1}(t,X(t))\bigg)}}dt + E\bigg((\hat{X}(T)-X(T))^{'}\hat{p}(T)\bigg). \end{eqnarray} We now expand the above equation (\ref{4.6}) term by term. For the first term in this equation we use the definition of $\mathcal{H}$ as in (\ref{3.4}) to obtain \begin{eqnarray}\label{4.7} &&E\int_{0}^{T}{\bigg(f_{1}(t,\hat{X}(t))-f_{1}(t,X(t))\bigg)}dt \nonumber\\ &=& E\int_{0}^{T}\bigg(\mathcal{H}(t,\hat{X}(t),\hat{u}(t),\theta(t),\hat{p}(t),\hat{q}(t),\hat{\eta}(t,\gamma))\nonumber \\ &-&\mathcal{H}(t,{X}(t),{u}(t),\theta(t),{p}(t),{q}(t),{\eta}(t,\gamma))\bigg)dt \nonumber \\ &-&E\int_{0}^{T}\bigg[\bigg(b(t,\hat{X}(t))-b(t,{X}(t))\nonumber\\ &-&\int_{\Gamma}{\bigg(g(t,\hat{X}(t-),\hat{u}(t-),\theta(t-),\gamma)-g(t,{X}(t-),{u}(t-),\theta(t-),\gamma)\bigg)}\pi(d\gamma)\bigg)\hat{p}(t)\nonumber\\ &+&tr\bigg((\sigma(t,\hat{X}(t))-\sigma(t,X(t)))^{'}\hat{q}(t)\bigg)\nonumber \\ &+&\int_{\Gamma}(g(t,\hat{X}(t-),\hat{u}(t-),\theta(t-),\gamma)-g(t,{X}(t-),{u}(t-),\theta(t-),\gamma))^{'}\eta(t,\gamma)\pi(d\gamma)\bigg]dt. \nonumber \\ \end{eqnarray} To expand the second term on the right hand side of (\ref{4.6}) we begin by applying the integration by parts formula to get, \begin{eqnarray*} (\hat{X}(T)-X(T))^{'}\hat{p}(T)&=& \int_{0}^{T}{(\hat{X}(t)-X(t))^{'}}d\hat{p}(t)\\ &+&\int_{0}^{T}{\hat{p}^{'}(t)d(\hat{X}(t)-X(t))}+[\hat{X}-X,\hat{p}](T). \end{eqnarray*} Substitute for $X$, $\hat{X}$ and $\hat{p}$ from (\ref{3.1}) and (\ref{3.5}) to obtain, \begin{eqnarray*} &&(\hat{X}(T)-X(T))^{'}\hat{p}(T) \\ &=&\int_{0}^{T}(\hat{X}(t)-X(t))^{'}\bigg(-\nabla_{x}\mathcal{H}(t,\hat{X}({t}),\hat{u}({t}),\hat{p}({t}),\hat{q}(t),\hat{\eta}({t,\gamma}))dt+\hat{q}^{'}(t)dW(t)\\ &+&\int_{\Gamma}{\hat{\eta}(t,\gamma)\tilde{N}(dt,d\gamma)}+\int_{\mathbb{R}}{\hat{\tilde{\eta}}(t,z)\tilde{N}_{1}(dt,dz)}\bigg)\\ &+&\int_{0}^{T}\hat{p}^{'}(t)\bigg\{\bigg(\bigg(b(t,\hat{X}(t))-b(t,X(t))\bigg) -\int_{\Gamma}\bigg(g(t,\hat{X}(t),\hat{u}(t-),\theta({t-}),\gamma)\\ &-&g(t,{X}(t-),u(t-),\theta({t-}),\gamma)\bigg)\pi(d\gamma)\bigg)dt\\ &+&\bigg(\sigma(t,\hat{X}(t))-\sigma(t,X(t))\bigg)^{'}dW(t)\\ &+&\int_{\Gamma}{\bigg(g(t,\hat{X}(t-),\hat{u}(t-),\theta({t-}),\gamma)-g(t,{X}(t-),u(t-),\theta({t-}),\gamma)\bigg)}\tilde{N}(dt,d\gamma)\bigg\}\\ &+&\int_{0}^{T}\bigg[tr\bigg(\hat{q}^{'}(t)\bigg(\sigma(t,\hat{X}(t))-\sigma(t,X(t))\bigg)\bigg)\\ &+&\int_{\Gamma}\bigg({g(t,\hat{X}(t),\hat{u}(t-),\theta({t-}),\gamma)-g(t,{X}(t),u(t-),\theta({t-}),\gamma)}\bigg)^{'}\eta(t,\gamma)\pi(d\gamma)\bigg]dt. \end{eqnarray*} Due to integrability conditions ({4.1})-({4.4}), the integral with respect to the Brownian motion and the Poisson random measure are square integrable martingales which are null at the origin. Thus taking expectations we obtain \begin{eqnarray*} E\bigg((\hat{X}(T)&-&X(T))^{'}\hat{p}(T)\bigg) \\ &=&\int_{0}^{T}(\hat{X}(t)-X(t))^{'}\bigg(-\nabla_{x}\mathcal{H}(t,\hat{X}({t}),\hat{u}({t}),\hat{p}({t}),\hat{q}(t),\hat{\eta}({t,\gamma}))\bigg)dt\\ &+&\int_{0}^{T}\bigg[\hat{p}^{'}(t)\bigg(b(t,\hat{X}(t))-b(t,X(t)) -\int_{\Gamma}\bigg(g(t,\hat{X}(t-),\hat{u}(t-),\theta({t-}),\gamma)\\ &-&g(t,{X}(t),u(t-),\theta({t-}),\gamma)\bigg)\pi(d\gamma)\bigg)\\ &+&\int_{0}^{T} tr\bigg(\hat{q}^{'}(t)(\sigma(t,\hat{X}(t))-\sigma(t,X(t)))\bigg)\\ &+&\int_{\Gamma}{\bigg(\bigg({g(t,\hat{X}(t-),\theta({t-}),u(t-),\gamma)-g(t,{X}(t-),\theta({t-}),u(t-),\gamma)}\bigg)^{'}\eta(t,\gamma))\bigg)\pi(d\gamma)}\bigg]dt. \end{eqnarray*} \begin{eqnarray*} \end{eqnarray*} Substitute the last equation and (\ref{4.7}) into the inequality (\ref{4.6}) to find after cancellation that \begin{eqnarray}\label{4.8} J(\hat{u})-J(u) & \geq & E\int_{0}^{T}\bigg(\mathcal{H}(t,\hat{X}(t),\hat{u}(t),\theta(t),\hat{p}(t),\hat{q}(t),\hat{\eta}(t,\gamma))-\mathcal{H}(t,{X}(t),{u}(t),\theta(t),{p}(t),{q}(t),\eta(t,\gamma))\nonumber\\ &-&(\hat{X}(t)-X(t))^{'}\nabla_{x}\mathcal{H}(t,\hat{X}(t),\hat{u}(t),\theta(t),\hat{p}(t),\hat{q}(t),\hat{\eta}(t,\gamma))\bigg)dt. \end{eqnarray} We can show that the integrand on the RHS of (\ref{4.8}) is non-negative a.s. for each $t \in [0,T]$ by fixing the state of the semi-Markov process and then using the assumed concavity of $\hat{\mathcal{H}}(x)$, we apply the argument in Framstad et al. \cite{Fr} . This gives $J(\hat{u}) \geq J(u)$ and $\hat{u}$ is an optimal control.$\qed$ \section{Connection to the Dynamic programming} We show the connection between the stochastic maximum principle and dynamic programming principle for the semi-Markov modulated regime switching jump diffusion. This tantamounts to explicitly showing connection between the value function $V(t,x,i,y)$ of the control problem and the adjoint processes $p(t), q(t)$ ,$\eta(t,\gamma)$ and $\tilde{\eta}(t,z)$. In order to apply the dynamic programming principle we put the problem into a Markovian framework by defining \begin{eqnarray}\label{5.1} J^{u}(t,x,i,y) \triangleq E^{X(t)=x,\theta(t)=i,Y(t)=y}[\int_{t}^{T}{f_{1}(t,X({t}),u(t),\theta({t}),Y({t}))dt+f_{2}(X(T),\theta(T),Y(T))}]. \end{eqnarray} and put \begin{eqnarray}\label{5.2} V(t,x,i,y)=\sup_{u \in \mathcal{A}(T)}J^{u}(t,x,i,y)~~~~\forall~~(t,x,i,y) \in [0,T] \times \mathbb{R}^{r} \times \mathcal{X}\times \mathbb{R}_{+}. \end{eqnarray}\\ {\bf Theorem 5.1}~~\textit{Assume that $V(\cdot,\cdot,i,\cdot)\in \mathcal{C}^{1,{3},1}([0,T]\times \mathbb{R}^{r}\times \mathcal{X}\times \mathbb{R}_{+})$ for each $i,j \in \mathcal{X}$ and that there exists an optimal Markov control $\hat{u}(t,x,i,y)$ for (\ref{5.2}), with the corresponding solution $\hat{X}=X^{(\hat{u})}$. Define \begin{eqnarray}\label{5.3} p_{k}(t) &\triangleq & \frac{\partial V}{\partial x_{k}}(t,\hat{X}(t),\theta(t),Y(t)). \end{eqnarray} \begin{eqnarray}\label{5.4} q_{kl}(t) &\triangleq & \sum_{i=1}^{r}{\sigma_{il}(t,\hat{X}(t),\hat{u}(t),\theta(t))\frac{\partial^{2} V}{{\partial x_{i}}{\partial x_{k}}}(t,\hat{X}(t),\theta(t),Y(t))}. \end{eqnarray} \begin{eqnarray}\label{5.5} \eta^{(k)}(t,\gamma) &\triangleq & \frac{\partial V}{\partial x_{k}}(t,\hat{X}(t),j,Y(t))-\frac{\partial V}{\partial x_{k}}(t,\hat{X}(t),i,Y(t)). \end{eqnarray} \begin{eqnarray}\label{5.6} \tilde{\eta}^{(k)}(t,z) &\triangleq & \frac{\partial V}{\partial x_{k}}(t,\hat{X}(t-),\theta(t-)+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y({t-}),z))\nonumber \\ &-&\frac{\partial V}{\partial x_{k}}(t,\hat{X}({t-}),\theta({t-}),Y({t-})). \end{eqnarray} {for each $(k,l =1,...,r)$. Also we assume that the coefficients $b(t,x,u,i)$, $\sigma(t,x,u,i)$ and $g(t,x,u,i,\gamma)$ belong to $C^{1}_{b}(\mathbb{R}^{r})$.} Then $p(t), q(t), \eta(t,\gamma)$ and $\tilde{\eta}(t,z)$ solves the adjoint equation (\ref{3.5}).}\\\\ We prove this theorem by using the following Ito's formula.\\ {\bf Theorem 5.2}~~{Suppose $r $ dimensional process $X(t)=(X_{1}(t),...,X_{r}(t))$ or $\{X_{g}(t)\} $ indexed by $(g=1,2,...,r)$ satisfies the following equation, \begin{eqnarray*} dX_{g}(t)=b_{g}(t,X(t),u(t),\theta(t))dt+\sum_{m=1}^{r}{\sigma_{gm}(t,X(t),u(t),\theta(t))}dW_{m}(t)+\int_{\Gamma}{g_{g}(t,X(t-),u(t),\theta(t-),\gamma)}{N}(dt,d\gamma). \end{eqnarray*} for some $X(0)= x_{0} \in \mathbb{R}^{r}~~~a.s.$ . Further let us assume that the coefficients $b, \sigma, g$ satisfies the conditions of Assumption (A1).\\ Let $ V(\cdot,\cdot,i,\cdot)~\in~C^{1,{3},1}([0,T] \times \mathbb{R}^{r}\times \mathcal{X}\times \mathbb{R}_{+})$. Then the generalized Ito's formula is given by \begin{eqnarray*} &&V(t,X({t}),\theta({t}),Y({t}))- V(t,x,\theta,y)=\int_{0}^{t}{G V(s,X({s}),\theta({s}),Y({s}))ds}\\ &+&\int_{0}^{t}{(\nabla_{x} V(s,X({s}),\theta({s}),Y({s})))'\sigma(s,X({s}),\theta({s}))dW({s})} \\ &+& \int_{0}^{t}\int_{\Gamma}[V(s,X({s-})+g(s,X({s-}),u(s),\theta({s-}),\gamma),\theta({s-}),Y({s-}))\\ &-&V(s,X({s-}),\theta({s-}),Y({s-}))]\tilde{N}(ds,d\gamma) \\ &+&\int_{0}^{t}\int_{\mathbb{R}}[V(s,X({s-}),\theta({s-})+\bar{h}(\theta({s-}),Y({s-}),z),Y({s-})-\bar{g}(\theta({s-}),Y({s-}),z))\\ &-& V(s,X({s-}),\theta({s-}),Y({s-}))]\tilde{N}_{1}(ds,dz), \end{eqnarray*} where the local martingale terms are explicitly defined as \\ \begin{eqnarray*} dM_{1}(t)&\triangleq &{(\nabla_{x} V(t,X({t}),\theta({t}),Y({t})))'\sigma(t,X({t}),u(t),\theta({t}))dW_{t}},\\ dM_{2}(t)&\triangleq &\int_{\Gamma}{[V(t,X({t-})+g(t,X({t-}),u(t),\theta({t-}),\gamma),\theta({t-}),Y({t-}))-V(t,X({t-}),\theta({t-}),Y({t-}))]\tilde{N}(dt,d\gamma)},\\ dM_{3}(t)&\triangleq &\int_{\mathbb{R}}[V\bigg(t,X({t-}),\theta({t-})+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y(t-),z)\bigg)\\ &-&V(t,X({t-}),\theta({t-}),Y({t-}))]\tilde{N_{1}}(dt,dz), \end{eqnarray*} for \begin{eqnarray*} G V(t,x,i,y)&=&\frac{\partial V(t,x,i,y)}{\partial t}\\&+&\frac{1}{2}\sum_{g,l=1}^{r}{a_{gl}(t,x,i)\frac{\partial V(t,x,i,y)}{\partial x_{g}\partial x_{l}}}\\&+&\sum_{g=1}^{r}{b_{g}(t,x,i)\frac{\partial V(t,x,i,y)}{\partial x_{g}}} \\ &+&\frac{\partial V(t,x,i,y)}{\partial y}\\&+&\frac{f^{h}(y|i)}{1-F^{h}(y|i)}\sum_{j \neq i,j \in \mathcal{X},i=1}^{M}{p_{ij}[V(t,x,j,0)-V(t,x,i,y)]} \\ &+&\lambda{\int_{\Gamma}{({V(t,x+g(t,x,i,\gamma),i,y)}-{V(t,x,i,y)})}\pi(d\gamma)},\\ \forall~t~\in~[0,T]~,x \in \mathbb{R}^{r}, (i = 1,....,M),~y \in \mathbb{R}_{+}. \end{eqnarray*} } {\textit{Proof}}~~ For details refer to Theorem 5.1 in Ikeda and Watanabe \cite{IW}. $\qed$\\ {\textit{Proof of Theorem 5.1}}~~From the standard theory of the Dynamic programming the following HJB equation holds: \begin{eqnarray*} \frac{\partial V}{\partial t}(t,x,i,y)+\sup_{u \in \mathcal{U}}\{f_{1}(t,x,u,i,y)+\mathcal{A}^{u}V(t,x,i,y)\}=0,\\ V(T,x,i,y)=f_{2}(x,i,y). \end{eqnarray*} where $\mathcal{A}^{u}$ is the infinitesimal generator and the supremum is attained by $\hat{u}(t,x,i,y)$. Define \begin{eqnarray*} F(t,x,u,i,y)=f_{1}(t,x,u,i,y)+\frac{\partial V}{\partial t}(t,x,i,y)+\mathcal{A}^{u}V(t,x,i,y). \end{eqnarray*} We assume that $f_{1}$ is differentiable w.r.t to $x$. We use the Ito's formula as described in Theorem 5.2 to get, \begin{eqnarray} F(t,x,u,i,y)&=&f_{1}(t,x,u,i,y)+\frac{\partial V}{\partial t}(t,x,i,y)\nonumber \\ &+&\sum_{k=1}^{r}{\frac{\partial V}{\partial x_{k}}(t,x,i,y)b_{k}(t,x,u,i) }+ \frac{1}{2}\sum_{k=1}^{r}\sum_{l=1}^{r}{\frac{\partial^{2}V}{\partial x_{k} \partial x_{l}}}(t,x,i,y)\sum_{i=1}^{r}{\sigma_{ki}(t,x,u,i)\sigma_{li}(t,x,u,i)}\nonumber \\ &+&\sum_{j \neq i,i=1}^{M}{\frac{p_{ij}f^{h}(y|i)}{1-F^{h}(y|i)}}{(V(t,x,j,0)-V(t,x,i,y))}+\frac{\partial V}{\partial y}(t,x,i,y) \nonumber \\ &+& \lambda \int_{\Gamma}{(V(t,x+g(t,x,u,i,\gamma),i,y)-V(t,x,i,y))}\pi(d\gamma). \end{eqnarray} Differentiate $F(t,x,\hat{u}(t,x,i,y),i,y)$ with respect to $x_{g}$ and evaluate at $x=\hat{X}(t)$, $i=\theta(t)$ and $y=Y(t)$, we get, \begin{eqnarray}\label{5.8} 0&=&\frac{\partial f_{1}}{\partial x_{g}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t),Y(t))\nonumber\\ &+& \frac{\partial^{2}V}{\partial x_{g} \partial t}(t,\hat{X}(t),\theta(t),Y(t))+\sum_{k=1}^{r}{\frac{\partial^{2}V}{\partial x_{g} \partial x_{k}}(t,\hat{X}(t),\theta(t),Y(t))b_{k}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}\nonumber\\ &+&\sum_{k=1}^{r}{\frac{\partial V}{\partial x_{k}}(t,\hat{X}(t),\theta(t),Y(t))\frac{\partial b_{k}}{\partial x_{g}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}\nonumber\\ &+&\frac{1}{2}\sum_{k=1}^{r}\sum_{l=1}^{r}{\frac{\partial^{3}V}{\partial x_{g} \partial x_{k} \partial x_{l}}(t,\hat{X}(t),\theta(t),Y(t))}\nonumber\\&\times&{\sum_{i=1}^{r}{\sigma_{k,i}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))\sigma_{l,i}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}}\nonumber\\ &+&\frac{1}{2}\sum_{k=1}^{r}\sum_{l=1}^{r}{\frac{\partial^{2}V}{\partial x_{k} \partial x_{l} }(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t),Y(t))}\nonumber\\&\times&{\frac{\partial}{\partial x_{g}}\sum_{i=1}^{r}{\sigma_{k,i}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))\sigma_{l,i}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}}\nonumber \\ &+&\sum_{j \neq i, j \in \mathcal{X}}^{M}{\frac {p_{ij}f^{h}(y|i)}{1-F^{h}(y|i)}\bigg(\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),j,0)-\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),i,y)\bigg)}\nonumber\\ &+&\lambda \int_{\Gamma}\bigg({\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t)+g(t,\hat{X}(t),\theta(t),\gamma),\theta(t),Y(t))-\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),\theta(t),Y(t))}\bigg)\pi(d\gamma). \end{eqnarray} Next define, $Y_{g}=\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),\theta(t),Y(t))$ for ($g =1,...,r$). By Ito's formula (Theorem 5.2) we obtain the dynamics of $Y_{g}(t)$ as follows, \begin{eqnarray*} dY_{g}(t)&=&\bigg\{\frac{\partial^{2}V}{\partial {x_{g}} \partial{t}}(t,\hat{X}(t),\theta(t),Y(t))+\sum_{k=1}^{r}{\frac{\partial^{2}V}{\partial {x_{g}} \partial{x_{k}}}(t,\hat{X}(t),\theta(t),Y(t))b_{k}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}\nonumber\\ &+&\frac{1}{2}\sum_{k=1}^{r}\sum_{l=1}^{r}{\frac{\partial^{3}V}{\partial x_{g} \partial x_{k} \partial x_{l}}(t,\hat{X}(t),\theta(t),Y(t))}\nonumber\\&\times&{\sum_{i=1}^{r}{\sigma_{ki}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t)) \times \sigma_{li}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}}\\ &+&\sum_{j \neq i, j =1}^{M}{\frac {p_{ij} f^{h}(y|i)}{1-F^{h}(y|i)}(\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),j,0)-\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),i,y))}\nonumber \\ &+&\lambda \int_{\Gamma}{\bigg(\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t)+g(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t),\gamma),\theta(t),Y(t))}\\&-&{\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t),\theta(t),Y(t))\bigg)}\pi(d\gamma)\bigg\}dt\\ &+&\sum_{k=1}^{r}\frac{{\partial^{2}V}}{{\partial x_{g} \partial x_{k}}}(t,\hat{X}(t),\theta(t),Y(t))\sum_{j=1}^{r}{\sigma_{kj}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))dW_{j}(t)}\nonumber \\ &+&\int_{\Gamma}\bigg\{\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t-)+g(t,\hat{X}(t-),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t-),\gamma),\theta(t-),Y(t-))\\ &-&\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t-),\theta(t-),Y(t-))\bigg\}\tilde{N}(dt,d\gamma)\\ &+&\int_{\mathbb{R}}\bigg\{\frac{\partial V}{\partial x_{g}}((t,X(t-),\theta(t-)+\bar{h}(\theta(t-),Y(t-),z),Y(t-)-\bar{g}(\theta(t-),Y(t-),z)))\\&-&\frac{\partial {V}}{\partial x_{g}}(t,\hat{X}(t-),\theta(t-),Y(t-))\bigg\}{\tilde{N}_{1}(dt,dz)}. \end{eqnarray*} We substitute $\frac{\partial^{2}V}{\partial{x_{g}}\partial t } $ from (\ref{5.8}) to get, \begin{eqnarray}\label{5.9} dY_{g}(t)&=& -\frac{\partial f_{1}}{\partial x_{g}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t),Y(t)))\nonumber\\ &-&\sum_{k=1}^{r}{\frac{\partial V}{\partial x_{k}}(t,\hat{X}(t),\theta(t),Y(t))\frac{\partial b_{k}}{\partial {x_{g}}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}\nonumber \\ &-&\frac{1}{2}\sum_{k=1}^{r}\sum_{l=1}^{r}{\frac{\partial^{2}V}{\partial x_{k}\partial x_{l}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t),Y(t))}\nonumber\\&\times&{\frac{\partial}{\partial x_{g}}(\sum_{k=1}^{r}{\sigma_{ki}(t,\hat{X}(t),\theta(t))}{ \sigma_{li}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))})}\nonumber\\ &+& \sum_{k=1}^{r}{\frac{\partial^{2}V}{\partial x_{g}\partial x_{k}}(t,\hat{X}(t),\theta(t),Y(t))\sum_{j=1}^{r}{\sigma_{kj}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))dW_{j}(t)}}\nonumber\\ &+&\int_{\Gamma}\bigg\{(\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t-)+g(t,X(t-),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t-),\gamma),\theta(t-),Y(t-))\nonumber\\ &-&\frac{\partial V}{\partial x_{g}}(t,\hat{X}(t-),\theta(t-),Y(t-)))\bigg\}\tilde{N}(dt,d\gamma)\nonumber\\ &+&\int_{\mathbb{R}}\bigg\{\frac{\partial V}{\partial x_{g}}((t,X(t-),\theta(t-)+\bar{h}(\theta(t-),Y(t-),z),Y(t-)-\bar{g}(\theta(t-),Y(t-),z)))\nonumber\\&-&\frac{\partial {V}}{\partial x_{g}}(t,\hat{X}(t-),\theta(t-),Y(t-))\bigg\}{\tilde{N}_{1}(dt,dz)}. \end{eqnarray} We have the following identity, \begin{eqnarray}\label{5.10} &&\frac{1}{2}\sum_{k=1}^{r}\sum_{l=1}^{r}{\frac{\partial^{2}V}{\partial x_{k}\partial x_{l}}(t,\hat{X}(t),\theta(t),Y(t))}\nonumber\\&\times&{\frac{\partial}{\partial x_{g}}\bigg(\sum_{i=1}^{r}{\sigma_{ki}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))\sigma_{li}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}\bigg)}\nonumber \\ &=&\sum_{k=1}^{r}\sum_{l=1}^{r}\sum_{i=1}^{r}{\sigma_{il}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t)){\frac{\partial^{2}V}{\partial x_{i}\partial x_{k}}(t,\hat{X}(t),\theta(t),Y(t))}}\nonumber\\&\times&{{\frac{\partial \sigma_{kl}}{\partial x_{g}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t))}}. \end{eqnarray} Next, from (\ref{3.4}) we obtain, \begin{eqnarray}\label{5.11} &&\frac{\partial \mathcal{H}}{\partial x_{g}}(t,X(t),u(t),\theta(t),Y(t),p(t),q(t),\eta(t,\gamma))\nonumber \\&=&\frac{\partial f_{1}}{\partial x_{g}}(t,\hat{X}(t),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t),Y(t))\nonumber\\ &+&\sum_{i=1}^{r}\bigg(\frac{\partial b_{i}}{\partial x_{g}}(t,\hat{X}(t-),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t-))\nonumber\\&-&\int_{\Gamma}{\frac{\partial g_{i}}{\partial x_{g}}(t,X(t-),\hat{u}(t,\hat{X}(t),\theta(t),Y(t)),\theta(t-),\gamma)}\pi(d\gamma)\bigg)p_{i}(t)+tr(\frac{\partial \sigma^{'}(t,x,\hat{u},\theta(t))}{\partial x_{g}}q)\nonumber\\ &+&{\sum_{i=1}^{r}\int_{\Gamma}{\frac{\partial g_{i}}{\partial x_{g}}(t,X(t-),\theta(t-),\gamma)}\pi(d\gamma)(\eta^{(g)}_{i}(t,\gamma))}. \end{eqnarray} We also note that \begin{eqnarray*} tr(\frac{\partial \sigma^{'}(t,x,u,i)}{\partial x_{g}}q)&=&\sum_{l=1}^{r}{[\frac{\partial \sigma^{'}(t,x,u,i)}{\partial x_{g}}q ]_{ll}}\\ &=&\sum_{l=1}^{r}\sum_{k=1}^{r}q_{k,l}\frac{\partial \sigma_{kl}}{\partial x_{g} }(t,x,u,i). \end{eqnarray*} Substitute (\ref{5.3})-(\ref{5.6}) and (\ref{5.11}) gives, \begin{eqnarray} dY_{g}(t)&=&-\frac{\partial\mathcal{H}}{\partial x_{g}}(t,X(t),u(t),\theta(t),Y(t),p(t),q(t),\eta(t,\gamma))dt+\sum_{j=1}^{r}{q_{gj}}(t)dW_{j}(t)\nonumber\\ &+&\int_{\Gamma}\eta(t,\gamma) \tilde{N}(dt,d\gamma)+\int_{\mathbb{R}}\tilde{\eta}(t,z) \tilde{N}_{1}(dt,dz). \end{eqnarray} Since $Y_{g}(t)=p_{g}(t)$ for each $g=1,...,r$, we have shown that $p(t),q(t),\eta(t,\gamma)$ and $\tilde{\eta}(t,z)$ solve the adjoint equation (\ref{3.5}). $\qed$\\ \section{Applications} We illustrate the theory developed towards applying it to some key financial wealth optimization problems. For an early motivation on applying sufficient maximum principle, we first consider wealth dynamics to follow semi-Markov modulated diffusion (no jumps case) and apply it towards the risk-sensitive control portfolio optimization problem. We follow it up by illustrating an application of semi-Markov modulated jump-diffusion wealth dynamics to a quadratic loss minimization problem. Unless otherwise stated, all the processes defined in this section are one dimensional.\\ {\bf Risk-sensitive control portfolio optimization}~~Let us consider a financial market consisting of two continuously traded securities namely the risk less bond and a stock. The dynamics of the riskless bond is known to follow \begin{eqnarray*} dS_{0}(t)=r(t,\theta(t-))S_{0}(t)dt~~~S_{0}(0)=1. \end{eqnarray*} where $r(t,\theta(t))$ is the risk-free interest rate at time $t$ and is modulated by an underlying semi-Markov process as described earlier. The dynamics of the stock price is given as \begin{eqnarray*} dS_{1}(t)=S_{1}(t)[(\mu(t,\theta(t-)))dt+\sigma(t,\theta(t-))dW(t)], \end{eqnarray*} where $(\mu(t,\theta(t-)))$ is the instantaneous expected rate of return and as usual $\sigma(t,\theta(t-))$ is the instantaneous volatility rate. The stock price process is thus driven by a 1-d Brownian motion. We denote the wealth of the investor to be $X(t) \in \mathbb{R}$ at time $t$. He holds $\theta_{1}(t)$ units of stock and $\theta_{0}(t)=1-\theta_{1}(t)$ units is held in the riskless bond market. From the self-financing principle (refer Karatzas and Shreve \cite{KS}), the wealth process follows the dynamics given as, \begin{eqnarray*} dX(t)=(r(t,\theta(t-))X(t)+h(t)\sigma(t,\theta(t-))\bar{m}(t,\theta(t-)))dt+h(t)\sigma(t,\theta(t-))dW(t)~~~X(0)=x, \end{eqnarray*} where $h(t)=\theta_{1}(t)S_{1}(t)$, { $\bar{m}(t,i)=\frac{\mu(t,i)-r(t,i)}{\sigma(t,i)} \geq 0$ and the variables $r(t,i), b(t,i)$ and $\sigma(t,i),$ and $\sigma^{-1}(t,i)$ for each $i \in \mathcal{X}$ are measurable and uniformly bounded in $t \in [0,T]$}. { Also $h(\cdot)$ occuring in the drift and diffusion term in above dynamics of $X(t)$ satisfies the following conditions\\ 1. $E[\int_{0}^{T}{h^{2}(t)dt}]< \infty$\\ 2. $E[\int_{0}^{T}{|r(t,\theta(t-))X(t)+h(t)\sigma(t,\theta(t-))\bar{m}(t,\theta(t-))|}dt+\int_{0}^{T}{h^{2}(t)\sigma^{2}(t,\theta(t-))}dt]< \infty$\\ 3. The SDE for $X$ has a unique strong solution.\\ These conditions on $h(\cdot)$ are needed in order to prevent doubling strategies which otherwise would yield arbitrary profit at time $T$ for an investor.}\\ \indent In a classical risk-sensitive control optimization problem, the investor aims to maximize over some admissible class of portfolio $\mathcal{A}(T)$ the following risk-sensitive criterion given by \begin{eqnarray*} J(\hat{h}(\cdot),x)&=&\max_{h \in \mathcal{A}(T)}{\frac{1}{\gamma}}\mathbb{E}[{X(T)}^{\gamma}|X(0)=x,\theta(0)=i,Y(0)=y],~~~\gamma \in (1,\infty) \\ &=& -\min_{h \in \mathcal{A}(T)} \frac{1}{\gamma} \mathbb{E}[{X(T)}^{\gamma}|X(0)=x,\theta(0)=i,Y(0)=y], \end{eqnarray*} where the exogenous parameter $\gamma$ is the usual risk-sensitive criterion that describes the risk attitude of an investor. Thus the optimal expected utility function depends on $\gamma$ and is a generalization of the traditional stochastic control approach to utility optimization in the sense that now the degree of risk aversion of the investor is explicitly parameterized through $\gamma$ rather than importing it in the problem via an exogeneous utility function. See Whittle \cite{Wh} for general overview on risk-sensitive control optimization. We now use the sufficient maximum principle (Theorem 4.1). Set the control problem $u(t)\triangleq h(t)$.\\ The corresponding Hamiltonian (for the non-jump case)(\ref{3.4}) becomes, \begin{eqnarray*} \mathcal{H}(t,x,u,i,p,q)=(r(t,i)x+u\sigma(t,i)\bar{m}(t,i))p+u\sigma(t,i)q. \end{eqnarray*} The adjoint process (\ref{3.5}) is given by \begin{eqnarray}\label{6.1} dp(t)&=&-r(t,\theta(t-))p(t)dt+q(t)dW(t)+\int_{\mathbb{R}}{\tilde{\eta}(t,z)\tilde{N}_{1}(dt,dz)},\nonumber \\ p(T)&=&X(T)^{\gamma-1}~~~~a.s.. \end{eqnarray} We need to determine $p(t),q(t)$ and $\eta(t,z)$ in (\ref{6.1}). Going by the terminal condition $p(T)$ we observe that the adjoint process $p$ is the first derivative of $(x^{\gamma})$. Hence we assume that $p(t)$ defined as, \begin{eqnarray*} p(t)=(X(t))^{\gamma-1}e^{{\phi(t,\theta(t),Y(t))}}. \end{eqnarray*} where $\phi(T,\theta(T)=i,Y(T))=0~~~a.s.$ for each $i \in \{1,...,M\}$. Using the Ito's formula we get, \begin{eqnarray}\label{6.2} &&\frac{dp(t)}{p(t)}=\sum_{i=1}^M {1}_{\theta(t-)=i}\bigg((\gamma-1)\bigg\{(r(t,\theta(t-))+\frac{u(t)\sigma(t,\theta(t-))\bar{m}(t,\theta(t-))}{X(t)}\bigg)\nonumber\\ &+&\frac{1}{2}(\gamma-1)(\gamma-2)\sigma^{2}(t,\theta(t-))\frac{u^{2}(t)}{X^{2}(t)} \nonumber \\ &+&\phi_{t}(t,\theta(t-),y)+\phi_{y}(t,\theta(t-),y)+\frac{f^{h}(y|\theta(t-)=i)}{1-F^{h}(y|\theta(t-)=i)}\sum_{j \neq i}{p_{ij}(\phi(t,j,0)-\phi(t,\theta(t-),y))}\bigg\}dt \nonumber \\ &+&{(\gamma-1)\frac{u(t)}{X(t)}\sigma(t,\theta(t-))}dW(t) \nonumber \\ &+&\int_{\mathbb{R}}\bigg(\phi(t,X(t-),\theta(t-)+\bar{h}(\theta(t-),Y(t-),z),Y(t-)-\bar{g}(\theta(t-),Y(t-),z))\nonumber\\ &-&\phi(t,\theta(t-),Y(t-))\bigg)\tilde{N}_{1}(dt,dz).\nonumber\\ \end{eqnarray} Comparing the coefficient of (\ref{6.2}) with that in (\ref{6.1}) we get \begin{eqnarray}\label{6.3} -r(t,\theta(t-))&=&\sum_{i=1}^M {1}_{\theta(t-)=i}\bigg((\gamma-1)\bigg(r(t,\theta(t-))+\frac{u(t)\sigma(t,\theta(t-))\bar{m}(t,i)}{X(t)}\bigg)+\frac{1}{2}(\gamma-1)(\gamma-2)\frac{u^{2}(t)}{X^{2}(t)} \nonumber \\ &+& \phi_{t}(t,\theta(t-),y)+\phi_{y}(t,\theta(t-),y)+\frac{f^{h}(y|i)}{1-F^{h}(y|\theta(t-)=i)}\sum_{j \neq i}{p_{ij}(\phi(t,j,0)-\phi(t,\theta(t-),y))}\bigg).\nonumber\\ \end{eqnarray} \begin{eqnarray}\label{6.4} {q}(t)=(\gamma-1)\frac{u(t)}{X(t)}\sigma(t,\theta(t-)){p}(t). \end{eqnarray} \begin{eqnarray}\label{6.5} \tilde{\eta}(t,z)&=&\bigg(\phi(t,\theta(t-)+\bar{h}(\theta(t-),Y(t-),z),Y(t-)-\bar{g}(\theta(t-),Y(t-),z))\nonumber \\ &-&\phi(t,\theta(t-),Y(t-))\bigg)p(t). \end{eqnarray} Let $\hat{u} \in \mathcal{A}(T)$ be a candidate optimal control corresponding to the wealth process $\hat{X}$ and the adjoint triplet ($\hat{p},\hat{q},\hat{\eta}$), then from the Hamiltonian (\ref{3.4}) for all $u \in \mathbb{R}$ we have \begin{eqnarray}\label{6.6} \mathcal{H}(t,\hat{X}(t),u,\theta(t),\hat{p}(t),\hat{q}(t))=\bigg(r(t,\theta(t))\hat{X}(t)+u\sigma(t,\theta(t)) \bar{m}(t,\theta(t))\bigg)\hat{p}(t)+u\sigma(t,\theta(t))\hat{q}(t). \end{eqnarray} As this is a linear function of $u$, we guess that the coefficient of $u$ vanishes at optimality, which results in the equality \begin{eqnarray}\label{6.7} \bar{m}(t,\theta(t-))\hat{p}(t)+\hat{q}(t)=0. \end{eqnarray} Substitute equation (\ref{6.7}) in (\ref{6.4}) to obtain the expression for the control as \begin{eqnarray}\label{6.8} \hat{u}(t)=\frac{\bar{m}(t,\theta(t-))}{(1-\gamma)\sigma(t,\theta(t-))}\hat{X}(t). \end{eqnarray} We now aim to determine the explicit expression for ${p}(t)$ which is only possible if we can determine what $\phi(t,\theta(t),Y(t))$ is. We substitute $\hat{u}$ from above and input it in equation (\ref{6.3}) to get \begin{eqnarray}\label{6.9} 0&=&\gamma r(t,\theta(t-))-{\bar{m}^{2}(t,\theta(t-))}+\frac{(2-\gamma)}{(1-\gamma)}\frac{\bar{m}^{2}(t,\theta(t-))}{2\sigma^{2}(t,\theta(t-))}\nonumber\\ &+& \phi_{t}(t,\theta(t-),y)+\phi_{y}(t,\theta(t-),y)+\frac{f^{h}(y|\theta(t-)=i)}{1-F^{h}(y|\theta(t-)=i)}\sum_{i=1,j \neq i}^{M}{p_{ij}(\phi(t,j,0)-\phi(t,\theta(t-),y))}.\nonumber\\ \end{eqnarray} with terminal boundary condition given as $\phi(T,\theta(T),Y(T))=0$~~~a.s. Consider the process \begin{eqnarray}\label{6.10} \tilde{\phi}(t,\theta(t),Y(t)) \triangleq E\bigg[\exp\bigg(\int_{t}^{T}\bigg\{\gamma r(s,\theta(s))-{\bar{m}^{2}(s,\theta(s))}+\frac{(2-\gamma)}{(1-\gamma)}\frac{\bar{m}^{2}(s,\theta(s))}{{2\sigma^{2}(s,\theta(s))}}\bigg\}ds\bigg)|{\theta(t-)=i,Y(t-)=y}\bigg].\nonumber\\ \end{eqnarray} We aim to show that $\phi=\tilde{\phi}$. For the same we define the following martingale, \begin{eqnarray}\label{6.11} R(t)\triangleq E\bigg[\exp\bigg(\int_{0}^{T}{\bigg\{\gamma r(s,\theta(s))-{\bar{m}^{2}(s,\theta(s))}+\frac{(2-\gamma)}{(1-\gamma)}\frac{\bar{m}^{2}(s,\theta(s))}{{2\sigma^{2}(s,\theta(s))}}\bigg\}}ds\bigg)|\mathcal{F}_{t}^{\theta,y}\bigg], \end{eqnarray} where $\mathcal{F}_{\tau}^{\theta,y} \triangleq \sigma\{\theta{(\tau)},Y(\tau), \tau \in [0,t]\}$ augmented with $\mathbb{P}$ null sets is the filtration generated by the processes $\theta(t)$ and $Y(t)$. From the $\{\mathcal{F}_{t}^{\theta,y}\}$-martingale representation theorem, there exist $\{\mathcal{F}_{t}^{\theta,y}\}$-previsible, square integrable process $\nu(t,i,y)$ such that \begin{eqnarray}\label{6.12} R(t)=R(0)+\int_{0}^{t}\int_{\mathbb{R}}{\nu(\tau,\theta(\tau-),Y(\tau-))}\tilde{N}_{1}(d\tau,dz). \end{eqnarray} By positivity of $R(t)$ we can define $\hat{\nu}(\tau,\theta(\tau-),Y(\tau-)) \triangleq (\nu(\tau,\theta(\tau-),Y(\tau-)))R^{-1}(\tau-)$ so that \begin{eqnarray}\label{6.13} R(t)=R(0)+{\int_{0}^{t}\int_{\mathbb{R}}{R(\tau-)\hat{\nu}(\tau,\theta(\tau-),Y(\tau-))}\tilde{N}_{1}(d\tau,dz)}. \end{eqnarray} From the definition of $\tilde{\phi}$ in (\ref{6.10}) and the definition of $R$ in (\ref{6.11}) it is easy to see that we have the following relationship \begin{eqnarray}\label{6.14} R(t)=\tilde{\phi}(t,\theta(t),Y(t))\exp\bigg\{\int_{0}^{t}(\gamma r(s,\theta(s))-{\bar{m}^{2}(s,\theta(s))}+\frac{(2-\gamma)}{(1-\gamma)}\frac{\bar{m}^{2}(s,\theta(s))}{{2\sigma^{2}(s,\theta(s))}})ds\bigg\},\nonumber\\ ~~\forall~t~\in~[0,T]. \end{eqnarray} Using the Ito's expansion of $\tilde{\phi}(t,\theta(t),Y(t))$ to the RHS of (\ref{6.14}) followed up by comparing it with martingale representation of $R(t)$ in (\ref{6.12}) we get $\phi:= \tilde{\phi}$. We can thus substitute $ \hat{q}$ and $\hat{\tilde{\eta}}$ in expression (\ref{6.4}),(\ref{6.5}) in lieu of $q$ and $\tilde{\eta}(t,z)$ respectively. With the choice of control $\hat{u}$ given by (\ref{6.8}) and boundedness condition on the market parameters $r,\mu$ and $\sigma$, the conditions in Theorem 4.1 are satisfied and hence $\hat{u}(t)$ is an optimal control process and the explicit representation of $\hat{p}$ is given by \begin{eqnarray*} \hat{p}(t)=(X(t))^{\gamma-1}e^{E[\exp(\int_{t}^{T}{\gamma r(s,\theta(s))-{\bar{m}^{2}(s,\theta(s))}+\frac{(2-\gamma)}{(1-\gamma)}\frac{\bar{m}^{2}(s,\theta(s))}{{2\sigma^{2}(s,\theta(s))}}ds|}{\theta(t-)=i,Y(t-)=y})]}. \end{eqnarray*} {\bf Quadratic loss minimization}~~ We now provide an example related to quadratic loss minimization where the portfolio wealth process is given by \begin{eqnarray}\label{6.15} dX^{h}({t})&=&\bigg(r({t},\theta(t))X^{h}(t)+h(t)\sigma(t,\theta(t))\bar{m}(t,\theta(t))-h(t)\int_{\Gamma}{g(t,X^{h}(t),\theta(t),\gamma)\pi(d\gamma)}\bigg)dt\nonumber\\ &+&h(t)\sigma(t,\theta(t))dW(t) + h(t)\int_{\Gamma}{g(t,X^{h}(t),\theta(t),\gamma)\tilde{N}(dt,d\gamma)}, \nonumber \\ X^{h}(0)&=&x_{0}~~a.s. \end{eqnarray} {where the market price of risk is defined as $\bar{m}({t},i,y)=\sigma^{-1}(t,i)(b(t,i)-r(t,i))$. {As like earlier example , we have that $\bar{m}(t,i) \geq 0$ and that the variables $r(t,i), b(t,i)$, $\sigma(t,i)$ , $\sigma^{-1}(t,i)$ and $g(t,x,i,\gamma)$ for each $i \in \mathcal{X}$ are measurable and uniformly bounded in $t \in [0,T]$. We assume that $g(t,x,i,\gamma)>-1$ for each $i \in \mathcal{X}$ and for a.a. $t,x,\gamma$. This insures that $X^{h}(t)>0$ for each $t$. We further assume following conditions for each $i \in \mathcal{X}$\\ 1. $E[\int_{0}^{T}{h^{2}(t)dt}]< \infty.$\\ 2. $E[\int_{0}^{T}{|r(t,i)X(t)+h(t)\sigma(t,i)\bar{m}(t,i)|}dt+\int_{0}^{T}{h^{2}(t)\sigma^{2}(t,i)}dt+\int_{0}^{T}{h^{2}(t)g^{2}(t,X(t),i,\gamma)}dt]< \infty.$\\ 3. $t \rightarrow \int_{\mathbb{R}}{h^{2}(t)g^{2}(t,x,i,\gamma)\pi(d\gamma)}$ is bounded. \\ 4. the SDE for $X$ has a unique strong solution.\\ } The portfolio process $h(\cdot)$ satisfying the above four conditions is said to be admissible and belongs to $\mathcal{A}(T)$ (say). } We consider the problem of finding an admissible portfolio process $h \in \mathcal{A}(T)$ such that \begin{eqnarray*} \inf_{h \in \mathcal{A}(T)}{E[(X^{h}(T)-d)^{2}]}, \end{eqnarray*} over all $h \in \mathcal{A}(T)$. Set the control process $u(t) \triangleq h(t)$ and $X(t) \triangleq X^{h}(t)$. For this example the Hamiltonian (\ref{3.4}) becomes \begin{eqnarray}\label{6.16} \mathcal{H}(t,x,h,i,y,p,q,\eta)&=&\bigg[r(t,i)x+u\sigma(t,i)\bar{m}(t,i)-u\int_{\Gamma}{g(t,x,i,\gamma)\pi(d\gamma)}\bigg]p+u\sigma(t,i)q \nonumber \\ &+&\bigg(u\int_{\Gamma}{g(t,x,i,\gamma)}\pi(d\gamma) \bigg)\eta , \end{eqnarray} and the adjoint equations are for all time $t \in [0,T)$, \begin{eqnarray}\label{6.17} dp(t)&=&-r(t,\theta(t-))p(t)dt+q(t)dW(t)+\int_{\Gamma}{\eta(t,\gamma)\tilde{N}(dt,d\gamma)}+\int_{\mathbb{R}}{\tilde{\eta}(t,z)\tilde{N}_{1}(dt,dz)}, \nonumber \\ p(T)&=&-2X(T)+2d ~~a.s. \end{eqnarray} We seek to determine $p(t),q(t), \eta(t,\gamma)$ and $\tilde{\eta}(t,z)$ in (\ref{6.17}). Going by (\ref{6.17}) we assume that , \begin{eqnarray}\label{6.18} p(t)=\phi(t,\theta(t),Y(t))X(t)+\psi(t,\theta(t),Y(t)). \end{eqnarray} with the terminal boundary conditions being \begin{eqnarray}\label{6.19} \phi(T,i,y)=-2~~~~~~~~~~~~\psi(T,i,y)=2d~~~~\forall ~i~\in~\mathcal{X}. \end{eqnarray} For the sake of convenience we again rewrite the following Ito's formula for a function $f(t,\theta(t),y(t))\in \mathcal{C}^{1,2,1}$ given as \begin{eqnarray}\label{6.20} &&df(t,\theta(t),Y(t))=\bigg(\frac{\partial f(t,\theta(t),Y(t))}{\partial {t}}+\frac{(f^{h}(y/i))}{(1-F^{h}(y/i))}\sum_{j \neq i, j=1}^{M}p_{\theta(t-)=i,j}[f(t,j,0)-f(t,\theta(t-),y)]\nonumber\\ &+&\frac{\partial f(t,\theta(t),Y(t))}{\partial y}\bigg)dt\nonumber \\ &+&\int_{\mathbb{R}}{[f(t,\theta({t-})+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y({t-}),z))-f(t,\theta({t-}),Y({t-}))]\tilde{N}_{1}(dt,dz)}.\nonumber \\ \end{eqnarray} We apply the Ito's product rule to (\ref{6.18}) to obtain \begin{eqnarray}\label{6.21} dp({t})&=&X({t-})d\phi(t,\theta(t-),Y(t))+\phi(t,\theta(t-),Y(t))dX(t)+d\phi(t,\theta(t-),Y(t))dX(t)+d\psi(t)\nonumber\\ &=& \sum_{i=1}^{M}{1_{\theta_{t-}=i}}\bigg\{X(t-)\bigg(\phi(t,\theta(t-),y)r(t,\theta(t-))+\phi_{t}(t,\theta(t-),Y(t))+\phi_{y}(t,\theta(t-),Y(t))\nonumber \\ &+&\sum_{i=1,j \neq i}^{M}{p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)}(\phi(t,j,0)-\phi(t,\theta(t-),Y(t)))}\bigg)+u(t)\phi(t,\theta(t-),Y(t))\sigma(t,\theta(t-))\bar{m}(t,\theta(t-))\nonumber \\ &-&u(t)\phi(t,\theta(t-),Y(t))\int_{{\Gamma}}{g(t,X(t),\theta(t-),\gamma)\pi(d\gamma)}+\psi_{t}(t,\theta(t-),Y(t))+\psi_{y}(t,\theta(t-),Y(t))\nonumber\\ &+&\sum_{i=1,i \neq j}^{M}{p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)}[\psi(t,j,0)-\psi(t,\theta(t-)=i,Y(t))]}\bigg\}dt \nonumber \\ &+&u(t)\phi(t,\theta({t-}),Y({t}))\sigma(t,\theta({t-}))dW({t})+u(t)\phi(t,\theta({t-}),Y({t-}))\int_{{\Gamma}}{g(t,X(t-),\theta({t-}),\gamma) \tilde{N}(dt,d\gamma)}\nonumber\\ &+&\int_{\mathbb{R}}\bigg[X(t-)(\phi(t,\theta({t-})+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y({t-}),z))-\phi(t,\theta({t-}),Y({t-})))\nonumber \\ &+&\psi(t,\theta({t-})+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y({t-}),z))-\psi(t,\theta({t-}),Y({t-}))\bigg]\tilde{N}_{1}(dt,dz). \nonumber \\ \end{eqnarray} Comparing coefficients with (\ref{6.17}) we obtain three equations given as \begin{eqnarray}\label{6.22} &-&r(t,\theta({t-}))p(t-)\nonumber\\&=&\sum_{i=1}^{M}{1_\{{\theta_{t-}=i},Y(t-)=y}\}\bigg\{X(t-)\bigg(\phi(t,\theta({t-}),Y(t))r(t,\theta({t-}))+\phi_{t}(t,\theta({t-}),Y(t))+\phi_{y}(t,\theta({t-}),Y(t))\nonumber\\ &+&\sum_{i=1,j \neq i}^{M}{p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)}(\phi(t,j,0)-\phi(t,\theta({t-}),Y(t)))}\bigg) +u(t)\phi(t,\theta({t-}),Y(t))\sigma(t,\theta({t-}))\bar{m}(t,\theta({t-}))\nonumber\\ &-&u(t)\phi(t,\theta({t-}),Y(t))\int_{\Gamma}{g(t,x,\theta({t-}),\gamma)\pi(d\gamma)} +\psi_{t}(t,\theta({t-})),Y(t)+\psi_{y}(t,\theta({t-}),Y(t))\nonumber\\ &+&\sum_{i \neq j}^{M}{p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)}[\psi(t,j,0)-\psi(t,\theta({t-}),Y(t))]}\bigg\}.\nonumber\\ \end{eqnarray} \begin{eqnarray}\label{6.23} q(t)=u(t)\phi(t,\theta({t-}),Y({t-}))\sigma(t,\theta({t-})). \end{eqnarray} \begin{eqnarray}\label{6.24} \eta(t,\gamma)=u(t)\phi(t,\theta({t-}),Y(t-))g(t,X(t-),\theta({t-}),\gamma). \end{eqnarray} \begin{eqnarray}\label{6.25} \tilde{\eta}(t,z)&=&X(t-)(\phi(t,\theta({t-})+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y({t-}),z))-\phi(t,\theta({t-}),Y({t-})))\nonumber \\ &+&\psi(t,\theta({t-})+\bar{h}(\theta({t-}),Y({t-}),z),Y({t-})-\bar{g}(\theta({t-}),Y({t-}),z))-\psi(t,\theta({t-}),Y({t-})).\nonumber\\ \end{eqnarray} Let $\hat{u} \in \mathcal{A}(T)$ be a candidate optimal control corresponding to the wealth process $\hat{X}(T)$ and the adjoint triplet ($\hat{p},\hat{q},\hat{\eta},\hat{\tilde{\eta}}$). Then from the Hamiltonian (\ref{3.4}) for all $u \in \mathcal{A}(T)$ we have \begin{eqnarray}\label{6.26} \mathcal{H}(t,\hat{X}(t),u,\theta(t),\hat{p}(t),\hat{q}(t),\hat{\eta}(t))&=&\bigg(r(t,\theta(t))\hat{X}(t)+u \sigma(t,\theta(t))\bar{m}(t,\theta(t))\nonumber\\ &-&u\int_{\Gamma}{g(t,\hat{X}(t-),\theta(t-),\gamma)\pi{d(\gamma)}}\bigg)\hat{p}(t)\nonumber\\ &+&u\sigma(t,\theta(t))\hat{q}(t)+\bigg(u\int_{\Gamma}{g(t,\hat{X}(t-),\theta(t-),\gamma)\pi(d{\gamma})}\bigg)\hat{\eta}(t,\gamma).\nonumber\\ \end{eqnarray} As this is a linear function of $u$, we guess that the coefficient of $u$ vanishes at optimality, which results in the following equality \begin{eqnarray}\label{6.27} \hat{q}(t)&=&\bigg(-\bar{m}(t,\theta({t-}))+\frac{1}{\sigma(t,\theta({t-}))}\int_{\Gamma}{g(t,\hat{X}(t),\theta(t),\gamma)\pi(d\gamma)}\bigg)\hat{p}(t)\nonumber\\ &-&\frac{1}{\sigma(t,\theta({t-}))}\int_{\Gamma}{(g^{'}(t,\hat{X}(t),\theta(t),\gamma))\pi(d\gamma)\hat{\eta}(t,\gamma)}.\nonumber\\ \end{eqnarray} Also substituting (\ref{6.27}) for $\hat{q}(t)$ in (\ref{6.23}) and using (\ref{6.18}) and(\ref{6.24}) we get, \begin{eqnarray}\label{6.28} \hat{u}(t)=\frac{\tilde{\Lambda}(t)}{\Lambda(t)}(\hat{X}(t)+\phi^{-1}(t,\theta({t-}),y)\psi(t,\theta({t-}),y)), \end{eqnarray} where \begin{eqnarray}\label{6.29} \tilde{\Lambda}(t)={-\bar{m}(t,\theta({t-}))\sigma(t,\theta({t-}))+\int_{\Gamma}{g(t,X(t),\theta({t-}),\gamma)}\pi(d\gamma)}. \nonumber \\ \Lambda(t)={\sigma}^{2}(t,\theta({t-}))+\phi(t,\theta({t-}),Y(t))\int_{\Gamma}{g^{'}(t,X(t),\theta({t-}),\gamma)g(t,X(t),\theta({t-}),\gamma)}\pi(d\gamma). \end{eqnarray} To find the optimal control it remains to find $\phi$ and $\psi$. To do so set $X(t):= \hat{X}(t), u(t):=\hat{u}(t)$ and $p(t):=\hat{p}(t)$ in (\ref{6.22}) and then substitute for $\hat{p}(t)$ in (\ref{6.18}) and $\hat{u}(t)$ from (\ref{6.28}) . As this result is linear in $\hat{X}(t)$ we compare the coefficient on both side of the resulting equation to get following two equations namely, \begin{eqnarray}\label{6.30} 0&=&2r\phi(t,i,Y(t))+\phi_{t}(t,i,Y(t))+\phi_{y}(t,i,Y(t))+\sum_{i \neq j,i=1}^{M}{p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)}}{(\phi(t,j,0)-\phi(t,i,Y(t)))}\nonumber\\ &+& \frac{\tilde{\Lambda}(t)}{\Lambda(t)}\sigma(t,i)\bar{m}(t,i)\phi(t,i,Y(t))-\frac{\tilde{\Lambda}(t)}{\Lambda(t)}\phi(t,i,Y(t))\int_{\Gamma}{g(t,X(t),i,\gamma)}\pi (d\gamma). \end{eqnarray} \begin{eqnarray}\label{6.31} 0&=&r\psi(t,i,Y(t))+\psi_{t}(t,i,Y(t))+\psi_{y}(t,i,Y(t))+\sum_{ i \neq j,i=1}^{M}{p_{ij}\frac{f^{h}(y/i)}{1-F^{h}(y/i)}(\psi(t,j,0)-\psi(t,i,Y(t)))}\nonumber\\ &+&\frac{\tilde{\Lambda}(t)}{\Lambda(t)}\sigma(t,i)\bar{m}(t,i)\psi(t,i,Y(t))-\frac{\tilde{\Lambda}(t)}{\Lambda(t)}\psi(t,i,y)\int_{\Gamma}g(t,X(t),i,\gamma) \pi d(\gamma).\nonumber\\ \end{eqnarray} with terminal boundary conditions given by (\ref{6.19}). Consider the following process \begin{eqnarray}\label{6.32} \tilde{\phi}(t,i,y)=-2E\bigg[\exp\bigg\{\int_{t}^{T}\bigg(2r(s,\theta({s-}))+\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\sigma(s,\theta({s-}))\bar{m}(s,\theta({s-}))\nonumber\\ -\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\int_{\Gamma}{g(s,X(s),\theta({s-}),\gamma)\pi(d\gamma)}\bigg)ds\bigg\}|{(\theta(s-)=i,Y(t)=y)}\bigg].\nonumber\\ \end{eqnarray} \begin{eqnarray}\label{6.33} \tilde{\psi}(t,i,y)&=&2dE\bigg[\exp\bigg\{\int_{t}^{T}\bigg(r(\theta({s-}),s)+\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\sigma(s,\theta({s-}))\bar{m}(s,\theta({s-}))\nonumber\\ &-&\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\int_{\Gamma}{g(s,X(s),\theta({s-}),\gamma)\pi(d\gamma)}\bigg)ds\bigg\}\bigg|{(\theta(s-)=i,Y(s)=y)}\bigg].\nonumber\\ \end{eqnarray} We aim to show that $\phi=\tilde{\phi}$ and $\psi=\tilde{\psi}$. We define the following martingales: \begin{eqnarray}\label{6.34} R(t)= \resizebox{.9\hsize}{!}{$E\bigg[\exp\bigg\{\int_{0}^{T}\bigg(2r(s,\theta(s-))+\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\sigma(s,\theta(s-))\bar{m}(s,\theta(s-)) -\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\int_{\Gamma}{g(s,X(s),\theta(s-),\gamma)\pi(d\gamma)}\bigg)ds\bigg\}|\mathcal{F}_{t}^{\theta,y}\bigg]$},\nonumber\\ \end{eqnarray} \begin{eqnarray}\label{6.35} S(t)= \resizebox{.9\hsize}{!}{$E\bigg[\exp\bigg\{\int_{0}^{T}\bigg(r(s,\theta(s-))+\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\sigma(s,\theta(s-))\bar{m}(s,\theta(s-)) -\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\int_{\Gamma}{g(s,X(s),\theta(s-),\gamma)}\pi(d\gamma)\bigg)ds\bigg\}|\mathcal{F}_{t}^{\theta,y}\bigg]$},\nonumber\\ \end{eqnarray} where $\mathcal{F}_{t}^{\theta,y}$ is defined as usual. We follow steps similar to that as seen in Example 1 and conclude that $\phi=\tilde{\phi}$ and $\psi=\tilde{\psi}$ by using joint-Markov property of ($\theta(t),Y(t)$), to obtain the following expression for the control $\hat{u}(t)$ given as \begin{eqnarray*} \hat{u}(t)= \resizebox{.9\hsize}{!}{$\frac{\tilde{\Lambda}(t)}{\Lambda(t)}\bigg(\hat{X}(t) -\frac{d E\bigg[\exp\bigg\{\int_{t}^{T}(r(s,\theta(s-))+\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\sigma(s,\theta(s-))\bar{m}(s,\theta(s-)) -\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\int_{\Gamma}{g(s,X(s),\theta(s-),\gamma)\pi(d\gamma)})ds\bigg\}|{(\theta(t-)=i,Y(t)=y)}\bigg]}{E\bigg[\exp\bigg\{\int_{t}^{T}(2r(s,\theta(s-))+\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\sigma(s,\theta(s-))\bar{m}(s,\theta(s-)) -\frac{\tilde{\Lambda}(s)}{\Lambda(s)}\int_{\Gamma}{g(s,X(s),\theta(s-),\gamma)\pi(d\gamma)})ds\bigg\}|{(\theta(t)=i,Y(t)=y)}\bigg]}\bigg)$}. \end{eqnarray*} For the choice of the control parameter and the boundedness conditions on the market parameters $r,b$,$\sigma$ and $g$, the conditions of Theorem 4.1 are satisfied and hence $\hat{u}$ is the optimal control process. \end{document}

\begin{document} \title{Discrete dynamical systems in group theory} arithmetic{}bstract{ In this expository paper we describe the unifying approach for many known entropies in Mathematics developed in \cite{DGV1}. First we give the notion of semigroup entropy $h_{\mathfrak{S}}:{\mathfrak{S}}\to\mathbb R_+$ in the category ${\mathfrak{S}}$ of normed semigroups and contractive homomorphisms, recalling also its properties from \cite{DGV1}. For a specific category $\mathfrak X$ and a functor $F:\mathfrak X\to {\mathfrak{S}}$ we have the entropy $h_F$, defined by the composition $h_F=h_{\mathfrak{S}}\circ F$, which automatically satisfies the same properties proved for $h_{\mathfrak{S}}$. This general scheme permits to obtain many of the known entropies as $h_F$, for appropriately chosen categories $\mathfrak X$ and functors $F:\mathfrak X\to {\mathfrak{S}}$. In the last part we recall the definition and the fundamental properties of the algebraic entropy for group endomorphisms, noting how its deeper properties depend on the specific setting. Finally we discuss the notion of growth for flows of groups, comparing it with the classical notion of growth for finitely generated groups.} \section{Introduction} This paper covers the series of three talks given by the first named author at the conference ``Advances in Group Theory and Applications 2011'' held in June, 2011 in Porto Cesareo. It is a survey about entropy in Mathematics, the approach is the categorical one adopted in \cite{DGV1} (and announced in \cite{D}, see also \cite{LoBu}). We start Section 4 recalling that a \emph{flow} in a category $\mathfrak X$ is a pair $(X,\phi)$, where $X$ is an object of $\mathfrak X$ and $\phi: X\to X$ is a morphism in $\mathfrak X$. A morphism between two flows $\phi: X\to X$ and $\psi: Y\to Y$ is a morphism $arithmetic{}lpha: X \to Y$ in $\mathfrak X$ such that the diagram $$\xymatrix{Xarithmetic{}r[r]^{arithmetic{}lpha} arithmetic{}r[d]_{\phi}&Yarithmetic{}r[d]^{\psi}\\Xarithmetic{}r[r]^{arithmetic{}lpha}& Y.}$$ commutes. This defines the category $\mathbf{Flow}_{\mathfrak X}$ of flows in $\mathfrak X$. To classify flows in $\mathfrak X$ up to isomorphisms one uses invariants, and entropy is roughly a numerical invariant associated to flows. Indeed, letting $\mathbb R_{\geq 0} = \{r\in \mathbb R: r\geq 0\}$ and $\mathbb R_+= \mathbb R_{\geq 0}\cup \{\infty\}$, by the term \emph{entropy} we intend a function \begin{equation}\label{dag} h: \mathbf{Flow}_{\mathfrak X}\to \mathbb R_+, \end{equation} obeying the invariance law $h(\phi) = h(\psi)$ whenever $(X,\phi)$ and $(Y,\psi)$ are isomorphic flows. The value $h(\phi)$ is supposed to measure the degree to which $X$ is ``scrambled" by $\phi$, so for example an entropy should assign $0$ to all identity maps. For simplicity and with some abuse of notations, we adopt the following \noindent\textbf{Convention.} If $\mathfrak X$ is a category and $h$ an entropy of $\mathfrak X$, writing $h: {\mathfrak X}\to \mathbb R_+$ we always mean $h: \mathbf{Flow}_{\mathfrak X}\to \mathbb R_+$ as in \eqref{dag}. The first notion of entropy in Mathematics was the measure entropy $h_{mes}$ introduced by Kolmogorov \cite{K} and Sinai \cite{Sinai} in 1958 in Ergodic Theory. The topological entropy $h_{top}$ for continuous self-maps of compact spaces was defined by Adler, Konheim and McAndrew \cite{AKM} in 1965. Another notion of topological entropy $h_B$ for uniformly continuous self-maps of metric spaces was given later by Bowen \cite{B} (it coincides with $h_{top}$ on compact metric spaces). Finally, entropy was taken also in Algebraic Dynamics by Adler, Konheim and McAndrew \cite{AKM} in 1965 and Weiss \cite{W} in 1974; they defined an entropy $\mathrm{ent}$ for endomorphisms of torsion abelian groups. Then Peters \cite{P} in 1979 introduced its extension $h_{alg}$ to automorphisms of abelian groups; finally $h_{alg}$ was defined in \cite{DG} and \cite{DG-islam} for any group endomorphism. Recently also a notion of algebraic entropy for module endomorphisms was introduced in \cite{SZ}, namely the algebraic $i$-entropy $\mathrm{ent}_i$, where $i$ is an invariant of a module category. Moreover, the adjoint algebraic entropy $\mathrm{ent}^\star$ for group endomorphisms was investigated in \cite{DGV} (and its topological extension in \cite{G}). Finally, one can find in \cite{AZD} and \cite{DG-islam} two ``mutually dual'' notions of entropy for self-maps of sets, namely the covariant set-theoretic entropy $\mathfrak h$ and the contravariant set-theoretic entropy $\mathfrak h^*$. The above mentioned specific entropies determined the choice of the main cases considered in this paper. Namely, $\mathfrak X$ will be one of the following categories (other examples can be found in \S\S \ref{NewSec1} and \ref{NewSec2}): \begin{itemize} \item[(a)] $\mathbf{Set}$ of sets and maps and its non-full subategory $\mathbf{Set}_{\mathrm{fin}}$ of sets and finite-to-one maps (set-theoretic entropies $\mathfrak h$ and $\mathfrak h^*$ respectively); \item[(b)] $\mathbf{CTop}$ of compact topological spaces and continuous maps (topological entropy $h_{top}$); \item[(c)] $\mathbf{Mes}$ of probability measure spaces and measure preserving maps (measure entropy $h_{mes}$); \item[(d)] $\mathbf{Grp}$ of groups and group homomorphisms and its subcategory $\mathbf{AbGrp}$ of abelian groups (algebraic entropy $\mathrm{ent}$, algebraic entropy $h_{alg}$ and adjoint algebraic entropy $\mathrm{ent}^\star$); \item[(e)] $\mathbf{Mod}_R$ of right modules over a ring $R$ and $R$-module homomorphisms (algebraic $i$-entropy $\mathrm{ent}_i$). \end{itemize} Each of these entropies has its specific definition, usually given by limits computed on some ``trajectories'' and by taking the supremum of these quantities (we will see some of them explicitly). The proofs of the basic properties take into account the particular features of the specific categories in each case too. It appears that all these definitions and basic properties share a lot of common features. The aim of our approach is to unify them in some way, starting from a general notion of entropy of an appropriate category. This will be the semigroup entropy $h_{\mathfrak{S}}$ defined on the category ${\mathfrak{S}}$ of normed semigroups. In Section \ref{sem-sec} we first introduce the category ${\mathfrak{S}}$ of normed semigroups and related basic notions and examples mostly coming from \cite{DGV1}. Moreover, in \S \ref{preorder-sec} (which can be avoided at a first reading) we add a preorder to the semigroup and discuss the possible behavior of a semigroup norm with respect to this preorder. Here we include also the subcategory $\mathfrak L$ of ${\mathfrak{S}}$ of normed semilattices, as the functors given in Section \ref{known-sec} often have as a target actually a normed semilattice. In \S \ref{hs-sec} we define explicitly the semigroup entropy $h_{\mathfrak{S}}: {\mathfrak{S}} \to \mathbb R_+$ on the category ${\mathfrak{S}}$ of normed semigroups. Moreover we list all its basic properties, clearly inspired by those of the known entropies, such as Monotonicity for factors, Invariance under conjugation, Invariance under inversion, Logarithmic Law, Monotonicity for subsemigroups, Continuity for direct limits, Weak Addition Theorem and Bernoulli normalization. Once defined the semigroup entropy $h_{\mathfrak{S}}:{\mathfrak{S}}\to \mathbb R_+$, our aim is to obtain all known entropies $h:{\mathfrak X} \to \mathbb R_+$ as a composition $h_F:=h_{\mathfrak{S}} \circ F$ of a functor $F: \mathfrak X\to {\mathfrak{S}}$ and $h_{\mathfrak{S}}$: \begin{equation*} \xymatrix@R=6pt@C=37pt {\mathfrak{X}arithmetic{}r[dd]_{F}arithmetic{}r[rrd]^{h=h_F} & & \\ & & \mathbb R_+ \\ {\mathfrak{S}}arithmetic{}r[rru]_{h_{\mathfrak{S}}} & & } \end{equation*} This is done explicitly in Section \ref{known-sec}, where all specific entropies listed above are obtained in this scheme. We dedicate to each of them a subsection, each time giving explicitly the functor from the considered category to the category of normed semigroups. More details and complete proofs can be found in \cite{DGV1}. These functors and the entropies are summarized by the following diagram: \begin{equation*} \xymatrix@-1pc{ &&&\mathbf{Mes}arithmetic{}r@{-->}[ddddr]|-{\mathfrak{mes}}arithmetic{}r[ddddddr]|-{h_{mes}}& &\mathbf{AbGrp}arithmetic{}r[ddddddl]|-{\mathrm{ent}}arithmetic{}r@{-->}[ddddl]|-{\mathfrak{sub}}&&\\ & &\mathbf{CTop}arithmetic{}r@{-->}[dddrr]|-{\mathfrak{cov}}arithmetic{}r[dddddrr]|-{h_{top}}& & & & \mathbf{Grp}arithmetic{}r@{-->}[dddll]|-{\mathfrak{pet}}arithmetic{}r[dddddll]|-{h_{alg}} & &\\ & \mathbf{Set}arithmetic{}r@{-->}[ddrrr]|-{\mathfrak{atr}}arithmetic{}r[ddddrrr]|-{\mathfrak h} && & & & & \mathbf{Grp}arithmetic{}r@{-->}[ddlll]|-{\mathfrak{sub}^\star}arithmetic{}r[ddddlll]|-{\mathrm{ent}^\star} \\ \mathbf{Set}_\mathrm{fin}arithmetic{}r@{-->}[drrrr]|-{\mathfrak{str}}arithmetic{}r[dddrrrr]|-{\mathfrak h^*} && && & & & &\mathbf{Mod}_Rarithmetic{}r@{-->}[dllll]|-{\mathfrak{sub}_i}arithmetic{}r[dddllll]|-{\mathrm{ent}_i} \\ & && & \mathfrak S arithmetic{}r[dd]|-{h_\mathfrak S} && & \\ \\ & && &{\mathbb R_+} & & & } \end{equation*} In this way we obtain a simultaneous and uniform definition of all entropies and uniform proofs (as well as a better understanding) of their general properties, namely the basic properties of the specific entropies can be derived directly from those proved for the semigroup entropy. The last part of Section \ref{known-sec} is dedicated to what we call Bridge Theorem (a term coined by L. Salce), that is roughly speaking a connection between entropies $h_1:\mathfrak X_1 \to \mathbb R_+$ and $h_2:\mathfrak X_2 \to \mathbb R_+$ via functors $\varepsilon: \mathfrak X_1 \to \mathfrak X_2$. Here is a formal definition of this concept: \begin{Definition}\label{BTdef} Let $\varepsilon: \mathfrak X_1 \to \mathfrak X_2$ be a functor and let $h_1:\mathfrak X_1 \to \mathbb R_+$ and $h_2:\mathfrak X_2 \to \mathbb R_+$ be entropies of the categories $\mathfrak X_1 $ and $ \mathfrak X_2$, respectively (as in the diagram below). \begin{equation*}\label{Buz} \xymatrix@R=6pt@C=37pt {\mathfrak{X}_1arithmetic{}r[dd]_{\varepsilon}arithmetic{}r[rrd]^{h_{1}} & & \\ & & \mathbb R_+ \\ \mathfrak{X}_2arithmetic{}r[rru]_{h_{2}} & & } \end{equation*}We say that the pair $(h_1, h_2)$ satisfies the \emph{weak Bridge Theorem} with respect to the functor $\varepsilon$ if there exists a positive constant $C_\varepsilon$, such that for every endomorphism $\phi$ in $\mathfrak X_1$ \begin{equation}\label{sBT} h_2(\varepsilon(\phi)) \leq C_\varepsilon h_1(\phi). \end{equation} If equality holds in \eqref{sBT} we say that $(h_1,h_2)$ satisfies the \emph{Bridge Theorem} with respect to $\varepsilon$, and we shortly denote this by $(BT_\varepsilon)$. \end{Definition} In \S \ref{BTsec} we discuss the Bridge Theorem passing through the category ${\mathfrak{S}}$ of normed semigroups and so using the new semigroup entropy. This approach permits for example to find a new and transparent proof of Weiss Bridge Theorem (see Theorem \ref{WBT}) as well as for other Bridge Theorems. A first limit of this very general setting is the loss of some of the deeper properties that a specific entropy may have. So in the last Section \ref{alg-sec} for the algebraic entropy we recall the definition and the fundamental properties, which cannot be deduced from the general scheme. We start Section 4 recalling the Algebraic Yuzvinski Formula (see Theorem \ref{AYF}) recently proved in \cite{GV}, giving the values of the algebraic entropy of linear transformations of finite-dimensional rational vector spaces in terms of the Mahler measure. In particular, this theorem provides a connection of the algebraic entropy with the famous Lehmer Problem. Two important applications of the Algebraic Yuzvinski Formula are the Addition Theorem and the Uniqueness Theorem for the algebraic entropy in the context of abelian groups. In \S \ref{Growth-sec} we describe the connection of the algebraic entropy with the classical topic of growth of finitely generated groups in Geometric Group Theory. Its definition was given independently by Schwarzc \cite{Sch} and Milnor \cite{M1}, and after the publication of \cite{M1} it was intensively investigated; several fundamental results were obtained by Wolf \cite{Wolf}, Milnor \cite{M2}, Bass \cite{Bass}, Tits \cite{Tits} and Adyan \cite{Ad}. In \cite{M3} Milnor proposed his famous problem (see Problem \ref{Milnor-pb} below); the question about the existence of finitely generated groups with intermediate growth was answered positively by Grigorchuk in \cite{Gri1,Gri2,Gri3,Gri4}, while the characterization of finitely generated groups with polynomial growth was given by Gromov in \cite{Gro} (see Theorem \ref{GT}). Here we introduce the notion of finitely generated flows $(G,\phi)$ in the category of groups and define the growth of $(G,\phi)$. When $\phi=\mathrm{id}_G$ is the identical endomorphism, then $G$ is a finitely generated group and we find exactly the classical notion of growth. In particular we recall a recent significant result from \cite{DG0} extending Milnor's dichotomy (between polynomial and exponential growth) to finitely generated flows in the abelian case (see Theorem \ref{DT}). We leave also several open problems and questions about the growth of finitely generated flows of groups. The last part of the section, namely \S \ref{aent-sec}, is dedicated to the adjoint algebraic entropy. As for the algebraic entropy, we recall its original definition and its main properties, which cannot be derived from the general scheme. In particular, the adjoint algebraic entropy can take only the values $0$ and $\infty$ (no finite positive value is attained) and we see that the Addition Theorem holds only restricting to bounded abelian groups. A natural side-effect of the wealth of nice properties of the entropy $h_F=h_{\mathfrak{S}}\circ F$, obtained from the semigroup entropy $h_{\mathfrak{S}}$ through functors $F:\mathfrak X\to {\mathfrak{S}}$, is the loss of some entropies that do not have all these properties. For example Bowen's entropy $h_B$ cannot be obtained as $h_F$ since $h_B(\phi^{-1})= h_B(\phi)$ fails even for the automorphism $\phi: \mathbb R \to \mathbb R$ defined by $\phi(x)= 2x$, see \S \ref{NewSec2} for an extended comment on this issue; there we also discuss the possibility to obtain Bowen's topological entropy of measure preserving topological automorphisms of locally compact groups in the framework of our approach. For the same reason other entropies that cannot be covered by this approach are the intrinsic entropy for endomorphisms of abelian groups \cite{DGSV} and the topological entropy for automorphisms of locally compact totally disconnected groups \cite{DG-tdlc}. This occurs also for the function $\phi \mapsto \log s(\phi)$, where $s(\phi)$ is the scale function defined by Willis \cite{Willis,Willis2}. The question about the relation of the scale function to the algebraic or topological entropy was posed by T. Weigel at the conference; these non-trivial relations are discussed for the topological entropy in \cite{BDG}. \section{The semigroup entropy}\label{sem-sec} \subsection{The category ${\mathfrak{S}}$ of normed semigroups} We start this section introducing the category ${\mathfrak{S}}$ of normed semigroups, and other notions that are fundamental in this paper. \begin{Definition}\label{Def1} Let $(S,\cdot)$ be a semigroup. \begin{itemize} \item[(i)] A \emph{norm} on $S$ is a map $v: S \to \mathbb R_{\geq 0}$ such that \begin{equation*} v(x \cdot y) \leq v(x) + v(y)\ \text{for every}\ x,y\in S. \end{equation*} A \emph{normed semigroup} is a semigroup provided with a norm. If $S$ is a monoid, a \emph{monoid norm} on $S$ is a semigroup norm $v$ such that $v(1)=0$; in such a case $S$ is called \emph{normed monoid}. \item[(ii)] A semigroup homomorphism $\phi:(S,v)\to (S',v')$ between normed semigroups is \emph{contractive} if $$v'(\phi(x))\leq v(x)\ \text{for every}\ x\in S.$$ \end{itemize} \end{Definition} Let ${\mathfrak{S}}$ be the category of normed semigroups, which has as morphisms all contractive semigroup homomorphisms. In this paper, when we say that $S$ is a normed semigroup and $\phi:S\to S$ is an endomorphism, we will always mean that $\phi$ is a contractive semigroup endomorphism. Moreover, let $\mathfrak M$ be the non-full subcategory of ${\mathfrak{S}}$ with objects all normed monoids, where the morphisms are all (necessarily contractive) monoid homomorphisms. We give now some other definitions. \begin{Definition} A normed semigroup $(S,v)$ is: \begin{itemize} \item[(i)] \emph{bounded} if there exists $C\in \mathbb N_+$ such that $v(x) \leq C$ for all $x\in S$; \item[(ii)]\emph{arithmetic} if for every $x\in S$ there exists a constant $C_x\in \mathbb N_+$ such that $v(x^n) \leq C_x\cdot \log (n+1)$ for every $n\in\mathbb N$. \end{itemize} \end{Definition} Obviously, bounded semigroups are arithmetic. \begin{Example}\label{Fekete} Consider the monoid $S = (\mathbb N, +)$. \begin{itemize} \item[(a)] Norms $v$ on $S$ correspond to {subadditive sequences} $(a_n)_{n\in\mathbb N}$ in $ \mathbb R_+$ (i.e., $a_{n + m}\leq a_n + a_m$) via $v \mapsto (v(n))_{n\in\mathbb N}$. Then $\lim_{n\to \infty} \phirac{a_n}{n}= \inf_{n\in\mathbb N} \phirac{a_n}{n}$ exists by Fekete Lemma \cite{Fek}. \item[(b)] Define $v: S \to \mathbb R_+$ by $v(x) = \log (1+ x)$ for $x\in S$. Then $v$ is an arithmetic semigroup norm. \item[(c)] Define $v_1: S \to \mathbb R_+$ by $v_1(x) = \sqrt x$ for $x\in S$. Then $v_1$ is a semigroup norm, but $(S, + , v_1)$ is not arithmetic. \item[(d)] For $a\in \mathbb N$, $a>1$ let $v_a(n) = \sum_i b_i$, when $n= \sum_{i=0}^k b_ia^i$ and $0\leq b_i < a$ for all $i$. Then $v_a$ is an arithmetic norm on $S$ making the map $x\mapsto ax$ an endomorphism in ${\mathfrak{S}}$. \end{itemize} \end{Example} \subsection{Preordered semigroups and normed semilattices}\label{preorder-sec} A triple $(S,\cdot,\leq)$ is a \emph{preordered semigroup} if the semigroup $(S,\cdot)$ admits a preorder $\leq$ such that $$x\leq y\ \text{implies}\ x \cdot z \leq y \cdot z\ \text{and}\ z \cdot x \leq z \cdot y\ \text{for all}\ x,y,z \in S.$$ Write $x\sim y$ when $x\leq y$ and $y\leq x$ hold simultaneously. Moreover, the \emph{positive cone} of $S$ is $$P_+(S)=\{a\in S:x\leq x \cdot a \ \text{and}\ x\leq a\cdot x\ \text{for every}\ x\in S\}.$$ A norm $v$ on the preordered semigroup $(S,\cdot,\leq)$ is \emph{monotone} if $x\leq y$ implies $v(x) \leq v(y)$ for every $x,y \in S$. Clearly, $v(x) = v(y)$ whenever $x \sim y$ and the norm $v$ of $S$ is monotone. Now we propose another notion of monotonicity for a semigroup norm which does not require the semigroup to be explicitly endowed with a preorder. \begin{Definition} Let $(S,v)$ be a normed semigroup. The norm $v$ is \emph{s-monotone} if $$\max\{v(x), v(y)\}\leq v(x \cdot y)\ \text{for every}\ x,y \in S.$$ \end{Definition} This inequality may become a too stringent condition when $S$ is close to be a group; indeed, if $S$ is a group, then it implies that $v(S) = \{v(1)\}$, in particular $v$ is constant. If $(S,+,v)$ is a commutative normed monoid, it admits a preorder $\leq^a$ defined for every $x,y\in S$ by $x\leq^a y$ if and only if there exists $z\in S$ such that $x+z=y$. Then $(S,\cdot,\leq)$ is a {preordered semigroup} and the norm $v$ is s-monotone if and only if $v$ is monotone with respect to $\leq^a$. The following connection between monotonicity and s-monotonicity is clear. \begin{Lemma} Let $S$ be a preordered semigroup. If $S=P_+(S)$, then every monotone norm of $S$ is also s-monotone. \end{Lemma} A \emph{semilattice} is a commutative semigroup $(S,\vee)$ such that $x\vee x=x$ for every $x\in S$. \begin{Example} \begin{itemize} \item[(a)] Each lattice $(L, \vee, \wedge)$ gives rise to two semilattices, namely $(L, \vee)$ and $(L, \wedge)$. \item[(b)] A filter $\mathcal F$ on a given set $X$ is a semilattice with respect to the intersection, with zero element the set $X$. \end{itemize} \end{Example} Let ${\mathfrak{L}}$ be the full subcategory of ${\mathfrak{S}}$ with objects all normed semilattices. Every normed semilattice $(L,\vee)$ is trivially arithmetic, moreover the canonical partial order defined by $$x\leq y\ \text{if and only if}\ x\vee y=y,$$ for every $x,y\in L$, makes $L$ also a partially ordered semigroup. Neither preordered semigroups nor normed semilattices are formally needed for the definition of the semigroup entropy. Nevertheless, they provide significant and natural examples, as well as useful tools in the proofs, to justify our attention to this topic. \subsection{Entropy in ${\mathfrak{S}}$}\label{hs-sec} For $(S,v)$ a normed semigroup $\phi:S\to S$ an endomorphism, $x\in S$ and $n\in\mathbb N_+$ consider the \emph{$n$-th $\phi$-trajectory of $x$} $$T_n(\phi,x) = x \cdot\phi(x)\cdot\ldots \cdot\phi^{n-1}(x)$$ and let $$c_n(\phi,x) = v(T_n(\phi,x)).$$ Note that $c_n(\phi,x) \leq n\cdot v(x)$. Hence the growth of the function $n \mapsto c_n(\phi,x)$ is at most linear. \begin{Definition} Let $S$ be a normed semigroup. An endomorphism $\phi:S\to S$ is said to have \emph{logarithmic growth}, if for every $x\in S$ there exists $C_x\in\mathbb N_+$ with $c_n(\phi,x) \leq C_x\cdot \log (n+1)$ for all $n\in\mathbb N_+$. \end{Definition} Obviously, a normed semigroup $S$ is arithmetic if and only if $\mathrm{id}_{S}$ has logarithmic growth. The following theorem from \cite{DGV1} is fundamental in this context as it witnesses the existence of the semigroup entropy; so we give its proof also here for reader's convenience. \begin{Theorem}\label{limit} Let $S$ be a normed semigroup and $\phi:S\to S$ an endomorphism. Then for every $x \in S$ the limit \begin{equation}\label{hs-eq} h_{{\mathfrak{S}}}(\phi,x):= \lim_{n\to\infty}\phirac{c_n(\phi,x)}{n} \end{equation} exists and satisfies $h_{{\mathfrak{S}}}(\phi,x)\leq v(x)$. \end{Theorem} \begin{proof} The sequence $(c_n(\phi,x))_{n\in\mathbb N_+}$ is subadditive. Indeed, \begin{align*} c_{n+m}(\phi,x)&= v(x\cdot\phi(x)\cdot\ldots\cdot\phi^{n-1}(x)\cdot\phi^{n}(x)\cdot\ldots\cdot\phi^{n+m-1}(x))\\ &=v((x\cdot\phi(x)\cdot\ldots\cdot\phi^{n-1}(x))\cdot\phi^{n}(x\cdot\ldots\cdot\phi^{m-1}(x))) \\ &\leq c_n(\phi,x)+v(\phi^{n}(x\cdot\ldots\cdot\phi^{m-1}(x))) \\ &\leq c_n(\phi,x)+v(x\cdot\ldots\cdot\phi^{m-1}(x))=c_n(\phi,x)+c_m(\phi,x). \end{align*} By Fekete Lemma (see Example \ref{Fekete} (a)), the limit $\lim_{n\to\infty} \phirac{c_n(\phi,x)}{n}$ exists and coincides with $\inf_{n\in\mathbb N_+} \phirac{c_n(\phi,x)}{n}$. Finally, $h_{{\mathfrak{S}}}(\phi,x)\leq v(x)$ follows from $c_n(\phi,x) \leq n v(x)$ for every $n\in\mathbb N_+$. \end{proof} \begin{Remark} \begin{itemize} \item[(a)] The proof of the existence of the limit defining $h_{{\mathfrak{S}}}(\phi,x) $ exploits the property of the semigroup norm and also the condition on $\phi$ to be contractive. For an extended comment on what can be done in case the function $v: S \to \mathbb R_+$ fails to have that property see \S \ref{NewSec1}. \item[(b)] With $S = (\mathbb N,+)$, $\phi = \mathrm{id}_\mathbb N$ and $x=1$ in Theorem \ref{limit} we obtain exactly item (a) of Example \ref{Fekete}. \end{itemize} \end{Remark} \begin{Definition}\label{SEofEndos} Let $S$ be a normed semigroup and $\phi:S\to S$ an endomorphism. The \emph{semigroup entropy} of $\phi$ is $$h_{{\mathfrak{S}}}(\phi)=\sup_{x\in S}h_{{\mathfrak{S}}}(\phi,x).$$ \end{Definition} If an endomorphism $\phi:S\to S$ has logarithmic growth, then $h_{{\mathfrak{S}}}(\phi) = 0$. In particular, $h_{{\mathfrak{S}}}(\mathrm{id}_S)=0$ if $S$ is arithmetic. Recall that an endomorphism $\phi:S\to S$ of a normed semigroup $S$ is \emph{locally quasi periodic} if for every $x\in S$ there exist $n,k\in\mathbb N$, $k>0$, such that $\phi^n(x)=\phi^{n+k}(x)$. If $S$ is a monoid and $\phi(1)=1$, then $\phi$ is \emph{locally nilpotent} if for every $x\in S$ there exists $n\in\mathbb N_+$ such that $\phi^n(x)=1$. \begin{Lemma}\label{locally} Let $S$ be a normed semigroup and $\phi:S\to S$ an endomorphism. \begin{itemize} \item[(a)]If $S$ is arithmetic and $\phi$ is locally periodic, then $h_{\mathfrak{S}}(\phi)=0$. \item[(b)] If $S$ is a monoid and $\phi(1)=1$ and $\phi$ is locally nilpotent, then $h_{\mathfrak{S}}(\phi)=0$. \end{itemize} \end{Lemma} \begin{proof} (a) Let $x\in S$, and let $l,k\in\mathbb N_+$ be such that $\phi^l(x)=\phi^{l+k}(x)$. For every $m\in\mathbb N_+$ one has $$T_{l+mk}(\phi,x)=T_l(\phi,x)\cdot T_m(\mathrm{id}_S,y) = T_l(\phi,x)\cdot y^m,$$ where $y=\phi^l(T_k(\phi,x))$. Since $S$ is arithmetic, there exists $C_x\in \mathbb N_+$ such that \begin{equation*}\begin{split} v(T_{l+mk}(\phi,x)) = v(T_l(\phi,x)\cdot y^m) \leq \\ v(T_l(\phi,x)) + v( y^m) \leq v(T_l (\phi,x)) + C_x\cdot\log (m+1), \end{split}\end{equation*} so $\lim_{m\to \infty} \phirac{v(T_{l+mk}(\phi,x))}{l+mk}=0$. Therefore we have found a subsequence of $(c_n(\phi,x))_{n\in\mathbb N_+}$ converging to $0$, hence also $h_{\mathfrak{S}}(\phi,x)=0$. Hence $h_{\mathfrak{S}}(\phi)=0$. (b) For $x\in S$, there exists $n\in\mathbb N_+$ such that $\phi^n(x)=1$. Therefore $T_{n+k}(\phi,x)=T_n(\phi,x)$ for every $k\in\mathbb N$, hence $h_{\mathfrak{S}}(\phi,x)=0$. \end{proof} We discuss now a possible different notion of semigroup entropy. Let $(S,v)$ be a normed semigroup, $\phi:S\to S$ an endomorphism, $x\in S$ and $n\in\mathbb N_+$. One could define also the ``left'' $n$-th $\phi$-trajectory of $x$ as $$T_n^{\#}(\phi,x)=\phi^{n-1}(x)\cdot\ldots\cdot\phi(x)\cdot x,$$ changing the order of the factors with respect to the above definition. With these trajectories it is possible to define another entropy letting $$h_{\mathfrak{S}}^{\#}(\phi,x)=\lim_{n\to\infty}\phirac{v(T_n^{\#}(\phi,x))}{n},$$ and $$h_{\mathfrak{S}}^{\#}(\phi)=\sup\{h_{\mathfrak{S}}^{\#}(\phi,x):x\in S\}.$$ In the same way as above, one can see that the limit defining $h_{\mathfrak{S}}^{\#}(\phi,x)$ exists. Obviously $h_{\mathfrak{S}}^{\#}$ and $h_{\mathfrak{S}}$ coincide on the identity map and on commutative normed semigroups, but now we see that in general they do not take always the same values. Item (a) in the following example shows that it may occur the case that they do not coincide ``locally'', while they coincide ``globally''. Moreover, modifying appropriately the norm in item (a), J. Spev\'ak found the example in item (b) for which $h_{\mathfrak{S}}^{\#}$ and $h_{\mathfrak{S}}$ do not coincide even ``globally''. \begin{Example} Let $X=\{x_n\}_{n\in\mathbb Z}$ be a faithfully enumerated countable set and let $S$ be the free semigroup generated by $X$. An element $w\in S$ is a word $w=x_{i_1}x_{i_2}\ldots x_{i_m}$ with $m\in\mathbb N_+$ and $i_j\in\mathbb Z$ for $j= 1,2, \ldots, m$. In this case $m$ is called the {\em length} $\ell_X(w)$ of $w$, and a subword of $w$ is any $w'\in S$ of the form $w'=x_{i_k}x_{i_k+1}\ldots x_{i_l}$ with $1\le k\le l\le n$. Consider the automorphism $\phi:S\to S$ determined by $\phi(x_n)=x_{n+1}$ for every $n\in\mathbb Z$. \begin{itemize}\label{ex-jan} \item[(a)] Let $s(w)$ be the number of adjacent pairs $(i_k,i_{k+1})$ in $w$ such that $i_k<i_{k+1}$. The map $v:S\to\mathbb R_+$ defined by $v(w)=s(w)+1$ is a semigroup norm. Then $\phi:(S,v)\to (S,v)$ is an automorphism of normed semigroups. It is straightforward to prove that, for $w=x_{i_1}x_{i_2}\ldots x_{i_m}\in S$, \begin{itemize} \item[(i)] $h_{\mathfrak{S}}^\#(\phi,w)=h_{\mathfrak{S}}(\phi,w)$ if and only if $i_1>i_m+1$; \item[(ii)] $h_{\mathfrak{S}}^\#(\phi,w)=h_{\mathfrak{S}}(\phi,w)-1$ if and only if $i_m=i_1$ or $i_m=i_1-1$. \end{itemize} Moreover, \begin{itemize} \item[(iii)]$h_{\mathfrak{S}}^\#(\phi)=h_{\mathfrak{S}}(\phi)=\infty$. \end{itemize} In particular, $h_{\mathfrak{S}}(\phi,x_0)=1$ while $h_{\mathfrak{S}}^\#(\phi,x_0)=0$. \item[(b)] Define a semigroup norm $\nu: S\to \mathbb R_+$ as follows. For $w=x_{i_1}x_{i_2}\ldots x_{i_n}\in S$ consider its subword $w'=x_{i_k}x_{i_{k+1}}\ldots x_{i_l}$ with maximal length satisfying $i_{j+1}=i_j+1$ for every $j\in\mathbb Z$ with $k\le j\le l-1$ and let $\nu(w)=\ell_X(w')$. Then $\phi:(S,\nu)\to (S,\nu)$ is an automorphism of normed semigroups. It is possible to prove that, for $w\in S$, \begin{enumerate} \item[(i)] if $\ell_X(w)=1$, then $\nu(T_n(\phi,w))=n$ and $\nu(T^\#_n(\phi,w))=1$ for every $n\in\mathbb N_+$; \item[(ii)] if $\ell_X(w)=k$ with $k>1$, then $\nu(T_n(\phi,w))< 2k$ and $\nu(T^\#_n(\phi,w))< 2k $ for every $n\in\mathbb N_+$. \end{enumerate} From (i) and (ii) and from the definitions we immediately obtain that \begin{itemize} \item[(iii)] $h_\mathfrak{S}(\phi)=1\neq 0=h^\#_\mathfrak{S}(\phi)$. \end{itemize} \end{itemize} \end{Example} We list now the main basic properties of the semigroup entropy. For complete proofs and further details see \cite{DGV1}. \begin{Lemma}[Monotonicity for factors] Let $S$, $T$ be normed semigroups and $\phi: S \to S$, $\psi:T\to T$ endomorphisms. If $arithmetic{}lpha:S\to T$ is a surjective homomorphism such that $arithmetic{}lpha \circ \psi = \phi \circ arithmetic{}lpha$, then $h_{{\mathfrak{S}}}(\phi) \leq h_{{\mathfrak{S}}}(\psi)$. \end{Lemma} \begin{proof} Fix $x\in S$ and find $y \in T$ with $x= arithmetic{}lpha(y)$. Then $c_n(x, \phi) \leq c_n(\psi, y)$. Dividing by $n$ and taking the limit gives $h_{{\mathfrak{S}}}(\phi,x) \leq h_{{\mathfrak{S}}}(\psi,y)$. So $h_{{\mathfrak{S}}}(\phi,x)\leq h_{{\mathfrak{S}}}(\psi)$. When $x$ runs over $S$, we conclude that $h_{{\mathfrak{S}}}(\phi) \leq h_{{\mathfrak{S}}}(\psi)$. \end{proof} \begin{Corollary}[Invariance under conjugation] Let $S$ be a normed semigroup and $\phi: S \to S$ an endomorphism. If $arithmetic{}lpha:T\to S$ is an isomorphism, then $h_{{\mathfrak{S}}}(\phi)=h_{{\mathfrak{S}}}(arithmetic{}lpha\circ\phi\circarithmetic{}lpha^{-1})$. \end{Corollary} \begin{Lemma}[Invariance under inversion]\label{inversion} Let $S$ be a normed semigroup and $\phi:S\to S$ an automorphism. Then $h_{{\mathfrak{S}}}(\phi^{-1})=h_{{\mathfrak{S}}}(\phi)$. \end{Lemma} \begin{Theorem}[Logarithmic Law] Let $(S,v)$ be a normed semigroup and $\phi:S\to S$ an endomorphism. Then $$ h_{{\mathfrak{S}}}(\phi^{k})\leq k\cdot h_{{\mathfrak{S}}}(\phi) $$ for every $k\in \mathbb N_+$. Furthermore, equality holds if $v$ is s-monotone. Moreover, if $\phi:S\to S$ is an automorphism, then $$h_{{\mathfrak{S}}}(\phi^k) = |k|\cdot h_{{\mathfrak{S}}}(\phi)$$ for all $k \in \mathbb Z\setminus\{0\}$. \end{Theorem} \begin{proof} Fix $k \in \mathbb N_+$, $x\in S$ and let $y= x\cdot\phi(x)\cdot\ldots\cdot\phi^{k-1}(x)$. Then \begin{align*} h_{\mathfrak{S}}(\phi^k)\geq h_{\mathfrak{S}}(\phi^k, y)&=\lim_{n\to\infty} \phirac{c_{n}(\phi^k,y)}{n}=\lim_{n\to \infty} \phirac{v (y\cdot \phi^k(y)\cdot\ldots \cdot\phi^{(n-1)k}(y)) }{n}=\\ &= k \cdot \lim_{n\to \infty} \phirac{c_{nk}(\phi,x)}{nk}=k\cdot h_{\mathfrak{S}}(\phi,x). \end{align*} This yields $h_{\mathfrak{S}}(\phi^k)\geq k\cdot h_{\mathfrak{S}}(\phi,x)$ for all $x\in S$, and consequently, $h_{\mathfrak{S}}(\phi^k)\geq k\cdot h_{\mathfrak{S}}(\phi)$. Suppose $v$ to be s-monotone, then \begin{equation*}\begin{split} h_{\mathfrak{S}}(\phi,x)=\lim_{n\to \infty} \phirac{v(x\cdot\phi (x)\cdot\ldots\cdot\phi^{nk-1}(x))}{n\cdot k} \geq \\ \lim_{n\to\infty} \phirac{ v(x\cdot\phi^k(x)\cdot\ldots\cdot(\phi^k)^{n-1}(x))}{n\cdot k}= \phirac{h_{\mathfrak{S}}(\phi^k,x)}{k} \end{split}\end{equation*} Hence, $k\cdot h_{\mathfrak{S}}(\phi)\geq h_{\mathfrak{S}}(\phi^k,x)$ for every $x\in S$. Therefore, $k\cdot h_{\mathfrak{S}}(\phi)\geq h_{\mathfrak{S}}(\phi^k)$. If $\phi$ is an automorphism and $k\in\mathbb Z\setminus\{0\}$, apply the previous part of the theorem and Lemma \ref{inversion}. \end{proof} The next lemma shows that monotonicity is available not only under taking factors: \begin{Lemma}[Monotonicity for subsemigroups] Let $(S,v)$ be a normed semigroup and $\phi:S\to S$ an endomorphism. If $T$ is a $\phi$-invariant normed subsemigroup of $(S,v)$, then $h_{{\mathfrak{S}}}(\phi)\geq h_{{\mathfrak{S}}}(\phi\restriction_{T})$. Equality holds if $S$ is ordered, $v$ is monotone and $T$ is cofinal in $S$. \end{Lemma} Note that $T$ is equipped with the induced norm $v\restriction_T$. The same applies to the subsemigroups $S_i$ in the next corollary: \begin{Corollary}[Continuity for direct limits] Let $(S,v)$ be a normed semigroup and $\phi:S\to S$ an endomorphism. If $\{S_i: i\in I\}$ is a directed family of $\phi$-invariant normed subsemigroup of $(S,v)$ with $ S =\varinjlim S_i$, then $h_{{\mathfrak{S}}}(\phi)=\sup h_{{\mathfrak{S}}}(\phi\restriction_{S_i})$. \end{Corollary} We consider now products in ${\mathfrak{S}}$. Let $\{(S_i,v_i):i\in I\}$ be a family of normed semigroups and let $S=\prod_{i \in I}S_i$ be their direct product in the category of semigroups. In case $I$ is finite, then $S$ becomes a normed semigroup with the $\max$-norm $v_{\prod}$, so $(S,v_{\prod})$ is the product of the family $\{S_i:i\in I\}$ in the category ${\mathfrak{S}}$; in such a case one has the following \begin{Theorem}[Weak Addition Theorem - products]\label{WAT} Let $(S_i,v_i)$ be a normed semigroup and $\phi_i:S_i\to S_i$ an endomorphism for $i=1,2$. Then the endomorphism $\phi_1 \times \phi_2$ of $ S _1 \times S_2$ has $h_{\mathfrak{S}}(\phi_1 \times \phi_2)= \max\{ h_{\mathfrak{S}}(\phi_1),h_{\mathfrak{S}}(\phi_2)\}$. \end{Theorem} If $I$ is infinite, $S$ need not carry a semigroup norm $v$ such that every projection $p_i: (S,v) \to (S_i,v_i)$ is a morphism in ${\mathfrak{S}}$. This is why the product of the family $\{(S_i,v_i):i\in I\}$ in ${\mathfrak{S}}$ is actually the subset $$S_{\mathrm{bnd}}=\{x=(x_i)_{i\in I}\in S: \sup_{i\in I}v_i(x_i)\in\mathbb R\}$$ of $S$ with the norm $v_{\prod}$ defined by $$v_{\prod}(x)=\sup_{i\in I}v_i(x_i)\ \text{for any}\ x=(x_i)_{i\in I}\in S_{\mathrm{bnd}}.$$ For further details in this direction see \cite{DGV1}. \subsection{Entropy in $\mathfrak M$} We collect here some additional properties of the semigroup entropy in the category $\mathfrak M$ of normed monoids where also coproducts are available. If $(S_i,v_i)$ is a normed monoid for every $i\in I$, the direct sum $$S= \bigoplus_{i\in I} S_i =\{(x_i)\in \prod_{i\in I}S_i: |\{i\in I: x_i \ne 1\}|<\infty\}$$ becomes a normed monoid with the norm $$v_\oplus(x) = \sum_{i\in I} v_i(x_i)\ \text{for any}\ x = (x_i)_{i\in I} \in S.$$ This definition makes sense since $v_i$ are monoid norms, so $v_i(1) = 0$. Hence, $(S,v_\oplus)$ becomes a coproduct of the family $\{(S_i,v_i):i\in I\}$ in $\mathfrak M$. We consider now the case when $I$ is finite, so assume without loss of generality that $I=\{1,2\}$. In other words we have two normed monoids $(S_1,v_1)$ and $(S_2,v_2)$. The product and the coproduct have the same underlying monoid $S=S_1\times S_2$, but the norms $v_\oplus$ and $v_{\prod}$ in $S$ are different and give different values of the semigroup entropy $h_{\mathfrak{S}}$; indeed, compare Theorem \ref{WAT} and the following one. \begin{Theorem}[Weak Addition Theorem - coproducts] Let $(S_i,v_i)$ be a normed monoid and $\phi_i:S_i\to S_i$ an endomorphism for $i=1,2$. Then the endomorphism $\phi_1 \oplus \phi_2$ of $S _1 \oplus S_2$ has $h_{\mathfrak{S}}(\phi_1 \oplus \phi_2)= h_{\mathfrak{S}}(\phi_1) + h_{\mathfrak{S}}(\phi_2)$. \end{Theorem} For a normed monoid $(M,v) \in \mathfrak M$ let $B(M)= \bigoplus_\mathbb N M$, equipped with the above coproduct norm $v_\oplus(x) = \sum_{n\in\mathbb N} v(x_n)$ for any $x=(x_n)_{n\in\mathbb N}\in B(M)$. The \emph{right Bernoulli shift} is defined by $$\beta_M:B(M)\to B(M), \ \beta_M(x_0,\dots,x_n,\dots)=(1,x_0,\dots,x_n,\dots),$$ while the \emph{left Bernoulli shift} is $${}_M\beta:B(M)\to B(M),\ {}_M\beta(x_0,x_1,\dots,x_n,\dots)=(x_1,x_2, \dots,x_n,\dots).$$ \begin{Theorem}[Bernoulli normalization] Let $(M,v)$ be a normed monoid. Then: \begin{itemize} \item[(a)] $h_{\mathfrak{S}} (\beta_M)=\sup_{x\in M}v(x)$; \item[(b)] $h_{\mathfrak{S}}({}_M\beta) = 0$. \end{itemize} \end{Theorem} \begin{proof} (a) For $x\in M$ consider $\underline{x}=(x_n)_{n\in\mathbb N}\in B(M)$ such that $x_0=x$ and $x_n=1$ for every $n\in\mathbb N_+$. Then $v_\oplus(T_n(\beta_M,\underline{x}))=n\cdot v(x)$, so $h_{\mathfrak{S}}(\beta_M,\underline{x})=v(x)$. Hence $h_{\mathfrak{S}}(\beta_M)\geq \sup_{x\in M}v(x)$. Let now $\underline{x}=(x_n)_{n\in\mathbb N}\in B(M)$ and let $k\in\mathbb N$ be the greatest index such that $x_k\neq 1$; then \begin{equation*}\begin{split} v_\oplus(T_n(\beta_M,\underline{x}))= \sum_{i=0}^{k+n} v(T_n(\beta_M,\underline{x})_i)\leq\\ \sum_{i=0}^{k-1} v(x_0\cdot\ldots\cdot x_i) + (n-k)\cdot v(x_1\cdot\ldots\cdot x_k)+\sum_{i=1}^{k} v(x_i\cdot\ldots\cdot x_k). \end{split}\end{equation*} Since the first and the last summand do not depend on $n$, after dividing by $n$ and letting $n$ converge to infinity we obtain $$h_{\mathfrak{S}}(\beta_M,\underline{x})=\lim_{n\to \infty} \phirac{v_\oplus(T_n(\beta_M,\underline{x}))}{n}\leq v(x_1\cdot\ldots\cdot x_k)\leq \sup_{x\in M}v(x).$$ (b) Note that ${}_M\beta$ is locally nilpotent and apply Lemma \ref{locally}. \end{proof} \subsection{Semigroup entropy of an element and pseudonormed semigroups}\label{NewSec1} One can notice a certain asymmetry in Definition \ref{SEofEndos}. Indeed, for $S$ a normed semigroup, the local semigroup entropy defined in \eqref{hs-eq} is a two variable function $$h_{{\mathfrak{S}}}: \mathrm{End}(S) \times S \to \mathbb R_+.$$ Taking $h_{{\mathfrak{S}}}(\phi)=\sup_{x\in S}h_{{\mathfrak{S}}}(\phi,x)$ for an endomorphism $\phi\in\mathrm{End}(S)$, we obtained the notion of semigroup entropy of $\phi$. But one can obviously exchange the roles of $\phi$ and $x$ and obtain the possibility to discuss the entropy of an element $x\in S$. This can be done in two ways. Indeed, in Remark \ref{Asymm} we consider what seems the natural counterpart of $h_{{\mathfrak{S}}}(\phi)$, while here we discuss a particular case that could appear to be almost trivial, but actually this is not the case, as it permits to give a uniform approach to some entropies which are not defined by using trajectories. So, by taking $\phi=\mathrm{id}_S$ in \eqref{hs-eq}, we obtain a map $h_{\mathfrak{S}}^0:S\to\mathbb R_+$: \begin{Definition} Let $S$ be a normed semigroup and $x\in S$. The \emph{semigroup entropy} of $x$ is $$ h_{{\mathfrak{S}}}^0(x):=h_{{\mathfrak{S}}}(\mathrm{id}_S,x) = \lim_{n\to\infty} \phirac{v(x^n)}{n}. $$ \end{Definition} We shall see now that the notion of semigroup entropy of an element is supported by many examples. On the other hand, since some of the examples given below cannot be covered by our scheme, we propose first a slight extension that covers those examples as well. Let ${\mathfrak S}^*$ be the category having as objects of all pairs $(S,v)$, where $S$ is a semigroup and $v:S \to \mathbb R_+$ is an \emph{arbitrary} map. A morphism in the category ${\mathfrak S}^*$ is a semigroup homomorphism $\phi: (S,v) \to (S',v')$ that is contracting with respect to the pair $v,v'$, i.e., $v'(\phi(x)) \leq v(x)$ for every $x\in S$. Note that our starting category ${\mathfrak S}$ is simply a full subcategory of ${\mathfrak S}^*$, having as objects those pairs $(S,v)$ such that $v$ satisfies (i) from Definition \ref{Def1}. These pairs were called normed semigroups and $v$ was called a semigroup norm. For the sake of convenience and in order to keep close to the current terminology, let us call the function $v$ in the larger category ${\mathfrak S}^*$ a \emph{semigroup pseudonorm} (although, we are imposing no condition on $v$ whatsoever). So, in this setting, one can define a local semigroup entropy $h_{{\mathfrak{S}}^*}: \mathrm{End}(S) \times S \to \mathbb R_+$ following the pattern of \eqref{hs-eq}, replacing the limit by $$h_{{\mathfrak{S}}^*} (\phi,x)=\limsup_{n\to \infty}\phirac{v(T_n(\phi,x))}{n}.$$ In particular, $$h_{{\mathfrak{S}}^*}^0(x)=\limsup_{n\to \infty}\phirac{v(x^n)}{n}.$$ Let us note that in order to have the last $\limsup$ a limit, one does not need $(S,v)$ to be in ${\mathfrak{S}}$, but it suffices to have the semigroup norm condition (i) from Definition \ref{Def1} fulfilled only for products of powers of the same element. We consider here three different entropies, respectively from \cite{MMS}, \cite{FFK} and \cite{Silv}, that can be described in terms of $h_{\mathfrak{S}}^0$ or its generalized version $h_{{\mathfrak{S}}^*}^0$. We do not go into the details, but we give the idea how to capture them using the notion of semigroup entropy of an element of the semigroup of all endomorphisms of a given object equipped with an appropriate semigroup (pseudo)norm. \begin{itemize} \item[(a)] Following \cite{MMS}, let $R$ be a Noetherian local ring and $\phi:R\to R$ an endomorphism of finite length; moreover, $\lambda(\phi)$ is the length of $\phi$, which is a real number $\geq 1$. In this setting the entropy of $\phi$ is defined by $$h_\lambda(\phi)=\lim_{n\to \infty}\phirac{\log\lambda(\phi^n)}{n}$$ and it is proved that this limit exists. Then the set $S=\mathrm{End}_{\mathrm{fl}}(R)$ of all finite-length endomorphisms of $R$ is a semigroup and $\log\lambda(-)$ is a semigroup norm on $S$. For every $\phi\in S$, we have $$ h_\lambda(\phi)=h_{\mathfrak{S}}(\mathrm{id}_S,\phi)=h_{{\mathfrak{S}}}^0(\phi). $$ In other words, $h_\lambda(\phi)$ is nothing else but the semigroup entropy of the element $\phi$ of the normed semigroup $S=\mathrm{End}_{\mathrm{fl}}(R)$. \item[(b)] We recall now the entropy considered in \cite{Silv}, which was already introduced in \cite{BV}. Let $t\in\mathbb N_+$ and $\varphi:\mathbb P^t\to\mathbb P^t$ be a dominant rational map of degree $d$. Then the entropy of $\varphi$ is defined as the logarithm of the dynamical degree, that is $$ h_\delta (\varphi)=\log \delta_\phi=\limsup_{n\to \infty}\phirac{\log\deg(\varphi^n)}{n}. $$ Consider the semigroup $S$ of all dominant rational maps of $\mathbb P^n$ and the function $\log\deg(-)$. In general this is only a semigroup pseudonorm on $S$ and $$h_{{\mathfrak{S}}^*}^0(\varphi)=h_\delta(\varphi).$$ Note that $\log\deg(-)$ is a semigroup norm when $\varphi$ is an endomorphism of the variety $\mathbb P^t$. \item[(c)] We consider now the growth rate for endomorphisms introduced in \cite{Bowen} and recently studied in \cite{FFK}. Let $G$ be a finitely generated group, $X$ a finite symmetric set of generators of $G$, and $\varphi:G\to G$ an endomorphism. For $g\in G$, denote by $\ell_X(g)$ the length of $g$ with respect to the alphabet $X$. The growth rate of $\varphi$ with respect to $x\in X$ is $$\log GR(\varphi,x)=\lim_{n\to \infty}\phirac{\log \ell_X(\varphi^n(x))}{n}$$ (and the growth rate of $\varphi$ is $\log GR(\varphi)=\sup_{x\in X} \log GR(\varphi,x)$). Consider $S=\mathrm{End}(G)$ and, fixed $x\in X$, the map $\log GR(-,x)$. As in item (b) this is only a semigroup pseudonorm on $S$. Nevertheless, also in this case the semigroup entropy $$\log GR(\varphi,x)=h_{{\mathfrak{S}}^*}^0(\varphi).$$ \end{itemize} \begin{Remark}\label{Asymm} For a normed semigroup $S$, let $h_{{\mathfrak{S}}}: \mathrm{End}(S) \times S \to \mathbb R_+$ be the local semigroup entropy defined in \eqref{hs-eq}. Exchanging the roles of $\phi\in \mathrm{End}(S)$ and $x\in S$, define the \emph{global semigroup entropy} of an element $x\in S$ by $$ h_{{\mathfrak{S}}}(x)=\sup_{\phi \in \mathrm{End}(S)}h_{{\mathfrak{S}}}(\phi,x). $$ Obviously, $h_{{\mathfrak{S}}}^0(x) \leq h_{{\mathfrak{S}}}(x)$ for every $x\in S$. \end{Remark} \section{Obtaining known entropies}\label{known-sec} \subsection{The general scheme} Let $\mathfrak X$ be a category and let $F:\mathfrak X\to {\mathfrak{S}}$ be a functor. Define the entropy $$h_{F}:\mathfrak X\to \mathbb R_+$$ on the category $\mathfrak X$ by $$h_{F}(\phi)=h_{{\mathfrak{S}}}(F(\phi)),$$ for any endomorphism $\phi: X \to X$ in $\mathfrak X$. Recall that with some abuse of notation we write $h_{F}:\mathfrak X\to \mathbb R_+$ in place of $h_{F}:\mathrm{Flow}_\mathfrak X\to \mathbb R_+$ for simplicity. Since the functor $F$ preserves commutative squares and isomorphisms, the entropy $h_{F}$ has the following properties, that automatically follow from the previously listed properties of the semigroup entropy $h_{\mathfrak{S}}$. For the details and for properties that need a further discussion see \cite{DGV1}. Let $X$, $Y$ be objects of $\mathfrak X$ and $\phi:X\to X$, $\psi:Y\to Y$ endomorphism in $\mathfrak X$. \begin{itemize} \item[(a)][Invariance under conjugation] If $arithmetic{}lpha:X\to Y$ is an isomorphism in $\mathfrak X$, then $h_{F}(\phi)=h_{F}(arithmetic{}lpha\circ\phi\circarithmetic{}lpha^{-1})$. \item[(b)][Invariance under inversion] If $\phi:X\to X$ is an automorphism in $\mathfrak X$, then $h_{F}(\phi^{-1})=h_{F}(\phi)$. \item[(c)][Logaritmic Law] If the norm of $F(X)$ is $s$-monotone, then $h_{F}(\phi^{k})=k\cdot h_{F}(\phi)$ for all $k\in \mathbb N_+$. \end{itemize} Other properties of $h_{F}$ depend on properties of the functor $F$. \begin{itemize} \item[(d)][Monotonicity for invariant subobjects] If $F$ sends subobject embeddings in $\mathfrak X$ to embeddings in ${\mathfrak{S}}$ or to surjective maps in ${\mathfrak{S}}$, then, if $Y$ is a $\phi$-invariant subobject of $X$, we have $h_{F}(\phi\restriction_Y)\leq h_{F}(\phi)$. \item[(e)][Monotonicity for factors] If $F$ sends factors in $\mathfrak X$ to surjective maps in ${\mathfrak{S}}$ or to embeddings in ${\mathfrak{S}}$, then, if $arithmetic{}lpha:T\to S$ is an epimorphism in $\mathfrak X$ such that $arithmetic{}lpha \circ \psi = \phi \circ arithmetic{}lpha$, then $h_F(\phi) \leq h_F(\psi)$. \item[(f)][Continuity for direct limits] If $F$ is covariant and sends direct limits to direct limits, then $h_F(\phi)=\sup_{i\in I} h_F(\phi\restriction_{X_i})$ whenever $X=\varinjlim X_i$ and $X_i$ is a $\phi$-invariant subobject of $X$ for every $i\in I$. \item[(g)][Continuity for inverse limits] If $F$ is contravariant and sends inverse limits to direct limits, then $h_F(\phi)=\sup_{i\in I} h_F(\overline\phi_i)$ whenever $X=\varprojlim X_i$ and $(X_i,\phi_i)$ is a factor of $(X,\phi)$ for every $i\in I$. \end{itemize} In the following subsections we describe how the known entropies can be obtained from this general scheme. For all the details we refer to \cite{DGV1} \subsection{Set-theoretic entropy} In this section we consider the category $\mathbf{Set}$ of sets and maps and its (non-full) subcategory $\mathbf{Set}_\mathrm{fin}$ having as morphisms all the finitely many-to-one maps. We construct a functor $\mathfrak{atr}:\mathbf{Set}\to{\mathfrak{S}}$ and a functor $\mathfrak{str}: \mathbf{Set}_\mathrm{fin} \to {\mathfrak{S}}$, which give the set-theoretic entropy $\mathfrak h$ and the covariant set-theoretic entropy $\mathfrak h^*$, introduced in \cite{AZD} and \cite{DG-islam} respectively. We also recall that they are related to invariants for self-maps of sets introduced in \cite{G0} and \cite{AADGH} respectively. A natural semilattice with zero, arising from a set $X$, is the family $({\mathcal S}(X),\cup)$ of all finite subsets of $X$ with neutral element $\emptyset$. Moreover the map defined by $v(A) = |A|$ for every $A\in\mathcal S(X)$ is an s-monotone norm. So let $\mathfrak{atr}(X)=(\mathcal S(X),\cup,v)$. Consider now a map $\lambda:X\to Y$ between sets and define $\mathfrak{atr}(\lambda):\mathcal S(X)\to \mathcal S(Y)$ by $A\mapsto \lambda(A)$ for every $A\in\mathcal S(X)$. This defines a covariant functor $$\mathfrak{atr}: \mathbf{Set} \to {\mathfrak{S}}$$ such that $$h_{\mathfrak{atr}}=\mathfrak h.$$ Consider now a finite-to-one map $\lambda:X\to Y$. As above let $\mathfrak{str}(X)=(\mathcal S(X),\cup,v)$, while $\mathfrak{str}(\lambda):\mathfrak{str}(Y)\to\mathfrak{str}(X)$ is given by $A \mapsto\lambda^{-1}(A)$ for every $A\in\mathcal S(Y)$. This defines a contravariant functor $$ \mathfrak{str}: \mathbf{Set}_\mathrm{fin}\to{\mathfrak{S}} $$ such that $$ h_{\mathfrak{str}}=\mathfrak h^*. $$ \subsection{Topological entropy for compact spaces} In this subsection we consider in the general scheme the topological entropy $h_{top}$ introduced in \cite{AKM} for continuous self-maps of compact spaces. So we specify the general scheme for the category $\mathfrak X=\mathbf{CTop}$ of compact spaces and continuous maps, constructing the functor $\mathfrak{cov}:\mathbf{CTop}\to{\mathfrak{S}}$. For a topological space $X$ let $\mathfrak{cov}(X)$ be the family of all open covers $\mathcal U$ of $X$, where it is allowed $\emptyset\in\mathcal U$. For ${\cal U}, {\cal V}\in \mathfrak{cov}(X)$ let ${\cal U} \vee {\cal V}=\{U\cap V: U\in {\cal U}, V\in {\cal V}\}\in \mathfrak{cov}(X)$. One can easily prove commutativity and associativity of $\vee$; moreover, let $\mathcal E=\{X\}$ denote the trivial cover. Then \begin{center} $(\mathfrak{cov}(X), \vee, \mathcal E)$ is a commutative monoid. \end{center} For a topological space $X$, one has a natural preorder ${\cal U} \prec{\cal V} $ on $\mathfrak{cov}(X)$; indeed, ${\cal V}$ \emph{refines} ${\cal U}$ if for every $V \in {\cal V}$ there exists $U\in {\cal U}$ such that $V\subseteq U$. Note that this preorder has bottom element $\mathcal E$, and that it is not an order. In general, ${\cal U} \vee {\cal U} \ne {\cal U} $, yet ${\cal U} \vee {\cal U} \sim {\cal U} $, and more generally \begin{equation}\label{vee} {\cal U} \vee {\cal U} \vee \ldots \vee {\cal U} \sim {\cal U}. \end{equation} For $X$, $Y$ topological spaces, a continuous map $\phi:X\to Y$ and ${\cal U}\in \mathfrak{cov} (Y)$, let $\phi^{-1}({\cal U})=\{\phi^{-1}(U): U\in {\cal U}\}$. Then, as $\phi^{-1}({\cal U} \vee {\cal V})= \phi^{-1}({\cal U})\vee \phi^{-1}({\cal V})$, we have that $\mathfrak{cov} (\phi): \mathfrak{cov} (Y)\to \mathfrak{cov} (X)$, defined by ${\cal U} \mapsto \phi^{-1}({\cal U})$, is a semigroup homomorphism. This defines a contravariant functor $\mathfrak{cov}$ from the category of all topological spaces to the category of commutative semigroups. To get a semigroup norm on $\mathfrak{cov}(X)$ we restrict this functor to the subcategory ${\mathbf{CTop}}$ of compact spaces. For a compact space $X$ and ${\cal U}\in \mathfrak{cov}(X)$, let $$M({\cal U})=\min\{|{\cal V}|: {\cal V}\mbox{ a finite subcover of }{\cal U}\}\ \text{and}\ v({\cal U})=\log M({\cal U}).$$ Now \eqref{vee} gives $v({\cal U} \vee {\cal U} \vee \ldots \vee {\cal U}) = v({\cal U})$, so \begin{center} $(\mathfrak{cov}(X), \vee, v)$ is an arithmetic normed semigroup. \end{center} For every continuous map $\phi:X\to Y$ of compact spaces and ${\cal W}\in \mathfrak{cov}(Y)$, the inequality $v(\phi^{-1}({\cal W}))\leq v({\cal W})$ holds. Consequently \begin{center} $\mathfrak{cov}(\phi): \mathfrak{cov} (Y)\to \mathfrak{cov} (X)$, defined by ${\cal W}\mapsto\phi^{-1}({\cal W})$, is a morphism in ${\mathfrak{S}}$. \end{center} Therefore the assignments $X \mapsto \mathfrak{cov}(X)$ and $\phi\mapsto\mathfrak{cov}(\phi)$ define a contravariant functor $$\mathfrak{cov}:\mathbf{CTop}\to {\mathfrak{S}}.$$ Moreover, $$h_{\mathfrak{cov}}=h_{top}.$$ Since the functor $\mathfrak{cov}$ takes factors in ${\mathbf{CTop}}$ to embeddings in ${\mathfrak{S}}$, embeddings in ${\mathbf{CTop}}$ to surjective morphisms in ${\mathfrak{S}}$, and inverse limits in ${\mathbf{CTop}}$ to direct limits in ${\mathfrak{S}}$, we have automatically that the topological entropy $h_{top}$ is monotone for factors and restrictions to invariant subspaces, continuous for inverse limits, is invariant under conjugation and inversion, and satisfies the Logarithmic Law. \subsection{Measure entropy} In this subsection we consider the category ${\mathbf{MesSp}}$ of probability measure spaces $(X, \mathfrak B, \mu)$ and measure preserving maps, constructing a functor $\mathfrak{mes}:{\mathbf{MesSp}}\to{\mathfrak{S}}$ in order to obtain from the general scheme the measure entropy $h_{mes}$ from \cite{K} and \cite{Sinai}. For a measure space $(X,\mathfrak{B},\mu)$ let $\mathfrak{P}(X)$ be the family of all measurable partitions $\xi=\{A_1,A_2,\ldots,A_k\}$ of $X$. For $\xi, \eta\in \mathfrak{P}(X)$ let $\xi \vee \eta=\{U\cap V: U\in \xi, V\in \eta\}$. As $\xi \vee \xi = \xi$, with zero the cover $\xi_0=\{X\}$, \begin{center} $(\mathfrak{P}(X),\vee)$ is a semilattice with $0$. \end{center} Moreover, for $\xi=\{A_1,A_2,\ldots,A_k\}\in \mathfrak{P}(X)$ the \emph{entropy} of $\xi$ is given by Boltzmann's Formula $$ v(\xi)=-\sum_{i=1}^k \mu(A_k)\log \mu(A_k). $$ This is a monotone semigroup norm making $\mathfrak{P}(X)$ a normed semilattice and a normed monoid. Consider now a measure preserving map $T:X\to Y$. For a cover $\xi=\{A_i\}_{i=1}^k\in \mathfrak{P}(Y)$ let $T^{-1}(\xi)=\{T^{-1}(A_i)\}_{i=1}^k$. Since $T$ is measure preserving, one has $T^{-1}(\xi)\in \mathfrak{P}(X)$ and $\mu (T^{-1}(A_i)) = \mu(A_i)$ for all $i=1,\ldots,k$. Hence, $v(T^{-1}(\xi)) = v(\xi)$ and so \begin{center} $\mathfrak{mes}(T):\mathfrak{P}(Y)\to\mathfrak{P}(X)$, defined by $\xi\mapsto T^{-1}(\xi)$, is a morphism in ${\mathfrak{L}}$. \end{center} Therefore the assignments $X \mapsto\mathfrak{P}(X)$ and $T\mapsto\mathfrak{mes}(T)$ define a contravariant functor $$\mathfrak{mes}:{\mathbf{MesSp}}\to{\mathfrak{L}}.$$ Moreover, $$h_{\mathfrak{mes}}=h_{mes}.$$ The functor $\mathfrak{mes}:{\mathbf{MesSp}}\to{\mathfrak{L}}$ is covariant and sends embeddings in ${\mathbf{MesSp}}$ to surjective morphisms in ${\mathfrak{L}}$ and sends surjective maps in ${\mathbf{MesSp}}$ to embeddings in ${\mathfrak{L}}$. Hence, similarly to $h_{top}$, also the measure-theoretic entropy $h_{mes}$ is monotone for factors and restrictions to invariant subspaces, continuous for inverse limits, is invariant under conjugation and inversion, satisfies the Logarithmic Law and the Weak Addition Theorem. In the next remark we briefly discuss the connection between measure entropy and topological entropy. \begin{Remark} \begin{itemize} \item[(a)] If $X$ is a compact metric space and $\phi: X \to X$ is a continuous surjective self-map, by Krylov-Bogolioubov Theorem \cite{BK} there exist some $\phi$-invariant Borel probability measures $\mu$ on $X$ (i.e., making $\phi:(X,\mu) \to (X,\mu)$ measure preserving). Denote by $h_\mu$ the measure entropy with respect to the measure $\mu$. The inequality $h_{\mu}(\phi)\leq h_{top}(\phi)$ for every $\mu \in M(X,\phi)$ is due to Goodwyn \cite{Goo}. Moreover the \emph{variational principle} (see \cite[Theorem 8.6]{Wa}) holds true: $$h_{top}(\phi)= \sup \{h_{\mu}(\phi): \mu\ \text{$\phi$-invariant measure on $X$}\}.$$ \item[(b)] In the computation of the topological entropy it is possible to reduce to surjective continuous self-maps of compact spaces. Indeed, for a compact space $X$ and a continuous self-map $\phi:X\to X$, the set $E_\phi(X)=\bigcap_{n\in\mathbb N}\phi^n(X)$ is closed and $\phi$-invariant, the map $\phi\restriction_{E_\phi(X)}:E_\phi(X)\to E_\phi(X)$ is surjective and $h_{top}(\phi)=h_{top}(\phi\restriction_{E_\phi(X)})$ (see \cite{Wa}). \item[(c)] In the case of a compact group $K$ and a continuous surjective endomorphism $\phi:K\to K$, the group $K$ has its unique Haar measure and so $\phi$ is measure preserving as noted by Halmos \cite{Halmos}. In particular both $h_{top}$ and $h_{mes}$ are available for surjective continuous endomorphisms of compact groups and they coincide as proved in the general case by Stoyanov \cite{S}. In other terms, denote by $\mathbf{CGrp}$ the category of all compact groups and continuous homomorphisms, and by $\textbf{CGrp}_e$ the non-full subcategory of $\textbf{CGrp}$, having as morphisms all epimorphisms in $\textbf{CGrp}$. So in the following diagram we consider the forgetful functor $V: \textbf{CGrp}_e\to \textbf{Mes}$, while $i$ is the inclusion of $\textbf{CGrp}_e$ in $\textbf{CGrp}$ as a non-full subcategory and $U:\mathbf{CGrp}\to \mathbf{Top}$ is the forgetful functor: \begin{equation*} \xymatrix{ \textbf{CGrp}_earithmetic{}r[r]^{i}arithmetic{}r[d]^V & \textbf{CGrp}arithmetic{}r[r]^{U}& \textbf{Top} \\ \textbf{Mes} } \end{equation*} For a surjective endomorphism $\phi$ of the compact group $K$, we have then $h_{mes}(V(\phi))=h_{top}(U(\phi))$. \end{itemize} \end{Remark} \subsection{Algebraic entropy} Here we consider the category $\mathbf{Grp}$ of all groups and their homomorphisms and its subcategory $\mathbf{AbGrp}$ of all abelian groups. We construct two functors $\mathfrak{sub}:\mathbf{AbGrp}\to{\mathfrak{L}}$ and $\mathfrak{pet}:\mathbf{Grp}\to{\mathfrak{S}}$ that permits to find from the general scheme the two algebraic entropies $\mathrm{ent}$ and $h_{alg}$. For more details on these entropies see the next section. Let $G$ be an abelian group and let $({\cal F}(G),\cdot)$ be the semilattice consisting of all finite subgroups of $G$. Letting $v(F) = \log|F|$ for every $F \in {\cal F}(G)$, then \begin{center} $({\cal F}(G),\cdot,v)$ is a normed semilattice \end{center} and the norm $v$ is monotone. For every group homomorphism $\phi: G \to H$, \begin{center} the map ${\cal F}(\phi): {\cal F}(G) \to {\cal F}(H)$, defined by $F\mapsto \phi(F)$, is a morphism in ${\mathfrak{L}}$. \end{center} Therefore the assignments $G\mapsto {\cal F}(G)$ and $\phi\mapsto {\cal F}(\phi)$ define a covariant functor $$\mathfrak{sub}: {\mathbf{AbGrp}} \to {\mathfrak{L}}.$$ Moreover $$h_{\mathfrak{sub}}=\mathrm{ent}.$$ Since the functor $\mathfrak{sub}$ takes factors in $\mathbf{AbGrp}$ to surjective morphisms in ${\mathfrak{S}}$, embeddings in $\mathbf{AbGrp}$ to embeddings in ${\mathfrak{S}}$, and direct limits in $\mathbf{AbGrp}$ to direct limits in ${\mathfrak{S}}$, we have automatically that the algebraic entropy $\mathrm{ent}$ is monotone for factors and restrictions to invariant subspaces, continuous for direct limits, invariant under conjugation and inversion, satisfies the Logarithmic Law. For a group $G$ let $\mathcal{H}(G)$ be the family of all finite non-empty subsets of $G$. Then $\mathcal{H}(G)$ with the operation induced by the multiplication of $G$ is a monoid with neutral element $\{1\}$. Moreover, letting $v(F) = \log |F|$ for every $F \in \mathcal{H}(G)$ makes $\mathcal{H}(G)$ a normed semigroup. For an abelian group $G$ the monoid $\mathcal{H}(G)$ is arithmetic since for any $F\in \mathcal{H}(G)$ the sum of $n$ summands satisfies $|F + \ldots + F|\leq (n+1)^{|F|}$. Moreover, $(\mathcal{H}(G),\subseteq)$ is an ordered semigroup and the norm $v$ is $s$-monotone. For every group homomorphism $\phi:G \to H$, \begin{center} the map $\mathcal{H}(\phi): \mathcal{H}(G) \to \mathcal{H}(H)$, defined by $F\mapsto \phi(F)$, is a morphism in ${\mathfrak{S}}$. \end{center} Consequently the assignments $G \mapsto (\mathcal{H}(G),v)$ and $\phi\mapsto \mathcal{H}(\phi)$ give a covariant functor $$\mathfrak{pet}:\mathbf{Grp}\to {\mathfrak{S}}.$$ Hence $$h_{\mathfrak{pet}}=h_{alg}.$$ Note that the functor $\mathfrak{sub}$ is a subfunctor of $\mathfrak{pet}: {\mathbf{AbGrp}} \to{\mathfrak{S}}$ as $ {\cal F}(G) \subseteq {\cal H}(G)$ for every abelian group $G$. As for the algebraic entropy $\mathrm{ent}$, since the functor $\mathfrak{pet}$ takes factors in $\mathbf{Grp}$ to surjective morphisms in ${\mathfrak{S}}$, embeddings in $\mathbf{Grp}$ to embeddings in ${\mathfrak{S}}$, and direct limits in $\mathbf{Grp}$ to direct limits in ${\mathfrak{S}}$, we have automatically that the algebraic entropy $h_{alg}$ is monotone for factors and restrictions to invariant subspaces, continuous for direct limits, invariant under conjugation and inversion, satisfies the Logarithmic Law. \subsection{$h_{top}$ and $h_{alg}$ in locally compact groups}\label{NewSec2} As mentioned above, Bowen introduced topological entropy for uniformly continuous self-maps of metric spaces in \cite{B}. His approach turned out to be especially efficient in the case of locally compact spaces provided with some Borel measure with good invariance properties, in particular for {continuous endomorphisms of locally compact groups provided with their Haar measure}. Later Hood in \cite{hood} extended Bowen's definition to uniformly continuous self-maps of arbitrary uniform spaces and in particular to continuous endomorphisms of (not necessarily metrizable) locally compact groups. On the other hand, Virili \cite{V} extended the notion of algebraic entropy to continuous endomorphisms of locally compact abelian groups, inspired by Bowen's definition of topological entropy (based on the use of Haar measure). As mentioned in \cite{DG-islam}, his definition can be extended to continuous endomorphisms of arbitrary locally compact groups. Our aim here is to show that both entropies can be obtained from our general scheme in the case of measure preserving topological automorphisms of locally compact groups. To this end we recall first the definitions of $h_{top}$ and $h_{alg}$ in locally compact groups. Let $G$ be a locally compact group, let $\mathcal C(G)$ be the family of all compact neighborhoods of $1$ and $\mu$ be a right Haar measure on $G$. For a continuous endomorphism $\phi: G \to G$, $U\in\mathcal C(G)$ and a positive integer $n$, the $n$-th cotrajectory $C_n(\phi,U)=U\cap \phi^{-1}(U)\cap\ldots\cap\phi^{-n+1}(U)$ is still in $\mathcal C(G)$. The topological entropy $h_{top}$ is intended to measure the rate of decay of the $n$-th cotrajectory $C_n(\phi,U)$. So let \begin{equation} H_{top}(\phi,U)=\limsup _{n\to \infty} - \phirac{\log \mu (C_n(\phi,U))}{n}, \end{equation} which does not depend on the choice of the Haar measure $\mu$. The \emph{topological entropy} of $\phi$ is $$ h_{top}(\phi)=\sup\{H_{top}(\phi,U):U\in\mathcal C(G)\}. $$ If $G$ is discrete, then $\mathcal C(G)$ is the family of all finite subsets of $G$ containing $1$, and $\mu(A) = |A|$ for subsets $A$ of $G$. So $H_{top}(\phi,U)= 0$ for every $U \in \mathcal C(G)$, hence $h_{top}(\phi)=0$. To define the algebraic entropy of $\phi$ with respect to $U\in\mathcal C(G)$ one uses the {$n$-th $\phi$-trajectory} $T_n(\phi,U)=U\cdot \phi(U)\cdot \ldots\cdot \phi^{n-1}(U)$ of $U$, that still belongs to $\mathcal C(G)$. It turns out that the value \begin{equation}\label{**} H_{alg}(\phi,U)=\limsup_{n\to \infty} \phirac{\log \mu (T_n(\phi,U))}{n} \end{equation} does not depend on the choice of $\mu$. The \emph{algebraic entropy} of $\phi$ is $$ h_{alg}(\phi)=\sup\{H_{alg}(\phi,U):U\in\mathcal C(G)\}. $$ The term ``algebraic'' is motivated by the fact that the definition of $T_n(\phi,U)$ (unlike $C_n(\phi,U)$) makes use of the group operation. As we saw above \eqref{**} is a limit when $G$ is discrete. Moreover, if $G$ is compact, then $h_{alg}(\phi)=H_{alg}(\phi,G)=0$. In the sequel, $G$ will be a locally compact group. We fix also a measure preserving topological automorphism $\phi: G \to G$. To obtain the entropy $h_{top}(\phi)$ via semigroup entropy fix some $V\in \mathcal C(G)$ with $\mu(V)\leq 1$. Then consider the subset $$ \mathcal C_0(G)=\{U\in \mathcal C(G): U \subseteq V\}. $$ Obviously, $\mathcal C_0(G)$ is a monoid with respect to intersection, having as neutral element $V$. To obtain a pseudonorm $v$ on $\mathcal C_0(G)$ let $v(U) = - \log \mu (U)$ for any $U \in \mathcal C_0(G)$. Then $\phi$ defines a semigroup isomorphism $\phi^\#: \mathcal C_0(G)\to \mathcal C_0(G)$ by $\phi^\#(U) = \phi^{-1}(U)$ for any $U\in \mathcal C_0(G)$. It is easy to see that $\phi^\#: \mathcal C_0(G)\to \mathcal C_0(G)$ is a an automorphism in ${\mathfrak{S}}^*$ and the semigroup entropy $h_{{\mathfrak{S}}^*}(\phi^\#)$ coincides with $h_{top}(\phi)$ since $H_{top}(\phi,U) \leq H_{top}(\phi,U')$ whenever $U \supseteq U'$. To obtain the entropy $h_{alg}(\phi)$ via semigroup entropy fix some $W\in \mathcal C(G)$ with $\mu(W)\geq 1$. Then consider the subset $$ \mathcal C_1(G)=\{U\in \mathcal C(G): U \supseteq W\} $$ of the set $\mathcal C(G)$. Note that for $U_1, U_2 \in \mathcal C_1(G)$ also $U_1U_2 \in \mathcal C_1(G)$. Thus $\mathcal C_1(G)$ is a semigroup. To define a pseudonorm $v$ on $\mathcal C_1(G)$ let $v(U) = \log \mu (U)$ for any $U \in \mathcal C_1(G)$. Then $\phi$ defines a semigroup isomorphism $\phi_\#: \mathcal C_1(G)\to \mathcal C_1(G)$ by $\phi_\#(U) = \phi(U)$ for any $U\in \mathcal C_1(G)$. It is easy to see that $\phi_\#: \mathcal C_1(G)\to \mathcal C_1(G)$ is a morphism in ${\mathfrak{S}}^*$ and the semigroup entropy $h_{{\mathfrak{S}}^*}(\phi_\#)$ coincides with $h_{alg}(\phi)$, since $ \mathcal C_1(G)$ is cofinal in $ \mathcal C(G)$ and $H_{alg}(\phi,U) \leq H_{alg}(\phi,U')$ whenever $U \subseteq U'$. \begin{Remark} We asked above the automorphism $\phi$ to be ``measure preserving". In this way one rules out many interesting cases of topological automorphisms that are not measure preserving (e.g., all automorphisms of $\mathbb R$ beyond $\pm \mathrm{id}_\mathbb R$). This condition is imposed in order to respect the definition of the morphisms in ${\mathfrak{S}}^*$. If one further relaxes this condition on the morphisms in ${\mathfrak{S}}^*$ (without asking them to be contracting maps with respect to the pseudonorm), then one can obtain a semigroup entropy that covers the topological and the algebraic entropy of arbitrary topological automorphisms of locally compact groups (see \cite{DGV} for more details). \end{Remark} \subsection{Algebraic $i$-entropy} For a ring $R$ we denote by $\mathbf{Mod}_R$ the category of right $R$-modules and $R$-module homomorphisms. We consider here the algebraic $i$-entropy introduced in \cite{SZ}, giving a functor ${\mathfrak{sub}_i}:\mathbf{Mod}_R\to {\mathfrak{L}}$, to find $\mathrm{ent}_i$ from the general scheme. Here $i: \mathbf{Mod}_R \to \mathbb R_+$ is an invariant of $\mathbf{Mod}_R$ (i.e., $i(0)=0$ and $i(M) = i(N)$ whenever $M\cong N$). Consider the following conditions: \begin{itemize} \item[(a)] $i(N_1 + N_2)\leq i(N_1) + i(N_2)$ for all submodules $N_1$, $N_2$ of $M$; \item[(b)] $i(M/N)\leq i(M)$ for every submodule $N$ of $M$; \item[(b$^*$)] $i(N)\leq i(M)$ for every submodule $N$ of $M$. \end{itemize} The invariant $i$ is called \emph{subadditive} if (a) and (b) hold, and it is called \emph{preadditive} if (a) and (b$^*$) hold. For $M\in\mathbf{Mod}_R$ denote by ${\cal L}(M)$ the lattice of all submodules of $M$. The operations are intersection and sum of two submodules, the bottom element is $\{0\}$ and the top element is $M$. Now fix a subadditive invariant $i$ of $\mathbf{Mod}_R$ and for a right $R$-module $M$ let $${\cal F}_i(M)=\{\mbox{submodules $N$ of $M$ with }i(M)< \infty\},$$ which is a subsemilattice of ${\cal L}(M)$ ordered by inclusion. Define a norm on ${\cal F}_i(M)$ setting $$v(H)=i(H)$$ for every $H \in {\cal F}_i(M)$. The norm $v$ is not necessarily monotone (it is monotone if $i$ is both subadditive and preadditive). For every homomorphism $\phi: M \to N$ in $\mathbf{Mod}_R$, \begin{center} ${\cal F}_i(\phi): {\cal F}_i(M) \to {\cal F}_i(N)$, defined by ${\cal F}_i(\phi)(H) =\phi(H)$, is a morphism in ${\mathfrak{L}}$. \end{center} Moreover the norm $v$ makes the morphism ${\cal F}_i(\phi)$ contractive by the property (b) of the invariant. Therefore, the assignments $M \mapsto {\cal F}_i(M)$ and $\phi\mapsto {\cal F}_i(\phi)$ define a covariant functor $$\mathfrak{sub}_i:\mathbf{Mod} _R\to {\mathfrak{L}}.$$ We can conclude that, for a ring $R$ and a subadditive invariant $i$ of $\mathbf{Mod}_R$, $$h_{\mathfrak{sub}_i}=\mathrm{ent}_i.$$ If $i$ is preadditive, the functor ${\mathfrak{sub}_i}$ sends monomorphisms to embeddings and so $\mathrm{ent}_i$ is monotone under taking submodules. If $i$ is both subadditive and preadditive then for every $R$-module $M$ the norm of ${\mathfrak{sub}_i}(M)$ is s-monotone, so $\mathrm{ent}_{i}$ satisfies also the Logarithmic Law. In general this entropy is not monotone under taking quotients, but this can be obtained with stronger hypotheses on $i$ and with some restriction on the domain of ${\mathfrak{sub}_i}$. A clear example is given by vector spaces; the algebraic entropy $\mathrm{ent}_{\dim}$ for linear transformations of vector spaces was considered in full details in \cite{GBS}: \begin{Example} Let $K$ be a field. Then for every $K$-vector space $V$ let ${\cal F}_d(M)$ be the set of all finite-dimensional subspaces $N$ of $M$. Then $({\cal F}_d(V),+)$ is a subsemilattice of $({\cal L}(V),+)$ and $v(H)=\dim H$ defines a monotone norm on ${\cal F}_d(V)$. For every morphism $\phi: V \to W$ in $\mathbf{Mod}_K$ \begin{center} the map ${\cal F}_d(\phi): {\cal F}_d(V) \to {\cal F}_d(W)$, defined by $H\mapsto\phi(H)$, is a morphism in ${\mathfrak{L}}$. \end{center} Therefore, the assignments $M \mapsto {\cal F}_d(M)$ and $\phi\mapsto {\cal F}_d(\phi)$ define a covariant functor $$\mathfrak{sub}_d:\mathbf{Mod}_K\to {\mathfrak{L}}.$$ Then $$h_{\mathfrak{sub}_d}=\mathrm{ent}_{\dim}.$$ Note that this entropy can be computed ad follows. Every flow $\phi: V \to V$ of $\mathbf{Mod}_K$ can be considered as a $K[X]$-module $V_\phi$ letting $X$ act on $V$ as $\phi$. Then $h_{\mathfrak{sub}_d}(\phi)$ coincides with the rank of the $K[X]$-module $V_\phi$. \end{Example} \subsection{Adjoint algebraic entropy} We consider now again the category $\mathbf{Grp}$ of all groups and their homomorphisms, giving a functor $\mathfrak{sub}^\star:\mathbf{Grp}\to {\mathfrak{L}}$ such that the entropy defined using this functor coincides with the adjoint algebraic entropy $\mathrm{ent}^\star$ introduced in \cite{DGS}. For a group $G$ denote by ${\cal C}(G)$ the family of all subgroups of finite index in $G$. It is a subsemilattice of $({\cal L}(G), \cap)$. For $N\in{\cal C}(G)$, let $$v(N) = \log[G:N];$$ then \begin{center} $({\cal C}(G),v)$ is a normed semilattice, \end{center} with neutral element $G$; moreover the norm $v$ is monotone. For every group homomorphism $\phi: G \to H$ \begin{center} the map ${\cal C}(\phi): {\cal C}(H) \to {\cal C}(G)$, defined by $N\mapsto \phi^{-1}(N)$, is a morphism in ${\mathfrak{S}}$. \end{center} Then the assignments $G\mapsto{\cal C}(G)$ and $\phi\mapsto{\cal C}(\phi)$ define a contravariant functor $$\mathfrak{sub}^\star:\mathbf{Grp}\to {\mathfrak{L}}.$$ Moreover $$h_{\mathfrak{sub}^\star}=\mathrm{ent}^\star.$$ There exists also a version of the adjoint algebraic entropy for modules, namely the adjoint algebraic $i$-entropy $\mathrm{ent}_i^\star$ (see \cite{Vi}), which can be treated analogously. \subsection{Topological entropy for totally disconnected compact groups} Let $(G,\tau)$ be a totally disconnected compact group and consider the filter base ${\cal V}_G(1)$ of open subgroups of $G$. Then \begin{center} $({\cal V}_G(1), \cap)$ is a normed semilattice \end{center} with neutral element $G \in {\cal V}_G(1)$ and norm defined by $v_o(V)=\log [G:V]$ for every $V\in{\cal V}_G(1)$. For a continuous homomorphism $\phi: G\to H$ between compact groups, \begin{center} the map ${\cal V}_H(1)\to {\cal V}_G(1)$, defined by $V \mapsto \phi^{-1}(V)$, is a morphism in ${\mathfrak{L}}$. \end{center} This defines a contravariant functor $$\mathfrak{sub}_o^\star:\mathbf{TdCGrp}\to{\mathfrak{L}},$$ which is a subfunctor of $\mathfrak{sub}^\star$. Then the entropy $h_{\mathfrak{sub}^\star_o}$ coincides with the restriction to $\mathbf{TdCGrp}$ of the topological entropy $h_{top}$. This functor is related also to the functor $\mathfrak{cov}:\mathbf{TdCGrp} \to{\mathfrak{S}}$. Indeed, let $G$ be a totally disconnected compact group. Each $V\in {\cal V}_G(1)$ defines a cover ${\cal U}_V=\{x\cdot V\}_{x\in G}$ of $G$ with $v_o(V)=v({\cal U}_V)$. So the map $V \mapsto {\cal U}_V$ defines an isomorphism between the normed semilattice $\mathfrak{sub}_o^\star(G)={\cal V}_G(1)$ and the subsemigroup $\mathfrak{cov}_s(G)=\{{\cal U}_V:V \in {\cal V}_G(1)\}$ of $\mathfrak{cov}(G)$. \subsection{Bridge Theorem}\label{BTsec} In Definition \ref{BTdef} we have formalized the concept of Bridge Theorem between entropies $h_1:\mathfrak X_1 \to \mathbb R_+$ and $h_2:\mathfrak X_2 \to \mathbb R_+$ via functors $\varepsilon: \mathfrak X_1 \to \mathfrak X_2$. Obviously, the Bridge Theorem with respect to the functor $\varepsilon$ is available when each $h_i$ has the form $h_i= h_{F_i}$ for appropriate functors $F_i: \mathfrak{X}_i \to {\mathfrak{S}}$ ($i= 1,2$) that commute with $\varepsilon$ (i.e., $F_1 = F_2 \varepsilon$), that is $$ h_2(\varepsilon(\phi))= h_1(\phi)\ \mbox{ for all morphisms $\phi$ in }\ \mathfrak X_1. $$ Actually, it is sufficient that $F_i$ commute with $\varepsilon$ ``modulo $h_{\mathfrak{S}}$" (i.e., $h_{\mathfrak{S}} F_1 = h_{\mathfrak{S}} F_2 \varepsilon$) to obtain this conclusion: \begin{equation}\label{Buzz} \xymatrix@R=6pt@C=37pt { \mathfrak{X}_1arithmetic{}r[dd]_{\varepsilon}arithmetic{}r[dr]^{F_1}arithmetic{}r@/^2pc/[rrd]^{h_{1}} & & \\ & {{\mathfrak{S}}}arithmetic{}r[r]|-{ {h_{\mathfrak{S}}}}&\mathbb R^+ \\ \mathfrak{X}_2arithmetic{}r[ur]_{F_2}arithmetic{}r@/_2pc/[rru]_{h_{2}} & & } \end{equation} In particular the Pontryagin duality functor {$\ \widehat{}: {\mathbf{AbGrp}} \to {\mathbf{CAbGrp}}$} connects the category of abelian groups and that of compact abelian groups so connects the respective entropies $h_{alg}$ and $h_{top}$ by a Bridge Theorem. Taking the restriction to torsion abelian groups and the totally disconnected compact groups one obtains: \begin{Theorem}[Weiss Bridge Theorem]\emph{\cite{W}}\label{WBT} Let $K$ be a totally disconnected compact abelian group and $\phi: K\to K$ a continuous endomorphism. Then $h_{top}(\phi) = \mathrm{ent}(\widehat \phi)$. \end{Theorem} \begin{proof} Since totally disconnected compact groups are zero-dimensional, every open finite cover $\mathcal U$ of $K$ admits a refinement consisting of clopen sets in $K$. Moreover, since $K$ admits a local base at 0 formed by open subgroups, it is possible to find a refinement of $\mathcal U$ of the form $\mathcal U_V$ for some open subgroup $ \mathcal V$. This proves that $\mathfrak{cov}_s(K)$ is cofinal in $\mathfrak{cov}(K)$. Hence, we have $$ h_{top}(\phi)=h_{\mathfrak{S}}(\mathfrak{cov}(\phi))=h_{\mathfrak{S}}(\mathfrak{cov}_s(\phi)). $$ Moreover, we have seen above that $\mathfrak{cov}_s(K)$ is isomorphic to $\mathfrak{sub}^\star_o(K)$, so one can conclude that $$h_{\mathfrak{S}}(\mathfrak{cov}_s(\phi))=h_{\mathfrak{S}}(\mathfrak{sub}^\star_o (\phi)).$$ Now the semilattice isomorphism $L\to \mathcal F(\widehat K)$ given by $N \mapsto N^\perp$ preserves the norms, so it is an isomorphism in ${\mathfrak{S}}$. Hence $$h_{\mathfrak{S}}(\mathfrak{sub}^\star_o (\phi))=h_{\mathfrak{S}}(\mathfrak{sub}(\widehat \phi))$$ and consequently $$h_{top}(\phi)= h_{\mathfrak{S}}(\mathfrak{sub}(\widehat \phi))=\mathrm{ent}(\widehat \phi).$$ \end{proof} The proof of Weiss Bridge Theorem can be reassumed by the following diagram. \begin{equation*} \xymatrix@R=6pt@C=37pt { (\widehat K,\widehat\phi)arithmetic{}r[r]^{\mathfrak{sub}}arithmetic{}r@/^4.5pc/[rrrddd]^{h_{\mathfrak{sub}}}&(({\cal F}(\widehat K),+);\mathfrak{sub}(\widehat \phi))arithmetic{}r[dd]_{\widehat{}}arithmetic{}r[dddrr]|-{h_{\mathfrak{S}}}& &\\ & & &\\ &((\mathfrak{sub}^\star_o(K),\cap);\mathfrak{sub}^\star_o(\phi))arithmetic{}r[dd]_{\gamma} & & \\ & & & \mathbb R^+ \\ &((\mathfrak{cov}_{s}(K),\vee);\phi)arithmetic{}r@{^{(}->}[dd]_{\iota} & & \\ & & &\\ (K,\phi)arithmetic{}r@/^3pc/[uuuuuu]^{\widehat{}\;\;}arithmetic{}r[r]_{\mathfrak{cov}}arithmetic{}r@/_4.5pc/[rrruuu]_{h_{\mathfrak{cov}}}&((\mathfrak{cov}(K),\vee);\mathfrak{cov}(\phi))arithmetic{}r[uuurr]|-{h_{\mathfrak{S}}} & & } \end{equation*} Similar Bridge Theorems hold for other known entropies; they can be proved using analogous diagrams (see \cite{DGV1}). The first one that we recall concerns the algebraic entropy $\mathrm{ent}$ and the adjoint algebraic entropy $\mathrm{ent}^\star$: \begin{Theorem} Let $\phi: G\to G$ be an endomorphism of an abelian group. Then $\mathrm{ent}^\star(\phi) = \mathrm{ent}(\widehat\phi)$. \end{Theorem} The other two Bridge Theorems that we recall here connect respectively the set-theoretic entropy $\mathfrak h$ with the topological entropy $h_{top}$ and the contravariant set-theoretic entropy $\mathfrak h^*$ with the algebraic entropy $h_{alg}$. We need to recall first the notion of generalized shift, which extend the Bernoulli shifts. For a map $\lambda:X\to Y$ between two non-empty sets and a fixed non-trivial group $K$, define $\sigma_\lambda:K^Y \to K^X$ by $\sigma_\lambda(f) = f\circ \lambda $ for $f\in K^Y$. For $Y = X$, $\lambda$ is a self-map of $X$ and $\sigma_\lambda$ was called \emph{generalized shift} of $K^X$ (see \cite{AADGH,AZD}). In this case $\bigoplus_X K$ is a $\sigma_\lambda$-invariant subgroup of $K^X$ precisely when $\lambda$ is finitely many-to-one. We denote $\sigma_\lambda\restriction_{\bigoplus_XK}$ by $\sigma_\lambda^\oplus$. Item (a) in the next theorem was proved in \cite{AZD} (see also \cite[Theorem 7.3.4]{DG-islam}) while item (b) is \cite[Theorem 7.3.3]{DG-islam} (in the abelian case it was obtained in \cite{AADGH}). \begin{Theorem} \emph{\cite{AZD}} Let $K$ be a non-trivial finite group, let $X$ be a set and $\lambda:X\to X$ a self-map. \begin{itemize} \item[(a)]Then $h_{top}(\sigma_\lambda)=\mathfrak h(\lambda)\log|K|$. \item[(b)] If $\lambda$ is finite-to-one, then $h_{alg}(\sigma_\lambda^\oplus)=\mathfrak h^*(\lambda)\log|K|$. \end{itemize} \end{Theorem} In terms of functors, fixed a non-trivial finite group $K$, let $\mathcal F_K: \mathbf{Set}\to \mathbf{TdCGrp}$ be the functor defined on flows, sending a non-empty set $X$ to $K^X$, $\emptyset$ to $0$, a self-map $\lambda:X\to X$ to $\sigma_\lambda:K^Y\to K^X$ when $X\ne \emptyset$. Then the pair $(\mathfrak h, h_{top})$ satisfies $(BT_{\mathcal F_K})$ with constant $\log |K|$. Analogously, let $\mathcal G_K: \mathbf{Set}_\mathrm{fin}\to \mathbf{Grp}$ be the functor defined on flows sending $X$ to $\bigoplus_X K$ and a finite-to-one self-map $\lambda:X\to X$ to $\sigma_\lambda^\oplus:\bigoplus_X K\to \bigoplus_X K$. Then the pair $(\mathfrak h^*, h_{alg})$ satisfies $(BT_{\mathcal G_K})$ with constant $\log |K|$. \begin{Remark} At the conference held in Porto Cesareo, R. Farnsteiner posed the following question related to the Bridge Theorem. Is $h_{top}$ studied in non-Hausdorff compact spaces? The question was motivated by the fact that the prime spectrum $\mathrm{Spec}(A)$ of a commutative ring $A$ is usually a non-Hausdorff compact space. Related to this question and to the entropy $h_\lambda$ defined for endomorphisms $\phi$ of local Noetherian rings $A$ (see \S \ref{NewSec1}), one may ask if there is any relation (e.g., a weak Bridge Theorem) between these two entropies and the functor $\mathrm{Spec}$; more precisely, one can ask whether there is any stable relation between $h_{top}(\mathrm{Spec}(\phi))$ and $h_\lambda(\phi)$. \end{Remark} \section{Algebraic entropy and its specific properties}\label{alg-sec} In this section we give an overview of the basic properties of the algebraic entropy and the adjoint algebraic entropy. Indeed, we have seen that they satisfy the general scheme presented in the previous section, but on the other hand they were defined for specific group endomorphisms and these definitions permit to prove specific features, as we are going to briefly describe. For further details and examples see \cite{DG}, \cite{DGS} and \cite{DG-islam}. \subsection{Definition and basic properties} Let $G$ be a group and $\phi:G\to G$ an endomorphism. For a finite subset $F$ of $G$, and for $n\in\mathbb N_+$, the \emph{$n$-th $\phi$-trajectory} of $F$ is \begin{equation*}\label{T_n} T_n(\phi,F)=F\cdot\phi(F)\cdot\ldots\cdot\phi^{n-1}(F); \end{equation*} moreover let \begin{equation}\label{gamma} {\gamma_{\phi,F}(n)}=|T_n(\phi,F)|. \end{equation} The \emph{algebraic entropy of $\phi$ with respect to $F$} is \begin{equation*}\label{H} H_{alg}(\phi,F)={\lim_{n\to \infty}\phirac{\log \gamma_{\phi,F}(n) }{n}}; \end{equation*} This limit exists as $H_{alg}(\phi,F)=h_{\mathfrak{S}}(\mathcal H(\phi),F)$ and so Theorem \ref{limit} applies (see also \cite{DG-islam} for a direct proof of the existence of this limit and \cite{DG} for the abelian case). The \emph{algebraic entropy} of $\phi:G\to G$ is $$ h_{alg}(\phi)=\sup\{H_{alg}(\phi,F): F\ \text{finite subset of}\ G\}=h_{\mathfrak{S}}(\mathcal H(\phi)). $$ Moreover $$ \mathrm{ent}(\phi)=\sup\{H_{alg}(\phi,F): F\ \text{finite subgroup of}\ G\}. $$ If $G$ is abelian, then $\mathrm{ent}(\phi)=\mathrm{ent}(\phi\restriction_{t(G)})= h_{alg}(\phi\restriction_{t(G)})$. Moreover, $h_{alg}(\phi) = \mathrm{ent}(\phi)$ if $G$ is locally finite, that is every finite subset of $G$ generates a finite subgroup; note that every locally finite group is obviously torsion, while the converse holds true under the hypothesis that the group is abelian (but the solution of Burnside Problem shows that even groups of finite exponent fail to be locally finite). For every abelian group $G$, the identity map has $h_{alg}(\mathrm{id}_G)=0$ (as the normed semigroup $\mathcal H(G)$ is arithmetic, as seen above). Another basic example is given by the endomorphisms of $\mathbb Z$, indeed if $\phi: \mathbb Z \to \mathbb Z$ is given by $\phi(x) = mx$ for some positive integer $m$, then $h_{alg}(\phi) = \log m$. The fundamental example for the algebraic entropy is the right Bernoulli shift: \begin{Example}\label{shift}(Bernoulli normalization) Let $K$ be a group. \begin{itemize} \item[(a)] The \emph{right Bernoulli shift} $\beta_K:K^{(\mathbb N)}\to K^{(\mathbb N)}$ is defined by $$(x_0,\ldots,x_n,\ldots)\mapsto (1,x_0,\ldots,x_n,\ldots).$$ Then $h_{alg}(\beta_K)=\log|K|$, with the usual convention that $\log|K|=\infty$ when $K$ is infinite. \item[(b)] The \emph{left Bernoulli shift} ${}_K\beta:K^{(\mathbb N)}\to K^{(\mathbb N)}$ is defined by $$(x_0,\ldots,x_n,\ldots)\mapsto (x_1,\ldots,x_{n+1},\ldots).$$ Then $h_{alg}({}_K\beta)=0$, as ${}_K\beta$ is locally nilpotent. \end{itemize} \end{Example} The following basic properties of the algebraic entropy are consequences of the general scheme and were proved directly in \cite{DG-islam}. \begin{fact}\label{properties} \emph{Let $G$ be a group and $\phi:G\to G$ an endomorphism. \begin{itemize} \item[(a)]\emph{[Invariance under conjugation]} If $\phi=\xi^{-1}\psi\xi$, where $\psi:H\to H$ is an endomorphism and $\xi:G\to H$ isomorphism, then $h_{alg}(\phi) = h_{alg}(\psi)$. \item[(b)]\emph{[Monotonicity]} If $H$ is a $\phi$-invariant normal subgroup of the group $G$, and $\overline\phi:G/H\to G/H$ is the endomorphism induced by $\phi$, then $h_{alg}(\phi)\geq \max\{h_{alg}(\phi\restriction_H),h_{alg}(\overline{\phi})\}$. \item[(c)]\emph{[Logarithmic Law]} For every $k\in\mathbb N$ we have $h_{alg}(\phi^k) = k \cdot h_{alg}(\phi)$; if $\phi$ is an automorphism, then $h_{alg}(\phi)=h_{alg}(\phi^{-1})$, so $h_{alg}(\phi^k)=|k|\cdot h_{alg}(\phi)$ for every $k\in\mathbb Z$. \item[(d)]\emph{[Continuity]} If $G$ is direct limit of $\phi$-invariant subgroups $\{G_i : i \in I\}$, then $h_{alg}(\phi)=\sup_{i\in I}h_{alg}(\phi\restriction_{G_i}).$ \item[(e)]\emph{[Weak Addition Theorem]} If $G=G_1\times G_2$ and $\phi_i:G_i\to G_i$ is an endomorphism for $i=1,2$, then $h_{alg}(\phi_1\times\phi_2)=h_{alg}(\phi_1)+h_{alg}(\phi_2)$. \end{itemize} } \end{fact} As described for the semigroup entropy in the previous section, and as noted in \cite[Remark 5.1.2]{DG-islam}, for group endomorphisms $\phi:G\to G$ it is possible to define also a ``left'' algebraic entropy, letting for a finite subset $F$ of $G$, and for $n\in\mathbb N_+$, $$T_n^\#(\phi,F)=\phi^{n-1}(F)\cdot\ldots\cdot\phi(F)\cdot F,$$ $$H^\#_{alg}(\phi,F)={\lim_{n\to \infty}\phirac{\log |T^\#_n(\phi,F)|}{n}}$$ and $$h_{alg}^\#(\phi)=\sup\{H^\#_{alg}(\phi,F): F\ \text{finite subset of}\ G\}.$$ Answering a question posed in \cite[Remark 5.1.2]{DG-islam}, we see now that $$h_{alg}(\phi)=h_{alg}^\#(\phi).$$ Indeed, every finite subset of $G$ is contained in a finite subset $F$ of $G$ such that $1\in F$ and $F={F^{-1}}$; for such $F$ we have $$H_{alg}(\phi,F)=H_{alg}^\#(\phi,F),$$ since, for every $n\in\mathbb N_+$, \begin{equation*}\begin{split} T_n(\phi,F)^{-1}=\phi^{n-1}(F)^{-1}\cdot\ldots\cdot\phi(F)^{-1}\cdot F^{-1}=\\ \phi^{n-1}(F^{-1})\cdot\ldots\cdot\phi(F^{-1})\cdot F^{-1}=T_n^\#(\phi,F) \end{split}\end{equation*} and so $|T_n(\phi,F)|=|T_n(\phi,F)^{-1}|=|T_n^\#(\phi,F)|$. \subsection{Algebraic Yuzvinski Formula, Addition Theorem and Uni\-que\-ness}\label{ab-sec} We recall now some of the main deep properties of the algebraic entropy in the abelian case. They are not consequences of the general scheme and are proved using the specific features of the algebraic entropy coming from the definition given above. We give here the references to the papers where these results were proved, for a general exposition on algebraic entropy see the survey paper \cite{DG-islam}. The next proposition shows that the study of the algebraic entropy for torsion-free abelian groups can be reduced to the case of divisible ones. It was announced for the first time by Yuzvinski \cite{Y1}, for a proof see \cite{DG}. \begin{Proposition}\label{AA_} Let $G$ be a torsion-free abelian group, $\phi:G\to G$ an endomorphism and denote by $\widetilde\phi$ the (unique) extension of $\phi$ to the divisible hull $D(G)$ of $G$. Then $h_{alg}(\phi)=h_{alg}(\widetilde\phi)$. \end{Proposition} Let $f(t)=a_nt^n+a_1t^{n-1}+\ldots+a_0\in\mathbb Z[t]$ be a primitive polynomial and let $\{\lambda_i:i=1,\ldots,n\}\subseteq\mathbb C$ be the set of all roots of $f(t)$. The \emph{(logarithmic) Mahler measure} of $f(t)$ is $$m(f(t))= \log|a_n| + \sum_{|\lambda_i|>1}\log |\lambda_i|.$$ The Mahler measure plays an important role in number theory and arithmetic geometry and is involved in the famous Lehmer Problem, asking whether $\inf\{m(f(t)):f(t)\in\mathbb Z[t]\ \text{primitive}, m(f(t))>0\}>0$ (for example see \cite{Ward0} and \cite{Hi}). If $g(t)\in\mathbb Q[t]$ is monic, then there exists a smallest positive integer $s$ such that $sg(t)\in\mathbb Z[t]$; in particular, $sg(t)$ is primitive. The Mahler measure of $g(t)$ is defined as $m(g(t))=m(sg(t))$. Moreover, if $\phi:\mathbb Q^n\to \mathbb Q^n$ is an endomorphism, its characteristic polynomial $p_\phi(t)\in\mathbb Q[t]$ is monic, and the Mahler measure of $\phi$ is $m(\phi)=m(p_\phi(t))$. The formula \eqref{yuzeq} below was given a direct proof recently in \cite{GV}; it is the algebraic counterpart of the so-called Yuzvinski Formula for the topological entropy \cite{Y1} (see also \cite{LW}). It gives the values of the algebraic entropy of linear transformations of finite dimensional rational vector spaces in terms of the Mahler measure, so it allows for a connection of the algebraic entropy with Lehmer Problem. \begin{Theorem}[Algebraic Yuzvinski Formula] \label{AYF}\emph{\cite{GV}} Let $n\in\mathbb N_+$ and $\phi:\mathbb Q^n\to\mathbb Q^n$ an endomorphism. Then \begin{equation}\label{yuzeq} h_{alg}(\phi)=m(\phi). \end{equation} \end{Theorem} The next property of additivity of the algebraic entropy was first proved for torsion abelian groups in \cite{DGSZ}, while the proof of the general case was given in \cite{DG} applying the Algebraic Yuzvinski Formula. \begin{Theorem}[Addition Theorem]\emph{\cite{DG}}\label{AT} Let $G$ be an abelian group, $\phi:G\to G$ an endomorphism, $H$ a $\phi$-invariant subgroup of $G$ and $\overline\phi:G/H\to G/H$ the endomorphism induced by $\phi$. Then $$h_{alg}(\phi)=h_{alg}(\phi\restriction_H)+ h_{alg}(\overline\phi).$$ \end{Theorem} Moreover, uniqueness is available for the algebraic entropy in the category of all abelian groups. As in the case of the Addition Theorem, also the Uniqueness Theorem was proved in general in \cite{DG}, while it was previously proved in \cite{DGSZ} for torsion abelian groups. \begin{Theorem}[Uniqueness Theorem]\label{UT}\emph{\cite{DG}} The algebraic entropy $$h_{alg}:\mathrm{Flow}_\mathbf{AbGrp}\to\mathbb R_+$$ is the unique function such that: \begin{itemize} \item[(a)] $h_{alg}$ is invariant under conjugation; \item[(b)] $h_{alg}$ is continuous on direct limits; \item[(c)] $h_{alg}$ satisfies the Addition Theorem; \item[(d)] for $K$ a finite abelian group, $h_{alg}(\beta_K)=\log|K|$; \item[(e)] $h_{alg}$ satisfies the Algebraic Yuzvinski Formula. \end{itemize} \end{Theorem} \subsection{The growth of a finitely generated flow in $\mathbf{Grp}$}\label{Growth-sec} In order to measure and classify the growth rate of maps $\mathbb N \to \mathbb N$, one need the relation $\preceq$ defined as follows. For $\gamma, \gamma': \mathbb N \to \mathbb N$ let $\gamma \preceq \gamma'$ if there exist $n_0,C\in\mathbb N_+$ such that $\gamma(n) \leq \gamma'(Cn)$ for every $n\geq n_0$. Moreover $\gamma\sim\gamma$ if $\gamma\preceq\gamma'$ and $\gamma'\preceq\gamma$ (then $\sim$ is an equivalence relation), and $\gamma\prec\gamma'$ if $\gamma\preceq\gamma'$ but $\gamma\not\sim\gamma'$. For example, for every $arithmetic{}lpha, \beta\in\mathbb R_{\geq0}$, $n^arithmetic{}lpha\sim n^\beta$ if and only if $arithmetic{}lpha=\beta$; if $p(t)\in\mathbb Z[t]$ and $p(t)$ has degree $d\in\mathbb N$, then $p(n)\sim n^d$. On the other hand, $a^n\sim b^n$ for every $a,b\in\mathbb R$ with $a,b>1$, so in particular all exponentials are equivalent with respect to $\sim$. So a map $\gamma: \mathbb N \to \mathbb N$ is called: \begin{itemize} \item[(a)] \emph{polynomial} if $\gamma(n) \preceq n^d$ for some $d\in\mathbb N_+$; \item[(b)] \emph{exponential} if $\gamma(n) \sim 2^n$; \item[(c)] \emph{intermediate} if $\gamma(n)\succ n^d$ for every $d\in\mathbb N_+$ and $\gamma(n)\prec 2^n$. \end{itemize} Let $G$ be a group, $\phi:G\to G$ an endomorphism and $F$ a non-empty finite subset of $G$. Consider the function, already mentioned in \eqref{gamma}, $$ \gamma_{\phi,F}:\mathbb N_+\to\mathbb N_+\ \text{defined by}\ \gamma_{\phi,F}(n)=|T_n(\phi,F)|\ \text{for every}\ n\in\mathbb N_+. $$ Since $$ |F|\leq\gamma_{\phi,F}(n)\leq|F|^n\mbox{ for every }n\in\mathbb N_+, $$ the growth of $\gamma_{\phi,F}$ is always at most exponential; moreover, $H_{alg}(\phi,F)\leq \log |F|$. So, following \cite{DG0} and \cite{DG-islam}, we say that $\phi$ has \emph{polynomial} (respectively, \emph{exponential}, \emph{intermediate}) \emph{growth at $F$} if $\gamma_{\phi,F}$ is polynomial (respectively, exponential, intermediate). Before proceeding further, let us make an important point here. All properties considered above concern practically the $\phi$-invariant subgroup $G_{\phi,F}$ of $G$ generated by the trajectory $T(\phi, F) = \bigcup_{n\in\mathbb N_+} T_n(\phi, F)$ and the restriction $\phi\restriction_{G_{\phi,F}}$. \begin{Definition} We say that the flow $(G, \phi)$ in $\mathbf{Grp}$ is \emph{finitely generated} if $G = G_{\phi,F}$ for some finite subset $F$ of $G$. \end{Definition} Hence, all properties listed above concern finitely generated flows in $\mathbf{Grp}$. We conjecture the following, knowing that it holds true when $G$ is abelian or when $\phi=\mathrm{id}_G$: if the flow $(G,\phi)$ is finitely generated, and if $G = G_{\phi,F}$ and $G = G_{\phi,F'}$ for some finite subsets $F$ and $F'$ of $G$, then $\gamma_{\phi,F}$ and $\gamma_{\phi,F'}$ have the same type of growth. In this case the growth of a finitely generated flow $G_{\phi,F}$ would not depend on the specific finite set of generators $F$ (so $F$ can always be taken symmetric). In particular, one could speak of growth of a finitely generated flow without any reference to a specific finite set of generators. Nevertheless, one can give in general the following \begin{Definition} Let $(G,\phi)$ be a finitely generated flow in $\mathbf{Grp}$. We say that $(G,\phi)$ has \begin{itemize} \item[(a)] \emph{polynomial growth} if $\gamma_{\phi,F}$ is polynomial for every finite subset $F$ of $G$; \item[(b)] \emph{exponential growth} if there exists a finite subset $F$ of $G$ such that $\gamma_{\phi,F}$ is exponential; \item[(c)] \emph{intermediate growth} otherwise. \end{itemize} We denote by $\mathrm{Pol}$ and $\mathrm{Exp}$ the classes of finitely generated flows in $\mathbf{Grp}$ of polynomial and exponential growth respectively. Moreover, $\mathcal M=\mathrm{Pol}\cup\mathrm{Exp}$ is the class of finitely generated flows of non-intermediate growth. \end{Definition} This notion of growth generalizes the classical one of growth of a finitely generated group given independently by Schwarzc \cite{Sch} and Milnor \cite{M1}. Indeed, if $G$ is a finitely generated group and $X$ is a finite symmetric set of generators of $G$, then $\gamma_X=\gamma_{\mathrm{id}_G,X}$ is the classical \emph{growth function} of $G$ with respect to $X$. For a connection of the terminology coming from the theory of algebraic entropy and the classical one, note that for $n\in\mathbb N_+$ we have $T_n(\mathrm{id}_G,X)=\{g\in G:\ell_X(g)\leq n\}$, where $\ell_X(g)$ is the length of the shortest word $w$ in the alphabet $X$ such that $w=g$ (see \S \ref{NewSec1} (c)). Since $\ell_X$ is a norm on $G$, $T_n(\mathrm{id}_G,X)$ is the ball of radius $n$ centered at $1$ and $\gamma_X(n)$ is the cardinality of this ball. Milnor \cite{M3} proposed the following problem on the growth of finitely generated groups. \begin{problem}[Milnor Problem]\label{Milnor-pb}{\cite{M3}} Let $G$ be a finitely generated group and $X$ a finite set of generators of $G$. \begin{itemize} \item[(i)] Is the growth function $\gamma_X$ necessarily equivalent either to a power of $n$ or to the exponential function $2^n$? \item[(ii)] In particular, is the {growth exponent} $\delta_G=\limsup_{n\to \infty}\phirac{\log\gamma_X(n)}{\log n}$ either a well defined integer or infinity? For which groups is $\delta_G$ finite? \end{itemize} \end{problem} Part (i) of Problem \ref{Milnor-pb} was solved negatively by Grigorchuk in \cite{Gri1,Gri2,Gri3,Gri4}, where he constructed his famous examples of finitely generated groups $\mathbb G$ with intermediate growth. For part (ii) Milnor conjectured that $\delta_G$ is finite if and only if $G$ is virtually nilpotent (i.e., $G$ contains a nilpotent finite-index subgroup). The same conjecture was formulated by Wolf \cite{Wolf} (who proved that a nilpotent finitely generated group has polynomial growth) and Bass \cite{Bass}. Gromov \cite{Gro} confirmed Milnor's conjecture: \begin{Theorem}[Gromov Theorem]\label{GT}\emph{\cite{Gro}} A finitely generated group $G$ has polynomial growth if and only if $G$ is virtually nilpotent. \end{Theorem} The following two problems on the growth of finitely generated flows of groups are inspired by Milnor Problem. \begin{problem} Describe the permanence properties of the class $\mathcal M$. \end{problem} Some stability properties of the class $\mathcal M$ are easy to check. For example, stability under taking finite direct products is obviously available, while stability under taking subflows (i.e., invariant subgroups) and factors fails even in the classical case of identical flows. Indeed, Grigorchuk's group $\mathbb G$ is a quotient of a finitely generated free group $F$, that has exponential growth; so $(F,\mathrm{id}_F) \in \mathcal M$, while $(\mathbb G, \mathrm{id}_{\mathbb G})\not \in\mathcal M$. Furthermore, letting $G = \mathbb G \times F$, one has $(G,\mathrm{id}_G) \in \mathcal M$, while $(\mathbb G, \mathrm{id}_{\mathbb G})\not \in\mathcal M$, so $\mathcal M$ is not stable even under taking direct summands. On the other hand, stability under taking powers is available since $(G,\phi) \in \mathcal M$ if and only if $(G,\phi^n) \in \mathcal M$ for $n\in\mathbb N_+$. \begin{problem}\label{Ques4} \begin{itemize} \item[(i)] Describe the finitely generated groups $G$ such that $(G,\phi)\in\mathcal M$ for every endomorphism $\phi:G\to G$. \item[(ii)] Does there exist a finitely generated group $G$ such that $(G,\mathrm{id}_G)\in\mathcal M$ but $(G,\phi)\not\in\mathcal M$ for some endomorphism $\phi:G\to G$? \end{itemize} \end{problem} In item (i) of the above problem we are asking to describe all finitely generated groups $G$ of non-intermediate growth such that $(G,\phi)$ has still non-intermediate growth for every endomorphism $\phi:G\to G$. On the other hand, in item (ii) we ask to find a finitely generated group $G$ of non-intermediate growth that admits an endomorphism $\phi:G\to G$ of intermediate growth. The basic relation between the growth and the algebraic entropy is given by Proposition \ref{exp} below. For a finitely generated group $G$, an endomorphism $\phi$ of $G$ and a pair $X$ and $X'$ of finite generators of $G$, one has $\gamma_{\phi,X}\sim\gamma_{\phi,X'}$. Nevertheless, $H_{alg}(\phi,X)\neq H_{alg}(\phi,X')$ may occur; in this case $(G,\phi)$ has necessarily exponential growth. We give two examples to this effect: \begin{Example}\label{exaAugust} \begin{itemize} \item[(a)] {\cite{DG-islam}} Let $G$ be the free group with two generators $a$ and $b$; then $X=\{a^{\pm 1},b^{\pm 1}\}$ gives $H_{alg}(\mathrm{id}_G,X)=\log 3$ while for $X'=\{a^{\pm 1},b^{\pm 1},(ab)^{\pm 1}\}$ we have $H_{alg}(\mathrm{id}_G,X')=\log 4$. \item[(b)] Let $G = \mathbb Z$ and $\phi: \mathbb Z \to \mathbb Z$ defined by $\phi(x) = mx$ for every $x\in \mathbb Z$ and with $m>3$. Let also $X= \{0,\pm 1\}$ and $X'= \{0,\pm 1, \ldots \pm m\}$. Then $H_{alg}(\phi,X) \leq \log |X| =\log 3$, while $H_{alg}(\phi,X')= h_{alg}(\phi)= \log m$. \end{itemize} \end{Example} \begin{Proposition}\label{exp}\emph{\cite{DG-islam}} Let $(G,\phi)$ be a finitely generated flow in {\bf Grp}. \begin{itemize} \item[(a)]Then $h_{alg}(\phi)>0$ if and only if $(G,\phi)$ has exponential growth. \item[(b)]If $(G,\phi)$ has polynomial growth, then $h_{alg}(\phi)=0$. \end{itemize} \end{Proposition} In general the converse implication in item (b) is not true even for the identity. Indeed, if $(G,\phi)$ has intermediate growth, then $h_{alg}(\phi)=0$ by item (a). So for Grigorchuk's group $\mathbb G$, the flow $(\mathbb G,\mathrm{id}_\mathbb G)$ has intermediate growth yet $h_{alg}(\mathrm{id}_\mathbb G)=0$. This motivates the following \begin{Definition}\label{MPara} Let $\mathcal G$ be a class of groups and $\Phi$ be a class of morphisms. We say that the pair $(\mathcal G, \Phi)$ satisfies Milnor Paradigm (briefly, MP) if no finitely generated flow $(G,\phi)$ with $G\in\mathcal G$ and $\phi\in\Phi$ can have intermediate growth. \end{Definition} In terms of the class $\mathcal M$, $$(\mathcal G, \Phi)\ \text{satisfies MP if and only if }\ (\mathcal G, \Phi)\in \mathcal M\ (\phiorall G\in\mathcal G)(\phiorall \phi\in\Phi) .$$ Equivalently, $(\mathcal G, \Phi)$ satisfies MP when $h_{alg}(\phi)=0$ always implies that $(G,\phi)$ has polynomial growth for finitely generated flows $(G,\phi)$ with $G\in\mathcal G$ and $\phi\in\Phi$. In these terms Milnor Problem \ref{Milnor-pb} (i) is asking whether the pair $(\mathbf{Grp},\mathcal{I}d)$ satisfies MP, where $\mathcal I d$ is the class of all identical endomorphisms. So we give the following general open problem. \begin{problem}\label{PB0} \begin{itemize} \item[(i)] Find pairs $(\mathcal G,\Phi)$ satisfying MP. \item[(ii)] For a given $\Phi$ determine the properties of the largest class $\mathcal G_\Phi$ such that $(\mathcal G_\Phi, \Phi)$ satisfies MP. \item[(iii)] For a given $\mathcal G$ determine the properties of the largest class $\Phi_{\mathcal G}$ such that $(\mathcal G, \Phi_{\mathcal G})$ satisfies MP. \item[(iv)] Study the Galois correspondence between classes of groups $\mathcal G$ and classes of endomorphisms $\Phi$ determined by MP. \end{itemize} \end{problem} According to the definitions, the class $\mathcal G_{\mathcal{I} d}$ coincides with the class of finitely generated groups of non-intermediate growth. The following result solves Problem \ref{PB0} (iii) for $\mathcal G=\mathbf{AbGrp}$, showing that $\Phi_\mathbf{AbGrp}$ coincides with the class $\mathcal E$ of all endomorphisms. \begin{Theorem}[Dichotomy Theorem]\emph{\cite{DG0}}\label{DT} There exist no finitely generated flows of intermediate growth in $\mathbf{AbGrp}$. \end{Theorem} Actually, one can extend the validity of this theorem to nilpotent groups. This leaves open the following particular case of Problem \ref{PB0}. We shall see in Theorem \ref{osin} that the answer to (i) is positive when $\phi=\mathrm{id}_G$. \begin{question}\label{Ques1} Let $(G,\phi)$ be a finitely generated flow in $\mathbf{Grp}$. \begin{itemize} \item[(i)] If $G$ is solvable, does $(G,\phi)\in\mathcal M$? \item[(ii)] If $G$ is a free group, does $(G,\phi)\in\mathcal M$? \end{itemize} \end{question} We state now explicitly a particular case of Problem \ref{PB0}, inspired by the fact that the right Bernoulli shifts have no non-trivial quasi-periodic points and they have uniform exponential growth (see Example \ref{bern}). In \cite{DG0} group endomorphisms $\phi:G\to G$ without non-trivial quasi-periodic points are called algebraically ergodic for their connection (in the abelian case and through Pontryagin duality) with ergodic transformations of compact groups. \begin{question}\label{Ques2} Let $\Phi_0$ be the class of endomorphisms without non-trivial quasi-periodic points. Is it true that the pair $(\mathbf{Grp},\Phi_0)$ satisfies MP? \end{question} For a finitely generated group $G$, the \emph{uniform exponential growth rate} of $G$ is defined as $$ \lambda(G)=\inf\{H_{alg}(\mathrm{id}_G,X):X\ \text{finite set of generators of}\ G\} $$ (see for instance \cite{dlH-ue}). Moreover, $G$ has \emph{uniform exponential growth} if $\lambda(G)>0$. Gromov \cite{GroLP} asked whether every finitely generated group of exponential growth is also of uniform exponential growth. This problem was recently solved by Wilson \cite{Wilson} in the negative. Since the algebraic entropy of a finitely generated flow $(G,\phi)$ in $\mathbf{Grp}$ can be computed as $$ h_{alg}(\phi)=\sup\{H_{alg}(\phi,F): F\ \text{finite subset of $G$ such that $G=G_{\phi,F}$}\}, $$ one can give the following counterpart of the uniform exponential growth rate for flows: \begin{Definition} For $(G,\phi)$ be a finitely generated flow in $\mathbf{Grp}$ let $$ \lambda(G,\phi)=\inf\{H_{alg}(\phi,F): F\ \text{finite subset of $G$ such that $G=G_{\phi,F}$} \}. $$ The flow $(G,\phi)$ is said to have \emph{uniformly exponential growth} if $\lambda(G,\phi)>0$. Let $\mathrm{Exp}_\mathrm u$ be the subclass of $\mathrm{Exp}$ of all finitely generated flows in $\mathbf{Grp}$ of uniform exponential growth. \end{Definition} Clearly $\lambda(G,\phi)\leq h_{alg}(\phi)$, so one has the obvious implication \begin{equation}\label{GP} h_{alg}(\phi)=0\ \mathbb Rightarrow\ \lambda(G,\phi)=0. \end{equation} To formulate the counterpart of Gromov's problem on uniformly exponential growth it is worth to isolate also the class $\mathcal W$ of the finitely generated flows in $\mathbf{Grp}$ of exponential but not uniformly exponential growth (i.e., $\mathcal W=\mathrm{Exp}\setminus \mathrm{Exp}_\mathrm u$). Then $\mathcal W$ is the class of finitely generated flows $(G,\phi)$ in $\mathbf{Grp}$ for which \eqref{GP} cannot be inverted, namely $h_{alg}(\phi)> 0=\lambda(G,\phi)$. We start stating the following problem. \begin{problem} Describe the permanence properties of the classes $\mathrm{Exp}_\mathrm u$ and $\mathcal W$. \end{problem} It is easy to check that $\mathrm{Exp}_\mathrm u$ and $\mathcal W$ are stable under taking direct products. On the other hand, stability of $\mathrm{Exp}_\mathrm u$ under taking subflows (i.e., invariant subgroups) and factors fails even in the classical case of identical flows. Indeed, Wilson's group $\mathbb W$ is a quotient of a finitely generated free group $F$, that has uniform exponential growth (see \cite{dlH-ue}); so $(F,\mathrm{id}_F)\in \mathrm{Exp}_\mathrm u$, while $(\mathbb W, \mathrm{id}_{\mathbb W})\in\mathcal W$. Furthermore, letting $G = \mathbb W \times F$, one has $(G,\mathrm{id}_G)\in \mathrm{Exp}_\mathrm u$, while $(\mathbb W, \mathrm{id}_{\mathbb W})\in\mathcal W$, so $\mathrm{Exp}_\mathrm u$ is not stable even under taking direct summands. In the line of MP, introduced in Definition \ref{MPara}, we can formulate also the following \begin{Definition}\label{GPara} Let $\mathcal G$ be a class of groups and $\Phi$ be a class of morphisms. We say that the pair $(\mathcal G, \Phi)$ satisfies Gromovr Paradigm (briefly, MP), if every finitely generated flow $(G,\phi)$ with $G\in\mathcal G$ and $\phi\in\Phi$ of exponential growth has has uniform exponential growth. \end{Definition} In terms of the class $\mathcal W$, $$ (\mathcal G, \Phi)\ \text{satisfies GP if and only if }\ (\mathcal G, \Phi)\not\in \mathcal M\ (\phiorall G\in\mathcal G)(\phiorall \phi\in\Phi) . $$ In these terms, Gromov's problem on uniformly exponential growth asks whether the pair $(\mathbf{Grp}, \mathcal I d)$ satisfies GP. In analogy to the general Problem \ref{PB0}, one can consider the following obvious counterpart for GP: \begin{problem}\label{PB1} \begin{itemize} \item[(i)] Find pairs $(\mathcal G, \Phi)$ satisfying GP. \item[(ii)] For a given $\Phi$ determine the properties of the largest class $\mathcal G_\Phi$ such that $(\mathcal G_\Phi,\Phi)$ satisfies GP. \item[(iii)] For a given $\mathcal G$ determine the properties of the largest class $\Phi_{\mathcal G}$ such that $(\mathcal G,\Phi_\mathcal G)$ satisfies GP. \item[(iv)] Study the Galois correspondence between classes of groups $\mathcal G$ and classes of endomorphisms $\Phi$ determined by GP. \end{itemize} \end{problem} We see now in item (a) of the next example a particular class of finitely generated flows for which $\lambda$ coincides with $h_{alg}$ and they are both positive, so in particular these flows are all in $\mathrm{Exp}_\mathrm u$. In item (b) we leave an open question related to Question \ref{Ques2}. \begin{Example}\label{bern} \begin{itemize} \item[(a)] For a finite group $K$, consider the flow $(\bigoplus_\mathbb N K,\beta_K)$. We have seen in Example \ref{shift} that $h_{alg}(\beta_K)=\log|K|$. In this case we have $\lambda(\bigoplus_\mathbb N K,\beta_K)=\log|K|$, since a subset $F$ of $\bigoplus_\mathbb N K$ generating the flow $(\bigoplus_\mathbb N K,\beta_K)$ must contain the first copy $K_0$ of $K$ in $\bigoplus_\mathbb N K$, and $H_{alg}(\beta_K,K_0)=\log|K|$. \item[(b)] Is it true that $\lambda(G,\phi) = h_{alg}(\phi) > 0$ for every finitely generated flow $(G,\phi)$ in $\mathbf{Grp}$ such that $\phi \in \Phi_0$? In other terms, we are asking whether all finitely generated flows $(G,\phi)$ in $\mathbf{Grp}$ with $\phi\in\Phi_0$ have uniform exponential growth (i.e., are contained in $\mathrm{Exp}_\mathrm u$). \end{itemize} \end{Example} One can also consider the pairs $(\mathcal G, \Phi)$ satisfying the conjunction MP \& GP. For any finitely generated flow $(G,\phi)$ in $\mathbf{Grp}$ one has \begin{equation}\label{osin-eq} (G,\phi)\ \text{has polynomial growth}\ \ \buildrel{(1)}\over\Longrightarrow\ h_{alg}(\phi)=0\ \ \buildrel{(2)}\over\Longrightarrow\ \lambda(G,\phi)=0. \end{equation} The converse implication of (1) (respectively, (2)) holds for all $(G,\phi)$ with $G\in\mathcal G$ and $\phi\in\Phi$ precisely when the pair $(\mathcal G, \Phi)$ satisfies MP (respectively, GP). Therefore, the pair $(\mathcal G, \Phi)$ satisfies the conjunction MP \& GP precisely when the three conditions in \eqref{osin-eq} are all equivalent (i.e., $\lambda(G,\phi)=0 \mathbb Rightarrow (G,\phi)\in \mathrm{Pol}$) for all finitely generated flows $(G,\phi)$ with $G\in\mathcal G$ and $\phi\in\Phi$. A large class of groups $\mathcal G$ such that $(\mathcal G, \mathcal I d)$ satisfies MP \& GP was found by Osin \cite{O} who proved that a finitely generated solvable group $G$ of zero uniform exponential growth is virtually nilpotent, and recently this result was generalized in \cite{O1} to elementary amenable groups. Together with Gromov Theorem and Proposition \ref{exp}, this gives immediately the following \begin{Theorem}\label{osin} Let $G$ be a finitely generated elementary amenable group. The following conditions are equivalent: \begin{itemize} \item[(a)] $h_{alg}(\mathrm{id}_G)=0$; \item[(b)] $\lambda(G)=0$; \item[(c)] $G$ is virtually nilpotent; \item[(d)] $G$ has polynomial growth. \end{itemize} \end{Theorem} This theorem shows that the pair $\mathcal G=\{\mbox{elementary amenable groups}\}$ and $\Phi =\mathcal{I} d$ satisfies simultaneously MP and GP. In other words it proves that the three conditions in \eqref{osin-eq} are all equivalent when $G$ is an elementary amenable finitely generated group and $\phi=\mathrm{id}_G$. \subsection{Adjoint algebraic entropy}\label{aent-sec} We recall here the definition of the adjoint algebraic entropy $\mathrm{ent}^\star$ and we state some of its specific features not deducible from the general scheme, so beyond the ``package" of general properties coming from the equality $\mathrm{ent}^\star=h_{\mathfrak{sub}^\star}$ such as Invariance under conjugation and inversion, Logarithmic Law, Monotonicity for factors (these properties were proved in \cite{DG-islam} in the general case and previously in \cite{DGS} in the abelian case applying the definition). In analogy to the algebraic entropy $\mathrm{ent}$, in \cite{DGS} the adjoint algebraic entropy of endomorphisms of abelian groups $G$ was introduced ``replacing" the family $\mathcal F(G)$ of all finite subgroups of $G$ with the family $\mathcal C(G)$ of all finite-index subgroups of $G$. The same definition was extended in \cite{DG-islam} to the more general setting of endomorphisms of arbitrary groups as follows. Let $G$ be a group and $N\in \mathcal C(G)$. For an endomorphism $\phi:G\to G$ and $n\in\mathbb N_+$, the \emph{$n$-th $\phi$-cotrajectory of $N$} is $$C_n(\phi,N)=N\cap\phi^{-1}(N)\cap\ldots\cap\phi^{-n+1}(N).$$ The \emph{adjoint algebraic entropy of $\phi$ with respect to $N$} is $$ H^\star(\phi,N)={\lim_{n\to \infty}\phirac{\log[G:C_n(\phi,N)]}{n}}. $$ This limit exists as $H^\star(\phi,N)=h_{\mathfrak{S}}(\mathcal C(\phi),N)$ and so Theorem \ref{limit} applies. The \emph{adjoint algebraic entropy of $\phi$} is $$\mathrm{ent}^\star(\phi)=\sup\{H^\star(\phi,N):N\in\mathcal C(G)\}.$$ The values of the adjoint algebraic entropy of the Bernoulli shifts were calculated in \cite[Proposition 6.1]{DGS} applying \cite[Corollary 6.5]{G0} and the Pontryagin duality; a direct computation can be found in \cite{G}. So, in contrast with what occurs for the algebraic entropy, we have: \begin{Example}[Bernoulli shifts]\label{beta*} For $K$ a non-trivial group, $$\mathrm{ent}^\star(\beta_K)=\mathrm{ent}^\star({}_K\beta)=\infty.$$ \end{Example} As proved in \cite{DGS}, the adjoint algebraic entropy satisfies the Weak Addition Theorem, while the Monotonicity for invariant subgroups fails even for torsion abelian groups; in particular, the Addition Theorem fails in general. On the other hand, the Addition Theorem holds for bounded abelian groups: \begin{Theorem}[Addition Theorem]\label{AT*} Let $G$ be a bounded abelian group, $\phi:G\to G$ an endomorphism, $H$ a $\phi$-invariant subgroup of $G$ and $\overline\phi:G/H\to G/H$ the endomorphism induced by $\phi$. Then $$\mathrm{ent}^\star(\phi)=\mathrm{ent}^\star(\phi\restriction_H)+\mathrm{ent}^\star(\overline\phi).$$ \end{Theorem} The following is one of the main results on the adjoint algebraic entropy proved in \cite{DGS}. It shows that the adjoint algebraic entropy takes values only in $\{0,\infty\}$, while clearly the algebraic entropy may take also finite positive values. \begin{Theorem}[Dichotomy Theorem]\label{dichotomy}\emph{\cite{DGS}} Let $G$ be an abelian group and $\phi:G\to G$ an endomorphism. Then \begin{center} either $\mathrm{ent}^\star(\phi)=0$ or $\mathrm{ent}^\star(\phi)=\infty$. \end{center} \end{Theorem} Applying the Dichotomy Theorem and the Bridge Theorem (stated in the previous section) to the compact dual group $K$ of $G$ one gets that for a continuous endomorphism $\psi$ of a compact abelian group $K$ either $\mathrm{ent} (\psi)=0$ or $\mathrm{ent}(\psi)=\infty$. In other words: \begin{Corollary} If $K$ is a compact abelian group, then every endomorphism $\psi:K\to K$ with $0 < \mathrm{ent} (\psi) < \infty$ is discontinuous. \end{Corollary} \end{document}

\begin{document} \title[Limits and Transitivity in Free Groups]{Limit Sets and Internal Transitivity in Free Group Actions} \author[K. Binder]{Kyle Binder} \address[K. Binder]{Baylor University, Waco TX, 76798} \email[K. Binder]{Kyle\_Binder@baylor.edu} \author[J. Meddaugh]{Jonathan Meddaugh} \address[J. Meddaugh]{Baylor University, Waco TX, 76798} \email[J. Meddaugh]{Jonathan\_Meddaugh@baylor.edu} \subjclass[2010]{37B50, 37B10, 37B20, 54H20} \keywords{shadowing, pseudo-orbit tracing property, topological dynamics} \begin{abstract} It has been recently shown that, under appropriate hypotheses, the $\omega$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the action of a finitely generated free group or monoid. We give general definitions for several types of limit sets and analogous notions of internal transitivity. We then demonstrate that these limit sets are completely characterized by internal transitivity properties in shifts of finite type and general dynamical systems exhibiting a form of the shadowing property. \end{abstract} \maketitle \section{Introduction} The study of group and semi-group actions on compact metric spaces is a natural generalization of the study of traditional discrete dynamical systems. Unsurprisingly, as the groups of interest increase in complexity, basic results from the standard theory of topological dynamical systems (i.e $\mathbb Z$ and $\mathbb N$ actions) become difficult to generalize or fail entirely. What is particularly surprising, however, is just how quickly things begin to fall apart. Even in the case of the action of the group $\mathbb Z^d$ for $d\geq 2$, simple questions regarding entropy are difficult to answer even in the simplified context of shift spaces \cite{hochman1, hochman2}. However, there are aspects of these actions that have been studied with some success. Of particular interest to this paper are generalizations of the known relationship between $\omega$-limit sets and internal chain transitivity, especially under the additional hypothesis of shadowing. In particular, it has long been known that $\omega$-limit sets are internally chain transitive \cite{Hirsch}, and that in certain categories of dynamical systems, the converse is also true including shifts of finite type, topologically hyperbolic maps, certain Julia sets of quadratic polynomials and Axiom A diffeomorphisms \cite{BGKR-omega, BGOR-DCDS, BR-ToAppear, BMR-ToAppear, Bowen}. More recently, Meddaugh and Raines demonstrated that under the assumption that the system has the \emph{shadowing property}, the collection of internally chain transitive sets is precisely the closure of the collection of $\omega$-limit sets in the Hausdorff topology \cite{MR}. Thus, in situations in which it is known, a priori, that the collection of $\omega$-limit sets is closed in this topology (interval maps are an important example of such a situation \cite{Blokh}), the shadowing property is sufficient to establish that internal chain transitivity coincides precisely with the property of being an $\omega$-limit set. This connection is further explored in \cite{GoodMeddaugh-ICT}, in which variations of the shadowing property are explored to more precisely characterize the necessary conditions under which this equivalence occurs. There have been a number of attempts to extend these sorts of results to the broader context of group actions. In particular, several authors have considered the structure of limit sets for $\mathbb Z^d$ actions, specifically for subshifts of $\Sigma^{\mathbb Z^d}$ where $\Sigma$ is a finite alphabet. Among the first generalizations along theses lines is the work of Oprocha \cite{oprochalimitsets, oprochashadowingmulti} in which notions of shadowing and limit sets for these sorts of actions are explored. The connection between limit sets and internal transitivity was explored by Meddaugh and Raines \cite{MR-Zd}. An important aspect of this body of work is that for multi-dimensional shift spaces, there is no unique natural analog for forward iteration, so many types of limit sets arise, and correspondingly, many notions of internal transitivity. One other common aspect of these results is that the problem of \emph{completability} arises quite frequently. This is not surprising, as it is related to tiling problems, in which this issue is well-known problem \cite{Berger}. It is at this point that we propose that $\mathbb Z^d$ actions are not necessarily the most natural successors to $\mathbb Z$ actions in terms of complexity. In this paper, we examine the action of finitely generated free groups and monoids on compact metric spaces. We examine notions of limit sets and establish that there are internal transitivity properties that characterize them in the context of shifts of finite type. We then introduce a generalization of the notion of shadowing to these actions and establish that a subshift has shadowing precisely when it is a shift of finite type. This parallels the known result for $\mathbb Z$ actions as demonstrated by Walters \cite{WaltersFiniteType}. We then go on to demonstrate that under the assumption of shadowing, we witness analogs to the results of Meddaugh and Raines \cite{MR}, and, under an appropriate analog of \emph{asymptotic shadowing}, we recover generalizations of the characterization of limit sets in terms of internal transitivity as developed by Good and Meddaugh \cite{GoodMeddaugh-ICT}. \section{Preliminaries} Let $G$ be a group or monoid and $X$ a compact metric space. A continuous left $G$-action on $X$ is a function $f:G\times X\to X$ that satisfies the following conditions: for each $g\in G$, the function $f_g$ defined by $f_g(x)=f(g,x)$ is continuous, for the identity $e\in G$, $f_e$ is the identity on $X$, and for $g,h\in G$, $f_{gh}=f_h\circ f_g$. For $n\in\mathbb{N} $, let $ F $ denote the free group on the $n$ generators $\{s_0,\ldots s_{n-1}\}$ and $ S=\{s_0,\ldots s_{n-1},s_0^{-1},\ldots s_{n-1}^{-1}\} $ be the set of all generators and their inverses. The set of reduced words of $F$ is the set $W=\{e\}\cup\{w_0w_1\cdots w_k\in S^\omega: w_i\neq w_{i+1}^{-1} \textrm{ for } i<k\}$ with $e$ denoting the empty word. Each element of $F$ has a unique representative in $W$ and the group operation of $F$ is realized in $W$ by concatenation followed by cancellation. We define the \emph{length} of an element $ u \in F $ to be the number of letters in its reduced representation. We denote this by $ \left| u \right| $. The identity of $ F $ is associated with the empty word $ e $ and has length $ 0 $. Finally, we will say for two elements of $ F $ with reduced representations $ u = u_{0} \dots u_{n} $ and $ v = v_{0} \dots v_{n+k} $ , that $ u $ is a \emph{prefix} of $ v $ if $ u_{i} = v_{i} $ for $ 0 \leq i \leq n $. Additionally, we will take the identity $ e $ to be a prefix of every element of $ F $. It will also be necessary to consider the infinite words of $F$. In particular, we define the set $W_\infty=\{\seq{w_i}_{i\in\omega}: w_i\neq w_{i+1}^{-1} \textrm{ for } i\in\omega\}$. Let $ u, w $ be two words (either finite or infinite) with length at least $ n $. We say $ u|_{n} = w|_{n} $ if $ u_{i} = w_{i} $ for $ 0 \leq i < n $. For $ n \in \mathbb{N} $, let $ H $ denote the free monoid on the $ n $ generators $ P = \{s_0,\ldots s_{n-1}\}$. In this case, $H$ coincides with the set of words of $\{e\}\cup\{w_0w_1\cdots w_k \in P^{\omega} \}$ with $ e $ denoting the empty word as every element of $ H $ has a unique representation, and the binary operation is simply concatenation. The collection of infinite words of $ H $ is the set $ H_{\infty} = \{\seq{w_i}_{i\in\omega}: w_{i} \in P\} $. The length of elements, prefixes, and restrictions $ w_{|n} $ are defined the same as in the free group case. For the sake of generality, we take $ G $ be either a free group or monoid with $ W $ the set of words, $ W_{\infty} $ the set of infinite words, and $ S $ the set of generators (with inverses in the group case). For a finite alphabet $ \mathcal{A} $, \emph{the full shift of $ \mathcal{A} $ over $ G $} (denoted $ \mathcal{A}^{G} $) is the set of all functions $ x: G \rightarrow \mathcal{A} $. There is a natural $G$-action $\sigma$ on $\mathcal A^{G}$ defined as follows. For every $ s \in S $ there is an associated \emph{shift map} $ \sigma_{s}: \mathcal{A}^{G} \rightarrow \mathcal{A}^{G} $ defined by $ \sigma_{s}(x)(u) = x(su) $. For $ v = v_{0} \dots v_{n} \in G $, we define $ \sigma_{v}(x) = \sigma_{v_{n}} \circ \dots \circ \sigma_{v_{0}}(x) $. Thus $ \sigma_{v}(x)(u) = x(vu) $. For fixed $ G $, define $ \Sigma^{n} = \left\{u \in G : \left|u \right| < n \right\} $. We can place a metric on $ \mathcal{A}^{G} $ defined by $ d(x,y) = \inf\left( \left\{ 2^{-n} : x|_{\Sigma^{n}} = y|_{\Sigma^{n}} \right\}\cup\{1\}\right) $. It is easy to see that under the topology induced by this metric, $\mathcal A^{G}$ is compact and the action $\sigma$ is continuous. In particular, if $d(x,y) = 2^{-n} $ then $ d(\sigma_{i}(x), \sigma_{i}(y)) \leq 2^{-n + 1} $. Furthermore, for $ u \in G $, if $ d(x,y) = 2^{-n} $ then $ d(\sigma_{u}(x), \sigma_{u}(y)) \leq \min \left\{ 2^{-m + \left| u \right|}, 1 \right\} $. An \emph{n-block} is a function $ B_{n}: \Sigma^{n} \rightarrow \mathcal{A} $. An element $ x \in \mathcal{A}^{G} $ is said to contain $ B_{n} $ if there exists a $ u \in G $ such that $ \sigma_{u}(x)|_{\Sigma^{n}} = B_{n} $. Let $ \mathcal{F} $ be a collection of m-blocks where m is allowed to range over the integers. We can create a subspace of $ \mathcal{A}^{G} $ defined as $ X_{\mathcal{F}} = \left\{x \in \mathcal{A}^{G} : x \; \textrm{does not contain any} \; B \in \mathcal{F} \right\} $. $ X_{\mathcal{F}} $ is compact and invariant under the shift maps. In this case, $ \mathcal{F} $ is called a \emph{set of forbidden blocks}. Any invariant and compact subspace of $ \mathcal{A}^{G} $ can be expressed as $ X_{\mathcal{F}} $ for some set $ \mathcal{F} $. This is called a \emph{shift space}. In the case that $ \mathcal{F} $ is finite, the shift space is a \emph{shift of finite type} (SFT). If $ M $ is the maximal integer such that some $ B \in \mathcal{F} $ is an M-block, $ Y $ is called an \emph{M-step shift of finite type}. In this case, we can assume without loss of generality that every element of $ \mathcal{F} $ is an M-block. \begin{figure} \caption{A representation of an element of $ \left\{0,1 \right\} \label{fig:M1} \end{figure} In a metric space $ X $, the distance between a point $ x \in X $ and closed subset $ A \subseteq X $ we define as $ d(x,A) = \inf_{a \in A}d(x,a) $. This gives rise to the \emph{Hausdorff distance} between two closed subsets $ A, B \subseteq X $ defined as \[ d_{H}(A,B) = \max\{\sup_{a \in A}d(a,B), \sup_{b \in B}d(b,A)\} .\] \section{$\omega$-limit Sets for Group Actions}\label{Omega} In the context of continuous group or monoid actions on a compact metric space $X$, there is no clear \emph{best} analogue to the notion of an $\omega$-limit set. Principal to this ambiguity is that there is more than one `future' to consider. This problem is especially apparent in the action of groups, but is still present even if we restrict our attention to monoids. In \cite{MR-Zd} and \cite{oprochalimitsets}, the authors explore some of these notions in the context of $\mathbb Z^d$ actions. Perhaps the most general notion of limit sets in this context are the semigroup limit sets as defined by Barros and Souza \cite{Barros2010}. \begin{definition} Let $S$ be a semigroup acting continuously on $X$ and let $\mathcal F$ be a family of subsets of $S$. For $x\in X$, we define \[\omega(x,\mathcal F)=\bigcap_{A\in\mathcal F}\overline{\{f_s(x) : s\in A\}}\] \end{definition} It is immediate that $\omega(x,\mathcal F)$ is a compact subset of $X$. If $\mathcal F$ is taken to be a filter base (for any $A,B\in\mathcal F$, there exists $C\in\mathcal F$ with $C\subseteq A\cap B$), then $\omega(x,\mathcal F)$ is nonempty, and under appropriate assumptions, it is also invariant \cite{Barros2012}. For the purposes of the following sections, we consider only the action of free groups and free monoids with finitely many generators. In particular, let $G$ be the free group or monoid on $n$ generators, with $S$ the set of generators (and inverses in the case $ G $ is a free group) and let $G$ act continuously on the compact metric space $X$ by the assignment $u\in G\mapsto f_u:X\to X$. We also take $W$ to be the set of reduced words in $G$, and $W_\infty$ the set of infinite reduced words. In this setting, there are a number of specific limit set types that warrant particular attention. The first of these is acquired by taking $\mathcal{F}$ to be the family of subsets of words with minimum lengths. \begin{definition} For $ x \in X $, $ \omega(x) = \bigcap_{n \in \omega} \overline {\left\{ f_{u}(x) : \left| u \right| > n \right\}} $. \end{definition} The equivalent metric formulation is given in the following lemma. \begin{lem} $ \omega(x) = \left\{y \in X : \forall \, n \in \mathbb{N} \; \; \exists \, \left| u_{n} \right| > n \; \; s.t. \; \; d(f_{u_{n}}(x), y) < \frac{1}{n} \right\} $. \end{lem} \begin{proof} Let $ y \in \omega(x) $. For $ n \in \mathbb{N} $ there exists a sequence $ \left\{ u_{i} \right\}_{i \in \mathbb{N}} \subseteq G $ with $ u_{i} > n $ such that $ \left\{ f_{u_{i}}(x) \right\} $ converges to $ y $, as $ y \in \overline{ \left\{ f_{u}(x): \left| u \right| > n \right\} }$. Thus we can choose $ \left| u_{n} \right| > n $ with $ d (f_{u_{n}}(x), y) < \frac{1}{n} $. Now let $ y \in \left\{ y \in X : \forall \, n \in \mathbb{N} \; \; \exists \, \left| u_{n} \right| > n \; \; s.t. \; \; d(f_{u_{n}}(x), y) < \frac{1}{n} \right\} $. For $ n \in \mathbb{N} $ choose $ \left| u_{n} \right| > n $ with $ d(f_{u_{n}}(x), y) < \frac{1}{n} $. Thus, for $ m \geq n $, $f_{u_{m}}(x) \in \left\{f_{u}(x):\left| u \right| > n \right\} $, and $ \left\{ f_{u_{i}}(x) \right\}_{i = m}^{\infty} $ converges to $ y $. Therefore $ y \in \overline{\left\{f_{u}(x): \left| u \right| > n \right\}} $. \end{proof} \begin{lem} $ \omega(x) $ is invariant. \end{lem} \begin{proof} Let $ y \in \omega(x) $ and $ i \in S $. By the uniform continuity of $ f_{i} $, there exists $ \delta_{n} > 0 $ such that $ d(f_{i}(x), f_{i}(y)) < \frac{1}{n} $ if $ d(x, y) < \delta_{n} $. As $ y \in \omega(x) $, there is a sequence $ \left\{ u_{j} \right\} $ increasing in length such that $ d(y, f_{u_{j}}(x)) < \delta_{j} $. Hence, $ d(f_{i}(y), f_{u_{j}i}(x)) < \frac{1}{j} $ for all $ j $ so $ y \in \omega(x) $. \end{proof} This notion of limit set captures all possible `futures' of $x$ under the action of $G$, completely discarding any notion of direction. In contrast, the following type of limit sense has a very strong sense of directionality, capturing only those behaviors of the orbit of $x$ realized along a particular trajectory. \begin{definition} For $ x \in X $ and $ w \in W_{\infty} $, $ \omega_{w}(x) = \bigcap_{n \in N} \overline {\left\{ f_{w_{0} \dots w_{k}}(x) : k > n \right\}}$. \end{definition} Here, we take the family $\mathcal{F}$ to be the collection of finite prefixes of the infinite word $w$. As expected, $\omega_w(x)$ is compact and non-empty. And, while it is not necessarily invariant, it does still have a limited level of invariance. \begin{thm} In the case of a free group action, for $ y \in \omega_{w}(x) $ there exists $ i \neq j \in S $ such that $ f_{i}(y), \; f_{j}(y) \in \omega_{w}(x) $. \end{thm} \begin{proof} Let $ y \in \omega_{w}(x) $ be given. By the uniform continuity of the maps $ f_{i}$, for every $ n \in \mathbb{N} $ there is $ k_{n} $ such that if $ d(x,z) < \frac{1}{k_{n}} $, then $ d(f_{i}(x), f_{i}(z)) < \frac{1}{n} $ for all $ i \in S $. For $ n \in \mathbb{N} $ choose $ m_{n} > n + 1 $ such that $ d(f_{w_{0} \dots w_{m_{n}}}(x), y) < \frac{1}{k_{m}} $. Choose $ i \in S $ such that $ i = w_{m_{n} + 1} $ for infinitely many $ n $. We can then choose $ j $ so $ j = w_{m_{n}}^{-1} $ for infinitely many $ n $ with $ w_{m_{n} + 1} = i $. Thus $ i \neq j $. By passing to a subsequence if necessary, we can assume $ w_{m_{n}} = i^{-1}$ and $ w_{m_{n}+1}= j$. Notice then that $ d(f_{w_{0} \dots w_{l_{n}}}(x), y) < \frac{1}{k_{n}} $. Thus $ d(f_{j}(f_{w_{0} \dots w_{m_{n}}}(x)), f_{j}(y)) = d(f_{w_{0} \dots w_{m_{n} - 1}}(x), f_{j}(y)) < \frac{1}{n} $. Similarly, $ d(f_{w_{0} \dots w_{m_{n} + 1}}(x), f_{i}(y)) < \frac{1}{n} $. As $ m_{n} - 1 > n $, $ f_{i}(y) $, $ f_{j}(y) \in \omega_{w}(x) $. \end{proof} Because there are not inverses in a free monoid, we obtain a slightly weaker result in this context. \begin{cor} In the case of a free monoid action, for $ y \in \omega_{w}(x) $ there exists $ i \in S $ such that $ f_{i}(y) \in \omega_{w}(x) $ and there exists $z\in\omega_w(x)$ and $j\in S$ with $y=f_j(z)$. \end{cor} A similar limit set with a less rigid, but still quite strong, notion of directionality is the following. Here, the family $\mathcal F$ is the collection of subsets of words that share prefixes of specified lengths with $w$. \begin{definition} For $ x \in X $ and $ w \in W_{\infty} $, $ \omega_{F_{w}}(x) = \bigcap_{n \in N} \overline {\left\{ f_{u}(x) : u|_{n} = w|_{n} \right\} } $ \end{definition} Again this is compact and non-empty and has an equivalent analytic definition. \begin{lem} $ \omega_{F_{w}}(x) = \left\{y \in X : \forall \; m \; \exists \, u \in G \; s.t. \; d(f_{u}(x), y)) < \frac{1}{m} \textrm{ and } u_{|m} = w_{|m} \right\} $. \end{lem} A principal benefit of this looser notion of directionality is that we recover invariance. \begin{thm} $ \omega_{F_{w}}(x) $ is invariant. \end{thm} \begin{proof} For $ m \in \mathbb{N} $ choose $ k_{m} > m $ such that for $ i \in S $, if $ d(y,z) < \frac{1}{k_{m}} $ then $ d(f_{i}(y), f_{i}(z)) < \frac{1}{m} $. Now find $ u \in G $ such that $ d(f_{u}(x), y) < \frac{1}{k_{m}} $. Thus $ d( f_{ui}(x), f_{i}(y)) < \frac{1}{m} $. As $ k_{m} > m $, $ ui_{|m} = w_{|m} $. Therefore $ f_{i}(y) \in \omega_{F_{w}}(x) $. \end{proof} Among these limit set types, we have the following relationships. \begin{thm} For all $w\in W_\infty$ and all $x\in X$, we have \[\omega_{w}(x) \subseteq \omega_{F_{w}}(x) \subseteq \omega(x).\] \end{thm} \begin{proof} The first inclusion follows from the observation that $ \left\{ f_{w_{0} \dots w_{k}}(x) : k > n \right\} \subseteq \left\{ f_{u}(x) : u_{|n} = w_{|n} \right\} $. The second inclusion similarly follows from the observation that $ \left\{ f_{u}(x) : u_{|n} = w_{|n} \right\}\subseteq \left\{ f_{u}(x) : \left| u \right| > n \right\}$. \end{proof} \section{Analogues of internal chain transitivity} As explored in a number of papers \cite{BGOR-DCDS, BGKR-omega, BMR-Variations, BDG-tentmaps, GoodMeddaugh-ICT}, there is a strong connection between limit sets and notions of transitivity. It is therefore not surprising that there are connections between analogous notions in the dynamics of group actions on compact metric spaces. In particular, we begin by seeking analogues of chain transitivity that will characterize the classes of limit sets described in Section \ref{Omega}. For notation, let $ \mathfrak{W}_w $ denote the collection of all $ \omega_{w}$-limit sets and $ \mathfrak{W}_{F_{w}} $ the collection of all $ \omega_{F_{w}}$-limit sets in $ X $. \begin{definition} Given $ \epsilon > 0 $ and an element $ u = u_{1} \dots u_{n} \in W $, an $ \epsilon $\emph{-chain indexed by u} is a sequence $ \left\{x_{1}, \dots, x_{n+1} \right\} $ of $ X $ such that $ d(f_{u_{i}}(x_{i})), x_{i+1}) < \epsilon $. \end{definition} \begin{definition} A closed subset $Y$ of $X$ is \emph{internally chain transitive} ($Y\in ICT$) if for every $ x, y \in Y $ and $ \epsilon > 0 $ there is a $ u \in W $ and $ \epsilon $-chain indexed by $ u $ with $ x_{1} = x $ and $ x_{n+1} = y $. \end{definition} \begin{thm} $ ICT $ is closed. \end{thm} \begin{proof} Let $ Y \in \overline{ICT} $, $x,y \in Y $ and $ \epsilon > 0 $. By the uniform continuity of $ f_{i} $ there is a $ \frac{\epsilon}{3} > \delta > 0 $ such that $ d(a,b) < \delta $ implies $ d(f_{i}(a), f_{i}(b)) < \frac{\epsilon}{3} $. Choose $ B \in ICT $ with $ d_{H}(A,B) < \delta $ and $ x', y' \in B $ with $ d(x,x') < \delta $ and $ d(y,y') < \delta $. There is a $ \delta $-chain indexed by $ u $ from $ x' $ to $ y' $ in $ B $. Say this chain is $ \left\{ x_{1}, \dots , x_{n+1} \right\} $. For $ 2 \leq i \leq n $ choose $ z_{i} \in Y $ with $ d(z_{i}, x_{i}) < \delta $ and let $ z_{1} = x $, $ z_{n+1} = y $. It follows that $ \left\{z_{1}, \dots, z_{n+1} \right\} $ is an $ \epsilon $-chain from $ x $ to $ y $ as $ d(f_{u_i}(z_{i}), z_{i+1}) \leq d(f_{u_i}(z_{i}), f_{u_i}(x_{i})) + d(f_{u_i}(x_{i}), x_{i+1}) + d(x_{i+1}, z_{i+1}) < \epsilon $. Therefore $ Y \in ICT $. \end{proof} We have the following results correlating limit sets with internal chain transitivity. \begin{lem}\label{omegaSufClose} For every $ \epsilon > 0 $, $ w = w_{1}w_{2}\dots \in W_\infty $, and $ x \in X $ there exists $ N_{\epsilon} \in \mathbb{N} $ such that for $ n > N_{\epsilon} $ $ d(f_{w_{1}\dots w_{n}}(x), \omega_{w}(x)) < \epsilon $. \end{lem} \begin{proof} Suppose to the contrary. Then we have an increasing sequence of integers $ \left\{m_{n}\right\} $ with $ d(f_{w_{1} \dots w_{m_{n}}}(x), \omega_{w}(x)) > \epsilon $. By passing to a subsequence if necessary, $ \left\{f_{w_{1} \dots w_{m_{n}}} (x) \right\} $ converges to a point $ y \in \omega_{w}(x) $. However, $ d(y, \omega_{w}(x)) \geq \epsilon $, a contradiction. \end{proof} \begin{thm} $ \mathfrak{W}_{w} \subseteq ICT $. \end{thm} \begin{proof} Let $ \omega_{w}(x) \in \mathfrak{W}_{w} $, $ y,z \in \omega_{w}(x) $, and $ \epsilon > 0 $. By the uniform continuity of $ f_{i} $, there is a $\frac{\epsilon}{3} > \delta > 0 $ such that if $ d(p, q) < \delta $ then $ d(f_{u}(p), f_{u}(q)) < \frac{\epsilon}{3} $ for $ u \in S $. Let $ N_{\delta} $ be given by the previous lemma. Find $ n > m > N_{\delta} $ such that $ d(f_{w_{1} \dots w_{m}}(x), y) < \delta $ and $ d(f_{w_{1} \dots w_{n}}(x), z) < \delta $. Let $k=n-m$ and fix $ t_{1}\dots t_{k} = w_{m+1} \dots w_{n} $. Set $ x_{0} = y $, $ x_{k} = z $. For $1<i< k $ choose $ x_{i} $ so $ d(x_{i}, \sigma_{w_{1} \dots w_{n + i -1}}(x)) < \delta $. We claim that $ d(f_{t_{i}}(x_{i}), x_{i+1}) < \epsilon $. $ d(x_{i}, f_{w_{1} \dots w_{n + i -1}}(x)) < \delta $ so $d(f_{t_{i}}(x_{i}), f_{w_{1} \dots w_{n + i}}(x)) < \frac{\epsilon}{3}$. Thus $ d(f_{t_{i}}(x_{i}), x_{i+1}) < d(f_{t_{i}}(x_{i}), f_{w_{1} \dots w_{n + i}}(x)) + d(x_{i+1}, f_{w_{1} \dots w_{n + i}}(x)) < \epsilon$. Therefore $ \omega_{w}(x) $ is internally chain transitive. \end{proof} \begin{lem}\label{omegaFSufClose} Given $ x \in X $ and $ w \in W $, for every $ \epsilon > 0 $ there exists $ N_{\epsilon} \in \mathbb{N} $ such that for all $ u \in G $ with $ u_{|n} = w_{|n} $ for $ n > N_{\epsilon} $, $ d(f_{u}(x), \omega_{F_{w}}(x)) < \epsilon $. \end{lem} \begin{proof} Suppose not. Then for every $ n > N $ there is a $ u_{n} \in G $ with $ u_{n |n} = w_{|n} $ such that $ d(f_{u_{n}}(x), \omega_{F_{w}}(x)) > \epsilon $. By passing to a subsequence if necessary, these $ f_{u_{n}}(x) $ converge to a point $ y \in \omega_{F_{w}}(x) $. However, $ d(y, \omega_{F_{w}}(x)) > \epsilon $, a contradiction. \end{proof} With this lemma, we get the following theorem using the same technique as in the analogous theorem for $ \omega_{w}(x) $. \begin{thm} For an $F$-action on $X$, $ \mathfrak{W}_{F_{w}} \subseteq ICT$. \end{thm} It should be noted that when $G$ is a group, internal chain transitivity is a rather weak condition. In particular, since there is no inherent `direction' for chains, any finite segment of an orbit is an internally chain transitive set. In particular, we fail to see the expected equivalence of being a limit set and being internally chain transitive. Even in the context of full shifts, internal chain transitivity alone is not enough to characterize $ \omega_{w}(x) $. \begin{example} Let $ X $ be the full-shift on the alphabet $ \left\{0,1,2\right\} $ over $ F_{2} $. Let $ Y \subset X = \left\{x_{0}, x_{1}, \dots \right\}$ where $ x_{0} $ is 1 for elements of the form $ a^{m}b^{n} $ ($ n > 0 $) and 0 elsewhere, $ x_{1} = \sigma_{b^{-1}}(x_{0}) $. $ x_{2} $ is the same as $ x_{0} $ except $ 1 $ is 2. For $ i > 2 $, $ x_{2+i} = \sigma_{a^{i}}(x_{2})$. It is not hard to see that this is ICT. Getting to and from $ x_{0} $ and $ x_{1} $ is just a shift as is $ x_{2} $ to $ x_{2+i} $. To get from $ x_{1} $ to $ x_{2} $ we just need to get $ i $ sufficiently large so that $ d(\sigma_{a}(x_{1}), x_{2+i}) < \epsilon $ and then shift to get to $ x_{2} $. However, $ Y $ cannot be expressed as $ \omega_{w}(x) $ for any $ w, x $. Note that for $ \epsilon < 2^{-2} $ any $ \epsilon $-chain ending at $ x_{0} $ must be indexed by a word ending in $ b $ and any $ \epsilon $-chain beginning at $ x_{0} $ and going to $ x_{i} $ ($ i > 0 $) must begin with $ b^{-1} $. However, we claim this can never be the case for an element in $ \omega_{w}(x) $. Let $ x \in X $, $ 2^{-2} > \epsilon > 0 $, $ w = w_{1}w_{2} \dots \in W_{\infty} $ be given, and choose $ N_{\epsilon} \in \mathbb{N} $ such that for $ m > N_{\epsilon} $, $ d(\sigma_{w_{1} \dots w_{m}}(x), \omega_{w}(x)) < \epsilon $. Let $ \frac{\epsilon}{3} > \delta > 0$ so that if $ d(x, y)< \delta $ $ d(\sigma_{i}(x), \sigma_{i}(y)) < \frac{\epsilon}{3} $. Find $ k > N_{\epsilon} $ so that $ d(\sigma_{w_{1} \dots w_{k}}(x), x_{1}) < \delta $, $ m > k $ with $ d(\sigma_{w_{1} \dots w_{m}}(x), x_{0}) < \delta $ and $ l > m $ with $ d(\sigma_{w_{1} \dots w_{l}}(x), x_{1}) < \delta $. As in the proof that $ \omega_{w}(x) $ is ICT, we can get an $ \epsilon $-chain between $ x_{1} $ and $ x_{0} $ indexed by the word $ w_{m+1}\dots s_{k} $ and a $ \epsilon $-chain between $ x_{0} $ and $ x_{1} $ indexed by $ w_{k+1} \dots w_{l} $. As $ w_{k} $, $ w_{k+1} $ are in a reduced word, $ w_{k} \neq w_{k+1}^{-1} $. This then shows that our example cannot be expressed as an $ \omega_w $-limit set. \end{example} With this example, we see that a stronger form of chain transitivity is necessary to characterize $\omega_w$-limit sets for free group actions. \begin{definition} Let $ Y$ be a closed subset of $X$. $ Y $ is \emph{consistently internally chain transitive} ($Y\in CICT$) if for every $ x \in Y $ there are $ i(x), t(x) \in S $ with $ i(x) \neq t(x)^{-1} $ such that for every $ \epsilon > 0 $ and $ x, y \in Y $ there is an $ \epsilon $-chain between $ x $ and $ y $ indexed by a word starting with $ i(x) $ and ending with $ t(y) $. \end{definition} It is immediate from this definition that $CICT \subseteq ICT$. Also, in the context of free monoids, $ CICT $ is equal to $ ICT $ as there are no inverses. In the case of $ ICT $, it was relatively easy to show the set is closed. However, the analogous result for $ CICT $ has an added subtlety. Not only must there be $ \delta $-chains, but also each point $ x $ has an associated initial and terminal index $ i(x) $ and $ t(x) $. For $ Y \in \overline{CICT} $ and $ x \in Y $, it is intuitive to define $ i(x), j(x) $ to match those of points in $ CICT $ sets that converge to $ x $. However, defining $ i(x), j(x) $ for all $ x $ in this manner may not imply $ i(x), t(y) $ interact in the necessary way for any pair $ x, y $. In order to properly choose $ i(x), t(x) $, we must appeal to properties of $ X $ being a compact metric space, namely that $ X $ is separable. \begin{thm} $ CICT $ is closed. \end{thm} \begin{proof} Let $ A \in \overline{CICT} $ and choose a countable, dense subset of $ A $, $ \Lambda = \left\{ x_{1}, x_{2}, \dots \right\} $. Choose a sequence $ \left\{ B_{i} \right\}_{i \in \mathbb{N}} \subseteq CICT $ so that $ d_{H}(A, B_{n}) < \frac{1}{n} $. For each $ x_{n} $ and $ k \in \mathbb{N} $ choose $ x_{n}^{k} \in B_{k} $ with $ d(x_{n}, x_{n}^{k}) < \frac{1}{k} $. For $ x_{1} $ we can choose $ i(x_{1}), j(x_{1}) $ such that $ i(x_{1}) = i(x_{1}^{k}) $ and $ j(x_{1}) = j(x_{1}^{k}) $ for infinitely many $ k $. By passing to a subsequence $ \left\{ k_{m}(1) \right\}$ of $ \mathbb{N}$, we can assume $ i(x_{1}) = i(x_{1}^{k_{m}(1)}) $ and $ j(x_{1}) = j(x_{1}^{k_{m}(1)}) $ for all $ k_{m}(1)$ in the sequence. By induction, we can choose $ i(x_{n}), t(x_{n})$ and a subsequence $ \left\{ k_{m}(n) \right\}$ of $ \left\{ k_{m}(n-1) \right\}$ so that $ i(x_{n}) = i(x_{n}^{k_{m}(n)}) $ and $ j(x_{n}) = j(x_{n}^{k_{m}(n)}) $. For $ x \in A \backslash \Lambda $, choose a sequence $ \left\{ x_{n_k} \right\} \subseteq \Lambda $ that converges to $ x $. Choose $ i(x), j(x) $ so that $ i(x) = i(x_{n_{k}}) $ and $ j(x) = j(x_{n_{k}}) $ for infinitely many $ k $. Let $ x, y \in A $ and $ \epsilon > 0 $ be given. By the uniform continuity of the maps $ f_{i} $ there is a $ \frac{\epsilon}{3} > \delta > 0 $ such that $ d(p,q) < \delta $ implies $ d(f_{i}(p), f_{i}(q)) < \frac{\epsilon}{3} $. Choose $ x_{n}, x_{l} \in \Lambda $ with $ d(x, x_{n}), d(y, x_{l}) < \frac{\delta}{2} $, and $ i(x_{n}) = i(x) $, $ j(x_{k}) = j(y) $. Without loss of generality, assume $ l > n $. Choose $ k_{m}(l) > \frac{2}{\delta}$. By our choice, $i(x_{n}) = i(x_{n}^{k_{m}(l)})$, $j(x_{l}) = j(x_{l}^{k_{m}(l)})$, $ d(x,x_{n}^{k_{m}(l)}), d(y,x_{l}^{k_{m}(l)}) < \delta$, and there is a $ \delta $-chain in $ B_{k_{m}(l)}$ from $ x_{n}^{k_{m}(l)} $ to $ x_{l}^{k_{m}(l)} $ indexed by a word beginning with $ i(x_{n}^{k_{m}(l)})$ and ending with $j(x_{l}^{k_{m}(l)})$. Replacing the first $ x_{n}^{k_{m}(l)} $ with $ x $ and the last $ x_{l}^{k_{m}(l)} $ with $ y $, and every other element of the chain with an element of $ A $ within $ \delta$ gives an $ \epsilon $-chain from $ x $ to $ y $ indexed by a word beginning with $ i(x) $ and ending with $ t(y) $. Thus $ A \in CICT $. \end{proof} \begin{lem} $ \mathfrak{W}_{w} \subseteq CICT$. \end{lem} \begin{proof} Fix $ w \in W_{\infty} $ and $ x \in X$. Choose $ \frac{1}{3n} > \delta_{n} > 0 $ so that for $ d(y,z) < \delta $, $ d(f_{i}(y), f_{i}(z) ) < \frac{1}{3n} $ and an increasing sequence of integers $ \{N_{n}\} $ so that if $ m > N_{n} $ $ d(f_{w_{1} \dots w_{m}}(x), \omega_{w}(x)) < \delta_{n} $. For $ y \in \omega_{w}(x)$, choose an increasing sequence of integers $ \left\{k_{n}(y)\right\} $ with \newline $ d(f_{w_{1}\dots w_{k_{n}(y)}}(x), y ) < \delta_{n} $. By passing to a subsequence if necessary, we can assume $ w_{k_{i}(y)} = w_{k_{i+1}(y)} $ and $ w_{1+k_{i}(y)} = w_{1+k_{i+1}(y)} $ for $ i \in \mathbb{N} $. Set $ t(y) = w_{k_{i}(y)} $ and $ i(y) = w_{1+k_{i}(y)} $. Note that $ i(y) \neq t(y)^{-1} $ as both are consecutive letters in a reduced word. For given $ y, z \in \omega_{w}(x)$ and $ \epsilon > 0 $, find $ \frac{1}{n} < \epsilon $. Choose $ N_{n} < k_{l}(y) < k_{m}(z) $. Just as in the proof that $ \omega_{w}(x) $ is ICT we can find a $ \epsilon $-chain in $ \omega_{w}(x) $ between $ y $ and $ z $ indexed by $ w_{1+k_{l}(y)} \dots w_{k_{m}(z)} $. \end{proof} In shifts of finite type $X$, for a given $Y\in CICT$, we can explicitly construct a word $w\in W_\infty$ and $x\in X$ with $Y=\omega_w(x)$. The following lemma helps to verify our construction. \begin{lem}\label{shiftPseudoOrbit} Let $ X $ be a shift space and fix a function $ \mathcal{O}: G \rightarrow X $. Suppose $ u \in G $ and $ m \in \mathbb{N} $ with the property that for $ v \in \Sigma^{m-1} $ and $ i \in S $, we have \[ d(\sigma_{i}(\mathcal{O}(uv)), \mathcal{O}(uvi)) < 2^{-m} .\] Then $\mathcal{O}(u)(v) = \mathcal{O}(uv)(e) $. \end{lem} \begin{proof} Fix $ v = v_{0} \dots v_{n} \in \Sigma^{m-1} $. As $ d( \sigma_{v_{0}}(\mathcal{O}(u)), \mathcal{O}(uv_{0}) ) < 2^{-m} $, the uniform continuity of the shift maps gives $ d( \sigma_{v_{0}v_{1}}(\mathcal{O}(u)), \sigma_{v_{1}}(\mathcal{O}(uv_{0})) ) < 2^{-m + 1} $. Since $ d( \sigma_{v_{1}}(\mathcal{O}(uv_{0})), \mathcal{O}(uv_{0}v_{1})) < 2^{-m} $ we have $ d( \sigma_{v_{0}v_{1}}(\mathcal{O}(u)), \mathcal{O}(uv_{0}v_{1})) < 2^{-m + 1} $. By induction we see $ d( \sigma_{v_{0}\dots v_{i}}(\mathcal{O}(u)), \mathcal{O}(uv_{0}\dots v_{i})) < 2^{-m + i} $. Therefore $ d( \sigma_{v_{0}\dots v_{n}}(\mathcal{O}(u)), \mathcal{O}(uv_{0}\dots v_{n})) < 2^{-m + n} \leq 2^{-1} $. This implies that $ \sigma_{v}(\mathcal{O}(u))(e) = \mathcal{O}(uv)(e) $. The left-hand side of this equation is equal to $ \mathcal{O}(u)(v)$, so our lemma holds. \end{proof} \begin{thm}\label{CICT} If $ X $ is a shift of finite type and $ Y \in CICT$, then $ Y = \omega_{w}(x)$ for some $ x \in X $ and $ w \in W_{\infty} $. \end{thm} \begin{proof} Let $ X $ be $ M $-step. For $ k \geq M + 1 $, let $ \left\{x^{k}_{i}\right\}_{i=0}^{n_{k}} \subseteq Y $ be sequence that $ 2^{-k} $ covers $Y$. As $ Y \in CICT $, there is a $ 2^{-k} $-chain from $x^{k}_{i} $ to $ x^{k}_{i+1} $ indexed by $ u_{i} $ where $ u_{i} $ begins with $ i(x^{k}_{i}) $ and ends with $ t(x^{k}_{i+1}) $. By concatenating these chains, we can get a $ 2^{-k} $-chain $ \left\{y^{k}_{0}, \dots, y^{k}_{n_{k}} \right\} $ from $ x^{k}_{0} $ to $ x^{k}_{n_{k}} $ indexed by $ w_{k} = v^{k}_{1} \dots v^{k}_{m_{k}} $ such that $ v^{k}_{1} = i(x^{k}_{0}) $, $ v^{k}_{m_{k}} = t(x^{k}_{n_{k}}) $, and for every $ i $ there is an $ n $ such that $ x^{k}_{i} = y^{k}_{n} $. We also have a $ 2^{-k-1} $-chain $ \left\{z^{k}_{0}, \dots, z^{k}_{l_{k}} \right\} $ from $ x^{k}_{n_{k}} $ to $ x^{k+1}_{0} $ indexed by $ w'_{k} = v'_{1} \dots v'_{l_{k}} $ with $ v'_{1} = i(x^{k}_{n_{k}}) $ and $ v'_{l_{k}} = i(x^{k+1}_{0}) $. Concatenating $ w = w_{M+1}w'_{M+1}w_{M+2}w'_{M+2} \dots $ and \newline $ \left\{y^{M+1}_{0}, \dots, y^{M+1}_{n_{M+1}}, z^{M+1}_{1}, \dots, z^{M+1}_{l_{M+1}}, y^{M+2}_{1}, \dots, y^{M+2}_{n_{M+2}}, \dots \right\} $, yields a sequence \newline $ \left\{z_{0}, z_{1}, \dots \right\} \subseteq Y $ and $ w = t_{1}t_{2} \dots \in W_{\infty} $ such that for all $ i \in \mathbb{N} $ $ d(\sigma_{t_{i+1}}(z_{i}), z_{i+1}) < 2^{-M-1} $ and for $ n > M+1 $ there is $ k_{n} $ such that for $ m > k_{n} $, $ d(\sigma_{t_{m}+1}(z_{m}), z_{m+1}) < 2^{-n} $. In order to keep track of these points and their interrelations, we define a function $ \mathcal{O}: G \rightarrow Y $. Set $ \mathcal{O}(t_{1}\dots t_{m}) = z_{m} $ and $ \mathcal{O}(e) = z_{0} $. For notation, let $ t_{0} = e $. For all other $ v \in G $ let $ n_{v} $ be the largest integer such that $ t_{0} \dots t_{n_{v}} $ is a prefix of $ v = t_{0} \dots t_{n_{v}} v' $. Then let $ \mathcal{O}(v) = \sigma_{v'}(z_{n_{v}}) $. By our construction, $ \mathcal{O} $ has the properties that: \begin{enumerate} \item For $ u \in G $, $ v \in \Sigma^{M} $, and $ i \in S $, $ d(\sigma_{i}(\mathcal{O}(uv)), \mathcal{O}(uvi)) < 2^{-M-1} $. \item For every $ n \in N $ there is $ k_{n} \in \mathbb{N} $ such that for $ \left| u \right| > k_{n} $, $ v \in \Sigma^{n} $, and $ i \in R $, $ d(\sigma_{i}(\mathcal{O}(uv)), \mathcal{O}(uvi)) < 2^{-n-1} $. \end{enumerate} Define $ x: G \rightarrow \mathcal{A} $ by $ x(u) = \mathcal{O}(u)(e) $. We claim that $ x \in X $ and that $ \omega_{w}(x) = A $. By property (1) and Lemma \ref{shiftPseudoOrbit}, every $ M $-block in $ x $ is the central $M$-block for some $ \mathcal{O}(v) $. As every $ \mathcal{O}(v) \in X $, this implies $ x \in X $. Property (2) implies that $ d(\sigma(x), \mathcal{O}(v)) < 2^{-n} $ for $ \left| v \right| > k_{n} $. Let $ y \in Y $ and $ n > \mathbb{N} $. Find $ k, i, m $ such that $ d(y, x^{k}_{i}) < 2^{-n-1} $, $ \mathcal{O}(u_{1} \dots u_{m}) = x^{k}_{i} $, with $ m > k_{n+1} $. Therefore $ d(y, \sigma_{u_{1} \dots u_{m}}(x)) < 2^{-n} $, so $ y \in \omega_{w}(x) $. Now suppose $ y \in \omega_{w}(x) $. For $ n \in \mathbb{N} $ find $ m > k_{n+1} $ so that $ d(y, \sigma_{u_{1} \dots u_{m}}(x)) < 2^{-n-1}$ and $ d(\sigma_{u_{1} \dots u_{m}}(x), \mathcal{O}(u_{1} \dots u_{m})) < 2^{-n-1} $. Therefore $ d(y, \mathcal{O}(u_{1} \dots u_{m})) < 2^{-n} $. As $ \mathcal{O}(u_{1} \dots u_{m} ) \in Y $ by construction and $ Y $ is closed, $ y \in Y $. Therefore $ Y = \omega_{w}(x) $. \end{proof} \begin{cor} Let $X$ be a shift of finite type over $G$ . Then $\mathfrak W_w = CICT$. If $G$ is a free monoid, then $\mathfrak W_w= ICT$. \end{cor} \begin{definition} Given a metric space $ X $ and $ \delta > 0 $, a \emph{$ \delta $-G-pseudo orbit} is a function $ \mathcal{O}: G \rightarrow ֤֤֤X $ such that for $ u \in G $ and $ i \in S $ we have $ d(f_{i}(\mathcal{O}(u)), \mathcal{O}(ui) ) < \delta $. \end{definition} \begin{definition} Let $Y$ be a closed subset of $X$. $ Y $ is \emph{internally block transitive} ($Y \in IBT$) if for every $ \delta > 0 $ $ x_{1}, \dots , x_{n} \in Y $ there is a $ \delta $-G-pseudo orbit $ \mathcal{O} $ of $ Y $ containing each $ x_{i} $. \end{definition} The notion of internally block transitive is related to that of mesh transitivity explored in \cite{MR-Zd}. The choice of the word block in this terminology might seem spurious, but it is worth noting that the conditions on the pseudo-orbit are finite in nature and in the context of a free group or monoid, a $ \delta$-G-pseudo orbit can be easily generated by a extending a block function $ B: \Sigma_{n} \to Y $ with $ d(B(ui), \sigma_{i}(B(u))< \delta$ for $ u \in G $ and $ i \in S $. \begin{thm}\label{IBTClosed} For any $ X $, $ IBT $ is closed. \end{thm} \begin{proof} Suppose that $ Y \in \overline{IBT} $ and $ \epsilon > 0 $. By uniform continuity, choose $ \frac{\epsilon}{3} > \delta > 0 $ so the $ d(p,q) < \delta $ implies $ d(f_{i}(p), f_{i}(q)) < \frac{\epsilon}{3} $. Choose $ x_{1}, \dots , x_{n} \in Y $ and $ B \in IBT $ with $ d_{H}(Y, B) < \delta $. Find $ y_{i} \in B $ with $ d(x_{i}, y_{i}) < \delta $ and let $ \mathcal{O} $ be a $ \delta $-G-pseudo orbit of $ B $ containing $ y_{1}, \dots , y_{n} $. We create a pseudo orbit $ \mathcal{O}' $ in $ A $ by replacing every $ y_{i} $ in $ \mathcal{O} $ with $ x_{i} $ and replacing any other $ y \in B $ with $ x \in A $ so that $ d(x,y) < \delta $. By the choice of $ \delta $, it is easy to see that $ \mathcal{O}'$ is an $ \epsilon $-G-pseudo orbit containing $ x_{0}, \dots, x_{n} $. Therefore $ A \in IBT $ and $ IBT $ is closed. \end{proof} \begin{thm} If $ Y \in IBT $, $ Y $ is invariant. \end{thm} \begin{proof} Let $ y \in Y $ and consider $ f_{i}(y) $. For $ n \in \mathbb{N} $ there is a $ \frac{1}{n} $-G-pseudo orbit $ \mathcal{O}_{n} $ containing $ y $. Say $ \mathcal{O}_{n}(u_{n}) = y $ for all $ n $. Then $ d(f_{i}(\mathcal{O}_{n}(u_{n})), \mathcal{O}_{n}(u_{n}i) ) < \frac{1}{n}$. Hence $ d(f_{i}(y), \mathcal{O}_{n}(u_{n}i) ) < \frac{1}{n} $. As $ \mathcal{O}_{n}(u_{n}i) \in Y $ and $ Y $ is closed, $ f_{i}(y) \in Y $. \end{proof} With this definition, it is not guaranteed that we can concatenate these $ \delta $-G-pseudo orbits together in a meaningful way. The following definitions will allow us sufficient conditions for which we have meaningful concatenation. The first is most applicable to free group actions, the second to free monoid actions. \begin{definition} If $ Y\in IBT$, an element $ y \in Y $ is \emph{i,j final} if for every $ \delta > 0 $, $ x_{1}, \dots, x_{n} \in Y $ there is a $ \delta $-G-pseudo orbit $ \mathcal{O} $ of $ Y $ and indexes $ u_{k} $ such that $ \mathcal{O}(u_{k}) = x_{k} $ and $ u_{k} \neq u_{m} $ for $ k \neq m $ and indexes $ u_{i}, u_{j} $ with $ \mathcal{O}(u_{i}) = \mathcal{O}(u_{j}) = y $, $ u_{i}, u_{j} $ end in $ i \neq j $ respectively and $ u_{i}, u_{j} $ are not prefixes of each other or any $ u_{k} $. \end{definition} \begin{definition} Let $Y$ be a closed subset of $X$. $Y\in IBT^*$ if and only if $Y\in IBT $ and there exists $y\in Y$ that is $i,j$-final. \end{definition} \begin{definition} If $ Y \in IBT $, an element $ y \in Y $ is \emph{final} if for every $ \delta > 0 $, $ x_{1}, \dots, x_{n} \in Y $ there is a $ \delta $-G-pseudo orbit $ \mathcal{O} $ of $ Y $ and indexes $ u_{k} $ and $ u_{y} $ such that $ \mathcal{O}(u_{k}) = x_{k} $, $ \mathcal{O}(e) = \mathcal{O}(u_{y}) = y $ with $ u_{y} $ not a prefix of any $ u_{k} $. \end{definition} \begin{definition} Let $Y$ be a closed subset of $X$. $Y\in IBT^{\circ}$ if and only if $Y\in IBT $ and there exists $y\in Y$ that is final. \end{definition} \begin{thm}\label{IBT*closed} $ IBT^{*} $ is closed. \end{thm} \begin{proof} Let $ Y \in \overline{IBT^{*}} $ and let $ \left\{ B_{n} \right\}_{n \in \mathbb{N}} \subseteq IBT^{*} $ converge to $ Y $. In each $ B_{n} $ there a point $ x_{n} $ that is $ i_{n}, j_{n} $ final. Choose $ i, j $ so that $ i = i_{n}, j= j_{n} $ infinitely often. By passing to a subsequence if necessary, we can assume $ i = i_{n}, j = j_{n} $ for all $ n $. Choose $ x \in X $ so $ \left\{ x_{n} \right\}_{n \in \mathbb{N}} $ converges to $ x $. By a similar technique to Theorem \ref{IBTClosed}, it is not hard to see $ Y \in IBT^{*} $ with $ x $ being $ i,j $ final. \end{proof} \begin{thm} $ IBT^{\circ} $ is closed. \end{thm} \begin{proof} This follows from the technique used in Theorem \ref{IBTClosed}. \end{proof} \begin{lem} Let $ w \in W_{\infty} $, $ x \in X $ be given. For a free group action, $ \omega_{F_{w}}(x) \in IBT^{*}$ with $ y \in \omega_{w}(x) $ i,j-final for some $ i, j $. \end{lem} \begin{proof} Let $ x_{1}, \dots, x_{n} \in \omega_{F_{w}}(x) $, $ y \in \omega_{F_{w}}(x) $, and $ \delta > 0 $ be given. By the uniform continuity of the maps $ f_{i} $, find $ \frac{\delta}{3} > \eta > 0 $ such that if $ d(x,y) < \eta $ then $ d(f_{i}(x), f_{i}(y)) < \frac{\delta}{3} $ for $ i \in S $. As $ y \in \omega_{w}(x) $ and $ \omega_{w}(x) $ is CICT, let $ i = i(y)^{-1} $ and $ j = t(y) $, so $ i \neq j $. Now we can find $ w_{1} \dots w_{k_{i}}w_{k_{i}+1} \in F $ with $ i = w_{k_{i}+1}^{-1} $, $ d(y, f_{w_{1} \dots w_{k_{i}}}(x)) < \eta $ and $ k_{i} > N_{\eta} $ given in Lemma \ref{omegaFSufClose}. For $ x_{1} $ we can find $ w_{1} \dots w_{k_{1}}v_{x_{1}} \in F $ with $ k_{1} > k_{i} + 1 $ such that $ d(x_{1}, f_{w_{1} \dots w_{k_{1}}v_{x_{1}}}(x)) < \eta $ and $ v_{x_{1}} $ does not begin with $ w_{k_{1} + 1} $. By induction we can find $ k_{m+1} > k_{m} $, $ v_{x_{m+1}} \in F $ such that $ d(x_{m+1}, f_{w_{1} \dots w_{k_{m+1}}v_{x_{m+1}}}(x)) < \eta $ and $ v_{x_{m+1}} $ does not begin with $ w_{k_{m+1}+1} $. Then we can get $ k_{j} > k_{n} $ with $ w_{k_{j}} = j $ and $ d(y, f_{w_{1} \dots w_{k_{j}}}(x)$. Now define $ \mathcal{O}: F \rightarrow \omega_{F_{w}}(x) $ by $ \mathcal{O}(w_{k_{i}+ 2} \dots w_{k_{m}}v_{x_{m}}) = x_{m} $, $ \mathcal{O}(w_{k_{i}+ 1}^{-1}) = \mathcal{O}(w_{k_{i}+ 2} \dots w_{k_{j}}) = y $. For $ u = w_{k_{i}+ 1}^{-1}v $, define $ \mathcal{O}(u) = f_{v}(y) $ and for all other $ u $ that has not yet been defined, we can choose $ z \in \omega_{w}(x) $ with $ d(z, f_{w_{1}\dots w_{k_{i}+1}u}(x)) < \eta $ and let $ \mathcal{O}(u) = z $. By construction $ \mathcal{O} $ is an $ \delta $-pseudo orbit. For the indexes, let $ u_{i} = w_{k_{i} + 1}^{-1} $, $ u_{j} = w_{k_{i} + 2} \dots w_{k_{j}} $, and $ u_{m} = w_{k_{i} + 2} \dots w_{k_{m}}v_{x_{m}} $. By construction $ \mathcal{O}(u_{m}) = x_{m} $ and $ u_{k} \neq u_{m} $ for $ k \neq m $, $ \mathcal{O}(u_{i}) = \mathcal{O}(u_{j}) = y $, $ u_{i}, u_{j} $ end in $ i, j $ respectively and $ u_{i}, u_{j} $ are not prefixes of each other or any $ u_{m} $. \end{proof} \begin{lem} Let $ w \in W_{\infty} $, $ x \in X $ be given. For a free monoid action, $ \omega_{F_{w}}(x) \in IBT^{\circ}$ with $ y \in \omega_{w}(x) $ final. \end{lem} \begin{proof} Let $ x_{1}, \dots, x_{n} \in \omega_{F_{w}}(x) $, $ y \in \omega_{F_{w}}(x) $, and $ \delta > 0 $ be given. By the uniform continuity of the maps $ f_{i} $, find $ \frac{\delta}{3} > \eta > 0 $ such that if $ d(x,y) < \eta $ then $ d(f_{i}(x), f_{i}(y)) < \frac{\delta}{3} $ for $ i \in S $. Now we can find $ w_{1} \dots w_{k} $ such that $ d(y, f_{w_{1} \dots w_{k}}(x)) < \eta $. Find $ k_{1} > k > N_{\eta}$ from Lemma \ref{omegaFSufClose} and $ u^{1} = u_{1} \dots u_{m} $ with $ u_{1} \neq w_{k_{1}+1} $ and $ d(x_{1}, f_{w_{1} \dots w_{k_{1}}u^{1}}(x)) < \eta$. Inductively find $ k_{i} > k_{i-1} $ and $ u^{k} = u_{1} \dots u_{m} $ with $ u_{1} \neq w_{k_{i}+1} $ and $ d(x_{i}, f_{w_{1} \dots w_{k_{i}}u^{k}}(x)) < \eta $. Finally, find $ k_{y} > k_{n} $ and $ u^{y} = u_{1} \dots u_{m} $ with $ u_{1} \neq w_{k_{n}+1} $ and $ d(y, f_{w_{1}\dots w_{k_{y}}u^{y}}, y) < \eta $. Define $ \mathcal{O}: H \rightarrow \omega_{w}(x) $ by $ \mathcal{O}(e) = \mathcal{O}(u^{y}) = y $, $ \mathcal{O}(u^{k}) = x_{k} $ and for all other $ u $ choose $ z \in \omega_{w}(x) $ so that $ d(f_{w_{1} \dots w_{k}u}(x), z) < \eta $. This satisfies the requirements for $ IBT^{\circ} $. \end{proof} \begin{thm}\label{IBTgroup} Suppose $ X \in \mathcal{A}^{F} $ is SFT with largest forbidden block size M, and let $ Y \subseteq X $ be IBT, invariant, and compact with some $ y \in Y $ i,j-final. Then $ Y = \omega_{F_{w}}(\bar{x}) $ for some $ w \in W $ and $ \bar{x} \in X $. \end{thm} \begin{proof} For $ n > M $, choose $ \left\{x_{1}^{n}, \dots, x_{k_{n}}^{n} \right\} \subseteq Y $ a $ 2^{-n} $ cover of $ Y $. By assumption, we can find a $ 2^{-n} $-pseudo orbit $ \mathcal{O}_{n} $ with indexes $ u_{i}^{n}, u_{j}^{n} \in \Sigma^{k_{n}} $ ending in $ i $ and $ j $ respectively such that $ \mathcal{O}_{n}(u_{i}^{n}) = \mathcal{O}_{n}(u_{j}^{n}) = y $ and indexes $ u_{m}^{n} $ such that $ \mathcal{O}_{n}(u_{m}^{n}) = x_{m}^{n} $ and $ u_{i}^{n}, u_{j}^{n} $ are not prefixes of $ u_{m}^{n} $ for all $ m $. We will inductively construct a function $ \mathcal{O}: F \rightarrow Y $ and a word $ w \in W_{\infty} $. First we construct $ w $. Define $ w_{1} = u_{j}^{M+1} $ and for $ n > 1 $, $ w_{n} = w_{n-1}(u_{i}^{M+n})^{-1}u_{j}^{M+n} $. By induction we will prove $ w_{n} $ ends with $ j $ and begins with $ w_{n-1} $. Note $ u_{j}^{M+1} $ ends with $ j $ so $ w_{1} $ ends with $ j $. Suppose for our inductive step that $ w_{n-1} $ ends with $ j $. As $ u_{i}^{n} $ ends with $ i $ and is not a prefix of $ u_{j}^{n} $ and $ u_{j}^{n} $ ends with $ j $ we have $ (u_{i}^{n})^{-1}u_{j}^{n} $ beginning with $ i^{-1} $ and ending with $ j $. Therefore $ w_{n} $ ends with $ j $. Finally, because $ i \neq j $, $ w_{n} $ begins with $ w_{n-1} $. Define $ w = \lim_{n \rightarrow \infty} w_{n} $. For ease in constructing $ \mathcal{O} $, define $ \mathcal{O}'_{n} $ to be $ \mathcal{O}_{n} $ restricted to elements of $ \Sigma^{k_{n}} $ which do not have $ u_{i}^{n}, u_{j}^{n} $ as a proper prefix. Let $ \mathcal{D}_{n} $ be the domain of $ \mathcal{O}'_{n} $. For $ u \in \mathcal{D}_{M+1} $ define $ \mathcal{O}(u) = \mathcal{O}'_{M+1}(u) $. Then for $ u \in \mathcal{D}_{M+n} $ define $ \mathcal{O}(w_{n-1}(u_{i}^{M+n})^{-1}u) = \mathcal{O}_{M+n}'(u) $. We will show that this step is well-defined. Suppose for some $ n < m $ there is $ v, v' $ in $ \mathcal{D}_{n}, \mathcal{D}_{m} $ respectively such that $ w_{n-1}(u_{i}^{M+n})^{-1}v = w_{m-1}(u_{i}^{M+m})^{-1}v' $. Write $ w_{m-1} = w_{n-1}(u_{i}^{M+n})^{-1}u_{j}^{M+n} \cdots (u_{i}^{M+m-1})^{-1}u_{j}^{M+m-1} $. \newline Thus $ (u_{i}^{M+n})^{-1}v = (u_{i}^{M+n})^{-1}u_{j}^{M+n} \cdots (u_{i}^{M+m-1})^{-1}u_{j}^{M+m-1}(u_{i}^{M+m})^{-1}v'$. \newline Therefore $ v = u_{j}^{M+n} \cdots (u_{i}^{M+m-1})^{-1}u_{j}^{M+m-1}(u_{i}^{M+m})^{-1}v' $. As $ v $ does not contain $ u_{j}^{M+n} $ as a proper prefix, $ v = u_{j}^{M+n} $, $ m = n + 1 $ and $ v' = u_{i}^{M+n+1} $. In this case, $ \mathcal{O}(w_{n-1}(u_{i}^{M+n})^{-1}u_{j}^{M+n}) = \mathcal{O}_{M+n}'(v) = y $ and $ \mathcal{O}(w_{n}(u_{i}^{M+n+1})^{-1}u_{i}^{M+n+1}) = \mathcal{O}_{M+n+1}'(u_{i}^{M+n+1}) = y $. Thus $ \mathcal{O} $, so far as it has been defined is well-defined. To complete the construction of $ \mathcal{O} $, for $ u \in F $ with $ \mathcal{O}(u) $ not already defined, let $ u' $ be the largest prefix of $ u $ with $ \mathcal{O}(u') $ already defined. Such $ u' $ always exists as $ \mathcal{O}(e) $ is already defined. Then define $ \mathcal{O}(u) = \sigma_{u'^{-1}u}(\mathcal{O}(u')) $. Thus we have defined $ \mathcal{O}: F \rightarrow Y $. From the construction, By our construction, $ \mathcal{O} $ has the properties that: \begin{enumerate} \item For $ u \in F $, $ v \in \Sigma^{M} $, and $ i \in S $, $ d(\sigma_{i}(\mathcal{O}(uv)), \mathcal{O}(uvi)) < 2^{-M-1} $. \item For every $ n \in N $ there is $ k_{n} \in \mathbb{N} $ such that for $ \left| u \right| > k_{n} $, $ v \in \Sigma^{n} $, and $ i \in S $, $ d(\sigma_{i}(\mathcal{O}(uv)), \mathcal{O}(uvi)) < 2^{-n-1} $. \end{enumerate} Just as in Theorem \ref{CICT}, by defining $ \bar{x} $ by $ \bar{x}(v) = \mathcal{O}(v)(e) $ these properties imply that $ \bar{x} \in X $ and $ d(\sigma(\bar{x}), \mathcal{O}(v)) < 2^{-n} $ for $ \left| v \right| > k_{n} $. Finally, we show that $ Y = \omega_{F_{w}}(\bar{x}) $. First $ \omega_{F_{w}}(\bar{x}) \subseteq Y $ as every N-block in $ \bar{x} $ is an N-block of an element of $ A $. Now let $ z \in Y $ and $ n \in \mathbb{N} $. We must find a $ u \in F $ with $ u_{|n} = w_{|n} $ and $ d(\sigma_{u}(\bar{x}), z) < \frac{1}{n} $. Note $ \left| w_{n} \right| \geq n $. Find $ x_{m}^{M+n+1} $ such that $ d(x_{m}^{M+n}, z) < 2^{-(M+n+1)} $. Let $ u = w_{n}(u_{i}^{M+n})^{-1}u_{m}^{M+n+1} $. Thus $ u_{|n} = w_{|n} $ and $ d(\sigma_{u}(\bar{x}), x_{m}^{M+n+1}) < 2^{-(M+n+1)} $. Therefore $ d(\sigma_{u}(\bar{x}), z) < 2^{-(M+n+1)} < \frac{1}{n} $. Thus $ Y = \omega_{F_{w}}(\bar{x}) $. \end{proof} Note that this construction depended only upon the properties of $ A \in IBT^{*} $. As long as one replaces the shift maps $ \sigma $ with just a more general function $ f $, the construction works in all other $ IBT^{*} $ sets with $ F $-actions. \begin{cor} Let $X$ be a shift of finite type over a free group $G$. Then $\mathfrak W_{F_w}=IBT^*$. \end{cor} Using a slightly different, yet more straightforward, construction, we obtain the same result for monoid actions. \begin{thm}\label{IBTmonoid} Suppose $ X \in \mathcal{A}^{H} $ is SFT with largest forbidden block size M, and let $ Y \subseteq X $ be IBT, invariant, and compact with some $ y \in Y $ final. Then $ Y = \omega_{F_{w}}(\bar{x}) $ for some $ w \in W_{\infty} $ and $ \bar{x} \in X $. \end{thm} \begin{proof} For $ n >M $, choose $ \left\{x_{1}^{n}, \dots, x_{k_{n}}^{n} \right\} \subseteq Y $ a $ 2^{-n} $ cover of $ Y $. By assumption, we can find a $ 2^{-n} $-pseudo orbit $ \mathcal{O}_{n} $ with indexes $ u_{y}^{n}, u_{m}^{n} $ such that $ \mathcal{O}(e) = \mathcal{O}(u^{n}_{y}) = y$, $ \mathcal{O}_{n}(u_{m}^{n}) = x_{m}^{n} $ and $ u_{y}^{n} $ is not a prefix of $ u_{m}^{n} $ for all $ m $. Define $ w = u_{y}^{M+1}u_{y}^{M+2} \dots $. For notation, define $ u_{y}^{0} = e $. For $ u \in H $, find the maximal $ m $ such that $ u = u_{y}^{0} \dots u_{y}^{m}u' $ and define $ \mathcal{O}(u) = \mathcal{O}_{m+1}(u') $. Letting $ \bar{x}(v) = \mathcal{O}(v)(e) $, the same reasoning as above gives $ \omega_{F_{w}}(\bar{x}) = Y $. \end{proof} \begin{cor} Let $X$ be a shift of finite type over a free monoid $H$. Then $\mathfrak W_{F_w}=IBT^\circ$. \end{cor} \section{Shadowing in Group Actions} In the previous section, we developed some connections between limit sets and internal transitivity properties in the context of shifts of finite type over finitely generated free groups. It is not surprising, given the results concerning these relationships in $\mathbb Z$- and $\mathbb N$-actions \cite{MR, BDG-tentmaps, BGKR-omega, GoodMeddaugh-ICT} that we are able to find analogous results outside of shift spaces. In order to demonstrate these properties, we first need to have an appropriate notion of shadowing for group actions. \begin{definition} For $ \epsilon > 0 $ a function $ \mathcal{O}: G \rightarrow ֤֤֤X $ is \emph{$ \epsilon $-shadowed by $x \in X $} if $ d(f_{u}(x), \mathcal{O}(u)) < \epsilon $ for all $ u \in G $. \end{definition} \begin{definition} A $G$-action on a compact metric space $X$ has the \emph{G-shadowing property} if for every $ \epsilon > 0 $ there exists $ \delta > 0 $ such that every $ \delta $-$G$-pseudo orbit is $ \epsilon $-shadowed by a point in $ X $. \end{definition} \begin{lem} If $ X $ is not an SFT, then for every $ n \in \mathbb{N} $ there is $ m > n $ such that there is a forbidden m-block of $ X $ such that every sub-block is not forbidden. \end{lem} \begin{proof} We will prove the contrapositive. Write $ X = X_{\mathcal{F}} $ for some set of forbidden blocks $ \mathcal{F} $. Let $ m $ be the largest integer such that there is an m-block $ B \in \mathcal{F} $ such that every sub-block of $ B $ is not in $ \mathcal{F} $. Let $ \mathcal{F}' = \left\{B \in \mathcal{F} : B \; \text{is a k-block for } \; k \leq m \right\} $. Note that $ \mathcal{F}' $ is finite as there only finitely many k-blocks for $ k \leq m $. We claim that $ X_{\mathcal{F}} = X_{\mathcal{F}'} $. As $ \mathcal{F}' \subseteq \mathcal{F} $, $ X_{\mathcal{F}'} \supseteq X_{\mathcal{F}} $. Now suppose $ x \notin X_{\mathcal{F}} $. Thus $ x $ contains a forbidden k-block $ B $ in $ \mathcal{F} $. If $ k \leq m $, $ B \in \mathcal{F}' $ so $ x \notin X_{\mathcal{F}'} $. If $ k > m $ then $ B $ contains a forbidden l-block for $ l < k $. By induction, $ B $ contains a forbidden l-block for $ l \leq m $. In either case, $ x $ contains a block forbidden in $ \mathcal{F}' $ so $ x \notin X_{\mathcal{F}'} $. As $ X = X_{\mathcal{F}} = X_{\mathcal{F}'} $, $ X $ is SFT. \end{proof} \begin{lem} Suppose $ \mathcal{O} $ is a $ 2^{-m} $-pseudo orbit. Then for $ u \in F $ and $ v \in \Sigma^{m-1} $, $ \mathcal{O}(u)(v) = \mathcal{O}(uv)(e) $. \end{lem} \begin{proof} This result follows from the same argument as Lemma \ref{shiftPseudoOrbit}. \end{proof} \begin{thm} A shift space $ X $ is an SFT if and only if $ X $ has the shadowing property. \end{thm} \begin{proof} Suppose $ X $ is an M-step SFT and let $ \epsilon > 0 $ be given. Choose $ k > M $ so $ 2^{-k} < \epsilon $. We claim every $ 2^{-k - 1} $-pseudo orbit can be $ \epsilon $-shadowed. Let $ \mathcal{O} $ be such a pseudo orbit. Construct $ x \in \mathcal{A}^{F} $ by $ x(u) = \mathcal{O}(u)(e) $. Let $ u \in F $ and $ v \in \Sigma^{k} $. By definition, $ \sigma_{u}(x)(v) = x(uv) = \mathcal{O}(uv)(1) $. By the previous lemma, the right-hand term equals $ \mathcal{O}(u)(v) $. Thus $ \sigma_{u}(x)_{|\Sigma^{k}} = \mathcal{O}(u)_{|\Sigma^{k}} $ so $ d(\sigma_{u}(x), \mathcal{O}(u)) < 2^{-k} $. This implies $ x $ $ \epsilon $-shadows $ \mathcal{O} $.Furthermore, $ x \in X $ as every M-block in $ x $ is an M-block in an element of $ X $, thus $ x $ contains no forbidden blocks. Now suppose $ X $ is not an SFT. Let $ \epsilon = 2^{-1} $ and suppose $ X $ has the shadowing property. Thus there is a $ \delta > 0 $ such that every $ \delta $-pseudo orbit can be $ \epsilon $-shadowed. Choose $ m $ with $ 2^{-m} < \delta $. By a previous lemma, there is a $ k > m + 2 $ such that $ X $ has a forbidden k-block $ B $ with all sub-blocks of $ B $ not forbidden. For $ i \in S $ let $ B_{i} $ be the (k-1)-block of $ B $ centered at $ i $. As these are not forbidden, there exists an $ x_{i} \in X $ such that $ x_{i |\Sigma^{k-1}} = B_{i} $. Let $ B_{1} = B_{|\Sigma^{k-1}} $ and $ x_{1} \in X $ such that $ x_{1|\Sigma^{k-1}} = B_{1} $. For $ u \in F $, define $ \mathcal{O}(iu) = \sigma_{u}(x_{i}) $ and $ \mathcal{O}(e) = x_{1} $. For $ i \in S $, $ d(\sigma_{i^{-1}}(x_{i}), x_{1}) < 2^{-k + 2} $ as $ B_{i} $ and $ B_{0} $ overlap on a (k-2)-block. For $ i,j \in S $ and $ u \in F $ with $ iuj \neq 1 $, $ \mathcal{O}(iuj) = \sigma_{uj}(x_{i}) = \sigma_{j}(\sigma_{u}(x_{i})) = \sigma_{j}(\mathcal{O}(iu)) $. Therefore $ \mathcal{O} $ is a $ \delta $-pseudo orbit. Suppose that $ x \in X $ $ \epsilon $-shadows $ \mathcal{O} $. Then for $ u \in F $, $ d(\sigma_{u}(x), \mathcal{O}(u) ) < \epsilon $. Particularly, this implies $ x(u) = \mathcal{O}(u)(e) $. We claim that $ x $ contains $ B $. Clearly, $ x(e) = B(e) $. For $ i \in S $ and $ u \in \Sigma^{k-1} $, $ x(iu) = \sigma_{iu}(x)(e) = \mathcal{O}(iu)(e) = \sigma_{u}(x_{i})(e) = x_{i}(u) = \mathcal{O}(i)(u) = B_{i}(u) = B(iu) $. Therefore our claim is correct, hence $ \mathcal{O} $ cannot be $ \epsilon $-shadowed, contradicting the assumption of $ X $ having the shadowing property. \end{proof} \begin{thm}\label{CICTgeneral} For an $G$-action on $X$ with $G$-shadowing, then $CICT=\overline{\mathfrak{W}_w}$. \end{thm} \begin{proof} Let $ Y \in CICT $. For $ n \in \mathbb{N} $, find $ \delta_{n} $ such that any $ \delta_{n} $-pseudo orbit can $ \frac{1}{n} $ shadowed. Define $ \mathcal{O} $ as follows. Fix $ k_{0} > \frac{1}{\delta_{n}} $. For $ k > k_{0} $ let $ \left\{x^{k}_{i}\right\}_{i=0}^{n_{k}} \subseteq Y $ be sequence that $ \frac{1}{k} $ covers $Y$. As $ Y $ is CICT, there is a $ \frac{1}{k} $-chain from $x^{k}_{i} $ to $ x^{k}_{i+1} $ indexed by $ u_{i} $ where $ u_{i} $ begins with $ i(x^{k}_{i}) $ and ends with $ t(x^{k}_{i+1}) $. By concatenating these chains, we can get a $ \frac{1}{k} $-chain $ \left\{y^{k}_{0}, \dots, y^{k}_{n_{k}} \right\} $ from $ x^{k}_{0} $ to $ x^{k}_{n_{k}} $ indexed by $ w_{k} = v^{k}_{1} \dots v^{k}_{m_{k}} $ such that $ v^{k}_{1} = i(x^{k}_{0}) $, $ v^{k}_{m_{k}} = t(x^{k}_{n_{k}}) $, and for every $ i $ there is an $ n $ such that $ x^{k}_{i} = y^{k}_{n} $. We also have a $ \frac{1}{k+1} $-chain $ \left\{z^{k}_{0}, \dots, z^{k}_{l_{k}} \right\} $ from $ x^{k}_{n_{k}} $ to $ x^{k+1}_{0} $ indexed by $ w'_{k} = v'_{1} \dots v'_{l_{k}} $ with $ v'_{1} = i(x^{k}_{n_{k}}) $ and $ v'_{l_{k}} = i(x^{k+1}_{0}) $. Concatenating $ w = w_{k_{0}}w'_{k_{0}}w_{k_{0}+1}w'_{k_{0} + 1} \dots $ and \newline $ \left\{y^{k_{0}}_{0}, \dots, y^{k_{0}}_{n_{k_{0}}}, z^{k_{0}}_{1}, \dots, z^{k_{0}}_{l_{k_{0}}}, y^{k_{0}+1}_{1}, \dots, y^{k_{0}+1}_{n_{k_{0}+1}}, \dots \right\} $, yields a sequence \newline $ \left\{z_{0}, z_{1}, \dots \right\} \subseteq Y $ and $ w = t_{1}t_{2} \dots \in W_{\infty} $ such that for all $ i \in \mathbb{N} $ $ d(f_{t_{i+1}}(z_{i}), z_{i+1}) < \delta_{n} $ and for $ n > \frac{1}{\delta_{n}} $ there is $ k_{n} $ such that for $ m > k_{n} $, $ d(f_{t_{m}+1}(z_{m}), z_{m+1}) < \frac{1}{n} $. To construct the pseudo orbit, set $ \mathcal{O}(t_{1}\dots t_{m}) = z_{m} $ and $ \mathcal{O}(e) = z_{0} $. For notation, let $ t_{0} = 1 $. For all other $ v \in F $ let $ n_{v} $ be the largest integer such that $ t_{0} \dots t_{n_{v}} $ is a prefix of $ v = t_{0} \dots t_{n_{v}}v'$. Then let $ \mathcal{O}(v) = f_{v'}(z_{m}) $. Let $ x $ be a point that shadows this pseudo-orbit. We want to show that $ d_{H}(\omega_{w}(x), Y) < \frac{1}{n} $. For any element of $ y \in \omega_{w}(x) $, $ d(y, Y) < \frac{1}{n} $ by the nature of the construction. If $ a \in Y $, then there is a prefix $ v_{m}$ of $ w_{n} $ with $ d(a, \mathcal(O)(v_{m}) ) < \frac{1}{m} $. By definition, $ \left\{ f_{v_{m}}(x) \right\} $ converges to a point $ z \in \omega_{w}(x) $. Thus $ d(a, z) < \frac{1}{n} $. Therefore $ d_{H}(\omega_{w}(x_{n}), Y) < \frac{1}{n} $. As $ n $ was arbitrary, $ Y \in \overline{\mathfrak{W}_w}$ and $ CICT \subseteq \overline{\mathfrak{W}_w} $. We have already shown that $ \mathfrak{W}_w \subseteq CICT $ and $ CICT $ is closed; thus $ \overline{\mathfrak{W}_w} \subseteq CICT $. \end{proof} \begin{figure} \caption{One step of the construction.} \label{fig:M2} \end{figure} With a similar construction, we obtain an analogous result for $ IBT $ sets. \begin{thm}\label{IBTgeneralgroup} For an $F$-action on $X$ with $F$-shadowing, $IBT^*=\overline{\mathfrak{W}_{F_w}}$. \end{thm} \begin{proof} Let $ Y \in IBT^{*} $. For $ k \in \mathbb{N} $, find $ \delta_{k} $ such that any $ \delta_{k} $-pseudo orbit can $ \frac{1}{k} $ shadowed. Define $ \mathcal{O} $ as follows. Fix $ k_{0} > \frac{1}{\delta_{k}} $. For $ k \geq k_{0} $ choose $ \left\{x_{1}^{n}, \dots, x_{k_{n}}^{n} \right\} \subseteq Y $ a $ \frac{1}{n} $ cover of $ Y $. By assumption, we can find a $ \frac{1}{n}$-pseudo orbit $ \mathcal{O}_{n} $ with indexes $ u_{i}^{n}, u_{j}^{n} \in \Sigma^{k_{n}} $ ending in $ i $ and $ j $ respectively such that $ \mathcal{O}_{n}(u_{i}^{n}) = \mathcal{O}_{n}(u_{j}^{n}) = y $ and indexes $ u_{m}^{n} $ such that $ \mathcal{O}_{n}(u_{m}^{n}) = x_{m}^{n} $ and $ u_{i}^{n}, u_{j}^{n} $ are not prefixes of $ u_{m}^{n} $ for all $ m $. We will inductively construct $ \mathcal{O}: F \rightarrow Y $ and a word $ w \in W_{\infty} $. First we construct $ w $. Define $ w_{0} = u_{j}^{k_{0}} $ and for $ n > 0 $, $ w_{n} = w_{n-1}(u_{i}^{k_{0}+n})^{-1}u_{j}^{k_{0}+n} $. By induction we will prove $ w_{n} $ ends with $ j $ and begins with $ w_{n-1} $. Note $ u_{j}^{k_{0}} $ ends with $ j $ so $ w_{0} $ ends with $ j $. Suppose for our inductive step that $ w_{n-1} $ ends with $ j $. As $ u_{i}^{n} $ ends with $ i $ and is not a prefix of $ u_{j}^{n} $ and $ u_{j}^{n} $ ends with $ j $ we have $ (u_{i}^{n})^{-1}u_{j}^{n} $ beginning with $ i^{-1} $ and ending with $ j $. Therefore $ w_{n} $ ends with $ j $. Finally, because $ i \neq j $, $ w_{n} $ begins with $ w_{n-1} $. Define $ w = \lim_{n \rightarrow \infty} w_{n} $. For ease in constructing $ \mathcal{O} $, define $ \mathcal{O}'_{n} $ to be $ \mathcal{O}_{n} $ restricted to elements of $ \Sigma^{k_{n}} $ which do not have $ u_{i}^{n}, u_{j}^{n} $ as a proper prefix. Let $ \mathcal{D}_{n} $ be the domain of $ \mathcal{O}'_{n} $. For $ u \in \mathcal{D}_{k_{0}} $ define $ \mathcal{O}(u) = \mathcal{O}'_{M}(u) $. Then for $ u \in \mathcal{D}_{k-{0}+n} $ define $ \mathcal{O}^{k_{0}}(w_{n-1}(u_{i}^{k_{0}+n})^{-1}u) = \mathcal{O}_{k_{0}+n}'(u) $. We will show that this step is well-defined. Suppose for some $ n < m $ there is $ v, v' $ in $ \mathcal{D}_{n}, \mathcal{D}_{m} $ respectively such that $ w_{n-1}(u_{i}^{k_{0}+n})^{-1}v = w_{m-1}(u_{i}^{k_{0}+m})^{-1}v' $. Write $ w_{m-1} = w_{n-1}(u_{i}^{k_{0}+n})^{-1}u_{j}^{k_{0}+n} \cdots (u_{i}^{k_{0}+m-1})^{-1}u_{j}^{k_{0}+m-1} $. \newline Thus $ (u_{i}^{k_{0}+n})^{-1}v = (u_{i}^{k_{0}+n})^{-1}u_{j}^{k_{0}+n} \cdots (u_{i}^{k_{0}+m-1})^{-1}u_{j}^{k_{0}+m-1}(u_{i}^{k_{0}+m})^{-1}v'$. \newline Therefore $ v = u_{j}^{k_{0}+n} \cdots (u_{i}^{k_{0}+m-1})^{-1}u_{j}^{k_{0}+m-1}(u_{i}^{k_{0}+m})^{-1}v' $. As $ v $ does not contain $ u_{j}^{k_{0}+n} $ as a proper prefix, $ v = u_{j}^{k_{0}+n} $, $ m = n + 1 $ and $ v' = u_{i}^{k_{0}+n+1} $. In this case, $ \mathcal{O}(w_{n-1}(u_{i}^{k_{0}+n})^{-1}u_{j}^{k_{0}+n}) = \mathcal{O}_{k_{0}+n}'(v) = y $ and $ \mathcal{O}(w_{n}(u_{i}^{k_{0}+n+1})^{-1}u_{i}^{k_{0}+n+1}) = \mathcal{O}_{k_{0}+n+1}'(u_{i}^{k_{0}+n+1}) = y $. Thus $ \mathcal{O} $, so far as it has been defined is well-defined. To complete the construction of $ \mathcal{O} $, for $ u \in F $ with $ \mathcal{O}(u) $ not already defined, let $ u' $ be the largest prefix of $ u $ with $ \mathcal{O}(u') $ already defined. Such $ u' $ always exists as $ \mathcal{O}(e) $ is already defined. Then define $ \mathcal{O}(u) = \sigma_{u'^{-1}u}(\mathcal{O}(u')) $. Thus we have defined $ \mathcal{O}: F \rightarrow Y $. From the construction, it is easy to see that $ \mathcal{O} $ is a $ \delta_{k} $-pseudo orbit. Let $ x \in X $ $ \frac{1}{k} $ shadow $ \mathcal{O} $. By an analogous argument to \ref{CICTgeneral}, $ d_{H}(\omega_{F_{w}}(x), Y) < \frac{1}{k} $. Thus $ Y \in IBT^{*}$. \end{proof} \begin{thm}\label{IBTgeneralmonoid} For an $H$-action on $X$ with $H$-shadowing, $IBT^{\circ}=\overline{\mathfrak{W}_{F_w}}$. \end{thm} \begin{proof} Let $ Y \in IBT^{\circ} $. For $ k \in \mathbb{N} $, find $ \delta_{k} $ such that any $ \delta_{k} $-pseudo orbit can $ \frac{1}{k} $ shadowed. Define $ \mathcal{O} $ as follows. Fix $ k_{0} > \frac{1}{\delta_{k}} $. For $ n \geq k_{0} $ choose $ \left\{x_{1}^{n}, \dots, x_{k_{n}}^{n} \right\} \subseteq Y $ a $ \frac{1}{n} $ cover of $ Y $. By assumption, we can find a $ \frac{1}{n} $-pseudo orbit $ \mathcal{O}_{n} $ with indexes $ u_{y}^{n}, u_{m}^{n} $ such that $ \mathcal{O}_{n}(e) = \mathcal{O}_{n}(u^{n}_{y}) = y$, $ \mathcal{O}_{n}(u_{m}^{n}) = x_{m}^{n} $ and $ u_{y}^{n} $ is not a prefix of $ u_{m}^{n} $ for all $ m $. Define $ w = u_{y}^{k_{0}}u_{y}^{k_{0}} \dots $. For notation, define $ u_{y}^{k_{0}-1} = e $. For $ u \in H $, find the maximal $ m $ such that $ u = u_{y}^{k_{0}-1} \dots u_{y}^{m}u' $ and define $ \mathcal{O}(u) = \mathcal{O}_{m+1}(u') $. Clearly, $ \mathcal{O} $ is a $ \delta_{k} $-H-pseudo orbit. By the same argument as above, choosing $ x $ to $ \frac{1}{k} $-shadow $ \mathcal{O} $ implies $ d_{H}(\omega_{F_{w}}(x), Y) < \frac{1}{k} $. Thus $ Y \in IBT^{\circ}$. \end{proof} \begin{definition} An \emph{asymptotic F-pseudo orbit} is a function $ \mathcal{O}: F \rightarrow X $ such that for every $ \delta > 0 $ there is an integer $ n $ such that for $ \left| w \right| > n $ and $ u\in S $, $ d(f_{u}(\mathcal{O}(w)), \mathcal{O}(wu)) < \delta $. \end{definition} \begin{definition} A function $ \mathcal{O}: F \rightarrow X $ is \emph{asymptotically shadowed} if there is an $ x \in X $ such that for every $ \epsilon > 0 $ there is an integer $ n $ such that for $ \left| w \right| > n $ $ d(f_{w}(x), \mathcal{O}(w)) < \epsilon $. \end{definition} It is not the case that in a shift of finite type that every asymptotic pseudo orbit can be asymptotically shadowed. Consider for instance the shift of finite type of $ \left\{0,1\right\}^{F} $ given by forbidding any 0 adjacent to a 1. This shift of finite type has two elements: $ x_{0} $ and $ x_{1} $, the constant maps of 0 and 1 respectively. We can construct an asymptotic pseudo orbit by first choosing $ j \in S $, then defining $ \mathcal{O}(ju) = x_{0} $ for $ u \in F $, $ \mathcal{O}(iu) = x_{1} $ for $ i \neq j \in S $ and $ u \in F $, and $ \mathcal{O}(1) = x_{1} $. Suppose that some $ y $ in the subshift asymptotically shadows $ \mathcal{O} $. Then $ y $ must contain both 0 and 1, meaning it contains a 0 adjacent to a 1, so $ y $ is not in the subshift. This contradicts $ \mathcal{O} $ being asymptotically shadowed. \begin{thm} If $ X $ is an m-step SFT, every asymptotic, $ 2^{-m-1} $ pseudo-orbit can be asymptotically shadowed. \end{thm} \begin{proof} Let $ \mathcal{O} $ be such a pseudo orbit. Construct $ x $ by $ x(u) = \mathcal{O}(u)(1) $. By the previous theorem, $ x \in X $. For $ k > m + 1 $ find $ l_{k} $ such that for $ \left| u \right| > l_{k} $, $ d(\sigma_{i}(\mathcal{O}(u)), \mathcal{O}(ui)) < 2^{-k} $ for all $ i \in S $. We claim that for $ \left| u \right| > l_{k+1} + k $, $ d(\sigma_{u}(x), \mathcal{O}(u)) < 2^{-k} $. Notice by the choice of $ u $ that $ \mathcal{O}_{|u \Sigma^{k}} $ is a finite portion of a $ 2^{-k-1} $ pseudo orbit. Hence by a previous lemma, $ \mathcal{O}(u)(v) = \mathcal{O}(uv)(e) $ for $ v \in \Sigma^{k} $. Thus $ \sigma_{u}(x)(v) = x(uv) = \mathcal{O}(uv)(1) = \mathcal{O}(u)(v) $. Hence $ d(\sigma_{u}(x), \mathcal{O}(u)) < 2^{-k} $, so our claim is correct. Therefore $ x $ asymptotically shadows $ \mathcal{O} $. \end{proof} As it happens, this form of shadowing is sufficient for our purposes. \begin{definition} A $G$-action on a compact metric spaces has the \emph{weak} $G$-asymptotic shadowing property if there exists $\delta>0$ such that every asymptotic $G$-pseudo-orbit which is also a $\delta$-$G$-pseudo-orbit is asymptotically shadowed. \end{definition} \begin{thm} For an $G$-action on $X$ with weak $G$-asymptotic shadowing, $ \mathfrak{W}_{w}(X) = CICT $. \end{thm} \begin{proof} It remains to show that $ CICT \subseteq \mathfrak{W}_{w}(X) $. Let $ A \in CICT $ and find $ \delta > 0 $ that witnesses the asymptotic shadowing property. Choose $ k > \frac{1}{\delta} $ and let $ \mathcal{O} $ and $ w $ be as defined in Theorem \ref{CICTgeneral}. By construction, $ \mathcal{O} $ is an asymptotic $ \delta $-F-pseudo orbit. If $ x \in X $ asymptotically shadows $ \mathcal{O} $, it is not difficult to see that $ A = \omega_{w}(x) $. \end{proof} If instead we use the constructions given in Theorems \ref{IBTgeneralgroup} and \ref{IBTgeneralmonoid}, we obtain the following results. \begin{thm} For an $G$-action on $X$ with weak $G$-asymptotic shadowing, \\ $ \mathfrak{W}_{F_{w}}(X) = IBT^{*} $. \end{thm} \begin{thm} For an $H$-action on $X$ with weak $H$-asymptotic shadowing, \\ $ \mathfrak{W}_{F_{w}}(X) = IBT^{\circ} $. \end{thm} \end{document}

\begin{document} \renewcommand{\bfseries}{\bfseriesseries} \renewcommand{\scshape}{\scshapeshape} \title[Sectional category and The Fixed Point Property] {Sectional category and The Fixed Point Property \\ } \author{Cesar A. Ipanaque Zapata} \address{Departamento de Matem\'{a}tica,UNIVERSIDADE DE S\~{A}O PAULO INSTITUTO DE CI\^{E}NCIAS MATEM\'{A}TICAS E DE COMPUTA\c{C}\~{A}O - USP , Avenida Trabalhador S\~{a}o-carlense, 400 - Centro CEP: 13566-590 - S\~{a}o Carlos - SP, Brasil} \curraddr{Departamento de Matem\'{a}ticas, CENTRO DE INVESTIGACI\'{O}N Y DE ESTUDIOS AVANZADOS DEL I. P. N. Av. Instituto Polit\'{e}cnico Nacional n\'{u}mero 2508, San Pedro Zacatenco, Mexico City 07000, M\'{e}xico} \email{cesarzapata@usp.br} \author{Jes\'{u}s Gonz\'{a}lez} \address{Departamento de Matem\'{a}ticas, CENTRO DE INVESTIGACI\'{O}N Y DE ESTUDIOS AVANZADOS DEL I. P. N. Av. Instituto Polit\'{e}cnico Nacional n\'{u}mero 2508, San Pedro Zacatenco, Mexico City 07000, M\'{e}xico} \email{jesus@math.cinvestav.mx} \subjclass[2010]{Primary 55M20, 55R80, 55M30; Secondary 68T40} \keywords{Fixed point property, Configuration spaces, Sectional category, Motion planning problem} \thanks {The first author would like to thank grant\#2018/23678-6, S\~{a}o Paulo Research Foundation (FAPESP) for financial support.} \begin{abstract} For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $\pi_{\red{2},1}^X:F(X,2)\to X$, and the fixed point property (FPP) for self-maps on $X$. Explicitly, we demonstrate that a space $X$ has the FPP if and only if 2 is the minimal cardinality of open covers $\{U_i\}$ of $X$ such that each $U_i$ admits a continuous local section for $\pi_{\red{2},1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics. \end{abstract} \maketitle \section{Introduction, \blue{outline and main results}} \green{A topological theory of motion planning was initiated } \def\red{} \def\green{} \def\blue{{in~\cite{farber2003topological}. As a result, Farber's topological complexity of the space of states of an autonomous agent and, more generally, the sectional number of a map} are numerical invariants } \def\red{} \def\green{} \def\blue{{appearing naturally in the emerging field of topological robotics} (see \cite{pavesic} or \cite{pavesic2019}).} \green{Let $X$ be a topological space and $k\geq 1$. The ordered configuration space of $k$ distinct points on $X$ (see \cite{fadell1962configuration}) is the topological space \[F(X,k)=\{(x_1,\ldots,x_k)\in X^k\mid ~~x_i\neq x_j\text{ whenever } i\neq j \},\] topologised as a subspace of the Cartesian power $X^k$. For $k\geq r\geq 1,$ there is a natural projection $\pi_{k,r}^X\colon F(X,k) \to F(X,r)$ given by $\pi_{k,r}^X(x_1,\ldots,x_r,\ldots,x_k)=(x_1,\ldots,x_r)$.} \green{The study of sectional number and topological complexity for the map $\pi_{k,r}^X$ is still non-existent and, in fact, this work takes a first step in this direction. Several examples are presented to illustrate } \def\red{} \def\green{} \def\blue{{the} result } \def\red{} \def\green{} \def\blue{{arising in this field.}} } \def\red{} \def\green{} \def\blue{{In more detail,} a topological space $X$ has \textit{the fixed point property} (FPP) if, for every continuous self-map $f$ of $X$, there is a point $x$ of $X$ such that $f(x)=x$. \blue{} \def\red{} \def\green{} \def\blue{{We} address the natural question of whether (and how) the FPP can be characterized in the category of Hausdorff spaces and continuous maps.} Such characterizations are known in smaller, more restrictive categories. \blue{For} instance, Fadell proved in 1969 (see \cite{fadell1970} for references) that, in the category of connected compact metric ANRs: \begin{itemize} \item If $X$ is a Wecken space, \blue{then} $X$ has the FPP if and only if $N(f)\neq 0$ for every self-map $f:X\to X$. \item If $X$ is a Wecken space satisfying the Jiang condition, $J(X)=\pi_1(X)$, then $X$ has the FPP if and only if $L(f)\neq 0$ for every self-map $f:X\to X$. \end{itemize} In this work we characterize the FPP \blue{within the category of Hausdorff spaces, and in terms of sectional number}. \green{Indeed, we demonstrate that a space $X$ has the FPP if and only if the sectional number $sec\hspace{.1mm}(\pi_{2,1}^X)$ } \def\red{} \def\green{} \def\blue{{equals 2} (Theorem \ref{characterizacao-ppf}). } \def\red{} \def\green{} \def\blue{{As a result,} we give an alternative proof of the fact that the real projective plane } \def\red{} \def\green{} \def\blue{{has} the FPP (Example \ref{rp2}).} As shown in Section \ref{kr-robot}, a particularly interesting feature of our characterization comes from its connection to current research trends in topological robotics. \green{On the other hand, the study of the Nielsen root number and the minimal root number for the map $\pi_{k,r}^X$ is still non-existent. This problem belongs to } \def\red{} \def\green{} \def\blue{{the so-called \emph{unstable case} in} the general problem } \def\red{} \def\green{} \def\blue{{of} coincidence theory (see } \def\red{} \def\green{} \def\blue{{\cite[Section~7]{goncalves2005}}). } \def\red{} \def\green{} \def\blue{{We} provide conditions in terms of the minimal root number of $\pi_{2,1}^X$ } \def\red{} \def\green{} \def\blue{{for} $X$ } \def\red{} \def\green{} \def\blue{{to} have the FPP (Proposition } \def\red{} \def\green{} \def\blue{{\ref{conditions}}). } \def\red{} \def\green{} \def\blue{{In addition, we} prove that the Nielsen root number $NR(\pi_{k,r}^X,a)$ is at most one (Proposition \ref{nrn-pi}).} \green{The paper is organized as follows: In Section \ref{secrt}, we recall the notions of minimal root number $MR[f,a]$ and the Nielsen root number $NR(f,a)$. In Section \ref{sn}, we recall the notion of Schwarz genus, standard sectional number and basic results about these numerical invariants. } \def\red{} \def\green{} \def\blue{{Our goal is to} study the sectional number for the projection map $\pi_{k,r}^X$. In particular, we demonstrate that a space $X$ has the FPP if and only if the sectional number $sec\hspace{.1mm}(\pi_{2,1}^X)$ } \def\red{} \def\green{} \def\blue{{equals 2} (Theorem \ref{characterizacao-ppf}). In Section \ref{tc-map}, we recall the notion of topological complexity for a map and basic results about these numerical invariant. As applications of our results, in Section \ref{kr-robot}, we study a particular problem in robotics. } \colorado{The authors of this paper deeply thank the referee for very valuable comments and timely corrections on previous versions of the work.} \section{Root theory}\label{secrt} In this section we give a brief exposition of standard mathematical topics in Root theory: the minimal root number and the Nielsen root number $NR(f,a)$. Our exposition is by no means complete, as we limit our attention to concepts that appear in geometrical and topological questions. More technical details can be found in standard works on root theory, like \cite{brooks1970number} or \cite{brown1999middle}. Let $f:X\to Y$ be a continuous map between topological spaces, and fix $a\in Y$. A point $x\in X$ such that $f(x)=a$ is called a \textit{root} of $f$ at $a$. In Nielsen root theory, by analogy with Nielsen fixed-point theory, the roots of $f$ at $a$ are grouped into Nielsen classes, a notion of essentiality is defined, and the Nielsen root number $NR(f,a)$ is defined to be the number of essential root classes. The Nielsen root number is a homotopy invariant and measures the size of the root set in the sense that $$NR(f,a)\leq MR[f,a]:=\min\{\mid g^{-1}(a)\mid\hspace{.2mm}\colon~~g\simeq f\}.$$ The number $MR[f,a]$ is called \textit{the minimal root number} for $f$ at $a$. A classical result of Wecken states that $NR(f,a)$ is in fact a sharp lower bound in the homotopy class of $f$ for many spaces, in particular, for compact manifolds of dimension at least $3$. Thus, in \blue{such cases}, the vanishing of $NR(f,a)$ is sufficient to deform a map $f$ to be \blue{root-free}. Among the central problems in Nielsen root theory (or the theory of root classes) are: \begin{itemize} \item the computation of $NR(f,a)$, \item the realization of $NR(f,a)$, i.e., deciding when \blue{the equality} $NR(f,a)=MR[f,a]$ holds. \end{itemize} \subsection{The Nielsen root number $NR(f,a)$} We recall from \cite{brooks1970number} the Nielsen root number $NR(f,a)$. Let $f:X\to Y$ be a continuous map between path-connected topological spaces, and choose a point $a\in Y$. Assume that the set of roots $f^{-1}(a)$ is non empty. Two such roots $x_0$ and $x_1$ are \textit{equivalent} if there is a path $\alpha:[0,1]\to X$ from $x_0$ to $x_1$ such that the loop $f\circ\alpha$ represents the trivial element in $\pi_1(Y,a)$. This is indeed an equivalence relation, and an equivalence class is called a \textit{root class}. Suppose $H:X\times [0,1]\to Y$ is a homotopy. Then a root $x_0\in H_0^{-1}(a)$ is said to be \textit{$H$-related} to a root $x_1\in H_1^{-1}(a)$ if and only if there is a path $\alpha:[0,1]\to X$ from $x_0$ to $x_1$ such that the loop $\beta:[0,1]\to Y,~\beta(t)=H(\alpha(t),t)$ represents the trivial element in $\pi_1(Y,a)$. Note that a root $x_0$ of $f:X\to Y$ is equivalent to another root $x_1$ if and only if $x_0$ is related to $x_1$ by the constant homotopy at $f$. A root $x_0\in f^{-1}(a)$ is said to be \textit{essential} if and only if for any homotopy $H:X\times [0,1]\to Y$ beginning at $f$, there is a root $x_1\in H_1^{-1}(a)$ to which $x_0$ is $H$-related. If one root in a root class is essential, then all other roots in that root class are essential \blue{too}, and we say that the root class itself is \text{essential}. The number of essential root classes is called the \textit{Nielsen number} of $(f,a)$ and is denoted by $NR(f,a)$. The number $NR(f,a)$ is a lower bound for the number of solutions of $f(x)=a$. If $f^\prime$ is homotopic to $f$ then $NR(f,a)=NR(f^\prime,a)$. Furthermore, $NR(f,a)\leq MR[f,a]$. The order of the cokernel of the fundamental group homomorphism $f_\#:\pi_1(X)\to \pi_1(Y)$ is denoted by $R(f)$, that is, \[R(f)=\left\lvert \dfrac{\pi_1(Y)}{f_\#(\pi_1(X))}\right\rvert;\] it depends only on the homotopy class of $f$. There are always at most $R(f)$ root classes of $f(x)=a$, in particular, $R(f)\geq NR(f,a)$. \begin{example} If $f_\#:\pi_1(X)\to \pi_1(Y)$ is an epimorphism, $ NR(f,a)\leq 1$. In particular, if $Y$ is simply connected, then $ NR(f,a)\leq 1$. \end{example} \section{\red{Sectional number}}\label{sn} \green{In this section we recall the notion of Schwarz genus } \def\red{} \def\green{} \def\blue{{together with} basic results } \def\red{} \def\green{} \def\blue{{from \cite{schwarz1958genus}} about this numerical invariant . Note that the notion of genus in Schwarz's paper \cite{schwarz1958genus} is } \def\red{} \def\green{} \def\blue{{given} for a fibration. We shall follow the terminology in \cite{pavesic2019} and refer to this notion as the Schwarz genus of a \colorado{continuous map}. Also, we recall from \cite{pavesic2019} the notion of standard sectional number.} Let $p:E\to B$ be a \colorado{continuous map}. A \textit{(homotopy) cross-section} or \textit{section} of $p$ is a (homotopy) right inverse of $p$, i.e., a map $s:B\to E$, such that $p\circ s = 1_B$ ($p\circ s \simeq 1_B$). Moreover, given a subspace $A\subset B$, a \textit{(homotopy) local section} of $p$ over $A$ is a (homotopy) section of the restriction map $p_|:p^{-1}(A)\to A$, i.e., a map $s:A\to E$, such that $p\circ s$ is (homotopic to) the inclusion $A\hookrightarrow B$. We recall the following definitions. \begin{definition} \begin{enumerate} \item The (standard) \textit{sectional number} of a \colorado{continuous map} $p\colon E\to B$, $sec\hspace{.1mm}(p)$, is the minimal \blue{cardinality of} open \blue{covers} of $B$, such that each element \blue{of the cover} admits a continuous local section to $p$. \item The \textit{sectional category} of $p$, also called Schwarz genus of $p$, and denoted by $secat(p),$ is the minimal \blue{cardinality of open covers of $B$, such that each element of the cover admits a continuous homotopy local section to $p$.} \end{enumerate} \end{definition} \azuloso{Note that $p$ is surjective whenever $sec\hspace{.1mm}(p)<\infty$. The corresponding assertion for $secat(p)$ may fail.} \begin{remark}\label{secat-sec} We have $secat(p)\leq sec\hspace{.1mm}(p)$. Furthermore, if $p$ is a fibration then $sec\hspace{.1mm}(p) = secat(p)$. \end{remark} \begin{lemma}\label{prop-sectional-category}\cite{schwarz1958genus} Let $p:E\to B$ be a continuous map and $R$ be a commutative ring with unit. If there exist cohomology classes $\alpha_1,\ldots,\alpha_k\in H^\ast(B;R)$ with $p^\ast(\alpha_1)=\cdots=p^\ast(\alpha_k)=0$ and $\alpha_1\cup\cdots\cup \alpha_k\neq 0$, then $sec\hspace{.1mm}(p)\geq k+1$. \end{lemma} } \def\red{} \def\green{} \def\blue{{A few observations worth keeping in mind in what follows are:} \begin{itemize} \item } \def\red{} \def\green{} \def\blue{{If} $B$ is path-connected (this case will appear in our work), we have that $\alpha\in H^\ast(B;R)$, $\alpha\neq 0$ with $p^\ast(\alpha)=0$ implies $\alpha\in \widetilde{H}^\ast(B;R)$. \item } \def\red{} \def\green{} \def\blue{{If} $p:E\to B$ } \def\red{} \def\green{} \def\blue{{is} a continuous map } \def\red{} \def\green{} \def\blue{{and} $p_\ast:H_\ast(E;R)\to H_\ast(B;R)$ or $p_\#:\pi_\ast(E)\to \pi_\ast(B)$ are not su\red{r}jective then $sec\hspace{.1mm}(p)\geq 2$. \item Let $p:E\to B$ be a continuous map. If $p$ has a section $s:B\to E$, then $p\circ s=1_B$ and $s^\ast\circ p^\ast=1_{H^\ast(B;R)}$. In particular, $p^\ast:H^\ast(B;R)\to H^\ast(E;R)$ is a monomorphism. \end{itemize} The following statement is well-known. \begin{lemma}\label{pullback}\cite{schwarz1958genus} Let $p:E\to B$ be a continuous map. If the following square \begin{eqnarray*} \xymatrix{ E^\prime \ar[r]^{\,\,} \ar[d]_{p^\prime} & E \ar[d]^{p} & \\ B^\prime \ar[r]_{\,\, f} & B &} \end{eqnarray*} is a pullback. Then $sec\hspace{.1mm}(p^\prime)\leq sec\hspace{.1mm}(p)$. \end{lemma} \colorado{We recall the pathspace construction from \cite[pg. 407]{hatcheralgebraic}. For a continuous map $f:X\to Y$, consider the space \begin{equation*} E_f=\{(x,\gamma)\in X\times PY\mid~\gamma(0)=f(x)\}. \end{equation*} The map \begin{equation*} \rho_f:E_f\to Y,~(x,\gamma)\mapsto \rho_f(x,\gamma)=\gamma(1), \end{equation*} is a fibration. \azuloso{Further,} the projection over the first coordinate $E_f\to X,~(x,\gamma)\mapsto x$ is a homotopy equivalence with homotopy inverse $c:X\to E_f$ given by $x\mapsto (x,\gamma_{f(x)})$, where $\gamma_{f(x)}$ is the constant path at $f(x)$. \azuloso{This factors} an arbitrary map $f:X\to Y$ as the composition $X\stackrel{c}{\to} E_f\stackrel{\rho_f}{\to} Y$ of a homotopy equivalence and a fibration.} \azuloso{For convenience, we record the following standard properties:} \begin{proposition}\label{secat-pf-equal-secat-f} \begin{enumerate} \item \azuloso{For a continuous map $f:X\to Y$, ${secat}(\rho_f)= {secat}(f).$} \item \colorado{If $f\simeq g$, then \azuloso{${secat}(f)={secat}(g).$}} \end{enumerate} \end{proposition} Next, we recall the notion of LS category which, in our setting, is one greater than that given in \cite{cornea2003lusternik}. For example, the category of a contractible space is one. \begin{definition} The \textit{Lusternik-Schnirelmann category} (LS category) or category of a topological space $X$, denoted cat$(X)$, is the least integer $m$ such that $X$ can be covered by $m$ open sets, all of which are contractible within $X$. \end{definition} We have $\text{cat}(X)=1$ iff $X$ is contractible. The LS category is a homotopy invariant, i.e., if $X$ is homotopy equivalent to $Y$ (which we shall denote by $X\simeq Y$), then $\text{cat}(X)=\text{cat}(Y)$. \colorado{The following lemma \azuloso{generalizes} Proposition 9.14 from \cite{cornea2003lusternik}.} \begin{lemma}\label{prop-secat-map} \colorado{Let $p:E\to B$ be a continuous map. \begin{enumerate} \item If $p$ is a fibration, then $sec\hspace{.1mm}(p)\leq \azuloso{\text{cat}}(B)$. In particular, for any continuous map $f:X\to Y$, we have $secat(f)\leq \text{cat}(Y)$. \item If $p$ is nulhomotopic, then $secat(p)=\text{cat}(B).$ \end{enumerate}} \end{lemma} \begin{proof} \colorado{The first part of item $(1)$ was proved in \cite[Proposition 9.14]{cornea2003lusternik}. For the second part of item $2$, by Proposition~\ref{secat-pf-equal-secat-f}, we have $secat(f)=secat(\rho_f)$ and thus $secat(f)\leq \text{cat}(Y)$.} \colorado{Item $2$ follows easily, because the inequality $\text{cat}(B)\leq secat(\rho_p)$ holds for any nulhomotopic map $p:E\to B$.} \end{proof} \subsection{Configuration spaces}\label{secconfespa} Let $X$ be a topological space and $k\geq 1$. The \textit{ordered configuration space} of $k$ distinct points on $X$ (see \cite{fadell1962configuration}) is the topological space \[F(X,k)=\{(x_1,\ldots,x_k)\in X^k\mid ~~x_i\neq x_j\text{ whenever } i\neq j \},\] topologised as a subspace of the Cartesian power $X^k$. For $k\geq r\geq 1,$ there is a natural projection \blue{$\pi_{k,r}^X\colon F(X,k) \to F(X,r)$ given by $\pi_{k,r}^X(x_1,\ldots,x_r,\ldots,x_k)=(x_1,\ldots,x_r)$.} \begin{lemma}[Fadell-Neuwirth fibration \cite{fadell1962configuration}] \label{TFN} Let $M$ be a connected $m-$dimensional topological manifold without boundary, where $m\geq 2$. \blue{For $k> r\geq 1$, the map} $\pi_{k,r}^M:F(M,k)\to F(M,r)$ is a locally trivial bundle with fiber $F(M-Q_r, k-r)$, \red{where $Q_r\subset M$ is a finite subset with $r$ elements}. In particular, $\pi_{k,r}^M$ is a fibration. \end{lemma} \red{\blue{The boundary restriction in Lemma~\ref{TFN} is important, for} $\pi_{k,r}^X:F(\blue{M},k)\to F(\blue{}M,r)$ } \def\red{} \def\green{} \def\blue{{might fail to be} a fibration if $\blue{M}$ is a manifold with boundary. \blue{This can be seen} by considering, for example, the manifold $\mathbb{D}^2$, with $k=2$ and $r=1$, \blue{for} the fibre $\mathbb{D}^2-\{(0,0)\}$ is not homotopy equivalent to the fibre $\mathbb{D}^2-\{(1,0)\}$.} \begin{proposition}\label{nrn-pi} Let $M$ be a connected $m-$dimensional topological manifold without boundary, where $m\geq 2$. \blue{For $k> r\geq 1$, the projection} $\pi_{k,r}^M:F(M,k)\to F(M,r)$ has Nielsen root number $NR(\pi_{k,r}^M,a)\leq 1$ for any $a\in F(M,r)$. \end{proposition} \begin{proof} The map $\pi_{k,r}^M:F(M,k)\to F(M,r)$ is a fibration with fiber $F(M-Q_r, k-r)$. We note that $F(M-Q_r, k-r)$ is path-connected. By the long exact homotopy sequence of the fibration $\pi_{k,r}^M$, we have the induced homomorphism $(\pi_{k,r}^M)_\#:\pi_1F(M,k)\to \pi_1F(M,r)$ is an epimorphism. Then, $R(\pi_{k,r}^M)=1$ and thus the Nielsen root number $NR(\pi_{k,r}^M,a)\leq 1$ for any $a\in F(M,r)$. \end{proof} } \def\red{} \def\green{} \def\blue{{Note that $MR[\pi_{k,1}^X,a]=0$ (in particular $NR(\pi_{k,1}^X,a)=0$) for any contractible space $X$.} \begin{proposition}\label{secop-pi-k-1}[Key lemma] For any $k\geq 2$ and $X$ a Hausdorff space, we have $sec\hspace{.1mm}(\pi_{k,1}^X)\leq k$. \end{proposition} \begin{proof} Fix an element $(p_1,\ldots,p_k)\in F(X,k)$. For each $i=1,\ldots,k$, set \[U_i:=X-\{p_1,\ldots,p_{i-1},p_{i+1},\ldots,p_k\}\] and \blue{let} $s_i:U_i\longrightarrow F(X,k)$ \blue{be} given by $s_i(x):=(x,p_1,\ldots,p_{i-1},p_{i+1},\ldots,p_k)$. We note that each $U_i$ is open (because $X$ is Hausdorff) and each $s_i$ is a local section of $\pi_{k,1}^X$. Furthermore,, $X=U_1\cup\cdots\cup U_k$. Thus, $sec\hspace{.1mm}(\pi_{k,1}^X)\leq k.$ \end{proof} \begin{remark} \colorado{\azuloso{Using} Lemma~\ref{prop-secat-map} \azuloso{we see that,} for any $k\geq2$, \begin{equation}\label{dosconds} \mbox{$\pi^X_{k,1}\simeq\star\;\;$ and $\;\;secat\hspace{.1mm}(\pi_{k,1}^X)=1\;\;$ if and only if $\;\;X\simeq\star$.} \end{equation} The most appealing situation of \azuloso{(\ref{dosconds})} holds for $k=2$, as in fact $sec\hspace{.1mm}(\pi_{2,1}^X)\in\{1,2\}$, in view of Proposition~\ref{secop-pi-k-1}. Indeed, it would be interesting to know whether there is a space $X$ for which $\pi^X_{2,1}$ is a nulhomotopic fibration having $sec\hspace{.1mm}(\pi_{2,1}^X)=2$. Such a space would have to be a non-contractible co-H-space of topological complexity 2 or 3 (see \azuloso{Proposition~\ref{nul-homotopy-implie-cat2}} and Remark~\ref{aaaaa}) and, more relevantly for the purposes of this paper, would have to satisfy the fixed point property ---see Definition~\ref{defifpp} and Theorem~\ref{characterizacao-ppf} below.} \end{remark} \begin{definition}\label{defifpp} A topological space $X$ has \textit{the fixed point property} (FPP) if for every continuous self-map $f$ of $X$ there is a point $x$ of $X$ such that $f(x)=x$. \end{definition} \begin{example} It is well known that the unit disc $D^m=\{x\in\mathbb{R}^m:~\parallel x\parallel\leq 1\}$ has the FPP (The Brouwer's fixed point theorem). The \red{real, complex and quaternionic} projective spaces, $\mathbb{RP}^{n}, \mathbb{CP}^{n}$ and $\mathbb{HP}^{n}$ have the FPP \red{when $n$ is even} (see \cite{hatcheralgebraic}). For the particular case, $\mathbb{RP}^{2}$, see Example \ref{rp2}. \end{example} Note that the map $\pi_{2,1}^X:F(X,2)\to X$ admits a cross-section if and only if there exists a fixed point free self-map $f:X\to X$. Thus, we have the following theorem. \begin{theorem}\label{characterizacao-ppf}[\blue{Main} theorem] Let $X$ be a Hausdorff space. The space $X$ has the FPP if and only if $sec\hspace{.1mm}(\pi_{2,1}^X)=2$. \end{theorem} \begin{proof} Suppose \blue{that} $X$ has the FPP, then $sec\hspace{.1mm}(\pi^{\red{X}}_{2,1})\geq 2$ \blue{so,} by the Key Lemma (Proposition \ref{secop-pi-k-1}), $sec\hspace{.1mm}(\pi_{2,1}^X)=2$. Suppose \blue{now that} $sec\hspace{.1mm}(\pi_{2,1}^X)=2$, \blue{so in} particular $sec\hspace{.1mm}(\pi_{2,1}^X)\neq 1$. Hence, $X$ has the FPP. \end{proof} \begin{example} No nontrivial topological group $G$ has the FPP. Indeed, the map $s:G\to F(G,2),~g\mapsto (g,g_1g)$ (for some fixed $g_1\neq e\in G$) is a cross-section for $\pi_{2,1}^G:F(G,2)\to G$. The self-map $G\to G,~g\mapsto g_1g$ is fixed point free. \end{example} \begin{example} We recall that the odd-dimensional projective spaces $\mathbb{RP}^{2n+1}$ has not the FPP, because there is a continuous self-map $h:\mathbb{RP}^{2n+1}\to \mathbb{RP}^{2n+1},$ given by the formula $h([x_1:y_1:\cdots:x_{n+1}:y_{n+1}])=[-y_1:x_1:\cdots:-y_{n+1}:x_{n+1}]$, without fixed point. Thus, $sec\hspace{.1mm}(\pi_{2,1}^{\mathbb{RP}^{2n+1}})=1.$ On the other hand, we know that an even-dimensional projective spaces $\mathbb{RP}^{2n}$ has the FPP. Thus, $sec\hspace{.1mm}(\pi_{2,1}^{\mathbb{RP}^{2n}})=2$. Analogous facts hold for complex and quaternionic projective spaces. \end{example} \begin{example} The spheres $S^n$ does not have the FPP, because the antipodal map $A:S^n\to S^n,~x\mapsto -x$ has not fixed points. Thus, $sec\hspace{.1mm}(\pi_{2,1}^{S^n})=1.$ \end{example} \begin{example} We know that any closed surface $\Sigma$, except for the projective plane $\Sigma\neq\mathbb{RP}^2$, has not the FPP. Thus, $sec\hspace{.1mm}(\pi_{2,1}^{\Sigma})=1.$ \end{example} \begin{corollary}\label{suficiente-ppf} Let $X$ be a Hausdorff space. If there exist $\alpha\in H^\ast(X;R)$ with $\alpha\neq 0$ and $(\pi_{2,1}^X)^\ast(\alpha)=0\in H^\ast(F(X,2);R)$, that is, if the induced homomorphism $(\pi_{2,1}^X)^\ast:H^\ast(X;R)\to H^\ast(F(X,2);R)$ is not injective, then $sec\hspace{.1mm}(\pi_{2,1}^X)= 2$. In particular, $X$ has the FPP. \end{corollary} \begin{proof} From Lemma \ref{prop-sectional-category}, $sec\hspace{.1mm}(\pi_{2,1}^X)\geq 1+1=2$. Then, by Proposition \ref{secop-pi-k-1}, $sec\hspace{.1mm}(\pi_{2,1}^X)= 2$. Thus, the result follows from Theorem \ref{characterizacao-ppf}. \end{proof} } \def\red{} \def\green{} \def\blue{{The converse of Corollary \ref{suficiente-ppf} is not true. For example, we recall that the unit disc $D^{m}:=\{x\in \mathbb{R}^m\mid~\parallel x\parallel\leq 1\}$ has the FPP (from the Brouwer`s fixed point theorem) and thus $sec\hspace{.1mm}(\pi_{2,1}^{D^{m}})=2.$ However, $\widetilde{H}^\ast(D^{m};R)=0$.} \begin{corollary} Let $X$ be a Hausdorff space. If the induced homomorphisms $(\pi_{2,1}^X)_\ast:H_\ast(F(X,2);R)\to H_\ast(X;R)$ or $(\pi_{2,1}^X)_\#:\pi_\ast(F(X,2))\to \pi_\ast(X)$ are not surjective, then $sec\hspace{.1mm}(\pi_{2,1}^X)= 2$. In particular, $X$ has the FPP. \end{corollary} \begin{example}\label{rp2} It is easy to see that $\pi_2(F(\mathbb{RP}^2,2))=0$ is trivial and $\pi_2(\mathbb{RP}^2)=\mathbb{Z}$. Then the induced homomorphism $(\pi_{2,1}^{\mathbb{RP}^2})_\#:\pi_2(F(\mathbb{RP}^2,2))\to \pi_2(\mathbb{RP}^2)$ is not surjective, and thus $sec\hspace{.1mm}(\pi_{2,1}^{\mathbb{RP}^2})= 2$. In particular, $\mathbb{RP}^2$ has the FPP. This part can also be proved by employing Lefschetz`s fixed point theorem. \end{example} \begin{remark} For $k\geq l\geq r$, consider the following diagram \begin{eqnarray*} \xymatrix{ F(X,k) \ar[r]^{\pi_{k,l}^X}\ar[d]_{\pi_{k,r}^X} & F(X,l)\ar[dl]^{\pi_{l,r}^X} \\ F(X,r) } \end{eqnarray*} It is easy to see that if $\pi_{l,r}^X\simeq$ \red{const}, then $\pi_{k,r}^X\simeq$ \red{const} for any $k\geq l\geq r$. Moreover, we have $MR[\pi_{l,r}^X,a]\geq MR[\pi_{k,r}^X,a] \text{ for any } k\geq l\geq r$. \end{remark} \begin{proposition}\label{conditions} \colorado{Let $X$ be a connected CW complex with $MR(\pi_{2,1}^X,x_0)=0$. Assume that there exist $\alpha\in \widetilde{H}^\ast(X;R)$ with $\alpha\neq 0$ and $i^\ast(\alpha)=0\in \widetilde{H}^\ast(X-\{x_0\};R)$ for some $x_0\in X$, that is, $i^\ast:\widetilde{H}^\ast(X;R)\to \widetilde{H}^\ast(X-\{x_0\};R)$ is not injective, where $i:X-\{x_0\}\hookrightarrow X$ is the inclusion map.} Then $sec\hspace{.1mm}(\pi_{2,1}^X)=2$. In particular, $X$ has the FPP. \end{proposition} \begin{proof} From $MR(\pi_{2,1}^X,x_0)=0$, there exist a continuous map $\varphi:F(X,2)\to X$ such that $\varphi^{-1}(x_0)=\emptyset$ and $\varphi\simeq\pi_{2,1}^X$. We have the \blue{homotopy} commutative diagram \begin{eqnarray*} \xymatrix{ F(X,2) \ar[r]^{\pi_{2,1}^X}\ar[d]_{\varphi} & X\\ X-\{x_0\}.\ar[ur]^{i} } \end{eqnarray*} The fact $\pi_{2,1}^X\simeq i\circ\varphi$ implies $\varphi^\ast\circ i^\ast=(\pi_{2,1}^X)^\ast$. In particular, $(\pi_{2,1}^X)^\ast(\alpha)=\varphi^\ast\circ i^\ast(\alpha)=0$. Therefore, there exist $\alpha\in \widetilde{H}^\ast(X;R)$ with $\alpha\neq 0$ and $(\pi_{2,1}^X)^\ast(\alpha)=0\in \widetilde{H}^\ast(F(X,2);R)$, then $sec\hspace{.1mm}(\pi_{2,1}^X)=2$. \end{proof} \begin{example} For $\pi_{2,1}^{S^2\vee S^1}:F(S^2\vee S^1,2)\to S^2\vee S^1$, we have $MR[\pi_{2,1}^{S^2\vee S^1},x_0]\geq 1$ for any $x_0\in S^2\vee S^1$. Indeed, we \colorado{show below that} $sec\hspace{.1mm}(\pi_{2,1}^{S^2\vee S^1})=1$. Also, there exist $\alpha\in \widetilde{H}^1(S^2\vee S^1;R)$ with $\alpha\neq 0$ and $i^\ast(\alpha)=0\in \widetilde{H}^1(S^2;R)$, \colorado{so Proposition~\ref{conditions} yields} $MR[\pi_{2,1}^{S^2\vee S^1},x_0]\neq 0$, \colorado{as asserted. Now, in order to construct a cross-section for $\pi_{2,1}^{S^2\vee S^1}$, it suffices to exhibit a selfmap $f\colon S^2\vee S^1\to S^2\vee S^1$ with no fixed points. Think of $S^2\vee S^1$ as $\left(S^2\times \{b_0\}\right)\cup \left(\{a_0\}\times S^1\right)$, where $a_0=(1,0,0)$ and $b_0=(1,0)$. Then the required map $f$ is} given by the formulae \begin{align*} \colorado{f(a,b_0)} & = \colorado{(a_0,\gamma(a_1)), \mbox{ for any $a=(a_1,a_2,a_3)\in S^2$, and}}\\ \colorado{f(a_0,b)} &=\colorado{(a_0,-b), \text{ for any } b\in S^1,} \end{align*} \colorado{where $\gamma\colon [-1,1]$ is a path in $S^1$ from $b_0$ to $-b_0$.} \end{example} We next relate our results to Farber's topological complexity, a homotopy invariant of $X$ introduced in \cite{farber2003topological}. Let $PX$ denote the space of all continuous paths $\gamma: [0,1] \to X$ in $X$ and $e_{0,1}: PX \to X \times X$ denote the map associating to any path $\gamma\in PX$ the pair of its initial and end points, i.e., $e_{0,1}(\gamma)=(\gamma(0),\gamma(1))$. Equip the path space $PX$ with the compact-open topology. \begin{definition}\cite{farber2003topological} The \textit{topological complexity} of a path-connected space $X$, denoted by TC$(X)$, is the least integer $m$ such that the cartesian product $X\times X$ can be covered with $m$ open subsets $U_i$ such that, for any $i = 1, 2, \ldots, m$, there exists a continuous local section $s_i:U_i \to PX$ of $e_{0,1}$, that is, $e_{0,1}\circ s_i = id$ over $U_i$. If no such $m$ exists, we set TC$(X)=\infty$. \end{definition} We have $\text{TC}(X)=1$ if and only if $X$ is contractible. The TC is a homotopy invariant, i.e., if $X\simeq Y$ then $\text{TC}(X)=\text{TC}(Y)$. Moreover, $\text{cat}(X)\leq \text{TC}(X)\leq 2\text{cat}(X)-1$ for any path-connected CW complex $X$. \begin{proposition}\label{nul-homotopy-implie-cat2} Let $X$ be a non-contractible path-connected \colorado{CW complex}. If $\pi_{2,1}^X\simeq x_0$ for some $x_0\in X$ then $X-\{x_0\}$ is contractible in $X$. Furthermore $\text{cat}(X)=2$, \colorado{$\text{TC}(X)\in\{2,3\}$} and \colorado{$sec(\pi_{2,1}^X)=secat(\pi_{2,1}^X)=2$}. \end{proposition} \begin{proof} Let $H:F(X,2)\times [0,1]\to X$ be a homotopy between $\pi_{2,1}^X$ and $x_0$. Set $G:(X-\{x_0\})\times [0,1]\to X$ given by the formula $G(x,t)=H((x,x_0),t)$. We have $G(x,0)=x$ and $G(x,1)=x_0$ for any $x\in X-\{x_0\}$. Thus $X-\{x_0\}$ is contractible in $X$. } \def\red{} \def\green{} \def\blue{{This obviously yields cat$(X)=2$, as well as $2\leq\text{TC}(X)\leq3$. \colorado{Furthermore, we have $2\geq sec(\pi_{2,1}^X)\geq secat(\pi_{2,1}^X)=\text{cat}(X)=2$.} } \end{proof} \begin{remark}\label{aaaaa} It is well known that $\text{cat}(X)=2$ corresponds to the case in which $X$ is a co-H-space. This is a large class of spaces } \def\red{} \def\green{} \def\blue{{including} all suspensions. In addition there are well-known examples of co-H-spaces that are not suspensions. In particular, a potential space satisfying the hypothesis in Proposition~\ref{nul-homotopy-implie-cat2} must be a co-H-space \colorado{of topological complexity 2 or 3 and would have to satisfy the fixed point property, but cannot be a closed smooth manifold. This last condition follows from the positive solution to the topological Poincaré conjecture.} \end{remark} \begin{remark}\cite{fadell1962configuration} Let $X$ be a topological space. The map $\pi_{k,1}^X:F(X,k)\longrightarrow X$ has a continuous section, i.e., $sec\hspace{.1mm}(\pi_{k,1}^X)=1$ if and only if there exist $k-1$ fixed point free continuous self-maps $f_2,\ldots,f_{k}:X\longrightarrow X$ which are non-coincident, that is, $f_i(x)\neq f_j(x)$ for any $i\neq j$ and $x\in X$. \end{remark} \begin{example} Let $G$ be a topological group with \blue{cardinality} $|G|\geq k$. Then $sec\hspace{.1mm}(\pi_{k,1}^G)=1$, because the map $s:G\to F(G,2),~g\mapsto (g,g_1g,\ldots,g_{k-1}g)$ is a cross section for $\pi_{k,1}^G$ (for some fixed $(g_1,\ldots,g_{k-1})\in F(G-\{e\},k-1)$). \end{example} \begin{example}\cite{fadell1962configuration} Let $M$ be a topological manifold without boundary and let $Q_m\subset M$ be a finite subset with $m$ elements. Then $sec\hspace{.1mm}(\pi_{k,1}^{M-Q_m})=1$ for any $m\geq 1$. \end{example} \begin{proposition}\label{secop-pi-k-r} \blue{Let $X$ be a Hausdorff space.} For any $k>r\geq 1$, we have $sec\hspace{.1mm}(\pi_{k,r}^X)\leq \binom{k}{r}$, where $\binom{k}{r}=\dfrac{k!}{r!(k-r)!}$, the standard binomial coefficient. \end{proposition} \begin{proof} Let $(p_1,\ldots,p_k)\in F(X,k)$ be a fixed $k-$tuple. Set $Q_k:=\{p_1,\ldots,p_k\}$ and for each $I_r\subseteq Q_k$ with $| I|=r$, let $Q_{I_r}:=Q_k-I_r=\{p_{j_1},\ldots,p_{j_{k-r}}\}$, where $j_1<\cdots<j_{k-r}$. Set $U_{I_r}:=F(M-Q_{I_r},r)$, and \blue{let} $s_{I_r}:U_{I_r}\to F(X,k)$ \blue{be} given by $s_{I_r}(x_1,\ldots,x_r):=(x_1,\ldots,x_r,p_{j_1},\ldots,p_{j_{k-r}})$. We note $U_{I_r}$ is open in $F(M,r)$ and each $s_I$ is a local section of $\pi_{k,r}^X$. Furthermore, $F(M,r)=\bigcup_{I\subseteq Q_k,~\mid I\mid=r}U_I$. Then, $sec\hspace{.1mm}(\pi_{k,r}^X)\leq \binom{k}{r}$. \end{proof} \begin{corollary}\label{general-case} Let $M$ be a connected topological manifold without boundary of dimension at least two. Then $sec\hspace{.1mm}(\pi_{k,r}^M)\leq \min\{\binom{k}{r},cat(F(M,r))\}$. \end{corollary} \begin{example}\label{sec-contractible} Let $M$ be a contractible topological manifold without boundary of dimension at least two. Then $sec\hspace{.1mm}(\pi_{k,1}^M)\leq \min\{k,cat(M)=1\}=1$. In particular, $M$ does not have the FPP. \end{example} \begin{remark}\label{diagram-spheres} \blue{From} the diagram \begin{eqnarray*} \xymatrix{ F(X,k) \ar[r]^{\pi_{k,k-1}^X}\ar[d]_{\pi_{k,1}^X} & F(X,k-1)\ar[dl]^{\pi_{k-1,1}^X} \\ X } \end{eqnarray*} \blue{it is clear} that, if $\pi_{k,1}^X$ admits a section, then $\pi_{k-1,1}^X$ also admits a section. Thus $sec\hspace{.1mm}(\pi_{k,1}^X)=1$ implies $sec\hspace{.1mm}(\pi_{r,1}^x)=1$ for any $r\leq k$. Furthermore, when $X=S^d$, \blue{the corresponding} diagram \begin{eqnarray*} \xymatrix{ F(S^d,k) \ar[r]^{\pi_{k,2}^{S^d}}\ar[d]_{\pi_{k,1}^{S^d}} & F(S^d,2)\ar[dl]^{\pi_{2,1}^{S^d}} \\ S^d } \end{eqnarray*} and \blue{the fact} that $\pi_{2,1}^{S^d}$ always admits a section \blue{imply that,} if $\pi_{k,2}^{S^d}$ admits a section, \blue{then}, so does $\pi_{k,1}^{S^d}$. The converse is also true, i.e., if $\pi_{k,1}^{S^d}$ admits a section, \blue{then} so does $\pi_{k,2}^{S^d}$ \cite{fadell1962configuration}. \end{remark} \begin{proposition}\label{sec-even-spheres} \red{If} $k>2$ and $d$ even, \red{then} $sec\hspace{.1mm}(\pi_{k,r}^{S^d})=cat(F(S^d,r))=2$, for \blue{$r\in\{1,2\}$.} \end{proposition} \begin{proof} First, we show that $sec\hspace{.1mm}(\pi_{k,r}^{S^d})\geq 2$ for any $d$ even, $k\geq 3$ and $r=1$ or $2$. \red{By} the above diagrams, it suffices to show that $sec\hspace{.1mm}(\pi_{3,1}^{S^d})\geq 2$ for $d$ even, that is, $\pi_{3,1}^{S^d}$ does not admit a cross section. If a cross section existed, it would generate a map $f:S^d\to S^d$ such that $f(x)\neq x$ and $f(x)\neq -x$ for any $x\in S^d$. \red{Indeed, suppose that $\pi_{3,1}^{S^d}$ admits a section, it implies that $\pi_{3,2}^{S^d}$ admits a section (see the last part of Remark \ref{diagram-spheres}), say $s:F(S^d,2)\to F(S^d,3)$. Recall that a section $\sigma$ to $\pi_{2,1}^{S^d}$ is given by the formulae $\sigma(x)=(x,-x)$ for any $x\in S^d$. Take $f=p_{3}\circ s\circ\sigma$, where $p_3$ is the projection on the third coordinate.} Since $f(x)\neq -x$ for every $x\in S^d$ it is easy to see that $f\simeq 1$ and $f$ has degree one and hence fixed points which is a contradiction (We recall that if $f:S^d\to S^d$ has not fixed points then $f$ is homotopic to the antipodal map and $f$ has degree $(-1)^{d+1}$). Thus, $\pi_{3,1}^{S^d}$ does not admit a cross section. From Proposition \ref{general-case}, $sec\hspace{.1mm}(\pi_{k,r}^{S^d})\leq \min\{\binom{k}{r},cat(F(S^d,r))=cat(S^d)=2\}$. Then $sec\hspace{.1mm}(\pi_{k,r}^{S^d})=2 \text{ for any } k\geq 3, r\in \{1,2\}$ ($d$ even). \end{proof} \begin{proposition}\label{prop-prop} \begin{enumerate} \item \red{If $L$ is a deformation retract of $X$}, then $\red{secat}(\pi_{k,1}^L)\geq \red{secat}(\pi_{k,1}^X)$. \item \cite{fadell1962configuration} If $M$ is \red{a smooth manifold} and admits a non-vanishing vector field, then $sec\hspace{.1mm}(\pi_{k,1}^M)=1$ for every $k$. \end{enumerate} \end{proposition} \begin{proof} $(1)$ Let $r:X\to L$ be a deformation retraction, i.e., $r\circ i=1_L$ and $i\circ r\simeq 1_X$, where $i:L\to X$ is the inclusion map. We have the following commutative diagram \begin{eqnarray*} \xymatrix{ F(L,k) \ar[r]^{i^k}\ar[d]_{\pi_{k,1}^L} & F(X,k)\ar[d]^{\pi_{k,1}^X} \\ L \ar[r]^{i} & X } \end{eqnarray*} Suppose $U\subset L$ is an open set of $L$ with \red{homotopy} local section $s:U\to F(L,k)$ of $\pi_{k,1}^L$. Set $V=r^{-1}(U)\subset X$ and consider $\sigma:V\to F(X,k)$ given by $\sigma=i^k\circ s\circ r$. \[ \xymatrix{ r^{-1}(U) \ar[r]^{r}\ar@{->}@/_20pt/[rrr]_{\sigma} & U \ar[r]^{s} & F(L,k) \ar[r]^{i^k} & F(X,k)} \] We have that $\sigma$ is a \red{homotopy} local section of $\pi_{k,1}^X$. Therefore, $\red{secat}(\pi_{k,1}^L)\geq \red{secat}(\pi_{k,1}^X)$. \end{proof} \red{From (\cite{fadell1962configuration}, Theorem $5$-$(b)$) we have if $L\subset X$ is a retract and $\pi_{k,1}^L$ admits a cross-section then $\pi_{k,1}^X$ admits a cross-section. The statement from Proposition \ref{prop-prop} does not hold when $L$ is a retract. For example, $X=S^d$ (with $d\geq 2$ even) and $L=S^d_{-}=\{(x_1,\ldots,x_{d+1})\in S^d:~~x_{d+1}\leq 0\}$. Note that $S^d_{-}$ is a retraction of $S^d$, the retraction map is given by $r:S^d\to S^d_{-}$, $r(x)=x$ if $x\in S^d_{-}$ and $r(x)=(x_1,\ldots,x_d,-x_{d+1})$ if $x_{d+1}\geq 0$. We have $S^d_{-}$ is contractible, indeed it is homeomorphic to the $d-$dimensional closed unit disc $\mathbb{D}^d$ and thus $secat(\pi_{k,1}^L)=1$. Here we consider $k>2$ and thus $secat(\pi_{k,1}^X)=2$ (see Proposition \ref{sec-even-spheres} and Remark \ref{secat-sec}).} \begin{corollary}\cite{fadell1962configuration} If $M$ is compact and the first Betti number of $M$ does not vanish, then $sec\hspace{.1mm}(\pi_{k,1}^M)=1$ for every $k$. \end{corollary} \begin{corollary}\cite{fadell1962configuration}\label{sec-odd-manifolds} If $M$ is an odd-dimensional differentiable manifold, \blue{then} $sec\hspace{.1mm}(\pi_{k,1}^M)=1$ for every $k$. \end{corollary} \begin{corollary}\label{sec-odd-spheres} \red{If} $k>2$ and $d$ odd, \red{then} $sec\hspace{.1mm}(\pi_{k,r}:F(S^d,k)\to F(S^d,r))=1$ for \blue{$r\in\{1,2\}$.} \end{corollary} \begin{proof} \red{\blue{This} follows from Corollary \ref{sec-odd-manifolds} and the last part of Remark \ref{diagram-spheres}.} \end{proof} \section{Topological complexity of a map}\label{tc-map} Recall that $PE$ denotes the space of all continuous paths $\gamma: [0,1] \longrightarrow E$ in $E$ and $e_{0,1}: PE \longrightarrow E \times E$ denotes the map associating to any path $\gamma\in PE$ the pair of its initial and end points $\pi(\gamma)=(\gamma(0),\gamma(1))$. Equip the path space $PE$ with the compact-open topology. Let $p:E\to B$ be a \colorado{continuous map}, and let $e_p:PE\to E\times B,~e_p=(1\times p)\circ e_{0,1}$. \begin{definition} The \textit{topological complexity} of the map $p$, denoted by TC$(p)$, is the sectional number $sec\hspace{.1mm}(e_p)$ of the map $e_p$, that is, the least integer $m$ such that the cartesian product $E\times B$ can be covered with $m$ open subsets $U_i$ such that for any $i = 1, 2, \ldots , m$ there exists a continuous local section $s_i : U_i \longrightarrow PE$ of $e_{P}$, that is, $e_{P}\circ s_i = id$ over $U_i$. If no such $m$ exists we set TC$(p)=\infty$. \end{definition} We use a definition of topological complexity which generally is not the same that given in \cite{pavesic2019}. However, under certain conditions, these two definitions coincides (see \cite{pavesic2019}). The proof of the following statement proceeds by analogy with \cite{pavesic2019}. \begin{proposition} For a map $p:E\to B$, we have $\text{TC}(p)\geq\max\{ \text{cat}(B),sec\hspace{.1mm}(p)\}$. \end{proposition} \begin{proof} Let $U\subset E\times B$ be an open subset and $s:U\to PE$ be a partial section of $e_p$. Fix $x_0\in E$ and consider the inclusion $i_0:B\to E\times B$, given as $i_0(b)=(x_0,b)$. Set $V=i_0^{-1}(U)\subset B$, it is an open subset of $B$. Consider the map $H:V\times [0,1]\to B$ given by $H(b,t)=p(s(x_0,b)(t))$. It is easy to check that $H$ is a null-homotopy. We conclude that $\text{TC}(p)\geq cat(B)$. On the other hand, consider the map $\sigma:V\to E$ defined by $\sigma(b)=s(x_0,b)(1)$. One can easily see that $\sigma$ is a partial section over $V$ to $p$. Therefore, $\text{TC}(p)\geq sec\hspace{.1mm}(p)$. \end{proof} The proof of the following statement proceeds by analogy with \cite{pavesic2019}. \begin{proposition}\label{section-ineq} Consider the diagram of maps $E^\prime\stackrel{p^\prime}{\to} E\stackrel{p}{\to} B\stackrel{p^{\prime\prime}}{\to}B^\prime$. If $p$ admits a section, then \begin{itemize} \item[$a$)] $\text{TC}(p^{\prime\prime})\leq \text{TC}(p^{\prime\prime} p)$. \item[$b$)] $\text{TC}(p p^\prime)\leq \text{TC}(p^\prime)$. \end{itemize} In particular, $\text{TC}(B)\leq\text{TC}(p)\leq \text{TC}(E)$. \end{proposition} \begin{proof} Let $s:B\to E$ be a section to $p$. $a)$ Suppose $\alpha_{p^{\prime\prime} p}:U\to PE$ is a partial section of $e_{p^{\prime\prime} p}$ over $U\subset E\times B^\prime$. Set $V:=(s\times 1_{B^\prime})^{-1}(U)\subset B\times B^\prime$. Then we can define the continuous map $\alpha_{p^{\prime\prime}}:V\to PB$ by \[\alpha_{p^{\prime\prime}}(b,b^\prime)(t):=\begin{cases} b, & \hbox{for $0\leq t\leq \frac{1}{2}$;} \\ p(\alpha_{p^{\prime\prime} p}(s(b),b^\prime)(2t-1)), & \hbox{for $\frac{1}{2}\leq t\leq 1$.} \end{cases}\] Since $\alpha_{p^{\prime\prime}}$ is a partial section of $e_{p^{\prime\prime}}$ over $V$, we conclude that $\text{TC}(p^{\prime\prime})\leq \text{TC}(p^{\prime\prime} p)$. $b)$ Let $\alpha_{p^\prime}:U\to PE^\prime$ be a partial section to $e_{p^\prime}:PE^\prime\to E^\prime\times E$ over $U\subset E^\prime\times E$. Set $V:=(1_{E^\prime}\times s)^{-1}(U)\subset E^\prime\times B$ and define the continuous map $\alpha_{pp^\prime}:V\to PE^\prime$ given by $\alpha_{pp^\prime}(e^\prime,b):=\alpha_{p^\prime}(e^\prime,s(b))$. It follows that $\alpha_{pp^\prime}$ is a partial section of $e_{pp^\prime}$ over $V$. This implies $\text{TC}(p p^\prime)\leq \text{TC}(p^\prime)$. \end{proof} \begin{theorem}\label{tc-implies-fpp} Let $X$ be a Hausdorff space. \begin{enumerate} \item If $X$ has the FPP, then $\text{TC}(\pi_{k,1}^X)\geq \max\{ \text{cat}(X),2\}$ for any $k\geq 2$. \item If $\text{TC}(\pi_{2,1}^X)<\text{TC}(X)$ or $\text{TC}(\pi_{2,1}^X)> \text{TC}(F(X,2))$, then $sec\hspace{.1mm}(\pi_{2,1}^X)=2$. In particular, $X$ has the FPP. \item If $X$ is a non-contractible space which does not have the FPP, then the configuration space $F(X,2)$ is not contractible. \end{enumerate} \end{theorem} \begin{proof} } \def\red{} \def\green{} \def\blue{{$(1)$:} We have $\text{TC}(\pi_{k,1})\geq sec\hspace{.1mm}(\pi_{k,1})\geq 2$. We recall that, $sec\hspace{.1mm}(\pi_{2,1})=2$ implies $sec\hspace{.1mm}(\pi_{k,1})\geq 2$, for any $k\geq 2$. } \def\red{} \def\green{} \def\blue{{$(2)$:} This follows from Proposition \ref{section-ineq}. } \def\red{} \def\green{} \def\blue{{$(3)$:} By Proposition \ref{section-ineq}, we have $1<\text{TC}(X)\leq \text{TC}(\pi_{2,1})\leq \text{TC}(F(X,2))$ and thus $F(X,2)$ is not contractible. \end{proof} } \def\red{} \def\green{} \def\blue{{Item (3) in Theorem \ref{tc-implies-fpp}} gives a partial generalization of the work } \def\red{} \def\green{} \def\blue{{in~\cite{zapata2017non}.} \begin{example} We know that the unit disc $D^m$ has the FPP. Then $\text{TC}(\pi_{k,1}^{D^m})\geq 2$, for any $k\geq 2$. \end{example} The following Lemma generalizes the statement given in (\cite{pavesic2019}, pg. 19). \begin{lemma}\label{general-pullback} If $p:E\to B$ is a fibration and $p^\prime:B\to B^\prime$ is a continuous map, then the following diagram is a pullback \begin{eqnarray*} \xymatrix{ PE \ar[r]^{\,\,p_{\#}} \ar[d]_{e_{p^\prime p}} & PB \ar[d]^{e_{p^\prime}} & \\ E\times B^\prime \ar[r]_{\,\, p\times 1_{B^\prime}} & B\times B^\prime &} \end{eqnarray*} \end{lemma} \begin{proof} For any $\beta:X\to PB$ and any $\alpha:X\to E\times B^\prime$ satisfying $e_{p^\prime}\circ\beta=(p\times 1_{B^\prime})\circ\alpha$, we will check that there exists $H:X\to PE$ such that $e_{p^\prime\circ p}\circ H=\alpha$ and $p_{\#}\circ H=\beta$. \begin{eqnarray*} \xymatrix{ X \ar@/^10pt/[drr]^{\,\,\beta} \ar@/_10pt/[ddr]_{\alpha} \ar@{-->}[dr]_{H} & & &\\ & PE \ar[r]^{\,\,p_{\#}} \ar[d]^{e_{p^\prime\circ p}} & PB \ar[d]^{e_{p^\prime}} & \\ & E\times B^\prime \ar[r]_{\quad p\times 1_{B^\prime}\quad} & B\times B^{\prime} &} \end{eqnarray*} Indeed, note that we have the following commutative diagram: \begin{eqnarray*} \xymatrix{ X \ar[r]^{\,\,p_{1}\circ\alpha} \ar[d]_{i_0} & E \ar[d]^{p} \\ X\times I \ar[r]_{\,\,\beta} & B} \end{eqnarray*} where $p_1$ is the projection onto the first coordinate. Because $p$ is a fibration, there exists $H:X\times I\to E$ satisfying $H\circ i_0=p_1\circ\alpha$ and $p\circ H=\beta$, thus we does. \end{proof} The following statement was proved in \cite{pavesic2019}; we give an \blue{elementary} proof in our context. \begin{proposition} If $p:E\to B$ is a fibration, then $\text{TC}(p^\prime p)\leq \text{TC}(p^\prime)$ for any $p^\prime:B\to B^\prime$. In particular, $\text{TC}(p)\leq \text{TC}(B)$. \end{proposition} \begin{proof} Since $p:E\to B$ is a fibration, the following diagram is a pullback (see Lemma \ref{general-pullback}) \begin{eqnarray*} \xymatrix{ PE \ar[r]^{\,\,p_{\#}} \ar[d]_{e_{p^\prime p}} & PB \ar[d]^{e_{p^\prime}} & \\ E\times B^\prime \ar[r]_{\,\, p\times 1_{B^\prime}} & B\times B^\prime &} \end{eqnarray*} This implies $\text{TC}(p^\prime p)=sec\hspace{.1mm}(e_{p^\prime p})\leq sec\hspace{.1mm}(e_{p^\prime})= \text{TC}(p^\prime)$. \end{proof} \begin{corollary} If $p:E\to B$ is a fibration that admits a section, then $\text{TC}(p)=\text{TC}(B)$. In particular, $\text{TC}(p)=1$ if and only if $B$ is contractible. \end{corollary} \section{The $(k,r)$ robot motion planning problem}\label{kr-robot} In this section we use the results above within a particular problem in robotics. Recall that, in general terms, the \textit{configuration space} or \textit{state space} of a system $\mathcal{S}$ is defined as the space of all possible states of $\mathcal{S}$ (see \cite{latombe2012robot} or \cite{lavalle2006planning}). Investigation of the problem of simultaneous collision-free motion planning for a multi-robot system consisting of $k$ distinguishable robots, each with state space $X$, leads us to study the ordered configuration space $F(X,k)$ of $k$ distinct points on $X$. \red{Recall the definition of \blue{the} ordered configuration \blue{space}} $F(X,k)$ in Subsection~\ref{secconfespa}. Note that the \red{$i$-th} coordinate of a point $(x_1,\ldots,x_n)\in F(X,k)$ represents the configuration of the \red{$i$-th} moving object, so that the condition $x_i\neq x_j$ reflects the collision-free requirement. \textit{The $(k,r)$ robot motion planning problem} consists in controlling simultaneously these $k$ robots without collisions, where one is interested in the initial positions of the $k$ robots and \textit{only interested in the final position of the first $r$ robots ($k\geq r$)} (see Figure \ref{fig1}). \begin{figure} \caption{The $(2,1)$ robot motion planning problem: we need to move Robots $1$ and $2$, simultaneously and avoiding collisions, from the initial positions $(a_1,a_2)$ to a final position $b_1$ of Robot $1$. We are only interested in the final position of the first robot.} \label{fig1} \end{figure} \textit{An algorithm} for the $(k,r)$ robot motion planning problem is a function which assigns to any pair of configurations $(A,B)\in F(X,k)\times F(X,r)$ consisting of an initial state $A=(a_1,\ldots,a_k)\in F(X,k)$ and a desired state $B=(b_1,\ldots,b_r)\in F(X,r)$, a continuous motion of the system starting at the initial state $A$ and ending at the desired state $B$ (see Figure \ref{fig2}). \begin{figure} \caption{An algorithm for the $(2,1)$ robot motion planning problem} \label{fig2} \end{figure} The central problem of modern robotics, \textit{the motion planning problem}, consists of finding a motion planning algorithm. We note that an algorithm to the $(k,r)$ robot motion planning problem is a (not necessarily continuous) section $s:F(X,k)\times F(X,r)\to PF(X,k)$ of the map $$e_{\pi_{k,r}^X}:PF(X,k)\to F(X,k)\times F(X,r),~e_{\pi_{k,r}^X}(\alpha)=(\alpha(0),\pi_{k,r}^X\alpha(1)),$$ where $\pi_{k,r}^X:F(X,k)\to F(X,r)$ is the projection of the first $r$ coordinates. A motion planning algorithm $s$ is called \textit{continuous} if and only if $s$ is continuous. Absence of continuity will result in instability of the behavior of the motion planning. In general, there is not a global continuous motion planning algorithm, and only local continuous motion plans may be found. This fact gives, in a natural way, the use of the numerical invariant TC$(\pi_{k,r}^X)$. Recall that TC$(\pi_{k,r}^X)$ is the minimal number of \textit{continuous} local motion plans to $e_{\pi_{k,r}^X}$ (i.e., continuous local sections for $e_{\pi_{k,r}^X}$), which are needed to construct an algorithm for autonomous motion planning of the $(k,r)$ robot motion planning problem. Any motion planning algorithm $s:=\{s_i:U_i\to PE\}_{i=1}^{n}$ is called \textit{optimal} if $n=\text{TC}(\pi_{k,r}^X)$. \begin{theorem} Let $M$ be a connected topological manifold without boundary of dimension at least $2$, and let $\pi_{k,r}^X:F(M,k)\to F(M,r)$ be the Fadell-Neuwirth fibration. \begin{enumerate} \item If $M$ does not have the FPP, then $\text{TC}(\pi_{2,1}^M)=\text{TC}(M).$ Hence the complexity \red{of} the $(2,1)$ robot motion planning problem is the same \red{as} the complexity \red{of} the manifold $M$. More general, if $sec\hspace{.1mm}(\pi_{k,r}^M)=1$, then $\text{TC}(\pi_{k,r}^M)=\text{TC}(F(M,r)).$ \item If $M$ has the FPP, then $\max\{2,\text{cat}(M)\}\leq \text{TC}(\pi_{k,1}^M)\leq\text{TC}(M)$, for any $k\geq 2$. In particular, $M$ is not contractible. \end{enumerate} \end{theorem} \begin{example} We recall that the $n$-dimensional sphere $S^n$ does not have the FPP. Then, $$\text{TC}(\pi_{2,1}^{S^n})=\text{TC}(S^n)=\begin{cases} 2, & \hbox{for $n$ odd;} \\ 3, & \hbox{for $n$ even.} \end{cases} $$ Furthermore, we have that any contractible topological manifold $M$ without boundary does not have the FPP. Hence, $\text{TC}(\pi_{2,1}^M)=\text{TC}(M)=1$. \end{example} \begin{example} \begin{itemize} \item The odd-dimensional projective spaces $\mathbb{RP}^m$ \red{do} not have the FPP, then $\text{TC}(\pi_{2,1}^{\mathbb{RP}^m})=\text{TC}(\mathbb{RP}^m)$. By \cite{farber2003topologicalproject}, the topological complexity $\text{TC}(\mathbb{RP}^m)$ for any $m\neq 1,3,7$, coincides with the smallest integer $k$ such that the projective space $\mathbb{RP}^m$ admits an immersion into $\mathbb{R}^{k-1}$. \item It is known that any closed surface $\Sigma$ except the projective plane $\Sigma\neq\mathbb{RP}^2$, does not have the FPP. Thus, $\text{TC}(\pi_{2,1}^{\Sigma})=\text{TC}(\Sigma).$ \item We have that the projective plane $\mathbb{RP}^2$ has the FPP. Furthermore, it is well known $\text{cat}(\mathbb{RP}^2)=3$ and $\text{TC}(\mathbb{RP}^2)=4$ \cite{farber2003topologicalproject}. Then, $3=\text{cat}(\mathbb{RP}^2)\leq\text{TC}(\pi_{k,1}^{\mathbb{RP}^2})\leq \text{TC}(\mathbb{RP}^2)=4$, for $k\geq 2$. \item For any connected compact Lie group, the Fadell-Neuwirth fibration $$\pi_{k,k-1}^{G\times \mathbb{R}^m}:F(G\times \mathbb{R}^m,k)\to F(G\times \mathbb{R}^m,k-1)$$ admits a continuous section (for $m\geq 2$). Then $\text{TC}(\pi_{k,k-1}^{G\times \mathbb{R}^m})=\text{TC}(F(G\times \mathbb{R}^m,k-1))$. By \cite{zapata2019cat}, the topological complexity $\text{TC}(F(G\times \mathbb{R}^m,2))=2\text{TC}(G)$. Hence, $\text{TC}(\pi_{3,2}^{G\times \mathbb{R}^m})=2\text{TC}(G)=2\text{cat}(G)$. \item Any connected Lie group has not the FPP and $\text{cat}(G)=\text{TC}(G)$. Then, $\text{TC}(\pi_{2,1}^G)=\text{TC}(G)=\text{cat}(G)$. In general, $\text{TC}(\pi_{k,1}^G)=\text{TC}(G)=\text{cat}(G)$ for any $k\geq 2$. \end{itemize} \end{example} \begin{example} \begin{itemize} \item We have $sec\hspace{.1mm}(\pi_{k,r}^{S^d})=cat(F(S^d,r))=2$, for $k\geq 3$, $d$ even, and $r=1,2$. Then $2=sec\hspace{.1mm}(\pi_{k,r}^{S^d})\leq\text{TC}(\pi_{k,r}^{S^d})\leq \text{TC}(F(S^d,r))=\text{TC}(S^d)=3$. \item For any $k\geq 2$ and $d$ odd, and $r=1,2$. We have $sec\hspace{.1mm}(\pi_{k,r}^{S^d})=1.$ Hence, $\text{TC}(\pi_{k,r}^{S^d})=\text{TC}(F(S^d,r))=\text{TC}(S^d)=2$. \end{itemize} \end{example} \begin{proposition}\cite{pavesic2019} Let $p:E\to B$ be a fibration between ANR spaces. Then \[\text{cat}(B)\leq\text{TC}(p)\leq\min\{ \text{cat}(E)+\text{cat}(E)sec\hspace{.1mm}(p)-1, \text{TC}(B),\text{cat}(E\times B)\}.\] In particular, $\text{TC}(p)=1$ if and only if $B$ is contractible. \end{proposition} \begin{theorem} Let $M$ be a connected topological manifold without boundary of dimension at least $2$. If $M$ has the FPP, then $$\max\{2,\text{cat}(M)\}\leq\text{TC}(\pi_{2,1}^M)\leq \min\{ 3\text{cat}(F(M,2))-1, \text{TC}(M),\text{cat}(F(M,2)\times M)\}.$$ \end{theorem} \end{document}

\begin{document} \begin{abstract} We answer two questions about the topology of end spaces of infinite type surfaces and the action of the mapping class group that have appeared in the literature. First, we give examples of infinite type surfaces with end spaces that are not self-similar, but a unique maximal type of end, either a singleton or Cantor set. Secondly, we use an argument of Tsankov to show that the ``local complexity" relation $\preceq$ on end types gives an equivalence relation that agrees with the notion of being locally homeomorphic. \end{abstract} \title{Two results on end spaces of infinite type surfaces} \section{Introduction} The paper \cite{largescale} introduced the notion of {\em self-similar} end spaces for infinite type surfaces, and proved that a self-similar end space necessarily contains a unique {\em maximal} type of end (with respect to the partial order defined below), with the set of ends of this type either a singleton or a Cantor set. This notion has turned out to be a useful one and it has appeared many times in the literature, see for instance \cite{BV, GRV, HQR, MT}. A partial converse to this statement was proved in \cite[Prop. 4.8]{largescale}. Namely, under the additional hypothesis that an infinite type surface $\Sigma$ contains no nondisplaceable subsurfaces, it was shown that the end space of $\Sigma$ is self-similar if and only if the set of maximal ends is either a singleton or a Cantor set of points of the same type. However, the necessity of this extra hypothesis (no nondisplaceable subsurfaces) was not discussed there, raising the question of whether it could be eliminated. This appeared as Question 1.4 in \cite{LL}, and Remark 6.2 in \cite{LV}. Here we answer the question, showing the strict converse (without extra assumptions) to \cite[Prop. 4.8]{largescale} is false: \begin{theorem} \label{thm:main} There exist examples of surfaces that have non self-similar end spaces with a unique maximal type of end and set of maximal ends homeomorphic to a singleton or to a Cantor set. \end{theorem} The partial order defined in \cite[Section 4]{largescale} is as follows. For $x,y \in E$, we say $y \preceq x$ if every neighborhood of $x$ contains a point locally homeomorphic to $y$, and we say $x$ and $y$ are of the same type if $x \preceq y$ and $y \preceq x$, so that $\preceq$ descends to a partial order on types. Informally, we think of $\preceq$ as describing the local complexity of an end. Of course, if $h(x) = y$ for some homeomorphism $h$ of $E$, then $x$ and $y$ are of the same type. Here we prove the converse to this statement, following an argument of T. Tsankov. This answers another question from \cite{largescale}. \begin{theorem} \label{thm:order} If $x \preceq y$ and $y \preceq x$, then there is a mapping class (equivalently, a homeomorphism) $h$ of $\Sigma$ taking $x$ to $y$. \end{theorem} The outline of the paper is as follows. First, we provide details on a local construction of non-comparable points (an idea sketched loosely in \cite{largescale}) as a tool towards the proof of Theorem \ref{thm:main}. We then prove Theorem \ref{thm:main} building examples first in the case where the maximal end is a singleton, followed by the Cantor set case. Finally, the proof of Theorem \ref{thm:order} is given in Section \ref{sec:order}. \section{Toolkit for Theorem \ref{thm:main}: Non-comparable points} \label{Sec:non-comparable} For a surface $\Sigma$, we denote the space of ends of $\Sigma$ by $E(\Sigma)$ or simply $E$. We define an equivalence relation on the end space by saying points $x,x' \in E$ are \emph{locally homeomorphic} if there exists some clopen neighborhood of $x$ in $E$ that is homeomorphic to some clopen neighborhood of $x'$ via a homeomorphism taking $x$ to $x'$. This is equivalent to saying that there is a homeomorphism of $\Sigma$ such the the induced map on the end space sends $x$ to $x'$. For an end $x$, we let $\Accu(x)$ denote the set of accumulation points of the set of all ends locally homeomorphic to $x$. We work with the relation $\preceq$ on points of $E$ as given above in the introduction. One may equivalently define $y \preceq x$ if $x \in \Accu(y)$. We say $x$ and $y$ are {\em of the same type} if $x \preceq y$ and $y \preceq x$, so that $\preceq$ descends to a partial order on types. We say $x$ is a {\em maximal type} if $x \preceq y$ implies $y \preceq x$ and denote the set of maximal points in $E$ by $\mathcal{M}(E)$. We say $x,y \in E$ are non-comparable if neither $x \preceq y$ nor $y \preceq x$ holds. See \cite{largescale} for more details and discussion. The first building block in our construction is a sequence of surfaces $D_n$ indexed by $n \in \mathbb{N}$, each with one boundary component, such that $D_n$ contains a unique maximal end $z_n$ and for all $i \neq j$ the ends $z_i$ and $z_j$ are non-comparable (the reader should picture $D_i$ and $D_j$ as disjoint subsurfaces of $\Sigma$). Note that a construction such as this is not possible when the surface is planar and has a countable number of ends. A classical result of Mazurkiewicz and Sierpinski \cite{MS} states that, for any surface with a countable set of ends, there exists a countable ordinal $\alpha$ such that the end space $E$ is homeomorphic to the ordinal $\omega^\alpha \cdot m + 1$ where $m$ is a positive integer. The assumption that $E$ has one maximal point implies that $m=1$. Now assume $D, D'$ are two genus zero surfaces with one boundary and a countable end space ($E$ and $E'$) such that that each end space has one maximal end ($x\in E$ and $x'\in E'$). Then their end spaces are respectively homeomorphic to $\omega^\alpha + 1$ and $\omega^{\alpha'}+ 1$ for some countable ordinals $\alpha$ and $\alpha'$. Now if $\alpha \leq \alpha$ then $x \preceq x'$, which means $x$ and $x'$ are comparable. We carry out the construction in both remaining cases, namely, when the set of ends is uncountable and the surface is planar (the proof easily generalizes to non-planar surfaces), and when the set of ends is countable and the surface is non-planar. \subsection*{Uncountable planar case} Let $D$ be a disc, let $C_n =Q_n \cup C \subset D$ be the union of a countable set $Q_n$ and a Cantor set $C$, with Cantor-Bendixson rank $n$ such that, for each derived set of $C_n$ that has isolated points, the accumulation set of the isolated points contains the Cantor set. For example, one may take the $C$ to be the standard middle-thirds Cantor set, and insert in each missing interval a set homeomorphic to $\omega^n+1$ to form $Q_n$. Now for each $C_n$, select a single point $z_n$ and let $C'_n$ be another Cantor set contained in $D$ so that $C_n \cap C'_n = \{z_n\}$. Puncturing $D$ along $C_n \cup C'_n$ gives a surface $D_n$ with one boundary component such that $z_n$ is the unique maximal end. By construction, $z_i$ and $z_j$ are non-comparable when $i \neq j$. \subsection*{Countable non-planar case} Let $D$ be a disk and let $\alpha$ and $\beta$ be two countable ordinals with $\beta < \alpha$. Let $E_\alpha$ be a subset of $D$ homeomorphic to $\omega^\alpha + 1$ and denote its (unique) maximal point by $z_{\alpha, \beta}$. Now, consider a closed subset $E_\beta \subset E_\alpha$ homeomorphic to $\omega^\beta+1$ where $z_{\alpha,\beta}$ is again the maximal point of $E_\beta$. For every isolated point $y$ of $E_\beta$ remove a disk around $y$ (keeping these disks pairwise disjoint) and glue back in a one-ended, infinite genus surface with one boundary component. We also puncture $D$ along the remaining points of $E_\alpha$ to obtain a surface $D_{\alpha, \beta}$. The point $z_{\alpha, \beta}$ is the unique maximal end of this surface. Moreover, for two pairs or countable ordinals $(\alpha, \beta)$ and $(\alpha', \beta')$ (satisfying $\beta < \alpha$ and $\beta' < \alpha'$) if $\alpha \geq \alpha'$ and $\beta < \beta'$ then $D_{\alpha, \beta}$ and $D_{\alpha', \beta'}$ are non-comparable. In fact, no end of $D_{\alpha, \beta}$ is of the same type as $z_{\alpha', \beta'}$ and vice versa. Hence we can, for example, fix $\alpha$ and vary $\beta$ to get a countable family of surfaces with one boundary where the maximal points are non-comparable. \subsection*{Uncountably many non-comparable points} It is also possible for a surface to contain uncountably many non-comparable points. For example, let $\Sigma$ be a sphere minus a Cantor set. Visualize $\Sigma$ as a union of pairs of pants. Enumerate the pairs of pants, remove a disk from each pair of pants, and glue back in a copy of $D_n$ to the $n$-th pair of pants. Call the resulting surface $\Sigma'$. Then all the ends of $\Sigma'$ coming from $\Sigma$ are non-comparable since small enough neighborhoods of any two such ends contain non-comparable points. \section{Proof of Theorem \ref{thm:main}} Now we construct the surface that will furnish the examples needed for Theorem \ref{thm:main}. We give the construction first for the case where $\mathcal{M}(E)$ is a singleton. We then modify the construction to produce examples where $\mathcal{M}(E)$ is a Cantor set. Start with a flute surface, meaning a sphere punctured along a sequence of points $p_1, p_2, ... $ accumulating at an end $p_\infty$. For each $i \neq \infty$, replace a neighborhood of the puncture $p_i$ with a Cantor tree $T_i$. We think of $T_i$ as a union of pants surfaces, indexed by finite binary strings, so that the pants indexed by a string $s_1... s_n$ has cuffs glued to the pants indexed by $s_1 ... s_{n-1}$, $s_1 ... s_n 0$, and $s_1 ... s_n 1$, and the first pair of pants $P_\emptyset$ is glued on where the original puncture was removed. Now for each $i$, we will replace a countable set of discs in $T_i$ with discs homeomorphic to copies of the previously constructed discs $D_n$ (from either construction in the previous section), according to the following recipe. For tree $T_i$, place copies of $D_1, D_2, \ldots D_i$ on the first pants surface, $P_\emptyset$, and place a copy of $D_k$ on each pants indexed by a word of length $k-i$. Thus, for each $k\geq i$ there are $2^{k-i}$ copies of $D_k$ on $T_i$. Call the resulting punctured surface $S$. An illustration is given in Figure \ref{fig:one_end}. \begin{figure*} \caption{Construction of the surface with unique maximal end.} \label{fig:one_end} \end{figure*} Note that all of the ends of each of the trees $T_i$ are pairwise locally homeomorphic. The end $p_\infty$ of our surface $S$ is the unique accumulation point of these tree ends, so it is the unique maximal end. We will now show that the end space of $S$ is not self-similar. Let $E_i$ denote the end space of the tree $T_i$. Consider the decomposition of the end space $E_1 \sqcup (E - E_1)$. Since $E_1$ does not contain a maximal end, to show the end space of the surface is not self-similar, it suffices to show that its complement contains no homeomorphic copy of $E_1$. Suppose for contradiction that we could find such. Note the the sets $U_i := \bigcup_{n=i}^\infty E_n$ form a neighborhood basis of $p_\infty$ in the end space. Since $p_\infty$ is the unique maximal end and $E_1$ is closed, any homeomorphic copy of $E_1$ must avoid some neighborhood of $p_\infty$ so is contained in a finite union $E_2 \cup E_3 \cup \ldots \cup E_N$. By construction, $E_1$ contains $2^{N}$ locally homeomorphic copies of the end $z_{N+1}$. But $E_2 \cup E_3 \cup \ldots \cup E_N$ contains $\sum_{i=1}^{N-1} 2^i < 2^N$ copies of $z_{N+1}$. A contradiction. Thus, $E_1$ cannot be mapped into its complement, so the end space is not self-similar. \subsection*{Cantor set case} A variation on the construction above can be used to produce a non self-similar surface with a unique maximal type and a Cantor set of maximal ends. First, following a similar procedure to the construction of the punctured trees $T_i$, for each $i \in \mathbb{N}$ we can build a Cantor tree $T'_i$ with a single boundary component that contains one copy of each of the discs $D_1$, $D_2$, ... $D_i$, and for each $k>i$ contains $2^{(2^{k-i})}$ copies of $D_k$, with each end of the tree locally homeomorphic. Now instead of starting with the flute, start with a Cantor tree constructed of pairs of pants indexed by binary strings, with the first pair of pants $P_\emptyset$ capped off by a disc on one of its boundary components. From each pair of pants indexed by a string of length $i$, remove a disc and glue in a copy of $T'_{i+1}$ to it along its boundary. In particular, $T'_1$ is glued to the first pair of pants indexed by the empty string. In the resulting surface, the ends of the original Cantor tree are precisely the maximal ends, forming a Cantor set of maximal ends of a single type. We claim again that this is not self-similar. To see this, let $E_1$ denote the end space of $T'_1$ and consider the decomposition of its end space into $E_1 \sqcup (E-E_1)$. Suppose for contradiction that $E-E_1$ contained a homeomorphic copy of $E_1$. As before, since $E_1$ and the set of maximal ends are both closed, the homeomorphic image of $E_1$ avoids some neighborhood of the maximal ends, so is contained in a union of end spaces of trees homeomorphic to $T'_i$ for a {\em bounded} set of indices $i$. We consider the maximal such index $N$, and again count copies of ends of type $z_{N+1}$. Without loss of generality, we may take $N\geq 4$. The set $E_1$ contains $2^{(2^{N})}$ copies of $z_{N+1}$. Since our surface is constructed using $2^{k-1}$ copies of each tree $T'_k$, the number of copies of $z_{N+1}$ in the union of all trees $T'_k$ for $2 \leq k \leq N$ is equal to \[ 2 \cdot 2^{(2^{N-1})} + 2^2 \cdot 2^{(2^{N-2})} + \ldots + 2^{N-1}\cdot 2^{(2^{N-N+1})} \] Set $j = 2^{N-1} + 1$. Then this sum is bounded above by \[ 2^j + 2^{j-1} + \ldots + 2^{j-N+2} < 2^{(2^N)} \] which gives the desired contradiction. \section{Proof of Theorem \ref{thm:order}} \label{sec:order} We now give the proof of Theorem \ref{thm:order}, following an argument of T. Tsankov. The key ingredient is the following zero-one law for Baire sets invariant under certain actions of Polish groups. \begin{theorem} \label{thm:zeroone}(See \cite[Theorem 8.46]{Kechris}.) Let $G$ be a group of homeomorphisms of a Baire space $X$, and assume that for all open $U, V \subset X$ there exists $g \in G$ with $gU \cap V \neq \emptyset$. Suppose that $A \subset X$ is a $G$-invariant set with the {\em Baire property}, meaning it differs from an open set (in the sense of symmetric difference) by a meager set. Then $A$ is either meager or has meager compliment in $X$. \end{theorem} \begin{proof}[Proof of Theorem \ref{thm:order}] Suppose $x \preceq y$ and $y \preceq x$. Let $H_x$ denote the ends that are locally homeomorphic to $x$ and $H_y$ the ends locally homeomorphic to $y$. If $H_x$ is finite, then it is easy to see that $H_x = H_y$. Otherwise, we have $\overline{H_x} = \overline{H_y}$, and every point of $\overline{H_x}$ is an accumulation point, therefore $\overline{H_x} \subset E$ is homeomorphic to a Cantor set. Denote this cantor set by $C$. This set $C$ is preserved by the action of $\mcg(\Sigma)$ on the end space. We claim the following: if $z \in C$ has a dense orbit in $C$ under $\mcg(\Sigma)$, then this orbit is comeager in $C$. Thus, there is only one such orbit, showing that $x$ is locally homeomorphic to $y$. That is, the claim proves the Theorem. To do this, let $G \subset \Homeo(C)$ denote the quotient of $\mcg(\Sigma)$ defined by restricting the action to $C$. Since $\mcg(\Sigma)$ is a Polish group and the kernel of the restriction map is closed, so is its quotient $G$. Furthermore, the action of $G$ on $C$ has the topological transitivity property that for every open $U$ and $U'$ in $C$, there exists some $g \in G$ with $gU \cap U' \neq \emptyset$, since both $U$ and $U'$ intersect $H_x$. Finally, any orbit of $G$ is a set with the Baire property, being the continuous image of a Polish space, so one may thus apply the topological zero-one law (Theorem \ref{thm:zeroone}) and conclude that a non-meager orbit is necessarily comeager in $C$. Thus, it suffices to show that any point $z$ with a dense orbit has a non-meager orbit. By a condition of Kechris-Rosendal \cite[Prop 3.2]{KR}, for this it suffices to show that for any open subgroup $V$ of $G$, the orbit $Vz$ is somewhere dense in $C$. Let $V$ be an open subgroup. Without loss of generality, we may take $V$ to be the subgroup consisting of homeomorphisms preserving a fixed decomposition of $C$ into finitely many clopen sets $C = X_1 \sqcup \ldots \sqcup X_n$ (possibly permuting the clopen sets) since such open subgroups form a basis for the topology of the homeomorphisms of the Cantor set. Again, without loss of generality, assume $z \in X_1$. We will show that $Vz$ is dense in $X_1$. Given some open set $U \subset X_1$, there exists $g \in G$ such that $gz \in U$. We assume also $gz \neq z$. Let $W$ be a clopen neighborhood of $z$ in $E$ small enough $W \cap C \subset X_1$, $gW \cap C \subset X_1$ and $gW \cap W = \emptyset$. Now define a homeomorphism $h$ of $E$ to agree with $g$ on $W$, agree with $g^{-1}$ on $gW$, and restrict to the identity elsewhere. Then the restriction of $h$ to $C$ (i.e. the image of $h$ in $G$) lies in $V$, and $h(z) = g(z) \in U$. This is what we needed to show. \end{proof} \begin{remark} This proof came about as a response to the question: {\em does a homeomorphism $h$ of $\{0,1\}^{\mathbb{Z}}$ that set-wise preserves each periodic orbit of the full shift necessarily preserve all orbits of the shift?} Tsankov \cite{Tsankov} answered this question in the negative, showing that, as in the proof above, there is only a single dense orbit under the group of periodic-orbit preserving maps of $\{0,1\}^{\mathbb{Z}}$. The argument above is a direct translation to this setting. \end{remark} \end{document}

\begin{document} \title[Kenig-Pipher condition and absolute continuity]{Failure to slide: a brief note on the interplay between the Kenig-Pipher condition and the absolute continuity of elliptic measures} \author{B. Poggi} \address{Bruno Giuseppe Poggi Cevallos \\ School of Mathematics \\ University of Minnesota \\ Minneapolis, MN 55455, USA} \email{poggi008@umn.edu} \maketitle \date{\today} \keywords{} \begin{abstract} In this note, we explore some consequences of the Modica-Mortola construction of a singular elliptic measure, as regards the link between the quantitative absolute continuity ($A_{\infty}$) of their approximations and the suitability of a well-known tool, the so-called Kenig-Pipher condition ($\operatorname{KP}$). The Kenig-Pipher condition is used to ascertain absolute continuity in the presence of some mild regularity of the coefficient matrix. We perform some modifications of the Modica-Mortola example to show the following two statements: (a) there are sequences of matrices for which both $\operatorname{KP}$ and the $A_{\infty}$ condition break down in the limit. (b) there are sequences of matrices for which $\operatorname{KP}$ breaks down but $A_{\infty}$ is preserved in the limit. \end{abstract} { \hypersetup{linkcolor=toc} \tableofcontents } \hypersetup{linkcolor=hyperc} \mathcal{S}ection{Introduction} In this note, we explore some consequences of the Modica-Mortola construction \cite{mm} of a singular elliptic measure, as regards the link between the quantitative absolute continuity of their approximations and the suitability of a well-known tool, the so-called Kenig-Pipher condition. The Modica-Mortola example, in conjunction with the Caffarelli-Fabes-Kenig example \cite{cfk}, were the first (and concurrent) constructions of an elliptic measure singular with respect to the surface measure of a smooth domain. The former uses an approximation procedure, lacunary sequences, and Riesz products, while the latter relies on the theory of quasi-conformal mappings. The interest in evidencing such cases had been aroused since Dahlberg \cite{dah1} proved a few years earlier that the elliptic measure for the Laplacian $L=-\Delta$ was absolutely continuous with respect to the surface measure of the unit ball. On the other hand, Caffarelli-Fabes-Mortola-Salsa \cite{cfms} had shown that all elliptic measures were doubling, precluding the existence of trivial examples. Ever since then, understanding the precise relationship between the coefficients of a divergence-form elliptic operator and the absolute continuity of the elliptic measure has been an ongoing and lively area of research, whose review we defer to any one of the many contemporaneous papers in the landscape. We do bring attention to one of the landmark results in the literature, \cite{kp3}, in which it is shown that quantifiable absolute continuity of the elliptic measure on the unit ball can be ascertained when the gradient of the coefficient matrix satisfies a Carleson-measure type condition. Their condition has come to be known as the \emph{Kenig-Pipher condition}, and its connection to the absolute continuity of elliptic measures has been seen to be remarkably robust in the past decade. Our aim here is modest. We will adapt the game played by Modica-Mortola to \begin{enumerate}[(a)] \item provide a sequence of $A_{\infty}$ elliptic measures for whom their Kenig-Pipher condition breaks down in the limit, but which nevertheless converge weakly to a singular elliptic measure, and \item provide a sequence of $A_{\infty}$ elliptic measures for whom their Kenig-Pipher condition breaks down in the limit, but which converge to an absolutely continuous elliptic measure. \end{enumerate} For case (a), we also argue that the placement in $A_{\infty}$ of the approximating measures degenerates. This shows that, in a sense (see Remark \mathbb{R}f{rm.top}), the Modica-Mortola matrix lies at the boundary of the set of matrices with $A_{\infty}$ elliptic measures. \mathcal{S}ection{The Modica-Mortola example: an exposition}\label{sec.mm} Let us first supply a brief exposition of the construction in \cite{mm}. In preparation, note that if $A, B$ are two positive functions, we say that $A\approx B$ if there exists a constant $C\geq1$ such that \[ \frac1CB\leq A\leq C B. \] If $C$ depends on some parameter $\beta$, then we make the dependence on $\beta$ explicit by using $A\approx_{\beta} B$ instead. We reserve the notation $dx$ for the Lebesgue measure on an interval. The idea of the construction is to jam the boundary with very thin layers consisting of a material with highly oscillatory periodic anisotropy. Let \[ A(x,y)=\begin{pmatrix}1&0\\0&\alpha(x,y)\end{pmatrix},\qquad A_j(x,y)=\begin{pmatrix}1&0\\0&\alpha_j(x,y)\end{pmatrix},~j\in\bb N,~ (x,y)\in\bb R^2, \] where \[ \alpha(x,y)=\begin{cases}\phi_1(x),\qquad\text{if }|y|\geq1/k_1,\\[2mm]\psi(k_{j+1}y)\phi_{j+1}(x)+(1-\psi(k_{j+1}y))\phi_j(x),\qquad\text{if }\frac1{k_{j+1}}\leq|y|<\frac1{k_j}, j=1,2,\ldots,\\[2mm]1,\qquad\text{if }y=0,\end{cases} \] \[ \alpha_j(x,y)=\begin{cases}\alpha(x,y),\qquad\text{if }|y|\geq1/k_j,\\[2mm]\phi_j(x)\qquad\text{if }|y|<1/k_j,\end{cases} \] \begin{equation}\label{eq.phij} \phi_j(x)=1+\frac1{4\pi\mathcal{S}qrt j}\mathbb{C}s(2\pi h_jx),\qquad j\in\bb N, \end{equation} and $\{h_j\}$, $\{k_j\}$ are suitable lacunary sequences (see, for instance, \cite{zyg}) of positive integers which can be chosen to satisfy that \begin{equation}\label{eq.lac} h_{j+1}\geq4h_j,\qquad k_{j+1}\geq2k_j, \qquad\text{and } h_j\geq jk_j\footnote{The condition $h_j\geq jk_j$ does not appear in \cite{mm}; however, it is clear by their inductive construction of the sequences in pages 16-17 that we may choose the sequences in this way, since we choose $h_{j+1}$ large based on already having chosen $k_{j+1}$. Note that asking this condition simplifies the proof of Proposition \mathbb{R}f{prop.break} below, but is probably not strictly required to achieve the result.}, \end{equation} and $\psi\in C_c^{\infty}(\bb R)$ is a smooth cut-off satisfying $\psi(t)=\psi(-t), 0\leq\psi(t)\leq1$ for all $t\in\bb R$, $\psi(t)=1$ if $|t|\leq1$, and $\psi(t)=0$ if $|t|\geq2$. Observe that $\alpha\in C^0(\bb R^2)\cap C^{\infty}(\{(x,y)\in\bb R^2:y\neq0\})$ and $\frac12\leq\alpha(x,y)\leq\frac32$ for every $(x,y)\in\bb R^2$. Note that $\alpha_j\in C^{\infty}(\bb R^2)$ for each $j\in\bb N$ and $\alpha_j$ converges pointwise uniformly in $\bb R^2$ to $\alpha$. Let $\Omega$ be an open, bounded subset of the upper-half plane $\bb R\times\bb R_+$, with smooth boundary and such that \[ [-10,20]\times\{0\}\mathcal{S}ubset\partial\Omega, \] and let $L_j:=-\operatorname{div }A_j\nabla$ and $L:=-\operatorname{div }A\nabla$ be operators on $\Omega$ with corresponding elliptic measures $\{\Omegaega_j^P\}_{P\in\Omega}$ and $\{\Omegaega^P\}_{P\in\Omega}$ on $\partial\Omega$. Henceforth, we fix an arbitrary $P\in\Omega$ and write $\Omegaega_j=\Omegaega_j^P$, $\Omegaega=\Omegaega^P$. We say that a measure $\mu$ is \emph{singular} with respect to a measure $\nu$ if $\mu$ is not absolutely continuous with respect to $\nu$. The main result in \cite{mm} is \begin{theorem}[A singular elliptic measure; \cite{mm}]\label{thm.main} The probability measure $\Omegaega$ on $\partial\Omega$ is singular with respect to the surface measure. \end{theorem} \noindent\emph{Proof.} In this situation, $\Omegaega_j^P$ converges weakly in the sense of measures to $\Omegaega^P$ on $\partial\Omega$ as $j\rightarrow\infty$ (cf. Lemma 1 in \cite{mm}). For each $j\in\bb N$, let $g_j=g_j(\cdot,P)$ be the \emph{Green function} for the Dirichlet problem for the operator $L_j$ on $\Omega$, with pole $P$. By the Green representation formula, for each $\chi\in C(\partial\Omega)$ we have the identity \begin{equation}\label{eq.greenformula2} \int_{\partial\Omega}\chi\,d\Omegaega_j=\int_{\partial\Omega}\chi (A_j\nabla g_j)\cdot\hat n\,d\mathcal{S}igma, \end{equation} where $\hat n$ is the unit normal vector on $\partial\Omega$ pointing inward and $\mathcal{S}igma$ is the surface measure on $\partial\Omega$, which is the $(n-1)$-dimensional Hausdorff measure on $\partial\Omega$ (up to a dimensional constant). Since $\Omega$ is a smooth bounded domain and each $A_j$ is smooth in $\bb R^2$, the classical Hopf lemma and the fact that $\alpha_j\approx 1$ imply that $(A\nabla g_j)\cdot\hat n\approx_j1$, so that $\Omegaega_j$ and $\mathcal{S}igma$ are mutually absolutely continuous. Let $\m K_j=\m K_j^P$ be the \emph{Poisson kernel} of $\Omegaega_j$ with pole $P$, which is the Radon-Nikodym derivative of $\Omegaega_j$ with respect to $\mathcal{S}igma$. We have concluded that \[ \m K_j\approx_j1,\qquad\text{on }\partial\Omega. \] Observe that by specializing (\mathbb{R}f{eq.greenformula2}) to the case when $\chi\in C_c([-1,1]\times\{0\})$, we procure the identity \begin{equation}\label{eq.greenformula1} \int_{\partial\Omega}\chi\,d\Omegaega_j=\int_{-1}^1\chi(x,0)\phi_j(x)\frac{\partial g_j}{\partial y}(x,0)\,dx, \end{equation} so that pointwise almost everywhere on $[-1,1]\times\{0\}$, we have the representation \[ \m K_j=\phi_j\frac{\partial g_j}{\partial y}. \] We will now sketch the fact that for each $j\geq2$, we may choose the lacunary sequences $\{h_j\}$, $\{k_j\}$ so that \begin{equation}\label{eq.likeriesz} \frac{\partial g_j}{\partial y}(x,0)\approx\prod_{i=1}^{j-1}\phi_i(x)=:\m R_{j-1}(x),\qquad x\in[-1,1], \end{equation} whence we summarily deduce that \begin{equation}\label{eq.likerieszk} \m K_j\approx\m R_j \end{equation} on $[-1,1]\times\{0\}$. We call $\m R_j$ a \emph{Riesz product} (see \cite{zyg}, Chapter V, Section 7). Let \[ M:=\min_{|x|\leq1}\frac{\partial g_1}{\partial y}(x,0) \] and observe that, by the Hopf lemma, $M>0$. The sequences $\{h_j\}, \{k_j\}$ are chosen so that the estimate \begin{equation}\label{eq.greendecay} \max_{|x|\leq1}\mathcal{B}ig|\frac{\partial g_{j+1}}{\partial y}(x,0)-\phi_j(x)\frac{\partial g_j}{\partial y}(x,0)\mathcal{B}ig|\leq M4^{-j-1} \end{equation} holds for all $j\in\bb N$. That this can be done is much of the program in \cite{mm}, and thus we leave the study of this technology to them. Consider for each $j\in\bb N$ the function \[ w_j(x):=\frac{\frac{\partial g_{j+1}}{\partial y}(x,0)}{\m R_j(x)},\qquad |x|\leq 1,~j\in\bb N, \] which can easily be rewritten as \[ w_j(x)=\frac{\partial g_1}{\partial y}(x,0)+\mathcal{S}um_{i=1}^j\frac{\frac{\partial g_{i+1}}{\partial y}(x,0)-\phi_i(x)\frac{\partial g_i}{\partial y}(x,0)}{\m R_i(x)}. \] From the above equality, (\mathbb{R}f{eq.greendecay}), and the fact that $\phi_j\geq\frac12$ for each $j\in\bb N$, it follows that $\{w_j\}$ is a Cauchy sequence in $C[-1,1]$, whence there exists $w\in C[-1,1]$ so that $w_j\rightarrow w$ in $C[-1,1]$, and moreover \[ \max_{|x|\leq1}|w_j(x)-w(x)|\leq M2^{-n-2},\qquad\text{for each }j\in\bb N, \] \[ w\geq\frac34M,\qquad\text{on }[-1,1]. \] These computations prove (\mathbb{R}f{eq.likeriesz}) and therefore (\mathbb{R}f{eq.likerieszk}). We now borrow from \cite{zyg} Chapter V, Section 7, Lemma 7.5, the wisdom that these \emph{Riesz products are the partial sums of a Fourier-Stieltjes series of a non-decreasing (non-constant) continuous function $F$ on $[-1,1]$, whose derivative is $0$ almost everywhere}. In particular, $\m R_j\,dx$ converges weakly in the sense of measures to a singular measure $dF$ on $[-1,1]$. Since we have (\mathbb{R}f{eq.likerieszk}) and the fact that $\Omegaega_j\rightarrow\Omegaega$ weakly in the sense of measures, it follows that $\Omegaega$ is singular, as desired. {$\mathcal{S}quare$} \mathcal{S}ection{The Modica-Mortola approximations, their absolute continuity, and the Kenig-Pipher condition} Let $\Omega\mathcal{S}ubset\bb R^2$ be as above. Given $X\in\Omega$, denote by $\delta(X)$ the distance from $X$ to $\partial\Omega$. For an elliptic matrix $\m A$ of bounded real measurable coefficients in $\Omega$, we say that $\m A$ satisfies the \emph{Kenig-Pipher condition} if the quantity \[ \n P(\m A):=\mathcal{S}up_{{\tiny\begin{matrix}q\in\partial\Omega\\0<r<\operatorname{diam}(\Omega)\end{matrix}}}\frac1r\int\!\!\!\int_{B(q,r)\cap\Omega}\mathcal{B}ig(\mathcal{S}up_{Y\in B(X,\frac{\delta(X)}2)}|\nabla\m A(Y)|^2\delta(Y)\mathcal{B}ig)\,dX \] is finite. It turns out that if the matrix $\m A$ satisfies the Kenig-Pipher condition, then the elliptic measure $\Omegaega_{\m A}$ associated to the operator $L=-\operatorname{div}\m A\nabla$ is absolutely continuous with respect to the surface measure. In fact, one can say more: in this case, the absolute continuity can be \emph{quantified} using the theory of Muckenhoupt $A_p$ weights (see \cite{ste} for definitions and the basic results). Thus, it can be shown that if $\m A$ satisfies the Kenig-Pipher condition, then $\Omegaega_{\m A}\in A_{\infty}(\mathcal{S}igma)$. We now briefly summarize some results regarding the Muckenhoupt $A_{\infty}$ class and Reverse H\"older classes which we shall later use. The characterization $\Omegaega_{\m A}\in A_{\infty}(\mathcal{S}igma)$ is equivalent to (and, therefore, may be defined as) the condition that $\Omegaega_{\m A}$ is absolutely continuous with respect to the surface measure $\mathcal{S}igma$ and such that the Poisson kernel $\m K_{\m A}$ lies in $RH_q$ for some $q>1$, where $RH_q$ is the space of non-negative weights satisfying a \emph{Reverse-H\"older inequality}: a non-negative weight $w$ on $\partial\Omega$ lies in $RH_q$ for $q>1$ if there exists a constant $C\geq1$ such that for all surface balls $\Delta=B\cap\partial\Omega$ ($B$ is an $n-$dimensional ball centered on $\partial\Omega$), the estimate \begin{equation}\label{eq.rhcond} \mathcal{B}ig(\frac1{\mathcal{S}igma(\Delta)}\int_{\Delta}w^q\,d\mathcal{S}igma\mathcal{B}ig)^{\frac1q}\leq C\frac1{\mathcal{S}igma(\Delta)}\int_{\Delta}w\,d\mathcal{S}igma \end{equation} holds. We let $|w|_{RH_q}$ be the infimum of the set of all possible constants $C$ such that (\mathbb{R}f{eq.rhcond}) holds. Moreover, we have the following direct characterization of $A_{\infty}$ weights\footnote{here, there is a slight abuse of notation, as we consider both $A_{\infty}$ for measures and for weights; these are essentially equivalent, however.}: a non-negative weight $w$ lies in $A_{\infty}$ if and only if the quantity \begin{equation} |w|_{A_{\infty}}:=\mathcal{S}up_{\Delta\mathcal{S}ubset\partial\Omega}\mathcal{B}ig\{\mathcal{B}ig(\frac1{\mathcal{S}igma(\Delta)}\int_{\Delta}w\,d\mathcal{S}igma\mathcal{B}ig)\exp\mathcal{B}ig(\frac1{\mathcal{S}igma(\Delta)}\int_{\Delta}\log w^{-1}\,d\mathcal{S}igma\mathcal{B}ig)\mathcal{B}ig\} \end{equation} is finite. We call $|w|_{A_\infty}$ the \emph{$A_{\infty}$ constant of w}. The limiting case of the Reverse H\"older classes is the space $RH_{L\log L}$ of weights which satisfy the reverse Jensen's inequality for the function $x\log x$: \[ \mathcal{V}ert w\mathcal{V}ert_{(L\log L,\frac{d\mathcal{S}igma|_{\Delta}}{\mathcal{S}igma(\Delta)})}\leq C\frac1{\mathcal{S}igma(\Delta)}\int_{\Delta}w\,d\mathcal{S}igma, \] and we call $|w|_{RH_{L\log L}}$ the infimum of the set of all possible $C$ such that the above inequality holds. We have that $A_{\infty}=RH_{L\log L}=\bigcup_{q>1}RH_q$. Using the notation of the previous section, note that for each fixed $j\in\bb N$, $A_j$ trivially satisfies the Kenig-Pipher condition. Indeed, since $A_j$ is smooth on $\bb R^2$, there exists a constant $C_j$ such that $|\nabla A_j|\leq C_j$ on $\overline{\Omega}$, whence \[ \n P(A_j)\lesssim_j\diam(\Omega)^2<\infty. \] This fact provides a second easy proof of the fact that for each $j\in\bb N$, $\Omegaega_j\in A_{\infty}(\mathcal{S}igma)$. The main calculations of this note follow. First, we check directly that the approximations $A_j$ of the Modica-Mortola example break the Kenig-Pipher condition ``in the limit''. \begin{proposition}[Breaking of the Kenig-Pipher condition on a sliding scale]\label{prop.break} There exists a sequence $\{A_j\}$ of diagonal elliptic matrices on $\Omega$ with smooth, bounded, real coefficients in $\Omega$ and uniformly continuous on $\overline\Omega$, and there exists a diagonal elliptic matrix $A$ on $\Omega$ with smooth, bounded, real coefficients in $\Omega$ and uniformly continuous on $\overline\Omega$, such that $A_j\rightarrow A$ pointwise uniformly on $\overline\Omega$, $\Omegaega_A$ is singular with respect to the surface measure, and \begin{equation}\label{eq.result} \n P(A_j)\gtrsim j. \end{equation} \end{proposition} \noindent\emph{Proof.} Let $\{A_j\}$, $A$ and all other variables be defined as in the previous section. Without loss of generality we may assume that $\Omega$ contains the square $[-1,2]\times[0,3]$. Fix $j\in\bb N$, and reckon the elementary estimates \begin{multline}\nonumber \n P(A_j)\geq\mathcal{S}up_{{\tiny\begin{matrix}x_0\in[0,1]\\0<r\leq\frac1{2k_j}\end{matrix}}}\frac1r\int_{x_0-r}^{x_0+r}\int_0^{\mathcal{S}qrt{r^2-(x-x_0)^2}}\mathcal{B}ig(\mathcal{S}up_{(x_2,y_2)\in B((x,y),\frac y2)}\mathcal{B}ig|\frac{\partial\alpha_j}{\partial x}(x_2,y_2)\mathcal{B}ig|^2y_2\mathcal{B}ig)\,dy\,dx\\ \geq\frac{h_j^2}{8j} \mathcal{S}up_{x_0\in[0,1]}\frac1{r_j}\int_{x_0-r_j}^{x_0+r_j}\int_0^{\mathcal{S}qrt{r_j^2-(x-x_0)^2}}y\mathcal{B}ig(\mathcal{S}up_{x_2\in(x-\frac y2,\, x+\frac y2)}|\mathcal{S}in(h_jx_2)|^2 \mathcal{B}ig)\,dy\,dx, \end{multline} where $r_j:=\frac1{2k_j}$. We may assume that both $h_j$ and $k_j$ are large, and $h_j\gg k_j$ (a fact afforded by virtue of the choice $h_j\geq jk_j$). Given $x_0\in[0,1]$, observe that \[ \mathcal{S}up_{x_2\in(x-\frac y2,\, x+\frac y2)}|\mathcal{S}in(h_jx_2)|^2 =1 \] for each $(x,y)$ in the set \begin{multline*} S:=\mathcal{B}ig\{(x,y):x\in(x_0-r_j,x_0-\tfrac{r_j}2)~,~y\in\big(\tfrac12\mathcal{S}qrt{r_j^2-(x-x_0)^2},\mathcal{S}qrt{r_j^2-(x-x_0)^2}\big)\mathcal{B}ig\}\\ \mathcal{S}ubset\big\{(x,y):x\in(x_0-r_j,x_0+r_j)~,~y\in\big(0,\mathcal{S}qrt{r_j^2-(x-x_0)^2}\big)\big\}, \end{multline*} because $\mathcal{S}in(h_j\cdot)$ is oscillating rapidly in an interval of length roughly $1/k_j$. Therefore, \begin{multline} \n P(A_j)\geq\frac{h_j^2}{8jr_j} \mathcal{S}up_{x_0\in[0,1]} \int_{x_0-r_j}^{x_0-\frac12r_j}\int_{\frac12\mathcal{S}qrt{r_j^2-(x-x_0)^2}}^{\mathcal{S}qrt{r_j^2-(x-x_0)^2}}\,y\,dy\,dx\\ \geq\frac{3h_j^2}{64jr_j} \mathcal{S}up_{x_0\in[0,1]} \int_{x_0-r_j}^{x_0-\frac12r_j}(r_j^2-(x-x_0)^2)\,dx \geq c\frac{h_j^2}{j}r_j^2\\ \geq c\frac1j\mathcal{B}ig(\frac{h_j}{k_j}\mathcal{B}ig)^2\geq cj. \end{multline} where $c\in(0,1)$ is a small fixed quantity. {$\mathcal{S}quare$} By using the method of proof above, it is clear that we may also directly show that $\n P(A)=+\infty$. By virtue of Theorem \mathbb{R}f{thm.main}, we may deduce heuristically that the $A_{\infty}$ constant of $\Omegaega_j^P$ must blow up as $j\rightarrow\infty$. We now present a rigorous description of this fact. \begin{proposition}[Degeneracy of the quantitative absolute continuity]\label{prop.deg} For any $q>1$, $|\m K_j|_{RH_q}$ goes to infinity as $j\rightarrow\infty$. Moreover, $|\m K_j|_{A_{\infty}}$ goes to infinity as $j\rightarrow\infty$. \end{proposition} \noindent\emph{Proof.} We show the first statement. Suppose otherwise, so that there exists $q>1$ and a constant $C_q$ such that for each $j\in\bb N$ and each surface ball $\Delta\mathcal{S}ubset\partial\Omega$, the estimate (\mathbb{R}f{eq.rhcond}) holds with $w\equiv\m K_j$ and $C\equiv C_q$. In particular, by setting $\Delta=[0,1]\times\{0\}$, we have that (\mathbb{R}f{eq.rhcond}) reduces to \[ \mathcal{V}ert\m K_j\mathcal{V}ert_{L^q[0,1]}\leq C_q\mathcal{V}ert\m K_j\mathcal{V}ert_{L^1[0,1]},\qquad j\in\bb N. \] Using (\mathbb{R}f{eq.likerieszk}), the above estimate implies that \begin{equation}\label{eq.compact} \mathcal{V}ert\m R_j\mathcal{V}ert_{L^q[0,1]}\lesssim_q\mathcal{V}ert\m R_j\mathcal{V}ert_{L^1[0,1]},\qquad j\in\bb N. \end{equation} In fact, we have that (see \cite{cfa} page 233) \begin{equation}\label{eq.l11} \mathcal{V}ert\m R_j\mathcal{V}ert_{L^1[0,1]}=1,\qquad j\in\bb N. \end{equation} Therefore, from (\mathbb{R}f{eq.compact}) we deduce that \begin{equation}\label{eq.bddq} \mathcal{V}ert\m R_j\mathcal{V}ert_{L^q[0,1]}\lesssim_q1, \end{equation} which in particular implies that the family $\{\m R_j\}$ is uniformly integrable on $[0,1]$. Recall that (\cite{zyg}, Chapter V, Section 7, Theorem 7.7) $\m R_j\rightarrow0$ pointwise a.e. on $[0,1]$. Then by (\mathbb{R}f{eq.bddq}) and the de la Vall\'ee Poussin criterion for equiintegrability (see \cite{bogachev} Volume I, Theorem 4.5.9), we must conclude by the Vitali Convergence Theorem that $\m R_j\rightarrow0$ in $L^1[0,1]$ as $j\rightarrow\infty$, but this stands in direct contradiction to (\mathbb{R}f{eq.l11}). The first desired statement follows. By the same technique as above, we can verify that $|\n K_j|_{RH_{L\log L}}\rightarrow\infty$ as $j\rightarrow\infty$, and this must imply (quantitatively \cite{br}, actually) that $|\n K_j|_{A_{\infty}}\rightarrow\infty$. {$\mathcal{S}quare$} Next, let us tweak some parameters in the Modica-Mortola construction to obtain \begin{proposition}[Degeneracy of the Kenig-Pipher condition while $A_{\infty}$ is preserved]\label{prop.kenigbreaksdown} There exists a sequence $\{A_j\}$ of diagonal elliptic matrices on $\Omega$ with smooth, bounded, real coefficients in $\Omega$ and uniformly continuous on $\overline\Omega$, and there exists a diagonal elliptic matrix $A$ on $\Omega$ with smooth, bounded, real coefficients in $\Omega$ and uniformly continuous on $\overline\Omega$, such that $A_j\rightarrow A$ pointwise uniformly on $\overline\Omega$, $\Omegaega_A$ is absolutely continuous with respect to the surface measure $\mathcal{S}igma$, and \begin{equation}\label{eq.result2} \n P(A_j)\gtrsim j,\qquad\n P(A)=+\infty. \end{equation} \end{proposition} \noindent\emph{Proof.} Consider the following modifications: First, we use the formula \[ \phi_j=1+\frac1{4\pi j}\mathbb{C}s(2\pi h_jx),\qquad j\in\bb N \] for $\phi_j$ instead of the formula (\mathbb{R}f{eq.phij}) in Section \mathbb{R}f{sec.mm}. Second, we ask that the lacunary sequences $\{h_j\}$, $\{k_j\}$ satisfy the additional stronger estimate \[ h_j\geq j^3 k_j,\qquad j\in\bb N. \] See the footnote to (\mathbb{R}f{eq.lac}). With these changes in mind and following the argument for the proof of Theorem \mathbb{R}f{thm.main}, we still conclude as before that the measures $\Omegaega_j$ converge weakly to a measure $\Omegaega$, and that the corresponding Riesz products $\m R_j(x)=\prod_{i=1}^j\phi_i$ form the partial sums of a Fourier-Stieltjes series for a non-decreasing continuous function $F$ on $[-1,1]$. Moreover, we may mimic the proof of Proposition \mathbb{R}f{prop.break} and easily deduce (\mathbb{R}f{eq.result2}) accordingly. On the other hand, in this situation, the amplitude coefficients $\frac1{2j}$ of the Riesz Products are such that the sum of their squares is finite. According to \cite{zyg} Chapter V, Section 6, Lemma 6.5, it follows that $\{\m R_j\}$ is a uniformly bounded sequence in $L^2[0,1]$ which converges pointwise a.e. on $[0,1]$ to $F'$. Per the Vitali Convergence Theorem, we must conclude that $\m R_j\rightarrow F'$ strongly in $L^1[0,1]$, whence the Fundamental Theorem of Calculus applies. Consequently, $dF=F'dx$ on $[0,1]$. Since $\m K_j\approx\m R_j$, it finally follows that $\Omegaega\ll dx$, and we do remark that $\Omegaega\in A_{\infty}$ holds via the Vitali Convergence Theorem and the Reverse H\"older inequality. {$\mathcal{S}quare$} \begin{remark}\label{rm.top} Let us reframe the above results as follows. Define $\bb M$ as the Fr\'echet space of $n\times n$ matrix functions in $C^0(\overline{\Omega})\cap C^{\infty}(\Omega)$ with the usual topology. Designate $\operatorname{KP}\mathcal{S}ubset\bb M$ as the subset of such matrices which satisfy the Kenig-Pipher condition, and \footnote{As is well-known, such matrices are regular for the Dirichlet problem; this is the impetus for our notation.} $\operatorname{D}\mathcal{S}ubset\bb M$ as the subset of such matrices for whom the associated elliptic measure lies in $A_{\infty}$. Let $\partial=\partial_{\bb M}$ be the boundary operator on $\bb M$. In this setting, note that the theorem of Kenig-Pipher \cite{kp3} is the statement that $\operatorname{KP}\mathcal{S}ubset\operatorname{D}$. Hence $\partial\operatorname{KP}\mathcal{S}ubset\overline{\operatorname{D}}$. What we have done in the previous propositions is to parse the relationship between $\partial\operatorname{KP}$ and $\overline{\operatorname{D}}$ more delicately. Indeed, observe that for $A\in\partial\operatorname{KP}$, we must necessarily have $\n P(A)=+\infty$, and if $A'\in\partial\operatorname{D}$, then it must be the case that $\Omegaega_{A'}$ is not $A_{\infty}$. Proposition \mathbb{R}f{prop.break} gives that \[ \partial\operatorname{KP}\cap\partial\operatorname{D}\neq\varnothing, \] which is not too surprising in light of the \cite{cfk} and \cite{mm} examples. On the other hand, Proposition \mathbb{R}f{prop.kenigbreaksdown} yields that \[ \partial\operatorname{KP}\cap \operatorname{D}\neq\varnothing, \] which prohibits any quantifiable ``equivalence'' between the Kenig-Pipher condition and the $A_{\infty}$ property. \end{remark} {} \end{document}

\begin{document} \title{Descriptional Complexity of Winning Sets of Regular Languages} \begin{abstract} We investigate certain word-construction games with variable turn orders. In these games, Alice and Bob take turns on choosing consecutive letters of a word of fixed length, with Alice winning if the result lies in a predetermined target language. The turn orders that result in a win for Alice form a binary language that is regular whenever the target language is, and we prove some upper and lower bounds for its state complexity based on that of the target language. \end{abstract} \section{Introduction} \label{sec:Intro} Let us define a word-construction game of two players, Alice and Bob, as follows. Choose a number $n \in \mathbb{N}$, a set of binary words $T \subseteq \{0, 1\}^n$ called the \emph{target set} and a word $w \in \{A, B\}^n$ called the \emph{turn order}, where $A$ stands for Alice and $B$ for Bob. The players construct a word $v \in \{0, 1\}^n$ so that, for each $i = 0, 1, \ldots, n-1$ in this order, the player specified by $w_i$ chooses the symbol $v_i$. If $v \in T$, then Alice wins the game, and otherwise Bob wins. The existence of a winning strategy for Alice depends on both the target set and the turn order. We fix the target set $T$ and define its \emph{winning set} $W(T)$ as the set of those words over $\{A,B\}$ that result in Alice having a winning strategy. We extend this definition to languages $L \subseteq \{0, 1\}^*$ by considering each length separately, so that $W(L) \subseteq \{A, B\}^*$ can also contain words of variable lengths. Winning sets were defined under this name in~\cite{salo2014playing} in the context of symbolic dynamics, but they have been studied before that under the name of \emph{order-shattering sets} in~\cite{Anstee2002,Friedl2003}. The winning set has several interesting properties: it is downward closed in the index-wise partial order induced by $A < B$ (as changing an $A$ to a $B$ always makes the game easier for Alice) and it has the same cardinality as the target set. This latter property was used in~\cite{peltomaki2019winning} to study the growth rates of substitutive subshifts. If the target language $L$ is regular, then so is $W(L)$, as it can be recognized by an alternating finite automaton, which only recognizes regular languages~\cite{Chandra1981}. Thus we can view $W$ as an operation on the class of regular languages, and in this article we study its state complexity in the general case and in several subclasses. In our construction the alternating automaton has the same state set as the original DFA, so our setting resembles parity games, where two players construct a path in a finite automaton~\cite{Zielonka1998}. The main difference is that in a parity game, the player who chooses the next move is the owner of the current state, whereas in our word-construction game it is determined by the turn order word. In the general case, the size of the minimal DFA for $W(L)$ can be doubly exponential in that of $L$. We derive a lower, but still superexponential, upper bound for bounded regular languages (languages that satisfy $L \subseteq w_1^* w_2^* \cdots w_k^*$ for some words $w_i$). We also study certain bounded permutation invariant languages, where membership is defined only by the number of occurrences of each symbol. In particular, we explicitly determine the winning sets of the languages $L_k = (0^* 1)^k 0^*$ of words with exactly $k$ occurrences of $1$. In this article we only consider the binary alphabet, but we note that the definition of the winning set can be extended to languages $L \subseteq \Sigma^*$ over an arbitrary finite alphabet $\Sigma$ in a way that preserves the properties of downward closedness and $|L| = |W(L)|$. The turn order word is replaced by a word $w \in \{1, \ldots, |\Sigma|\}^*$. On turn $i$, Alice chooses a subset of size $w_i$ of $\Sigma$, and Bob chooses the letter $v_i$ from this set. \section{Definitions} We present the standard definitions and notations used in this article. For a set $\Sigma$, we denote by $\Sigma^*$ the set of finite words over it, and the length of a word $w \in \Sigma^n$ is $|w| = n$. The notation $|w|_a$ means the number of occurrences of symbol $a \in \Sigma$ in $w$. The empty word is denoted by $\lambda$. For a language $L \subseteq \Sigma^*$ and $w \in \Sigma^*$, denote $w^{-1} L = \{ v \in \Sigma^* \;|\; w v \in L \}$. A finite state automaton is a tuple $\mathcal{A} = (Q, \Sigma, q_0, \delta, F)$ where $Q$ is a finite state set, $\Sigma$ a finite alphabet, $q_0 \in Q$ the initial state, $\delta$ is the transition function and $F \subseteq Q$ is the set of final states. The language accepted from state $q \in Q$ is denoted $\mathcal{L}_q(\mathcal{A}) \subseteq \Sigma^*$, and the language of $\mathcal{A}$ is $\mathcal{L}(\mathcal{A}) = \mathcal{L}_{q_0}(\mathcal{A})$. The type of $\delta$ and the definition of $\mathcal{L}(\mathcal{A})$ depend on which kind of automaton $\mathcal{A}$ is. \begin{itemize} \item If $\mathcal{A}$ is a deterministic finite automaton, or DFA, then $\delta : Q \times \Sigma \to Q$ gives the next state from the current state and an input symbol. We extend it to $Q \times \Sigma^*$ by $\delta(q, \lambda) = q$ and $\delta(q, s w) = \delta(\delta(q, s), w)$ for $q \in Q$, $s \in \Sigma$ and $w \in \Sigma^*$. The language is defined by $\mathcal{L}_q(\mathcal{A}) = \{ w \in \Sigma^* \;|\; \delta(q, w) \in F \}$. \item If $\mathcal{A}$ is a nondeterministic finite automaton, or NFA, then $\delta : Q \times \Sigma \to 2^Q$ gives the set of possible next states. We extend it to $Q \times \Sigma^*$ by $\delta(q, \lambda) = \{q\}$ and $\delta(q, s w) = \bigcup_{p \in \delta(q, s)} \delta(p, w)$ for $q \in Q$, $s \in \Sigma$ and $w \in \Sigma^*$. The language is defined by $\mathcal{L}_q(\mathcal{A}) = \{ w \in \Sigma^* \;|\; \delta(q, w) \cap F \neq \emptyset \}$. \end{itemize} An NFA can be converted into an equivalent DFA by the standard subset construction. Two states $p, q \in Q$ of $\mathcal{A}$ are equivalent, denoted $p \sim q$, if $\mathcal{L}_p(\mathcal{A}) = \mathcal{L}_q(\mathcal{A})$. Every regular language $L \subseteq \Sigma^*$ is accepted by a unique DFA with the minimal number of states, which are all nonequivalent, and every other DFA that accepts $L$ has an equivalent pair of states. Two words $v, w \in \Sigma^*$ are congruent by $L$, denoted $v \equiv_L w$, if for all $u_1, u_2 \in \Sigma^*$ we have $u_1 v u_2 \in L$ iff $u_1 w u_2 \in L$. They are right-equivalent, denoted $v \sim_L w$, if for all $u \in \Sigma^*$ we have $v u \in L$ iff $w u \in L$. The set of equivalence classes $\Sigma^* / {\equiv_L}$ is the syntactic monoid of $L$, and if $L$ is regular, then it is finite. In that case the equivalence classes of $\sim_L$ can be taken as the states of the minimal DFA of $L$. Let $\mathcal{P} : 2^{\Sigma^*} \to 2^{\Sigma^*}$ be an operation on languages, which may not be defined everywhere. The (regular) state complexity of $\mathcal{P}$ is the function $f : \mathbb{N} \to \mathbb{N}$, where $f(n)$ is the maximal number of states in a minimal automaton of $\mathcal{P}(\mathcal{L}(\mathcal{A}))$ for an $n$-state DFA $\mathcal{A}$. \section{Winning Sets} In this section we define winning sets of binary languages, present the construction of the winning set of a regular language using alternating automata, and prove some general lemmas. We defined the winning set informally at the beginning of Section~\ref{sec:Intro}. Now we give a more formal definition which does not explicitly mention games. \begin{definition}[Winning Set] Let $n \in \mathbb{N}$ and $T \subseteq \{0,1\}^n$ be arbitrary. The \emph{winning set} of $T$, denoted $W(T) \subseteq \{A, B\}^n$, is defined inductively as follows. If $n = 0$, then $T$ is either the empty set or $\{\lambda\}$, and $W(T) = T$. If $n \geq 1$, then $W(T) = \{ A w \;|\; w \in W(0^{-1} T) \cup W(1^{-1} T) \} \cup \{ B w \;|\; w \in W(0^{-1} T) \cap W(1^{-1} T) \}$. For a language $L \subseteq \{0, 1\}^*$, we define $W(L) = \bigcup_{n \in \mathbb{N}} W(L \cap \{0,1\}^n)$. \end{definition} The idea is that for Alice to win on a turn order of the form $A w$, she has to choose either $0$ or $1$ as the first letter $v_0$ of the constructed word $v$, and then follow a winning strategy on the target set $v_0^{-1} T$ and turn order $w$. On a word $B w$, Alice should have a winning strategy on $v_0^{-1} T$ and $w$ no matter which letter Bob chooses as $v_0$. In the next result, a language $L$ over a linearly ordered alphabet $\Sigma$ is \emph{downward closed} if $v \in L$, $w \in \Sigma^{|v|}$ and $w_i \leq v_i$ for each $i = 0, \ldots, |v|-1$ always implies $w \in L$. \begin{proposition}[Propositions~3.8 and~5.4 in~\cite{salo2014playing}] The winning set $W(L)$ is downward closed (with respect to the order $A < B$) and satisfies $|W(L)| = |L|$. If $L$ is a regular language, then $W(L)$ is also regular. \end{proposition} From an DFA $\mathcal{A}$, we can easily construct an alternating automaton for $W(\mathcal{A})$, with the same states. $B$ letters are handled with universal transitions and $A$ with existential transitions. We don't give an explicit construction of this alternating automaton, but we work on a corresponding NFA described in the next definition. \begin{definition}[Winning Set Automaton] \label{winningSetAuto} Let $\mathcal{A} = (Q, \{0,1\}, q_0, \delta, F)$ be a binary DFA. We define a ``canonical'' NFA for $W(\mathcal{L}(\mathcal{A}))$ as follows. The states are subsets of $Q$. From a state $S \subseteq Q$, reading $B$ leads to the set containing all the successors in $\mathcal{A}$ of elements of $S$. Reading $A$ leads nondeterministically to all sets containing for each element of $S$, either its successor when reading $0$, or the one when reading $1$. The only initial state is $\{q_0\}$, and final states are all subsets of $F$. We usually work on the determinization of this NFA, which we denote by $W(\mathcal{A}) = (2^{2^Q}, \{A, B\}, \{\{q_0\}\}, F_W, \delta_W)$. Here $F_W = \{ \gst{S} \in 2^{2^Q} \;|\; \exists S \in \gst{S} : S \subseteq F \}$. A state $\gst{S}$ of $W(\mathcal{A})$ is called a \emph{game state}. It represents a situation where Alice can force the game to be in one of the sets $S \in \gst{S}$, and Bob can choose the actual state $q \in S$. The transition function $\delta_W $ is defined by \begin{align*} \delta_W (\{S\},A) & = \{\{\delta(q,f(q)) \;|\; q \in S\} \;|\; f : S \rightarrow \{0,1\}\} \\ \delta_W (\{S\},B) & = \{\{\delta(q,b) \;|\; q \in S, b \in \{0,1\}\}\} \end{align*} and $\delta_W (\gst{S}, c) = \bigcup_{S \in \gst{S}} \delta_W (\{S\},c)$ for a game state $\gst{S}$ and $c \in \{A, B\}$. \iffalse $\delta_W (\{\emptyset\},A) = \{\emptyset\}$ $\delta_W (\{ S \cup \{q\}\},A) = \{ S' \cup \{\delta(q,b) \} | S' \in \delta_W (S,A), b \in \{0,1\}\}$ $\delta_W (\{\{q\}\},A) = \{ \{delta(q,0) \}, \{delta(q,1) \} \}$\\ $\delta_W (\{\{q\}\},B) = \{ \{delta(q,0), delta(q,1) \} \}$ \fi \end{definition} The following observations follow easily from the definition of $W(L)$. \begin{lemma} \label{basicProperties} Let $\mathcal{A}$ be a DFA with alphabet $\{0,1\}$, and $W(\mathcal{A})$ the winning set DFA from Definition~\ref{winningSetAuto}, and $\delta_W$ the iterated transition function for $W(\mathcal{A})$. Let $\gst{P},\gst{R},\gst{S},\gst{T}$ be game states of $W(\mathcal{A})$, $P,R,S,T,V \subseteq 2^Q$ sets of states, and $w$ a word over $\{A,B\}$. \begin{roster} \item \label{unionEquiv} If $\delta_W (\gst{P},w) = \gst{R}$ and $\delta_W (\gst{S},w) = \gst{T}$ then $\delta_W (\gst{P} \cup \gst{S}, w) = \gst{R} \cup \gst{T}$. \item \label{unionInsideEquiv} If $R \in \delta_W( \{S\},w )$ and $V \in \delta_W( \{T\},w )$, then some $P \in \delta_W( \{S \cup T\},w )$ satisfies $P \subseteq R \cup V$. Conversely, for each $P \in \delta_W(\{S \cup T\}, w)$ there exist $R \in \delta_W( \{S\},w )$ and $V \in \delta_W( \{T\},w )$ with $P = R \cup V$. \item \label{subsetEquiv} If $S,R \in \gst{T}$ and $S \subseteq R$, then $\gst{T} \sim \gst{T} \setminus \{R\}$. \item \label{removeStateEquiv} If $S \in \gst{S}$ and some $q \in S$ has no path to a final state, then $\gst{S} \sim \gst{S} \setminus \{S\}$. \item \label{finalSinkEquiv} If $S \in \gst{S}$ and there is a sink state $q \in S \cap F$, then $\gst{S} \sim ( \gst{S} \setminus \{S\}) \cup \{S \setminus \{q\} \}$. \item \label{smallWordEquiv} If $\gst{S} = \gst{R} \cup \{S\}$ and the shortest path from some $q \in S$ to an final state in $\mathcal{A}$ has length $\ell$, then for all $w \in \{A,B\}^{< \ell}$, $\delta_W(\gst{S},w)$ is final iff $\delta(\gst{R},w)$ is. \end{roster} \end{lemma} \begin{lemma} \label{lemmasEquivIncl} Recall the assumptions of Lemma~\ref{basicProperties}. \begin{roster} \item \label{equivChainDoubleIncl} Suppose that for every $S \in \gst{S}$ there exists $R \in \gst{R}$ with $R \subseteq S$, and reciprocally. Then $\gst{S} \sim \gst{R}$. \item \label{equivChainSingletons} Let $v, w \in \{A, B\}^*$. If for all $q \in Q$, the game states $\delta_W (\{\{q\}\},v)$ and $\delta_W (\{\{q\}\},w)$ are either both accepting or both rejecting, then $v \equiv_{W(\mathcal{L}(\mathcal{A}))} w$. \end{roster} \end{lemma} \begin{proof} \begin{roster} \item Let $w \in \{A,B\}^*$ be such that $\delta_W(\gst{S}, w)$ is accepting. Then some set $T \in \delta_W(\gst{S}, w)$ consists of accepting states of $\mathcal{A}$. By Lemma~\ref{basicProperties}\ref{unionEquiv} there exists $S \in \gst{S}$ with $T \in \delta_W(\{S\}, w)$. Let $R \in \gst{R}$ be such that $R \subseteq S$. Then there exists $V \in \delta_W(\gst{R}, w)$ with $V \subseteq T$ by Lemma~\ref{basicProperties}\ref{unionInsideEquiv}, so that $\delta_W(\gst{R}, w)$ is also accepting. \item Let $\gst{S} \in 2^{2^Q}$ be a game state and suppose $\delta_W(\gst{S}, v)$ is accepting, so there exists $P \in \delta_W(\gst{S}, v)$ consisting of accepting states of $\mathcal{A}$. By Lemma~\ref{basicProperties}\ref{unionEquiv} we have $\delta_W(\gst{S}, v) = \bigcup_{S \in \gst{S}} \delta_W(\{S\}, v)$, and similarly for $w$, so we may assume $\gst{S} = \{S\}$ is a singleton. By Lemma~\ref{basicProperties}\ref{unionInsideEquiv}, for each $q \in S$ there exists $R_q \in \delta_W(\{\{q\}\}, v)$ such that $P = \bigcup_{q \in S} R_q$. In particular each $R_q$ consists of accepting states of $\mathcal{A}$, so each $\delta_W(\{\{q\}\}, v)$ is accepting. Then $\delta_W(\{\{q\}\}, w)$ is also accepting, so there exists $T_q \in \delta_W(\{\{q\}\}, w)$ with $T_q \subseteq F$. By Lemma~\ref{basicProperties}\ref{unionInsideEquiv} there exists $P' \in \delta_W(\{S\}, w)$ with $P' \subseteq \bigcup_{q \in S} T_q \subseteq F$, and then $\delta_W(\{S\}, w)$ is accepting. This shows $v \equiv w$. \end{roster} \end{proof} \begin{proposition} \label{prop:upperBound} Let $\mathcal{A}$ an $n$-state DFA. The number of states in the minimal DFA for $W(\mathcal{L}(\mathcal{A}))$ is at most the Dedekind number $D(n)$. \end{proposition} \begin{proof} The Dedekind number $D(n)$ is the number of antichains in $2^{2^Q}$ by inclusion, and every game state is equivalent to an antichain by Lemma 3.1(f). \end{proof} Note that the growth of $D(n)$ is doubly exponential in $n$. We have computed the exact state complexity of the winning set operation for DFAs with at most $5$ states; the $6$-state case is no longer feasible with our program and computational resources. The sequence begins with $1, 4, 16, 62, 517$. \section{Doubly exponential lower bound} In this section we present the construction of a family of automata for which the number of states in the minimal winning set automaton is asymptotically optimal, that is to say doubly exponential. The idea is to reach any desired antichain of subsets of a special subset of states by reading the appropriate word, and then to make sure these game states are nonequivalent by reading a word which leads to acceptance only if the game state is the wanted one (apart from some technical details). To do this we split the automaton into several components. First we present a ``subset factory gadget'' that allows to make any desired set of the form $\{ S\}$ where $S$ is a subset of a specific length-$n$ path in the automaton. This gadget will be used several times to accumulate subsets in the game state. Then we present a ``testing gadget'' allowing to distinguish between a doubly exponential number of game states. The construction of $W(\mathcal{A})$ in Definition~\ref{winningSetAuto} shows that the labels of the transitions are not important with regard to the winning set language that is obtained from it. In this section we define automata by describing their graphs, and a node with two outgoing transitions can have them arbitrary labeled by $0$ and $1$. \begin{lemma}[Subset factory gadget] \label{lem:subsetFactoryGadget} Let $\mathrm{GenSubset}_n$ be the graph in Figure~\ref{subsetFactoryGadget}. For $i \in \{1, \ldots, n\}$, denote $o_i = e_{2n + i -1}$ (successors of the $c_i$). For all $S \in 2^{\{1,\ldots,n\}}$ there exists $w ^\mathrm{subset}_S \in \{A,B\}^{2n} $ such that $\delta_W ( \{\{b_1\}\}, w ^\mathrm{subset}_S) ) \sim \{ \{ o_i \;|\; i \in S \} \} $ for every DFA over $\{0,1\}$ that contains $\mathrm{GenSubset}_n$ as a subgraph. \begin{figure} \caption{\label{subsetFactoryGadget} \label{subsetFactoryGadget} \end{figure} \end{lemma} \begin{proof} Denote $f_i = e_{3 i - 2} $. For $i \in \{1, \ldots, n\}$ and $S \subseteq \{1, \ldots, i-1\}$, denote $S_i = \{e_{2 i - 4 + j} \;|\; j \in S\}$. Consider the game state $\gst{R}(i,S) = \{\{b_i\} \cup S_i\}$. If the automaton reads $A B$, the resulting game state is \[ \delta_W(\gst{R}(i,S), A B) = \{\{s_i, e_{3 i}\} \cup S_{i+1}, \{b_{i+1}\} \cup S_{i+1}\} \sim \gst{R}(i+1, S) . \] In the case of $B A$ we instead have \[ \delta_W(\gst{R}(i,S), B A) = \{ \{b_{i+1}, s_i\} \cup S_{i+1}, \{b_{i+1}, f_i\} \cup S_{i+1} \} \sim \gst{R}(i+1, S \cup \{i\}) . \] In both cases the final steps follow from Lemma~\ref{basicProperties}\ref{removeStateEquiv}. By Lemma~\ref{basicProperties}\ref{finalSinkEquiv} we also have $\gst{R}(n+1, S) \sim \{S_{n+1}\}$ since $b_{n+1}$ is an accepting sink state. Take $w^\mathrm{subset}_S$ as the concatenation $w_1 w_2 \dots w_n$ where $w_i = BA$ if $i \in S$, and $w_i = AB$ if $i \notin S$. This word satisfies the claim, since $\delta_W(\{\{b_1\}\}, w^\mathrm{subset}_S) = \{S_{n+1}\} = \{\{o_i \;|\; i \in S\}\}$. \end{proof} \begin{lemma}[Game state factory gadget]\label{subsetFactory} Let $\mathrm{GenState}_n$ be the graph in figure~\ref{stateFactoryGadget} and $\mathcal{A}$ any DFA over $\{0,1\}$ that contains it. For all $\gst{S} = \{ S_1, \ldots, S_\ell \}$ where each $S_i \subseteq \{r_1,\ldots, r_n\}$, there exists $w^\mathrm{gen}_{\gst{S}} \in \{A,B\}^{\ell (3 n + 1)}$, and a game state $\gst{S}'$ that does not contain a subset of the states of $\mathrm{GenState}_n$, such that $\delta_W (\{\{a_1\}\},w^\mathrm{gen}_{\gst{S}}) \sim \gst{S} \cup \{\{a_1\}\} \cup \gst{S}'$. \begin{figure} \caption{\label{stateFactoryGadget} \label{stateFactoryGadget} \end{figure} \end{lemma} \begin{proof} The idea is that previously made subsets will rotate in the rightmost cycle. Meanwhile, a singleton set will rotate in the left cycle, initiating from the state $a_1$ the creation of a new subset by reading the letter $A$. This new set is created in the subset factory component and joins the previously made sets in the rightmost cycle. Suppose we have reached a game state of the form $\gst{R} = \{ \{a_1\},\{r_i \;|\; i \in S_1\} , \ldots, \{r_i \;|\; i \in S_k\}\} \cup \gst{S}'$ where $\gst{S}'$ does not contain any subset of $\mathrm{GenState}_n$. We prove that by reading $A w^\mathrm{subset}_{S_{k+1}} A^n$, we reach a game state of the form $\{ \{a_1\},\{r_i \;|\; i \in S_1\} , \ldots, \{r_i \;|\; i \in S_{k+1}\}\} \cup \gst{S}''$. We analyze the elements of $\gst{R}$ separately. \begin{itemize} \item Because $|A w^{subset}_{S_{k+1}}A^{n}| = 3n+1$ is the size of the rightmost cycle, we have $\delta_W (\{ \{r_i \;|\; i \in S_j\} \}, A w^{subset}_{S_{k+1}} A^{n}) \sim \{ \{r_i : i \in S_j\}\}$ for each $j \leq k$. \item The game state $\{\{a_1\}\}$ first evolves into $\delta_W(\{\{a_1\}\}, A) = \{ \{a_2\},\{b_1\}\}$. The component $\{\{a_2\}\}$ becomes $\{\{a_1\}\}$ when we read $w^{subset}_{S_{k+1}}A^{n}$. As for $\{\{b_1\}\}$, Lemma~\ref{subsetFactory} gives $\delta_W(\{\{b_1\}\}, w^{subset}_{S_{k+1}} ) = \{ \{ o_i \;|\; i\in S_{k+1}\}\}$, and then $\delta_W(\{ \{ o_i \;|\; i\in S_{k+1}\} \}, A^n ) = \{ \{ r_{i} \;|\; i \in S_{k+1}\} \} \cup \gst{S}''$ where every set in $\gst{S}''$ contains a state outside of $\mathrm{GenState}_n$. \item The game state $\gst{S}'$ evolves into some $\gst{S}'''$ each of whose sets contains a state not in $\mathrm{GenState}_n$, since the gadget cannot be re-entered. \end{itemize} By Lemma~\ref{basicProperties}\ref{unionEquiv} we have $\delta_W (\gst{R}, A w^\mathrm{subset}_{S_{k+1}} A^{n}) \sim \{ \{a_1\},\{r_{i} \;|\; i \in S_1\}, \ldots, \{r_{i} \;|\; i \in S_{k+1}\}\} \cup \gst{S}'' \cup \gst{S}'''$. We obtain $w^\mathrm{gen}_{\gst{S}}$ as the concatenation of these words. \end{proof} \begin{lemma}[Testing gadget] \label{lem:testingGadget} Let $\mathrm{Testing}_n$ be the graph in Figure~\ref{measuringGadget}. \begin{roster} \item For all $P \subseteq \{1, \dots, n\}$ there exists $w ^\mathrm{test}_P \in \{A,B\}^{n} $ such that for each $I \subseteq \{1, \dots, n\}$, the game state $\delta_W ( \{\{ q_i \;|\; i \in I\}\}, w^\mathrm{test}_P)$ is accepting iff $I \subseteq P$. \item Let $V$ be the set of nodes of the graph $\mathrm{Testing}_n$. Then for all $\gst{S} \in 2^{2^{V}}$ and $w\in \{A, B\}^{\geq 2n}$, the game state $\delta_W (\gst{S},w)$ is not accepting. \end{roster} \begin{figure} \caption{\label{measuringGadget} \label{measuringGadget} \end{figure} \end{lemma} \begin{proof} \begin{roster} \item For $I \subseteq \{1, \ldots, 2 n\}$, denote $S_I = \{ q_i \;|\; i \in I \}$. If $2 n \notin I$, let $J = \{ i+1 \;|\; i \in I \}$. A simple case analysis together with Lemma~\ref{basicProperties}\ref{removeStateEquiv} and~\ref{finalSinkEquiv} shows that $\delta_W(\{S_I\}, A) \sim \{S_J\}$ and \[ \delta_W(\{S_I\}, B) = \begin{cases} \emptyset & \mbox{if } n \in I, \\ \{S_J\} & \mbox{otherwise.} \end{cases} \] Take \[ w^\mathrm{test}_P[i] = \begin{cases} A &\mbox{if } n-i+1 \in P, \\ B &\mbox{otherwise. } \end{cases} \] Then $\gst{S} = \delta_W (\{S_I\}, w^\mathrm{test}_P)$ is accepting if and only if $\gst{S} \sim \{\{q_{i+n} \;|\; i \in I\}\}$. This is equivalent to $w[n-i+1] = A$ for all $i \in I$, i.e. $I \subseteq P$. \item If $\gst{S} \in 2^{2^{V}}$ and $w$ with $|w| \geq 2n$, then every $S \in \delta_W (\gst{S},w)$ satisfies $S \subseteq \{r,r'\}$. \end{roster} \end{proof} \begin{theorem} For each $n > 0$ there exists a DFA $\mathcal{A}_n$ over $\{0,1\}$ with $15n + 3$ states such that the minimal DFA for $W(\mathcal{L}(\mathcal{A}_n))$ has a least $D(n)$ states. \end{theorem} Together with Proposition~\ref{prop:upperBound}, this implies that the state complexity of $W$ restricted to regular languages grows doubly exponentially. \begin{proof} Let $\mathcal{A}_n$ be the DFA obtained by combining $\mathrm{Testing}_n$ with the outgoing arrow of $\mathrm{GenState}_n$ and assigning $a_1$ as the initial state. For an antichain $\gst{S}$ on the powerset of $\{r_1, \ldots, r_n\}$, let $X_\gst{S} = \delta_W (\{\{a_1\}\}, w^\mathrm{gen}_\gst{S})$. By Lemma~\ref{subsetFactory} we have $X_\gst{S} \sim \{\{a_1\}\} \cup \gst{S} \cup \gst{S}'$ where each set in $\gst{S}'$ contains a state of $\mathrm{Testing}_n$. By definition, $X_\gst{S}$ is an accessible state of $W(\mathcal{A})$, and we show that distinct antichains $\gst{S}$ result in nonequivalent states. Let $P \subseteq \{1, \ldots, n\}$ and consider the game state $X'_\gst{S} = \delta_W(X_\gst{S}, A^{n+1} w^\mathrm{test}_P)$. We claim that $X'_\gst{S}$ is accepting iff some element of $\gst{S}$ is a subset of $\{ r_i \;|\; i \in P \}$. By Lemma~\ref{basicProperties}\ref{unionEquiv} we may analyze the components of $X_\gst{S}$ separately. \begin{itemize} \item Since the shortest path from $a_1$ to an accepting state has lenght $2 n + 2$ and $|A^{n+1} w^\mathrm{test}_P| = 2 n + 1$, we can ignore it by Lemma~\ref{basicProperties}\ref{smallWordEquiv}. \item Since each set of $\gst{S}'$ contains a state of $\mathrm{Testing}_n$ and $|A^{n+1} w^\mathrm{test}_P| \geq 2 n$, by Lemma~\ref{lem:testingGadget} the game state $\delta_W(\gst{S}', A^{n+1} w^\mathrm{test}_P)$ is not accepting. \item The game state $\delta_W(\gst{S}, A^{n+1})$ consists of the sets $\{ q_i \;|\; r_i \in S \}$ for $S \in \gst{S}$, as well as sets that contain at least one element of $\{r_{n+1}, \ldots, r_{2 n}\}$. We can ignore the latter by Lemma~\ref{basicProperties}\ref{smallWordEquiv}. Lemma~\ref{lem:testingGadget} shows that the former sets produce an accepting game state in $X'_\gst{S}$ iff some $S \in \gst{S}$ is a subset of $\{ r_i \;|\; i \in P \}$. \end{itemize} The Dedekind number $D(n)$ is the number of antichains on the powerset of $\{1, \ldots, n\}$, so we have found $D(n)$ nonequivalent states in $W(\mathcal{A})$. \end{proof} \section{Case of the bounded regular languages} In this section we prove an upper bound on the complexity of the winning set of a bounded regular language. Our proof technique is based on tracing the evolution of individual states of a DFA $\mathcal{A}$ in the winning set automaton $W(\mathcal{A})$ when reading several $A$-symbols in a row. \begin{definition}[Histories of Game States] \label{def:History} Let $\mathcal{A} = (Q, \{0,1\}, q_0, \delta, F)$ be a DFA. Let $\gst{S} \in 2^{2^Q}$ be a game state of $W(\mathcal{A})$, and for each $i \geq 0$, let $\gst{S}_i \sim \delta_W(\gst{S}, A^i)$ be the game state with all supersets removed as per Lemma~\ref{basicProperties}\ref{subsetEquiv}. A \emph{history function} for $\gst{S}$ is a function $h$ that associates to each $i > 0$ and each set $S \in \gst{S}_i$ a \emph{parent set} $h(i, S) \in \gst{S}_{i-1}$, and to each state $q \in S$ a set of \emph{parent states} $h(i, S, q) \subseteq h(i, S)$ such that \begin{itemize} \item $S \in \delta_W(\{h(i,S)\}, A)$ for each $i$, \item $h(i,S)$ is the disjoint union of $h(i, S, q)$ for $q \in S$, and \item $\{q\} \in \delta_W(\{h(i,S,q)\}, A)$ for all $q \in S$. \end{itemize} The \emph{history} of a set $S \in \gst{S}_i$ from $i$ under $h$ is the sequence $S_0, S_1, \ldots, S_i = S$ with $S_{j-1} = h(j, S_j)$ for all $0 < j \leq i$. A history of a state $q \in S$ in $S$ under $h$ is a sequence $q_0, \ldots, q_i = q$ with $q_{j-1} \in h(j, S_j, q_j)$ for all $0 < j \leq i$. \end{definition} A game state can have several different history functions, and each of them defines a history for each set $S$. A state of $S$ can have several histories under a single history function. These histories are consistent with themselves and each other. The proof of the main result of this section is based on the idea of choosing a ``good'' history function. Note that we have defined the history function only for sequences of $A$-symbols, since this simplifies the definition and histories with $B$-symbols are not used in the proof. For the rest of this section, we fix an $n$-state DFA $\mathcal{A} = (Q, \{0,1\}, q_0, \delta, F)$ that recognizes a bounded binary language and has disjoint cycles. Let the lengths of the cycles be $k_1, \ldots, k_p$, and let $\ell$ be the number of states not part of any cycle. We define a preorder ${\leq}$ on the state set $Q$ by reachability: $p \leq q$ holds if and only if there is a path from $p$ to $q$ in $\mathcal{A}$. The notation $p < q$ means $p \leq q$ and $q \not\leq p$. For two history functions $h, h'$ of a game state $\gst{S}$, we write $h \leq h'$ if for each $i > 0$, each $S \in \gst{S}_i$ and each $q \in S$, there exists a function $f : h(i, S, q) \to h'(i, S, q)$ with $p \leq f(p)$ for all $p \in h(i, S, q)$. This defines a preorder on the set of history functions of $\gst{S}$. We write $h < h'$ if $h \leq h'$ and $h' \not\leq h$. A history function $h$ is \emph{minimal} if there exists no history function $h'$ with $h' < h$. Intuitively, a minimal history function is one where the histories of states stay in the early cycles of $\mathcal{A}$ as long as possible. \begin{lemma} \label{existsLocallyMinimal} Each game state $\gst{S} \in 2^{2^Q}$ has at least one minimal history function. \end{lemma} \begin{proof} For each $i > 0$ and $S \in \gst{S}_i$, the set of possible choices for the parent $h(i, S)$ of $S$ and the parent set $h(i, S, q)$ of each state $q \in S$ is finite, and the choice is independent of the respective choices for other sets $S' \in \gst{S}_{i'}$ with $S' \neq S$ or $i' \neq i$. If we choose the parents that are minimal with respect to ${\leq}$ for each set, the resulting history function is minimal. \end{proof} \begin{lemma} \label{lem:ChangeHistory} Let $\gst{S} \in 2^{2^Q}$ be any game state of $W(\mathcal{A})$. Then there exist $k \leq \mathrm{lcm}(k_1, \ldots, k_p) + 2 n + \max_{x \neq y} \mathrm{lcm} (k_x, k_y)$ and $m \leq \mathrm{lcm}(k_1, \ldots, k_p)$ such that $\delta_W(\gst{S}, A^k) \sim \delta_W(\gst{S}, A^{k+m})$. \end{lemma} \begin{proof} Denote the cycles of $\mathcal{A}$ by $C_1, \ldots, C_p$, so that $|C_i| = k_i$ for each $i$. Let $h$ be a minimal history function of $\gst{S}$, given by Lemma~\ref{existsLocallyMinimal}. Define $\gst{S}_i$ for $i \geq 0$ as in Definition~\ref{def:History}. Let $t \geq 0$, $S \in \gst{S}_t$ and $q \in S$ be arbitrary, and let $S_0, \ldots, S_t = S$ and $q_0, \ldots, q_t = q$ be their histories under $h$. The history of $q$ travels through some of the cycles of $\mathcal{A}$, never entering the same cycle twice. We split the sequence $q_0, \ldots, q_n$ into words over $Q$ as $u_0 v_1^{p_1} u_1 v_2^{p_2} u_2 \cdots v_r^{p_r} u_r$, where \begin{itemize} \item each $p_j \geq 1$, \item each $v_j$ consists of the states of some cycle, which we may assume is $C_j$, repeated exactly once, \item the $u_j$ do not repeat states and each $u_j$ does not contain any states from $C_{j+1}$. \end{itemize} Intuitively, $v_j$ represents a phase of the history where the state stays in a cycle for several loops, and the $u_j$ represent transitions from one loop to another. Each $u_j$ ends right before the time step when the history of $q$ enters the loop $C_{j+1}$. It may share a nonempty prefix with $v_j$. We claim that $p_i k_i \leq \max_{x \neq y} \mathrm{lcm} (k_x, k_y)$ holds for all $1 < i \leq r$. Assume the contrary. Since $k_i = |v_i|$, we have in particular $p_i |v_i| > \mathrm{lcm}(|v_{i-1}|, |v_i|)$ for some $i$, so that $a |v_{i-1}| = b |v_i|$ holds for some $a > 0$ and $0 < b \leq p_i$. Denote $s = |u_0 v_1^{p_1} \cdots u_{i-1}|$, which is the time step after which the history of $q$ enters the repetitive portion of the previous loop $C_{i-1}$. Denote $K = |v_{i-1}^a u_{i-1}|$ and $q'_{s+1}, q'_{s+2}, \ldots, q'_{s + K} = v_{i-1}^a u_{i-1}$. Note that we may have $q'_{s+j} = q_{s+j}$ for some $1 \leq j < K$, but $q'_{s+K} < q_{s+K}$ since the former is not in $C_i$ while the latter is. In $\mathcal{A}$ we have transitions from $q_s$ to both $q_{s+1}$ and $q'_{s+1}$, and from each $q'_{s+j}$ to $q'_{s+j+1}$, as well as from $q'_{s+K}$ to $q_{s+K+1}$. For $q \leq j \leq K$ the game state $\delta_W(\gst{S}, A^{s+j})$ contains $S'_{s+j} =: (S_{s+j} \setminus \{q_{s+j}\}) \cup \{q'_{s+j}\}$. We also have $S_{s+K+1} \in \delta_W(\{S'_{s+K}\}, A)$. There are now two possibilities. If $q'_{s+K} \in S_{s+K}$, then $S'_{s+K} \in \delta_W(\gst{S}, A^{s+K})$ is a proper subset of $S_{s+K}$, which contradicts our choice of $\gst{S}_{s+K}$ as a version of $\delta_W(\gst{S}, A^{s+K})$ with all proper supersets removed. If $q'_{s+K} \notin S_{s+K}$, then we may define a new history function $h'$ by defining $h'(s+K+1, S) = S'_{a+K}$, $h'(s+K+1, S, q) = (h(s+K+1, S, q) \setminus \{q_{s+K}\}) \cup \{q'_{s+K}\}$, and $h'(t, S', q') = h(t, S', q')$ for all other choices of $t$, $S'$ and $q'$. Then the function $f : h'(s+K+1, S, q) \to h(s+K+1, S, q)$ defined by $f(q'_{s+K}) = q_{s+K}$ and $f(q') = q'$ for other $q' \in h'(s+K+1, S, q)$ shows $h' < h$, which contradicts the local minimality of $h$. We have now shown $p_i k_i \leq \max_{x \neq y} \mathrm{lcm} (k_x, k_y)$ for all $1 < i \leq r$. Denote $L = \mathrm{lcm}(k_1, \ldots, k_p)$. If $t \geq L + 2 \ell + p \cdot \max_{x \neq y} \mathrm{lcm} (k_x, k_y)$, then $p_1 k_1 \geq L + \ell$, which implies $q_\ell = q_{\ell+L}$ (note that $|u_0| \leq \ell$, so that $q_\ell, q_{\ell + L} \in C_1$). Since this holds for every history of every state of $S$ under $h$ and each state of each set $S_i$ for $i \leq t$ can be chosen as $q_i$ for some $q \in S$, we have $S_\ell = S_{\ell+L}$. Then $S \in \delta_W(\{S_{\ell+L}\}, A^{t-\ell-L}) = \delta_W(\{S_\ell\}, A^{t-\ell-L})$, so in particular $S \in \delta_W(\gst{S}, A^{t-L}) \sim \gst{S}_{t-L}$. On the other hand, $S \in \delta_W(\{S_\ell\}, A^{t-\ell}) = \delta_W(\{S_{\ell+L}\}, A^{t-\ell})$, so $S \in \delta_W(\gst{S}, A^{t+L}) \sim \gst{S}_{t+L}$. Since $S \in \gst{S}_t$ was arbitrary, we have $\gst{S}_t \sim \gst{S}' \subseteq \gst{S}_{t-L}$ and $\gst{S}_t \sim \gst{S}'' \subseteq \gst{S}_{t+L}$ for some game states $\gst{S}', \gst{S}'' \in 2^{2^Q}$. By considering $t+L$ instead of $t$ and doing the same analysis, we obtain $\gst{S}_t \sim \gst{S}_{t+L}$. \end{proof} \begin{theorem} Let $\mathcal{A}$ be an $n$-state DFA that recognizes a bounded binary laguage. Then there is a partition $\ell + k_1 + \cdots + k_p = n$ such that the minimal DFA for $W(\mathcal{L}(\mathcal{A}))$ has at most $\sum_{m = 0}^{\ell + p + 1} (p \cdot \max_{x \neq y} \mathrm{lcm}(k_x, k_y) + 2 \ell + 2 \mathrm{lcm}(k_1, \ldots, k_p))^m$ states. \end{theorem} \begin{proof} Denote the minimal DFA for $W(\mathcal{L}(\mathcal{A}))$ by $\mathcal{B}$. We may assume that $\mathcal{A}$ is minimal, and then it has disjoint cycles. Let $k_1, \ldots, k_p$ be the lengths of the cycles and $\ell$ the number of remaining states, and denote $P = p \cdot \max_{x \neq y} \mathrm{lcm}(k_x, k_y) + 2 \ell + 2 \mathrm{lcm}(k_1, \ldots, k_p)$. Then the language of $W(\mathcal{A})$ only contains words that have at most $\ell + p$ occurrences of $B$: in a game whose turn order has more $B$s than that, Bob can win by choosing to leave a cycle whenever possible, since the $\ell$ states outside the cycles can never be returned to. Consider a word $w = A^{t_0} B A^{t_1} B \cdots B A^{t_m}$ with $0 \leq m \leq \ell + p$. If $t_i \geq P$ for some $i$, then Lemma~\ref{lem:ChangeHistory} implies $\delta_W(\gst{S}, A^{t_i}) \sim \delta_W(\gst{S}, A^t)$ for the game state $\gst{S} = \delta_W(\{\{q_0\}\}, A^{t_0} B \cdots A^{t_{i-1}} B)$ and some $t < t_i$. Thus the number of distinct states of $\mathcal{B}$ reachable by words of this form is at most $P^{m+1}$. The claim directly follows. \end{proof} The state complexity implied by the result (the maximum of the expression taken over all partitions of $n$) is at least $n^n$. In particular, it grows superexponentially. We don't know whether the actual complexity of the winning set operation on bounded regular languages is exponential or not. If we combine the gadgets $\mathrm{GenSubset}_n$ and $\mathrm{Testing}_n$, the resulting DFA recognizes a language whose winning set requires at least $2^n$ states, so for finite regular languages (and therefore for bounded regular languages) the state complexity of the winning set is at least exponential. \section{Chain automata} In this section we investigate a family of binary automata consisting of a chain of states with a self-loop on each state. More formally, a \emph{chain automaton} is a DFA $\mathcal{A} = (Q, \{0,1\}, q_0, \delta, F)$ where $Q = \{0, 1, \ldots, m+p-1\}$, $q_0 = 0$, $\delta(i, 0) = i$ and $\delta(i, 1) = i+1$ for all $i \in Q$ except $\delta(m+p-1, 1) = m$. The automaton is \emph{$1$-bounded} if $p = 0$ and the state $m-1$ is not final. See Figure~\ref{chainGeneral} for a $1$-bounded chain automaton. It is easy to see that chain automata recognize exactly the regular languages $L$ such that $w \in L$ depends only on $|w|_1$, and the $1$-bounded subclass recognizes those where $|w|_1$ is also bounded. Of course, the labels of the transitions have no effect on the winning set $W(\mathcal{L}(\mathcal{A}))$ so the results of this section apply to every DFA with the structure of a chain automaton. \begin{figure} \caption{\label{chainGeneral} \label{chainGeneral} \end{figure} \begin{lemma} \label{chainEquivWords} Let $\mathcal{A}$ be an $n$-state chain automaton, and denote $\equiv_{W(\mathcal{L}(\mathcal{A}))}$ by $\equiv$. \begin{roster} \item \label{chain1} For every state $q \in Q$ and every $S \in \delta_W(\{\{q\}\}, A B)$, there exists $R \in \delta_W(\{\{q\}\}, B A)$ with $R \subseteq S$. \item \label{chain2} For all $k \in \mathbb{N}$, $B^k A^k B^{k+1} \equiv B^{k+1} A^k B^k$. \item \label{chain3} For all $k \in \mathbb{N}$, $A^{k+1} B^k A^k \equiv A^k B^k A^{k+1}$. \item \label{chain4} $A^{n-1} \equiv A^n$ and $B^{n-1} \equiv B^n$. \end{roster} \end{lemma} The intuition for~\ref{chain1} is that $B A$ produces game states that are better for Alice than $A B$, since Alice can undo any damage Bob just caused. \begin{proof} Label the states of $\mathcal{A}$ by $\{0, 1, \ldots, m+p-1\}$ as in the definition of chain automata. For \ref{chain1}, \ref{chain2} and \ref{chain3} we ``unroll'' the loop $m, m+1, \ldots, m+p-1$ to obtain an equivalent automaton with an infinite chain of states, simplifying the arguments. We also argue in terms of the NFA for $W(\mathcal{L}(\mathcal{A}))$, saying that ``we produce a set $R \subseteq Q$ from $S \subseteq Q$ by reading $w \in \{A,B\}^*$'' if $R \in \delta_W(\{S\}, w)$. In this formalism \ref{chain1} means that for any set produced from $\{q\}$ by reading $A B$, we can produce a subset of it by reading $B A$. To see what sets can be produced, we use a spacetime diagram (Figure~\ref{ABinclBA}) where the time increases to the south. In this diagram, by reading a $B$, a selected state will spread south and southeast. By reading an $A$, we have both possibilities, resulting in multiple sets. The claim follows directly. \begin{figure} \caption{\label{ABinclBA} \label{ABinclBA} \end{figure} We now prove \ref{chain2}. By Lemma~\ref{lemmasEquivIncl}\ref{equivChainSingletons} it is enough to consider singleton states $\{q\}$, and without loss of generality we assume $q = 0$. Because of (a) it is sufficient to prove that for every set obtained from $B^{k+1} A^k B^k$, we can produce a subset of it by reading $B^k A^k B^{k+1}$. After reading $B^{k+1}$, we have the interval $\{0, \dots k+1\}$. After $A^k$ we get a set included in $\{0,\dots, 2k+2\}$ where the distance between every two consecutive elements is less than $k$. After reading $B^k$ the gaps are filled and we get an interval containing $ \{k, \dots, 2k+2\}$. When reading $B^k A^k B^{k+1}$ we do the following: get $\{0, \dots k\}$ with $B^k$, make every element go to position $k$ with $A^k$, and extend with $B^{k+1}$ to get the interval $\{k, \dots, 2k+2\}$. For \ref{chain3}, for the same reason as previously it is sufficient to prove that for every set obtained from $\{0\}$ by reading $A^k B^k A^{k+1}$, we can produce a subset by reading $A^{k+1} B^k A^k$. First we prove that by reading $A^{k+1} B^k A^k$ we can get any singleton set $\{k\}, \{k+1\}, \ldots, \{2k+1\}$: By reading $A^{k+1} B^k$, we can get any singleton set between $0$ and $k+1$ and expand it to have any interval of length $k+1$ between $0$ and $2k+1$. Then by reading $A^k$ we can have a singleton state at the end position of the interval, that is between $k$ and $2k+1$. See Figure~\ref{AnABnAn}. \begin{figure} \caption{\label{AnABnAn} \label{AnABnAn} \end{figure} Now we prove that every set obtained by reading $A^k B^k A^{k+1}$ has at least one of $k, \ldots, 2k+1$. After reading $A^k B^k$ we have any interval of size $k+1$ between $0$ and $2k$. The state $k$ is always in the interval, and must go somewhere between positions $k$ and $2k+1$ after $A^{k+1}$. As for \ref{chain4}, reading $A^{n-1}$ or $A^n$ in any singleton game state $\{q\}$ produces exactly the states $\{q\}, \{q+1\}, \ldots, \{m-1\}$ and the loop $\{m\}, \ldots, \{m+p-1\}$. Likewise, reading $B^{n-1}$ or $B^n$ produces $\{q, q+1, \ldots, m-1\} \cup \{m, \ldots, m+p-1\}$. \end{proof} \begin{theorem} \label{chainBoundedBound} Let $\mathcal{A}$ be a $1$-bounded chain automaton with $n$ states. The number of states in the minimal DFA of $W(\mathcal{L}(\mathcal{A}))$ is $O(n^{1/5} e^{4 \pi \sqrt{\frac{n}{3}}})$. \end{theorem} \begin{proof} Since $\mathcal{A}$ does not accept any word with $n$ or more $1$-symbols, $W(\mathcal{L}(\mathcal{A}))$ contains no word with $n$ or more $B$-symbols. The equivalences $B^k A^k B^{k+1} \equiv B^{k+1} A^k B^k$ and $A^{k+1} B^k A^k \equiv A^k B^k A^{k+1}$ for $k \geq 0$ that follow from Lemma~\ref{chainEquivWords} allow us to rewrite every word of $W(\mathcal{L}(\mathcal{A}))$ in the form $A^{n_1} B^{n_2} A^{n_3} B^{n_4} \cdots A^{n_{2r-1}} B^{n_{2r}}$ where the sequence $n_1, \ldots, n_{2r}$ is first nondecreasing and then nonincreasing, and $n_2 + n_4 + \cdots + n_{2r} < n$. With Lemma~\ref{chainEquivWords}\ref{chain4} we can also guarantee $n_1, n_3, \ldots, n_{2r-1} < n$, so that $\sum_i n_i < 4 n$. In \cite{auluck1951some}, Auluck showed that the number $Q(m)$ of partitions $m = n_1 + \ldots n_r$ of an integer $m$ that are first nondecreasing and then nonincreasing is $\Theta(m^{-4/5} e^{2 \pi \sqrt{m/3}})$. Of course, $v \equiv w$ implies $v \sim w$. Thus the number of non-right-equivalent words for $W(\mathcal{L}(\mathcal{A}))$, and the number of states in its minimal DFA, is at most $1 + \sum_{m = 0}^{4 n-1} Q(m) = O(n^{1/5} e^{4 \pi \sqrt{\frac{n}{3}}})$. \end{proof} \section{Case study: exact number of $1$-symbols} In the previous section we proved a bound for the complexity of the winning set of a bounded permutation invariant language. Here we study a particular case, the language of words with exactly $n$ ones, or $L= (0^*1)^n 0^*$. We not only compute the number of states in the minimal automaton (which is cubic in $n$), but also describe the winning set. Throughout the section $\mathcal{A}$ is the minimal automaton for $L$, described in Figure \ref{01s01s}. For $S \subseteq Q$, we denote $\overline{S} = \{\min(S), \min(S) + 1, \ldots, \max(S)\}$, and for any game state $\gst{S}$ of $W(\mathcal{A})$, denote $\overline{\gst{S}} = \{ \overline{S} \;|\; S \in \gst{S}\}$. \begin{figure} \caption{\label{01s01s} \label{01s01s} \end{figure} \begin{lemma} \label{lem:stateIsInterval} Each game state $\gst{S}$ of $W(\mathcal{A})$ is equivalent to $\overline{\gst{S}}$. \end{lemma} \begin{proof} We prove by induction that for every $w\in \{A,B\}^*$, the game state $\delta_W(\gst{S},w)$ is final iff $\delta_W ( \overline{\gst{S}},w)$ is. \begin{itemize} \item For the empty word $\lambda$, $\delta_W (\gst{S}, \lambda) = \gst{S}$ is final iff $\{n\} \in \gst{S}$ iff $\{n\} \in \overline{\gst{S}} = \delta_W(\overline{\gst{S}}, \lambda)$. \item For $w= B v$, it's easy to see that $ \delta_W (\overline{\gst{S}}, B) = \overline{ \delta_W (\gst{S},B)}$. By the induction hypothesis, $ \delta_W (\gst{S}, B v) = \delta(\delta_W (\gst{S}, B), v)$ is final iff $ \delta_W (\overline{\delta_W (\gst{S}, B)}, v) = \delta_W (\delta_W (\overline{\gst{S}}, B), v) = \delta_W (\overline{\gst{S}}, B v)$ is. \item For $w= A v$, we have $ \overline{\delta_W (\overline{\gst{S}}, A)} = \overline{ \delta_W (\gst{S}, A)}$: For each set $S \in \gst{S}$, we focus on a set $R$ that can be obtained by a combination of choices by reading one $A$. The set $\overline{R}$ is only defined by the leftmost and rightmost elements in $R$. From $\overline{S}$, by making the same choices for the leftmost and rightmost elements we can produce the same leftmost and rightmost elements as in $R$ to obtain a set equivalent to $\overline{R}$. By the induction hypothesis, the game state $\delta_W (\gst{S}, A v) = \delta_W (\delta_W (\gst{S}, A), v)$ is final iff $\delta_W (\overline{\delta_W (\gst{S}, A)}, v) = \delta_W (\overline{\delta_W (\overline{\gst{S}}, A)}, v)$ is. Again by the induction hypothesis, this is equivalent to $\delta_W (\delta_W (\overline{\gst{S}}, A), v) = \delta_W(\overline{\gst{S}}, A v)$ being final. \end{itemize} \end{proof} \begin{lemma} \label{lem:formOfReachables} Let $T$ be the set of integer triples $(i, \ell, N)$ with $0 \leq i \leq n$, $1 \leq \ell \leq n-i+1$ and $1 \leq N \leq n-i-\ell+2$. For $(i, \ell, N) \in T$, let \[ \gst{S}(i, \ell, N) := \{ \{i, \ldots, i+\ell-1\}, \{i+1, \ldots, i+\ell\}, \ldots, \{i+N-1, \ldots, i+\ell+N-2\} \}. \] \begin{roster} \item \label{form1} Each reachable game state of $W(\mathcal{A})$ is equivalent to some $\gst{S}(i, \ell, N)$ for $(i, \ell, N) \in T$, or to $\emptyset$. \item \label{form2} The game states $\gst{S}(i,\ell,N)$ for $(i, \ell, N) \in T$ are nonequivalent. \item \label{form3} Every $\gst{S}(i,\ell,N)$ for $(i, \ell, N) \in T$ is equivalent to some reachable game state. \end{roster} \end{lemma} \begin{proof} We first make the following remarks which follow from Lemma~\ref{lem:stateIsInterval}. Let $S = \{ i, \ldots, j\}$. If $i=j$, then $\delta_W (\{S\}, A) \sim \{ \{i\}, \{i+1\} \}$, otherwise $\delta_W (\{S\}, A) \sim \{ \{i+1, \dots, j\} \}$. In both cases we also have $\delta_W (\{S\}, B) \sim \{ \{i, \dots, j+1\} \}$. We prove~\ref{form1}. From $\gst{S}(i,\ell,N)$, if we read $B$, we have $\gst{S}(i,\ell+1,N)$. If we read $A$ and $\ell=1$, we get $\gst{S}(i,1,N+1)$. If we read $A$ and $\ell > 1$, we obtain $\gst{S}(i+1,\ell-1,N)$. The claim follows since the initial game state is $\gst{S}(0, 1, 1)$, and if a set in the game state contains the state $n+1$, it is equivalent to $\emptyset$. For~\ref{form2}, we first distinguish game states with different lengths $\ell<\ell'$. By reading $A^\ell (B A)^k$ for a suitable $k \geq 0$ we reach a final game state from $\gst{S}(i, \ell, N)$ but not from any $\gst{S}(i', \ell', N')$ with $\ell < \ell'$. Now we suppose we have game states $\gst{S}(i,\ell,N)$ and $\gst{S}(i',\ell,N')$. By reading $A^{\ell-1}$, we get respectively $\gst{S}(i+\ell-1,1,N)$ and $\gst{S}(i'+\ell-1,1,N')$, which are intervals of singleton sets. Since these distributions of singleton sets are different, we can read $(B A)^k$ for a suitable $k \geq 0$ to obtain a final game state from one of them but not the other. For~\ref{form3}, reading $(B A)^i A^{N-1} B^{\ell-1}$ leads to a game state equivalent to $\gst{S}(i,\ell,N)$. \end{proof} The game state $\gst{S}(i,\ell,N)$ is an interval of intervals, where $i$ is the leftmost position of the first interval, $\ell$ is the common length of the intervals, and $N$ is the number of intervals. It is easy to see that reading $B A$ from $\gst{S}(i,\ell,N)$ produces a game state equivalent to $\gst{S}(i+1,\ell,N)$. In other ords, $BA$ can be use to ``make the game state go forward'' in the chain automaton. \begin{proposition} \label{lem:formOfH} The minimal automaton for $W(L)$ has $\frac{n^3}{6}+n^2+\frac{11n}{6}+2$ states. \end{proposition} \begin{proof} The parameter $i$ can vary from $0$ to $n$, $\ell$ from $1$ to $n-i+1$, and $N$ from $1$ to $n-i-\ell + 2$. We need to consider one more state for the sink state $\emptyset$. In total, there are $\frac{n^3}{6}+n^2+\frac{11n}{6}+2$ states. \end{proof} \begin{proposition} $W(L)$ is exactly the set of words $w \in \{A, B\}^*$ such that $|w|_A \geq n$, $|w|_B \leq n$, and every suffix $v$ of $w$ satisfies $|v|_A \geq |v|_B$. \end{proposition} \begin{proof} Every word of $W(L)$ has at least $n$ letters $A$ to let Alice win against Bob when he only plays $0$s. Similary it must have a most $n$ letters $B$ to let Alice wins when Bob only plays $1$s. Only game states of the form $\gst{S}(i,1,N)$ can be accepting. Since doing $A$ decreases the parameter $\ell$ by one, and $B$ increases $\ell$ by one, words of $W(L)$ must have after each $B$ an associated $A$ somewhere in the word. This is equivalent to the suffix condition. Conversely, if a word $w$ has this property, since $B A$ makes the whole game state move forward along the chain automaton, we can move all occurrences of $B A$ to the beginning of the word to obtain $w'$, which is in $W(L)$ iff $w$ is. Then $w' = (B A)^{k_1} A^{k_2} B^{k_3}$ for some $k_1, k_2, k_3 \geq 0$. Because of the suffix condition, there are no $B$ at the end of $w'$, so $k_3 = 0$. We also have $|w'|_A = |w|_A \geq n$ and $|w'|_B = |w|_B \leq n$, so $k_1+k_2 \geq n$ and $k_1 \leq n$. This means $\delta_W(\gst{S}(0,1,1), w') = \gst{S}(k_1, 1, k_2)$ so we have a game state with singletons and one of them is at position $n$, hence it is final and $w', w \in W(L)$. \end{proof} \section{A context-free language} In this section we prove that the winning set operator does not in general preserve context-free languages by studying the winning set of the Dyck langage. For better readability, $0$ stands for the opening parenthesis $($ and $1$ stands for the closing parenthesis $)$. \begin{proposition} \label{DyckWinningNotCFL} Denote by $D$ the Dyck language. The winning shift $W(D)$ is not context-free. \end{proposition} \begin{proof} Suppose by contradiction that $W(D)$ is context-free. Take $L = W(D) \cap (AA)^*(BB)^*(AA)^*$ which is context-free by intersection of a regular language and a context-free language. We claim that $L = \{ A^{2i} B^{2j} A^{2k} \;|\; i \geq j, k\geq 2j \}$. First, when Bob plays $2j$ times in a row, he can close $2j$ parentheses. Alice must play at least $2j$ times before that and open at least $2j$ parentheses in order to have a chance to win. But then Bob can open $2j$ parentheses instead of closing them, which means that when Alice plays a second time, she has to be able to close $4j$ parentheses. This means the right hand side contains $L$. Let then $w = A^{2i} B^{2j} A^{2k}$ with $i \geq j$ and $k \geq 2j$. A winning strategy for Alice on $w$ is to do first play ``01'' $i-j$ times and then ``0'' $2j$ times so that there are $2j$ parentheses left to be closed. After Bob plays $2j$ times, there is an even number $2h$ of parenthesis to be closed, smaller than $4j$. Then Alice wins by playing ``1'' $2h$ times and ``01'' $k-h$ times, which is legal because $h \leq 2j \leq k$. We apply Ogden's lemma on $L$ to show that it's not context-free. Take $N$ obtained from the lemma. Consider the the word $w = A^{2N} B^{2N} A^{4N} \in W(L)$ where we mark the $2N$ occurrences of $B$. By the lemma, $w$ can be written as $xuyvz$ so that $x u^n y v^n z \in L$ for all $n \geq 0$ and ($x$ and $u$ and $y$) or ($y$ and $v$ and $z$) contains at least one marked position. Because $u$ and $v$ can be repeated, one of them only contains $B$s and the other contains only $A$s or only $B$s. But then in the words $x u^n y v^n z$, as $n$ increases, the number of $A$s in the left part or in the right part remains constant while the number of $B$s increases. For large enough $n$ this contradicts $x u^n y v^n z \in L$. \end{proof} \end{document}

\begin{document} \title{Noncommutative Local Systems} \setlength{\parindent}{0pt} \begin{center} \author{ {\textbf{Petr R. Ivankov*}\\ e-mail: * monster.ivankov@gmail.com \\ } } \end{center} \noindent \paragraph{} Gelfand - Na\u{i}mark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theorem \cite{karoubi:k} states that complex topological $K$-theory coincides with $K$-theory of $C^*$-algebras. This article is concerned with generalization of local systems. The classical construction of local system implies an existence of a path groupoid. However the noncommutative geometry does not contain this object. There is a construction of local system which uses covering projections. Otherwise a classical (commutative) notion of a covering projection has a noncommutative generalization. A generalization of noncommutative covering projections supplies a generalization of local systems. \tableofcontents \section{Motivation. Preliminaries} \paragraph{} Local system examples arise geometrically from vector bundles with flat connections, and from topology by means of linear representations of the fundamental group. Generalization of local systems requires a generalization of a topological space given by the Gelfand-Na\u{i}mark theorem \cite{arveson:c_alg_invt} which states the correspondence between locally compact Hausdorff topological spaces and commutative $C^*$-algebras. \begin{thm}\label{gelfand-naimark}\cite{arveson:c_alg_invt} Let $A$ be a commutative $C^*$-algebra and let $\mathcal{X}$ be the spectrum of A. There is the natural $*$-isomorphism $\gamma:A \to C_0(\mathcal{X})$. \end{thm} \paragraph{} So any (noncommutative) $C^*$-algebra may be regarded as a generalized (noncommutative) locally compact Hausdorff topological space. We would like to generalize a notion of a local system. A classical notion of local system uses a fundamental groupoid. \begin{thm} \cite{spanier:at} For each topological space there is a category $\mathscr{P}(\mathcal{X})$ whose objects are points of $\mathcal{X}$, whose morphisms from $x_0$ to $x_1$ are the path classes with $x_0$ as origin and $x_1$ as end, and whose composite is the product of path classes. \end{thm} \begin{defn} \cite{spanier:at} The category $\mathscr{P}(\mathcal{X})$ is called the {\it category of path classes} of $\mathcal{X}$ or the {\it fundamental groupoid}. \end{defn} \begin{defn} \cite{spanier:at} A {\it local system} on a space $\mathcal{X}$ is a covariant functor from fundamental groupoid of $\mathcal{X}$ to some category. For any category $\mathscr{C}$ there is a category of local systems on $\mathcal{X}$ with values in $\mathscr{C}$. Two local systems are said to be {\it equivalent} if they are equivalent objects in this category. \end{defn} \paragraph{} Otherwise it is known that any connected gruopoid is equivalent to a category with single object, i.e. a groupoid is equivalent to a group which is regarded as a category. Any groupoid can be decomposed into connected components, therefore any local system corresponds to representations of groups. It means that in case of linearly connected space $\mathcal{X}$ local systems can be defined by representations of fundamental group $\pi_1(\mathcal{X})$. Otherwise there is an interrelationship between fundamental group and covering projections. This circumstance supplies a following definition \ref{borel_const_comm} and a lemma \ref{borel_local_system_app} which do not explicitly uses a fundamental groupoid. \begin{defn}\label{borel_const_comm}\cite{davis_kirk_at} Let $p : \mathcal{P} \to \mathcal{B}$ be a principal $G$-bundle. Suppose $G$ acts on the left on a space $\mathcal{F}$, i.e. an action $G \times \mathcal{F} \to \mathcal{F}$ is given. Define the {\it Borel construction} \begin{equation*} \mathcal{P} \times_G \mathcal{F} \end{equation*} to be the quotient space $\mathcal{P} \times \mathcal{F} / \approx$ where \begin{equation*} \left(x, f\right) \approx \left(xg, g^{-1}f\right). \end{equation*} \end{defn} We next give one application of the Borel construction. Recall that a local coefficient system is a fiber bundle over $B$ with fiber $A$ and structure group $G$ where $A$ is a (discrete) abelian group and G acts via a homomorphism $G \to \mathrm{Aut}(A)$. \begin{lem}\label{borel_local_system_app}\cite{davis_kirk_at} Every local coefficient system over a path-connected (and semilocally simply connected) space $B$ is of the form \begin{tikzpicture}\label{borel_local_comm} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { A & \widetilde{\mathcal{B}} \times_{\pi_1(\mathcal{B})}A \\ & B \\}; \path[-stealth] (m-1-1) edge node [left] {} (m-1-2) (m-1-2) edge node [right] {$q$} (m-2-2); \end{tikzpicture} i.e., is associated to the principal $\pi_1(\mathcal{B})$-bundle given by the universal cover $\widetilde{\mathcal{B}}$ of $\mathcal{B}$ where the action is given by a homomorphism $\pi_1(\mathcal{B}) \to \mathrm{Aut}(A)$. \end{lem} In lemma \ref{borel_local_comm} the $\mathcal{B}$ is a topological space, the $\widetilde{\mathcal{B}}$ means the universal covering space of $\mathcal{B}$, $\pi = \pi_1(\mathcal{B})$ is the fundamental group of the $\mathcal{B}$. The $\pi$ group equals to group of covering transformations $G(\widetilde{\mathcal{B}}, \mathcal{B})$ of the universal covering space. So above construction does not need fundamental groupoid, it uses a covering projection and a group of covering transformations. However noncommutative generalizations of these notions are developed in \cite{ivankov:infinite_cov_pr}. So local systems can be generalized. We may summarize several properties of the Gelfand - Na\u{i}mark correspondence with the following dictionary. \newline \break \begin{tabular}{|c|c|} \hline TOPOLOGY & ALGEBRA\\ \hline Locally compact space & $C^*$-algebra\\ Covering projection & Noncommutative covering projection \\ Group of covering transformations & Noncommutative group of covering transformations \\ Local system & ? \\ \hline \end{tabular} \newline \newline \break This article assumes elementary knowledge of following subjects: \begin{enumerate} \item Set theory \cite{halmos:set}, \item Category theory \cite{spanier:at}, \item Algebraic topology \cite{spanier:at}, \item $C^*$-algebras and operator theory \cite{pedersen:ca_aut}, \item Differential geometry \cite{koba_nomi:fgd}, \item Spectral triples and their connections \cite{connes:c_alg_dg,connes:ncg94,varilly:noncom,varilly_bondia}. \end{enumerate} The terms "set", "family" and "collection" are synonyms. Following table contains used in this paper notations. \newline \begin{tabular}{|c|c|} \hline Symbol & Meaning\\ \hline \\ $A^G$ & Algebra of $G$ invariants, i.e. $A^G = \{a\in A \ | \ ga=a, \forall g\in G\}$\\ $\mathrm{Aut}(A)$ & Group * - automorphisms of $C^*$ algebra $A$\\ $B(H)$ & Algebra of bounded operators on Hilbert space $H$\\ $\mathbb{C}$ (resp. $\mathbb{R}$) & Field of complex (resp. real) numbers \\ $C(\mathcal{X})$ & $C^*$ - algebra of continuous complex valued \\ & functions on topological space $\mathcal{X}$\\ $C_0(\mathcal{X})$ & $C^*$ - algebra of continuous complex valued \\ & functions on topological space $\mathcal{X}$\\ & functions on topological space $\mathcal{X}$\\ $G(\widetilde{\mathcal{X}} | \mathcal{X})$ & Group of covering transformations of covering projection \\ & $\widetilde{\mathcal{X}} \to \mathcal{X}$ \cite{spanier:at}\\ $H$ &Hilbert space \\ $M(A)$ & A multiplier algebra of $C^*$-algebra $A$\\ $\mathscr{P}(\mathcal{X})$ & Fundamental groupoid of a topological space $\mathcal{X}$\\ $U(H) \subset \mathcal{B}(H) $ & Group of unitary operators on Hilbert space $H$\\ $U(A) \subset A $ & Group of unitary operators of algebra $A$\\ $U(n) \subset GL(n, \mathbb{C}) $ & Unitary subgroup of general linear group\\ $\mathbb{Z}$ & Ring of integers \\ $\mathbb{Z}_m$ & Ring of integers modulo $m$ \\ $\Omega$ & Natural contravariant functor from category of commutative \\ & $C^*$ - algebras, to category of Hausdorff spaces\\ \hline \end{tabular} \section{Noncommutative covering projections} \paragraph{} In this section we recall the described in \cite{ivankov:infinite_cov_pr} construction of a noncommutative covering projection. Instead the expired "rigged space" notion we use the "Hilbert module" one. \subsection{Hermitian modules and functors} \begin{defn} \cite{rieffel_morita} Let $B$ be a $C^*$-algebra. By a (left) {\it Hermitian $B$-module} we will mean the Hilbert space $H$ of a non-degenerate *-representation $A \rightarrow B(H)$. Denote by $\mathbf{Herm}(B)$ the category of Hermitian $B$-modules. \end{defn} \paragraph{} Let $A$, $B$ be $C^*$-algebras. In this section we will study some general methods for construction of functors from $\mathbf{Herm}(B)$ to $\mathbf{Herm}(A)$. \begin{defn} \cite{rieffel_morita} Let $B$ be a $C^*$-algebra. By (right) {\it pre-$B$-Hilbert module} we mean a vector space, $X$, over complex numbers on which $B$ acts by means of linear transformations in such a way that $X$ is a right $B$-module (in algebraic sense), and on which there is defined a $B$-valued sesquilinear form $\langle,\rangle_X$ conjugate linear in the first variable, such that \begin{enumerate} \item $\langle x, x \rangle_B \ge 0$ \item $\left(\langle x, y \rangle_X\right)^* = \langle y, x \rangle_X$ \item $\langle x, yb \rangle_X = \langle x, y \rangle_Xb$. \end{enumerate} \end{defn} \begin{empt} It is easily seen that if we factor a pre-$B$-Hilbert module by subspace of the elements $x$ for which $\langle x, x \rangle_X = 0$, the quotient becomes in a natural way a pre-$B$-Hilbert module having the additional property that inner product is definite, i.e. $\langle x, x \rangle_X > 0$ for any non-zero $x\in X$. On a pre-$B$-Hilbert module with definite inner product we can define a norm $\|\|$ by setting \begin{equation}\label{rigged_norm_eqn} \|x\|=\|\langle x, x \rangle_X\|^{1/2}. \end{equation} From now we will always view a pre-$B$-Hilbert module with definite inner product as being equipped with this norm. The completion of $X$ with this norm is easily seen to become again a pre-$B$-Hilbert module. \end{empt} \begin{defn} \cite{rieffel_morita} Let $B$ be a $C^*$-algebra. By a {\it Hilbert $B$-module} we will mean a pre-$B$-Hilbert module, $X$, satisfying the following conditions: \begin{enumerate} \item If $\langle x, x \rangle_X\ = 0$ then $x = 0$, for all $x \in X$ \item $X$ is complete for the norm defined in (\ref{rigged_norm_eqn}). \end{enumerate} \end{defn} \begin{exm}\label{fin_rigged_exm} Let $A$ be a $C^*$-algebra and a finite group acts on $A$, $A^G$ is the algebra of $G$-invariants. Then $A$ is a Hilbert $A^G$-module on which is defined following $A^G$-valued form \begin{equation}\label{inv_scalar_product} \langle x, y \rangle_A = \frac{1}{|G|} \sum_{g \in G} g(x^*y). \end{equation} Since given by \ref{inv_scalar_product} sum is $G$-invariant we have $ \langle x, y \rangle_A \in A^G$. \end{exm} \paragraph{} Viewing a Hilbert $B$-module as a generalization of an ordinary Hilbert space, we can define what we mean by bounded operators on a Hilbert $B$-module. \begin{defn}\cite{rieffel_morita} Let $X$ be a Hilbert $B$-module. By a {\it bounded operator} on $X$ we mean a linear operator, $T$, from $X$ to itself which satisfies following conditions: \begin{enumerate} \item for some constant $k_T$ we have \begin{equation}\nonumber \langle Tx, Tx \rangle_X \le k_T \langle x, x \rangle_X, \ \forall x\in X, \end{equation} or, equivalently $T$ is continuous with respect to the norm of $X$. \item there is a continuous linear operator, $T^*$, on $X$ such that \begin{equation}\nonumber \langle Tx, y \rangle_X = \langle x, T^*y \rangle_X, \ \forall x, y\in X. \end{equation} \end{enumerate} It is easily seen that any bounded operator on a $B$-Hilbert module will automatically commute with the action of $B$ on $X$ (because it has an adjoint). We will denote by $\mathcal{L}(X)$ (or $\mathcal{L}_B(X)$ there is a chance of confusion) the set of all bounded operators on $X$. Then it is easily verified than with the operator norm $\mathcal{L}(X)$ is a $C^*$-algebra. \end{defn} \begin{defn}\cite{pedersen:ca_aut} If $X$ is a Hilbert $B$-module then denote by $\theta_{\xi, \zeta} \in \mathcal{L}_B(X)$ such that \begin{equation}\nonumber \theta_{\xi, \zeta} (\eta) = \zeta \langle\xi, \eta \rangle_X , \ (\xi, \eta, \zeta \in X) \end{equation} Norm closure of a generated by such endomorphisms ideal is said to be the {\it algebra of compact operators} which we denote by $\mathcal{K}(X)$. The $\mathcal{K}(X)$ is an ideal of $\mathcal{L}_B(X)$. Also we shall use following notation $\xi\rangle \langle \zeta \stackrel{\text{def}}{=} \theta_{\xi, \zeta}$. \end{defn} \begin{defn}\cite{rieffel_morita}\label{corr_defn} Let $A$ and $B$ be $C^*$-algebras. By a {\it Hilbert $B$-$A$-correspondence} we mean a Hilbert $B$-module, which is a left $A$-module by means of *-homomorphism of $A$ into $\mathcal{L}_B(X)$. \end{defn} \begin{empt}\label{herm_functor_defn} Let $X$ be a Hilbert $B$-$A$-correspondence. If $V\in \mathbf{Herm}(B)$ then we can form the algebraic tensor product $X \otimes_{B_{\mathrm{alg}}} V$, and equip it with an ordinary pre-inner-product which is defined on elementary tensors by \begin{equation}\nonumber \langle x \otimes v, x' \otimes v' \rangle = \langle \langle x',x \rangle_B v, v' \rangle_V. \end{equation} Completing the quotient $X \otimes_{B_{\mathrm{alg}}} V$ by subspace of vectors of length zero, we obtain an ordinary Hilbert space, on which $A$ acts (by $a(x \otimes v)=ax\otimes v$) to give a *-representation of $A$. We will denote the corresponding Hermitian module by $X \otimes_{B} V$. The above construction defines a functor $X \otimes_{B} -: \mathbf{Herm}(B)\to \mathbf{Herm}(A)$ if for $V,W \in \mathbf{Herm}(B)$ and $f\in \mathrm{Hom}_B(V,W)$ we define $f\otimes X \in \mathrm{Hom}_A(V\otimes X, W\otimes X)$ on elementary tensors by $(f \otimes X)(x \otimes v)=x \otimes f(v)$. We can define action of $B$ on $V\otimes X$ which is defined on elementary tensors by \begin{equation}\nonumber b(x \otimes v)= (x \otimes bv) = x b \otimes v. \end{equation} \end{empt} \subsection{Galois correspondences} \begin{defn}\label{herm_a_g_defn} Let $A$ be a $C^*$-algebra, $G$ is a finite or countable group which acts on $A$. We say that $H \in \mathbf{Herm}(A)$ is a {\it $A$-$G$ Hermitian module} if \begin{enumerate} \item Group $G$ acts on $H$ by unitary $A$-linear isomorphisms, \item There is a subspace $H^G \subset H$ such that \begin{equation}\label{g_act} H = \bigoplus_{g\in G}gH^G. \end{equation} \end{enumerate} Let $H$, $K$ be $A$-$G$ Hermitian modules, a morphism $\phi: H\to K$ is said to be a $A$-$G$-morphism if $\phi(gx)=g\phi(x)$ for any $g \in G$. Denote by $\mathbf{Herm}(A)^G$ a category of $A$-$G$ Hermitian modules and $A$-$G$-morphisms. \end{defn} \begin{rem} Condition 2 in the above definition is introduced because any topological covering projection $\widetilde{\mathcal{X}} \to \mathcal{X}$ commutative $C^*$ algebras $C_0\left(\widetilde{\mathcal{X}}\right)$, $C_0\left(\mathcal{X}\right)$ satisfies it with respect to the group of covering transformations $G(\widetilde{\mathcal{X}}| \mathcal{X})$. \end{rem} \begin{defn} Let $H$ be $A$-$G$ Hermitian module, $B\subset M(A)$ is sub-$C^*$-algebra such that $(ga)b = g(ab)$, $b(ga) = g(ba)$, for any $a\in A$, $b \in B$, $g \in G$. There is a functor $(-)^G: \mathbf{Herm}(A)^G \to\mathbf{Herm}(B)$ defined by following way \begin{equation} H \mapsto H^G. \end{equation} This functor is said to be the {\it invariant functor}. \end{defn} \begin{defn} Let $_AX_B$ be a Hilbert $B$-$A$ correspondence, $G$ is finite or countable group such that \begin{itemize} \item $G$ acts on $A$ and $X$, \item Action of $G$ is equivariant, i.e. $g (a\xi) = (ga) (g\xi)$ , and $B$ invariant, i.e. $g(\xi b)=(g\xi)b$ for any $\xi \in X$, $b \in B$, $a\in A$, $g \in G$, \item Inner-product of $G$ is equivariant, i.e. $\langle g\xi, g \zeta\rangle_X = \langle\xi, \zeta\rangle_X$ for any $\xi, \zeta \in X$, $g \in G$. \end{itemize} Then we say that $_AX_B$ is a {\it $G$-equivariant Hilbert $B$-$A$-correspondence}. \end{defn} \paragraph{} Let $_AX_B$ be a $G$-equivariant Hilbert $B$-$A$-correspondence. Then for any $H\in \mathbf{Herm}(B)$ there is an action of $G$ on $X\otimes_B H$ such that \begin{equation*} g \left(x \otimes \xi\right) = \left(x \otimes g\xi\right). \end{equation*} \begin{defn}\label{inf_galois_defn} Let $_AX_B$ be a $G$-equivariant Hilbert $B$-$A$-correspondence. We say that $_AX_B$ is {\it $G$-Galois Hilbert $B$-$A$-correspondence} if it satisfies following conditions: \begin{enumerate} \item $X \otimes_B H$ is a $A$-$G$ Hermitian module, for any $H \in \mathbf{Herm}(B)$, \item A pair $\left(X \otimes_B -, \left(-\right)^G\right)$ such that \begin{equation}\nonumber X \otimes_B -: \mathbf{Herm}(B) \to \mathbf{Herm}(A)^G, \end{equation} \begin{equation}\nonumber (-)^G: \mathbf{Herm}(A)^G \to \mathbf{Herm}(B). \end{equation} is a pair of inverse equivalence. \end{enumerate} \end{defn} Following theorem is an analog of to theorems described in \cite{miyashita_infin_outer_gal}, \cite{takeuchi:inf_out_cov}. \begin{thm}\cite{ivankov:infinite_cov_pr}\label{main_lem} Let $A$ and $\widetilde{A}$ be $C^*$-algebras, $_{\widetilde{A}}X_A$ be a $G$-equivariant Hilbert $A$-$\widetilde{A}$-correspondence. Let $I$ be a finite or countable set of indices, $\{e_i\}_{i\in I} \subset M(A)$, $\{\xi_i\}_{i\in I} \subset \ _{\widetilde{A}}X_A$ such that \begin{enumerate} \item \begin{equation}\label{1_mb} 1_{M(A)} = \sum_{i\in I}^{}e^*_ie_i, \end{equation} \item \begin{equation}\label{1_mkx} 1_{M(\mathcal{K}(X))} = \sum_{g\in G}^{} \sum_{i \in I}^{}g\xi_i\rangle \langle g\xi_i , \end{equation} \item \begin{equation}\label{ee_xx} \langle \xi_i, \xi_i \rangle_X = e_i^*e_i, \end{equation} \item \begin{equation}\label{g_ort} \langle g\xi_i, \xi_i\rangle_X=0, \ \text{for any nontrivial} \ g \in G. \end{equation} \end{enumerate} Then $_{\widetilde{A}}X_A$ is a $G$-Galois Hilbert $A$-$\widetilde{A}$-correspondence. \end{thm} \begin{defn} Consider a situation from the theorem \ref{main_lem}. Let us consider two specific cases \begin{enumerate} \item $e_i \in A$ for any $i \in I$, \item $\exists i \in I \ e_i \notin A$. \end{enumerate} Norm completion of the generated by operators \begin{equation*} g\xi_i^* \rangle \langle g \xi_i \ a; \ g \in G, \ i \in I, \ \begin{cases} a \in M(A), & \text{in case 1},\\ a \in A, & \text{in case 2}. \end{cases} \end{equation*} algebra is said to be the {\it subordinated to $\{\xi_i\}_{i \in I}$ algebra}. If $\widetilde{A}$ is the subordinated to $\{\xi_i\}_{i \in I}$ then \begin{enumerate} \item $G$ acts on $\widetilde{A}$ by following way \begin{equation*} g \left( \ g'\xi_i \rangle \langle g' \xi_i \ a \right) = gg'\xi_i \rangle \langle gg' \xi_i \ a; \ a \in M(A). \end{equation*} \item $X$ is a left $A$ module, moreover $_{\widetilde{A}}X_A$ is a $G$-Galois Hilbert $A$-$\widetilde{A}$-correspondence. \item There is a natural $G$-equivariant *-homomorphism $\varphi: A \to M\left(\widetilde{A}\right)$, $\varphi$ is equivariant, i.e. \begin{equation} \varphi(a)(g\widetilde{a})= g \varphi(a)(\widetilde{a}); \ a \in A, \ \widetilde{a}\in \widetilde{A}. \end{equation} \end{enumerate} A quadruple $\left(A, \widetilde{A}, _{\widetilde{A}}X_A, G\right)$ is said to be a {\it Galois quadruple}. The group $G$ is said to be a {\it group Galois transformations} which shall be denoted by $G\left(\widetilde{A}\ | \ A\right)=G$. \end{defn} \begin{rem} Henceforth subordinated algebras only are regarded as noncommutative generalizations of covering projections. \end{rem} \begin{defn} If $G$ is finite then bimodule $_{\widetilde{A}}X_A$ can be replaced with $_{\widetilde{A}}\widetilde{A}_A$ where product $\langle \ , \ \rangle_{\widetilde{A}}$ is given by \eqref{inv_scalar_product}. In this case a Galois quadruple $\left(A, \widetilde{A}, _{\widetilde{A}}X_A, G\right)=\left(A, \widetilde{A}, _{\widetilde{A}}A_A, G\right)$ can be replaced with a {\it Galois triple} $\left(A, \widetilde{A}, G\right)$. \end{defn} \subsection{Infinite noncommutative covering projections} \paragraph{} In case of commutative $C^*$-algebras definition \ref{inf_galois_defn} supplies algebraic formulation of infinite covering projections of topological spaces. However I think that above definition is not a quite good analogue of noncommutative covering projections. Noncommutative algebras contains inner automorphisms. Inner automorphisms are rather gauge transformations \cite{gross_gauge} than geometrical ones. So I think that inner automorphisms should be excluded. Importance of outer automorphisms was noted by Miyashita \cite{miyashita_fin_outer_gal,miyashita_infin_outer_gal}. It is reasonably take to account outer automorphisms only. I have set more strong condition. \begin{defn}\label{gen_in_def}\cite{rieffel_finite_g} Let $A$ be $C^*$-algebra. A *-automorphism $\alpha$ is said to be {\it generalized inner} if it is given by conjugating with unitaries from multiplier algebra $M(A)$. \end{defn} \begin{defn}\label{part_in_def}\cite{rieffel_finite_g} Let $A$ be $C^*$ - algebra. A *- automorphism $\alpha$ is said to be {\it partly inner} if its restriction to some non-zero $\alpha$-invariant two-sided ideal is generalized inner. We call automorphism {\it purely outer} if it is not partly inner. \end{defn} Instead definitions \ref{gen_in_def}, \ref{part_in_def} following definitions are being used. \begin{defn} Let $\alpha \in \mathrm{Aut}(A)$ be an automorphism. A representation $\rho : A\rightarrow B(H)$ is said to be {\it $\alpha$ - invariant} if a representation $\rho_{\alpha}$ given by \begin{equation*} \rho_{\alpha}(a)= \rho(\alpha(a)) \end{equation*} is unitary equivalent to $\rho$. \end{defn} \begin{defn} Automorphism $\alpha \in \mathrm{Aut}(A)$ is said to be {\it strictly outer} if for any $\alpha$- invariant representation $\rho: A \rightarrow B(H) $, automorphism $\rho_{\alpha}$ is not a generalized inner automorphism. \end{defn} \begin{defn}\label{nc_fin_cov_pr_defn} A Galois quadruple $\left(A, \widetilde{A}, _{\widetilde{A}}X_A, G\right)$ (resp. a triple $\left(A, \widetilde{A}, G\right)$) with countable (resp. finite) $G$ is said to be a {\it noncommutative infinite (resp. finite) covering projection} if action of $G$ on $\widetilde{A}$ is strictly outer. \end{defn} \section{Noncommutative generalization of local systems} \begin{defn}\label{loc_sys_defn} Let $A$ be a $C^*$-algebra, and let $\mathscr{C}$ be a category. A {\it noncommutative local system} contains following ingredients: \begin{enumerate} \item A noncommutative covering projection $\left(A, \widetilde{A}, _{\widetilde{A}}X_A, G\right)$ (or $\left(A, \widetilde{A}, G\right)$), \item A covariant functor $F: G \to \mathscr{C}$, \end{enumerate} where $G$ is regarded as a category with a single object $e$, which is the unity of $G$. Indeed a local system is a group homomorphism $G \to \mathrm{Aut}(F(e))$. \end{defn} \begin{exm} If $\mathcal{X}$ is a linearly connected space then there is the equivalence of categories $\mathscr{P}(\mathcal{X}) \approx \pi_1(\mathcal{X})$. Let $F: \mathscr{P}(\mathcal{X})\to \mathscr{C}$ is a local system then there is an object $A$ in $\mathscr{C}$ such that $F$ is uniquely defined by a group homomorphism $f: \pi_1(\mathcal{X}) \to \mathrm{Aut}(A)$. Let $G=\pi_1(\mathcal{X})/\mathrm{ker}f$ be a factor group and let $\widetilde{\mathcal{X}} \to \mathcal{X}$ be a covering projection such that $G(\widetilde{\mathcal{X}} | \mathcal{X})\approx G$. Then there is a natural group homomorphism $G \to \mathrm{Aut}(A)$ which can be regarded as covariant functor $G \to \mathscr{C}$. If $\mathcal{X}$ is locally compact and Hausdorff than from \cite{ivankov:infinite_cov_pr} it follows that there is a noncommutative covering projection $\left(C_0(\mathcal{X}), C_0(\mathcal{\widetilde{X}}), \ _{C_0(\mathcal{X})}X_{C_0(\mathcal{\widetilde{X}})} \ , G\right)$. So a noncommutative local system is a generalization of a commutative one. \end{exm} \section{Noncommutative bundles with flat connections} \subsection{Cotensor products} \begin{empt} {\it Cotensor products associated with Hopf algebras}. Let $H$ be a Hopf algebra over a commutative ring $k$, with bijective antipode $S$. We use the Sweedler notation \cite{karaali:ha} for the comultiplication on $H$: $\Delta(h)= h_{(1)}\otimes h_{(2)}$. $\mathcal{M}^H$ (respectively ${}^H\mathcal{M}$) is the category of right (respectively left) $H$-comodules. For a right $H$-coaction $\rho$ (respectively a left $H$-coaction $\lambda$) on a $k$-module $M$, we denote $$\rho(m)=m_{[0]}\otimes m_{[1]}\quad \ \mathrm{and} \ \quad\lambda(m)=m_{[-1]}\otimes m_{[0]}.$$ Let $M$ be a right $H$-comodule, and $N$ a left $H$-comodule. The cotensor product $M\square_H N$ is the $k$-module \begin{equation}\label{cotensor_hopf} M\square_H N= \left\{\sum_i m_i\otimes n_i\in M\otimes N~|~\sum_i \rho(m_i)\otimes n_i= \sum_i m_i\otimes \lambda(n_i)\right\}. \end{equation} If $H$ is cocommutative, then $M\square_H N$ is also a right (or left) $H$-comodule. \end{empt} \begin{empt} {\it Cotensor products associated with groups}. Let $G$ be a finite group. A set $H = \mathrm{Map}(G, \mathbb{C})$ has a natural structure of commutative Hopf algebra (See \cite{hajac:toknotes}). Addition (resp. multiplication) on $H$ is pointwise addition (resp. pointwise multiplication). Let $\delta_g\in H, ( g \in G)$ be such that \begin{equation}\label{group_hopf_action_rel} \delta_g(g')\left\{ \begin{array}{c l} 1 & g'=g\\ 0 & g' \ne g \end{array}\right. \end{equation} Comultiplication $\Delta: H \rightarrow H \otimes H$ is induced by group multiplication \begin{equation}\nonumber \Delta f(g) = \sum_{g_1 g_2 = g} f(g_1) \otimes f(g_2); \ \forall f \in \mathrm{Map}(G, \mathbb{C}), \ \forall g\in G. \end{equation} i.e. \begin{equation}\nonumber \Delta \delta_g = \sum_{g_1 g_2 = g} \delta_{g_1} \otimes \delta_{g_2}; \ \forall g\in G, \end{equation} Let $M$ (resp. $N$) be a linear space with right (resp. left) action of $G$ then \begin{equation}\label{cotensor_g} M\square_{\mathrm{Map}(G,\mathbb{C})}N = \left\{\sum_i m_i\otimes n_i\in M\otimes N~|~\sum_i m_i g\otimes n_i= \sum_i m_i\otimes gn_i;~\forall g\in G\right\}. \end{equation} Henceforth we denote by $M\square_GN$ a cotensor product $M\square_{\mathrm{Map}(G,\mathbb{C})}N$. \end{empt} \subsection{Bundles with flat connections in differential geometry}\label{fvb_dg} \paragraph{} I follow to \cite{koba_nomi:fgd} in explanation of the differential geometry and flat bundles. \begin{prop}\label{comm_cov_mani}(Proposition 5.9 \cite{koba_nomi:fgd}) \begin{enumerate} \item Given a connected manifold $M$ there is a unique (unique up to isomorphism) universal covering manifold, which will be denoted by $\widetilde{M}$. \item The universal covering manifold $\widetilde{M}$ is a principal fibre bundle over $M$ with group $\pi_1(M)$ and projection $p: \widetilde{M} \to M$, where $\pi_1(M)$ is the first homotopy group of $M$. \item The isomorphism classes of covering spaces over $M$ are in 1:1 correspondence with the conjugate classes of subgroups of $\pi_1(M)$. The correspondence is given as follows. To each subgroup $H$ of $\pi_1(M)$, we associate $E=\widetilde{M}/H$. Then the covering manifold $E$ corresponding to $H$ is a fibre bundle over $M$ with fibre $\pi_1(M)/H$ associated with the principal bundle $\widetilde{M}(M, \pi_1(M))$. If $H$ is a normal subgroup of $\pi_1(M)$, $E=\widetilde{M}/H$ is a principal fibre bundle with group $\pi_1(M)/H$ and is called a regular covering manifold of $M$. \end{enumerate} \end{prop} \paragraph{}Let $\Gammamma$ be a flat connection $P(M, G)$, where $M$ is connected and paracompact. Let $u_0\in P$; $M^*=P(u_0)$, the holonomy bundle through $u_0$; $M^*$ is a principal fibre bundle over $M$ whose structure group is the holomomy group $\Phi(u_0)$. In \cite{koba_nomi:fgd} is explained that $\Phi(u_0)$ is discrete, and since $M^*$ is connected, $M^*$ is a covering space of $M$. Set $x_0=\pi(u_0)\in M$. Every closed curve of $M$ starting from $x_0$ defines, by means of the parallel displacement along it, an element of $\Phi(u_0)$. In \cite{koba_nomi:fgd} it is explained that the same element of the first homotopy group $\pi_1(M, x_0)$ give rise to the same element of $\Phi(u_0)$. Thus we obtain a homomorphism of $\pi_1(M, x_0)$ onto $\Phi(u_0)$. Let $N$ be a normal subgroup of $\Phi(u_0)$ and set $M'=M^*/N$. Then $M'$ is principal fibre bundle over $M$ with structure group $\Phi(u_0)/N$. In particular $M'$ is a covering space of $M$. Let $P'(M',G)$ be the principal fibre bundle induced by covering projection $M'\to M$. There is a natural homomorphism $f: P' \to P$ \cite{koba_nomi:fgd}. \begin{prop}\label{flat_dg_prop} (Proposition 9.3 \cite{koba_nomi:fgd}) There exists a unique connection $\Gammamma'$ in $P'(M',G)$ which is is mapped into $\Gammamma$ by homomorphism $f: P'\to P$. The connection $\Gammamma'$ is flat. If $u'_0$ is a point of $P'$ such that $f(u'_0)=u_0$, then the holonomy group $\Phi(u'_0)$ of $\Gammamma'$ with reference point $u'_0$ is isomorphically mapped onto $N$ by $f$. \end{prop} \begin{empt}\label{dg_fl_con_ingr}{\it Construction of flat connections} Let $M$ be a manifold. Proposition \ref{flat_dg_prop} supplies construction of flat bundle $P(M,G)$ which imply following ingredients: \begin{enumerate} \item A covering projection $M'\to M$. \item A principal bundle $P'(M', G)$ with a flat connection $\Gammamma$. \end{enumerate} \end{empt} \begin{empt}\label{can_fl_conn}{\it Associated vector bundle}. A principal bundle $P(M,G)$ and a flat connection $\Gammamma$ are given by these ingredients. If $G$ acts on $\mathbb{C}^n$ then there is an associated with $P(M,G)$ vector fibre $\mathcal{F}$ bundle with a standard fibre $\mathbb{C}^n$. A space $F$ of continuous sections of $\mathcal{F}$ is a finitely generated projective $C(M)$-module. See \cite{koba_nomi:fgd}. \end{empt} \begin{empt}\label{can_fl_bundle}{\it Canonical flat connection and flat bunles}. There is a specific case of flat principal bundle such that $P'=M'\times G$ and $\Gammamma$ is a canonical flat connection \cite{koba_nomi:fgd}. In this case the existence of $P(M,G)$ depends only on $\pi_1(M)$ and does not depend on differential structure of $M$. \end{empt} \begin{empt}\label{comm_fund_k}{\it Local systems and $K$-theory}. If $R(G)$ is the group representation ring and $R_0(G)$ is a subgroup of zero virtual dimension then there is a natural homomorphism $R_0(G) \to K^0(M)$ described in \cite{gilkey:odd_space,wolf:const_curv}. \end{empt} \subsection{Topological noncommutative bundles with flat connections} \paragraph{} There are noncommutative generalizations of described in \ref{fvb_dg} constructions. According to Serre Swan theorem \cite{karoubi:k} any vector bundle over space $\mathcal{X}$ corresponds to a projective $C_0(\mathcal{X})$ module. \begin{defn} Let $\left(A, \widetilde{A}, G\right)$ be a finite noncommutative covering projection. According to definition \ref{loc_sys_defn} any group homomorphism $G \to U(n)$ is a local system. There is a natural linear action of $G$ on $\mathbb{C}^n$, and $\widetilde{A}\square_G\mathbb{C}^n$ is a left $A$-module which is said to be a {\it topological noncommutative bundle with flat connection}. \end{defn} \begin{lem} Let $\left(A, \widetilde{A}, G\right)$ be a finite noncommutative covering projection, and let $P = \widetilde{A}\square_G\mathbb{C}^n$ be a topological noncommutative bundle with flat connection. Then $P$ is a finitely generated projective left and right $A$-module. \end{lem} \begin{proof} According to definition $\widetilde{A}$ is a left finitely generated projective $A$-module. A left $A$-module $\widetilde{A}\otimes_{\mathbb{C}}\mathbb{C}^n$ is also finitely generated and projective because $\widetilde{A}\otimes_{\mathbb{C}}\mathbb{C}^n \approx \widetilde{A}^n$. There is a projection $p: \widetilde{A}\otimes_{\mathbb{C}}\mathbb{C}^n \to \widetilde{A}\otimes_{\mathbb{C}}\mathbb{C}^n$ given by: \begin{equation*} p(a \otimes x) = \frac{1}{|G|}\sum_{g\in G} ag \otimes g^{-1}x. \end{equation*} The image of $p$ is $P$, therefore $P$ is projective left $A$-module. Similarly we can prove that $P$ is a finitely generated projective right $A$-module \end{proof} \begin{exm} Let $M$ be a differentiable manifold $M'\to M$ is a covering projection $P'=M' \times U(n)$ is a principal bundle with a canonical flat connection $\Gammamma'$. So there are all ingredients of \ref{dg_fl_con_ingr}. So we have a principal bundle $P(M, U(n))$ with a flat connection $\Gammamma$. There is a noncommutative covering projection $\left(C(M), C(M'), _{C(M')}X_{C(M)}, G\right)$. Let $\mathcal{F}$ (resp. $\mathcal{F'}$) be a vector bundle associated with $P(M,U(n))$ (resp. $P(M',U(n))$), and let $F$ (resp. $F'$) be a projective finitely generated $C(M)$ (resp. $C(M')$ module which corresponds to $\mathcal{F}$ (resp. $\mathcal{F'}$). Then we have $F = C(M')\square_GF'$, i.e. $F$ is a topological flat bundle. \end{exm} \begin{rem} Since existence of $P(M, U(n)$ depend on topology of $M$ only we use a notion "topological noncommutative bundle with flat connection" is used for its noncommutative generalization. \end{rem} \begin{exm}\label{nc_torus_fin_cov} Let $A_{\theta}$ be a noncommutative torus $\left(A_{\theta}, A_{\theta'}, \mathbb{Z}_m\times\mathbb{Z}_n\right)$ a Galois triple described in \cite{ivankov:infinite_cov_pr}. Any group homomorphism $\mathbb{Z}_m\times\mathbb{Z}_n\to U(1)$ induces a topological noncommutative flat bundle. \end{exm} \subsection{General noncommutative bundles with flat connections} \begin{empt}\label{n_f_b_constr} A vector fibre bundle with a flat connection is not necessary a topological bundle with flat connection, since proposition \ref{fvb_dg} and construction \ref{dg_fl_con_ingr} does not require it. However general case of \ref{fvb_dg} and construction \ref{dg_fl_con_ingr} have a noncommutative analogue. The analogue requires a noncommutative generalization of differentiable manifolds with flat connections. Generalization of a spin manifold is a spectral triple \cite{connes:c_alg_dg,connes:ncg94,varilly:noncom,varilly_bondia}. First of all we generalize the proposition \ref{comm_cov_mani}. Suppose that there is a spectral triple $(\mathcal{B}, H, D)$ such that \begin{itemize} \item $\mathcal{B} \subset B$ is a pre-$C^*$-algebra which is a dense subalgebra in $B$. \item there is a faithful representation $B \to B(H)$. \end{itemize} Let $\left(B, A, G\right)$ be a finite noncommutative covering projection. According to 8.2 of \cite{ivankov:infinite_cov_pr} there is the spectral triple $(\mathcal{A}, A \otimes_BH, \widetilde{D} )$ such that \begin{itemize} \item $\mathcal{A} \subset A$ is a pre-$C^*$-algebra which is a dense subalgebra of $A$. \item $\widetilde{D}gh = g\widetilde{D}h$, for any $g \in G$, $h \in \Dom \widetilde{D}$. \end{itemize} Let $\mathcal{F}$ be a finite projective right $\mathcal{B}$-module with a flat connection $\nabla: \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{B}} \Omega^1(\mathcal{B})$. Let $\mathcal{E} = \mathcal{F}\otimes_{\mathcal{B}} \mathcal{A}$ be a projective finitely generated $\mathcal{A}$-module and the action of $G$ on $\mathcal{E}$ is induced by the action of $G$ on $\mathcal{A}$. According to \cite{ivankov:infinite_cov_pr} connection $\nabla$ can be naturally lifted to $\widetilde{\nabla}: \mathcal{E}\to \mathcal{E} \otimes_{\mathcal{B}} \Omega^1(\mathcal{B})$. Let $\mathcal{E}'$ be an isomorphic to $\mathcal{E}$ as $\mathcal{A}$-module and there is an action of $G$ on $\mathcal{E}'$ such that \begin{equation}\label{twisted_act} g(xa)=(gx)(ga); \ \forall x \in \mathcal{E}, \ \forall a \in \mathcal{A}, \ \forall g \in G. \end{equation} Different actions of $G$ give different $\mathcal{B}$-modules $\mathcal{F} = \mathcal{E}\square_G \mathcal{A}$, $\mathcal{F}' = \mathcal{E}'\square_G \mathcal{A}$. Both $\mathcal{F}$ and $\mathcal{F}'$ can be included into following sequences \begin{equation}\label{seqf} \mathcal{F} \xrightarrow{i} \mathcal{E} \xrightarrow{p} \mathcal{F}, \end{equation} \begin{equation*} \mathcal{F}' \xrightarrow{i'} \mathcal{E}' \xrightarrow{p'} \mathcal{F}'. \end{equation*} These sequences induce following \begin{equation}\label{seqfo} \mathcal{F} \otimes_{\mathcal{B}} \Omega^1(\mathcal{B}) \xrightarrow{i \otimes \mathrm{Id}_{\Omega^1(\mathcal{B})}} \mathcal{E}\otimes_{\mathcal{B}}\Omega^1(\mathcal{B}) \xrightarrow{p \otimes \mathrm{Id}_{\Omega^1(\mathcal{B})}} \mathcal{F}\otimes_{\mathcal{B}}\Omega^1(\mathcal{B}), \end{equation} \begin{equation*} \mathcal{F}' \otimes_{\mathcal{B}} \Omega^1(\mathcal{B}) \xrightarrow{i' \otimes \mathrm{Id}_{\Omega^1(\mathcal{B})}} \mathcal{E}'\otimes_{\mathcal{B}}\Omega^1(\mathcal{B}) \xrightarrow{p' \otimes \mathrm{Id}_{\Omega^1(\mathcal{B})}} \mathcal{F}'\otimes_{\mathcal{B}}\Omega^1(\mathcal{B}), \end{equation*} The connection $\nabla$ is given by \begin{equation*} \nabla p(x) = \left(p \otimes \mathrm{Id}_{\Omega^1(\mathcal{B})}\right)\left(\widetilde{\nabla}(x)\right); \ x \in \mathcal{E}. \end{equation*} From \eqref{seqf} and \eqref{seqfo} it follows that if $y\in \mathcal{F}$ then $\nabla y$ does not depend on $x \in \mathcal{E}$ such that $y=p(x)$. Similarly there is a flat connection $\nabla': \mathcal{F}' \to \mathcal{F}' \otimes_{\mathcal{B}} \Omega^1(\mathcal{B})$ given by \begin{equation*} \nabla' p'(x) = \left(p' \otimes \mathrm{Id}_{\Omega^1(\mathcal{B})}\right)\left(\widetilde{\nabla'}(x)\right); \ x \in \mathcal{E}'. \end{equation*} Following table explains a correspondence between the proposition \ref{comm_cov_mani} and the above construction. \newline \newline \break \begin{tabular}{|c|c|} \hline DIFFERENTIAL GEOMETRY & SPECTRAL TRIPLES\\ \hline Manifold $M$ & Spectral triple $(\mathcal{B}, H, D)$\\ The covering manifold $E$ & Spectral triple $(\mathcal{A}, \mathcal{A} \otimes_{\mathcal{B}}H, \widetilde{D} )$ \\ A regular covering projection $E\to M$ & A noncommutative covering projection $(B, A, G)$ \\ Group of covering transformations $\pi_1(M)/H$ & Group of noncommutative covering transformations $G$ \\ A connection on vector fibre bundle $F\to M$ & An operator $\nabla: \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{B}} \Omega^1(\mathcal{B})$ \\ \hline \end{tabular} \newline \newline \break \end{empt} \begin{exm} Let $(\mathcal{A}_{\theta}, H, D)$ be a spectral triple associated to a noncommutative torus $A_{\theta}$ generated by unitary elements $u,v\in A_{\theta}$. Let $\mathcal{F} = \mathcal{A}^4_{\theta}$ be a free module and let $e_1,..., e_4 \in \mathcal{F}$ be its generators. Let $\nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1(\mathcal{A}_{\theta})$ be a connection given by \begin{equation*} \nabla e_1 = c_u e_2 \otimes du, \ \nabla e_2 = -c_u e_1 \otimes du, \ \nabla e_3 = c_v e_4 \otimes dv, \ \nabla e_4 = -c_v e_3 \otimes dv. \end{equation*} where $c_u, c_v \in \mathbb{R}$. According to \cite{ivankov:nc_wilson_lines} the connection $\nabla$ is flat. Let $\left(A_{\theta}, A_{\theta'}, \mathbb{Z}_m\times\mathbb{Z}_n\right)$ a Galois triple from example \ref{nc_torus_fin_cov}. This data induces a spectral triple $\left(\mathcal{A}_{\theta'}, \mathcal{A}_{\theta'}\otimes_{\mathcal{A}_{\theta}}H, D\right)$. If $\mathcal{E} = \mathcal{F} \otimes_{\mathcal{A}_{\theta}}\mathcal{A}_{\theta'}$ then \begin{equation} \mathcal{E} \approx \mathcal{A}_{\theta'}\otimes \mathbb{C}^{4} \approx \mathcal{A}_{\theta}\otimes \mathbb{C}^{4nm} \end{equation} and there is a natural connection $\widetilde{\nabla}: \mathcal{E} \to \mathcal{E}\otimes_{\mathcal{A}_{\theta}}\Omega^1\left(\mathcal{A}_{\theta}\right)$. Let $\rho : \mathbb{Z}_m\times\mathbb{Z}_n \to U(4)$ be a nontrivial representation. There is an action of $\mathbb{Z}_m\times\mathbb{Z}_n$ on $\mathcal{E}' = \mathcal{A}_{\theta'}\otimes \mathbb{C}^{4}$ given by \begin{equation*} g (a \otimes x) = ga \otimes \rho(g)x; \ a \in \mathcal{A}_{\theta}, \ x \in \mathbb{C}^4. \end{equation*} which satisfies \eqref{twisted_act}. Then $\mathcal{F}' = \mathcal{A}_{\theta'}\square_{\mathbb{Z}_m\times\mathbb{Z}_n}\mathcal{E}'$ is a finitely generated $A_{\theta}$ module with a connection $\nabla': \mathcal{F}' \to \mathcal{F} \otimes \Omega^1(\mathcal{A}_{\theta})$ given by the construction \ref{n_f_b_constr}. \end{exm} \subsection{Noncommutative bundles with flat connections and $K$-theory} \paragraph{}A homomorphism $R_0(G) \to K^0(M)$ from \ref{comm_fund_k} can be generalized. Let $\left(A, \widetilde{A}, G\right)$ be a finite noncommutative covering projection and $\rho: G \to U(n)$ is a representation, $\mathrm{triv}_n: G \to U(n)$ is the trivial representation. Suppose that an action if $G$ on $\mathbb{C}^n$ is given by $\rho$. Then a homomorphism $R_0(G) \to K(A)$ is given by \begin{equation*} [\rho] - \left[\mathrm{triv}_n\right] \mapsto \left[\widetilde{A}\square_G\mathbb{C}^n\right] - \left[A^n\right]. \end{equation*} \section{Noncommutative generalization of Borel construction} \begin{empt} There is a noncommutative generalization of the Borel construction \ref{borel_const_comm} \end{empt} \begin{defn}\label{borel_const_ncomm} Let $A$, $B$ be $C^*$-algebras, let $G$ be a group which acts on both $A$ and $B$. Let $A\otimes_{\mathbb{C}} B$ is any tensor product such that $A\otimes_{\mathbb{C}} B$ is a $C^*$-algebra. The norm closure of generated by \begin{equation*} C = \left\{\sum_i a_i\otimes b_i\in A\otimes_{\mathbb{C}} B~|~\sum_i a_i g\otimes b_i= \sum_i a_i\otimes gb_i;~\forall g\in G\right\}. \end{equation*} subalgebra is said to be a {\it cotensor product of $C^*$-algebras}. Denote by $A \square_G B$ the cotensor product. \end{defn} \begin{rem} We do not fix a type of a tensor product because different applications can use different tensor products (See \cite{bruckler:tensor}). \end{rem} \begin{exm} Let $\mathcal{X}$, $\mathcal{Y}$ be locally compact Hausdorff spaces and let $G$ be a finite or countable group which acts on both $\mathcal{X}$ and $\mathcal{Y}$. Suppose that action on $\mathcal{X}$ (resp. $\mathcal{Y}$) is right (resp. left). Then there is natural right (resp. left) action of on $C_0(\mathcal{X})$ (resp. $C_0(\mathcal{Y})$). From \cite{bruckler:tensor} it follows that the minimal and the maximal norm on $C_0(\mathcal{X}) \otimes_{\mathbb{C}} C_0(\mathcal{Y})$ coincide. It is well known that $C_0(\mathcal{X}\times\mathcal{Y}) \approx C_0(\mathcal{X}) \otimes_{\mathbb{C}} C_0(\mathcal{Y})$. Let $\mathcal{Z} = \mathcal{X}\times\mathcal{Y} / \approx$ where $\approx$ is given by \begin{equation*} (xg, y) \approx (x, g^{-1}y). \end{equation*} It is clear that $C_0(\mathcal{Z}) \approx C_0(\mathcal{X}) \square_G C_0(\mathcal{Y})$. \end{exm} \begin{defn} Let $\left(A, \widetilde{A}, _{\widetilde{A}}X_A, G\right)$ be a Galois quadruple such that there is right action of $G$ of $\widetilde{A}$ and left action of $G$ on $C^*$-algebra $B$. A cotensor product $\widetilde{A}\square_G B$ is said to be a {\it noncommutative Borel construction}. \end{defn} \begin{exm} Let $p: \widetilde{\mathcal{B}}\to \mathcal{B}$ be a topological normal covering projection of locally compact topological spaces, and $G = G(\widetilde{\mathcal{B}}| \mathcal{B})$ is a group of covering transformations. Then $p$ is a principal $G(\widetilde{\mathcal{B}}| \mathcal{B})$-bundle. Let $\mathcal{F}$ be a locally compact topological space with action of $G$ on it. Then there is a natural isomorphism with the $C^*$-algebra of a topological Borel construction \begin{equation*} C_0(\widetilde{\mathcal{B}}\times_G\mathcal{F})\approx C_0(\widetilde{\mathcal{B}})\square_GC_0(\mathcal{F}). \end{equation*} \end{exm} \end{document}

{\beta}gin{document} \title[]{Edgeworth expansions for independent bounded integer valued random variables.} \vskip 0.1cm \author{Dmitry Dolgopyat and Yeor Hafouta} \dedicatory{ } \maketitle {\beta}gin{abstract} We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the underlying partial sums modulo some $h\in {\mathbb N}$ and the uniform distribution could fail to be $o({\sigma}ma_N^{1-r})$, where ${\sigma}ma_N$ is the standard deviation of the partial sum. Second, this distribution could have the required closeness but this closeness is unstable, in the sense that it could be destroyed by removing finitely many terms. In the first case, the expansion of order $r$ fails. In the second case it may or may not hold depending on the behavior of the derivatives of the characteristic functions of the summands whose removal causes the break-up of the uniform distribution. We also show that a quantitative version of the classical Prokhorov condition (for the strong local central limit theorem) is sufficient for Edgeworth expansions, and moreover this condition is, in some sense, optimal. \end{abstract} \tableofcontents \section{Introduction.}{\lambda}bel{SecInt} Let $X_1,X_2,...$ be a uniformly bounded sequence of independent integer-valued random variables. Set $S_N=X_1+X_2+...+X_N$, $V_N=V(S_N)={\theta}xt{Var}(S_N)$ and ${\sigma}_N=\sqrt{V_N}$. Assume also that $V_N\to\infty$ as $N\to\infty$. Then the central limit theorem (CLT) holds true, namely the distribution of $(S_N-{\mathbb E}(S_N))/{\sigma}_N$ converges to the standard normal distribution as $N\to\infty$. Recall that the local central limit theorem (LLT) states that, uniformly in $k$ we have \[ {\mathbb P}(S_N=k)=\frac1{\sqrt{2\pi}{\sigma}_N}e^{-\left(k-{\mathbb E}(S_N)\right)^2/2V_N}+o({\sigma}_N^{-1}). \] This theorem is also a classical result, and it has origins in De Moivre-Laplace theorem. The stable local central limit theorem (SLLT) states that the LLT holds true for any integer-valued square integrable independent sequence $X_1',X_2',...$ which differs from $X_1,X_2,...$ by a finite number of elements. We recall a classical result due to Prokhorov. {\beta}gin{theorem} \cite{Prok} {\lambda}bel{ThProkhorov} The SLLT holds iff for each integer $h>1$, {\beta}gin{equation}{\lambda}bel{Prokhorov} \sum_n {\mathbb P}(X_n\neq m_n {\theta}xt{ mod } h)=\infty \end{equation} where $m_n=m_n(h)$ is the most likely residue of $X_n$ modulo $h$. \end{theorem} We refer the readers' to \cite{Rozanov, VMT} for extensions of this result to the case when $X_n$'s are not necessarily bounded (for instance, the result holds true when $\displaystyle \sup_n\|X_n\|_{L^3}<\infty$). Related results for local convergence to more general limit laws are discussed in \cite{D-MD, MS74}. The above result provides a necessary and sufficient condition for the SLLT. It turns out that the difference between LLT and SLLT is not that big. {\beta}gin{proposition} {\lambda}bel{PrLLT-SLLT} Suppose $S_N$ obeys LLT. Then for each integer $h\geq2$ at least one of the following conditions occur: either (a) $\displaystyle \sum_n {\mathbb P}(X_n\neq m_n(h) {\theta}xt{ mod } h)=\infty$. or (b) $\exists j_1, j_2, \dots, j_k$ with $k<h$ such that $\displaystyle \sum_{s=1}^k X_{j_s}$ mod $h$ is uniformly distributed. In that case for all $N\geq \max(j_1, \dots, j_k)$ we have that $S_N$ mod $h$ is uniformly distributed. \end{proposition} Since we could not find this result in the literature we include the proof in Section \ref{FirstOrder}. Next, we provide necessary and sufficient conditions for the regular LLT. We need an additional notation. Let $\displaystyle K=\sup_n\|X_n\|_{L^\infty}$. Call $t$ {\theta}xtit{resonant} if $t=\frac{2\pi l}{m}$ with $0<m\leq 2K$ and $0\leq l<m.$ {\beta}gin{theorem} {\lambda}bel{ThLLT} The following conditions are equivalent: (a) $S_N$ satisfies LLT; (b) For each $\xi\in {\mathbb R}\setminus {\mathbb Z}$, $\displaystyle \lim_{N\to\infty} {\mathbb E}\left(e^{2\pi i \xi S_N}\right)=0$; (c) For each non-zero resonant point $\xi$, $\displaystyle \lim_{N\to\infty} {\mathbb E}\left(e^{2\pi i \xi S_N}\right)=0$; (d) For each integer $h$ the distribution of $S_N$ mod $h$ converges to uniform. \end{theorem} The proof of this result is also given in Section \ref{FirstOrder}. We refer the readers to \cite{Do, DS} for related results in more general settings. The local limit theorem deals with approximation of $P(S_N=k)$ up to an error term of order $o({\sigma}_N^{-1})$. Given $r\geq1$, the Edgeworth expansion of order $r$ holds true if there are polynomials $P_{b, N}$, whose coefficients are uniformly bounded in $N$ and their degrees do no depend on $N,$ so that uniformly in $k\in{\mathbb Z}$ we have that {\beta}gin{equation}{\lambda}bel{EdgeDef} {\mathbb P}(S_N=k)=\sum_{b=1}^r \frac{P_{b, N} (k_N)}{{\sigma}ma_N^b}\mathfrak{g}(k_N)+o({\sigma}ma_N^{-r}) \end{equation} where $k_N=\left(k-{\mathbb E}(S_N)\right)/{\sigma}_N$ and $\mathfrak{g}(u)=\frac{1}{\sqrt{2\pi}} e^{-u^2/2}. $ In Section \ref{Sec5} we will show, in particular, that Edgeworth expansions of any order $r$ are unique up to terms of order $o({\sigma}_N^{-r}),$ and so the case $r=1$ coincides with the LLT. Edgeworth expansions for discrete (lattice-valued) random variables have been studied in literature for iid random variables \cite[Theorem 4.5.4]{IL} \cite[Chapter VII]{Pet75}, (see also \cite[Theorem 5]{Ess}), homogeneous Markov chains \cite[Theorems 2-4]{Nag2}, decomposable statistics \cite{MRM}, or { dynamical systems \cite{FL} with good spectral properties such as expanding maps. Papers \cite{Bor16, GW17} discuss the rate of convergence in the LLT. Results for non-lattice variables were obtained in \cite{Feller, BR76, CP, Br} (which considered random vectors) and \cite{FL} (see also \cite{Ha} for corresponding results for random expanding dynamical systems). In this paper we obtain analogues of Theorems \ref{ThProkhorov} and \ref{ThLLT} for higher order Edgeworth expansions for independent but not identically distributed integer-valued uniformly bounded random variables. We begin with the following result. {\beta}gin{theorem} {\lambda}bel{ThEdgeMN} Let $\displaystyle K=\sup_j\|X_j\|_{L^\infty}.$ For each $r\in{\mathbb N}$ there is a constant $R\!\!=\!\!R(r, K)$ such that the Edgeworth expansion of order $r$ holds~if \[ M_N:=\min_{2\leq h\leq 2K}\sum_{n=1}^N{\mathbb P}(X_n\neq m_n(h) {\theta}xt{ mod } h)\geq R\ln V_N. \] In particular, $S_N$ obeys Edgeworth expansions of all orders if $$ \lim_{N\to\infty} \frac{M_N}{\ln V_N}=\infty. $$ \end{theorem} The number $R(r,K)$ can be chosen according to Remark \ref{R choice}. This theorem is a quantitative version of Prokhorov's Theorem \ref{ThProkhorov}. We observe that logarithmic in $V_N$ growth of various non-perioidicity characteristics of individual summands are often used in the theory of local limit theorems (see e.g. \cite{Mal78, MS70, MPP}). We will see from the examples of Section \ref{ScExamples} that this result is close to optimal. However, to justify the optimality we need to understand the conditions necessary for the validity of the Edgeworth expansion. {\beta}gin{theorem}{\lambda}bel{r Char} For any $r\geq1$, the Edgeworth expansion of order $r$ holds if and only if for any nonzero resonant point $t$ and $0\leq\ell<r$ we have \[ \bar \Phi_{N}^{(\ell)}(t)=o\left({\sigma}_N^{\ell+1-r}\right). \] where $\bar\Phi_{N}(x)={\mathbb E}[e^{ix (S_N-{\mathbb E}(S_N))}]$ and $\bar\Phi_{N}^{(\ell)}(\cdot)$ is its $\ell$-th derivative. \end{theorem} This result\sout{s} generalizes Theorem \ref{ThLLT}, however in contrast with that theorem, in the case $r>1$ we also need to take into account the behavior of the derivatives of the characteristic function at nonzero resonant points. The values of the characteristic function at the resonant points $2\pi l/m$ have clear probabilistic meaning. Namely, they control the rate equidistribution modulo $m$ (see part (d) of Theorem \ref{ThLLT} or Lemma \ref{LmUnifFourier}). Unfortunately, the probabilistic meaning of the derivatives is less clear, so it is desirable to characterize the validity of the Edgeworth expansions of orders higher than 1 without considering the derivatives. Example \ref{ExUniform} shows that this is impossible without additional assumptions. Some of the reasonable additional conditions are presented below. We start with the expansion of order 2. {\beta}gin{theorem}{\lambda}bel{Thm SLLT VS Ege} Suppose $S_N$ obeys the SLLT. Then the following are equivalent: (a) Edgeworth expansion of order 2 holds; (b) $|\Phi_N(t)|=o({\sigma}ma_N^{-1})$ for each nonzero resonant point $t$; (c) For each $h\leq 2K$ the distribution of $S_N$ mod $h$ is $o({\sigma}ma_N^{-1})$ close to uniform. \end{theorem} Corollary \ref{CorNoDer} provides an extension of Theorem \ref{Thm SLLT VS Ege} for expansions of an arbitrary order $r$ under an additional assumption that $\displaystyle \varphi:=\min_{t\in{\mathcal{R}}}\inf_{n}|\phi_n(t)|>0$, where ${\mathcal{R}}$ is the set of all nonzero resonant points. The latter condition implies in particular, that for each $\ell$ there is a uniform lower bound on the distance between the distribution of $X_{n_1}+X_{n_2}+\dots +X_{n_\ell}$ {\theta}xt{mod }$m$ and the uniform distribution, when $\{n_1, n_2,\dots ,n_\ell\} \in{\mathbb N}^\ell$ and $m\geq2$. Next we discuss an analogue of Theorem \ref{ThProkhorov} for expansions of order higher than 2. It requires a stronger condition which uses an additional notation. Given $j_1, j_2,\dots,j_s$ with $j_l\in [1, N]$ we write $$ S_{N; j_1, j_2, \dots,j_s}=S_N-\sum_{l=1}^s X_{j_l}. $$ Thus $S_{N; j_1, j_2, \dots ,j_s}$ is a partial sum of our sequence with $s$ terms removed. We will say that $\{X_n\}$ {\em admits an Edgeworth expansion of order $r$ in a superstable way} (which will be denoted by $\{X_n\}\in EeSs(r)$) if for each ${\bar s}$ and each sequence $j_1^N, j_2^N,\dots ,j_{s_N}^N$ with $s_N\leq {\bar s}$ there are polynomials $P_{b, N}$ whose coefficients are $O(1)$ in $N$ and their degrees do not depend on $N$ so that uniformly in $k\in{\mathbb Z}$ we have that {\beta}gin{equation}{\lambda}bel{EdgeDefSS} {\mathbb P}(S_{N; j_1^N, j_2^N, \dots, j_{s_N}^N}=k)=\sum_{b=1}^r \frac{P_{b, N} (k_N)}{{\sigma}ma_N^b} \mathfrak{g}(k_N)+o({\sigma}ma_N^{-r}) \end{equation} and the estimates in $O(1)$ and $o({\sigma}ma_N^{-r})$ are uniform in the choice of the tuples $j_1^N, \dots ,j_{s_N}^N.$ That is, by removing a finite number of terms we can not destroy the validity of the Edgeworth expansion (even though the coefficients of the underlying polynomials will of course depend on the choice of the removed terms). Let $\Phi_{N; j_1, j_2,\dots, j_s}(t)$ be the characteristic function of $S_{N; j_1, j_2,\dots, j_s}.$ {\beta}gin{remark} We note that in contrast with SLLT, in the definition of the superstrong Edgeworth expansion one is only allowed to remove old terms, but not to add new ones. This difference in the definition is not essential, since adding terms with sufficiently many moments (in particular, adding bounded terms) does not destroy the validity of the Edgeworth expansion. See the proof of Theorem \ref{Thm Stable Cond} (i) or the second part of Example \ref{ExNonAr}, starting with equation \eqref{Convolve}, for details. \end{remark} {\beta}gin{theorem}{\lambda}bel{Thm Stable Cond} (1) $S_N\in EeSs(1)$ (that is, $S_N$ satisfies the LLT in a superstable way) if and if it satisfies the SLLT. (2) For arbitrary $r\geq 1$ the following conditions are equivalent: (a) $\{X_n\}\in EeSs(r)$; (b) For each $j_1^N, j_2^N,\dots ,j_{s_N}^N$ and each nonzero resonant point $t$ we have $\Phi_{N; j_1^N, j_2^N,\dots, j_{s_N}^N}(t)=o({\sigma}ma_N^{1-r});$ (c) For each $j_1^N, j_2^N,\dots ,j_{s_N}^N$, and each $h \leq 2K$ the distribution of $S_{N; j_1^N, j_2^N,\dots, j_{s_N}^N}$ mod $h$ is $o({\sigma}ma_N^{1-r})$ close to uniform. \end{theorem} To prove the above results we will show that for any order $r$, we can always approximate $\mathbb{P}(S_N=k)$ up to an error $o({\sigma}ma_N^{-r})$ provided that instead of polynomials we use products of regular and the trigonometric polynomials. Those products allow us} to take into account possible oscillatory behavior of $P(S_N=k)$ when $k$ belongs to different residues mod $h$, where $h$ is denominator of a {\em resonant frequency.} When $M_N\geq R V_N$ for $R$ large enough, the new expansion coincides with the usual Edgeworth expansions. We thus derive that the condition $M_N\geq R\ln V_N$ is in a certain sense optimal. \section{Main result}{\lambda}bel{Main R} Let $X_1,X_2,...$ be a sequence of independent integer-valued random variables. For each $N\in{\mathbb N}$ we set $\displaystyle S_N=\sum_{n=1}^N X_n$ and $V_N={\theta}xt{Var}(S_N)$. We assume in this paper that $\displaystyle \lim_{N\to\infty}V_N=\infty$ and that there is a constant $K$ such that $$\sup_n \|X_n\|_{L^\infty}\leq K. $$ Denote ${\sigma}ma_N=\sqrt{V_N}.$ For each positive integer $m$, let $ q_n(m)$ be the second largest among $$\displaystyle \sum_{l\equiv j {\theta}xt{ mod }m} {\mathbb P}(X_n=l)={\mathbb P}(X_n\equiv j {\theta}xt{ mod }m),\,j=1,2,...,m$$ and $j_n(m)$ be the corresponding residue class. Set $$ M_N(m)=\sum_{n=1}^N q_n(m)\quad{\theta}xt{and}\quad M_N=\min_m M_N(m).$$ {\beta}gin{theorem}{\lambda}bel{IntIndThm} There $\exists J=J(K)<\infty$ and polynomials $P_{a, b, N}$, where $a\in 0, \dots ,J-1,$ $b\in \mathbb{N}$, with degrees depending only on $b$ but not on $a, K$ or on any other characteristic of $\{X_n\}$, such that the coefficients of $P_{a,b, N}$ are uniformly bounded in $N$, and, for any $r\geq1$ uniformly in $k\in{\mathbb Z}$ we have $${\mathbb P}(S_N=k)-\sum_{a=0}^{J-1} \sum_{b=1}^r \frac{P_{a, b, N} ((k-a_N)/{\sigma}ma_N)}{{\sigma}ma_N^b} \mathfrak{g}((k-a_N)/{\sigma}ma_N) e^{2\pi i a k/J} =o({\sigma}ma_N^{-r}) $$ where $a_N={\mathbb E}(S_N)$ and $\mathfrak{g}(u)=\frac{1}{\sqrt{2\pi}} e^{-u^2/2}. $ Moreover, $P_{0,1,N}\equiv1$, and given $K, r$, there exists $R=R(K,r)$ such that if $M_N\geq R \ln V_N$ then we can choose $P_{a, b, N}=0$ for $a\neq 0.$ \end{theorem} We refer the readers to (\ref{r=1}) for more details on these expansions in the case $r=1$, and to Section \ref{FirstOrder} for a discussion about the relations with local limit theorems. The resulting expansions in the case $r=2$ are given in (\ref{r=2'}). We note that the constants $J(K)$ and $R(K,r)$ can be recovered from the proof of Theorem \ref{IntIndThm}. {\beta}gin{remark} Since the coefficients of the polynomials $P_{a,b,N}$ are uniformly bounded, the terms corresponding to $b=r+1$ are of order $O({\sigma}_N^{-(r+1)})$ uniformly in $k$. Therefore, in the $r$-th order expansion we actually get that the error term is $O({\sigma}_N^{-(r+1)})$. \end{remark} {\beta}gin{remark} In fact, the coefficients of the polynomials $P_{a,b,N}$ for $a>0$ are bounded by a constant times $(1+M_N^{q})e^{-c_0 M_N}$, where $c_0>0$ depends only on $K$ and $q\geq 0$ depends only on $r$ and $K$. Therefore, these coefficient are small when $M_N$ is large. When $M_N\geq R(r,K)\ln V_N$ these coefficients become of order $o({\sigma}^{-r}_N).$ Therefore, they only contribute to the error term, and so we can replace them by $0$, as stated in Theorem \ref{IntIndThm}. \end{remark} {\beta}gin{remark} As in the derivation of the classical Edgeworth expansion, the main idea of the proof of Theorem \ref{IntIndThm} is the stationary phase analysis of the characteristic function. However, in contrast with the iid case there may be resonances other than 0 which contribute to the oscillatory terms in the expansion. Another interesting case where the classical Edgeworth analysis fails is the case of iid terms where the summands are non-arithmetic but take only finitely many values. It is shown in \cite{DF} that in that case, the leading correction to the Edgeworth expansion also comes from resonances. However, in the case studied in \cite{DF} the geometry of resonances is more complicated, so in contrast to our Theorem \ref{IntIndThm}, \cite{DF} does not get the expansion of all orders. \end{remark} \section{Edgeworth expansions under quantitative Prokhorov condition.} {\lambda}bel{ScEdgeLogProkh} In this section we prove Theorem \ref{ThEdgeMN}. In the course of the proof we obtain the estimates of the characteristic function on intervals not containing resonant points which will also play an important role in the proof of Theorem \ref{IntIndThm}. The proof of Theorem \ref{IntIndThm} will be completed in Section \ref{ScGEE} where we analyze additional contribution coming from nonzero resonant points which appear in the case $M_N\leq R \ln {\sigma}_N.$ Those contributions constitute the source of the trigonometric polynomials in the generalized Edgeworth expansions. \subsection{Characterstic function near 0.} Here we recall some facts about the behavior of the characteristic function near 0, which will be useful in the proofs of Theorems \ref{ThEdgeMN} and \ref{IntIndThm}. The first result holds for general uniformly bounded sequences $\{X_n\}$ (which are not necessarily integer-valued). {\beta}gin{proposition}{\lambda}bel{PropEdg} Suppose that $\displaystyle \lim_{N\to\infty} {\sigma}_N=\infty$, where ${\sigma}_N\!=\!\sqrt{V_N}= \sqrt{V(S_N)}$. Then for $k=1,2,3,...$ there exists a sequence of polynomials $(A_{k,N})_N$ whose degree $d_k$ depends only on $k$ so that for any $r\geq 1$ there is ${\delta}_r>0$ such that for all $N\geq1$ and $t\in[-{\delta}_r{\sigma}_N,{\delta}_r{\sigma}_N]$, {\beta}gin{equation}{\lambda}bel{FinStep.0} {\mathbb E}\left(e^{it(S_N-{\mathbb E}(S_N))/{\sigma}_N}\right)=e^{-t^2/2}\left(1+\sum_{k=1}^r \frac{A_{k,N}(t)}{{\sigma}_N^k}+\frac{t^{r+1}}{{\sigma}_N^{r+1}} O(1)\right). \end{equation} Moreover, the coefficients of $A_{k,N}$ are algebraic combinations of moments of the $X_m$'s and they are uniformly bounded in $N$. Furthermore {\beta}gin{equation}{\lambda}bel{A 1,n .1} A_{1,N}(t)= -\frac{i}{6}{\gamma}_N t^3\,\,{\theta}xt{ and }\,\,A_{2,N}(t)={\Lambda}mbda_{4}(\bar S_N){\sigma}_N^{-2}\frac{t^4}{4!}-\frac{1}{36}{\gamma}_N^2 t^6 \end{equation} where $\bar S_N=S_N-{\mathbb E}(S_N)$, ${\gamma}_N={\mathbb E}[(\bar S_N)^3]/{\sigma}_N^2$ and ${\Lambda}mbda_{4}(\bar S_N)$ is the fourth comulant of $\bar S_N$. \end{proposition} The proof is quite standard, so we just sketch the argument. The idea is to fix some $B_2>B_1>0$, and to partition $\{1,...,N\}$ into intervals $I_1,...,I_{m_N}$ so that $\displaystyle B_1\leq {\theta}xt{Var}(S_{I_l})\leq B_2 $ where for each $l$ we set $\displaystyle S_{I_l}=\sum_{j\in I_l}X_i$. It is clear that $m_N/{\sigma}_N^2$ is bounded away from $0$ and $\infty$ uniformly in $N$. Recall next that there are constants $C_p$, $p\geq2$ so that for any $n\geq1$ and $m\geq 0$ we have {\beta}gin{equation} {\lambda}bel{CenterMoments} \left\|\sum_{j=n}^{n+m}\big(X_j-{\mathbb E}(X_j)\big)\right\|_{L^p}\leq C_p\left(1+\left\|\sum_{j=n}^{n+m}\big(X_j-{\mathbb E}(X_j)\big)\right\|_{L^2}\right). \end{equation} This is a consequence of the multinomial theorem and some elementary estimates, and we refer the readers to either Lemma 2.7 in \cite{DS}, or Theorem 6.17 in \cite{Pel} for such a result in a much more general settings. Using the latter estimates we get that the $L^p$-norms of $S_{I_l}$ are uniformly bounded in $l$. This reduces the problem to the case when the variance of $X_n$ is uniformly bounded from below, and all the moments of $X_n-{\mathbb E}(X_n)$ are uniformly bounded. In this case, the proposition follows by considering the Taylor expansion of the function $\ln {\mathbb E}\big(e^{it(S_N-{\mathbb E}(S_N))/{\sigma}_N}\big)+\frac12t^2$, see \cite[\S XVI.6]{Feller}. {\beta}gin{proposition} {\lambda}bel{PrHalf} Given a square integrable random variable $X$, let $\bar{X}=X-{\mathbb E}(X).$ Then for each $h\in\mathbb{R}$ we have $$\left|{\mathbb E}(e^{ih \bar X})-1\right|\leq \frac12 h^2V(X).$$ \end{proposition} {\beta}gin{proof} Set $\varphi(h)={\mathbb E}(e^{ih \bar X})$. Then by the integral form of the second order Taylor reminder we have $$ |\varphi(h)-\varphi(0)-h\varphi'(0)|=|\varphi(h)-\varphi(0)|=\left|\int_0^h(t-h)\varphi''(t)dt\right|$$ $\hskip4.7cm \displaystyle \leq V(X)\int_{0}^{|h|}(|h|-t)dt= \frac12 h^2V(X). $ \end{proof} \subsection{ Non resonant intervals.} {\lambda}bel{SSNonRes} As in almost all the proofs of the LLT, the starting point in the proof of Theorem \ref{ThEdgeMN} (and Theorem \ref{IntIndThm}) is that for $k, N\in{\mathbb N}$ we have {\beta}gin{equation} {\lambda}bel{EqDual} 2\pi {\mathbb P}(S_N=k)=\int_{0}^{2\pi} e^{-itk}{\mathbb E}(e^{it S_N})dt. \end{equation} Denote ${\mathbb T}={\mathbb R}/2\pi {\mathbb Z}.$ Let $$ \Phi_N(t)={\mathbb E}(e^{it S_N})=\prod_{n=1}^N \phi_n(t) \quad {\theta}xt{where}\quad \phi_n(t)={\mathbb E}(e^{it X_n}).$$ Divide ${\mathbb T}$ into intervals $I_j$ of small size ${\delta}ta$ such that each interval contains at most one resonant point and this point is strictly inside $I_j.$ We call an interval resonant if it contains a resonant point inside. Then {\beta}gin{equation}{\lambda}bel{SplitInt} 2\pi {\mathbb P}(S_N=k)=\sum_{j}\int_{I_j} e^{-itk}{\mathbb E}(e^{it S_N})dt. \end{equation} We will consider the integrals appearing in the above sum individually. {\beta}gin{lemma}{\lambda}bel{Step1} There are constants $C,c>0$ which depend only on ${\delta}$ and $K$ so that for any non-resonant interval $I_j$ and $N\geq1$ we have $$ \int_{I_j} |\Phi_N(t)| dt\leq C e^{-c V_N}. $$ \end{lemma} {\beta}gin{proof} Let ${\hat q}_n, {\bar q}_n$ be the largest and the second largest values of ${\mathbb P}(X_n=j)$ and let ${\hat j}_n, {\bar j}_n$ be the corresponding values. Note that {\beta}gin{equation} {\lambda}bel{phiNSum} \phi_n(t)={\hat q}_n e^{it {\hat j}_n}+{\bar q}_n e^{it {\bar j}_n}+\sum_{l\neq {\hat j}_n, {\bar j}_n} {\mathbb P}(X_n=l) e^{i t l}. \end{equation} Since $I_j$ is non resonant, the angle between $e^{it {\hat j}_n}$ and $e^{it {\bar j}_n}$ is uniformly bounded from below. Indeed if this was not the case we would have $t {\bar j}_n-t{\hat j}_n\approx 2\pi l_n$ for some $l_n\in{\mathbb Z}.$ Then $t\approx \frac{2\pi l_n}{m_n}$ where $m_n={\bar j}_n-{\hat j}_n$ contradicting the assumption that $I_j$ is non-resonant. Accordingly $\exists c_1>0$ such that $\displaystyle \left|e^{it {\hat j}_n}+e^{it {\bar j}_n}\right|\leq 2-c_1. $ Therefore $$ \left|{\hat q}_n e^{it {\hat j}_n}+{\bar q}_n e^{it {\bar j}_n}\right| \leq { ({\hat q}_n-{\bar q}_n)+{\bar q}_n \left| e^{it {\hat j}_n}+e^{it {\bar j}_n} \right|} \leq {\hat q}_n+{\bar q}_n-2c_1 {\bar q}_n. $$ Plugging this into \eqref{phiNSum}, we conclude that $|\phi_n(t)|\leq 1-2c_1 {\bar q}_n$ for $t\in I_j.$ Multiplying these estimates over $n$ and using that $1-x\leq e^{-x},\, x>0$, we get $$ |\Phi_N(t)|\leq e^{-2c_1\sum_n {\bar q}_n}. $$ Since $V(X_n)\leq c_2 {\bar q}_n$ for a suitable constant $c_2$ we can rewrite the preceding as {\beta}gin{equation}{\lambda}bel{NonResDec} |\Phi_N(t)|\leq e^{-c_3V_N},\, c_3>0. \end{equation} Integrating over $I_j$ we obtain the result. \end{proof} \subsection{Prokhorov estimates} Next we consider the case where $I_j$ contains a nonzero resonant point $t_j=\frac{2\pi l}{m}.$ {\beta}gin{lemma}{\lambda}bel{Step2} There is a constant $c_0$ which depends only on $K$ so that for any nonzero resonant point $t_j=2\pi l/m$ we have {\beta}gin{equation}{\lambda}bel{Roz0} \sup_{t\in I_j}|{\mathbb E}(e^{it S_N})|\leq e^{-c_0 M_N(m)}. \end{equation} Thus, for any $r\geq1$ there is a constant $R=R(r,K)$ such that if $M_N(m)\geq R \ln V_N$, then the integral $\int_{I_j} e^{-itk}{\mathbb E}(e^{it S_N})dt$ is $o({\sigma}ma_N^{-r})$ uniformly in $k$, and so it only contributes to the error term. \end{lemma} {\beta}gin{proof} The estimate (\ref{Roz0}) follows from the arguments in \cite{Rozanov}, but for readers' convenience we recall its proof. Let $X$ be an integer-valued random variable so that $\|X\|_{L^\infty}\leq K$. Let $t_0=2\pi l/m$ be a nonzero resonant point, where $\gcd(l,m)=1$. Let $t\in{\mathbb T}$ be so that {\beta}gin{equation} {\lambda}bel{Neart0} |t-t_0|\leq{\delta}, \end{equation} where ${\delta}$ is a small positive number. Let $\phi_X(\cdot)$ denote the characteristic function of $X$. Since $x\leq e^{x-1}$ for any real $x$ we have \[ |\phi_X(t)|^2\leq e^{|\phi(t)|^2-1}. \] Next, we have \[ |\phi_X(t)|^2-1=\phi(t)\phi(-t)-1=\sum_{j=-2K}^{2K}\sum_s\tilde P_j \left[\cos(t_j)-1\right] \] where \[ \tilde P_j=\sum_{s}{\mathbb P}(X=s){\mathbb P}(X=j+s). \] Fix some $-2K\leq j\leq 2K$. We claim that if ${\delta}$ in \eqref{Neart0} is small enough and $j\not\equiv 0{\theta}xt{ mod }m$ then for each integer $w$ we have $|t-2\pi w/j|\geq {\varepsilon}_0$ for some ${\varepsilon}_0>0$ which depends only on $K$. This follows from the fact that $-2K\leq j\leq 2K$ and that $2\pi w/j\not=t_0$ (and there is a finite number of resonant points). Therefore, \[ \cos(tj)-1\leq -{\delta}_0 \] for some ${\delta}_0>0$. On the other hand, if $j=km$ for some integer $k$ then with $w=lk$ we have {\beta}gin{eqnarray*} \cos(tj)-1=-2\sin^2(tj/2)=-2\sin^2\left((tj-2\pi w)/2\right)\\=-2\sin^2\left(j(t-t_0)/2\right)\leq -{\delta}_1(t-t_0)^2 \end{eqnarray*} for some ${\delta}_1>0$ (assuming that $|t-t_0|$ is small enough). We conclude that \[ |\phi_X(t)|^2-1\leq -{\delta}_0\sum_{j\in A}\tilde P_j-{\delta}_1(t-t_0)^2\sum_{j\in B}\tilde P_j \] where $A=A(X)$ is the set of $j$'s between $-2K$ and $2K$ so that $j\not\equiv 0{\theta}xt{ mod }m$ and $B=B(X)$ is its complement in ${\mathbb Z}\cap[-2K,2K]$. Let $s_0$ be the most likely residue of $X$ mod $m$ and $s_1$ be the second most likely residue class. Since $$ {\mathbb P}(X\equiv s_0 {\theta}xt{ mod }m)\geq \frac{1}{m} \quad{\theta}xt{and}\quad {\mathbb P}(X\equiv s_1 {\theta}xt{ mod }m)=q_m(X) $$ it follows that $\displaystyle \sum_{j\in A} \tilde P_j\geq \frac{q_m(X)}{m}.$ Combining this with the trivial bound $\displaystyle \sum_{j\in B} \tilde P_j\geq {\mathbb P}^2(X\equiv s_0)\geq \frac{1}{m^2}$ we obtain $$ |\phi_X(t)|\leq \exp-\left[\frac12\left(\frac{{\delta}_0 q_m(X)}{m} + \frac{{\delta}_1 (t-t_0)^2}{m^2}\right)\right]. $$ Applying the above with $t_0=t_j$ and $X=X_n$, $1\leq n\leq N$ we get that {\beta}gin{equation}{\lambda}bel{CharEstRoz} |\Phi_N(t)|\leq e^{-c_0M_N(m)-\bar{c}_0 N(t-t_j)^2 }\leq e^{-c_0M_N(m)} \end{equation} where $c_0$ is some constant. \end{proof} {\beta}gin{remark} Using the first inequality in (\ref{CharEstRoz}) and arguing as in \cite[page 264]{Rozanov}, we can deduce that there are positive constants $C, c_1, c_2$ such that {\beta}gin{equation}{\lambda}bel{RozArg} \int_{I_j}|{\mathbb E}(e^{it S_N})|dt\leq C\left(e^{-c_1{\sigma}_N}+\frac{e^{-c_2M_N(m)}}{{\sigma}_N}\right). \end{equation} This estimate plays an important role in the proof of the SLLT in \cite{Rozanov}, but for our purposes a weaker bound \eqref{Roz0} is enough. Note also that in order to prove (\ref{Roz0}) we could have just used the trivial inequality $\cos(t_j)-1\leq 0$ when $j\equiv 0{\theta}xt{ mod }m$, but we have decided to present this part from \cite{Rozanov} in full. \end{remark} {\beta}gin{remark}{\lambda}bel{R choice} Let $d_{{\mathcal{R}}}$ be the minimal distance between two different resonant points. Then, when ${\delta}<2d_{{\mathcal{R}}}$, we can take ${\delta}_0=1-\cos(d_{{\mathcal{R}}})$ in the proof of Lemma \ref{Step2}. Therefore, we can take $c_0=\frac{1-\cos(d_{{\mathcal{R}}})}{4K}$ in \eqref{Roz0}. Hence Lemma \ref{Step2} holds with $R(r,K)=\frac{r+1}{2c_0}.$ \end{remark} \vskip0.2cm \subsection{ Proof of Theorem \ref{ThEdgeMN}} {\lambda}bel{Cmplt1} Fix some $r\geq1$. Lemmas \ref{Step1} and \ref{Step2} show that if $M_N\geq R(r,K)\ln V_N$, then all the integrals in the right hand side of (\ref{SplitInt}) are of order $o({\sigma}_N^{-r})$, except for the one corresponding to the resonant point $t_j=0$. That is, for any ${\delta}>0$ small enough, uniformly in $k$ we have $$2\pi {\mathbb P}(S_N=k)=\int_{-{\delta}}^{\delta} e^{-ih k}\Phi_N(h)dh+o({\sigma}_N^{-r}). $$ In order to complete the proof of Theorem \ref{ThEdgeMN}, we need to expand the above integral. Making a change of variables $h\to h/{\sigma}_N$ and using Proposition \ref{PropEdg}, we conclude that if ${\delta}$ is small enough then $$ \int_{-{\delta}}^{\delta} e^{-ih k}\Phi_N(h)dh=$$ $${\sigma}_N^{-1}\int_{-{\delta}{\sigma}_N}^{{\delta}{\sigma}_N}e^{-ihk_N}e^{-h^2/2}\left(1+\sum_{u=1}^r \frac{A_{u,N}(h)}{{\sigma}_N^k}+\frac{h^{r+1}}{{\sigma}_N^{r+1}} O(1)\right)dh $$ where $k_N=\left(k-{\mathbb E}(S_N)\right)/{\sigma}_N$. Since the coefficients of the polynomials $A_{u,N}$ are uniformly bounded in $N$, we can just replace the above integral with the corresponding integral over all ${\mathbb R}$ (i.e. replace $\pm{\delta}{\sigma}_N$ with $\pm\infty$). Now the Edgeworth expansions are achieved using that for any nonnegative integer $q$ we have that $(it)^qe^{-t^2/2}$ is the Fourier transform of the $q$-th derivative of ${\theta}xtbf{n}(t)=\frac{1}{\sqrt{2\pi}}e^{-t^2/2}$ and that for any real $a$, {\beta}gin{equation}{\lambda}bel{Fourir} \int_{-\infty}^\infty e^{-iat}\widehat{{\theta}xtbf{n}^{(q)}}(t) dt={\theta}xtbf{n}^{(q)}(a)=\frac{1}{\sqrt{2\pi}}(-1)^{q}H_q(a)e^{-a^2/2} \end{equation} where $H_q(a)$ is the $q$-th Hermite polynomial. \section{Generalized Edgeworth expansions: Proof of Theorem Theorem~ \ref{IntIndThm}} {\lambda}bel{ScGEE} \subsection{Contributions of resonant intervals.} Let $r\geq1$. As in the proof of Theorem \ref{ThEdgeMN}, our starting point is the equality {\beta}gin{equation} {\lambda}bel{EqDual} 2\pi {\mathbb P}(S_N=k)=\int_{0}^{2\pi} e^{-itk}{\mathbb E}(e^{it S_N})dt=\sum_{j}\int_{I_j} e^{-itk}{\mathbb E}(e^{it S_N})dt \end{equation} which holds for any $k\in{\mathbb N}$. We will consider the integrals appearing in the above sum individually. By Lemma \ref{Step1} the integrals over non-resonant intervals are of order $o({\sigma}_N^{-r})$, and so they can be disregarded. Moreover, in \S \ref{Cmplt1} we have expanded the integral over the resonant interval containing $0$. Now we will see that in the case $M_N< R(r,K) \ln V_N$ the contribution of nonzero resonant points need not be negligible. Let $t_j=\frac{2\pi l}{m}$ be a nonzero resonant point so that $M_N(m)<R(r,K) \ln V_N$ and let $I_j$ be the resonant interval containing it. Theorem~\ref{IntIndThm} will follow from an appropriate expansion of the integral $$\int_{I_j} e^{-itk}{\mathbb E}(e^{it S_N})dt.$$ We need the following simple result, which for readers' convenience is formulated as a lemma. {\beta}gin{lemma}{\lambda}bel{EpsLem} There exists ${\bar\varepsilon}>0$ so that for each $n\geq1$ with $q_n(m)\leq {\bar\varepsilon}$ we have $|\phi_n(t_j)|\geq\frac12$. In fact, we can take ${\bar\varepsilon}=\frac1{4m}$. \end{lemma} {\beta}gin{proof} Recall that $t_j=2\pi l/m$. The lemma follows since for any random variable $X$ we have $\displaystyle |{\mathbb E}(e^{i t_j X})|=$ $$\left|e^{it_js(m,X)}-\sum_{u\not\equiv s(m,X){\theta}xt{ mod m}}\big(e^{it_j s(m,X)}-e^{it_j u}\big)P(X\equiv u{\theta}xt{ mod } m)\right| $$ $$ \geq 1-2mq(m,X) $$ where $s(m,X)$ is the most likely value of $X{\theta}xt{ mod }m$ and $q(m,X)$ is the second largest value among $P(X\equiv u{\theta}xt{ mod } m)$, $u=0,1,2,...,m-1$. Therefore, we can take ${\bar\varepsilon}=\frac1{4m}$. \end{proof} Next, set ${\bar\varepsilon}=\frac{1}{8K}$ and let $N_0=N_0(N,t_j,{\bar\varepsilon})$ be the number of all $n$'s between $1$ to $N$ so that $q_n(m)>{\bar\varepsilon}$. Then $N_0\leq \frac{R \ln V_N}{{\bar\varepsilon}}$ because $M_N(m)\leq R \ln V_N.$ By permuting the indexes $n=1,2,...,N$ if necessary we can assume that $q_n(m)$ is non increasing. Let $N_0$ be the largest number such that $q_{N_0}\geq {\bar\varepsilon}.$ Decompose {\beta}gin{equation} {\lambda}bel{NPert-Pert} \Phi_N(t)=\Phi_{N_0}(t) \Phi_{N_0, N}(t) \end{equation} where $\displaystyle \Phi_{N_0, N}(t)=\prod_{n=N_0+1}^N \phi_n(t).$ {\beta}gin{lemma}{\lambda}bel{Step3} If the length ${\delta}$ of $I_j$ is small enough then for any $t=t_j+h\in I_j$ and $N\geq1$ we have \[ \Phi_{N_0,N}(t)=\Phi_{N_0,N}(t_j)\Phi_{N_0,N}(h) \Psi_{N_0,N}(h) \] where $$ \Psi_{N_0,N}(h)=\exp\left[O(M_N(m))\sum_{u=1}^\infty (O(1))^u h^u\right]. $$ \end{lemma} {\beta}gin{proof} Denote $$ \mu_n={\mathbb E}(X_n), \quad {\bar X}_n=X_n-\mu_n, \quad {\bar\phi}_n(t)={\mathbb E}(e^{it {\bar X}_n}).$$ Let $j_n(m)$ be the most likely residue mod $m$ for $X_n.$ Decompose $$ {\bar X}_n=s_n+Y_n+Z_n$$ where $Z_n\in m{\mathbb Z}$, $s_n=j_n(m)-\mu_n$, so that ${\mathbb P}(Y_n\neq 0)\leq m q_n(m).$ Then for $t=t_j+h$, {\beta}gin{equation} {\lambda}bel{IndCharTaylor} {\bar\phi}_n(t)=e^{i t_j s_n} {\mathbb E}\left(e^{i t_j Y_n} e^{ih {\bar X}_n}\right)={\bar\phi}_n(t_j) \psi_n(h) \end{equation} where $$ \psi_n(h)= \left(1+ \frac{i h {\mathbb E}(e^{i t_j Y} {\bar X}_n)-\frac{h^2}{2} {\mathbb E}(e^{i t_j Y} ({\bar X}_n)^2)+\dots}{{\mathbb E}(e^{i t_j Y_n})}\right) . $$ Next, using that for any $x\in(-1,1)$ we have $$1+x=e^{\ln (1+x)}=e^{x-x^2/2+x^3/3-...}$$ we obtain that for $h$ close enough to $0$, {\beta}gin{equation}{\lambda}bel{Expand} \psi_n(h)= \exp\left(\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k}\left(\frac1{{\mathbb E}(e^{it_jY_n})}\sum_{q=1}^\infty \frac{(ih)^q}{q!}{\mathbb E}(e^{it_j Y_n}(\bar X_n)^q)\right)^k\right) \end{equation} {\beta}gin{eqnarray} =\exp\left(\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k}\sum_{1\leq j_1,...,j_k}\frac1{({\mathbb E}(e^{it_jY_n}))^k}\prod_{r=1}^{k}\frac{(ih)^{j_r}}{j_r!}{\mathbb E}(e^{it_j Y_n}(\bar X_n)^{j_r})\right)\nonumber\\= \exp\left(\sum_{u=1}^\infty\left(\sum_{k=1}^{u}\frac{(-1)^{k+1}}{k}\sum_{j_1+...+j_k=u}\,\prod_{r=1}^{k}\frac{{\mathbb E}(e^{it_j Y_n}(\bar X_n)^{j_r})}{{\mathbb E}(e^{it_j Y_n})j_r!}\right)(ih)^u\right).\nonumber \end{eqnarray} Observe next that \[ {\mathbb E}[e^{it_j Y_n}(\bar X_n)^{j_r}]={\mathbb E}\left[(e^{it_j Y_n}-1)\big((\bar X_n)^{j_r}-{\mathbb E}[(\bar X_n)^{j_r}]\big)\right]+ {\mathbb E}[(\bar X_n)^{j_r}]{\mathbb E}(e^{it_j Y_n}) \] and so with $C=2K$, we have \[ \frac{{\mathbb E}[e^{it_j Y_n}(\bar X_n)^{j_r}]}{{\mathbb E}(e^{it_j Y_n})}=O(q_n(m))O(C^{j_r})+{\mathbb E}[(\bar X_n)^{j_r}]. \] Plugging this into (\ref{Expand}) and using that for $h$ small enough, {\beta}gin{equation*} \exp\left[ \sum_{u=1}^\infty\left(\sum_{k=1}^u\frac{(-1)^{k+1}}k\sum_{j_1+...+j_k=u}\prod_{r=1}^k\frac{{\mathbb E}({\bar X}_n^{j_r})}{j_r!}\right)(ih)^u\right]={\mathbb E}\left(e^{ih {\bar X}_n}\right) \end{equation*} we conclude that $$ \psi_n(h)={\mathbb E}(e^{ih\bar X_n}) \exp\left[\sum_{u=1}^\infty(O(1))^uO(q_n(m))h^u\right].$$ Therefore, \[ \Phi_{N_0,N}(t)=\Phi_{N_0,N}(t_j)\Phi_{N_0,N}(h) \Psi_{N_0,N}(h) \] where $\displaystyle \Psi_{ N_0, N}(h)=\exp\left[O(M_N(m))\sum_{u=1}^\infty (O(1))^u h^u\right]. $ \end{proof} {\beta}gin{remark}{\lambda}bel{CoeffRem} We will see in \S\ref{Fin} that the coefficients of the polynomials appearing in Theorem~\ref{IntIndThm} depend on the coefficients of the power series $\Psi_{N_0, N}(h)$ (see, in particular, \eqref{Cj(k)}). The first term in this series is $\displaystyle ih \sum_{n={N_0+1}}^N a_{n,j}$, where {\beta}gin{equation}{\lambda}bel{a n,j} a_{n,j}=\frac{{\mathbb E}[(e^{it_j Y_n}-1)\bar X_n]}{{\mathbb E}(e^{i t_j Y_n})}=\frac{{\mathbb E}(e^{it_j X_n}\bar X_n)}{{\mathbb E}(e^{it_j X_n})} \end{equation} while the second term is $\displaystyle \frac{h^2}{2} \sum_{n={N_0+1}}^N b_{n,j}$, where {\beta}gin{equation} {\lambda}bel{SecondTerm} b_{n,j}=\frac{{\mathbb E}[(e^{it_j Y_n}-1)\bar X_n]^2}{{\mathbb E}(e^{i t_j Y_n})^2}-\frac{{\mathbb E}[(e^{it_j Y_n}-1)((\bar X_n)^2-V(X_n))]}{{\mathbb E}(e^{i t_j Y_n})} \end{equation} $$ =a_{n,j}^2-\frac{{\mathbb E}\big(e^{it_j X_n}(\bar X_n)^2\big)}{{\mathbb E}(e^{it_j X_n})}. $$ In Section \ref{Sec2nd} we will use (\ref{a n,j}) to compute the coefficients of the polynomials from Theorem \ref{IntIndThm} in the case $r=2$, and \eqref{SecondTerm} is one of the main ingredients for the computation in the case $r=3$ (which will not be explicitly discussed in this manuscript). \end{remark} The next step in the proof of Theorem \ref{IntIndThm} is the following. {\beta}gin{lemma}{\lambda}bel{Step4} For $t=t_j+h\in I_j$ we can decompose {\beta}gin{equation}{\lambda}bel{S4} \Phi_{N_0}(t)=\Phi_{N_0}(t_j+h)=\sum_{l=0}^L \frac{\Phi_{N_0}^{(l)}(t_j)}{l!} h^l+O\left((h \ln V_N)^{L+1}\right). \end{equation} \end{lemma} {\beta}gin{proof} The lemma follows from the observation that the derivatives of $\Phi_{N_0}$ satisfy $|\Phi_{N_0}^{(k)}(t)|\leq O(N_0^k)\leq (C \ln V_N)^k$. \end{proof} \subsection{Completing the proof}{\lambda}bel{Fin} Recall \eqref{EqDual} and consider a resonant interval $I_j$ which does not contain $0$ such that $M_N(m)\leq R\ln{\sigma}_N$. Set $U_j=[-u_j,v_j]=I_j-t_j$. Let $N_0$ be as described below Lemma \ref{EpsLem}. Denote {\beta}gin{equation} {\lambda}bel{SNN0} S_{N_0,N}=S_N-S_{N_0},\, S_0=0, \end{equation} $$V_{N_0,N}={\theta}xt{Var}(S_N-S_{N_0})=V_N-V_{N_0}\quad{\theta}xt{and} \quad {\sigma}_{N_0,N}=\sqrt{V_{N_0,N}}.$$ Then {\beta}gin{equation}{\lambda}bel{Vars} V_{N_0,N}=V_N+O(\ln V_N)=V_N(1+o(1)). \end{equation} Denote $h_{N_0,N}=h/{\sigma}_{N_0,N}.$ By \eqref{FinStep.0}, if $|h_{N, N_0}|$ is small enough then {\beta}gin{equation}{\lambda}bel{FinStep} {\mathbb E}(e^{ih_{N_0,N} S_{N_0,N}})= \end{equation} $$e^{ih_{N_0,N}{\mathbb E}(S_{N_0,N})}e^{-h^2/2}\left(1+\sum_{k=1}^r \frac{A_{k,N_0,N}(h)}{{\sigma}_{N_0,N}^k}+\frac{h^{r+1}}{{\sigma}_{N_0,N}^{r+1}} O(1)\right)$$ where $A_{k,N_0,N}$ are polynomials with bounded coefficients (the degree of $A_{k,N_0,N}$ depends only on $k$). Let us now evaluate $\displaystyle \int_{I_j}e^{-itk}{\mathbb E}(e^{it S_N})dt. $ By Lemma \ref{Step3}, {\beta}gin{equation} {\lambda}bel{ResInt} \int_{I_j}e^{-itk}\Phi_N(t)dt= \end{equation} $$ e^{-it_j k}\Phi_{N_0,N}(t_j)\int_{U_j}e^{-ihk}\Phi_{N_0}(t_j+h)\Phi_{N_0,N}(h) \Psi_{N_0, N}(h)\; dh. $$ Therefore, it is enough to expand the integral on the RHS of \eqref{ResInt}. Fix a large positive integer $L$ and plug \eqref{S4} into \eqref{ResInt}. Note that for $N$ is large enough, $h_0$ small enough and $|h|\leq h_0$, Proposition \ref{PrHalf} and \eqref{Vars} show that there exist positive constants $c_0, c$ such that {\beta}gin{equation}{\lambda}bel{expo} |\Phi_{N_0,N}(h)|=|{\mathbb E}(e^{ih S_{N_0,N}})|\leq e^{-c_0(V_N-V_{N_0})h^2}\leq e^{-cV_N h^2}. \end{equation} Thus, the contribution coming from the term $O\left((h \ln V_N)^{L+1}\right)$ in the right hand side of (\ref{S4}) is at most of order \[ V_N^{R{\delta}}(\ln V_n)^{L+1}\int_{-\infty}^\infty h^{L+1}e^{-c V_N h^2}dh \] where ${\delta}$ is the diameter of $I_j$. Changing variables $x={\sigma}_N h$, where ${\sigma}_N=\sqrt{V_N}$ we get that the latter term is of order $(\ln V_n)^{L+1}{\sigma}_{N}^{-(L+1-2R{\delta})}$ and so when $L$ is large enough we get that this term is $o({\sigma}_N^{-r-1})$ (alternatively, we can take $L=r$ and ${\delta}$ to be sufficiently small). This means that it is enough to expand each integral of the form {\beta}gin{equation} {\lambda}bel{HLInt} \int_{U_j}e^{-ih k}h^l\Phi_{N_0,N}(h) \Psi_{N_0, N}(h) dh \end{equation} where $l=0,1,...,L$ (after changing variables the above integral is divided by ${\sigma}_{N_0,N}^{l+1}$). Next, Lemma \ref{Step3} shows that for any $d\in {\mathbb Z}$ we have {\beta}gin{equation} {\lambda}bel{EqOrderD} \Psi_{N_0, N}(h) =1+\sum_{u=1}^{d}C_{w,N}h^u+h^{d+1}O(1+M_N(m)^{d+1})|V_N|^{O(|h|)}, \end{equation} where $C_{w,N}=C_{w,N, t_j}$ are $O(M_{N}^u(m))=O((\ln V_N)^{u})$. Note that, with $a_{n,j}$ and $b_{n,j}$ defined in Remark \ref{CoeffRem}, we have {\beta}gin{equation}{\lambda}bel{C 1 N} C_{1,N}=i\sum_{n=N_0+1}^{N}a_{n,j} \end{equation} and \[ C_{2,N}=\frac12\sum_{n=N_0+1}^{N}b_{n,j}-\frac 12\left(\sum_{n=N_0+1}^{N}a_{n,j}\right)^2. \] Take $d$ large enough and plug \eqref{EqOrderD} into \eqref{HLInt}. Using (\ref{expo}), we get again that the contribution of the term $$h^{d+1}O(1+M_N(m)^{d+1})|V_N|^{O(|h|)}h^l\Phi_{N_0,N}(h)$$ to the above integral is $o({\sigma}_N^{-r})$. Thus, it is enough to expand each term of the form \[ \int_{U_j}e^{-ih k}h^{q}\Phi_{N_0,N}(h)dh \] where $0\leq q\leq L+d$. Using (\ref{FinStep}) and making the change of variables $h\to h/{\sigma}_{N_0,N}$ it is enough to expand {\beta}gin{equation}{\lambda}bel{MainInt} \int_{-\infty}^\infty e^{-ih (k-{\mathbb E}[S_{N_0,N}])/{\sigma}_{N_0,N}}h^qe^{-h^2/2}\left(1+\sum_{w=1}^r \frac{A_{w,N_0,N}(h)}{{\sigma}_{N_0,N}^w}+\frac{h^{r+1}}{{\sigma}_{N_0,N}^{r+1}} O(1)\right) \frac{dh}{{\sigma}_{N_0, N}}. \end{equation} This is achieved by using that $(it)^qe^{-t^2/2}$ is the Fourier transform of the $q$-th derivative of ${\theta}xtbf{n}(t)=\frac{1}{\sqrt{2\pi}}e^{-t^2/2}$ and that for any real $a$, {\beta}gin{equation}{\lambda}bel{Fourir} \int_{-\infty}^\infty e^{-iat}\widehat{{\theta}xtbf{n}^{(q)}}(t) dt={\theta}xtbf{n}^{(q)}(a)=\frac{1}{\sqrt{2\pi}}(-1)^{q}H_q(a)e^{-a^2/2} \end{equation} where $H_q(a)$ is the $q$-th Hermite polynomial. Note that in the above expansion we get polynomials in the variable $\displaystyle k_{N_0,N}=\frac{k-{\mathbb E}[S_N-S_{N_0}]}{{\sigma}_{N,N_0}}$, not in the variable $k_N=\frac{k-{\mathbb E}(S_N)}{{\sigma}_N}$. Since $k_{N_0,N}=k_N{\alpha}_{N_0,N}+O(\ln {\sigma}_N/{\sigma}_N)$, where ${\alpha}_{N_0,N}={\sigma}_{N}/{\sigma}_{N_0,N}=O(1)$, the binomial theorem shows that such polynomials can be rewritten as polynomials in the variable $k_N$ whose coefficients are uniformly bounded in $N$. We also remark that in the above expansions we get the exponential terms \[ e^{-\frac{(k-a_{N_0,N})^2}{2(V_N-V_{N_0})}} \quad{\theta}xt{where}\quad a_{N_0,N}={\mathbb E}[S_N-S_{N_0}] \] and not $e^{-(k-a_N)^2/2V_N}$ (as claimed in Theorem \ref{IntIndThm}). In order to address this fix some ${\varepsilon}<1/2$. Note that for $|k-a_{N_0,N}|\geq V_N^{\frac12+{\varepsilon}}$ we have \[ e^{-\frac{(k-a_{N_0,N})^2}{2(V_N-V_{N_0})}}=o(e^{-cV_N^{2{\varepsilon}}}) \quad {\theta}xt{and} \quad e^{-\frac{(k-a_{N_0,N})^2}{2V_N}}=o(e^{-cV_N^{2{\varepsilon}}}) {\theta}xt{ for some }c>0. \] Since both terms are $o({\sigma}_N^{-s})$ for any $s$, it is enough to explain how to replace $\displaystyle e^{-\frac{(k-a_{N_0,N})^2}{2(V_N-V_{N_0})}} $ with $\displaystyle e^{-\frac{(k-a_{N})^2}{2V_N}}$ when $|k-a_{N_0,N}|\leq V_N^{\frac12+{\varepsilon}}$ (in which case $\displaystyle |k-a_N|=O(V_N^{\frac12+{\varepsilon}})$). For such $k$'s we can write {\beta}gin{equation}{\lambda}bel{ExpTrans1} \exp\left[-\frac{(k-a_{N_0,N})^2}{2(V_N-V_{N_0})}\right]= \end{equation} $$ \exp\left[-\frac{(k-a_{N_0,N})^2}{2V_N}\right]\; \exp\left[-\frac{(k-a_{N_0,N})^2 V_{N_0}}{2V_N(V_N-V_{N_0})}\right]. $$ Since $\displaystyle \frac{(k-a_{N_0,N})^2 V_{N_0}}{2V_N(V_N-V_{N_0})} =O\left(V_N^{-(1-3{\varepsilon})}\right)$, for any $d_1$ we have {\beta}gin{equation}{\lambda}bel{ExTrans1.1} \exp\left[-\frac{(k-a_{N_0,N})^2 V_{N_0}}{2V_N(V_N-V_{N_0})}\right]= \end{equation} $$\sum_{j=0}^{d_1}\frac{V_{N_0}^j}{2^j(V_N-V_{N_0})^j j!} \left(\frac{(k-a_{N_0,N})^2}{{\sigma}_N^2}\right)^j+O(V_N^{-(d_1+1)(2-3{\varepsilon})}).$$ Note that (using the binomial formula) the first term on the above right hand side is a polynomial of the variable $(k-a_{N})/{\sigma}_N$ whose coefficients are uniformly bounded in $N$. Next we analyze the first factor in the RHS of \eqref{ExpTrans1}. As before, it is enough to consider $k$'s such that $|k-a_N|\leq V_N^{\frac12+{\varepsilon}}$ for a sufficiently small ${\varepsilon}.$ We have {\beta}gin{equation}{\lambda}bel{Centring} \exp\left[-\frac{(k-a_{N,N_0})^2}{2V_N}\right]= \end{equation} $$ \exp\left[-\frac{(k-a_N)^2}{2V_N}\right] \exp\left[-\frac{2(k-a_N)a_{N_0}+a_{N_0}^2}{2V_N}\right]. $$ Note that $\frac{(k-a_N)a_{N_0}+a_{N_0}^2}{2V_N} =k_N{\beta}ta_{N_0,N}+\theta_{N_0,N}$, where $${\beta}ta_{N_0,N}=\frac{a_{N_0}}{2{\sigma}_N}=O\left(\frac{\ln {\sigma}_N}{{\sigma}_N}\right) \;{\theta}xt{ and }\; \theta_{N_0,N}=\frac{a_{N_0}^2}{2V_N}=O\left(\frac{\ln^2{\sigma}_N}{V_N}\right).$$ Approximating $e^{\frac{(k-a_N)a_{N_0}+a_{N_0}^2}{2V_N}}$ by a polynomial of a sufficiently large degree $d_2$ in the variable $\frac{(k-a_N)a_{N_0}+a_{N_0}^2}{2V_N}$ completes the proof of existence of polynomials $P_{a,b,N}$ claimed in the theorem (the Taylor reminder in the last approximation is of order $\displaystyle O\left(V_N^{-d_2(\frac12-{\varepsilon})}\right)$, so we can take $d_2=4(r+1)$ assuming that ${\varepsilon}$ is small enough). Finally, let us show that the coefficients of the polynomials $P_{a,b,N}$ constructed above are uniformly bounded in $N$. In fact, we will show that for each nonzero resonant point $t_j=2\pi l/m$, the coefficients of the polynomials coming from integration over $I_j$ are of order $$O\left((1+M_N^{q_0}(m))e^{-M_N(m)}\right),$$ where $q_0=q_0(r)$ depends only on $r$. Observe that the additional contribution to the coefficients of the polynomials coming from the transition between the variables $k_N$ and $k_{N_0,N}$ is uniformly bounded in $N$. Hence we only need to show that the coefficients of the (original) polynomials in the variable $k_{N_0,N}$ are uniformly bounded in $N$. The possible largeness of these coefficient can only come from the terms $C_{u,N,t_j}$, for $u=0,1,2,...,d$ which are of order $M_N^u(m)$, respectively. However, the corresponding terms are multiplied by terms of the form $\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)$ for certain $\ell$'s which are uniformly bounded in $N$ (see also \eqref{Cj(k)}). We conclude that there are constants $W_j\in{\mathbb N}$ and $a_j\in{\mathbb N}$ which depend only on $t_j$ and $r$ so that the coefficients of the resulting polynomials are composed of a sum of at most $W_j$ terms of order $(M_N(m))^{a_j}\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)$, where $\ell\leq E(r)$ for some $E(r)\in{\mathbb N}$. Next, we have {\beta}gin{equation} {\lambda}bel{DerComb} \Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j)= \end{equation} $$\sum_{n_1,\dots, n_k\leq N_0; \atop \ell_1+\dots+\ell_k=\ell} {\gamma}ma_{\ell_1,\dots, \ell_k} \left(\prod_{q=1}^k \phi_{n_q}^{(\ell_q)}(t_j) \right) \left[\prod_{n\leq N, \; n\neq n_k} \phi_n(t_j)\right]$$ where ${\gamma}ma_{\ell_1,\dots, \ell_k}$ are bounded coefficients of combinatorial nature. Using (\ref{Roz0}) we see that for each $n_1,\dots, n_k$ the product in the square brackets is at most $C e^{-c_0 M_N(m)+O(1)}$ for some $C, c_0>0$. Hence $$|\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j)|\leq \hat{C} N_0^{\ell} \; e^{-c_0 M_N(m)},\,\,\hat C>0.$$ Now, observe that the definition of $N_0$ gives $M_N(m)\geq {\varepsilon}_0 N_0$, ${\varepsilon}_0>0$. Therefore $\displaystyle |\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j)|\leq C_0 M_N^\ell (m)e^{-c_0 M_N(m)}$, and so each one of the above coefficients is of order $M_N^{\ell'}(m)e^{-c_0 M_N(m)}$ for some $\ell'$ which does not depend on $N$. \qed {\beta}gin{remark} The transition between the variables $k_{N_0,N}$ and $k_{N}$ changes the monomials of the polynomials $P_{a,b,N}$, $a\not=0$ coming from integration over $I_j$, for $t_j\not=0$ into monomials of the form $\frac{c_N a_{N_0}^{j_1}{\sigma}_{N_0}^{j_2}k_N^{j_3}}{{\sigma}_N^{u}}$ for some bounded sequence $(c_N)$, $j_1,j_2,j_3\geq0$ and $u\in{\mathbb N}$. As we have explained, the coefficients of these monomials are uniformly bounded. Still, it seems more natural to consider such monomials as part of the polynomial $P_{a,b+u,N}$. In this case we still get polynomials with bounded coefficients since $a_{N_0}$ and ${\sigma}_{N_0}$ are both $O(N_0)$, $N_0=O(M_{N}(m))$ and $c_N$ contains a term of the form $\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j)$. \end{remark} {\beta}gin{remark} As can be seen from the proof, the resulting expansions might contain terms corresponding to ${\sigma}_N^{-s}$ for $s>r$. Such terms can be disregarded. For $\frac{|k-a_N|}{{\sigma}_N}\leq V_N^{\varepsilon}$ this follows because the coefficients of our exapansions are $O(1)$ and for $\frac{|k-a_N|}{{\sigma}_N}\geq V_N^{\varepsilon}$ this follows from \eqref{expo}. In practice, some of the polynomials $P_{a,b, N}$ with $b\leq r$ might have coefficients which are $\displaystyle o({\sigma}_N^{b-r})$ (e.g. when $b+u>r$ in the last remark) so they also can be disregarded. The question when the terms $P_{a,b, N}$ may be disregarded is in the heart of the proof of Theorem \ref{r Char} given in the next section. \end{remark} \subsection{A summary} The proofs of Proposition \ref{PrLLT-SLLT}, Theorem \ref{ThLLT} and Theorem \ref{r Char} will be based on careful analysis of the formulas of the polynomials from Theorem \ref{IntIndThm}. For this purpose, it will be helpful to summarize the main conclusions from the proof of Theorem \ref{IntIndThm}. Let $r\geq1$ and $t_j=2\pi l/m$ be a nonzero resonant point. Then the arguments in the proof of Theorem \ref{IntIndThm} yield that the contribution to the expansion coming from $t_j$ is {\beta}gin{equation}{\lambda}bel{Cj(k)} {\theta}xtbf{C}_j(k):= \end{equation} $$ e^{-it_j k}\Phi_{N_0,N}(t_j)\sum_{s\leq r-1}\left(\sum_{u+l=s}\frac{\Phi_{N_0}^{(l)}(t_j)C_{u,N}}{l!}\right)\int_{U_j}e^{-ihk}h^{s}\Phi_{N_0,N}(h)dh $$ where $U_j=I_j-t_j$, $C_{u, N}$ are given by \eqref{EqOrderD} and $C_{0,N}=1$. When $t_j=0$ then it is sufficient to consider only $s=0$, $N_0=0$ and the contribution is just the integral \[ \int_{-{\delta}}^{\delta} e^{-ihk}\Phi_{N}(h)dh \] where ${\delta}$ is small enough. As in (\ref{MainInt}), changing variables we can replace the integral corresponding to $h^s$ with {\beta}gin{eqnarray}{\lambda}bel{s} {\sigma}_{N_0,N}^{-s-1}\int_{-\infty}^\infty e^{-ih (k-{\mathbb E}[S_{N_0,N}])/{\sigma}_{N_0,N}}h^s e^{-h^2/2}\\\times\left(1+\sum_{w=1}^r \frac{A_{w,N_0,N}(h)}{{\sigma}_{N_0,N}^w}+\frac{h^{r+1}}{{\sigma}_{N_0,N}^{r+1}} O(1)\right)dh.\nonumber \end{eqnarray} After that was established, the proof was completed using \eqref{Fourir} and some estimates whose whose purpose was to make the transition between the variables $k_{N_0,N}$ and $k_N$. \section{Uniqueness of trigonometric expansions.}{\lambda}bel{Sec5} In several proofs we will need the following result. {\beta}gin{lemma}{\lambda}bel{Lemma} Let $r\geq1$ and $d\geq0$. Set ${\mathcal{R}}_0={\mathcal{R}}\cup\{0\}$ where ${\mathcal{R}}$ is the set of nonzero resonant points. For any $t_j\in{\mathcal{R}}_0$, let $A_{0,N}(t_j)$,...,$A_{d,N}(t_j)$ be sequences so that, uniformly in $k$ such that $\displaystyle k_N=\frac{k-{\mathbb E}(S_N)}{{\sigma}_N}=O(1)$ we have \[ \sum_{t_j\in{\mathcal{R}}_0}e^{-it_j k}\left(\sum_{m=0}^{d}k_N^m A_{m,N}(t_j)\right) =o({\sigma}_N^{-r}). \] Then for all $m$ and $t_j$ {\beta}gin{equation}{\lambda}bel{A def} A_{m,N}(t_j)=o({\sigma}_N^{-r}). \end{equation} In particular the polynomials from the definition of the (generalized) Edgeworth expansions are unique up to terms of order $o({\sigma}_N^{-r})$. \end{lemma} {\beta}gin{proof} The proof is by induction on $d$. Let us first set $d=0$. Then, for any $k\in{\mathbb N}$ we have {\beta}gin{equation}{\lambda}bel{d=0} \sum_{t_j\in{\mathcal{R}}_0}e^{-it_j k}A_{0,N}(t_j)=o({\sigma}_N^{-r}). \end{equation} Let $T$ be the number of nonzero resonant points, and let us relabel them as $\{x_1,...,x_T\}$. Consider the vector $$\mathfrak{A}_N=(A_{0,N}(0), A_{0,N}(x_1),...,A_{0,N}(x_T)).$$ Let ${\mathcal V}$ be the transpose of the Vandermonde matrix of the distinct numbers ${\alpha}_j=e^{-ix_j}, j=0,1,2,...,T$ where $x_0:=0$. Then ${\mathcal V}$ is invertible and by considering $k=0,1,2,...,T$ in (\ref{d=0}) we see that (\ref{d=0}) holds true if and only if \[ \mathfrak{A}_N={\mathcal V}^{-1}o({\sigma}_N^{-r})=o({\sigma}_N^{-r}). \] Alternatively, let $Q$ be the least common multiple of the denominators of $t_j\in{\mathcal{R}}$. Let $a_N(p)=A_{0,N}(2\pi p/Q)$ if $2\pi p/Q$ is a resonant point and $0$ otherwise. Then for $m=0,1,...,Q-1$ we have \[ \hat a_N(m):=\sum_{p=0}^{Q-1}a_N(p)e^{-2\pi pm/Q}=o({\sigma}_N^{-r}). \] Therefore, by the inversion formula of the discrete Fourier transform, \[ a_N(p)=Q^{-1}\sum_{m=0}^{Q-1} \hat a_N(m) e^{2\pi i m p/Q}=o({\sigma}_N^{-r}). \] Assume now that the theorem is true for some $d\geq 0$ and any sequences functions $A_{0,N}(t_j),...,A_{d,N}(t_j)$. Let $A_{0,N}(t_j),...,A_{d+1,N}(t_j)$ be sequences so that uniformly in $k$ such that $\displaystyle k_N:=\frac{k-{\mathbb E}(S_N)}{{\sigma}_N}=O(1)$ we have {\beta}gin{equation}{\lambda}bel{I} \sum_{t_j\in{\mathcal{R}}_0}e^{-it_j k}\left(\sum_{m=0}^{d+1}k_N^m A_{m,N}(t_j)\right)=o({\sigma}_N^{-r}). \end{equation} Let us replace $k$ with $k'=k+[{\sigma}_N]Q$, where $Q$ is the least common multiply of all the denominators of the nonzero $t_j$'s. Then $e^{-it_jk}=e^{-it_j k'}$. Thus, \[ \sum_{t_j\in{\mathcal{R}}_0}e^{-it_j k}\left(\sum_{m=0}^{d+1}(k_N'^m-k_N^{m})A_{m,N}(t_j)\right)=o({\sigma}_N^{-r}). \] Set $L_N=[{\sigma}_N]Q/{\sigma}_N\thickapprox Q$. Then the LHS above equals \[ L_N\sum_{t_j\in{\mathcal{R}}_0}e^{-it_j k}\left(\sum_{s=0}^{d}k_N^s{\mathcal A}_{s,N}(t_j)\right) \] where \[ {\mathcal A}_{s,N}(t_j)=\sum_{m=s+1}^{d+1}A_{m,N}(t_j)L_N^{m-s-1}. \] By the induction hypothesis we get that \[ {\mathcal A}_{s,N}(t_j)=o({\sigma}_N^{-r}) \] for any $s=0,1,...,d$. In particular \[ {\mathcal A}_{d,N}(t_j)=A_{d+1,N}(t_j)=o({\sigma}_N^{-r}). \] Substituting this into \eqref{I} we can disregard the last term $A_{d+1,N}(t_j)$. Using the induction hypothesis with $A_{0,N}(t_j),A_{1,N}(t_j),...,A_{d,N}(t_j)$ we obtain \eqref{A def}. \end{proof} \section{First order expansions}{\lambda}bel{FirstOrder} In this section we will consider the case $r=1$. By \eqref{Cj(k)} and \eqref{s}, we see that the contribution coming from the integral over $I_j$ is \[ {\sigma}_{N_0,N}^{-1}e^{-it_j k}\Phi_{N}(t_j)\sqrt {2\pi} e^{-k_{N_0,N}^2/2}+o({\sigma}_N^{-1}) \] where $k_{N_0,N}=(k-{\mathbb E}(S_{N_0,N}))/{\sigma}_{N_0,N}$. Now, using the arguments at the end of the proof of Theorem \ref{IntIndThm} when $r=1$ we can just replace $e^{-k_{N_0,N}^2/2}$ with $e^{-(k-{\mathbb E}(S_N))^2/2V_N}$ (since it is enough to consider the case when $k_{N_0,N}$ and $k_{0,N}$ are of order $V_N^{\varepsilon}$). Therefore, taking into account that ${\sigma}_{N_0,N}^{-1}-{\sigma}_{N}^{-1}=O({\sigma}_N^{-2}N_0)$ we get that {\beta}gin{equation}{\lambda}bel{r=1} \sqrt 2\pi{\mathbb P}(S_N=k)= \end{equation} $$\left(1+\sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\Phi_{N}(t_j)\right){\sigma}_{N}^{-1}e^{-(k-{\mathbb E}[S_N])^2/2V_N}+o({\sigma}ma_N^{-1}). $$ Here ${\mathcal{R}}$ is the set of all nonzero resonant points $t_j=2\pi l_j/m_j$. Indeed \sout{for} the contribution of the resonant points satisfying $M_N(m_j)\leq R(r,K)\ln V_N$ is analyzed in \S \ref{Fin}. The contribution of the other nonzero resonant points $t$ is $o({\sigma}_N^{-1})$ due to \eqref{Roz0} in Section \ref{ScEdgeLogProkh}. In particular, \eqref{Roz0} implies that $\Phi_N(t)=o({\sigma}_N^{-1})$ so adding the points with $M_N(m_j)\geq R(r,K)\ln V_N$ only changes the sum in the RHS of \eqref{r=1} by $o({\sigma}ma_N^{-1}). $ {\beta}gin{corollary}{\lambda}bel{FirstOrCor} The local limit theorem holds if and only if $\displaystyle \max_{t\in R}|\Phi_N(t)|=o(1)$. \end{corollary} {\beta}gin{proof} It follows from (\ref{r=1}) that the LLT holds true if and only if for any $k$ we have $$ \sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\Phi_{N}(t_j)=o(1). $$ Now, the corollary follows from Lemma \ref{Lemma}. \end{proof} Before proving Theorem \ref{ThLLT} we recall a standard fact which will also be useful in the proofs of Theorems \ref{Thm SLLT VS Ege} and \ref{Thm Stable Cond}. {\beta}gin{lemma} {\lambda}bel{LmUnifFourier} Let $\{\mu_N\}$ be a sequence of measures probability measures on ${\mathbb Z}/m{\mathbb Z}$ and $\{{\gamma}ma_N\}$ be a positive sequence. Then $\mu_N(a)=\frac{1}{m}+O({\gamma}ma_N)$ for all $a\in {\mathbb Z}/m{\mathbb Z}$ if and only iff $\hat\mu_N(b)=O({\gamma}ma_N)$ for all $b\in \left({\mathbb Z}/m{\mathbb Z}\right)\setminus \{0\}$ where $\hat\mu$ is the Fourier transform of $\mu.$ \end{lemma} {\beta}gin{proof} If $\mu_N(a)=\frac{1}{m}+O({\gamma}ma_n)$ then $$ \hat\mu_N(b)=\sum_{a=0}^{m-1} \mu_N(a) e^{2\pi i ab/m}= \sum_{a=0}^{m-1} \frac{1}{m} e^{2\pi iab/m}+O({\gamma}ma_N)=O({\gamma}ma_N).$$ Next $\hat\mu_N(0)=1$ since $\mu_N$ are probabilities. Hence if $\hat\mu_N(b)=O({\gamma}ma_N)$ for all $b\in \left({\mathbb Z}/m{\mathbb Z}\right)\setminus \{0\}$ then $$ \mu_N(a)=\frac{1}{m} \sum_{b=0}^{m-1} \hat\mu_N(b) e^{-2\pi i ba/m}= \frac{1}{m} \left[1+\sum_{b=1}^{m-1} \hat\mu_N(b) e^{-2\pi i ba/m}\right]=\frac{1}{m}+O({\gamma}ma_N)$$ as claimed. \end{proof} {\beta}gin{proof}[Proof of Theorem \ref{ThLLT}] The equivalence of conditions (b) and (c) comes from the fact that for non-resonant points the characteristic function decays faster than any power of ${\sigma}ma_N$ (see \eqref{NonResDec}). The equivalence of (a) and (c) is due to Corollary \ref{FirstOrCor}. Finally, the equivalence between (c) and (d) comes from Lemma \ref{LmUnifFourier}. \end{proof} {\beta}gin{remark} Theorem \ref{ThLLT} can also be deduced from \cite[Corollary 1.4]{Do}. Indeed the corollary says that either the LLT holds or there is an integer $h\in (0, 2K)$ and a bounded sequence $\{a_N\}$ such that the limit $$ {\mathbf{p}}(j)=\lim_{N\to\infty} {\mathbb P}(S_N-a_N=j {\theta}xt{ mod } h) $$ exists and moreover if $k-a_n\equiv j {\theta}xt{ mod }h$ then $$ {\sigma}ma_N {\mathbb P}(S_N=k)={\mathbf{p}}(j) h \mathfrak{g}\left(\frac{k-{\mathbb E}(S_N)}{{\sigma}ma_N}\right)+o\left({\sigma}ma_N^{-1}\right).$$ Thus in the second case the LLT holds iff ${\mathbf{p}}(j)=\frac{1}{h}$ for all $j$ which is equivalent to $S_N$ being asymptotically uniformly distributed mod $h$ and also to the Fourier transform of ${\mathbf{p}}(j)$ regarded as the measure on ${\mathbb Z}/(h{\mathbb Z})$ being the ${\delta}ta$ measure at 0. Thus the conditions (a), (c) and (d) of the theorem are equivalent. Also by the results of \cite[Section 2]{Do} (see also [\S 3.3.2]\cite{DS}) if ${\mathbb E}\left(e^{i\xi S_N}\right)$ does not converge to 0 for some non zero $\xi$ then $\displaystyle \left(\frac{2\pi}{\xi} \right) {\mathbb Z}\bigcap 2\pi {\mathbb Z}$ is a lattice in ${\mathbb R}$ which implies that $\xi$ is resonant, so condition (b) of the theorem is also equivalent to the other conditions. \end{remark} {\beta}gin{proof}[Proof of Proposition \ref{PrLLT-SLLT}.] Let $S_N$ satisfy LLT. Fix $m\in \mathbb{N}$ and suppose that $\displaystyle \sum_n q_n(m)<\infty.$ Let $s_n$ be the most likely residue of $X_n$ mod $m$. Then for $t=\frac{2\pi l}{m}$ we have $$\phi_n(t)=e^{i t s_n}-\sum_{j\not\equiv s_n\; {\theta}xt{mod}\; m} {\mathbb P}(X_n\equiv j\; {\theta}xt{mod m})\left(e^{its_n}-e^{itj}\right),$$ so that $1\geq |\phi_n(t)|\geq 1-2 m q_n(m).$ It follows that for each ${\varepsilon}>0$ there is $N({\varepsilon})$ such that $\displaystyle \left|\prod_{n=N({\varepsilon})+1}^\infty \phi_n(t)\right|>1-{\varepsilon}. $ Applying this for ${\varepsilon}=\frac{1}{2}$ we have {\beta}gin{equation} {\lambda}bel{PhiNHalf} \frac{1}{2}\leq\liminf_{N\to\infty} \left|\Phi_{N(1/2), N}(t)\right|\neq 0. \end{equation} On the other hand the LLT implies that {\beta}gin{equation} {\lambda}bel{PhiLim} \lim_{N\to\infty} \Phi_N(t)=0. \end{equation} Since $\Phi_N=\Phi_{N(1/2)} \Phi_{N(1/2), N}$, \eqref{PhiNHalf} and \eqref{PhiLim} imply that $\Phi_{N(1/2)}(t)=0.$ Since $\displaystyle \Phi_{N(1/2)}\left(\frac{2\pi l}{m}\right)=\prod_{n=1}^{N(1/2)} \phi_n\left(\frac{2\pi l}{m}\right)$ we conclude that there exists $n_l\leq N(1/2)$ such that $\phi_{n_l}(\frac{2\pi l}{m})=0.$ Hence $Y=X_{n_1}+X_{n_2}+\dots X_{n_{m-1}}$ satisfies ${\mathbb E}\left(e^{2\pi i (k/m)Y}\right)=0$ for $k=1,\dots m-1.$ By Lemma \ref{LmUnifFourier} both $Y$ and $S_N$ for $N\geq N(1/2)$ are uniformly distributed. This proves the proposition. \end{proof} \section{Characterizations of Edgeworth expansions of all orders.} {\lambda}bel{ScCharacterization} \subsection{Derivatives of the non-perturbative factor.} Next we prove the following result. {\beta}gin{proposition}{\lambda}bel{Thm} Fix $r\geq1,$ and assume that $M_N\leq R(r, K)\ln{\sigma}_N$ (possibly along a subsequence). Then Edgeworth expansions of order $r$ hold true (i.e. (\ref{EdgeDef}) holds for such $N$'s) iff for each $t_j\in{\mathcal{R}}$ and $0\leq\ell<r$ (along the underlying subsequence) we have {\beta}gin{equation}{\lambda}bel{Cond} {\sigma}_N^{r-\ell-1}\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)=o(1). \end{equation} \end{proposition} {\beta}gin{proof} First, in view of \eqref{Cj(k)} and \eqref{expo}, it is clear that the condition (\ref{Cond}) is sufficient for expansions of order $r$. Let us now prove that the condition (\ref{Cond}) is necessary for the expansion of order $r.$ We will use induction on $r$. For $r=1$ (see \eqref{r=1}) our expansions read \[ {\mathbb P}(S_N=k)={\sigma}_N^{-1}e^{-k_N^2/2} \left[1+\sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\Phi_N(t_j)\right]+o({\sigma}_N^{-1}). \] Therefore if \[ {\mathbb P}(S_N=k)={\sigma}_N^{-1}e^{-k_N^2/2}P_N(k_N)+o({\sigma}_N^{-1}) \] for some polynomial $P_N$ then Lemma \ref{Lemma} tells us that, in particular $\Phi_{N}(t_j)=o(1)$ for each $t_j\in{\mathcal{R}}$. Let us assume now that the necessity part in Proposition \ref{Thm} holds for $r'=r-1$ and prove that it holds for $r$. We will use the following lemma. {\beta}gin{lemma}{\lambda}bel{LemInd} Assume that for some $t_j\in{\mathcal{R}}$, {\beta}gin{equation}{\lambda}bel{Ind} {\sigma}_{N}^{r-2-l}\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(l)}(t_j)=o(1),\, l=0,1,...,r-2. \end{equation} Then, up to an $o({\sigma}_N^{-r})$ error term, the contribution of $t_j$ to the generalized Edgeworh expansions of order $r$ is {\beta}gin{equation}{\lambda}bel{Val0} e^{-it_j k}e^{-k_{N}^2/2}\left(\frac{\Phi_{N_0}(t_j)}{{\sigma}_N}+\sum_{q=2}^{r}\frac{\mathscr H_{N,q}(k_{N})}{{\sigma}_N^{q}}\right) \end{equation} with {\beta}gin{equation} {\lambda}bel{DefCH} \mathscr H_{N,q}(x)=\mathscr H_{N,q}(x;t_j)= \end{equation} $$ {\mathcal H}_{N,q,1}(x)+{\mathcal H}_{N,q,2}(x)+{\mathcal H}_{N,q,3}(x)+{\mathcal H}_{N,q,4}(x) $$ where $$ {\mathcal H}_{N, q,1}(x)= \frac{(i)^{q-1}H_{q-1}(x)\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(q-1)}(t_j)}{(q-1)!},$$ $$ {\mathcal H}_{ N,q, 2}(x)= \frac{(i)^{q-1}H_{q-1}(x)\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(q-2)}(t_j)C_{1,N,t_j}}{(q-2)!}, $$ $${\mathcal H}_{N,q,3}(x)= \frac{a_{N_0}(i)^{q-2}H'_{q-2}(x)\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(q-2)}(t_j)}{(q-2)!},$$ $$ {\mathcal H}_{N,q,4}(x)=-\frac{xa_{N_0}(i)^{q-2}H_{q-2}(x)\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(q-2)}(t_j)}{(q-2)!}, $$ and $H_q$ are Hermite polynomials. Here $C_{1,N,t_j}$ is given by \eqref{C 1 N} when $M_N(m)\leq R(K, r)\ln{\sigma}_N$, and $C_{1,N,t_j}=0$ when $M_N(m)>R(K,r)\ln{\sigma}_N.$ (Note that in either case $C_{1,N,t_j}=O(M_N(m))=O(\ln{\sigma}_N)$). As a consequence, when the Edgeworth expansions of order $r$ hold true and \eqref{Ind} holds, then uniformly in $k$ so that $k_N=O(1)$ we have {\beta}gin{equation}{\lambda}bel{Val} \frac{\Phi_{N}(t_j)}{{\sigma}_N}+\sum_{q=2}^{r}\frac{\mathscr H_{N,q}(k_{N};t_j)}{{\sigma}_N^{q}}=o({\sigma}_N^{-r}). \end{equation} \end{lemma} The proof of the lemma will be given in \S \ref{SSSummingUp} after we finish the proof of Proposition \ref{Thm}. By the induction hypothesis the condition \eqref{Ind} holds true. Let us prove now that for $\ell=0,1,2,...,r-1$ and $t_j\in{\mathcal{R}}$ we have \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)=o({\sigma}_N^{-r+1+\ell}). \] Let us write \[ \frac{\Phi_{N}(t_j)}{{\sigma}_N}+\sum_{q=2}^{r}\frac{\mathscr H_{N,q}(k_{N})}{{\sigma}_N^{q}}=\sum_{m=0}^{r-1}k_N^m A_{m,N}(t_j). \] Applying Lemmas \ref{Lemma} and \ref{LemInd} we get that \[ A_{m,N}(t_j)=o({\sigma}_N^{-r}) \] for each $0\leq m\leq r-1$ and $t_j\in{\mathcal{R}}$. Fix $t_j\in{\mathcal{R}}$. Using Lemma \ref{LemInd} and the fact that the Hermite polynomials $H_{u}$ have the same parity as $u$ and that their leading coefficient is $1$ we have {\beta}gin{eqnarray} {\lambda}bel{AEq1} A_{r-1,N}(t_j)={\sigma}_N^{-r}(i)^{r-2}\Bigg(i\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-1)}(t_j)/(r-1)\\+\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-2)}(t_j)(iC_{1,N}-a_{N_0})\Bigg)=o({\sigma}_N^{-r}) \notag \end{eqnarray} and {\beta}gin{eqnarray} {\lambda}bel{AEq2} \quad A_{r-2,N}(t_j)={\sigma}_N^{-r+1}(i)^{r-3}\Bigg(i\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-2)}(t_j)/(r-2)\\+\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-3)}(t_j)(iC_{1,N}-a_{N_0})\Bigg)=o({\sigma}_N^{-r}). \notag \end{eqnarray} Since $\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-3)}(t_j)=o({\sigma}_N^{-1})$, \eqref{AEq2} yields \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-2)}(t_j)=o({\sigma}_N^{-1}\ln{\sigma}_N). \] Plugging this into \eqref{AEq1} we get \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-1)}(t_j)=o(1). \] Therefore we can just disregard $\mathscr H_{r,N}(k_N;t_j)$ since its coefficients are of order $o({\sigma}_N^{-r})$. Since the term ${\mathcal H}_{r,N}(k_N;t_j)$ no longer appears, repeating the above arguments with $r-1$ in place of $r$ we have {\beta}gin{eqnarray*} A_{r-3,N}(t_j)={\sigma}_N^{-r+2}(i)^{r-4}\Bigg(i\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-3)}(t_j)/(r-3)\\+\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-4)}(t_j)(iC_{1,N}-a_{N_0})\Bigg)=o({\sigma}_N^{-r}). \end{eqnarray*} Since $\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-4)}(t_j)=o({\sigma}_N^{-2})$, the above asymptotic equality yields that \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-3)}(t_j)=o({\sigma}_N^{-2}\ln{\sigma}_N). \] Plugging this into \eqref{AEq2} we get \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(r-2)}(t_j)=o({\sigma}_N^{-1}). \] Hence, we can disregard also the term $\mathscr H_{r-1,N}(k_N;t_j)$. Proceeding this way we get that $\displaystyle \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)=o({\sigma}_N^{\ell+1-r}) $ for any $0\leq \ell<r$. \qed Before proving Lemma \ref{LemInd}, let us state the following result, which is a consequence of Proposition \ref{Thm} and \eqref{Roz0}. {\beta}gin{corollary}{\lambda}bel{CorNoDer} Suppose that for each nonzero resonant point $t$ we have $\displaystyle \inf_{n}|\phi_n(t)|>0$. Then for any $r$, the sequence $S_N$ obeys Edgeworth expansions of order $r$ if and only if $\Phi_N(t)=o({\sigma}_N^{1-r})$ for each nonzero resonant point $t$. \end{corollary} \subsection{Proof of Lemma \ref{LemInd}.} {\lambda}bel{SSSummingUp} {\beta}gin{proof} First, because of \eqref{Ind}, for each $0\leq s\leq r-1,$ the terms indexed by $l<s-1$ in \eqref{Cj(k)}, are of order $o({\sigma}_N^{-r})$ and so they can be disregarded. Therefore, we need only to consider the terms indexed by $l=s$ and $l=s-1$. For such $l$, using again \eqref{Ind} we can disregard all the terms in \eqref{s} indexed by $w\geq1$, since the resulting terms are of order $o({\sigma}_N^{-w-r}\ln{\sigma}_N)=o({\sigma}_N^{-r})$. Now, since ${\sigma}_{N_0,N}^{-1}-{\sigma}_{N}^{-1}=O(V_{N_0}/{\sigma}_N^3)$ we can replace ${\sigma}_{N_0,N}$ with ${\sigma}_{N}^{-1}$ in \eqref{Cj(k)}, as the remaining terms are of order $o({\sigma}_N^{-r-1})$. Therefore, using \eqref{Fourir} we get the following contributions from $t_j\in{\mathcal{R}}$, {\beta}gin{equation*} e^{-it_j k}\; e^{-k_{N_0,N}^2/2}\left(\frac{\Phi_{N}(t_j)}{{\sigma}_N}+\sum_{q=2}^{r}\frac{{\mathcal H}_{N,q}(k_{N_0,N})}{{\sigma}_N^{q}}\right) \end{equation*} where ${\mathcal H}_{N,q}(x)={\mathcal H}_{N,q,1}(x)+{\mathcal H}_{N,q,2}(x)$ and ${\mathcal H}_{N,q,j}, j=1,2$ are defined after \eqref{DefCH}. Note that when $x=O(1)$ and $q<r$, {\beta}gin{equation}{\lambda}bel{Order} \frac{{\mathcal H}_{N,q,1}(x)}{{\sigma}_N^q}=o({\sigma}_N^{-r+1})\,\,{\theta}xt { and }\,\,\frac{{\mathcal H}_{N,q,2}(x)}{{\sigma}_N^q}=o({\sigma}_N^{-r}\ln{\sigma}_N). \end{equation} while when $q=r$, {\beta}gin{equation}{\lambda}bel{Order.1} \frac{{\mathcal H}_{N,r,1}(x)}{{\sigma}_N^r}=O({\sigma}_N^{-r}\ln{\sigma}_N)\,\,{\theta}xt { and }\,\,\frac{{\mathcal H}_{N,r,2}(x)}{{\sigma}_N^r}=o({\sigma}_N^{-r}\ln{\sigma}_N). \end{equation} Next \[ k_{N_0,N}=(1+\rho_{N_0,N})k_N+\frac{a_{N_0}}{{\sigma}_N}+\theta_{N_0,N} \] where $\rho_{N_0,N}={\sigma}_{N}/{\sigma}_{N_0,N}-1=O(\ln{\sigma}_N/{\sigma}^2_N)$ and $$\theta_{N_0,N}=a_{N_0}\left(\frac{1}{{\sigma}_{N_0,N}}-\frac{1}{{\sigma}_N}\right)=O(\ln^2{\sigma}_N/{\sigma}_N^3).$$ Hence, when $|k_{N_0,N}|\leq{\sigma}_N^{{\varepsilon}}$ (and so $\displaystyle k_N=O({\sigma}_N^{{\varepsilon}})$) for some ${\varepsilon}>0$ small enough then for each $m\geq 1$ we have \[ k_{N_0,N}^m=k_{N}^m+mk_{N}^{m-1}a_{N_0}/{\sigma}_N+o({\sigma}_N^{-1}). \] Therefore, \eqref{Order} and \eqref{Order.1} show that upon replacing $H_{q-1}(k_{N_0,N})$ with $H_{q-1}(k_{N})$ the only additional term is \[ \frac{a_{N_0}(i)^{q-1}H'_{q-1}(k_N)\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(q-1)}(t_j)}{(q-1)!{\sigma}_N^{q+1}} \] for $q=2,3,...,r-1$. We thus get that the contribution of $t_j$ is {\beta}gin{equation*} e^{-it_j k} \;e^{-k_{N_0,N}^2/2}\left(\frac{\Phi_{N}(t_j)}{{\sigma}_N}+\sum_{q=2}^{r}\frac{{\mathcal C}_{N,q}(k_{N})}{{\sigma}_N^{q}}\right) \end{equation*} where {\beta}gin{equation*} {\mathcal C}_{N,q}(x)={\mathcal H}_{N,q}(x)+ \frac{a_{N_0}(i)^{q-2}H'_{q-2}(x)\Phi_{N_0,N}(t_j) \Phi_{N_0}^{(q-2)}(t_j)}{(q-2)!}. \end{equation*} Note that ${\mathcal C}_{N,2}(\cdot)={\mathcal H}_{N,2}(\cdot)$. Finally, we can replace $e^{-k_{N_0,N}^2/2}$ with \[ (1-k_Na_{N_0}/{\sigma}_N)e^{-k_{N}^2/2} \] since all other terms in the transition between $e^{-k_{N_0,N}^2/2}$ to $e^{-k_{N}^2/2}$ are of order $o({\sigma}_N^{-1})$ (see (\ref{ExpTrans1}) and (\ref{ExTrans1.1})). The term $-k_Na_{N_0}/{\sigma}_N$ shifts the $u$-th order term to the $u+1$-th term, $u=1,2,...,r-1$ multiplied by $-k_Na_{N_0}$. Next, relying on \eqref{Order} and \eqref{Order.1} we see that after multiplied by $k_Na_{N_0}/{\sigma}_N$, the second term ${\mathcal H}_{N,q,2}(k_N)$ from the definition of ${\mathcal H}_{N,q}(k_N)$ is of order $o({\sigma}_N^{-r-1}\ln^2{\sigma}_N){\sigma}_N^q$ and so this product can be disregarded. Similarly, we can ignore the additional contribution coming from multiplying the second term from the definition of ${\mathcal C}_{N,q}(k_N)$ by $-k_Na_{N_0}/{\sigma}_N$ (since this term is of order $o({\sigma}_N^{-r}\ln{\sigma}_N){\sigma}_N^q$). We conclude that, up to a term of order $o({\sigma}_N^{-r})$, the total contribution of $t_j$ is {\beta}gin{equation*} e^{-it_j k} \; e^{-k_{N}^2/2}\left(\frac{\Phi_{N}(t_j)}{{\sigma}_N}+\sum_{q=2}^{r}\frac{\mathscr H_{N,q}(k_{N}; t_j)}{{\sigma}_N^{q}}\right) \end{equation*} where $\displaystyle \mathscr H_{N,q}(x;t_j)={\mathcal C}_{N,q}(x)-\frac{x a_{N_0}(i)^{q-2}H_{q-2}(x) \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(q-2)}(t_j)}{(q-2)!} $ which completes the proof of \eqref{Val0}. Next we prove \eqref{Val}. On the one hand, by assumption we have Edgeworth expansions or order $r$, and, on the other hand, we have the expansions from Theorem \ref{IntIndThm}. Therefore, the difference between the two must be $o({\sigma}_N^{-r})$. Since the usual Edgeworth expansions contain no terms corresponding to nonzero resonant points, applying Lemma \ref{Lemma} and \eqref{Val0} we obtain \eqref{Val}. \end{proof} Note that the formulas of Lemma \ref{LemInd} together with already proven Proposition \ref{Thm} give the following result. {\beta}gin{corollary} {\lambda}bel{CrFirstNonEdge} Suppose that $\mathbb{E}(S_N)$ is bounded, $S_N$ admits the Edgeworth expansion of order $r-1$, and, either (a) for some ${\bar\varepsilon}\leq 1/(8K)$ we have $N_0=N_0(N,t_j,{\bar\varepsilon})=0$ for each each nonzero resonant point $t_j$, or (b) $\displaystyle \varphi:=\min_{t\in{\mathcal{R}}}\inf_{n}|\phi_n(t)|>0$. Then $$ \sqrt{2\pi} \mathbb{P}(S_N=k)$$ $$=e^{-k_N^2/2} \left[{\mathcal E}_r(k_N)+ \sum_{t_j\in{\mathcal{R}}}\left(\frac{\Phi_N(t_j)}{{\sigma}_N}+\frac{ik_N C_{1,N,t_j}\Phi_N(t_j)}{{\sigma}_N^2}\right) e^{-i t_j k}\right]+o({\sigma}ma_N^{-r})$$ where ${\mathcal E}_r(\cdot)$ is the Edgeworth polynomial of order $r$ (i.e. the contribution of $t=0$), and we recall that $$iC_{1,N,t_j}=-\sum_{n=1}^N\frac{{\mathbb E}(e^{it_j X_n}\bar X_n)}{{\mathbb E}(e^{it_j X_n})}.$$ \end{corollary} {\beta}gin{proof} Part (a) holds since under the assumption that $N_0=0$ all terms ${\mathcal H}_{N,q,j}$ in \eqref{DefCH} except ${\mathcal H}_{N, 2, 2}$ vanish. Part (b) holds since in this case the argument proceeds similarly to the proof of Theorem \ref{IntIndThm} if we set $N_0=0$ for any $t_j$ (since we only needed $N_0$ to obtain a positive lower bound on $|\phi_n(t_j)|$ for $t_j\in{\mathcal{R}}$ and $N_0<n\leq N$). \end{proof} {\beta}gin{remark} Observe that $\displaystyle {\sigma}_N^{-1}\gg |C_{1, N, {t_j}}|{\sigma}_N^{-2},$ so if the conditions of the corollary are satisfied but $|\Phi_N(t_j)|\leq c {\sigma}_N^{1-r}$ (possibly along a subsequence), then the leading correction to the Edgeworth expansion comes from $$ e^{-k_N^2/2} \sum_{t_j\in{\mathcal{R}}}\left(\frac{\Phi_N(t_j)}{{\sigma}_N} \right).$$ Thus Corollary \ref{CrFirstNonEdge} strengthens Corollary \ref{CorNoDer} by computing the leading correction to the Edgeworth expansion when the expansion does not hold. \end{remark} \subsection{Proof of Theorem \ref{r Char}} We will use the following. {\beta}gin{lemma}{\lambda}bel{Lem} Let $t_j$ be a nonzero resonant point, $r>1$ and suppose that $M_N\leq R\ln{\sigma}_N$, $R=R(r,K)$ and that $|{\mathbb E}(S_N)|=O(\ln{\sigma}_N)$. Then (\ref{Cond}) holds for all $0\leq \ell<r$ if and only if {\beta}gin{equation}{\lambda}bel{CondDer} |\Phi_{N}^{(\ell)}(t_j)|=o\left({\sigma}_N^{-r+\ell+1}\right) \end{equation} for all $0\leq \ell<r$. \end{lemma} {\beta}gin{proof} Let us first assume that (\ref{Cond}) holds. Recall that {\beta}gin{equation} {\lambda}bel{PhiNProduct} \Phi_N(t)=\Phi_{N_0}(t) \Phi_{N_0, N}(t) \end{equation} with {\beta}gin{equation} {\lambda}bel{TripleProduct} \Phi_{N_0, N}(t)=\Phi_{N_0, N} (t_j)\Phi_{N_0, N}(h) \Psi_{N_0, N}(h) \end{equation} where $t=t_j+h$ and {\beta}gin{equation} {\lambda}bel{TripleFactorization} \Psi_{N_0, N}(h)=\exp\left[O(M_N(m))\sum_{u=1}^\infty (O(1))^u h^u\right]. \end{equation} For $\ell=0$ the result reduces to \eqref{PhiNProduct}. For larger $\ell$'s we have {\beta}gin{equation}{\lambda}bel{Relation} \Phi_{N}^{(\ell)}(t_j)=\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)+\sum_{k=0}^{\ell-1}\binom{\ell}{k}\Phi_{N_0,N}^{(\ell-k)}(t_j)\Phi_{N_0}^{(k)}(t_j). \end{equation} Fix some $k<\ell$. Then by \eqref{TripleProduct}, {\beta}gin{eqnarray*} \Phi_{N_0, N}^{(\ell-k)}(t_j)=\Phi_{N_0,N}(t_j)\sum_{u=0}^{\ell-k}\binom{\ell-k}{u}\Phi_{N_0,N}^{(u)}(0)\Psi_{N_0,N}^{(\ell-k-u)}(0)\\=O(\ln^{\ell-k}{\sigma}_N)\Phi_{N_0,N}(t_j) \end{eqnarray*} where we have used that $\bar S_{N_0,N}=S_{N_0,N}-{\mathbb E}(S_{N_0,N})$ satisfies $|{\mathbb E}[(\bar S_{N_0,N})^q]|\leq C_q{\sigma}_{N_0,N}^{2q}$, (see \eqref{CenterMoments}). Therefore {\beta}gin{equation}{\lambda}bel{Thereofre} \Phi_{N_0,N}^{(\ell-k)}(t_j)\Phi_{N_0}^{(k)}(t_j)=O(\ln^{\ell-k}{\sigma}_N)\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(k)}(t_j). \end{equation} Finally, by (\ref{Cond}) we have \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(k)}(t_j)=o({\sigma}_N^{k+1-r}) \] and so, since $k<\ell$, \[ \Phi_{N_0}^{(\ell-k)}(t_j)\Phi_{N_0}^{(k)}(t_j)=o({\sigma}_N^{\ell+1-r}). \] This completes the proof that \eqref{CondDer} holds. Next, suppose that (\ref{CondDer}) holds for each $0\leq\ell<r$. Let use prove by induction on $\ell$ that {\beta}gin{equation}{\lambda}bel{Above} |\Phi_{N_0,N}\Phi_{N_0}^{(\ell)}(t_j)|=o\left({\sigma}_N^{-r+\ell+1}\right). \end{equation} For $\ell=1$ this follows from \eqref{Relation}. Now take $\ell>1$ and assume that \eqref{Above} holds with $k$ in place of $\ell$ for each $k<\ell$. Then by (\ref{Relation}), (\ref{Thereofre}) and the induction hypothesis we get that \[ \Phi_{N}^{(\ell)}(t_j)=\Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)+o({\sigma}_N^{\ell+1-r}). \] By assumption we have $\Phi_{N}^{(\ell)}(t_j)=o({\sigma}_N^{\ell+1-r})$ and hence \[ \Phi_{N_0,N}(t_j)\Phi_{N_0}^{(\ell)}(t_j)=o({\sigma}_N^{\ell+1-r}) \] as claimed. \end{proof} Theorem \ref{r Char} in the case $M_N\leq R\ln {\sigma}ma_N$ follows now by first replacing $X_n$ with $X_n-c_n$, where $(c_n)$ is a bounded sequence of integers so that ${\mathbb E}[S_N-C_N]=O(1)$, where {\beta}gin{equation} {\lambda}bel{CNInteger} C_N=\sum_{j=1}^n c_j \end{equation} (see Lemma 3.4 in \cite{DS}), and then applying Lemma~\ref{Lem} and Proposition~\ref{Thm}. It remains to consider the case when $M_N(m)\geq {\bar R} \ln {\sigma}ma_N$ where ${\bar R}$ is large enough. In that case, by Theorem \ref{ThEdgeMN}, the Edgeworth expansion of order $r$ hold true, and so, after the reduction to the case when ${\mathbb E}(S_N)$ is bounded, it is enough to show that $\displaystyle \Phi_N^{(\ell)}(t_j)=o\left({\sigma}ma_N^{-r+\ell+1}\right)$ for all $0\leq \ell<r.$ By the arguments of Lemma \ref{Lem} it suffices to show that for each $0\leq \ell<r$ we have $\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j) =o({\sigma}ma_N^{-r})$. To this end we write $$\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j) =\sum_{n_1,\dots, n_k\leq N_0; \atop \ell_1+\dots+\ell_k=\ell} {\gamma}ma_{\ell_1,\dots, \ell_k} \left(\prod_{q=1}^k \phi_{n_q}^{(\ell_q)}(t_j) \right) \left[\prod_{n\leq N, \; n\neq n_k} \phi_n(t_j)\right]$$ where ${\gamma}ma_{\ell_1,\dots, \ell_k}$ are bounded coefficients of combinatorial nature. Using (\ref{Roz0}) we see that for each $n_1,\dots, n_k$ the product in the square brackets is at most $C e^{-c_0 M_N(m)+O(1)}$ for some $C, c_0>0$. Hence $$|\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j)|\leq \hat{C} N_0^{\ell} \; e^{-c M_N(m)}.$$ It remains to observe that the definition of $N_0$ gives $M_N(m)\geq \hat{\varepsilon} N_0.$ Therefore $\displaystyle |\Phi_{N_0}^{(\ell)}(t_j)\Phi_{N_0, N}(t_j)|\leq C^* M_N^\ell (m) \; e^{-c M_N(m)}=o({\sigma}ma_N^{-r})$ provided that $M_N\geq {\bar R}\ln {\sigma}ma_N$ for ${\bar R}$ large enough. \qed \section{Edgeworth expansions and uniform distribution.} \subsection{Proof of Theorem \ref{Thm SLLT VS Ege}} {\lambda}bel{SSEdgeR=2} In view of Proposition \ref{Thm} with $r=2$, it is enough to show that if $\Phi_{N}(t_j)=o({\sigma}ma_N^{-1})$ then the SLLT implies that {\beta}gin{equation} {\lambda}bel{C11-C02} |\Phi_{N_0,N}(t_j)\Phi_{N_0}'(t_j)|=o(1) \end{equation} for any non-zero resonant point $t_j$ (note that the equivalence of conditions (b) and (c) of the theorem follows from Lemma \ref{LmUnifFourier}). Denote $\displaystyle \Phi_{N; k}(t)=\prod_{l\neq k, l\leq N} \phi_l(t)$. Let us first assume that $\phi_k(t_j)\not=0$ for all $1\leq k\leq N$. Then $\phi_k'(t_j) \Phi_{N; k}(t_j)=\phi_k'(t_j)\Phi_N(t_j)/\phi_k(t_j)$. Let ${\varepsilon}_N=\frac{\ln{\sigma}_N}{{\sigma}_N}$. If for all $1\leq k\leq N_0$ we have $|\phi_k(t_j)|\geq {\varepsilon}_N$ then $$ \left|\Phi_{N_0,N}(t_j)\Phi_{N_0}'(t_j)\right|=\left|\sum_{k=1}^{N_0}\phi_k'(t_j) \Phi_{N; k}(t_j)\right|\leq|\Phi_N(t_j)|\sum_{k=1}^{N_0}|\phi'_k(t_j)/\phi_k(t_j)| $$ $$ \leq C{\varepsilon}_N^{-1} N_0 |\Phi_N(t_j)| \leq C'{\sigma}_N|\Phi_N(t_j)|\to 0 {\theta}xt{ as }N\to\infty $$ where we have used that $N_0=O(\ln V_N)$. Next suppose there is at least one $1\leq k\leq N_0$ such that $|\phi_k(t_j)|<{\varepsilon}_N$. Let us pick some $k=k_N$ with the latter property. Then for any $k\not=k_N$, $1\leq k\leq N_0$ we have \[ |\phi_k'(t_j) \Phi_{N; k}(t_j)|\leq C|\phi_{k_N}(t_j)|<C{\varepsilon}_N. \] Therefore, \[ \left|\sum_{k\not=k_N,\,1\leq k\leq N_0}\phi_k'(t_j) \Phi_{N; k}(t_j)\right|\leq \frac{C'\ln^2 {\sigma}_N}{{\sigma}_N}=o(1). \] It follows that {\beta}gin{equation} {\lambda}bel{SingleTerm} \Phi_{N_0,N}(t_j)\Phi_{N_0}'(t_j)=\Phi_{N; k_N}(t_j) \phi_{k_N}'(t_j)+o(1). \end{equation} Next, in the case when $\phi_{k_0}(t_j)=0$ for some $1\leq k_0\leq N_0$, then \eqref{SingleTerm} clearly holds true with $k_N=k_0$ since all the other terms vanish. In summary, either \eqref{C11-C02} holds or we have \eqref{SingleTerm}. In the later case, using (\ref{Roz0}) we obtain {\beta}gin{equation} {\lambda}bel{PhiKBound} \left|\mathbb{E}\left(e^{i t_j S_{N; k_N}}\right)\right|\leq e^{-c_2\sum_{s\not=k_N, 1\leq s\leq N}q_s(m)}=e^{-c_2M_{N}(m)-q_{k_N}(m)} \end{equation} where $S_{N; k}=S_N-X_k$, and $c_2>0$ depends only on $K$. Since the SLLT holds true, $M_{N}$ converges to $\infty$ as $N\to\infty$. Taking into account that $0\leq q_{k_N}(m)\leq1$ we get that the left hand side of \eqref{PhiKBound} converges to $0$, proving \eqref{C11-C02}. \end{proof} \subsection{Proof of Theorem \ref{Thm Stable Cond}} We start with the proof of part (1). Assume that the LLT holds true in a superstable way. Let $X_1',X_2',...$ be a square integrable integer-valued independent sequence which differs from $X_1,X_2,...$ by a finite number of elements. Then there is $n_0\in{\mathbb N}$ so that $X_n=X'_n$ for any $n>n_0$. Set $\displaystyle S_N'=\sum_{n=1}^N X'_n$, $Y=S'_{n_0}$ and $Y_N=Y{\mathbb I}(|Y|<{\sigma}_N^{1/2+{\varepsilon}})$, where ${\varepsilon}>0$ is a small constant. By the Markov inequality we have $$ {\mathbb P}(|Y|\geq {\sigma}_N^{1/2+{\varepsilon}})={\mathbb P}(|Y|^2\geq {\sigma}_N^{1+2{\varepsilon}})\leq\|Y\|_{L^2}^2{\sigma}_N^{-1-2{\varepsilon}}=o({\sigma}_N^{-1}). $$ Therefore, for any $k\in{\mathbb N}$ and $N>n_0$ we have {\beta}gin{eqnarray*} {\mathbb P}(S'_N=k)={\mathbb P}(S_{N;1,2,...,n_0}+Y_N=k)+o({\sigma}_N^{-1})\\={\mathbb E}[{\mathbb P}(S_{N;1,2,...,n_0}=k-Y_N|X_1',...,X_{n_0}')]+o({\sigma}_N^{-1})\\= {\mathbb E}[P_{N:n_1,...,n_0}(k-Y_N)]+o({\sigma}_N^{-1}) \end{eqnarray*} where $P_{N:n_1,...,n_0}(s)={\mathbb P}(S_{N;1,2,...,n_0}=s)$ for any $s\in{\mathbb Z}$. Since the LLT holds true in a superstable way, we have, uniformly in $k$ and the realizations of $X_1',...,X_{n_0}'$ that $$ P_{N:n_1,...,n_0}(k-Y_N)=\frac{e^{-(k-Y_N-{\mathbb E}(S_N))^2/(2V_N)}}{\sqrt{2\pi}{\sigma}_N}+o({\sigma}_N^{-1}). $$ Therefore, {\beta}gin{equation}{\lambda}bel{Above1} {\mathbb P}(S'_N=k)= \end{equation} $$ \frac{e^{-(k-{\mathbb E}(S_N))^2/2V_N}}{\sqrt{2\pi}{\sigma}_N} {\mathbb E}\big(e^{-(k-{\mathbb E}(S_N))Y_N/V_N+Y_N^2/(2V_N)}\big)+o({\sigma}_N^{-1}). $$ Next, since $|Y_N|\leq {\sigma}_N^{1/2+{\varepsilon}}$ we have that $\|Y_N^2/(2V_N)\|_{L^\infty}\leq {\sigma}_{N}^{2{\varepsilon}-1}$, and so when ${\varepsilon}<1/2$ we have $\|Y_N^2/2V_N\|_{L^\infty}=o(1)$. Recall that $k_N=(k-{\mathbb E}(S_N))/{\sigma}_N$. Suppose first that $|k_N|\geq {\sigma}_N^{{\varepsilon}}$ with ${\varepsilon}<1/4.$ Since $$\big|(k-{\mathbb E}(S_N))Y_N/V_N\big|\leq |k_N|{\sigma}_N^{{\varepsilon}-\frac12},$$ we get that the RHS of \eqref{Above1} is $o({\sigma}_N^{-1})$ (uniformly in such $k$'s). On the other hand, if $|k_N|<{\sigma}_N^{{\varepsilon}}$ then $${\mathbb E}\big(e^{-(k-{\mathbb E}(S_N))Y_N/V_N+Y_N^2/2V_N}\big)=1+o(1)$$ (uniformly in that range of $k$'s). We conclude that, uniformly in $k$, we have $$ {\mathbb P}(S'_N=k)=\frac{e^{-(k-{\mathbb E}(S_N))^2/(2V_N)}}{\sqrt{2\pi}{\sigma}_N}+o({\sigma}_N^{-1}). $$ Lastly, since $\displaystyle \sup_{N}|{\mathbb E}(S_N)-{\mathbb E}(S_N')|\!\!<\!\infty$ and $\displaystyle \sup_{N}|{\theta}xt{Var}(S_N)-{\theta}xt{Var}(S_N')|\!<\!\!\infty,$ $$ {\mathbb P}(S'_N=k)=\frac{e^{-(k-{\mathbb E}(S'_N))^2/(2V'_N)}}{\sqrt{2\pi}{\sigma}'_N}+o(1/{\sigma}_N') $$ where $V_N'={\theta}xt{Var}(S_N')$ and ${\sigma}_N'=\sqrt{V_N'}$. Conversely, if the SLLT holds then $M_N(h)\to \infty$ for each $h\geq2$. Now if $t$ is a nonzero resonant point with denominator $h$ then \eqref{Roz0} gives $$ |\Phi_{N; j_1^{N}, j_2^{N},\dots ,j_{s_N}^{N}}(t)|\leq C e^{-c M_N(h)+\bar C {\bar s}},\, \,C,\bar C>0 $$ for any choice of $j_1^N,...,j_{s_N}^N$ and $\bar s$ with $s_N\leq \bar s$. Since the RHS tends to 0 as $N\to\infty$, $\{X_n\}\in EeSS(1)$ completing the proof of part (1). For part (2) we only need to show that (a) is equivalent to (b) as the equivalence of (b) and (c) comes from Lemma \ref{LmUnifFourier}. By replacing again $X_n$ with $X_n-c_n$ it is enough to prove the equivalency in the case when ${\mathbb E}(S_N)=O(1)$. The proof that (a) and (b) are equivalent consists of two parts. The first part is the following statement whose proof is a straightforward adaptation of the proof of Theorem \ref{r Char} and is therefore omitted. {\beta}gin{proposition} $\{X_n\}\in SsEe(r)$ if and only if for each ${\bar s}$, each sequence $j_1^N, j_2^N, \dots ,j_{s_N}^N$ with $s_N\leq {\bar s}$, each $\ell<r$ and each $t\in{\mathcal{R}}$ we have {\beta}gin{equation} {\lambda}bel{PhiDerFM} \Phi^{(\ell)}_{N; j_1^N, j_2^N, \dots, j_{s_N}^N}(t)= o({\sigma}ma_N^{\ell+1-r}). \end{equation} \end{proposition} Note that the above proposition shows that the condition $\displaystyle \Phi_{N; j_1^N, j_2^N, \dots ,j_{s_N}^N}(t)= o({\sigma}ma^{1-r}_{N})$ is necessary. The second part of the argument is to show that if $$ \Phi_{N; j_1^N, j_2^N, \dots ,j_{s_N}^N}(t)= o({\sigma}ma^{1-r}_{N}) $$ holds for every finite modification of $S_N$ with $s_N\leq {\bar s}+\ell$ (uniformly) then \eqref{PhiDerFM} holds for every modifications with $s_N\leq {\bar s}$ so that the condition $\displaystyle \Phi_{N; j_1^N, j_2^N, \dots ,j_{s_N}^N}(t)= o({\sigma}ma^{1-r}_{N})$ is also sufficient. To this end we introduce some notation. Fix a nonzero resonant point $t=\frac{2\pi l}{m}.$ Let $\check\Phi_{N}$ be the characteristic function of the sum $\check S_{N}$ of all $X_n$'s such that $1\leq n\leq N$, $n\not\in\{j_1^{N},j_2^{N},...,j_{s_N}^N\}$ and $q_{n}(m)\geq\bar{\varepsilon}ilon$. Let $\check N$ be the number of terms in $\check S_N.$ Denote $\tilde S_{N}=S_{N;j_1^N,j_2^N,\dots,j_{s_N}^{N}}-\check S_{N}$ and let $\tilde\Phi_N(t)$ be the characteristic function of $\tilde S_N.$ Similarly to the proof of Theorem \ref{r Char} it suffices to show that for each $\ell<r$ $$ \left| \check \Phi_N^{(\ell)} \tilde \Phi_N(t)\right|=o({\sigma}ma_N^{1+\ell-r}) $$ and, moreover, we can assume that $M_N(m)\leq {\bar R} \ln {\sigma}ma_N$ so that $\check N=O(\ln {\sigma}ma_N).$ We have (cf. \eqref{DerComb}) , $$ \check \Phi_N^{(\ell)} \tilde \Phi_N(t) =\sum_{\substack{n_1, \dots ,n_k; \;\\ \ell_1+\dots+\ell_k=\ell}} \prod_{w=1}^k {\gamma}ma_{\ell_1,\dots, \ell_k} \phi_{n_w}^{(\ell_w)}(t_j) \prod_{n\not\in\{n_1,n_2 \dots ,n_k, j_1^N,j_2^N \dots, j_{s_N}^N\}} \phi_n(t_j)$$ where the summation is over all tuples $n_1,n_2, \dots ,n_k$ such that $q_{n_w}(m)\geq\bar{\varepsilon}ilon$. Note that the absolute value of each term in the above sum is bounded by $C |\Phi_{N; n_1, \dots ,n_k, j_1\dots, j_{s_N}^N}(t_j)|=o({\sigma}ma_N^{1-r}).$ It follows that the whole sum is $$ o\left({\sigma}ma_N^{1-r} \check N^\ell\right) = o\left({\sigma}ma_N^{1-r} \ln^\ell {\sigma}ma_N \right) $$ completing the proof. \qed {\beta}gin{remark} Lemma \ref{LmUnifFourier} and Theorem \ref{r Char} show that the convergence to uniform distribution on any factor $\mathbb{Z}/h \mathbb{Z}$ with the speed $o({\sigma}_N^{1-r})$ is necessary for Edgeworth expansion of order $r.$ This is quite intuitive. Indeed calling $\mathscr E_r$ the Edgeworth function of order $r$, (i.e. the contribution from zero), then it is a standard result from numerical integration (see, for instance, \cite[Lemma A.2]{DNP}) that for each $s\in \mathbb{N}$ and each $j\in \mathbb{Z}$ $$ \sum_{k\in \mathbb{Z}} h \mathscr E_r\left(\frac{j+hk}{\sqrt{{\sigma}_N}}\right) =\int_{-\infty}^\infty \mathscr E_r(x) dx+o\left({\sigma}_N^{-s}\right)= 1+o\left({\sigma}_N^{-s}\right) $$ where in the last inequality we have used that the non-constant Hermite polynomials have zero mean with respect to the standard normal law (since they are orthogonal to the constant functions). However, using this result to show that $$\displaystyle \sum_{k\in \mathbb{Z}} \mathbb{P}(S_N=j+kh)=\frac{1}{h}+o\left({\sigma}_N^{1-r}\right) $$ requires a good control on large values of $k.$ While it appears possible to obtain such control using the large deviations theory it seems simpler to estimate the convergence rate towards uniform distribution from our generalized Edgeworth expansion. \end{remark} \section{Second order expansions}{\lambda}bel{Sec2nd} In this section we will compute the polynomials in the general expansions in the case $r=2$. First, let us introduce some notations which depend on a resonant point $t_j.$ Let $t_j=2\pi l_j/m_j$ be a nonzero resonant point such that $M_N(m_j)\leq R(2,K)\ln V_N$ where $R(2,K)$ is specified in Remark \ref{R choice}. Let $\check\Phi_{j,N}$ be the characteristic function of the sum $\check S_{j,N}$ of all $X_n$'s such that $1\leq n\leq N$ and $q_{n}(m_j)\geq\bar{\varepsilon}ilon=\frac1{8K}$. Note that $\check S_{j,N}$ was previously denoted by $S_{N_0}$. Let $\tilde S_{N,j}=S_N-\check S_{N,j}$ and denote by $\tilde \Phi_{N,j}$ its characteristic function. (In previous sections we denoted the same expression by $S_{N_0,N}$, but here we want to emphasize the dependence on $t_j$.) Let ${\gamma}ma_{N,j}$ be the ratio between the third moment of $\tilde S_{N,j}-{\mathbb E}(\tilde S_{N,j})$ and its variance. Recall that by \eqref{CenterMoments} $|{\gamma}ma_{N,j}|\leq C$ for some $C$. Also, let $C_{1,N,t_j}$ be given by (\ref{C 1 N}), with the indexes rearranged so that the $n$'s with $q_n(m)\geq\bar{\varepsilon}$ are the first $N_0$ ones ($C_{1,N,t_j}$ is at most of order $M_N(m)=O(\ln V_{N})$). For the sake of convenience, when either $t_j=0$ or $M_N(m_j)\geq R(2,K)\ln V_N$ we set $C_{1,N,t_j}=0$, $\tilde S_{N,j}=S_N$ and $\check S_{N,j}=0$. In this case $\tilde\Phi_{N,j}=\tilde\Phi_{N}$ and $\check\Phi_{N,j}\equiv 1$. Also denote $k_N=(k-{\mathbb E}(S_N))/{\sigma}_N,$ $\bar S_N=S_N-{\mathbb E}(S_N)$, and ${\gamma}ma_N={\mathbb E}(\bar S_N^3)/V_N$, (${\gamma}ma_N$ is bounded). {\beta}gin{proposition}{\lambda}bel{2 Prop} Uniformly in $k$, we have {\beta}gin{eqnarray}{\lambda}bel{r=2'} \sqrt {2\pi}{\mathbb P}(S_N=k)=\Big(1+\sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\Phi_{N}(t_j)\Big){\sigma}_{N}^{-1}e^{-k_N^2/2}\\-{\sigma}_{N}^{-2}e^{-k_N^2/2}\left({\gamma}ma_N k_N^3/6+\sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\tilde\Phi_{N,j}(t_j)P_{N,j}(k_N)\right)\nonumber\\+o({\sigma}ma_N^{-2})\nonumber \end{eqnarray} where $$ P_{N,j}(x)=\big(\check\Phi_{N,j}(t_j)(iC_{1,N,t_j}-{\mathbb E}(\check S_{N,j}))+i\check\Phi_{N,j}'(t_j)\big)x+\check\Phi_{N,j}(t_j){\gamma}ma_{N,j}x^3/6. $$ \end{proposition} {\beta}gin{proof} Let $t_j=\frac{2\pi l}{m}$ be a resonant point with $M_N(m)\leq R(2,K) \ln V_N.$ Recall that ${\theta}xtbf{C}_j(k)$ are given by \eqref{Cj(k)}. First, in order to compute the term corresponding to ${\sigma}_{N_0,N}^{-2}$ we need only to consider the case $s\leq1$ in (\ref{s}). Using \eqref{A 1,n .1} we end up with the following contribution of the interval containing $t_j$, $$ \sqrt{(2\pi)^{-1}}e^{-it_j k}\tilde \Phi_{N,j}(t_j){\sigma}_{N_0,N}^{-1}\Bigg(\int_{-\infty}^{\infty}e^{-ih(k-{\mathbb E}[\tilde S_{N,j}])/{\sigma}_{N,j}}e^{-h^2/2}dh$$ $$+{\sigma}_{N,j}^{-1}\int_{-\infty}^\infty e^{-ih(k-{\mathbb E}[\tilde S_{N,j}])/{\sigma}_{N,j}}\left(\frac{ih^3}{6}{\mathbb E}\left[\big(\tilde S_{N,j}-{\mathbb E}(\tilde S_{N,j})\big)^3\right] {\sigma}_{N,j}^{-3}\right)dh$$ $$+{\sigma}_{N,j}^{-1}(C_{1,N}\check\Phi_{N,j}(t_j)+\check\Phi_{N,j}'(t_j))\int_{-\infty}^\infty e^{-ih(k-{\mathbb E}(\tilde S_{N,j}))/{\sigma}_{N,j}}he^{-h^2/2}dh\Bigg)$$ $$=e^{-it_j k}\tilde\Phi_{N,j}(t_j)\sqrt{(2\pi)^{-1}} e^{-k_{N,j}^2/2}{\sigma}_{N,j}^{-1}\Big(\check\Phi_{N,j}(t_j)+i\big(C_{1,N,t_j}\check\Phi_{N,j}(t_j)+\check\Phi_{N,j}'(t_j)\big)$$ $$\times k_{N,j}{\sigma}_{N,j}^{-1}+\check\Phi_{N,j}(t_j)(k_{N,j}^3-3k_{N,j}){\gamma}ma_{N,j}{\sigma}_{N,j}^{-1}/6\Big) $$ where ${\sigma}_{N,j}=\sqrt{V(\tilde S_{N,j})}$, $k_{N,j}=(k-{\mathbb E}(\tilde S_{N,j}))/{\sigma}_{N,j}$ and ${\gamma}ma_{N,j}= \frac{{\mathbb E}[(\tilde S_{N,j}-{\mathbb E}(\tilde S_{N,j}))^3]}{{\sigma}_{N,j}^2}$ (which is uniformly bounded). As before we shall only consider the case where $|k_N|\leq V_N^{\varepsilon}$ with ${\varepsilon}=0.01$ since otherwise both the LHS and the RHS \eqref{r=2'} are $O({\sigma}_N^{-r})$ for all $r.$ Then, the last display can be rewritten as $I+I\!\!I$ where {\beta}gin{equation} I=\frac{e^{-i t_j k }}{\sqrt{2\pi} {\sigma}_{N,j}}\; e^{-k_{N,j}^2/2}\; \Phi_N(t_j); \end{equation} $$ I\!\!I =\frac{e^{-i t_j k }}{\sqrt{2\pi} {\sigma}^2_{N,j}}\; e^{-k_{N,j}^2/2}\times$$ $$ \left[\Phi_N(t_j) \left(i C_{1, N,t_j} k_{N,j}+\frac{{\gamma}ma_{N,j}}{6} \left(k_{N,j}^3-3 k_{N,j}\right)\right)+i\check\Phi_{N,j}'(t_j) \tilde\Phi_{N,j}(t_j) k_{N,j} \right]. $$ In the region $|k_N|\leq V_N^{\varepsilon}$ we have $$ I= \frac{e^{-i t_j k }}{\sqrt{2\pi} {\sigma}_{N}}\; e^{-k_{N}^2/2}\; \left[1-q_{N,j} k_N \right]\Phi_N(t_j)+ o\left({\sigma}_N^{-2}\right)$$ where $$q_{N,j}={\mathbb E}(\check S_{N,j})/{\sigma}_{N,j}={\mathbb E}(\check S_{N,j})/{\sigma}_N+O(\ln{\sigma}_N/{\sigma}_N^3)= O(\ln{\sigma}_N/{\sigma}_N) $$ while $$ I\!\!I=\frac{e^{-i t_j k }}{\sqrt{2\pi} {\sigma}^2_{N}}\; e^{-k_{N}^2/2}\times$$ $$ \left[\Phi_N(t_j) \left(i C_{1, N,t_j} k_{ N}+\frac{{\gamma}_{N,j} \left(k_{N}^3-3 k_{N}\right)}{6} \right)+i\check\Phi_{N,j}'(t_j) \tilde\Phi_{N,j}(t_j) k_{N} \right]$$ $$+o\left({\sigma}_N^{-2}\right).$$ This yields (\ref{r=2'}) with ${\mathcal{R}}_N$ in place of ${\mathcal{R}}$, where ${\mathcal{R}}_N$ is the set on nonzero resonant points $t_j=2\pi l/m$ such that $M_N(m)\leq R(2,K)\ln V_N$. Next, \eqref{Roz0} shows that if $M_N(m)\geq R(2,k)\ln V_N$ then \[ \sup_{t\in I_j}|\Phi_{N}(t)|\leq e^{-c_0M_N(m)}=o({\sigma}_N^{-2}) \] and so the contribution of $I_j$ to the right hand side of \eqref{r=2'} is $o({\sigma}_N^{-2}).$ Finally, the contribution coming from $t_j=0$ is \[ e^{-k_N^2/2}\left({\sigma}_N^{-1}+{\sigma}_N^{-2}{\gamma}ma_N^3 k_N^3/6\right) \] and the proof of the proposition is complete. \end{proof} {\beta}gin{remark}{\lambda}bel{Alter 2nd Order} Suppose that $M_N(m)\geq R(2,K)\ln V_N$ and let $N_0$ is the number of $n$'s between $1$ to $N$ so that $q_n(m)\geq \frac{1}{8K}$. Then using (\ref{Roz0}) we also have $$ |\check\Phi'_{N,j}(t_j)\tilde \Phi_{N,j}(t_j)|\leq\sum_{n\in{\mathcal B}_{{\bar\varepsilon}}(m)}|{\mathbb E}[X_ne^{it_j X_n}]|\cdot|\Phi_{N;n}(t_j)| $$ $$ \leq CN_0(N,t_j,{\bar\varepsilon})e^{-c_0 M_{N}(m)}\leq C'M_{N}(m)e^{-c_0 M_{N}(m)}, $$ where $${\mathcal B}_{N,{\bar\varepsilon}}(m)=\{1\leq n\leq N: q_n(m)>{\bar\varepsilon}\}.$$ Since $M_N(m)\geq R(2,K)\ln V_N$, for any $0<c_1<c_0$, when $N$ is large enough we have $$M_{N}(m)e^{-c_0 M_{N}(m)}\leq C_1e^{-c_1M_N(m)}=o({\sigma}_N^{-2}).$$ Similarly, $|{\mathbb E}(\check S_{N,j})\Phi_N(t_j)|=o({\sigma}_N^{-2})$ and $$C_{1,N,t_j}\Phi_{N}(t_j)=O(M_N(m))\Phi_{N}(t_j)=o({\sigma}_N^{-2}).$$ Therefore we get (\ref{r=2'}) when $\tilde S_{N,j}$ and $\check S_{N,j}$ are defined in the same way as in the case $M_N(m)\leq R(2,K)\ln V_N$. \end{remark} Under additional assumptions the order 2 expansion can be simplified. {\beta}gin{corollary} {\lambda}bel{CrR2-SLLT} If $S_N$ satisfies SLLT then $$ \sqrt {2\pi}{\mathbb P}(S_N=k)=\frac{e^{-k_N^2/2}}{{\sigma}_N} \left(1+\sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\Phi_{N}(t_j) -\frac{{\gamma}ma_N k_N^3}{6{\sigma}_N}\right) +o({\sigma}ma_N^{-2}). $$ \end{corollary} {\beta}gin{proof} The estimates of \S \ref{SSEdgeR=2} together with \eqref{Roz0} show that if $S_N$ satisfies the SLLT then for all $j$ $$(1+M_N(m))\tilde\Phi_{N,j} (t_j) \check \Phi_{N,j}(t_j)=o(1) {\theta}xt{ and } \tilde\Phi_{N,j} (t_j) \check \Phi'_{N,j}(t_j)=o(1).$$ Thus all terms in the second line of \eqref{r=2'} except the first one make a negligible contribution, and so they could be omitted. \end{proof} Next, assume that $S_N$ satisfies the LLT but not SLLT. According to Proposition \ref{PrLLT-SLLT}, in this case there exists $m$ such that $M_N(m)$ is bounded and for $k=1,\dots ,m-1$ there exists $n=n(k)$ such that $\phi_{n}(k/m)=0$. Let ${\mathcal{R}}_s$ denote the set of nonzero resonant points $t_j=\frac{2\pi k}{m}$ so that $M_N(m)$ is bounded and $\phi_{\ell_j}(t_j)=0$ for unique $\ell_j.$ {\beta}gin{corollary}{\lambda}bel{CrR2Z} Uniformly in $k$, we have {\beta}gin{eqnarray*} \sqrt {2\pi}{\mathbb P}(S_N=k)=\Big(1+\sum_{t_j\in {\mathcal{R}}}e^{-it_j k}\Phi_{N}(t_j)\Big){\sigma}_{N}^{-1}e^{-k_N^2/2}\\-{\sigma}_{N}^{-2}e^{-k_N^2/2}\left({\gamma}ma_N k_N^3/6+ \sum_{t_j\in {\mathcal{R}}_s}i e^{-it_j k}\Phi_{N; \ell_j}(t_j) \phi_{\ell_j}'(t_j) k_N \right)+o({\sigma}ma_N^{-2}).\nonumber \end{eqnarray*} \end{corollary} {\beta}gin{proof} As in the proof of Corollary \ref{CrR2-SLLT} we see that the contribution of the terms with $k/m$ with $M_N(m)\to\infty$ is negligible. Next, for terms in ${\mathcal{R}}_s$ the only non-zero term in the second line in \eqref{r=2'} corresponds to $\Phi_{N; \ell_j}(t_j) \phi_{\ell_j}'(t_j)$ while for the resonant points such that $\phi_{\ell}(t_j)=0$ for two different $\ell$s all terms vanish. \end{proof} \section{Examples.} {\lambda}bel{ScExamples} {\beta}gin{example} {\lambda}bel{ExNonAr} Suppose $X_n$ are iid integer valued with step $h>1.$ That is there is $s\in {\mathbb Z}$ such that ${\mathbb P}(X_n\in s+h {\mathbb Z})=1$ and $h$ is the smallest number with this property. In this case \cite[Theorem 4.5.4]{IL} (see also \cite[Theorem 5]{Ess}) shows that there are polynomials $P_b$ such that {\beta}gin{equation} {\lambda}bel{EssEdge} {\mathbb P}(S_N=k)= \sum_{b=1}^r \frac{P_{b} ((k-{\mathbb E}[S_N])/{\sigma}ma_N)}{{\sigma}ma_N^b} \mathfrak{g}\left(\frac{k-{\mathbb E}(S_N)}{{\sigma}ma_N}\right) +o({\sigma}ma_N^{-r}) \end{equation} for all $k\in sN+h{\mathbb Z}.$ Then $$ \sum_{a=0}^{h-1} \sum_{b=1}^r e^{2\pi i a(k-sN)/h} \frac{P_{b} ((k-{\mathbb E}[S_N])/{\sigma}ma_N)}{{\sigma}ma_N^b} \mathfrak{g}((k-{\mathbb E}(S_N))/{\sigma}ma_N)$$ provides $\displaystyle o({\sigma}ma_N^{-r})$ approximation to ${\mathbb P}(S_N=k)$ which is valid for {\em all} $k\in {\mathbb Z}.$ Next let ${\bar S}_N=X_0+S_N$ where $X_0$ is bounded and arithmetic with step 1. Then using the identity {\beta}gin{equation} {\lambda}bel{Convolve} {\mathbb P}({\bar S}_N=k)=\sum_{u\equiv k-s N{\theta}xt{ mod } h} {\mathbb P}(X_0=u) {\mathbb P}(S_N=k-u), \end{equation} \noindent invoking \eqref{EssEdge} and expanding $\displaystyle \mathfrak{g}\left(\frac{k-u-{\mathbb E}(S_N)}{{\sigma}ma_N}\right)$ in the Taylor series about $\frac{k-{ {\mathbb E}(S_N)}}{{\sigma}ma_N}$ we conclude that there are polynomials $P_{b,j}$ such that we have for $k\in j+h{\mathbb Z}$, $$ {\mathbb P}({\bar S}_N=k)= \sum_{b=1}^r \frac{P_{b, j} ((k-{\mathbb E}[S_N])/{\sigma}ma_N)}{{\sigma}ma_N^b} \mathfrak{g}\left(\frac{k-{\mathbb E}(S_N)}{{\sigma}ma_N}\right) +o({\sigma}ma_N^{-r}). $$ Again $$ \sum_{a=0}^{h-1} \sum_{j=0}^{h-1} e^{2\pi i a (k-j)/h} \sum_{b=1}^r \frac{P_{b,j} ((k-{\mathbb E}[S_N])/{\sigma}ma_N)}{{\sigma}ma_N^b} \mathfrak{g}\left(\frac{k-{\mathbb E}(S_N)}{{\sigma}ma_N}\right) $$ provides the oscillatory expansion valid for all integers. \end{example} {\beta}gin{example} {\lambda}bel{ExUniform} Our next example is a small variation of the previous one. Fix a positive integer $m.$ Let $X'$ be a random variable such that $X'$ mod $m$ is uniformly distributed. Then its characteristic function satisfies $\phi_{X'}(\frac{2\pi a}{m})=0$ for $a=1, \dots, m-1.$ We also assume that $\phi_{X'}'(\frac{2\pi a}{m})\neq 0$ for $a$ as above (for example one can suppose that $X'$ takes the values $Lm$, $1, 2, \dots ,m-1$ with equal probabilities where $L$ is a large integer). Let $X''$ take values in $m{\mathbb Z}$ and have zero mean. We also assume that $X''$ does not take values at $m_0{\mathbb Z}$ for a larger $m_0$. Then $q(X'',m_0)>0$ for any $m_0\not=m$. Fix $r\in \mathbb{N}$ and let $$X_n={\beta}gin{cases} X' & n\leq r, \\ X'' & n>r. \end{cases}$$ Then $M_N(m_0)$ grows linearly fast in $N$ if $m_0\not=m$ and $M_N(m)$ is bounded in $N$. We claim that $S_N$ admits the Edgeworth expansion of order $r$ but does not admit Edgeworth expansion of order $r+1.$ The first statement holds due to Theorem \ref{r Char}, since $\Phi^{(\ell)}_N(\frac{2\pi a}{m})=0$ for each $a\in {\mathbb Z}$ and each $\ell<r.$ On the other hand, since $\Phi_{N}^{(\ell)}(\frac{2\pi a}m)=0$ for any $\ell<r$, using Lemma \ref{Lem} we see that the conditions of Lemma \ref{LemInd} are satisfied with $r+1$ in place of $r$. Moreover, with $t_j=2\pi a/m$, $a\not=0$ we have ${\mathcal H}_{N,r+1,s}(x,t_j)\equiv0$ for any $q\leq r+1$ and $s=2,3,4$ while ${\mathcal H}_{N,q,w}(x,t_j)\equiv0$ for any $q\leq r$ and $w=1,2,3,4$. Furthermore, when $N\geq r$ we have {\beta}gin{eqnarray*} {\mathcal H}_{N,r+1,1}(x;t_j)=\frac{i^{r}H_{r}(x)\big(\phi_{X''}(2\pi a/m)\big)^{N-r}\Phi_{r}^{(r)}(2\pi a/m)}{r!}\\=(i)^{r}H_{r}(x)\big(\Phi_{X'}'(2\pi a/m)\big)^r. \end{eqnarray*} We conclude that $$ {\mathbb P}(S_N=k)$$ $$=\frac{e^{-k_N^2/2}}{\sqrt{2\pi}} \left[{\mathcal E}_{r+1}(k_N)+ \frac{i^r}{{\sigma}_N^{r+1}} \sum_{a=1}^{m-1} e^{-2\pi i ak/m} \left(\phi_{X'}'\left(\frac{2\pi a}{m}\right) \right)^r H_{r}(k_N) \right] $$ $$ +o({\sigma}_N^{-r-1}) $$ where ${\mathcal E}_{r+1}$ the Edgeworth polynomial (i.e. the contribution of 0) and $H_{r}(x)$ is the Hermite polynomial. Observe that since the uniform distribution on ${\mathbb Z}/m{\mathbb Z}$ is shift invariant, $S_N$ are uniformly distributed mod $m$ for all $N\in {\mathbb N}.$ This shows that for $r\geq 1$, one can not characterize Edgeworth expansions just in term of the distributions of $S_N$ mod $m$, so the additional assumptions in Theorems \ref{Thm SLLT VS Ege} and \ref{Thm Stable Cond} are necessary. Next, consider a more general case where for each $n$, $X_n$ equals in law to either $X'$ or $X''$, however, now we assume that $X'$ appears infinitely often. In this case $S_N$ obeys Edgeworth expansions of all orders since for large $N$, $\Phi_N(t)$ has zeroes of order greater $N$ at all points of the form $\frac{2\pi a}{m},$ $ a=1, \dots, m-1.$ In fact, the Edgeworth expansions hold in the superstable way since removing a finite number of terms does not make the order of zero to fall below $r.$ \end{example} {\beta}gin{example}{\lambda}bel{Eg1} Let $\mathfrak{p}_n=\min(1, \frac{\theta}{n})$ and let $X_n$ take value $0$ with probability $\mathfrak{p}_n$ and values $\pm 1$ with probability $\frac{1-\mathfrak{p}_n}{2}.$ In this example the only non-zero resonant point is $\pi=2\pi\times \frac{1}{2}.$ Then for small $\theta$ the contributions of $P_{1, b, N}$ (the only non-zero $a$ is $1$) are significant and as a result $S_N$ does not admit the ordinary Edgeworth expansion. Increasing $\theta$ we can make $S_N$ to admit Edgeworth expansions of higher and higher orders. Namely we get that for large $n,$ $\displaystyle \phi_n(\pi)=\frac{2\theta}{n}-1. $ Accordingly $$ \ln (-\phi_n(\pi))=-\frac{2\theta}{n}+O\left(\frac{1}{n^2}\right). $$ Now the asymptotic relation $$ \sum_{n=1}^N \frac{1}{n}=\ln N+{\mathfrak{c}}+O\left(\frac{1}{N}\right),$$ where ${\mathfrak{c}}$ is the Euler-Mascheroni constant, implies that that there is a constant ${\Gamma}ma(\theta)$ such that $$ \Phi_N(\pi)=\frac{(-1)^N e^{{\Gamma}ma(\theta)}}{N^{2\theta}}\left(1+O(1/N)\right).$$ Therefore $S_N$ admits the Edgeworth expansions of order $r$ iff $\displaystyle \theta>\frac{r-1}{4}.$ Moreover, if $\displaystyle \theta\in \left(\frac{r-2}{4}, \frac{r-1}{4}\right],$ then Corollary \ref{CrFirstNonEdge} shows that $$ \mathbb{P}(S_N=k)=\frac{e^{-k_N^2/2}}{\sqrt{2\pi}} \left[{\mathcal E}_{r}(k_N)+ \frac{(-1)^{N+k} e^{{\Gamma}ma(\theta) }}{N^{2\theta+(1/2)}}+O\left(\frac{1}{N^{2\theta+1} }\right)\right] $$ where ${\mathcal E}_r$ is the Edgeworth polynomial of order $r.$ In particular if $\theta\in (0, 1/4)$ then using that {\beta}gin{equation} {\lambda}bel{VarNearN} V_N=N+O(\ln N)=N\left(1+O\left(\frac{\ln N}{N}\right)\right) \end{equation} and hence {\beta}gin{equation} {\lambda}bel{SDNearN} {\sigma}ma_N=\sqrt{N}\left(1+O\left(\frac{\ln N}{N}\right)\right) \end{equation} we conclude that $$ \mathbb{P}(S_N=k)=\frac{e^{-k^2/(2N)}}{\sqrt{2\pi}} \left[\frac{1}{\sqrt{N}}+ \frac{(-1)^{N+k} e^{{\Gamma}ma(\theta) }}{N^{2\theta+(1/2)}}+O\left(\frac{1}{N^{2\theta+1}}\right)\right]. $$ Next, take $\mathfrak{p}_n=\min\left(1, \frac{\theta}{n^2}\right)$. Then the SLLT does not hold, since the Prokhorov condition fails. Instead we have (\ref{r=1}) with ${\mathcal{R}}=\{\pi\}$. Namely, uniformly in $k$ we have {\beta}gin{equation*} \sqrt 2\pi{\mathbb P}(S_N=k)=\left(1 +(-1)^k\prod_{u=1}^{N}\left(2\mathfrak{p}_u-1\right)\right){\sigma}_{N}^{-1}e^{-k^2/2V_N}+o({\sigma}ma_N^{-1}). \end{equation*} Next, $\mathfrak{p}_u$ is summable and moreover \[ \prod_{u=1}^{N}(2\mathfrak{p}_u-1)=(-1)^N U(1+O(1/N)) \] where $\displaystyle U=\prod_{n=1}^{\infty}(1-2\mathfrak{p}_u)$. We conclude that {\beta}gin{equation} {\lambda}bel{A1Ex3} \sqrt 2\pi{\mathbb P}(S_N=k)=\left(1+(-1)^{k+N}U\right){\sigma}_{N}^{-1}e^{-k^2/2V_N}+ O\left({\sigma}ma_N^{-2}\right) \end{equation} uniformly in $k$. In this case the usual LLT holds true if and only if $U=0$ in agreement with Proposition \ref{PrLLT-SLLT}. In fact, in this case we have a faster rate of convergence. To see this we consider expansions of order $2$ for $\mathfrak{p}_n$ as above. We observe that $q_m(2)=\mathfrak{p}_n$ for large $n.$ Thus \[ |{\mathbb E}(e^{\pi iX_n})|=1-2 \mathfrak{p}_n \] and so $|{\mathbb E}(e^{\pi X_n})|\geq \frac12$ when $n\geq N_{\theta}$ for some minimal $N_{\theta}$. Therefore, we can take $N_0=N_{\theta}$. Note also that we have $Y_n=X_n{\theta}xt{ mod }2-1$. We conclude that for $n>N_0$ we have \[ a_{n}=a_{n,j}=\frac{{\mathbb E}[((-1)^{Y_n}-1)X_n]}{{\mathbb E}[(-1)^{Y_n}]}=0 \] and so the term $C_{1,N}$ vanishes. Next, we observe that \[ {\gamma}ma_{N,j}=\frac{\sum_{n=N_0+1}^{N}{\mathbb E}(X_n^3)}{\sum_{n=N_0+1}^{N}(1-\mathfrak{p}_n)}=0. \] Finally, we note that ${\mathbb E}[(-1)^{X_n}X_n]=0$, and hence $\Phi_{N_0}'(\pi)=0$. Therefore, the second term in (\ref{r=2'}) vanishes and we have {\beta}gin{equation*} \sqrt 2\pi{\mathbb P}(S_N=k)=\left(1 +(-1)^{k+N} U \right){\sigma}_{N}^{-1}e^{-k^2/(2V_N)}+ O\left({\sigma}ma_N^{-3}\right). \end{equation*} Taking into account \eqref{VarNearN} and \eqref{SDNearN} we obtain $$ \sqrt 2\pi{\mathbb P}(S_N=k)=\frac{1 +(-1)^{k+N} U}{\sqrt{N}} \; e^{-k^2/(2N)}+ O\left(N^{-3/2}\right). $$ In particular, \eqref{A1Ex3} holds with the stronger rate $O\left({\sigma}ma_N^{-3}\right)$. \end{example} {\beta}gin{example}{\lambda}bel{NonSym r=2 Eg} The last example exhibited significant simplifications. Namely, there was only one resonant point, and, in addition, the second term vanished due to the symmetry. We now show how a similar analysis could be performed when the above simplifications are not present. Let us assume that $X_n$ takes the values $-1,0$ and $3$ with probabilities $a_n,b_n$ and $c_n$ so that $a_n+b_n+c_n=1$. Let us also assume that $ b_n<\frac18$ and that $a_n,c_n\geq \rho>0$ for some constant $\rho$. Then $$V(X_n)=9(c_n-c_n^2)+6a_nc_n+(a_n-a_n^2)\geq 6 \rho^2$$ \noindent and so $V_N$ grows linearly fast in $N$. Next, since we can take $K=3$, the denominators $m$ of the nonzero resonant points can only be $2,3,4,5$ or $6$. An easy check shows that for $m=3,5,6$ we have $q_n(m)\geq \rho$, and that for $m=2,4$ we have $q_n(m)=b_n$. Therefore, for $m=3,5,6$ we have $M_N(m)\geq \rho N$, and so we can disregard all the nonzero resonant points except for $\pi/2,\pi$ and $3\pi/2$. For the latter points we have {\beta}gin{equation} {\lambda}bel{Res1} \phi_n\left(\frac{\pi}{2}\right)=b_n-i(1-b_n), \end{equation} {\beta}gin{equation} {\lambda}bel{Res2} \phi_n(\pi)=2b_n-1,\quad \phi_n\left(\frac{3\pi}{2}\right)=b_n+i(1-b_n). \end{equation} Hence, denoting $\eta_n=b_n(1-b_n),$ we have {\beta}gin{equation*} \left|\phi_n\left(\frac{\pi}{2}\right)\right|^2=\left|\phi_n\left(\frac{3\pi}{2}\right)\right|^2=1-2\eta_n, \quad \big|\phi_n(\pi)\big|^2=1-4\eta_n. \end{equation*} Since we suppose that $\eta_n\leq b_n<\frac{1}{8}$ it follows that $1-4\eta_n\geq \frac12$. Then for the above three resonant points we can take $N_0=0$. Now Proposition~\ref{Thm} and a simple calculation show that for any $r$ we get the Edgeworth expansions of order $r$ if and only if \[ \prod_{n=1}^N(1-2\eta_n)= o\left(N^{1-r}\right). \] Let us focus for the moment on the case when $b_n={\gamma}ma/n$ for $n$ large enough where ${\gamma}ma>0$ is a constant. Rewriting \eqref{Res1}, \eqref{Res2} as {\beta}gin{equation} {\lambda}bel{PhiNRatio} \frac{\phi_n\left(\frac{\pi}{2}\right)}{-i}=(1-b_n)+ib_n, \quad \frac{\phi_n\left(\frac{3\pi}{2}\right)}{i}=(1-b_n)-ib_n, \quad \end{equation} $$ -\phi_n(\pi)=1-2 b_n $$ and, using that the condition $b_n<\frac{1}{8}$ implies that that $\phi_n(t)\neq 0$ for all $n\in \mathbb{N}$ and all $t\in \left\{\frac{\pi}{2}, \pi, \frac{3\pi}{2}\right\}$, we conclude similarly to Example \ref{Eg1} that there are non-zero complex numbers ${\kappa}ppa_1, {\kappa}ppa_3$ and a non-zero real number ${\kappa}ppa_2$ such that $$ \Phi_N\left(\frac{\pi}{2}\right)=\frac{(-i)^N {\kappa}ppa_1}{N^{\gamma}ma} e^{i {\gamma}ma \ln N } \left(1+O\left(\frac1N\right)\right), $$ $$ \Phi_N\left(\frac{3\pi}{2}\right)=\frac{i^N {\kappa}ppa_3}{N^{\gamma}ma} e^{-i {\gamma}ma \ln N } \left(1+O\left(\frac1N\right)\right), $$ $$ \Phi_N(\pi)=(-1)^N \frac{{\kappa}ppa_2e^{2{\gamma}ma w_N}}{N^{2{\gamma}ma}} \left(1+O\left(\frac1N\right)\right). $$ It follows that $S_N$ admits Edgeworth expansion of order $r$ iff ${\gamma}ma>\frac{r-1}{2}.$ In fact if $\frac{r-1}{2}<{\gamma}ma\leq \frac{r}{2}$ then Corollary \ref{CrFirstNonEdge} shows that $$ \mathbb{P}(S_N=k)= \frac{e^{-k_N^2/2}}{\sqrt{2\pi}} \Bigg[{\mathcal E}_{r}(k_N)+ \frac{{\kappa}ppa_1 e^{i{\gamma}ma \ln N}} {N^{\gamma}ma {\sigma}_N} + \frac{{\kappa}ppa_3 e^{-i{\gamma}ma \ln N}} {N^{\gamma}ma {\sigma}_N} +O\left(N^{-\eta}\right)\Bigg] $$ where ${\mathcal E}_r$ is the Edgeworth polynomial of order $r$ and $\eta\!\!=\!\!\min\left(2{\gamma}ma, \frac{r}{2}\right)+\frac{1}{2}.$ To give a specific example, let us suppose that $\frac{1}{2}\leq {\gamma}ma<1$ and that $E(X_n)=0$ which means that {\beta}gin{equation} {\lambda}bel{Ex4-A-C} a_n=\frac{3(1-b_n)}{4}, \quad c_n=\frac{1-b_n}{4}. \end{equation} Then {\beta}gin{equation} {\lambda}bel{Ex4M2M3} V_N=3N-3{\gamma}ma \ln N+O(1), \quad E(S_N^3)=6N-6{\gamma}ma \ln N+O(1), \end{equation} so Proposition \ref{2 Prop} gives $$ \sqrt{2\pi} \mathbb{P}(S_N=k)=$$ $$e^{-k^2/6 N} \left[\frac{1}{\sqrt{3N}} \left(1+\frac{{\kappa}ppa_1 i^{k-N} e^{i{\gamma}ma \ln N} +{\kappa}ppa_3 i^{N-k} e^{-i{\gamma}ma \ln N}} {N^{\gamma}ma}\right)- \frac{k^3}{81\sqrt{3 N^5}} \right] $$ $$ +O\left(N^{-3/2}\right) $$ Next, let us provide the second order trigonometric expansions under the sole assumption that $1-4\eta_n\geq \frac12$ and $a_n,c_n\geq \rho$. As we have mentioned, we only need to consider the nonzero resonant points $\pi/2,\pi,3\pi/2$ and for these points we have $N_0=0$. Therefore, the term involving the derivative in the right hand side of (\ref{r=2'}) vanishes. Now, a direct calculation shows that \[ C_{1,N,\pi}=\sum_{n=1}^{N}\frac{{\mathbb E}(e^{i\pi X_n}\bar X_n)}{{\mathbb E}(e^{i\pi X_n})}=2\sum_{n=1}^{N}\frac{(a_n-3c_n)b_n}{2b_n-1} \] and $$ C_{1,N,\pi/2}=\sum_{n=1}^N\frac{(a_n-3c_n)(1+i)b_n}{b_n-i(1-b_n)},\quad C_{1,N,3\pi/2}=\sum_{n=1}^N\frac{(a_n-3c_n)(1-i)b_n}{b_n+i(1-b_n)}. $$ Note that $3c_n-a_n={\mathbb E}(X_n)$. Set $$ {\Gamma}ma_{1,N}=\prod_{n=1}^{N}(b_n-i(1-b_n)), \quad {\Gamma}ma_{2,N}=\prod_{n=1}^N(2b_n-1), \quad {\Gamma}ma_{3,N}=\prod_{n=1}^{N}(b_n+i(1-b_n)). $$ Then ${\Gamma}ma_{s,N}={\mathbb E}(e^{\frac{s\pi i}{2} S_N})$. We also set $$ \Theta_{s,N}=C_{1,N,s\pi/2}{\Gamma}ma_{s,N},\,s=1,2,3 $$ and $${\Gamma}ma_{N}(k)=\sum_{j=1}^{3}e^{-j\pi i k/2}{\Gamma}ma_{j,N}, \quad \Theta_{N}(k)=\sum_{j=1}^{3}e^{-j\pi i k/2}\Theta_{j,N}.$$ Then by Proposition \ref{2 Prop} and Remark \ref{Alter 2nd Order}, uniformly in $k$ we have {\beta}gin{equation}{\lambda}bel{EgGen} \sqrt{2\pi}P(S_N=k)={\sigma}_N^{-1}\left(1+{\Gamma}ma_N(k)\right)e^{-k_N^2/2} \end{equation} $$ -{\sigma}_N^{-2}\left(k_N^3 T_N\big(1+{\Gamma}ma_{N}(k)\big)+ik_N\Theta_{N}(k)\right)e^{-k_N^2/2}+o({\sigma}_N^{-2}) $$ where $T_N=\frac{\displaystyle \sum_{n=1}^{N} {\mathbb E}(\bar X_n^3)}{6V_N}$, $\bar X_n=X_n-{\mathbb E}(X_n)$. Let us now consider a more specific situation. Namely we suppose that $b_n=\frac{{\gamma}ma}{n^{3/2}}$ for large $n$ and that $E(X_n)=0.$ Then \eqref{Ex4-A-C} shows that $C_{1, N, s\pi/2}=0.$ Next \eqref{PhiNRatio} gives $$ \frac{\Phi_N\left(\frac{\pi}{2}\right)}{(-i)^N}= \prod_{n=1}^N \left[(1-b_n)+ib_n\right]= \frac{{\bar k}appa_1}{\displaystyle \prod_{n=N+1}^\infty\left[(1-b_n)+ib_n\right]}$$ $$= {\bar k}appa_1 \left(1+\frac{2 {\gamma}ma(1-i)}{\sqrt{N}}+O\left(\frac{1}{N}\right)\right) $$ where $\displaystyle {\bar k}appa_1=\prod_{n=1}^\infty \left[(1-b_n)+ib_n\right].$ Likewise $$ \frac{\Phi_N\left(\frac{3\pi}{2}\right)}{i^N}= {\bar k}appa_3 \left(1+\frac{2 {\gamma}ma(1+i)}{\sqrt{N}}+O\left(\frac{1}{N}\right)\right) $$ and $$ \frac{\Phi_N\left(\pi\right)}{(-1)^N}= {\bar k}appa_2 \left(1+\frac{4 {\gamma}ma}{\sqrt{N}}+O\left(\frac{1}{N}\right)\right). $$ Taking into account \eqref{Ex4M2M3} we can reduce \eqref{EgGen} to the following expansion $$ \sqrt{2\pi}{\mathbb P}(S_N=k)=e^{-k^2/6N} \left[ \frac{1}{\sqrt{3N}} \left(1+\sum_{s=1}^3 {\bar k}appa_s i^{s(k-N)} \right)\right. $$ $$\left.+\frac{1}{N}\left( -\frac{\tilde k_N^3}{3} + \sum_{s=1}^3 {\bar k}appa_s i^{s(k-N)} \left(\frac{2{\gamma}ma(1-i^{-s}) }{\sqrt{3}}-\frac{\tilde k_N^3}{3} \right)\right) \right]+O\left(\frac{1}{N^{3/2}}\right) $$ where $\tilde k_N=k/\sqrt{3N}$. \end{example} {\beta}gin{example} Let $X'$ take value $\pm 1$ with probability $\frac{1}{2}$, $X''$ take values $0$ and $1$ with probability $\frac{1}{2}$, and $X^{{\delta}ta}$, ${\delta}\in[0,1]$ be the mixture of $X'$ and $X''$ with weights ${\delta}ta$ and $1-{\delta}ta.$ Thus $X^{\delta}ta$ take value $-1$ with probability $\frac{{\delta}ta}{2},$ the value $0$ with probability $\frac{1-{\delta}ta}{2}$ and value $1$ with probability $\frac{1}{2}$. Therefore, ${\mathbb E}(e^{\pi i X^{\delta}ta})=-{\delta}$. We suppose that $X_{2m}$ and $X_{2m-1}$ have the same law which we call $Y_m.$ The distribution of $Y_m$ is defined as follows. Set $k_j=3^{3^j}$, and let $Y_{k_j}$ have the same distribution as $X^{{\delta}_j}$ where ${\delta}_j=\frac{1}{\sqrt{k_{j+1}}}$. When $m\not\in\{k_j\}$ we let $Y_m$ have the distribution of $X'$. It is clear that $V_N$ grows linearly fast in $N$. Note also that ${\mathbb E}(e^{\pi i Y_m})=-{\delta}_j$ when $m=k_j$ for some $j$, and otherwise ${\mathbb E}(e^{\pi i Y_m})=-1$. Now, take $N\in{\mathbb N}$ such that $N>2k_2$, and let $J_N$ be so that $2k_{J_N}\leq N<2k_{J_N+1}$. Then $$ |\Phi_{N}(\pi)|\leq \prod_{j=1}^{J_N}(k_{j+1})^{-1}. $$ Since $k_{J_N}\leq \frac{N}2<k_{J_N+1}$ and $k_j=(k_{j+1})^{1/3}$ we have $k_{J_N+1}^{-1}\leq 2N^{-1}$ and $k_{J_N+1-m}\leq 2^{3^{-m}} N^{-3^{-m}}$ for any $0<m\leq J_N$. Denote $\displaystyle {\alpha}pha_N=\sum_{j=1}^{J_N-1}3^{-j}.$ Since ${\alpha}pha_N>1/3$ we get that \[ |\Phi_{N}(\pi)|\leq 2^{3/2}N^{-{\alpha}pha_N} =o(N^{-1-1/3}). \] Similarly, for each $j_1, j_2\leq N$, {\beta}gin{equation}{\lambda}bel{2nd} |\Phi_{N: j_1}(\pi)|\leq 2^{3/2} N^{-1/2-{\alpha}pha_N}=o(N^{-1/2-1/3}) \end{equation} and {\beta}gin{equation}{\lambda}bel{3rd} |\Phi_{N: j_1,j_2}(\pi)|\leq 2^{3/2} N^{-{\alpha}pha_N}=o(N^{-1/3}). \end{equation} Indeed, the largest possible values are obtained for $j_1=2k_{J_N}$ (or $j_1=2k_{J_{N+1}}-1$ if it is smaller than $N+1$) and $j_2=2k_{J_N}-1$ (or $j_2=2k_{J_{N}}$). Using the same estimates as in the proof of of Theorem \ref{Thm Stable Cond} we conclude from (\ref{2nd}) that $\displaystyle \Phi_N'(\pi)=o\left(1/\sqrt{N}\right)$ and we conclude from \eqref{3rd} that $\displaystyle \Phi_N''(\pi)=o(1).$ It follows from Lemma \ref{Lem} and Proposition \ref{Thm} that $S_N$ satisfies an Edgeworth expansion of order 3. The same conclusion holds if we remove a finite number of terms from the beginning of the sequence $\{X_n\}$ because the smallness of $\Phi_N(\pi)$ comes from the terms $X_{2 k_{j}-1} $ and $X_{2 k_j}$ for arbitrary large $j$'s. On the other hand $$ \left|\Phi_{2k_j; 2k_j, 2k_j-1, 2k_{j-1}, 2k_{j-1}-1}(\pi)\right|= \prod_{s=2}^{j-1} \left(3^{-3^{s}}\right)$$ $$= 3^{-(3^j-9)/2}=\frac{3^{9/2}}{\sqrt{k_j}} \gg \frac{1}{k_j}=3^{-3^j}. $$ It follows that $S_{2k_j; 2k_j, 2k_j-1, 2k_{j-1}, 2k_{j-1}-1}$ does not obey the Edgeworth expansion of order 3. Accordingly, stable Edgeworth expansions need not be superstable if $r=3.$ A similar argument allows to construct examples showing that those notions are different for all $r>2.$ \end{example} \section{Extension for uniformly bounded integer-valued triangular arrays} In this section we will describe our results for arrays of independent random variables. We refer to \cite{Feller}, \cite{Mu84, Mu91} and \cite{Dub}, \cite{VS}, \cite{Pel1} and \cite{DS} for results for triangular arrays of inhomogeneous Markov chains. Example where Markov arrays appear naturally include the theory of large deviations for inhomogeneous systems (see \cite{SaSt91, PR08, FH} and references wherein), random walks in random scenery \cite{CGPPS, GW17}, and statistical mechanics \cite{LPRS}. Let $X_n^{(N)},\,1\leq n\leq L_N$ be a triangular array such that for each fixed $N$, the random variables $X_n^{(N)}$ are independent and integer valued. Moreover, we assume that $$K:=\sup_{N}\sup_{n}\|X_n\|_{L^\infty}<\infty.$$ For each $N$ we set $\displaystyle S_N=\sum_{n=1}^{L_N}X_n^{(N)}$. Let $V_N={\theta}xt{Var}(S_N)$. We assume that $V_N\to\infty$, so that, by Lindenberg--Feller Theorem, the sequence $(S_N-{\mathbb E}(S_N))/{\sigma}_N$ obeys the CLT, where ${\sigma}_N=\sqrt{V_N}$. We say that the array $X_n^{(N)}$ obeys the SLLT if for any $k$ the LLT holds true for any uniformly square integrable array $Y_n^{(N)},\,1\leq n\leq L_N$, so that $Y_n^{(N)}=X_n^{(N)}$ for all but $k$ indexes $n$. Set $$ M_N:=\min_{2\leq h\leq 2K}\sum_{n=1}^{L_N}P(X_n\neq m_n^{(N)}(h) {\theta}xt{ mod } h)\geq R\ln V_N $$ where $m_n^{(N)}(h)$ is the most likely value of $X_n^{(N)}$ modulo $h$. Observe now that the proofs of Proposition \ref{PropEdg} and Lemmas \ref{Step1}, \ref{Step2}, \ref{Step3} and \ref{Step4} proceed exactly the same for arrays. Therefore, all the arguments in the proof of Theorem \ref{IntIndThm} proceed the same for arrays instead of a fixed sequence $X_n$. That is, we have {\beta}gin{theorem}{\lambda}bel{IntIndThmAr} There $\exists J=J(K)<\infty$ and polynomials $P_{a, b, N}$ with degrees depending only on $a$ and $b$, whose coefficients are uniformly bounded in $N$ such that, for any $r\geq1$ uniformly in $k\in{\mathbb Z}$ we have $${\mathbb P}(S_N=k)-\sum_{a=0}^{J-1} \sum_{b=1}^r \frac{P_{a, b, N} ((k-a_N)/{\sigma}ma_N)}{{\sigma}ma_N^b} \mathfrak{g}((k-a_N)/{\sigma}ma_N) e^{2\pi i a k/J} =o({\sigma}ma_N^{-r}) $$ where $a_N={\mathbb E}(S_N)$ and $\mathfrak{g}(u)=\frac{1}{\sqrt{2\pi}} e^{-u^2/2}. $ Moreover, $P_{0,1,N}\equiv1$ and given $K, r$, there exists $R=R(K,r)$ such that if $M_N\geq R \ln V_N$ then we can choose $P_{a, b, N}=0$ for $a\neq 0.$ \end{theorem} All the formulas for the coefficients of the polynomials $P_{a,b,N}$ remain the same in the arrays setup. In particular, we get that, uniformly in $k$ we have {\beta}gin{equation}{\lambda}bel{r=1 Ar} {\mathbb P}(S_N=k)=\left(1+\sum_{t\in{\mathcal{R}}}e^{-itk}\Phi_N(t)\right)e^{-k_N^2/2}{\sigma}_N^{-1}+o({\sigma}_N^{-1}) \end{equation} where $\Phi_N(t)={\mathbb E}(e^{it S_N})$. Next, our version for Proposition \ref{PrLLT-SLLT} for arrays is as follows. {\beta}gin{proposition} {\lambda}bel{PrLLT-SLLT Ar} Suppose $S_N$ obeys LLT. Then for each integer $h\geq2$, at least one of the following conditions occur: \vskip0.2cm either (a) $\displaystyle \lim_{N\to\infty}\sum_{n=1}^{L_{N}} {\mathbb P}(X_n\neq m_n^{(N)}(h) {\theta}xt{ mod } h)=\infty$. \vskip0.2cm or (b) there exists a subsequence $N_k$, numbers $s\in{\mathbb N}$ and ${\varepsilon}_0>0$ and indexes $1\leq j_1^{k},...,j_{s_k}^{k}\leq L_{N_k}$, $s_k\leq s$ so that the distribution of $\displaystyle \sum_{u=1}^{s_k}X^{(N_k)}_{j_u}$ converges to uniform ${\theta}xt{mod }h$, and the distance between the distribution of $\displaystyle S_{N_k}-\sum_{q=1}^{s_k}X^{(N_k)}_{j_q}$ and the uniform distribution ${\theta}xt{mod }h$ is at least ${\varepsilon}_0$. \end{proposition} {\beta}gin{proof} First, by \eqref{r=1 Ar} and Lemma \ref{Lemma} if the LLT holds then for any nonzero resonant point $t$ we have $\displaystyle \lim_{N\to\infty}|\Phi_N(t)|=0$. Now, if (a) does not hold true then there is a subsequence $N_k$ so that $\displaystyle \sum_{n=1}^{L_{N_k}}q(X_n^{(N_k)},h)\leq C$, where $C$ is some constant. Set $\displaystyle q_n^{(N_k)}(h)=q\left(X_n^{(N_k)},h\right)$. Then there are at most $8hC$ $n$'s between $1$ and $L_{N_k}$ so that $q_n^{(N_k)}(h)>\frac1{8 h}$. Let us denote these $n$'s by $n_{1,k},...,n_{s_k,k}$, $s_k\leq 8h C$. Next, for any $n$ and a nonzero resonant point $t=2\pi l/h$ we have {\beta}gin{equation} {\lambda}bel{PhiQArr} |\phi_n^{(N_k)}(t)|\geq 1-2hq_n^{(N_k)}(h) \geq e^{-2{\gamma}ma h q_n^{(N_k)}} \end{equation} where $\phi_n^{(N_k)}$ is the characteristic function of $X_n^{(N_k)}$ and ${\gamma}ma$ is such that for $\theta\in [0, 1/4]$ we have $1-\theta\geq e^{-{\gamma}ma \theta}.$ We thus get that {\beta}gin{equation}{\lambda}bel{C1.} \prod_{n\not\in\{n_{u,k}\}}|\phi_{n}^{(N_k)}(t)|\geq\prod_{n\not\in\{n_{u,k}\}}(1-2hq_n^{(N_k)}(h))\geq C_0 \end{equation} where $C_0>0$ is some constant. Therefore, $$ |\Phi_{N_k}(t)|\geq \prod_{u=1}^{s_k}|\phi_{n_{u,k}}^{(N_k)}(t)|\cdot C_0 $$ and so we must have {\beta}gin{equation}{\lambda}bel{C2.} \lim_{k\to\infty}\prod_{u=1}^{s_k}|\phi_{n_{u,k}}^{(N_k)}(t)|=0. \end{equation} Now (b) follows from \eqref{C1.}, \eqref{C2.} and Lemma \ref{LmUnifFourier}. \end{proof} Using (\ref{r=1 Ar}) we can now prove a version of Theorem \ref{ThProkhorov} for arrays. {\beta}gin{theorem}{\lambda}bel{ThProkhorovAr} The SLLT holds iff for each integer $h>1$, {\beta}gin{equation}{\lambda}bel{Prokhorov Ar} \lim_{N\to\infty}\sum_{n=1}^{L_N} {\mathbb P}(X_n^{(N)}\neq m_n {\theta}xt{ mod } h)=\infty \end{equation} where $m_n=m_n^{(N)}(h)$ is the most likely residue of $X_n^{(N)}$ modulo $h$. \end{theorem} {\beta}gin{proof} First, the arguments in the proof of (\ref{Roz0}) show that there are constants $c_0,C>0$ so that for any nonzero resonant point $t=2\pi l/h$ we have {\beta}gin{equation}{\lambda}bel{ROZ} |\Phi_N(t)|\leq Ce^{-c_0M_N(h)},\quad {\theta}xt{where}\quad M_N(h):=\sum_{n=1}^{L_N}q(X_n^{(N)},h). \end{equation} Let us assume that \eqref{Prokhorov Ar} holds for all integers $h>1$. Consider $s_N$--tuples $1\leq j_1^N,...,j_{s_N}^N\leq L_N$, where $s_N\leq \bar s$ is bounded in $N$. Then by applying \eqref{ROZ} with $\displaystyle \tilde S_N=S_N-\sum_{l=1}^{s_N}X_{j_l^N}^{(N)}$ we have {\beta}gin{equation}{\lambda}bel{TILD} \lim_{N\to\infty}|{\mathbb E}(e^{it\tilde S_N})|=0. \end{equation} Now, arguing as in the proof of Theorem \ref{Thm Stable Cond}(1), given a uniformly square integrable array $Y_n^{(N)}$ as in the definition of the SLLT, we still have \eqref{r=1 Ar}, even though the new array is not necessarily uniformly bounded. Applying (\ref{TILD}) we see that for any nonzero resonant point $t$ we have $$ \lim_{N\to\infty}\left|{\mathbb E}\left(\exp\left[it \sum_{n=1}^{L_N}Y_n^{(N)}\right]\right)\right|=0 $$ and so $\displaystyle S_N Y:=\sum_{n=1}^{L_N}Y_n^{(N)}$ satisfies the LLT. Now let us assume that $M_N(h)\not\to\infty$ for some $2\leq h\leq 2K$ (it not difficult to see that \eqref{Prokhorov Ar} holds for any $h>2K$). In other words after taking a subsequence we have that $M_{N_k}(h)\leq L$ for some $L<\infty.$ The proof of Proposition \ref{PrLLT-SLLT Ar} shows that there $s<\infty$ such that after possibly removing terms $n_{1, k}, n_{2, k},\dots ,n_{s_k, k}$ with $s_k\leq s$ we can obtain that $q_n^{(N_k)}(h) \leq \frac{1}{8h},$ $n\not\in\{n_{j, k}\}$. In this case \eqref{PhiQArr} shows that for each $\ell$ $$ |\Phi_{N_k; n_{1, k} , \dots, n_{s_k, k}}(2\pi\ell/h)|\geq e^{-2{\gamma}ma L}. $$ By Proposition \ref{PrLLT-SLLT Ar}, $S_{N_k; n_{1, k} , \dots, n_{s_k, k}}$ does not satisfy the LLT. \end{proof} Next, all the other arguments in our paper proceed similarly for arrays since they essentially rely only on the specific structure of the polynomials from Theorem \ref{IntIndThm}. For the sake of completeness, let us formulate the main (remaining) results here. {\beta}gin{theorem}{\lambda}bel{ThLLT Ar} The following conditions are equivalent: (a) $S_N$ satisfies LLT; (b) For each $\xi\in {\mathbb R}\setminus {\mathbb Z}$, $\displaystyle \lim_{N\to\infty} {\mathbb E}\left(e^{2\pi i \xi S_N}\right)=0$; (c) For each non-zero resonant point $\xi$, $\displaystyle \lim_{N\to\infty} {\mathbb E}\left(e^{2\pi i \xi S_N}\right)=0$; (d) For each integer $h$ the distribution of $S_N$ mod $h$ converges to uniform. \end{theorem} {\beta}gin{theorem} {\lambda}bel{ThEdgeMN Ar} For each $r$ there is $R=R(r, K)$ such that the Edgeworth expansion of order $r$ holds true if $M_N\geq R\ln V_N$. In particular, $S_N$ obeys Edgeworth expansions of all orders if $$ \lim_{N\to\infty} \frac{M_N}{\ln V_N}=\infty. $$ \end{theorem} {\beta}gin{theorem}{\lambda}bel{r Char Ar} For any $r\geq1$, the Edgeworth expansion of order $r$ holds if and only if for any nonzero resonant point $t$ and $0\leq\ell<r$ we have \[ \bar \Phi_{N}^{(\ell)}(t)=o\left({\sigma}_N^{\ell+1-r}\right) \] where $\bar\Phi_{N}(x)={\mathbb E}[e^{ix (S_N-{\mathbb E}(S_N))}]$. \end{theorem} {\beta}gin{theorem}{\lambda}bel{Thm SLLT VS Ege Ar } Suppose $S_N$ obeys the SLLT. Then the following are equivalent: (a) Edgeworth expansion of order 2 holds; (b) $|\Phi_N(t)|=o({\sigma}ma_N^{-1})$ for each nonzero resonant point $t$; (c) For each $h\leq 2K$ the distribution of $S_N$ mod $h$ is $o({\sigma}ma_N^{-1})$ close to uniform. \end{theorem} Next, we say that an array $\{X_n^{(N)}\}$ {\em admits an Edgeworth expansion of order $r$ in a superstable way} (denoted by $\{X_n^{(N)}\}\in EeSs(r)$) if for each ${\bar s}$ and each sequence $j_1^N, j_2^N,\dots ,j_{s_N}^N$ with $s_N\leq {\bar s}$ and $j_i^N\leq L_N$ there are polynomials $P_{b, N}$ whose coefficients are $O(1)$ in $N$ and their degrees do not depend on $N$ so that uniformly in $k\in{\mathbb Z}$ we have that {\beta}gin{equation}{\lambda}bel{EdgeDefSS Ar} {\mathbb P}(S_{N; j_1^N, j_2^N, \dots,j_{{s_N}^N}}=k)=\sum_{b=1}^r \frac{P_{b, N} (k_N)}{{\sigma}ma_N^b} \mathfrak{g}(k_N)+o({\sigma}ma_N^{-r}) \end{equation} and the estimates in $O(1)$ and $o({\sigma}ma_N^{-r})$ are uniform in the choice of the tuples $j_1^N, \dots ,j_{s_N}^N.$ Let $\Phi_{N; j_1, j_2,\dots, j_s}(t)$ be the characteristic function of $S_{N; j_1, j_2,\dots, j_s}.$ {\beta}gin{theorem}{\lambda}bel{Thm Stable Cond Ar} (1) $S_N\in EeSs(1)$ (that is, $S_N$ satisfies the LLT in a superstable way) if and if it satisfies the SLLT. (2) For arbitrary $r\geq 1$ the following conditions are equivalent: (a) $\{X_n^{(N)}\}\in EeSs(r)$; (b) For each $j_1^N, j_2^N,\dots ,j_{s_N}^N$ and each nonzero resonant point $t$ we have $\Phi_{N; j_1^N, j_2^N,\dots, j_{s_N}^N}(t)=o({\sigma}ma_N^{1-r});$ (c) For each $j_1^N, j_2^N,\dots ,j_{s_N}^N$, and each $h\leq 2K$ the distribution of $S_{N; j_1^N, j_2^N,\dots, j_{s_N}^N}$ mod $h$ is $o({\sigma}ma_N^{1-r})$ close to uniform. \end{theorem} {\beta}gin{thebibliography}{Bow75} \itemsep= amount \bibitem{BR76} R.N. Bhattacharya, R. Ranga Rao {\em Normal Approximation and Asymptotic Expansions,} Wiley, New York-London-Sydney-Toronto (1976) xiv+274 pp. \bibitem{Bor16} A. A. Borovkov {\em Refinement and generalization of the integro-local Stone theorem for sums of random vectors,} Theory Probab. Appl. {\bf 61} (2017) 590--612. \bibitem{Br} E. Breuillard, {\em Distributions diophantiennes et theoreme limite local sur ${\mathbb R}^d$}, Probab. Th. Rel. 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\betaegin{document} \thetaitle{Non-malleable encryption of quantum information} \alphauthor{Andris Ambainis} \alphaffiliation{Department of Computer Science, University of Latvia, Raina bulv. 19, Riga, LV-1586, Latvia} \alphaffiliation{Department of Combinatorics and Optimization \&{} Institute for Quantum Computing, University of Waterloo} \alphauthor{Jan Bouda} \alphaffiliation{Faculty of Informatics, Masaryk University, Botanick\'{a} 68a, 602\,00 Brno, Czech Republic} \alphauthor{Andreas Winter} \alphaffiliation{Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.} \alphaffiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543} \deltaate{3 February 2009} \betaegin{abstract} We introduce the notion of \epsilonmph{non-malleability} of a quantum state encryption scheme (in dimension $d$): in addition to the requirement that an adversary cannot learn information about the state, here we demand that no controlled modification of the encrypted state can be effected. We show that such a scheme is equivalent to a \epsilonmph{unitary 2-design} [Dankert \epsilonmph{et al.}], as opposed to normal encryption which is a unitary 1-design. Our other main results include a new proof of the lower bound of $(d^2-1)^2+1$ on the number of unitaries in a 2-design [Gross \epsilonmph{et al.}], which lends itself to a generalization to approximate 2-design. Furthermore, while in prime power dimension there is a unitary 2-design with $\leq d^5$ elements, we show that there are always approximate 2-designs with $O(\epsilonpsilon^{-2} d^4 \log d)$ elements. \epsilonnd{abstract} \muaketitle \sigmaection*{INTRODUCTION} The ordinary (and in terms of secret key length, optimal) encryption of quantum states on $n$ qubits is by applying a randomly chosen tensor product of Pauli operators (including the identity). This requires $2n$ bits of shared secret randomness, corresponding to the $4^n$ Pauli operators. (More generally, for states on a $d$-dimensional system, one can use the elements of the discrete Weyl group -- up to global phases -- of which there are $d^2$.) This is perfectly secure in the sense that the state the adversary can intercept is, without her knowing the key, always the maximally mixed state. For perfectly secure encryption with random unitaries, it was shown in~\chiite{Ambainis+Mosca...-Priva_quant_chann:2000} that $2n$ bits of secret key are also necessary for $n$ qubits. The lower bound of $2$ bits of key per qubit continues to hold even for $\epsilonpsilon$-approximate encryption (up to expressions in $\epsilonpsilon$), but there it becomes relevant how the approximation is defined --- whether it randomizes entangled states or not [see Eq.~\epsilonqref{eq:enc-scheme-2-correct} and \epsilonqref{eq:enc-scheme-2-naive} below]. In~\chiite{Hayden+Leung...-Rando_quant_state:2003} it was shown that in the latter case one gets away with $n+o(n)$ key bits for arbitrary $n$-qubit states; their construction was derandomized later in~\chiite{Ambainis.Smith-Smallpseudo-randomfamilies-2004} and~\chiite{Dickinson.Nayak-ApproximateRandomizationof-2006}. However, even perfectly secure encryption allows for a different sort of intervention by the adversary: she can, without ever attempting to learn the message, change the plaintext by effecting certain dynamics on the encrypted state. Consider briefly the classical one-time pad, i.e. an $n$-bit message XORed with a random $n$-bit string: by flipping a bit of the ciphertext, an adversary can effectively flip any bit of the recovered plaintext. In the quantum case, due to the (anti-)commutation relations of the Pauli operators, by applying to the ciphertext (encrypted state) some Pauli, she forces that the decrypted state is the plaintext modified by that Pauli: for an $n$-qubit state $\kappaet{\varphi}$, any adversary's Pauli operator $Q$ and secret key Pauli $P_k$, the decrypted state is \[ P_k^\deltaagger Q P_k \kappaet{\varphi} = \zetaeta Q\kappaet{\varphi}, \] with some (unimportant) global phase $\zetaeta = \zetaeta(P,Q)$. This is evidently an undesirable property of a encryption scheme, and can be classically addressed e.g. by authenticating the message as well as encrypting it. Interestingly, in the above quantum message case, it was shown in~\chiite{Barnum+Crepeau...-Authenticatio_of_q_mes:2002} that authenticating quantum messages is at least as expensive as encrypting them (it actually encrypts the message as well): one needs $2$ bits of shared secret key for each qubit authenticated, even in the approximate setting considered in~\chiite{Barnum+Crepeau...-Authenticatio_of_q_mes:2002}. Classical non-malleable cryptosystems include both symmetric and asymmetric encryption schemes, bit commitment, zero knowledge proofs and others~\chiite{DDN01}. Here we will introduce a formal definition of perfect non-malleability of a quantum state encryption scheme (NMES), i.e. resistance against predictable modification of the plaintext, as well as of two notions of approximate encryption with approximate non-malleability. We show that a unitary non-malleable channel is equivalent to unitary $2$-design in the sense of Dankert \epsilonmph{et al.}~\chiite{Dankert.Cleve.ea-ExactandApproximate-2006}. We use this fact to design an exact ideal non-malleable encryption scheme requiring $5\log d$ bits of key. Also, the lower bound of Gross \epsilonmph{et al.}~\chiite{Gross.Audenaert.ea:Evenlydistributedunitaries:-2007} for unitary $2$-designs applies for perfect NMES; we give a new proof of their result that at least $(d^2-1)^2+1$ unitaries are required, which also yields a more general lower bound of $(4-O(\epsilonpsilon)) \log d$ on the \epsilonmph{entropy} of an approximate unitary $2$-design. Finally we demonstrate that approximate NMES (unitary 2-designs) exist which require only $4\log d+\log\log d+ O(\log 1/\epsilonpsilon)$ bits of key. \sigmaection{General Model of Encryption} Suppose Alice wants to send a secret quantum message to Bob, say an arbitrary state $\rhoho \in \chiB(\chiH)$, a Hilbert space of dimension $d$. For this purpose they will use a encryption scheme with pre-shared secret key $K$ as follows. $K$ is distributed according to some probability distribution $p_K(k)$ and for each $k$ there is a pair of c.p.t.p.~(completely positive and trace preserving) maps \[ E_k:{\chial B}({\chial H}) \longrightarrow {\chial B}({\chial H}') \thetaext{ and } D_k:{\chial B}({\chial H}') \longrightarrow {\chial B}({\chial H}) \] for encryption and decryption. The combined effect of en- and decryption, averaged over all keys, is described by a c.p.t.p. map (noisy quantum channel) $R:{\chial B}({\chial H}) \longrightarrow {\chial B}({\chial H})$, acting on operators on ${\chial H}$ as \[ R(\rhoho) = \sigmaum_k p_K(k) D_k\betaigl( E_k(\rhoho) \betaigr). \] Similarly, for an adversary who intercepts the encrypted state but doesn't know the secret key, we have an average channel $R':{\chial B}({\chial H}) \longrightarrow {\chial B}({\chial H}')$, \[ R'(\rhoho) = \sigmaum_k p_K(k) E_k(\rhoho). \] Loosely speaking, the quality of the scheme is described by two parameters: first, the reliability, i.e.~how close $R$ is to the ideal channel; secondly, the secrecy, i.e.~how close $R'$ is to a constant (meaning a map taking all input states to a fixed output state). In an ideal scheme, $R=\id$ and $R'=\thetaext{const.}$, i.e.~there is a state $\xii_0$ on ${\chial H}'$, such that \betaegin{align} \label{eq:ideal-enc-scheme-1} \forall \rhoho &\quad R(\rhoho) = \rhoho, \\ \label{eq:ideal-enc-scheme-2} \forall \rhoho &\quad R'(\rhoho) = \xii_0. \epsilonnd{align} The issue of approximate performance is a little bit tricky: whereas for the reliability of communication there is essentially one notion, namely, for $\deltaelta > 0$, \betaegin{equation} \label{eq:enc-scheme-1} \forall \rhoho \quad \nuorm{\rhoho - R(\rhoho)}_1 \le \deltaelta, \thetaag{1'} \epsilonnd{equation} there are two asymptotically radically different notions of secrecy. One is the ``naive'' one \betaegin{equation} \label{eq:enc-scheme-2-naive} \forall \rhoho \quad \betaigl\| R'(\rhoho) - \xii_0 \betaigr\|_1 \leq \epsilonpsilon \thetaag{2'} \epsilonnd{equation} that does not randomize entangled states when applied locally. The ``correct'' (composable!) definition takes into account the possibility to apply $R'$ to part of an entangled state: \betaegin{equation} \label{eq:enc-scheme-2-correct} \forall \rhoho_{12} \quad \betaigl\| (R'\omegax\id)\rhoho_{12} - \xii_0\omegax\rhoho_2 \betaigr\|_1 \leq \epsilonpsilon. \thetaag{2''} \epsilonnd{equation} We note that the two conditions coincide in the ideal case $\epsilonpsilon=0$. The minimal key length required for (approximate) encryption reflects whether Eq.~\epsilonqref{eq:enc-scheme-2-naive} or Eq.~\epsilonqref{eq:enc-scheme-2-correct} is used. In the former case $\log d$ bits of key are necessary, and $\log d+o(\log d)$ bits of key are sufficient~\chiite{Hayden+Leung...-Rando_quant_state:2003,Ambainis.Smith-Smallpseudo-randomfamilies-2004} to randomize quantum system of dimension $d$, while in the latter case the key length essentially coincides with the exact encryption case and equals $(2-O(\epsilonpsilon))\log d$~\chiite{Ambainis+Mosca...-Priva_quant_chann:2000}. \sigmaection{Non-malleability} There is, of course, a simple scheme of encryption that implements an ideal scheme: on $n$ qubits, use a key of length $2n$ and apply an independent random Pauli operator to each qubit. (More generally, in dimension $d$, the key identifies one of the $d^2$ discrete Weyl operators made up of the basis shift and phase shift operators.) The adversary evidently cannot see any information about the plaintext state, but she can use the ciphertext in another way: by modulating the ciphertext with an arbitrary Pauli operation, she can effectively implement this Pauli transformation on the plaintext state. We shall show that this is not at all a necessary feature of any encryption scheme. There are, however, always two possible actions for the adversary (and their arbitrary convex combination). Namely, not to interfere at all, resulting in correct decryption of the state $\rhoho$ sent; or interception of the ciphertext and its replacement by a state $\epsilonta_0$ on ${\chial H}'$, resulting in Bob always decrypting the constant state $\rhoho_0 = \sigmaum_k p_K(k) D_k(\epsilonta_0)$. In other words, assuming the adversary implements an arbitrary quantum channel, i.e.~a completely positive and trace non-increasing ({c.p.t.$\leq${}}) map $\Lambdaambda:{\chial B}({\chial H}') \longrightarrow {\chial B}({\chial H}')$, the class of \epsilonmph{effective channels} on the plaintext she can realize, namely all channels \betaegin{equation*}\betaegin{split} {\widetilde\Lambdaambda}: {\chial B}({\chial H}) &\longrightarrow {\chial B}({\chial H}) \thetaext{ s.t.}\\ \rhoho &\longmapsto \sigmaum_k p_K(k) D_k\Bigl( \Lambdaambda\betaigl( E_k(\rhoho) \betaigr) \Bigr), \epsilonnd{split}\epsilonnd{equation*} will include all convex combinations of the identity (up to approximation as specified by $\epsilonpsilon$) and the completely forgetful channels $\langle \rhoho_0 \rhoangle$ mapping all inputs to the state $\rhoho_0 = \sigmaum_k p_K(k) D_k(\epsilonta_0)$, with arbitrary $\epsilonta_0$. We call an encryption scheme \epsilonmph{(perfectly) non-malleable}, if these are the only effective channels the adversary can realize, i.e.~if for every $\Lambdaambda$, ${\widetilde\Lambdaambda}$ is in the semi-linear span of $\id$ and the $\langle \rhoho_0 \rhoangle$, \betaegin{equation} \label{eq:ideal-tres-3} {\widetilde\Lambdaambda} \in {\chial C} := {\omegaperatorname{semi-lin}\,}\left( \{\id\} \chiup \left\{ \langle \rhoho_0 \rhoangle : \rhoho \muapsto \rhoho_0 = \sigmaum_k p_K(k) D_k(\epsilonta_0) \rhoight\} \rhoight), \epsilonnd{equation} with ${\omegaperatorname{semi-lin}\,}$ being the semi-linear hull, i.e. with any family of elements it also contains all their linear combinations, subject to complete positivity of the resulting operator. [Clearly, in the above the convex hull can be realized by an adversary; however, in general the full semi-linear hull is accessible; e.g.~for the Haar measure on the unitary group -- and infinite key -- the only constant channel is $\langle \thetaau \rhoangle$, with the maximally mixed state $\thetaau = \frac{1}{d}\1$, cf.~the beginning of the next section, in particular eqs.~(\rhoef{eq:channels})--(\rhoef{eq:semilinear-example}) On the other hand, any traceless unitary by the adversary results in the effective channel ${\widetilde\Lambdaambda}(\rhoho) = \frac{1}{d^2-1}\left(d^2\thetaau-\rhoho\rhoight)$.] Also, a word on why we demand this for all {c.p.t.$\leq${}}\ maps, which is a strictly larger class than c.p.t.p.: note that the adversary could implement an \epsilonmph{instrument}~\chiite{DaviesLewis:operational}, which is a resolution of a c.p.t.p.~map into {c.p.t.$\leq${}}\ ones. One of them will act randomly, but the adversary can learn which one, so could effectively correlate herself with the effective channel ${\widetilde\Lambdaambda}$. As before, this is to be understood up to approximations: for every effective channel ${\widetilde\Lambdaambda}$ there is $\Thetaheta \in {\chial C}$ such that \betaegin{equation} \label{eq:tres-3-naive} \forall \rhoho \quad \betaigl\| {\widetilde\Lambdaambda}(\rhoho) - \Thetaheta(\rhoho) \betaigr\|_1 \leq \thetaheta. \thetaag{3'} \epsilonnd{equation} However, again the ``correct'' (composable) definition has to take into account the possibility of applying the effective channels to part of an entangled state: \betaegin{equation} \label{eq:tres-3-correct} \forall \rhoho_{12} \quad \betaigl\| ({\widetilde\Lambdaambda}\omegax\id)\rhoho_{12} - (\Thetaheta\omegax\id)\rhoho_{12} \betaigr\|_1 \leq \thetaheta. \thetaag{3''} \epsilonnd{equation} We call the scheme \epsilonmph{strictly non-malleable}, if Eq.~(\rhoef{eq:ideal-tres-3}) or (\rhoef{eq:tres-3-naive}) or (\rhoef{eq:tres-3-correct}) holds for some set ${\chial C}' = {\omegaperatorname{semi-lin}\,}\betaigl\{ \id, \langle\rhoho_0\rhoangle \betaigr\}$ instead of ${\chial C}$. (In other words, there is essentially only one constant channel in ${\chial C}$, independent of $\epsilonta_0$.) Perfect non-malleability then corresponds to $\thetaheta = 0$, in either Eq.~(\rhoef{eq:tres-3-naive}) or (\rhoef{eq:tres-3-correct}) \sigmaection{Main Results} In this paper we restrict ourselves to the ``minimal'' case, when ${\chial H}' = {\chial H}$ is a $d$-dimensional Hilbert space, and to perfect transmission, i.e.~Eq.~(\rhoef{eq:ideal-enc-scheme-1}). This entails that $E_k$ is conjugation by a unitary $U_k$, while $D_k$ is simply the inverse, i.e.~conjugation by $U_k^\deltaagger$: \[ E_k(\rhoho) = U_k \rhoho U_k^\deltaagger,\quad D_k(\sigmaigma) = U_k^\deltaagger \sigmaigma U_k. \] Since convex combinations of unitary conjugation channels are unital, in an encryption scheme all input states are encrypted as the maximally mixed state $\xii_0 = \thetaau := \frac{1}{d}\1$ in Eqs.~(\rhoef{eq:ideal-enc-scheme-2}), (\rhoef{eq:enc-scheme-2-naive}) and (\rhoef{eq:enc-scheme-2-correct}). (For a more general discussion see~\chiite{BoudaZiman:Optimalityofprivate-2007}.) This means that the adversary can always implement channels \betaegin{equation} \label{eq:channels} \Thetaheta \in \chiC' = {\omegaperatorname{semi-lin}\,}\{ \id, \langle \thetaau \rhoangle \}, \epsilonnd{equation} where $\langle \thetaau \rhoangle$ is the completely depolarizing channel. Conversely, we demand that these are the only ones she can achieve: for every {c.p.t.$\leq${}}\ map $\Lambdaambda$, we demand that the effective channel ${\widetilde\Lambdaambda} \in \chiC'$, with \[ {\widetilde\Lambdaambda}(\rhoho) = \sigmaum_k p_K(k) U_k^\deltaagger \betaigl( \Lambdaambda(U_k \rhoho U_k^\deltaagger) \betaigr) U_k. \] This can be conveniently re-expressed using the Choi-Jamio\l{}kowski operators~\chiite{Choi:matrix,Jamiolkowski-Lineartransformationswhich-1972}: for the maximally entangled state $\Pihi_d = \frac{1}{d}\sigmaum_{i,j=0}^{d-1}\kappaet{ii}\!\betara{jj}$ on two systems labelled $1$ and $2$, let $\omegamega = J_\Lambdaambda := (\Lambdaambda \omegatimes \id)\Pihi_d$. Note that $\thetar J_\Lambdaambda \leq 1$ and that $\Lambdaambda$ can be be recovered from the Choi-Jamio\l{}kowski operator as follows: \betaegin{equation} \label{eq:CJ-inverse} \Lambdaambda(\rhoho) = d \thetar_2\betaigl( (\1\omegatimes\rhoho^\thetaop)J_\Lambdaambda \betaigr), \epsilonnd{equation} where $\rhoho^\thetaop$ is the transpose operator of $\rhoho$ with respect to the basis $\{ \kappaet{i} \}_{i=0}^{d-1}$. The image of the set $\chiC'$ under the Choi-Jamio\l{}kowski isomorphism is the set of bipartite positive operators \betaegin{equation} \label{eq:semilinear-example} (\chiC' \omegatimes \id)\Pihi_d = {\omegaperatorname{semi-lin}\,}\{ \Pihi_d, \thetaau\omegatimes\thetaau \} = \RR_{\gammaeq 0}\Pihi_d + \RR_{\gammaeq 0}(\1-\Pihi_d) =: \chiI, \epsilonnd{equation} which are (up to normalization) just the so-called \epsilonmph{isotropic states}. Note that these are exactly the (semidefinite) operators invariant under conjugation with $U\omegax\chiocon{U}$, and that integration over the Haar measure ${\rhom d}U$ implements the projection into $\chiI$: for every operator $X$, \betaegin{equation} \label{eq:iso-twirl} \int {\rhom d}U (U\omegax\chiocon{U}) X (U\omegax\chiocon{U})^\deltaagger = \alpha\Pihi_d + \beta(\1-\Pihi_d), \thetaext{ with } \alpha = \thetar X\Pihi_d,\ \beta=\frac{1}{d^2-1}\thetar X(\1-\Pihi_d). \epsilonnd{equation} The c.p.t.p.~mapping from $X$ to the above average is known as the $U\omegax\chiocon{U}$-twirl, denoted $\chiT_{U\omegax\chiocon{U}}$. On the other hand, exploiting the symmetry $\Pihi_d = (U\omegax\chiocon{U})\Pihi_d(U\omegax\chiocon{U})^\deltaagger$, we can write the Choi-Jamio\l{}kowski operator of the effective channel, \[\betaegin{split} \widetilde\omegamega &= ({\widetilde\Lambdaambda} \omegatimes \id)\Pihi_d \\ &= \sigmaum_k p_K(k) (U_k\omegax\1)^\deltaagger \Bigl[ (\Lambdaambda\omegax\id) \betaigl( (U\omegax\1)\Pihi_d(U\omegax\1)^\deltaagger\betaigr) \Bigr] (U_k\omegax\1) \\ &= \sigmaum_k p_K(k) (U_k\omegax\1)^\deltaagger \Bigl[ (\Lambdaambda\omegax\id) \betaigl( (\1\omegax U_k^\thetaop)\Pihi_d(\1\omegax U_k^\thetaop)^\deltaagger\betaigr) \Bigr] (U_k\omegax\1) \\ &= \sigmaum_k p_K(k) (U_k\omegax\chiocon{U_k})^\deltaagger \betaigl[ (\Lambdaambda\omegax\id) \Pihi_d \betaigr] (U_k\omegax\chiocon{U_k}) \\ &= \sigmaum_k p_K(k) (U_k\omegax\chiocon{U_k})^\deltaagger \omegamega (U_k\omegax\chiocon{U_k}) =: \chiT(\omegamega), \epsilonnd{split}\] where $\chiT$ is manifestly a c.p.t.p.~map. The condition that $\{ p_K(k), U_k \}$ forms a perfect NMES is now concisely expressed as $\chiT = \chiT_{U\omegax\chiocon{U}}$. This is precisely the condition for a so-called \epsilonmph{unitary 2-design} \chiite{Dankert.Cleve.ea-ExactandApproximate-2006}, see also \chiite{Gross.Audenaert.ea:Evenlydistributedunitaries:-2007}. Note that modulo a partial transpose, the $U\omegax\chiocon{U}$-twirl is equivalent to the more familiar $U\omegax U$-twirl \[ \chiT_{U\omegax U}(X) = \int {\rhom d}U (U\omegax U) X (U\omegax U)^\deltaagger = \alpha F + \beta (\1-F), \] with the swap (or flip) operator $F = \sigmaum_{i,j=0}^{d-1} \kappaet{ij}\!\betara{ji}$, mapping density operators to \epsilonmph{Werner states}~\chiite{Werner-QuantumStateswith-1989}. Thus we have proved, \betaegin{theorem} \label{thm:TRES-is-2design} Every perfect non-malleable encryption scheme is a unitary $2$-design. \qed \epsilonnd{theorem} \betaegin{corollary} \label{cor:TRES-implies-encryption} Any perfect non-malleable encryption scheme, i.e., an ensemble of unitaries $\{ p_K(k), U_k \}$ satisfying ${\widetilde\Lambdaambda} \in \chiC'$, is automatically an ideal encryption scheme, i.e.~Eq.~(\rhoef{eq:ideal-enc-scheme-2}) holds. \epsilonnd{corollary} \betaegin{proof} By Theorem~\rhoef{thm:TRES-is-2design} a perfect NMES is a unitary $2$-design. But then it is automatically a unitary $1$-design, meaning that for all $\rhoho$, $\sigmaum_k p_K(k) U_k \rhoho U_k^\deltaagger = \thetaau$, which is precisely Eq.~(\rhoef{eq:ideal-enc-scheme-2}). \epsilonnd{proof} \betaegin{theorem} \label{thm:lowerbounds} Every perfect non-malleable encryption scheme $\{ p_K(k), U_k \}$ requires at least $(d^2-1)^2+1$ unitaries. Furthermore, every $\thetaheta$--NMES as in Eq.~(\rhoef{eq:tres-3-correct}) with $\thetaheta \leq 1/e$ satisfies \[ H(p_K) \gammaeq H_2\left(\frac{1}{d^2}\rhoight) + 2\left(1-\frac{1}{d^2}\rhoight)\log(d^2-1) - 4\thetaheta\log d - H_2(\thetaheta) \gammaeq (4-O(\thetaheta)) \log d, \] where $H_2(x) = -x\log x - (1-x)\log(1-x)$ is the binary entropy. \epsilonnd{theorem} \betaegin{remark} In the light of Theorem~\rhoef{thm:TRES-is-2design}, the first part amounts to a demonstration that $2$-designs have to have at least $(d^2-1)^2+1$ unitaries; this was proved by Gross \epsilonmph{et al.}~\chiite{Gross.Audenaert.ea:Evenlydistributedunitaries:-2007}, but we give a different, direct, proof below. It seems that it is conjectured that in fact the better lower bound $d^2(d^2-1)$ holds in general -- which is true for so-called ``Clifford twirls'', and tight in some dimensions~\chiite{Chau:UnconditionallySecureKey-2005,Gross.Audenaert.ea:Evenlydistributedunitaries:-2007}. \epsilonnd{remark} \muedskip \betaegin{proof} Consider the Choi-Jamio\l{}kowski operator of $\chiT$, labeling the systems $1$, $2$, $1'$ and $2'$, and with the maximally entangled state understood between systems $12$ and $1'2'$: \[ \Omegamega_{U\omegax\chiocon{U}} := (\chiT_{U\omegax\chiocon{U}}^{12} \omegax \id^{1'2'})\Pihi_{d^2} = \frac{1}{d^2}\Pihi_d^{12} \omegax \Pihi_d^{1'2'} + \frac{1}{d^2(d^2-1)} (\1-\Pihi_d)^{12} \omegax (\1-\Pihi_d)^{1'2'}. \] On the other hand, for the first part of the theorem this has to be equal to \[ \Omegamega := (\chiT^{12}\omegax\id^{1'2'})\Pihi_{d^2} = \sigmaum_{k=1}^N p_K(k) (U_k^1 \omegax \chiocon{U}_k^2 \omegax \1^{1'2'}) \Pihi_{d^2} (U_k^1 \omegax \chiocon{U}_k^2 \omegax \1^{1'2'})^\deltaagger. \] Comparing ranks of the two right hand side expressions reveals immediately $N \gammaeq (d^2-1)^2+1$. \muedskip For the entropy statement in the approximate case, we note that by Eq.~(\rhoef{eq:tres-3-correct}), $\| \Omegamega - \Omegamega_{U\omegax\chiocon{U}} \|_1 \leq \thetaheta$, so by Fannes' inequality~\chiite{Fannes:inequality} and Schur concavity of the entropy~\chiite{Wehrl:review}, \[ H(p_K) \gammaeq S(\Omegamega) \gammaeq S(\Omegamega_{U\omegax\chiocon{U}}) - \thetaheta\log d^4 - H_2(\thetaheta), \] and we are done. \epsilonnd{proof} \betaegin{theorem}[Chau~\chiite{Chau:UnconditionallySecureKey-2005}, Gross \epsilonmph{et al.}~\chiite{Gross.Audenaert.ea:Evenlydistributedunitaries:-2007}] \label{thm:d5} If $d=p^n$ is a prime power, then there exists a perfect non-malleable encryption scheme with $d^5-d^3$ unitaries, meaning that the key length is $\leq 5 \log d$. In fact, such a scheme is obtained as the uniform ensemble over a particular subgroup of the Clifford group (i.e., the normalizer) of the $n$-th power Heisenberg-Weyl (aka generalised Pauli) group ${\chial P}_p^{\omegax n}$, where ${\chial P}_p$ is the group generated by the discrete Weyl operators \[ X_p = \sigmaum_{j=0}^{p-1} \kappaet{j\!+\!1 \muod p}\!\betara{j},\quad Z_p = \sigmaum_{k=0}^{p-1} e^{2\pii i k/p}\piroj{k}. \] \epsilonnd{theorem} \betaegin{proof} Apart from Chau~\chiite{Chau:UnconditionallySecureKey-2005} see Gross \epsilonmph{et al.}~\chiite{Gross.Audenaert.ea:Evenlydistributedunitaries:-2007}, as well as the crisp presentation of Grassl~\chiite{Grassl:6-SIC}. \epsilonnd{proof} \muedskip \betaegin{remark} We note that in even prime power dimension, the cardinality of the subgroup can be reduced to $(d^5-d^3)/8$. Furthermore, Chau~\chiite{Chau:UnconditionallySecureKey-2005} showed that for several small dimensions the minimum $d^4-d^2$ is attainable; see also Gross \epsilonmph{et al.}~\chiite{Gross.Audenaert.ea:Evenlydistributedunitaries:-2007} for another example of $2(d^4-d^2)$. \epsilonnd{remark} \betaegin{theorem} \label{thm:approx-2-design} For $0< \thetaheta \leq 1/2$ there exists a $\thetaheta$-NMES with $O(\thetaheta^{-2}d^4\log d)$ unitaries, i.e.~with key requirement of $4 \log d + \log\log d + O\left(\log\frac{1}{\thetaheta}\rhoight)$ bits. In fact, Eq.~(\rhoef{eq:tres-3-correct}) holds in the stronger form \betaegin{equation} \label{eq:tres-3-strong} (1-\thetaheta) \Thetaheta \leq {\widetilde\Lambdaambda} \leq (1+\thetaheta) \Thetaheta. \thetaag{$3^*$} \epsilonnd{equation} \epsilonnd{theorem} \betaegin{proof} Start from any exact unitary $2$-design, such as the unitary group with Haar measure, or the Clifford group or one of its admissible subgroups. We shall select $U_1,\ldots U_N$ independently at random from that chosen 2-design, and show that Eq.~(\rhoef{eq:tres-3-strong}) is true with high probability as soon as $N \gammag \thetaheta^{-2}d^4\log d$; which of course implies that there exist a particular selection of an ensemble $\{ 1/N, U_k \}_{k=1}^N$ satisfying (\rhoef{eq:tres-3-strong}). In fact, it is sufficient to show that for $\chiT(\omegamega) = \frac{1}{N} \sigmaum_{k=1}^N (U_k\omegax\chiocon{U}_k) \omegamega (U_k\omegax\chiocon{U}_k)^\deltaagger$, \[ (1-\thetaheta)\chiT_{U\omegax\chiocon{U}} \leq \chiT \leq (1+\thetaheta)\chiT_{U\omegax\chiocon{U}}, \] which in turn is equivalent to the corresponding statement for the Choi-Jamio\l{}kowski states -- compare Eq.~(\rhoef{eq:CJ-inverse}): \[ (1-\thetaheta)\Omegamega_{U\omegax\chiocon{U}} \leq \Omegamega \leq (1+\thetaheta)\Omegamega_{U\omegax\chiocon{U}}, \] where \betaegin{align*} \Omegamega_{U\omegax\chiocon{U}} &= (\chiT_{U\omegax\chiocon{U}}^{12} \omegax \id^{1'2'})\Pihi_{d^2} = \frac{1}{d^2}\Pihi_d^{12} \omegax \Pihi_d^{1'2'} + \frac{1}{d^2(d^2-1)} (\1-\Pihi_d)^{12} \omegax (\1-\Pihi_d)^{1'2'}, \\ \Omegamega &= (\chiT^{12}\omegax\id^{1'2'})\Pihi_{d^2} = \frac{1}{N} \sigmaum_{k=1}^N (U_k^1 \omegax \chiocon{U}_k^2 \omegax \1^{1'2'}) \Pihi_{d^2} (U_k^1 \omegax \chiocon{U}_k^2 \omegax \1^{1'2'})^\deltaagger. \epsilonnd{align*} Now $\Omegamega$ is a random variable, in fact an average of $N$ independent, identically distributed terms \( X_k := (U_k^1 \omegax \chiocon{U}_k^2 \omegax \1^{1'2'}) \Pihi_{d^2} (U_k^1 \omegax \chiocon{U}_k^2 \omegax \1^{1'2'})^\deltaagger \) with expectation $\EE X_k = \EE \Omegamega = \Omegamega_{U\omegax\chiocon{U}}$. All $X_k$ are bounded between $0$ and $\1$, so the technical result from~\chiite{AhlswedeWinter-ID} applies, the \epsilonmph{operator Chernoff bound}, yielding (with a universal constant $c>0$) \[ \Pir\betaigl\{ (1-\thetaheta)\Omegamega_{U\omegax\chiocon{U}} \leq \Omegamega \leq (1+\thetaheta)\Omegamega_{U\omegax\chiocon{U}} \betaigr\} \gammaeq 1 - 2d^4 e^{-c \thetaheta^2 N/d^4}, \] which implies the claim. \epsilonnd{proof} \sigmaection{Discussion} We have introduced the cryptographic primitive of a non-malleable quantum state encryption scheme. While many questions remain open, we have shown that every such scheme based on random unitaries is a unitary 2-design, showing in particular that every such scheme must use $4\log d$ bits of key, as opposed to the well-known $2\log d$ necessary and sufficient for quantum state encryption~\chiite{Ambainis+Mosca...-Priva_quant_chann:2000}. This situation essentially persists even if we relax the non-malleability to being approximate. On the other hand, there exists an exact construction based on the Jacobi subgroup of the Clifford group in dimension $d$, which requires $5\log d$ bits of key, and we show a new randomized construction requiring only $(4+o(1))\log d$ bits of key. We leave open the question of finding an explicit description of such a scheme, as well as that of finding an exact unitary 2-design with only $O(d^4)$ elements. What we also leave open is the perhaps more pressing problem of relaxing the condition that encryption is done by unitaries. Giving up this restriction results in an advantage in key size, see the work of Barnum \epsilonmph{et al.}~\chiite{Barnum+Crepeau...-Authenticatio_of_q_mes:2002}. More precisely, these authors show how using $2n+O(s)$ bits of secret key to encrypt $n-s$ qubits into $n$ qubits results in a $\thetaheta$-NMES with $\thetaheta = 2^{-O(s)}$. In our setting this can be understood as only using $d_0 < d$ of the Hilbert space dimensions for quantum information. Then, to transmit a state in the $d_0$-dimensional space ${\chial H}_0 \sigmaubset {\chial H}$, first $s$ key bits are used to specify a unitary rotation $V_\epsilonll$ of ${\chial H}$, and then the familiar further $2\log d$ bits of key are used to encrypt ${\chial H}$. If the $V_\epsilonll$ ($\epsilonll=1,\ldots,2^s$) are ``sufficiently random'' and $2^s \gammaeq d/d_0$ then it can be shown that while the adversary can implement certain effective channels on ${\chial H}$, for most $\epsilonll$ this will map the state significantly outside of ${\chial H}_\epsilonll := V_\epsilonll {\chial H}_0$. \alphacknowledgments JB and AW thank the Perimeter Institute for Theoretical Physics for its hospitality during a visit in 2006, where the present work was conceived. JB acknowledges support of the Hertha Firnberg ARC stipend program, and grant projects GA\v{C}R 201/06/P338, GA\v{C}R 201/07/0603 and MSM0021622419. 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\begin{equation}gin{document} \title{Causal Incompleteness: \A New Perspective on Quantum Non-locality} \begin{equation}gin{abstract} The mathematical notion of incompleteness (eg of rational numbers, Turing-computable functions, and arithmetic proof) does not play a key role in conventional physics. Here, a reformulation of the kinematics of quantum theory is attempted, based on an inherently granular and discontinuous state space, in which the quantum wavefunction is associated with a finite set of finite bit strings, and the unitary transformations of complex Hilbert space are reformulated as finite permutation and related operators incorporating complex and hyper-complex structure. Such a reformulation, consistent with Wheeler's `It from Bit' programme, provides the basis for a novel interpretation of the Bell theorem: that the experimental violation of the Bell inequalities reveals the inevitable incompleteness of the causal structure of physical theory. The kinematic reformulation of quantum theory so developed, provides a new perspective on the age-old dichotomy of free will versus determinism. \end{abstract} \section{Introduction} \label{sec:introduction} It is often said that the most profound theorem of 20th Century mathematics concerns the incompleteness of arithmetic proof. The basis of G\"{o}del's theorem, via its use of the Cantor diagonal slash, is directly related to the incompleteness of the rational numbers and the Turing-computable functions. Thus incompleteness is rather fundamental in mathematics; and since mathematics completely underpins our scientific understanding of the physical world, one might ask whether incompleteness has any role to play in physical theory. The potential role of incompleteness in conventional physics is masked by its generic use of continuum equations. Here a novel interpretation of Bell's eponymous theorem of quantum physics is discussed, based on an attempt to recast the complex Hilbert space of quantum theory, using granular, discontinuous mathematics. In this interpretation, the experimental violation of the Bell inequalities reveals, not the type of non-local causality which mainstream physics regards as bizarre but unavoidable, but rather the inevitable incompleteness of the causal structure which may be inferred from such theory. The conventional proof of Bell's theorem presumes that the settings of measuring instruments can be treated as free variables. That is, for a given entangled particle pair, it is assumed that the causal consequences of choosing alternate instrument settings on measurement outcomes are well defined. Bell(1993) himself realised that this issue was not metaphysically clear cut, since the world is given to us once only: `we cannot repeat an experiment changing just one variable; the hands of the clock will have moved and the moons of Jupiter'. However, for Bell, the existence or otherwise of free variables was something to be inferred from the mathematical structure of physical theory, rather than from metaphysical analysis. Hence, for example, if \begin{equation} \label{eq:det} \dot{\mathbf{X}}=\mathbf{F}[\mathbf{X}] \end{equation} denotes a conventional continuum evolution equation, such as occurs in standard quantum theory, electromagnetism, general relativity as so on, then (\ref{eq:det}) determines not only how a given initial state vector $\mathbf{X}(0)$ evolves at future times $t>0$, but also the causal consequences at $t>0$, of a hypothetical perturbation $\delta X$, for example to one of $\mathbf{X}(0)$'s components. In this respect, most physicists (the author included) would agree with Bell(1993)that his eponymous theorem `is primarily an analysis of certain kinds of physical theory', metaphysical concerns notwithstanding. On the other hand, the conventional non-locally causal interpretation of Bell's theorem, that the influence of some freely-chosen remote instrument setting can propagate through space at superluminal speed, remains as bizarre and incomprehensible today as when it was first propounded. Although there have been other proposals to understand the Bell theorem, such as backwards-in-time causality (Price, 1996), one might ask whether there exist classes of theory, formulated using less conventional mathematical structures to that of (\ref{eq:det}) above, for which the existence of an unrestricted set of causal consequences between alternate detector orientations and measurement outcomes, cannot be assumed. The purpose of this paper is to analyse `certain kinds of physical theory` for which the freedom to perturb mathematically the values of certain key variables, is determined by the underlying mathematical structure of the theory's state space. Of particular relevance here will be systems whose state space is generically granular and discontinuous (that is, state space cannot be `continued' by Cauchy-sequence methods). In Section \ref{sec:toy} an idealised model is outlined as the basis for a discussion of the standard EPR-Bohm-Bell experiment, in which the role of causal incompleteness is made explicit. Here we make use of an elementary property of the cosine function (albeit one that the author is unaware of having been used before in physics): if $0<\cos \theta< 1$ is rational, $\theta/\pi$ does not have a finite binary expansion: see Appendix 1 for a simple proof of this. In Section \ref{sec:q} is discussed a potential reformulation, granular and discontinuous, of the kinematics of conventional quantum theory. In this reformulation, the wavefunction becomes a set of finite bit strings and unitary transforms become permutation and related operators with inherent complex and hyper-complex structure. This kinematic reformulation may be of direct interest in quantum information theory as a novel attempt to define physical reality consistent with J.A. Wheeler's `It from Bit' programme (Wheeler, 1994). The analysis of causal incompleteness and the Bell theorem is discussed in section \ref{sec:toy} is relation to this reformulation. Some discussion of the notion of causal incompleteness in the context of the age-old cognitive dichotomy of free-will versus determinism, is given in Section \ref{sec:meta}. It is suggested that the type of mathematically-developed causal incompleteness discussed in sections \ref{sec:toy} and \ref{sec:q}, may present a new perspective on the age-old dichotomy of free will and determinism (Kane, 2002). \section{Quantum Entanglement and Causal Incompleteness: A Simplified Model} \label{sec:toy} The main goal of this paper is to outline a possible reformulation of the kinematics of quantum theory, in which the incompleteness of causal structure can be made explicit. Before doing so, a simplified deterministic model is developed, consistent with two-particle quantum entanglement statistics, which illustrates the potential role of causal incompleteness in the interpretation of the Bell theorem. Linkage between this simplified model and the proposed reformulation of the kinematics of quantum theory is developed in the next section. \subsection{Preliminaries} Let $\mathcal{N}_0$ denote a dyadic rational, ie a member of the set $\mathbbm{Q}_2$ of numbers with finite binary expansion. For example, let the binary expansion of $\mathcal{N}_0$ agree with the first $2^N$ bits of the binary Champernowne number $=.11011100101\ldots$, formed by concatenating the natural numbers $1,2,3,4\ldots$ in binary representation. With $c_n$ denoting the $n$th bit of $\mathcal{N}_0$, let $a_n=2c_n-1 \in \{1,-1\}$. Then, with $\{\lambda_n=\frac{n-1}{2^N}\pi: 1\le n \le 2^N\}$ denoting a finite subset of the unit semi-circle $0 \le \lambda \le \pi$, define \begin{equation} \label{eq:s} S_0(\lambda_n)=a_n \ \ \ \ \ \ S_0(\lambda_n+\pi)=-a_n. \end{equation} For sufficiently large $N$, $S_0$ is defined on arbitrarily-dense subsets of the unit circle. More generally, let $\mathcal{N}_{\theta}$ be the dyadic rational obtained by flipping ($0 \rightarrow 1, 1 \rightarrow 0$) every $1/\sin^2 \frac{\theta}{2}$th bit of $\mathcal{N}_0$, where $\cos \theta \in \mathbbm{Q}_2$. Hence, for example, with $\mathcal{N}_0$ based on the Champernowne number, then \begin{equation} \mathcal{N}_{\pi/2}=.10001001111\ldots \ \ \ \mathcal{N}_{\pi}=.00100011010\ldots \end{equation} Let $c_n(\theta)$ denote the corresponding bits of $\mathcal{N}_{\theta}$, $a_n(\theta)=2c_n(\theta)-1$, and define \begin{equation} \label{eq:s2} S_{\theta}(\lambda_n)=a_n(\theta) \ \ \ \ \ \ S_{\theta}(\lambda_n+\pi)=-a_n(\theta) \end{equation} Since $\mathcal{N}_0$ is based on a normal number (Hardy and Wright, 1979), for large enough $N$ the values of either $S_0$ or $S_{\theta}(\lambda)$, sampled over $S_0$-defined points in any subset of the unit circle, comprise equal numbers of +1's and -1's. Moreover, by construction, the corresponding sample coefficient of correlation \begin{equation} \label{eq:correal} C(\theta)=\langle \ S_0(\lambda)S_{\theta}(\lambda+\pi)\rangle =-\cos\theta \end{equation} \subsection{A Specific Reality} \label{sec:reality} We now add some interpretational baggage. Imagine two experimenters, each with Stern-Gerlach detectors, measuring the spin of entangled particle pairs in a standard EPR-Bohm-Bell experiment. The orientation of experimenter 1's detector is defined to be the $z$-axis; the orientation of experimenter 2's detector is at angle $\theta$ to the $z$ axis (corresponding to a rotation about the particle beam axis). Let $S_0(\lambda)$ and $S_{\theta}(\lambda+\pi)$ determine the measurement outcomes for a given entangled particle pair labelled by $\lambda$. (Nb if $\mathcal{M}_0=\mathcal{N}_{\theta}$ had been used in place of $\mathcal{N}_0$ as the generating base-2 normal number, then experimenter 1's spin measurement outcomes would be determined by $\mathcal{M}_{\theta}$.) Consider a $\theta(t)$ in which the experimenters' detectors have relative orientation $\theta=\theta_A$ when $t_1<t<t_2$ and $\theta=\theta_B$ when $t_3<t<t_4$. We have \begin{equation}gin{eqnarray} \label{eq:corr} C(\theta_A)&=&\langle \ S_0(\lambda)S_{\theta_A}(\lambda+\pi)\ \rangle=-\cos\theta_A \nonumber \\ C(\theta_B)&=&\langle \ S_0(\lambda)S_{\theta_B}(\lambda+\pi)\ \rangle=-\cos \theta_B \end{eqnarray} consistent with quantum experimentation. The values $S_0(\lambda)$ and $S_{\theta}(\lambda)$ associated with such a $\theta(t)$ are referred to as a `specific reality'. The space $\mathcal{U}(\mathcal{N}, \theta(t))$ of such `specific realities' can be generated, as far as this idealised model is concerned, by varying over continguous length-$2^N$ segments $\mathcal{N}$ of the Champernowne number, and timeseries $\theta(t)$ where, for all $t$, $\cos\theta(t) \in \mathbbm{Q}_2$. \subsection{Causal Extension of the Specific Reality} In the introduction, it was noted that (\ref{eq:det}) determined not only the evolution $\mathbf{X}(t)$ from some specific initial state, but also the causal effect on $\mathbf{X}(t)$ of a perturbation $\delta \mathbf{X}$ to that initial state. If the system is ergodic, then this causal effect is, in principle, determined from knowledge of the evolution $\mathbf{X}(t)$. In keeping with our intuition that the experimenters are free to choose individually the orientations of their detectors, it can similarly be asked whether the functions which describe the `specific reality' above, $S_0(\lambda)$ and $S_{\theta}(\lambda)$, also provide the information required to describe the causal effect on measurement outcome, of hypothetical perturbations $\delta\theta_1$ and $\delta\theta_2$ to experimenter 1 and 2's actual detector orientations. To this end, consider the functions \begin{equation}gin{eqnarray} \label{eq:cf} Sp_1(\delta\theta_1, \lambda)&=&\ \ S_0(\lambda-\delta\theta_1)\nonumber\\ Sp_2(\delta\theta_2, \lambda)&=&-S_{\theta}(\lambda-\delta\theta_2) \end{eqnarray} written in conventional local hidden-variable form. When $\delta\theta_1=\delta \theta_2=0$, $Sp_1$ and $Sp_2$ describe the specific reality above, ie \begin{equation}gin{eqnarray} Sp_1(0, \lambda)&=&\ \ S_0(\lambda)\nonumber\\ Sp_2(0, \lambda)&=&-S_{\theta}(\lambda)=S_{\theta}(\lambda+\pi) \end{eqnarray} Hence, assume that $Sp_1(\delta\theta_1, \lambda)$ determines the spin value of one particle of an entangled particle pair labelled by $\lambda$ under a hypothetical perturbation $\delta\theta_1$ to the orientation of experimenter 1's detector. Similarly, let $Sp_2(\delta\theta_2, \lambda)$ determine the spin value of the other entangled particle under a hypothetical perturbation $\delta\theta_2$ to the orientation of experimenter 2's detector. That is, we can think of (\ref{eq:cf}) as defining a pair of lists which give the causal consequences on measurement outcome of hypothetical perturbations to detector orientations. Since, for $N \rightarrow \infty$, $Sp_1$ and $Sp_2$ are defined on uniformly dense subsets of the circle, the functions $Sp_1$ and $Sp_2$ would, for sufficiently large $N$, appear to accommodate any hypothetical alternate choices of orientation experimenter 1 and 2 would care to make. From the construction of $S_0$ and $S_{\theta}$ above, a necessary condition that $Sp_1$ and $Sp_2$ be defined is that both $(\lambda-\delta\theta_1)/\pi$ and $(\lambda-\delta\theta_2/)\pi$ belong to $\mathbbm{Q}_2$. On the other hand, in order that $Sp_1$ and $Sp_2$ describe one of the specific realities of section \ref{sec:reality}, the cosine of the hypothetical relative orientation $\Delta \theta = \theta+(\delta\theta_2-\delta\theta_1)$ must be dyadic rational. There are certainly occasions where (\ref{eq:cf}) returns the correct quantum correlations when $\delta \theta_1 \ne 0$. For example, for all $\delta\theta_1=\delta\theta_2=\delta\theta'$, $\cos \Delta \theta = \cos \theta$ which by construction belongs to $\mathbbm{Q}_2$. In this situation, the hypothetical correlation $\langle \ Sp_1(\delta \theta', \lambda)Sp_2(\delta \theta', \lambda)\ \rangle$ is invariant under hypothetical identical perturbations $\delta \theta'$ to the orientations of both detectors. On the other hand, in general, it cannot be assumed that $\cos \Delta \theta \in \mathbbm{Q}_2$, the implications of which are discussed in the next section. \subsection{Bell's Theorem and Causal Incompleteness} As written, (\ref{eq:cf}) is local in terms of the hypothetical detector perturbations; $Sp_1$ does not depend on $\delta\theta_2$, and $Sp_2$ does not depend on $\delta\theta_1$. Does this imply that correlation statistics derived from (\ref{eq:cf}) must satisfy a Bell inequality? In order to derive a Bell inequality from the statistics of the two experiments with $\theta=\theta_A$ and $\theta=\theta_B$, we need to assume what, following EPR, are usually called `Reality Conditions'. For the first experiment where $\theta=\theta_A$, the relevant Reality Condition states that if a hypothetical perturbation $\delta\theta_1=\theta_A$ to the orientation of experimenter 1's detector were to have aligned experimenter 1's detector with experimenter 2's detector, then experimenter 1 would have measured exactly the opposite of experimenter 2, ie \begin{equation} \label{eq:reality} Sp_1(\theta_A, \lambda)=-Sp_2(0, \lambda) \end{equation} For the second experiment ($\theta=\theta_B$), the Reality Condition similarly requires \begin{equation} \label{eq:reality2} Sp_1(\theta_B, \lambda)=-Sp_2(0, \lambda) \end{equation} The anti-correlations expressed in (\ref{eq:reality}) and (\ref{eq:reality2}) should be contrasted with the anti-correlations, which by the definitions of $S_0(\lambda)$ and $S_{\theta}(\lambda+\pi)$, are guaranteed in the subset of occasions when $\theta=0$ (ie both detectors aligned with the $z$ axis) within the specific reality defined by $\theta=\theta(t)$. The difference between these two situations is exactly equivalent to the difference between the counterfactual and regularity definitions of causality (Menzies, 2001) as first enunciated by the philosopher David Hume. If (\ref{eq:reality}) and (\ref{eq:reality2}) are assumed, then the Bell inequalities follow by standard text-book analysis (eg Rae, 1992). However, in the present case, it must be asked whether (\ref{eq:reality}) and (\ref{eq:reality2}) are consistent with the global constraint $\cos \Delta \theta \in \mathbbm{Q}_2$. Consider (\ref{eq:reality}) in particular. Putting $\delta\theta_1=\theta_A$, $\delta\theta_2=0$ in (\ref{eq:cf}), we have \begin{equation}gin{eqnarray} \label{eq:crunch} Sp_1(\theta_A, \lambda)&=&S_0(\lambda-\theta_A)\nonumber\\ Sp_2(0, \lambda)&=&-S_{\theta_A}(\lambda) \end{eqnarray} But now the result in Appendix A becomes relevant. Since $\cos\theta_A$ is required to be dyadic rational, $\theta_A$ cannot be a dyadic rational fraction of $\pi$. Now, from (\ref{eq:crunch}), $Sp_2$ is only defined if $\lambda$ is a dyadic rational fraction of $\pi$. Hence, if $\lambda$ is a dyadic rational fraction of $\pi$, and $\theta_A$ not, then $(\lambda-\theta_A)/\pi\ \notin \mathbbm{Q}_2$. Hence, for given $\lambda$, ie entangled particle pair, $Sp_1(\theta_A, \lambda)$ and $Sp_2(0, \lambda)$ are not simultaneously defined (reminiscent of the Principle of Complementarity), no matter how large is $N$. Alternatively, $Sp_1(\theta_A, \lambda)$ and $Sp_2(0,\lambda)$ are not contained in the lists of causal relations defined by (\ref{eq:cf}), even, as $N \rightarrow \infty$, this list becomes infinitely long. A similar conclusion holds for (\ref{eq:reality2}). Hence, we cannot derive a Bell inequality from the correlation statistics associated with (\ref{eq:cf}). Is it not instead possible to derive a Bell inequality using in (\ref{eq:reality}) a hypothetical perturbation $\delta\theta$ which is a good dyadic rational approximation $\theta'_A$ to $\theta_A$? Since, the hidden-variable model (\ref{eq:cf}) is generically discontinuous, it is not possible. That is to say, if we consider a Cauchy sequence $\{\theta'_A, \theta''_A, \theta'''_A,\ldots\}$ of dyadic rational perturbations which converge on $\theta_A$, the corresponding sequence $Sp_1(\theta'_A, \lambda), Sp_1(\theta''_A, \lambda), Sp_1(\theta'''_A, \lambda), \ldots$ will not converge to some well-defined value $Sp_1(\theta_A, \lambda)$. Hence although a hidden-variable model has been defined, whose support is as dense as we like on the cirle, generating in the limit $N \rightarrow \infty$ an infinite set of causal relationships between measurement outcome and hypothetical perturbation to detector orientation, the causal structure is not sufficiently comprehensive to be able to derive a Bell inequality. Could quantum theory be recast in such a form as to be able to exploit this result? \section{It from Bit - Towards A Theory of Quantum Beables} \label{sec:q} In this section an attempt to reformulate the kinematics of quantum theory as a generically granular and discontinuous theory is outlined, in order to exploit the notion of causal incompleteness, as discussed above. In this reformulation, the quantum wavefunction $|\psi\rangle$ is a set of encoded bit strings, and the unitary transformations are self-similar permutation and related operators with complex and hyper-complex structure. As such, the wavefunction is literally identified with `information', consistent with J.A.Wheeler's (1994) `It from Bit' aphorism, and Bell's notion of beables. There is no additional collapse hypothesis. The reformulation renders the state space of the wavefunction as inherently granular and discontinuous, yielding a mathematical structure from which the notion of incompleteness, discussed above, is manifest. In this reformulation, the wavefunction of an elementary 2-level system will be identified with the single bit string \begin{equation} \label{eq:s3} \mathcal{S}=\{a_1, a_2, a_3 \ldots, a_{2^N}\} \end{equation} where $a_i \in \{1,-1\}$ and $N \gg 1$ denotes the number of such 2-state systems in the universe. The wavefunction of the universe as a whole is given by $N$ such bit strings; equivalently, a single rational $\mathcal{R}_U$ defined from a length-$2^N$ string comprising base-$2^N$ digits. The entanglement structure of the universe is defined by non-zero coefficients of correlation between individual bit strings; equivalently, in unequal frequencies of occurrence of the digits in the base-$2^N$ expansion of $\mathcal{R}_U$. Following Bohm (1980), this non-normal number structure could be referred to as the `implicate order', whereas the apparently random sequence of bits in any one string could be referred to as the `explicate order'. In the following, some emphasis is placed on ensuring that the proposed reformulation can correctly account for the `vastness' of Hilbert space. \subsection{Permutation Operators with Complex and Hyper-Complex Structure} One of the key features of quantum theory is that its state space is complex. Here we introduce complex structure through permutation operators $\mathbf{E}$, acting on bit strings $\mathcal{S}$, which satisfy \begin{equation} \label{eq:sqrt} \mathbf{E}^2(\mathcal{S})=-\mathcal{S}=\{-a_1, -a_2, -a_3 \ldots, -a_{2^N}\}. \end{equation} We start by defining such complex structures starting with the simplest $N=1$ `universe', and build up complexity for higher $N$. \subsubsection{N=1} With $\mathcal{S}=\{a_1,a_2\}$, define \begin{equation} \label{eq:i} \mathbf{i} (\mathcal{S}) =\{a_2, -a_1\} \end{equation} so that $\mathbf{E}=\mathbf{i}$ satisfies (\ref{eq:sqrt}). It is convenient to rewrite (\ref{eq:i}) as $\mathbf{i}(\mathcal{S}) =\{a_1, a_2\}i$ interpreting $\{a_1, a_2\}$ as a row vector, and \begin{equation} \label{eq:im} i=\left( \begin{equation}gin{array}{cc} 0&1\\ -1&0 \end{array} \right) \end{equation} The coefficient of correlation between $\mathcal{S}$ and $\mathbf{i}(\mathcal{S})$ is equal to zero. \subsubsection{N=2} With $\mathcal{S}=\{a_1,a_2, a_3, a_4\}$, define \begin{equation}gin{eqnarray} \mathbf{e}_1(\mathcal{S})&=&\{-a_3, -a_4, a_1, a_2\}\nonumber\\ \mathbf{e}_2(\mathcal{S})&=&\{-a_4, a_3, -a_2, a_1\}\nonumber\\ \mathbf{e}_3(\mathcal{S})&=&\{-a_2, a_1, a_4, -a_3\} \end{eqnarray} In matrix notation, this can be written, for $j=1,2,3$, as \begin{equation} \mathbf{e}_j (\mathcal{S}) =\{a_1, a_2, a_3, a_4\}e_j \end{equation} where $e_j$ are $4\times 4$ matrices in block $2 \times 2$ form \begin{equation} \label{eq:m2} e_1=\left( \begin{equation}gin{array}{cc} 0&1\\ -1&0 \end{array}\right),\ e_2=\left( \begin{equation}gin{array}{cc} 0&i\\ i&0 \end{array}\right),\ e_3=\left( \begin{equation}gin{array}{cc} i&0\\ 0&-i \end{array}\right) \end{equation} These matrices satisfy the laws of quaternionic multiplication, ie \begin{equation} e_j e_j = e_1 e_2 e_3 = -\mathrm{Id} \end{equation} and, hence, in particular, each $\mathbf{E}=\mathbf{e}_j$ satisfies (\ref{eq:sqrt}). Note that $e_1$ has the same block form as $i$ in (\ref{eq:im}). With $\mathbf{e}_0$ equal to the identity, the coefficient of correlation between any pair of sequences $(\mathbf{e}_j(\mathcal{S}), \mathbf{e}_k(\mathcal{S}))$, with $0 \le j,k \le 3$, is equal to zero. \subsubsection{N=3} Based on the quaternions above, we can, by self-similarity, construct 7 independent square-root-of-minus-one permutation operators acting on 8-element sequences $\mathcal{S}=\{a_1, a_2, \ldots a_8\}$, ie for $j=1,2 \ldots 8$, \begin{equation} \mathbf{E}_j(\mathcal{S})=\{a_1, a_2, \ldots, a_8\} E_j \end{equation} where $E_j$ are $8\times8$ matrices in block $2 \times 2$ form \begin{equation}gin{eqnarray} \label{eq:m3} E_1&=&\left( \begin{equation}gin{array}{cc} 0&\ \ 1\\ -1&\ 0 \end{array}\right),\ \nonumber \\ E_2=\left( \begin{equation}gin{array}{cc} 0&e_1\\ e_1&0 \end{array}\right), \ E_3&=&\left( \begin{equation}gin{array}{cc} 0&\ e_2\\ \ e_2&0 \end{array}\right),\ E_4=\left( \begin{equation}gin{array}{cc} 0&e_3\\ e_3&0 \end{array}\right) \nonumber\\ E_5=\left( \begin{equation}gin{array}{cc} e_1&0\\ 0&-e_1 \end{array}\right), \ E_6&=&\left( \begin{equation}gin{array}{cc} e_2&0\\ 0&-e_2 \end{array}\right),\ E_7=\left( \begin{equation}gin{array}{cc} e_3&0\\ 0&-e_3 \end{array}\right) \end{eqnarray} Note that $E_1$ has the same block form as $i$ in (\ref{eq:im}), and each $E_j$, $j>1$ belongs to one of the pure imaginary quaternion triples $\{E_1, E_2, E_5\}$, $\{E_1,E_3,E_6\}$ and $\{E_1, E_4, E_7\}$. With $\mathbf{E}_0$ equal to the identity, the coefficient of correlation between any pair of sequences $(\mathbf{E}_j(\mathcal{S}), \mathbf{E}_k(\mathcal{S})$, $0\le j,k \le 7$ is equal to zero. \subsubsection{Arbitrary $N$} The construction above can be continued, by self similarity, to $N=4, 5 \ldots$. For arbitary $N$ we have $2^N-1$ square-root-of-minus-one permutation matrices $E_1, E_2 \ldots, {E}_{2^N-1}$, each of which can be written as a $2 \times 2$ block matrix, with blocks representing $2^{N-1}\times 2^{N-1}$ matrices. The square roots are orthogonal to one another, and to the identity $E_0$ in the sense that the coefficient of correlation between any pair $(\mathbf{E}_j(\mathcal{S}), \mathbf{E}_k(\mathcal{S})$, $0\le j,k \le 2^N-1$, is equal to zero. By self-similarity, each of the $M<N$th sets of square roots of minus one, eg (\ref{eq:m2}) and (\ref{eq:m3}), are embedded in this larger $N$th set. Focus now on the matrix $E_1 \equiv E$ which has the special block form \begin{equation} \label{eq:E} E=\left( \begin{equation}gin{array}{cc} 0&1\\ -1&0 \end{array} \right) \end{equation} similar to (\ref{eq:im}), but where `$0$' and `$1$' denote the $2^{N-1} \times 2^{N-1}$ zero and identity matrices, respectively. $E$ has a square root which can be written as the $4\times4$ block matrix \begin{equation} E^{1/2}= \left( \begin{equation}gin{array}{cccc} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ -1&0&0&0 \end{array} \right) \end{equation} where `$0$' and `$1$' now denote the $2^{N-2} \times 2^{N-2}$ zero and identity matrices, respectively. In turn, $E^{1/2}$ has a square root which can be written as the $8\times8$ block matrix \begin{equation} E^{1/4}= \left( \begin{equation}gin{array}{cccccccc} 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\\ -1&0&0&0&0&0&0&0 \end{array} \right) \end{equation} where `$0$' and `$1$' now denote the $2^{N-3} \times 2^{N-3}$ zero and identity matrices, respectively. This procedure can be continued until we reach the $2^N$th root given by the $2^N \times 2^N$ matrix \begin{equation} \label{eq:root} E^{1/2^N}=\left( \begin{equation}gin{array}{cccccc} 0&1&\ &\ &\ldots&0\\ 0&0&1 &\ &\ &0\\ 0&0&0&1&\ &0\\ \ &\ &\ &\ &\ddots&\ \\ 0&0&0&0&\ldots&1\\ -1&0&0&0&\ldots&0 \end{array} \right) \end{equation} where `$0$' and `$1$' are scalars. Clearly $E^{1/2^N}$ is a $2^{N+2}$th root of unity and therefore generates a cyclic group of order $2^{N+2}$, a finite sub-group of $U(1)$. Applied to a bit string $\mathcal{S}=\{a_1, a_2, \ldots, a_{2^N}\}$, $E^{1/2^N}(\mathcal{S})=\{-a_{2^N}, a_1, a_2, \ldots, a_{2^N-1}\}$; that is, $E^{1/2^N}$ brings to the front, the (negation of the) trailing bit of $\mathcal{S}$, cf the discussion in section \ref{sec:meta}. \subsection{Towards A Granular Reformulation of Complex Hilbert Space} Here the results above are applied to a possible kinematic reformulation of quantum theory. Some preliminary ideas on such a reformulation were first presented in Palmer (2003). A more complete exposition will appear elsewhere. \subsubsection{1 Qubit} \label{sec:1qubit} The general 1-qubit state in quantum theory is given, for example, by \begin{equation} \label{eq:qubit} |\psi\rangle=p_0|\uparrow\rangle+p_1|\downarrow\rangle) \end{equation} where $p_0, p_1 \in \mathbbm{C}$ satisfy $|p_0|^2+|p_1|^2=1$. The Hilbert space of such a qubit is therefore three dimensional, including the phase degree of freedom \begin{equation} \label{eq:gphase} |\psi\rangle \mapsto e^{i\phi}|\psi\rangle \end{equation} which in quantum theory is viewed as `irrelevant' since the value $\phi$ does not affect the statistics of measurement outcomes. In the proposed reformulation proposed here, $|\psi\rangle \mapsto \mathcal{S}$, see (\ref{eq:s3}). The elements of $\mathcal{S}$ can be associated with what, operationally, are measurement outcomes, but which, following Bell, might be better called `beables'; $\mathcal{S}$ can therefore be thought of as a time series of such beables, whose first element is the beable corresponding to `now'. To account for the three degrees of freedom of complex Hilbert space, recall from (\ref{eq:m2}) that, in an $N$ qubit universe, any of the $2^N-2$ square roots of minus one, ${E}_j \ j>1$, is automatically part of the (pure imaginary) quaternionic triple $\{E, E_j, {E}_{j+2^{N-1}-1}\}$ if $j \le 2^{N-1}$, or the triple $\{E, E_{j-2^{N-1}+1}, E_j\}$ if $j>2^{N-1}$, where $E$ is given by (\ref{eq:E}). We build the reformulation of the qubit state (\ref{eq:qubit}) around one such quaternion triple, written in the generic form $\{E, E_a, E_b\}$. Then, \begin{equation}gin{eqnarray} |\uparrow\rangle &\mapsto& \{1,1,\ldots,1\}\nonumber\\ |\uparrow\rangle+|\downarrow\rangle &\mapsto& \mathbf{E}_a ( \{1,1,\ldots,1\})\nonumber\\ |\downarrow\rangle &\mapsto& \{0,0,\ldots,0\}. \end{eqnarray} Let us start with the degree of freedom associated with the phase transformation (\ref{eq:gphase}), here reformulated as $\mathcal{S} \mapsto \mathbf{E}^{2\phi/\pi}(\mathcal{S})$. Consistent with the invariance of the qubit wavefunction under a global phase transformation in standard quantum theory, here a qubit is regarded as an equivalence class of bit strings, where two bit strings $\mathcal{S},\ \mathcal{S}'$ in the class are related by $\mathcal{S}'=\mathbf{E}^{\alpha}(\mathcal{S})$, $\alpha \in \mathbbm{Q}_2$. To reformulate the `nontrivial' unitary transformations associated with the remaining two degrees of freedom, those which transform one qubit state into a physically-inequivalent qubit state, we define the notion of addition of sequences eg as in \begin{equation} \label{eq:superp} \mathcal{S}=(\cos\theta\ \mathbf{E}_a + \sin\theta\ \mathbf{E}_b) (\{1,1,\ldots,1\}) \end{equation} as follows. If $\cos\theta \in \mathbbm{Q}_2$, then the $n$th element of $\mathcal{S}$ is equal to the $n$th element of $\mathbf{E}_a(\{1,1,\ldots,1\})$ if the non-zero element in the $n$th row of $E^{\cos\theta}$ is a `1', and otherwise is equal to the $n$th element of $\mathbf{E}_b(\{1,1,\ldots,1\}$. The following properties of the defined `addition' operation are easy to show: \begin{equation}gin{itemize} \item When $\theta=0$, $\mathcal{S}=\mathbf{E}_a(\{1,1,\ldots,1\})$ ;when $\theta=\pi/2$, $\mathcal{S}=\mathbf{E}_b(\{1,1,\ldots,1\})$ \item With $\cos\theta \in \mathbbm{Q}_2$, the coefficient of correlation between $\mathcal{S}$ and $\mathbf{E}_a(\{1,1,\ldots,1\})$ is equal to $\cos\theta$. With $\theta$ varying uniformly in time, the phenomenon of wave interference is manifest. \item If $0<\cos\theta < \pi/2 \in \mathbbm{Q}_2$, then $\sin\theta \notin \mathbbm{Q}_2$ and vice versa; see Appendix A. \item If $\sin\theta \in \mathbbm{Q}_2$ then $\mathcal{S}$ is as (\ref{eq:superp}), with $\mathbf{E}_a$ swapped with $\mathbf{E}_b$. \item The definition is distributive in the sense that \begin{equation} \mathbf{E}^{\alpha}(\mathcal{S})=(\cos\theta\ \mathbf{E}^{\alpha}\mathbf{E}_a + \sin\theta\ \mathbf{E}^{\alpha}\mathbf{E}_b) \{1,1,\ldots,1\} \end{equation} \end{itemize} On this basis, we can write, for example \begin{equation}gin{eqnarray} |\uparrow\rangle+e^{i\theta}|\downarrow\rangle &\mapsto& \mathbf{E}_a (\cos\theta\ \mathbf{E}_0+\sin\theta\ \mathbf{E}\mathbf{E}_0)(\{1,1,\ldots,1\})\nonumber\\ &=& (\cos\theta\ \mathbf{E}_a + \sin\theta\ \mathbf{E}_a\mathbf{E}) (\{1,1,\ldots,1\})\nonumber\\ &=& (\cos\theta\ \mathbf{E}_a + \sin\theta\ \mathbf{E}_b) (\{1,1,\ldots,1\})=\mathcal{S} \end{eqnarray} from (\ref{eq:superp}). A key point in this proposed reformulation of Hilbert space, is that the notion of `adding wavefunctions' does not imply `superposition' of states, with all the (Schr\"{o}dinger's cat) paradoxes that that implies. Rather, the $n$th element (beable) of the bit string `$\mathcal{A}+i\mathcal{B}$' is formed from the $n$th elements of the bit strings $\mathcal{A}$ and $\mathcal{B}$, taking account of the hyper-complex structure of the associated permutation operators $\mathbf{E}_j$. Because of this complex structure, the $n$th elements of $\mathcal{A}+i\mathcal{B}$ need equal neither the $n$th elements of $\mathcal{A}$ nor $\mathcal{B}$. Note that no additional collapse hypothesis is required to obtain definite beable elements. \subsubsection{2 Qubits} In quantum theory, the general 2-qubit state is given by \begin{equation} \label{eq:2qubit} |\psi\rangle=p_0|\uparrow\uparrow\rangle+p_1|\uparrow\downarrow\rangle +p_2|\downarrow\uparrow\rangle+p_3|\downarrow\downarrow\rangle \end{equation} where $p_i \in \mathbbm{C}$ satisfy $|p_0|^2+|p_1|^2 +|p_2|^2+|p_3|^2=1$, giving a Hilbert space with seven degrees of freedom, including the global phase degree of freedom, modulo which gives the standard complex-three dimensional projective Hilbert space $\mathbbm{CP}_3$. In the reformulation proposed here, the wavefunction of a 2-qubit state (in a universe of $N$ qubits) is given by two $2^N$-long bit strings $\mathcal{S}_1$ and $\mathcal{S}_2$. Here the seven degrees of freedom are represented by recalling from (\ref{eq:m3}) that any ${E}_j \ j>1$ belongs to a set of seven square roots of minus one whose elements can be listed as $\{E, E_A, E_B, E_C, E_D, E_E, E_F\}$. As before we use roots of $E$ to represent global $`U(1)'$ invariance. Consistent with this, the coefficient of correlation between $\mathcal{S}_1$ and $\mathcal{S}_2$ is invariant under the global transformation \begin{equation} \label{eq:zphase} \mathcal{S}_1 \mapsto \mathbf{E}^{\alpha}(\mathcal{S}_1), \ \mathcal{S}_2 \mapsto \mathbf{E}^{\alpha}(\mathcal{S}_2) \end{equation} where $\alpha \in \mathbbm{Q}_2$. The remaining six degrees of freedom can be described by considering bit strings which combine (in the sense defined by (\ref{eq:superp})), the seven `basis' sequences \begin{equation}gin{eqnarray} \{1,1,&\ldots&,1\},\nonumber\\ \mathbf{E}_A(\{1,1,\ldots,1\} ), \mathbf{E}_B(\{1,1,&\ldots&,1\} ), \mathbf{E}_C(\{1,1,\ldots,1\} ),\nonumber\\ \mathbf{E}_D(\{1,1,\ldots,1\} ), \mathbf{E}_E(\{1,1,&\ldots&,1\} ), \mathbf{E}_F(\{1,1,\ldots,1\} ) \end{eqnarray} Representing the wavefunction $|\psi\rangle$ for two qubits as the pair $\{\mathcal{S}_1, \mathcal{S}_2\}$, the qubits will be said to be entangled if $\mathcal{S}_1$ is correlated with $\mathcal{S}_2$. Consider, for example \begin{equation}gin{eqnarray} \label{eq:entangled} \mathcal{S}_1&=&\mathbf{E}_A(\{1,1,\ldots,1\})\nonumber \\ \mathcal{S}_2&=&(\cos\theta\ \mathbf{E}_A + \sin\theta\ \mathbf{E}_D) \{1,1,\ldots,1\} \end{eqnarray} By the discussion in section \ref{sec:1qubit}, if $\cos\theta \in \mathbbm{Q}_2$ then the correlation between $\mathcal{S}_1$ and $\mathcal{S}_2$ is equal to $\cos\theta$. \subsubsection{$N$ Qubits} Continuing to larger $N$, the wavefunction of an entire universe of $N$ qubits is represented by a set $\{\mathcal{S}_1, \mathcal{S}_2, \ldots, \mathcal{S}_N\}$ of bit strings each of length $2^N$ - equivalently, as discussed above, as a rational $\mathcal{R}_U$ with $2^N$ base-$2^N$ digits. In standard quantum theory, the Hilbert space of $N$ qubits has dimension $2^N-1$, including one global phase degree of freedom. In the reformulation, we have the set $\{E, E_2 \ldots E_{2^N-1}\}$ of $2^N-1$ square roots on minus one. As before, the the global phase degree of freedom is represented by the roots $E^{\alpha}$ of $E$. The remaining degrees of freedom are associated with linear combinations of the basis sequences \begin{equation}gin{eqnarray} \{1,1,&\ldots&,1\},\nonumber\\ \mathbf{E}_2(\{1,1,\ldots,1\} ), \mathbf{E}_3(\{1,1,&\ldots&,1\} ), \ldots \mathbf{E}_{2^N-1}(\{1,1,\ldots,1\} ) \end{eqnarray} as in (\ref{eq:superp}). It can be asked whether the proposed reformulation is testably different from quantum theory. One interesting fact, which may be relevant in this respect, emerges at the level of $4$ qubits. Unlike the state space of $1$, $2$ or $3$ qubits, the Hilbert space $\mathbbm{S}^{31}$ of 4 qubits in standard quantum theory cannot be Hopf fibrated, due to a theorem of Adams and Atiyah (1966). As Bernevig and Chen (2003) note, the failure of the Hilbert space to fibrate appears to lead to fundamental difficulties in describing the entanglement structure of 4 or more qubits. By contrast, the present theory is constrained by neither the continuum properties of hyper-complex algebraic fields, nor their corresponding topological spaces. It is interesting to note that $4$-qubit structures are needed to describe the quantum of gravity. This indicates that the proposed reformulation may more readily incorporate the effects of gravity than does does conventional quantum theory. \subsubsection{Relation to the Idealised Model of Quantum Measurement} The construction of $S_0$ and $S_{\theta}$ in section \ref{sec:toy} is an idealisation of the hyper-complex permutation operators developed here. A more precise linkage bewteen $S_0$ and $S_{\theta}$ and the proposed reformulation of quantum theory, can be given as follows. In terms of the proposed 2-qubit reformulation of Hilbert space put $|\psi\rangle \mapsto \{\mathcal{S}, \mathcal{S}_{\theta}\}$, where \begin{equation}gin{eqnarray} \mathcal{S}&=&\mathbf{E}_a\{1,1,\ldots,1\}\nonumber\\ \mathcal{S}_{\theta}&=&(\cos \theta \ \mathbf{E}_a+\sin\theta \ \mathbf{E}_b)\{1,1,\ldots,1\}, \end{eqnarray} where $\{E, E_a, E_b\}$ denotes a pure imaginary quaternionic triple in the space of $2^N-1$ hyper-complex permutation operators. Using (\ref{eq:root}), define \begin{equation}gin{eqnarray} \mathbf{E}^{\alpha}(\mathcal{S})&=&\{a_1, a_2,\ldots, a_{2^N}\}\nonumber\\ \mathbf{E}^{\alpha}(\mathcal{S}_{\theta})&=&\{a_1(\theta), a_2(\theta), \ldots, a_{2^N}(\theta)\} \end{eqnarray} and put \begin{equation}gin{eqnarray} S_0(\frac{\alpha\pi}{2})&=&a_1 \nonumber\\ S_{\theta}(\frac{\alpha\pi}{2})&=&a_1(\theta) \end{eqnarray} that is, $S_{\theta}(\alpha\pi/2)$ denotes the leading bit of $\mathbf{E}^{\alpha}(\mathcal{S}_{\theta})$. When $\theta=0$, then $S_0$ and $S_{\theta}$ are identical. When $\theta=\pi/2$ then $\mathcal{S}_{\theta}=\mathbf{E}_b\{1,1,\ldots,1\}=\mathbf{E}\mathbf{E}_a\{1,1,\ldots,1\}=\mathbf{E}(\mathcal{S})$. Hence, $S_{\pi/2}(\alpha\pi/2)=S_0(\pi/2+\alpha\pi/2)$. When $\theta=\pi$, then $\mathcal{S}_{\theta}=-\mathbf{E}_a\{1,1,\ldots, 1\}=\mathbf{E}^2(\mathcal{S})$, hence $S_{\pi}(\alpha\pi/2)=S_0(\pi+\alpha\pi/2)$. When $\theta=3\pi/2$ then $\mathcal{S}_{\theta}=-\mathbf{E}_b\{1,1,\ldots,1\}=\mathbf{E}^3(\mathcal{S})$. Hence, $S_{3\pi/2}(\alpha\pi/2)=S_0(3\pi/2+\alpha\pi/2)$. For all other values of $\theta$, $\mathcal{S}_{\theta}$ is never equal to $\mathbf{E}^{\alpha}(\mathcal{S})$ for any $\alpha$ - since $\mathbf{E}^{\alpha}$ induces a cyclic displacement of the elements of $\mathcal{S}$, there is no $\alpha$ where $\mathbf{E}^{\alpha}(\mathcal{S})$ is partially correlated with $\mathcal{S}$. That is, the values of $\theta$ for which $\mathcal{S}_{\theta}$ belongs to the qubit equivalence class $\{\mathbf{E}^{\alpha}(\mathcal{S}):\ \alpha \in \mathbbm{Q}_2\}$, are precisely the values $\theta$ for which $\theta$ is a dyadic rational multiple of $\pi$ and $\cos\theta$ is dyadic rational ie $\{0,\pi/2, \pi, 3\pi/2\}$. This result links the idealised model in section \ref{sec:toy} with the proposed reformulation of quantum theory, and makes the result on causal incompleteness relevant to the interpretation of Bell's theorem in (this reformulation of) quantum theory. \section{Incompleteness and the Metaphysics of Free Will} \label{sec:meta} It has been proposed that inherent mathematical incompleteness (of the type describing the sets of rational numbers, Turing-computable functions, arithmetic proofs and so on) provides a new interpretation of the experimental violation of the Bell inequalities, one that does not invoke or require non-local causality. This interpretation applies to a class of physical theory for which state space is inherently granular and discontinuous. Within such class of theory, the freedom of experimenters to choose measurement settings is tempered by the granular structure of state space. The proposed interpretation of the Bell inequalities is that they reveal the inevitable mathematical incompleteness of the causal structure underlying such theory. The kinematic structure of quantum theory has been reformulated so that it then belongs to this class of theory. It is well known that violation of the Bell inequalities can be `explained' if the notion of free will is completely rejected. This is generally considered an unsatisfactory explanation, for reasons which go under the general description `conspiratorial'. In an attempt to clarify this issue, Bell (1993) considers a deterministic dynamical system which replaces the whimsical experimenter. This system selects between two possible outputs $a$ or $a'$ on the basis of the parity of the digit in the millionth decimal place of some input variable. Then, fixing $a$ or $a'$ fixes something about the input - ie whether the millionth digit is odd or even. Bell's objection to a deterministic explanation of the violation of his eponymous inequalities is this: this peculiar piece of information, the millionth digit, is unlikely to be the vital piece of information for any distinctively different purpose ie it is otherwise rather useless. However, in the reformulation of quantum kinematics, the wavefunction of the universe is constructed from finite $N$ bit strings each of length $2^N$, and the equivalent of unitary transformations involve permutation and related operators acting on these bit strings. Typically these permutation operators, represented as matrices, have terms on the anti-diagonal. For example, the global phase operator $\mathbf{E}^{1/2^N}$, see (\ref{eq:root}), acting on some sequence $\mathcal{S}$ brings to the front of the sequence, an element that was previously at the back of the sequence. That is, at the heart of our reformulation of the complex Hibert space of quantum theory, are operators whose action is very similar, in essence, to Bell's deterministic dynamical system. The entanglement structure of the universe is given by precise intricate relationships between the bits of the different bit strings. In this perspective, bits near the end of a bit string are no less `vital' for `distinctively different purposes' than bits near the front of the bit string. Like a Sudoku puzzle, violate one piece of the structure (either at the beginning or end of a bit string) and you violate the structure everywhere. Rather, the real difficulty with deterministic explanations of the violation of the Bell inequalities is their contradiction of our strong intuition that the experiment could have been performed differently. In Bell's example above, our intuition suggests the input could have been otherwise, at least as far as the trailing digits of the input number are concerned. Any explanation of the violation of the Bell inequality which does not address this deeply-held feeling, is not likely to be accepted. In the current proposal, our intuition about free will is not rejected \emph{per se}, but rather (as far as the EPR-Bohm-Bell experiments are concerned) is derived from the computational properties of the models $Sp_1$ and $Sp_2$, see (\ref{eq:cf}). Hence, our intuition infers that the experimenter could have chosen from an arbitarily dense set of alternative detector orientations to the one actually chosen, and, from the properties of $Sp_1$, this belief is not inconsistent with the (proposed reformulation of the) laws of physics. Since $Sp_1(0, \lambda)$ defines `reality', $Sp_1(\pi, \lambda)$ defines an alternate world precisely anitcorrelated with reality, and $Sp_1(\pi/2, \lambda)$ and $Sp_1(3\pi/2, \lambda)$ define alternate worlds uncorrelated with reality. However, if $Sp_1(\delta\theta_1, \lambda)$ solved algorithmically for a perturbed orientation such as required to derive a Bell inequality, $Sp_1$ would never halt. That is, in circumstances where our intuition might contradict physics, our intuition, acting computationally, would never be able to ascertain what the relevant measurement outcome would have been. On the other hand, our cognitive reasoning is (from time to time, at least) able to transcend such a purely computational perspective. Does an awareness of algorithmic incompleteness imply that non-computability is a feature of such cognitive reasoning, and by implication a feature of the laws of physics, as has been suggested by Penrose (1994)? In fact, it would be hard to reconcile this notion with this paper's underlying premise that the granular reformulation of quantum theory is ultimately finite (with the wavefunction of the universe being given by a sequence of $2^N$ base-$2^N$ digits, for some very large but nevertheless finite $N$). A possible alternative suggestion, therefore, is that an awareness of algorithmic incompleteness may instead arise from some cognitive ability to jump (perhaps involuntarily) between computationally-inequivalent finite systems. For example, in the model developed here, a finite division of the circle based on angular segments which are equal dyadic rational fractions of $\pi$, is not equivalent to a finite division of the cirle based on angular segments whose cosines are equal dyadic rational fractions ie based on equal divisions of the diameter. From an awareness of both finite cyclotomies one can recognise the inability of one to contain the other. In conclusion, it is ironic, perhaps, that exploitation of mathematical incompleteness in physical theory may turn out to be the key notion that allows EPR's goal to be achieved, of developing a theory of the quantum that is more physically complete than standard quantum theory. \appendix \section{A fundamental property of the cosine function} The discussion in section \ref{sec:toy} uses a rather basic property of the cosine function, albeit one rarely (if ever?) used in physics. For completeness, we give a simple proof of this property, based on unpublished work by Jahnel (2004). It can be seen as a special example from the theory of trigonometric diophantine equations (Conway and Jones, 1976). \textbf{Theorem} Let $0<\theta/\pi<1/2 \in \mathbbm{Q}_2$, then $\cos \theta \notin \mathbbm{Q}_2$. With $0<\theta/\pi<1/2 \in \mathbbm{Q}_2$, assume that $2\cos \theta =a/b$ is rational, where $a,b \in \mathbbm{Z}, b \ne 0$ have no common factors. Using the identity \begin{equation} 2\cos 2\theta = (2\cos\theta)^2-2 \end{equation} we have \begin{equation} 2\cos 2\theta=\frac{a^2-2b^2}{b^2} \end{equation} Now $a^2-2b^2$ and $b^2$ have no common factors, since if $p$ were a prime number dividing both, $p|b^2 \Rightarrow p|b$ and $p|(a^2-2b^2) \Rightarrow p|a$, a contradiction. Hence, if $b \ne \pm1$, then the denominators in $2\cos\theta$, $2\cos2\theta$, $2\cos4\theta$, $2\cos8\theta\ldots$ get bigger and bigger without limit. On the other hand, $\theta/\pi=m/n$ which means that the sequence $(2\cos 2^k \theta)_{k\in \mathbbm{N}}$ admits at most $n$ values. Hence we have contradiction. Hence $b=\pm1$. Hence $\cos\theta=0, \pm1/2, \pm1$. No $0<\theta/\pi<1/2 \in \mathbbm{Q}_2$ has $\cos\theta$ with these values. QED. Finally note that Pythagorean integer triples $\{x,y,z\}$ satisfying $x^2+y^2=z^2$ can be parametrised as $x=2uv, y=u^2-v^2, z=u^2+v^2$ where $(u, v)$ are integers without common factor and of different parity (eg Hardy and Wright, 1979). Hence, if $0 < \theta < \pi/2$ and both $\cos \theta$ and $\sin \theta$ are rational, then $\cos \theta = 2uv/(u^2+v^2)$ and $\sin\theta=(u^2-v^2)/(u^2+v^2)$. Since $u$ and $v$ are of different parity, then $u^2+v^2$ cannot be divided by 2. Hence $\cos \theta$ and $\sin\theta$ cannot both be dyadic rational. \section*{References} \begin{equation}gin{description} \item Adams, F.J. and Atiyah, M.F.A. 1966: On K-theory and Hopf invariant. Quarterly J. Math., 17, 31-8. \item Bernevig, B.A. and Chen H.-D., 2003: Geometry of the three-qubit state, entanglement and division algebras. J.Phys A, 36, 8325-8339. \item Bell, J.S., 1993: Free variables and local causality. In `Speakable and unspeakable in quantum mechanics.' Cambridge University Press. 212pp. \item Bohm, D., 1980: Wholeness and the implicate order. Routledge and Kegan Paul, London. \item Conway, J.H. and Jones, A.J., 1976: Trigonometric Diophantine Equations. Acta Arithmetica, 30, 229-240. \item Hardy, G.H. and Wright, E.M., 1979: The Theory of Numbers. Oxford University Press. \item Jahnel, J., 2005: When does the (co)-sine of a rational angle give a rational number? Available online at www.uni-math.gwdg.de/jahnel/linkstopaperse.html \item Kane, R., 2002: The Oxford Handbook on Free Will. Oxford University Press. 638pp \item Menzies, P. 2001: Counterfactual Theories of Causation. The Stanford Encyclopedia of Philosophy (Spring 2001 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/spr2001/entries/causation-counterfactual/ \item Palmer, T.N., 2004: A granular permutation-based representation of complex numbers and quanternions: elements of a possible realistic quantum theory. Proc. Roy. Soc., 60A, 1039-1055. \item Penrose, R., 1994: Shadows of the mind. Oxford University Press. Oxford. 450pp \item Price, H., 1996: Time's Arrow and Archimedes Point. Oxford University Press. 306pp \item Rae, A.I.M., 1992: Quantum Mechanics. Institute of Physics. Bristol. \item Wheeler, J.A., 1994: In `Physical Origins of Time Asymmetry, ed J.J. Halliwell, J.Peres-Mercader and W.H. Zurek, pp. 1-29. Cambridge University Press. \end{description} \end{document}

\begin{enumerate}gin{document} \begin{enumerate}gin{center}{\Large\bf A Generalised Harbourne-Hirschowitz Conjecture} \end{center} \begin{enumerate}gin{center}J. E. Alexander\end{center} \begin{enumerate}gin{abstract}We give a generalised Harbourne-Hirschowitz conjecture which suggests a test for determining when a linear system on a generic rational surface separates $k$-clusters. In particular when it is base point free or very ample.\footnote{Mathematical rewiews subject classification 14C20, 14E25, 14J26}\end{abstract} \section{Introduction} Recall the general context in which the Harbourne-Hirschowitz conjecture is formulated. This is best presented as a conjecture for generic rational surfaces which we define as follows. Fix a field $k$ and for an integer $r\geq 0$, let $\Lambda_r$ be the residual field at the generic point of $(\sbm P_k^2)^r$. We then have the blowing-up $\pi_r : X_r\longrightarrow \sbm P^2_{\Lambda_r}$ in the tautological sequence of points $x_1,\ldots ,x_r$ in the projective plane $\sbm P_{\Lambda_r}^2$. One says that $X_r$ is the generic rational surface of rank $r$, or the blowing-up of $\sbm P^2_k$ in $r$ generic points. As usual, a curve $E$ on $X_r$ with $E\simeq \sbm P^1$ and $E ^2=-1$ is called an exceptional curve on $X_r$. Since $X_r$ is rational, for any divisor class $H$ on $X_r$, either $h^0(X_r,\fasm O(H))=0$ or $h^2(X_r,\fasm O(H))=0$. One says that an effective divisor class $H$ is {\em non-special} if $h^1(X_r,\fasm O(H))=0$. In slightly different forms, \cite{Har3} Harbourne and \cite{Hir1} Hirschowitz conjectured that (see \cite{AHi}, \cite{CM1}, \cite{CM2}, \cite{GLS} for work on the conjecture) \noindent{\bf Conjecture (H-H)} (Harbourne-Hirschowitz) {\em The effective divisors $H$ on $X_r$ that are non-special are those that satisify $H.E\geq -1$ for all exceptional curves $E$ on $X_r$. } This can also be stated in the following form : if $H$ is an effective, special divisor on $X_r$ then $H=2E+H^{\prime}$ where $H^{\prime}.E\leq 0$. In particular $H$ is non-reduced. \noindent{\bf Remark.} More recently, it has been pointed out by Ciliberto and Miranda that this conjecture is in fact equivalent to an older conjecture of Segre which says that a special effective divisor is non-reduced. Our purpose here is to propose a conjecture in the same vein for separation properties such as base point freeness and very ampleness. In \cite{D'A-H}, when studying certain classes of very ample and base point free divisors, the authors tried unsuccessfully to propose a suitable conjecture. In \cite{Har2}, \cite{Har3} and \cite{Di Roc}, the respective authors showed that in the case $r\leq 9$, base point freeness, very ampleness and more general separation properties were characterised by intersection numbers on a small class of curves, but no general conjecture was forthcoming. In accordance with known results for $r\leq 9$ the following conjecture says that if a divisor class $H$ with sufficiently many sections does not separate $k$-clusters then there is an integral (even smooth and irreducible) curve $E$ such that $H$ does not separate some $k$-cluster in $E$. Furthermore, evidence would suggest that this failure must be detectable numerically. Before formulating the conjecture we need the following \begin{enumerate}gin{defn} {\em A $k$-{\em cluster} $Z$ on $X_r$ is any finite closed subscheme of $X$ of length (i.e. degree) $k$. We say that an effective divisor class $H$ on $X$ separates $k$-clusters if $h^0(X_r, I_Z(H))=0$.}\end{defn} \noindent{\bf Conjecture 1.} {\em Let $k\geq 0$ be an integer. If $H$ is an effective divisor class on $X_r$ such that $\gchi (X_r,\fasm O(H))\geq 3k$, then $H$ separates $k$-clusters if the following (necessary) conditions are satisfied : for all integral curves $E$ of genus $a$, if $0\leq a\leq k$ then $H.E\geq 2a-1+k$ and if $k< a \leq \frac{4k}{3}$ then $H.E\geq a+2k-1$.} A weakness in this formulation of the conjecture is that it would appear to be necessary to test on all integral classes of low genus. In fact for $r\geq 3$ one can use tha Weyl group action on $X_r$ to obtain a {\em standard} representative of classes $H$ that satisfy $H.E\geq 0$ for all exceptional classes $E$. One can then show that the conjecture is equivalent to testing on a finite set of isolated, integral, standard classes of low genus. In \S2, we give the structure theorem for effective divisors on $X_r$ as implied by H-H. In \S3, we give the motivation and assumptions underlying the conjecture. In \S4, we review the Weyl group action and the notion of a standard class. While the notion is not new, I have not seen mention in the literature of the caracterisation of these classes as minimising for the intersection product in all semi-standard orbits (prop. \ref{min}).This observation allows the reformulation of the conjecture given in \S4.3. In the remainder of the introduction we make some preliminary remarks. \subsection{Preliminary remarks} \begin{enumerate} \item If $k=0$, conjecture 1 is just the Harbourne-Hirshowitz conjecture for the natural cohomologie of $H$. For $k>0$, it corresponds to the surjectivity of the canonical map \begin{enumerate}gin{equation}\langlebel{rest}H^0(X_r,\fasm O(H))\longrightarrow H^0(X_r,\fasm O_Z(H)) \end{equation} for all $k$-clusters, so that it is equivalent to $|H|$ being base point free (resp. very ample) when $k=1$ (resp. $k=2$). Note that in the latter cases ($k=1,2$) we only require that $H.E\leq 2a-1+k$ for $0\leq a\leq k$. \item The conjecture is true for $r\leq 9$. This is best seen using the equivalence of conjectures 1 and 2. When $k=0,1,2$ this was proven in \cite{Har2} Harbourne and, for $k\geq 3$, $r\leq 8$ in \cite{Di Roc} Di Rocco, but any reader will see for themselves that her proof using the general vanishing theorem [S] goes over to $r=9$. \item The condition $\gchi (X_r,\fasm O(H))$ is the natural condition for a {\em general} $H$ to separate $k$-clusters in the following sense. Consider the canonical diagramme $$\begin{enumerate}gin{diagram}[size=2em,postscript] Z&\rInto&\mbox{Hilb}^k(X_r/\Lambda)\times_{\Lambda}X_r&\rTo ^{p}&X_r\\ &&\dTo_q&&\\ && \mbox{Hilb}^k(X_r/\Lambda)&&\end{diagram}$$ where $Z$ is the tautological subscheme. We then have the canonical vector bundle map $$H^0(X_r,\fasm O(H))\otimes\fasm O_{\mbox{Hilb}^k(X_r)}\longrightarrow q _{\star} \fasm O_Z(p ^{\star} H) $$ giving the map (\ref{rest}) at each point of the Hilbert scheme. The left hand side has rank $n=h^0(X_r,\fasm O(H))$ and the right hand side has rank $k$, so that the degeneracy locus is empty or has codimension $\leq n-k+1$. If the map is to be nowhere degenerate in general (i.e. without further numerical conditions being applied) then we need $n-k+1>\mbox{dim Hilb}^k(X_r)=2k$, giving the condition $h^0(X_r,\fasm O(H))\geq 3k$. In view of the Harbourne-Hirschowitz conjecture, we should have $h^1(X_r,\fasm O(H))=0$ and since $h^2(X_r,\fasm O(H))=0$ for an effective divisor on the rational surface $X_r$, this should be the same thing as $\gchi (X_r,\fasm O(H))\geq 3k$. \end{enumerate} \section{About isolated curves} \subsection{Harbourne-Hirschowitz and the structure of effective classes} \begin{enumerate}gin{defn}\langlebel{isolated}{\em An effective divisor $E$ on $X_r$ satisfying $E ^2 = a-1=E.K$; $a\geq 0$; will be called an isolated curve of genus $a$. {\em Note that exceptional curves are just isolated curves of genus zero}.}\end{defn} If one supposes that H-H holds, one gets a precise description of the effective classes. In particular, one gets that for $a\geq 2$, an isolated curve of genus $a$ is reduced and irreducible, and isolated in its linear system. \begin{enumerate}gin{prop}(Bossini)\langlebel{str} If H-H holds, then for any effective divisor class $H$ on $X_r$, the generic curve $C$ in $|H|$ can be decomposed as an orthogonal sum $A+n_1F_1+\cdots +n_rF_r$, where each $F_i$ is an exceptional curve, $n_i>0$ and $A$ is effective and satisfies $p(A)\geq 0$, $A^2\geq p(C)-1$ and $A ^2\geq 0$ if $p(A)=0$; $p$ being the arithmetic genus. If $A ^2>0$ then $A$ is reduced and irreducible. If $A ^2=0$ then we have one of the following two cases \begin{enumerate} \item $A ^2=0$, $A.K=-2b$ ($b>0$) in which case $A=bE$, where $E$ is a pencil class (definition \ref{classes}) \item $A^2=0=A.K$ in which case $A=mE$ ($m>0$) where $E$ is a smooth irreducible elliptic classe which, with respect to a suitable exceptional configuration (see \S~\ref{excep}), can be written $E\equiv 3E_0-E_1-\cdots -E_9$. \end{enumerate} \end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ One can write $C=A+\sum_{i=1}^s n_iE_i$, where the $E_i$ are the reduced, irreducible components of $C$ which are rational of self-intersection $<0$. . Now, it is known that the only integral rational curves on $X_r$ having self-intersection $<0$ are the exceptional curves. In fact if $W$ is such a curve with $W^2\leq 2$, then $W.(-K)<0$ and, specialising to the situation where all points are generic on a smooth plane cubic, $W$ must specialise to an effective divisor having a smooth elliptic component, but this is impossible for $W$ rational. We conclude that each $E_i$ is exceptional. Since each $E_i$ is isolated we have $E_i.E_j=0$ otherwise $\mbox{dim}|E_i+E_j|>0$ and the generic curve in $|E_i+E_j|$ does not decompose. As well $A$ is non-special by H-H, because it is positive on all exceptional curves. The isolation of $\sum_{i=1}^s n_iE_i$ in $|H|$ gives $h^0(X_r,\fasm O(A))=h^0(X_r,\fasm O(C))=h^0(X_r,\fasm O(A+E_i))$ which shows that $h^0(E_i,\fasm O_{E_i}(A+E_i))=0$ and finally that $A.E_i\leq 0$, but $A.E_i\geq 0$ in any case. Now if $A=A_1+A_2$ where $A_2$ is a reduced and irreducible curve and $A_1$ is an effective divisor, then $A_2$, not being rational of negative self-intersection, is non-special by H-H. This implies that $A_2^2\geq p(A_2)-1\geq 0$ if $p(A_2)>0$ and $A_2^2\geq 0$ if $p(A_2)=0$, where $p$ denotes the arithmetic genus. If $A_2^2=0$ then $A_2$ is a pencil class or an isolated integral elliptic class. Since this is true for all integral components $A_2$ of $A$, we conclude that $A ^2\geq 0$ and even $A ^2\geq p(A)-1$ (using the standard formulas for $p$). As such $A ^2\geq 0$ and $A ^2>0$ unless $A$ is a pencil class or an isolated elliptic class. Now using the fact that $A_1$ and $A_2$ are non-special (H-H), and the surjectivity of the map $|A_1|\times|A_2|\longrightarrow|A_1+A_2|$, we find by H-H $$\begin{array}{lll} \mbox{dim}\, |A_1|+\mbox{dim}\, |A_2|&\geq &\mbox{dim}\, |A_1+A_2|\\ &=&\gchi (X_r,\fasm O(A_1+A_2))-1\\ &=&\gchi (X_r,\fasm O(A_1))-1 +\gchi (X_r,\fasm O(A_2))-1 +A_1A_2\\ &=&\mbox{dim}\, |A_1|+\mbox{dim}\, |A_2|+A_1.A_2\\ \end{array}$$ so that $A_1.A_2=0$. By the algebraic index theorem, since we know that $A_i ^2\geq 0$, $i=1,2$, we must have $A_1^2=A_2^2=A_1.A_2=0$ and $C=bA_2$ (by the Cauchy-Schwartz inequality after writing $A_1$ and $A_2$ in an exceptional configuration) with $A_2$ a pencil class, or $A_2$ an isolated integral elliptic class, hence equivalent to $3E_0-E_1-\cdots -E_9$ by lemma \ref{bossi}.\end{proof} It is conjectured that all isolated curves $E$ of genus $a\geq 2$ are in fact smooth as well (see remark~\ref{ice}). \subsection{The ample classes} \begin{enumerate}gin{prop} If H-H holds then a divisor class $H$ on $X_r$ is ample if and only if $H^2>0$ and $H.E>0$ for all exceptional curves $E$ on $X_r$. If $H$ is standard and $H ^2>0$, and $H.E\leq 0$ for some reduced and irreducible curve $E$ on $X_r$, then $E$ is exceptional and $H.E=0$.\end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ If $H$ is standard then $h^2(X_r,\fasm O(H))=0$ and the condition $H^2>0$ shows that $aH$ is effective for $a>\!\!>0$. We can thus suppose $H$ effective, hence reduced and irreducible by (\ref{str}). If $E$ is a reduced and irreducible curve then $H.E\geq 0$, but if $H.E=0$ then $E^2 < 0$ by the algebraic index theorem and by (\ref{str}) again, this implies that $E$ is exceptional. \end{proof} In this direction one might consult \cite{Xu}. \section{The conditions of the conjecture} \subsection{The conditions $H.E\geq 2a-1+k$, $a\leq k$ and $H.E\geq a-1+2k$, $k<a\leq \frac{4}{3}k$ of the conjecture} Throughout this section let $H$ be an effective divisor class on $X_r$ ($r\geq 3$) such that $\gchi (\fasm O(H))\geq 3k$ and $H.F\geq k-1$ for all exceptional curves $F$ sur $X_r$. It has already been said that the underlying assumption in conjecture 1. is that if $|H|$ does not separate $k$-clusters then this because it does not separate some $k$-cluster in an integral (even smooth) curve on $X_r$. It is also supposed that if the map \begin{enumerate}gin{equation}\langlebel{reste}\rho : H^0(X_r,\fasm O(H))\longrightarrow H^0(X_r,\fasm O_E(H))\end{equation} has rank $\geq 2k$ then $|H|$ separates $k$-clusters in $E$. We will see that if H-H holds and $\fasm O_E(H)$ is special, or if $H-E$ is effective and special, then (\ref{reste}) always has rank $\geq 2k$. In the remaining cases (\ref{reste}) is surjective and there are two possibilities. Firstly, if $k>a$ then it is well known that $\fasm O_E(H)$ separates $k$-clusters if and only if $H.E\geq 2a-1+2k$, but in this case $h^0(\fasm O_E(H))$ can be less than $2k$. Secondly, if $k\leq a$, a necessary condition for $\fasm O_E(H)$ to separate $k$-clusters is that $H.E\geq a-2+2k$, and this is sufficient if $\fasm O_E(H)$ is a general invertible sheaf of degree $H.E$. We are thus supposing in the conjecture that $\fasm O_E(H)$ is general in this sense. This said we must prove the \begin{enumerate}gin{prop} Suppose that H-H holds. Then the image of (\ref{reste}) has dimension $<2k$ only if $\fasm O_E(H)$ is non-special. When $\fasm O_E(H)$ is non-special, $a>k$ and $H.E\leq a-2+2k$ one has $4k\geq 3a$.\end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ Firstly, if H-H holds, $E$ is non-special. There are two more or less obvious cases where the image of (\ref{reste}) has dimension $\geq 2k$. These are when $H-E$ is not effective (obvious) and when $H-E$ is effective and special. In fact, in the second case, if H-H holds, then by (\ref{str}) we can write $H-E=A+\sum_i n_iF_i$ as an orthogonal sum of an effective standard divisor $A$ and multiples $n_i>0$ of exceptional curves $F_i$ with at least one of the $n_i\geq 2$. Since $k-1\leq H.F_i=E.F_i-n_i$ one finds $E.F_i\geq n_i+k-1$ and $$\begin{array}{lll} & & h^0(X_r,\fasm O(H))-h^0(X_r,\fasm O(H-E)) \\ &= & \overline{\gchi} (H)-\overline{\gchi} (A) \\ &= & \overline{\gchi} (A)+\overline{\gchi} (E+\sum_i n_iF_i)+A.E- \overline{\gchi} (A) \\ &= & -\sum_i n_i(n_i-1)/2+\sum_i n_iE. F_i +A.E \\ &\geq& \sum_i n_i(n_i-1)/2+k( \sum_i n_i)+A.E \\ &\geq & 2k \end{array}$$ as required. Moreover, since $H.F\geq 0$ for all exceptional curves $F$ on $X_r$, $H$ is non-special. Also if $\fasm O_E(H)$ is special, $h^2(\fasm O_E(H-E))>0$, implying that $H-E$ is not effective, so that the image of (\ref{reste}) is at least $2k$ dimensionl in this case as well. We can thus suppose henceforth that $H-E$ and $\fasm O_E(H)$ are effective and non-special and we will show that if $a>k$ and $H.E\leq a-2+2k$ one has $4k\geq 3a$. \noindent{\bf Claim.} In this case, $H-3E$ is not effective, $\fasm O_E(H-E)$ is non-special, and $\fasm O_E(H-2E)$ is special. This gives the desired inequality as follows using Riemann-Roch and Cliffird's theorem $$\begin{array}{lll}\frac{1}{2} (E.(E-2H))+1&\geq & h^0(\fasm O_E(H-2E))\\ &\geq & h^0(\fasm O(H-2E)) \\ &\geq & h^0(\fasm O(H))- h^0(\fasm O_E(H))- h^0(\fasm O_E(H-E))\\ &=& h^0(\fasm O(H))-2E.H+E ^2 +2(a-1)\end{array}$$ so that $$5E.H +2\geq 2h^0(\fasm O(H))+4E ^2+4(a-1)$$ and using $E ^2\geq a-1$ (\ref{st}) and $a-2+2k\geq H.E$, we get $4k\geq 3a$. \noindent{\bf Proof of the claim.} Since $(H-3E).E = H.E-3E ^2 \leq a-2+2k-3(a-1)=2(k-a)+1<0$ and $E ^2\geq 0$ it follows that $H-3E$ is not effective. If $\fasm O_E(H-E)$ is special, then $h^2(\fasm O(H-2E))=h^1(\fasm O_E(H-E))\neq 0$, so that $H-2E$ is not effective and by Clifford's theorem $$k+1\leq h^0(\fasm O(H-E))\leq h^0(\fasm O_E(H-E))\leq\frac{1}{2}(E.H-E ^2)+1\leq\frac{1}{2} (a-2+2k-a+1)+1= k+\frac{1}{2}$$ From this contradiction we conclud that $\fasm O_E(H-E)$ is non-special. Finally, $$\begin{array}{lll}\overline{\gchi} (\fasm O(H-3E))&=& \overline{\gchi} (\fasm O(H))-\overline{\gchi}(\fasm O(3E))-3E.(H-3E)\\ &=& \overline{\gchi} (\fasm O(H))-3\overline{\gchi}(\fasm O(E))+6E ^2-3E.H\\ &\geq & 3k+6(a-1)-3(a-2+2k)\\ &=& 3(a-k) \; >\; 0 \end{array}$$ so that $h^2(\fasm O(H-3E))>0$ and $\fasm O_E(H-2E)$ is special. \end{proof} \section{\langlebel{excep}Translation into standard divisors} As we will see below, standard divisors are essentially those that test positive on all exceptional curves and include all integral classes that are not exceptional. The more general semi-standard classes include all efective classes on an $X_r$. What we want here is threefold: (1) an intrinsic characterisation of the (semi)-standard classes, (2) a standard way of writing such a class with respect to a suitable exceptional configuration, (3) the unicity of this expression (but not of the exceptional configuration). I have found no reference in the literature to parts (1) and (3), but (2) has been used extensively since Noether. We thus begin by defining the classic $\bm E$-(semi)-standard classes for a fixed exceptional configuration $\bm E$ (definition~\ref{standard}), and then show that this is equivalent to an intrinsic definition (proposition~\ref{st}). Finally in proposition~\ref{es}, we deal with the unicity via a minimising characterisation of the exceptional configurations that give $\bm E$-(semi)-standard representations of (semi)-standard classes. This gives corollary~\ref{min} which is essential for the reformulation of the conjecture. Before giving the definitions let us recall \cite{Hir2} that the Weyl groupe $W_r$ of the surface $X_r$ (see \cite{Har3} and \cite{Dol}) can be viewed as a groupe of $k$-automorphismes of $X_r$. There is an induced action of $W_r$ on $\mbox{Pic}(X_r)$ which stabilises the intersection form and fixes the canonical class. The latter action is faithful so that $W_r$ is identified with a subgroup of the orthogonal group $O_r$ of $(\mbox{Pic}(X_r), \langle\; ,\; \rangle)$ where $\langle\; ,\; \rangle$ is the intersection form. This induced action on $\mbox{Pic}(X_r)$ is simply transitive on the {\em exceptional configurations} which are by definition the orthogonal sequences of effective, irreducible classes $$\bm E=(E_0,E_1,\ldots ,E_r)$$ satisfying $$E_0 ^2=1\quad , \quad E_0.K_r=-3\quad , \quad E_j ^2=-1=E.K_r\quad ,\quad E_i.E_j=0\quad , \quad 0\leq i<j\leq r$$ In particulier $E_i\simeq \sbm P^1$ for $i=0,1,\ldots ,r$. For any fixed exceptional configuration $\bm E$ as above, the group $W_r$ is generated by the orthogonal reflexions $\sigma_i=\mbox{id}+\langle r_i,\; \rangle r_i$ where $$r_0=E_0-E_1-E_2-E_3,\hspace{1ex} r_i=E_i-E_{i+1},\hspace{1ex} i=1,\ldots ,r-1$$ Central to the proof of these results is Noether's inequality (see \cite{Dol} § 5), which says that if $r\geq 3$, then a sequence of non-negative integers $d,m_1\geq m_2\geq \cdots \geq m_r$ satisfying $$d^2-m_1^2-\cdots -m_r^2=-1=-3d+m_1+\cdots +m_r$$ also satisfies $d< m_1+m_2+m_3$. This inequality can also be used to show that for $r\geq 3$ any irreducible rational class $E$ (i.e. $E$ is irreducible, effective and $E ^2=-2-E.K_r\geq -1$ lies in the $W_r$ orbit of one and one only of the following classes \begin{enumerate}gin{equation}\langlebel{rc}E_r,\hspace{.25ex}E_0-E_1,\hspace{.25ex}E_0,\hspace{.25ex}2E_0,\hspace{.25ex} dE_0-(d-1)E_1-E_2,\hspace{.25ex} dE_0-(d-1)E_1,\quad d\geq 2\end{equation} Note that these are uniquely determined by $E ^2$ except for the pair $2E_0$ and $3E_0-2E_1-E_2$. \begin{enumerate}gin{defn}\langlebel{classes}{\em An irreducible rational classe $E$ on $X_r$ with $E ^2=0$ (resp. $E ^2=1$, resp. $E ^2=2$) will be called a pencil class (resp. line class, resp. quadratic class). Note that this is $W_r$-equivalent to $E_0-E_1$ (resp. $E_0$, resp. $2E_0-E_1-E_2$).} \end{defn} It is also true that $W_r$ acts transitively on the orthogonal sequences of exceptional curves of length $\neq r-1$. When the length is $r-1$ there are two orbits which, for a fixed exceptional configuration $\bm E=(E_0,E_1,\ldots ,E_r)$, have representatives of the form $(E_2,\ldots ,E_r)$ et $(E_0-E_1-E_2,E_3,\ldots ,E_r)$. These correspond respectively to sequences of length $r-1$ that can be extended and those that cannot. If we forget the ordering, then the non-extendable orthogonal sequences of exceptional curves correspondent to quadratic classes $E$ which induce birational morphismes of $X_r$ to a smooth quadrique $Q$ in $\sbm P ^3$, contracting the exceptional curves in such a non-extendable sequence. We define standard classes as follows \begin{enumerate}gin{defn}\langlebel{standard} {\em A divisor class $H$ on $X_r$ is said to be $\bm E$-{\em standard} for an exceptional configuration $\bm E=(E_0,E_1,\ldots ,E_r)$ if $$H\equiv dE_0-m_1E_1-\cdots -m_rE_r$$ where $d\geq m_1\geq \cdots \geq m_r\geq 0$, $d\geq m_1+m_2$ and $d\geq m_1+m_2+m_3$. We say that $H$ is $\bm E$-{\em semi-standard} if $d\geq 0$, $d\geq m_1+m_2+m_3$ et $d\geq m_1\geq m_2\geq \cdots \geq m_r$ (but some $m_i$ may be $<0$). We will say that a divisor class $H$ on $X_r$ is {\em standard} (resp. {\em semi-standard}) if it is $\bm E$-standard (resp. $\bm E$-semi-standard) for some exceptional configuration $\bm E$ on $X_r$.} \end{defn} \begin{enumerate}gin{exmp} The class $5E_0-2E_1-2E_2-2E_3$ is not $\bm E$-standard, but is standard as one can see by making the standard quadratic transformation by the reflection with root $r_0$.\end{exmp} \subsection{generating classes} It is obvious and was pointed out by Harbourne that the $\bm E$-standard classes are precisely those that can be expressed as non-negative sums \begin{enumerate}gin{equation}\langlebel{gcp}H\equiv aE_0 + b(E_0-E_2)+b(2E_0-E_1-E_2)+\sum_{i=3}^r\alpha_iC_i\end{equation} where $a,b,c,\alpha_i\geq 0$ and $C_i=-K_r+E_{i+1}+\cdots +E_r$ ($i\geq 3$). \begin{enumerate}gin{defn}{\em We say that the classes $E_0$, $E_0-E_1$, $2E_0-E_1-E_2$, $C_i$ ($i\geq 3$) are the (standard) {\em generating classes} of the exceptional configuration $\bm E$. The first three generating classes are called the rational generating classes while the $C_i$, which have arithmetic genus one, are called the elliptic generating classes.}\end{defn} \begin{enumerate}gin{rem} {\em As we saw above the rational generating classes of self intersection 0,1,2 each form a single orbit under the Weyl group action. The same is true for the elliptic generating classes for each self intersection number $\leq 6$ since $W_r$ fixes $K_r$ and acts transitively on the orthogonal sequences of exceptional curves of length $\neq r-1$.} \end{rem} \begin{enumerate}gin{lem}\langlebel{bossi}(Bossini) If $C$ is an $\bm E$-standard class satisfying $C^2=0=C.K_r$, then $C\equiv \alpha_9C_9$. \end{lem} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ The class $C$ is effective and it is clear that $C.E>0$ on all rational generating classes and $C.C_i\geq 0$ since $C.C_r=C.-K_r=0$. As well $C.C_i>0$ for $i\leq 8$ since any effective divisor that is a proper component of a curve in $|C_i|$ is rational. As such, $C\equiv \sum_{i\geq 9} \alpha_i C_i$. If $C\neq \alpha_9C_9$ then $C^2<0$.\end{proof} \subsection{semi-standard classes} An $\bm E$-semi-standard class $H$ has one of two forms reflecting the two orbits of orthogonal sequences of exceptional curves of length $r-1$. In fact, either $H=(dE_0-m_1E_1-\cdots -m_sE_s)+n_{s+1}E_{s+1}+\cdots +n_rE_r$ where the first part is standard and $n_i=-m_i>0$, or $m_3<0$, $m_1,m_2\geq 0$, $d < m_1+m_2$ and $$\begin{array}{lll} H&\equiv&(2d-m_1-m_2)E_0-(d-m_2)E_1-(d-m_1)E_2+(m_1+m_2-d)(E_0-E_1-E_2)\\ &&\; + n_3E_3+\cdots +n_rE_r\end{array}$$ where $n_i=-m_i>0$. As such any $\bm E$-semi-standard class, can be written in the form $A+\sum_i n_iF_i$, where $A$ is $\bm E$-standard, $n_i>0$ and the $F_i$ are a familly of orthogonal exceptional curves. \begin{enumerate}gin{prop} \langlebel{st} For $r\geq 2$, a divisor class $H$ on $X_r$ is standard (resp. semi-standard) if and only if $H.E\geq 0$ for all exceptional curves $E$ on $X_r$ (resp. $H.E\geq 0$ for all line and pencil classes $E$ on $X_r$). A classe $H$ is semi-standard if and only if it is an orthogonal sum of the form $H=A+\sum n_iF_i$, where $A$ is standard, $n_i>0$, and each $F_i$ is an exceptional curve. \end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ Firstly, if $H$ is non-negative on all exceptional classes, then choosing any line class $E_0$, we obtain an exceptional configuration $\bm E$, with $H=dE_0-m_1E_1-\cdots -m_rE_r$, $m_i\geq 0$ and $(d-m_1-m_2)=H.(E_0-E_1-E_2)\geq 0$, so that $H.E_0=d\geq 0$ on all line classes $E_0$. Suppose then that $H$ is non negative on all line classes. In this case we can choose a line class $E_0$ such that $H.E_0=d$ is the minimum for all line classes on $X_r$ and by suitably ordering the exceptional curves contracted by $E_0$ we obtain an ecxceptional configuration $\bm E=(E_0,E_1,\ldots ,E_r)$ with $$H=dE_0-m_1E_1-\cdots -m_rE_r$$ $m_1\geq \cdots \geq m_r$ and, if $r\geq 3$, $2d-m_1-m_2-m_3=H.(2E_0-E_1-E_2-E_3)\geq H.E_0=d$ by the minimality of $H.E_0$ over all line classes. When $H$ is positive on all exceptional curves we have $m_r\geq 0$, $d\geq m_1+m_2\geq m_1$ and when $H$ is positive on all pencil classes we have $d\geq m_1$. Conversely, if $H=dE_0-m_1E_1-\cdots -m_rE_r$ is $\bm E$-standard, we can write $$H\equiv aE_0 + b(E_0-E_2)+b(2E_0-E_1-E_2)+\sum_{i=3}^r\alpha_iC_i$$ as in (\ref{gcp}) and it suffices to note that each of the generating classes is positive on all exceptional curves. If $H$ is $\bm E$-semi-standard, then as we saw above, $H$ is an orthogonal sum $H\equiv A+\sum_in_iF_i$ where $A$ is $\bm E$-standard, $n_i>0$ and the $F_i$ are exceptional. Conversely every such class is non-negative on all line and pencil classes. \end{proof} \begin{enumerate}gin{cor}\langlebel{effst} On $X_r$ every effective class is semi-standard and every integral class that is not exceptional is standard.\end{cor} \begin{enumerate}gin{rem}\langlebel{ice}{\em We have defined isolated curves of genus $a\geq 1$ to be reduced and irreducible curves $E$ with $E ^2=a-1=E.K_r$. The preceeding proposition says that if H-H holds then for $a\geq 2$, we could define these to be standard classes with $E ^2=a-1=E.K_r$. The isolated curves of genus $a\leq 4$ were completely classified in \cite{Bos} and \cite{Mig}. The only standard classe with $E ^2=0=E.K_r$ is $C_9$, while the only standard classes with $E ^2=1=E.K_r$ are \begin{enumerate}gin{equation}\langlebel{isol}\begin{array}{lll}G_1&=&4E_0-2E_1-E_2-\cdots -E_{12}\\ G_2&=&6E_0-2E_1-\cdots -2E_8-E_9-E_{10}-E_{11}\\ G_3&=&9E_0-3E_1-\cdots -3E_8-2E_9-2E_{10}\end{array}\end{equation} and it has been proven that for $a=2,3,4$ all standard classes with $E ^2=a-1=E.K_r$ contain a unique curve and it is smooth and irreducible. }\end{rem} \begin{enumerate}gin{prop}\langlebel{es} Let $H\equiv dE_0-m_1E_1-\cdots -m_rE_r$ be an $\bm E$-standard class, then \begin{enumerate} \item for $i\geq 1$, $m_i+\cdots +m_r$ is the minimum value of $H.(G_i+\cdots +G_r)$ for all extendable orthogonal sequences of exceptional curves $G_i,\ldots ,G_r$. If $i\geq 3$ this is the minimum value on all sequences of exceptional curves, orthogonal or not. \item $d-m_1-m_2+m_3+\cdots +m_{r}$ is the minimum value of $H.(G_2+\cdots +G_r)$ on all unextendable orthogonal sequences of exceptional curves $G_2,\ldots ,G_r$. \end{enumerate} In particular, if $H$ is a standard class which can be written in the $\bm E$-standard form $dE_0-m_1E_1-\cdots -m_rE_r$, then the sequence $d,m_1,\ldots ,m_r$ is unique, independent of the exceptional configuration $\bm E$ in which $H$ is $\bm E$-standard. \end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ As we saw in (\ref{gcp}), we can write $H$ uniquely in the form $$H\equiv aE_0 + b(E_0-E_2)+c(2E_0-E_1-E_2)+\sum_{i=3}^r\alpha_iC_i$$ For $i\geq 3$ we have $m_i=\alpha_i+\cdots +\alpha_r$. If $E$ is an exceptional curve $$H.E\geq \alpha_i E.C_i+\cdots + \alpha_r E.C_r$$ and $E.C_i=1+E(E_{i+1}+\cdots +E_r)$, so that if $H.E < m_i$ then $E=E_j$ for some $j>i$. This shows that for {\em all} sequences $G_i,\ldots ,G_r$ ($3\leq i\leq r$) of exceptional curves, $H.(G_i+\cdots +G_r)\geq H.(E_i+\cdots +E_r)$. If $E$ is an exceptional curve and $m_3<H.E< m_2=m_3+c$ then $c>0$, $E. (2E_0-E_1-E_2)=0$ et $E\neq E_i$ for $i=3,\ldots ,r$. As such $E=E_0-E_1-E_2$. It follows that for extendable (resp. unextendable) sequences $G_2, \ldots , G_r$ of orthogonal exceptional curves, the minimum value of $H.(G_2+\cdots +G_r)$ is attained on the sequence $G_i=E_i$; $i=2,\ldots ,r$ (resp. $G_2=E_0-E_1-E_2$, $G_i=E_i$; $i=3,\ldots ,r$). If $E$ is an exceptional curve and $m_2<H.E< m_1=m_2+b$, then $b>0$, $E\neq E_j$; $j=2,\ldots ,r$; et $(E_0-E_1)E=0$. In this case $E=E_0-E_1-E_j$ for some $j>1$ and $E,E_2,\ldots ,E_r$ is not an orthogonal sequence. It follows that for orthogonal sequences of length $r$, $G_1,\ldots ,G_r$ of exceptional curves, the minimum value of $H.(G_1+\cdots +G_r)$ is attained on the sequence $(E_1,\ldots ,E_r)$. \end{proof} \begin{enumerate}gin{cor} The decomposition of a semi-standard class $D$ as an orthogonal sum $A+\sum_i n_iF_i$, where $A$ is standard, the $F_i$ are exceptional and $n_i>0$, is unique.\end{cor} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ If $B+\sum_i m_iG_i$ is another such decomposition, then, $B$ being standard, $D.F_i<0$ implies that $F_i.G_j<0$ for some $j$. Since the $F_i$ and $G_i$ are smooth and irreducible classes, we can suppose that $F_i=G_i$, then $n_i=D.F_i=m_i$.\end{proof} \begin{enumerate}gin{prop}\langlebel{min} Fix an exceptional configuration $\bm E=(E_0,E_1,\cdots ,E_r)$ on $X_r$. Let $H\equiv dE_0-m_1E_1-\cdots -m_rE_r$ be an $\bm E$-standard divisor and let $D$ be a semi-standard divisor on $X_r$. Then for $\sigma\in W_r$, the minimum value of $H.\sigma(D)$ occurs exactly when $\sigma(D)$ is $\bm E$-semi-standard. Otherwise said, if $\bm F=(F_0,F_1,\cdots ,F_r)$ is an exceptional configuration for which $D = d^{\prime} F_0 - m^{\prime}_1 F_1-\cdots - m^{\prime}_r F_r$ is $\bm F$-semi-standard and $D^{\prime} = d^{\prime} E_0 - m^{\prime}_1 E_1-\cdots - m^{\prime}_r E_r$ then $$H.D\geq H.D^{\prime}$$ \end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ Write $H$ as $$H\equiv aE_0 + b(E_0-E_2)+c(2E_0-E_1-E_2)+\sum_{i=3}^r\alpha_iC_i$$ and $D$ as an orthogonal sum (\ref{st}) $D=A+\sum_i ^r n_jG_j$ ($i\geq 2$) where $A$ is $\bm F$-standard, $n_i>0$ and $G_i,\ldots ,G_r$ is an orthogonal sequence of exceptional curves with either $F_j=E_j$ for $j=i,\ldots ,r$, or $i=2$, $G_2=F_0-F_1-F_2$, $G_j=F_j$ for $j=3,\ldots ,r$. By (\ref{es}), the proposition holds when $A=0$, so it suffices to prove the proposition for $D$ standard and even for $D$ one of the $\bm F$-standard generating classes. By (\ref{es}), the proposition holds for $D=-K_r-E_i-\cdots -E_r$ ($i\geq 3$) and by the symmetry of the proposition it suffices to suppose that $H$ (resp. $D$) is one of the rational generating classes. The system $|E_0|$ (resp. $|2E_0-E_1-E_2|$) is positive on all effective classes other than exceptional curves, so we need only show that $(2E_0-E_1-E_2).F_0\geq 2$ and $(2E_0-E_1-E_2).(2F_0-F_1-F_2)\geq2$. This follows from the fact that the only effective divisors with $(2E_0-E_1-E_2).G\leq 1$ move in a linear system of projective dimension at most one.\end{proof} When $r\leq 1$, there is only one exceptional configuration so that $\bm E$-standard and standard are equivalent. \begin{enumerate}gin{rem}\langlebel{minrat} {\em It follows from proposition~\ref{es} that if $r\geq 3$ then $H.E_r$ is the minimum value of $H.E$ on all exceptional curves $E$. It then follows from proposition~\ref{min} and the list of effective $\bm E$-standard rational classes~(\ref{rc}), that $H.E_r$ is the minimum on all effective rational classes.}\end{rem} \subsection{reformulation of the conjecture} \begin{enumerate}gin{defn}\langlebel{nd}{\em For an exceptional configuration $\bm E=(E_0,E_1,\ldots ,E_r)$ on $X_r$ we extend the definition of $\bm E$-standard class to include, for any $\delta\geq 0$, those classes $E$ on $X_{r+\delta}$ which for the canonical extension $\bm E^{\prime}=(E_0,E_1,\ldots ,E_r,E_{r+1},\ldots ,E_{r+\delta})$ of the exceptional configuration to $X_{r+\delta}$ is $\bm E^{\prime}$ standard on $X_{r+\delta}$. An $\bm E$-standard isolated curve of genus $a$ will then be an integral $\bm E$-standard class (for this extended definition) $E$ such that $E ^2=a-1=E.K_{r+\delta}$.}\end{defn} \begin{enumerate}gin{rem}{\em Note that if $H$ is a semi-standard or standard class on $X_r$ then so is its pull back (also denoted by $H$) to $X_{r+\delta}$. }\end{rem} We can now reformulate conjecture 1. in the following form \noindent{\bf Conjecture 2.} {\em Let $k>0$ be an integer and let $\bm E=(E_0,E_1,\ldots ,E_r)$ be an exceptional configuration on $X_r$; $r\geq 3$; and let $H\equiv dE_0-m_1E_1-\cdots -m_rE_r$ be an $\bm E$-standard class satisfying $\gchi (X_r,\fasm O(H))\geq 3k$ and $m_r\geq k-1$, then $H$ is non-special and separates $k$-clusters if the following necessary conditions are satisfied : $H.E\geq 2a-1+k$ for all $\bm E$-standard isolated curves of genus $a$ ($1\leq a\leq k$)and $H.E\geq a-1+2k$ for all $\bm E$-standard isolated curves of genus $a$ ($\frac{4}{3}k\geq a >k$)} \begin{enumerate}gin{prop} Conjecture 2 is equivalent to conjecture 1. \end{prop} \begin{enumerate}gin{proof}$\!\!\!\!\!.$ We have already noted in~\ref{minrat}, that for $r\geq 3$, $H.E_r$ is the minimum value of $H.E$ for all effective rational classes on $X_r$. It therefore suffices to show that if the stated condition holds on all $\bm E$-standard isolated curves of genus $a$, $1\leq a\leq k$ and $\frac{4}{3}k\geq a >k$, then it holds for all integral curves of such genus. An integral curve $F$ on $X_r$ of genus $a\geq 1$ is standard by~\ref{effst} and $H.F$ is minimum when $F$ is $\bm E$-semi-standard. If $F$ is $E$-standard, but not isolated, then letting $j=\mbox{min}\left\{i|1\leq i\leq r \mbox{ and } F.E_i=0\right\}$, the class $F^{\prime} =F-E_i-\cdots -E_{i+s}$ is an $\bm E$-standard isolated curve of genus $a>0$ for some $s\geq 0$ and satisfies $F^{\prime}.H\leq F.H$. Hence $H.F$ takes its minimum value on the $\bm E$-standard isolated curves of genus $a$.\end{proof} \begin{enumerate}gin{rem}\begin{enumerate} \item {\em To see that one does have to test in the range $\frac{4}{3}k\geq a>k$ when $k\geq 3$, consider $H=13E_0-9E_1-2E_2-\cdots -2E_{18}$ which is constructed from the isolated hyperelliptic curve of genus 4, $E=6E_0-4E_1-E_2\cdots -E_{18}$. We have $\gchi (\fasm O(H))=3.3$ and $H.E=8<4-1+2.3$.} \item {\em The following example shows that it is necessary to consider isolated curves which may not lie on $X_r$. It suffices to consider $kC_8$ on $X_8$. For any $k\geq 6$, $\gchi (kC_8)\geq 3k$, but $(kC_8).C_8=k<k+1$. However with the definitions introduced in (\ref{nd}) this can be detected on the isolated curve $C_9$ on $X_9$.} \end{enumerate} \end{rem} \section{Further motivation} Here we show that the general adjunction theorems imply the conjecture under much heavier restrictions on $H$. Recall the \begin{enumerate}gin{prop}(see \cite{B-S}, Theorem 2.1) Let $H$ be a divisor class on a smooth surface $S$ such that $H-K$ is nef and big and $ (H-K)^2\geq 4k+1$. Then $H$ separates $k$-clusters unless there exists an effective divisor $D$ on $S$ of arithmetic genus $p$ such that $H-K-2D$ is effective, \begin{enumerate}gin{equation}\langlebel{B-S} H.D \leq 2p-2+k\quad\mbox{ and }\quad 2p-2+D^2<H.D <2k+D.K\end{equation} \end{prop} Now suppose that H-H holds and let $H$ be a divisor class on $X_r$ such that $\gchi (X_r, \fasm O(H))\geq 3k$, $H.F\geq k-1$ for all exceptional curves and $H.F\geq k+1$ for all integral elliptic curves $F$. Suppose further that $H-K_r$ is nef and big and that $(H-K_r)^2\geq 4k+1$. Let $D$ be an effective divisor on $X_r$ satisfying (\ref{B-S}) so that $D=E+\sum_{i=1}^s n_iF_i$ is an orthogonal sum of a standard class $E$ and multiples $n_i>0$ of exceptional curves $F_i$. By (\ref{B-S}), we find $$ H.E +\sum_{i=1}^s n_iH.F_i\leq 2(p(E)-n_i(n_i+1)/2-1)-2+k$$ so that $H.E\leq 2p(E)-2$ and $E$ is neither elliptic nor rational, hence is integral of genus $a\geq 2$. As well, using the effectiveness of $H-K-2D$, we get $(H-K-2E-2(\sum_{i=1}^s n_iF_i)).E\geq 0$ so that $(H-K).E>2E ^2$. Now by the last part of (\ref{B-S}) $2k>(H-K).E+(H-K).(\sum_{i=1}^s n_iF_i)\geq 2 E ^2$, so that $k\geq a$ in accordance with the conjecture. \section{The not very ample standard classes} In this section we look at the $\bm E$-standard classes $H$ on $X_r$ with $m_r\geq 1$, $\gchi (X_r,\fasm O(H))\geq 6$ and $H.E< 2a-1+k$ for some isolated standard class $E$ of genus $a=1$ or $2$ for $k=1,2$; i.e. the standard classes which have sufficient sections and test positive on all exceptional curves, but are not base point free or not very ample because this is not the case for their restriction to an isolated curve of genus one or two. We will use the list (\ref{isol}) of isolated, genus two curves $G_i$. Let $H\equiv dE_0-m_1E_1-\cdots -m_rE_r$ be an $\bm E$-standard class with $m_r>0$ and $\gchi (X_r,\fasm O(H))\geq 6$. The only isolated $\bm E$-standard curve of genus one is $C_9=3E_0-E_1-\cdots -E_9$. To determine the required classes we can suppose that $r=9$. One easily sees that the standard classes with $H.C_9\leq 1$ and $\gchi(X_r,\fasm o(H))\geq 3$ is $H\equiv mC_9+E_9$ which has $\gchi=m$, and that those with $\gchi (X_r,\fasm O(H))\geq 6$ et $H.C_9\leq 2$, are $$H\equiv mC_9+E_9;\;\; H\equiv mC_9+2E_9;\;\; H\equiv mC_9+E_8+E_9$$ which have $\gchi$, $m+1$, $2m$ and $2m+1$ respectively. Let $E$ be an $\bm E$-standard isolated curves of genus $2$ (\ref{isol}) such that $H.E\leq 4$. If $H.E\leq 3$ then $H-4E$ is efective and negative on $E$ which is an ample divisor! We can thus suppose $H.E= 4$. In this case $H-2E$ is effective. If $H-3E$ is effective, then $H=3E+A$ with $1=E.A>A ^2$. As such, either $H=3E+F$ where $F$ is exceptional and $E.F=1$, or $H-3E$ is standard and isolated. In the; latter case, either $H=4E$ or $H=3E+A$ where $A$ is an isolated elliptic curve with $E.A=1$.(eg. $E=6E_0-2E_1-\cdots -2E_8-E_9-E_10-E_11$, $A=C_9$). If $H-3E$ is not effective, then $\overline{\gchi}(H)=5$ and $H=2E+A$ where $E.A=2$. There are two cases. In the first case, $A$ can be decomposed as a sum $A=A_1+A_2$ of reduced and irreducible divisors $A_i$ with $1=E.A_i>A_i ^2\geq -1$ so that the $A_i$ are exceptional or isolated and elliptic (eg. $E=6E_0-2E_1-\cdots -2E_8-E_9-E_{10}-E_{11}$ and $$A_1=C_9, A_2=E_{10}\quad \mbox{ or }\quad A_1=E_9,A_2=E_{10})$$ In the second case $H=2E+A$ where $A$ is reduced and irreducible, and $4=(E.A)^2>A^2\geq -1$, in which case, $A$ is an exceptional curve or an isolated curve of genus $a=$1,2 or 3. One has examples with ($E=G_1, A=C_9$), ($E=G_2, A=G_3$) and the pair $E=G_2$ $$A=12E_0-4E_1-\cdots -4E_8-2E_9-2E_{10}-2E_{11} $$ \begin{enumerate}gin{thebibliography}{ab0} \addcontentsline{toc}{chapter}{Bibliography} \bibitem{AHi} Alexander, J. and Hirschowitz, A. : {\em Polynomial interpolation in several variables.} J. Algebraic Geometry 4 (1995) 201-222. \bibitem{B-S} Baltrametti, M. and Sommese, A.J. : {\em The adjunction theory of complex projective varieties. } de Gruyter Expositions in Mathematics, 16. Walter de Gruyter \& Co., Berlin, 1995. \bibitem{Bos} Bossini, S. : Thése, Université de Nice 1989. \bibitem{CM1} Ciliberto, C. and Miranda, R. : {\em Degenerations of planar linear systems. }, Journal Reine Angew. Math., no. 501, (1998), 191-220. \bibitem{CM2} Ciliberto, C. and Miranda, R. : {\em Linear systems of plane curves with base points of equal multiplicity }, duke e-print math/9804018, to appear in Trans. Amer. Math. 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Ann. no. 272, (1985), 139-153. \bibitem{Har3} Harbourne, B. : {\em Complete linear systems on rational surfaces.} Trans. Amer. math. Soc., vol. 289, no. 1, (1985), 213-226. \bibitem{Mig}Mignon, T. : {\em These} Nice 1997. \bibitem {GLS} Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin : {\em Plane curves of minimal degree with prescribed singularities.} alg-geom/9704010 \bibitem{Xu} Xu, G. : {\em Ample line bundles on smooth surfaces. } J. reine angew. Math. 469, 199-209 (1995). \end{thebibliography} \begin{enumerate}gin{flushleft} James E. Alexander\\ Dept. de mathématiques\\ Faculté des Sciences\\ 2, bd Lavoisier\\ 49045 ANGERS, FRANCE $\,$\\ jea@univ-angers.fr \end{flushleft} \end{document}

\begin{document} \begin{center}{\large\bf Glauberman correspondents and extensions of nilpotent block algebras} \large\bf Lluis Puig and Yuanyang Zhou \end{center} \leqnsert\footins{{\cal S}criptsize 2000 {\leqt Mathematics Subject Classification}. Primary 20C15, 20C20 {\cal S}mallskip The second author is supported by Program for New Century Excellent Talents in University and by NSFC (No. 11071091).} \begin{abstract} The main purpose of this paper is to prove that the extensions of a nilpotent block algebra and its Glauberman correspondent block algebra are Morita equivalent under an additional group-theoretic condition (see Theorem 1.6); in particular, Harris and Linckelman's theorem and Koshitani and Michler's theorem are covered (see Theorems £7.5 and £7.6). The ingredient to carry out our purpose is the two main results in K\"ulshammer and Puig's work {\leqt Extensions of nilpotent blocks\/}; we actually revisited them, giving completely new proofs of both and slightly improving the second one (see Theorems £3.5 and £3.14). \end{abstract} \noindent{\bf\large 1. Introduction} \noindent{\bf 1.1.}\quad Let ${\cal O}$ be a complete discrete valuation ring with an algebraically closed residue field $k$ of characteristic $p$ and a quotient field ${\cal K}$ of characteristic 0. In addition, ${\cal K}$ is also assumed to be big enough for all finite groups that we consider below. Let $H$ be a finite group. We denote by ${\rm Irr}_{\cal K}(H)$ the set of all irreducible characters of $H$ over ${\cal K}$. Let $A$ be another finite group and assume that there is a group homomorphism $A\rightarrow {\rm Aut}(H)$. Such a group $H$ with an $A$-action is called an $A$-group. We denote by $H^A$ the subgroup of all $A$-fixed elements in $H$. Clearly $A$ acts on ${\rm Irr}_{\cal K}(H)$. We denote by ${\rm Irr}_{\cal K}(H)^A$ the set of all $A$-fixed elements in ${\rm Irr}_{\cal K}(H)$. Assume that $A$ is solvable and the order of $A$ is coprime to the order of $H$. By \cite[Theorem 13.1]{I}, there is a bijection $$\pi(H,\,A): {\rm Irr}_{\cal K}(H)^A \rightarrow {\rm Irr}_{\cal K}(H^A)$$ such that \noindent{\bf 1.1.1.} For any normal subgroup $B$ of $A$, the bijection $\pi(H,\, B)$ maps ${\rm Irr}_{{\cal K}}(H)^A$ to ${\rm Irr}_{{\cal K}}(H^B)^A$, and in ${\rm Irr}_{{\cal K}}(H)^A$ we have $$\pi(H,\,A) = \pi(H^B,\,A/B) \circ \pi(H,\,B)\,.$$ \noindent{\bf 1.1.2.} If $A$ is a $q$-group for some prime $q$, then for any $\frak c\frak hi\leqn {\rm Irr}_{{\cal K}}(H)^A$, the corresponding irreducible character $\pi(H,\,A)(\frak c\frak hi)$ of $G^A$ is the unique irreducible constituent of ${\rm Res}^H_{H^A}(\frak c\frak hi)$ occurring with a multiplicity coprime to~$q$. \noindent The character $\pi(H,\,A)(\frak c\frak hi)$ of $H^A$ is called the {\leqt Glauberman correspondent\/} of the character $\frak c\frak hi$ of $H$. \noindent{\bf 1.2.}\quad For any central idempotent $c$ of ${\cal O} H$, we denote by ${\rm Irr}_{{\cal K}}(H, c)$ the set of all irreducible characters of $H$ provided by some ${\cal K} Hc$-module. Let $b$ be a block of $H$ --- namely $b$ is a primitive central idempotent of ${\cal O} H\,;$ then ${\cal O} Hb$ is called the {\leqt block algebra\/} corresponding to $b$. Assume that $A$ stabilizes the block $b$ and centralizes a defect group of $b$. Then, by \cite[Proposition 1 and Theorem 1]{W}, $A$ stabilizes all characters of ${\rm Irr}_{\cal K}(H, b)$ and there is a unique block ${\leqt w}(b)$ of ${\cal O} (H^A)$ such that $${\rm Irr}_{{\cal K}}(H^A, {\leqt w}(b))=\pi(H,\,A)({\rm Irr}_{\cal K}(H, b))\,;$$ moreover ,there is a perfect isometry (see \cite{B1}) $$R_H^b: {\cal R}_{\cal K} (H, b)\rightarrow {\cal R}_{\cal K} (H^A, {\leqt w}(b))$$ such that $R_H^b(\frak c\frak hi)=\pm\pi(H,\,A)(\frak c\frak hi)$ for any $\frak c\frak hi\leqn {\rm Irr}_{{\cal K}}(H, b)$, where we denote by ${\cal R}_{\cal K} (H, b)$ and ${\cal R}_{\cal K} (H^A, {\leqt w}(b))$ the additive groups generated by ${\rm Irr}_{{\cal K}}(H, b)$ and ${\rm Irr}_{{\cal K}}(H^A, {\leqt w}(b))$. Such a block ${\leqt w}(b)$ is called the {\leqt Glauberman correspondent\/} of $b$ (see \cite{W}). Since a perfect isometry between blocks is often nothing but the character-theoretic `shadow' of a derived equivalence, it seems reasonable to ask whether there is a derived equivalence between a block and its Glauberman correspondent. In the last few years, some Morita equivalences between $b$ and ${\leqt w}(b)$ were found in the cases where $H$ is $p$-solvable or the defect groups of $b$ are normal in $H$, which supply Glauberman correspondences from ${\rm Irr}_{{\cal K}}(H, b)$ to ${\rm Irr}_{{\cal K}}(H^A, {\leqt w}(b))$ (see \cite{H}, \cite{KG} and \cite{H1}); moreover, all these Morita equivalences between $b$ and ${\leqt w}(b)$ are {\leqt basic} in the sense of~\cite{P1}. \noindent{\bf 1.3.}\quad By induction, the groups $H$ and $H^A$ and the blocks $b$ and ${\leqt w}(b)$ in the main results of \cite{H}, \cite{KG} and \cite{H1} can be reduced to the situation where, for some $A$-stable normal subgroup $K$ of $H\,,$ we have $H=H^A {\,\hskip-1pt\cdot\hskip-1pt\,} K$ , the block $b$ is an $H$-stable block of $K$ with trivial or central defect group, and the block ${\leqt w}(b)$ is an $H^A$-stable block of $K^A$ with trivial or central defect group. Recall that the block $b$ of $H$ is called {\leqt nilpotent\/} (see \cite{P4}) if the quotient group $N_H(R_\varepsilon)/C_H(R)$ is a $p$-group for any local pointed group $R_\varepsilon$ on ${\cal O} H b$. Blocks with trivial or central defect group are nilpotent and therefore in these situations ${\cal O} Hb$ and ${\cal O} (H^A){\leqt w}(b)$ are extensions of the nilpotent block algebras ${\cal O} K b$ and ${\cal O} K^A{\leqt w}(b)$ respectively. K\"ulshammer and Puig already precisely described the algebraic structure of extensions of nilpotent block algebras (see \cite{KP} or Section 3 below) and these results can be applied to blocks of $p$-solvable groups (see \cite{P2}) and to blocks with normal defect groups (see \cite{R, K}). Thus, it is reasonable to seek a common generalization of the main results of \cite{H, KG, H1} in the setting of extensions of nilpotent block algebras. \noindent{\bf 1.4.}\quad Let $G$ be another finite $A$-group having $H$ as an $A$-stable normal subgroup and consider the $A$-action on $H$ induced by the $A$-group $G$. We assume that $A$ stabilizes $b$ and denote by $N$ the stabilizer of $b$ in $G$. Clearly $N$ is $A$-stable. Set \begin{center}$ c={\rm Tr}_N^G (b)$ and $\alpha=\{c\}$\quad ; \end{center} then the idempotent $c$ is $A$-stable and $\alpha$ is an $A$-stable point of $G$ on the group algebra ${\cal O} H$ (the action of $G$ on ${\cal O} H$ is induced by conjugation). In particular, $G_\alpha$ is a pointed group on ${\cal O} H$. Let~$P$ be a defect group of $G_\alpha\,;$ then, by \cite[Proposition 5.3]{KP}, $Q=P\cap H$ is a defect group of the block $b$ of~$H$. \noindent{\bf Theorem 1.5.}\quad {\leqt Assume that $A$ centralizes $P$, that $A$ is solvable and that the orders of $G$ and $A$ are coprime. Set ${\leqt w}(c)={\rm Tr}_{N^A}^{G^A}({\leqt w}(b))$ and ${\leqt w}(\alpha)=\{{\leqt w}(c)\}$. Then, ${\leqt w}(\alpha)$ is a point of $G^A$ on the group algebra ${\cal O} (H^A)$ and $P$ is a defect group of the pointed group $(G^A)_{{\leqt w}(\alpha)}$ on ${\cal O} (H^A)\,.$ Moreover, if $G=H{\,\hskip-1pt\cdot\hskip-1pt\,} G^A$ and the block $b$ of $H$ is nilpotent, we have \begin{center}${\rm Irr}_{{\cal K}}(G, c)={\rm Irr}_{{\cal K}}(G, c)^A$ and $\pi(G, A)({\rm Irr}_{{\cal K}}(G, c))={\rm Irr}_{{\cal K}}(G^A, {\leqt w}(c))$. \end{center}} The following theorem shows that there is a ``{\leqt basic\/}" Morita equivalence between ${\cal O} G c$ and ${\cal O} G^A {\leqt w}(c)$; that is to say, this Morita equivalence induces basic Morita equivalences~\cite{P1} between corresponding block algebras. \noindent{\bf Theorem 1.6.}\quad {\leqt Assume that $A$ centralizes $P$, that $A$ is solvable and that the orders of $G$ and $A$ are coprime. Set ${\leqt w}(c)={\rm Tr}_{N^A}^{G^A}({\leqt w}(b))$. Assume that $G=G^A{\,\hskip-1pt\cdot\hskip-1pt\,} H$ and that the block $b$ is nilpotent. Then, there is an ${\cal O} (H\times H^A)$-module $M$ inducing a basic Morita equivalence between ${\cal O} Hb$ and ${\cal O} (H^A) {\leqt w}(b)\,,$ which can be extended to the inverse image $K$ in $N\times N^A$ of the ``diagonal'' subgroup of $N/H\times N^A/H^A$ in such a way that ${\rm Ind}^{G\times G^A}_K (M)$ induces a ``basic" Morita equivalence between ${\cal O} Gc$ and ${\cal O} (G^A){\leqt w}(c)$. } \noindent{\bf Remark 1.7.}\quad Since $G=H{\,\hskip-1pt\cdot\hskip-1pt\,} G^A$, we have $N=H{\,\hskip-1pt\cdot\hskip-1pt\,} N^A$ and then the inclusion $N^A{\cal S}ubset N$ induces a group isomorphism $N/H\cong N^A/H^A$. We use pointed groups introduced by Lluis Puig. For more details on pointed groups, readers can see \cite{P5} or Paragraph 2.5 below. In Section 2, we introduce some notation and terminology. Section 3 revisits K\"ulshammer and Puig's main results on extensions of nilpotent blocks; the proof of the existence and uniqueness of the finite group $L$ (see \cite[Theorem 1.8]{KP} and Theorem 3.5 below) is dramatically simplified; actually, Corollary 3.14 below slightly improves \cite[Theorem 1.12]{KP}; explicitly, $S_\gamma$ in Corollary 3.14 is unique up to determinant one. With the Glauberman correspondents of blocks due to Watanabe, in Section 4 we define Glauberman correspondents of extensions of blocks and compare the local structures of extensions of blocks and their Glauberman correspondents. By Puig's structure theorem of nilpotent blocks, there is a bijection between the sets of irreducible characters of the nilpotent block $b$ of $H$ and of its defect $Q$; in Section 5, for a suitable local point $\delta$ of $Q\,,$ we prove that this bijection preserves $N_G(Q_\delta)$-actions on these sets. As a consequence, we obtain an $N_G(Q_\delta)$-stable irreducible character $\frak c\frak hi$ of $H$ such that $\frak c\frak hi$ lifts the unique irreducible Brauer character of the nilpotent block $b$ of $H$ and that the Glauberman correspondent character $\pi(H, A)(\frak c\frak hi)$ lifts the unique irreducible Brauer character of the Glauberman correspondent block ${\leqt w}(b)$ of $H^A$ (see Lemma 5.6). Obviously, $N$ stabilizes the unique simple module in the nilpotent block $b$ of $H$; with this $N$-stable ${\cal O} H b$-simple module, we construct an $A$-stable $k^*$-group ${\cal S}kew3\hat {\bar N}^{^k}$ (see £2.3 and £3.13 below); since $N^A$ stabilizes the unique simple module of the nilpotent block ${\leqt w}(b)$ of $H^A\,,$ a $k^*$-group $\,\widehat{\overline{\! N^A}}^k$ is similarly constructed. In Section 6, we prove that $\,\widehat{\overline{\! N^A}}^k$ and $({\cal S}kew3\hat{\bar N}^{^k})^A$ are isomorphic as $k^*$-groups (see Theorem 6.4). In Section 7, we use the improved version of K\"ulshammer and Puig's main result to prove our main theorem 1.6. \eject \vskip 1cm \noindent{\bf\large 2. Notation and terminology} \noindent{\bf 2.1.}\quad Throughout this paper, all ${\cal O}$-modules are ${\cal O}$-free finitely generated --- except in 2.4 below; all ${\cal O}$-algebras have identity elements, but their subalgebras need not have the same identity element. Let $\cal A$ be an ${\cal O}$-algebra; we denote by ${\cal A}^\circ\,,$ ${\cal A}^*$, $Z({\cal A})$, $J({\cal A})$ and $1_{\cal A}$ the opposite ${\cal O}\hbox{-}$algebra of ${\cal A}\,,$ the multiplicative group of all invertible elements of ${\cal A}$, the center of~${\cal A}$, the radical of ${\cal A}$ and the identity element of ${\cal A}$ respectively. Sometimes we write $1$ instead of $1_{\cal A}\,.$ For any abelian group $V$, ${\rm id }_V$ denotes the identity automorphism on $V$. Let ${\cal B}$ be an ${\cal O}$-algebra; a homomorphism ${\cal F}: {\cal A}\rightarrow {\cal B}$ of ${\cal O}$-algebras is said to be an {\leqt embedding\/} if ${\cal F}$ is injective and we have $${\cal F}({\cal A})={\cal F}(1_{\cal A}){\cal B}{\cal F}(1_{\cal A})\quad .$$ Let $S$ be a set and $G$ be a group acting on $S$. For any $g\leqn G$ and $s\leqn S$, we write the action of $g$ on $s$ as $s{\,\hskip-1pt\cdot\hskip-1pt\,}g$. \noindent{\bf 2.2.}\quad Let $X$ be a finite group. An $X$-interior ${\cal O}$-algebra ${\cal A}$ is an ${\cal O}$-algebra ${\cal A}$ endowed with a group homomorphism $\rho:X\rightarrow {\cal A}^*$; for any $x, y\leqn X $ and $a\leqn {\cal A}$, we write $\rho(x)a\rho(y)$ as $x{\,\hskip-1pt\cdot\hskip-1pt\,} a{\,\hskip-1pt\cdot\hskip-1pt\,} y$ or $xay$ if there is no confusion. Let $\varrho: Y\rightarrow X$ be a group homomorphism; the ${\cal O}$-algebra ${\cal A}$ with the group homomorphism $\rho\circ\varrho: Y\rightarrow {\cal A}^*$ is an $Y$-interior ${\cal O}$-algebra and we denote it by ${\rm Res}_{\varrho}({\cal A})$. Let ${\cal A}'$ be another $X$-interior ${\cal O}$-algebra; an ${\cal O}$-algebra homomorphism ${\cal F}:{\cal A}\rightarrow {\cal A}'$ is said to be a homomorphism of $X$-interior ${\cal O}$-algebras if for any $x, y\leqn X $ and any $a\leqn {\cal A}$, we have ${\cal F}(xay)=x{\cal F}(a)y$. The tensor product ${\cal A}\bigotimes_{\cal O} {\cal A}'$ of ${\cal A}$ and ${\cal A}'$ is an $X$-interior ${\cal O}$-algebra with the group homomorphism $$ X\rightarrow ({\cal A}\otimes_{\cal O} {\cal A}')^*,\quad x\mapsto x1_{\cal A}\otimes x1_{{\cal A}'}\quad .$$ Let $Z$ be a subgroup of $X$ and let ${\cal B}$ be an ${\cal O} Z$-interior algebra. Obviously, the left and right multiplications by ${\cal O} Z$ on ${\cal B}$ define an $({\cal O} Z, {\cal O} Z)$-bimodule structure on ${\cal B}$. Set $${\rm Ind}_Z^X ({\cal B})={\cal O} X\otimes_{{\cal O} Z} {\cal B}\otimes_{{\cal O} Z} {\cal O} X $$ and then this the $({\cal O} X, {\cal O} X)$-bimodule ${\rm Ind}_Z^X ({\cal B})$ becomes an $X$-interior ${\cal O}$-algebra with the product $$(x\otimes b\otimes y)(x'\otimes b'\otimes y') =\cases{x\otimes b{\,\hskip-1pt\cdot\hskip-1pt\,} y x'{\,\hskip-1pt\cdot\hskip-1pt\,}b'\otimes y'&if $yx'\leqn Z$\cr {}&{}\cr 0 &otherwise\cr}$$ for any $x, y, x', y'\leqn X$ and any $b,b'\leqn {\cal B}\,,$ and with the homomorphism ${\cal O} X\rightarrow {\rm Ind}_Z^X ({\cal B})$ mapping $x\leqn X$ onto ${\cal S}um_y xy\otimes 1\otimes y^{-1}$, where $y$ runs over a set of representatives for left cosets of $Z$ in $X$. \noindent{\bf 2.3.}\quad A $k^*$-group with $k^*$-quotient $X$ is a group $\hat X$ endowed with an injective group homomorphism $\theta: k^*\rightarrow Z(\hat X)$ together with an isomorphism $\hat X/\theta(k^*)\cong X$; usually we omit to mention $\theta$ and the quotient $X=\hat X/\theta(k^*)$ is called the $k^*$-quotient of $\hat X$, writing $\lambda{\,\hskip-1pt\cdot\hskip-1pt\,} \hat x$ instead of $\theta(\lambda)\hat x$ for any $\lambda\leqn k^*$ and any $\hat x\leqn \hat X$. We denote by $\hat Y$ the inverse image of $Y$ in $\hat X$ for any subset $Y$ of~$X$ and, if no precision is needed, we often denote by $\hat x$ some lifting of an element $x\leqn X$. We denote by $\hat X^\circ$ the $k^*$-group with the same underlying group $\hat X$ endowed with the group homomorphism $\theta^{-1}: k^*\rightarrow Z(\hat X),\, \lambda\mapsto \theta(\lambda)^{-1}$. Let $\vartheta: Z\rightarrow X$ be a group homomorphism; we denote by ${\rm Res}_{\vartheta}(\hat X)$ the $k^*$-group formed by the group of pairs $(\hat x, y)\leqn \hat X\times Z$ such that $\vartheta(y)$ is the image of $\hat x$ in $X$, endowed with the group homomorphism mapping $\lambda\leqn k^*$ on $(\theta(\lambda), 1)$; up to suitable identifications, $Z$ is the $k^*$-quotient of ${\rm Res}_{\vartheta}(\hat X)$. Let $\hat U$ be another $k^*$-group with $k^*$-quotient $U$. A group homomorphism $\phi: \hat X\rightarrow \hat U$ is a homomorphism of $k^*$-groups if $\phi(\lambda{\,\hskip-1pt\cdot\hskip-1pt\,} \hat x)=\lambda{\,\hskip-1pt\cdot\hskip-1pt\,} \phi(\hat x)$ for any $\lambda\leqn k^*$ and $\hat x\leqn \hat X$. For more details on $k^*$-groups, please see \cite[\S 5]{P6}. \noindent{\bf 2.4.}\quad Let $\hat X$ be a $k^*$-group with $k^*$-quotient $X$. By \cite[Charpter II, Proposition 8]{S}, there exists a canonical decomposition ${\cal O}^*\cong (1+J({\cal O}))\times k^*$, thus $k^*$ can be canonically regarded as a subgroup of ${\cal O}^*$. Set $${\cal O}_*\hat X={\cal O}\otimes_{{\cal O} k^*}{\cal O} \hat X \quad, $$ where the left ${\cal O} k^*$-module ${\cal O}\hat X$ and the right ${\cal O} k^*$-module ${\cal O}$ are defined by the left and right multiplication by $k^*$ on $\hat X$ and ${\cal O} ^*$ respectively. It is straightforward to verify that the ${\cal O}$-module ${\cal O}_*\hat X$ is an ${\cal O}$-algebra with the distributive multiplication $$(a_1\otimes\hat x_1)(a_2\otimes\hat {x}_2) =a_1a_2\otimes\hat x_1\hat{x}_2$$ for any $a_1,a_2\leqn {\cal O}$ and any $\hat x_1,\hat{x}_2 \leqn \hat X$. \noindent{\bf 2.5.}\quad Let ${\cal A}$ be an $X$-algebra over ${\cal O}$; that is to say, ${\cal A}$ is endowed with a group homomorphism $\psi: X\rightarrow {\rm Aut}({\cal A})$, where ${\rm Aut}({\cal A})$ is the group of all ${\cal O}$-automorphisms of $A\,;$ usually, we omit to mention $\psi\,.$ For any subgroup $Y$ of $X$, we denote by ${\cal A}^Y$ the ${\cal O}$-subalgebra of all $Y$-fixed elements in ${\cal A}$. A {\leqt pointed group\/} $Y_\beta$ on ${\cal A}$ consists of a subgroup $Y$ of $X$ and of an $({\cal A}^Y)^*$-conjugate class $\beta$ of primitive idempotents of ${\cal A}^Y$. We often say that $\beta$ is a {\leqt point\/} of $Y$ on ${\cal A}$. Obviously, $X$ acts on the set of all pointed groups on ${\cal A}$ by the equality $(Y_\beta)^x=Y^x_{\psi(x^{-1})(\beta)}$ and we denote by $N_X(Y_\beta)$ the stabilizer of $Y_\beta$ in $X$ for any pointed group $Y_\beta$ on ${\cal A}$. Another pointed group $Z_\gamma$ is said {\leqt contained in\/} $Y_\beta$ if $Z\leq Y$ and there exist some $i\leqn \beta$ and $j\leqn \gamma$ such that $ij=ji=j$. For a subgroup $U$ of $G$, set $${\cal A}(U)=k\otimes _{\cal O} ({\cal A}^U/{\cal S}um_V {\cal A}^U_V)\quad$$ where $V$ runs over the set of proper subgroups of $U$ and ${\cal A}^U_V$ is the image of the relative trace map ${\rm Tr}_V^U: {\cal A}^V\rightarrow {\cal A}^U$; the canonical surjective homomorphism ${\rm Br}^{\cal A}_U: {\cal A}^U\rightarrow {\cal A}(U)$ is called the {\leqt Brauer homomorphism\/} of the $X$-algebra ${\cal A}$ at $U$. When ${\cal A}$ is equal to the group algebra ${\cal O} X$, the homomorphism $kC_X(U)\rightarrow {\cal A}(U)$ sending $x\leqn C_X(U)$ onto the image of $x$ in ${\cal A}(U)$ is an isomorphism, through which we identify ${\cal A}(U)$ with $kC_X(U)$. A pointed group $U_\gamma$ on $\cal A$ is said {\leqt local\/} if the image of $\gamma$ in ${\cal A}(U)$ is not equal to $\{0\}\,,$ which forces $U$ to be a $p$-group; then, a local pointed group $U_\gamma$ is said a {\leqt defect pointed group\/} of a pointed group $Y_\beta$ on $\cal A$ if $U_\gamma\leq Y_\beta$ and we have $\beta{\cal S}ubset {\rm Tr}_U^Z({\cal A}^U{\,\hskip-1pt\cdot\hskip-1pt\,} \gamma{\,\hskip-1pt\cdot\hskip-1pt\,} A^U)$, where ${\cal A}^U{\,\hskip-1pt\cdot\hskip-1pt\,} \gamma{\,\hskip-1pt\cdot\hskip-1pt\,} A^U$ is the ideal of ${\cal A}^U$ generated by~$\gamma$. Let $c$ be a block of $X\,;$ then $\{c\}$ is a point of $X$ on ${\cal O} X$ and if $P_\gamma$ is a defect pointed group of $X_{\{c\}}$ then $P$ is a defect group of $c$. \vskip 1cm \noindent{\bf\large 3. Extensions of nilpotent blocks revisited} In this section, we assume that ${\cal O}$ is a complete discrete valuation ring with an algebraically closed residue field of characteristic $p$. \noindent {\bf £3.1.}\quad Let $G$ be a finite group, $H$ be a normal subgroup of $G$ and $b$ be a block of $H$ over~${\cal O}$. Denote by $N$ the stabilizer of $b$ in $G$ and set $\bar N=N/H$. Obviously, $\beta=\{b\}$ is a point of $H$ and $N$ on ${\cal O} H$ and there is a unique pointed group $G_\alpha$ on ${\cal O} H$ such that $$H_\beta\leq N_\beta\leq G_\alpha\quad .$$ Let $Q_\delta$ be a defect pointed group of~$H_\beta$ and $P_\gamma$ be a defect pointed group of $N_\beta$ such that $Q_\delta\leq P_\gamma\,;$ by \cite[Proposition 5.3]{KP}, we have $Q=P\cap H$ and, since we have \cite[1.7]{KP} $${\cal O} G\,{\rm Tr}_N^G( b)\cong {\rm Ind}_N^G ({\cal O} N b) \quad ,$$ it is easily checked that $P_\gamma$ is also a {\leqt defect pointed group\/} of $G_\alpha$ \cite[1.12]{P3}. Assume that the block $b$ is {\leqt nilpotent}; it follows from \cite[Proposition~6.5]{KP} that $b$ remains a {nilpotent block\/} of $H{\,\hskip-1pt\cdot\hskip-1pt\,} R$ for any subgroup~$R$ of~$P\,,$ and from \cite[Theorem~6.6]{KP} that there is a unique {local point\/} $\varepsilon$ of $R$ on ${\cal O} H$ such that~$R_\varepsilon\leq P_\gamma\,.$ \noindent {\bf £3.2.}\quad Set ${\cal A} = {\cal O} Nb$ and ${\cal B} = {\cal O} Hb\,$. Choosing $j\leqn \delta$ and $i\leqn \gamma$ such that $ij=ji=j\,,$ we set \begin{center} ${\cal A}_\gamma=({\cal O} G)_\gamma=i{\cal A} i$, ${\cal B}_\gamma=({\cal O} H)_\gamma=i{\cal B}i$ and ${\cal B}_\delta=({\cal O} H)_\delta=j{\cal B}j$. \end{center} Then ${\cal A}_\gamma$ is a $P$-interior algebra with the group homomorphism $P\rightarrow {\cal A}_\gamma^*$ mapping $u$ onto $ui$ for any $u\leqn P$, ${\cal B}_\gamma$ is a $P$-stable subalgebra of ${\cal A}_{\gamma}$ and ${\cal B}_\delta$ is a $Q$-interior algebra with the group homomorphism $Q\rightarrow {\cal B}_\delta^*$ mapping $v\leqn Q$ onto $vj$ for any $v\leqn Q$. Clearly $\cal A$ is an $N/H$-graded algebra with the $\bar x$-component ${\cal O} H x b$, where $\bar x\leqn N/H$ and $x$ is a representative of $\bar x$ in $N$. Since $i$ belongs to the $1$-component $\cal B$, ${\cal A}_\gamma$ is an $N/H$-graded algebra with the $\bar x$-component $i({\cal O} H x)i$. \noindent {\bf £3.3.}\quad In \cite{KP} K\"ulshammer and Puig describe the structure of any block of $G$ lying over $b$ in terms of a new finite group $L$ which need not be {involved\/} in $G$ \cite[Theorem~1.8]{KP}. More explicitly, $L$ is a {group extension\/} of $\bar N$ by~$Q$ holding {strong uniqueness\/} properties. In order to prove these properties, in \cite{KP} the group $L$ is exhibited inside a suitable ${\cal O}\hbox{-}$algebra \cite[Theorem~8.13]{KP}, demanding a huge effort. But, as a matter of fact, these properties can be obtained {directly\/} from the so-called {local structure\/} of $G$ over ${\cal O} H b\,,$ a fact that we only have understood recently. Then, with these uniqueness properties in hand, the second main result \cite[Theorem~1.12]{KP} follows quite easily. With the notation and framework of \cite{KP}, we completely develop both new proofs. \noindent {\bf £3.4.}\quad Denote by ${\cal E}_{(b,H,G)}$ the category --- called the {\leqt extension category\/} associated to $G\,,$ $H$ and $b$ --- where the objects are all the subgroups of $P$ and, for any pair of subgroups $R$ and $T$ of $P\,,$ the morphisms from $T$ to $R$ are the pairs $(\psi_x,\bar x)$ formed by an injective group homomorphism $\psi_x\,\colon T\to R$ and an element $\bar x$ of $\bar N$ both determined by an element $x\leqn N$ fulfilling $T_\nu\leq (R_\varepsilon)^x$ where $\varepsilon$ and $\nu$ are the respective local points of $R$ and~$T$ on ${\cal O} H$ determined by~$P_\gamma$ --- in general, we should consider the {\leqt $(b,N)\hbox{-}$Brauer pairs\/} over the {$p\hbox{-}$permutation $N\hbox{-}$algebra ${\cal O} H b$\/} [5,~Definition~1.6 and Theorem~1.8] but, in our situation, they coincide with the {local pointed groups\/} over this $N\hbox{-}$algebra. The composition in ${\cal E}_{(b,H,G)}$ is determined by the composition of group homomorphisms and by the product in $\bar N \,.$ \noindent {\bf Theorem £3.5.}\quad {\leqt There is a triple formed by a finite group $L$ and by two group homomorphisms $$\tau : P\longrightarrow L\quad and\quad \bar\pi : L\longrightarrow \bar N \leqno £3.5.1\phantom{.}$$ such that $\tau$ is injective, that $\bar\pi$ is surjective, that we have ${\rm Ker}(\bar\pi) = \tau (Q)$ and $\bar\pi\big(\tau (u)\big) =\bar u$ for any $u\leqn P\,,$ and that these homomorphisms induce an equivalence of categories $${\cal E}_{(b,H,G)} \cong {\cal E}_{(1,\tau (Q),L)}. \leqno £3.5.2$$ {\cal S}mallskip \noindent Moreover, for another such a triple $L'\,,$ $\tau'$ and $\bar\pi'\,,$ there is a group isomorphism $\lambda\,\colon L\cong L'\,,$ unique up to conjugation, fulfilling $$\lambda\circ\tau = \tau'\quad and \quad \bar\pi'\circ\lambda = \bar\pi\,.$$\/} \par \noindent {\bf Proof:} Set $Z = Z(Q)\,,$ $M = N_G (Q_\delta)$ and ${\cal E} = {\cal E}_{(b,H,G)}\,,$ denote by ${\cal E} (R,T)$ the set of ${\cal E}\hbox{-}$morphisms from $T$ to $R\,,$ and write ${\cal E} (R)$ instead of ${\cal E} (R,R)\,;$ by the very definition of the category ${\cal E}\,,$ we have the exact sequence $$1\longrightarrow C_H (Q)\longrightarrow M\longrightarrow {\cal E} (Q) \longrightarrow 1 ; $$ it is clear that $M$ contains $P$ and that we have $C_H (Q)\cap P = Z\,;$ moreover, denoting by ${\cal E}_P (Q)$ the image of $P$ in ${\cal E} (Q)\,,$ it is easily checked from \cite[~Proposition~5.3]{KP} that ${\cal E}_P (Q)$ is a Sylow $p\hbox{-}$subgroup of~${\cal E} (Q)\,.$ {\cal S}mallskip We claim that the element $\bar h$ induced by $P$ in the second cohomology group ${{\cal B}bb H}^2 \big({\cal E}_P (Q),Z\big)$ belongs to the image of~${{\cal B}bb H}^2 ({\cal E} (Q),Z)\,.$ Indeed, according to in~ \cite[~Ch.~XII,~Theorem~10.1]{CE}, it suffices to prove that, for any subgroup $R$ of $P$ containing~$Z$ and any $(\varphi_x,\bar x)\leqn {\cal E} (Q)$ such that $$(\varphi_x,\bar x)\circ{\cal E}_{\! R}(Q)\circ (\varphi_x,\bar x)^{-1}\leq {\cal E}_{\!P} (Q), \leqno £3.5.3$$ the restriction ${\rm res}_{(\varphi_x,\bar x)} (\bar h)$ of $\bar h$ {via\/} the conjugation by $(\varphi_x,\bar x)$ and the element of ${{\cal B}bb H}^2\big({\cal E}_R(Q), Z \big)$ determined by $R$ coincide; actually, we may assume that $R$ contains $Q\,.$ Thus, $x$ normalizes $Q_\delta$ and inclusion~£3.5.3 forces $$C_H(Q){\,\hskip-1pt\cdot\hskip-1pt\,} R \leq \big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} P\big)^x; $$ in particular, respectively denoting by $\lambda$ and $\mu$ the points of $C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} P$ and $C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R$ on ${\cal O} H$ such that $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} P\big)_\lambda$ and $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)_\mu$ contain $Q_\delta$ \cite[Lemma~3.9]{P4}, by uniqueness we have $$\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)_\mu\leq \big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} P\big)_\lambda $$ and, with the notation above, it follows from \cite[~Proposition~3.5]{KP} that $P_\gamma$ and $R_\varepsilon$ are {defect pointed groups\/} of the respective pointed groups $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} P\big)_\lambda$ and $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)_\mu\,;$ consequently, there is $z\leqn C_H (Q)$ fulfilling $R_\varepsilon \leq (P_\gamma)^{zx}$ \cite[~Theorem~1.2]{P5}. That is to say, the conjugation by $zx$ induces a group homomorphism $R\to P$ mapping $Z$ onto $Z$ and inducing the element $(\psi_{zx},\overline{zx})$ of ${\cal E} (P,R)$ which extends $(\varphi_x,\bar x)\,,$ so that the map $${\rm res}_{(\varphi_x,\bar x)} : {\cal B}bb H^2\big({\cal E}_P (Q),Z\big)\longrightarrow {\cal B}bb H^2\big({\cal E}_R (Q),Z\big) $$ sends $\bar h$ to the element of~${{\cal B}bb H}^2\big({\cal E}_R(Q), Z \big)$ determined by~$R$ \cite[~Chap.~XIV, Theorem~4.2]{CE}. {\cal S}mallskip In particular, the corresponding element of ${{\cal B}bb H}^2 \big({\cal E} (Q),Z\big)$ determines a group extension $$1\longrightarrow Z\buildrel \tau \over\longrightarrow L \buildrel \pi \over \longrightarrow {\cal E} (Q)\longrightarrow 1 $$ and, since $\bar h\leqn {\cal B}bb H^2\big({\cal E}_P (Q),Z\big)$ is the image of this element, there is a {group extension\/} homomorphism $\tau\,\colon P\to L$ \cite[~Chap.~XIV, Theorem~4.2]{CE}; it is clear that $\tau$ is injective and, since ${\cal E}_P (Q)$ is a Sylow $p$-subgroup of ${\cal E} (Q)\,,$ ${\rm Im}(\tau)$ is a Sylow $p\hbox{-}$subgroup of $L\,;$ moreover, since $N = H{\,\hskip-1pt\cdot\hskip-1pt\,} M$ \cite[Theorem~1.2]{P5}, we have $$\bar N \cong M/C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} Q\cong {\cal E} (Q)/{\cal E}_Q (Q) ;$$ in particular, $\pi$ determines a group homomorphism $\bar\pi\,\colon L\to \bar N$ and, since $\tau$ is a {group extension\/} homomorphism, we get $\bar\pi\big(\tau (u)\big) = \bar u$ for any $u\leqn P$ and may choose $\pi$ in such a way that we have $$y\tau (v)y^{-1} = \tau\big(\varphi_x (v)\big) \leqno £3.5.4\phantom{.}$$ for any $y\leqn L$ and any $v\leqn Q$ where $\pi (y) = (\varphi_x,\bar x)$ for some $x\leqn N\,.$ Then, we claim that, up to a suitable modification of our choice of $\tau\,,$ the group $L$ endowed with $\tau$ and $\bar\pi$ fulfills the conditions above; set $\hat{\cal E} = {\cal E}_{(1,\tau (Q),L)}$ for short. {\cal S}mallskip For any pair of subgroups $R$ and $T$ of $P$ containing~$Q\,,$ since we have $H\cap R = Q = H\cap T\,,$ denoting by $\varepsilon$ and $\nu$ the respective local points of~$R$ and $T$ such that $P_\gamma$ contains $R_\varepsilon$ and $T_\nu\,,$ these local pointed groups contain~$Q_\delta$ and, in particular, any ${\cal E}\hbox{-}$morphism $$(\psi_x,\bar x) : T\longrightarrow R $$ determines an element $(\varphi_x,\bar x)$ of ${\cal E} (Q)$ fulfilling $$(\varphi_x,\bar x)\circ {\cal E}_T (Q)\circ (\varphi_x,\bar x)^{-1} {\cal S}ubset {\cal E}_R (Q) \quad .$$ Thus, for any $y\leqn L$ such that $\pi (y) = (\varphi_x,\bar x)\,,$ we have $$y\,\tau (T)\,y^{-1}\leq \tau (R) \quad ;$$ more precisely, for any $w\leqn T$ and any $v\leqn Q\,,$ from equality~£3.5.4 we get $$y\,\tau (v^w)\,y^{-1} = \tau \big(\varphi_x (v^w)\big) = \tau \big(\varphi_x(v)\big)^{\tau (\psi_x (w))} \quad ;$$ moreover, since $x T x^{-1}\leq R\,,$ we have $$\bar\pi \big(y\,\tau (w)\,y^{-1}\big) = \bar x\,\bar w\, \bar x^{-1} = \bar\pi {\cal B}ig(\tau (\psi_x (w)){\cal B}ig) \quad .$$ Hence, for any $w\leqn T$ and a suitable $\theta_x (w)\leqn Z\,,$ we get $$y\,\tau \big(w\,\theta_x (w)\big)\,y^{-1} = \tau (\psi_x (w)) \quad .$$ {\cal S}mallskip Conversely, since $R$ and $T$ have a unique (local) point on ${\cal O} Q\,,$ any $\hat{\cal E}\hbox{-}$morphism from $T$ to $R$ induced by an element $y$ of $L$ determines an element $\pi (y) = (\varphi_x,\bar x)$ of ${\cal E} (Q)\,,$ for some $x\leqn N\,,$ which still fulfills $$(\varphi_x,\bar x)\circ {\cal E}_T (Q)\circ (\varphi_x,\bar x)^{-1} {\cal S}ubset {\cal E}_R (Q) \quad ;$$ thus, as above, $x$ normalizes $Q_\delta$ and this inclusion forces $$C_H(Q){\,\hskip-1pt\cdot\hskip-1pt\,} T \leq \big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)^x \quad .$$ Once again, respectively denoting by $\lambda$ and $\mu$ the points of $C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R$ and $C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T$ on ${\cal O} H$ such that $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)_\lambda$ and $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T\big)_\mu$ contain $Q_\delta$ \cite[ Lemma~3.9]{P4}, and by $\varepsilon$ and $\nu$ the local points of $R$ and $T$ on ${\cal O} H$ such that $P_\gamma$ contains $R_\varepsilon$ and~$T_\nu\,,$ it follows from \cite[~Proposition~3.5]{KP} that $R_\varepsilon$ and $T_\nu$ are defect pointed groups of the respective pointed groups $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)_\lambda$ and $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T\big)_\mu\,;$ since by uniqueness we have $$\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T\big)_\mu\leq \big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} R\big)_\lambda ,$$ there is $z\leqn C_H (Q)$ fulfilling $T_\nu \leq (R_\varepsilon)^{zx}$ \cite[Theorem~1.2]{P5}. That is to say, the conjugation by $zx$ induces a group homomorphism $\psi_{zx}\,\colon T\to R$ mapping $Z$ onto $Z$ and inducing the element $(\psi_{zx},\overline{zx})$ of ${\cal E} (R,T)$ which extends $(\varphi_x,\bar x)\,;$ hence, as above, for any $w\leqn T$ and a suitable $\theta_y (w)\leqn Z$ we get $$y\,\tau \big(w\,\theta_y (w)\big)\,y^{-1} = \tau (\psi_{zx} (w)). \leqno £3.5.5$$ {\cal S}mallskip We claim that, for a suitable choice of $\tau\,,$ the elements $\theta_x (w)$ and $\theta_y (w)$ are always trivial; then, the equivalence of categories~£3.5.2 will be an easy consequence of the above correspondences. Above, for any~$y\leqn L$ such that $\tau (T){\cal S}ubset \tau (R)^y$ we have found an element $\big(\psi_y,\bar\pi (y)\big)\leqn {\cal E} (R,T)$ lifting~$\pi (y)$ in such a way that, for any~$w\leqn T\,,$ we have $$\tau \big(w\,\theta_y (w)\big) = \tau\big(\psi_y (w)\big)^y \leqno £3.5.6\phantom{.}$$ for a suitable $\theta_y (w)\leqn Z\,;$ note that, according to equality~£3.5.4, for any $v\leqn Q$ we have $\theta_y (v) = 1\,,$ and whenever $y$ belongs to $\tau (R)$ we may choose $\psi_y$ in such a way that $\theta_y (w) = 1\,.$ {\cal S}mallskip In this situation, for any $w,w'\leqn T\,,$ we get \begin{eqnarray*} \tau \big(ww'\theta_y (ww')\big) &=& \tau \big(\psi_y (ww')\big)^y \\ &=& \tau\big(\psi_y (w) \big)^y\tau\big(\psi_y (w') \big)^y \\ &=& \tau \big(w\,\theta_y (w)\big)\tau \big(w'\theta_y (w')\big) \\ &=& \tau\big(w\,\theta_y (w)\,w'\theta_y (w')\big) \\ &=& \tau \big(ww'\theta_y (w)^{w'}\theta_y (w')\big) \end{eqnarray*} and therefore, since $\tau$ is injective, we still get $$\theta_y (ww') = \theta_y (w)^{w'}\theta_y (w') \quad ;$$ in particular, for any $z\leqn Z$ we have $$\theta_y (wz) = \theta_y (w)^z\,\theta_y (z) = \theta_y (w) \quad .$$ In other words, the map $\theta_y$ determines a $Z\hbox{-}$valued $1\hbox{-}${cocycle\/} from the image $\tilde T$ of~$T$ in $\widetilde{\rm Aut}(Q) = {\rm Out} (Q)\,.$ {\cal S}mallskip Actually, the {cohomology class\/}~$\bar\theta_y$ of this $1\hbox{-}$cocycle does not depend on the choice of~$\psi_y\,;$ indeed, if another choice $\psi'_y$ determines~$\theta'_y\,\colon T\to Z$ then we clearly have $\psi'_y (T) = \psi_y (T)$ and, according to our argument above, there is $z\leqn C_H (Q)$ such that $$(T_\nu)^z = T_\nu\quad{\rm and}\quad \psi'_y = \psi_y\circ \kappa_z \quad ,$$ where $\kappa_z\,\colon T\to T$ denotes the conjugation by $z\,;$ actually, we still have $$[z,T]\leq H\cap T = Q \quad .$$ But, since $T_\nu$ is a defect pointed group of $\big(C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T\big)_\mu$ and, according to [4,~Theorem~1.2] and \cite[Proposition~6.5]{KP}, $\mu$ determines a {nilpotent block\/} of the group $C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T\,,$ we have $N_{C_H (Q){\,\hskip-1pt\cdot\hskip-1pt\,} T} (T_\nu) = C_H (T){\,\hskip-1pt\cdot\hskip-1pt\,} T\,.$ Thus, $z$ belongs to~$Z{\,\hskip-1pt\cdot\hskip-1pt\,} C_H (T)$ and we actually may assume that $z$ belongs to $Z\,.$ {\cal S}mallskip In this case, it follows from equality~£3.5.6 applied twice that \begin{eqnarray*} \tau \big(w\,\theta'_y (w)\big) &=& \tau\big(\psi'_y (w)\big)^y \\ &=& \tau\big(\psi_y (z w z^{-1})\big)^y \\ &=& \tau \big((z w z^{-1})\, \theta_y (z w z^{-1})\big) \end{eqnarray*} for any $w\leqn T$ and, since $\theta_y (z w z^{-1}) =\theta_y (w)$ and $\tau$ is injective, we get $$\theta'_y (w)\theta_y (w)^{-1} = w^{-1}z w z^{-1} = (z^{-1})^w z \quad .$$ Consequently, denoting by ${\cal T}_{\!L}$ the category where the objects are the subgroups of $\tau (P)$ and the set of morphisms ${\cal T}_{\! L} \big(\tau (R),\tau (T)\big)$ from $\tau (T)$ to $\tau (R)$ is just the corresponding {\leqt transporter\/} in $L\,,$ the correspondence sending an element $y\leqn {\cal T}_{\! L} \big(\tau (R),\tau (T)\big)$ to the cohomology class $\bar\theta_y$ of $\theta_y$ determines a map $$ \bar\theta_{_{R,T}} : {\cal T}_{\! L} \big(\tau (R),\tau (T)\big)\longrightarrow {{\cal B}bb H}^1 (\tilde T,Z) \quad .$$ {\cal S}mallskip Moreover, if $U$ is a subgroup of $P$ containing $Q$ and $t$ an element of~$L$ fulfilling $\tau (U){\cal S}ubset \tau (T)^t\,,$ as above we can choose $\big(\psi_t,\bar\pi (t)\big)\leqn {\cal E} (T,U)$ lifting~$\pi (t)$ in such a way that, for any~$u\leqn U\,,$ we have $$\tau \big(u\,\theta_t (u)\big) = \tau\big(\psi_t (u)\big)^t $$ for a suitable $\theta_t (u)\leqn Z\,;$ then, the composition $\big(\psi_y,\bar\pi (y)\big)\circ\big(\psi_t,\bar\pi (t)\big)$ lifts $\pi (yt)$ and, for any $u\leqn U\,,$ we may assume that (cf.~£3.5.4) \begin{eqnarray*} \tau \big(u\,\theta_{yt} (u)\big) &=& \tau\big((\psi_y\circ\psi_t) (u)\big)^{yt} \\ &=& \tau {\cal B}ig(\psi_t (u)\,\theta_y\big(\psi_t (u)\big){\cal B}ig)^t \\ &=& \tau \big(u\,\theta_t (u)\big) \tau {\cal B}ig(\theta_y \big(\psi_t (u)\big){\cal B}ig)^t \\ &=& \tau \bigg(u\,\theta_t (u)\,\pi (t)^{-1} {\cal B}ig(\theta_y \big(\psi_t (u)\big){\cal B}ig)\bigg)\quad ; \end{eqnarray*} finally, since $\tau$ is injective, using {additive notation\/} in $Z$ we get $$\theta_{yt} (u) = \theta_t (u) + \pi (t)^{-1}{\cal B}ig(\theta_y \big(\psi_t (u)\big){\cal B}ig) \quad .$$ {\cal S}mallskip Hence, denoting by $\tilde t$ the image of $t$ in $\widetilde{\rm Aut}(Q)$ and by $\psi_{\tilde t}\,\colon \tilde U\to \tilde T$ and ${\cal Z}(\tilde t)\,\colon Z\cong Z$ the corresponding group homomorphisms, we get the~{$1\hbox{-}$cocycle condition\/} $$\bar\theta_{yt} = \bar\theta_t + {{\cal B}bb H}^1 \big(\psi_{\tilde t}, {\cal Z} (\tilde t)\big) (\bar\theta_y) \quad ; \leqno £3.5.7$$ in particular, since $\theta_y (w) = 0$ whenever $y\leqn\tau (R)\,,$ it is easily checked from this condition that $\bar\theta_y$ only depends on the class of $y$ in the {\leqt exterior quotient\/} $$\tilde{\cal T}_{\! L} \big(\tau (R),\tau (T)\big) = \tau (R)\backslash {\cal T}_{\! L} \big(\tau (R),\tau (T)\big) .$$ Thus, respectively denoting by $\tilde L\,,$ $\tilde R\,,$ $\tilde T$ and $\tilde P$ the images of $L\,,$ $\tau(R)\,,$ $\tau(T)$ and~$\tau (P)$ in $\widetilde{\rm Aut}(Q)\,,$ the map $\bar\theta_{_{R,T}}$ above admits a factorization $${\cal S}kew4\tilde{\bar\theta}_{_{\tilde R,\tilde T}} : \tilde {\cal T}_{\!\tilde L} (\tilde R,\tilde T)\longrightarrow {{\cal B}bb H}^1 \big(\tilde T,Z\big) .$$ {\cal S}mallskip That is to say, let us consider the {exterior quotient\/} $\tilde{\cal T}_{\!\tilde L}$ of the category ${\cal T}_{\!\tilde L}$ and the {contravariant\/} functor $${\frak h^1_Z }: \tilde{\cal T}_{\!\tilde L}\longrightarrow {\cal A}b $$ to the category of Abelian groups ${\cal A}b$ mapping~$\tilde T$ on~${{\cal B}bb H}^1 \big(\tilde T,Z\big)\,;$ then, identifying the $\tilde{\cal T}_{\!\tilde L}\hbox{-}$morphism $\tilde y\leqn \tilde {\cal T}_{\!\tilde L} (\tilde R,\tilde T)$ with the obvious {$\tilde{\cal T}_{\!\tilde L}\hbox{-}$chain\/} ${\cal D}elta_1\longrightarrow \tilde{\cal T}_{\tilde L}$ --- the {functor\/} from the category ${\cal D}elta_1\,,$ formed by the objects $0$ and $1$ and a non-identity morphism $0\bullet 1$ from $0$ to $1\,,$ mapping $0$ on $T\,,$ $1$ on $R$ and $0\bullet 1$ on~$\tilde y$ --- the family $\bar \theta = \{\bar\theta_{\tilde y}\}_{\tilde y}\,,$ where $\tilde y$ runs over the set of all the $\tilde{\cal T}_{\!\tilde L}\hbox{-}$morphisms, defines a {$1\hbox{-}$cocycle\/} from $\tilde {\cal T}_{\!\tilde L}$ to ${\frak h^1_Z}$ since equalities~£3.5.7 guarantee that the {differential map\/} sends $\bar\theta$ to zero. {\cal S}mallskip We claim that this {$1\hbox{-}$cocycle\/} is a {$1\hbox{-}$coboundary\/}; indeed, for any subgroup~$\tilde R$ of $\tilde P\,,$ choose a set of representatives $\tilde X_{\tilde R}{\cal S}ubset \tilde L$ for the set of {double classes\/} $\tilde P\backslash \tilde L/\tilde R$ and, for any $\tilde n\leqn \tilde X_{\tilde R}\,,$ set $\tilde R_{\tilde n} = \tilde R\cap P^{\tilde n}\,,$ consider the $\tilde{\cal T}_{\tilde L}\hbox{-}$morphisms $\tilde n\,\colon \tilde R_{\tilde n}\to \tilde P$ and $\tilde\leqmath_{\tilde R_{\tilde n}}^{\tilde R} \,\colon \tilde R_{\tilde n} \to \tilde R$ respectively determined by $\tilde n$ and by the trivial element of~$\tilde L\,,$ and denote by $$({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{\tilde R_{\tilde n}}^{\tilde R}) : {{\cal B}bb H}^1 \big(\tilde R_{\tilde n},Z\big)\longrightarrow {{\cal B}bb H}^1 \big(\tilde R,Z\big) $$ the corresponding {transfer homomorphism\/} \cite[~Ch.~XII,~\S8]{CE}; then, setting $$\bar{\cal S}igma_{\tilde R} = {\vert P\vert \over \vert L\vert}{\,\hskip-1pt\cdot\hskip-1pt\,} {\cal S}um_{\tilde n\leqn \tilde X_{\tilde R}} \big(({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{\tilde R_{\tilde n}}^{\tilde R})\big) (\bar\theta_{\tilde n}) \quad ,$$ we claim that, for any ${\tilde y}\leqn\bar{\cal T}_{\!\tilde L}(\tilde R,\tilde T)\,,$ we have $$\bar\theta_{\tilde y} = \bar{\cal S}igma_{\tilde T} - \big({\frak h^1_Z} (\tilde y)\big)(\bar{\cal S}igma_{\tilde R}) \quad . \leqno £3.5.8$$ {\cal S}mallskip Indeed, note that ${\frak h^1_Z} (\tilde y)$ is the composition of the restriction {via\/} the $\tilde{\cal T}_{\tilde L}\hbox{-}$morphism $$\tilde\leqmath_{\tilde y\tilde T \tilde y^{-1}}^{\tilde R} : \tilde y\,\tilde T \,\tilde y^{-1}\longrightarrow \tilde R $$ determined by the trivial element of $L\,,$ with the conjugation determined by~$\tilde y\,,$ which we denote by~${\frak h^1_Z} (\tilde y_*)\,;$ thus, by the corresponding {Mackey equalities\/} \cite[Ch.~XII,~Proposition~9.1]{CE}, we get \begin{eqnarray*} &{\frak h^1_Z} (\tilde y){\cal B}ig({\cal S}um_{\tilde n\leqn \tilde X_R} \big(({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{\tilde R_{\tilde n}}^{\tilde R})\big) (\bar\theta_{\tilde n}){\cal B}ig) \\ =& {\frak h^1_Z} (\tilde y_*){\cal B}ig({\cal S}um_{\tilde n\leqn \tilde X_{\tilde R}}\, {\cal S}um_{\tilde r\leqn \tilde Y_{\tilde n}} \big(({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{\tilde P^{\tilde n \tilde r} \,\cap\, \tilde y\,\tilde T\, \tilde y^{-1}}^{\tilde y\, \tilde T\,\tilde y^{-1}}) \circ {\frak h^1_Z} (\tilde r)\big) (\bar\theta_{\tilde n}){\cal B}ig) \\ =& {\cal S}um_{\tilde n\leqn \tilde X_{\tilde R}}\, {\cal S}um_{\tilde r\leqn \tilde Y_{\tilde n}} \big(({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{\tilde P^{\tilde n\tilde r \tilde y} \,\cap\,\tilde T}^{\tilde T}) \circ {\frak h^1_Z} (\tilde r\tilde y)\big) (\bar\theta_{\tilde n})\quad , \end{eqnarray*} where, for any $\tilde n\leqn X_{\tilde R}\,,$ the subset $\tilde Y_{\tilde n}{\cal S}ubset \tilde R$ is a set of representatives for the set of {double classes\/} $\tilde R_{\tilde n} \backslash \tilde R/\,\tilde y\,\tilde T\, \tilde y^{-1}$ and, for any $\tilde r\leqn \tilde Y_{\tilde n}\,,$ we consider the $\tilde{\cal T}_{\tilde L}\hbox{-}$morphisms $$\tilde r: \tilde P^{\tilde n \tilde r} \cap \tilde y\,\tilde T\,\tilde y^{-1} \longrightarrow \tilde R_{\tilde n}\quad{\rm and}\quad \tilde r\tilde y : \tilde P^{\tilde n\tilde r \tilde y}\cap\tilde T \longrightarrow \tilde R_{\tilde n} \quad .$$ {\cal S}mallskip Moreover, setting $\tilde m = \tilde n\tilde r\tilde y$ for~$\tilde n\leqn \tilde X_{\tilde R}$ and $\tilde r\leqn \tilde Y_{\tilde n}\,,$ since we assume that $\theta_{\tilde r} = 0\,,$ it follows from equality~£3.5.7 that $$ \big({\frak h^1_Z} (\tilde r\tilde y)\big) (\bar\theta_{\tilde n}) = \bar\theta_{\tilde m} - \bar\theta_{\tilde r\tilde y} =\bar\theta_{\tilde m} - \big({\frak h^1_Z}(\tilde\leqmath_{\tilde T_{\tilde m}}^{\tilde T})\big)(\bar\theta_{\tilde y}) \quad ;$$ thus, choosing $\tilde X_{\tilde T} = \bigsqcup_{\,\tilde n\leqn \tilde X_{\tilde R}} \tilde n\,\tilde Y_{\tilde n}\,\tilde y\,,$ we get \cite[~Ch.~XII,~\S8.(6)]{CE} \begin{eqnarray*} \bar{\cal S}igma_{\tilde T} - \big({\frak h^1_Z} (\tilde y)\big) (\bar{\cal S}igma_{\tilde R}) &=& {\vert P\vert \over \vert L\vert}{\,\hskip-1pt\cdot\hskip-1pt\,} {\cal S}um_{\tilde m\leqn \tilde X_{\tilde T}} \big(({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{ \tilde T_{\tilde m}}^{\tilde T})\big){\cal B}ig(\bar\theta_{\tilde m} - \big({\frak h^1_Z} (\tilde r\tilde y)\big) (\bar\theta_{\tilde n}){\cal B}ig) \\ &=& {\vert P\vert \over \vert L\vert}{\,\hskip-1pt\cdot\hskip-1pt\,} {\cal S}um_{\tilde m\leqn \tilde X_{\tilde T}} \big(({\frak h}^1_Z)^{^{\!\circ}} (\tilde\leqmath_{\tilde T_{\tilde m}}^{\tilde T})\big){\cal B}ig(\big({\frak h^1_Z}(\tilde\leqmath_{\tilde T_{\tilde m}}^{\tilde T})\big) (\bar\theta_{\tilde y}){\cal B}ig) \\ &=& {\cal S}um_{\tilde m\leqn \tilde X_{\tilde T}}{\vert\tilde T/ \tilde T_{\tilde m}\vert \over\vert\tilde L/\tilde P\vert}{\,\hskip-1pt\cdot\hskip-1pt\,} \bar\theta_{\tilde y}=\bar\theta_{\tilde y} \quad . \end{eqnarray*} {\cal S}mallskip In particular, for any subgroup $\tilde R$ of $\tilde P\,,$ we get $$\bar {\cal S}igma_{\tilde R} = \big({\frak h^1_Z}(\tilde\leqmath_{\tilde R}^{\tilde P})\big)(\bar {\cal S}igma_{\tilde P}) $$ and the element $\bar{\cal S}igma_{\tilde P}\leqn {\cal B}bb H^1(\tilde P,Z)$ can be lifted to a $1\hbox{-}$cocycle ${\cal S}igma_{\tilde P}\,\colon \tilde P\to Z$ which determines a group automorphism ${\cal S}igma\,\colon P\cong P$ mapping $u\leqn P$ on $u\,{\cal S}igma_{\tilde P}(\tilde u)$ where $\tilde u$ denotes the image of $u$ in $\tilde P\,;$ moreover, according to equality~£3.5.8, in \noindent £3.5.5 we may choose $$\theta_y (w) = {\cal S}igma_{\tilde P}(\tilde w)\big(\pi (y)\big)^{-1}{\cal B}ig({\cal S}igma_{\tilde P} \big(\widetilde{\psi_y (w)}\big){\cal B}ig)^{-1} .$$ Hence, replacing $\tau$ by $\hat\tau = \tau\circ{\cal S}igma\,,$ the maps $\pi$ and $\hat\tau$ still fulfill the conditons above and, for any $w\leqn T\,,$ in~equality~£3.5.6 we get \begin{eqnarray*} \tau\big(\psi_y (w)\big)^y &=& \tau \big(w\,\theta_y (w)\big) \\ &=& \tau\bigg(w\big(w^{-1}{\cal S}igma (w)\big)\big(\pi (y)\big)^{-1} {\cal B}ig(\psi_y (w)^{-1} {\cal S}igma \big(\psi_y (w)\big){\cal B}ig)^{-1}\bigg) \\ &=& \tau \bigg({\cal S}igma (w)\big(\pi (y)\big)^{-1}{\cal B}ig({\cal S}igma \big(\psi_y (w) \big)^{-1}\psi_y (w){\cal B}ig)\bigg) \\ &=& \hat\tau (w)\tau {\cal B}ig({\cal S}igma\big(\psi_y (w)\big)^{-1} \psi_y (w){\cal B}ig)^y \\ &=& \hat\tau (w) \hat\tau\big(\psi_y (w)^{-1}\big)^y \tau\big(\psi_y (w) \big)^y \end{eqnarray*} so that, as announced, we obtain $$\hat\tau\big(\psi_y (w)\big)^y = \hat\tau (w) \quad .$$ {\cal S}mallskip In conclusion, we get a functor from $\hat{\cal E}$ to ${\cal E}$ mapping any $\hat {\cal E}\hbox{-}$morphism $$(\kappa_y,\bar y) : \hat\tau (T)\longrightarrow \hat\tau (R) $$ induced by an element $y$ of $L\,,$ where $\kappa_y$ denotes the corresponding conjugation by~$y$ which actually fulfills $\hat\tau (Q{\,\hskip-1pt\cdot\hskip-1pt\,} T)\leq \big(\hat\tau (Q{\,\hskip-1pt\cdot\hskip-1pt\,} R)\big)^y\,,$ on the ${\cal E}\hbox{-}$morphism $$\big(\psi_y,\bar\pi (y)\big) : T\longrightarrow R $$ where $\psi_y\,\colon T\to R$ is the group homomorphism determined by the equality $$\hat\tau_R \circ\psi_y = \kappa_y\circ \hat\tau_T \quad ,$$ $\hat\tau_R$ and $\hat\tau_T$ denoting the respective restrictions of $\hat\tau$ to $R$ and $T\,;$ indeed, it is clear that this correspondence maps the composition of $\hat{\cal E}\hbox{-}$morphisms on the corresponding composition of ${\cal E}\hbox{-}$morphisms. Moreover, it is clear that this functor is {faithful\/}, and it follows from our argument above that any ${\cal E}\hbox{-}$morphism $$(\psi_x,\bar x) : T\longrightarrow R $$ comes from an $\hat{\cal E}\hbox{-}$morphism from $\hat\tau (T)$ to $\hat\tau (R)\,.$ {\cal S}mallskip Moreover, for another triple $L'\,,$ $\tau'$ and $\bar\pi'$ fulfilling the above conditions, the corresponding equivalences of categories~£3.5.2 induce an equivalence of categories $$\hat{\cal E}\cong {\cal E}_{(1,\tau' (Q),L')} = {\cal E}' \quad ;\leqno £3.5.9$$ in particular, we have a group homomorphism $$\bar{\cal S}igma : L\longrightarrow \hat{\cal E} \big(\hat\tau (Q)\big)\cong {\cal E}' \big(\tau' (Q)\big)\cong L'/\tau' (Z) $$ and we claim that Lemma~£3.6 below applies to the finite groups $L$ and $L'\,,$ with the Sylow $p\hbox{-}$subgroup $\hat\tau (P)$ of $L\,,$ the Abelian normal $p\hbox{-}$group $\tau' (Z)$ of~$L'$ and the group homomorphism $\bar{\cal S}igma\,\colon L\to L'/\tau' (Z)$ above; indeed, the group homomorphism $\hat\tau (P)\to L'$ mapping $\hat\tau (u)$ on $\tau' (u)\,,$ for any $u\leqn P\,,$ clearly lifts the restriction of $\bar{\cal S}igma$ and it is easily checked from the equi-valence~£3.5.9 that it fulfills condition~£3.6.1 below. Consequently, the last statement immediately follows from this lemma. We are done. \noindent {\bf Lemma~£3.6.}\quad {\leqt Let $L$ be a finite group, $M$ a group, $Z$ a normal Abelian $p'\hbox{-}$divisible subgroup of $M$~and $\bar{\cal S}igma\,\colon L\to \bar M = M/Z$ a group homomorphism. Assume that, for a Sylow $p\hbox{-}$subgroup $P$ of $L\,,$ there exists a group homomorphism $\tau\,\colon P\to M$ lifting the restriction of $\bar{\cal S}igma$ to $P$ and fulfilling the following condition {\cal S}mallskip \noindent {\bf £3.6.1}\quad For any subgroup $R$ of $P$ and any $x\leqn L$ such that $R^x{\cal S}ubset P\,,$ there is $y\leqn M$ such that $\bar{\cal S}igma (x) = \bar y$ and $\tau (u^x) = \tau (u)^y$~for any $u\leqn R\,.$ {\cal S}mallskip \noindent Then, there is a group homomorphism ${\cal S}igma\,\colon L\to M$ lifting $\bar{\cal S}igma$ and extending~$\tau\,.$ Moreover, if ${\cal S}igma'\,\colon L\to M$ is a group homomorphism which lifts $\bar{\cal S}igma$ and extends~$\tau\,,$ then there is $z\leqn Z$ such that ${\cal S}igma' (x) = {\cal S}igma (x)^z$ for any $x\leqn L\,.$\/} \noindent {\bf Proof:} It is clear that $\bar{\cal S}igma$ determines an action of $L$ on $Z$ and it makes sense to consider the {cohomology groups\/} ${\cal B}bb H^n (L,Z)$ and ${\cal B}bb H^n (P,Z)$ for any $n$ in~${\cal B}bb N\,.$ But, $M$ determines an element $\bar\mu$ of~${{\cal B}bb H}^2 (\bar M,Z)$ \cite[~Chap.~XIV, Theorem~4.2]{CE} and if there is a group homomorphism $\tau\,\colon P\to M$ lifting the restriction of~$\bar{\cal S}igma$ then the corresponding image of $\bar\mu$ in ${{\cal B}bb H}^2 (P,Z)$ has to be zero \cite[Chap.~XIV, Theorem~4.2]{CE}; thus, since the restriction map $${{\cal B}bb H}^2 (L,Z)\longrightarrow {{\cal B}bb H}^2 (P,Z) $$ is injective \cite[~Ch.~XII,~Theorem~10.1]{CE}, we also get $$\big({{\cal B}bb H}^2 (\bar{\cal S}igma,{\rm id}_Z)\big)(\bar\mu) = 0 $$ and therefore there is a group homomorphism ${\cal S}igma\,\colon L\to M$ lifting $\bar{\cal S}igma\,.$ {\cal S}mallskip At this point, the {difference\/} between $\tau$ and the restriction of ${\cal S}igma$ to $P$ defines a {$1\hbox{-}$cocycle\/} $\theta\,\colon P\to Z$ and, for any subgroup $R$ of $P$ and any $x\leqn L$ such that $R^x{\cal S}ubset P\,,$ it follows from condition~£3.6.1 that, for a suitable $y\leqn M$ fulfilling $\bar y = \bar{\cal S}igma (x)\,,$ for any $u\leqn R$ we have \begin{eqnarray*} \theta (u^x) &=& \tau (u^x)^{-1}{\cal S}igma (u^x)\\ & =& \tau (u^{-1})^y{\cal S}igma (u)^{{\cal S}igma (x)} \\ &=& \tau (u^{-1})^y \tau (u)^{{\cal S}igma (x)}\theta (u)^{{\cal S}igma (x)} \\ &=& {\cal B}ig(\big(y{\cal S}igma (x)^{-1}\big)^{-1}\big(y{\cal S}igma (x)^{-1}\big)^{\tau (u)}\theta (u){\cal B}ig)^{{\cal S}igma (x)}\quad ; \end{eqnarray*} consequently, since the map sending $u\leqn R$ to $$\big(y{\cal S}igma (x)^{-1}\big)^{-1}\big(y{\cal S}igma (x)^{-1}\big)^{\tau (u)}\leqn Z $$ is a {$1\hbox{-}$coboundary\/}, the cohomology class $\bar\theta$ of $\theta$ is $L\hbox{-}${stable\/}, and it follows again from \cite[~Ch.~XII,~Theorem~10.1]{CE} that it is the restriction of a suitable element $\bar \eta\leqn {{\cal B}bb H}^1 (L,Z)\,;$ then, it suffices to modify ${\cal S}igma$ by a representative of~$\bar\eta$ to~get a new group homomorphism ${\cal S}igma'\,\colon L\to M$ lifting $\bar{\cal S}igma$ and extending~$\tau\,.$ {\cal S}mallskip Now, if ${\cal S}igma'\,\colon L\to M$ is a group homomorphism which lifts $\bar{\cal S}igma$ and extends~$\tau\,,$ the element ${\cal S}igma' (x){\cal S}igma (x)^{-1}$ belongs to $Z$ for any $x\leqn L$ and thus, we get a {$1\hbox{-}$cocycle\/} $\lambda\, \,\colon L\to Z$ mapping $x\leqn L$ on ${\cal S}igma' (x){\cal S}igma (x)^{-1}\,,$ which vanish over~$P\,;$ hence, it is a {$1\hbox{-}$coboundary\/} \cite[~Ch.~XII,~Theorem~10.1]{CE} and therefore there exists $z\leqn Z$ such that $$\lambda (x) = z^{-1}{\cal S}igma (x) z{\cal S}igma (x)^{-1} $$ so that we have ${\cal S}igma' (x) ={\cal S}igma (x)^z$ for any $x\leqn L\,.$ We are done. \noindent {\bf £3.7.}\quad Since $Q$ normalizes a unitary {full matrix\/} ${\cal O}\hbox{-}$subalgebra $T$ of ${\cal B}_\delta$ such that \cite[~Theorem~1.6]{P4} $${\cal B}_\delta\cong T\,Q\quad{\rm and}\quad {\rm rank}_{\cal O} (T)\equiv 1 \bmod p \quad ,\leqno £3.7.1$$ the action of $Q$ on $T$ admits a unique lifting to a group homomorphism \cite[1.8]{P4} $$Q\longrightarrow {\rm Ker}({\rm det}_T) \quad ;$$ hence, we have $${\cal B}_\delta\cong T\otimes_{\cal O} {\cal O} Q$$ and therefore ${{\cal B}}_\delta$ admits a unique two-sided ideal ${\frak n}_\delta$ such that, considering ${\cal B}_\delta/{\frak n}_\delta$ as a $Q$-interior ${\cal O}$-algebra, there is an isomorphism $${\cal B}_\delta/{\frak n}_\delta\cong T$$ of $Q$-interior ${\cal O}$-algebras. Then, a canonical {embedding\/} $f_\delta\,\colon {\cal B}_\delta\to {\rm Res}_Q^H ({\cal B})$ \cite[~2.8]{P4} and the ideal ${\frak n}_\delta$ determine a two-sided ideal ${\frak n}$ of $\cal B$ such that $S = {\cal B}/{\frak n}$ is also a {full matrix\/} ${\cal O}\hbox{-}$algebra. \noindent {\bf Proposition~£3.8.} {\leqt With the notation above, the action of $N$ on $\cal B$ stabilizes~${\frak n}\,.$\/} \noindent {\bf Proof:} Since we have $N = H{\,\hskip-1pt\cdot\hskip-1pt\,} N_G (Q_\delta)\,,$ for the first statement we may consider $x\leqn N_G (Q_\delta)\,;$ then, denoting by ${\cal S}igma_x$ the automorphism of $Q$ induced by the conjugation by $x\,,$ it is clear that the isomorphism $$f_x : {\rm Res}_{{\cal S}igma_x}\big({\rm Res}_Q^H ({\cal B})\big)\cong {\rm Res}_Q^H ({\cal B}) $$ of $Q$-interior algebras mapping $a\leqn \cal B$ on $a^x$ induces a commutative diagram of {\leqt exterior\/} homomorphisms of $Q$-interior algebras \cite[2.8]{P4} $$\matrix{{\rm Res}_{{\cal S}igma_x}\big({\rm Res}_Q^H ({\cal B})\big)&\buildrel \tilde f_x\over\cong &{\rm Res}_Q^H ({\cal B})\cr \hskip-10pt{{\cal S}criptstyle \tilde f_\delta}\hskip4pt\uparrow&\phantom{{\cal B}ig\uparrow}&\uparrow\hskip4pt {{\cal S}criptstyle \tilde f_\delta}\hskip-10pt\cr {\rm Res}_{{\cal S}igma_x} ({\cal B}_\delta)&\buildrel (\tilde f_x)_\delta\over\cong& {\cal B}_\delta\cr} \quad ;$$ moreover, the uniqueness of ${\frak n}_\delta$ clearly implies that this ideal is stabilized by~$(\tilde f_x)_\delta\,;$ consequently, ${\frak n} $ is still stabilized by~$\tilde f_x\,.$ \noindent {\bf £3.9.} In particular, $N$ acts on the {full matrix\/} ${\cal O}\hbox{-}$algebra $S$ and therefore the action on $S$ of any element $x\leqn N$ can be lifted to a suitable $s_x\leqn S^*\,;$ thus, setting ${\rm r }= {\rm rank}_{\cal O}(S)\,,$ denoting by $\bar H$ the image of $H$ in $S^*$ and considering a finite extension ${\cal O}'$ of ${\cal O}$ containing the group $U$ of $\vert H\vert\hbox{-}$th roots of unity and the ${\rm r}\hbox{-}$th roots of ${\rm det}_S (s_x)$ for any $x\leqn N\,,$ since ${\rm r}$ divides $\vert H\vert\,,$ the {\leqt pull-back\/} $$\matrix{N &\longrightarrow & {\rm Aut}({\cal O}'\otimes_{\cal O} S)\cr \uparrow&\phantom{\big\uparrow}&\uparrow\cr \hat N &\longrightarrow &(U\otimes\bar H){\,\hskip-1pt\cdot\hskip-1pt\,}{\rm Ker}({\rm det}_{{\cal O}'\otimes_{\cal O} S})\cr}$$ determines a central extension $\hat N$ of $N$ by $U\,,$ which clearly does not depend on the choice of ${\cal O}'\,;$ moreover, the inclusion $H\leq N$ and the structural group homomorphism $H\to ({\cal O}'\otimes_{\cal O} S)^*$ induces an injective group homomorphism $H\to \hat N$ with an image which is a normal subgroup of~$\frak c\frak heck N$ and has a {trivial\/} intersection with the image of $U$ --- we identify this image with $H$ and set $${\cal S}kew3\hat{\bar N} = \hat N/H\quad .$$ We will consider the $H\hbox{-}$interior $N\hbox{-}$algebras (see \cite[2.1]{P7}) $$\hat{\cal A} = S^\circ\otimes_{\cal O} {\cal A}\quad{\rm and}\quad \hat{\cal B} = S^\circ\otimes_{\cal O} {\cal B}$$ and note that ${\cal O}'\otimes \hat {\cal A}$ actually has an $\hat N\hbox{-}$interior algebra structure. \noindent {\bf £3.10.}\quad On the other hand, since $b$ is also a {nilpotent\/} block of the group $H{\,\hskip-1pt\cdot\hskip-1pt\,} P\,,$ it is easily checked that \cite[1.9]{P4} $${\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)b\big/J\big({\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)b\big)\cong k\otimes_{\cal O} S \quad ;$$ moreover, since the inclusion map ${\cal O} H\to {\cal O} (H{\,\hskip-1pt\cdot\hskip-1pt\,} P)$ is a {semicovering of $P\hbox{-}$algebras\/} \cite[Example~3.9, 3.10 and~Theorem~3.16]{KP}, we can identify $\gamma$ with a local point of $P$ on ${\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)b$. Set ${\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)_\gamma=i({\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P))i$ and $S_\gamma=\bar\leqmath S\bar\leqmath$, where $\bar\leqmath$ is the image of $i$ in $S\,;$ then, as in £3.7 above, we have an isomorphism of $P$-interior algebras \cite[Theorem~1.6]{P4} $${\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)_\gamma\cong S_\gamma\, P \quad ,\leqno £3.10.1$$ $S_\gamma$ is actually a {\leqt Dade $P\hbox{-}$algebra\/} --- namely, a {full matrix\/} $P\hbox{-}$algebra over ${\cal O}$ where $P$ stabilizes an ${\cal O}\hbox{-}$basis containing the unity element --- such that ${\rm rank}_{\cal O}(S_\gamma)\equiv 1\,\, {\rm mod}\,\, p$, and the action of $P$ on $S_\gamma$ can be uniquely lifted to a group homomorphism $P\to {\rm Ker}({\rm det}_{S_\gamma})$ \cite[1.8]{P4}, so that isomorphism~£3.10.1 becomes $${\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)_\gamma\cong S_\gamma\otimes_{\cal O} {\cal O} P¡£ \quad .\leqno £3.10.2$$ \noindent {\bf Proposition~£3.11.}\quad {\leqt With the notation above, the structural homomorphism ${\cal B}_\gamma\to S_\gamma$ of $P\hbox{-}$algebras is a strict semicovering.\/} \noindent {\bf Proof:} It follows from isomorphism~£3.10.2 that the canonical homomorphism of $P\hbox{-}$algebras $${\cal O}(H{\,\hskip-1pt\cdot\hskip-1pt\,} P)_\gamma\longrightarrow S_\gamma \leqno £3.11.1\phantom{.}$$ admits a $P\hbox{-}$algebra section mapping $s\leqn S_\gamma$ on the image of $s\otimes 1$ by the inverse of that isomorphism, which proves that the $P$-interior algebra homomorphism~£3.11.1 is a {covering\/} \cite[4.14 and Example~4.25]{P4}; thus, since the inclusion map ${\cal O} H\to {\cal O} (H{\,\hskip-1pt\cdot\hskip-1pt\,} P)$ is a semicovering of $P\hbox{-}$algebras, the canonical homomorphism of $P\hbox{-}$algebras $${\cal B}_\gamma = ({\cal O} H)_\gamma\longrightarrow S_\gamma $$ remains a {semicovering\/} \cite[~Proposition~3.13]{KP}; moreover, since ${\frak n}\leq J(\cal B)\,,$ it is a {strict semicovering\/} \cite[~3.10]{KP}. \noindent {\bf £3.12.}\quad Consequently, it easily follows from \cite[~Theorem~3.16]{KP} and \cite[~Proposition~5.6]{P4} that we still have a {strict semicovering\/} homomorphism of $P\hbox{-}$algebras $$(S_\gamma)^\circ\otimes_{\cal O} {\cal B}_\gamma\longrightarrow (S_\gamma)^\circ\otimes_{\cal O} S_\gamma\cong {\rm End}_{\cal O} (S_\gamma) \quad ;\leqno £3.12.1$$ hence, denoting by $\hat \gamma$ the local point of $P$ over $(S_\gamma)^\circ\otimes_{\cal O} {\cal B}_\gamma$ determined by~$\gamma\,,$ the image of $\hat\gamma$ in $(S_\gamma)^\circ\otimes_{\cal O} S_\gamma$ is contained in the corresponding local point of $P$ and therefore we get a {strict semicovering\/} homomorphism \cite[~5.7]{P4} $$\hat{\cal B}_{\hat\gamma}\longrightarrow {\cal O} \cong ((S_\gamma)^\circ\otimes_{\cal O} S_\gamma)_{\hat\gamma}$$ of $P\hbox{-}$algebras; that is to say, any $\hat\leqmath\leqn \hat \gamma$ is actually a primitive idempotent in~$\hat{\cal B}$ and therefore, for any local pointed group $R_{\hat\varepsilon}$ over $\hat{\cal B}$ contained in~$P_{\hat\gamma}\,,$ it also belongs to $\hat\varepsilon\,;$ in particular, denoting by $\hat \delta$ the local point of $Q$ over $(S_\gamma)^\circ\otimes_{\cal O} {\cal B}_\gamma$ determined by $\delta\,,$ we clearly have $\hat{\cal B}_{\hat\delta} = \hat\leqmath\hat{\cal B}\hat\leqmath\cong {\cal O} Q$ (cf.~£3.7.1). \noindent {\bf £3.13.}\quad As~in~\cite[~2.11]{KP}, we consider the $P$-interior algebra $\hat{\cal A}_{\hat\gamma} = \hat\leqmath\hat{\cal A}\hat\leqmath\,;$ since $\cal A$ is an $N/H$-graded algebra, $\hat{\cal A}_{\hat\gamma}$ is also an $N/H$-graded algebra. On the other hand, since ${\cal O}'/J({\cal O}')\cong k\,,$ we get a group homomorphism $\varpi\,\colon U\to k^*$ and, setting ${\cal D}elta_\varpi (U) = \{(\varpi (\xi),\xi^{-1})\}_{\xi\leqn U}\,,$ we obtain the obvious $k^*\hbox{-}$group $${\cal S}kew3\hat{\bar N}^{^k} =( k^*\times {\cal S}kew3\hat{\bar N})/{\cal D}elta_\varpi (U) \quad ; $$ then, with the notation of Theorem~£3.5, we set \cite[~5.7]{P6} $$\hat L = {\rm Res}_{\bar\pi} ({\cal S}kew3\hat{\bar N}^{^k}) \quad ;\leqno £3.13.1$$ thus, ${\cal O}_*\hat L^{^\circ}$ becomes a $P$-interior algebra {via\/} the lifting $\hat\tau\,\colon P\to \hat L^{^\circ}$ of the group homomorphism $\tau\,\colon P\to L\,,$ and it has an obvious $L/\tau(Q)$-graded algebra structure. The group homomorphism $\bar \pi$ induces a group isomorphism $L/\tau(Q)\cong N/H$, through which we identify $L/\tau(Q)$ and $N/H\,,$ so that ${\cal O}_*\hat L^{^\circ}$ becomes an $N/H$-graded algebra. \noindent {\bf Theorem~£3.14.}\quad {\leqt With the notation above, we have a $P$-interior and $N/H$-graded algebra isomorphism $\hat{\cal A}_{\hat\gamma}\cong {\cal O}_*\hat L^{^\circ}\,.$ } \noindent {\bf Proof:} Choosing $\hat\leqmath\leqn \hat\gamma\,,$ we consider the groups $$M = N_{(\hat\leqmath\hat{\cal A} \hat\leqmath)^*} (Q{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath)/k^*{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath\quad{\rm and}\quad Z = \big((\hat\leqmath\hat{\cal B}\hat\leqmath)^Q\big)^*\! \big/k^*{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath\cong 1 + J\big(Z ({\cal O} Q)\big) \quad ;$$ it is clear that $Z$ is a normal Abelian $p'\hbox{-}$divisible subgroup of $M\,,$ and we set~$\bar M = M/Z\,.$ In order to apply Lemma~£3.6, let $R$ be a subgroup of $P$ and $y$ an element of $L$ such that $\tau (R)\leq \tau (P)^y\,;$ since $\tau (Q)$ is normal in~$L\,,$ we actually may assume that $R$ contains $Q\,.$ According to the equivalence of categories~£3.5.2, denoting by $\varepsilon$ the unique local point of $R$ on $\cal B$ fulfilling $R_\varepsilon\leq P_\gamma$ \cite[~Theorem~6.6]{KP}, there is $x_y\leqn N$ such that $$\bar x_y = \bar\pi (y)\quad,\quad R_\varepsilon\leq (P_\gamma)^{x_y} \quad{\rm and}\quad \tau ({}^{x_y}v) ={}^y\tau (v) \hbox{\ \ for any $v\leqn R$} \quad ;\leqno £3.14.1$$ in particular, $x_y$ normalizes $Q_\delta\,.$ {\cal S}mallskip By Proposition 3.11, a local pointed group $R_\varepsilon$ on $\cal B$ such that $$Q_\delta\leq R_\varepsilon\leq P_\gamma$$ determines a local pointed group $R_{\tilde \varepsilon}$ on $S$ through the composition $${\cal B}_\gamma\longrightarrow S_\gamma\hookrightarrow S$$ (see \cite[Proposition 3.15]{KP}). Since $S_\gamma$ has a $P$-stable ${\cal O}$-basis, $S_\varepsilon$ still has a $R$-stable ${\cal O}$-basis and, by \cite[Theorem 5.3]{P4}, there are unique local pointed groups $R_{\tilde\varepsilon}$ on $S_\varepsilon$ and $R_{\hat\varepsilon}$ on $\hat{\cal B}$ such that $\hat l(\tilde l\otimes l)=\hat l=(\tilde l\otimes l)\hat l$ for suitable $l\leqn \varepsilon$, $\tilde l\leqn \tilde\varepsilon$ and $\hat l\leqn \hat\varepsilon\,;$. then, we claim that $R_{\hat\varepsilon}\leq (P_{\hat\gamma})^{x_y}$ and that $x_y$ stabilizes~$Q_{\hat\delta}\,.$ Indeed, since $(R_\varepsilon)^{x_y^{-1}}\leq P_\gamma$, we have $(R_{\tilde\varepsilon})^{x_y^{-1}}\leq P_{\tilde\gamma}$ and then it follows from \cite[Proposition 5.6]{P4} that we have $(R_{\hat\varepsilon})^{x_y^{-1}}\leq P_{\hat\gamma}$ or, equivalently, $R_{\hat\varepsilon}\leq (P_{\hat\gamma})^{x_y}\,;$ moreover, since $\delta$ is the unique local point of $Q$ such that $Q_\delta$ is contained in~$P_\gamma\,,$ again by \cite[Proposition 5.6]{P4} we can easily conclude that $x_y$ stabilizes $Q_{\hat\delta}\,.$ {\cal S}mallskip In particular, since the image of $\hat\leqmath^{\,x_y}$ in $\hat{\cal B} (R_{\hat\varepsilon})$ is not zero [14,~2.7] and since $\hat\leqmath$ is primitive in $\hat{\cal B}\,,$ $\hat\leqmath^{\,x_y}$ belongs to $\hat\varepsilon$ and therefore, since $\hat\leqmath$ also belongs to $\hat\varepsilon\,,$ there is $\hat a_y\leqn (\hat{\cal B}^R)^*$ such that $\hat\leqmath^{\,x_y} = \hat\leqmath^{\,\hat a_y}\,;$ choose $s_y\leqn S^*$ lifting the action of $x_y$ on $S$ and set $\hat x_y = s_y\otimes x_y\,,$ so that we have $$\hat\leqmath^{\,x_y} = (\hat x_y)^{-1} \hat\leqmath\, \hat x_y \quad ;$$ then, since $\hat x_y$ and $\hat a_y$ normalize~$Q\,,$ the element $\hat x_y \hat a_y^{-1}$ of $\hat A$ normalizes $Q{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath$ and therefore $\hat x_y \hat a_y^{-1}\hat\leqmath$ determines an element $m_y$ of~$M\,.$ We claim that the image $\bar m_y$ of $m_y$ in $\bar M$ only depends on~$y\leqn L$ and that, in the case where $R_\varepsilon =Q_\delta\,,$ this correspondence determines a group homomorphism $$\bar{\cal S}igma : L\longrightarrow \bar M \quad .$$ {\cal S}mallskip Indeed, if $x'\leqn N$ still fulfills conditions~£3.14.1 then we necessarily have $x' = x_y\,z$ for some $z\leqn C_H (R)$ and therefore it suffices to choose the element $\hat a_y{\,\hskip-1pt\cdot\hskip-1pt\,} z$ of $(\hat B^R)^*$ in the definition above. On the other hand, if $\hat a'\leqn (\hat B^R)^*$ still fulfills $\hat\leqmath^{\,\hat x_y} = \hat\leqmath^{\,\hat a'}$ then we clearly have $\hat a' = \hat c\,\hat a_y$ for some $\hat c\leqn (\hat B^R)^*$ centralizing $\hat\leqmath\,,$ so that $\hat c\,\hat\leqmath$ belongs to $(\hat\leqmath\hat B\hat\leqmath)^Q\,;$ hence, the image of $\hat x_y\hat a_y^{-1}\hat c^{-1}\hat\leqmath$ in $\bar M$ coincides with $\bar m_y\,.$ Moreover, in the case where $R_\varepsilon =Q_\delta\,,$ for any element $y'$ in $L$ we clearly can choose $\hat x_{yy'} = \hat x_y\, \hat x_{y'}\,;$ then, we have $$\hat\leqmath^{\,\hat x_{yy'}} = (\hat\leqmath^{\,\hat a_y})^{\hat x_{y'}} = \hat\leqmath^{\hat x_{y'}{\,\hskip-1pt\cdot\hskip-1pt\,} (\hat a_y)^{\hat x_{y'}}} = \hat\leqmath^{\hat a_{y'}(\hat a_y)^{\hat x_{y'}}}$$ and therefore, since $\hat a_{y'}(\hat a_y)^{\hat x_{y'}}$ still belongs to $(\hat B^Q)^*\,,$ we clearly can choose $\hat a_{yy'} = \hat a_{y'}(\hat a_y)^{\hat x_{y'}}\,,$ so that we get $$\hat x_{yy'}{\,\hskip-1pt\cdot\hskip-1pt\,} \hat a_{yy'}^{-1}\hat\leqmath = \hat x_y\,\hat x_{y'}{\,\hskip-1pt\cdot\hskip-1pt\,} \big(\hat a_{y'} (\hat a_y)^{\hat x_{y'}}\big)^{-1}\hat\leqmath = (\hat x_y{\,\hskip-1pt\cdot\hskip-1pt\,} \hat a_y^{-1}\hat\leqmath)(\hat x_{y'}{\,\hskip-1pt\cdot\hskip-1pt\,} \hat a_{y'}^{-1}\hat\leqmath)$$ which implies that $\bar m_{yy'} = \bar m_y\,\bar m_{y'}\,.$ This proves our claim. {\cal S}mallskip In particular, for any $u\leqn P\,,$ we can choose $x_{\tau (u)} = u$ and $\hat a_{\tau (u)} = 1\,;$ moreover, since the action of $P$ on $S_\gamma$ can be lifted to a unique group~homomorphism $\varrho \,\colon P\to {\rm Ker}({\rm det}_{S_\gamma})$ \cite[~1.8]{P4}, we may choose $\hat x_{\tau (u)} = \varrho (u)\otimes u\,;$ then, it is clear that the correspondence $\tau^*$ mapping $\tau (u)$ on the image of $(\varrho (u)\otimes u) \hat\leqmath$ in $M$ defines a group homomorphism from $\tau (P)\leq L$ to~$M$ lifting the corresponding restriction of $\bar{\cal S}igma\,.$ {\cal S}mallskip Finally, we claim that $\tau^*$ fulfills condition~£3.6.1; indeed, coming back to the general inclusion $\tau (R)\leq \tau (P)^y$ above, we clearly have $\bar{\cal S}igma (y) = \bar m_y$ and, according to the right-hand equalities in~£3.14.1, for any $v\leqn R$ we get $$\tau^*\big(\tau (v)^y\big) = v^{x_y}{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath = (v{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath)^{m_y} =\tau^*\big(\tau (v)\big)^{m_y} \quad .$$ Consequently, it follows from Lemma~£3.6 that $\bar{\cal S}igma$ can be lifted to a group homomorphism ${\cal S}igma\,\colon L\to M$ extending $\tau^*\,;$ moreover, the inverse image of~${\cal S}igma (L)$ in $N_{(\hat\leqmath\hat{\cal A} \hat\leqmath)^*} (Q{\,\hskip-1pt\cdot\hskip-1pt\,}\hat\leqmath)$ is a $k^*\hbox{-}$group which is clearly contained in $$\hat N{\,\hskip-1pt\cdot\hskip-1pt\,}({\cal O}'^*\otimes 1){\cal S}ubset {\cal O}'\otimes_{\cal O} \hat{\cal A} \quad ;$$ hence, according to definition~£3.13.1, ${\cal S}igma$ still can be lifted to a $k^*\hbox{-}$group homomorphism $$\hat{\cal S}igma : \hat L^{^\circ}\longrightarrow N_{(\hat\leqmath\hat{\cal A} \hat\leqmath)^*} (Q{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath) $$ mapping $\tau (u)$ on $u{\,\hskip-1pt\cdot\hskip-1pt\,} \hat\leqmath$ for any $u\leqn P\,;$ hence, we get a $P$-interior and $N/H$-graded algebra homomorphism $${\cal O}_*\hat L^{^\circ}\longrightarrow \hat{\cal A}_{\hat\gamma} \quad .\leqno £3.14.2$$ We claim that homomorphism £3.14.2 is an isomorphism. {\cal S}mallskip Indeed, denoting by $X\leq N_G (Q_\delta)$ a set of representatives for $\bar N = N/H\,,$ it is clear that we have $${\cal A} = \bigoplus_{x\leqn X} x{\,\hskip-1pt\cdot\hskip-1pt\,} {\cal B} $$ and therefore we still have $$\hat{\cal A} = S\otimes_{\cal O} {\cal A} = \bigoplus_{x\leqn X} (s_x\otimes x) (S\otimes_{\cal O} {\cal B}) = \bigoplus_{x\leqn X} (s_x\otimes x) \hat{\cal B} \quad ;$$ moreover, choosing as above an element $\hat a_x\leqn (\hat{\cal B}^Q)^*$ such that $\hat\leqmath^{\, x} = \hat\leqmath^{\,\hat a_x}\,,$ it is clear that $(s_x\otimes x) \hat a_x^{-1}\hat{\cal B} =(s_x\otimes x) \hat{\cal B} $ for any $x\leqn X$ and therefore we get $$\hat{\cal A}_{\hat\gamma} = \hat\leqmath \hat{\cal A}\hat\leqmath = \bigoplus_{x\leqn X} ((s_x\otimes x) \hat a_x^{-1}\hat\leqmath) (\hat\leqmath\hat{\cal B}\hat\leqmath) \quad ;$$ thus, since we know that $\hat\leqmath\hat{\cal B}\hat\leqmath\cong {\cal O} Q$ and that $L/\tau (Q)\cong \bar N\,,$ denoting by $Y\leq L$ a set of representatives for $L/\tau (Q)$ and by $\hat y$ a lifting of $y\leqn Y$ to $\hat L\,,$ we still get $$\hat{\cal A}_{\hat\gamma} \cong \bigoplus_{y\leqn Y} \hat{\cal S}igma (\hat y)\, {\cal O} Q $$ which proves that homomorphism~£3.14.2 is an isomorphism. \noindent {\bf Corollary~£3.15.}\quad {\leqt With the notation above, we have a $P$-interior and $N/H$-graded algebra isomorphism ${\cal A}_\gamma\cong S_\gamma\otimes_{\cal O} {\cal O}_*\hat L^{^\circ}\,.$\/} \noindent {\bf Proof:} Since $\hat{\cal A} = S^\circ\otimes_{\cal O} {\cal A}$ and we have a $P$-interior algebra embedding ${\cal O}\to S_\gamma\otimes_{\cal O} S_\gamma^\circ$ \cite[~5.7]{P4}, we still have the following commutative diagram of {\leqt exterior\/} $P$-interior algebra embeddings and homomorphisms \cite[2.10]{KP} $$\matrix{&{\cal A}_\gamma&\longrightarrow&\hskip-20pt S_\gamma\otimes_{\cal O} S_\gamma^\circ \otimes_{\cal O} {\cal A}_\gamma&\cr &\hskip-20pt\nearrow&\nearrow\hskip-60pt&\uparrow\cr {\cal B}_\gamma&\longrightarrow& S_\gamma\otimes_{\cal O} S_\gamma^\circ \otimes_{\cal O} {\cal B}_\gamma& S_\gamma\otimes_{\cal O} \hat{\cal A}_{\hat\gamma}&\hskip-20pt\cong&\hskip-20pt S_\gamma\otimes_{\cal O} {\cal O}_*\hat L^{^\circ}\cr &&&\hskip-40pt\nearrow&\nearrow\hskip-20pt&\cr &&S_\gamma\otimes_{\cal O} \hat{\cal B}_{\hat\gamma}\hskip-40pt&\hskip-20pt\cong&\hskip-20pt S_\gamma\otimes_{\cal O} {\cal O} Q\cr} \quad ;\leqno £3.15.1$$ moreover, since the unity element is primitive in $(S_\gamma)^P$ and the kernel of the canonical homomorphism $$(S_\gamma\otimes_{\cal O} {\cal O} Q)^P\longrightarrow (S_\gamma)^P $$ is contained in the radical, the unity element is primitive in $(S_\gamma\otimes_{\cal O} {\cal O} Q)^P$ too; since $P$ has a unique local point over $ S_\gamma\otimes_{\cal O} S_\gamma^\circ \otimes_{\cal O} {\cal A}_\gamma$ \cite[Proposition~5.6]{P4}, from diagram~£3.15.1 we get the announced isomorphism. \noindent {\bf £3.16.}\quad Let us take advantage of this revision to correct the erroneous proof of~\cite[1.15.1]{KP}. Indeed, as proved in Proposition~£3.11 above, we have a {strict covering\/} of $Q\hbox{-}$interior $k\hbox{-}$algebras $$k\otimes_{\cal O} {\cal B}_\delta\longrightarrow k\otimes_{\cal O} S_\delta \leqno £3.16.1$$ but {\leqt not\/} a {strict covering\/} $k\otimes_{\cal O} {\cal B}\longrightarrow k\otimes_{\cal O} S$ of $H\hbox{-}$interior $k\hbox{-}$algebras as stated in~\cite[1.15]{KP}; however, it follows from \cite[~2.14.4 and Lemma~9.12]{P6} that the isomorphism ${\cal B}_\delta (Q_\delta)\cong S_\delta (Q)$ induced by homomorphism~£3.16.1 \cite[4.14]{P4} forces the {embedding\/} ${\cal B}(Q_\delta)\to S (Q_{\bar\delta})$ where $\bar\delta$ denotes the local point of $Q$ over $S$ determined by $\delta\,;$ hence, we still have the isomorphism \cite[1.15.5]{KP} which allows us to complete the argument. \vskip 1cm \noindent{\bf\large 4. Extensions of Glauberman correspondents of blocks} In this section, we continue to use the notation in Paragraph 3.1, namely ${\cal O}$ is a complete discrete valuation ring with an algebraically closed residue field $k$ of characteristic $p\,;$ moreover we assume that its quotient field ${\cal K}$ has characteristic 0 and is big enough for all finite groups that we will consider; this assumption is kept throughout the rest of this paper. \noindent{\bf 4.1.}\quad Let $A$ be a cyclic group of order $q$, where $q$ is a power of a prime. Assume that $G$ is an $A$-group, that $H$ is an $A$-stable normal subgroup of~$G$ and that $b$ is $A$-stable. Note that, in this section, $b$ is not necessarily nilpotent. Assume that $A$ and $G$ have coprime orders; by \cite[Theorem 1.2]{P5}, $G$~acts transitively on the set of all defect groups of $G_\alpha$ and, obviously, $A$ also acts on this set; hence, since $A$ and $G$ have coprime orders, by \cite[Lemma 13.8 and Corollary 13.9]{I} $A$ stabilizes some defect group of $G_\alpha$ and $G^A$ acts transitively on the set of them. Similarly, $A$ stabilizes some defect group of $N_\beta$ and $N^A$ acts transitively on the set of them. Thus, we may assume that $A$ stabilizes $P\leq N$ and actually we ssume that $A$ centralizes~$P\,;$ recall that~$Q=P\cap H$. \noindent{\bf 4.2.}\quad Clearly $H^A$ is normal in $G^A$. We claim that $N^A$ is the stabilizer of ${\leqt w}(b)$ in $G^A$. Indeed, for any $x\leqn G^A$, $b^x$ is a block of $H$ and $Q^x$ is a defect group of $b^x$; since $A$ stabilizes $b^x$ and centralizes~$Q^x$, ${\leqt w}(b^x)$ makes sense. Note that $G$ acts on ${\rm Irr}_{\cal K}(H)\,,$ that $G^A$ acts on ${\rm Irr}_{\cal K}(H^A)$ and that the Glauberman correspondence $\pi(G, A)$ is compatible with the obvious actions of $G^A$ on ${\rm Irr}_{\cal K}(H)$ and ${\rm Irr}_{\cal K}(H^A)$. So we have \begin{eqnarray*} {\rm Irr}_{\cal K}(H^A, {\leqt w}(b^x)) &=& \pi(H, A)({\rm Irr}_{\cal K}(H, b^x)) \\ &=& \pi(H, A)({\rm Irr}_{\cal K}(H,b)^x )\\ &=& (\pi(H, A)({\rm Irr}_{\cal K}(H, b)))^x \\ &=& {\rm Irr}_{\cal K}(H^A,{\leqt w}(b))^x\,; \end{eqnarray*} in particular, we get ${\leqt w}(b^x)={\leqt w}(b)^x$ and therefore we have ${\leqt w}(b)^x={\leqt w}(b)$ if and only if $x$ belongs to $N^A$. We set \begin{center}${\leqt w}(c)={\rm Tr}^{G^A}_{N^A}({\leqt w}(b))$, ${\leqt w}(\beta)=\{{\leqt w}(b)\}$ and ${\leqt w}(\alpha)=\{{\leqt w}(c)\}$\quad. \end{center} Then ${\leqt w}(\beta)$ is a point of $N^A$ on ${\cal O} (H^A)$, ${\leqt w}(\alpha)$ is a point of $G^A$ on ${\cal O} (H^A)$, we have $(N^A)_{{\leqt w}(\beta)}\leq (G^A)_{{\leqt w}(\alpha)}$ and any defect group of $(N^A)_{{\leqt w}(\beta)}$ is a defect group of $(G^A)_{{\leqt w}(\alpha)}$. \noindent{\bf 4.3.}\quad Let ${\frak B}$ and ${\leqt w}({\frak B})$ be the respective sets of $A$-stable blocks of $G$ covering $b$ and of $G^A$ covering ${\leqt w}(b)\,.$ Take $e\leqn {\frak B}\,;$ since $P$ is a defect group of $G_\alpha$ and $c$ fulfills $ec=e$, $e$ has a defect group contained in $P$ and therefore, since $A$ centralizes $P\,,$ $e$ has a defect group centralized by~$A\,;$ hence, by \cite[Proposition 1 and Theorem 1]{W}, ${\leqt w}(e)$ makes sense and $A$ stabilizes all the characters in ${\rm Irr}_{\cal K}(G, e)\,;$ that is to say, $A$ stabilizes all the characters of blocks in ${\frak B}$. Moreover, by \cite[Theorem 13.29]{I}, ${\leqt w}(e)$ belongs to ${\leqt w}({\frak B})$. \noindent{\bf Proposition 4.4.}\quad {\leqt The map ${\leqt w}: {\frak B}\rightarrow {\leqt w}({\frak B}),\, e\mapsto {\leqt w}(e)$ is bijective and we have \begin{center}${\rm Irr}_{\cal K}(G^A, {\leqt w}(c))=\pi(G,\,A)({\rm Irr}_{\cal K}(G, c)^A)$\quad .\end{center} } {\cal S}mallskip\noindent{\leqt Proof.}\quad Assume that $g\leqn {\frak B}$ and ${\leqt w}(e)={\leqt w}(g)\,;$ then there exist $\frak c\frak hi\leqn {\rm Irr}_{\cal K}(G, e)$ and $\phi\leqn {\rm Irr}_{\cal K}(G, g)$ such that $\pi(G, A)(\frak c\frak hi)=\pi(G, A)(\phi)$; but this contradicts the bijectivity of the Glauberman correspondence. Therefore the map ${\leqt w}$ is injective. Take $h\leqn {\leqt w}({\frak B})\,;$ then $h$ covers ${\leqt w}(b)$ and so there exist $\zeta\leqn {\rm Irr}_{\cal K}(G^A, h)$ and $\eta\leqn {\rm Irr}_{\cal K}(H^A, {\leqt w}(b))$ such that $\eta$ is a constituent of ${\rm Res}^{G^A}_{H^A}(\zeta)$. Set $$\theta=(\pi(G, A))^{-1}(\zeta)\quad{\rm and}\quad \vartheta=(\pi(H, A))^{-1}(\eta) \quad ;$$ by \cite[Theorem 13.29]{I}, $\vartheta$ is a constituent of ${\rm Res}^G_H(\theta)\,;$ let $l$ be the block of~$G$ acting as the identity map on a ${\cal K} G$-module affording $\theta\,;$ then $l$ covers $b$ and we have ${\leqt w}(l)=h$. Finally, we have \begin{eqnarray*} \pi(G,\,A)({\rm Irr}_{\cal K}(G, c)^A) &=& \pi(G,\,A)(\cup_{e\leqn {\frak B}}{\rm Irr}_{\cal K}(G, e)) \\ &=& \cup_{{\leqt w}(e)\leqn {\leqt w}({\frak B})}{\rm Irr}_{\cal K}(G^A, {\leqt w}(e)) \\ &=& {\rm Irr}_{\cal K}(G^A, {\leqt w}(c)) \end{eqnarray*} \noindent{\bf Proposition 4.5.}\quad {\leqt $P$ is a defect group of the pointed group $(G^A)_{{\leqt w}(\alpha)}$.} {\cal S}mallskip\noindent{\leqt Proof.}\quad It suffices to show that $P$ is a defect group of~$(N^A)_{{\leqt w}(\beta)}$ (cf.~£3.1); thus, without loss of generality, we can assume that $G=N$. Obviously, $A$ stabilizes $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$ and $b$ is the unique block of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$ covering the block $b$ of~$H\,;$ since $P$ is a defect group of $G_\alpha$ and $N_\beta$, $P$ is maximal in $N$ such that ${\rm Br}_P^{{\cal O} H}(b)\neq 0\,;$ thus $P$ is maximal in $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$ such that ${\rm Br}_P^{{\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H)}(b)\neq 0\,;$ therefore $P$ is a defect group of $b$ as a block of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$. Since $A$ centralizes $P$, the Glauberman correspondent $b'$ of $b$ as a block of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$ makes sense; moreover by Proposition 4.4, $b'$ covers ${\leqt w}(b)$. Since ${\leqt w}(b)$ is the unique block of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H^A$ covering the block ${\leqt w}(b)$ of $H^A$, $b'={\leqt w}(b)$, and then, by \cite[Theorem 1]{W}, $P$ is a defect group of ${\leqt w}(b)$ as a block of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H^A$; in particular, ${\rm Br}_P^{{\cal O} (H^A)}({\leqt w}(b))\neq 0$. Since $P$ is a defect group of $G_\alpha$, by \cite[Theorem 5.3]{KP} the image of $P$ in the quotient group $N/H$ is a Sylow $p$-subgroup of $N/H\,;$ but, the inclusion map $N^A\hookrightarrow N$ induces a group isomorphism $N^A/H^A\cong (N/H)^A\,;$ hence, the image of $P$ in $N^A/H^A$ is a Sylow $p$-subgroup of $N^A/H^A\,;$ then, by \cite[Theorem 5.3]{KP} again, $P$ is a defect group of $(N^A)_{{\leqt w}(\alpha)}$. \noindent{\bf 4.6.}\quad We may assume that $A$ stabilizes $P_\gamma\,;$ then $A$ stabilizes $Q_\delta$ too (see \cite[Proposition 5.5]{KP}). Let $R$ be a subgroup such that $Q\leq R\leq P$ and $R_\varepsilon$ a local pointed group on ${\cal O} H$ contained in~$P_\gamma$. Since $A$ stabilizes $P_\gamma$ and centralizes $P$, $A$ centralizes $R$ and then, by \cite[Proposition 5.5]{KP}, it stabilizes~$R_\varepsilon$. Since~${\rm Br}_R^{{\cal O} H}(\varepsilon)$ is a point of $kC_H(R)$, then there is a unique block $b_\varepsilon$ of ${\cal O} C_H(R)$ such that ${\rm Br}_R^{{\cal O} H}(b_\varepsilon\varepsilon)={\rm Br}_R^{{\cal O} H}(\varepsilon)$ and, by \cite[Lemma 2.3]{Z}, $C_Q(R)$ is a defect group of $b_\varepsilon$; in particular, $b_\varepsilon$ is nilpotent. Obviously, $A$ centralizes $C_Q(R)$ and, since $A$ stabilizes $R_\varepsilon$ and thus it stabilizes $b_\varepsilon$, ${\leqt w}(b_\varepsilon)$ makes sense; moreover, ${\leqt w}(b_\varepsilon)$ is nilpotent and, since we have $$C_{H^A}(R)=C_{C_H(R)}(A)\quad ,$$ there is a unique local point ${\leqt w}(\varepsilon)$ of $R$ on ${\cal O} (H^A)$ such that \begin{center}${\rm Br}_R^{{\cal O} (H^A)}({\leqt w}(\varepsilon){\leqt w}(b_\varepsilon))={\rm Br}_R^{{\cal O} (H^A)}({\leqt w}(\varepsilon))\quad .$\end{center} \noindent{\bf Proposition 4.7.}\quad {\leqt $P_{{\leqt w}(\gamma)}$ is a defect pointed group of $(G^A)_{{\leqt w}(\alpha)}$ and $Q_{{\leqt w}(\delta)}$ is a defect pointed group of $(H^A)_{\{{\leqt w}(b)\}}$.} {\cal S}mallskip\noindent{\leqt Proof.}\quad By \cite[Proposition 2.8]{P7}, the inclusion map ${\cal O} H\hookrightarrow {\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,}H)$ is actually a strict semicovering $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$-algebra homomorphism; hence, $\gamma$ determines a unique local point $\gamma'$ of $P$ on ${\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H)$ such that $\gamma {\cal S}ubset \gamma'$. Obviously, $b$ is a block of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$. Since $\beta$ is also a point of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$ on ${\cal O} H$ and $P_\gamma$ is also a defect pointed group of the pointed group $(P{\,\hskip-1pt\cdot\hskip-1pt\,} H)_\beta$ on ${\cal O} H$, by \cite[Corollary 6.3]{KP} $P_{\gamma'}$ is a defect pointed group of the pointed group $(P{\,\hskip-1pt\cdot\hskip-1pt\,} H)_\beta$ on ${\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H)$. Let $b_{\gamma'}$ be the block of $C_{P{\,\hskip-1pt\cdot\hskip-1pt\,} H}(P)$ such that $${\rm Br}_P^{{\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H)}(b_{\gamma'}\gamma')={\rm Br}_P^{{\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H)}(\gamma')\quad ;$$ then $Z(P)$ is a defect group of $b_{\gamma'}$ and therefore ${\leqt w}(b_{\gamma'})$ makes sense. Obviously, $b_{\gamma'}$ covers $b_\gamma$ and thus ${\leqt w}(b_{\gamma'})$ covers ${\leqt w}(b_{\gamma})$ (see Proposition 4.4); but, since ${\leqt w}(b)$ is also the Glauberman correspondent of the block $b$ of $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$ (see the first paragraph of the proof of Proposition 4.5), by \cite[Proposition 4]{W} we have \begin{center}${\rm Br}_P^{{\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H^A)}({\leqt w}(b){\leqt w}(b_{\gamma'}))={\rm Br}_P^{{\cal O} (P{\,\hskip-1pt\cdot\hskip-1pt\,} H^A)}({\leqt w}(b_{\gamma'}))$ \quad ;\end{center} this forces ${\rm Br}_P^{{\cal O} (H^A)}({\leqt w}(b){\leqt w}(b_{\gamma}))={\rm Br}_P^{{\cal O} (H^A)}({\leqt w}(b_{\gamma}))$, which implies that $$P_{{\leqt w}(\gamma)}\leq (P{\,\hskip-1pt\cdot\hskip-1pt\,} H^A)_{{\leqt w}(\beta)}\leq (G^A)_{{\leqt w}(\alpha)}\quad ;$$ hence, by Proposition 4.5, $P_{{\leqt w}(\gamma)}$ is a defect pointed group of $(G^A)_{{\leqt w}(\alpha)}$. The statement that $Q_{{\leqt w}(\delta)}$ is a defect pointed group of $(H^A)_{\{{\leqt w}(b)\}}$ is clear. \noindent{\bf Lemma 4.8.}\quad {\leqt Let $R_\varepsilon$ and $T_\eta$ be local pointed groups on ${\cal B}$ such that $R$ is normal in $T$ and that we have $Q_\delta\leq R_\varepsilon\leq P_\gamma$ and $Q_\delta\leq T_\eta\leq P_\gamma$. Then, we have $R_\varepsilon\leq T_\eta$ if and only if we have $${\rm Br}_T^{{\cal O} C_H(R)}(b_\eta b_\varepsilon)={\rm Br}_T^{{\cal O} C_H(R)}(b_\gamma)\quad .$$} \par\noindent{\leqt Proof.}\quad Obviously, ${\cal B}$ is a $p$-permutation $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$-algebra (see \cite[Def. 1.1]{BP1}) by $P{\,\hskip-1pt\cdot\hskip-1pt\,} H$-conjugation and $(T, {\rm Br}_T^{{\cal B}}(b_{\eta}))$ and $(R, {\rm Br}_R^{{\cal B}}(b_{\varepsilon}))$ are $(b, P{\,\hskip-1pt\cdot\hskip-1pt\,} H)\hbox{-}$Brauer pairs (see \cite[Def. 1.6]{BP1}). Moreover $T$ stabilizes $b_\varepsilon$, and $\eta$ and $\varepsilon$ are the unique local points of $T$ and $R$ on ${\cal B}$ (see \cite[Proposition 5.5]{KP}) such that \begin{center}${\rm Br}_T^{{\cal B}}(\eta){\rm Br}_T^{{\cal B}}(b_{\eta})={\rm Br}_T^{{\cal B}}(\eta)$ and ${\rm Br}_R^{{\cal B}}(\varepsilon){\rm Br}_R^{{\cal B}}(b_{\varepsilon})={\rm Br}_R^{{\cal B}}(\varepsilon)\quad .$\end{center} Assume that $R_\varepsilon\leq T_\eta\,;$ then, there are $h\leqn \eta$ and $l\leqn \varepsilon$ such that $hl=l=lh\,;$ thus, we have $${\rm Br}_R^{{\cal B}}(hl)={\rm Br}_R^{{\cal B}}(l)\quad{\rm and}\quad {\rm Br}_R^{{\cal B}}(h){\rm Br}_R^{{\cal B}}(b_{\varepsilon})\neq 0\quad .$$ Then, it follows from \cite[Def. 1.7]{BP1} that $$(R, {\rm Br}_R^{{\cal B}}(b_{\varepsilon})){\cal S}ubset (T,{\rm Br}_T^{{\cal B}}(b_{\eta}))$$ and from \cite[Theorem 1.8]{BP1} that we have ${\rm Br}_T^{{\cal O} C_{H}(R)}(b_{\eta} b_{\varepsilon})={\rm Br}_T^{{\cal O} C_{H}(R)}(b_{\eta})$. Conversely, if we have $${\rm Br}_T^{{\cal O} C_{H}(R)}(b_{\eta} b_{\varepsilon})={\rm Br}_T^{{\cal O} C_{H}(R)}(b_{\eta})$$ then, by \cite[Theorem 1.8]{BP1} we still have ${\rm Br}_R^{{\cal B}}(e_{\varepsilon} h)={\rm Br}_R^{{\cal B}}(h)$ for any $h\leqn \eta$; hence, by the lifting theorem for idempotents, we get $R_{\varepsilon}\leq T_\eta$. Let $\cal R$ be a Dedekind domain of characteristic 0, $\pi$ be a finite set of prime numbers such that $l\cal R\neq \cal R$ for all $l\leqn \pi\,,$ and $X$ and $Y$ be finite groups with $X$ acting on $Y$. We consider the group algebra ${\cal R} Y$ and set $$Z_{\rm id}({\cal R} Y)=\oplus {\cal R} c$$ where $c$ runs over all central primitive idempotents of ${\cal R}Y$. Obviously, $X$ acts on $Z_{\rm id}({\cal R} Y)$ and, in the case that $X$ is a solvable $\pi$-group, Lluis Puig exhibes a $\cal R$-algebra homomorphism ${\mathcal G l}^Y_X: Z_{\rm id}({\cal R} X)\rightarrow Z_{\rm id}({\cal R} Y^X)$ (see \cite[Theorem 4.6]{P7}), which unifies the usual Brauer homomorphism and the Glauberman correspondence of characters --- called the {\leqt Brauer-Glauberman correspondence\/}. \noindent{\bf Proposition 4.9.}\quad {\leqt Let $R_\varepsilon$ and $T_\eta$ be local pointed groups on ${\cal B}$ such that $Q_\delta\leq R_\varepsilon\leq P_\gamma$ and that $Q_\delta\leq T_\eta\leq P_\gamma$. Then $R_\varepsilon\leq T_\eta$ and $R_{{\leqt w}(\varepsilon)}\leq T_{{\leqt w}(\eta)}$ are equivalent to each other. } \noindent{\leqt Proof.}\quad By induction we can assume that $R$ is normal and maximal in $T$; in particular, the quotient $T/R$ is cyclic. In this case, it follows from Lemma 4.8 that the inclusion $R_{{\leqt w}(\varepsilon)}\leq T_{{\leqt w}(\eta)}$ is equivalent to $${\rm Br}_T^{{\cal O} C_{H^A}(R)}({\leqt w}(b_\varepsilon){\leqt w}(b_\eta)) ={\rm Br}_T^{{\cal O} C_{H^A}(R)}({\leqt w}(b_\eta)) \quad .\leqno £4.9.1$$ Let ${\mathbb Z}$ be the ring of all rational integers and $S$ be the complement set of $p{\mathbb Z}\cup q\mathbb Z$ in $\mathbb Z\,;$ then $S$ is a multiplicatively closed set in $\mathbb Z$. We take the localization $S^{-1}\mathbb Z$ of $\mathbb Z$ at $S$ and regard it as a subring of ${\cal K}\,;$ since we assume that $\cal K$ is big enough for all finite groups we consider, we can assume that $\cal K$ contains an $|H|$-th primitive root $\omega$ of unity and we set \begin{center}${\cal R}=(S^{-1}{\mathbb Z})[\omega]\quad .$\end{center} Then $\cal R$ is a Dedekind domain (see \cite[Example 2 in Page 96 and Exercise 1 in Page 99]{AM}) and given a prime $l$, we have $l\cal R\neq \cal R$ if and only if $l=p$ or $l=q$. We consider the group algebra ${\cal R}C_H(R)$ and the obvious action of $(T\times A)/R\cong (T/R)\times A$ on it. Since $\cal R$ contains an $|H|$-th primitive unity root $\omega$, the blocks $b_\varepsilon$, $b_\eta$, ${\leqt w}(b_\varepsilon)$ and ${\leqt w}(b_\eta)$ respectively belong to $$Z_{\rm id}({\cal R}C_H(R))\;\;,\;\; Z_{\rm id}({\cal R}C_H(T))\;\;,\;\; Z_{\rm id}({\cal R}C_{H^A}(R))\quad{\rm and}\quad Z_{\rm id}({\cal R}C_{H^A}(T)) $$ (see \cite[Charpter IV, Lemma 7.2]{F}); then, by \cite[Corollary 5.9]{P7}, we have $${\mathcal G l}^{C_{H}(R)}_{A}(b_\varepsilon)={\leqt w}(b_\varepsilon)\quad{\rm and}\quad {\mathcal G l}^{C_{H}(T)}_{A}(b_\eta)={\leqt w}(b_\eta) \quad .$$ If $R_\varepsilon\leq T_\eta$, by Lemma 4.8 we have the equality $${\rm Br}_{T}^{{\cal O} C_H(R)}(b_\varepsilon b_\eta)={\rm Br}_{T}^{{\cal O} C_H(R)}(b_\eta)$$ which is equivalent to ${\mathcal G l}^{C_H(R)}_{T/R}(b_\varepsilon)b_\eta=b_\eta$ (see \cite[4.6.1 and the proof of Corollary 3.6]{P7}). Then by \cite[4.6.2]{P7}, we have \begin{eqnarray*} {\leqt w}(b_\eta) &=& {\mathcal G l}^{C_H(T)}_{A}(b_\eta)= {\mathcal G l}^{C_H(T)}_{A}({\mathcal G l}^{C_H(R)}_{T/R} (b_\varepsilon)b_\eta) \\&=& {\mathcal G l}^{C_H(R)}_{(T/R)\times A}(b_\varepsilon) {\mathcal G l}^{C_H(T)}_{A}(b_\eta) \\ &=&{\mathcal G l}^{C_{H^A}(R)}_{T/R} ({\mathcal G l}^{C_{H}(R)}_{A}(b_\varepsilon)) {\mathcal G l}^{C_H(T)}_{A}(b_\eta)\\ &=&{\mathcal G l}^{C_{H^A}(R)}_{T/R} ({\leqt w}(b_\varepsilon)) {\leqt w}(b_\eta)\quad. \end{eqnarray*} which is equivalent again to equality~£4.9.1 above (see \cite[4.6.1 and the proof of Corollary 3.6]{P7} and therefore it implies $R_{{\leqt w}(\varepsilon)}\leq T_{{\leqt w}(\eta)}$. The prove that $R_{{\leqt w}(\varepsilon)}\leq T_{{\leqt w}(\eta)}$ implies $R_\varepsilon\leq T_\eta$ is similar. \noindent{\bf 4.10.}\quad The assumptions and consequences above are very scattered; we collect them in this paragraph, so that readers can easily find them and we can conveniently quote them later. Let $A$ be a cyclic group of order~$q$, where $q$ is a prime number; we assume that $G$ is an $A$-group, that $H$ is an $A$-stable normal subgroup of $G\,,$ that $b$ is $A$-stable, that $A$ centralizes $P$ and stabilizes $P_\gamma\,,$ and that $A$ and $G$ have coprime orders. Without loss of generality, we may assume that $P\leq N$. Then, $A$ centralizes $Q$ and stabilizes $Q_\delta\,,$ so that the Glauberman correspondent ${\leqt w}(b)$ of the block $b$ makes sense; moreover, the block ${\leqt w}(b)$ determines two pointed group $(N^A)_{{\leqt w}(\beta)}$ and $(G^A)_{{\leqt w}(\alpha)}$ such that $(N^A)_{{\leqt w}(\beta)}\leq (G^A)_{{\leqt w}(\alpha)}$ (see them in Paragraph 4.2)., and the local pointed groups $P_\gamma$ and $Q_\delta$ determine respective defect pointed groups $P_{{\leqt w}(\gamma)}$ and $Q_{{\leqt w}(\delta)}$ of $(G^A)_{{\leqt w}(\alpha)}$ and $(H^A)_{{\leqt w}(\beta)}$ (see Paragraph 4.6 and Proposition 4.7); actually, by Proposition 4.9, we have $Q_{{\leqt w}(\delta)}\leq P_{{\leqt w}(\gamma)}$. Take ${\leqt w}(i)\leqn {\leqt w}(\gamma)$ and ${\leqt w}(j)\leqn {\leqt w}(\delta)\,,$ and set \begin{center} $({\cal O} G^A)_{{\leqt w}(\gamma)}={\leqt w}(i)({\cal O} G^A){\leqt w}(i)$ , $({\cal O} H^A)_{{\leqt w}(\gamma)}={\leqt w}(i)({\cal O} H^A){\leqt w}(i)$\\ and $({\cal O} H^A)_{{\leqt w}(\delta)}={\leqt w}(j)({\cal O} H^A){\leqt w}(j)\quad ;$\end{center} then, $({\cal O} G^A)_{{\leqt w}(\gamma)}$ is a $P$-interior and $(N^A/H^A)$-graded algebra; moreover, the $Q$-interior algebra $({\cal O} H^A)_{{\leqt w}(\delta)}$ with the group homomorphism $$Q\longrightarrow ({\cal O} H^A)_{{\leqt w}(\delta)}^*\quad , \quad u\mapsto u{\leqt w}(j)$$ is a source algebra of the block algebra ${\cal O} H^A{\leqt w}(b)$ (see \cite{P5}). \vskip 1cm \noindent{\bf\large 5. A Lemma} From now on, we use the notation and assumption in Paragraphs 3.1, 3.2 and 4.10; in particular, we assume that the block $b$ of $H$ is nilpotent. Obviously, $N_G(Q_\delta)$ acts on ${\rm Irr}_{\cal K}(H, b)$ and ${\rm Irr}_{\cal K}(Q)$ via the corresponding conjugation conjugation. Since $b$ is nilpotent, there is an explicit bijection between ${\rm Irr}_{\cal K}(H, b)$ and ${\rm Irr}_{\cal K}(Q)$ (see \cite[Theorem 52.8]{T}); in this section, we will show that this bijection is compatible with the $N_G(Q_\delta)$-actions; our main purpose is to obtain Lemma 5.6 below as a consequence of this compatibility. \noindent{\bf 5.1.}\quad For any $x\leqn N_G(Q_\delta)$, $xjx^{-1}$ belongs to $\delta$ and thus there is some invertible element $a_x\leqn {\cal B}^Q$ such that $xjx^{-1}=a_x ja_x^{-1}\,;$ let us denote by $X$ the set of all elements $(a_x^{-1}x)j$ such that $a_x$ is invertible in ${\cal B}^Q$ and we have $xjx^{-1}=a_x ja_x^{-1}$ when $x$ runs over $N_G(Q_\delta)$. Set \begin{center}$E_G(Q_\delta)= N_G(Q_\delta)/QC_H(Q)\quad ;$\end{center} then, the following equality $${\cal B}ig((a_x^{-1}x)j{\cal B}ig){\,\hskip-1pt\cdot\hskip-1pt\,} {\cal B}ig((a_y^{-1}y)j{\cal B}ig)= {\cal B}ig((a_x^{-1}xa_y^{-1}x^{-1})xy{\cal B}ig)j$$ shows that $X$ is a group with respect to the multiplication and it is easily checked that $Q{\,\hskip-1pt\cdot\hskip-1pt\,} ({\cal B}_\delta^Q)^*$ is normal in $X$ and that the map $$E_G(Q_\delta)\longrightarrow X/Q({\cal B}_\delta^Q)^*\leqno 5.1.1$$ sending the coset of $x\leqn N_G(Q_\delta)$ in $N_G(Q_\delta)/QC_H(Q)$ to the coset of $(a_x^{-1}x)j$ in $X/Q({\cal B}_\delta^Q)^*$ is a group isomorphism. \noindent{\bf 5.2.}\quad We denote by $Y$ the set of all such elements $a_x^{-1}x$ when $x$ runs over $N_G(Q_\delta)$ and $a_x$ over the invertible element of ${\cal B}^Q$ such that $a_x^{-1}x$ commutes with $j$. As in 5.1, it is easily checked that $Y$ is a group with respect to the multiplication $$(a_x^{-1}x){\,\hskip-1pt\cdot\hskip-1pt\,} (a_y^{-1}y)= (a_x^{-1}xa_y^{-1}x^{-1})xy \quad,$$ that $Y$ normalizes $Q{\,\hskip-1pt\cdot\hskip-1pt\,} (({\cal O} H)^Q)^*$ and that the map $$E_G(Q_\delta)\longrightarrow {\cal B}ig(Y{\,\hskip-1pt\cdot\hskip-1pt\,} Q{\,\hskip-1pt\cdot\hskip-1pt\,} ({\cal B}^Q)^*{\cal B}ig){\cal B}ig/{\cal B}ig(Q{\,\hskip-1pt\cdot\hskip-1pt\,} ({\cal B}^Q)^*{\cal B}ig) \leqno 5.2.1$$ sending the coset of $x\leqn N_G(Q_\delta)$ to the coset of $a_x^{-1}x$ in the right-hand quotient is a group isomorphism. \noindent{\bf 5.3.}\quad Let $I$ and $J$ be the sets of isomorphism classes of all simple ${\cal K}\otimes_{\cal O} {\cal B}\hbox{-}$ and ${\cal K}\otimes_{\cal O} {\cal B}_\delta\hbox{-}$modules respectively. Cleraly, $Y$ acts on $I\,;$ but, since $Y\cap ({\cal B}^Q)^*$ acts trivially on $I$, the action of $Y$ on $I$ induces an action of $E_G(Q_\delta)$ on $I$ through isomorphism 5.2.1; actually, this action coincides with the action of $E_G(Q_\delta)$ on ${\rm Irr}_{\cal K}(H, b)$ induced by the $N_G(Q_\delta)$-conjugation. Similarly, $X$ acts on $J$ and this action of $X$ on $J$ induces an action of $E_G(Q_\delta)$ on $J$ through isomorphism 5.1.1. But, by \cite[Corollary 3.5]{P5}, the functor $M\mapsto j{\,\hskip-1pt\cdot\hskip-1pt\,} M$ is an equivalence between the categories of finitely generated ${\cal B}$- and ${\cal B}_\delta$-modules, which induces a bijection between the sets $I$ and $J$. Then, since $Y$ commutes with $j$ and the map $$Y\longrightarrow X\quad ,\quad y\mapsto yj$$ is a group homomorphism, it is easily checked that this bijection is com-patible with the actions of $E_G(Q_\delta)$ on $I$ and $J$. \noindent{\bf 5.4.}\quad Recall that (cf.~£3.7) $${\cal B}_\delta\cong T\otimes_{\cal O} {\cal O} Q \leqno 5.4.1$$ where $T = {\rm End}_{\cal O} (W)$ for an endo-permutation ${\cal O} Q$-module $W$ such that the determinant of the image of any element of $Q$ in is one; in this case, the ${\cal O} Q$-module $W$ with these properties is unique up to isomorphism. Then, for any simple ${\cal K}\otimes_{\cal O} {\cal B}_\delta$-module $V$ there is a ${\cal K} Q$-module $V_W$, unique up to isomorphism, such that $$V\cong W\otimes_{\cal O} V_W$$ as ${\cal K}\otimes_{\cal O} {\cal B}_\delta$-modules; moreover the correspondence $$V\mapsto V_W\leqno 5.4.2$$ determines a bijection between $J$ and the set of isomorphism classes of all simple ${\cal K} Q$-modules. Now, the composition of this bijection with the bijection between isomorphism classes in 5.3 is a bijection from $I$ to the set of isomorphism classes of all simple ${\cal K} Q$-modules; translating this bijection to characters, we obtain a bijection $${\rm Irr}_{\cal K}(H, b)\longrightarrow {\rm Irr}_{\cal K} (Q)\quad ,\quad \frak c\frak hi_\lambda\mapsto \lambda \quad ;\leqno 5.4.3$$ let us denote by $\frak c\frak hi\leqn {\rm Irr}_{\cal K}(H, b)$ the image ofthe trivial character of $Q\,.$ \noindent{\bf 5.5.}\quad Moreover, the $N_G(Q_\delta)$-conjugation induces an action of $E_G(Q_\delta)$ on the set of isomorphism classes of all simple ${\cal K} Q$-modules and we claim that, for any simple ${\cal K}\otimes_{\cal O} {\cal B}_\delta$-module $V$ and any $\bar x\leqn E_G(Q_\delta)\,,$ we have a ${\cal K} Q$-module isomorphism $$^{\bar x}(V_W)\cong (^{\bar x}V)_W \quad ;\leqno 5.5.1$$ in particular, bijection 5.4.2 is compatible with the actions of $E_G(Q_\delta)$ on $J$ and on the set of isomorphism classes of simple ${\cal K} Q$-modules. Indeed, let $x$ be a lifting of $\bar x$ in $N_G(Q_\delta)$ and denote by $\varphi_x$ the isomorphism $$Q\cong Q\quad ,\quad u\mapsto xux^{-1} \quad ;$$ take a lifting $y=a_x^{-1}xj$ of $\bar x$ in $X$ through isomorphism 5.1.1; since the conjugation by $y$ stabilizes ${\cal B}_\delta$, the map $$f_y: {\cal B}_\delta\cong {\rm Res}_{\varphi_x}({\cal B}_\delta)\quad ,\quad a\mapsto yay^{-1}$$ is a $Q$-interior algebra isomorphism; then, by \cite[Corollary 6.9]{P4}, we can modify $y$ with a suitable element of $({\cal B}_\delta^Q)^*$ in such a way that $f_y$ stabilizes~$T\,;$ in this case, the restriction of $f_y$ to $T$ has to be inner and thus we have $W\cong {\rm Res}_{f_y}(W)$ as ${\rm T}\hbox{-}$modules. Moreover, since the action of $Q$ on $T$ can be uniquely lifted to a $Q$-interior algebra structure such that the determinant of the image of any $u\leqn Q$ in $T$ is one, $f_y$ also stabilizes the image of $Q$ in~$T\,;$ more precisely, $f_y$ maps the image of $u\leqn Q$ onto the image of $\varphi_x (u)\,.$ The claim follows. \noindent{\bf Lemma 5.6.}\quad {\leqt With the notation above, {\cal S}mallskip\noindent{\bf 5.6.1.}\quad The irreducible character $\frak c\frak hi$ is $N_G(Q_\delta)$-stable and its restriction to the set $H_{p'}$ of all $p$-regular elements of $H$ is the unique irreducible Brauer character of $H\,.$ {\cal S}mallskip\noindent{\bf 5.6.2.}\quad The Glauberman correspondent $\phi$ of $\frak c\frak hi$ is $N_{G^A}(Q_{{\leqt w}(\delta)})$-stable and its restriction to the set $H^A_{p'}$ of all $p$-regular elements of $H^A$ is the unique irreducible Brauer character of $H^A\,.$} {\cal S}mallskip\noindent{\leqt Proof.}\quad It follows from £5.3 and £5.5 that the bijection~£5.4.3 is compatible with the actions of $E_G(Q_\delta)$ in ${\rm Irr}_{\cal K}(H, b)$ and ${\rm Irr}_{\cal K} (Q)\,;$ hence, $\frak c\frak hi$ is $E_G(Q_\delta)$-stable and thus $N_G(Q_\delta)$-stable. Since $\phi$ is the unique irreducible constituent of ${\rm Res}^H_{H^A}(\frak c\frak hi)$ occurring with a multiplicity coprime to $q$ and $N_{G^A}(Q_{{\leqt w}(\delta)})$ is contained in $N_G(Q_\delta)$, $\phi$ has to be $N_{G^A}(Q_{{\leqt w}(\delta)})$-stable. By the very definition of the bijection 5.4.3, the restriction of $\frak c\frak hi$ to $H_{p'}$ is the unique Brauer character of $H$. Since the perfect isometry $R_H^b$ between $ {\cal R}_{\cal K} (H, b)$ and ${\cal R}_{\cal K} (H^A, {\leqt w}(b))$ maps $\psi\leqn I$ onto $\pm\pi(H, A)(\psi)$ and the blocks $b$ and ${\leqt w}(b)$ are nilpotent, by \cite[Theorem 4.11]{B} the decomposition matrices of $b$ and ${\leqt w}(b)$ are the same if the characters indexing their columns correspond to each other by the Glauberman correspondence; hence, the restriction of~$\phi$ to $H^A_{p'}$ is the unique Brauer character of $H^A$. \vskip 1cm \noindent{\bf\large 6. A $k^*$-group isomorphism $({\cal S}kew3\hat {\bar N}^{^k})^A\cong \,\widehat{\overline{\!N^A}}^{k}$} \noindent{\bf 6.1.}\quad Let $xH$ be an $A$-stable coset in $\bar N$. We consider the action of $H\rtimes A$ on $xH$ defined by the obvious action of $A$ on $xH$ and the right multiplication of $H$ on $xH\,;$ since $A$ and $G$ have coprime orders, it follows from \cite[Lemma 13.8 and Corollary 13.9]{I} that $xH\cap N^A$ is non-empty and that $H^A$ acts transitively on it; consequently, we have $\bar N^A= (H{\,\hskip-1pt\cdot\hskip-1pt\,} N^A)/H$ and the inclusion $N^A{\cal S}ubset N$ induces a group isomorphism $$\overline{ \!N^A}\cong \bar N^A= (H{\,\hskip-1pt\cdot\hskip-1pt\,} N^A)/H \quad .\leqno 6.1.1 $$ Note that if $G=H {\,\hskip-1pt\cdot\hskip-1pt\,} G^A$ then we have $\bar N^A= \bar N\,.$ \noindent{\bf 6.2.}\quad It follows from Lemma~£5.6 that $N = H{\,\hskip-1pt\cdot\hskip-1pt\,} N_G (Q_\delta)$ stabilizes $\frak c\frak hi$ and actually the central extension ${\cal S}kew3\hat{\bar N}$ of $\bar N$ by $U$ in~£3.9 above is nothing but the so-called {\leqt Clifford extension\/} of $\bar N$ over $\frak c\frak hi\,;$ moreover, since $A$ and $U$ also have coprime orders, we can prove as above that ${\cal S}kew3\hat{\bar N}^A$ is a central extension of $\bar N^A$ by~$U\,,$ which is the {\leqt Clifford extension\/} of $\bar N^A$ over $\frak c\frak hi\,.$ Since the Glauberman correspondent ${\leqt w}(b)$ is nilpotent, we can repeat all the above constructions for $G^A$, $H^A\,,$ ${\leqt w}(b)$ and $N^A\,;$ then, denoting by $U_A$ the group of $\vert H^A\vert\hbox{-}$th roots of unity, we obtain a central extension $\,\widehat{\overline{\!N^A}}$ of $\,\overline{ \!N^A} =\bar N^A$ by~$U_A\,,$ which is the {\leqt Clifford extension\/} of $\bar N^A$ over $\phi\,;$ moreover, note that $U_A$ is contained in~$U\,.$ \noindent{\bf 6.3.}\quad At this point, it follows from \cite[Corollary 4.16]{P8} that there is an extension group isomorphism $$\hat N^A\cong (U\times \,\widehat{\!N^A})/{\cal D}elta_{-1} (U_A) \leqno £6.3.1$$ where we are setting ${\cal D}elta_{-1}(U_A) = \{(\xi^{-1},\xi)\}_{\xi\leqn U_A}\,;$ moreover, according to \cite[Remark 4.17]{P7}, this isomorphism is defined by a sequence of Brauer homomorphisms --- in different characteristics --- and, in particular, it is quite clear that it maps any $y\leqn H\leq \hat N^A$ in the classes of $(1,y)$ in the right-hand member, so that isomorphism~£6.3.1 induces a new extension group isomorphism $${\cal S}kew3\hat{\bar N}^A\cong (U\times \,\widehat{\overline{\!N^A}})/{\cal D}elta_{-1} (U_A) \quad .$$ Consequently, denoting by $\varpi_A\,\colon U_A\to k^*$ the restriction of $\varpi\,,$ we get a $k^*\hbox{-}$group isomorphism \begin{eqnarray*}({\cal S}kew3\hat {\bar N}^{^k})^A &=& {\cal B}ig((k^*\times {\cal S}kew3\hat {\bar N})/{\cal D}elta_\varrho (U){\cal B}ig)^A \cong (k^*\times {\cal S}kew3\hat {\bar N}^A)/{\cal D}elta_\varrho (U) \\ &\cong &(k^*\times \,\widehat{\overline{\!N^A}})/{\cal D}elta_{\varrho_A} (U_A) \,= \,\widehat{\overline{\!N^A}}^k \end{eqnarray*} as announced. \noindent{\bf Remark 6.4.}\quad Note that if $G=H{\,\hskip-1pt\cdot\hskip-1pt\,} G^A$ then we have ${\cal S}kew3\hat {\bar N}^A= {\cal S}kew3\hat {\bar N}$. \vskip 1cm \noindent{\bf\large 7. Proofs of Theorems 1.5 and 1.6} \noindent{\bf 7.1.}\quad The first statement in Theorem~£1.5 follows from Propositions~£4.4 and~£4.5. From now on, we assume that the block $b$ of $H$ is nilpotent; thus, the Glauberman correspondent ${\leqt w}(b)$ is also nilpotent and $({\cal O} G^A){\leqt w}(c)$ is an extension of the nilpotent block algebra $({\cal O} H^A){\leqt w}(b)$. This section will be devoted to comparing the extensions ${\cal O} G c$ and ${\cal O} G^A{\leqt w}(c)$ of the nilpotent block algebras ${\cal O} H b$ and ${\cal O} H^A{\leqt w}(b)$. Applying Theorem 3.5 to the finite groups $G^A$ and $H^A$ and the nilpotent block ${\leqt w}(b)$ of $H^A$, we get a finite group $L^A$ and respective injective and surjective group homomorphisms $$\tau^A: P\longrightarrow L^A\quad{\rm and}\quad \bar\pi^A: L^A\longrightarrow \,\overline{\!N^A}$$ such that $\bar\pi^A(\tau^A(u))=\bar u$ for any $u\leqn P$, that ${\rm Ker}(\bar\pi^A)=\tau^A(Q)$ and that they induce an equivalence of categories $${\cal E}_{({\leqt w}(b),\, H^A,\, G^A)}\cong {\cal E}_{(1,\, \tau^A(Q),\, L^A)} \quad .$$ Similarly, we et $\widehat{ L^A}= {\rm res}_{\bar \pi^A}(\,\widehat{\overline{\!N^A}}^k)$ and denote by $\widehat{ \tau^A}\,\colon P\to \widehat{ L^A}$ the lifting of~$\tau^A\,;$ then, by Corollary 3.15, there is a $P$-interior full matrix algebra ${\leqt w}(S_\gamma)$ such that we have an isomorphism $$({\cal O} (G^A))_{{\leqt w}(\gamma)}\cong {\leqt w}(S_\gamma)\otimes_{{\cal O} } {\cal O}_*\widehat{L^A}^\circ\quad \leqno{7.1.1}$$ of both $P$-interior and $N^A/H^A$-graded algebras. \noindent{\bf Lemma 7.2.}\quad {\leqt Assume that $G=H{\,\hskip-1pt\cdot\hskip-1pt\,} G^A\,.$ Then we have $N=H{\,\hskip-1pt\cdot\hskip-1pt\,} N^A\,,$ the inclusion $N^A{\cal S}ubset N$ induces a group isomorphism $\,\overline{\!N^A}\cong \bar N$ and there is a group isomorphism $${\cal S}igma: L^A\cong L$$ such that ${\cal S}igma\circ\tau^A=\tau$ and $\bar\pi\circ{\cal S}igma=\bar\pi^A$. } \noindent{\leqt Proof.}\quad For any subgroups $R$ and $T$ of $P$ containing $Q$, let us denote by $${\cal E}_{(b,\, H,\, G)}(R, T)\quad{\rm and}\quad {\cal E}_{({\leqt w}(b),\, H^A,\, G^A)}(R, T)$$ the respective sets of ${\cal E}_{(b,\, H,\, G)}\hbox{-}$ and ${\cal E}_{({\leqt w}(b),\, H^A,\, G^A)}\hbox{-}$morphisms from $T$ to $R\,;$ since $A$ acts trivially in ${\cal E}_{(b,\, H,\, G)}(R, T)$, by \cite[Lemma 13.8 and Corollary 13.9]{I} each morphism in ${\cal E}_{(b,\, H,\, G)}(R, T)$ is induced by some element in $N^A\,;$ moreover, if $T_\nu$ and $R_\varepsilon$ are local pointed groups contained in $P_\gamma\,,$ it follows from Proposition 4.9 that we have $T_\nu\leq (R_\varepsilon)^x$ for some $x\leqn N^A$ if and only if we have $T_{{\leqt w}(\nu)}\leq (R_{{\leqt w}(\varepsilon)})^x$. Therefore, we get $${\cal E}_{(b,\, H,\, G)}(T, R)={\cal E}_{({\leqt w}(b),\, H^A,\, G^A)}(T, R) \quad . $$ At this point, it is easy to check that $L$, $\tau$ and $\bar\pi$ fulfill the conditions in Theorem 3.5 with respect to $G^A$, $H^A$ and the nilpotent block ${\leqt w}(b)$. Then this lemma follows from the uniqueness part in Theorem 3.5. \noindent{\bf Lemma 7.3.}\quad {\leqt Assume that $G=H{\,\hskip-1pt\cdot\hskip-1pt\,} G^A\,.$ Then there is a $k^*$-group isomorphism $\hat {\cal S}igma: \widehat{ L^A}\cong \hat L$ lifting ${\cal S}igma$ and fulfilling $\hat{\cal S}igma\circ \widehat{\tau^A }= \hat\tau\,.$ In particular, we have $${\rm Irr}_{{\cal K}}(G, c)={\rm Irr}_{{\cal K}}(G, c)^A\quad .$$} \par\noindent{\leqt Proof.}\quad The first statement is an easy consequence of £6.3 and Lemma 7.2; then, the last equality follows from Corollary~£3.15. \noindent{\bf 7.4.} {\leqt Proof of Theorem 1.6.}\quad Firstly we consider the case where the block $b$ of $H$ is not stabilized by~$G\,;$ then we have an isomorphism $${\rm Ind}^G_{N}({\cal O} N b)\cong {\cal O} G b$$ of ${\cal O} G$-interior algebras mapping $1\otimes a\otimes 1$ onto $a$ for any $a\leqn {\cal O} N b$ and an isomorphism $${\rm Ind}^{G^A}_{N^A}({\cal O} (N^A ){\leqt w}(b))\cong {\cal O} (G^A){\leqt w}(b)$$ of ${\cal O} (G^A)$-interior algebras mapping $1\otimes a\otimes 1$ onto $a$ for any $a\leqn {\cal O} (N^A) {\leqt w}(b)$. Suppose that an ${\cal O}(N^A\times N)$-module $M$ induces a Morita equivalence from ${\cal O} (N^A) {\leqt w}(b)$ to ${\cal O} N b$. Then it is easy to see that the ${\cal O}(G^A\times G)$-module ${\rm Ind}^{G^A\times G}_{N^A\times N} (M)$ induces a Morita equivalence from ${\cal O} Gc$ to ${\cal O} (G^A) {\leqt w}(c)\,.$ So, we can assume that $G= N$ and then we have $G^A=N^A\,.$ {\cal S}mallskip By Corollary 3.15, there exists an isomorphism of both $(N/H)$-graded and $P$-interior algebras $$({\cal O} G)_\gamma\cong S_\gamma\otimes_{{\cal O} } {\cal O}_*\hat L^{^\circ}\quad ; \leqno{7.4.1}$$ denote by $V_\gamma$ an ${\cal O} P$-module such that ${\rm End}_{\cal O}(V_\gamma)\cong S_\gamma\,;$ choosing $i\leqn \gamma$ and assuming that $({\cal O} G)_\gamma = i({\cal O} G)i\,,$ we know that the ${\cal O} Gb\otimes_{\cal O} ({\cal O} G)_\gamma^\circ\hbox{-}$module $({\cal O} G)i$ determines a Morita equivalence from ${\cal O} Gb$ to $({\cal O} G)_\gamma\,,$ whereas the $({\cal O} G)_\gamma\otimes_{\cal O} {\cal O}_*\hat L\hbox{-}$module $V_\gamma\otimes_{\cal O} {\cal O}_*\hat L^{^\circ}$ determines a Morita equivalence from $({\cal O} G)_\gamma$ to~${\cal O}_*\hat L^{^\circ}\,,$ so that the ${\cal O} Gb\otimes_{\cal O} {\cal O}_*\hat L\hbox{-}$ module $$({\cal O} G)i\otimes_{({\cal O} G)_\gamma} (V_\gamma\otimes_{\cal O} {\cal O}_*\hat L^{^\circ}) \cong ({\cal O} G)i\otimes_{S_\gamma} V_\gamma$$ determines a Morita equivalence from ${\cal O} Gb$ to~${\cal O}_*\hat L^{^\circ}\,.$ {\cal S}mallskip Similarly, choosing $j\leqn \delta$ such that $ji = j = ij\,,$ assuming that $j({\cal O} H)j = ({\cal O} H)_\delta$ and setting $j{\,\hskip-1pt\cdot\hskip-1pt\,} V_\gamma = V_\delta\,,$ so that $S_\delta = {\rm End}_{\cal O} (V_\delta)\,,$ the ${\cal O} Hb\otimes_{\cal O} {\cal O} Q\hbox{-}$ module $$({\cal O} H)j\otimes_{({\cal O} H)_\delta} (V_\delta\otimes_{\cal O} {\cal O} Q) \cong ({\cal O} H)j\otimes_{S_\delta} V_\delta$$ determines a Morita equivalence from ${\cal O} Hb$ to~${\cal O} Q\,.$ {\cal S}mallskip Analogously, with evident notation, the ${\cal O} (G^A) w(b)\otimes_{\cal O} {\cal O}_*\,\widehat{\!L^A}\hbox{-}$ module $${\cal O} (G^A) w(i) \otimes_{w(S_\gamma)} w(V_\gamma)$$ determines a Morita equivalence from ${\cal O} (G^A)(b)$ to~${\cal O}_*\,\widehat{L^A}^\circ\,,$ whereas the ${\cal O} (H^A) w(b)\otimes_{\cal O} {\cal O} Q\hbox{-}$module $${\cal O} (H^A)w(j)\otimes_{w(S_\delta)} w(V_\delta)$$ determines a Morita equivalence from ${\cal O} (H^A) w(b)$ to~${\cal O} Q\,.$ {\cal S}mallskip Consequently, identifying $\,\widehat{L^A}$ with $\hat L$ through the isomorphism $\hat{\cal S}igma$ (cf. Lemma~£7.3), the ${\cal O} (G\times G^A)\hbox{-}$module $$D= (({\cal O} G)i\otimes_{S_\gamma} V_\gamma)\otimes_{{\cal O}_*\hat L} (w(V_\gamma)^\circ \otimes_{w(S_\gamma)} w(i) {\cal O} (G^A))$$ determines a Morita equivalence from ${\cal O} Gb$ to~${\cal O} (G^A) w(b)\,,$ whereas the ${\cal O} (H\times H^A)\hbox{-}$module $$M = (({\cal O} H)j\otimes_{S_\delta} V_\delta)\otimes_{{\cal O} Q } (w(V_\delta)^\circ \otimes_{w(S_\delta)} w(j){\cal O} (H^A))$$ determines a Morita equivalence from ${\cal O} Hb$ to~${\cal O} (H^A) w(b)\,.$ {\cal S}mallskip Moreover, since we have the obvious inclusions $$({\cal O} H)j{\cal S}ubset ({\cal O} G)i\quad ,\quad S_\delta {\cal S}ubset S_\gamma\quad{\rm and}\quad V_\delta {\cal S}ubset V_\gamma \quad ,$$ it is easily checked that we have $$({\cal O} H)j\otimes_{S_\delta} V_\delta\cong ({\cal O} H)i\otimes_{S_\gamma} V_\gamma{\cal S}ubset ({\cal O} G)i\otimes_{S_\gamma} V_\gamma \quad ;\leqno £7.4.2$$ in particular, we have an evident section $$({\cal O} G)i\otimes_{S_\gamma} V_\gamma\longrightarrow ({\cal O} H)j\otimes_{S_\delta} V_\delta$$ which is actually an ${\cal O} Hb\otimes_{\cal O} {\cal O} Q\hbox{-}$module homomorphism. Similarly, we have a split ${\cal O} (H^A) w(b)\otimes_{\cal O} {\cal O} Q\hbox{-}$module monomorphism $${\cal O} (H^A)w(j)\otimes_{w(S_\delta)} w(V_\delta)\longrightarrow {\cal O} (G^A) w(i) \otimes_{w(S_\gamma)} w(V_\gamma) \quad .\leqno £7.4.3$$ {\cal S}mallskip In conclusion, the ${\cal O} Hb\otimes_{\cal O} {\cal O} Q\hbox{-}$ and ${\cal O} (H^A) w(b)\otimes_{\cal O} {\cal O} Q\hbox{-}$module homomorphisms~£7.4.2 and~£7.4.3, together with the inclusion ${\cal O} Q{\cal S}ubset {\cal O} \hat L\,,$ determine an ${\cal O} (H\times H^A)\hbox{-}$module homomorphism $$M\longrightarrow {\rm Res}_{H\times H^A}^{G\times G^A} (D) \leqno £7.4.4$$ which actually admits a section too. Now, denoting by $K$ the inverse image in $G\times G^A$ of the ``diagonal'' subgroup of $(G/H)\times (G^A/H^A)\,,$ we claim that the product by $K$ stabilizes the image of $M$ in $D\,,$ so that $M$ can be extended to an ${\cal O} K\hbox{-}$module. {\cal S}mallskip Actually, we have $$K = (H\times H^A){\,\hskip-1pt\cdot\hskip-1pt\,}{\cal D}elta (N_{G^A}(Q_\delta)) \quad ,$$ so that it suffices to prove that the image of $M$ is stable by multiplication by ${\cal D}elta (N_{G^A}(Q_\delta))\,.$ Given $x\leqn N_{G^A}(Q_\delta)$, there are some invertible elements $a_x\leqn ({\cal O} H)^Q$ and $b_x\leqn ({\cal O} (H^A))^Q$ such that $$xjx^{-1} = a_xj a_x^{-1}\quad{\rm and}\quad x w(j)x^{-1} = b_x w(j )b_x^{-1}$$ and therefore $a_x^{-1}x$ and $b_x^{-1}x$ respectively centralize $j$ and $w(j)\,,$ so that $a_x^{-1}xj$ and $b_x^{-1}x w(j)$ respectively belong to $({\cal O} G)_\delta$ and to $({\cal O} G^A)_{w(\delta)}\,;$ but, according to isomorphisms~£7.4.1 and~£7.1.1, we have $G/H\hbox{-}$ and $G^A/H^A\hbox{-}$gra-ded isomorphisms $$({\cal O} G)_\delta\cong S_\delta\otimes_{\cal O} {\cal O}_*\hat L^\circ\quad{\rm and}\quad ({\cal O} G^A)_{w(\delta)}\cong w(S_\delta)\otimes_{\cal O} {\cal O}_*\hat L^\circ $$ where we are setting $w(S_\delta) = w(j)w(S_\gamma)w(j)\,.$ {\cal S}mallskip Hence, identifying with each other both members of these isomorphisms and modifying if necessary our choice of $a_x\,,$ for some $s_x\leqn S_\delta\,,$ $t_x\leqn w(S_\delta)$ and $\hat y_x\leqn \hat L^\circ\,,$ we get $$a_x^{-1}x j = s_x\otimes \hat y_x\quad{\rm and}\quad b_x^{-1}x w(j) = t_x\otimes \hat y_x \quad .$$ Thus, setting $w(V_\delta) = w(j)w(V_\gamma)\,,$ for any $a\leqn ({\cal O} H)j\,,$ any $b\leqn ({\cal O} H^A)w(j)\,,$ any $v\leqn V_\delta$ and any $w\leqn w(V_\delta)\,,$ in $D$ we have \begin{eqnarray*}(x,x)\!\!\!\!&{\,\hskip-1pt\cdot\hskip-1pt\,}&\!\!\!\!(a\otimes v)\otimes (w\otimes b) = (x a\otimes v)\otimes (w\otimes bx^{-1})\\ &=& (x ax^{-1}a_x (a_x^{-1}xj)\otimes v)\otimes (w\otimes (w(j)x^{-1}b_x)b_x^{-1}xbx^{-1})\\ &=&(x ax^{-1}a_x \otimes s_x{\,\hskip-1pt\cdot\hskip-1pt\,}v){\,\hskip-1pt\cdot\hskip-1pt\,} \hat y_x\otimes \hat y_x^{-1}{\,\hskip-1pt\cdot\hskip-1pt\,}(w{\,\hskip-1pt\cdot\hskip-1pt\,}t_x^{-1}\otimes b_x^{-1}xbx^{-1})\\ &=&(x ax^{-1}a_x \otimes s_x{\,\hskip-1pt\cdot\hskip-1pt\,}v) \otimes (w{\,\hskip-1pt\cdot\hskip-1pt\,}t_x^{-1}\otimes b_x^{-1}xbx^{-1})\quad ; \end{eqnarray*} since $x ax^{-1}a_x $ and $b_x^{-1}xbx^{-1}$ respectively belong to $({\cal O} H)j$ and $w(j)({\cal O} H^A)\,,$ this proves our claim. {\cal S}mallskip Finally, since homomorphism~£7.4.4 actually becomes an ${\cal O} K\hbox{-}$module homomorphism, it induces an ${\cal O} (G\times G^A)\hbox{-}$module homomorphism $${\rm Ind}_K^{G\times G^A}(M)\longrightarrow D$$ which is actually an isomorphism as it is easily checked. We are done. The following theorem is due to Harris and Linckelmann (see \cite{H}). \noindent{\bf Theorem 7.5.}\quad {\leqt Let $G$ be an $A$-group and assume that $G$ is a finite $p$-solvable group and $A$ is a solvable group of order prime to $|G|$. Let $b $ be an $A$-stable block of $G$ over ${\cal O}$ with a defect group $P$ centralized by $A$ and denote by ${\leqt w}(b)$ the Glauberman correspondent of the block $b$. Then the block algebras ${\cal O} Gb$ and ${\cal O} (G^A){\leqt w}(b)$ are basically Morita equivalent. } \noindent{\leqt Proof.}\quad By \cite[Theorem 5.1]{H}, we can assume that $b$ is a $G\rtimes A$-stable block of ${\rm O}_{p'}(G)$, where ${\rm O}_{p'}(G)$ is the maximal normal $p'$-subgroup of $G$. Clearly $b$ as a block of ${\rm O}_{p'}(G)$ is nilpotent and thus ${\cal O} Gb$ is an extension of the nilpotent block algebra ${\cal O} {\rm O}_{p'}(G) b$. By \cite[Theorem 5.1]{H} again, ${\leqt w}(b)$ is a $G^A$-stable block of ${\rm O}_{p'}(G^A)$ and thus is nilpotent; thus ${\cal O} (G^A) {\leqt w}(b)$ is an extension of the nilpotent block algebra ${\cal O} {\rm O}_{p'}(G^A) {\leqt w}(b)$. By \cite[Theorem 4.1]{H}, ${\leqt w}(b)$ is also the Glauberman correspondent of $b$ as a block of ${\rm O}_{p'}(G)$. Then, by Theorem 1.6, the block algebras ${\cal O} Gb$ and ${\cal O} (G^A){\leqt w}(b)$ are basically Morita equivalent. The following theorem is due to Koshitani and Michler (see \cite{KG}). \noindent{\bf Theorem 7.6.}\quad {\leqt Let $G$ be an $A$-group and assume that $A$ is a solvable group of order prime to $|G|$. Let $b $ be an $A$-stable block of $G$ over ${\cal O}$ with a defect group $P$ centralized by $A$ and denote by ${\leqt w}(b)$ the Glauberman correspondent of the block $b$. Assume that $P$ is normal in $G$. Then, the block algebras ${\cal O} Gb$ and ${\cal O} (G^A{)\leqt w}(b)$ have isomorphic source algebras. } \noindent{\leqt Proof.}\quad Since $P$ is normal in $G$, by \cite[2.9]{AB} there is a block $b_P$ of $C_G(P)$ such that $b={\rm Tr}^G_{G_{b_P}}(b_P)$, where $G_{b_P}$ is the stabilizer of $b_P$ in $G$. Since $A$ and $G$ have coprime orders, by \cite[Lemma 13.8 and Corollary 13.9]{I}, $b_P$ can be chosen such that $A$ stabilizes $b_P$. Since $P$ is the unique defect group of $b$, $P$ has to be contained in $G_{b_P}\,;$ then by \cite[Proposition 5.3]{KP}, the intersection $Z(P)=P\cap C_G(P)$ is the defect group of $b_P$ and, in particular, $b_P$ is nilpotent. Thus the block ${\cal O} G b$ is an extension of the nilpotent block algebra~${\cal O}(P{\,\hskip-1pt\cdot\hskip-1pt\,} C_G(P))b_P$ and, in particular, we have $\bar N \cong E_G (P_\gamma)\,.$ The Glauberman correspondent of $b_P$ makes sense and by \cite[Proposition 4]{W}, we have $${\leqt w}(b)={\rm Tr}^{G^A}_{(G^A)_{{\leqt w}(b_P)}}({\leqt w}(b_P)) \quad .$$ Since ${\leqt w}(b_P)$ has defect group $Z(P)$, it is also nilpotent and thus ${\cal O} (G^A){\leqt w}(b)$ is an extension of the nilpotent block algebra ${\cal O}(P{\,\hskip-1pt\cdot\hskip-1pt\,} C_{G^A}(P)){\leqt w}(b_P)\,;$ once again, we have $\bar N ^A\cong E_{G^A} (P_{w(\gamma)})\,.$ {\cal S}mallskip On the other hand, since $P$ is normal in $G\,,$ it follows from \cite[Proposition 14.6]{P6} that $$({\cal O} G)_\gamma\cong {\cal O}_*(P \rtimes \hat E_G (P_\gamma))\quad{\rm and}\quad ({\cal O} G^A)_{w(\gamma)}\cong {\cal O}_*(P \rtimes \hat E_{G^A} (P_{w(\gamma)})) \quad ;$$ but, it follows from~£6.3 that we have a $k^*\hbox{-}$group isomorphism $$\hat E_G (P_\gamma)\cong \hat E_{G^A} (P_{w(\gamma)}) \quad .$$ We are done. Lluis Puig CNRS, Institut de Math\'ematiques de Jussieu 6 Av Bizet, 94340 Joinville-le-Pont, France {\cal S}mallskip puig@math.jussieu.fr Yuanyang Zhou Department of Mathematics and Statistics Central China Normal University Wuhan, 430079 P.R. China {\cal S}mallskip zhouyy74@163.com \end{document}

\begin{document} \title {All integral slopes can be Seifert fibered slopes\\for hyperbolic knots} \author{Kimihiko Motegi\\Hyun-Jong Song} \address{Department of Mathematics, Nihon University\\Tokyo 156-8550, Japan} \secondaddress{Division of Mathematical Sciences, Pukyong National University\\599-1 Daeyondong, Namgu, Pusan 608-737, Korea} \asciiaddress{Department of Mathematics, Nihon University\\Tokyo 156-8550, Japan\\and\\Division of Mathematical Sciences, Pukyong National University\\599-1 Daeyondong, Namgu, Pusan 608-737, Korea} \gtemail{\mailto{motegi@math.chs.nihon-u.ac.jp}{\rm\qua and\qua}\mailto{hjsong@pknu.ac.kr}} \asciiemail{motegi@math.chs.nihon-u.ac.jp, hjsong@pknu.ac.kr} \begin{abstract} Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the $3$-sphere $S^3$? It is conjectured that if $r$-surgery on a hyperbolic knot in $S^3$ yields a Seifert fiber space, then $r$ is an integer. We show that for each integer $n \in \mathbb{Z}$, there exists a tunnel number one, hyperbolic knot $K_n$ in $S^3$ such that $n$-surgery on $K_n$ produces a small Seifert fiber space. \end{abstract} \asciiabstract{ Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such that n-surgery on K_n produces a small Seifert fiber space.} \primaryclass{57M25, 57M50} \keywords{Dehn surgery, hyperbolic knot, Seifert fiber space, surgery slopes} \maketitle {\small\it This paper is dedicated to Donald M. Davis on the occasion of his 60th birthday.\leftskip25pt\rightskip25pt\par} \section{Introduction} Let $K$ be a knot in the $3$-sphere $S^3$ with a tubular neighborhood $N(K)$. Then the set of \textit{slopes} for $K$ (i.e., $\partial N(K)$-isotopy classes of simple loops on $\partial N(K)$) is identified with $\mathbb{Q} \cup \{\infty \}$ using preferred meridian-longitude pair so that a meridian corresponds to $\infty$. A slope $\gamma$ is said to be \textit{integral} if a representative of $\gamma$ intersects a meridian exactly once, in other words, $\gamma$ corresponds to an integer under the above identification. In the following, we denote by $(K; \gamma)$ the $3$-manifold obtained from $S^3$ by Dehn surgery on a knot $K$ with slope $\gamma$, i.e., by attaching a solid torus to $S^3-$int$N(K)$ in such a way that $\gamma$ bounds a meridian disk of the filled solid torus. If $\gamma$ corresponds to $r \in \mathbb{Q} \cup \{ \infty \}$, then we identify $\gamma$ and $r$ and write $(K; r)$ for $(K; \gamma)$. \par We denote by $\mathcal{L}$ the \textit{set of lens slopes} $\{r \in \mathbb{Q}\ |\ \exists$ hyperbolic knot $K \subset S^3$ such that $(K;r)$ is a lens space$\}$, where $S^3$ and $S^2 \times S^1$ are also considered as lens spaces. Then the cyclic surgery theorem \cite{CGLS} implies that $\mathcal{L} \subset \mathbb{Z}$. A result of Gabai \cite[Corollary 8.3]{Ga} shows that $0 \not\in \mathcal{L}$, a result of Gordon and Luecke \cite{G-Lu} shows that $\pm 1 \not\in \mathcal{L}$. In \cite{KM} Kronheimer and Mrowka prove that $\pm 2 \not\in \mathcal{L}$. Furthermore, a result of Kronheimer, Mrowka, Ozsv\'ath and Szab\'o \cite{KMOS} implies that $\pm 3, \pm 4 \not\in \mathcal{L}$. Besides, Berge \cite[Table of Lens Spaces]{Berge} suggests that if $n \in \mathcal{L}$, then $|n| \ge 18$ and not every integer $n$ with $|n| \ge 18$ appears in $\mathcal{L}$. Fintushel and Stern \cite{FS} had shown that $18$-surgery on the $(-2, 3, 7)$ pretzel knot yields a lens space. \textit{Which slope $($rational number$)$ can or cannot appear in the set of Seifert fibered slopes $\mathcal{S} = \{r \in \mathbb{Q}\ |\ \exists$ hyperbolic knot $K \subset S^3$ such that $(K;r)$ is Seifert fibered$\}$?} It is conjectured that $\mathcal{S} \subset \mathbb{Z}$ \cite{Go}. The purpose of this paper is to prove: \begin{THM} \label{Seifert slopes} For each integer $n \in \mathbb{Z}$, there exists a tunnel number one, hyperbolic knot $K_n$ in $S^3$ such that $(K_n; n)$ is a small Seifert fiber space $($i.e., a Seifert fiber space over $S^2$ with exactly three exceptional fibers$)$. \end{THM} \begin{rem} Since $K_n$ has tunnel number one, it is embedded in a genus two Heegaard surface of $S^3$ and strongly invertible \cite[Lemma 5]{Mor}. See \cite[Question 3.1]{MMM}. \end{rem} Theorem \ref{Seifert slopes}, together with the previous known results, shows: \begin{CO} \label{LZS} $\mathcal{L} \subsetneqq \mathbb{Z} \subset \mathcal{S}$. \end{CO} \begin{rems}$\phantom{99}$ (1)\qua For the \textit{set of reducing slopes} $\mathcal{R} = \{r \in \mathbb{Q}\ |\ \exists$ hyperbolic knot $K \subset S^3$ such that $(K;r)$ is reducible$\}$, Gordon and Luecke \cite{G-Lu1} have shown that $\mathcal{R} \subset \mathbb{Z}$. In fact, the cabling conjecture \cite{GS} asserts that $\mathcal{R} = \emptyset$. (2)\qua For the \textit{set of toroidal slopes} $\mathcal{T} = \{r \in \mathbb{Q}\ |\ \exists$ hyperbolic knot $K \subset S^3$ such that $(K;r)$ is toroidal$\}$, Gordon and Luecke \cite{G-Lu2} have shown that $\mathcal{T} \subset \mathbb{Z}/2$ (integers or half integers). In \cite{Tera}, Teragaito shows that $\mathbb{Z} \subset \mathcal{T}$ and conjectures that $\mathcal{T} \subsetneqq \mathbb{Z}/2$. \end{rems} \textbf{Acknowledgements}\qua We would like to thank the referee for careful reading and useful comments. \newline The first author was partially supported by Grant-in-Aid for Scientific Research (No.\ 15540095), The Ministry of Education, Culture, Sports, Science and Technology, Japan. \newline \section{Hyperbolic knots with Seifert fibered surgeries} Our construction is based on an example of a longitudinal Seifert fibered surgery given in \cite{IMS}. Let $k \cup c$ be a $2$-bridge link given in Figure \ref{fig:knotKn}, and let $K_n$ be a knot obtained from $k$ by $\frac{1}{-n+4}$-surgery along $c$. \begin{figure} \caption{$K_n$} \label{fig:knotKn} \end{figure} We shall say that a Seifert fiber space is of \textit{type} $S^2(n_1, n_2, n_3)$ if it has a Seifert fibration over $S^2$ with three exceptional fibers of indices $n_1, n_2$ and $n_3$ $(n_i \ge 2)$. Since $K_4$ is unknotted, $(K_4; 4)$ is a lens space $L(4, 1)$. For the other $n$'s, we have: \begin{LM} \label{Seifert} $(K_n; n)$ is a small Seifert fiber space of type $S^2(3, 5, |4n-15|)$ for any integer $n \ne 4$. \end{LM} \begin{proof} Since the linking number of $k$ and $c$ is one (with suitable orientations), $(K_n ; n)$ has surgery descriptions as in Figure \ref{fig:description}. \begin{figure} \caption{Surgery descriptions of $(K_n; n)$} \label{fig:description} \end{figure} Let us take the quotient by the strong inversion of $S^3$ with an axis $L$ as shown in Figure \ref{fig:msequence1}. \begin{figure}\label{fig:msequence1} \end{figure} Then we obtain a branch knot $b'$ which is the image of the axis $L$. The Montesinos trick (\cite{Mon}, \cite{Bl}) shows that $-\frac{1}{2}, -1, \frac{3n-11}{-n+4}$ and $1$-surgery on $t_1, t_2, c$ and $k$ in the upstairs correspond to $-\frac{1}{2}, -1, \frac{3n-11}{-n+4}$ and $1$-untangle surgery on $b'$ in the downstairs, where an \textit{$r$-untangle surgery} is a replacement of $\frac{1}{0}$-untangle by $r$-untangle. (We adopt Bleiler's convention \cite{Ble} on the parametrization of rational tangles.) These untangle surgeries convert $b'$ into a link $b$ (Figure \ref{fig:msequence1}). \begin{figure} \caption{Continued from Figure \ref{fig:msequence1} \label{fig:msequence2} \end{figure} Following the sequence of isotopies in Figures \ref{fig:msequence1} and \ref{fig:msequence2}, we obtain a Montesinos link $M(\frac{2}{5}, -\frac{2}{3}, \frac{n-4}{4n-15})$. Since $(K_n ; n)$ is the double branched cover of $S^3$ branched over the Montesinos link $M(\frac{2}{5}, -\frac{2}{3}, \frac{n-4}{4n-15})$, $(K_n; n)$ is a Seifert fiber space of type $S^2(3, 5, |4n-15|)$ as desired. \end{proof} \begin{LM} \label{hyperbolic} The knot $K_n$ is hyperbolic if $n \ne 3, 4, 5$. \end{LM} \begin{proof} Note that the $2$-bridge link given in Figure \ref{fig:knotKn} is not a $(2, p)$-torus link, and hence by \cite{Me} it is a hyperbolic link. If $n \ne 3, 4, 5$, then $|-n+4|>1$ and it follows from \cite[Theorem 1]{AMM1} (also \cite[Theorem 1.2]{AMM3}) that $K_n$ is a hyperbolic knot. See also \cite[Corollary A.2]{G-Lu3}, \cite[Theorem 1.2]{MM3} and \cite[Theorem 1.1]{AMM2}. \end{proof} \begin{rem} It follows from \cite{Ma}, \cite{KMS} that $K_n$ is a nontrivial knot except when $n = 4$. An experiment using Weeks' computer program ``SnapPea" \cite{W} suggests that $K_3$ and $K_5$ are hyperbolic, but we will not use this experimental results. \end{rem} \begin{LM} \label{tunnel} The knot $K_n$ has tunnel number one for any integer $n \ne 4$. \end{LM} \begin{proof} Since the link $k \cup c$ is a two-bridge link, the tunnel number of $k \cup c$ is one with unknotting tunnel $\tau$; A regular neighborhood $N(k \cup c \cup \tau)$ is a genus two handlebody and $S^3 - \mathrm{int}N(k \cup c \cup \tau)$ is also a genus two handlebody, see Figure \ref{fig:tunnel}. \begin{figure}\label{fig:tunnel} \end{figure} Then the general fact below (in which $k \cup c$ is not necessarily a two-bridge link) shows that the tunnel number of $K_n$ is less than or equal to one. Since our knot $K_n$ $(n \ne 4)$ is knotted in $S^3$, the tunnel number of $K_n$ is one. \end{proof} \begin{CL} \label{tunnel1} Let $k \cup c$ be a two component link in $S^3$ which has tunnel number one. Assume that $c$ is unknotted in $S^3$. Then every knot obtained from $k$ by twisting along $c$ has tunnel number at most one. \end{CL} \begin{proof} Let $\tau$ be an unknotting tunnel and $V$ a regular neighborhood of $k \cup c \cup \tau$ in $S^3$; $V$ is a genus two handlebody. Since $\tau$ is an unknotting tunnel for $k \cup c$, by definition, $W = S^3 - \mathrm{int}V$ is also a genus two handlebody. Take a small tubular neighborhood $N(c) \subset \mathrm{int}V$ and perform $-\frac{1}{n}$-surgery on $c$ using $N(c)$. Then we obtain a knot $k_n$ as the image of $k$ and obtain a genus two handlebody $V(c; -\frac{1}{n})$. Note that $V(c; -\frac{1}{n})$ and $W$ define a genus two Heegaard splitting of $S^3$, see Figure \ref{fig:arc}, where $c_n^*$ denotes the core of the filled solid torus. \begin{figure}\label{fig:arc} \end{figure} Then it is easy to see that an arc $\tau_n$ given by Figure \ref{fig:arc} is an unknotting tunnel for $k_n$ as desired. \end{proof} Now we are ready to prove Theorem \ref{Seifert slopes}. Lemmas \ref{Seifert}, \ref{hyperbolic} and \ref{tunnel} show that our knots $K_n$ enjoy the required properties, except for $n = 3, 4, 5$. To prove Theorem \ref{Seifert slopes}, we find hyperbolic knots $K'_n$ so that $(K'_n; n)$ is Seifert fibered for $n = 3, 4, 5$ (instead of showing that $K_3$, $K_5$ are hyperbolic). As the simplest way, let $K'_3$, $K'_4$ and $K'_5$ be the mirror image of $K_{-3}$, $K_{-4}$ and $K_{-5}$, respectively. Since $K_{-3}$, $K_{-4}$ and $K_{-5}$ are tunnel number one, hyperbolic knots by Lemmas \ref{hyperbolic} and \ref{tunnel}, their mirror images $K'_3$, $K'_4$ and $K'_5$ are also tunnel number one, hyperbolic knots. It is easy to observe that $(K'_3; 3)$ (resp. $(K'_4; 4)$, $(K'_5; 5)$) is the mirror image of $(K_{-3}; -3)$ (resp. $(K_{-4}; -4)$, $(K_{-5}; -5)$). By Lemma \ref{Seifert}, $(K_{-3}; -3)$, $(K_{-4}; -4)$ and $(K_{-5}; -5)$ are Seifert fibered, and hence $(K'_3; 3)$, $(K'_4; 4)$ and $(K'_5; 5)$ are also Seifert fibered. Putting $K_n$ as $K'_n$ for $n = 3, 4, 5$, we finish a proof of Theorem \ref{Seifert slopes}. \hspace*{\fill} $\qed$ \section{Identifying exceptional fibers} In \cite{MM3}, Miyazaki and Motegi conjectured that if $K$ admits a Seifert fibered surgery, then there is a trivial knot $c \subset S^3$ disjoint from $K$ which becomes a Seifert fiber in the resulting Seifert fiber space, and verified the conjecture for several Seifert fibered surgeries \cite[Section 6]{MM3}, see also \cite{EM}. Furthermore, computer experiments via ``SnapPea" \cite{W} suggest that such a knot $c$ is realized by a short closed geodesic in the hyperbolic manifold $S^3 - K$, for details see \cite[Section 9]{MM3}, \cite{Mot2}. In this section, we verify the conjecture for Seifert fibered surgeries given in Theorem \ref{Seifert slopes}. Recall that $K_n$ is obtained from $k$ by $\frac{1}{-n+4}$-surgery on the trivial knot $c$ (i.e., $(n-4)$-twist along $c$), see Figure \ref{fig:knotKn}. Denote by $c_n$ the core of the filled solid torus. Then $K_n \cup c_n$ is a link in $S^3$ such that $c_n$ is a trivial knot. \begin{LM} \label{fiber} After $n$-surgery on $K_n$, $c_n$ becomes an exceptional fiber of index $|4n-15|$ in the resulting Seifert fiber space $(K_n; n)$. \end{LM} \begin{proof} Following the sequences given by Figures \ref{fig:msequence1} and \ref{fig:msequence2}, we have a Montesinos link with three arcs $\gamma$, $\tau_1$ and $\tau_2$ as in Figure \ref{fig:positions}, where $n = 1$ in the final Montesinos link, and $\gamma$, $\tau_1$, $\tau_2$ and $\kappa$ are the images of $c$, $t_1$, $t_2$ and $k$, respectively. \begin{figure} \caption{Positions of exceptional fibers} \label{fig:positions} \end{figure} From Figure \ref{fig:positions} we recognize that $t_1, t_2$ and $c$ become exceptional fibers of indices $5$, $3$ and $|4n-15|$, respectively in $(K_n; n)$. \end{proof} For $n \ne 3, 4, 5$, $c_n$ becomes an exceptional fiber of index $|4n-15|$, which is the unique maximal index, in $(K_n; n)$. Experiments via ``SnapPea" \cite{W} suggest that $c_n$ is a shortest closed geodesic in $S^3 - K_n$ $(n \ne 3, 4, 5)$. For sufficiently large $|n|$, hyperbolic Dehn surgery theorem \cite{T1}, \cite{T2} shows that $c_n$ is the unique shortest closed geodesic in $S^3 - K_n$. \par Let us assume that $n = 3, 4, 5$. Then we have put $K_n$ as the mirror image of $K_{-n}$ in the proof of Theorem \ref{Seifert slopes}. Let $k' \cup c'$ be the mirror image of the link $k \cup c$. Then $K_n$ is obtained also from $k'$ by $\frac{1}{-n-4}$-surgery on $c'$ (i.e., $(n+4)$-twist along $c'$); we denote the core of the filled solid torus by $c'_n$. Note that there is an orientation reversing diffeomorphism from $(K_{-n}; -n)$ to $(K_n; n)$ sending $c_{-n}$ (regarded as a fiber in $(K_{-n}; -n)$) to $c'_n$ (regarded as a fiber in $(K_n; n)$). Thus the above observation implies that $c'_n$ becomes an exceptional fiber of index $|4n+15|$, which is the unique maximal index, in $(K_n; n)$ $(n = 3, 4, 5)$. \Addresses\recd \end{document}

\begin{document} \begin{frontmatter} \title{A parallel-in-time fixed-stress splitting method for Biot's consolidation model} \author[Bergen_address]{Manuel Borregales\corref{mycorrespondingauthor}}\ead{Manuel.Borregales@uib.no} \author[Bergen_address]{Kundan Kumar}\ead{Kundan.Kumar@uib.no} \author[Bergen_address]{Florin Adrian Radu}\ead{Florin.Radu@uib.no} \author[UZ_address]{Carmen Rodrigo}\ead{carmenr@unizar.es} \author[CWI_address]{Francisco Jos\'e Gaspar}\ead{F.J.Gaspar@cwi.nl} \address[Bergen_address]{Department of Mathematics, University of Bergen, All\'egaten 41, 50520 Bergen, Norway} \address[CWI_address]{CWI, Centrum Wiskunde \& Informatica, Science Park 123, P.O. Box 94079, 1090 Amsterdam, The Netherlands} \address[UZ_address]{IUMA and Department of Applied Mathematics, University of Zaragoza, Mar\'ia de Luna, 3, 50018 Zaragoza, Spain} \cortext[mycorrespondingauthor]{Corresponding author} \begin{abstract} In this work, we study the parallel-in-time iterative solution of coupled flow and geomechanics in porous media, modelled by a two-field formulation of the Biot's equations. In particular, we propose a new version of the fixed stress splitting method, which has been widely used as solution method of these problems. This new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization. We present a rigorous convergence analysis of the method and a numerical experiment to demonstrate the robust behaviour of the algorithm. \end{abstract} \begin{keyword} Iterative fixed-stress split scheme \sep parallel-in-time \sep Biot's model \MSC[2010] 00-01\sep 99-00 \end{keyword} \end{frontmatter} \section{Introduction} The coupled poroelastic equations describe the behaviour of fluid-saturated porous materials undergoing deformation. Such coupling has been intensively investigated, starting from the pioneering one-dimensional work of Terzaghi \cite{terzaghi}, which was extended to a more general three-dimensional theory by Biot \cite{Biot1, Biot2}. Biot's model was originally developed to study geophysical applications such as reservoir geomechanics, however, nowadays it is widely used in the modeling of many applications in a great variety of fields, ranging from geomechanics and petroleum engineering, to biomechanics or food processing. There is a vast literature on Biot's equations and their existence, uniqueness, and regularity, see Showalter \cite{showalter}, Phillips and Wheeler \cite{phillips1} and the references therein. Reliable numerical methods for solving poroelastic problems are needed for the accurate solution of multi-physics phenomena appearing in different application areas. In particular, the solution of the large linear systems of equations arising from the discretization of Biot's model is the most consuming part when real simulations are performed. For this reason, a lot of effort has been made in the last years to design efficient solution methods for these problems. Two different approaches can be adopted, the so-called monolithic or fully coupled methods and the iterative coupling methods. The monolithic approach consists of solving the linear system simultaneously for all the unknowns. The challenge here, is the design of efficient preconditioners to accelerate the convergence of Krylov subspace methods and the design of efficient smoothers in a multigrid framework. Recent advances in both directions can be found in \cite{Bergamaschi2007, Ferronato2010, Gaspar2004, Luo2017} and the references therein. These methods usually provide unconditional stability and convergence. Iterative coupling methods, however, solve sequentially the equations for fluid flow and geomechanics, at each time step, until a converged solution within a prescribed tolerance is achieved. They offer several attractive features as their flexibility, for example, since they allow to link two different codes for fluid flow and geomechanics for solving the coupled poroelastic problems. The design of iterative schemes however is an important consideration for an efficient, convergent, and robust algorithm. The most used iterative coupling methods are the drained and undrained splits, which solve the mechanical problem first, and the fixed-strain and fixed-stress splits, which on the contrary solve the flow problem first \cite{Kim2009, Kim_PhD}. Among iterative coupling schemes, the fixed stress splitting method is the most widely used. This sequential-implicit method basically consists in solving the flow problem first fixing the volumetric mean total stress, and then the mechanics part is solved from the values obtained at the previous flow step. In the last years, a lot of research has been done on this method. The unconditional stability of the fixed-stress split method is shown in \cite{Kim2011} using a von Neumann analysis. In addition, stability and convergence of the fixed-stress split method have been rigorously established in \cite{Mikelic}. Recently, in \cite{Both2017} the authors have proven the convergence of the fixed-stress split method in energy norm for heterogeneous problems. Estimates for the case of the multirate iterative coupling scheme are obtained in \cite{Almani}, where multiple finer time steps for flow are taken within one coarse mechanics time step, exploiting the different time scales for the mechanics and flow problems. In \cite{Bause}, the convergence of this method is demonstrated in the fully discrete case when space-time finite element methods are used. In \cite{Castelleto2015}, the authors present a very interesting approach which consists to re-interpret the fixed-stress split scheme as a preconditioned-Richardson iteration with a particular block-triangular preconditioning operator. Recently, in \cite{Gaspar_Rodrigo2017} an inexact version of the fixed-stress split scheme has been successfully proposed as smoother in a geometric multigrid framework, which provides an efficient monolithic solver for Biot's problem. All the previously mentioned algorithms are based on a time-marching approach, in which each time step is solved after the other in a sequential manner, and therefore they do not allow the parallelization of the temporal variable. Time parallel time integration methods, however, are receiving a lot of interest nowadays because of the advent of massively parallel systems with thousands of cores, permitting to reduce drastically the computing time \cite{Gander2015}. Due to the mixed elliptic-parabolic structure of Biot's problem, the development of parallel-in time algorithms is not intuitive. In the present work, we introduce a very simple version of the fixed-stress splitting method for the poroelasticity problem which allows an easy parallelization in time. We further analyze the method and show that it is convergent. Techniques similar with the ones from \cite{Mikelic, Both2017, Bause} are used. For completeness, in Section 3, we include a new proof for the convergence of the fixed-stress split algorithm in the semidiscrete case. The theoretical results are sustained by numerical computations. The remainder of the paper is organized as follows. In Section \ref{sec:model} we briefly introduce the poroelasticity model and present the considered finite element discretizations. Section \ref{sec:fixed_stress} is devoted to the description of the classical fixed-stress split algorithm. In Section \ref{sec:parallel_fixed}, the parallel-in-time new approach based on the fixed-stress split algorithm is presented and its convergence analysis is derived. Section \ref{sec:numerical} illustrates the robustness of the proposed parallel-in-time fixed-stress split method through a numerical experiment. Finally, some conclusions are drawn in Section \ref{sec:conclusion}. \section{Mathematical model and discretization}\label{sec:model} The equations describing poroelastic flow and deformation are derived from the principles of fluid mass conservation and the balance of forces on the porous matrix. More concretely, according to Biot's theory~\cite{Biot1,Biot2}, and assuming $\Omega$ a bounded open subset of $ {\mathbb R}^d,\; d \leq 3$, with regular boundary $\Gamma$, the consolidation process must satisfy the following system of partial differential equations: \begin{eqnarray} \mbox{\rm equilibrium equation:} & & -{\rm div} \, {\boldsymbol \sigma}' + \alpha \nabla \, p = \rho {\bm g}, \quad {\rm in} \, \Omega, \label{eq11} \\ \mbox{\rm constitutive equation:} & & \bm{\sigma}' = 2 G {\boldsymbol \varepsilon}(\bm{u}) + \lambda\ddiv(\bm{u}) {\bm I}, \quad {\rm in} \, \Omega, \label{eq12} \\ \mbox{\rm compatibility condition:} & & {\boldsymbol \varepsilon}({\bm u}) = \frac{1}{2}(\nabla {\bm u} + \nabla {\bm u}^t), \quad {\rm in} \, \Omega, \label{eq13} \\ \mbox{\rm Darcy's law:} & & {\bm q} = - \frac{1}{\mu_f} {\bm K} \left(\nabla p - \rho_f {\bm g} \right), \quad {\rm in} \, \Omega, \label{eq14} \\ \mbox{\rm continuity equation:} & & \frac{\partial}{\partial t} \left( \frac{1}{\beta} p + \alpha \nabla \cdot \, \bm u \right) + \nabla \cdot \, {\bm q} = f, \quad {\rm in} \, \Omega, \label{eq15} \end{eqnarray} where $\boldsymbol I$ is the identity tensor, ${\bm u}$ is the displacement vector, $p$ is the pore pressure, ${\boldsymbol \sigma}'$ and ${\boldsymbol \varepsilon}$ are the effective stress and strain tensors for the porous medium, ${\bm g}$ is the gravity vector, ${\bm q}$ is the percolation velocity of the fluid relative to the soil, $\mu_f$ is the fluid viscosity and $\boldsymbol K$ is the absolute permeability tensor. The Lam\'{e} coefficients, $\lambda$ and $G$, can be also expressed in terms of the Young's modulus $E$ and the Poisson's ratio $\nu$ as $\lambda = E \nu / ((1-2 \nu )(1+\nu))$ and $G= E/(2+2\nu)$. The bulk density $\rho$ is related to the densities of the solid ($\rho_s$) and fluid ($\rho_f$) phases as $\rho = \phi \rho_f + (1-\phi) \rho_s$, where $\phi$ is the porosity. $\beta$ is the Biot modulus and $\alpha$ is the Biot coefficient given by $\alpha = 1 - K_b/K_s,$ where $K_b$ is the drained bulk modulus , and $K_s$ is the bulk modulus of the solid phase. If considering the displacements of the solid matrix ${\bm u}$ and the pressure of the fluid $p$ as primary variables, we obtain the so-called two-field formulation of the Biot's consolidation model. With this idea in mind, combining equations \eqref{eq11}-\eqref{eq15}, the mathematical model can be written as \begin{eqnarray} && -\ddiv\bm{\sigma}' + \alpha \nabla p = \rho {\bm g},\bm quad \bm{\sigma}' = 2 G \ {\boldsymbol \varepsilon}(\bm{u}) + \lambda\ddiv(\bm{u}) \bm{I}, \label{two-field1}\\ && \frac{\partial}{\partial t} \left( \frac{1}{\beta} p + \alpha \nabla \cdot \, \bm u \right) - \nabla \cdot \, \left( \frac{1}{\mu_f} {\bm K} \left(\nabla p - \rho_f {\bm g} \right)\right) = f.\label{two-field2} \end{eqnarray} The most important feature of this mathematical model is that the equations are strongly coupled. Here, the Biot parameter $\alpha$ plays the role of coupling parameter between these equations. In order to ensure the existence and uniqueness of solution, we must supplement the system with appropriate boundary and initial conditions. For instance, \begin{equation}\label{bound-cond} \begin{array}{ccccc} p = 0, & & \quad \boldsymbol \sigma' \, {\bm n} &=& {\bm 0}, \quad \hbox {on }\Gamma _t, \\ {\bm u} = {\bm 0}, & & \quad \displaystyle {\bm K} \left(\nabla p - \rho_f {\bm g} \right) \cdot {\bm n} &=& 0, \quad \hbox {on } \Gamma _c, \end{array} \end{equation} where ${\bm n}$ is the unit outward normal to the boundary and $\Gamma_t \cup \Gamma_c = \Gamma$, with $\Gamma_t$ and $\Gamma_c$ disjoint subsets of $\Gamma$ having non null measure. For the initial time, $t=0$, the following condition is fulfilled \begin{equation}\label{ini-cond} \left (\frac{1}{\beta} p + \alpha \nabla \cdot {\bm u} \right)\, (\bm{x},0)=0, \, \bm{x} \in\Omega. \end{equation} \subsection{Semi-discretization in space} To introduce the spatial discretization of the Biot model, we choose the finite element method. We define the standard Sobolev spaces ${\bm V} = \{{\bm u}\in (H^1(\Omega))^n \ | \ {\bm u}|_{\Gamma _c} = {\bm 0} \},$ and $Q = \{p \in H^1(\Omega) \ | \ p|_{\Gamma _t} = 0\}$, with $H^1(\Omega)$ denoting the Hilbert subspace of $L_2(\Omega)$ of functions with first weak derivatives in $L_2(\Omega)$. Then, we introduce the variational formulation for the two-field formulation of the Biot's model as follows: For each $t \in (0,T]$, find $({\bm u}(t) ,p(t)) \in {\bm V} \times Q$ such that \begin{eqnarray} && a(\bm{u}(t),\bm{v}) - \alpha(p(t),\ddiv \bm{v}) = (\rho \bm g,\bm{v}), \quad \forall \ \bm{v}\in \bm V, \label{variational1}\\ && \alpha(\ddiv \partial_t{\bm{u}(t)},q) + \frac{1}{\beta}(\partial_t p(t),q) + b(p(t),q) = (f,q) + ( {\bm K}\mu_f^{-1} \rho_f {\bm g},\nabla q ), \quad \forall \ q \in Q,\label{variational2} \end{eqnarray} where $(\cdot,\cdot)$ is the standard inner product in the space $L_2(\Omega)$, and the bilinear forms $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$ are given as \begin{eqnarray*}\label{bilinear} a(\bm{u},\bm{v}) &=& 2 G \int_{\Omega}{\boldsymbol \varepsilon}(\bm{u}):{\boldsymbol \varepsilon}(\bm{v}) \, {\rm d} \Omega + \lambda\int_{\Omega} \ddiv\bm{u}\ddiv\bm{v} \, {\rm d} \Omega, \\ b(p,q) &=& \int_{\Omega} \frac{{\bm K}}{\mu_f} \nabla p \cdot \nabla q \, {\rm d} \Omega. \end{eqnarray*} Finally, the initial condition is given by \begin{equation} \left (\frac{1}{\beta} p(0) + \alpha \nabla \cdot {\bm u}(0),q \right) = 0, \quad \forall \ q \in L_2(\Omega). \end{equation} It is important to consider a finite element pair of spaces ${\bm V}_h \times Q_h$ satisfying an inf-sup condition. One very simple choice would be the stabilized P1-P1 scheme firstly introduced in \cite{Aguilar2008} and widely analyzed in \cite{RGHZ2016}, in which ${\bm V}_h $ consists of the space of piece-wise (with respect to a triangulation ${\cal T}_h$) linear continuous vector valued functions on $\Omega$ and the space $Q_h$ consists of piece-wise linear continuous scalar valued functions. Other choices would be P2-P1, that is, piece-wise quadratic continuous vector valued functions for displacements and piece-wise linear continuous scalar valued functions for pressure, widely studied by Murad and Loula \cite{MuradLoula92, MuradLoula94, MuradLoulaThome}; or the so-called MINI element \cite{RGHZ2016} in which the only difference is that ${\bm V}_h = {\bm V}_l \oplus {\bm V}_b$, where ${\bm V}_l$ the space of piece-wise linear continuous vector valued functions and ${\bm V}_b$ is the space of bubble functions. Discrete inf-sup stability conditions and convergence results for the stabilized P1-P1 and the MINI element were recently derived in \cite{RGHZ2016}. The semi-discretized problem can be written as follows: For each $t\in (0,T]$, find $({\bm u}_h(t) ,p_h(t)) \in {\bm V}_h \times Q_h$ such that \begin{eqnarray} && a(\bm{u}_h(t),\bm{v}_h) - \alpha(p_h(t),\ddiv \bm{v}_h) = (\rho \bm g,\bm{v}_h), \quad \forall \ \bm{v}_h \in \bm V_h, \label{semi_discrete_variational1}\\ && \alpha(\ddiv \partial_t{\bm{u}_h(t)},q_h) + \frac{1}{\beta} (\partial_t{p_h(t)},q_h) + b(p_h(t),q_h) = (f_h,q_h) + ( {\bm K}\mu_f^{-1} \rho_f {\bm g},\nabla q_h ), \quad \forall \ q_h \in Q_h,\label{semi_discrete_variational2} \end{eqnarray} giving rise to the following algebraic/differential equations system, \begin{equation}\label{system_ODES} \left[ \begin{array}{cc} 0 & 0 \\ B & M_p \end{array} \right] \left[ \begin{array}{c} \dot{u}_h \\ \dot{p}_h \end{array} \right] + \left[ \begin{array}{cc} A & B^t \\ 0 & - C \end{array} \right] \left[ \begin{array}{c} {u}_h \\ {p}_h \end{array} \right] = \left[ \begin{array}{c} {g}_h \\ {\widetilde{f}}_h \end{array} \right], \end{equation} where we have denoted $\dot{u}_h \equiv \partial_t {\bm{u}_h(t)}$ and $\dot{p}_h \equiv \partial_t{p_h(t)}$. \begin{remark} We wish to emphasize that the solver based on the fixed stress split method, which we are going to propose in this work, can be applied to other different discretizations of the problem as mixed finite-elements or finite volume schemes, for example. \end{remark} \section{The fixed-stress split algorithm for the semi discretized problem}\label{sec:fixed_stress} A popular alternative for solving the poroelasticity problem in an iterative manner is to solve first the flow problem supposing a constant volumetric mean total stress, and once the flow problem is solved, the mechanic problem is then exactly solved. This is the so-called fixed-stress split method. More concretely, the volumetric mean total stress is defined as the mean of the trace of the total stress tensor, that is $\sigma_v = {\rm tr}({\bm \sigma}) /3$, and it is related to the volumetric strain $\varepsilon_v = {\rm tr}(\varepsilon)$ as $\sigma_v = K_b \varepsilon_v - \alpha p.$ By using this relation, we write the flow equation in terms of the volumetric mean total stress instead of the volumetric strain, \begin{equation}\label{flow2_equation} \left(\frac{1}{\beta} + \frac{\alpha^2}{K_b} \right) \frac{\partial p}{\partial t} + \frac{\alpha}{K_b} \frac{\partial \sigma_v}{\partial t} - \nabla \cdot \, \left( \frac{1}{\mu_f} {\bm K} \left(\nabla p - \rho_f {\bm g} \right)\right) = f. \end{equation} Then, the fixed-stress split scheme is based on solving the flow equation considering known the volumetric mean total stress, \begin{equation}\label{flow3_equation} \left(\frac{1}{\beta} + \frac{\alpha^2}{K_b} \right) \frac{\partial p}{\partial t} - \nabla \cdot \, \left( \frac{1}{\mu_f} {\bm K} \left(\nabla p - \rho_f {\bm g} \right)\right) = f - \alpha \frac{\partial} {\partial t} (\nabla \cdot \, {\bf u} ) + \frac{\alpha^2}{K_b} \frac{\partial p}{\partial t}. \end{equation} Finally, instead of the physical parameter $\frac{\alpha^2}{K_b}$, a general parameter $L$ to fix can be considered, obtaining \begin{equation}\label{flow4_equation} \left(\frac{1}{\beta} + L \right) \frac{\partial p}{\partial t} - \nabla \cdot \, \left( \frac{1}{\mu_f} {\bm K} \left(\nabla p - \rho_f {\bm g} \right)\right) = f - \alpha \frac{\partial} {\partial t} (\nabla \cdot \, {\bf u} ) + L \frac{\partial p}{\partial t}. \end{equation} Then, given an initial guess $({\bm u}_h^0(t),p_h^0(t))$, the fixed-stress split algorithm gives us a sequence of approximations $({\bm u}_h^i(t),p_h^i(t))$, $i \geq 1$ as follows: \\ \noindent {\bf Step 1:} Given $({\bm u}_h^{i-1}(t),p_h^{i-1}(t)) \in {\bm V}_h \times Q_h$, find $p_h^i(t) \in Q_h$ such that \begin{eqnarray} && (\frac{1}{\beta} +L) (\partial_t{p_h^i(t)},q_h) + b(p_h^i(t),q_h) + \alpha(\ddiv \partial_t{\bm{u}^{i-1}_h(t)},q_h) = L (\partial_t{p_h^{i-1}(t)},q_h) + \nonumber \\ & & \bm quad \bm quad (f_h,q_h) + ( {\bm K}\mu_f^{-1} \rho_f {\bm g},\nabla q_h ), \quad \forall \ q_h \in Q_h.\label{discrete_variational_split_pressure} \end{eqnarray} {\bf Step 2:} Given $p_h^{i}(t) \in Q_h$, find ${\bm u}_h^{i}(t) \in {\bm V}_h$ such that \begin{equation} \label{discrete_variational_split_displacement} a(\bm{u}_h^i(t),\bm{v}_h) = \alpha(p_h^i(t),\ddiv \bm{v}_h) + (\rho \bm g,\bm{v}_h), \quad \forall \ \bm{v}_h \in \bm V_h. \end{equation} The algorithm starts with an initial approximation $({\bm u}_h^0(t),p_h^0(t))$ defined along the whole time-interval. A natural choice is to take this approximation constant and equal to the values specified by the initial condition, $({\bm u}_h^0(t),p_h^0(t)) = (\bm u_0, p_0), \; t \in (0,T].$ \subsection{Convergence analysis in the semidiscrete case} Let $\delta \bm{u}_h^i(t) = \bm{u}_h^i(t) - \bm{u}_h^{i-1}(t)$ and $\delta p_h^i(t) = p_h^i(t)-p_h^{i-1}(t)$ denote the difference between two succesive approximations for displacements and for pressure, respectively. \begin{thm} The fixed-stress split method given in \eqref{discrete_variational_split_pressure}-\eqref{discrete_variational_split_displacement} converges for any $L \ge \frac{\alpha^2}{2(\frac{2G}{d}+\lambda)}$. There holds \begin{equation} \label{thm_contraction_semidiscrete} \int_0^t \| \partial_t \delta p_h^i(s) \|^2 \, {\rm d} s \leq \frac{L}{(\frac{1}{\beta}+L)} \int_0^t \| \partial_t \delta p_h^{i-1}(s) \|^2 \, {\rm d} s. \end{equation} \end{thm} \begin{proof} We take the time derivative of the difference of two successive iterates of the mechanic equation \eqref{discrete_variational_split_displacement} and test the resulting equation by ${\bf v}_h = \partial_t \delta {\bf u}_h^{i-1}$ to get \begin{equation} \label{test_mech} 2 G ({\boldsymbol \varepsilon}( \partial_t \delta {\bf u}_ h^{i}),{\boldsymbol \varepsilon}( \partial_t \delta {\bf u}_h^{i-1})) + \lambda (\nabla \cdot \partial_t \delta {\bf u}_ h^{i},\nabla \cdot \partial_t \delta {\bf u}_ h^{i-1}) - \alpha (\partial_t \delta p_h^i,\nabla \cdot \partial_t \delta {\bf u}_ h^{i-1}) = 0. \end{equation} By taking the difference between two successive iterates of the flow equation \eqref{discrete_variational_split_pressure} and testing with $q_h = \partial_t \delta p_h^i$, we obtain \begin{equation} \label{test_flow} \frac{1}{\beta} \| \partial_t \delta p_h^i \|^2 + L (\partial_t(\delta p_h^i - \delta p_h^{i-1}), \partial_t \delta p_h^i) + b( \delta p_h^i,\partial_t \delta p_h^i) +\alpha (\nabla \cdot \partial_t \delta {\bf u}_ h^{i-1},\partial_t \delta p_h^i) = 0. \end{equation} After summing up equations \eqref{test_mech} and \eqref{test_flow}, and using the identities $$ (\sigma,\xi) = \frac{1}{4} \| \sigma+\xi \|^2-\frac{1}{4} \| \sigma-\xi \|^2, \quad (\sigma-\xi,\sigma) = \|\sigma\|^2 - \| \xi \|^2+\| \sigma-\xi \|^2, $$ one has \begin{eqnarray} & & \frac{G}{2} \| {\boldsymbol \varepsilon} ( \partial_t \delta {\bf u}_h^{i} + \partial_t \delta {\bf u}_h^{i-1}) \|^2 + \frac{\lambda}{4} \| \nabla \cdot ( \partial_t \delta {\bf u}_h^{i} + \partial_t \delta {\bf u}_h^{i-1}) \|^2 + \frac{1}{\beta} \| \partial_t \delta p_h^i \|^2 + \frac{1}{2} \frac{{\rm d}}{{\rm d} t} \| \delta p_h^i \|_B^2 \nonumber \\ & & + \frac{L}{2} (\| \partial_t \delta p_h^i \|^2 - \| \partial_t \delta p_h^{i-1} \|^2 + \| \partial_t \delta p_h^i - \partial_t \delta p_h^{i-1}\|^2) = \frac{G}{2} \| {\boldsymbol \varepsilon} ( \partial_t \delta {\bf u}_h^{i} - \partial_t \delta {\bf u}_h^{i-1}) \|^2 + \nonumber \\ & & \frac{\lambda}{4} \| \nabla \cdot ( \partial_t \delta {\bf u}_h^{i} - \partial_t \delta {\bf u}_h^{i-1}) \|^2. \label{intermediate_formula} \end{eqnarray} Next, we consider the time derivative of the difference of two successive iterates of the mechanic equation \eqref{discrete_variational_split_displacement} and test by ${\bf v}_h = \partial_t \delta {\bf u}_h^{i} - \partial_t \delta {\bf u}_h^{i-1}$. By applying the Cauchy-Schwarz inequality, it follows \begin{equation} \label{ineq_div_pressure} \| \nabla \cdot ( \partial_t \delta {\bf u}_h^{i} - \partial_t \delta {\bf u}_h^{i-1}) \| \leq \frac{\alpha}{\frac{2G}{d}+\lambda} \| \partial_t \delta p_h^i - \partial_t \delta p_h^{i-1}\|. \end{equation} Inserting equality \eqref{test_mech} into equation \eqref{intermediate_formula} and by applying Cauchy-Schwarz and \eqref{ineq_div_pressure} inequalities, we obtain \begin{eqnarray} & & \frac{G}{2} \| {\boldsymbol \varepsilon} ( \partial_t \delta {\bf u}_h^{i} + \partial_t \delta {\bf u}_h^{i-1}) \|^2 + \frac{\lambda}{4} \| \nabla \cdot ( \partial_t \delta {\bf u}_h^{i} + \partial_t \delta {\bf u}_h^{i-1}) \|^2 + \frac{1}{\beta} \| \partial_t \delta p_h^i \|^2 + \frac{1}{2} \frac{{\rm d}}{{\rm d} t} \| \delta p_h^i \|_B^2 \nonumber \\ & & + \frac{L}{2}(\| \partial_t \delta p_h^i \|^2 + \| \partial_t \delta p_h^i - \partial_t \delta p_h^{i-1}\|^2) \leq \frac{L}{2} \| \partial_t \delta p_h^{i-1} \|^2 + \frac{\alpha^2}{4(\frac{2G}{d}+\lambda)} \| \partial_t \delta p_h^i - \partial_t \delta p_h^{i-1}\|^2. \nonumber \end{eqnarray} Discarding the first three positive terms, taking $L \ge \frac{\alpha^2}{2(\frac{2G}{d}+\lambda)}$, and integrating from $0$ to $t$ we finally obtain \eqref{thm_contraction_semidiscrete}. It implies that the scheme is a contraction and therefore convergent. This completes the proof. \end{proof} \begin{remark} It is easy to see that the fixed-stress split method in the semidiscrete case is an iterative method based on a suitable splitting for solving the differential/algebraic equation system \eqref{system_ODES}. In detail, the iterative method can be written in the form \begin{equation} \left[\! \begin{array}{cc} 0 & 0 \\ 0 & (1+L) M_p \end{array} \!\right] \left[\! \begin{array}{c} \dot{u}_h^i \\ \dot{p}_h^i \end{array} \!\right] \!+\! \left[\! \begin{array}{cc} A & B^t \\ 0 & - C \end{array} \!\right] \left[\! \begin{array}{c} {u}_h^i \\ {p}_h^i \end{array} \!\right] \!= \!\left[\! \begin{array}{cc} 0 & 0 \\ -B & L M_p \end{array} \!\right] \left[\! \begin{array}{c} \dot{u}_h^{i-1} \\ \dot{p}_h^{i-1} \end{array} \!\right] \!+\! \left[\! \begin{array}{c} {g}_h \\ {\widetilde{f}}_h \end{array} \!\right]. \label{splitting_DAE} \end{equation} \end{remark} \section{The parallel in time fixed-stress split algorithm for the fully discretized problem} \label{sec:parallel_fixed} \subsection{Parallel-in-time algorithm} For time discretization we use the backward Euler method on a uniform partition $\left\{t_0, t_1, \ldots, t_N\right\}$ of the time interval $(0,T]$ with constant time-step size $\tau$, $N \tau = T$. Then, we have the following fully discrete scheme corresponding to \eqref{semi_discrete_variational1}-\eqref{semi_discrete_variational2}: For $n = 1, 2, \ldots, N$, find $({\bm u}_h^n ,p_h^n) \in {\bm V}_h \times Q_h$ such that \begin{eqnarray} && a(\bm{u}_h^n,\bm{v}_h) - \alpha(p_h^n,\ddiv \bm{v}_h) = (\rho \bm g,\bm{v}_h), \quad \forall \ \bm{v}_h \in \bm V_h, \label{total_discrete_variational1}\\ && \alpha(\ddiv \bar{\partial}_t{\bm{u}_h^n},q_h) + \frac{1}{\beta} ( \bar{\partial}_t{p_h^n},q_h) + b(p_h^n,q_h) = (f_h^n,q_h) + ( {\bm K}\mu_f^{-1} \rho_f {\bm g},\nabla q_h ), \; \forall \ q_h \in Q_h,\label{total_discrete_variational2} \end{eqnarray} where $\bar{\partial}_t{\bm{u}_h^n} := (\bm{u}_h^n - \bm{u}_h^{n-1})/\tau$ and $\bar{\partial}_t{p_h^n} := (p_h^n - p_h^{n-1})/\tau$. \\ We now discuss a parallel-in time version of the fixed-stress split method. This algorithm arises in a natural way from the iterative method \eqref{splitting_DAE} by discretizing in time. In this way, given an initial guess $\{({\bm u}_h^{n,0},p_h^{n,0}), n = 0,1, \ldots, N\}$, the new fixed-stress split algorithm gives us a sequence of approximations $\{({\bm u}_h^{n,i},p_h^{n,i}), n = 0,1, \ldots,N\}$, $i \geq 1$, as follows: \\ \noindent {\bf Step 1:} Given $\{({\bm u}_h^{n,i-1},p_h^{n,i-1}), n = 0,1, \ldots,N\}$, find $p_h^{n,i} \in Q_h, n = 1, \ldots,N,$ such that \begin{eqnarray} && \left(\frac{1}{\beta} +L\right) \left(\frac{p_h^{n,i}-p_h^{n-1,i}}{\tau},q_h\right) + b(p_h^{n,i},q_h) = \alpha \left(\ddiv \frac{\bm{u}_h^{n,i-1}-\bm{u}_h^{n-1,i-1}}{\tau},q_h\right) + \nonumber \\ & & \bm quad \bm quad L \left(\frac{p_h^{n,i-1}-p_h^{n-1,i-1}}{\tau},q_h\right)+ (f_h^n,q_h) + ( {\bm K}\mu_f^{-1} \rho_f {\bm g},\nabla q_h ), \quad \forall \ q_h \in Q_h, \label{total_discrete_variational_split_pressure} \\ & & p_h^{0,i} = p_0. \nonumber \end{eqnarray} {\bf Step 2:} Given $p_h^{n,i} \in Q_h, n = 1, \ldots,N,$ find ${\bm u}_h^{n,i} \in {\bm V}_h, n = 1, \ldots,N,$ such that \begin{equation} \label{total_discrete_variational_split_displacement} a(\bm{u}_h^{n,i},\bm{v}_h) = \alpha(p_h^{n,i}, \ddiv \bm{v}_h) + (\rho \bm g,\bm{v}_h), \quad \forall \ \bm{v}_h \in \bm V_h. \end{equation} \begin{remark} We wish to emphasize that the proposed method is parallel-in-time in contrast to the classical sequential fixed-stress split method based on time-stepping, given as follows: \\ \hrule \noindent For $n = 1, 2, \ldots, N$\\ [-3.5ex] \; For $i=1,2,\ldots$\\ [-3.5ex] \quad \quad {\bf Step 1:} Given $({\bm u}_h^{n,i-1},p_h^{n,i-1}) \in {\bm V}_h \times Q_h$, find $p_h^{n,i} \in Q_h$ such that \begin{eqnarray*} && \left(\frac{1}{\beta} +L\right) \left(\frac{p_h^{n,i}-p_h^{n-1,i}}{\tau},q_h\right) + b(p_h^{n,i},q_h) = \alpha \left(\ddiv \frac{\bm{u}_h^{n,i-1}-\bm{u}_h^{n-1,i-1}}{\tau},q_h\right) + \nonumber \\ & & \bm quad \bm quad L \left(\frac{p_h^{n,i-1}-p_h^{n-1,i-1}}{\tau},q_h\right)+ (f_h^n,q_h) + ( {\bm K}\mu_f^{-1} \rho_f {\bm g},\nabla q_h ), \quad \forall \ q_h \in Q_h, \label{total_discrete_variational_split_pressure_2} \end{eqnarray*} \quad \quad {\bf Step 2:} Given $p_h^{n,i} \in Q_h$, find ${\bm u}_h^{n,i} \in {\bm V}_h$ such that \begin{equation*} \label{total_discrete_variational_split_displacement_2} a(\bm{u}_h^{n,i},\bm{v}_h) = \alpha(p_h^{n,i}, \ddiv \bm{v}_h) + (\rho \bm g,\bm{v}_h), \quad \forall \ \bm{v}_h \in \bm V_h. \end{equation*} \; End \\ [-3.5ex] \noindent End\\ \hrule Notice that in the Step 2 of the new algorithm, $N-1$ independent elliptic problems can be solved in parallel. \end{remark} \subsection{Convergence analysis} Let $\delta \bm{u}_h^{n,i} = \bm{u}_h^{n,i} - \bm{u}_h^{n,i-1}$ and $\delta p_h^{n,i} = p_h^{n,i}-p_h^{n,i-1}$ denote the difference between two succesive approximations for displacements and for pressure, respectively. \begin{thm} The fixed-stress split method given in \eqref{total_discrete_variational_split_pressure}-\eqref{total_discrete_variational_split_displacement} is convergent for any stabilization parameter $L \ge \frac{\alpha^2}{2(\frac{2G}{d}+\lambda)}$. There holds \begin{equation} \label{thm_contraction_fullydiscrete} \sum_{n=1}^N \tau \| \bar{\partial}_t \delta p_h^{n,i} \|^2 \, \leq \frac{L}{(\frac{1}{\beta}+L)} \sum_{n=1}^N \tau \| \bar{\partial}_t \delta p_h^{n,i-1} \|^2. \end{equation} \end{thm} \begin{proof} We take the difference of two successive iterates of the mechanic equation \eqref{total_discrete_variational_split_displacement} and test the resulting equation by ${\bf v}_h = \bar{\partial}_t \delta {\bf u}_h^{n,i-1}$ to get for $n=1, 2,\ldots, N,$ \begin{equation} \label{test_mech_fully} 2 G ({\boldsymbol \varepsilon}( \bar{\partial}_t \delta {\bf u}_ h^{n,i}),{\boldsymbol \varepsilon}( \bar{\partial}_t \delta {\bf u}_h^{n,i-1})) + \lambda (\nabla \cdot \bar{\partial}_t \delta {\bf u}_ h^{n,i},\nabla \cdot \bar{\partial}_t \delta {\bf u}_ h^{n,i-1}) - \alpha (\bar{\partial}_t \delta p_h^{n,i},\nabla \cdot \bar{\partial}_t \delta {\bf u}_ h^{n,i-1}) = 0. \end{equation} By taking the difference between two successive iterates of the flow equation \eqref{total_discrete_variational_split_pressure} and testing with $q_h = \bar{\partial}_t \delta p_h^{n,i}$, we obtain \begin{equation} \label{test_flow_fully} \frac{1}{\beta} \| \bar{\partial}_t \delta p_h^{n,i} \|^2 + L (\bar{\partial}_t (\delta p_h^{n,i} - \delta p_h^{n,i-1}), \bar{\partial}_t \delta p_h^{n,i}) + b( \delta p_h^{n,i},\bar{\partial}_t \delta p_h^{n,i}) +\alpha (\nabla \cdot \bar{\partial}_t \delta {\bf u}_h^{n,i-1},\bar{\partial}_t \delta p_h^{n,i}) = 0. \end{equation} After summing up equations \eqref{test_mech_fully} and \eqref{test_flow_fully}, and using the identities $$ (\sigma,\xi) = \frac{1}{4} \| \sigma+\xi \|^2-\frac{1}{4} \| \sigma-\xi \|^2, \quad (\sigma-\xi,\sigma) = \|\sigma\|^2 - \| \xi \|^2+\| \sigma-\xi \|^2, $$ one has \begin{eqnarray} & & \frac{G}{2} \| {\boldsymbol \varepsilon} ( \bar{\partial}_t \delta {\bf u}_h^{n,i} + \bar{\partial}_t \delta {\bf u}_h^{n,i-1}) \|^2 + \frac{\lambda}{4} \| \nabla \cdot ( \bar{\partial}_t \delta {\bf u}_h^{n,i} + \bar{\partial}_t \delta {\bf u}_h^{n,i-1}) \|^2 + \frac{1}{\beta} \| \bar{\partial}_t \delta p_h^{n,i} \|^2 \nonumber \\ & & + \frac{L}{2} (\| \bar{\partial}_t \delta p_h^{n,i} \|^2 + \| \bar{\partial}_t \delta p_h^{n,i} - \bar{\partial}_t \delta p_h^{n,i-1}\|^2) + \frac{1}{2\tau} (\| \delta p_h^{n,i} \|_B^2 + \| \delta p_h^{n,i} - \delta p_h^{n-1,i} \|_B^2) \nonumber \\ & & = \frac{G}{2} \| {\boldsymbol \varepsilon} ( \bar{\partial}_t \delta {\bf u}_h^{n,i} - \bar{\partial}_t \delta {\bf u}_h^{n,i-1}) \|^2 + \frac{\lambda}{4} \| \nabla \cdot ( \bar{\partial}_t \delta {\bf u}_h^{n,i} - \bar{\partial}_t \delta {\bf u}_h^{n,i-1}) \|^2 + \frac{L}{2} \| \bar{\partial}_t \delta p_h^{n,i-1} \|^2 \nonumber \\ & & + \frac{1}{2\tau} \| \delta p_h^{n-1,i} \|_B^2. \label{intermediate_formula_fully} \end{eqnarray} Next, we consider the difference of two successive iterates of the mechanic equation \eqref{total_discrete_variational_split_displacement} and test by ${\bf v}_h = \bar{\partial}_t \delta {\bf u}_h^{n,i} - \bar{\partial}_t \delta {\bf u}_h^{n,i-1}$ to get \begin{equation}\label{equality_2_fully} 2 G \| {\boldsymbol \varepsilon}( \bar{\partial}_t \delta {\bf u}_ h^{n,i}- \bar{\partial}_t \delta {\bf u}_ h^{n,i-1}) \|^2 + \lambda \| \nabla \cdot ( \bar{\partial}_t \delta {\bf u}_ h^{n,i}- \bar{\partial}_t \delta {\bf u}_ h^{n,i}) \|^2 = \alpha ( \bar{\partial}_t \delta p_h^{n,i}- \bar{\partial}_t \delta p_h^{n,i-1},\nabla \cdot ( \bar{\partial}_t \delta {\bf u}_h^{n,i} - \bar{\partial}_t \delta {\bf u}_h^{n,i-1})). \end{equation} From this equality, by applying Cauchy-Schwarz inequality, it is easy to see \begin{equation} \label{ineq_div_pressure_fully} \| \nabla \cdot ( \bar{\partial}_t \delta {\bf u}_h^{n,i} - \bar{\partial}_t \delta {\bf u}_h^{n,i-1}) \| \leq \frac{\alpha}{\frac{2G}{d}+\lambda} \| \bar{\partial}_t \delta p_h^{n,i} - \bar{\partial}_t \delta p_h^{n,i-1}\|. \end{equation} Inserting equality \eqref{equality_2_fully} into equation \eqref{intermediate_formula_fully} and by applying Cauchy-Schwarz inequality and \eqref{ineq_div_pressure_fully}, we obtain \begin{eqnarray} & & \frac{G}{2} \| {\boldsymbol \varepsilon} ( \partial_t \delta {\bf u}_h^{n,i} + \partial_t \delta {\bf u}_h^{n,i-1}) \|^2 + \frac{\lambda}{4} \| \nabla \cdot ( \partial_t \delta {\bf u}_h^{n,i} + \partial_t \delta {\bf u}_h^{n,i-1}) \|^2 + \frac{1}{\beta} \| \partial_t \delta p_h^{n,i} \|^2 \nonumber \\ & & + \frac{L}{2}(\| \partial_t \delta p_h^{n,i} \|^2 + \| \partial_t \delta p_h^{n,i} - \partial_t \delta p_h^{n,i-1}\|^2) + \frac{1}{2\tau} (\| \delta p_h^{n,i} \|_B^2 + \| \delta p_h^{n,i} - \delta p_h^{n-1,i} \|_B^2) \nonumber \\ & & \leq \frac{L}{2} \| \partial_t \delta p_h^{n,i-1} \|^2 + \frac{1}{2\tau} \| \delta p_h^{n-1,i} \|_B^2 + \frac{\alpha^2}{4(\frac{2G}{d}+\lambda)} \| \partial_t \delta p_h^{n,i} - \partial_t \delta p_h^{n,i-1}\|^2. \nonumber \end{eqnarray} Discarding positive terms, taking $L \ge \frac{\alpha^2}{2(\frac{2G}{d}+\lambda)}$, and summing up from $n=1$ to $N$, we finally obtain \eqref{thm_contraction_fullydiscrete}. This implies that the scheme is a contraction and therefore convergent. This completes the proof. \end{proof} \begin{remark} Notice that the values of parameter $L$ result to be the same as in the classical fixed-stress split scheme. \end{remark} \section{Numerical experiment}\label{sec:numerical} In this section, we present a numerical experiment with the purpose of illustrating the performance of the parallel-in-time fixed stress splitting (PFS) method described in Section \ref{sec:parallel_fixed}. We compare the PFS method with the classical fixed stress splitting (FS), see e.g. \cite{Both2017}. As test problem, we use Mandel's problem, which is a well-established 2D benchmark problem with a known analytical solution \cite{abousleiman1996mandel,Mandel}. This problem is very often used in the community for verifying the implementation and the performance of the numerical schemes, see e.g. \cite{phillips1,Kim2009,RGHZ2016,Wan_2015}. We implemented the problem in the open-source software package deal.II \cite{DealII}. Mandel's problem consists in a poroelastic slab of extent $2a$ in the $x$ direction, $2b$ in the $y$ direction, and infinitely long in the z-direction, and is sandwiched between two rigid impermeable plates (see Figure \ref{xMandelProblem}). At time $t=0$, a uniform vertical load of magnitude $2F$ is applied and an equal, but upward force is applied to the bottom plate. This load is supposed to remain constant. The domain is free to drain and stress-free at $x=\pm a$. Gravity is neglected. For the numerical solution, the symmetry of the problem allows us to use a quarter of the physical domain as computational domain (see Figure \ref{tMandel2Problem2}). Moreover, the rigid plate condition is enforced by adding constrained equations so that vertical displacement $u_y(b,t)$ on the top is equal to a known constant value. \begin{figure} \caption{Mandel's problem physical domain.} \label{xMandelProblem} \caption{Mandel's problem quarter domain} \label{tMandel2Problem2} \caption{Mandel's problem} \label{MandelProblem} \end{figure} The application of a load (2F) causes an instantaneous and uniform pressure increase throughout the domain \cite{GayX}; this is predicted theoretically \cite{abousleiman1996mandel} and it can be used as an initial condition \begin{align} p(x,y,0)&=\frac{F B(1+v_u)}{3a}, \nonumber\\ \bm u(x,y,0)&= \begin{pmatrix} \displaystyle \frac{F v_u x}{2G}, & \displaystyle \frac{-F b(1-v_u)y}{2G a} \end{pmatrix}^\top,\nonumber \end{align} being $B$ the Skempton's coefficient, that for our problem is $B=0.8333$, and $\nu_u = \frac{3\nu + B(1-2 \nu)}{3-B(1-2\nu)}$ the undrained Poisson's ratio. The boundary conditions are specified in Table \ref{BoundaryCondition} and the input parameters for Mandel's problem are listed in Table \ref{Parameter}. For all cases, we use as convergence criterion for the schemes $10^{-8} \norm{\delta p^{n,i}} +10^2\norm{\delta {\bm u}^{n,i}} \leq 10^{-8}$, due to the different orders of magnitude among the primary variables. \begin{table}[h!] \centering \caption{Boundary conditions for Mandel's problem} \resizebox{.55\textwidth}{!}{ \begin{tabular}{ l l l } \hline Boundary & Flow & Mechanics \\ \specialrule{.1em}{.05em}{.05em} $x=0$ & $ \bm q \cdot {\bf n}= 0$ & $\bm u \cdot {\bf n} = 0$ \\ $y=0$ & $ \bm q \cdot {\bf n}= 0$ & $\bm u \cdot {\bf n} = 0$ \\ $x=a$ & $p=0$ & $ {\bf \sigma} \cdot {\bf n} = 0$ \\ $y=b$ & $ \bm q \cdot {\bf n} = 0$ & $ {\bf \sigma}_{12}=0$; $\bm u \cdot {\bf n} = u_y(b,t)$ \\ \hline \end{tabular}} \label{BoundaryCondition} \end{table} \begin{table}[h!] \centering \caption{Input parameter for Mandel's problem} \label{Parameter} \resizebox{.95\textwidth}{!}{ \begin{tabular}{ l l c | llc} \hline Symbol & Quantity & Value & Symbol & Quantity & Value \\ \specialrule{.1em}{.05em}{.05em} a & Dimension in $x$ & 100 m& b & Dimension in $y$ & 10 m \\ K & Permeability & 100 D & $\mu_f$ & Dynamic viscosity & 10 $cp$ \\ $\alpha$ & Biot's constant & 1.0 & $\beta$ & Biot's modulus & $1.65\times 10^{10}$ Pa \\ $\nu$& Poisson's ratio & $0.4$ & $E$& Young's modulus & $ 5.94\times 10^9 Pa$ \\ $B$& Skempton coefficient & $0.83333$ & $\nu_u$& Undrained Poisson's ratio & $ 0.44$ \\ $c$ &Diffusivity coefficient& 46.526 $m^2/s$ & $F$ & Force intensity & $6.8 \times 10^8 N/m$\\ $h_x$& Grid spacing in $x$ & 2.5 m & $h_y$& Grid spacing in $y$ & 0.25 m\\ $\tau$& Time step & 1 s & $T$& Total simulation time & 32 s\\ \hline \end{tabular} } \end{table} \begin{figure} \caption{Pressure solutions} \label{solutiona} \caption{Displacement solutions} \label{solutionb} \caption{Comparison of numerical and analytical solutions of the (a) pore pressure and (b) displacements for Mandel's problem in different times with $\nu = 0.2$.} \label{solution} \end{figure} In Figure \ref{solution}, the numerical and the analytical solutions of Mandel's problem are depicted for different values of time. There is a very good match between both solutions for all cases. Moreover, the results demonstrate the Mandel-Cryer effect, first showing a pressure raise during the first 20 seconds and then, a sudden dissipation throughout the domain. The number of iterations for the parallel fixed stress splitting method and the classical fixed stress splitting method are reported in Figure \ref{Test_Iterations} for different values of parameter $L$ and various values of $\nu$. We remark a very similar behaviour of the two methods, with the optimal stabilization parameter $L$ being in this case the physical one $L_{phy} := \alpha^2 / \left(\frac{2G}{d}+\lambda\right)$, see e.g. \cite{Kim2009, Mikelic, Both2017}. \begin{figure} \caption{Performance of the splitting schemes PFS and FS for different values of $L$, $\tau = 1 [s]$, $h = 0.25[m]$. Both schemes has the same optimum value $L_{opt} \label{Test_Iterations} \end{figure} Further, in Figure \ref{Timing} the CPU time reduction for the parallel splitting (PFS) method compared with the classical fixed stress (FS) method is reported in percentage for different values of the Poisson's ratio (see Figure \ref{Timingcomparison}) and for different time steps (see Figure \ref{Timingcomparison_dt}). The CPU time reduction is estimated by the ratio between the CPU time of PFS and the CPU time of FS. We use parallelism on shared memory machines. In this regard, PFS is set to use one processor to solve the flow problem and up to eight processors in parallel to solve the mechanics, while FS is set to use one processor to solve flow and mechanics. For the smallest Poisson's ration ($\nu = 0.4$), PFS is $1\%$ slower than the FS when one processor is used. However, PFS is $10\%$ faster when four processors are used. Still, we notice that the CPU time reduction is limited to the CPU time of the flow problem. In Table \ref{CPU_time_comsumtion} we report the percentages of CPU which are devoted for solving the flow and the mechanic problems for different Poisson's ratio values $\nu$. As expected, we clearly observe that when solving the mechanics requires more time, the PFS method becomes more efficient. A reduction of CPU time up to 50\% (for $\nu \rightarrow 0.5$) is observed (see Figure \ref{Timingcomparison}). Nevertheless, due to the relative small size of the problem, by increasing the number of processors to more than four we do not see a further significant reduction of the CPU time. This is due to the time lost in the communication between the processors. Finally, we remark that the mesh size and the time step $\tau$ does not influence the number of iterations. This can be seen in Table \ref{DT_iterations}, where we provide the number of iterations for both algorithms, varying the space and time discretization parameters. However, Figure \ref{Timingcomparison_dt} shows that the CPU time of PFS slightly improves for smaller time steps $\tau$. This is in agreement with the theory. \begin{figure} \caption{CPU time reduction for different values of Poisson's ratio $\nu$; $\tau = 1 [s]$, $h = 0.25[m]$. } \label{Timingcomparison} \caption{CPU time reduction for different time step sizes; $\nu =0.49999$, $h = 0.25[m]$. } \label{Timingcomparison_dt} \caption{Parallel fixed stress splitting CPU times reduction. } \label{Timing} \end{figure} \begin{table}[htb] \centering \caption{CPU time consumption} \label{CPU_time_comsumtion} \resizebox{.55\textwidth}{!}{ \begin{tabular}{ lcc} $\nu$ & Flow problem & Mechanic problem \\ \hline 0.4 & $82\%$&$18\%$\\ 0.49& $76\%$&$24\%$\\ 0.499& $68\%$&$32\%$\\ 0.4999& $47\%$&$53\%$\\ 0.49999& $34\%$&$66\%$\\ \hline \end{tabular} } \end{table} \begin{table}[htb] \centering \caption{Number of iterations for different values of $\tau$,$h$, $\nu$} \label{DT_iterations} \resizebox{.65\textwidth}{!}{ \begin{tabular}{ lcc} $\nu = 0.49999$\\ \hline $\tau [s]$ & PFS & FS \\ \hline 1.000 & 2&2.10\\ 0.500& 2&2.03\\ 0.250& 2&2.02\\ 0.125& 2&2.01\\ \hline \end{tabular} \begin{tabular}{ lcc} $\nu = 0.499$\\ \hline $h[m]$ & PFS & FS \\ \hline 0.5000 & 3&3.20\\ 0.2500 & 3&3.20\\ 0.1250 & 3&3.19\\ 0.0625& 3&3.19\\ \hline \end{tabular}} \end{table} \section{Conclusions} \label{sec:conclusion} We considered the quasi-static Biot's model in the two-field formulation and presented a new fixed stress type splitting method for solving it. The main benefit of the new method is that the mechanics can be solved in a parallel-in-time manner, taking advantage of the current massively parallel computers. We have rigorously analyzed the convergence of the method. If the stabilization term $L$ is chosen big enough, the method is shown to be convergent. The theoretical results are indicating a similar behaviour with the classical fixed stress splitting method (in terms of convergence rate and stabilization parameter). We further performed numerical tests by using the Mandel benchmark problem. The numerical results are confirming a similar behaviour of the parallel fixed stress and the classical fixed stress method in terms of number of iterations and optimal stabilization parameter. With respect to the CPU time, we observe that the new scheme is very efficient (up to 50\% reduction of the CPU time) for problems where solving the mechanics requires more time than solving the flow (in the case of Mandel's problem for Poisson's ratio close to five). Nevertheless, the parallel implementation has still to be optimized. Up to date, only relatively small problems could be solved and therefore the new parallel method remained efficient only for not too many processors (less than eight). {\bf Acknowledgements} The work of F.~A. Radu and K. Kumar was partially supported by the NFR - Toppforsk project TheMSES \#250223. The work of F.~J. Gaspar is supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement NO 705402, POROSOS. The research of C.~Rodrigo is supported in part by the Spanish project FEDER /MCYT MTM2016-75139-R and the DGA (Grupo consolidado PDIE). \end{document}

\begin{equation}\labelgin{document} \title{Unsaturated deformable porous media flow \\ with phase transition\footnote{The financial supports of the FP7-IDEAS-ERC-StG \#256872 (EntroPhase), of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR ``Matematica d'Eccellenza in biologia ed ingegneria come accelleratore di una nuona strateGia per l'ATtRattivit\`a dell'ateneo pavese'', and of the GA\v CR Grant GA15-12227S and RVO: 67985840 are gratefully acknowledged. The paper also benefited from the support the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for ER.}} \author{Pavel Krej\v c\'{\i} \thanks{Institute of Mathematics, Czech Academy of Sciences, \v Zitn\'a~25, CZ-11567~Praha 1, Czech Republic, E-mail {\tt krejci@math.cas.cz}.}\,\,, Elisabetta Rocca \thanks{Dipartimento di Matematica, Universit\`a degli Studi di Pavia and IMATI-C.N.R., Via Ferrata 5, I-27100 Pavia, Italy, E-mail {\tt elisabetta.rocca@unipv.it}.} \,, and J\"urgen Sprekels \thanks{Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse~39, D-10117 Berlin, Germany, E-mail {\tt sprekels@wias-berlin.de}, and Department of Mathematics, Humboldt-Universit\"at zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany.} } \maketitle \begin{equation}\labelgin{abstract}\noindent In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates. \end{abstract} \section*{Introduction}\label{int} A model for fluid flow in partially saturated porous media with thermomechanical interaction was proposed and analyzed in \cite{ak,dkr1, dkr2}. Here, we extend the model by including the effects of freezing and melting of the fluid in the pores. Typical examples, in which such situations arise, are related to groundwater flows and to the freezing-melting cycles of water sucked into the pores of concrete. Notice that the latter process forms one of the main reasons for the degradation of concrete in buildings, bridges, and roads. However, many of the governing effects in concrete like the multi-component microstructure, the breaking of pores, chemical reactions, the hysteresis of the saturation-pressure curves, and the occurrence of shear stresses, are still neglected in our model. While often in continuum models for three-component and multi-component porous media the intention is to describe the propagation of sound waves in these media (e.g., \cite{albers,wil}), we investigate in the present paper -- instead of using partial balance equations for each component -- a continuum model combining the principles of conservation of mass and momentum with the first and the second principles of thermodynamics. In, e.~g., \cite{albers}, flow in porous media is described in Eulerian coordinates in order to incorporate, for example, the effects of fast convection. Here, instead, we assume that slow diffusion is dominant, and choose the Lagrangian description as in \cite{ak, dkr1, wil}. The resulting system of coupled ODEs and PDEs then appears to be a nonlinear extension of the linear model in \cite{show}, referred to as a simplified Biot system, to the case when also the occurrence of temperature changes and phase transitions is taken into account. In addition to the model studied in \cite{dkr1}, we include here the effects of freezing and melting. The idea is the following. The pores in the matrix material contain a mixture of H\!${}_2$\!O and gas, and H\!${}_2$\!O itself is a mixture of the liquid (water) and the solid phase (ice). That is, in addition to the other physical quantities like capillary pressure, displacement, and absolute temperature, we need to consider the evolution of a phase parameter $\chi$ representing the relative proportion of water in the H\!${}_2$\!O part and its influence on pressure changes due to the different mass densities of water and ice. Unlike in \cite{ak,dkr1,dkr2,ss}, we do not consider hysteresis in the model. We believe that the mathematical results can be extended to the case of capillary hysteresis as in \cite{ak,dkr1,dkr2}. In our model without shear stresses, elastoplastic hysteresis effects as in \cite{ak,dkr1,ss} cannot occur. As it will be detailed in Section~\ref{mod}, we assume that the deformations are small, so that $\mathrm{\,div\,} u$ is the relative local volume change, where $u$ represents the displacement vector. Moreover, we assume that the volume of the matrix material does not change during the process, and thus the volume and mass balance equations with Darcy's law for the water flux lead to a nonlinear degenerate parabolic equation for the capillary pressure, see~\eqref{e4b}. In the equation of motion, we take into account the pressure components due to phase transition and temperature changes, and we further simplify the system in order to make it mathematically tractable by assuming that the process is quasistatic and the shear stresses are negligible. The problem of existence of solutions for the coupled system without this assumption is open and, in our opinion, very challenging. Finally, we use the balance of internal energy and the entropy inequality to derive the dynamics for absolute temperature and phases; they turn out to be, respectively, a parabolic equation for the temperature with highly nonlinear right-hand side (quadratic in the derivatives) and an ordinary differential inclusion for the phase parameter $\chi$. Finally, let us note that -- in order to model the freezing and melting phenomena in the pores -- we have borrowed here some ideas from our earlier publications on freezing and melting in containers filled with water with rigid, elastic, or elastoplastic boundaries (cf. \cite{mb,kr,krsbottle,krsgrav,krsWil}). It was shown there how important it is to account for the difference in specific volumes of water and of ice. There is an abundant classical mathematical literature on phase transition processes, see, e.g., the monographs \cite{bs}, \cite{fremond}, \cite{visintin}, and the references therein. It seems, however, that only few publications take into account that the mass densities and specific volumes of the phases differ. In \cite{fr1}, the authors proposed to interpret a phase transition process in terms of a balance equation for macroscopic motions, and to include the possibility of voids. Well-posedness of an initial-boundary value problem associated with the resulting PDE system was proved there, and the case of two different densities $\varrho_1$ and $\varrho_2$ for the two substances undergoing phase transitions was pursued in \cite{fr2}. Let us also mention the papers \cite{rr1, rr2, RocRos12, RocRos14} dealing with macroscopic stresses in phase transitions models, where the different properties of the viscous (liquid) and elastic (solid) phases were taken into account and the coexisting viscous and elastic properties of the system were given a distinguished role, under the working assumption that they indeed influence the phase transition process. The model studied there includes inertia, viscous, and shear viscosity effects (depending on the phases). This is reflected in the analytical expressions of the associated PDEs for the strain $u$ and the phase parameter $\chi$: the $\chi$-dependence, e.~g., in the stress-strain relation, leads to the possible degeneracy of the elliptic operator therein. Finally, we can quote in this framework the model analyzed in~\cite{kss2} and \cite{kss}, which pertains to nonlinear thermoviscoplasticity: in the spatially one-dimensional case, the authors prove the global well-posedness of a PDE system that both incorporates hysteresis effects and models phase change but, however, does not display a degenerating character. Another coupled system for temperature, displacement, and phase parameter has been derived in order to model the full thermomechanical behavior of shape memory alloys. A long list of references for further developments can be found in the monographs \cite{fremond} and \cite{visintin}. The paper is organized as follows: in the next Section~\ref{mod}, we derive the model in full generality from the basic principles of continuum thermodynamics. In Section~\ref{mat}, we state the mathematical problem, the main assumptions on the data, and the main Theorem \ref{t1}, the proof of which is split into Sections~\ref{cut}, \ref{apr}, and \ref{proo}. The steps of the proof are as follows: we first cut off some of the pressure and temperature dependent terms in the system in Section~\ref{cut} by means of a cut-off parameter $R$ and solve the related problem employing a special Galerkin approximation scheme. Then, in Section~\ref{apr}, we first prove the positivity of the temperature by means of a maximum principle technique, and then we perform the -- independent of $R$ -- estimates on the system. They mainly consist of: the energy estimate, the so-called Dafermos estimate (with negative small powers of the temperature), Moser-type and then higher-order estimates for the capillary pressure and for the temperature. This allows us in Section~\ref{proo} to pass to the limit in the cut-off system as $R\to\infty$, which will conclude the proof of the existence result. \section{The model}\label{mod} We consider a connected domain $\Omega\subset \mathbb{R}^3$ filled by a deformable matrix material with pores containing a mixture of H\!${}_2$\!O and gas, where we assume that H\!${}_2$\!O may appear in one of the two phases: water or ice. We also assume that the volume of the solid matrix remains constant during the process, and let $c_s \in (0,1)$ be the relative proportion of solid in the total reference volume. We denote, for $x\in \Omega$ and time $t\in [0,T]$, \begin{equation}\labelgin{description} \item $W(x,t)\in [0,1]$ ... relative proportion of H\!${}_2$\!O in the total pore volume; \item $A(x,t)\in [0,1]$ ... relative proportion of gas in the total pore volume; \item $\chi(x,t)\in [0,1]$ ... relative proportion of water in the H\!${}_2$\!O part; \item $\xi(x,t)$ ... mass flux vector; \item $p(x,t)$ ... capillary pressure; \item $u(x,t)$ ... displacement vector; \item $\sigma(x,t)$ ... stress tensor; \item $\theta(x,t)$ ... absolute temperature. \end{description} Then $\chi W$ represents the relative proportion of water in the total pore volume, and $(1-\chi) W$ represents the relative proportion of ice in the total pore volume. We assume that the deformations are small, so that $\mathrm{\,div\,} u$ is the relative local volume change. By hypothesis, the volume of the matrix material does not change, so that the volume balance reads \begin{equation}\label{e1} W(x,t) + A(x,t) + c_s = 1+\mathrm{\,div\,} u(x,t). \end{equation} For $A$, we assume the functional relation \begin{equation}\label{e2} A = 1-c_s - \varphi(p)\,, \end{equation} where $\varphi$ is an increasing function that satisfies $\varphi(-\infty) = \varphi^\flat \in (0,1)$ and $\varphi(\infty) = 1-c_s$, $\varphi^\flat + c_s < 1$. This means that the porous medium cannot be made completely dry by thermomechanical processes alone. Combining \eqref{e1} with \eqref{e2}, we obtain that \begin{equation}\label{e1a} W = \varphi(p) + \mathrm{\,div\,} u\,. \end{equation} \subsection{Mass balance}\label{mass} Consider an arbitrary control volume $V \subset \Omega$. The water content in $V$ is given by the integral $\int_V \rho_L \chi W \,\mathrm{d} x$, where $\rho_L$ is the water mass density, and the ice content is $\int_V \rho_S (1-\chi) W \,\mathrm{d} x$, where $\rho_S$ is the ice mass density. The mass conservation principle then reads \begin{equation}\label{e3} \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_V \rho_L \chi W \,\mathrm{d} x + \int_{\partial V} \xi\cdot n\,\mathrm{d} s(x) = -\frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_V \rho_S (1-\chi) W \,\mathrm{d} x\,, \end{equation} where $n$ the unit outward normal vector to $\partial V$. In differential form, we obtain \begin{equation}\label{e4} \rho_L (\chi W)_t + \mathrm{\,div\,} \xi = -\rho_S ((1-\chi) W)_t\,. \end{equation} The right-hand side of \eqref{e4} is the positive or negative liquid water source due to the solidification or melting of the ice. We assume the water flux in the form of the Darcy law \begin{equation}\label{e5} \xi = - \mu(p) \nabla p, \end{equation} with a proportionality factor $\mu(p)>0$. This, \eqref{e1a}, and \eqref{e4}, yield the equation \begin{equation}\label{e4b} \big((\chi{+}\rho^*(1{-}\chi))(\varphi(p) + \mathrm{\,div\,} u)\big)_t - \frac{1}{\rho_L}\mathrm{\,div\,} (\mu(p)\nabla p) = 0, \end{equation} with $\rho^* = \rho_S/\rho_L \in (0,1)$. \subsection{Equation of motion}\label{moti} The equation of motion is considered in the form \begin{equation}\label{mo1} \rho_M u_{tt} - \mathrm{\,div\,} \sigma = g\,, \end{equation} where $\rho_M$ is the mass density of the matrix material, $\sigma$ is the stress tensor, and $g$ is a volume force acting on the body (e.~g., gravity). For $\sigma$, we prescribe the constitutive equation \begin{equation}\label{mo2} \sigma = B\varepsilon_t + A\varepsilon + \big((\chi{+}\rho^*(1{-}\chi))(\lambda \mathrm{\,div\,} u - p) - \begin{equation}\labelta(\theta-\theta_c)\big)\delta\,, \end{equation} where $\varepsilon = \nabla_s u :=\frac12 (\nabla u + \nabla u^T)$ is the small strain tensor, $\delta$ is the Kronecker tensor, $B$ is a symmetric positive definite viscosity tensor, $A$ is the symmetric positive definite elasticity tensor of the matrix material, $\lambda>0$ is the bulk elasticity modulus of water, $\theta>0$ is the absolute temperature, $\theta_c>0$ is a fixed referential temperature, and $\begin{equation}\labelta \in \mathbb{R}$ is the relative solid-liquid thermal expansion coefficient. The term $\,(\chi{+}\rho^*(1{-}\chi))(\lambda \mathrm{\,div\,} u - p)\,$ accounts for the pressure component due to the phase transition. \subsection{Energy and entropy balance}\label{ener} We have to derive formulas for the densities of internal energy $U$ and entropy $S$ such that the energy balance balance equation and the Clausius--Duhem inequality hold for all processes. Let $q$ be the heat flux vector, and let $V \subset \Omega$ be again an arbitrary control volume. The total internal energy in $V$ is $\int_V U \,\mathrm{d} x$, and the total mechanical power $Q(V)$ supplied to $V$ equals $$ Q(V) = \int_V \sigma:\varepsilon_t \,\mathrm{d} x - \int_{\partial V} \frac{1}{\rho_L} p\, \xi\cdot n \,\mathrm{d} s(x)\,, $$ where $\xi$ is the fluid mass flux \eqref{e5}. We thus have that \begin{equation}\label{e11} \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_V U\,\mathrm{d} x + \int_{\partial V} q\cdot n \,\mathrm{d} s(x) = \int_V \sigma:\varepsilon_t \,\mathrm{d} x - \int_{\partial V} \frac{1}{\rho_L} p\,\xi \cdot n \,\mathrm{d} s(x)\,. \end{equation} Again, by the Gauss formula, we obtain the energy balance equation in differential form, namely \begin{equation}\label{e12} U_t + \mathrm{\,div\,} q = \sigma:\varepsilon_t - \frac{1}{\rho_L} \mathrm{\,div\,}( p\xi)\,. \end{equation} The internal energy and entropy densities $U$ and $S$, as well as the heat flux vector $q$, have to be chosen in order to satisfy, for all processes, the Clausius--Duhem inequality \begin{equation}\label{e13} S_t + \mathrm{\,div\,}\big(\frac{q}{\theta}\big) \ge 0, \end{equation} or, taking into account the energy balance \eqref{e12}, \begin{equation}\label{e14} U_t - \theta S_t + \frac{q\cdot\nabla \theta}{\theta} \le \sigma:\varepsilon_t - \frac{1}{\rho_L} \mathrm{\,div\,}( p\xi)\,. \end{equation} We consider $\varepsilon, \chi, p, \theta$ as state variables and $U, S$ as state functions, independent of $\nabla\theta$. Hence, as a consequence of \eqref{e14}, two inequalities have to hold separately for all processes, namely \begin{equation}\label{e15} q\cdot\nabla \theta \le 0\,, \quad U_t - \theta S_t \le \sigma:\varepsilon_t - \frac{1}{\rho_L} \mathrm{\,div\,}( p\xi)\,. \end{equation} For simplicity, we assume Fourier's law for the heat flux, \begin{equation}\label{e16} q = -\kappa(\theta) \nabla\theta, \end{equation} with the heat conductivity coefficient $\kappa = \kappa(\theta) >0$. We further introduce the free energy $F$ by the formula $F = U - \theta S$, so that, in terms of $F$, the second inequality in \eqref{e15} takes the form \begin{equation}\label{e17} \quad F_t + \theta_t S \le \sigma:\varepsilon_t + \frac{1}{\rho_L} \mathrm{\,div\,}( p\mu(p)\nabla p)\,. \end{equation} We claim that the right choice of $F$ for \eqref{e17} to hold is given by \begin{equation}\labelr \nonumber F &=& \frac12 A\varepsilon:\varepsilon + (\chi{+}\rho^*(1{-}\chi))\left(V(p)+ \frac{\lambda}{2} (\mathrm{\,div\,} u)^2\right)\\ \label{e18f} && +\, L\chi \left(1 - \frac{\theta}{\theta_c}\right) -\begin{equation}\labelta(\theta - \theta_c)\mathrm{\,div\,} u + F_0(\theta) + I(\chi),\\ \label{e18s} S &=& -\frac{\partial F}{\partial\theta} = \frac{L}{\theta_c}\chi + \begin{equation}\labelta \mathrm{\,div\,} u - F'_0(\theta), \end{equation}r where \begin{equation}\label{e18a} V(p) = p \varphi(p) - \Phi(p)\,, \quad \Phi(p) = \int_0^p \varphi(\tau) \,\mathrm{d} \tau\,, \end{equation} $F_0(\theta)$ is a purely caloric component of $F$, $L>0$ is the latent heat, and $I$ is the indicator function of the interval $[0,1]$. It remains to check that if we choose the phase dynamics equation in the form \begin{equation}\label{e6} \gamma(\theta) \chi_t+ \partial I(\chi) \ni (1-\rho^*)\left(\Phi(p)+ p\mathrm{\,div\,} u - \frac{\lambda}{2}(\mathrm{\,div\,} u)^2\right) + L\left(\frac{\theta}{\theta_c}-1\right) \end{equation} with a coefficient $\gamma(\theta)>0$, then \eqref{e17} holds for all processes. Indeed, by \eqref{e18f}--\eqref{e18s} and \eqref{e4b} we have that \begin{equation}\labelrs F_t + \theta_t S &=& A\varepsilon: \varepsilon_t + (\chi{+}\rho^*(1{-}\chi))(V'(p)p_t + \lambda \mathrm{\,div\,} u \mathrm{\,div\,} u_t)\\ &&+\, (1 - \rho^*)\chi_t \left(V(p)+ \frac{\lambda}{2} (\mathrm{\,div\,} u)^2\right) + L\chi_t \left(1 - \frac{\theta}{\theta_c}\right)- \begin{equation}\labelta(\theta-\theta_c)\mathrm{\,div\,} u_t\,,\\ \sigma:\varepsilon_t &=& B \varepsilon_t:\varepsilon_t + A\varepsilon: \varepsilon_t + (\chi{+}\rho^*(1{-}\chi))(\lambda \mathrm{\,div\,} u \mathrm{\,div\,} u_t - p \mathrm{\,div\,} u_t)\\ && -\, \begin{equation}\labelta(\theta-\theta_c)\mathrm{\,div\,} u_t\,,\\ \frac{1}{\rho_L}\mathrm{\,div\,} (p \mu(p)\nabla p) &=& \frac{1}{\rho_L} \mu(p)|\nabla p|^2 + p (\chi{+}\rho^*(1{-}\chi))(\varphi'(p)p_t + \mathrm{\,div\,} u_t)\\ && +\, p(1 - \rho^*)\chi_t (\varphi(p) + \mathrm{\,div\,} u)\,. \end{equation}rs Hence (note that $V(p) - p\varphi(p) = -\Phi(p)$), \begin{equation}\labelr \nonumber F_t + \theta_t S - \sigma:\varepsilon_t - \frac{1}{\rho_L}\mathrm{\,div\,} (p \mu(p)\nabla p) &=& - B \varepsilon_t:\varepsilon_t - \frac{1}{\rho_L} \mu(p)|\nabla p|^2 \\ \nonumber && \hspace{-25mm} +\, \chi_t \left(L\left(1 - \frac{\theta}{\theta_c}\right) + (1 - \rho^*)\left( \frac{\lambda}{2} (\mathrm{\,div\,} u)^2 - \Phi(p) - p \mathrm{\,div\,} u\right)\right)\\ \label{xx} &=& - B \varepsilon_t:\varepsilon_t - \frac{1}{\rho_L} \mu(p)|\nabla p|^2 - \gamma(\theta)\chi_t^2 \le 0, \end{equation}r by virtue of \eqref{e6}, so that \eqref{e17} holds. Now observe that \begin{equation}\labelr \nonumber U &=& F + \theta S\\ \nonumber &=& \frac12 A\varepsilon:\varepsilon + (\chi{+}\rho^*(1{-}\chi))\left(V(p)+ \frac{\lambda}{2} (\mathrm{\,div\,} u)^2\right)\\ \label{e20} &&+\, L\chi + \begin{equation}\labelta\theta_c\mathrm{\,div\,} u + F_0(\theta)-\theta F_0'(\theta) + I(\chi)\,. \end{equation}r The derivative of the purely caloric component $F_0(\theta) - \theta F'_0(\theta)$ is the specific heat capacity $c(\theta) = -\theta F''(\theta)$. Assuming that $c(\theta) = c_0$ is a positive constant, we obtain that $F_0(\theta) = - c_0\theta \log(\theta/\theta_c)$ up to a linear function, and \begin{equation}\label{e20a} U = \frac12 A\varepsilon:\varepsilon + (\chi{+}\rho^*(1{-}\chi))\left(V(p)+ \frac{\lambda}{2} (\mathrm{\,div\,} u)^2\right) + L\chi + \begin{equation}\labelta\theta_c\mathrm{\,div\,} u + c_0 \theta + I(\chi)\,. \end{equation} We now rewrite Eq.~\eqref{e12} in a more suitable form, using \eqref{xx}. We have \begin{equation}\labelr \nonumber 0 &=& U_t + \mathrm{\,div\,} q - \sigma:\varepsilon_t - \frac{1}{\rho_L} \mathrm{\,div\,}( p\mu(p)\nabla p)\\ \nonumber &=& (F+\theta S)_t + \mathrm{\,div\,} q - \sigma:\varepsilon_t - \frac{1}{\rho_L} \mathrm{\,div\,}( p\mu(p)\nabla p)\\ \label{ene4} &=& - B \varepsilon_t:\varepsilon_t - \frac{1}{\rho_L} \mu(p)|\nabla p|^2 - \gamma(\theta)\chi_t^2 + \theta S_t + \mathrm{\,div\,} q\,, \end{equation}r which yields the identity \begin{equation}\label{ene5} c_0 \theta_t - \mathrm{\,div\,}(\kappa(\theta)\nabla\theta) = B \varepsilon_t:\varepsilon_t +\frac{1}{\rho_L} \mu(p)|\nabla p|^2 + \gamma(\theta)\chi_t^2 - \frac{L}{\theta_c}\theta\chi_t - \begin{equation}\labelta \theta \mathrm{\,div\,} u_t\,. \end{equation} \section{The mathematical problem}\label{mat} We consider the system \begin{equation}\labelgin{align}\label{ae1} &\big((\chi{+}\rho^*(1{-}\chi))(\varphi(p) + \mathrm{\,div\,} u)\big)_t \,=\, \frac{1}{\rho_L}\mathrm{\,div\,} (\mu(p)\nabla p)\,,\\ \label{ae2} &\,\rho_M u_{tt} \,=\, \mathrm{\,div\,} \sigma + g\,,\\[2mm] &\hspace*{8.7mm}\sigma \,=\, B\nabla_s u_t + A\nabla_s u + ((\chi{+}\rho^*(1{-}\chi))(\lambda \mathrm{\,div\,} u - p) \,-\, \begin{equation}\labelta(\theta-\theta_c))\delta\,,\label{ae3} \end{align} \begin{equation}\labelgin{align} \label{ae4} &\hspace*{8.5mm}\gamma(\theta) \chi_t+ \partial I(\chi) \,\ni\, (1-\rho^*)\left(\Phi(p)+ p\mathrm{\,div\,} u - \frac{\lambda}{2}(\mathrm{\,div\,} u)^2\right) \,+\, L\left(\frac{\theta}{\theta_c}-1\right),\\[2mm] &\label{ae5} c_0 \theta_t - \mathrm{\,div\,}(\kappa(\theta)\nabla\theta)\,=\, B \nabla_s u_t:\nabla_s u_t +\frac{1}{\rho_L} \mu(p)|\nabla p|^2 + \gamma(\theta)\chi_t^2 \,-\, \frac{L}{\theta_c}\theta\chi_t - \begin{equation}\labelta \theta \mathrm{\,div\,} u_t\,, \end{align} for the unknown functions $p,u,\chi,\theta$, coupled with the boundary conditions \begin{equation}\labelr\label{be1} u &\!\!=\!\!& 0\,,\\ \label{be2} \xi\cdot n &\!\!=\!\!& \alpha(x) (p - p^*)\,,\\ \label{be3} q \cdot n &\!\!=\!\!& ^{(m)}ega(x) (\theta - \theta^*)\,, \end{equation}r on $\partial\Omega$, where $p^*$ is a given outer pressure, $\theta^*$ is a given outer temperature, $\alpha(x) \ge 0$ is the permeability of the boundary, and $^{(m)}ega(x) \ge 0$ is the heat conductivity of the boundary. We can also simplify the problem by assuming that water is incompressible. This corresponds to the choice $\lambda = 0$, whence the system becomes \begin{equation}\labelr\label{le1} \big((\chi{+}\rho^*(1{-}\chi))(\varphi(p) + \mathrm{\,div\,} u)\big)_t &\!\!=\!\!& \frac{1}{\rho_L}\mathrm{\,div\,} (\mu(p)\nabla p)\,, \\ \label{le2} \rho_M u_{tt} &\!\!=\!\!& \mathrm{\,div\,} \sigma + g\,,\\ \label{le3} \sigma &\!\!=\!\!& B\nabla_s u_t + A\nabla_s u - (p (\chi{+}\rho^*(1{-}\chi)) - \begin{equation}\labelta(\theta-\theta_c))\delta\,,\qquad\\ \label{le4} \gamma(\theta) \chi_t+ \partial I(\chi) &\!\!\ni\!\!& (1-\rho^*)\left(\Phi(p)+ p\mathrm{\,div\,} u\right) + L\left(\frac{\theta}{\theta_c}-1\right),\\ \nonumber c_0 \theta_t - \mathrm{\,div\,}(\kappa(\theta)\nabla\theta) &\!\!=\!\!& B \nabla_s u_t:\nabla_s u_t +\frac{1}{\rho_L} \mu(p)|\nabla p|^2 + \gamma(\theta)\chi_t^2\\ \label{le5} &&-\, \frac{L}{\theta_c}\theta\chi_t - \begin{equation}\labelta \theta \mathrm{\,div\,} u_t\,. \end{equation}r We further simplify the system by assuming that the process is quasistatic and that the shear stresses are negligible. Then \eqref{le2}--\eqref{le3} can be reduced to \begin{equation}\labelr \label{le2a} 0 &\!\!=\!\!& \mathrm{\,div\,} \sigma + g\,,\\ \label{le3a} \sigma &\!\!=\!\!& (\nu \mathrm{\,div\,} u_t + \lambda_M \mathrm{\,div\,} u - p (\chi{+}\rho^*(1{-}\chi)) - \begin{equation}\labelta(\theta-\theta_c))\delta\,. \end{equation}r Assuming that the force $g$ admits a potential $G$, that is, $g = \nabla G$, this yields \begin{equation}\label{le6} \nu \mathrm{\,div\,} u_t + \lambda_M \mathrm{\,div\,} u - p (\chi{+}\rho^*(1{-}\chi)) - \begin{equation}\labelta(\theta-\theta_c) \,=\, -G + H(t)\,, \end{equation} where $H(t)$ is an ``integration constant'', $\nu$ is the bulk viscosity coefficient, and $\lambda_M$ is the bulk elasticity modulus of the matrix material. In view of the boundary condition \eqref{be1}, we have that \begin{equation}\label{le7} H(t)\, =\, - \frac{1}{|\Omega|}\int_{\Omega} (p (\chi{+}\rho^*(1{-}\chi)) + \begin{equation}\labelta(\theta-\theta_c) -G)(x,t)\,\mathrm{d} x\,. \end{equation} With the new unknown function $w = \mathrm{\,div\,} u$, which represents the {\em relative volume change\/}, the system \eqref{le1}--\eqref{le5} then becomes \begin{equation}\labelr\label{lu1} \big((\chi{+}\rho^*(1{-}\chi))(\varphi(p) + w)\big)_t &\!\!=\!\!& \frac{1}{\rho_L}\mathrm{\,div\,} (\mu(p)\nabla p)\,, \\[1mm] \label{lu2} \nu w_t + \lambda_M w &\!\!=\!\!& p (\chi{+}\rho^*(1{-}\chi)) + \begin{equation}\labelta(\theta-\theta_c) - G + H(t)\,,\\[1mm] \label{lu4} \gamma(\theta) \chi_t+ \partial I(\chi) &\!\!\ni\!\!& (1-\rho^*)\left(\Phi(p)+ p w\right) + L\left(\frac{\theta}{\theta_c}-1\right),\\[1mm] \label{lu5} c_0 \theta_t - \mathrm{\,div\,}(\kappa(\theta)\nabla\theta) &\!\!=\!\!& \nu w_t^2 +\frac{1}{\rho_L} \mu(p)|\nabla p|^2 + \gamma(\theta)\chi_t^2 - \frac{L}{\theta_c}\theta\chi_t - \begin{equation}\labelta \theta w_t\,. \end{equation}r We prescribe the initial conditions \begin{equation}\labelgin{align} \label{inip} p(x,0) &\,=\, p^0(x)\,,\\ \label{iniu} w(x,0) &\,=\, w^0(x)\,,\\ \label{inich} \chi(x,0) &\,=\, \chi^0(x)\,,\\ \label{init} \theta(x,0) &\,=\, \theta^0(x)\,. \end{align} The weak formulation of Problem \eqref{lu1}--\eqref{lu5} reads as follows: \begin{equation}\labelr\label{wu1} \int_{\Omega}\left(((\chi{+}\rho^*(1{-}\chi)) (\varphi(p)+w))_t \eta +\frac{1}{\rho_L}\mu(p)\nabla p\cdot\nabla \eta\right)\,\mathrm{d} x &\!\!=\!\!& \int_{\partial\Omega} \alpha(x)(p^*-p)\eta \,\mathrm{d} s(x),\qquad \\ \label{wu2} \nu w_t + \lambda_M w - p (\chi{+}\rho^*(1{-}\chi)) - \begin{equation}\labelta(\theta-\theta_c) &\!\!=\!\!& - G + H(t)\quad\mbox{a.~e.},\\ \label{wu4} \gamma(\theta) \chi_t + \partial I(\chi) - (1-\rho^*)(\Phi(p)+pw) &\!\!\ni\!\!& L\left(\frac{\theta}{\theta_c}-1\right) \quad\mbox{a.~e.},\\ \label{wu5} \int_{\Omega}\left(c_0\theta_t - \gamma(\theta)\chi_t^2 + \frac{L}{\theta_c}\theta\chi_t - \nu w_t^2 + \begin{equation}\labelta \theta w_t\right) \zeta\,\mathrm{d} x && \\ \nonumber +\int_{\Omega}\left(- \frac{1}{\rho_L}\mu(p)|\nabla p|^2 \zeta + \kappa(\theta) \nabla \theta\cdot\nabla \zeta\right)\,\mathrm{d} x &=& \int_{\partial\Omega} ^{(m)}ega(x)(\theta^*-\theta)\zeta \,\mathrm{d} s(x)\,, \end{equation}r almost everywhere in $(0,T)$ and for all test functions $\eta\in W^{1,2}(\Omega)$ and $\zeta \in W^{1,q^*}(\Omega)$, with some $q^* > 1$ that will be specified below in Theorem \ref{t1}. \vspace*{3mm} \begin{equation}\labelgin{hypothesis}\label{h1} {\rm We fix a time interval $[0,T]$ and assume that the data of Problem \eqref{wu1}--\eqref{wu5} have the following properties:} \begin{equation}\labelgin{itemize} \item[{\rm (i)}] $\gamma : [0,\infty) \to [0,\infty)$ {\rm is continuous; $\exists 0 < c_\gamma < C_\gamma: c_\gamma(1+\theta) \le \gamma(\theta) \le C_\gamma(1+\theta)$ for all $\theta \ge 0$;} \item[{\rm (ii)}] $\kappa : [0,\infty) \to [0,\infty)$ {\rm is continuous; $\exists 0<c_\kappa< C_\kappa$, $0< a < 1$, $a< \hat a < \frac{16}{5} + \frac65 a: c_\kappa (1+ \theta^{1+a}) \le \kappa(\theta) \le C_\kappa (1+ \theta^{1+\hat a})$ for all $\theta\ge 0$;} \item[{\rm (iii)}] {\rm $\theta^0 \in W^{1,2}(\Omega) \cap L^\infty(\Omega)$, $\theta^*\in L^\infty(\partial\Omega\times (0,T))$, $\theta^*_t\in L^2(\partial\Omega\times (0,T))$, $\exists \bar \theta>0: \theta^0(x) \ge \bar\theta$, $\theta^*(x,t) \ge \bar \theta$;} \item[{\rm (iv)}] $\exists\, 0 < \hat\delta \le \delta < 1/4,\ \exists\, 0<c_\varphi< C_\varphi$ {\rm such that for all $p\in \mathbb{R}$ we have that\\$c_\varphi \max\{1,|p|\}^{-1-\delta} \le \varphi'(p) \le C_\varphi \max\{1,|p|\}^{-1-\hat\delta}$;} \item[{\rm (v)}] {\rm $\exists 0 < c_\mu < C_\mu: c_\mu \le \mu(p) \le C_\mu$ for all $p \in \mathbb{R}$;} \item[{\rm (vi)}] {\rm $p^0 \in W^{1,2}(\Omega) \cap L^\infty(\Omega)$, $p^*\in L^\infty(\partial\Omega\times (0,T)) \cap L^2(0,T; W^{1,2}(\partial\Omega))$, $p^*_t\in L^2(\partial\Omega\times (0,T))$;} \item[{\rm (vii)}] {\rm $w^0, \chi^0\in L^\infty(\Omega)$, $\chi^0(x) \in [0,1]$ a.\,e., $\int_{\Omega} w^0(x)\,\mathrm{d} x = 0$; \item[{\rm (viii)}] $G\in L^\infty(\Omega\times (0,T))$, $G_t\in L^2(\Omega\times (0,T))$;} \item[{\rm (ix)}] {\rm $\Omega \subset \mathbb{R}^3$ is a bounded connected set of class $C^{1,1}$, $\alpha: \partial\Omega \to [0,\infty)$ is Lipschitz continuous, $^{(m)}ega \in L^\infty(\partial\Omega)$, $^{(m)}ega(x) \ge 0$ a.\,e., $\int_{\partial\Omega} \alpha(x)\,\mathrm{d} s(x) >0$, $\int_{\partial\Omega} ^{(m)}ega(x)\,\mathrm{d} s(x) >0$.} \end{itemize} \end{hypothesis} It is worth noting that it follows from \eqref{lu2} and (vii), using the definition of the functions $G$ and $H$, that \begin{equation}\labelgin{equation} \label{mean} \int_{\Omega} w(x,t)\,\mathrm{d} x\,=\,\int_{\Omega} w^0(x)\,\mathrm{d} x\,=\,0 \quad\hbox{\ for}\ all\, t\in [0,T]. \end{equation} The main result of this paper is the following existence result. \begin{equation}\labelgin{theorem}\label{t1} Let Hypothesis \ref{h1} hold true. Then there exists a solution $(p,w,\chi,\theta)$ to the system \eqref{inip}--\eqref{wu5}, \eqref{le7}, with the regularity \begin{equation}\labelgin{align} \label{regu1} &p \in L^\infty(\Omega\times (0,T)), \quad p_t, \nabla\theta \in L^2(\Omega\times (0,T)), \quad \nabla p \in L^\infty(0,T;L^2(\Omega)),\\[1mm] \label{regu2} &\theta, w_t \in L^{\bar p}(\Omega\times (0,T)), \quad w,\chi_t \in L^\infty(0,T;L^{\bar p}(\Omega)) \,\mbox{ for \,$\bar p < 8+a$},\\[1mm] \label{regu3} &\theta_t \in L^2(0,T;W^{-1,q^*}(\Omega)) \mbox{ with \,$q^*>1$\, given by } (\ref{qstar}). \end{align} \end{theorem} The proof of Theorem \ref{t1} will be divided into several steps which each constitutes a new section in this paper. \section{Cut-off system}\label{cut} The strategy for solving Problem \eqref{wu1}--\eqref{wu5} and proving Theorem \ref{t1} is the following: we choose a parameter $R>0$ and first solve a cut-off system with the intention to let $R$ tend to infinity. More precisely, for $R>0$ and $z \in \mathbb{R}$ we denote by $$Q_R(z) = \max\{-R, \min\{z, R\}\}$$ the projection onto $[-R,R]$, and set $$P_R(z) = z - Q_R(z).$$ We further denote \begin{equation}\label{ce1} \varphi_R(p) = \varphi(p) + P_R(p)\,, \quad \Phi_R(p) = \int_0^p \varphi_R(\tau) \,\mathrm{d} \tau\,, \quad V_R(p) = p \varphi_R(p) - \Phi_R(p)\,, \end{equation} and \begin{equation}\label{ce2} \gamma_R(p,\theta) = \left(1+ (p^2 - R^2)^+\right)\gamma(Q_R(\theta^+)), \end{equation} for $p,\theta \in \mathbb{R}$, and replace \eqref{wu1}--\eqref{wu5} by the cut-off system \begin{equation}\labelr\label{cu1} \int_{\Omega}\left(((\chi{+}\rho^*(1{-}\chi)) (\varphi_R(p)+w))_t \eta +\frac{1}{\rho_L}\mu(p)\nabla p\cdot\nabla \eta\right)\,\mathrm{d} x &\!\!=\!\!& \int_{\partial\Omega} \alpha(x)(p^*-p)\eta \,\mathrm{d} s(x),\qquad \\[2mm] \label{cu2} \nu w_t + \lambda_M w - p (\chi{+}\rho^*(1{-}\chi)) - \begin{equation}\labelta(Q_R(\theta^+)-\theta_c) &\!\!=\!\!& - G + H_R(t)\quad\mbox{a.~e.},\\[2mm] \label{cu4} \gamma_R(p,\theta) \chi_t + \partial I(\chi) - (1-\rho^*)(\Phi_R(p)+pw) &\!\!\ni\!\!& L\left(\frac{Q_R(\theta^+)}{\theta_c}-1\right)\quad\mbox{a.~e.}, \end{equation}r \vspace*{-9mm} \begin{equation}\labelgin{align} &\int_{\Omega}\left(c_0\theta_t \zeta + \kappa(Q_R(\theta^+)) \nabla \theta\cdot\nabla \zeta\right)\,\mathrm{d} x \, - \int_{\Omega}\left(\frac{1}{\rho_L}\mu(p)Q_R(|\nabla p|^2) + \gamma_R(p,\theta)\chi_t^2 + \nu w_t^2\right) \zeta \,\mathrm{d} x \nonumber\\ \label{cu5} &\quad+\int_{\Omega} Q_R(\theta^+)\left(\frac{L}{\theta_c}\chi_t + \begin{equation}\labelta w_t\right) \zeta\,\mathrm{d} x \,\,=\,\, \int_{\partial\Omega} ^{(m)}ega(x)(\theta^*-\theta)\zeta \,\mathrm{d} s(x), \end{align} for all test functions $\eta,\zeta \in W^{1,2}(\Omega)$, with \begin{equation}\label{le7a} H_R(t) = - \frac{1}{|\Omega|}\int_{\Omega} (p (\chi{+}\rho^*(1{-}\chi)) + \begin{equation}\labelta(Q_R(\theta^+)-\theta_c) -G)\,\mathrm{d} x\,. \end{equation} For the system \eqref{cu1}--\eqref{le7a}, we prove the following result. \begin{equation}\labelgin{proposition}\label{t2} Let Hypothesis \ref{h1} hold and let $R>0$ be given. Then there exists a solution $(p,w,\chi,\theta)$ to \eqref{cu1}--\eqref{le7a}, \eqref{inip}--\eqref{init} with the regularity $p,w,\chi,\theta,w_t \in L^q(\Omega; C[0,T])$ for $1\le q < 3$, $p_t,\theta_t \in L^2(\Omega\times (0,T))$, and $\nabla p, \nabla\theta, \chi_t \in L^\infty(0,T;L^2(\Omega))$. \end{proposition} \begin{equation}\labelgin{pf}{Proof of Proposition \ref{t2}}\ Let $$M(p) := \int_0^p \mu(\tau)\,\mathrm{d} \tau , \quad K_R(\theta) := \int_0^\theta \kappa(Q_R(\tau^+))\,\mathrm{d} \tau,$$ and set $v = M(p)$, $z = K_R(\theta)$. Then \eqref{cu1}--\eqref{cu5} is transformed into the system \begin{equation}\labelr\label{ku1} \int_{\Omega}\left(((\chi{+}\rho^*(1{-}\chi))(\varphi_R(p)+w))_t \eta + \frac{1}{\rho_L}\nabla v\cdot\nabla \eta\right)\,\mathrm{d} x &\!\!=\!\!& \int_{\partial\Omega} \alpha(x)(p^*-p)\eta \,\mathrm{d} s(x),\qquad \\[2mm] \label{ku2} \nu w_t + \lambda_M w - p (\chi{+}\rho^*(1{-}\chi)) - \begin{equation}\labelta(Q_R(\theta^+)-\theta_c) &\!\!=\!\!& - G + H_R(t)\quad\mbox{a.~e.},\\ \label{ku4} \gamma_R(p,\theta) \chi_t + \partial I(\chi) - (1-\rho^*)(\Phi_R(p)+pw) &\!\!\ni\!\!& L\left(\frac{Q_R(\theta^+)}{\theta_c}-1\right)\quad\mbox{a.~e.}, \end{equation}r \vspace*{-9mm} \begin{equation}\labelgin{align} \nonumber &\int_{\Omega}\left(c_0\theta_t \zeta + \nabla z\cdot\nabla \zeta\right)\,\mathrm{d} x \,- \int_{\Omega}\left(\frac{1}{\rho_L}\mu(p)Q_R(|\nabla p|^2) + \gamma_R(p,\theta)\chi_t^2 + \nu w_t^2\right) \zeta \,\mathrm{d} x \\[1mm] \label{ku5} &\quad+\int_{\Omega} Q_R(\theta^+)\left(\frac{L}{\theta_c}\chi_t + \begin{equation}\labelta w_t\right) \zeta\,\mathrm{d} x \,\,=\,\, \int_{\partial\Omega} ^{(m)}ega(x)(\theta^*-\theta)\zeta \,\mathrm{d} s(x), \end{align} which we solve by Galerkin approximations. To this end, let $\{e_k; k=0,1,\dots\}$ denote the complete orthonormal system of eigenfunctions of the problem \begin{equation}\label{eigen} -\Delta e_k = \lambda_k e_k \ \mbox{ in\ } \Omega \,, \quad \nabla e_k \cdot n = 0 \ \mbox{ on\ } \partial\Omega\,. \end{equation} We approximate $v$ and $z$ by the finite sums \begin{equation}\label{ge1} v^{(n)}(x,t) = \sum_{k=0}^n v_k(t) e_k(x)\,, \quad z^{(n)}(x,t) = \sum_{k=0}^n z_k(t) e_k(x)\,, \end{equation} where $v_k, z_k, w^{(n)}, \chi^{(n)}$ satisfy the system \begin{equation}\labelr \nonumber &&\hspace{-16mm}\int_{\Omega}\left(((\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)}))(\varphi_R(p^{(n)})+w^{(n)}))_t e_k + \frac{1}{\rho_L}\nabla v^{(n)}\cdot\nabla e_k\right)\,\mathrm{d} x \\ \label{gu1} &=& \int_{\partial\Omega} \alpha(x)(p^*-p^{(n)}) e_k \,\mathrm{d} s(x), \quad k=0,1, \dots, n,\\[2mm] \nonumber &&\hspace{-16mm}\nu w^{(n)}_t + \lambda_M w^{(n)} - p^{(n)}(\chi^{(n)} {+} \rho^*(1{-}\chi^{(n)})) - \begin{equation}\labelta(Q_R((\theta^{(n)})^+)-\theta_c) \\[1mm] \label{gu2} &=& - G + H^{(n)}_R(t)\quad\mbox{a.~e.},\\[2mm] \nonumber &&\hspace{-16mm} \gamma_R(p^{(n)},\theta^{(n)}) \chi^{(n)}_t + \partial I(\chi^{(n)}) - (1-\rho^*)(\Phi_R(p^{(n)})+p^{(n)} w^{(n)})\\[1mm] \label{gu4} &\ni& L\left(\frac{Q_R((\theta^{(n)})^+)}{\theta_c}-1\right)\quad\mbox{a.~e.},\\[2mm] \nonumber &&\hspace{-16mm}\int_{\Omega}\left(c_0\theta^{(n)}_t e_k + \nabla z^{(n)}\cdot\nabla e_k\right) + Q_R((\theta^{(n)})^+)\left(\frac{L}{\theta_c}\chi^{(n)}_t + \begin{equation}\labelta w^{(n)}_t\right) e_k\,\mathrm{d} x \\ \nonumber &&- \int_{\Omega}\left(\frac{1}{\rho_L}\mu(p)Q_R(|\nabla p^{(n)}|^2) + \gamma_R(p^{(n)},\theta^{(n)})(\chi^{(n)}_t)^2 + \nu (w^{(n)}_t)^2\right) \zeta \,\mathrm{d} x \\ \label{gu5} &=& \int_{\partial\Omega} ^{(m)}ega(x)(\theta^*-\theta^{(n)})e_k \,\mathrm{d} s(x), \end{equation}r with $p^{(n)} := M^{-1}(v^{(n)})$, $\theta^{(n)} := K_R^{-1}(z^{(n)})$, and \begin{equation}\label{le7c} H^{(n)}_R(t) := - \frac{1}{|\Omega|}\int_{\Omega} (p^{(n)}(\chi^{(n)} + \rho^*(1{-}\chi^{(n)})) + \begin{equation}\labelta(Q_R((\theta^{(n)})^+)-\theta_c) -G)\,\mathrm{d} x\,, \end{equation} and with the initial conditions \begin{equation}\labelgin{align} \label{inipn} v_k(0) &= \int_{\Omega} M(p^0(x)) e_k(x)\,\mathrm{d} x\,,\\ \label{initn} z_k(0) &= \int_{\Omega} K_R(\theta^0(x)) e_k(x)\,\mathrm{d} x\,,\\ \label{iniun} w^{(n)}(x,0) &= w^0(x)\,,\\ \label{inichn} \chi^{(n)}(x,0) &= \chi^0(x)\,. \end{align} This is an easy ODE system that admits a unique solution on some interval $[0,T_n) \subset [0,T]$. Moreover, the solution $w^{(n)}$ of \eqref{gu1} enjoys the explicit representation \begin{equation}\labelr\nonumber && w^{(n)}(x,t) \,=\, \hbox{\rm e}^{-(\lambda_M/\nu)t} w^0(x)+\frac{1}{\nu}\int_0^t \hbox{\rm e}^{(\lambda_M/\nu)(t'-t)}(-G + H^{(n)}_R)(x,t')\,\mathrm{d} t'\\ \label{n1} &&\hspace{5mm} +\, \frac{1}{\nu} \int_0^t \hbox{\rm e}^{(\lambda_M/\nu)(t'-t)} \left(p^{(n)}(\chi^{(n)} {+} \rho^*(1{-}\chi^{(n)})) + \begin{equation}\labelta(Q_R((\theta^{(n)})^+)-\theta_c)\right)(x,t')\,\mathrm{d} t'.\qquad \end{equation}r Also \eqref{gu4} is of a standard form, namely, \begin{equation}\label{n2} \chi^{(n)}_t + \partial I(\chi^{(n)}) \ni F^{(n)}, \end{equation} with \begin{equation}\label{n3} F^{(n)} = (1-\rho^*) \frac{\Phi_R(p^{(n)})+p^{(n)} w^{(n)}}{\gamma_R(p^{(n)},\theta^{(n)})} + \frac{L(Q_R((\theta^{(n)})^+) -\theta_c)}{\theta_c\gamma_R(p^{(n)},\theta^{(n)})}, \end{equation} or, equivalently, \begin{equation}\label{n3a} \chi^{(n)} \in [0,1]\,, \quad (F^{(n)} - \chi^{(n)}_t)(\chi^{(n)} - \tilde\chi) \ge 0 \ \mbox{ a.\,e. }\ \hbox{\ for}\ all \tilde\chi \in [0,1]. \end{equation} By virtue of \eqref{n1}--\eqref{n3}, we have for all $(x,t) \in \Omega\times (0,T_n)$ the inequalities \begin{equation}\label{ge3} \left. \begin{equation}\labelgin{array}{rcl} |w^{(n)}(x,t)|+|\chi^{(n)}_t(x,t)| &\le& C_R \Big(1+\int_0^t|p^{(n)}(x,t')|\,\mathrm{d} t' + \int_0^t\int_{\Omega}|p^{(n)}(x',t')|\,\mathrm{d} x'\,\mathrm{d} t'\Big)\\[2mm] |w^{(n)}_t(x,t)| &\le& C_R \Big(1+|p^{(n)}(x,t)| + \int_0^t|p^{(n)}(x,t')|\,\mathrm{d} t'\\ &&+\, \int_{\Omega} |p^{(n)}(x',t)|\,\mathrm{d} x' + \int_0^t\int_{\Omega}|p^{(n)}(x',t')|\,\mathrm{d} x'\,\mathrm{d} t'\Big) \end{array} \right\} \end{equation} where, here and in the following, $C_R>0$ denote constants which possibly depend on $R$ and on the data, but not on $n$. We now derive some a priori estimates for the solutions to the Galerkin system. To begin with, we first test \eqref{gu1} by $v_k(t)$ and sum over $k=1, \dots, n$ to obtain the identity (note that $v^{(n)} = M(p^{(n)})$, by definition) \begin{equation}\labelr\nonumber &&\hspace{-12mm} (1-\rho^*) \int_{\Omega} \chi^{(n)}_t(\varphi_R(p^{(n)}) + w^{(n)}) M(p^{(n)})\,\mathrm{d} x + \int_{\Omega} (\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)}))\varphi_R(p^{(n)})_t M(p^{(n)})\,\mathrm{d} x \\ \nonumber &&+\,\int_{\Omega}(\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)})) w^{(n)}_t M(p^{(n)})\,\mathrm{d} x + \frac{1}{\rho_L}\int_{\Omega}|\nabla v^{(n)}|^2 \,\mathrm{d} x\\ \label{gn1} &&+\, \int_{\partial\Omega} \alpha(x)(p^{(n)}-p^*)M(p^{(n)})\,\mathrm{d} s(x) = 0. \end{equation}r We rewrite the first term of \eqref{gn1}, using the identity \begin{equation}\labelr\nonumber \int_{\Omega} \chi^{(n)}_t(\varphi_R(p^{(n)}) + w^{(n)}) M(p^{(n)})\,\mathrm{d} x &=& \int_{\Omega} \chi^{(n)}_t(\Phi_R(p^{(n)}) + p^{(n)} w^{(n)}) \frac{M(p^{(n)})}{p^{(n)}}\,\mathrm{d} x\\ \label{gn2} &&+\, \int_{\Omega} \chi^{(n)}_t V_R(p^{(n)})\frac{M(p^{(n)})}{p^{(n)}} \,\mathrm{d} x. \end{equation}r From \eqref{gu4} it follows that for a.\,e. $(x,t) \in \Omega\times (0,T)$ we have, by Young's inequality, \begin{equation}\labelr\nonumber &&\hspace{-12mm}(1-\rho^*)\chi^{(n)}_t(\Phi_R(p^{(n)}) + p^{(n)} w^{(n)}) \frac{M(p^{(n)})}{p^{(n)}}\\ \nonumber &=& \frac{M(p^{(n)})}{p^{(n)}}\left( \gamma_R(p^{(n)},\theta^{(n)}) \bigl|\chi^{(n)}_t\bigr|^2 + \frac{L}{\theta_c}(\theta_c - Q_R((\theta^{(n)})^+)) \chi_t^{(n)}\right)\\ \label{gn3} &\ge& \frac{c_{\mu}}{2} \gamma_R(p^{(n)},\theta^{(n)}) \bigl|\chi^{(n)}_t\bigr|^2 - C_R\,. \end{equation}r The second term in \eqref{gn1} can be rewritten as \begin{equation}\labelr\nonumber \int_{\Omega} (\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)}))\varphi_R(p^{(n)})_t M(p^{(n)})\,\mathrm{d} x &=& \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega} (\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)})) V_{R,M}(p^{(n)})\,\mathrm{d} x\\ \label{gn4} &&-\, (1-\rho^*)\int_{\Omega} \chi^{(n)}_t V_{R,M}(p^{(n)})\,\mathrm{d} x, \end{equation}r where we denote \begin{equation}\label{gn5} V_{R,M}(p) = \int_0^p \varphi_R'(\tau) M(\tau)\,\mathrm{d} \tau\,. \end{equation} We see, in particular, that there exist constants $ 0 < c_{R,\mu} < C_{R,\mu}$ such that $$ c_{R,\mu} p^2 \le V_{R,M}(p) \le C_{R,\mu} p^2 $$ for all $p \in \mathbb{R}$. Combining \eqref{gn2}--\eqref{gn5} with \eqref{gn1}, we obtain that \begin{equation}\labelr\nonumber &&\hspace{-12mm}\frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega} (\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)})) V_{R,M}(p^{(n)})\,\mathrm{d} x + \frac{c_{\mu}}{2} \int_{\Omega}\gamma_R(p^{(n)},\theta^{(n)}) \bigl|\chi^{(n)}_t\bigr|^2 \,\mathrm{d} x + \frac{1}{\rho_L}\int_{\Omega}|\nabla v^{(n)}|^2 \,\mathrm{d} x\\ \nonumber &&+\, \int_{\partial\Omega} \alpha(x)(p^{(n)}-p^*)M(p^{(n)})\,\mathrm{d} s(x)\\ \nonumber &\le& C_R \,- \int_{\Omega}(\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)})) w^{(n)}_t M(p^{(n)})\,\mathrm{d} x\\ \label{gn6} &&+\, (1-\rho^*)\int_{\Omega} \chi^{(n)}_t \left(V_{R,M}(p^{(n)}) - V_R(p^{(n)})\frac{M(p^{(n)})}{p^{(n)}}\right)\,\mathrm{d} x. \end{equation}r By \eqref{ge3} and H\"older's inequality, we have that \begin{equation}\label{gn6a} \left|\int_{\Omega}(\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)})) w^{(n)}_t M(p^{(n)})(x,t)\,\mathrm{d} x\right| \le C_R \int_{\Omega} \left(|p^{(n)}|^2(x,t) + \int_0^t |p^{(n)}|^2(x,t') \,\mathrm{d} t'\right)\,\mathrm{d} x, \end{equation} and, similarly, \begin{equation}\labelr \nonumber &&\hspace{-12mm}\left|\int_{\Omega} \chi^{(n)}_t \left(V_{R,M}(p^{(n)}) - V_R(p^{(n)})\frac{M(p^{(n)})}{p^{(n)}}\right)\,\mathrm{d} x\right| \le C_R \int_{\Omega} |\chi^{(n)}_t| |p^{(n)}|^2 \,\mathrm{d} x \\[1mm] \nonumber &\le& \frac{c_{\mu}}{4} \int_{\Omega}\gamma_R(p^{(n)},\theta^{(n)}) \bigl|\chi^{(n)}_t\bigr|^2 \,\mathrm{d} x + C_R \int_{\Omega} \frac{|p^{(n)}|^4}{\gamma_R(p^{(n)},\theta^{(n)})}\,\mathrm{d} x \\ [1mm]\label{gn7} &\le& \frac{c_{\mu}}{4} \int_{\Omega}\gamma_R(p^{(n)},\theta^{(n)}) \bigl|\chi^{(n)}_t\bigr|^2 \,\mathrm{d} x + C_R \int_{\Omega} \left(1+|p^{(n)}|^2\right)\,\mathrm{d} x. \end{equation}r Let $[0,T_n)$ denote the maximal interval of existence of our solution. Using \eqref{gn6}--\eqref{gn7}, and Gronwall's lemma, we thus can infer that \begin{equation}\label{gn8} \mathop{{\rm sup\,ess}\,}_{t\in (0,T_n)}\int_{\Omega} |p^{(n)}|^2(x,t)\,\mathrm{d} x +\int_0^{T_n}\int_{\Omega} |\nabla p^{(n)}|^2\,\mathrm{d} x\,\mathrm{d} t + \int_0^{T_n}\int_{\partial\Omega} \alpha(x)|p^{(n)}|^2 \,\mathrm{d} s(x)\,\mathrm{d} t \le C_R\,. \end{equation} In particular, the Galerkin solution exists globally, and for every $n \in \mathbb{N}$ we have $T_n=T$. In what follows, we denote by $|\cdot|_p$ the norm in $L^p(\Omega)$, by $\|\cdot\|_p$ the norm in $L^p(\Omega\times (0,T))$, by $\|\cdot\|_{\partial\Omega, p}$ the norm in $L^p(\partial\Omega\times (0,T))$, and by $\|\cdot\|_{W^{\ell,p}(\Omega)}$ the norm in $W^{\ell,p}(\Omega)$ for $\ell \in \mathbb{N}$ and $1 \le p \le \infty$. Let us recall the Gagliardo-Nirenberg inequality \begin{equation}\label{gn} |u|_q \le C\left(|u|_s + |u|_s^{1-\rho}|\nabla u|_p^\rho\right), \end{equation} with $$ \rho = \frac{\frac 1s - \frac1q}{\frac 1s + \frac 1N - \frac 1p}\,\,, $$ which is valid for all $1\le s<q$, $1/q > (1/p) - (1/N)$, every bounded open set $\Omega \subset \mathbb{R}^N$ with Lipschitzian boundary, and every function $u \in W^{1,p}(\Omega)$. For $t\in (0,T)$, $N=3$, $s = p = 2$, and $q=4$, we have, in particular, $$ |p^{(n)}(t)|_4 \le C\left(|p^{(n)}(t)|_2 + |p^{(n)}(t)|_2^{1/4} |\nabla p^{(n)}(t)|_2^{3/4}\right). $$ Hence, by \eqref{gn8}, \begin{equation}\label{gn9} \int_0^T \left(\int_{\Omega} |p^{(n)}(x,t)|^4 \,\mathrm{d} x\right)^{2/3}\,\mathrm{d} t \le C_R\,, \end{equation} independently of $n$. Next, we test \eqref{gu1} by $\dot v_k(t)$ and sum over $k=0,1,\dots n$ to obtain the identity \begin{equation}\labelr\nonumber &&\hspace{-16mm}\frac{\,\mathrm{d}}{\,\mathrm{d} t} \left(\frac1{2\rho_L}\int_{\Omega} |\nabla v^{(n)}|^2\,\mathrm{d} x + \int_{\partial\Omega} \alpha(x)(\hat M(v^{(n)})-p^* v^{(n)})\,\mathrm{d} s(x)\right)\\ \nonumber && +\int_{\Omega}\left(((\chi^{(n)}{+}\rho^*(1{-}\chi^{(n)}))(\varphi_R(p^{(n)})+w^{(n)}))_t v_t^{(n)}\right)\,\mathrm{d} x\\ \label{gd0} &=& - \int_{\partial\Omega} \alpha(x) p^*_t v^{(n)} \,\mathrm{d} s(x), \end{equation}r where $\hat M' = M^{-1}$. We have the pointwise lower bound $$ \varphi_R(p^{(n)})_t v^{(n)}_t \ge C_R |v^{(n)}_t|^2, $$ and thus, by \eqref{gd0} and H\"older's inequality, we have for all $t \in [0,T]$ that \begin{equation}\labelr\nonumber &&\hspace{-16mm}\int_0^t\int_{\Omega} |v^{(n)}_t|^2\,\mathrm{d} x\,\mathrm{d} t' + \left(\int_{\Omega} |\nabla v^{(n)}|^2(x,t) \,\mathrm{d} x + \int_{\partial\Omega} \alpha(x)|v^{(n)}|^2(x,t)\,\mathrm{d} s(x)\right) \\ \label{gd1} &\le& C_R \left(1+ \int_0^t\int_{\Omega} (|w^{(n)}_t|^2 + |\chi^{(n)}_t|^2 |w^{(n)}|^2)\,\mathrm{d} x\,\mathrm{d} t' +\int_0^t\int_{\partial\Omega} \alpha(x)|v^{(n)}|^2\,\mathrm{d} s(x)\,\mathrm{d} t'\right).\quad \end{equation}r A bound for $\|w^{(n)}_t\|_2^2$ follows from \eqref{ge3} and \eqref{gn8}. Moreover, owing to \eqref{ge3}, we have for $t\in (0,T)$ that $$ \int_{\Omega} |\chi^{(n)}_t|^2 |w^{(n)}|^2 (x,t)\,\mathrm{d} x \le C_R\left(1 + \int_{\Omega} \left(\int_0^t |p^{(n)}(x,t')|\,\mathrm{d} t'\right)^4\,\mathrm{d} x\right). $$ We use the Minkowski inequality in the form $$ \left(\int_{\Omega} \left(\int_0^t |p^{(n)}(x,t')|\,\mathrm{d} t'\right)^4\,\mathrm{d} x\right)^{1/4} \le \int_0^t \left(\int_{\Omega} |p^{(n)}(x,t')|^4\,\mathrm{d} x\right)^{1/4} \,\mathrm{d} t' $$ to check that $$ \int_{\Omega} |\chi^{(n)}_t|^2 |w^{(n)}|^2 (x,t)\,\mathrm{d} x \le C_R\left(1 + \left(\int_0^t \left(\int_{\Omega} |p^{(n)}(x,t')|^4\,\mathrm{d} x\right)^{1/4} \,\mathrm{d} t'\right)^4\right) \le C_R, $$ by virtue of \eqref{gn9}. Then \eqref{gd1} and the Gronwall argument imply that \begin{equation}\label{ge2} \|v^{(n)}(t)\|_{W^{1,2}(\Omega)}^2 + \int_0^t |v^{(n)}_t(t')|_2^2\,\mathrm{d} t' \le C_R, \end{equation} whence also \begin{equation}\label{ge2a} \|p^{(n)}(t)\|_{W^{1,2}(\Omega)}^2 + \int_0^t |p^{(n)}_t(t')|_2^2\,\mathrm{d} t' \le C_R \end{equation} for $t \in (0,T)$. We continue by testing \eqref{gu5} by $\dot z_k(t)$ and summing over $k=0,1,\dots n$. Note that, thanks to \eqref{n2}--\eqref{n3}, \eqref{ge3}, and \eqref{gn8}, we have that $$ \gamma_R(p^{(n)},\theta^{(n)})(\chi^{(n)}_t(x,t))^2 + \nu (w^{(n)}_t(x,t))^2 \le C_R \left(1+|p^{(n)}(x,t)| + \int_0^t|p^{(n)}(x,t')|\,\mathrm{d} t'\right)^3 $$ for a.\,e. $(x,t) \in \Omega \times (0,T)$. This yields the inequality \begin{equation}\labelrs &&\hspace{-16mm}\frac{\,\mathrm{d}}{\,\mathrm{d} t} \left(\frac12\int_{\Omega} |\nabla z^{(n)}|^2\,\mathrm{d} x + \int_{\partial\Omega} ^{(m)}ega(x)(\hat K_R(v^{(n)})-\theta^* v^{(n)})\,\mathrm{d} s(x)\right) + \frac12 \int_{\Omega} c_0\theta^{(n)}_t z^{(n)}_t \,\mathrm{d} x\\ &\le& \int_{\partial\Omega} ^{(m)}ega(x) |\theta^*_t| |z^{(n)}| \,\mathrm{d} s(x) + C_R\int_{\Omega} \left(1+|p^{(n)}(x,t)| + \int_0^t|p^{(n)}(x,t')|\,\mathrm{d} t'\right)^6\,\mathrm{d} x, \end{equation}rs where $\hat K_R' = K_R^{-1}$. Using \eqref{ge2a} and the Sobolev embedding theorem, we obtain, as before, that \begin{equation}\label{ge2b} \|\theta^{(n)}(t)\|_{W^{1,2}(\Omega)}^2 + \int_0^t |\theta^{(n)}_t(t')|_2^2\,\mathrm{d} t' \le C_R \end{equation} for $t \in (0,T)$. Hence, there exist a subsequence of $\{(p^{(n)}, \theta^{(n)}): n \in \mathbb{N}\}$, which is again indexed by $n$, and functions $p, \theta$, such that \begin{equation}\labelrs p^{(n)}_t \to p_t, \ \theta^{(n)}_t \to \theta_t, && \mbox{weakly in } \ L^2(\Omega\times (0,T)),\\ \nabla p^{(n)} \to \nabla p, \ \nabla \theta^{(n)} \to \nabla \theta, && \mbox{weakly-star in } \ L^\infty(0,T;L^2(\Omega)),\\ p^{(n)} \to p, \ \theta^{(n)} \to \theta, && \mbox{strongly in } \ L^q(\Omega; C[0,T]) \ \hbox{\ for}\ \ 1\le q < 3,\\ \end{equation}rs where we used the compact embedding $W^{1,2}(\Omega\times (0,T))\hookrightarrow\hookrightarrow L^q(\Omega;C[0,T])$ for $1\le q<3$, see \cite{bin}. We now check that the sequences $\{w^{(n)}\}, \{\chi^{(n)}\}, \{w^{(n)}_t\}, \{\chi^{(n)}_t\}$ converge strongly in appropriate function spaces and that the limit functions satisfy the system \eqref{cu1}--\eqref{le7a}. Passing again to a subsequence if necessary, we may fix a set $\Omega' \subset \Omega$ with meas$(\Omega\setminus\Omega') = 0$ such that \begin{equation}\label{n4} \lim_{n\to \infty} \sup_{t\in [0,T]}|p^{(n)}(x,t) - p(x,t)| = 0\,, \quad \lim_{n\to \infty} \sup_{t\in [0,T]}|\theta^{(n)}(x,t) - \theta(x,t)| = 0, \quad \hbox{\ for}\ all\, x \in \Omega', \end{equation} and such that the functions $t \mapsto p(x,t)$ and $t \mapsto \theta(x,t)$ belong to $C[0,T]$ for all $x \in \Omega'$. In particular, we can define the real numbers $$ \widetilde p(x) := \sup_{t\in [0,T]}|p(x,t)|\,, \quad \widetilde \theta(x): = \sup_{t\in [0,T]}|\theta(x,t)|, \ \hbox{\ for}\ x \in \Omega'\,. $$ Let $x \in \Omega'$ be arbitrarily fixed now. Then there is some $n_0(x) \in \mathbb{N}$ such that for $n > n_0(x)$ we have $|p^{(n)}(x,t)| \le 2\widetilde p(x)$ and $|\theta^{(n)}(x,t)| \le 2\widetilde \theta(x)$, for all $t \in [0,T]$ and $x \in \Omega'$. For $n,m \in \mathbb{N}$, $n,m > n_0(x)$, we have by \eqref{n1} for $t \in [0,T]$ and $x \in \Omega'$ that \begin{equation}\label{n5} |w^{(n)}(x,t) - w^{(m)}(x,t)| \le C_R(1+\widetilde p(x))\int_0^t (|\chi^{(n)} - \chi^{(m)}| + |p^{(n)} - p^{(m)}| + |\theta^{(n)} - \theta^{(m)}|)(x,t')\,\mathrm{d} t'. \end{equation} Hence, with the notation of \eqref{n3}, \begin{equation}\labelr\nonumber &&\hspace{-16mm}\int_0^t|F^{(n)}(x,t') - F^{(m)}(x,t')|\,\mathrm{d} t'\\ \label{n6} &\le& C_R(1+\widetilde p(x))^2\int_0^t (|\chi^{(n)} - \chi^{(m)}| + |p^{(n)} - p^{(m)}| + |\theta^{(n)} - \theta^{(m)}|)(x,t')\,\mathrm{d} t'. \end{equation}r The well-known $L^1$-Lipschitz continuity result for variational inequalities (see, e.\,g., \cite[Theorem 1.12]{cmuc}) tells us that \begin{equation}\label{n7} \int_0^t |\chi^{(n)}_t - \chi^{(m)}_t|(x,t')\,\mathrm{d} t' \le 2 \int_0^t|F^{(n)}(x,t') - F^{(m)}(x,t')|\,\mathrm{d} t'. \end{equation} Since $\{p^{(n)}(x,\cdot)\}$ and $\{\theta^{(n)}(x,\cdot)\}$ converge uniformly for each $x \in \Omega'$, we may apply the Gronwall argument to conclude that $\{\chi^{(n)}(x,\cdot)\}$, $\{w^{(n)}(x,\cdot)\}$, $\{w_t^{(n)}(x,\cdot)\}$ are Cauchy sequences in $C[0,T]$ and that $\{\chi_t^{(n)}(x,\cdot)\}$ is a Cauchy sequence in $W^{1,1}(0,T)$, for every $x \in \Omega'$. Hence, there exist functions $\chi, w : \Omega'\times (0,T)$ such that, as $n\to\infty$, \begin{equation}\label{n8} \sup_{t \in [0,T]}|w^{(n)}(x,t) - w(x,t)| \to 0\,, \ \ \sup_{t \in [0,T]}|\chi^{(n)}(x,t) - \chi(x,t)| \to 0\,, \ \ \sup_{t \in [0,T]}|w_t^{(n)}(x,t) - w_t(x,t)| \to 0 \end{equation} and \begin{equation}\label{n9} \int_0^T|\chi^{(n)}_t(x,t) - \chi_t(x,t)| \,\mathrm{d} t \to 0, \end{equation} for all $x \in \Omega'$. Since $|\chi^{(n)}|, |w^{(n)}|, |\chi^{(n)}_t|, |w^{(n)}_t|$ admit a pointwise upper bound \eqref{ge3} in terms of convergent sequences in $L^q(\Omega; C[0,T])$ for $1\le q < 3$, we can use the Lebesgue Dominated Convergence Theorem to conclude that \begin{equation}\label{n10} w^{(n)} \to w\,, \quad w_t^{(n)} \to w_t\,, \quad \chi^{(n)} \to \chi, \quad \mbox{ strongly in } \ L^q(\Omega; C[0,T])\,, \end{equation} and \begin{equation}\label{n11} \chi^{(n)}_t \to \chi_t \quad \mbox{ strongly in } \ L^1(\Omega\times (0,T)). \end{equation} Moreover, from H\"older's inequality we obtain that \begin{equation}\labelrs &&\hspace{-12mm}\int_0^T\int_{\Omega} |\chi^{(n)}_t - \chi_t|^2\,\mathrm{d} x \,\mathrm{d} t = \int_0^T\int_{\Omega} |\chi^{(n)}_t - \chi_t|^{1/3} |\chi^{(n)}_t - \chi_t|^{5/3}\,\mathrm{d} x \,\mathrm{d} t\\ && \le\, \left(\int_0^T\int_{\Omega} |\chi^{(n)}_t - \chi_t|\,\mathrm{d} x \,\mathrm{d} t\right)^{1/3} \left(\int_0^T\int_{\Omega} |\chi^{(n)}_t - \chi_t|^{5/2}\,\mathrm{d} x \,\mathrm{d} t\right)^{2/3} \to 0, \end{equation}rs by virtue of \eqref{n11} and \eqref{ge3}. We can therefore pass to the limit in \eqref{gu1}--\eqref{le7c}, where \eqref{gu4} is interpreted as the variational inequality \eqref{n3a}, and check that its limit is the desired solution to \eqref{cu1}--\eqref{le7a}. \end{pf} \section{Estimates independent of $R$}\label{apr} In this section, we derive estimates for the solutions to \eqref{cu1}--\eqref{cu5} which are independent of the cut-off parameter $R$. In the entire section, we denote by $C$ positive constants which may depend on the data of the problem, but not on $R$. \subsection{Positivity of temperature}\label{pos} For every nonnegative test function $\zeta \in W^{1,2}(\Omega)$ we have, by virtue of \eqref{cu5}, and using the fact that $\gamma_R(p,\theta) \ge c_\gamma>0$, \begin{equation}\label{pe25} \int_{\Omega}\left(c_0\theta_t \zeta + \kappa(Q_R(\theta^+)) \nabla \theta\cdot\nabla \zeta\right)\,\mathrm{d} x + \int_{\partial\Omega} ^{(m)}ega(x)(\theta-\theta^*) \zeta \,\mathrm{d} s(x) \ge - C\int_{\Omega} (Q_R(\theta^+))^2 \zeta \,\mathrm{d} x \end{equation} with a constant $C$ depending only on the constants $L, \theta_c, \begin{equation}\labelta, \nu, c_\gamma$. Let $\psi$ be the solution of the equation \begin{equation}\label{epsi} c_0 \dot\psi(t) + C\psi^2(t) = 0\,, \quad \psi(0) = \bar\theta\,. \end{equation} Then \begin{equation}\label{epsi2} \psi(t) = \frac{\bar\theta c_0}{c_0 + \bar\theta C t}\,\,, \end{equation} and we have \begin{equation}\labelr \nonumber &&\hspace{-16mm}\int_{\Omega}\left(c_0(\psi- \theta)_t \zeta + \kappa(Q_R(\theta^+)) \nabla(\psi- \theta)\cdot\nabla \zeta\right)\,\mathrm{d} x\, - \int_{\partial\Omega} ^{(m)}ega(x)(\theta-\theta^*)\zeta \,\mathrm{d} s(x)\\ \label{pe25b} &\le& C\int_{\Omega} ((Q_R(\theta^+))^2 - \psi^2) \zeta \,\mathrm{d} x \end{equation}r for every nonnegative test function $\zeta \in W^{1,2}(\Omega)$. In particular, for $\zeta(x,t) = (\psi(t) - \theta(x,t))^+$, we obtain that \begin{equation}\label{pe25c} \frac{\,\mathrm{d}}{\,\mathrm{d} t}\frac{c_0}{2}\int_{\Omega}((\psi- \theta)^+)^2\,\mathrm{d} x + \int_{\partial\Omega} ^{(m)}ega(x)(\theta^*-\theta)(\psi- \theta)^+ \,\mathrm{d} s(x) \le C\int_{\Omega} ((Q_R(\theta^+))^2 - \psi^2)(\psi- \theta)^+ \,\mathrm{d} x\,. \end{equation} From Hypothesis 2.1\,(iii), we obtain for all values of $x$ and $t$ that $$ (\theta^*-\theta)(\psi- \theta)^+ \ge 0\,, \quad ((Q_R(\theta^+))^2 - \psi^2)(\psi- \theta)^+ = (Q_R(\theta^+) - \psi)(Q_R(\theta^+) + \psi)(\psi- \theta)^+ \le 0\,, $$ and from \begin{equation}\label{pe25d} \frac{\,\mathrm{d}}{\,\mathrm{d} t}\frac{c_0}{2}\int_{\Omega}((\psi- \theta)^+)^2\,\mathrm{d} x \le 0\,, \quad (\psi- \theta)^+(x,0) = 0, \end{equation} we conclude that, independently of $R>0$, \begin{equation}\label{pos0} \theta(x,t) \ge \psi(t) \ge \frac{\bar\theta c_0}{c_0 + \bar\theta C T} > 0 \quad \mbox{ for all }\ x \ \mbox{ and } \ t. \end{equation} \subsection{Energy estimate}\label{enes} We test \eqref{cu1} by $\eta=p$, \eqref{cu5} by $\zeta = 1$, and sum up. With the notation \eqref{ce1}, we use the identities \begin{equation}\labelr\label{en5} \int_{\Omega}((\chi{+}\rho^*(1{-}\chi))\varphi_R(p))_t p \,\mathrm{d} x &=& \frac{\,\mathrm{d}}{\,\mathrm{d} t}\int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) V_R(p)\,\mathrm{d} x + \int_{\Omega}(1-\rho^*)\Phi_R(p)\chi_t \,\mathrm{d} x\,,\qquad\\ \label{en5a} \int_{\Omega}((\chi{+}\rho^*(1{-}\chi)) w)_t p \,\mathrm{d} x &=& \int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) w_t p \,\mathrm{d} x + \int_{\Omega}(1-\rho^*)wp\chi_t \,\mathrm{d} x\,,\qquad\\ \label{en6} \chi_t \left(\gamma_R(p,\theta)\chi_t - L\frac{Q_R(\theta)}{\theta_c}\right) &=& -L\chi_t + (1-\rho^*)(\Phi_R(p)+pw)\chi_t, \end{equation}r which follow from \eqref{cu4}, and we obtain that \begin{equation}\labelr \nonumber &&\frac{\,\mathrm{d}}{\,\mathrm{d} t}\int_{\Omega} \left(c_0\theta + L\chi + (\chi{+}\rho^*(1{-}\chi)) V_R(p)+ \frac{\lambda_M}{2} w^2 + \begin{equation}\labelta\theta_c w\right)\,\mathrm{d} x \\ \label{en1} && \qquad + \int_{\partial\Omega} \left(^{(m)}ega(x)(\theta-\theta^*) + \alpha(x)(p-p^*)p\right)\,\mathrm{d} s(x) \le \int_{\Omega} w_t(H_R(t) - G)\,\mathrm{d} x. \end{equation}r Note that $V_R'(p) = p \varphi_R'(p)$, $V_R(0) = 0$, so that $V_R(p) > 0$ for all $p \ne 0$. Furthermore, $$ \int_{\Omega} w_t H_R(t)\,\mathrm{d} x = H_R(t) \int_{\Omega} w_t \,\mathrm{d} x = 0, $$ so that \eqref{en1} can be written as \begin{equation}\labelr \nonumber &&\frac{\,\mathrm{d}}{\,\mathrm{d} t}\int_{\Omega} \left(c_0\theta + L\chi + (\chi{+}\rho^*(1{-}\chi)) V_R(p)+ \frac{\lambda_M}{2} w^2 + (\begin{equation}\labelta\theta_c+G) w\right )\,\mathrm{d} x \\ \label{en1a} && \qquad + \int_{\partial\Omega} \left(^{(m)}ega(x)(\theta-\theta^*) + \alpha(x)(p-p^*)p\right)\,\mathrm{d} s(x) \le \int_{\Omega} G_t w\,\mathrm{d} x. \end{equation}r By Gronwall's argument and Hypothesis \ref{h1}, we thus have \begin{equation}\label{en0} \mathop{{\rm sup\,ess}\,}_{t\in (0,T)} \int_{\Omega} (\theta + V_R(p) + w^2)\,\mathrm{d} x + \int_0^T\int_{\partial\Omega}(^{(m)}ega(x)\theta + \alpha(x) p^2)\,\mathrm{d} s(x)\,\mathrm{d} t \le C\,. \end{equation} \subsection{The Dafermos estimate}\label{dafe} We denote $\hat\theta = Q_R(\theta) = Q_R(\theta^+)$ and rewrite \eqref{cu5} in the form \begin{equation}\labelgin{align} &\int_{\Omega}\left(c_0\theta_t \zeta + \kappa(\hat\theta) \nabla \theta\cdot\nabla \zeta\right)\,\mathrm{d} x\, - \int_{\Omega}\left(\frac{1}{\rho_L}\mu(p)Q_R(|\nabla p|^2) + \gamma_R(p,\theta)\chi_t^2 + \nu w_t^2\right) \zeta \,\mathrm{d} x \nonumber\\ \label{cu5a} &+\int_{\Omega} \hat\theta\left(\frac{L}{\theta_c}\chi_t + \begin{equation}\labelta w_t\right) \zeta\,\mathrm{d} x \,=\, \int_{\partial\Omega} ^{(m)}ega(x)(\theta^*-\theta)\zeta \,\mathrm{d} s(x), \end{align} for every $\zeta \in W^{1,2}(\Omega)$. We test \eqref{cu5a} by $\zeta = -\hat\theta^{-a}$. This yields the identity \begin{equation}\labelr \nonumber &&\hspace{-12mm}\int_{\Omega}\frac{a\kappa(\hat\theta)}{\hat\theta^{1+a}} |\nabla \hat\theta|^2\,\mathrm{d} x + \int_{\Omega}\hat\theta^{-a}\left(\frac{1}{\rho_L}\mu(p)Q_R(|\nabla p|^2) + \gamma_R(p,\theta)\chi_t^2 + \nu w_t^2\right) \,\mathrm{d} x \\ \label{cu5b} &=&\int_{\Omega} \hat\theta^{1-a}\left(\frac{L}{\theta_c}\chi_t + \begin{equation}\labelta w_t\right) \,\mathrm{d} x +\int_{\partial\Omega} ^{(m)}ega(x)(\theta-\theta^*)\hat\theta^{-a} \,\mathrm{d} s(x) + \frac{c_0}{1-a}\frac{\,\mathrm{d}}{\,\mathrm{d} t}\int_{\Omega}\hat\theta^{1-a}\,\mathrm{d} x.\qquad \end{equation}r By Hypothesis \ref{h1}\,(ii), we have $\frac{\kappa(\hat\theta)}{\hat\theta^{1+a}} \ge c_\kappa$. Furthermore, H\"older's and Young's inequalities give the estimate \begin{equation}\label{de0} \int_{\Omega} \hat\theta^{1-a}\left(|\chi_t| + |w_t|\right) \,\mathrm{d} x \le \frac C\tau \int_{\Omega} \hat\theta^{2-a}\,\mathrm{d} x + \tau \int_{\Omega} \hat\theta^{-a}\left(\chi_t^2 + w_t^2\right) \,\mathrm{d} x \end{equation} for every $\tau > 0$. This and \eqref{en0} yield the estimate \begin{equation}\label{de1} \int_0^T\int_{\Omega}|\nabla\hat\theta(t)|^2\,\mathrm{d} x\,\mathrm{d} t \le C\left(1 + \int_0^T\int_{\Omega} \hat\theta^{2-a}\,\mathrm{d} x\,\mathrm{d} t\right)\,. \end{equation} {}From the Gagliardo-Nirenberg inequality \eqref{gn} with $s=1$, $p=2$, and $N=3$, we obtain that \begin{equation}\label{de1a} |\hat\theta(t)|_q \le C\left(1+ |\nabla\hat\theta(t)|_2^\rho\right), \end{equation} with $\rho = (6/5(1 - (1/q))$, where we used \eqref{en0} once more. In particular, for every $q \le 8/3$, we have by \eqref{de1} and \eqref{de1a} that \begin{equation}\label{de2a} \int_0^T|\hat\theta(t)|_q^{5q/3(q-1)}\,\mathrm{d} t \le C\left(1+ \int_0^T|\nabla\hat\theta(t)|_2^2\,\mathrm{d} t\right) \le C\left(1 + \int_0^T|\hat\theta|_{2-a}^{2-a}\,\mathrm{d} x\,\mathrm{d} t\right)\,. \end{equation} Moreover, using \eqref{de2} first for $q = 2-a$ and then for $q=8/3$, we obtain that \begin{equation}\label{de2} \int_0^T|\hat\theta(t)|_{8/3}^{8/3}\,\mathrm{d} t + \int_0^T|\nabla\hat\theta(t)|_2^2\,\mathrm{d} t \le C, \end{equation} independently of $R$. \subsection{Estimates for the capillary pressure}\label{capi} We choose an even function $b:\mathbb{R} \to (0,\infty)$ such that the functions $b$ and $p \mapsto p b(p)$ are Lipschitz continuous and such that $b'(p) \ge 0$ for $p > 0$. Then, owing to \eqref{ge2a}, $\eta = p b(p)$ is an admissible test function in \eqref{cu1}, and the term under the time derivative has the form \begin{equation}\labelr \nonumber &&\int_{\Omega} ((\chi{+}\rho^*(1{-}\chi)) (\varphi_R(p)+w))_t p b(p) \,\mathrm{d} x \,= \int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) \varphi_R(p)_t p b(p) \,\mathrm{d} x\\ \label{bbe1} && \quad +\,\int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) w_t p b(p) \,\mathrm{d} x + (1-\rho^*)\int_{\Omega} \chi_t \left(\varphi_R(p) + w\right)p b(p) \,\mathrm{d} x. \end{equation}r For $p \in \mathbb{R}$, we put \begin{equation}\labelgin{align*} V_b(p) &\,:=\, \int_0^p \varphi'(\tau)\, \tau\, b(\tau) \,\mathrm{d} \tau,\quad \hat P_{R,b}(p) \,:=\, \int_0^p P'_R(\tau)\, \tau\, b(\tau) \,\mathrm{d}\tau,\\ \Psi_{R,b}(p) &\,:=\, \varphi_R(p) p b(p) - \hat P_{R,b}(p) - V_b(p) - \Phi_R(p) b(p) \,=\, \int_0^p V_R(\tau) \,\tau\, b'(\tau) \,\mathrm{d}\tau\,. \end{align*} Then $V_b(p) > 0$ and $\Psi_{R,b}(p) \ge 0$ for all $p\ne 0$, and \eqref{bbe1} can be rewritten as \begin{equation}\labelr \nonumber &&\int_{\Omega} ((\chi{+}\rho^*(1{-}\chi)) (\varphi_R(p)+w))_t p b(p) \,\mathrm{d} x = \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) (\hat P_{R,b}(p) + V_b(p)) \,\mathrm{d} x\\ \label{bbe2} && \quad +\,\int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) w_t p b(p) \,\mathrm{d} x + (1-\rho^*)\int_{\Omega} \chi_t \left((\Phi_R(p) + wp)b(p) + \Psi_{R,b}(p)\right)\,\mathrm{d} x.\qquad \end{equation}r Owing to \eqref{en6}, we have, with the notation from Subsection \ref{dafe}, that $$ (1-\rho^*)\chi_t (\Phi_R(p) + wp)\, =\, \gamma_R(p,\theta) \chi_t^2 \,+\, \frac{L}{\theta_c} (\theta_c - \hat\theta) \chi_t\, \ge\, \frac12 \gamma_R(p,\theta) \chi_t^2 - C (1+ \hat\theta) $$ with a constant $C>0$ independent of $R$. Similarly, $$ \left|\int_{\Omega} \chi_t \Psi_{R,b}(p)\,\mathrm{d} x \right| \le \frac14 \int_{\Omega} \gamma_R(p,\theta) \chi_t^2 b(p) \,\mathrm{d} x + C \int_{\Omega} \frac{\Psi_{R,b}^2(p)}{b(p)\gamma_R(p,\theta)}\,\mathrm{d} x. $$ We have, by definition, that $\Psi_{R,b}(p) \le V_R(p) b(p)$, hence $$ \frac{\Psi_{R,b}^2(p)}{b(p)\gamma_R(p,\theta)} \,\le\, C\frac{V_{R}^2(p) b(p)}{\gamma_R(p,\theta)}\, \le\, C b(p)\frac{(V(p) + \frac12(p^2 - R^2)^+)^2}{1 + (p^2 - R^2)^+} \,\le\, C p^2 b(p) $$ independently of $R$. We conclude that \begin{equation}\labelr \nonumber &&\int_{\Omega} ((\chi{+}\rho^*(1{-}\chi)) (\varphi_R(p)+w))_t p b(p) \,\mathrm{d} x \ge \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) (\hat P_{R,b}(p) + V_b(p)) \,\mathrm{d} x\\ \label{bbe3} && \quad +\,\frac14 \int_{\Omega} \gamma_R(p,\theta) \chi_t^2 b(p) \,\mathrm{d} x - C \int_{\Omega} \left(1+ |w_t| + |p| + \hat\theta\right) |p| b(p) \,\mathrm{d} x.\qquad \end{equation}r {}From \eqref{cu1}, with $\eta = p b(p)$, we thus obtain, in particular, that \begin{equation}\labelr \nonumber &&\hspace{-10mm} \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) (\hat P_{R,b}(p)+V_b(p)) \,\mathrm{d} x + \int_{\Omega} \mu(p)(p b'(p) + b(p))|\nabla p|^2 \,\mathrm{d} x \\ \label{bbe4} && +\, \int_{\partial\Omega} \alpha(x) (p-p^*) p b(p)\,\mathrm{d} s(x) \le C \int_{\Omega} \left(1{+}|w_t|{+}|p|{+}\hat\theta\right) |p| b(p) \,\mathrm{d} x\,, \end{equation}r with a constant $C>0$ which is independent of both $b$ and $R$. To estimate the right-hand side of \eqref{bbe4}, we first notice that $|H(t)| \le C(1 + \int_{\Omega} |p|\,\mathrm{d} x)$, and from \eqref{cu2} and Hypothesis \ref{h1}\,(viii) we obtain the pointwise bounds \begin{equation}\labelr \label{c3} |w(x,t)| &\le& C\left(1 + \int_0^t(|p(x,{t'})| + \hat\theta(x,{t'}))\,\mathrm{d}{t'} + \int_0^t\int_{\Omega}|p(x',{t'})|\,\mathrm{d} x'\,\mathrm{d}{t'}\right),\\ \label{c3a} |w_t(x,t)| &\le& |w(x,t)|+ C\left(1 + |p(x,t)| + \hat\theta(x,t)+ \int_{\Omega}|p(x',t)|\,\mathrm{d} x' \right)\,. \end{equation}r In particular, for $b(p) \equiv 1$ we have $\Psi_{R,b}(p) = 0$ and $V_b=V,$ and it follows from \eqref{bbe4}--\eqref{c3} that \begin{equation}\labelr\label{esti} \nonumber &&\hspace{-16mm} \int_{\Omega} ((\chi{+}\rho^*(1{-}\chi)) V(p))(x,t)\,\mathrm{d} x + \int_0^t\int_{\Omega} \mu(p)|\nabla p|^2(x,t')\,\mathrm{d} x\,\mathrm{d}{t'} + \int_0^t\int_{\partial\Omega} \alpha(x) |p|^{2}(x,t')\,\mathrm{d} s(x)\,\mathrm{d}{t'}\\ \label{c4} &\le& C \left(1+ \int_0^t\int_{\Omega} \left( \hat\theta |p| + |p|^2\right)\,\mathrm{d} x\,\mathrm{d}{t'}\right). \end{equation}r We have, by Hypothesis \ref{h1}\,(iv), that $V(p) \ge c_{\varphi}(|p|^{1-\delta} - \delta)/(1-\delta)$. The energy estimate \eqref{en0} then yields that \begin{equation}\label{c4a} \int_{\Omega} |p|^{1-\delta}(x,t)\,\mathrm{d} x \le C\,. \end{equation} Moreover, by \eqref{de2}, $\hat\theta$ is bounded in $L^{8/3}(\Omega\times (0,T))$. We thus obtain from \eqref{c4} that \begin{equation}\labelr \nonumber &&\hspace{-16mm} \int_{\Omega} ((\chi{+}\rho^*(1{-}\chi)) V(p))(x,t)\,\mathrm{d} x + \int_0^t\int_{\Omega} \mu(p)|\nabla p|^2\,\mathrm{d} x\,\mathrm{d}{t'} + \int_0^t\int_{\partial\Omega} \alpha(x) |p|^{2}\,\mathrm{d} s(x)\,\mathrm{d}{t'}\\ \label{c5} &\le& C \left(1+ \int_0^t\int_{\Omega} |p|^2\,\mathrm{d} x\,\mathrm{d}{t'}\right)\,. \end{equation}r Furthermore, by Hypothesis \ref{h1}\,(ix), $\Omega$ is connected, and $\int_{\partial\Omega}\alpha(x)\,\mathrm{d} s(x)>0$. This implies that there exists a constant $C_\Omega>0$, which depends only on $\Omega$, such that, a.~e. in $(0,T)$, \begin{equation}\label{c6} C_\Omega\,\|p\|_{W^{1,2}(\Omega)}^2\,\le\,\int_{\partial\Omega} \alpha(x) |p|^{2}\,\mathrm{d} s(x) + \int_{\Omega} \mu(p)|\nabla p|^2\,\mathrm{d} x\,. \end{equation} Moreover, we infer from H\"older's inequality that \begin{equation}\label{c6a} \int_{\Omega} |p|^2\,\mathrm{d} x = \int_{\Omega} |p|^{(1-\delta)/2}|p|^{(3+\delta)/2}\,\mathrm{d} x \le \left(\int_{\Omega} |p|^{1-\delta}\,\mathrm{d} x\right)^{1/2} \left(\int_{\Omega} |p|^{3+\delta}\,\mathrm{d} x\right)^{1/2}\,. \end{equation} Hence, by \eqref{c4a}, \begin{equation}\labelgin{align} \label{c6b} &\int_0^t\int_{\Omega} |p|^2\,\mathrm{d} x\,\mathrm{d}{t'} \le C\int_0^t\left(\int_{\Omega} |p|^{3+\delta}\,\mathrm{d} x\right)^{1/2}\,\mathrm{d}{t'} \le C\left(\int_0^t\left(\int_{\Omega} |p|^{3+\delta}\,\mathrm{d} x\right)^{2/(3+\delta)}\,\mathrm{d}{t'} \right)^{(3+\delta)/4} \nonumber \\ &\quad\le\,C\left(\int_0^t |p(t')|_{3+\delta}^2\,\mathrm{d} t'\right)^{(3+\delta)/4}\,. \end{align} Since $\delta<1$, we have the embedding inequality $$ |p(t)|_{3+\delta}^2 \le C\,\|p(t)\|_{W^{1,2}(\Omega)}^2, $$ so that from \eqref{c6b} it follows that \begin{equation}\label{c7} \int_0^t\int_{\Omega}|p|^2\,\mathrm{d} x\,\mathrm{d} {t'} \,\le\, C\left(\int_0^t\|p({t'})\|_{W^{1,2}(\Omega)}^2\,\mathrm{d} {t'}\right)^{(3+\delta)/4}. \end{equation} Employing Young's inequality, we therefore conclude from \eqref{esti} and \eqref{c6} that \begin{equation}\label{c7a} \|p\|_{L^2(0,T;W^{1,2}(\Omega))} \le C. \end{equation} Moreover, for $(x,t) \in \Omega\times (0,T)$, $q\ge 1$ and $s > 1$, we have $$ |p(x,t)|^q = |p(x,t)|^{(1-\delta)/s} |p(x,t)|^{q-(1-\delta)/s}, $$ whence, by H\"older's inequality with $s' = s/(s-1)$, $$ |p(t)|_q^q = \left(\int_{\Omega}|p(x,t)|^{1-\delta}\,\mathrm{d} x\right)^{1/s} \left(\int_{\Omega}|p(x,t)|^{(q-(1-\delta)/s)s'}\,\mathrm{d} x\right)^{1/s'}. $$ We thus obtain from \eqref{c4a} and \eqref{c7a} that \begin{equation}\label{c8} \int_0^T |p(t)|_q^q \,\mathrm{d} t \le C, \end{equation} provided that $(q-(1-\delta)/s)s' \le 6$ and $s' \ge 3$. In other words, $$ q \le \frac{1-\delta}{s} + \frac{6}{s'} = \frac{5+\delta}{s'} + 1-\delta \le \frac{5+\delta}{3} + 1-\delta, $$ and the maximal admissible value for $q$ in \eqref{c8} is given by \begin{equation}\label{c9} q = \frac{8 - 2\delta}{3}\,. \end{equation} Let now the function $b$ in \eqref{bbe4} be arbitrary. For $p\in \mathbb{R}$, we put $\hat b(p): = \int_0^p \tau\,b(\tau) \,\mathrm{d} \tau$. Then $\hat b$ is convex, and we have the inequality \begin{equation}\label{hatb} \hat b(p) - \hat b(p^*) \le (p-p^*) \hat b'(p) = (p-p^*) p b(p). \end{equation} {}From \eqref{bbe4}, \eqref{de2}, \eqref{c3}, \eqref{c8}, \eqref{c9}, and \eqref{hatb} it follows that there exists a function $h\in L^q(\Omega\times (0,T))$ such that $$ \|h\|_q \le C, $$ with a constant $C>0$ independent of $b$ and $R$, as well as \begin{equation}\labelr \nonumber &&\hspace{-10mm} \frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega} (\chi{+}\rho^*(1{-}\chi))(\hat P_{R,b}(p)+ V_b(p)) \,\mathrm{d} x + c_{\mu} \int_{\Omega} b(p)|\nabla p|^2 \,\mathrm{d} x +\int_{\partial\Omega}\alpha(x)\hat b(p)\,\mathrm{d} s(x)\\ \label{bbe5} &\le& \int_{\partial\Omega}\alpha(x)\hat b(p^*)\,\mathrm{d} s(x) + \int_{\Omega} h |p| b(p) \,\mathrm{d} x. \end{equation}r Integration of \eqref{bbe5} in time, using the fact that $\chi+\rho^*(1-\chi)\ge\rho^*>0$, yields the estimate \begin{equation}\labelgin{align} \nonumber & \int_{\Omega} V_b(p)(x,t) \,\mathrm{d} x + \int_0^T\int_{\Omega} b(p)|\nabla p|^2 \,\mathrm{d} x \,\mathrm{d} {t'} +\int_0^T\int_{\partial\Omega}\alpha(x)\hat b(p)\,\mathrm{d} s(x)\,\mathrm{d} {t'}\\ \label{bbe6} &\le\, C\left(\int_{\Omega} V_b(p)(x,0) \,\mathrm{d} x + \int_0^T\int_{\partial\Omega}\alpha(x)\hat b(p^*)\,\mathrm{d} s(x)\,\mathrm{d}{t'} + \left(\int_0^T\int_{\Omega} (|p| b(p))^{q'} \,\mathrm{d} x \,\mathrm{d} {t'}\right)^{1/q'}\right), \end{align} for all $t \in [0,T]$, with $q' = \frac{q}{q-1} = \frac{8-2\delta}{5-2\delta}$. Now let $k>0$ be given, and let $\{b_n\}_{n \in \mathbb{N}}$ be a sequence of even, smooth, bounded functions which are nondecreasing in $(0,\infty)$ and such that $b_n(p) \nearrow |p|^{2k}$ locally uniformly in $\mathbb{R}$. Then $(|p| b_n(p))^{q'} \nearrow |p|^{(1+2k)q'}$ locally uniformly. From \eqref{c8} we know that $p\in L^q(\Omega\times (0,T))$, where $q$ is given by \eqref{c9}. Hence, the integral on the right-hand side of \eqref{bbe6} is meaningful if $(1+2k)q'\le q$, that is, if $3k \le 1 - \delta$. In particular, thanks to Hypothesis \ref{h1}\,(iv), $k=\delta$ is an admissible choice. We continue by induction. To this end, assume that \begin{equation}\label{bbe7} \int_0^T \int_{\Omega} |p|^{(1+2k)q'} \,\mathrm{d} x \,\mathrm{d} t =: J_k < \infty \end{equation} holds true for some $k \ge \delta$. Using the denotations $$V_{b_n}(p):=\int_0^p\varphi'(\tau)\,\tau\,b_n(\tau)\,\mathrm{d}\tau,\quad \hat b_n(p):=\int_0^p\tau\,b_n(\tau)\,\mathrm{d}\tau \quad \mbox{for $\,n\in\mathbb{N}$}, $$ we can estimate the terms occurring on the right-hand side of \eqref{bbe6} for $n\in\mathbb{N}$ as follows: \begin{equation}\labelrs \int_{\Omega} V_{b_n}(p)(x,0) \,\mathrm{d} x &\le& C|p^0|_\infty^{2k + 1 - \hat\delta}\,,\\ \int_0^T\int_{\partial\Omega}\alpha(x)\hat b_n(p^*)\,\mathrm{d} s(x)\,\mathrm{d}{t'} &\le& C\|p^*\|_{\partial\Omega,\infty}^{2k+2}\,. \end{equation}rs Put $E = \max\{1, |p^0|_\infty, \|p^*\|_{\partial\Omega,\infty}\}$. Then \eqref{bbe6} can for $n\in\mathbb{N}$ be rewritten as \begin{equation}\label{bbe8} \int_{\Omega} V_{b_n}(p)(x,t) \,\mathrm{d} x + \int_0^T\!\!\!\int_{\Omega} b_n(p)|\nabla p|^2 \,\mathrm{d} x \,\mathrm{d} {t'} +\int_0^T\!\!\!\int_{\partial\Omega}\!\!\alpha(x)\hat b_n(p)\,\mathrm{d} s(x)\,\mathrm{d} {t'} \le C\max\{E^{2k+2}, J_k^{1/q'}\}, \end{equation} independently of $k$, $R$, and $n$. By virtue of Fatou's lemma, we can take the limit as $n \to \infty$ to obtain that \eqref{bbe8} holds true for $b(p) = |p|^{2k}$. Using the estimate \begin{equation}\label{bbe9} V_b(p) \ge \frac{c_{\varphi}}{2k+1-\delta}\left(|p|^{2k + 1 -\delta} - \frac{1+\delta}{2k+2}\right), \end{equation} we thus have shown that \begin{equation}\labelr \nonumber &&\hspace{-12mm}\frac{1}{2k+1}\int_{\Omega} |p|^{2k + 1 -\delta}(x,t) \,\mathrm{d} x + \int_0^T\int_{\Omega} |p|^{2k} |\nabla p|^2 \,\mathrm{d} x \,\mathrm{d} {t'} +\frac{1}{2k+2}\int_0^T\int_{\partial\Omega}\alpha(x)|p|^{2k+2}\,\mathrm{d} s(x)\,\mathrm{d} {t'}\\ \label{bbe10} &\le& C\max\{E^{2k+2}, J_k^{1/q'}\}. \end{equation}r \subsection{Moser iterations} \label{mose} We first recall a technical lemma proved in \cite[Lemma~3.1]{kg}. \begin{equation}\labelgin{lemma}\label{l1} Let $\Omega \subset \mathbb{R}^N$ be a bounded Lipschitzian domain, $N\ge 2$. Moreover, let $q_0 = (N+2)/2$, $q_0' = (N+2)/N$, and suppose that the real numbers $s, r$ satisfy the inequalities \begin{equation}\label{m1} \frac12 \le s \le r \le \frac{N+2s}{N+2} \le 1\,. \end{equation} Furthermore, assume that a function $v \in L^2(0,T;W^{1,2}(\Omega))$ satisfies for a.~e. $t \in (0,T)$ the inequality \begin{equation}\label{m2} |v(t)|_{2s}^{2s} + \int_0^T |v({t'})|^2_{W^{1,2}(\Omega)}\,\mathrm{d} {t'} \le A \max\left\{B, \|v\|_{2rq'}\right\}^{2r}, \end{equation} for some $q' < q_0'$, $A\ge 1$, and $B\ge 1$. Then there exists a constant $C\ge 1$, which is independent of the choice of $v$, $B$, and $A$, such that \begin{equation}\label{m3} \|v\|_{2rq_0'} \le C A^{1/(2r)} \max \left\{B,\|v\|_{2rq'}\right\}. \end{equation} \end{lemma} We now apply Lemma \ref{l1} to the inequality \eqref{bbe10} with $q$ given by \eqref{c9}. Put $v_k := p |p|^{k}$. Then \eqref{bbe10} can be rewritten, using H\"older's inequality, as \begin{equation}\label{m12} |v_k(t)|_{2s}^{2s} + \int_0^T |v_k({t'})|^2_{W^{1,2}(\Omega)}\,\mathrm{d} {t'} \le (k+1)^2 A \max\left\{E^{k+1}, \|v_k\|_{2rq'}\right\}^{2r} \end{equation} with $$ 2s = \frac{2k + 1 - \delta}{k+1}\,, \quad 2r = \frac{2k + 1}{k+1}, \quad q' = \frac{q}{q-1}\,, $$ and with a constant $A\ge 1$ depending only on the initial and boundary data. We see that the hypothesis \eqref{m1} of Lemma \ref{l1} is fulfilled whenever $k \ge \delta$. The assertion of Lemma \ref{l1} then ensures that \begin{equation}\label{m14} \|v_k\|_{2rq_0'} \le C ((k+1)^2 A)^{1/(2r)} \max \left\{E^{k+1},\|v_k\|_{2rq'}\right\}, \end{equation} which entails that \begin{equation}\label{m15} \max\left\{E,\|p\|_{(2k+1)q_0'}\right\} \le C^{1/(k+1)} ((k+1)^2 A)^{1/(2k+1)}\max\left\{E,\|p\|_{(2k+1)q'}\right\}. \end{equation} By induction, we check that the choice $b(p) = |p|^{2k}$ is justified for every $k \ge \delta$. Moreover, we set $\widetilde \nu := (q_0'/q') - 1 > 0$ and define the sequence $\{k_j\}_{j\ge 0}$ by the formula \begin{equation}\label{m16} 2k_j + 1 = (2\delta +1)(1+\widetilde\nu)^j. \end{equation} Set $D_j := \max\left\{E,\|p\|_{(2k_j+1)q_0'}\right\}$. Then \eqref{m15} takes the form \begin{equation}\label{m17} D_j \le C^{1/(k_j+1)} ((k_j+1)^2 A)^{1/(2k_j+1)} D_{j-1} \ \hbox{\ for}\ \ j\in \mathbb{N}, \end{equation} and therefore, \begin{equation}\label{m18} \log D_j - \log D_{j-1} \le \frac{1}{k_j+1} \log C + \frac{1}{2k_j+1} \log((k_j+1)^2 A). \end{equation} We have $k_0 = \delta$ and $D_0 \le C$, by \eqref{c8}--\eqref{c9} and the condition $\delta<1/4$ in Hypothesis \ref{h1}\,(iv). The series on the right-hand side of \eqref{m18} is convergent, and we thus have $$ D_j \le D_0 \prod_{j=1}^\infty C^{1/(k_j+1)^2}((k_j+1)^2 A)^{1/(2k_j+1)} \le C^* $$ with a constant $C^*$ independent of $j$, which enables us to conclude that \begin{equation}\label{m19} \|p\|_\infty \le C^*. \end{equation} \subsection{Higher order estimates for the capillary pressure}\label{high} We aim at taking the limit as $R\nearrow \infty$ in \eqref{cu1}--\eqref{le7a}. Hence, we can restrict ourselves to parameter values $R > C^*$ with $C^*$ from \eqref{m19} and rewrite \eqref{cu1}--\eqref{le7a} in the form \begin{equation}\labelr\label{hu1} \int_{\Omega}\left(((\chi{+}\rho^*(1{-}\chi)) (\varphi(p)+w))_t \eta +\frac{1}{\rho_L}\mu(p)\nabla p\cdot\nabla \eta\right)\,\mathrm{d} x &=& \int_{\partial\Omega} \alpha(x)(p^*-p)\eta \,\mathrm{d} s(x),\qquad \\ \label{hu2} \nu w_t + \lambda_M w - p(\chi + \rho^*(1{-}\chi)) - \begin{equation}\labelta(\hat\theta-\theta_c) &=& - G + H_R(t)\quad a.~e.,\\ \label{hu4} \gamma(\hat\theta) \chi_t + \partial I(\chi) - (1-\rho^*)(\Phi(p)+pw) &\ni& L\left(\frac{\hat\theta}{\theta_c}-1\right)\quad a.~e., \\ \nonumber \int_{\Omega}\left(c_0\theta_t \zeta + \kappa(\hat\theta) \nabla \theta\cdot\nabla \zeta\right)\,\mathrm{d} x + \int_{\partial\Omega} ^{(m)}ega(x)(\theta-\theta^*)\zeta \,\mathrm{d} s(x) &=& \frac{1}{\rho_L} \int_{\Omega} \mu(p)Q_R(|\nabla p|^2)\zeta\,\mathrm{d} x\\ \label{hu5} &&\hspace{-110mm} +\,\int_{\Omega}\Big(\chi_t\big((1{-}\rho^*)(\Phi(p) + pw)-L\big) + w_t((\chi{+}\rho^*(1{-}\chi)) p {-} \lambda_M w {-} \begin{equation}\labelta\theta_c {-} G {+} H_R(t))\Big)\zeta\,\mathrm{d} x\quad \end{equation}r for every test functions $\eta,\zeta \in W^{1,2}(\Omega)$, with $\hat\theta = Q_R(\theta)$ and \begin{equation}\label{hu6} H_R(t) = - \frac{1}{|\Omega|}\int_{\Omega} (p (\chi{+}\rho^*(1{-}\chi)) + \begin{equation}\labelta(\hat\theta -\theta_c) -G)(x,t)\,\mathrm{d} x\,. \end{equation} We test \eqref{hu1} by $\eta = \mu(p) p_t$, which is an admissible choice by \eqref{ge2a}. Then \begin{equation}\labelr\nonumber &&\hspace{-12mm}\int_{\Omega} (\chi{+}\rho^*(1{-}\chi)) \varphi'(p)\mu(p)|p_t|^2 \,\mathrm{d} x +\frac{1}{2\rho_L}\frac{\,\mathrm{d}}{\,\mathrm{d} t} \int_{\Omega}\mu^2(p)|\nabla p|^2\,\mathrm{d} x + \int_{\partial\Omega} \alpha(x)(p-p^*)\mu(p) p_t \,\mathrm{d} s(x)\\ \label{he1} &=& \int_{\Omega} \left((1-\rho^*)\chi_t w + (\chi{+}\rho^*(1{-}\chi)) w_t\right)\mu(p) p_t\,\mathrm{d} x. \end{equation}r Note that by Hypothesis \ref{h1}\,(iv) and \eqref{m19}, we have $$ \varphi'(p) \ge \frac{c_{\varphi}}{\max\{1, C^*\}^{1+\delta}}. $$ We set \begin{equation}\label{mu} \hat\mu(p) = \int_0^p \tau \mu(\tau) \,\mathrm{d} \tau, \quad M(p) = \int_0^p \mu(\tau)\,\mathrm{d}\tau, \end{equation} and integrate \eqref{he1} in time to obtain the estimate \begin{equation}\labelr\nonumber &&\hspace{-12mm}\int_0^{t}\int_{\Omega} |p_t|^2 \,\mathrm{d} x\,\mathrm{d} {t'} + \int_{\Omega}|\nabla p|^2(x,t)\,\mathrm{d} x + \int_{\partial\Omega} \alpha(x)\hat\mu(p)(x,t) \,\mathrm{d} s(x)\\ \nonumber &\le& C\Bigg({1\,+}\int_{\partial\Omega}\alpha(x)M(p)|p^*|(x,t) \,\mathrm{d} s(x) + \int_0^{t} \int_{\partial\Omega}\alpha(x)M(p)|p^*_t|(x,t') \,\mathrm{d} s(x)\,\mathrm{d} t'\\ \nonumber &&+\, \int_0^{t}\int_{\Omega} (|\chi_t w| + |w_t|)|p_t|\,\mathrm{d} x \,\mathrm{d}{t'}\Bigg)\\ \label{he2} &\le& C\left(1 + \int_0^{t}\int_{\Omega} (|\chi_t w| + |w_t|)|p_t|\,\mathrm{d} x \,\mathrm{d}{t'}\right) \end{equation}r for all $t \in [0,T]$, whence we infer that \begin{equation}\labelr\nonumber &&\hspace{-12mm}\int_0^{t}\int_{\Omega} |p_t|^2 \,\mathrm{d} x\,\mathrm{d} {t'} + \int_{\Omega}|\nabla p|^2(x,t)\,\mathrm{d} x + \int_{\partial\Omega} \alpha(x)\hat\mu(p)(x,t) \,\mathrm{d} s(x)\\ \label{he3} &\le& C\left(1 + \int_0^{t}\int_{\Omega} (|\chi_t w|^2 + |w_t|^2)\,\mathrm{d} x \,\mathrm{d}{t'}\right). \end{equation}r By virtue of \eqref{c3}--\eqref{c3a}, \eqref{hu4}, and \eqref{m19}, we have the pointwise bounds \begin{equation}\labelr \label{he4} |w(x,t)| &\le& C \left(1+ \int_0^t \hat\theta(x,{t'})\,\mathrm{d}{t'}\right),\\ \label{he5} |\chi_t(x,t)| &\le& C(1+ |w(x,t)|) \,\le\, C \left(1+ \int_0^t \hat\theta(x,{t'})\,\mathrm{d}{t'}\right),\\ \label{he6} |w_t(x,t)| &\le& C \left(1+ \hat\theta(x,t) + \int_0^t \hat\theta(x,{t'})\,\mathrm{d}{t'}\right). \end{equation}r By \eqref{de2} and the Sobolev embedding theorem, we know that $\hat\theta$ is bounded in $L^{8/3}(\Omega\times (0,T)) \cap L^2(0,T; L^6(\Omega))$. Let us recall again the Minkowski inequality $$ \left(\int_{\Omega} \left(\int_0^t \hat\theta (x,{t'}) \,\mathrm{d}{t'}\right)^6\,\mathrm{d} x\right)^{1/6} \le \int_0^t \left(\int_{\Omega} \hat\theta^6(x,{t'})\,\mathrm{d} x\right)^{1/6} \,\mathrm{d}{t'}\,, $$ which implies that \begin{equation}\labelr \label{he4a} \int_{\Omega} \left(|w(x,t)|^6 + |\chi_t(x,t)|^6\right) \,\mathrm{d} x &\le& C \quad \mbox{ for a.\,e. } t \in (0,T),\\ \label{he5a} \|w_t\|_{8/3} &\le& C. \end{equation}r Hence, the right-hand side of \eqref{he3} is bounded independently of $R$, and we have for all $t \in [0,T]$ that \begin{equation}\label{he7} \int_0^t\int_{\Omega} |p_t|^2 \,\mathrm{d} x\,\mathrm{d} {t'} + \int_{\Omega}|\nabla p|^2(x,t)\,\mathrm{d} x + \int_{\partial\Omega} \alpha(x)\hat\mu(p)(x,t) \,\mathrm{d} s(x) \le C\,. \end{equation} Now let $M(p)$ be as in \eqref{mu}. By \eqref{he4a}--\eqref{he7}, and by comparison in \eqref{hu1}, the term $\Delta M(p)$ is bounded in $L^{2}(\Omega\times (0,T))$, independently of $R$. In terms of the new variable $\tilde p = M(p)$, the boundary condition \eqref{be2} is nonlinear, and the $W^{2,2}$-regularity of $ M(p)$ follows from considerations similar to those used in the proof of \cite[Theorem 4.1]{kpa}, inspired by \cite{jn}. We thus may employ the Gagliardo-Nirenberg inequality \eqref{gnm} in the form \begin{equation}\label{gnm} |\nabla M(p)(t)|_q \le C\left(|\nabla M(p)(t)|_2+|\nabla M(p)(t)|_2^{1-\rho}|\Delta M(p)(t)|_2^\rho\right) \end{equation} with $\rho = 3(\frac12 - \frac{1}{q})$. Together with \eqref{he7}, we conclude that \begin{equation}\label{he8a} \int_0^T|\nabla p(t)|_q^s \,\mathrm{d} t \le C \quad \hbox{\ for}\ q \in (2,6] \ \mbox{ and }\ \frac{1}{q} + \frac{2}{3s} = \frac12. \end{equation} In particular, for $s=4$ and $s=q$ we obtain, respectively, \begin{equation}\label{he8} \int_0^T|\nabla p(t)|_3^4 \,\mathrm{d} t \le C\,, \quad \|\nabla p\|_{10/3} \le C. \end{equation} \subsection{Higher order estimates for the temperature}\label{temp} The previous estimates \eqref{he4a}--\eqref{he5a} and \eqref{he8} entail that \eqref{hu5} has the form \begin{equation}\label{te1} \int_{\Omega}\left(c_0\theta_t \zeta + \kappa(\hat\theta) \nabla \theta\cdot\nabla \zeta\right)\,\mathrm{d} x + \int_{\partial\Omega} ^{(m)}ega(x)(\theta-\theta^*)\zeta \,\mathrm{d} s(x) = \int_{\Omega} \tilde F \zeta\,\mathrm{d} x \end{equation} for every $\zeta \in W^{1,2}(\Omega)$, with a function $\tilde F$ such that \begin{equation}\label{te2} \|\tilde F\|_{5/3} \le C\,, \quad \int_0^T|\tilde F(t)|_{3/2}^2 \,\mathrm{d} t \le C\,, \end{equation} independently of $R$. Assume now that for some $p_0 \ge 8/3$ we have \begin{equation}\label{te3} \|\hat\theta\|_{p_0} \le C. \end{equation} We know that this is true for $p_0 = 8/3$ by virtue of \eqref{de2}. Set $r_0 = 2p_0/5$. Then we may put $\zeta = \hat\theta^{r_0}$ in \eqref{te1} and obtain, using Hypothesis \ref{h1}\,(ii), that \begin{equation}\label{te4} \frac{1}{r_0+1}\int_{\Omega} \hat\theta^{r_0 + 1}(x,t)\,\mathrm{d} x + r_0 \int_0^t\int_{\Omega} \hat\theta^{r_0 + a} |\nabla\hat\theta|^2\,\mathrm{d} x\,\mathrm{d}{t'} \le C\,. \end{equation} We now denote $$ v = \hat\theta^p\,, \quad p = 1 + \frac{r_0+a}{2}\,, \quad s = \frac{r_0 + 1}{p}\,, $$ and rewrite \eqref{te4} as \begin{equation}\label{te5} \int_{\Omega} |v|^{s}(x,t)\,\mathrm{d} x + \int_0^t\int_{\Omega} |\nabla v|^2\,\mathrm{d} x\,\mathrm{d}{t'} \le C(r_0 + 1)\,. \end{equation} By the Gagliardo-Nirenberg inequality \eqref{gn}, we have $\|v\|_q \le C(r_0 + 1)$ for $q = 2 + \frac{2s}{3}$. Hence, \begin{equation}\label{te6} \|\hat\theta\|_{p_1} \le C(r_0 + 1) \quad \hbox{\ for}\ \ p_1 = pq = \frac{2p_0}{3} + \frac83 + a\,. \end{equation} We now proceed by induction according to the recipe $ p_{j+1} = \frac{2p_j}{3} + \frac83 + a\,,\ r_j = \frac{2p_j}{5}\,. $ We have $\lim_{j\to \infty} p_j = 8 + 3a$. After finitely many steps, we may stop the algorithm and put $\bar p := p_j < 8 + 3a$ with \begin{equation}\label{te7} \|\hat\theta\|_{\bar p} + \mathop{{\rm sup\,ess}\,} |\hat\theta(t)|_{\bar r+ 1} \le C\,,\quad \bar r = \frac{2\bar p}{5} > \hat a, \end{equation} with the constant $\hat a$ introduced in Hypothesis \ref{h1}\,(ii). By Proposition \ref{t2}, we may test \eqref{te1} by $\theta$, which yields \begin{equation}\label{te8} \int_{\Omega} \theta^{2}(x,t)\,\mathrm{d} x + \int_0^t\int_{\Omega} \kappa(\hat\theta) |\nabla\theta|^2\,\mathrm{d} x\,\mathrm{d}{t'} + \int_0^t\int_{\partial\Omega} ^{(m)}ega(x) \theta^2 \,\mathrm{d} s(x)\,\mathrm{d}{t'} \le C \|\theta\|_{5/2}\,. \end{equation} Using the Gagliardo-Nirenberg inequality again, for instance, we conclude that \begin{equation}\label{te9} \int_{\Omega} \theta^{2}(x,t)\,\mathrm{d} x + \int_0^t\int_{\Omega} \kappa(\hat\theta) |\nabla\theta|^2\,\mathrm{d} x\,\mathrm{d}{t'} + \int_0^t\int_{\partial\Omega} ^{(m)}ega(x) \theta^2 \,\mathrm{d} s(x)\,\mathrm{d}{t'} \le C\,. \end{equation} This enables us to derive an upper bound for the integral $\int_{\Omega} \kappa(\hat\theta) \nabla\theta\cdot\nabla\zeta \,\mathrm{d} x$, which we need for getting an estimate for $\theta_t$ from the equation \eqref{te9}. We have, by H\"older's inequality and Hypothesis \ref{h1}\,(ii), that \begin{equation}\labelr \nonumber \int_{\Omega} |\kappa(\hat\theta) \nabla\theta\cdot\nabla\zeta| \,\mathrm{d} x &=& \int_{\Omega} |\kappa^{1/2}(\hat\theta) \nabla\theta\cdot\kappa^{1/2}(\hat\theta)\nabla\zeta| \,\mathrm{d} x\\ \label{e715} &\le& C\left(\int_{\Omega}\kappa(\hat\theta) |\nabla\theta|^2\,\mathrm{d} x\right)^{1/2} \left(\int_{\Omega} \hat\theta^{1+\hat a}|\nabla\zeta|^2\,\mathrm{d} x\right)^{1/2}. \end{equation}r We now choose $\hat q > 1$ such that $(1+\hat a)\hat q = 1+\bar r$, where $\bar r$ is defined in \eqref{te7}. Choosing now \begin{equation}\label{qstar} q^* = \frac{2 \hat q}{\hat q - 1}, \end{equation} we obtain from H\"older's inequality that \begin{equation}\label{e714} \int_{\Omega} \hat\theta^{1+\hat a}|\nabla\zeta|^2\,\mathrm{d} x \le \left(\int_{\Omega} \hat\theta^{1+\bar r} \,\mathrm{d} x\right)^{1/\hat q} \left(\int_{\Omega} |\nabla\zeta|^{q^*}\,\mathrm{d} x\right)^{2/q^*} \le C \left(\int_{\Omega} |\nabla\zeta|^{q^*}\,\mathrm{d} x\right)^{2/q^*}\,, \end{equation} by virtue of \eqref{te7}. Eq.~\eqref{e715} then yields the bound \begin{equation}\label{e716} \int_{\Omega} |\kappa(\hat\theta) \nabla\theta\cdot\nabla\zeta| \,\mathrm{d} x \le C\left(\int_{\Omega}\kappa(\hat\theta) |\nabla\theta|^2\,\mathrm{d} x\right)^{1/2} \left(\int_{\Omega} |\nabla\zeta|^{q^*}\,\mathrm{d} x\right)^{1/q^*}. \end{equation} Hence, by \eqref{te9}, \begin{equation}\label{e717} \int_0^T\int_{\Omega} |\kappa(\hat\theta) \nabla\theta\cdot\nabla\zeta| \,\mathrm{d} x \,\mathrm{d} t \le C \|\zeta\|_{L^2(0,T;W^{1,q^*}(\Omega))}\,. \end{equation} {}From \eqref{te2} it follows that testing with $\zeta \in L^2(0,T;W^{1,q^*}(\Omega))$ is admissible. We thus obtain from \eqref{te1} that \begin{equation}\label{e718} \int_0^T\int_{\Omega} \theta_t \zeta \,\mathrm{d} x \,\mathrm{d} t \le C \|\zeta\|_{L^2(0,T;W^{1,q^*}(\Omega))}\,. \end{equation} \section{Proof of Theorem \ref{t1}} \label{proo} Let $R_i \nearrow \infty$ be a sequence such that $R_1 > C^*$ with $C^*$ as in \eqref{m19}, and let $(p,w,\chi,\theta) = (p^{(i)}, w^{(i)}, \chi^{(i)}, \theta^{(i)})$ be solutions of \eqref{hu1}--\eqref{hu6} corresponding to $R = R_i$, with $\hat\theta = \hat\theta^{(i)} = Q_{R_i}(\theta^{(i)})$ and test functions $\eta,\zeta \in W^{1,2}(\Omega)$. Our aim is to check that at least a subsequence converges as $i \to \infty$ to a solution of \eqref{wu1}--\eqref{wu5}, \eqref{le7}, with test functions $\eta\in W^{1,2}(\Omega)$, $\zeta \in W^{1,q^*}(\Omega)$ with $q^*$ as in Theorem \ref{t1}. First, for the capillary pressure $p = p^{(i)}$ we have the estimates \eqref{m19}, \eqref{he7}, \eqref{he8}, which imply that, passing to a subsequence if necessary, \begin{equation}\labelrs p^{(i)} \to p && \mbox{strongly in } L^r(\Omega\times (0,T)) \ \mbox{ for every } \ r\ge 1\,,\\ p_t^{(i)} \to p_t && \mbox{weakly in } L^2(\Omega\times (0,T))\,,\\ \nabla p^{(i)} \to \nabla p && \mbox{strongly in } L^r(\Omega\times (0,T)) \ \mbox{ for every } \ 1 \le r < \frac{10}{3}\,. \end{equation}rs We easily show that \begin{equation}\label{pr2} Q_{R_i}\bigl(|\nabla p^{(i)}|^2\bigr) \to |\nabla p|^2 \ \mbox{ strongly in } L^r(\Omega\times (0,T)) \ \mbox{ for every } \ 1 \le r < \frac{5}{3}\,. \end{equation} Indeed, let $\Omega^{(i)}_T \subset \Omega\times (0,T)$ be the set of all $(x,t) \in \Omega\times (0,T)$ such that $|\nabla p^{(i)}(x,t)|^2 > R_i$. By \eqref{he8}, we have $$ C \ge \int_0^T\int_{\Omega} |\nabla p^{(i)}(x,t)|^{10/3} \,\mathrm{d} x\,\mathrm{d} t \ge \iint_{\Omega^{(i)}_T}|\nabla p^{(i)}(x,t)|^{10/3} \,\mathrm{d} x\,\mathrm{d} t \ge |\Omega^{(i)}_T| R_i^{5/3}\,, $$ hence $|\Omega^{(i)}_T| \le C R_i^{-5/3}$. For $r < \frac{5}{3}$, we use H\"older's inequality to get the estimate \begin{equation}\labelrs &&\hspace{-12mm}\int_0^T\int_{\Omega} \left|Q_{R_i}(|\nabla p^{(i)}|^2) - |\nabla p^{(i)}|^2\right|^r\,\mathrm{d} x\,\mathrm{d} t = \iint_{\Omega^{(i)}_T}\left|R_i - |\nabla p^{(i)}|^2\right|^r\,\mathrm{d} x\,\mathrm{d} t \le \iint_{\Omega^{(i)}_T} |\nabla p^{(i)}|^{2r}\,\mathrm{d} x\,\mathrm{d} t \\ &\le& \left(\iint_{\Omega^{(i)}_T} |\nabla p^{(i)}|^{10/3}\,\mathrm{d} x\,\mathrm{d} t\right)^{3r/5} |\Omega^{(i)}_T|^{1-(3r/5)}\,, \end{equation}rs and \eqref{pr2} follows. For the temperature $\theta = \theta^{(i)}$, we proceed in a similar way. From the compactness result in \cite[Theorem 5.1]{li}, it follows that, for a subsequence, $$ \theta^{(i)} \to \theta \ \mbox{ strongly in } L^2(\Omega\times (0,T)). $$ Furthermore, by \eqref{te7}, $\hat\theta^{(i)}$ are uniformly bounded in $L^{r}(\Omega\times (0,T))$ for every $r < 8+3a$. A similar argument as above yields that $$ \hat\theta^{(i)} \to \theta \ \mbox{ strongly in } L^r(\Omega\times (0,T)) \ \mbox{ for every } \ 1 \le r < 8+3a\,. $$ Indeed, by \eqref{te9} and \eqref{e718}, \begin{equation}\labelrs \theta_t^{(i)} \to \theta_t && \mbox{weakly in } L^2(0,T; W^{-1,q^*}(\Omega))\,,\\ \nabla \theta^{(i)} \to \nabla \theta && \mbox{weakly in } L^2(\Omega\times (0,T))\,. \end{equation}rs The strong convergences of $w^{(i)} \to w$, $w^{(i)}_t \to w_t$, $\chi^{(i)} \to \chi$, $\chi^{(i)}_t \to \chi_t$ are handled using the estimates \eqref{he4}--\eqref{he6} similarly as in the proof of Proposition \ref{t2} at the end of Section \ref{cut}. 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\begin{document} \title[One group of inequalities]{One group of inequalities with altitudes and medians in triangle} \author{Zhivko Zhelev} \address{Zhivko Zhelev\newline \mathcal{H}space*{5mm}Department of Mathematics and Informatics,\newline \mathcal{H}space*{5mm}Trakia University, Rektorat, AF, 367A,\newline \mathcal{H}space*{5mm}6003 St. Zagora,\newline \mathcal{H}space*{5mm}Bulgaria.\newline \mathcal{H}space*{5mm}Email: \tt{zhelev@uni-sz.bg}} \date{} \begin{abstract} In the article we prove some inequalities that contain relations between altitudes and medians in triangle. At least one of these inequalities has not been considered in the literature before and the main theorem has also not been proved elsewhere in that form.\\ \indent Some immediate corollaries have been presented as well. \end{abstract} \keywords{medians, altitudes, inequality. \newline \mathcal{H}space*{5mm}2000 {\it Mathematics Subject Classification.} 51M04, 51M05, 51M16.} \maketitle \indent Geometry of the triangle is a realm of elementary geometry where interested new results pop up all the time. There are plenty of theorems concerning geometry of the triangle, including hundreds of geometric inequalities (see for example \cite{Bottema}). The following result consists of two inequalities which look quite pleasant but the second one turned out to be extremely difficult to tackle. In fact we prove the following \begin{thm} Let $\triangle ABC$ be an arbitrary triangle with sides $a$, $b$ and $c$. Let also $h_a, h_b, h_c$ and $m_a, m_b, m_c$ be the altitudes and medians to these sides respectively. Then the following two inequalities hold: \begin{eqnarray} ah_a+bh_b+ch_c\le\mathcal{S}qrt{bc}h_a+\mathcal{S}qrt{ac}h_b+\mathcal{S}qrt{ab}h_c\\ am_a+bm_b+cm_c\le\mathcal{S}qrt{bc}m_a+\mathcal{S}qrt{ac}m_b+\mathcal{S}qrt{ab}m_c . \end{eqnarray} \end{thm} \indent As far as the author knows, the second inequality has not been proved yet. But different random checks of values of the triangle sides have not brought any counterexamples.\\ \indent Our proof is based on one and two variable functions theory which we consider an impediment since the problem formulated above is in the area of the elementary mathematics.\\ \indent First we formulate some lemmas which are more or less obvious. \begin{Lemma}[{\bf{Arithmetic-Geometric-Mean Inequality}}] Let $a_1, a_2,\ldots a_n\in\mathbb{R}$ and $a_i\ge 0$, $i=1,\ldots ,n$. Then the following inequality is true: \begin{equation} \dfrac{a_1+a_2+\cdots + a_n}{n}\ge \root n\of{a_1a_1\cdots a_n} . \end{equation} \end{Lemma} \underline{\bf{Proof.}} See [4, p. 18-19] and \cite{Nelson} for a detailed proof.$\mathcal{H}fill\mathcal{S}quare$ \begin{Lemma} If $a+b+c>0$, then $a^3+b^3+c^3\ge 3abc$. \end{Lemma} \underline{\bf{Proof.}} From Lemma 1 ($n=3$), we get that $a_1+a_2+a_3\ge 3\root 3\of{a_1a_2a_3}$. Replacing $a_1$ with $a^3$, $a_2$ with $b^3$ and $a_3$ with $c^3$, we get that $a^3+b^3+c^3\ge 3\root 3\of{a^3b^3c^3}=3abc$., q. e. d.$\mathcal{H}fill\mathcal{S}quare$\\ \underline{\bf{Proof of the main theorem.}} \\ In order to prove (1) we use that $ah_a=bh_b=ch_c=2S_{\triangle ABC}$. Then $$ \begin{array}{c} ah_a+bh_b+ch_c-\mathcal{S}qrt{bc}h_a-\mathcal{S}qrt{ac}h_b-\mathcal{S}qrt{ab}h_c= 6S-\dfrac{2S\mathcal{S}qrt{bc}}{a}-\dfrac{2S\mathcal{S}qrt{ac}}{b}-\dfrac{2S\mathcal{S}qrt{ab}}{c}=\\ \\ 2S\left (3-\dfrac{\mathcal{S}qrt{bc}}{a}-\dfrac{\mathcal{S}qrt{ac}}{b}-\dfrac{\mathcal{S}qrt{ab}}{c}\right ). \end{array} $$ Therefore, $$ \begin{array}{c} ah_a+bh_b+ch_c\le\mathcal{S}qrt{bc}h_a+\mathcal{S}qrt{ac}h_b+\mathcal{S}qrt{ab}h_c\iff\left (3-\dfrac{\mathcal{S}qrt{bc}}{a}-\dfrac{\mathcal{S}qrt{ac}}{b}-\dfrac{\mathcal{S}qrt{ab}}{c}\right )\le 0\\ \\ \iff 3abc\le (bc)^{\frac{3}{2}}+(ac)^{\frac{3}{2}}+(ab)^{\frac{3}{2}},\,\,a>0,\,\, b>0,\,\, c>0. \end{array} $$ \indent By the substitution $x=\mathcal{S}qrt{bc}$, $y=\mathcal{S}qrt{ac}$, $z=\mathcal{S}qrt{ab}$, we get that $$ 3abc\le (bc)^{\frac{3}{2}}+(ac)^{\frac{3}{2}}+(ab)^{\frac{3}{2}}\iff 3xyz\le x^3+y^3+z^3,\,\,x>0,\,\, y>0,\,\, z>0. $$ \indent Last inequality is exactly Lemma 2 and the proof of (1) is completed.\\ \indent Now, without loss of generality, we can assume for $\triangle ABC$, that $a\ge b\ge c$. Three main cases are possible: \begin{itemize} \item[$\dagger )$] $\triangle ABC$ is equilateral, i. e. $a=b=c$; \item[$\dagger\dagger )$] $\triangle ABC$ is isosceles, i. e. $a=b>c$; \item[$\dagger\dagger\dagger )$] $\triangle ABC$ is an arbitrary triangle, i. e. $a>b>c$. \end{itemize} \indent First, let's rewrite (2) in the form \begin{equation} (a-\mathcal{S}qrt{bc})m_a+(b-\mathcal{S}qrt{ac})m_b+(c-\mathcal{S}qrt{ab})m_c\le 0. \end{equation} \underline{\bf{First case.}} If $\triangle ABC$ is equilateral, then $a-\mathcal{S}qrt{bc}=b-\mathcal{S}qrt{ac}=c-\mathcal{S}qrt{ab}=0$ and (4) is fulfilled as an equality. We will see below that this case is in fact the extremal case for our problem.\\ \underline{\bf{Second case.}}Now let $\triangle ABC$ be isosceles and we may assume $a=b>c>0$. On the other hand we have that \begin{equation} \begin{array}{c} m_a=\dfrac{1}{2}\mathcal{S}qrt{2b^2+2c^2-a^2}=\dfrac{1}{2}\mathcal{S}qrt{a^2+2c^2}=m_b,\\ m_c=\dfrac{1}{2}\mathcal{S}qrt{2a^2+2b^2-c^2}=\dfrac{1}{2}\mathcal{S}qrt{4a^2-c^2}. \end{array} \end{equation} \indent Then using (5), (4) is transformed into \begin{equation} (a-\mathcal{S}qrt{ac})\mathcal{S}qrt{a^2+2c^2}\le\dfrac{a-c}{2}\mathcal{S}qrt{4a^2-c^2},\quad a>c. \end{equation} We will prove (6). After some tedious computations, we get consequently: $$ \begin{array}{c} (a-\mathcal{S}qrt{ac})\mathcal{S}qrt{a^2+2c^2}\le\dfrac{a-c}{2}\mathcal{S}qrt{4a^2-c^2}\iff\\ \\ \mathcal{S}qrt{a}(\mathcal{S}qrt{a}-\mathcal{S}qrt{c})\mathcal{S}qrt{a^2+2c^2}\le\dfrac{(\mathcal{S}qrt{a}-\mathcal{S}qrt{c})(\mathcal{S}qrt{a}+\mathcal{S}qrt{c})}{2} \mathcal{S}qrt{4a^2-c^2}\iff\\ \\ 2\mathcal{S}qrt{a^3+2ac^2}\le (\mathcal{S}qrt{a}+\mathcal{S}qrt{c})\mathcal{S}qrt{4a^2-c^2}\iff\\ \\ \dfrac{4(a^3+2ac^2)}{4a^2-c^2}\le a+c+2\mathcal{S}qrt{ac}\iff\dfrac{4a^3+8ac^2-4a^3+ac^2-4a^2c+c^3}{4a^2-c^2}\le 2\mathcal{S}qrt{ac}\iff\\ \\ \left (\dfrac{c^3+9ac^2-4a^2c}{4a^2-c^2}\right )^2\le 4ac\iff\\ \\ c^6+81a^2c^4+16a^4c^2+18ac^5-8a^2c^4-72a^3c^3\le 64a^5c+4ac^5-32a^3c^3\iff\\ c^6+14ac^5+73a^2c^4-40a^3c^3+16a^4c^2-64a^5c\le 0\iff\\ c(c^5+14ac^4+73a^2c^3-40a^3c^2+16a^4c-64a^5)\le 0\iff\\ c^5+14ac^4+73a^2c^3-40a^3c^2+16a^4c-64a^5\le 0\iff\\ \\ \left (\dfrac{c}{a}\right )^5+14\left (\dfrac{c}{a}\right )^4+ 73\left (\dfrac{c}{a}\right )^3-40\left (\dfrac{c}{a}\right )^2+16\left (\dfrac{c}{a}\right )-64\le 0. \end{array} $$ \indent In the last expression, let $\dfrac{c}{a}=t\in (0,1)$, and it follows that $$ \begin{array}{c} t^5+14t^4+73t^3-40t^2+16t-64\le 0\iff\\ \\ \underbrace{(t-1)}_{<0}\underbrace{(t^4+15t^3+88t^2+48t+64)}_{>0\,\,\,\mbox{if}\,\,\, t>0}\le 0. \end{array} $$ But the last one is obviously true and that proves (6) in this case.$\mathcal{H}fill\mathcal{S}quare$\\ \underline{\bf{Third case.}} Let now $\triangle ABC$ be an arbitrary triangle and let $a>b>c>0$. We rewrite (4) in the form \begin{equation} \dfrac{1}{2}(a-\mathcal{S}qrt{bc})\mathcal{S}qrt{2b^2+2c^2-a^2}+\dfrac{1}{2}(b-\mathcal{S}qrt{ac}) \mathcal{S}qrt{2a^2+2c^2-b^2}+\dfrac{1}{2}(c-\mathcal{S}qrt{ab})\mathcal{S}qrt{2a^2+2b^2-c^2}\le 0 \end{equation} or \begin{equation} \begin{array}{c} \dfrac{1}{2}a^2\underbrace{\left [\left (1-\mathcal{S}qrt{\dfrac{b}{a}}\mathcal{S}qrt{\dfrac{c}{a}}\right )\mathcal{S}qrt{2\left (\dfrac{b}{a}\right )^2+2\left (\dfrac{c}{a}\right )^2-1}+ \left (\dfrac{b}{a}-\mathcal{S}qrt{\dfrac{c}{a}}\right )\mathcal{S}qrt{2+2\left (\dfrac{c}{a}\right )^2-\left (\dfrac{b}{a}\right )^2}\right ]}_ {F\left (\dfrac{b}{a},\dfrac{c}{a}\right ):=F(x,y)}+\\ \underbrace{\dfrac{1}{2}a^2\left [\left (\dfrac{c}{a}-\mathcal{S}qrt{\dfrac{b}{a}}\right )\mathcal{S}qrt{2+2\left (\dfrac{b}{a}\right )^2-\left (\dfrac{c}{a}\right )^2}\right ]}_{F\left (\dfrac{b}{a},\dfrac{c}{a}\right ):=F(x,y)}\le 0, \end{array} \end{equation} \noindent and therefore $\dfrac{1}{2}a^2F(x,y)\le 0\iff F(x,y)\le 0$.\\ \indent Two variable function $F(x,y)$ defined above, we name \textit{devil-fish function} and the surface this function plots in $\mathbb{R}^3$ -- \textit{devil-fish surface}.\\ \indent In our new notation inequality (4) transforms into the extremal problem \begin{equation} \begin{array}{c} \displaystyle{\max_{(x,y)\in M\mathcal{S}ubset\mathbb{R}^2}F(x,y)},\mbox{where}\\ \\ M=\{(x,y)\in\mathbb{R}^2|\,\,0\le y\le x\le 1,\,\,x+y\ge 1\}, \end{array} \end{equation} \noindent where $M$ is geometrically a right-angle triangle and the last inequality follows from the fact that $a+b>c$ in an arbitrary triangle.\\ \indent It is a straightforward check that the devil-fish function is well-defined on $M$. Therefore, in order to prove (4), we have to prove that $\displaystyle{\max_{(x,y)\in M}F(x,y)}=0$.\\ Note that $M$ is a compact set and since $M$ is a continuous function on $M$, it follows from some classical results in the real mathematical analysis, that $F(x,y)$ reaches its maximum there. In order to find that maximal point we consider the interior and the boundary of $M$ separately, $M=\mbox{int}\, M\cup\partial M$.\\ \indent Using some calculating programs such as \textit{Maple} (computations can be done manually but that can turn into an extremely tedious hardwork), we find that $$ \begin{array}{c} \dfrac{\partial F(x,y)}{\partial x}=-\dfrac{y\mathcal{S}qrt{2x^2+2y^2-1}}{2\mathcal{S}qrt{xy}}+\dfrac{2(1-\mathcal{S}qrt{xy})x}{\mathcal{S}qrt{2x^2+2y^2-1}}+\mathcal{S}qrt{2+2y^2-x^2}-\\ \\ \dfrac{(x-\mathcal{S}qrt{y})x}{2+2y^2-x^2}-\dfrac{2+2x^2-y^2}{2\mathcal{S}qrt{x}}+\dfrac{2(y-\mathcal{S}qrt{x})x}{\mathcal{S}qrt{2+2x^2-y^2}}, \end{array} $$ \noindent and since the devil-fish function is a symmetric function, i. e. $F(x,y)=F(y,x)$, it follows immediately that $$ \begin{array}{c} \dfrac{\partial F(x,y)}{\partial y}=-\dfrac{x\mathcal{S}qrt{2y^2+2x^2-1}}{2\mathcal{S}qrt{xy}}+\dfrac{2(1-\mathcal{S}qrt{xy})y}{\mathcal{S}qrt{2y^2+2x^2-1}}+\mathcal{S}qrt{2+2x^2-y^2}-\\ \\ \dfrac{(y-\mathcal{S}qrt{x})y}{2+2x^2-y^2}-\dfrac{2+2y^2-x^2}{2\mathcal{S}qrt{y}}+\dfrac{2(x-\mathcal{S}qrt{y})y}{\mathcal{S}qrt{2+2y^2-x^2}}. \end{array} $$ \indent Then we solve the system $$\left| \begin{array}{c}\dfrac{\partial F}{\partial x}=0\\ \\ \dfrac{\partial F}{\partial y}=0\end{array}, \right. $$ \noindent which gives us two points: $M_1(0,9238127491\ldots ,0,1660179102\ldots )$ and $M_2(1,1)$. Note that $M_1, M_2\in M$ and $M_1\in\mbox{int}\,M$, à $M_2\in\partial M$. \indent Again using some software, we get for the devil-fish function's hessian: $$ H(x,y):=\dfrac{\partial^2 F}{\partial x^2}\cdot\dfrac{\partial^2 F}{\partial y^2}-\left [\dfrac{\partial^2F}{\partial x\partial y}\right ]^2_{|M_2(1,1)}=\left (-\dfrac{3\mathcal{S}qrt{3}}{2}\right )\left (-\dfrac{3\mathcal{S}qrt{3}}{2}\right )-\left (\dfrac{3\mathcal{S}qrt{3}}{4}\right )^2=\left (\dfrac{9}{4}\right )^2>0 $$ \noindent and since $\dfrac{\partial^2F}{\partial x^2}_{|M_2(1,1)}=-\dfrac{3\mathcal{S}qrt{3}}{2}<0$, it follows that point $M_2(1,1)$ is a possible point of maximum over $M$. For completeness we need to check what is going on the boundary because $M_2\in\partial M$. But that is straightforward and this checking consists of the cases $\{x=0\}$, $\{x=y\}$, $\{x=1\}$, and $\{x+y=1\}$. This leads to the fact that the point in question is in fact an absolute maximum in $M$. This yields $$ \mathcal{S}up_{(x,y)\in M} F(x,y)=\max_{(x,y)\in M} F(x,y)=F(1,1)=0. $$ Additionally, $M_1$ is a point of a local minimum and moreover: $$ F(x,y)_{|M_1}=F(0,9238127491\ldots ,0,1660179102\ldots )=-0,4280657968\ldots $$ The devil-fish surface can be seen below: \begin{figure} \caption{Devil-fish surface} \end{figure} Using software such as \textit{Maple} and \textit{Mathematica}, one can get different views to that surface: \begin{figure} \caption{Different views of the devil-fish surface} \end{figure} Therefore all the cases are done and the theorem is proved.$\mathcal{H}fill\mathcal{S}quare$\\ \noindent The following immediate corollary is true: \begin{Corollary} Let $a,b$ and $c$ be the sides of a triangle $\triangle ABC$ and let also $m_a,m_b$ è $m_c$ be the medians to these sides respectively. Then \begin{itemize} \item[a)] $(2p-3a)m_a+(2p-3b)m_b+(2p-3c)m_c\ge 0,\,\,\mbox{where}\,\,p:=\dfrac{a+b+c}{2}.$ \item[b)]$\dfrac{m_a}{m_c}\le\dfrac{\mathcal{S}qrt{ab}+\mathcal{S}qrt{ac}+\mathcal{S}qrt{bc}}{a+b+c}\le 1,\,\,a\ge b\ge c$. \end{itemize} \end{Corollary} \end{document}

\begin{document} \sloppy \title[Invariance principle for tempered fractional time series models]{Invariance principle for tempered fractional time series models} \author{Farzad Sabzikar} \address{Farzad Sabzikar, Department of Statistics and Probability, Michigan State University, East Lansing MI 48823} \email{sabzika2@stt.msu.edu} \begin{abstract} Autoregressive tempered fractionally integrated moving average (ARTFIMA) time series is a useful model for velocity data in turbulence flows. In this paper, we obtain an invariance principle for the partial sum of an ARTFIMA process. The limiting process is called tempered Hermite process of order one, $THP^{1}$, which is well-defined for any $H>\frac{1}{2}$. When $\frac{1}{2}<H<1$, we develop the Wiener integral with respect to $THP^{1}$ to provide the sufficient condition for the convergence \begin{equation*} n^{-H}\sum_{k=0}^{+\infty}f\Big(\frac{k}{n}\Big)X^{\frac{\lambda}{n}}_{k}\rightarrow \int_{\mathbb R}f(u)Z^{1}_{H,\lambda}(du) \end{equation*} in distribution, as $n\to\infty$, where $X_{k}$ is an ARTFIMA time series and $Z^{1}_{H,\lambda}$ is $THP^{1}$. \end{abstract} \maketitle \section{Introduction} The motivation of this work comes from the application of stochastic processes in the theory of turbulence. Kolmogorov \cite{KolmogorovFBM, Friedlander} proposed a model for the energy spectrum of turbulence in the inertial range, predicting that the spectrum $f(k)$ would follow a power law $f(k)\propto k^{-5/3}$ where $k$ is the frequency. Figure 1 illustrates the complete Kolmogorov spectral model for turbulence, and the power law approximation in the inertial range. Large eddies are produced in the low frequency range. In the inertial range, larger eddies are continuously broken down into smaller eddies, until they eventually dissipate, in the high frequency range. \begin{figure} \caption{Kolmogorov spectral density (solid line) and power law approximation in the inertial range (dotted line), from \cite{Meerschaertsabzikarkumarzeleki} \label{fig4} \end{figure} The autoregressive tempered fractionally integrated moving average (ARTFIMA) time series modifies the coefficient of an autoregressive tempered fractionally integrated moving average (ARFIMA) model by multiplying an exponential tempering factor. The spectral density of the ARTFIMA $(0,\alpha,\lambda,0)$ is proportional to $\left|e^{-(\lambda+ik)}-1\right|^{-2\alpha}\approx (\lambda^2+k^2)^{-\alpha}$ when $k,\lambda$ are sufficiently small. For small values of the tempering parameter $\lambda$, the spectral density of an ARTFIMA $(0,\alpha,\lambda,0)$ time series grows like $k^{-2\alpha}$ as $|k|$ decreases, but remains bounded as $|k|\to 0$, in agreement with the general theory of turbulence illustrated in Figure 1. We refer the reader to \cite{Meerschaertsabzikarkumarzeleki} to see the application of the ARTFIMA time series for turbulence in geophysical flows. As it was mentioned, the ARTFIMA time series can be a useful discrete time stochastic model for turbulence. In this paper, we are interested to answer several questions which are related with the ARTFIMA time series. The first question is: \begin{description} \item[1] Assume $X_{k}$ follows an ARTFIMA time series model. When do we have an invariance principle \begin{equation*} n^{-H}\sum_{k=0}^{[nt]}X_{k}\mathbb Rightarrow Y(t) \end{equation*} and what is the limiting process $Y$? \end{description} We prove that $\{Y(t)\}_{t\geq 0}$ is a Gaussian process which interpolates between fractional Brownian motion (FBM) and the standard Ornstein Uhlenbeck process with the time domain representation \begin{equation*} Y(t):=Z^{1}_{H,\lambda}(t)={\int_{\mathbb R}\int_{0}^{t}\Big({(s-y)_{+}^{H-\frac{3}{2}}}e^{-\lambda(s-y)_{+}}\Big)\ ds\ B(dy)}, \end{equation*} where $(x)_{+}=xI(x>0)$, $B(dy)$ is an independently scattered Gaussian random measure on $\mathbb R$ with control measure $\sigma^2\ dx$, $H>\frac{1}{2}$ and $\lambda>0$. The process $Z^{1}_{H,\lambda}$ is called tempered Hermite process of order one, $THP^{1}$. We called $Z^{1}_{H,\lambda}$ as tempered Hermite process of order one since it can be extended to \begin{equation*}\label{eq:THPdefnorderk} Z_{H,\lambda}^{k}(t):={\int_{\mathbb R^k}^{'}\int_{0}^{t}\Big(\prod^{k}_{i=1}{(s-y_i)_{+}^{-(\frac{1}{2}+\frac{1-H}{k})}}e^{-\lambda(s-y_i)_{+}}\Big)\ ds\ B(dy_1)\ldots B(dy_k)} \end{equation*} for any $k\geq 2$. The prim on the integral sign shows that one does not integrate on diagonals where $y_{i}=y_{j}$, $i\neq j$. The Hermite process \cite{Taqqu,Dobrushinmajor} is a special case of $\{Z_{H,\lambda}^{k}\}$ with $\lambda=0$. Unlike the Hermite process, tempered Hermite process of order $k$ is well-defined also for any $H>\frac{1}{2}$ because the exponential tempering keeps the integrand in $L^{2}(\mathbb R^k)$. The second question is naturally the extension of the first question: \begin{description} \item[2] Suppose $f$ is a deterministic function. What is the sufficient condition for the convergence \begin{equation*} n^{-H}\sum_{k=0}^{+\infty}f\Big(\frac{k}{n}\Big)X_{k}\rightarrow \int_{\mathbb R}f(u)Z^{1}_{H,\lambda}(du) \end{equation*} in distribution, as $n\to\infty$? \end{description} In order to provide the sufficient condition for the convergence (in distribution) in the second question, we first require to develop the Wiener integral $\int_{\mathbb R}f(u)Z^{1}_{H,\lambda}(du)$, where $f$ is a deterministic functions in an appropriate space and $\frac{1}{2}<H<1$. Our approach to develop the Wiener integral with respect to $Z^{1}_{H,\lambda}$ is based on tempered fractional calculus. In fact, we show that a representation of $Z^{1}_{H,\lambda}$ based on fractional calculus. That is, \begin{equation*}\label{eq:connection with TFI} Z^{1}_{H,\lambda}(t) =\Gamma(H-\frac{1}{2})\int_{-\infty}^{+\infty}\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}{\mathbf 1}_{[0,t]}\Big)(x)\ B(dx), \end{equation*} where $H>\frac{1}{2}$ and $\Big({\mathbb{I}}^{\alpha,\lambda}_{-}f\Big)(x)$ is tempered fractional integral of order $\alpha$ of a function $f$. We refer the reader to \cite{MeerschaertsabzikarSPA} for more details on tempered fractional integrals and derivatives. This representation enables us to characterize the classes of deterministic functions $f$ for which the Wiener integral $\int_{\mathbb R}f\ Z^{1}_{H,\lambda}(du)$ is well defined. On the other hand, we need to study the asymptotic behavior of the spectral density of the ARTFIMA $(0,\alpha,\lambda,0)$ for low frequency. Therefor, we can prove the convergence in distribution that we had in the second question. The paper is organized as follows. In Section \ref{sec2}, we define tempered Hermite process of order one, $\{Z^{1}_{H,\lambda}\}$, using a time domain representation, and we develop the spectral domain representation of $\{Z^{1}_{H,\lambda}\}$. In Section \ref{sec3}, we answer the second question by investigating the Wiener integral with respect to tempered Hermite process of order one. In Section \ref{sec4}, we recall the definition of the autoregressive tempered fractionally integrated moving average (ARTFIMA) time series and some of its basic properties such as the covariance function and spectral density. The answer of the first and third question are given in Section \ref{sec5}. Some definitions and lemmas which are related to fractional calculus are contained in Appendix. \section{Time and spectral domain representations }\label{sec2} In this section, we define tempered Hermite processes of order one, $THP^{1}$. We start with the time domain representation. Let $\{B(t)\}_{t\in \mathbb R}$ be a real-valued Brownian motion on the real line, a process with stationary independent increments such that $B(t)$ has a Gaussian distribution with mean zero and variance $|t|$ for all $t\in\mathbb R$. Define an independently scattered Gaussian random measure $B(dx)$ with control measure $m(dx)=dx$ by setting $B[a,b]=B(b)-B(a)$ for any real numbers $a<b$, and then extending to all Borel sets. Since Brownian motion sample paths are almost surely of unbounded variation, the measure $B(dx)$ is not almost surely $\sigma$-additive, but it is a $\sigma$-additive measure in the sense of mean square convergence. Then the stochastic integrals $I(f):=\int f(x)B(dx)$ are defined for all functions $f:\mathbb R\to\mathbb R$ such that $\int f(x)^2dx<\infty$, as Gaussian random variables with mean zero and covariance $\mathbb E[I(f)I(g)]=\int f(x)g(x)dx$. See for example \cite[Chapter 3]{SamorodnitskyTaqqu} or \cite[Section 7.6]{FCbook}. \begin{defn}\label{defTHP} Given an independently scattered Gaussian random measure $B(dx)$ on $\mathbb R$ with control measure $\sigma^2 dx$, $H>\frac{1}{2}$, $\lambda>0$, the stochastic integral \begin{equation}\label{eq:THPdefn} Z_{H,\lambda}^{1}(t):={\int_{\mathbb R}\int_{0}^{t}\Big({(s-y)_{+}^{H-\frac{3}{2}}}e^{-\lambda(s-y)_{+}}\Big)\ ds\ B(dy)} \end{equation} where $(x)_{+}=xI(x>0)$, will be called a {\it tempered Hermit process} of order one \textrm{$(THP^1)$}. \end{defn} The next lemma shows that $Z_{H,\lambda}^{1}(t)$ is well-defined for any $t>0$. \begin{lem}\label{lem:g squar integrable} The function \begin{equation}\label{eq:integrand} g_{H,\lambda,t}(y):=\int_{0}^{t}(s-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(s-y)_{+}}\ ds \end{equation} is square integrable over the entire real line for any $H>\frac{1}{2}$ and $\lambda>0$. \end{lem} \begin{proof} The proof is similar to \cite[Theorem 3.5]{BaiTaqqu}. To show that $g_{H,\lambda,t}(y)$ is square integrable over the entire real line, we write \begin{equation*} \begin{split} \int_{\mathbb R}g_{H,\lambda,t}(y)^{2}\ dy&=\int_{\mathbb R}\int_{0}^{t}\int_{0}^{t}(s_1-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(s_1-y)_{+}}(s_2-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(s_2-y)_{+}}ds_1\ ds_2 \ dy\\ &=2\int_{0}^{t}ds_1\int_{s_1}^{t}ds_2\int_{\mathbb R}(s_1-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(s_1-y)_{+}}(s_2-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(s_2-y)_{+}} \ dy\\ &=2\int_{0}^{t}ds\int_{0}^{t-s}du\int_{\mathbb R}(w)_{+}^{H-\frac{3}{2}}e^{-\lambda(w)_{+}}(w+u)_{+}^{H-\frac{3}{2}}e^{-\lambda(w+u)_{+}} \ dw\\ &\qquad (s=s_1, u=s_2-s_1, w=s_1-y)\\ &=2\int_{0}^{t}ds\int_{0}^{t-s}u^{2H-2}e^{-\lambda u}du\int_{0}^{+\infty}{x}^{H-\frac{3}{2}}(1+x)^{H-\frac{3}{2}}e^{-2\lambda ux}\ dx\\ &=\frac{2\Gamma(H-\frac{1}{2})}{\sqrt{\pi}(2\lambda)^{H-1}}\int_{0}^{t}ds\int_{0}^{t-s}u^{H-1}K_{1-H}(\lambda u)\ du, \end{split} \end{equation*} where we applied a standard integral formula \cite[Page 344]{Gradshteyn} \begin{equation}\label{eq:standard integral formula} \int_{0}^{\infty}x^{\nu-1}(x+\beta)^{\nu-1}e^{-\mu x}\ dx=\frac{1}{\sqrt{\pi}}\big(\frac{\beta}{\mu}\big)^{\nu-\frac{1}{2}}e^{\frac{\beta\mu}{2}} \Gamma(\nu)K_{\frac{1}{2}-\nu}(\frac{\beta\mu}{2}), \end{equation} for $|\arg \beta|<\pi$, $Re \mu>0$, $Re \nu>0$. Here $K_{\nu}(x)$ is modified Bessel function of the second kind (see Appendix for more details about $K_{\nu}(x)$). Next, we need to show that the last integrals is finite for any $H>\frac{1}{2}$. First, assume $\frac{1}{2}<H<1$. In that case, $K_{1-H}(\lambda u)\sim u^{H-1}$ as $u\to 0$ (\cite[Chapter 9]{abramowitz}), and hence the integrand $u^{H-1}K_{1-H}(\lambda u)\sim u^{2H-2}$ ,as $u\to 0$, which is integrable provided that $H>\frac{1}{2}$. Now, let $H>1$. In the later case, $K_{1-H}(\lambda u)\sim u^{1-H}$ as $u\to 0$ and therefore the integrands $u^{H-1}K_{1-H}(\lambda u)\sim C$, $C$ is a constant, which in integrable and this completes the proof. \end{proof} \begin{rem} {\emph {When $\lambda=0$, the right-hand side of \eqref{eq:THPdefn} is a fractional Brownian motion (FBM), a self-similar Gaussian stochastic process with Hurst scaling index $H$ (e.g., see \cite{EmbrechtsMaejima}). When $\lambda=0$ and $H>1$, the right-hand side of \eqref{eq:THPdefn} does not exist, since the integrand is not in $L^{2}(\mathbb{R})$. However, $THP^1$ with $\lambda>0$ and $H>1$ is well-defined, because the exponential tempering keeps the integrand in $L^{2}(\mathbb{R})$.}} \end{rem} We now compute the covariance function $R(t,s)=\mathbb E[Z^{1}_{H,\lambda}(t)Z^{1}_{H,\lambda}(s)]$ of $THP^1$. \begin{prop}\label{prop:THPcovariance} The process $Z^{1}_{H,\lambda}$ given by \eqref{eq:THPdefn} has the covariance function \begin{equation}\label{eq:THPacvf} R(t,s)=\frac{2\Gamma(H-\frac{1}{2})}{\sqrt{\pi}(2\lambda)^{H-1}} \int_{0}^{t}\int_{0}^{s}|u-v|^{H-1}K_{1-H}(\lambda|u-v|)dv\ du. \end{equation} \end{prop} \begin{proof} The proof is similar to that of Lemma \ref{lem:g squar integrable}. By applying Fubini and the It\^{o} isometry, we have \begin{equation*} \begin{split} R(t,s)&=2\int_{\mathbb R}\int_{0}^{t}\int_{0}^{s}(u-y)_{+}^{H-\frac{3}{2}} (v-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(u-y)_{+}}e^{-\lambda(v-y)_{+}}dv\ du\ dy\\ &=2\int_{0}^{t}\int_{0}^{s}\int_{-\infty}^{v} (u-y)^{H-\frac{3}{2}}(v-y)^{H-\frac{3}{2}}e^{\lambda (u-y)}e^{-\lambda(v-y)} dy\ dv\ du\\ &\qquad (\textrm{assume}\quad v<u)\\ &=2\int_{0}^{t}\int_{0}^{s}(u-v)^{2H-2}e^{-\lambda (u-v)}\int_{0}^{+\infty} {x}^{H-\frac{3}{2}}(1+x)^{H-\frac{3}{2}}e^{-2\lambda (u-v)x} dx\ dv\ du\\ &\qquad (u-y=x(u-v)+u-v)\\ &=\frac{2\Gamma(H-\frac{1}{2})}{\sqrt{\pi}(2\lambda)^{H-1}} \int_{0}^{t}\int_{0}^{s}(u-v)^{H-1}K_{1-H}(\lambda(u-v))dv\ du, \end{split} \end{equation*} where we applied \eqref{eq:standard integral formula} to get the last integral. Hence \begin{equation*} R(t,s)=\frac{2\Gamma(H-\frac{1}{2})}{\sqrt{\pi}(2\lambda)^{H-1}} \int_{0}^{t}\int_{0}^{s}|u-v|^{H-1}K_{1-H}(\lambda|u-v|)dv\ du. \end{equation*} By using the same argument in Lemma \ref{lem:g squar integrable}, one can show that the covariance is finite for any $H>\frac{1}{2}$. \end{proof} The next results shows that $THP^1$ has a nice scaling property, involving both the time scale and the tempering. Here the symbol $\triangleq $ indicates the equivalence of finite dimensional distributions. \begin{prop}\label{prop:sssi} The process $Z^{1}_{H,\lambda}$ given by \eqref{defTHP} has stationary increments, such that \begin{equation}\label{eq:scalingTHP} \left\{Z_{H,\lambda}^{1}(ct)\right\}_{t\in\mathbb R}{\triangleq}\left\{c^{H}Z_{H,c\lambda}^{1}(t)\right\}_{t\in\mathbb R} \end{equation} for any scale factor $c>0$. \end{prop} \begin{proof} Since $B(dy)$ has control measure $m(dy)=\sigma^2 dy$, the random measure $B(c\ dy)$ has control measure $c^{1/2} \sigma^2dy$. Given $t_j$, $j=1,\dots,n$, a change of variables ${s=cs'}$ and ${y=cy'}$ then yields \begin{equation*} \begin{split} &\left(Z_{H,\lambda}^{1}(ct_j):j=1,\ldots,n\right)\\ &=\left({\int_{\mathbb R}\int_{0}^{ct_j}\Big({(s-y)_{+}^{H-\frac{3}{2}}}e^{-\lambda(s-y)_{+}}\Big)ds\ B(dy)}\right)\\ &=\left({\int_{\mathbb R}\int_{0}^{t_j}\Big({(cs'-cy)_{+}^{H-\frac{3}{2}}}e^{-\lambda(cs'-cy)_{+}}\Big)\ c\ ds'\ B(dy)}\right)\\ &\triangleq c^{H}\left({\int_{\mathbb R}\int_{0}^{t_j}\Big({(s'-y')_{+}^{H-\frac{3}{2}}}e^{-\lambda c(s'-y')_{+}}\Big)\ ds'\ B(c\ dy')}\right)\\ &=\left(c^{H}Z_{H,c\lambda}^{1}(t_j):j=1,\ldots,n\right) \end{split} \end{equation*} so that \eqref{eq:scalingTHP} holds. Suppose now $s_j<t_j$, and change variables $x=x'+s$, $y=s+y'$ to get \begin{equation*} \begin{split} &\left(Z_{H,\lambda}^{1}(t_j)-Z_{H,\lambda}^{1}(s_j):j=1,\ldots,n\right)\\ &=\left({\int_{\mathbb R}\int_{s_j}^{t_j}\Big({(x-y)_{+}^{H-\frac{3}{2}}}e^{-\lambda(x-y)_{+}}\Big)dx B(dy)}\right)\\ &=\left({\int_{\mathbb R}\int_{0}^{t_j-s_j}\Big({(x'+s-y)_{+}^{H-\frac{3}{2}}}e^{-\lambda(x'+s-y)_{+}}\Big)dx' B(dy)}\right)\\ &\triangleq \left({\int_{\mathbb R}\int_{0}^{t_j-s_j}\Big({(x'-y')_{+}^{H-\frac{3}{2}}}e^{-\lambda(x'-y')_{+}}\Big)dx' B(dy')}\right)\\ &=\left(Z_{H,\lambda}^{1}(t_j-s_j):j=1,\ldots,n\right) \end{split} \end{equation*} which shows that $THP^1$ has stationary increments. \end{proof} We next give another representation of $THP^1$ which is called the spectral domain representation. Let $\hat B_1$ and $\hat B_2$ be independent Gaussian random measures with $\hat B_{1}(A)=\hat B_{1}(-A)$, $\hat B_{2}(A)=-\hat B_{2}(-A)$ and $\mathbb E[(\hat B_{i}(A))^2]=m(A)/2$, where $m(dx)=\sigma^2 dx$, and define the complex-valued Gaussian random measure $\hat B=\hat B_{1}+i\hat B_{2}$. If $f(x)$ is a complex-valued function of $x$ real such that its Fourier transform $\hat f(\omega):=(2\pi)^{-1/2}\int e^{i\omega x}f(x)\,dx$ exists and $\int |\hat f(\omega)|^2 d\omega<\infty$, we define the stochastic integral $\hat I(\hat f)=\int \hat f(\omega)\hat B(d\omega):=\int \hat f_1(\omega) \hat B_1(d\omega)-\int \hat f_2(\omega)\hat B_2(d\omega)$, where $\hat f=\hat f_1+i\hat f_2$ is separated into real and imaginary parts. Then $\hat I(\hat f)$ is a Gaussian random variable with mean zero, such that $\mathbb E[\hat I(\hat f)\hat I(\hat g)]=\int \hat f(\omega)\overline{\hat g(k)}\,d\omega$ for all such functions, and the Parseval identity $\int f(x)g(x)\,dx=\int \hat f(\omega) \overline{\hat g(\omega)}\,d\omega$ implies that $(\int f(x)B(dx),\int g(x)B(dx))\triangleq (\int \hat f(\omega)\hat B(d\omega),\int \hat g(\omega)\hat B(d\omega))$, see Proposition 7.2.7 in \cite{SamorodnitskyTaqqu}. \begin{prop}\label{prop:THPdefharmo} The process $Z^{1}_{H,\lambda}$ given by \eqref{defTHP} has the spectral domain representation \begin{equation}\label{eq:THPdefharmo} Z^{1}_{H,\lambda}(t)\triangleq \frac{1}{C(H)}\int_{\mathbb R} \frac{e^{i\omega t}-1}{i\omega} (\lambda+i\omega)^{\frac{1}{2}-H} \widehat{B}(d\omega), \end{equation} where $C(H)=\frac{\sqrt{2\pi}}{\Gamma(H-\frac{1}{2})}$. \end{prop} \begin{proof} To show that the stochastic integral \eqref{eq:THPdefharmo} exists, note that $\Big|\frac{e^{i\omega t}-1}{i\omega}(\lambda+i\omega)^{\frac{1}{2}-H}\Big|^2$ is bounded for $\omega\rightarrow 0$ and behaves like $|\omega|^{-1-2H}$ , as $\omega\rightarrow \infty$, which is integrable provided that $H>0$. Observe that the function $g_{H,\lambda,t}$, given by \eqref{eq:integrand}, has the Fourier transform \begin{equation*} \begin{split} \widehat{g_{H,\lambda,t}}(\omega)&= \frac{1}{\sqrt{2\pi}}\int_{\mathbb R}e^{i\omega y} \int_{0}^{t}(s-y)^{H-\frac{3}{2}}_{+}e^{-\lambda(s-y)_{+}}\ ds\ dy\\ &=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}e^{i\omega y}\int_{\mathbb R} (s-y)^{H-\frac{3}{2}}e^{-\lambda(s-y)}{\mathbf 1}\{0<s<t\}{\mathbf 1}\{s-y>0\}\ ds\ dy\\ &=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}e^{i(s-u)\omega}\int_{\mathbb R} {u}^{H-\frac{3}{2}}e^{-\lambda u}{\mathbf 1}\{0<s<t\}{\mathbf 1}\{u>0\}\ ds\ du\\ &=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}e^{is\omega}{\mathbf 1}\{0<s<t\}\int_{\mathbb R} {u}^{H-\frac{3}{2}}e^{-(\lambda+i\omega)u}{\mathbf 1}\{u>0\}\ du\ ds\\ &=\frac{\Gamma(H-\frac{1}{2})}{\sqrt{2\pi}}\frac{e^{i\omega t}-1}{i\omega} (\lambda+i\omega)^{\frac{1}{2}-H} \end{split} \end{equation*} provided that $ H>\frac{1}{2}$ and then by applying \eqref{eq:THPdefn} \begin{equation*} \begin{split} Z^{1}_{H,\lambda}(t)&=\int_{-\infty}^{+\infty}g_{H,\lambda,t}(x)B(dx)\\ &\triangleq \int^{+\infty}_{-\infty}\widehat{g_{H,\lambda,t}}(\omega){\hat{B}}(d\omega)= \frac{1}{C(H)}\int^{+\infty}_{-\infty}\frac{e^{i\omega t}-1}{i\omega}(\lambda+i\omega)^{\frac{1}{2}-H}{\hat{B}}(d\omega) \end{split} \end{equation*} which is equivalent to \eqref{eq:THPdefharmo}. \end{proof} \begin{rem} {\emph{We called the process $Z^{1}_{H,\lambda}$ as the tempered Hermite process of order one, since it is a special case of the following stochastic process which is called tempered Hermite process of order $k$: \begin{equation}\label{eq:THPdefnorderk} Z_{H,\lambda}^{k}(t):={\int_{\mathbb R^k}\int_{0}^{t}\Big(\prod^{k}_{i=1}{(s-y_i)_{+}^{-(\frac{1}{2}+\frac{1-H}{k})}}e^{-\lambda(s-y_i)_{+}}\Big)\ ds\ B(dy_1) \ldots B(dy_k)} \end{equation} for any $k\geq 1$ and $H>\frac{1}{2}$. It is easy to check that $Z_{H,\lambda}^{k}$ has stationary increment with the scaling property given by \eqref{eq:scalingTHP}. Moreover, one can verify that $Z_{H,\lambda}^{k}$ has the spectral domain representation \begin{equation}\label{eq:THPdefharmo orderk} Z^{k}_{H,\lambda}(t)=c(H,k)\int_{\mathbb R^k}^{''} \frac{e^{it(\omega_1+\ldots+\omega_k)}-1}{i(\omega_1+\ldots+\omega_k)} \prod_{j=1}^{k}(\lambda+i\omega_j)^{-\big(\frac{1}{2}-\frac{1-H}{k}\big)} \widehat{B}(d\omega_1)\ldots\widehat{B}(d\omega_k), \end{equation} where $c(H,k)=\Big(\frac{\Gamma(\frac{1}{2}-\frac{1-H}{k})}{\sqrt{2\pi}}\Big)^{k}$ is a constant depending on $H$ and $k$. The double prim on the integral indicates that one does not integrate on diagonals where $\omega_i=\omega_j$, $i\neq j$. In this paper, we just consider tempered Hermite process of order one.}} \end{rem} Finally, we close this section with introducing tempered Hermite noise which is the increment of tempered Hermite process of order one. Given a $THP^{1}$ ,\eqref{eq:THPdefn}, we define tempered Hermite noise (THN) \begin{equation}\label{eq:THNdef} X_{n}=Z^{1}_{H,\lambda}(n+1)-Z_{H,\lambda}^{1}(n)\quad\text{for integers $0<n<\infty$.} \end{equation} It follows easily from \eqref{eq:THPdefn} that THN has the time domain representation \begin{equation}\label{eq:TFGNmoving} X_n= \int_{\mathbb R}\int_{n}^{n+1}(s-y)_{+}^{\frac{3}{2}-H}e^{-\lambda(s-y)_{+}}ds\ B(dy). \end{equation} Using \eqref{eq:THPdefharmo}, it also follows that THN has the spectral domain representation, \begin{equation}\label{eq:THNharmonizable} X_n= \frac{1}{C(H)}\int_{\mathbb R}e^{in\omega} \frac{e^{i\omega}-1}{i\omega}({\lambda+i\omega})^{\frac{1}{2}-H}\widehat{B}(d\omega). \end{equation} It follows from \eqref{eq:THNharmonizable} that {\rm THN} is a stationary Gaussian time series with mean zero and covariance function \begin{equation}\label{eq:THNacvf} r(n):=\mathbb{E}[X_0 X_n]=\frac{\sigma^2}{C(H)^{2}} \int_{\mathbb R}e^{in\omega}\Big|\frac{e^{i\omega}-1}{i\omega}\Big|^{2}({\lambda^2+\omega^2})^{\frac{1}{2}-H}(d\omega). \end{equation} \begin{prop}\label{PropSpedDens} {\rm THP} \eqref{eq:THNdef} has the spectral density \begin{equation}\label{eq:specdensTHP} h(\omega)=\frac{1}{C(H)^2}\Big|\frac{e^{i\omega}-1}{i\omega}\Big|^2 \sum^{+\infty}_{\ell=-\infty} \sigma^2{[\lambda^2+(\omega+2\pi \ell)^2]}^{\frac{1}{2}-H}. \end{equation} \end{prop} \begin{proof} Recall that the spectral density \begin{equation}\label{eq:spectralsum} h(\omega)=\frac{1}{2\pi}\sum^{+\infty}_{j=-\infty}{e^{-i\omega n}}{r(n)}\quad\text{and}\quad r(n)=\int^{\pi}_{-\pi}{e^{i\omega n}}{h(\omega)}d{\omega} . \end{equation} Apply \eqref{eq:THNacvf} to write \begin{equation}\label{specdenscalc} \begin{split} r(n) =&\frac{\sigma^2}{C(H)^2}\int^{+\infty}_{-\infty}e^{i\omega n}\Big|\frac{e^{i\omega}-1}{i\omega}\Big|^{2}(\lambda^2+\omega^2)^{\frac{1}{2}-H}d\omega\\ =&\frac{1}{C(H)^{2}}\int^{+\pi}_{-\pi}e^{i\omega n}\Big|\frac{e^{i\omega}-1}{i\omega}\Big|^{2} \sum^{+\infty}_{\ell=-\infty}{\sigma^2}{[\lambda^2+(\omega+2\pi \ell)^2]}^{\frac{1}{2}-H}d\omega \end{split} \end{equation} and then it follows from \eqref{eq:spectralsum} that the spectral density of THN is given by \eqref{eq:specdensTHP}. \end{proof} \begin{rem}\label{remLowk} {\emph {Extending the definition \eqref{eq:THNdef} to all $n$ real positive, we obtain the continuous parameter THN \[X_{t}=Z^{1}_{H,\lambda}(t+1)-Z^{1}_{H,\lambda}(t) .\] The spectral domain representation of this process is given by \eqref{eq:THNharmonizable} with $n$ replaced by $t$, and the proof of Proposition \ref{PropSpedDens} implies that $X_t$ has spectral density \begin{equation}\label{eq:specdensTHPcont} h(\omega)=\frac{\sigma^2}{C(H)^{2}}\Big|\frac{e^{i\omega}-1}{i\omega}\Big|^{2} {[\lambda^2+\omega^2]}^{\frac{1}{2}-H} \end{equation} for all real $\omega$. The fact that $\big|\frac{e^{i\omega}-1}{i\omega}\big|$ is bounded as $\omega\to 0$ yields the low frequency approximation \begin{equation}\label{eq:spectralowferequncy} h(\omega)\approx \frac{\sigma^2}{C(H)^{2}}{(\lambda^2+\omega^2)}^{\frac{1}{2}-H}. \end{equation} By taking $H=\frac{4}{3}$ in \eqref{eq:spectralowferequncy}, we get $h(\omega)\approx \omega^{-5/3}$ which is the spectral model suggested by Kolmogorov \cite{KolmogorovFBM, Friedlander} for the energy spectrum of turbulence in the inertial range. The spectral density of THN has some applications in turbulent flows \cite{Meerschaertsabzikarkumarzeleki} }}. \end{rem} \section{Wiener integrals with respect to tempered Hermite process of order one}\label{sec3} In order to get the main results of this paper, Section \ref{sec5} , we need to develop the Wiener integrals with respect to $Z^{1}_{H,\lambda}$. We consider two cases: \begin{itemize} \item $\frac{1}{2}<H<1$, $\lambda>0$ \item $H>1$, $\lambda>0$ \end{itemize} We start with the first case. We first establish a link between $Z^{1}_{H,\lambda}$ and tempered fractional calculus. \begin{lem}\label{lem:connection with TFI} For a tempered Hermite process of order one given by \eqref{eq:THPdefn}, $THP^1$, with $\lambda>0$, we have: \begin{equation}\label{eq:connection with TFI} Z^{1}_{H,\lambda}(t) =\Gamma(H-\frac{1}{2})\int_{-\infty}^{+\infty}\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}{\mathbf 1}_{[0,t]}\Big)(x)\ B(dx) \end{equation} where $H>\frac{1}{2}$. \end{lem} \begin{proof} Write the kernel function from \eqref{eq:integrand} in the form \begin{equation*} \begin{split} g_{H,\lambda,t}(x) &=\int^{t}_{0}(s-x)^{H-\frac{3}{2}}_{+}e^{-\lambda(s-x)_{+}}\ ds\\ &=\int^{+\infty}_{-\infty}{\mathbf 1_{[0,t]}}(s)(s-x)^{H-\frac{3}{2}}_{+}e^{-\lambda(s-x)_{+}}\ ds\\ &=\Gamma(H-\frac{1}{2})\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}{\mathbf 1}_{[0,t]}\Big)(x) \end{split} \end{equation*} which gives the desired result. \end{proof} Next we discuss a general construction for stochastic integrals with respect to $THP^{1}$. For a standard Brownian motion $\{B(t)\}_{t\in\mathbb R}$ on $(\Omega,{\mathcal F},P)$, the stochastic integral ${{\mathcal I}}(f):=\int f(x)B(dx)$ is defined for any $f\in L^2(\mathbb R)$, and the mapping $f\mapsto {{\mathcal I}}(f)$ defines an isometry from $L^2(\mathbb R)$ into $L^2(\Omega)$, called the {\it It\^{o} isometry}: \begin{equation}\label{eq:ItoIsometry} \ip{{\mathcal I}(f)}{{\mathcal I}(g)}_{L^2(\Omega)}={\rm Cov}[{\mathcal I}(f),{\mathcal I}(g)]=\int f(x)g(x)\,dx=\ip fg_{L^2(\mathbb R)} . \end{equation} Since this isometry maps $L^2(\mathbb R)$ onto the space $\overline{\rm Sp}(B)=\{{{\mathcal I}}(f):f\in L^2(\mathbb R)\}$, we say that these two spaces are isometric. For any elementary function (step function) \begin{equation}\label{eq:elementarydefn} f(u)=\sum^{n}_{i=1}a_{i}{\mathbf 1_{[t_{i},t_{i+1})}(u)}, \end{equation} where $a_i,t_i$ are real numbers such that $t_i<t_{j}$ for $i<j$, it is natural to define the stochastic integral \begin{equation}\label{eq:THPintegraldefn} {{\mathcal I}}^{\alpha,\lambda}(f)=\int_{\mathbb{R}}f(x)Z^{1}_{H,\lambda}(dx)=\sum^{n}_{i=1}a_i \left[Z^{1}_{H,\lambda}(t_{i+1})-Z^{1}_{H,\lambda}(t_{i})\right], \end{equation} and then it follows immediately from \eqref{eq:connection with TFI} that for $f\in {\mathcal E}$, the space of elementary functions, the stochastic integral \[{{\mathcal I}}^{\alpha,\lambda}(f)=\int_{\mathbb{R}}f(x)Z^{1}_{H,\lambda}(dx)={\Gamma(H-\frac{1}{2})}\int_{\mathbb{R}} \Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x)\ B(dx) \] is a Gaussian random variable with mean zero, such that for any $f,g\in {\mathcal E}$ we have \begin{equation}\label{eq:innerproductTFI} \begin{split} &\ip{{{\mathcal I}}^{\alpha,\lambda}(f)}{{{\mathcal I}}^{\alpha,\lambda}(g)}_{L^2(\Omega)} =\mathbb{E}\left(\int_{\mathbb{R}}f(x)Z^{1}_{H,\lambda}(dx)\int_{\mathbb{R}}g(x)Z^{1}_{H,\lambda}(dx)\right) \\ &=\Gamma(H-\frac{1}{2})^2\int_{\mathbb{R}} \Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x) \Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}g\Big)(x)\ dx , \end{split} \end{equation} in view of \eqref{eq:connection with TFI} and the It\^{o} isometry \eqref{eq:ItoIsometry}. The linear space of Gaussian random variables $\left\{{{\mathcal I}}^{\alpha,\lambda}(f),f\in\mathcal{E}\right\}$ is contained in the larger linear space \begin{equation} \overline{\rm Sp}(Z^{1}_{H,\lambda})=\left\{X:{{\mathcal I}}^{\alpha,\lambda}(f_n)\rightarrow X\ \textrm{in $L^2(\Omega)$ for some sequence $(f_n)$ in $\mathcal{E}$}\right\}. \end{equation} An element $X\in\overline{\rm Sp}(Z^{1}_{H,\lambda})$ is mean zero Gaussian with variance \begin{equation*} {\operatorname{Var}}(X)=\lim_{n\to\infty} {\operatorname{Var}}[{{\mathcal I}}^{\alpha,\lambda}(f_n)] , \end{equation*} and $X$ can be associated with an equivalence class of sequences of elementary functions $(f_n)$ such that ${{\mathcal I}}^{\alpha,\lambda}(f_n)\to X$ in $L^{2}(\mathbb{R})$. If $[f_X]$ denotes this class, then $X$ can be written in an integral form as \begin{equation}\label{eq:stochasticintegraldefn} X=\int_{\mathbb{R}}[f_{X}] dZ^{1}_{H,\lambda} \end{equation} and the right hand side of \eqref{eq:stochasticintegraldefn} is called the stochastic integral with respect to $THP^{1}$ on the real line (see, for example, Huang and Cambanis \cite{Huang}, page 587). In the special case of a Brownian motion $\lambda=0, H=\frac{1}{2}$, ${{\mathcal I}}^{\alpha,\lambda}(f_n)\rightarrow X$ along with the It\^{o} isometry \eqref {eq:ItoIsometry} implies that $(f_n)$ is a Cauchy sequence, and then since $L^2(\mathbb R)$ is a (complete) Hilbert space, there exists a unique $f\in L^2(\mathbb R)$ such that $f_n\to f$ in $L^2(\mathbb R)$, and we can write $X=\int_{\mathbb{R}} f(x) B(dx)$. However, if the space of integrands is not complete, then the situation is more complicated. Here we investigate stochastic integrals with respect to $THP^{1}$ based on time domain representation. Equation \eqref{eq:innerproductTFI} suggests the appropriate space of integrands for $THP^{1}$, in order to obtain a nice isometry that maps into the space $\overline{\rm Sp}(Z^{1}_{H,\lambda})$ of stochastic integrals. \begin{thm}\label{thm:stochasticcalculus for TFI and moving general} Given $\frac{1}{2}<H<1$ and $\lambda>0$, the class of functions \begin{equation}\label{eq:A1 star class} {\mathcal{A}}_{1}:=\left\{f\in L^{2}(\mathbb{R}):\int_{\mathbb{R}} \left|\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x)\right|^{2}dx<\infty\right\}, \end{equation} is a linear space with inner product \begin{equation}\label{eq:productTFIf} \begin{split} {\langle f,g \rangle}_{{\mathcal{A}}_{1}} &:={\langle F,G\rangle}_{L^{2}(\mathbb{R})} \end{split} \end{equation} where \begin{equation}\label{eq:defnF}\begin{split} F(x)&=\Gamma(H-\frac{1}{2})\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x)\\ G(x)&=\Gamma(H-\frac{1}{2})\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}g\Big)(x) . \end{split}\end{equation} The set of elementary functions ${\mathcal{E}}$ is dense in the space ${\mathcal{A}}_{1}$. The space ${\mathcal{A}}_{1}$ is not complete. \end{thm} \begin{proof} The proof is similar to \cite[Theorem 3.5]{MeerschaertsabzikarSPA}. To show that ${\mathcal{A}}_{1}$ is an inner product space, we will check that ${\langle f,f \rangle}_{{\mathcal{A}}_{1}}=0$ implies $f=0$ almost everywhere. If ${\langle f,f \rangle}_{{\mathcal{A}}_{1}}=0$, then in view of \eqref{eq:productTFIf} and \eqref{eq:defnF} we have ${\langle F,F\rangle}_{2}=0$, so $F(x)=\Gamma(H-\frac{1}{2})\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x)=0$ for almost every $x\in\mathbb R$. Then \begin{equation}\label{eq1:stochasticcalculus for TFI and moving general} \Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x)=0\quad\text{for almost every $x\in\mathbb R$.} \end{equation} Apply ${\mathbb{D}}^{H-\frac{1}{2},\lambda}_{-}$ to both sides of equation \eqref{eq1:stochasticcalculus for TFI and moving general} and use Lemma \ref{lem:inversoperator} to get $f(x)=0$ for almost every $x\in\mathbb R$, and hence ${\mathcal{A}}_{1}$ is an inner product space. Next, we want to show that the set of elementary functions ${\mathcal E}$ is dense in ${\mathcal{A}}_{1}$. For any $f\in{\mathcal{A}}_{1}$, we also have $f\in{L}^{2}(\mathbb{R})$, and hence there exists a sequence of elementary functions $(f_n)$ in $L^2(\mathbb R)$ such that $\|f-f_n\|_2\to 0$. But \begin{equation*} \|f-f_n\|_{{\mathcal{A}}_{1}}={\langle f-f_n,f-f_n \rangle}_{{\mathcal{A}}_{1}}={\langle F-F_n,F-F_n\rangle}_2=\|F-F_n\|_2, \end{equation*} where $ F_n(x)=\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}{f_n}\Big)(x) $ and $F(x)$ is given by \eqref{eq:defnF}. Lemma \ref{lem:TFI and Lp} implies that \begin{equation*} \|f-f_n\|_{{\mathcal{A}}_{1}}=\left\|F-F_n\right\|_{2}=\|{\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}(f-f_n)\|_{2}\leq C\|f-f_n\|_{2} \end{equation*} for some $C>0$, and since $\|f-f_n\|_2\to 0$, it follows that the set of elementary functions is dense in ${{\mathcal{A}}_{1}}$. Finally, we provide an example to show that ${\mathcal{A}}_{1}$ is not complete. The functions \begin{equation*} \widehat{f_n}(\omega)=|\omega|^{-p}\mathbf 1_{\{1<|\omega|<n\}}(\omega),\ p>0, \end{equation*} are in $L^{2}(\mathbb{R})$, $\overline{\widehat{f_n}(\omega)}=\widehat{f_n}(-\omega)$, and hence they are the Fourier transforms of functions $f_n\in L^{2}(\mathbb{R})$. Apply Lemma \ref{lem:FourierTFI} to see that the corresponding functions $F_n(x)=\Gamma(H-\frac{1}{2})\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f_n\Big)(x)$ from \eqref{eq:defnF} have Fourier transform \begin{equation}\begin{split}\label{eq:A1Fdef} {\mathcal{F}}[F_n](\omega) &=\Gamma(H-\frac{1}{2})(\lambda+i\omega)^{\frac{1}{2}-H}\hat f_n(\omega). \end{split}\end{equation} Since $\frac{1}{2}-H<0$, it follows that \[\|F_n\|_2^2=\|\hat F_n\|_2^2=\Gamma(H-\frac{1}{2})^2\int_{-\infty}^{\infty}\left|\widehat{f_n}(\omega)\right|^{2}(\lambda^2+\omega^2)^{\frac{1}{2}-H}<\infty\] for each $n$, which shows that $f_n\in {\mathcal{A}}_{1}$. Now it is easy to check that $f_n-f_m\to 0$ in ${{\mathcal{A}}_{1}}$, as $n,m\to\infty$, whenever $p>1-H$, so that $(f_n)$ is a Cauchy sequence. Choose $p=\frac{1}{2}$ and suppose that there exists some $f\in{{\mathcal{A}}_{1}}$ such that $\|f_n- f\|_{{\mathcal{A}}_{1}}\to 0$ as $n\to\infty$. Then \begin{equation}\label{eq:a3} \int_{-\infty}^{\infty}\left|\widehat{f_n}(\omega)-\widehat{f}(\omega)\right|^{2}(\lambda^2+\omega^2)^{\frac{1}{2}-H}\to 0 \end{equation} as $n\to\infty$, and since, for any given $m\geq 1$, the value of $\widehat{f_n}(\omega)$ does not vary with $n>m$ whenever $\omega\in [-m,m]$, it follows that $\hat{f}(\omega)=|\omega|^{-\frac{1}{2}}1_{\{|\omega|>1\}}$ on any such interval. Since $m$ is arbitrary, it follows that $\hat{f}(\omega)=|\omega|^{-\frac{1}{2}}1_{\{|\omega|>1\}}$, but this function is not in $L^{2}(\mathbb{R})$, so $\hat{f}(\omega)\notin {{\mathcal{A}}_{1}}$, which is a contradiction. Hence ${{\mathcal{A}}_{1}}$ is not complete, and this completes the proof. \end{proof} \begin{rem} {\emph{It follows from Lemma \ref{lem:TFI and Lp} that ${\mathcal{A}}_{1}$ contains every function in $L^{2}(\mathbb{R})$, and hence they are the same set, but endowed with a different inner product.}} \end{rem} We now define the stochastic integral with respect to $THP^{1}$ for any function in ${\mathcal{A}}_{1}$ in the case where $\frac{1}{2}<H<1$. \begin{defn}\label{defn:TFI of general f} For any $\frac{1}{2}<H<1$ and $\lambda>0$, we define \begin{equation}\label{eq:TFIof f resp $THP^{1}$ genral} \int_{\mathbb{R}}f(x)Z^{1}_{H,\lambda}(dx):={\Gamma(H-\frac{1}{2})}\int_{\mathbb{R}} \Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)(x)\ B(dx) \end{equation} for any $f\in{\mathcal{A}}_{1}$. \end{defn} \begin{thm}\label{thm:SLRDisometric} For any $\frac{1}{2}<H<1$ and $\lambda>0$, the stochastic integral ${\mathcal I}^{\alpha,\lambda}$ in \eqref{eq:TFIof f resp $THP^{1}$ genral} is an isometry from ${\mathcal{A}}_{1}$ into $\overline{\rm Sp}(Z^{1}_{H,\lambda})$. Since ${\mathcal{A}}_{1}$ is not complete, these two spaces are not isometric. \end{thm} \begin{proof} It follows from Lemma \ref{lem:TFI and Lp} that the stochastic integral \eqref{eq:TFIof f resp $THP^{1}$ genral} is well-defined for any $f\in{\mathcal{A}}_{1}$. Proposition 2.1 in Pipiras and Taqqu \cite{PipirasTaqqu} implies that, if $\mathcal{D}$ is an inner product space such that $(f,g)_{\mathcal{D}}=\ip{{{\mathcal I}}^{\alpha,\lambda}(f)}{{{\mathcal I}}^{\alpha,\lambda}(g)}_{L^2(\Omega)}$ for all $f,g\in\mathcal{E}$, and if $\mathcal{E}$ is dense $\mathcal{D}$, then there is an isometry between $\mathcal{D}$ and a linear subspace of $\overline{\rm Sp}(Z^{1}_{H,\lambda})$ that extends the map $f\to{{\mathcal I}}^{\alpha,\lambda}(f)$ for $f\in\mathcal{E}$, and furthermore, $\mathcal{D}$ is isometric to $\overline{\rm Sp}(Z^{1}_{H,\lambda})$ itself if and only if $\mathcal{D}$ is complete. Using the It\^{o} isometry and the definition \eqref{eq:TFIof f resp $THP^{1}$ genral}, it follows from \eqref{eq:productTFIf} that for any $f,g\in{{\mathcal{A}}_{1}}$ we have \[{\langle f,g \rangle}_{{\mathcal{A}}_{1}}={\langle F,G\rangle}_{L^{2}(\mathbb{R})} =\ip{{{\mathcal I}}^{\alpha,\lambda}(f)}{{{\mathcal I}}^{\alpha,\lambda}(g)}_{L^2(\Omega)} ,\] and then the result follows from Theorem \ref{thm:stochasticcalculus for TFI and moving general}. \end{proof} We now apply the spectral domain representation of $THP^{1}$ to investigate the stochastic integral with respect to $THP^{1}$. Apply the Fourier transform of an indicator function to write this spectral domain representation in the form \begin{equation*}\label{eq:$THP^{1}$defharmo and indicator*} Z^{1}_{H,\lambda}(t)=\Gamma(H-\frac{1}{2})\int^{+\infty}_{-\infty} \widehat{\mathbf 1}_{[0,t]}(\omega)(\lambda+i\omega)^{\frac{1}{2}-H}\hat{B}(d\omega). \end{equation*} It follows easily that for any elementary function \eqref{eq:elementarydefn} we may write \begin{equation}\label{eq:$THP^{1}$defharmo and elementary} {\mathcal I}^{\alpha,\lambda}(f)=\Gamma(H-\frac{1}{2})\int^{\infty}_{-\infty} \widehat{f}(\omega)(\lambda+i\omega)^{\frac{1}{2}-H}\hat{B}(d\omega) , \end{equation} and then for any elementary functions $f$ and $g$ we have \begin{equation}\label{eq:inner harmo} \ip{{\mathcal I}^{\alpha,\lambda}(f)}{{\mathcal I}^{\alpha,\lambda}(g)}_{L^2(\Omega)} =\Gamma(H-\frac{1}{2}) \int_{-\infty}^{\infty}\widehat{f}(\omega)\overline{\widehat{g}(\omega)}(\lambda^2+\omega^2)^{\frac{1}{2}-H}d\omega . \end{equation} \begin{thm}\label{thm:A2space} For any $\frac{1}{2}<H<1$ and $\lambda>0$, the class of functions \begin{equation}\label{eq:A3class} \mathcal{A}_{2}:=\left\{f\in L^{2}(\mathbb{R}):\int \left|\widehat{f}(\omega)\right|^{2}(\lambda^2+\omega^2)^{\frac{1}{2}-H}\ d\omega<\infty\right\} , \end{equation} is a linear space with the inner product \begin{equation}\label{eq:productharmo} {\langle f,g \rangle}_{{\mathcal{A}}_{2}}=\Gamma(H-\frac{1}{2})^2 \int_{-\infty}^{+\infty}\widehat{f}(\omega)\overline{\widehat{g}(\omega)}(\lambda^2+\omega^2)^{\frac{1}{2}-H}d\omega . \end{equation} The set of elementary functions ${\mathcal{E}}$ is dense in the space ${\mathcal{A}}_2$. The space ${\mathcal{A}}_2$ is not complete. \end{thm} \begin{proof} The proof combines Theorem \ref{thm:stochasticcalculus for TFI and moving general} and using the Plancherel Theorem. Since $H>\frac{1}{2}$, the function $(\lambda^2+\omega^2)^{\frac{1}{2}-H}$ is bounded by a constant $C(H,\lambda)$ that depends only on $H$ and $\lambda$, so for any $f\in L^{2}(\mathbb{R})$ we have \begin{equation}\label{eq:A3toL2} \int_{\mathbb{R}} \left|\widehat{f}(\omega)\right|^{2}(\lambda^2+\omega^2)^{\frac{1}{2}-H}\ d\omega\leq C(H,\lambda) \int_{\mathbb{R}} \left|\widehat{f}(\omega)\right|^{2}\ d\omega <\infty \end{equation} and hence $f\in \mathcal{A}_{2}$. Since $\mathcal{A}_{2}\subset L^{2}(\mathbb{R})$ by definition, this proves that $L^{2}(\mathbb{R})$ and $\mathcal{A}_{2}$ are the same set of functions, and then it follows from Lemma \ref{lem:TFI and Lp} that $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are the same set of functions. Observe that $\varphi_f=\Big({\mathbb{I}}^{H-\frac{1}{2},\lambda}_{-}f\Big)$ is again a function with Fourier transform \[\hat \varphi_f=(\lambda+i\omega)^{\frac{1}{2}-H}\hat f .\] Then it follows from the Plancherel Theorem that \begin{equation*}\begin{split} \ip fg_{{\mathcal{A}}_{1}}=\Gamma(H-\frac{1}{2})^2\ip {\varphi_f}{\varphi_g}_2&=\Gamma(1-\alpha)^2\ip {\hat\varphi_f}{\hat\varphi_g}_2\\ &=\Gamma(1-\alpha)^2\int_{-\infty}^{+\infty}\widehat{f}(\omega)\overline{\widehat{g}(\omega)}(\lambda^2+\omega^2)^{\frac{1}{2}-H}d\omega=\ip fg_{{\mathcal{A}}_{2}} \end{split}\end{equation*} and hence the two inner products are identical. Then the conclusions of Theorem \ref{thm:A2space} follow from Theorem \ref{thm:stochasticcalculus for TFI and moving general}. \end{proof} \begin{defn}\label{defn:stochIntFourier} For any $H>\frac{1}{2}$ and $\lambda>0$, we define \begin{equation}\label{eq:stochIntFourier} {\mathcal I}^{\alpha,\lambda}(f)=\Gamma(1-\alpha)\int^{\infty}_{-\infty} \widehat{f}(\omega)(\lambda+i\omega)^{\frac{1}{2}-H}\hat{B}(d\omega) \end{equation} for any $f\in{\mathcal{A}}_{2}$. \end{defn} \begin{thm}\label{thm:SLRDisometric2} For any $H>\frac{1}{2}$ and $\lambda>0$, the stochastic integral ${\mathcal I}^{\alpha,\lambda}$ in \eqref{eq:stochIntFourier} is an isometry from ${\mathcal{A}}_{2}$ into $\overline{\rm Sp}(Z^{1}_{H,\lambda})$. Since ${\mathcal{A}}_{2}$ is not complete, these two spaces are not isometric. \end{thm} \begin{proof} The proof of Theorem \ref{thm:A2space} shows that $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are identical when $H>\frac{1}{2}$. Then the result follows immediately from Theorems \ref{thm:SLRDisometric}. \end{proof} Now, we consider the second case that we mentioned at the beginning of this section. we will show that $Z^{1}_{H,\lambda}$ is a continuous semimartingale with finite variation and hence one can define stochastic integrals $I(f):=\int f(x)Z^{1}_{H,\lambda}(dx)$ in the standard manner, via the It\^{o} stochastic calculus (e.g., see Kallenberg \cite[Chapter 15]{Kallenberg}). \begin{thm}\label{thm:THPsemimartingale} A tempered Hermite process of order one $\{Z^{1}_{H,\lambda}(t)\}_{t\geq 0}$ with $H>1$ and $\lambda>0$ is a continuous semimartingale with the canonical decomposition \begin{equation}\label{thm:THPsemimartingaleEq} Z^{1}_{H,\lambda}(t)=\int_{0}^{t} M_{H,\lambda}(s)\ ds \end{equation} where \begin{equation}\label{eq:THPdef2} M_{H,\lambda}(s):={\int^{+\infty}_{-\infty}(s-y)_{+}^{H-\frac{3}{2}} e^{-\lambda(s-y)_{+}}\ B(dy)} . \end{equation} Moreover, $\{Z^{1}_{H,\lambda}(t)\}_{t\geq 0}$ is a finite variation process. \end{thm} \begin{proof} Let $\{{\mathcal{F}}^{B}_{t}\}_{t\geq 0}$ be the $\sigma$-algebra generated by $\{B_s:0\leq s\leq t\}$. Given a function $g:{\mathbb R}\to {\mathbb R}$ such that $g(t)=0$ for all $t<0$, and \begin{equation}\label{gtCheridito} g(t)=C+\int_{0}^{t}h(s)\ ds,\quad\text{for all $t>0$}, \end{equation} for some $C\in\mathbb{R}$ and some $h\in L^{2}(\mathbb{R})$, a result of Cheridito \cite[Theorem 3.9]{Cheridito} shows that the Gaussian stationary increment process \begin{equation}\label{CheriditoDef} Y^{g}_{t}:=\int_{\mathbb{R}}[g(t-y)-g(-y)]\ B(dy),\ t\geq 0 \end{equation} is a continuous $\{{\mathcal{F}}^{B}_{t}\}_{t\geq 0}$ semimartingale with canonical decomposition \begin{equation} Y^{g}_{t}=g(0)B_{t}+\int_{0}^{t}\int_{-\infty}^{s}h(s-y)B(dy)ds , \end{equation} and conversely, that if \eqref{CheriditoDef} defines a semimartingale on $[0,T]$ for some $T>0$, then $g$ satisfies these properties. Define $g(t)=0$ for $t\leq 0$ and \begin{equation} g(t):=\int_{0}^{t}s^{H-\frac{3}{2}}e^{-\lambda s}\ ds\quad\text{for $t>0$.} \end{equation} It is easy to check that the function $g(t-y)-g(-y)$ is square integrable over the entire real line for any $H>\frac{1}{2}$ and $\lambda>0$ (See Lemma \ref{lem:g squar integrable}). Next observe that \eqref{gtCheridito} holds with $C=0$, $h(s)=0$ for $s<0$ and \begin{equation} h(s):=s^{H-\frac{3}{2}}e^{-\lambda s}\in L^{2}(\mathbb{R}) \end{equation} for any $H>1$ and $\lambda>0$. Then it follows from \cite[Theorem 3.9]{Cheridito} that $THP^{1}$ is a continuous semimartingale with canonical decomposition \begin{equation} \begin{split} Z^{1}_{H,\lambda}&=\int_{-\infty}^{+\infty}\int_{0}^{t}(s-y)_{+}^{H-\frac{3}{2}} e^{-\lambda(s-y)_{+}}\ ds\ B(dy)\\ &=\int_{0}^{t}\int_{-\infty}^{+\infty}(s-y)_{+}^{H-\frac{3}{2}} e^{-\lambda(s-y)_{+}} B(dy)\ ds\\ \end{split} \end{equation} which reduces to \eqref{thm:THPsemimartingaleEq}. Since $C=0$, Theorem 3.9 in \cite{Cheridito} implies that $\{Z^{1}_{H,\lambda}(t)\}$ is a finite variation process. \end{proof} \begin{rem} {\emph{ When $H=\frac{3}{2}$ and $\lambda>0$, the Gaussian stochastic process \eqref{eq:THPdef2} is an Ornstein-Uhlenbeck process. When $H>1$ and $\lambda>0$, it is a one dimensional Mat\'ern stochastic process \cite{Banerjee,Gneiting,Handcock}, also called a ``fractional Ornstein-Uhlenbeck process'' in the physics literature \cite{lim}. It follows from Knight \cite[Theorem 6.5]{Knight} that $M_{H,\lambda}(t)$ is a semimartingale in both cases.}} \end{rem} Cheridito \cite[Theorem 3.9]{Cheridito} provides a necessary and sufficient condition for the process \eqref{CheriditoDef} to be a semimartingale, and then it is not hard to check that $THP^{1}$ is {\em not a semimartingale} in the remaining case when $\frac{1}{2}<H<1$. \section{ARTFIMA time series; Definition and basic properties}\label{sec4} In this section, we first recall the definition of the autoregressive tempered fractionally integrated moving average (ARTFIMA) time series and some of its basic properties such as the covariance function and spectral density. The tempered fractional difference operator is defined by: \begin{equation}\label{eq:TFdiffDef} \Delta^{\alpha,\lambda}_h f(x)=\sum_{j=0}^\infty w_j e^{-\lambda jh}f(x-jh) \quad\text{with}\quad w_j:=(-1)^j\binom \alpha j=\frac{(-1)^j\Gamma(1+\alpha)}{j!\Gamma(1+\alpha-j)} \end{equation} for $\alpha>0$ and $\lambda>0$, where $\Gamma(\cdot)$ is the Euler gamma function. If $\lambda=0$ and $\alpha$ is a positive integer, then equation \eqref{eq:TFdiffDef} reduces to the usual definition of the fractional difference operator. The ARMA$(p,q)$ model, which combines an autoregression of order $p$ with a moving average of order $q$, is defined by \begin{equation}\label{eq:ARMA} X_t-\sum_{j=1}^p \phi_jX_{t-j}=Z_t+\sum_{i=1}^q \theta_i Z_{t-i} \end{equation} where $\{Z_t\}$ is an i.i.d.\ sequence of uncorrelated random variables (white noise). We now recall the definition of the ARTFIMA $(p,\alpha,\lambda,q)$. \begin{defn} The discrete time stochastic process $\{X_t\}$ is called an {\it autoregressive tempered fractional integrated moving average} , ${ARTFIMA}\ (p,\lambda,\alpha,q)$, if \begin{equation} \Delta^{\alpha,\lambda}_{1}X{_t}=\sum_{j=0}^\infty w_j e^{-\lambda jh}X_{t-j}, \end{equation} follows an ARMA$(p,q)$ model. \end{defn} Let $\{X_t\}$ be an ARTFIMA $(0,\lambda,\alpha,0)$ process. Then, \begin{equation*} X_t=\Delta^{-\alpha,\lambda}_1 Z_{t}=\sum_{j=0}^{\infty}(-1)^j e^{-\lambda j}\binom{-\alpha}{j}Z_{t-j},\\ \end{equation*} where $\Delta^{-\alpha,\lambda}_1$ is the inverse operator of $\Delta^{\alpha,\lambda}_1$ and can be defined by \eqref{eq:TFdiffDef}. In other words, $X_t=\Delta^{-\alpha,\lambda}_1 Z_t$, is a tempered fractionally integrated ARMA$(p,q)$ model. The fractional integration operator $\Delta^{-\alpha,\lambda}_1$, the inverse of $\Delta^{\alpha,\lambda}_1$, is also defined by \eqref{eq:TFdiffDef}. We refer the reader to \cite{TFC} to find more details about the tempered fractional difference operator. In this paper we are interested in the case $ARTFIMA\ (0,\alpha,\lambda,0)$. \begin{rem} {\emph {Since $\{Z_t\}$ is stationary and \begin{equation*} \sum_{j=0}^{\infty}(-1)^j e^{-\lambda j}\binom{-\alpha}{j}=(1-e^{-\lambda})^{-\alpha}<\infty \end{equation*} for any $\alpha>0$ and $\lambda>0$, Proposition 3.1.2 in \cite{BrockwellDavisTSTM} implies that the series \begin{equation*} X_t=\Delta^{-\alpha,\lambda}_1 Z_{t}=\sum_{j=0}^{\infty}(-1)^j e^{-\lambda j}\binom{-\alpha}{j}Z_{t-j} \end{equation*} is stationary and converges absolutely with probability one.}} \end{rem} \begin{rem} {\emph{Peiris \cite{Peiris} has proposed a generalized autoregressive GAR$(p)$ time series model $(1-\beta B)^\alpha X_t=Z_t$ for applications in finance, where $|\beta|<1$. Taking $\beta=e^{-\lambda}$ we obtain the ARTFIMA$(0,\alpha,\lambda,0)$ model.}} \end{rem} We next state the spectral density and covariance function of ARTFIMA $(0,\alpha,\lambda,0)$. \begin{thm}\label{thm:Proprties} Let $\{X_t\}$ be an ARTFIMA $(0,\alpha,\lambda,0)$ times series. \begin{description} \item [a] $\{X_t\}$ has the spectral density \begin{equation}\label{eq:spectral} h(\omega)=\frac{\sigma^2}{2\pi}\Big|1-e^{-(\lambda+i\omega)}\Big|^{-2\alpha}, \end{equation} for $-\pi\leq \omega\leq \pi$. \item [b] The covariance function of $\{X_t\}$ is \begin{equation}\label{eq:covarianceartfima} \gamma_k=\mathbb{E}(X_t X_{t+k})=\frac{\sigma^2}{2\pi}\frac{e^{-\lambda k}\Gamma(\alpha+k)}{\Gamma(\alpha)k!}\ {_2F_1(\alpha;k+\alpha;k+1;e^{-2\lambda})}, \end{equation} where $_2F_1(a;b;c;z)=\sum_{j=0}^{\infty}\frac{\Gamma(a+j)\Gamma(b+j)\Gamma(c)z^j}{\Gamma(a)\Gamma(b)\Gamma(c+j)\Gamma(j+1)}$ is the hypergeometric function. \end{description} \end{thm} \begin{proof} (a) Writing $X_t=\psi_{\lambda}(B)Z_t$, we have $\psi_{\lambda}(B)=(1-e^{-\lambda}B)^{-\alpha}$. Then the general theory of linear filters implies that $X_t$ has spectral density $f_X(k)=|\Psi(e^{-ik})|^2f_Z(k)$ using the complex absolute value (e.g., see \cite{BrockwellDavisTSTM}). That is \begin{equation*} \begin{split} h(\omega)&=\frac{\sigma^2}{2\pi}\psi_{\lambda}(e^{i\omega})\psi_{\lambda}(e^{-i\omega})\\ &=\frac{\sigma^2}{2\pi}\left(1-2e^{-\lambda}\cos{\omega}+e^{-2\lambda}\right)^{-\alpha}\\ &=\Big|1-e^{-(\lambda+i\omega)}\Big|^{-2\alpha} \end{split} \end{equation*} and this gives \eqref{eq:spectral}. In order to show $(b)$, we have \begin{equation*} \begin{split} \gamma_k &=\int_{-\pi}^{\pi}\cos{(k\omega)}h(\omega)\ d\omega\\ &=\int_{-\pi}^{\pi}\frac{\sigma^2}{2\pi}\frac{\cos{(k\omega)}}{\left(1-2e^{-\lambda}\cos{\omega}+e^{-2\lambda}\right)^{\alpha}}\ d\omega\\ &=\int_{0}^{2\pi}\frac{\sigma^2}{2\pi}\frac{(-1)^{k}\cos{(k\omega')}}{\left(1-2e^{-\lambda}\cos{\omega}+e^{-2\lambda}\right)^{\alpha}}\ d\omega'\ [\omega':=\omega+\pi]\\ &=\sigma^2\frac{e^{-\lambda k}\Gamma(k+\alpha)}{\Gamma(\alpha)k!}\ {_2F_1(\alpha;k+\alpha;k+1;e^{-2\lambda})}, \end{split} \end{equation*} where we applied the following integral formula: \begin{equation*} \frac{1}{2\pi}\int_{0}^{2\pi}\frac{\cos{k\omega'}}{\left(1-2z\cos{\omega}+z^2\right)^{\alpha}}\ d\omega' =\frac{z^k \Gamma(k-\alpha)}{\Gamma(\alpha)k!}{_2F_1(\alpha;k+\alpha;k+1;z^2)} \end{equation*} and hence we proved part (b). \end{proof} The next lemma gives the spectral representation of the ARTFIMA $(0,\alpha,\lambda,0)$. We will use this lemma in the next section. \begin{lem}\label{lem:spectralrepresntation ARTFIMA} Let $X^{\lambda}_{k}$ be the ARTFIMA $(0,\alpha,\alpha,0)$ time series such that $X^{\lambda}_{k}=\sum_{j=0}^{\infty}(-1)^j e^{-\lambda j}\binom{-\alpha}{j}Z_{t-j}$. Then, $X^{\lambda}_{k}$ has the spectral representation \begin{equation*} X^{\lambda}_{k}=\int_{-\pi}^{\pi}e^{ik\nu}\ dW_{\lambda}(\nu), \end{equation*} where $dW_{\lambda}(\nu)=\Big(1-e^{-(\lambda+i\nu)}\Big)^{-\alpha}dW$ and $\{W(\nu), -\pi\leq \nu\leq\pi\}$ is a right-continuous orthogonal increment process. \end{lem} \begin{proof} Suppose $\{Z_t\}$ has the spectral representation $Z_t=\int_{-\pi}^{\pi}e^{ik\nu}dW(\nu)$, where $\{W(\nu), -\pi\leq \nu\leq\pi\}$ is a right-continuous orthogonal increment process. Then Theorem 4.10.1 in \cite{BrockwellDavisTSTM} implies that $X^{\lambda}_{k}$ has the spectral representation \begin{equation*} \begin{split} X^{\lambda}_{k}&=\int_{-\pi}^{\pi}e^{ik\nu}\sum_{j=0}^{\infty}(-1)^j e^{-\lambda j}\binom{-\alpha}{j}e^{-ij\nu}\ dW(\nu)\\ &=\int_{-\pi}^{\pi}e^{ik\nu}\Big(1-e^{-(\lambda+i\nu)}\Big)^{-\alpha}\ dW(\nu), \end{split} \end{equation*} and this completes the proof. \end{proof} The next lemma gives the asymptotic result of the covariance function of the ARTFIMA $(0,\alpha,\lambda,0)$. \begin{lem}\label{lem:asymARTFIMA} Let $\{X_t\}$ be an ARTFIMA $(0,\alpha,\lambda,0)$ times series with the covariance function $\gamma_k=\mathbb{E}(X_t X_{t+k})$ given by \eqref{eq:covarianceartfima}. Then \begin{equation*} \gamma_k\sim \frac{\sigma^2}{2\pi}\frac{1}{\Gamma(\alpha)}e^{-\lambda k}k^{\alpha-1}(1-e^{-2\lambda})^{-\alpha} \end{equation*} as $|k|\to\infty$. \end{lem} \begin{proof} By applying \eqref{eq:covarianceartfima} and the fact that $\frac{\Gamma(a+k)}{\Gamma(b+k)}\sim k^{a-b}$, as $k\to\infty$, we have \begin{equation*} \begin{split} \gamma_k&=\sigma^2\frac{e^{-\lambda k}\Gamma(k+\alpha)}{\Gamma(\alpha)k!}\ {_2F_1(\alpha;k+\alpha;k+1;e^{-2\lambda})}\\ &=\sigma^2\frac{e^{-\lambda k}\Gamma(k+\alpha)}{\Gamma(\alpha)k!}\sum_{j=0}^{\infty}\frac{\Gamma(\alpha+j)\Gamma(k+\alpha+j)\Gamma(k+1)e^{-2\lambda j}}{\Gamma(\alpha)\Gamma(k+\alpha)\Gamma(k+j+1)(j)!}\\ & \sim \sigma^2\frac{1}{\Gamma(\alpha)}e^{-\lambda k}k^{\alpha-1} \sum_{j=0}^{\infty}\frac{\Gamma(\alpha+j)e^{-2\lambda j}}{\Gamma(\alpha)(j)!}\\ &=\sigma^2\frac{1}{\Gamma(\alpha)}e^{-\lambda k}k^{\alpha-1}(1-e^{-2\lambda})^{-\alpha}, \end{split} \end{equation*} which gives the desired result. \end{proof} \begin{rem} {\emph{ The ARTFIMA $(0,\alpha,\lambda,0)$ is short memory process, since by Lemma \ref{lem:asymARTFIMA} one can show that $\sum_{k=0}^{\infty}\gamma_k<\infty$. }} \end{rem} \section{Weak Convergence Results}\label{sec5} We now in a position to answer the first and second question. We start with the first one. Assume $H=\frac{1}{2}+\alpha$ for $\alpha>0$ and let $\{Z_j\}_{j\in\mathbb{Z}}$ be a sequence of independent and identically distributed random variables mean zero and variance one . Define the random variables \begin{equation}\label{eq:X definition} Y^{\frac{\lambda}{n}}_{k}:=\sum_{j\in\mathbb{Z}}C^{\frac{\lambda}{n}}_{j} Z_{k-j},\quad k=1,2,\ldots \end{equation} where \begin{equation}\label{eq:Cdef} C^{\frac{\lambda}{n}}_{j}= \begin{cases}\frac{1}{\Gamma(\alpha)} j^{\alpha-1}e^{-\frac{\lambda}{n}j} &\mbox{if } j\geq 1 \\ 0 & \mbox{if } j\leq 0. \end{cases} \end{equation} For $t\geq 0$, we define $S^{\frac{\lambda}{n}}(t)$ as the partial sum of $\{X^{\frac{\lambda}{n}}_{k}\}$, \begin{equation}\label{eq:S definition} S^{\frac{\lambda}{n}}(t):=\sum_{k=1}^{[t]}Y^{\frac{\lambda}{n}}_{k}+(t-[t])Y^{\frac{\lambda}{n}}_{[t]+1},\quad t\geq 0, \end{equation} where $[t]$ is the largest integer less than or equals $t$ and $\sum_{k=1}^{0}=0$. We also define \begin{equation}\label{eq:Xi definition} \xi^{\frac{\lambda}{n}}_{m}(t):=\sum_{j=1-m}^{[t]-m}C^{\frac{\lambda}{n}}_{j}+(t-[t])C^{\frac{\lambda}{n}}_{t+1-m}, \end{equation} for $m\in\mathbb{Z}$ and $t\geq 0$, where $\sum_{j=1-m}^{-m}=0$. Then we have from \eqref{eq:X definition} and \eqref{eq:S definition}, \begin{equation}\label{eq:S and Xi connection} S^{\frac{\lambda}{n}}(t)=\sum_{m\in\mathbb{Z}}\xi^{\frac{\lambda}{n}}_{m}(t) Z_m. \end{equation} On the other hand, \begin{equation}\label{eq:Xi and coefficient connection} \begin{split} \xi^{\frac{\lambda}{n}}_{m}(nt)&=\sum_{j=1-m}^{[nt]-m}C^{\frac{\lambda}{n}}_{j} \sim \int_{-m}^{[nt]-m}j^{\alpha-1}e^{-\frac{\lambda}{n}j}\ dj\\ &=\Big(\frac{n}{\lambda}\Big)^{\alpha}\int_{-\frac{\lambda m}{n}}^{\frac{\lambda }{n}([nt]-m)}{\omega}^{\alpha-1}e^{-\omega}\ d\omega\\ &=\Big(\frac{n}{\lambda}\Big)^{\alpha}\Big[\gamma(\alpha,\frac{\lambda }{n}([nt]-m))-\gamma(-\frac{\lambda m}{n})\Big], \end{split} \end{equation} when $m$ is negative and $|m|$ is large. From \eqref{eq:S and Xi connection} and \eqref{eq:Xi and coefficient connection} we have: \begin{lem} For any $\theta_1,\theta_2,\ldots,\theta_p$, $t_1,t_2,\ldots,t_p\geq 0$, we have \begin{equation*} n^{-2H}\sum_{m\in\mathbb{Z}}\Big|\sum_{r=1}^{p}\theta_r \xi^{\frac{\lambda}{n}}_{m}(nt)\Big|^{2} \rightarrow \int_{-\infty}^{+\infty}\Big|\sum_{r=1}^{p}\theta_r \int_{0}^{t}(s-y)_{+}^{H-\frac{3}{2}}e^{-\lambda(s-y)_{+}}\ ds \Big|^{2}\ dy \end{equation*} as $n\to \infty$. \end{lem} \begin{proof} By applying \eqref{eq:Xi and coefficient connection}, we get: \begin{equation*} \begin{split} &n^{-2H}\sum_{m\in\mathbb{Z}}\Big|\sum_{r=1}^{p}\theta_{r}\xi^{\frac{\lambda}{n}}_{m}(nt_r) \Big|^2\\ &\sim n^{-2H}\Big(\frac{n}{\lambda}\Big)^{2\alpha}\sum_{m\in\mathbb{Z}}\Big|\sum_{r=1}^{p}\theta_{r}\Big[ \gamma(\alpha,\frac{\lambda }{n}([nt]-m))-\gamma(\alpha,-\frac{\lambda m}{n})\Big]\Big|^2\\ &=\Big(\frac{1}{\lambda}\Big)^{2\alpha}n^{2\alpha-2H+1}\ n^{-1}\sum_{m\in\mathbb{Z}}\Big|\sum_{r=1}^{p}\theta_{r}\Big[ \gamma(\alpha,\frac{\lambda }{n}([nt]-m))-\gamma(-\frac{\lambda m}{n})\Big]\Big|^2\\ &\rightarrow \Big(\frac{1}{\lambda}\Big)^{2\alpha}\int_{\mathbb R}\Big|\sum_{r=1}^{p}\theta_{r}\int_{-\lambda y}^{\lambda t_r- \lambda y} x_{+}^{\alpha-1}e^{-(x)_{+}}\ dx\Big|^{2}\ dy,\ ({\textrm{as}}\quad n\to\infty),\\ &\qquad\qquad ({\textrm{define}}\quad \lambda s=x+\lambda y)\\ &=\int_{\mathbb R}\Big|\sum_{r=1}^{p}\theta_{r}\int_{0}^{t_r} (s-y)_{+}^{\alpha-1}e^{-\lambda (s-y)_{+}}\ ds\Big|^{2}\ dy\\ &=\int_{\mathbb R}\Big|\sum_{r=1}^{p}\theta_{r}\int_{0}^{t_r} (s-y)_{+}^{H-\frac{3}{2}}e^{-\lambda (s-y)_{+}}\ ds\Big|^{2}\ dy \end{split} \end{equation*} and this completes the proof. \end{proof} \begin{thm}\label{thm:fdd theorem} The finite dimensional distribution of $n^{-H}S^{\frac{\lambda}{n}}(nt)$ converge in distribution to $Z^{1}_{H,\lambda}(t)$, given by \eqref{eq:THPdefn}, as $n\to\infty$. That is \begin{equation*} \Big(n^{-H}S^{\frac{\lambda}{n}}(nt_1),n^{-H}S^{\frac{\lambda}{n}}(nt_2),\ldots,n^{-H}S^{\frac{\lambda}{n}}(nt_p)\Big) \to \Big(Z^{1}_{H,\lambda}(t_1),Z^{1}_{H,\lambda}(t_2),\ldots,Z^{1}_{H,\lambda}(t_p)\Big) \end{equation*} as $n\to\infty$. \end{thm} \begin{proof} Let $\theta_1,\theta_2,\ldots,\theta_p$, $t_1,t_2,\ldots,t_p\geq 0$. Then by computing the characteristic function of $n^{-H}\sum_{r=1}^{p}\theta_r S^{\frac{\lambda}{n}(nt_r)}$ we get \begin{equation}\label{eq:converegence chf1} \begin{split} \mathbb{E}\Big[\exp\{in^{-H}\sum_{r=1}^{p}\theta_r S^{\frac{\lambda}{n}(nt_r)}\}\Big] &=\mathbb{E}\Big[\exp\{in^{-H}\sum_{m\in\mathbb{Z}}\sum_{r=1}^{p} \theta_r \xi^{\frac{\lambda}{n}}_{m}(nt_r)Z_m\}\Big]\\ &=\prod_{m\in\mathbb{Z}}\Big[\exp\{in^{-H}\sum_{r=1}^{p} \theta_r \xi^{\frac{\lambda}{n}}_{m}(nt_r)Z_m\}\Big]\\ &=\prod_{m\in\mathbb{Z}}\exp\{{-n^{-2H}|\sum_{r=1}^{p}\theta_r \xi^{\frac{\lambda}{n}}_{m}(nt_r)|^{2}}\}\\ &=\exp\{{-\sum_{m\in\mathbb{Z}}n^{-2H}|\sum_{r=1}^{p}\theta_r \xi^{\frac{\lambda}{n}}_{m}(nt_r)|^2}\}. \end{split} \end{equation} Taking the limit of \eqref{eq:converegence chf1} yields \begin{equation*} \begin{split} &\lim_{n\to\infty}\mathbb{E}\Big[\exp\{i n^{-H}\sum_{r=1}^{p}\theta_r S^{\frac{\lambda}{n}(nt_r)}\}\Big]= \exp\Big[-\lim_{n\to\infty}n^{-2H}\sum_{m\in\mathbb{Z}}\Big|\sum_{r=1}^{p}\theta_r \xi^{\frac{\lambda}{n}}_{m}(nt_r)\Big|^2\Big]\\ &=\exp\Big\{\int_{\mathbb R}\Big|\sum_{r=1}^{p}\theta_{r}\int_{0}^{t_r} (s-y)_{+}^{H-\frac{3}{2}}e^{-\lambda (s-y)_{+}}\ ds\Big|^{2}\ dy\Big\}\\ &=\mathbb{E}\Big[\exp\{{i\sum_{r=1}^{p}\theta_r Z^{1}_{H,\lambda}(t_r)}\}\Big] \end{split} \end{equation*} as $n\to \infty$ and this completes the proof. \end{proof} \begin{thm}\label{thm:tightnessTheorem1} Let $\{Z_j\}_{j\in\mathbb{Z}}$ be a sequence of i.i.d random variables with mean zero and finite variance. Then $n^{-H}S^{\frac{\lambda}{n}}(nt)$ converges weakly to $Z^{1}_{H,\lambda}(t)$, given by \eqref{eq:THPdefn}, in $C[0,1]$ ,as $n\to\infty$ $(C[0,1]$ is the space of all continuous functions defined on $[0,1])$. That is \begin{equation} n^{-H}S^{\frac{\lambda}{n}}(nt)\mathbb Rightarrow Z^{1}_{H,\lambda}(t), \end{equation} where $\mathbb Rightarrow$ means weak convergence in $C[0,1]$. \end{thm} \begin{proof} In Theorem \ref{thm:fdd theorem}, We have shown the finite dimensional convergence of $n^{-H}S^{\frac{\lambda}{n}}(nt)$ to $Z^{1}_{H,\lambda}(t)$. Therefore, here, we just need to prove the tightness of $n^{-H}S^{\frac{\lambda}{n}}(nt)$. We show that for $0\leq t_1 \leq t_2\leq 1$, \begin{equation} \mathbb{E}\Big[\Big|n^{-H}S^{\frac{\lambda}{n}}(nt_2)-n^{-H}S^{\frac{\lambda}{n}}(nt_1)\Big|^{2}\Big]\leq C |t_2-t_1|^{2H}, \end{equation} where $C$ is a constant. First apply \eqref{eq:Xi and coefficient connection} to get \begin{equation}\label{eq:tightness1} \begin{split} &\sum_{m\in\mathbb{Z}}|\xi^{\frac{\lambda}{n}}_{m}(nt_2)-\xi^{\frac{\lambda}{n}}_{m}(nt_1)|\\ &\leq \Big(\frac{n}{\lambda}\Big)^{2\alpha}\int_{\mathbb R}\Big|\int_{\frac{\lambda}{n}([nt_1]-x)}^{\frac{\lambda}{n}([nt_2]-x)}\omega^{\alpha-1}_{+}e^{-(\omega)_{+}}\ d\omega\Big|^{2}\ dx\\ &=\int_{\mathbb R}\Big|\int_{[nt_1]-x}^{[nt_2]-x}y^{\alpha-1}_{+}e^{-\frac{\lambda}{n}(y)_{+}}\ dy\Big|^{2}\ dx\qquad (y:=\frac{n\omega}{\lambda})\\ &=\int_{\mathbb R}\Big|\int_{0}^{[nt_2]-[nt_1]}\Big(z+(ns-x)\Big)^{\alpha-1}_{+}e^{-\frac{\lambda}{n}(z+(ns-x))_{+}}\ dz\Big|^{2}\ dx\qquad (z:=y-(ns-x))\\ &\leq C\int_{\mathbb R}\Big|(nt_2-x)_{+}^{\alpha}-(nt_1-x)_{+}^{\alpha}\Big|^{2}. \end{split} \end{equation} Maejima \cite{Maejima} proved that \begin{equation}\label{eq:maejima upper bound} \int_{\mathbb R}\Big||nt_2-x|^{\alpha}-|nt_1-x|^{\alpha}\Big|^{2}\leq n^{1+2\alpha}(t_2-t_1)^{1+2\alpha}. \end{equation} Hence by applying \eqref{eq:tightness1} and \eqref{eq:maejima upper bound} we have \begin{equation*} \begin{split} &\mathbb{E}\Big[\Big|n^{-H}\Big(S^{\frac{\lambda}{n}}(nt_2)-S^{\frac{\lambda}{n}}(nt_1)\Big)\Big|^{2}\Big]\\ &=n^{-2H}\mathbb{E}\Big(\sum_{m\in\mathbb{Z}}\Big|\Big(\xi^{\frac{\lambda}{n}}_{m}(nt_2)- \xi^{\frac{\lambda}{n}}_{m}(nt_1)\Big)Z_m\Big|^{2} \Big)\\ &=n^{-2H}\sigma^2\sum_{m\in\mathbb{Z}}\Big|\xi^{\frac{\lambda}{n}}_{m}(nt_2)-\xi^{\frac{\lambda}{n}}_{m}(nt_1)\Big|^{2}\\ &\leq C\sigma^2 n^{-2H}n^{1+2\alpha}|t_2-t_1|^{1+2\alpha}\quad (H=\alpha+\frac{1}{2})\\ &\leq C|t_2-t_1|^{2H}. \end{split} \end{equation*} Thus the tightness of $n^{-H}S^{\frac{\lambda}{n}}(nt)$ follows from Theorem 12.3 in \cite{Bill} and this completes the proof. \end{proof} \begin{thm}\label{thm:ARTFIMACONVERGENCE} Let $\alpha>0$ and $X^{\lambda}_{k}$ be the ARTFIMA $(0,\alpha,\lambda,0)$. Suppose \begin{equation*} T^{\frac{\lambda}{n}}(t):=\sum_{k=1}^{[t]}X^{\frac{\lambda}{n}}_{k}+(t-[t])X^{\frac{\lambda}{n}}_{[t]+1},\quad t\geq 0. \end{equation*} Then, \begin{equation*} n^{-H}T^{\frac{\lambda}{n}}(nt)\mathbb Rightarrow Z^{1}_{H,\lambda}(t) \end{equation*} as $n\to\infty$ in $C[0,1]$. \end{thm} \begin{proof} It follows from Stirling's approximation that \begin{equation*} \omega^{\frac{\lambda}{n}}_{j}=(-1)^{j}\binom{-\alpha}{j}e^{-\frac{\lambda}{n}}\sim \frac{\alpha}{\Gamma(1+\alpha)}j^{\alpha-1}e^{-\frac{\lambda}{n}} =C^{\frac{\lambda}{n}}_{j} \quad\text{as $j\to\infty$,} \end{equation*} where $C^{{\lambda}}_{j}$ is from \eqref{eq:Cdef}, see \cite[p.\ 24]{FCbook}. Hence for any $\varepsilonilon>0$ there exists some positive integer $N$ such that \begin{equation}\label{eq:bounds} (1-\varepsilonilon)C^{\frac{\lambda}{n}}_{j}<\omega^{\frac{\lambda}{n}}_{j}<(1+\varepsilonilon)C^{\frac{\lambda}{n}}_{j} \end{equation} for all $j>N$. It follows that \begin{equation}\label{eq:upperboundresult} \begin{split} \sum_{j=0}^{\infty}|\omega^{\frac{\lambda}{n}}_{j}|^{2}&\leq \Big[\sum_{j=0}^{N}|\omega^{\frac{\lambda}{n}}_{j}|^{2}+(1+\varepsilonilon)^{2} \sum_{j=N+1}^{\infty}|C^{\frac{\lambda}{n}}_{j}|^{2}\Big]\\ &\leq\Big[\sum_{j=0}^{N}|\omega^{\frac{\lambda}{n}}_{j}|^{2}+(1+\varepsilonilon)^{2} \sum_{j=0}^{\infty}|C^{\frac{\lambda}{n}}_{j}|^{2}\Big]\\ \end{split} \end{equation} and consequently, \begin{equation}\label{eq:chflower} \begin{split} \mathbb{E}\Big[\exp\{i\theta \Delta^{-\alpha,\frac{\lambda}{n}}_{1}Z_t\}\Big]&= \exp\left\{-\theta^2\sigma^2\sum_{j=0}^{\infty}\Big|{\omega}^{\frac{\lambda}{n}}_{j}\Big|^{2}\right\}\\ &\geq C_{1}\exp\left\{-(1+\varepsilonilon^2)\theta^2\sigma^2\sum_{j=0}^{\infty}\Big|{C}^{\frac{\lambda}{n}}_{j}\Big|^{2}\right\}\\ &= C_{1}\mathbb{E}\Big[\exp\{i(1+\varepsilonilon)\theta X^{\frac{\lambda}{n}}_{t}\}\Big], \end{split} \end{equation} where $C_1=\exp\left\{-\theta^2\sigma^2\sum_{j=0}^{N}\Big|{\omega}^{\frac{\lambda}{n}}_{j}\Big|^{2}\right\}$ is a finite positive constant. Similarly, \begin{equation}\label{eq:lowerboundresult} \sum_{j=0}^{\infty}|\omega^{\frac{\lambda}{n}}_{j}|^{2}\geq \Big[\sum_{j=0}^{N}|\omega^{\frac{\lambda}{n}}_{j}|^{2}+(1-\varepsilonilon)^{2} \sum_{j=N+1}^{\infty}|C^{\frac{\lambda}{n}}_{j}|^{2}\Big],\\ \end{equation} so that \begin{equation}\label{eq:chfupper} \begin{split} \mathbb{E}\Big[\exp\{i\theta \Delta^{-\alpha,\frac{\lambda}{n}}_{1}Z_t\}\Big]&= \exp\left\{-\theta^2\sigma^2\sum_{j=0}^{\infty}\Big|{\omega}^{\frac{\lambda}{n}}_{j}\Big|^{2}\right\}\\ &\leq C_{2}\exp\left\{-(1-\varepsilonilon^2)\theta^2\sigma^2\sum_{j=0}^{\infty}\Big|{C}^{\frac{\lambda}{n}}_{j}\Big|^{2}\right\}\\ &=C_{2}\mathbb{E}\Big[\exp\{i(1-\varepsilonilon)\theta X^{\frac{\lambda}{n}}_{t}\}\Big], \end{split} \end{equation} where $C_2=\exp\left\{-\theta^2\sigma^2\sum_{j=0}^{N}\Big|{\omega}^{\frac{\lambda}{n}}_{j}\Big|^{2}\right\}$ is a finite positive constant. From \eqref{eq:chflower} and \eqref{eq:chfupper} we have : \begin{equation}\label{eq:lowerupper chf} C_{1}\mathbb{E}\Big[\exp\{i(1+\varepsilonilon)\theta X^{\frac{\lambda}{n}}_{t}\}\Big]\leq \mathbb{E}\Big[\exp\{i\theta \Delta^{-\alpha,\frac{\lambda}{n}}_{1}Z_t\}\Big]\leq C_{2}\mathbb{E}\Big[\exp\{i(1-\varepsilonilon)\theta X^{\frac{\lambda}{n}}_{t}\}\Big] \end{equation} for any $\varepsilon>0$. The proof now follows from \eqref{eq:lowerupper chf} and Theorem \ref{thm:fdd theorem} and Theorem \eqref{thm:tightnessTheorem1} by letting $\varepsilonilon\to 0$. \end{proof} Next, we answer the second question that we had in the introduction. Our approach follows that of Pipiras and Taqqu \cite{PipirasTaqqu2}. For $m\in\mathbb{N}\cup\{\infty\}$, we define the approximation \begin{equation*} f^{+}_{n,m}=\sum_{j=0}^{m}f\Big(\frac{j}{n}\Big)1_{[\frac{j}{n},\frac{(j+1)}{n}]}, \qquad f^{-}_{n,m}=\sum_{j=-m}^{-1}f\Big(\frac{j}{n}\Big)1_{[\frac{j}{n},\frac{(j+1)}{n}]}, \end{equation*} \begin{equation*} f^{+}_{n}=f^{+}_{n,\infty}, \qquad f^{-}_{n}=f^{+}_{n,\infty},\qquad f_{m}=f^{+}_{n}+f^{-}_{n}. \end{equation*} The following theorem answers the third question that we had in the introduction. \begin{thm}\label{thm:the third question} Let $\alpha>0$ and $X^{\lambda}_{j}$ be the ARTFIMA $(0,\alpha,\lambda,0)$ times series. Suppose also that the following, condition $A$, is satisfied: \begin{equation*} Condition A:\quad f,f^{\pm}_{n}\in\mathcal{A}_{2}, \|f^{\pm}_{n}-f^{\pm}_{n,m}\|_{\mathcal{A}_{2}}\to 0\quad as\ m\to\infty, \|f-f_n\|_{\mathcal{A}_{2}}\to 0\quad as\ n\to\infty. \end{equation*} Then, \begin{equation*} n^{-H}\sum_{k=0}^{+\infty}f\Big(\frac{k}{n}\Big)X^{\frac{\lambda}{n}}_{k}\rightarrow \int_{\mathbb R}f(u)Z^{1}_{H,\lambda}(du) \end{equation*} in distribution as $n\to\infty$. \end{thm} \begin{proof} For the proof, we suppose \begin{equation*} W^{\frac{\lambda}{n}}_{n}=\frac{1}{n^{\frac{1}{2}+\alpha}}\sum_{j=-\infty}^{+\infty}f\Big(\frac{j}{n}\Big)X^{\frac{\lambda}{n}}_{j},\quad \ W=\int_{\mathbb R}f(u)Z^{1}_{H,\lambda}(du). \end{equation*} The Wiener integral $W$ is well-defined, since $f\in\mathcal{A}_{2}$. To show that the series $W^{\frac{\lambda}{n}}_{n}$ is well-defined, apply the spectral representation of $X^{\frac{\lambda}{n}}_{k}$ by Lemma \ref{lem:spectralrepresntation ARTFIMA} and write \begin{equation*} \begin{split} \frac{1}{n^{\frac{1}{2}+\alpha}}\sum_{j=0}^{m}f\Big(\frac{j}{n}\Big)X^{\frac{\lambda}{n}}_{j}&= \frac{1}{n^{\frac{1}{2}+\alpha}}\int_{-\pi}^{\pi}\Big(\sum_{j=0}^{m}f\Big(\frac{j}{n}\Big)e^{ijx}\Big)dZ_{\frac{\lambda}{n}}(x)\\ &=\frac{1}{n^{\frac{1}{2}+\alpha}}\int_{\mathbb R}\Big(\sum_{j=0}^{m}f\Big(\frac{j}{n}\Big)e^{\frac{ij\omega}{n}}\Big)1_{[-\pi n,\pi n]}(\omega)dZ_{\frac{\lambda}{n}}\Big(\frac{\omega}{n}\Big)\\ &=\frac{1}{n^{\alpha}-\frac{1}{2}}\int_{\mathbb R}\Bigg(\sum_{j=0}^{m}f\Big(\frac{j}{n}\Big)\frac{e^{\frac{i(j+1)\omega}{n}}-e^{\frac{ij\omega}{n}}}{i\omega} \Bigg)\frac{\frac{i\omega}{n}}{e^{\frac{i\omega}{n}}-1}1_{[-\pi n,\pi n]}(\omega)dZ_{\frac{\lambda}{n}}\Big(\frac{\omega}{n}\Big)\\ &=\frac{1}{n^{\alpha}-\frac{1}{2}}\int_{\mathbb R}\widehat{f_{n,m}}(\omega)\frac{\frac{i\omega}{n}}{e^{\frac{i\omega}{n}}-1}1_{[-\pi n,\pi n]}(\omega)dZ_{\frac{\lambda}{n}}\Big(\frac{\omega}{n}\Big) \end{split} \end{equation*} and hence we get \begin{equation*} \begin{split} \mathbb{E}\Bigg|\frac{1}{n^{\frac{1}{2}+\alpha}}\sum_{j=0}^{m}f\Big(\frac{j}{n}\Big)X^{\frac{\lambda}{n}}_{j}\Bigg|^{2} &=\int_{\mathbb R}\Big|\widehat{f_{n,m}}(\omega)\Big|^{2}\Bigg| \frac{\frac{i\omega}{n}}{e^{\frac{i\omega}{n}}-1}\Bigg|^{2}1_{[-\pi n,\pi n]}(\omega) \frac{1}{n^{2\alpha}}h_{\frac{\lambda}{n}}\Big(\frac{\omega}{n}\Big)\ d\omega\\ &\leq C\int_{\mathbb R}\Big|\widehat{f_{n,m}}(\omega)\Big|^{2}\frac{1}{n^{2\alpha}}\Bigg(\frac{\lambda^2}{n^2}+\frac{\omega^2}{n^2}\Bigg)^{-\alpha}\ d\omega.\\ &=C\|f^{+}_{n,m}\|^{2}_{\mathcal{A}_{2}} \end{split} \end{equation*} Then, by Condition $A$, \begin{equation*} \mathbb{E}\Bigg|\frac{1}{n^{\frac{1}{2}+\alpha}}\sum_{j=m_1+1}^{m_2}f\Big(\frac{j}{n}\Big)X^{\frac{\lambda}{n}}_{j}\Bigg|^{2}\leq C\|f^{+}_{n,m_2}-f^{+}_{n,m_1}\|^{2}_{\mathcal{A}_{2}}\to 0 \end{equation*} as $m_1,m_2\to \infty$. We next show that $W^{\frac{\lambda}{n}}_{n}$ convergence to $W$ in distribution. Recall from Theorem \ref{thm:A2space} that the set of elementary functions are dense in $\mathcal{A}_{2}$ and then there exists a sequence of elementary functions $f^{l}$ such that $\|f^{l}-f\|_{\mathcal{A}_{2}}\to 0$, as $l\to\infty$. Now, assume \begin{equation*} W^{\frac{\lambda}{n},l}_{n}=\frac{1}{n^{\frac{1}{2}+\alpha}}\sum_{j=-\infty}^{+\infty}f^{l}\Big(\frac{j}{n}\Big)X^{\frac{\lambda}{n}}_{j},\quad \ W^{l}=\int_{\mathbb R}f^{l}(u)Z^{1}_{H,\lambda}(du). \end{equation*} Observe that $W^{\frac{\lambda}{n},l}_{n}$ is well-define, since $W^{\frac{\lambda}{n},l}_{n}$ has a finite number elements and the elementary function $f^{l}$ is in $\mathcal{A}_{2}$. According to Theorem 4.2. in Billingsley \cite{Bill}, the series $W^{\frac{\lambda}{n}}_{n}$ convergence in distribution to $W$ if \begin{description} \item[Step 1] $W^{l}\to W$, as $l\to\infty$, \item[Step 2] for all $l\in\mathbb{N}$, $W^{\frac{\lambda}{n},l}_{n}\to W^{l}$, as $n\to\infty$, \item[Step 3] $\limsup_{l}\limsup_{n}\mathbb{E}\Big|W^{\frac{\lambda}{n},l}_{n}-W^{\frac{\lambda}{n}}_{n}\Big|^{2}=0$. \end{description} Step $1$: The random variables $W^{l}$ and $W$ have normal distribution with mean zero and finite variance $\|f^{l}\|_{\mathcal{A}_{2}}$ and $\|f\|_{\mathcal{A}_{2}}$, respectively (See Theorem \ref{thm:A2space} and Definition \ref{defn:stochIntFourier}). Therefore $\mathbb{E}\Big|W^{l}-W\Big|^{2}=\|f^{l}-f\|_{\mathcal{A}_{2}}\to 0$ as $l\to\infty$. Step $2$: Observe that $W^{\frac{\lambda}{n},l}_{n}=\int_{\mathbb R}f^{l}(u)T^{\frac{\lambda}{n}}(du)$. Because $f^l$ is an elementary function, then the integral $W^{\frac{\lambda}{n},l}_{n}$ depends on the process $T^{\frac{\lambda}{n}}$ through a finite number of the points only. Now, Theorem \ref{thm:fdd theorem} and Theorem \ref{thm:ARTFIMACONVERGENCE} imply that $W^{\frac{\lambda}{n},l}_{n}\to W^{l}$, in distribution ,as $n\to\infty$, for all $i\in\mathbb{N}$. Step $3$: For this step, we follow the same way as Pipiras and Taqqu did in Theorem 3.2 \cite{PipirasTaqqu2}. We have $\mathbb{E}\Big|W^{\frac{\lambda}{n},l}_{n}-W^{\frac{\lambda}{n}}_{n}\Big|^{2}\leq C\|f^{l}_{n}-f_{n}\|_{\mathcal{A}_{2}}$, where \begin{equation*} f^{l}_{n}:=\sum_{j}f^{l}\Big(\frac{j}{n}\Big)1_{(\frac{j}{n},\frac{(j+1)}{n})}(u). \end{equation*} Note that $f^{l}$ is an elementary function and therefore $\widehat{f^{l}_{n}}$ converges to $\widehat{f^{l}}$ at every point and $\Big|\widehat{f^{l}_{n}}(\omega)-\widehat{f^{l}}(\omega)\Big|\leq \widehat{g^{l}}(\omega)$ uniformly in $n$, for some function $\widehat{g^{l}}(\omega)$ which is bounded by $C_1$ and $C_2|\omega|^{-1}$ for all $\omega\in\mathbb R$ (See Theorem 3.2. in \cite{PipirasTaqqu2} for more details). Let $\mu^{\alpha}(d\omega)=|\omega|^{-2\alpha}d\omega$ and $\mu^{\alpha}_{\lambda}(d\omega)=|\lambda^2+\omega^2|^{-2\alpha}d\omega$ be the measures on the real line for $\alpha>0$. Then apply the dominated converges theorem to see that \begin{equation*} \begin{split} \|f^{l}_{n}-f^{l}\|^{2}_{\mathcal{A}_{2}}&=\|\widehat{f^{l}_{n}}-\widehat{f^{l}}\|_{L^{2}(\mathbb R,\mu^{\alpha}_{\lambda})}\\ &\leq \|\widehat{f^{l}_{n}}-\widehat{f^{l}}\|_{L^{2}(\mathbb R,\mu^{\alpha})}\to 0, \end{split} \end{equation*} as $n\to\infty$. Hence by Condition $A$, the $\limsup_{n}\mathbb{E}\Big|W^{\frac{\lambda}{n},l}_{n}-W^{\frac{\lambda}{n}}_{n}\Big|^{2}\leq C\|f^{l}-f\|^{2}_{\mathcal{A}_{2}}\to 0$ as $l\to\infty$ and this completes the proof. \end{proof} \begin{rem} {\emph{The result of Theorem \ref{thm:the third question} can also be derived by the following condition: \begin{equation*} Condition B:\quad f,f^{\pm}_{n}\in\mathcal{A}_{1}, \|f^{\pm}_{n}-f^{\pm}_{n,m}\|_{\mathcal{A}_{1}}\to 0\quad as\ m\to\infty, \|f-f_n\|_{\mathcal{A}_{1}}\to 0\quad as\ n\to\infty. \end{equation*} Since we have \begin{equation*}\begin{split} \ip fg_{{\mathcal{A}}_{1}}=\Gamma(H-\frac{1}{2})^2\ip {\varphi_f}{\varphi_g}_2&=\Gamma(1-\alpha)^2\ip {\hat\varphi_f}{\hat\varphi_g}_2\\ &=\Gamma(1-\alpha)^2\int_{-\infty}^{+\infty}\widehat{f}(\omega)\overline{\widehat{g}(\omega)}(\lambda^2+\omega^2)^{\frac{1}{2}-H}d\omega=\ip fg_{{\mathcal{A}}_{2}} \end{split}\end{equation*} by the Plancherel Theorem.}} \end{rem} \section{Appendix} Here we recall the definitions of tempered fractional integrals and derivatives and their properties that we used in the pervious sections. \begin{defn}\label{defn:Tempered fractional integral} For any $f\in{L}^{p}({\mathbb{R}})$ (where $1\leq p<\infty$), the positive and negative tempered fractional integrals are defined by \begin{equation}\label{eq:positivetempered fractional integral} {\mathbb I}^{\alpha,\lambda}_{+} f(t)=\frac{1}{\Gamma(\alpha)}\int_{-\infty}^{+\infty} f(u)(t-u)_{+}^{\alpha-1}e^{-\lambda(t-u)_{+}}du \end{equation} and \begin{equation}\label{eq:negativetempered fractional integral} {\mathbb I}^{\alpha,\lambda}_{-} f(t)=\frac{1}{\Gamma(\alpha)}\int_{-\infty}^\infty f(u)(u-t)_{+}^{\alpha-1}e^{-\lambda(u-t)_{+}}du \end{equation} respectively, for any $\alpha>0$ and $\lambda >0$, where ${\Gamma(\alpha)=\int_{0}^{+\infty}e^{-x}x^{\alpha-1}dx}$ is the Euler gamma function, and $(x)_{+}=x I(x>0)$. \end{defn} When $\lambda=0$ these definitions reduce to the (positive and negative) Riemann-Liouville fractional integral \cite{FCbook,oldham,Samko}, which extends the usual operation of iterated integration to a fractional order. When $\lambda=1$, the operator \eqref{eq:positivetempered fractional integral} is called the Bessel fractional integral \cite[Section 18.4]{Samko}. We state the following lemma without the proof. We refer the reader to see Lemma 2.2 in \cite{MeerschaertsabzikarSPA}. \begin{lem}\label{lem:TFI and Lp} For any $\alpha>0$, $\lambda>0$, and $p\geq 1$, ${\mathbb I}^{\alpha,\lambda}_{\pm}$ is a bounded linear operator on $L^{p}(\mathbb{R})$ such that \begin{equation}\label{TFIbound} \|{\mathbb I}^{\alpha,\lambda}_{\pm} f\|_{p}\leq \lambda^{-\alpha}\|f\|_{p} \end{equation} for all $f\in L^{p}(\mathbb{R})$. \end{lem} Next we discuss the relationship between tempered fractional integrals and Fourier transforms. Recall that the Fourier transform \[{\mathcal{F}}[f](\omega)=\hat{f}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{i\omega x}f(x)dx\] for functions $f\in L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$ can be extended to an isometry (a linear onto map that preserves the inner product) on $L^{2}(\mathbb{R})$ such that \begin{equation}\label{eq:FouriertransformL2} \widehat{f}(\omega)=\lim_{n\to\infty}\frac{1}{\sqrt{2\pi}}\int_{-n}^{n}e^{-i\omega x}f(x)\ dx \end{equation} for any $f\in L^{2}(\mathbb{R})$, see for example \cite[Theorem 6.6.4]{Kierat}. \begin{lem}\label{lem:FourierTFI} For any $\alpha>0$ and $\lambda>0$ we have \begin{equation}\label{TFI-FTeq} {\mathcal{F}}[{\mathbb I}^{\alpha,\lambda}_{\pm} f](\omega)=\hat{f}{(\omega)}(\lambda\pm i\omega)^{-\alpha} \end{equation} for all $f\in{L}^{1}({\mathbb{R}})$ and all $f\in{L}^{2}({\mathbb{R}})$. \end{lem} \begin{proof} See Lemma 6.6 in \cite{MeerschaertsabzikarSPA}. \end{proof} Next we consider the inverse operator of the tempered fractional integral, which is called a tempered fractional derivative. For our purposes, we only require derivatives of order $0<\alpha<1$, and this simplifies the presentation. \begin{defn}\label{TFDdef} The positive and negative tempered fractional derivatives of a function $f:\mathbb R\to\mathbb R$ are defined as \begin{equation}\label{eq:temperedfractionalderivativepositive} {\mathbb{D}}^{\alpha,\lambda}_{+}f(t)={\lambda}^{\alpha}f(t)+\frac{\alpha}{\Gamma(1-\alpha)}\int_{-\infty}^{t}\frac{f(t)-f(u)}{(t-u)^{\alpha+1}} \,e^{-\lambda(t-u)}du. \end{equation} and \begin{equation}\label{eq:temperedfractionalderivativenegative} {\mathbb{D}}^{\alpha,\lambda}_{-}f(t)={\lambda}^{\alpha}f(t)+\frac{\alpha}{\Gamma(1-\alpha)}\int_{t}^{+\infty}\frac{f(t)-f(u)} {(u-t)^{\alpha+1}}\,e^{-\lambda(u-t)}du \end{equation} respectively, for any $0<\alpha<1$ and any $\lambda>0$. \end{defn} If $\lambda=0$, the definitions \eqref{eq:temperedfractionalderivativepositive} and \eqref{eq:temperedfractionalderivativenegative} reduce to the positive and negative Marchaud fractional derivatives \cite[Section 5.4]{Samko}. Note that tempered fractional derivatives cannot be defined pointwise for all functions $f\in L^{p}(\mathbb{R})$, since we need $|f(t)-f(u)|\to 0$ fast enough to counter the singularity of the denominator $(t-u)^{\alpha+1}$ as $u\to t$. We can extend the definition of tempered fractional derivatives to a suitable class of functions in $L^2(\mathbb R)$. For any $\alpha>0$ and $\lambda>0$ we may define the fractional Sobolev space \begin{equation}\label{def:fracSovolev} W^{\alpha,2}(\mathbb R):=\{f\in L^2(\mathbb R):\int_{\mathbb R} (\lambda^2+\omega^2)^{\alpha}|\hat f(\omega)|^2\,d\omega<\infty\} , \end{equation} which is a Banach space with norm $\|f\|_{\alpha,\lambda}=\|(\lambda^2+\omega^2)^{\alpha/2}\hat f(\omega)\|_2$. The space $W^{\alpha,2}(\mathbb R)$ is the same for any $\lambda>0$ (typically we take $\lambda=1$) and all the norms $\|f\|_{\alpha,\lambda}$ are equivalent, since $1+\omega^2\leq\lambda^2+\omega^2\leq \lambda^2(1+\omega^2)$ for all $\lambda\geq 1$, and $\lambda^2+\omega^2\leq1+\omega^2\leq \lambda^{-2}(1+\omega^2)$ for all $0<\lambda<1$. \begin{defn}\label{TFDdef2} The positive (resp., negative) tempered fractional derivative ${\mathbb D}^{\alpha,\lambda}_{\pm}f(t)$ of a function $f\in W^{\alpha,2}(\mathbb R)$ is defined as the unique element of $L^2(\mathbb R)$ with Fourier transform $\widehat{f}{(\omega)}(\lambda\pm i\omega)^{\alpha}$ for any $\alpha>0$ and any $\lambda>0$. \end{defn} \begin{lem}\label{lem:inversoperator} For any $\alpha>0$ and $\lambda>0$, we have \begin{equation}\label{eq:DIfisf} {\mathbb D}^{\alpha,\lambda}_{\pm}{\mathbb I}^{\alpha,\lambda}_{\pm}f(t)=f(t) \end{equation} for any function $f\in L^{2}(\mathbb{R})$, and \begin{equation}\label{eq:IDfisf} {\mathbb I}^{\alpha,\lambda}_{\pm}{\mathbb D}^{\alpha,\lambda}_{\pm}f(t)=f(t) \end{equation} for any $f\in W^{\alpha,2}(\mathbb R)$. \end{lem} \begin{proof} Given $f\in L^{2}(\mathbb{R})$, note that $g(t)={\mathbb I}^{\alpha,\lambda}_{\pm}f(t)$ satisfies $\hat g(k)=\hat{f}{(\omega)}(\lambda\pm i\omega)^{-\alpha}$ by Lemma \ref{lem:FourierTFI}, and then it follows easily that $g\in W^{\alpha,2}(\mathbb{R})$. Definition \ref{TFDdef2} implies that \begin{equation}\label{eq:FouriertransformD} {\mathcal F}[{\mathbb D}^{\alpha,\lambda}_{\pm}{\mathbb I}^{\alpha,\lambda}_{\pm}f](\omega)={\mathcal F}[{\mathbb D}^{\alpha,\lambda}_{\pm}g](\omega) =\widehat{g}(\omega)(\lambda\pm i\omega)^{\alpha} =\hat f(\omega) , \end{equation} and then \eqref{eq:DIfisf} follows using the uniqueness of the Fourier transform. The proof of \eqref{eq:IDfisf} is similar. \end{proof} Here we collect some well known facts about the modified Bessel function of the second kind and we refer the reader to (Chapter 9, \cite{abramowitz}) for more details. The modified Bessel function $K_{\nu}(x)$ is regular function of $x$. It satisfies the following simple inequality \begin{equation*} K_{\nu}(x)>0\quad\text{for all $x>0$, for all $\nu\in\mathbb R$} \end{equation*} and it has the following asymptotic expansion: \begin{equation*} K_{\nu}(x)\sim 2^{|\nu|-1}\Gamma(|\nu|)x^{-|\nu|}\qquad (\nu\neq 0) \end{equation*} as $x\to 0$. \end{document}

\mathbf{b}egin{document} \title{Log-concavity for series in reciprocal gamma functions and applications} \mathbf{b}egin{center} \parbox{12cm}{ \small\textbf{Abstract.} Euler's gamma function $\Gamma(x)$ is logarithmically convex on $(0,\infty)$. Additivity of logarithmic convexity implies that the function $x\to\sum{f_k\Gamma(x+k)}$ is also log-convex (assuming convergence) if the coefficients are non-negative. In this paper we investigate the series $\sum{f_k\Gamma(x+k)^{-1}}$, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the sum is log-concave if $kf_k^2\geq(k+1)f_{k-1}f_{k+1}$ and is discrete Wright log-concave if $f_k^2\geq{f_{k-1}f_{k+1}}$. We conjecture that the latter condition is in fact sufficient for the log-concavity of the sum. We exemplify our general theorems by deriving known and new inequalities for the modified Bessel, Kummer and generalized hypergeometric functions and their parameter derivatives.} \end{center} \mathbf{b}igskip Keywords: \emph{Gamma function, log-concavity, Tur\'{a}n inequality, hypergeometric functions, modified Bessel function, Kummer function} \mathbf{b}igskip MSC2010: 26A51, 33C20, 33C15, 33C05 \mathbf{b}igskip \paragraph{1. Introduction.} A positive continuous function $f$ defined on a real interval $I\subseteq\mathbf{R}$ is said to be \textbf{Jensen log-concave} if \mathbf{b}egin{equation}\label{eq:log-conc} f(\mu+h)^2\geq f(\mu)f(\mu+2h) \end{equation} for all $h>0$ and all $\mu$ such that $[\mu,\mu+2h]\subseteq{I}$. If inequality (\ref{eq:log-conc}) is strict the function $f$ is called strictly Jensen log-concave. If the sign of the inequality is reversed one talks about Jensen log-convexity (or strict Jensen log-convexity). For continuous functions Jensen log-concavity is equivalent to log-concavity, i.e. concavity of $\log(f)$ (but is weaker in general). It is also equivalent to the seemingly stronger inequality \mathbf{b}egin{equation}\label{eq:wright-log-conc} \phi_{h,s}(\mu)=f(\mu+h)f(\mu+s)-f(\mu)f(\mu+h+s)\geq 0 ~\text{for all}~~h,s>0, \end{equation} which expresses the fact that $\mu\to f(\mu+h)/f(\mu)$ is non-increasing for any $h>0$. We tacitly assume here and below that all arguments lie in $I$. A function satisfying (\ref{eq:wright-log-conc}) is called \textbf{Wright log-concave} \cite[Chapter~I.4]{MPF}. For comparisons of these notions and their higher order analogues see also the recent paper \cite{NRW}. One is also frequently encountered with the situation when (\ref{eq:log-conc}) or (\ref{eq:wright-log-conc}) only holds for integer values of $h$. We will express this fact by saying that $f$ is \textbf{discrete log-concave} or \textbf{discrete Wright log-concave}, respectively. In this case, however, we only have the implication (\ref{eq:wright-log-conc})$\mathbf{R}ightarrow$(\ref{eq:log-conc}), while the reverse implication is not true even for continuous functions. We note that $f$ is discrete Wright log-concave if and only if \mathbf{b}egin{equation}\label{eq:disc-wright} \phi_{1,s}(x)=f(\mu+1)f(\mu+s)-f(\mu)f(\mu+s+1)\geq{0} ~\text{for all}~~s>0. \end{equation} Indeed, (\ref{eq:disc-wright}) says that $f(\mu+1)/f(\mu)$ is non-increasing, so that $f(\mu+2)/f(\mu+1)$ is again non-increasing implying that their product $f(\mu+2)/f(\mu)$ is non-increasing, i.e. satisfies (\ref{eq:wright-log-conc}) with $h=2$. In a similar fashion (\ref{eq:wright-log-conc}) holds for all integer values of $h$ which is discrete Wright log-concavity by definition. Discrete Jensen log-concavity and log-convexity are also frequently referred to as ''Tur\'{a}n type inequalities'' following the classical result of Paul Tur\'{a}n for Legendre polynomials \cite{Turan}. If $f:\mathbb{N}_0\to{\mathbf{R}_+}$ is a sequence, then discrete log-concavity reduces to the inequality $f_k^2\geq{f_{k-1}f_{k+1}}$, $k\in\mathbb{N}$. We additionally require that the sequence $\{f_k\}_{k=0}^{\infty}$ is non-trivial and has no internal zeros: $f_{N}=0~\mathbf{R}ightarrow~f_{N+i}=0$ for all $i\in\mathbb{N}_{0}$. Such sequences are also known as $PF_2$ (P\'{o}lya frequency sub two) or doubly positive \cite{Karlin}. Clearly, if $f$ is (Jensen or Wright) log-concave then $1/f$ is (Jensen or Wright) log-convex. Notwithstanding the simplicity of this relation, several important properties of log-concavity and log-convexity differ. Particularly important is that log-convexity is additive while log-concavity is not. Further, log-convexity is a stronger property than convexity whereas log-concavity is weaker than concavity. In \cite{KS} the second author and Sergei Sitnik initiated the investigation of the following problem: under what conditions on non-negative sequence $\{f_k\}$ and the numbers $a_i$, $b_j$ the function \mathbf{b}egin{equation}\label{eq:problem1} \mu\to f(\mu;x):=\sum\limits_{k=0}^{\infty}f_k\frac{\prod_{i=1}^{n}\Gamma(a_i+\mu+\varepsilon_ik)} {\prod_{j=1}^{m}\Gamma(b_j+\mu+\varepsilon_{n+j}k)}x^k \end{equation} is (discrete) log-concave or log-convex? Here $\Gamma$ is Euler's gamma function and $\varepsilon_l$ can be $1$ or $0$. In \cite{KS} the authors treated the cases of (\ref{eq:problem1}) with $n=1$, $m=0$, $\varepsilon_1=1$; $n=m=1$, $\varepsilon_1=1$, $\varepsilon_2=0$; and $n=m=1$, $\varepsilon_1=0$, $\varepsilon_2=1$. Of course, the log-convexity cases are nearly trivial but the results of \cite{KS} go beyond log-convexity by showing that the function $\phi_{h,s}(\mu;x)$ on the left-hand side of (\ref{eq:wright-log-conc}) has non-positive Taylor coefficients in powers of $x$, so that $x\to-\phi_{h,s}(\mu;x)$ is absolutely monotonic. According to Hardy, Littlewood and P\'{o}lya theorem \cite[Proposition~2.3.3]{NP} this also implies that this function is multiplicatively concave: $\phi_{h,s}(\mu;x^{\lambda}y^{1-\lambda})\geq{\phi_{h,s}(\mu;x)^{\lambda}\phi_{h,s}(\mu;y)^{1-\lambda}}$ for $\lambda\in[0,1]$. In this paper we treat the case $n=0$, $m=1$, $\varepsilon_1=1$. We get slightly different results depending on conditions imposed on the sequence $\{f_{k}\}$. If $f_{k}$ is log-concave without internal zeros (i.e. doubly positive) we prove discrete Wright log-concavity of the series (\ref{eq:problem1}). We conjecture that the true log-concavity holds but we were unable to demonstrate this. If $\{f_kk!\}$ is doubly positive we show that (\ref{eq:problem1}) is log-concave. We do so by establishing non-negativity of the Taylor coefficients of $\phi_{h,s}(\mu;x)$ in powers of $x$ (either for all or only for integer $h>0$). Again, by Hardy, Littlewood and P\'{o}lya theorem this implies that $x\to\phi_{h,s}(\mu;x)$ is multiplicatively convex. The paper is organized as follows: in section~2 we collect several lemmas repeatedly used in the proofs; section~3 comprises two theorems constituting the main content of the paper; in section~4 we give applications to Bessel, Kummer and generalized hypergeometric functions and relate them to several previously known results. \paragraph{2. Preliminaries.} We will need several lemmas which we prove in this section. First, we formulate an elementary inequality we will repeatedly use below. \mathbf{b}egin{lemma}\label{lm:uvrs} Suppose $u,v,r,s>0$, $u=\max(u,v,r,s)$ and $uv>rs$. Then $u+v>r+s$. \end{lemma} \textbf{Proof.} Indeed, dividing by $u$ we can rewrite the required inequality as $r'+s'<1+v'$, where $r'=r/u\in(0,1)$, $s'=s/u\in(0,1)$, $v'=v/u\in(0,1)$. Since $r's'<v'$ by $rs<uv$, the required inequality follows from the elementary inequality $r'+s'<1+r's'$. ~~$\square$ Lemma~\ref{lm:uvrs} is a particular case of a much more general result on logarithmic majorization - see \cite[2.A.b]{MOA}. In the next lemma we say that a sequence has no more than one change of sign if the pattern is $(--\cdots--00\cdots00++\cdots++)$, where zeros and minus signs may be omitted. \mathbf{b}egin{lemma}\label{lm:sum} Suppose $\{f_k\}_{k=0}^{n}$ has no internal zeros and $f_k^2\geq{f_{k-1}f_{k+1}}$, $k=1,2,\ldots,n-1$. If the real sequence $A_0,A_1,\ldots,A_{[n/2]}$ satisfying $A_{[n/2]}>0$ and $\sum\limits_{0\leq{k}\leq{n/2}}\!\!\!\!A_k\geq{0}$ has no more than one change of sign, then \mathbf{b}egin{equation}\label{eq:keysum} \sum\limits_{0\leq{k}\leq{n/2}}f_{k}f_{n-k}A_k\geq{0}. \end{equation} Equality is only attained if $f_k=f_0\mathbf{a}lpha^k$, $\mathbf{a}lpha>0$, and $\sum\limits_{0\leq{k}\leq{n/2}}\!\!\!\!A_k=0$. \end{lemma} \textbf{Proof.} Suppose $f_k>0$, $k=s,\ldots,p$, $s\geq{0}$, $p\leq{n}$. Log-concavity of $\{f_k\}_{k=0}^{n}$ clearly implies that $\{f_{k}/f_{k-1}\}_{k=s+1}^{p}$ is decreasing, so that for $s+1\leq{k}\leq{n-k+1}\leq{p+1}$ $$ \frac{f_k}{f_{k-1}}\geq\frac{f_{n-k+1}}{f_{n-k}}~\Leftrightarrow~f_{k}f_{n-k}\geq f_{k-1}f_{n-k+1}. $$ Since $k\leq{n-k+1}$ is true for all $k=1,2,\ldots,[n/2]$, the weights $f_{k}f_{n-k}$ assigned to negative $A_k$s in (\ref{eq:keysum}) are smaller than those assigned to positive $A_k$s leading to (\ref{eq:keysum}). The equality statement is obvious.~~~$\square$ \mathbf{b}egin{lemma}\label{lm:gammasum} Suppose $m$ is non-negative integer. The inequality \mathbf{b}egin{equation}\label{eq:gammasum} \sum\limits_{k=0}^{m}\frac{1}{k!(m-k)!} \left(\frac{1}{\Gamma(k+\mu+a)\Gamma(m-k+\mu+b)}-\frac{1}{\Gamma(m-k+\mu)\Gamma(k+\mu+a+b)}\right)\geq{0} \end{equation} holds for all $a,b\geq{0}$, $\mu\geq{-1}$ and such that $\mu+a\geq{0}$, $\mu+b\geq{0}$ . Equality is only attained if $ab=0$. \end{lemma} \textbf{Proof.} Using the easily verifiable relations $$ (c)_k=\frac{\Gamma(c+k)}{\Gamma(c)},~~(m-k)!=\frac{(-1)^km!}{(-m)_k}~~\text{and}~~(c)_{m-k}=\frac{(-1)^k(c)_m}{(1-c-m)_k} $$ we obtain \mathbf{b}egin{multline*} \sum\limits_{k=0}^{m}\frac{1}{k!(m-k)!\Gamma(k+\mathbf{a}lpha)\Gamma(m-k+\mathbf{b}eta)}=\frac{1}{m!\Gamma(\mathbf{a}lpha)\Gamma(\mathbf{b}eta)} \sum\limits_{k=0}^{m}\frac{(-1)^k(-m)_k}{k!(\mathbf{a}lpha)_k(\mathbf{b}eta)_{m-k}} \\ =\frac{1}{m!\Gamma(\mathbf{a}lpha)\Gamma(\mathbf{b}eta)(\mathbf{b}eta)_m}\sum\limits_{k=0}^{m}\frac{(-m)_k(1-\mathbf{b}eta-m)_k}{k!(\mathbf{a}lpha)_k} =\frac{(\mathbf{a}lpha+\mathbf{b}eta+m-1)_m}{m!\Gamma(\mathbf{a}lpha)(\mathbf{a}lpha)_m\Gamma(\mathbf{b}eta)(\mathbf{b}eta)_m} \\ =\frac{\Gamma(\mathbf{a}lpha+\mathbf{b}eta+2m-1)}{\Gamma(\mathbf{a}lpha+m)\Gamma(\mathbf{b}eta+m)\Gamma(\mathbf{a}lpha+\mathbf{b}eta+m-1)m!}, \end{multline*} where we have used the Chu-Vandermonde identity \cite[Corollary~2.2.3]{AAR} $$ \sum\limits_{k=0}^{m}\frac{(-m)_k(a)_k}{(c)_kk!}=\frac{(c-a)_m}{(c)_m}. $$ This leads to an explicit evaluation of the left hand side of (\ref{eq:gammasum}) as $$ \frac{\Gamma(2\mu+a+b+2m-1)}{\Gamma(2\mu+a+b+m-1)m!}\left( \frac{1}{\Gamma(m+\mu+a)\Gamma(m+\mu+b)}-\frac{1}{\Gamma(m+\mu)\Gamma(m+\mu+a+b)} \right) $$ For $a,b,\mu>0$, the required inequality reduces to log-convexity of $\Gamma(x)$ for $x>0$ \cite[Corollary~1.2.6]{AAR}. If $ab=0$ we clearly get the equality. If $a,b>0$, $\mu=m=0$, the second term in parentheses disappears and (\ref{eq:gammasum}) holds strictly. If $m=0$, $-1\leq\mu<0$ and $\mu+a\geq{0}$, $\mu+b\geq{0}$ then the term outside the parentheses reduces to 1 while the second term inside parentheses is strictly negative (since $\mu+a+b>0$), so that the sum is strictly positive. If $m\geq{1}$ then $\mu+m\geq{0}$ and we are back in the previous situation.~~$\square$ \mathbf{b}egin{lemma}\label{lm:rec-gammas} Suppose $m\geq{0}$ is an integer. Then for all complex $\mathbf{b}eta$ and $\mu$ \mathbf{b}egin{multline}\label{eq:rec-gamma-sum} S_m(\mu,\mathbf{b}eta):=\sum\limits_{k=0}^{m}\left\{\frac{1}{\Gamma(k+\mu+1)\Gamma(m-k+\mu+\mathbf{b}eta)} -\frac{1}{\Gamma(k+\mu)\Gamma(m-k+\mu+\mathbf{b}eta+1)}\right\} \\ =\frac{(\mu+\mathbf{b}eta)_{m+1}-(\mu)_{m+1}}{\Gamma(\mu+m+1)\Gamma(\mu+\mathbf{b}eta+m+1)}. \end{multline} \end{lemma} \textbf{Proof.} We have \mathbf{b}egin{multline*} S_m(\mu,\mathbf{b}eta)=\frac{1}{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta)\Gamma(\mu)\Gamma(\mu+\mathbf{b}eta+1)} \sum\limits_{k=0}^{m}\left\{\frac{\Gamma(\mu)\Gamma(\mu+\mathbf{b}eta+1)}{(\mu+1)_k(\mu+\mathbf{b}eta)_{m-k}} -\frac{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta)}{(\mu)_{k}(\mu+\mathbf{b}eta+1)_{m-k}}\right\} \\ =\frac{1}{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta+1)} \sum\limits_{k=0}^{m}\left\{\frac{\mu+\mathbf{b}eta}{(\mu+1)_k(\mu+\mathbf{b}eta)_{m-k}} -\frac{\mu}{(\mu)_{k}(\mu+\mathbf{b}eta+1)_{m-k}}\right\} \\ =\frac{1}{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta+1)} \sum\limits_{k=0}^{m}\frac{1}{(\mu)_k(\mu+\mathbf{b}eta)_{m-k}} \left\{\frac{\mu(\mu+\mathbf{b}eta)}{\mu+k} -\frac{\mu(\mu+\mathbf{b}eta)}{\mu+\mathbf{b}eta+m-k}\right\} \\ =\frac{1}{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta+1)} \sum\limits_{k=0}^{m}\frac{\mathbf{b}eta+m-2k}{(\mu+1)_k(\mu+\mathbf{b}eta+1)_{m-k}}. \end{multline*} Writing $$ u_k=\frac{1}{(\mu+1)_{k-1}(\mu+\mathbf{b}eta+1)_{m-k}},~1\leq{k}\leq{m},~~ u_0=\frac{\mu}{(\mu+\mathbf{b}eta+1)_{m}},~u_{m+1}=\frac{\mu+\mathbf{b}eta}{(\mu+1)_{m}}, $$ we get a telescoping sum, since $$ u_{k+1}-u_{k}=\frac{1}{(\mu+1)_{k}(\mu+\mathbf{b}eta+1)_{m-k-1}}-\frac{1}{(\mu+1)_{k-1}(\mu+\mathbf{b}eta+1)_{m-k}} =\frac{\mathbf{b}eta+m-2k}{(\mu+1)_k(\mu+\mathbf{b}eta+1)_{m-k}} $$ for $k=0,1,\ldots,m$, so that $$ \sum\limits_{k=0}^{m}(u_{k+1}-u_{k})=u_{m+1}-u_{0}=\frac{\mu+\mathbf{b}eta}{(\mu+1)_{m}}-\frac{\mu}{(\mu+\mathbf{b}eta+1)_{m}} $$ and $$ \frac{1}{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta+1)} \sum\limits_{k=0}^{m}\frac{\mathbf{b}eta+m-2k}{(\mu+1)_k(\mu+\mathbf{b}eta+1)_{m-k}}= \frac{(\mu+\mathbf{b}eta)_{m+1}-(\mu)_{m+1}}{\Gamma(\mu+m+1)\Gamma(\mu+\mathbf{b}eta+m+1)}.~\square $$ The following is a straightforward consequence of formula (\ref{eq:rec-gamma-sum}). \mathbf{b}egin{corol}\label{cr:rec-gammas} If $\mu\geq{-1}$, $\mathbf{b}eta\geq{0}$ and $\mu+\mathbf{b}eta\geq{0}$ then $S_{m}(\mu,\mathbf{b}eta)\geq{0}$. The inequality is strict unless $\mathbf{b}eta=0$. \end{corol} \paragraph{3. Main results.} In this section we prove two general theorems for series in reciprocal gamma functions. The power series expansions in this section are understood as formal, so that no questions of convergence are discussed. In applications the radii of convergence will usually be apparent. The results of this section are exemplified by concrete special functions in the subsequent section. \mathbf{b}egin{theo}\label{th:gammadenom} Suppose $\{f_n\}_{n=0}^{\infty}$ is a non-trivial non-negative log-concave sequence without internal zeros. Then the function \mathbf{b}egin{equation}\label{eq:func} \mu\mapsto f(\mu,x)=\sum\limits_{n=0}^{\infty}\frac{f_nx^n}{n!\Gamma(\mu+n)}, \end{equation} is strictly log-concave on $(0,\infty)$ for each fixed $x\geq{0}$. Moreover, the function $$ \varphi_{a,b,\mu}(x):=f(a+\mu,x)f(b+\mu,x)-f(a+b+\mu,x)f(\mu,x)=\sum\limits_{m=0}^{\infty}\varphi_mx^m $$ has positive power series coefficients $\varphi_m>0$ for $\mu\geq{-1}$, $a,b>0$ and $\mu+a\geq{0}$, $\mu+b\geq{0}$ so that $\varphi_{a,b,\mu}(x)$ is absolutely monotonic and multiplicatively convex on $(0,\infty)$. \end{theo} \noindent\textbf{Proof.} Cauchy product yields $$ \varphi_m=\sum\limits_{k=0}^{m}\frac{f_kf_{m-k}}{k!(m-k)!}\left(\frac{1}{\Gamma(k+\mu+a)\Gamma(m-k+\mu+b)}-\frac{1}{\Gamma(m-k+\mu)\Gamma(k+\mu+a+b)}\right). $$ Further, by Gauss summation (the first term is combined with the last, the second with the second last etc.) we can write $\varphi_m$ in the form \mathbf{b}egin{equation}\label{eq:repr} \varphi_m=\sum\limits_{k=0}^{[m/2]}\frac{f_kf_{m-k}}{k!(m-k)!}M_k(a,b,\mu), \end{equation} where for $k<m/2$ \mathbf{b}egin{multline*} M_k(a,b,\mu)=\underbrace{[\Gamma(k+\mu+a)\Gamma(m-k+\mu+b)]^{-1}}_{=u} +\underbrace{[\Gamma(k+\mu+b)\Gamma(m-k+\mu+a)]^{-1}}_{=v} \\[0pt] -\underbrace{[\Gamma(m-k+\mu)\Gamma(k+\mu+a+b)]^{-1}}_{=r} -\underbrace{[\Gamma(k+\mu)\Gamma(m-k+\mu+a+b)]^{-1}}_{=s} \end{multline*} and for $k=m/2$ $$ M_k=[\Gamma(m/2+\mu+a)\Gamma(m/2+\mu+b)]^{-1}-[\Gamma(m/2+\mu)\Gamma(m/2+\mu+a+b)]^{-1}. $$ Under assumptions on $\mu,a,b$ made in the theorem \mathbf{b}egin{equation}\label{pos} \sum\limits_{k=0}^{[m/2]}\frac{M_k(a,b,\mu)}{k!(m-k)!}\geq{0} \end{equation} according to Lemma~\ref{lm:gammasum}. Write $M_k:=M_k(a,b,\mu)$ for brevity. We aim to show that the sequence $\{M_k\}_{k=0}^{[m/2]}$ has no more than one change of sign with $M_{[m/2]}>0$ in order to apply Lemma~\ref{lm:sum}. Due to log-convexity of $\Gamma(x)$ the ratio $x\mapsto \Gamma(x+\mathbf{a}lpha)/\Gamma(x)$ is strictly increasing on $(0,\infty)$ when $\mathbf{a}lpha>0$. This immediately implies $M_{[m/2]}>0$ and the following inequalities \mathbf{b}egin{equation}\label{vu} v\geq u \Leftrightarrow \frac{\Gamma(m-k+\mu+b)}{\Gamma(m-k+\mu+a)}\geq \frac{\Gamma(k+\mu+b)}{\Gamma(k+\mu+a)}~-~\text{true for}~k\leq{m-k}, \end{equation} \mathbf{b}egin{equation}\label{us} u > s \Leftrightarrow \frac{\Gamma(m-k+\mu+a+b)}{\Gamma(m-k+\mu+b)}> \frac{\Gamma(k+\mu+a)}{\Gamma(k+\mu)}~-~\text{true for}~k\leq{m-k}, \end{equation} \mathbf{b}egin{equation}\label{vr} v\geq r \Leftrightarrow \frac{\Gamma(k+\mu+a+b)}{\Gamma(k+\mu+b)} \geq \frac{\Gamma(m-k+\mu+a)}{\Gamma(m-k+\mu)}~-~\text{true for}~(m-b)/2\leq k, \end{equation} \mathbf{b}egin{equation}\label{rv} r\geq v \Leftrightarrow \frac{\Gamma(m-k+\mu+a)}{\Gamma(m-k+\mu)}\geq \frac{\Gamma(k+\mu+a+b)}{\Gamma(k+\mu+b)}~-~\text{true for}~k\leq(m-b)/2. \end{equation} If $(m-b)/2\leq k<m/2$ then the sum of (\ref{us}) and (\ref{vr}) yields $M_k=u+v-r-s>0$. If $k<(m-b)/2$ then it follows from (\ref{vu}), (\ref{us}) and (\ref{rv}) that $r>v>u>s$ (equality cannot be attained in (\ref{vu}) and (\ref{rv}) under this restriction on $k$). We will change notation to simplify writing: $\mathbf{a}lpha = \mu, \mathbf{b}eta = b + \mu, \delta = a$. According the hypothesis of the theorem we have $\mathbf{b}eta>\mathbf{a}lpha>0, \, \delta>0$. We will show now that if $M_k<0 \Leftrightarrow u+v<r+s$ for some $0<k<(m-b)/2$ then $M_{k-1}<0$ as well. Indeed, using $z\Gamma(z)=\Gamma(z+1)$ we can write $M_{k-1}$ in the following form $$ M_{k-1}(\delta)= \frac{k-1+\mathbf{a}lpha+\delta}{m-k+\mathbf{b}eta}u+\frac{k-1+\mathbf{b}eta}{m-k+\mathbf{a}lpha+\delta}v-\frac{k-1+\mathbf{b}eta+\delta}{m-k+\mathbf{a}lpha}r-\frac{k-1+\mathbf{a}lpha}{m-k+\mathbf{b}eta+\delta}s. $$ Treating $u, v, r, s$ as constants we see that $M_{k-1}(0)<0$ by forming a combination of $r>v$ and $r+s>v+u$ with positive coefficients: $(C_1+C_2)r+C_2s>(C_1+C_2)v+C_2u$, where $$ C_1+C_2=\frac{k-1+\mathbf{b}eta}{m-k+\mathbf{a}lpha}>C_2=\frac{k-1+\mathbf{a}lpha}{m-k+\mathbf{b}eta}. $$ Further, differentiating with respect to $\delta$, we get $$ M'_{k-1}(\delta)= \frac{1}{m-k+\mathbf{b}eta}u-\frac{k-1+\mathbf{b}eta}{(m-k+\mathbf{a}lpha+\delta)^2}v-\frac{1}{m-k+\mathbf{a}lpha}r+\frac{k-1+\mathbf{a}lpha}{(m-k+\mathbf{b}eta+\delta)^2}s, $$ which is obviously negative since $r>u$ (by (\ref{rv}) and (\ref{vu})) and $v>s$ (by (\ref{vu}) and (\ref{us})). This shows that $M_{k-1}< 0$ and hence $\{M_k\}_{k=0}^{[m/2]}$ has no more than one change of sign. Applying Lemma~\ref{lm:sum} with $A_k=M_k/(k!(m-k)!)$ we conclude that $\varphi_m>0$. Multiplicative convexity follows by Hardy, Littlewood and P\'{o}lya theorem \cite[Proposition~2.3.3]{NP}. ~~$\square$ \par\refstepcounter{theremark}\textbf{Remark \mathbf{a}rabic{theremark}.} If $\{f_n\}_{n=0}^{\infty}$ is log-convex then $\varphi_m$ can take both signs. \mathbf{b}egin{corol}\label{cr:compl-mon} Assume the series in $(\ref{eq:func})$ converges for all $x\geq{0}$. Then $\varphi_{a,b,\mu}(1/y)$ is completely monotonic and log-convex on $[0,\infty)$, so that there exists a non-negative measure $\tau$ supported on $[0,\infty)$ such that \mathbf{b}egin{equation}\label{eq:compl-mon} \varphi_{a,b,\mu}(x)=\int\limits_{[0,\infty)}e^{-t/x}d\tau(t). \end{equation} \end{corol} \textbf{Proof}. According to \cite[Theorem~3]{MS} a convergent series of completely monotonic with non-negative coefficients is again completely monotonic. This implies that $y\to\varphi_{a,b,\mu}(1/y)$ is completely monotonic, so that the above integral representation follows by Bernstein's theorem \cite[Theorem~1.4]{SSV}. Log-convexity follows from complete monotonicity according to \cite[Exersice 2.1(6)]{NP}.~~$\square$ In the next corollary we adopt the convention $\Gamma(-1)=-\infty$, $\Gamma(0)=+\infty$. \mathbf{b}egin{corol}\label{cr:f-twosided} Under hypotheses and notation of Theorem~\ref{th:gammadenom} \mathbf{b}egin{equation}\label{eq:f-twosided} \frac{\Gamma(a+\mu)\Gamma(b+\mu)}{\Gamma(\mu)\Gamma(a+b+\mu)}<\frac{f(\mu,x)f(a+b+\mu,x)}{f(a+\mu,x)f(b+\mu,x)}<1~\text{for all}~x\geq{0}. \end{equation} If $\mu=0$ or $\mu=-1$ we additionally require that $x\ne0$ otherwise the left inequality becomes equality. \end{corol} \textbf{Proof.} The estimate from above is a restatement of Theorem~\ref{th:gammadenom} since it is equivalent to $\phi_{\mathbf{b}eta,\mu}(x)>0$. The estimate from below is obvious for $\mu=-1$ for we have zero or negative number (if $a=1$ or $b=1$) on the left and a positive number on the right for $x>0$. The remaining proof will be divided into two cases (I) $\mu\geq{0}$; and (II)$-1<\mu<0$, $\mu+a\geq{0}$, $\mu+b\geq{0}$ (recall that $a,b>0$ by hypotheses of Theorem~\ref{th:gammadenom}). In case (I) the left-hand inequality in (\ref{eq:f-twosided}) follows from strict log-convexity of $\mu\to\Gamma(\mu)f(\mu,x)$ which has been proved in \cite[Theorem~3]{KS} (where one has to take account of the formula $(\mu)_k=\Gamma(\mu+k)/\Gamma(\mu)$). In case (II) $\Gamma(\mu)<0$ and the left-hand inequality in (\ref{eq:f-twosided}) reduces to $$ \Gamma(a+\mu)f(a+\mu,x)\Gamma(b+\mu)f(b+\mu,x)>\Gamma(\mu)f(\mu,x)\Gamma(a+b+\mu)f(a+b+\mu,x). $$ This inequality follows by observing that $\Gamma(\mu)f(\mu,x)=\sum_{n=0}^{\infty}f_nx^n(n!(\mu)_n)^{-1}$ and $$ \sum\limits_{k=0}^{m}\frac{f_{k}f_{m-k}}{k!(m-k)!} \left\{\frac{1}{(a+\mu)_k(b+\mu)_{m-k}}-\frac{1}{(\mu)_k(a+b+\mu)_{m-k}}\right\}>0, $$ since for $k=1,2,\ldots,m$ $(\mu)_k<0$ and for $k=0$ $(b+\mu)_{m}<(a+b+\mu)_{m}$.~~$\square$ \mathbf{b}egin{corol}\label{cr:phi-below} Under hypotheses and notation of Theorem~\ref{th:gammadenom} and for all $x\geq{0}$ $$ f(a+\mu,x)f(b+\mu,x)-f(a+b+\mu,x)f(\mu,x)\geq f_0^2\left[\frac{1}{\Gamma(\mu+a)\Gamma(\mu+b)}-\frac{1}{\Gamma(\mu)\Gamma(\mu+a+b)}\right] $$ with equality only at $x=0$. \end{corol} \textbf{Proof.} Indeed, the claimed inequality is just $\varphi_{a,b,\mu}(x)\geq\varphi_{a,b,\mu}(0)$ which is true by Theorem~\ref{th:gammadenom}.~~$\square$ \par\refstepcounter{theremark}\textbf{Remark \mathbf{a}rabic{theremark}.} Corollaries~\ref{cr:f-twosided} and \ref{cr:phi-below} imply by elementary calculation the following bounds for the so called ``generalized Turanian'' $\Delta_{\varepsilon}(\mu,x):=f(\mu,x)^2-f(\mu+\varepsilon,x)f(\mu-\varepsilon,x)$: \mathbf{b}egin{equation}\label{eq:genTuranian} A_\varepsilon(\mu)f_0^2\leq\Delta_{\varepsilon}(\mu,x)\leq B_\varepsilon(\mu)f(\mu,x)^2, \end{equation} where $\mu-\varepsilon\geq{-1}$, $\mu\geq{0}$, $x\geq{0}$ and \mathbf{b}egin{equation}\label{eq:AB} A_\varepsilon(\mu)=\frac{\Gamma(\mu-\varepsilon)\Gamma(\mu+\varepsilon)-\Gamma(\mu)^2} {\Gamma(\mu-\varepsilon)\Gamma(\mu+\varepsilon)\Gamma(\mu)^2},~~~~ B_\varepsilon(\mu)=\frac{\Gamma(\mu-\varepsilon)\Gamma(\mu+\varepsilon)-\Gamma(\mu)^2} {\Gamma(\mu-\varepsilon)\Gamma(\mu+\varepsilon)}. \end{equation} In particular, if $\varepsilon=1$ the bounds (\ref{eq:genTuranian}) simply to ($\mu\geq{0}$, $x\geq{0}$) \mathbf{b}egin{equation}\label{eq:Turanian} \frac{f_0^2}{\mu\Gamma(\mu)^2}\leq f(\mu,x)^2-f(\mu+1,x)f(\mu-1,x)\leq \frac{1}{\mu}f(\mu,x)^2,~~~x\geq{0},~\mu\geq{0}. \end{equation} Theorem~\ref{th:gammadenom} can be reformulated in terms of the numbers $g_n:=f_n/n!$. The hypotheses of the theorem require then that these numbers satisfy $$ g_n^2\geq\frac{n+1}{n} g_{n-1}g_{n+1} $$ - a condition stronger then log-concavity. If we weaken it to log-concavity we are only able to prove discrete Wright log-concavity of $\mu\to\,f(\mu,x)$ in the next theorem. We conjecture below that the adjective ``discrete'' is actually redundant. \mathbf{b}egin{theo}\label{th:gammadenom1} Suppose $\{g_n\}_{n=0}^{\infty}$ is a non-trivial non-negative log-concave sequence without internal zeros. Then the function \mathbf{b}egin{equation}\label{eq:g-def} \mu\to g(\mu,x)=\sum\limits_{n=0}^{\infty}\frac{g_nx^n}{\Gamma(\mu+n)}, \end{equation} is strictly discrete Wright log-concave on $(0,\infty)$ for each fixed $x\geq{0}$. Moreover, the function $$ \lambda_{\mathbf{b}eta,\mu}(x):=g(\mu+1,x)g(\mu+\mathbf{b}eta,x)-g(\mu,x)g(\mu+\mathbf{b}eta+1,x)=\sum\limits_{m=0}^{\infty}\lambda_mx^m $$ has positive power series coefficients $\lambda_m>0$ for each $\mu\geq{-1}$ and $\mathbf{b}eta>0$ such that $\mu+\mathbf{b}eta\geq{0}$. This implies that $x\to\lambda_{\mathbf{b}eta,\mu}(x)$ is absolutely monotonic and multiplicatively convex on $(0,\infty)$. \end{theo} \textbf{Proof.} Pursuing the same line of argument as in Theorem~\ref{th:gammadenom} we have by the Cauchy product and the Gauss summation: $$ \lambda_m=\sum\limits_{k=0}^{[m/2]}g_kg_{m-k}M_k(1,\mathbf{b}eta,\mu), $$ where the numbers $M_k$ are defined in the proof of Theorem~\ref{th:gammadenom}, below formula (\ref{eq:repr}). Under assumptions on $\mu$ and $\mathbf{b}eta$ made in the theorem we have $$ \sum\limits_{k=0}^{[m/2]}M_k(1,\mathbf{b}eta,\mu)=S_m(\mu,\mathbf{b}eta)>0 $$ according to Corollary~\ref{cr:rec-gammas}. Further, it has been shown in the course of the proof of Theorem~\ref{th:gammadenom} that the sequence $\{M_k\}_{k=0}^{[m/2]}$ has no more than one change of sign with $M_{[m/2]}>0$. Hence by Lemma~\ref{lm:sum} we conclude that $\lambda_m>0$ implying discrete Wright log-concavity of $\mu\to{g(\mu,x)}$ for each $x\geq{0}$ and absolutely monotonicity of $x\to\lambda_{\mathbf{b}eta,\mu}(x)$. Multiplicative convexity follows by Hardy, Littlewood and P\'{o}lya theorem \cite[Proposition~2.3.3]{NP}. ~~$\square$ \mathbf{b}egin{corol}\label{cr:g-compl-mon} Assume the series in $(\ref{eq:g-def})$ converges for all $x\geq{0}$. Then $\lambda_{\mathbf{b}eta,\mu}(1/y)$ is completely monotonic and log-convex on $[0,\infty)$, so that there exists a non-negative measure $\tau$ supported on $[0,\infty)$ such that \mathbf{b}egin{equation}\label{eq:g-compl-mon} \lambda_{\mathbf{b}eta,\mu}(x)=\int\limits_{[0,\infty)}e^{-t/x}d\tau(t). \end{equation} \end{corol} \mathbf{b}egin{corol}\label{cr:g-twosided} Under hypotheses and notation of Theorem~\ref{th:gammadenom1} \mathbf{b}egin{equation}\label{eq:g-twosided} \frac{\mu}{\mathbf{b}eta+\mu}<\frac{g(\mu,x)g(1+\mathbf{b}eta+\mu,x)}{g(1+\mu,x)g(\mathbf{b}eta+\mu,x)}<1~\text{for all}~x\geq{0}. \end{equation} \end{corol} \textbf{Proof.} The estimate from above is a restatement of Theorem~\ref{th:gammadenom1} since it is equivalent to $\lambda_{\mathbf{b}eta,\mu}(x)>0$. The estimate from below is obvious for $\mu=-1$ for we a negative number on the left and a positive number on the right. The remaining proof will be divided into two cases (I) $\mu\geq{0}$; and (II)$-1<\mu<0$, $\mu+\mathbf{b}eta\geq{0}$ (recall that $\mathbf{b}eta>0$ by hypotheses of Theorem~\ref{th:gammadenom1}). In case (I) the left-hand inequality in (\ref{eq:g-twosided}) follows from strict log-convexity of $\mu\to\Gamma(\mu)g(\mu,x)$ which, in view of $(\mu)_k=\Gamma(\mu+k)/\Gamma(\mu)$, has been proved in \cite[Theorem~3]{KS}. In case (II) the left-hand inequality in (\ref{eq:g-twosided}) can be rewritten as $$ \Gamma(1+\mu)g(1+\mu,x)\Gamma(\mathbf{b}eta+\mu)g(\mathbf{b}eta+\mu,x)>\Gamma(\mu)g(\mu,x)\Gamma(1+\mathbf{b}eta+\mu)g(1+b+\mu,x). $$ This inequality follows by observing that $\Gamma(\mu)g(\mu,x)=\sum_{n=0}^{\infty}g_n(\mu)_n^{-1}$ and $$ \sum\limits_{k=0}^{m}g_{k}g_{m-k} \left\{\frac{1}{(1+\mu)_k(\mathbf{b}eta+\mu)_{m-k}}-\frac{1}{(\mu)_k(1+\mathbf{b}eta+\mu)_{m-k}}\right\}>0, $$ since $(\mu)_k<0$ for $k=1,2,\ldots,m$ and $(\mathbf{b}eta+\mu)_{m}<(1+\mathbf{b}eta+\mu)_{m}$ for $k=0$.~~$\square$ \mathbf{b}egin{corol}\label{cr:lambda-below} Under hypotheses and notation of Theorem~\ref{th:gammadenom1} and for all $x\geq{0}$ $$ g(\mu+1,x)g(\mu+\mathbf{b}eta,x)-g(\mu,x)g(\mu+\mathbf{b}eta+1,x)\geq \frac{g_0^2\mathbf{b}eta}{\Gamma(\mu+1)\Gamma(\mu+\mathbf{b}eta+1)} $$ with equality only at $x=0$. \end{corol} \par\refstepcounter{theremark}\textbf{Remark \mathbf{a}rabic{theremark}.} Since we have only proved discrete Wright log-concavity in Theorem~\ref{th:gammadenom1} we cannot make any statements about the ``generalized Turanian'' $g(\mu,x)^2-g(\mu+\varepsilon,x)g(\mu-\varepsilon,x)$. We can assert, however, that the standard Turanian satisfies the following bounds similar to those in (\ref{eq:Turanian}) \mathbf{b}egin{equation}\label{eq:Turanian1} \frac{g_0^2}{\mu\Gamma(\mu)^2}\leq g(\mu,x)^2-g(\mu+1,x)g(\mu-1,x)\leq \frac{1}{\mu}g(\mu,x)^2,~~~x\geq{0},~\mu\geq{0}. \end{equation} \textbf{Conjecture~1.} Under hypotheses of Theorem~\ref{th:gammadenom1} the function $\mu\to{g(\mu,x)}$ is log-concave on $(0,\infty)$ for each fixed $x\geq{0}$. Moreover, the function $$ x\to g(\mu+\mathbf{a}lpha,x)g(\mu+\mathbf{b}eta,x)-g(\mu,x)g(\mu+\mathbf{a}lpha+\mathbf{b}eta,x) $$ has positive power series coefficients for $\mu\geq{-1}$ and $\mu+\mathbf{a}lpha\geq{0}$, $\mu+\mathbf{b}eta\geq{0}$, where $\mathbf{a}lpha,\mathbf{b}eta>0$. The above conjecture is equivalent to the assertion that \mathbf{b}egin{equation*} \sum\limits_{k=0}^{m}\left\{\frac{1}{\Gamma(k+\mu+\mathbf{a}lpha)\Gamma(m-k+\mu+\mathbf{b}eta)} -\frac{1}{\Gamma(k+\mu)\Gamma(m-k+\mu+\mathbf{a}lpha+\mathbf{b}eta)}\right\}>0 \end{equation*} which extends Corollary~\ref{cr:rec-gammas}. \paragraph{4. Applications and relation to other work.} We start with the well-studied case of the modified Bessel function. Even for this classical case we can add to the current knowledge. \textbf{Example~1}. The modified Bessel function is defined by the series \cite[formula~(4.12.2)]{AAR} $$ I_{\nu}(u)=\sum\limits_{n\geq0}\frac{(u/2)^{2n+\nu}}{n!\Gamma(n+\nu+1)}. $$ Hence, if we set $f_n=1$ $\forall n$, $x=(u/2)^{2}$ and $\mu=\nu+1$ in Theorem~\ref{th:gammadenom} and use $\partial^2_\nu\log(u/2)^{\nu}=0$ we immediately conclude that $\nu\to{I_{\nu}(u)}$ is log-concave on $(-1,\infty)$ for each fixed $u>0$. Moreover, for any $\nu\geq{-1}$ and $\nu-\varepsilon\geq{-2}$ the ``generalized Turanian'' \mathbf{b}egin{equation}\label{eq:I1} u\to\Delta_{\varepsilon}(\nu,u):=(I_{\nu}(u))^2-I_{\nu+\varepsilon}(u)I_{\nu-\varepsilon}(u) \end{equation} has positive power series coefficients, is multiplicatively convex and according to (\ref{eq:genTuranian}) satisfies \mathbf{b}egin{equation}\label{eq:I2} (u/2)^{2\nu}A_\varepsilon(\nu+1)\leq\Delta_{\varepsilon}(\nu,u)\leq B_\varepsilon(\nu+1)(I_{\nu}(u))^2,~~u\geq{0}, \end{equation} where $A_\varepsilon$ and $B_\varepsilon$ are defined in (\ref{eq:AB}). In particular for $\varepsilon=1$ we get for $\nu\geq{-1}$: \mathbf{b}egin{equation}\label{eq:I3} \frac{(u/2)^{2\nu}}{(\nu+1)\Gamma(\nu+1)^2}\leq(I_{\nu}(u))^2-I_{\nu+1}(u)I_{\nu-1}(u) \leq\frac{1}{\nu+1}(I_{\nu}(u))^2. \end{equation} All the more, the function $(u/2)^{-2\nu}\Delta_{\varepsilon}(\nu,u)$ admits representation (\ref{eq:compl-mon}). Proofs of various forms of log-concavity of $I_{\nu}(u)$ have a long history. The discrete log-concavity, $I_{\nu-1}(x)I_{\nu+1}(x)\leq{(I_{\nu}(x))^2}$, and the right-hand side of (\ref{eq:I3}) for $\nu\geq{0}$ were probably first demonstrated in 1951 by Thiruvenkatachar and Nanjundiah \cite{TN}. In fact, our method here is an extension of their approach, so that they could have proved the log-concavity of $\nu\to I_{\nu}(x)$. The discrete log-concavity was rediscovered by Amos \cite{Am} in 1974 and later by Joshi and Bissu \cite{JB} in 1991 with different proofs. Their paper also gives a proof of the right-hand side of (\ref{eq:I3}) for $\nu\geq{0}$. Finally, Lorch in \cite{Lo} and later Baricz in \cite{Baricz10H} showed the log-concavity of $\nu\to I_{\nu}(x)$ on $(-1,\infty)$ and demonstrated the positivity of the function (\ref{eq:I1}) for $\nu>-1/2$ and small $\varepsilon$. He also conjectured that the positivity remains true for $\nu>-1$ and $\varepsilon\in(0,1]$. Baricz \cite{Baricz10} demonstrated the Lorch's conjecture for $\varepsilon=1$ and extended the right-hand side of (\ref{eq:I3}) to $\nu>-1$. Our results here not only confirm Lorch's conjecture but also refine and strengthen it by proving (\ref{eq:I2}) and the positivity of the power series coefficients of (\ref{eq:I1}). Various extensions and a related results can also be found in \cite{Baricz10,Baricz12,Segura}. We note that many proofs use special properties of the modified Bessel functions, like differential-recurrence relations, zeros etc. Theorem~\ref{th:gammadenom} and its corollaries show that it is in fact the structure of the power series that is responsible for the bounds (\ref{eq:I2}) and (\ref{eq:I3}). \textbf{Example~2}. In his 1993 preprint \cite{Sitnik} Sitnik, among other things, proved the inequality $$ R_{n}^2(x)>R_{n-1}(x)R_{n+1}(x), ~~x>0,~~n=1,2,\ldots, $$ where $$ R_{n}(x)=e^{x}-\sum\limits_{k=0}^{n}\frac{x^k}{k!}=\frac{x^{n+1}}{(n+1)!}{_1F_1}(1;n+2;x) $$ is the exponential remainder. We can generalize this function as follows $$ R_{\eta,\nu}(x)=\frac{x^{\nu+1}}{\Gamma(\nu+2)}{_1F_1}(\eta;\nu+2;x) =x^{\nu+1}\sum\limits_{k=0}^{\infty}\frac{(\eta)_kx^k}{\Gamma(\nu+2+k)k!}. $$ It is straightforward to check that the sequence $g_k=(\eta)_k/k!$ is log-concave iff $\eta\geq{1}$. Then according to Theorem~\ref{th:gammadenom1} the function $\nu\to R_{\eta,\nu}(x)$ is discrete Wright log-concave on $(-2,\infty)$ for each fixed $\eta\geq{1}$, $x>0$ and $$ x\to R_{\eta,\nu+1}(x)R_{\eta,\nu+\mathbf{b}eta}(x)-R_{\eta,\nu}(x)R_{\eta,\nu+\mathbf{b}eta+1}(x) $$ has positive power series coefficients for $\nu\geq-3$, $\nu+\mathbf{b}eta\geq-2$, $\mathbf{b}eta>0$. Moreover, $$ \frac{x^{2\nu+2}}{(\nu+2)\Gamma(\nu+2)^2}\leq R_{\eta,\nu}(x)^2-R_{\eta,\nu+1}(x)R_{\eta,\nu-1}(x)\leq \frac{1}{\nu+2}R_{\eta,\nu}(x)^2,~~~x\geq{0},~\nu\geq{-2}. $$ \textbf{Example~3}. In addition to the results for the Kummer function presented in Example~2 above we can derive bounds for its logarithmic derivative. The logarithmic derivatives of the Kummer function plays an important role in some probabilistic applications - see \cite{SK}. Let us use abbreviated notation $F(a;b;x)={_1F_1}(a;b;x)$. The following contiguous relations are easy to check (recall that $F'(a;b;x)=(a/b)F(a+1;b+1;x)$): \mathbf{b}egin{equation}\label{eq:con1} aF(a;b;x)-aF(a+1;b;x)+xF'(a;b;x)=0, \end{equation} \mathbf{b}egin{equation}\label{eq:con2} abF(a+1;b;x)=b(a+x)F(a;b;x)-(b-a)xF(a;b+1;x), \end{equation} \mathbf{b}egin{equation}\label{eq:con3} b(b-1)(F(a;b-1;x)-F(a;b;x))-axF(a+1;b+1;x)=0. \end{equation} Dividing (\ref{eq:con2}) by $b$ and substituting $aF(a+1;b;x)$ into (\ref{eq:con1}) we get after simplification and dividing by $\Gamma(b+1)$: $$ \frac{1}{\Gamma(b+1)}F(a;b+1;x)=\frac{F(a;b;x)-F'(a;b;x)}{(b-a)\Gamma(b)} $$ From (\ref{eq:con3}) we obtain: $$ \frac{1}{\Gamma(b-1)}F(a;b-1;x)=\frac{1}{\Gamma(b)}(xF'(a;b;x)+(b-1)F(a;b;x)) $$ Thus we get the following expression for the Turanian: \mathbf{b}egin{multline*} \frac{F(a;b;x)^2}{\Gamma(b)^2}-\frac{F(a;b+1;x)F(a;b-1;x)}{\Gamma(b+1)\Gamma(b-1)} \\ =\frac{1}{\Gamma(b)^2(b-a)}\left\{-(a-1)F(a;b;x)^2+xF'(a;b;x)^2+(b-x-1)F(a;b;x)F'(a;b;x)\right\} \end{multline*} Hence, inequality (\ref{eq:Turanian}) becomes ($x>0$, $b>0$, $a\geq{1}$): $$ \frac{1}{b\Gamma(b)^2}< \frac{-(a-1)F(a;b;x)^2+xF'(a;b;x)^2+(b-x-1)F(a;b;x)F'(a;b;x)}{\Gamma(b)^2(b-a)} <\frac{1}{b\Gamma(b)^2}F(a;b;x)^2, $$ which leads to ($F\equiv{F(a;b;x)}$, $F'\equiv{F'(a;b;x)}$): $$ 0<\frac{-(a-1)+x(F'/F)^2+(b-x-1)(F'/F)}{(b-a)}<\frac{1}{b}. $$ Solving these quadratic inequalities we arrive at $$ \frac{x+1-b+\sqrt{(x+1-b)^2+4x(a-1)}}{2x}<\frac{F'(a;b;x)}{F(a;b;x)} <\frac{x+1-b+\sqrt{(x+1-b)^2+4xa(b-1)/b}}{2x} $$ for $x>0$ and $b>a\geq{1}$. The upper and lower bounds interchange if $x>0$, $a\geq{1}$ and $0<b<a$: $$ \frac{x+1-b+\sqrt{(x+1-b)^2+4xa(b-1)/b}}{2x}<\frac{F'(a;b;x)}{F(a;b;x)} <\frac{x+1-b+\sqrt{(x+1-b)^2+4x(a-1)}}{2x}. $$ These bounds are quite precise numerically especially when $a$ and $b$ are close. \textbf{Example~4}. The generalized hypergeometric function is defined by the series \mathbf{b}egin{equation}\label{eq:pFqdefined} {_{p}F_q}\left(\left.\!\!\mathbf{b}egin{array}{c} a_1,a_2,\ldots,a_p\\ b_1,b_2,\ldots,b_q\end{array}\right|z\!\right):=\sum\limits_{n=0}^{\infty}\frac{(a_1)_n(a_2)_n\cdots(a_{p})_n}{(b_1)_n(b_2)_n\cdots(b_q)_nn!}z^n, \end{equation} where $(a)_0=1$, $(a)_n=a(a+1)\cdots(a+n-1)$, $n\geq{1}$, denotes the rising factorial. The series (\ref{eq:pFqdefined}) converges in the entire complex plane if $p\leq{q}$ and in the unit disk if $p=q+1$. In the latter case its sum can be extended analytically to the whole complex plane cut along the ray $[1,\infty)$ \cite[Chapter~2]{AAR}. Applications of Theorems~\ref{th:gammadenom} and \ref{th:gammadenom1} to generalized hypergeometric function is largely based on the following lemma. \mathbf{b}egin{lemma}\label{lm:HVVKS} Denote by $e_k(x_1,\ldots,x_q)$ the $k$-th elementary symmetric polynomial, $$ e_0(x_1,\ldots,x_q)=1,~~~e_k(x_1,\ldots,x_q)=\!\!\!\!\!\!\!\!\sum\limits_{1\leq{j_1}<{j_2}\cdots<{j_k}\leq{q}} \!\!\!\!\!\!\!\!x_{j_1}x_{j_2}\cdots{x_{j_k}},~~k\geq{1}. $$ Suppose $q\geq{1}$ and $0\leq{r}\leq{q}$ are integers, $a_i>0$, $i=1,\ldots,q-r$, $b_i>0$, $i=1,\ldots,q$, and \mathbf{b}egin{equation}\label{eq:symmetric-chain1} \frac{e_q(b_1,\ldots,b_q)}{e_{q-r}(a_1,\ldots,a_{q-r})}\leq \frac{e_{q-1}(b_1,\ldots,b_q)}{e_{q-r-1}(a_1,\ldots,a_{q-r})}\leq\cdots \leq\frac{e_{r+1}(b_1,\ldots,b_q)}{e_{1}(a_1,\ldots,a_{q-r})}\leq e_{r}(b_1,\ldots,b_q). \end{equation} Then the sequence of hypergeometric terms \emph{(}if $r=q$ the numerator is $1$\emph{)}, $$ f_n=\frac{(a_1)_n\cdots(a_{q-r})_n}{(b_1)_n\cdots(b_q)_n}, $$ is log-concave, i.e. $f_{n-1}f_{n+1}\leq{f_n^2}$, $n=1,2,\ldots$ It is strictly log-concave unless $r=0$ and $a_i=b_i$, $i=1,\ldots,q$. \end{lemma} The proof of this lemma for $r=0$ can be found in \cite[Theorem~4.4]{HVV} and \cite[Lemma~2]{KS}. The latter reference also explains how to extend the proof to general $r$ (see the last paragraph of \cite{KS}). This leads immediately to the following statements. \mathbf{b}egin{theo} Let $0\leq{p}\leq{q}$ be integers. Denote $$ f(\nu,x):=\frac{1}{\Gamma(\nu)}{_pF_{q+1}}(a_{1},\ldots,a_{p};\nu,b_{1},\ldots,b_{q};x) $$ and suppose that parameters $(a_1,\ldots,a_p)$, $(b_1,\ldots,b_q)$ satisfy \emph{(\ref{eq:symmetric-chain1})}. Then the function $f(\nu,x)$ satisfies Theorem~\ref{th:gammadenom} and Corollaries~\ref{cr:compl-mon}-\ref{cr:phi-below}. \end{theo} \mathbf{b}egin{theo} Let $0\leq{p}\leq{q+1}$ be integers. Denote $$ g(\nu,x):=\frac{1}{\Gamma(\nu)}{_{p}F_{q+1}}(a_{1},\ldots,a_{p};\nu,b_{1},\ldots,b_{q};x) $$ and suppose that parameters $(a_1,\ldots,a_p)$, $(1,b_1,\ldots,b_q)$ satisfy \emph{(\ref{eq:symmetric-chain1})}. Then the function $g(\nu,x)$ satisfies Theorem~\ref{th:gammadenom1} and Corollaries~\ref{cr:g-compl-mon}-\ref{cr:lambda-below}. \end{theo} \textbf{Example~5}. Our last example is non-hypergeometric. Consider the parameter derivative of the regularized Kummer function: $$ \frac{\partial}{\partial{a}}\frac{1}{\Gamma(b)}{_1F_1}(a;b;x)= \sum\limits_{k=0}^{\infty}\frac{(\psi(a+k)-\psi(a))(a)_k}{\Gamma(b+k)k!}x^k $$ (this function cannot be expressed as hypergeometric function of one variable). If $a\geq{1}$ then the sequence $$ h_k=\frac{(\psi(a+k)-\psi(a))(a)_k}{k!} $$ is log-concave, since \mathbf{b}egin{multline*} h_k^2-h_{k-1}h_{k+1}=\frac{(a)_{k-1}(a)_{k}}{(k-1)!k!}\times \\ \left\{\frac{a+k-1}{k}(\psi(a+k)-\psi(a))^2-\frac{a+k}{k+1}(\psi(a+k-1)-\psi(a))(\psi(a+k+1)-\psi(a))\right\}>0. \end{multline*} The last inequality holds because $y\to{\psi(a+y)-\psi(a)}$ is concave according to the Gauss formula~\cite[Theorem~1.6.1]{AAR} $$ (\psi(a+y)-\psi(a))''_y=\psi''(a+y)=-\int\limits_{0}^{\infty}\frac{t^2e^{-t(a+y)}}{1-e^{-t}}dt<0 $$ and hence is log-concave while $(a+k-1)/k>(a+k)/(k+1)$ if $a\geq{1}$. 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\begin{document} \title{Integers Represented as a Sum of Primes and Powers of Two} \author{D.R. Heath-Brown and J.-C. Puchta\\ Mathematical Institute, Oxford} \date{} \maketitle \section{Introduction} It was shown by Linnik [10] that there is an absolute constant $K$ such that every sufficiently large even integer can be written as a sum of two primes and at most $K$ powers of two. This is a remarkably strong approximation to the Goldbach Conjecture. It gives us a very explicit set $\cl{K}(x)$ of integers $n\le x$ of cardinality only $O((\log x)^K)$, such that every sufficiently large even integer $N\le x$ can be written as $N=p+p'+n$, with $p,p'$ prime and $n\in\cl{K}(x)$. In contrast, if one tries to arrange such a representation using an interval in place of the set $\cl{K}(x)$, all known results would require $\cl{K}(x)$ to have cardinality at least a positive power of $x$. Linnik did not establish an explicit value for the number $K$ of powers of 2 that would be necessary in his result. However, such a value has been computed by Liu, Liu and Wang [12], who found that $K=54000$ is acceptable. This result was subsequently improved, firstly by Li [8] who obtained $K=25000$, then by Wang [18], who found that $K=2250$ is acceptable, and finally by Li [9] who gave the value $K=1906$. One can do better if one assumes the Generalized Riemann Hypothesis, and Liu, Liu and Wang [13] showed that $K=200$ is then admissible. The object of this paper is to give a rather different approach to this problem, which leads to dramatically improved bounds on the number of powers of 2 that are required for Linnik's theorem. \begin{theorem} Every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. \end{theorem} \begin{theorem} Assuming the Generalized Riemann Hypothesis, every sufficiently large even integer is a sum of two primes and exactly 7 powers of 2. \end{theorem} We understand that Ruzsa and Pintz have, in work in preparation, given an independent proof of Theorem 2, and have established a version of Theorem 1 requiring only 8 powers of 2. Indeed, already in 2000, Pintz had announced the values $K=12$ unconditionally, and $K=10$ on the Generalized Riemann Hypothesis. Although we have not seen an account of this work, we understand that that our approach is different in a number of respects. We should also report that Elsholtz, in unpublished work, has shown that one can obtain $K=12$ in Theorem 1, by a variant of our method. He does this by improving our constant $2.7895$ in (25) to $2.96169$, by using $D=21$, and replacing our estimate (41) for $C_2$ by $C_2\le 1.992$. Previous workers have based their line of attack on a proof of Linnik's theorem due to Gallagher [3]. Let $\varpi$ be a small positive constant. Set \begin{equation} S(\alpha)=\sum_{\varpi N<p\le N}e(\alpha p), \end{equation} where $e(x):=\exp(2\pi ix)$, and \[T(\alpha)=\sum_{1\le\nu\le L}e(\alpha 2^{\nu}),\;\;\; L=[\frac{\log N/2K}{\log 2}].\] As in earlier proofs of Linnik's Theorem we shall use estimates for ${\rm meas}(\cl{A}_{\lambda})$, where \[\cl{A}_{\lambda}=\{\alpha\in[0,1]: |T(\alpha)|\ge\lambda L\}.\] In \S 7 we shall bound ${\rm meas}(\cl{A}_{\lambda})$ by a new method, suggested to us by Professor Keith Ball. This provides the following estimates. \begin{lemma} We have \[{\rm meas}(\cl{A}_{\lambda})\ll N^{-E(\lambda)}\] with $E(0.722428)>1/2$ and $E(0.863665)>109/154$. \end{lemma} We are extremely grateful to Professor Ball for suggesting his alternative approach to us. An earlier version of this paper used a completely different technique to bound $E(\lambda)$ and showed that one can take \[E(\lambda)\geq 0.822\lambda^2 +o(1)\] as $N\rightarrow\infty$. This sufficed to establish Theorems 1 and 2 with 24 and 9 powers of 2 respectively. For comparison with Lemma 1, the best bound for $E(\lambda)$ in the literature is due to Liu, Liu and Wang [11; Lemma 3], and states that \[E(1-\eta)\le 1-F(\frac{2+\sqrt{2}}{4}\eta)-F(1-\frac{2+\sqrt{2}}{4}\eta) +o(1)\] for $\eta<(7e)^{-1}$, where $F(x)=x(\log x)/(\log 2)$. The estimate provided by Lemma 1 will be injected into the circle method, where it will be crucial in bounding the minor arc contribution. On the major arcs we shall improve on Gallagher's analysis so as to show that hypothetical zeros close to $\sigma=1$ play no r\^{o}le. Thus, in contrast to previous workers, we will have no need for explicit numerical zero-free regions for $L$-functions. Naturally this produces a considerable simplification in the computational aspects of our work. Thus it is almost entirely the values of the constants in Lemma 1 which determine the number of powers of 2 appearing in Theorems 1 and 2. The paper naturally divides into two parts, one of which involves the circle method and zeros of $L$-functions, and the other of which is devoted to the proof of Lemma 1. We begin with the former. One remark about notation is in order. At various stages in the proof, numerical upper bounds on $\varpi$ will be required. Since we shall always take $\varpi$ to be sufficiently small, we shall assume that any such bound is satisfied. Moreover, since $\varpi$ is to be thought of as fixed, we will allow the implied constants in the $O(\ldots)$ and $\ll$ notations to depend on $\varpi$. \section{The Major Arcs} We shall follow the method of Gallagher [3; \S 1] closely. We choose a parameter $P$ in the range $1\le P\le N^{2/5}$ and define the major arcs ${\mathfrak M}$ as the set of $\alpha\in[0,1]$ for which there exist $a\in\mathbb{Z}$ and $q\in \mathbb{N}$ such that $q\le P$ and \[|\alpha-\frac{a}{q}|\le\frac{P}{qN}.\] If $\chi$ is a character to modulus $q$, we write \[c_n(\chi)=\sum_{a=1}^{q}\chi(a)e(\frac{an}{q})\] and \[\tau(\chi)=\sum_{a=1}^{q}\chi(a)e(\frac{a}{q}).\] Moreover we put \[A(\chi,\beta)=\sum_{\varpi N<p\le N}\chi(p)e(\beta p)\] and \[I_{n,s}(\chi,\chi')=\int_{-P/sN}^{P/sN}A(\chi,\beta) A(\chi',\beta)e(-\beta n)d\beta.\] If $\chi$ is a character to a modulus $r|q$ we also write $\chi_q$ for the induced character modulo $q$, and if $\chi,\chi'$ are characters to moduli $r$ and $r'$ respectively, we set \[J_n(\chi,\chi')=\twosum{q\le P}{[r,r']|q}\frac{1}{\phi(q)^2} c_n(\chi_q\chi'_q)\tau(\overline{\chi_q})\tau(\overline{\chi'_q}) I_{n,q}(\chi,\chi').\] Then, by a trivial variant of the argument leading to Gallagher [3; (3)], we find that \begin{equation} \int_{{\mathfrak M}}S(\alpha)^2 e(-\alpha n)d\alpha= \sum_{\chi,\chi'}J_n(\chi,\chi')+O(P^{5/2}), \end{equation} for any integer $n$, the sum being over primitive characters $\chi,\chi'$ to moduli $r,r'$ for which $[r,r']\le P$. In what follows we shall take $1\le n\le N$. To estimate the contribution from a particular pair of characters $\chi,\chi'$ we put \[A_q(\chi)=\{\int_{-P/qN}^{P/qN}|A(\chi,\beta)|^2 d\beta\}^{1/2}\] and \[C_n(\chi,\chi')=\twosum{q\le P}{[r,r']|q}\frac{1}{\phi(q)^2} |c_n(\chi_q\chi'_q)\tau(\overline{\chi_q})\tau(\overline{\chi'_q})|.\] Note that what Gallagher calls $||A(\chi)||$ is our $A_1(\chi)$. We have $A_q(\chi)\le A_m(\chi)$ whenever $m\le q$. Then, as in Gallagher [3; (4)] we find \begin{equation} |J_n(\chi,\chi')|\le C_n(\chi,\chi')A_{[r,r']}(\chi)A_{[r,r']}(\chi'). \end{equation} It is in bounding $C_n(\chi,\chi')$ that there is a loss in Gallagher's argument. Let $r''$ be the conductor of $\chi\chi'$, and write $m=[r,r']$. moreover, for any positive integers $a$ and $n$ we write \[a_n=\frac{a}{(a,n)}.\] Then Gallagher shows that \[C_n(\chi,\chi')\le (rr'r'')^{1/2}\sum_{q\le P,\,m|q}(\phi(q)\phi(q_n))^{-1},\] where $q/m$ is square-free and coprime to $m$. Moreover we have $r''|m_n$. It follows that \[C_n(\chi,\chi')\le \frac{(rr'r'')^{1/2}}{\phi(m)\phi(m_n)} \sum_{(s,m)=1}\mu^2 (s)/\phi(s)\phi(s_n).\] The sum on the right is \[\prod_{p\, | \hspace{-1.1mm}/\, mn}(1+\frac{1}{(p-1)^2}) \prod_{p|n, p\, | \hspace{-1.1mm}/\, m}(1+\frac{1}{(p-1)})\ll \prod_{p|n, p\, | \hspace{-1.1mm}/\, m}\frac{p}{(p-1)},\] and \[\frac{m}{\phi(m)}\prod_{p|n, p\, | \hspace{-1.1mm}/\, m}\frac{p}{(p-1)}\le \frac{n}{\phi(n)}\frac{m_n}{\phi(m_n)}.\] We therefore deduce that \[C_n(\chi,\chi')\ll \frac{(rr'r'')^{1/2}}{m}\frac{m_n}{\phi^2 (m_n)} \frac{n}{\phi(n)}.\] Now if $p^e||r$ and $p^f||r'$, then $p^{|e-f|}|r''$, since $r''$ is the conductor of $\chi\chi'$. (Here the notation $p^e||r$ means, as usual, that $p^e|r$ and $p^{e+1}\, | \hspace{-1.1mm}/\, r$.) We therefore set \begin{equation} h=(r,r')\;\;\;\mbox{and}\;\;\; r=hs,\; r'=hs', \end{equation} so that $ss'|r''$ and $m=hss'$. Since \[\frac{m_n}{\phi^2(m_n)}\ll m_n ^{\varpi-1}\] we therefore have \[\frac{(rr'r'')^{1/2}}{m}\frac{m_n}{\phi^2 (m_n)}\ll (ss')^{-1/2}{r''}^{1/2}m_n^{\varpi-1}.\] Now, using the bounds $r''\le m_n$ and $ss'\le r''$, we find that \begin{eqnarray*} \frac{(rr'r'')^{1/2}}{m}\frac{m_n}{\phi^2 (m_n)}&\ll& (ss')^{-1/2}{r''}^{1/2}{r''}^{\varpi-1}\\ &=&(ss')^{-1/2}{r''}^{\varpi-1/2}\\ &\ll & (ss')^{\varpi-1}. \end{eqnarray*} Alternatively, using only the fact that $m_n\ge r''$, we have \begin{eqnarray*} \frac{(rr'r'')^{1/2}}{m}\frac{m_n}{\phi^2 (m_n)}&\ll & (ss')^{-1/2}m_n^{1/2}m_n^{\varpi-1}\\ &\ll & m_n^{\varpi-1/2}. \end{eqnarray*} These estimates produce \[C_n(\chi,\chi')\ll \min\{(ss')^{\varpi-1}\,,\,m_n^{\varpi-1/2}\} \frac{n}{\phi(n)}.\] On combining this with the bounds (2) and (3) we deduce the following result. \begin{lemma} Suppose that $P\le N^{2/5-\varpi}$. Then \[\int_{{\mathfrak M}}S(\alpha)^2 e(-\alpha n)d\alpha=J_n(1,1)+ O(\frac{n}{\phi(n)}S_n)+O(N^{1-\varpi}),\] where \[S_n=\sum_{\chi,\chi'}A_{[r,r']}(\chi) A_{[r,r']}(\chi')\min\{(ss')^{\varpi-1}\,,\,m_n^{-1/3}\},\] the sum being over primitive characters, not both principal, of moduli $r,r'$, with $[r,r']\le P$. \end{lemma} We have next to consider $A_m(\chi)$. According to the argument of Montgomery and Vaughan [15; \S 7] we have \[A_m(\chi)\ll N^{1/2}\max_{\varpi N<x\le N}\max_{0<h\le x}(h+mN/P)^{-1} |\sum_{x}^{x+h}\chi(p)|.\] Note that we have firstly taken account of the restriction in (1) to primes $p>\varpi N$, and secondly replaced $(h+N/P)^{-1}$ as it occurs in Montgomery and Vaughan, by the smaller quantity $(h+mN/P)^{-1}$. The argument of [15; \S 7] clearly allows this. By partial summation we have \[\sum_{x}^{x+h}\chi(p)\ll (\log x)^{-1}\max_{0<j\le h}\sum_{x}^{x+j}\chi(p)\log p.\] Moreover, a standard application of the `explicit formula' for $\psi(x,\chi)$ produces the estimate \[\sum_{x}^{x+j}\chi(p)\log p\ll N^{1/2+3\varpi}(\log N)^2+ \sum_{\rho}|\frac{(x+j)^{\rho}}{\rho}-\frac{x^{\rho}}{\rho}|,\] where the sum over $\rho$ is for zeros of $L(s,\chi)$ in the region \[\beta\ge\frac{1}{2}+3\varpi,\;\;\;|\gamma|\le N.\] When $\chi$ is the trivial character we shall include the pole $\rho=1$ amongst the `zeros'. Since $j\le h$ and \[\frac{(x+j)^{\rho}}{\rho}-\frac{x^{\rho}}{\rho}\ll \min\{jN^{\beta-1}\,,\,N^{\beta}|\gamma|^{-1}\},\] we find that \[A_m(\chi)\ll \frac{P}{m}N^{4\varpi}+\frac{N^{1/2}}{\log N} \{\max_{0<h\le N}(h+mN/P)^{-1} \sum_{\rho}N^{\beta-1}\min\{h\,,\,N|\gamma|^{-1}\}.\] However we have \[\min\{\frac{h}{h+H}\,,\,\frac{A}{h+H}\}\le\min\{1\,,\,\frac{A}{H}\}\] whenever $h,H,A>0$. Applying this with $H=mN/P$ and $A=N|\gamma|^{-1}$, we deduce that \begin{equation} A_m(\chi)\ll \frac{P}{m}N^{4\varpi}+\frac{N^{1/2}}{\log N} \sum_{\rho}N^{\beta-1}\min\{1\,,\,Pm^{-1}|\gamma|^{-1}\}. \end{equation} \section{The Sum $S_n$} In order to investigate the sum $S_n$ we decompose the available ranges for $r,r'$ and the corresponding zeros $\rho,\rho'$ into (overlapping) ranges \begin{equation} \left\{\begin{array}{cc}R\le r\le RN^{\varpi},&\;\;\;R'\le r'\le R'N^{\varpi},\\ T-1\le |\gamma|<TN^{\varpi},&\;\;\; T'-1\le |\gamma'|<T'N^{\varpi}. \end{array}\right. \end{equation} Clearly $O(1)$ such ranges suffice to cover all possibilities, so it is enough to consider the contribution from a fixed range of the above type. Throughout this section we shall follow the convention that $\rho=1$ is to included amongst the `zeros' corresponding to the trivial character. Let $N(\sigma,\chi,T)$ denote as usual, the number of zeros $\rho$ of $L(s,\chi)$, in the region $\beta\ge \sigma$, $|\gamma|\le T$, and let $N(\sigma,r,T)$ be the sum of $N(\sigma,\chi,T)$ for all characters $\chi$ of conductor $r$. Since \[N^{\beta-1}=N^{3\varpi-1/2}+\int_{1/2+3\varpi}^{\beta}N^{\sigma-1} (\log N)d\sigma\] for $\beta\ge 1/2+3\varpi$, we find that \begin{equation} \sum_{\rho}N^{\beta-1}\ll N^{6\varpi-1/2}RT+I(r)\log N, \end{equation} where the sum is over zeros of $L(s,\chi)$ for all $\chi$ of conductor $r$, subject to $T-1\le |\gamma|\le TN^{\varpi}$, and were \[I(r)=\int_{1/2+3\varpi}^{1}N^{\sigma-1}N(\sigma,r,TN^{\varpi})d\sigma.\] In view of the minimum occuring in (5) it is convenient to set \[m(R,T)=\min(1,\frac{P}{RT}).\] We now insert (7) into (5) so that, for given $r,r'$, the range (6) contributes to \[\sum_{\chi\!\!\!\pmod{r}}A_m(\chi)\] a total \begin{eqnarray} &\ll& \phi(r)\frac{P}{m}N^{4\varpi}+\frac{N^{1/2}}{\log N}m(R,T)N^{6\varpi-1/2}RT +N^{1/2}m(R,T)I(r)\nonumber\\ &\ll& PN^{6\varpi}+N^{1/2}m(R,T)I(r). \end{eqnarray} Similarly, for the double sum \[\sum_{\chi\!\!\!\pmod{r}}\sum_{\chi'\!\!\!\pmod{r'}}A_m(\chi)A_m(\chi')\] the contribution is \begin{equation} \begin{array}{ll}\ll & P^{2}N^{12\varpi}+PN^{1/2+6\varpi}m(R,T)I(r)\\ &{}+PN^{1/2+6\varpi}m(R',T')I(r')+Nm(R,T)m(R',T')I(r)I(r'). \end{array} \end{equation} We then sum over $r,r'$ using the following lemma. \begin{lemma} Let \[\max_{r\le R}N(\sigma,r,T)=N_1(R),\;\;\; \max_{r'\le R'}N(\sigma',r',T')=N_1(R'),\] and \[\sum_{r\le R}N(\sigma,r,T)=N_2(R),\;\;\; \sum_{r'\le R'}N(\sigma',r',T')=N_2(R').\] In the notation of (4) we have \begin{eqnarray} \lefteqn{\sum_{r\le R}\sum_{r'\le R'} N(\sigma,r,T)N(\sigma',r',T')(ss')^{\varpi-1}}\hspace{2cm}\\ &\ll & \{N_1(R)N_2(R)N_1(R')N_2(R')\}^{1/2+2\varpi},\nonumber \end{eqnarray} for $1/2\le\sigma,\sigma'\le 1$. Moreover, if \[P\le N^{45/154-4\varpi},\] then \begin{equation} \sum_{r,r'}m(R,T)m(R',T') N(\sigma,r,TN^{\varpi})N(\sigma',r',T'N^{\varpi})(ss')^{\varpi-1}, \end{equation} \begin{equation}\ll N^{(1-\varpi)(1-\sigma)+(1-\varpi)(1-\sigma')} \end{equation} for $1/2+3\varpi\le\sigma,\sigma'\le 1$, where the summation is for $R\le r\le RN^{\varpi}$ and $R'\le r'\le R'N^{\varpi}$. \end{lemma} We shall prove this at the end of this section. Henceforth we shall assume that $P\le N^{45/154-4\varpi}$. For suitable values of $\eta$ in the range \begin{equation} 0\le\eta\le \log\log N \end{equation} we shall define $\cl{B}(\eta)$ to be the set of characters $\chi$ of conductor $r\le P$, for which the function $L(s,\chi)$ has at least one zero in the region \[\beta>1-\frac{\eta}{\log N},\;\;\; |\gamma|\le N.\] According to our earlier convention the trivial character is always in $\cl{B}(\eta)$. Now, if we restrict attention to pairs $\chi,\chi'$ for which $\chi\not\in\cl{B}(\eta)$ we have \begin{eqnarray*} \lefteqn{\sum_{R\le r\le RN^{\varpi}}\sum_{R'\le r'\le R'N^{\varpi}} Nm(R,T)m(R',T')I(r)I(r')(ss')^{\varpi-1}}\hspace{3cm}\\ &\ll & \int_{1/2+3\varpi}^{1-\eta/\log N}\int_{1/2+3\varpi}^{1} N^{1-\varpi(1-\sigma)-\varpi(1-\sigma')}d\sigma' d\sigma\\ &\ll & N^{1-\varpi\eta/\log N}(\log N)^{-2}\\ &=& e^{-\varpi\eta}N(\log N)^{-2}. \end{eqnarray*} Terms for which $\chi\in\cl{B}(\eta)$ but $\chi'\not\in\cl{B}(\eta)$ may be handled similarly. This concludes our discussion of the final term in (9) for the time being. To handle the third term in (9) we use the zero density estimate \begin{equation} \sum_{r\le R}N(\sigma,r,T)\ll (R^2 T)^{\kappa(\sigma)(1-\sigma)}, \end{equation} where \begin{equation} \kappa(\sigma)=\left\{\begin{array}{cc} \frac{3}{2-\sigma}+\varpi, & \frac{1}{2}\le\sigma\le\frac{3}{4}\\ \frac{12}{5}+\varpi, & \frac{3}{4}\le\sigma\le 1.\end{array}\right. \end{equation} This follows from results of Huxley [5], Jutila [7; Theorem 1] and Montgomery [14; Theorem 12.2]. For each fixed value of $r'$ we have \begin{eqnarray*} \sum_{r} (ss')^{\varpi-1}&\le& \sum_{h|r'} (r'/h)^{\varpi-1}\sum_{s\le P/h}s^{\varpi-1}\\ &\ll &\sum_{h|r'} (r'/h)^{\varpi-1}(P/h)^{\varpi}\\ &\ll & N^{\varpi}. \end{eqnarray*} The contribution of the third term in (9) to $S_n$ is therefore \[\ll PN^{1/2+5\varpi}m(R',T')\sum_{r'}I(r').\] However the bound (14) shows that \[m(R',T')\sum_{r'}N(\sigma,r',TN^{\varpi})\ll \min\{1\,,\,\frac{P}{R'T'}\} ({R'}^2 N^{2\varpi}T'N^{\varpi})^{\kappa(\sigma)(1-\sigma)}.\] Since \[0\le \kappa(\sigma)(1-\sigma)\le 1\] in the range $1/2+\varpi\le\sigma\le 1$, this is \[\ll (P^2 N^{3\varpi})^{\kappa(\sigma)(1-\sigma)}.\] Moreover, if $P\le N^{45/154-4\varpi}$, then \[(P^2 N^{3\varpi})^{\kappa(\sigma)(1-\sigma)}N^{\sigma-1}\le N^{f(\sigma)}\] with \begin{eqnarray*} f(\sigma)&=&(\frac{45}{77}\kappa(\sigma)-1)(1-\sigma)\\ &\le& (\frac{45}{77}\{\frac{12}{5}+\varpi\}-1)(1-\sigma)\\ &\le& (\frac{31}{77}+\varpi)(1-\sigma)\\ &\le& (\frac{31}{77}+\varpi)\frac{1}{2}\\ &\le& \frac{31}{154}+\varpi. \end{eqnarray*} It follows that the contribution of the third term in (9) to $S_n$ is \[\ll PN^{1/2+6\varpi}.N^{31/154+\varpi}\ll N^{1-\varpi}.\] The second term may of course be handled similarly. Finally we deal with the first term of (9) which produces a contribution to $S_n$ which is \begin{eqnarray*} &\ll & P^{2}N^{12\varpi}\sum_{r,r'}(ss')^{\varpi-1}\\ &\ll & P^{2}N^{12\varpi}\sum_{ss'h\le P}(ss')^{\varpi-1}\\ &\ll & P^{2}N^{12\varpi}\sum_{ss'\le P}P(ss')^{\varpi-2}\\ &\ll & P^{3}N^{12\varpi}\\ &\ll & N^{1-\varpi}, \end{eqnarray*} for $P\le N^{45/154-4\varpi}$. We summarize our conclusions thus far as follows. \begin{lemma} If $P\le N^{45/154-4\varpi}$ then \[S_n\le \sum_{\chi,\chi'\in\cl{B}(\eta)}A_m(\chi)A_m(\chi')m_n^{-1/3} +O(e^{-\varpi\eta}N(\log N)^{-2}).\] \end{lemma} To handle the characters in $\cl{B}(\eta)$ we use the zero-density estimate \begin{equation} N(\sigma,r,T)\ll (rT)^{\kappa(\sigma)(1-\sigma)}, \end{equation} with $\kappa(\sigma)$ given by (15). This also follows from work of Huxley [5], Jutila [7; Theorem 1] and Montgomery [14; Theorem 12.1]. Thus \begin{eqnarray*} m(R,T)N(\sigma,r,TN^{\varpi})&\ll & \max\{1\,,\,\frac{P}{RT}\}(rTN^{\varpi})^{\kappa(\sigma)(1-\sigma)}\\ &\ll &(PN^{2\varpi})^{\kappa(\sigma)(1-\sigma)}\\ &\ll &(PN^{2\varpi})^{(12/5+\varpi)(1-\sigma)}\\ &\ll &N^{(1-\varpi)(1-\sigma)} \end{eqnarray*} for $P\le N^{45/154-4\varpi}$. We deduce that \[m(R,T)I(r)\ll (\log N)^{-1}.\] It follows from (8) that \[A_m(\chi)\ll N^{1/2}(\log N)^{-1}.\] We also note that \[\#\cl{B}(\eta)\ll\sum_{r}N(1-\frac{\eta}{\log N},r,N)\ll (P^2 N)^{3\eta/\log N}\ll e^{6\eta},\] by (14), since $\kappa(\sigma)\le 3$ for all $\sigma$. We therefore have the following facts. \begin{lemma} If $\chi\in\cl{B}(\eta)$, we have $A_m(\chi)\ll N^{1/2}(\log N)^{-1}$. Moreover, we have $\#\cl{B}(\eta)\ll e^{6\eta}$. \end{lemma} We end this section by establishing Lemma 3. We shall suppose, as we may by the symmetry, that \begin{equation} N_2(R)N_1(R')\le N_2(R')N_1(R). \end{equation} Let $U\ge 1$ be a parameter whose value will be assigned in due course, see (18). For those terms of the sum (10) in which $ss'\ge U$ we plainly have a total \[\le \sum_{r\le R}\sum_{r'\le R'}N(\sigma,r,T)N(\sigma',r',T')U^{\varpi-1} \ll N_2(R)N_2(R')U^{\varpi-1}.\] On the other hand, when $ss'<U$ we observe that, for fixed $s,s'$ we have \begin{eqnarray*} \sum_{h}N(\sigma,hs,T)N(\sigma',hs',T')&\ll& \sum_{h}N(\sigma,hs,T)N_1(R')\\ &\ll& \sum_{r}N(\sigma,r,T)N_1(R')\\ &\ll& N_2(R)N_1(R').\\ \end{eqnarray*} On summing over $s$ and $s'$ we therefore obtain a total \[\ll N_2(R)N_1(R')\sum_{ss'\le U}(ss')^{\varpi-1}\ll N_2(R)N_1(R')U^{2\varpi}.\] It follows that the sum (10) is \[\ll N_2(R)\{N_2(R')U^{2\varpi-1}+N_1(R')U^{2\varpi}\}.\] We therefore choose \begin{equation} U=N_2(R')/N_1(R'), \end{equation} whence the sum (10) is \begin{eqnarray*} &\ll& N_2(R)N_1(R')U^{2\varpi}\\ &\ll& N_2(R)N_1(R')\{N_1(R)N_2(R)N_1(R')N_2(R')\}^{2\varpi}\\ &\ll& \{N_2(R)N_1(R')N_2(R')N_1(R)\}^{1/2} \{N_1(R)N_2(R)N_1(R')N_2(R')\}^{2\varpi}\\ \end{eqnarray*} in view of (17). This produces the required bound. To establish (12) we shall bound $N_1(R)$ and $N_1(R')$ using (16). Moreover to handle $N_2(R)$ and $N_2(R')$ we shall use the estimate \[\sum_{r\le R}N(\sigma,r,T)\ll \left\{\begin{array}{cc} (R^2 T)^{\kappa(\sigma)(1-\sigma)},\; &\; \frac{1}{2}+\varpi\le\sigma\le \frac{23}{38}\\ (R^2 T^{6/5})^{\lambda(1-\sigma)},\; &\; \frac{23}{38}<\sigma\le 1,\end{array}\right.\] where \[\lambda=\frac{20}{9}+\varpi.\] This follows from (14) and (15) along with Heath-Brown [4; Theorem 2] and Jutila [7; Theorem 1]. We now see that the sum (11) may be estimated as \begin{equation} \ll m(R,T)R^aT^c.m(R',T'){R'}^b{T'}^d. N^{e}, \end{equation} say, where \[a=\left\{\begin{array}{cc} 3\kappa(\sigma)(1-\sigma)(\frac{1}{2}+2\varpi), \; &\; \frac{1}{2}+3\varpi\le\sigma\le \frac{23}{38}\\ \{\kappa(\sigma)+2\lambda\}(1-\sigma)(\frac{1}{2}+2\varpi),\; &\; \frac{23}{38}<\sigma\le 1,\end{array}\right.\] and \[c=\left\{\begin{array}{cc} 2\kappa(\sigma)(1-\sigma)(\frac{1}{2}+2\varpi) ,\; &\; \frac{1}{2}+3\varpi\le\sigma\le \frac{23}{38}\\ \{\kappa(\sigma)+6\lambda/5\}(1-\sigma)(\frac{1}{2}+2\varpi),\; &\; \frac{23}{38}<\sigma\le 1,\end{array}\right.\] and similarly for $b$ and $d$. Moreover we may take \[e=6\varpi(1-\sigma)+6\varpi(1-\sigma').\] It therefore follows that $0\le c,d< 1$, whence (19) is maximal for $T=P/R$ and $T'=P/R'$. Similarly we have $a\ge c$ and $b\ge d$. Thus, after substituting $T=P/R$ and $T'=P/R'$ in (19), the resulting expression is increasing with respect to $R$ and $R'$, and hence is maximal when $R=R'=P$. We therefore see that (20) is \[\ll P^{a+b}N^e.\] Finally one can check that \[(\frac{45}{154}-4\varpi)a\le (1-7\varpi)(1-\sigma),\] and similarly for $b$. This suffices to establish the bound (12) for $P\le N^{45/154-4\varpi}$. \section{Summation Over Powers of 2} In this section we consider the major arc integral \[\int_{{\mathfrak M}}S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha,\] where we now assume $N$ to be even. According to Lemmas 2 and 4 we have \begin{eqnarray} \int_{{\mathfrak M}}S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha&=& \Sigma_0+O(e^{-\varpi\eta}N(\log N)^{-2}\Sigma_1)\nonumber\\ &&\hspace{1cm}+O(N(\log N)^{-2}\Sigma_2), \end{eqnarray} where \[\Sigma_0=\sum_{n} J_n(1,1),\] \[\Sigma_1=\sum_{n}\frac{n}{\phi(n)}\] and \[\Sigma_2=\sum_{\chi,\chi'\in\cl{B}(\eta)}\sum_{n} \frac{n}{\phi(n)}m_n^{-1/3}.\] In each case the sum over $n$ is for values \begin{equation} n=N-\sum_{j=1}^{K}2^{\nu_j}. \end{equation} We begin by considering the main term $\Sigma_0$. We put \[T(\beta)=\sum_{\varpi N<m\le N}\frac{e(\beta m)}{\log m}\] and \[R(\beta)=S(\beta)-T(\beta).\] We also set \[||R||=\int_{-P/N}^{P/N}|R(\beta)|^2 d\beta\] and \[J(n)=\twosum{\varpi<m_1,m_2<N}{m_1+m_2=n}(\log m_1)^{-1}(\log m_2)^{-1}.\] Then, as in Gallagher [3; (11)], we have \begin{eqnarray} J_n(1,1)&=&J(n)\cl{S}(n) +O(N(\log N)^{-2}\frac{n}{\phi(n)}d(n)\frac{\log P}{P})\nonumber\\ &&\hspace{1cm}+O(\frac{n}{\phi(n)}\{N^{1/2}(\log N)^{-1}||R||+||R||^2\}), \end{eqnarray} where \[\cl{S}(n)=\prod_{p|n}(\frac{p}{p-1})\prod_{p\, | \hspace{-1.1mm}/\, n}(1-\frac{1}{(p-1)^2}).\] In analogy to (5) we have \[||R||\ll PN^{4\varpi}+\frac{N^{1/2}}{\log N} \sum_{\rho}N^{\beta-1}\min\{1\,,\,P|\gamma|^{-1}\},\] where the sum over $\rho$ is for zeros of $\zeta(s)$ in the region \[\beta\ge\frac{1}{2}+3\varpi,\;\;\;|\gamma|\le N.\] We split the range for $|\gamma|$ into $O(1)$ overlapping intervals \[T-1\le |\gamma|\le TN^{\varpi},\] and find, as in (8) that each range contributes \[\ll PN^{4\varpi}+N^{1/2}\min\{1\,,\,\frac{P}{T}\} \{N^{6\varpi-1/2}T+\int_{1/2+3\varpi}^{1}N^{\sigma-1}N(\sigma,1,TN^{\varpi})d\sigma\}\] to $||R||$. Using the case $R=1$ of (14), together with Vinogradov's zero-free region \[\sigma\ge 1-\frac{c_0}{(\log T)^{3/4}(\log\log T)^{3/4}}\] (see Titchmarsh [16; (6.15.1)]), we find that this gives \[||R||\ll N^{1/2}(\log N)^{-10},\] say, for $P\le N^{45/154-4\varpi}$. The error terms in (22) are therefore $O(N(\log N)^{-9})$. We also note that \begin{eqnarray*} J(n)&=&(\log N)^{-2}\#\{m_1,m_2:\varpi N<m_1,m_2\le N,\,m_1+m_2=n\}\\ &&\hspace{3cm}+O(N(\log N)^{-3})\\ &=&(\log N)^{-2}R(n)+O (N(\log N)^{-3}), \end{eqnarray*} where \[R(n)=\left\{\begin{array}{cc} 2N-n,& (1+\varpi)N\le n\le 2N,\\ n-2\varpi N,& 2\varpi N\le n\le (1+\varpi)N,\\ 0, & \mbox{otherwise}.\end{array}\right.\] In particular, we have $R(N-m)=(1-2\varpi)N(\log N)^{-2}+O(m(\log N)^{-2})$ for $1\le m\le N$. Since \[\cl{S}(n)\ll\frac{n}{\phi(n)}\ll\log\log N, \] we find, on taking $n$ of the form (21), that \[\sum_{n}J(n)\cl{S}(n)=(1-2\varpi)N(\log N)^{-2}\sum_{n}\cl{S}(n)+O(N(\log N)^{K-5/2})\] for $K\ge 2$, whence \[\Sigma_0=(1-2\varpi)N(\log N)^{-2}\sum_{n}\cl{S}(n)+ O(N(\log N)^{K-5/2}).\] Since the numbers $n$ are all even, we have \[\cl{S}(n)=2C_0\prod_{p|n, p\not=2}\frac{p-1}{p-2}=2C_0\sum_{d|n}k(d),\] where \begin{equation} C_0=\prod_{p\not=2}(1-\frac{1}{(p-1)^2}) \end{equation} and $k(d)$ is the multiplicative function defined by taking \begin{equation} k(p^e)=\left\{\begin{array}{cc} 0,\; & p=2\;\mbox{or}\;e\ge 2,\\ (p-2)^{-1},\;& \mbox{otherwise.} \end{array}\right. \end{equation} For any odd integer $d$ we shall define $\varepsilon(d)$ to be the order of 2 in the multiplicative group modulo $d$, and we shall set \[H(d;N,K)=\#\{(\nu_1,\ldots,\nu_K): 1\le\nu_i\le\varepsilon(d),\, d|N-\sum 2^{\nu_i}\}.\] Then for any fixed $D$ we have \begin{eqnarray*} \sum_{n}\cl{S}(n)&=&2C_0\sum_{d}k(d)\#\{n:d|n\}\\ &\ge&2C_0\sum_{d\le D}k(d)\#\{n:d|n\}\\ &\ge&2C_0\sum_{d\le D}k(d)H(d;N,K)[L/\varepsilon(d)]^K\\ &\ge&\{1+O((\log N)^{-1})\}2C_0L^K \sum_{d\le D}k(d)H(d;N,K)\varepsilon(d)^{-K}. \end{eqnarray*} We shall take $D=5$. We trivially have $\varepsilon(1)=1$ and $H(1;N,K)=1$ for all $N$ and $K$. When $d=3$ or $d=5$ the powers of 2 run over all non-zero residues modulo $d$, and it is an easy exercise to check that \[H(d;N,K)=\left\{\begin{array}{cc} \frac{1}{d}\{(d-1)^K-(-1)^K\}, & d\, | \hspace{-1.1mm}/\, N\\ \frac{1}{d}\{(d-1)^K+(-1)^K (d-1)\}, & d|N.\end{array}\right.\] Thus if $K\ge 7$ we have \[H(3;N,K)\varepsilon(3)^{-K}\ge \frac{1}{3}(1-2^{-6})\] and \[H(5;N,K)\varepsilon(5)^{-K}\ge \frac{1}{5}(1-4^{-6}),\] whence \[2\sum_{d\le D}k(d)H(d;N,K)\varepsilon(d)^{-K}\ge 2.7895\] for any choice of $N$. We therefore conclude that \begin{equation} \Sigma_0\ge 2.7895(1-2\varpi)C_0N(\log N)^{-2}L^K+O(N(\log N)^{K-5/2}), \end{equation} providing that $K\ge 9$. To bound $\Sigma_1$ we note that \[\frac{n}{\phi(n)}\ll\prod_{p|n,\,p\not=2}(1+\frac{1}{p})= \sum_{q|n,\,2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q}.\] We deduce that \[\Sigma_1\ll \sum_{q\le N,\,2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q}\#\{n:\,q|n\}.\] However, if $q$ is odd, then \[\#\{\nu:0\le\nu\le L,\,2^{\nu}\equiv m\!\!\!\pmod{q}\}\ll 1+\frac{L}{\varepsilon(q)}.\] It follows that \[\#\{n:\,q|n\}\ll L^{K-1}+L^K /\varepsilon(q),\] whence \[\Sigma_{1}\ll (\log N)^K+ (\log N)^K\sum_{q\le N,\,2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q\varepsilon(q)}.\] To bound the final sum we call on the following simple result of Gallagher [3; Lemma 4] \begin{lemma} We have \[\sum_{\varepsilon(q)\le x}\frac{\mu^2(q)}{\phi^2(q)}q\ll\log x.\] \end{lemma} From this we deduce that \begin{equation} \sum_{x/2<\varepsilon(q)\le x}\frac{\mu^2(q)}{q\varepsilon(q)}\ll\frac{\log x}{x}. \end{equation} We take $x$ to run over powers of $2$ and sum the resulting bounds to deduce that \[\sum_{q\le N,\,2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q\varepsilon(q)}\ll 1,\] and hence that \begin{equation} \Sigma_{1}\ll (\log N)^K. \end{equation} Turning now to $\Sigma_2$, we fix a particular pair of characters $\chi,\chi'\in\cl{B}(\eta)$, and investigate \[\sum_{n}\frac{n}{\phi(n)}m_n^{-1/3}=\Sigma_2(\chi,\chi'),\] say. Let $m=[r,r']$ as usual, and write $m=2^{\mu}f$, with $f$ odd. Put $g=(f,n)$ so that \begin{equation} m_n\ge f_n=f/g, \end{equation} and consider \[\sum_{g|n}\frac{n}{\phi(n)}.\] As before we have \[\frac{n}{\phi(n)}\ll\sum_{q|n,\,2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q}.\] Terms $q$ with $q\ge d(n)$ can contribute at most $1$ in total, so that in fact \[\frac{n}{\phi(n)}\ll\sum_{q|n,\,2\, | \hspace{-1.1mm}/\, q, q\le d(n)}\frac{\mu^2(q)}{q}.\] Thus, if \[D=\max_{1\le n\le N}d(n),\] we deduce as before that \begin{eqnarray*} \sum_{g|n}\frac{n}{\phi(n)}&\ll&\sum_{q\le D,\,2\, | \hspace{-1.1mm}/\, q} \frac{\mu^2(q)}{q}\#\{n:\,[g,q]|n\}\\ &\ll &\sum_{q\le D,\,2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q}\{(\log N)^{K-1}+\frac{(\log N)^K}{\varepsilon([g,q])}\}. \end{eqnarray*} Here we note that \[\sum_{q\le D}q^{-1}\ll \log D\ll\frac{\log N}{\log\log N}.\] To deal with the remaining terms let $\xi$ be a positive parameter. Then \begin{eqnarray*} \sum_{\varepsilon(q)>\xi}\frac{\mu^2(q)}{q\varepsilon([g,q])}&\le & \sum_{\varepsilon(q)>\xi}\frac{\mu^2(q)}{q\varepsilon(q)}\\ &\ll& \frac{\log{\xi}}{\xi}, \end{eqnarray*} by (26). If $\varepsilon(q)\le \xi$ we note that \begin{equation} q\le 2^{\varepsilon(q)}-1,\;\; \mbox{for}\;\;q>1, \end{equation} so that $q\le 2^{\xi}$. Thus \begin{eqnarray*} \sum_{\varepsilon(q)\le\xi}\frac{\mu^2(q)}{q\varepsilon([g,q])}&\le & \sum_{q\le 2^{\xi}}\frac{\mu^2(q)}{q\varepsilon(g)}\\ &\le & \frac{\xi}{\varepsilon(g)}. \end{eqnarray*} On choosing $\xi=\sqrt{\varepsilon(g)}$ we therefore conclude that \[\sum_{2\, | \hspace{-1.1mm}/\, q}\frac{\mu^2(q)}{q\varepsilon([g,q])}\ll \frac{\log\varepsilon(g)}{\sqrt{\varepsilon(g)}},\] and hence that \[\sum_{g|n}\frac{n}{\phi(n)}\ll (\log N)^K \{(\log\log N)^{-1}+\varepsilon(g)^{-1/3}\}.\] It follows from (29) that $\varepsilon(g)\gg\log g$, and we now conclude that \[\sum_{g|n}\frac{n}{\phi(n)}\ll (\log N)^K \{(\log\log N)^{-1}+(\log g)^{-1/3}\}.\] We now observe from (28) that \[\Sigma_2(\chi,\chi')\le \sum_{n}\frac{n}{\phi(n)}(\frac{f}{(f,n)})^{-1/3}.\] Let $\tau\ge 1$ be a parameter to be fixed in due course. Then terms in which $(f,n)\le f/\tau$ contribute \[\le \tau^{-1/3}\sum_{n}\frac{n}{\phi(n)}=\tau^{-1/3}\Sigma_1 \ll \tau^{-1/3}(\log N)^K,\] by (27). The remaining terms contribute \begin{eqnarray*} &\le&\sum_{g|f,\,g\ge f/\tau}(f/g)^{-1/3}\sum_{g|n}\frac{n}{\phi(n)}\\ &\ll& \sum_{g|f,\,g\ge f/\tau}(f/g)^{-1/3} (\log N)^K \{(\log\log N)^{-1}+(\log g)^{-1/3}\}\\ &\ll& \sum_{g|f,\,g\ge f/\tau} (\log N)^K \{(\log\log N)^{-1}+(\log f)^{-1/3}\}\\ &\ll& \sum_{j|f,\,j\le \tau} (\log N)^K \{(\log\log N)^{-1}+(\log f)^{-1/3}\}\\ &\ll& \tau(\log N)^K \{(\log\log N)^{-1}+(\log f)^{-1/3}\}. \end{eqnarray*} We deduce that \[\Sigma_2(\chi,\chi')\ll\tau^{-1/3}(\log N)^K+ \tau(\log N)^K \{(\log\log N)^{-1}+(\log f)^{-1/3}\}.\] We therefore choose \[\tau=\{(\log\log N)^{-1}+(\log f)^{-1/3}\}^{-3/4},\] whence \begin{equation} \Sigma_2(\chi,\chi')\ll (\log N)^K\{(\log\log N)^{-1/4}+(\log f)^{-1/12}\}. \end{equation} In order to bound $f$ from below we note that, since $\chi,\chi'$ are not both trivial, we may suppose that $\chi$, say, is non-trivial. We then use a result of Iwaniec [6;~Theorem~2]. This shows that if $L(\beta+i\gamma,\chi)=0$, with $|\gamma|\le N$, and $\chi$ of conductor $r\le N$, then either $\chi$ is real, or \[1-\beta\gg \{\log d+(\log N\log\log N)^{3/4}\}^{-1},\] where $d$ is the product of the distinct prime factors of $r$. In our application we clearly have $f\ge d/2$, so that if $\chi$, say, is in $\cl{B}(\eta)$ we must have \[\frac{\eta}{\log N}\gg \{\log f+(\log N\log\log N)^{3/4}\}^{-1}\] if $\chi$ is not real. Thus, if we insist that $\eta\le (\log N)^{1/5}$ it follows that either \[\log f\gg\eta^{-1}\log N\gg (\log N)^{4/5},\] or $\chi$ is real. Of course if $\chi$ is real we will have $16\, | \hspace{-1.1mm}/\, r$, whence $f\gg r$. Moreover we will also have \[(\log N)^{4/5}\gg\frac{\eta}{\log N}\gg 1-\beta\gg r^{\varpi-1/2},\] so that $f\gg r\gg(\log N)^{3/2}$. Thus in either case we find that $\log f\gg\log\log N$, so that (30) yields \[ \Sigma_2(\chi,\chi')\ll (\log N)^K(\log\log N)^{-1/12}.\] In view of the bound for $\#\cl{B}(\eta)$ given in Lemma 5, we conclude that \begin{equation} \Sigma_2\ll e^{12\eta}(\log N)^K(\log\log N)^{-1/12}. \end{equation} We may now insert the bounds (25), (27) and (31) into (20) to deduce that \begin{eqnarray*} \int_{{\mathfrak M}}S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha &\ge &2.7895(1-2\varpi)C_0N(\log N)^{-2}L^K\\ &&\hspace{2mm}{}+O(N(\log N)^{K-5/2})\\ &&\hspace{4mm}{}+O(e^{-\varpi\eta}N(\log N)^{K-2})\\ &&\hspace{6mm}{}+O(e^{12\eta}N(\log N)^{K-2}(\log\log N)^{-1/12}). \end{eqnarray*} We therefore define $\eta$ by taking \[e^{\eta}=(\log\log N)^{1/145},\] so that $\eta$ satisfies the condition (13), and conclude as follows. \begin{lemma} If $p\le N^{45/154-4\varpi}$ and $K\ge 9$ we have \[\int_{{\mathfrak M}}S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha \ge 2.7895(1-3\varpi)C_0N(\log 2)^{-2}L^{K-2}\] for large enough $N$. \end{lemma} \section{A Mean Square Estimate} In this section we shall estimate the mean square \[J({\mathfrak m})=\int_{{\mathfrak m}}|S(\alpha)T(\alpha)|^2 d\alpha,\] where ${\mathfrak m}=[0,1]\setminus{\mathfrak M}$ is the set of minor arcs. Instead of this integral, previous researchers have worked with the larger integral \[J=\int_{0}^1 |S(\alpha)T(\alpha)|^2 d\alpha.\] Thus it was shown by Li [9; Lemma 6], building on work of Liu, Liu and Wang [13; Lemma 4] that \[J\le (24.95+o(1))\frac{C_0}{\log^2 2}N,\] In this section we shall improve on this bound, and give a lower bound for the corresponding major arc integral \[J({\mathfrak M})=\int_{{\mathfrak M}}|S(\alpha)T(\alpha)|^2 d\alpha.\] By subtraction we shall then obtain our bound for $J({\mathfrak m})$. We begin by observing that \[J=\sum_{\mu,\nu\le L}r(2^{\mu}-2^{\nu}),\] where \[r(n)=\#\{\varpi N<p_1,p_2\le N: n=p_1-p_2\}.\] Moreover, by Theorem 3 of Chen [2] we have \[r(n)\le C_0 C_1 h(n)\frac{N}{(\log N)^2},\] for $n\not=0$ and $N$ sufficiently large, where $C_0$ is given by (23), \begin{equation} C_1=7.8342, \end{equation} and \[h(n)=\prod_{p|n,\,p>2}(\frac{p-1}{p-2}).\] Observe that our notation for the constants that occur differs from that used by Liu, Liu and Wang, and by Li. Since $h(2^{\mu}-2^{\nu})=h(2^{\mu-\nu}-1)$ for $\mu>\nu$ we conclude, as in Liu, Liu and Wang [13; \S 3] and Li [9; \S 4] that \begin{equation} \sum_{\mu\not=\nu\le L}r(2^{\mu}-2^{\nu})\le 2C_0 C_1\frac{N}{(\log N)^2} \sum_{1\le l\le L}(L-l)h(2^l-1), \end{equation} while the contribution for $\mu=\nu$ is $L\pi(N)-L\pi(\varpi N)\le LN(\log N)^{-1}$, for large $N$. Now \[h(n)=\sum_{d|n}k(d),\] where $k(d)$ is the multiplicative function defined in (24). Thus \begin{eqnarray*} \sum_{1\le j\le J}h(2^j-1)&=& \sum_{d=1}^\infty k(d)\#\{j\le J: d|2^j-1\}\\ &=&\sum_{d=1}^\infty k(d)[\frac{J}{\varepsilon(d)}]. \end{eqnarray*} However $[\theta]=\theta+O(\theta^{1/2})$ for any real $\theta>0$, whence \begin{equation} \sum_{1\le j\le J}h(2^j-1)=C_2 J+O(J^{1/2}) \end{equation} with \begin{equation} C_2=\sum_{d=1}^\infty \frac{k(d)}{\varepsilon(d)}. \end{equation} Here we use the observation that the sum \[\sum_{d=1}^\infty \frac{k(d)}{\varepsilon(d)^{1/2}}\] is convergent, since Lemma 6 implies that \begin{equation} \sum_{x/2<\varepsilon(d)\le x}\frac{k(d)}{\varepsilon(d)^{1/2}}\ll x^{-1/2}\sum_{x/2<\varepsilon(d)\le x}\frac{\mu^2(d)d}{\phi^2(d)}\ll \frac{\log x}{x^{1/2}} \end{equation} for any $x\geq 2$. We may now use partial summation in conjunction with (34) to deduce that \[\sum_{1\le l\le L}(L-l)h(2^l-1)=C_2\frac{L^2}{2}+O(L^{3/2}),\] Thus, using (33) we reach the following result. \begin{lemma} We have \[J\le \{\frac{C_0 C_1 C_2}{\log^2 2}+\frac{1}{\log 2}+o(1)\}N,\] with the constants given by (23), (32) and (35). \end{lemma} We now turn to the integral $J({\mathfrak M})$. According to Lemma 3.1 of Vaughan [17], if \[|\alpha-\frac{a}{q}|\le\frac{\log x}{x},\;\;\;(a,q)=1,\] and $q\le 2\log x$, we have \[\sum_{p\le x}e(\alpha p)\log p=\frac{\mu(q)}{\phi(q)}v(\alpha-\frac{a}{q}) +O(x(\log x)^{-3}),\] with \[v(\beta)=\sum_{m\le x}e(\beta m).\] It follows by partial summation that \[S(\alpha)=\frac{\mu(q)}{\phi(q)}w(\alpha-\frac{a}{q}) +O(N(\log N)^{-4}),\] with \[w(\beta)=\sum_{\varpi N<m\le N}\frac{e(\beta m)}{\log m},\] providing that \begin{equation} |\alpha-\frac{a}{q}|\le\frac{\log N}{N},\;\;\;(a,q)=1 \end{equation} and $q\le\log N$. Then if $\mathfrak{a}$ denotes the set of $\alpha\in[0,1]$ for which such $a,q$ exist, we easily compute that \begin{eqnarray*} J(\mathfrak{M})&\ge&J(\mathfrak{a})\\ &=& \int_{{\mathfrak a}}|\frac{\mu(q)}{\phi(q)}w(\alpha-\frac{a}{q}) T(\alpha)|^2 d\alpha+O(N(\log N)^{-1}), \end{eqnarray*} where, for each $\alpha\in\mathfrak{a}$, we have taken $a/q$ to be the unique rational satisfying (37). By partial summation we have \[w(\beta)\ll (||\beta||\log N)^{-1},\] whence \[\int_{-(\log N)/N}^{(\log N)/N}|w(\beta)T(\frac{a}{q}+\beta)|^2 d\beta =\int_{-1/2}^{1/2}|w(\beta)T(\frac{a}{q}+\beta)|^2 d\beta+O(N(\log N)^{-1}). \] It follows that \[J(\mathfrak{a})= \sum_{q\le\log N}\sum_{(a,q)=1}\frac{\mu^2(q)}{\phi^2(q)} \int_{0}^{1}|w(\beta)T(\frac{a}{q}+\beta)|^2 d\beta+ O(N(\log N)^{-1}\log\log N).\] The integral on the right is \[\sum_{0\le\mu,\nu\le L}e(a(2^{\mu}-2^{\nu})/q)S(2^{\mu}-2^{\nu}),\] where \begin{eqnarray*} S(n)&=& \twosum{\varpi N<m_1,m_2\le N}{m_1-m_2=n}(\log m_1)^{-1}(\log m_2)^{-1}\\ &=&(\log N)^{-2}\#\{m_1,m_2:\varpi N<m_1,m_2\le N,\,m_1-m_2=n\}\\ &&\hspace{3cm}+O(N(\log N)^{-3})\\ &=&(\log N)^{-2}\max\{N(1-\varpi)-|n|\,,\,0\}+O (N(\log N)^{-3}). \end{eqnarray*} Thus \begin{equation} S(n)=(1-\varpi)N(\log N)^{-2}+O(|n|(\log N)^{-2})+O (N(\log N)^{-3}) \end{equation} for $n\ll N$. On summing over $a$ we now obtain \[J(\mathfrak{a})= \sum_{0\le\mu,\nu\le L}\sum_{q\le\log N}\frac{\mu^2(q)}{\phi^2(q)} c_q(2^{\mu}-2^{\nu})S(2^{\mu}-2^{\nu})+O(N(\log N)^{-1}\log\log N),\] where $c_q(n)$ is the Ramanujan sum. When $q$ is square-free we have $c_q(n)=\mu(q)\mu((q,n))\phi((q,n))$. Thus the error terms in (38) make a total contribution $O(N(\log N)^{-1}\log\log N)$ to $J(\mathfrak{a})$. Moreover \[\mu^2(q)c_q(n)=\mu(q)\sum_{d|(q,n)}\mu(d)d,\] whence \[\sum_{0\le\mu,\nu\le L}\mu^2(q)c_q(n)=\mu(q)\sum_{d|q}\mu(d)d \#\{\mu,\nu:\,1\le\mu,\nu\le L,\,d|2^{\mu}-2^{\nu}\}.\] If $d$ is odd we have \[\#\{\mu,\nu:\,1\le\mu,\nu\le L,\,d|2^{\mu}-2^{\nu}\}=L^2\varepsilon(d)^{-1}+O(L),\] while if $d$ is even, of the form $2e$ with $e$ odd, we have \[\#\{\mu,\nu:\,1\le\mu,\nu\le L,\,d|2^{\mu}-2^{\nu}\}=L^2\varepsilon(e)^{-1}+O(L).\] The error terms contribute $O(N(\log N)^{-1}\log\log N)$ to $J(\mathfrak{a})$, by (38), so that \[J(\mathfrak{a})=\frac{(1-\varpi)N}{(\log N)^2}L^2 \sum_{q\le\log N}\frac{\mu(q)}{\phi^2(q)}\sum_{d|q}\mu(d)d\varepsilon(d)^{-1} +O(N(\log N)^{-1}\log\log N),\] where $\varepsilon(d)$ is to be interpreted as $\varepsilon(e)$ when $d=2e$. Now \begin{eqnarray} \sum_{q\le\log N}\frac{\mu(q)}{\phi^2(q)}\sum_{d|q}\frac{\mu(d)d}{\varepsilon(d)}&=& \sum_{d\le\log N}\frac{\mu(d)d}{\varepsilon(d)} \twosum{q\le\log N}{d|q}\frac{\mu(q)}{\phi^2(q)}\nonumber\\ &=&\sum_{d\le\log N}\frac{\mu(d)d}{\varepsilon(d)} \sum_{j\le (\log N)/d}\frac{\mu(jd)}{\phi^2(jd)}\nonumber\\ &=&\sum_{d\le\log N}\frac{\mu^2(d)d}{\varepsilon(d)\phi^2(d)} \twosum{j\le (\log N)/d}{(j,d)=1}\frac{\mu(j)}{\phi^2(j)}\nonumber\\ &=&\sum_{d\le\log N}\frac{\mu^2(d)d}{\varepsilon(d)\phi^2(d)} \{\twosum{j=1}{(j,d)=1}^{\infty}\frac{\mu(j)}{\phi^2(j)} +O(\frac{d}{\log N})\}\nonumber\\ &=&\sum_{d\le\log N}\frac{\mu^2(d)d}{\varepsilon(d)\phi^2(d)} \prod_{p\, | \hspace{-1.1mm}/\, d}\{1-(p-1)^{-2}\}\nonumber\\ &&\hspace{1cm}+ O((\log N)^{-1}\sum_{d\le\log N}\frac{\mu^2(d)d^2}{\varepsilon(d)\phi^2(d)}). \end{eqnarray} If $d=2e$ with $e$ odd, we have \[\frac{\mu^2(d)d}{\varepsilon(d)\phi^2(d)} \prod_{p\, | \hspace{-1.1mm}/\, d}\{1-(p-1)^{-2}\}=2C_0 k(e)/\varepsilon(d),\] while if $d$ is odd we have \[\prod_{p\, | \hspace{-1.1mm}/\, d}\{1-(p-1)^{-2}\}=0,\] since the factor with $p=2$ vanishes. Moreover \[\sum_{d\gg\log N}\frac{k(d)}{\varepsilon(d)}\ll \frac{\log N}{\log\log N}\] by Lemma 6, applied as in (36). The leading term in (39) is therefore $2C_0 C_2+o(1)$, with $C_0$ and $C_2$ as in (23) and (35). To bound the error term we use Lemma 6, which shows that \[\twosum{X<d\le 2X}{x<\varepsilon(d)\le 2x}\frac{\mu^2(d)d^2}{\varepsilon(d)\phi^2(d)} \ll\frac{X\log x}{x}.\] According to (29) we must have $x\gg\log X$, so on summing as $x$ runs over powers of $2$ we obtain \[\sum_{X<d\le 2X}\frac{\mu^2(d)d^2}{\varepsilon(d)\phi^2(d)} \ll\frac{X\log\log X}{\log X}.\] Now, summing as $X$ runs over powers of $2$ we conclude that \[\sum_{d\le\log N}\frac{\mu^2(d)d^2}{\varepsilon(d)\phi^2(d)}\ll \frac{(\log N)(\log\log\log N)}{\log\log N}.\] We may therefore summarize our results as follows. \begin{lemma} We have \[ J(\mathfrak{M})\ge \{\frac{2(1-\varpi)C_0 C_2}{\log^2 2}+o(1)\}N,\] and hence \[J(\mathfrak{m})\le \{\frac{C_0(C_1-2+2\varpi)C_2}{\log^2 2} +\frac{1}{\log 2}+o(1)\}N,\] by Lemma 8. \end{lemma} It remains to compute the constants. We readily find \[\prod_{2<p\le 200000}(1-(p-1)^{-2})=0.6601...\] Since \[\prod_{p>K}(1-(p-1)^{-2})\ge\prod_{n=K}^{\infty}(1-n^{-2}) =1-K^{-1},\] we deduce that \begin{equation} C_0\ge 0.999995\times0.6601\ge 0.66. \end{equation} However the estimation of $C_2$ is more difficult. We set \[m=\prod_{e\le x}(2^e-1)\] and \[s(x)=\sum_{\varepsilon(d)\le x}k(d),\] whence \begin{eqnarray*} s(x)&\le&\sum_{d|m}k(d)\\ &=& h(m)\\ &=&\prod_{p|m,\,p>2}(\frac{p-1}{p-2})\\ &\le & \prod_{p>2}(\frac{(p-1)^2}{p(p-2)})\prod_{p|m}(\frac{p}{p-1})\\ &=& C_0^{-1}\frac{m}{\phi(m)}. \end{eqnarray*} Moreover we have $m/\phi(m)\le e^{\gamma}\log x$ for $x\ge 9$, as shown by Liu, Liu and Wang [13; (3.9)]. It then follows that \begin{eqnarray*} C_2&=&\int_{1}^{\infty}s(x)\frac{dx}{x^2}\\ &=&\int_{1}^{M}s(x)\frac{dx}{x^2}+\int_{M}^{\infty}s(x)\frac{dx}{x^2}\\ &\le&\sum_{\varepsilon(d)\le M}\int_{\varepsilon(d)}^{M}k(d)\frac{dx}{x^2}+ C_0^{-1}e^{\gamma}\int_{M}^{\infty}\log x\frac{dx}{x^2}\\ &\le &\sum_{\varepsilon(d)<M}k(d)(\frac{1}{\varepsilon(d)}-\frac{1}{M})+ 2.744(\frac{1+\log M}{M}) \end{eqnarray*} for any integer $M\ge 9$. We now set \[\sum_{\varepsilon(d)=e}k(d)=\kappa(e)\] so that \[\sum_{e|d}\kappa(e)=\sum_{\varepsilon(e)|d}k(e).\] However $\varepsilon(e)|d$ if and only if $e|2^d-1$. Thus \[\sum_{e|d}\kappa(e)=\sum_{e|2^d-1}k(e)=h(2^d-1).\] We therefore deduce that \[\kappa(e)=\sum_{d|e}\mu(e/d)h(2^d-1).\] This enables us to compute \[\sum_{\varepsilon(d)<M}k(d)(\frac{1}{\varepsilon(d)}-\frac{1}{M})= \sum_{m<M}\kappa(m)(\frac{1}{m}-\frac{1}{M})\] by using information on the prime factorization of $2^d-1$ for $d<M$. In particular, taking $M=20$ we find that \[\sum_{m<20}\kappa(m)(\frac{1}{m}-\frac{1}{20})=1.6659\ldots,\] and hence that \begin{equation} C_2\le\sum_{m<20}\kappa(m)(\frac{1}{m}-\frac{1}{20})+ 2.744(\frac{1+\log 20}{20})=2.2141\ldots \end{equation} For comparison with this upper bound for $C_2$ we note that \[C_2\ge\sum_{d\le 10000}k(d)/\varepsilon(d)=1.9326\ldots\] This latter figure is probably closer to the true value, but the discrepancy is small enough for our purposes. From (32), (40) and (41) we calculate that \[(C_1-2)C_2+C_0^{-1}\log 2\le 13.967,\] so that Lemma 9 yields the following bound. \begin{lemma} We have \[J(\mathfrak{m})\le \{13.968+o(1)\}C_0\frac{N}{\log^2 2}.\] \end{lemma} \section{Completion of the Proof} Let $R(N)$ denote the number of representations of $N$ as a sum of two primes and $K$ powers of $2$ in the ranges under consideration, so that \[R(N)=\int_0^1S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha.\] To estimate the minor arc contribution to $R(N)$ we first bound $S(\alpha)$. According to Theorem 3.1 of Vaughan [17] we have \[\sum_{p\le x}e(\alpha p)\log p\ll (\log x)^4\{xq^{-1/2}+x^{4/5}+x^{1/2}q^{1/2}\}\] if $|\alpha-a/q|\le q^{-2}$ with $(a,q)=1$. Thus if $\alpha\in\mathfrak{m}$ we may take $P\ll q\ll N/P$ to deduce that \[S(\alpha)\ll (\log N)^3\{N^{4/5}+NP^{-1/2}\}.\] Taking $P=N^{45/154-4\varpi}$, we obtain \[S(\alpha)\ll N^{263/308+3\varpi}.\] If one assumes the Generalized Riemann Hypothesis, we may apply Lemma 12 of Baker and Harman [1], which implies that \[\sum_{n\le x}\Lambda(n)e((\frac{a}{q}+\beta) n)\ll (\log x)^2\{q^{-1}\min(x,|\beta|^{-1})+x^{1/2}q^{1/2}+x(q|\beta|)^{1/2}\}\] when $|\beta|\le x^{-1/2}$. It follows by partial summation that \[S(\frac{a}{q}+\beta)\ll (\log N)\{q^{-1}\min(N,|\beta|^{-1}) +N^{1/2}q^{1/2}+N(q|\beta|)^{1/2}\}\] for $|\beta|\le N^{-1/2}$. According to Dirichlet's Approximation Theorem, we can find $a$ and $q$ with \[|\alpha-\frac{a}{q}|\le\frac{1}{qN^{1/2}},\;\;\;q\le N^{1/2}.\] Thus \[S(\alpha)\ll (\log N)N^{3/4}\] unless $q\le N^{1/4}$ and $|\alpha-a/q|\le q^{-1}N^{-3/4}$. Since $\alpha\in\mathfrak{m}$ and $P=N^{45/154-4\varpi}\ge N^{1/4}$, these latter conditions cannot hold. We therefore conclude that \[S(\alpha)\ll N^{\theta+o(1)}\] for $\alpha\in\mathfrak{m}$, where we take $\theta=263/308$ in general, and $\theta=3/4$ under the Generalized Riemann Hypothesis. We now have \begin{eqnarray*} \int_{{\mathfrak m}\cap\cl{A}_{\lambda}}S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha&\ll&{\rm meas}(\cl{A}_{\lambda})N^{2\theta+o(1)}L^K\\ &\ll& N^{-E(\lambda)+2\theta+o(1)}\\ &\ll&N, \end{eqnarray*} providing that $E(\lambda)>2\theta-1$. Thus, according to Lemma 1, we may take $\lambda=0.863665$ unconditionally, and $\lambda=0.722428$ under the Generalized Riemann Hypothesis. It remains to consider the set ${\mathfrak m}\setminus\cl{A}_{\lambda}$. Here we have \begin{eqnarray*} |\int_{{\mathfrak m}\setminus\cl{A}_{\lambda}}S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha|&\le& (\lambda L)^{K-2} \int_{{\mathfrak m}}|S(\alpha)T(\alpha)|^2 d\alpha\\ &\le& (\lambda L)^{K-2} 13.968\frac{C_0}{\log^2 2} N. \end{eqnarray*} Finally we compare this with the estimate for the major arc integral, given by Lemma 7, and conclude that \[\int_0^1 S(\alpha)^2 T(\alpha)^K e(-\alpha N)d\alpha >0\] providing that $N$ is large enough, $\varpi$ is small enough, and \[ 13.968\lambda^{K-2} < 2.7895. \] When $\lambda=0.863665$ this is satisfied for $K>12.991$, so that $K=13$ is admissible. Similarly, when $\lambda=0.722428$ one can take any $K>6.995$, so that $K=7$ is admissible. This completes the proof of our theorems, subject to Lemma 1. \section{Proof of Lemma 1} In this section we shall prove Lemma 1. We shall again use $\varpi$ to denote a small positive constant. We shall allow the constants implied by the $O(\ldots)$ and $\ll$ notations to depend on $\varpi$, although sometimes we shall mention the dependence explicitly for emphasis. As mentioned in the introduction, the method we shall adopt was suggested to us by Professor Keith Ball, and is based on the martingale method for proving exponential inequalities in probability theory. It is convenient to work with \[T_L(\alpha)=T(\alpha/2)=\sum_{0\le n\le L-1}e(\alpha 2^n)\] in place of $T(\alpha)$. Clearly we have \[{\rm meas}\{\alpha\in[0,1]: |T_L(\alpha)|\ge\lambda L\}= {\rm meas}(\cl{A}_{\lambda}).\] Let $M=1+[2\pi/\varpi]$ and suppose that $|T_L(\alpha)|\ge\lambda L$ with $\arg(T_L(\alpha))=\phi$. Write $m=[M\phi/2\pi]$ and $\rho_m=e(-m/M)$. Then \[|e^{-i\phi}-\rho_m|\le |\phi-\frac{2\pi m}{M}|\le \frac{2\pi}{M}\le\varpi,\] whence \begin{eqnarray*} {\rm Re}(\rho_m T_L(\alpha))&\ge& {\rm Re}(e^{-i\phi}T_L(\alpha))- \varpi|T_L(\alpha)|\\ &=&(1-\varpi)|T_L(\alpha)|\\ &\ge&(1-\varpi)\lambda L. \end{eqnarray*} It follows that \begin{eqnarray*} \lefteqn{{\rm meas}\{\alpha\in[0,1]: |T_L(\alpha)|\ge\lambda L\}} \hspace{2cm}\\ &\le& \sum_{m=0}^{M-1}{\rm meas}\{\alpha\in[0,1]: {\rm Re}(\rho_m T_L(\alpha)) \ge(1-\varpi)\lambda L\}\\ &\ll_{\varpi}&\sup_{|\rho|=1}{\rm meas}\{\alpha\in[0,1]: {\rm Re}(\rho T_L(\alpha))\ge(1-\varpi)\lambda L\}. \end{eqnarray*} We now set \[S(\xi,\rho,L)=\int_{0}^{1}\exp\{\xi{\rm Re}(\rho T_L(\alpha))\}d\alpha,\] for an arbitrary real $\xi>0$, whence \[S(\xi,\rho,L)\ge \exp\{\xi(1-\varpi)\lambda L\} {\rm meas}\{\alpha\in[0,1]: {\rm Re}(\rho T_L(\alpha))\ge(1-\varpi)\lambda L\}.\] It therefore follows that \begin{equation} {\rm meas}(\cl{A}_{\lambda})\ll \exp\{-\xi(1-\varpi)\lambda L\} \sup_{|\rho|=1}S(\xi,\rho,L). \end{equation} For any integer $h$, we have $T_L(\alpha)=T_{L-h}(2^h\alpha)+T_h(\alpha)$. Moreover, for any function $f$ we have \[\int_0^1 f(\alpha)d\alpha = \frac{1}{2^h}\int_0^{1} \sum_{r=0}^{2^h-1}f(\frac{\beta}{2^h}+\frac{r}{2^h})d\beta.\] It therefore follows that \[S(\xi,\rho,L)=\frac{1}{2^h}\int_0^{1}\sum_{r=0}^{2^h-1} \exp\{\xi{\rm Re}(\rho T_{L-h}(\beta+r))\} \exp\{\xi{\rm Re}(\rho T_{h}(\frac{\beta+r}{2^h}))\}d\beta.\] Since $T(\alpha)$ has period $1$ this becomes \[\int_0^{1}\exp\{\xi{\rm Re}(\rho T_{L-h}(\beta))\}\frac{1}{2^h} \sum_{r=0}^{2^h-1}\exp\{\xi{\rm Re}(\rho T_{h}(\frac{\beta+r}{2^h}))\} d\beta.\] If we now set \begin{equation} F(\xi,h)=\sup_{\beta\in[0, 1],\, |\rho|=1}\frac{1}{2^h} \sum_{r=0}^{2^h-1}\exp\{\xi{\rm Re}(\rho T_h(\frac{\beta + r}{2^h}))\} \end{equation} we deduce that \[S(\xi,\rho,L)\le S(\xi,\rho,L-h)F(\xi,h).\] Using this inductively we find that \[S(\xi,\rho,L)\le S(\xi,\rho,L-nh)F(\xi,h)^n,\] and taking $n=[L/h]$ we deduce that \[S(\xi,\rho,L)\ll_{\xi,h} F(\xi,h)^n\ll_{\xi,h} F(\xi,h)^{L/h}.\] When we combine this with (42) we deduce that \[{\rm meas}(\cl{A}_{\lambda})\ll_{\xi,h,\varpi} \exp\{-\xi(1-\varpi)\lambda L\} F(\xi,h)^{L/h}.\] It follows that we may take \[E(\lambda)=\frac{\xi\lambda}{\log 2}-\frac{\log F(\xi,h)}{h\log 2}- \frac{\varpi}{\log 2}\] for any $h\in\mathbb{N}$, any $\xi>0$ and any $\varpi>0$. We proceed to show that the supremum in (43) occurs at $\beta=0$ and $\rho=1$, whence \begin{equation} F(\xi,h)=\frac{1}{2^h}\sum_{r=0}^{2^h-1} \exp\{\xi{\rm Re}(T_h(\frac{r}{2^h}))\}. \end{equation} Since \[{\rm Re}(\rho T_h(\frac{\beta+r}{2^h}))=\frac{1}{2} \{\rho T_h(\frac{\beta+r}{2^h})+\overline{\rho}\,T_h(\frac{-\beta-r}{2^h})\},\] we find that \begin{eqnarray*} \lefteqn{\sum_{r=0}^{2^h-1}\exp\{\xi{\rm Re}(\rho T_h(\frac{\beta+r}{2^h}))\}} \hspace{2cm}\\ & = & \sum_{n=0}^\infty\frac{1}{2^n\cdot n!}\sum_{r=0}^{2^h-1} \xi^n \left(\rho T_h(\frac{\beta + r}{2^h}) + \overline{\rho}\, T_h(\frac{-\beta-r}{2^h})\right)^n. \end{eqnarray*} However \[\sum_{r=0}^{2^h-1} \left(\rho T_h(\frac{\beta + r}{2^h}) + \overline{\rho}\, T_h(\frac{-\beta-r}{2^h})\right)^n = \sum_{m=0}^n \left(\begin{array}{c}n\\m\end{array}\right) \rho^{2m-n} S(n,m,h,\beta),\] where \begin{equation} S(n,m,h,\beta)=\sum_{r=0}^{2^h-1} T_h(\frac{\beta+r}{2^h})^m T_h(\frac{-\beta-r}{2^h})^{n-m}. \end{equation} It follows that \begin{equation} F(\xi,h)\le \frac{1}{2^h}\sup_{\beta\in[0,1]} \sum_{n=0}^\infty\frac{1}{2^n\cdot n!} \xi^n \sum_{m=0}^n {n\choose m} |S(n,m,h,\beta)|. \end{equation} We now expand the powers of $T_h$ occurring in (45), and perform the summation over $r$. We then see that $S(n,m,h,\beta)$ is a sum of terms \[2^h\exp\{\beta(2^{a_1}+\ldots+2^{a_m}-2^{b_1}-\ldots-2^{b_{n-m}})\},\] over integer values $a_i,b_j$ between 0 and $h-1$, subject to the condition \[2^{a_1}+\ldots+2^{a_m}\equiv 2^{b_1}+\ldots+2^{b_{n-m}}\pmod{2^h}.\] It is now apparent that $|S(n,m,h,\beta)|\le S(n,m,h,0)$, whence (46) yields \begin{eqnarray*} F(\xi,h)&\le &\frac{1}{2^h}\sum_{n=0}^\infty\frac{1}{2^n\cdot n!} \xi^n \sum_{m=0}^n {n\choose m} S(n,m,h,0)\\ &=&\frac{1}{2^h}\sum_{r=0}^{2^h-1} \exp\{\xi{\rm Re}(T_h(\frac{r}{2^h}))\}, \end{eqnarray*} The assertion (44) now follows. Hence it remains to compute $F(\xi,h)$ using (44) and optimize for $\xi$ in (42). We have carried out the computations for $h=16$. Comparing the results for this value with the outcome for smaller values of $h$, it appears that the potential improvements obtainable by choosing $h$ larger than 16 are only small. After taking suitable care over rounding errors we find that we may take $\xi=1.181$ to get \[E(0.863665)>\frac{109}{154}+10^{-8}\] and $\xi=0.905$ to get \[E(0.722428)>\frac{1}{2}+10^{-8}.\] Using Mathematica 4.1 on a PC, computing the values $T_{16}(r/2^{16})$ for the integers $0\leq r\leq 2^{16}-1$ took about 7 minutes, and summing these values up to obtain $F(\xi, h)$ took 24 seconds for each of the two values of $\xi$. D.R. Heath-Brown Mathematical Institute, 24-29, St.Giles', Oxford OX1 3LB, ENGLAND rhb@maths.ox.ac.uk J.-C. Puchta Mathematical Institute, 24-29, St.Giles', Oxford OX1 3LB, ENGLAND puchta@maths.ox.ac.uk \end{document}

\begin{document} \setstcolor{red} \title{Digital-analog co-design of the Harrow-Hassidim-Lloyd algorithm} \author{Ana Martin} \email{Corresponding author: ana.martinf@ehu.eus} \affiliation{Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain} \affiliation{EHU Quantum Center, University of the Basque Country UPV/EHU, Bilbao, Spain} \affiliation{Quantum Mads, Uribitarte Kalea 6, 48001 Bilbao, Spain} \author{Ruben Ibarrondo} \affiliation{Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain} \affiliation{EHU Quantum Center, University of the Basque Country UPV/EHU, Bilbao, Spain} \author{Mikel Sanz} \email{Corresponding author: mikel.sanz@ehu.es} \affiliation{Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain} \affiliation{EHU Quantum Center, University of the Basque Country UPV/EHU, Bilbao, Spain} \affiliation{Ikerbasque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain} \affiliation{BCAM-Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao, Basque Country, Spain} \begin{abstract} The Harrow-Hassidim-Lloyd quantum algorithm was proposed to solve linear systems of equations $A\vec{x} = \vec{b}$ and it is the core of various applications. However, there is not an explicit quantum circuit for the subroutine which maps the inverse of the problem matrix $A$ into an ancillary qubit. This makes challenging the implementation in current quantum devices, forcing us to use hybrid approaches. Here, we propose a systematic manner to implement this subroutine, which can be adapted to other functions $f(A)$ of the matrix $A$, we present a co-designed quantum processor which reduces the depth of the algorithm, and we introduce its digital-analog implementation. The depth of our proposal scales with the precision $\epsilon$ as $\mathcal{O}(\epsilon^{-1})$, which is bounded by the number of samples allowed for a certain experiment. The co-design of the Harrow-Hassidim-Lloyd algorithm leads to a ``kite-like" architecture, which allows us to reduce the number of required SWAP gates. Finally, merging a co-design quantum processor architecture with a digital-analog implementation contributes to the reduction of noise sources during the experimental realization of the algorithm. \end{abstract} \maketitle \section{Introduction} \begin{figure*} \caption{{\bf Quantum circuit for the digital implementation of the HHL.} \label{Fig1} \end{figure*} The Harrow-Hassidim-Lloyd (HHL) quantum algorithm was proposed in 2009 \citep{HHL2009} to solve linear systems of equations: given a matrix $A$ and a vector $\vec{b}$, find a vector $\vec{x}$ such that $A\vec{x} = \vec{b}$. Due to the significance of linear systems of equations in several fields in science and engineering, this algorithm has attracted significant attention. However, one of the main issues with the HHL algorithm is, to our knowledge, that there is no explicit implementation for one of its key subroutines, known as ancilla quantum encoding (AQE). This subroutine stores the inversion of the eigenvalues of the problem matrix $A$, into the amplitude of an ancillary qubit. In the literature, this routine has been circumvented by employing different tricks. For instance, by using a previous knowledge about the eigenvalues of the matrix $A$. Then, it is straightforward to tailor the best set of rotations to map those values into the amplitude of the ancilla qubit. Several experimental implementations in different experimental platforms \citep{Photonic_HHL_2013, NMH_HHL_2014, Photonic_HHL_2014, JWP2017} have followed this approach to solve $2\times 2$ linear system of equations. There is an alternative implementation of the HHL for which no spectral information of the matrix $A$ is required. In Ref. \citep{LJL2019} a quantum-classical hybridization of the algorithm is proposed. They repeatedly perform the quantum phase estimation (QPE) to obtain a $n_R$-bit description of the eigenvalues of $A$. Then, they determine the simplest circuit implementation to perform the AQE, tailored for those eigenvalues. Once the AQE is determined, it is possible to perform the complete HHL algorithm. However, this alternative fails if the vector $|b\rangle$ is not efficiently prepared. Despite the intrinsic interest of these approaches, both of them jeopardize the advantage of the algorithm, since they presume previous knowledge about the eigenvalues of the matrix $A$. Algorithms such as the HHL are hard to implement in noisy intermediate-scale quantum (NISQ) devices. Indeed, current quantum computers are still not sufficiently robust against noise, which limits the depth of the algorithms that we can implement. These noise sources become much or less manageable depending on the computational paradigm. The most extended is the digital quantum computation (DQC) paradigm, in which the algorithm is decomposed into one- and two-qubit gates. Short depth quantum circuits have been implemented with this paradigm in several fields such as quantum machine learning \citep{ARSL2018, OSCSL2018}, finance \citep{YDING2019, PRICING2019, Javi2021}, open quantum systems \citep{SSSPS2016}, quantum chemistry \citep{GALMSSL2016}, or quantum field theories \citep{LSKDBLD2017}. In absence of quantum error correction, the main drawback of this paradigm is related to the cumulative noise that arises whenever a two-qubit gate is applied. To apply a two-qubit gate between qubits A and B, the natural interaction between them has to be enhanced, while keeping the rest of interactions suppressed. If the interaction with the rest of the qubits of the system is not correctly attenuated, then the logical two-qubit operation is not rightly implemented, leading to errors in our quantum routine. Using quantum control techniques, it is possible to mitigate the error of the two-qubit gates, but those techniques are not scalable. Thus, it seems natural to get advantage of the intrinsic interaction in the processor as the resource to perform quantum computing, avoiding the interaction suppression which leads to errors. That is precisely the main idea behind the digital-analog quantum computation (DAQC) paradigm. The DAQC paradigm makes use of the natural interaction among the elements that conform a quantum system to perform quantum simulations, together with single-qubit gates to change the state of a particular qubit. Consequently, DAQC merges the flexibility of digital quantum computation with the robustness of analog simulations \citep{Parra2018, LPSS2018, MLSS2020, Headley2020, Celeri2021, GMS2021}. A special case of two-qubit gate is the SWAP gate. This gate is not native in any quantum processor, thus it must be decomposed in terms of the intrinsic quantum gates, which is extremely expensive. Therefore, an adequate connectivity in the processor may dramatically reduce the use of SAWP gates, substantially decreasing the depth of the algorithm, which leads to an enhancement of the total fidelity. This is precisely the idea behind the co-design: build a quantum processor whose architecture is adapted to the connectivity of a concrete algorithm, simultaneously customizing the implementation of the algorithm in terms of the native interactions of the quantum platform. In this Article, we propose a quantum processor architecture that is tailored for the HHL algorithm. More specifically, we introduce a systematic manner to implement the AQE subroutine which is independent of the problem matrix $A$ and, thus, it does not require the classical hybridization of the HHL algorithm. Due to its relevance, we focus on the computation of $f(\lambda) = 1/\lambda$, but this subroutine can perform any other function $f(A)$, as long as it satisfies the conditions specified in the supplementary material of Ref. \citep{HHL2009}. This implementation requires an exponential number of two-qubit gates which is very demanding for the NISQ era and hinders us from implementing the fully quantum algorithm in present technology. The found precision of the algorithm depends on the precision of the estimation of the eigenvalues given the number of register qubits $n_R$ and the precision of the estimation of the found mean values $\langle c | f(A) | b\rangle$, which depends on the number of samples $N_s$. These errors are independent, consequently, it is useless to reduce one of them far beyond the other, so $n_R$ can be bounded by $\mathcal{O}\left(\log\left[\kappa \sqrt{N_s}\right]\right)$ in terms of $N_s$. Consequently, the depth of the AQE step depends on the precision, but not on the dimension of the problem matrix $A$. Taking into account the connectivity required by the algorithm, we propose a co-design ``kite-like" architecture for the quantum processor that reduces the number of SWAP gates, improving the implementation of the HHL algorithm in NISQ devices. Additionally, aiming at the experimental realization of the algorithm, we propose a digital-analog implementation in the kite-like quantum processor to perform the complete algorithm. An outline for our work is as follows. In Sec. \ref{section_II} we present a theoretical description of the HHL algorithm divided into three steps, as well as the digital implementation of each step (Sec. \ref{section_IIa}). Additionally, we describe how the systematic implementation of the AQE step can be achieved for both the HHL and the concrete problem of solving linear system of equations (Sec. \ref{section_IIb}). Afterwards, in Sec. \ref{section_III}, we propose a co-designed quantum processor architecture that can enhance the performance of the HHL. Finally. in Sec. \ref{section_IV}, we present the DAQC description for the HHL algorithm. The conclusions are in Sec. \ref{section_V}. \section{Fully quantum implementation of the HHL algorithm} \label{section_II} In this section, we review the HHL algorithm and describe its implementation under the DQC paradigm. We go into details on how to apply the AQE step, so that the algorithm can be performed without previous knowledge of the eigenvalues of the problem matrix $A$. For a $s$-sparse system matrix of size $N\times N$ and condition number $\kappa$, which is given by the ratio of the maximal and minimal singular values of $A$, the HHL algorithm can reach a desired computational accuracy $\epsilon$ within a running time of $\mathcal{O}\left(\log \left(N\right) s^2\kappa^2 \epsilon\right)$ under specific circumstances \citep{A2015}, comparing to the best known classical algorithm of $\mathcal{O}\left( N_s \kappa/\log \epsilon\right)$. The algorithm involves three sets of qubits: a single ancilla qubit which stores the inverse of the eigenvalues of the matrix problem in its amplitude, a register of $n_R$ qubits to encode the $n_R$-binary representation of the eigenvalues of the problem matrix, and a set of $n_M= \log_2 (\dim |b\rangle)$ memory qubits used to load the state $|b\rangle$ and store the output $|x\rangle$. The amount of qubits required to perform the HHL algorithm depends both on the size of the matrix $A$ and the precision one would like to reach. \subsection{Description of the HHL algorithm} \label{section_IIa} The HHL algorithm is divided in three steps: QPE to compute the eigenvalues of the problem matrix $A$, AQE to map the inverse of those eigenvalues into the amplitude of the ancilla qubit, and inverse QPE to reset the registers back to the ground state $|0\rangle^{\otimes n_R}_R$. In the following, we describe each subroutine in detail. In Fig. \ref{Fig1}(a) we show a scheme representation of the algorithm: the qubits are separated in the thee different sets and the steps are delimited in three colored boxes, one for each routine. Since the HHL can be employed as a subroutine of a bigger problem, it is reasonable to assume that the memory qubits are initialize in the state $|b\rangle_\text{M} = \sum_{i=1}^N b_i|i\rangle$, where $|i\rangle$ denotes the computational basis of the $n_M$ qubits, as a consequence of previous operations in the system. If this was not the case and if $b_i$ and $\sum_i |b_i|^2$ are efficiently computable, then it is possible to prepare $|b\rangle$ following the procedure described in Ref. \citep{GR2002}. Either way, the system is initially in the state $|0\rangle_\text{A} \otimes |0\rangle_\text{R}^{\otimes n_R} \otimes |b\rangle_\text{M}$. The first step of the algorithm is to apply QPE to compute the eigenvalues $\lambda_j$ of $A$ and encode them in a binary form into the state of the register qubits. Given $|b\rangle=\sum_{j=1}^N\alpha_j|u_j\rangle$, where $|u_j\rangle$ is the eigenvector basis of the matrix $A$, the state of the system after the QPE is $ |0\rangle_A\otimes \sum_{j=1}^N\sum_{k=0}^{2^{n_R}-1} \alpha_j\beta_{k|j}|k\rangle_R \otimes|u_j\rangle_M $, where the coefficient $\beta_{k|j}|k$ is defined as \begin{equation} \beta_{k|j}=\frac{1}{2^{n_R}}\sum_{y=0}^{2^{n_R}-1}e^{2\pi i y \left( \lambda_j - k/2^{n_R} \right)}. \end{equation} If all the eigenvalues $\lambda_j$ are perfectly $n_R$-estimated, we can relabel them as $\widetilde{\lambda_k}\equiv k/2^{n_R}$. So that $\beta_{k|j}=\delta_{\lambda_k,\lambda_j}$, and the final state of the system after the QPE is performed is as follows, \begin{equation} |0\rangle_\text{A}\otimes \sum_{j=1}^N\sum_{k=0}^{2^{n_R}-1} \alpha_j|\widetilde{\lambda_k}\rangle_\text{R} \otimes|u_j\rangle_\text{M} . \end{equation} Once the estimated eigenvalues are encoded in a state superposition of the register qubits, the AQE maps them into the amplitude of the ancillary qubit, so that the resulting state of the ancilla is \begin{equation} \sum_{j=1}^N\sum_{k=0}^{2^{n_R}-1}\left(\sqrt{1-\frac{C^2}{\widetilde{\lambda_k^2}}}|0\rangle_\text{A}+\frac{C}{\widetilde{\lambda_k}}|1\rangle_\text{A}\right)\alpha_j|\widetilde{\lambda_k}\rangle|u_j\rangle, \end{equation} with $C \leqslant 1$ being a normalization constant chosen to be $\mathcal{O}\left(1/\kappa\right)$. In this work we propose a systematic and fully quantum protocol to achieve this mapping without needing to know in advance the eigenvalues of $A$. We describe the process within the next subsection. Finally, the inverse QPE has to be performed in order to uncompute the $|\lambda_j\rangle$ on the registers and reset them to the initial $|0\rangle^{\otimes n_R}$ state. After this step, the system is found in the state \begin{equation}\label{EQ_final_state} \sum_{j=1}^N\sum_{k=0}^{2^{n_R}-1} \left(\sqrt{1-\frac{C^2}{\widetilde{\lambda_k^2}}}|0\rangle_\text{A}+\frac{C}{\widetilde{\lambda_k}}|1\rangle_\text{A}\right)\otimes|0\rangle_\text{R}\otimes \alpha_j |u_j\rangle_\text{M} \end{equation} To get the normalized solution of the linear equation, the ancillary qubit has to be measure in the $Z$-axis. If the outcome state of the ancilla qubit is $|1\rangle_\text{A}$, then the state describing the system successfully represents the solution of the linear equation as \begin{equation}\label{EQ_measurement} \frac{1}{C}\sum_{j=1}^N\frac{\alpha_j}{\widetilde{\lambda_k}}|u_j\rangle_\text{M}, \end{equation} up to a normalization factor. The pure state is obtained when the matrix $A$ is perfectly $n_R$-estimated. If there exists an eigenvalue of $A$ which is not perfectly $n_R$-estimated, then the total state of Eq. (\ref{EQ_final_state}) becomes a pure entangled state so that the state in Eq. (\ref{EQ_measurement}) turns into a mixed state. \begin{figure} \caption{{\bf DAQC decomposition of a $cZ$ gate.} \label{Fig2} \end{figure} \begin{figure*} \caption{{\bf Implementation of the AQE using DQC and DAQC techniques.} \label{Fig3} \end{figure*} \subsection{Ancilla quantum encoding} \label{section_IIb} The AQE maps the superposition state of the registers into the amplitude of the ancilla qubit by means of the application of different controlled rotations. These rotation operations control each register qubit and act on the ancilla. In previous literature, the angle of these rotations was computed using the eigenvalues of the matrix $A$, which requires a previous knowledge of this information either before formulating the problem \citep{JWP2017}, or by applying QPE \citep{LJL2019}. But, if we know this information beforehand, then the problem of solving the linear system of equation becomes trivial. In this section, we present a systematic way to perform the AQE operation without the necessity of knowing in advance further information about the problem matrix $A$. This way, our proposal constitute, to our knowledge, the first fully quantum explicit implementation of the HHL algorithm. To perform the AQE step, we will first rotate the system applying the unitary gate $V$, which is defined as \begin{equation}\label{eqV} V = \frac{1}{\sqrt{2}}\begin{pmatrix} -i & i\\ 1 & 1 \end{pmatrix}, \end{equation} and then, apply a series of rotations around the $Z$-axis in the ancilla qubit controlled by the state of the register qubits. This operation is defined as \begin{equation} U_{\text{AQE}} = \sum_{p=0}^{2^{n_R}-1} R_z\left(-\phi(p)\right)_\text{A} \otimes |\vec{b}_p\rangle\langle \vec{b}_p|_\text{R}, \end{equation} where, \begin{eqnarray} R_z \left(\phi(p)\right) &=& \begin{pmatrix} e^{-i\phi(p)/2} & 0\\ 0 & e^{i\phi(p)/2} \end{pmatrix}, \label{eqRz}\\ \phi(p) &=& \left\{ \begin{array}{lr} 0 & \text{if } p = 0\\ 2 \arcsin\frac{1}{p} & \text{otherwise} \end{array}\right. . \end{eqnarray} The bit-string $\vec{b}_p$ is the binary representation of the decimal number $p$, in other words, $p=\sum_{i=0}^{n_R} (\vec{b}_p)_i 2^{n_R-i}$. In Fig. \ref{Fig1}(b), we show an example of the AQE routine for a $n_R=4$ qubits case. This implementation of the AQE step, requires an exponential number of multi-controlled gates. Following Ref. \citep{MVBS2004}, it can be decompose in $4^{n_R}$ single-qubit gates and $4^{n_R} - 2^{n_R+1}$ CNOT gates, which might be too demanding for NISQ devices. Therefore, the depth of the AQE step depends on the number of register qubits, and this number is related to the precision and error of the HHL algorithm. In order to implement these multi-controlled operations employing only single- and two-qubit gates acting on the ancilla, we employ the procedure proposed in Ref. \citep{MVBS2004}. There, they provide an equivalent circuit which employs subsequent rotations with modified angles alternated with CNOT gates between the control and the target qubits. Consequently, the operation described in this section can be implemented with the circuit represented in Fig.\ref{Fig1}(d), where the angles $\theta_i$ are the modified angles of the single-qubit rotations related with the original angles by \begin{equation} \mqty(\theta_1 \\ \theta_2 \\ \vdots \\ \theta_{2^{n_R}}) = M^{-1} \mqty(\phi (0) \\ \phi(1) \\ \vdots \\ \phi(2^{n_R}-1)), \end{equation} where \begin{equation} M_{ij} = (-1)^{\text{bin}(i-1) \cdot g(j-1)} \text{ and } M^{-1} = \frac{1}{2^{n_R}} M^{T}, \end{equation} where $\text{bin}(i)$ is the $n_R$-bit binary representation of the integer $i$, $g(j)$ represents the $j$-th string of the binary reflected Gray code (counting from 0), and the dot represents the bit-wise product. Hence, substituting the expression for $\phi(b)$ we obtain \begin{equation}\label{eqThetaJ} \theta_i = \frac{1}{2^{n_R}} \sum_{j=1}^{2^{n_R}-1} (-1)^{\text{bin}(j) \cdot g(i-1)} \arcsin \frac{1}{j}. \end{equation} The expected error of the algorithm in terms of the final state is $|| |x\rangle - |\tilde{x}\rangle || = \mathcal{O}\left(\kappa/2^{n_R}\right)$, where $|x\rangle$ is the solution of the $A|x\rangle=|b\rangle$ problem and $|\bar{x}\rangle$ is the solution obtained by the algorithm. We now consider the scenario in which we are interested in estimating the expected value of an observable $\Omega$ with a limited number of samples $N_s$. The error scales as $\mathcal{O}\left( 1/\sqrt{N_s}\right)$. Thus, the expected error, $\epsilon$, of estimating an observable $\Omega$ in $|x\rangle$ by measuring $|\tilde{x}\rangle$ is \begin{equation}\label{eq12} \epsilon = \mathcal{O}\left(\frac{1}{\sqrt{N_s}} + \frac{\eta}{2^{n_R}}\right), \end{equation} where $\eta$ is a constant of $\mathcal{O}(\kappa)$, being $\kappa$ the condition number of the problem matrix $A$. If the number of samples is fixed in $N_s$ samples, then, the number of register qubits can be estimated by imposing that none of the summands in Eq. \eqref{eq12} is dominant. This implies that the sensible amount of register qubits $n_R$ is of $\mathcal{O}\left(\log\left[\kappa \sqrt{N_s}\right]\right)$, which only depends on the number of samples, but not on the size of the problem matrix $A$. In the {\it generalized}-HHL (gHHL), one wants to calculate $\vec{x} = f(A) \vec{b}$, where $f(A)$ is a matrix function. Then, our method can easily be adapted to fulfill this task. It is straightforward to show that the desired operation requires the AQE to perform the following operation on the ancilla qubit, \begin{equation} |0\rangle_\text{A} |\widetilde{\lambda_k}\rangle_\text{R} \xrightarrow{\text{AQE}} \left(\sqrt{1-f(\lambda^2)}|0\rangle_\text{A} + f(\lambda)|1\rangle_\text{A}\right)|\widetilde{\lambda_k}\rangle_\text{R}. \end{equation} To achieve this mapping, we would redefine the rotation angles of the multi-controlled operations as $\phi(p) = 2 \arcsin f(p/2^{n_R})$. However, the found bounds for the error are only satisfied if the function $f(\lambda)$ satisfies the condition \begin{equation} \left| \left( \frac{\text{d}f}{\text{d}x}\right)^2 \left(1+\frac{1}{1-f(x)^2}\right)\right|<\eta^2 \quad \text{in the interval } x\in [0,1), \end{equation} that can be derived form the Lemma 2 of the supplementary material of Ref. \citep{HHL2009}, where $\eta$ is an arbitrary constant of order $\mathcal{O}(1)$. This condition on the function $f(\lambda)$ implies that, in a realistic case, one would construct an auxiliary function $F$ that satisfy that bound and behaves as $f(\lambda)$ in a certain interval, which is a similar procedure to normalizing a vector to load it as a quantum state. \section{Co-designed processor architecture for the HHL} \label{section_III} If the architecture of the processor in which the algorithm is being implemented is not optimal, a significant amount of SWAP gates might be required in order to replaced the missed connections. This demand increases the depth of the quantum circuit, making its implementation in NISQ devices a challenging task. Having an optimized quantum processor architecture keeps the depth controlled and reduces the amount of SWAP gates required to replace the missed connections. Here, we present an architecture for the quantum processing unit to implement the gHHL algorithm. An optimized quantum processor architecture for the gHHL takes into account the dependence between the three sets of qubits in the different steps of the algorithm. Each set plays a different role in the algorithm and thus, require different connections: the register qubits are connected among them and, simultaneously, to the ancilla and memory qubits, whereas the ancilla and the memory sets need not to be connected directly. In Fig.\ref{Fig1} (e) we show how this leads to a ``kite-like'' architecture that satisfy the demanded connections of the qubit sets in the different steps of the gHHL. In the context of hardware implementation, it is worth noting that we are speaking about logical qubits. This means that, each logical qubit can be constituted by a group of physical qubits on which we perform the necessary computational operations. \section{Digital-analog implementation for the HHL algorithm} \label{section_IV} The DAQC paradigm combines analog blocks with digital steps to approximate any unitary with arbitrary precision. The digital steps are single-qubit gates and the analog blocks are constituted by the time evolution of the interaction Hamiltonian inherent to the quantum processor. By getting advantage of the natural interaction among qubits, the DAQC claims to be more resilient against noise than the fully digital one \citep{MLSS2020, GMS2021}. Here, we propose a DAQC implementation that takes into account all of the subroutines constituting the gHHL. As previously mentioned, it is possible to describe the gHHL in a succession of three steps (QPE, AQE and inverse QPE). At the same time, QPE can be described as well by two subroutines: the controlled-Hamiltonian evolution and the inverse quantum Fourier transform. The division of each step into more fundamental subroutines simplifies the description of the complete gHHL algorithm in the DAQC paradigm, since some of those subroutines has already been worked out in previous works \citep{LPSS2018, MLSS2020}. To perform the controlled-Hamiltonian evolution that constitutes the first part of the QPE, we assume that the problem matrix $A$ admits an $M$-body decomposition, where $M<<\log_2 \dim (A)$. Under this condition, it is possible to upload that matrix efficiently using the DAQC protocol as described in Ref. \citep{Parra2018}. The next step that conforms the QPE subroutine is the implementation of the inverse quantum Fourier transform on the register qubits. This step was explicitly described in Ref. \citep{MLSS2020}. Finally, in order to implement the AQE step in a digital-analog paradigm, it is useful to firstly decompose it into single-qubit rotations, namely Hadamard gates and rotations around the X-axis, and controlled-Z ($cZ$) gates, as shown in Fig. \ref{Fig3} (a). Then, we can get a step-wise DAQC (sDAQC) decomposition by constructing the Hamiltonian of the $cZ$ gates, which is a two-body Ising Hamiltonian of the form \begin{equation} H_\text{cZ} = -\frac{1}{2}Z\otimes\mathbb{1}+\frac{1}{4} Z\otimes Z. \end{equation} In the sDAQC protocol, the interaction among the qubits of the system are switched off before applying the digital steps, and switched on again immediately afterwards. This approach does not substantially improve the result of the digital approach since the errors induced by the attenuation of the natural interaction between the qubits are similar. A better alternative is the banged DAQC (bDAQC). In this paradigm, the interaction between the qubits is not turned off to perform the single-qubit gates, which are applied on top of the interaction Hamiltonian. Although this means that an intrinsic error is introduced, this error scales better with the number of qubits when the time for the single-qubit gates, $\Delta t$, is significantly smaller that the natural time scale of the analog blocks. In Fig. \ref{Fig2}, we show an scheme of both the sDAQC and bDAQC protocols for the $cZ$ gate. For simplicity, we will assume that the interaction between the qubits of our kite-like quantum processor is an homogeneous two-body Ising Hamiltonian, $H_{I}$. Then, every $cZ$ performed between the ancilla and one of the register qubits can be decomposed into two analog blocks in the following way \begin{eqnarray} U_\text{cZ} &=& e^{i\frac{\pi}{4} Z_\text{A}\otimes Z_\text{R}} \nonumber \\ &=& \left( \bigotimes_{\substack{k=1\\ k\neq i}}^{n_R}X_k\right) e^{i\frac{\pi}{8}H_I} \left( \bigotimes_{\substack{k=1\\ k\neq i}}^{n_R}X_k\right) e^{i\frac{\pi}{8}H_I}. \end{eqnarray} In Fig. \ref{Fig2}, we show a scheme of the sDAQC implementation of a $cZ$ gate. To implement the complete AQE using DAQC, we would decompose each $cZ$ in the digital and analog blocks described above. Each analog blocks would take the same amount of time $t$ to act, and the digital steps would take a fixed time $\Delta t$ to be applied. In Fig. \ref{Fig3} (b), we show the sDAQC implementation of the AQE. Only half of the procedure is represented since the other half is symmetrical. As we mention previously, in terms of scalability it is preferable to follow the bDAQC approach. For the sake of completeness, the explicit DAQC description of the AQE step comprises, at most, $2^{n_R}$ analog blocks and $(n_R+1) 2^{n_R}+1$ single-qubit gates. This description can be improved by employing optimization techniques. \section{Conclusions} \label{section_V} The implementation of the gHHL algorithm can be divided into three steps: QPE to compute the eigenvalues of the problem matrix $A$, AQE to map a function of the problem matrix $A$ into the amplitude of the ancillary qubit, and the inverse QPE to set the registers back to the ground state. The manner to perform the first and last steps is well described in the literature \citep{NCh2000}, whereas an explicit description of the AQE step has remained as an open question. In this work, we have shown how the AQE can be accomplished in a systematic manner which does not require the hybridization of the algorithm, making this the first fully quantum implementation of the gHHL, to our knowledge. Implementing the AQE routine using our protocol requires an exponential number of single- and two-qubit gates, what has hindered us from implementing it in NISQ devices. This number depends on the amount of register qubits of our setup and is related to the precision of the algorithm. Attending to the number of samples required for a particular experiment, it is possible to diminish the number of register qubits needed to perform the gHHL algorithm, more concrete, for a $N_s$-sample experiment, the number of register qubits needed would be of order $\mathcal{O}\left(\log\left[\kappa\sqrt{N_s}\right]\right)$. This means that the number of register qubits $n_R$ depends exclusively on the bond for $f(A)$ and the number of samples $N_s$, but it is independent form the dimension of the matrix $A$. In order to reduce the depth of the gHHL algorithm, we proposed a co-designed quantum processor tailored to the implementation of the gHHL. This structure significantly reduces the amount of SWAP gates needed to artificially connect the qubits of the system which are not physically connected but they are necessary to perform the algorithm. Thus, the co-design also reduces the depth of the algorithm. The kite-like quantum processor connects the register qubits with both the ancilla and the memory qubits, while keeping these last two sets of qubits disconnected from each other. There is another approach to the implementation of the gHHL that naturally arises from the idea of getting advantage of the connections naturally present in the quantum processor, and it is to implement the algorithm using DAQC techniques. In view of the results obtained in previous works about the simulation of quantum subroutines, such as the quantum Fourier transform, which forms part of the gHHL, it is expected that a DAQC implementation of the gHHL will produce better results in terms of noise scaling with the number of qubits \citep{MLSS2020, GMS2021}. For this reason, we propose a complete protocol for the implementation of the gHHL algorithm, which allows obtaining results when applied in a system with a high number of qubits. \section{Acknowledgments} The authors acknowledge financial support from the QUANTEK project from ELKARTEK program (KK-2021/00070), Spanish Ram\'on y Cajal Grant RYC-2020-030503-I and the project grant PID2021-125823NA-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and "ERDF Invest in your Future", as well as from QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, and the EU FET-Open projects Quromorphic (828826) and EPIQUS (899368). IQM Quantum Computers funded project ``Generating quantum algorithms and quantum processor optimization". R.I. acknowledges the support of the Basque Government Ph.D. grant PRE\_2021-1-0102. \end{document}

\begin{document} \title{On moduli subspaces of central extensions of rational H-spaces} \begin{abstract} We investigate the moduli sets of central extensions of H-spaces enjoying inversivity, power associativity and Moufang properties. By considering rational H-extensions, it turns out that there is no relationship between the first and the second properties in general. \end{abstract} \section{Introduction} We assume that a space has the homotopy type of a path connected CW-complexes with a nondegenerate base point $*$ and that all maps are based maps.\\ \indent An {\it H-space} is a space $X$ endowed with a map $\mu :X\times X \to X$ , called a multiplication, such that both the restrictions $\mu |_{X\times *}$ and $\mu |_{*\times X}$ are homotopic to the identity map of $X$. The multiplication naturally induces a binary operation on the homotopy set $[Y,X]$ for any space $Y$. In \cite{Jam1960}, James has proved that $[Y, X]$ is an algebraic loop; that is, it has a two sided unit element and for any elements $x$ and $y\in [Y,X]$, the equations $xa=y$ and $bx=y$ have unique solutions $a,b\in [Y,X]$.\\ \indent Loop theoretic properties of H-spaces have been considered by several authors, for example, Curjel \cite{Cur1968} and Norman \cite{Nor1963}. In \cite{AL1990}, Arkowitz and Lupton considered H-space structures on inversive, power associative and Moufang properties. They consider whether there exists an H-space structure which does not satisfy the properties. Thanks to general theory of algebraic loops, we see that Moufang property implies inversivity and power associativity. However, it is expected that there is no relationship between the latter two properties in general.\\ \indent In \cite{Kac1995}, Kachi introduced central extensions of H-spaces which are called central H-extensions; see Definition \ref{H-ex}. Roughly speaking, for a given homotopy associative and homotopy commutative H-space $X_{1}$ and an H-space $X_{2}$, a central H-extension of $X_{1}$ by $X_{2}$ is defined to be the product $X_{1}\times X_{2}$ with a twisted multiplication. He also gave a classification theorem for the extensions. In fact, a quotient set of an appropriate homotopy set classifies the equivalence classes of central H-extensions; see Theorem \ref{thm:Kac}. Such the quotient set is called {\it the moduli set} of H-extensions. Moreover, we refer to the subset of the moduli set corresponding the set of the equivalence classes of H-extensions enjoying a property $P$ via the bijection in the classifying theorem as {\it the moduli subset} of H-extensions associated with the property $P$.\\ \indent The objective of the paper is to investigate the moduli subsets of central extensions of H-spaces associated with inversive, power associative or Moufang property. If a given H-space is ${\mathbb Q}$-local, then the moduli set of its central H-extensions is endowed with a vector space structure over ${\mathbb Q}$. It turns out that the moduli subset mentioned above inherit the vector space structure. This fact enable us to compare such the moduli subsets as vector space, and also to measure the size of the moduli set with the dimension of the vector space. Thus our main theorem (Theorem \ref{lem:4.4}) deduces the following result. \begin{ass}\label{ex1} Let $S_{{\rm inv}}$, $S_{{\rm p.a}}$ and $S_{{\rm Mo}}$ be the moduli subsets of central H-extensions of the Eilenberg-MacLane space $K({\mathbb Q},n)$ by $K({\mathbb Q},m)$ associated with inversive, power associative and Moufang properties, respectively. \begin{enumerate} \item If $m$ is even, $n=km$ and $k\geq 4$ is even, then $S_{{\rm Mo}}$ is the proper subset of $S_{{\rm p.a}}$ and $S_{{\rm p.a}}$ is the proper subset of $S_{{\rm inv}}:$ $$S_{{\rm Mo}}\subsetneq S_{{\rm p.a}}\subsetneq S_{{\rm inv}}.$$ \item If $m$ is even, $n=km$ and $k\geq 5$ is odd, then $S_{{\rm Mo}}$ is the proper subset of $S_{{\rm p.a}}\cap S_{{\rm inv}}$. Moreover, $S_{{\rm p.a}}\cap S_{{\rm inv}}$ is a proper subset of $S_{{\rm p.a}}$ and $S_{{\rm inv}}:$ $$S_{{\rm Mo}}\subsetneq S_{{\rm p.a}} \cap S_{{\rm inv}} \subsetneq S_{{\rm p.a}} \text{ \ and \ } S_{{\rm p.a}} \cap S_{{\rm inv}} \subsetneq S_{{\rm inv}}.$$ \item Otherwise, $S_{{\rm inv}}=S_{{\rm p.a}}=S_{{\rm Mo}}$. \end{enumerate} \end{ass} \indent The organization of this paper is as follows. In Section 2, we will recall several fundamental definitions and facts on H-spaces and algebraic loops. The classification theorem of central H-extensions are also described. In Section 3, we will present necessary and sufficient conditions for central H-extensions to be inversive, power associative and Moufang. The conditions allows us to describe the moduli set of extensions enjoying each of the properties in terms of a homotopy set of maps. In Section 4, we will deal with rational H-spaces and the dimensions of the moduli spaces mentioned above. Moreover, several examples are presented. Assertion \ref{ex1} is proved at the end of Section 4. \section{Preliminaries} We begin by recalling the definition of an algebraic loop. \begin{defn}{\rm An} algebraic loop $(Q,\cdot )$ {\rm is a set $Q$ with a map \begin{eqnarray*} \cdot : Q\times Q\to Q ; \ (x,y)\longmapsto xy \end{eqnarray*} such that the equations $ax=b$ and $ya=b$ have unique solutions $x,y\in Q$ for all $a,b\in Q$. Moreover, there exists an element $e\in Q$ such that $xe=x=xe$ for all $x\in Q$. The element $e$ is called} unit. \end{defn} Algebraic loops with particular multiplications are considered. \begin{defn}\label{def:2.1.5} {\rm An algebraic loop $(Q,\cdot)$ is called to be} \begin{enumerate} \item inversive {\rm if, to any element $x$ of $Q$, there exists a unique element $x^{-1}$ of $Q$ such that the equalities $x^{-1}x=e=xx^{-1}$ hold,} \item power associative {\rm if $x(xx)=(xx)x$ for any $x\in Q$,} \item Moufang {\rm if $(x(yz))x=(xy)(zx)$ for any $x,y,z\in Q$,} \item symmetrically associative {\rm if $(xy)x=x(yx)$ for any $x,y\in Q$.} \end{enumerate} \end{defn} In Section 3, we will focus on symmetrically associative property to examine Moufang property. \begin{lem}{\rm \cite[Lemma 2A]{Bru1946}}\label{lem:2.3} If an algebraic loop $Q$ is Moufang, then it is inversive. \end{lem} Thus we have implication between properties for multiplication described in Definition \ref{def:2.1.5}. \[\xymatrix{ \fbox{power associative} & & \fbox{inversive}\\ \fbox{symmetrically associative} \ar@{=>}[u] & & \fbox{Moufang} \ar@{=>}[ll] \ar@{=>}[u]\\ }\] We bring the loop theoretic notion to the realm of H-spaces. \begin{defn} {\rm Let $(X,\mu)$ be an H-space. A} left inverse $l: X\to X$ {\rm and a} right inverse $r:X\to X$ {\rm of an H-space $(X,\mu)$ are maps such that $\mu (l\times id)\Delta \simeq * \simeq \mu(id\times r)\Delta$.} \end{defn} \begin{defn} {\rm An H-space $(X,\mu)$ is called to be} \begin{enumerate} \item inversive {\rm if the left inverse and the right inverse of $(X,\mu )$ are homotopic,} \item power associative {\rm if $\mu(\mu \times id)(\Delta \times id)\Delta \simeq \mu(id \times \mu)(\Delta \times id)\Delta$,} \item Moufang {\rm if $\mu(\mu \times \mu)\theta \simeq \mu(\mu \times id)(id \times \mu \times id)\theta $, where $\theta : X^{3}\to X^{4}$ is defined by $\theta (x,y,z)=(x,y,z,x),$} \item symmetrically associative {\rm if $\mu(\mu \times id)(id \times t )(\Delta \times id)\simeq \mu(id \times \mu)(id\times t)(\Delta \times id)$, where $t:X^{2}\to X^{2}$ is defined by $t(x,y)=(y,x)$}. \end{enumerate} \end{defn} The following proposition characterizes the above conditions on H-spaces by homotopy sets. \begin{prop} Let $(X,\mu)$ be an H-space. Then $(X,\mu)$ is inversive $($respectively, power associative, Moufang and symmetrically associative$)$ if and only if the homotopy set $[Y,X]$ is inversive $($respectively, power associative, Moufang and symmetrically associative$)$ for any space $Y$. \end{prop} \begin{proof}{\rm Suppose $X$ is inversive. Let $l$ and $r$ be a left inverse and a right inverse of $X$, respectively. Then, for any $[f]\in [Y,X]$, $[lf]$ is the inverse element of $[f]$. Conversely, if $[Y,X]$ is inversive for any space $Y$, then there exists $[\nu ]\in [X,X]$ such that $ [\nu][id_{X}]=*=[id_{X}][\nu]. $ A similar argument shows the other cases. }\end{proof} Here we recall the definition of the central extension of an H-spaces. \begin{defn}\label{H-ex}{\rm \cite{Kac1995} Let $(X_{1},\mu_{1})$ be a homotopy associative and homotopy commutative H-space and $(X_{2},\mu_{2})$ an H-space. An H-space $(X,\mu)$ is a} central H-extension of $(X_{1},\mu_{1})$ by $(X_{2},\mu_{2})$ {\rm if there exists a sequence of H-spaces \begin{eqnarray*} (X_{1},\mu_{1}) \stackrel{f_{1}}{\longrightarrow } ( X,\mu ) \stackrel{f_{2}}{\longrightarrow }(X_{2},\mu_{2}) \end{eqnarray*} such that the sequence \begin{eqnarray*} e \longrightarrow [Y,X_{1}] \stackrel{f_{1*}}{\longrightarrow } [Y,X] \stackrel{f_{2*}}{\longrightarrow } [Y,X_{2}] \longrightarrow e \end{eqnarray*} is exact as algebraic loops for any space $Y$ and the image of $f_{1*}$ is contained in the center of $[Y,X]$, where $f_{i*}$ is the algebraic loop homomorphism induced by $f_{i}$. } \end{defn} In order to describe the classification theorem for central H-extensions presented by Kachi, we recall the definition of H-deviations. Let $(X,\mu)$ be an H-space and $Y$ a space. For any $[f],[g]\in [Y,X]$, let $D(f,g)$ be a unique element of $[Y,X]$ such that $D(f,g)[g]=[f]$. \begin{defn}{\rm Let $(X,\mu)$ and $(Y,\mu')$ be H-spaces, and let $f:X\to Y$ be a map. An} H-deviation of the map f {\rm is an element \begin{eqnarray*} HD(f) \in [X\wedge X,Y] \end{eqnarray*} such that $q^{*}(HD(f))=D(f\mu , \mu'(f\times f))$, where $q:X\times X \to X\wedge X$ is the quotient map.} \end{defn} The existence of H-deviations is insured by the following lemma. \begin{lem}{\rm \cite[Lemma 1.3.5]{Zab1976}}\label{lem:2.2.2} Let $(X,\mu)$ be an H-space. Then for any spaces $Y,Z$, \begin{eqnarray*} e\longrightarrow [Y\wedge Z,X] \stackrel{q^{*}}{\longrightarrow } [Y\times Z,X]\stackrel{i^{*}}{\longrightarrow } [Y\vee Z,X]\longrightarrow e \end{eqnarray*} is the short exact sequence as algebraic loops, where $i:Y\vee Z\to Y\times Z$ is the inclusion and $q:Y\times Z\to Y\wedge Z$ is the quotient map. \end{lem} Since $i^{*}D(f\mu , \mu'(f\times f))=D(f\mu i , \mu'(f\times f)i)=e$, there exists an element $HD(f) \in [X\wedge X,Y]$ such that $q^{*}(HD(f))=D(f\mu , \mu'(f\times f))$. If $f\simeq f':X\to X'$, then $HD(f)=HD(f')$. Therefore, the H-deviation map \begin{eqnarray*} HD : [X,Y] \to [X\wedge X,Y] \end{eqnarray*} is defined by sending a class $[f]$ to the class $HD(f)$. Moreover, if $X$ is a homotopy associative and homotopy commutative H-space, then the H-deviation map is an algebraic loop homomorphism ( \cite[Corollary 2.4]{Kac1995}). \begin{defn}{\rm \cite{Kac1995} Two central H-extensions \begin{eqnarray*} (X,\mu _{X}) \stackrel{f_{i}}{\longrightarrow } (Z_{i},\mu_{i} ) \stackrel{g_{i}}{\longrightarrow }(Y,\mu_{Y}) \ (i=1,2) \end{eqnarray*} is said to be} equivalent {\rm if there exists an H-map $h:(Z_{1},\mu_{1}) \to (Z_{2},\mu_{2})$ such that the following diagram is homotopy commutative: $$ \begin{CD} X @>f_{1}>> Z_{1} @>g_{1}>> Y\\ @V=VV @VhVV @V=VV \\ X @>>f_{2}> Z_{2} @>>g_{2}> Y.\\ \end{CD} $$ } \end{defn} It is readily seen that this relation is an equivalence relation. We denote by $CH(X_{1},\mu_{1};X_{2},\mu_{2})$ the set of equivalence classes of central H-extensions of $(X_{1},\mu_{1})$ by $(X_{2},\mu_{2})$, and by $[(X,\mu),f_{1},f_{2}]$ the equivalence class of the central H-extension \begin{eqnarray*} (X_{1},\mu_{1}) \stackrel{f_{1}}{\longrightarrow } (X,\mu ) \stackrel{f_{2}}{\longrightarrow }(X_{2},\mu_{2}). \end{eqnarray*} \begin{thm}{\rm \cite[Theorem 4.3]{Kac1995}} \label{thm:Kac} Let $(X_{1},\mu_{1})$ be a homotopy associative and homotopy commutative H-space, and $(X_{2},\mu_{2})$ an H-space. Define \begin{eqnarray*} \Phi : [X_{2}\wedge X_{2},X_{1}]/\mathrm{Im} HD \longrightarrow CH(X_{1},\mu_{1};X_{2},\mu_{2}) \end{eqnarray*} by sending $[\omega ]\in [X_{2}\wedge X_{2},X_{1}]/\mathrm{Im} HD$ to $[(X_{1}\times X_{2},\mu_{\omega}),i_{1},p_{2}]\in CH(X_{1},\mu_{1};X_{2},\mu_{2})$, where the multiplication $\mu_{\omega}$ of $X_{1}\times X_{2}$ is defined by $\mu_{\omega}((x_{1},x_{2}),(y_{1},y_{2}))=(\mu_{1}(\mu_{1}(x_{1},y_{1}),\omega q(x_{2},y_{2})),\mu_{2}(x_{2},y_{2}))$, $q:X_{2}\times X_{2}\to X_{2}\wedge X_{2}$ is the quotient map. Then $\Phi $ is bijective. \end{thm} Thus, for an equivalence class ${\mathcal E}$ in $CH(X_{1},\mu_{1};X_{2},\mu_{2})$, there exists a map $\omega :X_{2}\wedge X_{2}\to X_{1}$ such that $\Phi [\omega ]=[{\mathcal E}]$, the map $\omega $ is called {\it the classifying map} of the central H-extensions ${\mathcal E}$. Moreover, we refer to the set $[X_{2}\wedge X_{2},X_{1}]/\mathrm{Im} HD$ as {\it the moduli set} of H-extensions of $(X_{1},\mu_{1})$ by $(X_{2},\mu_{2})$. \section{Moduli subsets of central H-extensions} We retain the notation and terminology described in the previous section. Let $(X_{1},\mu_{1})$ be a homotopy associative and homotopy commutative H-space, and $(X_{2},\mu_{2})$ an H-space. Then a central H-extension of $(X_{1},\mu_{1})$ by $(X_{2},\mu_{2})$ is of the form $(X_{1}\times X_{2}, \mu_{\omega})$. Let $i_{j}:X_{j}\to X_{1}\times X_{2}$ and $p_{j}:X_{1}\times X_{2}\to X_{j}$ denote the inclusions and the projections, respectively. Let $\Delta_{j}:X_{j}\to X_{j}\times X_{j}$ and $\Delta:X_{1}\times X_{2}\to (X_{1}\times X_{2})^{2}$ be diagonal maps. We denote by $id_{j}$ and $id$ the identity maps of $X_{j}$ and $X_{1}\times X_{2}$, respectively. \begin{prop}\label{prop:3.1} The central H-extension $(X_{1}\times X_{2},\mu_{\omega})$ is inversive if and only if $(X_{2},\mu_{2})$ is inversive and $\omega q(l_{2}\times id_{2})\Delta _{2}\simeq \omega q(id_{2} \times r_{2})\Delta _{2}$, where $l_{2}$ and $r_{2}$ are the left inverse and the right inverse of $(X_{2},\mu_{2})$. \end{prop} Before proving Proposition \ref{prop:3.1}, we prepare lemmas. \begin{lem}\label{lem:3.2} Let $(X,\mu)$, $(Y,\mu')$ be H-spaces, and let $f:X\to Y$ be a map. If $f$ is an H-map, then $fl\simeq l'f$, $fr\simeq r'f$, where $l,l'$ are left inverses of $(X,\mu),(Y,\mu')$ and $r,r'$ are right inverses of $(X,\mu),(Y,\mu')$ respectively. \end{lem} \begin{proof}{\rm We see that $[fl][f]=[*]=[l'f][f]$ in $[X,Y]$ since $f$ is an H-map. Hence, we get $[fl]=[l'f]$. Similarly, one obtains the equality $[fr]=[r'f]$.} \end{proof} \begin{lem}\label{lem:3.3} Let $Y,Z$ be spaces, and let $i:Y\to Y\times Z$, $p:Y\times Z\to Y$ be the inclusion and the projection. Then one has \begin{eqnarray*} \mathrm{Ker} \{ i^{*}:[Y\times Z,X]\to [Y,X] \} \cap \mathrm{Im} \{ p^{*}:[Y,X]\to [Y\times Z,X] \} = \{ e \} \end{eqnarray*} for any H-space $X$. \end{lem} \begin{proof}{\rm For any $[f]\in \mathrm{Ker} i^{*}\cap \mathrm{Im} p^{*}$, there is a map $g:Y\rightarrow X$ such that $f\simeq gq$. Since $[f]$ is in $\mathrm{Ker} i^{*}$, it follows that $*\simeq fi\simeq g$ and hence $f\simeq *$. } \end{proof} \noindent {\it Proof of Proposition \ref{prop:3.1}.} Let $l$ and $r$ be a left inverse and a right inverse of $(X_{1}\times X_{2},\mu_{\omega })$, respectively. Note that $l\simeq r$ if and only if $p_{j}l\simeq p_{j}r$ for $j=1,2$. In order to prove Proposition \ref{prop:3.1}, it is enough to show \begin{enumerate} \item $p_{1}l \simeq p_{1}r$ if and only if $\omega q(l_{2}\times id_{2})\Delta _{2}\simeq \omega q(id_{2} \times r_{2})\Delta _{2}$, \item $p_{2}l\simeq p_{2}r$ if and only if $(X_{2},\mu_{2})$ is inversive. \end{enumerate} Since $* \simeq p_{1}\mu_{\omega }(l\times id) \Delta = \mu_{1}(\mu_{1}(p_{1}l\times p_{1})\Delta\times \omega q(p_{2}l\times p_{2})\Delta)\Delta$, it follows that \begin{eqnarray*} [*]=([p_{1}l][p_{1}])[\omega q(p_{2}l\times p_{2})\Delta] \end{eqnarray*} in $[X_{1}\times X_{2},X_{1}]$. Similarly, we have \begin{eqnarray*} [*]=([p_{1}][p_{1}r])[\omega q(p_{2}\times p_{2}r)\Delta ] \end{eqnarray*} in $[X_{1}\times X_{2},X_{1}].$ Since $[X_{1}\times X_{2},X_{1}]$ is an abelian group, we see that $[p_{1}l]=[p_{1}r]$ if and only if $[\omega q(p_{2}l\times p_{2})\Delta]=[\omega q(p_{2}\times p_{2}r)\Delta].$ Lemma \ref{lem:3.2} allows us to obtain that \begin{eqnarray*} [\omega q(p_{2}l\times p_{2})\Delta]=[\omega q(l_{2}\times id_{2} )\Delta_{2}p_{2}] \end{eqnarray*} and \begin{eqnarray*} [\omega q(p_{2}\times p_{2}r)\Delta]=[\omega q(id_{2}\times r_{2})\Delta_{2}p_{2}]. \end{eqnarray*} If $[\omega q(l_{2}\times id_{2} )\Delta_{2}p_{2}]=[\omega q(id_{2}\times r_{2})\Delta_{2}p_{2}]$, then $[\omega q(l_{2}\times id_{2})\Delta_{2}] =[\omega q(l_{2}\times id_{2} )\Delta_{2}p_{2}i_{2}]=[\omega q(id_{2}\times r_{2})\Delta_{2}p_{2}i_{2}]= [\omega q(id_{2}\times r_{2})\Delta_{2}]$. Therefore we have the assertion $(1)$. Suppose that $p_{2}l\simeq p_{2}r$. Then $l_{2}\simeq p_{2}li_{2}\simeq p_{2}ri_{2}\simeq r_{2}$ and $(X_{2},\mu_{2})$ is inversive. Conversely, suppose that $(X_{2},\mu_{2})$ is inversive. Then it follows that $p_{2}li_{2}\simeq p_{2}ri_{2}$ and hence \begin{eqnarray*} [*]=[p_{2}li_{2}][p_{2}ri_{2}]^{-1}=[p_{2}li_{2}][r_{2}p_{2}ri_{2}]=i_{2}^{*}([p_{2}l][r_{2}p_{2}r]). \end{eqnarray*} Therefore we see that $[p_{2}l][r_{2}p_{2}r]$ is in $\mathrm{Ker} i_{2}^{*}.$ On the other hand, $[p_{2}l][r_{2}p_{2}r]=[l_{2}p_{2}][r_{2}r_{2}p_{2}]=p_{2}^{*}([l_{2}][r_{2}r_{2}])\in \mathrm{Im} p_{2}^{*}.$ In view of Lemma \ref{lem:3.3}, we have \begin{eqnarray*} [*] = [p_{2}l][r_{2}p_{2}r] =[p_{2}l][p_{2}r]^{-1}. \end{eqnarray*} Hence, it follows that $[p_{2}l]=[p_{2}r]$. We have the second assertion. \begin{prop}\label{prop:3.4} The central H-extension $(X_{1}\times X_{2},\mu_{\omega})$ is power associative if and only if $(X_{2},\mu_{2})$ is power associative and $\omega q(\mu_{2}\times id_{2})\overline{\Delta}_{2} \simeq \omega q(id_{2} \times \mu_{2})\overline{\Delta}_{2}$, where $\overline{\Delta}_{2}=(\Delta _{2} \times id_{2})\Delta_{2}$. \end{prop} \begin{proof}{\rm As in the proof of proposition \ref{prop:3.1}, it is enough to show the following: \begin{enumerate} \item $p_{1}\mu_{\omega }(\mu_{\omega}\times id)\overline{\Delta}\simeq p_{1}\mu_{\omega }(id \times\mu_{\omega})\overline{\Delta}$\\ if and only if $\omega q(\mu_{2}\times id_{2})\overline{\Delta}_{2} \simeq \omega q(id_{2} \times \mu_{2})\overline{\Delta}_{2}$, where $\overline{\Delta}=(\Delta \times id)\Delta$. \item $p_{2}\mu_{\omega }(\mu_{\omega}\times id)\overline{\Delta}\simeq p_{2}\mu_{\omega }(id \times\mu_{\omega})\overline{\Delta}$\\ if and only if $(X_{2},\mu_{2})$ is power associative. \end{enumerate} The statement $(1)$ is trivial. Since the map $p_{2}$ is an H-map, the assertion $(2)$ follows. } \end{proof} The same argument as in Proposition \ref{prop:3.4} does work to prove the following propositions. The details are left to the reader. \begin{prop}\label{prop:3.15} An H-space $(X_{1}\times X_{2},\mu_{\omega})$ is Moufang if and only if $(X_{2},\mu_{2})$ is Moufang and $[\mu_{1}(\omega q\times \omega q)\theta _{2}][\omega q(\mu_{2}\times \mu_{2})\theta _{2}]=[\mu_{1}(\omega q\times \omega q)(id_{2} \times (\mu_{2} \times id_{2} \times id_{2} )\Delta '_{2} )][\omega q (\mu_{2}(id_{2}\times \mu_{2})\times id_{2})\theta _{2}]\in [X_{2}^{3},X_{1}]$, where $\Delta'_{2}$ is the diagonal map of $X_{2}\times X_{2}$ and $\theta _{2}:X_{2}^{3}\longrightarrow X_{2}^{4}$ is defined by $\theta_{2} (x,y,z)=(x,y,z,x)$. \end{prop} \begin{prop}\label{prop:3.17} $(X_{1}\times X_{2},\mu_{\omega})$ is symmetrically associative if and only if $(X_{2},\mu_{2})$ is symmetrically associative and $\mu_{1}(\omega q\times \omega q)(t \times (id_{2}\times \mu_{2}t)(\Delta _{2}\times id_{2}))\Delta'_{2}\simeq \mu_{1}(\omega q\times \omega q)(id_{2}\times id_{2} \times (\mu_{2}\times id_{2})(id_{2}\times t)(\Delta_{2}\times id_{2}))\Delta'_{2}.$ \end{prop} Next, we consider the following subsets of the homotopy set $[X_{2}\wedge X_{2},X_{1}]$: \begin{align*} G_{{\rm inv}} &= \{ [\omega] \in [X_{2}\wedge X_{2},X_{1}] \ \mid \ \omega q(l_{2}\times id_{2})\Delta _{2}\simeq \omega q(id_{2} \times r_{2})\Delta _{2} \}, \\ G_{{\rm p.a}} &= \{ [\omega] \in [X_{2}\wedge X_{2},X_{1}] \ \mid \ \omega q(\mu_{2}\times id_{2})\overline{\Delta}_{2} \simeq \omega q(id_{2} \times \mu_{2})\overline{\Delta}_{2} \}, \\ G_{{\rm Mo}} &= \{ [\omega] \in [X_{2}\wedge X_{2},X_{1}] \ \mid \ \Gamma _{{\rm Mo}}(\omega )\simeq \Gamma '_{{\rm Mo}}(\omega ) \} \ \text{and} \\ G_{{\rm s.a}} &= \{ [\omega] \in [X_{2}\wedge X_{2},X_{1}] \ \mid \ \mu _{1}(\omega \times \omega )\Gamma _{{\rm s.a}}\simeq \mu _{1}(\omega \times \omega )\Gamma ' _{{\rm s.a}}\}, \end{align*} where \begin{align*} &\Gamma _{{\rm Mo}}(\omega )=\mu_{1}(\mu_{1}(\omega q\times \omega q)\theta _{2}\times \omega q(\mu_{2}\times \mu_{2})\theta _{2})\Delta_{X_{2}^{3}},\\ &\Gamma '_{{\rm Mo}}(\omega )=\mu_{1}(\mu_{1}(\omega q\times \omega q)(id_{2} \times (\mu_{2} \times id_{X_{2}^{2}} )\Delta '_{2} )\times \omega q (\mu_{2}(id_{2}\times \mu_{2})\times id_{2})\theta _{2})\Delta_{X_{2}^{3}}, \\ &\Gamma _{{\rm s.a}}=(q\times q)(t \times (id_{2} \times \mu_{2}t)(\Delta _{2}\times id_{2}))\Delta'_{2}\ \text{and}\\ &\Gamma ' _{{\rm s.a}}=(q\times q)(id_{2}\times id_{2} \times (\mu_{2}\times id_{2})(id_{2}\times t)(\Delta_{2}\times id))\Delta'_{2}. \end{align*} \begin{lem} The sets $G_{{\rm inv}}$, $G_{{\rm p.a}}$, $G_{{\rm Mo}}$ and $G_{{\rm s.a}}$ are subgroups of $[X_{2}\wedge X_{2},X_{1}]$. \end{lem} \begin{proof}{\rm For any $[\omega_{1}],[\omega_{2}]\in G_{{\rm inv}}$, we have \begin{align*} [\mu_{1}(\omega_{1}\times \omega_{2})\Delta''_{2} q(l_{2}\times id_{2})\Delta _{2}] &= [\omega_{1}q(l_{2}\times id_{2})\Delta_{2}][\omega_{2}q(l_{2}\times id_{2})\Delta_{2}] \\ &= [\omega_{1}q(id_{2}\times r_{2})\Delta_{2}][\omega_{2}q(id_{2}\times r_{2})\Delta_{2}] \\ &= [\mu_{1}(\omega_{1}\times \omega_{2})\Delta''_{2} q(id_{2}\times r_{2})\Delta _{2}] \end{align*} where $\Delta''_{2}$ is the diagonal map of $X_{2}\wedge X_{2}$. Hence $[\omega_{1}][\omega_{2}]\in G_{{\rm inv}}$. Let $l_{1}$ be a left inverse of $(X_{1},\mu_{1})$. Then we have \begin{eqnarray*} [ l_{1}\omega_{1}q(l_{2}\times id_{2})\Delta _{2}] = [l_{1}\omega_{1}q(id_{2}\times r_{2})\Delta _{2}]. \end{eqnarray*} Hence we obtain $[\omega_{1}]^{-1}=[l_{1}\omega_{1}]\in G_{{\rm inv}}$. Similarly, we can show that $G_{{\rm p.a}}$, $G_{{\rm Mo}}$ and $G_{{\rm s.a}}$ are subgroups of $[X_{2}\wedge X_{2},X_{1}]$.} \end{proof} Let $\Phi : [X_{2}\wedge X_{2},X_{1}]/\mathrm{Im} HD \to CH(X_{1},\mu_{1};X_{2},\mu_{2})$ be the bijection mentioned in Theorem \ref{thm:Kac}. \begin{lem}\label{lem:4.2} If $(X_{2},\mu _2)$ is inversive $($respectively, power associative, Moufang and symmetrically associative$)$, then $\mathrm{Im} HD \subset G_{{\rm inv}}$ $($respectively, $\mathrm{Im} HD \subset G_{{\rm p.a}}$, $\mathrm{Im} HD \subset G_{{\rm Mou.}}$ and $\mathrm{Im} HD \subset G_{{\rm s.a}})$. \end{lem} \begin{proof}{\rm For any $[\omega]\in \mathrm{Im} HD$, $\Phi ([\omega])=[(X_{1}\times X_{2},\mu_{1}\times \mu_{2}),i_{1},p_{2}]$. Since $(X_{2},\mu_{2})$ is inversive, $(X_{1}\times X_{2},\mu_{1}\times \mu_{2})$ is also inversive and $\omega q(l_{2}\times id_{2})\Delta _{2}\simeq \omega q(id_{2} \times r_{2})\Delta _{2}$ by Proposition \ref{prop:3.1}. Hence $[\omega]\in G_{{\rm inv}}$. A similar augment shows the other cases. } \end{proof} \begin{thm}\label{prop:4.1.3} The following statements hold. \ \\[-1.2em] \begin{enumerate} \item If $(X_{2},\mu _2)$ is inversive, then $\Phi _{{\rm inv}}:G_{{\rm inv}}/\mathrm{Im} HD\to \{ [(X,\mu),f_{1},f_{2}]\in CH(X_{1},\mu_{1};X_{2},\mu_{2}) \mid (X,\mu)$ is inversive $\}$, which is the restricted homomorphism of $\Phi $ to $G_{{\rm inv}}/\mathrm{Im} HD $, is bijective. \item If $(X_{2},\mu _2)$ is power associative, then $\Phi _{{\rm p.a}}:G_{{\rm p.a}}/\mathrm{Im} HD\to \{ [(X,\mu),f_{1},f_{2}]\in CH(X_{1},\mu_{1};X_{2},\mu_{2}) \mid (X,\mu)$ is power associative $\}$, which is the restricted homomorphism of $\Phi $ to $G_{{\rm p.a}}/\mathrm{Im} HD $, is bijective. \item If $(X_{2},\mu _2)$ is Moufang, then $\Phi _{{\rm Mo}}:G_{{\rm Mo}}/\mathrm{Im} HD\to \{ [(X,\mu),f_{1},f_{2}]\in CH(X_{1},\mu_{1};X_{2},\mu_{2}) \mid (X,\mu)$ is Moufang $\}$, which is the restricted homomorphism of $\Phi $ to $G_{{\rm Mo}}/\mathrm{Im} HD $, is bijective. \item If $(X_{2},\mu _2)$ is symmetrically associative, then $\Phi _{{\rm s.a}}:G_{{\rm s.a}}/\mathrm{Im} HD\to \{ [(X,\mu),f_{1},f_{2}]\in CH(X_{1},\mu_{1};X_{2},\mu_{2}) \mid (X,\mu)$ is symmetrically associative $\}$, which is the restricted homomorphism of $\Phi $ to $G_{{\rm s.a}}/\mathrm{Im} HD $, is bijective. \end{enumerate} \end{thm} \begin{proof}{\rm By Proposition \ref{prop:3.1}, we see that the map $\Phi_{{\rm inv}}$ is a well-defined homomorphism and hence it is injective. Let $(X,\mu)$ be an inversive central H-extension of $(X_{1},\mu_{1})$ by $(X_{2},\mu_{2})$. Theorem \ref{thm:Kac} yields that there exists a map $\omega :X_{2}\wedge X_{2}\to X_{1}$ such that $[(X,\mu),f_{1},f_{2}]=[(X_{1}\times X_{2},\mu_{\omega}),i_{1},p_{2}]$. Since $(X,\mu)$ is inversive, $(X_{1}\times X_{2},\mu_{\omega})$ is also inversive. Hence, $\omega q(l_{2}\times id_{2})\Delta _{2}\simeq \omega q(id_{2} \times r_{2})\Delta _{2}$ by Proposition \ref{prop:3.1}. Therefore $\Phi _{{\rm inv}}([\omega]) = [(X,\mu),f_{1},f_{2}]$ and $\Phi_{{\rm inv}}$ is surjective. In the same view, we can show that $\Phi _{{\rm p.a}}$, $\Phi _{{\rm Mo}}$ and $\Phi _{{\rm s.a}}$ are bijective. } \end{proof} We denote $G_{{\rm inv}}/\mathrm{Im} HD$, $G_{{\rm p.a}}/\mathrm{Im} HD$, $G_{{\rm Mo}}/\mathrm{Im} HD$ and $G_{{\rm s.a}}/\mathrm{Im} HD$ by $S_{{\rm inv}}$, $S_{{\rm p.a}}$, $S_{{\rm Mo}}$ and $S_{{\rm s.a}}$, respectively, and call {\it the moduli subspace} of the H-extensions associated with the corresponding properties. \section{The moduli spaces of central H-extensions of rational H-spaces} In this section, we will investigate the moduli set $G/\mathrm{Im} HD$ and its subsets $S_{{\rm inv}}$, $S_{{\rm p.a}}$, $S_{{\rm Mo}}$ and $S_{{\rm s.a}}$ in the rational cases. If $(X,\mu)$ is a ${\mathbb Q}$-local, simply-connected H-space and the homotopy group $\pi _{*}(X)$ are of finite type, then $H^{*}(X;{\mathbb Q})\cong \Lambda (x_{1},x_{2},\cdots )$ as an algebra, the free commutative algebra with basis $x_{1},x_{2},\cdots $ by Hopf's Theorem \cite[p.286]{Spa1966}. \begin{thm}{\rm \cite[Proposition 1]{Sch1984}} \label{thm:Sch} Let $(X,\mu)$ be a ${\mathbb Q}$-local, simply-connected H-space and $\pi _{*}(X)$ are of finite type, let $Y$ be a space. Then the canonical map \begin{eqnarray*} [Y,X]\longrightarrow \mathrm{Hom}_{{\rm Alg}}(H^{*}(X;{\mathbb Q} ),H^{*}(Y;{\mathbb Q} )), \ f\longmapsto H^{*}(f) \end{eqnarray*} is bijective. \end{thm} According to Arkowitz and Lupton \cite{AL1990}, we give $\mathrm{Hom}_{{\rm Alg}}(H^{*}(X;{\mathbb Q} ),H^{*}(Y;{\mathbb Q} ))$ an algebraic loop structure so that the above canonical map is an algebraic loop homomorphism. \begin{defn}{\rm Let $M=\Lambda (x_{i};i\in J )$ be a free commutative algebra generated by elements $x_{i}$ for $i\in J$. A homomorphism $\nu :M\to M\otimes M$ is called} a diagonal {\rm if the following diagram is commutative, where $\varepsilon :M\to {\mathbb Q}$ is augmentation:} \[\xymatrix{ & M \ar[d]^{\nu } \ar[ld]_{\cong } \ar[rd]^{\cong }& \\ {\mathbb Q} \otimes M & M\otimes M \ar[l]^{\varepsilon \otimes id} \ar[r]_{id\otimes \varepsilon } &M\otimes {\mathbb Q}. }\] \end{defn} Let $(X,\mu)$ be an H-space. Then $H^{*}(\mu):H^{*}(X;{\mathbb Q})\to H^{*}(X;{\mathbb Q})\otimes H^{*}(X;{\mathbb Q})$ is a diagonal. \begin{thm}{\rm \cite[Lemma 3.1]{AL1990}} \label{thm:AL} Let $M=\Lambda (x_{1},x_{2},\cdots )$ be a free commutative algebra with the diagonal map $\nu :M\to M\otimes M$, and $A$ a graded algebra. We define the product of $\mathrm{Hom}_{{\rm Alg}}(M,A)$ by \begin{eqnarray*} \alpha \cdot \beta = m(\alpha \otimes \beta )\nu , \end{eqnarray*} where $m$ is the product of $A$. Then $\mathrm{Hom}_{{\rm Alg}}(M,A)$ is an algebraic loop endowed with the product. \end{thm} It follows from the proof of Theorem \ref{thm:AL} elements $\gamma _{1}$ and $\gamma _{2}\in \mathrm{Hom}_{{\rm Alg}}(M,A)$ satisfy the condition $\gamma _{1}\cdot \alpha = \beta$ and $\alpha \cdot \gamma _{2} = \beta $, then \begin{align*} &\gamma _{1}(x_{i})=\beta (x_{i})-\alpha (x_{i})-m(\gamma _{1}\otimes \alpha )P(x_{i}) \ \text{and}\\ &\gamma _{2}(x_{i})=\beta (x_{i})-\alpha (x_{i})-m(\alpha \otimes \gamma _{2} )P(x_{i}), \end{align*} where $P(x_{i})=\nu (x_{i})-x_{i}\otimes 1 -1\otimes x_{i}.$ \begin{defn} A left inverse $\lambda$ $(${\rm respectively}, a right inverse $\rho )$ {\rm of an algebraic loop $\mathrm{Hom}_{{\rm Alg}}(M,A)$ are elements of $\mathrm{Hom}_{{\rm Alg}}(M,A)$ such that $\lambda \cdot id = e$ $($respectively, $id \cdot \rho=e )$, where $e$ is the unit of $\mathrm{Hom}_{{\rm Alg}}(M,A)$. } \end{defn} Let $A$ be a graded algebra and $M=\Lambda (x_{i};i\in J )$ a free commutative Hopf algebra for which each $x_{i}$ are primitive $($that is, $\nu (x_{i})=x_{i}\otimes 1+1\otimes x_{i}.)$ We consider the canonical isomorphism \begin{eqnarray*} \Psi : \mathrm{Hom}_{{\rm Alg}}(M, A) \stackrel{\cong }{\rightarrow } \mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{i};j\in J \rangle , A) \end{eqnarray*} which is defined by \begin{eqnarray*} \Psi (\alpha )(x_{i})=\alpha (x_{i}) \ {\rm for} \ \alpha \in \mathrm{Hom}_{{\rm Alg}}(M, A), \end{eqnarray*} where ${\mathbb Q} \langle x_{i};i\in J \rangle$ denotes the graded ${\mathbb Q}$-vector space with basis $x_{i}$ for $i\in J$. Let $\mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{i};i\in J \rangle , A)$ denote the set of ${\mathbb Q}$-linear maps from ${\mathbb Q} \langle x_{i};i\in J \rangle$ to $A$. Since $\mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{i};i\in J \rangle , A)$ is a ${\mathbb Q}$-vector space with respect to the canonical sum and the canonical scalar multiple, we can give the ${\mathbb Q}$-vector space structure to $\mathrm{Hom}_{{\rm Alg}}(M,A)$ via $\Psi $. We regard $\mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{i};i\in J \rangle , A)$ as an algebraic loop with the canonical sum. Then the isomorphism $\Psi $ is a morphism of algebraic loops. In fact, \begin{eqnarray*} \alpha \cdot \beta (x_{i}) = m(\alpha \otimes \beta )\nu (x_{i}) = \alpha (x_{i}) + \beta (x_{i}) = (\alpha +\beta )(x_{i}). \end{eqnarray*} Let $(X_{1},\mu_{1})$ be a ${\mathbb Q}$-local, simply-connected, homotopy associative and homotopy commutative H-space and the homotopy group of $X_{1}$ are of finite type, let $(X_{2},\mu_{2})$ be an H-space, and let $H^{*}(X_{1};{\mathbb Q})=\Lambda (x_{1},x_{2},\cdots )$. Since $(X_{1},\mu_{1})$ is a homotopy associative and homotopy commutative H-space, we see that each $x_{i}$ is primitive; see \cite[Corollary 4.18]{MM1965}. \begin{lem}\label{lem:3.3.1} Let $X$ and $Y$ be spaces. Then \begin{eqnarray*} H^{n}(X\wedge Y;{\mathbb Q}) \cong \{ x\otimes y \in H^{n}(X\times Y;{\mathbb Q}) \mid {\rm deg} \, x >0 , {\rm deg} \, y > 0 \} \ (n\geq 1). \end{eqnarray*} \end{lem} \begin{proof}{\rm For any $n\geq 1$, we consider the following commutative diagram: \[\xymatrix{ [X\wedge Y,K({\mathbb Q} ,n )] \ar[r]^{q^{*}} \ar[d]^{\cong } & [X\times Y, K({\mathbb Q} ,n)] \ar[r]^{i^{*}} \ar[d]^{\cong } & [X\vee Y,K({\mathbb Q} ,n)] \ar[d]^{\cong } \\ H^{n}(X\wedge Y;{\mathbb Q}) \ar[r]^{H^{n}(q)} & H^{n}(X\times Y;{\mathbb Q} ) \ar[r]^{H^{n}(i)} & H^{n}(X\vee Y ;{\mathbb Q} ), }\] where $K({\mathbb Q},n)$ is the Eilenberg-MacLane space. By Lemma \ref{lem:2.2.2}, $H^{n}(q)$ is injective and we have $$ H^{n}(X\wedge Y;{\mathbb Q})\cong \mathrm{Ker} H^{n}(i) = \{ x\otimes y \in H^{n}(X\times Y;{\mathbb Q}) \mid {\rm deg} \, x >0 , {\rm deg} \, y > 0 \}. $$ }\end{proof} Define the map \begin{eqnarray*} \overline{HD} : \mathrm{Hom}_{{\mathbb Q}}({\mathbb Q} \langle x_{1} ,x_{2},\cdots \rangle , H^{*}(X_{2} ;{\mathbb Q} ))\rightarrow \mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{1} ,x_{2},\cdots \rangle , H^{*}(X_{2}\wedge X_{2} ;{\mathbb Q} )) \end{eqnarray*} by \begin{eqnarray*} \overline{HD}(\alpha )(x_{i})= P_{2}(\alpha (x_{i})) \ {\rm for} \ \alpha \in \mathrm{Hom}_{{\mathbb Q}}({\mathbb Q} \langle x_{1} ,x_{2},\cdots \rangle , H^{*}(X_{2} ;{\mathbb Q} )), \end{eqnarray*} where $P_{2}(x) = H^{*}(\mu_{2})(x) -x\otimes 1 - 1\otimes x \ (x\in H^{*}(X_{2};{\mathbb Q}))$. Then we see that $\overline{HD}$ is a ${\mathbb Q}$-linear map. \begin{prop}\label{prop:4.4} The following diagram of algebraic loops is commutative$:$ \[\xymatrix{ \mathrm{Hom}_{{\mathbb Q}}({\mathbb Q} \langle x_{1},x_{2},\cdots \rangle , H^{*}(X_{2} ;{\mathbb Q} )) \ar[r]^{\hspace{-1.5em}\overline{HD}} & \mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{1},x_{2}, \cdots \rangle , H^{*}(X_{2}\wedge X_{2} ;{\mathbb Q} ))\\ [X_{2},X_{1}] \ar[r]_{HD} \ar[u]^{\Psi H^{*}}_{\cong } & [X_{2}\wedge X_{2},X_{1}] \ar[u]_{\Psi H^{*}}^{\cong }.\\ }\] \end{prop} \begin{proof}{\rm By using the canonical isomorphism $\Psi $, we see that for any $f \in [X_{2},X_{1}]$ there exists a unique element $\overline{D}(f ) \in \mathrm{Hom}_{{\mathbb Q}}(\langle x_{1} ,x_{2},\cdots \rangle , H^{*}(X_{2}\times X_{2} ;{\mathbb Q} )$ such that $\overline{D}(f ) + (H^{*}(f) \otimes H^{*}(f) )\Psi H^{*}(\mu_{1}) = H^{*}(\mu_{2}) \Psi H^{*}(f) $. Therefore, it follows that $\overline{D}(f)= \Psi H^{*}(D(f\mu_{1}, \mu_{2}(f\times f)))$. Observe that each $x_{i}$ is primitive. In view of the proof of Theorem \ref{thm:AL}, we have $$ \overline{D}(f)(x_{i}) = P_{2}(\Psi H^{*}(f)(x_{i})). $$ Lemma \ref{lem:3.3.1} implies that $\overline{D}(f)(x_{i})=H^{*}(q)\overline{HD}(\Psi H^{*}(f) )(x_{i})$. Hence, by the definition of an H-deviation, we obtain $$ H^{*}(q)\Psi H^{*}(HD(f)) =\Psi H^{*}(D(f\mu_{1},\mu_{2}(f\times f))) = H^{*}(q)\overline{HD}(\Psi H^{*}(f) ). $$ Since $H^{*}(q)$ is injective, it follows from Lemma \ref{lem:2.2.2} and Theorem \ref{thm:Sch} that $\Psi H^{*}(HD(f))=\overline{HD}(\Psi H^{*}(f))$.} \end{proof} Put $V=\mathrm{Hom}_{{\mathbb Q} }({\mathbb Q} \langle x_{1} ,x_{2},\cdots \rangle , H^{*}(X_{2}\wedge X_{2} ;{\mathbb Q} ))$ and set \begin{align*} V_{{\rm inv}} &= \{ \alpha \in V \ | \ m_{2}(\lambda _{2}\otimes id)H^{*}(q)\alpha = m_{2}(id\otimes \rho _{2})H^{*}(q)\alpha \},\\ V_{{\rm p.a}} &= \{ \alpha \in V \ | \ \overline{m}_{2}(H^{*}(\mu_{2})\otimes id)H^{*}(q)\alpha = \overline{m}_{2}(id\otimes H^{*}(\mu_{2}))H^{*}(q)\alpha \},\\ V_{{\rm Mo}} &= \{ \alpha \in V \ | \ \overline{\Gamma} _{{\rm Mo}}(\alpha )=\overline{\Gamma }' _{{\rm Mo}}(\alpha ) \},\\ V_{{\rm s.a}} &= \{ \alpha \in V \ | \ H^{*}(\Gamma _{{\rm s.a}})(\alpha \otimes \alpha) H^{*}(\mu_{1})=H^{*}(\Gamma' _{{\rm s.a}})(\alpha \otimes \alpha) H^{*}(\mu_{1}) \}, \end{align*} where $m_{2}$ is the product of $H^{*}(X_{2};{\mathbb Q})$, $\overline{m}_{2}=m_{2}(m_{2}\otimes id)$, and $\lambda _{2}, \rho _{2}$ are a left inverse and a right inverse of $\mathrm{Hom}_{{\rm Alg}}(H^{*}(X_{2};{\mathbb Q}),H^{*}(X_{2};{\mathbb Q}))$, respectively. Let $$ \overline{\Gamma} _{{\rm Mo}}(\alpha )=H^{*}(\Gamma _{{\rm Mo}}(\omega )), \ \overline{\Gamma}' _{{\rm Mo}}(\alpha )=H^{*}(\Gamma _{{\rm Mo}}(\omega )) $$ where $\omega :X_{2}\wedge X_{2}\to X_{1}$ satisfy $H^{*}(\omega)=\alpha $. Then, we see that $V_{{\rm inv}}$, $V_{{\rm p.a}}$, $V_{{\rm Mo}}$ and $V_{{\rm s.a}}$ are subspaces of $V$. \begin{thm}\label{lem:4.4} The following statements hold. \ \\[-1.2em] \begin{enumerate} \item If $(X_{2},\mu _2)$ is inversive, then $\mathrm{Im} \overline{HD} \subset V_{{\rm inv}}$ and the canonical map $S_{{\rm inv}} \to V_{{\rm inv}}/ \mathrm{Im} \overline{HD}$ is an isomorphism of algebraic loops. \item If $(X_{2},\mu _2)$ is power associative, then $\mathrm{Im} \overline{HD} \subset V_{{\rm p.a}}$ and the canonical map $S_{{\rm p.a}} \to V_{{\rm p.a}}/ \mathrm{Im} \overline{HD}$ is isomorphism of algebraic loops. \item If $(X_{2},\mu _2)$ is Moufang, then $\mathrm{Im} \overline{HD} \subset V_{{\rm Mo}}$ and the canonical map $S_{{\rm Mo}} \to V_{{\rm Mo}}/ \mathrm{Im} \overline{HD}$ is isomorphism of algebraic loops. \item If $(X_{2},\mu _2)$ is symmetrically associative, then $\mathrm{Im} \overline{HD} \subset V_{{\rm s.a}}$ and the canonical map $S_{{\rm s.a}} \to V_{{\rm s.a}}/ \mathrm{Im} \overline{HD}$ is isomorphism of algebraic loops. \end{enumerate} \end{thm} \begin{proof}{\rm The assertions follow from Lemma \ref{lem:4.2} and Proposition \ref{prop:4.4}. } \end{proof} \begin{lem}\label{lem:3.3.6} Let $V^{k}= \mathrm{Hom}_{{\mathbb Q}}({\mathbb Q}\langle x_{k} \rangle , H^{*}(X_{2}\wedge X_{2};{\mathbb Q}))$ for $k=1,2,\cdots$. Then the ${\mathbb Q}$-linear map \begin{eqnarray*} \xi :\displaystyle\bigoplus _{k\geq 1}V^{k} \longrightarrow V ; \ \ \xi (\alpha _{1}, \alpha _{2},\cdots ) (x_{k})=\alpha _{k}(x_{k}) \ (\alpha _{k}\in V^{k}) \end{eqnarray*} is an isomorphism. Moreover, let \begin{align*} & V_{{\rm inv}}^{k} = \{ \alpha \in V^{k} \mid m_{2}(\lambda _{2}\otimes id)H^{*}(q)\alpha = m_{2}(id\otimes \rho _{2})H^{*}(q)\alpha \} ,\\ & V_{{\rm p.a}}^{k} = \{ \alpha \in V^{k} \mid \overline{m}_{2}(H^{*}(\mu_{2})\otimes id)H^{*}(q)\alpha = \overline{m}_{2}(id\otimes H^{*}(\mu_{2}))H^{*}(q)\alpha \} ,\\ &V_{{\rm Mo}}^{k} = \{ \alpha \in V^{k} \ | \ \overline{\Gamma} _{{\rm Mo}}(\alpha )=\overline{\Gamma }' _{{\rm Mo}}(\alpha ) \} \ {\rm and}\\ &V_{{\rm s.a}}^{k} = \{ \alpha \in V^{k} \ | \ H^{*}(\Gamma _{{\rm s.a}})(\alpha \otimes \alpha) H^{*}(\mu_{1})=H^{*}(\Gamma' _{{\rm s.a}})(\alpha \otimes \alpha) H^{*}(\mu_{1})\} . \end{align*} We define a ${\mathbb Q}$-linear map \begin{eqnarray*} \overline{HD}^{ \hspace{0.15em} k} : \mathrm{Hom}_{{\mathbb Q}}({\mathbb Q} \langle x_{k} \rangle , H^{*}(X_{2} ;{\mathbb Q} ))\rightarrow V^{k} \end{eqnarray*} by \begin{eqnarray*} \overline{HD}^{\hspace{0.15em}i}(\alpha )(x_{k})= P_{2}(\alpha (x_{k})) , \ \alpha \in \mathrm{Hom}_{{\mathbb Q}}({\mathbb Q} \langle x_{k} \rangle , H^{*}(X_{2} ;{\mathbb Q} )). \end{eqnarray*} Then the restrictions \begin{align*} &\displaystyle\bigoplus _{k\geq 1}V_{{\rm inv}}^{k} \longrightarrow V_{{\rm inv}} & &\displaystyle\bigoplus _{k\geq 1}V_{{\rm p.a}}^{k} \longrightarrow V_{{\rm p.a}} &\displaystyle\bigoplus _{k\geq 1}V_{{\rm Mo}}^{k} \longrightarrow V_{{\rm Mo}} \\ &\displaystyle\bigoplus _{k\geq 1}V_{{\rm s.a}}^{k} \longrightarrow V_{{\rm s.a}} & &\displaystyle\bigoplus _{k\geq 1}\mathrm{Im} \overline{HD}^{ \hspace{0.15em} k} \longrightarrow \mathrm{Im} \overline{HD} & \end{align*} of $\xi $ to $\displaystyle\bigoplus _{k\geq 1}V_{{\rm inv}}^{k}$, $\displaystyle\bigoplus _{k\geq 1}V_{{\rm p.a}}^{k}$, $\displaystyle\bigoplus _{k\geq 1}V_{{\rm Mo}}^{k}$, $\displaystyle\bigoplus _{k\geq 1}V_{{\rm s.a}}^{k}$ and $\displaystyle\bigoplus _{k\geq 1}\mathrm{Im} \overline{HD}^{ \hspace{0.15em} k}$ are all isomorphisms. \end{lem} \begin{proof} {\rm The ${\mathbb Q}$-linear map \begin{eqnarray*} \xi ' :V\longrightarrow \displaystyle\bigoplus _{k\geq 1}V^{k}; \ \ \xi '(\alpha )=(\alpha i_{1},\alpha i_{2},\cdots ) \end{eqnarray*} is an inverse map of $\xi $, where $i_{k}:{\mathbb Q}\langle x_{k} \rangle \longrightarrow {\mathbb Q} \langle x_{1},x_{2},\cdots \rangle$ is the inclusion. } \end{proof} By Lemma \ref{lem:3.3.6}, for the study of $V_{{\rm inv}}/\mathrm{Im} \overline{HD}$, $V_{{\rm p.a}}/\mathrm{Im} \overline{HD}$, $V_{{\rm Mo}}/\mathrm{Im} \overline{HD}$ and $V_{{\rm s.a}}/\mathrm{Im} \overline{HD}$, it is enough to investigate these vector space in the case where $H^{*}(X_{1};{\mathbb Q})=\Lambda (x)$. In what follows, we give some examples of $V$, $V_{{\rm inv}}$, $V_{{\rm p.a}}$, $V_{{\rm Mo}}$, $V_{{\rm s.a}}$ and $\mathrm{Im} \overline{HD}$. \begin{ex}\label{ex:4.9} {\rm Let $X_{2}$ be a rational H-space such that $H^{*}(X_{2};{\mathbb Q})=\Lambda (y)$ and that ${\rm deg} \, y$ is odd. Then $X_{2}$ is a homotopy associative H-space for any multiplication. Then we have $V=V_{{\rm inv}}=V_{{\rm p.a}}=V_{{\rm Mo}}=V_{{\rm s.a}}$ and $\mathrm{Im} \overline{HD}=\{ 0 \}$. } \end{ex} \begin{ex}\label{ex:3.3.8}{\rm Let $X_{2}$ be a rational H-space such that $H^{*}(X_{2};{\mathbb Q})=\Lambda (y)$ and that ${\rm deg} \, y$ is even. Then $X_{2}$ is a homotopy associative H-space for any multiplication. Then one has the following:\\ 1. If ${\rm deg} \, x=2{\rm deg} \, y$, then $$V=V_{{\rm inv}}=V_{{\rm p.a}}=V_{{\rm Mo}}=V_{{\rm s.a}}=\mathrm{Im} \overline{HD}\cong {\mathbb Q}.$$ 2. If ${\rm deg} \, x=3{\rm deg} \, y$, then $$V\cong {\mathbb Q}^{2} \ \text{and} \ V_{{\rm inv}}=V_{{\rm p.a}}=V_{{\rm Mo}}=V_{{\rm s.a}} = \mathrm{Im} \overline{HD} \cong \{(r_{1},r_{2})\in {\mathbb Q}^{2} \mid r_{1}=r_{2}\}.$$ 3. If ${\rm deg} \, x = m{\rm deg} \, y$ and $m\geq 4$, then $V\cong {\mathbb Q} ^{m-1},$\\ \hspace{1em} $V_{{\rm inv}}$ $\cong \left\{ \begin{array}{l} V \ (m:{\rm even})\\ \Bigl\{ (r_{1},\cdots ,r_{m-1})\in {\mathbb Q}^{m-1} \mid \displaystyle\sum^{m-1}_{j=1}(-1)^{j+1}r_{j} =0\Bigr\} \ (m:{\rm odd}) \end{array} \right. $ \\ \hspace{1em}$V_{{\rm p.a}}\cong \Bigl\{ (r_{1},\cdots ,r_{m-1})\in {\mathbb Q}^{m-1} \mid \displaystyle\sum^{m-1}_{j=1}(2^{j}-2^{m-j})r_{j} =0\Bigr\}$\\ \hspace{1em}$V_{{\rm s.a}}$ $\cong \left\{ \begin{array}{l} \{ (r_{1},\cdots ,r_{m-1})\in {\mathbb Q}^{m-1} \mid r_{l}=r_{m-l}, \ l=1,\cdots , \frac{m-2}{2}\} \ (m:{\rm even})\\ \{ (r_{1},\cdots ,r_{m-1})\in {\mathbb Q}^{m-1} \mid r_{l}=r_{m-l}, \ l=1,\cdots , \frac{m-1}{2}\} \ (m:{\rm odd}) \end{array} \right. $\\ \hspace{1em}$\mathrm{Im} \overline{HD}\cong \{ (r_{1},\cdots ,r_{m-1})\in {\mathbb Q}^{m-1} \mid$\\ \hspace{10em} $(m-1)!r_{1}=\cdots =i!(m-i)!r_{i}=\cdots = (m-1)!r_{m-1} \}$ \\ 4. Other cases, $V=V_{{\rm inv}}=V_{{\rm p.a}}=V_{{\rm Mo}}=V_{{\rm s.a}}=\mathrm{Im} \overline{HD}=\{ 0 \}$. } \end{ex} \begin{proof} {\rm In the statement 2, we choose the basis $\sigma _{1}$ and $\sigma _{2}$ of V defined by \begin{eqnarray*} \sigma _{1}(x)= y\otimes y^{2} \ \text{and} \ \sigma _{2}(x)= y^{2}\otimes y. \end{eqnarray*} We define the isomorphism of ${\mathbb Q}$-vector spaces $f:V\to {\mathbb Q}^{2}$ by $f(\sigma _{1})=(1,0)$ and $f(\sigma _{2})=(0,1)$. Then, for any element $\alpha =r_{1}\sigma _{1}+r_{2}\sigma _{2} \ (r_{i}\in {\mathbb Q})$ of V, we have \begin{align*} &m_{2}(\lambda _{2}\otimes id)H^{*}(q)\alpha (x)- m_{2}(id\otimes \rho _{2})H^{*}(q)\alpha (x)= (-2r_{1}+2r_{2})y^{3}, \\ &\overline{m}_{2}(H^{*}(\mu_{2})\otimes id)H^{*}(q)\alpha (x)- \overline{m}_{2}(id\otimes H^{*}(\mu_{2}))H^{*}(q)\alpha (x)=(-2r_{1}+2r_{2})y^{3}. \end{align*} The element $\alpha $ is in $\mathrm{Im} \overline{HD}$ if and only if there exists $\beta $ in $\mathrm{Hom}_{{\mathbb Q}}({\mathbb Q} \langle x \rangle , H^{*}(X_{2} ;{\mathbb Q} ))$ such that $\alpha =\overline{HD}(\beta )$. Put $\beta (x)=ry^{3} \ (r\in {\mathbb Q})$, then we have $r_{1}=3r=r_{2}$. Hence, $V_{{\rm p.a}}=V_{{\rm Mo}}=V_{{\rm s.a}} = \mathrm{Im} \overline{HD}$. Therefore 2 is shown. We can prove the statements 1,3 and 4 similar to that computations. } \end{proof} \noindent {\it Proof of Assertion \ref{ex1}.} If $n=km$ and $k\geq 4$ is even, we have $$ V_{{\rm Mo}}/\mathrm{Im} HD \subseteq V_{{\rm s.a}}/\mathrm{Im} HD \subsetneq V_{{\rm p.a}}/\mathrm{Im} HD \subsetneq V/\mathrm{Im} HD =V_{{\rm inv}}/\mathrm{Im} HD$$ by Lemma \ref{lem:3.3.6} and Example \ref{ex:3.3.8}. Thus the statement (1) holds. A similar argument show the other cases. \qed \section{Acknowledgment} The author is deeply grateful to Katsuhiko Kuribayashi and Ryo Takahashi who provided helpful comments and suggestions. \end{document}

\begin{document} \title{Detecting non-decomposability of time evolution via extreme gain of correlations} \author{Tanjung Krisnanda} \affiliation{School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore} \author{Ray Ganardi} \affiliation{School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore} \author{Su-Yong Lee} \affiliation{School of Computational Sciences, Korea Institute for Advanced Study, Hoegi-ro 85, Dongdaemun-gu, Seoul 02455, Korea} \author{Jaewan Kim} \affiliation{School of Computational Sciences, Korea Institute for Advanced Study, Hoegi-ro 85, Dongdaemun-gu, Seoul 02455, Korea} \author{Tomasz Paterek} \affiliation{School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore} \affiliation{MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit, UMI 3654 Singapore, Singapore} \begin{abstract} Non-commutativity is one of the most elementary non-classical features of quantum observables. Here we propose a method to detect non-commutativity of interaction Hamiltonians of two probe objects coupled via a mediator. If these objects are open to their local environments, our method reveals non-decomposability of temporal evolution into a sequence of interactions between each probe and the mediator. The Hamiltonians or Lindblad operators can remain unknown throughout the assessment, we only require knowledge of the dimension of the mediator. Furthermore, no operations on the mediator are necessary. Technically, under the assumption of decomposable evolution, we derive upper bounds on correlations between the probes and then demonstrate that these bounds can be violated with correlation dynamics generated by non-commuting Hamiltonians, e.g., Jaynes-Cummings coupling. An intuitive explanation is provided in terms of multiple exchanges of a virtual particle which lead to the excessive accumulation of correlations. A plethora of correlation quantifiers are helpful in our method, e.g., quantum entanglement, discord, mutual information, and even classical correlation. Finally, we discuss exemplary applications of the method in quantum information: the distribution of correlations and witnessing dimension of an object. \end{abstract} \maketitle \section{Introduction} All classical observables are functions of positions and momenta. Since there is no fundamental limit on the precision of position and momentum measurement in classical physics, all classical observables are, in principle, measurable simultaneously. Quite differently, the Heisenberg uncertainty principle forbids simultaneous exact knowledge of quantum observables corresponding to position and momentum. The underlying non-classical feature is their non-commutativity: Any pair of non-commuting observables cannot be simultaneously measured to arbitrary precision, as first demonstrated by Robertson in his famous uncertainty relation~\cite{robertson1929}. Other examples of non-classical phenomena with underlying non-commutativity of observables include violations of Bell inequalities~\cite{landau1987,peres-book} or, more generally, non-contextual inequalities; e.g., see~\cite{thompson2016}. Here we describe a method to detect non-commutativity of interaction Hamiltonians, and generally non-decomposability of temporal evolution, from the dynamics of correlations. Consider the situation depicted in Fig.~\ref{FIG_setup}, where the probe systems $A$ and $B$ do not interact directly but only via the mediator $C$; i.e., there is no Hamiltonian term $H_{AB}$. In general, we allow all objects to be open systems and study whether the evolution operator cannot be represented by a sequence of operations between each probe and the mediator, i.e., $\Lambda_{BC}\Lambda_{AC}$ or in reverse order. For the special case in which all systems are closed, non-decomposability implies non-commutativity of interaction Hamiltonians, i.e., $[H_{AC},H_{BC}]\ne0$. Indeed, for commuting Hamiltonians, the unitary evolution operator is decomposable into $U_{BC}U_{AC}$, where, for example, $U_{AC} = \exp(- i t H_{AC})$ and we set $\hbar = 1$. We show that for decomposable evolution, correlations between $A$ and $B$ are bounded. We also show with concrete dynamics generated by non-commuting Hamiltonians that these bounds can be violated. The bounds derived depend solely on the dimensionality of $C$ and not on the actual form of the evolution operators. Hence, these operators can remain unknown throughout the assessment. This is a desired feature, as experimenters usually do not reconstruct the evolution operators via process tomography. It also allows applications of the method to situations where the physics is not understood to the extent that reasonable Hamiltonians or Lindblad operators can be written down. Furthermore, the assessment does not depend on the initial state of the tripartite system and does not require any operations on the mediator. It is therefore applicable to a variety of experimental situations; Refs.~\cite{rauschenbeutel2001,sahling2015,baart2017,hamsen} provide concrete examples. \begin{figure} \caption{Probe objects $A$ and $B$ individually interact with a mediator $C$, but not with each other ($A$, $B$, and $C$ could be open to their local environments). The coherent parts of the interactions are described by Hamiltonians $H_{AC} \label{FIG_setup} \end{figure} We begin by presenting the general bounds on the amount of correlations one can establish if the evolution is decomposable. It is shown that these bounds are generic and hold for a plethora of correlation quantifiers. We then calculate concrete bounds on exemplary quantifiers and show how they can be violated in a system of two fields coupled by a two-level atom. We discuss the origin of the violation in terms of ``Trotterized" evolution, where a virtual particle is exchanged between $A$ and $B$ multiple times if the Hamiltonians do not commute but only once if they do commute. Finally, we focus on immediate applications in quantum information and discuss the consequences of our findings for correlation distribution protocols and dimension witnesses. \section{Results} \subsection{General bounds} Consider the setup illustrated in Fig. \ref{FIG_setup}. System $C$, with finite dimension $d_C$, is mediating interactions between higher-dimensional systems $A$ and $B$. For simplicity we take $d_A=d_B > d_C$. We assume that there is no direct interaction between $A$ and $B$, such that the Hamiltonian of the whole tripartite system is of the form $H_{AC}+H_{BC}$ (local Hamiltonians $H_A$, $H_B$, and $H_C$ included). Our bounds follow from a generalization of the following simple observation. Consider, for the moment, the relative entropy of entanglement as the correlation quantifier \cite{vedral1997}. If the evolution is decomposable, it can be written as $\Lambda_{BC} \Lambda_{AC}$, or in reverse order. Therefore, it is as if particle $C$ interacted first with $A$ and then with $B$, a scenario similar to that in Refs.~\cite{cubitt2003,streltsov2012,chuan2012,edssexp1,edssexp2,edssexp3}. The first interaction can generate at most $\log_2(d_C)$ ebits of entanglement, whereas the second, in the best case, can swap all this entanglement. In the end, particles $A$ and $B$ gain at most $\log_2(d_C)$ ebits. The bound is indeed independent of the form of interactions. Furthermore, it is intuitively clear, as this is just the ``quantum capacity'' of the mediator. Now let us consider correlation quantifiers obtained in the so-called ``distance" approach~\cite{vedral1997, modi2010}. The idea is to quantify correlation $Q_{X:Y}$ in a state $\rho_{XY}$ as the shortest distance $D(\rho_{XY},\sigma_{XY})$ from $\rho_{XY}$ to a set of states $\sigma_{XY} \in \mathcal{S}$ without the desired correlation property, i.e., $Q_{X:Y} \equiv \inf_{\sigma_{XY}\in \mathcal{S}} D(\rho_{XY},\sigma_{XY})$. For example, the relative entropy of entanglement is given by the relative entropy of a state to the set of disentangled states~\cite{vedral1997}. It turns out that most such quantifiers are useful for the task introduced here. The conditions we require are that (i) $\mathcal{S}$ is closed under local operations $\Lambda_Y$ on $Y$, (ii) $D(\Lambda[\rho],\Lambda[\sigma]) \leq D(\rho, \sigma)$ (monotonicity), and (iii) $D(\rho_0, \rho_1) \leq D(\rho_0, \rho_2) + D(\rho_2, \rho_1)$ (triangle inequality). They are sufficient to prove the following theorem. \begin{theorem}\label{TH_cons} Suppose a correlation $Q_{X:Y}$ satisfies properties {\rm(i)--(iii)} listed above. If the evolution operator $\Lambda_{ABC}$ is decomposable into $\Lambda_{BC}\Lambda_{AC}$, then \begin{equation} Q_{A:B} (t) \leq I_{AC:B}(0) + \sup_{\ket{\psi}} Q_{A:C}, \label{TH_IQ_BOUND} \end{equation} where $I_{AC:B}(0) = \inf_{\sigma_{AC} \otimes \sigma_B} D(\rho, \sigma_{AC} \otimes \sigma_B)$, $\rho$ is the initial tripartite state, and the supremum of $Q_{A:C}$ is taken over pure states of $AC$. \end{theorem} \begin{proof} The proof is given in Appendix A. \end{proof} Note that although the relative entropy does not satisfy (iii) it still follows Theorem \ref{TH_cons} (cf. Lemma \ref{LM_re} in Appendix A). Correlations between probe $A$ and probe $B$ are therefore bounded by the maximal achievable correlation with the mediator, $\sup_{\ket{\psi}} Q_{A:C}$. The additional term $I_{AC:B}(0)$ reduces to the usual mutual information if $D(\rho_{XY},\sigma_{XY})$ is the relative entropy distance~\cite{modi2010} and characterizes the amount of total initial correlations between one of the probes and the rest of the system. Note that the bound is independent of time. This can be seen as a result of the effective description of such dynamics given by $\Lambda_{BC}\Lambda_{AC}$. The particle $C$ is exchanged between $A$ and $B$ only once, independently of the duration of the dynamics. In a typical experimental situation the initial state can be prepared as completely uncorrelated $\rho = \rho_A \otimes \rho_B \otimes \rho_C$, in which case Theorem~\ref{TH_cons} simplifies and the bound is given solely in terms of the ``correlation capacity'' of the mediator: \begin{equation} Q_{A:B} (t) \le \sup_{\ket{\psi}} Q_{A:C}. \label{EQ_CC} \end{equation} Clearly, the same bound holds for initial states of the form $\rho = \rho_{AC} \otimes \rho_B$. In Appendix B we show that, with this initial state, Eq.~(\ref{EQ_CC}) holds for any correlation quantifier that is monotonic under local operations $\Lambda_{BC}$, not necessarily based on the distance approach, e.g., any entanglement monotone. For initial states that are close to $\rho = \rho_{AC} \otimes \rho_B$ one can utilize the continuity of the von Neumann entropy \cite{fannes1973continuity} and see that $I_{AC:B}(0)$ in Eq. (\ref{TH_IQ_BOUND}) is indeed small. We can also ensure that the initial state is of the form $\rho = \rho_{AC} \otimes \rho_B$ by performing a correlation breaking channel on $B$ first. One example of such a channel is a measurement in the computational basis followed by a measurement in some complementary (say Fourier) basis. This implements the correlation breaking channel $(\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{AC} \otimes \Lambda_B) (\rho_{ABC}) = \rho_{AC} \otimes \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{d_B}$~\cite{footnote}. In this way, our method does not require any knowledge of the initial state and any operations on the mediator, similar in spirit to the detection of quantum discord of inaccessible objects in Ref.~\cite{krisnanda2017}. We now move to concrete correlation quantifiers and their correlation capacities. \subsection{Exemplary measures and bounds} We provide four correlation quantifiers which capture different types of correlations between quantum particles. All of them are shown to be useful in detecting non-decomposability. Mutual information is a measure of total correlations~\cite{groisman2005} and is defined as $I_{X:Y} = S_X + S_Y - S_{XY}$, where, e.g., $S_X$ is the von Neumann entropy of subsystem $X$. It can also be seen as a distance-based measure with the relative entropy as the distance and a set of product states $\sigma_X \otimes \sigma_Y$ as $\mathcal{S}$~\cite{modi2010}. The supremum in Eq. (\ref{EQ_CC}) is attained by the state (recall that $d_A > d_C$), \begin{equation} |\Psi \rangle = \frac{1}{\sqrt{d_C}} \sum_{j = 1}^{d_C} |a_j \rangle |c_j \rangle, \label{EQ_MAX_ENT} \end{equation} where $| a_j \rangle$ and $| c_j \rangle$ form orthonormal bases. One finds $\sup_{\ket{\psi}} I_{A:C} = 2 \log_2(d_C)$. An interesting quantifier in the context of non-classicality detection is the classical correlation in a quantum state. It is defined as mutual information of the classical state obtained by performing the best local von Neumann measurements on the original state $\rho$~\cite{terhal2002}, i.e., $C_{X:Y} = \sup_{\Pi_X \otimes \Pi_Y} I_{X:Y}(\Pi_X \otimes \Pi_Y(\rho) )$, where $\Pi_X \otimes \Pi_Y(\rho) = \sum_{xy} \proj{xy} \rho \proj{xy}$, and $\ket{x}$, $\ket{y}$ form orthonormal bases. The supremum of mutual information over classical states of $AC$ is $\log_2(d_C)$. Quantum discord is a form of purely quantum correlations that contain quantum entanglement. It can be phrased as a distance-based measure. In particular, we consider the relative entropy of discord~\cite{modi2010}, also known as the one-way deficit~\cite{horodecki2005}. It is an asymmetric quantity defined as $\Delta_{X|Y} = \inf_{\Pi_Y} S(\Pi_Y(\rho)) - S(\rho)$, where $\Pi_Y$ is a von Neumann measurement conducted on subsystem $Y$. The relative entropy of discord is maximized by the state (\ref{EQ_MAX_ENT}), for which we have $\sup_{\ket{\psi}} \Delta_{A|C} = \log_2(d_C)$. Our last example is negativity, a computable entanglement monotone~\cite{vidal2002}. For a bipartite system negativity is defined as $N_{X:Y} = (||\rho^{T_X}||_1-1)/2$, where $||.||_1$ denotes the trace norm and $\rho^{T_X}$ is a matrix obtained by partial transposition of $\rho$ with respect to $X$. Negativity is maximized by the state (\ref{EQ_MAX_ENT}), and the supremum reads $\sup_{\ket{\psi}} N_{A:C} = (d_C-1)/2$. Clearly, many other correlation quantifiers are suitable for our detection method because the assumptions behind Eqs.~(\ref{TH_IQ_BOUND}) and (\ref{EQ_CC}) are not demanding. In fact, one may wonder which correlations do not qualify for our method. A concrete example is the geometric quantum discord based on $p$-Schatten norms with $p > 1$, as it may increase under local operations on $BC$~\cite{piani2012,paula2013}. \subsection{Violations} We now demonstrate, with concrete dynamics generated by non-commuting Hamiltonians, that the bounds derived can be violated. We next discuss the origin of this violation. Consider a two-level atom $C$, i.e., $d_C=2$, mediating interactions between two cavity fields $A$ and $B$. A similar scenario has been considered and implemented, for example, in Refs. \cite{rauschenbeutel2001,messina2002,browne2003,hamsen}. The interaction between the atom and each cavity field is taken to follow the Jaynes-Cummings model, \begin{equation} H = g (\hat a\hat \sigma_+ +\hat a^{\dagger}\hat \sigma_-)+g (\hat b\hat \sigma_+ +\hat b^{\dagger}\hat \sigma_-), \label{EQ_JC} \end{equation} where $\hat a$ ($\hat b$) is the annihilation operator of field $A$ ($B$), while $\hat \sigma_+$ ($\hat \sigma_-$) is the raising (lowering) operator of the two-level atom. For simplicity, we have assumed that the interaction strengths between the two-level atom and the fields are the same. Note that $H$ is of the form $H_{AC}+H_{BC}$ with non-commuting components. The resulting correlation dynamics are plotted in Fig. \ref{FIG_dynamics}. Mutual information and negativity were calculated directly, whereas for the classical correlation and the relative entropy of discord, we provide the lower bounds $\tilde C_{A:B}$ and $-S_{A|B}$, respectively. $\tilde C_{A:B}$ is calculated as the mutual information of the state resulting from projective local measurements in the Fock basis (no optimization over measurements performed). The negative conditional entropy $-S_{A|B}$ is a lower bound on the distillable entanglement \cite{devetak2005distillation}, which in turn is a lower bound on the relative entropy of entanglement $E_{A:B}$ \cite{horodecki2000limits}. Therefore, we note the chain of inequalities $-S_{A|B} \le E_{A:B} \le \Delta_{A|B} \le I_{A:B}$, where the last two inequalities follow from \cite{modi2010}. Already these lower bounds can beat the limit set by decomposable evolution, and therefore, all mentioned correlations can detect non-decomposability of the evolution. Since we consider closed systems, this infers non-commutativity of the Jaynes-Cummings couplings. We also note another non-classical feature of the studied dynamics: since Fig. \ref{FIG_dynamics} shows entanglement gain, according to Ref.~\cite{krisnanda2017} there must be quantum discord $D_{AB|C}$ during the evolution. \begin{figure} \caption{Correlation dynamics with the Jaynes-Cummings model (solid curves) and the corresponding bounds for decomposable evolution (dashed lines). (a) Mutual information, (b) lower bound on the classical correlation (see the main text), (c) lower bound on the relative entropy of discord, and (d) negativity. In all cases, time is rescaled with the interaction strength $g$ and the initial state of $ABC$ is varied: $\ket{110} \label{FIG_dynamics} \end{figure} It is apparent that the detection is easier (faster and with more pronounced violation) with a higher number of photons in the initial states of the cavity fields. We offer an intuitive explanation. Consider, for example, $| m n 0 \rangle$ as the initial state of $ABC$. By defining $\hat \xi = (\hat a +\hat b )/\sqrt{2}$, the Hamiltonian of Eq. (\ref{EQ_JC}) becomes $\sqrt{2} g(\hat \xi \hat \sigma_+ + \hat \xi^{\dagger}\hat \sigma_-)$ and it is straightforward to obtain the unitary evolution~\cite{scully-book}. One finds that the quantum state of the fields oscillates incoherently between $\sum^{m+n}_{j=0}c_j(t) |j\rangle_A |m+n - j\rangle_B$ and $\sum^{m+n-1}_{j=0}d_j(t) |j\rangle_A |m+n-1-j\rangle_B$. Both of these states are superpositions of essentially $m+n$ bi-orthogonal terms giving rise to high entanglement and, therefore, also other forms of correlations. Figure~\ref{FIG_dynamics} illustrates that different correlation quantifiers have different detection capabilities and it is not clear at this stage whether there is a universal measure with which non-commutativity is detected, e.g., the fastest. For most initial states we studied mutual information detected non-commutativity the most rapidly, but there are exceptions, as shown by the black curve corresponding to the initial state $| 1 0 1\rangle$. With this initial state the mutual information never violates its bound, but the negativity does. \section{Discussion} Let us present the origin of the violation just observed. Since the total Hamiltonian is of the form $H_{AC} + H_{BC}$, the Suzuki-Trotter expansion of the resulting evolution is particularly illuminating, \begin{equation}\label{EQ_trotterexp} e^{i t (H_{AC} + H_{BC})} = \lim_{n \to \infty} \left( e^{i\Delta t H_{BC}} e^{i\Delta t H_{AC}} \right)^n, \end{equation} where $\Delta t = t / n$. If Hamiltonians do not commute, it is necessary to think about Eq. (\ref{EQ_trotterexp}) as $n$ sequences of pairwise interactions of $C$ with $A$ followed by $C$ with $B$, each for a time $\Delta t$. Each pair of interactions can only increase correlations up to the correlation capacity of the mediator, but their multiple use allows the accumulation of correlations beyond what is possible with commuting Hamiltonians. Recall that, in the latter case, we deal with only one exchange of system $C$, independently of the duration of dynamics. We stress that Trotterization is just a mathematical tool and in the laboratory system $C$ is continuously coupled to $A$ and $B$. It is rather as if a virtual particle $C$ were transmitted multiple times between $A$ and $B$, interacting with each of them for a time $\Delta t$. Our results imply that the non-commutativity (non-decomposability in general) is a desired feature of interactions in the task of correlation distribution, which is important for quantum information applications. As a contrasting physical illustration, we consider the strong dipole-dipole interactions in our field-atom-field example. The Hamiltonian reads \begin{equation} H^{\prime}=g (\hat a+\hat a^{\dagger}) (\hat \sigma_+ + \hat \sigma_-)+g (\hat b+\hat b^{\dagger})(\hat \sigma_+ + \hat \sigma_-), \label{EQ_DD} \end{equation} with commuting components, i.e., $[H_{AC},H_{BC}]=0$. One can verify that the results of this model are in agreement with all the bounds we derived. Furthermore, we prove in Appendix C that, with this coupling, the state of $AB$ at time $t$ is effectively given by a two-qubit separable state. This makes $N_{A:B}(t)=0$ and $I_{A:B}(t)\le 1$. Note the counter-intuitive result that strong interactions produce bounded correlations between the probes, while weak interactions (Jaynes-Cummings coupling) can increase the correlations above the bounds. We also note an application of our bounds to estimate the dimension of the mediator; see, e.g., Refs.~\cite{dimwit1,dimwit2,dimwit3} for other dimension witnesses. For decomposable evolution (including discrete sequential operators considered in Refs.~\cite{cubitt2003,streltsov2012,chuan2012,edssexp1,edssexp2,edssexp3}), the amount of correlation between the probes is bounded by the correlation capacity $\sup_{\ket{\psi}} Q_{A:C}$, which is a function of $d_C$. If one observes a $Q_{A:B}(t)$ value that is larger than the correlation capacity of a certain $d_C$, then the dimension of the mediator must be larger than $d_C$. Finally, we wish to discuss a scenario where the three systems are open to their own \emph{local} environments, as realized, e.g., in~\cite{hamsen}. We take the evolution following the master equation in Lindblad form, \begin{eqnarray} \dot \rho & = & -i[H_{AC}+H_{BC},\rho]+\sum_{X=A,B,C}L_X\rho, \label{EQ_open}\\ L_X\rho & = & \sum_k Q^X_k\rho Q^{X\dag}_k-\frac{1}{2}\{Q^{X\dag}_kQ^X_k,\rho\}, \nonumber \end{eqnarray} where the last term in (\ref{EQ_open}) is the incoherent part of the evolution and $L_X$ describes interactions of system $X$ with its local environment, i.e. the operators $Q^X_k$ act on system $X$ only. We denote $\mathcal{L}_{AC}=-i[H_{AC},\cdot]+L_A+L_C$ and $\mathcal{L}_{BC}=-i[H_{BC},\cdot]+L_B$. One readily verifies that if $[H_{AC},H_{BC}]=0$ and $[L_C,H_{BC}]=0$, we have commuting Lindblad operators $[\mathcal{L}_{AC},\mathcal{L}_{BC}]=0$. Note that, if one includes $L_C$ in $\mathcal{L}_{BC}$ instead, the second condition for commuting Lindblad operators now reads $[L_C,H_{AC}]=0$. For commuting Lindbladians, the corresponding evolution decomposes as $\Lambda_{BC} \Lambda_{AC}$, or in reverse order. Therefore, our bounds apply accordingly. Their violation implies that either the Hamiltonians do not commute or the operators describing dissipative channels on $C$ do not commute with $H_{AC}$ and $H_{BC}$. In particular, if $C$ is kept isolated so that its noise can be ignored, the violation of our bounds is solely the result of the non-commutativity of the Hamiltonians. \section{Conclusions} We linked non-commutativity of interaction Hamiltonians (non-decomposability of time evolution in general) to the amount of correlations that can be created in the associated dynamics. This led us to a method for detection of non-decomposability of evolution in a scenario where subsystem $C$ mediates interactions between $A$ and $B$ (all these objects can interact with their local environments). The method requires no explicit form of the evolution operators or knowledge of the initial state of the tripartite system. Non-decomposability is detected by observing violation of certain bounds on $AB$ correlations, as measured by most correlation quantifiers. Furthermore, no operation on $C$ is necessary at any time, which makes this strategy experimentally friendly. In particular, in addition to avoiding characterization of the interactions, the physics of $C$ can remain largely unknown---only its dimension should be identified.\\ \section*{Acknowledgments} We thank Alexander Streltsov and Kavan Modi for insightful discussions, and Matthew Lake for comments on the manuscript. S.-Y. L. would like to thank Changsuk Noh for useful comments. T. K. and T. P. thank Wies{\l}aw Laskowski for hospitality at the University of Gda{\'n}sk. This work is supported by Singapore Ministry of Education Academic Research Fund Tier 2 Project No. MOE2015-T2-2-034. \appendix \section{$\mbox{Proof of Theorem 1}$} \label{APP_TH_BOUND} For completeness let us begin with a useful lemma. \setcounter{theorem}{0} \begin{lemma}\label{LM_pre} For a measure of correlations $Q_{X:Y}$ between party $X$ and party $Y$ that is non-increasing under local operations on $Y$, the following property holds: $Q_{X:Y}$ is invariant under tracing-out of uncorrelated systems on the side of $Y$. \end{lemma} \begin{proof} Since the correlation measure is non-increasing under local operations on $Y$, tracing out an uncorrelated system on the side of $Y$ can only decrease the correlation. However, if the correlation is strictly decreasing, then there is a reverse process, i.e., attaching the uncorrelated system back and, therefore, increasing the correlation. Hence, the correlation $Q_{X:Y}$ has to be invariant under tracing-out of uncorrelated systems on $Y$. In fact, this is true for all reversible operations. \end{proof} Our main theorem is proven as follows. \setcounter{theorem}{0} \begin{theorem}\label{TH_cons} Consider a correlation measure $Q_{X:Y} \equiv \inf_{\sigma_{XY}\in \mathcal{S}} D(\rho_{XY},\sigma_{XY})$ satisfying the following properties: \begin{enumerate} \item[{\rm (i)}] $\mathcal{S}$ is closed under local operations $\Lambda_Y$ on $Y$; \item[{\rm (ii)}] $D(\Lambda[\rho],\Lambda[\sigma]) \leq D(\rho, \sigma)$; and \item[{\rm (iii)}] $D(\rho_0, \rho_1) \leq D(\rho_0, \rho_2) + D(\rho_2, \rho_1)$. \end{enumerate} If the evolution operator $\Lambda_{ABC}$ is decomposable into $\Lambda_{BC}\Lambda_{AC}$, then \begin{equation} Q_{A:B} (t) \leq I_{AC:B}(0) + \sup_{\ket{\psi}} Q_{A:C}, \end{equation} where $I_{AC:B}(0) = \inf_{\sigma_{AC} \otimes \sigma_B} D(\rho, \sigma_{AC} \otimes \sigma_B)$, $\rho$ is the initial tripartite state, and the supremum of $Q_{A:C}$ is taken over pure states of $AC$. \end{theorem} \begin{proof} Properties (i) and (ii), and the definition of $Q_{X:Y}$ as the shortest distance, imply that $Q_{X:Y}$ is nonincreasing under local operations on $Y$. Accordingly, the property proven in Lemma~\ref{LM_pre} applies. We have \begin{eqnarray} Q_{A:B} (t) &\leq& Q_{A:BC} \left(\Lambda_{BC} \Lambda_{AC} [\rho] \right) \label{APP_TH_S1} \\ &\le& Q_{A:BC} \left( \Lambda_{AC} [\rho] \right) \label{APP_TH_S2} \\ & \le & D \left(\Lambda_{AC} [\rho], \mu \right) \label{APP_TH_S3} \\ & \leq & D \left( \Lambda_{AC} [\rho], \Lambda_{AC} [\sigma^0_{AC}] \otimes \sigma^0_B \right) \nonumber \\ & + & D\left( \Lambda_{AC} [\sigma^0_{AC}] \otimes \sigma^0_{B}, \mu \right) \label{APP_TH_TRIAN}\\ & \le & D(\rho, \sigma^0_{AC} \otimes \sigma^0_B) \nonumber \\ &+ & D\left( \Lambda_{AC} [\sigma^0_{AC}] \otimes \sigma^0_{B}, \mu \right) \label{APP_TH_S4} \\ &=& I_{AC:B}(0) + Q_{A:BC} (\Lambda_{AC} [\sigma^0_{AC}] \otimes \sigma^0_{B} ) \label{APP_TH_S5} \\ &=& I_{AC:B}(0) + Q_{A:C} (\Lambda_{AC} [\sigma^0_{AC}] ) \label{APP_TH_S6} \\ &\leq& I_{AC:B}(0) +\sup_{\ket{\psi}} Q_{A:C}, \label{APP_TH_S7} \end{eqnarray} where the steps are justified as follows. In line (\ref{APP_TH_S1}) we have used the fact that $Q_{X:Y}$ is nonincreasing under local operations on $Y$ (tracing out $C$). Line (\ref{APP_TH_S2}) follows, as $Q_{A:BC}$ is nonincreasing under local operation $\Lambda_{BC}$. The next line, (\ref{APP_TH_S3}), utilizes the definition of $Q_{A:BC}$ as the shortest distance to the set of states $\mu \in \mathcal{S}_{A:BC}$. The inequality of (\ref{APP_TH_TRIAN}) follows from the triangle inequality (iii). Note that the first distance in (\ref{APP_TH_TRIAN}) does not depend on $\mu$ and at this point one can choose any $\sigma^0_{AC}$ and $\sigma^0_B$. The inequality (\ref{APP_TH_S4}) invokes property (ii). In (\ref{APP_TH_S5}), we have chosen $\sigma^0_{AC} \otimes \sigma^0_B$ as the closest product state to $\rho$ and $\mu$ as a state in $\mathcal{S}_{A:BC}$ closest to $\Lambda_{AC} [\sigma^0_{AC}] \otimes \sigma^0_{B}$. Line (\ref{APP_TH_S6}) uses the invariance of $Q_{A:BC}$ under tracing-out of the uncorrelated system $\sigma^0_B$. For the final inequality, we note that a correlation measure that is nonincreasing under local operations on at least one side must be maximal on pure states~\cite{streltsov2012general}. \end{proof} \setcounter{theorem}{1} \begin{lemma}\label{LM_re} The conclusion in Theorem \ref{TH_cons} still follows for the relative entropy as a distance measure. \end{lemma} \begin{proof} Let us begin with an identity, \begin{eqnarray} S(\rho||\sigma_X\otimes \sigma_Y)&=&\mbox{tr}(\rho \log{\rho}-\rho\log{\sigma_X \otimes \sigma_Y}) \nonumber \\ &=& \mbox{tr}(\rho \log{\rho}-\rho \log{\rho_X \otimes \rho_Y}) \nonumber \\ &&+\mbox{tr}(\rho \log{\rho_X \otimes \rho_Y} - \rho \log{\sigma_X \otimes \sigma_Y}) \nonumber \\ &=&S(\rho||\rho_X\otimes \rho_Y)+S(\rho_X||\sigma_X) \nonumber \\ &&+S(\rho_Y||\sigma_Y), \label{GG0} \end{eqnarray} where $\rho_X$ and $\rho_Y$ are the marginals of $\rho$ and we have used, for example, relation $\mbox{tr}(\rho \log{\sigma_X \otimes \sigma_Y}) = \mbox{tr}(\rho_X \log{\sigma_X}) + \mbox{tr}(\rho_Y \log{\sigma_Y})$. Although relative entropy satisfies (ii) \cite{uhlmann1977relative}, it is well known not to follow (iii). Therefore, starting from (\ref{APP_TH_S2}), we have \begin{eqnarray} &&Q_{A:BC} \left( \Lambda_{AC} [\rho] \right) \nonumber \\ &=&\inf_{\mu \in \mathcal{S}_{A:BC}} S \left(\Lambda_{AC}[\rho] || \mu \right) \label{GG01}\\ &\le& S(\Lambda_{AC}[\rho] || \mu_{AC}\otimes \mu_B) \label{GG1}\\ &=& S(\Lambda_{AC}[\rho] || \rho_{AC}^{\prime}\otimes \rho_B^{\prime}) \nonumber \\ &&+S(\rho_{AC}^{\prime} || \mu_{AC})+S(\rho_{B}^{\prime} || \mu_{B}) \label{GG2} \\ &=&I_{AC:B}(\Lambda_{AC}[\rho] )+Q_{A:C}(\rho_{AC}^{\prime}) \label{GG3} \\ &\le&I_{AC:B}(0)+\sup_{\ket{\psi}} Q_{A:C}, \end{eqnarray} where $\rho_{AC}^{\prime}$ and $\rho_B^{\prime}$ are marginals of $\Lambda_{AC}[\rho]$. The steps above are justified as follows. Line (\ref{GG1}) follows for any state of the form $\mu_{AC}\otimes \mu_B\in \mathcal{S}_{A:BC}$. We have used identity (\ref{GG0}) in line (\ref{GG2}). The equality (\ref{GG3}) uses the definition of mutual information as the relative entropy from a state to its marginals \cite{modi2010}. We have also chosen $\mu_{AC}$ as a state in $\mathcal{S}_{A:C}$ closest to $\rho_{AC}^{\prime}$ and $\mu_B=\rho_B^{\prime}$. The last line follows as mutual information is non-increasing under local operation $\Lambda_{AC}$ and the correlation $Q_{A:C}$ achieves the supremum on pure states. \end{proof} \section{$\mbox{Proof of Eq. (2) for correlations only}$\\ $\mbox{monotonic under local operations}\:\Lambda_{BC}$} \label{APP_NOND} \setcounter{theorem}{1} \begin{theorem}\label{TH_main} Suppose the initial state has the form $\rho = \rho_{AC} \otimes \rho_B$. If the evolution operator is decomposable into $\Lambda_{BC}\Lambda_{AC}$, then \begin{equation} Q_{A:B}(t)\le \sup_{\ket{\psi}} \:Q_{A:C} \end{equation} for all correlation measures, $Q$, non-increasing under local operations $\Lambda_{BC}$. \end{theorem} \begin{proof} For initial states of the form $\rho_{AC}\otimes \rho_B$ we have the following chain of arguments \begin{eqnarray} Q_{A:B}(t) & \le & Q_{A:BC}(t)\le Q_{A:BC}(\Lambda_{AC}[\rho]) \nonumber \\ &=& Q_{A:C}(\Lambda_{AC}[\rho]) \le \sup_{\ket{\psi}} Q_{A:C}, \end{eqnarray} where the steps are justified as follows. Since the action of tracing out (the, in general, correlated) system $C$ is a local operation on $BC$, we obtain the first inequality. The second inequality follows as the correlation is non-increasing under $\Lambda_{BC}$. As we start with the initial state $\rho_{AC}\otimes \rho_B$ and $\Lambda_{AC}$ does not act on $B$, system $B$ stays uncorrelated in $\Lambda_{AC}[\rho]$. Using Lemma \ref{LM_pre}, we have the equality. Finally, the correlation $Q_{A:C}$ is again maximal on pure states. \end{proof} \section{$\mbox{Proof of separability via dipole-dipole}$\\ $\mbox{coupling for particular initial states}$} \label{APP_N} Let us define $\hat{\xi} = (\hat a +\hat b )/\sqrt{2}$. The dipole-dipole Hamiltonian, Eq. (6), is reformulated as $H^{\prime}=\sqrt{2}g(\hat \xi +\hat \xi^{\dagger})\hat \sigma_x$, where $\hat \sigma_x=\hat \sigma_+ +\hat \sigma_-$ and $[\hat \xi, \hat \xi^{\dagger}]=\openone$. The unitary evolution operator is given by \begin{eqnarray}\label{EQ_hddunitary} \hat U_t&=&e^{-iH^{\prime}t} \\ &=&\frac{1}{2} [(\openone-\hat \sigma_x)e^{i\sqrt{2}gt(\hat \xi + \hat \xi^{\dagger})}+(\openone+\hat \sigma_x)e^{-i\sqrt{2}gt(\hat \xi + \hat \xi^{\dagger})}]\nonumber \\ &=&\frac{1}{2} [(\openone-\hat \sigma_x)\hat D_a(\alpha)\hat D_b(\alpha) +(\openone+\hat \sigma_x)\hat D_a(-\alpha)\hat D_b(-\alpha)], \nonumber \end{eqnarray} where $\alpha=igt$ and, e.g., $\hat D_a (\alpha)=\exp(\alpha\hat a^{\dagger}-\alpha^{\ast}\hat a)$. Given an initial state $|mn0\rangle$, the state at time $t$ reads \begin{eqnarray} | \psi_t \rangle & = & \frac{1}{4} [ (d^{(mn)}_{++}|D^{(m)}_+,D^{(n)}_+\rangle + d^{(mn)}_{--}|D^{(m)}_-,D^{(n)}_-\rangle )|0\rangle \nonumber \\ & - & (d^{(mn)}_{+-}|D^{(m)}_+,D^{(n)}_-\rangle+d^{(mn)}_{-+}|D^{(m)}_-,D^{(n)}_+\rangle)|1\rangle], \nonumber \end{eqnarray} where \begin{eqnarray} d^{(mn)}_{\pm\pm} & = & 2\sqrt{[1\pm e^{-2|\alpha|^2} L_m(4|\alpha|^2)][1\pm e^{-2|\alpha|^2} L_n(4|\alpha|^2)]},\nonumber \\ |D^{(n)}_{\pm}\rangle & = & \frac{1}{\sqrt{d^{(nn)}_{\pm\pm}}}[\hat{D}(\alpha)\pm\hat{D}(-\alpha)]|n\rangle . \nonumber \end{eqnarray} Note that $\langle D^{(n)}_+|D^{(n)}_-\rangle=0$ and $\langle D^{(n)}_{\pm}|D^{(n)}_{\pm}\rangle=1$. $L_n(|\alpha|^2)$ is the Laguerre polynomial, which comes from the relation $\langle n|\hat{D}(\alpha)|n\rangle=e^{-|\alpha|^2/2}L_n(|\alpha|^2)$. After tracing-out of the atomic mode $C$, the state of the fields is effectively given by a two-qubit state, \begin{eqnarray} \frac{1}{16} \begin{pmatrix} (d^{(mn)}_{++})^2 & 0& 0& d^{(mn)}_{++}d^{(mn)}_{--} \\ 0 & (d^{(mn)}_{+-})^2 & d^{(mn)}_{++}d^{(mn)}_{--} & 0 \\ 0 & d^{(mn)}_{++}d^{(mn)}_{--} & (d^{(mn)}_{-+})^2 & 0 \\ d^{(mn)}_{++}d^{(mn)}_{--} & 0 & 0 & (d^{(mn)}_{--})^2 \nonumber \end{pmatrix},\\ \end{eqnarray} which is positive under partial transposition and, hence, separable \cite{peres1996,horodecki1996m}. The same result follows for initial state $\ket{mn1}$. \end{document}

\begin{document} \newcommand{\diam} {\operatorname{diam}} \newcommand{\Scal} {\operatorname{Scal}} \newcommand{\scal} {\operatorname{scal}} \newcommand{\mathbb{R}ic} {\operatorname{Ric}} \newcommand{\Hess} {\operatorname{Hess}} \newcommand{\grad} {\operatorname{grad}} \newcommand{\mathbb{R}m} {\operatorname{Rm}} \newcommand{\mathbb{R}c} {\operatorname{Rc}} \newcommand{\mathbb{C}urv} {S_{B}^{2}\left( \mathfrak{so}(n) \right) } \newcommand{ \tr } {\operatorname{tr}} \newcommand{ \id } {\operatorname{id}} \newcommand{ \mathbb{R}iczero } {\stackrel{\circ}{\mathbb{R}ic}} \newcommand{ \ad } {\operatorname{ad}} \newcommand{ \Ad } {\operatorname{Ad}} \newcommand{ \dist } {\operatorname{dist}} \newcommand{ \rank } {\operatorname{rank}} \newcommand{\operatorname{Vol}}{\operatorname{Vol}} \newcommand{\operatorname{dVol}}{\operatorname{dVol}} \newcommand{ \zitieren }[1]{ \hspace{-3mm} \cite{#1}} \newcommand{ \pr }{\operatorname{pr}} \newcommand{\operatorname{diag}}{\operatorname{diag}} \newcommand{\mathcal{L}}{\mathcal{L}} \newcommand{\operatorname{av}}{\operatorname{av}} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{acknowledgment}[theorem]{Acknowledgment} \newtheorem{DefAndLemma}[theorem]{Definition and lemma} \newtheorem{questionroman}[theorem]{Question} \newenvironment{remarkroman}{\begin{remark} \normalfont }{\end{remark}} \newenvironment{exampleroman}{\begin{example} \normalfont }{\end{example}} \newenvironment{question}{\begin{questionroman} \normalfont }{\end{questionroman}} \renewcommand{(\alph{enumi})}{(\alph{enumi})} \newtheorem{maintheorem}{Theorem}[] \renewcommand*{\themaintheorem}{\Alph{maintheorem}} \newtheorem*{theorem*}{Theorem} \newtheorem*{corollary*}{Corollary} \newtheorem*{remark*}{Remark} \newtheorem*{example*}{Example} \newtheorem*{question*}{Question} \newcommand{\mathbb{R}}{\mathbb{R}} \newcommand{\mathbb{N}}{\mathbb{N}} \newcommand{\mathbb{Z}}{\mathbb{Z}} \newcommand{\mathbb{Q}}{\mathbb{Q}} \newcommand{\mathbb{C}}{\mathbb{C}} \newcommand{\mathbb{F}}{\mathbb{F}} \newcommand{\mathcal{X}}{\mathcal{X}} \newcommand{\mathcal{D}}{\mathcal{D}} \newcommand{\mathbb{C}ont}{\mathcal{C}} \begin{abstract} This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $\Ad_K$-invariant irreducible summands, the existence of continuous families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano-Dancer-Gallaugher-Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to B\"ohm. The techniques also apply to $m$-quasi-Einstein metrics. \end{abstract} \maketitle \section*{Introduction} A Riemannian manifold $(M,g)$ is called {\em Ricci soliton} if there exists a smooth vector field $X$ on $M$ and a real number $\varepsilon \in \mathbb{R}$ such that \begin{equation*} \mathbb{R}ic + \frac{1}{2} L_X g + \frac{\varepsilon}{2} g = 0, \end{equation*} where $L_X g$ denotes the Lie derivative of the metric $g$ with respect to $X.$ Ricci solitons are generalisations of Einstein manifolds and will be called {\em non-trivial} if $X$ is not a Killing vector field. If $X$ is the gradient of a smooth function $u \colon M \to \mathbb{R}$ then it is called a {\em gradient} Ricci soliton. It is called {\em shrinking}, {\em steady} or {\em expanding} depending on whether $\varepsilon<0,$ $\varepsilon=0$ or $\varepsilon>0.$ Ricci solitons were introduced by Hamilton \cite{HamiltonRFonSurfaces} as self-similar solutions to the Ricci flow and play an important role in its singularity analysis. This paper studies the Ricci soliton equation under the assumption of a large symmetry group. For example, Lauret \cite{LauretHomogeneousRS} has constructed non-gradient, homogeneous expanding Ricci solitons. However, Petersen-Wylie \cite{PWRigidityWithSymmetry} have shown that any homogeneous gradient Ricci soliton is rigid, i.e. it is isometric to a quotient of $N \times \mathbb{R}^k,$ where $N$ is Einstein with Einstein constant $\lambda$ and $\mathbb{R}^k$ is equipped with the Euclidean metric and soliton potential $\frac{\lambda}{2} |x|^2.$ Therefore it is natural to assume that the Ricci soliton is of {\em cohomogeneity one.} That is, a Lie group acts isometrically on $(M,g)$ and the generic orbit is of codimension one. This will be the setting of this paper. A systematic investigation was initiated by Dancer-Wang \cite{DWCohomOneSolitons} who set up the general framework. Previous examples include the first non-trivial compact Ricci soliton due to Cao \cite{CaoSoliton} and Koiso \cite{KoisoSoliton} or the examples of Feldman-Ilmanen-Knopf \cite{FIKSolitons}, which include the first non-compact shrinking Ricci soliton. It is worth noting that all of these examples, as well as their generalisations due to Dancer-Wang \cite{DWCohomOneSolitons}, are {\em K\"ahler.} In fact, all currently known non-trivial compact Ricci solitons are K\"ahler. On the other hand, Angenent-Knopf \cite{AngenentKnopfRSConicalSingNonuniqueness} constructed non-compact, non-K\"ahler shrinking Ricci solitons. Hamilton's cigar is also K\"ahler, whereas its higher dimensional analogue, the rotationally symmetric steady soliton on $\mathbb{R}^n$, $n>2,$ the Bryant soliton, is {\em non}-K\"ahler. By extending these examples in a series of papers and then in joint work with Buzano and Gallaugher, Dancer-Wang constructed steady and expanding Ricci solitons of multiple warped product type \cite{DWExpandingSolitons, DWSteadySolitons}, \cite{BDGWExpandingSolitons}, \cite{BDWSteadySolitons}. They also numerically investigated the case where the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $\Ad_K$-invariant irreducible real summands and found numerical evidence for the existence of continuous families of complete steady and expanding Ricci solitons on certain non-trivial vector bundles in \cite{BDGWExpandingSolitons, BDGWSteadySolitons}. This paper gives a rigorous construction thereof: Let $G$ be a compact Lie group and let $K \subset H \subset G$ be closed subgroups such that $H/K=S^{d_S}$. Then $H$ acts linearly on $\mathbb{R}^{d_S+1}$ and the associated vector bundle $G \times_H \mathbb{R}^{d_S+1}$ is a cohomogeneity one manifold. Examples where the Lie algebra of $G/K$ decomposes into two inequivalent $\Ad_K$-invariant irreducible real summands include the triples \begin{align} (G,H,K) & = (Sp(1) \times Sp(m+1), Sp(1) \times Sp(1) \times Sp(m), Sp(1) \times Sp(m)), \nonumber \\ (G,H,K) & = (Sp(m+1), Sp(1) \times Sp(m), U(1) \times Sp(m)), \label{GroupDiagrams} \\ (G,H,K) & = (Spin(9), Spin(8), Spin(7)). \nonumber \end{align} These examples come from the Hopf fibrations, cf. \cite{BesseEinstein}. In the first and third case, the associated vector bundle is diffeomorphic to $\mathbb{H}P^{m+1} \setminus \left\{ \text{ point } \right\}$ and $CaP^2 \setminus \left\{ \text{ point } \right\},$ respectively. The main theorem is the following: \begin{maintheorem} On $CaP^2 \setminus \left\{ \text{ point } \right\},$ $\mathbb{H}P^{m+1} \setminus \left\{ \text{ point } \right\}$ for $m \geq 1$ and on the vector bundle associated to $(G,H,K) = (Sp(m+1), Sp(1) \times Sp(m), U(1) \times Sp(m))$ for $m \geq 3,$ there exist a $1$-parameter family of non-homothetic complete steady and a $2$-parameter family of non-homothetic complete expanding Ricci solitons. The steady Ricci solitons are asymptotically paraboloid and thus non-collapsed. The expanding Ricci solitons are asymptotically conical. \label{MainTheoremTwoSummands} \end{maintheorem} Notice that non-trivial gradient steady and expanding Ricci solitons must be non-compact. Furthermore, due to Perelman's \cite{Perelman1} no-local collapsing theorem, blow up limits of finite time Ricci flow singularities are necessarily non-collapsed. The construction of the Ricci solitons in Theorem \ref{MainTheoremTwoSummands} partially carries over to the case of complex line bundles over Fano K\"ahler-Einstein manifolds, where Cao \cite{CaoSoliton} and Feldman-Ilmanen-Knopf \cite{FIKSolitons} previously constructed {\em K\"ahler} Ricci solitons. In contrast, Theorem \ref{MainTheoremRSOnLineBundles} exhibits continuous families of complete {\em non}-K\"ahler steady and expanding Ricci solitons. \begin{maintheorem} Let $(V,J,g)$ be a Fano K\"ahler-Einstein manifold of real dimension $d$. Suppose that the first Chern class is given by $c_1(V,J) = p \rho$ for an indivisible class $\rho \in H^2(V,J)$ and $\mathbb{R}ic_g = pg.$ For $q \in \mathbb{Z}$ let $\pi \colon P_q \to V$ be the principal circle bundle with Euler class $q \pi^{*} \rho$ and let $L_q$ be the total space of the associated complex line bundle. If $2p^2 > (d+2)q^2>0$ there exist a $1$-parameter family of non-homothetic complete steady Ricci solitons and a $2$-parameter family of non-homothetic complete expanding Ricci solitons on $L_q.$ In particular there exist non-K\"ahler Ricci solitons on $L_q.$ \label{MainTheoremRSOnLineBundles} \end{maintheorem} In the steady case these Ricci solitons were independently discovered by Stolarski \cite{StolarskiSteadyRSOnCxLineBundles} and Appleton \cite{AppletonSteadyRS}, who use different techniques. The proof of Theorem \ref{MainTheoremTwoSummands} establishes that the Ricci soliton metrics correspond to trajectories in a bounded region of a phase space, which implies completeness. This method also applies to Einstein metrics. In particular, in the situation of Theorem \ref{MainTheoremTwoSummands}, the methods of this paper provide an alternative construction of Ricci flat metrics and Einstein metrics with negative scalar curvature due to B\"ohm \cite{BohmNonCompactEinstein}, see also remark \ref{RemarkBoehmSetUpAndProofCompleteness}. The associated coordinate change moreover allows good control on the trajectories close to the singular orbit. In the Einstein case this also yields an alternative approach to the following result of B\"ohm \cite{BohmInhomEinstein, BohmNonCompactEinstein}: The two summands Einstein metrics converge to explicit solutions with conical singularities as the volume of the singular orbit tends to zero. In comparison to B\"ohm's work, the main technical simplification is that the methods of this paper do not use the Poincar\'e-Bendixson theorem, see remark \ref{RemarkConvergenceConeSolutions}. As an application, an analysis of the {\em Ricci flat} trajectories will be used to reconstruct Einstein metrics of positive scalar curvature due to B\"ohm \cite{BohmInhomEinstein}. The vector bundles associated to the two families of group diagrams in \eqref{GroupDiagrams} also admit {\em explicit} Ricci flat metrics in the lowest dimensional case $m=1.$ These are in fact of special holonomy $G_2$ and $Spin(7),$ respectively, and were discovered earlier by Bryant-Salamon \cite{BSExceptionalHolonomy} and Gibbons-Page-Pope \cite{GPPEinsteinOnSphereR3R4bundles}. However, it is worth noting that these metrics correspond to {\em linear} trajectories in the above phase space, see theorem \ref{ExplicitRFTrajectories}. The techniques in this paper moreover apply if the Bakry-\'Emery Ricci tensor $\mathbb{R}ic + \Hess u$ is replaced with the more general version $\mathbb{R}ic - \Hess u - \frac{1}{m} du \otimes du.$ For any $m \in (0, \infty]$ this leads to the notion of $m$-quasi-Einstein metrics, i.e. Riemannian manifolds which satisfy the curvature condition \begin{equation*} \mathbb{R}ic + \Hess u - \frac{1}{m} du \otimes du + \frac{\varepsilon}{2} g = 0 \end{equation*} for $u \in C^{\infty}(M)$ and $\varepsilon \in \mathbb{R}.$ These metrics play an important role in the study of Einstein warped products, cf. \cite{CaseSMMSAndQEM} or \cite{HPWUniquenessWarpedProductEinstein} and references therein. The initial value problem for cohomogeneity one $m$-quasi-Einstein manifolds will be discussed in the spirit of Eschenburg-Wang \cite{EWInitialValueEinstein} and Buzano \cite{BuzanoInitialValueSolitons}, see theorem \ref{QEMInitialValueTheorem}, and the $m$-quasi-Einstein analogue of Theorem \ref{MainTheoremTwoSummands} is proven in theorem \ref{TwoSummandsQEM}. Furthermore, the setting of $m$-quasi Einstein metrics allows a unified proof of the existence of Einstein metrics and Ricci soliton metrics on $\mathbb{R}^{d_1+1} \times M_2 \times \ldots \times M_r,$ for $d_1 \geq 1,$ where $(M_i, g_i)$ are Einstein manifolds with positive scalar curvature. This summaries earlier work due to B\"ohm \cite{BohmNonCompactEinstein}, Dancer-Wang \cite{DWSteadySolitons, DWExpandingSolitons} for $d_1 > 1$ and Buzano-Dancer-Gallaugher-Wang \cite{BDGWExpandingSolitons, BDWSteadySolitons} for $d_1 = 1:$ \begin{maintheorem} Let $M_2, \ldots, M_r$ be Einstein manifolds with positive scalar curvature and let $d_1 \geq 1$ and $m \in (0, \infty].$ Then there is an $(r -1)$-parameter family of non-trivial, non-homothetic, complete, smooth Bakry-\'Emery flat $m$-quasi-Einstein metrics and an $r$-parameter family of non-trivial, non-homothetic, complete, smooth $m$-quasi-Einstein metrics with quasi-Einstein constant $\frac{\varepsilon}{2} > 0$ on $\mathbb{R}^{d_1+1} \times M_2 \times \ldots \times M_r.$ \label{MainTheoremQEM} \end{maintheorem} \textit{Structure of the paper.} Section \ref{CohomOneRicciSolitonEQ} reviews the Ricci soliton equation on cohomogeneity one manifolds and recalls some structure theorems. Section \ref{SectionNewSolitons} focuses on the two summands case, with section \ref{SectionSolitonsFromCircleBundles} discussing the case of complex line bundles over Fano K\"ahler-Einstein manifolds. Completeness of the metrics in Theorem \ref{MainTheoremTwoSummands} is shown in section \ref{CompletenessTwoSummands} and the asymptotic behaviour is studied in section \ref{SectionTwoSummandsAsymptotics}. Applications to convergence to cone solutions and B\"ohm's Einstein metrics of positive scalar curvature follow in sections \ref{SectionConvergenceToConeSolutions} and \ref{SectionBohmEinsteinMetricsPosScal}, respectively. Finally, section \ref{SectionQuasiEinsteinMetrics} discusses $m$-quasi-Einstein metrics and the proof of Theorem \ref{MainTheoremQEM}. \textit{Acknowledgements.} I wish to thank my PhD advisor Prof. Andrew Dancer for constant support, helpful comments and numerous discussions. \section{The cohomogeneity one Ricci soliton equation} \label{CohomOneRicciSolitonEQ} \subsection{The general set-up} \label{SectionCohomOneSetUp} The general framework for cohomogeneity one Ricci solitons has been set up by Dancer-Wang \cite{DWCohomOneSolitons}: Let $(M,g)$ be a Riemannian manifold and let $G$ be a compact connected Lie group which acts isometrically on $(M,g).$ The action is of {\em cohomogeneity one} if the orbit space $M / G$ is one-dimensional. In this case, choose a unit speed geodesic $\gamma \colon I \to M$ that intersects all principal orbits perpendicularly. Let $K=G_{\gamma(t)}$ denote the principal isotropy group. Then $\Phi \colon I \times G/K \to M_0,$ $(t,gK) \mapsto g \cdot \gamma(t)$ is a $G$-equivariant diffeomorphism onto an open dense subset $M_0$ of $M$ and the pullback metric is of the form $\Phi^{*}g=dt^2 + g_t,$ where $g_t$ is a $1$-parameter family of metrics on the principal orbit $P=G/K.$ Let $N = \Phi_*( \frac{\partial}{\partial t})$ be a unit normal vector field and let $L_t = \nabla N$ denote the shape operator of the hypersurface $\Phi(\left\lbrace t\right\rbrace \times P).$ Via $\Phi,$ $L_t$ can be regarded as a one-parameter family of $G$-equivariant, $g_t$-symmetric endomorphisms of $TP$ which satisfies $\dot{g}_t = 2 g_t L_t.$ Similarly, let $\mathbb{R}ic_t$ be the Ricci curvature corresponding to $g_t.$ According to Eschenburg-Wang \cite{EWInitialValueEinstein} the Ricci curvature of the cohomogeneity one manifold $(M,g)$ is given by \begin{align*} \mathbb{R}ic(X,N) & = - g_t(\delta^{\nabla^t} L_t, X) - d ( \tr(L_t) ) (X), \\ \mathbb{R}ic(N,N) & = -\tr(\dot{L})-\tr(L_t^2), \\ \mathbb{R}ic(X,Y) & = -g_t(\dot{L}(X),Y) - \tr(L_t)g_t(L_t(X),Y) + \mathbb{R}ic_t(X,Y), \end{align*} where $X, Y \in TP,$ $ \delta^{\nabla^t} \colon T^*P \otimes TP \to TP$ is the codifferential, and $L_t$ is regarded as a $TP$-valued $1$-form on $TP.$ Dancer-Wang \cite{DWCohomOneSolitons} observed that, since $G$ is compact, any cohomogeneity one Ricci soliton induces a Ricci soliton with a $G$-invariant vector field. Hence, in the case of gradient Ricci solitons, the soliton potential can be assumed to be $G$-invariant. The gradient Ricci soliton equation $\mathbb{R}ic + \Hess u + \frac{\varepsilon}{2} g= 0$ then takes the form \begin{align} -( \delta^{\nabla^t}L_t)^{\flat} - d(\tr(L_t)) & = 0, \label{CohomOneRSa}\\ - \tr( \dot{L}_t) - \tr(L_t^2) + \ddot{u} + \frac{\varepsilon}{2} & =0, \label{CohomOneRSb}\\ - \dot{L}_t - (- \dot{u} + \tr(L_t)) L_t + r_t + \frac{\varepsilon}{2} \mathbb{I} & = 0, \label{CohomOneRSc} \end{align} where $r_t = g_t \circ \mathbb{R}ic_t$ is the Ricci endomorphism, i.e. $g_t(r_t(X),Y)=\mathbb{R}ic_t(X,Y)$ for all $X,Y \in TP.$ Conversely, the above system induces a gradient Ricci soliton on $I \times P$ provided that the metric $g_t$ is defined via $\dot{g}_t = 2 g_t L_t.$ The special case of constant $u$ recovers the cohomogeneity one Einstein equations. From now on, for simplicity, the $t$-dependence may not be stated explicitly. It is an immediate consequence of \eqref{CohomOneRSb} that the mean curvature with respect to the volume element $e^{-u}d \operatorname{Vol}_g$ is a Lyapunov function if $\varepsilon \leq 0.$ \begin{proposition} Fix $\varepsilon \leq 0.$ Then the generalised mean curvature $-\dot{u} + \tr(L)$ is monotonically decreasing along the flow of the cohomogeneity one Ricci soliton equation. \end{proposition} If the Ricci soliton metric is at least $C^3$-regular, then the second Bianchi identity implies that the {\em conservation law} \begin{equation} \ddot{u} + (-\dot{u}+\tr(L)) \dot{u} = C+ \varepsilon u \label{GeneralConservationLaw} \end{equation} has to be satisfied for some constant $C \in \mathbb{R}.$ Using the equations \eqref{CohomOneRSb} and \eqref{CohomOneRSc} it can be reformulated as \begin{equation} \tr (r) + \tr ( L^2 ) - \left( - \dot{u} + \tr \left( L \right) \right) ^2 + (n-1) \frac{\varepsilon}{2} = C +\varepsilon u. \label{ReformulatedGeneralConsLaw} \end{equation} Recall that the scalar curvature $R$ of a cohomogeneity one Riemannian manifold $(M^{n+1},g)$ is given by $R= \tr(r) - \tr(L^2)- \tr(L)^2 - 2 \tr(\dot{L}).$ Hence it follows with \eqref{CohomOneRSc} that the conservation law \eqref{ReformulatedGeneralConsLaw} is just the cohomogeneity one version of Hamilton's \cite{HamiltonSingularites} general identity $R + | \nabla u |^2 + \varepsilon u = \overline{C}$ for gradient Ricci solitons (where $\overline{C}= - C -\frac{n+1}{2} \varepsilon$). This also provides a formula for the scalar curvature in terms of the soliton potential: \begin{equation} R = - C - \varepsilon u - \dot{u}^2 - (n+1) \frac{\varepsilon}{2}. \label{ScalarCurvatureRicciSoliton} \end{equation} \subsection{Ricci solitons with a singular orbit} \label{MetricWithASingularOrbit} From now on, assume that there is a singular orbit $Q = G/H$ at $t=0.$ That is, the orbit at $t=0$ is of dimension strictly less than the dimension of the principal orbit, and let $H=G_{\gamma(0)}$ denote its isotropy group. Building up on an idea of Back \cite{BackLocalTheoryofEquiv}, see also \cite{EWInitialValueEinstein}, Dancer-Wang \cite{DWCohomOneSolitons} have shown that in the presence of a singular orbit, equation \eqref{CohomOneRSc} implies \eqref{CohomOneRSa} automatically, provided that the metric is at least $C^2$-regular and the soliton potential is of class $C^3.$ Moreover, if in this case the conservation law \eqref{GeneralConservationLaw} is satisfied, then equation \eqref{CohomOneRSb} holds as well. Conversely, any trajectory of the Ricci soliton equations \eqref{CohomOneRSb}, \eqref{CohomOneRSc} that describes a $C^3$-regular metric with a singular orbit has to satisfy the conservation law \eqref{ReformulatedGeneralConsLaw}. The initial value problem for gradient cohomogeneity one Ricci solitons has been considered by Buzano \cite{BuzanoInitialValueSolitons}. Extending Eschenburg-Wang's work \cite{EWInitialValueEinstein} in the Einstein case, under a simplifying, technical assumption, the initial value problem can be solved close to a singular orbit regardless of the soliton being shrinking, steady or expanding. However, the solution may not be unique. For a precise statement, see theorem \ref{QEMInitialValueTheorem}. Notice that $u(0)=0$ can be assumed, as the Ricci soliton equation is invariant under changing the potential by an additive constant. Furthermore, the existence of a singular orbit at $t=0$ imposes the smoothness condition $\dot{u}(0)=0$ on the soliton potential $u.$ If $d_S$ denotes the dimension of the collapsing sphere at the singular orbit, then the trace of the shape operator grows like $\tr(L) = \frac{d_S}{t} + O(t)$ as $t \to 0.$ Therefore the conservation law \eqref{GeneralConservationLaw} implies $\ddot{u}(0)= \frac{C}{d_S + 1}.$ To summarize: \begin{equation} u(0)=0, \ \ \ \dot{u}(0)=0, \ \ \ \ddot{u}(0)=\frac{C}{d_S +1}. \label{InitialConditionsPotentialFunction} \end{equation} The existence of a singular orbit has consequences for the behaviour of the soliton potential. Proposition \ref{PotentialFunctionOfExpandingRSAlongCohomOneFlow} below follows from \cite[Propositions 2.3 and 2.4]{BDWSteadySolitons} and \cite[Proposition 1.11]{BDGWExpandingSolitons}. It should be emphasised that the properties hold along the flow of the Ricci soliton equation and completeness of the metric is not required. \begin{proposition} Along any Ricci soliton trajectory with $\varepsilon \geq 0$ and $C<0$ in \eqref{InitialConditionsPotentialFunction} that corresponds to a cohomogeneity one manifold of dimension $n+1$ with a singular orbit at $t=0,$ for $t > 0$ and as long as the solution exists, the soliton potential satisfies $u(t),$ $\dot{u}(t)<0$ and also $\ddot{u}(t)<0$ if $\varepsilon >0$ or $\varepsilon = 0$ and $L_t \neq 0.$ Furthermore, if $\varepsilon = 0$ and $C \leq 0,$ there holds $\tr(L_t) \leq \frac{n}{t}$ for $t>0$ and as long as the solution exists. \label{PotentialFunctionOfExpandingRSAlongCohomOneFlow} \end{proposition} \begin{remarkroman} The quantity $\frac{\tr(L)}{-\dot{u}+\tr(L)}$ will appear frequently in later calculations. It is useful to note that it satisfies the differential equation \begin{equation*} \frac{d}{dt} \frac{\tr(L)}{-\dot{u}+\tr(L)} = \frac{1}{-\dot{u}+\tr(L)} \left\lbrace \left( \frac{\tr(L)}{-\dot{u}+\tr(L)} -1 \right)\left( \tr(L^2) - \frac{\varepsilon}{2} \right) + \ddot{u} \right\rbrace. \end{equation*} In particular, in the steady case, proposition \ref{PotentialFunctionOfExpandingRSAlongCohomOneFlow} shows that $\frac{\tr(L)}{-\dot{u}+\tr(L)}$ is monotonically decreasing as long as $-\dot{u}+\tr(L) >0.$ According to proposition \ref{CompleteSteadyRSAsymptotics} below, this is always true if the metric corresponds to a complete steady Ricci soliton. In this case, moreover, it follows that $\frac{\tr(L)}{-\dot{u}+\tr(L)} \to 0$ as $t \to \infty.$ \label{RemarkEvolutionOftrLOverGeneralisedMeanCurvature} \end{remarkroman} \subsection{Consequences of completeness} \label{SectionConsequencesOfCompleteness} If the solution corresponds to a non-trivial {\em complete} Ricci soliton metric, further restrictions on the asymptotics of the soliton potential and the metric are known. In the steady case, according to a result of Chen \cite{ChenStrongUniquenessRF}, the ambient scalar curvature of steady Ricci solitons satisfies $R \geq 0$ with equality if and only if the metric is Ricci flat. Then \eqref{ScalarCurvatureRicciSoliton} implies that $C \leq 0$ is a necessary for completeness and $C=0$ precisely corresponds to the Ricci flat case. Munteanu-Sesum \cite{MSgradientRicciSolitons} have shown that non-trivial complete steady Ricci solitons have at least linear volume growth and Buzano-Dancer-Wang used this to show in \cite[Proposition 2.4 and Corollary 2.6]{BDWSteadySolitons}: \begin{proposition} Along any trajectory which corresponds to a non-trivial {\em complete} steady cohomogeneity one Ricci soliton of dimension $n+1$ with a singular orbit at $t=0$ and integrability constant $C<0,$ the estimates \begin{align*} 0 < \tr(L) \leq \frac{n}{t} \ \text{ and } \ 0 < - \dot{u} \tr(L) < R < 2 \sqrt{-C} \frac{n}{t} + \frac{n^2}{t^2} \end{align*} hold for $t > 0$ and the soliton potential satisfies \begin{align*} -\dot{u}(t) \to \sqrt{-C} \ \text{ and } \ \ddot{u}(t) \to 0 \end{align*} as $t \to \infty.$ \label{CompleteSteadyRSAsymptotics} \end{proposition} In the case of expanding Ricci solitons, a similar result of Chen \cite{ChenStrongUniquenessRF} implies that the scalar curvature $R$ of a non-trivial, complete expanding Ricci soliton satisfies $R > - \frac{\varepsilon}{2}(n+1).$ It follows from \eqref{ScalarCurvatureRicciSoliton} that $0 \geq - \dot{u} ^2 > C + \varepsilon u$ holds on any complete expanding Ricci soliton. The smoothness condition \eqref{InitialConditionsPotentialFunction} at the singular orbit therefore requires $C<0$ as a {\em necessary} condition to construct non-trivial, complete expanding Ricci solitons. Conversely, Einstein metrics with negative scalar curvature correspond to trajectories with $C=0.$ Once the Ricci soliton is shown to be complete, it follows from results of Buzano-Dancer-Gallaugher-Wang \cite{BDGWExpandingSolitons} that any non-trivial, complete, gradient expanding Ricci soliton has at least logarithmic volume growth. This has consequences for the asymptotic behaviour of the soliton, see \cite[Equation (1.10) and Proposition 1.18]{BDGWExpandingSolitons}: There exists constants $a_0, a_1 >0$ and a time $t_0>0$ such that for all $t>t_0$ \begin{equation} | \tr(L_t) |< \sqrt{\frac{n}{2} \varepsilon} \ \text{ and } \ a_1 t + a_0 < - \dot{u}(t) < \frac{\varepsilon}{2} t + \sqrt{-C} \label{GeneralAsymptoticsExpandingRS} \end{equation} i.e. $- \dot{u}$ growths approximately linearly for $t$ large enough. \subsection{The B\"ohm functional} \label{BohmFunctionalSection} B\"ohm \cite{BohmNonCompactEinstein} introduced the functional $\mathscr{F}_0$ to the study of Einstein manifolds of cohomogeneity one. Subsequently it was considered by Dancer-Wang and their collaborators Buzano, Gallaugher and Hall in the context of cohomogeneity one Ricci solitons \cite{BDWSteadySolitons, BDGWExpandingSolitons, DHWShrinkingSolitons}. The significance of $\mathscr{F}_0$ lies in the fact that it is monotonic under mild assumptions. To define it, let $v(t) = \sqrt{\det g_t}$ denote the relative volume of the principal orbits and let $L^{(0)} = L - \frac{1}{n}\tr(L) \mathbb{I}$ denote the trace less part of the shape operator. Then the B\"ohm functional is given by \begin{equation} \mathscr{F}_0= v ^{\frac{2}{n}} \left( \tr(r_t) + \tr(( L^{(0)})^2 )\right). \label{BohmFunctional} \end{equation} The following proposition is due to Dancer-Hall-Wang \cite[Proposition 2.17]{DHWShrinkingSolitons}. \begin{proposition} Along the flow of a $C^3$-regular cohomogeneity one gradient Ricci soliton the B\"ohm functional $\mathscr{F}_0$ satisfies \begin{equation} \frac{d}{dt} \mathscr{F}_0 = - 2 v^{\frac{n}{2}} \tr((L^{(0)})^2) \left( -\dot{u} + \frac{n-1}{n} \tr(L) \right). \label{DerivativeOfBohmFunctional} \end{equation} \label{PropositionBohmFunctional} \end{proposition} \begin{remarkroman} The $C^3$-regularity condition guarantees that the conservation law \eqref{GeneralConservationLaw} is satisfied. On the other hand the existence of a singular orbit along the trajectory is not required to prove \eqref{DerivativeOfBohmFunctional}. \end{remarkroman} \section{New Examples of Ricci solitons} \label{SectionNewSolitons} \subsection{The geometric set-up} \label{SectionGeometricSetUp} Let $(M^{n+1},g)$ be a Riemannian manifold and suppose that $G$ is a compact connected Lie group which acts isometrically on $(M,g)$. Assume that the orbit space is a half open interval and let $K \subset H$ denote the isotropy groups of the principal and singular orbit, respectively. It follows that $M$ is diffeomorphic to the open disc bundle $G \times_H D^{d_S+1} \to G/H,$ where $D^{d_S+1}$ denotes the normal disc to the singular orbit $G/H$ and $S^{d_S} = H/K$ is the collapsing sphere. Conversely, let $G$ be a compact connected Lie group and let $K \subset H$ be closed subgroups such that $H /K$ is a sphere. Then $G \times_H \mathbb{R}^{d_S+1}$ is a cohomogeneity one manifold with principal orbit $G/K.$ Suppose that the non-principal orbit $G/H$ is singular, i.e. of dimension strictly less than $G/K.$ Choose a bi-invariant metric $b$ on $G$ which induces the metric of constant curvature $1$ on $H/K.$ The {\em two summands case} assumes that the space of $G$-invariant metrics on the principal orbit is two dimensional: Let $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ be an $\Ad(K)$-invariant decomposition of the Lie algebra of $G$ and suppose furthermore that $\mathfrak{p}$ decomposes into two inequivalent, $b$-orthogonal, irreducible $K$-modules, $\mathfrak{p} = \mathfrak{p}_1 \oplus \mathfrak{p}_2.$ In fact, $\mathfrak{p}_1$ can be identified with the tangent space to the collapsing sphere $S^{d_S} = H /K$ and $\mathfrak{p}_2$ with the tangent space of the singular orbit $Q=G/H.$ Let $g_S = b_{|\mathfrak{p}_1}$ and $g_Q = b_{|\mathfrak{p}_2}$ denote the induced metrics. Then, away from the singular orbit $Q,$ the metric on $M$ is given by \begin{equation} g_{M \setminus Q} = dt^2 +f_1(t)^2 g_S + f_2(t)^2 g_Q \label{TwoSummandsMetric} \end{equation} and the shape operator of the principal orbit takes the form \begin{align*} L_t=\left( \frac{\dot{f}_1}{f_1} \mathbb{I}_{d_1}, \frac{\dot{f}_2}{f_2} \mathbb{I}_{d_2} \right), \end{align*} where $d_1=d_S$ is the dimension of the collapsing sphere and $d_2$ is the dimension of the singular orbit. Furthermore, it follows from the theory of Riemannian submersions and the O'Neill calculus, cf. \cite{BohmInhomEinstein}, that the Ricci endomorphism takes the form \begin{align} r_t = \left( \left\lbrace \frac{A_1}{d_1} \frac{1}{f_1^2} + \frac{A_3}{d_1} \frac{f_1^2}{f_2^4} \right\rbrace \mathbb{I}_{d_1}, \left\lbrace \frac{A_2}{d_2} \frac{1}{f_2^2} - \frac{2 A_3}{d_2} \frac{f_1^2}{f_2^4} \right\rbrace \mathbb{I}_{d_2} \right). \label{RicciEndomorphismTwoSummands} \end{align} Here the constants $A_i \geq 0$ are defined as follows: $A_1 = d_1 (d_1 -1)$, $A_2 = d_2 \mathbb{R}ic^Q,$ where $\mathbb{R}ic^Q$ is the Einstein constant of the isotropy irreducible space $(Q,g_Q),$ and $A_3 = d_2 || A ||^2$, where $ || A || \geq 0$ appears naturally in the theory of Riemannian submersions, cf. \cite{BohmInhomEinstein}: Fix the background metric $g_P=g_S + g_Q$ on the principal orbit $P$ and let $\nabla^{g_P}$ be the corresponding Levi-Civita connection. If $H_1, \ldots, H_{d_2}$ is an orthonormal basis of horizontal vector fields with respect to the Riemannian submersion $(G/K,g_P) \to (G/H,g_Q)$, then $|| A ||^2 = \sum_{i=1}^{d_2} g_S( (\nabla_{H_1}^{g_P} H_i)_{|v}, (\nabla_{H_1}^{g_P} H_i)_{|v} )$ is the norm of an O'Neill tensor associated to the above Riemannian submersion, where $( \cdot )_{|v}$ denotes the projection onto the tangent space of the fibre $S^{d_1}=S^{d_S}.$ Warped product metrics with two homogeneous summands provide examples with $||A||=0.$ Examples with $||A|| > 0$ are given by the total spaces of non-trivial disc bundles which are induced by the Hopf fibrations, cf. \cite{BesseEinstein}. The following table, which lists the corresponding group diagrams and associated constants, is taken from \cite[Table 1]{BohmInhomEinstein}. \begin{table}[!ht] $\begin{array}{l|l|l|l|l} \text{} & \mathbb{C}P^{m+1} & \mathbb{H}P^{m+1} & F^{m+1} & CaP^2 \\ \hline G & U(m+1) & Sp(1) \times Sp(m+1) & Sp(m+1) & Spin(9) \\ H & U(1) \times U(m) & Sp(1) \times Sp(1) \times Sp(m) & Sp(1) \times Sp(m) & Spin(8) \\ K & U(m) & Sp(1) \times Sp(m) & U(1) \times Sp(m) & Spin(7) \\ d_1 & 1 & 3 & 2 & 7 \\ d_2 & 2m & 4m & 4m & 8 \\ ||A||^2 & 1 & 3 & 8 & 7 \\ \mathbb{R}ic^Q & 2m+2 & 4m+8 & 4m+8 & 28 \end{array}$ \caption{Group diagrams associated to Hopf fibrations\label{HopfFibrationsTable}} \end{table} The soliton potential $u$ will be assumed to be invariant under the action of $G$, $u=u(t),$ and $u(0)=0$ will be fixed. If $u$ satisfies the smoothness conditions \eqref{InitialConditionsPotentialFunction} and the functions $f_1,$ $f_2$ satisfy \begin{equation} f_1(0)=0, \ \dot{f}_1(0)=1 \ \text{ and } \ f_2(0)= \bar{f} > 0, \ \dot{f}_2(0)=0, \label{SmoothnessMetricTwoSummandsGeometricSetUpSection} \end{equation} then the work of Buzano \cite{BuzanoInitialValueSolitons} implies that there is a unique local solution of the Ricci soliton equations with these initial conditions, and it extends the soliton potential and the metric smoothly over the singular orbit. \begin{remarkroman} The two summands case is also the set-up for B\"ohm's work \cite{BohmInhomEinstein, BohmNonCompactEinstein} on Einstein manifolds. In fact, the Lyapunov function \eqref{LyapunovForNonTrivialBundles} is motivated by B\"ohm's work. In contrast, B\"ohm's construction relies on the Poincar\'e-Bendixson theorem. In the Ricci soliton case, however, the extra degree of freedom of the soliton potential does not allow a similar reduction of the Ricci soliton equations to a planar ODE and a new proof is required, see remark \ref{RemarkConvergenceConeSolutions}. Conversely, the methods of section \ref{CompletenessTwoSummands} recover B\"ohm's non-compact Einstein manifolds. \label{RemarkBoehmSetUpAndProofCompleteness} \end{remarkroman} \subsection{Qualitative ODE analysis} \label{CompletenessTwoSummands} The Ricci soliton equations for the two summands system can be read off from the discussion in section \ref{SectionGeometricSetUp} and equations \eqref{CohomOneRSb} and \eqref{CohomOneRSc}. However, in this form, the equations become singular at the singular orbit. Therefore, a rescaling will be introduced which smooths the Ricci soliton equation close to the initial value. It was effectively used by Dancer-Wang \cite{DWSteadySolitons} and is motivated by Ivey's work \cite{IveyNewExamplesRS}. Notice that under the coordinate change \begin{align} X_i & = \frac{1}{- \dot u + \tr(L)} \frac{\dot{f}_i}{f_i}, \ \ \ \ Y_i = \frac{1}{- \dot u + \tr(L)} \frac{1}{f_i}, \ \text{for} \ i=1,2, \label{RescaledTwoSummandsVariables} \\ \mathcal{L} & = \frac{1}{- \dot u + \tr(L)}, \ \ \ \ \ \ \frac{d}{ds} = \frac{1}{-\dot{u}+\tr(L)} \frac{d}{dt} \nonumber \end{align} the cohomogeneity one two summands Ricci soliton equations reduce to the ODE system \begin{align} X_1^{'} & = X_1 \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 -1 \right) + \frac{A_1}{d_1} Y_1^2+\frac{\varepsilon}{2} \mathcal{L}^2 + \frac{A_3}{d_1} \frac{Y_2^4}{Y_1^2}, \label{RescaledTwoSummandsODE} \\ X_2^{'} & = X_2 \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 -1 \right) + \frac{A_2}{d_2} Y_2^2+\frac{\varepsilon}{2} \mathcal{L}^2 - \frac{2 A_3}{d_2} \frac{Y_2^4}{Y_1^2}, \nonumber \\ Y_j^{'} & =Y_j \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 - X_j \right), \nonumber \\ \mathcal{L}^{'} & =\mathcal{L} \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 \right). \nonumber \end{align} Here and in the following, the $\frac{d}{ds}$ derivative is denoted by a prime $'.$ On the other hand, the $\frac{d}{dt}$ derivative will always correspond to a dot $\dot{}$ . To establish some basic properties of this ODE system, it will be enough to assume that $d_1, d_2 > 0,$ $A_1, A_2 > 0$ and $A_3 \geq 0.$ However, in the main body of the paper $d_1 > 1$ and $A_1, A_2, A_3 > 0$ will be assumed. \begin{remarkroman} (a) The case $A_3 = 0$ is already well understood from works on multiple warped products, see \cite{IveyNewExamplesRS}, \cite{GKExpandingRS}, \cite{DWExpandingSolitons, DWSteadySolitons}, \cite{BDGWExpandingSolitons}, \cite{BDWSteadySolitons} and \cite{AngenentKnopfRSConicalSingNonuniqueness}. (b) The case $d_1=1$ implies $A_1=0$ in geometric applications. In this case Cao-Koiso \cite{CaoSoliton},\cite{KoisoSoliton} and Feldman-Ilmanen-Knopf \cite{FIKSolitons} found explicit solutions to the associated {\em K\"ahler} Ricci solitons equations. {\em Non-K\"ahler} steady and expanding Ricci solitons will be constructed in section \ref{SectionSolitonsFromCircleBundles}. In the steady case these were independently found by Stolarski \cite{StolarskiSteadyRSOnCxLineBundles} and Appleton \cite{AppletonSteadyRS}, who use different techniques. \end{remarkroman} Notice that time, metric and soliton potential can be recovered from the ODE via \begin{align*} t(s) = t(s_0) + \int_{s_0}^{s} \mathcal{L}( \tau ) d \tau \ \text{ and } \ f_i = \frac{\mathcal{L}}{Y_i}, \ \text{for } \ i=1,2, \ \text{ and } \ \dot{u} = \frac{\sum_{i=1}^2 d_i X_i - 1}{\mathcal{L}}. \end{align*} In the new coordinate system, the smoothness conditions for the metric in \eqref{SmoothnessMetricTwoSummandsGeometricSetUpSection} and the soliton potential in \eqref{InitialConditionsPotentialFunction} correspond to the stationary point \begin{align} X_1 = Y_1 = \frac{1}{d_1} \ \text{ and } \ X_2 = Y_2 = 0 \ \text{ and } \ \mathcal{L} =0. \label{InitialCriticalPoint} \end{align} Trajectories emanating from \eqref{InitialCriticalPoint} will be parametrised so that \eqref{InitialCriticalPoint} corresponds to $s = - \infty.$ The conservation law \eqref{ReformulatedGeneralConsLaw} takes the form \begin{equation} \sum_{i=1}^2 d_i X_i^2 + \sum_{i=1}^2 A_i Y_i^2 - A_3 \frac{Y_2^4}{Y_1^2} + (n-1)\frac{\varepsilon}{2} \mathcal{L}^2 = 1 + \left( C + \varepsilon u \right) \mathcal{L}^2. \label{GeneralTwoSummandsConsLaw} \end{equation} Consider the functions \begin{align*} \mathcal{S}_1 & = \sum_{i=1}^2 d_i X_i^2 + \sum_{i=1}^2 A_i Y_i^2 - A_3 \frac{Y_2^4}{Y_1^2} + (n-1)\frac{\varepsilon}{2} \mathcal{L}^2 -1, \\ \mathcal{S}_2 & = \sum_{i=1}^2 d_i X_i -1. \end{align*} Notice that $\mathcal{S}_1$ occurs in the conservation law and $\mathcal{S}_2 = \frac{\dot{u}}{- \dot{u} + \tr(L)}$ encodes the derivative of the soliton potential in the rescaled coordinates. Fix $\varepsilon \geq 0$ and recall from section \ref{SectionConsequencesOfCompleteness} that $C \leq 0$ is a necessary condition to obtain trajectories that correspond to complete steady or expanding Ricci solitons and that $C=0$ is the Einstein case. Due to the initial conditions \eqref{InitialConditionsPotentialFunction} and proposition \ref{PotentialFunctionOfExpandingRSAlongCohomOneFlow}, the soliton potential satisfies $u, \dot{u} \leq 0$ if $C \leq 0,$ and away from the singular orbit equality can only occur in the Einstein case. Therefore, any trajectory with $\varepsilon \geq 0$ and $C \leq 0$ satisfies $\mathcal{S}_1, \mathcal{S}_2 \leq 0.$ Equality occurs at the initial stationary point \eqref{InitialCriticalPoint} and then {\em Einstein} trajectories lie in the locus \begin{equation} \left\{ \mathcal{S}_1 = 0\right\} \cap \left\{ \mathcal{S}_2 = 0\right\} \label{EinsteinLocus} \end{equation} whereas trajectories of {\em complete non-trivial} Ricci solitons are contained in the locus \begin{equation} \left\{ \mathcal{S}_1 < 0\right\} \cap \left\{ \mathcal{S}_2 < 0\right\}. \label{SolitonLocus} \end{equation} Conversely, trajectories in these loci correspond to Einstein metrics and non-trivial Ricci solitons. The invariance of the above loci for $\varepsilon \geq 0$ follows from the Ricci soliton ODE as a direct calculation verifies \begin{align*} \frac{1}{2} \frac{d}{ds} \mathcal{S}_1 & = \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 \right) \mathcal{S}_1 + \frac{\varepsilon}{2} \mathcal{L}^2 \cdot \mathcal{S}_2 , \\ \frac{d}{ds} \mathcal{S}_2 & = \mathcal{S}_1 + \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 - 1 \right) \mathcal{S}_2. \end{align*} Now the existence of trajectories which lie in one of the above loci and in the unstable manifold of the critical point \eqref{InitialCriticalPoint} will be discussed. Different trajectories will correspond to non-homothetic Einstein or Ricci soliton metrics. The linearisation of the Ricci soliton ODE at the initial stationary point \eqref{InitialCriticalPoint} is given by \begin{equation*} \begin{pmatrix} \frac{3}{d_1}-1 & 0 & \frac{2(d_1-1)}{d_1} & 0 & 0 \\ 0 & \frac{1}{d_1}-1 & 0 & 0 & 0 \\ \frac{1}{d_1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{d_1} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{d_1} \end{pmatrix}. \end{equation*} The corresponding eigenvalues are hence $\frac{2}{d_1},$ and both $\frac{1}{d_1}-1$ and $\frac{1}{d_1}$ appear twice. In particular, the critical point is {\em hyperbolic} if $d_1 >1.$ The corresponding eigenspaces are given by $E_{\frac{2}{d_1}} = \operatorname{span} \left\lbrace (2,0,1,0,0) \right\rbrace ,$ $E_{\frac{1}{d_1}-1}= \operatorname{span} \left\lbrace (0,1,0,0,0), (d_1-1,0,-1,0,0) \right\rbrace$ and $E_{\frac{1}{d_1}} = \operatorname{span} \left\lbrace (0,0,0,1,0), (0,0,0,0,1) \right\rbrace.$ Notice that the stationary point \eqref{InitialCriticalPoint} lies in the set $\left\{ \mathcal{S}_1 = 0\right\} \cap \left\{ \mathcal{S}_2 = 0\right\}.$ Furthermore, $\left\{ \mathcal{S}_1 = 0\right\}$ is a submanifold of $\mathbb{R}^5$ if $Y_1 \neq 0$ and its tangent space at \eqref{InitialCriticalPoint} is $\operatorname{span} \left\lbrace (1,0,d_1-1,0,0) \right\rbrace^{\perp}.$ Similarly, $\left\{ \mathcal{S}_2 = 0\right\}$ is a submanifold with tangent space $\operatorname{span} \left\lbrace (d_1,d_2,0,0,0) \right\rbrace^{\perp}$ at \eqref{InitialCriticalPoint}. Notice that both tangent spaces contain $E_{\frac{1}{d_1}}$ but not $E_{\frac{2}{d_1}}$ and that $E_{\frac{1}{d_1}} \oplus E_{\frac{2}{d_1}}$ is the tangent space to the unstable manifold. According to the above discussion, trajectories in the unstable manifold of \eqref{InitialCriticalPoint} that either remain in the set $\left\{ \mathcal{S}_1 = 0\right\} \cap \left\{ \mathcal{S}_2 = 0\right\}$ or flow into $\left\{ \mathcal{S}_1 < 0\right\} \cap \left\{ \mathcal{S}_2 < 0\right\}$ need to be considered. Notice, however, that if $\varepsilon = 0$ the ODE for $\mathcal{L}$ decouples. Hence, the soliton system effectively reduces to a system in $X_i, Y_i$ for $i=1,2.$ Counting trajectories with respect to the possibly reduced system then gives the following result. \begin{proposition} Suppose that $d_1 > 1.$ If $\varepsilon \neq 0,$ then there exists a $1$-parameter family of trajectories lying both in the unstable manifold of \eqref{InitialCriticalPoint} and the Einstein locus \eqref{EinsteinLocus} and a $2$-parameter family of trajectories lying both in the unstable manifold of \eqref{InitialCriticalPoint} and the Ricci soliton locus \eqref{SolitonLocus}. If $\varepsilon =0,$ then the unstable manifold of \eqref{InitialCriticalPoint} with respect to the reduced two summands ODE in $X_1, X_2$ and $Y_1, Y_2$ contains a unique trajectory lying in the Einstein locus \eqref{EinsteinLocus} and a $1$-parameter family of trajectories lying in the Ricci soliton locus \eqref{SolitonLocus}. These give rise to an (up to scaling) unique Ricci flat metric and a $1$-parameter family of Ricci solitons with soliton potential $u=0$ at the singular orbit. \label{NumberOfParameterFamilies} \end{proposition} Proposition \ref{NumberOfParameterFamilies} is in agreement with the theory of solutions to the initial value problem for cohomogeneity one Ricci solitons and Einstein metrics developed by Buzano \cite{BuzanoInitialValueSolitons} and Eschenburg-Wang \cite{EWInitialValueEinstein}, respectively. Their methods also carry over to the case $d_1 = 1.$ Notice that the ODE system \eqref{RescaledTwoSummandsODE} and the initial stationary point \eqref{InitialCriticalPoint} are invariant under changing the signs of $Y_2,$ $\mathcal{L}.$ Since $\mathcal{L}^{-1}=-\dot{u} + \tr(L) \to + \infty$ as $t \to 0,$ $\mathcal{L}>0$ will be assumed along the trajectories. The choice $f_2(0)= \bar{f}>0$ in \eqref{SmoothnessMetricTwoSummandsGeometricSetUpSection} implies $f_2(t)>0$ for small $t>0$ and thus $Y_2 > 0$ will be assumed. Recall that $\lim_{s \to - \infty} Y_1(s)=1.$ The ODEs for $Y_1, Y_2, \mathcal{L}$ imply that positivity of the variables is preserved along the flow. The following lemma shows a basic dynamical property of the Ricci soliton ODE and sets up the discussion of the long time behaviour. \begin{lemma} Let $\varepsilon \geq 0$ and consider a trajectory of the two summands Ricci soliton ODE that emanates from \eqref{InitialCriticalPoint} at $s = - \infty$ and enters either \eqref{EinsteinLocus} or \eqref{SolitonLocus}. Then there holds $X_1 >0$ for all finite $s$ and $X_2$ is positive for sufficiently negative $s.$ Moreover, suppose there is an $s_0 \in \mathbb{R}$ such that $X_2(s_0) <0.$ Then $X_2(s) < 0$ for all $s \geq s_0.$ \label{XVariablesPositiveInitially} \end{lemma} \begin{proof} Recall that $\lim_{s \to - \infty} X_1 = 1/d_1 >0$ and in particular $X_1$ is positive initially. If there is an $s \in \mathbb{R}$ such that $X_1(s) = 0,$ then $X_1^{'}(s)>0.$ By continuity this implies $X_1 > 0$ everywhere. The conservation law \eqref{GeneralTwoSummandsConsLaw} implies that $\sum_{i=1}^2 d_i X_i^2 -1 \leq A_3 \frac{Y_2^4}{Y_1^2} - \sum_{i=1}^2 A_i Y_i^2 < 0$ close to \eqref{InitialCriticalPoint} as $Y_1 \to \frac{1}{d_1}$ and $Y_2 \to 0.$ Similarly, $\frac{A_2}{d_2} Y_2^2 - \frac{2 A_3}{d_2} \frac{Y_2^4}{Y_1^2} >0$ for sufficiently negative times. If $X_2(s_0) < 0$ in this region, then the ODE \begin{align*} X_2^{'} = X_2 \left( \sum_{i=1}^2 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 -1 \right) + \frac{A_2}{d_2} Y_2^2+\frac{\varepsilon}{2} \mathcal{L}^2 - \frac{2 A_3}{d_2} \frac{Y_2^4}{Y_1^2} \end{align*} implies that $X_2^{'}(s_0) > 0$ as $\varepsilon \geq 0.$ In particular $X_2(s) \leq X_2(s_0) < 0$ for all $s \leq s_0.$ This contradicts $X_2 \to 0$ as $s \to - \infty.$ If the last statement is not true, then there exist $s_{*} < s^{*}$ such that $X_2 < 0$ on $(s_{*}, s^{*})$ and \begin{align*} X_2(s_{*}) & = 0 \ \text{ and } \ X_2^{'}(s_{*}) \leq 0, \\ X_2(s^{*}) & = 0 \ \text{ and } \ X_2^{'}(s^{*}) \geq 0. \end{align*} It follows that $ \frac{A_2}{d_2} Y_2^2(s_{*}) + \frac{\varepsilon}{2} \mathcal{L}^2(s_{*}) - 2 \frac{A_3}{d_2} \frac{Y_2^4}{Y_1^2}(s_{*}) \leq 0$ which is equivalent to \begin{equation*} \frac{A_2}{d_2} \leq \left[ 2 \frac{A_3}{d_2} \left( \frac{Y_2}{Y_1} \right) ^2 - \frac{\varepsilon}{2} \left( \frac{\mathcal{L}}{Y_2} \right) ^2 \right](s_{*}). \end{equation*} Similarly, the second condition implies the reverse inequality at $s^{*}.$ Therefore, \begin{align*} 0 & \leq \left[ 2 \frac{A_3}{d_2} \left( \frac{Y_2}{Y_1} \right) ^2 - \frac{\varepsilon}{2} \left( \frac{\mathcal{L}}{Y_2} \right) ^2 \right](s_{*}) - \left[ 2 \frac{A_3}{d_2} \left( \frac{Y_2}{Y_1} \right) ^2 - \frac{\varepsilon}{2} \left( \frac{\mathcal{L}}{Y_2} \right) ^2 \right](s^{*}) \\ & = \frac{d}{ds} \left[ 2 \frac{A_3}{d_2} \left( \frac{Y_2}{Y_1} \right) ^2 - \frac{\varepsilon}{2} \left( \frac{\mathcal{L}}{Y_2} \right) ^2 \right](\xi) \cdot (s_{*} - s^{*} ) \end{align*} for some $\xi \in (s_{*}, s^{*}).$ On the other hand, observe that \begin{align*} \frac{d}{ds} \frac{Y_2}{Y_1} = \frac{Y_2}{Y_1} (X_1-X_2) \ \text{ and } \ \frac{d}{ds} \frac{\mathcal{L}}{Y_2} = \frac{\mathcal{L}}{Y_2} X_2. \end{align*} Therefore, $X_2( \xi) < 0,$ $\varepsilon \geq 0$ and $s_{*} < s^{*}$ imply \begin{align*} 0 & \leq \frac{d}{ds} \left[ 2 \frac{A_3}{d_2} \left( \frac{Y_2}{Y_1} \right) ^2 - \frac{\varepsilon}{2} \left( \frac{\mathcal{L}}{Y_2} \right) ^2 \right](\xi) \cdot (s_{*} - s^{*} ) \\ & = \ 2 \ \left[ 2 \frac{A_3}{d_2} \left( \frac{Y_2}{Y_1} \right) ^2 (X_1-X_2) - \frac{\varepsilon}{2} \left( \frac{\mathcal{L}}{Y_2} \right) ^2 X_2 \right](\xi) \cdot (s_{*} - s^{*} ) < 0, \end{align*} which is a contradiction. \end{proof} \begin{remarkroman} In fact, the possibility that $X_2 < 0$ is the only obstruction to long time existence. Geometrically this says that along the trajectory of an incomplete metric the shape operator cannot remain positive definite. If $A_3=0,$ then $X_2 > 0$ is immediate and the Einstein and Ricci soliton loci \eqref{EinsteinLocus} and \eqref{SolitonLocus}, respectively, are bounded regions in phase space. Completeness of the metric then follows as in propositions \ref{CompletenessEpsZeroTwoSummands} and \ref{CompletenessEpsPosTwoSummands} below. Geometrically the case $A_3=0$ corresponds to the doubly warped product situation which was considered by Ivey \cite{IveyNewExamplesRS}, Gastel-Kronz \cite{GKExpandingRS}, Dancer-Wang \cite{DWExpandingSolitons, DWSteadySolitons} and Angenent-Knopf \cite{AngenentKnopfRSConicalSingNonuniqueness}. \end{remarkroman} If $A_3>0,$ notice that $X_2 > 0$ clearly holds as long as $\frac{Y_2}{Y_1} < \sqrt{\frac{A_2}{2 A_3}}.$ Therefore, the quotient \begin{equation*} \omega = \frac{Y_2}{Y_1}. \end{equation*} plays a central role in the discussion. Observe that $\omega$ satisfies \begin{align} \omega{'} = \omega (X_1-X_2). \label{DByDsOmega} \end{align} In fact this implies that the Ricci soliton equation is equivalent to an ODE system with polynomial right hand side. In order to obtain an a priori bound for $\omega,$ fix $d_1>1$ and consider the function \begin{align} \widehat{\mathcal{G}}(\omega) = \frac{A_1}{d_1} \frac{\omega^{2(d_1-1)}}{2(d_1-1)} - \frac{A_2}{d_2} \frac{\omega^{2d_1}}{2d_1} + A_3 \left( \frac{1}{d_1} + \frac{2}{d_2} \right) \frac{\omega^{2(d_1+1)}}{2(d_1+1)}. \label{FunctionGHatTwoSummandsCase} \end{align} Along trajectories of the two summands Ricci soliton ODE there holds \begin{align*} \frac{d}{ds} \widehat{\mathcal{G}}(\omega) = \omega^{2(d_1-1)} \left\{ \frac{A_1}{d_1} - \frac{A_2}{d_2} \omega^2 + A_3 \left( \frac{1}{d_1} + \frac{2}{d_2} \right) \omega^4 \right\} \left( X_1 - X_2 \right) \end{align*} and non-zero roots of $\widehat{\mathcal{G}}$ are of the form \begin{align*} \omega^2 = \frac{1}{2} \frac{A_2}{A_3} \frac{d_1+1}{2d_1+d_2} \left\lbrace 1 \pm \sqrt{1 - 4 \frac{A_1 A_3}{A_2^2} \frac{d_2 (2d_1+d_2)}{(d_1-1)(d_1 + 1)}} \right\rbrace. \end{align*} In particular, there exist two positive roots $0 < \hat{\omega}_1 < \hat{\omega}_2$ if and only if \begin{equation} \widehat{D} = \frac{A_2^2}{d_2^2} - 4 \frac{A_1}{d_1(d_1-1)} \frac{A_3}{d_2} \frac{d_1}{d_1+1} (2d_1 +d_2) > 0. \label{DefinitionDHat} \end{equation} Moreover, in this case, $\hat{\omega}_1^2 < \frac{A_2}{2 A_3}.$ \begin{proposition} Suppose that $d_1 > 1,$ $\widehat{D} >0$ and $\varepsilon \geq 0.$ Then the set \begin{align*} \left\{ \ X_2 > 0 \ \text{ and } \ 0 < \frac{Y_2}{Y_1} < \hat{\omega}_1 \ \right\} \end{align*} contains any trajectory of the two summands Ricci soliton ODE that emanates from \eqref{InitialCriticalPoint} and flows into either \eqref{EinsteinLocus} or \eqref{SolitonLocus}. \label{X2VariablePositive} \end{proposition} \begin{proof} The ODE for $X_2$ shows that $X_2$ remains positive if $\frac{Y_2^2}{Y_1^2} = \omega^2 < \frac{A_2}{2 A_3}.$ Since $\hat{\omega}_1^2 < \frac{A_2}{2 A_3},$ it suffices to show that $\omega < \hat{\omega}_1$ as long as $X_2 > 0.$ Consider the function \begin{align} \mathcal{K} = \frac{1}{2} \omega^{2(d_1-1)} \left( \frac{X_1 - X_2}{Y_1} \right)^2 - \widehat{\mathcal{G}} \left( \omega \right), \label{LyapunovForNonTrivialBundles} \end{align} which was introduced by B\"ohm in the Einstein case \cite{BohmInhomEinstein}. On the set $X_2 >0$ it is a Lyapunov function since \begin{align*} \frac{d}{ds} \mathcal{K} = \omega^{2(d_1-1)} \left( \frac{X_1 - X_2}{Y_1} \right)^2 \left\{ \sum_{i=1}^2 d_i X_i - 1 - (n-1) X_2 \right\} \end{align*} and $\sum_{i=1}^2 d_i X_i - 1 \leq 0$ holds in both loci. Notice that $\lim_{s \to - \infty} \mathcal{K} = 0$ and $\mathcal{K} \geq 0$ if $\frac{Y_2}{Y_1} = \omega = \hat{\omega}_1.$ However, $\mathcal{K}$ is non-increasing and strictly decreasing close to \eqref{InitialCriticalPoint}. This completes the proof. \end{proof} \begin{corollary} Suppose that $d_1 >1,$ $\widehat{D} > 0$ and $\varepsilon \geq 0.$ Then along trajectories emanating from \eqref{InitialCriticalPoint} and flowing into \eqref{EinsteinLocus} or \eqref{SolitonLocus} there holds $X_1,$ $X_2 >0$ for all finite times. Moreover, the variables $X_1, X_2$ and $Y_1, Y_2$ and $\omega$ are bounded, and if $\varepsilon >0$ then $\mathcal{L}$ is bounded too. In particular, the rescaled flow exists for all times. \label{FlowExistsForAllTimes} \end{corollary} \begin{proof} According to lemma \ref{XVariablesPositiveInitially} one has $X_1,$ $X_2 > 0$ initially and $X_1 > 0$ is preserved along the flow. Positivity of $X_2$ follows from proposition \ref{X2VariablePositive} and $X_1,$ $X_2$ remain bounded as $0 \leq d_1 X_1 + d_2 X_2 \leq 1$ due to \eqref{EinsteinLocus} and \eqref{SolitonLocus}. Then the ODE for $\mathcal{L}$ implies that $\mathcal{L}$ cannot blow up in finite time as $\varepsilon \geq 0.$ By the same argument, this also holds for $Y_1,$ $Y_2.$ Alternatively, it follows from the bound $\frac{Y_2^2}{Y_1^2} < \hat{\omega}_1^2 < \frac{A_2}{2 A_3}$ that $A_2 - A_3 \frac{Y_2^2}{Y_1^2} > \frac{A_2}{2}.$ The Einstein and Ricci soliton loci \eqref{EinsteinLocus} and \eqref{SolitonLocus} are therefore contained in the bounded region $\left\lbrace \ \sum_{i=1}^2 d_i X_i^2 + A_1 Y_1^2 + \frac{A_2}{2} Y_2^2 + (n-1) \frac{\varepsilon}{2} \mathcal{L}^2 \leq 1 \ \right\rbrace.$ By considering $\omega = \frac{Y_2}{Y_1}$ as an independent variable, one obtains an ODE system with polynomial right hand side. Since $\omega < \hat{\omega}_1$ is bounded, standard ODE theory implies that the flow exists for all times. \end{proof} In order to prove that the corresponding metrics are complete, it suffices to show that $t_{\max} = \infty.$ Recall from the coordinate change that \begin{equation} t(s) = t(s_0) + \int_{s_0}^s \mathcal{L} (\tau) d \tau. \label{TimeRescaling} \end{equation} Therefore it is necessary to estimate the asymptotic behaviour of $\mathcal{L}.$ This needs to be considered separately for the cases $\varepsilon = 0$ and $\varepsilon >0.$ \begin{proposition} Suppose that $d_1>1$ and $\widehat{D}>0.$ Then the corresponding steady Ricci soliton and Ricci flat metrics are complete. \label{CompletenessEpsZeroTwoSummands} \end{proposition} \begin{proof} A special feature of the case $\varepsilon = 0$ is that $\mathcal{L} = \frac{1}{-\dot{u}+\tr(L)}$ is in fact a {\em Lyapunov} function. As $\mathcal{L}$ becomes positive initially and is therefore monotonically increasing, it is bounded away from zero for $s \geq s_0$ and any $s_0 \in \mathbb{R}.$ Then the time rescaling \eqref{TimeRescaling} shows that $t \to \infty$ as $s \to \infty,$ i.e. the metrics are complete. \end{proof} \begin{remarkroman} The Ricci flat metrics in proposition \ref{CompletenessEpsZeroTwoSummands} have already been constructed by B\"ohm \cite{BohmNonCompactEinstein} by different means, see remark \ref{RemarkConvergenceConeSolutions}. \end{remarkroman} The cases of expanding Ricci solitons and Einstein metrics with negative scalar curvature correspond to $\varepsilon >0.$ It will be sufficient to have an upper bound on $\mathcal{L}$ to prove completeness of the metric. \begin{lemma} Let $d_1>1,$ $\widehat{D} > 0$ and $\varepsilon > 0.$ Then along trajectories that emanate from \eqref{InitialCriticalPoint} and flow into \eqref{EinsteinLocus} or \eqref{SolitonLocus} there holds \begin{equation*} 0 < \frac{\varepsilon}{2} \mathcal{L}^2 \leq \max \left\lbrace \frac{1}{d_1}, \frac{1}{d_2} \right\rbrace. \end{equation*} Moreover, in the Einstein case, given $s_0 \in \mathbb{R}$ there holds \begin{equation*} \frac{\varepsilon}{2} \mathcal{L}^2(s) \geq \min \left\lbrace \frac{\varepsilon}{2} \mathcal{L}^2(s_0), \frac{1}{n} \right\rbrace \end{equation*} for all $s \geq s_0.$ In particular, $\mathcal{L}(s)$ is bounded away from zero for all $s \geq s_0.$ \label{LemmaLBoundedAwayFromZero} \end{lemma} \begin{proof} Notice that $0 \leq \sum_{i=1}^2 d_i X_i^2 \leq \max \left\lbrace \frac{1}{d_1}, \frac{1}{d_2} \right\rbrace $ on $X_1, X_2 \geq 0$ and $\sum_{i=1}^2 d_i X_i \leq 1.$ Therefore, if there is an $s_0$ such that $\frac{\varepsilon}{2} \mathcal{L}^2(s_0) > \max \left\lbrace \frac{1}{d_1}, \frac{1}{d_2} \right\rbrace $ then $\mathcal{L}^{'}(s_0) < 0.$ This yields $\mathcal{L}(s) \geq \mathcal{L}(s_0)$ for all $s \leq s_0,$ which contradicts $\lim_{s \to - \infty} \mathcal{L} =0.$ To prove the second statement, suppose that $\frac{\varepsilon}{2} \mathcal{L}^2(s_0) < \frac{1}{n}.$ Since $\sum_{i=1}^2 d_i X_i = 1$ in the Einstein locus, one has $\sum_{i=1}^2 d_i X_i^2 \geq \frac{1}{n}$ and therefore $\mathcal{L}^{'}(s_0) > 0.$ Hence, $\frac{\varepsilon}{2}\mathcal{L}^2$ is monotonically increasing whenever it is less than $\frac{1}{n}.$ \end{proof} \begin{corollary} Suppose that $d_1>1$ and $\widehat{D}>0.$ Then the corresponding expanding Ricci solitons and Einstein metrics with negative scalar curvature are complete. \label{CompletenessEpsPosTwoSummands} \end{corollary} \begin{proof} Suppose for contradiction that $t_{\max} < \infty.$ Due to \eqref{TimeRescaling} this is equivalent to saying that $\Vert \mathcal{L} \Vert_{L^1(0, \infty)} < \infty.$ However, since $\mathcal{L}$ is bounded due to lemma \ref{LemmaLBoundedAwayFromZero}, this implies $\mathcal{L} \in L^2(0, \infty).$ Hence the ODE for $\mathcal{L}$ yields $\mathcal{L}(s) \geq \mathcal{L}(0) \exp \left( - \frac{\varepsilon}{2} \Vert \mathcal{L} \Vert_{L^2(0, \infty)} \right) > 0$ and $\mathcal{L}$ is bounded away from zero for $s \geq 0.$ However, this contradicts $\mathcal{L} \in L^1(0, \infty).$ \end{proof} \begin{remarkroman} The Einstein metrics of negative scalar curvature in corollary \ref{CompletenessEpsPosTwoSummands} have already been constructed by B\"ohm \cite{BohmNonCompactEinstein} by different means, see remark \ref{RemarkConvergenceConeSolutions}. \end{remarkroman} \subsection{Ricci solitons from circle bundles} \label{SectionSolitonsFromCircleBundles} The two summands case allows the possibility $d_1 = 1$ and $A_1=0.$ Geometrically this case is realised by manifolds which are foliated by principal circle bundles over a Fano K\"ahler-Einstein manifold $(V,J,g).$ In this setting, examples of {\em K\"ahler} Ricci solitons have been found by Cao-Koiso \cite{CaoSoliton},\cite{KoisoSoliton} and Feldman-Ilmanen-Knopf \cite{FIKSolitons}. {\em Non-K\"ahler} examples have also been constructed independently by Stolarski \cite{StolarskiSteadyRSOnCxLineBundles} and Appleton \cite{AppletonSteadyRS}. The precise geometric set-up is as follows: Recall that due to a theorem of Kobayashi \cite{KobayashiCompactFanos} any Fano manifold $V$ is simply connected and hence $H^2(V, \mathbb{Z})$ is torsion free. Therefore the first Chern class is $c_1(V, J)= p \rho$ for a positive integer $p$ and an indivisible class $\rho \in H^2(V, \mathbb{Z}).$ Suppose that the Ricci curvature of $(V,g)$ is normalised to be $\mathbb{R}ic = p g.$ If $\pi \colon P \to V$ is the principal circle bundle with Euler class $q \pi^{*} \rho$ for a non-zero integer $q \in \mathbb{Z} \setminus \left\{ 0 \right\}$ and $\theta$ the principal $S^1$-connection with curvature form $\Omega = q \pi^{*} \eta,$ where $\eta$ is the K\"ahler form associated to $g,$ then the Ricci soliton equation on $I \times P$ corresponding to the metric \begin{equation*} dt^2 + f_1^2(t) \theta \otimes \theta + f_2^2(t) \pi^{*} g \end{equation*} is described by the two summands system with $d_1=1,$ $d_2=d=\dim_{\mathbb{R}} V$ and $A_1=0,$ $A_2=d_2 p,$ $A_3 = \frac{d_2 q^2}{4}.$ Notice also that the structure of the ODE has changed since $A_1=0.$ If the smoothness conditions \eqref{SmoothnessMetricTwoSummandsGeometricSetUpSection} are satisfied, this construction induces a smooth metric on the associated complex line bundle over $V.$ Metrics whose curvature tensor is invariant under the complex structure are considered by Dancer-Wang \cite{DWCohomOneSolitons} in the Ricci soliton case and by Wang-Wang \cite{WWEinsteinS2Bundles} in the Einstein case. This condition is equivalent to saying that \begin{align*} \frac{\dot{f}_2^2}{f_2^2} - \frac{q^2}{4} \frac{f_1^2}{f_2^4} = \left( - \dot{u} + \tr(L) + \frac{\dot{f}_1}{f_1} \right) \frac{\dot{f}_2}{f_2} - \frac{p}{f_2^2} + \frac{\varepsilon}{2}. \end{align*} As a special case, the {\em K\"ahler} condition reads \begin{equation*} \frac{\dot{f}_2}{f_2}= - \frac{q}{2} \frac{f_1}{f_2^2} \end{equation*} and it is preserved by the flow. In both cases, the equations can actually be integrated {\em explicitly.} In order to investigate {\em non-K\"ahler} trajectories, the Ricci soliton ODE will be studied qualitatively as before. To adjust the argument in proposition \ref{X2VariablePositive} to the conditions $d_1=1$ and $A_1=0,$ adopt the convention $\frac{A_1}{d_1(d_1-1)} = 1.$ That is, consider \begin{align*} \widehat{\mathcal{G}}( \omega ) = \frac{1}{2} - \frac{p}{2} \omega^2 + \frac{d+2}{16}q^2 \omega^4 \ \text{ and } \ \mathcal{K} = \frac{1}{2} \left( \frac{X_1-X_2}{Y_1} \right) ^2 - \widehat{\mathcal{G}}( \omega ) \end{align*} and note that $\widehat{\mathcal{G}}$ has two positive roots $0<\hat{\omega}_1 < \hat{\omega}_2$ if $2p^2 > (d+2)q^2.$ Then the proof of proposition \ref{X2VariablePositive} shows \begin{proposition} Suppose that $d_1=1,$ $A_1=0$ and $2p^2 > (d+2)q^2 >0.$ If $\varepsilon \geq 0,$ the set \begin{align*} \left\lbrace X_2 > 0 \ \text{ and } \ 0 < \frac{Y_2}{Y_1} <\hat{\omega}_1 \right\rbrace \end{align*} contains any trajectory of the Ricci soliton ODE that emanates from \eqref{InitialCriticalPoint} and flows into either \eqref{EinsteinLocus} or \eqref{SolitonLocus}. \label{X2PositiveCircleBundles} \end{proposition} Completeness of the metric can then be established as in proposition \ref{CompletenessEpsZeroTwoSummands} and corollary \ref{CompletenessEpsPosTwoSummands}. Notice in particular that long time existence still follows from corollary \ref{FlowExistsForAllTimes}. As the proof shows, even though $Y_1$ is not controlled by the conservation law \eqref{GeneralTwoSummandsConsLaw} anymore since $A_1 =0,$ it cannot blow up in finite time. \begin{corollary} Let $d_1=1,$ $A_1 = 0,$ $2p^2 > (d+2)q^2 >0$ and $\varepsilon \geq 0.$ Then any trajectory of the Ricci soliton ODE which emanates from the critical point \eqref{InitialCriticalPoint} and lies in the Einstein locus \eqref{EinsteinLocus} or Ricci soliton locus \eqref{SolitonLocus} corresponds to a complete Einstein or Ricci soliton metric, respectively. \end{corollary} \begin{remarkroman} (a) Notice on the contrary that the construction of K\"ahler Ricci solitons due to Feldman-Ilmanen-Knopf \cite{FIKSolitons} requires the condition $-q=p$ in the steady case and $-q > p$ in the expanding case, see also \cite[Theorem 4.20 and Remark 4.21]{DWCohomOneSolitons}. For example, in the case of $\mathbb{C}P^n$ one has $p=\frac{d+2}{2}$ and one thus requires $p>q^2>0$ for the argument of proposition \ref{X2PositiveCircleBundles} to work. In particular, the K\"ahler examples due to Feldman-Ilmanen-Knopf are not covered by the corollary. In the case of $\mathbb{C}P^n,$ these K\"ahler Ricci soliton metrics have also been investigated by Chave-Valent \cite{CVQasuiEinsteinRenormalization}. (b) Explicit K\"ahler and non-K\"ahler Einstein metrics have already been described by Calabi \cite{CalabiKaehlerMetrics}, B{\'e}rard-Bergery \cite{BerardBergerySurDeNouvellesEintein}, Page-Pope \cite{PagePopeEinsteinMetricsOnCxLineBundles} and Wang-Wang \cite{WWEinsteinS2Bundles}. (c) If $q > p,$ Appleton \cite{AppletonSteadyRS} proves that there cannot exist a complete Ricci flat metric on the associated complex line bundle. In particular, the corresponding trajectory cannot satisfy the bound $ \omega < \frac{4p}{(d+2)q^2}$ for all times. \label{RemarkCircleBundleSolutions} \end{remarkroman} Observe that the initial stationary point \eqref{InitialCriticalPoint} is not hyperbolic in the case $d_1=1.$ Therefore a center manifold exists and the analysis before proposition \ref{NumberOfParameterFamilies} does not carry over. However, the work of Buzano \cite{BuzanoInitialValueSolitons} and Eschenburg-Wang \cite{EWInitialValueEinstein} still applies and the existence of Ricci soliton trajectories can be deduced, see also \cite{StolarskiSteadyRSOnCxLineBundles} or \cite{AppletonSteadyRS} for different arguments. Thus Theorem \ref{MainTheoremRSOnLineBundles} follows from the following result: \begin{theorem} Suppose that $d_1=1,$ $A_1=0$ and $2p^2 > (d+2)q^2 >0.$ If $\varepsilon = 0$ there exists a $1$-parameter family and if $\varepsilon > 0$ a $2$-parameter family of trajectories lying in both the unstable manifold of \eqref{InitialCriticalPoint} and the Ricci soliton locus \eqref{SolitonLocus}. In particular, these give rise to complete Ricci soliton metrics on the total spaces of the corresponding complex line bundles over Fano K\"ahler-Einstein manifolds. Similarly, there exist a (up to homotheties) unique complete Ricci flat metric and a $1$-parameter family of complete Einstein metrics with negative scalar curvature on these spaces. \label{SolitonsOnCircleBundles} \end{theorem} It follows from the work of Appleton \cite{AppletonSteadyRS} that the Ricci solitons in theorem \ref{SolitonsOnCircleBundles} are asymptotically conical. Furthermore, recall from remark \ref{RemarkCircleBundleSolutions} that the existence of Einstein metrics is well known. \subsection{Asymptotics} \label{SectionTwoSummandsAsymptotics} This section discusses the asymptotic behaviour of the metrics which were constructed in section \ref{CompletenessTwoSummands}. In particular it will be shown that the steady Ricci solitons are asymptotically paraboloid and the expanding Ricci solitons are asymptotically conical. \subsubsection{Cone solutions} \label{SectionConeSolutions} The concrete asymptotics of the metrics depend on the following well known construction, cf. \cite{BohmNonCompactEinstein} or \cite{DHWShrinkingSolitons}. \begin{proposition} Let $(P, g_E)$ be a homogeneous space with $Ric = (n-1) g_E.$ Then the metrics \begin{align*} dt^2 & + \sin^2(t) g_E \ \text{ for } \ t \in (0, \pi), \\ dt^2 + t^2 g_E & \ \text{ and } \ dt^2 + \sinh^2(t) g_E \ \text{ for } \ t > 0 \end{align*} define cohomogeneity one Einstein metrics on $(0, \frac{\pi}{2}) \times P$ and $(0, \infty) \times P$ with Einstein constant $- \frac{\varepsilon}{2} = n, 0, -n,$ respectively. Any of these solutions will be called a {\em cone solution.} Furthermore, the Ricci flat metrics together with the soliton potential $- \dot{u}(t) = \frac{\varepsilon}{2}t$ induce a shrinking or expanding Ricci soliton on $(0, \infty) \times P$ depending on whether $\varepsilon < 0$ or $\varepsilon >0.$ If $\varepsilon < 0$ these solutions are called {\em conical Gaussians.} \label{ExplicitConeSolutionsProposition} \end{proposition} The above metrics have conical singularities at the singular orbits unless each singular orbit consists of a point. In this case the metrics correspond to the standard metrics on $S^{n+1},$ $\mathbb{R}^{n+1}$ and $\mathbb{H}^{n+1},$ respectively. To obtain concrete formulae in the two summands case, the following definitions are required. \begin{definition} Positive solutions $(c_1, c_2)$ to the equations \begin{align} (n-1) d_1 = \frac{A_1}{c_1^2} + A_3 \frac{c_1^2}{c_2^4} \ \text{ and } \ (n-1) d_2 = \frac{A_2}{c_2^2} - 2 A_3 \frac{c_1^2}{c_2^4} \label{DefConeSolutionOne} \end{align} are called {\em cone solutions}. \label{DefinitionConeSolutions} \end{definition} \begin{remarkroman} If $A_3 >0,$ the cone solutions take the explicit form \begin{align*} c_1^2 & = \frac{1}{2d_1+d_2} \left( \frac{A_2^2 d_1 + 4 A_1 A_3 (2d_1+d_2)}{2 A_3 (n-1)(2d_1+d_2)} \mp \sqrt{D} \right), \\ c_2^2 & = \frac{1}{2d_1+d_2} \left( A_2 n \pm 2 A_3 (2d_1+d_2) \sqrt{D} \right), \end{align*} where the discriminant $D$ is given by \begin{equation} D = \left( \frac{A_2}{2 A_3} \frac{d_1}{2d_1+d_2} \right)^2 - \frac{A_1}{A_3} \frac{d_2}{2d_1+d_2}. \label{DiskriminatConeSolutions} \end{equation} Inserting the geometric definitions of the constants $A_1,$ $A_2,$ $A_3$ into \eqref{DiskriminatConeSolutions}, one obtains \begin{align*} D \geq 0 \ \text{ if and only if } \ \frac{(\mathbb{R}ic^{G/H})^2}{4 ||A||^2} \geq (2d_1+d_2) \frac{d_1-1}{d_1}. \end{align*} Suppose that there are two real cone solutions. For a cone solution $(c_1, c_2),$ set $\omega=\frac{c_1}{c_2}.$ Then the ordering $\omega_1 < \omega_2$ defines the {\em first} and {\em second} cone solution. In particular, if $\widehat{D}>0,$ cf. \eqref{DefinitionDHat}, there exist two cone solutions and it is easy to check that $\omega_1 < \hat{\omega}_1 < \omega_2 < \hat{\omega}_2$ in this case. This has also been observed by B\"ohm \cite{BohmInhomEinstein}. \label{ExplicitFormulaConeSolutions} \end{remarkroman} Let $D \geq 0$ and let $(c_1, c_2)$ be a cone solution as in definition \ref{DefinitionConeSolutions}. With the normalisation of the Einstein constant $- \frac{\varepsilon}{2} \in \left\{ -n, 0, n \right\},$ the two summands Einstein cone solutions of proposition \ref{ExplicitConeSolutionsProposition} take the form \begin{align} f_i(t)= c_i \sin(t) \ \text{ for } \ t \in (0, \pi) \label{ExplicitConeSolutionScalPos} \end{align} in the case of positive scalar curvature and \begin{align} f_i(t)= c_i t \ \text{ and } \ f_i(t)= c_i \sinh(t) \ \text{ for } \ t > 0 \label{ExplicitConeSolutionScalNonPos} \end{align} in the Ricci flat and negative scalar curvature case, respectively. Any cone solution is called {\em first} cone solution if that is the case for the pair $(c_1,c_2)$ as remark \ref{ExplicitFormulaConeSolutions}. \begin{example} \normalfont Recall the examples of group diagrams in table \ref{HopfFibrationsTable}, which induce the Hopf fibrations. In the $\mathbb{H}P^{m+1}$-example the cone solutions are \begin{align*} c_1^2 & = \frac{9+14m+4m^2}{(1+2m)(3+2m)^2} \ \text{ and } \ c_2^2 = \frac{9+14m+4m^2}{(1+2m)(3+2m)}, \\ c_1^2 & = c_2^2 = 1, \end{align*} in the $F^{m+1}$-example they are given by \begin{align*} c_1^2 & = \frac{(1+m)^2+m}{(1+m)^2(1+4m)} \ \text{ and } \ c_2^2 = 4 \frac{(1+m)^2 + m}{(2m+1)^2+m}, \\ c_1^2 & = \frac{1+m}{1+4m} \ \text{ and } \ c_2^2 = 4 c_1^2, \end{align*} and in the $CaP^2$-example they are $c_1^2 = \frac{57}{121},$ $c_2^2 = \frac{19}{11}$ and $c_1^2 = c_2^2 = 1.$ In all cases, the first pair also describes the first cone solution. \label{ExamplesConeSolutionsFromHopfFibrations} \end{example} The following elementary but useful characterisation of $\omega_1$ and $\omega_2$ is immediate from definition \ref{DefinitionConeSolutions} and remark \ref{ExplicitFormulaConeSolutions}. \begin{proposition} Let $D > 0.$ Then the two positive roots of the function \begin{equation} f( \omega ) = \frac{A_1}{d_1} - \frac{A_2}{d_2} \omega^2 + A_3 \left( \frac{1}{d_1} + \frac{2}{d_2} \right) \omega^4 \label{FunctionDeterminingSecondDerivativeOmega} \end{equation} are the ratios $\omega_1,$ $\omega_2$ of the first and second cone solution, respectively, i.e. \begin{align*} \omega_{1}^2 = \frac{A_2}{2A_3} \frac{d_1}{2d_1+d_2} - \sqrt{ D } \ \ \text{and} \ \ \omega_{2}^2 = \frac{A_2}{2A_3} \frac{d_1}{2d_1+d_2} + \sqrt{ D }. \end{align*} In particular, it follows that $\omega_1^2 < \frac{A_2}{4A_3}$ and $\omega_2^2 < \frac{A_2}{2A_3}.$ \label{CharacterisationOfConeSolutionRatio} \end{proposition} \subsubsection{Steady Ricci solitons} \label{SteadyRSAymptoticsSection} The rotationally symmetric Bryant soliton on $\mathbb{R}^n,$ $n \geq 3,$ is asymptotically paraboloid and therefore non-collapsed. It will be shown that this is also the case for the non-trivial steady Ricci solitons constructed in section \ref{CompletenessTwoSummands}. Recall from proposition \ref{CompleteSteadyRSAsymptotics} that on a complete, non-trivial cohomogeneity one steady Ricci soliton there holds $-\dot{u}(t) \to \sqrt{-C}$ as $t \to \infty$ and $0 < \tr(L) \leq \frac{n}{t}$ for $t >0.$ Therefore, if the shape operator remains positive definite, it follows that $\frac{\dot{f}_i}{f_i} \to 0$ as $t \to \infty.$ According to corollary \ref{FlowExistsForAllTimes}, this automatically holds in the two summands case if $d_1 > 1$ and $\widehat{D}>0.$ In order to obtain the concrete asymptotics of the metric if $A_3 >0$, an understanding of the long time behaviour of $\omega$ is essential: \begin{proposition} Let $d_1 >1,$ $\widehat{D}>0$ and $\varepsilon = 0.$ Then along trajectories of non-trivial steady Ricci solitons the limit $\omega_{\infty} = \lim_{t \to \infty} \omega(t)$ exists and $\lim_{t \to \infty} \dot{\omega}(t) = 0.$ \label{OmegaConverges} \end{proposition} \begin{proof} Let $v(t) = \sqrt{\det g_t} = f_1^{d_1}(t) f_2^{d_2}(t)$ denote the relative volume of the principal orbit and consider the variables $\frac{v^{1/n}}{f_i}$ and $\frac{v^{2/n} \dot{f}_i}{f_i}$ for $i=1,2.$ Observe that $\frac{v^{1/n}}{f_1} = \frac{1}{\omega^{d_2/n}}$ and $\frac{v^{1/n}}{f_2} = \omega^{d_1/n}.$ Therefore, the B\"ohm functional has the lower bound \begin{align*} \mathscr{F}_0 = v ^{\frac{2}{n}} \left( \tr(r_t) + \tr(( L^{(0)})^2 )\right) \geq v ^{\frac{2}{n}} \tr(r_t) = \frac{A_1}{\omega^{2 d_2/n}} + A_2 \omega^{2 d_1/n} - A_3 w^{2(2 d_1 + d_2)/n}. \end{align*} Since $\mathscr{F}_0$ is non-increasing, $v ^{\frac{2}{n}} \tr(r_t)$ is bounded from above for $t \geq t_0 > 0$ and hence $\omega$ is bounded away from zero for these $t.$ As $\omega < \hat{\omega}_1,$ the variables $\frac{v^{1/n}}{f_i}$ are hence bounded for $t \geq t_0.$ Furthermore, the variables $\frac{v^{2/n} \dot{f}_i}{f_i}$ satisfy the ODE system \begin{align*} \frac{d}{dt} \frac{v^{2/n} \dot{f}_1}{f_1} & = - ( - \dot{u} + \frac{n-2}{n} \tr(L) ) \frac{v^{2/n} \dot{f}_1}{f_1} + \frac{A_1}{d_1} \left( \frac{v^{1/n}}{f_1} \right)^2 + \frac{A_3}{d_1} \omega^2 \left( \frac{v^{1/n}}{f_2} \right)^2, \\ \frac{d}{dt} \frac{v^{2/n} \dot{f}_2}{f_2} & = - ( - \dot{u} + \frac{n-2}{n} \tr(L) ) \frac{v^{2/n} \dot{f}_2}{f_2} + \frac{A_2}{d_2} \left( \frac{v^{1/n}}{f_2} \right)^2 - \frac{2 A_3}{d_2} \omega^2 \left( \frac{v^{1/n}}{f_2} \right)^2. \end{align*} Due to the known asymptotics, the coefficient of $\frac{v^{2/n} \dot{f}_i}{f_i}$ tends to $- \sqrt{-C}$ and the remaining polynomial terms are bounded. Hence, by comparison, the variables $\frac{v^{2/n} \dot{f}_i}{f_i}$ remain bounded. Therefore, one can pass to the $\omega$-limit set $\Omega.$ Due to its monotonicity, $\mathscr{F}_0$ converges. Its derivative \eqref{DerivativeOfBohmFunctional} has to vanish on $\Omega$ and therefore the limiting value $\mathscr{F}_0 = (v ^{\frac{2}{n}} \tr(r))_{\infty}$ can be expressed in terms of $\omega$ as above. In particular, $\omega$ converges. The asymptotics of $\dot{\omega}$ simply follow from the ODE $\dot{\omega} = \omega \left\lbrace \frac{\dot{f}_1}{f_1} - \frac{\dot{f}_2}{f_2} \right\rbrace$ and the fact that $\frac{\dot{f}_i}{f_i} \to 0$ as $t \to \infty.$ \end{proof} \begin{remarkroman} It is also possible to derive an integral formula for $\dot{\omega}.$ Indeed, it is straightforward to check that \begin{align*} \frac{d}{dt} \left\lbrace \dot{\omega} e^{-u} f_1^{d_1-1} f_2^{d_2+1} \right\rbrace = f(\omega) e^{-u} f_1^{d_1-2} f_2^{d_2} , \end{align*} where $f(\omega)$ is defined in \eqref{FunctionDeterminingSecondDerivativeOmega}. If $d_1 > 1,$ it follows that \begin{align*} \dot{\omega}(t) = \frac{e^u(t)}{f_1^{d_1-1}(t) f_2^{d_2+1}(t)} \cdot \int_{0}^{t} f( \omega(s) ) e^{-u(s)} f_1^{d_1-2}(s) f_2^{d_2}(s) ds. \end{align*} Since $f_1, f_2$ are monotonic and $e^{u(t)} \int_{0}^t e^{-u(s)} ds \to \frac{1}{\sqrt{-C}}$ as $t \to \infty$ due to L'H\^{o}pital's rule, one has the bound $\dot{\omega}(t) \leq \overline{C} \cdot \frac{1}{f_1(t) f_2(t)}$ for some constant $\overline{C} > 0.$ \label{IntegralFormulaOmegaDot} \end{remarkroman} Now the asymptotics of the metric can be deduced: \begin{proposition} Let $d_1>1,$ $A_1>0, \widehat{D}>0$ and suppose that $(d_1+1) \ddot{u}(0) = C < 0.$ Then the corresponding two summands steady Ricci soliton metrics satisfy \begin{align*} - \dot{u}(t) \to \sqrt{-C} \ \text{ and } \ \frac{f_i^2(t)}{t} \to \frac{2}{\sqrt{-C}}(n-1) c_i^2 \end{align*} as $t \to \infty,$ where $(c_1, c_2)$ denotes the first cone solution. In particular, $\omega \to \omega_1$ as $t \to \infty.$ \label{SummarisingAsymptoticsSteadyRicciSoliton} \end{proposition} \begin{proof} Recall that $- \dot{u}(t) \to \sqrt{-C}$ as $t \to \infty$ due to proposition \ref{CompleteSteadyRSAsymptotics}. Notice that $f_1,$ $f_2$ satisfy \begin{align*} \ddot{f}_1 & = - ( - \dot{u} - d_2 \frac{\dot{\omega}}{\omega} ) \dot{f}_1 - (n-1) \frac{\dot{f}_1^2}{f_1} + \frac{A_1+A_3 \omega^4}{d_1 f_1}, \\ \ddot{f}_2 & = - ( - \dot{u} + d_1 \frac{\dot{\omega}}{\omega} ) \dot{f}_2 - (n-1) \frac{\dot{f}_2^2}{f_2} + \frac{A_2-2 A_3 \omega^2}{d_2 f_2}. \end{align*} As $\omega < \hat{\omega}_1 < \sqrt{\frac{A_2}{2 A_3}},$ $A_1 > 0$ and $\omega$ converges, both $f_1, f_2$ satisfy a differential equation of the form \begin{align*} \ddot{f} = - a_1 \dot{f} - (n-1) \frac{\dot{f}^2}{f} + \frac{a_2}{2 f}, \end{align*} where $a_i \colon [0,\infty) \to \mathbb{R}$ are smooth functions with $\lim_{t \to \infty} a_i(t) = a_i^{\ast} >0.$ Set $A= \min \lbrace a_1^{\ast}, a_2^{\ast}\rbrace.$ It is shown in \cite[Lemma 6.2]{AppletonSteadyRS} that for every $\varepsilon \in (0, A)$ and every solution $f: [0, \infty) \to \mathbb{R}$ with $f(0), \dot{f}(0)>0$ there exists $t_0>0$ such that \begin{align*} f(t_0)^2 + \gamma_{-} \left(1+ \varepsilon \right)^{-1} \left( t-t_0 \right) \leq f^2(t) \leq f(t_0)^2 + \gamma_{+} \left(t-t_0 \right) \end{align*} for all $t >t_0$, where $\gamma_{\pm} = \frac{a_2^{\ast} \pm \varepsilon}{a_1^{\ast} \mp \varepsilon}$. It follows that $\frac{\gamma_{1,{-}}}{\gamma_{2,{+}}} \leq \omega_{\infty}^2 \leq \frac{\gamma_{1,{+}}}{\gamma_{2,{-}}}$ for every sufficiently small $\varepsilon >0.$ In the limit as $\varepsilon \to 0$ one obtains equality and thus \begin{align*} \omega_{\infty}^2 =\frac{d_2}{d_1} \frac{A_1+A_3 \omega_{\infty}^4}{A_2 - 2 A_3 \omega_{\infty}^2}. \end{align*} In particular, $0 < \omega_{\infty} \leq \hat{\omega}_1$ is a root of $f(\omega)$ and due the characterisation \ref{CharacterisationOfConeSolutionRatio} of the cone solutions, it follows that $\omega_{\infty} = \omega_1$ is the ratio of the first cone solution. The asymptotic behaviour of $f_1,$ $f_2$ now follows with the formulae in definition \ref{DefinitionConeSolutions}. \end{proof} \begin{remarkroman} For $d_1 >1,$ $\widehat{D}>0$ and $\varepsilon = 0,$ the asymptotics of the rescaled Ricci soliton ODE of section \ref{CompletenessTwoSummands} are \begin{align*} X_1, X_2 \to 0 \ \text{ and } \ Y_1, Y_2 \to 0 \ \text{ and } \ \mathcal{L} \to \frac{1}{\sqrt{-C}} \end{align*} as $s \to \infty.$ \label{VariablesBoundedSteadySolitonTwoSummandsCase} \end{remarkroman} \subsubsection{Expanding Ricci solitons} \label{ExpandingRSAymptoticsSection} It will be shown that the expanding Ricci solitons are asymptotically conical at infinity and the soliton potential grows quadratically at infinity. Recall from \eqref{GeneralAsymptoticsExpandingRS} that on a complete, non-trivial cohomogeneity one expanding Ricci soliton $-\dot{u}$ is asymptotically linear and the mean curvature of the principal orbit is bounded. Furthermore, corollary \ref{FlowExistsForAllTimes} implies that the shape operator is positive definite in the two summands case. The definition of the rescaled variables in \eqref{RescaledTwoSummandsVariables} thus implies: \begin{proposition} Suppose that $d_1 >1$ and $\widehat{D} > 0$ and consider the flow of the Ricci soliton ODE in the phase space of expanding Ricci solitons. Then \begin{align*} X_1, X_2 \to 0 \ \text{ and } \ Y_1, Y_2 \to 0 \ \text{ and } \ \mathcal{L} \to 0 \end{align*} as $s \to \infty.$ \end{proposition} A modification of the discussion in \cite{DWExpandingSolitons} can now be used to deduce the claimed asymptotically conical geometry at infinity: \begin{proposition} Suppose that $d_1 >1$ and $\widehat{D} > 0.$ Then along trajectories corresponding to non-trivial expanding Ricci soliton metrics the soliton potential and shape operator satisfy \begin{align*} \frac{- \dot{u}(t)}{t} \to \frac{\varepsilon}{2} \ \text{ and } \ t \cdot L_t \to \mathbb{I}_n \end{align*} as $t \to \infty.$ \label{AsymptoticsOfExpandingTwoSummandsRS} \end{proposition} \begin{proof} Consider the ODE system \begin{align*} \frac{d}{ds} \frac{X_1}{\mathcal{L}^2} & = \left(- \sum_{i=1}^2 d_i X_i^2 -1 \right) \frac{X_1}{\mathcal{L}^2} + \frac{\varepsilon}{2} \left( 1 + X_1 \right) + \frac{A_1}{d_1} \left( \frac{Y_1}{\mathcal{L}} \right)^2 + \frac{A_3}{d_1} \left( \frac{Y_2}{\mathcal{L}} \right)^2 \left( \frac{Y_2}{Y_1} \right)^2, \\ \frac{d}{ds} \frac{X_2}{\mathcal{L}^2} & = \left(- \sum_{i=1}^2 d_i X_i^2 -1 \right) \frac{X_2}{\mathcal{L}^2} + \frac{\varepsilon}{2} \left( 1 + X_2 \right) + \frac{A_2}{d_2} \left( \frac{Y_2}{\mathcal{L}} \right)^2 - \frac{2 A_3}{d_2} \left( \frac{Y_2}{\mathcal{L}} \right)^2 \left( \frac{Y_2}{Y_1} \right)^2 \end{align*} and notice that $\frac{d}{ds} \frac{Y_i}{\mathcal{L}} = - \frac{Y_i}{\mathcal{L}} X_i$ implies that both limits $\hat{y}_i = \lim_{s \to \infty} \frac{Y_i}{\mathcal{L}} \in [0, \infty)$ exist. If $\hat{y}_1 = 0$ then, as $\omega = \frac{Y_2}{Y_1}$ remains bounded, one necessarily also has $\hat{y}_2 = 0.$ In this case one can proceed as in \cite[Lemma 3.15]{DWExpandingSolitons} to show that $\frac{X_i}{\mathcal{L}^2} \to \frac{\varepsilon}{2}$ as $s \to \infty$ because the extra terms involving $A_3$ tend to zero. Similarly, integrating the ODE for $\mathcal{L}$ implies $\mathcal{L}^2 \cdot s \to \frac{1}{\varepsilon}$ as $s \to \infty.$ Since $dt = \mathcal{L} ds,$ this yields $s \sim \frac{\varepsilon}{4} t^2$ and hence $\mathcal{L} \cdot t \to \frac{2}{\varepsilon}$ as $t \to \infty$. The claim then follows from the definition of the coordinate change in \eqref{RescaledTwoSummandsVariables}. It remains to rule out that possibly $\hat{y}_1 > 0.$ In this case the existence of $\omega_{\infty} = \lim_{s \to \infty} \frac{Y_2}{Y_1}$ is immediate. Hence the ODEs imply \begin{align*} \frac{X_1}{\mathcal{L}^2} \to \frac{\varepsilon}{2} + \frac{1}{d_1} \left( A_1 \hat{y}_1^2+A_3 \hat{y}_2^2 \omega_{\infty}^2 \right) \ \text{ and } \ \frac{X_2}{\mathcal{L}^2} \to \frac{\varepsilon}{2} + \frac{1}{d_2} \left( A_2 \hat{y}_2^2 - 2 A_3 \hat{y}_2^2 \omega_{\infty}^2 \right) \end{align*} as $s \to \infty$ and both limits are positive as $\varepsilon > 0$ and $\omega_{\infty}^2 < \frac{A_2}{2 A_3}.$ Set $\Lambda_i = \lim_{s \to \infty} \frac{X_i}{\mathcal{L}^2} > 0.$ It follows that \begin{align*} \frac{Y_i^{'}}{\mathcal{L}^{'}} = \frac{Y_i \left(\sum_{i=1}^{2} d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 - X_i\right)}{\mathcal{L}\left(\sum_{i=1}^{2} d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 \right)} \to \hat{y}_i \cdot \frac{\varepsilon + 2 \Lambda_i}{\varepsilon} \end{align*} as $s \to \infty,$ but if $\hat{y}_1 > 0$ L'H\^{o}pital's rule implies $\Lambda_1=0$ and thus a contradiction. \end{proof} \subsubsection{Ricci flat metrics} \label{SubSectionRFMetrics} The rescaled coordinates of section \ref{CompletenessTwoSummands} are particularly suited to analyse the Ricci flat trajectories. The induced Ricci flat metric is asymptotically conical and in fact is asymptotic to the first cone solution $f_i(t) = c_i t.$ This also follows from B\"ohm's \cite{BohmNonCompactEinstein} original construction, see remark \ref{RemarkConvergenceConeSolutions}. \begin{proposition} Let $d_1 >1$ and $\widehat{D}>0.$ Along trajectories of the Ricci flat system $X_i \to \frac{1}{n}$ and $Y_i \to \frac{1}{n c_i}$ as $s \to \infty,$ where $(c_1,c_2)$ denotes the first cone solution. \label{VariablesBoundedRicciFlatTwoSummandsCase} \end{proposition} \begin{proof} Recall from corollary \ref{FlowExistsForAllTimes} that the variables $X_i, Y_i$ for $i=1,2$ are all positive and bounded along the flow since $d_1 >1$ and $\widehat{D}>0.$ To deduce the asymptotics, consider the function \begin{equation*} \mathcal{G}= Y_1^{d_1} Y_2^{d_2}, \end{equation*} which is in fact the inverse of B\"ohm's Lyapunov \eqref{BohmFunctional} in the $X$-$Y$-coordinates. Its derivative is given by \begin{equation*} \mathcal{G}^{'} = n \mathcal{G} \left\{ \sum_{i=1}^2 d_i X_i^2 - \frac{1}{n} \right\} \end{equation*} and hence it is non-decreasing and bounded. Thus, it converges to a finite positive limit as $s \to \infty.$ This also shows that $Y_1, Y_2$ are bounded away from zero as $s \to \infty.$ Standard ODE theory now implies that the $\omega$-limit set $\Omega$ of the flow of $X_1, X_2,$ $Y_1, Y_2$ is non-empty, compact, connected and flow invariant. As $\mathcal{G}$ is monotonic and bounded, it must be constant on $\Omega.$ But since $d_1 X_1 + d_2 X_2 =1$ there holds $\mathcal{G}^{'} = 0$ if and only if $X_1 = X_2 = \frac{1}{n}.$ Moreover, this yields \begin{align*} 0 & = X_1^{'} = \frac{1}{n} \left( \frac{1}{n} - 1 \right) + \frac{A_1}{d_1} Y_1^2 + \ \frac{A_3}{d_1} \frac{Y_2^4}{Y_1^2}, \\ 0 & = X_2^{'} = \frac{1}{n} \left( \frac{1}{n} - 1 \right) + \frac{A_2}{d_2} Y_2^2 - 2 \frac{A_3}{d_2} \frac{Y_2^4}{Y_1^2} \end{align*} on the $\omega$-limit set. In particular, the pair $((n Y_1)^{-1}, (n Y_2)^{-1})$ satisfies the equations \eqref{DefConeSolutionOne} of the cone solutions. Since the bound $Y_2/Y_1 < \hat{\omega}_1$ holds along the flow and $\omega_1 < \hat{\omega}_1 < \omega_2 < \hat{\omega}_2,$ it follows that $Y_i \to \frac{1}{nc_i}$ as $s \to \infty,$ where $(c_1,c_2)$ describes the first cone solution. This completes the proof. \end{proof} The asymptotic behaviour of the metric can be deduced from $\dot{f}_i = \frac{X_i}{Y_i} \to c_i$ as $t \to \infty$. The metric is therefore asymptotically conical at infinity. \subsubsection{Ricci flat metrics: Explicit trajectories and rotational behaviour} \label{RicciFlatPlanarSystem} It is a special feature of the Ricci flat equation that it reduces to a planar system for the variables $X_1, Y_1$ as the variable $\mathcal{L}$ decouples completely. More generally, in the Einstein case there holds $X_2= \frac{1}{d_2}(1 - d_1 X_1)$ and the conservation law \eqref{GeneralTwoSummandsConsLaw} then determines $Y_2$ in terms of $X_1, Y_1$ and $\frac{\varepsilon}{2} \mathcal{L}^2.$ Explicitly, it is given by \begin{equation*} Y_2^2 = \frac{A_2}{2 A_3} Y_1^2 \pm \frac{1}{2 A_3} \sqrt{A_2^2 Y_1^4 + 4 A_3 \left( \sum_{i=1}^2 d_i X_i^2 +(n-1) \frac{\varepsilon}{2} \mathcal{L}^2 - 1 + A_1 Y_1^2 \right) Y_1^2 }. \end{equation*} Initially, $Y_2$ is given by the solution corresponding to '$-$' as $\lim_{s \to - \infty} Y_2 =0.$ Notice that the discriminant vanishes if and only if $Y_2^2/Y_1^2= \frac{A_2}{2 A_3}$ and recall that if $d_1 >1,$ $\widehat{D} > 0$ and $\varepsilon \geq 0$ the estimate $Y_2^2/Y_1^2 < \hat{\omega}_1^2 < \frac{A_2}{2 A_3}$ has been established. Hence, in this case, only the '$-$' solution is realised by the flow. Therefore, consider the ODE system \begin{align*} X_1^{'} = \ & \left( X_1 + \frac{1}{d_1} \right) \left( n \frac{d_1}{d_2} X_1^2 - 2 \frac{d_1}{d_2} X_1 + \frac{1}{d_2} - \frac{\varepsilon}{2} \mathcal{L}^2 - 1 \right) \\ & \ + \left( 2 A_1 + \frac{A_2^2}{2 A_3} \right) \frac{Y_1^2}{d_1}+ \frac{\varepsilon}{2}\left( 1 + \frac{n}{d_1} \right) \mathcal{L}^2 \\ & \ - \frac{A_2}{2 d_1 A_3} \sqrt{A_2^2 Y_1^4 + 4 A_3 \left( \sum_{i=1}^2 d_i X_i^2 +(n-1) \frac{\varepsilon}{2} \mathcal{L}^2 - 1 + A_1 Y_1^2 \right) Y_1^2 }, \\ Y_1^{'} = & \ Y_1 \left( n \frac{d_1}{d_2} X_1^2 - 2 \frac{d_1}{d_2} X_1 + \frac{1}{d_2} - \frac{\varepsilon}{2} \mathcal{L}^2 - X_1 \right), \\ \mathcal{L}^{'} = & \ \mathcal{L} \ \left( n \frac{d_1}{d_2} X_1^2 - 2 \frac{d_1}{d_2} X_1 + \frac{1}{d_2} - \frac{\varepsilon}{2} \mathcal{L}^2 \right). \end{align*} In the Ricci flat case this yields indeed a $2$-dimensional system for $X_1$ and $Y_1.$ Moreover, one has $\mathcal{L}(s) = \mathcal{L}(s_0) \exp \left[ \int_{s_0}^s \left( n \frac{d_1}{d_2} X_1^2 - 2 \frac{d_1}{d_2} X_1 + \frac{1}{d_2} \right) d \tau \right].$ Recall from proposition \ref{VariablesBoundedRicciFlatTwoSummandsCase} that one expects $(X_1,Y_1) \to (\frac{1}{n}, \frac{1}{n c_1})$ as $s \to \infty$ if the cone solutions are real. To study the dynamics of the planar $(X_1, Y_1)$-system close to the stationary point $(\frac{1}{n},\frac{1}{n c_1}),$ consider its linearisation at that point. It is described by the matrix \begin{equation*} \begin{pmatrix} -\frac{n-1}{n} & 2 \frac{c_1}{n}\left[ n-1 - 2 A_3 \frac{c_1^2}{c_2^4}\left( \frac{1}{d_1} + \frac{1}{d_2} \right) \right] \\ -\frac{1}{c_1 n} & 0 \end{pmatrix}. \end{equation*} The eigenvalues are the solutions to the quadratic equation \begin{equation*} \lambda^2 + \frac{n-1}{n} \lambda + \frac{2}{n^2} \left[ n-1 - 2 A_3 \frac{c_1^2}{c_2^4}\left( \frac{1}{d_1} + \frac{1}{d_2} \right) \right] = 0 \end{equation*} and it is therefore easy to deduce: \begin{corollary} The limiting point of the Ricci flat trajectories is a stable spiral if and only if \begin{equation} \frac{(n-1)(n-9)}{8} + 2 A_3 \frac{c_1^2}{c_2^4} \left( \frac{1}{d_1} + \frac{1}{d_2} \right) < 0. \end{equation} In particular, if $A_3 =0$ this is equivalent to $2 \leq n \leq 8.$ Otherwise, it is a stable node. \label{StableSpiral} \end{corollary} The reduction to the planar $(X_1,Y_1)$-system can also be used to describe explicit trajectories. Trajectories which correspond to smooth complete Ricci flat metrics must emanate from $( \frac{1}{d_1}, \frac{1}{d_1} )$ and are expected to converge to $( \frac{1}{n}, \frac{1}{n c_1} ).$ In low dimensional examples, these trajectories are actually realised by straight lines! This can be seen by introducing polar coordinates centred at $( \frac{1}{d_1}, \frac{1}{d_1} )$, and a straightforward calculation verifies that the angle remains constant. This provides a new coordinate representation of metrics of special holonomy considered by Bryant-Salamon \cite{BSExceptionalHolonomy} and Gibbons-Page-Pope \cite{GPPEinsteinOnSphereR3R4bundles}. \begin{theorem} On the open disc bundles associated to the group diagrams $G=Sp(2),$ $H=Sp(1) \times Sp(1),$ $K= U(1) \times Sp(1)$ and $G=Sp(1) \times Sp(2),$ $H=Sp(1) \times Sp(1) \times Sp(1),$ $K=Sp(1) \times Sp(1)$ the trajectories of the complete Ricci flat two summands metrics are line segments when represented in the above coordinate system. \label{ExplicitRFTrajectories} \end{theorem} \subsubsection{Einstein metrics with negative scalar curvature} \label{SubSectionAsymptoticsEinsteinScalNeg} It will be shown that in this case the B\"ohm functional $\mathscr{F}_0$ asymptotically approaches the value of the first cone solution, and hence work of B\"ohm implies that the metric is in fact asymptotic to the first cone solution $f_i(t) = c_i \sinh(t)$. \begin{proposition} Let $d_1 >1$ and $\widehat{D}>0.$ Then the asymptotic behaviour of trajectories corresponding to complete Einstein metrics with negative scalar curvature is given by \begin{align*} X_1, X_2 \to \frac{1}{n} \ \text{ and } \ Y_1, Y_2 \to 0, \ \omega = \frac{Y_2}{Y_1} \to \omega_1 \ \text{ and } \ \mathcal{L} \to \sqrt{\frac{2}{n \varepsilon}} \end{align*} as $s \to \infty,$ where $\omega_1 = \frac{c_1}{c_2}$ is the ratio of the first cone solution. Furthermore, $\mathscr{F}_0 \to n(n-1) c_{1}^{{2d_1}/{n}} c_{2}^{{2d_2}/{n}}$ as $s \to \infty,$ which is the value of $\mathscr{F}_0$ evaluated on the first cone solution $(c_1, c_2).$ \label{TwoSummandsEinsteinMetricsWithNegativeScalarCurvature} \end{proposition} \begin{proof} As in the proof of corollary \ref{FlowExistsForAllTimes}, introduce the variable $\omega = \frac{Y_2}{Y_1}$ in order to view the Ricci soliton equation as an ODE with polynomial right hand side. Furthermore, all variables remain bounded along the flow and hence the $\omega$-limit set $\Omega$ is non-empty, connected, compact and flow-invariant. Recall from lemma \ref{LemmaLBoundedAwayFromZero} that $\mathcal{L}(s)$ is bounded away from zero for $s \geq 0.$ As the quotients $\frac{Y_i}{\mathcal{L}}$ satisfy $\frac{d}{ds} \frac{Y_i}{\mathcal{L}} = - \frac{Y_i}{\mathcal{L}} X_i,$ they are monotonically decreasing and hence converge as $s \to \infty.$ Moreover, the quotients are well-defined on $\Omega.$ Therefore their derivatives vanish, which implies $Y_i \cdot X_i = 0$ on $\Omega.$ But due to the bounds on $\mathcal{L}$ in lemma \ref{LemmaLBoundedAwayFromZero}, $X_1$ is bounded away from zero and in particular is non-zero on $\Omega.$ This implies $0 < Y_2 < \hat{\omega}_1 Y_1 \to 0$ as $s \to \infty.$ Now consider the evolution of the B\"ohm functional $\mathscr{F}_0 = v ^{\frac{2}{n}} \left( \tr(r_t) + \tr(( L^{(0)})^2 )\right),$ which was introduced in \eqref{BohmFunctional}. In the current coordinate system it is given by \begin{align*} \mathscr{F}_0 & = \prod_{i=1}^2 Y_i^{-2d_i / n} \left\lbrace \sum_{i=1}^2 A_i Y_i^2 - A_3 \frac{Y_2^4}{Y_1^2} + \sum_{i=1}^2 d_i X_i^2 - \frac{1}{n}\left( \sum_{i=1}^2 d_i X_i \right) ^2 \right\rbrace \\ & = \prod_{i=1}^2 Y_i^{-2d_i / n} \left\lbrace A_1 Y_1^2 + Y_2^2 \left( A_2 - A_3 \frac{Y_2^2}{Y_1^2} \right) + \sum_{i=1}^2 d_i X_i^2 - \frac{1}{n} \right\rbrace. \end{align*} Observe that it is bounded from below by zero as $\frac{Y_2}{Y_1} = \omega < \hat{\omega}_1 < \frac{A_2}{2 A_3}.$ Furthermore, according to \eqref{DerivativeOfBohmFunctional}, $\mathscr{F}_0$ is non-increasing and therefore converges as $s \to \infty.$ However, for $\mathscr{F}_0$ to be finite on the $\omega$-limit set $\Omega,$ one has to have $\sum_{i=1}^2 d_i X_i^2 = \frac{1}{n},$ which forces $X_1 = X_2 = \frac{1}{n}$ in the Einstein locus $\sum_{i=1}^2 d_i X_i = 1$ as $X_1, X_2 \geq 0.$ Therefore $X_1$ is constant on $\Omega$ and then also $\mathcal{L}$ due to the ODE for $X_1.$ Finally, the ODE for $\mathcal{L}$ itself shows that $\frac{\varepsilon}{2} \mathcal{L}^2 = \frac{1}{n}$ on $\Omega.$ To deduce the asymptotic behaviour of $\omega$, first observe that the monotonicity of $\mathscr{F}_0$ and \begin{align*} \mathscr{F}_0 & = \frac{1}{\omega^{2 d_2 / n}} \left( A_1 + A_2 \omega^2 - A_3 \omega^4 \right) + v^{2/n} \tr( ( L^{(0)})^2) \\ & = \frac{A_1}{\omega^{2 d_2/n}} + A_2 \omega^{2 d_1/n} - A_3 w^{2(2 d_1 + d_2)/n} + v^{2/n} \tr( ( L^{(0)})^2) \end{align*} imply that $\omega$ is bounded away from zero for $t \geq t_0 > 0.$ Notice furthermore that \begin{align*} \frac{d}{dt} v^{2/n} \tr( ( L^{(0)})^2) & = \frac{d}{dt} \mathscr{F}_0 - \frac{d}{dt} \left\lbrace \frac{A_1}{\omega^{2 d_2/n}} + A_2 \omega^{2 d_1/n} - A_3 w^{2(2 d_1 + d_2)/n} \right\rbrace \\ & = -2 \frac{n-1}{n} v^{2/n} \tr( ( L^{(0)})^2) - \frac{2 d_1 d_2}{n} \omega^{-2 d_2/n-1} f(\omega), \end{align*} where the polynomial $f(\omega)$ is defined in \eqref{FunctionDeterminingSecondDerivativeOmega}. Therefore, $v^{2/n} \tr( ( L^{(0)})^2)$ can be treated as an independent variable, which is nonnegative, bounded by $\mathscr{F}_0$ and satisfies a well-defined ODE on the $\omega$-limit set $\Omega.$ Since $\mathscr{F}_0$ takes a finite value on $\Omega$ and $\frac{d}{dt} \mathscr{F}_0 = -2 \frac{n-1}{n} v^{2/n} \tr( ( L^{(0)})^2),$ it follows that $v^{2/n} \tr( ( L^{(0)})^2) \to 0$ as $t \to \infty.$ This in turn implies $f(\omega) \to 0$ and thus $\omega \to \omega_1$ as $t \to \infty$ due to proposition \ref{CharacterisationOfConeSolutionRatio}. This also implies $\mathscr{F}_0 \to \frac{1}{\omega_1^{2 d_2 / n}} \left( A_1 + A_2 \omega_1^2 - A_3 \omega_1^4 \right)$ as $t \to \infty,$ which is easily seen to be the value of the first cone solution by using the identities in definition \ref{DefinitionConeSolutions}. \end{proof} Notice that $\frac{\dot{f}_i}{f_i} = \frac{X_i}{\mathcal{L}} \to \sqrt{\frac{\varepsilon}{2n}}$ as $t \to \infty$ immediately implies that $f_1, f_2$ grow exponentially at infinity. In fact, the metric is asymptotic to the first cone solution at infinity. This follows from a more general result of B\"ohm \cite[Corollary 2.4]{BohmNonCompactEinstein}: If the scalar curvature of the principal orbit is positive and $\mathscr{F}_0$ is bounded from below, then any Einstein trajectory that takes a constant value on $\mathscr{F}_0$ is a cone solution. An argument specifically adapted to the two summands case is given in the proof of proposition \ref{ConvergenceToConeSolution}, see also remark \ref{RemarkConvergenceConeSolutions} (a). \subsection{Convergence to cone solutions} \label{SectionConvergenceToConeSolutions} The results in sections \ref{SubSectionRFMetrics} and \ref{SubSectionAsymptoticsEinsteinScalNeg} show that the non-compact Ricci flat metrics and Einstein metrics with negative scalar curvature of section \ref{CompletenessTwoSummands} are asymptotic to the cone solutions at infinity. In this section it will be shown that the asymptotics of the {\em Ricci flat} trajectories also imply that the metric actually converges to the cone solution as the volume of the singular tends to zero, i.e. as $f_2(0) = \bar{f} \to 0.$ In fact this follows for any sign of the Einstein constant and recovers convergence results due to B\"ohm \cite{BohmInhomEinstein, BohmNonCompactEinstein}. In comparison to B\"ohm's work, the main technical simplification is that the proof does not rely on the Poincar\'e-Bendixson theorem, see also remark \ref{RemarkConvergenceConeSolutions}. Recall from \eqref{TwoSummandsMetric} that the metric is given by \begin{equation*} g_{M \setminus Q} = dt^2 +f_1(t)^2 g_S + f_2(t)^2 g_Q \end{equation*} away from the singular orbits. It follows from the results of Eschenburg-Wang \cite{EWInitialValueEinstein} that there exists a unique one parameter family $c_{\bar{f}}(t) = (f_1,\dot{f}_1,f_2,\dot{f}_2)(t)$ of solutions to the Einstein equations \eqref{CohomOneRSb}, \eqref{CohomOneRSc} with initial condition $c_{\bar{f}}(0)=(0,1,\bar{f},0)$ for any $\bar{f}>0.$ Moreover, B\"ohm \cite{BohmInhomEinstein} has observed that it depends {\em continuously} on the initial condition $\bar{f}>0.$ Notice also that \eqref{CohomOneRSc} implies $(d_1+1) \ddot{f}_2(0) = \frac{\varepsilon}{2} \bar{f} + \frac{A_2}{d_2} \frac{1}{\bar{f}} >0$ if either $\varepsilon \geq 0$ or $\bar{f}^2< - \frac{2}{\varepsilon} \frac{A_2}{d_2}$ and $\varepsilon < 0.$ However, the equations are a priori not well defined if $\bar{f}=0.$ This singular condition corresponds geometrically to the collapse of the full principal orbit. To describe the behaviour of the Einstein equations as the volume of the singular orbit tends to zero more concretely, the following observation is key: In the $(X_i,Y_i, \mathcal{L})$-coordinate system defined in \eqref{RescaledTwoSummandsVariables}, the initial condition $(0,1,\bar{f},0)$ of the trajectory $c_{\bar{f}}$ corresponds to the stationary point \eqref{InitialCriticalPoint}, which is independent of $\bar{f}.$ Furthermore, the initial condition $f_2(0)=\bar{f}$ can be recovered via $\bar{f} = \lim_{s \to - \infty} \frac{\mathcal{L}}{Y_2}.$ In particular, $\bar{f} = 0$ is the limit of trajectories with $\mathcal{L} \equiv 0.$ However, the two coordinate systems are only equivalent along trajectories with $\mathcal{L} > 0.$ Nonetheless, due to the continuous dependence on the initial condition, any trajectory with $\mathcal{L} \equiv 0$ can hence be viewed as a continuous limit of Einstein trajectories. Hence, the collapse $\bar{f} \to 0$ is described in the $(X_i,Y_i, \mathcal{L})$-coordinates by the solution of the {\em Ricci flat} equations. By construction this solution lies in the unstable manifold of \eqref{InitialCriticalPoint} and due to proposition \ref{NumberOfParameterFamilies} it is indeed unique. Furthermore, due to the uniqueness of solutions $c_{\bar{f}}$ of the Einstein equations with initial condition $c_{\bar{f}}(0)=(0,1,\bar{f},0)$, one might expect that the limit as $\bar{f} \to 0$ is a cone solution. This intuition is confirmed in proposition \ref{ConvergenceToConeSolution}. In the case of Einstein metrics with positive scalar curvature, the proof of proposition \ref{ConvergenceToConeSolution} requires the concept of {\em maximal volume orbits:} Notice that the volume $V$ of the principal orbit satisfies $\dot{V}=V \tr(L),$ where $\tr(L)$ is the mean curvature. Along trajectories corresponding to Einstein metrics with positive scalar curvature, every critical point of $V$ is a maximum or a singular orbit is reached. Therefore, if the maximal volume orbit exists, it is unique and characterised by $\tr(L) = 0.$ In the two summands case, if $A_3 = 0,$ due to a result of B\"ohm \cite[section 4, (e)]{BohmInhomEinstein}, the maximal volume orbit always exists. An alternative argument is discussed below, mainly to introduce a natural coordinate system which extends past the maximal volume orbit. \begin{lemma} If $A_3 = 0$ and $\varepsilon <0,$ then any Einstein trajectory has a maximal volume orbit. \label{ExistenceMaxVolumeOrbit} \end{lemma} \begin{proof} In analogy to \eqref{RescaledTwoSummandsVariables}, introduce the variables \begin{equation} \widehat{X}_i = \frac{\dot{f}_i}{f_i}, \ \widehat{Y}_i = \frac{1}{f_i}, \ \text{ for } \ i=1,2, \ \text{ and } \ \widehat{\mathcal{L}} = {\tr(L)}. \label{HatVariables} \end{equation} Due to the assumption $A_3=0$ the two summands Einstein equations take the form \begin{align*} \frac{d}{dt} & \widehat{X}_i = - \widehat{X}_i \widehat{\mathcal{L}}+ \frac{A_i}{d_i} \widehat{Y}_i^2 + \frac{\varepsilon}{2} \\ \frac{d}{dt} & \widehat{Y}_i = - \widehat{X}_i \widehat{Y}_i, \\ \frac{d}{dt} & \widehat{\mathcal{L}} = \frac{\varepsilon}{2} - \sum_{i=1}^2 d_i \widehat{X}_i^2 \end{align*} and the conservation law is \begin{equation} \sum_{i=1}^2 d_i \widehat{X}_i^2 + \sum_{i=1}^2 A_i \widehat{Y}_i^2 + (n-1) \frac{\varepsilon}{2} = \widehat{\mathcal{L}}^2. \label{UnrescaledConsLaw} \end{equation} Notice that the time slice has not been rescaled and that the conservation law \eqref{UnrescaledConsLaw} and $\widehat{\mathcal{L}} = \sum_{i=1}^2 d_i \widehat{X}_i$ describe the rescaled Einstein locus \eqref{EinsteinLocus}. Clearly the above system is an ODE system with polynomial right hand side. In particular, a solution can only develop a finite time singularity if the norm of $( \widehat{X}_i, \widehat{Y}_i, \widehat{\mathcal{L}} )$ blows up. However, the conservation law \eqref{UnrescaledConsLaw} shows that this can only be the case if $\widehat{\mathcal{L}}$ blows up. At the first singular orbit, i.e. at time $t=0,$ one has $\widehat{\mathcal{L}}=+ \infty$ and $\widehat{\mathcal{L}}$ is strictly decreasing for all $t >0$ as $\varepsilon <0.$ Hence, the finite time singularity corresponds to $\widehat{\mathcal{L}}=- \infty$ and in particular there exists a time with $\tr(L) = \widehat{\mathcal{L}} = 0,$ the maximal volume orbit. \end{proof} From now on fix the normalisation $- \frac{\varepsilon}{2} \in \left\{ -n, 0, n \right\}$ of the Einstein constant $- \frac{\varepsilon}{2}$ and recall that in this case the corresponding cone solutions are given by \eqref{ExplicitConeSolutionScalPos}, \eqref{ExplicitConeSolutionScalNonPos}. The following proposition recovers the convergence results of B\"ohm \cite[Theorem 5.7]{BohmInhomEinstein}, \cite[Theorem 11.1]{BohmNonCompactEinstein}. \begin{proposition} Suppose that $d_1 >1$ and $A_3 = 0.$ As $\bar{f} \to 0,$ the solution $c_{\bar{f}}$ to the two summands Einstein equations converges to the first cone solution on every relatively compact subset of $(0, \pi)$ if $-\frac{\varepsilon}{2}=n$ and $(0, \infty)$ if $-\frac{\varepsilon}{2} \in \left\{ -n, 0 \right\},$ respectively. \label{ConvergenceToConeSolution} \end{proposition} \begin{proof} Recall that the limit trajectory with $\bar{f} = 0$ corresponds to a trajectory with $\mathcal{L} \equiv 0,$ more precisely the unique solution of the Ricci flat system in $X_i,$ $Y_i$ in the unstable manifold of \eqref{InitialCriticalPoint}. According to proposition \ref{VariablesBoundedRicciFlatTwoSummandsCase}, the Ricci flat trajectory asymptotically approaches the first cone solution, which takes the constant value $X_i = \frac{1}{n}$ and $Y_i=\frac{1}{nc_i}$ for $i=1,2.$ Notice that this is in fact the value at $t=0$ of all cone solutions. Therefore it will be called {\em base point} of the cone solution. If $\varepsilon \geq 0$ notice as in the proof of proposition \ref{TwoSummandsEinsteinMetricsWithNegativeScalarCurvature} that the variables $X_i, Y_i$ are bounded, that the B\"ohm functional $\mathscr{F}_0$ is bounded from below and non-increasing, and that it has a critical point on the cone solution. In fact, any Einstein trajectory that takes a constant value on $\mathscr{F}_0$ is a cone solution and $\mathscr{F}_0 = n(n-1) c_{1}^{{2d_1}/{n}} c_{2}^{{2d_2}/{n}}.$ However, since $A_3=0,$ the cone solution is unique and hence the minimum. If $-\frac{\varepsilon}{2}=n,$ then \eqref{DerivativeOfBohmFunctional} implies that $\mathscr{F}_0$ achieves its minimum along a trajectory $c_{\bar{f}}$ on the maximal volume orbit. On any maximal volume orbit the coordinates \eqref{HatVariables} satisfy $\sum_{i=1}^2 d_i \widehat{X}_i = \widehat{\mathcal{L}}=0$ and the conservation law \eqref{UnrescaledConsLaw} hence implies that the variables $\widehat{X}_i, \widehat{Y}_i$ are bounded. Thus, $\mathscr{F}_0 = n(n-1) \prod_{i=1}^2 \widehat{Y}_i^{-2d_i/n}$ has a minimum on the maximal volume orbit, which is achieved by the value of cone solution. However, $\mathscr{F}_0$ is constant on the cone solution and since the solution $c_{\bar{f}}$ approaches the base point of the cone solution as $\bar{f} \to 0$, the claim follows. \end{proof} \begin{remarkroman} (a) The simplifying assumption $A_3=0$ can be relaxed. For the geometric examples in \ref{ExamplesConeSolutionsFromHopfFibrations}, one can calculate directly that the first cone solution realises the minimum. So the exact same proof works if $\widehat{D}>0$ and $\varepsilon \geq 0$ due to proposition \ref{X2VariablePositive}. (b) The behaviour of the B\"ohm functional close to cone solutions was studied in a more general context in \cite{BohmNonCompactEinstein}. In particular, B\"ohm shows that any {\em stable} cone solution is a local attractor of the cohomogeneity one Einstein equations. In the two summands case, the cone solutions are stable if $d_1>1$. However, the cone solutions corresponding to the circle bundle construction of section \ref{SectionSolitonsFromCircleBundles} are {\em un}stable. (c) In the original proof, B\"ohm \cite{BohmInhomEinstein} uses a coordinate system specifically adapted to the cone solution to find a limit trajectory, which solves a planar ODE. The limit trajectory lies in a compact planar domain and the Poincar\'e-Bendixson theorem is applied to prove convergence to the base point. Stability of the first cone solution then follows via an attractor function, a version of which is \eqref{LyapunovForNonTrivialBundles} in the Einstein case. The planar ODE in B\"ohm's work is similar to the reduction of the Ricci flat equations to a planar ODE in section \ref{RicciFlatPlanarSystem}. However, in the Ricci soliton case, the extra degree of freedom of the soliton potential prevents a similar reduction and a different proof is required. (d) B\"ohm's \cite{BohmNonCompactEinstein} construction of the complete, non-compact Einstein metrics which were recovered in section \ref{CompletenessTwoSummands} relies on the above convergence result, i.e. on the fact that for $f_2(0)= \bar{f} \to 0$ the trajectories remain close to the cone solution and are thus defined for all times. The proof in section \ref{CompletenessTwoSummands} shows moreover that one obtains an Einstein metric for {\em all} $f_2(0)>0.$ Notice that in the Ricci flat case the metric is unique up to scaling. \label{RemarkConvergenceConeSolutions} \end{remarkroman} \subsection{B\"ohm's Einstein metrics of positive scalar curvature} \label{SectionBohmEinsteinMetricsPosScal} For the convenience of the reader, this section explains how the refined asymptotics of the {\em Ricci flat} equations in section \ref{RicciFlatPlanarSystem} and proposition \ref{ConvergenceToConeSolution} yield B\"ohm's \cite{BohmInhomEinstein} Einstein metrics of positive scalar curvature on $S^5, \ldots, S^9$ and other low dimensional spaces, including $S^2 \times S^3, \ldots, S^2 \times S^7$ or $S^4 \times S^5$. It should be emphasised that the overall strategy of the construction due to B\"ohm remains the same. \subsubsection{Symmetric solutions} \label{SymmetricSolutions} A solution $c_{\bar{f}} = (f_1,\dot{f}_1,f_2,\dot{f}_2)$ of the two summands Einstein equations with initial condition $c_{\bar{f}}(0)=(0,1,\bar{f},0)$ is called {\em symmetric} if there is $\tau>0$ such that $c_{\bar{f}}(\tau)=(0,-1,\bar{f},0)$. In fact, $c_{\bar{f}}$ is symmetric if and only if there exists $t_0>0$ such that $c_{\bar{f}}(t_0)=(f_1(t_0),0,f_2(t_0),0)$ with $f_1(t_0),f_2(t_0) >0.$ In particular, reflection along the maximal volume orbit, the unique orbit with $\tr(L)=0,$ is an isometry precisely for symmetric solutions. Moreover, since $\omega = \frac{f_1}{f_2}$ satisfies $\dot{\omega} = \omega( \frac{\dot{f}_1}{f_1} - \frac{\dot{f}_2}{f_2}),$ any symmetric solution is characterised by a critical point of $\omega$ on the maximal volume orbit. It is an important observation due to B\"ohm \cite[Lemma 4.2.1]{BohmInhomEinstein} that critical points of $\omega$ are {\em non-degenerate}. The non-degeneracy of the critical points of $\omega$ allows the application of the following general counting principle. \begin{lemma} Let $T_{\bar{f}}, \varepsilon_{\bar{f}}$ be continuous, positive functions of the real parameter $\bar{f}.$ Suppose that $c_{\bar{f}} \colon [0,T_{\bar{f}} + \varepsilon_{\bar{f}}) \to \mathbb{R}^n$ is a family of $C^{1}$-maps which depends continuously on $\bar{f}$ and $\omega \in C^{1}$ is a real valued map such that any critical point of $\omega = \omega \circ c_{\bar{f}}$ is non-degenerate and $\dot{\omega}(0) >0$ for all $\bar{f}.$ Let $\mathcal{C}(\bar{f}) = \mathcal{C}(\bar{f}, T_{\bar{f}})$ denote the number of critical points of $\omega$ along $c_f$ before $T_{\bar{f}}.$ Fix $\bar{f}_1 < \bar{f}_2.$ Then the following statements hold: \begin{enumerate} \item If $\dot{\omega}(T_{\bar{f}}) \neq 0$ for all $\bar{f} \in [\bar{f}_1, \bar{f}_2]$ then $\mathcal{C}(\bar{f})$ is constant on $[\bar{f}_1, \bar{f}_2].$ \item If $\bar{f}^{*}$ is the unique value of $\bar{f} \in [\bar{f}_1, \bar{f}_2]$ with $\dot{\omega}(T_{\bar{f}})=0$ then \begin{equation*} | \mathcal{C}(\bar{f}^{'}) - \mathcal{C}(\bar{f}^{''}) | \leq 1 \end{equation*} for all $\bar{f}^{'}, \bar{f}^{''} \in [\bar{f}_1, \bar{f}_2]$ with $\bar{f}^{'} < \bar{f}^{*} < \bar{f}^{''}.$ \end{enumerate} In particular, for any $\bar{f}^{'}, \bar{f}^{''} \in [\bar{f}_1, \bar{f}_2]$ there exist at least $| \mathcal{C}(\bar{f}^{'}) - \mathcal{C}(\bar{f}^{''}) | $ solutions with $\dot{\omega}(T_{\bar{f}})=0$ for $\bar{f} \in [\bar{f}^{'}, \bar{f}^{''}].$ \label{GeneralCountingArgument} \end{lemma} \begin{remarkroman} (a) In fact, if $c_{\bar{f}}$ is just continuous, the lemma can still be used to count roots of continuous functions along $c_{\bar{f}}.$ In this case $c_{\bar{f}}$ has to intersect the zero set of the function transversally. (b) B\"ohm proved the counting principle explicitly in the case where $T_{\bar{f}}$ is the time when the maximal volume orbit is reached, \cite[Lemmas 4.4 and 4.5]{BohmInhomEinstein}. More recently it was used by Foscolo-Haskins in the construction of nearly K\"ahler metrics, \cite[Lemma 7.2]{FHNearlyKaehler}. \end{remarkroman} \begin{theorem}[B\"ohm] Let $d_1 >1,$ $A_3=0$ and $-\frac{\varepsilon}{2} = n.$ If the dimension of the principal orbit satisfies $2 \leq n \leq 8,$ there exist infinitely many symmetric solutions to the two summands Einstein equations. \label{SymmetricEinsteinMetrics} \end{theorem} \begin{proof} Recall that symmetric solutions are induces by critical points of $\omega$ at the maximal volume orbit. With the normalisation $A_2 = d_2 (d_2-1)>0,$ i.e. in geometric applications $\mathbb{R}ic^Q = d_2-1 > 0,$ the metric of the round sphere $(f_1,f_2)(t) = (\sin(t), \cos(t))$ induces a solution to the two summands Einstein equations without any critical point of $\omega$ before the maximal volume orbit. According to lemma \ref{GeneralCountingArgument}, it suffices to show that there are trajectories with an arbitrarily high number of critical points of $\omega$ before the maximal volume orbit. Recall that the maximal volume orbit of a trajectory is achieved exactly when $\tr(L)=0.$ In the $(X_i,Y_i, \mathcal{L})$-coordinates \eqref{RescaledTwoSummandsVariables} this corresponds to the blow up time of $\mathcal{L}.$ In particular, critical points of $\omega$ which are detected by the rescaled system happen to be before the maximal volume orbit. Recall that $\omega{'} = \omega \left( X_1 - X_2 \right)$ and that every critical point in the rescaled variables also corresponds to a critical point of $\omega$ in the original time frame $t.$ Since the Einstein trajectories lie in the subvariety $d_1 X_1 + d_2 X_2 = 1,$ critical points occur if and only if $X_1= \frac{1}{n}.$ Recall that by proposition \ref{VariablesBoundedRicciFlatTwoSummandsCase} the trajectory of the {\em Ricci flat} system satisfies $X_i \to \frac{1}{n}$ and $Y_i \to \frac{1}{c_i n}$ where $(c_1,c_2)$ denotes the first cone solution. Moreover, observe that the Ricci flat system is realised by solutions to the two summands system for any value of $\varepsilon \in \mathbb{R}$ by the trajectory with $\mathcal{L} \equiv 0,$ as $\varepsilon$ and $\mathcal{L}$ only occur in the combination $\frac{\varepsilon}{2}\mathcal{L}^2.$ However, as explained in section \ref{SectionConvergenceToConeSolutions}, the limit $\mathcal{L} \equiv 0$ exactly corresponds to a smoothing of the trajectory $c_{\bar{f}}$ in the limit $\bar{f}=0.$ Due to the continuous dependence of the solution on the initial condition, for any $\varepsilon \in \mathbb{R}$ and $\bar{f} >0$ small enough, the solution to the two summands system approaches the base point of the first cone solution along a trajectory which is $C^{0}$-close to the Ricci flat trajectory $\gamma_{\text{RF}}$ of proposition \ref{VariablesBoundedRicciFlatTwoSummandsCase} with $\mathcal{L} \equiv 0,$ and then remains close to the actual cone solution in the sense of proposition \ref{ConvergenceToConeSolution}. The dimension assumption and corollary \ref{StableSpiral} imply that the projection of the Ricci flat trajectory $\gamma_{\text{RF}}$ onto the $(X_1, Y_1)$-plane rotates infinitely often around the stationary point $(\frac{1}{n}, \frac{1}{c_1 n}),$ which is the base point of the first cone solution. Hence, the variable $X_1$ takes the value $X_1 = \frac{1}{n}$ arbitrarily often, which implies that $\mathcal{C}(\bar{f}, T_{\bar{f}}) \to \infty$ as $\bar{f} \to 0,$ where $T_{\bar{f}}$ denotes the time of the maximal volume orbit. A direct computation of curvatures shows that the metrics are inhomogeneous and non-isometric, cf. \cite[section 6]{BohmInhomEinstein}. \end{proof} As an explicit application, theorem \ref{SymmetricEinsteinMetrics} recovers B\"ohm's Einstein metrics on certain low dimensional spaces \cite[Theorem 3.4]{BohmInhomEinstein}. \begin{corollary}[B\"ohm] Let $d_S >1$ and suppose that $Q$ is a compact, connected, isotropy irreducible homogeneous space of positive Ricci curvature and of dimension $d_Q.$ If $2 \le d_S, d_Q$ and $d_S + d_Q \leq 8,$ then there exist infinitely many non-isometric cohomogeneity one Einstein metrics of positive scalar curvature on $S^{d_S+1} \times Q.$ \end{corollary} \begin{remarkroman} By considering the linearisation of the Einstein equations along the cone solutions, B\"ohm was also able to construct a symmetric cohomogeneity one Einstein metric on $\mathbb{H}P^{2} \# \overline{\mathbb{H}P}^{2}.$ \end{remarkroman} \subsubsection{B\"ohm's Einstein metrics on low dimensional spheres} \label{EinsteinMetricsOnSpheres} Cohomogeneity one Einstein manifolds with singular orbits of (possibly different) dimensions $d_1,$ $d_2$ can be constructed via solutions $c_{\bar{f}} = (f_1,\dot{f}_1,f_2,\dot{f}_2)$ with $c_{\bar{f}}(0)=(0,1,\bar{f},0)$, $c_{\bar{f}}(\tau)=(\bar{f}^{'},0,0,-1)$ and $\bar{f}, \bar{f}^{'}, \tau>0$. In the case of $SO(d_1+1) \times SO(d_2+1)$-invariant doubly warped product metrics on spheres, the convergence theory from section \ref{SectionConvergenceToConeSolutions} can be applied to give B\"ohm's \cite{BohmInhomEinstein} inhomogeneous Einstein metrics on $S^5, \ldots, S^9.$ For the rest of the section, fix $A_3 = 0$ and normalise the Ricci curvature of the singular orbit to be $\mathbb{R}ic^Q = d_2-1,$ so that $A_i=d_i (d_i-1)$ holds for $i=1,2.$ As before, the trajectory $c_{\bar{f}}$ will always correspond to an Einstein metric on a tubular neighbourhood of a singular orbit of dimension $d_2$ and with a principal orbit of dimension $n=d_1+d_2.$ As in section \ref{SectionConvergenceToConeSolutions}, in the $(X_i, Y_i, \mathcal{L})$-coordinates the limit trajectory with $\bar{f} = 0$ corresponds to the unique solution of the Ricci flat system in the unstable manifold of \eqref{InitialCriticalPoint} with $\mathcal{L} \equiv 0$. Recall from proposition \ref{VariablesBoundedRicciFlatTwoSummandsCase} that this trajectory approaches the base point $(\frac{1}{n}, \frac{1}{n c_1})$ of the cone solution asymptotically. Under the dimensional assumptions $d_1 >1$ and $2 \leq n \leq 8,$ the base point $(\frac{1}{n}, \frac{1}{n c_1})$ is a stable spiral due to corollary \ref{StableSpiral}. As in the proof of theorem \ref{SymmetricEinsteinMetrics}, it follows from the continuous dependence on the initial value, that also $c_{\bar{f}},$ for $\bar{f}>0$ small enough, exhibits a rotational behaviour as it approaches the base point at $t=0$ of the first cone solution $\gamma.$ Proposition \ref{ConvergenceToConeSolution} then says that given any compact set $K \subset \subset (0, \pi),$ the trajectory $c_{\bar{f}}$ remains $C^{0}$-close to $\gamma$ on $K$ if $\bar{f}>0$ is small enough. In fact, $c_{\bar{f}}$ obeys a rotational behaviour in every slice around the cone solution as $\bar{f} \to 0.$ To make this precise, notice that the variables $(\widehat{X}_1,\widehat{Y}_1,\widehat{\mathcal{L}})$ of \eqref{HatVariables} form a local coordinate system along the Einstein trajectories away from the singular orbits. For example, the first cone solution has coordinates $(\cot(t), \frac{1}{c_1 \sin(t)}, n \cot(t)).$ One should think of $\widehat{\mathcal{L}}$ as the time variable. For any fixed value $\widehat{\mathcal{L}}$ and for $\bar{f} >0$ sufficiently small, the trajectory $c_{\bar{f}}$ intersects the $(\widehat{X}_1,\widehat{Y}_1,\widehat{\mathcal{L}})$-plane $P_{\widehat{\mathcal{L}}}$ in a unique point. As $\bar{f}>0$ varies, the intersection points describe a continuous curve in this plane. \begin{proposition} Let $A_3 = 0$ and $2 \leq n \leq 8.$ Then in any coordinate slice $P_{\widehat{\mathcal{L}}},$ the intersection points of $c_{\bar{f}}$ with a disc around the first cone solution $\gamma$ in $P_{\widehat{\mathcal{L}}}$ exhibit the same rotational behaviour as $\bar{f} \to 0.$ \end{proposition} This follows from the general counting principle \ref{GeneralCountingArgument} applied to the time $T_{\bar{f}}$ when $c_{\bar{f}}$ intersects the disc, the observation that $\mathcal{C}(c_{\bar{f}},T_{\bar{f}}) \to \infty$ as $\bar{f} \to 0,$ and lemma \ref{OccurrenceOfCriticalPoints} below. \begin{lemma} Along any trajectory $c_{\bar{f}},$ critical points of $\omega$ occur if and only if $\widehat{X}_1 = \frac{\widehat{\mathcal{L}}}{n}$ and $\omega$ is increasing if $\widehat{X}_1 > \frac{\widehat{\mathcal{L}}}{n}$ and decreasing if $\widehat{X}_1 < \frac{\widehat{\mathcal{L}}}{n}.$ Moreover, if $A_3 = 0,$ then the $\widehat{Y}_1$-coordinate of $\omega$ in $P_{\widehat{\mathcal{L}}}$ satisfies $\widehat{Y}_1(\omega) > \widehat{Y}_1(\gamma)$ at any maximum and $\widehat{Y}_1(\omega) < \widehat{Y}_1(\gamma)$ at any minimum, where $\gamma$ is the first cone solution. \label{OccurrenceOfCriticalPoints} \end{lemma} \begin{proof} The first statement follows from $\dot{\omega} = \omega ( \widehat{X}_1 - \widehat{X}_2 )$ and $\sum_{i=1}^2 d_i \widehat{X}_i = \widehat{\mathcal{L}}.$ If $A_3 = 0,$ the identity $\ddot{\omega} = \frac{A_1}{d_1} \widehat{Y}_1^2 - \frac{A_2}{d_2} \widehat{Y}_2^2$ holds at every critical point of $\omega.$ However, as $\dot{\omega} = \ddot{\omega} = 0$ only occurs on the cone solution, the claim follows. \end{proof} Suppose that $d_{\overline{F}}=(F_1,\dot{F}_1, F_2, \dot{F}_2)$ is also an Einstein trajectory which satisfies $d_{\overline{F}}(0)=(0,1,\overline{F},0)$ but instead induces a metric on a tubular neighbourhood of a singular orbit of dimension $d_2$ and with a principal orbit of dimension $n= d_1+d_2.$ Then the above considerations also apply to $d_{\overline{F}}.$ Clearly, the trajectories depend continuously on the parameters $d_1, d_2>1.$ Hence, in the dimension range $2 \leq n \leq 8,$ the trajectories $c_{\bar{f}}$ and $d_{\overline{F}}$ have the {\em same} rotational behaviour as they approach their respective base point of the cone solution at $t=0.$ Now consider the twisted trajectory $d_{\overline{F}}^{\text{twisted}}(t)=(F_2, - \dot{F}_2, F_1, - \dot{F}_1)(t).$ If $\tau >0$ is small enough, then $d_{\overline{F}}^{\text{twisted}}(\tau - t)$ is an actual solution to the Einstein equations due to the symmetries of the equations in $d_1, d_2$ as $A_3=0.$ That is, $d_{\overline{F}}^{\text{twisted}}(t)$ runs through the Einstein equations in `opposite direction', starting at $d_{\overline{F}}^{\text{twisted}}(0) = (\overline{F},0,0,-1)$ and then approaching the same cone solution as $c_{\bar{f}}$ but at the base point corresponding to $t = \pi.$ In particular, it has the {\em opposite} rotational behaviour to $c_{\bar{f}}.$ Due to proposition \ref{ConvergenceToConeSolution}, for $\bar{f}, \overline{F} >0$ small enough, both $c_{\bar{f}}$ and $d_{\overline{F}}^{\text{twisted}}$ intersect the plane $\{(\widehat{X}_1,\widehat{Y}_1, n)\},$ which is the slice of the maximal volume orbit of the cone solution, in a unique point. Since both trajectories in fact wind around the cone solution arbitrarily often as $\bar{f}, \overline{F} \to 0,$ respectively, and $c_{\bar{f}},$ $d_{\overline{F}}^{\text{twisted}}$ have the opposite rotational behaviour, there are infinitely many intersection points in this (or any other) slice. For any such, there exist $t_0, t_1 > 0$ such that the matching condition $c_{\bar{f}}(t_0) = d_{\overline{F}}^{\text{twisted}}(t_1)$ holds. Then \begin{align*} \tilde{c}_{\bar{f},\overline{F}}(t) = \begin{cases} c_{\bar{f}}(t) & \ \text{ for } \ 0 \leq t \leq t_0 \\ d_{\overline{F}}^{\text{twisted}}(t_0+t_1-t) & \ \text{ for } \ t_0 \leq t \leq t_0+t_1 \end{cases} \end{align*} satisfies $\tilde{c}_{\bar{f},\overline{F}}(0)=(0,1,\bar{f},0)$ and $\tilde{c}_{\bar{f},\overline{F}}(t_0+t_1)=(\overline{F},0,0,-1)$ and it is a {\em smooth} solution to the Einstein equation as required. Smoothness indeed follows from the uniqueness of solutions to ODEs with fixed initial conditions, since $\tilde{c}_{\bar{f},\overline{F}}$ clearly solves the Einstein equations on both intervals. In particular, any such pair $(\bar{f}, \overline{F})$ induces an Einstein metric $g{(\bar{f}, \overline{F})}$ on $S^{n+1}.$ \begin{remarkroman} A direct curvature computation shows that the metrics are indeed inhomogeneous. Moreover, if the metrics $g{(\bar{f}, \overline{F})}$ and $g{(\bar{f}^{'}, \overline{F}{'})}$ on $S^{n+1}$ are isometric, it follows that $\bar{f}=\bar{f}^{'}$ if $d_1 \neq d_2$ and $\bar{f}=\bar{f}^{'}$ or $\bar{f}=\overline{F}{'}$ if $d_1 = d_2,$ since isometries must map orbits onto orbits, see \cite[Section 7]{BohmInhomEinstein}. \end{remarkroman} This recovers B\"ohm's Einstein metrics on low dimensional spheres \cite[Theorem 3.6]{BohmInhomEinstein}: \begin{corollary}[B\"ohm] On $S^5$ and $S^6$ there exists one, on $S^7$ and $S^8$ there exist two, and on $S^9$ there exist three infinite families of non-isometric, strictly cohomogeneity one Einstein metrics of positive scalar curvature. \end{corollary} \section{Quasi-Einstein Metrics} \label{SectionQuasiEinsteinMetrics} \subsection{Introduction} \label{QEMIntroSection} In the study of smooth metric measure spaces the $m$-Bakry-\'Emery Ricci tensor $\mathbb{R}ic + \Hess u - \frac{1}{m} du \otimes du$ plays a central role, cf. \cite{CaseSMMSAndQEM}. It also naturally appears in the context of warped product Einstein manifolds, where it has led to the notion of $m$-{\em quasi-Einstein metrics} or $(\lambda, n+m)$-Einstein metrics in the terminology of He-Petersen-Wylie, cf. \cite{HPWUniquenessWarpedProductEinstein}: \begin{definition} Let $(M,g)$ be an $n$-dimensional Riemannian manifold, $u \in C^{\infty}(M)$ and $m \in (0,\infty].$ Then $(M,g,e^{-u} d \operatorname{Vol}_M)$ is called {\em $m$-quasi-Einstein manifold} if \begin{equation} \mathbb{R}ic + \Hess u - \frac{1}{m} du \otimes du + \frac{\varepsilon}{2} g = 0. \label{QEMequation} \end{equation} The sum $m+n$ is called {\em effective dimension} and $-\frac{\varepsilon}{2}$ is the {\em quasi-Einstein constant.} \end{definition} Kim-Kim \cite{KimKim} observed that any connected $m$-quasi-Einstein manifold with $m < \infty$ satisfies the following conservation law: There exists a constant $\mu \in \mathbb{R},$ called {\em characteristic constant,} such that \begin{equation} \mathcal{D}elta u - | \nabla u |^2 + m \mu e^{2u / m} + m \frac{\varepsilon}{2} = 0. \label{QEMConsLaw} \end{equation} In this case, Kim-Kim \cite{KimKim} proved that if $m>1$ is an integer and $(N^m,h)$ is Einstein with $\mathbb{R}ic_h = \mu h$, then the warped product \begin{equation} (M \times N, g + e^{-2u /m} h) \label{EQWarpedProductKimKim} \end{equation} is Einstein. Conversely, if $(M \times N, g + e^{-2u /m} h)$ is Einstein, then $(M,g,e^{-u} d \operatorname{Vol}_M)$ must be $m$-quasi-Einstein. This point of view on Einstein warped products was successfully used by Case-Shu-Wei \cite{CSWRigidityQEM} to show that any compact {\em K\"ahler} $m$-quasi-Einstein metric with $m < \infty$ is Einstein. In contrast, recall that all {\em known} non-trivial compact Ricci solitons are K\"ahler. Hall \cite{HallQEinstein} constructed $m$-quasi-Einstein metrics on total spaces of complex vector bundles associated to principal circle bundles over products of Fano K\"ahler-Einstein manifolds. Due to the induced hypersurface foliation their geometry can in fact be described using the cohomogeneity one equations from section \ref{CohomOneQEMsection}. The case of a single base factor is due to L\"u-Page-Pope \cite{LuPagePopeQEinstein}. Remarkably, the L\"u-Page-Pope metrics are conformally K\"ahler and the associated K\"ahler class is a multiple of the first Chern class as shown by Batat-Hall-Jizany-Murphy \cite{BHJMConfKahlerQEM}. \subsection{The initial value problem for cohomogeneity one quasi-Einstein metrics} \label{CohomOneQEMsection} The formulae for the Ricci curvature of a cohomogeneity one manifold in section \ref{SectionCohomOneSetUp} yield that the $m$-quasi-Einstein equation takes the form \begin{align} -( \delta^{\nabla^t}L_t)^{\flat} - d(\tr(L_t)) & = 0, \label{QEMequationA} \\ - \tr( \dot{L}_t) - \tr(L_t^2) + \ddot{u} - \frac{1}{m} \dot{u}^2 + \frac{\varepsilon}{2} & =0, \label{QEMequationB} \\ - \dot{L}_t - (- \dot{u} + \tr(L_t)) L_t + r_t + \frac{\varepsilon}{2} \mathbb{I} & = 0, \label{QEMequationC} \end{align} and the conservation law \eqref{QEMConsLaw} is given by \begin{equation} \ddot{u} + (- \dot{u} + \tr(L)) \dot{u} + m \mu e^{2u/m} + m \frac{\varepsilon}{2} = 0. \label{CohomOneQEMConsLaw} \end{equation} \begin{remarkroman} Notice that for $f=e^{-u/m}$ the conservation law is equivalent to \begin{align*} \frac{d}{dt} \frac{\dot{f}}{f} = - \left( m \frac{\dot{f}}{f} + \tr(L) \right) \frac{\dot{f}}{f} + \frac{\mu}{f^2} + \frac{\varepsilon}{2}, \end{align*} and hence it is the Einstein equation for the added factor in Kim-Kim's \cite{KimKim} warped product construction \eqref{EQWarpedProductKimKim}. \label{ConsLawIsEinsteinEQ} \end{remarkroman} The following proposition generalises an observation due to Back\cite{BackLocalTheoryofEquiv} in the Einstein case, see also \cite{EWInitialValueEinstein} and \cite{DWCohomOneSolitons}. \begin{proposition} Let $M$ be a connected manifold and $g$ a $C^2$-Riemannian metric on $M.$ Suppose that $G$ is a compact Lie group which acts isometrically and with cohomogeneity one on $(M,g)$ and that the action has a singular orbit. Let $u \in C^3(M)$ be $G$-invariant. Then \eqref{QEMequationC} implies \eqref{QEMequationA} and if the conservation law \eqref{CohomOneQEMConsLaw} is satisfied, then \eqref{QEMequationB} holds as well. \label{ReducedQEMsystem} \end{proposition} \begin{proof} The fact that \eqref{QEMequationC} implies \eqref{QEMequationA} follows as in the Ricci soliton case, cf. \cite[Proposition 3.19]{DWCohomOneSolitons}, as the equations are identical. Let $v_t$ be the relative volume of the principal orbit $(P,g_t).$ Then it follows that $\frac{d}{dt} v = \tr(L) v$ and due to \cite[Formula (3.16)]{DWCohomOneSolitons} there holds \begin{equation*} \frac{d}{dt} \left( v^2 \left( \mathbb{R}ic(N,N) + \frac{\varepsilon}{2} \right) \right) + v^2 \left( 2 \dot{u} \tr(L^2)+ \frac{d}{dt} \left( \dot{u}\tr(L) \right) \right) = 0. \end{equation*} By combining this with $\mathbb{R}ic(N,N)= - \tr(\dot{L}) - \tr(L^2)$ and the conservation law \eqref{CohomOneQEMConsLaw}, one obtains \begin{equation*} \frac{d}{dt} \left( v^2 \left( \mathbb{R}ic(N,N) + \ddot{u} - \frac{1}{m} \dot{u}^2 + \frac{\varepsilon}{2} \right) \right) = 2 \dot{u} v^2 \left( \mathbb{R}ic(N,N) + \ddot{u} - \frac{1}{m} \dot{u}^2 + \frac{\varepsilon}{2} \right). \end{equation*} Therefore $v^2 \left( \mathbb{R}ic(N,N) + \ddot{u} - \frac{1}{m} \dot{u}^2 + \frac{\varepsilon}{2} \right)$ is a multiple of $e^{2 u}$ which vanishes at the singular orbit, and thus vanishes identically. \end{proof} \begin{proposition} Let $M$ be a smooth manifold of dimension $\dim M \geq 3.$ Suppose that a solution of the $m$-quasi-Einstein equation on $M$ is given by a $C^2$-Riemannian metric $g$ and $u \in C^3(M).$ Then $g$ and $u$ are real analytic in harmonic and geodesic normal coordinates. \label{QEMregularity} \end{proposition} \begin{proof} The $m$-quasi-Einstein equation and the contracted second Bianchi identity give rise to the PDE \begin{align*} \mathbb{R}ic + \Hess u - \frac{1}{m} du \otimes du + \frac{\varepsilon}{2} g & = 0, \\ \mathcal{D}elta(du) + \mathbb{R}ic( \cdot, \grad u) - \frac{2}{m} \left( \mathcal{D}elta u \right) du & = 0 \end{align*} for $(g,u).$ Notice that the $\frac{1}{m}$-terms are of lower order and thus the principal symbol is the same as in the Ricci soliton case. Hence, $(g,u)$ is a solution of a quasi-linear elliptic system and the regularity analysis in\cite[Lemma 3.2]{DWCohomOneSolitons} carries over without any changes, see also \cite[Theorem 5.2]{dTKRegularity}. \end{proof} The initial value problem for $m$-quasi-Einstein metrics at a singular orbit can be solved analoguously to Buzano's \cite{BuzanoInitialValueSolitons} approach in the Ricci soliton case: Due to proposition \ref{ReducedQEMsystem} it suffices to consider \eqref{QEMequationC}, \eqref{CohomOneQEMConsLaw} and the relation $\dot{g}_t = 2 g_t L_t.$ Setting up an ODE system for $(g_t, L_t, u)$ as in \cite{BuzanoInitialValueSolitons}, one observes that the $-\frac{1}{m} \dot{u}^2$-term simply disappears in the error terms that occur in Buzano's proof because it is of lower order. In particular, the construction of a formal power series solution is unchanged. Due to the real analyticity of $m$-quasi-Einstein metrics as in proposition \ref{QEMregularity}, a theorem of Malgrange \cite[Theor{\`e}me 7.1]{MalgrangeEquationsDifferentielle} then yields a genuine solution. Alternatively, a Picard iteration may be applied as in \cite{EWInitialValueEinstein}. \begin{theorem} Let $G$ be a compact Lie group acting isometrically on a connected Riemannian manifold $(M,g)$ and suppose there exists a singular orbit $Q = G / H.$ Choose $q \in M$ such that $Q = G \cdot q$ and denote by $V = T_qM / T_qQ$ the normal space of $Q$ at q. Then $H$ acts linearly and orthogonally on $V$ and a tubular neighbourhood of $Q$ may be identified with its normal bundle $E = G \times_H V.$ The principal orbits are $P = G / K = G \cdot v$ for any $v \in V \setminus \left\lbrace 0\right\rbrace.$ These can be identified with the sphere bundle of $E$ (with respect to an $H$-invariant scalar product on $V$). Let $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{p}_{-}$ be a decomposition of the Lie algebra of $G$ where $\mathfrak{p}_{-}$ is an $Ad_H$-invariant complement of $\mathfrak{h}= \operatorname{Lie}(H).$ Assume that $V$ and $\mathfrak{p}_{-}$ have no common irreducible factors as $K$-representations. Then for any $\varepsilon \in \mathbb{R},$ any $m \in (0, \infty],$ any $G$-invariant metric $g_Q$ on $Q$ and any shape operator $L \colon E \to \operatorname{Sym}^2(T^{*}Q)$ there exists a G-invariant $m$-quasi-Einstein metric on some open disc bundle of $E$. \label{QEMInitialValueTheorem} \end{theorem} \begin{remarkroman} The assumption that $V$ and $\mathfrak{p}_{-}$ have no common irreducible factors as $K$-representations is primarily a technical simplification but as Eschenburg-Wang point out in \cite[Remark 2.7]{EWInitialValueEinstein} it is also natural in the context of the Kaluza-Klein construction. \end{remarkroman} \subsection{New quasi-Einstein metrics} The analysis of the two summands case in section \ref{CompletenessTwoSummands} can be adapted to the $m$-quasi-Einstein case for $m < \infty.$ Recall that the metric restricted to the principal orbit is given by $g_t = f_1(t)^2 g_S + f_2(t)^2 g_Q,$ and set $f_3(t) = e^{-u(t)/m}.$ Due to proposition \ref{ReducedQEMsystem}, it suffices to consider \eqref{QEMequationC} and \eqref{CohomOneQEMConsLaw}, and thus the two summands $m$-quasi-Einstein equations take the form \begin{align*} \frac{d}{dt} \left( \frac{\dot{f}_1}{f_1} \right) & = - \tr ( \widehat{L} ) \frac{\dot{f}_1}{f_1} + \frac{\varepsilon}{2} + \frac{A_1}{d_1} \frac{1}{f_1^2} + \frac{A_3}{d_1} \frac{f_1^2}{f_2^4}, \\ \frac{d}{dt} \left( \frac{\dot{f}_2}{f_2} \right) & = - \tr ( \widehat{L} ) \frac{\dot{f}_2}{f_2} + \frac{\varepsilon}{2} + \frac{A_2}{d_2} \frac{1}{f_2^2} - 2 \frac{A_3}{d_2} \frac{f_1^2}{f_2^4}, \\ \frac{d}{dt} \left( \frac{\dot{f}_3}{f_3} \right) & = - \tr ( \widehat{L} ) \frac{\dot{f}_3}{f_3} + \frac{\varepsilon}{2} + \frac{\mu}{f_3^2}, \end{align*} where $\widehat{L} = \operatorname{diag} \left( \frac{\dot{f}_1}{f_1} \mathbb{I}_{d_1}, \frac{\dot{f}_2}{f_2} \mathbb{I}_{d_2}, \frac{\dot{f}_3}{f_3} \mathbb{I}_{m} \right)$ corresponds to the shape operator in Kim-Kim's \cite{KimKim} warped product construction. Notice that $\widehat{L}$ is only well-defined if $m \in \mathbb{N},$ but its trace always is. Due to the regularity theorem \ref{QEMInitialValueTheorem} the metric can be smoothly extended over the singular orbit if the initial conditions \begin{align*} f_1(0)=0, \ \dot{f}_1(0)=1 \ \text{ and } \ f_2(0)= \bar{f} > 0, \ \dot{f}_2(0)=0 \end{align*} are imposed. Clearly one may fix $u(0)=0$ and then \begin{align*} f_3(0) = 1 \ \text{ and } \ \dot{f}_3(0) = 0 \ \text{ and } \ \ddot{f}_3(0) = \varepsilon + 2 \mu \end{align*} are the corresponding smoothness conditions for $f_3$. Fix $\varepsilon \geq 0$ and $\mu > 0$. It follows that $\dot{f}_i(t) > 0$ for $i=1, 2, 3$ and sufficiently small $t>0.$ In analogy to \eqref{RescaledTwoSummandsVariables}, set \begin{align*} \mathcal{L} = \frac{1}{\tr( \widehat{L} )}, \ \frac{d}{ds} = \mathcal{L} \cdot \frac{d}{dt} \ \text{and} \ X_i = \mathcal{L} \cdot \frac{\dot{f}_i}{f_i}, \ Y_i = \mathcal{L} \cdot \frac{1}{f_i} \ \text{for} \ i=1,2,3. \end{align*} In particular, $\mathcal{L}, X_i, Y_i$ are positive initially. Set $d_3 =m.$ Then $\sum_{i=1}^3 d_i X_i =1$ and the rescaled two summands $m$-quasi-Einstein equations take the form \begin{align*} X_1^{'} & = X_1 \left( \sum_{i=1}^3 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 -1 \right) + \frac{A_1}{d_1} Y_1^2+\frac{\varepsilon}{2} \mathcal{L}^2 + \frac{A_3}{d_1} \frac{Y_2^4}{Y_1^2}, \\ X_2^{'} & = X_2 \left( \sum_{i=1}^3 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 -1 \right) + \frac{A_2}{d_2} Y_2^2+\frac{\varepsilon}{2} \mathcal{L}^2 - \frac{2 A_3}{d_2} \frac{Y_2^4}{Y_1^2}, \\ X_3^{'} & = X_3 \left( \sum_{i=1}^3 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 -1 \right) + \mu Y_3^2+\frac{\varepsilon}{2} \mathcal{L}^2, \end{align*} \begin{align*} Y_j^{'} & =Y_j \left( \sum_{i=1}^3 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 - X_j \right) \ \text {for} \ j=1,2,3, \\ \mathcal{L}^{'} & =\mathcal{L} \left( \sum_{i=1}^3 d_i X_i^2 - \frac{\varepsilon}{2} \mathcal{L}^2 \right). \end{align*} It follows that $\mathcal{L},$ $X_i$, $Y_i >0$ holds along the flow, except possibly for $X_2.$ In the situations of proposition \ref{X2VariablePositive} and proposition \ref{X2PositiveCircleBundles}, the respective proofs carry over to show that both $X_2 > 0$ and $\frac{Y_2}{Y_1} < \hat{\omega}_1$ are preserved. Since $\hat{\omega}_1^2 < \frac{A_2}{2A_3},$ the conservation law \begin{align*} \sum_{i=1}^3 d_i X_i^2 + A_1 Y_1^2 + A_2 Y_2^2 + m \mu Y_3^2 - A_3 \frac{Y_2^4}{Y_1^2} + (n-1) \frac{\varepsilon}{2} \mathcal{L}^2 = 1 \end{align*} implies that $X_i$, $Y_i$ are bounded for $i=1,2,3.$ Thus $\mathcal{L}$ cannot blow up in finite time either. Completeness of the metric now follows as in proposition \ref{CompletenessEpsZeroTwoSummands} and corollary \ref{CompletenessEpsPosTwoSummands}. If $\varepsilon > 0$ and $\mu = 0,$ then $\mathcal{L},$ $X_i,$ $Y_i$ are bounded due to the conservation law, except possibly $Y_3$. However, the ODE for $Y_3$ implies that $Y_3$ cannot blow up in finite time and a similar argument applies. This shows: \begin{theorem} Let $d_1 \geq 1,$ $A_1=d_1(d_1-1)$ and $(d_1+1)A_2^2 > 4 d_1 d_2 (2d_1+d_2) A_3>0$ and fix $m>0.$ Then the associated two summands ODE gives rise to a $1$-parameter family of complete, non-trivial non-homothetic $m$-Bakry-\'Emery Ricci flat metrics and a $2$-parameter family of non-trivial, complete, non-homothetic $m$-quasi Einstein metrics with quasi-Einstein constant $- \frac{\varepsilon}{2} < 0,$ all of which have positive characteristic constant. Furthermore, there exists a $1$-parameter family of complete, non-trivial non-homothetic $m$-quasi-Einstein metrics with quasi-Einstein constant $- \frac{\varepsilon}{2} < 0$ and vanishing characteristic constant. \label{TwoSummandsQEM} \end{theorem} \begin{remarkroman} Case \cite{CaseNonExistenceQEM} has shown that any complete, non-trivial $m$-Bakry-\'Emery Ricci flat quasi-Einstein manifold has positive characteristic constant. \end{remarkroman} Notice that if $A_3 = 0$ and $m \in \mathbb{N},$ the above construction gives rise to a triply warped product Einstein metric. Multiple warped product Einstein metrics of nonpositive scalar curvature were constructed by B\"ohm \cite{BohmNonCompactEinstein} on $\mathbb{R}^{d_1+1} \times M_2 \times \ldots \times M_r$ if $d_1>1,$ for Einstein manifolds $(M_i,g_i)$ of positive scalar curvature $\mu_i >0.$ The corresponding steady and expanding Ricci solitons have been constructed by Dancer-Wang \cite{DWExpandingSolitons, DWSteadySolitons} who in joint work with Buzano and Gallaugher \cite{BDGWExpandingSolitons, BDWSteadySolitons} also settled the case $d_1 =1.$ Away from the singular orbit, on $(0, \infty) \times S^{d_1-1} \times M_2 \times \ldots \times M_r,$ the metrics are of the form $dt^2 + \sum_{i=1}^{r} f_i^2(t) g_i.$ Notice that the corresponding $m$-quasi-Einstein equations are \begin{align*} \frac{d}{dt} \frac{\dot{f}_i}{f_i} & = - (- \dot{u} + \tr(L) ) \frac{\dot{f}_i}{f_i} + \frac{\varepsilon}{2} + \frac{\mu_i}{f_i^2} \ \text{ for } \ i=1, \ldots, r, \end{align*} where $\mu_1 = d_1-1.$ If $m < \infty,$ then $f_{r+1} = e^{-u/m}$ satisfies an analogous equation, where $\mu_{r+1} > 0$ will be the characteristic constant of the induced $m$-quasi-Einstein metric. Due to the regularity theorem \ref{QEMregularity} one induces a smooth $m$-quasi-Einstein metric on the trivial $\mathbb{R}^{d_1+1}$-bundle over $M_2 \times \ldots \times M_r$ by imposing the initial conditions $f_1=0,$ $\dot{f}_1=1$ and $f_i>0$ for $i \geq 2$ at $t=0$ and by requiring that $f_i(t)$ for $i \geq 2$ and $u(t)$ are even. Set $d_{r+1} = m$ if $m<\infty$ and $d_{r+1}= \mu_{r+1}=0$ if $m=\infty.$ In terms of the rescaled coordinates $\mathcal{L},$ $X_i,$ $Y_i$ of \eqref{RescaledTwoSummandsVariables} the above initial conditions correspond to the stationary point \begin{equation*} X_1 = \frac{1}{d_1}, Y_1=\frac{1}{d_1} \ \text{and} \ X_i=Y_i=\mathcal{L} = 0 \ \text{for} \ i \geq 2 \end{equation*} of the Ricci soliton ODE \begin{align*} \mathcal{L}^{'} & = \mathcal{L} \left( \sum_{j=1}^{r+1} d_j X_j^2 - \frac{\varepsilon}{2} \mathcal{L}^2 \right), \\ X_i^{'} &= X_i \left( \sum_{j=1}^{r+1} d_j X_j^2 - \frac{\varepsilon}{2} \mathcal{L}^2 - 1 \right) + \frac{\varepsilon}{2} \mathcal{L}^2 + \mu_i Y_i^2, \\ Y_i^{'} &= Y_i \left( \sum_{j=1}^{r+1} d_j X_j^2 - \frac{\varepsilon}{2} \mathcal{L}^2 - X_i \right). \end{align*} Notice that $f_i(t)>0,$ $\dot{f}_i(t) >0$ for $t >0$ small and thus the rescaled coordinates $\mathcal{L},$ $X_i,$ $Y_i$ are also positive initially. Moreover, for $\varepsilon \geq 0$ positivity is preserved by the flow of the Ricci soliton ODE. Consider \begin{align*} \mathcal{S}_{1, m} = \sum_{i=1}^{r+1} d_i X_i^2 + \sum_{i=1}^{r+1} \mu_i Y_i^2 + (n-1)\frac{\varepsilon}{2} \mathcal{L}^2 -1 \ \text{ and } \ \mathcal{S}_{2,m} = \sum_{i=1}^{r+1} d_i X_i -1. \end{align*} In analogy to \eqref{EinsteinLocus}, \eqref{SolitonLocus} it follows that trajectories lying in the preserved locus $\left\{ \mathcal{S}_{1,m} = 0\right\} \cap \left\{ \mathcal{S}_{2,m} = 0\right\}$ correspond to non-trivial $m$-quasi-Einstein metrics for $m < \infty.$ Similarly, non-trivial Ricci soliton metrics correspond to trajectories in $\left\{ \mathcal{S}_{1,\infty} < 0\right\} \cap \left\{ \mathcal{S}_{2,\infty} < 0\right\}$ and Einstein metrics to trajectories in $\left\{ \mathcal{S}_{1,\infty} = 0\right\} \cap \left\{ \mathcal{S}_{2,\infty} = 0\right\}$. In all cases, if $\varepsilon \geq 0,$ the variables $X_i, Y_i \geq 0$ are bounded, except possibly $Y_1$ if $d_1=1$. However, the ODEs for $\mathcal{L},$ $Y_1$ show as before that $\mathcal{L},$ $Y_1$ cannot blow up in finite time. Completeness of the metric again follows as in proposition \ref{CompletenessEpsZeroTwoSummands} and corollary \ref{CompletenessEpsPosTwoSummands}. Thus this construction yields $m$-quasi-Einstein metrics on multiple warped products as in Theorem \ref{MainTheoremQEM}, and a unified proof of the works of B\"ohm \cite{BohmNonCompactEinstein} and Buzano-Dancer-Gallaugher-Wang \cite{DWSteadySolitons, DWExpandingSolitons, BDGWExpandingSolitons, BDWSteadySolitons}. \end{document}

\begin{document} \title {Marcel Riesz on N\"orlund Means} \vcrossingte{} \author[P.L. Robinson]{P.L. Robinson} \address{Department of Mathematics \\ University of Florida \\ Gainesville FL 32611 USA } \email[]{paulr@ufl.edu} \subjclass{} \keywords{} \begin{abstract} We note that the necessary and sufficient conditions established by Marcel Riesz for the inclusion of {\it regular} N\"orlund summation methods are in fact applicable quite generally. \end{abstract} \maketitle \section*{Introduction} \medbreak One of the simplest classes of summation methods for divergent series was introduced independently by N\"orlund [2] in 1920 and by Voronoi in 1901 with an annotated English translation [5] by Tamarkin in 1932. Explicitly, let $(p_n : n \geqslant 0)$ be a real sequence, with $p_0 > 0$ and with $p_n \geqslant 0$ whenever $n > 0$; when $n \geqslant 0$ let us write $P_n = p_0 + \cdots + p_n.$ To each sequence $s = (s_n : n \geqslant 0)$ is associated the sequence $N^p s$ of {\it N\"orlund means} defined by $$ (N^ps)_m = \frac{p_0 s_m + \cdots + p_m s_0}{p_0 + \cdots + p_m} = \frac{1}{P_m} \sum_{n = 0}^m p_{m - n} s_n.$$ We say that the sequence $s$ is $(N, p)$-{\it convergent} to $\sigma$ precisely when the sequence $N^p s$ converges to $\sigma$ in the ordinary sense, writing this as $$s \xrightarrow{(N, p)} \sigma$$ or as $s \rightarrow \sigma \; (N, p)$; viewing the formation of N\"orlund means as a summation method, when $(s_n : n \geqslant 0)$ happens to be the sequence of partial sums of the series $\sum_{n \geqslant 0} a_n$ we may instead say that this series is $(N, p)$-{\it summable} to $\sigma$ and write $$\sum_{n = 0}^{\infty} a_n = \sigma \: (N, p).$$ \medbreak An important question regarding N\"orlund summation methods (and summation methods in general) concerns inclusion. We say that $(N, q)$ {\it includes} $(N, p)$ precisely when each $(N, p)$-convergent sequence is $(N, q)$-convergent to the same limit; equivalently, when each $(N, p)$-summable series is $(N, q)$-summable to the same sum. This relationship will be symbolized by $(N, p) \rightsquigarrow (N, q)$. The important notion of regularity may be seen as a special case of inclusion: the N\"orlund method $(N, q)$ is said to be {\it regular} precisely when each ordinarily convergent sequence is $(N, q)$-convergent to the same limit; that is, precisely when $(N, u) \rightsquigarrow (N, q)$ where $u_0 = 1$ and where $u_n = 0$ whenever $n > 0$. Precise necessary and sufficient conditions for the {\it regular} N\"orlund method $(N, q)$ to include the {\it regular} N\"orlund method $(N, p)$ were determined by Marcel Riesz and communicated to Hardy in a letter, an extract from which appeared as [3]. The line of argument indicated by Riesz in his letter was amplified by Hardy in his classic treatise `{\it Divergent Series}' [1], which we recommend for further information regarding summation methods in general and N\"orlund methods in particular. \medbreak Our primary purpose here is to point out that the necessary and sufficient `Riesz' conditions in fact apply to N\"orlund methods quite generally, without regularity hypotheses. \medbreak \section*{Inclusive Riesz Conditions} \medbreak A celebrated theorem of Silverman, Steinmetz and Toeplitz gives necessary and sufficient conditions for a linear summation method to be regular, and proves to be very useful. The infinite matrix $[c_{m, n} : m, n \geqslant 0]$ yields a linear summation method $C$ by associating to each sequence $s = (s_n : n \geqslant 0)$ a corresponding sequence $t = (t_m : m \geqslant 0)$ given by $$t_m := \sum_{n = 0}^{\infty} c_{m, n} s_n$$ assumed convergent; to say that this linear summation method is {\it regular} is to say that, whenever the sequence $s$ is convergent, the sequence $t$ is convergent and $\lim_{m \rightarrow \infty} t_m = \lim_{n \rightarrow \infty} s_n$. The Silverman-Steinmetz-Toeplitz theorem may now be stated as follows. \medbreak \begin{theorem} \leftarrowbel{SST} The linear summation method $C$ with matrix $[c_{m, n} : m, n \geqslant 0]$ is regular precisely when each of the following conditions is satisfied: \\ {\rm (i)} there exists $H \geqslant 0$ such that for each $m \geqslant 0$ $$\sum_{n = 0}^{\infty} |c_{m, n}| \leqslant H;$$\\ {\rm (ii)} for each $n \geqslant 0$ $$\lim_{m \rightarrow \infty} c_{m, n} = 0;$$\\ {\rm (iii)} $$\lim_{m \rightarrow \infty} \sum_{n = 0}^{\infty} c_{m, n} = 1.$$ \end{theorem} \begin{proof} This appears conveniently as Theorem 2 in [1]. \end{proof} \medbreak Now, let $(N, p)$ and $(N, q)$ be N\"orlund summation methods, or N\"orlunds for short. As $p_0$ is nonzero, the (triangular Toeplitz) system $$q_n = k_0 p_n + \cdots + k_n p_0 \; \; \; (n \geqslant 0)$$ is solved (recursively) by a unique sequence $k = (k_n : n \geqslant 0)$ of comparison coefficients; by summation, it follows that whenever $n \geqslant 0$ also $$Q_n = k_0 P_n + \cdots + k_n P_0.$$ The comparison sequence $k$ generates a (formal) power series $$k(x) = \sum_{n \geqslant 0} k_n x^n$$ while the N\"orlund sequences $p$ and $q$ also generate their own power series; the convolution relation $q = k * p$ between sequences corresponds to the relation $$q(x) = k(x) p(x)$$ between generating functions. We remark that if the N\"orlunds $(N, p)$ and $(N, q)$ are regular, their power series $p(x)$ and $q(x)$ converge whenever $|x| < 1$; the nonvanishing of $p(0) = p_0$ then ensures that the power series $k(x)$ converges when $|x|$ is small. \medbreak The introduction of the sequence $(k_n : n \geqslant 0)$ of comparison coefficients facilitates the following convenient expression for the N\"orlund means determined by $(N, q)$ in terms of the N\"orlund means determined by $(N, p)$. \medbreak \begin{theorem} \leftarrowbel{NN} If $r = (r_n : n \geqslant 0)$ is any sequence then $$ (N^q r)_m = \sum_{n = 0}^{\infty} c_{m, n} (N^p r)_n$$ where if $n > m$ then $c_{m, n} = 0$ while if $n \leqslant m$ then $c_{m, n} = k_{m - n} P_n / Q_m$. \end{theorem} \begin{proof} Direct calculation: simply take the definition $$Q_m (N^q r)_m = q_0 r_m + \cdots + q_m r_0$$ and rearrange thus $$k_0 p_0 r_m + \cdots + (k_0 p_m + \cdots + k_m p_0) r_0 = k_0 (p_0 r_m + \cdots + p_m r_0) + \cdots + k_m (p_0 r_0)$$ to obtain $$Q_m (N^q r)_m = k_0 P_m (N^p r)_m + \cdots + k_m P_0 (N^p r)_0.$$ \end{proof} \medbreak We note that this result appears in the proof of [1] Theorem 19 but is there recorded only for regular N\"orlunds and established by comparing power series expansions; the argument presented here (essentially due to N\"orlund) is taken from [1] Theorem 17 and comes directly from the comparison coefficients without involving regularity. \medbreak The Riesz conditions ${\bf R}_{p q}$ associated to the N\"orlunds $(N, p)$ and $(N, q)$ may now be stated as follows: \medbreak ${\bf R}_{p q}^1$: there exists $H \geqslant 0$ such that for each $m \geqslant 0$ $$|k_0| P_m + \cdots + |k_m| P_0 \leqslant H Q_m;$$ \medbreak ${\bf R}_{p q}^2$: the sequence $(k_m / Q_m : m \geqslant 0)$ converges to zero. \medbreak As mentioned in the introduction, the fact that ${\bf R}_{p q}^1$ and ${\bf R}_{p q}^2$ are both necessary and sufficient for the inclusion $(N, p) \rightsquigarrow (N, q)$ between {\it regular} N\"orlunds appeared in [3] and was elaborated in [1] where it becomes Theorem 19. In what follows, we re-examine the line of argument taken in [3] and [1], deliberately stripping regularity hypotheses. \medbreak Henceforth, we shall write $C_{p q}$ for the linear summation method with matrix $[c_{m, n} : m, n \geqslant 0]$ expressing $(N, q)$ in terms of $(N, p)$ as in Theorem \ref{NN}. \medbreak On the one hand, we relate inclusion $(N, p) \rightsquigarrow (N, q)$ to regularity of $C_{p q}$. \medbreak \begin{theorem} \leftarrowbel{inclusionreg} The inclusion $(N, p) \rightsquigarrow (N, q)$ holds precisely when the linear summation method $C_{p q}$ is regular. \end{theorem} \begin{proof} Assume $(N, p) \rightsquigarrow (N, q)$. Let $s = (s_n : n \geqslant 0)$ be any sequence. Note that $s = N^p r$ for a unique sequence $r = (r_n : n \geqslant 0)$ found by recursively solving the triangular Toeplitz system $$P_n s_n = p_0 r_n + \cdots + p_n r_0 \; \; \; (n \geqslant 0).$$ According to Theorem \ref{NN}, if $m \geqslant 0$ then $$t_m := \sum_{n = 0}^{\infty} c_{m, n} s_n = \sum_{n = 0}^{\infty} c_{m, n} (N^p r)_n = (N^q r)_m.$$ Now, let $s \rightarrow \sigma$: then $N^p r \rightarrow \sigma$ (by choice of $r$) hence $r \xrightarrow{(N, p)} \sigma$ (by definition of $(N, p)$-convergence) so that $r \xrightarrow{(N, q)} \sigma$ (by the $(N, p) \rightsquigarrow (N, q)$ assumption) whence $N^q r \rightarrow \sigma$ (by definition of $(N, q)$-convegence); that is, $t \rightarrow \sigma$. This proves that $C_{p q}$ is regular. \medbreak Assume that $C_{p q}$ is regular. Let $r = (r_n : n \geqslant 0)$ be $(N, p)$-convergent to $\sigma$: then $N^p r \rightarrow \sigma$ so Theorem \ref{NN} and the regularity of $C_{p q}$ yield $N^q r \rightarrow \sigma$; thus, $r$ is $(N, q)$-convergent to $\sigma$ also. This proves $(N, p) \rightsquigarrow (N, q)$. \end{proof} \medbreak On the other hand, we relate regularity of $C_{p q}$ to the Riesz conditions ${\bf R}_{p q}$. \medbreak \begin{theorem} \leftarrowbel{regRiesz} The linear summation method $C_{p q}$ is regular precisely when the Riesz conditions ${\bf R}_{p q}^1$ and ${\bf R}_{p q}^2$ are satisfied. \end{theorem} \begin{proof} Assume $C_{p q}$ to be regular and invoke Theorem \ref{SST}. Part (i) furnishes $H \geqslant 0$ such that for each $m \geqslant 0$ $$\frac{|k_m| P_0 + \cdots + |k_0| P_m}{Q_m} = \sum_{n = 0}^{m} \Big|\frac{k_{m - n} P_n}{Q_m}\Big| = \sum_{n = 0}^{\infty} |c_{m, n}| \leqslant H$$ whence ${\bf R}_{p q}^1$ holds. Part (ii) says that $\lim_{m \rightarrow \infty} c_{m, n} = 0$ for each $n \geqslant 0$; in particular, $$0 = \lim_{m \rightarrow \infty} c_{m, 0} = \lim_{m \rightarrow \infty} \frac{k_m}{Q_m} P_0$$ whence ${\bf R}_{p q}^2$ holds. \medbreak Assume that ${\bf R}_{p q}^1$ and ${\bf R}_{p q}^2$ are satisfied. Theorem \ref{SST}(i) holds because $$\sum_{n = 0}^{\infty} |c_{m, n}| = \sum_{n = 0}^{m} \Big|\frac{k_{m - n} P_n}{Q_m}\Big| = \frac{|k_m| P_0 + \cdots + |k_0| P_m}{Q_m} \leqslant H$$ on account of ${\bf R}_{p q}^1$. Theorem \ref{SST}(ii) holds because $$|c_{m, n}| = \Big|\frac{k_{m - n} P_n}{Q_m}\Big| \leqslant \frac{|k_{m - n}|}{Q_{m - n}} P_n \rightarrow 0 \; \; {\rm as} \; \; m \rightarrow \infty$$ on account of ${\bf R}_{p q}^2$. Finally, Theorem \ref{SST}(iii) holds simply because of the relation $$Q_m = k_0 P_m + \cdots + k_m P_0.$$ Theorem \ref{SST} now guarantees that $C_{p q}$ is regular. \end{proof} \medbreak In conclusion, the Riesz conditions ${\bf R}_{p q}$ are both necessary and sufficient for the inclusion $(N, p) \rightsquigarrow (N, q)$ without any assumptions of regularity. \medbreak \begin{theorem} Let $(N, p)$ and $(N, q)$ be any N\"orlund methods. The inclusion $(N, p) \rightsquigarrow (N, q)$ holds precisely when the Riesz conditions ${\bf R}_{p q}^1$ and ${\bf R}_{p q}^2$ are satisfied. \end{theorem} \begin{proof} Simply combine Theorem \ref{inclusionreg} and Theorem \ref{regRiesz}. \end{proof} \bigbreak \begin{center} {\small R}{\footnotesize EFERENCES} \end{center} \medbreak [1] G.H. Hardy, {\it Divergent Series}, Clarendon Press, Oxford (1949). \medbreak [2] N.E. Norlund, {\it Sur une application des fonctions permutables}, Lunds Universitets \r{A}rsskrift (2) Volume 16 Number 3 (1920) 1-10. \medbreak [3] M. Riesz, {\it Sur l'\'equivalence de certaines m\'ethodes de sommation}, Proceedings of the London Mathematical Society (2) {\bf 22} (1924) 412-419. \medbreak [4] P.L. Robinson, {\it Finite N\"orlund Summation Methods}, arXiv 1712.06744 (2017). \medbreak [5] G.F. Voronoi, {\it Extension of the Notion of the Limit of the Sum of Terms of An Infinite Series}, Annals of Mathematics (2) Volume 33 Number 3 (1932) 422-428. \end{document}

\begin{document} \title[Potential well theory for DNLS]{Potential well theory for the derivative nonlinear Schr\"{o}dinger equation} \author{Masayuki Hayashi} \address{Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan} \curraddr{} \email{hayashi@kurims.kyoto-u.ac.jp} \thanks{} \subjclass[2010]{Primary 35Q55, 35Q51, 37K05; Secondary 35A15} \keywords{derivative nonlinear Schr\"{o}dinger equation, solitons, potential well, variational methods} \dedicatory{} \begin{abstract} We consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation} \leftabel{eq:1} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in {\mathbb R} \times {\mathbb R}, \ b \in {\mathbb R} . \end{equation} If $b=0$, this equation is known as a gauge equivalent form of well-known derivative nonlinear Schr\"{o}dinger equation (DNLS), which is mass critical and completely integrable. The equation \eqref{eq:1} can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data $u_0\in H^1(\mathbb{R})$ satisfies the mass condition $\| u_0\|_{L^2}^2 <4\pi$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on \eqref{eq:1} for general $b\in{\mathbb R}$, which is exactly corresponding to $4\pi$-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both $4\pi$-mass condition and algebraic solitons. \end{abstract} \maketitle \tableofcontents \numberwithin{equation}{section} \section{Introduction} \subsection{Setting of the problem} In this paper, we consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation} \leftabel{eq:1.1} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in {\mathbb R} \times {\mathbb R}, \ b \in {\mathbb R} . \end{equation} The equation \eqref{eq:1.1} is $L^2$-critical (mass critical) in the sense that the equation and $L^2$-norm are invariant under the scaling transformation \begin{align} \leftabel{eq:1.2} u_{\leftambda}(t,x)=\leftambda^{\frac{1}{2}} u(\leftambda^2 t,\leftambda x), \quad \leftambda >0. \end{align} This equation has the following conserved quantities: \begin{align*} \tag{Energy} E(u)&:=\frac{1}{2}\lefteft\| \partial_x u \rightight\|_{L^2}^2 - \frac{1}{4}\rightbra[i|u|^2\partial_xu ,u] -\frac{b}{6}\| u\|_{L^6}^6 , \\ \tag{Mass} M(u)&:= \| u \|_{L^2}^2, \\ \tag{Momentum} P(u)&:=\rightbra[i\partial_x u,u], \end{align*} where $\rightbra[\cdot,\cdot]$ is an inner product defined by \begin{align*} \rightbra[v ,w] :={\rightm Re}\int_{{\mathbb R}} v(x)\overline{w(x)} dx\quad\text{for}~v, w\in L^2({\mathbb R} ). \end{align*} We note that ({\rightm Re}f{eq:1.1}) can be rewritten as the following Hamiltonian form: \begin{align} \leftabel{eq:1.3} i \partial_t u= E ' (u). \end{align} When $b=0$, the equation \begin{equation*} \leftabel{DNLS} \tag{DNLS} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u =0 , \quad (t,x) \in {\mathbb R} \times {\mathbb R} \end{equation*} is known as a standard derivative nonlinear Schr\"{o}dinger equation (DNLS). This equation has several gauge equivalent forms. If we apply the following gauge transformation to the solution of \eqref{DNLS} \begin{align} \leftabel{eq:1.4} \psi (t,x) = u (t,x)\exp\left( -\frac{i}{2} \int_{-\infty}^{x} |u (t,x)|^2 dx\right) , \end{align} then $\psi$ satisfies the following equation: \begin{equation} \leftabel{eq:1.5} i \partial_t \psi + \partial_x^2 \psi +i \partial_x( |\psi |^{2} \psi )=0 , \quad (t,x) \in {\mathbb R} \times {\mathbb R}. \end{equation} The equation ({\rightm Re}f{eq:1.5}) originally appeared in plasma physics as a model for the propagation of Alfv\'{e}n waves in magnetized plasma (see \cite{MOMT76, M76}), and it is known to be completely integrable (see \cite{KN78}). The equation ({\rightm Re}f{eq:1.1}) can be considered as a generalized equation of ({\rightm Re}f{DNLS}) while preserving both $L^2$-criticality and Hamiltonian structure. We note that complete integrable structure is known only for the case $b=0$. In this paper we study \eqref{eq:1.1} by variational approach not depending on complete integrability, and that enables us to treat \eqref{eq:1.1} for general $b\in{\mathbb R}$ in a unified way. The main aim of this paper is to investigate the structure of ({\rightm Re}f{eq:1.1}) from the viewpoint of potential well theory. \subsection{DNLS and mass critical NLS} There is a large literature on the Cauchy problem for \eqref{DNLS}; see \cite{TF80, TF81, H93, HO92, T99, BL01, CKSTT01, CKSTT02, Wu13, Wu15, GW17, FHI17, H18, JLPS18, JLPS18a} and references therein. Here we are mainly interested in the results of energy space $H^1({\mathbb R})$. Hayashi and Ozawa \cite{HO92} proved that \eqref{DNLS} is globally well-posed in $H^1({\mathbb R})$ under the mass condition $M (u_0) < 2\pi$, where $u_0\in H^1({\mathbb R})$ is the initial data. The mass condition was recently improved by Wu \cite{Wu15} to $M(u_0) < 4\pi$. We note that the value $4\pi$ corresponds to the mass of algebraic solitons of ({\rightm Re}f{DNLS}) and algebraic solitons correspond to the threshold case in the existence of solitons. We will discuss the solitons for \eqref{DNLS} in more detail later. Fukaya, the author and Inui \cite{FHI17} gave a sufficient condition for global existence in $H^1({\mathbb R})$ covering Wu's result by variational approach. In particular they established the global results for the threshold case $M (u_0)=4\pi$ and $P(u_0)<0$, and for the oscillating data with arbitrarily large mass. We note that in these global results by PDE approach the class of the solution lies in $(C\cap L^{\infty})({\mathbb R}, H^1({\mathbb R}))$. Recently, in \cite{JLPS18} it was proved by inverse scattering approach that \eqref{DNLS} is globally well-posed for any initial data belonging to weighted Sobolev space $H^{2,2} ({\mathbb R} )$, where \begin{align*} H^{2,2} ({\mathbb R} ) := \left\{ u\in H^2 ({\mathbb R} )~;~\braket{\cdot}^2u\in L^2({\mathbb R} ) \right\}, \quad \braket{x}:=(1+x^2)^{1/2}. \end{align*} The class of the solution lies in $C({\mathbb R}, H^{2,2}({\mathbb R}))$ for the initial data $u_0\in H^{2,2} ({\mathbb R} )$. This is the strong result obtained by using complete integrable structure, but the global well-posedness in the energy space $H^1({\mathbb R})$ above the mass threshold $4\pi$ is not clear yet. We note that the algebraic solitons do not belong to $H^{2,2}({\mathbb R})$, but they belong to $H^1({\mathbb R})$. This fact implies that the difference of function spaces between $H^1({\mathbb R})$ and $H^{2,2}({\mathbb R})$ is a delicate issue for \eqref{DNLS}. \eqref{DNLS} is closely related to the focusing mass critical nonlinear Schr\"odinger equation in one space dimension.\footnote{It is also related to other mass critical dispersive equations such as the quintic Korteweg-de Vries equation and the modified Benjamin--Ono equation, but we will not discuss them further here.} Let us consider the following nonlinear Schr\"odinger equation: \begin{align} \leftabel{NLS} \tag{NLS} i\partial_t v+\partial_x^2 v+\frac{3}{16}|v|^4v=0, \quad (t,x) \in {\mathbb R} \times {\mathbb R}. \end{align} By using the following gauge transformation to the solution of \eqref{DNLS} \begin{align} \leftabel{eq:1.6} v(t,x) = u(t,x)\exp\left( \frac{i}{4} \int_{-\infty}^{x} |u(t,x)|^2 dx\right), \end{align} we have another gauge equivalent form: \begin{align} \leftabel{DNLS'} \tag{DNLS$'$} i \partial_t v + \partial_x^2 v +\frac{i}{2} |v|^{2} \partial_x v -\frac{i}{2}v^2\partial_x \overline{v} +\frac{3}{16}|v|^4v =0, \quad (t,x) \in {\mathbb R} \times {\mathbb R}. \end{align} The equations \eqref{NLS} and \eqref{DNLS'} have mass critical structure and the same conserved quantities in the forms of \begin{align} \tag{Energy} &{{\mathcal E}}(v)= \frac{1}{2}\| \partial_x v\|_{L^2}^2 -\frac{1}{32} \| v\|_{L^6}^6 ,\\ \tag{Mass} &{{\mathcal M}}(v) =\| v\|_{L^2}^2. \end{align} Moreover, they have the same standing wave solutions as $v_{\omega}(t,x)=e^{i\omega t}Q_{\omega}(x)$, where $\omega >0$ and $Q_{\omega}>0$ is the positive solution of \begin{align*} -Q'' +\omega Q -\frac{3}{16}Q^5=0, ~x\in {\mathbb R}. \end{align*} From the work of Weinstein \cite{W82}, we have the following sharp Gagliardo--Nirenberg inequality \begin{align} \leftabel{GN1} \frac{1}{32}\| f\|_{L^6}^6 \lefteq \frac{1}{2}\left( \frac{{\mathcal M}(f)}{{\mathcal M}(Q_{\omega})} \right)^2\| \partial_{x}f\|_{L^2}^2 ~\text{for all}~f\in H^1({\mathbb R}). \end{align} If the initial data $v_0\in H^1({\mathbb R})$ satisfies ${\mathcal M}(v_0)<{\mathcal M} (Q_{\omega}) =2\pi$, by \eqref{GN1} and conservation laws of the mass and the energy, we deduce that the corresponding $H^1({\mathbb R})$-solution of \eqref{DNLS'} or \eqref{NLS} exists globally in time, and satisfies \begin{align} \leftabel{eq:1.8} \frac{1}{2}\left( 1- \left( \frac{{\mathcal M}(v_0)}{{\mathcal M}(Q_{\omega})} \right)^2\right) \| \partial_x v(t)\|_{L^2}^2 \lefteq {\mathcal E}(v_0) ~\text{for all } t\in{\mathbb R}. \end{align} In such a way $2\pi$-mass condition for \eqref{DNLS} was established in \cite{HO92}. For the case of \eqref{NLS}, it is known that this mass condition is sharp, in the sense that for any $\rightho \geq 2\pi$, there exists initial data $v_0 \in H^1({\mathbb R})$ such that ${\mathcal M} (v_0) =\rightho$ and such that the corresponding $H^1({\mathbb R})$-solution to ({\rightm Re}f{NLS}) blows up in finite time. \subsection{Potential well theory for NLS} \leftabel{sec:1.3} Here we review the mass condition for \eqref{NLS} from the viewpoint of potential well theory. For $\omega>0$ we define the action functional by \begin{align*} {\mathcal S}_{\omega}(\varphi) ={\mathcal E} (\varphi)+\frac{\omega}{2} {\mathcal M} (\varphi). \end{align*} We note that $Q_{\omega}$ is a critical point of ${\mathcal S}_{\omega}$, i.e., ${\mathcal S}_{\omega}'(Q_{\omega}) =0$. We consider the following subsets of the energy space: \begin{align*} {\mathscr A}_{\omega}:=& \left\{ \varphi\in H^1({\mathbb R} ): {\mathcal S}_{\omega}(\varphi ) < {\mathcal S}_{\omega}(Q_{\omega})\right\}, \\ {\mathscr A} :=&\bigcup_{ \substack{\omega>0 } } {\mathscr A}_{\omega}. \end{align*} The set ${\mathscr A}$ describes the data below the ground state in the sense of action. Since ${\mathcal E}(Q_{\omega})=0$ and ${\mathcal M}(Q_{\omega})=2\pi$, the set ${\mathscr A}$ is decomposed into two disjoint sets as ${\mathscr A} = {\mathscr G} \cup {\mathscr B}$, where \begin{align*} {\mathscr G} &:=\left\{ \varphi \in H^1({\mathbb R}) : {\mathcal M}(\varphi) <2\pi \right\}, \\ {\mathscr B} &:=\left\{ \varphi \in H^1({\mathbb R}) : {\mathcal M}(\varphi) >2\pi , {\mathcal E} (\varphi) <0 \right\}. \end{align*} For \eqref{NLS} the global behavior of solutions to the initial data in ${\mathscr A}$ is well understood now. As can be seen above, the solution for the initial data in ${\mathscr G}$ is global, and furthermore scatters both forward and backward in time (see \cite{D15}). On the other hand, the solution for the initial data in ${\mathscr B}$ blows up both forward and backward in finite time (see \cite{OT91}). One can also give a variational characterization to the sets ${\mathscr G}$ and ${\mathscr B}$ as follows. We define the functional by ${\mathcal K}_{\omega}(\varphi)=\left.\frac{d}{d\leftambda}{\mathcal S}_{\omega}(\leftambda\varphi)\right|_{\leftambda =1}$ for $\omega >0$, and introduce the following subsets of the energy space: \begin{align*} {\mathscr A}_{\omega}^+ :=&\left\{ \varphi\in{\mathscr A}_{\omega} :{\mathcal K}_{\omega}(\varphi) \geq 0 \right\},\\ {\mathscr A}_{\omega}^- :=&\left\{ \varphi\in{\mathscr A}_{\omega} :{\mathcal K}_{\omega}(\varphi) < 0\right\},\\ {\mathscr A}^{+} :=&\bigcup_{ \substack{\omega>0 } } {\mathscr A}_{\omega}^{+}, ~ {\mathscr A}^{-} :=\bigcup_{ \substack{\omega>0 } } {\mathscr A}_{\omega}^{-}. \end{align*} Then, one can easily prove that the sets ${\mathscr A}_{\omega}^+$ and ${\mathscr A}_{\omega}^-$ are invariant under the flow of \eqref{NLS},\footnote{If the initial data in ${\mathscr A}_{\omega}^+$ (resp. ${\mathscr A}_{\omega}^-$), then the corresponding solution of \eqref{NLS} also belongs to ${\mathscr A}_{\omega}^+$ (resp. ${\mathscr A}_{\omega}^-$) as long as the solutions exists.} and that \begin{align} \leftabel{eq:1.9} {\mathscr G} ={\mathscr A}^+ , ~{\mathscr B} ={\mathscr A}^-. \end{align} The key point in the proof for this claim is variational characterization of $Q_{\omega}$ on the Nehari manifold: $\medmuskip=1mu \left\{ \varphi\in H^1({\mathbb R})\setminus\{ 0\}: {\mathcal K}_{\omega}(\varphi)=0\right\}$. We note that the set \begin{align*} {\mathscr A}^0= \left\{ \varphi\in H^1({\mathbb R}) :{\mathcal M}(\varphi)=2\pi, {\mathcal E}(\varphi)=0\right\} \end{align*} gives the boundary of both ${\mathscr G}$ and ${\mathscr B}$. By variational characterization of $Q_{\omega}$, this set is actually equal to the standing wave up to translations and phase shifts, i.e., \begin{align} \leftabel{eq:1.10} {\mathscr A}^0 = \left\{ e^{i\theta}Q_{\omega}(\cdot -y): \theta, y \in{\mathbb R}, \,\omega >0\right\} . \end{align} \subsection{Two types of solitons} \leftabel{sec:1.4} Despite many similarities to \eqref{NLS}, $2\pi$-mass condition is not sharp for \eqref{DNLS}. One of this reason comes from the difference of soliton structure. It is well-known that ({\rightm Re}f{DNLS}) has a two-parameter family of solitons in the form of $u_{\omega ,c}(x,t)=e^{i\omega t}\phi_{\omega ,c}(x-ct)$, where $-2\sqrt{\omega}<c\lefteq 2\sqrt{\omega}$, and $\phi_{\omega ,c}$ is the complex-valued solution of the equation \begin{align*} -\phi'' +\omega\phi +ic\phi' -i|\phi|^2\phi' =0, ~x\in {\mathbb R}. \end{align*} An interesting property for \eqref{DNLS} is that the equation has algebraic solitons (the case $c=2\sqrt{\omega}$) as well as bright solitons (the case $\omega >c^2/4$); see \cite{KN78}. Actually we have the following explicit decay of these solitons at infinity; \begin{align*} \begin{array}{ll} \text{if}~\omega >c^2/4, &|\phi_{\omega ,c}(x)| \sim e^{-\sqrt{4\omega -c^2}|x|}\\[5pt] \text{if}~c=2\sqrt{\omega}, &|\phi_{\omega ,c}(x)| \sim (c|x|)^{-1} \end{array} \text{as}~ |x| \gg 1. \end{align*} \indent We note that the following curve \begin{align} \leftabel{eq:1.11} {\mathbb R}^+ \ni\omega \mapsto (\omega , 2s\sqrt{\omega}) \in {\mathbb R}^2 ~\text{for}~s\in (-1,1] \end{align} gives the scaling of the solitons for \eqref{DNLS}. Indeed we have the following relation \begin{align} \leftabel{eq:1.12} \phi_{\omega ,2s\sqrt{\omega}}(x) =\omega^{1/4} \phi_{1,2s} (\sqrt{\omega}x), \end{align} which especially implies that the mass of the solitons is invariant on this curve. From the explicit formulae of the mass of the soliton (see \cite{CO06}), we deduce that the function \begin{align} \leftabel{eq:1.13} (-1, 1]\ni s\mapsto M(\phi_{1 ,2s} ) =8\tan^{-1} \sqrt{ \frac{1+s}{1-s} } \in (0,4\pi ] \end{align} is strictly increasing and surjective. In particular we note that the value $4\pi$ corresponds to the mass of algebraic solitons. This property is quite different from \eqref{NLS}. By using Galilean invariance of the equation \eqref{NLS}, one can generate a two-parameter family of solitary waves \begin{align*} v_{\omega ,c}(t,x) = e^{i\omega t+\frac{i}{2}cx-\frac{i}{4}c^2t}Q_{\omega}(x-ct) \end{align*} from standing waves, but their mass is always $2\pi$ for any $\omega >0$ and $c\in{\mathbb R}$. The stability of the solitons have been also studied in previous works. Colin and Ohta \cite{CO06} proved that all bright solitons for \eqref{DNLS} are orbitally stable. Their proof depends on variational arguments, which are closely related to the work of Shatah \cite{S83}. See also \cite{GW95} for partial results before \cite{CO06}. On the other hand, the orbital stability or instability for algebraic solitons is still an open problem. When $b>0$, the equation \eqref{eq:1.1} still has two types of solitons, but situation becomes different due to the focusing effect from the quintic term. Ohta \cite{O14} proved that for each $b>0$ there exists a unique $s^*=s^*(b) \in (0,1)$ such that the soliton $u_{\omega ,c}$ is orbitally stable if $-2\sqrt{\omega}<c<2s^*\sqrt{\omega}$, and orbitally unstable if $2s^*\sqrt{\omega}<c<2\sqrt{\omega}$. In \cite{NOW17} it was proved that the algebraic soliton $u_{\omega ,2\sqrt{\omega}}$ is orbitally unstable when $b>0$ is sufficiently small. If we observe the momentum of the solitons, the momentum is positive in the stable region, and negative in the unstable region; see Figure {\rightm Re}f{fig:1}. This indicates that the momentum of the soliton has an essential effect on the stability. In the borderline case $c=2s^*\sqrt{\omega}$, the momentum of the solitons is zero, and the orbital stability or instability in this case remains an open problem. \begin{figure} \caption{The stable/unstable region of solitons in the case $b>0$.} \end{figure} The solitons of \eqref{eq:1.1} in the case $b<0$ seem to have been little studied. Since the nonlinear term with derivative has a focusing effect, the equation \eqref{eq:1.1} still has a two-parameter family of the solitons even if the quintic term is defocusing. More precisely, we have the following result. \begin{proposition} \leftabel{prop:1.1} Let $b<0$. The equation \eqref{eq:1.1} has a two-parameter family of solitons $u_{\omega ,c}(x,t)=e^{i\omega t}\phi_{\omega ,c}(x-ct)$ if and only if $(\omega ,c)$ satisfies \begin{align} \leftabel{eq:1.14} \begin{array}{ll} \displaystyle\text{if}~b>-3/16,& \displaystyle -2\sqrt{\omega} <c\lefteq 2\sqrt{\omega} ,\\[3pt] \displaystyle\text{if}~b\lefteq -3/16,& \displaystyle -2\sqrt{\omega} <c<-2s_{\ast}\sqrt{\omega}, \end{array} \end{align} where $s_{\ast} :=\sqrt{ -\gamma /(1-\gamma)}$ and $\gamma :=1+\frac{16}{3}b$. \end{proposition} We note that the value $b=-3/16$ gives the turning point in the structure of the solitons. In particular algebraic solitons exist only for the case $b>-3/16$. In the case $b\lefteq -3/16$ the solitons still exist, but their velocity must be negative. We note that $0\lefteq s_{\ast} <1$ and $s_{\ast}\uparrow 1$ as $b\downarrow -\infty$. This means that as the defocusing effect is stronger, the existence region of solitons is narrower; see Figure {\rightm Re}f{fig:2}. \begin{figure} \caption{Existence region of solitons.} \end{figure} Similarly as \eqref{DNLS}, the curve \eqref{eq:1.12} gives the scaling of the solitons for the equation \eqref{eq:1.1}. For the variety of the mass we have the following result. \begin{proposition} \leftabel{prop:1.2} Let $b\in {\mathbb R}$. If $b>-3/16$, the function \begin{align*} (-1, 1]\ni s\mapsto M(\phi_{1 ,2s} ) \in \left( 0, \frac{4\pi}{\sqrt{\gamma} } \right] \end{align*} is strictly increasing and surjective. Similarly, if $b\lefteq -3/16$, the function \begin{align*} (-1, -s_*)\ni s\mapsto M(\phi_{1 ,2s} ) \in (0, \infty ) \end{align*} has the same property. \end{proposition} To examine the effect of the momentum is important in our analysis. We recall that the momentum of the solitons in the case $b\geq 0$ have the following property: \begin{align*} \begin{array}{ll} \displaystyle\text{if}~b=0,& P(\phi_{1,2s}) >0~\text{for}~s\in (-1,1)~\text{and}~P(\phi_{1,2})=0,\\[3pt] \displaystyle\text{if}~b>0,& P(\phi_{1,2s})>0~\text{for}~s\in (-1, s^*),~P(\phi_{1,2s^*})=0\\[3pt] &\text{and}~P(\phi_{1,2s})<0~\text{for}~s\in(s^{*},1]. \end{array} \end{align*} One can prove that $s^*(b)\to 1$ as $b\downarrow 0$ (see Remark {\rightm Re}f{rem:2.8}). In this sense we set $s^*(0):=1$. We note that the value $s^*$ is characterized by \begin{align} \leftabel{eq:1.15} P(\phi_{1,2s^*(b)}) =0 \quad \text{for all}~b\geq 0. \end{align} For the momentum of the solitons in the case $b<0$, we have the following result. \begin{proposition} \leftabel{prop:1.3} Let $b<0$. The momentum of all solitons for the equation \eqref{eq:1.1} is positive. \end{proposition} In Section {\rightm Re}f{sec:2} we study the solitons of \eqref{eq:1.1} in more detail, and give a proof for these propositions. \subsection{Main results} First we establish the mass condition for the equation ({\rightm Re}f{eq:1.1}). The local well-posedness in the energy space $H^1({\mathbb R})$ was obtained in \cite{HO94a, Oz96}. In \cite{Oz96} it was proved that \eqref{eq:1.1} was globally well-posed in $H^1({\mathbb R})$ under the mass condition \begin{align} \leftabel{eq:1.16} \begin{array}{ll} \displaystyle M(u_0) < \frac{2\pi}{\sqrt{\gamma}} &\text{if}~ b>0, \\[9pt] \displaystyle M(u_0) <2\pi &\text{if}~ b\lefteq 0, \end{array} \end{align} where we recall that $\gamma =1+\frac{16}{3}b$. This result is considered as a natural extension of $2\pi$-mass condition for \eqref{DNLS}.\footnote{Actually more general equation including \eqref{eq:1.1} is studied in \cite{Oz96}. } From the following energy form \begin{align} \leftabel{eq:1.17} E\left( {\mathcal G}_{-1/4}(u) \right) =\frac{1}{2}\| \partial_x u \|_{L^2}^2 -\frac{\gamma}{32}\| u\|_{L^6}^6, \end{align} where \begin{align*} {\mathcal G}_{a}(u)(t,x):=\exp\left( ia\int_{-\infty}^{x}|u(t,y)|^2 dy\right)u(t,x) \quad\text{for}~a\in{\mathbb R}, \end{align*} the mass condition seems to be necessary when $b>-3/16$. By using the sharp Gagliardo--Nirenberg inequality \eqref{GN1} and the conservation laws of mass and energy, $\frac{2\pi}{\sqrt{\gamma}}$-mass condition is obtained when $b>-3/16$. We note that the value $\frac{2\pi}{\sqrt{\gamma}}$ corresponds to the mass of the standing waves of \eqref{eq:1.1}, i.e., $\frac{2\pi}{\sqrt{\gamma}} =M(\phi_{\omega,0})$. Our first result gives the improvement of the mass condition in previous works. \begin{theorem} \leftabel{thm:1.4} Let $u_0\in H^1({\mathbb R})$ satisfy each of the following two cases\textup{:} \begin{enumerate}[\rightm (i)] \setlength{\itemsep}{3pt} \item If $b>0$, $M(u_0) < M(\phi_{1,2s^*})$, or $M(u_0) =M(\phi_{1,2s^*})$ and $P(u_0)<0$. \item If $-3/16<b\lefteq 0$, $M(u_0) < \frac{4\pi}{\gamma^{3/2}}$, or $M(u_0) =\frac{4\pi}{\gamma^{3/2}}$ and $P(u_0)<0$. \end{enumerate} Then the $H^1({\mathbb R})$-solution $u$ of \eqref{eq:1.1} with $u(0)=u_0$ exists globally both forward and backward in time. Moreover we have \begin{align*} \sup_{t\in{\mathbb R}} \| u(t)\|_{H^1}\lefteq C(\| u_0\|_{H^1})<\infty . \end{align*} \end{theorem} \begin{remark} \leftabel{rem:1.5} When $b\lefteq -3/16$, the equation \eqref{eq:1.1} is globally well-posed for any initial data $u_0\in H^1({\mathbb R})$. In particular the global result in the case $b=-3/16$ is compatible with Theorem {\rightm Re}f{thm:1.4}, since $\frac{4\pi}{\gamma^{3/2}}\uparrow\infty$ as $b\downarrow -3/16$. \end{remark} We recall that if $b>0$ the soliton $\phi_{1,2s^*}$ corresponds to borderline case in the stable/unstable region of solitons as in Figure {\rightm Re}f{fig:1}. By Proposition {\rightm Re}f{prop:1.2}, we have the following relation \begin{align*} \frac{2\pi}{\sqrt{\gamma}} =M(\phi_{1,0}) <M(\phi_{1,2s^*}) < M(\phi_{1,2}) = \frac{4\pi}{\sqrt{\gamma}}, \end{align*} which implies that our mass condition improves the one \eqref{eq:1.16}. We note that \begin{align*} M (\phi_{1,2s^*}) \to 4\pi ~\text{as}~b\downarrow 0, \end{align*} which follows from the claim that $s^*(b)\to 1$ as $b\downarrow 0$. This means that the mass condition in Theorem {\rightm Re}f{thm:1.4} is compatible with $4\pi$-mass condition for ({\rightm Re}f{DNLS}). The mass condition in the case $-3/16<b<0$ is more interesting. Since $0<\gamma <1$ in this case, the value $\frac{4\pi}{\gamma^{3/2}}$ is greater than $4\pi$. This means that $4\pi$-mass condition for ({\rightm Re}f{DNLS}) is improved due to the defocusing effect from the quintic term. Moreover, the value $\frac{4\pi}{\gamma^{3/2}}$ is even greater than the mass of algebraic solitons. Indeed, we have the following relation: \begin{align*} M (\phi_{1,2}) = \frac{4\pi}{\sqrt{\gamma}}< \frac{4\pi}{\gamma^{3/2}} =M(\phi_{1,2})+P(\phi_{1,2}), \end{align*} which indicates that positive momentum of algebraic solitons boosts the threshold value. Our next result is a global result for large data. If we consider sufficiently oscillating data, we obtain the global result for arbitrarily large mass: \begin{theorem} \leftabel{thm:1.6} Let $b>-3/16$. Given $\psi \in H^1({\mathbb R} )$, and set the initial data as $u_{0,\mu}=e^{i\mu x}\psi$. Then, there exists $\mu_0=\mu_0(\psi ) >0$ such that if $\mu\geq\mu_0$, then the $H^1({\mathbb R})$-solution $u_{\mu}$ of \eqref{eq:1.1} with $u_{\mu}(0)=u_{0,\mu}$ exists globally both forward and backward in time. Moreover we have \begin{align*} \sup_{t\in{\mathbb R}} \| u_{\mu}(t)\|_{H^1}\lefteq C(\| u_{0,\mu}\|_{H^1})<\infty . \end{align*} \end{theorem} This global result was first discovered in \cite{FHI17} for \eqref{DNLS}.\footnote{As seen in \cite{FHI17}, this global result still holds for the generalized derivative nonlinear Schr\"{o}dinger equation in $L^2$-supercritical setting. } It is worthwhile to compare the global results for the quadratic oscillating data in \eqref{NLS}. Cazenave and Weissler \cite{CW92} established global existence for oscillating data as follows: Given $\psi \in H^{1,1}({\mathbb R})$ which is defined by \begin{align*} H^{1,1} ({\mathbb R} ) := \left\{ u\in H^1 ({\mathbb R} )~;~\braket{\cdot}u\in L^2({\mathbb R} ) \right\}. \end{align*} Set the initial data as $u_{0,\beta} :=e^{i\frac{\beta |x|^2}{4}}\psi$. Then, there exists $\beta_0=\beta_0(\psi ) >0$ such that if $\beta\geq\beta_0$, the corresponding solution $u_{\beta}$ for \eqref{NLS} satisfies $C((0, \infty), H^{1,1}({\mathbb R}))$.\footnote{The solution $u_{\beta}$ also satisfies $u_{\beta}\in L^{\infty}((0, \infty), H^1 ({\mathbb R}))$.} We note that the quadratic oscillating data only yields global solutions forward in time. In general the solution $u_{\beta}$ may blow up in a finite negative time (see \cite[Remark 6.5.9]{C03}). The other important difference is that the oscillating factor in Theorem {\rightm Re}f{thm:1.6} comes from the change of the momentum, but on the other hand, the quadratic oscillating factor in \cite{CW92} comes from pseudo-conformal transformation. We note that \eqref{eq:1.1} has no Galilean or pseudo-conformal invariance. In particular, due to the lack of Galilean invariance it is reasonable to consider that the momentum of initial data essentially influences global properties of the solution to ({\rightm Re}f{eq:1.1}). The proofs of Theorem {\rightm Re}f{thm:1.4} and Theorem {\rightm Re}f{thm:1.6} are done by adapting variational arguments developed in the work \cite{FHI17}, and actually obtained from a more general result (Theorem {\rightm Re}f{thm:1.7} below). The key point in our approach is to give a variational characterization of the solitons on the Nehari manifold with respect to the action functional. However, for the equation \eqref{eq:1.1} when $b<0$, the quintic term $b|u|^4u$ becomes an obstacle to characterize the solitons. To overcome that we consider the following gauge equivalent form: \begin{align} \leftabel{ME} \tag{1.1$'$} i\partial_t v+\partial_x^2 v+\frac{i}{2}|v|^2\partial_x v-\frac{i}{2}v^2\partial_x\overline{v}+\frac{3}{16}\gamma |v|^4v=0,\quad (t,x) \in {\mathbb R} \times {\mathbb R}. \end{align} We note that \eqref{ME} is transformed from \eqref{eq:1.1} through the gauge transformation $v={\mathcal G}_{1/4}(u)$. When $b=0$ this equation is nothing but \eqref{DNLS'}. The equation \eqref{ME} has the following conserved quantities and solitons: \begin{align*} \tag{Energy} {\mathcal E} (v) &:=\frac{1}{2} \| \partial_x v\|_{L^2}^2-\frac{\gamma}{32} \| v\|_{L^6}^6 , \\ \tag{Mass} {\mathcal M} (v)&:=\| v \|_{L^2}^2, \\ \tag{Momentum} {\mathcal P} (v)&:=\rightbra[i\partial_x v,v] +\frac{1}{4} \| v\|_{L^4}^4,\\ \tag{Soliton} v_{\omega ,c}(t ,x) &:={\mathcal G}_{1/4}(u_{\omega ,c})(t,x)=e^{i\omega t}\varphi_{\omega ,c}(x-ct). \end{align*} We note that global well-posedness in $H^1({\mathbb R})$ for the equations \eqref{eq:1.1} and \eqref{ME} is equivalent since $u\mapsto{\mathscr G}_{1/4}(u)$ is locally Lipschitz continuous on $H^1({\mathbb R})$. From the energy formula of \eqref{ME} one can characterize solitons on the Nehari manifold if $b\geq -3/16$ (see Proposition {\rightm Re}f{prop:4.1}). Based on this variational characterization we formulate potential well theory with two parameters which is related to the classical work of Payne and Sattinger \cite{PS75}. We see that a two-parameter family of potential wells has rich and complex structure compared with the one-parameter one as in Section {\rightm Re}f{sec:1.3}. To state our main results, we prepare some notations. We define the action functional by \begin{align*} {\mathcal S}_{\omega ,c}(\varphi ) :={\mathcal E} (\varphi)+\frac{\omega}{2} {\mathcal M} (\varphi) +\frac{c}{2}{\mathcal P} (\varphi). \end{align*} We note that $\varphi_{\omega, c}$ is a critical point of ${\mathcal S}_{\omega,c}$, i.e., ${\mathcal S}_{\omega ,c}'(\varphi_{\omega ,c}) =0$. We also define the functional by ${\mathcal K}_{\omega ,c}(\varphi):=\left.\frac{d}{d\leftambda}{\mathcal S}_{\omega ,c}(\leftambda\varphi)\right|_{\leftambda =1}$. Similarly as in the case of \eqref{NLS}, we consider the following subsets of the energy space: \begin{align*} {\mathscr A}_{\omega ,c}:=& \left\{ \varphi\in H^1({\mathbb R} ): {\mathcal S}_{\omega, c}(\varphi ) < {\mathcal S}_{\omega ,c}(\varphi_{\omega, c})\right\}, \\ {\mathscr A}_{\omega, c}^+ :=&\left\{ \varphi\in{\mathscr A}_{\omega, c} :{\mathcal K}_{\omega, c}(\varphi) \geq 0 \right\},\\ {\mathscr A}_{\omega, c}^- :=&\left\{ \varphi\in{\mathscr A}_{\omega, c} :{\mathcal K}_{\omega ,c}(\varphi) < 0\right\}. \end{align*} Here we introduce the potential well along the scaling curve: \begin{align*} {\mathscr A}_{s}:= \bigcup_{ \substack{\omega>0 } } {\mathscr A}_{\omega ,2s\sqrt{\omega}},~ {\mathscr A}_{s}^{\pm}:= \bigcup_{ \substack{\omega>0 } } {\mathscr A}_{\omega ,2s\sqrt{\omega}}^{\pm} \quad\text{for}~s\in (-1,1]. \end{align*} We define the mass threshold value in Theorem {\rightm Re}f{thm:1.4} as \begin{align} \leftabel{eq:1.18} M^* =M^*(b):= \left\{ \begin{array}{ll} M(\phi_{1,2s^*(b)}) &\text{if}~b\geq 0,\\[3pt] M(\phi_{1,2})+P(\phi_{1,2}) &\text{if}~-3/16<b\lefteq 0. \end{array} \right. \end{align} We note that $M^*(0)$ is well defined since \begin{align*} M(\phi_{1,2s^*(0)})=M(\phi_{1,2})~\text{and}~P(\phi_{1,2})=0 \quad\text{when}~b=0. \end{align*} Our main result in this paper is the following classification of a two-parameter family of potential wells which covers Theorem {\rightm Re}f{thm:1.4} and Theorem {\rightm Re}f{thm:1.6}. \begin{theorem} \leftabel{thm:1.7} Let $b>-3/16$ and let $(\omega ,c)$ satisfy $-2\sqrt{\omega}<c\lefteq 2\sqrt{\omega}$. Then, each of ${\mathscr A}_{\omega ,c}^{+}$ and ${\mathscr A}_{\omega ,c}^{-}$ is invariant under the flow of \eqref{ME}. If $v_0 \in{\mathscr A}_{\omega ,c}^+$, then the $H^1({\mathbb R})$-solution $v$ of \eqref{ME} with $v(0)=v_0$ exists globally both forward and backward in time, and satisfies the following uniform estimate\textup{:} \begin{align} \leftabel{eq:1.19} \| \partial_x v\|_{L^{\infty}({\mathbb R} ,L^2)}^2 \lefteq 8{\mathcal S}_{\omega ,c}(v_0)+\frac{c^2}{2}{\mathcal M} (v_0). \end{align} Moreover the following statements hold\textup{:} \begin{enumerate}[\rightm (i)] \setlength{\itemsep}{3pt} \item For each $s\in (-1,1]$, ${\mathscr A}^+_{s}$ and ${\mathscr A}^-_s$ have no elements in common on the set $\{ \varphi\in H^1({\mathbb R}) : {\mathcal M}(\varphi ) \geq M^*\}$. \item If ${\mathcal M} (\varphi)<M^*$, or ${\mathcal M} (\varphi)=M^*$ and ${\mathcal P} (\varphi)<0$, then $\varphi\in{\mathscr A}^+_{s^*}$ if $b\geq 0$, or $\varphi\in{\mathscr A}^+_{1}$ if $-3/16< b\lefteq 0$. \item For given $\psi\in H^1({\mathbb R})\setminus\{ 0\}$ the following properties hold\textup{:} \begin{enumerate}[\rightm (a)] \item There exists $\mu_0 =\mu_0 (\psi)>0$ such that if $\mu\geq\mu_0$, then $e^{i\mu x}\psi\in {\mathscr A}^+_1$. \item There exist $\varepsilon\in (0,1)$ and large $\mu >0$ such that $e^{-i(1-\varepsilon )\mu x}\psi\in {\mathscr A}^-_{-(1-\varepsilon)}$, where $\varepsilon$ and $\mu$ depend on $\psi$. \end{enumerate} \item Assume ${\mathcal E}(\varphi)<0$. Then $\varphi\in\bigcap_{-1<s\lefteq 1}{\mathscr A}^-_s$. In particular, if $M(\varphi)\geq M^*$, then $\varphi\not\in\bigcup_{-1< s\lefteq 1}{\mathscr A}_s^+$.\footnote{The negative energy is possible only when ${\mathcal M}(\varphi)> \frac{2\pi}{\sqrt{\gamma}}$. Note that the following case \begin{align*} {\mathcal E} (\varphi) <0, {\mathcal M}(\varphi) =M^*~\text{and}~{\mathcal P} (\varphi)\lefteq 0 \end{align*} does not occur from the assertion (vi).} \item Assume ${\mathcal E}(\varphi) \geq 0$ and ${\mathcal M} (\varphi) \geq M^*$. If ${\mathcal P}(\varphi) \geq 0\,(\text{resp.}\,{\mathcal P}(\varphi)\lefteq 0)$, then $\varphi\not\in\bigcup_{0\lefteq s\lefteq 1}{\mathscr A}_s\,(\text{resp.}\,\varphi\not\in\bigcup_{-1<s\lefteq 0}{\mathscr A}_s)$. In particular, if ${\mathcal P}(\varphi)=0$, then $\varphi\not\in\bigcup_{-1< s\lefteq 1}{\mathscr A}_s$. \item Assume ${\mathcal M} (\varphi)=M^*$. Then the following properties hold\textup{:} \begin{enumerate}[\rightm(a)] \item When $b\geq 0$, ${\mathcal E}(\varphi)={\mathcal P}(\varphi)=0$ if and only if there exist $\theta , y\in{\mathbb R}$ and $\omega >0$ such that $\varphi =e^{i\theta}\varphi_{\omega ,2s^*\sqrt{\omega} }(\cdot -y)$. Moreover, there exists no $\varphi\in H^1({\mathbb R})$ such that ${\mathcal E}(\varphi)<0$ and ${\mathcal P}(\varphi)\lefteq 0$, or ${\mathcal E}(\varphi)\lefteq 0$ and ${\mathcal P}(\varphi) <0$. \item When $-3/16 <b<0$, there exists no $\varphi\in H^1({\mathbb R})$ such that ${\mathcal E}(\varphi)\lefteq 0$ and ${\mathcal P}(\varphi)\lefteq 0$. \end{enumerate} \end{enumerate} \end{theorem} \begin{remark} \leftabel{rem:1.8} Concerning the assertion (i), we recall that ${\mathscr A}^+$ and ${\mathscr A}^-$ for \eqref{NLS} in Section {\rightm Re}f{sec:1.3} are mutually disjoint. However, in general ${\mathscr A}_s^+$ and ${\mathscr A}_s^-$ have elements in common on the set $\{ \varphi\in H^1({\mathbb R}) : {\mathcal M}(\varphi ) < M^*\}$. For example it follows from the assertions (ii) and (iv) that \begin{align*} {\mathcal E}(\varphi)<0, ~\frac{2\pi}{\sqrt{\gamma}}<{\mathcal M}(\varphi)<M^* \Longrightarrow \begin{array}{ll} \varphi\in {\mathscr A}_{s^*}^+ \cap {\mathscr A}_{s^*}^- &\text{if}~b\geq0, \\ \varphi\in {\mathscr A}_{1}^+ \cap {\mathscr A}_{1}^- &\text{if}~-\frac{3}{16}<b\lefteq 0. \end{array} \end{align*} This gives a notable property of two-parameter family of potential wells, which is also closely related to the stability of solitons (see \cite{H19}). We note that the interval $(0,M^*(b))$ for $b\geq 0$ corresponds to the range of the mass of stable solitons. \end{remark} \begin{remark} \leftabel{rem:1.9} For given $\psi\in H^1({\mathbb R})$ and $c\in{\mathbb R}$, we have \begin{align} \leftabel{eq:1.20} {\mathcal E} (e^{icx}\psi)\sim c^2 , ~ {\mathcal P} (e^{icx}\psi)\sim -c\quad\text{as}~|c|\gg 1. \end{align} In particular one can see that oscillating factor in Theorem {\rightm Re}f{thm:1.6} causes to change the momentum to the negative direction. The assertion (iii-b) gives the counterpart of this global result, and implies that the oscillating direction is essential for generating global and bounded solutions. \end{remark} \begin{remark} \leftabel{rem:1.10} When $b=-3/16$, one can still prove that ${\mathscr A}_{\omega ,c}^{\pm}$ is invariant under the flow, and that for $v_0\in{\mathscr A}^+_{\omega ,c}$ the corresponding solution satisfies the uniform estimate \eqref{eq:1.19}. Moreover we have the following claim (Proposition {\rightm Re}f{prop:5.2}): \begin{align*} \bigcup_{-1< s<0}{\mathscr A}_s =\bigcup_{-1< s<0}{\mathscr A}_s^+=H^1({\mathbb R}). \end{align*} This gives the characterization by potential well theory to the global result in the case $b=-3/16$. \end{remark} Theorem {\rightm Re}f{thm:1.7} is the first classification theorem for a two-parameter family of potential wells. For \eqref{DNLS} the relation between $4\pi$-mass condition and ${\mathscr A}^+_{\omega ,c}$ was first pointed out in \cite{FHI17}, but in the present paper we give a characterization for ${\mathscr A}^-_{\omega ,c}$ as well as ${\mathscr A}^+_{\omega ,c}$. Adopting a family of potential wells along the scaling curve is a new idea, which is useful to examine the properties of potential wells. The assertions (ii) and (iii-a) give a representation by potential wells to the global results in Theorem {\rightm Re}f{thm:1.4} and Theorem {\rightm Re}f{thm:1.6}. It follows from Proposition {\rightm Re}f{prop:1.2} that ${\mathscr A}_s^+$ contains solitons with arbitrarily small mass,\footnote{Furthermore, one can prove that $\| \phi_{1,2s}\|_{H^1}\to 0$ as $s\downarrow -1$, which implies that \eqref{eq:1.1} has the solitons with arbitrarily small $H^1({\mathbb R})$-norm.} which implies that the solutions for the data in ${\mathscr A}_s^+$ do not scatter in general. Also, we note that the equation \eqref{eq:1.1} corresponds to the long range scattering, and it is known that modified scattering occurs for small data in weighted Sobolev spaces (see \cite{HO94modi, Oz96, GHLN13}). These properties give quite different situation from the set ${\mathscr A}^+$ for \eqref{NLS}. For \eqref{DNLS} it was proved in \cite{JLPS18a} that the soliton resolution holds for generic data in $H^{2,2}({\mathbb R})$, but the global dynamics in the energy space is still far from clear. The assertions (iv) and (v) show the optimality of the mass threshold value $M^*$ in the sense that for any $\rightho\geq M^*$ there exists $\varphi\in H^1({\mathbb R})$ such that ${\mathcal M} (\varphi)=\rightho$ and $\varphi\not\in {\mathscr A}^+_{s}$ for any $s\in (-1,1]$. The set of the data satisfying \begin{align} \leftabel{eq:1.21} {\mathcal E}(\varphi)<0, {\mathcal M}(\varphi)>M^* ~\text{or}~{\mathcal E}(\varphi)<0, {\mathcal M}(\varphi)=M^*, {\mathcal P} (\varphi)>0 \end{align} is an important subset contained in $\bigcap_{-1<s\lefteq 1}{\mathscr A}^-_s$, and has analogy with the set ${\mathscr B}$ for \eqref{NLS}. The set of the data satisfying \begin{align} \leftabel{eq:1.22} {\mathcal E}(\varphi)\geq 0, {\mathcal M}(\varphi) \geq M^*, {\mathcal P}(\varphi)=0 \end{align} gives a subset of the complement of ${\mathscr A}_s$ for any $s\in (-1,1]$. If we replace ${\mathcal P}(\varphi)=0$ by ${\mathcal P}(\varphi)\neq 0$, then this inclusion does not hold. Indeed, it follows from \eqref{eq:1.20} and the assertion (iii) that the oscillating data $e^{icx}\psi$ for large $|c|>0$ gives the counterexample. The assertion (vi) gives some constraint condition on ${\mathcal M}(\varphi )= M^*$. When $b\geq 0$ the following set \begin{align*} {\mathscr B}_0 :=\left\{ \varphi\in H^1({\mathbb R}) : {\mathscr M} (\varphi)=M^*, \,{\mathcal E} (\varphi)={\mathcal P} (\varphi)=0\right\} \end{align*} gives the boundary of both ${\mathscr A}_{s^*}^+$ and ${\mathscr A}_{s^*}^-$. From (vi-a) the set ${\mathscr B}_0$ corresponds to the borderline solitons with respect to stability/instability, i.e., \begin{align} \leftabel{eq:1.23} {\mathscr B}_0 = \left\{ e^{i\theta}\varphi_{\omega ,2s^*\sqrt{\omega}}(\cdot -y): \theta, y \in{\mathbb R}, \,\omega >0\right\}, \end{align} which gives analogy with the relation \eqref{eq:1.10} for \eqref{NLS}. On the other hand, when $-3/16<b<0$ the set ${\mathscr B}_{0}$ is empty. From Theorem {\rightm Re}f{thm:1.7} we see that the mass threshold value $M^*$ gives the turning point in the structure of potential wells. In this sense $M^*$ corresponds to the value $2\pi$ for \eqref{NLS}. Therefore, taking into account the mass critical structure of the equation, we conjecture that the mass condition in Theorem {\rightm Re}f{thm:1.4} is sharp. To state more precisely, let us say that $(GE)(u_0)$ holds for $u_0\in H^1({\mathbb R})$ if the $H^1({\mathbb R})$-solution $u$ of \eqref{eq:1.1} with $u(0)=u_0$ is global both forward and backward in time, and uniformly bounded in $H^1({\mathbb R})$, i.e., $u\in (C\cap L^{\infty})({\mathbb R} ,H^1({\mathbb R}))$. We define the positive value $m^*$ by \begin{align*} m^* := \sup \left\{ m>0 : \forall u_0 \in H^1({\mathbb R}), M(u_0)<m {\mathbb R}ightarrow (GE)(u_0)~\text{holds} \right\}. \end{align*} Then, our conjecture is organized as follows: \begin{conjecture} \leftabel{conj:1.11} Let $b>-3/16$. Then $m^*=M^*$. \end{conjecture} Theorem {\rightm Re}f{thm:1.4} implies that $m^*\geq M^*$. Although for \eqref{DNLS} global existence was proved for any initial data in $H^{2,2}({\mathbb R})$ in \cite{JLPS18}, we note that their results do not imply nonexistence of infinite time blow-up solutions, i.e., the $H^1({\mathbb R})$-norm of the solution may be unbounded in time. Also, existence/nonexistence of finite time blow-up solutions in $H^1({\mathbb R})$ is still an open problem. It is known that finite time blow-up occurs for \eqref{DNLS} on a bounded interval or on the half line, with Dirichlet boundary condition (see \cite{Tan04, Wu13}). The data satisfying the condition \eqref{eq:1.21} is a good candidate generating singular solutions. From Theorem {\rightm Re}f{thm:1.7} and analogy with \eqref{NLS}, one can say that this condition gives certain obstruction for generating global and bounded solutions. Related to Conjecture {\rightm Re}f{conj:1.11}, one can prove the following blow-up criterion: \begin{theorem} \leftabel{thm:1.12} Let the initial data $u_0\in H^1({\mathbb R})$ satisfy $M(u_0)=4\pi$. Suppose that the corresponding solution $u$ of \eqref{DNLS} blows up in time $T^*\in (0,\infty]$.\footnote{We say that the solution $u$ blows up in infinite time if $\leftim_{t\to\infty}\| \partial_x u(t)\|_{L^2}=\infty$.} Then, there exist functions $\theta (t)\in {\mathbb R}$ and $y(t)\in{\mathbb R}$ such that \begin{align} \leftabel{eq:1.24} u(t)-\frac{e^{i\theta (t)}}{\leftambda(t)^{1/2}}\phi_{1,2}\left( \frac{x-y(t)}{\leftambda (t)}\right) \to 0~\text{in}~H^1({\mathbb R})~\text{as}~t\to T^*, \end{align} where $\leftambda (t):=\| \partial_x\phi_{1,2}\|_{L^2} / \| \partial_x u(t) \|_{L^2}$. \end{theorem} This result is analogous to the one obtained by Weinstein \cite{W86} for \eqref{NLS}. The similar result of Theorem {\rightm Re}f{thm:1.12} was first obtained in \cite{KW18}. Here we give a simple alternative proof by using characterization on the Nehari manifold which is related to concentration compactness arguments in \cite{W86}. \subsection{Organization of the paper} The rest of this paper is organized as follows. In Section {\rightm Re}f{sec:2} we study the solitons of the equation \eqref{eq:1.1} and calculate the conserved quantities of them. In Section {\rightm Re}f{sec:3} we review the gauge transformation and the local well-posedness theory in the energy space. In Section {\rightm Re}f{sec:4} we give a variational characterization of two types of solitons in a unified way. In Section {\rightm Re}f{sec:5} we establish potential well theory for \eqref{ME} by applying the variational characterization, and give a proof of Theorem {\rightm Re}f{thm:1.7}. In Section {\rightm Re}f{sec:6} we organize potential well theory for \eqref{DNLS}, and give a proof of Theorem {\rightm Re}f{thm:1.12}. \section{Solitons and conserved quantities} \leftabel{sec:2} \subsection{A two-parameter family of solitons} \leftabel{sec:2.1} Here we formulate the solitons of ({\rightm Re}f{eq:1.1}) following \cite{CO06}. Consider solutions of ({\rightm Re}f{eq:1.1}) of the form \begin{align} \leftabel{eq:2.1} u_{\omega,c}(t,x)=e^{i\omega t} \phi_{\omega,c}(x-ct) \end{align} for $(\omega,c) \in{\mathbb R}^2$, and assume that $\phi_{\omega ,c} \in H^1({\mathbb R})$. It is clear that $\phi_{\omega ,c}$ must satisfy the following equation: \begin{align} \leftabel{eq:2.2} -\phi ''+ \omega \phi +ic \phi ' - i|\phi|^{2} \phi '-b|\phi |^4\phi =0, \quad x \in{\mathbb R}. \end{align} We note that the equation ({\rightm Re}f{eq:2.2}) can be rewritten as $S_{\omega ,c}' (\phi) =0$, where \begin{align} \leftabel{eq:2.3} S_{\omega ,c}(\phi ):=E(\phi )+\frac{\omega}{2} M(\phi )+\frac{c}{2}P(\phi ). \end{align} Applying the following gauge transformation to $\phi_{\omega ,c}$ \begin{align} \leftabel{eq:2.4} \phi_{\omega,c}(x) &= \Phi_{\omega,c}(x) \exp\left( i\frac{c}{2}x - \frac{i}{4} \int_{-\infty}^{x} \left|\Phi_{\omega,c}(y)\right|^2 dy \right), \end{align} then $\Phi_{\omega ,c}$ satisfies the following equation: \begin{align} \leftabel{eq:2.5} - \Phi ''+ \left(\omega- \frac{c^2}{4}\right) \Phi +\frac{1}{2}{\rightm Im}\left( \overline{\Phi}\Phi'\right) \Phi+ \frac{c}{2} |\Phi|^{2} \Phi - \frac{3}{16}\gamma |\Phi|^{4}\Phi =0, \quad x\in {\mathbb R}, \end{align} where $\gamma = 1+\frac{16}{3}b$. From $\Phi_{\omega ,c}\in H^1({\mathbb R})$ and the equation \eqref{eq:2.5}, one can show that ${\rightm Im}\left( \overline{\Phi_{\omega ,c}}\Phi_{\omega ,c}'\right)=0$ (see \cite[Lemma 2]{CO06}). Therefore $\Phi_{\omega ,c}$ satisfies the following elliptic equation with double power nonlinearity: \begin{align} \leftabel{eq:2.6} - \Phi ''+ \left(\omega- \frac{c^2}{4}\right) \Phi + \frac{c}{2} |\Phi|^{2} \Phi - \frac{3}{16}\gamma |\Phi|^{4}\Phi =0, \quad x\in {\mathbb R}. \end{align} The positive radial (even) solution of ({\rightm Re}f{eq:2.6}) is explicitly obtained as follows; if $\gamma >0$, \begin{align} \leftabel{eq:2.7} \Phi_{\omega,c}^2(x) &= \left\{ \begin{array}{ll} \thickmuskip=0mu \medmuskip=1mu \thinmuskip=0mu \displaystyle \frac{ 2(4\omega - c^2) }{\sqrt{c^2+\gamma (4\omega -c^2)} \cosh ( \sqrt{4\omega- c^2}x)-c } & \thickmuskip=2mu \displaystyle \text{if}~-2\sqrt{\omega}<c<2\sqrt{\omega}, \\ & \\ \medmuskip=1mu \displaystyle \frac{4c}{(cx)^2+\gamma} & \thickmuskip=3.2mu \displaystyle \text{if}~c=2\sqrt{\omega}, \end{array} \right. \end{align} or if $\gamma \lefteq 0$, \begin{align} \leftabel{eq:2.8} \Phi_{\omega,c}^2(x) &= \begin{array}{ll}\displaystyle \medmuskip=1mu \thinmuskip=0mu \frac{ 2(4\omega - c^2) }{\sqrt{c^2+\gamma (4\omega -c^2)} \cosh ( \sqrt{4\omega- c^2}x)-c } & \thickmuskip=2mu \medmuskip=1mu \displaystyle \text{if}~-2\sqrt{\omega} <c<-2s_{\ast}\sqrt{\omega}, \end{array} \end{align} where $s_{\ast} =s_{\ast}(\gamma)=\sqrt{ -\gamma /(1-\gamma)}$. We note that the condition \begin{align} \leftabel{eq:2.9} \begin{array}{ll} \displaystyle\text{if}~ \gamma >0\Leftrightarrow b>-3/16,& \displaystyle -2\sqrt{\omega} <c\lefteq 2\sqrt{\omega} ,\\[7pt] \displaystyle\text{if}~ \gamma \lefteq 0\Leftrightarrow b\lefteq -3/16,& \displaystyle -2\sqrt{\omega} <c<-2s_{\ast}\sqrt{\omega} \end{array} \end{align} is a necessary and sufficient condition for the existence of non-trivial solutions of ({\rightm Re}f{eq:2.6}) vanishing at infinity; see \cite{BeL83}. From ({\rightm Re}f{eq:2.1}), ({\rightm Re}f{eq:2.4}), ({\rightm Re}f{eq:2.7}) and ({\rightm Re}f{eq:2.8}), we obtain the following explicit formulae of solitons: \begin{align} \leftabel{eq:2.10} u_{\omega ,c}(t,x)=e^{i\omega t +i\frac{c}{2}(x-ct)-\frac{i}{4}\int_{-\infty}^{x-ct}|\Phi_{\omega ,c}(y)|^2 dy}\Phi_{\omega ,c}(x-ct). \end{align} \subsection{Mass of the solitons} \leftabel{sec:2.2} In this subsection we calculate the mass of the solitons. First we prepare the following elementary integration formulae: \begin{lemma} \leftabel{lem:2.1} Let $-1<\alpha $. Then we have \begin{align} \leftabel{eq:2.11} \int_{-\infty}^{\infty} \frac{dy}{\cosh y+\alpha } =\left\{ \begin{array}{lll} \displaystyle \frac{4}{ \sqrt{1-\alpha^2} }\tan^{-1} \sqrt{ \frac{1-\alpha}{1+\alpha} } &\displaystyle \text{\rightm if} &|\alpha |<1,\\[8pt] \displaystyle \quad 2 &\displaystyle \text{\rightm if} &\alpha =1, \\[2pt] \displaystyle \frac{2}{ \sqrt{\alpha^2 -1} } \leftog \left( \alpha +\sqrt{\alpha^2 -1}\right) &\displaystyle \text{\rightm if} &\alpha >1. \end{array} \right. \end{align} \end{lemma} \begin{proof} See the formula 3.513, 2 in \cite{GR07}. \end{proof} By using Lemma {\rightm Re}f{lem:2.1}, we have the following proposition. \begin{proposition} \leftabel{prop:2.2} Let $(\omega ,c)$ and $\gamma$ satisfy \eqref{eq:2.9}. Then the following properties hold\textup{:} \begin{enumerate}[\rightm (i)] \item When $\gamma >0$, we have \begin{align} \leftabel{eq:2.12} M\left( \phi_{\omega ,c} \right) = \left\{ \begin{array}{ll} \displaystyle \frac{8}{\sqrt{\gamma}} \tan^{-1} \sqrt{ \frac{1+\beta}{1-\beta} } &\displaystyle \text{\rightm if}~-2\sqrt{\omega} <c<2\sqrt{\omega}, \\[10pt] \displaystyle \frac{4\pi}{\sqrt{\gamma}} &\displaystyle\text{\rightm if}~c=2\sqrt{\omega}, \end{array} \right. \end{align} where $\beta$ is defined by \begin{align} \leftabel{eq:2.13} \beta =\beta(\omega ,c):= \frac{c}{ \sqrt{c^2+\gamma (4\omega -c^2)} }. \end{align} \item When $\gamma =0$, we have \begin{align} \leftabel{eq:2.14} M\left( \phi_{\omega ,c} \right) =\frac{4\sqrt{4\omega -c^2}}{-c}\quad \text{\rightm if}~-2\sqrt{\omega} <c<0. \end{align} \item When $\gamma <0$, we have \begin{align} \leftabel{eq:2.15} M\left( \phi_{\omega ,c} \right) =\frac{4}{\sqrt{-\gamma}} \leftog \left( \alpha +\sqrt{\alpha^2 -1}\right) \quad\text{\rightm if}~-2\sqrt{\omega} <c<-2s_{\ast}\sqrt{\omega}, \end{align} where $\alpha$ is defined by \begin{align} \leftabel{eq:2.16} \alpha =\alpha (\omega ,c):= \frac{-c}{ \sqrt{c^2+\gamma (4\omega -c^2)} }. \end{align} \end{enumerate} Moreover, if $\gamma >0$, the function \begin{align*} (-1, 1] \ni s \mapsto M\left(\phi_{1,2s} \right) \in \left( 0, \frac{4\pi}{ \sqrt{\gamma} }\right] \end{align*} is continuous, strictly increasing and surjective. Similarly, if $\gamma\lefteq 0$ the function \begin{align*} \left( -1, -s_{\ast} \right) \ni s \mapsto M\left(\phi_{1,2s}\right) \in ( 0, \infty ) \end{align*} has the same property. \end{proposition} \begin{proof} Let $(\omega ,c)$ and $\gamma$ satisfy \eqref{eq:2.9}. When $\omega >c^2/4$, from the explicit formulae of the solitons, we have \begin{align} \leftabel{eq:2.17} M\left( \phi_{\omega ,c} \right) =M\left(\Phi_{\omega ,c} \right)&= \int_{-\infty}^{\infty} \frac{ 2(4\omega -c^2)dx }{\sqrt{c^2+\gamma (4\omega -c^2)} \cosh ( \sqrt{4\omega- c^2}x)-c } \\ &= \frac{ 2\sqrt{4\omega -c^2} }{ \sqrt{c^2+\gamma (4\omega -c^2)} } \int_{-\infty}^{\infty} \frac{dy}{ \cosh y +\alpha }, \notag \end{align} where $\alpha$ is defined by ({\rightm Re}f{eq:2.16}). Case 1-1: $\gamma >0$ and $-2\sqrt{\omega} <c<2\sqrt{\omega}$. In this case we note that $|\alpha| <1$ and \begin{align} \leftabel{eq:2.18} 1-\alpha^2 &= 1-\frac{ c^2 }{ c^2+\gamma (4\omega -c^2) } =\frac{ \gamma (4\omega -c^2) }{ c^2+\gamma (4\omega -c^2) }. \end{align} Applying Lemma~{\rightm Re}f{lem:2.1} to ({\rightm Re}f{eq:2.17}), we have \begin{align} \leftabel{eq:2.19} M\left( \phi_{\omega ,c}\right) &= \frac{ 2\sqrt{4\omega -c^2} }{ \sqrt{c^2+\gamma (4\omega -c^2)} } \cdot \frac{4}{\sqrt{1-\alpha^2}} \tan^{-1} \sqrt{ \frac{1-\alpha}{1+\alpha} } \\ &= \frac{8}{ \sqrt{\gamma} }\tan^{-1} \sqrt{ \frac{1+\beta}{1-\beta} }, \notag \end{align} where $\beta:=-\alpha$. We note that the function $(\omega ,c)\mapsto \beta(\omega ,c)$ is constant on the scaling curve \eqref{eq:1.11}. For $s\in (-1,1]$ we have \begin{align*} \beta (s):=\beta (\omega ,2s\sqrt{\omega} ) &=\frac{s}{ \sqrt{s^2+\gamma (1-s^2)} }= \frac{ {\rightm sgn}~s}{ \sqrt{1+\gamma \left( \frac{1}{s^2} -1\right)} }. \end{align*} This shows that the function \begin{align} \leftabel{eq:2.20} (-1,1] \ni s \mapsto \beta(s) \in (-1,1] \end{align} is continuous, strictly increasing and surjective. Therefore, by ({\rightm Re}f{eq:2.19}) we obtain that the function \begin{align*} (-1, 1) \ni s \mapsto M\left( \phi_{1,2s} \right) \in \left( 0, \frac{4\pi}{ \sqrt{\gamma} }\right) \end{align*} has the same property. We also note that \begin{align} \leftabel{eq:2.21} \leftim_{s\to 1-0} M\left( \phi_{1,2s} \right) = \frac{4\pi}{ \sqrt{\gamma} }. \end{align} Case 1-2: $\gamma >0$ and $c=2\sqrt{\omega}$. From the explicit formulae of algebraic solitons, we have \begin{align} \leftabel{eq:2.22} M\left(\phi_{c^2/4 ,c}\right) =M\left(\Phi_{c^2/4 ,c}\right) &= \int_{-\infty}^{\infty} \frac{4c}{c^2x^2 +\gamma} dx =\frac{4\pi}{ \sqrt{\gamma} }. \end{align} From ({\rightm Re}f{eq:2.21}) and ({\rightm Re}f{eq:2.22}), we obtain that \begin{align} \leftabel{eq:2.23} \leftim_{s\to 1-0} M\left( \phi_{1,2s} \right) = M\left( \phi_{1,2}\right), \end{align} which completes the proof of the case $\gamma >0$. Case 2: $\gamma =0$ and $-2\sqrt{\omega} <c<0$. In this case we note that $\alpha =1$. From ({\rightm Re}f{eq:2.17}) and Lemma~{\rightm Re}f{lem:2.1}, we have \begin{align} \leftabel{eq:2.24} M\left( \phi_{\omega ,c}\right) &= \frac{ 2\sqrt{4\omega -c^2} }{ -c } \int_{-\infty}^{\infty} \frac{dy}{ \cosh y +1 } =\frac{4\sqrt{4\omega -c^2}}{-c}. \end{align} For $s\in (-1,0)$, we have \begin{align*} M\left( \phi_{1,2s} \right) &=\frac{4\sqrt{1-s^2}}{-s}=4\sqrt{\frac{1}{s^2}-1}, \end{align*} which yields that the function \begin{align} \leftabel{eq:2.25} (-1, 0) \ni s \mapsto M\left( \phi_{1,2s} \right) \in (0,\infty ) \end{align} is continuous, strictly increasing and surjective. Case 3: $\gamma <0$ and $-2\sqrt{\omega} <c<-2s_{\ast}\sqrt{\omega}$. In this case we note that $\alpha >1$. From Lemma~{\rightm Re}f{lem:2.1}, ({\rightm Re}f{eq:2.17}) and ({\rightm Re}f{eq:2.18}), we have \begin{align} \leftabel{eq:2.26} M\left( \phi_{\omega ,c}\right) &= \frac{ 2\sqrt{4\omega -c^2} }{ \sqrt{c^2+\gamma (4\omega -c^2)} } \int_{-\infty}^{\infty} \frac{dy}{ \cosh y +\alpha } \\ &= \frac{ 2\sqrt{4\omega -c^2} }{ \sqrt{c^2+\gamma (4\omega -c^2)} } \cdot \frac{2}{\sqrt{\alpha^2 -1}} \leftog \left( \alpha +\sqrt{\alpha^2 -1}\right) \notag\\ &= \frac{4}{\sqrt{-\gamma}} \leftog \left( \alpha +\sqrt{\alpha^2 -1}\right). \notag \end{align} We note that \begin{align} \leftabel{eq:2.27} \alpha (s) :=\alpha (\omega ,2s\sqrt{\omega} ) &= \frac{-s}{ \sqrt{(1-\gamma )s^2 +\gamma} }= \frac{1}{ \sqrt{1-\gamma +\gamma s^{-2} } }. \end{align} This yields that the function \begin{align*} \left( -1, -s_{\ast} \right) \ni s \mapsto \alpha (s) \in (1, \infty ) \end{align*} is continuous, strictly increasing and surjective. From the formula ({\rightm Re}f{eq:2.26}), we deduce that the function \begin{align} \leftabel{eq:2.28} \left( -1, -s_{\ast}\right) \ni s \mapsto M\left( \phi_{1,2s} \right) \in (0,\infty ) \end{align} has the same property. This completes the proof. \end{proof} \subsection{Momentum of the solitons} \leftabel{sec:2.3} In this subsection we calculate the momentum of the solitons. From the formula ({\rightm Re}f{eq:2.4}) of the solitons, we have \begin{align} \leftabel{eq:2.29} P(\phi _{\omega ,c})&={\rightm Re} \int_{{\mathbb R}} i\phi_{\omega ,c}' \overline{\phi_{\omega ,c}} dx =-\frac{c}{2}M(\Phi_{\omega ,c}) +\frac{1}{4} \| \Phi_{\omega ,c}\|_{L^4}^4. \end{align} To calculate the $L^4$-norm, we prepare the following elementary integration formulae. \begin{lemma} \leftabel{lem:2.3} Let $-1<\alpha $. Then we have \begin{align} \leftabel{eq:2.30} \thickmuskip=0mu \medmuskip=0mu \thinmuskip=0mu \int_{-\infty}^{\infty} \frac{dy}{(\cosh y+\alpha )^2} =\left\{ \begin{array}{ll} \thickmuskip=0mu \medmuskip=0mu \thinmuskip=0mu \displaystyle \frac{2}{1-\alpha^2} -\frac{4\alpha}{ (1-\alpha^2 )^{3/2} }\tan^{-1} \sqrt{ \frac{1-\alpha}{1+\alpha} } &\text{\rightm if}~~|\alpha |<1,\\[10pt] \displaystyle \quad \frac{2}{3} &\text{\rightm if}~~\alpha =1, \\[10pt] \thickmuskip=0mu \medmuskip=0mu \thinmuskip=0mu \displaystyle -\frac{2}{\alpha^2 -1}+\frac{2\alpha}{ (\alpha^2 -1)^{3/2} } \leftog \left( \alpha +\sqrt{\alpha^2 -1}\right) &\text{\rightm if}~\alpha >1. \end{array} \right. \end{align} \end{lemma} \begin{proof} Change variables $t=e^y$ and apply the formula 3.252, 4 in \cite{GR07}. \end{proof} By using Lemma {\rightm Re}f{lem:2.3}, we have the following proposition. \begin{proposition} \leftabel{prop:2.4} The momentum of the solitons is represented as follows\textup{;} if $\gamma >0$ and $-2\sqrt{\omega} <c\lefteq 2\sqrt{\omega}$, or if $\gamma <0$ and $-2\sqrt{\omega}<c<-2s_{\ast}\sqrt{\omega}$, we have \begin{align} \leftabel{eq:2.31} P(\phi_{\omega ,c}) = \frac{c}{2} \left( -1+ \frac{1}{\gamma} \right) M(\phi_{\omega ,c}) +\frac{2}{\gamma}\sqrt{4\omega -c^2}. \end{align} If $\gamma =0$ and $-2\sqrt{\omega}<c<0$, we have \begin{align} \leftabel{eq:2.32} P(\phi_{\omega ,c}) = - \frac{2\omega +c^2}{3c} M(\phi_{\omega ,c}) . \end{align} \end{proposition} \begin{remark} \leftabel{rem:2.5} The momentum is represented by the same formula in the cases $\gamma >0$ and $\gamma<0$ although each mass is represented by the different functions. \end{remark} \begin{proof} We only consider the case $\omega >c^2/4$. The case $c=2\sqrt{\omega}$ is calculated more easily. We note that $\Phi_{\omega ,c}^2(x)$ is rewritten as \begin{align*} \Phi_{\omega ,c}^2(x) &= \frac{ 2(4\omega -c^2) }{ \sqrt{c^2+\gamma (4\omega -c^2)} }\cdot\frac{1}{ \cosh ( \sqrt{4\omega- c^2}x) +\alpha }, \end{align*} where $\alpha$ is defined by ({\rightm Re}f{eq:2.16}). Then we have \begin{align} \leftabel{eq:2.33} \| \Phi_{\omega ,c}\|_{L^4}^4 &= \frac{ 4(4\omega -c^2)^{3/2} }{ c^2+\gamma (4\omega -c^2) } \int_{-\infty}^{\infty} \frac{dy}{ (\cosh y +\alpha )^2 }. \end{align} Case 1: $\gamma >0$ and $-2\sqrt{\omega} <c<2\sqrt{\omega}$. In this case we note that $|\alpha| <1$. From Lemma {\rightm Re}f{lem:2.3}, ({\rightm Re}f{eq:2.18}) and ({\rightm Re}f{eq:2.12}), we obtain that \begin{align} \leftabel{eq:2.34} \| \Phi_{\omega ,c}\|_{L^4}^4 &= \frac{ 4(4\omega -c^2)^{3/2} }{ c^2+\gamma (4\omega -c^2) }\cdot \left[ \frac{2}{1-\alpha^2} -\frac{4\alpha}{ (1-\alpha^2 )^{3/2} }\tan^{-1} \sqrt{ \frac{1-\alpha}{1+\alpha} }\right] \\ &=\frac{8}{\gamma} \sqrt{4\omega -c^2} +\frac{16c}{\gamma^{3/2}} \tan^{-1} \sqrt{ \frac{1+\beta}{1-\beta} } \notag\\ &=\frac{8}{\gamma} \sqrt{4\omega -c^2} +\frac{2c}{\gamma} M(\Phi_{\omega ,c}). \notag \end{align} From ({\rightm Re}f{eq:2.29}) and ({\rightm Re}f{eq:2.34}), we have \begin{align*} P(\phi_{\omega ,c}) &=-\frac{c}{2}M(\Phi_{\omega ,c}) +\frac{1}{4} \| \Phi_{\omega ,c}\|_{L^4}^4 \\ &=\frac{c}{2} \left( -1+ \frac{1}{\gamma} \right) M(\Phi_{\omega ,c}) + \frac{2}{\gamma}\sqrt{4\omega -c^2}. \end{align*} Case 2: $\gamma <0$ and $-2\sqrt{\omega} <c<0$. In this case we note that $\alpha =1$. By Lemma {\rightm Re}f{lem:2.3} and ({\rightm Re}f{eq:2.14}), we obtain that \begin{align} \leftabel{eq:2.35} \| \Phi_{\omega ,c}\|_{L^4}^4 &= \frac{ 4(4\omega -c^2)^{3/2} }{ c^2} \int_{-\infty}^{\infty} \frac{dy}{ (\cosh y +1 )^2 } \\ &= -\frac{2(4\omega -c^2)}{3c} M(\Phi_{\omega ,c}). \notag \end{align} From ({\rightm Re}f{eq:2.29}) and ({\rightm Re}f{eq:2.35}), we have \begin{align*} P(\phi_{\omega ,c}) &=-\frac{c}{2} M(\Phi_{\omega ,c}) +\frac{1}{4} \| \Phi_{\omega ,c}\|_{L^4}^4\\ &=- \frac{2\omega +c^2}{3c} M(\Phi_{\omega ,c}). \end{align*} Case 3: $\gamma >0$ and $-2\sqrt{\omega} <c<-2s_{\ast}\sqrt{\omega}$. In this case we note that $\alpha >1$. By Lemma {\rightm Re}f{lem:2.3}, ({\rightm Re}f{eq:2.18}) and ({\rightm Re}f{eq:2.15}), we obtain that \begin{align} \leftabel{eq:2.36} \| \Phi_{\omega ,c}\|_{L^4}^4 & = \frac{ 4(4\omega -c^2)^{3/2} }{ c^2+\gamma (4\omega -c^2) }\cdot \left[\! -\frac{2}{\alpha^2 -1} +\frac{2\alpha}{ (\alpha^2 -1)^{3/2} } \leftog\left( \alpha +\sqrt{\alpha^2 -1} \right) \!\right] \\%\! スペースを少し狭める &=\frac{8}{\gamma} \sqrt{4\omega -c^2} -\frac{8c}{(-\gamma )^{3/2}} \leftog\left( \alpha +\sqrt{\alpha^2 -1} \right) \notag\\ &=\frac{8}{\gamma} \sqrt{4\omega -c^2} -\frac{2c}{-\gamma} M(\Phi_{\omega ,c}). \notag \end{align} This is exactly the same as the formula ({\rightm Re}f{eq:2.34}). Hence the momentum has the same formula as the Case 1. This completes the proof. \end{proof} By the Pohozaev identity, the energy of the soliton is represented by the momentum. \begin{proposition} \leftabel{prop:2.6} Let $(\omega ,c)$ and $\gamma$ satisfy \eqref{eq:2.9}. Then we have \begin{align} \leftabel{eq:2.37} E(\phi_{\omega ,c}) =-\frac{c}{4} P(\phi_{\omega ,c}). \end{align} \end{proposition} \begin{proof} For completeness we give a proof. Let $\phi^{\leftambda}(x) =\leftambda^{1/2}\phi(\leftambda x)$ for $\leftambda >0$. Then we have \begin{align} \leftabel{eq:2.38} S_{\omega ,c} (\phi_{\omega ,c}^{\leftambda}) &= E(\phi_{\omega ,c}^{\leftambda}) +\frac{\omega}{2}M(\phi_{\omega ,c}^{\leftambda}) +\frac{c}{2}P (\phi_{\omega ,c}^{\leftambda}) \\ &=\leftambda^2 E(\phi_{\omega ,c}) +\frac{\omega}{2}M(\phi_{\omega ,c})+\frac{c\leftambda}{2}P(\phi_{\omega ,c}). \notag \end{align} Since $S_{\omega ,c}' (\phi_{\omega ,c}) =0$, we deduce that \begin{align*} 0= \left. \frac{d}{d\leftambda} S_{\omega ,c} (\phi_{\omega ,c}^{\leftambda}) \right|_{\leftambda =1} =2E(\phi_{\omega ,c}) + \frac{c}{2}P(\phi_{\omega ,c}). \end{align*} Hence the result follows. \end{proof} \subsection{Positivity of the momentum} \leftabel{sec:2.4} The effect of the momentum plays an essential role in the potential well theory. In this subsection we study the sign of the momentum of the soliton. For $(\omega ,c)$ satisfying \eqref{eq:2.9}, we rewrite $(\omega ,c) =(\omega ,2s\sqrt{\omega})$, where the parameter $s$ satisfies \begin{align} \leftabel{eq:2.39} \begin{array}{ll} \displaystyle\text{if}~b>-3/16,& \displaystyle -1 <s\lefteq 1,\\[7pt] \displaystyle\text{if}~b\lefteq -3/16,& \displaystyle -1 <s<-s_{\ast}. \end{array} \end{align} Since $P(\phi_{\omega ,2s\sqrt{\omega}}) =\sqrt{\omega}P(\phi_{1,2s})$, it is enough to check the sign of $P(\phi_{1,2s})$. \begin{figure} \caption{The function $s\mapsto P(\phi_{1,2s} \end{figure} \begin{proposition} \leftabel{prop:2.7} Let $s$ satisfy \eqref{eq:2.39}. Then the following properties hold\textup{:} \begin{enumerate}[\rightm (i)] \setlength{\itemsep}{2pt} \item If $b<0$, $P(\phi_{1,2s})>0$ for any $s$. \item If $b =0$, $P(\phi_{1,2s}) >0$ for $s\in (-1,1)$ and $P(\phi_{1,2})=0$. \item If $b>0$, there exists a unique $\thickmuskip=1mu s^{*}=s^*(b)\in (0,1)$ such that $\thickmuskip=0mu\thinmuskip=0mu P(\phi_{1,2s^*})=0$. Moreover, we have $P(\phi_{1,2s})>0$ for $s\in (-1, s^*)$ and $P(\phi_{1,2s})<0$ for $s\in(s^{*},1]$. \end{enumerate} \end{proposition} \begin{remark} \leftabel{rem:2.8} The existence of $s^*$ in (iii) was first proved in \cite{O14}. As in Figure {\rightm Re}f{fig:3}, the zero point of the function $s\mapsto P(\phi_{1,2s})$ moves to the right and converges to $1$ as $b\downarrow 0$. This remark is rigorously proved below. \end{remark} \begin{proof} From the formula \eqref{eq:2.29}, $P(\phi_{1,2s})$ is always positive when $s\lefteq 0$. Hence we need only consider the case $s>0$. (i) It is enough to consider the case $-3/16 <b<0$. First we note that the formula ({\rightm Re}f{eq:2.31}) is rewritten as \begin{align} \leftabel{eq:2.40} P(\phi_{1,2s}) =s\left( -1+\frac{1}{\gamma} \right) M(\phi_{1,2s}) +\frac{4}{\gamma}\sqrt{1-s^2}. \end{align} Since $-1+\frac{1}{\gamma}>0$, it follows from ({\rightm Re}f{eq:2.40}) that $P(\phi_{1,2s}) >0$ for $s\in(0,1]$. (ii) This is obvious from the formula ({\rightm Re}f{eq:2.40}). (iii) When $b>0$, we note that \begin{align*} &P(\phi_{1,0}) =\frac{4}{\gamma}>0, \\ &P(\phi_{1,2}) = \left( -1+\frac{1}{\gamma} \right) M(\phi_{1,2})= -\frac{4\pi\left(\gamma -1\right)}{\gamma^{3/2}} <0, \end{align*} and the function $[0,1]\ni s\mapsto P(\phi_{1,2s})$ is continuous and strictly decreasing. Therefore there exists $s^{*}\in (0,1)$ such that $P(\phi_{1,2s^*})=0$, $P(\phi_{1,2s})>0$ for $s\in (0, s^*)$ and $P(\phi_{1,2s})<0$ for $s\in(s^{*},1]$. This completes the proof. \end{proof} We define a function $(\omega ,c)\mapsto d(\omega ,c)$ by \begin{align} \leftabel{eq:2.41} d(\omega ,c) := S_{\omega ,c} (\phi_{\omega ,c}). \end{align} We note that $d(\omega ,2s\sqrt{\omega})=\omega d(1,2s)$. From Proposition {\rightm Re}f{prop:2.7} we obtain the following key lemma on the proof of Theorem {\rightm Re}f{thm:1.7}. \begin{lemma} \leftabel{lem:2.9} Let $s$ satisfy \eqref{eq:2.39}. Then the following properties hold\textup{:} \begin{enumerate}[\rightm (i)] \item If $b>0$, the function $(-1,1]\ni s \mapsto d(1,2s) $ is strictly increasing on $(-1,s^*)$ and strictly decreasing on $(s^* ,1]$. \item If $-3/16<b\lefteq 0$, the function $(-1,1]\ni s \mapsto d(1,2s) $ is strictly increasing. \item If $b\lefteq -3/16$, the function $(-1,-s_*)\ni s \mapsto d(1,2s) $ is strictly increasing. \end{enumerate} \end{lemma} \begin{proof} From the definition we have \begin{align*} d(1,2s) =S_{1,2s}(\phi_{1,2s})=E(\phi_{1,2s})+\frac{1}{2}M(\phi_{1,2s})+sP(\phi_{1,2s}). \end{align*} Since $S_{1,2s}'(\phi_{1,2s})=0$, we have \begin{align*} \frac{d}{ds} d(1,2s) =P(\phi_{1,2s}). \end{align*} Hence the result follows from Proposition {\rightm Re}f{prop:2.7}. \end{proof} \section{Gauge transformation and local well-posedness in $H^1({\mathbb R})$} \leftabel{sec:3} In this section we review the gauge transformation and the local well-posedness theory in the energy space. First we recall the result of local well-posedness for ({\rightm Re}f{eq:1.1}) in the energy space. \begin{theorem}[\cite{Oz96}] \leftabel{thm:3.1} For every $u_0 \in H^1 ({\mathbb R})$, there exist $0<T_{\rightm min}, T_{\rightm max}\lefteq \infty$ and a unique, maximal solution $u\in C((-T_{\rightm min},T_{\rightm max}) , H^1 ({\mathbb R}) )\cap L^4((-T_{\rightm min},T_{\rightm max}), W^{1,\infty}({\mathbb R} ))$ of \eqref{eq:1.1} with $u(0)=u_0$. Furthermore, the following properties hold\textup{:} \begin{enumerate}[\rightm (i)] \setlength{\itemsep}{3pt} \item If $T_{\rightm max}<\infty$\,{\rightm (}resp., if $T_{\rightm min}<\infty${\rightm )}, then $\|\partial_x u(t)\|_{L^2}\!\to\!\infty$ as $t\uparrow T_{\rightm max}$\, {\rightm (}resp., as $t\downarrow -T_{\rightm min}${\rightm )}. \item There is conservation of energy, mass and momentum; i.e., $E(u(t))=E(u_0)$, $M(u(t))=M(u_0 )$ and $P(u(t))=P(u_0)$ for all $t \in (-T_{\min}, T_{\max})$. \item Continuous dependence is satisfied in the following sense$;$ if $u_{0n}\to u_0$ in $H^1({\mathbb R} )$ and if $I\subset (-T_{\rightm min} (u_0 ) ,T_{\rightm max} (u_0) )$ is a closed interval, then the maximal solution $u_n$ of \eqref{eq:1.1} with $u_n(0)=u_{0n}$ is defined on $I$ for $n$ large enough and satisfies $u_n \to u$ in $C(I, H^1 ({\mathbb R}))$. \end{enumerate} \end{theorem} In \cite{Oz96} the proof of Theorem {\rightm Re}f{thm:3.1} is done by transforming the equation ({\rightm Re}f{eq:1.1}) into a new system of equations as follows; see also \cite{H93, HO92, HO94a}. For the solution $u$ of ({\rightm Re}f{eq:1.1}), we set \begin{align*} \varphi (t,x) &= \exp\left( \frac{i}{2}\int_{-\infty}^{x}|u(t,y)|^2 dy\right) u(t,x) ,\\ \psi (t,x) &= \exp\left( \frac{i}{2}\int_{-\infty}^{x}|u(t,y)|^2 dy\right) \partial_x u(t,x), \end{align*} then new functions $\varphi$ and $\psi$ formally satisfy \begin{align} \leftabel{eq:3.1} \left\{ \begin{array}{l} i\partial_t \varphi +\partial^2_x \varphi =i\varphi^2\overline{\psi} +f(\varphi ),\\[3pt] i\partial_t \psi +\partial^2_x \psi =-i\psi^2\overline{\varphi} +\partial_{\varphi}f(\varphi )\psi +\partial_{\overline{\varphi}}f(\varphi )\overline{\psi}, \end{array} \right. \end{align} where $f(\varphi )=-b|\varphi|^4\varphi$. Since the system ({\rightm Re}f{eq:3.1}) has no loss of derivatives unlike the original equation ({\rightm Re}f{eq:1.1}), one can solve the Cauchy problem by the fixed point argument. However, in order to construct the solution of ({\rightm Re}f{eq:1.1}) through the system, we need to solve the equation ({\rightm Re}f{eq:3.1}) under the constraint condition \begin{align*} \psi =\partial_x\varphi -\frac{i}{2}|\varphi|^2\varphi. \end{align*} This requires more or less complex calculation; see \cite{HO94a} for the details. We refer to \cite{HO16} for a more direct approach without using a system of equations. We note that the gauge transformation plays a key role when one transforms the equation ({\rightm Re}f{eq:1.1}) into a system of equations ({\rightm Re}f{eq:3.1}). Here we consider more general gauge transformations as seen in \cite{Wu13}. For $a\in{\mathbb R}$ we define ${\mathcal G}_a : H^1({\mathbb R})\to H^1({\mathbb R})$ by \begin{align} \leftabel{eq:3.2} {\mathcal G}_a (u)(t,x) =\exp\left( ia\int_{-\infty}^{x} |u(t,y)|^2dy \right) u(t,x). \end{align} By a direct computation we have the following result. \begin{proposition} \leftabel{prop:3.2} Let $a\in{\mathbb R}$, and let $u\in C((-T_{\rightm min},T_{\rightm max}) , H^1 ({\mathbb R}) )$ be a maximal solution of \eqref{eq:1.1}. Then $v={\mathcal G}_a (u) \in C((-T_{\rightm min},T_{\rightm max}) , H^1 ({\mathbb R}) )$, and satisfies the following equation \begin{align} \leftabel{eq:3.3} i\partial_t v+\partial^2_x v+(-2a+1)i|v|^2\partial_x v-2aiv^2\partial_x\overline{v}+\left( a^2+\frac{a}{2}+b\right) |v|^4v=0. \end{align} Moreover, the equation \eqref{eq:3.3} has the following conserved quantities\textup{:} \begin{align*} E_a (v) &=\frac{1}{2} \| \partial_x v\|_{L^2}^2+\left( a-\frac{1}{4}\right) \rightbra[i|v|^2\partial_x v, v] +\left( \frac{a^2}{2}-\frac{a}{4}-\frac{b}{6}\right) \| v\|_{L^6}^6 , \\ M_a (v)&= \| v \|_{L^2}^2, \\ P_a (v)&= \rightbra[i\partial_x v,v] +a\| v\|_{L^4}^4. \end{align*} \end{proposition} \begin{remark} We note that the functions $u$ and ${\mathcal G}_a(u)$ are defined on the same maximal interval. The well-posedness in $H^1({\mathbb R})$ for the equations \eqref{eq:1.1} and \eqref{eq:3.3} is equivalent since $u\mapsto{\mathscr G}_{a}(u)$ is locally Lipschitz continuous on $H^1({\mathbb R})$. \end{remark} It is important to choose the suitable parameter $a\in{\mathbb R}$ depending on the situation. If we set $a=1/2$, the term $i|v|^2\partial_x v$ is removed in ({\rightm Re}f{eq:3.3}) and it is useful when one treats the Fourier restriction norm (see \cite{T99, CKSTT01, CKSTT02}). When $a=1/4$ the interaction term with derivative in the energy is canceled out, which is useful to derive a mass condition by using sharp Gagliardo--Nirenberg inequalities (see \cite{HO92, Wu13, Wu15}). In this paper we apply the gauge transformation in the case $a=1/4$ for giving variational characterization of the solitons including the case $b<0$. By Proposition {\rightm Re}f{prop:3.2}, $v={\mathcal G}_{1/4}(u)$ satisfies the equation \begin{align} \leftabel{eq:3.4} i\partial_t v+\partial_x^2 v+\frac{i}{2}|v|^2\partial_x v-\frac{i}{2}v^2\partial_x\overline{v}+\frac{3}{16}\gamma |v|^4v=0, \quad \gamma =1+\frac{16}{3}b, \end{align} which is nothing but the equation \eqref{ME}. The conserved quantities of \eqref{eq:3.4} are as follows: \begin{align*} \tag{Energy} {\mathcal E} (v) &=E_{1/4}(v)=\frac{1}{2} \| \partial_x v\|_{L^2}^2-\frac{\gamma}{32} \| v\|_{L^6}^6 , \\ \tag{Mass} {\mathcal M} (v)&=M_{1/4}(v)=\| v \|_{L^2}^2, \\ \tag{Momentum} {\mathcal P} (v)&=P_{1/4}(v)=\rightbra[i\partial_x v,v] +\frac{1}{4} \| v\|_{L^4}^4. \end{align*} We note that the energy functional ${\mathcal E}(v)$ is nonnegative if $b \lefteq -3/16$. Hence we have the following result. \begin{proposition} \leftabel{prop:3.4} Let $b\lefteq -3/16$. For every $u_0 \in H^1({\mathbb R})$, the maximal $H^1({\mathbb R})$-solution $u$ of \eqref{eq:1.1} given by Theorem {\rightm Re}f{thm:3.1} is global and \begin{align*} \sup_{t\in{\mathbb R}} \| u(t)\|_{H^1}\lefteq C(\| u_0\|_{H^1})<\infty . \end{align*} \end{proposition} When $b>-3/16$, by applying the sharp Gagliardo--Nirenberg inequality \begin{align} \leftabel{eq:3.5} \| f\|_{L^6}^6 \lefteq \frac{4}{\pi^2}\| f\|_{L^2}^4\| \partial_{x}f\|_{L^2}^2\quad ( \Leftrightarrow\eqref{GN1} ), \end{align} we deduce that if the initial data $u_0\in H^1({\mathbb R})$ satisfying $\| u_0\|_{L^2}^2 <\frac{2\pi}{\sqrt{\gamma}}$, then the corresponding solution is global and bounded. A similar approach was originally taken in \cite{HO92, HO94a, Oz96}. Finally, we discuss the solitons of \eqref{eq:3.4}. Let $(\omega ,c )$ satisfy ({\rightm Re}f{eq:2.9}). The equation \eqref{eq:3.4} has a two-parameter family of solitons \begin{align} \leftabel{eq:3.6} v_{\omega ,c}(t ,x) ={\mathcal G}_{1/4}(u_{\omega ,c})(t,x)=e^{i\omega t}\varphi_{\omega ,c}(x-ct), \end{align} where $\varphi_{\omega ,c}$ is defined by \begin{align*} \varphi_{\omega ,c} (x) =e^{i\frac{cx}{2}}\Phi_{\omega ,c}(x). \end{align*} We note that $\varphi_{\omega ,c}$ satisfies the equation \begin{align} \leftabel{eq:3.7} -\varphi'' +\omega\varphi +ic\varphi' +\frac{c}{2}|\varphi |^2\varphi -\frac{3}{16}\gamma |\varphi|^4 \varphi =0, \quad x\in {\mathbb R}. \end{align} We note that \eqref{eq:3.7} is rewritten as ${\mathcal S}_{\omega ,c} ' (\varphi )=0$, where \begin{align*} {\mathcal S}_{\omega ,c} (\varphi ) ={\mathcal E} (\varphi )+\frac{\omega}{2}{\mathcal M} (\varphi ) +\frac{c}{2}{\mathcal P} (\varphi ). \end{align*} For the action functionals $S_{\omega ,c}$ and ${\mathcal S}_{\omega ,c}$, we have the following relation: \begin{align*} S_{\omega ,c}(u)={\mathcal S}_{\omega ,c} ({\mathcal G}_{1/4} (u) ) \quad \text{for any}~u\in H^1({\mathbb R}). \end{align*} In particular, we have \begin{align} \leftabel{eq:3.8} d(\omega ,c) =S_{\omega ,c} (\phi_{\omega ,c}) ={\mathcal S}_{\omega ,c} (\varphi_{\omega ,c}). \end{align} \section{Variational characterization} \leftabel{sec:4} In this section we give a variational characterization of the soliton $v_{\omega ,c}$ defined by ({\rightm Re}f{eq:3.6}). Here we assume that $\gamma$ and $(\omega ,c)$ satisfy \begin{align} \leftabel{eq:4.1} \begin{array}{ll} \displaystyle\text{if}~\gamma >0\Leftrightarrow b>-3/16,& \displaystyle -2\sqrt{\omega} <c\lefteq 2\sqrt{\omega} ,\\[7pt] \displaystyle\text{if}~\gamma = 0\Leftrightarrow b= -3/16,& \displaystyle -2\sqrt{\omega} <c<0. \end{array} \end{align} We prepare some notations. First we define function spaces by \begin{align} \leftabel{eq:4.2} \varphi \in X_{\omega,c} &\iff \left\{ \begin{array}{ll} \displaystyle \varphi \in H^1({\mathbb R}) & \displaystyle\text{if}~ \omega >c^2/4, \\[5pt] \displaystyle e^{-i\frac{cx}{2}}\varphi \in \dot{H}^1({\mathbb R}) \cap L^{4}({\mathbb R}) & \displaystyle\text{if}~c=2\sqrt{\omega}, \end{array} \right.\\[3pt] \| \varphi\|_{ X_{c^2/4,c} }&:=\| e^{-i\frac{c}{2}\cdot}\varphi\|_{\dot{H}^1\cap L^4}. \notag \end{align} Note that $H^1({\mathbb R}) \subset X_{c^2/4,c}$. We consider the functional ${\mathcal K}_{\omega ,c}(\varphi )=\left.\frac{d}{d\leftambda}{\mathcal S}_{\omega ,c}(\leftambda u)\right|_{\leftambda=1}$ which has the following explicit formula: \begin{align} \leftabel{eq:4.3} {\mathcal K}_{\omega ,c}(\varphi ) := \| \partial_x\varphi\|_{L^2}^2+\omega\|\varphi\|_{L^2}^2 +c\rightbra[i\partial_x\varphi, \varphi] +\frac{c}{2}\| \varphi\|_{L^4}^4 -\frac{3}{16}\gamma\| \varphi\|_{L^6}^6. \end{align} We consider the following minimization problem: \begin{align*} \mu (\omega ,c) :=\inf \left\{ {\mathcal S}_{\omega ,c}(\varphi) :\varphi \in X_{\omega ,c}\setminus \{ 0\} , {\mathcal K}_{\omega ,c}(\varphi )=0 \right\}. \end{align*} We define the sets ${\mathscr G}_{\omega ,c}$ and ${\mathscr M}_{\omega ,c}$ by \begin{align*} {\mathscr G}_{\omega ,c}&:=\left\{ \varphi \in X_{\omega ,c}\setminus\{ 0\} : {\mathcal S}_{\omega ,c}'(\varphi )=0\right\},\\ {\mathscr M}_{\omega ,c}&:=\left\{ \varphi \in X_{\omega ,c}\setminus\{ 0\} : {\mathcal S}_{\omega ,c}(\varphi) =\mu (\omega ,c), {\mathcal K}_{\omega ,c}(\varphi )=0 \right\}. \end{align*} ${\mathscr G}_{\omega ,c}$ is the set of nontrivial critical points of ${\mathcal S}_{\omega ,c}$, and ${\mathscr M}_{\omega ,c}$ is the set of minimizers of ${\mathcal S}_{\omega ,c}$ on the Nehari manifold. The main result in this section is the following result. \begin{proposition} \leftabel{prop:4.1} Let $\gamma$ and $(\omega ,c)$ satisfy \eqref{eq:4.1}. Then we have \begin{align} \leftabel{eq:4.4} {\mathscr G}_{\omega ,c}={\mathscr M}_{\omega ,c} =\left\{ e^{i\theta}\varphi_{\omega ,c}(\cdot -y):\theta\in[0,2\pi), y\in{\mathbb R} \right\}, \end{align} and $d(\omega ,c)=\mu (\omega ,c)$. \end{proposition} Our proof of Proposition {\rightm Re}f{prop:4.1} depends on concentration compactness arguments in \cite{CO06} (see \cite{FHI17} for the case $c=2\sqrt{\omega}$). For convenience of notations, we define \begin{align*} {\mathcal L}_{\omega ,c} (\varphi )&:=\|\partial_x\varphi\|_{L^2}^2+\omega\|\varphi\|_{L^2}^2 +c\rightbra[i\partial_x\varphi, \varphi] ,\\ {\mathcal I}_{\omega ,c}(\varphi )&:={\mathcal S}_{\omega,c}(\varphi )-\frac{1}{4}{\mathcal K}_{\omega ,c}(\varphi ) =\frac{1}{4}{\mathcal L}_{\omega ,c}(\varphi )+\frac{\gamma}{64}\|\varphi\|_{L^6}^6. \end{align*} First we prove the following lemma. \begin{lemma} \leftabel{lem:4.2} Let $\gamma$ and $(\omega ,c)$ satisfy \eqref{eq:4.1}. Then the following properties hold\textup{:} \begin{enumerate}[\rightm (i)] \item If $\omega >c^2/4$, there exists $C_1=C_1(\omega ,c)$ such that \begin{align*} {\mathcal L}_{\omega ,c}(\varphi ) \geq C_1\| \varphi\|_{H^1}^2~\text{for}~\varphi\in H^1({\mathbb R} ). \end{align*} \item $\mu (\omega ,c) >0$. \item If $\varphi \in X_{\omega ,c}$ satisfies ${\mathcal K}_{\omega ,c}(\varphi )<0$, then $\mu (\omega ,c)<{\mathcal I}_{\omega ,c}(\varphi)$. \end{enumerate} \end{lemma} \begin{proof} (i) See Lemma 7 (1) in \cite{CO06}. (ii) Case 1: $\omega >c^2/4$. Let $\varphi\in H^1({\mathbb R})\setminus\{0\}$ satisfy ${\mathcal K}_{\omega ,c}(\varphi )=0$. By (i), ({\rightm Re}f{eq:4.3}) and the Sobolev inequality, there exists $C_2>0$ such that \begin{align*} C_1\|\varphi\|_{H^1}^2\lefteq{\mathcal L}_{\omega ,c}(\varphi ) &=-\frac{c}{2}\|\varphi\|_{L^4}^4+\frac{3}{16}\gamma\|\varphi\|_{L^6}^6\\ &\lefteq\frac{|c|}{2}\|\varphi\|_{L^2}\| \varphi\|_{L^6}^3+\frac{3}{16}\gamma\| \varphi\|_{L^6}^6\\ &\lefteq\frac{C_1}{2}\| \varphi\|_{H^1}^2+C_2\| \varphi\|_{H^1}^6. \end{align*} This yields that $\| \varphi\|_{H^1}^4\geq \frac{C_1}{2C_2}$. Hence we have \begin{align*} \mu (\omega ,c)&=\inf\left\{ {\mathcal I}_{\omega ,c}(\varphi ):\varphi \in H^1({\mathbb R} )\setminus\{0\} , {\mathcal K}_{\omega ,c}(\varphi )=0 \right\}\\ &\geq\frac{1}{4}\inf\left\{ {\mathcal L}_{\omega ,c}(\varphi ):\varphi \in H^1({\mathbb R} )\setminus\{0\} , {\mathcal K}_{\omega ,c}(\varphi )=0 \right\}\\ &\geq\frac{C_1}{4}\sqrt{\frac{C_1}{2C_2}} >0. \end{align*} Case 2: $c=2\sqrt{\omega}$. In this case we have \begin{align} \leftabel{eq:4.5} {\mathcal L}_{\omega ,c}(\varphi ) =\left\| \partial_x\varphi -\frac{i}{2}c\varphi \right\|_{L^2}^2 +\left( \omega -\frac{c^2}{4} \right) \| \varphi \|_{L^2}^2=\left\| \partial_x\left( e^{-i\frac{cx}{2}}\varphi\right) \right\|_{L^2}^2>0 \end{align} for $\varphi \in X_{\omega ,c}\setminus\{ 0\}$. This yields that $\mu (\omega , c)\geq 0$. We prove $\mu(\omega ,c) >0$ by contradiction. Assume that $\mu (\omega ,c)=0$. Then one can take the minimizing sequence $\{ \varphi_n\} \subset X_{\omega ,c}\setminus\{ 0\}$ such that \begin{align} \leftabel{eq:4.6} {\mathcal S}_{\omega ,c} (\varphi_n ) \underset{n\to\infty}{\leftongrightarrow} 0, ~\text{and}~{\mathcal K}_{\omega ,c}(\varphi_n )=0~\text{for all}~n\in{\mathbb N}. \end{align} Since ${\mathcal S}_{\omega ,c}$ is rewritten as \begin{align} \leftabel{eq:4.7} {\mathcal S}_{\omega ,c}(\varphi )=\frac{1}{4}{\mathcal K}_{\omega ,c}(\varphi )+\frac{1}{4}{\mathcal L}_{\omega ,c}(\varphi ) +\frac{\gamma}{64}\| \varphi\|_{L^6}^6, \end{align} from ({\rightm Re}f{eq:4.5}) and ({\rightm Re}f{eq:4.6}) we obtain that \begin{align*} \left\| \partial_x\left( e^{-i\frac{cx}{2}}\varphi_n \right) \right\|_{L^2}, ~\|\varphi_n\|_{L^6} \leftongrightarrow 0 \end{align*} as $n\to\infty$. By using an elementary interpolation inequality \begin{align*} \| f\|_{L^{\infty}}^4 \lefteq 4\| f\|_{L^6}^3\| \partial_x f\|_{L^2}, \end{align*} we have $\| \varphi_n\|_{L^{\infty}}\to 0$ as $n\to\infty$. Hence we have \begin{align*} 0={\mathcal K}_{\omega ,c}(\varphi_n )&={\mathcal L}_{\omega ,c}(\varphi_n) +\frac{c}{2}\| \varphi_n\|_{L^4}^4 -\frac{3}{16}\gamma \| \varphi_n\|_{L^6}^6\\ &\geq\left( \frac{c}{2}-\frac{3}{16}\gamma\| \varphi_n\|_{L^{\infty}}^2\right) \|\varphi_n\|_{L^4}^4 >0 \end{align*} for large $n\in{\mathbb N}$, which is a contradiction with \eqref{eq:4.6}. (iii) Let $\varphi\in X_{\omega ,c}\setminus\{ 0\}$ satisfy ${\mathcal K}_{\omega ,c}(\varphi) <0$. Then there exists a unique $\leftambda _0 \in (0,1)$ such that ${\mathcal K}_{\omega ,c}(\leftambda_0\varphi ) =0$. From the definition of $\mu(\omega ,c)$, we have \begin{align*} \mu(\omega ,c) \lefteq {\mathcal I}_{\omega ,c}(\leftambda_0 \varphi ) =\frac{\leftambda_0^2}{4}{\mathcal L}_{\omega ,c}(\varphi )+\frac{\leftambda_0^6\gamma}{64}\| \varphi\|_{L^6}^6 <{\mathcal I}_{\omega ,c}(\varphi ). \end{align*} This completes the proof. \end{proof} By the standard ODE arguments (see e.g. \cite{C03, FHI17}), we have the following lemma. \begin{lemma} \leftabel{lem:4.3} Let $\gamma$ and $(\omega ,c)$ satisfy \eqref{eq:2.9}. Then we have \begin{align*} {\mathscr G}_{\omega ,c} =\left\{ e^{i\theta}\varphi_{\omega ,c}(\cdot -y):\theta\in[0,2\pi), y\in{\mathbb R} \right\}. \end{align*} \end{lemma} Next we prove the following result. \begin{lemma} \leftabel{lem:4.4} Let $\gamma$ and $(\omega ,c)$ satisfy \eqref{eq:4.1}. Assume that ${\mathscr M}_{\omega ,c}\neq \emptyset$. Then we have ${\mathscr G}_{\omega, c}={\mathscr M}_{\omega ,c}$. Moreover we have $d(\omega ,c)=\mu(\omega ,c)$. \end{lemma} \begin{proof} First we prove ${\mathscr M}_{\omega ,c}\subset {\mathscr G}_{\omega ,c}$. Let $\varphi \in{\mathscr M}_{\omega ,c}$. Since $\varphi$ is a minimizer on the Nehari manifold, there exists a Lagrange multiplier $\eta\in{\mathbb R}$ such that ${\mathcal S}_{\omega ,c}'(\varphi )=\eta{\mathcal K}_{\omega ,c}'(\varphi )$. Thus we have \begin{align*} 0={\mathcal K}_{\omega ,c}(\varphi )=\tbra[{\mathcal S}_{\omega ,c}'(\varphi ), \varphi] =\eta\tbra[{\mathcal K}_{\omega ,c}'(\varphi ), \varphi ]. \end{align*} By ${\mathcal K}_{\omega ,c}(\varphi )=0$ and $\varphi\neq 0$, we have \begin{align*} \tbra[{\mathcal K}_{\omega ,c}'(\varphi ),\varphi ]&=2{\mathcal L}_{\omega ,c}(\varphi )+2c\|\varphi\|_{L^4}^4 -\frac{9}{8}\gamma\| \varphi\|_{L^6}^6\\ &=-2{\mathcal L}_{\omega ,c}(\varphi )-\frac{3}{8}\gamma\|\varphi\|_{L^6}^6<0. \end{align*} This yields that $\eta=0$ and $\varphi\in{\mathscr G}_{\omega ,c}$, which implies ${\mathscr M}_{\omega ,c}\subset {\mathscr G}_{\omega ,c}$. Conversely, let $\varphi\in{\mathscr G}_{\omega ,c}$. By Lemma {\rightm Re}f{lem:4.3}, there exist $\theta_0\in[0,2\pi )$ and $y_0\in{\mathbb R}$ such that $\varphi =e^{i\theta_0}\varphi_{\omega ,c}(\cdot -y_0)$. Since ${\mathscr M}_{\omega ,c}\neq\emptyset$, we can take some $\psi\in{\mathscr M}_{\omega ,c}$. By ${\mathscr M}_{\omega ,c}\subset {\mathscr G}_{\omega ,c}$ and Lemma {\rightm Re}f{lem:4.3}, there exist $\theta_1\in[0,2\pi )$ and $y_1\in{\mathbb R}$ such that $\psi =e^{i\theta_1}\varphi_{\omega ,c}(\cdot -y_1)$. Thus we have \begin{align*} {\mathcal S}_{\omega ,c}(\varphi ) ={\mathcal S}_{\omega ,c}(\varphi_{\omega ,c}) ={\mathcal S}_{\omega ,c}(\psi ) = \mu (\omega ,c). \end{align*} This yields that $\varphi\in{\mathscr M}_{\omega ,c}$ since ${\mathcal K}_{\omega ,c}(\varphi )=\tbra[{\mathcal S}_{\omega ,c}'(\varphi ), \varphi]=0$. This completes the proof. \end{proof} To complete the proof of Proposition {\rightm Re}f{prop:4.1}, we need to prove that ${\mathscr M}_{\omega ,c} \neq\emptyset$. To this end, we prepare two useful lemmas on concentration compactness. \begin{lemma}[\cite{Lie83, BFV14}] \leftabel{lem:4.5} Let $p\geq 2$. Let $\{f_n\}$ be a bounded sequence in $\dot{H}^1({\mathbb R})\cap L^{p}({\mathbb R})$. Assume that there exists $q\in(p,\infty)$ such that $\leftimsup_{n \to \infty} \norm[f_n]_{L^q}>0$. Then, there exist $\{y_n\}\subset{\mathbb R}$ and $f \in \dot{H}^1({\mathbb R})\cap L^{p}({\mathbb R})\setminus \{0\}$ such that $\{f_n(\cdot-y_n)\}$ has a subsequence that converges to $f$ weakly in $\dot{H}^1({\mathbb R})\cap L^{p}({\mathbb R})$. \end{lemma} \begin{lemma}[{\cite{BL83}}] \leftabel{lem:4.6} Let $1\lefteq p < \infty$. Let $\{f_n\}$ be a bounded sequence in $L^p({\mathbb R})$ and $f_n \to f$ a.e. in ${\mathbb R}$ as $n\to \infty$. Then we have \begin{align*} \| f_n\|_{L^p}^p - \| f_n-f\|_{L^p}^p - \| f \|_{L^p}^p \to 0 \end{align*} as $n \to \infty$. \end{lemma} The assertion ${\mathscr M}_{\omega ,c}\neq\emptyset$ follows from the following stronger claim. \begin{proposition} \leftabel{prop:4.7} Let $\gamma$ and $(\omega ,c)$ satisfy \eqref{eq:4.1}. If a sequence $ \thickmuskip=3mu \{\varphi_n\}\subset X_{\omega,c}$ satisfies \begin{align} \leftabel{eq:4.8} {\mathcal S}_{\omega ,c}(\varphi_n ) \to\mu (\omega ,c)~\text{and}~{\mathcal K}_{\omega ,c}(\varphi_n ) \to 0~\text{as}~n\to\infty , \end{align} then there exist a sequence $\{ y_n\}\subset{\mathbb R}$ and $v\in{\mathscr M}_{\omega ,c}$ such that $\{ \varphi_n(\cdot -y_n)\}$ has a subsequence that converges to $v$ strongly in $X_{\omega ,c}$. \end{proposition} \begin{remark} \leftabel{rem:4.8} If we only prove that ${\mathscr M}_{\omega ,c}\neq\emptyset$, we may assume that ${\mathcal K}_{\omega ,c}(\varphi_n)=0$ for all $n\in{\mathbb N}$. However, when one studies stability problems around the solitons, it is essential to consider the minimizing sequence $\{ \varphi_n\}$ satisfying ${\mathcal K}_{\omega ,c}(\varphi_n)\neq 0$; see \cite{CO06, H19} or Section {\rightm Re}f{sec:6}. \end{remark} \begin{proof} {\bf Step 1.} $\{ \varphi_n\}$ is bounded in $X_{\omega ,c}$. If $\omega >c^2/4$, this follows from ({\rightm Re}f{eq:4.7}) and Lemma {\rightm Re}f{lem:4.2} (i). If $c=2\sqrt{\omega}$, from ({\rightm Re}f{eq:4.5}) and \eqref{eq:4.7} we obtain that \begin{align*} \sup_{n\in{\mathbb N}}\|\varphi_n\|_{L^6}^6,~ \sup_{n\in{\mathbb N}}\|\partial_x\left(e^{-i\frac{cx}{2}}\varphi_n\right)\|_{L^2}^2 <\infty . \end{align*} Since we have \begin{align} \leftabel{eq:4.9} {\mathcal K}_{\omega ,c}(\varphi_n )&={\mathcal L}_{\omega ,c} (\varphi_n ) +\frac{c}{2}\| \varphi_n\|_{L^4}^4 -\frac{3}{16}\gamma \| \varphi_n\|_{L^6}^6, \end{align} we deduce that $\{ \varphi_n\}$ is also bounded in $L^4({\mathbb R})$.\\ {\bf Step 2.} $\leftimsup_{n\to\infty}\| \varphi_n\|_{L^6} >0$. Suppose that $\leftim_{n\to\infty}\| \varphi_n\|_{L^6}=0$. If $\omega >c^2/4$, by the boundedness of $\{ \varphi_n\}$ in $L^2({\mathbb R})$ we have \begin{align*} \| \varphi_n\|_{L^4}^4 \lefteq\|\varphi_n\|_{L^2}\|\varphi_n\|_{L^6}^3 \underset{n\to\infty}{\leftongrightarrow} 0. \end{align*} From ({\rightm Re}f{eq:4.9}) we deduce that ${\mathcal L}_{\omega ,c} (\varphi_n )\to 0$. By ({\rightm Re}f{eq:4.7}), we have ${\mathcal S}_{\omega ,c}(\varphi_n) \to 0$, but this gives a contradiction with $\mu (\omega ,c)>0$. If $c=2\sqrt{\omega}$, from ({\rightm Re}f{eq:4.9}) we obtain that \begin{align*} {\mathcal L}_{\omega ,c}(\varphi_n ), ~\|\varphi_n\|_{L^4}^4 \underset{n\to\infty}{\leftongrightarrow} 0, \end{align*} which yields ${\mathcal S}_{\omega ,c}(\varphi_n) \to 0$ again. This gives a contradiction.\\ {\bf Step 3.} By Step 1, Step 2 and Lemma {\rightm Re}f{lem:4.5}, there exist $\{ y_n\}\subset{\mathbb R}$ and $v \in X_{\omega ,c}\setminus\{ 0\}$ such that a subsequence of $\{\varphi(\cdot -y_n)\}$ (we denote it by $\{ v_n\}$) converges to $v$ weakly in $X_{\omega ,c}$. Taking a subsequence if necessary, we have $v_n \to v$ a.e. in ${\mathbb R}$. By applying Lemma {\rightm Re}f{lem:4.6}, we have \begin{align} \leftabel{eq:4.10} &{\mathcal K}_{\omega ,c}(v_n)-{\mathcal K}_{\omega ,c}(v_n-v)-{\mathcal K}_{\omega ,c}(v)\leftongrightarrow 0,\\ \leftabel{eq:4.11} &{\mathcal I}_{\omega ,c}(v_n)-{\mathcal I}_{\omega ,c}(v_n-v)-{\mathcal I}_{\omega ,c}(v)\leftongrightarrow 0, \end{align} as $n\to\infty$.\\ {\bf Step 4.} ${\mathcal K}_{\omega ,c}(v)\lefteq 0$. Suppose that ${\mathcal K}_{\omega ,c}(v) >0$. By ${\mathcal K}_{\omega ,c}(v_n) \to 0$ and \eqref{eq:4.10}, we have \begin{align*} {\mathcal K}_{\omega ,c}(v_n -v) \to -{\mathcal K}_{\omega ,c}(v) <0. \end{align*} This implies that ${\mathcal K}_{\omega ,c}(v_n -v)<0$ for large $n\in{\mathbb N}$. Applying Lemma {\rightm Re}f{lem:4.2} (iii), we have $\mu(\omega ,c)<{\mathcal I}_{\omega ,c}(v_n -v)$ for large $n\in{\mathbb N}$. By \eqref{eq:4.8} we have ${\mathcal I}_{\omega ,c}(v_n) \to\mu (\omega ,c)$. Combined with ({\rightm Re}f{eq:4.11}), we have \begin{align*} {\mathcal I}_{\omega ,c}(v) =\leftim_{n\to\infty}\left\{{\mathcal I}_{\omega ,c}(v_n) -{\mathcal I}_{\omega ,c}(v_n-v)\right\} \lefteq \mu(\omega ,c) -\mu (\omega ,c) =0, \end{align*} which yields that $v=0$. This is a contradiction.\\ {\bf Step 5.} By Step 4, Lemma {\rightm Re}f{lem:4.2} (iii), and the weakly lower semicontinuity of ${\mathcal I}_{\omega ,c}$, we have \begin{align*} \mu (\omega ,c) \lefteq {\mathcal I}_{\omega ,c} (v) \lefteq \leftiminf_{n\to\infty} {\mathcal I}_{\omega ,c} (v_n) =\mu (\omega ,c). \end{align*} Thus we have ${\mathcal I}_{\omega ,c}(v)=\mu (\omega ,c)$. By Step 4 and Lemma {\rightm Re}f{lem:4.2} (iii), we have ${\mathcal K}_{\omega ,c}(v) =0$. Therefore $v\in{\mathscr M}_{\omega ,c}$. By ({\rightm Re}f{eq:4.11}) and ${\mathcal I}_{\omega ,c}(v)=\mu (\omega ,c)$, we have ${\mathcal I}_{\omega ,c}(v_n -v) \to 0$, which yields that $v_n\to v$ strongly in $X_{\omega ,c}$. This completes the proof. \end{proof} \section{A two-parameter family of potential wells} \leftabel{sec:5} In this section we prove Theorem {\rightm Re}f{thm:1.7}. We recall the following subsets of the energy space: \begin{align*} {\mathscr A}_{\omega ,c}=& \left\{ \varphi\in H^1({\mathbb R} ): {\mathcal S}_{\omega, c}(\varphi ) < d(\omega ,c)\right\}, \\ {\mathscr A}_{\omega, c}^+ =&\left\{ \varphi\in{\mathscr A}_{\omega, c} :{\mathcal K}_{\omega, c}(\varphi) \geq 0 \right\},\\ {\mathscr A}_{\omega, c}^- =&\left\{ \varphi\in{\mathscr A}_{\omega, c} :{\mathcal K}_{\omega ,c}(\varphi) < 0\right\}. \end{align*} First we prove that ${\mathscr A}_{\omega ,c}^{\pm}$ is invariant under the flow of \eqref{ME}. \begin{lemma} \leftabel{lem:5.1} Let $b \geq -3/16$ and $(\omega,c)$ satisfy \eqref{eq:4.1}. Then, each of ${\mathscr A}_{\omega ,c}^{+}$ and ${\mathscr A}_{\omega ,c}^{-}$ is invariant under the flow of \eqref{ME}. If the initial data $v_0 \in{\mathscr A}_{\omega ,c}^+$, then the corresponding solution is global, and satisfies the following uniform estimate\textup{:} \begin{align} \leftabel{eq:5.1} \| \partial_x v\|_{L^{\infty}({\mathbb R} ,L^2)}^2 \lefteq 8{\mathcal S}_{\omega ,c}(v_0)+\frac{c^2}{2}{\mathcal M} (v_0). \end{align} \end{lemma} \begin{proof} Assume that $v_0\in{\mathscr A}_{\omega ,c}^+$. Let $v\in C((-T_{\rightm min},T_{\rightm max}) , H^1 ({\mathbb R}) )$ be a maximal solution of \eqref{ME} with $v(0)=v_0$. If ${\mathcal K}_{\omega ,c}(v_0)=0$, by Proposition {\rightm Re}f{prop:4.1}, we have $v_0 =0$. By uniqueness we have $v(t)=0$ for all $t\in{\mathbb R}$. Consider the case ${\mathcal K}_{\omega ,c}(v_0)>0$. If there exists $t_{*}\in (-T_{\rightm min},T_{\rightm max})$ such that ${\mathcal K}_{\omega ,c}(v(t_{*}))=0$, the above argument gives that $v\equiv0$, which is a contradiction. Since the function $t\mapsto{\mathcal K}_{\omega ,c}(v(t))$ is continuous, we deduce that ${\mathcal K}_{\omega ,c}(v(t))>0$ for all $t\in (-T_{\rightm min},T_{\rightm max})$. This implies that ${\mathscr A}_{\omega ,c}^+$ is invariant under the flow of \eqref{ME}. Similarly one can prove that ${\mathscr A}_{\omega ,c}^-$ is also invariant. Next we prove that the initial data $v_0\in{\mathscr A}_{\omega ,c}^+$ generates global and bounded solutions. By ({\rightm Re}f{eq:4.7}) and $v(t)\in{\mathscr A}_{\omega ,c}^+$, we obtain that \begin{align*} {\mathcal S}_{\omega ,c}(v_0)&={\mathcal S}_{\omega ,c}(v(t))\\ &=\frac{1}{4}{\mathcal K}_{\omega ,c}(v(t))+\frac{1}{4}{\mathcal L}_{\omega ,c}(v(t)) +\frac{\gamma}{64}\| v(t)\|_{L^6}^6\\ &\geq\frac{1}{4}{\mathcal L}_{\omega ,c}(v(t))\\ &\geq \frac{1}{4}\left\|\partial_x\left( e^{-i\frac{cx}{2}}v(t)\right) \right\|_{L^2}^2 \end{align*} for all $t\in (-T_{\rightm min},T_{\rightm max})$. This implies that $T_{\rightm min}=T_{\rightm max}=\infty$. Moreover, we have \begin{align*} \| \partial_x v(t)\|_{L^2}^2&\lefteq\left( \left\| \partial_x v(t) -\frac{c}{2}iv(t)\right\|_{L^2}+\frac{|c|}{2}\| v(t)\|_{L^2}\right)^2 \\ &\lefteq 2\left\|\partial_x\left( e^{-i\frac{cx}{2}}v(t)\right) \right\|_{L^2}^2+\frac{c^2}{2}{\mathcal M} (v_0) \\ &\lefteq 8{\mathcal S}_{\omega ,c}(v_0)+\frac{c^2}{2}{\mathcal M} (v_0) \end{align*} for all $t\in{\mathbb R}$. This completes the proof. \end{proof} We are now in a position to complete the proof of Theorem {\rightm Re}f{thm:1.7}. For convenience we often use the notation $\mu :=\sqrt{\omega}$ in the proof. \begin{proof}[Proof of Theorem {\rightm Re}f{thm:1.7}] First we note that \begin{align} \leftabel{eq:5.2} \begin{array}{ll} \displaystyle\text{if}~b\geq 0,& \displaystyle \max_{-1 <s\lefteq 1}d(1,2s)=d(1,2s^*),\\[5pt] \displaystyle\text{if}~-3/16<b\lefteq 0,& \displaystyle \max_{-1 <s\lefteq 1}d(1,2s)=d(1,2), \end{array} \end{align} which follows from Lemma {\rightm Re}f{lem:2.9}. From \eqref{eq:3.8} and Proposition {\rightm Re}f{prop:2.6} we have \begin{align} \leftabel{eq:5.3} 2d(1,2s) =M(\phi_{1,2s})+sP(\phi_{1,2s}). \end{align} Therefore, from the definition of $M^*$, the relation \eqref{eq:5.2} is rewritten as \begin{align} \leftabel{eq:5.4} \max_{-1 <s\lefteq 1}2d(1,2s) = M^* (b)=M^* \end{align} for $b>-3/16$. (i) First we prove the claim on the set above the mass threshold $M^*$. Assume by contradiction that there exists $\varphi\in H^1({\mathbb R})$ such that ${\mathcal M}(\varphi)>M^*$ and $\varphi\in{\mathscr A}_s^+ \cap{\mathscr A}_s^-$ for some $s\in (-1,1]$. Then, there exist $\mu_1, \mu_2 >0$ such that \begin{align*} &{\mathcal S}_{\mu_i^2,2s\mu_i}(\varphi) <d(\mu_i^2, 2s\mu_i)\quad \text{for}~i=1,2,\\ &{\mathcal K}_{\mu_1^2 ,2s\mu_1}(\varphi) <0, ~{\mathcal K}_{\mu_2^2 ,2s\mu_2}(\varphi) >0. \end{align*} We may assume that $0<\mu_1<\mu_2$. Here we set the function $f_s:{\mathbb R}^+\to{\mathbb R}$ by \begin{align} \leftabel{eq:5.5} f_s(\mu):=&{\mathcal S}_{\mu^2,2s\mu}(\varphi) -d(\mu^2, 2s\mu)\\ =&{\mathcal E}(\varphi)+\frac{\mu^2}{2}\Bigl( {\mathcal M}(\varphi) -2d(1,2s)\Bigr) +s\mu{\mathcal P}(\varphi). \notag \end{align} From \eqref{eq:5.4} we have ${\mathcal M}(\varphi) -2d(1,2s)>0$, which yields that the function $f_s$ is strictly convex. In particular $J_s:=\{\mu>0 : f_s(\mu)<0\}$ is an open interval which contains $\mu_1$ and $\mu_2$. From the explicit formula of ${\mathcal K}_{\mu^2,2s\mu}(\varphi)$ (see \eqref{eq:4.3}), there exists a unique $\mu_0\in (\mu_1, \mu_2)$ such that ${\mathcal K}_{\mu_0^2,2s\mu_0}(\varphi)=0$. Since $\mu_0\in J_s$, in conclusion we deduce that there exists $\mu_0 >0$ such that \begin{align*} {\mathcal S}_{\mu_0^2,2s\mu_0}(\varphi) <d(\mu_0^2, 2s\mu_0), ~{\mathcal K}_{\mu_0^2,2s\mu_0}(\varphi)=0. \end{align*} However, from Proposition {\rightm Re}f{prop:4.1} this yields that $\varphi =0$, which is absurd. Therefore, ${\mathscr A}^+_s$ and ${\mathscr A}^-_s$ are mutually disjoint on $\{\varphi\in H^1({\mathbb R}): {\mathcal M}(\varphi)> M^*\}$. Next we consider the case ${\mathcal M}(\varphi)=M^*$. We only consider the case $b\geq 0$ since the case $-3/16<b<0$ is treated similarly. From \eqref{eq:5.4} we obtain that ${\mathcal M}(\varphi) >2d(1,2s)$ for any $s\in (-1,1]$ but $s\neq s^*$. Hence, when $s\neq s^*$, one can use the argument above in the same way. When $s=s^*$, the function \eqref{eq:5.5} in this case is equal to \begin{align*} f_{s^*}(\mu)=&{\mathcal S}_{\mu^2,2s^*\mu}(\varphi) -d(\mu^2, 2s^*\mu) ={\mathcal E}(\varphi) +s^*\mu{\mathcal P}(\varphi). \end{align*} From this formula, we deduce that $J_{s^*}$ is an open interval if it is not empty. Hence the argument above still holds in this case. This completes the proof of (i). (ii) Assume that ${\mathcal M} (\varphi)<M^*$, or ${\mathcal M} (\varphi)=M^*$ and ${\mathcal P} (\varphi)<0$. We note that for any $\varphi\in H^1({\mathbb R})\setminus\{ 0\}$ there exists large $\omega >0$ such that \begin{align} \leftabel{eq:5.6} {\mathcal K}_{\omega ,2s\sqrt{\omega}}(\varphi) &= \| \partial_x \varphi\|_{L^2}^2 + \omega\| \varphi\|_{L^2}^2 \\ &\quad +s \sqrt{\omega}\left( 2\rightbra[i\partial_x \varphi, \varphi] +\| \varphi\|_{L^4}^4 \right) -\frac{3}{16}\gamma\| \varphi\|_{L^6}^6> 0, \notag \end{align} where $\omega$ depends on $s$ and $\varphi$. We also note that \begin{align} \leftabel{eq:5.7} &{\mathcal S}_{\omega ,2s\sqrt{\omega}}(\varphi) < d(\omega ,2s\sqrt{\omega}) \Leftrightarrow {\mathcal E} (\varphi)+s\sqrt{\omega} {\mathcal P} (\varphi) < \frac{ \omega}{2} \bigl( 2d(1,2s)- {\mathcal M} (\varphi) \bigr) \end{align} for any $s\in (-1,1]$. When $b\geq 0$, if we set $s=s^*$, the last inequality in \eqref{eq:5.7} holds for large $\omega>0$ from \eqref{eq:5.2} and \eqref{eq:5.4}. Combined with ({\rightm Re}f{eq:5.6}), we deduce that $\varphi\in{\mathscr A}_{s^*}^+$. When $-3/16<b\lefteq 0$, if we set $s=1$, $\varphi\in{\mathscr A}_{1}^+$ is proved in the same way. (iii)(a) From the definition of ${\mathcal S}_{\omega ,c}$, we have \begin{align*} {\mathcal S}_{\mu^2,2\mu}(e^{i\mu x}\psi )&=\frac{1}{2}{\mathcal L}_{\mu^2,2\mu}(e^{i\mu x}\psi) +\frac{\mu}{4}\| \psi \|_{L^4}^4-\frac{\gamma}{32}\| \psi\|_{L^6}^6\\ &=\frac{1}{2}\| \partial_x \psi\|_{L^2}^2 +\frac{\mu}{4}\| \psi\|_{L^4}^4-\frac{\gamma}{32}\| \psi \|_{L^6}^6. \end{align*} Since $d(\mu^2 ,2\mu)=\mu^2 d(1,2)$ and $d(1,2)>0$, we deduce that \begin{align*} {\mathcal S}_{\mu^2,2\mu}(e^{i\mu x}\psi ) < d(\mu^2 ,2\mu) \end{align*} for large $\mu>0$. Similarly, we have \begin{align*} {\mathcal K}_{\mu^2,2\mu}(e^{i\mu x}\psi) &={\mathcal L}_{\mu^2,2\mu}(e^{i\mu x}\psi) +\mu \| \psi\|_{L^4}^4-\frac{3}{16}\gamma\| \psi\|_{L^6}^6\\ &=\|\partial_x\psi \|_{L^2}^2 +\mu\| \psi \|_{L^4}^4-\frac{3}{16}\gamma\| \psi\|_{L^6}^6 > 0 \end{align*} for large $\mu>0$. This yields that $e^{i\mu x}\psi\in{\mathscr A}^+_1$. (b) First we note that \begin{align*} {\mathcal S}_{\mu^2,2s\mu}(e^{is\mu x}\psi )&=\frac{1}{2}\| \partial_x \psi\|_{L^2}^2 +\frac{\mu^2}{2} \left( 1-s^2\right) \| \psi\|_{L^2}^2+ \frac{s\mu}{4}\| \psi\|_{L^4}^4-\frac{\gamma}{32}\| \psi \|_{L^6}^6, \\ {\mathcal K}_{\mu^2,2s\mu}(e^{is\mu x}\psi )&=\| \partial_x \psi\|_{L^2}^2 +\mu^2 \left( 1-s^2\right) \| \psi\|_{L^2}^2+ s\mu\| \psi\|_{L^4}^4-\frac{3}{16}\gamma\| \psi \|_{L^6}^6 \end{align*} for any $s\in (-1,1]$. We fix large $\mu >0$ such that \begin{align*} {\mathcal S}_{\mu^2,-2\mu}(e^{-i\mu x}\psi )&=\frac{1}{2}\| \partial_x \psi\|_{L^2}^2 -\frac{\mu}{4}\| \psi\|_{L^4}^4-\frac{\gamma}{32}\| \psi \|_{L^6}^6<0,\\ {\mathcal K}_{\mu^2,-2\mu}(e^{-i\mu x}\psi )&=\| \partial_x \psi\|_{L^2}^2 -\mu\| \psi\|_{L^4}^4-\frac{3}{16}\gamma\| \psi \|_{L^6}^6<0. \end{align*} We note that \begin{align*} {\mathcal S}_{\mu^2,-2\mu}(e^{is\mu x}\psi )&=\leftim_{s\downarrow -1}{\mathcal S}_{\mu^2,2s\mu}(e^{is\mu x}\psi ),\\ {\mathcal K}_{\mu^2,-2\mu}(e^{-i\mu x}\psi )&=\leftim_{s\downarrow -1}{\mathcal K}_{\mu^2,2s\mu}(e^{is\mu x}\psi ), \end{align*} and $\leftim_{s\downarrow -1}d(1,2s)=0$. Therefore, there exists small $\varepsilon >0$ such that for any $s\in (-1,-1+\varepsilon)$ we have \begin{align*} {\mathcal S}_{\mu^2,2s\mu}(e^{is\mu x}\psi ) <d(\mu^2,2s\mu),~ {\mathcal K}_{\mu^2,2s\mu}(e^{is\mu x}\psi )<0. \end{align*} This yields that $e^{is\mu x}\psi\in{\mathscr A}^-_{s}$ for $s\in (-1,-1+\varepsilon)$. (iv) Assume ${\mathcal E}(\varphi)<0$. We note that the functional ${\mathcal K}_{\omega ,c}$ is rewritten as \begin{align} \leftabel{eq:5.8} {\mathcal K}_{\omega ,c}(\varphi) =6{\mathcal E}(\varphi) -2\| \partial_x\varphi\|_{L^2}^2+\omega\|\varphi\|_{L^2}^2 +c\rightbra[i\partial_x\varphi, \varphi] +\frac{c}{2}\| \varphi\|_{L^4}^4 . \end{align} From \eqref{eq:5.7} and this formula, we deduce that for each $s\in (-1,1]$ there exists small $\omega >0$ such that \begin{align*} {\mathcal S}_{\omega ,2s\sqrt{\omega}}(\varphi) < d(\omega ,2s\sqrt{\omega}) , ~ {\mathcal K}_{\omega ,2s\sqrt{\omega}}(\varphi)<0. \end{align*} This yields that $\varphi\in\bigcap_{-1<s\lefteq 1}{\mathscr A}^-_s$. If we assume further that ${\mathcal M} (\varphi)\geq M^*$, it follows from (i) that $\varphi\notin\bigcup_{-1< s\lefteq 1}{\mathscr A}_s^+$. (v) Assume by contradiction that $\varphi\in\bigcup_{0\lefteq s\lefteq 1}{\mathscr A}_s$ under the assumption \begin{align*} {\mathcal E} (\varphi) \geq0, {\mathcal M}(\varphi) \geq M^*~\text{and}~{\mathcal P} (\varphi)\geq 0. \end{align*} Then, there exist $s_0\in [0,1]$ and $\omega_0 >0$ such that ${\mathcal S}_{\omega_0 ,2s_0\sqrt{\omega_0}}(\varphi) < d(\omega_0 ,2s_0\sqrt{\omega_0})$. This is equivalent that \begin{align*} {\mathcal E} (\varphi)+\frac{ \omega_0}{2} \bigl( {\mathcal M} (\varphi) -2d(1,2s_0) \bigr) +s_0\sqrt{\omega_0} {\mathcal P} (\varphi) < 0. \end{align*} But this is absurd, since ${\mathcal M} (\varphi) -2d(1,2s_0)\geq 0$ from \eqref{eq:5.4}. In the same way one can prove that \begin{align*} {\mathcal E} (\varphi) \geq0, {\mathcal M}(\varphi) \geq M^*~\text{and}~{\mathcal P} (\varphi)\lefteq 0 \Longrightarrow\varphi\notin \bigcup_{-1< s\lefteq 0}{\mathscr A}_s. \end{align*} (vi)(a) Let $b\geq 0$. Assume that ${\mathcal M}(\varphi)=M^*$, ${\mathcal E}(\varphi)\lefteq 0$ and ${\mathcal P}(\varphi)\lefteq 0$ except for the case ${\mathcal E}(\varphi)={\mathcal P}(\varphi)=0$. We note that the function $f_{s^*}$ defined by \eqref{eq:5.5} has the following formula: \begin{align*} f_{s^*}(\mu) ={\mathcal E}(\varphi) +s^*\mu{\mathcal P}(\varphi). \end{align*} From the assumption we note that $\{ \mu>0:f_{s^*}(\mu)<0\}={\mathbb R}^+$. From the formulae \eqref{eq:5.6} and \eqref{eq:5.8}, we deduce that \begin{align} \leftabel{eq:5.9} \begin{array}{ll} {\mathcal K}_{\mu^2 ,2s^*\mu}(\varphi) >0 ~&\text{for large}~\mu>0, \\ {\mathcal K}_{\mu^2 ,2s^*\mu}(\varphi) <0 ~&\text{for small}~\mu>0. \end{array} \end{align} Therefore, there exists $\mu_0 >0$ such that \begin{align*} {\mathcal S}_{\mu_0^2,2s^*\mu_0}(\varphi) <d(\mu_0^2, 2s^*\mu_0), ~{\mathcal K}_{\mu_0^2,2s^*\mu_0}(\varphi)=0. \end{align*} However, from Proposition {\rightm Re}f{prop:4.1} this yields that $\varphi =0$, which is absurd. Now we consider the case ${\mathcal M}(\varphi)=M^*$ and ${\mathcal E}(\varphi)={\mathcal P}(\varphi)=0$. In this case we have $f_{s^*}\equiv 0$, which is equivalent that ${\mathcal S}_{\mu^2,2s^*\mu}(\varphi) =d(\mu^2, 2s^*\mu)$ for any $\mu>0$. Since the statement \eqref{eq:5.9} still holds in this case, we deduce that there exists a unique $\mu_0 >0$ such that \begin{align*} {\mathcal S}_{\mu_0^2,2s^*\mu_0}(\varphi) =d(\mu_0^2, 2s^*\mu_0), ~{\mathcal K}_{\mu_0^2,2s^*\mu_0}(\varphi)=0. \end{align*} From Proposition {\rightm Re}f{prop:4.1} again, there exist $\theta , y\in{\mathbb R}$ such that $\varphi =e^{i\theta}\varphi_{\mu_0^2 ,2s^*\mu_0 }(\cdot -y)$. This completes the proof of (vi-a). (b) In the same way as in the proof of (vi-a), we deduce that there exist no $\varphi\in H^1({\mathbb R})$ such that ${\mathcal M}(\varphi)=M^*$, ${\mathcal E}(\varphi)\lefteq 0$ and ${\mathcal P}(\varphi)\lefteq 0$ except for the case ${\mathcal E}(\varphi)={\mathcal P}(\varphi)=0$. We consider the case ${\mathcal M}(\varphi)=M^*$ and ${\mathcal E}(\varphi)={\mathcal P}(\varphi)=0$. Similarly, one can prove that there exists a unique $\mu_0 >0$ such that \begin{align*} {\mathcal S}_{\mu_0^2,2\mu_0}(\varphi) =d(\mu_0^2, 2\mu_0), ~{\mathcal K}_{\mu_0^2,2\mu_0}(\varphi)=0. \end{align*} From Proposition {\rightm Re}f{prop:4.1}, there exist $\theta , y\in{\mathbb R}$ such that $\varphi =e^{i\theta}\varphi_{\mu_0^2 ,2\mu_0 }(\cdot -y)$. But this is a contradiction, since ${\mathcal P}(\varphi_{1,2})>0$ from Proposition {\rightm Re}f{prop:1.3}. This completes the proof. \end{proof} In the case $b=-3/16$, we have the following result. \begin{proposition} \leftabel{prop:5.2} Let $b=-3/16$. Then we have \begin{align*} \bigcup_{-1< s<0}{\mathscr A}_s =\bigcup_{-1< s<0}{\mathscr A}_s^+=H^1({\mathbb R}) . \end{align*} \end{proposition} \begin{proof} Given $\varphi\in H^1({\mathbb R})$. From \eqref{eq:5.3} and Proposition {\rightm Re}f{prop:2.4} we have \begin{align*} d(1,2s) =\frac{1-s^2}{3}M(\phi_{1,2s}) \end{align*} for $s\in(-1,0)$. It follows from Proposition {\rightm Re}f{prop:2.2} that $d(1,2s)\to\infty$ as $s\to 0-$. Hence, for $\varphi\in H^1({\mathbb R})$ one can take $s_0\in (-1,0)$ such that \begin{align} \leftabel{eq:5.10} 2\,d(1,2s_0)- {\mathcal M} (\varphi) >0. \end{align} We note that \begin{align*} {\mathcal S}_{\omega ,2s_0\sqrt{\omega}}(\varphi) < d(\omega ,2s_0\sqrt{\omega}) \Leftrightarrow {\mathcal E} (\varphi)+{s_0\sqrt{\omega}}{\mathcal P} (\varphi) < \frac{\omega}{2} \left( 2\,d(1,2s_0)- {\mathcal M} (\varphi) \right) . \end{align*} By \eqref{eq:5.10} the last inequality holds for large $\omega >0$. Combined with \eqref{eq:5.6}, we deduce that $\varphi\in{\mathscr A}_{s_0}^+$. This completes the proof. \end{proof} \section{Potential well theory for the Hamiltonian form} \leftabel{sec:6} When $b\geq 0$ one can give a variational characterization of solitons to the equation \eqref{eq:1.1} on the Nehari manifold. Based on this variational characterization one can establish potential well theory for \eqref{eq:1.1} similarly as in the case of \eqref{ME}. We recall that the equation \eqref{eq:1.1} has Hamiltonian structure \eqref{eq:1.3} which is useful when one studies problems of solitons and related topics. Hence it would be worthwhile to restate potential well theory for the Hamiltonian form. Here we only consider the case of \eqref{DNLS} for simplicity. After we organize potential well theory for \eqref{DNLS}, we give a proof of Theorem {\rightm Re}f{thm:1.12}. Let us consider the case $b=0$. We define the functional by $K_{\omega ,c}(\varphi):=\left.\frac{d}{d\leftambda}S_{\omega ,c}(\leftambda\varphi)\right|_{\leftambda =1}$, which has the following explicit formula: \begin{align} \leftabel{eq:6.1} K_{\omega ,c}(\varphi) =\| \partial_x\varphi\|_{L^2}^2+\omega\|\varphi\|_{L^2}^2 +c\rightbra[i\partial_x\varphi, \varphi] -\rightbra[ i|\varphi|^2\partial_x\varphi, \varphi]. \end{align} We consider the following subsets of the energy space: \begin{align*} {\mathscr K}_{\omega ,c}:=& \left\{ \varphi\in H^1({\mathbb R} ): S_{\omega, c}(\varphi ) < S_{\omega ,c}(\varphi_{\omega, c})\right\}, \\ {\mathscr K}_{\omega, c}^+ :=&\left\{ \varphi\in{\mathscr K}_{\omega, c} :K_{\omega, c}(\varphi) \geq 0 \right\}, {\mathscr K}_{\omega, c}^- :=\left\{ \varphi\in{\mathscr K}_{\omega, c} :K_{\omega ,c}(\varphi) < 0\right\},\\ {\mathscr K}_{s}:=& \bigcup_{ \substack{\omega>0 } } {\mathscr K}_{\omega ,2s\sqrt{\omega}},~ {\mathscr K}_{s}^{\pm}:= \bigcup_{ \substack{\omega>0 } } {\mathscr K}_{\omega ,2s\sqrt{\omega}}^{\pm} \quad\text{for}~s\in (-1,1]. \end{align*} Now we can rewrite Theorem {\rightm Re}f{thm:1.7} as the following claim for \eqref{DNLS}: \begin{theorem} \leftabel{thm:6.1} Let $(\omega ,c)$ satisfy $-2\sqrt{\omega}<c\lefteq 2\sqrt{\omega}$. Then, each of ${\mathscr K}_{\omega ,c}^{+}$ and ${\mathscr K}_{\omega ,c}^{-}$ is invariant under the flow of \eqref{DNLS}. If $u_0 \in{\mathscr K}_{\omega ,c}^+$, then the $H^1({\mathbb R})$-solution $u$ of \eqref{DNLS} with $u(0)=u_0$ exists globally both forward and backward in time, and satisfies the following uniform estimate\textup{:} \begin{align} \leftabel{eq:6.2} \| \partial_x v\|_{L^{\infty}({\mathbb R} ,L^2)}^2 \lefteq 8S_{\omega ,c}(u_0)+\frac{c^2}{2} M (u_0). \end{align} Moreover the following statements hold\textup{:} \begin{enumerate}[\rightm (i)] \setlength{\itemsep}{3pt} \item For each $s\in (-1,1]$, ${\mathscr K}^+_{s}$ and ${\mathscr K}^-_s$ have no elements in common on the set $\{ \varphi\in H^1({\mathbb R}) : M(\varphi ) \geq 4\pi\}$. \item If $M (\varphi)<4\pi$, or $M (\varphi)=4\pi$ and $P (\varphi)<0$, then $\varphi\in{\mathscr K}^+_{1}$. \item For given $\psi\in H^1({\mathbb R})\setminus\{ 0\}$ the following properties hold\textup{:} \begin{enumerate}[\rightm (a)] \item There exists $\mu_0 =\mu_0 (\psi)>0$ such that if $\mu\geq\mu_0$, then $e^{i\mu x}\psi\in {\mathscr K}^+_1$. \item There exist $\varepsilon\in (0,1)$ and large $\mu >0$ such that $e^{-i(1-\varepsilon )\mu x}\psi\in {\mathscr K}^-_{-(1-\varepsilon)}$, where $\varepsilon$ and $\mu$ depend on $\psi$. \end{enumerate} \item Assume $E(\varphi)<0$. Then $\varphi\in\bigcap_{-1<s\lefteq 1}{\mathscr K}^-_s$. In particular, if $M(\varphi)\geq 4\pi$, then $\varphi\not\in\bigcup_{-1< s\lefteq 1}{\mathscr K}_s^+$. \item Assume $E(\varphi) \geq 0$ and $M (\varphi) \geq 4\pi$. If $P(\varphi) \geq 0\,(\text{resp.}\,P(\varphi)\lefteq 0)$, then $\varphi\not\in\bigcup_{0\lefteq s\lefteq 1}{\mathscr K}_s\,(\text{resp.}\,\varphi\not\in\bigcup_{-1<s\lefteq 0}{\mathscr K}_s)$. In particular, if $P(\varphi)=0$, then $\varphi\not\in\bigcup_{-1< s\lefteq 1}{\mathscr K}_s$. \item Assume $M (\varphi)=4\pi$. $E(\varphi)=P(\varphi)=0$ if and only if there exist $\theta , y\in{\mathbb R}$ and $\omega >0$ such that $\varphi =e^{i\theta}\phi_{\omega ,2\sqrt{\omega} }(\cdot -y)$. Moreover, there exists no $\varphi\in H^1({\mathbb R})$ such that $E(\varphi)<0$ and $P(\varphi)\lefteq 0$, or $E(\varphi)\lefteq 0$ and $P(\varphi) <0$. \end{enumerate} \end{theorem} Theorem {\rightm Re}f{thm:6.1} characterizes $4\pi$-mass condition for \eqref{DNLS} from the viewpoint of potential well theory. We note that algebraic solitons give the boundary of both ${\mathscr K}^+_1$ and ${\mathscr K}^-_1$. This is a notable property of algebraic solitons, which is analogous to the one of standing waves for \eqref{NLS}. We also note that Theorem {\rightm Re}f{thm:1.12} gives another interesting property of algebraic solitons. For the proof of Theorem {\rightm Re}f{thm:1.12}, we use the following lemma on the concentration compactness, which is corresponding to Proposition {\rightm Re}f{prop:4.7}. \begin{proposition}[\cite{CO06, FHI17}] \leftabel{prop:6.2} Let $(\omega ,c)$ satisfy $-2\sqrt{\omega}<c\lefteq 2\sqrt{\omega}$. Let $X_{\omega ,c}$ be defined by \eqref{eq:4.2}. If a sequence $ \thickmuskip=3mu \{\varphi_n\}\subset X_{\omega,c}$ satisfies \begin{align*} S_{\omega ,c}(\varphi_n ) \to d(\omega ,c)~\text{and}~K_{\omega ,c}(\varphi_n ) \to 0~\text{as}~n\to\infty , \end{align*} then there exist a sequence $\{ y_n\}\subset{\mathbb R}$ and $\theta_0,y_0\in{\mathbb R}$ such that $\{ \varphi_n(\cdot -y_n)\}$ has a subsequence that converges to $e^{i\theta_0}\phi_{\omega ,c}(\cdot -y_0)$ strongly in $X_{\omega ,c}$. \end{proposition} \begin{proof}[Proof of Theorem {\rightm Re}f{thm:1.12}] It is enough to prove that for any sequence ${t_n}$ such that $t_n\to T^*$, there exist a subsequence (denote it again by $\{ t_n\}$), and sequences $y(t_n)$ and $\theta(t_n)$ such that \begin{align} \leftabel{eq:6.3} e^{i\theta(t_n)}u_{\leftambda (t_n)}(t_n, \cdot +y(t_n))\to \phi_{1,2}~\text{in}~H^1({\mathbb R})~\text{as}~t_n\to T^*, \end{align} where $u_{\leftambda(t)}(t,x) :=\leftambda(t)^{1/2}u(t, \leftambda(t)x)$. Since $\leftambda (t_n)\to 0$ as $t_n\to T^*$, we obtain that \begin{align*} E(u_{\leftambda(t_n)}(t_n))&=\leftambda (t_n)^2E(u_0) \to 0,\\ P(u_{\leftambda(t_n)}(t_n))&=\leftambda (t_n)P(u_0) \to 0, \end{align*} and $M(u_{\leftambda(t_n)}(t_n))=M(u_0)=4\pi$ for any $n\in{\mathbb N}$. Hence we have \begin{align} \leftabel{eq:6.4} S_{1,2}(u_{\leftambda (t_n)}(t_n))&=E(u_{\leftambda(t_n)}(t_n))+\frac{1}{2}M(u_{\leftambda(t_n)}(t_n))+P(u_{\leftambda(t_n)}(t_n))\\ &\underset{t_n\to T^*}{\leftongrightarrow}\frac{1}{2}\cdot 4\pi =d(1,2). \notag \end{align} We note that the functional $K_{\omega ,c}$ is rewritten as \begin{align*} K_{\omega ,c}(\varphi) =-\|\partial_x\varphi\|_{L^2}^2 +4E(\varphi)+\omega M(\varphi)+cP(\varphi). \end{align*} Since $E(\phi_{1,2})=P(\phi_{1,2})=0$, we have \begin{align*} 0=K_{1 ,2}(\phi_{1,2})=-\|\partial_x\phi_{1,2}\|_{L^2}^2 +M(\phi_{1,2}). \end{align*} Therefore, we deduce that \begin{align} \leftabel{eq:6.5} K_{1,2}(u_{\leftambda (t_n)}(t_n))&=-\leftambda(t_n)^2\| \partial_xu(t_n) \|_{L^2}^2+ 4E(u_{\leftambda(t_n)}(t_n))\\ &\quad +M(u_{\leftambda(t_n)}(t_n) )+2P(u_{\leftambda(t_n)}(t_n)) \notag\\ &\underset{t_n\to T^*}{\leftongrightarrow} -\|\partial_x\phi_{1,2}\|_{L^2}^2 +M(\phi_{1,2})=0. \notag \end{align} Therefore, by \eqref{eq:6.4}, \eqref{eq:6.5} and Proposition {\rightm Re}f{prop:6.2}, there exist a subsequence of $\{ t_n\}$, $\theta (t_n)$ and $y(t_n)$ such that \begin{align*} g(t_n):=e^{i\theta(t_n)}u_{\leftambda (t_n)}(t_n, \cdot +y(t_n))\underset{t_n\to T^*}{\leftongrightarrow} \phi_{1,2}~\text{in}~X_{1,2}. \end{align*} In particular we have \begin{align} \leftabel{eq:6.6} e^{-ix}g(t_n)\to e^{-ix}\phi_{1,2}~\text{in}~\dot{H}^1({\mathbb R}), \\ \leftabel{eq:6.7} e^{-ix}g(t_n)\rightightharpoonup e^{-ix}\phi_{1,2}~\text{weakly in}~L^2({\mathbb R}). \end{align} From the mass conservation, we note that \begin{align} \leftabel{eq:6.8} M(g(t_n))=M(u_{\leftambda_n}(t_n))=4\pi=M(\phi_{1,2}) \end{align} for any $n\in{\mathbb N}$. Combined with \eqref{eq:6.7}, we obtain that \begin{align} \leftabel{eq:6.9} e^{-ix}g(t_n)\to e^{-ix}\phi_{1,2}~\text{strongly in}~L^2({\mathbb R}). \end{align} From \eqref{eq:6.6} and \eqref{eq:6.9}, we obtain \eqref{eq:6.3}. This completes the proof. \end{proof} \section*{Acknowledgments} The results of this paper were mostly obtained when the author was a PhD student at Waseda University. The author would like to thank his thesis adviser Tohru Ozawa for constant encouragements. The author is also grateful to Noriyoshi Fukaya and Takahisa Inui for useful discussions, and to Nobu Kishimoto, Kenji Nakanishi, and Yoshio Tsutsumi for helpful comments. This work was supported by JSPS KAKENHI Grant Numbers JP17J05828, JP19J01504, and Top Global University Project, Waseda University. \end{document}

\begin{document} \title{Robust Positivity Problems for Linear Recurrence Sequences} \begin{abstract} Linear Recurrence Sequences (LRS) are a fundamental mathematical primitive for a plethora of applications such as the verification of probabilistic systems, model checking, computational biology, and economics. Positivity (are all terms of the given LRS non-negative?) and Ultimate Positivity (are all but finitely many terms of the given LRS non-negative?) are important open number-theoretic decision problems. Recently, the robust versions of these problems, that ask whether the LRS is (Ultimately) Positive despite small perturbations to its initialisation, have gained attention as a means to model the imprecision that arises in practical settings. However, the state of the art is ill-equipped to reason about imprecision when its extent is explicitly specified. In this paper, we consider Robust Positivity and Ultimate Positivity problems where the neighbourhood of the initialisation, expressed in a natural and general format, is also part of the input. We contribute by proving sharp decidability results: decision procedures at orders our techniques are unable to handle for general LRS would entail significant number-theoretic breakthroughs. \end{abstract} \section{Introduction} \label{section:intro} A real Linear Recurrence Sequence (LRS) of order $\kappa$ is an infinite sequence of real numbers $(u_0, u_1, u_2, \dots)$ having the following property: there exist $\kappa$ real constants $a_{0}, \dots, a_{\kappa-1}$, with $a_0 \ne 0$ such that for all $n \ge 0$: \begin{equation} u_{n+\kappa} = a_{\kappa-1}u_{n+\kappa-1} + \dots a_0 u_n. \end{equation} The constants $a_0, \dots, a_{\kappa-1}$ define the linear recurrence relation $\mathbf{a}$; they are also associated with the characteristic polynomial $ X^\kappa - a_{\kappa-1}X^{\kappa-1} - \dots - a_1X - a_0. $ The initial terms $u_0, \dots, u_{\kappa-1}$ are collectively denoted as the initialisation $\mathbf{c}$. An LRS is uniquely specified by $(\mathbf{a}, \mathbf{c})$. The best-known example is the Fibonacci sequence $\seq{0, 1, 1, 2, 3, 5, 8, \dots}$, satisfying the recurrence relation $u_{n+2} = u_{n+1} + u_n$: it is named after Leonardo of Pisa, who used it to model the population growth of rabbits. LRS have been extensively studied, and found several mathematical and scientific applications since. The monograph of Everest \textit{et al.} \cite{Everest2003RecurrenceS} is a comprehensive treatise on the mathematical aspects of Recurrence Sequences. Important number-theoretic decision problems for Linear Recurrence Sequences include Positivity (is $u_n \ge 0$ for all $n$?), Ultimate Positivity (is $u_n \ge 0$ for all but finitely many $n$?) and the closely related Skolem Problem (is $u_n = 0$ for some $n$?). We remark that a Positive LRS is necessarily Ultimately Positive. These problems have applications in software verification, probabilistic model checking, discrete dynamic systems, theoretical biology, and economics. Decidability has been open for decades, with breakthroughs in restricted settings: Mignotte \textit{et al.\ }\cite{mignotte} and Vereshchagin \cite{vereshchagin} independently proved the Skolem Problem to be decidable up to order $4$. Ouaknine and Worrell \cite{joeljames3} showed Positivity and Ultimate Positivity are decidable up to order $5$ but number-theoretically hard at order $6$. For \textit{simple} LRS (those whose characteristic polynomials have no repeated roots), they showed that Positivity is decidable up to order $9$ \cite{ouaknine2014positivity} and Ultimate Positivity is decidable at all orders \cite{ouaknine2014ultimate}. These results were originally proven for LRS specified by \textit{rational} recurrences and initialisations, but can be generalised to real algebraic input as well. In this paper, we focus on Positivity and Ultimate Positivity for \textbf{sequences defined by real algebraic input}. In contrast, the \textit{uninitialised} variants of these problems are far more tractable. Braverman \cite{Braverman06} and Tiwari \cite{Tiwari04} consider whether \textit{every} possible initialisation keeps the sequence Positive, and decide so in $\mathsf{PTIME}$. More recently, this result has been extended to processes with choices \cite{AGV18}. We argue that practical applications need a middle ground: recurrence relations that arise in practice need to be contextualised by actual instances of sequences; however, considering \textit{precise} initialisations does not account for inherently imprecise real world measurements, and the requirement of safety margins. We thus study robust variants: given a recurrence and an initialisation, do all initialisations in a neighbourhood satisfy (Ultimate) Positivity? \paragraph*{Related Work} In this paper, we focus on the neighbourhood-of-initialisation notion of robustness, which was first introduced in \cite{originalstacs}, and more comprehensively treated in \cite{originalarxiv}. Works with a more control-theoretic flavour include \cite{rounding20}, which allows for rounding at every step before applying the recurrence; in the same vein, \cite{pseudo21} allows for $\varepsilon$-disturbances at every step of the sequence. Our notion of robustness has been considered in \cite{originalstacs,originalarxiv,pseudo21}, however, these works primarily concern themselves with simply deciding whether there \textit{exists} a neighbourhood around the given point that satisfies Positivity, or whether there \textit{exists} a tolerance $\varepsilon$ such that the sequence avoids a region despite $\varepsilon$-disturbances at every step. Although they do identify that robust problems are hard when the neighbourhood is given as input, in the absence of decidability results, their hardness results are not sharp. There are, of course, broader approaches to model and reason about imprecision: \cite{N21} considers a model of computation that can take arbitrary real numbers as input, thereby allowing imprecision in both the initialisation and the recurrence. Even in this setting, the focus is on whether the decision is locally constant in \textit{some} neighbourhood of the given instance of the Positivity Problem, as opposed to whether the decision holds for an entire \textit{given} neighbourhood. \paragraph*{Our contribution} We address the gap in the robustness state of the art by exploring the frontiers of decidability when the neighbourhood is given as input. Concretely, our input consists of a linear recurrence relation $\mathbf{a}$ and a neighbourhood of initialisations centred around $\mathbf{c}$. Our problem is to decide whether all initialisations in the \textbf{given} neighbourhood result in (Ultimately) Positive sequences. When neighbourhoods are expressly given as input, their geometry plays a critical role in the decision procedure. The notion of neighbourhoods that we primarily focus on is based on the $\ell^2$-norm. We seek to slightly generalise the Euclidean $\varepsilon$-ball. More specifically, we use the Mahalanobis distance to define neighbourhoods. Our parameter is the positive definite matrix $\mathbf{S}$, and the neighbourhood of $\mathbf{c}$ it specifies is the set of all points $\mathbf{c'} \in \mathbb{R}^{\kappa}$ such that $(\mathbf{c'} - \mathbf{c})^T\mathbf{S}(\mathbf{c'} - \mathbf{c}) \le 1$. The size of neighbourhoods is usually parametrised by an $\varepsilon$: in our case, we can account for it by simply scaling $\mathbf{S}$. In the statistical context, $\mathbf{S}$ is the inverse of a covariance matrix; and thus, our formulation is a rather natural way of capturing noise and measurement errors in the input, whose components may often be correlated. Our \textbf{novelty}, to the best of our knowledge, lies in identifying a general and practical way of explicitly specifying neighbourhoods, and establishing the \textbf{first decidability results} in such a setting, albeit at low orders or subject to spectral constraints. As first discussed in \cite[Section 5]{joeljames3}, solving decision problems on Linear Recurrence Sequences in full generality is an endeavour fraught with number-theoretic hardness. Decision procedures for Positivity problems for LRS of higher order would allow number theorists to compute properties of irrational numbers that are considered inaccessible to contemporary techniques. These include the Diophantine approximation type, which intuitively describes the quality of the ``best'' rational approximation of a given irrational number, and the Lagrange constant, which intuitively describes how well increasingly precise rational approximations of a given irrational number converge. We justify the inability of our techniques to handle LRS of higher orders by \textbf{reducing the computation of Diophantine approximation types and Lagrange constants} to robust Positivity problems for LRS of lower orders than ever before. \begin{table}[H] \begin{tabular}{|l|r|r|r|} \hline & \multicolumn{2}{c|}{\bf Decidability Proof} & \\ \hline \textbf{Problem:} $\mathbf{S}$-\textbf{Robust}& \textbf{General}& {\bf Simple} &{\bf Hardness} \\ \hline Positivity & order $\le 4$ & order $\le 5$ &Diophantine hard at order 5\\ Uniform Ultimate Positivity & order $\le 4$ & \textbf{all orders} & Lagrange hard at order 5 \\ Non-uniform Ultimate Positivity & order $\le 4$ & order $\le 4$ & \cite{joeljames3,originalarxiv}: Lagrange hard at order 6 \\ \hline \end{tabular} \caption{Main results, summarised. The distinction between uniform and non-uniform refers to whether the threshold index for certifying Ultimate Positivity must be common for the entire neighbourhood.} \end{table} \paragraph*{Structure of the paper} The exponential polynomial closed form is an invaluable tool in the study of LRS, and we devote \S\ref{section:solspace} to its exposition. This equips us to introduce our Robust Positivity Problems and intuit their decidability proofs in \S\ref{section:problems}. Linear Recurrences and Diophantine Approximation are intrinsically connected: number-theoretic results form the basis of decision procedures; open problems are a yardstick for hardness reductions. We survey the number theory relevant to us in \S\ref{section:diophantine}. We then prove our decidability results in the technical \S\ref{section:decidability} and \S\ref{section:decidability2}, and present our hardness reduction in \S\ref{section:hardness}. We provide concluding perspective in \S\ref{section:perspective}. We refer the reader to Appendix \ref{appendix:prelims} for a summary of the standard notation and prerequisites we use. \section{The exponential polynomial closed form} \label{section:solspace} We begin by discussing the exponential polynomial closed form, a perspective that is routinely leveraged to study the behaviour of Linear Recurrence Sequences. Simple LRS (no repeated characteristic roots) have the closed form \begin{equation} \label{eq:exppoly} u_n = \sum_j w_j \rho_j^n + \sum_j (z_j \gamma_j^n + \bar{z_j}\bar{\gamma_j}^n) \end{equation} where each $\rho_j, \gamma_j, \bar{\gamma_j}$ are distinct roots of the characteristic polynomial. By straightforward arithmetic on the above expression, we can see that if $(u_n)_{n\in \mathbb{N}}, (v_n)_{n\in \mathbb{N}}$ are simple LRS with sets of characteristic roots $U$ and $V$ respectively, then \begin{itemize} \item $r_n = u_n + v_n$ is a simple LRS, whose set of roots is $U \cup V$. \item $r_n = u_n \cdot v_n$ is a simple LRS, whose set of roots is $\{\gamma_1\gamma_2: \gamma_1 \in U, \gamma_2 \in V\}$. \end{itemize} In general, one can encode a linear recurrence $\mathbf{a}$ as a $\kappa \times \kappa$ companion matrix $\mathbf{A}$, and interpret the initialisation $\mathbf{c}$ as a vector. Then, $u_n$ is given by the first coordinate of $\mathbf{A}^n\mathbf{c}$, i.e. \begin{equation} \label{eq:companion} \begin{bmatrix} u_n \\ u_{n+1} \\ \vdots \\ u_{n+\kappa-1} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \dots & \vdots \\ a_0 & a_1 & a_2 & \dots & a_{\kappa-1} \end{bmatrix}^n \begin{bmatrix} u_0 \\ u_{1} \\ \vdots \\ u_{\kappa-1} \end{bmatrix}. \end{equation} Let $\mathbf{e_1}^T$ denote the row vector $\begin{bmatrix}1 & 0 & \dots & 0\end{bmatrix}$. We can thus write $u_n = \mathbf{e_1}^T\mathbf{A}^n\mathbf{c}$. It is now a standard fact that LRS have the following \textbf{real exponential polynomial} closed form \begin{equation} \label{eq:realexppoly} u_n = \left(\sum_{j=1}^{k_1}\sum_{\ell = 0}^{m_j-1} z_{j\ell}\rho_j^n n^\ell\right) + \left(\sum_{j=k_1 + 1}^{k_2} \sum_{\ell = 0}^{m_j-1} (x_{j\ell} \cos n\theta_j + y_{j\ell}\sin n\theta_j)\rho_j^n n^\ell\right) \end{equation} where $\rho_j$ (alternately, $\rho_j e^{i\theta_j}$) are roots of the characteristic polynomial defined by $\mathbf{a}$, each with multiplicity $m_j$. The coefficients $z_{j\ell}, x_{j\ell}, y_{j\ell}$ each depend linearly on $\mathbf{c}$. $u_n$ can thus be equivalently expressed as the inner (dot) product $\seq{\mathbf{p}, \mathbf{q_n}}$ where $\{\mathbf{q_n}\}_{n\in \mathbb{N}}$ is the sequence of vectors of terms that occur in the exponential polynomial expression, and $\mathbf{p}$ is the vector of corresponding coefficients. The choice of $\{\mathbf{q_n}\}_{n\in \mathbb{N}}$ can differ in ``phase'': one can replace $\cos n\theta, \sin n\theta$ by $\cos (n\theta-\varphi), \sin(n\theta-\varphi)$ for some choice of $\varphi$, and adjust the corresponding coefficients in $\mathbf{p}$ accordingly. Roots such that $|\rho_j|$ is the largest are called \textbf{dominant}. The growth rate of a term in the above expression is governed by $\rho_j^n n^\ell$. Terms with the fastest growth are called \textbf{dominant terms}, and they drive the asymptotic behaviour of the LRS. A standard, intuitive prerequisite for Ultimate Positivity is that the leading terms in the exponential polynomial expression must include one that is real and strictly positive, otherwise their dominant contribution oscillates between positive and negative. It is formalised by applying \cite[Lemma 4]{Braverman06} to the dominant terms in expression \ref{eq:realexppoly} and arguing that the contribution from the remaining terms vanishes asymptotically. \begin{proposition} \label{prop:folklore} If the characteristic polynomial has no real dominant root of maximum multiplicity, then in any full-dimensional neighbourhood of initialisations, there exists an initialisation, such that the sequence has infinitely many positive terms, and infinitely many negative terms. \end{proposition} \textbf{Henceforth, we assume that the characteristic polynomial has a real positive dominant root of maximum multiplicity, for otherwise the answer to Ultimate Positivity is trivially NO.} We define $\seq{\mathbf{p}, \mathbf{q_n}}_{dom}$ to be the normalised contribution of the dominant terms in the exponential polynomial solution. That is, if the dominant growth rate is $\rho^n n^\ell$, we pick terms with that growth rate, and divide their contribution by $\rho^n n^\ell$. For example, if $ u_n = p_1 2^n + p_2 2^n\cos (n\theta -\varphi) + p_3 2^n \sin (n\theta-\varphi) + p_4 $ then $$\seq{\mathbf{p}, \mathbf{q_n}}_{dom} = p_1 + p_2 \cos (n\theta-\varphi) + p_3 \sin (n\theta-\varphi).$$ We define $ \mu(\mathbf{c}) = \liminf_{n \in \mathbb{N}} \seq{\mathbf{p}, \mathbf{q_n}}_{dom}. $ Note that $\mu$ is an intrinsic property of the initialisation $\mathbf{c}$ and the sequence it generates, and hence is invariant under the choice of ``phase shift'' $\varphi$ while defining $\{\mathbf{q_n}\}_{n\in \mathbb{N}}$. In our example, it is $p_1 - \sqrt{p_2^2 + p_3^2}$. \section{Robust Positivity Problems} \label{section:problems} In this paper, we shall focus on defining and tackling Robust Positivity problems. Our input consists of a linear recurrence relation $\mathbf{a}$, an initialisation $\mathbf{c}$, and a positive definite matrix $\mathbf{S}$ that is used to define a neighbourhood around $\mathbf{c}$. \textbf{All input is real algebraic.}\footnotemark \footnotetext{The field of algebraic numbers $\overline{\mathbb{Q}}$ is the algebraic closure of the rationals $\mathbb{Q}$. Arithmetic and polynomial factoring over $\overline{\mathbb{Q}}$ can be performed with exact precision. We use $\mathbb{A}$ to denote the field of real algebraic numbers, and refer the reader to Appendix \ref{appendix:prelims} for an initiation to these number fields.} \begin{problem}[$\mathbf{S}$-Robust Positivity] \label{prob:rrobpos} Decide whether for all $\mathbf{c'}$ such that $(\mathbf{c'} - \mathbf{c})^T\mathbf{S}(\mathbf{c'} - \mathbf{c}) \le 1$, the LRS $(\mathbf{a}, \mathbf{c'})$ is positive. \end{problem} \begin{problem}[$\mathbf{S}$-Robust Uniform Ultimate Positivity] \label{prob:rrobuniultpos} Decide whether there exists an $N$ such that for all $\mathbf{c'}$ with $(\mathbf{c'} - \mathbf{c})^T\mathbf{S}(\mathbf{c'} - \mathbf{c}) \le 1$, the LRS $(\mathbf{a}, \mathbf{c'})$ is positive from the $N^{th}$ term onwards. \end{problem} We can switch the order in which $N$ and $\mathbf{c'}$ are quantified, and query a weaker notion of Robust Ultimate Positivity: \begin{problem}[$\mathbf{S}$-Robust Non-uniform Ultimate Positivity] \label{prob:rrobnonuniultpos} Decide whether for all $\mathbf{c'}$ with $(\mathbf{c'} - \mathbf{c})^T\mathbf{S}(\mathbf{c'} - \mathbf{c}) \le 1$ , there exists an $N$ such that the LRS $(\mathbf{a}, \mathbf{c'})$ is positive from the $N^{th}$ term onwards. \end{problem} The attentive reader might have already noticed that we depart from convention and specify neighbourhoods as \textit{closed} balls. Although \cite{originalarxiv} does not solve the problems we consider in this paper, it makes crucial observations about the geometry: for Problems \ref{prob:rrobpos} and \ref{prob:rrobuniultpos}, there is no difference between open and closed balls. On the other hand, Problem \ref{prob:rrobnonuniultpos} becomes considerably easier with open balls, and its decidability in this case is tackled in \cite{originalarxiv} itself. \subsection{Uniform Variants: The foundation} \label{section:uniformfoundation} In general, an arbitrary point $\mathbf{c'}$ is expressed as $\mathbf{c} + \mathbf{d}$, where $\mathbf{d} \in \mathcal{P}$, a full-dimensional neighbourhood symmetric about the origin. Observe equation \ref{eq:companion}. The $n^{th}$ term of the LRS is non-negative throughout the neighbourhood if and only if for all $d \in \mathcal{P}$, $ \mathbf{e_1}^T \mathbf{A}^n (\mathbf{c + d}) \ge 0. $ We can use the symmetry of $\mathcal{P}$ about the origin to rewrite the above as \begin{equation} \label{eq:illustrate} \mathbf{e_1}^T \mathbf{A}^n \mathbf{c}\ge \max_{\mathbf{d} \in \mathcal{P}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{d} \ge 0. \end{equation} As a simple illustration, assume that the neighbourhood is defined by a polytope rather than a positive definite matrix. This situation arises, for instance, when the metric is based on the $\ell^1$- or $\ell^\infty$-norm, as opposed to the $\ell^2$-norm. In this simple example, $\mathcal{P}$ is a polytope, hence $\mathbf{e_1}^T\mathbf{A}^n\mathbf{d}$ is maximised at one of the finitely many corners $\{\mathbf{d_1}, \dots, \mathbf{d_k}\}$. Thus, Robust (Uniform Ultimate) Positivity is decided by using the state of the art \cite{joeljames3} to check the (Ultimate) Positivity of each of the LRS $(\mathbf{a}, \mathbf{c+d_i})$. The geometry of our setting is not simple enough to allow such a straightforward approach. \textbf{The overview of our approach to Problems \ref{prob:rrobpos} and \ref{prob:rrobuniultpos} is as follows.} \begin{enumerate} \item Decide (constructively for Problem \ref{prob:rrobpos}) whether there exists an $N_1$ such that $\mathbf{e_1}^T \mathbf{A}^n \mathbf{c} \ge 0$ for all $n > N_1$. If $N_1$ is explicitly required, the state of the art is able to tackle LRS of order $\le 5$ \cite{joeljames3} and Simple LRS of order $\le 9$ \cite{ouaknine2014positivity}. In the non-constructive case, it can further handle all simple LRS \cite{ouaknine2014ultimate}. \item Use linear-algebraic arguments to define a real algebraic LRS $(v_n)_{n=0}^\infty$, such that $v_n \ge 0$ if and only if $|\mathbf{e_1}^T \mathbf{A}^n \mathbf{c}|\ge \max_{\mathbf{d} \in \mathcal{P}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{d}$. \item Decide (constructively for Problem \ref{prob:rrobpos}) whether there exists $N_2$ such that $v_n \ge 0$ for all $n > N_2$. Positivity throughout the neighbourhood is thus guaranteed beyond step $N = \max(N_1, N_2)$. If either $N_1$ or $N_2$ does not exist, then Robust Ultimate Positivity, and hence Robust Positivity, does not hold. \item \textbf{Only for Problem \ref{prob:rrobpos}:} Explicitly check inequality \ref{eq:illustrate} for $n \le N$. \end{enumerate} Our novelty lies in Step 2 and identifying when Step 3 can be implemented. We now discuss how we perform Steps 2 and 4 when $\mathcal{P} = \mathcal{B}_\mathbf{S}$, a neighbourhood of vectors $\mathbf{d}$ such that $\mathbf{d}^T\mathbf{S}\mathbf{d} \le 1$. The defining parameter $\mathbf{S}$ is a real algebraic positive definite matrix. We note that since $\mathbf{S}$ is positive definite, it can be factored as $\mathbf{G}^T\mathbf{G}$, where $\mathbf{G}$ is a real algebraic invertible matrix. We denote $\mathbf{Gd} = \mathbf{f}$. We argue that $\mathbf{G}^{-1}$ bijectively maps the Euclidean unit ball $\mathcal{B}$ to $\mathcal{B}_\mathbf{S}$. The bijection is clear from the invertibility of the matrix. Suppose $\mathbf{d} = \mathbf{G}^{-1}\mathbf{f}$, where $\mathbf{f} \in \mathcal{B}$, i.e. $\mathbf{f}^T\mathbf{f} \le 1$. Then $\mathbf{d}^T\mathbf{Sd} = \mathbf{d}^T\mathbf{G}^T\mathbf{Gd} = \mathbf{f}^T\mathbf{f} \le 1.$ Hence, \begin{equation} \label{eq:bijectivemap} \max_{\mathbf{d} \in \mathcal{B}_\mathbf{S}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{d} = \max_{\mathbf{f} \in \mathcal{B}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{G}^{-1}\mathbf{f}. \end{equation} $\mathcal{B}$ is a convex set; thus a linear function will necessarily be maximised at its boundary, i.e. when $||\mathbf{f}|| = 1$. The linear function $\mathbf{h}^T\mathbf{f}$ is maximised over the unit Euclidean ball when $\mathbf{f}$ is aligned along $\mathbf{h}$; the maximum value is $||\mathbf{h}||$. We can thus perform Step 4 because \begin{equation} \max_{\mathbf{d} \in \mathcal{B}_\mathbf{S}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{d} = \left|\left|\left( \mathbf{e_1}^T\mathbf{A}^n\mathbf{G}^{-1} \right)^T\right|\right|. \end{equation} For Step 2, we need a necessary and sufficient condition for $|\mathbf{e_1}^T \mathbf{A}^n \mathbf{c}|\ge \max_{\mathbf{d} \in \mathcal{P}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{d}$, in terms of the positivity of an LRS at iterate $n$. We simply square both sides of the inequality, and transfer all terms to the left: \begin{equation} \label{eq:critical} (\mathbf{e_1}^T \mathbf{A}^n \mathbf{c})^2 - (\mathbf{e_1}^T \mathbf{A}^n \mathbf{g_1})^2 - \dots - (\mathbf{e_1}^T \mathbf{A}^n \mathbf{g_\kappa})^2 \ge 0. \end{equation} Crucially, $\mathbf{g_1}, \dots, \mathbf{g_\kappa}$ are the linearly independent columns of the invertible $\mathbf{G}^{-1}$. Only Step 3 remains to be addressed: we must (constructively) decide whether there exists $N_2$ such that the previous inequality holds for all $n > N_2$. In \S\ref{section:decidability}, we give the technical details, thus proving our first main decidability result. \begin{theorem}[First Main Decidability Result] \label{thm:decide} Problem \ref{prob:rrobuniultpos} is decidable for simple LRS. Problem \ref{prob:rrobpos} is decidable for simple LRS up to order 5. Problems \ref{prob:rrobpos} and \ref{prob:rrobuniultpos} are decidable for general LRS up to order 4. \end{theorem} \subsection{The non-uniform variant: An overview} \label{section:nonuniformoverview} As discussed at length in \cite{originalstacs,originalarxiv}, $\mu(\mathbf{c'})= \liminf_{n \in \mathbb{N}} \seq{\mathbf{p'}, \mathbf{q_n}}_{dom} \ge 0$ is necessary for the Ultimate Positivity of $\mathbf{c'}$; $\mu(\mathbf{c'}) > 0$ is sufficient. \textbf{Our strategy for Problem \ref{prob:rrobnonuniultpos} is as follows.} \begin{enumerate} \item Use the First Order Theory of the Reals to check that $\mu(\mathbf{c'}) \ge 0$ for all $\mathbf{c'}$ in the given neighbourhood, and detect the critical boundary cases when $\mu(\mathbf{c'}) = 0$. \item Exploit the low dimensionality to decide the critical boundary cases when $\mu(\mathbf{c'}) = 0$. \end{enumerate} \begin{figure} \caption{Visual intuition. The region $\mu \ge 0$ is defined by the intersection of halfspaces. The orientation of the neighbourhood relative to this region is deduced with the First Order Theory of the Reals. When there are finitely many halfspaces, the critical case is marked by the ball being tangent to the separating hyperplane(s) at finitely many discrete points. In low dimensions, Ultimate Positivity can be decided for these boundary cases using existing techniques. When there are infinitely many halfspaces, they carve out a region that resembles a cone. The neighbourhood can either touch the cone as before, or be nestled in it, having a continuous, connected region of tangency. In the latter case, Robust Ultimate Positivity can be handled with number-theoretic arguments in the low-dimensional setting.} \label{fig:geometricpicture} \end{figure} We adopt this strategy (see Figure \ref{fig:geometricpicture}) and prove our second decidability result in \S\ref{section:decidability2}. \begin{theorem}[Second Decidability Result] \label{thm:decide2} Problem \ref{prob:rrobnonuniultpos} is decidable up to order 4. \end{theorem} \section{Diophantine Approximation} \label{section:diophantine} We justify the inability of our techniques to generalise to LRS of higher order by establishing a connection to a number-theoretic hurdle: that of Diophantine Approximation. Diophantine Approximation is a vast and active number-theoretic field of research, one of whose concerns is the approximation of reals by rational numbers. A key tool in this regard is the partial fraction expansion $[a_0; a_1, a_2, \dots]$ of an irrational $t$: $$ t = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{\dots}}} $$ where $a_0, a_1, a_2, \dots \in \mathbb{N}$. Truncating this expansion at progressively greater depths yields a series of increasingly accurate approximations. The quality of the rational approximation depends not only on its accuracy but also on the size of the denominator. As discussed in the Introduction, evaluating the quality of the approximation, or that of the convergence, seems inaccessible to contemporary number theory. The above intuition about the quality of the approximation is captured in the following definition of $L(t)$, the (homogenous) Diophantine approximation type: \begin{equation} L(t) = \inf\left\{c \in \mathbb{R} : \left|t - \frac{p}{q}\right| < \frac{c}{q^2} \text{ for some } p, q\in \mathbb{Z}\right\}. \end{equation} Similarly, the quality of the convergence is formalised by defining $L_{\infty}(t)$, the (homogenous) Lagrange constant: \begin{equation} L_\infty(t) = \inf\left\{c \in \mathbb{R} : \left|t - \frac{p}{q}\right| < \frac{c}{q^2} \text{ for infinitely many } p, q\in \mathbb{Z}\right\}. \end{equation} For technical purposes, we use an equivalent definition that relates to the continued fraction perspective, and allows for a slight generalisation. We follow Lagarias and Shallit’s terminology \cite{dio-constants} and use $[x]$ to denote the shortest distance from $x$ to an integer; while $[x]_b$ denotes the shortest distance from $x$ to an integer multiple of $b \in \mathbb{R}$. It is easy to observe the property $[x]_b = b[x/b]$. \begin{definition}[Diophantine Approximation Type] \label{def:L} The homogenous Diophantine approximation type $L(t)$ is defined to be $\inf_{n \in \mathbb{N}_{>0}} n[nt]$. The inhomogeneous Diophantine approximation type $L(t, s)$ is defined to be $\inf_{n \in \mathbb{N}_{>0}} n[nt - s]$, $s \notin \mathbb{Z} + t\mathbb{Z}$. \end{definition} \begin{definition}[Lagrange constant] \label{def:Linfty} The homogenous Lagrange constant $L_\infty(t)$ is defined to be $\liminf_{n \in \mathbb{N}} n[nt]$. The inhomogeneous Lagrange constant $L_\infty(t, s)$ is defined to be\\ $\liminf_{n \in \mathbb{N}} n[nt - s]$, $s \notin \mathbb{Z} + t\mathbb{Z}$. \end{definition} From the definitions, it is clear that $0 \le L(t) \le L_\infty(t)$. Due to the work of Khintchine \cite{khintchine}, it is known that these constants lie between $0$ and ${1}/{\sqrt{5}}$. In our setting, the irrational $t$ comes from the argument $\theta$ of the characteristic root $\rho e^{i\theta}$ of the LRS: $t = \theta/2\pi$. The following observation hints at how the correspondence could further extend to Positivity problems. It is pivotal to our novel low-dimensional decidability result for Uniform Robustness. \begin{lemma} \label{eq:quadraticdecay} For all $\theta$ that are not rational multiples of $2\pi$, and all $\varphi$, there exist infinitely many $n\in \mathbb{N}$ such that $ 1 - \cos(n\theta- \varphi) \le \frac{1}{2}\left[n\theta - \varphi \right]_{2\pi}^2 = 2\pi^2[nt -s]^2 \le \frac{2\pi^2}{5n^2}. $ \end{lemma} The properties of the LRS are driven by whether the characteristic root $\rho e^{i\theta}$ is a root of unity, i.e. $\theta$ is a rational multiple of $2\pi$. If yes, decision procedures are often much simpler; if not, we appeal to the number theory discussed in this section. One can detect whether an algebraic characteristic root is a root of unity by brute enumeration. \begin{lemma} \label{lemma:rootofunity} Let $\alpha$ be an algebraic number of degree $d$. Then if $\alpha$ is a $k^{th}$ root of unity, $k \le 2d^2$. \end{lemma} \begin{proof} The degree of the $k^{th}$ root of unity is precisely $\Phi(k)$, where $\Phi$ denotes the Euler totient function. $\Phi(k) \ge \sqrt{k/2}$. The desired inequality follows. \end{proof} We record a number-theoretic fact which describes the density of the integer multiples of an irrational $x$ modulo $1$ in the unit interval: its proof relies on continued fraction expansions and the Ostrowski numeration system \cite{bourla2016ostrowski,berthe2022dynamics}, and is deferred to Appendix \ref{appendix:ostrowski}. This result is decisive when considering Non-Uniform Robustness. \begin{lemma} \label{lemma:existsreal} For every irrational number $x$, strictly decreasing real positive function $\psi$, and interval $\mathcal{I} = [a, b] \subset [0, 1], ~ a \ne b$, there exists $y_0 \in \mathcal{I}$ such that $[nx - y_0] < \psi(n)$ for infinitely many even $n$, and $y_1 \in \mathcal{I}$ such that $[nx - y_1] < \psi(n)$ for infinitely many odd $n$. \end{lemma} The familiar density theorem is an immediate corollary of the above powerful result. Indeed, we can consider an interval of length $\varepsilon/2$, and take $\psi(n) = \varepsilon/2$. \begin{lemma} \label{lemma:density} Let $x$ be irrational, and $y \in [0, 1)$. For every $\varepsilon > 0$, there exist infinitely many even $n$, and infinitely many odd $n$ such that $[nx - y] < \varepsilon$. \end{lemma} The following application of the density theorem is central to the computation of the discrete $\mu(\mathbf{c}) = \liminf_{n\in \mathbb{N}}\seq{\mathbf{p}, \mathbf{q_n}}_{dom}$. \begin{lemma} \label{eq:liminfmin} Suppose $\theta$ is not a rational multiple of $2\pi$. Let $h_1(t)$ be continuous with period $2\pi$. Then $ \liminf_n \left(h_1(n\theta) + h_2(-1^n)\right) = \min_{t \in [0, 2\pi], b\in \{-1, 1\}} \left(h_1(t) + h_2(b)\right). $ \end{lemma} We note that despite the observations and results mentioned in the preceding discussion, the Diophantine approximation type and Lagrange constant of most transcendental numbers are unknown. For instance, computing $L_\infty(\pi)$ is a longstanding and mathematically interesting open problem. We refer the reader to \cite[Section 5]{joeljames3} for a cursory survey of the history of relevant developments in the field of Diophantine approximation. This source reduces the computation of the constants $L(t)$ and $L_\infty(t)$ discussed above to the non-robust variants of Positivity problems for LRS of order 6. In \S\ref{section:hardness}, we prove analogous hardness results for robust Positivity problems of order 5. To that end, we define a similar class of transcendental numbers relevant to our reduction. Let \begin{equation} \mathcal A=\{p+q i \in \mathbb{C} \mid p,q \in \mathbb{A}, p^2+q^2=1, \forall n.~(p + qi)^n \ne 1\} \end{equation} i.e., the set $\mathcal A$ consists of algebraic numbers on the unit circle in $\mathbb{C}$, none of which are roots of unity. In particular, writing $p+q i= e^{i 2 \pi \theta}$, we have that $\theta \notin \mathbb{Q}$. We denote: \begin{equation} \label{eq:keyset} \mathcal{T} = \left\{ \theta \in (- 1/2, 1/2] \mid e^{2 \pi i \theta} \in \mathcal{A}\right\}. \end{equation} The set $\mathcal{T}$ is dense in $(- \frac 1 2, \frac 1 2]$. In general, we don't have a method to compute $L(\theta)$ or $L_\infty(\theta)$ for $\theta \in \mathcal{T}$, or approximate them to arbitrary precision. \begin{definition}[Number-theoretic hardness] \label{def:hardness} Let $\mathcal{T}$ be as above. A decision problem is said to be $\mathcal{T}$-Diophantine hard (resp.\ $\mathcal{T}$-Lagrange hard), if its decidability entails that given any $t \in \mathcal{T}$ and $\varepsilon > 0$, one can compute $\ell$ such that $|\ell - L(t)| < \varepsilon$ (resp.\ $|\ell - L_\infty(t)| < \varepsilon$). \end{definition} \begin{theorem}[Main Hardness Result] \label{thm:hardness} Problem \ref{prob:rrobpos} (resp.\ Problem \ref{prob:rrobuniultpos}) is $\mathcal{T}$-Diophantine hard (resp.\ $\mathcal{T}$-Lagrange hard) at order 5. \end{theorem} As noted in \cite{originalarxiv}, in view of the Lagrange hardness (Definition \ref{def:hardness}) of Ultimate Positivity at order 6 \cite{joeljames3}, Problem \ref{prob:rrobnonuniultpos}, which asks whether the given neighbourhood consists entirely of initialisations that produce an Ultimately Positive sequence, is also Lagrange hard at order 6. The idea is to use the existing reduction from the computation of Lagrange constants to Ultimate Positivity, and extend it to the robust variant: one simply constructs a neighbourhood of initialisations that has the hard instance of Ultimate Positivity on its surface, but otherwise lies entirely in the region where Ultimate Positivity is guaranteed. \begin{theorem} \label{thm:hardness2} Problem \ref{prob:rrobnonuniultpos} is $\mathcal{T}$-Lagrange hard at order 6. \end{theorem} \section{Decidability of Uniform Robustness} \label{section:decidability} \label{thm:abelian} In this section, we prove Theorem \ref{thm:decide} by showing that we can implement Step 3 of the overview in \S\ref{section:uniformfoundation}: (constructively) decide whether there exists $N_2$ such that for all $n > N_2$, \begin{equation} \label{eq:criticalcopy} (\mathbf{e_1}^T \mathbf{A}^n \mathbf{c})^2 - (\mathbf{e_1}^T \mathbf{A}^n \mathbf{g_1})^2 - \dots - (\mathbf{e_1}^T \mathbf{A}^n \mathbf{g_4})^2 \ge 0. \end{equation} \begin{theorem}[First Main Decidability Result, restated] \label{thm:decidecopy} Problem \ref{prob:rrobuniultpos} (Robust Uniform Ultimate Positivity) is decidable for simple LRS. Problem \ref{prob:rrobpos} (Robust Positivity) is decidable for simple LRS up to order 5. Problems \ref{prob:rrobpos} and \ref{prob:rrobuniultpos} are decidable for general LRS up to order 4. \end{theorem} \subsection{Simple LRS} We begin by treating simple LRS. The goal is to show that the current state of the art is equipped to handle instances relevant to this setting. Recall the discussion on the point-wise sums of products of simple LRS, surrounding equation \ref{eq:exppoly}. If the original LRS is simple, then inequality \ref{eq:criticalcopy} is also an instance of Ultimate Positivity for simple LRS; indeed, its input can be seen to be real algebraic. In case we are only interested in Robust Ultimate Positivity, the non-constructive decision procedure \cite{ouaknine2014ultimate} suffices, because it completely solves Ultimate Positivity for simple LRS. As a corollary of the proof of the decidability of Positivity of simple LRS up to order 9 \cite{ouaknine2014positivity}, Ultimate Positivity for simple LRS is \textit{constructively} decidable if one of the following holds: \textbf{(a)} all characteristic roots have the same modulus; {\bf(b)} there are at most three pairs of complex conjugates among the dominant (maximal modulus) characteristic roots. We argue that for the original simple LRS $(u_n)_n$, $5$ is the highest order that guarantees that at least one of the conditions holds for the resulting simple LRS $(v_n)_n$ in inequality \ref{eq:criticalcopy}. For this, we recall the property discussed after equation \ref{eq:exppoly}: if $U$ is the set of characteristic roots of $(u_n)_n$, then the set of characteristic roots of $v_n = u_n^2$ is $V = \{\lambda_1\lambda_2: \lambda_1, \lambda_2 \in U\}$. By Proposition \ref{prop:folklore}, $U$ contains a real positive dominant root $\rho$. It is clear that the dominant roots of $V$ result from, and only from multiplying together pairs of dominant roots from $U$. If $U$ does not have complex dominant roots, neither does $V$. If $\lambda \in U$ is a dominant complex root, then $\lambda\bar\lambda = \rho^2$. If $U = \{\rho, \lambda_1, \lambda_2, \bar{\lambda_1}, \bar{\lambda_2}\}$, all dominant, then all roots of $V$ are dominant, and condition \textbf{(a)} is met. The only remaining case is that $U$ has one pair of complex conjugates among its dominant roots: the scenario that results in most dominant roots in V is $U_{dom} = \{\rho, -\rho, \lambda, \bar{\lambda}\}$. Then, the dominant roots in $V$ are $\{\rho^2, -\rho^2, \lambda^2, \bar\lambda^2, \pm \rho\lambda, \pm \rho\bar\lambda\}$: three conjugate pairs, and condition \textbf{(b)} is met. Finally, we record that order 5 is maximal: consider $U = \{\rho, \lambda_1, \lambda_2, \bar{\lambda_1}, \bar{\lambda_2}, \alpha\}$ with $\alpha$ non-dominant. Then $V$ has five pairs of complex conjugates among its dominant roots, along with the presence of non-dominant roots. \subsection{Non-simple LRS} We treat order $4$ LRS: our techniques naturally apply to lower orders too. We make extensive use of the real exponential polynomial closed form \ref{eq:realexppoly} and the surrounding discussion. The key lies in expressing the critical inequality \ref{eq:criticalcopy} as \begin{equation} \label{eq:start} \seq{\mathbf{p}, \mathbf{q_n}}^2 - \seq{\mathbf{b_1}, \mathbf{q_n}}^2 - \dots - \seq{\mathbf{b_4}, \mathbf{q_n}}^2 \ge 0 ~~\Leftrightarrow~~ \seq{\mathbf{x}, \mathbf{r_n}} \ge 0 \end{equation} and choosing $\{\mathbf{q_n}\}_{n\in\mathbb{N}}$ judiciously. If all the characteristic roots of the original LRS are real, then $\mathbf{q_n}$ is free of trigonometric terms, and hence so is $\mathbf{r_n}$. Thus $\seq{\mathbf{x}, \mathbf{r_n}}$ is also an LRS with all real characteristic roots, and constructively deciding the existence of $N_2$ is easily done through elementary growth arguments. We shall thus assume the presence of a pair of complex conjugates among the characteristic roots. As discussed through Proposition \ref{prop:folklore}, any decision regarding Ultimate Positivity is NO in the absence of a real positive dominant root. At order $4$, this means that there is \textbf{exactly one pair of complex conjugates} among the roots. We further assume, without loss of generality, that \textbf{the real positive dominant root is unity}. We shall also assume \textbf{non-degeneracy}, i.e.\ the ratio of any pair of distinct roots of the characteristic polynomial is not a root of unity. This can be detected, courtesy Lemma \ref{lemma:rootofunity}. In our restricted setting, degeneracy can arise because: (a) $-1$ is a characteristic root; (b) a characteristic root is of the form $\rho e^{2\pi i \cdot \frac{\ell}{k}}$, i.e. a scaled $k^{th}$ root of unity. In this case, any LRS $\seq{\mathbf{v}, \mathbf{q_n}}$ with roots $\{1, \alpha, \rho e^{\pm 2\pi i \cdot \frac{\ell}{k}}\}$ can be decomposed as the interleaving of $2k$ real LRS, each with characteristic roots $\{1, \rho^{2k}\} \cup \{\alpha^{2k}\}$. The only possibility, therefore, is that the characteristic roots are $1, 1, \gamma, \bar{\gamma}$. Let $0 < |\gamma| = \rho \le 1$, where $\gamma = \rho e^{i\theta}$ is not a scaled root of unity. We take inequality \ref{eq:start} as the starting point for our computations. Let $\mathbf{q_n} = \begin{bmatrix} n & 1 & \rho^n\cos(n\theta - \varphi) & \rho^n\sin(n\theta -\varphi) \end{bmatrix}^T$. Let $\mathbf{u_1}^T, \dots, \mathbf{u_4}^T$ be the rows of the \textbf{invertible} matrix $\begin{bmatrix} \mathbf{b_1}& \dots & \mathbf{b_4}\end{bmatrix}$. The table below shows the terms and coefficients on simplifying inequality \ref{eq:start}. \begin{table}[H] \begin{tabular}{|l|l|l|} \hline \textbf{Term}& \textbf{Coefficient}& {\bf Explicitly} \\ \hline $n^2$ & $z_2$ & $p_1^2 - \seq{\mathbf{u_1}, \mathbf{u_1}}$ \\ \hline $n$ & $z_1$ & $2p_1p_2 - 2\seq{\mathbf{u_1}, \mathbf{u_2}}$ \\ \hline $1$ & $z_0$ & $p_2^2 - \seq{\mathbf{u_2}, \mathbf{u_2}} $ \\ \hline $n\rho^n\cos (n\theta - \varphi)$ & $x_2$ & $2p_1p_3 - 2\seq{\mathbf{u_1}, \mathbf{u_3}}$ \\ \hline $n\rho^n\sin (n\theta - \varphi)$ & $y_2$ & $2p_1p_4 - 2\seq{\mathbf{u_1}, \mathbf{u_4}}$ \\ \hline $\rho^n\cos (n\theta-\varphi)$ & $x_1$ & $2p_2p_3 - 2\seq{\mathbf{u_2}, \mathbf{u_3}}$ \\ \hline $\rho^n\sin (n\theta-\varphi)$ & $y_1$ & $2p_2p_4 - 2\seq{\mathbf{u_2}, \mathbf{u_4}}$ \\ \hline $\rho^{2n}$ & $w$ & $\frac{1}{2}(p_3^2 + p_4^2) - \frac{1}{2} (\seq{\mathbf{u_3}, \mathbf{u_3}} + \seq{\mathbf{u_4}, \mathbf{u_4}})$ \\ \hline $\rho^{2n}\cos (2n\theta - 2\varphi)$ & $x_0$ & $\frac{1}{2}(p_3^2 - p_4^2) - \frac{1}{2}(\seq{\mathbf{u_3}, \mathbf{u_3}} - \seq{\mathbf{u_4}, \mathbf{u_4}})$ \\ \hline $\rho^{2n}\sin (2n\theta - 2\varphi)$ & $y_0$ & $2p_3p_4 - 2\seq{\mathbf{u_3}, \mathbf{u_4}}$ \\ \hline \end{tabular} \end{table} If $\rho < 1$, then the dominant growth rate for the problem to be non-trivial is $n^2, n, 1, $ or $\rho^{2n}$. The former cases can be solved with straightforward growth arguments, while the last case results in an order 3 LRS that can easily be dealt with \cite{ouaknine2014positivity,joeljames3}. We thus assume $\rho = 1$. Again, if $z_2 \ne 0$, then decidability is trivial because the dominant growth rate of $n^2$ is dictated by a single term; hence we assume $z_2 = 0$. In this case, there are two groups of terms, based on growth rate: one with $n$, the other with $1$. To study these groups, we define \begin{align} f(t) &= z_1 + x_2 \cos(t -\varphi) + y_2\sin(t - \varphi) \\ g(t) &= z_0 + w + x_1\cos(t - \varphi) + y_1\sin(t - \varphi) + x_0 \cos(2t - 2\varphi) + y_0\sin(2t-2\varphi) \end{align} Since $\theta$ is not a rational multiple of $2\pi$, $\{n\theta \text{ mod } 2\pi\}$ is dense in $[0, 2\pi]$, and we invoke Lemma \ref{eq:liminfmin} to deduce \begin{equation} \liminf_{n\in \mathbb{N}} f(n\theta) = \min_{t \in [0, 2\pi]} f(t) = z_1 -\sqrt{x_2^2 + y_2^2} = \mu. \end{equation} If $\mu < 0$, then the critical inequality $nf(n\theta) + g(n\theta) \ge 0$ will be violated infinitely often. If $\mu > 0$, we can compute an $N_2$ beyond which it is guaranteed to be satisfied. We thus concern ourselves with the case where $\mu = 0$. Recall the discussion around $\mu$ when its concept was first defined after Proposition \ref{prop:folklore}: it is an intrinsic property of the problem itself, and invariant under the ``phase'' $\varphi$ chosen in the basis of solutions. We thus assume that $\varphi$ is chosen in such a way that the minimum is attained at $\varphi$, i.e. $f(\varphi) = 0$. This choice can be made by applying the trigonometric identity $\cos(a - b) = \cos a \cos b + \sin a \sin b$ to $f(t)$. This means that $y_2 = 0$, and we choose $-z_1 = x_2 < 0$. Now, if $g(\varphi) > 0$, we compute a positive lower bound on $f(t)$ for $t$ such that $g(t) < 0$. This then results in an $N_2$ beyond which $nf(n\theta) + g(n\theta) \ge 0$ is guaranteed. If $g(\varphi) < 0$, then the inequality has infinitely many violations. This is due to Lemma \ref{eq:quadraticdecay}, which asserts that there are infinitely many $n$ for which $f(n\theta) \le 2\pi^2z_1/5n^2$. These $n$ are necessarily such that $n\theta$ is close to $\varphi$, and the negativity of $g(n\theta)$ is thus decisive. The final case that remains is $g(\varphi) = 0$. We argue that remarkably, it does not arise at all! \begin{lemma} If $z_2 = \mu=0$, it cannot be the case that $f(\varphi) = g(\varphi) = 0$. \end{lemma} \begin{proof} Suppose, for the sake of contradiction, the scenario actually occurs. This means that \\ $ z_2 = z_1 + x_2 = z_0 + w + x_1 + x_0 = 0. $ From the table, these respectively imply \begin{align*} p_1^2 &= \seq{\mathbf{u_1}, \mathbf{u_1}}, \\ p_1(p_2 + p_3) &= \seq{\mathbf{u_1}, \mathbf{u_2} + \mathbf{u_3}}, \\ (p_2 + p_3)^2 &= \seq{\mathbf{u_2} + \mathbf{u_3}, \mathbf{u_2} + \mathbf{u_3}}. \end{align*} This implies that $|\seq{\mathbf{u_1}, \mathbf{u_2} + \mathbf{u_3}}| = ||\mathbf{u_1}||\cdot||\mathbf{u_2} + \mathbf{u_3}||$, i.e. $\mathbf{u_1}$ is a scaled multiple of $\mathbf{u_2} + \mathbf{u_3}$. This contradicts the fact that the rows of the invertible $\begin{bmatrix} \mathbf{b_1}& \dots & \mathbf{b_4}\end{bmatrix}$ are linearly independent, and we're done. \end{proof} \section{Non-uniform Robustness: Decidability at order four} \label{section:decidability2} In this section, we prove Theorem \ref{thm:decide2}. The techniques naturally apply to lower orders, and we omit their explicit treatment. Recall the critical condition from our overview in \S\ref{section:nonuniformoverview}: \begin{equation} \mu(\mathbf{c'}) = \liminf_{n\in \mathbb{N}}\seq{\mathbf{p'}, \mathbf{q_n}}_{dom} \ge 0 \end{equation} for all $\mathbf{c'}$ in the neighbourhood is necessary for the decision to be YES; the inequality holding strictly is sufficient. Critical cases arise when the surface of the neighbourhood touches the region where $\mu = 0$, and the non-dominant terms, if any, can potentially have a negative contribution. We demonstrate that these can be detected and dealt with. Since Proposition \ref{prop:folklore} guarantees the existence of a real positive dominant term, $\seq{\mathbf{p}, \mathbf{q_n}}_{dom}$ can only be of one of the following forms: {\bf(a)} $z$; {\bf(b)} $z + w(-1)^n$; {\bf(c)} $z + x\cos n\theta + y\sin n\theta$; {\bf(d)} $z + x\cos n\theta + y\sin n\theta + w(-1)^n$, where $x, y, z, w$ are linear in the initialisation $\mathbf{c}$. Cases (a), (b), and (c), (d) where $\theta$ is a rational multiple of $2\pi$ (detected with Lemma \ref{lemma:rootofunity}) are the easiest. The region $\mu \ge 0$ is carved out by \textit{finitely} many halfspaces, defined by separating hyperplanes of the form $z + bw + c_0 x + s_0 y = 0$. By elementary linear algebra and co-ordinate geometry (e.g.\. by working in a basis where the neighbourhood is a perfect hypersphere), one can determine whether $\mu > 0$ for the entire neighbourhood, or whether $\mu < 0$ for some points in the neighbourhood, or whether the neighbourhood touches a hyperplane. Each hyperplane has at most one point of tangency, whose algebraic coordinates can be solved for. These critical points are low-dimensional instances of Ultimate Positivity, and can be decided with the state of the art \cite{ouaknine2014ultimate}. We therefore assume that $\theta$ is not a rational multiple of $2\pi$, and we are in Case (c) or (d). We apply Lemma \ref{eq:liminfmin}, we get that \begin{equation} \mu(\mathbf{c}) = \liminf_{n \in \mathbb{N}} \seq{\mathbf{p}, \mathbf{q_n}} = \min_{t \in \mathbb{R}, b \in \{\pm 1\}} z + x\cos t + y\sin t + wb = z - \sqrt{x^2 + y^2} - |w|. \end{equation} If we are in Case (d), there are no dominant roots, and $\mu \ge 0$ throughout the neighbourhood is necessary as well as sufficient for the decision to be YES. This is an algebraic condition, and can be checked using the First Order Theory of the Reals.\footnotemark \footnotetext{$|w|$ is expressed with $\exists r.~r^2 = w^2 \land r \ge 0$; a similar trick works for $\sqrt{x^2 + y^2}$.} Case (c) remains. $\seq{\mathbf{q_n}, \mathbf{p}} = z + x\cos n\theta + y\sin n\theta + w\alpha^n$, where $0 < |\alpha| < 1$. As discussed, we can use the First Order Theory of the Reals to check the sufficient $\mu > 0$, and the necessary $\mu \ge 0$ throughout the neighbourhood. We consider the scenario where the necessity check succeeds, but the sufficiency check fails. The decision can be NO only if there are points on the surface of the neighbourhood where $\mu = 0$, and the non-dominant $w\alpha^n$ can make a negative contribution. We describe how these points are found and analysed. First, we observe that the region $\mu \ge 0$ is given by the cone $z - \sqrt{x^2 + y^2} \ge 0$. It can be intuited as being carved out by a continuum of hyperplanes $z + x\cos\phi + y\sin\phi = 0$. We encode the above discussion to find the critical points with the following first order formula with free variable $c$, which stands for $\cos \phi$ \begin{equation} \label{eq:intersection} \chi_1(c):= \exists s \exists \mathbf{c'}.~ (\mathbf{c'} - \mathbf{c})^T\mathbf{S}(\mathbf{c'} - \mathbf{c}) = 1 \land z' + cx' + sy' = 0 \land c^2 + s^2 = 1 \land w' \sim 0. \end{equation} In the above $\sim$ is $\ne$ if the non-dominant root $\alpha < 0$, and is $<$ if $\alpha > 0$. We can use Theorem \ref{thm:renegar} to get an equivalent quantifier free formula: this comprises purely of polynomial (in-)equalities in the free variable $c$. The set of $c$, and hence $\cos \phi$, satisfying these, consists of finitely many intervals. Of course, Ultimate Positivity is guaranteed when this set is empty: it means there are no points threatening to violate Ultimate Positivity. We first dispose of the case where all intervals consist of single points. Consider an interval $\{c_0\}$ consisting of a single point. This is illustrated by the case of the ball touching the cone in Figure \ref{fig:geometricpicture}. Due to its origins and discrete occurrence, $c_0$ must be a root of a polynomial obtained by quantifier elimination on $\chi_1$, and is hence algebraic. The corresponding critical point is the point of tangency of the neighbourhood with a hyperplane with a real algebraic equation. Thus, it generates a real algebraic instance of Ultimate Positivity, which can be decided with the techniques of \cite{ouaknine2014ultimate}. If, however, the set of $c$ satisfying $\chi_1$ consists of intervals that have more than one point, then the techniques of \cite{ouaknine2014ultimate} to decide Ultimate Positivity for a single point with algebraic coordinates are no longer accessible. This situation is illustrated by the case of the ball nestled in cone in Figure \ref{fig:geometricpicture}. Let $[\phi_1, \phi_2]$ be an interval of $\phi$ such that: a) all values of $c$ between $\cos\phi_1$ and $\cos\phi_2$ satisfy $\chi_1$, b) The corresponding witnesses $z'$ are at most $z_0$, and c) The corresponding witnesses $w'$ have magnitude at least some fixed $w_0$. Then, we must have for each $\phi$ (and corresponding $z(c), x = -cz, y = -z\sin \phi, w)$) in this interval, the following inequality is violated only finitely often: \begin{equation} z - z\cos(n\theta - \phi) + w\alpha^n \ge 0. \end{equation} We consider an even weaker inequality, which, in this context, we argue is bound to be violated infinitely often: \begin{equation} z_0[n\theta - \phi]_{2\pi}^2 \ge 2w_0\alpha^n. \end{equation} The argument hinges on Lemma \ref{lemma:existsreal}, which we restate: \begin{lemma} \label{lemma:existsreal3} For every irrational number $x$, strictly decreasing real positive function $\psi$, and interval $\mathcal{I} = [a, b] \subset [0, 1], ~ a \ne b$, there exists $y_0 \in \mathcal{I}$ such that $[nx - y_0] < \psi(n)$ for infinitely many even $n$, and $y_1 \in \mathcal{I}$ such that $[nx - y_1] < \psi(n)$ for infinitely many odd $n$. \end{lemma} Now, if $\alpha < 0$, we use Lemma \ref{lemma:existsreal3} on the irrational $\theta/2\pi$, and the decreasing $\sqrt{\frac{w_0 |\alpha|^n}{2\pi^2z_0}}$ to argue that there exists a $\phi$ in the desired interval, such that the weaker inequality will be violated for infinitely many $n$ of the the appropriate parity. Thus, we can return NO if we are in the case where the set of $c$ satisfying $\chi_1$ (equation \ref{eq:intersection}) consists of intervals that contain more than a single point. \section{Uniform Robustness: Hardness at order five} \label{section:hardness} We shall prove Theorem \ref{thm:hardness} in this section. That is, given $t \in \mathcal{T}, s$ as defined in equation \ref{eq:keyset}, we shall give rational $\mathbf{a}, \mathbf{c}$ such that varying $\mathbf{S} =\mathbf{G}^T\mathbf{G}$ while invoking $\mathbf{S}$-Robust Positivity decision procedures will enable us to approximate $L(t, s)$ and $L_\infty(t, s)$ to arbitrary precision. We assume $t, s$ are specified by $\cos \theta, \cos \varphi \in \mathbb{A}$, such that $2\pi t = \theta, 2\pi s = \varphi$. Our LRR $\mathbf{a}$ is such that the roots of the characteristic polynomial are $1, 1, 1, e^{2\pi it}, e^{-2\pi i t}$, i.e. the characteristic polynomial is $ (X- 1)^3(X^2 - 2X\cos\theta + 1) $. Here, $ u_n = \mathbf{e_1}^T\mathbf{A}^n\mathbf{c} = \seq{\mathbf{p}, \mathbf{q_n}} $. For the problem instance we create in our reduction, we choose $\mathbf{p} = \begin{bmatrix}r & 0 & 1+\frac{r}{2} & -1 & 0 \end{bmatrix}^T$, where $r$ is a parameter we use to tune our guess for $L(t, s)$ and $L_\infty(t, s)$; we choose $\mathbf{q_n}^T = \mathbf{e_1}^T\mathbf{A}^n\mathbf{V} = \begin{bmatrix}n^2 & n & 1 & \cos(2\pi(nt-s)) & \sin(2\pi(nt-s))\end{bmatrix}$. We choose $\mathbf{S} = \mathbf{G}^T\mathbf{G}/r^2$, where $\mathbf{G} = \mathbf{V}^{-1}$. Thus, our critical inequality is \begin{equation} \label{eq:corereduction} \seq{\mathbf{p}, \mathbf{q_n}} = \mathbf{e_1}^T\mathbf{A}^n\mathbf{c} \ge \max_{d \in \mathcal{B}_\mathbf{S}} \mathbf{e_1}^T\mathbf{A}^n\mathbf{d} = ||r(\mathbf{e_1}^T\mathbf{A}^n\mathbf{G}^{-1})^T|| = r||\mathbf{q_n}||. \end{equation} Let $\Psi(n, r)$ denote the proposition $ \seq{\mathbf{p}, \mathbf{q_n}} \ge r||\mathbf{q_n}||$. Our reduction works by proving that for any guess $r>0$, given $\varepsilon>0$, we can compute an $N$ such that for all $n \ge N$ \begin{align} \label{eq:property1} &\Psi(n, r) \Rightarrow n[nt - s] > \frac{(1-\varepsilon)\sqrt{7r}}{4\pi}. \\ \label{eq:property2} \neg &\Psi(n, r) \Rightarrow n[nt - s] < \frac{\sqrt{7r}}{(1-\varepsilon)4\pi}. \end{align} To compute $L_\infty(t, s) = \liminf_{n \in \mathbb{N}} n[nt-s]$ by increasingly precise approximations, we query Robust Uniform Ultimate Positivity: does $\Psi(n,r)$ hold for all but finitely many $n$? If the decision is YES, then we use property \ref{eq:property1} to argue that for any $\varepsilon$, $n[nt-s]$ exceeds $\frac{(1-\varepsilon)\sqrt{7r}}{4\pi}$ for all but finitely many $n$, hence $L_\infty(t, s)$ must be at least $\frac{\sqrt{7r}}{4\pi}$. Conversely, if the decision is NO, we use property \ref{eq:property2} to deduce that for any $\varepsilon$, $n[nt-s]$ falls short of $\frac{\sqrt{7r}}{(1-\varepsilon)4\pi}$ for infinitely many $n$, hence $L_\infty(t, s)$ must be at most $\frac{\sqrt{7r}}{4\pi}$. By definition, $L(t,s) = \inf_{n\in\mathbb{N}_{>0}}n[nt-s]$. Given the guess $r$, precision $\varepsilon$, the corresponding $N$, and oracle access to whether $\Psi(n, r)$ holds for all $n \ge N$, it follows from properties \ref{eq:property1} and \ref{eq:property2} that we can resolve the dichotomy between $\inf_{n \ge N}n[nt-s] \ge \frac{(1-\varepsilon)\sqrt{7r}}{4\pi}$ and $\inf_{n \ge N}n[nt-s] \le \frac{\sqrt{7r}}{(1-\varepsilon)4\pi}$. By explicitly computing $n[nt-s]$ for the prefix $n < N$ to arbitrary precision, one has a procedure for approximating $L(t, s)$. We now explain how we use Robust Positivity as an oracle to decide whether $\Psi(n, r)$ holds for all $n \ge N$. Note that as it is, our query specifies a recurrence $\mathbf{a}$, an initialisation $\mathbf{c}$, a neighbourhood defined by $\mathbf{S}$ asks for the Robust Positivity of a \textit{suffix} of the sequence, as opposed to the entire sequence. We create a new instance with updated $\mathbf{c'}$ and $\mathbf{S'}$ to implement the shift: \begin{align} \forall n\ge N.~ \mathbf{e_1}^T\mathbf{A}^n\mathbf{c} \ge \max_{\mathbf{d} \in \mathcal{B}_\mathbf{S}}\mathbf{e_1}^T\mathbf{A}^n\mathbf{d} ~&\Leftrightarrow~ \forall n.~ \mathbf{e_1}^T\mathbf{A}^n(\mathbf{A}^N\mathbf{c}) \ge \max_{\mathbf{d} \in \mathcal{B}_\mathbf{S}}\mathbf{e_1}^T\mathbf{A}^n(\mathbf{A}^N\mathbf{d}) \\ &\Leftrightarrow~ \forall n.~ \mathbf{e_1}^T\mathbf{A}^n(\mathbf{c'}) \ge \max_{\mathbf{d} \in \mathcal{B}_\mathbf{S'}}\mathbf{e_1}^T\mathbf{A}^n(\mathbf{d'}). \end{align} It is clear that $\mathbf{c'} = \mathbf{A}^n\mathbf{c}$. Using the same reasoning as we did in the derivation of equation \ref{eq:bijectivemap}, we argue $\mathbf{S'} = (\mathbf{A}^{-N})^T\mathbf{S}\mathbf{A}^{-N}$. The reduction is thus complete, but for the proof of properties \ref{eq:property1} and \ref{eq:property2}. By definition, $\Psi(n, r)$ holds if and only if $rn^2 + \frac{r}{2} + 1 - \cos 2(\pi(nt-s)) \ge r\sqrt{n^4 + n^2 + 2} $. Through elementary algebraic manipulations, we can alternately group the terms as \begin{equation} \label{eq:pivotal} 1 - \cos (2\pi (nt-s)) \ge \frac{r}{2}\left(\frac{7n^2 + 14}{(n^2 + \sqrt{n^4 + n^2 + 2})(n^2 +4+ \sqrt{n^4 + n^2 + 2})}\right) = r\cdot Q(n) \end{equation} We note that in the limit, the ratio of $Q(n)$ to $7/8n^2$ tends to $1$ from below. On the other hand, for small values of $x$, the expression $x^2/2$ is a close over-approximation for $1 -\cos x$. We capture the crucial interdependence in the following technical lemma. \begin{lemma} \label{lemma:numerical} Let $r > 0$. For every $\varepsilon > 0$, we can compute $N$ such that \begin{enumerate} \item For all $n\ge N$, $Q(n) > {7(1-\varepsilon)^2}/{8n^2}$. \item $1 - \cos x < 7r/{8N^2} ~\Rightarrow~ 1- \cos x \ge (1 - \varepsilon)^2x^2/2$. \end{enumerate} \end{lemma} For some $r, \varepsilon$, let $N$ be computed by Lemma \ref{lemma:numerical}. Consider $n \ge N$. In case $\Psi(n, r)$ holds, property \ref{eq:property1} follows by considering the beginning and end of the chain of inequalities \begin{equation} {2\pi^2[nt-s]^2} = \frac{[2\pi(nt-s)]^2_{2\pi}}{2} \ge 1 - \cos(2\pi(nt-s)) \ge r\cdot Q(n) > \frac{7r(1-\varepsilon)^2}{8n^2}. \end{equation} Similarly, if $\neg\Psi(n, r)$ holds, we can use Lemma \ref{lemma:numerical} to construct the chain \begin{equation} 2\pi^2(1-\varepsilon)^2 [nt-s]^2= \frac{(1-\varepsilon)^2 [2\pi(nt-s)]_{2\pi}^2}{2} \le 1 - \cos(2\pi(nt-s)) < r\cdot Q(n) < \frac{7r}{8N^2}. \end{equation} \section{Extensions and Perspective} \label{section:perspective} We note that our techniques for $\mathbf{S}$-Robust Non-uniform Ultimate Positivity hinge on the First Order Theory of the Reals. Observe that this was rather agnostic to the exact shape of the neighbourhood: we can easily extend the same techniques to arbitrary \textit{semi-algebraic} neighbourhoods. As outlined at the outset, we contributed towards a sharp and comprehensive picture of what is \textit{decidable} about Robust Positivity Problems for real algebraic Linear Recurrence Sequences. We find it remarkable that number-theoretic analyses involving Diophantine approximation, which usually show up in the context of hardness, also play a significant role in our \textit{decidability} proofs! However, a rather conspicuous gap in our picture is the status of $\mathbf{S}$-Robust Non-uniform Ultimate Positivity at order $5$: this seems to require even more delicate analysis. An obvious, but possibly tedious future direction would be to tie up the book-keeping loose ends, and meticulously account for the complexity of our techniques. We chose to work with algebraic numbers; in settings involving rational numbers where scaling to integers and accessing an $\mathsf{PosSLP}$ oracle is viable, the complexity usually lies in $\mathsf{PSPACE}$. However, this might blow up significantly in the absence of efficient positivity testing for a different class of arithmetic circuit. At a higher level, we note that we chose our norm to be based on the standard matrix inner product. It is interesting to investigate what kinds of decidability and hardness results hold for neighbourhoods specified using different norms. Perhaps, results could be universal across a wider class of norms, and there could be a profound underlying linear-algebraic reason whose discovery would be mathematically significant. In the grand Formal Methods scheme, the study of Hyperproperties \cite{hyperproperties} is an exciting natural way robustness problems for Linear Dynamical Systems could fit in. Hyperlogics reason about sets of traces of an infinite time system, rather than a single trace. They gained importance as a means to verify security in view of attacks like Meltdown and Spectre. A quintessential hyperproperty, for instance, would specify a reasonable notion of \textit{indistinguishability} of traces. In that regard, our notions of $\mathbf{S}$-Robust Positivity and $\mathbf{S}$-Robust Uniform Ultimate Positivity bear striking resemblance. Exploring deeper connections is a fascinating future research avenue. \appendix \section{Appendix: Notation and Prerequisites} \label{appendix:prelims} For the purposes of discussing robustness, we shall use $\mathcal{B}$ to denote the unit Euclidean ball in $\mathbb{R}^\kappa$, centred at the origin. Similarly, we use $\mathcal{B}_{\mathbf{S}}$ to denote the set of $\mathbf{d}$ such that $\mathbf{d}^T\mathbf{Sd} \le 1$. For real column vectors $\mathbf{x}, \mathbf{y}$, we use $\seq{\mathbf{x}, \mathbf{y}}$ to denote the inner product $\mathbf{x}^T\mathbf{y} = \mathbf{y}^T\mathbf{x}$. The notation $||\mathbf{x}||$ denotes the standard $\ell^2$-norm $\sqrt{\seq{\mathbf{x}, \mathbf{x}}}$. Throughout this paper, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ respectively denote the natural numbers, integers, rationals, reals, and complex numbers. $\alpha \in \mathbb{C}$ is said to be algebraic if it is a root of a polynomial with integer coefficients. Algebraic numbers form an algebraically closed field, denoted by $\overline{\mathbb{Q}}$. We denote the field of real algebraic numbers by $\mathbb{A}$. This Appendix contains a brief initiation to this number field $\mathbb{A}$ and $\overline{\mathbb{Q}}$. The key takeaways are that the usual arithmetic as well as polynomial root computation can be carried out with perfect precision, and that the First Order Theory of the Reals $\seq{\mathbb{R}; +, \cdot, \ge, 0, 1}$ is a decidable logical system powerful enough to fit our purposes. \subsection{Algebraic Numbers: Arithmetic} For an algebraic number $\alpha$, its defining polynomial $p_\alpha$ is the unique polynomial in $\mathbb{Z}[X]$ of least degree such that the GCD of its coefficients is $1$ and $\alpha$ is one of its roots. Given a polynomial $p \in \mathbb{Z}[X]$, we denote the length of its representation by $\text{size}(p)$; its height, denoted by $H(p)$, is the maximum absolute value of the coefficients of $p$; $d(p)$ denotes the degree of $p$. The height $H(\alpha)$ and degree $d(\alpha)$ of $\alpha$ are defined to be the height and degree of $p_\alpha$. For any $p \in \mathbb{Z}[X]$, the distance between distinct roots is effectively lower bounded in terms of its degree and height \cite{mignottecon}. This bound allows one to represent an algebraic number $\alpha$ as a 4-tuple $(p,a,b,r)$ where $p$ is the defining polynomial, and $a+bi$ is a rational approximation of sufficient precision $r\in\mathbb{Q}$. We use $\text{size}{\alpha}$ to denote the size of this representation, i.e., number of bits needed to write down this 4-tuple. Given a polynomial $p\in \mathbb{Z}[X]$, one can compute its roots in polynomial time \cite{findroots1operate1}. Recently, implementations of algorithms to factor polynomials in $\overline{\mathbb{Q}}[X]$ have been verified \cite{factor-algebraic}. Given $\alpha$, $\beta$ two algebraic numbers, one can always compute the representations of $\alpha+\beta$, $\alpha\beta$, $\frac 1 \alpha$, $\Re(\alpha)$, $\Im(\alpha), |\alpha|$, and decide $\alpha = \beta$, $\alpha > \beta$ in polynomial time wrt the size of their representations. \cite{findroots1operate1,findroots2operate2}. \subsection{First Order Theory of the Reals} This logical theory reasons about the universe of real numbers, and is denoted $\seq{\mathbb{R}; +, \cdot, \ge, 0, 1}$. That is, variables take real values; terms can be added and multiplied, we have the comparison predicate, and direct access to the constants $0$ and $1$. Thus, our propositional atoms are inequalities involving polynomials with integer coefficients. With existential quantifiers and polynomials, we can thus express algebraic constants too. Formally, we have access to only the existential quantifier, negation, and disjunction; however, this can express the universal quantifier and all other Boolean connectives as well. Variables are either quantified or free. Remarkably, the First Order Theory of the Reals admits quantifier elimination: for any formula $\chi(\mathbf{x})$, whose free variables are $\mathbf{x}$, there exists an \textbf{equivalent} formula $\psi(\mathbf{x})$ that does not contain any quantified variables. The following result is relevant to us. \begin{theorem}[Renegar \cite{renegar}] \label{thm:renegar} Let $M \in \mathbb{N}$ be fixed. Let $\chi(\mathbf{x})$ be a formula with fewer than $M$ variables in total. There exists a procedure that returns an equivalent quantifier-free formula $\psi(\mathbf{x})$ in disjunctive normal form. This procedure runs in time polynomial in the size of the representation of $\chi$. \end{theorem} \section{Appendix: Ostrowski Numeration System} \label{appendix:ostrowski} In this appendix, we prove Lemma \ref{lemma:existsreal}. We state number-theoretic properties of the continued fraction representation and Ostrowski Numeration System without proof. We refer the reader to \cite{bourla2016ostrowski} for a more detailed exposition, and we closely follow the discussion surrounding \cite[Propositions 1.1, 2.1]{berthe2022dynamics} in our own proof. We first prove a slightly simpler statement. \begin{lemma} \label{lemma:existsreal2} For every irrational number $x$, strictly decreasing real positive function $\psi$, and interval $\mathcal{I} = [\alpha, \beta] \subset [0, 1], ~ \alpha \ne \beta$, there exists $y \in \mathcal{I}$ such that $[nx - y] < \psi(n)$ for infinitely many $n$. \end{lemma} \begin{proof} Without loss of generality, we can assume that $x \in (0, 1)$. Consider the continued fraction representation of $x$: $[0; a_1, a_2, a_3, \dots]$ $$ x = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots}}}} $$ where $a_1, a_2, a_3, \dots \in \mathbb{N}$. Let the rational approximation of $x$ obtained by truncating the expansion at the $k^{th}$ level be $\frac{p_k}{q_k}$, i.e. $\frac{p_1}{q_1} = \frac{1}{a_1}$, and so on. Let $\theta_k = q_k x -p_k$. We have that $|\theta_k| = (-1)^k\theta_k$. It is well known that $|\theta_k| < 1/q_k$. We define $q_{-1} = p_0 := 0$, and $p_{-1} = q_0 := 1$, so that for $k \ge 1$, the following recurrences hold: $$ p_k = a_kp_{k-1} + p_{k-2}, ~ q_k = a_kq_{k-1} + q_{k-2} $$ We thus have that $q_k \ge \left(\frac{1 + \sqrt{5}}{2}\right)^k = \phi^k$. \begin{proposition}[\cite{berthe2022dynamics}] \label{prop:absconv} Let irrational $x$ and its continued fraction representation $[0; a_1, a_2, a_3, \dots]$ be as above. The infinite series $$ \sum_{i=1}^\infty a_i |\theta_{i-1}| $$ converges. \end{proposition} \begin{proposition}[Ostrowski Numeration System, \cite{berthe2022dynamics}] \label{prop:numsys} Every real number $y \in [0, 1)$ can be written uniquely in the form $$ y = \sum_{i=1}^\infty b_i |\theta_{i-1}| = \sum_{i=1}^\infty (-1)^{i-1}b_i \theta_{i-1} $$ where $b_i \in \mathbb{N}$ $b_i \le a_i$ for all $i \ge 1$. If for some $i$, $a_i = b_i$, then $b_{i+1} = 0$. $a_i \ne b_i$ for infinitely many odd, and infinitely many even indices $i$. \end{proposition} We prove Lemma \ref{lemma:existsreal2} by using the free choice of $b_i$ in this system to construct appropriate $y$. We first handle the issue of placing $y$ in the correct interval $[\alpha, \beta]$. Let $\beta - \alpha = \delta$. We use Proposition \ref{prop:absconv} to argue that there exists a suffix of the infinite series, such that changing the suffix does not change the real number it represents by more than $\delta/2$. Then, we can simply fix the corresponding prefix of $(\alpha + \beta)/2$ to be the prefix of $y$. Once this prefix is locked in, our strategy is to set $b_i$ to $0$ in even positions, and $1$ in some odd positions, to ensure that for sufficiently large $k$, $n_k = \sum_{i=1}^k b_i (-1)^{i-1}q_{i-1}$ is positive, and increasing in $k$. Now, notice that since $b_i, p_i$ are all integers, for any $y$, \begin{align*} [n_kx - y] &= \left[\sum_{i=1}^k b_i (-1)^{i-1}q_{i-1}x - \sum_{i=1}^k b_i (-1)^{i-1}p_{i-1} - y\right] \\ &= \left[\sum_{i=1}^k b_i (-1)^{i-1}\theta_{i-1} - y\right] \\ &= \left[- \sum_{i=k+1}^\infty b_i (-1)^{i-1}\theta_{i-1} \right] = \sum_{i=k+1}^\infty b_i |\theta_{i-1} |\\ &< \sum_{i=k+1}^\infty b_i \frac{1}{q_{i-1}} \le \sum_{i=k+1}^\infty b_i \frac{1}{\phi^{i-1}} \le \frac{c}{\phi^k} \end{align*} Note that the last constant $c$ can be set independently of the choice of which $b_i$ are $1$, and which are $0$: it comes from the convergence of the geometric sum. We now make the choice of where to set $b_i=1$. To conclude the proof, we shall show that given a decreasing function $\psi$, we can ensure that for infinitely many distinct $n_k$, $$[n_k x - y] < \frac{c}{\phi^k} \le \psi(n_k) = \psi\left(\sum_{i=1}^k b_i (-1)^{i-1}q_{i-1}\right)$$. The first inequality is guaranteed. Suppose the second inequality does not hold. Then, from $i = k$ onwards, we keep assigning $b_i := 0$. This holds $n_k$ constant as $k$ increases, but decreases $\frac{c}{\phi^k}$. Eventually, the second inequality will indeed hold. After this point, for the next odd $i$, we can set $b_i$ to $1$, and get a new $n_k$. We continue this ad infinitum, and we are done. \end{proof} Now, to get infinitely many \textit{even} $n$, apply Lemma \ref{lemma:existsreal2} with $2x$, $[a, b]$, $\psi_0(n) = \psi(2n)$. For some choice of $y$, there will be infinitely many $n$ such that $[2nx - y] < \psi_0(n) = \psi(2n)$. To get infinitely many \textit{odd} $n$, we can take a subset of the interval, and shift it by $x$. Take $\psi_1(n) = \psi(2n+1)$. For some choice of $y - x$, there will be infinitely many $n$ such that $[n(2x) - (y-x)] = [(2n+1)x - y] < \psi_1(n) = \psi(2n+1)$. \end{document}

\begin{document} \title{ALLSAT compressed with wildcards: An invitation for C-programmers} \author{Marcel Wild} \maketitle \begin{quote} {\bf Abstract} The model set of a general Boolean function in CNF is calculated in a compressed format, using wildcards. This novel method can be explained in very visual ways. Preliminary comparison with existing methods (BDD's and ESOPs) looks promising but our algorithm begs for a C encoding which would render it comparable in more systematic ways. \end{quote} \section{Introduction} By definition for us the {\it ALLSAT} problem is the task to enumerate all models of a Boolean function $\varphi=\varphi(x_1,...,x_t)$. In our article $\varphi$ is given by a CNF $C_1\wedge\ldots C_s$ with clauses $C_i$. The Boolean functions can be of a specific kind (e.g. Horn formulae), or they can be general Boolean functions. The article in front of you is one in a planned series\footnote{Article [W] contains a tentative account of the planned topics in the series, and it reviews wildcard-related previous publications of the author. The appeal of the article in your hands is its no-fuzz approach (for Theorems look in [W]) and its strong visual component.} of articles dedicated to the general theme of 'ALLSAT compressed with wildcards'. While much research has been devoted to SATISFIABILITY, the ALLSAT problem commanded less attention. The seemingly first systematic comparison of half a dozen methods is carried out in the article of Toda and Soh [TS]. It contains the following, unsurprising finding. If there are billions of models then the algorithms that put out their models one-by-one, stand no chance against the only competitor offering compression. The latter is a method of Toda (referenced in [TS]) that is based on Binary Decision Diagrams (BDD); see [K] for an introduction to BDD's. Likewise the method propagated in the present article has the potential for compression. Whereas BDDs achieve their compression using the common don't-care symbol $\ast$ (to indicate bits free to be 0 or 1) our method employs three further kinds of wildcards, and is entirely different from BDDs. Referring to these wildcards we call it the $men$-algorithm. In a nutshell, the $men$-algorithm retrieves the model set $Mod(\varphi)$ by {\it imposing} one clause after the other: (1)\quad $\{0,1\}^t\supseteq Mod(C_1)\supseteq Mod(C_1\wedge C_2)\supseteq \cdots\supseteq Mod(C_1\wedge\ldots\wedge C_n)=Mod(\varphi)$ The Section break up is as follows. In Section 2 we visualize the core maneuver for achieving (1). It will turn out that the intermediate stages of shrinking $\{0,1\}^t$ to $Mod(\varphi)$ do not exactly match the $n+1$ idealized stages $Mod(C_1\wedge\ldots\wedge C_k)$ in $(1)$. Section 3 starts with a well-known Boolean tautology, which for $k=2$ is $x_1\vee x_2\leftrightarrow x_1\vee(\overline{x_1}\wedge x_2)$. Generally the $k$ terms to the right of $\leftrightarrow$ are mutually exclusive, i.e. their model sets are {\it disjoint}. The problem of keeping systems $r_i$ of bitstrings disjoint upon imposing clauses on them, is the core technical difficulty of the present article. It will be handled by wildcards that adapt well to the above tautology. While in Section 3 only positive, or only negative clauses are considered (leading to dual kinds of wildcards), both kinds occur {\it together} in Section 4. This requires a third type of wildcard, which in turn makes the systems $r_i$ more intricate. Fortunately (Section 5) this doesn't get out of hand. Being able to alternately impose positive clauses like $x_1\vee x_2$ and negative clauses like $\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}$, does not enable us to impose the {\it mixed} clause $x_1\vee x_2\vee\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}$. But it certainly helps (Section 6). After the brief technical Section 7, in Section 8 we carry out the $men$-algorithm on some random moderate-size Boolean functions, and observe that the compression achieved compares favorably to BDD's and ESOP's. We calculate the latter two by using the commands {\tt expr2bdd} of Python and {\tt BooleanConvert} of Mathematica. Of course only systematic\footnote{The $men$-algorithm awaits implementation in either high-end Mathematica-code or in C. As to Mathematica, this remains the only programming language I master. If any reader wants to implement in C the $men$-algorithm, e.g. as a PhD topic, then he/she is welcome to seize this offer on a silver platter. The benefit (as opposed to pointless coding efforts with Mathematica) is that the $men$-algorithm coded in C or C+ becomes comparable to the methods evaluated in [TS], and possibly others.} experiments will show the precise benefits and deficiencies of the three methods. \section{ Visualization of the LIFO-stack and the Core Maneuver} {\bf 2.1} For the time being it suffices to think of a {\it 012men-row} as a row (=vector) $r$ that contains some of the symbols, $0, 1, 2,m,e,n$. Any such $r$ of length $t$ represents a certain set of length $t$ bitstrings. (This will be fully explained and motivated in later Sections). As a sneak preview, the number of length 10 bitstrings represented by $r=(2,m,e,m,1,n,e,e,1,n)$ is 84. We say that $r$ is $\varphi$-{\it infeasible} with respect to a 10-variate Boolean function $\varphi$ if no bitstring in $r$ satisfies $\varphi$. Otherwise $r$ is called $\varphi$-{\it feasible}. If {\it all} bitstrings in $r$ satisfy a Boolean formula $\psi$ then we say that $r$ {\it fulfills} $\psi$. {\bf 2.2} The input for the $men$-algorithm is any Boolean function $\varphi:\{0,1\}^t\to \{0,1\}$ given in CNF format $C_1\wedge C_2\wedge\ldots\wedge C_s$. The output of the men-algorithm is the {\it model set} $Mod(\varphi)$, i.e. the set of bitstrings ${\bf x}$ with $\varphi({\bf x})=1$. Here $Mod(\varphi)$ comes as a disjoint union of 012men-rows. If there is no ambiguity we may simply speak of rows instead of 012men-rows. The basic supporting data-structure is a Last-In-First-Out (LIFO) stack, filled with changing 012men-rows. (It is well known that LIFO amounts to DFS=Depth-First-Search of a tree, but the author prefers the LIFO point of view.) At the beginning the only 012men-row in the LIFO stack is $(2,2,...,2)$, thus the powerset $\{0,1\}^t$, see (1). Suppose that by induction we obtained a LIFO stack as shown in Figure 1a (so each $*$ is one of the symbols $0,1,2,m,e,n$). \includegraphics[scale=1.0]{menAlgoFig1} {\sl Figure 1a: LIFO stack before imposing $C_9$ \hspace*{1.1cm} Figure 1b: LIFO stack after imposing $C_9$} The top row $r$ fulfills $C_1\wedge ... \wedge C_8$, but not yet $C_9$, which hence is the {\it pending clause}. Similarly the other rows have pending clauses as indicated in the last column. To {\it impose} $C_9$ upon $r$ means replacing $r$ by a few successor rows $r_i$, called the {\it sons} of $r$, whose union is disjoint and contains exactly those bitstrings in $r$ that satisfy $C_9$. This maneuver is the core novel ingredient of the $men$-algorithm (as opposed to LIFO or SAT-solvers which are present in every decent ALLSAT algorithm [TS]). Sections 3 to 6 deliver the details of how the sons $r_i$ get calculated. As shown in Section 5 the number of sons is bounded by the length of the imposed clause. For now we illustrate the core maneuver with the Venn diagram in Figure 2. By assumption $r\subseteq Mod(C_1\wedge ... \wedge C_8)$ but $r\not\subseteq Mod(C_9)$. The part $r\setminus Mod(C_9)$ 'melts away' and the remainder of $r$ gets decomposed into four {\it candidate sons} $r_1$ to $r_4$. Having discarded the $\varphi$-infeasible row $r_3$ (more details in 2.2.1) we turn to $r_1,\ r_2,\ r_4$. They all fullfil $C_9$ by consruction. Say $r_2$ does not fulfil $C_{10}$. Then its pending clause is $C_{10}$. Say $r_4$ happens to fulfill $C_{10}$ to $C_{13}$ but not $C_{14}$. Then its pending clause is $C_{14}$. Say $r_1$ happens to fulfill $C_1$ up to $C_s$. Then $r_1$ is {\it final} in the sense that $r_1\subseteq Mod(\varphi)$. One then removes $r_1$ from the LIFO stack and outputs (or stores) it as part of the required compressed delivery of $Mod(\varphi)$. The rows $r_2,\ r_4$ are the {\it sons} of $r$ and take its place (in any order) on top of the LIFO stack, see Figure 1b. This finishes the imposition of $C_9$ upon $r$. \includegraphics[scale=1.0]{menAlgoFig2} {\sl Figure 2: Visualization of the core maneuver} {\bf 2.2.1.} That $r_3$ is $\varphi$-infeasible can be detected as follows. Translate $r_3$ into a Boolean CNF $\sigma$. (As a sneak preview, if $r_3=(e,0,e,1,e)$, then $\sigma\ =\ (x_1\vee x_3\vee x_5)\wedge \overline{x_2}\wedge x_4$.) Evidently $r_3$ is $\varphi$-infeasible, if and only if $\varphi\wedge\sigma$ is insatisfiable. This can be determined with any off-the-shelf SAT-solver. In contrast, determining the pending clause of a row works fast because for {\it any} 012men-row $r$ and {\it any} given clause $C$ it is straightforward (Section 7) to check whether or not $r'$ fulfills $C$. {\bf 2.3} By induction at all stages the union $U$ of all final rows and of all rows in the LIFO stack is disjoint and contains $Mod(\varphi)$. Whenever the pending clause of any top row $r$ gets imposed on $r$, a nonempty part of $r$ melts away, and so the new set $U$ strictly shrinks. Hence the procedure ends in finite time. Specifically, once the LIFO stack becomes empty, the set $U$ equals the disjoint union of all final rows, which in turn equals $Mod(\varphi)$. See Section 6 for carrying out all of this with a concrete Boolean function $\varphi$. \section{The Flag of Bosnia and its higher level variants} {\bf 3.1} The real Flag of Bosnia\footnote{Strictly speaking Bosnia should be Bosnia-Herzegowina, but this long name gets too clumsy. Other national flags, such as the Flag of Papua (used in previous publications), have similar patterns but miss out on relevant details.} (FoB) features a white main diagonal, the lower triangle is blue, and the upper triangle yellow. Using $0,1,2$ as colors the two kinds of FoBes we care about are rendered in Figure 3 and 4. Here {\it Type} 1 and {\it Type} 0 refers to the color of the diagonal. \includegraphics[scale=0.58]{menAlgoFig3and4} {\sl Figure 3: FoB of Type 1 \hspace*{1.7cm} Figure 4: FoB of Type 0.} The FoB of Type 1 visualizes in obvious ways the righthand side of the well-known tautology (2)\qquad $(x_1\vee x_2\vee x_3\vee x_4)\ \leftrightarrow\ x_1\,\vee\,(\overline{x_1}\wedge x_2)\,\vee\, (\overline{x_1}\wedge\overline{x_2}\wedge x_3)\,\vee\,(\overline{x_1}\wedge\overline{x_2}\wedge\overline{x_3}\wedge x_4)$ The dimension $4\times 4$ generalizes to any $k\times k$, but only $k\ge 2$ will be relevant. It is essential that the four clauses on the right in $(2)$ are mutually disjoint, i.e. their conjunction is insatisfiable. Equation (2) (for any $k\ge 2$) is the key for many methods that {\it orthogonalize} an arbitrary DNF into an {\it exclusive sums of products (ESOP)}; see [B,p.327]. It will be essential for us as well, but we orthogonalize CNF's, not DNF's. What is more, our use of wildcards results into 'fancy kinds of ESOPs', i.e. disjoint unions of 012men-rows. As in previous publications we prefer to write 2 for the common don't-care symbol $\ast$. Thus the $012$-{\it row} $(2,0,1,2,1)$ by definition is the set of bitstrings $\{({\bf 0},0,1,{\bf 0},1),({\bf 0},0,1,{\bf 1},1),({\bf 1},0,1,{\bf 0 },1),({\bf 1},0,1,{\bf 1},1)\}$. Thus in view of (2) the model set of $x_1\vee x_2\vee x_3\vee x_4$ is the disjoint union of the four $012$-rows constituting the FoB in Figure 3. This matches the row-wise cardinality count: $8+4+2+1=2^4-1$. Dually the FoB of Type 0 in Figure 4 visualizes the tautology (3)\qquad $(\overline{x_1}\vee \overline{x_2}\vee \overline{x_3}\vee \overline{x_4})\ \leftrightarrow\ \overline{x_1}\vee(x_1\wedge \overline{x_2})\vee(x_1\wedge x_2\wedge \overline{x_3})\vee (x_1\wedge x_2\wedge x_3\wedge \overline{x_4})$ {\bf 3.2} More original than writing 2 instead of $\ast$, is it to dismiss the whole FoB in Figure 3 and replace it by the single wildcard $(e,e,e,e)$ which by {\it definition}\footnote{Surprisingly, this idea seems to be new. Information to the contrary is welcome. The definition generalizes to tuplets $(e,e...,e)$ of length $t\ge 2$. For simplicity we sometimes strip $(e,e,...,e)$ to $ee...e$. Observe that a single $e$ (which we forbid) would amount to $1$.} is the set of all length 4 bitstrings ${\bf x}=(x_1,x_2,x_3,x_4)$ with 'at least one 1'. In other words, only (0,0,0,0) is forbidden. Thus e.g $(1,e,0,e)$ is the set of bitstrings $\{(1,{\bf 1},0,{\bf 0}),\ (1,{\bf 0},0,{\bf 1}),\ (1,{\bf 1},0,{\bf 1})\}$. If several $e$-wildcards occur, they need to be distinguished by subscripts. For instance the $012e$-{\it row} $r_1$ in Figure 5a represents the model set of the CNF (4)\qquad $(x_1\vee x_2\vee x_3\vee x_4)\wedge(x_5\vee x_6\vee x_7\vee x_8)$. The $e$ symbols need {\it not be contiguous}. But for better visualization our examples tend to clump $e$-symbols with the same subscript. Not all symbols $0,1,2,e$ need to occur in a $012e$-row. In other words, $012$-rows are special cases of $012e$-rows. {\bf 3.3} The fewest number of disjoint $012$-rows required to represent the single row $r_1$ seems to be a hefty sixteen. These 012-rows are obtained by 'multiplying out' two FoBes of Type 1. Thus the $e$-wildcard boosts compression. But can the $e$-formalism handle overlapping clauses? It is here where the FoBes dismissed in 3.2 get vindicated, but they need to reinvent themselves as 'Meta-FoBes'. To fix ideas, let ${\cal F}:=Mod(C_1\wedge C_2\wedge C_3)\subseteq Mod(C_1\wedge C_2)=r_1$, where (5)\qquad $C_1\wedge C_2\wedge C_3:=(x_1\vee x_2\vee x_3\vee x_4)\wedge(x_5\vee x_6\vee x_7\vee x_8)\wedge(x_3\vee x_4\vee x_5\vee x_6)$. We claim that ${\cal F}$ is the disjoint union of the two $012e$-rows $r_2$ and $r_3$ in Figure 5a, and shall refer to the framed part as a {\it Meta-FoB} (of dimensions $2\times 2$). Specifically, the bitstrings $(x_3,x_4,x_5,x_6)$ satisfying the overlapping clause $x_3\vee x_4\vee x_5\vee x_6$ are collected in $(e,e,e,e)$ and come in two sorts. The ones with $x_3=1$ or $x_4=1$ are collected in $(e,e,2,2)$, and the other ones are in $(0,0,e,e)$. These two quadruplets constitute, up to some adjustments, the two rows of our Meta-FoB. The first adjustment is that the right half of $(e,e,2,2)$ gets erased by the left part of the old constraint $(e_2,e_2,e_2,e_2)$ in $r_1$. The further adjustments do not concern the shape of the Meta-FoB per se, but rather are {\it repercussions} caused by the Meta-FoB outside of it. Namely, $(e_1,e_1,e_1,e_1)$ in $r_1$ splits into $(2,2,e,e)$ (left half of $r_2$) and $(e_1,e_1,0,0)$ (left part of $r_3$). It should be clear why $(e_2,e_2,e_2,e_2)$ in $r_1$ transforms differently: It stays the same in $r_2$ (as noticed already), and it becomes $(e,e,2,2)$ in $r_3$. Because of its diagonal entries (shaded) our Meta-FoB is\footnote{Generally the lengths of the diagonal entries $e...e$ match the cardinalities of the traces of the overlapping clause. For instance, imposing $x_4\vee x_5$ instead of $x_3\vee x_4\vee x_5\vee x_6$ triggers the Meta-FoB of Type $(e,1)$ in Figure 5b. We keep the terminology Type $(e,1)$ despite the fact that all diagonal entries are $1$. Confusion with FoBes of Type 1 (Figure 3)is unlikely.} a Meta-FoB of {\it Type $(e,1)$}. Why defining $r_2$ and $r_3$ in such complicated ways? Isn't $r_2\uplus r_3$ just the same as $r_1\setminus (e_1,e_1,0,0,0,0,e_2,e_2)$? Yes it is (and it matches 180+36=225-9), but the $men$-algorithm only digests set systems rendered as disjoint set-unions, it cannot handle set-differences. \includegraphics[scale=0.46]{menAlgoFig5} {\sl \hspace*{0.5cm} Figure 5a: Meta-FoB of Type $(e,1)$ \hspace*{2cm} Fig. 5b: Small Meta-FoB of Type $(e,1)$ } {\bf 3.4} In dual fashion we define a second wildcard $(n,n,...,n)$ as the set of all length $t$ bitstrings that have 'at least one 0' (where $t$ is the number of $n$'s). We define $012n$-{\it rows} dually to 012e-rows. Mutatis mutandis the same arguments as above show that by using a dual Meta-FoB of Type $(n,0)$ one can impose $(n,n,...,n)$ upon disjoint constraints $(n_i,n_i,...,n_i)$. See Figure 6 which shows that the model set of (5')\qquad $(\overline{x_1}\vee\overline{x_2} \vee \overline{x_3}\vee \overline{x_4})\wedge(\overline{x_5}\vee\overline{x_6} \vee \overline{x_7}\vee \overline{x_8})\wedge(\overline{x_3}\vee\overline{x_4} \vee \overline{x_5}\vee \overline{x_6})$ can be represented as disjoint union of the two $012n$-rows $r_2$ and $r_3$. \includegraphics[scale=0.49]{menAlgoFig6} {\sl Figure 6: Meta-FoB of Type $(n,0)$} \section{Positive and negative clauses simultaneously} New issues arise if $nnnn$ (or dually $eeee$) needs to be imposed on {\it distinct} types of wildcards, say $n_1n_1n_1n_1$ and $e_1e_1e_1e_1$ as occuring in row $r_1$ of Figure 7. Specifically, let $n_1n_1n_1n_1$ model $C_1\,=\,\overline{x_1}\vee\overline{x_2}\vee\overline{x_3}\vee\overline{x_4}$, let $e_1e_1e_1e_1$ model $C_2=x_5\vee x_6\vee x_7\vee x_8$, and $nnnn$ model the overlapping clause $C_3\,=\,\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}\vee\overline{x_6}$. We need to sieve the model set ${\cal F}:=Mod(C_1\wedge C_2\wedge C_3)$ from $r_1:=Mod(C_1\wedge C_2)$. Thus we need to represent $\{{\bf x}\in r_1:\ {\bf x}\ {\it satisfies}\ C_3\}$ in compact format. To do so write $r_1=r_2\uplus r_2'$, where $r_2:=\{{\bf x}\in r_1:\ {\bf x}\ {\it satisfies}\ \overline{x_3}\vee\overline{x_4}\}\ =\ \{{\bf x}\in r_1:\ x_3=0\,{\it or}\, x_4=0\}$, $r_2':=\{{\bf x}\in r_1:\ {\bf x}\ {\it violates}\ \overline{x_3}\vee \overline{x_4}\}\ =\ \{{\bf x}\in r_1:\ x_3=x_4=1\}$. It follows that $r_2\subseteq {\cal F}$ and in fact (6)\qquad ${\cal F}\ =\ r_2\uplus \{{\bf x}\in r_2':\ {\bf x}\ {\it satisfies}\ C_3\}$ \includegraphics[scale=0.53]{menAlgoFig7} {\sl Figure 7: Meta-FoB of Type $(n,m,0)$} It is clear that $r_2$ and $r_2'$ can be rendered as in Figure 7. For instance ${\bf x}=(0,0,1,1,1,1,1,1)$ is in $r_2'$ but does not satisfy $C_3$. How can one represent the rightmost set in (6) in a useful format? To do so we define a third wildcard $$(m,m,\ldots,m):=(e,e,\ldots,e)\cap(n,n,\ldots,n)$$ In other words, $(m,m,\ldots,m)$ is the set of all bitstrings with 'at least one 1 and at least one 0'. A moment's thought shows that the rightmost set in (6) is the disjoint union of $r_3$ and $r_4$ in Figure 7. The framed part in Figure 7 constitutes a Meta-FoB of {\it Type} $(n,m,0)$, i.e. all diagonal entries are $n$ or $m$ or $0$. While $ee...e$ and $nn...n$ are duals of each other, $mm...m$ is selfdual. Hence in a dual way we can impose $ee...e$ (matching $x_3\vee x_4\vee x_5\vee x_6$) upon $r_1$ by virtue of a Meta-FoB of {\it Type} $(e,m,1)$. This is carried out in Figure 8. We note in passing that the choice of letters $n$ and $m$ stems from 'nul' and 'mixed' respectively. The letter $e$ stems from 'eins' which is German for 'one'. \includegraphics[scale=0.48]{menAlgoFig8} {\sl Figure 8: Meta-FoB of Type $(e,m,1)$} \section{Imposing a positive or negative clause upon a 012men-row} Having a third wildcard $mm...m$ proved to be useful in Section 4, but the prize is that we need to cope with general $012men$-{\it rows} (defined in the obvious way) and impose $nn...n$ or $ee...e$ (or even $mm...m$) upon them! Fortunately imposing $mm...m$ won't be necessary and the imposition of $nn...n$ or $ee...e$ upon a 012men-row can be achieved using Meta-FoBes of Type $(n,m,0)$ and $(e,m,1)$ respectively. In Figure 9 the imposition of $\overline{x_3}\vee\overline{x_4}\vee\overline{x_6}\vee\overline{x_7}\cdots\vee\overline{x_{14}} $ upon the $012men$-row $r_1$ is carried out (thus $nn...n$ has length 11 viewing that $\overline{x_5}$ is omitted). This boils down to the imposition of the shorter clause $\overline{x_6}\vee\overline{x_7}\cdots\vee\overline{x_{14}}$ since each ${\bf x}\in r_1$ has $x_3=x_4=1$. We omit the details of why the Meta-FoB of Type $(n,m,0)$, and its repercussions outside, look the way they look. For the most part this should be self-explanatory in view of our deliberations so far. \includegraphics[scale=0.77]{menAlgoFig9} {\sl Figure 9: Another Meta-FoB of Type $(n,m,0)$} {\bf 5.1} But let us add a few comments in a different vein. Notice that $\overline{x_6}\vee\overline{x_7}\cdots\vee\overline{x_{14}}$ has 9 literals whereas the induced Meta-FoB has 7 rows. Generally speaking the shaded rectangles in any Meta-FoB arising from imposing a positive or negative clause upon $r$, are of dimensions $1\times t$ (any $t\ge 1$ can occur) and $2\times 2$. This implies that the number of rows in such a Meta-FoB (=number of sons of $r$) is at most the number of literals in that clause. Although imposing a {\it mixed} clause is more difficult (Section 6), it is easy to see that the number of literals remains an upper bound to the number of sons. {\bf 5.2} Imposing the corresponding {\it positive} clause $x_3\vee x_4\vee\cdots\vee x_{14}$ upon $r_1$ would be trivial since each bitstring ${\bf x}$ in $r_1$ satisfies this clause in view of $x_3=x_4=1$. Energetic readers may enjoy imposing the shorter clause $ x_6\vee\cdots\vee x_{14}$ (thus $ee...e$) upon $r_1$ by virtue of a Meta-FoB of Type $(e,m,1)$. In contrast, we dissuade imposing $mm...m$ upon $r_1$ by virtue of some novel Meta-FoB because for the time being\footnote{This concerns our present focus on arbitrary CNFs. For special types of CNFs, e.g. such that the presence of $x_i\vee x_j\vee\cdots\vee x_k$ implies the presence of $\overline{x_i}\vee \overline{x_j}\vee\cdots\vee\overline{x_k}$, imposing $mm...m$ may well be beneficial.} only our capability (to be honed in Section 6) to either impose $ee...e$ or $nn...n$ matters. This skill, as well as a clever ad hoc maneuver, will suffice to impose any {\it mixed} clause upon any 012men-row. \section{Handling general (=mixed) clauses} Let us embark on the compression of the model set of the CNF with clauses \begin{description} \item{(7) }$\ C_1=\overline{x_1}\vee\overline{x_2}\vee\overline{x_3},\hspace{0.6cm} C_2=x_4\vee x_5\vee x_6\vee x_7,\hspace{0.6cm} C_3=\overline{x_8}\vee\overline{x_9}\vee\overline{x_{10}}$, \item{} \hspace{0.6cm} $C_4=\overline{x_2}\vee\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}\vee x_6\vee x_7\vee x_8\vee x_9,\hspace{0.6cm} C_5=x_1\vee\overline{x_3}\vee\overline{x_4}\vee \overline{x_6}\vee\overline{x_7}$ \end{description} It is clear that $r_1$ in Figure 10 compresses the model set of $C_1\wedge C_2\wedge C_3$. Hence the pending clause of $r_1$ is $C_4$. In order to sieve ${\cal F}:=Mod(C_1\wedge C_2\wedge C_3\wedge C_4)$ from $r_1=Mod(C_1\wedge C_2\wedge C_3)$ we first split $r_1$ as $r_1=r_2\uplus r_2'$ where \begin{description} \item{(8) } $r_2:=\,\{{\bf x}\in r_1:\ {\bf x}\ {\it satisfies}\ \overline{x_2}\vee\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}\ \}\ $ {\it and} \item{} \hspace{0.7cm} $r_2':=\,\{{\bf x}\in r_1:\ {\bf x}\ {\it violates}\ \overline{x_2}\vee\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}\ \}$. \end{description} Then we have $r_2\subseteq {\cal F}$, and (akin to (6)) in fact $(9)\quad {\cal F}=r_2\uplus \{{\bf x}\in r_2':\ {\bf x}\ {\it satisfies}\ x_6\vee x_7\vee x_8\vee x_9\ \}$. Similar to (6), but more demanding, both parts on the right in (9) must now be rewritten as disjoint union of $012men$-rows. {\bf 6.1} Enter the 'ad hoc maneuver' mentioned above: Roughly speaking both bitstring systems $r_2$ and $r_2'$ temporarily morph into 'overloaded' 012men-rows. The latter will morph back, one after the other in 6.1.2 and 6.1.3, in disjoint collections of (ordinary) 012men-rows. Two definitions are in order. If in a 012men-row $r$ we bar any symbols, then the obtained {\it overloaded Type A row} by definition consists of the bitstrings in $r$ that feature at least one $0$ on a barred location. It follows that $r_2$ equals the overloaded Type A row with the same name in Figure 10. Similarly, if in a row $r$ we encircle, respectively decorate with stars, nonempty disjoint sets of symbols, then the obtained {\it overloaded Type B row} by definition consists of the bitstrings in $r$ that feature 1's at all encircled locations, and feature at least one 1 on the starred locations. It follows that the rightmost set in (9) equals the overloaded Type B row $r_3$ in Figure 10. We shall see that merely starring symbols (omitting encircling) also comes up. The definition of such an {\it overloaded Type C row} is as expected. {\bf 6.1.2} As to turning $r_2$ and $r_3$ into ordinary $012men$-rows, we first look at $r_2$, while carrying along the overloaded row $r_3$. Transforming $r_2$ simply amounts to impose the negative part $\overline{x_2}\vee\overline{x_3}\vee\overline{x_4}\vee\overline{x_5}$ of clause $C_4$ upon $r_1$, and hence works with the Meta-FoB of Type $(n,m,0)$ that stretches over $r_4$ to $r_6$. As to $r_5$, it fulfills $C_5$ (since each ${\bf x}\in C_5$ has $x_4=0$), and so is final and leaves the LIFO stack (Section 2). {\bf 6.1.3} As to transforming $r_3$, the first step is to replace the encircled symbols by 1's and to record the ensuing repercussions. Some starred symbols may change in the process but they must {\it keep their star}. The resulting overloaded Type C row still represents the {\it same} set of bitstrings $r_3$. The second step is to impose the positive part $x_6\vee x_7\vee x_8\vee x_9$ of $C_4$ by virtue of a Meta-FoB, see $r_7$ to $r_9$ in Figure 10. \includegraphics[scale=0.77]{menAlgoFig10} {\sl Figure 10: The $men$-algorithm in action. Snapshots of the LIFO stack.} {\bf 6.1.4} In likewise fashion (details left to the reader) the algorithm proceeds in Figure 11. Observe that in Figure 11 we {\it permuted} the columns in order to better visualize the imposition of clause $C_5$. Note that $r_{10},\ r_{11}$ are overloaded rows of Type A and B. The $men$-algorithm ends after the last row in the LIFO stack gets removed. \includegraphics[scale=0.77]{menAlgoFig11} {\sl Figure 11: Further snapshots of the LIFO stack.} Altogether there are ten (disjoint) final rows $r_5,\ r_8,\ r_9,\ r_{12},\ r_{13},\ r_{14},\ r_{11},\ r_{15},\ r_{16},\ r_{17}$. Their union is $Mod(\varphi)$, which hence is of cardinality $$|Mod(\varphi)|=21+1+4+420+14+168+14+28+21+14=695$$ \section{Testing whether a 012men-row fulfills a clause} Here we verify the claim made in 2.2 that checking whether a 012men-row $r$ fulfills a clause $C$ is straightforward. Indeed, focusing on the most elaborate case of a {\it mixed} clause $C$ the following holds. \begin{description} \item[(10)] If $C=x_1\vee\cdots\vee x_s\vee \overline{x_{s+1}}\vee\cdots\vee \overline{x_t}$ and $r=(a_1,..,a_s,a_{s+1},..,a_t,\ldots)$ then $r$ fulfills $C$ iff one of these cases occurs: \begin{description} \item[(i)] For some $1\le j\le s$ one has $a_j=1$; \item[(ii)] $\{1,\ldots,s\}$ contains the position-set of a full $e$-wildcard or full $m$-wildcard; \item[(iii)] For some $s+1\le j\le t$ one has $a_j=0$; \item[(iv)] $\{s+1,\ldots,t\}$ contains the position-set of a full $n$-wildcard or full $m$-wildcard; \end{description} \end{description} {\it Proof of} $(10)$. It is evident that each of $(i)$ to $(iv)$ individually implies that all bitstrings ${\bf x}\in r$ satisfy $C$. Conversely suppose that $(i)$ to $(iv)$ are false. We must pinpoint a bitstring in $r$ that violates $C$. To fix ideas, consider $r$ of length 18 and the clause $C=x_1\vee\cdots\vee x_6\vee \overline{x_7}\vee\cdots\vee\overline{x_{13}}$. (For readibility the disjunctions $\vee$ are omitted in Figure 12.) Properties $(i)$ to $(iv)$ are false for $C$. For instance the position-set $\{6,7,8\}$ of $m_1m_1m_1$ is neither contained in $\{1,\ldots,s\}$ nor in $\{s+1,\ldots,t\}$. That it is contained in their union is irrelevant. One checks that $r_{\rm vio}\subseteq r$ and that each bitstring $x\in r_{\rm vio}$ violates $C$. \includegraphics[scale=0.73]{menAlgoFig12} {\sl Figure 12: The $012men$-row $r$ does not fulfill clause $C$.} \section{ Comparison with BDD's and ESOP's} We reiterate from Section 1 that the $men$-algorithm has not yet been implemented. Therefore we content ourselves to take two medium-size random CNFs and hand-calculate what the $men$-algorithm does with them. We compare the outcome with two competing paradigms; ESOP's in 8.2, and BDD's in 8.3. But first we warm up in 8.1 by looking how ESOP and BDD handle $Mod(\mu_t)$ for $\mu_t=(x_1\vee\cdots\vee x_t)\wedge(\overline{x_1}\vee\cdots\vee\overline{x_t})$. Recall that the $men$-algorithm achieves optimal compression here: $Mod(\mu_t)=(m,m,\ldots,m)$. {\bf 8.1} One checks that the 012-rows of the Table on the right of Figure 13 constitute an ESOP of $\mu_5$. Let us verify that the BDD on the left in Figure 13 also yields $\mu_5$. As for any BDD, each nonleaf node A yields 'its own' Boolean function (on a subset of the variables). For instance, there are two nodes labelled with $x_2$. The left, call it A, yields a Boolean function $\alpha(x_2,x_3,x_4,x_5)$ whose model set is the disjoint union of the four 012-rows in the top square in the Table on the right. For instance, the bitstring $(0,0,1,0)$ belongs to $(0,0,1,2)$, and indeed it triggers (in the usual way, [K]) a path that leads from A to $\top$. Similarly the right node labelled $x_2$, call it B, yields some Boolean function $\beta(x_2,x_3,x_4,x_5)$ whose model set is the disjoint union of the four 012-rows in the bottom square in the Table on the right. It is now evident that whole Table represents the model set of the whole BDD, thus $Mod(\mu_5)$. Conversely, as is well known [B,p.327], each BDD gives rise\footnote{Unfortunately Mathematica does not openly support BDDs, and so the author had to turn to Python for that purpose. In general the Python BDD's yielded quite different ESOPs than Mathematica's ESOP-command. In the unlikely case that the latter is based on BDD's (I didn't manage to find out) this must be due to different variable orderings.} to an ESOP. It is easy to calculate its exclusive products, and even easier to to predict their number. See [W] for details. \includegraphics[scale=1.1]{menAlgoFig13} {\sl Figure 13: Each BDD readily yields an ESOP} {\bf 8.2} Consider this CNF: \begin{description} \item{(11)} $\ \varphi_1=(x_5\vee x_7\vee x_{10}\vee\overline{x_2}\vee\overline{x_4})\wedge(x_1\vee x_2\vee x_9\vee\overline{x_7}\vee\overline{x_5})$ \item{} \hspace{1.6cm} $\wedge\ (x_2\vee x_3\vee x_7\vee\overline{x_4}\vee\overline{x_9})\wedge(x_8\vee x_9\vee x_{10}\vee\overline{x_4}\vee\overline{x_9})$ \end{description} All clauses have 3 positive and 2 negative literals, which were randomly chosen (but avoiding $x_i\vee\overline{x_i}$) from a set of 20 literals. Table 14 shows the fourteen rows that the men-algorithm produces to compresses $Mod(\varphi_1)$. One reads off that $|Mod(\varphi_1)|=16+48+\cdots+18=898$. \includegraphics[scale=0.83]{menAlgoFig14} {\sl Table 14: Applying the men-algorithm to $\varphi_1$ in (11).} Using the Mathematica-command {\tt BooleanConvert} (option "ESOP") transforms (11) to an ESOP $(\overline{x_4}\wedge x_9)\vee(x_1\wedge\overline{x_4}\wedge x_8\wedge\overline{ x_9})\vee\cdots$, which amounts to a union $(2,2,2,0,2,2,2,2,1,2)\cup $ $(1,2,2,0,2,2,2,1,0,2)\cup\cdots$ of 23 disjoint 012-rows. We note that the ESOP algorithm is quite sensitive\footnote{And so is the men-algorithm. For both methods, no attempt to optimize clause order has been made.} to the order of clauses. Incidentally the 23 rows above stem from one of the optimal permutations of clauses; the worst would yield 36 rows. Adding the random clause $(x_5\vee x_6\vee x_8\vee\overline{x_3}\vee\overline{x_9})$ to $\varphi_1$ triggers twenty six 012men-rows, but between 27 and 56 many 012-rows with the ESOP-algorithm. The second example in (12) has longer clauses, all of them either positive or negative (for ease of hand-calculation). Long clauses make our wildcards more effective still. \begin{description} \item{(12)} $\varphi_2=(x_3\vee x_4\vee x_6\vee x_7\vee x_9\vee x_{14}\vee x_{15}\vee x_{16}\vee x_{17}\vee x_{18})$ \item{} \hspace{1.4cm} $\wedge\ (\overline{x_3}\vee\overline{x_5}\vee\overline{x_8}\vee\overline{x_9}\vee\overline{x_{11}}\vee\overline{x_{12}}\vee\overline{x_{13}}\vee\overline{x_{14}}\vee\overline{x_{15}}\vee\overline{x_{17}})$ \item{} \hspace{1.4cm} $\wedge\ (x_1\vee x_4\vee x_5\vee x_6\vee x_9\vee x_{12}\vee x_{14}\vee x_{15}\vee x_{17}\vee x_{18}) $ \item{} \hspace{1.4cm} $\wedge\ (\overline{x_1}\vee\overline{x_2}\vee\overline{x_3}\vee\overline{x_8}\vee\overline{x_{11}}\vee\overline{x_{13}}\vee\overline{x_{14}}\vee\overline{x_{16}}\vee\overline{x_{17}}\vee\overline{x_{18}})$ \item{} \hspace{1.4cm} $\wedge\ (x_2\vee x_3\vee x_7\vee x_8\vee x_{11}\vee x_{13}\vee x_{14}\vee x_{16}\vee x_{17}\vee x_{18}) $ \end{description} Table 15 shows the ten rows the men-algorithm uses to compress $Mod(\varphi_2)$. In contrast the ESOP-algorithm uses between 85 and 168 many 012-rows, depending on the order of the clauses. \includegraphics[scale=1.04]{menAlgoFig15} {\sl Table 15: Applying the men-algorithm to $\varphi_2$ in (12).} {\bf 8.3} As to BDD's, one of many\footnote{Recall that the size of a BDD greatly depends on the chosen variable order. The variable order can be optimized in intelligent ways [K] but that costs time. The author does not know whether Python 3.5.2 embarks on such manoevers.} BDD's of $\varphi_2$ is rendered in Figure 16 below. It has 60 nodes and induces (in the way sketched in 8.1) an ESOP with 173 exclusive products. \includegraphics[scale=0.99]{bigbdd.JPG} {\sl Figure 16: Some BDD of $\varphi_2$.} \section*{References} \begin{enumerate} \item[{[B]}] E. Boros, Orthogonal forms and shellability, Section 7 in: Boolean Fuctions (ed. Y. Crama, P.L. Hammer), Enc. Math. Appl. 142, Cambridge University Press 2011. \item[{[K]}] D. Knuth, The art of computer programming, Volume 4 (Preprint), Section 7.14: Binary decision diagrams, Addison-Wesley 2008. \item [{[TS]}] Takahisa Toda and Takehide Soh. 2016. Implementing Efficient All Solutions SAT Solvers. J. Exp. Algorithmics 21, Article 1.12 (2016), 44 pages. DOI: https://doi.org/10.1145/2975585 \item[{[W]}] M. Wild, ALLSAT compressed with wildcards: Converting CNFs to orthogonal DNFs, ResearchGate. \end{enumerate} \end{document}

\begin{document} \begin{abstract} In recent work, Cuntz, Deninger and Laca have studied the Toeplitz type C*-algebra associated to the affine monoid of algebraic integers in a number field, under a time evolution determined by the absolute norm. The KMS equilibrium states of their system are parametrized by traces on the C*-algebras of the semidirect products $J_\gamma \rtimes \ok^*$ resulting from the multiplicative action of the units $\ok^*$ on integral ideals $J_\gamma$ representing each ideal class $\gamma \in \Cl_K$. At each fixed inverse temperature $\beta >2$, the extremal equilibrium states correspond to extremal traces of $C^*(J_\gamma \rtimes \ok^*)$. Here we undertake the study of these traces using the transposed action of $\ok^*$ on the duals $\hat{J}_\gamma$ of the ideals and the recent characterization of traces on transformation group C*-algebras due to Neshveyev. We show that the extremal traces of $C^*(J_\gamma \rtimes \ok^*)$ are parametrized by pairs consisting of an ergodic invariant measure for the action of $\ok^*$ on $\hat{J}_\gamma$ together with a character of the isotropy subgroup associated to the support of this measure. For every class $\gamma$, the dual group $\hat{J}_\gamma$ is a $d$-torus on which $\ok^*$ acts by linear toral automorphisms. Hence, the problem of classifying all extremal traces is a generalized version of Furstenberg's celebrated $\times_2$ $\times_3$ conjecture. We classify the results for various number fields in terms of ideal class group, degree, and unit rank, and we point along the way the trivial, the intractable, and the conjecturally classifiable cases. At the topological level, it is possible to characterize the number fields for which infinite $\ok^*$-invariant sets are dense in $\hat{J}_\gamma$, thanks to a theorem of Berend; as an application we give a description of the primitive ideal space of $C^*(J_\gamma \rtimes \ok^*)$ for those number fields. \end{abstract} \maketitle \section{Introduction} Let $K$ be an algebraic number field and let $\OO_{\! K}$ denote its ring of integers. The associated multiplicative monoid $\OO_{\! K}^\times := \OO_{\! K} \setminus \{0\}$ of nonzero integers acts by injective endomorphisms on the additive group of $\OO_{\! K}$ and gives rise to the semi-direct product $\OO_{\! K}^\timesox$, the affine monoid (or `$b+ax$ monoid') of algebraic integers in $K$. Let $\{\xi_{(x,w)}: (x,w) \in \OO_{\! K}^\timesox\}$ be the standard orthonormal basis of the Hilbert space $\ell^2(\OO_{\! K}^\timesox)$. The left regular representation $L$ of $ \OO_{\! K}^\timesox$ by isometries on $\ell^2(\OO_{\! K}^\timesox)$ is determined by $L_{(b,a)} \xi_{(x,w)} = \xi_{(b+ax,aw)}$. In \cite{CDL}, Cuntz, Deninger and Laca studied the Toeplitz-like C*-algebra $\mathfrak{T} [\OO_{\! K}] := C^*(L_{(b,a)}: (b,a) \in \OO_{\! K}^\timesox)$ generated by this representation and analyzed the equilibrium states of the natural time evolution $\sigma$ on $\mathfrak{T} [\OO_{\! K}]$ determined by the absolute norm $N_a := | \OO_{\! K}^\times/(a)|$ via \[ \sigma _t (L_{(b,a)}) = N_a^{it} L_{(b,a)} \qquad a\in \OO_{\! K}^\times, \ \ t\in \mathbb R. \] One of the main results of \cite{CDL} is a characterization of the simplex of KMS equilibrium states of this dynamical system at each inverse temperature $\beta \in (0,\infty]$. Here we will be interested in the low-temperature range of that classification. To describe the result briefly, let $\ok^*$ be the group of units, that is, the elements of $\OO_{\! K}^\times$ whose inverses are also integers, and recall that by a celebrated theorem of Dirichlet, $\ok^* \cong W_K \times \mathbb Z^{r+s-1}$, where $W_K$ (the group of roots of unity in $\ok^*$) is finite, $r$ is the number of real embeddings of $K$, and $s$ is equal to half the number of complex embeddings of $K$. Let $\Cl_K $ be the ideal class group of $K$, which, by definition, is the quotient of the group of all fractional ideals in $K$ modulo the principal ones, and is a finite abelian group. For each ideal class $\gamma \in \Cl_K$ let $J_\gamma \in \gamma$ be an integral ideal representing $\gamma$. By \cite[Theorem 7.3]{CDL}, for each $\beta > 2$ the KMS$_\beta$ states of $C^*(\OO_{\! K}^\timesox)$ are parametrized by the tracial states of the direct sum of group C*-algebras $\bigoplus_{\gamma\in\Cl_K} C^*(J_\gamma\rtimes \ok^*)$, where the units act by multiplication on each ideal viewed as an additive group. It is intriguing that exactly the same direct sum of group C*-algebras also plays a role in the computation of the $K$-groups of the semigroup C*-algebras of algebraic integers in the work of Cuntz, Echterhoff and Li, see e.g. \cite[Theorem 8.2.1]{CEL}. Considering as well that the group of units and the ideals representing different ideal classes are a measure of the failure of unique factorization into primes in $\OO_{\! K}$, we feel it is of interest to investigate the tracial states of the C*-algebras $C^*(J_\gamma\rtimes \ok^*)$ that arise as a natural parametrization of KMS equilibrium states of $C^*(\OO_{\! K}^\timesox)$. This work is organized as follows. In Section \ref{FromKMS} we review the phase transition from \cite{CDL} and apply a theorem of Neshveyev's to show in \thmref{thm:nesh} that the extremal KMS states arise from ergodic invariant probability measures and characters of their isotropy subgroups for the actions $\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \hat J_\gamma$ of units on the duals of integral ideals. We begin Section \ref{unitaction} by showing that for imaginary quadratic fields, the orbit space of the action of units is a compact Hausdorff space that parametrizes the ergodic invariant probability measures. All other number fields have infinite groups of units leading to `bad quotients' for which noncommutative geometry provides convenient tools of analysis. Units act by toral automorphisms and so the classification of equilibrium states is intrinsically related to the higher-dimensional, higher-rank version of the question, first asked by H. Furstenberg, of whether Lebesgue measure is the only nonatomic ergodic invariant measure for the pair of transformations $\times 2$ and $\times 3$ on $\mathbb R/\mathbb Z$. Once in this framework, it is evident from work of Sigmund \cite{Sig} and of Marcus \cite{Mar} on partially hyperbolic toral automorphisms and from the properties of the Poulsen simplex \cite{LOS}, that for fields whose unit rank is $1$, which include real quadratic fields, there is an abundance of ergodic measures, \proref{poulsen}, and hence of extremal equilibrium states, see also \cite{katz}. We also show in this section that there is solidarity among integral ideals with respect to the ergodicity properties of the actions of units, \proref{oneidealsuffices}. In Section \ref{berendsection}, we look at the topological version of the problem and we identify the number fields for which \cite[Theorem 2.1]{B} can be used to give a complete description of the invariant closed sets. In \thmref{conjecturalclassification} we summarize the consequences, for extremal equilibrium at low temperature, of the current knowledge on the generalized Furstenberg conjecture. For fields of unit rank at least $2$ that are not complex multiplication fields, i.e. that have no proper subfields of the same unit rank, we show that if there is an extremal KMS state that does not arise from a finite orbit or from Lebesgue measure, then it must arise from a zero-entropy, nonatomic ergodic invariant measure; it is not known whether such a measure exists. For complex multiplication fields of unit rank at least $2$, on the other hand, it is known that there are other measures, arising from invariant subtori. As a byproduct, we also provide in \proref{ZWclaim} a proof of an interesting fact stated in \cite{ZW}, namely the units acting on algebraic integers are generic among toral automorphism groups that have Berend's ID property. We conclude our analysis in Section \ref{prim} by computing the topology of the quasi-orbit space of the action $\ok^*\mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}$ for number fields satisfying Berend's conditions. As an application we also obtain an explicit description of the primitive ideal space of the C*-algebra $C(\OO_{\! K} \rtimes \ok^*)$, \thmref{primhomeom}. For the most part, sections \ref{unitaction} and \ref{berendsection} do not depend on operator algebra considerations other than for the motivation and the application, which are discussed in sections \ref{FromKMS} and \ref{prim}. \noindent{\sl Acknowledgments:} This research started as an Undergraduate Student Research Award project and the authors acknowledge the support from the Natural Sciences and Engineering Research Council of Canada. We would like to thank Martha {\L}\k{a}cka for pointing us to \cite{LOS}, and we also especially thank Anthony Quas for bringing Z. Wang's work \cite{ZW} to our attention, and for many helpful comments, especially those leading to Lemmas \ref{partition} and \ref{Anthony'sLemma1}. \section{From KMS states to invariant measures and isotropy}\label{FromKMS} Our approach to describing the tracial states of the C*-algebras $\bigoplus_{\gamma\in\Cl_K} C^*(J_\gamma\rtimes \ok^*)$ is shaped by the following three observations. First, the tracial states of a group C*-algebra form a Choquet simplex \cite{thoma}, so it suffices to focus our attention on the {\em extremal traces}. Second, there is a canonical isomorphism $C^*(J\rtimes \ok^*) \cong C^*(J)\rtimes \ok^* $, which we may combine with the Gelfand transform for $C^*(J)$, thus obtaining an isomorphism of $C^*(J\rtimes \ok^*)$ to the transformation group C*-algebra $C(\hat{J})\rtimes \ok^*$, associated to the transposed action of $\ok^*$ on the continuous complex-valued functions on the compact dual group $\hat{J}$. Specifically, the action of $\ok^*$ on $\hat{J}$ is determined by \begin{equation}\label{actiononjhat} (u \cdot \chi )(j):= \chi( u j), \qquad u \in \ok^*, \ \ \chi \in \hat{J}, \ \ j\in J, \end{equation} or by $\langle j, u \cdot \chi \rangle = \langle u j, \chi\rangle$, if we use $\langle \ , \ \rangle$ to denote the duality pairing of $J$ and $\hat J$. Third, this puts the problem of describing the tracial states squarely in the context of Neshveyev's characterization of traces on crossed products, so our task is to identify and describe the relevant ingredients of this characterization. In brief terms, when \cite[Corollary 2.4]{nes} is interpreted in the present situation, it says that that for each integral ideal $J$, the extremal traces on $C(\hat{J})\rtimes \ok^*$ are parametrized by triples $(H,\chi,\mu)$ in which $H$ is a subgroup of $\ok^*$, $\chi$ is a character of $H$, and $\mu$ is an ergodic $\ok^*$-invariant measure on $\hat{J}$ such that the set of points in $\hat{J}$ whose isotropy subgroups for the action of $\ok^*$ are equal to $H$ has full $\mu$ measure. Recall that, by definition, an $\ok^*$-invariant probability measure $\mu$ on $\widehat{J}$ is \emph{ergodic invariant} for the action of $\ok^*$ if $\mu(A) \in\{0,1\}$ for every $\ok^*$-invariant Borel set $A\subset \hat{J} $. Our first simplification is that the action of $\ok^*$ on $\hat J$ automatically has $\mu$-almost everywhere constant isotropy with respect to each ergodic invariant probability measure $\mu$. \begin{lemma}\label{automaticisotropy} Let $K$ be an algebraic number field with ring of integers $\OO_{\! K}$ and group of units $\ok^*$ and let $J$ be a nonzero ideal in $\OO_{\! K}$. Suppose $\mu$ is an ergodic $\ok^*$-invariant probability measure on $\hat J$. Then there exists a unique subgroup $H_\mu$ of $\ok^*$ such that the isotropy group $(\ok^*)_\chi:= \{u \in \ok^*: u\cdot \chi = \chi\}$ is equal to $H_\mu$ for $\mu$-a.a. characters $\chi\in \hat J$. \end{lemma} \begin{proof} For each subgroup $H \leq \ok^*$, let $M_H := \{\chi \in \hat J \mid (\ok^*)_\chi = H\}$ be the set of characters of $J$ with isotropy equal to $H$. Since the isotropy is constant on orbits, each $M_H$ is $\ok^*$-invariant, and clearly the $M_H$ are mutually disjoint. By Dirichlet's unit theorem $\ok^* \cong W_K \times \mathbb Z^{r+s-1}$ with $W_K$ finite, and $r$ and $2s$ the number of real and complex embeddings of $K$, respectively. Thus every subgroup of $\ok^*$ is generated by at most $|W_K| + (r+s-1)$ generators, and hence $\ok^*$ has only countably many subgroups. Thus $\{M_H: H \leq \ok^*\}$ is a countable partition of $\hat J$ into subsets of constant isotropy. We claim that each $M_H$ is a Borel measurable set in $\hat J$. To see this, observe: \begin{align*} M_H =&\{ \chi \in \hat J: u\cdot \chi = \chi \text{ for all }u \in H\text{ and } u\cdot \chi \neq \chi \text{ for all }u \in \ok^*\setminus H\}\\ =& \Big(\bigcap\limits_{u \in H} \{\chi \in \hat J \mid \chi^{-1}(u\cdot\chi)=1\} \Big)\bigcap \Big( \bigcap\limits_{u \in \ok^*\setminus H} \{\chi \in \hat J \mid \chi^{-1}(u\cdot \chi)\ne 1\}\Big) \end{align*} because $u\cdot \chi = \chi$ iff $\chi^{-1}(u\cdot\chi)=1$. Since the map $\chi \mapsto\chi^{-1}(u\cdot \chi)$ is continuous on $\hat J$, the sets in the first intersection are closed and those in the second one are open. By above, the intersection is countable, so $M_H$ is Borel-measurable, as desired. For every Borel measure $\mu$ on $\hat J$, we have \[ \sum\limits_{H \leq \ok^*} \mu (M_H) = \mu\Big(\bigcup\limits_{H \leq \ok^*} M_H \Big) = 1, \] so at least one $M_H$ has positive measure. If $\mu$ is ergodic $\ok^*$-invariant, then there exists a unique ${H_\mu \leq \ok^*}$ such that $\mu(M_{H_\mu}) = 1$ and thus $H_\mu$ is the (constant) isotropy group of $\mu$-a.a points $\chi \in \hat J$. \end{proof} Since each ergodic invariant measure determines an isotropy subgroup, the characterization of extremal traces from \cite[Corollary 2.4]{nes} simplifies as follows. \begin{theorem}\label{thm:nesh} Let $K$ be an algebraic number field with ring of integers $\OO_{\! K}$ and group of units $\ok^*$ and let $J$ be a nonzero ideal in $\OO_{\! K}$. Denote the standard generating unitaries of $C^*(J\rtimes \ok^*)$ by $\delta_j$ for $j\in J$ and $\nu_u$ for $u \in \ok^*$. Then for each extremal trace $\tau$ on $C^*(J\rtimes \ok^*)$ there exists a unique probability measure $\mu_\tau$ on $\hat J$ such that \begin{equation} \label{mufromtau} \int_{\hat J} \langlej, x\rangle d\mu_\tau(x) = \tau(\delta_j) \quad \text{ for } j \in J. \end{equation} The probability measure $\mu_\tau $ is ergodic $\ok^*$-invariant, and if we denote by $H_{\mu_\tau}$ its associated isotropy subgroup from \lemref{automaticisotropy}, then the function $\chi_\tau$ defined by $\chi_\tau(h):= \tau(\nu_h)$ for $h \in H_{\mu_\tau}$ is a character on $H_{\mu_\tau}$. Furthermore, the map $\tau \mapsto (\mu_\tau,\chi_\tau)$ is a bijection of the set of extremal traces of $C^*(J\rtimes \ok^*)$ onto the set of pairs $(\mu, \chi)$ consisting of an ergodic $\ok^*$-invariant probability measure $\mu$ on $\hat J$ and a character $\chi \in \widehat H_\mu$. The inverse map $(\mu,\chi) \mapsto \tau_{(\mu,\chi)}$ is determined by \begin{equation} \label{muchi-parameters} \tau_{(\mu,\chi)}(\delta_j \nu_u) = \begin{cases}\displaystyle\chi(u)\int_{\hat J} \langlej, x\rangled\mu(x) &\text{ if $u \in H_\mu$}\\ 0 &\text{ otherwise,}\end{cases} \end{equation} for $j\in J$ and $u\in \ok^*$. \end{theorem} \begin{proof} Recall that equation \eqref{actiononjhat} gives the continuous action of $\ok^*$ by automorphisms of the compact abelian group $\hat J$ obtained on transposing the multiplicative action of $\ok^*$ on $J$. There is a corresponding action $\alpha$ of $\ok^*$ by automorphisms of the C*-algebra $C(\hat J)$ of continuous functions on $\hat{J}$; it is given by $\alpha_u(f) (\chi) = f(u^{-1} \cdot \chi)$. The characterization of traces \cite[Corollary 2.4]{nes} then applies to the crossed product $C(\hat J) \rtimes_\alpha \ok^*$ as follows. For a given extremal tracial state $\tau$ of $C^*(J\rtimes \ok^*)$ there is a probability measure $\mu_\tau$ on $\hat J$ that arises, via the Riesz representation theorem, from the restriction of $\tau$ to $C^*(J) \cong C(\hat J)$ and is characterized by its Fourier coefficients in equation \eqref{mufromtau}. By \lemref{automaticisotropy}, there is a subset of $\hat J$ of full $\mu_\tau$ measure on which the isotropy subgroup is automatically constant, and is denoted by $H_{\mu_\tau}$. The unitary elements $\nu_u$ generate a copy of $C^*(\ok^*)$ inside $C(\hat J) \rtimes_\alpha \ok^*$ and the restriction of $\tau$ to these generators determines a character $\chi_\tau$ of $H_{\mu_\tau}$ given by $\chi_\tau(u) := \tau(\nu_u)$. See the proof of \cite[Corollary 2.4]{nes} for more details. By \lemref{automaticisotropy}, the condition of almost constant isotropy is automatically satisfied for every ergodic invariant measure on $\hat J$, hence every ergodic invariant measure arises as $\mu_\tau$ for some extremal trace $\tau$. The parameter space for extremal tracial states is thus the set of all pairs $(\mu,\chi)$ consisting of an ergodic $\ok^*$-invariant probability measure $\mu$ on $\hat J$ and a character $\chi$ of the isotropy subgroup $H_\mu$ of $\mu$. Formula \eqref{muchi-parameters} is a particular case of the formula in \cite[Corollary 2.4]{nes} with $f$ equal to the character function $f(\cdot) = \langlej,\cdot\rangle$ on $\hat J$ associated to $j\in J$. Since for a fixed $u \in \ok^*$ the right hand side of \eqref{muchi-parameters} is a continuous linear functional of the integrand and the character functions span a dense subalgebra, this particular case is enough to imply \begin{equation} \tau_{(\mu,\chi)}(f\nu_u) = \begin{cases}\displaystyle\chi(u)\int_{\hat J} f(x)d\mu(x) &\text{ if $u \in H_\mu$}\\ 0 &\text{ otherwise,}\end{cases} \end{equation} for every $f\in C(\hat J)$. \end{proof} \section{The action of units on integral ideals}\label{unitaction} Combining \cite[Theorem 7.3]{CDL} with \thmref{thm:nesh} above, we see that for $\beta> 2$, the extremal KMS$_\beta$ equilibrium states of the system $(\mathfrak{T} [\OO_{\! K}], \sigma)$ are indexed by pairs $(\mu,\kappa)$ consisting of an ergodic invariant probability measure $\mu$ and a character $\kappa$ of its isotropy subgroup relative to the action of the unit group $\ok^*$ on a representative of each ideal class. If the field $K$ is imaginary quadratic, that is, if $r=0$ and $s=1$, then the group of units is finite, consisting exclusively of roots of unity. In this case, things are easy enough to describe because the space of $\ok^*$-orbits in $\hat{J}$ is a compact Hausdorff topological space. \begin{proposition} \label{imaginaryquadratic}Suppose $K$ is an imaginary quadratic number field, let $J \subset \OO_{\! K}$ be an integral ideal and write $W_K$ for the group of units. Then the orbit space $W_K \backslash \hat J $ is a compact Hausdorff space and the closed invariant sets in $\hat J$ are indexed by the closed sets in $W_K \backslash \hat J $. Moreover, the ergodic invariant probability measures on $\hat J$ are the equiprobability measures on the orbits and correspond to unit point masses on $W_K\backslash \hat J$. \end{proposition} \begin{proof} Since $W_K$ is finite, distinct orbits are separated by disjoint invariant open sets, so the quotient space $W_K \backslash \hat J $ is a compact Hausdorff space. Since $\hat J$ is compact, the quotient map $q: \hat J \to W_K \backslash \hat J $ given by $q(\chi ) := W_K \cdot \chi$ is a closed map by the closed map lemma, and so invariant closed sets in $\hat J$ correspond to closed sets in the quotient. For each probability measure $\mu$ on $\hat J$, there is a probability measure $\tilde\mu$ on $W_K \backslash \hat J$ defined by \[\tilde\mu(E) := \mu(q^{-1}(E))\quad \text{ for each measurable } E\subseteq W_K \backslash \hat J.\] This maps the set of $W_K$-invariant probability measures on $\hat J$ onto the set of all probability measures on $W_K \backslash \hat J $. Ergodic invariant measures correspond to unit point masses on $W_K\backslash \hat J$, and their $W_K$-invariant lifts are equiprobability measures on single orbits in $\hat J$. \end{proof} As a result we obtain the following characterization of extremal KMS equilibrium states. \begin{corollary} Suppose $K$ is an imaginary quadratic algebraic number field and let $J_\gamma$ be an integral ideal representing the ideal class $\gamma\in \Cl_K$. For $\beta>2$, the extremal KMS$_\beta$ states of the system $(\mathfrak T[\OO_{\! K}], \sigma^N)$ are parametrized by the triples $(\gamma, W\cdot \chi, \kappa)$, where $\gamma\in \Cl_K$, $\chi$ is a point in $ \hat J_\gamma$, with orbit $W\cdot \chi$ and $\kappa$ is a character of the isotropy subgroup of $\chi$. \end{corollary} Before we discuss invariant measures and isotropy for fields with infinite group of units, we need to revisit a few general facts about the multiplicative action of units on the algebraic integers and, more generally, on the integral ideals. The concise discussion in \cite{ZW} is particularly convenient for our purposes. As is customary, we let $d = [K:\mathbb Q]$ be the {\em degree} of $K$ over $\mathbb Q$. The number $r$ of real embeddings and the number $2s$ of complex embeddings satisfy $r+2s=d$. We also let $n = r+s -1$ be the {\em unit rank} of $K$, namely, the free abelian rank of $\ok^*$ according to Dirichlet's unit theorem. We shall denote the real embeddings of $K$ by $\sigma_j:K \to \mathbb R$ for $j = 1, 2, \cdots r$ and the conjugate pairs of complex embeddings of $K$ by $\sigma_{r +j}, \sigma_{r+s+j} : K \to \mathbb C$ for $ j = 1, \cdots, s$. Thus, there is an isomorphism \[ \sigma: K \leftarrowimes_\mathbb Q \mathbb R \to \mathbb R^r \times \mathbb C^s \] such that \[ \sigma (k\leftarrowimes x) = (\sigma_1( k) x, \sigma_2(k) x, \cdots, \sigma_r(k) x;\, \sigma_{r+1}(k) x, \cdots, \sigma_{r+s}(k) x ). \] The ring of integers $\OO_{\! K}$ is a free $\mathbb Z$-module of rank $d$, and thus $\OO_{\! K}\leftarrowimes_\mathbb Z \mathbb R \cong \mathbb R^d \cong \mathbb R^r \oplus \mathbb C^s$. We temporarily fix an integral basis for $\OO_{\! K}$, which fixes an isomorphism $\theta: \OO_{\! K} \to \mathbb Z^d$. Then, at the level of $\mathbb Z^d$, the action of each $u \in \ok^*$ is implemented as left multiplication by a matrix $A_u \in GL_d(\mathbb Z)$. Moreover, once this basis has been fixed, the usual duality pairing $\langle \mathbb Z^d, \mathbb R^d/\mathbb Z^d \rangle $ given by $\langle n, t \rangle = \exp{2\pi i (n \cdot t)} $, with $n\in \mathbb Z^d$, $t\in \mathbb R^d$ and $n\cdot t = \sum_{j=1}^d n_j t_j$, gives an isomorphism of $\mathbb R^d/\mathbb Z^d $ to $\widehat{\OO}_{\! K}$, in which the character $\chi_t\in \widehat{\OO}_{\! K}$ corresponding to $t\in \mathbb R^d/\mathbb Z^d$ is given by $\chi_t(x) = \exp{2\pi i (\theta(x) \cdot t)} $ for $x\in \OO_{\! K}$. Thus, the action of a unit $u\in \ok^*$ is \[(u\cdot \chi_t)(x) = \chi_t(u\cdot x) = \exp{2\pi i (A_u \theta(x) \cdot t)} = \exp{2\pi i ( \theta(x) \cdot A_u^T t)}.\] This implies that the action $\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}$ is implemented, at the level of $\mathbb R^d/\mathbb Z^d$, by the representation $\rho: \ok^* \to GL_d(\mathbb Z)$ defined by $\rho(u) = A_u^{T}$, cf. \cite[Theorem 0.15]{Wal}. Similar considerations apply to the action of $\ok^*$ on $\hat J$ for each integral ideal $J \subset \OO_{\! K}$, giving a representation $\rho_J: \ok^* \to GL_d(\mathbb Z)$. For ease of reference we state the following fact about this matrix realization $\rho_J$ of the action of $\ok^*$ on $\hat{J}$. \begin{proposition}\label{diagonalization} The collection of matrices $\{\rho(u) : u \in \ok^*\}$ is simultaneously diagonalizable (over $\mathbb C$) , and for each $u\in \ok^*$ the eigenvalue list of $\rho(u)$ is the list of its archimedean embeddings $\sigma_k(u): k = 1, 2, \cdots r+2s$. \end{proposition} See e.g. the discussion in \cite[Section 2.1]{LW}, and \cite[Section 2.1]{ZW} for the details. Multiplication of complex numbers in each complex embedding is regarded as the action of 2x2 matrices on $\mathbb R+i\mathbb R \cong \mathbb R^2$, and the 2x2 blocks corresponding to complex roots simultaneously diagonalize over $\mathbb C^d$. The self duality of $\mathbb R^r\oplus \mathbb C^s $ can be chosen to be compatible with the isomorphism mentioned right after (2.1) in \cite{LW} and with multiplication by units. See also \cite[Ch7]{KlausS}. When the number field $K$ is not imaginary quadratic, then $\ok^*$ is infinite and so the analysis of orbits and invariant measures is much more subtle; for instance, most orbits are infinite, some are dense, and the orbit space does not have a Hausdorff topology. We summarize for convenience of reference the known basic general properties in the next proposition. \begin{proposition}\label{orbitsandisotropy} Let $K$ be a number field with $\operatorname{rank}(\ok^*) \geq 1$, and let $J$ be an ideal in $\OO_{\! K}$. Then normalized Haar measure on $\hat J$ is ergodic $\ok^*$-invariant, and for each $\chi \in \hat J$, \begin{enumerate} \item the orbit $\ok^*\cdot \chi$ is finite if and only if $\chi$ corresponds to a point with rational coordinates in the identification $\hat J \cong \mathbb R^d /\mathbb Z^d$; in this case the corresponding isotropy subgroup is a full-rank subgroup of $\ok^*$; \item the orbit $\ok^*\cdot \chi$ is infinite if and only if $\chi $ corresponds to a point with at least one irrational coordinate in $\mathbb R^d /\mathbb Z^d$; \item the characters $\chi$ corresponding to points $(w_1,w_2,\ldots,w_d) \in \mathbb R^d$ such that the numbers $1, w_1, w_2, \ldots w_d$ are rationally independent have trivial isotropy. \end{enumerate} \end{proposition} \begin{proof} By \proref{diagonalization}, for each $u\in \ok^*$, the eigenvalues of the matrix $\rho(u)$ encoding the action of $u$ at the level of $\mathbb R^d/\mathbb Z^d$ are precisely the various embeddings of $u$ in the archimedean completions of $K$. Since $\operatorname{rank}(\ok^*) \geq 1$, there exists a non-torsion element $u \in \ok^*$, whose eigenvalues are not roots of unity. Hence normalized Haar measure is ergodic for the action of $\{\rho(u): u\in\ok^*\}$ by \cite[Corollary 1.10.1]{Wal} and the first assertion now follows from \cite[Theorem 5.11]{Wal}. The isotropy is a full rank subgroup of $\ok^*$ because $|\ok^*/(\ok^*)_x| = |\ok^*\cdot x| < \infty$. Let $w = (w_1, w_2, \cdots, w_d)$ be a point in $\mathbb R^d/\mathbb Z^d$ such that $1, w_1, \ldots, w_d$ are rationally independent. Suppose $w$ is a fixed point for the matrix $\rho(u) \in GL_d(\mathbb Z)$ acting on $\mathbb R^d/\mathbb Z^d$. Then $\rho(u) w = w$ (mod $\mathbb Z^d$) and hence $(\rho(u) - I)w \in \mathbb Z^d$, i.e. \[[(\rho(u)-I)w]_i = \sum\limits_{j=1}^d (\rho(u)-I)_{ij}w_j \in \mathbb Z\] for all $1 \le i \le d$. Since $(\rho(u)-I)_{ij} \in \mathbb Z$ for all $i,j$, the rational independence of $1, w_1, \ldots, w_d$ implies that $\rho(u) = I$, so $u=1$, as desired. \end{proof} We see next that for the number fields with unit rank 1 there are many more ergodic invariant probability measures on $\widehat{\OO}_{\! K}$ than just Haar measure and measures supported on finite orbits. In fact, a smooth parametrization of these measures and of the corresponding KMS equilibrium states of $(\mathfrak T[\OO_{\! K}], \sigma)$ seems unattainable. \begin{proposition}\label{poulsen} Suppose the number field $K$ has unit-rank equal to $1$, namely, $K$ is real quadratic, mixed cubic, or complex quartic. Then the simplex of ergodic invariant probability measures on $\widehat{\OO}_{\! K}$ is isomorphic to the Poulsen simplex \cite{LOS}. \end{proposition} \begin{proof} The fundamental unit gives a partially hyperbolic toral automorphism of $\widehat{\OO}_{\! K}$, for which Haar measure is ergodic invariant. By \cite{Mar,Sig}, the invariant probability measures of such an automorphism that are supported on finite orbits are dense in the space of all invariant probability measures. This remains true when we include the torsion elements of $\ok^*$. Since these equiprobabilities supported on finite orbits are obviously ergodic invariant and hence extremal among invariant measures, it follows from \cite[Theorem 2.3]{LOS} that the simplex of invariant probability measures on $\widehat{\OO}_{\! K}$ is isomorphic to the Poulsen simplex. \end{proof} For fields with unit rank at least $2$, whether normalized Haar measure and equiprobabilities supported on finite orbits are the only ergodic $\ok^*$-invariant probability measures is a higher-dimensional version of the celebrated Furstenberg conjecture, according to which Lebesgue measure is the only non-atomic probability measure on $\mathbb T = \mathbb R/\mathbb Z$ that is jointly ergodic invariant for the transformations $\times 2$ and $\times 3$ on $\mathbb R$ modulo $\mathbb Z$. As stated, this remains open, however, Rudolph and Johnson have proved that if $p$ and $q$ are multiplicatively independent positive integers, then the only probability measure on $\mathbb R/\mathbb Z$ that is ergodic invariant for $\times p$ and $ \times q$ and has non-zero entropy is indeed Lebesgue measure \cite{Rud,Joh}. Number fields always give rise to automorphisms of tori of dimension at least $2$, so, strictly speaking the problem in which we are interested does not contain Furstenberg's original formulation as a particular case. Nevertheless, the higher-dimensional problem is also interesting and open as stated in general, and there is significant recent activity on it and on closely related problems \cite{KS,KK, KKS}. In particular, see \cite{EL1} for a summary of the history and also a positive entropy result for higher dimensional tori along the lines of the Rudolph--Johnson theorem. We show next that the toral automorphism groups arising from different integral ideals have a solidarity property with respect to the generalized Furstenberg conjecture. \begin{proposition} \label{oneidealsuffices} If for some integral ideal $J$ in $\OO_{\! K}$ the only ergodic $\ok^*$-invariant probability measure on $\hat J$ having infinite support is normalized Haar measure, then the same is true for every integral ideal in $\OO_{\! K}$. \end{proposition} The proof depends on the following lemmas. \begin{lemma}\label{partition} Let $J\subseteq I$ be two integral ideals in $\OO_{\! K}$ and let $r: \hat I \to \hat J$ be the restriction map. Denote by $\lambda_{\hat I}$ normalized Haar measure on $\hat I$. For each $\gamma \in \hat J$, there exists a neighborhood $N$ of $\gamma$ in $\hat J$ and homeomorphisms $h_j$ of $N$ into $\hat I$ for $j = 1, 2, \ldots , | I/J |$, with mutually disjoint images and such that \begin{enumerate} \item $\lambda_{\hat I} (h_j(E)) = \lambda_{\hat I} ( h_k(E))$ for every measurable $E\subseteq N$ and $1\leq j, k \leq | I/J |$; \item $r\circ h_j = \operatorname {id}_N$; \item $r^{-1} ( E) = \bigsqcup_j h_j(E)$ for all $E \subseteq N$, that is, the $h_j$'s form a complete system of local inverses of $r$ on $N$. \end{enumerate} \end{lemma} \begin{proof} Let $J^\perp:= \{\kappa\in \hat I: \kappa (j) = 1, \forall j\in J\}$ be the kernel of the restriction map $r: \hat I \to \hat J$. Since $J^\perp$ is a subgroup of order $| I/J | <\infty$, and since $\hat I$ is Hausdorff, we may choose a collection $\{ A_\kappa: \kappa \in J^\perp\}$ of mutually disjoint open subsets of $\hat I$ such that $\kappa \in A_\kappa$ for each $\kappa\in J^\perp$. Define $B_1:= \bigcap_{\kappa \in J^\perp} \kappa^{-1} A_\kappa$ and for each $\kappa\in J^\perp$ let $B_\kappa := \kappa B_1$. Then $\{B_\kappa : \kappa \in J^\perp\}$ is a collection of mutually disjoint open sets such that $\kappa \in B_\kappa$ and $r(B_\kappa) = r(B_1)$ for every $\kappa \in J^\perp$. We claim that the restrictions $r: B_\kappa \to \hat J$ are homeomorphisms onto their image. Since the $B_\kappa$ are translates of $B_1$ and since $r$ is continuous and open, it suffices to verify that $r$ is injective on $B_1$. This is easy to see because if $r(\xi_1) = r(\xi_2)$ for two distinct elements $\xi_1,\xi_2$ of $B_1$, then $\xi_2 = \kappa \xi_1$ for some $\rho \in J^\perp \setminus \{1\}$, and this would contradict $B_1 \cap \kappa B_1 = \emptyset$. This proves the claim. We may then take $N: = \gamma \, r(B_1)$ and define $h_\rho := (r |_{B_\rho})^{-1}$, for which properties (1)-(3) are now easily verified. \end{proof} \begin{lemma}\label{Anthony'sLemma1} Let $X$ be a measurable space and let $T: X \to X$ be measurable. Suppose that $\lambda$ is an ergodic $T$-invariant probability measure on $X$. If $\mu$ is a $T$-invariant probability measure on $X$ such that $\mu \ll \lambda$, then $\mu = \lambda$. \end{lemma} \begin{proof} Fix $f \in L^\infty(\lambda)$ and define $(A_n f)(x) = \frac{1}{n} \sum\limits_{k=0}^{n-1} f(T^{k}x)$. Let $S = \{x \in X: (A_nf)(x) \to \int_X f d\lambda\}$. By the Birkhoff ergodic theorem, we have that $\lambda(S^c)=0$, and so $\mu(S^c)=0$ as well, that is, $(A_nf)(x) \to \int_X f d\lambda$ $\mu$-a.e.. Since $f \in L^\infty(\lambda)$ and $\mu \ll \lambda$, we have that $f \in L^\infty(\mu)$ as well, with $\|f\|_\infty^\mu \le \|f\|_\infty^\lambda$. Observe that $|(A_nf)(x)| \le \|f\|_\infty^\lambda$ for $\mu$-a.e. $x$, and so by the dominated convergence theorem, $\int_X A_n f d\mu \to \int_X\left(\int_X f d\lambda\right) d\mu = \int_X f d\lambda$, with the last equality because $\mu(X)=1$. Because $\mu$ is $T$-invariant, we have that $\int_X A_n f d\mu = \int_X f d\mu$ for all $n$. Combining this with the above implies that $\int_X f d\lambda = \int_X f d\mu$ for all $f \in L^\infty(\lambda)$. In particular, this holds for the indicator function of each measurable set, and so $\mu = \lambda.$ \end{proof} \begin{lemma} \label{fromItoJ} Let $J \subseteq I$ be two integral ideals in $\widehat{\OO}_{\! K}$ and let $r: \hat I \to \hat J$ be the restriction map. If $\mu$ is an ergodic $\OO_{\! K}^*$-invariant probability measure on $\hat I$, then $\tilde{\mu}:= \mu \circ r^{-1}$ is an ergodic invariant probability measure on $\hat J$. Moreover, the support of $\mu$ is finite if and only if the support of $\tilde \mu$ is finite. \end{lemma} \begin{proof} Assume $\mu$ is ergodic invariant on $\hat I$ and let $E \subseteq \hat J$ be an $\ok^*$-invariant measurable set. Since $r$ is $\ok^*$-equivariant, $r^{-1}(E)$ is also $\ok^*$-invariant so $\tilde{\mu}(E):=\mu(r^{-1}(E)) \in \{0,1\}$ because $\mu$ is ergodic invariant. Thus, $\tilde{\mu}$ is also ergodic invariant. The statement about the support follows immediately because $r$ has finite fibers. \end{proof} \begin{lemma}\label{liftingF} Suppose $J \subseteq I$ are integral ideals in $\OO_{\! K}$, and let $\lambda_{\hat J}, \lambda_{\hat I}$ be normalized Haar measures on $\hat J, \hat I$, respectively. If the only ergodic $\OO_{\! K}^*$-invariant probability measure on $\hat J$ with infinite support is $\lambda_{\hat J}$, then the only ergodic $\OO_{\! K}^*$-invariant probability measure on $\hat I$ with infinite support is $\lambda_{\hat I}$. \end{lemma} \begin{proof} Let $\mu$ be an ergodic $\OO_{\! K}^*$-invariant probability measure on $\hat I$ with infinite support. By \lemref{fromItoJ}, $\mu \circ r^{-1}$ is an ergodic $\OO_{\! K}^*$-invariant probability measure on $\hat J$ with infinite support, and so by assumption must equal $\lambda_{\hat J}$. In particular, $\lambda_{\hat I} \circ r^{-1} = \lambda_{\hat J}$. Since $\hat J$ is compact, the open cover $\{N_\gamma: \gamma \in \hat J\}$ given by the sets constructed in \lemref{partition} has a finite subcover, that is, there exist $\gamma_1, \ldots, \gamma_n \in \hat J$ so that $\hat J = \bigcup\limits_{k=1}^n N_{\gamma_k}$, where $N_{\gamma_k}$ is a neighborhood of $\gamma_k \in \hat J$ satisfying the conditions stated in \lemref{partition}, with corresponding maps $h_{\gamma_k}^{(j)}$, for $1 \le j \le |I/J|$ and $1 \le k \le n$. We will first show that if $B \subseteq \hat I$ is such that $r|_B$ is a homeomorphism with $r(B) \subseteq N_{\gamma_k}$ for some $k$, and if $\lambda_{\hat I}(B) = 0$, then $\mu(B) = 0$. Suppose $B$ is such a set and $\lambda_{\hat I}(B)=0$. By part (3) of \lemref{partition}, $r^{-1}(r(B)) = \bigsqcup\limits_{j=1}^{|I/J|} h_{\gamma_k}^{(j)}(r(B))$, so $r^{-1}(r(B))$ is a disjoint union of $|I/J|$ sets, all having the same measure under $\lambda_{\hat I}$. Moreover, there exists some $1 \le j \le |I/J|$ such that $h_{\gamma_k}^{(j)}(r(B)) = B$, because the $h_{\gamma_k}^{(j)}$'s form a complete set of local inverses for $r$, and $r$ is injective on $B$. Putting these together yields \[\lambda_{\hat I}(r^{-1}(r(B))) = |I/J| \lambda_{\hat I}(h_{\gamma_k}^{(j)}(r(B))) = |I/J|\lambda_{\hat I}(B) =0.\] Since $\mu \circ r^{-1} = \lambda_{\hat J} = \lambda_{\hat I} \circ r^{-1}$, this implies that $\mu(r^{-1}(r(B))) = 0$ as well, and since $B \subseteq r^{-1}(r(B))$, we have that $\mu(B) = 0$. Now, since $r: \hat I \to \hat J$ is a covering map, for each $\chi \in \hat I$, there exists an open neighbourhood $U_\chi$ of $\chi$ such that $r|_{U_\chi}$ is a homeomorphism. Let $1 \le k \le n$ be such that $r(\chi) \in N_{\gamma_k}$, and let $W_\chi:=U_\chi \cap r^{-1}(N_{\gamma_k})$. This forms another open cover of $\hat I$, and so by compactness of $\hat I$, there exists a finite subcover $W_1, \ldots, W_m$. Finally, let $A \subseteq \hat I$ be such that $\lambda_{\hat I}(A)=0$. Then $A \cap W_i$ is a set on which $r$ acts as a homeomorphism, and there exists $1 \le k \le n$ such that $r(A \cap W_i) \subseteq N_{\gamma_k}$. Thus, by the above, we conclude $\mu(A \cap W_i) = 0$ for all $1 \le i \le m$. Since these sets cover $A$, we have that $\mu(A)=0$, and hence $\mu \ll \lambda_{\hat I}$, as desired. By \lemref{Anthony'sLemma1} it follows that $\mu = \lambda_{\hat I}$. \end{proof} \begin{proof}[Proof of \proref{oneidealsuffices}:] Suppose $J$ is an integral ideal such that the only $\ok^*$-invariant probability measure on $\hat J$ having an infinite orbit is normalized Haar measure. By \lemref{liftingF} applied to the inclusion $J\subset \OO_{\! K}$, the only ergodic $\OO_{\! K}^*$-invariant probability measure on $\widehat{\OO}_{\! K}$ with infinite support is normalized Haar measure. Suppose now $I \subseteq \OO_{\! K}$ is an arbitrary integral ideal. Since the ideal class group is finite, a power of $I$ is principal and thus we may choose $q \in \mathcal O^\times_K$ such that $q \OO_{\! K} \subseteq I$. The action of $\OO_{\! K}^*$ on $\widehat{\OO}_{\! K}$ is conjugate to the action of $\OO_{\! K}^*$ on $\widehat{q\OO_{\! K}}$, and so the only ergodic $\OO_{\! K}^*$-invariant probability measure on $\widehat{q\OO_{\! K}}$ with infinite support is normalized Haar measure. Thus, by \lemref{liftingF} again with $\widehat{q\OO_{\! K}} \subseteq \hat I$, we conclude that the only ergodic $\OO_{\! K}^*$-invariant probability measure on $\hat I$ is $\lambda_{\hat I}$. \end{proof} In order to understand the situation for number fields with unit rank higher than $1$, we review in the next section the topological version of the problem of ergodic invariant measures, namely, the classification of closed invariant sets. \section{Berend's theorem and number fields}\label{berendsection} An elegant generalization to higher-dimensional tori of Furstenberg's characterization \cite[Theorem IV.1]{F} of closed invariant sets for semigroups of transformations of the circle was obtained by Berend \cite[Theorem 2.1]{B}. The fundamental question investigated by Berend is whether an infinite invariant set is necessarily dense, and his original formulation is for semigroups of endomorphisms of a torus. Here we are interested in the specific situation arising from an algebraic number field $K$ in which the units $\ok^*$ act by automorphisms on $\hat J$ for integral ideals $J \subseteq \OO_{\! K}$ representing each ideal class, so we paraphrase Berend's Property ID for the special case of a group action on a compact space. \begin{definition} (cf. \cite[Definition 2.1]{B}.) Let $G$ be a group acting on a compact space $X$ by homeomorphisms. We say that the action $G\mathrel{\reflectbox{$\righttoleftarrow$}} X$ {\em satisfies the ID property}, or that it has the {\em infinite invariant dense property}, if the only closed infinite $G$-invariant subset of $X$ is $X$ itself. \end{definition} The first observation is a topological version of the measure-theoretic solidarity proved in \proref{oneidealsuffices}; namely, if $K$ is a given number field, then the action $\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \hat J$ has the ID property either for all integral ideals $J$, or for none. \begin{proposition}\label{IDforallornone} Suppose $K$ is an algebraic number field, and let $J$ be an ideal in $\OO_{\! K}$. Then the action of $\ok^*$ on $\hat J$ is ID if and only if the action of $\ok^*$ on $\hat{\OO_{\! K}}$ is ID. \end{proposition} \begin{proof} Suppose first that $J_1 \subseteq J_2$ are ideals in $\OO_{\! K}$ and assume that the action of $\ok^*$ on $\hat{J}_2$ is ID. The restriction map $r: \hat J_2 \to \hat J_1$ is $\ok^*$-equivariant, continuous, surjective, and has finite fibers. Thus, if $E$ were a closed, proper, infinite $\ok^*$-invariant subset of $\hat{J}_1$, then $r^{-1}(E)$ would be a closed, proper, infinite $\ok^*$-invariant subset of $\hat{J}_2$, contradicting the assumption that the action of $\ok^*$ on $\hat{J}_2$ is ID. So no such set $E$ exists, proving that the action of $\ok^*$ on $\hat{J}_1$ is also ID. In particular, if the action of $\ok^*$ on $\hat{\OO_{\! K}}$ is ID, then the action on $\hat{J}$ is also ID for every integral ideal $J\subset \OO_{\! K}$. For the converse, recall that, as in the proof of \proref{oneidealsuffices}, there exists an integer $q \in \OO_{\! K}^\times$ such that $q\OO_{\! K} \subseteq J$, so we may apply the preceding paragraph to this inclusion. Since the action of $\ok^*$ on $\widehat{q\OO_{\! K}}$ is conjugate to that on $\widehat{\OO}_{\! K}$, this completes the proof. \end{proof} In order to decide for which number fields the action of units on the integral ideals is ID, we need to recast Berend's necessary and sufficient conditions in terms of properties of the number field. Recall that, by definition, a number field is called a {\em complex multiplication (or CM) field} if it is a totally imaginary quadratic extension of a totally real subfield. These fields were studied by Remak \cite{rem}, who observed that they are exactly the fields that have a {\em unit defect}, in the sense that they contain a proper subfield $L$ with the same unit rank. \begin{theorem}\label{berend4units} Let $K$ be an algebraic number field and let $J$ be an ideal in $\OO_{\! K}$. The action of $\ok^*$ on $\hat J$ is ID if and only if $K$ is not a CM field and $\operatorname{rank} \ok^* \geq 2$. \end{theorem} For the proof we shall need a few number theoretic facts. We believe these are known but we include the relatively straightforward proofs below for the convenience of the reader. \begin{lemma}\label{99percent} Suppose $\mathcal F$ is a finite family of subgroups of $\mathbb Z^d$ such that $\operatorname{rank}(F) < d$ for every $F\in \mathcal F$. Then there exists $m\in \mathbb Z^d$ such that $m+F$ is nontorsion in $\mathbb Z^d/F$ for every $F \in \mathcal F$. \end{lemma} \begin{proof} Recall that for each subgroup $F$ there exists a basis $\{n^F_j\}_{j= 1, 2, \ldots , d}$ of $\mathbb Z^d$ and integers $a_1, a_2, \ldots, a_{\operatorname{rank}(F)}$ such that \[ F = \textstyle\big\{ \sum_{i=1}^{\operatorname{rank}(F)} k_i n^F_i:\ k_i \in a_i\mathbb Z, \ 1\leq i \leq \operatorname{rank}(F)\big\}. \] The associated vector subspaces $S_F := \operatorname{span}_\mathbb R\{n^F_1, \ldots, n^F_{\operatorname{rank}(F)}\} $ of $\mathbb R^d$ are proper and closed so $\mathbb R^d \setminus \cup_F S_F$ is a nonempty open set, see e.g. \cite[Theorem 1.2]{rom}. Let $r$ be a point in $\mathbb R^d \setminus \cup_F S_F$ with rational coordinates. If $k$ denotes the l.c.m. of all the denominators of the coordinates of $r$, then $m := kr \in \mathbb Z^d$ and its image $m+F \in \mathbb Z^d/F$ is of infinite order for every $F$ because $m \notin S_F$. \end{proof} \begin{proposition}\label{unitdefect} Let $K$ be an algebraic number field. Then there exists a unit $u\in \ok^*$ such that $K = \mathbb Q(u^k)$ for every $k\in \N^\times$ if and only if $K$ is not a CM field. \end{proposition} \begin{proof}Assume first $K$ is not a CM field. Then $\operatorname{rank} \OO_{\!F}^* < \operatorname{rank} \ok^*$ for every proper subfield $F$ of $K$. Since there are only finitely many proper subfields $F$ of $K$, \lemref{99percent} gives a unit $u\in \ok^*$ with nontorsion image in $\ok^*/\OO_{\!F}^*$ for every $F$. Thus $u^k \notin F$ for every proper subfield $F$ of $K$ and every $k\in \mathbb N$. Assume now $K$ is a CM field, and let $F$ be a totally real subfield with the same unit rank as $K$ \cite{rem}. Then the quotient $\ok^*/\OO_{\!F}^*$ is finite and there exists a fixed integer $m$ such that $u^m \in F$ for every $u\in \ok^*$. \end{proof} \begin{lemma} \label{friday} Let $k$ be an algebraic number field with $\operatorname{rank} \ok^* \geq 1$. Then for every embedding $\sigma: k \to \mathbb C$, there exists $u \in \ok^* $ such that $|\sigma(u)| > 1$. \end{lemma} \begin{proof} Assume for contradiction that $\sigma$ is an embedding of $k$ in $\mathbb C$ such that $\sigma(\ok^* ) \subseteq \{z \in \mathbb C: |z|=1\}$. Let $K = \sigma(k)$ and let $U_K = \sigma(\ok^* )$. Then $K \cap \mathbb R$ is a real subfield of $K$ with $ U_{K\cap \mathbb R} = \{\pm1\}$, so $K \cap \mathbb R= \mathbb Q$. Also $K \cap \mathbb R$ is the maximal real subfield of $K$, and since we are assuming $\operatorname{rank} \ok^* \geq 1$, $K$ cannot be a CM field. To see this, suppose that $k$ were CM. Let $\ell \subseteq k$ be a totally real subfield such that $[k:\ell] = 2$. Since $\ell$ is totally real, $\sigma(\ell)\subseteq \mathbb R$, and since $K \cap \mathbb R= \mathbb Q$, it must be that $\sigma(\ell) = \mathbb Q$. Then $\ell = \mathbb Q$, so $k$ is quadratic imaginary, contradicting $\operatorname{rank} \ok^* \ge 1$. By \proref{unitdefect}, there exists $u \in U_K$ such that $K = \mathbb Q(u)$. Since $|u|=1$, we have that $\overline K =\mathbb Q(\overline u) = \mathbb Q(u^{-1})= \mathbb Q(u) = K$, so $K$ is closed under complex conjugation. Write $u = a + ib$. Then $u + \overline u = 2a \in K \cap \mathbb R= \mathbb Q$, so $a \in \mathbb Q$. Thus, $K = \mathbb Q(u) = \mathbb Q(ib).$ Since $|u|=1$, $a^2+b^2=1$, and so we have that \[\mathbb Q(ib) \cong \mathbb Q(\sqrt{-b^2}) \cong \mathbb Q(\sqrt{a^2-1}) \cong \mathbb Q\left(\sqrt{\frac{m^2-n^2}{n^2}}\right) \cong \mathbb Q(\sqrt{m^2-n^2}),\] where $a = m/n \in \mathbb Q$. Thus, $K$ is a quadratic field. But it cannot be quadratic imaginary because $\operatorname{rank} U_K \ge 1$, and it cannot be quadratic real because all the units lie on the unit circle. This proves there can be no such embedding. \end{proof} \begin{proof} [Proof of \thmref{berend4units}] By \proref{IDforallornone}, it suffices to prove the case $J = \OO_{\! K}$. Let $d = [K:\mathbb Q]$ and recall that $\widehat{\OO}_{\! K} \cong \mathbb T^d$. All we need to do is verify that Berend's necessary and sufficient conditions for ID \cite[Theorem 2.1]{B}, when interpreted for the automorphic action of $\ok^*$ on $\widehat{\OO}_{\! K}$, characterize non-CM fields of unit rank $2$ or higher. Since the action of $\ok^*$ by linear toral automorphisms $\rho(u)$ with $u\in \ok^*$ is faithful by \cite[p. 729]{KKS}, Berend's conditions are: \begin{enumerate} \item (totally irreducible) there exists a unit $u$ such that the characteristic polynomial of $\rho(u^n)$ is irreducible for all $n\in \mathbb N$; \item (quasi-hyperbolic) for every common eigenvector of $\{\rho(u): u\in \ok^*\}$, there is a unit $u\in \ok^*$ such that the corresponding eigenvalue of $\rho(u)$ is outside the unit disc; and \item (not virtually cyclic) there exist units $u,v\in \ok^*$ such that if $m,n \in \mathbb N$ satisfy $\rho(u^m) = \rho(v^n)$, then $m = n = 0$. \end{enumerate} Suppose first that the action of $\ok^*$ on $\OO_{\! K}$ is ID. By \cite[Theorem 2.1]{B} conditions (1) and (3) above hold, i.e. the action of $\ok^*$ on $\OO_{\! K}$ is totally irreducible and not virtually cyclic. By \proref{unitdefect}, $K$ is not a CM field and since $\rho:\ok^* \to GL_d(\mathbb Z)$ is faithful, (3) is a restatement of $\operatorname{rank}\ok^* \geq 2$. Suppose now that $K$ is not CM and has unit-rank at least $2$. By \proref{unitdefect}, there exists $u \in \ok^*$ such that $\mathbb Q(u^n) = K$ for every $n \in \mathbb N$. Hence the minimal polynomial of $\rho(u^n)$ has degree $d$, and so it coincides with the characteristic polynomial. This proves that condition (1) holds, i.e. the action of $\rho(u)$ is totally irreducible. We have already observed that condition (3) holds iff the unit rank of $K$ is at least $2$, so it remains to see that the hyperbolicity condition (2) holds too. In the simultaneous diagonalization of the matrix group $\rho(\ok^*)$, the diagonal entries of $\rho(u)$ are the embeddings of $u$ into $\mathbb R$ or $\mathbb C$, see e.g. \cite[p.729]{KKS}. Then condition (2) follows from \lemref{friday}. \end{proof} \begin{remark} Notice that for units acting on algebraic integers, Berend's hyperbolicity condition (2) is automatically implied by the rank condition (3). \end{remark} \begin{remark} Since the matrices representing the actions of $\ok^*$ on $\hat J$ and on $\hat \OO_{\! K}$ are conjugate over $\mathbb Q$, \proref{IDforallornone} can be derived from the implication (1)$\implies$(3) in \cite[Proposition 2.1]{KKS}. We may also see that the matrices implementing the action on $\hat J$ and on $\hat \OO_{\! K}$ have the same sets of characteristic polynomials, so the questions of expansive eigenvalues (condition (2)) and of total irreducibility are equivalent for the two actions. The third condition is independent of whether we look at $\hat J$ or $\hat \OO_{\! K}$, so this yields yet another proof of Proposition \ref{IDforallornone}. \end{remark} By \thmref{berend4units}, for each non-CM algebraic number field $K$ with unit rank at least $2$, the action $\ok^*$ on $\widehat{\OO}_{\! K}$, transposed as $\{\rho(u): u\in \ok^*\}$ acting on $\mathbb R^d/\mathbb Z^d$, is an example of an abelian toral automorphism group for which one may hope to prove that normalized Haar measure is the only ergodic invariant probability measure with infinite support. So it is natural to ask which groups of toral automorphisms arise this way. A striking observation of Z. Wang \cite[Theorem 2.12]{ZW}, see also \cite[Proposition 2.2]{LW}, states that every finitely generated abelian group of automorphisms of $\mathbb T^d$ that contains a totally irreducible element and whose rank is maximal and greater than or equal to $2$ arises, up to conjugacy, from a finite index subgroup of units acting on the integers of a non-CM field of degree $d$ and unit rank at least $2$, cf. \cite[Condition 1.5]{ZW}. We wish next to give a proof of the converse, which was also stated in \cite{ZW}. \begin{proposition} \label{ZWclaim} Suppose $G$ is an abelian subgroup of $SL_d(\mathbb Z)$ satisfying \cite[Condition 2.8]{ZW}. Specifically, suppose there exist \begin{itemize} \item a non-CM number field $K$ of degree $d$ and unit rank at least $2$; \item an embedding $\phi:G \to \ok^*$ of $G$ into a finite index subgroup of $\ok^*$; \item a co-compact lattice $\Gamma$ in $K\subset K\leftarrowimes_\mathbb Q \mathbb R\cong \mathbb R^d$ invariant under multiplication by $\phi(G)$; and \item a linear isomorphism $\psi: \mathbb R^d \to K\leftarrowimes_\mathbb Q \mathbb R\cong \mathbb R^d$ mapping $\mathbb Z^d$ onto $\Gamma$ that intertwines the actions $G\mathrel{\reflectbox{$\righttoleftarrow$}}\mathbb R^d$ and $\phi(G) \mathrel{\reflectbox{$\righttoleftarrow$}} (K\leftarrowimes_\mathbb Q \mathbb R)/\Gamma$. \end{itemize} Then $G $ satisfies \cite[Condition 1.5]{ZW}, namely \begin{enumerate} \item $\operatorname{rank}(G)\geq 2$; \item the action $g\mathrel{\reflectbox{$\righttoleftarrow$}} \mathbb R^d/\mathbb Z^d \cong \mathbb T^d$ is totally irreducible for some $g\in G$; \item $\operatorname{rank} G_1 = \operatorname{rank} G$ for each abelian subgroup $G_1 \subsetSL_d(\mathbb Z)$ containing~$G$. \end{enumerate} \end{proposition} \begin{proof} Suppose $K$ is a non-CM algebraic number field of degree $d$ with unit rank at least $2$, and assume $G$ is a subgroup of $SL_d(\mathbb Z)$ that satisfies the assumptions with respect to $K$. Part (1) of \cite[Condition 1.5]{ZW} is immediate, because $\phi(G)$ is of full rank in $\ok^*$. By \proref{unitdefect}, there exists a unit $u\in \ok^*$ such that the characteristic polynomial of $\rho(u^m)$ is irreducible over $\mathbb Q$ for all $m \in \mathbb N$. This is equivalent to the action of $u^m$ on $\widehat{\OO}_{\! K}$ being irreducible for all $m \in \mathbb N$, see, e.g. \cite[Proposition 3.1]{KKS}. Since $\phi(G)$ is of finite index in $\ok^*$, there exists $N\in \mathbb N$ such that $u^N \in \phi(G)$. We claim that $g := \phi^{-1}(u^N)$ is a totally irreducible element in $G\mathrel{\reflectbox{$\righttoleftarrow$}} \mathbb R^d/\mathbb Z^d$. To see this, it suffices to show that the characteristic polynomial of $g^k$ is irreducible over $\mathbb Q$ for every positive integer $k$. Since the linear isomorphism $\psi$ intertwines the actions $g^k\mathrel{\reflectbox{$\righttoleftarrow$}} \mathbb T^d$ and $\rho(\phi(g))^k \mathrel{\reflectbox{$\righttoleftarrow$}} (K\leftarrowimes_\mathbb Q\mathbb R)/\Gamma$, the characteristic polynomial of $g^k$ equals the characteristic polynomial of $\rho(\phi(g))^k = \rho(u^{kN})$, which is irreducible because it coincides with the characteristic polynomial of $u^{kN}$ as an element of the ring $\OO_{\! K}$. This proves part (2) of Condition 1.5. Suppose now that $G_1$ is an abelian subgroup of $SL_d(\mathbb Z)$ containing $G$ and apply the construction from \cite[Proposition 2.13]{ZW} (see also \cite{KlausS,EL1}) to the irreducible element $g\in G\subset G_1 \mathrel{\reflectbox{$\righttoleftarrow$}} \mathbb T^d$. Up to an automorphism, the resulting number field arising from this construction is $K= \mathbb Q(u^N)$, and the embedding $\phi_1: G_1 \to \ok^*$ is an extension of $\phi:G\to \ok^*$. Since $\phi(G)\subset \phi_1(G_1) \subset \ok^*$ and $\phi(G)$ is of finite index in $\ok^*$, \[\operatorname{rank} (G_1) = \operatorname{rank}\phi_1(G_1) = \operatorname{rank} \ok^* = \operatorname{rank} \phi(G) = \operatorname{rank} G,\] and this proves proves part (3) of Condition 1.5. \end{proof} As a consequence, we see that the action of units on the algebraic integers of number fields are generic for group actions with Berend's ID property in the following sense, cf. \cite{ZW,LW}. \begin{corollary} If $G$ is a finitely generated abelian subgroup of $SL_d(\mathbb Z)$ of torsion-free rank at least 2 that contains a totally irreducible element and is maximal among abelian subgroups of $SL_d(\mathbb Z)$ containing $G$, then $G$ is conjugate to a finite-index toral automorphism subgroup of the action of $\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}$ for a non-CM algebraic number field $K$ of degree $d$ and unit rank at least $2$. \end{corollary} Finally, we summarize what we can say at this point for equilibrium states of C*-algebras associated to number fields with unit rank strictly higher than one. If the generalized Furstenberg conjecture is verified, the following result would complete the classification started in \proref{imaginaryquadratic} and \proref{poulsen}. Let $K$ be a number field and for each $\gamma \in \Cl_K$ define $F_\gamma $ to be the set of all pairs $(\mu, \chi)$ with $\mu$ an equiprobability measure on a finite orbit of the action of $\ok^*$ in $\hat {J}_\gamma$, and $\chi \in \hat{H}_\mu$, where the $\mu$-a.e. isotropy group $H_\mu$ is a finite index subgroup of $\ok^*$. Also let $(\lambda_J, 1) $ denote the pair consisting of normalized Haar measure on $\hat{J}$ and the trivial character of its trivial a.e. isotropy group. Then the map $(\mu,\chi) \mapsto \tau_{\mu,\chi}$ from \thmref{thm:nesh} gives an extremal tracial state of $C^*(J_\gamma \rtimes \ok^*)$ for each pair $(\mu,\chi) \in F_\gamma \sqcup \{(\lambda_{J_\gamma}, 1) \}$. Recall that the map $\tau \mapsto \varphi_\tau$ from \cite[Theorem 7.3]{CDL} is an affine bijection of all tracial states of $\bigoplus_{\gamma\in \Cl_K} C^*(J_\gamma\rtimes \ok^*)$ onto $\mathcal K_\beta$, the simplex of KMS$_\beta$ equilibrium states of the system $(\mathfrak T[\OO_{\! K}], \sigma)$ studied in \cite{CDL}. \begin{theorem}\label{conjecturalclassification} Suppose $K$ is an algebraic number field with unit rank at least $2$ and define $\Phi:(\mu,\chi) \mapsto \varphi_{\tau_{\mu,\chi}}$ to be the composition of the maps from \thmref{thm:nesh} and from \cite[Theorem 7.3]{CDL}, assigning a state $\varphi_{\tau_{\mu,\chi}} \in \operatorname{Extr}(\mathcal K_\beta)$ to each pair $(\mu,\chi)$ consisting of an ergodic invariant probability measure $\mu$ in one of the $\hat{J}_\gamma$ and an associated character of the $\mu$-almost constant isotropy $H_\mu$. Let \[ F_K:= \bigsqcup_{\gamma \in \Cl_K} \big(F_\gamma \sqcup \{(\lambda_{J_\gamma}, 1) \}\big) \] be the set of pairs whose measure $\mu$ has finite support or is Haar measure. Then \begin{enumerate} \item if $K$ is a CM field, then the inclusion $\Phi(F_K) \subset \operatorname{Extr}(\mathcal K_\beta)$ is proper; and \item if $K$ is not a CM field, and if there exists $ \phi \in \operatorname{Extr}(\mathcal K_\beta) \setminus \Phi(F_K) $ then the measure $\mu$ on $\hat{J}_\gamma$ arising from $\phi$ has zero-entropy and infinite support. \end{enumerate} \end{theorem} \begin{proof} To prove assertion (1), recall that when $K$ is a CM field Berend's theorem implies that there are invariant subtori, which have ergodic invariant probability measures on the fibers, cf. \cite{KK,KS}. These measures give rise to tracial states and to KMS states not accounted for in $\Phi(F_K)$. Assertion (2) follows from \cite[Theorem 1.1]{EL1}. \end{proof} \section{Primitive ideal space}\label{prim} The computation of the primitive ideal spaces of the C*-algebras $C^*(J \rtimes \ok^*)$ associated to the action of units on integral ideals lies within the scope of Williams' characterization in \cite{DW}. We briefly review the general setting next. Let $G$ be a countable, discrete, abelian group acting continuously on a second countable compact Hausdorff space $X$. We define an equivalence relation on $X$ by saying that {\em $x$ and $y$ are equivalent} if $x$ and $y$ have the same orbit closure, i.e. if $\overline{G\cdot x} = \overline{G \cdot y}$. The equivalence class of $x$, denoted by $[x]$, is called the {\em quasi-orbit} of $x$, and the quotient space, which in general is not Hausdorff, is denoted by $\mathcal{Q}(G \mathrel{\reflectbox{$\righttoleftarrow$}} X)$ and is called the {\em quasi-orbit space}. It is important to distinguish the quasi-orbit of a point from the closure of its orbit, as the latter may contain other points with strictly smaller orbit closure. Let $\varepsilonilon_x$ denote evaluation at $x\in X$, viewed as a one-dimensional representation of $C(X)$. For each character $\kappa \in \hat G_x$, the pair $(\varepsilonilon_x,\kappa)$ is clearly covariant for the transformation group $(C(X), G_x)$, and the corresponding representation $\varepsilonilon_x\times \kappa$ of $C(X) \rtimes G_x$ gives rise to an induced representation $\operatorname{Ind}_{{G_x}}^G(\varepsilonilon_x\times \kappa)$ of $C(X) \rtimes G$, which is irreducible because $\varepsilonilon_x\times \kappa$ is. Since $G$ is abelian and the action is continuous, whenever $x$ and $y$ are in the same quasi-orbit, $[x] = [y]$, the corresponding isotropy subgroups coincide: $G_x = G_y$. Thus, we may consider an equivalence relation on the product $\mathcal{Q}(G \mathrel{\reflectbox{$\righttoleftarrow$}} X) \times \hat G$ defined by \[ ([x],\kappa) \sim ([y],\lambda) \quad \iff \quad [x]=[y] \text{ and } \kappa\restr{G_x} = \lambda\restr{G_x} . \] By \cite[Theorem 5.3]{DW}, the map $(x,\kappa) \mapsto \ker \operatorname{Ind}_{{G_x}}^G(\varepsilonilon_x\times \kappa)$ induces a homeomorphism of $(\mathcal{Q}(G \mathrel{\reflectbox{$\righttoleftarrow$}} X) \times \hat G)/_{\!\sim}$ onto the primitive ideal space of the crossed product $C(X)\rtimes G$, see e.g. \cite[Theorem 1.1]{primbc} for more details on this approach. We wish to apply the above result to actions $\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \hat J$ for integral ideals $J$ of non-CM number fields with unit rank at least $2$, as in \thmref{berend4units}. Notice that by \proref{orbitsandisotropy} if the orbit $\ok^*\cdot \chi$ is finite, then it is equal to the quasi-orbit $[\chi]$. The first step is to describe the quasi-orbit space for the action of units. We focus on the case $J = \OO_{\! K}$; ideals representing nontrivial classes behave similarly because of the solidarity established in \proref{IDforallornone}. \begin{proposition}\label{quasiorbitset} Suppose $K$ is a non-CM algebraic number field with unit rank at least $2$. Then the quasi-orbit space of the action $\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}$ is \[ \mathcal{Q}(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}) = \{[x]: |\ok^*\cdot x | < \infty\}\cup \{\omega_\infty\}. \] The point $\omega_\infty$ is the unique infinite quasi-orbit $[\alpha]$ of any $\alpha \in \widehat{\OO}_{\! K}\cong \mathbb R^d/\mathbb Z^d$ having at least one irrational coordinate. The closed proper subsets are the finite subsets all of whose points are finite (quasi-)orbits. Infinite subsets and subsets that contain the infinite quasi-orbit $\omega_\infty$ are dense in the whole space. \end{proposition} \begin{proof} By \thmref{berend4units}, the closure of each infinite orbit is the whole space. Thus, the points with infinite orbits collapse into a single quasi-orbit \[ \omega_\infty := \{x\in \widehat{\OO}_{\! K}: |\ok^* \cdot x| =\infty\} = \{x\in \widehat{\OO}_{\! K}: \overlineerline{\ok^* \cdot x} = \widehat{\OO}_{\! K}\}. \] That this is the set of points with at least one irrational coordinate is immediate from \cite[Theorem 5.11]{Wal}. When the orbit of $x$ is finite, it is itself a quasi-orbit, which we view as a point in $\mathcal{Q}(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$. In this case $x\in \widehat{\OO}_{\! K}$ has all rational coordinates. To describe the topology, recall that the quotient map $q: \widehat{\OO}_{\! K} \to \mathcal{Q}(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$ is surjective, continuous and open by the Lemma in page 221 of \cite{PG}, see also the proof of Proposition 2.4 in \cite{primbc}. Any two different finite quasi-orbits $[x]$ and $[y]$ are finite, mutually disjoint subsets of $\widehat{\OO}_{\! K}$ and as such can be separated by disjoint open sets $V$ and $W$, so that $[x]\subset V$ and $[y] \subset W$. Passing to the quotient space, we have $[x]\notin q(W)$ and $[y] \notin q(V)$, so $[x]$ and $[y]$ are $T_1$-separated, which implies that finite sets of finite quasi-orbits are closed in $\mathcal Q(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$. The singleton $\{\omega_\infty\}$ is dense in $\mathcal Q(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$ because every infinite orbit in $\widehat{\OO}_{\! K}$ is dense by \thmref{berend4units}. If $A$ is an infinite subset of $\mathcal Q(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$ consisting of finite quasi-orbits, then $\bigcup_{[x]\in A} [x]$ is an infinite invariant set in $\widehat{\OO}_{\! K}$, hence is dense by \thmref{berend4units}. This implies that $\omega_\infty$ is in the closure of $A$, and hence $A$ is dense in $\mathcal Q(\ok^* \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$. \end{proof} \begin{theorem} \label{primhomeom} Let $K$ be a non-CM algebraic number field with unit rank at least 2, and let $G = \ok^*$. The primitive ideal space of $C(\widehat{\OO}_{\! K})\rtimes G$ is homeomorphic to the space \begin{equation}\label{primdef} \bigsqcup\limits_{[x]} \left(\{[x]\} \times \hat{G}_x \right) \end{equation} in which a net $([x_\iota], \gamma_\iota)$ converges to $([x],\gamma)$ iff $[x_\iota] \to [x]$ in $\mathcal{Q}(G \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$ and $\gamma_\iota|_{G_x} \to \gamma|_{G_x}$ in $\hat{G}_x$. Notice that if $[x]$ is a finite quasi-orbit, then the net $\{[x_\iota] \}$ is eventually constant equal to $[x]$, and if $[x] = \omega_\infty$, then the condition $\gamma_\iota|_{G_{\omega_\infty}} \to \gamma|_{G_{\omega_\infty}}$ is trivially true because $G_{\omega_\infty} = \{1\}$. \end{theorem} \begin{proof} Consider the diagram below, where $f$ is the quotient map and the vertical map $g$ is defined by $g([([x],\gamma)]) = ([x], \gamma|_{G_x})$, where $[([x],\gamma)]$ denotes the equivalence class of $([x],\gamma)$ with respect to $\sim$. \[ \begin{tikzcd} \mathcal{Q}(G\mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}) \times \hat G \arrow{r}{f} \arrow[swap]{dr}{g\circ f} & \mathcal{Q}(G\mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}) \times \hat G/\sim \arrow{d}{g} \\ & \bigsqcup\limits_{[x]} \left(\{[x]\} \times \hat{G}_x \right) \end{tikzcd} \] By the fundamental property of the quotient map, see e.g. \cite[Theorem 9.4]{wil}, $g\circ f$ is continuous if and only if $g$ is continuous. It is clear that $g$ is a bijection. We show next that $g \circ f$ is continuous. Suppose that $([x_\iota], \gamma_\iota)$ is a net in $\mathcal{Q}(G\mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}) \times \hat G$ converging to $([x],\gamma)$. Then $[x_\iota] \to [x]$ in $\mathcal{Q}(G\mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K})$, and $\gamma_\iota \to \gamma$ in $\hat G$. Then clearly also $\gamma_\iota|_{G_x} \to \gamma|_{G_x}$ in $\hat{G}_x$ as well. Hence the net $g\circ f([x_\iota], \gamma_\iota)$ converges to $g\circ f([x],\gamma) = ([x], \gamma|_{G_x})$, as desired. It remains to show that $g^{-1}$ is continuous, or equivalently, that $g$ is a closed map. Suppose that $W \subseteq \mathcal{Q}(G\mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}) \times \hat G / \sim$ is closed, and suppose that $([x_\iota], \gamma_\iota)$ is a net in $g(W)$ converging to $([x], \gamma)$. Consider any net $(\tilde{\gamma_\iota})$ in $\hat{G}$ such that $\tilde{\gamma_\iota}|_{G_x} = \gamma_\iota$. By the compactness of $\hat{G}$, there exists a convergent subnet $\tilde{\gamma}_{\iota_\eta}$ with limit $\tilde{\gamma}$. Then $\tilde{\gamma}_{\iota_\eta}|_{G_x} \to \tilde{\gamma}|_{G_x}$ as well, so $\gamma_{\iota_\eta} \to \tilde{\gamma}|_{G_x}$. Since $\hat{G}_x$ is Hausdorff, limits are unique, and hence $\tilde{\gamma}|_{G_x} = \gamma$. The net $([x_{\iota_\eta}], \tilde{\gamma}_{\iota_\eta})$ converges to $([x], \tilde{\gamma})$ in $\mathcal{Q}(G \mathrel{\reflectbox{$\righttoleftarrow$}} \widehat{\OO}_{\! K}) \times \hat{G}$, and since $f$ is continuous, $f([x_{\iota_\eta}], \tilde{\gamma}_{\iota_\eta}) \to f([x],\tilde{\gamma})$. Moreover, $f([x_{\iota_\eta}], \tilde{\gamma}_{\iota_\eta}) = [([x_{\iota_\eta}], \tilde{\gamma}_{\iota_\eta})] \in W$ because $g$ is injective and $g([([x_{\iota_\eta}],\tilde{\gamma}_{\iota_\eta})]) = ([x_{\iota_\eta}],\gamma_{\iota_\eta}) \in g(W)$ by assumption. Since $W$ is closed, $[([x],\tilde{\gamma})] \in W$, and so its image $([x], \gamma) \in g(W)$, as desired. \end{proof} \begin{remark} Recall that $G \cong W \times \mathbb Z ^d$ with $W$ the roots of unity in $G$, and that the isotropy subgroup $G_x$ is constant on the quasi-orbit $[x]$ of $x$. If $[x]$ is finite, then $G_x$ is of full rank in $G$, and thus $G_x \cong V_x \times \mathbb Z^d$, with $V_x \subset W$ the torsion part of $G_x$. Hence, for every finite quasi-orbit $[x]$, we have $\hat{G_x} \cong \hat V_{[x]} \times \mathbb T^d$. Notice that $\hat V_{[x]} \cong V_{[x]}$ (noncanonically) because $V_x$ is finite. \end{remark} \end{document}

\begin{document} \title{Incentive Compatible Mechanism for Influential Agent Selection} \author{Xiuzhen Zhang \and Yao Zhang \and Dengji Zhao} \authorrunning{X. Zhang et al.} \institute{ShanghaiTech University, Shanghai, China \\ \email{\{zhangxzh1,zhangyao1,zhaodj\}@shanghaitech.edu.cn}} \maketitle \begin{abstract} Selecting the most influential agent in a network has huge practical value in applications. However, in many scenarios, the graph structure can only be known from agents' reports on their connections. In a self-interested setting, agents may strategically hide some connections to make themselves seem to be more important. In this paper, we study the incentive compatible (IC) selection mechanism to prevent such manipulations. Specifically, we model the progeny of an agent as her influence power, i.e., the number of nodes in the subgraph rooted at her. We then propose the Geometric Mechanism, which selects an agent with at least $1/2$ of the optimal progeny in expectation under the properties of incentive compatibility and fairness. Fairness requires that two roots with the same contribution in two graphs are assigned the same probability. Furthermore, we prove an upper bound of $1/(1+\ln 2)$ for any incentive compatible and fair selection mechanisms. \keywords{Incentive compatibility \and Mechanism Design \and Influence Approximation.} \end{abstract} \section{Introduction} The motivation for influential agent selection in a network comes from real-world scenarios, where networks are constructed from the following/referral relationships among agents and the most influential agents are selected for various purposes (e.g., information diffusion~\cite{kimura2007extracting} or opinion aggregation~\cite{mohammadinejad2019consensus}). However, in many cases, the selected agents are rewarded (e.g., coupons or prizes), and the network structures can only be known from their reports on their following relationships. Hence, agents have incentives to strategically misreport their relationships to make themselves selected, which causes a deviation from the optimal results. An effective selection mechanism should be able to prevent such manipulations, i.e., agents cannot increase their chances to be selected by misreporting, which is a key property called incentive compatibility. There have been many studies about incentive compatible selection mechanisms with different influential measurements for various purposes (e.g., maximizing the in-degrees of the selected agent~\cite{alon2011sum,fischer2015optimal,caragiannis2021impartial}). In this paper, we focus on selecting an agent with the largest progeny. For this purpose, the following two papers are the most related studies. Babichenko et al.~\cite{babichenko2018incentive} proposed the Two Path Mechanism based on random walks. Although their mechanism achieves a good approximation ratio of $2/3$ between the expected and the optimal influence in trees, it has no guaranteed performance in forests or general directed acyclic graphs (DAGs). Furthermore, Babichenko et al.~\cite{BabichenkoDT20} advanced these results by proposing another two selection mechanisms with an approximation ratio of about $1/3$ in forests. In these two papers, the authors assumed that agents can add their out-edges to any other agents in the network. This strong assumption limited the design of incentive compatible mechanisms. Also, in many cases, agents cannot follow someone they do not know. Therefore, we focus on the manipulation of hiding the connections they already have. In practice, it is possible that two agents know each other, but they are not connected. Then they are more than welcome to connect with each other, which is not harmful for the selection. Moreover, there still exists a noticeable gap between the approximation ratios of existing mechanisms and a known upper bound of $4/5$~\cite{BabichenkoDT20} for all incentive compatible selection mechanisms in forests. Therefore, by restricting the manipulations of agents, we want to investigate whether we can do better. Furthermore, the previous studies mainly explored the forests, while in this paper, we also looked at DAGs. A DAG forms naturally in many applications because there exist sequential orders for agents to join the network. Each agent can only connect to others who joined the network before her, e.g., a reference or referral relationship network. Then, in a DAG, each node represents an agent, and each edge represents the following relationship between two agents. In this setting, the action of each agent is to report a set of her out-edges, which can only be a subset of her true out-edges. The goal is to design selection mechanisms to incentivize agents to report their true out-edge sets. Besides the incentive compatibility, we also consider another desirable property called fairness. Fairness requires that two agents with the maximum progeny in two graphs share the same probability of being selected if their progeny make no difference in both graphs (the formal definition is given in Section 2). Then, we present an incentive compatible selection mechanism with an approximation ratio of $1/2$ and prove an upper bound of $1/(1+\ln 2)$ for any incentive compatible and fair selection mechanism. \subsection{Our Contributions} We focus on the incentive compatible selection mechanism in DAGs. It is natural to assign most of the probabilities to select agents with more progeny to achieve a good approximation ratio. Thus, we identify a special set of agents in each graph, called the influential set. Each agent in the set, called an influential node, is a root with the maximum progeny if deleting all her out-edges in the graph. They are actually the agents who have the chances to make themselves the optimal agent with manipulations. On the other hand, we also define a desirable property based on the graph structure, called fairness. It requires that the most influential nodes (the agents with the maximum progeny) in two graphs have the same probability to be selected if the number of nodes in the two graphs, the subgraphs constructed by the two nodes' progeny, and the influential sets are all the same. Based on these ideas, we propose the Geometric Mechanism, which only assigns positive probabilities to the influential set. Each influential node will be assigned a selection probability related to her ranking in the influential set. We prove that the Geometric Mechanism satisfies the properties of incentive compatibility and fairness and can select an agent with her progeny no less than $1/2$ of the optimal progeny in expectation. The approximation ratio of the previous mechanisms is at most $1/\ln 16$ $(\approx 0.36)$. Under the constraints of incentive compatibility and fairness, we also give an upper bound of $1/(1+\ln 2)$ for the approximation ratio of any selection mechanism, while the previous known upper bound for any incentive compatible selection mechanism is $4/5$. \subsection{Other Related Work} \subsubsection{Without the constraint of incentive compatibility.} Focusing on influence maximization, Kleinberg~\cite{kleinberg2007cascading} proposed two models for describing agents' diffusion behaviours in networks, i.e., the linear threshold model and the independent cascade model. It is proved to be NP-hard to select an optimal subset of agents in these two models. Following this, there are studies on efficient algorithms to achieve bounded approximation ratios between the selected agents and the optimal ones under these two models~\cite{zhang2016identifying,huang2020efficient,ko2018efficient,morone2015influence}. In cases where only one influential agent can be selected, the most related literature also studied methods to rank agents based on their abilities to influence others in a given network, i.e., their centralities in the network. A common way is to characterize their centralities based on the structure of the network. In addition to the classic centrality measurements (e.g., closeness and betweenness~\cite{kundu2011new,pal2014centrality}) or Shapley value based characterizations~\cite{narayanam2008determining}, there are also other ranking methods in real-world applications, such as PageRank \cite{page1999pagerank} where each node is assigned a weight according to its connected edges and nodes. \subsubsection{With the constraint of incentive compatibility.} In this setting, incentive compatible selection mechanisms are implemented in two ways: with or without monetary payments. The first kind of mechanism incentivizes agents to truthfully reveal their information by offering them payments based on their reports. For example, Narayanam et al.~\cite{narahari2011incentive} considered the influence maximization problem where the network structure is known to the planner, and each agent will be assigned a fixed positive payment based on influence probabilities they reported. With monetary incentives, there are also different mechanisms proposed to prevent agents from increasing their utilities by duplicating themselves or colluding together~\cite{emek2011mechanisms,shen2019multi,zhang2020sybil}. To achieve incentive compatible mechanisms without monetary incentives, the main idea of the existing work is to design probabilistic selection mechanisms and ensure that each agent's selection probability is independent of her report~\cite{alon2011sum,aziz2016strategyproof,fischer2015optimal}. For example, Alon et al.~\cite{alon2011sum} designed randomized selection mechanisms in the setting of approval voting, where networks are constructed from agents' reports. Our work belongs to this category. \section{The Model} Let $\mathcal{G}^n$ be the set of all possible directed acyclic graphs (DAGs) with $n$ nodes and $\mathcal{G} = \bigcup_{n \in \mathbb{N}^*}\mathcal{G}^n$ be the set of all directed acyclic graphs. Consider a network represented by a graph $G=(N,E) \in \mathcal{G}$, where $N = \{1,2,\cdots, n\}$ is the node set and $E$ is the edge set. Each node $i \in N$ represents an agent in the network and each edge $(i,j) \in E$ indicates that agent $i$ follows (quotes) agent $j$. Let $P_i$ be the set of agents who can reach agent $i$, i.e., for all agent $j \in P_i$, there exists at least one path from $j$ to $i$ in the network. We assume $i \in P_i$. Let $p_i = |P_i|$ be agent $i$'s progeny and $p^* = \max_{i\in N} |P_i|$ be the maximum progeny in the network. Our objective is to select the agent with the maximum progeny. However, we do not know the underlying network and can only construct the network from the following/referral relationships declared by all agents, i.e., their out-edges. In a game-theoretical setting, agents are self-interested. If we simply choose an agent $i \in N$ with the maximum progeny, agents who directly follow agent $i$ may strategically misreport their out-edges (e.g., not follow agent $i$) to increase their chances to be selected. Therefore, in this paper, our goal is to design a selection mechanism to assign each agent a proper selection probability, such that no agent can manipulate to increase her chance to be selected and it can provide a good approximation of the expected progeny in the family of DAGs. For each agent $i \in N$, her type is denoted by her out-edges $\theta_i= \{(i,j) \mid (i,j) \in E, j \in N\}$, which is only known to her. Let $\theta = (\theta_1,\cdots,\theta_n)$ be the type of all agents and $\theta_{-i}$ be the type of all agents expect $i$. Let $\theta_i'$ be agent $i$'s report to the mechanism and $\theta' = (\theta_1',\cdots, \theta_n')$ be the report profile of all agents. Note that agents do not know the others except for the agents they follow in the network. Then $\theta_i' \subseteq \theta_i$ should hold for all $i \in N$, which satisfies the Nested Range Condition~\cite{green1986partially} thus guarantees the revelation principles. Thereby, we focus on direct revelation mechanism design here. Let $\Phi(\theta_i)$ be the space of all possible report profiles of agent $i$ with true type $\theta_i$, i.e., $\Phi(\theta_i) = \{\theta_i' \mid \theta_i' \subseteq \theta_i\}$. Let $\Phi(\theta)$ be the set of all possible report profiles of all agents with true type profile $\theta$. Given $n$ agents, let $\Theta^n$ be the set of all possible type profile of $n$ agents. Given $\theta \in \Theta^n$ and a report profile $\theta' \in \Phi(\theta)$, let $G(\theta')= (N,E')$ be the graph constructed from $\theta'$, where $N = \{1,2,\cdots,n\}$ and $E' = \{(i,j) \mid i,j\in N, (i,j) \in \theta' \}$. Denote the progeny of agent $i$ in graph $G(\theta')$ by $p_i(\theta')$ and the maximum progeny in this graph by $p^*(\theta')$. We give a formal definition of a selection mechanism. \begin{definition} A selection mechanism $\mathcal{M}$ is a family of functions $f: \Theta^n \rightarrow [0,1]^{n}$ for all $n \in \mathbb{N}^*$. Given a set of agents $N$ and their report profile $\theta'$, the mechanism $\mathcal{M}$ will give a selection probability distribution on $N$. For each agent $i \in N$, denote her selection probability by $x_i(\theta')$. We have $x_i(\theta')\in [0,1]$ for all $i \in N$ and $\sum_{i \in N}x_i(\theta') \leq 1$. \end{definition} Next, we define the property of \emph{incentive compatibility} for a selection mechanism, which incentivizes agents to report their out-edges truthfully. \begin{definition}[Incentive Compatible] A selection mechanism $\mathcal{M}$ is \textbf{incentive compatible (IC)} if for all $N$, all $i \in N$, all $\theta \in \Theta^n$, all $\theta_{-i}' \in \Phi(\theta_{-i})$ and all $\theta_i' \in \Phi(\theta_i)$, $x_i((\theta_i,\theta_{-i}')) \geq x_i((\theta_i',\theta_{-i}'))$. \end{definition} An incentive compatible selection mechanism guarantees that truthfully reporting her type is a dominant strategy for all agents. An intuitive realization is a uniform mechanism where each agent gets the same selection probability. However, there exists a case where most of the probabilities are assigned to agents with low progeny, thus leading to an unbounded approximation ratio. We desire an incentive compatible selection mechanism to achieve a bounded approximation ratio for all DAGs. We call this property \emph{efficiency} and define the efficiency of a selection mechanism by its approximation ratio. \begin{definition} Given a set of agents $N = \{1,2,\cdots,n\}$, their true type profile $\theta \in \Theta^n$, the performance of an incentive compatible selection mechanism in the graph $G(\theta)$ is defined by \[R(G(\theta)) = \frac{\sum_{i \in N} x_i(\theta)p_i(\theta)}{p^*(\theta)}.\] We say an incentive compatible selection mechanism $\mathcal{M}$ is \textbf{efficient with an approximation ratio $r$} if for all $N$, all $\theta \in \Theta^n$, $R(G(\theta)) \geq r$. \end{definition} This property guarantees that the worst-case ratio between the expected progeny of the selected agent and the maximum progeny is at least $r$ for all DAGs. Without the constraint of incentive compatibility, an optimal selection mechanism will always choose the agent with the maximum progeny. While in the strategic setting, an agent with enough progeny can misreport to make herself the agent with the maximum progeny. We define such an agent as \textit{an influential node}. In a DAG, there can be multiple influential nodes. Thus we define them as \textit{the influential set}, denoted by $S^{inf.}$. For example, in the graph shown in Figure~\ref{fig:inf}, when removing agent $3$'s out-edge, agent $3$ will be the root with the maximum progeny, same for agents $1$ and $2$. The formal definitions are as follows. \begin{figure} \caption{An example for illustrating the definition of influential nodes: agents $1,2,3$ are the influential nodes and they form the influential set in the graph $G$. } \label{fig:inf} \end{figure} \begin{definition}\label{def:inf} For a set of agents $N=\{1,2,\cdots,n\}$, their true type profile $\theta \in \Theta^n$ and their report profile $\theta' \in \Phi(\theta)$, an agent $i \in N$ is an \textbf{influential node} in the graph $G(\theta')$ if $p_i ((\theta_{-i}',\emptyset)) \succ p_j((\theta_{-i}',\emptyset))$ for all $j \neq i \in N$, where $p_i\succ p_j$ if either $p_i > p_j$ or $p_i = p_j$ with $i<j$. \end{definition} \begin{definition}\label{def:infset} For a set of agents $N=\{1,2,\cdots,n\}$, their true type profile $\theta \in \Theta^n$ and their report profile $\theta' \in \Phi(\theta)$, the \textbf{influential set} in the graph $G(\theta')$ is the set of all influential nodes, denoted by $S^{inf.}(G(\theta')) = \{s_1,\cdots,s_m\}$, where $s_i \succ s_j$ holds if and only if $p_i \succ p_j$, $s_i \succ s_{j}$ holds for all $m\geq j> i \geq 1$ and $m = |S^{inf.}(G(\theta'))|$. \end{definition} According to the above definitions, we present three observations about the properties of influential nodes. \begin{observation}\label{ob:onepath} Given a set of agents $N = \{1,2,\cdots,n\}$, their true type $\theta \in \Theta^n$ and their report profile $\theta' \in \Phi(\theta)$, there must exist a path that passes through all agents in $S^{inf.}(G(\theta'))$ with an increasing order of their progeny. \end{observation} \begin{proof} Let the influential set be $S^{inf.}(G(\theta')) = \{s_1,\cdots,s_m\}$. The statement shows that agent $s_j$ is one of the progeny of agent $s_i$ for all $1 \leq i < j\leq m$, then we can prove it by contradiction. Assume that there doesn't exist a path passing through all agents in the influential set, then there must be an agent $j$ such that $s_j \notin P_{s_i}$ for all $1\leq i < j$. Since $s_i,s_j \in S^{inf.}(G(\theta'))$, for all $1\leq i < j$, we have \begin{align} p_{s_i}((\theta'_{-s_i},\emptyset)) &\succ p_{s_j}((\theta'_{-s_i},\emptyset)), \label{1} \\ p_{s_j}((\theta'_{-s_j},\emptyset)) &\succ p_{s_i}((\theta'_{-s_j},\emptyset)). \label{2} \end{align} We also have $p_{s_j}((\theta'_{-s_j},\emptyset)) = p_{s_j}((\theta'_{-s_i},\emptyset))$ and $p_{s_i}((\theta'_{-s_j},\emptyset)) = p_{s_i}((\theta'_{-s_i},\emptyset))$ since $s_j \notin P_{s_i}$ and there is no cycle in the graph. With the lexicographical tie-breaking way, the inequality \ref{1} and \ref{2} cannot hold simultaneously. Therefore, we get a contradiction. \end{proof} \begin{observation}\label{ob:mosti} Given a set of agents $N = \{1,2,\cdots, n\}$, their true type $\theta \in \Theta^n$ and their report profile $\theta' \in \Phi(\theta)$, let $S^{inf.}(G(\theta'))= \{s_1,\cdots,s_m\}$ be the influential set in the graph $G(\theta')$. Then, agent $s_1$ has no out-edges and she is the one with the maximum progeny, i.e., agent $s_1$ is \textbf{the most influential node}. \end{observation} \begin{proof} We prove this statement by contradiction. Assume that agent $s_1$ has at least one out-edge. Then there must exist an agent $i \in N$ such that $s_1 \in P_i$ and $p_i((\theta_{-i}',\emptyset)) \succ p_k((\theta_{-i}',\emptyset))$ for all $k \neq i$, otherwise there must exist an agent $j \in N$ such that $s_1 \notin P_j$ and $p_j((\theta_{-i}',\emptyset)) \succ p_i((\theta_{-i}',\emptyset))$, which means that $s_1 \notin S^{inf.}(G(\theta'))$ since $p_i((\theta_{-i}',\emptyset)) \succ p_{s_1}((\theta_{-i}',\emptyset))$. Thus, such an $i$ must exist when agent $s_1$ has out-edges. Now, we must have $i \in S^{inf.}(G(\theta'))$ and $p_i \succ p_{s_1}$, which contradicts with $p_{s_1} \succ p_{j}$ for all $j \in S^{inf.}(G(\theta'))$ and $j \neq s_1$. Then we can conclude that agent $s_1$ has no out-edges. Since $p_{s_1}((\theta_{-s_1}',\emptyset))$ $ \succ p_{k}((\theta_{-s_1}',\emptyset))$ for all $k \neq s_1$, we can get that agent $s_1$ has the maximum progeny in the graph $G(\theta')$ and she is the most influential node. \end{proof} \begin{observation}\label{ob:set} Given a set of agents $N = \{1,2,\cdots,n\}$, their true type profile $\theta \in \Theta^n$, for all agent $i \in N$, all $\theta_{-i}' \in \Phi(\theta_{-i})$, if agent $i$ is not an influential node in the graph $G((\theta_{-i}',\theta_i))$, she cannot make herself an influential node by misreporting. \end{observation} \begin{proof} Given other agents' report $\theta_{-i}'$, whether an agent $i$ can be an influential node depends on the relation between $p_i((\theta_{-i}',\emptyset))$ and $p_j((\theta_{-i}',\emptyset))$, rather than the out-edges reported by agent $i$. \end{proof} There is one additional desirable property we consider in this paper. Consider two graphs $G,G' \in \mathcal{G}^n$ illustrated in Figure~\ref{fig:consistency}, where they have the same influential set ($S^{inf.}(G) = S^{inf.}(G')$) and $s_1$ is the most influential node in both graphs. Additionally, the subgraphs constructed by agents in $P_{s_1}$ are the same in both $G$ and $G'$ (The red parts in Figure~\ref{fig:consistency}, represented by $G(s_1) = G'(s_1)$). The only difference between the two graphs lies in the edges that are not in the subgraphs constructed by agents in $P_{s_1}$ (The yellow parts in Figure~\ref{fig:consistency}). \begin{figure} \caption{Example for fairness: in graphs $G$ and $G'$, $S^{inf.} \label{fig:consistency} \end{figure} We can observe that $s_1$ and all her progeny have the same contributions in the two graphs intuitively. Therefore, it is natural to require that a selection mechanism assigns the same probability to $s_1$ in the two graphs. We call this property \emph{fairness} and give the formal definition as follows. \begin{definition}[Fairness] For a graph $G = (N,E) \in \mathcal{G}$, define a subgraph constructed by agent $i$'s progeny as $G(i) = (P_i,E_i)$, where $E_i = \{(j,k) \mid j,k \in P_i, (j,k) \in E\}$ and $i \in N$. A selection mechanism $\mathcal{M}$ is \textbf{fair} if for all $N$, for all $G, G' \in \mathcal{G}^n$ where $S^{inf.}(G) = S^{inf.}(G') = \{s_1,\cdots,s_m\}$ and $G(s_1) = G'(s_1)$, then $x_{s_1}(G) = x_{s_1}(G')$. \end{definition} \section{Geometric Mechanism}\label{mechanism} In this section, we present the Geometric Mechanism, denoted by $\mathcal{M}_G$. In Observation~\ref{ob:set}, an agent without enough progeny cannot make herself an influential node by reducing her out-edges. Therefore, to maximize the approximation ratio, we can just assign positive selection probabilities to agents in the influential set. This is the intuition of the Geometric Mechanism. \begin{framed} \noindent\textbf{Geometric Mechanism} \noindent\rule{\textwidth}{0.5pt} \begin{enumerate} \item Given the set of agents $N = \{1,2,\cdots,n\}$, their true type profile $\theta \in \Theta^n$ and their report profile $\theta' \in \Phi(\theta)$, find the influential set $S^{inf.}$ in the graph $G(\theta')$: \[S^{inf.}(G(\theta')) = \{ s_1,\cdots,s_m\},\] where $s_i \succ s_{i+1}$ for all $1 \leq i \leq m-1$. \item The mechanism gives the selection probability distribution on all agents as the following. \begin{align*} x_i = \begin{cases} 1/(2^{m-j+1}), & i = s_j,\\ 0, & i \notin S^{inf.}(G(\theta')). \end{cases} \end{align*} \end{enumerate} \end{framed} Note that the Geometric Mechanism assigns each influential node a selection probability related to her ranking in the influential set. Besides, an agent' probability is decreasing when her progeny is increasing. This is reasonable because if an influential node $j$ is one of the progeny of another influential node $i$, the contribution of agent $i$ partially relies on $j$. To guarantee efficiency and incentive compatibility simultaneously, we assign a higher probability to agent $j$ compared to agent $i$. We give an example to illustrate how our mechanism works below. \begin{example} Consider the network $G$ shown in Figure~\ref{fig:example}. \begin{figure} \caption{An example for the Geometric Mechanism.} \label{fig:example} \end{figure} We can observe that only agents $1$ and $2$ will have the largest progeny in the graphs when they have no out-edges respectively. Thus, the influential set is $S^{inf.}(G)=\{1,2\}$. Since $p_1\succ p_2$, then according to the probability assignment defined in the Geometric Mechanism, we choose agent $1$ with probability $1/4$, choose agent $2$ with probability $1/2$ and choose no agent with probability $1/4$. The expected progeny chosen by the Geometric Mechanism in this graph is \[ \mathbb{E}[p] = \frac{1}{2}\times 6 + \frac{1}{4} \times 8 = 5. \] On the other hand, the largest progeny is given by agent $1$, which is $8$, so that the expected ratio of the Geometric Mechanism in this graph is $5/8$. \end{example} Next, in Theorems~\ref{thm:GM p} and~\ref{thm:GM up}, we show that our mechanism satisfies the properties of incentive compatibility and fairness and has an approximation ratio of $1/2$ in the family of DAGs. \begin{theorem}\label{thm:GM p} In the family of DAGs, the Geometric Mechanism satisfies incentive compatibility and fairness. \end{theorem} \begin{proof} In the following, we give the proof for these properties separately. \noindent \textbf{Incentive Compatibility.} Given a set of agents $N = \{1,2,\cdots,n\}$, their true type $\theta \in \Theta^n$ and their report profile $\theta' \in \Phi(\theta)$, let $G(\theta')$ be the graph constructed by $\theta'$, and $S^{inf.}(G(\theta'))$ be the influential set in $G(\theta')$. To prove that the mechanism is incentive compatible, we need to show that $x_i((\theta_{-i}',\theta_i)) \geq x_i((\theta_{-i}',\theta_i'))$ holds for all agent $i \in N$. \begin{itemize} \item According to Observation~\ref{ob:set}, for agent $i \notin S^{inf.}(G((\theta_{-i}',\theta_i)))$, she cannot misreport to make herself be an influential node. Thus, her selection probability will always be zero. \item If agent $i \in S^{inf.}(G((\theta_{-i}',\theta_i)))$, she cannot misreport to make herself be out of the influential set. Suppose $ S^{inf.}(G((\theta_{-i}',\theta_i))) = \{s_1,\cdots,s_m\}$ and $i = s_l$, $1\leq l\leq m$. Denote the set of influential nodes in her progeny when she truthfully reports by $S_i((\theta_{-i}',\theta_i)) = \{j \in S^{inf.}(G((\theta_{-i}',\theta_i))) \mid p_i ((\theta_{-i}',\theta_i))$ $ \succ p_j((\theta_{-i}',\theta_i)) \}$. Then agent $i$'s selection probability in the graph $G((\theta_{-i}',\theta_i))$ is $x_i((\theta_{-i}',\theta_i)) = 1/({2^{m-l+1}}) = 1/(2^{|S_i((\theta_{-i}',\theta_i))|+1})$. When she misreports her type as $\theta_i' \subset \theta_i$, i.e., deleting a nonempty subset of her real out-edges, $p_j((\theta_{-j}',\emptyset)) \succ p_k((\theta_{-j}',\emptyset))$ still holds for all $j \in S_i((\theta_{-i}',\theta_i))$, all $k \in N$ and $k\neq j$. This can be inferred from Observation~\ref{ob:onepath}, agent $j$ is one of the progeny of agent $i$ for all $j \in S_i$. Thus, agent $i$'s report will not change agent $j$'s progeny. Moreover, some other agent $t \in P_i$ may become an influential node in the graph $G((\theta_{-i}',\theta_i'))$, since $\max_{k\in N, k\neq t}p_k((\theta_{-t}',\emptyset))$ may be decreased and $p_t((\theta_{-t}',\emptyset))$ keeps unchanged. Then we can get $S_i((\theta_{-i}',\theta_i)) \subseteq S_i((\theta_{-i}',\theta_i'))$, which implies that $x_i((\theta_{-i}',\theta_i)) = 1/{2^{|S_i((\theta_{-i}',\theta_i))|+1}} \geq x_i((\theta_{-i}',\theta_i'))= 1/{2^{|S_i((\theta_{-i}',\theta_i'))|+1}}$. \end{itemize} Thus, no agent can increase her probability by misreporting her type and the Geometric Mechanism satisfies incentive compatibility. \noindent \textbf{Fairness.} For any two graph $G,G' \in \mathcal{G}^n$, if their influential sets and the subgraphs constructed by the progeny of the most influential node are both the same, i.e., $S^{inf.}(G) = S^{inf.}(G') = \{s_1,\cdots,s_m\}$ and $G(s_1) = G'(s_1)$, according to the definition of Geometric Mechanism, agent $s_1$ will always get a selection probability of $1/2^m$. Therefore, the Geometric Mechanism satisfies fairness. \end{proof} \begin{theorem}\label{thm:GM up} In the family of DAGs, the Geometric Mechanism can achieve an approximation ratio of $1/2$. \end{theorem} \begin{proof} Given a graph $G = (N,E) \in \mathcal{G}$ and its influential set $S^{inf.}(G) = \{s_1,\cdots,s_m\}$, the maximum progeny is $p^* = p_{s_1}$. Then the expected ratio should be \begin{align*} R &= \frac{\mathbb{E}[p]}{p^*} = \frac{\sum_{i \in S^{inf.}(G) }x_i p_i}{p^*} \\ &= \frac{\sum_{i=1 }^m 1/(2^{m-i+1}) \cdot p_{s_i}}{p^*} \\ &= \sum_{i=2}^{m} \frac{1}{2^{m-i+1}} \cdot \frac{p_{s_i}}{p_{s_1}} + \frac{1}{2^m} \cdot \frac{p_{s_1}}{p_{s_1}} \\ &\geq \sum_{j = 1}^{m-1} \frac{1}{2^j} \cdot \frac{1}{2} + \frac{1}{2^m} \\ &= \frac{1}{2} - \frac{1}{2^m} + \frac{1}{2^m} = \frac{1}{2}. \end{align*} The inequality holds since $p_{s_i}/ p_{s_1} \geq \frac{1}{2}$ holds for all $1\leq i \leq m-1$. This can be inferred from Observation~\ref{ob:onepath}, agent $s_i$ is one of agent $s_1$'s progeny for all $i > 1$. If $p_{s_i}/ p_{s_1} < \frac{1}{2}$, then we will have $p_{s_i}((\theta_{-{s_i}},\emptyset)) \prec p_{s_1}((\theta_{-s_i},\emptyset))$, which contradicts with that $s_i \in S^{inf.}(G)$. The expected ratio holds for any directed acyclic graph, which means that \[r_{\mathcal{M}_G} = \min_{G \in \mathcal{G}} R(G) = \frac{1}{2}.\] Thus we complete the proof. \end{proof} \section{Upper Bound and Related Discussions}\label{charater} In this section, we further give an upper bound for any incentive compatible and fair selection mechanisms in Theorem~\ref{theorem:ub}. After that, we consider a special class of selection mechanisms, called root mechanisms (detailed in Section~\ref{openproblems}), which contains the Geometric Mechanism. Then, we propose two conjectures on whether root mechanisms and fairness will limit the upper bound of the approximation ratio. \subsection{Upper Bound}\label{ub} We prove an upper bound for any IC and fair selection mechanisms as below. \begin{theorem}\label{theorem:ub} For any incentive compatible and fair selection mechanism $\mathcal{M}$, $r_{\mathcal{M}} \leq \frac{1}{1+ \ln 2}$. \end{theorem} \begin{proof} Consider the graph $G = (N,E)$ shown in Figure~\ref{fig:upperbound}, the influential set in $G$ is $S^{inf.}(G) = \{{2k-1}, {2k-2}, \cdots, k\}$. When $k \rightarrow \infty$, for each agent $i$, $i\leq k-1$, their contributions can be ignored, it is without loss of generality to assume that they get a probability of zero, i.e., $x_{i}(G)=0$. Then, applying a generic incentive compatible and fair mechanism $\mathcal{M}$ in the graph $G$, assume that $x_{i}(G) = \beta_{i-k}$ is the selection probability of agent $i$, $ \beta_{i-k} \in [0,1]$ and $ \sum_{i = k}^{2k-1} \beta_{i-k} \leq 1$. For each agent $i \in N$, set $N_i = P_i(G)$, $N_{-i} = N\setminus N_i$, $E_i = \{(j,k) \mid j,k \in I_i, (j,k) \in E\}$ and $E_{-i} = E\setminus \{E_i \cup \theta_i\}$. Define a set of graphs $\mathcal{G}_i = \{G'= (G(i);G(-i)) \mid G(-i) = (N_{-i},E_{-i}'), E_{-i}' \subseteq E_{-i}\}$. Then for any graph $G' \in \mathcal{G}_i$, it is generated by deleting agent $i$'s out-edge and a subset of out-edges of agent $i$'s parent nodes, illustrated in Figure~\ref{fig:upperbound}. For any $i \geq k$ and any graph $G' \in \mathcal{G}_i$, the influential set in the graph $G'$ should be $S^{inf.}(G') = \{i,{i-1},\cdots,k\}$. \begin{figure} \caption{The upper part is the origin graph $G$. The bottom part is an example in $\mathcal{G} \label{fig:upperbound} \end{figure} To get the upper bound of the approximation ratio, we consider a kind of ``worst-case" graphs where the contributions of agents except influential nodes can be ignored when $k \rightarrow \infty$. Since the mechanism $\mathcal{M}$ satisfies the fairness, it holds that $x_{i}(G') = x_{i}(G'')$ for any two graphs $G', G'' \in \mathcal{G}_i$. Then for any graph $G' \in \mathcal{G}_k$, agent $k$ is assigned the same probability. Thus, we can find that in the graph set $\mathcal{G}_k$, the ``worst-case" graph $G_k$ is a graph where there are only edges between $k$ and $i$, $1\leq i \leq k-1$ (shown in Figure~\ref{fig:worstcase}). \begin{figure} \caption{The ``worst-case" graph $G_k$ in the set $\mathcal{G} \label{fig:worstcase} \end{figure} Since no matter how much the probability the mechanism assigns to other agents, the expected ratio for the graph $G_k$ approaches the probability $x_{k}(G_k)$ when $k \rightarrow \infty$, i.e., \[\lim_{k \rightarrow \infty} R(G_k) \leq \lim_{k \rightarrow \infty} x_{k}(G_k) + \frac{1}{k}\cdot (1-x_{k}(G_k)) = x_{k}(G_k).\] The inequality holds since $\sum_{i=1}^{2k-1} x_{i}(G_k) \leq 1$. Similarly, for any $k<j \leq 2k-2$, the ``worst-case" graph $G_j$ in $\mathcal{G}_j$ is the graph where the out-edge of agent $i$ is deleted for all $i \geq j$. When $k \rightarrow \infty$, the expected ratio in the graph $G_j$ is \[\lim_{k \rightarrow \infty} R(G_j) \leq \lim_{k \rightarrow \infty} \sum_{i=k}^j x_{i}(G_j)\cdot \frac{i}{j} + \frac{1}{j}\cdot \left(1-\sum_{i=k}^j x_{i}(G_j)\right) = \sum_{i=k}^j x_{i}(G_j)\cdot \frac{i}{j}.\] Therefore, in these ``worst-case" graphs, we assume that only influential nodes can be assigned positive probabilities. Suppose that for the graph $G_j$, $k \leq j \leq 2k-2$, an influential node $i$ gets a probability of $x_{i} = \beta_{i-k}^{(j-i)}$ for $k \leq i \leq j$. Since the mechanism $\mathcal{M}$ is incentive compatible, it holds that $x_{i}(G) \geq x_{i}(G')$ for all $G' \in \mathcal{G}_i$ and all $i \in N$. To maximize the expected progeny of the selected agent in all graphs, we can set $x_{i}(G') = x_{i}(G)$ for all $G' \in \mathcal{G}_i$ and all $i \in N$. Similarly, it also holds that $x_{i}(G'') \geq x_{i}(G')$ for any $i \in N$, any $G' \in \mathcal{G}_i$, any $G'' \in \mathcal{G}_j$ and $k \leq i < j \leq 2k-1$. When $k \rightarrow \infty$, we can compute the performance of the mechanism $\mathcal{M}$ in different graphs as the following. \begin{align*} &R(G_{j}) = \sum_{i=k}^{j} \beta_{i-k}^{(j-i)} \cdot \frac{i}{j}, k\leq j \leq 2k-2,\\ &R(G) = \sum_{i=k}^{2k-1} \beta_{i-k} \cdot \frac{i}{2k-1}, \end{align*} with $\beta_{i-k}^{(j-i)} \geq \beta_{i-k}^{(0)}$, $\beta_{i-k}^{(0)} = \beta_{i-k}$, $k\leq i \leq 2k-1$, $k\leq j \leq 2k-2$. The approximation ratio of the mechanism $\mathcal{M}$ should be at most the minimum of $R(G_j)$ and $R(G)$ for $k \leq j \leq 2k-2$, i.e., \begin{align*} r_{\mathcal{M}} \leq \min \left\{\beta_{0}^{(0)}, \beta_{0}^{(1)}\cdot \frac{k}{k+1} + \beta_{1}^{(0)}, \cdots, \beta_0\cdot \frac{k}{2k-1} + \beta_1\cdot \frac{k+1}{2k-1}+\cdots+\beta_{k-1}\right\}. \end{align*} Then we can choose $\beta_{i-k}^{(j-i)}$ to achieve the highest minimum expected ratio. We find that $r_{\mathcal{M}} \leq \frac{1}{1+\ln 2}$ and the equation holds when $k \rightarrow \infty$ and \begin{equation*} \begin{cases} \beta_{i-k}^{(j-i)} = \beta_{i-k}, \\ \beta_0 + \beta_1 + \cdots + \beta_{k-1} = 1, \\ \beta_0^{(0)} = \beta_0^{(1)}\cdot \frac{k}{k+1} + \beta_{1}^{(0)} = \cdots = \beta_0 \cdot \frac{k}{2k-1} + \beta_1\cdot \frac{k+1}{2k-1}+\cdots+\beta_{k-1}. \end{cases} \end{equation*} \end{proof} \subsection{Open Questions}\label{openproblems} Note that the approximation ratio of the Geometric Mechanism is close to the upper bound we prove in Section~\ref{ub}. However, there is still a gap between them. In this section, we suggest two open questions which narrow down the space for finding the optimal selection mechanism. \subsubsection{Root Mechanism.} Recall that our goal in this paper is to maximize the approximation ratio between the expected progeny of the selected agent and the maximum progeny. If requiring incentive compatibility, a selection mechanism cannot simply select the most influential node. However, we can identify a subset of agents who can pretend to be the most influential node. This is the influential set we illustrate in Definition~\ref{def:infset}, and we show that agents cannot be placed into the influential set by misreporting as illustrated in Observation~\ref{ob:set}. Utilizing this idea, we see that if we assign positive probabilities only to these agents, then the selected agent has a large progeny, and agents have less chance to manipulate. We call such mechanisms as \emph{root mechanisms}. \begin{definition} A root mechanism $\mathcal{M}_R$ is a family of functions $f_R: \Theta^n \rightarrow [0,1]^{n}$ for all $n \in \mathbb{N}^*$. Given a set of agents $N$ and their report profile $\theta'$, a root mechanism $\mathcal{M}_R$ only assigns positive selection probabilities to agents in the set $S^{inf.}(G(\theta'))$. Let $x_i(\theta')$ be the probability of selecting agent $i\in N$. Then $x_i(\theta') = 0$ for all $i \notin S^{inf.}(G(\theta'))$, $x_i(\theta') \in [0,1]$ for all $i \in N$ and $\sum_{i\in N} x_i(\theta') \leq 1$. \end{definition} It is clear that our Geometric Mechanism is a root mechanism, whose approximation ratio is not far from the upper bound of $1/(1+\ln 2)$. We conjecture that an optimal incentive compatible selection mechanism and an optimal incentive compatible root mechanism share the same approximation ratio bound. \begin{conjecture} If an optimal incentive compatible root mechanism $\mathcal{M}_R$ has an approximation ratio of $r_{\mathcal{M}_R}^*$, there does not exist other incentive compatible selection mechanism that can achieve a strictly better approximation ratio. \end{conjecture}\label{up root} \begin{proof}[Discussion] An optimal incentive compatible selection mechanism will usually try to assign more probabilities to agents with more progeny. Following this way, we assign zero probability to all agents who are not an influential node and find a proper probability distribution for the influential set, rather than giving non-zero probabilities to all agents. Since any agent who is not an influential node cannot make herself in the influential set when other agents' reports are fixed, this method will not cause a failure for incentive compatibility. \end{proof} \subsubsection{Fairness.}\label{consistency} Note that the upper bound of $1/(1+\ln 2)$ is for all incentive compatible and fair selection mechanisms. We should also consider whether an incentive compatible selection mechanism can achieve a better approximation ratio without the constraint of fairness. Here, we conjecture that an incentive compatible selection cannot achieve an approximation ratio higher than $1/(1+\ln 2)$ if the requirement of fairness is relaxed. \begin{conjecture} If an optimal incentive compatible and fair mechanism $\mathcal{M}$ can achieve an approximation ratio of $r_{\mathcal{M}}^*$, there does not exist other incentive compatible mechanism with a strictly higher approximation ratio. \end{conjecture} \begin{proof}[Discussion] Let $\mathcal{G}_f$ be a set of graphs where for any two graphs $G,G' \in \mathcal{G}_f$, their number of nodes, their influential sets $S^{inf.}(G) = S^{inf.}(G') = \{s_1,\cdots,s_m\}$ and the subgraphs constructed by agent $s_1$'s progeny are same. If an incentive compatible selection mechanism is not fair, there must exist such a set $\mathcal{G}_f$ where the mechanism fails fairness. Then the expected ratios in two graphs in $\mathcal{G}_f$ may be different, and the graph with a lower expected ratio might be improved since these two graphs are almost equivalent. One possible way for proving this conjecture is to design a function that reassigns probabilities for all graphs in $\mathcal{G}_f$ such that $x_{s_1}$ is the same for these graphs without hurting the property of incentive compatibility, and all graphs in $\mathcal{G}_f$ then share the same expected ratio without hurting the efficiency of the selection mechanism. \end{proof} \section{Conclusion} In this paper, we investigate a selection mechanism for choosing the most influential agent in a network. We use the progeny of an agent to measure her influential level so that there exist some cases where an agent can decrease her out-edges to make her the most influential agent. We target selection mechanisms that can prevent such manipulations and select an agent with her progeny as large as possible. For this purpose, we propose the Geometric Mechanism that achieves at least $1/2$ of the optimal progeny. We also show that no mechanism can achieve an expected progeny of the selected agent that is greater than $1/(1+\ln 2)$ of the optimal under the conditions of incentive compatibility and fairness. There are several interesting aspects that have not been covered in this paper. First of all, there is still a gap between the efficiency of our proposed mechanism and the given upper bound. One of the future work is to find the optimal mechanism if it exists. In this direction, we also leave two open questions for further investigations. Moreover, selecting a set of influential agents rather than a single agent is also important in real-world applications (e.g., ranking or promotion). So another future work is to extend our results to the settings where a set of $k$ ($k>1$) agents need to be selected. \end{document}

\begin{document} \title{Environment Assisted Metrology with Spin Qubits} \author{P. Cappellaro}\email{pcappell@mit.edu}\affiliation{Nuclear Science and Engineering Dept., Massachusetts Institute of Technology, Cambridge MA 02139 USA} \author{G. Goldstein} \affiliation{Department of Physics, Harvard University, Cambridge MA 02138 USA} \author{J. S. Hodges}\altaffiliation{Current address: Quantum Information Science Group, MITRE Corp. 260 Industrial Way West, Eatontown, NJ 07724, USA} \affiliation{Nuclear Science and Engineering Dept., Massachusetts Institute of Technology, Cambridge MA 02139 USA} \affiliation{Department of Physics, Harvard University, Cambridge MA 02138 USA} \author{L. Jiang} \affiliation{Institute for Quantum Information, California Institute of Technology, Pasadena CA 91125 USA} \author{J. R. Maze} \affiliation{Faculty of Physics, Pontificia Universidad Catolica de Chile, Santiago 7820436, Chile} \author{A. S. S{\o}rensen}\affiliation{QUANTOP, Niels Bohr Institute, Copenhagen University, DK 2100, Denmark} \author{M. D. Lukin}\affiliation{Department of Physics, Harvard University, Cambridge MA 02138 USA} \begin{abstract} We investigate the sensitivity of a recently proposed method for precision measurement [Phys. Rev. Lett. {\bf 106}, 140502 (2011)], focusing on an implementation based on solid-state spin systems. The scheme amplifies a quantum sensor response to weak external fields by exploiting its coupling to spin impurities in the environment. We analyze the limits to the sensitivity due to decoherence and propose dynamical decoupling schemes to increase the spin coherence time. The sensitivity is also limited by the environment spin polarization; therefore we discuss strategies to polarize the environment spins and present a method to extend the scheme to the case of zero polarization. The coherence time and polarization determine a figure of merit for the environment's ability to enhance the sensitivity compared to echo-based sensing schemes. This figure of merit can be used to engineer optimized samples for high-sensitivity nanoscale magnetic sensing, such as diamond nanocrystals with controlled impurity density. \end{abstract} \maketitle \section{Introduction}\label{Intro} Quantum metrology seeks to achieve precision measurements with an accuracy beyond the limits imposed by the central limit theorem~\cite{Giovannetti11} (the standard quantum limit, SQL). Although many proposals for achieving the quantum limits of sensitivity (as defined by the Heisenberg bounds) have been presented, they are often difficult to implement in practice. The main challenges arise from the deleterious effects of noise and decoherence on the (entangled) states required for quantum metrology and from the unavailability of the Hamiltonians and measurement strategies needed to create and readout these entangled states. We recently introduced a scheme~\cite{Goldstein11} that aims at overcoming these two challenges. We proposed to use the environment of the sensor as an additional resource for metrology and we showed how to achieve the desired interaction Hamiltonian using coherent control techniques. In this paper we focus on one possible implementation of this \textit{environment assisted metrology} (EAM) scheme -- a spin sensor embedded in a bath of other spins -- in order to derive more detailed results on the sensitivity achievable. In addition, we will analyze in depth the effects of decoherence and of finite polarization. The paper is organized as follows. In Section~\ref{EAM} we present the EAM scheme: the control sequence that achieves it and the sensitivity gain in the idealized situation of no decoherence. This restriction is lifted in Section~\ref{Decoherence}, where we analyze the effects of decoherence, both analytically and with numerical simulations. We further provide strategies to reduce the effects of decoherence. In section~\ref{Sensitivity} we use these results to derive limits of the proposed EAM strategy and compare them to usual strategies that do not take advantage of the environment. Since the sensitivity depends on the polarization of the spin environment, we propose in section~\ref{Polarization} schemes for polarizing these ancillary spins and we further extend the scheme to the case where no polarization is available. A second extension of the EAM method is presented in section~\ref{Spins}, where more general spin systems are studied. \section{The environment assisted metrology scheme}\label{EAM} We consider the metrology task of measuring a parameter $b$ via its interaction with a quantum probe. The task can be accomplished by using a Ramsey scheme, where a two level system is first prepared in a superposition of the two states, which then acquire a phase difference that is mapped onto the populations by a second pulse. An example of this scheme is magnetometry with solid-state spins~\cite{Taylor08}, where the probe interacts with the external magnetic field via a Hamiltonian ${\mathcal{H}}\propto bS_z$, acquiring a phase $\propto bt$ during the interrogation time $t$. Then, the bound to the sensitivity is set by the dephasing rate that limits the time the probe can interact with the external field associated to the parameter to be measured. Coherent control techniques can be used to isolate the probe from its environment, thus increasing the coherence time. If the environment interacts as well with the external field to be measured -- as it is the case for a spin bath -- a different strategy is possible: in Ref. \cite{Goldstein11} we showed that the spin environment can be used as a resource in this case, by mapping the phase acquired by the environment spins onto the probe spin before readout. Here we provide more details of the method presented in Ref.~\cite{Goldstein11} and consider several extension of the work. To this end we assume that the spin environment can be collectively controlled and partially polarized. These spins could thus be considered as an ancillary system. Still, since they cannot be addressed individually nor read out, they cannot be used directly as probes or in sequential adaptive schemes~\cite{Schaffry11,Giovannetti06,Higgins07}. In addition, because their couplings to the probe spin cannot be switched off, they are a cause of decoherence for the probe spin (as we will see in section \ref{Decoherence}) and thus they can be considered as environment. Nevertheless we show that one can make active use of these spins, to increase the sensitivity of a measurement. Ancillary qubits have been considered as a resource for parameter estimation~\cite{Boixo08} in a scheme inspired by the deterministic quantum computing with only one pure qubit (DQC1) model~\cite{Knill98}. In that scheme, the probe qubit is initially prepared in a superposition state, then the ancillary system interacts with the external parameter \textit{conditional} on the state of the probe, which is finally readout (see Fig.~\ref{fig:Circuit}.a). When the conditional evolution is given by the operator $U=e^{-ibt\sum_kI_z^k}$ (where $\vec{I}^k$ are the ancilla spin operators) the sensitivity achieves the SQL (scaling as 1$/\sqrt{n}$ where $n$ is the number of ancillary qubits) for ancillas in a completely mixed state~\cite{Boixo08} and the Heisenberg limit for pure state (scaling as $1/n$). In that case, it is convenient to read out the y-component of the probe spin, which gives a signal $S=\sin(nbt)$. Since the signal is enhanced by a factor of $n$ for small fields $nbt\ll1$ this yields an Heisenberg-limited sensitivity scaling as $1/n$. Indeed, for pure input states, the circuit creates an entangled state that provides a signal enhancement. Below we modify this scheme so that it can be implemented for realistic physical systems. In general, the ancillas dependence on the external parameter cannot be controlled by the probe spin, as it is implicitly assumed above. Thus it is necessary to intersperse the evolution under the interaction with the external field with C-Not gates (Fig.~\ref{fig:Circuit}.b). With this modification we achieve a similar evolution as before. However, even this simpler scheme cannot be easily implemented and is not compatible with our assumptions of limited control on the environment spins: if the ancillas are spins in the environment, it is not possible to control them individually, thus the C-Not gates cannot be implemented since the required interaction time for the C-Not operations will be different for the different spins. The key to using the environment spins -- with the corresponding limited control -- as a resource for parameter estimation is to realize that the scheme works also if the the controlled gates are not ideal $\pi$-rotations. The rotations can differ for different spins, as long as the state of the probe spin is flipped (Not gate) before the second set of controlled gates: this ensures that all the environment spins contributes constructively to the final phase, as we derive below. \begin{figure} \caption{\label{fig:Circuit} \label{fig:Circuit} \end{figure} We note that the spin flip of the probe achieves two other results: first, it makes the evolution insensitive to static noise (as produced for example by a very slowly varying spin bath) since the gate amount to a spin echo for the probe spin. Secondly, the echo pulse refocuses the entanglement created in the first half of the circuit; this operation cancels undesired terms in the signal that would arise when considering a more realistic scenario where both the external field and the couplings to the probe spin used to create controlled rotations are always present at the same time. The idealized scheme in Fig.~\ref{fig:Circuit} can be implemented in practice with realistic resources, with the EAM pulse sequence of Fig.~\ref{fig:Sequence}. We consider a system comprising a sensor spin ($S=1$) and environment spins ($I^k$), which in a convenient rotating frame on resonance with the $m_s=0\to1$ transition, is described by the Hamiltonian: \begin{equation} \begin{array}{l} {\mathcal{H}}=b(t)\left(\gamma_SS_z+\gamma_I \sum_k I_z^k\right)+\sum\lambda_k S_z I_z^k\\ \quad=\vert{0}\rangle\langle{0}\vert \left[b(t)\gamma_I\sum_k I_z^k\right]+\\ \qquad\vert{1}\rangle\langle{1}\vert \left[\gamma_S b(t)+\sum_k(\gamma_I b(t)+ \lambda_k) I_z^k\right], \end{array} \label{eq:Hamiltonian} \end{equation} where $b(t)$ is the external field to be measured, $\gamma_{S,I}$ are the gyromagnetic ratios of the probe and environment spins respectively, $\lambda_{k}$ are the dipole couplings between the sensor and environment spins, and $\vert{0}\rangle$ ($\vert{1}\rangle$) denotes the $m_s=0$ ($m_s=1$) eigenstate of the $S_z$ operator. We choose a spin-1 system for its analogy with Nitrogen-Vacancy centers in diamond~\cite{Jelezko06,Childress06} as they have emerged as good quantum probes of magnetic fields~\cite{Taylor08,Maze08,Balasubramanian08} for their controllability, optical readout and long coherence times. In addition Nitrogen paramagnetic impurities (P1 centers~\cite{Hanson08}) can act as the environment spins, since they can be collectively controlled~\cite{Delange10}. The choice of a spin-1 system is in addition important since the presence of an eigenstate with zero eigenvalue effectively allows shutting off the interaction between the probe spin and the environment spins at given times: this flexibility makes the EAM scheme easier to implement. We will lift this restriction and examine more general case in section~\ref{Spins}. \begin{figure} \caption{EAM pulse sequence: the vertical bars represent microwave pulses on resonance with the probe (top part of the figure) or environment spins (center), performing the labeled rotations. We assume that the field to be measured is an AC field synchronized with the pulse sequence as shown in the bottom of the figure.} \label{fig:Sequence} \end{figure} In the sequence in Fig. \ref{fig:Sequence} the probe spin undergoes a spin-echo sequence induced by pulses on resonance with the transition between the states $|0\rangle$ and $|1\rangle$ before being measured. For any given evolution of the environment, the signal can then be calculated from $S(t)=[1+\mathcal{S}(t)]/2$, with~\cite{Witzel06,Rowan65}: \begin{equation} \mathcal{S}(t)=\text{Im}\left[\tr{U_0U_1\rho_{\text{env}}U_0^\downarrowg U_1^\downarrowg}\right] \label{eq:pseudospin} \end{equation} Here the propagators $U_i=e^{-i{\mathcal{H}}_it}$ are defined as the evolution of the environment spins in the $m_s=i$ manifold, where ${\mathcal{H}}_0=b(t)\gamma_I\sum_k I_z^k$ and ${\mathcal{H}}_1=\gamma_S b(t)+\sum_k(\gamma_I b(t)+ \lambda_k) I_z^k$ (see Eq.~\ref{eq:Hamiltonian}). The pulsed evolution of the environment, giving the propagators $U_i$, can be most easily calculated in the \textit{toggling} frame~\cite{Haeberlen76}, the interaction frame defined by the control pulses. In this frame, the Hamiltonian (\ref{eq:Hamiltonian}) becomes piecewise time-dependent, with operators alternating between the z- and x-axis. The evolution for the sequence of Fig.~\ref{fig:Sequence} and the resulting signal (Eq.~\ref{eq:pseudospin}) can be calculated exactly in the case of a single ancilla. Here we will present only the result for many ancillas in the limit of \textit{small} field $b$, following the derivation of Ref.~\cite{Goldstein11}. We neglect for the moment the coupling of the sensor spin to the magnetic field and only keep first order term in the field. By expanding the exponentials, the only terms contributing to the signal are then \[ \begin{array}{l} \text{Im}\left[ \text{Tr}\left\{ e^{-i\tau/4\sum_k\lambda_kI^k_x}e^{-i\tau/4\sum_k\lambda_kI^k_z}\rho_{env}e^{i\tau/4\sum_k\lambda_kI^k_z}\times\right.\right. \\ \left.\left. e^{i\tau/4\sum_k\lambda_kI^k_x} ({-i}\overline{B}_2\tau\sum I^k_z) \right\}\right]=- \overline{B}_2\tau\sum_k\cos(\lambda \tau/4) \end{array} \] and \[ \begin{array}{l} \text{Im}\left[ \text{Tr}\left\{ e^{-i\tau/4\sum_k\lambda_kI^k_x}e^{-i\tau/4\sum_k\lambda_kI^k_z}(i \overline{B}_2\tau\sum I^k_z)\rho_{env}\times\right.\right. \\ \left.\left. e^{i\tau/4\sum_k\lambda_kI^k_z} e^{i\tau/4\sum_k\lambda_kI^k_x} \right\}\right]=\overline{B_{2}}\tau, \end{array} \] inserted dt in integrals where $\overline{B_{2}}=-\frac{1}{\tau}\int_{\frac{\tau}{2}}^{\frac{3\tau}{4}}b\left(t\right) dt$.\\ The signal is then given by $S=\half[1-\sin(\Phi)]$, with \begin{equation} \Phi=\gamma_S\overline{B_{1}}\left[1+2P\frac{\gamma_I\overline{B_{2}}}{\gamma_S\overline{B_{1}}}\sum_k \sin\left(\frac{\lambda_{k}\tau}8\right)^2\right], \label{eq:Phase} \end{equation} where $\overline{B_{1}}=\frac{1}{\tau}\left(\int_{0}^{\frac{\tau}{2}}b(t)dt-\int_{\frac{\tau}{2}}^{\tau}b(t)dt\right)$ is the contribution from the direct coupling of the sensor with the field and we have introduced the polarization $P\leq1$ of the environment spins, so that the initial state of each spin in the environment is $\rho_k=\Id/2+PI_z^k$. The factor in the square bracket is the amplification attained as compared to magnetometry performed via a spin echo~\cite{Taylor08}. We can always get an amplification, as $\sin\left(\frac{\lambda_{k}\tau}8\right)^2$ is non-negative and changing the pulse phases always ensures that $\gamma_I\overline{B_2}$ and $\gamma_S\overline{B_1}$ have the same sign. \begin{figure*} \caption{ Left: Simulations of signal decay for spin echo sequence (Red, dotted and dash-dotted lines) and EAM sequence (Black, dashed and solid lines). Right: Simulations of the two sequences with a WAHUHA sequence embedded in each time period (for 1 to 50 cycles). The time was normalized by the largest sensor-environment spin coupling, $\tau\sim[\pi/\lambda_{\rm max} \label{fig:Simulation} \end{figure*} For values of the couplings such that $\left|\lambda_{k}\tau\right|\gtrsim2\pi$, or \textit{strongly coupled} environment spins, the terms $\sin\left(\frac{\lambda_{k}\tau}8\right)^2$ average to $\half$. \textit{Weakly coupled} environment spins ($\lambda_{k}\leq1$) contribute instead with a factor $\propto\lambda_{k}^{2}$ and we obtain a total phase \begin{equation} \Phi=\gamma_S\overline{B_{1}}\tau\left[1+ P\frac{\gamma_I\overline{B_{2}}}{\gamma_S\overline{B_{1}}}\left(n_{sc}+2\sum \,^{'}(\lambda_k\tau/8)^2\right)\right], \label{eq:PhaseStrong} \end{equation} where $n_{sc}$ is the number of strongly coupled spins and the primed sum runs only over the weakly coupled spins (this last term can generally be neglected compared to the strongly coupled spin contribution). The sensitivity of the EAM scheme is easily calculated by noting that ideally the only noise contribution is the shot noise of the spin probe. For $\gamma_S=\gamma_I\equiv\gamma$ and assuming an oscillating field in phase with the echo sequence $b(t)=b_0\sin(2\pi t/\tau)$, the sensitivity~\cite{Bollinger96,Wineland94} per unit time $\eta=\frac{\Delta S}{\|\pdev S{b_0}\|}\sqrt{T}$ is \begin{equation} \eta=\frac{\pi}{C\gamma(2+\half P n_{sc})\sqrt{\tau}}, \label{eq:SensitivityIdeal} \end{equation} where we introduced the factor $C$~\cite{Taylor08} to include any non-ideality of the measurement procedure (here we assumed $T=N\tau$, with $N$ the number of repetitions of the measurement). The sensitivity scales as $1/n_{sc}$ achieving a Heisenberg-like scaling\footnote{Equation~(\ref{eq:SensitivityIdeal}) is valid only for $P\neq0$. We will analyze the case $P=0$ in Section~\ref{Polarization}.}. We note that even in this ideal case, there are two factors that reduce the sensitivity: a limited polarization of the environment spins and the reduction of the time during which the interaction with the external field is effective (because of the scheme proposed, a phase is acquired which is proportional to only $1/4^{th}$ of the total sequence time). The EAM scheme thus demonstrates that it is possible to attain nearly Heisenberg limited sensitivity for metrology with a new class of entangled states (other than squeezed or GHZ states) that as we will see in the following are more robust to decoherence. Furthermore, these states can be created with limited control resources, thus opening the possibility of using spins in the environment as a resource for metrology. \section{Decoherence}\label{Decoherence} The results in the previous section did not take into account the effects of decoherence caused both by the environment spins used as an ancillary system and by any other residual bath. In this section we will take these effects into account and show that even in the non-ideal case the EAM sequence can provide a sensitivity enhancement with respect to other control scenarios (such as a spin-echo) that only aim at refocusing the interaction of the probe spin with the environment spins. \subsection{Decoherence induced by the environment spins couplings} The interactions among environment spins hamper the EAM scheme in two ways. First, flip-flops of environment spins lead to a loss of coherence of the probe spin. This effect is the same that is observed during a spin echo, and we will show that the resulting coherence time $T_2$ is on the same order for the two sequences. Second, the interactions will also cause the environment spins to lose their internal phase coherence, resulting in a smaller accumulated phase $\Phi$. Still, this effect happens on a time scale $\tau_I$ given by the environment spin correlation time, which is usually longer than the probe coherence time, $\tau_I\geq T_2$. Thus the sensitivity is ultimately limited by $T_2$, as in the spin-echo case. Consider the system evolution as given by Eq.~\ref{eq:pseudospin} (for simplicity in the absence of the magnetic field $b$). Now the propagators are given by the Hamiltonian \begin{equation} \begin{array}{l} H= b\left(t\right)\left(\gamma_SS_{z}+\gamma_I \sum I_z^k\right)+\sum\lambda_kS_zI_z^k\\ \qquad+\sum\kappa_{jk}\left(3I_z^jI_z^k-\vec{I^j}\cdot\vec{I^k}\right) \end{array} \label{eq:FullHamiltonian} \end{equation} where $\kappa_{ij}$ are the intra-bath couplings given by the magnetic dipole interaction among spins. Because of the presence of the couplings, the evolution in the two halves of the sequence is no longer the same, thus the interaction between the probe spin and the environment spins can no longer be perfectly refocused. This effect, usually called spectral diffusion, is observed as well in spin echo experiments and lead to the coherence time $T_2$. The addition of a modulation of the environment spins is not expected to change substantially the coherence time, as hinted by the short time evolution expansion presented in Ref.~\cite{Goldstein11}. An exception is for a perfectly polarized bath: in that case, flip-flops are quenched in the spin-echo, but they are still allowed in the EAM scheme since they are enabled by the rotation of the spins during the protocol; the effect of flip-flop quenching is however noticeable only for very high polarization of the bath~\cite{Fischer09,Takahashi08}. From this argument we expect that one can have a similar interrogation time $\tau$ in Eq. (\ref{eq:SensitivityIdeal}) for the EAM scheme considered here as for a simple spin echo sequence. Unlike for different entangled states~\cite{Huelga97}, the enhancement from entanglement is therefore not counterbalanced by a decrease in the interrogation time $\tau$, and the EAM scheme does allow for a significant improvement of the sensitivity. We further verify this claim by simulations. We used the disjoint cluster approximation~\cite{Maze08b} to simulate the sequence in Fig.~\ref{fig:Sequence} for a system comprising the probe spin surrounded by an environment of 25-50 spins randomly positioned in a cube with sides of unit length. By averaging over many spatial distributions of the environment spins, the simulation converges quickly even for small cluster sizes and it gives information about the average coherence time~\cite{Yang08,Witzel10}. The system we consider is inspired by a NV center in a nano-crystal of diamond in the presence of P1 Nitrogen impurities~\cite{Goldstein11}, but the results are more generally valid. For comparison, we also simulated the evolution under a spin echo sequence. From the results in Fig.~\ref{fig:Simulation} we see that the coherence time is not qualitatively different for the two sequences. The figure shows in addition that the coherence time depends on the density of the environment spins, a fact that will be important in evaluating the sensitivity achievable with the EAM scheme. The second effect of the intra-bath couplings is to make the environment spin themselves loose their coherence, in a time on the order of their correlation time $\tau_I$, which is given by the rate of spin flip-flop driven by the dipolar interaction. If the environment spins are no longer in a coherent state, the phase they acquire does not add up constructively, resulting in a smaller phase $\Phi$. Still, this effect is comparable to the previous one, since the correlation time is at least on the same order of $T_2$. In addition to the environment spins that are used as ancillary sensors, the system could be in contact with an additional spin bath. For example, in the case of the NV center in diamond this bath is given by the $^{13}$C nuclear spins. The effects of this quasi-static bath are refocused by the $\pi$ pulse on the NV center and by the two $\pi/2$ pulses on the environment spins, which amount to a so-called ``Hahn echo''~\cite{Hahn50} sequence. Any residual decay is again comparable to what is observed in a simple spin echo for the probe spin. \subsection{Dynamical Decoupling}\label{sec:decoupling} An increase in the effective correlation time of the environment spins would be beneficial in two ways, by both increasing the coherence time of the probe spin, through its influence on the sensor spin $T_2$-time, and directly by improving the environment spin coherence. Dynamical decoupling schemes could achieve this goal. \begin{figure} \caption{ (color online) Embedding of a WAHUHA sequence in the EAM sequence. The WAHUHA is shown at the bottom, together with the ``direction'' of the Hamiltonian in the toggling frame. } \label{fig:WHH} \end{figure} The dipolar Hamiltonian can be refocused using homonuclear decoupling sequences such as the WAHUHA sequence~\cite{Waugh68}. The pulse modulation gives a time-dependent Hamiltonian for the spin-spin interaction that averages to zero over a cycle time $t_c$. If the modulation is fast compared to the couplings, the effective Hamiltonian over the cycle is well approximated by its average. A simple symmetrization of the pulse sequence~\cite{Burum79} can further cancel out the first order correction, leaving errors that are only quadratic in the product $\kappa t_c$ (and do not depend on the total evolution time, that could be given by many cycles)~\cite{Haeberlen76}. Fig.~\ref{fig:WHH} shows how to incorporate a WAHUHA sequence within the EAM sequence. We modified the phases of the pulses with respect to the original sequence in order to obtain an effective coupling between the probe and environment spins $\propto\frac{1}{\sqrt{3}}S_{z}\sum\lambda_{k}I_{z(x)}^{k}$ in the odd(even) time intervals. These phase changes do not affect the average of the dipolar Hamiltonian and hence the performance of the WAHUHA sequence. Unfortunately the modulation does not only averages out the dipolar Hamiltonian, but it also reduces the linear terms, by a factor $1/\sqrt{3}$. In many cases, the increase in coherence time more than compensate for this weighting factor. In Fig.~\ref{fig:Simulation} we simulated via the disjoint cluster method the coherence of the EAM and spin-echo sequences, while applying the WAHUHA sequence in between the pulses. Comparing the results obtained in the absence of dynamical decoupling, we see that the sequence is very effective in increasing the coherence time. A different strategy for directly increasing the probe spin $T_2$ is to use more than one $\pi$-pulse during the total sequence time~\cite{Taylor08}. This technique is inspired by concatenated dynamical decoupling schemes and in particular by the CPMG sequence~\cite{Carr54,Meiboom58}. More generally, these examples indicate that the EAM scheme can be combined with various forms of decoupling. \section{Sensitivity}\label{Sensitivity} In the previous section we saw that the coherence time (and hence the time during which the phase can be acquired) depends on the density of the environment spins. For the EAM sequence, the signal too depends on the environment spin density, since it determines how many environment spins are close enough to the probe spin to be considered ``strongly coupled''. Thus the optimal sensitivity arises from a compromise between the environment spin density and the interrogation time. Including the probe decoherence due to the environment spins, as well as other bath contributions, yielding a coherence time $T_2^B$, the sensitivity of Eq.~\ref{eq:SensitivityIdeal} becomes: \begin{equation} \eta=\frac{\pi e^{(\tau/T_2)^3}e^{(\tau/T_2^B)^3}}{C\gamma\sqrt{\tau}(2+\half P n_{sc})} \label{eq:SensitivityReal} \end{equation} The functional form we assumed for the decay is inspired by the measured behavior of NV centers in diamond~\cite{Childress06,Delange10} and usually arises from a Lorentzian spectrum of the bath. \begin{figure} \caption{ (Color online) Sensitivity of the EAM scheme normalized by the sensitivity of a spin-echo scheme in the absence of any environment spin, $\eta/\eta_{\text{echo} \label{fig:Sensitivity} \end{figure} The couplings between the probe and environment spins scales as $\lambda_k\sim\gamma^2/r_k^3$ (assuming dipolar interaction and setting $\gamma_I=\gamma_S=\gamma$ for simplicity), with $r_k$ the distance to the probe spin. Then, for a fixed duration $\tau$ of the EAM sequence, the number of ``strongly coupled'' spins $n_{sc}$ scales as $n_{sc}(\tau)\sim\gamma^2\rho \tau$, where $\rho$ is the density of the environment spins. The probe coherence time also scales with the density as $T_2\propto1/\rho$. The sensitivity is then a function of two parameters: how many polarized spins are strongly coupled in the coherence time $T_2$ and how much the coherence time is reduced with respect to the background bath coherence time by introducing the ancillary environment spins. We define a quantity $Q=P \rho \gamma^2 T_2$, which describes the ``quality'' of the environment spins. A second quantity describing the reduction in coherence time due to the ancillary environment spins is given by the ratio $r=T_2^B/T_2$. The EAM sensitivity then depends only on these two parameters and the bath coherence time, such that Eq.~(\ref{eq:SensitivityReal}) becomes \begin{equation} \eta=\frac{\pi e^{(1+r^3)(\tau/T_2^B)^3}}{C\gamma\sqrt{\tau}(2+\frac\tau{2T_2^B} r Q )} \label{eq:SensitivityParams} \end{equation} We can further optimize the sensitivity with respect to the interrogation time $\tau$ and compare it to the case where the field is measured by a probe spin (via a spin-echo sequence) in the presence of the background spin bath only (that is, no environment ancillary spins). In this case, the sensitivity is given by~\cite{Taylor08} \begin{equation} \eta_{\text{echo,1}}=\frac{\pi e^{(\tau/T_2^B)^3}}{2C\gamma\sqrt{\tau}} \label{eq:etaAC} \end{equation} \begin{figure} \caption{(Color online) Sensitivity ratio $\eta/\eta_{\text{echo} \label{fig:SensitivityRatio} \end{figure} As show in Fig.~\ref{fig:Sensitivity}, the EAM sensitivity as given by Eq. (\ref{eq:SensitivityParams}) improves up to $r=1$, where the decoherence due to the environment spins becomes more important than the background bath. The improvement depends on the ``quality'' $Q$, since for higher $Q$ there are more strongly coupled spins at a given $T_2$ time. It is then clear that there is an optimum number of environment spins that one would want to introduce in the system to obtain the optimal sensitivity. Alternatively, the quality $Q$ can be improved by increasing $T_2$ using the dynamical decoupling methods we introduced in section~\ref{sec:decoupling}. If the number of environment spins is instead fixed, we are interested in comparing the EAM and spin-echo scheme for a given system (e.g. a given nanocrystal of diamond). In Fig.~\ref{fig:SensitivityRatio} we plot the ratio of the EAM sensitivity to the spin-echo sensitivity : \begin{equation} \eta_{\text{echo}}=\frac{\pi e^{(1+r^3)(\tau/T_2^B)^3}}{2C\gamma\sqrt{\tau}} \label{eq:sensitivityechoenv} \end{equation} (in the figure this expression is optimized with respect to the interrogation time $\tau$). In the high $r$ limit the sensitivity ratio depends only on the $Q$ factor, as $\eta/\eta_{\text{echo}}\approx\frac{\sqrt[6]{e^2/3}}{ \left(1+2^{-7/3} Q\right)}$. To estimate the potential sensitivity improvement of the EAM method we express $Q$ in terms of measurable quantities. Specifically, we can write the environment quality as $Q=P\rho\gamma^2T_2\approx P\sqrt{\sum\lambda_k^2} T_2$, where we used the fact that $\gamma^2\rho\tau=n_{sc}(\tau)\lesssim\tau\sqrt{\sum\lambda_k^2} $ . The average distribution of couplings $M_2=\sqrt{\sum\lambda_k^2}$ is related to the second moment of the probe spin, which gives its dephasing time $M_2=1/T_2^*$. Then the sensitivity improvement is given by the ratio $T_2/T_2^*$, which can be quite large in many system. \section{Polarization and sensitivity}\label{Polarization} The sensitivity discussed in the previous section depends on the polarization of the environment spins. In this section we first propose methods for creating this polarization, under the assumption that the probe spin can be polarized at will. We then generalize the EAM scheme to the case where no polarization is available. This generalization will furthermore prove useful in the case where the field to be measured is affected by a random phase. \subsection{Polarizing the environment spins} In an environment assisted magnetometer working at room temperature, the environment spins will be in a thermal state, close to the maximally mixed state. Polarization need then to be created by relying on the probe spin and the Hamiltonian (Eq.~\ref{eq:FullHamiltonian}) that is required for the measurement scheme. To do this we assume that the probe spin can be repetitively polarized: this is the case e.g. for an NV center that can be polarized optically. Polarization could then be transferred to the spins in the environment by a swapping Hamiltonian such as ${\mathcal{H}}_{SW}\sim (S_xI_x+S_yI_y)$. Although this operator is contained in the dipole-dipole Hamiltonian, it is usually quenched in the rotating frame, if the energies of the two spin species are different. For example, in the case of NV and P1 spins, the zero-field splitting of the NV creates an energy mismatch. The swapping Hamiltonian can be reintroduced by inducing a Hartman-Hahn matching of the energies in the rotating frame under a continuous microwave irradiation~\cite{Hartman62,Weis06}. By adjusting the Rabi frequency and the offset, the two spin species are brought into resonance and spin flip-flops (allowing polarization transfer) are now allowed, leading to a buildup of polarization. The environment spins can then be polarized efficiently by alternating periods during which the probe spin is polarized and periods during which polarization exchange is driven by the microwave irradiation. The buildup of polarization can happen either via direct interaction between the probe spin and a spin in the environment, or indirectly via spin-diffusion~\cite{Khutsishvili66,Ramanathan08}. Since we are interested only in polarizing strongly coupled spins, the first process is dominant. Then, we can estimate the polarization time by the number of spins we want to polarize divided by their average coupling strength, $T_{pol}\sim n_{sc}(T)/\ave{\lambda}\approx n_{sc}(T)T_2^*$ (where we used $1/T_2^*=\sqrt{\sum\lambda_k^2}$ to estimate the average coupling strength, an upper bound for $T_{pol}$ would be more generally $T_{pol} \leq n_{sc}(T) T/\pi$). A different strategy to initialize the spin environment is measurement-based polarization~\cite{Togan11} with either feedback or adaptive schemes. Precise measurement of the local magnetic field created by the spin environment at the sensor spin location effectively determines the environment spin state, with an increasing knowledge of the magnetic field shift corresponding to a reduced spin-state distribution and hence higher polarization. The polarization time will reduce the achievable sensitivity per root Hz, $\eta$, since it increase the preparation time such that fewer measurement can be performed during a certain time interval. The exact sensitivity degradation will depend on many factors, e.g. the depolarization ($T_1$) time of the environment spins, which determines how often the preparation step needs to be repeated. \subsection{EAM with no polarization and phase error} In the discussion so far we assumed that the external field to be measured was either static or oscillating in phase with the control sequence. For $b(t)=b\cos(2\pi t/\tau)$, we obtained the signal $S=\half(1-\sin\Phi)$, where the phase is given by Eq.~(\ref{eq:Phase}) for the EAM scheme or by $\Phi=2\gamma b \tau/\pi$ for the spin-echo scheme. However, if the field has a random phase (or cannot be synchronized perfectly with the pulse sequence) the signal averaged over many runs goes to zero, as $\ave{S}=\half(1-\ave{\sin\Phi})=0$ if $\ave{\Phi}=0$. Furthermore, even higher momenta of the signal, $\ave{S^n}$ are zero; thus it is not possible to infer information about the stochastic field by this method. A possible solution is to change the phase of the final pulse~\cite{Meriles10} (or equivalently, to introduce an additional, known phase accumulation during the free evolution). Then the signal becomes \begin{equation} S=\half[1+\ave{\cos(\Phi+\theta)}] \label{eq:SignalNoPol} \end{equation} where $\theta$ is the phase difference between the initial and final pulse, $\Phi$ is the phase due to the field to be measured and we neglect any decay for simplicity. Since $\ave{\cos(\Phi+\theta)}=\ave{\cos\Phi}\cos\theta$, the maximum signal is obtained for $\theta=0$, or by setting the phase of the initial and final pulse to be equal. The phase $\Phi$ acquired in the modified EAM scheme (Fig.~\ref{fig:Sequence} with the last pulse along x) is different than that obtained in Eq.~(\ref{eq:Phase}). In the limit of small fields, we obtain the signal \begin{equation}\label{eq:SignalEAMNoPol}\begin{array}{l} S_x=1-\half(\frac{b t}{2\pi})^2\left(2+ \sum_k[1+\cos\left(\frac{\lambda_k t}4\right)^2] \sin \left(\frac{\lambda_k t}4\right)^2\right)\\ \quad \approx 1-\half\left(\frac{b t}{2\pi}\right)^2\left[2+ \frac34 n_{sc}\right], \end{array} \end{equation} where again we only summed over the ``strongly coupled'' environment spins. We note that the signal does not depend on the polarization of the environment spins (at least to first order in the polarization and to second order in the field $b$). Thus, even in the absence of any polarization it is possible to measure the external field (although not the sign of it). We compare the achievable sensitivity of the EAM and spin-echo method in the case where no polarization is present and the control sequence describe above is used. Optimizing the sensitivity with respect to the interrogation time $\tau$, we obtain the sensitivity for the spin echo sequence \begin{equation} \eta_{\text{echo},x}= \frac{\pi}{C\gamma\sqrt{\tau}} \label{eq:sensitivityechox} \end{equation} while for the EAM sequence we have \begin{equation} \eta_{\text{EAM},x}\approx \frac{\pi }{C\gamma\sqrt{\tau}\sqrt{1+\frac32 n_{sc}}} \label{eq:sensitivityeamx} \end{equation} In the case of zero polarization (or of a signal with a random phase) it is no longer possible to obtain a quantum enhancement and have a scaling proportional to $1/n_{sc}$ by exploiting the spins in the bath. Indeed if there is no polarization, no entanglement is created in the system, and no quantum enhancement of the sensitivity is expected\footnote{Note that our assumptions of no direct access to individual ancillary spins preclude using adaptive schemes, which have been shown in other conditions to achieve quantum enhancement of the sensitivity even without entanglement~\cite{Higgins07}.}. Nevertheless, for favorable conditions of the spin environment (high quality $Q$ and low ratio $r$) it might still be beneficial to use the EAM scheme instead of a simple spin-echo magnetometry because it allows for an improvement $\sim\sqrt{n_{sc}}$ by exploiting the unpolarized spins. \section{Extension to other spin probes}\label{Spins} In the previous sections we presented a scheme that relied on the fact that for one of the eigenstates of the probe spin ($\vert{0}\rangle$) the couplings to the environment spins was zero. It is possible to extend the EAM scheme to the case where the probe is a spin-1/2, but only if the environment-probe spin couplings are all of the same sign. Such a situation could for example be realized by considering a single quantum dot in the nuclear spin environment. The interaction between the central spin and the environment spins in this system is given by the contact interaction, whose strength depends mainly on the electronic spin wavefunction density and does not present the strong angular dependence of the dipolar interaction. To apply the EAM scheme with a spin-$1/2$ probe, we rotate the environment spins to be aligned along the $y$ axis before applying the sequence shown in Fig. (\ref{fig:Sequence}). For small fields, the additional phase acquired thanks to the environment spins is given by \begin{equation} \Phi_{1/2}\propto b \tau P\sum_k\sin\left(\frac{\lambda_k\tau}4\right)^3 \label{eq:Phase12} \end{equation} If all the couplings $\lambda_k$ are positives, the environment spins contributions add constructively and it is always possible to find a time $\tau$ s.t. there are $n_{sc}$ strongly coupled spins for which $0\leq\lambda_k \tau\leq4\pi$, so that $\sum_k\sin\left(\frac{\lambda_k\tau}4\right)^3\approx \frac{4}{3\pi}n_{sc}$. More generally, it is also possible to use probes with higher spins, selecting two of their eigenstates as the levels of interest, by driving transitions on resonance with their energy difference. If the two eigenstates $\ket{a}$ and $\ket{b}$ are such their eigenvalues have different absolute values, $|m_a|\neq|m_b|$ then we can apply the EAM sequence for any value of the couplings, as the phase enhancement will be $\Phi\propto \sum_k\sin\left(\frac{|m_a|-|m_b|}8\lambda_k \tau\right)^2$. Otherwise, one might use the modified scheme just presented in this section, if all the coupling constants are positive. As shown in this section, the scheme we introduced is quite flexible and can be applied to many different physical systems, beyond the one we focused on in this paper. Besides spin systems, the same ideas could for example find an implementation based on trapped ions~\cite{Goldstein11}. pace{-6pt} \section{Discussion and conclusion} In conclusion, we analyzed the EAM scheme introduced in Ref.~\cite{Goldstein11}, which aims at enhancing the sensitivity of a single solid-state spin magnetic field sensor, by exploiting the possibility to coherently control part of its spin environment. The environment spins act as sensitive probes of the external magnetic field, and their acquired phase is read out via the interaction with the sensor spin. Since the measurement scheme maintains roughly the same coherence times of spin-echo-based magnetometry and the noise is still the shot-noise of a single qubit, we achieve a quasi-Heisenberg limited sensitivity enhancement. We analyzed in detail the sources of decoherence and confirmed with numerical simulations that the sensor coherence time under the EAM scheme is comparable to the $T_2$ time under spin-echo, since the leading cause for decoherence has the same origin in the two cases. We further showed that dynamical decoupling schemes aimed at increasing the correlation time of the spin environment, by reducing the effects of intra-bath couplings, can be embedded in the measurement scheme and leads to longer coherence times and enhanced sensitivity. We extended the EAM scheme to the case where the environment spins are in a highly-mixed (zero-polarization) state, by appropriately modifying the detection sequence. This modified scheme achieve a classical scaling of the sensitivity, but can still be beneficial whenever the polarization methods we outlined cannot be applied or the AC field to be measured has a random phase. Our analysis finds that the sensitivity is determined by the ``quality'' of the environment, a parameter that takes into account a compromise between the number of strongly coupled environment spins with the reduced coherence time they entail. This result can be used to define the specifications of engineered systems with controlled densities of spin impurities for optimal sensitivity. \textbf{Acknowledgments}. This work was supported by the NSF, NIST, ARO (MURI - QuISM), DARPA QuASAR, the Packard Foundation and the Danish National Research Foundation. \end{document}

\begin{document} \begin{frontmatter} \title{Automated Evolutionary Approach for the Design of Composite Data-Driven Modelling Workflows} \author{Nikolay O. Nikitin} \ead{nnikitin@itmo.ru} \author{Pavel Vychuzhanin} \ead{pavel.vychuzhanin@itmo.ru} \author{Mikhail Sarafanov} \ead{mik_sar@itmo.ru} \author{Iana S. Polonskaia} \ead{ispolonskaia@itmo.ru} \author{Ilia Revin} \author{Irina V. Barabanova} \author{Gleb Maximov} \author{Anna V. Kalyuzhnaya} \author{Alexander Boukhanovsky} \address{ITMO University, Saint-Petersburg, Russia} \begin{abstract} In the paper, the automated approach for the data-driven design of composite modeling workflows is proposed. The approach combined key ideas of both automated machine learning and workflow management systems. It allows designing the pipelines with customizable graph-based structure, analyzes the obtained results, and reproduces them. The evolutionary approach is used for the flexible identification of pipeline structure. The additional algorithms for sensitivity analysis, atomization, and hyperparameter tuning are implemented to improve the effectiveness of the approach. Also, the software implementation on this approach is presented as an open-source framework. The set of experiments is conducted for the different datasets and tasks (classification, regression, time series forecasting). The obtained results confirm the correctness and effectiveness of the proposed approach in the comparison with the state-of-the-art competitors and baseline solutions. \end{abstract} \begin{keyword} AutoML\sep workflow\sep composite pipeline \sep evolutionary algorithms \sep WMS \end{keyword} \end{frontmatter} \linenumbers \section{Introduction} \textcolor{blue}{понятия воркфлоу, пайплайна, композитных приложений} Nowadays, data-driven modeling is a quite promising approach to solve various scientific and applied problems. To evaluate the complex modeling task (especially in the cloud and distributed environments), it is usually decomposed to a large number of interconnected blocks that are referred to as a workflow \cite{visheratin2016workflow}. The workflows that are specialized for modeling tasks are usually referred to as pipelines (linear, variable-shaped \cite{zoller2019benchmark} or composite models \cite{kovalchuk2018conceptual}). In complex systems, the modeling pipelines are usually used as part of the composite application that controls the evaluation of the models. \textcolor{blue}{Существует множество приложений этих механизмов для создания систем научных вычислений: всевозможные WMS. И основной тренд развития WMS (только сейчас он слегка замедлился, нет идей) — это автоматизация процесса построения WF под задачу.} Since the interconnections in the workflow can be relatively complex (an example is presented in Fig.~\ref{fig_wfs}a), there is a set of specialized workflow management systems (WMS). The main aim of the WMS is to automate the generation or adaptation of the workflow for a specific task. Despite the existence of different works devoted to this problem, appropriate practical implementation of the automation tool does not exist. \begin{figure*} \caption{The comparison of different representation workflows: a) the problem-independent representation of the common workflow b) the high-level structure of the modelling workflow (pipeline). c) the detailed structure of the composite pipeline for machine learning-based modelling. The criteria for the automated design of workflows are highlighted.} \label{fig_wfs} \end{figure*} \textcolor{blue}{Автоматизация построения WF под задачу должна исходить из некоторой идеи (что мы хотим) и некоторого процесса (что именно мы автоматизируем). Обычно автоматизируют процесс создания структуры графа, реже — выбора узлов. А опираются в основном на (а) методы планирования (минимизация времени работы) — как Денис, и (б) методы инженерии знаний (если удается описать логику приложений).} The task of workflow design automation can be divided into two conceptual aspects: the object for optimization and the method for it. The optimization object for the workflow is usually represented as the structure of the graph or the selection of the nodes in it. The possible methods are workflow scheduling \cite{chirkin2017execution, thekkepuryil2021effective} for execution time minimization or knowledge engineering (for the composite applications with an understandable internal logic) \cite{nasonov2018multi}. \textcolor{blue}{Если п. 3(а) в целом получается совсем неплохо (об этом диссеры Дениса Н. и Миши М.), то пункт 3(б) в целом НЕ работает. Т.к. никто не научился описывать задачу так, чтобы задать явно целевую функцию для автоматизации построения WF (в диссере Паши С. это как-то решалось, но требовало офигенных размеров графов знаний). Мы понимаем, что в целом для науки направление 3(б) нереализуемо. Но зато есть область знания, где целевая функция для процесса автоматизации построения WF может быть сформулирована «на раз». Это — ML.} However, it is a complicated task to formalize the objective function for workflow optimization (for example, expert-based identification of complex knowledge graphs can be required). It makes it impossible to automate the design for any custom workflow. However, specialized solutions for specific types of workflows can be proposed. A promising direction of research is the design of machine learning (ML) pipelines, which can be considered as a subclass of computational workflows. The main stages of the typical ML modeling pipeline are represented in Fig.~\ref{fig_wfs}b. This pipeline has a linear structure, that makes the automation task relatively simple. It can be solved using basic optimization methods, e.g. grid search \cite{zoller2019benchmark}. The difference from other types of workflows is that the objective for the ML pipelines can be effectively formalized as an estimation of modeling error. \textcolor{blue}{Классическая логика ML: как ставятся задачи, требующие порождения таких WF (что надо совмещать разные методы), как измерить результат (в каком виде целевая функция). Сослаться на то, как действуют люди на хакатонах по ML («переборный интеллект» для разных методов). However, the linear pipelines are mostly applicable to simple benchmarks \cite{AutoML2021}. In practice, it can be impossible to build an effective model of raw data using the pipeline with a static structure. There are a lot of different specialized blocks that can be involved in the pipelines in a parallel way. As an example, data preprocessing, feature selection, dimensionality reduction, gap filling, ensembling, post-processing blocks can be mentioned. An example of the structure of a graph-based (composite) pipeline is presented in Fig.~\ref{fig_wsf}c. It can be seen that there are a lot of mutually complementary blocks involved in the pipeline: e.g. different feature selection algorithms are applied to the same dataset in order to extract more useful features from the data. However, building composite pipelines (optimal selection and tuning of the blocks and connection between them) is a sophisticated and time-consuming task even for experts in data science. {\color{red} Сейчас уже есть WMS для ML, но — они абсолютно безмозглые (человек руками лепит граф, пример — MS Azure). А почему мы мы считаем, что именно для ML нужна автоматизация создания WF : (а) нет априорных предметных знаний, все на данных — потому часто переборный интеллект решает все; (б) есть абсолютно четкое количественное выражение целевой функции — метрики моделей (в) там наработано много разных библиотек (есть из чего выбирать реализации узлов); } This problem is further complicated by the fact that the existing implementations of the WMS for ML modeling pipelines have a lack of 'intelligent' automation and still require a lot of expert involvement. As an example, the AzureML \cite{team2016azureml} can be noted. At the same time, the design of a composite pipeline is a complex problem that involves experts in both data science and the target domain. It raises a lot of issues that make it too complicated to deliver the ML into real-world business processes. However, the design automation for ML workflows is a promising and achievable task due to a set of factors: \begin{itemize}[] \item[(a)] the ill-structured domain-specific knowledge usually does not have a critical influence in data-driven workflows; \item[(b)] the objective functions for the workflow optimization can be defined in an implicit and computable way; \item[(c)] there are a lot of ML algorithms implemented as open-source libraries, that allow selecting workflow blocks implementations from a large set of solutions. \end{itemize} There are a lot of Automated Machine learning (AutoML) approaches to the automation of ML models design \cite{AutoML2021}. The most common targets of automation are combined model selection and hyperparameter optimization (the CASH acronym is frequently used for it). Besides that, there are several approaches that also take the automation of the feature selection and data preprocessing into account. The design of ensembles of the ML models can also be automated using state-of-the-art frameworks. Also, there are a lot of tools that allow solving the Neural Architecture Search (NAS) problem and identifying the most suitable structure of a neural network for a specific problem \cite{elsken2019neural}. {\color{red} Пункты (а,б) уже являются логикой AutoML. Но вот пункт (в) к ней не вполне относится — потому что фреймворки AutoML в основном работают на собственных механизмах, а не собирают в кучу чужие сервисы. Зато пункт (в) — это классика распределенных (в широком смысле вычислений). Потому наша идея в статье — скрестить логику AutoML с логикой WMS (=ключевой топик FGCS), и посмотреть, что из этого получится. } However, most of the existing AutoML solutions are quite distanced from real business and industrial applications. Although there are several enterprise-related frameworks that exist, they do not always achieve the appropriate quality of the pipelines. The reason is that AutoML covers (a) and (b) points from the described list of potential effectiveness factors, but the point (c) - the flexible selection and design of building blocks - is still an unsolved problem here. At the same time, (c) is an essential component of many WMS. That is why we decided to develop a flexible approach that combines the main features of AutoML and WMS. It allows us to increase the quality of the modeling of different real-world processes using data-driven and hybrid models with a complex, heterogeneous structure. The second aspect is the open-source implementation of this approach, which makes it possible to use the proposed approach for different purposes in a transparent and reproducible way. For these reasons, the open-ended evolutionary automated modeling techniques and their software implementation in a (\url{https://github.com/nccr-itmo/FEDOT}) Framework are described. Different aspects of composite modeling are discussed: automation of pipelines design, analysis of the obtained pipelines, reproducibility of the pipelines. The paper is organized as follows: Section~\ref{sec_related} contains the analysis of the existing AutoML solution; Section~\ref{sec_problem} describes the problem statement for the data-driven models' design; Section~\ref{sec_approach} provides an extensive description of the proposed evolutionary approach to models' design; Section~\ref{sec_software} describes the open-source software implementation of the proposed approach as a part of the framework; Section~\ref{sec_exp} provides the results of the experiments for different benchmarks; Section~\ref{sec_disc} contains the discussions on the perspectives of modern AutoML and the main conclusions for the paper. \section{Related works} \label{sec_related} \subsection{Design of data-driven modeling pipelines} There are a lot of recipes for modeling, predicting, and forecasting various processes using available data and mathematical methods that allow building conceptual models on this data \cite{caldwell2013mathematical}. These models can have a different nature. They can be based on physical laws represented as equations that are known a priori, that allow reproducing the underlying nature of the target phenomena. If the specific physical law is unknown, equation discovery methods can be used \cite{hvatov2020}. However, in many real cases (e.g. modeling of technical or social systems) there is no physics-based assumption for the model design. In this case, statistical and machine learning (ML) methods have demonstrated their high effectiveness \cite{jordan2015machine} and now they are applicable in various fields \cite{das2017survey}. However, it is not enough just to select a machine learning method to build an effective model - the entire modeling workflow (pipeline) is required to process the data and models. For example, the quality of the obtained model largely depends on the choice of the data preprocessing approach \cite{zelaya2019towards} or the effectiveness of hyperparameters tuning \cite{probst2019tunability}. Also, there are not any universally applicable methods for data processing, model selection, or hyperparameters tuning. Due to the inhomogeneity, multi-modality, and stochasticity of the raw un-processed datasets, a promising way to improve the quality of the machine learning models is to build ensembles of models \cite{sagi2018ensemble} and combine imprecise predictions of several ML models (or the same model with different data as inputs) into one, better prediction. Different approaches to ensembling exist: stacking \cite{pavlyshenko2018using}, bagging and boosting \cite{gonzalez2020practical}, hybridization \cite{ardabili2019advances}. Bagging involves dividing the original training set into many sub-samples, each of which is used for teaching a separate model in an ensemble. As an example, the random forest model can be used: a lot independent decision trees are trained on bagging, and in the final forecast their results are combined. Boosting is based on sequential training of models, each of which is aimed at reducing the error of the predecessor. By default, this ensemble method does not impose restrictions on the types of models used, however, boosting ensembles based on decision trees has gained the greatest popularity in machine learning. The widely-used open-source solutions for boosting are XGBoost, LightGBM, and CatBoost. For stacking, the final ensemble can adopt a more complex structure, which can be represented as a directed graph, where nodes are ML models, and connections are data streams. For example, if the default bagging-based ensemble uses the voting classifier \cite{ruta2005classifier} to combine model forecasts, the voting logic can be implemented in a separate ML model. Ensemble modeling methods are also used in the fields of environmental process modeling. In this case, the main difference lies in the type of models being used, where numerical models based on the equations of mathematical physics can be included in ensembles. Models in the ensemble can vary not only according to data for training but also according to the internal configuration: meta-parameters of numerical models, boundary and initial conditions \cite{vychuzhanin2019robust}. The hybrid modeling is based on the ensembling of the data-driven and numerical models \cite{sun2020comprehensive}. This technology is used in various subject domains \cite{zhang2019hybrid}, where the results of mathematical physics models are improved using data-driven model predictions. However, one disadvantage that limits the use of this technology in composite modeling is the lack of a universal interface for domain-specific models, which significantly complicates their use in composite pipelines. This approach has a lot of implementations for different problems. For example, the described ensemble methods are implemented within the machine learning frameworks, e.g. Scikit-Learn \cite{pedregosa2011scikit}. However, the problem of their combination as a part of a modeling pipeline still has no satisfactory solution. In this paper, we decided to use the concept of a composite model (described in \cite{kalyuzhnaya2020automatic}) to represent the various structures of pipelines in a unified form. Data-driven models can not be distinguished from the corresponding pipeline, so the composite model and composite pipeline can be considered synonymous in this case. From the perspective of applied data science, the modeling approach as a numerical method is inseparable from its software implementation. That is why we devoted the second part of this section to the review of state-of-the-art frameworks and tools for data-driven pipelines design and evaluation. \subsection{Automated tools for pipeline design} In the field of modern ML it is hard even to estimate the number of various tools, frameworks, and platforms that allow building data-driven models of the different processes and events of the real or virtual world. In the frame of this paper, we decided to focus on the analysis of existing solutions of AutoML-related products from both scientific and enterprise domains. In this section, our goal was to identify the main advantages and disadvantages of existing tools of automated modeling in order to substantiate the relevance of the proposed approach to the model design. First of all, although many variations of the AutoML tools exist, many of them are based on the same ML frameworks that are used for the ML part of the modeling itself. It means that each AutoML tool can be described in the following way: item class of problems that can be solved (classification, regression, etc); types of data that can be processed (tables, images, texts, etc); parts of the ML pipeline that can be automated; types of models that can be used in the automated design; optimization approach used to automate the modeling; availability of the open-source implementation. The results of this comparison for a set of state-of-the-art solutions for pipeline automation are presented in Table~\ref{tab_automl_surv}. \begin{table*} \caption{The comparison of the main aspects of several state-of-the-art open-source AutoML tools. Short references to the repositories in \url{github.com} are provided.} \centering \refstepcounter{table} \label{tab_automl_surv} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{Tool} & \textbf{ Pipeline } & \begin{tabular}[c]{@{}c@{}}\textbf{ Optimisation}\\\textbf{algorithm }\end{tabular} & \textbf{ Input data } & \textbf{ Scaling } & \begin{tabular}[c]{@{}c@{}}\textbf{ Additional }\\\textbf{features }\end{tabular} & \textbf{ GitHub repository} \\ \hline TPOT & Variable & GP & Tabular & \begin{tabular}[c]{@{}c@{}}Multiprocessing,\\~Rapids\end{tabular} & \begin{tabular}[c]{@{}c@{}}Code \\generation\end{tabular} & \url{EpistasisLab/tpot} \\ \hline H2O & Fixed & Grid Search & Tabular, Texts & Hybrid & - & \url{h2oai/h2o-3} \\ \hline \begin{tabular}[c]{@{}c@{}}Auto\\Sklearn\end{tabular} & Fixed & SMAC & Tabular & - & - & \url{automl/auto-sklearn} \\ \hline ATM & Fixed & BTB & Tabular & Hybrid & - & \url{HDI-Project/ATM} \\ \hline \begin{tabular}[c]{@{}c@{}}Auto\\Gluon\end{tabular} & Fixed & Fixed & \begin{tabular}[c]{@{}c@{}}Tabular, Images,\\Texts\end{tabular} & - & \begin{tabular}[c]{@{}c@{}}NAS, \\AWS integration\end{tabular} & \url{awslabs/autogluon} \\ \hline LAMA & Fixed & Fixed, Optuna & Tabular & - & Profiling & \begin{tabular}[c]{@{}c@{}}\url{sberbank-ai/}, \\\url{LightAutoML}\end{tabular} \\ \hline NNI & Fixed & Bayes & Tabular, Images & \begin{tabular}[c]{@{}c@{}}Hybrid, \\Kubernetes\end{tabular} & NAS, WebUI & \url{microsoft/nni} \\ \hline \end{tabular} \end{table*} Also, there are a lot of more specific frameworks that are oriented towards using one single modeling task: the EPDE framework for the automated design of models based on differential equations \cite{maslyaev2020data}, the ClusterEnsembles framework for the automation of clustering \cite{strehl2002cluster}, AutoTS for time series forecasting \cite{khider2019autots}, etc. It can be seen that a large amount of well-developed frameworks exists. However, there is no ready-to-use solution that can be applied as a universal tool to automate the generation of a data-driven model with a complex structure. For this reason, we decided to develop an approach that allows solving the data-driven design problem in an automated, flexible, and interpretable way using multi-objective evolutionary optimization. To make the approach available for the broad community and adaptable to new tasks, we implemented it as a part of the open-source framework described in Section~\ref{sec_software}. \section{Problem statement} \label{sec_problem} Several different classes of modeling workflows can be distinguished. In \cite{zoller2019benchmark}, the fixed structure pipelines and variable shaped pipelines are discussed. However, the continuous development of the data-driven modeling approach raises a set of issues, that can be solved using pipelines with an even more complex and heterogeneous structure. Modeling workflows can be represented as graphs with different properties. The simplest linear pipelines can be described as a path graph, ensemble-based pipelines can be described as a minimum spanning graph, and composite pipelines are directed acyclic graphs with a single limitation - the existence of the final node (the destination for all data flows). This classification is detailed in Table~\ref{tab_graphs}. \begin{table}[h!] \renewcommand\tabularxcolumn[1]{m{#1}} \newcolumntype{Y}{>{\centering\arraybackslash}X} \centering \begin{tabularx}{\columnwidth}{|c|Y|Y|} \hline Structure & Graph type & Represented modelling workflow \\ \hline \begin{minipage}{.15\textwidth} \includegraphics[width=\linewidth]{images/g1.png} \end{minipage} & Path graph (only one parent for each node) & Linear pipelines \\ \hline \begin{minipage}{.15\textwidth} \includegraphics[width=\linewidth]{images/g2.png} \end{minipage} & Minimum spanning graph (all paths between all pairs of nodes have the same length) & Ensemble pipelines (low variability) \\ \hline \begin{minipage}{.15\textwidth} \includegraphics[width=\linewidth]{images/g3.png} \end{minipage} & Directed acyclic graph (with all paths finishing in the final node) & Composite pipelines (high variability) \\ \hline \end{tabularx} \caption{The description of different graph representations for modelling workflows} \label{tab_graphs} \end{table} In this paper, we propose to consider composite pipelines (also referred to as composite models \cite{kalyuzhnaya2020automatic} as a more perspective approach. A composite pipeline is a heterogeneous data-driven model with a graph-based internal structure, within which several atomic functional blocks (models and data-related operators) can be identified. The main concepts of well-known variable-shaped pipelines (e.g. implemented in \cite{olson2016tpot}) are close to composite pipelines. However, several differences should be noted. Firstly, composite pipelines can contain more than one model aimed for different purposes (classification, clustering, forecasting, etc.) and use data with different types (text, images, tables, etc.). For instance, some part of the composite pipeline is aimed to forecast time-series based on given prehistory. And the output is used as an additional feature for a final classification task. Thus, in composite pipelines, several data sources (even with different types) can be involved. Secondly, all predictive models and data processing algorithms are represented as functional blocks ${A}_{i}$ and processed in a unified form as a part of the pipeline. Therefore the proposed approach operates with any atomic blocks that are provided. Also, for composite pipelines it is possible to introduce a 'fractality' property: a composite model can be used as an atomic block thus it can be included in another composite model. From the mathematical point of view, the structure of composite pipeline $P$ can be described as a directed acyclic graph (DAG) $G$. In this case, ${P}^{G}$ can be formalized as follows: \begin{equation} \begin{aligned}[t] {{P}^{G}}=\left\langle {{V}_{i}},{{E}_{j}} \right\rangle =\left\langle {{A}_{i}},{{\{{{H}_{{{A}_{i}}}}\}}_{k}},{{E}_{j}} \right\rangle, \end{aligned} \end{equation} where $V$ are the vertices with a complex structure that can be represented as a tuple $\left\langle {{A}_{i}},{{\{{{H}_{{{A}_{i}}}}\}}_{k}}\right\rangle$ with modelling or preprocessing functional blocks ${A}_{i}$ and their hyper-parameters ${{\{{{H}_{{{A}_{i}}}}\}}_{k}}$. Directed edges $E$ represent the data flow between functional blocks. In this case, the optimization task for the pipeline structure can be described as follows: \begin{equation} {f}^{max}\left( {{P}^{*}} \right)=\underset{P}{\mathop{\max }}\,{f}\left( \mathbb{P}|{{T}_{gen}}\le {{\tau }_{g}} \right), \end{equation} where $f$ is the objective function that characterizes the modelling quality for the considered workflow and ${f}^{max}$ is the maximal value of the fitness function obtained during optimization, $\mathbb{P}$ is a set of possible pipeline structures (search space), ${{T}_{gen}}$ is time spent for workflow design, ${\tau}_{g}$ - is time limit. The application of the proposed approach to real-world problems is quite a promising research direction. However, the described integration of the WMS and AutoML features raises several issues that should be solved during the implementation of the approach. In the paper, we try to propose possible solutions to them and analyze their effectiveness. \subsection*{Issue 1: Do composite pipelines have an advantage against more simple pipelines in real tasks?} \label{iss_composite} There are a lot of examples of the modeling workflows application to different real-world problems. However, most of them are designed by domain experts or data scientists (using pipeline automation solutions that are close to WMS). At the same time, the most of AutoML tools allow building simple pipelines with near-fixed structure. Also, the practical applicability of the composite pipelines to the variety of modeling tasks (classification, regression, time series forecasting, clustering) is still unclear, since the impact of time and resources limitations can overcome the potential effort from the more flexible structure. The corresponding experimental results are not systematized in the literature. In order to substantiate the proposed approach, it is necessary to provide the experimental evaluation of the composite pipelines against baselines and alternative methods of pipeline design. \subsection*{Issue 2: Which blocks can be used as a part of a composite pipeline?} \label{iss_blocks} The graph-based representation of the composite pipeline allows representing a wide range of possible configurations of data-driven models \cite{qi2020graph}. However, the possible types of functional blocks are quite limited in the existing AutoML approaches. Otherwise, the WMS allows integrating different blocks into workflows, however, there is still no unified way to represent different models and operations that can be used during the automated design of the composite pipelines approach described in the paper. \subsection*{Issue 3: How to combine the AutoML and WMS approaches for the automated pipeline design?} \label{iss_design} There is no oblivious way to combine the best features of approaches implemented in existing AutoML and WMS. The tools for automated machine learning (AutoML) are mostly focused on linear or sub-linear pipelines (see Sec.~\ref{sec_related}), so the advanced approach (i.e. genetic algorithms) can be redundant \cite{zoller2021incremental}. The WMS is not aimed to solve the design problems in an automated way and is focused on decision-making support for the expert. So, there is no ready-to-use concept for an implementation of a hybrid automated modeling approach that can be efficiently applied to composite pipelines. To build the modeling workflows in an effective, robust, and controllable way, a suitable automatic method should be chosen or developed. Also, it should be applicable for all types of pipelines described in Table~\ref{tab_graphs} to avoid both over-complicated and over-simplified solutions. \subsection*{Issue 4: How to analyze the structure of pipeline?} \label{iss_interp} The WMS-derived approach to the pipeline design raises the structural analysis problem for the pipeline. The structure of the modeling pipeline consists of many different blocks and connections between them. If the composite pipeline is obtained using automated design methods (see \hyperref[iss_design]{Issue 3})), its analysis can be confusing for the expert. To simplify it, an automated approach that allows estimating the impact and importance of each structural block is required. The existing methods allow analyzing the hyperparameters of the models in blocks, but there is no such approach for structural analysis. \subsection*{Issue 5: How to tune the hyperparameters in composite pipelines?} \label{iss_tuing} Fine-tuning of the hyperparameters in the composite pipeline can be different from the same task for atomic machine learning models \cite{kalyuzhnaya2021towards} that are using in existing AutoML tools. As an example, the data preprocessing block also require appropriate tuning, but the quality metric evaluation can not be provided without connection with the models. To improve the efficiency of the approach described in the paper, a unified strategy of hyper-parameter tuning that can be effective for different modeling tasks and types of data is required. \subsection{Issue 6: How to reproduce the results of composite modeling?} \label{iss_repr} The reproducibility of machine learning pipelines and related experiments is a critical problem in many scientific and industrial applications \cite{sugimura2018building}. The complex structure of the composite pipelines raises the additional aspects of reproducibility and reputability problems. There are different ways or models export exist in the existing AutoML tools: code generation, binary files, etc. The WMS usually defines the domain-specific language for the description of the workflow. However, the application of full-scale custom DSL can be redundant for the modeling pipelines. So, the specialized method that allows exporting and importing the composite pipelines in a simple, human-understandable, and reliable way should be implemented. Thereby, the problem statement for the proposed approach to the composite data-driven modeling pipelines design consists of \hyperref[iss_design]{Issue 1})) - \hyperref[iss_design]{Issue 6})). The paper is devoted to the justification of the proposed solutions to these issues, as well as the description, analysis, and validation of the implemented approach. \section{Design of the composite data-driven workflows} \label{sec_approach} This section is devoted to the various aspects of the proposed approach, that allows designing the composite data-driven workflows in an automated way. Since it is based on a concept of a combination of best practices from both AutoML and WMS, both model-related and automation-related aspects of the pipeline design problem were considered as a subject for analysis, validation, and justification. \subsection{Pipelines, models, and operations} As it was noted in Sec.\ref{sec_problem}, the pipeline can be represented as an acyclic directed graph. The basic abstractions of the composite pipeline are: \begin{itemize} \item \textbf{Operation} - an operation is an action that is performed on the data: an algorithm for data transformation or machine learning model that produces predictions; \item \textbf{Node} is a container in which an operation is placed. There can only be one operation in one node. A \textbf{Primary} node accepts only raw data, and a \textbf{Secondary} node uses the predictions of the previous level nodes as predictors; \item \textbf{Chain} is the implementation of the modeling workflow (with defined fit/predict operations) represented as a graph that consists of nodes (as vertices) and data flows (as edges). \end{itemize} Generally, we divide the building block for pipelines into several types (Table~\ref{tab_operations}). It can be divided into two groups: a) models and data operations, that are typical blocks for AutoML techniques; b) task-specific models (specific statistical models, equation-based-models, etc) and operators for data flow (merging, target decomposition, data source, stochastic data generators, etc). The blocks for modeling contain operations that transform features into predictions. The data preprocessing operations can change features but do not approximate the relationship between them. The data flow operations can transform (or generate) both features and targets. \begin{table} \centering \caption{Types in the building blocks can be used in the proposed structure of the modeling workflow: a) blocks that usually using in the pipelines generated by AutoML; b) blocks usually using in the manually-designed modeling workflows} \label{tab_operations} \begin{tabular}{|c|c|c|c|} \hline \begin{tabular}[c]{@{}c@{}}Building \\blocks \\types\end{tabular} & Description & Example & Metadata \\ \hline \multicolumn{4}{|c|}{a) AutoML-specific types} \\ \hline \begin{tabular}[c]{@{}c@{}}ML \\models\end{tabular} & \begin{tabular}[c]{@{}c@{}}Transform \\features into \\predictions\end{tabular} & \begin{tabular}[c]{@{}c@{}}Ridge\\regression\end{tabular} & \begin{tabular}[c]{@{}c@{}}\#simple,\\\#linear, \\\#interpret-\\able\end{tabular} \\ \hline \begin{tabular}[c]{@{}c@{}}Data\\processing\\operations\end{tabular} & \begin{tabular}[c]{@{}c@{}}Modify \\data without \\prediction\end{tabular} & \begin{tabular}[c]{@{}c@{}}Outlier\\filtering\end{tabular} & \#non-linear \\ \hline \multicolumn{4}{|c|}{b) Workflow-specific types} \\ \hline \begin{tabular}[c]{@{}c@{}}Task-specific\\models\end{tabular} & \begin{tabular}[c]{@{}c@{}}Suitable for \\specific\\data types\\and tasks\end{tabular} & \begin{tabular}[c]{@{}c@{}}Equation-based\\runoff \\model\end{tabular} & \begin{tabular}[c]{@{}c@{}}\#interpret-\\able,\\\#time-series\\specific\end{tabular} \\ \hline \begin{tabular}[c]{@{}c@{}}Data flow \\operation\end{tabular} & \begin{tabular}[c]{@{}c@{}}Modify only \\target based \\on previous\\model \\predictions \\to estimate \\the errors\end{tabular} & \begin{tabular}[c]{@{}c@{}}Merge data \\from different \\sources\end{tabular} & \#non-default \\ \hline \end{tabular} \end{table} The description of the operations is stored in JSON files. Tags are used to control the "task to solve - operation to use" match. An example of several operations with the tags associated with it can be seen in Table~\ref{tab_operations}. The tag filtering mechanism allows flexible selecting of the appropriate blocks for composing pipelines. Thus, it is possible to use any combination of the proposed operations. For example, it is possible to create pipelines only with ML models, or only with linear models and simple preprocessing methods. In the proposed approach a uniform data flow strategy should be used regardless of the specific blocks in the pipeline. There are several ways to represent data flow in the pipeline. The first one is the blocks that use the parent models' predictions as an input. However, it restricts the shape of the obtained decision boundary, since the number of input features is quite small. The second implementation always passes the data to the input of any model (for example, this technique is implemented in the TPOT tool as a stacked estimation block). However, this approach has a lack of flexibility. For the genetic programming paradigm, we can implement input data as an independent pseudo-block. In this case, the input of any model can be enriched by the data (if it is useful for the increasing of the fitness metric value). The comparison of several strategies for management of data flows in the pipeline is provided in Fig.~\ref{fig_enrich}. \begin{figure*} \caption{The comparison of different strategies of data flow management for the secondary nodes: a) sequential ; b) direct; c) adaptive.} \label{fig_enrich} \end{figure*} Data operations can be considered as data-specific blocks that can potentially increase the accuracy of the final forecast. For time series, it can be moving average filtering, for tables - outliers detection based on the random sample consensus algorithm, or a recursive feature elimination (RFE) for feature selection. Also, normalization, dimensionality reduction, encoding, imputation, and other operations with data can be included in the pipeline design task as building blocks. For this reason, the strategy that allows processing various building blocks and connections between them (similar to WMS) is considered the most flexible and used as a part of the proposed approach for pipeline design. \subsection{Automated evolutionary design} \label{subsec_evo} The procedure of the automated design allows selection of model structure for certain input data automatically and consists of two steps: \begin{itemize} \item \textbf{Composition} - the process of finding a pipeline structure. By default, the framework has an evolutionary algorithm responsible for this, in which optimization is performed using genetic selection, crossover, and mutation operators (see Fig.~\ref{fig_evo}). At this stage, operations in nodes are changed, subtrees are removed, added, or extended for each individual in a population. Each operator has a performing probability. Hyperparameters of operations in nodes are also can be mutated. Mutation operators are generally subdivided into two groups: for exploration and for exploitation aims. The first group grows new parts of pipelines (sometimes replacing old selected parts). The second performs local changes or reducing pipelines' parts.; \item \textbf{Hyperparameter tuning} is a process in which the structure of a pipeline does not change, but only the hyperparameters in the nodes are tuned. This step is started after the composition is finished. \end{itemize} The high-level pseudocode of the evolutionary design approach is presented in Alg.~\ref{alg_gp}. It is necessary to load the database, define the search space, objectives, termination criteria, and evolutionary search algorithm hyperparameters before the optimization. By default, the first population of pipelines is generated randomly. However, it is possible to add existing hand-developed baselines or previously obtained AutoML solutions (using the same or another AutoML approach) as a first approximation to improve the convergence of the search process. We should make every attempt to reuse the existing knowledge because starting the search from scratch is usually expensive. During the entire evolutionary process, the algorithm measures objectives for each new generated pipeline (if it is not contained in cache). Each new composite model fitted with a training sample obtained predicted values on a test sample and evaluate metrics using them (see Alg.~\ref{alg_gp}, Evaluation procedure). The search process stops if one of the stop criteria is satisfied. Adaptive evolutionary schemes (as described in \cite{evans2020adaptive}) assume hyperparameters (such as population size, offspring size, rates of evolutionary operators,constraints, e.t.c.) adaptation during the evolution depending on the diversity of the population and its convergence speed (Alg.~\ref{alg_gp}, line~\ref{alg:hyperparams}). Besides the constraints (include structural-based, e.t.c.) that are considered in evolutionary operations, each new individual produced by crossover or mutation procedure is checked for the rules which are defined based on object representation type (e.g. DAG can not have isolated nodes). More details in Alg.~\ref{alg_gp}, line~\ref{alg:valid}. In Fig.~\ref{fig_evo} the scheme of the evolutionary design approach is presented. Each evolutionary procedure is demonstrated in simplified form to improve readability. In practice, few different types of each evolutionary operator can be used in the algorithm. In this case, operator type can be selected randomly with specified probabilities. Operators can have either equal probabilities or be assigned by the special evolutionary scheme. Such scheme performs the dynamic adaptation of operators’ probabilistic rates on the level of the population (\cite{semenkina2014hybrid}). In \cite{nikitin2020structural} examples of evolutionary operators for the graph structures (crossovers, mutations, and regularization) with a detailed description. \begin{figure*} \caption{The concept of the multi-objective evolutionary design of the composite machine learning workflows (pipelines).} \label{fig_evo} \end{figure*} \begin{algorithm*} \caption{Pseudocode of evolutionary algorithm that is implemented as a part of composition module} \begin{algorithmic}[1] \Procedure{EvolutionaryComposer}{} \State \underline{Input:} objectiveFunctions = $\left\{qualityObj, complexityObj, ...\right\}$, operations = $\left\{models, dataOperations, ... \right\}$, stopCriterions = \State = $\left\{timeLimit, generLimit, ...\right\}$, evoOperators = $\left\{crossoverTypes, mutationTypes, selectionTypes, regularTypes, ...\right\}$, \State constraints = $\left\{structuralConst, optimiserConst, ...\right\}$, operatorsRates, data, popSize, initialChains, evaluatedChainsCache \State \underline{Output:} array with nondominated models (Pareto frontier) \State $pop\gets \Call{InitPopulation}{models, popSize, structuralConst, initialChains} $ \State $pop\gets \Call{Evaluation}{pop, objectiveFunctions, data}$ \While{not \Call{Terminate}{stopCriterions}} \State $\triangleright$ \textit{In adaptive evo. schemes hyperparameters (e.g. popSize,rates,constraints, e.t.c.) update during evolution} \State \Call{UpdateHyperparams}{evolutionaryScheme} \label{alg:hyperparams} \State $pop \gets \Call{Regularization}{pop, regularTypes}$ \State $\triangleright$ \textit{Pareto frontier contains only one best individual in the case of a single-objective algorithm} \State $pareto, parents \gets \Call{UpdatePareto}{pop}, \Call{ParentsSelection}{pop, offspringSize, selectionTypes}$ \State $offspring \gets \Call{Reproduction}{parents, crossoverTypes, mutationTypes,operatorsRates, constraints, models}$ \State $pop \gets \Call{Evaluate}{offspring, objectiveFunctions, data, evaluatedChainsCache}$ \State $pop \gets \Call{Selection}{pop\cup offspring, popSize, selectionTypes}$ \EndWhile \State \textbf{return} $pareto$ \EndProcedure \Procedure{Evaluation}{} \State \underline{Input:} pop, objectiveFunction, data, cache \For {$ind$ in $pop$} \State $mlModel\leftarrow$ \Call{Convertor}{$ind$} \State $\triangleright$ \textit{Model fitting and obtaining its prediction} \State $prediction \leftarrow$ \Call{FitPredict}{$mlModel$,data, cache} \State $\triangleright$ \textit{Metrics evaluation and recording their values to corresponding fields of individual} \State \Call{MetricsEvaluation}{$ind$,$prediction$,$objectiveFunctions$ } \EndFor \EndProcedure \Procedure{Reproduction}{} \State \underline{Input:} parents, crossoverTypes, mutationTypes, operatorsRates = $\left\{crossoverRate, mutatioRate\right\}$, constraints, models \State \underline{Output:} array with new individuals \State $offspring \gets \left\{\right\}$ \For {$parent1, parent2$ in $parents$} \State $\triangleright$ \textit{Random choice of evolutionary operators types from avaliable} \State $crossover \gets \Call{RandomChoice}{crossoverTypes, crossoverRate}$ \State $mutation \gets \Call{RandomChoice}{mutationTypes, mutationRate}$ \State $\triangleright$ \textit{Each new individual is checked for the rules which are defined on the basis of object representation type} \While{not $\Call{Validation}{newInds}$} \label{alg:valid} \State $newInds \gets \Call{crossover}{parent1, parent2, constraints}$ \State $newInds \gets \Call{mutation}{newInds, constraints, models}$ \EndWhile \State $offspring \gets offspring\cup newInds$ \EndFor \State \textbf{return} $offspring$ \EndProcedure \label{evo_params_adapt} \end{algorithmic} \label{alg_gp} \end{algorithm*} \subsection{Hyperparameters tuning strategies} During the optimal chain structure search, the evolutionary algorithm uses mutation operators. With the help of one of these operators, the hyperparameters in the nodes can be configured. However, during the evolution process, it occurs very slowly, since, in addition to setting up hyperparameters, the search for the optimal chain topology and operation locations in the nodes is carried out. So, obtained pipeline after the evolutionary identification stage is still can be not optimal due to unconfigured hyperparameters. To achieve the appropriate modeling quality, the hyperparameter of each operation should be fine-tuned after composing. Such tuning can be conducted in different ways. Since the result of the pipeline, design has a graph-based structure, it was suggested to tune the nodes sequentially or simultaneously. At the same time, for tuning, regardless of the strategy, the "black box" Bayesian optimization algorithm was used. So, there are several candidate strategies was proposed: \begin{itemize} \item Serial isolated tuning - approach to tuning hyperparameters in nodes, where the hyperparameters are tuned sequentially for each node in the pipeline separately. In this case, the hyperparameter optimization is performed in isolation for each node. The data is not passed through the pipeline, but only through the corresponding subtree. The error is minimized at the specific node for which the parameters are selected. \item Chain simultaneous tuning - approach in which hyperparameter optimization is performed simultaneously for each node in the pipeline. Thus, the entire composite pipeline is optimized as a black-box. The error is minimized for the entire chain. \item Sequential tuning - approach is similar to "serial isolated tuning", but the difference is that if the hyperparameters are optimized at one node, the quality metric is recalculated for the entire pipeline. The error is minimized for the entire chain. \end{itemize} To compare these strategies, we considered the results of their experiments on two types of problems: classification and regression. For each of the two problems, three composite pipelines with different complexity were analyzed. Each pipeline was run on a specific example 100 times. The number of iterations for tuning is 100. The MAE and ROC-AUC metrics were measured before and after setting the parameters. You can see the results of comparing the algorithms for the regression problem in Fig.~\ref{fig_tuning_comprasion} \begin{figure} \caption{Comparison of the three proposed optimization algorithms in terms of improving the quality metric of the composite pipeline.} \label{fig_tuning_comprasion} \end{figure} In the regression problem, simultaneous tuning proved to be the most effective approach, as it provided the greatest increase in accuracy and provided more reliable solutions. For the classification problem, the sequential tuning and simultaneous tuning approaches showed the same result. Serial isolated tuning in both problems proved to be the most inaccurate approach, but the most computationally cheap of the presented ones. For these reasons, we used simultaneous tuning as a part of automated pipeline design during the experimental studies described in Sec.~\ref{sec_exp}. \subsection{Analysis and improvement of the pipeline structure using sensitivity analysis} Sensitivity analysis (SA) algorithms are an important instrument in the analysis of the models and modeling results. It can be used to estimate uncertainties in the input parameters that affect the output. SA can help to reduce the evaluation cost by removing the least important features from data. This approach is successfully used in different methods of feature importance estimation. However, SA also can be used to analyze the structure of composite pipelines obtained by automated methods. The analysis important since the pipelines has a heterogeneous graph-based structure containing different operations, which can be redundant or non-optimal. The evolutionary approach described in the Subsec~\ref{subsec_evo} can produce a lot of sub-optimal solutions that required additional post-processing. Also, it is useful to estimate the importance of each block in the pipeline. The idea is to analyze the impact (positive or negative) of each block of the pipeline for the specified modeling tasks. Since most of the AutoML solutions have linear pipelines, there is little coverage of such an analysis in the literature. The numerical results of SA represents the ratio of origin quality score to score with node deletion (or replacement) operations. It can be represented as follows: \begin{equation} \label{eq:sensitivity_index_del} {S}^{imp}_{i} = \frac{1}{N}(1 - {\sum_{n=1}^{N}\frac{F({P}^{G})}{F({P}^{G'})}}), \end{equation} where $F$ - objective function for pipeline (e.g. modelling error measure), ${P}^{G'}$ - pipeline with modified structure, $N$ - number of SA iterations, $i$ - index of node to analyze. The larger value of the ${S}^{imp}_{i}$ index, the more importance can be estimated for the $i$-th block in the pipeline. If the value of the structural importance is negative, it allows finding a modification of the pipeline that allows improving the overall quality score. It can be used to post-process the pipeline and improve its effectiveness. This approach allows us to conduct the analysis of the pipeline and obtain the estimations of importance for each node. The example is presented in Fig.~\ref{fig_sa}. \begin{figure} \caption{The example of importance analysis for the composite modelling pipeline. The numbers represents the values of ${S} \label{fig_sa} \end{figure} To confirm the correctness of the proposed SA-based approach for the pipeline analysis, we conducted several simple experiments (the SA was also used during benchmarking in Sec~\ref{sec_exp}). The experiments are devoted to the assessment of structural importance, which means quality change caused by the composite pipeline modification at the chosen node. The experiments were based on the credit scoring (classification)\cite{creditScoring} and cholesterol prediction (regression)\cite{cholesterolRegr} tasks with the static pipeline, static composite pipelines obtained manually and design by the evolutionary approach described in Subcec.~\ref{subsec_evo}. The results of the SA for the classification task show that in four of six cases deleting is not recommended. Moreover, the node replacement (with all possible models within the problem) is better, on average, only in one case, out of six. The detailed replacement analysis for each node shows how each pipeline changes quality compared to the original one. Considering only the number of failed changes, it allows inferring that in 50\% of cases, the original component is better than the new. The evolutionary obtained pipeline within 20 generations and populations in size of 20 shows approximately the same results. The regression problem considers static and automated model designs. The result shows that the pipeline obtained statically has low structural quality because SA results confirm that the removal or replacement of several nodes leads to an increase in the score. On the opposite side, the composed pipeline for the regression problem is more robust even on a low number of generations but still can be improved. Additionally, there was a hypothesis of node value conditioned by the number of connections it has. Perhaps, the more edges node has, the higher importance is. The averaged results for the comparison of different pipelines are presented in Tab.~\ref{tab_sa}. \begin{table} \centering \caption{Results of experiment with sensitivity analysis different implementation of pipelines. $S_{imp}^{avg}$ the value of sensitivity index averaged for all nodes, $N_{total}$ - number of nodes in pipeline, $N_{repl}$ - number of candidate nodes for replacement, $N_{del}$ - number of candidate nodes for deletion.} \label{tab_sa} \begin{tabular}{|c|c|c|c|c|c|} \hline Case & Pipeline & \begin{tabular}[c]{@{}c@{}}$S_{imp}^{avg}$\\\end{tabular} & $N_{del}$ & $N_{repl}$ & $N_{total}$ \\ \hline \multirow{3}{*}{Class} & Linear & 70 & 0 & 1 & 2 \\ \cline{2-6} & \begin{tabular}[c]{@{}c@{}}Composite\\static\end{tabular} & 40 & 2 & 1 & 5 \\ \cline{2-6} & \begin{tabular}[c]{@{}c@{}}Composite \\automated\end{tabular} & 60 & 0 & 1 & 7 \\ \hline \multirow{3}{*}{Reg} & Linear & 75 & 0 & 1 & 3 \\ \cline{2-6} & \begin{tabular}[c]{@{}c@{}}Composite\\static\end{tabular} & 55 & 2 & 2 & 5 \\ \cline{2-6} & \begin{tabular}[c]{@{}c@{}}Composite\\automated\end{tabular} & 65 & 1 & 0 & 4 \\ \hline \end{tabular} \end{table} The proposed implementation of the sensitivity analysis can contribute to composite pipeline structure explanation and also improve the modeling quality. The described approach is available as a part of the FEDOT framework and described in the tutorial at the framework official documentation\footnote{https://fedot.readthedocs.io/en/latest/fedot/features/sensitivity\_analysis.html}. \subsection{Reproducilibity of pipelines} The ability to reproduce the results of experiments in the field of computer science is essential both from theoretical and practical points of view \cite{peng2011reproducible}. The field of ML pipelines requires special attention to the reproducibility of experiments because the fitting of pipelines using large datasets takes a significant amount of time and is highly dependent on the technical capabilities of the system. There is a set of requirements that can be proposed in terms of composite pipelines reproducibility \cite{tatman2018practical}. It allows defining the set criteria for both data and ML models. The reproducibility for data point of view can be achieved by the saving of the sample used for training the models as a set of files. The reproducibility of ML models can be achieved by the preservation of the ML methods and models, the version of the programming language, the versions, and names of the external libraries. Also, the image of the virtual machine or a whole file system can be created \cite{pineau2020improving}. Most of the state-of-the-art AutoML frameworks have a quite simple implementation of the pipeline exports (as script or binary file with serialized model). For the WMS, the common way for workflow exporting is a description in the domain-specific language. We decided to combine the advantages of both approaches, so the proposed approach includes the specific features to ensure reproducibility. To provide a consistent import and export of pipelines in different systems, its structural representation is implemented in human-readable JSON format. The data exporting is implemented as an archive that contains both train and validation data. Also, the interface for creating and managing a virtual machine can be implemented if necessary. It makes it possible to create virtual containers with the necessary dependencies, that allow us to build composite pipelines and fit them on different datasets. \definecolor{background}{HTML}{FFFFFF} \lstdefinelanguage{json}{ basicstyle=\small\ttfamily, numberstyle=\scriptsize, stepnumber=1, numbersep=2pt, showstringspaces=false, breaklines=true, frame=lines, backgroundcolor=\color{background}, moredelim=**[is][\color{red}]{@}{@} } \begin{lstlisting}[language=json,mathescape=true,label={lst:fig_json},caption={Ensemble architecture in JSON fromat}] { # $\textit{TYPES OF THE NODES IN THE ENSEMBLE}$ "total_chain_operations": { "xgboost": 1, "scaling": 1, ... }, # $\textit{DEPTH OF THE ENSEMBLE}$ "depth": 3, # $\textit{ALL NODES IN THE ENSEMBLE}$ "nodes": [ { "operation_id": 1, "operation_type": "xgboost", "operation_name": "XGBClassifier", # $\textit{CUSTOM PARAMS}$ "custom_params": { "n_estimators": 250 }, # $\textit{FULL PARAMS (custom + default)}$ "params": { "learning_rate": 0.3, "max_depth": 6, "n_estimators": 250, ... }, # $\textit{PARENT NODES IDs}$ "nodes_from": [0], # $\textit{FITTED MODEL PATH}$ "fitted_operation_path": "fitted_operations/operation_1.pkl" }, ... ]} \end{lstlisting} The implemented module has a flexible functionality for the communication of the results and their reproducibility in other systems. The user can save the pipeline structure in JSON (Lst.~\ref{lst:fig_json}) format, which describes chain structure. In addition, users can save trained models, data, dependencies, logs, and a visual representation of the pipeline as shown in Fig.\ref{fig_json_atomized}. In this case, importing data is as easy as exporting it. Besides, the pipeline itself can be transformed into an atomized model that can be embedded in the new pipeline as a modeling block, as shown in Fig.\ref{fig_json_atomized}. This allows creating nested pipelines that can be used to achieve higher effectiveness for multi-scale tasks. \begin{figure} \caption{The implemented approach for reproducible of the composite pipelines. The example of atomization for pipeline is also presented.} \label{fig_json_atomized} \end{figure} The described atomization technique is also can be used to adapt the pipeline to the updated data sample. In comparison with the pipeline design from the scratch (or from the existing pipeline as an initial assumption), the atomization-based approach allows decreasing the dimensionality of the search space and achieve the appropriate results in shorted type The idea is described in Fig.~\ref{fig_atom}. \begin{figure*} \caption{The scheme of the proposed approach for adaptation of the composite pipelines to the new data using atomization. It allow decreasing the search space for the evolutionary optimization and reduce computational requirements.} \label{fig_atom} \end{figure*} The simple initial experimental evaluation was conducted for the classification task (credit scoring). The initial pipeline with five nodes was identified using evolutionary composer for the first 5000 rows of data (quality for validation sample is ROC AUC 0.834). Then, 10000 rows of data were added and optimization re-stared from different initial assumptions: previous pipeline and its atomized version). The atomization-based approach allows achieving higher quality (ROC AUC 0.848 vs 0.844) for lower time (3 min vs 8 min). These results were also confirmed for other datasets. For this reason, atomization was used as a part of evolutionary composer and used during experimental studies. \section{Software implementation} \label{sec_software} The proposed approach is implemented as an algorithmic core of the FEDOT framework. It was designed as a multi-purpose AutoML tool that allows us to identify a suitable machine learning pipeline for a given task and dataset in an automatic way. The framework is not focused on certain AutoML subtasks, such as data preprocessing, feature selection, or hyperparameters optimization, but allows one to solve a general task such as structure learning. In this case, for a given dataset the solution is presented by a directed acyclic graph (DAG) structure, where the nodes are ML models or data operations and the edges are dataflows between the nodes. The architecture of the framework is based on several principles. Firstly, the core logic (especially, the optimization process) should not be strictly dependent on a certain task. Secondly, there is an option to simply extend or replace most of the modules in the framework for the custom user requirements. In order to simplify the usage of the framework, a high-level Python API was implemented. Inspired by well-known frameworks, like H2O or AutoGluon, it is possible to obtain an ML pipeline with only several lines of code. However, the user can customize and instantiate FEDOT objects for the needs. \subsection{ML pipelines execution} In FEDOT, ML pipeline is considered as a \textsl{Chain} object. It is an abstraction of a DAG structure that provides interfaces for graph modifications and pipeline execution. Chain contains \textsl{Nodes} that isolate implementations of certain operations. The execution of the pipeline is initialized recursively from a final node, and the data flow is going in the opposite direction. Each node includes \textsl{Operation} object that is a base class of two interfaces - \textsl{Model} and \textsl{DataOperation}.In ML pipelines a particular model or data operation can be considered as building blocks. In general, model is a function that transform input data (features) into output (predictions) and has two methods - \textsl{fit} and \textsl{predict}. In turn, data operations modify only input data, like PCA, undersampling, or normalization. Also, the same operations can be implemented in various ways. Therefore we additionally designed a \textsl{EvaluationStrategy} logic layer in the framework. For instance, if the user wants to use a logistic regression from scikit-learn\footnote{https://github.com/scikit-learn/scikit-learn} framework, then it must be implemented as a \textsl{SklearnEvaluationStrategy} with logistic regression inside in FEDOT. \subsection{ML pipelines building and optimization} Pipeline structure optimization logic is embedded into a separate module. In FEDOT the entry point for this operation is a \textsl{Composer} interface. Thus, the optimization is called \textsl{composing}. The purpose of a Composer is to connect Chain-related logic and optimization algorithms. The last are implemented via sub-classes of \textsl{Optimiser} interface. For instance, previously mentioned GPComp algorithm is provided as a \textsl{GPOptimiser} class. The described design allows us to conduct the experiments with optimization algorithms of different nature without strong connectivity with the core of FEDOT. \begin{figure*} \caption{The structure of proposed solution for automated design of the modelling workflows. The implementation of this approach is available as a core of FEDOT framework} \label{fig_detailed_pipeline} \end{figure*} \section{Experimental studies for efficiency analysis of the proposed approached} \label{sec_exp} \subsection{Classification and regression benchmarks} To demonstrate the effectiveness of the evolutionary design of pipelines, an experiment was conducted using ten data sets for regression and classification problems (five data sets for each problem), obtained from the Penn Machine Learning Benchmarks repository\footnote{https://github.com/EpistasisLab/pmlb}. These data sets cover a broad range of applications, and combinations of categorical, ordinal, and continuous features. There are no missing values in these data sets. Selected data sets were split on training and test set in a ratio of 80/20. An example of running an experiment is located in the specified repository\footnote{https://github.com/ITMO-NSS-team/FEDOT-benchmarks/blob/master/experiments/four\_pipelines/run.py} The values of quality metrics are presented in Table~\ref{tab_automl_comprasion}. The experiment includes: \begin{itemize} \item Initialisation of several tools: FEDOT framework implementation (based on the proposed approach), TPOT and MLBox frameworks, baseline model (XGBoost); \item Evaluation of the automated pipeline design using each approach on the set of classification and regression benchmarks; \item Estimation of modeling error: MAE and RMSE are used as quality metrics for regression tasks and F1, ROC-AUC is used as a quality metric for classification tasks. \end{itemize} To initialize the setups for evolutionary tools (TPOT and FEDOT), the values of the following hyperparameters were selected: a maximum number of generations to create a composite pipeline – 200; population size – 10 individuals. The maximum time for evaluation was limited by 10 min for all approaches. The following hyperparameters values were selected for the baseline XGBoost model: maximum depth - 3; shrinkage coefficient of each tree contribution – 0.3; the number of boosting stages to perform – 300. The main meta-features of the selected data sets are shown in \ref{tab_dataset_features}. Imbalance shows a value of imbalance metric, where zero means that the data set is perfectly balanced, and the higher the value, the more imbalanced the data set. \begin{table} \centering \caption{Main meta-features of the PMLB datasets for regression (regr) and classification (clf) used during experimenents.} \label{tab_dataset_features} \begin{tabular}{|c|c|c|c|c|} \hline Dataset & $N_{samples}$ & $N_{feat}$ & Task & Imbalance \\ \hline Cpu\_small & 8192 & 12 & regr. & - \\ \hline Elusage & 55 & 2 & regr. & - \\ \hline Faculty & 50 & 4 & regr. & - \\ \hline C2\_250\_25 & 250 & 25 & regr. & - \\ \hline 1027\_ESL & 488 & 4 & regr. & - \\ \hline Magic & 19020 & 10 & binary clf. & 0.08 \\ \hline Labor & 57 & 16 & binary clf. & 0.08 \\ \hline Flare & 1066 & 10 & binary clf. & 0.43 \\ \hline Ionosphere & 351 & 34 & binary clf. & 0.08 \\ \hline Spect & 267 & 22 & binary clf. & 0.34 \\ \hline \end{tabular} \end{table} \begin{table*}[ht!] \centering \caption{Results of experiment and values of quality metrics for each of the proposed benchmark approaches. The standard deviation of the quality metrics (MAE, RMSE, F1, area under ROC curve) is estimated for the 20 independent runs.} \label{tab_automl_comprasion} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Quality\\metric\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Pipeline and \\framework\end{tabular}} & \multicolumn{5}{c|}{\textbf{\textbf{\textbf{\textbf{Regression data sets}}}}} \\ \cline{3-7} & & 1027\_ESL & 1096\_FacultySalaries & 227\_cpu\_small & 228\_elusage & 605\_fri\_c2\_250\_25 \\ \hline \multirow{4}{*}{MAE} & Static (XGBoost) & 0.441\textcolor[rgb]{0.2,0.2,0.2}{±0.010} & 1.597\textcolor[rgb]{0.2,0.2,0.2}{±0.027} & 2.006\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 10.484\textcolor[rgb]{0.2,0.2,0.2}{±0.037} & \textbf{0.349\textbf{±0.019}} \\ \cline{2-7} & \textbf{\textbf{Composite (FEDOT)}} & \textbf{0.430\textbf{±0.009}} & \textbf{1.120\textbf{±0.012}} & \textbf{1.931}\textcolor[rgb]{0.2,0.2,0.2}{\textbf{±0.007}} & \textbf{9.451\textbf{±0.020}} & 0.371\textcolor[rgb]{0.2,0.2,0.2}{±0.033} \\ \cline{2-7} & Variable (TPOT) & 0.432\textcolor[rgb]{0.2,0.2,0.2}{±0.004} & 1.508\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 2.064\textcolor[rgb]{0.2,0.2,0.2}{±0.011} & 11.051\textcolor[rgb]{0.2,0.2,0.2}{±0.031} & 0.373\textcolor[rgb]{0.2,0.2,0.2}{±0.014} \\ \cline{2-7} & Linear (MLBox) & 0.434\textcolor[rgb]{0.2,0.2,0.2}{±0.018} & 3.670\textcolor[rgb]{0.2,0.2,0.2}{±0.048} & 1.934\textcolor[rgb]{0.2,0.2,0.2}{±0.005} & 21.075\textcolor[rgb]{0.2,0.2,0.2}{±0.054} & 0.423\textcolor[rgb]{0.2,0.2,0.2}{±0.076} \\ \hline \multirow{4}{*}{RMSE} & Static (XGBoost) & 0.588\textcolor[rgb]{0.2,0.2,0.2}{±0.028} & 2.199\textcolor[rgb]{0.2,0.2,0.2}{±0.037} & 2.799\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 14.053\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & \textbf{0.448\textbf{±0.025}} \\ \cline{2-7} & \textbf{Composite (FEDOT)} & \textbf{0.566\textbf{±0.013}} & \textbf{1.403\textbf{±0.024}} & \textbf{2.759\textbf{±0.007}} & \textbf{11.558\textbf{±0.020}} & 0.466\textcolor[rgb]{0.2,0.2,0.2}{±0.035} \\ \cline{2-7} & Variable (TPOT) & 0.570\textcolor[rgb]{0.2,0.2,0.2}{±0.009} & 1.789\textcolor[rgb]{0.2,0.2,0.2}{±0.033} & 2.886\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 14.863\textcolor[rgb]{0.2,0.2,0.2}{±0.032} & 0.465\textcolor[rgb]{0.2,0.2,0.2}{±0.014} \\ \cline{2-7} & Linear (MLBox) & 0.628\textcolor[rgb]{0.2,0.2,0.2}{±0.052} & 4.894\textcolor[rgb]{0.2,0.2,0.2}{±0.061} & 2.793\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 26.141\textcolor[rgb]{0.2,0.2,0.2}{±2.27} & 0.505\textcolor[rgb]{0.2,0.2,0.2}{±0.089} \\ \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Quality\\metric\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Pipeline and\\framework\end{tabular}} & \multicolumn{5}{c|}{\textbf{\textbf{Classification data sets}}} \\ \cline{3-7} & & flare & ionosphere & labor & magic & spect \\ \hline \multirow{4}{*}{F1} & Static (XGBoost) & 0.299\textcolor[rgb]{0.2,0.2,0.2}{±0.110} & \textbf{0.939\textbf{±0.026}} & 0.895\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 0.793\textcolor[rgb]{0.2,0.2,0.2}{±0.013} & 0.837\textcolor[rgb]{0.2,0.2,0.2}{±0.077} \\ \cline{2-7} & \textbf{\textbf{Composite (FEDOT)}} & \textbf{0.332\textbf{\textbf{\textbf{±0.057}}}} & 0.919\textcolor[rgb]{0.2,0.2,0.2}{±0.026} & \textbf{0.931\textbf{±0.013}} & \textbf{0.817\textbf{\textbf{\textbf{±0.004}}}} & \textbf{0.893\textbf{±0.062}} \\ \cline{2-7} & Variable (TPOT) & 0.292\textcolor[rgb]{0.2,0.2,0.2}{±0.034} & 0.774\textcolor[rgb]{0.2,0.2,0.2}{±0.052} & 0.840\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 0.809\textcolor[rgb]{0.2,0.2,0.2}{±0.004} & 0.830\textcolor[rgb]{0.2,0.2,0.2}{±0.046} \\ \cline{2-7} & Linear (MLBox) & 0.181\textcolor[rgb]{0.2,0.2,0.2}{±0.097} & 0.933\textcolor[rgb]{0.2,0.2,0.2}{±0.088} & 0.840\textcolor[rgb]{0.2,0.2,0.2}{±0.069} & 0.812\textcolor[rgb]{0.2,0.2,0.2}{±0.017} & 0.845\textcolor[rgb]{0.2,0.2,0.2}{±0.071} \\ \hline \multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}ROC\\AUC\end{tabular}} & Static (XGBoost) & 0.693\textcolor[rgb]{0.2,0.2,0.2}{±0.025} & \textbf{0.956\textbf{±0.007}} & 0.923\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 0.915\textcolor[rgb]{0.2,0.2,0.2}{±0.015} & 0.736\textcolor[rgb]{0.2,0.2,0.2}{±0.075} \\ \cline{2-7} & \textbf{\textbf{\textbf{\textbf{Composite (FEDOT)}}}} & \textbf{0.708\textbf{±0.033}} & 0.951\textcolor[rgb]{0.2,0.2,0.2}{±0.011} & \textbf{0.958\textbf{±0.019}} & \textbf{0.930\textbf{±0.004}} & \textbf{0.779\textbf{±0.043}} \\ \cline{2-7} & Variable (TPOT) & 0.701\textcolor[rgb]{0.2,0.2,0.2}{±0.008} & 0.954\textcolor[rgb]{0.2,0.2,0.2}{±0.079} & 0.958\textcolor[rgb]{0.2,0.2,0.2}{±0.023} & 0.928\textcolor[rgb]{0.2,0.2,0.2}{±0.004} & 0.657\textcolor[rgb]{0.2,0.2,0.2}{±0.039} \\ \cline{2-7} & Linear (MLBox) & 0.509\textcolor[rgb]{0.2,0.2,0.2}{±0.012} & 0.907\textcolor[rgb]{0.2,0.2,0.2}{±1.036} & 0.515\textcolor[rgb]{0.2,0.2,0.2}{±0.058} & 0.856\textcolor[rgb]{0.2,0.2,0.2}{±0.026} & 0.628\textcolor[rgb]{0.2,0.2,0.2}{±0.031} \\ \hline \end{tabular} \end{table*} As we can see from Table~\ref{tab_automl_comprasion}, the results obtained during the experiments demonstrate the advantage of composite pipeline created by the proposed approach over competitors. The only exception is a single case for regression and classification problems respectively, where the maximum value of the quality metric was obtained using a static pipeline. However, it should be noted that in all 10 experiments proposed approach showed better results compared to the other AutoML frameworks. \subsection{Time series forecasting benchmarks} An important task in machine learning is time series forecasting. However, just a few libraries for automatic machine learning can successfully perform this task. The proposed approach can be also used for the automated design of the pipelines for the time series forecasting. During the experiments, a comparison with specialized libraries for automated time series forecasting (AutoTS~\cite{khider2019autots} and Prophet~\cite{taylor2018forecasting}) was conducted. For comparison, twelve time-series datasets obtained from the archive of Federal Reserve Economic Data (FRED) were used. Ten datasets had a length of 408 to 1028 elements - they are considered as examples of short series during experiments. Two of them had a length of 17000 elements - they are considered as examples of long series. The forecast horizon was changed from 10 to 100 elements with a step of ten elements for short series, and from 10 to 200 elements with the same step for long time series. For each forecast, the predicted and actual values were compared using the Mean Absolute Percentage Error (MAPE). The execution time of each library for finding a solution was also monitored and limited. In some cases, algorithms were run until convergence was achieved. The experimental results allow confirming that the proposed approach outperforms competitors (Table~\ref{ts_errors}) in most cases. \begin{table}[ht!] \caption{Results of an experiment with the time series forecasting The modeling error is represented as MAPE. $h$ is the forecast horizon. The quality metric (MAPE) is averaged for the 20 independent runs. The composite pipeline is obtained automatically using the proposed approach.} \label{ts_errors} \centering \begin{tabular}{|c|c|c|c|c|c|} \hline \multirow{3}{*}{Tool} & \multicolumn{5}{c|}{\textbf{\textbf{MAPE for validation part, \%}}} \\ \cline{2-6} & \multicolumn{2}{c|}{Short time series} & \multicolumn{2}{c|}{Long time series} & \multirow{2}{*}{Avg.} \\ \cline{2-5} & h $\leq$ 50 & h $\leq$ 100 & h $\leq$ 100 & h $\leq$ 200 & \\ \hline Prophet & 14.6 & 12.7 & 78.7 & 95.6 & 50.4 \\ \hline AutoTS & 7.1 & 11.7 & 25.9 & 18.7 & 15.8 \\ \hline \begin{tabular}[c]{@{}@{}c@{}}\textbf{Composite}\\\textbf{pipeline} \\\textbf{(FEDOT)}\end{tabular} & \textbf{6.8} & \textbf{10.7} & \textbf{9.6} & \textbf{6.6} & \textbf{8.4} \\ \hline \end{tabular} \end{table} As can be seen from Table~\ref{ts_errors}, the proposed approach demonstrates the significant advantage against state-of-the-art libraries for all forecasting horizons and time series. Fig.~\ref{ts_pipeline} demonstrates an example of the composing pipeline obtained for the time series forecasting task. As can be seen from the figure, the pipeline combines several preprocessing blocks and modeling blocks. \begin{figure} \caption{The details of the composed pipeline for time series forecasting task a) Structure of the pipeline; b) Comparison between forecast obtained with evolutionary obtained pipeline and competing libraries forecasts (for test part of time series)} \label{ts_pipeline} \end{figure} The scripts and data for described experiments with time series forecasting are available in the repository \footnote{\url{https://github.com/ITMO-NSS-team/Fedot-TS-Benchmark/tree/master/time_series_case_1}}. \section{Discussions and conclusions} \label{sec_disc} In the paper, the evolutionary approach for the automated design of the data-driven modeling workflow is proposed. It is based on the concept of combination of pipeline design methods used AutoML and workflow management systems (WMS): graph-based structure of pipeline, several types of building blocks, descriptive representation of the pipeline, spesialized methods for fine-tuning, structural explainability, etc. This approach allows building complex graph-based pipelines that consist of different building blocks: data-driven models, data prepossessing functions, task-specific models, data flow transformations. Each pipeline is designed individually for a specific task that allows reducing the time efforts and increases the modeling quality. The evolutionary optimizer is used to automate the pipeline design. To achieve better effectiveness and applicability of the proposed approach, additional procedures for sensitivity analysis and atomization are implemented. Besides the algorithmic development, we make this approach available as a part of the self-developed AutoML framework - FEDOT. It is an open-source tool that can be used under a free license. There is both a simple interface (API) for inexperienced users and a customizable API for experts are available. The experimental validation of the proposed approach was conducted on different tasks. The classification, regression, and time series forecasting benchmarks were evaluated for the set of state-of-the-art solutions and baselines. The FEDOT-based implementation of the evolutionary approach demonstrates a significant advantage over competitors. Essentially, we evaluated our approach only for a limited set of benchmarks and do not claim to have the superior approach for all tasks. In practice, there is a high variety of effective AutoML solutions exists. Some of them have flaws in technical aspects (such as support for scalability and distributed computing, Kubernetes, and integration with MLOps tools). In others, there are conceptual issues like oversimplified optimization algorithms or their interpretability. So, existing solutions have a lot to improve. For example, there are several areas and prospects of development of AutoML that a based on the concept of 'model factory' that can provide different solutions to the user depending on the given conditions: the types of data sets, forecasting horizons, the lifetime of the model, etc. The modeling pipelines hat models can be derived for different data sets: random samples, data within time ranges. It is also possible to obtain "short-lived" models on a current data slice. We aimed the described approach as a promising step to the applied implementation of this concept. The generative evolutionary design of the pipelines for the different tasks makes it possible to obtain a valuable contribution to the different fields. \section{Code and data availability} \label{sec_code} The software implementation of all described methods and algorithms are available in the open repository \url{https://github.com/nccr-itmo/FEDOT}. The reproducible setups for all presented experiments are available in the \url{https://github.com/ITMO-NSS-team/FEDOT-benchmarks}. \section{Acknowledgments} ... \end{document}

\begin{document} \ifarxiv \title{\TITEL} \PERSON \keywords{\KEYWORDS} \subjclass[2012]{[{\bf Theory of computation}]: Formal languages and automata theory---Automata over infinite objects} \titlecomment{This work is supported by the DFG research project GELO} \begin{abstract} \ABSTRACT \end{abstract} \fi \maketitle \section{Introduction} Finite automata play a crucial role in many areas of computer science. In particular, finite automata have been used to represent certain infinite structures. The basic notion of this branch of research is the class of automatic structures (cf. \cite{KhoussainovN94}). A structure is automatic if its domain as well as its relations are recognised by (synchronous multi-tape) finite automata processing finite words. This class has the remarkable property that the first-order theory of any automatic structure is decidable. One goal in the theory of automatic structures is a classification of those structures that are automatic (cf.~\cite{Delhomme04,KhoussainovRS05,KhoussainovNRS07,DBLP:journals/apal/KhoussainovM09,KuLiLo11}). Besides finite automata reading finite or infinite (i.e., $\omega$-shaped) words there are also finite automata reading finite or infinite \emph{trees}. Using such automata as representation of structures leads to the notion of tree-automatic structures \cite{Blumensath1999}. The classification of tree-automatic structures is less advanced but some results have been obtained in the last years (cf.~\cite{Delhomme04,Huschenbett13,KaLiLo12}). Schlicht and Stephan \cite{SchlichtS13} and Finkel and \Todorcevic \cite{FinkelT11} have started research on a new branch of automatic structures based on automata processing $\alpha$-words where $\alpha$ is some ordinal. An $\alpha$-word is a map $w\in\Sigma^\alpha$ for some finite alphabet $\Sigma$. We call $w$ a finite $\alpha$-word if there is one symbol $\diamond$ such that $w(\beta)=\diamond$ for all but finitely many ordinals $\beta < \alpha$. We call the structures represented by finite-word $\alpha$-automatic structures \automatic{\alpha}. Many of the fundamental results on automatic structures have analogues in the setting of \automatic{\alpha} structures. \begin{itemize} \item The first-order theory of every \automatic{\alpha} structure is decidable and the class of \automatic{\alpha} structures is closed under expansions by first-order definable relations for all $\alpha < \OmegaPlus$ \cite{HuKaSchl14}. \item Lifting a result of Blumensath \cite{Blumensath1999} from the word- and tree-automatic setting, there is an \automatic{\alpha} structure which is complete for the class of \automatic{\alpha} structures under first-order interpretations \cite{HuKaSchl14}. \item The sum-of-box-augmentation technique of Delhomm\'e\xspace \cite{Delhomme04} for tree-automatic structures has an analogue for ordinal-automatic structures which allows to classify all \automatic{\alpha} ordinals \cite{SchlichtS13} and give sharp bounds on the ranks of \automatic{\alpha} scattered linear orderings \cite{SchlichtS13} and well-founded order trees \cite{KartzowS13}. \item The word-automatic Boolean algebras \cite{KhoussainovNRS07} and the \automatic{\omega^n} Boolean algebras \cite{HuKaSchl14} have been classified. In contrast, a classification of the tree-automatic Boolean algebras is still open. \end{itemize} In summary one can say that all known techniques which allow to prove that a structure is not tree-automatic have known counterparts for ordinal-automaticity. The only exception to this rule has been Delhomm\'e\xspace's growth-rate-technique \cite{Delhomme04}. We close this gap by showing that the maximal growth rates of ordinal automatic structures also has a polynomial bound. This allows to show that the Rado graph is not \automatic{\alpha}. In fact, we show that the bound on the maximal growth-rate of \automatic{\alpha} structure that we provide is strictly smaller than the bound for tree-automatic and strictly greater than the bound for word-automatic structures. Exhibiting this fact, we provide a new example of a structure that is \automatic{\omega^2} but not word-automatic. This example also shows that our growth-rate bound for \automatic{\alpha} structure is essentially optimal. One of the long-standing open problems in the field of automatic structures is the question whether the field of the reals $\Struc{R} = (\mathbb{R}, +, \cdot, 0, 1)$ has a presentation based on finite automata. Due to cardinality reasons it is clear that this structure is not word- or tree-automatic. Recently, Zaid et al.~\cite{ZaidGKP14} have shown that $\Struc{R}$ (as well as every infinite integral domain) is not infinite-word-automatic. This leaves infinite-tree-automata as the last classical candidate that might allow to represent $\Struc{R}$. Note that the cardinality argument also shows that $\Struc{R}$ is not \automatic{\alpha} for all countable $\alpha$ (because the set of finite $\alpha$-words is countable). Nevertheless the set of finite $\omega_1$-words is uncountable whence $\Struc{R}$ may be a priori \automatic{\alpha} for some uncountable ordinal $\alpha$. Using the growth rate argument we can show that no infinite integral domain is \automatic{\alpha} for any ordinal $\alpha < \OmegaPlus$. Let us mention that it also remains open whether $\Struc{R}$ is automatic with respect to automata that also accept infinite $\alpha$-words for some $\alpha \geq \omega^2$. \subsection{Outline of the Paper} In the next section we recall the necessary definitions on \automatic{\alpha} structures and the fundamental notions concerning growth rates. In Section~\ref{sec:BasicResults} we recall basic results on \automatic{\alpha} structures which are needed to obtain the growth rate bound in Section~\ref{sec:GrowthRate}. Finally, Section~\ref{sec:Applications} contains applications of the growth rate argument to the random graph, integral domains and concludes with the construction of a new example of an \automatic{\omega^2} structure which is not word-automatic because its growth-rate exceeds the known bound for word-automatic structures. \section{Definitions} \label{sec:defs} \subsection{Ordinals} We identify an ordinal $\alpha$ with the set of smaller ordinals \mbox{$\{\beta \mid \beta < \alpha\}$}. We say $\alpha$ has \emph{countable cofinality} if $\alpha=0$ or there is a sequence $(\alpha_i)_{i\in\omega}$ of ordinals such that $\alpha = \sup \{\alpha_i+1\mid i\in\omega\}$. Otherwise we say $\alpha$ has \emph{uncountable cofinality}. We denote the first uncountable ordinal by $\omega_1$. Note that it is the first ordinal with uncountable cofinality. For every ordinal $\alpha$ and every $n\in \mathbb{N}$, let $\alpha_{\sim n}$ be the ordinal of the form $\alpha_{\sim n} = \omega^{n+1}\beta$ for some ordinal $\beta$ such that \begin{equation*} \alpha = \alpha_{\sim n}+\omega^n m_n + \omega^{n-1} m_{n-1}+ \dots + m_0 \end{equation*} for some natural numbers $m_0, \dots, m_n$. \subsection{Ordinal-Shaped Words} \emph{First of all, we agree on the following convention:} In this article, every alphabet $\Sigma$ contains a distinguished \emph{blank symbol} which is denoted by $\diamond_\Sigma$ or, if the alphabet is clear from the context, just by $\diamond$. Moreover, for alphabets $\Sigma_1,\dotsc,\Sigma_r$, the distinguished symbol of the alphabet $\Sigma_1 \times \dotsb \times \Sigma_r$ will always be $\diamond_{\Sigma_1 \times \dotsb \times \Sigma_r} = (\diamond_{\Sigma_1},\dotsc,\diamond_{\Sigma_r})$. For some limit ordinal $\beta\leq\alpha$ and a map \mbox{$w:\alpha+1\to A$} we introduce the following notation for the \emph{set of images cofinal in $\beta$}: \begin{equation*} \lim_{\beta} w:= \{a\in A \mid \forall{\beta'<\beta} \exists{\beta' < \beta'' < \beta} w(\beta'') = a\}. \end{equation*} \begin{definition} An \emph{$(\alpha)$-word (over $\Sigma$)} (called a finite $\alpha$ word over $\Sigma$) is a map $w\colon \alpha \to \Sigma$ whose \emph{support}, i.e., the set \begin{equation*} \supp(w) = \Set{\beta \in \alpha | w(\beta)\neq\diamond }, \end{equation*} is finite. The set of all $(\alpha)$-words over $\Sigma$ is denoted by $\FinWords{\Sigma}{\alpha}$. We write $\diamond^\alpha$ for the constantly $\diamond$ valued word $w:\alpha\to \Sigma$, $w(\beta)=\diamond$ for all $\beta<\alpha$. \end{definition} \begin{definition} If $\gamma \leq \delta \leq \alpha$ are ordinals and $w:\alpha\to \Sigma$ some $(\alpha)$-word, we denote by $w{\restriction}_{[\gamma,\delta)}$ the restriction of $w$ to the subword between position $\gamma$ (included) and $\delta$ (excluded). \end{definition} \subsection{Automata and Automatic Structures} \Buchi \cite{Buchi65} has already introduced automata that process $(\alpha)$-words. These behave like usual finite automata at successor ordinals while at limit ordinals a limit transition that resembles the acceptance condition of a Muller-automaton is used. \begin{definition} An \emph{ordinal automaton} is a tuple $(Q, \Sigma, I, F, \delta)$ where $Q$ is a finite set of states, $\Sigma$ a finite alphabet, $I\subseteq Q$ the initial states, $F\subseteq Q$ the final states and \begin{equation*} \delta \subseteq (Q\times \Sigma \times Q) \cup (2^Q\times Q) \end{equation*} is the transition relation. \end{definition} \begin{definition} A \emph{run} of $\Aut{A}$ on the $(\alpha)$-word $w \in \FinWords{\Sigma}{\alpha}$ is a map $r: \alpha+1 \to Q$ such that \begin{itemize} \item $ \left( r(\beta), w(\beta), r(\beta+1)\right)\in \Delta$ for all $\beta<\alpha$ \item $(\lim_{\beta} r, r(\beta))\in\Delta$ for all limit ordinals $\beta\leq\alpha$. \end{itemize} The run $r$ is \emph{accepting} if $r(0)\in I$ and $r(\alpha)\in F$. For $q,q' \in Q$, we write $q \run{w}{\Aut{A}} q'$ if there is a run $r$ of $\Aut{A}$ on $w$ with $r(0) = q$ and $r(\alpha) = q'$. \end{definition} In the following, we always fix an ordinal $\alpha$ and then concentrate on the set of $(\alpha)$-words that a given ordinal automaton accepts. In order to stress this fact, we will call the ordinal-automaton an $(\alpha)$-automaton. \begin{definition} Let $\alpha$ be some ordinal and $\Aut{A}$ be an $(\alpha)$-automaton. The \emph{$(\alpha)$-language} of $\Aut{A}$, denoted by $L_{\FinWords{\Sigma}{\alpha}}(\Aut{A})$, consists of all $(\alpha)$-words $w \in \FinWords{\Sigma}{\alpha}$ which admit an accepting run of $\Aut{A}$ on $w$. Whenever $\alpha$ is clear from the context, we may omit the subscript $\FinWords{\Sigma}{\alpha}$ and just write $L(\Aut{A})$ instead of $L_{\FinWords{\Sigma}{\alpha}}(\Aut{A})$. \end{definition} Automata on words (or infinite words or (infinite) trees) have been applied fruitfully for representing structures. This can be lifted to the setting of $(\alpha)$-words and leads to the notion of \automatic{\alpha} structures. In order to use $(\alpha)$-automata to recognise relations of $(\alpha)$-words, we need to encode tuples of $(\alpha)$-words by one $(\alpha)$-word: \begin{definition} Let $\Sigma$ be an alphabet and $r \in \mathbb{N}$. \begin{enumerate}[(1)] \item We regard any tuple $\bar{w} = (w_1,\dotsc,w_r) \in \bigl(\FinWords{\Sigma}{\alpha}\bigr)^r$ of $(\alpha)$-words over some alphabet $\Sigma$ as an $(\alpha)$-word $\bar{w} \in \FinWords{(\Sigma^r)}{\alpha}$ over the alphabet $\Sigma^r$ by defining \begin{equation*} \bar{w}(\beta) = \bigl(w_1(\beta),\dotsc,w_r(\beta)\bigr) \end{equation*} for each $\beta < \alpha$. \item An \emph{$r$-dimensional $(\alpha)$-automaton over $\Sigma$} is an $(\alpha)$-automaton $\Aut{A}$ over $\Sigma^r$. The $r$-ary relation on $\FinWords{\Sigma}{\alpha}$ \emph{recognised} by $\Aut{A}$ is denoted \begin{equation*} R(\Aut{A}) = \Set{\bar w \in \bigl(\FinWords{\Sigma}{\alpha}\bigr)^r | \bar w \in L(\Aut{A}) } \,. \end{equation*} \end{enumerate} \end{definition} Usually, this interpretation of $\bar{w}$ as an $(\alpha)$-word is called \emph{convolution} of $\bar{w}$ and denoted $\otimes \bar{w}$. For the sake of convenience, we just omit the symbol $\otimes$. \begin{definition} Let $\tau = \{R_1, R_2, \dots, R_m\}$ be a finite relational signature and let relation symbol $R_i$ be of arity $r_i$. A structure $\Struc{A}=(A,R^\Struc{A}_1,R^\Struc{A}_2, \dots, R^\Struc{A}_m)$ is \emph{\automatic{\alpha}} if there are an alphabet $\Sigma$ and $(\alpha)$-automata $\Aut{A}, \Aut{A}_\approx,\Aut{A}_1, \dots, \Aut{A}_m$ such that \begin{itemize} \item $\Aut{A}$ is an $(\alpha)$-automaton over $\Sigma$, \item for each $R_i\in \tau$, $\Aut{A}_i$ is an $r_i$-dimensional $(\alpha)$-automaton over $\Sigma$ recognising an $r_i$-ary relation $R(\Aut{A}_i)$ on $L(\Aut{A})$, \item $\Aut{A}_\approx$ is a $2$-dimensional $(\alpha)$-automaton over $\Sigma$ recognising a congruence relation $R(\Aut{A}_\approx)$ on the structure $\Struc{A}'= \left (L(\Aut{A}), L(\Aut{A}_1), \dotsc, L(\Aut{A}_m)\right)$, and \item the quotient structure $\Struc{A}'/R(\Aut{A}_\approx)$ is isomorphic to $\Struc{A}$, i.e., $\Struc{A}'/R(\Aut{A}_\approx)\cong \Struc{A}$. \end{itemize} In this situation, we call the tuple $(\Aut{A}, \Aut{A}_\approx,\Aut{A}_1, \dots, \Aut{A}_m)$ an \emph{\automatic{\alpha} presentation} of $\Struc{A}$. This presentation is said to be \emph{injective} if $L(\Aut{A}_\approx)$ is the identity relation on $L(\Aut{A})$. In this case, we usually omit $\Aut{A}_\approx$ from the tuple of automata forming the presentation. \end{definition} \subsection{Definitions Concerning Growth Rates} The basic idea behind the growth rate technique is the question how many elements of a structure can be distinguished using a fixed finite set of relations and a set of parameters which has $n$ elements. We call two elements $a$ and $b$ distinguishable by a $(1+p)$-ary relation $R$ with parameters from $E$ if there are $e_1, e_2, \dots, e_p\in E$ such that $(a, e_1, e_2, \dots, e_p)\in R$ while $(b, e_1, e_2, \dots, e_p)\notin R$. If $\lvert E \rvert = n$ and $R$ is some relation, it is clear that there are at most $2^{n^p}$ many elements that are pairwise distinguishable by $E$ with parameters from $E$. Delhomm\'e\xspace \cite{Delhomme04} has shown that for every tree-automatic relation $R$ there are always sets $E$ with $n$ elements such that there are at most $n^c$ pairwise distinguishable elements where $c$ is a constant only depending on $R$ (and not on $n$ or $E$). For word-automatic structures this bound even drops to $n\cdot c$. We now provide basic definitions that allow to derive a similar bound for \automatic{\alpha} structures. \begin{definition} Let $\Struc{A}$ be an \automatic{\alpha} structure with domain $A$ and $\Phi$ be a finite set of $(\alpha)$-automata such that each $\Aut{A}\in\Phi$ recognises a $1+p$-ary relation $R_{\Aut{A}} \subseteq A^{1+p}$. Let $E\subseteq A$ be a finite set and let $\Fam{F}$ be an infinite family of subsets of $A$ with $\emptyset\in\Fam{F}$. \begin{enumerate} \item For all $a,a'\in A$ we write $a\sim^\Phi_E a'$ if \begin{equation*} (a, e_1, \dots, e_p) \in R_{\Aut{A}} \Leftrightarrow (a', e_1, \dots, e_p) \in R_{\Aut{A}} \end{equation*} for all $e_1, \dots, e_p \in E$ and all $\Aut{A}\in \Phi$, i.e., $a$ and $a'$ are indistinguishable with the automata from $\Phi$ and parameters in $E$. \item We say $S\subseteq A$ is \emph{$E$-$\Phi$-free} if $a \not\sim^\Phi_E a'$ for all $a, a'\in S$. \item We say some set $G\subseteq E$ is maximal $E$-$\Phi$-free if $G$ is $E$-$\Phi$-free and there is no $E$-$\Phi$-free strict superset of $G$. \item For all $S\subseteq A$ we write $\lvert S\restriction{\Fam{F}} \rvert$ for \begin{equation*} \max\Set{\lvert F \rvert | F\in \Fam{F}\text{\ with }F\subseteq S}. \end{equation*} Set \begin{equation*} \nu^\Phi_\Fam{F}(E) = \min\Set{\lvert G\restriction{\Fam{F}} \rvert | G\text{\ maximal }E\text{-}\Phi\text{-free} }. \end{equation*} and for $n\in\mathbb{N}$, set \begin{equation*} \nu^\Phi_\Fam{F}(n) = \inf \Set{\nu^\Phi_\Fam{F}(E) | E\in\Fam{F}, \lvert E \rvert = n}\in \mathbb{N}\cup\{\infty\} \end{equation*} (where $\inf \emptyset = \infty$). \end{enumerate} \end{definition} $\nu_{\Fam{F}}^\Phi$ measures the minimal growth rate of sets definable from $\Phi$ with a finite set of parameters with respect to some infinite family $\Fam{F}$. In most applications $\Fam{F}$ can be defined to be the set of all subsets. In this case $\nu_{\Fam{F}}^\Phi$ just measures the growth rate of sets definable from $\Phi$ with a finite set of parameters. Let us comment on how such a function $\nu_{\Fam{F}}^\Phi$ is usually used. Typical results on growth rate are of the form ``there are infinitely many $n\in\mathbb{N}$ such that $\nu_{\Fam{F}}^\Phi(n) \leq p(n)$'' for a certain polynomial $p$. If $\Fam{F}$ is the set of all subsets of the domain of the given structures, this says that for infinitely many values of $n$ there is a subset $E$ of size $n$ such that every maximal $E$-$\Phi$ free set $G$ has size at most $p(n)$. \section{Basic Results} \label{sec:BasicResults} In this Section we cite some results from \cite{HuKaSchl14} that turn out to be useful in the following sections. \begin{proposition}[Proposition~3.6 of \cite{HuKaSchl14}] \label{prop:PumpingLemma} Let $\alpha\geq 1$ be an ordinal of countable cofinality and let $\Aut{A} = (S, \Sigma, I, F, \Delta)$ be an automaton with $\lvert S \rvert \leq m$. For all $s_0,s_1\in S$ and $\sigma\in\Sigma$, \begin{equation*} s_0 \run{\sigma^{\omega^m}}{\Aut{A}} s_1 \Longleftrightarrow s_0 \run{\sigma^{\omega^m\alpha}}{\Aut{A}} s_1. \end{equation*} \end{proposition} \begin{proposition}[cf.~Proposition~3.7 of \cite{HuKaSchl14}] \label{prop:PumpingLemmaUnc} Let $\alpha\geq 1$ be an ordinal of uncountable cofinality and let $\Aut{A} = (S, \Sigma, I, F, \Delta)$ be an automaton with $\lvert S \rvert \leq m$. For all $s_0,s_1\in S$ and $\sigma\in\Sigma$, \begin{equation*} s_0 \run{\sigma^{\omega_1}}{\Aut{A}} s_1 \Longleftrightarrow s_0 \run{\sigma^{\alpha}}{\Aut{A}} s_1. \end{equation*} \end{proposition} \begin{lemma}[Lemma~3.19 of \cite{HuKaSchl14}]\label{lem:WellOrder} For every finite alphabet $\Sigma$, there is an $\alpha$-automaton that recognises a well-order $\sqsubseteq$ on the set $\FinWords{\Sigma}{\alpha}$. Moreover the relation $\subseteq_\supp$ given by $w \subseteq_\supp v$ if and only if $\supp(w) \subseteq \supp(v)$ is \automatic{\alpha}. \end{lemma} \section{The Growth Rate Technique} \label{sec:GrowthRate} Delhomm\'e\xspace proved the following bounds on the growth rates of maximal $\Fam{F}$-$E$-$\Phi$-free sets in the word- and tree-automatic setting. \begin{proposition}[\cite{Delhomme04}] For each set $\Phi$ of word-automatic relations, there is a constant $k$ such that $\nu_{\Fam{F}}^\Phi (n) \leq k\cdot n$ for infinitely many $n\in \mathbb{N}$. For each set $\Phi$ of tree-automatic relations, there is a constant $k$ such that $\nu_{\Fam{F}}^\Phi(n) \leq n^k$ for infinitely many $n\in \mathbb{N}$. \end{proposition} The basic proof idea is to show that any $E$-$\Phi$-free set $G$ can be transformed into an $E$-$\Phi$-free set $G'$ such that $\lvert G\rvert = \lvert G' \rvert$ whose elements are all words (or trees) that have a domain that is similar to the union of the domains of all parameters from $E$. In order to prove a similar result we first provide a notion of having similar domains for $(\alpha)$-words. \begin{definition} Let $m\in\mathbb{N}$, $X$ a finite set of ordinals and $\beta$ and ordinal of the form \begin{equation*} \beta=\beta_{\sim m}+\omega^m n_m+\omega^{m-1} n_{m-1}+ \dots + n_0. \end{equation*} \begin{itemize} \item Let $U_m(\beta)$ denote the set of ordinals $\gamma=\gamma_{\sim m}+\omega^m l_m+\omega^{m-1} l_{m-1}+ \dots +l_0$ such that one of the following holds: \begin{itemize} \item $\gamma=\beta$, \item $\gamma_{\sim m}=\beta_{\sim m}$ and for $k$ maximal with $l_k\neq n_k$, we have $l_k\leq n_k+m$ and $l_i\leq m$ for all $i<k$, or \item $\gamma_{\sim m} = \beta_{\sim m} +\omega_1 c$ for some $1\leq c \leq m$ and $l_i \leq m$ for all $0\leq i \leq m$. \end{itemize} \item Let $U_m(X,\delta)= \left(\bigcup_{\gamma\in X\cup\{0, \delta\}} U_m(\gamma) \right) \cap \delta$. \item Let $U_m^1(X,\delta)=U_m(X,\delta)$ and $U_m^{i+1}(X,\delta)=U_m(U_m^i(X,\delta),\delta)$ for $i\in\mathbb{N}$. \end{itemize} \end{definition} A crucial observation is that, roughly speaking, there are few $(\alpha)$-words with support in $U^i_m(X, \alpha)$. Using the following abbreviations, we make this idea precise in the following lemma: \begin{enumerate} \item $c_m(\beta)=\max_{i\leq m} n_i$, \item $c_m(X)=\max_{\gamma\in X} c_m(\gamma)$, and \item $d_m(X)=|\{\gamma_{\sim m}\mid \gamma\in X\cup\{0\}\}|$. \end{enumerate} \begin{lemma} \label{lem:BoundsOnU} Suppose that $X$ is a finite set of ordinals and $i\geq 1$. Then \begin{align*} &\lvert U_m^i(X,\alpha)\rvert \leq (c_m(X\cup\{\alpha\})+im)^{m+1} \cdot (i \cdot m+1) \cdot d_m(X\cup\{\alpha\}). \end{align*} \end{lemma} \begin{proof} A simple induction shows that all $\gamma \in U_m^i(\beta)$ satisfy $\gamma_{\sim m} = \beta_{\sim m} + \omega_1 \cdot k$ for some $0\leq k \leq (i \cdot m)$. One also proves inductively that the coefficient of $\omega^j$ of an element of $U_m^i(X, \alpha)$ is bounded by $(c_m(w)+im)$. \end{proof} In this section, we fix an ordinal $\alpha$, and a finite set of $(\alpha)$-automata \begin{equation*} \Phi = (\Aut{A}_1, \Aut{A}_2, \dots, \Aut{A}_n). \end{equation*} Without loss of generality, each automaton has the same state set $Q$. We fix the constant $K = \lvert 2^{Q\times Q} \rvert^n +1$. The following proposition contains the main technical result that allows to use the relative growth technique for ordinal-automatic structures. This proposition implies that for all $E$ and $\Phi$ there is a $E$-$\Phi$-free set of maximal size with support in $U_K(\supp(E))$. Since $U_K(\supp(E))$ is small this provides an upper bound on the minimal size of maximal $E$-$\Phi$-free sets. \begin{proposition}\label{prop:RelativeGrowthsOrdinalAutomatic} Let $E\subseteq \FinWords{\Sigma}{\alpha}$ and $v\in\FinWords{\Sigma}{\alpha}$. There is a word $w\in U_K(\supp(E), \alpha)$ such that $v \sim^\Phi_E w$. \end{proposition} For better readability we first provide a simple tool for the proof \begin{lemma} Let $E\subseteq \FinWords{\Sigma}{\alpha}$ and $v\in\FinWords{\Sigma}{\alpha}$. Let $n\in\mathbb{N}$ and $\beta$ some ordinal such that $\beta+\omega^{n+1}\leq \alpha$. If there is an ordinal $\gamma$ such that $\gamma < \gamma + \omega^n (K-1) \leq \beta < \gamma+\omega^{n+1}$ and \begin{equation} \label{eq:suppEMpty} \supp(E) \cap [\gamma, \gamma+\omega^{n+1}) = \emptyset \end{equation} then there are natural numbers $n_1 <n_2 \leq K$ such that for \begin{equation*} w:= v \restriction{[0,\gamma+\omega^nn_1)} + v \restriction{[\gamma+\omega^n n_2, \alpha)} \end{equation*} $w \sim^\Phi_E v$, i.e., for all $\bar e \in E^k$ and $\Aut{A}\in \Phi$ \begin{equation*} \Aut{A} \text{\ accepts } v\otimes \bar e \text{\ iff } \Aut{A} \text{\ accepts } w\otimes \bar e \end{equation*} \end{lemma} \begin{proof} Set $I=\Set{1, 2, \dots, n}$. We define the function \begin{equation*} f:\Set{0, 1, \dots, K} \to 2^{Q\times Q \times I} \end{equation*} such that $f(j)$ contains $(q,p,i)$ if and only if there is a run of $\Aut{A}_i$ from state $q$ to state $p$ on $v\restriction{[\gamma, \gamma+\omega^n j)} \otimes \diamond^{\omega^n j}$. By choice of $K$ there are $n_1<n_2$ with $f(n_1) = f(n_2)$. Thus, \begin{align*} &q \run{v\otimes \bar e}{\Aut{A}_i} p\\ \xLeftrightarrow{\eqref{eq:suppEMpty}} &\exists{r,s} \left(q \run{(v\otimes \bar e)\restriction{[0, \gamma)}}{\Aut{A}_i} r \land r \run{v\restriction{[\gamma, \gamma+\omega^n n_2)} \otimes \diamond^{\omega^n n_2}}{\Aut{A}_i} s \land s \run{(v\otimes \bar e) \restriction{[\gamma+\omega^n n_2, \alpha)}}{\Aut{A}_i} p\right)\\ \xLeftrightarrow[= f(n_2)]{f(n_1) } &\exists{r,s} \left(q \run{(v\otimes \bar e)\restriction{[0, \gamma)}}{\Aut{A}_i} r \land r \run{v\restriction{[\gamma, \gamma+\omega^n n_1)} \otimes \diamond^{\omega^n n_1}}{\Aut{A}_i} s \land s \run{(v\otimes \bar e) \restriction{[\gamma+\omega^n n_2, \alpha)}}{\Aut{A}_i} p\right)\\ \xLeftrightarrow{\eqref{eq:suppEMpty}} &q \run{w\otimes \bar e}{\Aut{A}_i} p \end{align*} from which we immediately conclude that $v \sim^\Phi_E w$. \end{proof} \begin{proof} [Proof of Proposition~\ref{prop:RelativeGrowthsOrdinalAutomatic}] The proof is by outer induction on $\lvert \supp(v) \setminus U_K(\supp(E), \alpha) \rvert$ and by inner (transfinite) induction on \begin{equation*} \beta = \min( \supp(v) \setminus U_K(\supp(E), \alpha)). \end{equation*} Fix the presentation \begin{equation*} \beta = \beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ b_0 \end{equation*} with $b_0, \dots, b_K\in \mathbb{N}$ and proceed as follows. \begin{itemize} \item If there is some $n\leq K$ such that $b_n+1-K \geq 0$ and \begin{equation*} \begin{aligned} &(\supp(E)\cup\{\alpha\}) \cap [\epsilon_1, \epsilon_2) = \emptyset \text{\ where }\\ &\epsilon_1 = \beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ \omega^n (b_n+1-K) \text{\ and}\\ &\epsilon_2 = \beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ \omega^{n+1} (b_{n+1}+1), \end{aligned} \end{equation*} then we can apply the previous lemma and obtain a word $v'$ such that $v \sim^\Phi_E v'$ and \begin{equation*} \lvert \supp(v') \setminus U_K(\supp(E), \alpha) \rvert < \lvert \supp(v) \setminus U_K(\supp(E), \alpha) \rvert \end{equation*} or \begin{equation*} \beta' = \min( \supp(v') \setminus U_K(\supp(E), \alpha)). \end{equation*} has a presentation \begin{equation*} \beta' = \beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ \omega^{n+1} b_{n+1} + \omega^n b' + \omega^{n-1} b_{n-1} + \dots + b_0 \end{equation*} with $b' < b_n$. \item Assume that the conditions for the previous case are not satisfied. Either $b_i \leq K-1 $ for $0\leq i \leq K$ or there is a minimal $i\leq K$ such that $b_i \geq K$. We first show that the latter case cannot occur. Assuming $b_i \geq K$ we have \begin{equation*} \begin{aligned} &(\supp(E)\cup\{\alpha\}) \cap [\epsilon_1 , \epsilon_2) \neq \emptyset \text{\ where}\\ &\epsilon_1=\beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ \omega^i (b_i + 1 - K) \text{\ and}\\ &\epsilon_2= \beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ \omega^{i+1} (b_{i+1}+1). \end{aligned} \end{equation*} Thus, there is some $\gamma \in \supp(E)\cup\{\alpha\}$ with \begin{equation*} \gamma = \beta_{\sim K} + \omega^K b_K + \omega^{K-1} b_{K-1} +\dots+ \omega^{i+1} b_{i+1} + \omega^i c_i + \omega^{i-1} c_{i-1} + \dots + c_0 \end{equation*} such that $c_i + K-1 > b_i$ whence $\beta \in U_K(\gamma)\subseteq U_K(\supp(E), \alpha)$ contradicting the definition of $\beta$. Thus, we can assume that $b_i \leq K-1$ for all $0\leq i\leq K$. By definition of $\beta$ we conclude that $\beta_{\sim K} \neq \gamma_{\sim K} + \omega_1 c$ for $c\in\{0, 1, \dots, K\}$ for all $\gamma\in \supp(E)\cup \{0,\alpha\}$. We proceed with one of the following cases depending on the cofinality of $\beta_{\sim K}$. \begin{enumerate} \item If $\beta_{\sim K}$ has countable cofinality, let \begin{align*} &\gamma = \max( ( \supp(E \cup\{v\} ) \cap \beta_{\sim K} )+1 \\ &\delta = \min((\supp(E) \cap [\beta, \alpha)) \cup \{\alpha\}) \text{\ and}\\ &\delta' = \max( \supp(v) \cap \delta_{\sim K})+1. \end{align*} From the definition of $\beta$ it follows that $\gamma_{\sim K} < \beta_{\sim K} < \delta_{\sim K}$ and that $\sup(E) \cap [\gamma, \delta) = \emptyset$. Note that $[\gamma, \beta_{\sim K})$ is of shape $\omega^{K+1} \eta_1$ for some ordinal $\eta_1\geq 1$ of countable cofinality. By definition of $\delta'$, $[\delta', \delta_{\sim K})$ is of shape $\omega^{K+1} \eta_2$ for some ordinal $\eta_2\geq 1$. Choose an ordinal $\eta$ such that $[\beta_{\sim K}, \delta') + \eta$ is isomorphic to $[\gamma, \delta_{\sim K})$ and define \begin{equation*} w:= v\restriction{[0, \gamma)} + \diamond^{\omega^{K}} + v\restriction{[\beta_{\sim K}, \delta')} + \diamond^{\eta} + v\restriction{[\delta_{\sim K}, \alpha)}. \end{equation*} For all $\bar e \in E^k$ and $\Aut{A}\in A$ we conclude that \begin{align*} &q \run{v \otimes \bar e}{\Aut{A}} p \\ \Leftrightarrow& \exists{r,s,t,u \in Q} \left( \begin{alignedat}{3} &q \run{(v \otimes \bar e)\restriction{[0,\gamma)}}{\Aut{A}} r& &\land r \run{\diamond^{\omega^{K+1} \eta_1}}{\Aut{A}} s& &\land s \run{(v \otimes \bar e)\restriction{[\beta_{\sim K}, \delta')}}{\Aut{A}} t \\ && &\land t \run{\diamond^{\omega^{K+1}} \eta_2}{\Aut{A}} u & &\land u \run{(v \otimes \bar e)\restriction{[\delta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right) \\ \Leftrightarrow& \exists{r,s,t,u \in Q} \left( \begin{alignedat}{3} q \run{(v \otimes \bar e)\restriction{[0,\gamma)}}{\Aut{A}} r &\land r \run{\diamond^{\omega^{K}}}{\Aut{A}} s &&\land s \run{(v\otimes \bar e)\restriction{[\beta_{\sim K},\delta')}}{\Aut{A}} t\\ &\land t \run{\diamond^{\eta}}{\Aut{A}} u &&\land u \run{(v \otimes \bar e)\restriction{[\delta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right) \\ \Leftrightarrow & q \run{w\otimes \bar e}{\Aut{A}} p \end{align*} whence $w \sim^\Phi_E v$. Moreover \begin{equation*} \lvert \supp(w) \setminus U_K(\supp(E), \alpha) \rvert < \lvert \supp(v) \setminus U_K(\supp(E), \alpha) \rvert \end{equation*} because the letter at position $\beta$ in $v$ has been shifted to position \begin{equation*} \gamma_{\sim K} + \omega^K (b_K+1) + \omega^{K-1} b_{k-1} + \omega^{K-2} b_{k-2} + \dots + b_0. \end{equation*} Since all $b_i \leq K-1$ we conclude that this position belongs to $U_K(\supp(E), \alpha)$. \item If $\beta_{\sim K}$ has uncountable cofinality, Let \begin{align*} &\gamma = \max( \supp(E) \cap \beta_{\sim K} )+1 \\ &\delta = \min((\supp(E) \cap [\beta, \alpha)) \cup \{\alpha\}) \text{\ and}\\ &\delta' = \max( \supp(v) \cap \delta_{\sim K})+1. \end{align*} From the definition of $\beta$ and the uncountable cofinality of $\beta_{\sim K}$ it follows that $\gamma_{\sim K} +\omega_1 K < \beta_{\sim K} < \delta_{\sim K}$ and that $\sup(E) \cap [\gamma, \delta) = \emptyset$. Let \begin{equation*} \gamma' = \max( \supp(E\cup\{v\}) \cap \beta_{\sim K})+1 \end{equation*} and note that $\gamma' \leq \gamma_{\sim K} + \omega_1 (K+1)$ because $\gamma'\in U_K(E, \alpha)$ since $\beta$ has been chosen minimal. If $\gamma' \geq \gamma_{\sim K} + \omega_1 K$, then we do the following preparatory step that locally changes $v$ to some $v'$ such that $ v \sim^\Phi_E v'$ and shrinking the corresponding value of $\gamma'$. For this purpose, note that $e\restriction{[\gamma, \beta)} =\diamond^{[\gamma, \beta)}$ for all $e\in E$. By choice of $K$ there are numbers $i < j \leq K$ such that for all $\Aut{A}\in\Phi$, all $q,p\in Q$ and all $\bar e \in E^k$ we have \begin{equation}\label{eq:omega1ijRunsEquality} q \run{(v\otimes \bar e)\restriction{[\gamma, \gamma + \omega_1 i)}}{\Aut{A}} p \Longleftrightarrow q \run{(v\otimes \bar e)\restriction{[\gamma, \gamma + \omega_1 j)}}{\Aut{A}} p \end{equation} Choose an ordinal $\eta$ such that $\omega_1 \cdot (j-i) + [\gamma+\omega_1 j, \gamma') = [\gamma+\omega_1 j, \gamma') + \eta$ and set \begin{align*} v' = v\restriction{[0, \gamma_{\sim K} + \omega_1 i)} + v\restriction{[\gamma_{\sim K} + \omega_1 j, \gamma')} + \diamond^\eta + v\restriction{[\gamma', \alpha)}. \end{align*} Now $v \sim^\Phi_E v'$ because for all $\bar e \in E^k$ \begin{align*} &q \run{v \otimes \bar e}{\Aut{A}} p \\ \Leftrightarrow& \exists{r,s,t\in Q} \left( \begin{alignedat}{3} &q \run{(v \otimes \bar e)\restriction{[0,\gamma + \omega_1 j)}}{\Aut{A}} r &&\land r \run{(v \otimes \bar e)\restriction{[\gamma + \omega_1 j, \gamma')}}{\Aut{A}} s \\ \land\ &s \run{(v \otimes \bar e)\restriction{[\gamma', \beta_{\sim K})}}{\Aut{A}} t &&\land t \run{(v \otimes \bar e)\restriction{[\beta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right)\\ \xLeftrightarrow{Eq.~\eqref{eq:omega1ijRunsEquality}}& \exists{r,s,t\in Q} \left( \begin{alignedat}{3} &q \run{(v \otimes \bar e)\restriction{[0,\gamma + \omega_1 i)}}{\Aut{A}} r &&\land r \run{(v \otimes \bar e)\restriction{[\gamma + \omega_1 j, \gamma')}}{\Aut{A}} s \\ \land\ &s \run{\diamond^{[\gamma', \beta_{\sim K})}}{\Aut{A}} t &&\land t \run{(v \otimes \bar e)\restriction{[\beta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right)\\ \xLeftrightarrow{Prop.~\ref{prop:PumpingLemmaUnc}}& \exists{r,s,t\in Q} \left( \begin{alignedat}{3} &q \run{(v' \otimes \bar e)\restriction{[0,\gamma + \omega_1 i)}}{\Aut{A}} r &&\land r \run{(v \otimes \bar e)\restriction{[\gamma + \omega_1 j, \gamma')}}{\Aut{A}} s \\ \land\ &s \run{\diamond^{\eta+[\gamma', \beta_{\sim K})}}{\Aut{A}} t &&\land t \run{(v \otimes \bar e)\restriction{[\beta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right)\\ \Leftrightarrow & q \run{v'\otimes \bar e}{\Aut{A}} p \end{align*} Note that the definitions of $\beta$, $\gamma$, $\delta$ and $\delta'$ with respect to $v'$ agree with those for $v$. Thus, from now on we replace $v$ by $v'$ whence we can assume that $\gamma' < \gamma + \omega_1 K$. Note that $[\gamma', \beta_{\sim K})$ is of shape $\omega^{K+1} \eta_1$ for some ordinal $\eta_1\geq 1$ of uncountable cofinality. By definition of $\delta'$, $[\delta', \delta_{\sim K})$ is of shape $\omega^{K+1} \eta_2$ for some ordinal $\eta_2\geq 1$. Choose an ordinal $\eta$ such that $[\beta_{\sim K}, \delta') + \eta$ is isomorphic to $[\gamma' +\omega_1, \delta_{\sim K})$ and define \begin{equation*} w:= v\restriction{[0, \gamma')} + \diamond^{\omega_1} + v\restriction{[\beta_{\sim K}, \delta')} + \diamond^{\eta} + v\restriction{[\delta_{\sim K}, \alpha)}. \end{equation*} For all $\bar e \in E^k$ and $\Aut{A}\in A$ we conclude that \begin{align*} &q \run{v \otimes \bar e}{\Aut{A}} p \\ \Leftrightarrow& \exists{r,s,t,u\in Q} \left( \begin{alignedat}{3} q \run{(v \otimes \bar e)\restriction{[0,\gamma')}}{\Aut{A}} r &\land r \run{\diamond^{\omega^{K+1} \eta_1}}{\Aut{A}} s &&\land s \run{(v \otimes \bar e)\restriction{[\beta_{\sim K}, \delta')}}{\Aut{A}} t \\ &\land t \run{\diamond^{\omega^{K+1}} \eta_2}{\Aut{A}} u &&\land u \run{(v \otimes \bar e)\restriction{[\delta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right)\\ \Leftrightarrow& \exists{r,s,t,u\in Q} \left( \begin{alignedat}{2} q \run{(v \otimes \bar e)\restriction{[0,\gamma')}}{\Aut{A}} r &\land r \run{\diamond^{\omega_1}}{\Aut{A}} s &&\land s \run{(v \otimes \bar e)\restriction{[\beta_{\sim K}, \delta')}}{\Aut{A}} t\\ &\land t \run{\diamond^{\eta}}{\Aut{A}} u &&\land u \run{(v \otimes \bar e)\restriction{[\delta_{\sim K}, \alpha)}}{\Aut{A}} p \end{alignedat} \right) \\ \Leftrightarrow & q \run{w\otimes \bar e}{\Aut{A}} p \end{align*} whence $w \sim^\Phi_E v$. Moreover \begin{equation*} \lvert \supp(w) \setminus U_K(\supp(E), \alpha) \rvert < \lvert \supp(v) \setminus U_K(\supp(E), \alpha) \rvert \end{equation*} because the letter at position $\beta$ in $v$ has been shifted to position \begin{equation*} \gamma'_{\sim K}+\omega_1 + \omega^K b_K + \omega^{K-1} b_{k-1} + \omega^{K-2} b_{k-2} + \dots + b_0 \end{equation*} and since all $b_i<K-1$ we conclude that this is position is contained in $U_K(\supp(E), \alpha)$ because $\gamma_{\sim K} = \eta_{\sim K}+ \omega_1 c$ for some $\eta\in \supp(E)\cup\{0\}$ and some $c\in\{0, 1, \dots, K-1\}$. \qedhere \end{enumerate} \end{itemize} \end{proof} The previous result allows us to directly deduce the following bound on the growth rates of ordinal-automatic relations. \begin{theorem}\label{thm:GrowthRageOrdinalAutomatic} Fix an infinite family $\Fam{F}$ of sets of $(\alpha)$-words with $\emptyset\in\Fam{F}$. For every $c > 1$, $\nu^\Phi(n) \leq n^c$ for infinitely many $n\in \mathbb{N}$. \end{theorem} \begin{proof} Heading for a contradiction, assume that there is a natural number $N$ such that \begin{equation*} \forall{E\in\Fam{F} \text{\ with }\lvert E \rvert >N} \forall{G \text{\ maximal $E$-$\Phi$-free}} \exists{F\in\Fam{F}} (F\subseteq G \text{ and } \lvert F \rvert \geq \lvert E \rvert^c) \end{equation*} where $c > 1$. Take a finite set $F_0$ of parameters with $\lvert F_0 \rvert > N$. Having defined a finite set $F_{i-1}$ such that $\supp(F_{i-1})\in U_K^{i-1}(\supp(F_0))$ we can use the previous lemma to choose some maximal $F_{i-1}$-$\Phi$-free set $G_i$ with $\supp(G_i) \in U_K^{i}(\supp(F_0))$. By assumption, there is some $F_i\in\Fam{F}$ with $F_i\subseteq G_i$ and $\lvert F_i \rvert \geq \lvert F_{i-1} \rvert^c$. By induction we obtain $\lvert F_{i} \rvert \geq \lvert F_0 \rvert^{c^i}$. On the other hand, all elements of $F_i$ have support in $U_K^{i}(\supp(F_0))$ which by Lemma~\ref{lem:BoundsOnU} is at most of size \begin{equation*} (\lvert \Sigma \rvert +1)^{c_0 (iK+1) ( c_1 + K i)^K} \end{equation*} for some constants $c_0$ and $c_1$. Since $c^i$ grows faster than any polynomial in $i$ we have $c^i > c_0 (i+1) ( c_1 + K i)^K$ for some large $i$ which leads to a contradiction. \end{proof} \section{Applications} \label{sec:Applications} \subsection{Random Graph} \label{sec:RandomGraph} The \emph{random graph (or Rado graph)} $(V, E)$ is the unique countable graph that has the property that for any choice of finite subsets $V_0, V_1\subseteq V$ there is a node $v'$ which is adjacent to every element of $V_0$ but not adjacent to any element of $V_1$. \begin{theorem} Given an ordinal $\alpha$, the random graph is not \automatic{\alpha}. \end{theorem} \begin{proof} Heading for a contradiction assume that the random graph was \automatic{\alpha}. As shown by Delhomm\'e\xspace\cite{Delhomme04}, the random graph satisfies $\nu_{\Fam{F}}^\Phi(n) = 2^n$ for all $n\in\mathbb{N}$ where $\Phi$ consists of only one automaton recognising the edge relation of the random graph and $\Fam{F}$ contains all subsets of the domain of the random graph. This contradicts Proposition~\ref{prop:RelativeGrowthsOrdinalAutomatic}. \end{proof} \begin{remark} Similarly, taking $\Fam{F}$ to be the family of all antichains, one proves that the random partial order is not \automatic{\alpha} (cf.~\cite{KhoussainovNRS07} for an analogous result for automatic structures). \end{remark} \subsection{Integral Domains} \label{sec:IntegralDom} In this part we show that there is no infinite \automatic{\alpha} integral domain for any $\alpha < \OmegaPlus$. We cannot use the growth rate theorem directly but use a variant of its proof. The difference is that we do not use a fixed set of relations $\Phi$ when defining the sequence $(F_i)_{i\in \mathbb{N}}$ but in each step we take a different relation but ensure that we can still apply Proposition~\ref{prop:RelativeGrowthsOrdinalAutomatic} with a fixed constant $K$ in each step. This is ensured by using relations defined by a fixed automaton $\Aut{A}$ which has an additional parameter which is chosen very carefully. In fact, we follow the proof of Khoussainov et al. \cite{KhoussainovNRS07} from the automatic case. Let us recall the basic definitions and some observations from their proof. An \emph{integral domain} is a commutative ring with identity $(D,+, \cdot, 0, 1)$ such that $d \cdot e = 0 \Rightarrow d=0$ or $e=0$ for all $d,e\in D$. \begin{lemma}[cf.~Proof of Theorem~3.10 from \cite{KhoussainovNRS07}] Let $(D, +, \cdot, 0, 1)$ be an integral domain and $E\subseteq D$ a finite subset. There is some $d\in D$ such that for all $a_1, a_2, b_1, b_2 \in E$, if $a_1 d + b_1 = a_2 d + b_2$, then $a_1 = a_2$ and $b_1 = b_2$, i.e., the function $f_d:E^2 \to D$, $(e_1,e_2)\mapsto e_1 d+ e_2$ is injective. \end{lemma} \begin{proposition} Let $\Struc{A} = (D, +, \cdot, 0, 1)$ be an \automatic{\alpha} integral domain for some \mbox{$\alpha < \OmegaPlus$}.\footnote{Without loss of generality, we can assume that $\Struc{A}$ has a injective representation by the automata $(\Aut{A}_D, \Aut{A}_+, \Aut{A}_\cdot, \Aut{A}_0, \Aut{A}_1)$ such that $L(\Aut{A}_D) = D$, i.e., $D$ is a set of $(\alpha)$-words (cf.~\cite{HuKaSchl14}).} There is a constant $m$ such that for every finite set $X\subseteq \alpha$ of ordinals we have \begin{equation*} \lvert\Set{d\in D\mid \supp(d)\subseteq U_m(X, \alpha)}\rvert \geq \lvert\Set{d\in D\mid \supp(d)\subseteq X}\rvert^2. \end{equation*} \end{proposition} \begin{proof} As an abbreviation, we use the expression $x_1, \dots, x_n \subseteq_\supp y$ for \begin{equation*} x_1 \subseteq_\supp y \land \dots \land x_n\subseteq_\supp y. \end{equation*} Let $\psi(x, p)$ denote the formula \begin{equation*} x\in D \land \forall{a_1,a_2,b_1,b_2\subseteq_\supp p} \left( ( a_1 x+b_1 = a_2 x+b_2) \rightarrow (a_1 = a_2 \land b_1 = b_2) \right) \end{equation*} and $\psi_{\min}(x,p)$ the formula \begin{equation*} \psi(x,p) \land \forall y (\psi(y,p) \rightarrow x \sqsubseteq y) \end{equation*} where $\sqsubseteq$ denotes the \automatic{\alpha} well-order from Lemma ~\ref{lem:WellOrder}. Due to the previous lemma for every $p\in\FinWords{\Sigma}{\alpha}$ there is a unique $x$ satisfying $\psi_{\min}(x,p)$. Moreover, the map \mbox{$f:(a,b)\mapsto ax+b$} is injective when the domain is restricted to words with support contained in $\supp(p)$. Since $\sqsubseteq$ is \automatic{\alpha} and \automatic{\alpha} structures are close under first-order definitions, there is an automaton $\Aut{A}_\varphi$ corresponding to the following formula \begin{equation*} \varphi(p,a,b,c) = a\subseteq_\supp p \land b\subseteq_\supp p \land \exists x \left(\psi_{\min}(x,P) \land c= ax+b\right). \end{equation*} For each finite set $X$ of ordinals, choose an $(\alpha)$-word $p_X$ such that $\supp(p) = X$. Set \begin{align*} D_X &= \Set{d\in D | c\subseteq_\supp p} \text{\ and}\\ F_X &= \Set{c \in D | \exists{a,b\in\FinWords{\Sigma}{\alpha}} \Aut{A}_\varphi \text{\ accepts } (p_X, a, b, c)}. \end{align*} Since we are dealing with an injective presentation, Proposition~\ref{prop:RelativeGrowthsOrdinalAutomatic} implies that for every $a,b\in D_X$ there is some $c_{a,b}\in F_X$ such that $\Aut{A}_\varphi$ accepts $(p_X, a, b, c_{a,b})$ and $\supp(c_{a,b}) \subseteq U_K(\supp(p_X)\cup \supp(a) \cup \supp(c), \alpha) = U_K(X, \alpha)$. Moreover, $c_{a,b}=c_{a',b'}$ implies $a=a'$ and $b=b'$ whence we conclude that \begin{equation*} \lvert \Set{d\in D | \supp(d) \subseteq U_K(X, \alpha)}\rvert \geq \lvert F_X \rvert \geq \lvert D_X\rvert^2. \end{equation*} \end{proof} \begin{remark} We crucially rely on $\alpha < \OmegaPlus$ because otherwise we cannot be sure that there is an automaton corresponding to $\psi(x,p)$. \end{remark} \begin{corollary} Let $\alpha < \OmegaPlus$. There is no \automatic{\alpha} infinite integral domain. In particular, there is no \automatic{\alpha} infinite field. \end{corollary} \begin{proof} Assume $D$ is the domain of an \automatic{\alpha} infinite integral domain. Choose two elements $d_1\neq d_2$ from $D$ and let $X = \supp(d_1)\cup \supp(d_2)$. Set \begin{equation*} F_0 = \Set{d\in D | \supp(d)\subseteq \supp(d_1)\cup\supp(d_2)}. \end{equation*} Iterated application of the previous lemma yields that \begin{equation*} F_i:= \Set{d\in D | \supp(d)\subseteq U^i_K(\supp(F_0),\alpha)} \end{equation*} satisfies $\lvert F_{i+1} \rvert \geq \lvert F_i \rvert^2$. Since $\lvert F_0 \rvert \geq 2$ we conclude that $\lvert F_n\rvert \geq 2^{2^n}$ and $\supp(F_n)\subseteq U^n_K(X, \alpha)$. From Lemma~\ref{lem:BoundsOnU} we conclude that there are only $2^{p(n)}$ many elements in $D$ with support in $U^n_K(X, \alpha)$ for some polynomial $p(n)$ which results in a contradiction for large $n$. \end{proof} \section{Optimality of the Bound on the Growth-Rate} Recall that word-automatic structures satisfy that $\nu_{\Fam{F}}^{\Phi}(n) < n \cdot k$ for some constant $k$ where $\Fam{F}$ is a family as before and $\Phi$ is a finite set of word-automatic relations. In contrast, our bound for \automatic{\alpha} structures is only $n^k$ for every constant $k > 1$. In this section, we give an example that, if $\alpha\geq \omega^2$, then there are \automatic{\alpha} structures violating any bound of the form $n \cdot k$ for every constant $k$.\footnote{It is easily shown that for $\alpha<\omega^2$ every \automatic{\alpha} structure is word-automatic and vice versa whence the stronger bound from the word-automatic case applies.} \begin{definition} For every $n\in\mathbb{N}$, let \begin{equation*} D_n = \Set{\omega n_1 + n_2 | n_1 + n_2 \leq n} \end{equation*} and let $T_n$ be the set $(\omega^2)$-words over $\Sigma=\Set{a,b,\diamond}$ such that $w\in T_n$ if and only if $\supp(w) = D_n$. For $i\in\{a,b\}$, we also define functions \begin{align*} f_i:\FinWords{\Sigma}{\omega^2} \times \FinWords{\Sigma}{\omega^2} \to \FinWords{\Sigma}{\omega^2} \text{ by}\\ f_i(w,v)(\alpha) = \begin{cases} i &\text{if }\alpha=0,\\ w(\omega n_1 + n_2) &\text{if }\alpha = \omega n_1 + n_2+1 \\ v(\omega n_1) &\text{if } \alpha = \omega (n_1+1). \end{cases} \end{align*} \end{definition} It is not difficult to see that the graphs of $f_a$ and $f_b$ are \automatic{\omega^2} relations. Let $\Phi$ consist of two automata, one corresponding to the graph of $f_a$ and one corresponding to the graph of $f_b$. It is straightforward to verify that $T_{n+1} = f_a(T_n\times T_n) \cup f_b(T_n \times T_n)$. Moreover, since $f_a$ and $f_b$ are functions, it follows that any maximal $T_n$-$\Phi$-free set is of the form $T_{n+1}\cup \{w\}$ for some $(\omega^2)$-word $w\notin T_{n+1}$. A simple calculation shows that $\lvert D_n \rvert = \frac{(n+1)(n+2)}{2}$ whence $\lvert T_n \rvert = 2^{\frac{(n+1)(n+2)}{2}}$. \begin{corollary} Setting $\Fam{F}$ to be the family of the sets $T_n$ for every $n\in\mathbb{N}$, we obtain that $\nu_{\Fam{F}}^\Phi(m) = \begin{cases} m \cdot 2^{n+2} &\text{if } m = 2^{\frac{(n+1)(n+2)}{2}}, \\ \infty &\text{otherwise.} \end{cases} $ \end{corollary} \begin{proof} Just note that $\lvert T_{n+1} \rvert = 2^{\frac{(n+3)(n+2)}{2}} = 2^{\frac{(n+1)(n+2)}{2}+(n+2)} = \lvert T_n \rvert \cdot 2^{n+2}$. \end{proof} Since there for every constant $k$ there is some value $n_0\in\mathbb{N}$ such that $2^{n+2}\geq k$ this shows that $\nu_{\Fam{F}}^\Phi(m) \leq m \cdot k$ only holds for finitely many $m\in\mathbb{N}$. This shows that there is no word-automatic presentation of the \automatic{\omega^2} structure $(\FinWords{\Sigma}{\omega^2}, f_a, f_b)$ and that the bound in Theorem~\ref{thm:GrowthRageOrdinalAutomatic} cannot be replaced by $n\cdot k$. \end{document}

\begin{document} \title{\Large \bf Noise-driven bifurcations in a nonlinear Fokker--Planck system describing stochastic neural fields} \allowdisplaybreaks \begin{abstract} \noindent The existence and characterisation of noise-driven bifurcations from the spatially homogeneous stationary states of a nonlinear, non-local Fokker--Planck type partial differential equation describing stochastic neural fields is established. The resulting theory is extended to a system of partial differential equations modelling noisy grid cells. It is shown that as the noise level decreases, multiple bifurcations from the homogeneous steady state occur. Furthermore, the shape of the branches at a bifurcation point is characterised locally. The theory is supported by a set of numerical illustrations of the condition leading to bifurcations, the patterns along the corresponding local bifurcation branches, and the stability of the homogeneous state and the most prevalent pattern: the hexagonal one. \end{abstract} \pagenumbering{arabic} \section{Introduction} We establish the existence of noise-driven bifurcations from the spatially homogeneous steady states of the partial differential equation (PDE) \begin{equation}\label{eqn:2} \tau\dfrac{\partial \rho}{\partial t} = -\dfrac{\partial }{\partial s}\left[ \left( \Phi\left( \int_{\mathbb T^d} W(x-y)\int_0^{+\infty}s\rho(y,s,t)\diff s \diff y + B\right) -s \right)\rho(x,s,t) \right] + \sigma \dfrac{\partial^2 \rho}{\partial s^2}, \end{equation} which aims to describe a network of noisy neurons. In this nonlinear Fokker--Planck type model, $\rho(x,s,t)$ is at time $t$ the probability density of neurons at location $x\in \mathbb T^d$, with activity level $s\in[0,+\infty)$. Here, $\mathbb T^d$ is the $d$-dimensional square torus of length $L$ endowed with the quotient topology, $\mathbb T^d := \sfrac{\mathbb{R}^d}{( L \mathbb{Z})^d}$, as the neurons activity levels are assumed to be spatially connected on a toroidal geometry. The interactions between the neurons are modeled through the periodic connectivity function $W$ weighting the averaging in space of the mean activity levels of the neurons. The network of neurons is assumed to receive a constant external input through $B$, and the total input signal to each neuron is then controlled through the modulation function $\Phi$. The parameter $\tau$ is the system's relaxation time, and $\sigma$ determines the noise level in the network. To prohibit the activity level from becoming negative, a no-flux boundary condition is prescribed at $s=0$: \begin{equation*} \left( \Phi\left( \int_{\mathbb T^d} W(x-y)\int_0^{+\infty}s\rho(y,s,t)\diff s \diff y + B\right)\rho(x,s,t) - \sigma \dfrac{\partial \rho}{\partial s}(x,s,t) \right)\bigg|_{s=0} = 0, \end{equation*} for all $(x,t)\in\mathbb T^d\times\mathbb{R}_+$. Last, we make the assumption that the neurons are distributed homogeneously in space, thereby imposing that any initial datum $\rho^0$ satisfies \begin{equation}\label{eqn:mass_x} \forall x\in\mathbb T^d, \qquad \int_{0}^{+\infty} \rho^0(x,s)\diff s = \dfrac1{L^d}. \end{equation} Since there is no movement of neurons in space and owing to the no-flux boundary condition, it can be checked that for any smooth solution this property propagates in time to the density $\rho(x,\cdot,t)$. Our results also apply to the following generalisation of \eqref{eqn:2} : \begin{align}\label{eq:4PDE} \tau \frac{\partial \rho^\beta}{\partial t} = -\frac{\partial}{\partial s}\Bigg( \Big[\Phi^\beta(x,t) -s\Big] \rho^\beta \Bigg) + \sigma \frac{\partial^2 \rho^\beta}{\partial s^2}, \end{align} where $\Phi^\beta(x,t)$ is given by \begin{align}\label{eq:phi} \Phi^\beta(x,t) = \Phi \left(\frac{1}{4}\sum_{\beta'=1}^4 \int_{\mathbb T^d} W^{\beta'}(x-y) \int_{0}^\infty s \rho^{\beta'} (y,s,t)\, \diff s \diff y + B^\beta(t) \right), \end{align} for $\beta =1,2,3,4$, with corresponding no-flux boundary and mass normalisation conditions. In the case $d=2$, \eqref{eq:4PDE} models a network of noisy grid cells \cite{CHS}. Grid cells, discovered in \cite{gridcells}, are neurons which play a pivotal role in spatial representation by firing (emitting spikes) in a hexagonal pattern as a mammal moves around in an open environment, see \cite{tenyears} for a short summary. In addition to the mechanisms already incorporated in \eqref{eqn:2}, \eqref{eq:4PDE} also splits the neurons into four different groups, each having their orientation preference in physical space: $\beta =1,2,3,4$ (north, west, south, east). To model the observed behaviour of grid cells, an orientation preference dependent shift $r^\beta$ is included in the connectivity $W^\beta(x-y)=W(x-y-r^\beta)$, and \eqref{eq:4PDE} is connected with the movement of a mammal through the orientation and time dependent input $B^\beta$. We note however, that time dependent behaviour is beyond the scope of the present manuscript. In the rest of the manuscript we therefore assume $B^\beta(t) = B$, where $B$ is a constant, as was done in \cite{CHS}. In \cite{tenyears} understanding the effects of noise on networks of grid cells was emphasized as a challenge, and this sparked the development and initial study of \eqref{eq:4PDE} in \cite{CHS}. The model \eqref{eq:4PDE}, which is based on the ODE models of \cite{coueyetal,burakfiete}, has been rigorously shown to be the mean-field limit of a network of stochastic grid cells with Gaussian independent noise \cite{CCS21} by adapting Sznitman's coupling method \cite{Sznitman1991}. Furthermore, novel evidence pointing towards a toroidal connectivity for grid cells has recently been provided in \cite{Gardner2022} by using topological data analysis tools. The authors in \cite{CHS} showed the existence and uniqueness of spatially homogeneous stationary states for any given noise strength $\sigma >0$ to equations \eqref{eqn:2} and \eqref{eq:4PDE} under suitable assumptions on the model parameters. Notice that \eqref{eqn:2} and \eqref{eq:4PDE} share the same spatially homogeneous solutions. This one-parameter family of spatially homogeneous solutions will be denoted by $(\rho_\infty^\kappa, \kappa)$ where $\kappa=1/\sigma$ for ease of notation. The main contribution in \cite[Theorem 3.6]{CHS} implies a linear stability criteria for $\rho_\infty^\kappa$ as a solution to \eqref{eqn:2}. More precisely, $\rho_\infty^\kappa$ is linearly asymptotically stable in $L^2\big({\mathbb T}^d \times \mathbb{R}_+\big)$ for \eqref{eqn:2} as long as \begin{align} \label{eq:linearstabilitycondintro} \dfrac{\tilde W (k)}{\Theta(k)} < \frac{1}{\kappa L^{\frac d2} (\Phi_0^\kappa)' \int_0^\infty (s-L^d\bar{\rho}_\infty^\kappa)^2 \rho_\infty^\kappa \diff s}, \end{align} for all $k \in \mathbb{N}$ with $ (\Phi_0^\kappa)':=\Phi'(W_0 \bar\rho_\infty^\kappa + B)$, $\bar\rho_\infty^\kappa$ the mean in $s$ of $\rho_\infty^\kappa$, $W_0$ the integral of $W$ over $\mathbb T^d$, and $\tilde W (k)$ the Fourier mode of the periodic connectivity kernel $W$. Here, $\Theta(k)$ is a normalization factor for the Fourier coefficient of a periodic function in $L^2\big({\mathbb T}^d\big)$ whose precise definition is found in Section \ref{sec:prelim}. Moreover, \cite[Section 4]{CHS} provides numerical evidence of the existence of bifurcation branches emanating from the curve $(\rho_\infty^\kappa, \kappa)$ for \eqref{eq:4PDE} leading to hexagonal patterns in the case of a radially symmetric connectivity kernel $W$, which are reminiscent of similar stationary patterns for the Wilson--Cowan equations \cite{EC1979}, see \cite{Murray03}. In this work, we obtain sharp conditions for the appearance of local bifurcation branches from $(\rho_\infty^\kappa, \kappa)$ based on the Fourier modes of the connectivity kernel $W$ for both \eqref{eqn:2} and \eqref{eq:4PDE}. Our main result, stated in Main Theorem \ref{thm:mainmain}, shows that for any value of $\kappa$ such that there exists a unique Fourier mode $k^*$ of $W$ with \begin{align} \label{eq:branchintro} \dfrac{\tilde W (k^*)}{\Theta(k^*)} = q(\bar \rho_\infty^\kappa, \kappa):=\dfrac{1}{L^{\frac d2} (\Phi_0^\kappa)' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)}, \end{align} leads to a local bifurcation branch of spatially in-homogeneous stationary solutions to \eqref{eqn:2}. This result is illustrated in Fig. \ref{fig:intro-illustration} for a particular choice of the parameters in the model, see Section 5 for more details. We will show in Section 2 that $q(\bar \rho_\infty^\kappa, \kappa)$, the right hand side of \eqref{eq:branchintro}, coincides with the right hand side of \eqref{eq:linearstabilitycondintro}. As a byproduct, Main Theorem \ref{thm:mainmain} shows that the linear asymptotic stability condition \eqref{eq:linearstabilitycondintro} is sharp under suitable assumptions for a large family of connectivity functions $W$. In particular, we only require that the connectivity kernel $W$ is coordinate-wise even as opposed to the common assumption of a radially symmetric kernel in neural field models. The study of kernels being less symmetric is relevant from a neuroscience perspective as connections between neurons can deteriorate, or be locally disturbed by activity outside the network. Analogous results with a slightly different condition than \eqref{eq:branchintro}, involving the shifts $r^\beta$, hold for \eqref{eq:4PDE}, see Section 4. More precisely, we show that a rigorous study of the bifurcations of the one component model \eqref{eqn:2} is sufficient to understand the bifurcations of the four component model \eqref{eq:4PDE}, and that the conditions leading to bifurcations for \eqref{eq:4PDE} are a simple modification of the conditions for \eqref{eqn:2}. \begin{figure} \caption{Plots illustrating Main Theorem \ref{thm:mainmain} \label{fig:intro-illustration} \end{figure} The local bifurcation analysis is obtained by adapting the strategy of \cite{CGPS20} developed for nonlinear nonlocal McKean--Vlasov equations on the torus. The authors in \cite{CGPS20} used the classical approach of Crandall--Rabinowitz local bifurcation theorems in the right functional setting to show branching from constant stationary states. However, in our setting we have to deal with spatially homogeneous steady states that are not constants in $s$. We therefore apply Crandall--Rabinowitz arguments to the averages in the activity level of stationary states instead. We will see that the stationary states $\rho_\infty^\kappa$ implicitly depend on the bifurcation parameter $\kappa$, which leads to cumbersome expressions for the required Fr\'echet derivatives. Another difficulty is that there is no gradient flow structure for \eqref{eqn:2}, or in other words, no variational formulation in terms of steepest descent of free energies as in \cite{CGPS20}. Therefore, although the numerical bifurcation diagrams of \cite{CHS} suggest the existence of discontinuous phase transitions, we cannot adapt the tools of \cite{CGPS20} to rigorously discover them. We thus mainly focus on obtaining all local bifurcation branches emanating from the family $(\rho_\infty^\kappa, \kappa)$ in this work. The strategy of \cite{CGPS20} does enable us to establish the condition leading to bifurcation points \eqref{eq:branchintro} and the existence of certain branches, but is not sufficient for characterising multiple branches emanating from the same point when the bifurcation problem is equivariant, for example when the connectivity kernel is radially symmetric. For this, we need the equivariant branching lemma (see for example \cite[Theorem 2.3.2]{CL-equivariantbif} and the classification of 2D and 3D planforms in \cite{Dionne}). A recent relevant paper in this direction is \cite{FSV2022} where bifurcations with respect to a gain parameter of a Wilson--Cowan-type equation \cite{WC1, WC2} for the primary visual cortex are studied. Let us finally mention that noise driven patterns and bifurcations in related computational neuroscience models have been studied by several authors \cite{bressloff2012, TouboulPhysD,KE13,K14,MB, TouboulSIAM, B19, BAC19,RP19}, see \cite{FTC09,TCL16} for noise in the connectivity, largely from a stochastic viewpoint. Moreover, a rigorous derivation of mesoscopic models by coarsening from stochastic differential equations via mean-field limits have also been developed in \cite{MS02, FTC09,FI15,TouboulPhysD,TouboulSIAM,CT18}. These works deal with spatially extended systems of neural networks modelled by their voltage with random connectivity interactions using large deviation principles \cite{AG95,G97}. The outline of the manuscript is as follows. We start by introducing the setting in Section \ref{sec:prelim}, before providing a summary of the main results concerning the bifurcations of \eqref{eqn:2}. We end the section by linking it to the linear stability condition provided for \eqref{eq:4PDE} in \cite{CHS}. Section \ref{sec:bifurcations} is devoted to proving the main results, which includes establishing some results on the spatially homogeneous stationary state itself. An outline of the generalisation of the main results to the four component model \eqref{eq:4PDE} is provided in Section \ref{sec:4component}. The theoretical results are then illustrated through a series of plots in Section \ref{sec:numerics}, before providing a set of future perspectives in Section \ref{sec:future}. \section{Preliminaries, main results, and relation to stability}\label{sec:prelim} In this section we set the stage for establishing the existence of noise-driven bifurcations of \eqref{eqn:2} and \eqref{eq:4PDE}. First, we provide details concerning the spaces we will work in, before stating the main result for \eqref{eqn:2} in Main Theorem \ref{thm:mainmain}. We also discuss how these results are linked to the stability conditions for the Mckean-Vlasov equation in \cite{CGPS20} and the grid cell model in \cite{CHS}. \subsection{Characterisation of the stationary states} Henceforth, we denote $\mathbb{N}=\{0,1,\dots\}$, $\mathbb{R}_+^*=(0,+\infty)$, \[ \bar \rho(x) = \int_{0}^{+\infty} s\rho(x,s)\diff s \] the activity average of $\rho$ at $x \in \mathbb T^d $, and \[ W_0 = \int_{\mathbb T^d} W(x) \diff x. \] By setting the left hand side of \eqref{eqn:2} to zero and integrating with respect to $s$, we obtain that the stationary states satisfy \[ \sigma \partial_s \rho(x,s) = - \left(s - \Phi\left( W \ast \bar \rho (x) + B \right) \right) \rho(x,s), \] using the no-flux boundary condition. Here, $u\ast v : x\mapsto \int_{\mathbb T^d} u(x-y)v(y)\diff y$ denotes the convolution of $u$ and $v$ on $\mathbb T^d$. Thus, the stationary states must solve \begin{equation}\label{eq:stationarystateequ} \rho(x,s) = \dfrac{1}{Z_\rho} \exp\left( - \dfrac{ \big( s - \Phi\left( W\ast \bar \rho (x) + B \big) \right)^2}{2\sigma} \right), \end{equation} with, owing to \eqref{eqn:mass_x}, \[ Z_\rho = L^d \int_{0}^{+\infty} \exp\left( - \dfrac{ \big( s - \Phi\left( W\ast \bar \rho (x) + B \big) \right)^2}{2\sigma} \right) \diff s. \] For the sake of simplicity, let us denote \[ \kappa = \dfrac{1}{\sigma}, \qquad \Phi_{\bar\rho}(x) = \Phi(W \ast \bar\rho(x) + B),\qquad \Phi'_{\bar \rho}(x) = \Phi'(W \ast \bar\rho(x) + B). \] Then, any stationary state must be a zero of the functional \begin{equation*} \mathcal G(\rho ,\kappa) = \rho -\dfrac{1}{Z_\rho}\e^{ -\kappa \frac{ \left( s - \Phi_{\bar\rho} \right)^2}{2}} , \end{equation*} with \[ Z_\rho = L^d \int_{0}^{+\infty} \e^{ -\kappa \frac{ \left( s - \Phi_{\bar\rho} \right)^2}{2}} \diff s. \] Consequently, the average $x\mapsto\bar \rho(x)$ is a zero of the functional \begin{equation}\label{eq:onecompfunctional} \bar{\mathcal G}(\bar \rho ,\kappa) = \bar\rho -\dfrac{1}{Z_\rho} \int_0^{+\infty} s \e^{ -\kappa \frac{ \left( s - \Phi_{\bar\rho} \right)^2}{2}}\diff s, \end{equation} with \[ Z_\rho = L^d \int_{0}^{+\infty} \e^{ -\kappa \frac{ \left( s - \Phi_{\bar\rho} \right)^2}{2}} \diff s. \] A quick check will confirm that we can fully characterise the stationary states of \eqref{eqn:2} through considering the zeros of the functional for the averages, $\bar{\mathcal G}(\bar \rho ,\kappa)$. Finally, the short calculation \begin{align*} \rho(x,0) &= - \int_{0}^{+\infty} \dfrac{\partial \rho}{\partial s}(x,s) \diff s = \kappa \int_{0}^{+\infty} (s-\Phi_{\bar \rho}(x) )\rho(x,s) \diff s \\ & = \kappa \Big( \underbrace{\int_{0}^{+\infty} s \rho(x,s)\diff s}_{ = \bar\rho(x)} - \Phi_{\bar\rho}(x) \underbrace{\int_{0}^{+\infty} \rho(x,s)\diff s }_{=\frac1{L^d}} \Big), \end{align*} yields the following relation between the value at $s=0$ of the stationary states and the mean values, \begin{equation}\label{eqn:rho_0} \rho(x,0) = \left(\bar \rho(x) - \dfrac{\Phi_{\bar\rho}(x)}{L^d} \right)\kappa. \end{equation} This relation will appear throughout the manuscript. \subsection{Hilbert spaces and Fourier bases}\label{sec:basis} Throughout this manuscript, we use either the space $L^2(\mathbb T^d)$ or the Hilbert space $L^2_S(\mathbb T^d)$ of coordinate-wise even $L^2$-functions on $\mathbb T^d := \sfrac{\mathbb{R}^d}{( L \mathbb{Z})^d}, L>0$, that is to say \[ L^2_S(\mathbb T^d) = \left\{ \ u\in L^2(\mathbb T^d) \ \big| \ \forall i\in\llbracket 1,d \rrbracket,\ u(x_1,\dots,-x_i,\dots,x_d) = u(x_1,\dots,x_i,\dots,x_d)\ \ \text{a.e. in } \mathbb T^d \ \right\}. \] Similarly, for any $m\in \mathbb{N}$, we define the coordinate-wise even Sobolev space $H^m_S(\mathbb T^d) = H^m(\mathbb T^d)\cap L^2_S(\mathbb T^d)$. Following the reasons stated in \cite[Section 2.2]{FSV2022}, we will consider a connectivity kernel $W\in H^m_S(\mathbb T^d)$, with $2m > d$. The advantage is twofold. First, the smoothing property of the convolution and Sobolev embeddings imply that $W\ast h$ will be continuous for all $h\in L^2(\mathbb T^d)$. Second, it avoids pathological connectivity kernels with singularities. While working in the space $L^2_S(\mathbb T^d)$, we use the Hilbert basis $ \left(\omega_k\right)_{k\in \mathbb{N}^d} $ defined by \begin{equation}\label{eqn:HilbertElements} \omega_k(x) = \dfrac{\Theta(k)}{L^{\frac d2}} \prod_{i=1}^{d} \omega_{k_i}(x_i), \end{equation} with \[ \omega_{k_i}(x) = \left\{\begin{array}{ll} \cos\left(\dfrac{2\pi k_i}{L} x_i\right) & \mathrm{if}\ k_i > 0, \\ 1 & \mathrm{if}\ k_i=0, \end{array}\right. \qquad \mathrm{and} \qquad \Theta(k) = \prod_{i=1}^{d} \sqrt{2 - \delta_{k_i,0}}, \] where the Kronecker delta $\delta_{i,j}$ is 0 if $i\neq j$ and 1 if $i=j$. We assume that the connectivity function $W$ lies in $L^2_S(\mathbb T^d)$. The Fourier modes of $W$ are \[\tilde W(k) = \pscal{ W }{ \omega_k} = \int_{\mathbb T^d} W(x)\omega_k(x)\diff x , \quad k\in\mathbb{N}^d, \] Moreover, since $W\in L_S^2(\mathbb T^d)$, for all $g\in L_S^2(\mathbb T^d)$, \begin{equation*} W\ast g (x) = \int_{\mathbb T^d} W(x-y) g(y) \diff y = \dfrac{L^\frac d2}{\Theta(k)} \sum_{k\in \mathbb{N}^d} \tilde W(k) \tilde g(k) \omega_k(x). \end{equation*} In particular, for all $k\in\mathbb{N}^d$, \begin{equation}\label{trigo} W\ast \omega_k (x) = \dfrac{L^{\frac d2}\tilde W(k)}{\Theta(k)} \omega_k(x) . \end{equation} When we work in the larger space $L^2(\mathbb T^d)$, we extend the above Hilbert basis to multi-indexes in $\mathbb{Z}^d$: we consider the base $ \left(\omega_k\right)_{k\in \mathbb{Z}^d} $ defined by \eqref{eqn:HilbertElements}, \[ \omega_{k_i}(x) = \left\{\begin{array}{ll} \cos\left(\dfrac{2\pi k_i}{L} x_i\right) & \mathrm{if}\ k_i > 0, \\ 1 & \mathrm{if}\ k_i=0,\\ \sin\left(\dfrac{2\pi k_i}{L} x_i\right) & \mathrm{if}\ k_i < 0. \\ \end{array}\right. \qquad \mathrm{and} \qquad \Theta(k) = \prod_{i=1}^{d} \sqrt{2 - \delta_{k_i,0}}. \] The formula for the convolution with basis vectors then becomes \begin{equation}\label{trigo2} W\ast \omega_k (x) = \dfrac{L^{\frac d2}\tilde W(|k|)}{\Theta(|k|)} \omega_k(x), \quad |k| = (|k_1|,\dots,|k_d|) . \end{equation} \subsection{Assumptions and main theorem} We assume the following in order to perform the bifurcation analysis. \begin{hyp}[On $\Phi$, $W$ and $B$]\label{as:1} There exists $m\in\mathbb{N}$, such that $2m > d$ and $W\in H^m_S(\mathbb T^d)$. The function $\Phi$ is continuous on $\mathbb{R}$, $C^3$ on $\mathbb{R}^*$, increasing on $\mathbb{R}_+$ and $\Phi\equiv 0$ on $\mathbb{R}_-$. Moreover, $B > 0$ and $W_0 = \int_{\mathbb T^d} W(y)\diff dy < 0$. \end{hyp} We remind the reader that the existence and uniqueness of spatially homogeneous stationary states of \eqref{eqn:2} was proven in \cite[Prop. 3.1]{CHS} and hold under (less restrictive assumptions than) Assumption \ref{as:1}. Let $\rho_\infty^\kappa: s\mapsto \rho_\infty^\kappa(s)$ be the constant in space steady state associated with the parameter $\kappa>0$. Note that in particular $ \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)=0$. Whenever necessary, the dependence on $\kappa$ of $\rho_\infty$ will be highlighted with a superscript $\kappa$. Otherwise we will drop it. Denote \[ \Phi_0^\kappa = \Phi(W_0 \bar\rho_\infty^\kappa + B),\qquad (\Phi^\kappa_0)' = \Phi'(W_0 \bar\rho_\infty^\kappa + B), \qquad (\Phi^\kappa_0)'' = \Phi''(W_0 \bar\rho_\infty^\kappa + B). \] Then, applying \eqref{eqn:rho_0} to the spatially homogeneous steady state $\rho=\rho_\infty^\kappa$, we get the relation \begin{equation}\label{eqn:rho_inf_0} \rho_\infty^\kappa(0) = \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa. \end{equation} This relation is used throughout the manuscript. We are now ready to state the main result of this work. The different parts of the proofs are found in Section \ref{sec:bifurcations}. \begin{maintheorem}\label{thm:mainmain} Grant Assumption \ref{as:1}, and let $\kappa^*\in\left(\tfrac{2|W_0|^2}{L^{2d}\pi B^2},+\infty\right)$. Assume that there exists a unique $k^*\in\mathbb{N}^d$ such that \begin{equation}\label{eqn:W_tilde_equal_0} \dfrac{\tilde W (k^*)}{\Theta(k^*)} = \dfrac{1}{L^{\frac d2} (\Phi_0^{\kappa^*})' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^{\kappa^*} \left(\bar \rho_\infty^{\kappa^*} - \dfrac{ \Phi_0^{\kappa^*}}{L^d}\right)\kappa^* \right)}. \end{equation} Assume also that $ (\Phi_0^{\kappa^*})''\geq 0$. Then $(\bar\rho_\infty^{\kappa^*},\kappa^*)$ is a bifurcation point of $\bar{\mathcal G}(\bar\rho,\kappa)=0$. Moreover, in a neighbourhood $U\times (\kappa^*-\varepsilon,\kappa^*+\varepsilon)$ of $(\bar\rho_\infty^{\kappa^*},\kappa^*)$ in $L^2_S(\mathbb T^d)\times \mathbb{R}_+^*$, the average of the stationary state is either of the form $(\bar \rho_\infty^\kappa,\kappa)$ or on the non-homogeneous solution curve \begin{equation*} \{\ (\bar \rho_{\kappa(z)},\kappa(z)) \ | \ z\in(-\delta,\delta), \ (\bar \rho_{\kappa(0)},\kappa(0))=(\bar \rho_\infty^{\kappa^*},\kappa^*),\ \delta>0 \ \}, \end{equation*} with \begin{equation*} \bar \rho_{\kappa(z)}(x) = \bar \rho_\infty^{\kappa(z)} + z \omega_{k^*}(x) + o(z), \qquad x\in \mathbb T^d. \end{equation*} \end{maintheorem} The uniqueness condition on $k^*$ can be relaxed: if the bifurcation problem exhibits symmetries and $k^*$ is unique up to appropriate geometrical transformations, equivariant bifurcation theory provides the same result. We treat this general case in Section \ref{sec:high_dim}. More information about the shape of the branch can be found in Theorem \ref{thm:main_2} for the case of a unique $k^*$ and in Section \ref{sec:high_dim} for the case where it is unique up to symmetries. A relaxed convexity assumption on $\Phi$ can be found in Remark \ref{rem:convexity}. The right hand side of \eqref{eqn:W_tilde_equal_0} is strictly positive, as stated in the following lemma. \begin{lemma}\label{lm:g_eta} We have the relation \[ \dfrac1{L^d} - L^d\bar \rho_\infty \left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa = \frac{1}{L^d} g\left(\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa\right), \] where \[ g(\eta):= 1-\frac{2}{\sqrt{\pi}}\frac{\exp (-\eta^2)}{1+\erf (\eta)}\left[\frac{1}{\sqrt{\pi}}\frac{\exp (-\eta^2)}{1+\erf (\eta)}+\eta\right], \] with $\eta = \sqrt{\frac{\kappa}{2}} \Phi_0^\kappa$. Furthermore, the function $\eta\mapsto g(\eta)$ is increasing and for all $\eta \in \mathbb{R}_+$, \[ 1-\frac{2}{\pi} \leqslant g(\eta) < 1. \] \end{lemma} The function $\erf(z)$ is the standard error function, $\erf(z):=\tfrac{2}{\sqrt\pi}\int_{0}^{z} \e^{-s^2}\diff s$, and the proof of the lemma can be found in the appendix. As the right-hand side of \eqref{eqn:W_tilde_equal_0} is positive, there exists no bifurcations of the form described in the main theorem if all the Fourier modes of $W$ are negative. A similar conclusion was drawn in \cite{CGPS20} for the McKean--Vlasov equation, however with opposite sign. They introduce the notion of H-stability: a function $W\in L^2(\mathbb T^d)$ is said to be H-stable if all its Fourier modes are non-negative. The authors of \cite{CGPS20} prove, under mild assumptions, that for H-stable potentials there exists a unique stationary state which is globally asymptotically stable. In the setting of this manuscript, if $-W$ is H-stable, the hypotheses of Theorem \ref{thm:mainmain} cannot be satisfied. Proving that the constant in $x$ stationary state $\rho_\infty$ is asymptotically stable for the time-dependent system \eqref{eqn:2} when $-W$ is H-stable is an open question and the energy method utilised in \cite{CGPS20} is not applicable here. However, there exists a result on the linear stability of \eqref{eqn:2}. \subsection{Linear stability}\label{sec:linearstability} In \cite[Theorem 3.6]{CHS} it is shown that the homogeneous in space stationary state $\rho_\infty^\kappa$ is linearly asymptotically stable in $L^2\big({\mathbb T}^d \times \mathbb{R}_+\big)$ as long as \begin{align} \label{eq:linearstabilitycond1} \dfrac{\tilde W (k)}{\Theta(k)} < \frac{1}{\kappa (\Phi_0^\kappa)' L^{\frac d2} \int_0^\infty (s-L^d\bar{\rho}_\infty^\kappa)^2 \rho_\infty^\kappa \diff s} \end{align} for all $k \in \mathbb{N}$. In the following lemma, we show that when replacing the inequality with an equality in \eqref{eq:linearstabilitycond1}, yielding the value of $\kappa$ where $\rho_\infty$ is no longer linearly asymptotically stable, \eqref{eq:linearstabilitycond1} is identical to the bifurcation condition \eqref{eqn:W_tilde_equal_0}. For many problems, linear stability of a stationary state $\bar\rho_\infty$ is equivalent to the spectrum of $D_{\bar\rho} \bar {\mathcal G} (\bar\rho_\infty,\kappa)$ being located in the left complex half-space (see the discussion in \cite[Sec. I.7]{Kielhofer2012}). For some equations, for example semilinear equations, it is even possible to get nonlinear stability out of the study of the spectrum of $D_{\bar\rho} \bar {\mathcal G} (\bar\rho_\infty,\kappa)$. However, as \eqref{eqn:2} is both nonlinear and non-local, we cannot apply this general theory. The rigorous connection between the eigenvalues and linear stability for \eqref{eqn:2} is provided in the following lemma. \begin{lemma}\label{rem:linear} The linear asymptotic stability of $\rho_\infty$ in $L^2\big(\mathbb T^d \times \mathbb{R}_+\big)$ is lost at the smallest $\kappa$ for which there exists a $k\in\mathbb{N}^d$ such that $\tfrac{\tilde W (k)}{\Theta(k)}$ satisfies \eqref{eqn:W_tilde_equal_0}. \end{lemma} \begin{proof} The result follows by checking that the right hand side of \eqref{eq:linearstabilitycond1} is identical the right hand side of \eqref{eqn:W_tilde_equal_0}. Remember that $\int_0^\infty \rho_\infty \diff s = \frac{1}{L^d}$. Then, the denominator of the right hand side in \eqref{eq:linearstabilitycond1} is \begin{align*} L^{\frac d2} \kappa \int_0^\infty (s-L^d\bar{\rho}_\infty)^2 \rho_\infty \diff s & = L^{\frac d2} \kappa \left( \int_0^\infty s^2 \rho_\infty \diff s - 2L^d \bar{\rho}_\infty \int_0^\infty s \rho_\infty \diff s + L^{2d}\bar{\rho}_\infty ^2 \int_0^\infty \rho_\infty \diff s\right) \\ & =L^{\frac d2} \left(\kappa \int_0^\infty s^2 \rho_\infty \diff s - L^d \kappa \bar{\rho}_\infty^2 \right) \\ & = L^{\frac d2}\left( 2 \int_0^\infty \tfrac{\kappa}{2}(s- \Phi_0^\kappa)^2 \rho_\infty \diff s + \kappa 2 \Phi_0^\kappa\bar{\rho}_\infty -\frac{\kappa}{L^d} (\Phi_0^\kappa)^2 - L^d\kappa \bar{\rho}_\infty^2\right). \end{align*} Remembering the expression for $\rho_\infty$ in \eqref{eq:stationarystateequ} and for $\rho_\infty(0)$ in \eqref{eqn:rho_inf_0}, we compute the integral as follows: \begin{align*} 2 \int_0^\infty \tfrac{\kappa}{2}(s- \Phi_0^\kappa)^2 \rho_\infty \diff s & = \sqrt{\frac{2}{\kappa}} \frac{1}{2} \left(\frac{\sqrt{\pi}}{Z_\rho} \left(1 + \erf \left(\tfrac{\sqrt{\kappa} \Phi_0^\kappa}{\sqrt{2}} \right) \right) - 2\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa \rho_\infty (0) \right) \\ & = \sqrt{\frac{2}{\kappa}} \frac{1}{2}\left(\frac{\sqrt{2\kappa}}{L^d} - 2\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa \kappa \left(\bar{\rho}_\infty - \frac{ \Phi_0^\kappa}{L^d} \right) \right) \\ & = \frac{1}{L^d} - \Phi_0^\kappa \kappa \left(\bar{\rho}_\infty - \frac{ \Phi_0^\kappa}{L^d} \right). \end{align*} Thus, \begin{align*} L^{\frac d2} \kappa \int_0^\infty (s-\bar{\rho}_\infty)^2 \rho_\infty \diff s & = L^{\frac d2}\left(\frac{1}{L^d} - \Phi_0^\kappa \kappa \left(\bar{\rho}_\infty - \frac{ \Phi_0^\kappa}{L^d} \right) + \kappa 2 \Phi_0^\kappa\bar{\rho}_\infty -\frac{\kappa}{L^d} (\Phi_0^\kappa)^2 -L^d\kappa \bar{\rho}_\infty^2\right) \\ & = L^{\frac d2}\left(\frac{1}{L^d} - L^d\bar{\rho}_\infty \left(\bar{\rho}_\infty-\frac{ \Phi_0^\kappa}{L^d}\right) \kappa \right), \end{align*} which, when multiplied by $ (\Phi_0^\kappa)'$ is identical to the denominator of the right hand side of \eqref{eqn:W_tilde_equal_0}. \end{proof} Note that under Assumption \ref{as:1}, if $-W$ is H-stable, $\rho_\infty$ is a linearly stable stationary state of \eqref{eqn:2} for any $\kappa>0$. \section{Bifurcation analysis}\label{sec:bifurcations} This section is devoted to proving the main Theorem \ref{thm:mainmain}. We start by assuming that the connectivity $W$ is coordinate-wise even on the square, and look for coordinate-wise even, non-homogeneous in space stationary states following \cite{CGPS20}. In Section \ref{sec:high_dim} we consider possible additional symmetries of $W$. But first we provide results concerning the asymptotic behaviour of the homogeneous in space stationary state. \subsection{Asymptotic behaviour of the homogeneous in space stationary state} \begin{lemma}\label{lm:kappa_small} Grant Assumption \ref{as:1}. Then, we have \[\forall\kappa\in\left(0,\dfrac{2|W_0|^2}{L^{2d}\pi B^2}\right], \qquad \bar{\rho}_\infty^\kappa = \frac1{L^d}\sqrt{\dfrac{2}{\kappa\pi}}, \qquad \bar{\rho}_\infty^\kappa \geqslant \dfrac{B}{|W_0|}, \qquad \Phi_0^\kappa = (\Phi_0^{\kappa})' = 0. \] and \[ \forall\kappa\in\left(\dfrac{2|W_0|^2}{L^{2d}\pi B^2},+\infty\right), \qquad \bar{\rho}_\infty^\kappa < \dfrac{B}{|W_0|}. \] \end{lemma} \begin{proof} For a given value $\kappa$, by Proposition 3.1 of \cite{CHS}, we know that there exists a unique stationary state which is constant in space and thus a unique associated mean value $\bar\rho_\infty^\kappa$. If \begin{equation}\label{condkappa} \kappa \leqslant \dfrac{2|W_0|^2}{L^{2d}\pi B^2}, \end{equation} then we have \[ \frac1{L^d}\sqrt{\dfrac{2}{\kappa\pi}} \geqslant \dfrac{B}{|W_0|} \qquad \mathrm{and} \qquad \Phi\left(W_0 \frac1{L^d}\sqrt{\dfrac{2}{\kappa\pi}} + B \right) = 0. \] Then, if $ \Phi_0^\kappa=0$, \[ \bar \rho_\infty^\kappa = \dfrac{\displaystyleplaystyle \int_0^{+\infty} s \e^{-\frac\kappa2 s^2}\diff s}{L^d\sqrt{\frac{\pi}{2\kappa}}(1+\mathrm{erf}(0))} = \dfrac{1}{L^d\kappa} \sqrt{\dfrac{2\kappa}{\pi}} = \frac1{L^d}\sqrt{\dfrac{2}{\kappa \pi}}. \] Hence, if \eqref{condkappa} holds, then the unique constant in space smooth stationary state satisfies \[ \bar \rho_\infty^\kappa = \frac1{L^d}\sqrt{\dfrac{2}{\kappa\pi}}. \] Now, let $\kappa\in \left( \tfrac{2|W_0|^2}{L^{2d}\pi B^2},+\infty\right)$. Assume that \[ \bar{\rho}_\infty^\kappa \geqslant \dfrac{B}{|W_0|}. \] Then $ \Phi_0^\kappa=0$ and we have \[ \bar \rho_\infty^\kappa =\frac1{L^d} \sqrt{\dfrac{2}{\kappa \pi}} = \frac1{L^d}\sqrt{\dfrac2\pi} \sqrt{\dfrac{1}{\kappa}} <\frac1{L^d} \sqrt{\dfrac2\pi} \sqrt{\dfrac{L^{2d}\pi B^2}{2 |W_0|^2}} = \dfrac{B}{|W_0|}, \] which is a contradiction. \end{proof} \begin{lemma}\label{lm:der_bar_rho} Grant Assumptions \ref{as:1}. Then the function $\kappa\mapsto \bar\rho_\infty$ is differentiable and for all $\kappa\in(0,+\infty)$, \[ \dfrac{\diff \bar\rho_\infty}{\diff\kappa}(\kappa) = - \dfrac{1}{2\kappa } \dfrac{(1 + L^d \Phi_0^\kappa \bar\rho_\infty\kappa)\left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right)}{\left(1 - (\Phi_0^\kappa)' \left(\dfrac1{L^d} - L^d\bar\rho_\infty\left(\bar\rho_\infty-\dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa\right)W_0\right)}. \] \end{lemma} \begin{proof} The mean $\bar\rho_\infty$ of the constant stationary state is a solution to \begin{equation*} \tilde G(\bar\rho_\infty,\kappa) = 0, \end{equation*} where for all $y\in \mathbb{R}_+^*$ \[ \tilde G(y,\kappa) = y -\dfrac{1}{Z_\rho} \int_0^{+\infty} s \e^{ -\kappa \frac{ \left( s - \Phi(W_0 y + B) \right)^2}{2}}\diff s, \qquad Z_\rho = L^d \int_{0}^{+\infty} \e^{ -\kappa \frac{ \left( s - \Phi(W_0 y + B) \right)^2}{2}} \diff s. \] We have \begin{equation*} \dfrac{\partial \tilde G}{\partial y}(\bar\rho_\infty,\kappa) = \left(1 - (\Phi_0^\kappa)' \left(\dfrac1{L^d} - L^d\bar\rho_\infty\left(\bar\rho_\infty-\dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa\right)W_0\right), \end{equation*} and \begin{equation*} \dfrac{\partial \tilde G}{\partial \kappa}(\bar\rho_\infty,\kappa) = \dfrac{\partial \bar{\mathcal G}}{\partial \kappa}(\bar\rho_\infty,\kappa) = \dfrac{1}{2\kappa } (1 + L^d \Phi_0^\kappa \bar\rho_\infty\kappa)\left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right). \end{equation*} By the implicit function theorem, if \[ \dfrac{\partial \tilde G}{\partial y}(\bar\rho_\infty,\kappa) \neq 0, \] then \begin{equation*} \dfrac{\diff \bar\rho_\infty}{\diff\kappa}(\kappa) = -\dfrac{\dfrac{\partial \tilde G}{\partial \kappa}(\bar\rho_\infty,\kappa)}{\dfrac{\partial \tilde G}{\partial y}(\bar\rho_\infty,\kappa)}. \end{equation*} But by Assumption \ref{as:1}, $W_0<0$ and by Lemma \ref{lm:g_eta}, \[ \dfrac1{L^d} - L^d\bar\rho_\infty\left(\bar\rho_\infty-\dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa > 0, \] so $\dfrac{\partial \tilde G}{\partial y}(\bar\rho_\infty,\kappa) > 0$. Hence the result. \end{proof} \begin{lemma}\label{lm:inf_bound_rho} Grant Assumption \ref{as:1}. Then, there exists a value $\rho^*\in\mathbb{R}_+^*$ such that for all $\kappa\in\mathbb{R}_+^*$, $\bar\rho_\infty^\kappa \geqslant \rho^*$ and $\lim_{\kappa\to+\infty} \bar\rho_\infty^\kappa = \rho^*$. \end{lemma} \begin{proof} Assume there exists a sequence $\kappa_n$ such that \[ \lim_{n\to+\infty} \bar\rho_\infty^{\kappa_n} = 0. \] Then we have \[ \lim_{n\to+\infty} \Phi_0^{\kappa_n} = \Phi(B) > 0. \] But recall that \[ \rho_\infty^{\kappa_n}(0) = \left(\bar\rho_\infty^{\kappa_n}-\dfrac{ \Phi_0^{\kappa_n}}{L^d}\right)\kappa_n. \] For $n$ large enough, $ \rho_\infty^{\kappa_n}(0) < 0 $ which is impossible. Moreover, by Lemma \ref{lm:der_bar_rho} the function $\kappa\mapsto \bar\rho_\infty^\kappa$ is decreasing. Hence the result. \end{proof} \begin{lemma}\label{lm:equiv} Grant Assumption \ref{as:1}. Then, the quantity $\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}$ converges exponentially fast towards 0 when $\kappa$ tends to $+\infty$. Moreover, there exists $\Phi_* > 0$ such that $\lim_{\kappa\to+\infty} \Phi_0^\kappa(\kappa) = \Phi_*$ and \[ \bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\,\underset{\kappa\to+\infty}{\sim}\, \dfrac{\e^{-\frac{\kappa}2\Phi_*^2}}{L^d\sqrt{2\pi\kappa}}. \] \end{lemma} \begin{proof} By \eqref{eqn:rho_bar_eta}, \begin{equation}\label{eqn:equivalent} \bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d} = \dfrac{\sqrt2 \e^{-\frac\kappa2 (\Phi_0^\kappa)^2}}{ L^d \sqrt{\pi\kappa} \left(1+\mathrm{erf}\left(\tfrac{ \Phi_0^\kappa\sqrt{\kappa}}{\sqrt2}\right)\right)}. \end{equation} Since $ \Phi_0^\kappa$ is bounded, we have \begin{equation}\label{eqn:first_conv_0} \lim_{\kappa\to+\infty} \left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right) = 0. \end{equation} If $\liminf_{\kappa \to +\infty} \Phi_0^\kappa = 0$, then $\liminf_{\kappa \to +\infty} \bar\rho_\infty = 0$, which contradicts Lemma \ref{lm:inf_bound_rho}. Therefore, the convergence in \eqref{eqn:first_conv_0} is exponentially fast. Last, by Lemma \ref{lm:der_bar_rho}, $\bar \rho_\infty$ is decreasing, thereby $ \Phi_0^\kappa$ is non-increasing. But we also know that $ \Phi_0^\kappa$ is bounded from above by $\Phi(B)$. Hence, there exists a positive limit $\Phi_* > 0$ for $ \Phi_0^\kappa$ when $\kappa\to+\infty$. Using this fact and the properties of the error function in \eqref{eqn:equivalent}, we obtain the desired equivalent. \end{proof} \begin{remark} Since $\Phi_* = L^d \rho^*$, given a function $\Phi$ it may be possible to find those values explicitly. For example, if $\Phi(x)= \max(x,0)$ is the ReLU function, then \[ W_0 \rho^* + B = L^d \rho^*, \] and thus \[ \rho^* = \dfrac{B}{L^d+|W_0|},\qquad \Phi_* = \dfrac{L^d B}{L^d+|W_0|}. \] \end{remark} \subsection{Fréchet derivatives of the functional} In order to establish the existence of bifurcating branches from the branch of the spatially homogeneous stationary state, we need to compute a few Fréchet derivatives. \begin{lemma} Grant Assumption \ref{as:1}. Then $\bar G$ is $C^3$ Fréchet differentiable on $L^2(\mathbb T^d)\times (0,+\infty)$. Let $h_1\in L^2(\mathbb T^d)$, the first order Fréchet derivatives of $\bar{\mathcal G}$ at a point $(\bar \rho , \kappa)$ are \begin{equation*} D_{\bar \rho} \bar{\mathcal G}(\bar \rho ,\kappa)[h_1]\ =\ h_1 - \Phi'_{\bar\rho} \left (\dfrac 1 {L^d} - L^d\big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right)\kappa\right) \, W \ast h_1 \end{equation*} and \begin{equation}\label{eqn:D_kappa_full} D_\kappa \bar{\mathcal G}(\bar\rho ,\kappa) =\dfrac{1}{2\kappa }\bigg( \bar{\mathcal G}(\bar \rho,\kappa) + \left(1 + L^d\Phi_{\bar\rho}\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big)\kappa \right)\left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d}\right) \bigg) . \end{equation} Let $\rho_\infty^\kappa$ be a smooth constant in space steady state of \eqref{eqn:2} associated with a parameter $\kappa$. Then, \begin{equation*} D_{\bar \rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa ,\kappa)[h_1]\ =\ h_1 - (\Phi_0^\kappa)' \left (\dfrac 1 {L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) \, W \ast h_1, \end{equation*} and \begin{equation*} D_\kappa \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) = \dfrac{1}{2\kappa }\left(1 + L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) . \end{equation*} \end{lemma} \begin{proof} To prove smoothness of the functional, let us remark first that by Taylor expansion of $\Phi$, which is $C^3$, we have for all $h\in L^2(\mathbb T^d)$, \[ \Phi(W\ast(\bar\rho + h) + B) = \Phi_{\bar\rho} + \Phi_{\bar\rho}' W\ast h + \frac12\Phi_{\bar\rho}'' (W\ast h )^2 + \frac16\Phi_{\bar\rho}''' (W \ast h)^3 + o\left( \norme{ (W \ast h)^3 } \right).\] Note then, that for all $i\in\mathbb{N}$, $i\geq 1$, since $W, W\ast h \in H^k(\mathbb T^d) \subset L^\infty(\mathbb T^d)$ and \[ \norme{(W \ast h)^i}_{L^2} = \sqrt{ \int_{\mathbb T^d} \left(\int_{\mathbb T^d} W(x-y) h(y) \diff y\right)^{2i} \diff x } \leqslant \norme{W}_{L^\infty}^i \norme{h}_{L^1}^i \leqslant L^{\frac {id}2} \norme{W}_{L^\infty}^i \norme{h}_{L^2}^i. \] Hence the map $\bar\rho\mapsto \Phi(W\ast\bar\rho + B)$ is $C^3$ Fréchet differentiable. Denote $J(\bar\rho,\kappa | s) = \e^{ -\kappa \frac{ \left( s - \Phi_{\bar\rho} \right)^2}{2}}$. Since, by Sobolev embedding and properties of the convolution, $W\ast \bar \rho \in C^0(\mathbb T^d)$ and \[ \norme{W\ast \bar\rho}_{L^\infty} \leq \norme{W}_{L^\infty} \norme{ \bar \rho}_{L^1} \leqslant L^{\frac d2}\norme{W}_{L^\infty} \norme{ \bar \rho}_{L^2}, \] then for any $s,\kappa\in\mathbb{R}_+$, the functional $J(\cdot,\kappa|s)$ is locally bounded in $L^2(\mathbb T^d)$ and thus the functionals \[ (\bar\rho,\kappa) \mapsto \int_0^{+\infty} s J(\bar\rho,\kappa)\diff s \quad \mathrm{and} \quad (\bar\rho,\kappa) \mapsto \int_{0}^{+\infty} J(\bar\rho,\kappa) \diff s \] are Fréchet differentiable by dominated convergence. Hence, $\bar{\mathcal G}$ is $C^3$ Fréchet differentiable by arithmetic operations where all products and quotients involve strictly positive $C^3$ functions. Note that \[ \dfrac{1}{Z_\rho} \int_0^{+\infty} s \exp\left( - \kappa\dfrac{ \big(s - \Phi( W \ast \bar \rho + B )\big)^2 }{2} \right)\diff s = \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa). \] Let $h_1\in L^2(\mathbb T^d)$. The first derivative in $\bar \rho$ is \begin{multline*} D_{\bar \rho} \bar{\mathcal G}(\bar \rho ,\kappa)[h_1]\ =\ h_1 - \dfrac{\Phi_{\bar\rho}'}{Z_\rho} \int_{0}^{+\infty} s e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \kappa (s - \Phi_{\bar\rho} ) \diff s\, W \ast h_1 \\ + L^d( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)) \dfrac{\Phi_{\bar\rho}'}{Z_\rho} \int_0^{+\infty} e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \kappa (s - \Phi_{\bar\rho} ) \diff s\, W \ast h_1 \end{multline*} By integration by parts \[ \int_{0}^{+\infty} s e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \kappa (s - \Phi_{\bar\rho} ) \diff s = \dfrac{Z_\rho} {L^d}, \] and by direct integration \[ \int_0^{+\infty} e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \kappa (s - \Phi_{\bar\rho} ) \diff s = e^{-\frac\kappa2 \Phi_{\bar\rho}^2}. \] Hence, using that \[ \dfrac{e^{-\frac\kappa2 \Phi_{\bar\rho}^2}}{Z_\rho} = \rho(x,0) = \left(\bar \rho(x) - \dfrac{\Phi_{\bar\rho}(x)}{L^d} \right)\kappa, \] we obtain \begin{equation*} D_{\bar \rho} \bar{\mathcal G}(\bar \rho ,\kappa)[h_1]\ =\ h_1 - \Phi_{\bar\rho}' \left (\dfrac 1 {L^d} - L^d( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right)\kappa\right) \, W \ast h_1 \end{equation*} The first derivative in $\kappa$ is \begin{align*} D_\kappa \bar{\mathcal G}(\bar\rho ,\kappa) =& \dfrac{1}{Z_\rho}\bigg( \int_0^{+\infty} \dfrac s 2 \e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \big(s - \Phi_{\bar\rho}\big)^2 \diff s \\ &-\ L^d\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big) \int_{0}^{+\infty}\dfrac 12 \e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \big(s - \Phi_{\bar\rho}\big)^2 \diff s \bigg). \end{align*} We can write it in the form \begin{equation*} D_\kappa \bar{\mathcal G}(\bar \rho,\kappa)= \dfrac{1}{2Z_\rho}\int_0^{+\infty} e^{- \kappa\frac{ (s - \Phi_{\bar\rho}(x))^2 }{2}} (s - \Phi_{\bar\rho}(x))^2 \big(s- L^d \big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big) \big)\diff s . \end{equation*} By integration by part, we have \begin{align*} D_\kappa \bar{\mathcal G}(\bar \rho,\kappa) = \, & \dfrac{1}{2\kappa Z_\rho}\bigg( \left[ -e^{- \kappa\frac{ (s - \Phi_{\bar\rho}(x))^2 }{2}} (s - \Phi_{\bar\rho}(x))\big(s-L^d\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big)\big) \right]^{+\infty}_0\\ & + \int_0^{+\infty} e^{- \kappa\frac{ (s - \Phi_{\bar\rho}(x))^2 }{2}} \big(2s - \Phi_{\bar\rho}(x)- L^d \big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big) \big)\diff s \bigg). \end{align*} Moreover, \begin{align*} \dfrac{1}{Z_\rho}\bigg[ -e^{- \kappa\frac{ (s - \Phi_{\bar\rho}(x))^2 }{2}} \big(s - \Phi_{\bar\rho})&(s-L^d\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big)\big) \bigg]^{+\infty}_0 \\ &= L^d \Phi_{\bar\rho}(x) \big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big)\frac{e^{-\frac\kappa2 \Phi_{\bar\rho}(x)^2}}{Z_\rho} \\ & = L^d \Phi_{\bar\rho}(x) \big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big) \rho(x,0)\\ & = L^d\Phi_{\bar\rho}(x) \big(\bar\rho(x) -\bar{\mathcal G}(\bar\rho ,\kappa)(x)\big)\left(\bar \rho(x) - \dfrac{\Phi_{\bar\rho}(x)}{L^d} \right)\kappa, \end{align*} and using the definition of $Z_\rho$, \begin{align*} \dfrac{1}{Z_\rho}\int_0^{+\infty} e^{- \kappa\frac{ (s - \Phi_{\bar\rho})^2 }{2}} \big(2s - \Phi_{\bar\rho}- L^d\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big) \big)\diff s &= 2 \bar\rho - \dfrac{1}{L^d} \big( \Phi_{\bar\rho} + L^d\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big)\big) \\ &= \bar\rho - \dfrac{\Phi_{\bar\rho}}{L^d} +\bar{\mathcal G}(\bar\rho ,\kappa). \end{align*} We obtain \begin{align*} D_\kappa \bar{\mathcal G}(\bar \rho,\kappa) & = \dfrac{1}{2\kappa }\bigg( \bar{\mathcal G}(\bar \rho,\kappa) + \bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} + L^d\Phi_{\bar\rho} \big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d}\right)\kappa \bigg)\\ & = \dfrac{1}{2\kappa }\bigg( \bar{\mathcal G}(\bar \rho,\kappa) + \left(1 + L^d\Phi_{\bar\rho}\big(\bar\rho -\bar{\mathcal G}(\bar\rho ,\kappa)\big)\kappa \right)\left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d}\right) \bigg) . \end{align*} If we apply the first order derivatives on the constant stationary state $(\bar \rho_\infty^\kappa,\kappa)$ we obtain \begin{equation*} D_{\bar \rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa ,\kappa)[h_1]\ =\ h_1 - (\Phi_0^\kappa)' \left (\dfrac 1 {L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) \, W \ast h_1, \end{equation*} and \begin{equation*} D_\kappa \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) = \dfrac{1}{2\kappa } \left(1 + L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right). \end{equation*} \end{proof} We continue by differentiating the formula \eqref{eqn:D_kappa_full} with respect to $\bar\rho$ in order to obtain the cross derivative $D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho,\kappa)$. \begin{lemma}\label{lm:cross} For all $h_1\in L^2(\mathbb T^d)$, the second order Fréchet cross derivative of $\bar{\mathcal G}$ at a point $(\bar \rho , \kappa)$ is \begin{align*} D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho,\kappa) [h_1] = \, & \dfrac1{2\kappa} \bigg( D_{\bar \rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_1] + \left(1+ L^d \Phi_{\bar\rho}(\bar \rho - \bar{\mathcal G}(\bar \rho,\kappa))\kappa\right)\left(h_1 - \dfrac{\Phi'_{\bar\rho}}{L^d} W\ast h_1\right) \\ & +L^d\kappa\left( \Phi_{\bar\rho}' (\bar \rho - \bar{\mathcal G}(\bar \rho,\kappa)) W\ast h_1 + \Phi_{\bar\rho}(h_1-D_{\bar \rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_1]) \right)\left(\bar\rho - \dfrac{\Phi_{\bar\rho}}{L^d}\right) \bigg). \end{align*} Let $\rho_\infty^\kappa$ be a smooth constant in space steady state of \eqref{eqn:2} associated with a parameter $\kappa$. Then, \begin{align*} D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [h_1] = \,& \dfrac1{2\kappa} \bigg( D_{\bar \rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_1] + \left(1+ L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa\right)\left(h_1 - \dfrac{\Phi'_0}{L^d} W\ast h_1\right) \\ & +L^d\kappa\left( (\Phi_0^\kappa)' \bar \rho_\infty^\kappa W\ast h_1 + \Phi_0^\kappa(h_1-D_{\bar \rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_1]) \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) \bigg). \end{align*} \end{lemma} In the same way, we can compute the second and third order derivatives in $\bar\rho$. Denote \begin{align*} \Phi''_{\bar\rho} &= \Phi''( W\ast \bar\rho + B ), \qquad\qquad\qquad (\Phi_0^\kappa)'' = \Phi''( W_0 \bar\rho_\infty^\kappa + B ),\\ \Phi'''_{\bar\rho} &= \Phi'''( W\ast \bar\rho + B ),\qquad \mathrm{and} \qquad (\Phi^\kappa_0)''' = \Phi'''( W_0 \bar\rho_\infty^\kappa + B ). \end{align*} We have the following lemma. \begin{lemma}\label{lm:third} Let $\rho_\infty^\kappa$ be a smooth constant in space steady state of \eqref{eqn:2} associated with a parameter $\kappa$. Then, for all $h_1, h_2, h_3\in L^2(\mathbb T^d)$, \begin{align*} D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,&\kappa) [h_1, h_2] \\ = &\, - (\Phi^\kappa_0)'' \left (\dfrac 1 {L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa\right) \, W \ast h_1 \, W \ast h_2 \\ & + (\Phi^\kappa_0)' L^d \left( \big(h_2 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_2]\big) \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) + \bar \rho_\infty^\kappa \left(h_2 - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast h_2 \right) \right)\kappa\, W \ast h_1, \end{align*} and \begin{align*} D^3_{\bar\rho \bar\rho \bar\rho} &\bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [h_1, h_2, h_3] \\ = &\, - (\Phi^\kappa_0)''' \left (\dfrac 1 {L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa\right) \, W \ast h_1 \, W \ast h_2 \, W \ast h_3 \\ &+ (\Phi^\kappa_0)'' L^d \left( \big(h_3 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_3]\big) \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) + \bar \rho_\infty^\kappa\left(h_3 - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast h_3 \right) \right)\kappa\, W \ast h_1 \, W \ast h_2\\ & + (\Phi^\kappa_0)'' L^d \left( \big(h_2 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_2]\big) \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) + \bar \rho_\infty^\kappa \left(h_2 - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast h_2 \right) \right)\kappa\, W \ast h_1 \, W\ast h_3 \\ & + (\Phi^\kappa_0)' L^d \Bigg[ - D^2_{\bar\rho\bar \rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_2,h_3]\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) + \big(h_2 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_2]\big) \left(h_3 - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast h_3 \right) \\ & + \big(h_3 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[h_3]\big)\left(h_2 - \dfrac{(\Phi^\kappa_{0})'}{L^d}\, W\ast h_2 \right) - \bar \rho_\infty^\kappa \dfrac{ (\Phi_0^\kappa)''}{L^d}\, W\ast h_2 \, W\ast h_3\Bigg] \kappa\, W \ast h_1 . \end{align*} \end{lemma} \begin{proof} For all $h_1, h_2, h_3\in L^2(\mathbb T^d)$, the second and third order Fréchet derivative in $\bar\rho$ of $\bar{\mathcal G}$ at a point $(\bar \rho , \kappa)$ are \begin{align*} D^2_{\bar\rho \bar\rho} &\bar{\mathcal G}(\bar \rho,\kappa) [h_1, h_2] \\ =&\, - \Phi''_{\bar\rho} \left (\dfrac 1 {L^d} - L^d\big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right)\kappa\right) \, W \ast h_1 \, W \ast h_2 \\ &+ \Phi'_{\bar\rho} L^d \left( \big(h_2 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_2]\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right) + \big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big)\left(h_2 - \dfrac{\Phi_{\bar\rho}'}{L^d}\, W\ast h_2 \right) \right)\kappa\, W \ast h_1, \end{align*} and \begin{align*} D^3_{\bar\rho \bar\rho \bar\rho} & \bar{\mathcal G}(\bar \rho,\kappa) [h_1, h_2, h_3] \\ =& \, - \Phi'''_{\bar\rho} \left (\dfrac 1 {L^d} - L^d\big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right)\kappa\right) \, W \ast h_1 \, W \ast h_2 \, W \ast h_3 \\ &+ \Phi''_{\bar\rho} L^d \left( \big(h_3 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_3]\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right) + \big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big)\left(h_3 - \dfrac{\Phi_{\bar\rho}'}{L^d}\, W\ast h_3 \right) \right)\kappa\, W \ast h_1 \, W \ast h_2\\ & + \Phi''_{\bar\rho} L^d \big(h_2 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_2]\big) \left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right) \kappa\, W \ast h_1 \, W\ast h_3 \\ & + \Phi'_{\bar\rho} L^d \left( - D^2_{\bar\rho\bar \rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_2,h_3]\left(\bar \rho - \dfrac{\Phi_{\bar\rho}}{L^d} \right) + \big(h_2 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_2]\big) \left(h_3 - \dfrac{\Phi_{\bar\rho}'}{L^d}\, W\ast h_3 \right) \right) \kappa\, W \ast h_1 \\ & + \Phi''_{\bar\rho} L^d \big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big)\left(h_2 - \dfrac{\Phi_{\bar\rho}'}{L^d}\, W\ast h_2 \right) \kappa\, W \ast h_1 \, W\ast h_3 \\ & + \Phi'_{\bar\rho} L^d \left( \big(h_3 - D_{\bar\rho} \bar{\mathcal G}(\bar \rho,\kappa)[h_3]\big)\left(h_2 - \dfrac{\Phi_{\bar\rho}'}{L^d}\, W\ast h_2 \right) - \big( \bar \rho-\bar{\mathcal G}(\bar \rho ,\kappa)\big) \dfrac{\Phi_{\bar\rho}''}{L^d}\, W\ast h_2 \, W\ast h_3\right) \kappa\, W \ast h_1 . \end{align*} Letting $\bar \rho = \bar \rho_\infty ^\kappa$ yields the result. \end{proof} \subsection{Proof of the main results} We are now ready to rigorously state and prove the main results stated in Theorem \ref{thm:mainmain}. For convenience, the main theorem is split into two parts: Theorem \ref{thm:main} concerning the bifurcation points, and Theorem \ref{thm:main_2} characterising the corresponding bifurcation branches. The case of higher dimensional kernels of the $\bar\rho$ first derivative and additional symmetries is discussed in Subsection \ref{sec:high_dim}. We define the functional \begin{equation}\label{eq:onecompfunctionalH} \begin{array}{rccl} \mathcal H : & L_S^2(\mathbb T^d)\times \mathbb{R}_+^* & \to & L_S^2(\mathbb T^d) \\ & (\bar \rho,\kappa) &\mapsto & \bar {\mathcal G} (\bar \rho_\infty^\kappa + \bar\rho, \kappa). \end{array} \end{equation} which is the one we will apply the Crandall--Rabinowitz theorem to. For completeness, the Crandall--Rabinowitz theorem in the present context is stated in the appendix (Theorem \ref{thm:CR}). Note that for all $\kappa\in\mathbb{R}_+^*$, $\mathcal H(0,\kappa)=0$ and that for $\kappa>\tfrac{2|W_0|^2}{L^{2d}\pi B^2}$, owing to Assumption \ref{as:1}, the functional $\mathcal H$ satisfies all the smoothness hypotheses required. The Fréchet derivatives of $\mathcal H$ are \begin{align*} D_{\bar\rho} \mathcal H (\bar \rho, \kappa) & = D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa + \bar \rho,\kappa), \\ D_{\kappa} \mathcal H (\bar \rho, \kappa) & = D_{\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa + \bar \rho,\kappa) + D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa+\bar\rho,\kappa)\left[\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa)\right],\\ D^2_{\bar\rho \kappa} \mathcal H (\bar \rho, \kappa) & = D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa + \bar \rho,\kappa) + D^2_{\bar\rho\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa + \bar \rho,\kappa) \left[\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa), \cdot\right] ,\\ D_{\bar \rho \bar \rho} \mathcal H (\bar \rho, \kappa) & = D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa + \bar \rho,\kappa), \\ D_{ \bar \rho \bar \rho \bar \rho} \mathcal H (\bar \rho, \kappa) & = D^3_{\bar\rho \bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa + \bar \rho,\kappa). \end{align*} In particular, we have \begin{align}\label{eqn:der_H_0} D_{\bar\rho} \mathcal H (0, \kappa) & = D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa), \nonumber \\ D_{\kappa} \mathcal H (0, \kappa) & = D_{\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) + D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)\left[\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa)\right],\nonumber \\ D^2_{\bar\rho \kappa} \mathcal H (0, \kappa) & = D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)+ D^2_{\bar\rho\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) \left[\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa), \cdot\right],\\ D_{\bar \rho \bar \rho} \mathcal H (0, \kappa) & = D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa ,\kappa), \nonumber \\ D_{ \bar \rho \bar \rho \bar \rho} \mathcal H (0, \kappa) & = D^3_{\bar\rho \bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) \nonumber . \end{align} \begin{theorem}\label{thm:main} Grant Assumption \ref{as:1}. Let $\kappa\in\left(\tfrac{2|W_0|^2}{L^{2d}\pi B^2},+\infty\right)$. If there exists a unique $k^*\in\mathbb{N}^d$ such that \begin{equation}\label{eqn:W_tilde_equal} \dfrac{\tilde W (k^*)}{\Theta(k^*)} = \dfrac{1}{L^{\frac d2} (\Phi_0^\kappa)' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)}. \end{equation} and $ (\Phi_0^\kappa)''\geq 0$, then $(\bar\rho_\infty^\kappa,\kappa)$ is a bifurcation point of $\bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)=0$. \end{theorem} \begin{proof} \noindent \emph{i)} We first rewrite \[ D_{\bar\rho} \mathcal H(0,\kappa) = I - T, \] with $ T : L^2_S(\mathbb S) \to L^2_S(\mathbb S)$ defined by \[ T h_1 = (\Phi_0^\kappa)' \left (\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)\, W \ast h_1. \] First, we have to check that the operator $D_{\bar\rho} \mathcal H(0,\kappa)$ is not the identity, that is to say $T\neq 0$. We know that $ (\Phi_0^\kappa)'\neq 0$ by Assumption \ref{as:1} and Lemma \ref{lm:kappa_small}. Hence, by Lemma \ref{lm:g_eta}, $T\neq 0$. \noindent \emph{ii)} Let the basis of $L^2_S(\mathbb{T}^d )$ be as described in Section \ref{sec:basis}. By \eqref{trigo}, for all $k\in\mathbb{N}^d$, we have \begin{align*} \norme{ T \omega_k}_2^2 & = \norme{ (\Phi_0^\kappa)'\left (\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) \dfrac{\tilde W(k)}{\Theta(k)} \omega_k(\cdot) }_2^2\\ & = \left| (\Phi_0^\kappa)'\left (\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) \dfrac{\tilde W(k)}{\Theta(k)}\right|. \end{align*} Hence, the Hilbert--Schmidt norm \[ \norme{ T}_{HS}^2 = \sum_{k\in\mathbb{N}} \norme{ T \omega_k}_2^2,\] is finite and $ T$ is a Hilbert--Schmidt operator. Therefore, $D_{\bar\rho} \mathcal H(0,\kappa)$ is a Fredholm operator. \noindent \emph{iii)} Note that the mapping $z\mapsto I + z W\ast\cdot$ is norm-continous; indeed, for all $z_1,z_2\in\mathbb{R}$, \begin{equation*}\norme{(I + z_1 W\ast\cdot) - (I + z_2W\ast\cdot)} = | z_1-z_2 | \norme{W\ast \cdot}. \end{equation*} Hence, taking \[z= - (\Phi_0^\kappa)'\left (\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right),\] and applying the invariance by homotopy of the index of Fredholm operators, we have $\mathrm{Ind}(D_{\bar\rho} \mathcal H(0,\kappa)) = \mathrm{Ind}(I)=0$. \noindent \emph{iv)} We now diagonalise $D_{\bar \rho} \mathcal H(0,\kappa)$ with respect to the orthonormal basis defined in Section \ref{sec:basis}: \[ D_{\bar\rho} \mathcal H(0,\kappa)[\omega_{k}](x) = \left\{\begin{array}{ll} 1 - L^\frac d2 (\Phi_0^\kappa)' \left(\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)W_0,\qquad & if\ k=0,\\ \left( 1 - \dfrac{L^{\frac d2}F(k)}{\Theta(k)} \left (\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) \right) \omega_k(x), \qquad & otherwise, \end{array} \right. \] with $F(k) = (\Phi_0^\kappa)' \tilde W(k)$. Since $W_0<0$, and by Lemma \ref{lm:g_eta}, \[ 1 - L^\frac d2 (\Phi_0^\kappa)' \left(\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)W_0 > 0. \] By assumption \eqref{eqn:W_tilde_equal_0}, the kernel of $D_{\bar\rho} \mathcal H(0,\kappa)$ is thus one dimensional. \noindent\emph{v)} Last, we need to check that if $v\in \mathrm{Ker}\big(D_{\bar \rho} \mathcal H(0,\kappa)\big)$ and $\norme{v}=1$, then $D^2_{\bar\rho\kappa} \mathcal H(0,\kappa) [v]\notin \mathrm{Im}\big(D_{\bar \rho} \mathcal H(0,\kappa)\big)$. Since $W$ is symmetric, the operator $I-T$ is self-adjunct. Then, since $\mathrm{Im}(I-T)$ is closed, we have \[ \mathrm{Im}(I-T) = \mathrm{Ker}( (I-T)^*)^\perp = \mathrm{Ker}(I-T)^\perp. \] As a result, we just need to check that \[ \pscal{D^2_{\bar\rho\kappa} \mathcal H(0,\kappa) [v]}{ v } \neq 0.\] According to \eqref{eqn:der_H_0}, \begin{equation*} D^2_{\bar\rho \kappa} \mathcal H (0, \kappa)[v] = D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[v]+ D^2_{\bar\rho\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) \left[\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa), v\right]. \end{equation*} Applying the formula in Lemma \ref{lm:cross}, and noticing that $v\in \mathrm{Ker}\big(D_{\bar \rho} \bar{\mathcal G}(\bar\rho_\infty^\kappa,\kappa)\big)$, we obtain \begin{equation*} D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [v] = \dfrac{1}{2\kappa}\left(1+ L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa\right)\left(v - \dfrac{(\Phi^\kappa_0)'}{L^d} W\ast v\right) + \dfrac{L^d}{2} \left( (\Phi_0^\kappa)' \bar \rho_\infty^\kappa W\ast v + \Phi_0^\kappa v \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right). \end{equation*} Since $v\in \mathrm{Ker}\big(D_{\bar \rho} \bar{\mathcal G}(\bar \rho_\infty,\kappa)\big)$, we have \[ W\ast v = \dfrac{1}{ (\Phi_0^\kappa)' \left (\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)}v. \] Hence, \begin{align}\label{eqn:brick_1} \dfrac{1}{2\kappa}&\left(1+ L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa\right)\left(v - \dfrac{(\Phi^\kappa_0)'}{L^d} W\ast v\right) \nonumber \\ & = -\dfrac{1}{2\kappa}\left(1+ L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa\right) \dfrac{L^{2d}\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa}{L^d\left(\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa\right)} v\\ &= -\dfrac{L^d}{2}\left(1+ L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa\right) \dfrac{\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} v, \nonumber \end{align} and \begin{align}\label{eqn:brick_2} \dfrac{L^d}{2} & \left( (\Phi_0^\kappa)' \bar \rho_\infty^\kappa W\ast v + \Phi_0^\kappa v \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\nonumber \\ &= \dfrac{L^d}{2}\left( \dfrac{\bar\rho_\infty^\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} + \Phi_0^\kappa \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) v. \end{align} Collecting \eqref{eqn:brick_1} and \eqref{eqn:brick_2}, we obtain \begin{align*} D^2_{\bar\rho\kappa} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [v] & = \dfrac{L^d}{2}\left( \dfrac{\bar\rho_\infty^\kappa - \left(1+ L^d \Phi_0^\kappa\bar \rho_\infty^\kappa\kappa\right)\bar\rho_\infty^\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} + \Phi_0^\kappa \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) v\\ \nonumber & = \dfrac{L^d \Phi_0^\kappa}{2}\left( 1 - \dfrac{ L^d(\bar \rho_\infty^\kappa)^2\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} \right)\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) v . \end{align*} On the other hand, we have \begin{align*} D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)& \left[\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa), v\right] \\ = & \, \underbrace{- (\Phi^\kappa_0)'' \left (\dfrac 1 {L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa\right) \, W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) \, W \ast v}_{=:A_1} \\ & + \underbrace{(\Phi^\kappa_0)' L^d \left( \big(v - D_{\bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[v]\big) \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) + \bar \rho_\infty^\kappa \left(v - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast v \right) \right)\kappa\, W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) }_{=:A_2}, \end{align*} where \begin{equation*} A_1 = - \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa), \end{equation*} and \begin{gather*} \begin{aligned} A_2 & = (\Phi^\kappa_0)' L^d \left( \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)v + \bar \rho_\infty^\kappa \left(v - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast v \right) \right)\kappa\, W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa)\\ & = (\Phi^\kappa_0)' L^d \left( \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)v + \bar \rho_\infty^\kappa \dfrac{L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} \right)\kappa\, W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa)\, v \\ & = (\Phi^\kappa_0)' L^d \left( 1 - \dfrac{L^d(\bar \rho_\infty^\kappa)^2\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} \right) \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa)\, v . \end{aligned} \end{gather*} Therefore, \begin{align}\label{eqn:no_more_idea_for_labels_3} D^2_{\bar\rho \kappa} \mathcal H(0,\kappa) \left[ v\right] = & \left( \underbrace{ - \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa)}_{\leqslant 0} \right. \\ & \left. + \underbrace{L^d\left(\dfrac{ \Phi_0^\kappa}{2} + (\Phi_0^\kappa)' W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) \right)}_{>0}\! \underbrace{ \left( 1 - \dfrac{L^d(\bar \rho_\infty^\kappa)^2\kappa}{\tfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \tfrac{ \Phi_0^\kappa}{L^d}\right)\kappa} \right)}_{=:A_3}\! \underbrace{\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)}_{>0} \right) v. \nonumber \end{align} Hence, if $A_3$ is of negative sign, then $D^2_{\bar\rho\kappa} \mathcal H (0,\kappa) [v] \neq 0$. Moreover, $A_3<0$ if and only if \begin{equation*} L^d(\bar \rho_\infty^\kappa)^2\kappa > \dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa. \end{equation*} If we denote $\eta = \sqrt{\frac{\kappa}{2}} \Phi_0^\kappa$, then by Lemma \ref{lm:g_eta}, \[ \dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa = \frac{1}{L^d}\left( 1-\frac{2}{\sqrt{\pi}}\frac{\exp (-\eta^2)}{1+\erf (\eta)}\left[\frac{1}{\sqrt{\pi}}\frac{\exp (-\eta^2)}{1+\erf (\eta)}+\eta\right]\right) = \dfrac{g(\eta)}{L^d} ,\] and by \eqref{eqn:rho_bar_eta}, \begin{equation*} L^d (\bar\rho_\infty^\kappa)^2 \kappa = \dfrac{2}{L^d} \left( \dfrac{1}{\sqrt\pi}\frac{\exp (-\eta^2)}{1+\erf (\eta)} + \eta \right)^2. \end{equation*} Hence, $A_3 < 0$ if and only if \begin{equation*} w(\eta) := \left( \dfrac{1}{\sqrt\pi}\frac{\exp (-\eta^2)}{1+\erf (\eta)} + \eta \right)\left( \dfrac{2}{\sqrt\pi}\frac{\exp (-\eta^2)}{1+\erf (\eta)} + \eta \right) > \dfrac12, \end{equation*} where the function $w$ is the same as defined in \eqref{eqn:def_h}. In the proof of Lemma \ref{lm:g_eta}, we obtained that \[w(0)=\dfrac{2}{\pi}>\dfrac12,\] and that $w$ is an increasing function on $\mathbb{R}_+$. Therefore $A_3<0$, which in turn implies \[ \pscal{D^2_{\bar\rho\kappa} \mathcal H(0,\kappa) [v]}{ v } \neq 0.\] \end{proof} \begin{remark}\label{rem:convexity} The assumption $ (\Phi_0^\kappa)'' \geq 0$ in Theorem \ref{thm:main} can be relaxed. Indeed, in \eqref{eqn:no_more_idea_for_labels_3} \begin{align*} A_4:= - \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) + L^d\left(\dfrac{ \Phi_0^\kappa}{2} + (\Phi_0^\kappa)' W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) \right) A_3 \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) < 0, \end{align*} is sufficient for $ D^2_{\bar\rho \kappa} \mathcal H(0,\kappa) \left[ v\right] \neq 0$. By Lemma \ref{lm:g_eta} and \eqref{eqn:rho_bar_eta}, \begin{align*} A_3 = 2\left(\frac{1}{2}-\frac{(f(\eta)+\eta))^2}{g(\eta)} \right) \leq 2 \left(\frac{1}{2}-\frac{f(0)^2}{g(0)} \right) = \frac{\pi-4}{\pi-2}, \end{align*} as it can be checked that $\tfrac{(f(\eta)+\eta))^2}{g(\eta)}$ is an increasing function in $\eta$. Thus, \begin{align*} A_4 & \leq - \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) + L^d\left(\dfrac{ \Phi_0^\kappa}{2} + (\Phi_0^\kappa)' W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa} (\kappa)\right) \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) \frac{\pi-4}{\pi-2}\\ & \leq - \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) + L^d\dfrac{ \Phi_0^\kappa}{2} \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right) \frac{\pi-4}{\pi-2}, \end{align*} as, from the expression of $\tfrac{\diff \bar\rho_\infty}{\diff \kappa}$ in Lemma \ref{lm:der_bar_rho}, $\tfrac{\diff \bar\rho_\infty}{\diff \kappa}$ can range from $-\infty$ to $0$. By writing out the remaining $\tfrac{\diff \bar\rho_\infty}{\diff \kappa}$, gathering terms, and remembering that $\left(\bar \rho_\infty - \tfrac{ \Phi_0^\kappa}{L^d}\right)>0$ in the above inequality, we find that \begin{align*} \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{1}{2\kappa } \dfrac{(1 + L^d \Phi_0^\kappa \bar\rho_\infty^\kappa\kappa)}{1 - (\Phi_0^\kappa)'W_0 \tfrac{g(\eta)}{L^d}} + L^d\dfrac{ \Phi_0^\kappa}{2} \frac{\pi-4}{\pi-2} & \leq \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{1}{2\kappa } \dfrac{(1 + L^d \Phi_0^\kappa \bar\rho_\infty^\kappa\kappa)}{1 - \tfrac{ (\Phi_0^\kappa)'W_0}{L^d}(1-\tfrac{2}{\pi})} + L^d\dfrac{ \Phi_0^\kappa}{2} \frac{\pi-4}{\pi-2} \\ & \leq \dfrac{ (\Phi_0^\kappa)''W_0}{ (\Phi_0^\kappa)'} \dfrac{L^d}{2\kappa} \dfrac{\left(1 + \Phi_0^\kappa \left( \Phi_0^\kappa \kappa + \sqrt{\tfrac{2}{\pi \kappa}}\right)\right)}{L^d - (\Phi_0^\kappa)'W_0\left(1-\tfrac{2}{\pi}\right)} + L^d\dfrac{ \Phi_0^\kappa}{2} \frac{\pi-4}{\pi-2}\\ & < 0 \end{align*} will yield $A_4 < 0$. For the first inequality we again used the lower bound on $g(\eta)$ in Lemma \ref{lm:g_eta}, and for the second the relation \eqref{eqn:rho_0} together with a maximum of $\rho_\infty^\kappa(0)$. By inserting the lower bound of $\kappa$ from Lemma \ref{lm:kappa_small} and rearranging, we see that requiring \begin{align*} (\Phi_0^\kappa)'' > 2 \frac{4-\pi}{\pi-2} \frac{ \Phi_0^\kappa (\Phi_0^\kappa)'}{L^{2d}\pi B^2}\frac{L^d- (\Phi_0^\kappa)'W_0\left(1-\frac{2}{\pi} \right)}{1+ \Phi_0^\kappa\left( \Phi_0^\kappa + \frac{L^d B}{|W_0|}\right)} W_0, \end{align*} is sufficient for $ D^2_{\bar\rho \kappa} \mathcal H(0,\kappa) \left[ v\right] \neq 0$. \end{remark} \begin{theorem}[Characterisation of the branch]\label{thm:main_2} Let the assumptions of Theorem \ref{thm:main} hold. Then at the bifurcation point $(\bar\rho_\infty^{\kappa^*},\kappa^*)$, there exists a continuously differentiable curve of non-homogeneous solutions to $\bar{\mathcal{G}}(\bar \rho_\infty^\kappa, \kappa)=0$ ( $\bar{\mathcal{G}}$ is defined in \eqref{eq:onecompfunctional}), \begin{equation}\label{eq:nontrivial_rho} \{\ (\bar \rho_{\kappa(z)},\kappa(z)) \ | \ z\in(-\delta,\delta), \ (\bar \rho_{\kappa(0)},\kappa(0))=(\bar \rho_\infty^{\kappa^*},\kappa^*),\ \delta>0 \ \}, \end{equation} such that, for all $x\in \mathbb T^d$, \begin{equation*} \bar \rho_{\kappa(z)}(x) = \bar \rho_\infty^{\kappa(z)} + z \omega_{k^*}(x) + o(z), \end{equation*} where $o(z)\in \mathrm{span}(\omega_{k^*})^\perp$. In a neighbourhood $U\times (\kappa^*-\varepsilon,\kappa^*+\varepsilon)$ of $(\bar\rho_\infty^{\kappa^*},\kappa^*)$ in $L^2_S(\mathbb T^d)\times \mathbb{R}_+^*$, all the solutions are either of the form $(\bar \rho_\infty^\kappa,\kappa)$ or on the non-homogeneous curve \eqref{eq:nontrivial_rho}. Moreover, the function $z\mapsto \kappa(z)$ satisfies $\kappa'(0)=0$, and \begin{equation*} \kappa''(0) = -\dfrac13 \dfrac{\mathcal K_1 }{ \mathcal K_2}, \end{equation*} with \begin{align*} \mathcal K_1 = \Bigg( - \dfrac{(\Phi^\kappa_0)'''}{(\Phi^\kappa_0)'} &+ 2(\Phi^\kappa_0)'' L^d \mathcal K_3 \rho_\infty^\kappa(0) + (\Phi^\kappa_0)' L^d\Bigg[ -\left( \dfrac{2L^{\frac d2}\Theta(k^*)}{\tilde W(k^*)} - 2\dfrac{ (\Phi_0^\kappa)'}{L^d} + \bar\rho_\infty^\kappa(\Phi^\kappa_0)'' L^{\frac d2} \dfrac{\tilde W(k^*)}{\Theta(k^*)} \right)\kappa\\ &+ \frac{\rho_\infty^\kappa(0)}{\kappa}\left( \dfrac{(\Phi^\kappa_0)''}{ (\Phi_0^\kappa)'} - \Phi'_0 L^d \mathcal K_3 \rho_\infty^\kappa(0) \right) \Bigg]\Bigg) L^d\dfrac{\tilde W(k^*)^2}{\Theta(k^*)^2} \norme{\omega_{k^*}^2}^2_2, \end{align*} with \begin{align*} \rho_\infty^\kappa(0) = \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) \kappa > 0, \qquad \mathcal{K}_3 = 1-L^d\bar\rho_\infty^\kappa\kappa (\Phi_0^\kappa)' L^{\frac d2} \dfrac{\tilde W(k^*)}{\Theta(k^*)} < 0, \end{align*} and $\mathcal K_2 = \pscal{D^2_{\bar\rho \kappa} \mathcal H(0,\kappa) \left[ \omega_{k^*}\right]}{\omega_{k^*}}<0$, where $ D^2_{\bar\rho \kappa} \mathcal H(0,\kappa) \left[ \omega_{k^*} \right]$ is defined in equation \eqref{eqn:no_more_idea_for_labels_3}. \end{theorem} \begin{proof} Let us now characterise the branch of the bifurcation by computing $\kappa'(0)$ and $\kappa''(0)$. We drop the star in $k^*$ for the sake of readability and denote by $\omega_k$ the element spanning the kernel of $D_{\bar\rho} \bar {\mathcal G} (\bar\rho_\infty^\kappa,\kappa)$. Recall that \[ W\ast \omega_k = \dfrac{1}{ (\Phi_0^\kappa)' \left (\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)}\omega_k =: C_1 \omega_k, \] and that, as obtained in the proof of Theorem \ref{thm:main}, \begin{equation*} \omega_k - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast \omega_k = - \dfrac{L^d\bar \rho_\infty\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa}\,\omega_k =: - C_2\, \omega_k, \end{equation*} with $C_2$ a positive constant. According to Lemma \ref{lm:third} and denoting again \[A_3 = 1 - \dfrac{L^d(\bar \rho_\infty^\kappa)^2\kappa}{\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa}<0,\] we obtain \begin{equation*} D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [\omega_k, \omega_k] = \left[ -\dfrac{(\Phi^\kappa_0)''}{ (\Phi_0^\kappa)'} + \Phi'_0 L^d A_3\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa \right]C_1 \omega_k^2 \end{equation*} We have \[\pscal{\omega_k^2}{\omega_k} = \int_{\mathbb T^d}\omega_k^3(x)\diff x = 0.\] Therefore, $\pscal{D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty,\kappa) [\omega_k, \omega_k]}{\omega_k}=0$ and \begin{equation*} \kappa'(0) = -\dfrac12\dfrac{\pscal{D^2_{\bar\rho \bar\rho} \mathcal H(0,\kappa) [\omega_k, \omega_k]}{\omega_k}}{\pscal{D^2_{\bar\rho\kappa} \mathcal H(0,\kappa)[\omega_k] }{\omega_k}} = -\dfrac12\dfrac{\pscal{D^2_{\bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [\omega_k, \omega_k]}{\omega_k}}{\pscal{D^2_{\bar\rho\kappa} \mathcal H(0,\kappa)[\omega_k] }{\omega_k}} = 0 \end{equation*} This result was expected as it was unlikely that the bifurcation would be transcritical. Now, let us compute $\kappa''(0)$. We are going to use the formula \begin{equation*} \kappa''(0) = -\dfrac13 \dfrac{\pscal{D^3_{\bar\rho\bar\rho\bar\rho} \mathcal H(0,\kappa) [\omega_k,\omega_k,\omega_k]}{\omega_k} }{\pscal{D^2_{\rho\kappa} \mathcal H(0,\kappa)[\omega_k]}{\omega_k}}, \end{equation*} from Theorem \ref{thm:characterisation}. To compute the third order derivative term, we start with the general expression in Lemma \ref{lm:third} and we use the facts that $D^3_{\bar\rho\bar\rho\bar\rho} \mathcal H(0,\kappa) [\omega_k,\omega_k,\omega_k]=D^3_{\bar\rho\bar\rho\bar\rho} \bar {\mathcal G} (\bar\rho_\infty^\kappa,\kappa) [\omega_k,\omega_k,\omega_k]$ and that $D_{\bar\rho} \bar {\mathcal G} (\bar\rho_\infty^\kappa,\kappa)[\omega_k] = 0$: \begin{align*} D^3_{\bar\rho \bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [\omega_k, \omega_k, \omega_k] = & \, - (\Phi^\kappa_0)''' \left (\dfrac 1 {L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa\right) ( W \ast \omega_k )^3 \\ & + 2(\Phi^\kappa_0)'' L^d \left( \omega_k \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) + \bar \rho_\infty^\kappa\left(\omega_k - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast \omega_k \right) \right)\kappa (W \ast \omega_k)^2 \\ & + (\Phi^\kappa_0)' L^d \Bigg[ - D^2_{\bar\rho\bar \rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa)[\omega_k,\omega_k]\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) \\ & \quad + 2\omega_k \left(\omega_k - \dfrac{ (\Phi_0^\kappa)'}{L^d}\, W\ast \omega_k \right) - \bar \rho_\infty^\kappa \dfrac{ (\Phi_0^\kappa)''}{L^d}\, W\ast \omega_k \, W\ast \omega_k\Bigg] \kappa\, W \ast \omega_k . \end{align*} Hence, we obtain \begin{align*} & \pscal{D^3_{\bar\rho \bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [\omega_k, \omega_k, \omega_k]}{\omega_k} \\ & \qquad\qquad\quad = \Bigg( - \dfrac{(\Phi^\kappa_0)'''}{(\Phi^\kappa_0)'} {C_1}^2 + 2(\Phi^\kappa_0)'' L^d A_3 {C_1}^2 \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) \kappa \\ & \qquad\qquad\qquad + (\Phi^\kappa_0)' L^d\bigg[ -\left( 2 C_2 + \bar\rho_\infty^\kappa(\Phi^\kappa_0)'' {C_1}^2 \right)\kappa \\ & \qquad\qquad\qquad + \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\left( \dfrac{(\Phi^\kappa_0)''}{ (\Phi_0^\kappa)'} - \Phi'_0 L^d A_3\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa \right) \bigg]C_1\Bigg) \pscal{{\omega_k}^3}{\omega_k}. \end{align*} Now, notice that \[ C_1 = L^{\frac d2} \dfrac{\tilde W(k)}{\Theta(k)}, \qquad A_3 = 1-L^d\bar\rho_\infty^\kappa\kappa (\Phi_0^\kappa)' C_1 \qquad\mathrm{and}\qquad C_2 = 1 - \dfrac{ (\Phi_0^\kappa)'}{L^d}C_1 = 1-\dfrac{ (\Phi_0^\kappa)'\tilde W(k)}{L^{\frac d2}\Theta(k)}. \] Thus, \begin{align*} & \pscal{D^3_{\bar\rho \bar\rho \bar\rho} \bar{\mathcal G}(\bar \rho_\infty^\kappa,\kappa) [\omega_k, \omega_k, \omega_k]}{\omega_k} \\ & \qquad = \Bigg( - \dfrac{(\Phi^\kappa_0)'''}{(\Phi^\kappa_0)'} + 2(\Phi^\kappa_0)'' L^d \mathcal K_3 \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right) \kappa \\ & \qquad\quad + (\Phi^\kappa_0)' L^d\Bigg[ -\left( \dfrac{2L^{\frac d2}\Theta(k)}{\tilde W(k)} - 2\dfrac{ (\Phi_0^\kappa)'}{L^d} + \bar\rho_\infty^\kappa(\Phi^\kappa_0)'' L^{\frac d2} \dfrac{\tilde W(k)}{\Theta(k)} \right)\kappa\\ & \qquad\quad + \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\left( \dfrac{(\Phi^\kappa_0)''}{ (\Phi_0^\kappa)'} - (\Phi^\kappa_0)' L^d \mathcal K_3\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)\kappa \right) \Bigg]\Bigg) L^d\dfrac{\tilde W(k)^2}{\Theta(k)^2} \norme{\omega_k^2}^2_2, \end{align*} where \begin{align*} \mathcal K_3 = 1-L^d\bar\rho_\infty^\kappa\kappa (\Phi_0^\kappa)' L^{\frac d2} \dfrac{\tilde W(k)}{\Theta(k)}. \end{align*} \end{proof} Finding the sign of $\kappa''(0)$ (which determines whether the bifurcation is subcritical or supercritical) is hard as the formula involves a several terms, some of them being of unpredictable sign and scale, like $ (\Phi_0^\kappa)'''$. However, for certain choices of $\Phi$, this convoluted formula can be simplified, like we show in the following remark. \begin{remark} In the case of the ReLU function $\Phi(x)=(x)^+$, the expression for $\mathcal{K}_1$ simplifies and we can obtain the asymptotic sign of $\kappa''(0)$ for bifurcations happening close to $\kappa=+\infty$. More precisely, using the notation for constants in the previous proof, we have \[\kappa''(0) = -\dfrac{L^dC_1\kappa}3 \dfrac{ - 2 C_2 + L^d |A_3|\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)^2 }{\pscal{D^2_{\bar\rho\kappa} \mathcal H(0,\kappa)[\omega_k] }{\omega_k}}\] and we know that $C_2>0$ and $\pscal{D^2_{\bar\rho\kappa} \mathcal H(0,\kappa)[\omega_k] }{\omega_k} < 0$ according to the proof of Theorem \ref{thm:main}. Hence, the sign of $ \kappa''(0)$ depends on the sign of \[ L^d |A_3|\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)^2 - 2 C_2 = \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)L^d \left[ |A_3|\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)-\bar\rho_\infty^\kappa\kappa C_1\right], \] with $C_1>0$, and so it depends upon the sign of \[ |A_3|\left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d} \right)-\bar\rho_\infty^\kappa\kappa C_1. \] The positive term tends to 0 exponentially fast (Lemma \ref{lm:equiv}) when $\kappa$ tends to $+\infty$ and the negative term tends to $-\infty$ linearly (because $C_1$ and $\bar\rho_\infty$ tend to a finite positive limit by Lemmata \ref{lm:inf_bound_rho} and \ref{lm:equiv}). Therefore, past some threshold value for $\kappa$, all the bifurcations are subcritical. It is then likely that past this value, the bifurcation branches are nonlinearly unstable. This could explain why they were not witnessed in the numerical study of the bifurcation branches in \cite{CHS}, and strengthens the hypothesis of the existence of a hysteresis phenomena as suggested by the authors. \end{remark} \subsection{Higher dimensional kernels and equivariant bifurcations}\label{sec:high_dim} In order to satisfy the hypotheses of Theorem \ref{thm:mainmain} for some value $\kappa$, we need to ensure not only that for some $k^*$ the equality \eqref{eqn:W_tilde_equal} holds, but also that this $k^*$ is unique, that is to say: for all $k\in\mathbb{N}^d\setminus\{k^*\}$, \[ \frac{\tilde W(k^*)}{\Theta(k^*)}\neq \frac{\tilde W(k)}{\Theta(k)}. \] This is because the hypotheses of the Crandall--Rabinowitz theorem (Theorem \ref{thm:CR}) require that the kernel of the Fredholm operator $D_{\bar\rho} \mathcal H (0,\kappa) = D_{\bar\rho} \bar{\mathcal G} (\bar\rho_\infty,\kappa)$ is one-dimensional. However, when the kernel of the operator $D_{\bar\rho} \bar {\mathcal G}(\bar \rho_\infty,\kappa)$ is of dimension higher than one, a bifurcation can still arise. In the previous results, the choice of the space $L_S^2(\mathbb T^d)$ of coordinate-wise even functions was already playing two roles: to quotient out the symmetry implied by $W\in L_S^2(\mathbb T^d)$, and to avoid the problem of translation invariance of the stationary states. As noted in the introduction, a natural generalisation of this procedure is to use equivariant bifurcation theory \cite{CL-equivariantbif,Dionne,FSV2022} to take full advantage of all the symmetries of $W$. This can be applied to coordinate-wise even kernels, but it is particularly useful for radially symmetric ones. The approach is as follows. First, a closed subgroup of the Euclidean group $E_d$ acts from the left on a function space by \[ \eta \cdot h(x) = h(\eta^{-1}(x)). \] Here, the Euclidean group $E_d$ refers to the set of all affine isometries in $\mathbb{R}^d$. We still denote by $\mathcal H$ the same functional as in the previous subsection, but defined on $L^2(\mathbb T^d)$, where $\mathbb T^d$ is the square torus with length $L$, or a restriction of it. Given a lattice $\mathcal L$ on $\mathbb T^d$, we denote $\Gamma$ the largest subgroup of $E_d$ that acts on $\mathcal L$-periodic functions such that $\mathcal H(\cdot,\kappa)$ is equivariant under the action of $\Gamma$, \textit{i.e.} \[ \forall \eta \in\Gamma, \ \forall h\in L^2(\sfrac{\mathbb T^d}{\mathcal L}), \quad \mathcal H( \eta \cdot h,\kappa) = \eta \cdot \mathcal H(h,\kappa). \] Now, define \[ \mathcal V = \mathrm{Ker}\big( D_{\bar\rho} \mathcal H (0,\kappa) \big) \subset L^2(\mathbb T^d).\] For any isotropy subgroup $\Sigma$ for the action of $\Gamma$ on $\mathcal V$, denote \[ \mathrm{Fix}_{\mathcal V}( \Sigma ) = \{ h \in\mathcal V \mbox{ such that } \forall \eta\in\Sigma,\ \eta \cdot h = h \}. \] Then, up to technical conditions, whenever $\dim\big(\mathrm{Fix}_{\mathcal V}( \Sigma )\big) = 1$, by the equivariant branching lemma \cite[Theorem 2.3.2]{CL-equivariantbif}, there is a unique branch of steady-state solutions of \eqref{eq:4PDE} whose averages $\bar\rho$ have the symmetry $\Sigma$. An important consideration is that the fundamental cell of the lattice $\mathcal L$ must satisfy two matching conditions: the periodicity condition associated with the flat torus of length $L$ on which \eqref{eqn:2} is posed and the symmetry of the convolution on the torus with the kernel $W$. The easiest way to ensure both is to choose a square lattice adapted to the length $L$ of $\mathbb T^d$. \begin{theorem}\label{thm:ebl} Grant Assumption \ref{as:1}. Consider $\mathcal L$ the trivial $d-$dimensional cubic lattice of the flat torus generated by the vectors $(L,0,\dots,0), \dots, (0,\dots,0,L)$. Let $\Gamma$ be a subgroup of $E_d$. Assume $\mathcal H$ is $\Gamma-$equivariant on $L^{2}(\mathbb{T}^d)$ and that there exists a multi-index $k^*\in \mathbb N^d$ such that \begin{align*} \dfrac{\tilde W (k^*)}{\Theta(k^*)} =\dfrac{1}{L^{\frac d2} (\Phi_0^\kappa)' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)}, \end{align*} where $\kappa\in\mathbb{R}_+^*$ and $ (\Phi_0^\kappa)''\geq 0.$ Denote $\mathcal V = \mathrm{Ker}\big( D_{\bar\rho} \mathcal H (0,\kappa) \big) \subset L^{2}(\mathbb{T}^d)$. Then, for all isotropy subgroups $\Sigma$ of $\Gamma$, such that $ \mathrm{dim}\big(\mathrm{Fix}_{\mathcal V}( \Sigma ) \big) = 1$, there exists a unique branch of stationary states of \eqref{eqn:2} bifurcating from the trivial curve and whose average $\bar\rho$ in $s$ has the symmetry of $\Sigma$. \end{theorem} \begin{proof} In order to prove the result, we check the hypotheses of the equivariant branching lemma as stated in \cite[Theorem 2.3.2]{CL-equivariantbif}. The regularity required for the map $\mathcal H$ can be deduced from Lemma \ref{lm:der_bar_rho}. The fact that $0$ is an isolated eigenvalue with finite multiplicity stems from the same arguments as in the proof of Theorem \ref{thm:main}. Let us now consider the functional \eqref{eq:onecompfunctionalH} on the bigger space $L^2(\mathbb T^d)$. We prove that it is a Fredholm operator from $L^2(\mathbb T^d)$ to $L^2(\mathbb T^d)$ in the same way, and diagonalise $D_{\bar\rho} \mathcal H (0,\kappa)$ on the Hilbert basis of $L^2(\mathbb T^d)$ defined in Section \ref{sec:basis}; this time we use \eqref{trigo2} for the convolution. Similar computations show that the kernel $\mathcal V$ is generated by all Fourier modes $\omega_k(x)$ associated to a multi-index $k\in\mathbb{Z}^d$ such that \[ 1 - \dfrac{L^{\frac d2}(\Phi_0^\kappa)' \tilde W(|k|)}{\Theta(|k|)} \left (\dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) = 0, \] where $|k|=|(k_1,\dots,k_d)| = (|k_1|,\dots,|k_d|)$. The proof of the second order condition can be done in the same way as in the proof of Theorem \ref{thm:main}. \end{proof} Notice that, for a chosen $W$, the kernel $\mathcal V$ is now higher dimensional than when we were diagonalizing in $L^2_S(\mathbb T^d)$. Note also that a particular case of Theorem \ref{thm:ebl} is Theorem \ref{thm:main} when the only symmetries are reflections on each coordinate. The dimension of the Fix criterion then comes down to the uniqueness of the index $k^*$. The general result of Theorem \ref{thm:ebl} does not indicate whether the bifurcation is saddle-node, transcritical or pitchfork, but for a specific choice of symmetries of $W$ and of the isotropy group $\Sigma$, it is possible to check the needed hypotheses of Theorem 2.3.2 in \cite{CL-equivariantbif} to specify the type of bifurcation. In all the cases we have checked at hand, the bifurcations are of pitchfork type like in Theorem \ref{thm:main_2}. \begin{remark} In Theorem \ref{thm:ebl}, we consider the lattice generated by the vectors $(L,0,\dots,0)$, $\dots$, $(0,\dots,0,L)$ for the sake of clarity. Indeed, it allows us to obtain the same bifurcation condition \eqref{eqn:W_tilde_equal_0} as in Theorem \ref{thm:main}. The fact that additional bifurcation points are not appearing when we consider a finer lattice generated by the vectors $(L/n,0,\dots,0), \dots, (0,\dots,0,L/n)$, $n\in\mathbb{N}^*$ is not obvious due to the nonlinear aspects of the condition. However, thanks to Lemma \ref{lm:g_eta}, we can rewrite the $L$-dependency in a clearer form: \[ L^{\frac d2} (\Phi_0^\kappa)' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right) = \dfrac1{L^\frac d2} (\Phi_0^\kappa)' g\left( \sqrt{\frac \kappa 2 } \Phi_0^\kappa\right). \] Hence, taking a smaller square lattice of length $L/n$ would introduce a factor $n^\frac d2$ in the right-hand side of \eqref{eqn:W_tilde_equal_0}. The corresponding rescaling of the Fourier modes (note that since the integration area is smaller in the computation of $\Phi^\kappa_0$, $W$ has to be rescaled), introduces a factor $n^\frac d2$ in the left-hand side too. Hence, the bifurcation condition remains unchanged. \end{remark} \begin{remark}\label{rem:exchange} In the spirit of \cite[Remark 4.6]{CGPS20}, we can do another interesting particular case at hand. Let $\mathfrak{S}_d$ be the group of permutations of $\{1,\dots,d\}$. Introduce the equivalence relation in $\mathbb{N}^d$, $k \sim l$ if and only if there exists $\zeta\in \mathfrak{S}_d$ such that $k=\zeta(l)$, and denote $[k]$ the equivalence class of any $k\in \mathbb{N}^d$. Define the space of exchangeable coordinate functions \[ L^2_{ex}(\mathbb T^d) = \{ u\in L_S^2(\mathbb T^d) \ | \ \forall \zeta\in\mathfrak{S}_d, \forall x\in \mathbb T^d,\ u(x) = u(\zeta(x)) \}, \] endowed with the Hilbert basis $(\omega_{[k]})_{[k]\in \sfrac{\mathbb{N}^d}{\sim}}$ defined by $ \omega_{[k]}(x) = \frac{1}{\sqrt{\mathrm{card([k])}}} \sum_{l\in [k]} \omega_l(x)$, where $\sfrac{\mathbb N ^d}{\sim}$ denotes the associated quotient set containing all the equivalence classes. Then, there is a bifurcation point whenever condition \eqref{eqn:W_tilde_equal_0} in Theorem \ref{thm:main} is satisfied on a unique equivalence class $[k^*]$ and $(\Phi_0^\kappa)''\geq 0$ holds. In this case, the shape of the branch is given by \begin{equation*} \bar \rho_{\kappa(z)}(x) = \bar \rho_\infty^{\kappa(z)} + z \dfrac{1}{\sqrt{\mathrm{card([k])}}} \sum_{l\in [k]} \omega_l(x) + o(z), \qquad z\in(-\delta,\delta), \end{equation*} with the same values $\kappa'(0)=0$ and $\kappa''(0)$ as defined in Theorem \ref{thm:main_2}. \end{remark} The most interesting case in our application framework is dimension $d=2$. In this case, the Euclidean group associated with the trivial lattice on the torus is the compact semi-direct sum of the Dihedral group $D_4$ and the compact group of translations on $\mathbb T^2$, $D_4 \overset{.}{+} \mathbb T^2$, see for example \cite{Dionne}. If we assume that $W$ is radially symmetric, then $\mathcal H$ is $\Gamma-$equivariant with $\Gamma=D_4 \overset{.}{+} \mathbb T^2$. In order to classify the patterns we can obtain with Theorem \ref{thm:ebl}, the procedure developed in \cite{Dionne,Dionne2} consists of writing the kernel $\mathcal V \subset L^2(\mathbb T^d)$ as a direct sum of $\Gamma-$irreducible subspaces $\mathcal V = \mathcal V_1 \oplus \cdots \oplus \mathcal V_n$, which in turn implies, for any isotropy subgroup $\Sigma$ of $\Gamma$, \[ \mathrm{Fix}_{\mathcal V}( \Sigma ) = \mathrm{Fix}_{\mathcal V_1}( \Sigma )\oplus \cdots \oplus \mathrm{Fix}_{\mathcal V_n}( \Sigma ). \] Hence, it comes down to classifying $\Gamma$-irreducible representations of $\mathcal V$ depending on its dimension. In dimension 2, this work has already been done, for example in a general context in \cite{Dionne,Dionne2} or in the context of deterministic neural field models in \cite{FSV2022,VCF2015}. Our spectral computations in the proofs of Theorem \ref{thm:main} and Theorem \ref{thm:ebl} indicate that if $W$ is radially symmetric and $\tilde W(k)/\Theta(k)$ are unique up to permutations of $k$, then $\mathrm{dim}(\mathcal V) = 4$ or $\mathrm{dim}(\mathcal V) = 8$. In dimension 8, the full classification of isotropy subgroups of $\Gamma$ acting on a $8-$dimensional representation can be found in \cite[Table 3]{Dionne2}, with dimensions of the $\mathrm{Fix}$ set for each case. In dimension 4, there is a unique translation free irreducible representation. The reader can also find in \cite[Table 3]{Dionne} a general table relating planforms (patterns arising through equivariant bifurcations) to the type of lattice, the dimension of $\mathcal V$ and the isotropy subgroups. As explained in \cite[Section 2.a]{Dionne}, it is possible to consider only translation free subgroups by lowering the dimension if necessary, which we will show below in an example for stripes. Therefore, we can use our previous analysis of bifurcation points in terms of the Fourier modes of $W$, combined with \cite[Theorem 2.3]{Dionne}, in order to conclude that when the bifurcation conditions of Theorem \ref{thm:ebl} are satisfied, there are branches of stationary states from the spatially homogeneous one with the following symmetries: stripes, also called rolls \cite[Figure 4]{CHS}; simples squares \cite[Figure 1]{Dionne}; squares \cite[Figure 2]{Dionne}; and anti-squares \cite[Figure 3]{Dionne}. In Section \ref{sec:numerics} we comment on some of these patterns and numerically explore their presence and stability in the setting of \eqref{eqn:2}. In order to make the abstract procedure above more explicit, let us take $L=1$ and detail two examples for the setting in the left plot of Figure \ref{fig:radsym-vs-nonradsym} in Section \ref{sec:numerics}. Numerical results indicate that the first bifurcation points in this case are associated to the multi-indexes $k=(4,0)$, $k=(4,1)$ and $k=(3,3)$. \begin{example}\label{ex:1} Consider the bifurcation point $\kappa^*$ associated to the mode $k=(4,0)$. Then $\mathcal V$ is generated by the functions \[ \omega_{(4,0)} = \sqrt 2 \cos(8 \pi x_1) , \quad \omega_{(-4,0)} = \sqrt 2 \sin(8 \pi x_1),\quad \omega_{(0,4)} = \sqrt 2 \cos(8 \pi x_2), \quad \omega_{(0,-4)} = \sqrt 2 \sin(8 \pi x_2) . \] The subgroup $\Sigma$ generated by the rotation of angle $\pi$ and the circle $\mathbb S$ of torus translations on the $x_1$ axis (respectively the $x_2$ axis) fixes only the span of $\omega_{(0,4)}$ (resp. of $\omega_{(4,0)}$), so we can apply Theorem \ref{thm:ebl}. The branch having the symmetry of $\Sigma$, consists of horizontal (resp. vertical) stripes. Since this planform arises from a subgroup which is not translation free, it was expected that it would have a one dimensional shape. Another way to obtain this pattern is to apply Theorem \ref{thm:ebl} in $L^2_S(\mathbb T^2)$ instead of $L^2(\mathbb T^2)$; then, the kernel $\mathcal V$ is only two-dimensional, containing the cosines, and the subgroup generated by the reflection on $x_2$ (resp. on $x_1$) fixes only a one dimensional space and generates the horizontal (resp. vertical) stripes. The translation free subgroup $\Sigma=D_4$ fixes $\omega_{(4,0)}+\omega_{(0,4)}$, and the branch with this symmetry consists of simple squares, see the left plot in Figure \ref{fig:modepatterns}. Note that this branch could also be found with the approach of exchangable coordinates outlined in Remark \ref{rem:exchange}. \end{example} \begin{example}\label{ex:2} Consider the Fourier mode $k=(3,3)$. The kernel $\mathcal V$ is now generated by \begin{align*} &\omega_{(3,3)} = 2 \cos(6 \pi x_1)\cos(6\pi x_2), \quad \omega_{(3,-3)} = 2 \cos(6 \pi x_1)\sin(6\pi x_2),\\ &\omega_{(-3,3)} = 2 \sin(6 \pi x_1)\cos(6\pi x_2), \quad \omega_{(-3,-3)} = 2 \sin(6 \pi x_1)\sin(6\pi x_2). \end{align*} The translation free subgroup $\Sigma=D_4$ fixes only $\omega_{(3,3)}$, and hence gives rise to a bifurcating branch with the symmetry of simple squares, see the right plot in Figure \ref{fig:modepatterns}. This branch matches the pitchfork bifurcation in $L^2_S(\mathbb T^2)$ which Theorem \ref{thm:main} yields. \end{example} \begin{remark}\label{rem:higherdim} Finally, we highlight the following: \\ 1. If $\mathrm{Fix}_{\mathcal V}( \Sigma )$ is of higher dimension than 1, there could still be a bifurcation point there, but it is not detectable with the equivariant branching lemma.\\ 2. When $W$ is just component-wise even, we can still apply Theorem \ref{thm:ebl}, but $\Gamma$, and correspondingly the number of isotropy subgroups, will be smaller compared to the case where $W$ is radially symmetric. \end{remark} \subsection{Study of the bifurcations given a connectivity kernel} Another important question is to determine, given a connectivity kernel $W$, how many continuous bifurcations will arise from the constant stationary state and for which values of the parameter. Under a convexity assumption on $\Phi$, we can provide some answers. \begin{theorem}[Bifurcations given a connectivity kernel $W$]\label{thm:Given} Grant $\Phi''\geqslant 0$ and Assumption \ref{as:1}. For all $k\in\mathbb{N}^d\setminus\{0\}$ such that $\tilde W(k)/\Theta(k)$ is unique up to permutations of $k$, if \begin{equation}\label{eqn:given_potential} \dfrac{\tilde W(k)}{\Theta(k)} > \dfrac{L^{\frac d 2}}{\Phi'(W_0 \rho^* + B)}, \end{equation} where $\rho^*=\lim_{\kappa\to+\infty} \bar\rho_\infty$, then there exists a unique $\kappa$ such that $(\bar\rho_\infty^\kappa,\kappa)$ is a bifurcation point. \end{theorem} \begin{proof} According to Lemma \ref{lm:g_eta}, the bifurcation condition can be re-written in the form \[ \dfrac{\tilde W (k)}{\Theta(k)} = \dfrac{L^{\frac d2}}{ (\Phi_0^\kappa)' g\left(\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa\right)}, \] with the function $g$ being increasing and satisfying for all $\eta>0$, \[ 1-\frac{2}{\pi} = g(0) < g(\eta) < \lim_{\eta\to+\infty} g(\eta) = 1. \] By Lemma \ref{lm:der_bar_rho}, $\bar\rho_\infty$ is decreasing, and thus $ \Phi_0^\kappa$ is increasing. Since we assume $\Phi''\geqslant 0$, $\Phi'$ is increasing and thus $ (\Phi_0^\kappa)'$ is also increasing. We deduce from all this information that the function \[ \Psi : \kappa\mapsto \dfrac{L^{\frac d2}}{ (\Phi_0^\kappa)' g\left(\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa\right)} \] is decreasing. By continuity of $\Phi'$ and Lemma \ref{lm:kappa_small} and \ref{lm:inf_bound_rho}, we have \[ \lim_{\kappa\to \kappa_c} \Psi(\kappa) = +\infty \qquad \mathrm{and} \qquad \lim_{\kappa\to +\infty} \Psi(\kappa) = \dfrac{L^{\frac d 2}}{\Phi'(W_0 \rho^* + B)}, \] where $\kappa_c = \tfrac{2|W_0|^2}{L^{2d}\pi B^2}$ and $\rho^* = \lim_{\kappa\to +\infty} \bar \rho_\infty$. Note that the function $\Psi$ is not defined on $(0,\kappa_c)$ since $ (\Phi_0^\kappa)'=0$. Hence, for all $k\in\mathbb{N}^d$ satisfying the hypotheses, there is a unique intersection up to permutations of $k$, between the horizontal line $\tfrac{\tilde W (k)}{\Theta(k)}$ and the decreasing function $\Psi$ for some value $\kappa\in(\kappa_c,+\infty)$. We can then apply either Theorem \ref{thm:main} or the discussion in Subsection \ref{sec:high_dim} to prove that there is a bifurcation for this value of $\kappa$. \end{proof} \begin{remark} If $\Phi$ is not a $C^1$ function at point 0, for example the ReLU function $\Phi=(x)^+$, then the function \[ \Psi : \kappa\mapsto \dfrac{L^{\frac d2}}{ (\Phi_0^\kappa)' g\left(\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa\right)} \] does not satisfy $\lim_{\kappa\to \kappa_c} \Psi(\kappa) = 0$. However, if we have a left limit $\Phi'_{+} = \lim_{\kappa\to0^+} \Phi'(\kappa)$, and if $\Phi''$ exists on $\mathbb{R}^*$, we can still state a similar result with \begin{equation*} \dfrac{L^{\frac d 2}}{\Phi'(W_0 \rho^* + B)} < \dfrac{\tilde W(k)}{\Theta(k)} < \dfrac{\pi L^{\frac d 2}}{(\pi-2)\Phi'_{+}} \end{equation*} in lieu of \eqref{eqn:given_potential}. \end{remark} \begin{corollary}[Characterisation of the first bifurcation]\label{cor:first} Grant $\Phi''\geqslant 0$ and Assumption \ref{as:1}. Let $k^*\in\mathbb{N}$ be such that \[ \dfrac{\tilde W(k^*)}{\Theta(k^*)} = \max \left\{ \dfrac{\tilde W(k)}{\Theta(k)} \ | \ k\in\mathbb{N}. \right\}, \] If \[\dfrac{\tilde W(k^*)}{\Theta(k^*)} > \dfrac{L^{\frac d 2}}{\Phi'(W_0 \rho^* + B)}\] and if for all $k\in\mathbb{N}^d\setminus\{k^*\}$, either $ \tfrac{\tilde W(k^*)}{\Theta(k^*)}\neq \tfrac{\tilde W(k)}{\Theta(k)}$ or $k$ is a permutation of $k^*$, then there exists $\kappa^*$ such that $(\bar\rho_\infty^{\kappa^*},\kappa^*)$ is the first bifurcation point and $\kappa^*$ is the unique positive number such that \begin{equation}\label{eqn:W_tilde_equal2} \dfrac{\tilde W (k^*)}{\Theta(k^*)} = \dfrac{1}{L^{\frac d2} (\Phi_0^{\kappa^*})' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^{\kappa^*} \left(\bar \rho_\infty^{\kappa^*} - \dfrac{ \Phi_0^{\kappa^*}}{L^d}\right)\kappa^* \right)}. \end{equation} \end{corollary} Notice that the value $\kappa^*$ characterised by \eqref{eqn:W_tilde_equal2}, that yields the first bifurcation point in the above corollary, coincide with the very point where the PDE system looses its linear stability, as we discussed in Section \ref{sec:linearstability}. \section{Bifurcations of the full four component model} \label{sec:4component} Utilizing the details derived during the bifurcation analysis of the one component model \eqref{eqn:2}, we now sketch the procedure for finding bifurcation points of the full four component model \eqref{eq:4PDE}. First, note that when $B$ is constant in \eqref{eq:4PDE}, the PDEs for the four different directions $\beta = 1,2,3,$ and $4$, have the same right hand side. With $W\in L^2_S(\mathbb T^d)$, one can check that this system has the same spatially homogeneous stationary states as \eqref{eqn:2}. However, our study of the functional for the mean for the one component model \eqref{eq:onecompfunctional} does not fully characterise the bifurcations for the four component model \eqref{eq:4PDE}. This is due to the fact that the argument of $\Phi $ now is different: $W \ast \bar{\rho} (x) $ is replaced by $m(x) := \tfrac{1}{4}\sum_\beta W^\beta \ast \bar \rho^\beta (x)$, see \eqref{eq:phi}. To be more specific, each component $\rho^1, \rho^2, \rho^3$ and $\rho^4$, will satisfy the calculations in Section \ref{eq:stationarystateequ}, giving \begin{align}\label{eq:rhobetastat} \bar\rho^\beta = \dfrac{1}{Z_\rho} \int_0^{+\infty} s \e^{ -\kappa \frac{ \left( s - \Phi(m + B) \right)^2}{2}}\diff s, \qquad Z_\rho = L^d\int_0^{+\infty} \e^{ -\kappa \frac{ \left( s - \Phi(m + B) \right)^2}{2}}\diff s, \end{align} for each $\beta$. As a consequence, the stationary states are equal, $\rho^1(x)=\rho^2(x)=\rho^3(x)=\rho^4(x)$, since the value of $\Phi(x) = \Phi(m(x)+B)$ is identical in the four different directions. Assuming no spatial dependence of the stationary state and periodicity of $W$ then yields $\bar\rho^\beta = \bar\rho_\infty$, which is the unique spatially homogeneous zero of the functional $\bar{\mathcal{G}}(\bar \rho,\kappa)$ for the mean in the one component case \eqref{eq:onecompfunctional}. For spatially dependent stationary states, however, $\Phi(m(x)+B)$ in \eqref{eq:rhobetastat} and $\Phi(W \ast \bar\rho +B)$ when $\bar{\mathcal{G}}(\bar \rho,\kappa)=0$ in \eqref{eq:onecompfunctional} are not identical, which leads to minor modifications in the bifurcation conditions as we now show. We are only interested in bifurcations from the spatially homogeneous solutions, so it suffices to consider bifurcations of the spatially homogeneous stationary states of the sum $m$. The stationary states of $m$ are zeroes of the functional \begin{align*} \mathcal{Q}(m, \kappa) = m - \sum_\beta W^\beta \ast \left( \dfrac{1}{Z_\rho} \int_0^{+\infty} s \e^{ -\kappa \frac{ \left( s - \Phi(m + B) \right)^2}{2}}\diff s\right). \end{align*} We define the functional $\mathcal{V}(m, \kappa) :=\mathcal{Q}(m + m_\infty^\kappa, \kappa) = \mathcal{Q}(m + W_0\bar\rho_\infty^\kappa, \kappa)$, where $m_\infty^\kappa= W_0\bar\rho_\infty^\kappa$ is the spatially homogeneous stationary state for a given $\kappa$. Like in the one component case, we wish to apply the Crandall--Rabinowitz theorem (Theorem \ref{thm:CR}) to $\mathcal{V}(m, \kappa)$. This can be done by following the proof of Theorem \ref{thm:main} step by step, and making the necessary adjustments. Note that keeping the assumptions on $W$, it makes sense to consider the same space and basis as in the proof of Theorem \ref{thm:main}. Let $h_1,h_2\in L^2_S(\mathbb T^d)$. By following the calculations leading up to, and in the proof of, Theorem \ref{thm:main}, one can check that \begin{align}\label{eq:mderiv} D_m \mathcal{V}(0, \kappa)[h_1] = D_m \mathcal{Q}(W_0\bar \rho_\infty^\kappa, \kappa)[h_1] = h_1 -\frac{ (\Phi_0^\kappa)'}{4} \left(\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \frac{ \Phi_0^\kappa}{L^d}\right)\kappa\right) \sum_\beta W^\beta \ast h_1, \end{align} and \begin{align} \label{eq:mkderiv} D_{m\kappa}^2 & \mathcal{V}(0, \kappa)[h_2] \nonumber \\ = & \, D^2_{m\kappa} \mathcal Q(m_\infty^\kappa,\kappa)[h_2]+ D^2_{mm} \mathcal Q(m_\infty^\kappa,\kappa) \left[W_0\dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa), h_2\right] \nonumber \\ = & \, - (\Phi_0^\kappa)'L^d \left(\bar \rho_\infty^\kappa - \frac{ \Phi_0^\kappa}{L^d}\right)\frac{ \Phi_0^\kappa}{2}\left[ \frac{1}{L^d}g\left(\sqrt{\tfrac{\kappa}{2}} \Phi_0^\kappa\right) -L^d\kappa (\bar \rho_\infty^\kappa)^2 \right] \frac{1}{4} \sum_\beta W^\beta \ast h_2 \\ & + W_0 \dfrac{\diff \bar\rho_\infty}{\diff \kappa}(\kappa) \bigg[-\frac{ (\Phi_0^\kappa)''}{L^d}g\left(\sqrt{\tfrac{\kappa}{2}} \Phi_0^\kappa\right) \nonumber \\ & \qquad \qquad \qquad \,\,\ + ( (\Phi_0^\kappa)')^2L^d\kappa\left(\bar \rho_\infty^\kappa -\frac{ \Phi_0^\kappa}{L^d}\right)\!\!\!\left(\frac{1}{L^d}g\left(\sqrt{\tfrac{\kappa}{2}} \Phi_0^\kappa\right)-L^d\kappa (\bar \rho_\infty^\kappa)^2\right)\!\bigg] \!\frac{1}{4} \!\sum_\beta W^\beta\!\ast h_2 \nonumber \end{align} with $g(\eta)$ as defined in Lemma \ref{lm:g_eta}, \[ \frac{1}{L^d} g\left(\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa\right) = \dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa. \] Now, as in the proof of Theorem \ref{thm:main}, we let $h_1 = \omega_k$, where $\omega_k$ is in the orthonormal basis of $L^2_S(\mathbb T^d)$, in \eqref{eq:mderiv}. Using the fact that $W$ and the shifts $r^\beta$ are coordinate-wise even, we calculate \begin{align*} \sum_\beta W^\beta \ast \omega_k = & \, \sum_\beta \int_{\mathbb T^d} W(x-y-r^\beta) \tfrac{\Theta(k)}{L^{\frac d2}}\prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} y_i\right) \diff y \\ = & \, \sum_\beta \int_{\mathbb T^d} W(x-y-r^\beta) \tfrac{\Theta(k)}{L^{\frac d2}}\prod_{i=1}^{d}\bigg[\cos\left(\tfrac{2\pi k_i}{L} (y_i+r_i^\beta-x_i\right)\cos\left(\tfrac{2\pi k_i}{L} (x_i-r_i^\beta)\right) \\ & - \sin\left(\tfrac{2\pi k_i}{L} (y_i+r_i^\beta-x_i)\right)\sin\left(\tfrac{2\pi k_i}{L} (x_i-r^\beta)\right) \bigg] \diff y \\ = & \, \sum_\beta \int_{\mathbb T^d} W(x-y-r^\beta) \tfrac{\Theta(k)}{L^{\frac d2}}\prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} (y_i+r_i^\beta-x_i\right)\cos\left(\tfrac{2\pi k_i}{L} (x_i-r_i^\beta)\right) \diff y \\ = & \, \sum_\beta \int_{\mathbb T^d} W(x-y-r^\beta) \prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} (y_i+r_i^\beta-x_i\right) \diff y \tfrac{\Theta(k)}{L^{\frac d2}} \prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} (x_i-r_i^\beta)\right) \\ = & \, \tfrac{L^{\frac d2}}{\Theta(k)}\Tilde{W}(k) \sum_\beta \tfrac{\Theta(k)}{L^{\frac d2}} \prod_{i=1}^{d}\left[\cos\left(\tfrac{2\pi k_i}{L} x_i\right)\cos\left(\tfrac{2\pi k_i}{L} r_i^\beta\right)-\sin\left(\tfrac{2\pi k_i}{L} x_i\right)\sin\left(\tfrac{2\pi k_i}{L} r_i^\beta\right)\right] \\ = & \, \tfrac{L^{\frac d2}}{\Theta(k)}\Tilde{W}(k) \omega_k \sum_\beta \prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} r_i^\beta\right), \end{align*} where the last step follows due to the shifts being coordinate-wise even. Then, comparing the resulting expression with the bifurcation condition $\eqref{eqn:W_tilde_equal}$, we arrive at the slightly modified assumption \begin{align}\label{eq:4compbifurcation} \exists ! \ k^* \in \mathbb N^d \quad \mathrm{s.t.} \quad \dfrac{\tilde W (k^*)}{\Theta(k^*)} \frac{1}{4}\sum_\beta \prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} r_i^\beta\right) = \dfrac{1}{L^{\frac d2} (\Phi_0^\kappa)' \left( \dfrac1{L^d} - L^d\bar \rho_\infty^\kappa \left(\bar \rho_\infty^\kappa - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa \right)} \end{align} for $(m_\infty^\kappa,\kappa)=(W_0 \bar \rho_\infty^\kappa,\kappa)$ being a bifurcation point. Setting $h_2= v \in \mathrm{Ker}\big(D_{m} \mathcal V(0,\kappa)\big)$ in \eqref{eq:mkderiv}, we find that the assumptions needed for $D_{m\kappa}^2 \mathcal{V}(0, \kappa)\left[ v\right] \neq 0$ are exactly the same as for the one component model (cf. \eqref{eqn:no_more_idea_for_labels_3}). With $h_1=v$ in \eqref{eq:mderiv} one can easily check that $v \in \mathrm{Ker}\big(D_{m} \mathcal V(0,\kappa)\big)$ has to satisfy \begin{align*} \frac{1}{4}\sum_\beta W^\beta \ast v = \frac{1}{ (\Phi_0^\kappa)' \left(\dfrac{1}{L^d} - L^d\bar \rho_\infty^\kappa\left(\bar \rho_\infty^\kappa - \frac{ \Phi_0^\kappa}{L^d}\right)\kappa\right)}v, \end{align*} such that \begin{align*} D_{m\kappa}^2 \mathcal{V}(0, \kappa)[v] = D_{\bar \rho \kappa}^2 \mathcal{H}(0, \kappa)[v], \end{align*} where $D_{\bar \rho \kappa}^2 \mathcal{H}(0, \kappa)[v]$ can be found in \eqref{eqn:no_more_idea_for_labels_3}. Hence, Theorem \ref{thm:main} is valid, under the same assumptions on $\Phi$, for the four component model \eqref{eq:4PDE} when $\tilde W (k^*)$ in \eqref{eqn:W_tilde_equal} is exchanged with $$ \tilde W (k^*) \frac{1}{4}\sum_\beta\prod_{i=1}^{d}\cos\left(\tfrac{2\pi k_i}{L} r_i^\beta\right). $$ As highlighted in Subsection \ref{sec:linearstability} for \eqref{eqn:2}, this condition coincides with the linear stability threshold obtained in \cite[Theorem 3.6]{CHS} in the two dimensional case $d=2$ when the shifts are coordinate-wise even. Similarly, the higher order derivatives needed to characterise the shape of the branches in Theorem \ref{thm:main_2} are the same as in the one component case. Thus, Theorem \ref{thm:main_2} holds without modification. Finally, this can straightforwardly be extended to Theorem \ref{thm:ebl} with the slightly modified bifurcation condition. \section{The bifurcation condition and patterns in two dimensions}\label{sec:numerics} We now present supplementary insight about the location and shape of the bifurcations described theoretically in the previous sections in the case of two spatial dimensions ($d=2$). We also provide a brief comment on the stability of the spatially homogeneous state and on the stationary hexagonal pattern for \eqref{eqn:2} and \eqref{eq:4PDE} with radially symmetric kernels (see also \cite{CHS}). \begin{figure} \caption{Bifurcation points represented as the crossing of Fourier modes (the black horizontal lines) and the right hand side of \eqref{eqn:W_tilde_equal_0} \label{fig:radsym-vs-nonradsym} \end{figure} \subsection{The bifurcation condition} Theorems \ref{thm:main}, \ref{thm:ebl}, and \ref{thm:Given} give us sufficient conditions for bifurcations, but since the right-hand side of the condition \eqref{eqn:W_tilde_equal_0} depends nonlinearly on $\kappa$, it is not easy to locate the bifurcation points theoretically. Thus, we plot in Figure \ref{fig:radsym-vs-nonradsym}, for a choice of $\Phi$ and two choices of $W$, the right-hand side of this condition in blue as a function of $\kappa$ and the left-hand side as horizontal lines representing Fourier modes. Each crossing between the decreasing blue curve and the horizontal lines yields a possible bifurcation point as described by Theorem \ref{thm:main} and Theorem \ref{thm:ebl}. A zoom of the left plot of Figure \ref{fig:radsym-vs-nonradsym} was presented in Figure \ref{fig:intro-illustration} in the introduction. In all the plots, the vertical dashed line represents the linear stability condition from \cite{CHS}, recalled in Section \ref{sec:linearstability}, which coincides with the location of the first bifurcation (see Corollary \ref{cor:first}). The red line represents the critical value $\kappa_c = \tfrac{2|W_0|^2}{L^{2d}\pi B^2}$ (see Lemma \ref{lm:kappa_small}): no bifurcation can occur for $\kappa \leqslant \kappa_c$ for the right-hand side of the condition \eqref{eqn:W_tilde_equal_0} has the constant value $+\infty$ on $(0,\kappa_c]$. The left plot in Figure \ref{fig:radsym-vs-nonradsym} shows the case of a radially symmetric connectivity kernel. Here, each crossing yields a bifurcation point. The crossings with the dashed black lines denote that there is a unique branch in $L^2_S(\mathbb T^d)$ at that point according to Theorem \ref{thm:main}. Note that this does not imply that this branch is the only branch in $L^2(\mathbb T^d)$ at that point or that there are no coordinate-wise even patterns at the other bifurcation points. As described in Section \ref{sec:high_dim}, each crossing in the left plot of Figure \ref{fig:radsym-vs-nonradsym} gives rise to multiple branches, each with their square symmetry, emanating from the bifurcation point. It can be seen in the right plot of Figure \ref{fig:radsym-vs-nonradsym} that a lot more bifurcation points in $L^2_S(\mathbb T^d)$ are detected with Theorem \ref{thm:main} when $W$ is in $L^2_S(\mathbb T^d)$, but is not radially symmetric, compared to the case of the radially symmetric connectivity to the left. Note that also in this case Theorem \ref{thm:ebl} yields additional bifurcation branches, see Remark \ref{rem:higherdim}. Finally, we note that the only change of the plots in Figure \ref{fig:radsym-vs-nonradsym} in the case of the four component model \ref{eq:4PDE} is that the Fourier modes are re-scaled by a shift-dependent factor according to the bifurcation condition \eqref{eq:4compbifurcation} in Section \ref{sec:4component}. \subsection{Patterns and numerical exploration of stability} Here we plot examples of patterns along some of the bifurcation branches and numerically describe the loss of stability of the spatially homogeneous state of \eqref{eqn:2} through a simple bifurcation diagram in the case of the radially symmetric kernel of Figure \ref{fig:radsym-vs-nonradsym}. We do not provide colorbar scales for the plots in this section as we are only interested in the qualitative shape of the patterns, and not the quantitative aspects. \begin{figure} \caption{Patterns in $L^2_S(\mathbb{T} \label{fig:modepatterns} \end{figure} In Figure \ref{fig:modepatterns} the three functions $\omega(x)$ corresponding to the leftmost (first) crossings in the left plot of Figure \ref{fig:radsym-vs-nonradsym} according to Theorem \ref{thm:main} and Remark \ref{rem:exchange} are depicted. They correspond to the values or equivalence classes $[k^*] = ((4,0), (0,4))$, $[k^*]=((1,4), (4,1))$, and $k^*=(3,3)$ (from left to right in Figure \ref{fig:modepatterns}). Keeping $L=1$, the functions that we plot are \begin{equation*} \omega_{[(4,0)]}(x) = \frac{1}{\sqrt 2}\big(\cos(8\pi x) + \cos(8\pi y)\big), \qquad \omega_{[(1,4)]}(x)= \cos(8\pi x)\cos(2\pi x) + \cos(2\pi x)\cos(8\pi x), \end{equation*} and \begin{equation*} \omega_{(3,3)}(x) = 2\cos(6\pi x)\cos(6\pi y). \end{equation*} Note that the existence of these patterns could also be deduced from Theorem \ref{thm:ebl}, see Examples \ref{ex:1} and \ref{ex:2}. As remarked in Section \ref{sec:high_dim}, the shape of the patterns along the different branches have been characterised in two dimensions \cite{Dionne}, and, as already noted, with a square periodic lattice, there will be branches with patterns consisting of rolls/stripes, simple squares, super squares, or anti-squares. Restricting the functional \eqref{eq:onecompfunctionalH} to the square lattice defined by the periodicity of the problem as we did in Section \ref{sec:high_dim}, the characteristic hexagonal pattern obtained through time evolution of the PDE system \eqref{eq:4PDE} in \cite{CHS} in the case of a radially symmetric connectivity kernel will thus not appear along any of the corresponding bifurcation branches. This can heuristically be explained by the fact that a hexagonal pattern cannot be fitted periodically on the square while at the same time preserve any square symmetries---the ratio between the inradius and circumradius of a hexagon is $\frac{\sqrt{3}}{2}$. A simple Fourier decomposition of a hexagonal pattern which periodically fits on the square $[-0.5,0.5]^2$ is \begin{align}\label{eq:hex} \omega^{\mathrm{hex}}(x_1,x_2) = & \, \cos(6\pi x_1)\cos(6\pi x_2) + \cos(8\pi x_1)\cos(2\pi x_2) + \cos(2\pi x_1)\cos(8\pi x_2)\\ & + \sin(6\pi x_1)\sin(6\pi x_2) - \sin(8\pi x_1)\sin(2\pi x_2) - \sin(2\pi x_1)\sin(8\pi x_2), \nonumber \end{align} but this is not symmetric with respect to the corresponding periodicity lattice. One can check that this pattern is coordinate-wise even with respect to axes rotated clockwise by $\frac{\pi}{12}$. \begin{figure} \caption{Time-transient patterns of the mean by solving the PDE \eqref{eqn:2} \label{fig:timetransient} \end{figure} To illustrate that the hexagonal pattern numerically appears as a stable stationary state of \eqref{eqn:2}, a typical time evolution for the PDE \eqref{eqn:2} can be found in Figure \ref{fig:timetransient}. Here $\bar\rho(x,t)$ is plotted at different times for a $\kappa$ close to the three leftmost bifurcation points of the left plot in Figure \ref{fig:radsym-vs-nonradsym}. The time series is obtained by introducing a machine precision perturbation of spatially homogeneous initial data at one position and then using the same numerical method as in \cite{CHS} on a $64\times64$-grid. We observe that at first the patterns are coordinate-wise even and go through different shapes before stabilising into the bottom left pattern of Figure \ref{fig:timetransient}. The solution stays close to this pattern for some time. Eventually a symmetry breaking event happens (bottom center of Figure \ref{fig:timetransient}) and the solution stabilises into a hexagonal pattern (bottom right of Figure \ref{fig:timetransient}) which numerically has the same coordinate-wise even symmetry axes as \eqref{eq:hex}. As pointed out in \cite{CHS}, numerical experiments indicate that this pattern does not occur continuously as a stable bifurcation branch at a point on the spatially homogeneous branch, see Figure \ref{fig:bifurcationdiagram}. Figure \ref{fig:bifurcationdiagram} shows bifurcation diagrams with respect to $\kappa$. The diagrams are created as in \cite{CHS} for \eqref{eq:4PDE} by numerically evolving \eqref{eqn:2} up to stabilisation to a steady value for a given $\kappa$. Then, the stationary states for increasing or decreasing $\kappa$ are recursively computed for smaller (r2l) or larger (l2r) values of $\kappa$ by taking as initial data the already computed steady state. Indicated by a sudden jump in the difference between $\max_x \bar \rho (x)$ and $\min_x \bar \rho (x)$ in the left plot of Figure \ref{fig:bifurcationdiagram}, there is what appears to be a discontinuous phase transition between the spatially homogeneous steady state and the hexagonal steady state. This is also apparent in the bifurcation diagram of $\|\bar \rho\|_{L^2(\mathbb{T}^2)}$ to the right. Also note the different positions of the jumps for decreasing and increasing $\kappa$, which indicates a hysteresis phenomenon. Finally, the loss of linear stability (see Lemma \ref{rem:linear}) is to the right of both jumps (indicated by a red dot). This was also remarked in \cite{CHS} for \eqref{eq:4PDE}. \begin{figure} \caption{Bifurcation diagrams with the settings of the left plot in Figure \ref{fig:radsym-vs-nonradsym} \label{fig:bifurcationdiagram} \end{figure} \section{Perspectives}\label{sec:future} Although having taken a crucial step towards understanding the patterning of the stochastic neural field models under consideration, we have only touched the surface of what this framework allows to explore. One direction is to be even less restrictive with the symmetry assumptions on the connectivity kernel $W$. A further understanding of the bifurcations is needed to fully understand the emergence and stability of the hexagonal stationary state for \eqref{eqn:2} and \eqref{eq:4PDE} in the case of radially symmetric connectivity kernels. From the discussion in the previous section, it is clear that the systems possess numerically stable hexagonal steady states, and that no hexagonal steady states could be proven to exist when utilising the square periodicity of the problem. Two plausible explanations stand out regarding the emergence as the noise parameter $\sigma$ is varied. One is that there could be other bifurcations occurring along the spatially non-homogeneous branches leading to the hexagonal pattern. This conjecture is supported by the bifurcation diagrams in \cite[Figure 1]{VCF2015} for a related neural field model. This is however by no means evidence that this has to be the case for \eqref{eqn:2} as the model and bifurcation parameter in \cite{VCF2015} do differ from \eqref{eqn:2}. Furthermore, the stable patterns with square periodicity displayed in \cite{VCF2015} are only \emph{almost} hexagonal. The second is that the hexagonal pattern does in fact numerically appear as a local branch bifurcating continuously from the spatially homogeneous branch. A first thought is to again follow the approach of Dionne \cite{Dionne}, but this time restrict the functional to a hexagonal lattice. This is however complicated due to the convolution in \eqref{eqn:2}, which leads to the hexagonal pattern in Figure \ref{fig:timetransient}, having square periodicity. Regarding the time-dependent system \eqref{eqn:2}, we have yet to provide a complete existence and well-posedness study, which is challenging due to the nonlinear boundary condition. Furthermore, nonlinear stability is a completely open and tough question. Finding a solution is crucial to fully understand the dynamics, and this process will require sophisticated analytical tools. Finally, to extend the PDE models considered here to a more realistic setting, an investigation of space correlated noise and time delays is pertinent as well. \appendix \section{General results in bifurcation theory} This appendix is adapted from the material contained in \cite{Kielhofer2012} and is meant as a brief introduction to the bifurcation theory utilised to prove Theorem \ref{thm:main}. The corresponding equivariant bifurcation theory needed in the proof of Theorem \ref{thm:ebl} can be found for example in the book by Chossat and Lauterbach \cite{CL-equivariantbif}. Let $X,Y,Z$ be three real Banach spaces, $U\subset X$, $V \subset Y$ be two open sets and consider a continuous mapping $F:U\times V\to Z$. We provide in this appendix some abstract results about the problem \begin{equation}\label{eqn:bifurcation_general} \mathcal H(x,\kappa) = 0,\qquad x\in U,\ \kappa\in V. \end{equation} We see $X$ as a state of configurations for a system and $Y$ as a set of parameters. Given a solution $(x_0,\kappa_0)$ to \eqref{eqn:bifurcation_general}, we want to know when a small change of the parameters around $\kappa_0$ entails a significant change in the configuration. This can happen only when the implicit function theorem cannot be applied, for example when \[ D_x \mathcal H (x_0,\kappa_0) : X\to Z \] is not bijective. Let us consider the case of a one dimensional parameter space, $V\subset Y = \mathbb{R}$. We make assumptions on the existence of a ``trivial'' branch of solutions to \eqref{eqn:bifurcation_general} and on the smoothness of $\mathcal H$. \begin{hyp}\label{as:trivial} The open set $U$ contains 0 and for all $\kappa\in V$, $\mathcal H(0,\kappa)=0$. \end{hyp} Note that if the branch is of the form $(x(s),\kappa(s))$, with $x(s)$ non-constant, it is possible to rescale the functional $\mathcal H$ to obtain a trivial branch like in the assumption. It is what we do in the beginning of the proof of Theorem \ref{thm:main}. \begin{hyp}\label{as:smooth} We assume that $\mathcal H\in C^2(U\times V,Z)$. For some value $\kappa_0\in V$, we assume that $\mathcal H(\cdot,\kappa_0)$ is a Fredholm operator of index 0 such that \[ \dim\big(\ker(D_x \mathcal H(0,\kappa_0))\big) = \mathrm{codim}\big(\mathrm{range}(D_x \mathcal H(0,\kappa_0)\big) = 1 \] \end{hyp} We now state a result about the existence of a second branch of solutions crossing the trivial branch for some value $\kappa_0\in V$ of the parameter, which is one of the core results that we use in this article. It can be found in \cite[Th I.5.1.]{Kielhofer2012}. \begin{theorem}[Crandall--Rabinowitz Theorem]\label{thm:CR} Grant Assumptions \ref{as:trivial} and \ref{as:smooth}. Assume that \begin{equation*} \ker\big(D_x \mathcal H(0,\kappa_0)\big) = \mathrm{span}(\omega_0),\qquad \omega_0\in U,\ \norme{\omega_0}=1 \end{equation*} and \begin{equation*} D^2_{x\kappa} \mathcal H(0,\kappa_0) [\omega_0] \notin \mathrm{range}\big( D_x \mathcal H(0,\kappa_0) \big). \end{equation*} Then there exists a nontrivial continuously differentiable curve through $(0,\kappa_0)$ \begin{equation}\label{eq:nontrivial} \{\ (x(s),\kappa(s)) \ | \ s\in(-\delta,\delta), \ (x(0),\kappa(0))=(0,\kappa_0), \delta>0 \ \}, \end{equation} such that \begin{equation*} \forall s\in(-\delta,\delta),\quad \mathcal H(x(s),\kappa(s))=0, \end{equation*} and in a neighbourhood $U_1\times V_1 \subset U\times V$ of $(0,\kappa_0)$, all the solutions to \eqref{eqn:bifurcation_general} are either on the trivial solution line or on the nontrivial solution line \eqref{eq:nontrivial}. \end{theorem} We call the intersection $(0,\kappa_0)$ of these two curves a \textit{bifurcation point}. Note that the second order derivative $D^2_{x\kappa} \mathcal H(0,\kappa_0): \omega\mapsto D^2_{x\kappa} \mathcal H(0,\kappa_0)[\omega]$ while applied to the point $(0,\kappa_0)$ lies in $\mathcal L (X,Z)$ the set of bounded linear operators from $X$ to $Z$. With more derivatives of $\mathcal H$, it is possible to harvest more information about the local behaviour of the nontrivial curve of solutions. We state here a particular case of the general results from \cite[Secs. I.5--6]{Kielhofer2012}. \begin{theorem}[Characterisation of the branch]\label{thm:characterisation} Assume the hypotheses of Theorem \ref{thm:CR}, that $\mathcal H\in C^3(U\times V,Z)$, and that $X$ is a Hilbert space endowed with a scalar product $\pscal{\cdot}{\cdot}$. Then, the function $s\mapsto \kappa(s)$ is twice differentiable. The tangent vector to the nontrivial curve at the bifurcation point $(0,\kappa_0)$ is $(\omega_0, \kappa'(0))$, which means that there exists a bounded continuous function $r:\mathrm{span}(\omega_0)\times V \to L_S^2(\mathbb T^d)$ such that $r(0,0)=0$, and \begin{equation*} x(s) = s\omega_0 + r(s\,\omega_0,\kappa(s)),\qquad \lim_{s\to 0} \dfrac{\norme{r(s\,\omega_0,\kappa(s))}}{|s| + |\kappa(s) - \kappa(0)|}=0. \end{equation*} The value $\kappa'(0)$ is given by \begin{equation*} \kappa'(0) = -\dfrac12 \dfrac{\pscal{D^2_{xx} \mathcal H(0,\kappa_0) [\omega_0,\omega_0]}{\omega_0} }{\pscal{D^2_{x\kappa} \mathcal H(0,\kappa_0)[\omega_0]}{\omega_0}}. \end{equation*} Moreover, if $\kappa'(0)=0$, then \begin{equation*} \kappa''(0) = -\dfrac13 \dfrac{\pscal{D^3_{xxx} \mathcal H(0,\kappa_0) [\omega_0,\omega_0,\omega_0]}{\omega_0} }{\pscal{D^2_{x\kappa} \mathcal H(0,\kappa_0)[\omega_0]}{\omega_0}}, \end{equation*} and \begin{equation*} x(s) = s\omega_0 + o(s), \qquad s\in(-\delta,\delta). \end{equation*} \end{theorem} When $\kappa'(0)\neq0$, the bifurcation is called \textit{transcritical}. If $\kappa'(0)=0$, it is called a pitchfork bifurcation, which is called \textit{subcritical} when $\kappa''(0)<0$ and \textit{supercritical} when $\kappa''(0)>0$. \section{A supplementary proof} \begin{proof}[Proof of Lemma \ref{lm:g_eta}] Notice that by \eqref{eqn:rho_0}, \[\dfrac1{L^d} - L^d\bar \rho_\infty \left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa = \dfrac1{L^d} - L^d\bar \rho_\infty \rho_\infty(0), \] and using the expression for the stationary state \eqref{eq:stationarystateequ}, we have \[ \rho_\infty(0) = \dfrac{\sqrt{2\kappa} \e^{-\frac\kappa2 (\Phi_0^\kappa)^2}}{ L^d \sqrt{\pi} \left(1+\mathrm{erf}(\frac{ \Phi_0^\kappa\sqrt{\kappa}}{\sqrt2})\right)}. \] Moreover, using that \[ \int_0^{+\infty} \e^{-\frac\kappa2(s- \Phi_0^\kappa)^2}\diff s = \sqrt{\dfrac{\pi}{2\kappa}}\left(1+\mathrm{erf}\left( \dfrac{ \Phi_0^\kappa\sqrt\kappa}{\sqrt2} \right)\right), \] we compute \begin{align}\label{eqn:rho_bar_eta} \bar \rho_\infty & = \dfrac{\displaystyleplaystyle \int_0^{+\infty} (s- \Phi_0^\kappa) \e^{-\frac\kappa2(s- \Phi_0^\kappa)^2}\diff s + \Phi_0^\kappa\int_0^{+\infty} \e^{-\frac\kappa2(s- \Phi_0^\kappa)^2}\diff s }{\displaystyleplaystyle L^d \int_0^{+\infty} \e^{-\frac\kappa2(s- \Phi_0^\kappa)^2}\diff s } \nonumber \\ & = \dfrac{\e^{-\frac\kappa2 (\Phi_0^\kappa)^2}}{\displaystyleplaystyle\kappa L^d \int_0^{+\infty}\e^{-\frac\kappa2(s- \Phi_0^\kappa)^2}\diff s} + \dfrac{ \Phi_0^\kappa}{L^d} = \dfrac{\sqrt2 \e^{-\frac\kappa2 (\Phi_0^\kappa)^2}}{ L^d \sqrt{\pi\kappa} \left(1+\mathrm{erf}(\frac{ \Phi_0^\kappa\sqrt{\kappa}}{\sqrt2})\right)} + \dfrac{ \Phi_0^\kappa}{L^d} . \end{align} Hence, \begin{align*} \dfrac1{L^d} - L^d\bar \rho_\infty \left(\bar \rho_\infty - \dfrac{ \Phi_0^\kappa}{L^d}\right)\kappa & = \dfrac1{L^d}\left( 1 - \frac{2}{\sqrt{\pi}}\frac{\e^{-\frac\kappa2 (\Phi_0^\kappa)^2}}{1+\mathrm{erf}\left(\tfrac{ \Phi_0^\kappa\sqrt{\kappa}}{\sqrt2}\right)}\left[\frac{1}{\sqrt{\pi}}\frac{\e^{-\frac\kappa2 (\Phi_0^\kappa)^2}}{1+\mathrm{erf}\left(\tfrac{ \Phi_0^\kappa\sqrt{\kappa}}{\sqrt2}\right)}+\sqrt{\frac{\kappa}{2}} \Phi_0^\kappa\right] \right)\\ & = \frac{1}{L^d} g\left(\sqrt{\tfrac{\kappa}{2}} \Phi_0^\kappa\right). \end{align*} Let us denote \[ f(\eta) = \dfrac{1}{\sqrt\pi}\frac{\exp (-\eta^2)}{1+\erf (\eta)}. \] Since the error function $\erf(\eta)$ is differentiable on $\mathbb{R}$ and satisfies \[ \erf'(\eta)= \dfrac{2}{\sqrt\pi} \e^{-\eta^2}, \] then $f$ has the following property: \[ f'(\eta) = - 2 f(\eta) (f(\eta)+\eta).\] It follows that, for all $\eta\in\mathbb{R}_+$, \begin{equation*} g'(\eta) = 2f(\eta) \left[ 2(f(\eta) + \eta)(2f(\eta)+\eta) - 1\right]. \end{equation*} Let us denote \begin{equation}\label{eqn:def_h} w(\eta) = (f(\eta) + \eta)(2f(\eta)+\eta) \end{equation} for all $\eta\in\mathbb{R}_+$, \begin{align*} w'(\eta) & = \big(1 - 2 f(\eta))[f(\eta)+\eta]\big)(2f(\eta)+\eta) + (f(\eta)+\eta)\big( 1-4f(\eta)[f(\eta)+\eta] \big)\\ & = g(\eta)(2f(\eta)+\eta) + (2g(\eta)-1)(f(\eta)+\eta) \\ & = (4g(\eta)-1) f(\eta) + (3g(\eta)-1) \eta. \end{align*} We have \[ w(0) \,=\, \dfrac2\pi \, >\, \frac12,\] so $g'(0) > 0$ and by continuity of $g'$, $g$ is increasing on a neighbourhood of 0. Assume by contradiction that $g$ is not increasing on $\mathbb{R}_+$. Let $\bar\eta = \inf\{ \eta\in(0,+\infty) \ | \ g'(\eta) = 0 \}$. For all $\eta\in[0,\bar\eta]$ we have, \[ 4g(\eta)-1\, >\, 3g(\eta)-1 \, \geqslant\, 3\left(1-\frac{2}{\pi}\right)-1 \,>\, 0. \] Therefore, $w$ is increasing on $[0,\bar\eta]$. But then, \[ g'(\bar\eta) = 4f(\bar\eta) \left( w(\bar\eta) - \frac12\right) > 4f(\bar\eta) \left( w(0) - \frac12\right) > 0, \] which constitutes a contradiction. Therefore, $g$ is an increasing function on $\mathbb{R}_+$. As a consequence, for all $\eta\in\mathbb{R}_+$, \[ 1-\frac{2}{\pi}\, =\, g(0) \, \leqslant \, g(\eta) \, < \, \lim_{\eta\to +\infty} g(\eta)\, =\, 1. \] \end{proof} \end{document}

\begin{document} \begin{abstract} In this article, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying $\mathbb{U}(1)$ symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the $H^3$ level. In addition, we prove a blow-up criterium at the $H^2$ level. By doing so, we improve a result of Huneau and Luk in \cite{hunluk18} on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms. \varepsilonnd{abstract} \maketitle \tableofcontents \section{Introduction} \subsection{Presentation of the results} In this article, we are interesting in solving the Einstein vacuum equations \begin{equation*} R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0 \varepsilonnd{equation*} on a four-dimensional lorentzian manifold $(\mathcal{M}, ^{(4)}g)$, where $R_{\mu\nu}$ and $R$ are the Ricci tensor and the scalar curvature associated to $^{(4)}g$. We assume that the manifold $\mathcal{M}$ amits a translation Killing field, this symmetry being called the $\mathbb{U}(1)$ symmetry. Thanks to this symmetry, the $3+1$ Einstein vacuum equations reduce to the $2+1$ Einstein equations coupled with two scalar fields satisfying a wave map system: \begin{equation}\left\|abel{notre système} \left\|eft\{ \begin{array}{l} \Box_g \varphi =-\frac{1}{2}e^{-4\varphi}\partial^\right\|ho\Omegamega\partial_\right\|ho\Omegamega\\ \Box_g \Omegamega =4\partial^\right\|ho\Omegamega\partial_\right\|ho\varphi\\ R_{\mu\nu}(g) =2\partial_{\mu}\varphi\partial_{\nu}\varphi+\frac{1}{2}e^{-4\varphi}\partial_\mu\Omegamega\partial_\nu\Omegamega \varepsilonnd{array} \right\|ight. \varepsilonnd{equation} where $\varphi$ and $\Omegamega$ are the two scalar fields and $g$ is a $2+1$ lorentzian metric appearing in the decomposition of $^{(4)}g$ (see Section \right\|ef{section U(1) symmetry} for more details). The goal of this paper is to solve the previous system in an elliptic gauge. This particular choice of gauge for the $2+1$ spacetime will be precisely defined in Section \right\|ef{section geometrie}, but let us just say for now that it allows us to recast the Einstein equations as a system of \textit{semilinear elliptic equations} for the metric coefficients. This gauge is therefore especially useful for low-regularity problems, since it offers additional regularity for the metric. More precisely, we obtain two results on this system: local well-posedness with some precise smallness assumptions and a blow-up criterium. Both these results can be roughly stated as follows (see Theorems \right\|ef{theoreme principal} and \right\|ef{theo 2} for some precise statements): \begin{thm}[Rough version of Theorem \right\|ef{theoreme principal}]\left\|abel{rough theo 1} Given admissible initial data for $(\varphi,\Omegamega)$ large in $H^3$ and small enough in $W^{1,4}$ (the smallness threshold being independent of the potentially large $H^3$-norms), there exists a unique solution to \varepsilonqref{notre système} in the elliptic gauge on $[0,T]\times \mathbb{R}^2$ for some $T>0$ depending on the initial $H^3$-norms. \varepsilonnd{thm} \begin{thm}[Rough version of Theorem \right\|ef{theo 2}]\left\|abel{rough theo 2} If the time of existence $T$ of the solution obtained in Theorem \right\|ef{rough theo 1} is finite, then the $H^2$ norm of $(\varphi,\Omegamega)$ diverges at $T$ or the smallness in $W^{1,4}$ no longer holds. \varepsilonnd{thm} \subsection{Strategy of proof and main challenges} Let us briefly discuss the strategy employed to prove the two previous theorems and point out the main challenges we face. We adopt the same global strategy as in the work of Huneau and Luk \cite{hunluk18}, and we will discuss the differences and similarities with this article in Section \right\|ef{section comparaison}. \subsubsection{Theorem \right\|ef{rough theo 1}} In order to prove Theorem \right\|ef{rough theo 1}, we need to solve the Einstein equations in the elliptic gauge. As the name of the gauge suggests, the system \varepsilonqref{notre système} then reads \begin{equation}\left\|abel{système avec jauge} \left\|eft\{ \begin{array}{l} \Box_g U =(\partial U)^2 \\ \mathscr{D}elta g = (\partial U)^2 + (\partial g)^2 \varepsilonnd{array} \right\|ight. \varepsilonnd{equation} where $U$ denotes either $\varphi$ or $\Omegamega$ and in the second equation $g$ denotes any metric coefficient. One of the main challenges of solving such a system is therefore the inversion of the Laplacian operator on a unbounded set, here $\mathbb{R}^2$. Indeed this will imply that some of the metric coefficients, the lapse $N$ and the conformal factor $\gamma$ (see Section \right\|ef{section geometrie} for their definitions), presents some logarithmic growth at spacelike infinity. To counteract these growth, we work in the whole paper with weighted Sobolev spaces (see Definition \right\|ef{wss}). One major aspect of Theorem \right\|ef{rough theo 1} is that the smallness assumed for the initial data is only at the $W^{1,4}$ level, while their higher order norms can be arbitrarily large. It is quite unusual to require some smallness on the initial data to only prove \textit{local} existence, usually one would only ask for smallness on the time of existence. Here however, the smallness of the time of existence can only be of help when performing energy estimates for the hyperbolic part of \varepsilonqref{système avec jauge}. When dealing with the non-linearities in the elliptic part of \varepsilonqref{système avec jauge}, we rely on the smallness of the solution to close the hierarchy of estimates we introduce. However, one of the strength of our result is that the smallness of the initial data is only assumed for their first derivatives in $L^4$ topology. The higher order norm, i.e the $L^2$ norm of their second and third derivatives can be large, and this largeness is \textit{not} compensated by the smallness ot the initial data, which concretely means that the smallness threshold in Theorem \right\|ef{rough theo 1} doesn't depend on the $H^3$ norm of the initial data. This initial data regime (largeness in $H^3$ not compensated by smallness in $W^{1,4}$) is motivated by the main application of this article, namely to the construction of high-frequency spacetimes in the context of the Burnett conjecture in general relativity. See Huneau and Luk's article \cite{hunluk} for the application of the present article and \cite{bur89} for the original paper of Burnett. \par\left\|eavevmode\par Despite the particularities of the elliptic gauge we just discussed, the global strategy to solve the Einstein vacuum equations is standard: \begin{itemize} \item we first solve the \textit{constraint equations} and by doing so construct initial data for the metric on the slice $\{ t=0\}$ which in particular satisfies the gauge conditions, \item then, we solve a \textit{reduced system}, which in our case is a coupled system of elliptic, wave and transport equations, \item finally, we prove using the Bianchi identity that solving the reduced system actually implies the full Einstein vacuum equations and the propagation of the gauge conditions. \varepsilonnd{itemize} As a final comment, note that wave map structure of the hyperbolic part of \varepsilonqref{système avec jauge} plays no role in the proof of Theorem \right\|ef{rough theo 1}. \subsubsection{Theorem \right\|ef{rough theo 2}} Inversely, the wave map structure of the coupling between the wave equations for $\varphi$ and $\Omegamega$ is at the heart of Theorem's \right\|ef{rough theo 1} proof. This result basically means that the $H^2$ norm of the initial data controls the time of existence of the solution (as long as the smallness in $W^{1,4}$ holds), whereas we need $H^3$ regularity to prove local existence. This is not a consequence of the standard energy method for the wave equation, since in dimension 2 it only allows for $H^{2+\varepsilon}$ regularity. Reaching $H^2$ requires therefore to use another structure, in our case the wave map structure of the hyperbolic part of \varepsilonqref{système avec jauge}. Since the work of Choquet-Bruhat in \cite{CBwavemaps} it is well-known that we can associate to any wave map systems a \textit{third order energy estimate}, which we crucially use to reach $H^2$. As explain above, we rely on the smallness of the initial data to prove local-existence. This requirement has the following consequence: we are unable to prove local well-posedness at the $H^2$ level. Indeed, in order to obtain such a result, we would need (in addition to the third order energy estimate) to propagate the smallness in $W^{1,4}$ through the wave map system using only $H^2$ norms. This is not possible in dimension 2 using only energy estimates. Therefore, we need to assume that the $W^{1,4}$ smallness is propagated, which explains why we "only" prove a blow-up criterium and not local well-posedness at the $H^2$ level. \subsection{Relation to previous works} In this section we discuss the link of our work with the litterature. To say it briefly, the proof of Theorem \right\|ef{rough theo 1} draws from \cite{hunluk18} and the proof of Theorem \right\|ef{rough theo 2} uses tehniques from Choquet-Bruhat. \subsubsection{An improvement of \cite{hunluk18}}\left\|abel{section comparaison} This work has a lot of common points with the work of Huneau and Luk in \cite{hunluk18}, where they also study the system \varepsilonqref{notre système}. In this section, let us detail the similarities and differences between these two works. \par\left\|eavevmode\par The system actually solved in \cite{hunluk18} is the Einstein \textit{null dust} system in \textit{polarized} $\mathbb{U}(1)$ symmetry. The polarized assumption implies $\Omegamega=0$, and thus simplifies the hyperbolic part of the Einstein equations: a classical linear wave equation replaces our wave map system and its non-linear coupling associated to the non-polarized case we study here. The Einstein null dust system is a particular case of Einstein Vlasov system and is translated as follows: the system studied in \cite{hunluk18} is coupled with some transport equations for massless particles along null geodesics. This involves the solving of the eikonal equation and thus requires the use of the null structure in $2+1$ dimension to avoid a loss of derivatives. Since we solve the Einstein \textit{vacuum} equations, this difficulty disappears in our work. \par\left\|eavevmode\par As explained earlier in this introduction, the actual structure of the hyperbolic part of \varepsilonqref{notre système} doesn't influence the proof of the local existence of solutions. The proof given here nevertheless differs from the one of \cite{hunluk18} because of the differences in terms of regularity of the initial data. In \cite{hunluk18}, the initial data enjoy $H^4$ regularity and are small in $W^{1,\infty}$. This should be compared to our assumptions: $H^3$ regularity with smallness in $W^{1,4}$ only. Because of this fact, the hierarchy of estimates we introduce during the bootstrap argument differs from the one introduced in \cite{hunluk18}. \subsubsection{Symmetry and wave maps} As explained in the seventh chapter of the appendix of \cite{cho09}, the presence of a symmetry group acting on the spacetime generically implies the reduction of the Einstein vacuum equations into a coupled system between some Einstein-type equations and a wave map system. This is in particular the case for the $\mathbb{U}(1)$ symmetry. In \cite{Malone}, Moncrief performed this reduction and in \cite{CBM} Choquet-Bruhat and Moncrief prove local-existence at the $H^2$ level for a manifold of the form $\mathbb{R}_t \times \Sigma \times \mathbb{U}(1)$ where $\Sigma$ is a compact two-dimensional manifold. The compactness of $\Sigma$ allows them to use Schauder fixed point theorem, thus avoiding the need for some initial smallness. This has to be compared to the present work, where we need some initial smallness to solve the PDE system. As explained earlier in this introduction, the wave map structure is particularly important for the proof of Theorem \right\|ef{rough theo 2}. Indeed, as noted by Choquet-Bruhat in \cite{CBwavemaps} in the most general case, it is always possible to associate to any wave map system a third order energy estimate (see also \cite{MR2387237}). \section{Geometrical setting} In this section, we first introduce our notations, and then we present the $\mathbb{U}(1)$ symmetry and the elliptic gauge. \subsection{Notations} In this section we introduce the notations of this article. We will be working on $\mathcal{M}\vcentcolon=I\times\mathbb{R}^2$, where $I\subset\mathbb{R}$ is an interval. This space will be given a coordinate system $(t,x^1,x^2)$. We will use $x^i$ with lower case Latin index $i=1,2$ to denote the spatial coordinates. \paragraph{Convention with indices :} \begin{itemize} \item Lower case Latin indices run through the spatial indices 1, 2, while lower case Greek indices run through all the spacetime indices. Moreover, repeat indices are always summed over their natural range. \item Lower case Latin indices are always raised and lowered with respect to the standard Euclidean metric $\mathrm{d}elta_{ij}$, while lower case Greek indices are raised and lowered with respect to the spacetime metric $g$. \varepsilonnd{itemize} \paragraph{Differential operators :} \begin{itemize} \item For a function $f$ defined on $\mathbb{R}^{2+1}$, we set $\partial f=(\partial_t f,\nabla f)$, where $\nabla f$ is the usual spatial gradient on $\mathbb{R}^2$. Samewise, $\mathscr{D}elta$ denotes the standard Laplacian on $\mathbb{R}^2$. If $A=(a_1,a_2)$ and $B=(b_1,b_2)$ are two vectors of $\mathbb{R}^2$, we use the dot notation for their scalar product \begin{equation*} A\cdot B=a_1b_1+a_2b_2= \mathrm{d}elta^{ij}a_ib_j. \varepsilonnd{equation*} The notation $|\cdot|$ is reserved for the norm associated to this scalar product, meaning $|A|^2=A\cdot A$. \item $\mathcal{L}$ denotes the Lie derivatives, $D$ denotes the Levi-Civita connection associated to the spacetime metric $g$, and $\Box_g$ denotes the d'Alembertian operator on functions : \begin{equation*} \Box_gf=\frac{1}{\sqrt{|\mathrm{d}et(g)|}}\partial_{\mu}\left\|eft( \left\|eft(g^{-1}\right\|ight)^{\mu\nu}\sqrt{|\mathrm{d}et(g)|}\partial_{\nu}f\right\|ight). \varepsilonnd{equation*} \item $L$ denotes the euclidean conformal Killing operator acting on vectors on $\mathbb{R}^2$ to give a symmetric traceless (with respect to $\mathrm{d}elta$) covariant 2-tensor : \begin{equation*} (L\xi)_{ij}\vcentcolon=\mathrm{d}elta_{j\varepsilonll}\partial_i\xi^{\varepsilonll}+\mathrm{d}elta_{i\varepsilonll}\partial_j\xi^{\varepsilonll}-\mathrm{d}elta_{ij}\partial_k\xi^k. \varepsilonnd{equation*} \varepsilonnd{itemize} \paragraph{Functions spaces :} We will work with standard functions spaces $L^p$, $H^k$, $C^m$, $C^{\infty}_c$, etc., and assume their standard definitions. We use the following convention : \begin{itemize} \item All function spaces will be taken on $\mathbb{R}^2$ and the measures will be the 2D-Lebesgue measure. \item When applied to quantities defined on a spacetime $I\times\mathbb{R}^2$, the norms $L^p$, $H^k$, $C^m$ denote fixed-time norms. In particular, if in an estimate the time $t\in I$ in question is not explicitly stated, then it means that the estimate holds for all $t\in I$ for the time interval $I$ that is appropriate for the context. \varepsilonnd{itemize} We will also work in weighted Sobolev spaces, which are well-suited to elliptic equations. We recall here their definition, together with the definition of weighted Hölder space. The properties of these spaces that we need are listed in Appendix \right\|ef{appendix B}. We use the standard notation $\left\|angle x \right\|angle=\left\|eft( 1+|x|^2\right\|ight)^{\frac{1}{2}}$ for $x\in\mathbb{R}^2$. \begin{mydef}\left\|abel{wss} Let $m\in\mathbb{N}$, $1<p<\infty$, $\mathrm{d}elta\in\mathbb{R}$. The weighted Sobolev space $W^{m,p}_{\mathrm{d}elta}$ is the completion of $C^{\infty}_0$ under the norm \begin{equation*} \| u\|_{W^{m,p}_{\mathrm{d}elta}}=\sum_{|\beta|\left\|eq m} \left\|eft\| \left\|angle x \right\|angle^{\mathrm{d}elta+\beta}\nabla^{\beta}u \right\|ight\|_{L^p}. \varepsilonnd{equation*} We will use the notation $H^m_{\mathrm{d}elta}=W^{m,2}_{\mathrm{d}elta}$, $L^p_{\mathrm{d}elta}=W^{0,p}_{\mathrm{d}elta}$ and $W^{m,p}$ denotes the standard Sobolev spaces on $\mathbb{R}^2$. The weighted Hölder space $C^m_{\mathrm{d}elta}$ is the completion of $C^m_c$ under the norm \begin{equation*} \| u\|_{C^m_{\mathrm{d}elta}}=\sum_{|\beta|\left\|eq m} \left\|eft\| \left\|angle x \right\|angle^{\mathrm{d}elta+\beta}\nabla^{\beta}u \right\|ight\|_{L^{\infty}}. \varepsilonnd{equation*} For a covariant 2-tensor $A_{ij}$ tangential to $\mathbb{R}^2$, we use the convention : \begin{equation*} \| A\|_{X}=\sum_{i,j=1,2} \| A_{ij}\|_{X}, \varepsilonnd{equation*} where $X$ stands for any function spaces defined above. \varepsilonnd{mydef} We denote by $B_r$ the ball in $\mathbb{R}^2$ of radius $r$ centered at 0. \subsection{Einstein vacuum equations with a translation Killing field}\left\|abel{section U(1) symmetry} In this section, we present the $\mathbb{U}(1)$ symmetry. From now on, we consider a Lorentzian manifold $(I\times\mathbb{R}^3,^{(4)}g)$, where $I\subset\mathbb{R}$ is an interval, and $^{(4)}g$ is a Lorentzian metric, for which $\partial_3$ is a Killing field. Following the Appendix VII of \cite{cho09}, this is equivalent to say that $^{(4)}g$ has the following form : \begin{equation} ^{(4)}g=e^{-2\varphi}g+e^{2\varphi}(\mathrm{d} x^3+A_\alpha\mathrm{d} x^\alpha)^2, \varepsilonnd{equation} where $\varphi:I\times\mathbb{R}^2\left\|ongrightarrow\mathbb{R}$ is a scalar function, $g$ is a Lorentzian metric on $I\times\mathbb{R}^2$ and $A$ is a 1-form on $I\times\mathbb{R}^2$. The \textit{polarized} $\mathbb{U}(1)$ symmetry is the case where $A=0$. We extend $\varphi$ to a function on $I\times\mathbb{R}^3$ in such a way that $\varphi$ does not depend on $x^3$. Given this ansatz of the metric, the vector field $\partial_3$ is Killing and hypersurface orthogonal. Assuming that the metric $^{(4)}g$ satisfies the Einstein vacuum equations, i.e $R_{\mu\nu}(^{(4)}g)=0$, one can prove that there exists a function $\Omegamega$ such that \begin{equation*} F=-e^{-3\varphi}*\mathrm{d}\Omegamega \varepsilonnd{equation*} where $F_{\alpha\beta}=\partial_\alpha A_\beta-\partial_\beta A_\alpha$. \par\left\|eavevmode\par The Einstein vacuum equations for $(I\times\mathbb{R}^3,^{(4)}g)$ are thus equivalent to the following system of equations : \begin{equation}\left\|abel{EVE} \left\|eft\{ \begin{array}{l} \Box_g \varphi =-\frac{1}{2}e^{-4\varphi}\partial^\right\|ho\Omegamega\partial_\right\|ho\Omegamega\\ \Box_g \Omegamega =4\partial^\right\|ho\Omegamega\partial_\right\|ho\varphi\\ R_{\mu\nu}(g) =2\partial_{\mu}\varphi\partial_{\nu}\varphi+\frac{1}{2}e^{-4\varphi}\partial_\mu\Omegamega\partial_\nu\Omegamega \varepsilonnd{array}. \right\|ight. \varepsilonnd{equation} Solving the system \varepsilonqref{EVE} is the goal of this article. Note that the last equation of \varepsilonqref{EVE} is actually the Einstein equation $G_{\mu\nu}(g)=T_{\mu\nu}$ with the following stress-energy-momentum tensor : \begin{equation} T_{\mu\nu}=2\partial_{\mu}\varphi\partial_{\nu}\varphi-g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi+\frac{1}{2}e^{-4\varphi}\left\|eft( 2\partial_{\mu}\Omegamega\partial_{\nu}\Omegamega-g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\Omegamega\partial_{\beta}\Omegamega\right\|ight).\left\|abel{tenseur energie impulsion } \varepsilonnd{equation} \subsection{The elliptic gauge}\left\|abel{section geometrie} In this section, we present the elliptic gauge. We first write the $(2+1)$-dimensional metric $g$ on $\mathcal{M}\vcentcolon=I\times\mathbb{R}^2$ in the usual form : \begin{equation} g=-N^2\mathrm{d} t^2+\Bar{g}_{ij}\left\|eft(\mathrm{d} x^i+\beta^i\mathrm{d} t \right\|ight)\left\|eft(\mathrm{d} x^j+\beta^j\mathrm{d} t \right\|ight). \varepsilonnd{equation} Let $\Sigma_t\vcentcolon=\varepsilonnstq{(s,x)\in\mathcal{M}}{s=t}$ and $e_0\vcentcolon=\partial_t-\beta^i\partial_i$, which is a future directed normal to $\Sigma_t$. The function $N$ is called the lapse and the vector field $\beta$ is the shift. We introduce $\mathbf{T}\vcentcolon=\frac{e_0}{N}$, the unit future directed normal to $\Sigma_t$. We introduce the second fundamental form of the embedding $\Sigma_t\xhookrightarrow{}\mathcal{M}$ \begin{equation} K_{ij}\vcentcolon=-\frac{1}{2N}\mathcal{L}_{e_0}\Bar{g}_{ij}.\left\|abel{Kij} \varepsilonnd{equation} We decompose $K$ into its trace and traceless part : \begin{equation} K_{ij}=H_{ij}+\frac{1}{2}\Bar{g}_{ij}\tau,\left\|abel{def H} \varepsilonnd{equation} where $\tau=\mathrm{tr}_{\Bar{g}}K$. We introduce the following gauge conditions, which define the elliptic gauge : \begin{itemize} \item $\Bar{g}$ is conformally flat, i.e there exists a function $\gamma$ such that \begin{equation}\left\|abel{g bar} \Bar{g}_{ij}=e^{2\gamma}\mathrm{d}elta_{ij}. \varepsilonnd{equation} \item the hypersurfaces $\Sigma_t$ are maximal, which means that $K$ is traceless, i.e \begin{equation} \tau=0. \varepsilonnd{equation} \varepsilonnd{itemize} Thus, the metric takes the following form : \begin{equation} g=-N^2\mathrm{d} t^2+e^{2\gamma}\mathrm{d}elta_{ij}\left\|eft(\mathrm{d} x^i+\beta^i\mathrm{d} t \right\|ight)\left\|eft(\mathrm{d} x^j+\beta^j\mathrm{d} t \right\|ight).\left\|abel{metrique elliptique} \varepsilonnd{equation} The main computations in the elliptic gauge are performed in Appendix \right\|ef{appendix A}. They show that \varepsilonqref{EVE} is schematically of the form \begin{equation}\left\|abel{EVE2} \left\|eft\{ \begin{array}{l} \Box_g U =(\partial U)^2 \\ \mathscr{D}elta g = (\partial U)^2+ (\partial g)^2 \varepsilonnd{array} \right\|ight. \varepsilonnd{equation} where $U$ denotes either $\varphi$ or $\Omegamega$ and in the second equation $g$ denotes any metric coefficient. \section{Main results} \subsection{Initial data}\left\|abel{subsection initial data} We now describe our choice of initial data for the system \varepsilonqref{EVE}. We distinguish the \textit{admissible} initial data, and the \textit{admissible free} initial data. For the rest of this paper, we choose a fixed smooth cutoff function $\chi:\mathbb{R}\to\mathbb{R}$ such that $\chi_{|[-1,1]}=0$ and $\chi_{|[-2,2]}=0$. The notation $\chi\left\|n$ stands for the function $x\in\mathbb{R}^2\left\|ongmapsto \chi(|x|)\left\|n(|x|)$. \begin{mydef}[Admissible initial data] For $-1<\mathrm{d}elta<0$ and $R>0$, an admissible initial data set with respect to the elliptic gauge for \varepsilonqref{EVE} consists of \begin{enumerate} \item A conformally flat intrinsic metric $(e^{2\gamma}\mathrm{d}elta_{ij})_{|\Sigma_0}$ which admits a decomposition \begin{equation*} \gamma=-\alpha\chi\left\|n+\Tilde{\gamma}, \varepsilonnd{equation*} where $\alpha\geq 0$ is a constant and $\Tilde{\gamma}\in H^{4}_{\mathrm{d}elta}$. \item A second fundamental form $(H_{ij})_{|\Sigma_0}\in H^{3}_{\mathrm{d}elta+1}$ which is traceless. \item $\left\|eft( \mathbf{T}\varphi,\nabla\varphi \right\|ight)_{|\Sigma_0}\in H^2$, compactly supported in $B_R$. \item $\left\|eft( \mathbf{T}\Omegamega,\nabla\Omegamega \right\|ight)_{|\Sigma_0}\in H^2$, compactly supported in $B_R$. \item $\gamma$ and $H$ are required to satisfy the following constraint equations : \begin{align*} \partial^iH_{ij}&=-2e^{2\gamma}\mathbf{T}\varphi\partial_j\varphi-\frac{1}{2}e^{-4\varphi+2\gamma}\mathbf{T}\Omegamega\partial_j\Omegamega, \\ \mathscr{D}elta\gamma&=-\frac{e^{-2\gamma}}{2}\vert H\vert^2-e^{2\gamma}\left\|eft(\mathbf{T}\varphi\right\|ight)^2-\vert\nabla\varphi\vert^2-\frac{e^{-4\varphi}}{4}\left\|eft( e^{-2\gamma}\left\|eft(\mathbf{T}\Omegamega\right\|ight)^2+|\nabla\Omegamega|^2 \right\|ight). \varepsilonnd{align*} \varepsilonnd{enumerate} \varepsilonnd{mydef} We recall that the constraint equations for the Einstein Vacuum Equations are $G_{00}=T_{00}$ and $G_{0i}=T_{0i}$, where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor, and $T_{\mu\nu}$ is the stress-energy-momentum tensor associated to the matter fields $\varphi$ and $\Omegamega$, according to the RHS of system \varepsilonqref{EVE}. The fact that these equations reduces in the elliptic gauge to the previous equations on $H$ and $\gamma$ can be proved using the computations done in Appendix \right\|ef{appendix A}. \par\left\|eavevmode\par We define the notion of admissible free initial data as follows : \begin{mydef}[Admissible free initial data] We set $\mathring{\varphi}=e^{2\gamma}\mathbf{T}\varphi$ and $\mathring{\Omegamega}=e^{2\gamma}\mathbf{T}\Omegamega$, where $\gamma$ is as in \varepsilonqref{metrique elliptique}. For $-1<\mathrm{d}elta<0$ and $R>0$, an admissible free initial data set with respect to the elliptic gauge for \varepsilonqref{EVE} is given by $(\mathring{\varphi},\nabla\varphi)_{|\Sigma_0}\in H^2$ and $(\mathring{\Omegamega},\nabla\Omegamega)_{|\Sigma_0}\in H^2$, all compactly supported in $B_R$, satisfying \begin{equation}\left\|abel{orthogonality condition} \int_{\mathbb{R}^2}\left\|eft(2\mathring{\varphi}\partial_j\varphi+\frac{1}{2}e^{-4\varphi}\mathring{\Omegamega}\partial_j\Omegamega\right\|ight)\mathrm{d} x=0, \quad j=1,2. \varepsilonnd{equation} \varepsilonnd{mydef} The interest of the admissible free initial data is that we can construct from them a set of admissible initial data, which in particular satisfies the constraint equations. Note that instead of prescribing $\mathbf{T} \varphi$ and $\mathbf{T}\Omegamega$, we prescribe a suitable rescaled version of them, which allows the decoupling of the two constraint equations : we will first solve for $H$ and then for $\gamma$. \subsection{Statement of the theorems} The following is our main result on local well-posedness for \varepsilonqref{EVE}. \begin{thm}\left\|abel{theoreme principal} Let $-1<\mathrm{d}elta<0$ and $R>0$. Given an admissible free initial data set such that \begin{equation*} \left\|eft\| \mathring{\varphi} \right\|ight\|_{L^4}+\left\|eft\| \nabla\varphi \right\|ight\|_{L^4}+\left\|eft\| \mathring{\Omegamega} \right\|ight\|_{L^4}+\left\|eft\| \nabla\Omegamega \right\|ight\|_{L^4}\left\|eq \varepsilon, \varepsilonnd{equation*} and \begin{equation*} C_{high}\vcentcolon=\left\|eft\| \mathring{\varphi} \right\|ight\|_{H^2}+\left\|eft\| \nabla\varphi \right\|ight\|_{H^2}+\left\|eft\| \mathring{\Omegamega} \right\|ight\|_{H^2}+\left\|eft\| \nabla\Omegamega \right\|ight\|_{H^2}<\infty, \varepsilonnd{equation*} for any $C_{high}$, there exists a constant $\varepsilon_{0}=\varepsilon_{0}(\mathrm{d}elta,R)>0$ independent of $C_{high}$ and a time $T=T(C_{high},\mathrm{d}elta,R)>0$ such that, if $0<\varepsilon\left\|eq\varepsilon_{0}$, there exists a unique solution to \varepsilonqref{EVE} in elliptic gauge on $[0,T]\times\mathbb{R}^2$. Moreover, defining $\mathrm{d}elta'=\mathrm{d}elta-\varepsilon$, there exists a constant $C_h=C_h(C_{high},\mathrm{d}elta,R)>0$ such that \begin{itemize} \item The fields $\varphi$ and $\Omegamega$ satisfy for all $t\in [0,T]$ \begin{align*} \left\|eft\| \partial_t^2\varphi \right\|ight\|_{H^1}+\left\|eft\| \partial_t\varphi \right\|ight\|_{H^2}+\left\|eft\| \nabla\varphi \right\|ight\|_{H^2}&\left\|eq C_h,\\ \left\|eft\| \partial_t^2\Omegamega \right\|ight\|_{H^1}+\left\|eft\| \partial_t\Omegamega \right\|ight\|_{H^2}+\left\|eft\| \nabla\Omegamega \right\|ight\|_{H^2}&\left\|eq C_h, \varepsilonnd{align*} and their supports are both included in $ J^+\left\|eft(\Sigma_0\cap B_R\right\|ight)$, where $J^+$ denotes the causal future. \item The metric components $\gamma$ and $N$ can be decomposed as \begin{equation*} \gamma=-\alpha\chi\left\|n+\Tilde{\gamma},\quad N=1+N_a\chi\left\|n+\Tilde{N}, \varepsilonnd{equation*} with $\alpha\geq 0$ and $N_a(t)\geq 0$ a function of $t$ alone. \item $\gamma$, $N$ and $\beta$ satisfy the following estimates for all $t\in[0,T]$ : \begin{align*} |\alpha|+\left\|eft\|\Tilde{\gamma}\right\|ight\|_{H^4_{\mathrm{d}elta}}+\left\|eft\|\partial_t\Tilde{\gamma}\right\|ight\|_{H^3_{\mathrm{d}elta}}+\left\|eft\|\partial_t^2\Tilde{\gamma}\right\|ight\|_{H^2_{\mathrm{d}elta}} & \left\|eq C_h,\\ \left\|eft|N_a\right\|ight|+\left\|eft|\partial_tN_a\right\|ight|+\left\|eft|\partial_t^2N_a\right\|ight| & \left\|eq C_h,\\ \left\|eft\|\Tilde{N}\right\|ight\|_{H^4_{\mathrm{d}elta}}+\left\|eft\|\partial_t\Tilde{N}\right\|ight\|_{H^3_{\mathrm{d}elta}}+\left\|eft\|\partial_t^2\Tilde{N}\right\|ight\|_{H^2_{\mathrm{d}elta}} & \left\|eq C_h,\\ \left\|eft\| \beta \right\|ight\|_{H^4_{\mathrm{d}elta'}} +\left\|eft\| \partial_t\beta \right\|ight\|_{H^3_{\mathrm{d}elta'}}+\left\|eft\| \partial_t^2\beta \right\|ight\|_{H^2_{\mathrm{d}elta'}}&\left\|eq C_h. \varepsilonnd{align*} \item The following conservation laws hold : \begin{align} \int_{\mathbb{R}^2}\left\|eft(4e^{2\gamma}\mathbf{T}\varphi\partial_j\varphi+e^{-4\varphi+2\gamma}\mathbf{T}\Omegamega\partial_j\Omegamega\right\|ight)\mathrm{d} x&=0,\left\|abel{CL1}\\ \int_{\mathbb{R}^2}\left\|eft(2e^{-2\gamma}|H|^2+4e^{2\gamma}(\mathbf{T}\varphi)^2+e^{-4\varphi+2\gamma}(\mathbf{T}\Omegamega)^2+4|\nabla\varphi|^2+e^{-4\varphi}|\nabla\Omegamega|^2 \right\|ight)\mathrm{d} x &=4\alpha.\left\|abel{CL2} \varepsilonnd{align} \varepsilonnd{itemize} \varepsilonnd{thm} This theorem has the following corollary, which basically states that if we want to have $T=1$, it suffices to take $C_{high}$ small enough. We will omit the details of its proof because it is actually simplier than the proof of Theorem \right\|ef{theoreme principal}. \begin{coro}\left\|abel{CORO} Suppose the assumptions of Theorem \right\|ef{theoreme principal} hold. There exists $\varepsilon_{small}=\varepsilon_{small}(\mathrm{d}elta,R)>0$ such that if $C_{high}$ and $\varepsilon$ in Theorem \right\|ef{theoreme principal} satisfy \begin{equation*} C_{high},\varepsilon\left\|eq\varepsilon_{small}, \varepsilonnd{equation*} then the unique solution exists in $[0,1]\times\mathbb{R}^2$. Moreover, there exists $C_0=C_0(\mathrm{d}elta,R)$ such that all the estimates in Theorem \right\|ef{theoreme principal} hold with $C_h$ replaced by $C_0\varepsilon$. \varepsilonnd{coro} \par\left\|eavevmode\par We also prove the following theorem, which is a blow-up criterium (See the introduction of Section \right\|ef{section theo 2} for a discussion of this theorem) : \begin{thm}\left\|abel{theo 2} Let $T>0$ be the maximal time of existence of the solution of \varepsilonqref{EVE} obtained in Theorem \right\|ef{theoreme principal}. If $T<+\infty$ and $\varepsilon_0$ is small enough (still independent of $C_{high}$) then one of the following holds : \begin{enumerate}[label=\right\|oman*)] \item $\sup_{[0,T)} \left\|eft( \left\| \partial \varphi \right\|_{H^1} + \left\| \partial\Omegamega \right\|_{H^1} \right\|ight)=+\infty$, \item $\sup_{[0,T)} \left\|eft( \left\| \partial \varphi \right\|_{L^4} + \left\| \partial\Omegamega \right\|_{L^4} \right\|ight)>\varepsilon_0$, \varepsilonnd{enumerate} \varepsilonnd{thm} As said in the introduction, one major feature of these two theorems is that the smallness constant $\varepsilon_0$ does not depend on $C_{high}$. Note in contrast that the time of existence $T$ does depend on $C_{high}$. \subsection{The reduced system}\left\|abel{section reduced system} In order to solve the Einstein Equations, we will first solve the following system, which we call the reduced system. It is identical to the one introduced in \cite{hunluk18}. \begin{align} \mathscr{D}elta N&=e^{-2\gamma}N\vert H\vert^2+\frac{\tau^2}{2}e^{2\gamma}N+\frac{2e^{2\gamma}}{N}(e_0\varphi)^2+\frac{e^{2\gamma-4\varphi}}{2N}(e_0\Omegamega)^2,\left\|abel{EQ N} \\ L\beta&=2e^{-2\gamma}NH,\left\|abel{EQ beta} \\ N\tau&=-2e_0\gamma+\mathrm{div}\beta,\left\|abel{EQ tau} \\ e_0H_{ij}&=-2e^{-2\gamma}NH_i^{\;\,\varepsilonll}H_{j\varepsilonll}+\partial_{(j}\beta^kH_{i)k}-\frac{1}{2}(\partial_i\Bar{\Omegatimes}\partial_j)N+(\mathrm{d}elta_i^k\Bar{\Omegatimes}\partial_j\gamma)\partial_kN\nonumber\\&\qquad\qquad\qquad\qquad\qquad-(\partial_i\varphi\Bar{\Omegatimes}\partial_j\varphi)N-\frac{1}{4}e^{-4\varphi}(\partial_i\Omegamega\Bar{\Omegatimes}\partial_j\Omegamega)N,\left\|abel{EQ H} \\ \mathbf{T}^2\gamma-e^{-2\gamma}\mathscr{D}elta\gamma&=-\frac{\tau^2}{2}+\frac{1}{2}\mathbf{T}\left\|eft(\frac{\mathrm{div}(\beta)}{N}\right\|ight)+e^{-2\gamma}\left\|eft(\frac{\mathscr{D}elta N}{2N}+\vert\nabla\varphi\vert^2+\frac{1}{4}e^{-4\varphi}\left\|eft|\nabla\Omegamega \right\|ight|^2\right\|ight),\left\|abel{EQ gamma} \\ \mathbf{T}^2\varphi-e^{-2\gamma}\mathscr{D}elta \varphi&=\frac{e^{-2\gamma}}{N}\nabla \varphi\cdot\nabla N+\tau\mathbf{T}\varphi+\frac{1}{2}e^{-4\varphi}\left\|eft( (e_0\Omegamega)^2+|\nabla\Omegamega|^2 \right\|ight),\left\|abel{EQ ffi}\\ \mathbf{T}^2\Omegamega-e^{-2\gamma}\mathscr{D}elta \Omegamega&=\frac{e^{-2\gamma}}{N}\nabla \Omegamega\cdot\nabla N+\tau\mathbf{T}\Omegamega-4e_0\Omegamega e_0\varphi-4\nabla\Omegamega\cdot\nabla\varphi,\left\|abel{EQ omega} \varepsilonnd{align} where we use the notation $u_i\Bar{\Omegatimes} v_j=u_iv_j +u_jv_i-\mathrm{d}elta_{ij}u^kv_k$. Let us explain where equations \varepsilonqref{EQ N}-\varepsilonqref{EQ omega} come from : \begin{itemize} \item Considering \varepsilonqref{appendix R00} and \varepsilonqref{JSP}, the equation $R_{00}=T_{00}-g_{00}\mathrm{tr}_gT$ without the term in $e_0\tau$ gives \varepsilonqref{EQ N}. \item For $\beta$ and $\tau$, the equations \varepsilonqref{EQ beta} and \varepsilonqref{EQ tau} simply come from \varepsilonqref{appendix beta} and \varepsilonqref{appendix tau}. \item To obtain the equation for $H$, we basically take the traceless part of $R_{ij}$. More precisely, using \varepsilonqref{appendix Rij}, \varepsilonqref{appendix trace ricci}, \varepsilonqref{JSP} and \varepsilonqref{JSP 3} the equation \begin{equation*} R_{ij}-\frac{1}{2}\mathrm{d}elta_{ij}\mathrm{d}elta^{k\varepsilonll}R_{k\varepsilonll}=T_{ij}-g_{ij}\mathrm{tr}_gT-\frac{1}{2}\mathrm{d}elta_{ij}\mathrm{d}elta^{k\varepsilonll}\left\|eft(T_{k\varepsilonll}-g_{k\varepsilonll}\mathrm{tr}_gT \right\|ight) \varepsilonnd{equation*} gives \varepsilonqref{EQ H}. \item Considering \varepsilonqref{appendix trace ricci} and \varepsilonqref{JSP 2}, the equation $\mathrm{d}elta^{ij}R_{ij}=\mathrm{d}elta^{ij}(T_{ij}-g_{ij}\mathrm{tr}_gT)$ reads : \begin{equation*} \mathscr{D}elta\gamma=\frac{\tau^2}{2}e^{2\gamma}-\frac{e^{2\gamma}}{2}\mathbf{T}\tau-\frac{\mathscr{D}elta N}{2N}-\left\|eft|\nabla\varphi\right\|ight|^2-\frac{1}{4}e^{-4\varphi}\left\|eft|\nabla\Omegamega \right\|ight|^2. \varepsilonnd{equation*} Using \varepsilonqref{appendix tau}, we can compute $\mathbf{T}\tau$ and inject it in the previous equation to obtain \varepsilonqref{EQ gamma}. \item For the equation on the matter fields $\varphi$ and on $\Omegamega$, we simply use Proposition \right\|ef{appendix box} to rewrites $\Box_g\varphi$ and $\Box_g\Omegamega$. \varepsilonnd{itemize} After obtaining a solution to the reduced system, our next task will be to prove that this solution is in fact a solution of \varepsilonqref{EVE}. Note that $H$ and $\gamma$ no longer satisfy elliptic equations, whereas in the "full" Einstein equations in the elliptic gauge they do. We follow this strategy to avoid to propagate the two conservation laws \varepsilonqref{CL1} and \varepsilonqref{CL2}, which would have been essential for solving elliptic equations and obtain a suitable behavior at spacelike infinity for $H$ and $\gamma$. Since we assume these conservation laws to hold initially, we do obtain this behavior while solving the constraints equations. Therefore, only $N$ and $\beta$ satisfy elliptic equations, and the reduced system is a coupled hyperbolic-elliptic-transport system. \subsection{Outline of the proof} We briefly discuss the structure of this article, which aims at proving Theorems \right\|ef{theoreme principal} and \right\|ef{theo 2}. \par\left\|eavevmode\par First of all, in Section \right\|ef{initial data}, we solve the constraints equations. More precisely we prove that an admissible free initial data set gives rise to an actual admissible initial data set, thus satisfying the constraints equations. Then, we split the proof of Theorem \right\|ef{theoreme principal} into two parts : \begin{itemize} \item in Section \right\|ef{section solving the reduced system}, we solve the reduced system \varepsilonqref{EQ N}-\varepsilonqref{EQ omega} using an iteration scheme, with initial data given by Section \right\|ef{initial data}. During this iteration scheme, we first prove that our sequence of approximate solution is uniformaly bounded (see Section \right\|ef{uniforme boundedness}) and then that it is a Cauchy sequence (see Section \right\|ef{Cauchy}). \item in Section \right\|ef{section end of proof}, we prove that the solution to the reduced system is indeed a solution to $\varepsilonqref{EVE}$ and that it satisfies all the estimates stated in Theorem \right\|ef{theoreme principal}. \varepsilonnd{itemize} We prove Theorem \right\|ef{theo 2} in Section \right\|ef{section theo 2}, using a continuity argument based on a special energy estimate which suits the wave map structure of the coupled system satisfied by $\varphi$ and $\Omegamega$. \par\left\|eavevmode\par Finally, this article contains two appendices : \begin{itemize} \item Appendix \right\|ef{appendix A} presents the computations of the connection coefficients and the Ricci tensor in the elliptic gauge, as well as some formulae related to the stress-energy-momentum tensor. \item Appendix \right\|ef{appendix B} presents the main tools regarding the spaces $W^{m,p}_{\mathrm{d}elta}$ : embeddings results, product laws, and a theorem due to McOwen which allows us to solve elliptic equations on thoses spaces. It ends with some standard inequalities used in the proof. \varepsilonnd{itemize} \section{Initial data and the constraints equations}\left\|abel{initial data} In this section, we follow \cite{hun16} and discuss the initial data for the reduced system, and in particular we solve the constraints equations. More precisely, we will show that an admissible free initial data set gives rise to a unique admissible initial data set satisfying the constraint equations. We will then derive the initial data for $N$ and $\beta$ and prove their regularity properties. Note that, since $\mathring{\varphi}$ and $\mathring{\Omegamega}$ are prescribed, once we have the initial data for $N$, $\beta$ and $\gamma$, we obtain the initial data for $\partial_t\varphi$ and $\partial_t\Omegamega$. We will only care about highlighting the dependence on $\varepsilon$ and $C_{high}$ in the following estimates and will use the notation $\left\|esssim$ where the implicit constant only depends on $\mathrm{d}elta$, $R$ or on any constants coming from embeddings results. \par\left\|eavevmode\par Before we go into solving the constraint equations, let us prove a simple lemma which will allows us to deal with the $e^{-4\varphi}$ and $e^{\pm2\gamma}$ factors, which will occur many times in the equations. \begin{lem}\left\|abel{useful lem} Let $\gamma=-\alpha\chi\left\|n+\Tilde{\gamma}$ be a function on $\mathbb{R}^2$ such that $0\left\|eq\alpha\left\|eq 1$, $\|\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta}}\left\|eq 1$ and $\varphi\in H^3$ a compactly supported function on $\mathbb{R}^2$ such that $\left\|\varphi\right\|_{W^{1,4}}\left\|eq\varepsilon$. Then, for all functions $f$ on $\mathbb{R}^2$ and $\nu\in\mathbb{R}$, the following estimates holds for $k=0,1,2$ : \begin{align} \left\|eft\|e^{-2\gamma}f \right\|ight\|_{H^k_{\nu}}&\left\|esssim \|f\|_{H^k_{\nu+2\alpha}}\left\|abel{useful gamma 1},\\ \left\|eft\|e^{2\gamma}f \right\|ight\|_{H^k_{\nu}}&\left\|esssim \|f\|_{H^k_{\nu}}\left\|abel{useful gamma 1a},\\ \left\| e^{-4\varphi}f\right\|_{H^{k}_\nu}&\left\|esssim \left\| f\right\|_{H^{k}_\nu}+k\left\|\nabla\varphi\right\|_{H^2}\left\| f\right\|_{H^{k'-1}}+k(k-1)\left\|\nabla^2\varphi f\right\|_{L^2}\left\|abel{useful ffi 1}. \varepsilonnd{align} Moreover, if in addition $\|\nabla\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta'+1}}<\infty$, the following estimate holds : \begin{align} \left\|eft\|e^{-2\gamma}f \right\|ight\|_{H^3_{\nu}}&\left\|esssim \|f\|_{H^3_{\nu+2\alpha}}+\|\nabla\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta'+1}}\|f\|_{H^2_{\nu+2\alpha}},\left\|abel{useful gamma 2}\\ \left\|eft\|e^{2\gamma}f \right\|ight\|_{H^3_{\nu}}&\left\|esssim \|f\|_{H^3_{\nu}}+\|\nabla\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta'+1}}\|f\|_{H^2_{\nu}}.\left\|abel{useful gamma 2a} \varepsilonnd{align} \varepsilonnd{lem} \begin{proof} We recall the embedding $H^2_{\mathrm{d}elta}\xhookrightarrow{}L^{\infty}$, which implies that $\left\|eft| e^{-2\Tilde{\gamma}}\right\|ight|\left\|esssim 1$, which allows us to forget about these factors in the following computations. Similarly, we have $\left\|eft|e^{2\alpha\chi\left\|n} \right\|ight|\left\|esssim \left\|angle x\right\|angle^{2\alpha}$, which will be responsible for the change of decrease order (this remark also implies that proving \varepsilonqref{useful gamma 1} and \varepsilonqref{useful gamma 2} is enough to get \varepsilonqref{useful gamma 1a} and \varepsilonqref{useful gamma 2a}). Moreover, we only prove \varepsilonqref{useful gamma 2}, since it will be clear that its proof will include the proof of \varepsilonqref{useful gamma 1}. With these remarks in mind, we compute directly : \begin{align*} \left\|eft\| e^{-2\gamma}f \right\|ight\|_{H^3_{\nu}} & \left\|esssim \left\|eft\|f \right\|ight\|_{H^3_{\nu+2\alpha}}+ \left\|eft\|\nabla\gamma f \right\|ight\|_{L^2_{\nu+2\alpha+1}} + \left\|eft\|\nabla^2\gamma f \right\|ight\|_{L^2_{\nu+2\alpha+2}}\\&\qquad+ \left\|eft\|\left\|eft( \nabla\gamma\right\|ight)^2f \right\|ight\|_{L^2_{\nu+2\alpha+2}} +\left\|eft\| \nabla\gamma\nabla f\right\|ight\|_{L^2_{\nu+2\alpha+2}} +\left\|eft\|\nabla^2\gamma\nabla f \right\|ight\|_{L^2_{\nu+2\alpha+3}} \\&\qquad+\left\|eft\|\left\|eft(\nabla\gamma \right\|ight)^2\nabla f \right\|ight\|_{L^2_{\nu+2\alpha+3}}+\left\|eft\|\nabla\gamma\nabla^2 f \right\|ight\|_{L^2_{\nu+2\alpha+3}} \\&\qquad+\left\|eft\| \nabla^3\gamma f\right\|ight\|_{L^2_{\nu+2\alpha+3}}+\left\|eft\|\nabla\gamma\nabla^2\gamma f \right\|ight\|_{L^2_{\nu+2\alpha+3}}+\left\|eft\|\left\|eft(\nabla\gamma \right\|ight)^3f \right\|ight\|_{L^2_{\nu+2\alpha+3}} \varepsilonnd{align*} Because of $\left\|eft|\nabla^{a}(\chi\left\|n) \right\|ight|\left\|esssim \left\|angle x\right\|angle^{-|a|}$ (which is valid for every multi-index $ a\neq 0$), we can forget about the $\chi\left\|n$ part in $\gamma$ and pretend that $\gamma$ is replaced by $\Tilde{\gamma}$. Using the product estimate (see Proposition \right\|ef{prop prod}), we can deal with all these terms : \begin{center} \begin{tabular}{ c c } $\left\|eft\|\nabla\Tilde{\gamma} f \right\|ight\|_{L^2_{\nu+2\alpha+1}}\left\|esssim\|\nabla\Tilde{\gamma}\|_{H^1_{\mathrm{d}elta+1}}\|f\|_{H^1_{\nu+2\alpha+1}},\quad$ & $\left\|eft\|\nabla^2\Tilde{\gamma} f \right\|ight\|_{L^2_{\nu+2\alpha+2}}\left\|esssim \left\|eft\| \nabla^2\Tilde{\gamma}\right\|ight\|_{L^2_{\mathrm{d}elta+2}}\| f\|_{H^2_{\nu+2\alpha}}$, \\ $\left\|eft\|\left\|eft(\nabla\Tilde{\gamma}\right\|ight)^2 f \right\|ight\|_{L^2_{\nu+2\alpha+2}}\left\|esssim \|\nabla\Tilde{\gamma}\|_{H^1_{\mathrm{d}elta+1}}^2\|f\|_{H^2_{\nu+2\alpha}},\quad$ & $\left\|eft\|\nabla\Tilde{\gamma} \nabla f \right\|ight\|_{L^2_{\nu+2\alpha+2}}\left\|esssim\| \nabla\Tilde{\gamma}\|_{H^1_{\mathrm{d}elta+1}}\|\nabla f\|_{H^1_{\nu+2\alpha+1}}$, \\ $\left\|eft\|\nabla^2\Tilde{\gamma} \nabla f \right\|ight\|_{L^2_{\nu+2\alpha+3}}\left\|esssim \left\|eft\|\nabla^2\Tilde{\gamma} \right\|ight\|_{L^2_{\mathrm{d}elta+2}}\|\nabla f\|_{H^2_{\nu+2\alpha+1}},\quad$ & $\left\|eft\|\left\|eft(\nabla\Tilde{\gamma}\right\|ight)^2 \nabla f \right\|ight\|_{L^2_{\nu+2\alpha+3}}\left\|esssim \|\nabla\Tilde{\gamma}\|_{H^1_{\mathrm{d}elta+1}}^2\|\nabla f\|_{H^2_{\nu+2\alpha+1}},$ \\ $\left\|eft\|\nabla\Tilde{\gamma} \nabla^2f \right\|ight\|_{L^2_{\nu+2\alpha+3}}\left\|esssim \|\nabla\Tilde{\gamma}\|_{H^1_{\mathrm{d}elta+1}}\left\|eft\|\nabla^2 f\right\|ight\|_{H^1_{\nu+2\alpha+2}},\quad$ & $\left\|eft\|\nabla^3\Tilde{\gamma} f \right\|ight\|_{L^2_{\nu+2\alpha+3}}\left\|esssim \left\|eft\|\nabla^3\Tilde{\gamma}\right\|ight\|_{L^2_{\mathrm{d}elta'+3}}\|f\|_{H^2_{\nu+2\alpha}},$\\ $\left\|eft\|\nabla\Tilde{\gamma}\nabla^2\Tilde{\gamma} f \right\|ight\|_{L^2_{\nu+2\alpha+3}}\left\|esssim \|\nabla\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\|\nabla^2\Tilde{\gamma}f \right\|ight\|_{L^2_{\nu+2\alpha+2}},\quad$ & $\left\|eft\|\left\|eft(\nabla\Tilde{\gamma}\right\|ight)^3 f \right\|ight\|_{L^2_{\nu+2\alpha+3}}\left\|esssim \|\nabla\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\|\left\|eft(\nabla\Tilde{\gamma}\right\|ight)^2 f \right\|ight\|_{L^2_{\nu+2\alpha+3}}.$ \varepsilonnd{tabular} \varepsilonnd{center} Note that the last two estimates involve $\left\|eft\|\nabla^2\Tilde{\gamma}f \right\|ight\|_{L^2_{\nu+2\alpha+2}}$ and $ \left\|eft\|\left\|eft(\nabla\Tilde{\gamma}\right\|ight)^2 f \right\|ight\|_{L^2_{\nu+2\alpha+3}}$, which have already been estimated. Looking at these estimates, we see that the only ones which uses $\| \nabla\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta'+1}}$ are the three last ones. Those terms don't appear if we only differentiate twice or less, it is therefore clear why \varepsilonqref{useful gamma 1} is also proved. The proof of \varepsilonqref{useful ffi 1} is identical, using the embeddings $W^{1,4}\xhookrightarrow{}L^{\infty}$ and $H^2\xhookrightarrow{}L^{\infty}$. \varepsilonnd{proof} \subsection{The constraints equations}\left\|abel{section constraints equations} We are now ready to solve the constraints equations, which we rewrite in terms of $\mathring{\varphi}$ and $\mathring{\Omegamega}$ : \begin{align} \partial^iH_{ij}&=-2\mathring{\varphi}\partial_j\varphi-\frac{1}{2}e^{-4\varphi}\mathring{\Omegamega}\partial_j\Omegamega, \left\|abel{C1}\\ \mathscr{D}elta\gamma&=-e^{-2\gamma}\left\|eft(\mathring{\varphi}^2+\frac{1}{4}e^{-4\varphi}\mathring{\Omegamega}^2+\frac{1}{2}\vert H\vert^2\right\|ight)-\vert\nabla\varphi\vert^2-\frac{1}{4}e^{-4\varphi}|\nabla\Omegamega|^2.\left\|abel{C2} \varepsilonnd{align} \begin{lem}\left\|abel{CI sur H} The equation \varepsilonqref{C1} admits a unique solution $H\in H^3_{\mathrm{d}elta+1}$, a symmetric traceless covariant 2-tensor with $\Vert H\Vert_{H^1_{\mathrm{d}elta+1}}\left\|esssim \varepsilon^2$. \varepsilonnd{lem} \begin{proof} We look for a solution under the form $H=LY$ where $Y$ is a 1-form. We have $\partial^iH_{ij}=\mathscr{D}elta Y_j$ and $Y$ solves \begin{equation*} \mathscr{D}elta Y_j=-2\mathring{\varphi}\partial_j\varphi-\frac{1}{2}e^{-4\varphi}\mathring{\Omegamega}\partial_j\Omegamega. \varepsilonnd{equation*} Using the definition of $L$, it's easy to check that $LY$ is a traceless symmetric 2-tensor. We use the Theorem \right\|ef{mcowens 1} in the case $p=2$ and $m=0$, the range of the Laplacian is then the functions $f\in H^0_{\mathrm{d}elta+2}$ such that $\int f=0$. By assumption, $\int_{\mathbb{R}^2}\left\|eft(-2\mathring{\varphi}\partial_j\varphi-\frac{e^{-4\varphi}}{2}\mathring{\Omegamega}\partial_j\Omegamega\right\|ight)\mathrm{d} x=0$ and thanks to the support property, the Hölder inequality and \varepsilonqref{useful ffi 1} we have : \begin{equation*} \left\|eft\| \mathscr{D}elta Y_j\right\|ight\|_{ H^0_{\mathrm{d}elta+2}}\left\|esssim \left\|eft\| \mathring{\varphi}\partial_j\varphi\right\|ight\|_{ L^2}+\left\|eft\| \mathring{\Omegamega}\partial_j\Omegamega\right\|ight\|_{ L^2}\left\|esssim \left\|\mathring{\varphi}\right\|_{L^4}\left\|\nabla\varphi\right\|_{L^4}+\left\|\mathring{\Omegamega}\right\|_{L^4}\left\|\nabla\Omegamega\right\|_{L^4}\left\|esssim \varepsilon^2. \varepsilonnd{equation*} Thus, there exists a unique solution $Y_j\in H^2_{\mathrm{d}elta}$. Moreover we have $\Vert Y_j\Vert_{H^2_{\mathrm{d}elta}}\left\|eq \varepsilon^2$, which implies $\Vert H\Vert_{H^1_{\mathrm{d}elta+1}}\left\|eq \varepsilon^2$. We can improve the regularity of $H$, by noting that \begin{equation*} \Vert H\Vert_{H^3_{\mathrm{d}elta+1}}\left\|eq \Vert Y\Vert_{H^4_{\mathrm{d}elta}}\left\|esssim\Vert \mathring{\varphi}\nabla\varphi\Vert_{H^2}+\Vert \mathring{\Omegamega}\nabla\Omegamega\Vert_{H^2}\left\|esssim C_{high}^2. \varepsilonnd{equation*} In the last inequality we use the fact that in dimension 2, $H^2$ is an algebra. Our solution $H\in H^3_{\mathrm{d}elta+1}$ is unique, because of the following fact : if $H\in H^3_{\mathrm{d}elta+1}$ is a traceless symmetric divergence free 2-tensor, we have componentwise $\mathscr{D}elta H_{ij}=0$, which implies $H=0$, again thanks to Theorem \right\|ef{mcowens 1}. \varepsilonnd{proof} \begin{lem}\left\|abel{lem CI gamma} For $\varepsilon$ sufficiently small, the equation \varepsilonqref{C2} admits a unique solution $\gamma=-\alpha \chi\left\|n+\Tilde{\gamma}$ with $\Tilde{\gamma}\in H^4_{\mathrm{d}elta}$, $\Vert \Tilde{\gamma}\Vert_{H^2_{\mathrm{d}elta}}\left\|esssim \varepsilon^2$ and $0\left\|eq\alpha\left\|esssim \varepsilon^2$. \varepsilonnd{lem} \begin{proof} We are going to use a fixed point argument in $[0,\varepsilon]\times B_{H^2_{\mathrm{d}elta}}(0,\varepsilon)$. We define on this space the application $\phi:(\alpha^{(1)},\Tilde{\gamma}^{(1)})\left\|ongmapsto(\alpha^{(2)},\Tilde{\gamma}^{(2)})$, where $\gamma^{(2)}$ is the unique solution of \begin{equation} \mathscr{D}elta\gamma^{(2)}=-\vert\nabla\varphi\vert^2-\frac{1}{4}e^{-4\varphi}|\nabla\Omegamega|^2-e^{-2\gamma^{(1)}}\left\|eft( \frac{1}{2}\vert H\vert^2+\mathring{\varphi}^2+\frac{1}{4}e^{-4\varphi}\mathring{\Omegamega}^2\right\|ight),\left\|abel{contraction 1} \varepsilonnd{equation} with the notation $\gamma^{(i)}=-\alpha^{(i)}\chi(r)\left\|n(r)+\Tilde{\gamma}^{(i)}$. We want to prove that if $\varepsilon$ is small enough, $\phi$ is indeed a contraction. Let us show that the RHS of \varepsilonqref{contraction 1} is in $H^0_{\mathrm{d}elta+2}$. By assumption on $\varphi$ and $\Omegamega$ we can write, using Hölder's inequality and \varepsilonqref{useful ffi 1} : \begin{equation*} \left\|eft\| \vert \nabla \varphi \vert^2+\frac{1}{4}e^{-4\varphi}|\nabla\Omegamega|^2\right\|ight\|_{ H^0_{\mathrm{d}elta+2}}\left\|esssim \left\|eft\| \vert \nabla \varphi \vert^2\right\|ight\|_{ L^2}+\left\|eft\| \vert \nabla \Omegamega \vert^2\right\|ight\|_{ L^2}\left\|esssim \varepsilon^2. \varepsilonnd{equation*} For the term $e^{-2\gamma^{(1)}}|H|^2$, we use \varepsilonqref{useful gamma 1}, the product estimate and choose $\varepsilon$ small enough : \begin{equation*} \left\|eft\| e^{-2\gamma^{(1)}}|H|^2 \right\|ight\|_{ H^0_{\mathrm{d}elta+2}}\left\|esssim \left\|eft\| |H|^2\right\|ight\|_{H^0_{\mathrm{d}elta+2(1+\varepsilon)}} \left\|esssim \Vert H\Vert_{H^1_{\mathrm{d}elta+1}}^2\left\|esssim \varepsilon^4. \varepsilonnd{equation*} The last terms is handled with the same arguments : \begin{equation*} \left\|eft\| e^{-2\gamma^{(1)}}\left\|eft(\mathring{\varphi}^2+\frac{1}{4}e^{-4\varphi}\mathring{\Omegamega}^2 \right\|ight)\right\|ight\|_{H^0_{\mathrm{d}elta+2}} \left\|esssim \left\|eft\| \mathring{\varphi}^2\right\|ight\|_{L^2} +\left\|eft\| \mathring{\Omegamega}^2\right\|ight\|_{L^2}\left\|esssim \varepsilon^2. \varepsilonnd{equation*} We next prove the bound on $\alpha^{(2)} =-\frac{1}{2\pi}\int_{\mathbb{R}^2}(\text{RHS of \varepsilonqref{contraction 1}})$, its positivity being clear. We have \begin{align*} \left\|eft|\alpha^{(2)}\right\|ight| & \left\|esssim \left\|eft\| \nabla\varphi \right\|ight\|_{L^2}^2+ \left\|eft\| \nabla\Omegamega \right\|ight\|_{L^2}^2 + \left\|eft\|e^{-\gamma^{(1)}} H \right\|ight\|_{L^2}^2 + \left\|eft\| e^{-\gamma^{(1)}}\mathring{\varphi} \right\|ight\|_{L^2}^2+ \left\|eft\| e^{-\gamma^{(1)}-2\varphi}\Omegamega \right\|ight\|_{L^2}^2 \\& \left\|esssim\left\|eft\| \nabla\varphi \right\|ight\|_{L^4}^2+\left\|eft\| \nabla\Omegamega \right\|ight\|_{L^4}^2+ \left\|eft\|H\right\|ight\|_{H^0_{\varepsilon}}^2+\left\|eft\| \mathring{\varphi} \right\|ight\|_{L^4}^2+\left\|eft\| \mathring{\Omegamega} \right\|ight\|_{L^4}^2 \\& \left\|esssim \varepsilon^2, \varepsilonnd{align*} where we used Hölder's inequality, \varepsilonqref{useful gamma 1} (for the three last terms) and the support property of $\varphi$, $\mathring{\varphi}$, $\Omegamega$ and $\mathring{\Omegamega}$. In conclusion, thanks to Corollary \right\|ef{mcowens 2}, if $\varepsilon$ is small enough, $\phi$ is indeed an application from $[0,\varepsilon]\times B_{H^2_{\mathrm{d}elta}}(0,\varepsilon)$ to itself and we can prove in the same way that this is a contraction. \par\left\|eavevmode\par We can improve the regularity of $\Tilde{\gamma}$, using \varepsilonqref{useful gamma 1} and \varepsilonqref{useful ffi 1} : \begin{align*} \left\|eft\| \Tilde{\gamma}\right\|ight\|_{H^4_{\mathrm{d}elta}}&\left\|esssim \left\|eft\| e^{-2\gamma}H^2 \right\|ight\|_{ H^2_{\mathrm{d}elta+2}} + \left\|eft\| e^{-2\gamma}\mathring{\varphi}^2\right\|ight\|_{H^2}+ + \left\|eft\| \vert \nabla \varphi \vert^2\right\|ight\|_{ H^2} \\& \left\|esssim \left\|eft\| H^2 \right\|ight\|_{H^2_{\mathrm{d}elta+2\varepsilon+2}}+\left\|eft\| \mathring{\varphi}^2\right\|ight\|_{H^2}+\left\|eft\| \vert \nabla \varphi \vert^2\right\|ight\|_{ H^2} \\& \left\|esssim \|H\|_{H^3_{\mathrm{d}elta+1}}^2+C_{high}^2, \varepsilonnd{align*} where in the last inequality, we used the product estimate (with $\varepsilon$ small enough) for the first term and the algebraic structure of $H^2$ for the remaining terms. Thanks to Lemma \right\|ef{CI sur H}, the final quantity is finite, which concludes the proof. \varepsilonnd{proof} \subsection{Initial data to the reduced system} The equations satisfied by $N$ and $\beta$ are : \begin{align} \mathscr{D}elta N & =e^{-2\gamma}N\left\|eft(\vert H\vert^2+\mathring{\varphi}^2+\frac{1}{4}e^{-4\varphi}\mathring{\Omegamega}^2\right\|ight) , \left\|abel{CI sur N}\\ L\beta & =2e^{-2\gamma}NH.\left\|abel{CI sur beta} \varepsilonnd{align} The equation \varepsilonqref{CI sur N} comes from \varepsilonqref{EQ N} in the case $\tau=0$, and the equation \varepsilonqref{CI sur beta} comes from \varepsilonqref{appendix beta}. \begin{lem}\left\|abel{lem CI N} For $\varepsilon$ sufficiently small, the equation \varepsilonqref{CI sur N} admits a unique solution $N=1+N_a\chi\left\|n+\Tilde{N}$ with $\Tilde{N}\in H^4_{\mathrm{d}elta}$, $\Vert \Tilde{N}\Vert_{H^2_{\mathrm{d}elta}}\left\|esssim \varepsilon^2$ and $0\left\|eq N_a\left\|esssim \varepsilon^2$. \varepsilonnd{lem} \begin{proof} We look for a solution of the form $N=1+N_a\chi(r)\left\|n(r)+\Tilde{N}$, with $N_a\geq 0$. On the space $[0,\varepsilon]\times B_{H^2_{\mathrm{d}elta}}(0,\varepsilon)$, we define the application $\phi(N_a^{(1)},\Tilde{N}^{(1)})=(N_a^{(2)},\Tilde{N}^{(2)})$ where (with the notation $N^{(i)}=1+N_a^{(i)}\chi(r)\left\|n(r)+\Tilde{N}^{(i)}$), $N^{(2)}$ is the solution of \begin{equation} \mathscr{D}elta N^{(2)}=e^{-2\gamma}N^{(1)}(\vert H\vert^2+\mathring{\varphi}^2+\frac{1}{4}e^{-4\varphi}\mathring{\Omegamega}^2). \varepsilonnd{equation} Let's show that the RHS is in $H^0_{\mathrm{d}elta+2}$. Thanks to the support property of $\mathring{\varphi}$ and $\mathring{\Omegamega}$, the first term is handled quite easily using \varepsilonqref{useful gamma 1}, \varepsilonqref{useful ffi 1} and the fact that $\left\|eft\|N^{(1)} \right\|ight\|_{L^{\infty}(B_R)}\left\|esssim 1$ (note the embedding $H^2_\mathrm{d}elta\xhookrightarrow{}L^{\infty}$) : \begin{equation*} \left\|eft\| e^{-2\gamma}N^{(1)}\left\|eft(\mathring{\varphi}^2+\frac{1}{4}e^{-4\varphi}\mathring{\Omegamega}^2\right\|ight)\right\|ight\|_{H^0_{\mathrm{d}elta+2}}\left\|esssim \left\|eft\|N^{(1)} \right\|ight\|_{L^{\infty}(B_R)}\left\|eft(\left\|eft\|\mathring{\varphi}^2\right\|ight\|_{L^2}+\left\|\mathring{\Omegamega}^2\right\|_{L^2}\right\|ight)\left\|esssim\varepsilon^2. \varepsilonnd{equation*} Using again \varepsilonqref{useful gamma 1}, the fact that $\vert \chi\left\|n\vert\left\|esssim\left\|angle x\right\|angle^{\frac{\mathrm{d}elta+1}{2}} $, the embedding $H^2_{\mathrm{d}elta}\xhookrightarrow{}L^{\infty}$ (used for $\Tilde{N}^{(1)}$) and the product estimate, we handle the second term : \begin{align*} \left\|eft\| e^{-2\gamma}N^{(1)}|H|^2\right\|ight\|_{H^0_{\mathrm{d}elta+2}} & \left\|esssim\left\|eft\| N^{(1)}|H|^2\right\|ight\|_{H^0_{\mathrm{d}elta+2+2\varepsilon}}\nonumber\\ & \left\|esssim \left\|eft\| |H|^2\right\|ight\|_{H^0_{\mathrm{d}elta+2+2\varepsilon}}\left\|eft(1+\left\|eft\|\Tilde{N}^{(1)}\right\|ight\|_{H^2_{\mathrm{d}elta}}\right\|ight)+\varepsilon\left\|eft\| |H|^2\right\|ight\|_{H^0_{\mathrm{d}elta+2+2\varepsilon+\frac{\mathrm{d}elta+1}{2}}} \\&\left\|esssim(1+\varepsilon)\left\|eft\| H\right\|ight\|_{H^1_{\mathrm{d}elta+1}}^2 \\&\left\|esssim \varepsilon^4. \varepsilonnd{align*} We showed that, for $\varepsilon$ small enough, we have $\Vert\mathscr{D}elta N^{(2)}\Vert_{H^0_{\mathrm{d}elta+2}}\left\|esssim \varepsilon^2$. \\ We have : \begin{equation} 2\pi N_a^{(2)}=\int_{\mathbb{R}^2}e^{-2\gamma}N^{(1)}|H|^2+\int_{\mathbb{R}^2}e^{-2\gamma}N^{(1)}\mathring{\varphi}^2+\frac{1}{4}\int_{\mathbb{R}^2}e^{-2\gamma-4\varphi}N^{(1)}\mathring{\Omegamega}^2. \varepsilonnd{equation} If $\varepsilon$ is small enough, we have $N^{(1)}\geq 0$ (using the embedding $H^2_{\mathrm{d}elta}\xhookrightarrow{}L^{\infty}$) so that $N_a^{(2)}\geq 0$. With the same kind of arguments than previously, we can easily show that $N_a^{(2)}\left\|esssim\varepsilon^2$. \\ This concludes the fact that $\phi$ is well defined (providing $\varepsilon$ is small and thanks to Corollary \right\|ef{mcowens 2}), and that this is a contraction (the calculations are likewise, since the equation is linear). \par\left\|eavevmode\par We can improve the regularity of $\Tilde{N}$, using \varepsilonqref{useful gamma 1} and \varepsilonqref{useful ffi 1} : \begin{align} \left\|eft\|\Tilde{N} \right\|ight\|_{H^4_{\mathrm{d}elta}}&\left\|esssim \left\|eft\|e^{-2\gamma}N|H|^2 \right\|ight\|_{H^2_{\mathrm{d}elta+2}}+ \left\|eft\|e^{-2\gamma}N\mathring{\varphi}^2 \right\|ight\|_{H^2}+ \left\|eft\|e^{-2\gamma-4\varphi}N\mathring{\Omegamega}^2 \right\|ight\|_{H^2} \left\|abel{idem1} \\& \left\|esssim \left\|eft\| |H|^2\right\|ight\|_{H^2_{\mathrm{d}elta+2+2\varepsilon}}\left\|eft(1+\left\|eft\|\Tilde{N}^{(1)}\right\|ight\|_{H^2_{\mathrm{d}elta}}\right\|ight)+\varepsilon\left\|eft\| \chi\left\|n |H|^2\right\|ight\|_{H^2_{\mathrm{d}elta+2+2\varepsilon}} + \left\|eft\|N\right\|ight\|_{L^{\infty}(B_R)}\left\|eft(\left\|eft\| \mathring{\varphi}^2 \right\|ight\|_{H^2}+\left\|\mathring{\Omegamega}^2\right\|_{H^2}\right\|ight).\nonumber \varepsilonnd{align} Using $\vert \chi\left\|n\vert\left\|esssim\left\|angle x\right\|angle^{\frac{\mathrm{d}elta+1}{2}} $ and $\left\|eft| \nabla^a(\chi\left\|n)\right\|ight|\left\|esssim\left\|angle x\right\|angle^{-|a|}$ (for $a\neq0$), we easily show that $\left\|eft\| \chi\left\|n |H|^2\right\|ight\|_{H^2_{\mathrm{d}elta+2+2\varepsilon}}\left\|esssim \left\|eft\| |H|^2\right\|ight\|_{H^2_{\mathrm{d}elta+2+2\varepsilon+\frac{\mathrm{d}elta+1}{2}}}$ to obtain : \begin{equation*} \left\|eft\| \Tilde{N}\right\|ight\|_{H^4_{\mathrm{d}elta}}\left\|esssim (1+\varepsilon)\left\|eft\| H\right\|ight\|_{H^3_{\mathrm{d}elta+1}}^2+\|\mathring{\varphi}\|_{H^2}^2+\|\mathring{\Omegamega}\|_{H^2}^2, \varepsilonnd{equation*} which is finite, thanks to Lemma \right\|ef{CI sur H}. \varepsilonnd{proof} The following simple lemma will be useful in order to use Theorem \right\|ef{mcowens 1} for $\beta$ : \begin{lem}\left\|abel{divergence nulle} Let $m\in\mathbb{N}$, $\nu\in\mathbb{R}$ and $u=(u_1,u_2)$ be a fonction from $\mathbb{R}^2$ to $\mathbb{R}^2$ such that $u_i\in H^m_{\nu}$. If $m\geq 2$ and $\nu>0$, then \begin{equation*} \int_{\mathbb{R}^2}\mathrm{div}(u)=0 \varepsilonnd{equation*} \varepsilonnd{lem} \begin{proof} We fix $R>0$ and use the Stokes formula : \begin{align*} \bigg| \int_{B_R}\mathrm{div}(u)\bigg| = \bigg| \int_{\partial B_R}u.n\,\mathrm{d}\sigma\bigg| \left\|eq \int_{\partial B_R}\left\|angle x\right\|angle^{-\nu-1}\left\|angle x\right\|angle^{\nu+1}|u.n|\,\mathrm{d}\sigma \left\|esssim \Vert u\Vert_{C^0_{\nu+1}}R^{-\nu} \varepsilonnd{align*} If $m\geq 2$ and $\nu>0$ we have the Sobolev embeddings $H^m_{\nu}\subset C^0_{\nu+1}$, which concludes the proof since the last inequality implies \begin{equation*} \left\|im_{R\to +\infty}\int_{B_R}\mathrm{div}(u)=0. \varepsilonnd{equation*} \varepsilonnd{proof} \begin{lem}\left\|abel{lem CI beta} For $\varepsilon$ sufficiently small, the equation \varepsilonqref{CI sur beta} admits a unique solution $\beta\in H^4_{\mathrm{d}elta'}$ with $\Vert \beta\Vert_{H^2_{\mathrm{d}elta'}}\left\|esssim\varepsilon^2$. \varepsilonnd{lem} \begin{proof} We take the divergence of \varepsilonqref{CI sur beta} to obtain the following elliptic equation : \begin{equation} \mathscr{D}elta\beta_j=\partial^i(2Ne^{-2\gamma}H_{ij})\left\|abel{laplacien beta CI} \varepsilonnd{equation} Thanks to Lemma \right\|ef{divergence nulle}, $\int_{\mathbb{R}^2}\partial^i(2Ne^{-2\gamma}H_{ij})=0$ (the fact that $e^{-2\gamma}NH\in H^2_{\mathrm{d}elta'+1}$ will be proved in the sequel of this proof). Thus, in order to apply Theorem \right\|ef{mcowens 1}, it remains to show that $\Vert \partial^i(2Ne^{-2\gamma}H_{ij})\Vert_{H^0_{\mathrm{d}elta'+2}}\left\|esssim \varepsilon^2$. For that, we use \varepsilonqref{useful gamma 1}, $\varepsilon|\chi\left\|n|\left\|esssim\left\|angle x\right\|angle^{\frac{\varepsilon}{2}}$, Lemmas \right\|ef{CI sur H} and \right\|ef{lem CI N} : \begin{align*} \left\|eft\| \partial^i(2Ne^{-2\gamma}H_{ij})\right\|ight\|_{H^0_{\mathrm{d}elta'+2}} &\left\|esssim \left\|eft\| e^{-2\gamma}NH\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\\&\left\|esssim \left\|eft\| H\right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|eft(1+\left\|eft\|\Tilde{N} \right\|ight\|_{H^2_{\mathrm{d}elta}} \right\|ight)+\left\|eft\| H\right\|ight\|_{H^1_{\mathrm{d}elta'+1+C\varepsilon^2+\frac{\varepsilon}{2}}} \\& \left\|esssim \varepsilon^2, \varepsilonnd{align*} where in the last inequality, we take $\varepsilon$ such that $C\varepsilon^2\left\|eq\frac{\varepsilon}{2}$. Thus, we can apply Theorem \right\|ef{mcowens 1} to obtain the existence of a solution to \varepsilonqref{laplacien beta CI}. We can improve the regularity of this solution using \varepsilonqref{useful gamma 2} : \begin{align*} \left\|eft\|\beta \right\|ight\|_{H^4_{\mathrm{d}elta'}}&\left\|esssim \left\|eft\| e^{-2\gamma}NH\right\|ight\|_{H^3_{\mathrm{d}elta'+1}} \\& \left\|esssim\left\|eft\|NH \right\|ight\|_{H^3_{\mathrm{d}elta'+C\varepsilon^2+1}}+\left\|eft\|\nabla\Tilde{\gamma} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\| NH\right\|ight\|_{H^2_{\mathrm{d}elta'+C\varepsilon^2+1}} \\& \left\|esssim\left\|eft(1+\left\|eft\|\nabla\Tilde{\gamma} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \right\|ight)\left\|eft(\left\|eft\| H\right\|ight\|_{H^3_{\mathrm{d}elta+1}}\left\|eft(1+\left\|eft\|\Tilde{N} \right\|ight\|_{H^4_{\mathrm{d}elta}} \right\|ight)+\left\|eft\| H\right\|ight\|_{H^3_{\mathrm{d}elta'+1+C\varepsilon^2+\frac{\varepsilon}{2}}}\right\|ight). \varepsilonnd{align*} Taking $\varepsilon$ such that $C\varepsilon^2\left\|eq\frac{\varepsilon}{2}$, we conclude using Lemmas \right\|ef{CI sur H}, \right\|ef{lem CI gamma} and \right\|ef{lem CI N} that $\left\|eft\|\beta \right\|ight\|_{H^4_{\mathrm{d}elta'}}<\infty$. \par\left\|eavevmode\par It remains to show that our solution $\beta$ satisfies $L\beta=2Ne^{-2\gamma}H$. We have shown that $L\beta-2Ne^{-2\gamma}H$ is a covariant symmetric traceless divergence free 2-tensor, it implies that its components are harmonic, and thus vanishes (because they belong to $H^4_{\mathrm{d}elta'}$). We use the same argument to show that the solution is unique. \varepsilonnd{proof} In order to have $\tau_{|\Sigma_0}=0$, we must have the following : \begin{lem} We set $e_0\gamma=\frac{1}{2}\mathrm{div}(\beta)$. Then, we have $e_0\gamma\in H^3_{\mathrm{d}elta'+1}$ and $\Vert e_0\gamma\Vert_{H^1_{\mathrm{d}elta'+1}}\left\|eq\varepsilon^2$. \varepsilonnd{lem} \begin{proof} It follows directly from the estimates on $\beta$ proved in Lemma \right\|ef{lem CI beta} and from Lemma \right\|ef{B1}. \varepsilonnd{proof} We summarise in the next corollary our results about the constraints equations and the initial data : \begin{coro}\left\|abel{coro premiere section} For $\varepsilon$ sufficiently small depending only on $\mathrm{d}elta$, given a free initial data set, there exists an initial data set to the reduced system such that the constraints equations are satisfied and $\tau_{|\Sigma_0}=0$ . Moreover, we have the following estimates : \begin{itemize}[label=\textbullet] \item there exists $C>0$ depending only on $\mathrm{d}elta$ and $R$ such that : \begin{equation} \Vert H\Vert_{H^1_{\mathrm{d}elta+1}}+\vert\alpha\vert+\left\|\Tilde{\gamma}\right\|_{H^2_{\mathrm{d}elta}}+\Vert e_0\gamma\Vert_{H^1_{\mathrm{d}elta'+1}}+\vert N_a\vert+\left\|\Tilde{N}\right\|_{H^2_{\mathrm{d}elta}}+\Vert \beta\Vert_{H^2_{\mathrm{d}elta}}\left\|eq C\varepsilon^2\left\|abel{CI petit} \varepsilonnd{equation} \item there exists $C_i>0$ depending on $\mathrm{d}elta$, $R$ and $C_{high}$ such that : \begin{equation} \Vert H\Vert_{ H^3_{\mathrm{d}elta+1}}+\left\|\Tilde{\gamma}\right\|_{ H^4_{\mathrm{d}elta}}+\Vert e_0\gamma\Vert_{H^3_{\mathrm{d}elta'+1}}+\left\|\Tilde{N}\right\|_{ H^4_{\mathrm{d}elta}}+\Vert\beta\Vert_{ H^4_{\mathrm{d}elta'}}\left\|eq C_i\left\|abel{CI gros} \varepsilonnd{equation} \varepsilonnd{itemize} \varepsilonnd{coro} \section{Solving the reduced system}\left\|abel{section solving the reduced system} In this section, we solve the reduced system of equations introduced in Section \right\|ef{section reduced system} by an iteration methode. We first prove that we can construct a sequence, defined in Section \right\|ef{section iteration scheme} and bounded in a small space (this is done in Section \right\|ef{uniforme boundedness}). Then we prove in Section \right\|ef{Cauchy} that the sequence is Cauchy in a larger space, which will imply the existence and uniqueness of solutions to the reduced system of equations. \subsection{Iteration scheme}\left\|abel{section iteration scheme} In order to solve the reduced system \varepsilonqref{EQ N}-\varepsilonqref{EQ ffi}, we construct the sequence \begin{equation*} (N^{(n)}=1+N_a^{(n)}\chi\left\|n+\Tilde{N}^{(n)},\tau^{(n)},H^{(n)},\beta^{(n)}, \gamma^{(n)}=-\alpha\chi\left\|n+\Tilde{\gamma}^{(n)},\varphi^{(n)},\Omegamega^{(n)}) \varepsilonnd{equation*} iteratively as follows : for $n=1,2$, let $N^{(n)},\tau^{(n)},H^{(n)},\beta^{(n)}, \gamma^{(n)},\varphi^{(n)}$ be time-independent, with initial data as in Section \right\|ef{initial data}. For $n\geq 2$, given the $n$-th iterate, the $(n+1)$-st iterate is then defined by solving the following system with initial data as in Section \right\|ef{initial data} : \begin{align} \mathscr{D}elta N^{(n+1)}&=e^{-2\gamma^{(n)}}N^{(n)}\left\|eft| H^{(n)}\right\|ight|^2+\frac{\left\|eft(\tau^{(n)}\right\|ight)^2}{2}e^{2\gamma^{(n)}}N^{(n)}\nonumber\\&\qquad\quad\quad+\frac{2e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2+\frac{e^{2\gamma^{(n)}-4\varphi^{(n)}}}{2N^{(n)}}\left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2 \left\|abel{reduced system N}\\ L\beta^{(n+1)}&=2e^{-2\gamma^{(n)}}N^{(n)}H^{(n)} \left\|abel{reduced system beta}\\ \tau^{(n+1)}&=-2\mathbf{T}^{(n-1)}\gamma^{(n)}+\frac{\mathrm{div}\left\|eft(\beta^{(n)}\right\|ight)}{N^{(n-1)}} \left\|abel{reduced system tau}\\ e_0^{(n+1)}\left\|eft(H^{(n+1)}\right\|ight)_{ij}&=-2e^{-2\gamma^{(n)}}N^{(n)}\left\|eft(H^{(n)}\right\|ight)_i^{\;\,\varepsilonll}\left\|eft(H^{(n)}\right\|ight)_{j\varepsilonll}+\partial_{(j}\left\|eft(\beta^{(n)}\right\|ight)^k\left\|eft(H^{(n)}\right\|ight)_{i)k}\nonumber\\&\qquad-\frac{1}{2}\left\|eft(\partial_i\Bar{\Omegatimes}\partial_j\right\|ight)N^{(n)}+\left\|eft(\mathrm{d}elta_i^k\Bar{\Omegatimes}\partial_j\gamma^{(n)}\right\|ight)\partial_kN^{(n)}\left\|abel{reduced system H}\\&\qquad-\left\|eft(\partial_i\varphi^{(n)}\Bar{\Omegatimes}\partial_j\varphi^{(n)}\right\|ight)N^{(n)}-\frac{1}{4}e^{-4\varphi^{(n)}}\left\|eft(\partial_i\Omegamega^{(n)}\Bar{\Omegatimes}\partial_j\Omegamega^{(n)}\right\|ight)N^{(n)} \nonumber\\ \left\|eft(\mathbf{T}^{(n)}\right\|ight)^2\gamma^{(n+1)}-e^{-2\gamma^{(n)}}\mathscr{D}elta \gamma^{(n+1)}&=-\frac{\left\|eft(\tau^{(n)}\right\|ight)^2}{2}+\frac{1}{2N^{(n)}}e_0^{(n-1)}\left\|eft(\frac{\mathrm{div}\left\|eft(\beta^{(n)}\right\|ight)}{N^{(n-1)}}\right\|ight)\nonumber\\&\qquad\quad\quad+e^{-2\gamma^{(n)}}\left\|eft(\frac{\mathscr{D}elta N^{(n)}}{2N^{(n)}}+\left\|eft|\nabla\varphi^{(n)}\right\|ight|^2+\frac{1}{4}e^{-4\varphi^{(n)}}\left\|eft|\nabla\Omegamega^{(n)} \right\|ight|^2\right\|ight) \left\|abel{reduced system gamma}\\ \left\|eft(\mathbf{T}^{(n)}\right\|ight)^2\varphi^{(n+1)}-e^{-2\gamma^{(n)}}\mathscr{D}elta \varphi^{(n+1)}&=\frac{e^{-2\gamma^{(n)}}}{N^{(n)}}\nabla \varphi^{(n)}\cdot\nabla N^{(n)}+\frac{\tau^{(n)} e_0^{(n-1)}\varphi^{(n)}}{N^{(n)}}\nonumber\\&\qquad\qquad\qquad +\frac{1}{2}e^{-4\varphi^{(n)}}\left\|eft( \left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2+\left\|eft|\nabla\Omegamega^{(n)}\right\|ight|^2 \right\|ight)\left\|abel{reduced system fi}\\ \left\|eft(\mathbf{T}^{(n)}\right\|ight)^2\Omegamega^{(n+1)}-e^{-2\gamma^{(n)}}\mathscr{D}elta \Omegamega^{(n+1)}&=\frac{e^{-2\gamma^{(n)}}}{N^{(n)}}\nabla \Omegamega^{(n)}\cdot\nabla N^{(n)}+\frac{\tau^{(n)} e_0^{(n-1)}\Omegamega^{(n)}}{N^{(n)}}\nonumber\\&\qquad\qquad\qquad-4e_0^{(n-1)}\Omegamega^{(n)} e_0^{(n-1)}\varphi^{(n)}-4\nabla\Omegamega^{(n)}\cdot\nabla\varphi^{(n)},\left\|abel{reduced system omega} \varepsilonnd{align} This system is not a linear system in the $(n+1)$-th iterate, because of the term $e_0^{(n+1)}H^{(n+1)}$ in \varepsilonqref{reduced system H} (which contains $\beta^{(n+1)}\cdot\nabla H^{(n+1)}$). The local well-posedness of this system follows from the estimates we are about to prove. Note that we use the following notation : \begin{equation*} e_0^{(k)} = \partial_t - \nabla \beta^{(k)}\cdot \nabla \quad \text{and}\quad \mathbf{T}^{(k)}=\frac{e_0^{(k)}}{N^{(k)}}. \varepsilonnd{equation*} \subsection{Boundedness of the sequence}\left\|abel{uniforme boundedness} The first step is to show that the sequence is uniformly bounded in appropriate function spaces. We proceed by strong induction and suppose that the following estimates hold for all $k$ up to some $n\geq 2$ and for all $t\in [0,T]$. Here, $A_0\left\|l A_1\left\|l A_2\left\|l A_3\left\|l A_4$ are all sufficiently large constants independent of $\varepsilon$ or $n$ to be choosen later. We also set $\mathrm{d}elta'=\mathrm{d}elta-\varepsilon$ and take $\varepsilon$ small enough so that $-1<\mathrm{d}elta'$. We also choose $\left\|ambda>0$ a small constant such that $\left\|ambda<\mathrm{d}elta+1$. \begin{itemize}[label=\textbullet] \item $N^{(k)}$ is of the form $N^{(k)}=1+N_a^{(k)}\chi\left\|n+\Tilde{N}^{(k)}$ with $N_a^{(k)}\geq 0$ and \begin{align} \left\|eft| N_a^{(k)}\right\|ight|+\left\|eft\|\Tilde{N}^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta}}&\left\|eq \varepsilon\left\|abel{HR N 1},\\ \left\|eft|\partial_tN_a^{(k)}\right\|ight|+\left\|eft\|\Tilde{N}^{(k)}\right\|ight\|_{H^3_{\mathrm{d}elta}}+\left\|eft\|\partial_t\Tilde{N}^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta}}&\left\|eq 2C_i\left\|abel{HR N 2}, \\ \left\|eft\| \Tilde{N}^{(k)}\right\|ight\|_{H^4_{\mathrm{d}elta}} & \left\|eq A_2 C_i^2\left\|abel{HR N 3}. \varepsilonnd{align} \item $\beta^{(k)}$ satisfies \begin{align} \left\|eft\|\beta^{(k)}\right\|ight\|_{H^2_{\mathbf{\mathrm{d}elta'}}}&\left\|eq \varepsilon\left\|abel{HR beta 1},\\ \left\|eft\|\beta^{(k)}\right\|ight\|_{H^3_{\mathbf{\mathrm{d}elta'}}}&\left\|eq A_0 C_i\left\|abel{HR beta 2},\\ \left\|eft\| \nabla e_0^{(k-1)}\beta^{(k)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|eq C_i\left\|abel{HR beta 2.5},\\ \left\|eft\| e_0^{(k-1)}\beta^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta'}}&\left\|eq A_1C_i\left\|abel{HR beta 3},\\ \left\|eft\| e_0^{(k-1)}\beta^{(k)}\right\|ight\|_{H^3_{\mathrm{d}elta'}}&\left\|eq A_4C_i^2.\left\|abel{HR beta 4} \varepsilonnd{align} \item $H^{(k)}$ satisfies \begin{align} \left\|eft\| H^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}&\left\|eq 2C_i\left\|abel{HR H 1},\\ \left\|eft\| e_0^{(k)}H^{(k)}\right\|ight\|_{L^2_{1+\left\|ambda}}&\left\|eq \varepsilon,\left\|abel{HR H 1.5}\\ \left\|eft\| e_0^{(k)}H^{(k)}\right\|ight\|_{H^1_{\mathrm{d}elta+1}}&\left\|eq A_0C_i,\left\|abel{HR H 2}\\ \left\|eft\| e_0^{(k)}H^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}&\left\|eq A_3C_i^2.\left\|abel{HR H 3} \varepsilonnd{align} \item $\tau^{(k)}$ satisfies \begin{align} \left\|eft\| \tau^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}&\left\|eq A_1C_i\left\|abel{HR tau 1},\\ \left\|eft\| \partial_t\tau^{(k)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|eq A_2C_i\left\|abel{HR tau 2},\\ \left\|eft\| \partial_t\tau^{(k)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}&\left\|eq A_3C_i.\left\|abel{HR tau 3} \varepsilonnd{align} \item $\gamma^{(k)}$ is of the form $\gamma^{(k)}=-\alpha\chi\left\|n+\Tilde{\gamma}^{(k)}$ with $\alpha$ as previously and $\Tilde{\gamma}^{(k)}$ satisfies \begin{align} \sum_{|\alpha|\left\|eq 2} \left\|eft\Vert\mathbf{T}^{(k-1)}\nabla^{\alpha}\Tilde{\gamma}^{(k)}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}+\left\|eft\| \nabla \Tilde{\gamma}^{(k)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}&\left\|eq 8 C_i,\left\|abel{HR gamma 1} \\ \left\|eft\Vert\partial_t\left\|eft(\mathbf{T}^{(k-1)}\Tilde{\gamma}^{(k)}\right\|ight)\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1}}&\left\|eq A_0C_i,\left\|abel{HR gamma 2} \\ \left\|eft\Vert\partial_t\left\|eft(\mathbf{T}^{(k-1)}\Tilde{\gamma}^{(k)}\right\|ight)\right\|ight\Vert_{H^1_{\mathrm{d}elta'+1}}&\left\|eq A_2C_i.\left\|abel{HR gamma 3} \varepsilonnd{align} \item $\varphi^{(k)}$ and $\Omegamega^{(k)}$ are compactly supported in \begin{equation*} \varepsilonnstq{(t,x)\in[0,T]\times\mathbb{R}^2}{\vert x\vert\left\|eq R+C_s(1+R^{\varepsilon})t}, \varepsilonnd{equation*} where $C_s>0$ is to be choosen in Lemma \right\|ef{support fi n+1}. Choosing $T$ smaller if necessary, we assume that the above set is a subset of $[0,T]\times B_{2R}$. Moreover, the following estimates hold : \begin{align} \left\|eft\|\partial_t\varphi^{(k)}\right\|ight\|_{H^2}+\left\|eft\|\nabla\varphi^{(k)}\right\|ight\|_{H^2}+\left\|eft\Vert\partial_t\left\|eft(\mathbf{T}^{(k-1)}\varphi^{(k)}\right\|ight)\right\|ight\Vert_{H^1}&\left\|eq A_0C_i,\left\|abel{HR fi 1}\\ \left\|eft\|\partial_t\Omegamega^{(k)}\right\|ight\|_{H^2}+\left\|eft\|\nabla\Omegamega^{(k)}\right\|ight\|_{H^2}+\left\|eft\Vert\partial_t\left\|eft(\mathbf{T}^{(k-1)}\Omegamega^{(k)}\right\|ight)\right\|ight\Vert_{H^1}&\left\|eq A_0C_i,\left\|abel{HR omega 1} \varepsilonnd{align} \varepsilonnd{itemize} Recalling the statement of Theorem \right\|ef{theoreme principal}, $C_{high}$ is a potentially large constant on which $T$ can depend, but $\varepsilon_{0}$ has to be independent of $C_{high}$ and $C_i$ (which, as explained in Corollary \right\|ef{coro premiere section}, depends on $C_{high}$). Therefore, in the following estimates, we will keep trace of $C_i$, and $\varepsilon C_i$ is not a small constant. We will use the symbol $\left\|esssim$ where the implicit constants are independent of $A_0$, $A_1$, $A_2$, $A_3$, $A_4$ and $C_i$ and use $C$ as the notation for such a constant. Moreover, $C(A_i)$ will denote a constant depending on $A_i$, but not on $C_i$. At the end of the proof, we will choose $A_0$, $A_1$, $A_2$, $A_3$ and $A_4$ such that $C(A_i)\left\|l A_{i+1}$ for all $i=0,\mathrm{d}ots,3$. \par\left\|eavevmode\par Our goal now is to prove that all this estimates are still true for the next iterate. For most of these, we will in fact show that they hold with better constants on the RHS. \subsubsection{Preliminary estimates} The next result will be very useful in the sequel : \begin{prop}\left\|abel{commutation estimate} The following estimate holds : \begin{equation*} \left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eq 9C_i.\left\|abel{commutation estimate eq} \varepsilonnd{equation*} \varepsilonnd{prop} \begin{proof} In view of \varepsilonqref{HR gamma 1}, we have to commute $\mathbf{T}^{(n-1)}$ with $\nabla^{\alpha}$ (for $|\alpha|\left\|eq 2$). Indeed : \begin{equation*} \left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eq \sum_{|\alpha|\left\|eq 2}\left\|eft( \left\|eft\Vert\mathbf{T}^{(n-1)}\nabla^{(\alpha)}\Tilde{\gamma}^{(n)}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}+\left\|eft\|\left\|eft[ \mathbf{T}^{(n-1)},\nabla^{\alpha}\right\|ight]\Tilde{\gamma}^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha|}}\right\|ight) \varepsilonnd{equation*} Using the commutation formula $[e_0^{(n-1)},\nabla]=\nabla\beta^{(n-1)}\nabla$, we compute \begin{equation*} \left\|eft[\mathbf{T}^{(n-1)},\nabla\right\|ight] \Tilde{\gamma}^{(n)} =\frac{\nabla\beta^{(n-1)}}{N^{(n-1)}}\nabla \Tilde{\gamma}^{(n)}-\frac{\nabla N^{(n-1)}}{N^{(n-1)}}\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \varepsilonnd{equation*} We need smallness for the metric component so we use on one hand \varepsilonqref{HR beta 1} the product estimate, the fact that $|\frac{1}{N^{(n-1)}}|\left\|esssim 1$ and on the other hand the fact that $|\nabla(\chi\left\|n)|\left\|esssim \left\|angle x\right\|angle^{-1}$ and \varepsilonqref{HR N 1} to write \begin{align*} \left\|eft\|\left\|eft[ \mathbf{T}^{(n-1)},\nabla\right\|ight]\Tilde{\gamma}^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta'+2}}&\left\|esssim \left\|eft\|\nabla\beta^{(n-1)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\left\|eft\|\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}+\left\|eft|N_a^{(n-1)}\right\|ight|\left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1}} \\ & \quad+ \left\|eft\|\nabla\Tilde{N}^{(n-1)}\right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\\ &\left\|esssim \varepsilon\left\|eft(\left\|eft\|\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}+ \left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}\right\|ight)+\varepsilon\left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}. \varepsilonnd{align*} Now we compute $\left\|eft[\mathbf{T}^{(n-1)},\nabla^2\right\|ight] \Tilde{\gamma}^{(n)}$ : \begin{align*} \left\|eft[\mathbf{T}^{(n-1)},\nabla^2\right\|ight] \Tilde{\gamma}^{(n)}&=2\frac{\nabla\beta^{(n-1)}}{N^{(n-1)}}\nabla^2 \Tilde{\gamma}^{(n)}-2\frac{\nabla N^{(n-1)}}{N^{(n-1)}}\mathbf{T}^{(n-1)}\nabla\Tilde{\gamma}^{(n)} +\left\|eft( \frac{\nabla^2\beta^{(n-1)}}{N^{(n-1)}}+\frac{\nabla N^{(n-1)}\nabla\beta^{(n-1)}}{\left\|eft(N^{(n-1)}\right\|ight)^2}\right\|ight)\nabla \Tilde{\gamma}^{(n)}\\ &-\left\|eft(\frac{\nabla^2 N^{(n-1)}}{N^{(n-1)}}+\left\|eft(\frac{\nabla N^{(n-1)}}{N^{(n-1)}} \right\|ight)^2 \right\|ight)\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}. \varepsilonnd{align*} Using the product estimate and $|\frac{1}{N^{(n-1)}}|\left\|esssim 1$ we do the following : \begin{align*} &\left\|eft\|\left\|eft[ \mathbf{T}^{(n-1)},\nabla^2\right\|ight] \Tilde{\gamma}^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+3}}& \\&\left\|esssim \left\|eft\| \nabla^2\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}} \left\|eft\| \nabla\beta^{(n-1)} \right\|ight\|_{H^1_{\mathrm{d}elta'+1}} +\left\|eft\|N_a^{(n-1)}\nabla(\chi\left\|n)\mathbf{T}^{(n-1)}\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+3}} +\left\|eft\| \nabla\Tilde{N}^{(n-1)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|eft\|\mathbf{T}^{(n-1)}\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}} \\ & \qquad+\left\|eft\| \nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft(\left\|eft\| \nabla^2\beta^{(n-1)} \right\|ight\|_{L^2_{\mathrm{d}elta'+2}}+\left\|eft\| \nabla N^{(n-1)}\nabla\beta^{(n-1)} \right\|ight\|_{L^2_{\mathrm{d}elta'+2}} \right\|ight)\\ & \qquad+ \left\|eft\| \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft( \left\|eft\| \nabla^2N^{(n-1)} \right\|ight\|_{L^2_{\mathrm{d}elta+2}} +\left\|eft\| \left\|eft(\nabla\Tilde{N}^{(n-1)}\right\|ight)^2+N_a^{(n-1)}\nabla(\chi\left\|n)\nabla\Tilde{N}^{(n-1)} \right\|ight\|_{L^2_{\mathrm{d}elta+2}} \right\|ight)\\ &\qquad+\left\|eft\| \left\|eft( N_a^{(n-1)}\nabla(\chi\left\|n)\right\|ight)^2\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+3}}. \varepsilonnd{align*} Now using \varepsilonqref{HR N 1}, \varepsilonqref{HR beta 1}, $|\nabla(\chi\left\|n)|\left\|esssim \left\|angle x\right\|angle^{-1}$ we have : \begin{align*} \left\|eft\|\left\|eft[ \mathbf{T}^{(n-1)},\nabla^2\right\|ight] \Tilde{\gamma}^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+3}} \left\|esssim & \;\varepsilon\left\|eft( \sum_{|\alpha|\left\|eq 2} \left\|eft\Vert\mathbf{T}^{(n-1)}\nabla^{\alpha}\Tilde{\gamma}^{(n)}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}+\left\|eft\|\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\right\|ight) \\ & +\varepsilon\left\|eft\| \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}+\varepsilon\left\|eft\| \mathbf{T}^{(n-1)}\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}} \varepsilonnd{align*} It remains to deal with the last term in this last inequality. Using the same type of arguments as above we can show that : \begin{equation*} \left\|eft\|\mathbf{T}^{(n-1)}\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}}\left\|esssim \left\|eft\|\mathbf{T}^{(n-1)}\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+2}} + \left\|eft\|\mathbf{T}^{(n-1)}\nabla^2\Tilde{\gamma}^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+3}} +\varepsilon\left\|eft\| \nabla^2\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}} \varepsilonnd{equation*} Summarising, we get : \begin{align*} \left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \left\|esssim & (1+\varepsilon)\left\|eft(\sum_{|\alpha|\left\|eq 2} \left\|eft\Vert\mathbf{T}^{(n-1)}\nabla^{\alpha}\Tilde{\gamma}^{(n)}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}+\left\|eft\|\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\right\|ight)\\&\qquad+\varepsilon \left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \varepsilonnd{align*} By choosing $\varepsilon$ small enough, we can absorb the last term of the RHS into the LHS and using \varepsilonqref{HR gamma 1} we finally prove the desired result. \varepsilonnd{proof} We continue with a propagation of smallness result. \begin{prop}\left\|abel{prop propsmall} The following estimates hold for $T$ sufficiently small and $C_p>0$ a constant depending on $\mathrm{d}elta$ and $R$ only : \begin{align} \left\|eft\|\partial_t\varphi^{(n)}\right\|ight\|_{L^4}+\left\|eft\|\nabla\varphi^{(n)}\right\|ight\|_{L^4}+\left\|eft\|\partial_t\Omegamega^{(n)}\right\|ight\|_{L^4}+\left\|eft\|\nabla\Omegamega^{(n)}\right\|ight\|_{L^4}&\left\|eq C_p\varepsilon\left\|abel{propsmall fi},\\ \left\|eft\| H^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta+1}}&\left\|eq C_p\varepsilon^2\left\|abel{propsmall H},\\ \left\|eft\| \Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'}}&\left\|eq C_p \varepsilon^2\left\|abel{propsmall gamma},\\ \left\|eft\|\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+1}}&\left\|eq C_p\varepsilon^2\left\|abel{propsmall eo gamma},\\ \left\|eft\|\tau^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}&\left\|eq C_p\varepsilon^2.\left\|abel{propsmall tau} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} By Corollary \right\|ef{coro premiere section}, all these quantities satisfy the desired smallness estimates at $t=0$. The fact that these estimates are true for all $t\in[0,T]$ will then follow from calculus inequalities of the type \begin{equation*} \sup_{s\in[0,T]}\|u\|_{W^{m,p}_{\varepsilonta}}(s)\left\|eq C'\left\|eft( \|u\|_{W^{m,p}_{\varepsilonta}}(0)+\int_0^T\|\partial_tu\|_{W^{m,p}_{\varepsilonta}}(s)\mathrm{d} s\right\|ight). \varepsilonnd{equation*} Therefore, it remains to show that the $\partial_t$ derivatives (we recall that $\partial_t=e_0^{(n)}+\beta^{(n)}\cdot\nabla=e_0^{(n-1)}+\beta^{(n-1)}\cdot\nabla$) of all these terms in the relevant norms are bounded by a constant depending on $A_0$, $A_1$, $A_2$, $A_3$, $A_4$ or $C_i$, and then to choose $T$ small enough. We proceed as follows : \begin{itemize} \item for $\nabla\varphi^{(n)}$ and $\nabla\Omegamega^{(n)}$, we use the embedding $H^1\xhookrightarrow{}L^4$ and \varepsilonqref{HR fi 1} : \begin{equation} \left\|eft\|\partial_t\nabla\varphi^{(n)}\right\|ight\|_{L^4}\left\|esssim \left\|eft\|\nabla\partial_t\varphi^{(n)}\right\|ight\|_{H^1}\left\|esssim \left\|eft\|\partial_t\varphi^{(n)}\right\|ight\|_{H^2}\left\|esssim A_0C_i,\left\|abel{propsmall a} \varepsilonnd{equation} and we do the same for $\nabla\Omegamega^{(n)}$, using \varepsilonqref{HR omega 1}. \item for $\partial_t\varphi^{(n)}$ and $\partial_t\Omegamega^{(n)}$, we use the support property of $\varphi^{(n)}$, the embedding $H^1\xhookrightarrow{}L^4$, \varepsilonqref{HR fi 1}, \varepsilonqref{HR beta 1}, \varepsilonqref{HR beta 3}, \varepsilonqref{HR N 2} and \varepsilonqref{propsmall a} : \begin{align*} \left\|eft\|\partial_t^2\varphi^{(n)}\right\|ight\|_{L^4}&\left\|eq \left\|eft\|N^{(n-1)}\partial_t\left\|eft(\mathbf{T}^{(n-1)}\varphi^{(n)}\right\|ight)\right\|ight\|_{L^4}+\left\|eft\|\mathbf{T}^{(n-1)}\varphi^{(n)}\partial_tN^{(n-1)}\right\|ight\|_{L^4}+ \left\|eft\|\partial_t\left\|eft( \beta^{(n)}\cdot\nabla\varphi^{(n)}\right\|ight)\right\|ight\|_{L^4}\\ & \left\|esssim \left\|eft\|\partial_t\left\|eft(\mathbf{T}^{(n-1)}\varphi^{(n)}\right\|ight)\right\|ight\|_{H^1}+\left\|eft\|\partial_t\varphi^{(n)}\right\|ight\|_{H^2}+ \left\| \partial_t\beta^{(n)} \right\|_{H^2} \left\|eft\|\nabla\varphi^{(n)}\right\|ight\|_{H^2}+ \left\|eft\|\partial_t\nabla\varphi^{(n)}\right\|ight\|_{L^4}\\& \left\|esssim A_0C_i+ \varepsilon A_1C_i, \varepsilonnd{align*} and we do the same for $\partial_t\Omegamega^{(n)}$, using \varepsilonqref{HR omega 1}. \item for $H^{(n)}$, we use \varepsilonqref{HR H 1}, \varepsilonqref{HR H 2}, \varepsilonqref{HR beta 1} and the product estimate : \begin{equation*} \left\|eft\|\partial_t H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|eq \left\|eft\| e_0^{(n)}H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}+\left\|eft\|\beta^{(n)}\nabla H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|esssim C_i. \varepsilonnd{equation*} \item for $\Tilde{\gamma}^{(n)}$, we use \varepsilonqref{commutation estimate eq}, \varepsilonqref{HR beta 1} and \varepsilonqref{HR gamma 1} : \begin{align*} \left\|eft\| \partial_t\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'}}&\left\|eq \left\|eft\| N^{(n-1)}\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'}}+\left\|eft\|\beta^{(n-1)}\cdot\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'}}\\ & \left\|esssim\left\|eft\| \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}+ \left\|eft\|\nabla\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\|\beta^{(n-1)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\\ &\left\|esssim C_i+A_0C_i^2. \varepsilonnd{align*} \item for $\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}$ and $\tau^{(n)}$, we simply use \varepsilonqref{HR gamma 3} and \varepsilonqref{HR tau 3}, which give directly the result. \varepsilonnd{itemize} \varepsilonnd{proof} \subsubsection{Elliptic estimates} We begin with the two elliptic equations (the ones for $N$ and $\beta$). These are the most difficult to handle, because we can't rely on the smallness of a time parameter and therefore have to keep properly trace of the $\varepsilon$, $C_i$ and $A_i$. \begin{prop}\left\|abel{hr+1 N prop} For $n\geq 2$, $N^{(n+1)}$ admits a decomposition \begin{equation*} N^{(n+1)}=1+N_a^{(n+1)}\chi\left\|n+\Tilde{N}^{(n+1)}, \varepsilonnd{equation*} with $N_a^{(n+1)}\geq 0$ and such that \begin{align} \left\|eft| N_a^{(n+1)}\right\|ight|+\left\|eft\|\Tilde{N}^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta}}&\left\|esssim \varepsilon^2\left\|abel{HR+1 N 1},\\ \left\|eft|\partial_tN_a^{(n+1)}\right\|ight|+\left\|eft\|\Tilde{N}^{(n+1)}\right\|ight\|_{H^3_{\mathrm{d}elta}}+\left\|eft\|\partial_t\Tilde{N}^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta}}&\left\|esssim \varepsilon C(A_3)C_i\left\|abel{HR+1 N 2}, \\ \left\|eft\| \Tilde{N}^{(n+1)}\right\|ight\|_{H^4_{\mathrm{d}elta}}&\left\|esssim \varepsilon^2C(A_2)C_i^2+C(A_0)C_i^2.\left\|abel{HR+1 N 3} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} We claim that : \begin{equation*} \left\|eft\| \text{RHS of \varepsilonqref{reduced system N}} \right\|ight\|_{L^2_{\mathrm{d}elta+2}}\left\|eq C\varepsilon^2. \varepsilonnd{equation*} Except for the term $e^{2\gamma^{(n)}}N^{(n)}\left\|eft( \tau^{(n)}\right\|ight)^2$, all the terms in \varepsilonqref{reduced system N} can be estimated in an identical manner as in Lemma \right\|ef{lem CI N}, except that we estimate the norms using Proposition \right\|ef{prop propsmall} instead of using the assumtions on the reduced data and the estimates in Lemmas \right\|ef{CI sur H} and \right\|ef{lem CI gamma}. It therefore remains to control $e^{2\gamma^{(n)}}N^{(n)}\left\|eft( \tau^{(n)}\right\|ight)^2$. Using \varepsilonqref{useful gamma 1a} and \varepsilonqref{HR N 1}, we see that $\left\|eft\|e^{2\gamma^{(n)}} N^{(n)}\right\|ight\|_{C^0_\varepsilon}\left\|esssim 1$. We finally use the product estimate and \varepsilonqref{propsmall tau} to handle $\left\|eft( \tau^{(n)}\right\|ight)^2$ : \begin{equation*} \left\|eft\|e^{2\gamma^{(n)}}N^{(n)}\left\|eft( \tau^{(n)}\right\|ight)^2 \right\|ight\|_{L^2_{\mathrm{d}elta+2}} \left\|esssim \left\|eft\| e^{2\gamma^{(n)}} N^{(n)}\right\|ight\|_{C^0_\varepsilon} \left\|eft\| \tau^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+1}}^2\left\|esssim \varepsilon^4. \varepsilonnd{equation*} This proves the claim. Applying Corollary \right\|ef{mcowens 2} to $N^{(n+1)}-1$ yields the existence of the decomposition of $N^{(n+1)}$, as well as the estimate \varepsilonqref{HR+1 N 1}. \par\left\|eavevmode\par We now turn to the proof of \varepsilonqref{HR+1 N 2}. To obtain the $H^3_{\mathrm{d}elta}$ bound for $\Tilde{N}^{(n+1)}$, we need to control the RHS of \varepsilonqref{reduced system N} in $H^1_{\mathrm{d}elta+2}$ : \begin{itemize} \item for the term $e^{-2\gamma^{(n)}}N^{(n)}|H^{(n)}|^2$, we do exactly the same calculations as in \varepsilonqref{idem1}, but in $H^1_{\mathrm{d}elta+2}$ instead of $H^2_{\mathrm{d}elta+2}$. In contrast to \varepsilonqref{idem1}, here we have less liberty to bound the term $|H^{(n)}|^2$ (because we need $C_i$ and not $C_i^2$ bounds), therefore we use \varepsilonqref{HR H 1} and \varepsilonqref{propsmall H} to write \begin{equation*} \left\| e^{-2\gamma^{(n)}}N^{(n)}\left\|eft|H^{(n)}\right\|ight|^2\right\|_{H^1_{\mathrm{d}elta+2}} \left\|esssim\left\|eft\| \left\|eft|H^{(n)}\right\|ight|^2\right\|ight\|_{H^1_{\mathrm{d}elta+2+2\varepsilon+\frac{\mathrm{d}elta+1}{2}}}\left\|esssim \left\|eft\|H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|eft\|H^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta+1}}\left\|esssim \varepsilon^2C_i. \varepsilonnd{equation*} \item for the term $e^{2\gamma^{(n)}}N^{(n)}( \tau^{(n)})^2$, we note that $\tau^{(n)}$ and $H^{(n)}$ satisfy the exact same estimate (according to \varepsilonqref{HR H 1}, \varepsilonqref{HR tau 1}, \varepsilonqref{propsmall H} and \varepsilonqref{propsmall tau}), except for a slight difference of weights ($\mathrm{d}elta'$ instead of $\mathrm{d}elta$) and constants ($A_1$ compared to 2). Therefore we treat this term exactly as the previous one and omit the details. \item we now discuss the term $\frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2$. Since the smallness for $e_0^{(n-1)}\varphi^{(n)}$ is at the $L^4$-level (thanks to \varepsilonqref{propsmall fi}), any spatial derivative of $e_0^{(n-1)}\varphi^{(n)}$ destroys the $\varepsilon$-smallness, and therefore we have to be precise. Thanks to \varepsilonqref{useful gamma 1a}, we can forget the $e^{2\gamma^{(n)}}$ factor, thanks to \varepsilonqref{HR N 1} we have $\left\|eft| \frac{1}{N^{(n)}}\right\|ight|\left\|esssim 1$ (we also forget about $\nabla(\chi\left\|n)$) and thanks to \varepsilonqref{HR N 2} and the embedding $H^2_{\mathrm{d}elta+1}\xhookrightarrow{}L^{\infty}$ we have $\left\|eft\|\nabla \Tilde{N}^{(n)}\right\|ight\|_{L^{\infty}}\left\|esssim C_i$ : \begin{align*} \left\|eft\|\frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{H^1_{\mathrm{d}elta+2}}&\left\|esssim \left\|eft\|\left\|eft( e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{L^2}\left\|eft( 1+ \left\|eft\|\nabla \Tilde{N}^{(n)}\right\|ight\|_{L^{\infty}}\right\|ight) +\left\|eft\|e_0^{(n-1)}\varphi^{(n)}\nabla\left\|eft( e_0^{(n-1)}\varphi^{(n)}\right\|ight) \right\|ight\|_{L^2}\\ &\left\|esssim \left\|eft\| e_0^{(n-1)}\varphi^{(n)} \right\|ight\|_{L^4}^2\left\|eft( 1+ \left\|eft\|\nabla \Tilde{N}^{(n)}\right\|ight\|_{L^{\infty}}\right\|ight)+\left\| e_0^{(n-1)}\varphi^{(n)} \right\|_{L^4}\left\| e_0^{(n-1)}\varphi^{(n)} \right\|_{H^2}, \varepsilonnd{align*} where in the last inequality we used Hölder's inequality and the Sobolev injection $H^1\xhookrightarrow{}L^4$. We now use \varepsilonqref{propsmall fi} and \varepsilonqref{HR fi 1} to obtain : \begin{align*} \left\|eft\|\frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{H^1_{\mathrm{d}elta+2}}&\left\|esssim \varepsilon^2(1+C_i)+\varepsilon A_0C_i\left\|esssim \varepsilon C(A_0)C_i. \varepsilonnd{align*} \item the term $\frac{e^{2\gamma^{(n)}-4\varphi^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2$ is handled in a similar way, using first \varepsilonqref{useful ffi 1} to get ride of the $e^{-4\varphi^{(n)}}$ factor, and then using \varepsilonqref{HR omega 1} instead of \varepsilonqref{HR fi 1}. \varepsilonnd{itemize} This concludes the proof of the estimate $\left\|eft\| \Tilde{N}^{(n+1)}\right\|ight\|_{H^3_{\mathrm{d}elta}}\left\|esssim \varepsilon C(A_1)C_i$. \par\left\|eavevmode\par We now turn to the estimate for $\partial_tN^{(n+1)}$, including both $\partial_tN_a^{(n+1)}$ and $\partial_t\Tilde{N}^{(n+1)}$. Since the RHS of \varepsilonqref{reduced system N} is differentiable in $t$, it is easy to see that $\partial_tN^{(n+1)}=\partial_tN_a^{(n+1)}\chi\left\|n+\partial_t\Tilde{N}^{(n+1)}$ is the solution given by Corollary \right\|ef{mcowens 2} to the equation \begin{equation*} \mathscr{D}elta f=\partial_t(\text{RHS of \varepsilonqref{reduced system N}}). \varepsilonnd{equation*} Therefore, to finish the proof of \varepsilonqref{HR+1 N 2}, it suffices to bound the integral of $\partial_t(\text{RHS of \varepsilonqref{reduced system N}})$ with respect to $\mathrm{d} x$ and to bound $\partial_t(\text{RHS of \varepsilonqref{reduced system N}})$ in $L^2_{\mathrm{d}elta+2}$. Since the estimate for $\partial_t\tau^{(n)}$ are worse than those for $\partial_tH^{(n)}$, and those for $\tau^{(n)}$ and $H^{(n)}$ are similar, we will treat the term $\partial_t \left\|eft( e^{2\gamma^{(n)}}N^{(n)}( \tau^{(n)})^2\right\|ight) $ and leave the easier term $\partial_t \left\|eft( e^{2\gamma^{(n)}}N^{(n)}| H^{(n)}|^2\right\|ight) $ to the reader. We use \varepsilonqref{useful gamma 1a} for the $e^{2\gamma^{(n)}}$ factor and the fact that $|\chi\left\|n|\left\|esssim \left\|angle x\right\|angle^{\varepsilon}$ : \begin{align*} \left\|eft\|\partial_t \left\|eft( e^{2\gamma^{(n)}}N^{(n)}( \tau^{(n)})^2\right\|ight) \right\|ight\|_{L^2_{\mathrm{d}elta+2}} &\left\|esssim\left\|eft\|N^{(n)} \right\|ight\|_{C^0_{\varepsilon}} \left\|eft( \left\|eft\|\tau^{(n)}\partial_t\tau^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta+2+\varepsilon}}+\left\|eft\|\partial_t\Tilde{\gamma}^{(n)}( \tau^{(n)})^2 \right\|ight\|_{L^2_{\mathrm{d}elta+2+\varepsilon}}\right\|ight)\\&\qquad +\left\|eft|\partial_tN_a^{(n)} \right\|ight|\left\|eft\| ( \tau^{(n)})^2\right\|ight\|_{L^2_{\mathrm{d}elta+2+3\varepsilon}} +\left\|eft\| \partial_t\Tilde{N}^{(n)}( \tau^{(n)})^2\right\|ight\|_{L^2_{\mathrm{d}elta+2+2\varepsilon}} \\& \left\|esssim\varepsilon^2C(A_3)C_i. \varepsilonnd{align*} where in the last inequality we use $\left\|eft\|N^{(n)} \right\|ight\|_{C^0_{\varepsilon}}\left\|esssim 1$ (which comes from \varepsilonqref{HR N 1}), $\left\|eft\| \partial_t\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|esssim C_i$ (wich comes from \varepsilonqref{commutation estimate eq}), \varepsilonqref{HR N 2} and the product estimate together with \varepsilonqref{propsmall tau} (for $(\tau^{(n)})^2$). For $\tau^{(n)}\partial_t\tau^{(n)}$, we use the Hölder's inequality ($L^4_{\mathrm{d}elta'+2}\times L^4_{\mathrm{d}elta'+1}\xhookrightarrow{}L^2_{\mathrm{d}elta+2+\varepsilon}$), the embedding $H^1_{\mathrm{d}elta'+1}\xhookrightarrow{}L^4_{\mathrm{d}elta'+1}$, \varepsilonqref{propsmall tau} and \varepsilonqref{HR tau 3}. We now turn to the compactly supported term $\partial_t\left\|eft(\frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2\right\|ight)$. We use \varepsilonqref{useful gamma 1a} for the $e^{2\gamma^{(n)}}$ factor and $\left\|eft|\frac{1}{N^{(n)}}\right\|ight|+\left\|eft\| \chi\left\|n\right\|ight\|_{L^{\infty}(B_{2R})}\left\|esssim 1$ : \begin{align*} \left\|eft\| \partial_t\left\|eft(\frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2\right\|ight)\right\|ight\|_{L^2} &\left\|esssim \left\|eft\|e_0^{(n-1)}\varphi^{(n)}\partial_t\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight) \right\|ight\|_{L^2}+\left\|eft\|\partial_t\Tilde{\gamma}^{(n)}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{L^2}\\&\qquad+\left\|eft\|\partial_t\Tilde{N}^{(n)}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{L^2}+\left\|eft|\partial_tN_a^{(n)}\right\|ight|\left\|eft\|\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{L^2} \\&\left\|esssim \left\|eft\|e_0^{(n-1)}\varphi^{(n)} \right\|ight\|_{L^4}\left\|eft(C_i\left\|eft\|e_0^{(n-1)}\varphi^{(n)} \right\|ight\|_{L^4}+ \left\|eft\|\partial_t\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight) \right\|ight\|_{H^1}\right\|ight) \left\|esssim \varepsilon C_i, \varepsilonnd{align*} where in the last inequality we use $\left\|eft\| \partial_t\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|esssim C_i$, \varepsilonqref{HR N 2} and the embedding $L^4\times H^1\xhookrightarrow{}L^2$. The term $\partial_t\left\|eft(\frac{e^{2\gamma^{(n)}-4\varphi^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2\right\|ight)$ is handled in the same way, using \varepsilonqref{useful ffi 1} and \varepsilonqref{propsmall fi} to get rid of the $e^{-4\varphi^{(n)}}$. Combining all these estimates concludes the proof of \varepsilonqref{HR+1 N 2}. \par\left\|eavevmode\par We now turn to the proof of \varepsilonqref{HR+1 N 3}. To obtain the $H^4_{\mathrm{d}elta}$ bound for $\Tilde{N}^{(n+1)}$, we need to control the RHS of \varepsilonqref{reduced system N} in $H^2_{\mathrm{d}elta+2}$. Since we already know that the RHS of \varepsilonqref{reduced system N} is in $H^1_{\mathrm{d}elta+2}$ with bound $\varepsilon C(A_1)C_i$, it remains to bound the $L^2_{\mathrm{d}elta+4}$ norm of the second derivative the RHS of \varepsilonqref{reduced system N} : \begin{itemize} \item for the term $e^{-2\gamma^{(n)}}N^{(n)}|H^{(n)}|^2$, we first use \varepsilonqref{useful gamma 1}, and then the embedding $H^1_{\mathrm{d}elta+1}\xhookrightarrow{}L^4_{\mathrm{d}elta+1}$, \varepsilonqref{HR H 1}, \varepsilonqref{propsmall H}, \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2} and \varepsilonqref{HR N 3} : \begin{align*} \left\| \nabla^2\left\|eft(e^{-2\gamma^{(n)}}N^{(n)}|H^{(n)}|^2\right\|ight) \right\|_{L^2_{\mathrm{d}elta+4}} &\left\|esssim \left\|\nabla^2N^{(n)}(H^{(n)})^2 \right\|_{L^2_{\mathrm{d}elta+4}} + \left\| \nabla N^{(n)}H^{(n)}\nabla H^{(n)} \right\|_{L^2_{\mathrm{d}elta+4}} \\&\qquad+\left\| N^{(n)} H^{(n)} \nabla^2H^{(n)} \right\|_{L^2_{\mathrm{d}elta+4}} +\left\| N^{(n)} (\nabla H^{(n)})^2 \right\|_{L^2_{\mathrm{d}elta+4}} \\& \left\|esssim\varepsilon^2 C(A_2) C_i^2 \varepsilonnd{align*} where we used $L^\infty$ bounds for $N^{(n)}$, $\nabla N^{(n)}$ and $\nabla^2 N^{(n)}$ (see \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2} and \varepsilonqref{HR N 3} respectively). Note that for $N$ the logarithmic growth is handled by adding a small weight. We also used \varepsilonqref{HR H 1} and \varepsilonqref{propsmall H} and the product law to handle the $H^{(n)}$ terms. \item for the term $e^{2\gamma^{(n)}}N^{(n)}( \tau^{(n)})^2$, we note that $\tau^{(n)}$ and $H^{(n)}$ satisfy the exact same estimate (according to \varepsilonqref{HR H 1}, \varepsilonqref{HR tau 1}), except for a slight difference of weights ($\mathrm{d}elta'$ instead of $\mathrm{d}elta$) and constants ($A_1$ compared to 2). Therefore we treat this term exactly as the previous one and omit the details. \item we next discuss the compactly supported term $\frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2$. We first use \varepsilonqref{useful gamma 1} : \begin{align*} \left\|eft\|\nabla^2\left\|eft( \frac{e^{2\gamma^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2\right\|ight) \right\|ight\|_{L^2} &\left\|esssim \left\|eft\| e_0^{(n-1)}\varphi^{(n)}\nabla^2\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight) \right\|ight\|_{L^2}+\left\|eft\|\left\|eft(\nabla\left\|eft( e_0^{(n-1)}\varphi^{(n)}\right\|ight)\right\|ight)^2 \right\|ight\|_{L^2}\\&+\left\|eft\|\nabla N^{(n)} e_0^{(n-1)}\varphi^{(n)}\nabla\left\|eft(e_0^{(n-1)}\varphi^{(n)}\right\|ight)\right\|ight\|_{L^2}+\left\|eft\|\nabla^2N^{(n)}\left\|eft( e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{L^2}\\&+\left\|eft\|\left\|eft( \nabla N^{(n)}\right\|ight)^2\left\|eft( e_0^{(n-1)}\varphi^{(n)}\right\|ight)^2 \right\|ight\|_{L^2} \\& \left\|esssim\varepsilon^2C(A_2)C_i^2+C(A_0)C_i^2, \varepsilonnd{align*} where we used \varepsilonqref{HR N 2}, \varepsilonqref{HR fi 1} and \varepsilonqref{propsmall fi}. The idea is to use $L^{\infty}$-bounds for $\nabla N^{(n)}$ and $\nabla^2 N^{(n)}$ and the Hölder's inequality to deal with the product of terms depending on $\varphi^{(n)}$. \item the term $\frac{e^{2\gamma^{(n)}-4\varphi^{(n)}}}{N^{(n)}}\left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2$ is handled in a similar way, but we have to be careful about the case where two derivatives hit $e^{-4\varphi^{(n)}}$. Using \varepsilonqref{useful ffi 1}, \varepsilonqref{useful gamma 1a} and $1\left\|esssim N^{(n)}$, this leads to estimating the following term : \begin{align*} \left\|\nabla^2\varphi^{(n)}\left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2 \right\|_{L^2}\left\|esssim \left\| \nabla^2\varphi^{(n)}\right\|_{L^4}\left\| e_0^{(n-1)}\Omegamega^{(n)}\right\|_{L^4}\left\| e_0^{(n-1)}\Omegamega^{(n)}\right\|_{L^\infty}\left\|esssim \varepsilon C(A_0)C_i^2, \varepsilonnd{align*} where we used \varepsilonqref{HR fi 1}, \varepsilonqref{HR omega 1} and \varepsilonqref{propsmall fi}. \varepsilonnd{itemize} This concludes the proof of \varepsilonqref{HR+1 N 3}. \varepsilonnd{proof} The following lemma will allow us to estimate the $H^1$ norm of solutions of elliptic equations. \begin{lem} Let $\xi=(\xi^1,\xi^2)$ a vector field on $\mathbb{R}^2$, for all $\sigma<1$ the following holds : \begin{align} \left\|\nabla\xi \right\|_{L^2_{\sigma}} \left\|esssim \left\| L\xi\right\|_{L^2_{1}}\left\|abel{killing operator H1}. \varepsilonnd{align} \varepsilonnd{lem} \begin{proof} We set $A_{ij}\vcentcolon=(L\xi)_{ij}$ and take the divergence to obtain $\mathscr{D}elta\xi^i=\mathrm{d}elta^{ij}\partial^kA_{kj}$. Let $w(x)=\left\|angle x\right\|angle^{2\sigma}$. We multiply this equation by $w\xi^{\varepsilonll}$, contract it with $\mathrm{d}elta_{i\varepsilonll}$ and integrate over $\mathbb{R}^2$ to get (after integrating by parts) : \begin{equation*} \mathrm{d}elta_{i\varepsilonll}\int_{\mathbb{R}^2}\nabla(w\xi^{\varepsilonll})\cdot\nabla\xi^i\mathrm{d} x=\int_{\mathbb{R}^2}\partial^k(w\xi^{\varepsilonll})A_{k\varepsilonll}\mathrm{d} x, \varepsilonnd{equation*} which becomes \begin{equation*} \left\| \nabla\xi\right\|_{L^2_{\sigma}}^2=\frac{1}{2} \int_{\mathbb{R}^2}\mathscr{D}elta w|\xi|^2\mathrm{d} x + \int_{\mathbb{R}^2}w\partial^k\xi^{\varepsilonll}A_{k\varepsilonll}\mathrm{d} x +\int_{\mathbb{R}^2}\partial^k w\xi^{\varepsilonll}A_{k\varepsilonll}\mathrm{d} x . \varepsilonnd{equation*} Using the Cauchy-Schwarz inequality and the trick $ab\left\|eq \varepsilonta a^2+\frac{1}{\varepsilonta} b^2$, we have \begin{equation*} \int_{\mathbb{R}^2}w\partial^k\xi^{\varepsilonll}A_{k\varepsilonll}\mathrm{d} x\left\|esssim \left\|\nabla\xi \right\|_{L^2_{\sigma}} \left\| A \right\|_{L^2_{\sigma}}\left\|esssim \varepsilonta\left\|\nabla\xi \right\|_{L^2_{\sigma}}^2+\frac{1}{\varepsilonta}\left\| A \right\|_{L^2_{\sigma}}^2. \varepsilonnd{equation*} We note that $|\nabla w|\left\|esssim \left\|angle x \right\|angle^{2\sigma-1}$ and $|\mathscr{D}elta w|\left\|esssim \left\|angle x \right\|angle^{2\sigma-2}$, which imply that \begin{equation*} \int_{\mathbb{R}^2}\partial^k w\xi^{\varepsilonll}A_{k\varepsilonll}\mathrm{d} x \left\|esssim \left\|\xi \right\|_{L^2_{\sigma-1}} \left\| A \right\|_{L^2_{\sigma}}\left\|esssim \left\|\xi \right\|_{L^2_{\sigma-1}}^2+\left\| A \right\|_{L^2_{\sigma}}^2\quad\text{and}\quad \frac{1}{2} \int_{\mathbb{R}^2}\mathscr{D}elta w|\xi|^2\mathrm{d} x \left\|esssim \left\|\xi \right\|_{L^2_{\sigma-1}}^2. \varepsilonnd{equation*} Thus, \begin{equation*} \left\| \nabla\xi\right\|_{L^2_{\sigma}}^2 \left\|esssim \left\|\xi \right\|_{L^2_{\sigma-1}}^2+\left\|eft( 1+\frac{1}{\varepsilonta}\right\|ight)\left\| A \right\|_{L^2_{\sigma}}^2+\varepsilonta\left\|\nabla\xi \right\|_{L^2_{\sigma}}^2. \varepsilonnd{equation*} We take $\varepsilonta$ small enough in order to absorb $\varepsilonta\left\|\nabla\xi \right\|_{L^2_{\sigma}}^2$ into the LHS. Taking the square root of the inequality we obtained, we get : \begin{equation*} \left\| \nabla\xi\right\|_{L^2_{\sigma}} \left\|esssim \left\|\xi \right\|_{L^2_{\sigma-1}}+\left\| A \right\|_{L^2_{\sigma}}. \varepsilonnd{equation*} It remains to show that $\left\|\xi \right\|_{L^2_{\sigma-1}}\left\|esssim \left\| A\right\|_{L^2_1}$. For that, we start by using Lemma \right\|ef{prop holder 2} : $\sigma<1$ so there exists $r>2$ such that $\sigma<\frac{2}{r}<1$. According to Lemma \right\|ef{prop holder 2}, we have $\left\|\xi \right\|_{L^2_{\sigma-1}}\left\|esssim \left\|\xi\right\|_{L^r}$. Recalling that $\mathscr{D}elta\xi^i=\mathrm{d}elta^{ij}\partial^kA_{kj}$ we have : \begin{equation*} \xi^i(x)=\frac{\mathrm{d}elta^{ij}}{2\pi}\int_{\mathbb{R}^2}\left\|n|x-y|\partial^kA_{kj}\mathrm{d} y=-\frac{\mathrm{d}elta^{ij}}{2\pi}\int_{\mathbb{R}^2}\frac{y^k-x^k}{|x-y|^2}A_{kj}\mathrm{d} y.\ \varepsilonnd{equation*} Therefore we can use the Hardy-Littlewood-Sobolev inequality (Proposition \right\|ef{prop HLS}), that \begin{equation} \left\|\xi \right\|_{L^r}\left\|esssim \left\| A*\frac{1}{|\cdot|} \right\|_{L^r} \left\|esssim \left\| A\right\|_{L^{\frac{2r}{2+r}}} . \varepsilonnd{equation} We again use Lemma \right\|ef{prop holder 2} to get the embedding $L^2_1\xhookrightarrow{}L^{\frac{2r}{2+r}}$ (recall that $r>2$), which conclude the proof of \varepsilonqref{killing operator H1}. \varepsilonnd{proof} \begin{prop}\left\|abel{hr+1 beta prop} For $n\geq 2$, the following estimates hold : \begin{align} \left\|eft\|\beta^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta'}}&\left\|esssim \varepsilon^2\left\|abel{HR+1 beta 1},\\ \left\|eft\|\beta^{(n+1)}\right\|ight\|_{H^3_{\mathrm{d}elta'}}&\left\|esssim C_i\left\|abel{HR+1 beta 2},\\ \left\|eft\|\nabla e_0^{(n)}\beta^{(n+1)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|esssim \varepsilon C_i\left\|abel{HR+1 beta 2.5},\\ \left\|eft\| e_0^{(n)}\beta^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta'}}&\left\|esssim A_0C_i\left\|abel{HR+1 beta 3},\\ \left\|eft\| e_0^{(n)}\beta^{(n+1)}\right\|ight\|_{H^3_{\mathrm{d}elta'}}&\left\|esssim A_3C_i^2.\left\|abel{HR+1 beta 4} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} In view of Proposition \right\|ef{prop propsmall} and \varepsilonqref{HR N 1}, the existence and uniqueness of $\beta^{(n+1)}$ and the estimate \varepsilonqref{HR+1 beta 1} can be proven in exactly the same manner as in Lemma \right\|ef{CI sur beta} and we omit the details. \par\left\|eavevmode\par We begin by the proof of \varepsilonqref{HR+1 beta 2}. We take the divergence of \varepsilonqref{reduced system beta} to get \begin{equation} \mathscr{D}elta(\beta^{(n+1)})^i=2\mathrm{d}elta^{i\varepsilonll}\mathrm{d}elta^{jk}\partial_k\left\|eft( e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_{j\varepsilonll} \right\|ight).\left\|abel{laplacien beta} \varepsilonnd{equation} Note that the RHS has 0 mean (by Lemma \right\|ef{divergence nulle}) and therefore by Theorem \right\|ef{mcowens 1}, in order to prove \varepsilonqref{HR+1 beta 2}, it suffices to bound the RHS of \varepsilonqref{laplacien beta} in $H^1_{\mathrm{d}elta'+2}$ by $CC_i$. Using \varepsilonqref{useful gamma 1}, \varepsilonqref{HR H 1}, \varepsilonqref{HR N 1} and $\varepsilon\left\|eft|\chi\left\|n \right\|ight|\left\|esssim \left\|angle x \right\|angle^{\frac{\varepsilon}{2}}$ and taking $\varepsilon$ small enough, we get : \begin{align} \left\|eft\|\partial_k\left\|eft( e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_{j\varepsilonll}\right\|ight)\right\|ight\|_{H^1_{\mathrm{d}elta'+2}}&\left\|esssim \left\|eft\|N^{(n)}H^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+2\varepsilon^2+1}} \left\|abel{NnHn}\\ &\left\|esssim \left\|eft\| H^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}\left\|eft( 1+ \left\|eft\|\Tilde{N}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta}} \right\|ight)+ \left\|eft\| H^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+2\varepsilon^2+\frac{\varepsilon}{2}+1}}\nonumber\\& \left\|esssim C_i.\nonumber \varepsilonnd{align} We now turn to the proof of \varepsilonqref{HR+1 beta 2.5}. We have $e_0^{(n)}\beta^{(n+1)}=\partial_t\beta^{(n+1)}-\beta^{(n)}\cdot\nabla\beta^{(n+1)}$. Using \varepsilonqref{HR beta 1}, \varepsilonqref{HR+1 beta 1} and the product estimate we have \begin{equation*} \left\| \beta^{(n)}\cdot\nabla\beta^{(n+1)}\right\|_{H^1_{\mathrm{d}elta'}} \left\|esssim \left\|\beta^{(n)} \right\|_{H^2_{\mathrm{d}elta'}}\left\| \beta^{(n+1)}\right\|_{H^2_{\mathrm{d}elta'}} \left\|esssim \varepsilon^2. \varepsilonnd{equation*} Applying $\partial_t$ to \varepsilonqref{reduced system beta}, $\partial_t\beta^{(n+1)}$ satisfy $(L\partial_t\beta^{(n+1)})_{ij}=2\partial_t(e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_{ij})$. We apply \varepsilonqref{killing operator H1} with $\sigma =\mathrm{d}elta'+1$ and use $|\chi\left\|n|\left\|esssim \left\|angle x\right\|angle^{\varepsilonta}$ (where $\varepsilonta$ is as small as we want) : \begin{align*} \left\|\nabla\partial_t\beta^{(n+1)} \right\|_{L^2_{\mathrm{d}elta'+1}} &\left\|esssim \left\|\partial_t\left\|eft(e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_{ij}\right\|ight) \right\|_{L^2_{1}} \\&\left\|esssim \left\| \partial_t\Tilde{\gamma}^{(n)}H^{(n)}\right\|_{L^2_{1+2\varepsilon^2+\varepsilonta}}+\left\| \partial_t\Tilde{\gamma}^{(n)}\Tilde{N}^{(n)}H^{(n)}\right\|_{L^2_{1+2\varepsilon^2}} + \left\|\partial_t N_a^{(n)}H^{(n)} \right\|_{L^2_{1+2\varepsilon^2+\varepsilonta}} \\&\qquad+ \left\| \partial_t\Tilde{N}^{(n)}H^{(n)} \right\|_{L^2_{1+2\varepsilon^2}} + \left\| \partial_tH^{(n)} \right\|_{L^2_{1+2\varepsilon^2+\varepsilonta}}+ \left\| \Tilde{N}^{(n)}\partial_tH^{(n)} \right\|_{L^2_{1+2\varepsilon^2}} \\& \left\|esssim \varepsilon(1+ C_i), \varepsilonnd{align*} where used \varepsilonqref{HR gamma 1}, \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2}, \varepsilonqref{propsmall H} and \varepsilonqref{HR H 1.5} (with $\varepsilon$ and $\varepsilonta$ small enough, depending on $\left\|ambda$). We now turn to the proof of \varepsilonqref{HR+1 beta 3} and \varepsilonqref{HR+1 beta 4}. Applying $e_0^{(n)}$ to \varepsilonqref{laplacien beta}, we show that the following equation is satisfied : \begin{equation} \mathscr{D}elta(e_0^{(n)}\beta^{(n+1)})^i=2\mathrm{d}elta^{i\varepsilonll}\mathrm{d}elta^{jk}e_0^{(n)}\partial_k\left\|eft( e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_{j\varepsilonll} \right\|ight)+\left\|eft[ \mathscr{D}elta,e_0^{(n)} \right\|ight](\beta^{(n+1)})^i=\vcentcolon I+II.\left\|abel{laplacien e0 beta} \varepsilonnd{equation} It's easy to check the RHS of \varepsilonqref{laplacien e0 beta} has 0 mean, as a consequence we can apply Theorem \right\|ef{mcowens 1}, so that in order to prove the estimate \varepsilonqref{HR+1 beta 3}, it suffices to bound the RHS of \varepsilonqref{laplacien e0 beta} in $L^2_{\mathrm{d}elta'+2}$ by $C_i$ : \begin{itemize} \item For $I$, we first commute $\nabla$ and $e_0^{(n)}$ : \begin{equation} | I| \left\|esssim \left\|eft|\nabla e_0^{(n)}\left\|eft( e^{-2\gamma^{(n)}}N^{(n)}H^{(n)} \right\|ight) \right\|ight|+\left\|eft| \nabla\beta^{(n)}\right\|ight| \left\|eft|\nabla\left\|eft( e^{-2\gamma^{(n)}}N^{(n)}H^{(n)} \right\|ight) \right\|ight|.\left\|abel{commutation estimate''} \varepsilonnd{equation} It implies, using \varepsilonqref{useful gamma 1} : \begin{align} \| I\|_{L^2_{\mathrm{d}elta'+2}} & \left\|esssim \left\|eft\| (e_0^{(n)}\gamma^{(n)})N^{(n)}H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2\varepsilon^2+1}}+\left\|eft\|(e_0^{(n)}N^{(n)})H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2\varepsilon^2+1}}\left\|abel{I L^2}\\&\qquad+\left\|eft\|N^{(n)}(e_0^{(n)}H^{(n)}) \right\|ight\|_{H^1_{\mathrm{d}elta'+2\varepsilon^2+1}}+\left\|eft\|\nabla\beta^{(n)} \right\|ight\|_{L^{\infty}} \left\|eft\|N^{(n)}H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2\varepsilon^2+1}}.\nonumber \varepsilonnd{align} Thanks to \varepsilonqref{HR beta 2} and \varepsilonqref{propsmall H}, the last term is bounded by $\varepsilon A_0 C_i$. Thanks to \varepsilonqref{HR H 2}, the third term is handled as $N^{(n)}H^{(n)}$ in \varepsilonqref{NnHn} and is therefore bounded by $A_0C_i$. Thanks to \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2}, \varepsilonqref{HR beta 1}, \varepsilonqref{HR H 1} and \varepsilonqref{propsmall H}, the second term is bounded by $C_i$. The first term is similar to the second, and actually easier to bound, so we omit the details. We have shown that $\| I\|_{L^2_{\mathrm{d}elta'+2}}\left\|esssim A_0C_i$. \item For $II$, we use the following commutation estimate : \begin{equation} \left\|eft| \left\|eft[ \mathscr{D}elta,e_0^{(n)} \right\|ight](\beta^{(n+1)})^i \right\|ight| \left\|esssim \left\|eft|\nabla\beta^{(n)} \right\|ight|\left\|eft|\nabla^2\beta^{(n+1)} \right\|ight|+\left\|eft|\nabla^2\beta^{(n)} \right\|ight|\left\|eft|\nabla\beta^{(n+1)} \right\|ight|.\left\|abel{commutation estimate'} \varepsilonnd{equation} Now, using in addition \varepsilonqref{HR beta 1}, \varepsilonqref{HR beta 2}, \varepsilonqref{HR+1 beta 1} and \varepsilonqref{HR+1 beta 2} and the product estimate : \begin{align*} \left\|eft\| II\right\|ight\|_{L^2_{\mathrm{d}elta'+2}} &\left\|esssim \left\|eft\|\nabla\beta^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\left\|eft\|\nabla^2\beta^{(n+1)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}}+\left\|eft\| \nabla^2\beta^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+2}}\left\|eft\|\nabla\beta^{(n+1)} \right\|ight\|_{H^1_{\mathrm{d}elta'+1}} \\&\left\|esssim\varepsilon A_0C_i. \varepsilonnd{align*} \varepsilonnd{itemize} Similarly, in order to prove \varepsilonqref{HR+1 beta 4}, we have to bound the RHS of \varepsilonqref{laplacien e0 beta} in $H^1_{\mathrm{d}elta'+2}$ by $CC_i^2$ : \begin{itemize} \item For $I$, we again use \varepsilonqref{commutation estimate''}. Instead of using $L^{\infty}$ bounds for $\nabla\beta^{(n)}$, we use the product estimate and then \varepsilonqref{useful gamma 1}. For the terms where $e_0^{(n)}$ appears, we simply use \varepsilonqref{useful gamma 1} : \begin{align*} \| I\|_{H^1_{\mathrm{d}elta'+2}} & \left\|esssim \left\|eft\| (e_0^{(n)}\gamma^{(n)})N^{(n)}H^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+2\varepsilon^2+1}}+\left\|eft\|(e_0^{(n)}N^{(n)})H^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+2\varepsilon^2+1}}\\&\qquad+\left\|eft\|N^{(n)}(e_0^{(n)}H^{(n)}) \right\|ight\|_{H^2_{\mathrm{d}elta'+2\varepsilon^2+1}}+\left\|eft\|\nabla\beta^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\| N^{(n)}H^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+2\varepsilon^2+1}}.\nonumber \varepsilonnd{align*} The last term is handled thanks to \varepsilonqref{HR beta 2} and \varepsilonqref{NnHn} and is indeed bounded by $A_0C_i^2$. The third term is similar to the last one (because $e_0^{(n)}H^{(n)}$ satisfies \varepsilonqref{HR H 3}) and is therefore handled as in \varepsilonqref{NnHn}, finally it is bounded by $A_3C_i^2$. We handled the two first terms as we did in \varepsilonqref{I L^2}, using \varepsilonqref{HR H 1} instead of \varepsilonqref{propsmall H}, this change explains why we get $C_i^2$ instead of $C_i$. \item For $II$, we use again the commutation estimate \varepsilonqref{commutation estimate'}, \varepsilonqref{HR beta 2} and \varepsilonqref{HR+1 beta 2} and the product estimate : \begin{align*} \left\|eft\| \left\|eft[ \mathscr{D}elta,e_0^{(n)} \right\|ight](\beta^{(n+1)})^i\right\|ight\|_{L^2_{\mathrm{d}elta'+2}} &\left\|esssim \left\|eft\|\nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\|\nabla^2\beta^{(n+1)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}}+\left\|eft\| \nabla^2\beta^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+2}}\left\|eft\|\nabla\beta^{(n+1)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \\&\left\|esssim A_0^2C_i^2. \varepsilonnd{align*} \varepsilonnd{itemize} \varepsilonnd{proof} We have finished all the elliptic estimates, in the sequel we deal with evolution equations, and we will use the freedom of taking $T$ as small as we want, in order to recover our estimates. \subsubsection{The transport equation and $\tau^{(n+1)}$} We begin this section by prooving the estimates on $H^{(n+1)}$. We first prove a technical lemma about the transport equation : \begin{lem}\left\|abel{transport inegalite} Let $\sigma\in\mathbb{R}$. If $f$ and $h$ satisfy \begin{equation*} e_0^{(n+1)}f=h, \varepsilonnd{equation*} then, \begin{equation*} \sup_{t\in[0,T]}\| f\|_{L^2_{\sigma}}(t)\left\|eq 2 \| f\|_{L^2_{\sigma}}(0)+2\sqrt{T}\sup_{t\in[0,T]}\| h\|_{L^2_{\sigma}}(t) . \varepsilonnd{equation*} \varepsilonnd{lem} \begin{proof}Let $w(x)=\left\|angle x\right\|angle^{2\sigma}$. We multiply the equation $e_0^{(n+1)}f=h$ by $w f$ and integrate over $\mathbb{R}^2$. Writing $e_0^{(n+1)}=\partial_t-\beta^{(n+1)}\cdot\nabla$, we get : \begin{equation*} \frac{\mathrm{d}}{\mathrm{d} t}\left\|eft(\|f\|_{L^2_{\sigma}}^2\right\|ight)=2\int_{\mathbb{R}^2}w fh\,\mathrm{d} x+\int_{\mathbb{R}^2}w \beta^{(n+1)}\cdot\nabla \left\|eft(f^2\right\|ight)\,\mathrm{d} x. \varepsilonnd{equation*} We integrate by part the last term in order to get : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}\left\|eft(\|f\|_{L^2_{\sigma}}^2\right\|ight)&=2\int_{\mathbb{R}^2}w fh\,\mathrm{d} x-\int_{\mathbb{R}^2} f^2\mathrm{div}\left\|eft(w\beta^{(n+1)}\right\|ight)\mathrm{d} x . \varepsilonnd{align*} For the last term, we use \varepsilonqref{HR+1 beta 2} (and the embedding $H^2_{\mathrm{d}elta'+1}\xhookrightarrow{}C^0_1$) and $\left\|eft| \nabla w\right\|ight|\left\|esssim\frac{w}{\left\|angle x \right\|angle}$ to obtain : \begin{align*} -\int_{\mathbb{R}^2} f^2\mathrm{div}\left\|eft(w\beta^{(n+1)}\right\|ight)\mathrm{d} x \left\|esssim C_i \| f\|_{L^2_{\sigma}} \varepsilonnd{align*} For the first term, we simply use the Cauchy-Scwharz inequality and $2ab\left\|eq a^2+b^2$ to obtain : \begin{equation*} 2\int_{\mathbb{R}^2}w f h\,\mathrm{d} x\left\|eq 2\left\|eft(\int_{\mathbb{R}^2}w f^2 \mathrm{d} x\right\|ight)^{\frac{1}{2}}\left\|eft(\int_{\mathbb{R}^2}w h^2 \mathrm{d} x\right\|ight)^{\frac{1}{2}}\left\|eq \| f\|_{L^2_{\sigma}}^2+ \| h\|_{L^2_{\sigma}}^2. \varepsilonnd{equation*} Summarising, we get : \begin{equation*} \frac{\mathrm{d}}{\mathrm{d} t}\left\|eft(\|f\|_{L^2_{\sigma}}^2\right\|ight)\left\|eq C(C_i) \| f\|_{L^2_{\sigma}}^2+ \| h\|_{L^2_{\sigma}}^2. \varepsilonnd{equation*} We apply Gronwall's Lemma, take $T$ small enough and use $\sqrt{a^2+b^2}\left\|eq a+b$ to get : \begin{equation*} \sup_{t\in[0,T]}\| f\|_{L^2_{\sigma}}(t)\left\|eq 2 \| f\|_{L^2_{\sigma}}(0)+2\sqrt{T}\sup_{t\in[0,T]}\| h\|_{L^2_{\sigma}}(t) . \varepsilonnd{equation*} \varepsilonnd{proof} \begin{prop}\left\|abel{HR+1 H prop} For $n\geq 2$, the following estimates hold : \begin{align} \left\|eft\Vert H^{(n+1)}\right\|ight\Vert_{H^2_{\mathrm{d}elta+1}}&\left\|eq 2C_i\left\|abel{HR+1 H 1},\\ \left\|eft\Vert e_0^{(n+1)}H^{(n+1)}\right\|ight\Vert_{L^2_{1+\left\|ambda}}&\left\|esssim \varepsilon^2,\left\|abel{HR+1 H 1.5}\\ \left\|eft\Vert e_0^{(n+1)}H^{(n+1)}\right\|ight\Vert_{H^1_{\mathrm{d}elta+1}}&\left\|esssim C_i,\left\|abel{HR+1 H 2}\\ \left\|eft\Vert e_0^{(n+1)}H^{(n+1)}\right\|ight\Vert_{H^2_{\mathrm{d}elta+1}}&\left\|esssim A_2C_i^2.\left\|abel{HR+1 H 3} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} To prove \varepsilonqref{HR+1 H 1.5} we just bound the RHS of \varepsilonqref{reduced system H} in $L^2_{\mathrm{d}elta+1}$ using the weighted product estimates $H^1\times H^1\xhookrightarrow{} L^2$ and using weighted $L^{\infty}$ estimates for $N^{(n)}$ in the first and last terms. More concretely, we use \varepsilonqref{HR N 1}, \varepsilonqref{propsmall H}, \varepsilonqref{HR beta 1}, \varepsilonqref{propsmall gamma}, \varepsilonqref{propsmall fi}, \varepsilonqref{useful gamma 1} and \varepsilonqref{useful ffi 1}, and we recall that $\left\|ambda<\mathrm{d}elta+1$ : \begin{align*} \left\|eft\Vert e_0^{(n+1)}H^{(n+1)}\right\|ight\Vert_{L^2_{1+\left\|ambda}} &\left\|esssim \left\| N^{(n)} (H^{(n)})^2 \right\|_{L^2_{1+\left\|ambda}} + \left\|\nabla\beta^{(n)} H^{(n)} \right\|_{L^2_{1+\left\|ambda}}+\left\| \nabla^2 N^{(n)}\right\|_{L^2_{1+\left\|ambda}} \\&\qquad+ \left\| \nabla\gamma^{(n)}\nabla N^{(n)}\right\|_{L^2_{1+\left\|ambda}} + \left\| N^{(n)} (\nabla \varphi^{(n)})^2 \right\|_{L^2}+ \left\| N^{(n)} (\nabla \Omegamega^{(n)})^2 \right\|_{L^2}\\&\left\|esssim \varepsilon^2. \varepsilonnd{align*} We continue by the proof of \varepsilonqref{HR+1 H 2} and \varepsilonqref{HR+1 H 3}, which amounts to bounding the $H^1_{\mathrm{d}elta+1}$ and $H^2_{\mathrm{d}elta+1}$ norms of the RHS of \varepsilonqref{reduced system H}. First notice that the terms $e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_i^{\;\,\varepsilonll}(H^{(n)})_{j\varepsilonll}$, $(\partial_i\varphi^{(n)}\Bar{\Omegatimes}\partial_j\varphi^{(n)})N^{(n)}$ and $(\partial_i\Omegamega^{(n)}\Bar{\Omegatimes}\partial_j\Omegamega^{(n)})N^{(n)}$ are analogous to terms in \varepsilonqref{reduced system N} (because $\nabla\varphi^{(n)}$ and $\partial_t\varphi^{(n)}$ satisfy the same estimates, samewise for $\Omegamega^{(n)}$), and can be treated as in Proposition \right\|ef{hr+1 N prop}. We recall the estimates obtained : \begin{align*} \left\| e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_i^{\;\,\varepsilonll}(H^{(n)})_{j\varepsilonll} \right\|_{H^1_{\mathrm{d}elta+1}} & \left\|esssim \varepsilon^2 C_i,\\ \left\| e^{-2\gamma^{(n)}}N^{(n)}(H^{(n)})_i^{\;\,\varepsilonll}(H^{(n)})_{j\varepsilonll}\right\|_{H^2_{\mathrm{d}elta+1}} & \left\|esssim \varepsilon^2 C(A_2) C_i^2,\\ \left\|(\partial_i\varphi^{(n)}\Bar{\Omegatimes}\partial_j\varphi^{(n)})N^{(n)} \right\|_{H^1_{\mathrm{d}elta+1}}+\left\| e^{-4\varphi}(\partial_i\Omegamega^{(n)}\Bar{\Omegatimes}\partial_j\Omegamega^{(n)})N^{(n)} \right\|_{H^1_{\mathrm{d}elta+1}} & \left\|esssim \varepsilon C(A_0)C_i,\\ \left\|(\partial_i\varphi^{(n)}\Bar{\Omegatimes}\partial_j\varphi^{(n)})N^{(n)} \right\|_{H^2_{\mathrm{d}elta+1}}+\left\| e^{-4\varphi}(\partial_i\Omegamega^{(n)}\Bar{\Omegatimes}\partial_j\Omegamega^{(n)})N^{(n)} \right\|_{H^2_{\mathrm{d}elta+1}} & \left\|esssim \varepsilon^2 C(A_2) C_i^2+C(A_0)C_i^2. \varepsilonnd{align*} The remaining terms are treated as follows : \begin{itemize} \item We first use \varepsilonqref{HR H 1} and \varepsilonqref{HR beta 1} and the product estimate : \begin{align*} \left\|eft\|\partial_{(j}(\beta^{(n)})^k(H^{(n)})_{i)k} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}& \left\|esssim \left\|eft\|H^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}\left\|eft\|\beta^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|esssim \varepsilon C_i. \varepsilonnd{align*} Then we use \varepsilonqref{propsmall H}, \varepsilonqref{HR H 1} and \varepsilonqref{HR beta 2} and the product estimate : \begin{align*} \left\|eft\|\partial_{(j}(\beta^{(n)})^k(H^{(n)})_{i)k} \right\|ight\|_{H^2_{\mathrm{d}elta+1}}&\left\|esssim \left\|eft\| \nabla\beta^{(n)}H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}+\left\|eft\| \nabla^3\beta^{(n)}H^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta+3}}\\&\qquad+\left\|eft\| \nabla^2\beta^{(n)}\nabla H^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta+3}}+\left\|eft\| \nabla\beta^{(n)}\nabla^2H^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta+3}} \\&\left\|esssim\left\|eft\|\nabla\beta^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft\|H^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta+1}} + \left\|\nabla^3\beta^{(n)} \right\|_{L^2_{\mathrm{d}elta'+3}}\left\| H^{(n)} \right\|_{H^2_{\mathrm{d}elta+1}} \\&\qquad+\left\|\nabla^2\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'+2}} \left\|\nabla H^{(n)} \right\|_{H^1_{\mathrm{d}elta+2}} +\left\|eft\| \nabla\beta^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \left\|eft\|\nabla^2H^{(n)}\right\|ight\|_{L^2_{\mathrm{d}elta+3}}\\ &\left\|esssim \varepsilon^2 A_0 C_i+A_0C_i^2. \varepsilonnd{align*} \item We use \varepsilonqref{HR N 2} and the fact that $\left\|angle x\right\|angle^{\alpha}\in L^2$ if and only if $\alpha<-1$ : \begin{equation*} \left\|eft\| (\partial_i\Bar{\Omegatimes}\partial_j)N^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|eq \left\|eft\|\nabla^2\Tilde{N}^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta+1}}+\left\|eft| N_a^{(n)} \right\|ight|\left\|eft\|\nabla^2(\chi\left\|n)\right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|esssim C_i. \varepsilonnd{equation*} We then use \varepsilonqref{HR N 3} for the $H^2$ estimate : \begin{equation*} \left\|eft\| (\partial_i\Bar{\Omegatimes}\partial_j)N^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta+1}} \left\|eq \left\|eft\|\nabla^2\Tilde{N}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}+\left\|eft| N_a^{(n)} \right\|ight|\left\|eft\|\nabla^2(\chi\left\|n)\right\|ight\|_{H^2_{\mathrm{d}elta+1}}\left\|esssim A_2 C_i^2. \varepsilonnd{equation*} \item For the following term, we get both $H^1$ and $H^2$ estimates by using \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2} and \varepsilonqref{HR gamma 1} and the product estimate : \begin{align*} \left\|eft\| (\mathrm{d}elta_i^k\Bar{\Omegatimes}\partial_j\gamma^{(n)})\partial_kN^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}&\left\|eq \left\|eft\|\nabla\Tilde{\gamma}^{(n)}\nabla\Tilde{N}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}+|\alpha|\left\|eft\|\nabla(\chi\left\|n)\nabla\Tilde{N}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}\\&\quad+\left\|eft|N_a^{(n)}\right\|ight|\left\|eft\|\nabla(\chi\left\|n)\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}+|\alpha|\left\|eft|N_a^{(n)}\right\|ight|\left\|eft\|(\nabla(\chi\left\|n))^2\right\|ight\|_{H^2_{\mathrm{d}elta+1}} \\&\left\|esssim \left\|eft\|\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}} \left\|eft\|\nabla\Tilde{N}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}} +\varepsilon\left\|eft( \left\|eft\|\nabla\Tilde{N}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta}}+\left\|eft\|\nabla\Tilde{\gamma}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta}} \right\|ight)\\&\qquad+ \varepsilon^2\left\|eft\|\left\|angle x\right\|angle^{\mathrm{d}elta-1}\right\|ight\|_{L^2} \\&\left\|esssim\varepsilon C_i^2. \varepsilonnd{align*} \varepsilonnd{itemize} We now prove \varepsilonqref{HR+1 H 1}. We recall the following commutation formula : \begin{align*} \left\|eft|\left\|eft[ e_0^{(n+1)},\nabla \right\|ight]H^{(n+1)}\right\|ight|&\left\|esssim \left\|eft|\nabla\beta^{(n+1)} \right\|ight|\left\|eft|\nabla H^{(n+1)} \right\|ight|,\\ \left\|eft|\left\|eft[ e_0^{(n+1)},\nabla^2 \right\|ight]H^{(n+1)}\right\|ight|&\left\|esssim \left\|eft|\nabla\beta^{(n+1)} \right\|ight|\left\|eft|\nabla^2 H^{(n+1)} \right\|ight|+\left\|eft|\nabla^2\beta^{(n+1)} \right\|ight|\left\|eft|\nabla H^{(n+1)} \right\|ight|. \varepsilonnd{align*} Hence, using \varepsilonqref{HR+1 beta 2} : \begin{equation} \left\|eft\|e_0^{(n+1)}\nabla^{\alpha}H^{(n+1)}_{ij} \right\|ight\|_{L^2_{\mathrm{d}elta+1+|\alpha|}}\left\|esssim \left\|eft\|e_0^{(n+1)}H^{(n+1)}_{ij} \right\|ight\|_{H^2_{\mathrm{d}elta+1}}+C_i\left\|eft\|H^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}\left\|abel{transport H} \varepsilonnd{equation} where $|\alpha|\left\|eq 2$. We apply the Lemma \right\|ef{transport inegalite} with $\sigma=\mathrm{d}elta+1+|\alpha|$ and $f=\nabla^{\alpha} H^{(n+1)}$ : \begin{align*} \sup_{t\in[0,T]}\left\|eft\| \nabla^{\alpha} H^{(n+1)} \right\|ight\|_{L^2_{\mathrm{d}elta+1+|\alpha|}}(t) &\left\|eq 2\left\|eft\| \nabla^{\alpha} H^{(n+1)} \right\|ight\|_{L^2_{\mathrm{d}elta+1+|\alpha|}}(0)+2\sqrt{T}\sup_{t\in[0,T]}\left\|eft\|e_0^{(n+1)}\nabla^{\alpha}H^{(n+1)}_{ij} \right\|ight\|_{L^2_{\mathrm{d}elta+1+|\alpha|}} \\& \left\|esssim 2\left\|eft\| \nabla^{\alpha} H^{(n+1)} \right\|ight\|_{L^2_{\mathrm{d}elta+1+|\alpha|}}(0) +2\sqrt{T}C_i^2+2C_i\sqrt{T}\left\|eft\|H^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}, \varepsilonnd{align*} where in the last inequality we use \varepsilonqref{transport H} and \varepsilonqref{HR+1 H 3}. We sum over all $|\alpha|\left\|eq 2$ and absorb the term $\left\|eft\|H^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta+1}}$ of the RHS into the LHS (choosing $T$ small enough). Recalling that $\left\|eft\| H^{(n+1)} \right\|ight\|_{H^2_{\mathrm{d}elta+1}}(0)\left\|eq C_i$ ends the proof of \varepsilonqref{HR+1 H 1}. \varepsilonnd{proof} Next, we prove the estimates for $\tau^{(n+1)}$, gathered in the following proposition : \begin{prop}\left\|abel{hr+1 tau prop} For $n\geq 2$, the following estimates hold : \begin{align} \left\|eft\| \tau^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}&\left\|esssim A_0C_i\left\|abel{HR+1 tau 1},\\ \left\|eft\| \partial_t\tau^{(n+1)}\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|esssim A_1C_i\left\|abel{HR+1 tau 2},\\ \left\|eft\| \partial_t\tau^{(n+1)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}&\left\|esssim A_2 C_i\left\|abel{HR+1 tau 3}. \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} In view of $\varepsilonqref{reduced system tau}$, the estimates for $\tau^{(n+1)}$ can be obtained by directly controlling \begin{equation*} -2\mathbf{T}^{(n-1)}\gamma^{(n)}+\frac{\mathrm{div}\left\|eft(\beta^{(n)}\right\|ight)}{N^{(n-1)}} . \varepsilonnd{equation*} We bound the two terms separately, using first \varepsilonqref{HR beta 2}, \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2} and $\left\|eft|\frac{1}{N^{(n-1)}} \right\|ight|\left\|esssim 1$ : \begin{align*} \left\|eft\| \frac{\mathrm{div}\left\|eft(\beta^{(n)}\right\|ight)}{N^{(n-1)}} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} & \left\|esssim \left\|eft\| \beta^{(n)}\right\|ight\|_{H^3_{\mathrm{d}elta'}} + \left\| \nabla\Tilde{N}^{(n-1)}\nabla \beta^{(n)} \right\|_{L^2_{\mathrm{d}elta'+2}} \\&\qquad+ \left\|\nabla^2\Tilde{N}^{(n-1)}\nabla \beta^{(n)} \right\|_{L^2_{\mathrm{d}elta'+3}}+\left\|\nabla\Tilde{N}^{(n-1)}\nabla^2 \beta^{(n)} \right\|_{L^2_{\mathrm{d}elta'+3}}\\ &\left\|esssim \left\|eft\| \beta^{(n)}\right\|ight\|_{H^3_{\mathrm{d}elta'}}+ \left\| \nabla\Tilde{N}^{(n-1)}\right\|_{H^1_{\mathrm{d}elta+1}}\left\|\nabla\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'+1}} \\& \qquad +\left\|\nabla^2\Tilde{N}^{(n-1)}\right\|_{H^1_{\mathrm{d}elta+2}} \left\|\nabla\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'+1}}+\left\| \nabla\Tilde{N}^{(n-1)}\right\|_{H^1_{\mathrm{d}elta+1}}\left\| \nabla^2\beta^{(n)}\right\|_{H^1_{\mathrm{d}elta'+2}} \\& \left\|esssim (1+\varepsilon)A_0C_i. \varepsilonnd{align*} Using in addition Proposition \right\|ef{commutation estimate} we get : \begin{align*} \left\|eft\|\mathbf{T}^{(n-1)}\gamma^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \left\|eq \left\|eft\| \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} +|\alpha|\left\|eft\|\frac{\beta^{(n-1)}\cdot\nabla(\chi\left\|n)}{N^{(n-1)}} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|esssim (1+\varepsilon)C_i, \varepsilonnd{align*} which concludes the proof of \varepsilonqref{HR+1 tau 1}. \par\left\|eavevmode\par We now turn to the estimates concerning $\partial_t\tau^{(n+1)}$, which has the following expression : \begin{equation*} \partial_t\tau^{(n+1)}=-2\partial_t\left\|eft( \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight)+2\alpha\nabla(\chi\left\|n)\cdot\partial_t\left\|eft( \frac{\beta^{(n-1)}}{N^{(n-1)}} \right\|ight)+\partial_t\left\|eft( \frac{\mathrm{div}(\beta^{(n)})}{N^{(n-1)}} \right\|ight) \varepsilonnd{equation*} By \varepsilonqref{HR gamma 2}, $\left\|eft\|\partial_t( \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)})\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}\left\|eq A_0 C_i$. Then, we note, that thanks to \varepsilonqref{HR beta 1} and \varepsilonqref{HR beta 3}, we have $\|\partial_t\beta^{(n)}\|_{H^1_{\mathrm{d}elta'+1}}\left\|esssim A_1C_i$ (and the same with $n$ replaced by $n-1$). For the second term, we do the following : \begin{align} \left\|eft\| 2\alpha\nabla(\chi\left\|n)\cdot\partial_t\left\|eft( \frac{\beta^{(n-1)}}{N^{(n-1)}} \right\|ight)\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|esssim\;\varepsilon\left\|eft( \left\|eft\|\partial_t\beta^{(n-1)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}+\left\|eft| \partial_tN_a^{(n-1)} \right\|ight|\left\|eft\|\beta^{(n-1)}\right\|ight\|_{L^2_{\mathrm{d}elta'}}+\left\|eft\|\partial_t\Tilde{N}^{(n-1)}\right\|ight\|_{L^2_{\mathrm{d}elta}}\left\|eft\|\beta^{(n-1)}\right\|ight\|_{L^{\infty}}\right\|ight)\nonumber\\&\left\|esssim \varepsilon A_1C_i,\left\|abel{lkj} \varepsilonnd{align} where we used \varepsilonqref{HR N 2} and \varepsilonqref{HR beta 1}. The third term is very similar : \begin{align*} \left\|eft\| \partial_t\left\|eft( \frac{\mathrm{div}(\beta^{(n)})}{N^{(n-1)}} \right\|ight)\right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|esssim \left\|eft\|\partial_t\beta^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'}}+ \left\|eft\|\nabla\beta^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\left\|eft( \left\|eft\|\partial_t\Tilde{N}^{(n-1)}\right\|ight\|_{H^1_{\mathrm{d}elta}} +\left\|eft| \partial_tN_a^{(n-1)} \right\|ight|\right\|ight)\\& \left\|esssim A_1C_i. \varepsilonnd{align*} This finishes the proof of \varepsilonqref{HR+1 tau 2}. \par\left\|eavevmode\par We now turn to the proof of \varepsilonqref{HR+1 tau 3}. In view of \varepsilonqref{HR+1 tau 2}, we just have to bound $\|\nabla \partial_t\tau^{(n+1)}\|_{L^2_{\mathrm{d}elta'+2}}$ by $C_i^2$. We have the following expression : \begin{align*} \nabla \partial_t\tau^{(n+1)}&=-2\nabla\partial_t\left\|eft( \mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight)+2\alpha\nabla^2(\chi\left\|n)\cdot \partial_t\left\|eft( \frac{\beta^{(n-1)}}{N^{(n-1)}} \right\|ight)+2\alpha\nabla(\chi\left\|n)\partial_t\left\|eft( \frac{\nabla\beta^{(n-1)}}{N^{(n-1)}} \right\|ight)\\ &\quad -2\alpha\nabla(\chi\left\|n)\partial_t\left\|eft( \frac{\beta^{(n-1)} \nabla N^{(n-1)}}{\left\|eft(N^{(n-1)}\right\|ight)^2} \right\|ight)+ \partial_t\left\|eft( \frac{\mathrm{div}(\nabla\beta^{(n)})}{N^{(n-1)}} \right\|ight)-\partial_t\left\|eft( \frac{\nabla N^{(n-1)}\mathrm{div}(\beta^{(n)})}{\left\|eft(N^{(n-1)}\right\|ight)^2} \right\|ight) \\&=\vcentcolon I + II + III+IV+V+VI. \varepsilonnd{align*} The term $I$ is easily handle thanks to \varepsilonqref{HR gamma 3} : we have $\left\|eft\|I\right\|ight\|_{L^2_{\mathrm{d}elta'+2}}\left\|eq A_2 C_i$. For the other terms, we make the following remarks : \begin{itemize} \item the term $VI$ is worse than the term $IV$, \item the term $V$ is worse than the terms $II$ and $III$. \varepsilonnd{itemize} Thus, it only remains to bound the terms $V$ and $VI$, for which we use \varepsilonqref{HR N 2}, \varepsilonqref{HR beta 1} and \varepsilonqref{HR beta 3} : \begin{align*} \left\| V\right\|_{L^2_{\mathrm{d}elta'+2}} &\left\|esssim \left\|\partial_t\beta^{(n)} \right\|_{H^2_{\mathrm{d}elta'}} +\left\|\nabla^2\beta^{(n)} \right\|_{L^2_{\mathrm{d}elta'+2}}\left\|\partial_t\Tilde{N}^{(n-1)} \right\|_{H^2_{\mathrm{d}elta}}\left\|esssim A_1C_i+\varepsilon C_i\left\|esssim A_1 C_i.\\ \left\| VI \right\|_{L^2_{\mathrm{d}elta'+2}} & \left\|esssim \left\|\nabla\Tilde{N}^{(n-1)} \right\|_{H^1_{\mathrm{d}elta+1}}\left\|\partial_t\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'}}+\left\|\nabla\partial_t\Tilde{N}^{(n-1)} \right\|_{H^1_{\mathrm{d}elta+1}}\left\|\nabla\beta^{(n-1)} \right\|_{H^1_{\mathrm{d}elta'+1}}\\&\qquad +\left\|\partial_t\Tilde{N}^{(n-1)} \right\|_{H^2_{\mathrm{d}elta}}\left\|\nabla\Tilde{N}^{(n-1)} \right\|_{H^1_{\mathrm{d}elta+1}}\left\|\nabla\beta^{(n-1)} \right\|_{H^1_{\mathrm{d}elta'+1}} \\&\left\|esssim \varepsilon C_i. \varepsilonnd{align*} This concludes the proof of \varepsilonqref{HR+1 tau 3}. \par\left\|eavevmode\par \varepsilonnd{proof} \subsubsection{Energy estimate for $\Box_{g^{(n)}}$} In this section, we establish the usual energy estimate for the operator $\Box_{g^{(n)}}$. \begin{lem}\left\|abel{inegalite d'energie lemme 1} Let $\sigma\in\mathbb{R}$. If $h$ is a solution of \begin{equation}\left\|abel{partie principale de box} \left\|eft(\mathbf{T}^{(n)} \right\|ight)^2h-e^{-2\gamma^{(n)}}\mathscr{D}elta h=f, \varepsilonnd{equation} then, if $T$ is sufficiently small, we have for all $t\in[0,T]$ \begin{align} \left\|eft\Vert\mathbf{T}^{(n)}h\right\|ight\Vert_{L^2_{\sigma}}(t)&+\left\|eft\Vert e^{-\gamma^{(n)}}\nabla h\right\|ight\Vert_{L^2_{\sigma}}(t)\nonumber\\& \left\|eq 2\left\|eft( \left\|eft\Vert\mathbf{T}^{(n)}h\right\|ight\Vert_{L^2_{\sigma}}(0)+\left\|eft\Vert e^{-\gamma^{(n)}}\nabla h\right\|ight\Vert_{L^2_{\sigma}}(0)+\sqrt{2T}\sup_{s\in[0,T]}\left\|eft\| fN^{(n)}\right\|ight\|_{L^2_{\sigma}}\right\|ight).\left\|abel{inégalité d'énergie} \varepsilonnd{align} \varepsilonnd{lem} \begin{proof} Let $w(x)=\left\|angle x \right\|angle^{2\sigma}$. We multiply the equation by $w e_0^{(n)}h$ and we integrate over $\mathbb{R}^2$ with respect to $\mathrm{d} x$. After integration by parts we obtain : \begin{equation} \int_{\mathbb{R}^2}\frac{w}{2}e_0^{(n)}\left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2\mathrm{d} x+\int_{\mathbb{R}^2}\nabla h\cdot\nabla\left\|eft(e^{-2\gamma^{(n)}}w e_0^{(n)}h\right\|ight)\mathrm{d} x=\int_{\mathbb{R}^2}w f e_0^{(n)}h\,\mathrm{d} x.\left\|abel{IPP} \varepsilonnd{equation} We define the energy $E(t)\vcentcolon=\int_{\mathbb{R}^2}w\left\|eft( \left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2+e^{-2\gamma^{(n)}}|\nabla h|^2\right\|ight)(t,x)\mathrm{d} x$ and compute its time derivative, writing $\partial_t=e_0^{(n)}+\beta^{(n)}\cdot\nabla$ and integrating by parts the terms coming from $\beta^{(n)}\cdot\nabla$ : \begin{align*} \frac{\mathrm{d} E}{\mathrm{d} t}(t) = \int_{\mathbb{R}^2} we_0^{(n)}\left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2\mathrm{d} x + \int_{\mathbb{R}^2} w e_0^{(n)}& \left\|eft(e^{-2\gamma^{(n)}} |\nabla h|^2 \right\|ight) \mathrm{d} x \\&-\int_{\mathbb{R}^2}\mathrm{div} (w\beta^{(n)})\left\|eft( \left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2+e^{-2\gamma^{(n)}}|\nabla h|^2\right\|ight)\mathrm{d} x \varepsilonnd{align*} We now use \varepsilonqref{IPP} to express the first integral in $\frac{\mathrm{d} E}{\mathrm{d} t}$ : \begin{align*} \frac{\mathrm{d} E}{\mathrm{d} t}(t)& =2\int_{\mathbb{R}^2}w f e_0^{(n)}h\,\mathrm{d} x - 2\int_{\mathbb{R}^2}\nabla h\cdot\nabla\left\|eft(e^{-2\gamma^{(n)}}w e_0^{(n)}h\right\|ight)\mathrm{d} x \\& \quad + \int_{\mathbb{R}^2} w e_0^{(n)} \left\|eft(e^{-2\gamma^{(n)}} |\nabla h|^2 \right\|ight) \mathrm{d} x -\int_{\mathbb{R}^2}\mathrm{div} (w\beta^{(n)})\left\|eft( \left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2+e^{-2\gamma^{(n)}}|\nabla h|^2\right\|ight)\mathrm{d} x \varepsilonnd{align*} We now expand the second integral and commute $\nabla$ and $e_0^{(n)}$ : \begin{align*} - 2\int_{\mathbb{R}^2}\nabla h\cdot\nabla\left\|eft(e^{-2\gamma^{(n)}}w e_0^{(n)}h\right\|ight)\mathrm{d} x & =- \int_{\mathbb{R}^2}w e^{-2\gamma^{(n)}} e_0^{(n)}\left\|eft( |\nabla h|^2 \right\|ight) \mathrm{d} x +2 \int_{\mathbb{R}^2} w e^{-2\gamma^{(n)}} \partial_i h \nabla h \cdot \nabla\beta^{(n)i} \mathrm{d} x \\& \quad -2 \int_{\mathbb{R}^2} e^{-2\gamma^{(n)}}e_0^{(n)}h \nabla h\cdot\nabla w \mathrm{d} x -2 \int_{\mathbb{R}^2} w e_0^{(n)} h\nabla h\cdot\nabla\left\|eft( e^{-2\gamma^{(n)}}\right\|ight) \mathrm{d} x \varepsilonnd{align*} With this, we see that the $\partial h \partial^2h$ terms in $\frac{\mathrm{d} E}{\mathrm{d} t}$ cancel each other. Thus, we obtain the following energy equality : \begin{align} \frac{\mathrm{d} E}{\mathrm{d} t}(t)& = -\int_{\mathbb{R}^2}\mathrm{div} (w\beta^{(n)})\left\|eft( \left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2+e^{-2\gamma^{(n)}}|\nabla h|^2\right\|ight)\mathrm{d} x \nonumber +2 \int_{\mathbb{R}^2} w e^{-2\gamma^{(n)}} \partial_i h \nabla h \cdot \nabla\beta^{(n)i} \mathrm{d} x\\& \quad -2 \int_{\mathbb{R}^2} e^{-2\gamma^{(n)}}e_0^{(n)}h \nabla h\cdot\nabla w \mathrm{d} x + 2\int_{\mathbb{R}^2}w f e_0^{(n)}h\,\mathrm{d} x + R_{\gamma^{(n)}}(t) \left\|abel{energy equality} \varepsilonnd{align} with \begin{equation*} R_{\gamma^{(n)}}(t)\vcentcolon=-2\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}}w|\nabla h|^2\partial_t\gamma^{(n)}\mathrm{d} x+ 4\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}}w e_0^{(n)}h\nabla h\cdot\nabla\gamma^{(n)}\mathrm{d} x-2\int_{\mathbb{R}^2} e^{-2\gamma^{(n)}}w|\nabla h|^2\beta^{(n)}\cdot \nabla\gamma^{(n)}\mathrm{d} x. \varepsilonnd{equation*} This sort of remainder contains all the term involving derivatives of $\gamma^{(n)}$, therefore it would vanish if $e^{-2\gamma^{(n)}}$ didn't appear in \varepsilonqref{partie principale de box}. We then show that the first three integrals in \varepsilonqref{energy equality} can be bounded by $E(t)$ : \begin{itemize} \item Let's show that $|\mathrm{div}(w\beta^{(n)})|\left\|eq C(C_i)w$. We have $\mathrm{div}(w\beta^{(n)})=w\mathrm{div}(\beta^{(n)})+\nabla w\cdot\beta^{(n)}$. We have $|\nabla w|\left\|esssim\frac{ w}{\left\|angle x\right\|angle}$ and $\beta^{(n)}$ bounded so $\nabla w\cdot\beta^{(n)}$ is indeed bounded by $w$. For $w\mathrm{div}(\beta^{(n)})$, we use the embedding $H^2_{\mathrm{d}elta'+1}\xhookrightarrow{}L^{\infty}$ and the estimate \varepsilonqref{HR beta 2}. This shows that \begin{equation*} -\int_{\mathbb{R}^2}\mathrm{div} (w\beta^{(n)})\left\|eft( \left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2+e^{-2\gamma^{(n)}}\nabla h\right\|ight)\mathrm{d} x\left\|esssim A_0C_iE(t). \varepsilonnd{equation*} \item Let's show that $|e^{-\gamma^{(n)}}N^{(n)}\nabla w|\left\|esssim w$. We have $|e^{-\gamma^{(n)}}|\left\|esssim \left\|angle x\right\|angle^{\varepsilon^2}$ and $|N^{(n)}|\left\|esssim\left\|angle x\right\|angle^{\frac{1}{2}}$ so $|e^{-\gamma^{(n)}}N^{(n)}\nabla w|\left\|esssim w \left\|angle x\right\|angle^{\varepsilon^2-\frac{1}{2}}\left\|esssim w$, providing $\varepsilon$ is small. This allows us to do the following (using $2ab\left\|eq a^2+b^2$) : \begin{equation*} -2\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}}e_0^{(n)}h\nabla h\cdot\nabla w\,\mathrm{d} x \left\|esssim \int_{\mathbb{R}^2} w\left\|eft|\frac{e_0^{(n)}h}{N^{(n)}}\right\|ight|e^{-\gamma^{(n)}}|\nabla h|\,\mathrm{d} x\left\|esssim E(t). \varepsilonnd{equation*} \item We already used the fact that $\nabla\beta^{(n)}$ is bounded by $A_0C_i$ so we simply do \begin{equation*} 2\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}}w\partial_ih\nabla h\cdot\nabla\beta^{(n)i}\,\mathrm{d} x\left\|esssim A_0C_i\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}}w|\nabla h|^2\,\mathrm{d} x \left\|esssim A_0C_iE(t). \varepsilonnd{equation*} \varepsilonnd{itemize} We now show that $R_{\gamma^{(n)}}(t)$ can also be bounded by $E(t)$ : \begin{itemize} \item Let's show that $\partial_t\gamma^{(n)}$ is bounded. Since $\alpha$ doesn't depend on time, we have $\partial_t\gamma^{(n)}=N^{(n-1)}\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)}+\beta^{(n)}\cdot\nabla\Tilde{\gamma}^{(n)}$. For the first term we use the Proposition \right\|ef{commutation estimate} and the embedding $H^2_{\mathrm{d}elta'+1}\xhookrightarrow{}C^0_1$ (together with the fact that $|N^{(n-1)}|\left\|esssim\left\|angle x\right\|angle$). For the second term we simply use the embedding $H^2_{\mathrm{d}elta'}\xhookrightarrow{}L^{\infty}$. We thus get \begin{equation*} 2\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}} w|\nabla h|^2\partial_t\gamma^{(n)}\mathrm{d} x \left\|eq C(C_i) E(t). \varepsilonnd{equation*} \item Let's show that $e^{-\gamma^{(n)}}N^{(n)}\nabla\gamma^{(n)}$ is bounded. We only deal with the $\chi\left\|n$ part of $N^{(n)}$ (since $\Tilde{N}^{(n)}$ is bounded), and only with the $\nabla\Tilde\gamma^{(n)}$ part in $\nabla\gamma^{(n)}$ (because $\nabla(\chi\left\|n)$ decrease more than $e^{-\gamma^{(n)}}$). Using $|\chi\left\|n|\left\|esssim \left\|angle x\right\|angle^{\varepsilon^2}$ and $e^{-\gamma^{(n)}}\left\|esssim \left\|angle x\right\|angle^{\varepsilon^2}$, we write \begin{equation*} \left\|eft|e^{-\gamma^{(n)}}\chi\left\|n\nabla\Tilde{\gamma}^{(n)}\right\|ight|\left\|esssim \|\nabla\Tilde{\gamma}^{(n)}\|_{C^0_{2\varepsilon^2}}. \varepsilonnd{equation*} If $\varepsilon$ is small enough we have the embedding $H^2_{\mathrm{d}elta'+1}\xhookrightarrow{}C^0_{2\varepsilon^2}$ which together with \varepsilonqref{HR gamma 1} allows us to say \begin{equation*} 4\int_{\mathbb{R}^2}e^{-2\gamma^{(n)}}w e_0^{(n)}h\nabla h\cdot\nabla\gamma^{(n)}\mathrm{d} x \left\|esssim C(C_i) \int_{\mathbb{R}^2}w\left\|eft|\frac{e_0^{(n)}h}{N^{(n)}}\right\|ight|e^{-\gamma^{(n)}}|\nabla h|\,\mathrm{d} x \left\|eq C(C_i) E(t). \varepsilonnd{equation*} \item We already used multiple times that $\beta^{(n)}$ and $\nabla\gamma^{(n)}$ are bounded (by $\varepsilon$ and $C(C_i)$ respectively), and thus \begin{equation*} 2\int_{\mathbb{R}^2} e^{-2\gamma^{(n)}}w|\nabla h|^2\beta^{(n)}\cdot \nabla\gamma^{(n)}\mathrm{d} x\left\|eq C(C_i) E(t). \varepsilonnd{equation*} \varepsilonnd{itemize} For the last integral in \varepsilonqref{energy equality} we apply Cauchy-Schwarz inequality : \begin{equation*} 2\int_{\mathbb{R}^2}w f e_0^{(n)}h\,\mathrm{d} x\left\|eq 2\left\|eft(\int_{\mathbb{R}^2}w f^2 N^{(n)2}\mathrm{d} x\right\|ight)^{\frac{1}{2}}\left\|eft(\int_{\mathbb{R}^2}w \left\|eft(\mathbf{T}^{(n)}h\right\|ight)^2 \mathrm{d} x\right\|ight)^{\frac{1}{2}}\left\|eq E(t)+\int_{\mathbb{R}^2}w f^2 N^{(n)2}\mathrm{d} x . \varepsilonnd{equation*} Summarising all the estimates, we get : \begin{equation*} \frac{\mathrm{d} E}{\mathrm{d} t}(t)\left\|eq C(C_i)E(t)+\int_{\mathbb{R}^2}w f^2 N^{(n)2}(t,x)\,\mathrm{d} x . \varepsilonnd{equation*} We apply Gronwall's inequality with $T$ sufficiently small to obtain \begin{equation*} E(t)\left\|eq 2\left\|eft( E(0)+T\sup_{s\in[0,T]}\int_{\mathbb{R}^2}w f^2 N^{(n)2}(s,x)\,\mathrm{d} x\right\|ight) . \varepsilonnd{equation*} We recognize in $E(t)$ a weighted Sobolev norm, and using inequality such as $\frac{1}{\sqrt{2}}(a+b)\left\|eq \sqrt{a^2+b^2}\left\|eq a+b$, we obtain the inequality of the lemma. \varepsilonnd{proof} \begin{lem}\left\|abel{inegalite d'energie lemme} If $h$ is a solution of \varepsilonqref{partie principale de box} then, if $T$ is sufficiently small, we have for all $t\in[0,T]$ \begin{align} &\sum_{|\alpha|\left\|eq 2} \left\|eft( \left\|eft\Vert\mathbf{T}^{(n)}\nabla^{\alpha}h\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}(t)+\left\|eft\Vert e^{-\gamma^{(n)}}\nabla (\nabla^{\alpha}h)\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}(t)\right\|ight)\nonumber \\ & \left\|eq 3\sum_{|\alpha|\left\|eq 2} \left\|eft( \left\|eft\Vert\mathbf{T}^{(n)}\nabla^{\alpha}h\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}(0)+\left\|eft\Vert e^{-\gamma^{(n)}}\nabla (\nabla^{\alpha}h)\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}(0)\right\|ight)+C(C_i)\sqrt{T}\sup_{s\in[0,T]}\left\|eft\| fN^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}(s). \left\|abel{inegalite d'energie equation} \varepsilonnd{align} \varepsilonnd{lem} \begin{proof} For the sake of clarity, we set \begin{equation*} \mathcal{E}^{(n)}[h](t)=\sum_{|\alpha|\left\|eq 2} \left\|eft( \left\|eft\Vert\mathbf{T}^{(n)}\nabla^{\alpha}h\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}(t)+\left\|eft\Vert e^{-\gamma^{(n)}}\nabla (\nabla^{\alpha}h)\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}(t)\right\|ight) \varepsilonnd{equation*} If $h$ satisfies \varepsilonqref{partie principale de box}, then, applying $\nabla^{\alpha}$ to the equation, we show that $\nabla^{\alpha}h$ satisfies \begin{equation} \left\|eft( \mathbf{T}^{(n)} \right\|ight)^2\nabla^{\alpha}h-e^{-2\gamma^{(n)}}\mathscr{D}elta (\nabla^{\alpha}h)=\nabla^{\alpha}f+\left\|eft[\nabla^{\alpha},e^{-2\gamma^{(n)}}\right\|ight]\mathscr{D}elta h+\left\|eft[\left\|eft( \mathbf{T}^{(n)} \right\|ight)^2,\nabla^{\alpha}\right\|ight]h. \left\|abel{eq derivee} \varepsilonnd{equation} Thanks to the previous lemma, in order to prove \varepsilonqref{inegalite d'energie equation}, we have to bound $\sum_{|\alpha|\left\|eq 2}\Vert (\text{RHS of \varepsilonqref{eq derivee}})\times N^{(n)}\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}$. \begin{itemize} \item First step is bounding $\Vert N^{(n)}\nabla^{\alpha}f \Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}$ (using the fact that $\frac{1}{N^{(n)}}\in L^{\infty}$, $|\nabla^{\alpha}(\chi\left\|n)|\left\|eq \left\|angle x\right\|angle^{-|\alpha|}$, the product estimates and \varepsilonqref{HR N 2}). If $|\alpha|=1$ : \begin{align*} \Vert N^{(n)}\nabla^{\alpha}f \Vert_{L^2_{\mathrm{d}elta'+2}} &\left\|esssim \Vert\nabla^{\alpha}(fN^{(n)})\Vert_{L^2_{\mathrm{d}elta'+2}}+ \Vert f\nabla^{\alpha} N^{(n)}\Vert_{L^2_{\mathrm{d}elta'+2}}\\ &\left\|esssim \Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}+\Vert N^{(n)} f\nabla^{\alpha}(\chi\left\|n)\Vert_{L^2_{\mathrm{d}elta'+2}}+\Vert N^{(n)} f\nabla^{\alpha}\Tilde{N}^{(n)}\Vert_{L^2_{\mathrm{d}elta'+2}}\\ &\left\|esssim \Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}} \left\|eft( 2+ \Vert \nabla^{\alpha}\Tilde{N}^{(n)}\Vert_{H^{2}_{\mathrm{d}elta+1}}\right\|ight)\\ &\left\|eq C(C_i)\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}. \varepsilonnd{align*} If $|\alpha|=2$, then there exist $\alpha_1,\alpha_2$ with $|\alpha_1|=|\alpha_2|=1$ such that : \begin{equation*} \Vert N^{(n)}\nabla^{\alpha}f\Vert_{L^2_{\mathrm{d}elta'+3}}\left\|esssim \Vert \nabla^{\alpha}(fN^{(n)}) \Vert_{L^2_{\mathrm{d}elta'+3}}+\Vert f\nabla^{\alpha}N^{(n)} \Vert_{L^2_{\mathrm{d}elta'+3}}+\Vert \nabla^{\alpha_1}N^{(n)}\nabla^{\alpha_2}f \Vert_{L^2_{\mathrm{d}elta'+3}}. \varepsilonnd{equation*} The two first terms can be handled as in the case $|\alpha|=1$. For the last term we do the following : \begin{align*} \Vert \nabla^{\alpha_1}N^{(n)}\nabla^{\alpha_2}f \Vert_{L^2_{\mathrm{d}elta'+3}} & \left\|esssim \Vert N^{(n)} \nabla^{\alpha_1}N^{(n)}\nabla ^{\alpha_2}f \Vert_{L^2_{\mathrm{d}elta'+3}}\\ &\left\|esssim \Vert N^{(n)}\nabla^{\alpha_2}f \Vert_{L^2_{\mathrm{d}elta'+2}}\Vert \nabla^{\alpha_1}N^{(n)} \Vert_{H^2_{\mathrm{d}elta+1}}\\ &\left\|eq C(C_i)\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}, \varepsilonnd{align*} where in the last inequality we use the calculation of the $|\alpha|=1$ case. Summarising, we get : \begin{equation} \sum_{|\alpha|\left\|eq 2}\|N^{(n)}\nabla^{\alpha}f \|_{L^2_{\mathrm{d}elta'+1+|\alpha|}}\left\|eq C(C_i)\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}\left\|abel{first step}. \varepsilonnd{equation} \item Second step is bounding $\left\|eft\| N^{(n)}\left\|eft[\nabla^{\alpha},e^{-2\gamma^{(n)}}\right\|ight]\mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha|}}$. If $|\alpha|=1$ we have $\left\|eft[\nabla^{\alpha},e^{-2\gamma^{(n)}}\right\|ight]\mathscr{D}elta h=-2e^{-2\gamma^{(n)}}\nabla^{\alpha}\gamma^{(n)}\mathscr{D}elta h$. Using the fact that $\left\|eft|\Tilde{N}^{(n)}\right\|ight|\left\|eq\varepsilon$, \varepsilonqref{useful gamma 1}, $|\chi\left\|n|\left\|esssim\left\|angle x\right\|angle^{\varepsilon^2}$ and the expression of $\gamma^{(n)}$ we have \begin{align*} \left\|eft\| e^{-2\gamma^{(n)}}N^{(n)}\nabla^{\alpha}\gamma^{(n)}\mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+2}} & \left\|esssim \left\|eft\| \nabla^{\alpha}\gamma^{(n)}\mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+2+3\varepsilon^2}} \\ & \left\|esssim\left\|eft\| \nabla^{\alpha}(\chi\left\|n)\mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+2+3\varepsilon^2}}+\left\|eft\| \nabla^{\alpha}\Tilde{\gamma}^{(n)}\mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+2+3\varepsilon^2}}. \varepsilonnd{align*} Since $|\alpha|=1$, we have $|\nabla^{\alpha}(\chi\left\|n)|\left\|esssim \left\|angle x \right\|angle^{-1}$ and $\nabla^{\alpha}\Tilde{\gamma}^{(n)}\in H^2_{\mathrm{d}elta'+1}$ which embeds in $C^0_1$. This implies : \begin{equation*} \left\|eft\| e^{-2\gamma^{(n)}}N^{(n)}\nabla^{\alpha}\gamma^{(n)}\mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+2}} \left\|eq C(C_i) \|\mathscr{D}elta h\|_{L^2_{\mathrm{d}elta'+1+3\varepsilon^2}}\left\|eq C(C_i)\sum_{|\alpha'|=1}\left\|eft\Vert e^{-\gamma^{(n)}}\nabla (\nabla^{\alpha'}h)\right\|ight\Vert_{L^2_{\mathrm{d}elta'+2}}, \varepsilonnd{equation*} where in the last inequality, we used that $1\left\|esssim |e^{-\gamma^{(n)}}|$ and took $\varepsilon$ small enough. If $|\alpha|=2$, then there exist $\alpha_1,\alpha_2$ with $|\alpha_1|=|\alpha_2|=1$ such that : \begin{align*} \left\|eft\| N^{(n)}\left\|eft[\nabla^{\alpha},e^{-2\gamma^{(n)}}\right\|ight]\mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+3}} &\left\|esssim \left\|eft\|e^{-2\gamma^{(n)}}N^{(n)}\nabla^{\alpha}\gamma^{(n)} \mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+3}}+\left\|eft\|e^{-2\gamma^{(n)}}N^{(n)}\nabla^{\alpha_1}\gamma^{(n)} \nabla^{\alpha_2}\mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+3}}\\ &\quad + \left\| e^{-2\gamma^{(n)}} N^{(n)} \nabla^{\alpha_1}\gamma^{(n)}\nabla^{\alpha_2}\gamma^{(n)}\mathscr{D}elta h \right\|_{L^2_{\mathrm{d}elta'+3}} \\& \left\|esssim \left\|eft\|\nabla^{\alpha}\gamma^{(n)} \mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+3+3\varepsilon^2}}+\left\|eft\|\nabla^{\alpha_1}\gamma^{(n)} \nabla^{\alpha_2}\mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+3+3\varepsilon^2}} \\&\quad + \left\| \nabla^{\alpha_1}\gamma^{(n)}\nabla^{\alpha_2}\gamma^{(n)}\mathscr{D}elta h \right\|_{L^2_{\mathrm{d}elta'+3+3\varepsilon^2}} \varepsilonnd{align*} For the first term, we use the fact that $|\nabla^{\alpha}(\chi\left\|n)|\left\|esssim \left\|angle x \right\|angle^{-2}$, $\nabla^{\alpha}\Tilde{\gamma}^{(n)}\in H^1_{\mathrm{d}elta'+2}$ (since $|\alpha|=2$) and the product estimate to get : \begin{equation*} \left\|eft\|\nabla^{\alpha}\gamma^{(n)} \mathscr{D}elta h\right\|ight\|_{L^2_{\mathrm{d}elta'+3+3\varepsilon^2}}\left\|eq C(C_i) \|\mathscr{D}elta h\|_{H^1_{\mathrm{d}elta'+2}}\left\|eq C(C_i)\sum_{|\alpha'|=1,2}\left\|eft\|e^{-\gamma^{(n)}}\nabla(\nabla^{\alpha'}h)\right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha'|}}. \varepsilonnd{equation*} For the second term, we again use that $\nabla^{\alpha_1}\Tilde{\gamma}^{(n)},\nabla^{\alpha_1}(\chi\left\|n)\in C^0_1$ (since $|\alpha_1|=1$) to get \begin{equation*} \left\|eft\|\nabla^{\alpha_1}\gamma^{(n)} \nabla^{\alpha_2}\mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+3+3\varepsilon^2}}\left\|eq C(C_i) \|\nabla^{\alpha_2}\mathscr{D}elta h\|_{L^2_{\mathrm{d}elta'+2+ 3\varepsilon^2}}\left\|eq C(C_i)\sum_{|\alpha'|=2}\left\|eft\|e^{-\gamma^{(n)}}\nabla(\nabla^{\alpha'}h)\right\|ight\|_{L^2_{\mathrm{d}elta'+3}}. \varepsilonnd{equation*} The third term is easier to handle than the first one. Summarising, we get : \begin{equation} \sum_{|\alpha|\left\|eq 2}\left\|eft\| N^{(n)}\left\|eft[\nabla^{\alpha},e^{-2\gamma^{(n)}}\right\|ight]\mathscr{D}elta h \right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha|}}\left\|eq C(C_i) \mathcal{E}^{(n)}[h].\left\|abel{second step} \varepsilonnd{equation} \item Third step is bounding $\left\|eft\| N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla^{\alpha}\right\|ight]h \right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha|}}$. Given the expression of $\mathcal{E}^{(n)}[h]$, we are allowed to bound this term by norms involving $\mathbf{T}^{(n)}\nabla^{\mu}$, $\nabla^{\nu}$ (for $|\mu|\left\|eq 2$ and $|\nu|\left\|eq 3$) and $\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2$. The strategy is then to express $N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla \right\|ight]h$ and $N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla^2 \right\|ight]h$ in terms of those operators acting on $h$, using the commutation formula \begin{equation*} \left\|eft[\mathbf{T}^{(n)},\nabla\right\|ight] h =\frac{\nabla\beta^{(n)}}{N^{(n)}}\nabla h-\frac{\nabla N^{(n)}}{N^{(n)}}\mathbf{T}^{(n)}h. \varepsilonnd{equation*} Doing so, we find the following formula (we don't write the irrelevant numerical constants) : \begin{align} N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla \right\|ight]h & =\left\|eft(e_0^{(n)}\left\|eft(\frac{\nabla\beta^{(n)}}{N^{(n)}}\right\|ight) +\frac{\left\|eft(\nabla\beta^{(n)}\right\|ight)^2}{N^{(n)}} \right\|ight)\nabla h +\left\|eft(\frac{\nabla\beta^{(n)}\nabla N^{(n)}}{N^{(n)}} +e_0^{(n)}\left\|eft( \frac{\nabla N^{(n)}}{N^{(n)}} \right\|ight) \right\|ight)\mathbf{T}^{(n)}h\left\|abel{commutateur 1}\\& \qquad\qquad+2\nabla\beta^{(n)}\mathbf{T}^{(n)}\nabla h -2\nabla N^{(n)}\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2h.\nonumber \varepsilonnd{align} We recall that $\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2h=e^{-2\gamma^{(n)}}\mathscr{D}elta h+f$, so that the $\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2$ term in \varepsilonqref{commutateur 1} has already been estimate during the two first steps. The coefficients in front of $\nabla h$ and $\mathbf{T}^{(n)} h$ are all in $\mathcal{C}^1_0$ except the two involving $e_0\nabla N^{(n)}$, for wich we use the product law $H^1\times H^1$ and \varepsilonqref{HR N 2} : \begin{align*} \left\| e_0^{(n)}\left\|eft( \frac{\nabla N^{(n)}}{N^{(n)}} \right\|ight)\mathbf{T}^{(n)}h \right\|_{L^2_{\mathrm{d}elta'+2}} & \left\|esssim \left\|\nabla \partial_t N^{(n)} \right\|_{H^1_{\mathrm{d}elta+1}} \left\| \mathbf{T}^{(n)} h \right\|_{H^1_{\mathrm{d}elta'+1}} \left\|esssim C(C_i)\mathcal{E}^{(n)}[h]. \varepsilonnd{align*} We only need to bound the coefficient in front of $\mathbf{T}^{(n)}\nabla$ in $L^{\infty}$, which is easily thanks to \varepsilonqref{HR N 3}. This allows us to handle the case $|\alpha|=1$ : \begin{align} &\left\|eft\| N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla^{\alpha}\right\|ight]h \right\|ight\|_{L^2_{\mathrm{d}elta'+2}}\nonumber\\&\left\|eq C(C_i)\left\|eft( \sum_{|\alpha'|\left\|eq 1} \left\|eft( \left\|eft\Vert\frac{e_0^{(n)}\nabla^{\alpha'}h}{N^{(n)}}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha'|}}+\left\|eft\| e^{-\gamma^{(n)}}\nabla (\nabla^{\alpha'}h)\right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha'|}} \right\|ight)+\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}} \right\|ight).\left\|abel{alpha 1} \varepsilonnd{align} Before turning to the case $|\alpha|=2$, let's remark that, in view of \varepsilonqref{eq derivee}, so far we have proved that \begin{equation} \left\|eft\| \left\|eft( \mathbf{T}^{(n)}\right\|ight)^2\nabla h\right\|ight\|_{L^2_{\mathrm{d}elta'+2}}\left\|eq C(C_i) \left\|eft( \mathcal{E}^{(n)}[h]+\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}\right\|ight).\left\|abel{L carré} \varepsilonnd{equation} This means that, even if $\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2\nabla h$ doesn't appear in the expression of $\mathcal{E}^{(n)}[h]$, we are allowed to use it in the sequel of the third step. We now turn to the case $|\alpha|=2$ and push our calculations further to get (we still don't write the irrelevant numerical constants) : \begin{align} & N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla^2 \right\|ight]h =\nabla N^{(n)}\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2\nabla h +\nabla N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla \right\|ight]h \nonumber\\ & \left\|eft( N^{(n)}\nabla\mathbf{T}^{(n)}\left\|eft( \frac{\nabla\beta^{(n)}}{N^{(n)}}\right\|ight)+N^{(n)}\nabla\left\|eft(\left\|eft(\frac{\nabla\beta^{(n)}}{N^{(n)}}\right\|ight)^2\right\|ight) +\frac{\nabla N^{(n)}\left\|eft(\nabla\beta^{(n)}\right\|ight)^2}{N^{(n)} }+ \nabla\beta^{(n)}\mathbf{T}^{(n)}\left\|eft(\frac{\nabla N^{(n)}}{N^{(n)}}\right\|ight)\right\|ight)\nabla h \nonumber\\ &+\left\|eft( e_0^{(n)}\left\|eft(\frac{\nabla\beta^{(n)}}{N^{(n)}} \right\|ight)+\frac{\left\|eft(\nabla\beta^{(n)}\right\|ight)^2}{N^{(n)}}\right\|ight)\nabla^2 h +N^{(n)}\nabla\left\|eft(\frac{\nabla N^{(n)}}{N^{(n)}}\right\|ight) \left\|eft( \mathbf{T}^{(n)}\right\|ight)^2 h + \nabla\beta^{(n)}\mathbf{T}^{(n)}\nabla^2 h\nonumber\\ & +\left\|eft( N^{(n)}\nabla\left\|eft(\frac{\nabla\beta^{(n)}\nabla N^{(n)}}{N^{(n)}}\right\|ight)+N^{(n)}\nabla\mathbf{T}^{(n)}\left\|eft( \frac{\nabla N^{(n)}}{N^{(n)}}\right\|ight) +\frac{\nabla \beta^{(n)}\left\|eft(\nabla N^{(n)}\right\|ight)^2}{N^{(n)} }+ \nabla N^{(n)}\mathbf{T}^{(n)}\left\|eft(\frac{\nabla N^{(n)}}{N^{(n)}}\right\|ight) \right\|ight)\mathbf{T}^{(n)} h\nonumber\\ & +\left\|eft( \nabla\beta^{(n)}\nabla N^{(n)} +e_0^{(n)}\left\|eft( \frac{\nabla N^{(n)}}{N^{(n)}}\right\|ight)+N^{(n)}\nabla\left\|eft( \frac{\nabla\beta^{(n)}}{N^{(n)}} \right\|ight)+\frac{\nabla\beta^{(n)}\nabla N^{(n)}}{N^{(n)}} \right\|ight)\mathbf{T}^{(n)}\nabla h\left\|abel{commutateur 2}. \varepsilonnd{align} We need to estimate the $L^2_{\mathrm{d}elta'+3}$ norm of \varepsilonqref{commutateur 2}. The term $\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2 h$ has already been handled since $h$ satisfies \varepsilonqref{partie principale de box}. Since $\nabla N^{(n)}\in C^0_1$ we can use $\varepsilonqref{L carré}$ to estimate the term $\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2\nabla h$. With the same argument, using \varepsilonqref{alpha 1} we handle the $\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla \right\|ight]h$ term. Thanks to \varepsilonqref{HR beta 2} and \varepsilonqref{HR beta 3}, the coefficients in front of $\mathbf{T}^{(n)}\nabla^2h$ and $\nabla^2h$ are in the appropriate weighted $L^\infty$-based spaces ($L^\infty$ and $C^0_1$ respectively). The only problematic terms are the ones where two spatial derivatives hit $\beta^{(n)}$ or when at least one spatial derivative and $\mathbf{T}^{(n)}$ hit $N^{(n)}$. For them, we use the product estimate (see Proposition \right\|ef{prop prod}). Let us give two examples, the first one using the embedding $H^1\times H^1\xhookrightarrow{}L^2$ (with appropriate weights) : \begin{align*} \left\| \nabla^2\beta^{(n)} \mathbf{T}^{(n)}\nabla h \right\|_{L^2_{\mathrm{d}elta'+3}} & \left\|esssim \left\|\nabla^2\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'+2}} \left\| \mathbf{T}^{(n)}\nabla h \right\|_{H^1_{\mathrm{d}elta'+2}} \left\|esssim C(C_i) \mathcal{E}^{(n)}[h]. \varepsilonnd{align*} The second example uses the embedding $L^2\times H^2\xhookrightarrow{}L^2$ (with appropriate weights) : \begin{align*} \left\| \nabla \mathbf{T}^{(n)} \nabla N^{(n)}\mathbf{T}^{(n)}h\right\|_{L^2_{\mathrm{d}elta'+3}} \left\|esssim \left\|eft( 1+ \left\| \nabla^2 \partial_t \Tilde{N}^{(n)} \right\|_{L^2_{\mathrm{d}elta+2}} \right\|ight) \left\| \mathbf{T}^{(n)} h \right\|_{H^2_{\mathrm{d}elta'+1}} \left\|esssim C(C_i) \mathcal{E}^{(n)}[h]. \varepsilonnd{align*} This allows us to handle entirely the case $|\alpha|=2$. Summarising the third step, we get : \begin{equation} \sum_{|\alpha|\left\|eq 2}\left\|eft\| N^{(n)}\left\|eft[\left\|eft( \mathbf{T}^{(n)}\right\|ight)^2,\nabla^{\alpha}\right\|ight]h \right\|ight\|_{L^2_{\mathrm{d}elta'+1+|\alpha|}} \left\|eq C(C_i)\left\|eft( \mathcal{E}^{(n)}[h]+\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}\right\|ight).\left\|abel{third step} \varepsilonnd{equation} \varepsilonnd{itemize} Combining \varepsilonqref{first step}, \varepsilonqref{second step} and \varepsilonqref{third step}, we get for all $t\in [0,T]$ : \begin{equation} \mathcal{E}^{(n)}[h](t)\left\|eq 2 \mathcal{E}^{(n)}[h](0)+C(C_i)\sqrt{T}\left\|eft( \sup_{s\in[0,T]}\Vert fN^{(n)}\Vert_{H^2_{\mathrm{d}elta'+1}}(s)+\mathcal{E}^{(n)}[h](t)\right\|ight). \varepsilonnd{equation} By choosing $T$ sufficiently small, we can absorb the term $\mathcal{E}^{(n)}[h](t)$ of the RHS into the LHS and conclude the proof of the lemma. \varepsilonnd{proof} With this energy estimate, we are ready to prove estimates on $\Tilde{\gamma}^{(n+1)}$, $\varphi^{(n+1)}$ and $\Omegamega^{(n+1)}$. The spatial term in the energy $\mathcal{E}^{(n)}[h]$ is different from what appear in \varepsilonqref{HR gamma 1}, but \varepsilonqref{propsmall gamma} implies that $1\left\|esssim e^{-\gamma^{(n)}}$ so we will get back the estimates we want for $\Tilde{\gamma}^{(n+1)}$, $\varphi^{(n+1)}$ and $\Omegamega^{(n+1)}$ if we bound $\mathcal{E}^{(n)}$, using Lemma \right\|ef{inegalite d'energie lemme}. \subsubsection{Hyperbolic estimates} We use our energy estimate to prove the estimates on $\Tilde{\gamma}^{(n+1)}$. Since we are not getting $\gamma^{(n+1)}$ from an elliptic equation, we cannot obtain the decomposition $\gamma^{(n+1)}=-\alpha\chi\left\|n+\Tilde{\gamma}^{(n+1)}$ directly. Our strategy is to solve for $\gamma^{(n+1)}+\alpha\chi\left\|n$ to artificially recover our decomposition after having set $\Tilde{\gamma}^{(n+1)}\vcentcolon=\gamma^{(n+1)}+\alpha\chi\left\|n$. For the sake of clarity, we gather in the following lemma the estimates of the extra terms due to $\alpha\chi\left\|n$ : \begin{lem}\left\|abel{op chi ln lemme} We set $\Psi^{(n)}= N^{(n)}\left\|eft(\left\|eft(\mathbf{T}^{(n)}\right\|ight)^2(\chi\left\|n)- e^{-2\gamma^{(n)}}\mathscr{D}elta (\chi\left\|n) \right\|ight)$. For $n\geq 2$, the following estimate hold : \begin{align} \left\|eft\| \Psi^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}&\left\|eq C(C_i),\left\|abel{op chi ln 1}\\ \left\|eft\| \Psi^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'+1}}&\left\|eq A_0C_i.\left\|abel{op chi ln 3} \varepsilonnd{align} \varepsilonnd{lem} \begin{proof} To prove \varepsilonqref{op chi ln 1}, we don't need to be very precise about the dependence on $C_i$ of the bound, so we don't give many details. For the first part of $\Psi^{(n)}$ : \begin{align*} \left\|eft\| N^{(n)}\left\|eft(\mathbf{T}^{(n)}\right\|ight)^2(\chi\left\|n)\right\|ight\|_{H^2_{\mathrm{d}elta'+1}} & \left\|esssim \left\|eft\| \partial_t\left\|eft( \frac{\beta^{(n)}}{N^{(n)}}\right\|ight) \right\|ight\|_{H^2_{\mathrm{d}elta'}}+\left\|eft\|\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'}}\left\|eft( \left\|eft\|\frac{\beta^{(n)}}{N^{(n)}} \right\|ight\|_{H^2_{\mathrm{d}elta'-1}}+\left\|eft\|\nabla\left\|eft( \frac{\beta^{(n)}}{N^{(n)}} \right\|ight) \right\|ight\|_{H^2_{\mathrm{d}elta'}} \right\|ight)\\ &\left\|eq C(C_i). \varepsilonnd{align*} For the second part of $\Psi^{(n)}$, we notice that $\mathscr{D}elta(\chi\left\|n)=\mathscr{D}elta(\chi)\left\|n+\nabla\chi\cdot\nabla\left\|n$ is a smooth compactly supported function (its support is included in $B_2$) and in particular belongs to all $C^k$ spaces, using \varepsilonqref{useful gamma 1} and \varepsilonqref{HR N 1} : \begin{align*} \left\|eft\|e^{-2\gamma^{(n)}} N^{(n)}\mathscr{D}elta(\chi\left\|n) \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} & \left\|esssim\left\|eft\|\mathscr{D}elta(\chi\left\|n) \right\|ight\|_{C^2} \left\|eft\| N^{(n)} \right\|ight\|_{H^2(B_2)}\left\|esssim 1. \varepsilonnd{align*} We now turn to the proof of \varepsilonqref{op chi ln 3}. For the first part of $\Psi^{(n)}$, we use \varepsilonqref{HR beta 1} (and the product estimate $H^2_{\mathrm{d}elta'}\times H^1_{\varepsilonta}\xhookrightarrow{}H^1_{\varepsilonta}$), \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2} and \varepsilonqref{HR beta 3}, and actually the only term that will bring some $C_i$ are $\partial_t\beta^{(n)}$ and $\partial_t N^{(n)}$ : \begin{align*} \left\|eft\| N^{(n)}\left\|eft(\mathbf{T}^{(n)}\right\|ight)^2(\chi\left\|n)\right\|ight\|_{H^1_{\mathrm{d}elta'+1}} & \left\|esssim \left\|eft\|\partial_t\left\|eft(\frac{\beta^{(n)}}{N^{(n)}} \right\|ight)\right\|ight\|_{H^1_{\mathrm{d}elta'}}+\left\|eft\|\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'}}\left\|eft( \left\|eft\|\frac{\beta^{(n)}}{N^{(n)}} \right\|ight\|_{H^1_{\mathrm{d}elta'-1}}+\left\|eft\|\nabla\left\|eft( \frac{\beta^{(n)}}{N^{(n)}} \right\|ight) \right\|ight\|_{H^1_{\mathrm{d}elta'}} \right\|ight) \\&\left\|esssim A_0C_i+\varepsilon. \varepsilonnd{align*} For the second part of $\Psi^{(n)}$, we again use the properties of $\mathscr{D}elta(\chi\left\|n)$, \varepsilonqref{useful gamma 1} and \varepsilonqref{HR N 1} : \begin{align*} \left\|eft\|e^{-2\gamma^{(n)}} N^{(n)}\mathscr{D}elta(\chi\left\|n) \right\|ight\|_{H^1_{\mathrm{d}elta'+1}} &\left\|esssim \left\|eft\|\mathscr{D}elta(\chi\left\|n) \right\|ight\|_{C^1} \left\|eft\| N^{(n)} \right\|ight\|_{H^1(B_2)}\left\|esssim 1. \varepsilonnd{align*} \varepsilonnd{proof} \begin{prop}\left\|abel{hr+1 gamma prop} For $n\geq 2$ the following estimates hold : \begin{align} \sum_{|\alpha|\left\|eq 2} \left\|eft\Vert\mathbf{T}^{(n)}\nabla^{\alpha}\Tilde{\gamma}^{(n+1)}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}+\left\|eft\|\nabla\Tilde{\gamma}^{(n+1)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}&\left\|eq 8 C_i,\left\|abel{HR+1 gamma 1}\\ \left\|eft\| \partial_t\left\|eft(\mathbf{T}^{(n)}\Tilde{\gamma}^{(n+1)}\right\|ight) \right\|ight\|_{L^2_{\mathrm{d}elta'+1}}&\left\|esssim C_i,\left\|abel{HR+1 gamma 2}\\ \left\|eft\| \partial_t\left\|eft(\mathbf{T}^{(n)}\Tilde{\gamma}^{(n+1)}\right\|ight)\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}&\left\|esssim A_1C_i.\left\|abel{HR+1 gamma 3} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} The strategy is to recover the decomposition of $\gamma^{(n+1)}$ by setting $\Tilde{\gamma}^{(n+1)}\vcentcolon=\gamma^{(n+1)}+\alpha\chi\left\|n$. In view of \varepsilonqref{reduced system gamma}, $\Tilde{\gamma}^{(n+1)}$ is solution of \begin{equation} \left\|eft(\mathbf{T}^{(n)}\right\|ight)^2\Tilde{\gamma}^{(n+1)}-e^{-2\gamma^{(n)}}\mathscr{D}elta \Tilde{\gamma}^{(n+1)}=\left\|eft( \text{RHS of \varepsilonqref{reduced system gamma}} \right\|ight)+ \alpha\frac{\Psi^{(n)}}{N^{(n)}}.\left\|abel{eq gamma tilde} \varepsilonnd{equation} In order to prove \varepsilonqref{HR+1 gamma 1} and in view of \varepsilonqref{inegalite d'energie equation}, we have to bound $\left\|eft\| \text{RHS of \varepsilonqref{eq gamma tilde}}\times N^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}$, and thanks to the factor $\sqrt{T}$, we don't need to worry about the bounds. Thanks to \varepsilonqref{op chi ln 1}, it remains to deal with the RHS of \varepsilonqref{reduced system gamma} mutltiplied by $N^{(n)}$, which gives the following expression : \begin{align*} -\frac{N^{(n)}\left\|eft(\tau^{(n)}\right\|ight)^2}{2}+\frac{1}{2}e_0^{(n-1)}\left\|eft(\frac{\mathrm{div}(\beta^{(n)})}{N^{(n-1)}}\right\|ight)&+e^{-2\gamma^{(n)}}\frac{\mathscr{D}elta N^{(n)}}{2}+e^{-2\gamma^{(n)}}N^{(n)}|\nabla\varphi^{(n)}|^2\\&+\frac{1}{4}e^{-2\gamma^{(n)}-4\varphi^{(n)}}N^{(n)} |\nabla\Omegamega^{(n)}|^2=\vcentcolon I+II+III+IV+V. \varepsilonnd{align*} \begin{itemize} \item For $I$, we mainly use the fact that $H^2_{\mathrm{d}elta'+1}$ is an algebra and \varepsilonqref{HR tau 1} (and the product estimate to deal with $\chi\left\|n$) : \begin{equation*} \|I\|_{H^2_{\mathrm{d}elta'+1}}\left\|esssim \left\|eft\| \tau^{(n)}\right\|ight\|^2_{H^2_{\mathrm{d}elta'+1}}\left\|eft( 1+ \left\|eft\|\Tilde{N}^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\right\|ight)\left\|esssim C(C_i). \varepsilonnd{equation*} \item For $II$, we use the computations already performed about $\left\|eft[\mathbf{T}^{(n-1)},\nabla \right\|ight]$, \varepsilonqref{HR N 2} (which implies that $\left\|eft|\frac{1}{N^{(n-1)}}\right\|ight|$ and $\left\|eft\|\nabla N^{(n-1)}\right\|ight\|_{C^1}$ are bounded) to get rid of the $\frac{1}{N^{(n-1)}}$ factors, in order to get : \begin{align} \| II\|_{H^2_{\mathrm{d}elta'+1}} & \left\|esssim \left\|eft\|e_0^{(n-1)}N^{(n-1)} \nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}+\left\|eft\| \mathbf{T}^{(n-1)}\beta^{(n)} \right\|ight\|_{H^3_{\mathrm{d}elta'}}\left\|abel{II}\\&\qquad +\left\|eft\|\nabla N^{(n-1)}\mathbf{T}^{(n-1)}\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}+\left\|eft\|\nabla\beta^{(n-1)}\nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\nonumber. \varepsilonnd{align} Using \varepsilonqref{HR beta 4}, it's easy to see that $\left\|eft\| \mathbf{T}^{(n-1)}\beta^{(n)} \right\|ight\|_{H^3_{\mathrm{d}elta'}}\left\|eq C(C_i)$, and we recall the embedding $H^3_{\mathrm{d}elta'}\xhookrightarrow{}H^2_{\mathrm{d}elta'+1}$. Using $|\nabla(\chi\left\|n)|\left\|esssim \left\|angle x\right\|angle^{-1}$ and \varepsilonqref{HR N 2}, we see that $\left\|eft\|\nabla N^{(n-1)}\right\|ight\|_{ H^3_{\mathrm{d}elta}}\left\|esssim C_i$ and thus we use the product estimate to write : \begin{equation*} \left\|eft\| \mathbf{T}^{(n-1)}\beta^{(n)} \right\|ight\|_{H^3_{\mathrm{d}elta'}}+\left\|eft\|\nabla N^{(n-1)}\mathbf{T}^{(n-1)}\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \left\|esssim \left\|eft\| \mathbf{T}^{(n-1)}\beta^{(n)} \right\|ight\|_{H^3_{\mathrm{d}elta'}}\left\|eft( 1+\left\|eft\|\nabla N^{(n-1)} \right\|ight\|_{H^2_{\mathrm{d}elta}} \right\|ight) \left\|eq C(C_i). \varepsilonnd{equation*} Using $\nabla\beta^{(n)}\in H^3_{\mathrm{d}elta'+1}\xhookrightarrow{}H^2_{\mathrm{d}elta'+2}$, $\varepsilonqref{HR N 2}$, and $\nabla N^{(n-1)}\in H^2_{\mathrm{d}elta}$ : \begin{align*} \left\|eft\|e_0^{(n-1)}N^{(n-1)} \nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}& \left\|esssim \left\|eft| \partial_t N_a^{(n-1)}\right\|ight|\left\|eft\| \chi\left\|n \nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}+\left\|eft\|\partial_t\Tilde{N}^{(n-1)}\nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\\&\qquad+\left\|eft\|\beta^{(n-1)}\nabla N^{(n-1)}\nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+1}} \\& \left\|esssim\left\|eft(\left\|eft| \partial_t N_a^{(n-1)}\right\|ight|+\left\|eft\|\partial_t\Tilde{N}^{(n-1)}\right\|ight\|_{H^2_{\mathrm{d}elta}}\right\|ight)\left\|eft\| \nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+2}}\\&\qquad+\left\|eft\|\beta^{(n-1)}\right\|ight\|_{H^2_{\mathrm{d}elta'+2}}\left\|eft\|\nabla N^{(n-1)}\right\|ight\|_{H^2_{\mathrm{d}elta}}\left\|eft\|\nabla\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'+2}} \\& \left\|eq C(C_i). \varepsilonnd{align*} The last term in \varepsilonqref{II} doesn't present any difficulty and we get $\| II\|_{H^2_{\mathrm{d}elta'+1}}\left\|eq C(C_i)$. \item For $III$, we first notice that $\mathscr{D}elta(\chi\left\|n)=\mathscr{D}elta(\chi)\left\|n+\nabla\chi\cdot\nabla\left\|n$ is a smooth compactly supported function and therefore belongs to all $H^k$ spaces. We then use \varepsilonqref{useful gamma 1} and \varepsilonqref{HR N 3} : \begin{align*} \|III\|_{H^2_{\mathrm{d}elta'+1}} &\left\|esssim \left\|eft| N_a^{(n)}\right\|ight|\left\|eft\|\mathscr{D}elta(\chi\left\|n) \right\|ight\|_{H^2}+\left\|eft\|\mathscr{D}elta\Tilde{N}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta+1}}\left\|eq C(C_i). \varepsilonnd{align*} \item For $IV$, thanks to the support property of $\varphi^{(n)}$, we don't worry about the decrease of our functions. We simply use \varepsilonqref{useful gamma 1}, \varepsilonqref{HR N 3} (which implies that $\left\|eft\| N^{(n)}\right\|ight\|_{C^2(B_{2R})}\left\|eq C(C_i)$) and the fact that $H^2$ is an algebra : \begin{align*} &\| IV\|_{H^2}\left\|esssim \left\|eft\| N^{(n)}\right\|ight\|_{C^2(B_{2R})} \left\|eft\|\nabla\varphi^{(n)} \right\|ight\|_{H^2}^2\left\|eq C(C_i). \varepsilonnd{align*} \item For $V$, we do as for $IV$, using in addition \varepsilonqref{useful ffi 1}, which implies that it remains to deal with the following term : \begin{align*} \left\|\nabla^2\varphi^{(n)}|\nabla\Omegamega^{(n)}|^2 \right\|_{L^2} \left\|esssim \left\| \nabla^2\varphi^{(n)}\right\|_{L^2}\left\|\nabla\Omegamega^{(n)} \right\|_{H^2}^2\left\|eq C(C_i). \varepsilonnd{align*} \varepsilonnd{itemize} Using Lemma \right\|ef{inegalite d'energie lemme}, we get for all $t\in [0,T]$ : \begin{equation*} \mathcal{E}^{(n)}\left\|eft[ \Tilde{\gamma}^{(n+1)} \right\|ight](t)\left\|eq 3\, \mathcal{E}^{(n)}\left\|eft[ \Tilde{\gamma}^{(n+1)} \right\|ight](0)+C(C_i)\sqrt{T}.\left\|abel{blabla} \varepsilonnd{equation*} It remains to show that $\mathcal{E}^{(n)}\left\|eft[ \Tilde{\gamma}^{(n+1)} \right\|ight](0)$ is bounded by $C_i$. Therefore, the following calculations will be performed on $\Sigma_0$ and we can forget about the indices $(n)$ or $(n+1)$ and use the estimates \varepsilonqref{CI petit} and \varepsilonqref{CI gros}, which are more comfortable. Using the calculations performed in Proposition \right\|ef{commutation estimate}, we show that : \begin{align} \sum_{|\alpha|\left\|eq 2} \left\|eft\Vert\mathbf{T}\nabla^{\alpha}\Tilde{\gamma}\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}} \left\|eq (1+C\varepsilon)\| \mathbf{T}\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta+1}}+C\varepsilon\mathcal{E}\left\|eft[ \Tilde{\gamma} \right\|ight](0)+C\varepsilon\|\Tilde{\gamma}\|_{H^4_{\mathrm{d}elta}}.\left\|abel{E0 temps} \varepsilonnd{align} Using the same ideas as in the second step of the proof of Lemma \right\|ef{inegalite d'energie lemme}, we show that : \begin{align} \sum_{|\alpha|\left\|eq 2} \left\|eft\Vert e^{-\gamma}\nabla( \nabla^{\alpha}\Tilde{\gamma})\right\|ight\Vert_{L^2_{\mathrm{d}elta'+1+|\alpha|}}\left\|eq (1+C\varepsilon)\|\Tilde{\gamma}\|_{H^4_{\mathrm{d}elta}}.\left\|abel{E0 espace} \varepsilonnd{align} Putting together \varepsilonqref{E0 temps} and \varepsilonqref{E0 espace} and using \varepsilonqref{CI petit} and \varepsilonqref{CI gros}, we get \begin{equation*} \mathcal{E}\left\|eft[ \Tilde{\gamma} \right\|ight](0) \left\|eq (1+C\varepsilon) \left\|eft( \| \mathbf{T}\Tilde{\gamma}\|_{H^2_{\mathrm{d}elta+1}} +\|\Tilde{\gamma}\|_{H^4_{\mathrm{d}elta}}\right\|ight)+C\varepsilon\mathcal{E}\left\|eft[ \Tilde{\gamma} \right\|ight](0) \left\|eq2 (1+C\varepsilon)C_i +C\varepsilon\mathcal{E}\left\|eft[ \Tilde{\gamma} \right\|ight](0). \varepsilonnd{equation*} We can absorb the last term of the RHS into the LHS by choosing $\varepsilon$ small enough. Taking $T$ small enough and remembering that $1\left\|esssim e^{-\gamma^{(n)}}$, we finish the proof of \varepsilonqref{HR+1 gamma 1}. \par\left\|eavevmode\par We now turn to the proof of \varepsilonqref{HR+1 gamma 2} and \varepsilonqref{HR+1 gamma 3} which amounts to estimating $\partial_t\left\|eft(\mathbf{T}^{(n)}\Tilde{\gamma}^{(n+1)}\right\|ight)$, which, thanks to \varepsilonqref{eq gamma tilde}, has the following expression : \begin{align} \partial_t\left\|eft(\mathbf{T}^{(n)}\Tilde{\gamma}^{(n+1)}\right\|ight)&=e^{-2\gamma^{(n)}}N^{(n)}\mathscr{D}elta\Tilde{\gamma}^{(n+1)}-\frac{N^{(n)}\left\|eft(\tau^{(n)}\right\|ight)^2}{2}+\frac{1}{2}e_0^{(n-1)}\left\|eft(\frac{\mathrm{div}(\beta^{(n)})}{N^{(n-1)}}\right\|ight)\\&\qquad+e^{-2\gamma^{(n)}}\frac{\mathscr{D}elta N^{(n)}}{2}+e^{-2\gamma^{(n)}}N^{(n)}\left\|eft|\nabla\varphi^{(n)}\right\|ight|^2\\&\qquad+\frac{1}{4}e^{-2\gamma^{(n)}-4\varphi^{(n)}}N^{(n)} |\nabla\Omegamega^{(n)}|^2+ \alpha \Psi^{(n)}\\&=\vcentcolon I+II+III+IV+V+VI+VII. \varepsilonnd{align} The term $VII$ is handled thanks to \varepsilonqref{op chi ln 3}. For the remainings terms, we first bound their $L^2_{\mathrm{d}elta'+1}$ norms with $C_i$, and then the $H^1_{\mathrm{d}elta'+1}$ norms of their derivatives by $C_i^2$. \begin{itemize} \item For $I$, we first perform the $H^1$ estimate, using \varepsilonqref{useful gamma 1}, $\varepsilon|\chi\left\|n|\left\|esssim \left\|angle x \right\|angle^{\varepsilon}$, \varepsilonqref{HR N 1} and \varepsilonqref{HR+1 gamma 1} : \begin{align*} \| I\|_{H^1_{\mathrm{d}elta'+1}}&\left\|esssim \left\|eft\| N^{(n)}\mathscr{D}elta\Tilde{\gamma}^{(n+1)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\left\|esssim \left\|eft\|\mathscr{D}elta\Tilde{\gamma}^{(n+1)} \right\|ight\|_{H^1_{\mathrm{d}elta'+2}}\left\|eft(1+\left\|eft\|\Tilde{N}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta}} \right\|ight)\left\|esssim C_i. \varepsilonnd{align*} To get the $L^2$ estimate, we simply use the embeddings $H^1_{\mathrm{d}elta'+1}\xhookrightarrow{}L^2_{\mathrm{d}elta'+1}$. \item For $II$, we first use \varepsilonqref{HR N 1} and \varepsilonqref{propsmall tau} : \begin{equation*} \|II\|_{L^2_{\mathrm{d}elta'+1}}\left\|esssim \left\|eft\|\tau^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}^2\left\|eft(1+ \left\| \Tilde{N}^{(n)}\right\|_{H^2_{\mathrm{d}elta}} \right\|ight)\left\|esssim \varepsilon^2. \varepsilonnd{equation*} For the $H^1$ estimate, we use \varepsilonqref{HR N 1}, \varepsilonqref{propsmall tau} and \varepsilonqref{HR tau 1} : \begin{equation*} \|II\|_{H^1_{\mathrm{d}elta'+1}}\left\|esssim \left\|eft\|\tau^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta'+1}}\left\|eft\|\tau^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'+1}}\left\|eft(1+ \left\| \Tilde{N}^{(n)}\right\|_{H^2_{\mathrm{d}elta}} \right\|ight)\left\|esssim \varepsilon A_1C_i. \varepsilonnd{equation*} \item For $III$, we use \varepsilonqref{HR beta 2.5}, \varepsilonqref{HR beta 1} and \varepsilonqref{HR N 2} : \begin{align*} \left\| III \right\|_{L^2_{\mathrm{d}elta'+1}} &\left\|esssim \left\|\nabla\partial_t\beta^{(n)} \right\|_{L^2_{\mathrm{d}elta'+1}} + \left\| \nabla\beta^{(n)}\right\|_{H^1_{\mathrm{d}elta'+1}}\left\|\partial_t\Tilde{N}^{(n-1)}\right\|_{H^2_{\mathrm{d}elta}}+ \left\|\beta^{(n-1)}\right\|_{H^2_{\mathrm{d}elta'}}\left\|\nabla\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'+1}}\\&\left\|esssim C_i. \varepsilonnd{align*} For the $H^1$ estimate, we use : \begin{align*} \left\| III \right\|_{H^1_{\mathrm{d}elta'+1}}& \left\|esssim \left\| \nabla\partial_t\beta^{(n)}\right\|_{H^1_{\mathrm{d}elta'+1}} + \left\|\nabla\beta^{(n)}\partial_t\Tilde{N}^{(n-1)} \right\|_{H^1_{\mathrm{d}elta'+1}} + \left\| \beta^{(n-1)}\nabla\beta^{(n)}\right\|_{H^1_{\mathrm{d}elta'+1}} \\ &\left\|esssim \left\| \nabla\partial_t\beta^{(n)}\right\|_{H^1_{\mathrm{d}elta'+1}} + \left\|\nabla\beta^{(n)}\right\|_{H^1_{\mathrm{d}elta'+1}} \left\| \partial_t\Tilde{N}^{(n-1)}\right\|_{H^2_{\mathrm{d}elta}} +\left\|\beta^{(n-1)} \right\|_{H^2_{\mathrm{d}elta'}} \left\|\nabla\beta^{(n)} \right\|_{H^1_{\mathrm{d}elta'+1}} \\&\left\|esssim A_1 C_i + \varepsilon C_i +\varepsilon^2. \varepsilonnd{align*} \item For $IV$, we recall that $\mathscr{D}elta(\chi\left\|n)$ is a smooth compactly supported function. We only perform the $H^1$ estimate, because the $L^2$ estimate will be a consequence of the embedding $H^1_{\mathrm{d}elta'+1}\xhookrightarrow{}L^2_{\mathrm{d}elta'+1}$. We use \varepsilonqref{useful gamma 1} and \varepsilonqref{HR N 2} : \begin{equation*} \| IV\|_{H^1_{\mathrm{d}elta'+1}} \left\|esssim \left\|eft| N_a^{(n)}\right\|ight|\left\|eft\|\mathscr{D}elta(\chi\left\|n) \right\|ight\|_{H^1}+\left\|eft\|\mathscr{D}elta\Tilde{N}^{(n)}\right\|ight\|_{H^1_{\mathrm{d}elta+1}}\left\|esssim C_i. \varepsilonnd{equation*} \item For $V$, we don't care about the decrease of our functions, thanks to the support property of $\varphi^{(n)}$. We first use \varepsilonqref{useful gamma 1} and the fact that $N^{(n)}\in L^{\infty}(B_{2R})$ (which comes from \varepsilonqref{HR N 1}), the Hölder's inequality and \varepsilonqref{propsmall fi} : \begin{equation*} \| V\|_{L^2}\left\|eq \left\|eft\|N^{(n)} \right\|ight\|_{L^{\infty}(B_{2R})} \left\|eft\|\nabla\varphi^{(n)} \right\|ight\|_{L^4}^2\left\|esssim \varepsilon^2. \varepsilonnd{equation*} For the $H^1$ estimate, we use \varepsilonqref{useful gamma 1}, \varepsilonqref{HR N 1}, \varepsilonqref{HR N 2} and \varepsilonqref{HR fi 1} : \begin{align*} \| \nabla V\|_{L^2}&\left\|esssim \left\|\nabla\varphi^{(n)}\nabla^2\varphi^{(n)} \right\|_{L^2}+ \left\|\nabla\Tilde{N}^{(n)} \left\|eft|\nabla\varphi^{(n)}\right\|ight|^2\right\|_{L^2} \\& \left\|esssim \left\| \nabla^2\varphi^{(n)} \right\|_{H^1} \left\| \nabla\varphi^{(n)}\right\|_{L^4} + \left\| \nabla\Tilde{N}^{(n)}\right\|_{H^2_{\mathrm{d}elta+1}}\left\|eft\|\nabla\varphi^{(n)} \right\|ight\|_{L^4}^2 \\& \left\|esssim\varepsilon A_0C_i. \varepsilonnd{align*} \item For $VI$, we do as for $V$, using in addition \varepsilonqref{useful ffi 1}. \varepsilonnd{itemize} \varepsilonnd{proof} We are now interesting in proving estimates for $\varphi^{(n+1)}$ and $\Omegamega^{(n+1)}$. We first prove their support property : \begin{lem}\left\|abel{support fi n+1} There exists $C_s>0$ such that for $\varepsilon$, $T$ sufficiently small (depending on $R$), $\varphi^{(n+1)}$ is supported in \begin{equation*} \varepsilonnstq{(t,x)\in[0,T]\times\mathbb{R}^2}{\vert x\vert\left\|eq R+C_s(1+R^{\varepsilon})t}. \varepsilonnd{equation*} In particular, choosing $T$ small enough, $\mathrm{supp}(\varphi^{(n+1)})\subset [0,T]\times B_{2R}$. \varepsilonnd{lem} \begin{proof} Since the initial data for $\varphi^{(n+1)}$ and $\partial_t\varphi^{(n+1)}$ are compactly supported and $\Box_{g^{(n)}}\varphi^{(n+1)}$ is compactly supported in \begin{equation*} A\vcentcolon=\varepsilonnstq{(t,x)\in[0,T]\times\mathbb{R}^2}{\vert x\vert \left\|eq R+C_s(1+R^{\varepsilon})t} \varepsilonnd{equation*} we juste have to show that $\partial A$ is a spacelike hypersurface. We set $f(x,t)=-|x|+C_s(1+R^{\varepsilon})t$, in order to have $\partial A=f^{-1}(-R)$. Thus, we have to show that $(g^{(n)})^{-1}(\mathrm{d} f,\mathrm{d} f)$ is non-positive on this hypersurface. We have $\mathrm{d} f=-\frac{x_i}{|x|}\mathrm{d} x^i+C_s(1+R^{\varepsilon})\mathrm{d} t$, which implies : \begin{align*} \left\|eft(g^{(n)}\right\|ight)^{-1}(\mathrm{d} f,\mathrm{d} f)&=\frac{x_ix_j}{|x|^2}\left\|eft(g^{(n)}\right\|ight)^{ij}+C_s^2(1+R^{\varepsilon})^2\left\|eft(g^{(n)}\right\|ight)^{tt}-\frac{2x_i}{|x|}C_s(1+R^{\varepsilon})\left\|eft(g^{(n)}\right\|ight)^{it}\\ &=e^{-2\gamma^{(n)}}-\left\|eft(\frac{x\cdot\beta^{(n)}}{|x|N^{(n)}}\right\|ight)^2-\left\|eft( \frac{C_s(1+R^{\varepsilon})}{N^{(n)}}\right\|ight)^2-\frac{2(x\cdot\beta^{(n)})C_s(1+R^{\varepsilon})}{(N^{(n)})^2|x|} \varepsilonnd{align*} We have $e^{-2\gamma^{(n)}}\left\|esssim \left\|angle x\right\|angle^{2\varepsilon^2}$, $\frac{|x\cdot\beta^{(n)}|}{(N^{(n)})^2|x|}+\left\|eft(\frac{x\cdot\beta^{(n)}}{|x|N^{(n)}}\right\|ight)^2\left\|esssim \varepsilon$, so choosing the parameters appropriately, one easily sees that $(g^{(n)})^{-1}(\mathrm{d} f,\mathrm{d} f)$ is non-positive on the hypersurface. \varepsilonnd{proof} \begin{prop}\left\|abel{hr+1 fi prop} For $n\geq 2$, the following estimates holds : \begin{align} \left\|eft\|\partial_t\varphi^{(n+1)}\right\|ight\|_{H^2}+\left\|eft\|\nabla\varphi^{(n+1)}\right\|ight\|_{H^2} & \left\|esssim C_i ,\left\|abel{esti ondes phi 1}\\ \left\|eft\Vert \partial_t\left\|eft( \mathbf{T}^{(n)}\varphi^{(n+1)} \right\|ight) \right\|ight\Vert_{H^1} & \left\|esssim C_i.\left\|abel{esti ondes phi 2} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} First, note that since $\varphi^{(n+1)}$ is compactly supported in $B_{2R}$ for all time by previous lemma, we do not need to worry about the spatial decay in this proof. We recall the wave equation satisfied by $\varphi^{(n+1)}$ : \begin{align*} \left\|eft( \mathbf{T}^{(n)} \right\|ight)^2\varphi^{(n+1)}-e^{-2\gamma^{(n)}}\mathscr{D}elta \varphi^{(n+1)}&=\frac{e^{-2\gamma^{(n)}}}{N^{(n)}}\nabla \varphi^{(n)}\cdot\nabla N^{(n)}+\frac{\tau^{(n)} e_0^{(n-1)}\varphi^{(n)}}{N^{(n)}}\\&\qquad+\frac{1}{2}e^{-4\varphi^{(n)}}\left\|eft( \left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2+\left\|eft|\nabla\Omegamega^{(n)}\right\|ight|^2 \right\|ight). \varepsilonnd{align*} Using our energy estimate for this wave equation we see that to prove the first part of the proposition we have to bound \begin{equation*} \left\|eft\Vert e^{-2\gamma^{(n)}}\nabla \varphi^{(n)}\cdot\nabla N^{(n)}\right\|ight\Vert_{H^2}+\left\|eft\Vert \tau^{(n)} e_0^{(n-1)}\varphi^{(n)}\right\|ight\Vert_{H^2}+\left\| e^{-4\varphi^{(n)}}N^{(n)}(e_0^{(n-1)}\Omegamega^{(n)})^2 \right\|_{H^2} +\left\| e^{-4\varphi^{(n)}}N^{(n)} |\nabla\Omegamega^{(n)}|^2\right\|_{H^2} . \varepsilonnd{equation*} We mainly use the fact that in dimension 2, $H^2$ is an algebra. Noting that every norm is not taking on the whole space but only on $B_{2R}$, using \varepsilonqref{useful gamma 1} and \varepsilonqref{useful ffi 1} and thanks to the estimates made on the $n$-th iterate, it's easy to see that this quantity is bounded by some constant $C(A_i,C_i)$. We also recall that : \begin{equation*} \mathcal{E}^{(n)}\left\|eft[ \varphi^{(n+1)} \right\|ight](0)\left\|esssim C_i. \varepsilonnd{equation*} Thanks to the Lemma \right\|ef{inegalite d'energie lemme}, if $T$ is small enough, we have for all $t\in[0,T]$ \begin{equation*} \mathcal{E}^{(n)}\left\|eft[ \varphi^{(n+1)} \right\|ight](t)\left\|esssim C_i. \varepsilonnd{equation*} Thanks to the support property of $\varphi^{(n+1)}$, the fact that $1\left\|esssim e^{-\gamma^{(n)}}$ and $|N^{(n)}|\left\|esssim 1$ (on $B_{2R}$), we have : \begin{equation} \left\|\partial_t\varphi^{(n+1)}\right\|_{H^2}+\left\|\nabla\varphi^{(n+1)}\right\|_{H^2}\left\|esssim \mathcal{E}^{(n)}\left\|eft[ \varphi^{(n+1)} \right\|ight]+C_i.\left\|abel{commutation energie} \varepsilonnd{equation} which concludes the proof of \varepsilonqref{esti ondes phi 1}. \par\left\|eavevmode\par We next prove the estimate \varepsilonqref{esti ondes phi 2}. We use the equation satisfied by $\varphi^{(n+1)}$ to express the term we want to estimate: \begin{align*} \partial_t\left\|eft( \mathbf{T}^{(n)}\varphi^{(n+1)} \right\|ight) & =e^{-2\gamma^{(n)}}N^{(n)}\mathscr{D}elta\varphi^{(n+1)}+e^{-2\gamma^{(n)}}\nabla \varphi^{(n)}\cdot\nabla N^{(n)}+\tau^{(n)} e_0^{(n-1)}\varphi^{(n)}+\beta^{(n)}\cdot\nabla\left\|eft( \mathbf{T}^{(n)}\varphi^{(n+1)} \right\|ight)\\&\qquad+\frac{1}{2}e^{-4\varphi^{(n)}}N^{(n)} \left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2+\frac{1}{2}e^{-4\varphi^{(n)}}N^{(n)}\left\|eft|\nabla\Omegamega^{(n)}\right\|ight|^2 \\ & =\vcentcolon I+II+III+IV+V+VI. \varepsilonnd{align*} Thus, it remains to bound those terms by $C_i$ in $H^1$, and the main difficulty is avoiding any $C_i^2$ bound. We mainly use the embedding of $H^1$ in $L^q$ for all $q\geq 2$ and the Hölder inequality, in particular the $L^4\times L^4\xhookrightarrow{}L^2$ and $L^8\times L^8\xhookrightarrow{}L^4$ case (note that in the following we do not write down the factors that are trivially in $L^{\infty}$) : \begin{itemize}[label=\textbullet] \item for $I$, the only issues are the terms where $\Tilde{N}^{(n)}$ or $\Tilde{\gamma}^{(n)}$ get one derivative : \begin{align*} \|I\|_{H^1} & \left\|esssim \Vert\nabla\varphi^{(n+1)}\Vert_{H^2}+\|\nabla\Tilde{N}^{(n)}\mathscr{D}elta\varphi^{(n+1)}\|_{L^2}+\|\nabla\Tilde{\gamma}^{(n)}\mathscr{D}elta\varphi^{(n+1)}\|_{L^2}\\ & \left\|esssim \Vert\nabla\varphi^{(n+1)}\Vert_{H^2}\left\|eft(1+\|\nabla\Tilde{N}^{(n)}\|_{H^1}+\|\nabla\Tilde{\gamma}^{(n)}\|_{H^1}\right\|ight)\\ & \left\|esssim C_i. \varepsilonnd{align*} \item for $II$, we forget about the $\chi\left\|n$ in $N^{(n)}$, which is less problematic than $\Tilde{N}^{(n)}$ : \begin{align*} \|II\|_{H^1} & \left\|esssim\|\nabla\varphi^{(n)}\cdot\nabla N^{(n)}\|_{L^2}+\|\nabla^2\varphi^{(n)}\nabla N^{(n)}\|_{L^2}+\|\nabla\varphi^{(n)}\nabla^2 N^{(n)}\|_{L^2}+\|\nabla\Tilde{\gamma}^{(n)}\nabla\varphi^{(n)}\nabla N^{(n)}\|_{L^2}\\ & \left\|esssim\|\nabla\varphi^{(n)}\|_{L^4}\|\Tilde{N}^{(n)}\|_{H^4}+\|\nabla\varphi^{(n)}\|_{H^2}\|\Tilde{N}^{(n)}\|_{H^2}\left\|eft( 1+\|\Tilde{\gamma}^{(n)}\|_{H^2}\right\|ight)\\ & \left\|esssim C_i. \varepsilonnd{align*} \item for $III$, we use \varepsilonqref{HR tau 1} when no derivatives hits $e_0^{(n-1)}\varphi^{(n)}$ and \varepsilonqref{propsmall tau} when one derivative hits $e_0^{(n-1)}\varphi^{(n)}$ : \begin{align*} \| III\|_{H^1}&\left\|esssim \|\tau^{(n)}\|_{H^2}\left\|eft( \Vert\partial_t\varphi^{(n)}\Vert_{L^4}+\Vert\nabla\varphi^{(n)}\Vert_{L^4}\left\|eft( 1+\left\|eft\| \nabla\beta^{(n-1)}\right\|ight\|_{H^1}\right\|ight) \right\|ight)\\& \quad+\|\tau^{(n)}\|_{H^1}\left\|eft( \|\nabla^2\varphi^{(n)}\|_{H^1}+\|\partial_t\nabla\varphi^{(n)}\|_{H^1}\right\|ight)\\& \left\|esssim C_i. \varepsilonnd{align*} \item for $IV$, we just notice that, applying the same type of arguments as in Proposition \right\|ef{commutation estimate}, it's easy to deduce from the first part of this proof that $\left\|eft\|\mathbf{T}^{(n)}\varphi^{(n+1)}\right\|ight\|_{H^2}\left\|esssim C_i$ : \begin{align*} \|IV\|_{H^1} & \left\|esssim \left\|eft\|\mathbf{T}^{(n)}\varphi^{(n+1)}\right\|ight\|_{H^1}+ \left\|eft\|\nabla\beta^{(n)}\nabla\left\|eft( \mathbf{T}^{(n)}\varphi^{(n+1)} \right\|ight)\right\|ight\|_{L^2}+\left\|eft\|\beta^{(n)}\nabla^2\left\|eft( \mathbf{T}^{(n)}\varphi^{(n+1)} \right\|ight)\right\|ight\|_{L^2}\\ & \left\|esssim \left\|eft\|\mathbf{T}^{(n)}\varphi^{(n+1)}\right\|ight\|_{H^2}\left\|eft( 1+\|\nabla\beta^{(n)}\|_{H^1}\right\|ight)\\ & \left\|esssim C_i. \varepsilonnd{align*} \item for $V$, we use first \varepsilonqref{useful ffi 1} and the fact that $N^{(n)}$ and $\nabla N^{(n)}$ are bounded, and then \varepsilonqref{propsmall fi} and \varepsilonqref{HR omega 1} : \begin{align*} \left\| V\right\|_{H^1} & \left\|esssim \left\|eft( 1 + \left\| \nabla\Tilde{N}^{(n)} \right\|_{H^2} \right\|ight) \left\| \left\|eft(e_0^{(n-1)}\Omegamega^{(n)}\right\|ight)^2 \right\|_{L^2}+ \left\| e_0^{(n-1)}\Omegamega^{(n)}\nabla e_0^{(n-1)}\Omegamega^{(n)} \right\|_{L^2} \\& \left\|esssim C_i \left\| e_0^{(n-1)}\Omegamega^{(n)}\right\|_{L^4}^2+ \left\| e_0^{(n-1)}\Omegamega^{(n)}\right\|_{L^4}\left\|\nabla e_0^{(n-1)}\Omegamega^{(n)}\right\|_{H^1} \\&\left\|esssim \varepsilon^2 C_i+\varepsilon C_i \varepsilonnd{align*} \item for $VI$, we do as for $V$, since $\nabla\Omegamega^{(n)}$ and $e_0^{(n-1)}\Omegamega^{(n)}$ satisfy the same estimates. \varepsilonnd{itemize} \varepsilonnd{proof} \begin{prop}\left\|abel{hr+1 omega prop} For $n\geq 2$, the following estimates holds : \begin{align} \left\|eft\|\partial_t\Omegamega^{(n+1)}\right\|ight\|_{H^2}+\left\|eft\|\nabla\Omegamega^{(n+1)}\right\|ight\|_{H^2} & \left\|esssim C_i ,\left\|abel{esti ondes omega 1}\\ \left\|eft\Vert \partial_t\left\|eft( \mathbf{T}^{(n)}\Omegamega^{(n+1)} \right\|ight) \right\|ight\Vert_{H^1} & \left\|esssim C_i.\left\|abel{esti ondes omega 2} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} The proof of Proposition \right\|ef{hr+1 omega prop} uses the same estimates as the one of Proposition \right\|ef{hr+1 fi prop} (since $\varphi^{(n)}$ and $\Omegamega^{(n)}$ satisfy the same estimates), so we omit the details. \varepsilonnd{proof} Looking at the estimates we proved for the $(n+1)$-th iterate in Propositions \right\|ef{hr+1 N prop}, \right\|ef{hr+1 beta prop}, \right\|ef{HR+1 H prop}, \right\|ef{hr+1 tau prop}, \right\|ef{hr+1 gamma prop}, \right\|ef{hr+1 fi prop} and \right\|ef{hr+1 omega prop} we see that in order to recover the estimates \varepsilonqref{HR N 1}-\varepsilonqref{HR fi 1}, we have to choose the constants $A_0$, $A_1$, $A_2$, $A_3$ and $A_4$ such that $C(A_i)\left\|l A_{i+1}$ for all $i=0,\mathrm{d}ots,3$ and $\varepsilon$ small, depending on the $A_i$ constants. We make such a choice. This concludes the proof of the fact claimed above : the sequence constructed in Section \right\|ef{section iteration scheme} is uniformly bounded. Moreover, the bounds \varepsilonqref{HR N 1}-\varepsilonqref{HR omega 1} hold for every $k\in\mathbb{N}$. and for every $t\in[0,T]$. \subsection{Convergence of the sequence}\left\|abel{Cauchy} In this section, we show that the sequence we constructed in fact converges to a limit in larger functional spaces than those used in the previous sequence, where we only proved boundedness. To this end, we will show that the sequence is a Cauchy sequence. We introduce the following distances, as in \cite{hunluk18} : \begin{align} d_1^{(n)} &\vcentcolon= \left\|eft\|\Tilde{\gamma}^{(n+1)}-\Tilde{\gamma}^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta'}} +\left\|eft\|\partial_t\left\|eft(\Tilde{\gamma}^{(n+1)}-\Tilde{\gamma}^{(n)} \right\|ight) \right\|ight\|_{L^2_{\mathrm{d}elta'}}+\left\|eft\|H^{(n+1)}-H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}}\nonumber\\&\qquad+\left\|eft\|\tau^{(n+1)}-\tau^{(n)} \right\|ight\|_{L^2_{\mathrm{d}elta'+1}}+\left\|eft\|\partial_t \left\|eft(\varphi^{(n+1)}-\varphi^{(n)} \right\|ight) \right\|ight\|_{H^1} +\left\|eft\|\nabla\left\|eft(\varphi^{(n+1)}-\varphi^{(n)}\right\|ight) \right\|ight\|_{H^1} \\&\qquad +\left\|eft\|\partial_t \left\|eft(\Omegamega^{(n+1)}-\Omegamega^{(n)} \right\|ight) \right\|ight\|_{H^1} +\left\|eft\|\nabla\left\|eft(\Omegamega^{(n+1)}-\Omegamega^{(n)}\right\|ight) \right\|ight\|_{H^1},\nonumber\\ d_2^{(n)} &\vcentcolon= \left\|eft| N_a^{(n+1)}-N_a^{(n)}\right\|ight|+\left\|eft\|\Tilde{N}^{(n+1)}-\Tilde{N}^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta}}+\left\|eft\| \beta^{(n+1)}-\beta^{(n)}\right\|ight\|_{H^2_{\mathrm{d}elta'}},\\ d_3^{(n)} &\vcentcolon= \left\|eft\| \partial_t \left\|eft(\mathbf{T}^{(n)}\varphi^{(n+1)}-\mathbf{T}^{(n-1)}\varphi^{(n)} \right\|ight)\right\|ight\|_{L^2}+\left\|eft\| \partial_t \left\|eft(\mathbf{T}^{(n)}\Omegamega^{(n+1)}-\mathbf{T}^{(n-1)}\Omegamega^{(n)} \right\|ight)\right\|ight\|_{L^2},\\ d_4^{(n)} &\vcentcolon= \left\|eft\|e_0^{(n+1)}H^{(n+1)}-e_0^{(n)}H^{(n)} \right\|ight\|_{H^1_{\mathrm{d}elta+1}},\\ d_5^{(n)} &\vcentcolon= \left\|eft\|\partial_t \left\|eft(\mathbf{T}^{(n)}\Tilde{\gamma}^{(n+1)}-\mathbf{T}^{(n-1)}\Tilde{\gamma}^{(n)} \right\|ight) \right\|ight\|_{L^2_{\mathrm{d}elta'}}+\left\|eft\|\partial_t\left\|eft(\tau^{(n+1)}-\tau^{(n)} \right\|ight) \right\|ight\|_{L^2_{\mathrm{d}elta'+1}},\\ d_6^{(n)} &\vcentcolon= \left\|eft|\partial_t\left\|eft(N_a^{(n+1)}-N_a^{(n)} \right\|ight) \right\|ight|+\left\|eft\|\partial_t\left\|eft(\Tilde{N}^{(n+1)}-\Tilde{N}^{(n)} \right\|ight) \right\|ight\|_{H^2_{\mathrm{d}elta}}+\left\|eft\|e_0^{(n)}\beta^{(n+1)}-e_0^{(n-1)}\beta^{(n)} \right\|ight\|_{H^2_{\mathrm{d}elta'}}. \varepsilonnd{align} The goal is to show that each series $\sum_{n\left\|eq 0}d_i^{(n)}$ is converging. This is a consequence of the following Proposition. At this low-level of regularity, its proof is identical to the corresponding one done in \cite{hunluk18} (see Proposition $8.19$ and Corollary $8.20$ in this article). \begin{prop}\left\|abel{distance prop} If $T$ and $\varepsilon$ is small enough (where $\varepsilon$ does not depend on $C_i$), the following bounds hold for every $n\geq 3$ : \begin{align*} d_1^{(n)} + d_2^{(n)} + d_3^{(n)} + d_4^{(n)} + d_5^{(n)} + d_6^{(n)} \left\|esssim 2^{-n}. \varepsilonnd{align*} \varepsilonnd{prop} It shows that in the function spaces involved in the definition of the distances $d^{(n)}_i$, the sequence we constructed is Cauchy and therefore convergent to some \begin{equation}\left\|abel{strong limit} (N=1+N_a\chi\left\|n+\Tilde{N},\tau,H,\beta,\gamma=-\alpha\chi\left\|n+\Tilde{\gamma},\varphi,\Omegamega). \varepsilonnd{equation} Since the sequence is bounded in a smaller space, we can find a subsequence weakly converging to some limit, which has to coincide with the strong limit \varepsilonqref{strong limit}. Consequently, \varepsilonqref{strong limit} satisfies the estimates \varepsilonqref{HR N 1}-\varepsilonqref{HR omega 1}, from which we can prove that it is a solution to the reduced system \varepsilonqref{EQ N}-\varepsilonqref{EQ omega}. If there are two solutions to the reduced system, we can control their difference using the distances $d^{(n)}_i$ and arguing as in Proposition \right\|ef{distance prop} we show that these two solutions coincide. This proves the uniqueness of solution to the reduced system. We summarize this discussion in the following corollary : \begin{coro}\left\|abel{coro reduced sytem} Given the initial conditions in Section \right\|ef{initial data}, there exists a unique solution \begin{equation} (N,\beta,\tau,H,\gamma,\varphi,\Omegamega)\left\|abel{sol} \varepsilonnd{equation} to the reduced system \varepsilonqref{EQ N}-\varepsilonqref{EQ omega} such that : \begin{itemize} \item $\gamma$ and $N$ admit the decompositions \begin{equation*} \gamma=-\alpha\chi\left\|n+\Tilde{\gamma},\qquad N=1+N_a\chi\left\|n+\Tilde{N}, \varepsilonnd{equation*} where $\alpha\geq0$ is a constant, $N_a(t)\geq 0$ is a function of $t$ alone and \begin{equation*} \Tilde{\gamma}\in H^3_{\mathrm{d}elta'}, \quad \mathbf{T}\Tilde{\gamma}\in H^2_{\mathrm{d}elta'+1}, \quad \partial_t\mathbf{T}\Tilde{\gamma}\in H^1_{\mathrm{d}elta'+1}, \quad \Tilde{N}\in H^4_{\mathrm{d}elta}, \quad \partial_t\Tilde{N}\in H^2_{\mathrm{d}elta}, \varepsilonnd{equation*} with estimates depending on $C_i$, $\mathrm{d}elta$ and $R$. \item $(\beta,\tau,H)$ are in the following spaces : \begin{equation*} \beta\in H^3_{\mathrm{d}elta'},\quad e_0\beta\in H^3_{\mathrm{d}elta'},\quad \tau\in H^2_{\mathrm{d}elta'+1},\quad \partial_t\tau\in H^1_{\mathrm{d}elta'+1}, \quad H,e_0H\in H^2_{\mathrm{d}elta'+1}, \varepsilonnd{equation*} with estimates depending on $C_i$, $\mathrm{d}elta$ and $R$. \item The smallness conditions in \varepsilonqref{HR N 1} and \varepsilonqref{HR beta 1} and Proposition \right\|ef{prop propsmall} hold (without the $(n)$). \varepsilonnd{itemize} \varepsilonnd{coro} \section{End of the proof of Theorem \right\|ef{theoreme principal} }\left\|abel{section end of proof} In this section we conclude the proof of Theorem \right\|ef{theoreme principal} in two steps. As a first step, we show that the unique solution of the reduced system obtained in Corollary \varepsilonqref{coro reduced sytem} is actually a solution of the full system \varepsilonqref{EVE}. As we will see in Proposition \right\|ef{Gij et Goo}, this involves among other things propagating the gauge condition $\tau =0$ (the condition $\Bar{g}=e^{2\gamma}\mathrm{d}elta$ is also a gauge condition but we don't need to propagate it). As in the harmonic gauge, this step is done using the Bianchi equation and the constraint equations. While in the harmonic gauge the Bianchi equation implies a second order hyperbolic system for the gauge, here we obtain a transport system (see Proposition \right\|ef{usage de bianchi}). In a second step, we prove the remaining estimates stated in Theorem \right\|ef{theoreme principal}, i.e the $H^4$ norm of the metric coefficients with a loss of one regularity order for each time derivative. For this, we use the full Einstein equations in the elliptic gauge, thanks to the first step. \subsection{Solving the Einstein equations} In order to solve \varepsilonqref{EVE} in the elliptic gauge, it only remains to prove that $G_{\mu\nu}=T_{\mu\nu}$ (the wave equations for $\varphi$ and $\Omegamega$ being already included into the reduced system) and that $\tau=0$. To define properly the tensors $G$ and $T$ we need to define a metric. Let $g$ be the metric on $\mathbb{R}^2\times [0,T]$ defined by the geometric quantities $N$, $\gamma$ and $\beta$ (obtained from \varepsilonqref{sol}) as in \varepsilonqref{metrique elliptique}. To compute the Einstein tensor of $g$, we need the second form fundamental and its traceless part. We define $K$ with $H$, $\gamma$ and $\tau$ (obtained from \varepsilonqref{sol}) according to \varepsilonqref{def H} and \varepsilonqref{g bar}. Thanks to \varepsilonqref{EQ beta} and \varepsilonqref{EQ tau} we have \begin{align*} K_{ij}=H_{ij}+\frac{1}{2}e^{2\gamma}\tau\mathrm{d}elta_{ij} = -\frac{1}{2N}e_0\left\|eft( e^{2\gamma}\right\|ight)\mathrm{d}elta_{ij}+\frac{e^{2\gamma}}{2N}\left\|eft( \partial_i\beta_j+\partial_j\beta_i \right\|ight). \varepsilonnd{align*} By \varepsilonqref{seconde forme fonda}, this proves that $K_{ij}$ is the second fundamental form of $\Sigma_t$. On the other hand, by \varepsilonqref{EQ tau}, we know that $\tau$ is the mean curvature of $\Sigma_t$. This implies that $H_{ij}$ is the traceless part of $K_{ij}$ with respect to $\Bar{g}=e^{2\gamma}\mathrm{d}elta$. We also define the tensor $T$ with $g$ and $(\varphi,\Omegamega)$ (obtained from \varepsilonqref{sol}) according to \varepsilonqref{tenseur energie impulsion }. We can now use both our computations in the elliptic gauge and the reduced system to compute $G_{00}-T_{00}$ and $G_{ij}-T_{ij}$. \begin{prop}\left\|abel{Gij et Goo} Given a solution to \varepsilonqref{EQ N}-\varepsilonqref{EQ omega}, the Einstein tensor in the basis $(e_0,\partial_i)$ is given by : \begin{align} G_{00}&=\frac{N}{2}e_0\tau+T_{00},\left\|abel{G 00}\\ G_{ij}&=\frac{e^{2\gamma}e_0\tau}{2N}\mathrm{d}elta_{ij}+T_{ij}.\left\|abel{G ij} \varepsilonnd{align} Moreover, we have $D^{\mu}T_{\mu\nu}=0$. \varepsilonnd{prop} \begin{proof} In this proof we just have to put together our calculations about $R_{\mu\nu}$ and $T_{\mu\nu}$ perdormed in Appendix \varepsilonqref{appendix A} and the reduced system \varepsilonqref{EQ N}-\varepsilonqref{EQ omega}. Note that putting \varepsilonqref{EQ tau} and \varepsilonqref{EQ gamma} together gives back an elliptic equation satisfied by $\gamma$ : \begin{equation} \mathscr{D}elta\gamma=\frac{\tau^2}{2}e^{2\gamma}-\frac{e^{2\gamma}}{2N}e_0\tau-\frac{\mathscr{D}elta N}{2N}-\left\|eft|\nabla\varphi\right\|ight|^2-\frac{1}{4}e^{-4\varphi}\left\|eft|\nabla\Omegamega \right\|ight|^2.\left\|abel{vraie eq sur gamma} \varepsilonnd{equation} In order to compute $G_{\mu\nu}$, we need the scalar curvature $R$, which, thanks to \varepsilonqref{EQ N}, \varepsilonqref{vraie eq sur gamma} and \varepsilonqref{appendix R}, has the following expression : \begin{equation*} R= -\mathbf{T}\tau+2e^{-2\gamma}|\nabla\varphi|^2-\frac{2}{N^2}(e_0\varphi)^2+\frac{1}{2}e^{-2\gamma-4\varphi}|\nabla\Omegamega|^2-\frac{1}{2N^2}e^{-4\varphi}(e_0\Omegamega)^2. \varepsilonnd{equation*} We also recall the expression of $g_{\mu\nu}$ in the $(e_0,\partial_i)$ basis : $g_{00}=-N^2$, $g_{ij}=e^{2\gamma}\mathrm{d}elta_{ij}$ and $g_{0i}=0$. Since $N$ satisfies \varepsilonqref{EQ N} and thanks to \varepsilonqref{appendix R00} we get : \begin{align*} G_{00}&=R_{00}-\frac{1}{2}g_{00}R\\ &=\frac{N}{2}e_0\tau+(e_0\varphi)^2+N^2e^{-2\gamma}|\nabla\varphi|^2+\frac{1}{4}e^{-4\varphi}\left\|eft( (e_0\Omegamega)^2+e^{-2\gamma}N^2|\nabla\Omegamega|^2\right\|ight), \varepsilonnd{align*} which, looking at \varepsilonqref{T 00}, gives \varepsilonqref{G 00}. Thanks to \varepsilonqref{EQ H} and \varepsilonqref{appendix Rij} we get : \begin{equation*} R_{ij}=\mathrm{d}elta_{ij}\left\|eft(-\mathscr{D}elta\gamma+\frac{\tau^2}{2}e^{2\gamma}-\frac{e^{2\gamma}}{2}\mathbf{T}\tau-\frac{\mathscr{D}elta N}{2N}\right\|ight)+2\partial_i\varphi\partial_j\varphi-\mathrm{d}elta_{ij}|\nabla\varphi|^2+\frac{1}{2}e^{-4\varphi}\partial_i\Omegamega\partial_j\Omegamega-\frac{1}{4}e^{-4\varphi}\mathrm{d}elta_{ij}|\nabla\Omegamega|^2, \varepsilonnd{equation*} which, using \varepsilonqref{vraie eq sur gamma} gives $R_{ij}=2\partial_i\varphi\partial_j\varphi+\frac{1}{2}e^{-4\varphi}\partial_i\Omegamega\partial_j\Omegamega$. It gives us \begin{equation*} G_{ij}=\frac{1}{2}e^{2\gamma}\mathbf{T}\tau\mathrm{d}elta_{ij}+2\partial_i\varphi\partial_j\varphi+\frac{e^{2\gamma}}{N^2}(e_0\varphi)^2\mathrm{d}elta_{ij}-|\nabla\varphi|^2\mathrm{d}elta_{ij}+\frac{1}{4}e^{-4\varphi}\left\|eft( 2\partial_i\Omegamega\partial_j\Omegamega+\frac{e^{2\gamma}}{N^2}(e_0\Omegamega)^2\mathrm{d}elta_{ij}-|\nabla\Omegamega|^2\mathrm{d}elta_{ij}\right\|ight), \varepsilonnd{equation*} which, looking at \varepsilonqref{T ij}, gives \varepsilonqref{G ij}. The conservation law $D^{\mu}T_{\mu\nu}=0$ is just a consequence of \varepsilonqref{EQ ffi}, \varepsilonqref{EQ omega} and \varepsilonqref{divergence de T}. \varepsilonnd{proof} By Proposition \right\|ef{Gij et Goo}, in order to show that a solution to \varepsilonqref{EQ N}-\varepsilonqref{EQ omega} is indeed a solution to \varepsilonqref{EVE} it remains to show that $\tau=0$ and $G_{0i} - T_{0i}=0$. These will be shown simultaneously and the Bianchi identities \begin{equation*} D^{\mu}G_{\mu\nu}=0 \varepsilonnd{equation*} are used in the following proposition to obtain a coupled system for this two quantities. For the sake of clarity, we use the following notations : \begin{equation*} A_i\vcentcolon=G_{0i}-T_{0i},\qquad B_i\vcentcolon=G_{0i}-T_{0i}-\frac{N}{2}\partial_i\tau, \varepsilonnd{equation*} and $\mathrm{div}(A)=\mathrm{d}elta^{ij}\partial_iA_j$. The important remark about these quantities is that if we manage to show that $e_0\tau=0$, $A_i=0$ and $B_i=0$, we first have $G_{0i}-T_{0i}=0$, which, looking at the expression of $B_i$ implies that $\nabla\tau =0$, which, in addition to $e_0\tau=0$ and $\tau_{|\Sigma_0}=0$ implies that $\tau=0$ in the whole space-time. \begin{prop}\left\|abel{usage de bianchi} The quantities $A_i$, $B_i$ and $e_0\tau$ satisfy the following coupled system : \begin{align} e_0A_i &=\frac{N}{2}\partial_ie_0\tau+\frac{\partial_iN}{2}e_0\tau+\left\|eft(\mathbf{T} N+N\tau\right\|ight)A_i+ \partial_i\beta^j A_j ,\left\|abel{coupled system 1}\\ e_0B_i & = \frac{\partial_iN}{2}e_0\tau+N\tau A_i+\mathbf{T} NB_i+\partial_i\beta^jB_j,\left\|abel{coupled system 3}\\ e_0\left\|eft( e_0\tau\right\|ight)&=2e^{-2\gamma}N\mathrm{div}(A)+2e^{-2\gamma}\mathrm{d}elta^{ij}\partial_iNA_j+(2N\tau+\mathbf{T} N) e_0\tau .\left\|abel{coupled system 2} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} The equation \varepsilonqref{coupled system 3} follows from \varepsilonqref{coupled system 1} and we omit the proof, which is a direct computation. Thanks to the previous proposition and the Bianchi identity, we have $D^{\mu}(G_{\mu\nu}-T_{\mu\nu})=0$. We first prove \varepsilonqref{coupled system 1}. By \varepsilonqref{covariante 1} and \varepsilonqref{covariante 2}, \begin{align} D_0(G_{0i}-T_{0i}) & = e_0(G_{0i}-T_{0i})-\mathbf{T} N(G_{0i}-T_{0i})-e^{-2\gamma}\mathrm{d}elta^{jk}N\partial_kN(G_{ij}-T_{ij}) \nonumber\\&\quad-\frac{\partial_iN}{N}(G_{00}-T_{00})-\frac{1}{2}\left\|eft(2\mathrm{d}elta^{j}_ie_0\gamma+\partial_i\beta^j-\mathrm{d}elta_{ik}\mathrm{d}elta^{j\varepsilonll}\partial_{\varepsilonll}\beta^k \right\|ight)(G_{0j}-T_{0j}) \nonumber\\ & =e_0(G_{0i}-T_{0i})-\mathbf{T} N(G_{0i}-T_{0i})-\partial_iNe_0\tau\left\|abel{DoAoi}\\&\quad-\frac{1}{2}\left\|eft(2\mathrm{d}elta^{j}_ie_0\gamma+\partial_i\beta^j-\mathrm{d}elta_{ik}\mathrm{d}elta^{j\varepsilonll}\partial_{\varepsilonll}\beta^k \right\|ight)(G_{0j}-T_{0j}),\nonumber \varepsilonnd{align} where in the last equality we have used \varepsilonqref{G 00} and \varepsilonqref{G ij}. Similarly, by \varepsilonqref{covariante 4} and \varepsilonqref{G 00}-\varepsilonqref{G ij}, \begin{align} g^{jk}D_j(G_{ki}-T_{ki}) = \frac{1}{2}\partial_i\mathbf{T}\tau-\frac{\tau}{N}(G_{0i}-T_{0i})-\frac{1}{2N^2}\left\|eft(2\mathrm{d}elta_i^ke_0\gamma-\mathrm{d}elta_{\varepsilonll}^k\partial_i\beta^{\varepsilonll}-\mathrm{d}elta_{i\varepsilonll}\mathrm{d}elta^{jk}\partial_j\beta^{\varepsilonll} \right\|ight)(G_{0k}-T_{0k}) \left\|abel{DjAji} \varepsilonnd{align} Thanks to $D^{\mu}(G_{\mu i}-T_{\mu i})=0$, we have \begin{equation*} \text{\varepsilonqref{DoAoi}}-N^2\times\text{\varepsilonqref{DjAji}}=0, \varepsilonnd{equation*} which, after some straightforward simplifications, gives exactly \varepsilonqref{coupled system 1}. We now prove \varepsilonqref{coupled system 2}. By \varepsilonqref{covariante 1} and \varepsilonqref{G 00}, \begin{equation} D_0(G_{00}-T_{00})=\frac{N}{2}e_0(e_0\tau)-\frac{e_0 N}{2}e_0\tau-2e^{-2\gamma}\mathrm{d}elta^{ij}N\partial_iN(G_{0j}-T_{0j}).\left\|abel{DoAoo} \varepsilonnd{equation} On the other hand, by \varepsilonqref{covariante 3}-\varepsilonqref{covariante 4} and \varepsilonqref{G 00}-\varepsilonqref{G ij}, \begin{equation} g^{ij}D_i(G_{j0}-T_{j0})=e^{-2\gamma}\mathrm{d}elta^{ij}\partial_{i}(G_{j0}-T_{j0})-e^{-2\gamma}\mathrm{d}elta^{ij}\frac{\partial_iN}{N}(G_{j0}-T_{j0})+\tau e_0\tau.\left\|abel{DiAi0} \varepsilonnd{equation} Thanks to $D^{\mu}(G_{\mu 0}-T_{\mu 0})=0$, we have \begin{equation*} -\frac{1}{N}\times\text{\varepsilonqref{DoAoo}}+N\times\text{\varepsilonqref{DiAi0}}=0, \varepsilonnd{equation*} which, after some straightforward simplifications, gives exactly \varepsilonqref{coupled system 2}. \varepsilonnd{proof} \begin{prop} Suppose the solution to \varepsilonqref{EQ N}-\varepsilonqref{EQ omega} as constructed in Section \right\|ef{section solving the reduced system} arises from initial data with $\tau_{|\Sigma_0}=0$ and that the constraint equations are initially satisfied, then the solution satisfies \begin{align*} \tau&=0,\\ G_{0i}&=T_{0i}. \varepsilonnd{align*} As a consequence, the solution to \varepsilonqref{EQ N}-\varepsilonqref{EQ omega} is indeed a solution to \varepsilonqref{EVE}. \varepsilonnd{prop} \begin{proof} We set the following energy : \begin{equation*} E(t)\vcentcolon=\left\|eft\| e_0\tau\right\|ight\|_{L^2}^2+\sum_{i=1,2}\left\|eft(\left\|eft\|2 e^{-\gamma}A_i\right\|ight\|_{L^2}^2+\left\|eft\| B_i\right\|ight\|_{L^2}^2 \right\|ight). \varepsilonnd{equation*} We first note that $E(0)=0$ because our solution arises from initial date satisfying the constraint equations (which implies that $(G_{0i}-T_{0i})_{|\Sigma_0}=0$) and because $\tau_{|\Sigma_0}=0$. Our goal is to show that $E(t)=0$ for all $t\in[0,T]$. We first multiply \varepsilonqref{coupled system 3} by $B_i$ and sum over $i=1,2$ the two equations we obtain. We integrate over $\mathbb{R}^2$ and write $e_0=\partial_t-\beta\cdot\nabla$ to obtain (after an integration by part on the last term) : \begin{align*} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\sum_{i=1,2}\left\|eft\| B_i\right\|ight\|_{L^2}^2&=\int_{\mathbb{R}^2}\sum_{i=1,2}\frac{\partial_iN}{2}B_ie_0\tau+\int_{\mathbb{R}^2}\sum_{i=1,2}N\tau B_iA_i+\int_{\mathbb{R}^2}\sum_{i=1,2}\mathbf{T} NB_i^2\\&\qquad+\int_{\mathbb{R}^2}\sum_{i=1,2}\partial_i\beta^jB_iB_j-\int_{\mathbb{R}^2}\sum_{i=1,2}\frac{1}{2}\mathrm{div}(\beta)B_i^2. \varepsilonnd{align*} Using Corollary \right\|ef{coro reduced sytem} and Proposition \right\|ef{embedding}, we see that the quantities $\nabla N$, $N\tau$, $\mathbf{T} N$ and $\nabla\beta$ are bounded (for $N\tau$ and $N\partial_iN$, we use the decay property of $\tau$ and $\partial_i N$ to deal with the logarithmic growth of $N$). Using the trick $2ab\left\|eq a^2+b^2$ and the fact that $1\left\|esssim e^{-\gamma}$, we get : \begin{equation} \frac{\mathrm{d}}{\mathrm{d} t}\sum_{i=1,2}\left\|eft\| B_i\right\|ight\|_{L^2}^2\left\|eq C E(t).\left\|abel{coupled energy 1} \varepsilonnd{equation} Similarly, multiplying \varepsilonqref{coupled system 2} by $e_0\tau$ , we get : \begin{align} \frac{\mathrm{d}}{\mathrm{d} t}\left\|eft\|e_0\tau \right\|ight\|_{L^2}^2 & = 4\int_{\mathbb{R}^2}e^{-2\gamma}N\,\mathrm{div}(A)e_0\tau+2\int_{\mathbb{R}^2}\sum_{i=1,2}e^{-2\gamma}e_0\tau\partial_iN A_i+2\int_{\mathbb{R}^2}(2N\tau+\mathbf{T} N)\left\|eft( e_0\tau\right\|ight)^2\nonumber\\&\qquad\qquad-\int_{\mathbb{R}^2}\mathrm{div}(\beta)\left\|eft( e_0\tau\right\|ight)^2 \nonumber\\&=-4\int_{\mathbb{R}^2}e^{-2\gamma}NA_i\partial_ie_0\tau+O(E(t)),\left\|abel{coupled energy 2} \varepsilonnd{align} where we integrated by part the first term and bound the other terms just as we did for $\|B_i\|_{L^2}$, mainly using Corollary \right\|ef{coro reduced sytem}. Now, writing $\partial_t=e_0+\beta\cdot\nabla$ and integrating by part, we get : \begin{align*} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\sum_{i=1,2}\|e^{-\gamma} A_i\|_{L^2}^2&=\int_{\mathbb{R}^2}\sum_{i=1,2}e^{-2\gamma}A_ie_0A_i-\frac{1}{2}\int_{\mathbb{R}^2}\sum_{i=1,2}\mathrm{div}\left\|eft(e^{-2\gamma}\beta\right\|ight)A_i^2+\frac{1}{2}\int_{\mathbb{R}^2}\sum_{i=1,2}\partial_t(-2\Tilde{\gamma})e^{-2\gamma}A_i^2 \\& =\int_{\mathbb{R}^2}\sum_{i=1,2}e^{-2\gamma}A_ie_0A_i+O(E(t)). \varepsilonnd{align*} Using \varepsilonqref{coupled system 1}, we thus get : \begin{align} \frac{\mathrm{d}}{\mathrm{d} t}\sum_{i=1,2}\|2e^{-\gamma} A_i\|_{L^2}^2= 4\int_{\mathbb{R}^2}\sum_{i=1,2}e^{-2\gamma}NA_i\partial_ie_0\tau+O(E(t)).\left\|abel{coupled energy 3} \varepsilonnd{align} Looking at \varepsilonqref{coupled energy 2} and \varepsilonqref{coupled energy 3}, we see that our choice of scaling in the expression of $E(t)$ implies a cancellation and we finally get, recalling \varepsilonqref{coupled energy 1} : \begin{equation*} \frac{\mathrm{d}}{\mathrm{d} t}E(t)\left\|eq CE(t) \varepsilonnd{equation*} which, using the Gronwall's Lemma and $E(0)=0$, implies that $E(t)=0$ for all $t\in [0,T]$, which implies the desired result. \varepsilonnd{proof} \subsection{Improved regularity} To conclude the proof of the Theorem \right\|ef{theoreme principal}, it only remains to prove the bounds stated in this theorem. Notice that some of the estimates are already obtained in Corollary \right\|ef{coro reduced sytem}. This improvement of regularity is due to the fact that we now know that the solution of the reduced system is also a solution to the system \varepsilonqref{EVE}, and therefore all the metric components solves elliptic equations. \begin{lem}\left\|abel{lemme elliptique} The metric components $N$, $\gamma$ and $\beta$ satisfy the following elliptic equations : \begin{align} \mathscr{D}elta N & =e^{-2\gamma}N|H|^2+\frac{2e^{2\gamma}}{N}(e_0\varphi)^2+\frac{e^{2\gamma-4\varphi}}{2N}(e_0\Omegamega)^2,\left\|abel{elliptic N}\\ \mathscr{D}elta\gamma & = -|\nabla\varphi|^2-\frac{1}{4}e^{-4\varphi}|\nabla\Omegamega|^2-\frac{e^{2\gamma}}{N^2}(e_0\varphi)^2-\frac{e^{2\gamma-4\varphi}}{4N^2}(e_0\Omegamega)^2-\frac{e^{-2\gamma}}{2}|H|^2,\left\|abel{elliptic gamma}\\ \mathscr{D}elta\beta^j & =\mathrm{d}elta^{jk}\mathrm{d}elta^{i\varepsilonll}(L\beta)_{ik}\left\|eft(\frac{\partial_{\varepsilonll}N}{2N}-\partial_{\varepsilonll}\gamma \right\|ight)-2\mathrm{d}elta^{kj}e_0\varphi\partial_k\varphi-\frac{1}{2}e^{-4\varphi}\mathrm{d}elta^{kj}e_0\Omegamega\partial_k\Omegamega.\left\|abel{elliptic beta} \varepsilonnd{align} \varepsilonnd{lem} \begin{proof} Since we solved \varepsilonqref{EVE}, we have $R_{00}=T_{00}-g_{00}\mathrm{tr}_{g}T$, which, according to \varepsilonqref{appendix R00} and \varepsilonqref{JSP}, easily implies \varepsilonqref{elliptic N}. Using \varepsilonqref{appendix R00}, \varepsilonqref{appendix R} and the fact that $\tau=0$, we get that \begin{equation*} G_{00}=N^2e^{-2\gamma}\left\|eft(- \mathscr{D}elta\gamma-\frac{e^{-2\gamma}}{2}|H|^2\right\|ight), \varepsilonnd{equation*} Using \varepsilonqref{T 00} and the fact that $G_{00}=T_{00}$ we get \varepsilonqref{elliptic gamma}. The equation $R_{0j}=2e_0\varphi\partial_j\varphi+\frac{1}{2}e^{-4\varphi}e_0\Omegamega\partial_j\Omegamega$ and the fact that $\tau=0$ together with \varepsilonqref{appendix R0j} and \varepsilonqref{appendix beta} gives \varepsilonqref{elliptic beta}. \varepsilonnd{proof} In the following proposition, we state and prove the missing estimates : \begin{prop} Taking $\varepsilon_0$ smaller if necessary, the following estimates hold : \begin{align} \left\|eft\|\Tilde{\gamma}\right\|ight\|_{H^4_{\mathrm{d}elta}}+ \left\|eft\| \beta \right\|ight\|_{H^4_{\mathrm{d}elta'}} & \left\|eq C_h,\left\|abel{improved regularity 1}\\ \left\|eft\|\partial_t\Tilde{\gamma}\right\|ight\|_{H^3_{\mathrm{d}elta}}+ \left\|eft\|\partial_t\Tilde{N}\right\|ight\|_{H^3_{\mathrm{d}elta}}+\left\|eft\| \partial_t\beta \right\|ight\|_{H^3_{\mathrm{d}elta'}} & \left\|eq C_h,\left\|abel{improved regularity 2}\\ \left\|eft|\partial_t^2N_a\right\|ight|+\left\|eft\|\partial_t^2\Tilde{N}\right\|ight\|_{H^2_{\mathrm{d}elta}}+\left\|eft\| \partial_t^2\beta \right\|ight\|_{H^2_{\mathrm{d}elta'}}+\left\|eft\|\partial_t^2\Tilde{\gamma}\right\|ight\|_{H^2_{\mathrm{d}elta}} & \left\|eq C_h.\left\|abel{improved regularity 3} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} The idea is just to apply Corollary \right\|ef{mcowens 2} to the equations \varepsilonqref{elliptic N}-\varepsilonqref{elliptic gamma}-\varepsilonqref{elliptic beta}, after having proven, using the regularity obtained in Corollary \right\|ef{coro reduced sytem}, that the RHS of these equations are in the appropriate spaces. For the estimates involving time derivatives, we proceed in the same way after having differenciated once or twice the equations \varepsilonqref{elliptic N}-\varepsilonqref{elliptic gamma}-\varepsilonqref{elliptic beta}. We omit the details, since the computations are straightforward (mainly because now we don't have to worry about the constants in the estimates). \varepsilonnd{proof} This concludes the proof of Theorem \right\|ef{theoreme principal}. \section{Proof of Theorem \right\|ef{theo 2}}\left\|abel{section theo 2} \subsection{Almost $H^2$ well-posedness} At this stage, thanks to Theorem \right\|ef{theoreme principal}, we proved that the system \varepsilonqref{EVE} is well posed locally in time with initial data $(\partial\varphi,\partial\Omegamega)\in H^2$. The next step would be to consider initial data $(\partial\varphi,\partial\Omegamega)$ which are only in $H^1$. In order to obtain well-posedness in this setting, we could regularize the initial data with a sequence $(\partial\varphi_n,\partial\Omegamega_n)\in H^2$ to which we can apply Theorem \right\|ef{theoreme principal}, thus obtaining a sequence of solution to \varepsilonqref{EVE} on $[0,T_n)$. A priori, if $(\partial\varphi,\partial\Omegamega)$ only belongs to $H^1$, the $H^2$ norm of $(\partial\varphi_n,\partial\Omegamega_n)$ explodes as $n$ tends to $+\infty$ and therefore the sequence $(T_n)_{n\in\mathbb{N}}$ converges to 0, forbidding us to define a limit on some non-trivial interval. To prevent this to happen, we need to prove that the $H^2$ and $L^4$ estimates of each $(\partial\varphi_n,\partial\Omegamega_n)$ can be propagated on some fixed interval using only their $H^1$ norm (which are bounded by the $H^1$ norm of the initial data) using the system that $\varphi$ and $\Omegamega$ solve, i.e the system \varepsilonqref{WM ffi}-\varepsilonqref{WM omega} below. As we will see in the rest of this section, it is possible to improve the $H^2$ norm. But unfortunately, we can't improve the $L^4$ estimates using only the $H^1$ norm and the system \varepsilonqref{WM ffi}-\varepsilonqref{WM omega}. Note that this difficulty already occured in the proof of Theorem \right\|ef{theoreme principal} but we bypassed it by taking advantage of the smallness of the time of existence (see Proposition \right\|ef{prop propsmall}), something that we cannot do in this approximation procedure. Therefore, we can't prove local well-posedness at the $H^2$ level. Instead we prove a blow-up criterium, meaning that the $L^4$ estimates that we can't propagate is assumed to hold from the start. It only remains to improve the $H^2$ estimates. \subsection{The wave map structure} To prove Theorem \right\|ef{theo 2}, we argue by contradiction and assume throughout this section that the following statements both hold on $[0,T)$ : \begin{align} \left\| \partial \varphi \right\|_{H^1} + \left\| \partial\Omegamega \right\|_{H^1}&\left\|eq C_0,\left\|abel{propa H1}\\ \left\| \partial \varphi \right\|_{H^1} + \left\| \partial\Omegamega \right\|_{L^4}&\left\|eq \varepsilon_0,\left\|abel{propa L4} \varepsilonnd{align} for some $C_0>0$, and $\varepsilon_0>0$ defined in Theorem \right\|ef{theoreme principal}, and where $T$ is the maximal time of existence of a solution to \varepsilonqref{EVE}. The goal is to show that we can actually bound the $H^2$ norm of $\partial\varphi$ and $\partial\Omegamega$ on $[0,T)$, and hence up to $T$, using \varepsilonqref{propa H1}. Then, using in addition \varepsilonqref{propa L4} and applying Theorem \right\|ef{theoreme principal}, we construct a solution of \varepsilonqref{EVE} beyond the time $T$. This would contradict the maximality of $T$, and thus prove Theorem \right\|ef{theo 2}. \par\left\|eavevmode\par In order to estimate the $H^2$ norm of $\partial\varphi$ and $\partial\Omegamega$ using \varepsilonqref{propa H1}, we are going to use the wave map structure of the coupled wave equations solved by $\varphi$ and $\Omegamega$, which we recall : \begin{align} \Box_g\varphi&=-\frac{1}{2}e^{-4\varphi}\partial^\right\|ho\Omegamega\partial_\right\|ho\Omegamega,\left\|abel{WM ffi}\\ \Box_g\Omegamega&=4\partial^\right\|ho\Omegamega\partial_\right\|ho\varphi.\left\|abel{WM omega} \varepsilonnd{align} We also recall the expression of the operator $\Box_g$ in the case $\tau=0$ : \begin{equation} \Box_g f= -\mathbf{T}^2f+\frac{e^{-2\gamma}}{N}\mathrm{div}(N\nabla f),\left\|abel{expression de box} \varepsilonnd{equation} where $f$ is any function on $\mathcal{M}$. Note the following notation for the rest of this section : $U$ stands for $\varphi$ or $\Omegamega$, $g$ stands for any metric coefficient, meaning $N$, $\gamma$ and $\beta$. \subsubsection{The naive energy estimate} We want to control the $H^2$ norm of $\partial U$. As $U$ satisfies a wave equation, we could use Lemma \right\|ef{inegalite d'energie lemme}. With our formal notation, this wave equation writes \begin{equation*} \Box_g U= g^{-1}(\partial U)^2. \varepsilonnd{equation*} Thus, Lemma \right\|ef{inegalite d'energie lemme} would basically implies that \begin{align*} \left\| \partial\nabla^2 U \right\|_{L^2}^2 & \left\|esssim C_{high}^2 + \int_0^t \left\| \partial\nabla^2 U \nabla^2\left\|eft( g^{-1}(\partial U)^2\right\|ight) \right\|_{L^1} \\& \left\|esssim C_{high}^2 +\int_0^t\left\|(\partial\nabla^2 U)^2\partial U \right\|_{L^1} + \cdots, \varepsilonnd{align*} where the dots reprensent term easily bounded by $\left\|\partial U \right\|_{H^2}^2$. The problem is that, using only \varepsilonqref{propa H1} and \varepsilonqref{propa L4}, the term $\left\|(\partial\nabla^2 U)^2\partial U \right\|_{L^1}$ cannot be bounded by $\left\|\partial U \right\|_{H^2}^2$, it requires necessarily $\left\|\partial U \right\|_{H^2}^{2+\varepsilonta}$ with $\varepsilonta>0$. Thus, a continuity argument, aiming at proving boundedness in $H^2$, would be impossible to carry out. Therefore, we need to use deeper the structure of the coupled equations \varepsilonqref{WM ffi} and \varepsilonqref{WM omega}. This structure will allows us to define a third order energy, which will have the property of avoiding $\left\|\partial U \right\|_{H^2}^{2+\varepsilonta}$ terms into the energy estimate. \subsubsection{The third order energy} The system \varepsilonqref{WM ffi}-\varepsilonqref{WM omega} has actually more structure than we could expect : it is a wave map system, as shown in \cite{Malone}. More precisely, if we consider the map $u=(\varphi,\Omegamega)$, then $u$ is an harmonic map from $([0,T)\times \mathbb{R}^3,g)$ to $(\mathbb{R}^2, h)$ with $h$ being the following metric : \begin{equation*} 2(\mathrm{d} x)^2 + \frac{1}{2}e^{-4x}(\mathrm{d} y)^2. \varepsilonnd{equation*} For those wave map systems, Choquet-Bruhat in \cite{CBwavemaps} noted that we can define a third order energy, which in our case is \begin{equation*} \mathscr{E}_3\vcentcolon=\mathscr{E}_3^\varphi+\mathscr{E}_3^\Omegamega, \varepsilonnd{equation*} with \begin{align*} \mathscr{E}_3^\varphi & \vcentcolon= \int_{\mathbb{R}^2} 2 \left\|eft[ \frac{1}{N^2}\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight)^2 +e^{-2\gamma}\left\|eft|\nabla\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega \nabla\Omegamega \right\|ight|^2\right\|ight]\mathrm{d} x, \\ \mathscr{E}_3^\Omegamega &\vcentcolon = \int_{\mathbb{R}^2}\frac{1}{2}e^{-4\varphi}\left\|eft[ \frac{1}{N^2}\left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2\partial_j\partial_i\varphi e_0\Omegamega \right\|ight)^2 \right\|ight.\\&\qquad\qquad\qquad\qquad\qquad \left\|eft.+e^{-2\gamma}\left\|eft|\nabla\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \nabla\varphi-2\partial_j\partial_i\varphi \nabla\Omegamega \right\|ight|^2\right\|ight] \mathrm{d} x . \varepsilonnd{align*} Our goal is to show that we can estimate $\mathscr{E}_3$ by $\left\|\partial U \right\|_{H^2}^{2}$. We start by commuting $\Box_g$ with $\partial_i\partial_j$ to obtain : \begin{align} \Box_g\partial_j\partial_i\varphi+e^{-4\varphi}g^{\alpha\beta}\partial_\alpha\partial_j\partial_i\Omegamega\partial_\beta\Omegamega & = F^\varphi_{ij}\left\|abel{WM dd ffi},\\ \Box_g\partial_j\partial_i\Omegamega - 4g^{\alpha\beta}\partial_\alpha\partial_j\partial_i\Omegamega\partial_\beta\varphi- 4g^{\alpha\beta}\partial_\alpha\Omegamega\partial_\beta\partial_j\partial_i\varphi&= F^\Omegamega_{ij},\left\|abel{WM dd omega} \varepsilonnd{align} where we set \begin{align} F^\varphi_{ij} & \vcentcolon= \left\|eft[\Box_g ,\partial_j\partial_i \right\|ight]\varphi+-\frac{1}{2}\partial_i\partial_j\left\|eft(e^{-4\varphi}g^{\alpha\beta} \right\|ight)\partial_\alpha\Omegamega\partial_\beta\Omegamega\nonumber \\&\qquad\qquad\qquad-\partial_{(i}\left\|eft(e^{-4\varphi}g^{\alpha\beta} \right\|ight)\partial_\alpha\partial_{j)}\Omegamega\partial_\beta\Omegamega - e^{-4\varphi}g^{\alpha\beta}\partial_\alpha\partial_i\Omegamega\partial_\beta\partial_j\Omegamega,\left\|abel{F ffi}\\ F^\Omegamega_{ij} & \vcentcolon=\left\|eft[\Box_g ,\partial_j\partial_i \right\|ight]\Omegamega+ 4\partial_i\partial_jg^{\alpha\beta}\partial_\alpha\Omegamega\partial_\beta\varphi\nonumber\\&\qquad +4\partial_{(i}g^{\alpha\beta}\partial_\alpha\Omegamega\partial_\beta\partial_{j)}\varphi+4\partial_{(i}g^{\alpha\beta}\partial_\alpha\partial_{j)}\Omegamega\partial_\beta\varphi+4g^{\alpha\beta}\partial_\alpha\partial_{(i}\Omegamega\partial_\beta\partial_{j)}\varphi.\left\|abel{F omega} \varepsilonnd{align} We also define the following quantity : \begin{align}\left\|abel{def Reste} \mathscr{R}&\vcentcolon = \left\|\partial_t\gamma \right\|_{L^\infty}\left\|eft(\left\|\partial \nabla^2 U\right\|_{L^2}^2 + \left\|\partial U\nabla^2 U \right\|_{L^2}^2\right\|ight)\nonumber \\& \quad \; +\left\|eft(\left\|\partial \nabla^2 U\right\|_{L^2} + \left\|\partial U\nabla^2 U \right\|_{L^2}\right\|ight) \left\|eft( \left\|\nabla ^2 U (\partial U)^2 \right\|_{L^2}+\left\| (\partial U)^3\right\|_{L^2}+\left\| F^U\right\|_{L^2}\right\|ight.\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left\|eft.+\left\| (\partial\nabla U)^2 \right\|_{L^2} +\left\|\nabla g\nabla^3U \right\|_{L^2} + \left\|\nabla g\nabla U\nabla^2 U \right\|_{L^2} \right\|ight) \nonumber \varepsilonnd{align} where by $F^U$ we mean either $F^\varphi$ or $F^\Omegamega$. For clarity, the computations for the time derivative of the energy $\mathscr{E}_3$ are done in Appendix \right\|ef{appendix C}, where we prove the following proposition. \begin{prop}\left\|abel{dernier coro} The energy $\mathscr{E}_3$ satisfies \begin{equation*} \frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3=O(\mathscr{R}(t)). \varepsilonnd{equation*} \varepsilonnd{prop} This proposition shows the interest of the energy $\mathscr{E}_3$ : its time derivative do not include terms of the form $\left\|(\partial\nabla^2U)^2\partial U \right\|_{L^1}$, unlike the usual energy estimate of Lemma \right\|ef{inegalite d'energie lemme}. \subsection{Continuity argument} Before starting the continuity argument, we need to show that $\mathscr{R}$ can be bounded by $\left\| \partial U \right\|_{H^2}^2$ (Lemmas \right\|ef{estimation F} and \right\|ef{majorer R}) and to compare $\mathscr{E}_3$ with $\left\| \partial U \right\|_{H^2}^2$ (Lemma \right\|ef{comparaison}). To this end, we will use the following key estimates : \begin{align} \left\| u \right\|_{L^4}&\left\|esssim \left\| u\right\|_{L^2}^{\frac{1}{2}}\left\| u\right\|_{H^1}^{\frac{1}{2}},\left\|abel{estimate A}\\ \left\| u\right\|_{L^\infty}&\left\|esssim \left\| u\right\|_{L^2}^{\frac{1}{2}}\left\| \nabla^2u\right\|_{L^2}^{\frac{1}{2}}.\left\|abel{estimate B} \varepsilonnd{align} Both are consequences of the Gagliardo-Nirenberg interpolation inequality, see Proposition \right\|ef{GN}. We will use without mention the fact that $\left\|\varphi\right\|_{L^2}+ \left\| \Omegamega\right\|_{L^2}\left\|esssim \left\|\varphi\right\|_{L^4}+ \left\| \Omegamega\right\|_{L^4}\left\|esssim \varepsilon_0$ (since $\varphi$ and $\Omegamega$ are compactly supported function and because of \varepsilonqref{propa L4}), and also the fact that $\left\| g\right\|_{H^2}\left\|esssim \varepsilon_0$. We also need to estimate $\nabla^3g$. For this, we apply the usual elliptic estimate to the equation $\mathscr{D}elta g = (\nabla g)^2 + (\partial U)^2$ (this is the type of equations solved by the metric coefficients in the elliptic gauge, see Lemma \right\|ef{lemme elliptique}). It first gives : \begin{align*} \left\| \nabla g \right\|_{W^{2,\frac{4}{3}}} & \left\|esssim \left\| \nabla^2 g \nabla g \right\|_{L^\frac{4}{3}} + \left\| \partial \nabla U \partial U \right\|_{L^\frac{4}{3}} \\& \left\|esssim \left\|\nabla g \right\|_{H^1}^2 + \left\|\partial U \right\|_{H^1}^2 \varepsilonnd{align*} where we used Hölder's inequality $L^2\times L^4 \xhookrightarrow{} L^\frac{4}{3}$ and the embedding $H^1\xhookrightarrow{}L^4$. The embedding $W^{2,\frac{4}{3}}\xhookrightarrow{}L^\infty$ then gives : \begin{equation} \left\| \nabla g \right\|_{L^\infty} \left\|esssim \varepsilon_0^2 + C_0^2. \left\|abel{L infini dg} \varepsilonnd{equation} The $L^2$ elliptic estimate implies : \begin{align*} \left\| \nabla g \right\|_{H^2} & \left\|esssim \left\| \nabla^2 g \nabla g \right\|_{L^2} + \left\| \partial \nabla U \partial U \right\|_{L^2} \\ & \left\|esssim \varepsilon_0 \left\| \nabla g \right\|_{H^2} + \varepsilon_0 C_0 \left\| \partial U\right\|_{H^2}^\frac{1}{2} \varepsilonnd{align*} where we used $\left\| g \right\|_{H^2}\left\|esssim \varepsilon_0$, the Hölder's inequality and \varepsilonqref{estimate A}. Taking $\varepsilon_0$ small enough this gives : \begin{equation} \left\| \nabla g \right\|_{H^2}\left\|esssim C(C_0)\left\|\partial U \right\|_{H^2}^\frac{1}{2}. \left\|abel{H 2 dg} \varepsilonnd{equation} In the sequel, we will commute without mention $\partial$ and $\nabla$ since $[e_0,\nabla]=\nabla\beta \nabla$ and $\nabla\beta$ can be bounded using \varepsilonqref{L infini dg}. \subsubsection{The energy $\mathscr{R}$} We start by the estimates for $F^U$ : \begin{lem}\left\|abel{estimation F} There exists $C(C_0)>0$ such that \begin{align*} \left\| F^\varphi_{ij}\right\|_{L^2}+\left\| F^\Omegamega_{ij}\right\|_{L^2}& \left\|esssim C(C_0)\left\|\partial U \right\|_{H^2}. \varepsilonnd{align*} \varepsilonnd{lem} \begin{proof} The expressions of $F^\varphi_{ij}$ and $F^\Omegamega_{ij}$ are given by \varepsilonqref{F ffi} and \varepsilonqref{F omega}. We start by estimate the commutator $[\Box_g,\nabla^2]U$. Looking at the expression \varepsilonqref{expression de box}, we start by the spatial part of $\Box_g$ : \begin{align*} \left\| \left\|eft[ \nabla^2, g\nabla (g\nabla\cdot) \right\|ight] U \right\|_{L^2} & \left\|esssim \left\| g\nabla^3 g \nabla U \right\|_{L^2} + \left\| \nabla g \nabla^2 g \nabla U \right\|_{L^2} + \left\| g\nabla^2 g \nabla^2 U \right\|_{L^2} + \left\| (\nabla g)^2 \nabla^2 U \right\|_{L^2} \\&\quad + \left\| g\nabla g \nabla^3 U \right\|_{L^2}. \varepsilonnd{align*} For the last two terms, we simply bound $\nabla g$ using \varepsilonqref{L infini dg} and put $\partial U$ in $H^2$ : \begin{align*} \left\| (\nabla g)^2 \nabla^2 U \right\|_{L^2} + \left\| g\nabla g \nabla^3 U \right\|_{L^2} & \left\|esssim C(C_0) \left\|eft( \left\| \nabla^2 U \right\|_{L^2} + \left\| \nabla^3 U \right\|_{L^2} \right\|ight). \varepsilonnd{align*} We do the same for the second term, using in addition $\left\|\nabla^2 g \right\|_{L^2}\left\|esssim 1$ and the embedding $H^2\xhookrightarrow{}L^\infty$ : \begin{align*} \left\| \nabla g \nabla^2 g \nabla U \right\|_{L^2} & \left\|esssim C(C_0) \left\| \nabla U \right\|_{L^\infty}. \varepsilonnd{align*} We deal with the third term using first the Hölder's inequality, the embedding $H^1\xhookrightarrow{}L^4$, \varepsilonqref{H 2 dg} and \varepsilonqref{estimate A} : \begin{align*} \left\| g\nabla^2 g \nabla^2 U \right\|_{L^2} & \left\|esssim \varepsilon_0 \left\| \nabla^2g \right\|_{H^1} \left\| \nabla^2U \right\|_{L^4} \\& \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2}^{\frac{1}{2}} \left\| \nabla^2 U \right\|_{H^1}^{\frac{1}{2}}. \varepsilonnd{align*} For the first term, we put $\nabla^3g$ in $L^2$ and $\nabla U$ in $L^\infty$, and then use \varepsilonqref{H 2 dg} and \varepsilonqref{estimate B} : \begin{align*} \left\| g\nabla^3 g \nabla U \right\|_{L^2} & \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2}^{\frac{1}{2}} \left\| \nabla U \right\|_{H^2}^{\frac{1}{2}}. \varepsilonnd{align*} Summarizing everything we obtain : \begin{equation} \left\| \left\|eft[ \nabla^2, g\nabla (g\nabla\cdot) \right\|ight] U \right\|_{L^2} \left\|esssim C(C_0) \left\|\partial U \right\|_{H^2}.\left\|abel{spatial} \varepsilonnd{equation} We now estimate the contribution of $\mathbf{T}^2$ to the commutator. We have : \begin{align*} \left\| \left\|eft[ \nabla^2, \mathbf{T}^2 \right\|ight] U \right\|_{L^2} & \left\|esssim \left\| \nabla^2 g \partial_t^2 U \right\|_{L^2} + \left\| \nabla g \nabla \partial_t^2 U \right\|_{L^2} + \left\|\nabla^2\partial_t g\partial U \right\|_{L^2} +\left\|\nabla^2g \nabla \partial U \right\|_{L^2} + \left\|\nabla g \nabla^2\partial U \right\|_{L^2} \varepsilonnd{align*} The last two terms have already been estimated during the proof of \varepsilonqref{spatial}. For the first two terms, we use the equation $\Box_g U= (\partial U)^2$ satisfied by $U$ to express $\partial_t^2 U$. It shows that \begin{align} |\partial_t^2 U| & \left\|esssim \left\|eft( 1 + |\nabla g| \right\|ight) |\partial U |^2 + |g\nabla^2U| \left\|esssim C(C_0) \left\|eft( |\partial U |^2 + |\nabla^2U|\right\|ight)\left\|abel{dt2} \varepsilonnd{align} where we also used \varepsilonqref{L infini dg}. We can put $\nabla^2g$ in $L^2$ and $\partial U$ in $L^\infty$ using \varepsilonqref{estimate B} (note that the second term has already been estimated) : \begin{align*} \left\| \nabla^2 g \partial_t^2 U \right\|_{L^2} & \left\|esssim \left\|\nabla^2g (\partial U)^2\right\|_{L^2} + \left\| \nabla^2g \nabla^2 U\right\|_{L^2} \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2} \varepsilonnd{align*} The equation $\Box_g U= (\partial U)^2$ also gives us \begin{equation*} |\nabla \partial_t^2 U|\left\|esssim |\nabla \partial U \partial U | + |\nabla^2 g \nabla U| + |\nabla^3 U| + |\nabla g \nabla^2 U|. \varepsilonnd{equation*} Because of \varepsilonqref{L infini dg}, $\left\| \nabla g \nabla \partial_t^2 U \right\|_{L^2} \left\|esssim C(C_0)\left\| \nabla \partial_t^2 U \right\|_{L^2} $ and the previous estimate shows therefore that all the terms in $\left\| \nabla g \nabla \partial_t^2 U \right\|_{L^2} $ have already been estimated. This gives : \begin{equation*} \left\| \nabla g \nabla \partial_t^2 U \right\|_{L^2} \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2}. \varepsilonnd{equation*} It remains to deal with the term involving $\partial_t g$. This quantity satisfies the following equation : \begin{equation*} \mathscr{D}elta \partial_t g = \nabla \partial_t g \nabla g + \partial_t^2 U \nabla U + \partial_t U \nabla \partial_t U. \varepsilonnd{equation*} The usual elliptic estimates gives us \begin{align*} \left\|\partial_t g \right\|_{H^2} & \left\|esssim \left\| \nabla \partial_t g \nabla g \right\|_{L^2} + \left\|\partial_t^2 U \nabla U \right\|_{L^2} + \left\|\partial_t U \nabla \partial_t U \right\|_{L^2} \\& \left\|esssim \varepsilon_0 \left\| \nabla \partial_t g \right\|_{H^1} + C(C_0) \left\| \partial U \right\|_{H^2}^\frac{1}{2} \varepsilonnd{align*} where we used \varepsilonqref{estimate A} and \varepsilonqref{estimate B}. Taking $\varepsilon_0$ small enough, this shows that $\left\|\partial_t g \right\|_{H^2}\left\|esssim C(C_0) \left\| \partial U \right\|_{H^2}^\frac{1}{2}$. With this, we estimate the remaining term in the commutator $\left\|eft[ \nabla^2,\mathbf{T}^2\right\|ight]$ using in addition \varepsilonqref{estimate B} : \begin{align*} \left\|\nabla^2\partial_t g\partial U \right\|_{L^2} & \left\|esssim \left\| \nabla^2\partial_t g \right\|_{L^2} \left\| \partial U \right\|_{L^\infty} \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2} \varepsilonnd{align*} Thus, we obtain : \begin{equation} \left\| \left\|eft[ \nabla^2, \mathbf{T}^2 \right\|ight] U \right\|_{L^2} \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2}.\left\|abel{time} \varepsilonnd{equation} Putting \varepsilonqref{spatial} and \varepsilonqref{time} together we finally obtain : \begin{equation*} \left\| \left\|eft[ \Box_g, \nabla^2 \right\|ight] U \right\|_{L^2} \left\|esssim C(C_0) \left\| \partial U \right\|_{H^2}. \varepsilonnd{equation*} The lemma is actually proved because all the remaining terms in $F^U$ have already been estimated in the proof of \varepsilonqref{spatial} and \varepsilonqref{time}. \varepsilonnd{proof} We now estimate $\mathscr{R}$ : \begin{lem}\left\|abel{majorer R} There exists $C'(C_0)>0$ such that \begin{equation*} \mathscr{R}\left\|eq C'(C_0) \left\| \partial U \right\|_{H^2}^2. \varepsilonnd{equation*} \varepsilonnd{lem} \begin{proof} First, note that the previous lemma handles the term $\left\| F^U \right\|_{L^2}$. Most of the remaining terms in $\mathscr{R}$ can simply be estimated using the Hölder's inequality, \varepsilonqref{estimate A} and \varepsilonqref{estimate B} : \begin{align*} \left\|\partial U \nabla^2 U \right\|_{L^2} & \left\|esssim C_0 \left\|\partial U \right\|_{H^2}, \\ \left\| \nabla^2U (\partial U)^2\right\|_{L^2} & \left\|esssim \varepsilon_0 C_0 \left\|\partial U \right\|_{H^2}, \\ \left\| (\partial U)^3\right\|_{L^2} & \left\|esssim \varepsilon_0^2 \left\| \partial U \right\|_{H^2}, \\ \left\| (\partial\nabla U)^2 \right\|_{L^2} & \left\|esssim C_0 \left\| \partial U \right\|_{H^2}, \\ \left\| \nabla g \nabla U \nabla^2 U \right\|_{L^2} & \left\|esssim \varepsilon_0C(C_0)\left\|\partial U \right\|_{H^2} . \varepsilonnd{align*} Let us give the details only for the last one. We bound $\nabla g$ with \varepsilonqref{L infini dg}, and then we the Hölder's inequality $L^4 \times L^4\xhookrightarrow{} L^2$ and the embedding $H^1\xhookrightarrow{} L^4$ : \begin{align*} \left\| \nabla g \nabla U \nabla^2 U \right\|_{L^2} & \left\|esssim \left\| \nabla g \right\|_{L^\infty} \left\| \nabla U \right\|_{L^4} \left\| \nabla^2 U \right\|_{L^4}\left\|esssim \varepsilon_0C(C_0)\left\|\partial U \right\|_{H^2} . \varepsilonnd{align*} Because of \varepsilonqref{appendix tau} and the gauge condition $\tau=0$ we have $|\partial_t\gamma|\left\|esssim |\nabla g|$ so we estimate $\left\| \partial_t\gamma \right\|_{L^\infty}$ with \varepsilonqref{L infini dg}. Samewise with \varepsilonqref{L infini dg} we estimate the very last term appearing in $\mathscr{R}$ : \begin{align*} \left\| \nabla g \nabla^3 U \right\|_{L^2} \left\|esssim (\varepsilon_0^2+C_0^2) \left\| \partial U \right\|_{H^2}. \varepsilonnd{align*} \varepsilonnd{proof} In the next lemma, we compare $\mathscr{E}_3$ with the $H^2$ norm of $\partial U$. We omit the proof since all the terms involved have been already estimated in the two previous lemmas. \begin{lem}\left\|abel{comparaison} There exists $K(C_0)>0$ such that \begin{align*} \mathscr{E}_3& \left\|eq K(C_0) \left\| \partial U \right\|_{H^2}^2,\\ \left\| \nabla^2\partial U \right\|_{L^2}^2& \left\|eq K(C_0)\mathscr{E}_3 +\varepsilon_0^2K(C_0) \left\|\partial U \right\|_{H^2}^2. \varepsilonnd{align*} \varepsilonnd{lem} \subsubsection{Conclusion} Putting everything together, we can now complete the continuity argument by propagating the $H^2$ regularity. We consider the following bootstrap assumption : \begin{equation} \left\| \partial U \right\|_{H^2}(t) \left\|eq C_1\varepsilonxp(C_1t),\left\|abel{bootstrap H2} \varepsilonnd{equation} with $C_1>0$ to be chosen later. Let $T_0<T$ be the maximal time such that \varepsilonqref{bootstrap H2} holds for all $0\left\|eq t\left\|eq T_0$. Note that if $C_1$ is large enough we have $T_0>0$, since $\partial\varphi$ and $\partial\Omegamega$ are initially in $H^2$. \begin{prop} If $\varepsilon_0$ is small enough (still independent of $C_{high}$) and $C_1$ is large enough, the following holds on $[0,T_0]$ : \begin{equation} \left\| \partial U \right\|_{H^2}(t) \left\|eq \frac{1}{2}C_1\varepsilonxp(C_1t). \varepsilonnd{equation} \varepsilonnd{prop} \begin{proof} The $H^1$ norm of $\partial U$ is already controled, so it suffices to prove the bound stated in the proposition for $\left\| \nabla^2\partial U \right\|_{L^2}$. For this, we use the Proposition \right\|ef{dernier coro}, which implies that for $t\in[0,T_0]$ (we also use Lemma \right\|ef{comparaison} and \varepsilonqref{bootstrap H2}) : \begin{align*} \left\| \nabla^2\partial U \right\|_{L^2}^2(t) & \left\|eq K^2\left\|\partial U \right\|_{H^2}^2(0)+\varepsilon_0^2K \left\|\partial U \right\|_{H^2}^2(t)+CK\int_0^t\mathscr{R}(s)\mathrm{d} s \\&\left\|eq K^2C_{high}^2+\varepsilon_0^2K C_1^2\varepsilonxp(2C_1t)+CK\int_0^t\mathscr{R}(s)\mathrm{d} s, \varepsilonnd{align*} for some $C>0$ given by Proposition \right\|ef{dernier coro}. We now use Lemma \right\|ef{majorer R} : \begin{align*} \left\| \nabla^2\partial U \right\|_{L^2}^2(t) & \left\|eq K^2C_{high}^2+\varepsilon_0^2K C_1^2\varepsilonxp(2C_1t)+CKC'(C_0)\int_0^t\left\|\partial U \right\|_{H^2}^2(s)\mathrm{d} s \\& \left\|eq K^2C_{high}^2+\varepsilon_0^2K C_1^2\varepsilonxp(2C_1t) + \frac{1}{2}CKC'(C_0)C_1\varepsilonxp(2C_1t). \varepsilonnd{align*} We now choose $C_1\geq \max\left\|eft(3CKC'(C_0),\sqrt{6}KC_{high}\right\|ight)$ and $\varepsilon_0\left\|eq \frac{1}{\sqrt{6K}}$, so that each term of the previous inequality is bounded by $\frac{1}{6}C_1^2\varepsilonxp(2C_1t)$. This concludes the proof. \varepsilonnd{proof} By continuity of the quantities involved, the previous proposition contradicts the maximality of $T_0$, and thus proves that $T_0=T$. As explained at the beginning of Section \right\|ef{section theo 2}, this concludes the proof of Theorem \right\|ef{theo 2}. \appendix \section{Computations in the elliptic gauge}\left\|abel{appendix A} In this section, we collect some computations for the spacetime metric in the elliptic gauge defined in Section \right\|ef{section geometrie}. See also \cite{hunluk18}. \subsection{Connection coefficients}\left\|abel{connection coefficients} The 2+1 metric $g$ has the form \begin{equation*} g=-N^2\mathrm{d} t+\Bar{g}_{ij}\left\|eft(\mathrm{d} x^i+\beta^i\mathrm{d} t\right\|ight)\left\|eft(\mathrm{d} x^j+\beta^j\mathrm{d} t\right\|ight), \varepsilonnd{equation*} with $\Bar{g}=e^{2\gamma}\mathrm{d}elta$. In the basis $(e_0,\partial_i)$, we have $g_{00}=-N^2$, $g_{0i}=0$ and $g_{ij}=e^{2\gamma}\mathrm{d}elta_{ij}$, which gives $\mathrm{d}et(g)=-e^{4\gamma}N^2$. In the basis $(\partial_t,\partial_i)$ we have : \begin{equation}\left\|abel{inverse de g} g^{-1}=\frac{1}{N^2} \begin{pmatrix} -1 & \beta^1 & \beta^2 \\ \beta^1 & N^2e^{-2\gamma}-\left\|eft(\beta^1\right\|ight)^2 & -\beta^1\beta^2 \\ \beta^2 & -\beta^1\beta^2 & N^2e^{-2\gamma}-\left\|eft(\beta^2\right\|ight)^2 \varepsilonnd{pmatrix} \varepsilonnd{equation} This allows us to compute $\Box_gh$ for $h$ a function on $\mathcal{M}$ : \begin{prop}\left\|abel{appendix box} If $h$ is a function on $\mathcal{M}$, we have \begin{align*} \Box_gh & =-\mathbf{T}^2h+\frac{e^{-2\gamma}}{N}\mathrm{div}(N\nabla h)+\tau\mathbf{T} h \\&=-\mathbf{T}^2h+e^{-2\gamma}\mathscr{D}elta h+\frac{e^{-2\gamma}}{N}\nabla h\cdot\nabla N+\tau\mathbf{T} h. \varepsilonnd{align*} \varepsilonnd{prop} \begin{proof} By definition of $\Box_g$, we have : \begin{align*} \Box_g h & = \frac{1}{\sqrt{|\mathrm{d}et(g)|}}\partial_\beta\left\|eft(g^{\beta\alpha}\sqrt{|\mathrm{d}et(g)|}\partial_\alpha h \right\|ight) \\& = \frac{e^{-2\gamma}}{N}\partial_t \left\|eft(\frac{e^{2\gamma}}{N} \left\|eft( -\partial_t h +\beta^1\partial_1h+\beta^2\partial_2h \right\|ight) \right\|ight) \\& \quad + \frac{e^{-2\gamma}}{N} \partial_1 \left\|eft( \frac{e^{2\gamma}}{N} \left\|eft( \beta^1\partial_t h+\left\|eft(N^2e^{-2\gamma}-\left\|eft(\beta^1\right\|ight)^2\right\|ight) \partial_1 h -\beta^1\beta^2\partial_2h \right\|ight) \right\|ight) \\& \quad + \frac{e^{-2\gamma}}{N} \partial_2 \left\|eft( \frac{e^{2\gamma}}{N} \left\|eft( \beta^2\partial_t h -\beta^1\beta^2\partial_1h+\left\|eft(N^2e^{-2\gamma}-\left\|eft(\beta^2\right\|ight)^2\right\|ight) \partial_2 h \right\|ight) \right\|ight), \varepsilonnd{align*} where we used the expression of $g^{-1}$ in the basis $(\partial_t,\partial_i)$ (see expression \varepsilonqref{inverse de g}). By rearranging the terms, we get : \begin{align*} \Box_g h & =- \frac{1}{N}\partial_t\mathbf{T} h -\frac{2\partial_t\gamma}{N}\mathbf{T} h + \frac{e^{-2\gamma}}{N}\mathrm{div}\left\|eft( e^{2\gamma}\mathbf{T} h\beta + N \nabla h \right\|ight) \\& = -\mathbf{T}^2h+\frac{e^{-2\gamma}}{N}\mathrm{div}(N\nabla h)+\left\|eft(-2\mathbf{T}\gamma+\frac{\mathrm{div}(\beta)}{N} \right\|ight)\mathbf{T} h. \varepsilonnd{align*} This proves the proposition, by looking at \varepsilonqref{appendix tau}. \varepsilonnd{proof} We now compute the connection coefficients for the metric \varepsilonqref{metrique elliptique} in the basis $(e_0,\partial_i)$. Notice that $[e_0,\partial_i]=\partial_i\beta^j\partial_j$. Using this, we compute : \begin{align*} g(D_0e_0,e_0)&=\frac{1}{2}e_0g_{00}=-\frac{1}{2}e_0(N^2)=-Ne_0N,\\ g(D_ie_0,e_0)&=\frac{1}{2}\partial_ig_{00}=-\frac{1}{2}\partial_i(N^2)=-N\partial_iN,\\ g(D_0e_0,\partial_i)&=-\frac{1}{2}\partial_ig_{00}-g(e_0,[e_0,\partial_i])=N\partial_iN,\\ g(D_ie_0,\partial_j)&=\frac{1}{2}\left\|eft(e_0g_{ij}-g(\partial_i,[e_0,\partial_j])-g(\partial_j,[e_0,\partial_i])\right\|ight)=\frac{e^{2\gamma}}{2}\left\|eft(2e_0\gamma\mathrm{d}elta_{ij}-\partial_i\beta^k\mathrm{d}elta_{jk}-\partial_j\beta^k\mathrm{d}elta_{ik}\right\|ight), \\ g(D_0\partial_i,e_0)&=\frac{1}{2}\partial_ig_{00}+g(e_0,[e_0,\partial_i])=-N\partial_iN,\\ g(D_0\partial_i,\partial_j)&=\frac{1}{2}\left\|eft(e_0g_{ij}+g(\partial_i,[\partial_j,e_0])+g(\partial_j,[e_0,\partial_i])\right\|ight)=\frac{e^{2\gamma}}{2}\left\|eft(2e_0\gamma\mathrm{d}elta_{ij}+\partial_i\beta^k\mathrm{d}elta_{jk}-\partial_j\beta^k\mathrm{d}elta_{ik}\right\|ight),\\ g(D_i\partial_j,e_0)&=\frac{1}{2}\left\|eft(-e_0g_{ij}-g(\partial_i,[\partial_j,e_0])+g(\partial_j,[e_0,\partial_i])\right\|ight)=-\frac{e^{2\gamma}}{2}\left\|eft(2e_0\gamma\mathrm{d}elta_{ij}-\partial_i\beta^k\mathrm{d}elta_{jk}-\partial_j\beta^k\mathrm{d}elta_{ik}\right\|ight),\\ g(D_i\partial_j,\partial_k)&=e^{2\gamma}\left\|eft(\mathrm{d}elta_{ik}\partial_j\gamma+\mathrm{d}elta_{jk}\partial_i\gamma-\mathrm{d}elta_{ij}\mathrm{d}elta^{\varepsilonll}_k\partial_{\varepsilonll}\gamma \right\|ight).\\ \varepsilonnd{align*} The first two expressions are derived using $X(g(Y,Z))=g(D_XY,Z)+g(Y,D_XZ)$ and the other ones with the Koszul formula : \begin{equation*} 2g(D_VW,X)=Vg(W,X)+Wg(X,V)-Xg(V,W)-g(V,[W,X])+g(W,[X,V])+g(X,[V,W]). \varepsilonnd{equation*} From the above calculations, we obtain \begin{align} D_0e_0 &=\mathbf{T} N e_0 +e^{-2\gamma}\mathrm{d}elta^{ij}N\partial_iN\partial_j , \left\|abel{covariante 1}\\ D_0\partial_i &=\partial_iN\mathbf{T}+\frac{1}{2}\left\|eft(2\mathrm{d}elta^{j}_ie_0\gamma+\partial_i\beta^j-\mathrm{d}elta_{ik}\mathrm{d}elta^{j\varepsilonll}\partial_{\varepsilonll}\beta^k \right\|ight)\partial_j ,\left\|abel{covariante 2}\\ D_ie_0 &=\partial_iN\mathbf{T}+\frac{1}{2}\left\|eft(2\mathrm{d}elta^{j}_ie_0\gamma-\partial_i\beta^j-\mathrm{d}elta_{ik}\mathrm{d}elta^{j\varepsilonll}\partial_{\varepsilonll}\beta^k \right\|ight)\partial_j ,\left\|abel{covariante 3}\\ D_i\partial_j &=\frac{e^{2\gamma}}{2N}\left\|eft(2\mathrm{d}elta_{ij}e_0\gamma-\left\|eft(\partial_i\beta^k \right\|ight)\mathrm{d}elta_{jk}-\left\|eft(\partial_j\beta^k \right\|ight)\mathrm{d}elta_{ik} \right\|ight)\mathbf{T}+\left\|eft(\mathrm{d}elta^k_i\partial_j\gamma+\mathrm{d}elta^k_j\partial_i\gamma-\mathrm{d}elta_{ij}\mathrm{d}elta^{k\varepsilonll}\partial_{\varepsilonll}\gamma\right\|ight)\partial_k.\left\|abel{covariante 4} \varepsilonnd{align} \subsection{Decomposition of the Ricci tensor}\left\|abel{ricci tensor} \begin{prop} Given $g$ of the form \varepsilonqref{metrique elliptique}, we have the following identities : \begin{align} K_{ij}&=-\frac{\mathrm{d}elta_{ij}}{2}\mathbf{T}\left\|eft( e^{2\gamma}\right\|ight)+\frac{e^{2\gamma}}{2N}\left\|eft(\partial_i\beta_j+\partial_j\beta_i\right\|ight),\left\|abel{seconde forme fonda}\\ H_{ij}&=\frac{e^{2\gamma}}{2N}(L\beta)_{ij},\left\|abel{appendix beta}\\ \tau &=-2\mathbf{T}\gamma+\frac{\mathrm{div}\left\|eft(\beta\right\|ight)}{N}.\left\|abel{appendix tau} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} The equation \varepsilonqref{seconde forme fonda} follows from \varepsilonqref{Kij}, and \varepsilonqref{appendix beta} and \varepsilonqref{appendix tau} follow from \varepsilonqref{seconde forme fonda}. \varepsilonnd{proof} \begin{prop} Given $g$ of the form \varepsilonqref{metrique elliptique}, the components of the Ricci tensor in the basis $(e_0,\partial_i)$ are given by \begin{align} R_{ij} &=\mathrm{d}elta_{ij}\left\|eft(-\mathscr{D}elta\gamma+\frac{\tau^2}{2}e^{2\gamma}-\frac{e^{2\gamma}}{2}\mathbf{T}\tau-\frac{\mathscr{D}elta N}{2N}\right\|ight) -\mathbf{T} H_{ij}-2e^{-2\gamma}H_i^{\;\varepsilonll}H_{j\varepsilonll}\left\|abel{appendix Rij}\\& \quad+\frac{1}{N}\left\|eft( \partial_j\beta^kH_{ki}+\partial_i\beta^kH_{kj}\right\|ight)-\frac{1}{N}\left\|eft( \partial_i\partial_jN-\frac{1}{2}\mathrm{d}elta_{ij}\mathscr{D}elta N-\left\|eft( \mathrm{d}elta_i^k\partial_j\gamma+\mathrm{d}elta_j^k\partial_i\gamma-\mathrm{d}elta_{ij}\mathrm{d}elta^{\varepsilonll k}\partial_{\varepsilonll}\gamma \right\|ight)\partial_k N \right\|ight)\nonumber,\\ R_{0j} &= N\left\|eft(\frac{1}{2}\partial_j\tau-e^{-2\gamma}\partial^iH_{ij} \right\|ight),\left\|abel{appendix R0j}\\ R_{00} &= N\left\|eft(e_0\tau-e^{-4\gamma}N\left\|eft|H\right\|ight|^2-\frac{N\tau^2}{2}+e^{-2\gamma}\mathscr{D}elta N \right\|ight)\left\|abel{appendix R00}. \varepsilonnd{align} Moreover, \begin{align} \mathrm{d}elta^{ij}R_{ij}&= 2\left\|eft(-\mathscr{D}elta\gamma+\frac{\tau^2}{2}e^{2\gamma}-\frac{e^{2\gamma}}{2}\mathbf{T}\tau-\frac{\mathscr{D}elta N}{2N}\right\|ight),\left\|abel{appendix trace ricci}\\ R&=-2\mathbf{T}\tau +\frac{3}{2}\tau^2 + e^{-4\gamma}\left\|eft|H\right\|ight|^2-2e^{-2\gamma}\frac{\mathscr{D}elta N}{N} - 2e^{-2\gamma}\mathscr{D}elta\gamma.\left\|abel{appendix R} \varepsilonnd{align} \varepsilonnd{prop} \begin{proof} From Chapter 6 of \cite{cho09}, we have \begin{align} R_{ij} & = \Bar{R}_{ij}+K_{ij}\mathrm{tr}_{\Bar{g}}K-2K_i^{\;\varepsilonll}K_{j\varepsilonll}- N^{-1}\left\|eft( \mathcal{L}_{e_0}K_{ij}+D_i\partial_jN\right\|ight),\left\|abel{Rij CB}\\ R_{0j} & = N\left\|eft( \partial_j( \mathrm{tr}_{\Bar{g}}K)-D_{\varepsilonll}K^{\varepsilonll}_{\;j}\right\|ight) ,\left\|abel{R0j CB}\\ R_{00} & = N\left\|eft( e_0(\mathrm{tr}_{\Bar{g}}K)-N|K|^2+\mathscr{D}elta_{\Bar{g}} N \right\|ight),\left\|abel{R00 CB} \varepsilonnd{align} where $\Bar{D}$, $\Bar{R}_{ij}$ and $\mathscr{D}elta_{\Bar{g}}$ are defined with respect to $\Bar{g}$. First, by \varepsilonqref{def H} and the connection coefficients computations, \varepsilonqref{Rij CB} becomes \begin{align} R_{ij} & =-\mathrm{d}elta_{ij}\mathscr{D}elta\gamma+\tau\left\|eft( H_{ij}+\frac{1}{2}e^{2\gamma}\mathrm{d}elta_{ij}\tau\right\|ight)-2e^{-2\gamma}\left\|eft( H_i^{\;\varepsilonll} +\frac{1}{2}e^{2\gamma}\mathrm{d}elta_i^{\varepsilonll}\tau\right\|ight) \left\|eft( H_{j\varepsilonll} +\frac{1}{2}e^{2\gamma}\mathrm{d}elta_{j\varepsilonll}\tau\right\|ight) \left\|abel{Rij inter}\\&\quad -\frac{1}{N} \left\|eft( \mathcal{L}_{e_0}K_{ij}+\partial_i\partial_jN- \left\|eft( \mathrm{d}elta_i^k\partial_j\gamma+\mathrm{d}elta_j^k\partial_i\gamma-\mathrm{d}elta_{ij}\mathrm{d}elta^{k\varepsilonll}\partial_{\varepsilonll}\gamma\right\|ight) \partial_kN \right\|ight). \nonumber \varepsilonnd{align} To proceed, we compute $\mathcal{L}_{e_0}K_{ij}$ by considering $H_{ij}$ and $\tau$ : \begin{align*} \mathcal{L}_{e_0}H_{ij}&=e_0H_{ij}-\partial_j\beta^kH_{ki}-\partial_i\beta^kH_{kj},\\ \mathcal{L}_{e_0}(\tau\Bar{g}_{ij})&=e^{2\gamma}\mathrm{d}elta_{ij}e_0\tau-2N\tau K_{ij}. \varepsilonnd{align*} Therefore, using \varepsilonqref{def H} and plugging $\mathcal{L}_{e_0}K_{ij}$ into \varepsilonqref{Rij inter}, we obtain \varepsilonqref{appendix Rij}. The expression of $R_{0j}$ in \varepsilonqref{appendix R0j} follows from \varepsilonqref{R0j CB} and the fact that for any covariant symmetric 2-tensor $A_ij$, \begin{equation*} \Bar{g}^{ik}\Bar{D}_kA_{ij}=e^{-2\gamma}\partial^iA_ij-\partial_j\gamma\mathrm{tr}_{\Bar{g}}A. \varepsilonnd{equation*} Using \varepsilonqref{R00 CB} and the conformal invariance of the Laplacian we easily get \varepsilonqref{appendix R00}. To prove \varepsilonqref{appendix trace ricci}, we first note that \begin{equation*} \mathrm{d}elta^{ij}(\partial_j\beta^k H_{ki}+\partial_i\beta^k H_{kj})=H_{ij}(L\beta)^{ij}. \varepsilonnd{equation*} Combining this with \varepsilonqref{appendix beta}, we obtain \begin{equation*} \mathrm{d}elta^{ij}\left\|eft( -2e^{-2\gamma}H_i^{\;\varepsilonll}H_{j\varepsilonll}+\frac{1}{N}(\partial_j\beta^k H_{ki}+\partial_i\beta^k H_{kj}) \right\|ight)=0. \varepsilonnd{equation*} Taking the trace of \varepsilonqref{appendix Rij} and using this identity yield \varepsilonqref{appendix trace ricci}. Finally, by putting \varepsilonqref{metrique elliptique}, \varepsilonqref{appendix R00} and \varepsilonqref{appendix trace ricci} we easily get \varepsilonqref{appendix R}. \varepsilonnd{proof} \subsection{The stress-energy-momentum tensor}\left\|abel{T mu nu} Define $T_{\mu\nu}$ by \begin{equation*} T_{\mu\nu}=2\partial_{\mu}\varphi\partial_{\nu}\varphi-g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi+\frac{1}{2}e^{-4\varphi}\left\|eft( 2\partial_{\mu}\Omegamega\partial_{\nu}\Omegamega-g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\Omegamega\partial_{\beta}\Omegamega\right\|ight). \varepsilonnd{equation*} \begin{prop} The following identities are satisfied (with respect to the $(e_0,\partial_i)$ basis) : \begin{align} T_{00}&= (e_0\varphi)^2+e^{-2\gamma}N^2|\nabla\varphi|^2+\frac{1}{4}e^{-4\varphi}\left\|eft( (e_0\Omegamega)^2+e^{-2\gamma}N^2|\nabla\Omegamega|^2\right\|ight),\left\|abel{T 00}\\ T_{0j}&= 2e_0\varphi\partial_j\varphi+\frac{1}{2}e^{-4\varphi}e_0\Omegamega\partial_j\Omegamega,\left\|abel{T 0j}\\ T_{ij}&= 2\partial_i\varphi\partial_j\varphi+\frac{e^{2\gamma}}{N^2}(e_0\varphi)^2\mathrm{d}elta_{ij}-|\nabla\varphi|^2\mathrm{d}elta_{ij}\left\|abel{T ij}\\&\qquad+\frac{1}{4}e^{-4\varphi}\left\|eft( 2\partial_i\Omegamega\partial_j\Omegamega+\frac{e^{2\gamma}}{N^2}(e_0\Omegamega)^2\mathrm{d}elta_{ij}-|\nabla\Omegamega|^2\mathrm{d}elta_{ij}\right\|ight),\nonumber\\ \mathrm{tr}_gT &= -g^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi-\frac{1}{4}e^{-4\varphi}g^{\alpha\beta}\partial_{\alpha}\Omegamega\partial_{\beta}\Omegamega,\\ T_{00}-g_{00}\mathrm{tr}_gT &= 2\left\|eft(e_0\varphi\right\|ight)^2+\frac{1}{2}e^{-4\varphi}(e_0\Omegamega)^2,\left\|abel{JSP}\\ T_{ij}-g_{ij}\mathrm{tr}_gT &= 2\partial_{i}\varphi\partial_{j}\varphi+\frac{1}{2}e^{-4\varphi}\partial_{i}\Omegamega\partial_{j}\Omegamega,\left\|abel{JSP 2}\\ \mathrm{d}elta^{ij}\left\|eft( T_{ij}-g_{ij}\mathrm{tr}_gT\right\|ight) & =2\left\|eft|\nabla\varphi \right\|ight|^2+\frac{1}{2}e^{-4\varphi}\left\|eft|\nabla\Omegamega \right\|ight|^2,\left\|abel{JSP 3}\\ D^{\mu}T_{\mu\nu}&=2(\Box_g\varphi)\partial_{\nu}\varphi+\frac{1}{2}e^{-4\varphi}(\Box_g\Omegamega)\partial_{\nu}\Omegamega \left\|abel{divergence de T}\\&\qquad-e^{-4\varphi}\partial^\mu\varphi \left\|eft( 2\partial_{\mu}\Omegamega\partial_{\nu}\Omegamega-g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\Omegamega\partial_{\beta}\Omegamega\right\|ight) .\nonumber \varepsilonnd{align} \varepsilonnd{prop} \section{Weighted Sobolev spaces}\left\|abel{appendix B} Here are some results about weighted Sobolev spaces on $\mathbb{R}^2$, which are systematically used during the proof. Most of them can be found in the Appendix I of \cite{cho09}. \begin{lem}\left\|abel{B1} Let $m\geq 1$, $p\in[1,\infty)$ and $\mathrm{d}elta\in\mathbb{R}$, then \begin{align*} \|\nabla u\|_{W^{m-1,p}_{\mathrm{d}elta+1}} & \left\|esssim \| u\|_{W^{m,p}_{\mathrm{d}elta}},\\ \|\nabla u\|_{C^{m-1}_{\mathrm{d}elta}} & \left\|esssim \| u\|_{C^{m}_{\mathrm{d}elta+1}}. \varepsilonnd{align*} \varepsilonnd{lem} We have an easy embedding result, which is a straightforward application of the Hölder’s inequality : \begin{lem}\left\|abel{prop holder 2} If $1\left\|eq p_1\left\|eq p_2\left\|eq \infty$ and $\mathrm{d}elta_2-\mathrm{d}elta_1>2\left\|eft( \frac{1}{p_1}-\frac{1}{p_2} \right\|ight)$, then we have the continuous embedding \begin{equation*} L^{p_2}_{\mathrm{d}elta_2}\xhookrightarrow{}L^{p_1}_{\mathrm{d}elta_1}. \varepsilonnd{equation*} \varepsilonnd{lem} Next, we have Sobolev embedding theorems for weighted Sobolev spaces : \begin{prop}\left\|abel{embedding} Let $s,m\in\mathbb{N}\cup\{0\}$, $1<p<\infty$. \begin{itemize} \item If $s>\frac{2}{p}$ and $\beta\left\|eq \mathrm{d}elta+\frac{2}{p}$, then we have the continuous embedding \begin{equation*} W^{s+m,p}_{\mathrm{d}elta}\xhookrightarrow{}C^m_{\beta}. \varepsilonnd{equation*} \item If $s<\frac{2}{p}$, then we have the continuous embedding \begin{equation*} W^{s+m,p}_{\mathrm{d}elta}\xhookrightarrow{}W^{m,\frac{2p}{2-sp}}_{\mathrm{d}elta+s}. \varepsilonnd{equation*} \varepsilonnd{itemize} \varepsilonnd{prop} We will also need a product estimate. \begin{prop}\left\|abel{prop prod} Let $s,s_1,s_2\in\mathbb{N}\cup\{0\}$, $p\in[1,\infty]$, $\mathrm{d}elta,\mathrm{d}elta_1,\mathrm{d}elta_2\in\mathbb{R}$ such that $s\left\|eq\min(s_1,s_2)$, $s<s_1+s_2-\frac{2}{p}$ and $\mathrm{d}elta<\mathrm{d}elta_1+\mathrm{d}elta_2+\frac{2}{p}$. Then we have the continuous multiplication property \begin{equation*} W^{s_1,p}_{\mathrm{d}elta_1}\times W^{s_2,p}_{\mathrm{d}elta_2} \xhookrightarrow{}W^{s,p}_{\mathrm{d}elta}. \varepsilonnd{equation*} \varepsilonnd{prop} The following simple lemma will be useful as well. \begin{lem} Let $\alpha\in\mathbb{R}$ and $g\in L^{\infty}_{loc}$ such that $|g(x)|\left\|esssim \left\|angle x \right\|angle^{\alpha}$. Then the multiplication by $g$ map $L^2_{\mathrm{d}elta+\alpha}$ to $L^2_{\mathrm{d}elta}$ with operator norm bounded by $\sup_{x\in\mathbb{R}^2}\frac{|g(x)|}{\left\|angle x\right\|angle^{\alpha}}$. \varepsilonnd{lem} The next result, which is due to McOwen, concerns the invertibility of the Laplacian on weighted Sobolev spaces. Its proof can be found in \cite{mco79}. \begin{thm}\left\|abel{mcowens 1} Let $m,s\in\mathbb{N}\cup\{0\}$ and $-1+m<\mathrm{d}elta<m$. The Laplace operator $\mathscr{D}elta:H^{s+2}_{\mathrm{d}elta}\left\|ongrightarrow H^{s}_{\mathrm{d}elta+2}$ is an injection with closed range \begin{equation*} \varepsilonnstq{f\in H^{s}_{\mathrm{d}elta+2}}{ \forall v\in \cup_{i=0}^m\mathcal{H}_i,\; \int_{\mathbb{R}^2}fv=0}, \varepsilonnd{equation*} where $\mathcal{H}_i$ is the set of harmonic polynomials of degree $i$. Moreover, $u$ obeys the estimate \begin{equation*} \| u\|_{H^{s+2}_{\mathrm{d}elta}}\left\|eq C(\mathrm{d}elta,m,p)\|\mathscr{D}elta u\|_{H^{s}_{\mathrm{d}elta+2}}. \varepsilonnd{equation*} \varepsilonnd{thm} The following is a corollary of Theorem \right\|ef{mcowens 1} : \begin{coro}\left\|abel{mcowens 2} Let $-1<\mathrm{d}elta<0$ and $f\in H^0_{\mathrm{d}elta+2}$. Then there exists a solution u of \begin{equation*} \mathscr{D}elta u=f, \varepsilonnd{equation*} which can be written \begin{equation*} u=\frac{1}{2\pi}\left\|eft(\int_{\mathbb{R}^2}f\right\|ight)\chi(|x|)\left\|n(|x|)+v, \varepsilonnd{equation*} where $\chi$ is as in Section \right\|ef{subsection initial data} and $\|v\|_{H^2_{\mathrm{d}elta}}\left\|eq C(\mathrm{d}elta)\|f\|_{H^0_{\mathrm{d}elta+2}}$. \varepsilonnd{coro} We will also use some classical inequalities, which we recall here, even if they are not related to weighted Sobolev spaces. The proof of the next property can be found in Appendix A of \cite{tao06}. \begin{prop}\left\|abel{littlewood paley} If $s\in\mathbb{N}$, then \begin{equation*} \| uv\|_{H^s}\left\|esssim \|u\|_{H^s}\|v\|_{L^{\infty}}+\|v\|_{H^s}\|u\|_{L^{\infty}}. \varepsilonnd{equation*} \varepsilonnd{prop} We recall the Hardy-Littlewood-Sobolev inequality : \begin{prop}\left\|abel{prop HLS} If $0<\alpha<2$ and $1<p<r<\infty$ and $\frac{1}{r}=\frac{1}{p}-\frac{\alpha}{2}$, then \begin{equation*} \left\| u* \frac{1}{|\cdot|^{2-\alpha}} \right\|_{L^r}\left\|esssim \left\| u \right\|_{L^p}. \varepsilonnd{equation*} \varepsilonnd{prop} We recall the Gagliardo-Nirenberg inequality, for which a proof can be found in \cite{fri69} : \begin{prop}\left\|abel{GN} Let $1\left\|eq q,r\left\|eq +\infty$, $m\in\mathbb{N}^*$. Let $\alpha\in\mathbb{R}$ and $j\in\mathbb{N}$ such that \begin{equation*} \frac{j}{m}\left\|eq\alpha\left\|eq 1. \varepsilonnd{equation*} Then : \begin{equation*} \left\| \nabla^j u \right\|_{L^p} \left\|esssim \left\| \nabla ^mu\right\|_{L^r}^\alpha \left\| u \right\|_{L^q}^{1-\alpha}, \varepsilonnd{equation*} with \begin{equation*} \frac{1}{p}=\frac{j}{2}+\left\|eft(\frac{1}{r}-\frac{m}{2} \right\|ight)\alpha+\frac{1-\alpha}{q}. \varepsilonnd{equation*} \varepsilonnd{prop} \section{Third order energy estimate}\left\|abel{appendix C} In this section, we prove Proposition \right\|ef{dernier coro}. We split the proof into two lemmas : their goal is to point out the dependence of $\frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\varphi$ and $\frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\Omegamega$ on non-linear terms in $\partial\nabla^2 U$. \begin{lem} The energy $\mathscr{E}_3^\varphi$ satisfies \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\varphi & = \int_{\mathbb{R}^2}2e^{-4\varphi}e_0 \partial_j\partial_i\varphi \left\|eft( -\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\Omegamega\right\|ight)\mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}\nabla \partial_j\partial_i\varphi\cdot \left\|eft( e_0\partial_j\partial_i\Omegamega \nabla\Omegamega-e_0\Omegamega\nabla \partial_j\partial_i\Omegamega \right\|ight)\mathrm{d} x +O(\mathscr{R}(t)). \varepsilonnd{align*} \varepsilonnd{lem} \begin{proof} We split $\mathscr{E}_3^\varphi$ into two parts $A^\varphi+B^\varphi$ : \begin{equation*} \mathscr{E}_3^\varphi =\underbrace{\int_{\mathbb{R}^2} \frac{2}{N^2}\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight)^2\mathrm{d} x}_{A^\varphi\vcentcolon=} +\underbrace{\int_{\mathbb{R}^2}2e^{-2\gamma}\left\|eft|\nabla\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega \nabla\Omegamega \right\|ight|^2\mathrm{d} x}_{B^\varphi\vcentcolon=}. \varepsilonnd{equation*} We start with $A^\varphi$, by writing $\partial_t=e_0+\beta\cdot\nabla$. Note that if for some function $f$ we have $\left\| \nabla g f \right\|_{L^1}=O(\mathscr{R})$, then by integration by parts we have : \begin{equation*} \int_{\mathbb{R}^2}\partial_tf\mathrm{d} x = \int_{\mathbb{R}^2}e_0f\mathrm{d} x+\int_{\mathbb{R}^2}\beta\cdot\nabla f\mathrm{d} x = \int_{\mathbb{R}^2}e_0f\mathrm{d} x - \int_{\mathbb{R}^2}\mathrm{div}(\beta)f\mathrm{d} x= \int_{\mathbb{R}^2}e_0f\mathrm{d} x+O(\mathscr{R}). \varepsilonnd{equation*} Therefore, in what follows, we can forget about the $\beta\cdot\nabla$-part in $\partial_t$, which only contributes to $O(\mathscr{R})$. We now compute : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}A^\varphi & = \int_{\mathbb{R}^2}4\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight)\left\|eft( \mathbf{T}^2\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega \mathbf{T}^2\Omegamega \right\|ight)\mathrm{d} x \\&\quad +O(\mathscr{R}(t)). \varepsilonnd{align*} We then replace terms involving $\mathbf{T}^2$ according to \varepsilonqref{expression de box}, and then replace $\Box_g \partial_j\partial_i\varphi$ according to \varepsilonqref{WM dd ffi} ($F^\varphi_{ij}$ and $\partial_j\partial_i\Omegamega\Box_g\Omegamega$ only contributes to $O(\mathscr{R})$) : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}A^\varphi& = \int_{\mathbb{R}^2}4\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight) \left\|eft[ -\Box_g \partial_j\partial_i\varphi+\frac{e^{-2\gamma}}{N}\mathrm{div}(N\nabla \partial_j\partial_i\varphi)+\frac{1}{2}e^{-4\varphi}\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega \right\|ight. \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \left\|eft. -\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega \Box_g\Omegamega +\frac{e^{-2\gamma}}{2N}e^{-4\varphi}\partial_j\partial_i\Omegamega \mathrm{div}(N\nabla\Omegamega) \right\|ight]\mathrm{d} x+O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}4\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight) \left\|eft[ -\frac{1}{2}e^{-4\varphi}\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\Omegamega\right\|ight. \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \left\|eft. +\frac{e^{-2\gamma}}{N}\mathrm{div}(N\nabla \partial_j\partial_i\varphi)+\frac{e^{-2\gamma}}{2N}e^{-4\varphi}\partial_j\partial_i\Omegamega \mathrm{div}(N\nabla\Omegamega) \right\|ight]\mathrm{d} x+O(\mathscr{R}(t)). \varepsilonnd{align*} We integrate by parts the terms with a divergence and expand : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}A^\varphi & = \int_{\mathbb{R}^2}4\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight) \left\|eft( -\frac{1}{2}e^{-4\varphi}\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\Omegamega\right\|ight)\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}4e^{-2\gamma}\nabla \partial_j\partial_i\varphi\cdot \nabla\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight)\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\nabla\Omegamega\cdot\nabla\left\|eft( \partial_j\partial_i\Omegamega \left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight)\right\|ight)\mathrm{d} x+O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}4\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight) \left\|eft( -\frac{1}{2}e^{-4\varphi}\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\Omegamega\right\|ight)\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}4e^{-2\gamma}\nabla \partial_j\partial_i\varphi\cdot\nabla e_0 \partial_j\partial_i\varphi \mathrm{d} x -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\varphi\cdot\nabla (\partial_j\partial_i\Omegamega e_0\Omegamega)\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}(\nabla\Omegamega\cdot\nabla \partial_j\partial_i\Omegamega )\left\|eft(e_0\partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega e_0\Omegamega \right\|ight)\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\partial_j\partial_i\Omegamega \nabla\Omegamega\cdot\nabla e_0\partial_j\partial_i\varphi\mathrm{d} x - \int_{\mathbb{R}^2}e^{-8\varphi}e^{-2\gamma}\partial_j\partial_i\Omegamega \nabla\Omegamega\cdot\nabla(\partial_j\partial_i\Omegamega e_0\Omegamega)\mathrm{d} x +O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}2e_0\partial_j\partial_i\varphi \left\|eft( -e^{-4\varphi}\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\Omegamega\right\|ight)\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\Omegamega\nabla \partial_j\partial_i\varphi\cdot\nabla \partial_j\partial_i\Omegamega \mathrm{d} x -\int_{\mathbb{R}^2}4e^{-2\gamma}\nabla \partial_j\partial_i\varphi\cdot\nabla e_0 \partial_j\partial_i\varphi \mathrm{d} x \\& \quad-\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\partial_j\partial_i\Omegamega \nabla\Omegamega\cdot\nabla e_0\partial_j\partial_i\varphi\mathrm{d} x +O(\mathscr{R}(t)) \varepsilonnd{align*} We now deal with $B^\varphi$ : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}B^\varphi & = \int_{\mathbb{R}^2}4e^{-2\gamma} \left\|eft( \nabla \partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega \nabla\Omegamega \right\|ight)\cdot e_0\left\|eft( \nabla \partial_j\partial_i\varphi+\frac{1}{2}e^{-4\varphi}\partial_j\partial_i\Omegamega \nabla\Omegamega \right\|ight)\mathrm{d} x +O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}e_0\partial_j\partial_i\Omegamega \nabla\partial_j\partial_i\varphi\cdot \nabla\Omegamega \mathrm{d} x \\&\quad+ \int_{\mathbb{R}^2}4e^{-2\gamma}\nabla \partial_j\partial_i\varphi\cdot \nabla e_0 \partial_j\partial_i\varphi \mathrm{d} x +\int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}\partial_j\partial_i\Omegamega \nabla\Omegamega\cdot\nabla e_0\partial_j\partial_i\varphi \mathrm{d} x +O(\mathscr{R}(t)). \varepsilonnd{align*} We see that the terms which contains $\nabla e_0 \partial_j\partial_i\varphi $ in $A^\varphi$ and $B^\varphi$ cancel each other, and that every terms wich are linear in $\partial \nabla^2 U$ only contribute to $O(\mathscr{R}(t))$, so that : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\varphi & = \int_{\mathbb{R}^2}2e^{-4\varphi}e_0 \partial_j\partial_i\varphi \left\|eft( -\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\Omegamega+e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\Omegamega\right\|ight)\mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}\nabla \partial_j\partial_i\varphi\cdot \left\|eft( e_0\partial_j\partial_i\Omegamega \nabla\Omegamega-e_0\Omegamega\nabla \partial_j\partial_i\Omegamega \right\|ight)\mathrm{d} x +O(\mathscr{R}(t)). \varepsilonnd{align*} \varepsilonnd{proof} \begin{lem} The energy $\mathscr{E}_3^\Omegamega$ satisfies \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\Omegamega & =\int_{\mathbb{R}^2}2e^{-4\varphi}e_0\partial_j\partial_i\Omegamega \left\|eft( \mathbf{T}\partial_j\partial_i\varphi\mathbf{T}\Omegamega -e^{-2\gamma}\nabla\partial_j\partial_i\varphi\cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad +\int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}\nabla\partial_j\partial_i\Omegamega \cdot \left\|eft( e_0\Omegamega\nabla\partial_j\partial_i\varphi-e_0\partial_j\partial_i\varphi\nabla\Omegamega \right\|ight) \mathrm{d} x+O(\mathscr{R}(t)). \varepsilonnd{align*} \varepsilonnd{lem} \begin{proof} The proof of this lemma is very similar to the one of the previous lemma, except that we also differenciate the coefficient $e^{-4\varphi}$ in the energy $\mathscr{E}_3^\Omegamega$. We split $\mathscr{E}_3^\Omegamega$ into two parts $A^\Omegamega+B^\Omegamega$ : \begin{align*} \mathscr{E}_3^\varphi & = \underbrace{\int_{\mathbb{R}^2} \frac{1}{2N^2}e^{-4\varphi} \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2\partial_j\partial_i\varphi e_0\Omegamega \right\|ight)^2\mathrm{d} x}_{A^\Omegamega\vcentcolon=} \\&\qquad\qquad\qquad\qquad+\underbrace{\int_{\mathbb{R}^2}\frac{1}{2}e^{-4\varphi}e^{-2\gamma}\left\|eft|\nabla\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \nabla\varphi-2\partial_j\partial_i\varphi \nabla\Omegamega \right\|ight|^2\mathrm{d} x}_{B^\Omegamega\vcentcolon=}. \varepsilonnd{align*} We start by $A^\Omegamega$ : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}A^\Omegamega & = \int_{\mathbb{R}^2}e^{-4\varphi} \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight)\left\|eft( \mathbf{T}^2\partial_j\partial_i\Omegamega -2\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\varphi -2\partial_j\partial_i\Omegamega \mathbf{T}^2\varphi\right\|ight.\\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left\|eft.-2\mathbf{T} \partial_j\partial_i\varphi \mathbf{T}\Omegamega -2 \partial_j\partial_i\varphi \mathbf{T}^2\Omegamega \right\|ight) \mathrm{d} x \\&\quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e_0\varphi \left\|eft( \mathbf{T} \partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \mathbf{T}\varphi-2 \partial_j\partial_i\varphi \mathbf{T}\Omegamega \right\|ight)^2\mathrm{d} x +O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}e^{-4\varphi} \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight) \left\|eft[ -\Box_g\partial_j\partial_i\Omegamega +\frac{e^{-2\gamma}}{N}\mathrm{div}(N\nabla \partial_j\partial_i\Omegamega ) + 2\partial_j\partial_i\Omegamega \Box_g\varphi \right\|ight. \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \left\|eft.- \frac{2e^{-2\gamma}}{N}\partial_j\partial_i\Omegamega \mathrm{div}(N\nabla\varphi) +2 \partial_j\partial_i\varphi \Box_g\Omegamega \right\|ight. \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left\|eft.- \frac{2e^{-2\gamma}}{N} \partial_j\partial_i\varphi \mathrm{div}(N\nabla\Omegamega) -2\mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\varphi-2\mathbf{T} \partial_j\partial_i\varphi \mathbf{T}\Omegamega \right\|ight]\mathrm{d} x \\&\qquad\qquad\qquad-\int_{\mathbb{R}^2}2e^{-4\varphi}e_0\varphi \left\|eft( \mathbf{T} \partial_j\partial_i\Omegamega \right\|ight)^2\mathrm{d} x+O(\mathscr{R}(t)). \varepsilonnd{align*} We integrate by parts the terms with a divergence (note that we differenciate the $e^{-4\varphi}$, but the one with $\mathrm{div}(N\nabla\partial_j\partial_i\Omegamega)$ in front is the only divergence term which gives a main term) : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}A^\Omegamega & = \int_{\mathbb{R}^2}2e^{-4\varphi} \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight)\left\|eft( \mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\varphi+ \mathbf{T} \partial_j\partial_i\varphi \mathbf{T}\Omegamega \right\|ight. \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left\|eft. -2e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\varphi-2e^{-2\gamma}\nabla \partial_j\partial_i\varphi \cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight)\mathrm{d} x \\&\quad+\int_{\mathbb{R}^2}4e^{-4\varphi}e^{-2\gamma}\nabla\varphi\cdot\nabla\partial_j\partial_i\Omegamega \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight) \mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\nabla\varphi\cdot \nabla\left\|eft( \partial_j\partial_i\Omegamega \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight)\right\|ight)\mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\nabla\Omegamega\cdot\nabla \left\|eft( \partial_j\partial_i\varphi \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight)\right\|ight)\mathrm{d} x \\& \quad-\int_{\mathbb{R}^2}2e^{-4\varphi}e_0\varphi \left\|eft( \mathbf{T} \partial_j\partial_i\Omegamega \right\|ight)^2\mathrm{d} x+O(\mathscr{R}(t)). \varepsilonnd{align*} We now expand all the terms and note again that the linear terms in $\partial\nabla^2 U$ only contribute to $O(\mathscr{R}(t))$ : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}A^\Omegamega & = \int_{\mathbb{R}^2}2e^{-4\varphi} \left\|eft( e_0\partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega e_0\varphi-2 \partial_j\partial_i\varphi e_0\Omegamega \right\|ight) \left\|eft( \mathbf{T} \partial_j\partial_i\Omegamega \mathbf{T}\varphi+ \mathbf{T} \partial_j\partial_i\varphi \mathbf{T}\Omegamega \right\|ight. \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \left\|eft.-2e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\varphi-2e^{-2\gamma}\nabla \partial_j\partial_i\varphi \cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\varphi|\nabla \partial_j\partial_i\Omegamega|^2 \mathrm{d} x +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\Omegamega\nabla \partial_j\partial_i\Omegamega \cdot\nabla \partial_j\partial_i\varphi \mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}6e^{-4\varphi}e^{-2\gamma}(\nabla\varphi\cdot\nabla \partial_j\partial_i\Omegamega ) e_0\partial_j\partial_i\Omegamega \mathrm{d} x +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}(\nabla\Omegamega\cdot\nabla \partial_j\partial_i\varphi ) e_0\partial_j\partial_i\Omegamega \mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma} \partial_j\partial_i\varphi \nabla\Omegamega\cdot\nabla e_0\partial_j\partial_i\Omegamega \mathrm{d} x -\int_{\mathbb{R}^2}e^{-4\varphi}e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla e_0\partial_j\partial_i\Omegamega \mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}\partial_j\partial_i\Omegamega \nabla\varphi\cdot\nabla e_0\partial_j\partial_i\Omegamega \mathrm{d} x -\int_{\mathbb{R}^2}2e^{-4\varphi}e_0\varphi \left\|eft( \mathbf{T} \partial_j\partial_i\Omegamega \right\|ight)^2\mathrm{d} x+O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}2e^{-4\varphi} e_0\partial_j\partial_i\Omegamega \left\|eft( \mathbf{T} \partial_j\partial_i\varphi \mathbf{T}\Omegamega +e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\varphi-e^{-2\gamma}\nabla \partial_j\partial_i\varphi \cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\varphi|\nabla \partial_j\partial_i\Omegamega|^2 \mathrm{d} x +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\Omegamega\nabla \partial_j\partial_i\Omegamega \cdot\nabla \partial_j\partial_i\varphi \mathrm{d} x \\&\quad +\int_{\mathbb{R}^2}e^{-4\varphi}e^{-2\gamma}(-\nabla \partial_j\partial_i\Omegamega +2\partial_j\partial_i\Omegamega \nabla\varphi+2 \partial_j\partial_i\varphi \nabla\Omegamega)\cdot \nabla e_0\partial_j\partial_i\Omegamega \mathrm{d} x +O(\mathscr{R}(t)) \varepsilonnd{align*} We now deal with $B^\Omegamega$ : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}B^\Omegamega & =\int_{\mathbb{R}^2}e^{-4\varphi}e^{-2\gamma}\left\|eft(\nabla \partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \nabla\varphi-2 \partial_j\partial_i\varphi \nabla\Omegamega \right\|ight)\cdot e_0\left\|eft(\nabla \partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \nabla\varphi-2 \partial_j\partial_i\varphi \nabla\Omegamega \right\|ight)\mathrm{d} x \\&\quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\varphi\left\|eft|\nabla \partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \nabla\varphi-2 \partial_j\partial_i\varphi \nabla\Omegamega \right\|ight|^2\mathrm{d} x \\& = \int_{\mathbb{R}^2}e^{-4\varphi}e^{-2\gamma}(\nabla \partial_j\partial_i\Omegamega -2\partial_j\partial_i\Omegamega \nabla\varphi-2 \partial_j\partial_i\varphi \nabla\Omegamega)\cdot \nabla e_0\partial_j\partial_i\Omegamega \mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\partial_j\partial_i\Omegamega\nabla \partial_j\partial_i\Omegamega\cdot \nabla\varphi\mathrm{d} x -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0 \partial_j\partial_i\varphi\nabla \partial_j\partial_i\Omegamega \cdot \nabla\Omegamega\mathrm{d} x \\&\quad- \int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\varphi|\nabla \partial_j\partial_i\Omegamega |^2\mathrm{d} x+O(\mathscr{R}(t)). \varepsilonnd{align*} We see that the terms which contains $\nabla e_0 \partial_j\partial_i\Omegamega $ in $A^\Omegamega$ and $B^\Omegamega$ cancel each other, therefore : \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\Omegamega & = \int_{\mathbb{R}^2}2e^{-4\varphi} e_0\partial_j\partial_i\Omegamega \left\|eft( \mathbf{T} \partial_j\partial_i\varphi \mathbf{T}\Omegamega +e^{-2\gamma}\nabla \partial_j\partial_i\Omegamega \cdot\nabla\varphi-e^{-2\gamma}\nabla \partial_j\partial_i\varphi \cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad +\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\Omegamega\nabla \partial_j\partial_i\Omegamega \cdot\nabla \partial_j\partial_i\varphi \mathrm{d} x -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0\partial_j\partial_i\Omegamega\nabla \partial_j\partial_i\Omegamega\cdot \nabla\varphi\mathrm{d} x \\& \quad -\int_{\mathbb{R}^2}2e^{-4\varphi}e^{-2\gamma}e_0 \partial_j\partial_i\varphi\nabla \partial_j\partial_i\Omegamega \cdot \nabla\Omegamega\mathrm{d} x +O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}2e^{-4\varphi}e_0\partial_j\partial_i\Omegamega \left\|eft( \mathbf{T}\partial_j\partial_i\varphi\mathbf{T}\Omegamega -e^{-2\gamma}\nabla\partial_j\partial_i\varphi\cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad +\int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}\nabla\partial_j\partial_i\Omegamega \cdot \left\|eft( e_0\Omegamega\nabla\partial_j\partial_i\varphi-e_0\partial_j\partial_i\varphi\nabla\Omegamega \right\|ight) \mathrm{d} x+O(\mathscr{R}(t)) \\& = \int_{\mathbb{R}^2}2e^{-4\varphi}e_0\partial_j\partial_i\Omegamega \left\|eft( \mathbf{T}\partial_j\partial_i\varphi\mathbf{T}\Omegamega -e^{-2\gamma}\nabla\partial_j\partial_i\varphi\cdot\nabla\Omegamega \right\|ight) \mathrm{d} x \\& \quad +\int_{\mathbb{R}^2}2e^{-2\gamma}e^{-4\varphi}\nabla\partial_j\partial_i\Omegamega \cdot \left\|eft( e_0\Omegamega\nabla\partial_j\partial_i\varphi-e_0\partial_j\partial_i\varphi\nabla\Omegamega \right\|ight) \mathrm{d} x+O(\mathscr{R}(t)). \varepsilonnd{align*} Adding the two previous lemmas, we see that the main parts of $\frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\varphi$ and $\frac{\mathrm{d}}{\mathrm{d} t}\mathscr{E}_3^\Omegamega$ cancel each other, and we obtain Proposition \right\|ef{dernier coro}. \varepsilonnd{proof} \begin{thebibliography}{10} \bibitem{bur89} Gregory~A. 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\begin{document} \begin{abstract} In this paper we consider the steady water wave problem for waves that possess a merely $L_r-$integrable vorticity, with $r\in(1,\infty)$ being arbitrary. We first establish the equivalence of the three formulations--the velocity formulation, the stream function formulation, and the height function formulation-- in the setting of strong solutions, regardless of the value of $r$. Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small amplitude capillary and capillary-gravity water waves with a $L_r-$integrable vorticity. \varepsilonnd{abstract} \maketitle \mathbb{S}ection{Introduction}\langlebel{Sec:1} We consider the classical problem of traveling waves that propagate at the surface of a two-dimensional inviscid and incompressible fluid of finite depth. Our setting is general enough to incorporate the case when the vorticity of the fluid is merely $L_r-$integrable, with $r>1$ being arbitrary. The existence of solutions of the Euler equations in ${\mathbb R}^n$ describing flows with an unbounded vorticity distribution has been addressed lately by several authors, cf. \cite{ Ke11, MB02, Vi00} and the references therein, whereas for traveling free surface waves in two-dimensions there are so far no existence results which allow for a merely $L_r$-integrable vorticity. In our setting, the hydrodynamical problem is modeled by the steady Euler equations, to which we refer to as the {\varepsilonm velocity formulation}. For classical solutions and in the absence of stagnation points, there are two equivalent formulations, namely the {\varepsilonm stream function} and the {\varepsilonm height function} formulation, the latter being related to the semi-Lagrangian Dubreil-Jacotin transformation. This equivalence property stays at the basis of the existence results of classical solutions with general H\"older continuous vorticity, cf. \cite{CoSt04} for gravity waves and \cite{W06b, W06a} for waves with capillarity. Very recent, taking advantage of the weak formulation of the governing equations, it was rigorously established that there exist gravity waves \cite{CS11} and capillary-gravity waves \cite{CM13xx, MM13x} with a discontinuous and bounded vorticity. The waves found in the latter references are obtained in the setting of strong solutions when the equations of motion are satisfied in $L_r,$ $r>2$ in \cite{CS11}, respectively $L_\infty$ in \cite{CM13xx, MM13x}. The authors of \cite{CS11} also prove the equivalence of the formulations in the setting of $L_r$-solutions, under the restriction that $r>2$. Our first main result, Theorem \maxthop{\rm Re}\nolimitsf{T:EQ}, establishes the equivalence of the three formulations for strong solutions that possess Sobolev and weak H\"older regularity. For this we rely on the regularity properties of such solutions, cf. \cite{AC11, EM13x, MM13x}. This equivalence holds for gravity, capillary-gravity, and pure capillary waves without stagnation points and with a $L_r$-integrable vorticity, without making any restrictions on $r\in(1,\infty).$ The equivalence result Theorem \maxthop{\rm Re}\nolimitsf{T:EQ} stays at the basis of our second main result, Theorem \maxthop{\rm Re}\nolimitsf{T:MT}, where we establish the existence of small amplitude capillary and capillary-gravity water waves having a $L_r-$integrable vorticity distribution for any $r\in(1,\infty).$ On physical background, studying waves with an unbounded vorticity is relevant in the setting of small-amplitude wind generated waves, when capillarity plays an important role. These waves may possess a shear layer of high vorticity adjacent to the wave surface \cite{O82, PB74}, fact which motivates us to consider unbounded vorticity distributions. {Moreover, an unbounded vorticity at the bed is also physically relevant, for example when describing turbulent flows along smooth channels (see the empirical law on page 109 in \cite{B62}).} In contrast to the irrotational case when, in the absence of an underlying current, the qualitative features of the flow are well-understood \cite{C12, Co06, DH07}, in the presence of a underlying, even uniform \cite{CoSt10, HH12, SP88}, current many aspects of the flow are more difficult to study, or are even untraceable, and one has to rely often on numerical simulations, cf. \cite{KO1, KO2, SP88}. For example, by allowing for a discontinuous vorticity, the latter studies display the influence of a favorable or adverse wind on the amplitude of the waves, or describe extremely high rotational waves and the the flow pattern of waves with eddies. The rigorous existence of waves with capillarity was obtained first in the setting of irrotational waves \cite{MJ89, JT85, JT86, RS81} and it was only recently extended to the setting of waves with constant vorticity and stagnation points \cite{ CM13a, CM13, CM13x} (see also \cite{CV11}). In the context of waves with a general H\"older continuous \cite{W06b, W06a} or discontinuous \cite{CM13xx, MM13x} vorticity the existence results are obtained by using the height function formulation and concern only small amplitude waves without stagnation points. Theorem \maxthop{\rm Re}\nolimitsf{T:MT}, which is the first rigorous existence result for waves with unbounded vorticity, is obtained by taking advantage of the weak interpretation of the height function formulation. More precisely, recasting the nonlinear second-order boundary condition on the surface into a nonlocal and nonlinear equation of order zero enables us to introduce the notion of weak (which is shown to be strong) solution for the problem in a suitable analytic setting. By means of local bifurcation theory and ODE techniques we then find local real-analytic curves consisting, with the exception of a single laminar flow solution, only of non-flat symmetric capillary (or capillary-gravity) water waves. The methods we apply are facilitated by the presence of capillary effects (see e.g. the proof of Lemma \maxthop{\rm Re}\nolimitsf{L:4}), though not on the value of the surface tension coefficient, the existence question for pure gravity waves with unbounded vorticity being left as an open problem. The outline of the paper is as follows: we present in Section \maxthop{\rm Re}\nolimitsf{Sec:2} the mathematical setting and establish the equivalence of the formulations in Theorems \maxthop{\rm Re}\nolimitsf{T:EQ}. We end the section by stating our main existence result Theorem \maxthop{\rm Re}\nolimitsf{T:MT}, the Section \maxthop{\rm Re}\nolimitsf{Sec:3} being dedicated to its proof. \mathbb{S}ection{Classical formulations of the steady water wave problem and the main results}\langlebel{Sec:2} Following a steady periodic wave from a reference frame which moves in the same direction as the wave and with the same speed $c$, the equations of motion are the steady-state Euler equations \begin{subequations}\langlebel{eq:P} \begin{equation}\langlebel{eq:Euler} \left\{ \begin{array}{rllll} ({u}-c) { u}_x+{ v}{ u}_y&=&-{ P}_x,\\ ({ u}-c) { v}_x+{ v}{ v}_y&=&-{ P}_y-g,\\ { u}_x+{v}_y&=&0 \varepsilonnd{array} \right.\qquad \text{in $\Omega_\varepsilonta,$} \varepsilonnd{equation} with $x$ denoting the direction of wave propagation and $y$ being the height coordinate. We assumed that the free surface of the wave is the graph $y=\varepsilonta(x),$ that the fluid has constant unitary density, and that the flat fluid bed is located at $y=-d$. Hereby, $\varepsilonta$ has zero integral mean over a period and $d>0$ is the average mean depth of the fluid. Moreover, $\Omega_\varepsilonta $ is the two-dimensional fluid domain \[ \Omega_\varepsilonta:=\{(x,y)\,:\,\text{$ x\in\mathbb{S} $ and $-d<y<\varepsilonta(x)$}\}, \] with $\mathbb{S}:={\mathbb R}/(2\partiali{\mathbb Z})$ denoting the unit circle. This notation expresses the $2\partiali$-periodicity in $x $ of $\varepsilonta,$ of the velocity field $(u, v), $ and of the pressure $P$. The equations \varepsilonqref{eq:Euler} are supplemented by the following boundary conditions \begin{equation}\langlebel{eq:BC} \left\{ \begin{array}{rllll} P&=&{P}_0-\mathbb{S}igma\varepsilonta''/(1+\varepsilonta'^2)^{3/2}&\text{on $ y=\varepsilonta(x)$},\\ v&=&({ u}-c) \varepsilonta'&\text{on $ y=\varepsilonta(x)$},\\ v&=&0 &\text{on $ y=-d$}, \varepsilonnd{array} \right. \varepsilonnd{equation} the first relation being a consequence of Laplace-Young's equation which states that the pressure jump across an interface is proportional to the mean curvature of the interface. We used $ P_0$ to denote the constant atmospheric pressure and $\mathbb{S}igma>0$ is the surface tension coefficient. Finally, the vorticity of the flow is the scalar function \begin{equation*} \omega:= { u}_y-{ v}_x\qquad\text{in $\Omega_\varepsilonta$.} \varepsilonnd{equation*} \varepsilonnd{subequations} The velocity formulation \varepsilonqref{eq:P} can be re-expressed in terms of the stream function $\partialsi $, which is introduced via the relation $\nabla \partialsi=(-v,u-c)$ in $\Omega_\varepsilonta$, cf. Theorem \maxthop{\rm Re}\nolimitsf{T:EQ}, and it becomes a free boundary problem \begin{equation}\langlebel{eq:psi} \left\{ \begin{array}{rllll} \Delta \partialsi&=&\gamma(-\partialsi)&\text{in}&\Omega_\varepsilonta,\\ \displaystyle|\nabla\partialsi|^2+2g(y+d)-2\mathbb{S}igma\frac{\varepsilonta''}{(1+\varepsilonta'^2)^{3/2}}&=&Q&\text{on} &y=\varepsilonta(x),\\ \partialsi&=&0&\text{on}&y=\varepsilonta(x),\\ \partialsi&=&-p_0&\text{on} &y=-d. \varepsilonnd{array} \right. \varepsilonnd{equation} Hereby, the constant $p_0<0$ represents the relative mass flux, $Q\in{\mathbb R}$ is related to the total head, and the function $\gamma:(p_0,0)\to{\mathbb R}$ is the vorticity function, that is \begin{equation}\langlebel{vor} \omega(x,y)=\gamma(-\partialsi(x,y)) \varepsilonnd{equation} for $(x,y)\in \Omega_\varepsilonta.$ The equivalence of the velocity formulation \varepsilonqref{eq:P} and of the stream function formulation \varepsilonqref{eq:psi} in the setting of classical solutions without stagnation points, that is when \begin{equation}\langlebel{SC} u-c<0\qquad\text{in $\overline \Omega_\varepsilonta$} \varepsilonnd{equation} has been established in \cite{Con11, CoSt04}. We emphasize that the assumption \varepsilonqref{SC} is crucial when proving the existence of the vorticity function $\gamma$. Additionally, the condition \varepsilonqref{SC} guarantees in the classical setting considered in these references that the semi-hodograph transformation $\Phi:\overline\Omega_\varepsilonta\to\overline\Omega$ given by \begin{equation}\langlebel{semH} \Phi(x,y):=(q,p)(x,y):=(x,-\partialsi(x,y))\qquad \text{for $(x,y)\in\overline\Omega_\varepsilonta$}, \varepsilonnd{equation} where $\Omega:=\mathbb{S}\times(p_0,0),$ is a diffeomorphism. This property is used to show that the previous two formulations \varepsilonqref{eq:P} and \varepsilonqref{eq:psi} can be re-expressed in terms of the so-called height function $h:\overline \Omega\to{\mathbb R}$ defined by \begin{equation}\langlebel{hodo} h(q,p):=y+d \qquad\text{for $(q,p)\in\overline\Omega$}. \varepsilonnd{equation} More precisely, one obtains a quasilinear elliptic boundary value problem \begin{equation}\langlebel{PB} \left\{ \begin{array}{rllll} (1+h_q^2)h_{pp}-2h_ph_qh_{pq}+h_p^2h_{qq}-\gamma h_p^3&=&0&\text{in $\Omega$},\\ \displaystyle 1+h_q^2+(2gh-Q)h_p^2-2\mathbb{S}igma \frac{h_p^2h_{qq}}{(1+h_q^2)^{3/2}}&=&0&\text{on $p=0$},\\ h&=&0&\text{on $ p=p_0,$} \varepsilonnd{array} \right. \varepsilonnd{equation} the condition \varepsilonqref{SC} being re-expressed as \begin{equation}\langlebel{PBC} \min_{\overline \Omega}h_p>0. \varepsilonnd{equation} The equivalence of the three formulations \varepsilonqref{eq:P}, \varepsilonqref{eq:psi}, and \varepsilonqref{PB} of the water wave problem, when the vorticity is only $ L_r-$integrable, has not been established yet for the full range $r\in(1,\infty)$. In the context of strong solutions, when the equations of motion are assumed to hold in $L_r$, there is a recent result \cite[Theorem 2]{CS11} established in the absence of capillary forces. This result though is restricted to the case when $r>2,$ this condition being related to Sobolev's embedding $W^2_r\hookrightarrow C^{1+\alpha}$ in two dimensions. In the same context, but for solutions that possess weak H\"older regularity, there is a further equivalence result \cite[Theorem 1]{VZ12}, but again one has to restrict the range of H\"older exponents. Our equivalence result, cf. Theorem \maxthop{\rm Re}\nolimitsf{T:EQ} and Remark \maxthop{\rm Re}\nolimitsf{R:-2} below, is true for all $r\in(1,\infty)$ and was obtained in the setting of strong solutions that possess, additionally to Sobolev regularity, weak H\"older regularity, the H\"older exponent being related in our context to Sobolev's embedding in only one dimension. This result enables us to establish, cf. Theorem \maxthop{\rm Re}\nolimitsf{T:MT} and Remark \maxthop{\rm Re}\nolimitsf{R:0}, the existence of small-amplitude capillary-gravity and pure capillary water waves with $L_r-$integrable vorticity function for any $r\in(1,\infty).$ We denote in the following by $\mathop{\rm tr}\nolimits_0$ the trace operator with respect to the boundary component $p=0$ of $\overline\Omega,$ that is $\mathop{\rm tr}\nolimits_0v=v(\cdot,0)$ for all $v\in C(\overline\Omega).$ In the following, we use several times the following product formula \begin{equation}\langlebel{PF} \qquad \partial(uv)=u\partial v+v\partial u\qquad\text{for all $u,v\in W^1_{1,loc}$ with $uv, u\partial v+v\partial u\in L_{1,loc},$} \varepsilonnd{equation} cf. relation (7.18) in \cite{GT01}. \begin{thm}[Equivalence of the three formulations]\langlebel{T:EQ} Let $r\in(1,\infty)$ be given and define $\alpha=(r-1)/r\in(0,1).$ Then, the following are equivalent \begin{itemize} \item[$(i)$] the height function formulation together with \varepsilonqref{PBC} for $h\in C^{1+\alpha}(\overline\Omega)\cap W^2_r(\Omega)$ such that $\mathop{\rm tr}\nolimits_0 h \in W^2_r(\mathbb{S})$, and $\gamma\in L_r((p_0,0))$; \item[$(ii)$] the stream function formulation for $\varepsilonta\in W^2_r(\mathbb{S})$, $\partialsi\in C^{1+\alpha}(\overline\Omega_\varepsilonta)\cap W^2_r(\Omega_\varepsilonta)$ satisfying $\partialsi_y<0$ in $\overline\Omega_\varepsilonta$, and $\gamma\in L_r((p_0,0))$; \item[$(iii)$] the velocity formulation together with \varepsilonqref{SC} for $u,v, P\in C^{\alpha}(\overline\Omega_\varepsilonta)\cap W^1_r(\Omega_\varepsilonta),$ and $\varepsilonta\in W^2_r(\mathbb{S}).$ \varepsilonnd{itemize} \varepsilonnd{thm} \begin{rem}\langlebel{R:-2} Our equivalence result is true for both capillary and capillary-gravity water waves. Moreover, it is also true for pure gravity waves when the proof is similar, with modifications just when proving that $(iii)$ implies $(i)$: instead of using \cite[Theorem 5.1]{MM13x} one has to rely on the corresponding regularity result established for gravity waves, cf. Theorem 1.1 in \cite{EM13x}. We emphasize also that the condition $\mathop{\rm tr}\nolimits_0 h \in W^2_r(\mathbb{S})$ requested at $(i)$ is not a restriction. In fact, as a consequence of $h\in C^{1+\alpha}(\overline\Omega)\cap W^2_r(\Omega)$ being a strong solution of \varepsilonqref{PB}-\varepsilonqref{PBC} for $\gamma\in L_r((p_0,0))$, we know that the wave surface and all the other streamlines are real-analytic curves, cf. \cite[Theorem 5.1]{MM13x} and \cite[Theorem 1.1]{EM13x}. Particularly, $\mathop{\rm tr}\nolimits_0 h$ is a real-analytic function, i.e. $\mathop{\rm tr}\nolimits_0 h\in C^\omega(\mathbb{S})$. Furthermore, in view of the same references, all weak solutions $h\in C^{1+\alpha}(\overline\Omega)$ of \varepsilonqref{PB}, cf. Definition \maxthop{\rm Re}\nolimitsf{D:1} (or \cite{CS11} for gravity waves), satisfy $h \in W^2_r(\mathbb{S})$. \varepsilonnd{rem} \begin{proof}[Proof of Theorem \maxthop{\rm Re}\nolimitsf{T:EQ}] Assume first $(i)$ and let \begin{align}\langlebel{DEF} d:=\frac{1}{2\partiali}\int_0^{2\partiali} \mathop{\rm tr}\nolimits_0 h\, dq\in(0,\infty)\qquad\text{and}\qquad \varepsilonta:=\mathop{\rm tr}\nolimits_0 h-d\in W^2_r(\mathbb{S}). \varepsilonnd{align} We prove that there exists a unique function $\partialsi\in C^{1+\alpha}(\overline\Omega_\varepsilonta)$ with the property that \begin{align}\langlebel{PP1} y+d-h(x,-\partialsi(x,y))=0\qquad\text{for all $(x,y)\in\overline\Omega_\varepsilonta.$} \varepsilonnd{align} To this end, let $H:\mathbb{S}\times{\mathbb R}\to{\mathbb R}$ to be a continuous extension of $h$ to $\mathbb{S}\times{\mathbb R}$, having the property that $H(q,\cdot)\in C^1({\mathbb R})$ is strictly increasing and has a bounded derivative for all $q\in\mathbb{S}.$ Moreover, define the function $F:\mathbb{S}\times{\mathbb R}\times{\mathbb R}\to {\mathbb R}$ by setting \begin{align*} F(x,y,p)=y+d-H(x,p). \varepsilonnd{align*} For every fixed $x\in\mathbb{S}$, we have \[\text{$F(x,\cdot,\cdot)\in C^1({\mathbb R}\times{\mathbb R},{\mathbb R}),$ \quad $F(x,\varepsilonta(x),0)=0$,\quad and $F_p(x,\cdot,\cdot)=-H_p(x,\cdot)<0$. }\] Using the implicit function theorem, we find a $C^1-$function $\partialsi(x,\cdot):(\varepsilonta(x)-\varepsilon,\varepsilonta(x)+\varepsilon)\to{\mathbb R}$ with the property that \begin{align*} F(x,y,-\partialsi(x,y))=0 \qquad\text{for all $y\in (\varepsilonta(x)-\varepsilon,\varepsilonta(x)+\varepsilon)$}. \varepsilonnd{align*} As $\partialsi_y(x,y)=1/F_p(x,y,-\partialsi(x,y))$, we deduce that $\partialsi(x,\cdot) $ is a strictly decreasing function which maps, due to the boundedness of $H_p(x,\cdot)$, bounded intervals onto bounded intervals. Therefore, $\partialsi(x,\cdot) $ can be defined on $(-\infty,\varepsilonta(x)]$. In view of $F(x,-d,p_0)=0,$ we get that $\partialsi(x,-d)=-p_0$ for each $x\in\mathbb{S}.$ Observe also that, due to the periodicity of $H$ and $\varepsilonta$, $\partialsi$ is $2\partiali-$periodic with respect to $x$, while, because use of $F\in C^1(\mathbb{S}\times{\mathbb R}\times[p_0,0]),$ we have $\partialsi\in C^1(\Omega_\varepsilonta)$. Since the relation \varepsilonqref{PP1} is satisfied in $\overline\Omega_\varepsilonta$, it is easy to see now that in fact $\partialsi\in C^{1+\alpha}(\overline\Omega_\varepsilonta).$ In order to show that $\partialsi$ is the desired stream function, we prove that $\partialsi\in W^2_r(\Omega_\varepsilonta).$ Noticing that the relation \varepsilonqref{PP1} yields \begin{align}\langlebel{RE1} \partialsi_y(x,y)=-\frac{1}{h_p(x,-\partialsi(x,y))}\qquad\text{and}\qquad \partialsi_x(x,y)=\frac{h_q(x,-\partialsi(x,y))}{h_p(x,-\partialsi(x,y))} \varepsilonnd{align} in $\overline\Omega_\varepsilonta,$ the variable transformation \varepsilonqref{semH}, integration by parts, and the fact that $h$ is a strong solution of \varepsilonqref{PB} yield \begin{align*} \Delta\partialsi[\widetilde\partialhi]=& -\int_{\Omega_\varepsilonta}\left(\partialsi_y\widetilde\partialhi_y+\partialsi_x\widetilde\partialhi_x\right)\, d(x,y)=-\int_{\Omega}\left(h_q\partialhi_q-\frac{1+h_q^2}{h_p}\partialhi_p\right)\, d(q,p)\\ =&\int_\Omega\left(h_{qq}-\frac{2h_qh_{pq}}{h_p}+\frac{(1+h_q^2)h_{pp}}{h_p^2}\right)\partialhi\, d(q,p)=\int_\Omega(\gamma \partialhi)h_p\, d(q,p) \\ =&\int_{\Omega_\varepsilonta} \gamma(-\partialsi)\widetilde \partialhi \, d(x,y) \varepsilonnd{align*} for all $\widetilde \partialhi\in C^\infty_0(\Omega_\varepsilonta),$ whereby we set $\partialhi:=\widetilde\partialhi\circ\Phi^{-1}.$ This shows that $\Delta\partialsi=\gamma(-\partialsi)\in L_r(\Omega_\varepsilonta)$. Taking into account that $\partialsi(x,y)=p_0(y-\varepsilonta(x))/(d+\varepsilonta(x))$ for $(x,y)\in\partial\Omega_\varepsilonta,$ whereby in fact $\varepsilonta\in C^\omega(\mathbb{S}),$ c.f. Remark \maxthop{\rm Re}\nolimitsf{R:-2}, we find by elliptic regularity, cf. e.g. \cite[Theorems 3.6.3 and 3.6.4]{CW98}, that $\partialsi\in W^2_r(\Omega_\varepsilonta).$ It is also easy to see that $(\varepsilonta,\partialsi)$ satisfy also the second relation of \varepsilonqref{eq:psi}, and this completes our arguments in this case. We now show that $(ii)$ implies $(iii)$. To this end, we define \begin{align}\langlebel{PP2} (u-c,v)&:=(\partialsi_y,-\partialsi_x)\qquad\text{and}\qquad P:=-\frac{|\nabla\partialsi|^2}{2}-g(y+d)-\Gammaamma(-\partialsi)+P_0+\frac{Q}{2}, \varepsilonnd{align} where $\Gammaamma$ is given by \begin{equation}\langlebel{E:G}\Gammaamma(p):=\int_0^p\gamma(s)\, ds\qquad\text{for $p\in[p_0,0].$}\varepsilonnd{equation} Clearly, we have that $u,v\in C^{\alpha}(\overline\Omega_\varepsilonta) \cap W^1_r(\Omega_\varepsilonta) $ and $\Gammaamma\in C^\alpha([p_0,0])\cap W^1_r((p_0,0)).$ Moreover, because $\partialsi\in C^{1+ \alpha}(\overline\Omega_\varepsilonta)\cap W^2_r(\Omega_\varepsilonta),$ the formula \varepsilonqref{PF} shows that $|\nabla\partialsi|^2\in W^1_r(\Omega_\varepsilonta),$ and therefore also $P\in C^{\alpha}(\overline\Omega_\varepsilonta)\cap W^1_r(\Omega_\varepsilonta).$ The boundary conditions \varepsilonqref{eq:BC} are easy to check. Furthermore, the conservation of mass equation is a direct consequence of the first relation of \varepsilonqref{PP2}. We are left with the conservation of momentum equations. Therefore, we observe the function $\Gammaamma(-\partialsi)$ is differentiable almost everywhere and its partial derivatives belong to $L_r(\Omega_\varepsilonta)$, meaning that $\Gammaamma(-\partialsi)\in W^1_r(\Omega_\varepsilonta)$, cf. \cite{DD12}, the gradient $\nabla(\Gammaamma(-\partialsi))$ being determined by the chain rule. Taking now the weak derivative with respect to $x$ and $y$ in the second equation of \varepsilonqref{PP2}, respectively, we obtain in view of \varepsilonqref{PF}, the conservation of momentum equations. We now prove that $(iii)$ implies $(ii)$. Thus, choose $u,v, P\in C^{\alpha}(\overline\Omega_\varepsilonta)\cap W^1_r(\Omega_\varepsilonta) $ and $\varepsilonta\in W^2_r(\mathbb{S}) $ such that $(\varepsilonta, u-c, v,P)$ is a solution of the velocity formulation. We define \begin{equation} \partialsi(x,y):=-p_0+\int_{-d}^{y} (u(x,s)-c)\, ds\qquad\text{for $(x,y)\in\overline\Omega_\varepsilonta,$} \varepsilonnd{equation} with $p_0$ being a negative constant. It is not difficult to see that the function $\partialsi$ belongs to $ C^{1+\alpha}(\overline\Omega_\varepsilonta)\cap W^2_r(\Omega_\varepsilonta) $ and that it satisfies $\nabla\partialsi=(-v,u-c).$ The latter relation allows us to pick $p_0$ such that $\partialsi=0$ on $y=\varepsilonta(x).$ Also, we have that $\partialsi=-p_0$ on the fluid bed. We next show that the vorticity of the flow satisfies the relation \varepsilonqref{vor} for some $\gamma\in L_r((p_0,0)).$ To this end, we proceed as in \cite{ BM11} and use the property that the mapping $\Phi$ given by \varepsilonqref{semH} is an isomorphism of class $C^{1+\alpha}$ to compute that \begin{align*} \partial_q (\omega\circ \Phi^{-1})[ \partialhi]=&\int_{\Omega_\varepsilonta} (v_x-u_y)((u-c) \widetilde\partialhi_x+v\widetilde\partialhi_y)\, d(x,y) \varepsilonnd{align*} for all $ \partialhi\in C^\infty_0(\Omega).$ Again, we set $\widetilde\partialhi :=\partialhi\circ\Phi\in C^{1+\alpha}_0(\overline\Omega_\varepsilonta).$ Since our assumption $(iii)$ implies that $(u-c)^2$ and $v^2$ belong to $W^1_r(\Omega_\varepsilonta)$, cf. \varepsilonqref{PF}, density arguments, \varepsilonqref{eq:Euler}, and integration by parts yield \begin{align*} \partial_q (\omega\circ \Phi^{-1})[ \partialhi]=&\int_{\Omega_\varepsilonta} ((u-c)v_x+vv_y)\widetilde\partialhi_x\, d(x,y)-\int_{\Omega_\varepsilonta} ((u-c)u_x+vu_y)\widetilde\partialhi_y\, d(x,y)\\[1ex] =&-\int_{\Omega_\varepsilonta} (P_y+g)\widetilde\partialhi_x\, d(x,y)+\int_{\Omega_\varepsilonta} P_x\widetilde\partialhi_y\, d(x,y)=0. \varepsilonnd{align*} Consequently, there exists $\gamma\in L_r((p_0,0))$ with the property that $\omega\circ\Phi^{-1}=\gamma$ almost everywhere in $\Omega$. This shows that \varepsilonqref{vor} is satisfied in $L_r(\Omega_\varepsilonta).$ Next, we observe that the same arguments used when proving that $(ii)$ implies $(iii)$ yield that the energy \[ E:=P+\frac{|\nabla\partialsi|^2}{2}+g(y+d)+\Gammaamma(-\partialsi) \] is constant in $\overline\Omega_\varepsilonta.$ Defining $Q:=2(E-P_0),$ one can now easily see that $(\varepsilonta,\partialsi)$ satisfies \varepsilonqref{eq:psi}, and we have established $(ii)$. In the final part of the proof we assume that $(ii)$ is satisfied and we prove $(i)$. Therefore, we let $h:\overline\Omega\to{\mathbb R}$ be the mapping defined by \varepsilonqref{hodo} (or equivalently \varepsilonqref{PP1}). Then, we get that $ h\in C^{1+\alpha}(\overline\Omega)$ verifies the relations \varepsilonqref{PP1} and \varepsilonqref{RE1}. Consequently, $\mathop{\rm tr}\nolimits_0 h\in W^2_r(\mathbb{S})$ and one can easily see that the boundary conditions of \varepsilonqref{PB} and \varepsilonqref{PBC} are satisfied. In order to show that $h$ belongs to $W^2_r(\Omega)$ and it also solves the first relation of \varepsilonqref{PB}, we observe that the first equation of \varepsilonqref{eq:psi} can be written in the equivalent form \begin{equation}\langlebel{eq:psi2} (\partialsi_x\partialsi_y)_x+\frac{1}{2}\left(\partialsi_y^2-\partialsi_x^2\right)_y+(\Gammaamma(-\partialsi))_y=0\qquad\text{in $L_r(\Omega_\varepsilonta)$}. \varepsilonnd{equation} Therewith and using the change of variables \varepsilonqref{semH}, we find \begin{align*} &\int_\Omega\frac{h_q}{h_p} \partialhi_q-\left(\Gammaamma+\frac{1+h_q^2}{2h_p^2}\right)\partialhi_p\, d(q,p)\\[1ex] &=-\int_{\Omega_\varepsilonta}\left((\partialsi_x\partialsi_y)_x+\frac{1}{2}\left(\partialsi_y^2-\partialsi_x^2\right)_y+(\Gammaamma(-\partialsi))_y\right)\widetilde\partialhi\, d(x,y)=0, \varepsilonnd{align*} for all $\partialhi\in C^1_0(\Omega)$ and with $\widetilde\partialhi :=\partialhi\circ\Phi\in C^{1+\alpha}_0(\overline\Omega_\varepsilonta)$. Hence, $h\in C^{1+\alpha}(\overline\Omega)$ is a weak solution of the height function formulation, cf. Definition \maxthop{\rm Re}\nolimitsf{D:1}. We are now in the position to use the regularity result Theorem 5.1 in \cite{MM13x} which states that the distributional derivatives $\partial_q^mh$ also belong to $C^{1+\alpha}(\overline\Omega)$ for all $m\geq1.$ Particularly, setting $m=1$, we find that $h_p$ is differentiable with respect to $q$ and $\partial_q(h_p)=\partial_p(h_q)\in C^\alpha(\overline\Omega).$ Exploiting the fact that $h$ is a weak solution, we see that the distributional derivatives \[ \partial_q\left(\Gamma+\frac{1+h_q^2}{2h_p^2}\right),\quad \partial_p\left(\Gamma+\frac{1+h_q^2}{2h_p^2}\right)=\partial_q\left( \frac{h_q}{h_p}\right) \] belong both to $C^\alpha(\overline\Omega)\mathbb{S}ubset L_r(\Omega).$ Additionally, $1+h_q^2\in C^{1+\alpha}(\overline\Omega)$ and regarding $\Gamma$ as an element of $ W^1_r(\Omega)$, we obtain \[ \frac{1 }{h_p^2}\in W^1_r(\Omega). \] Because $h$ satisfies \varepsilonqref{PBC} and recalling that $h_p$ is a bounded function, \cite[Theorem 7.8]{GT01} implies that $h_p\in W^1_r(\Omega).$ Hence, $h\in C^{1+\alpha}(\overline\Omega)\cap W^2_r(\Omega)$ and it is not difficult to see that $h$ satisfies the first equation of \varepsilonqref{PB} in $L_r(\Omega)$, cf. \varepsilonqref{PF}. This completes our arguments. \varepsilonnd{proof} We now state our main existence result. \begin{thm}[Existence result]\langlebel{T:MT} We fix $r\in(1,\infty)$ , $p_0\in(-\infty,0),$ and define the H\"older exponent $\alpha:=(r-1)/r\in(0,1).$ We also assume that the vorticity function $\gamma$ belongs to $L_r((p_0,0)).$ Then, there exists a {positive integer $N$} such that for each {integer $n\geq N$} there exists a local real-analytic curve $ {\mathcal{C}_{n}}\mathbb{S}ubset C^{1+\alpha}(\overline\Omega)$ consisting only of strong solutions of the problem \varepsilonqref{PB}-\varepsilonqref{PBC}. Each solution $h\in {\mathcal{C}_{n}}$, {$n\geq N,$} satisfies additionally \begin{itemize} \item[$(i)$] $h\in W^2_r(\Omega)$, \item[$(ii)$] $h(\cdot,p)$ is a real-analytic map for all $p\in[p_0,0].$ \varepsilonnd{itemize} Moreover, each curve ${\mathcal{C}_{n}}$ contains a laminar flow solution and all the other points on the curve describe waves that have minimal period $2\partiali/n $, only one crest and trough per period, and are symmetric with respect to the crest line. \varepsilonnd{thm} \begin{rem}\langlebel{R:0} While proving Theorem \maxthop{\rm Re}\nolimitsf{T:MT} we make no restriction on the constant $g$, meaning that the result is true for capillary-gravity waves but also in the context of capillary waves (when we set $g=0$). Sufficient conditions which allow us to choose {$N=1$} in Theorem \maxthop{\rm Re}\nolimitsf{T:MT} can be found in Lemma \maxthop{\rm Re}\nolimitsf{L:9}. Also, if $\gamma\in C((p_0,0)),$ the solutions found in Theorem \maxthop{\rm Re}\nolimitsf{T:MT} are classical as one can easily show that, additionally to the regularity properties stated in Theorem \maxthop{\rm Re}\nolimitsf{T:EQ}, we also have $h\in C^2(\Omega),$ $\partialsi\in C^2(\Omega_\varepsilonta)$, and $(u,v,P)\in C^1(\Omega_\varepsilonta)$. \varepsilonnd{rem} \mathbb{S}ection{Weak solutions for the height function formulation}\langlebel{Sec:3} This last section is dedicated to proving Theorem \maxthop{\rm Re}\nolimitsf{T:MT}. Therefore, we pick $r\in(1,\infty)$ and let $\alpha=(r-1)/r\in(0,1)$ be fixed in the remainder of this paper. The formulation \varepsilonqref{PB} is very useful when trying to determine classical solution of the water wave problem \cite{W06b, W06a}. However, when the vorticity function belongs to $L_r((p_0,0)),$ $r\in(1,\infty),$ the curvature term and the lack of regularity of the vorticity function gives rise to several difficulties when trying to consider the equations \varepsilonqref{PB} in a suitable (Sobolev) analytic setting. For example, the trivial solutions of \varepsilonqref{PB}, see Lemma \maxthop{\rm Re}\nolimitsf{L:LFS} below, belong merely to $W^2_r(\Omega)\cap C^{1+\alpha}(\overline\Omega).$ When trying to prove the Fredholm property of the linear operator associated to the linearization of the problem around these trivial solutions, one has to deal with an elliptic equation in divergence form and having coefficients merely in $W^1_r(\Omega)\cap C^{\alpha}(\overline\Omega),$ cf. \varepsilonqref{L1}. The solvability of elliptic boundary value problems in $W^2_r(\Omega)$ requires in general though more regularity from the coefficients. Also, the trace $\mathop{\rm tr}\nolimits_0 h_{qq}$ which appears in the second equation of \varepsilonqref{PB} is meaningless for functions in $W^2_r(\Omega).$ Nevertheless, using the fact that the operator $(1-\partial_q^2):H^2(\mathbb{S})\to L_2(\mathbb{S})$ is an isomorphism and the divergence structure of the first equation of \varepsilonqref{PB}, that is \[\left(\frac{h_q}{h_p}\right)_q-\left(\Gammaamma+\frac{1+h_q^2}{2h_p^2}\right)_p=0\qquad\text{in $\Omega$,}\] with $\Gamma$ being defined by the relation \varepsilonqref{E:G}, one can introduce the following definition of a weak solution of \varepsilonqref{PB}. \begin{defn}\langlebel{D:1} A function $h\in C^{1}(\overline\Omega)$ which satisfies \varepsilonqref{PBC} is called a {\varepsilonm weak solution} of \varepsilonqref{PB} if we have \begin{subequations}\langlebel{WF} \begin{align} h+(1-\partial_q^2)^{-1}\mathop{\rm tr}\nolimits_0\left( \frac{\left(1+h_q^2+(2gh-Q)h_p^2\right)(1+h_q^2)^{3/2}}{2\mathbb{S}igma h_p^2}-h\right)=&0\qquad\text{on $p=0$;}\langlebel{PB0}\\[1ex] h=&0\qquad\text{on $p=p_0$;}\langlebel{PB1} \varepsilonnd{align} and if $h$ satisfies the following integral equation \begin{equation}\langlebel{PB2} \int_\Omega\frac{h_q}{h_p}\partialhi_q-\left(\Gammaamma+\frac{1+h_q^2}{2h_p^2}\right)\partialhi_p\, d(q,p)=0\qquad\text{for all $\partialhi\in C^1_0(\Omega)$.} \varepsilonnd{equation} \varepsilonnd{subequations} \varepsilonnd{defn} Clearly, any strong solution $h\in C^{1+\alpha}(\overline\Omega)\cap W^2_r(\Omega)$ with $\mathop{\rm tr}\nolimits_0 h \in W^2_r(\mathbb{S})$ is a weak solution of \varepsilonqref{PB}. Furthermore, because of \varepsilonqref{PB0}, any weak solution of \varepsilonqref{PB} has additional regularity on the boundary component $p=0,$ that is $\mathop{\rm tr}\nolimits_0 h\in C^2(\mathbb{S}).$ The arguments used in the last part of the proof of Theorem \maxthop{\rm Re}\nolimitsf{T:EQ} show in fact that any weak solution $h$ which belongs to $C^{1+\alpha}(\overline\Omega)$ is a strong solution of \varepsilonqref{PB} (as stated in Theorem \maxthop{\rm Re}\nolimitsf{T:EQ} $(i)$). The formulation \varepsilonqref{WF} has the advantage that in can be recast as an operator equation in a functional setting that enables us to use bifurcation results to prove existence of weak solutions. To present this setting, we introduce the following Banach spaces: \begin{align*} X&\!:=\!\left\{\widetilde h\in C^{1+\alpha}_{2\partiali/n}(\overline\Omega)\,:\, \text{$\widetilde h$ is even in $q$ and $\widetilde h\big|_{p=p_0}=0$}\right\},\\ Y_1&\!:=\!\{f\in\mathcal{D}'(\Omega)\,:\, \text{$f=\partial_q\partialhi_1+\partial_p\partialhi_2$ for $\partialhi_1,\partialhi_2\in C^\alpha_{2\partiali/n}(\overline\Omega)$ with $\partialhi_1$ odd and $\partialhi_2$ even in $q$}\},\\ Y_2&\!:=\!\{\varphi\in C^{1+\alpha}_{2\partiali/n}(\mathbb{S})\,:\, \text{$\varphi$ is even}\}, \varepsilonnd{align*} the positive integer $n\in{\mathbb N}$ being fixed later on. The subscript $2\partiali/n$ is used to express $2\partiali/n-$periodic in $q$. We recall that $Y_1$ is a Banach space with the norm \[ \|f\|_{Y_1}:=\inf\{\|\partialhi_1\|_\alpha+\|\partialhi_2\|_\alpha\,:\, f=\partial_q\partialhi_1+\partial_p\partialhi_2\}. \] In the following lemma we determine all laminar flow solutions of \varepsilonqref{WF}. They correspond to waves with a flat surface $\varepsilonta=0$ and having parallel streamlines. \begin{lemma}[Laminar flow solutions]\langlebel{L:LFS} Let $\Gammaamma_{M}:=\max_{[p_0,0]}\Gammaamma$. For every $\langlembda\in(2\Gamma_M,\infty)$, the function $H(\cdot;\langlembda)\in W^2_r((p_0,0))$ with \[ H(p;\langlembda):=\int_{p_0}^p \frac{1}{\mathbb{S}qrt{\langlembda-2\Gammaamma(s)}}\, ds\qquad\text{for $p\in[p_0,0]$} \] is a weak solution of \varepsilonqref{WF} provided that \[ Q=Q(\langlembda):=\langlembda+2g\int_{p_0}^0\frac{1}{\mathbb{S}qrt{\langlembda-2\Gammaamma(p)}}\, dp. \] There are no other weak solutions of \varepsilonqref{WF} that are independent of $q$. \varepsilonnd{lemma} \begin{proof} It readily follows from \varepsilonqref{PB2} that if $H$ is a weak solution of \varepsilonqref{WF} that is independent of the variable $q$, then $2\Gammaamma+1/H_p^2=0$ in $\mathcal{D}'((p_0,0)).$ The expression for $H$ is obtained now by using the relation \varepsilonqref{PB1}. When verifying the boundary condition \varepsilonqref{PB0}, the relation $(1-\partial_q^2)^{-1}\xi=\xi$ for all $\xi\in{\mathbb R}$ yields that $Q$ has to be equal with $Q(\langlembda).$ \varepsilonnd{proof} Because $H(\cdot;\langlembda)\in W^2_r((p_0,0))$, we can interpret by means of Sobolev's embedding $H(\cdot;\langlembda)$ as being an element of $X.$ We now are in the position of reformulating the problem \varepsilonqref{WF} as an abstract operator equation. Therefore, we introduce the nonlinear and nonlocal operator $\mathcal{F}:=(\mathcal{F}_1,\mathcal{F}_2):(2\Gamma_M,\infty)\times X\to Y:=Y_1\times Y_2$ by the relations \begin{align*} \mathcal{F}_1(\langlembda,{\widetilde h})\!:=\!&\left(\frac{{\widetilde h}_q}{H_p+{\widetilde h}_p}\right)_q-\left(\Gammaamma+\frac{1+{\widetilde h}_q^2}{2(H_p+{\widetilde h}_p)^2}\right)_p,\\ \mathcal{F}_2(\langlembda,{\widetilde h})\!:=\!&\mathop{\rm tr}\nolimits_0{\widetilde h}+(1-\partial_q^2)^{-1}\mathop{\rm tr}\nolimits_0\!\left(\!\frac{\left(1+{\widetilde h}_q^2+(2g(H+{\widetilde h})-Q) (H_p+{\widetilde h}_p)^2\right)(1+{\widetilde h}_q^2)^{3/2}}{2\mathbb{S}igma(H_p+{\widetilde h}_p)^2}-{\widetilde h}\!\right) \varepsilonnd{align*} for $(\langlembda,{\widetilde h})\in (2 \Gamma_M,\infty)\times X,$ whereby $H=H(\cdot;\langlembda)$ and $Q=Q(\langlembda)$ are defined in Lemma \maxthop{\rm Re}\nolimitsf{L:LFS}. The operator $\mathcal{F}$ is well-defined and it depends real-analytically on its arguments, that is \begin{align}\langlebel{BP0} \mathcal{F}\in C^\omega((2\Gammaamma_M,\infty)\times X, Y). \varepsilonnd{align} With this notation, determining the weak solutions of the problem \varepsilonqref{PB} reduces to determining the zeros $(\langlembda,{\widetilde h})$ of the equation \begin{align}\langlebel{BP} \mathcal{F}(\langlembda,{\widetilde h})=0\qquad\text{in $Y$ } \varepsilonnd{align} for which ${\widetilde h}+H(\cdot;\langlembda)$ satisfies \varepsilonqref{PBC}. From the definition of $\mathcal{F}$ we know that the laminar flow solutions of \varepsilonqref{PB} correspond to the trivial solutions of $\mathcal{F}$ \begin{align}\langlebel{BP1} \mathcal{F}(\langlembda,0)=0\qquad\text{for all $\langlembda\in(2 \Gamma_M,\infty).$} \varepsilonnd{align} Actually, if $(\langlembda,{\widetilde h})$ is a solution of \varepsilonqref{BP}, the function $h:={\widetilde h}+H(\cdot;\langlembda)\in X$ is a weak solution of \varepsilonqref{PB} when $Q=Q(\langlembda)$, provided that ${\widetilde h}$ is sufficiently small in $C^1(\overline\Omega).$ In order to use the theorem on bifurcation from simple eigenvalues due to Crandall and Rabinowitz \cite{CR71} in the setting of \varepsilonqref{BP}, we need to determine special values of $\langlembda$ for which the Fr\'echet derivative $\partial_{\widetilde h}\mathcal{F}(\langlembda,0)\in\mathcal{L}(X,Y)$, defined by \[ \partial_{\widetilde h}\mathcal{F}(\langlembda,0)[w]:=\lim_{\varepsilon\to0}\frac{\mathcal{F}(\langlembda,\varepsilon w)-\mathcal{F}(\langlembda,0)}{\varepsilon}\qquad\text{for $w\in X$,} \] is a Fredholm operator of index zero with a one-dimensional kernel. To this end, we compute that $\partial_{\widetilde h} \mathcal{F} (\langlembda,0)=:(L,T)$ with $L\in\mathcal{L}(X,Y_1) $ and $T\in\mathcal{L}(X,Y_2)$ being given by \begin{equation}\langlebel{L1} \begin{aligned} Lw:=& \left(\frac{w_q}{H_p}\right)_q+\left(\frac{w_p}{H_p^3}\right)_p,\\ Tw:=&\mathop{\rm tr}\nolimits_0 w+(1-\partial_q^2)^{-1} \mathop{\rm tr}\nolimits_0 \left(\frac{gw-\langlembda^{3/2}w_p}{\mathbb{S}igma}-w\right) \varepsilonnd{aligned}\qquad\quad\text{for $w\in X,$} \varepsilonnd{equation} and with $H=H(\cdot;\langlembda)$ as in Lemma \maxthop{\rm Re}\nolimitsf{L:LFS}. We now study the properties of the linear operator $\partial_{\widetilde h} \mathcal{F} (\langlembda,0)$, $\langlembda>2\Gamma_M.$ Recalling that $H\in C^{1+\alpha}([p_0,0]),$ we obtain together with \cite[Theorem 8.34]{GT01} the following result. \begin{lemma}\langlebel{L:2} The Fr\' echet derivative $\partial_{\widetilde h} \mathcal{F}(\langlembda,0)\in\mathcal{L}(X,Y)$ is a Fredholm operator of index zero for each $\langlembda\in(2\Gamma_M,\infty).$ \varepsilonnd{lemma} \begin{proof} See the proof of Lemma 4.1 in \cite{MM13x}. \varepsilonnd{proof} In order to apply the previously mentioned bifurcation result, we need to determine special values for $\langlembda$ such that the kernel of $\partial_{\widetilde h} \mathcal{F}(\langlembda,0)$ is a subspace of $X$ of dimension one. To this end, we observe that if $0\neq w\in X$ belongs to the kernel of $\partial_{\widetilde h} \mathcal{F}(\langlembda,0)$, the relation $Lw=0$ in $Y_1$ implies that, for each $k\in{\mathbb N},$ the Fourier coefficient \[ w_k(p):=\langle w(\cdot, p)|\cos(kn\cdot)\rangle_{L_2}:=\int_0^{2\partiali} w(q,p)\cos(knq)\, dq\qquad\text{for $p\in[p_0,0]$} \] belongs to $C^{1+\alpha}([p_0,0])$ and solves the equation \begin{align}\langlebel{EQ:M} \left(\frac{w_k'}{H_p^3}\right)'-\frac{(kn)^2w_k}{H_p}=0\qquad\text{in $\mathcal{D}'((p_0,0)).$} \varepsilonnd{align} Additionally, multiplying the relation $Tw=0$ by $\cos(knq)$ we determine, in virtue of the symmetry of the operator $(1-\partial_q^2)^{-1},$ that is \begin{align*} \langle f|(1-\partial_q^2)^{-1}g\rangle_{L_2}=\langle (1-\partial_q^2)^{-1}f|g\rangle_{L_2}\qquad\text{for all $f,g\in L_2(\mathbb{S})$}, \varepsilonnd{align*} a further relation \[(g+\mathbb{S}igma (kn)^2)w_k(0)=\langlembda^{3/2}w_k'(0).\] Finally, because of $w\in X$, we get $w_k(p_0)=0$. Since $W^1_r((p_0,0))$ is an algebra for any $r\in(1,\infty),$ cf. \cite{A75}, it is easy to see that $w_k$ belongs to $ W^2_r((p_0,0))$ and that it solves the system \begin{equation}\langlebel{E:m} \left\{ \begin{array}{rlll} (a^3(\langlembda) w')'-\mu a(\langlembda)w&=&0 &\text{in $L_r((p_0,0))$,}\\ (g+\mathbb{S}igma\mu)w(0)&=&\langlembda^{3/2}w'(0),\\ w(p_0)&=&0, \varepsilonnd{array}\right. \varepsilonnd{equation} when $\mu=(kn)^2.$ For simplicity, we set $a(\langlembda):=a(\langlembda;\cdot):=\mathbb{S}qrt{\langlembda-2\Gammaamma}\in W^1_r((p_0,0)).$ Our task is to determine special values for $\langlembda$ with the property that the system \varepsilonqref{E:m} has nontrivial solutions, which form a one-dimensional subspace of $W^2_r((p_0,0))$, { only for $\mu=n^2.$} Therefore, given $(\langlembda,\mu)\in(2 \Gamma_M,\infty)\times[0,\infty),$ we introduce the Sturm-Liouville type operator $R_{\langlembda,\mu}:W^2_{r,0} \to L_r((p_0,0))\times {\mathbb R}$ by \begin{equation*} R_{\langlembda,\mu}w:= \begin{pmatrix} (a^3(\langlembda) w')'-\mu a(\langlembda)w\\ (g+\mathbb{S}igma\mu)w(0)-\langlembda^{3/2}w'(0) \varepsilonnd{pmatrix}\qquad\text{for $w\in W^2_{r,0},$} \varepsilonnd{equation*} whereby $W^2_{r,0}:=\{w\in W^2_r((p_0,0))\,:\, w(p_0)=0\}.$ Additionally, for $(\langlembda,\mu)$ as above, we let $v_i\in W^2_r((p_0,0))$, with $v_i:=v_i(\cdot;\langlembda,\mu)$, denote the unique solutions of the initial value problems \begin{equation}\langlebel{ERU} \left\{\begin{array}{lll} (a^3(\langlembda) v_1')'-\mu a(\langlembda)v_1=0\qquad \text{in $L_r((p_0,0))$},\\[1ex] v_1(p_0)=0,\quad v_1'(p_0)=1, \varepsilonnd{array} \right. \varepsilonnd{equation} and \begin{equation}\langlebel{ERUa} \left\{\begin{array}{lll} (a^3(\langlembda)v_2')'-\mu a(\langlembda)v_2=0\qquad \text{in $L_r((p_0,0))$},\\[1ex] v_2(0)=\langlembda^{3/2},\quad v_2'(0)=g+\mathbb{S}igma\mu. \varepsilonnd{array} \right. \varepsilonnd{equation} Similarly as in the bounded vorticity case $\gamma\in L_\infty((p_0,0)) $ considered in \cite{MM13x}, we have the following property. \begin{prop}\langlebel{P:2} Given $(\langlembda,\mu)\in(2\Gamma_M,\infty)\times[0,\infty),$ $R_{\langlembda,\mu}$ is a Fredholm operator of index zero and its kernel is at most one-dimensional. Furthermore, the kernel of $R_{\langlembda,\mu}$ is one-dimensional exactly when the functions $v_i$, $i=1,2,$ given by \varepsilonqref{ERU} and \varepsilonqref{ERUa}, are linearly dependent. In the latter case we have $\mathop{\rm Ker}\nolimits R_{\langlembda,\mu}=\mathbb{S}pa\{v_1\}.$ \varepsilonnd{prop} \begin{proof} First of all, $R_{\langlembda,\mu}$ can be decomposed as the sum $R_{\langlembda,\mu}=R_I+R_c$, whereby \[ R_Iw:= \begin{pmatrix} (a^3(\langlembda)w')'-\mu a(\langlembda)w\\ -\langlembda^{3/2}w'(0) \varepsilonnd{pmatrix} \qquad \text{and}\qquad R_cw:= \begin{pmatrix} 0\\ (g+\mathbb{S}igma\mu) w(0) \varepsilonnd{pmatrix} \] for all $w\in W^2_{r,0}.$ It is not difficult to see that $R_c$ is a compact operator. Next, we show that $R_I:W^2_{r,0} \to L_r((p_0,0))\times {\mathbb R}$ is an isomorphism. Indeed, if $w\in W^2_{r,0}$ solves the equation $R_Iw=(f,A), $ with $ (f,A)\in L_r((p_0,0))\times {\mathbb R}$, then, since $W^2_r((p_0,0))\hookrightarrow C^{1+\alpha}([p_0,0]),$ we have \begin{equation}\langlebel{VF} \int_{p_0}^0\left(a^3(\langlembda)w'\varphi'+\mu a(\langlembda)w\varphi\right)dp=-A\varphi(0)-\int_{p_0}^0 f\varphi\, dp \varepsilonnd{equation} for all $\varphi\in H_*:=\{w\in W^1_2((p_0,0))\,:\, w(p_0)=0\}$. The right-hand side of \varepsilonqref{VF} defines a linear functional in $\mathcal{L}(H_*,{\mathbb R})$ and that the left-hand side corresponds to a bounded bilinear and coercive functional in $H_*\times H_*.$ Therefore, the existence and uniqueness of a solution $w\in H_*$ of \varepsilonqref{VF} follows from the Lax-Milgram theorem, cf. \cite[Theorem 5.8]{GT01}. In fact, one can easily see that $w_*\in W^2_{r,0}$, so that $R_I$ is indeed an isomorphism. That the kernel of $R_{\langlembda,\mu}$ is at most one-dimensional can be seen from the observation that if $w_1,w_2\in W^2_r((p_0,0))$ are solutions of $(a^3(\langlembda) w')'-\mu a(\langlembda)w=0$, then \begin{equation}\langlebel{BV}a^3(\langlembda)(w_1w_2'-w_2w_1')=const. \qquad\text{in $[p_0,0]$}.\varepsilonnd{equation} Particularly, if $w_1, w_2\in W^2_{r,0},$ we obtain, in view of $a(\langlembda)>0 $ in $[p_0,0],$ that $w_1$ and $w_2$ are linearly dependent. To finish the proof, we notice that if the functions $v_1$ and $v_2$, given by \varepsilonqref{ERU} and \varepsilonqref{ERUa}, are linearly dependent, then they both belong to $\mathop{\rm Ker}\nolimits R_{\langlembda,\mu}.$ Moreover, if $0\neq v\in \mathop{\rm Ker}\nolimits R_{\langlembda,\mu},$ the relation \varepsilonqref{BV} yields that $v$ is collinear with both $v_1 $ and $v_2$, argument which completes our proof. \varepsilonnd{proof} In view of the Proposition \maxthop{\rm Re}\nolimitsf{P:2}, we are left to determine $(\langlembda,\mu)\in(2\Gamma_M,\infty)\times[0,\infty)$ for which the Wronskian \[ W(p;\langlembda,\mu):=\left| \begin{array}{lll} v_1&v_2\\ v_1'&v_2' \varepsilonnd{array} \right| \] vanishes on the entire interval $[p_0,0].$ Recalling \varepsilonqref{BV}, we arrive at the problem of determining the zeros of the real-analytic (\varepsilonqref{ERU} and \varepsilonqref{ERUa} can be seen as initial value problems for first order ordinary differential equations) function $W(0;\cdot,\cdot):(2\Gamma_M,\infty)\times [0,\infty)\to{\mathbb R}$ defined by \begin{equation}\langlebel{DEFG} W(0;\langlembda,\mu):=\langlembda^{3/2}v_1'(0;\langlembda,\mu)-(g+\mathbb{S}igma\mu)v_1(0;\langlembda,\mu). \varepsilonnd{equation} We emphasize that the methods used in \cite{CM13xx, MM13x, W06b, W06a} in order to study the solutions of $W(0;\cdot,\cdot)=0$ cannot be used for general $L_r-$integrable vorticity functions. Indeed, the approach {chosen in the context of classical $C^{2+\alpha}-$solutions} in \cite{W06b, W06a} is based on regarding the Sturm-Liouville problem \varepsilonqref{E:m} as a non standard eigenvalue problem (the boundary condition depends on the eigenvalue $\mu$). For this, the author of \cite{W06b, W06a} introduces a Pontryagin space with a indefinite inner product and uses abstract results pertaining to this setting. In our context such considerations are possible only when restricting $r\geq 2.$ On the other hand, the methods used in \cite{CM13xx, MM13x} are based on direct estimates for the solution of \varepsilonqref{ERU}, but these estimates rely to a large extent on the boundedness of $\gamma.$ Therefore, we need to find a new approach when allowing for general $L_r-$integrable vorticity functions. {Our strategy is as follows: in a first step we find a constant $\langlembda_0\geq 2\Gamma_M $ such that the function $ W(p;\langlembda,\cdot)$ changes sign on $(0,\infty)$ for all $\langlembda>\langlembda_0,$ cf. Lemmas \maxthop{\rm Re}\nolimitsf{L:1} and \maxthop{\rm Re}\nolimitsf{L:4}. For this, the estimates established in Lemma \maxthop{\rm Re}\nolimitsf{L:3} within the setting of ordinary differential equations are crucial. In a second step, cf. Lemmas \maxthop{\rm Re}\nolimitsf{L:5} and \maxthop{\rm Re}\nolimitsf{L:6}, we prove that $ W(p;\langlembda,\cdot)$ changes sign exactly once on $(0,\infty)$, the particular value where $ W(p;\langlembda,\cdot)$ vanishes being called $\mu(\langlembda).$ The properties of the mapping $\langlembda\mapsto \mu(\langlembda)$ derived in Lemma \maxthop{\rm Re}\nolimitsf{L:6} are the core of the analysis of the kernel of $\partial_{\widetilde h}\mathcal{F}(\langlembda,0).$ } As a first result, we state the following lemma. \begin{lemma}\langlebel{L:1} There exists a unique minimal $\langlembda_0\geq 2\Gamma_M$ such that $W(0;\langlembda,0)>0$ for all $\langlembda>\langlembda_0.$ \varepsilonnd{lemma} \begin{proof} First, we note that given $(\langlembda,\mu)\in(2\Gamma_M,\infty)\times[0,\infty)$, the function $v_1$ satisfies the following integral relation \begin{equation}\langlebel{v1} v_1(p)=\int_{p_0}^p\frac{a^3(\langlembda;p_0)}{a^3(\langlembda;{s})}\, d{s}+\mu\int_{p_0}^p\frac{1}{a^3(\langlembda;s)}\int_{p_0}^sa(\langlembda;r)v_1(r)\, dr\, ds\qquad\text{for $p\in[p_0,0].$} \varepsilonnd{equation} Particularly, $v_1$ is a strictly increasing function on $[p_0,0]$. Furthermore, since $a(\langlembda;0)=\langlembda^{1/2},$ we get \begin{align*} W(0;\langlembda,0)=&a^3(\langlembda;p_0)-g\int_{p_0}^0\frac{a^3(\langlembda;p_0)}{a^3(\langlembda;p)}\, dp=a^3(\langlembda;p_0)\left(1-g\int_{p_0}^0\frac{1}{a^3(\langlembda;p)}\, dp\right)\to_{\langlembda\to\infty}\infty. \varepsilonnd{align*} This proves the claim. \varepsilonnd{proof} We note that if $g=0,$ then $\langlembda_0=2\Gamma_M.$ In the context of capillary-gravity water waves it is possible to choose, in the case of a bounded vorticity function, $\langlembda_0>2\Gamma_M$ as being the unique solution of the equation $W(0;\langlembda_0,0)=0$. In contrast, for certain unbounded vorticity functions $\gamma\in L_r((p_0,0)),$ with $r\in(1,\infty),$ the latter equation has no zeros in $(2\Gamma_M,\infty).$ Indeed, if we set $\gamma(p):=\delta(-p)^{-1/(kr)}$ for $p\in(p_0,0),$ where $\delta>0$ and $k,r\in(1,3) $ satisfy $kr<3,$ then $\gamma\in L_r((p_0,0))$ and, for sufficiently large $\delta $ (or small $p_0$), we have \begin{align*} \inf_{\langlembda>2\Gamma_M} W(0;\langlembda,0)>0. \varepsilonnd{align*} This property leads to restrictions on the wavelength of the water waves bifurcating from the laminar flow solutions found in Lemma \maxthop{\rm Re}\nolimitsf{L:LFS}, cf. Proposition \maxthop{\rm Re}\nolimitsf{P:3}. The estimates below will be used in Lemma \maxthop{\rm Re}\nolimitsf{L:4} to bound the integral mean and the first order moment of the solution $v_1$ of \varepsilonqref{ERU} on intervals $[p_1(\mu),0]$ with $p_1(\mu)\nearrow0$ as $\mu\to\infty.$ \begin{lemma}\langlebel{L:3} Let $p_1\in(p_0,0)$, $A, B\in(0,\infty),$ and $(\langlembda,\mu)\in(2\Gamma_M,\infty)\times[0,\infty)$ be fixed and define the positive constants \begin{equation}\langlebel{constants} \begin{aligned} &\underline C:=\min_{p\in[p_1,0]}\frac{a^3(\langlembda;p_1)}{a^3(\langlembda;p)},\quad \overline C:=\max_{p\in[p_1,0]}\frac{a^3(\langlembda;p_1)}{a^3(\langlembda;p)},\\ &\underline D:=\min_{s,p\in[p_1,0]}\frac{a(\langlembda;s)}{a^3(\langlembda;p)}, \quad \overline D:=\max_{s,p\in[p_1,0]}\frac{a(\langlembda;s)}{a^3(\langlembda;p)}. \varepsilonnd{aligned} \varepsilonnd{equation} Then, if $v\in W^2_r((p_1,0))$ is the solution of \begin{equation}\langlebel{EEE} \left\{\begin{array}{lll} (a^3(\langlembda) v')'-\mu a(\langlembda)v=0\qquad \text{in $L_r((p_1,0))$},\\[1ex] v(p_1)=A,\quad v'(p_1)=B, \varepsilonnd{array} \right. \varepsilonnd{equation} we have the following estimates \begin{align} \int_{p_1}^0v(p)\, dp\!\leq& -\frac{A\mu^{-1/2}\mathbb{S}inh(p_1\mathbb{S}qrt{\overline D}\mu^{1/2})}{\mathbb{S}qrt{\overline D}}+\frac{B\overline C\mu^{-1}\left(\cosh(p_1\mathbb{S}qrt{\overline D}\mu^{1/2})-1\right)}{\overline D},\langlebel{FE1}\\[1ex] \int_{p_1}^0(-p)v(p)\, dp\!\geq& \frac{A\mu^{-1 }\left(\cosh(p_1\mathbb{S}qrt{\underline D}\mu^{1/2})-1\right)}{\underline D}+\frac{B\underline C\mu^{-1}\!\left(\mathbb{S}qrt{\underline D}p_1 - \mathbb{S}inh(p_1\mathbb{S}qrt{\underline D}\mu^{1/2}) \mu^{-1/2}\right)}{\underline D^{3/2}}\langlebel{FE2}. \varepsilonnd{align} \varepsilonnd{lemma} \begin{proof} It directly follows from \varepsilonqref{EEE} that \begin{align}\langlebel{Der} v'(p)=\frac{a^3(\langlembda;p_1)}{a^3(\langlembda;p)}B+\mu\int_{p_1}^p\frac{a (\langlembda;s)}{a^3(\langlembda;p)}v(s)\, ds\qquad\text{for all $p\in[p_1,0],$} \varepsilonnd{align} and therefore \[ v'(p)\leq B\overline C+\mu \overline D\int_{p_1}^pv(s)\, ds \qquad\text{in $p\in[p_1,0],$} \] cf. \varepsilonqref{constants}. Letting now $\overline u:[p_0,0]\to{\mathbb R}$ be the function defined by \[ \overline u(p):=\int_{p_1}^pv(s)\, ds\qquad\text{for $p\in[p_1,0],$} \] we find that $\overline u\in W^3_r((p_0,0))$ solves the following problem \begin{equation*} \overline u''-\mu \overline D\overline u\leq B\overline C\quad \text{in $(p_1,0)$},\qquad \overline u(p_1)=0,\, \, \overline u'(p_1)=A. \varepsilonnd{equation*} It is not difficult to see that $\overline u\leq \overline z$ on $[p_1,0],$ where $\overline z$ denotes the solution of the initial value problem \begin{equation*} \overline z''-\mu \overline D \overline z= B\overline C\quad \text{in $(p_1,0)$},\qquad \overline z(p_1)=0,\, \, \overline z'(p_1)=A. \varepsilonnd{equation*} The solution $\overline z$ of this problem can be determined explicitly \begin{align*} \overline z(p)=\frac{A\mathbb{S}inh(\mathbb{S}qrt{\overline D}\mu^{1/2}(p-p_1))}{\mathbb{S}qrt{\overline D\mu}}+\frac{B\overline C\left(\cosh(\mathbb{S}qrt{\overline D}\mu^{1/2}(p-p_1))-1\right)}{\overline D\mu}, \qquad p\in[p_1,0], \varepsilonnd{align*} which gives, in virtue of $\overline u(0)\leq \overline z(0),$ the first estimate \varepsilonqref{FE1}. In order to prove the second estimate \varepsilonqref{FE2}, we first note that integration by parts leads us to \begin{align*} \int_{p_1}^0(-p)v(p)\,dp =\int_{p_1}^0\int_{p_1}^p v(s)\, ds\,dp \qquad\text{in $[p_1,0],$} \varepsilonnd{align*} so that it is natural to define the function $\underline u:[p_0,0]\to{\mathbb R}$ by the relation \[ \underline u(p):=\int_{p_1}^p\int_{p_1}^rv(s)\, ds\, dr\qquad\text{for $p\in[p_1,0].$} \] Recalling \varepsilonqref{Der}, we find similarly as before that \[ v'(p)\geq B\underline C+\mu \underline D\int_{p_1}^pv(s)\, ds \qquad\text{in $p\in[p_1,0],$} \] and integrating this inequality over $(p_1,p)$, with $p\in(p_1,0)$, we get \begin{align*} v(p)\geq A+B\underline C(p-p_1)+\mu \underline D\int_{p_1}^p\int_{p_1}^rv(s)\, ds\, dr \qquad\text{in $p\in[p_1,0].$} \varepsilonnd{align*} Whence, $\underline u\in W^4_r((p_0,0))$ solves the problem \begin{equation*} \underline u''-\mu \underline D\,\underline u\geq A+B\underline C(p-p_1)\quad \text{in $(p_1,0)$},\qquad \underline u(p_1)=0,\, \, \underline u'(p_1)=0. \varepsilonnd{equation*} As the right-hand side of the above inequality is positive, we find that $\underline u\geq \underline z$ on $[p_1,0],$ where $\underline z$ stands now for the solution of the problem \begin{equation*} \underline z''-\mu \underline D \, \underline z= A+B\underline C(p-p_1)\quad \text{in $(p_1,0)$},\qquad \underline z(p_1)=0,\, \, \underline z'(p_1)=0. \varepsilonnd{equation*} One can easily verify that $\underline z$ has the following expression \begin{align*} \underline z(p)=&\frac{A\left(\cosh(\mathbb{S}qrt{\underline D}\mu^{1/2}(p-p_1))-1\right)}{ \underline D\mu }\\ &+\frac{B\underline C\left( \underline D^{-1/2}\mathbb{S}inh(\mathbb{S}qrt{\underline D}\mu^{1/2}(p-p_1))\mu^{-1/2}-(p-p_1)\right)}{\underline D\mu} \varepsilonnd{align*} for $ p\in[p_1,0] $, and, since $\underline u(0)\geq \underline z(0),$ we obtain the desired estimate \varepsilonqref{FE2}. \varepsilonnd{proof} The estimates \varepsilonqref{FE1} and \varepsilonqref{FE2} are the main tools when proving the following result. \begin{lemma}\langlebel{L:4} Given $\langlembda> 2\Gamma_M,$ we have that \begin{align}\langlebel{ES} \lim_{\mu\to\infty} W(0;\langlembda,\mu)=-\infty. \varepsilonnd{align} \varepsilonnd{lemma} \begin{proof} Recalling the relations \varepsilonqref{DEFG} and \varepsilonqref{v1}, we write $W(0;\langlembda,\mu)=T_1+\mu T_2,$ whereby we defined \begin{align*} T_1&:=a^3(\langlembda;p_0)\left(1-(g+\mathbb{S}igma\mu)\int_{p_0}^0\frac{1}{a^3(\langlembda;p)}\, dp\right),\\ T_2&:=\int_{p_0}^0a(\langlembda;p)v_1(p)\, dp-(g+\mathbb{S}igma\mu)\int_{p_0}^0\frac{1}{a^3(\langlembda;s)}\int_{p_0}^sa(\langlembda;r)v_1(r)\, dr\, ds. \varepsilonnd{align*} Because $a(\langlembda)$ is a continuous and positive function that does on depend on $\mu$, it is easy to see that $T_1\to-\infty$ as $\mu\to\infty.$ In the remainder of this proof we show that \begin{equation}\langlebel{QE1} \lim_{\mu\to\infty} T_2=-\infty. \varepsilonnd{equation} In fact, since $a(\langlembda)$ is bounded from below and from above in $(0, \infty)$, we see, by using integration by parts, that \varepsilonqref{QE1} holds provided that there exists a constant $\beta\in(0,1)$ such that \begin{equation}\langlebel{QE2} \lim_{\mu\to\infty} \left( \int_{p_0}^0v_1(p)\, dp-\mu^{\beta}\int_{p_0}^0(-p)v_1(p)\, dp\right)=-\infty. \varepsilonnd{equation} We now fix $\beta\in(1/2,1)$ and prove that \varepsilonqref{QE2} is satisfied if we make this choice for $\beta.$ Therefore, we first choose $\gamma\in(1/2,\beta)$ with \begin{align}\langlebel{ch} \frac{2\beta-1}{2\gamma-1}=4. \varepsilonnd{align} Because for sufficiently large $\mu$ we have \begin{align*} \int_{p_0}^{-\mu^{-\gamma}}v_1(p)\, dp-\mu^{\beta}\int_{p_0}^{-\mu^{-\gamma}}(-p)v_1(p)\, dp\leq&\int_{p_0}^{-\mu^{-\gamma}}v_1(p)\, dp-\mu^{\beta}\int_{p_0}^{-\mu^{-\gamma}}\mu^{-\gamma}v_1(p)\, dp\\[1ex] =&(1-\mu^{\beta-\gamma})\int_{p_0}^{-\mu^{-\gamma}}v_1(p)\, dp\to_{\mu\to\infty}-\infty, \varepsilonnd{align*} we are left to show that \begin{align}\langlebel{QE3} \limsup_{\mu\to\infty}\left(\int_ {-\mu^{-\gamma}}^0v_1(p)\, dp-\mu^{\beta}\int_{-\mu^{-\gamma}}^0(-p)v_1(p)\, dp\right)<\infty. \varepsilonnd{align} The difficulty of showing \varepsilonqref{QE2} is mainly caused by the fact that the function $v_1$ grows very fast with $\mu.$ However, because the volume of the interval of integration in \varepsilonqref{QE3} decreases also very fast when $\mu\to\infty$, the estimates derived in Lemma \maxthop{\rm Re}\nolimitsf{L:3} are accurate enough to establish \varepsilonqref{QE3}. To be precise, for all $\mu>(-1/p_0)^{1/\gamma}$, we set $p_1:=-\mu^{-\gamma}$, $A:=v_1(p_1),$ $B:=v_1'(p_1)$, and obtain that the solution $v_1$ of \varepsilonqref{EEE} satisfies \begin{align}\langlebel{QE4} \int_ {-\mu^{-\gamma}}^0v_1(p)\, dp-\mu^{\beta}\int_{-\mu^{-\gamma}}^0(-p)v_1(p)\, dp\leq \frac{A \mathbb{S}inh(\mathbb{S}qrt{\overline D}\mu^{1/2-\gamma})}{\underline D\mu^{1/2}}E_1+\frac{B\underline C}{\underline D\mu}E_2, \varepsilonnd{align} whereby $A, B, \overline C,\underline C,\overline D,\underline D$ are functions of $\mu$ now, cf. \varepsilonqref{constants}, and \begin{align*} E_1&:=\frac{\underline D}{\mathbb{S}qrt{\overline D}}- \mu^{\beta-1/2 }\frac{ \cosh( \mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})-1 }{ \mathbb{S}inh(\mathbb{S}qrt{\overline D}\mu^{1/2-\gamma})},\\[1ex] E_2&:= \frac{\overline C\underline D}{\underline C\overline D} \left(\cosh(\mathbb{S}qrt{\overline D}\mu^{1/2-\gamma})-1\right) -\mu^{\beta -\gamma}\left(\frac{\mathbb{S}inh(\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})}{\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma}}-1\right). \varepsilonnd{align*} Recalling that $\gamma>1/2$ and that $A$,$B,$ $\overline C,\underline C,\overline D,\underline D$ are all positive, it suffices to show that $E_1$ and $E_2$ are negative when $\mu$ is large. In order to prove this property, we infer from \varepsilonqref{constants} that, as $\mu\to\infty,$ we have \[ \overline D\to \langlembda^{-1},\qquad \overline D\to \langlembda^{-1},\qquad \overline C\to 1, \qquad \underline C\to1. \] Moreover, using the substitution $t:=\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma} $ and l'Hospital's rule, we find \begin{align*} \lim_{\mu\to\infty}E_1=&1-\lim_{\mu\to\infty}\mu^{\beta-1/2 }\frac{ \cosh( \mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})-1 }{ \mathbb{S}inh(\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})}\frac{\mathbb{S}inh(\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})}{ \mathbb{S}inh(\mathbb{S}qrt{\overline D}\mu^{1/2-\gamma})}\\[1ex] =&1-\lim_{\mu\to\infty}\mu^{\beta-1/2 }\frac{ \cosh( \mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})-1 }{ \mathbb{S}inh(\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})} =1-\frac{1}{\langlembda^2} \lim_{t\mathbb{S}earrow0}\frac{ \cosh( t)-1 }{ t^4\mathbb{S}inh(t)}=-\infty, \varepsilonnd{align*} cf. \varepsilonqref{ch}, and by similar arguments \begin{align*} \lim_{\mu\to\infty}E_2=&-\lim_{\mu\to\infty}\mu^{\beta -\gamma}\left(\frac{\mathbb{S}inh(\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma})}{\mathbb{S}qrt{\underline D}\mu^{1/2-\gamma}}-1\right)=-\frac{1}{\langlembda^{3/2}} \lim_{t\mathbb{S}earrow0}\frac{ \mathbb{S}inh( t)-t }{ t^4}=-\infty. \varepsilonnd{align*} Hence, the right-hand side of \varepsilonqref{QE4} is negative when $\mu$ is sufficiently large, fact which proves the desired inequality \varepsilonqref{QE3}. \varepsilonnd{proof} Combining the Lemmas \maxthop{\rm Re}\nolimitsf{L:1} and \maxthop{\rm Re}\nolimitsf{L:4}, we see that the equation $W(0;\cdot,\cdot)=0$ has at least a solution for each $\langlembda>\langlembda_0.$ Concerning the sign of the first order derivatives $W_\langlembda(0;\cdot,\cdot)$ and $W_\mu(0;\cdot,\cdot)$ at the zeros of $W(0;\cdot,\cdot)$, which will be used below to show that $W(0;\cdot,\cdot)$ has a unique zero for each $\langlembda>\langlembda_0$, the results established for a H\"older continuous \cite{W06b, W06a} or for a bounded vorticity function \cite{CM13xx, MM13x} extend also to the case of a $L_r$-integrable vorticity function, without making any restriction on $r\in(1,\infty).$ \begin{lemma}\langlebel{L:5} Assume that $(\overline\langlembda,\overline\mu)\in(\langlembda_0,\infty)\times(0,\infty)$ satisfies $W(0; \overline\langlembda,\overline\mu)=0.$ Then, we have \begin{align}\langlebel{slim} W_\langlembda(0; \overline\langlembda,\overline\mu)>0\qquad\text{and}\qquad W_\mu(0; \overline\langlembda,\overline\mu)<0. \varepsilonnd{align} \varepsilonnd{lemma} \begin{proof} The Proposition \maxthop{\rm Re}\nolimitsf{P:2} and the discussion following it show that $\mathop{\rm Ker}\nolimits R_{\overline\langlembda,\overline\mu} =\mathbb{S}pa\{v_1\}$, whereby $v_1:=v_1(\cdot;\overline\langlembda,\overline\mu)$. To prove the first claim, we note that the algebra property of $W^1_r((p_0,0))$ yields that the partial derivative $v_{1,\langlembda}:=\partial_\langlembda v_{1}(\cdot,\overline\langlembda,\overline\mu)$ belongs to $W^2_r((p_0,0))$ and solves the problem \begin{equation}\langlebel{v1l} \left\{\begin{array}{lll} (a^{3}(\overline\langlembda)v_{1,\langlembda}')'-\overline\mu a(\overline\langlembda) v_{1,\langlembda}= -(3a^2(\overline\langlembda)a_{\langlembda}(\overline\langlembda)v_1')'+\overline\mu a_{\langlembda}(\overline\langlembda)v_1\qquad\text{in $ L_{r}((p_0,0))$,}\\[1ex] v_{1,\langlembda}(p_0)= v_{1,\langlembda}'(p_0)=0, \varepsilonnd{array}\right. \varepsilonnd{equation} where $a_{\langlembda}(\overline\langlembda)=1/(2a(\overline\langlembda))$. Because of the embedding $W^2_r((p_0,0))\hookrightarrow C^{1+\alpha}([p_0,0]),$ we find, by multiplying the differential equation satisfied by $v_1$, cf. \varepsilonqref{ERU}, with $v_{1,\langlembda}$ and the first equation of \varepsilonqref{v1l} with $v_1$, and after subtracting the resulting relations the first claim of \varepsilonqref{slim} \begin{align*} W_{\langlembda}(0;\overline\langlembda,\overline\mu)&=\overline\langlembda^{3/2}v_{1,\langlembda}'(0)+\frac{3}{2}\overline\langlembda^{1/2}v_1'(0)-(g+\mathbb{S}igma\overline\mu)v_{1,\langlembda}(0)\\ &=\frac{1}{v_1(0)}\left(\int_{p_0}^{0}\frac{3a(\overline\langlembda)}{2} v_1'^{ 2}+\frac{\overline\mu}{2a(\overline\langlembda)}v_1^2\, dp\right)>0.\varepsilonnd{align*} For the second claim, we find as above that $v_{1,\mu}:=\partial_\mu v_{1}(\cdot,\overline\langlembda,\overline\mu)\in W^2_r((p_0,0))$ is the unique solution of the problem \begin{equation}\langlebel{v1mu} \left\{\begin{array}{lll}(a^{3}(\overline\langlembda)v_{1,\mu}')'-\overline\mu a(\overline\langlembda) v_{1,\mu}=a(\overline\langlembda)v_1\qquad \text{in $ L_{r}((p_0,0))$},\\[1ex] v_{1,\mu}(p_0)=v_{1,\mu}'(p_0)=0. \varepsilonnd{array}\right. \varepsilonnd{equation} Also, if we multiply the differential equation satisfied by $v_1$ with $v_{1,\mu}$ and the first equation of \varepsilonqref{v1mu} with $v_1$, we get after building the difference of these relations \begin{align*} \int_{p_0}^0\!a(\overline\langlembda)v_1^2\, dp=\overline\langlembda^{3/2}\!v_{1,\mu}'(0)v_1(0)-\overline\langlembda^{3/2}\!v_1'(0)v_{1,\mu}(0)=v_1(0)\left(\!\overline\langlembda^{3/2}v_{1,\mu}'(0)-(g+\mathbb{S}igma\overline \mu)v_{1,\mu}(0)\!\right)\!, \varepsilonnd{align*} the last equality being a consequence of the fact that $v_1$ and $v_2:=v_2(\cdot;\overline\langlembda,\overline\mu) $ are collinear for this choice of the parameters. Therefore, we have \begin{align}\langlebel{qqq} W_\mu(0;\overline\langlembda,\overline\mu)=\overline\langlembda^{3/2}v_{1,\mu}'(0)-\mathbb{S}igma v_1(0)-(g+\mathbb{S}igma\overline\mu)v_{1,\mu}(0)=\frac{1}{v_1(0)} \left(\int_{p_0}^0a(\overline\langlembda)v_1^2\, dp-\mathbb{S}igma v_1^2(0)\right). \varepsilonnd{align} In order to determine the sign of the latter expression, we multiply the first equation of \varepsilonqref{ERU} by $v_1$ and get, by using once more the collinearity of $v_1$ and $v_2,$ that \[ \int_{p_0}^0a(\overline\langlembda)v_1^2\, dp-\mathbb{S}igma v_1^2(0)=\frac{1}{\overline\mu}\left(gv_1^2(0)-\int_{p_0}^0a^3(\overline\langlembda)v_1'^2\, dp\right). \] If $g=0$, the latter expression is negative and we are done. On the other hand, if we consider gravity effects, because of $\overline\mu>0,$ it is easy to see that $a^{3/2}(\overline\langlembda)v_1'$ and $a^{-3/2}(\overline\langlembda)$ are linearly independent functions, fact which ensures together with Lemma \maxthop{\rm Re}\nolimitsf{L:1} and with H\"older's inequality that \begin{align*} gv_1^2(0)&=g\left(\int_{p_0}^0a^{3/2}(\overline\langlembda)v_1'\frac{1}{a^{3/2}(\overline\langlembda)}\, dp\right)^2\\ &<g\left(\int_{p_0}^0a^{3 }(\overline\langlembda)v_1'^2 \, dp\right)\left(\int_{p_0}^0 \frac{1}{a^{3 }(\overline\langlembda)}\, dp\right)\leq \int_{p_0}^0a^{3 }(\overline\langlembda)v_1'^2 \, dp, \varepsilonnd{align*} and the desired claim follows from \varepsilonqref{qqq}. \varepsilonnd{proof} We conclude with the following result. \begin{lemma}\langlebel{L:6} Given $\langlembda>\langlembda_0,$ there exists a unique zero $\mu=\mu(\langlembda)\in (0,\infty)$ of the equation $W(0;\langlembda,\mu(\langlembda))=0.$ The function \[\mu:(\langlembda_0,\infty)\to(\inf_{(\langlembda_0,\infty)}\mu(\langlembda),\infty),\qquad \langlembda\mapsto\mu(\langlembda)\] is strictly increasing, real-analytic, and bijective. \varepsilonnd{lemma} \begin{proof} Given $\langlembda>\langlembda_0,$ it follows from the Lemmas \maxthop{\rm Re}\nolimitsf{L:1} and \maxthop{\rm Re}\nolimitsf{L:4} that there exists a constant $\mu(\langlembda)>0$ such that $W(0;\langlembda,\mu(\langlembda))=0.$ The uniqueness of this constant, and the real-analyticity and the monotonicity of $\langlembda\mapsto\mu(\langlembda)$ follow readily from Lemma \maxthop{\rm Re}\nolimitsf{L:5} and the implicit function theorem. To complete the proof, let us assume that we found a sequence $\langlembda_n\to\infty$ such that $(\mu(\langlembda_n))_n$ is bounded. Denoting by $v_{1n}$ the (strictly increasing) solution of \varepsilonqref{ERU} when $(\langlembda,\mu)=(\langlembda_n,\mu(\langlembda_n)),$ we infer from \varepsilonqref{v1} that there exists a constant $C>0$ such that \[ v_{1n}(p)\leq C\left(1+\int_{p_0}^pv_{1n}(s)\, ds\right)\qquad\text{for all $n\geq 1$ and $p\in[p_0,0].$} \] Gronwall's inequality yields that the sequence $(v_{1n})_n$ is bounded in $C([p_0,0])$ and, together with \varepsilonqref{v1}, we find that \[0=W(0;\langlembda_n,\mu(\langlembda_n))\geq a^3(\langlembda_n;p_0)-(g+\mathbb{S}igma\mu(\langlembda_n))v_{1n}(0)\underlinederset{n\to\infty}\to\infty.\] This is a contradiction, and the proof is complete. \varepsilonnd{proof} We choose now the integer $N$ from Theorem \maxthop{\rm Re}\nolimitsf{T:MT}, to be {the smallest positive integer which} satisfies \begin{equation}\langlebel{eq:rest} { N^2}>\inf_{(\langlembda_0,\infty)}\mu(\langlembda). \varepsilonnd{equation} Invoking Lemma \maxthop{\rm Re}\nolimitsf{L:6}, we {find a sequence $(\langlembda_n)_{n\geq N}\mathbb{S}ubset (\langlembda_0,\infty)$ having the properties that $\langlembda_n\nearrow\infty$ and \begin{equation}\langlebel{eq:sec} \text{$\mu(\langlembda_n)=n^2$ \qquad for all $n\geq N.$} \varepsilonnd{equation}} We conclude the previous analysis with the following result. \begin{prop}\langlebel{P:3} Let {$N\in{\mathbb N}$ be defined by \varepsilonqref{eq:rest}. Then, for each $n\geq N $, the Fr\'echet derivative $\partial_{\widetilde h} \mathcal{F}(\langlembda_n,0)\in\mathcal{L}(X,Y)$, with $\langlembda_n$ defined by \varepsilonqref{eq:sec}, is a Fredholm operator of index zero with a one-dimensional kernel $\mathop{\rm Ker}\nolimits\partial_{\widetilde h} \mathcal{F}(\langlembda_n,0)=\mathbb{S}pa\{w_n\}$, whereby $w_n\in X$ is the function $w_n(q,p):=v_1(p;\langlembda_n,n^2)\cos(nq)$ for all $(q,p)\in\overline\Omega.$} \varepsilonnd{prop} \begin{proof} The result is a consequence of the Lemmas \maxthop{\rm Re}\nolimitsf{L:2} and \maxthop{\rm Re}\nolimitsf{L:6}, and of Proposition \maxthop{\rm Re}\nolimitsf{P:2}. \varepsilonnd{proof} In order to apply the theorem on bifurcations from simple eigenvalues to the equation \varepsilonqref{BP}, we still have to verify the transversality condition \begin{equation}\langlebel{eq:TC} \partial_{\langlembda {\widetilde h}}\mathcal{F}(\langlembda_n,0)[w_n]\notin\mathop{\rm Im}\nolimits \partial_{\widetilde h} \mathcal{F}(\langlembda_n,0) \varepsilonnd{equation} for $n\geq N.$ \begin{lemma}\langlebel{L:TC} The transversality condition \varepsilonqref{eq:TC} is satisfied for all {$n\geq N$}. \varepsilonnd{lemma} \begin{proof} The proof is similar to that of the Lemmas 4.4 and 4.5 in \cite{MM13x}, and therefore we omit it. \varepsilonnd{proof} We come to the proof of our main existence result. \begin{proof}[Proof of Theorem \maxthop{\rm Re}\nolimitsf{T:MT}] {Let $N$ be defined by \varepsilonqref{eq:rest}, and let $(\langlembda_n)_{n\geq N}\mathbb{S}ubset (\langlembda_0,\infty)$ be the sequence defined by \varepsilonqref{eq:sec}.} Invoking the relations \varepsilonqref{BP0}, \varepsilonqref{BP1}, the Proposition \maxthop{\rm Re}\nolimitsf{P:3}, and the Lemma \maxthop{\rm Re}\nolimitsf{L:TC}, we see that all the assumptions of the theorem on bifurcations from simple eigenvalues of Crandall and Rabinowitz \cite{CR71} are satisfied for the equation \varepsilonqref{BP} at each of the points $\langlembda=\langlembda_n,$ $n\geq N.$ Therefore, for each $n\geq N$, there exists $\varepsilon_n>0$ and a real-analytic curve \[\text{$(\widetilde \langlembda_n,{\widetilde h}_n):(\langlembda_n-\varepsilon_n,\langlembda_n+\varepsilon_n)\to (2\Gamma_M,\infty)\times X,$ }\] consisting only of solutions of the problem \varepsilonqref{BP}. Moreover, as $s\to0$, we have that \begin{equation}\langlebel{asex} \widetilde\langlembda_n(s)=\langlembda_n+O(s)\quad \text{in ${\mathbb R}$},\qquad {\widetilde h}_n(s)=sw_n+O(s^2)\quad \text{in $X$}, \varepsilonnd{equation} whereby $w_n\in X$ is the function defined in Proposition \maxthop{\rm Re}\nolimitsf{P:3}. Furthermore, in a neighborhood of $(\langlembda_n,0),$ the solutions of \varepsilonqref{BP} are either laminar or are located on the local curve $(\widetilde\langlembda_n,{\widetilde h}_n)$. The constants $\varepsilon_n$ are chosen sufficiently small to guarantee that $H(\cdot;\widetilde\langlembda_n(s))+{\widetilde h}_n(s)$ satisfies \varepsilonqref{PBC} for all $|s|<\varepsilon_n$ and all $n\geq N.$ For each integer $n\geq N,$ the curve ${\mathcal{C}_{n}}$ mentioned in Theorem \maxthop{\rm Re}\nolimitsf{T:MT} is parametrized by $[s\mapsto H(\cdot;\widetilde\langlembda_n(s))+{\widetilde h}_n(s)]\in C^\omega((-\varepsilon_n,\varepsilon_n),X).$ We pick now a function $h$ on one of the local curves ${\mathcal{C}_{n}}$. In order to show that this weak solution of \varepsilonqref{WF} belongs to $ W^2_r(\Omega)$, we first infer from Theorem 5.1 in \cite{MM13x} that the distributional derivatives $\partial_q^mh$ also belong to $C^{1+\alpha}(\overline\Omega)$ for all $m\geq1.$ Using the same arguments as in the last part of the proof of Theorem \maxthop{\rm Re}\nolimitsf{T:EQ}, we find that $h\in C^{1+\alpha}(\overline\Omega)\cap W^2_r(\Omega)$ satisfies the first equation of \varepsilonqref{PB} in $L_r(\Omega)$. Because $(1-\partial_q^2)^{-1}\in \mathcal{L}(C^\alpha(\mathbb{S}), C^{2+\alpha}(\mathbb{S}))$, the equation \varepsilonqref{PB0} yields that $\mathop{\rm tr}\nolimits_0 h\in C^{2+\alpha}(\mathbb{S})$, and therefore $h$ is a strong solution of \varepsilonqref{PB}. Moreover, by \cite[Corollary 5.2]{MM13x}, result which shows that the regularity properties of the streamlines of classical solutions \cite{Hen10, DH12} persist even for weak solutions with merely integrable vorticity, $[q\mapsto h(q,p)]$ is a real-analytic map for any $p\in[p_0,0]$. Finally, because of \varepsilonqref{asex}, it is not difficult to see that any solution $h=H(\cdot;\widetilde \langlembda_n(s))+{\widetilde h}_n(s)\in{\mathcal{C}_{n}},$ with $s\neq0 $ sufficiently small, corresponds to waves that possess a single crest per period and which are symmetric with respect to the crest (and trough) line. \varepsilonnd{proof} As noted in the discussion following Lemma \maxthop{\rm Re}\nolimitsf{L:1}, when $r\in(1,3),$ there are examples of vorticity functions $\gamma\in L_r((p_0,0))$ for which the mapping $\langlembda\mapsto\mu(\langlembda)$ defined in Lemma \maxthop{\rm Re}\nolimitsf{L:6} is bounded away from zero on $(\langlembda_0,\infty)$. This property imposes restrictions (through the { positive integer $N$}) on the wave length of the water waves solutions bifurcating from the laminar flows, cf. Theorem \maxthop{\rm Re}\nolimitsf{T:MT}. The lemma below gives, in the context of capillary-gravity waves, sufficient conditions which ensure that $\mu:(\langlembda_0,\infty)\to(0,\infty)$ is a bijective mapping, which corresponds to the choice $N=1$ in Theorem \maxthop{\rm Re}\nolimitsf{T:MT}, situation when no restrictions are needed. On the other hand, when considering pure capillary waves and if $\mu:(\langlembda_0,\infty)\to(0,\infty)$ is a bijective mapping, then necessarily $\Gammaamma_M=\Gamma(p_0),$ and the problems \varepsilonqref{ERU} and \varepsilonqref{ERUa} become singular as $\langlembda\to \langlembda_0=2\Gamma_M.$ Therefore, finding sufficient conditions in this setting appears to be much more involved. \begin{lemma}\langlebel{L:9} Let $r\geq3$, $\gamma\in L_r((p_0,0))$ and assume that $g>0$. Then, $\langlembda_0>2\Gamma_M$ and {the integer $N$ in Theorem \maxthop{\rm Re}\nolimitsf{T:MT} satisfies $N=1$}, provided that \begin{align}\langlebel{eq.condCG} \int_{p_0}^0a(\langlembda_0)\left(\int_{p_0}^p\frac{1}{a^3(\langlembda_0;s)}\, ds\right)^2\, dp<\frac{\mathbb{S}igma}{g^2}. \varepsilonnd{align} \varepsilonnd{lemma} \begin{proof} Let us assume that $\Gamma(p_1)=\Gamma_M$ for some $p_1\in[p_0,0)$ (the case when $p_1=0$ is similar). Then, if $\delta<1$ is such that $p_1+\delta<0,$ we have \begin{align*} \lim_{\langlembda\mathbb{S}earrow2\Gamma_M}\int_{p_0}^0 \frac{dp}{a^3(\langlembda;p)}&=\lim_{\varepsilon\mathbb{S}earrow 0}\int_{p_0}^0\frac{dp}{\mathbb{S}qrt{\varepsilon+2(\Gammaamma(p_1)-\Gammaamma(p))}^3}\geq c\lim_{\varepsilon\mathbb{S}earrow 0}\int_{p_1}^{p_1+\delta}\frac{dp}{\varepsilon^{3/2}+\left|\int_{p_1}^p\gamma(s)\, ds\right|^{3/2}}\\ & \geq c\lim_{\varepsilon\mathbb{S}earrow 0}\int_{p_1}^{p_1+\delta}\!\frac{dp}{\varepsilon^{3/2}+\left\| \gamma \right\|_{L_r}^{3/2}|p-p_1|^{3\alpha/2}}\geq c\lim_{\varepsilon\mathbb{S}earrow 0}\int_{p_1}^{p_1+\delta}\!\frac{dp}{\varepsilon+p-p_1}=\infty \varepsilonnd{align*} with $\alpha=(r-1)/r$ and with $c$ denoting positive constants that are independent of $\varepsilon$. We have used the relation $3\alpha/2\geq1$ for $r\geq3.$ In view of Lemma \maxthop{\rm Re}\nolimitsf{L:1}, we find that $\langlembda_0>2\Gamma_M$ is the unique zero of $W(0;\cdot,0).$ Recalling now \varepsilonqref{qqq} and the relation \varepsilonqref{v1}, one can easily see, because of $W(0;\langlembda_0,0)=0,$ that the condition \varepsilonqref{eq.condCG} yields $W_\mu(0;\langlembda_0,0)<0$. Since Lemma \maxthop{\rm Re}\nolimitsf{L:6} implies $W(0;\langlembda_0, \inf_{(\langlembda_0,\infty)}\mu)=0,$ the relation $W_\mu(0;\langlembda_0,0)<0$ together with Lemma \maxthop{\rm Re}\nolimitsf{L:5} guarantee that $\inf_{(\langlembda_0,\infty)}\mu=0$. This proves the claim. \varepsilonnd{proof} \begin{thebibliography}{10} \bibitem{A75} R.~A. Adams. \newblock {\varepsilonm {Sobolev spaces}}. \newblock Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. \newblock Pure and Applied Mathematics, Vol. 65. \bibitem{B62} B.~T. 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\begin{document} \title{On Carlson's depth conjecture in group cohomology} \author[D.~J. Green]{David J. Green} \address{Dept of Mathematics \\ Univ.\@ of Wuppertal \\ D--42097 Wuppertal \\ Germany} \email{green@math.uni-wuppertal.de} \subjclass{Primary 20J06} \date{5 June 2002} \begin{abstract} We establish a weak form of Carlson's conjecture on the depth of the mod-$p$ cohomology ring of a $p$-group. In particular, Duflot's lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups. The proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map. We demonstrate the existence of systems of parameters (so-called polarised systems) which are particularly well adapted to this comodule structure. \end{abstract} \maketitle \section*{Introduction} \noindent Let $G$ be a finite $p$-group and $C$ its greatest central elementary abelian subgroup. Write $k$ for the prime field $\f$. Cohomology will always be with coefficients in~$k$. Denote by $r$ the $p$-rank of~$G$, and by $z$ the $p$-rank of $C$. Following Broto and Henn~\cite{BrHe:Central} we shall exploit the fact that the multiplication map $\mu \colon G \times C \rightarrow G$, $(g,c) \mapsto g.c$ is a group homomorphism. The main result of this paper is as follows: \begin{theorem} \label{theorem:introED1} Suppose that $G$ is a $p$-group whose centre has $p$-rank $z$. Then the following statements are equivalent: \begin{enumerate} \item The mod-$p$ cohomology ring $\coho{G}$ is not detected on the centralizers of its rank $z+1$ elementary abelian subgroups. \item There is an associated prime $\mathfrak{p}$ such that $\coho{G}/\mathfrak{p}$ has dimension $z$. \item The depth of $\coho{G}$ equals $z$. \end{enumerate} \end{theorem} \noindent This is a special case of a conjecture due to Carlson~\cite{Carlson:DepthTransfer}, reproduced here as Conjecture~\ref{conjecture:ED}. Recall that Duflot proved in~\cite{Duflot:Depth} that $z$ is a lower bound for the depth. So this result characterises the cases where Duflot's lower bound is tight. Theorem~\ref{theorem:introED1} is proved as Theorem~\ref{theorem:ED1}. The proof rests upon the existence of \emph{polarised systems}: homogeneous systems of parameters for $\coho{G}$ which are particularly well adapted to the $\coho{C}$-comodule structure. There are two extreme types of behaviour which a cohomology class $x \in \coho{G}$ can demonstrate under the comodule structure map $\mu^*$: one extreme is that $\operatorname{Res}_C(x)$ is nonzero, and so $\mu^*(x) = 1 \otimes \operatorname{Res}_C(x) + \text{other terms}$. The other extreme is that $x$ is primitive, meaning that $\mu^*(x) = x \otimes 1$. Roughly speaking, a polarised system of parameters is one consisting solely of elements which each exhibit one or the other of these extreme kinds of behaviour. The precise definition, which ensures that each such system is a detecting sequence for the depth of $\coho{G}$, is given in Definition~\ref{definition:polarised}\@. Polarised systems of parameters always exist, as is proved in Theorem~\ref{theorem:existence}\@. \noindent In addition to Theorem~\ref{theorem:introED1} we also prove a weak form of the general case of Carlson's conjecture. This is done in Theorem~\ref{theorem:polarisedEqualities}, which includes the following statement: \begin{theorem} \label{theorem:introMyEDgen} Let $\mathbb{Z}eta_1$, \dots, $\mathbb{Z}eta_z$, $\kappa_1$, \dots, $\kappa_{r-z}$ be a polarised system of parameters for $\coho{G}$. Then $\coho{G}$ has depth $z + \sa$, where $\sa \in \{0,\ldots,r-z\}$ is defined by \[ \sa = \max \{ s \leq r-z \mid \text{$\kappa_1,\ldots,\kappa_s$ is a regular sequence in $\coho{G}$} \} \, . \] \end{theorem} \section{Primitive comodule elements} \label{section:primitive} \noindent Group multiplication $\mu \colon G \times C \rightarrow G$ is a group homomorphism. As observed by Broto and Henn~\cite{BrHe:Central}, this means that $\coho{G}$ inherits the structure of a comodule over the coalgebra $\coho{C}$. Recall that $x \in \coho{G}$ is called a primitive comodule element if $\mu^*(x) = x \otimes 1 \in \coho{G \times C} \cong \coho{G} \otimes_k \coho{C}$. As the comodule structure map $\mu^*$ is simultaneously a ring homomorphism, it follows that the primitives form a subalgebra $\PH{G}$ of $\coho{G}$. As the quotient map $G \rightarrow G/C$ coequalises $\mu$ and projection onto the first factor of $G \times C$, it follows that the image of inflation from $\coho{G/C}$ is contained in $\PH{G}$. \begin{lemma} \label{lemma:quotientComodule} Suppose that $I$ is a homogeneous ideal in $\coho{G}$ which is generated by primitive elements. Then \[ \mu^*(I) \subseteq I \otimes_k \coho{C} \, , \] and so $\mu^*$ induces a ring homomorphism \[ \lambda \colon \coho{G}/I \rightarrow \coho{G}/I \otimes_k \coho{C} \] which induces an $\coho{C}$-comodule structure on $\coho{G}/I$. \end{lemma} \begin{proof} For $x \in I$ and $y \in \coho{G}$ one has $\mu^*(xy) = \mu^*(x) \mu^*(y) = (x \otimes 1) \mu^*(y) \in I \otimes_k \coho{C}$. \end{proof} \begin{lemma} \label{lemma:myBrotoCarlsonHenn} Suppose that $\mathbb{Z}eta_1, \ldots, \mathbb{Z}eta_t$ is a sequence of homogeneous elements of $\coho{G}$ whose restrictions form a regular sequence in $\coho{C}$, and suppose that $I$~is an ideal in $\coho{G}$ generated by primitive elements. Then $\mathbb{Z}eta_1, \ldots, \mathbb{Z}eta_t$ is a regular sequence for the quotient ring $\coho{G}/I$. \end{lemma} \begin{proof} Carlson's proof for the case $I=0$ (Proposition 5.2 of~\cite{Carlson:Problems}) generalises easily. Denote by $R$ the polynomial algebra $k[\mathbb{Z}eta_1, \ldots, \mathbb{Z}eta_t]$. The map $\lambda$~of Lemma~\ref{lemma:quotientComodule} induces an $R$-module structure on $\coho{G}/I \otimes_k \coho{C}$, and $\lambda$~is a split monomorphism of $R$-modules, the splitting map being induced by projection onto the first factor $G \times C \rightarrow G$. So as an $R$-module $\coho{G}/I$ is a direct summand of $\coho{G}/I \otimes_k \coho{C}$. The result will therefore follow if we can show that $\coho{G}/I \otimes_k \coho{C}$ is a free $R$-module. To see that $\coho{G}/I \otimes_k \coho{C}$ is indeed a free $R$-module set \[ F_i := \sum_{j \geq i} \coho[j]{G}/(I \cap \coho[j]{G}) \] and observe that $F_i \otimes_k \coho{C}$ is an $R$-submodule of $\coho{G}/I \otimes_k \coho{C} = F_0 \otimes_k \coho{C}$. Projection $G \times C \rightarrow C$ makes $F_i \otimes_k \coho{C} / F_{i+1} \otimes_k \coho{C}$ a free $\coho{C}$-module. Now for $x \in F_i$, $y \in \coho{C}$ and $\theta \in R$ we have $\theta.(x \otimes y) \in x \otimes (\operatorname{Res}^G_C \theta). y + F_{i+1} \otimes_k \coho{C}$. So the $R$-module structure on $F_i \otimes_k \coho{C} / F_{i+1} \otimes_k \coho{C}$ is induced by the restriction map $R \rightarrow \coho{G} \rightarrow \coho{G}/I \rightarrow \coho{C}$ from the free $\coho{C}$-structure. But $\coho{C}$ is a free $R$-module by Theorem 10.3.4 of \cite{Evens:book}, because the restrictions of $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_t$ form a regular sequence. So $F_i \otimes_k \coho{C} / F_{i+1} \otimes_k \coho{C}$ is a free $R$-module for all~$i$, whence it follows that $F_0 \otimes_k \coho{C} / F_i \otimes_k \coho{C}$ is a free $R$-module for all~$i$. As the degree of each homogeneous element of $F_i \otimes_k \coho{C}$ is at least~$i$, it follows that $\coho{G}/I \otimes_k \coho{C}$ is itself a free $R$-module. \end{proof} \begin{corollary} \label{coroll:myBrotoCarlsonHenn} Let $G$ be a $p$-group whose centre has $p$-rank~$z$. Suppose that there is a length~$s$ regular sequence in $\coho{G}$ which consists entirely of primitive elements. Then the depth of $\coho{G}$ is at least $z + s$. \end{corollary} \begin{proof} Let $I$ be the ideal generated by the primitive elements in the regular sequence. Let $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$ be elements of $\coho{G}$ whose restrictions form a homogeneous system of parameters for $\coho{C}$: it is well known that such sequences exist. Now apply Lemma~\ref{lemma:myBrotoCarlsonHenn}. One concrete example of such classes $\mathbb{Z}eta_i$ is obtained as follows. Let $\rho_G$ be the regular representation of~$G$, and $\mathbb{Z}eta_i$ the Chern class $c_{p^n - p^{n-i}}(\rho_G)$ for $1 \leq i \leq z$, where $p^n$ is the order of~$G$. Then $\rho_G$ restricts to~$C$ as $|G:C|$ copies of the regular representation $\rho_C$, whence $\operatorname{Res}_C(\mathbb{Z}eta_i) = c_{p^z - p^{z-i}}(\rho_C)^{|G:C|}$. But the $c_{p^z - p^{z-i}}(\rho_C)$ are (up to a sign) the Dickson invariants. See the proof of Theorem~\ref{theorem:existence} for more details. \end{proof} \section{Polarised systems of parameters} \label{section:polarised} \noindent We shall now give the definition of a polarised system of parameters, the key definition of this paper. In fact we shall introduce two closely related concepts: the axioms for a polarised system (Definition~\ref{definition:polarised}) are easily checked in practice, whereas the special polarised systems of Definition~\ref{definition:specialPolarised} have precisely the properties we shall need to investigate depth. Lemma~\ref{lemma:polarisedDefinitions} shows that the two concepts are more or less interchangeable. \begin{definition} \label{definition:ACG} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$. Denote by $C$ the greatest central elementary abelian subgroup of~$G$, and set \begin{align*} \mathcal{A}^C(G) & := \{ V \leq G \mid \text{$V$ is elementary abelian and contains~$C$} \} \, , \\ \mathcal{A}^C_d(G) & := \{ V \in \mathcal{A}^C(G) \mid \text{$V$ has $p$-rank $d$} \} \, \\ \mathcal{H}^C_d(G) & := \{ C_G(V) \mid V \in \mathcal{A}^C_d(G) \} \, . \end{align*} So $\mathcal{A}^C(G)$ is the disjoint union of the $\mathcal{A}^C_{z+s}(G)$ for $0 \leq s \leq r-z$. \end{definition} \begin{definition} \label{definition:polarised} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$. Recall that inflation map $\operatorname{Inf} \colon \coho{V/C} \rightarrow \coho{V}$ is a split monomorphism for each $V \in \mathcal{A}^C(G)$, and so its image $\operatorname{Im} \operatorname{Inf}$ is isomorphic to $\coho{V/C}$. A system $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1,\ldots,\kappa_{r-z}$ of homogeneous elements of $\coho{G}$ shall be called a polarised system of parameters if the following four axioms are satisfied. \begin{description} \item[(PS1)] $\operatorname{Res}_C(\mathbb{Z}eta_1)$, \dots, $\operatorname{Res}_C(\mathbb{Z}eta_z)$ is a system of parameters for $\coho{C}$. \item[(PS2)] $\operatorname{Res}_V(\kappa_j)$ lies in $\operatorname{Im} \operatorname{Inf}$ for each $1 \leq j \leq r-z$ and for each $V \in \mathcal{A}^C(G)$. \item[(PS3)] For each $V \in \mathcal{A}^C(G)$, the restrictions $\operatorname{Res}_V(\kappa_1), \ldots, \operatorname{Res}_V(\kappa_s)$ constitute a system of parameters for $\operatorname{Im} \operatorname{Inf}$. Here $z+s$ is the rank of~$V$. \item[(PS4)] $\operatorname{Res}_V(\kappa_j) = 0$ for $V \in \mathcal{A}^C_{z+s}(G)$ with $0 \leq s < j \leq r-z$. \end{description} \end{definition} \begin{remark} Polarised systems of parameters always exist, as we shall see in Theorem~\ref{theorem:existence}\@. Observe that Axiom (PS1) involves only the $\mathbb{Z}eta_i$, whereas the remaining axioms involve only the $\kappa_j$. Basically Axiom (PS1) says that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$ is a regular sequence which can be detected on the centre, (PS2) says that the $\kappa_j$ are primitive after raising to suitably high $p$th powers, and (PS3) says that the $\kappa_j$ together with the $\mathbb{Z}eta_i$ will form a detecting sequence for the depth of $\coho{G}$. Axiom (PS4) is a more technical condition which we shall only use once: it is needed in Lemma~\ref{lemma:polarisedDefinitions} to show that, after raising to a suitably high $p$th power, each $\kappa_j$ is a sum of transfer classes as required by Axiom (PS5) below. \end{remark} \begin{lemma} Polarised systems of parameters for $\coho{G}$ are indeed systems of parameters. \end{lemma} \begin{proof} Let $V \in \mathcal{A}^C_{z+s}(G)$. The restrictions of $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1,\ldots, \kappa_s$ constitute a system of parameters for~$\coho{V}$ by (PS1) and (PS3)\@. Hence $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1, \ldots, \kappa_{r-z}$ are algebraically independent over~$k$, for we may choose $V$ to have $p$-rank~$r$. Now let $\gamma$ be a homogeneous element of $\coho{G}$. For $V \in \mathcal{A}^C(G)$ there is a monic polynomial $f_V(x)$ with coefficients in $k[\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1, \ldots, \kappa_{r-z}]$ such that $f_V(\gamma)$ has zero restriction to~$V$. Taking the product of all such polynomials one obtains a polynomial $f(x)$ such that $f(\gamma)$ has zero restriction to each maximal elementary abelian subgroup of~$G$. So $f(\gamma)$ is nilpotent by a well-known result of Quillen. \end{proof} \begin{definition} \label{definition:specialPolarised} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$. A system $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1,\ldots,\kappa_{r-z}$ of homogeneous elements of $\coho{G}$ shall be called a special polarised system of parameters if it satisfies the following five axioms: (PS1), (PS3), (PS4) and \begin{description} \item[(PS$\mathbf{2'}$)] $\kappa_j$ is a primitive element of the $\coho{C}$-comodule $\coho{G}$ for each $1 \leq j \leq r-z$. \item[(PS5)] $\kappa_j$ lies in $\sum_{H \in \mathcal{H}^C_{z+i}(G)} \operatorname{Tr}^G_H(\coho{H})$ for each $1 \leq i \leq j \leq r-z$. \end{description} \end{definition} \begin{lemma} \label{lemma:polarisedDefinitions} Axiom (PS$2'$) implies Axiom (PS2), and so every special polarised system is a polarised system of parameters. Conversely for each polarised system $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1, \ldots, \kappa_{r-z}$ there is a nonnegative integer $N$ such that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1^{p^N}, \ldots, \kappa_{r-z}^{p^N}$ is a special polarised system of parameters for $\coho{G}$. \end{lemma} \begin{proof} Let $V \in \mathcal{A}^C(G)$. Restriction from $\coho{G}$~to $\coho{V}$ is a map of $\coho{C}$-comodules and so sends primitive elements to primitive elements. But the subalgebra of primitive elements of $\coho{V}$ coincides with the image of inflation from $V/C$. So (PS$2'$) implies (PS2). Now suppose $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1, \ldots, \kappa_{r-z}$ is a polarised system for $\coho{G}$. For each $1 \leq j \leq r-z$ the restriction of $\kappa_j$ to each $V \in \mathcal{A}^C(G)$ is primitive by (PS2)\@. Hence the element $\mu^*(\kappa_j) - \kappa_j \otimes 1$ of $\coho{G} \otimes \coho{C}$ has zero restriction to every maximal elementary abelian subgroup and is therefore nilpotent. For fixed $1 \leq i \leq r-z$, denote by $\mathcal{K}$ the set consisting of those subgroups $K$~of $G$ such that $C_G(K)$ is not $G$-conjugate to any subgroup of any $H \in \mathcal{H}^C_{z+i}$. Following Carlson (Proof of Corollary 2.2~of \cite{Carlson:DepthTransfer}), observe that every $K \in \mathcal{K}$ has $p$-rank less than $z+i$. Moreover every $K \in \mathcal{K}$ is contained in $K_C = \langle K,C \rangle$, and $K_C$~itself lies in~$\mathcal{K}$. So $\operatorname{Res}_{K_C}(\kappa_j) = 0$ for all $j \geq i$ by (PS4)\@. Hence each such $\kappa_j$ lies in the radical ideal $\sqrt{J'}$, where $J'$ is the ideal $\bigcap_{K \in \mathcal{K}} \ker \operatorname{Res}_K$. So by Benson's result on the image of the transfer map (Theorem~1.1 of \cite{Benson:ImTr}) the $\kappa_j$ also lie in $\sqrt{J}$, where $J$ is the ideal $\sum_{H \in \mathcal{H}^C_{z+i}(G)} \operatorname{Tr}^G_H(\coho{H})$. \end{proof} \begin{remark} The above proof is the only time we shall make use of the Axiom (PS4)\@. In particular, the results of \S\ref{section:specialPolarisedDepth} do not depend on (PS4)\@. I do not know whether or not (PS4) is a consequence of the remaining axioms for a special polarised system of parameters. Axiom (PS5) will be used in Lemma~\ref{lemma:kappaAssocPrime} to prove the existence of an associated prime ideal with desirable properties. \end{remark} \begin{lemma} \label{lemma:genBrotoHenn} Suppose that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1,\ldots,\kappa_{r-z}$ is a polarised system of parameters for $\coho{G}$. Let $0 \leq s \leq r-z$. Then the sequence $\mathbb{Z}eta_1, \ldots, \mathbb{Z}eta_z, \kappa_1, \ldots, \kappa_s$ is regular in $\coho{G}$ if and only if the sequence $\kappa_1, \ldots, \kappa_s$ is regular in $\coho{G}$. \end{lemma} \begin{proof} Recall that regular sequences may be permuted at will. Moreover, replacing one element of a sequence by its $p$th power has no effect on whether the sequence is regular or not. Hence by Lemma~\ref{lemma:polarisedDefinitions} it suffices to prove the result for special polarised systems. So we may assume that the given sequence is a special polarised system of parameters. Let $I$ be the homogeneous ideal $I$ in $\coho{G}$ generated by $\kappa_1, \ldots, \kappa_s$. By Axiom (PS$2'$) this ideal is generated by primitive elements. Also, the restrictions of $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$ form a regular sequence in $\coho{C}$ by Axiom~(PS1)\@. Therefore Lemma~\ref{lemma:myBrotoCarlsonHenn} tells us that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$ constitute a regular sequence for $\coho{G}/I$. \end{proof} \section{Three depth-related numbers} \label{section:threeNumbers} \noindent In this section we shall introduce three numbers $\tauH$, $\taua$~and $\tauaH$, each of which is an approximation to the depth $\tau$ of $\coho{G}$. \begin{definition} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$. Write $\tau$ for the depth of $\coho{G}$ and set \[ \tauH := \max \{ d \in \{z, \ldots, r\} \mid \text{The family $\mathcal{H}^C_d(G)$ detects $\coho{G}$} \} \, . \] \end{definition} \noindent In~\cite{Carlson:DepthTransfer} Carlson formulates the following conjecture: \begin{conjecture}[Carlson] \label{conjecture:ED} The number~$\tauH$ coincides with the depth $\tau$~of $\coho{G}$. Moreover, $\coho{G}$ has an associated prime $\mathfrak{p}$ such that $\dim \coho{G}/\mathfrak{p} = \tau$. \end{conjecture} \noindent In fact, Carlson formulates the conjecture not just for $p$-groups, but for arbitrary finite groups. In this article however we only consider $p$-groups. In Theorem~\ref{theorem:ED1} we shall prove a special case of this conjecture, after deriving a partial result for the general case in Theorem~\ref{theorem:polarisedEqualities}. For this we need two more depth-related numbers. \begin{definition} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$, and let $\mathfrak{a} = (\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1,\ldots,\kappa_{r-z})$ be a polarised system of parameters for $\coho{G}$\@. Define $\taua$ to be $z + \sa$, where $\sa$ is the largest $s \in \{0, \ldots, r-z\}$ such that $\kappa_1,\ldots, \kappa_s$ is a regular sequence in $\coho{G}$. Let $\SaH$~be the subset of $\{0, \ldots, r-z\}$ such that $s$~lies in $\SaH$ if and only if the restriction map \[ \coho{G}/(\kappa_1,\ldots,\kappa_{s-1}) \rightarrow \prod_{H \in \mathcal{H}^C_{z+s}} \coho{H}/(\operatorname{Res}_H \kappa_1,\ldots, \operatorname{Res}_H \kappa_{s-1}) \] is injective\footnote{So $\SaH$ always contains~$0$, and $1$~lies in $\SaH$ if and only if the family $\mathcal{H}^C_{z+1}$ detects $\coho{G}$.}. Define $\saH := \max \SaH$ and $\tauaH := z + \saH$. \end{definition} \begin{lemma} \label{lemma:tauCdash} Let $\mathfrak{a} = (\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1,\ldots,\kappa_{r-z})$ be a polarised system of parameters for $\coho{G}$\@. If $s > 0$ lies in $\SaH$ then the family $\mathcal{H}^C_{z + s}(G)$ detects $\coho{G}$ and $s-1$ lies in~$\SaH$. Therefore $\tauH \geq \tauaH$ and $\SaH = \{ 0, \ldots, \saH \}$. \end{lemma} \noindent For the proof we shall need an elementary fact about regular sequences. \begin{lemma} \label{lemma:genGrComm} Suppose that $R,S$ are connected graded commutative $k$-algebras and that $f \colon R \rightarrow S$ is an algebra homomorphism which respects the grading. Suppose further that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d$ is a family of homogenous positive-degree elements of $R$ satisfying the following conditions: \begin{enumerate} \item $f(\mathbb{Z}eta_1),\ldots,f(\mathbb{Z}eta_d)$ is a regular sequence in~$S$. \item The induced map $f_d \colon R/(\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d) \rightarrow S/(f(\mathbb{Z}eta_1),\ldots,f(\mathbb{Z}eta_d))$ is an injection. \end{enumerate} Then $f \colon R \rightarrow S$ is an injection. \end{lemma} \begin{proof} It suffices to prove the case $r=1$. Write $\mathbb{Z}eta$~for $\mathbb{Z}eta_1$. Let $a \not = 0$ be an element of $\ker(f)$ whose degree is as small as possible. Since $f(a) = 0$ in $S/(f(\mathbb{Z}eta))$ it follows that there is an $a' \in R$ with $a = a' \mathbb{Z}eta$. Since $f(a) = 0$ and $f(\mathbb{Z}eta)$ is regular it follows that $a' \in \ker(f)$, contradicting the minimality of $\deg(a)$. \end{proof} \begin{proof}[Proof of Lemma~\ref{lemma:tauCdash}] Apply Lemma~\ref{lemma:genGrComm} to the family $\kappa_1,\ldots, \kappa_{s-1}$ with $R = \coho{G}$, $S = \prod_{\mathcal{H}^C_{z + s}} \coho{H}$ and $f$ the product of the restriction maps. Because $s \in \SaH$ the induced map of quotients is an injection. By Axiom~(PS3) the restrictions of $\kappa_1,\ldots, \kappa_{s-1}$ form a regular sequence in $\coho{V}$ for each $V \in \mathcal{A}^C_{z + s}(G)$, and so by \cite[Prop.~5.2]{Carlson:Problems} they form a regular sequence in $\coho{H}$ for each $H \in \mathcal{H}^C_{z + s}$. Hence the restrictions form a regular sequence in~$S$, and so the family $\mathcal{H}^C_{z + s}$ detects $\coho{G}$. If instead we just invoke the first step in the proof of Lemma~\ref{lemma:genGrComm}, we see that the $\coho{H}/(\operatorname{Res} \kappa_1,\ldots,\operatorname{Res} \kappa_{s-2})$ with $H \in \mathcal{H}^C_{z + s}$ detect $\coho{G}/(\kappa_1,\ldots,\kappa_{s-2})$. \end{proof} \section{Depth and special polarised systems} \label{section:specialPolarisedDepth} \noindent The following fact from commutative algebra is well known. \begin{lemma} \label{lemma:assocPrime} Let $A$ be a graded commutative ring and $M$ a Noetherian graded $A$-module. Suppose that $\mathfrak{p}$ is an associated prime of~$M$, and that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d$ is a regular sequence for $M$. Then $M/(\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d)M$ has an associated prime~$\mathfrak{q}$ containing $\mathfrak{p}$. \end{lemma} \begin{proof} It suffices to prove the case $d=1$. Write $\mathbb{Z}eta$~for $\mathbb{Z}eta_1$. Pick $x \in M$ with $\operatorname{Ann}_A(x) = \mathfrak{p}$. If $x$ lies in $\mathbb{Z}eta M$ then there is an $x' \in M$ with $\mathbb{Z}eta x' = x$. As $\mathbb{Z}eta$ is regular it follows that $\operatorname{Ann}_A(x') = \mathfrak{p}$ too, so replace $x$~by $x'$. This can only happen finitely often, as $M$ is Noetherian and $Ax$ is strictly contained in $Ax'$. So we may assume that $x$~does not lie in $\mathbb{Z}eta M$, which means that the image of $x$~in $M/\mathbb{Z}eta M$ is nonzero and annihilated by $\mathfrak{p}$. \end{proof} \begin{lemma} \label{lemma:kappaAssocPrime} Let $\mathfrak{a} = (\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1,\ldots,\kappa_{r-z})$ be a special polarised system of parameters for $\coho{G}$, and $I$ the ideal generated by the $\kappa_j$ with $j \leq \saH$. Then the $\coho{G}$-module $\coho{G}/I$ has an associated prime $\mathfrak{p}$ which contains $\kappa_1$, \dots, $\kappa_{r-z}$. \end{lemma} \begin{proof} Set $s = \saH + 1$. The result is trivial if $\saH = r-z$, so we may assume that $s \leq r-z$. By definition of $\saH$, the family $\mathcal{H} = \mathcal{H}^C_{z + s}$ does not detect the quotient $\coho{G}/I$. Pick a class $x \in \coho{G}$ which does not lie in the ideal $I$, but whose restriction to each $H \in \mathcal{H}$ does lie in the ideal $\coho{H} . \operatorname{Res}_H(I)$. Let $A$ be the ideal of classes in $\coho{G}$ which annihilate the image of $x$~in the quotient $\coho{G}/I$. For any $j \geq s$ we have $\kappa_j \in \sum_{H \in \mathcal{H}} \operatorname{Tr}^G_H \coho{H}$ by Axiom~(PS5), say $\kappa_j = \sum_H \operatorname{Tr}_H \gamma_H$. So $\kappa_j x = \sum_H \operatorname{Tr}_H (\gamma_H \operatorname{Res}_H(x))$ by Frobenius reciprocity. Now by assumption $\operatorname{Res}_H(x)$ lies in the ideal generated by $\operatorname{Res}_H(\kappa_1)$, \dots, $\operatorname{Res}_H(\kappa_{s-1})$; and this by a second application of Frobenius reciprocity means that $\kappa_j x$ lies in the ideal $I$. So $\kappa_j \in A$ for all $j \geq s$, which means that the $\coho{G}$-module $\coho{G}/I$ has an associated prime $\mathfrak{p}$ containing $\kappa_1,\ldots,\kappa_{r-z}$. \end{proof} \begin{theorem} \label{theorem:specialPolarisedEqualities} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$, and let $ \mathfrak{a} = (\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1,\ldots,\kappa_{r-z})$ be a special polarised system of parameters for $\coho{G}$. Then the numbers $\taua$~and $\tauaH$ both coincide with the depth $\tau$~of $\coho{G}$. \end{theorem} \begin{proof} We shall prove that $\tau \geq \taua \geq \tauaH = \tau$. Each $\kappa_j$ is primitive by Axiom~(PS$2'$), so $\tau \geq \taua$ by Corollary~\ref{coroll:myBrotoCarlsonHenn}. $\taua \geq \tauaH$: Suppose that $s \in \SaH$ and $\kappa_1$, \dots, $\kappa_{s-1}$ is a regular sequence in $\coho{G}$. If $\kappa_1,\ldots,\kappa_s$ is not a regular sequence then there is some nonzero $x \in \coho{G}/(\kappa_1,\ldots,\kappa_{s-1})$ annihilated by $\kappa_s$. Since $s \in \SaH$ there is some $H \in \mathcal{H}^C_{z + s}$ such that $\operatorname{Res}_H(x)$ is nonzero in $\coho{H}/(\operatorname{Res}_H \kappa_1,\ldots,\operatorname{Res}_H \kappa_{s-1})$. But this is a contradiction since (as in the proof of Lemma~\ref{lemma:tauCdash}) the restrictions of $\kappa_1,\ldots,\kappa_s$ form a regular sequence in $\coho{H}$. By induction on~$s$ we deduce that $\taua \geq \tauaH$. $\tauaH = \tau$: Set $s = \saH$ and denote by $I$ the ideal $(\kappa_1,\ldots,\kappa_s)$ of $\coho{G}$. By Lemma~\ref{lemma:kappaAssocPrime} the $\coho{G}$-module $\coho{G}/I$ has an associated prime~$\mathfrak{p}$ which contains $\kappa_1$, \dots, $\kappa_{r-z}$. As $\taua \geq \tauaH$ the sequence $\kappa_1,\ldots,\kappa_s$ is regular in $\coho{G}$, so by Lemma~\ref{lemma:genBrotoHenn} the sequence $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1$, \dots, $\kappa_s$ is regular in $\coho{G}$. Therefore the sequence $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$ is regular for $\coho{G}/I$, and so (by Lemma~\ref{lemma:assocPrime}) the $\coho{G}$-module $\coho{G}/(\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z,\kappa_1,\ldots,\kappa_s)$ has an associated prime $\mathfrak{q}$ containing all elements of the homogeneous system of parameters $\mathbb{Z}eta_1$, \dots, $\mathbb{Z}eta_z$, $\kappa_1$, \dots, $\kappa_{r-z}$ for $\coho{G}$. So the depth of this quotient module is zero. But every regular sequence in $\coho{G}$ can be extended to a length~$\tau$ regular sequence (see \cite[\S4.3--4]{Benson:PolyInvts}, for example). So $\tau = z + s = \tauaH$. \end{proof} \section{Depth and polarised systems} \label{section:polarisedDepth} \noindent In this section we shall remove the requirement in Theorem~\ref{theorem:specialPolarisedEqualities} that the polarised system be special. \begin{theorem} \label{theorem:polarisedEqualities} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank $z$, and let $ \mathfrak{a} = (\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1,\ldots,\kappa_{r-z})$ be a polarised system of parameters for $\coho{G}$. Then the numbers $\taua$~and $\tauaH$ both coincide with the depth $\tau$~of $\coho{G}$. \end{theorem} \noindent For the proof we shall need one further fact about regular sequences. \begin{lemma} \label{lemma:liftingInjections} Suppose that $R,S$ are connected graded commutative $k$-algebras and that $f \colon R \rightarrow S$ is an algebra homomorphism which respects the grading. Suppose further that $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d$ is a family of homogenous positive-degree elements of $R$ satisfying the following conditions: \begin{enumerate} \item $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d$ is a regular sequence in~$R$. \item The induced map $\fui[n] \colon R/(\mathbb{Z}eta_1^{n_1},\ldots,\mathbb{Z}eta_d^{n_d}) \rightarrow S/(f(\mathbb{Z}eta_1)^{n_1},\ldots,f(\mathbb{Z}eta_d)^{n_d})$ is an injection for certain positive integers $n_1,\ldots,n_d$. \end{enumerate} Then the induced map $\fui[1] \colon R/(\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_d) \rightarrow S/(f(\mathbb{Z}eta_1), \ldots, f(\mathbb{Z}eta_d))$ is an injection. \end{lemma} \begin{proof} Pick $x \in R$ with $\fui[1](x) = 0$. Then $\fui[n](\mathbb{Z}eta_1^{n_1-1} \ldots \mathbb{Z}eta_d^{n_d-1} x) = 0$ and so \begin{equation} \label{eqn:peelOff} \text{$\mathbb{Z}eta_1^{n_1-1} \ldots \mathbb{Z}eta_d^{n_d-1} x$ lies in the ideal $(\mathbb{Z}eta_1^{n_1}, \ldots, \mathbb{Z}eta_d^{n_d})$ of $R$.} \end{equation} Set $\mathbb{Z}eta' := \mathbb{Z}eta_1^{n_1-1} \ldots \mathbb{Z}eta_{d-1}^{n_{d-1}-1}$. Then there are $a_1, \ldots, a_d \in R$ such that $\mathbb{Z}eta' \mathbb{Z}eta_d^{n_d-1} x = \mathbb{Z}eta_1^{n_1} a_1 + \cdots + \mathbb{Z}eta_d^{n_d} a_d$, whence $\mathbb{Z}eta_d^{n_d-1} (\mathbb{Z}eta' x - \mathbb{Z}eta_d a_d) \in (\mathbb{Z}eta_1^{n_1}, \ldots, \mathbb{Z}eta_{d-1}^{n_{d-1}})$. As the sequence $\mathbb{Z}eta_1^{n_1}, \ldots, \mathbb{Z}eta_{d-1}^{n_{d-1}}, \mathbb{Z}eta_d^s$ is regular in~$R$ for $s \geq 1$ we deduce that $\mathbb{Z}eta' x \in (\mathbb{Z}eta_1^{n_1}, \ldots, \mathbb{Z}eta_{d-1}^{n_{d-1}}, \mathbb{Z}eta_d)$. So we have reduced Eqn.~\eqref{eqn:peelOff} to the case $n_d = 1$ without altering the remaining $n_t$. As regular sequences may be permuted at will we deduce that $x \in (\mathbb{Z}eta_1, \ldots, \mathbb{Z}eta_d)$. \end{proof} \begin{proof}[Proof of Theorem~\ref{theorem:polarisedEqualities}] Arguing exactly as in the proof of Theorem~\ref{theorem:specialPolarisedEqualities} one shows that $\taua \geq \tauaH$. By Lemma~\ref{lemma:polarisedDefinitions} there is an integer $N \geq 0$ such that the system of parameters $\mathfrak{b} = (\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z, \kappa_1^{p^N}, \ldots, \kappa_{r-z}^{p^N})$ is special polarised. From the definition one sees that $\taua = \taua[b]$. So by Theorem~\ref{theorem:specialPolarisedEqualities} it suffices to prove that $\tauaH \geq \tauaH[b]$. But this is a consequence of Lemma~\ref{lemma:liftingInjections} applied to the sequence $\kappa_1, \ldots, \kappa_{\saH[b]-1}$ with $R = \coho{G}$, $S = \prod_{\mathcal{H}} \coho{H}$ and $f$~the restriction map, where $\mathcal{H} = \mathcal{H}^C_{z+\saH[b]}(G)$ and $n_i = p^N$ for all~$i$. \end{proof} \section{Dickson invariants} \label{section:Dickson} \noindent Let $V$ be an $m$-dimensional $k$-vector space. We shall make extensive use of the Dickson invariants, the polynomial generators of the ring of $\text{\sl GL}(V)$-invariants in $k[V]$. See Benson's book~\cite{Benson:PolyInvts} for proofs of the properties of these invariants. Denote by $f_V$ the polynomial in $k[V][X]$ defined as follows: \begin{equation} \label{eqn:fVdef} f_V(X) = \prod_{v \in V} (X - v) \, . \end{equation} Recall that Dickson proved there are homogeneous polynomials $D_s(V)$ for $1 \leq s \leq m$ such that \begin{equation} \label{eqn:fV} f_V(X) = \sum_{s = 0}^m (-1)^s D_s(V) X^{p^{m-s}} \, , \end{equation} where $D_0(V) = 1$. The sequence $D_1(V), \ldots, D_m(V)$ is regular in $k[V]$, and the invariant ring $k[V]^{\text{\sl GL}(V)}$ is the polynomial algebra $k[D_1(V), \ldots, D_m(V)]$. If $\pi \colon V \rightarrow U$ is projection onto a codimension~$\ell$ subspace, then the induced map $k[V] \rightarrow k[U]$ sends \begin{equation} \label{eqn:DicksonRes} D_s(V) \mapsto \begin{cases} D_s(U)^{p^{\ell}} & \text{if $s \leq \dim(U)$,} \\ 0 & \text{otherwise.} \end{cases} \end{equation} \section{Existence of polarised systems} \label{section:existence} \noindent For each elementary abelian $p$-group $V$ we shall embed $k[V^*]$ in $\coho{V}$ by identifying $V^*$ with the image of the Bockstein map $\coho[1]{V} \rightarrow \coho[2]{V}$. \subsection{A construction using the norm map} Let $G$ be a $p$-group of order $p^n$ and $p$-rank $r$ whose centre has $p$-rank~$z$. We shall only be interested in the case $r > z$. Let $U_1, \ldots, U_K$ be representatives of the $G$-orbits in $\mathcal{A}^C_{z+1}(G)$, which is a $G$-set via the conjugation action. For each $U \in \mathcal{A}^C_{z+1}(G)$ pick a basis element $x_U$ for the one-dimensional subspace $\operatorname{Ann}(C)$ of $U^*$, and observe that $x_U^{p-1}$ is independent of the basis element chosen. As before, view $U^*$ as a subspace of $\coho[2]{U}$. Define $\Theta \in \coho{G}$ by \begin{equation} \label{eqn:ThetaDef} \Theta = \prod_{i = 1}^K \mathcal{N}^G_{U_i} \left(1 + x_{U_i}^{p-1}\right)^{|G : N_G(U_i)|} \, . \end{equation} Now consider the restriction $\operatorname{Res}_V(\Theta)$ for $V \in \mathcal{A}^C(G)$. By the Mackey formula \[ \operatorname{Res}_V (\Theta) = \prod_{i = 1}^K \prod_{g \in U_i \setminus G / V} \mathcal{N}^V_{U_i^g \cap V} \, g^* \operatorname{Res}^{U_i}_{U_i \cap {}^g V} \left(1 + x_{U_i}^{p-1}\right)^{|G : N_G(U_i)|} \, . \] The intersection $U_i \cap {}^g V$ always contains $C$, the largest central elementary abelian subgroup of~$G$. Conversely the intersection equals $C$ (and $x_{U_i}$ therefore restricts to zero) unless $U' = U_i^g$ lies in~$V$, in which case $g^* x_{U_i}^{p-1} = x_{U'}^{p-1}$. Moreover, the number of double cosets $U_i g V$ leading to this $U'$ is $|N_G(U_i)|/|V|$ and every $U'$~in $\mathcal{A}^C_{z+1}(V) := \{ U \in \mathcal{A}^C_{z+1}(G) \mid U \leq V \}$ occurs for some~$i$. So \begin{equation} \label{eqn:ResV_Theta} \operatorname{Res}_V(\Theta) = \prod_{U' \in \mathcal{A}^C_{z+1}(V)} \mathcal{N}_{U'}^V (1 + x_{U'}^{p-1})^{|G : V|} \, . \end{equation} In particular for $U \in \mathcal{A}^C_{z+1}(G)$ one has \begin{equation} \label{eqn:ResU_Theta} \operatorname{Res}_U(\Theta) = 1 + x_U^{(p-1)p^{n-(z+1)}} \, . \end{equation} Let $\eta \in \coho[2(p-1)p^{n-(z+1)}]{G}$ be the homogeneous component of $\Theta$ in this degree. As the norm map from $\coho{U'}$~to $\coho{V}$ is a ring homomorphism (see \cite[Proposition~6.3.3]{Evens:book}), we deduce from Eqn.~\eqref{eqn:ResV_Theta} that \begin{equation} \label{eqn:ResV_eta} \operatorname{Res}_V(\eta) = \mathrm{H}at{\eta}^{|G:V|} \quad \text{for} \quad \mathrm{H}at{\eta} = \sum_{U' \in \mathcal{A}^C_{z+1}(V)} \mathcal{N}_{U'}^V x_{U'}^{p-1} \, . \end{equation} Denote by $W = W(V)$ the subspace $\operatorname{Ann}(C)$~of $V^*$. Then $\mathrm{H}at{\eta}$ lies in $k[W]$, since $\mathcal{N}_{U'}^V (x_{U'})$ is the product of all $\phi \in V^*$ with $\operatorname{Res}_{U'} (\phi) = x_{U'}$. Moreover $\mathrm{H}at{\eta}$ is by construction a $\text{\sl GL}(W)$-invariant, so a scalar multiple of $D_1(W)$ for degree reasons. By considering the restriction to any $U \in \mathcal{A}^C_{z+1}(V)$, we deduce from Eqn.~\eqref{eqn:DicksonRes} that \begin{equation} \label{eqn:ResV_eta_Dickson} \operatorname{Res}^G_V(\eta) = D_1(W)^{|G : V|} \, . \end{equation} \subsection{The existence proof} \begin{theorem} \label{theorem:existence} Let $G$ be a $p$-group of order $p^n$ and $p$-rank $r$ whose centre has $p$-rank~$z$. For $1 \leq i \leq z$ define $\mathbb{Z}eta_i \in \coho{G}$ by $\mathbb{Z}eta_i = c_{p^n - p^{n-i}}(\rho_G)$, a Chern class of the regular representation of~$G$. If $z < r$ define $\eta \in \coho{G}$ as above and homogeneous elements $\kappa_1, \kappa_2, \ldots, \kappa_{r-z} \in \coho{G}$ as follows: \[ \kappa_j := \mathcal{P}^{p^{n - z + j - 4}} \cdots \mathcal{P}^{p^{n - z - 1}} \mathcal{P}^{p^{n - z - 2}} (\eta) \in \coho[2(p^{n - z} - p^{n - z - j})]{G} \] for $1 \leq j \leq r - z$. Then for each $1 \leq j \leq r-z$ and for each $V \in \mathcal{A}^C_{z+s}(G)$ one has \begin{equation} \label{eqn:existence} \operatorname{Res}^G_V(\kappa_j) = \begin{cases} D_j(W)^{|G:V|} & \text{if $j \leq s$, and} \\ 0 & \text{otherwise.} \end{cases} \end{equation} Here, $W$~is the subspace of $V^*$ which annihilates~$C$. Then $\mathbb{Z}eta_1, \ldots, \mathbb{Z}eta_z, \kappa_1, \ldots, \kappa_{r-z}$ is a polarised system of parameters for $\coho{G}$. So $\coho{G}$ has both polarised and special polarised systems of parameters. \end{theorem} \begin{proof} Equation~\eqref{eqn:existence} holds for $\kappa_1 = \eta$ by Eqn.~\eqref{eqn:ResV_eta_Dickson}. The general case of Eqn.~\eqref{eqn:existence} follows from Eqn.~\eqref{eqn:DicksonRes} and the action of the Steenrod algebra on the Dickson invariants (see~\cite{Wilkerson:Dickson}). Axioms (PS2) and (PS4) follow immediately from Eqn.~\eqref{eqn:existence}. Axiom (PS3) holds because the Dickson invariants form a regular sequence. Observe that $\rho_G$ restricts to~$C$ as $p^{n-z}$ copies of the regular representation $\rho_C$. So the total Chern class $c(\rho_G)$ restricts to $C$ as $c(\rho_C)^{p^{n-z}}$, meaning that $\operatorname{Res}_C(\mathbb{Z}eta_i) = c_{p^z - p^{z-i}}(\rho_C)^{p^{n-z}}$ for $1 \leq i \leq z$. In view of Eqn.~\eqref{eqn:fVdef} one has $c(\rho_C) = f_{C^*}(1)$ and hence $c_{p^z - p^{z - i}}(\rho_C) = (-1)^i D_i(C^*)$ by Equation~\eqref{eqn:fV}, a well known observation due originally to Milgram. So $\operatorname{Res}_C(\mathbb{Z}eta_i) = (-1)^i D_i(C^*)^{|G:C|}$, which means that Axiom (PS1) is satisfied and so a polarised system of parameters has been constructed. Hence special polarised systems of parameters exist too, by Lemma~\ref{lemma:polarisedDefinitions}. \end{proof} \section{Tightness of Duflot's lower bound} \label{section:DuflotTight} \noindent Recall that Duflot's Theorem~\cite{Duflot:Depth} states that the depth of $\coho{G}$ is at least~$z$. \begin{theorem} \label{theorem:ED1} Let $G$ be a $p$-group of $p$-rank $r$ whose centre has $p$-rank~$z$. Then the following statements are equivalent: \begin{enumerate} \item \label{enum:notDetected} The mod-$p$ cohomology ring $\coho{G}$ is not detected on the family $\mathcal{H}^C_{z+1}(G)$. \item \label{enum:assocPrime} $\coho{G}$ has an associated prime $\mathfrak{p}$ such that the dimension of $\coho{G}/\mathfrak{p}$ is $z$. \item \label{enum:oneK1} There is a polarised system of parameters $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1,\ldots,\kappa_{r-z}$ for $\coho{G}$ such that $\kappa_1$ is a zero divisor in $\coho{G}$. \item \label{enum:allK1} If $\mathbb{Z}eta_1,\ldots,\mathbb{Z}eta_z$, $\kappa_1,\ldots,\kappa_{r-z}$ is a polarised system of parameters for $\coho{G}$ then $\kappa_1$ is a zero divisor in $\coho{G}$. \item \label{enum:depthz} The depth of $\coho{G}$ equals $z$. \end{enumerate} \end{theorem} \begin{proof} Carlson proved in~\cite{Carlson:DepthTransfer} that \eqref{enum:notDetected} implies \eqref{enum:depthz}\@. A standard commutative algebra argument shows that \eqref{enum:assocPrime} implies \eqref{enum:depthz}\@. We saw in Lemma~\ref{lemma:genBrotoHenn} that \eqref{enum:depthz} implies \eqref{enum:allK1}, and \eqref{enum:oneK1} follows from \eqref{enum:allK1} by the existence result Theorem~\ref{theorem:existence}. Now let $\mathfrak{a}$ be a polarised system of parameters satisfying Statement~\eqref{enum:oneK1}, which is equivalent to $\taua = z$. So $\tauaH = z$ by Theorem~\ref{theorem:polarisedEqualities}\@. As in the proof of that theorem there is a special polarised system of parameters~$\mathfrak{b}$ which satisfies $\tauaH[b] = \tauaH = z$, so \eqref{enum:assocPrime} follows by Lemma~\ref{lemma:kappaAssocPrime}. Finally consider the definition of~$\tauaH$. If $\tauaH = z$ then $\mathcal{H}^C_{z+1}(G)$ does not detect $\coho{G}$, yielding~\eqref{enum:notDetected}\@. \end{proof} \section{An example} Let $G$ be the extraspecial $p$-group $p^{1+2n}_+$ with $n \geq 1$. This group has order $p^{2n+1}$ and $p$-rank $n+1$. Its centre has $p$-rank~$1$. If $p$~is odd $G$ has exponent~$p$. The mod-$p$ cohomology ring $\coho{G}$ is Cohen-Macaulay for $p=2$ by Theorem~4.6 of~\cite{Quillen:Extraspecial}, for Quillen computes the cohomology ring as the quotient of a polynomial algebra by a regular sequence. Also, Milgram and Tezuka showed in~\cite{MilgramTezuka} that the cohomology ring is Cohen-Macaulay for $G = 3^{1+2}_+$. From now on assume that $p$ is odd, with $n \geq 2$ if $p = 3$. Then by a result of Minh~\cite{Minh:EssExtra}, there are essential classes. For such groups the centre $C$ is cyclic of order~$p$, and the set $\mathcal{H}^C_2(G)$ of centralisers coincides with the set of maximal subgroups. Consequently $\mathcal{H}^C_{z+1}(G)$ does not detect $\coho{G}$, and so $\coho{G}$ has depth~$1$ by the part of Theorem~\ref{theorem:ED1} proved by Carlson in~\cite{Carlson:DepthTransfer}. Now let $V$ be a rank $n+1$ elementary abelian subgroup of~$G$, and let $\mathrm{H}at{\rho}$ be a one-dimensional ordinary representation of~$V$ whose restriction to $C$ is not trivial. Let $\rho$ be the induced representation of~$G$. Then $\rho$ is an irreducible representation of degree~$p^n$, and its character restricts to each $U \in \mathcal{A}^C_{n+1}(G)$ as the sum of all degree one characters whose restrictions to $C$ coincide with the restriction of the character of~$\mathrm{H}at{\rho}$. Set $\mathbb{Z}eta_1 := c_{p^n}(\rho)$ and $\kappa_j := c_{p^n - p^{n-j}}(\rho)$ for $1 \leq j \leq n$. Then $\mathbb{Z}eta_1, \kappa_1, \ldots, \kappa_n$ satisfies the axioms for a polarised system of parameters for $\coho{G}$. So combining Minh's result with Theorem~\ref{theorem:ED1} one deduces that $\kappa_1$ has nontrivial annihilator in $\coho{G}$. Conversely a direct proof of this fact would yield a new proof of Minh's result. If $n=1$ and $p > 3$ it is known (see~\cite{Leary:integral}) that $c_2(\rho)$ is a nonzero essential class which annihilates~$\kappa_1$. \end{document}

\begin{document} \begin{frontmatter} \title{Cancellability and Regularity of Operator Connections} \author{Pattrawut Chansangiam} \ead{kcpattra@kmitl.ac.th} \address{Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand.} \begin{abstract} An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above and the transformer inequality. In this paper, we introduce and characterize the concepts of cancellability and regularity of operator connections with respect to operator monotone functions, Borel measures and certain operator equations. In addition, we investigate the existence and the uniqueness of solutions for such operator equations. \end{abstract} \begin{keyword} operator connection \sep operator monotone function \sep operator mean \MSC[2010]47A63 \sep 47A64 \end{keyword} \end{frontmatter} \section{Introduction} A general theory of connections and means for positive operators was given by Kubo and Ando \cite{Kubo-Ando}. Let $B(\mathcal{H})$ be the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$. The set of positive operators on $\mathcal{H}$ is denoted by $B(\mathcal{H})^+$. Denote the spectrum of an operator $X$ by $\Sp(X)$. For Hermitian operators $A,B \in B(\mathcal{H})$, the partial order $A \leqslant B$ means that $B-A \in B(\mathcal{H})^+$. The notation $A>0$ suggests that $A$ is a strictly positive operator. A \emph{connection} is a binary operation $\,\sigma\,$ on $B(\mathcal{H})^+$ such that for all positive operators $A,B,C,D$: \begin{enumerate} \item[(M1)] \emph{monotonicity}: $A \leqslant C, B \leqslant D \implies A \,\sigma\, B \leqslant C \,\sigma\, D$ \item[(M2)] \emph{transformer inequality}: $C(A \,\sigma\, B)C \leqslant (CAC) \,\sigma\, (CBC)$ \item[(M3)] \emph{continuity from above}: for $A_n,B_n \in B(\mathcal{H})^+$, if $A_n \downarrow A$ and $B_n \downarrow B$, then $A_n \,\sigma\, B_n \downarrow A \,\sigma\, B$. Here, $A_n \downarrow A$ indicates that $A_n$ is a decreasing sequence and $A_n$ converges strongly to $A$. \end{enumerate} Two trivial examples are the left-trivial mean $\omegaega_l : (A,B) \mapsto A$ and the right-trivial mean $\omegaega_r: (A,B) \mapsto B$. Typical examples of a connection are the sum $(A,B) \mapsto A+B$ and the parallel sum \begin{align*} A \,:\,B = (A^{-1}+B^{-1})^{-1}, \quad A,B>0, \end{align*} the latter being introduced by Anderson and Duffin \cite{Anderson-Duffin}. From the transformer inequality, every connection is \emph{congruence invariant} in the sense that for each $A,B \geqslant 0$ and $C>0$ we have \begin{align*} C(A \,\sigma\, B)C \:=\: (CAC) \,\sigma\, (CBC). \end{align*} A \emph{mean} is a connection $\sigma$ with normalized condition $I \,\sigma\, I = I$ or, equivalently, fixed-point property $A \,\sigma\, A =A$ for all $A \geqslant 0$. The class of Kubo-Ando means cover many well-known operator means in practice, e.g. \begin{itemize} \item $\alpha$-weighted arithmetic means: $A \triangledown_{\alpha} B = (1-\alpha)A + \alpha B$ \item $\alpha$-weighted geometric means: $A \#_{\alpha} B = A^{1/2} ({A}^{-1/2} B {A}^{-1/2})^{\alpha} {A}^{1/2}$ \item $\alpha$-weighted harmonic means: $A \,!_{\alpha}\, B = [(1-\alpha)A^{-1} + \alpha B^{-1}]^{-1}$ \item logarithmic mean: $(A,B) \mapsto A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}$ where $f: \mathbb{R}^+ \to \mathbb{R}^+$, $f(x)=(x-1)/\log{x}$, $f(0) \equiv 0$ and $f(1) \equiv 1$. Here, $\mathbb{R}^+=[0, \infty)$. \end{itemize} It is a fundamental that there are one-to-one correspondences between the following objects: \begin{enumerate} \item[(1)] operator connections on $B(\mathcal{H})^+$ \item[(2)] operator monotone functions from $\mathbb{R}^+$ to $\mathbb{R}^+$ \item[(3)] finite (positive) Borel measures on $[0,1]$ \item[(4)] monotone (Riemannian) metrics on the smooth manifold of positive definite matrices. \end{enumerate} Recall that a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ is said to be \emph{operator monotone} if \begin{align*} A \leqslant B \implies f(A) \leqslant f(B) \end{align*} for all positive operators $A,B \in B(\mathcal{H})$ and for all Hilbert spaces $\mathcal{H}$. This concept was introduced in \cite{Lowner}; see also \cite{Bhatia,Hiai,Hiai-Yanagi}. A remarkable fact is that (see \cite{Hansen-Pedersen}) a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ is operator monotone if and only if it is \emph{operator concave}, i.e. \begin{align*} f((1-\alpha)A + \alpha B) \:\geqslant\: (1-\alpha) f(A) + \alpha f(B), \quad \alpha \in (0,1) \end{align*} holds for all positive operators $A,B \in B(\mathcal{H})$ and for all Hilbert spaces $\mathcal{H}$. A connection $\sigma$ on $B(\mathcal{H})^+$ can be characterized via operator monotone functions as follows: \begin{thm}[\cite{Kubo-Ando}] \label{thm: Kubo-Ando f and sigma} Given a connection $\sigma$, there is a unique operator monotone function $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying \begin{align*} f(x)I = I \,\sigma\, (xI), \quad x \geqslant 0. \end{align*} Moreover, the map $\sigma \mapsto f$ is a bijection. \end{thm} We call $f$ the \emph{representing function} of $\sigma$. A connection also has a canonical characterization with respect to a Borel measure via a meaningful integral representation as follows. \begin{thm}[\cite{Pattrawut}] \label{thm: 1-1 conn and measure} Given a finite Borel measure $\mu$ on $[0,1]$, the binary operation \begin{align} A \,\sigma\, B = \int_{[0,1]} A \,!_t\, B \,d \mu(t), \quad A,B \geqslant 0 \label{eq: int rep connection} \end{align} is a connection on $B(\mathcal{H})^+$. Moreover, the map $\mu \mapsto \sigma$ is bijective, in which case the representing function of $\sigma$ is given by \begin{align} f(x) = \int_{[0,1]} (1 \,!_t\, x) \,d \mu(t), \quad x \geqslant 0. \label{int rep of OMF} \end{align} \end{thm} We call $\mu$ the \emph{associated measure} of $\sigma$. A connection is a mean if and only if $f(1)=1$ or its associated measure is a probability measure. Hence every mean can be regarded as an average of weighted harmonic means. From \eqref{eq: int rep connection} and \eqref{int rep of OMF}, $\sigma$ and $f$ are related by \begin{align} f(A) \:=\: I \,\sigma\, A, \quad A \geqslant 0. \label{eq: f(A) = I sm A} \end{align} A connection $\sigma$ is said to be \emph{symmetric} if $A \,\sigma\, B = B \,\sigma\, A$ for all $A,B \geqslant 0$. The notion of monotone metrics arises naturally in quantum mechanics. A metric on the differentiable manifold of $n$-by-$n$ positive definite matrices is a continuous family of positive definite sesqilinear forms assigned to each invertible density matrix in the manifold. A monotone metric is a metric with contraction property under stochastic maps. It was shown in \cite{Petz} that there is a one-to-one correspondence between operator connections and monotone metrics. Moreover, symmetric metrics correspond to symmetric means. In \cite{Hansen}, the author defined a symmetric metric to be \emph{nonregular} if $f(0)=0$ where $f$ is the associated operator monotone function. In \cite{Gibilisco}, $f$ is said to be \emph{nonregular} if $f(0) = 0$, otherwise $f$ is \emph{regular}. It turns out that the regularity of the associated operator monotone function guarantees the extendability of this metric to the complex projective space generated by the pure states (see \cite{Petz-Sudar}). In the present paper, we introduce the concept of cancellability for operator connections in a natural way. Various characterizations of cancellability with respect to operator monotone functions, Borel measures and certain operator equations are provided. It is shown that a connection is cancellable if and only if it is not a scalar multiple of trivial means. Applications of this concept go to certain operator equations involving operator means. We investigate the existence and the uniqueness of such equations. It is shown that such equations are always solvable if and only if $f$ is unbounded and $f(0)=0$ where $f$ is the associated operator monotone function. We also characterize the condition $f(0)=0$ for arbitrary connections without assuming the symmetry. Such connection is said to be nonregular. \section{Cancellability of connections} Since each connection is a binary operation, we can define the concept of cancellability as follows. \begin{defn} A connection $\sigma$ is said to be \begin{itemize} \item \emph{left cancellable} if for each $A>0, B \geqslant 0$ and $C \geqslant 0$, \begin{align*} A \,\sigma\, B = A \,\sigma\, C \implies B=C \end{align*} \item \emph{right cancellable} if for each $A>0, B \geqslant 0$ and $C \geqslant 0$, \begin{align*} B \,\sigma\, A = C \,\sigma\, A \implies B=C \end{align*} \item \emph{cancellable} if it is both left and right cancellable. \end{itemize} \end{defn} \begin{lem} \label{lem: nonconstant OM} Every nonconstant operator monotone function from $\mathbb{R}^+$ to $\mathbb{R}^+$ is injective. \end{lem} \begin{proof} Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a nonconstant operator monotone function. Suppose there exist $b>a \geqslant 0$ such that $f(a)=f(b)$. Since $f$ is monotone increasing (in usual sense), $f(x)=f(a)$ for all $a \leqslant x \leqslant b$ and $f(y) \geqslant f(b)$ for all $y \geqslant b$. Since $f$ is operator concave, $f$ is concave in usual sense and hence $f(x)=f(b)$ for all $x \geqslant b$. The case $a=0$ contradicts the fact that $f$ is nonconstant. Consider the case $a>0$. The monotonicity of $f$ yields $f(x) \leqslant f(a)$ for all $0 \leqslant x \leqslant a$. The differentiability of $f$ implies that $f(x)=f(a)$ for all $x \geqslant 0$, again a contradiction. \end{proof} A similar result for this lemma under the restriction that $f(0)=0$ was obtained in \cite{Molnar}. The left-cancellability of connections is characterized as follows. \begin{thm} \label{thm: cancallability 1} Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent: \begin{enumerate} \item[(1)] $\sigma$ is left cancellable ; \item[(2)] for each $A \geqslant 0$ and $B \geqslant 0$, $I \sigma A = I \sigma B \implies A=B$ ; \item[(3)] $\sigma$ is not a scalar multiple of the left-trivial mean ; \item[(4)] $f$ is injective, i.e., $f$ is left cancellable in the sense that \begin{align*} f \circ g = f \circ h \:\implies\: g=h \quad ; \end{align*} \item[(5)] $f$ is a nonconstant function ; \item[(6)] $\mu$ is not a scalar multiple of the Dirac measure $\delta_0$ at $0$. \end{enumerate} \end{thm} \begin{proof} Clearly, (1) $\mathbb{R}ightarrow$ (2) $\mathbb{R}ightarrow$ (3) and (4) $\mathbb{R}ightarrow$ (5). For each $k \geqslant 0$, it is straightforward to show that the representing function of the connection \begin{align*} k \omegaega_l \;:\; (A,B) \mapsto kA \end{align*} is the constant function $f \equiv k$ and its associated measure is given by $k\delta_0$. Hence, we have the implications (3) $\Leftrightarrow$ (5) $\Leftrightarrow$ (6). By Lemma \ref{lem: nonconstant OM}, we have (5) $\mathbb{R}ightarrow$ (4). (4) $\mathbb{R}ightarrow$ (2): Assume that $f$ is injective. Consider $A \geqslant 0$ and $B \geqslant 0$ such that $I \sigma A = I \sigma B$. Then $f(A) = f(B)$ by \eqref{eq: f(A) = I sm A}. Since $f^{-1} \circ f (x) =x$ for all $x \in \mathbb{R}^+$, we have $A=B$. (2) $\mathbb{R}ightarrow$ (1): Let $A>0, B\geqslant 0$ and $C \geqslant 0$ be such that $A \,\sigma\, B = A \,\sigma\, C$. By the congruence invariance of $\sigma$, we have \begin{align*} A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}} \:=\: A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} C A^{-\frac{1}{2}}) A^{\frac{1}{2}} \end{align*} and thus $I \,\sigma\, A^{-\frac{1}{2}} B A^{-\frac{1}{2}} = I \,\sigma\, A^{-\frac{1}{2}} C A^{-\frac{1}{2}} $. Now, the assumption (2) implies $$ A^{-\frac{1}{2}} B A^{-\frac{1}{2}} \:=\: A^{-\frac{1}{2}} C A^{-\frac{1}{2}}, $$ i.e., $B=C$. \end{proof} Recall that the \emph{transpose} of a connection $\sigma$ is the connection \begin{align*} (A,B) \;\mapsto B \,\sigma\, A. \end{align*} If $f$ is the representing function of $\sigma$, then the representing function of the transpose of $\sigma$ is given by the \emph{transpose} of $f$ (see \cite{Kubo-Ando}), defined by \begin{align*} x \mapsto xf(1/x), \quad x>0. \end{align*} A connection is \emph{symmetric} if it coincides with its transpose. \begin{thm} Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent: \begin{enumerate} \item[(1)] $\sigma$ is right cancellable ; \item[(2)] for each $A \geqslant 0$ and $B \geqslant 0$, $A \sigma I = B \sigma I \implies A=B$ ; \item[(3)] $\sigma$ is not a scalar multiple of the right-trivial mean ; \item[(4)] the transpose of $f$ is injective ; \item[(5)] $f$ is not a scalar multiple of the identity function $x \mapsto x$ ; \item[(6)] $\mu$ is not a scalar multiple of the Dirac measure $\delta_1$ at $1$. \end{enumerate} \end{thm} \begin{proof} It is straightforward to see that, for each $k\geqslant 0$, the representing function of the connection \begin{align*} k \omegaega_r \;:\; (A,B) \mapsto kB \end{align*} is the function $x \mapsto kx$ and its associated measure is given by $k\delta_1$. The proof is done by replacing $\sigma$ with its transpose in Theorem \ref{thm: cancallability 1}. \end{proof} \begin{remk} The injectivity of the transpose of $f$ does not imply the surjectivity of $f$. To see that, take $f(x)=(1+x)/2$. Then the transpose of $f$ is $f$ itself. \end{remk} The following results are characterizations of cancellability of connections. \begin{cor} Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent: \begin{enumerate} \item[(1)] $\sigma$ is cancellable ; \item[(2)] $\sigma$ is not a scalar multiple of the left/right-trivial mean ; \item[(3)] $f$ and its transpose are injective ; \item[(4)] $f$ is neither a constant function nor a scalar multiple of the identity function; \item[(5)] $\mu$ is not a scalar multiple of $\delta_0$ or $\delta_1$. \end{enumerate} In particular, every nontrivial mean is cancellable. \end{cor} \begin{remk} The ``order cancellability" does not hold for general connections, even if we restrict them to the class of means. For each $A,B>0$, it is not true that the condition $I \,\sigma\, A \leqslant I \,\sigma\, B$ or the condition $A \,\sigma\, I \leqslant B \,\sigma\, I$ implies $A \leqslant B$. To see this, take $\sigma$ to be the geometric mean. It is not true that $A^{1/2} \leqslant B^{1/2}$ implies $A \leqslant B$ in general. \end{remk} \section{Applications to certain operator equations} Cancellability of connections can be restated in terms of the uniqueness of certain operator equations as follows. A connection $\sigma$ is left cancellable if and only if \begin{quote} for each given $A>0$ and $B \geqslant 0$, if the equation $A \,\sigma\, X =B$ has a solution, then it has a unique solution. \end{quote} The similar statement for right-cancellability holds. In this section, we investigate the existence and the uniqueness of the operator equation $A \,\sigma\, X =B$. \begin{thm} \label{thm: operator eq 1} Let $\sigma$ be a connection which is not a scalar multiple of the left-trivial mean. Let $f$ be its representing function. Given $A>0$ and $B \geqslant 0$, the operator equation \begin{align*} A \, \sigma X \, \:=\: B \end{align*} has a (positive) solution if and only if $\Sp (A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) \subseteq \mathbb{R}ange (f)$. In fact, such a solution is unique and given by \begin{align*} X \:=\: A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}}. \end{align*} \end{thm} \begin{proof} Suppose that there is a positive operator $X$ such that $A \,\sigma\, X =B$. The congruence invariance of $\sigma$ yields \begin{align*} A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) A^{\frac{1}{2}} \:=\: B. \end{align*} The property \eqref{eq: f(A) = I sm A} now implies \begin{align*} f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \:=\: I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}}\:=\: A^{-\frac{1}{2}} B A^{-\frac{1}{2}}. \end{align*} By spectral mapping theorem, \begin{align*} \Sp (A^{-\frac{1}{2}} B A^{-\frac{1}{2}} ) =\Sp\left(f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \right) = f\left(\Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}})\right) \subseteq \mathbb{R}ange (f). \end{align*} Conversely, suppose that $\Sp (A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) \subseteq \mathbb{R}ange (f)$. Since $\sigma \neq k \omegaega_l$ for all $k \geqslant 0$, we have that $f$ is nonconstant by Theorem \ref{thm: cancallability 1}. It follows that $f$ is injective by Lemma \ref{lem: nonconstant OM}. The assumption yields the existence of the operator $X \equiv A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}}$. We obtain from the property \eqref{eq: f(A) = I sm A} that \begin{align*} A \,\sigma\, X \:&=\: A \,\sigma\, A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}} \\ \:&=\: A^{\frac{1}{2}} \left[I \,\sigma\, f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) \right] A^{\frac{1}{2}} \\ \:&=\: A^{\frac{1}{2}} f\left(f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}\right) A^{\frac{1}{2}} \\ \:&=\: B \end{align*} The uniqueness of a solution follows from the left-cancellability of $\sigma$. \end{proof} Similarly, we have the following theorem. \begin{thm} Let $\sigma$ be a connection which is not a scalar multiple of the right-trivial mean. Given $A>0$ and $B \geqslant 0$, the operator equation \begin{align*} X \,\sigma A \, \:=\: B \end{align*} has a (positive) solution if and only if $\Sp (A^{-1/2} B A^{-1/2}) \subseteq \mathbb{R}ange (g)$, here $g$ is the representing function of the transpose of $\sigma$. In fact, such a solution is unique and given by \begin{align*} X \:=\: A^{1/2} g^{-1}(A^{-1/2}BA^{-1/2}) A^{1/2}. \end{align*} \end{thm} \begin{thm} \label{cor: f is unbounded and f(0)=0} Let $\sigma$ be a connection with representing function $f$. Then the following statements are equivalent: \begin{enumerate} \item[(1)] $f$ is unbounded and $f(0)=0$ ; \item[(2)] $f$ is surjective, i.e., $f$ is right cancellable in the sense that \begin{align*} g \circ f = h \circ f \:\implies\: g=h \quad ; \end{align*} \item[(3)] the operator equation \begin{align} A \,\sigma X \, \:=\: B \label{eq: A sm X =B} \end{align} has a unique solution for any given $A>0$ and $B \geqslant 0$. \end{enumerate} Moreover, if (1) holds, then the solution of \eqref{eq: A sm X =B} varies continuously in each given $A>0$ and $B \geqslant 0$, i.e. the map $(A,B) \mapsto X$ is separately continuous with respect to the strong-operator topology. \end{thm} \begin{proof} (1) $\mathbb{R}ightarrow$ (2): This follows directly from the intermediate value theorem. (2) $\mathbb{R}ightarrow$ (3): It is immediate from Theorem \ref{thm: operator eq 1}. (3) $\mathbb{R}ightarrow$ (1): Assume (3). The uniqueness of solution for the equation $A \,\sigma\, X =B$ implies the left-cancellability of $\sigma$. By Theorem \ref{thm: cancallability 1}, $f$ is injective. The assumption (3) implies the existence of a positive operator $X$ such that \begin{align*} f(X) \:=\: I \,\sigma\, X \:=\: 0. \end{align*} The spectral mapping theorem implies that $f(\ld)=0$ for all $\lambda\in \Sp(X)$. Since $f$ is injective, we have $\Sp(X) =\{\ld\}$ for some $\lambda\in \mathbb{R}^+$. Suppose that $\ld>0$. Then there is $\epsilonsilon >0$ such that $X > \epsilon I >0$. It follows from the monotonicity of $\sigma$ that \begin{align*} 0 \:=\: I \,\sigma\, X \:\geqslant\: I \,\sigma\, \epsilon I \:=\: f(\epsilon)I, \end{align*} i.e. $f(\epsilon)=0$. Hence, $\epsilon = \ld$. Similarly, since $X > (\epsilon/2) I >0$, we have $\epsilon = 2 \ld$, a contradiction. Thus, $\ld=0$ and $f(0)=0$. Now, let $k>0$. The assumption (3) implies the existence of $X \geqslant 0$ such that $I \,\sigma\, X =kI$. Since $f(X)=kI$, we have $f(\ld)=k$ for all $\lambda\in \Sp(X)$. Since $\Sp(X)$ is nonempty, there is $\lambda\in \Sp(X)$ such that $f(\ld)=k$. Therefore, $f$ is unbounded. Assume that (1) holds. Then the map $(A,B) \mapsto X$ is well-defined. Recall that if $A_n \in B(\mathcal{H})^+$ converges strongly to $A$, then $\phii(A_n)$ converges strongly to $\phii(A)$ for any continuous function $\phii$. It follows that the map $$(A,B) \mapsto X = A^{\frac{1}{2}} f^{-1}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}}) A^{\frac{1}{2}} $$ is separately continuous in each variable. \end{proof} \begin{ex} Consider the quasi-arithmetic power mean $\#_{p,\alpha}$ with exponent $p \in [-1,1]$ and weight $\alpha \in (0,1)$, defined by \begin{align*} A \,\#_{p,\alpha}\, B \:=\: \left[(1-\alpha)A^p + \alpha B^p \right]^{1/p}. \end{align*} Its representing function of this mean is given by \begin{align*} f_{p,\alpha}(x) \:=\: (1-\alpha+\alpha x^p)^{1/p}. \end{align*} The special cases $p=1$ and $p=-1$ are the $\alpha$-weighted arithmetic mean and the $\alpha$-weighted harmonic mean, respectively. The case $p=0$ is defined by continuity and, in fact, $\#_{0,\alpha} = \#_{\alpha}$ and $f_{0,\alpha}(x)=x^{\alpha}$. Given $A>0$ and $B \geqslant 0$, consider the operator equation \begin{align} A \,\#_{p,\alpha}\, X \:=\: B. \label{eq weighted GM} \end{align} \underline{The case $p=0$ :} Since the range of $f_{0,\alpha}(x)=x^{\alpha}$ is $\mathbb{R}^+$, the equation \eqref{eq weighted GM} always has a unique solution given by \begin{align*} X \:=\: A^{1/2} (A^{-1/2}BA^{-1/2})^{1/\alpha} A^{1/2} \:\equiv \: A \,\#_{1/\alpha}\, B. \end{align*} \underline{The case $0<p\leqslant 1$ :} The range of $f_{p,\alpha}$ is the interval $[(1-\alpha)^{1/p}, \infty)$. Hence, the equation \eqref{eq weighted GM} is solvable if and only if $\Sp(A^{-1/2} B A^{-1/2}) \subseteq [(1-\alpha)^{1/p}, \infty)$, i.e., $B \geqslant (1-\alpha)^{1/p} A$. \underline{The case $-1 \leqslant p< 0$ :} The range of $f_{p,\alpha}$ is the interval $[0, (1-\alpha)^{1/p})$. Hence, the equation \eqref{eq weighted GM} is solvable if and only if $\Sp(A^{-1/2} B A^{-1/2}) \subseteq [0, (1-\alpha)^{1/p})$, i.e., $B < (1-\alpha)^{1/p} A$. For each $p \in [-1,0) \cup (0,1]$ and $\alpha \in (0,1)$, we have \begin{align*} f_{p,\alpha}^{-1} (x) \:=\: \left(1-\frac{1}{\alpha} + \frac{1}{\alpha} x^p \right)^{1/p}. \end{align*} Hence, the solution of \eqref{eq weighted GM} is given by \begin{align*} X \:=\: \left[(1-\frac{1}{\alpha})A^p + \frac{1}{\alpha} B^p \right]^{1/p} \:\equiv\: A \;\#_{p, \frac{1}{\alpha}}\; B. \end{align*} \end{ex} \begin{ex} Let $\sigma$ be the logarithmic mean with representing function \begin{align*} f(x) \:=\: \frac{x-1}{\log x}, \quad x>0. \end{align*} Here, $f(0) \equiv 0$ by continuity. We see that $f$ is unbounded. Thus, the operator equation $A \,\sigma X \, =B$ is solvable for all $A>0$ and $B \geqslant 0$. \end{ex} \begin{ex} Let $\eta$ be the dual of the logarithmic mean, i.e., \begin{align*} \eta \,:\, (A,B) \mapsto \left(A^{-1} \, \sigma \, B^{-1}\right)^{-1} \end{align*} where $\sigma$ denotes the logarithmic mean. The representing function of $\eta$ is given by \begin{align*} f(x) \:=\: \frac{x}{x-1} \log{x}, \quad x>0. \end{align*} We have that $f(0) \equiv 0$ by continuity and $f$ is unbounded. Therefore, the operator equation $A \,\eta X \, =B$ is solvable for all $A>0$ and $B \geqslant 0$. \end{ex} \section{Regularity of connections} In this section, we consider the regularity of connections. \begin{thm} \label{thm: regularity} Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent. \begin{enumerate} \item[(1)] $f(0)=0$ ; \item[(2)] $\mu(\{0\}) = 0$ ; \item[(3)] $I \,\sigma\, 0 = 0$ ; \item[(4)] $A \,\sigma\, 0 =0$ for all $A \geqslant 0$ ; \item[(5)] for each $A \geqslant 0$, $0 \in \Sp(A) \Longrightarrow 0 \in \Sp(I \,\sigma\, A)$ ; \item[(6)] for each $A,X \geqslant 0$, $0 \in \Sp(A) \Longrightarrow 0 \in \Sp(X \,\sigma\, A)$. \end{enumerate} \end{thm} \begin{proof} From the integral representation \eqref{int rep of OMF}, we have \begin{align} f(x) \:=\: \mu(\{0\}) + \mu(\{1\})x + \int_{(0,1)} (1\,!_t\, x)\,d\mu(t), \quad x\geqslant 0, \label{eq: int rep of OMR expand} \end{align} i.e. $f(0)=\mu(\{0\})$. From the property \eqref{eq: f(A) = I sm A}, we have $I \,\sigma\, 0 =f(0)I$. Hence, (1)-(3) are equivalent. It is clear that (4) $\mathbb{R}ightarrow$ (3) and (6) $\mathbb{R}ightarrow$ (5). (3) $\mathbb{R}ightarrow$ (4): Assume that $I \,\sigma\, 0=0$. For any $A>0$, we have by the congruence invariance that \begin{align*} A \,\sigma\, 0 \:=\: A^{\frac{1}{2}} (I \,\sigma\, 0) A^{\frac{1}{2}} \:=\: 0. \end{align*} For general $A \geqslant 0$, we have $(A + \epsilon I) \,\sigma\, 0 =0$ for all $\epsilon>0$ by the previous claim and hence $A \,\sigma\, 0 =0$ by the continuity from above. (5) $\mathbb{R}ightarrow$ (1): We have $0 \in \Sp(I \,\sigma\, 0) = \Sp(f(0)I) = \{f(0)\}$, i.e. $f(0)=0$. (1) $\mathbb{R}ightarrow$ (6): Assume $f(0)=0$. Consider $A \geqslant 0$ such that $0 \in \Sp(A)$, i.e. $A$ is not invertible. Assume first that $X>0$. Then \begin{align*} X \,\sigma\, A \:=\: X^{\frac{1}{2}} (I \,\sigma\, X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) X^{\frac{1}{2}} \:=\: X^{\frac{1}{2}} f(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) X^{\frac{1}{2}}. \end{align*} Since $X^{-\frac{1}{2}} A X^{-\frac{1}{2}}$ is not invertible, we have $0 \in \Sp(X^{-\frac{1}{2}} A X^{-\frac{1}{2}})$ and hence by spectral mapping theorem \begin{align*} 0 \:=\: f(0) \in f \left(\Sp(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) \right) \:=\: \Sp\left(f(X^{-\frac{1}{2}} A X^{-\frac{1}{2}}) \right). \end{align*} This implies that $X \,\sigma\, A$ is not invertible. Now, consider $X \geqslant 0$. The previous claim shows that $(X+I)\,\sigma\, A$ is not invertible. Since $X \,\sigma\, A \leqslant (X+I) \,\sigma\, A$, we conclude that $X \,\sigma\, A$ is not invertible. \end{proof} We say that a connection $\sigma$ is \emph{nonregular} if one (thus, all) of the conditions in Theorem \ref{thm: regularity} holds, otherwise $\sigma$ is \emph{regular}. Hence, regular connections correspond to regular operator monotone functions and regular monotone metrics. \begin{remk} Recall from \cite{Pattrawut} that every connection $\sigma$ can be written as the sum of three connections. The \emph{singularly discrete part} is a countable sum of weighted harmonic means with nonnegative coefficients. The \emph{absolutely continuous part} is a connection admitting an integral representation with respect to Lebesgue measure $m$ on $[0,1]$. The \emph{singularly continuous part} is associated to a continuous measure mutually singular to $m$. Hence, a connection, whose associated measure has no singularly discrete part, is nonregular. \end{remk} \begin{remk} Let $\sigma$ be a connection with representing function $f$ and associated measure $\mu$. Let $g$ be the representing function of the transpose of $\sigma$. From \eqref{eq: int rep of OMR expand}, \begin{align*} g(0) = \lim_{x \to 0^+} xf(\dfrac{1}{x}) = \lim_{x \to \infty} \frac{f(x)}{x} = \mu(\{1\}). \end{align*} Thus, the transpose of $\sigma$ is nonregular if and only if $\mu(\{1\})=0$. \end{remk} \begin{thm} \label{thm: regularity of mean} The following statements are equivalent for a mean $\sigma$. \begin{enumerate} \item[(1)] $\sigma$ is nonregular ; \item[(2)] $I \,\sigma\, P = P$ for each projection $P$. \end{enumerate} \end{thm} \begin{proof} (1) $\mathbb{R}ightarrow$ (2): Assume that $f(0)=0$ and consider a projection $P$. Since $f(1)=1$, we have $f(x)=x$ for all $x \in \{0,1\}\supseteq \Sp(P)$. Thus $I \,\sigma\, P = f(P)=P$. (2) $\mathbb{R}ightarrow$ (1): We have $0=I \,\sigma\, 0=f(0)I$, i.e. $f(0)=0$. \end{proof} To prove the next result, recall the following lemma. \begin{lem}[\cite{Kubo-Ando}] If $f: \mathbb{R}^+ \to \mathbb{R}^+$ is an operator monotone function such that $f(1)=1$ and $f$ is neither the constant function $1$ nor the identity function, then \begin{enumerate} \item[1)] $0<x<1 \implies x<f(x)<1$ \item[2)] $1<x \implies 1<f(x)<x$. \end{enumerate} \end{lem} \begin{thm} Let $\sigma$ be a nontrivial mean. For each $A \geqslant 0$, if $I \,\sigma\, A =A$, then $A$ is a projection. Hence, the following statements are equivalent: \begin{enumerate} \item[(1)] $\sigma$ is nonregular ; \item[(2)] for each $A \geqslant 0$, $A$ is a projection if and only if $I \,\sigma\, A = A$. \end{enumerate} \end{thm} \begin{proof} Since $I \,\sigma\, A=A$, we have $f(A)=A$ by \eqref{eq: f(A) = I sm A}. Hence $f(x)=x$ for all $x \in \Sp(A)$ by the injectivity of the continuous functional calculus. Since $\sigma$ is a nontrivial mean, the previous lemma forces that $\Sp(A) \subseteq \{0,1\}$, i.e. $A$ is a projection. \end{proof} \begin{thm} Under the condition that $\sigma$ is a left-cancellable connection with representing function $f$, the following statements are equivalent: \begin{enumerate} \item[(1)] $\sigma$ is nonregular. \item[(2)] The equation $f(x)=0$ has a solution $x$. \item[(3)] The equation $f(x)=0$ has a unique solution $x$. \item[(4)] The only solution to $f(x)=0$ is $x=0$. \item[(5)] For each $A>0$, the equation $A \,\sigma\, X =0$ has a solution $X$. \item[(6)] For each $A>0$, the equation $A \,\sigma\, X =0$ has a unique solution $X$. \item[(7)] For each $A>0$, the only solution to the equation $A \,\sigma\, X =0$ is $X=0$. \item[(8)] The equation $I \,\sigma\, X =0$ has a solution $X$. \item[(9)] The equation $I \,\sigma\, X =0$ has a unique solution $X$. \item[(10)] The only solution to the equation $I \,\sigma\, X =0$ is $X=0$. \end{enumerate} Similar results for the case of right-cancellability hold. \end{thm} \begin{proof} It is clear that (1) $\mathbb{R}ightarrow$ (2), (6) $\mathbb{R}ightarrow$ (5) and (10) $\mathbb{R}ightarrow$ (8). Since $f$ is injective by Theorem \ref{thm: cancallability 1}, we have (2) $\mathbb{R}ightarrow$ (3) $\mathbb{R}ightarrow$ (4). (4) $\mathbb{R}ightarrow$ (10): Let $X \geqslant 0$ be such that $I \,\sigma\, X =0$. Then $f(X)=0$ by \eqref{eq: f(A) = I sm A}. By spectral mapping theorem, $f(\Sp(X)) =\{0\}$. Hence, $\Sp(X)=\{0\}$, i.e. $X=0$. (8) $\mathbb{R}ightarrow$ (9): Consider $X \geqslant 0$ such that $I \,\sigma\, X =0$. Then $f(X)=0$. Since $f$ is injective with continuous inverse, we have $X=f^{-1}(0)$. (9) $\mathbb{R}ightarrow$ (6): Use congruence invariance. (5) $\mathbb{R}ightarrow$ (7): Let $A>0$ and consider $X \geqslant 0$ such that $A \,\sigma\, X=0$. Then $A^{\frac{1}{2}} (I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) A^{\frac{1}{2}} =0$, i.e. $f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) = I \,\sigma\, A^{-\frac{1}{2}} X A^{-\frac{1}{2}} =0$. Hence, \begin{align*} f\left( \Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \right) = \Sp\left( f(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) \right) =\{0\}. \end{align*} Suppose there exists $\lambda\in \Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}})$ such that $\ld>0$. Then $f(0)<f(\ld)=0$, a contradiction. Hence, $\Sp(A^{-\frac{1}{2}} X A^{-\frac{1}{2}}) =\{0\}$, i.e. $A^{-\frac{1}{2}} X A^{-\frac{1}{2}} =0$ or $X=0$. (7) $\mathbb{R}ightarrow$ (1): We have $f(0)I = I \,\sigma\, 0 =0$, i.e. $f(0)=0$. \end{proof} \end{document}

\begin{document} \title{{\bf Global Solution of Atmospheric Circulation Models with Humidity Effect} \thanks{Foundation item: the National Natural Science Foundation of China (NO. 11271271).}} \author{{\sl Hong Luo} \\{\small College of Mathematics and Software Science, Sichuan Normal University,} \\ {\small Chengdu, Sichuan 610066, China }} \date{} \maketitle \begin{minipage}{5.5 in} \noindent{\bf Abstract:}\,\ \ The atmospheric circulation models are deduced from the very complex atmospheric circulation models based on the actual background and meteorological data. The models are able to show features of atmospheric circulation and are easy to be studied. It is proved that existence of global solutions to atmospheric circulation models with the use of the $T$-weakly continuous operator. {\bf Key Words:}\ \ Atmospheric Circulation Models; Humidity Effect; Global Solution {\bf 2000 Mathematics Subject Classification:} \ 35A01, 35D30, 35K20\\ \end{minipage} \section{Introduction} Mathematics is a summary and abstraction of the production practices and the natural sciences, and is a powerful tool to explain natural phenomena and reveal the laws of nature as well. Atmospheric circulation is one of the main factors affecting the global climate, so it is very necessary to understand and master its mysteries and laws. Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies. On the contrary, it is also the important result of these physical transports, balance and conversion. Thus it is of necessity to study the characteristics, formation, preservation, change and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human's understanding of nature, but also the helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources. \par The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments. The atmosphere or the ocean or the coupled atmosphere and ocean can be viewed as an initial and boundary value problem \cite{Phillips} \cite{Rossby}, or an infinite dimensional dynamical system \cite{Lions1}\cite{Lions2}\cite{Lions3}. These phenomena involve a broad range of temporal and spatial scales \cite{Charney2}. For example, according to \cite{von Neumann}, the motion of the atmosphere can be divided into three categories depending on the time scale of the prediction. They are motions corresponding respectively to the short time, medium range and long term behavior of the atmosphere. The understanding of these complicated and scientific issues necessitate a joint effort of scientists in many fields. Also, as \cite{von Neumann} pointed out, this difficult problem involves a combination of modeling, mathematical theory and scientific computing. Some authors have studied the atmospheric motions viewed as an infinite dimensional dynamical system. In \cite{Lions1}, the authors study as a first step towards this long-range project which is widely considered as the basic equations of atmospheric dynamics in meteorology, namely the primitive equations of the atmosphere. The mathematical formulation and attractors of the primitive equations, with or without vertical viscosity, are studied. First of all, by integrating the diagnostic equations they present a mathematical setting, and obtain the existence and time analyticity of solutions to the equations. They then establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems. In \cite{Li}, based on the complete dynamical equations of the moist atmospheric motion, the qualitative theory of nonlinear atmosphere with dissipation and external forcing and its applications are systematically discussed by new theories and methods on the infinite dimensional dynamical system. In \cite{Wang}, by Lions theorem in the Hilbert space, the existence and uniqueness of the weak solution of water vapour-equation with the first boundary-value problem are proven, and the scheme of the finite-element method according to the weak solution is proposed. In \cite{Huang}, the model of climate for weather forecast is studied, and the existence of the weak solution is proved by Galerkin method. The asymptotic behaviors of the weak solution is described by the trajectory attractors. In this article, Atmospheric circulation equations with humidity effect is considered, which is different from the previous research. Previous studies are based on two kinds of equations, one is about the heat and humidity transfer(\cite{Li},\cite{Wang}), without considering the diffusion of heat and of humidity; the other is p-coordinates equation in(\cite{Lions1},\cite{Huang}), being used to consider the horizontal movement of the atmosphere, and the vertical direction of the velocity is transformed into pressure and topography. Atmospheric circulation equations with humidity effect, derivation from the original equations in \cite{Richardson}, considering the diffusion of heat and of humidity, not be restricted by p-coordinates, can be deformed based on the different concerns. In the last part of the article, Existence of global solutions to the atmospheric circulation models is obtained, which implies that atmospheric circulation has its own running way as humidity source and heat source change, and confirms that the atmospheric circulation models are reasonable. The paper is organized as follows. In Section 2 we present derivation of atmospheric circulation models. In Section 3, we prove that the atmospheric circulation models possess global solutions by using space sequence method. \section{Derivation of Atmospheric Circulation Models with \\Humidity Effect} \subsection{Original Model} The hydrodynamical equations governing the atmospheric circulation are the Navier-Stokes equations with the Coriolis force generated by the earth¡¯s rotation, coupled with the first law of thermodynamics. \par Let $(\varphi, \theta, r)$ be the spheric coordinates, where $\varphi$ represents the longitude, $\theta$ the latitude, and $r$ the radial coordinate. The unknown functions include the velocity field $u=(u_\varphi, u_\theta, u_r)$, the temperature function $T$, the humidity function $q$, the pressure $p$ and the density function $\rho$. Then the equations governing the motion and states of the atmosphere consist of the momentum equation, the continuity equation, the first law of thermodynamic, and the diffusion equation for humidity, and the equation of state (for ideal gas): \begin{eqnarray} \label{eqa1} \rho[\frac{\partial u}{\partial t}+\nabla_u u+2\overrightarrow{\Omega} \times u]+\nabla p+\overrightarrow{\kappa} \rho g=\mu \Delta u, \end{eqnarray} \begin{eqnarray} \label{eqa2} \frac{\partial \rho}{\partial t}+div(\rho u)=0, \end{eqnarray} \begin{eqnarray} \label{eqa3} \rho c_v[\frac{\partial T}{\partial t}+u\cdot\nabla T] +p div u=Q+\kappa_T \Delta T, \end{eqnarray} \begin{eqnarray} \label{eqa4} \rho[\frac{\partial q}{\partial t}+u \cdot \nabla_q]=G+\kappa_q \Delta q, \end{eqnarray} \begin{eqnarray} \label{eqa5} p=R_0\rho T, \end{eqnarray} Where $-\infty <\varphi <+\infty$, $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, $r_0<r<r_0+h$, $r_0$ is the radius of the earth, $h$ is the height of the troposphere, $\Omega$ is the earth's rotating angular velocity, $g$ is the gravitative constant, $\mu, \kappa_T, \kappa_q, c_v, R_0$ are constants, $Q$ and $G$ are heat and humidity sources, and $\overrightarrow{\kappa} = (0, 0, 1)$. The differential operators used are as follows: \par 1. The gradient and divergence operators are given by: $$ \nabla=(\frac{1}{r\cos\theta}\frac{\partial}{\partial \varphi}, \frac{1}{r} \frac{\partial}{\partial \theta}, \frac{\partial}{\partial r}), $$ $$ div u=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 u_r)+\frac{1}{r \cos\theta} \frac{\partial (u_\theta \cos \theta)}{\partial \theta}+\frac{1}{r \cos\theta} \frac{\partial u_\varphi}{\partial \varphi}, $$ \par 2. In the spherical geometry, although the Laplacian for a scalar is different from the Laplacian for a vectorial function, we use the same notation $\Delta$ for both of them: $$ \Delta u=( \Delta u_\varphi+\frac{2}{r^2 \cos\theta} \frac{\partial u_r }{\partial \varphi}+\frac{2 \sin \theta}{r^2 \cos^2 \theta} \frac{\partial u_\theta}{\partial \varphi}- \frac{u_\varphi}{r^2 \cos^2 \theta}, $$ $$ \Delta u_\theta+\frac{2}{r^2} \frac{\partial u_r }{\partial \theta}-\frac{u_\theta}{r^2 \cos^2 \theta}- \frac{2 \sin\theta}{r^2 \cos^2 \theta}\frac{u_\varphi}{\partial \varphi}, $$ $$ \Delta u_r+\frac{2 u_r}{r^2}+\frac{2}{r^2 \cos \theta} \frac{\partial (u_\theta \cos\theta)}{\partial \theta}- \frac{2}{r^2 \cos \theta}\frac{u_\varphi}{\partial \varphi}), $$ $$ \Delta f=\frac{1}{r^2 \cos\theta} \frac{\partial f}{\partial\theta}(\cos\theta \frac{\partial}{\partial \theta})+\frac{1}{r^2 \cos^2 \theta} \frac{\partial^2 f}{\partial \varphi^2}+\frac{1}{r^2} \frac{\partial f}{\partial r}(r^2 \frac{\partial}{\partial r}), $$ \par 3. The convection terms are given by $$ \nabla_u u=(u\cdot \nabla u_\varphi+\frac{u_\varphi u_r}{r}-\frac{u_\varphi u_\theta}{r} \tan \theta, $$ $$ u\cdot \nabla u_\theta+\frac{u_\theta u_r}{r}+\frac{u^2_\varphi}{r} \tan \theta, u\cdot \nabla u_r-\frac{u^2_\varphi+ u^2_\theta}{r}), $$ \par 4. The Coriolis term $2 \overrightarrow{\Omega}\times u$ is given by $$ 2 \overrightarrow{\Omega}\times u=2 \Omega(\cos \theta u_r-\sin\theta u_\theta, \sin \theta u_{\theta}, -\cos\theta u_\varphi), $$ Here $\overrightarrow{\Omega}$ is the angular velocity vector of the earth, and $\Omega$ is the magnitude of the angular velocity. \par They are supplemented with the following initial value conditions \begin{eqnarray} \label{eqa100} (u, T, q ) = (\varphi_{10},\varphi_{20}, \varphi_{30}) \quad at \quad t = 0. \end{eqnarray} \par Boundary conditions are needed at the top and bottom boundaries $(r_0, r_0+h)$. At the top and bottom boundaries $(r=r_0,r_0+h)$, either the Dirichlet boundary condition or the free boundary condition is given \begin{eqnarray} \label{eqa101} (\hbox{Dirichlet})\quad \left\{ \begin{array}{ll} (u, T,q ) =(0, T_0, q_0), & r=r_0,\\ (u, T,q )=(0, T_1, q_1), & r=r_0+h, \end{array} \right. \end{eqnarray} \begin{eqnarray} \label{eqa102} (\hbox{free})\quad \left\{ \begin{array}{ll} (u, T,q ) =(0, T_0, q_0), & r=r_0,\\ (u_r,T,q )=(0,T_1, q_1),\quad \frac{\partial(u_\varphi, u_\theta)}{\partial r} = 0 & r=r_0+h, \end{array} \right. \end{eqnarray} \par For $\varphi$, periodic condition are usually used, for any integers $k_1\in Z$ \begin{eqnarray} \label{eqa103} (u, T, q )(\varphi + 2k_1 \pi, \theta , r) = (u, T, q )(\varphi + 2k_1 \pi, \theta , r). \end{eqnarray} \par Because $(\varphi, \theta, r)$ are in a circular field with $C^\infty$ boundary, the space domain is taken as $(0,2\pi) \times (-\frac{\pi}{2}, \frac{\pi}{2}) \times (r_0,r_0+h)$ and periodic condition is written as \begin{eqnarray} \label{eqa104} (u, T, q )(0,\theta, r) = (u, T, q )(2\pi,\theta, r). \end{eqnarray} For simplicity, we study the problem with the Dirichlet boundary conditions, and all results hold true as well for other combinations of boundary conditions. Atmospheric convection equations can be read as (\ref{eqa1})-(\ref{eqa101}), (\ref{eqa104}). \par The above equations were basically the equations used by L. F. Richardson in his pioneering work \cite{Richardson}. However, they are in general too complicated to conduct theoretical analysis. As practiced by the earlier workers such as J. Charney, and from the lessons learned by the failure of Richardson's pioneering work, one tries to be satisfied with simplified models approximating the actual motions to a greater or lesser degree instead of attempting to deal with the atmosphere in all its complexity. By starting with models incorporating only what are thought to be the most important of atmospheric influences, and by gradually bringing in others, one is able to proceed inductively and thereby to avoid the pitfalls inevitably encountered when a great many poorly understood factors are introduced all at once. The simplifications are usually done by taking into consideration of some main characterizations of the large-scale atmosphere. One such characterization is the small aspect ratio between the vertical and horizontal scales, leading to hydrostatic equation replacing the vertical momentum equation. The resulting system of equation are called the primitive equations (PEs); see among others \cite{Lions1}. The another characterization of the large scale motion is the fast rotation of the earth, leading to the celebrated quasi-geostrophic equations \cite{Charney1}. \par For convenience of research, the approximations we adopt involves the following components: \par First, we often use Boussinesq assumption, where the density is treated as a constant except in the buoyancy term and in the equation of state. \par Second, because the air is generally compressible, we do not use the equation of state for ideal gas, rather, we use the following empirical formula, which can by considered as the linear approximation of \begin{eqnarray} \label{eqa6} \rho=\rho_0[1-\alpha_T(T-T_0)+\alpha_q(q-q_0)], \end{eqnarray} where $\rho_0$ is the density at $T = T_0$ and $q = q_0$, and $\alpha_T$ and $\alpha_q$ are the coefficients of thermal and humidity expansion. \par Third, since the aspect ratio between the vertical scale and the horizontal scale is small, the spheric shell the air occupies is treated as a product space $S^2_{r_0}\times(r_0, r_0+ h)$. This approximation is extensively adopted in geophysical fluid dynamics. Under the above simplification, we have the following equations governing the motion and states of large scale atmospheric circulations: \begin{eqnarray} \label{eqa7} \frac{\partial u}{\partial t}+\nabla_u u=\nu \Delta u-2\overrightarrow{\Omega} \times u-\frac{1}{\rho_0}\nabla p-[1-\alpha_T(T-T_0)+\alpha_q(q-q_0)] g e_z, \end{eqnarray} \begin{eqnarray} \label{eqa8} \frac{\partial T}{\partial t}+(u\cdot\nabla) T =Q+\kappa_T \Delta T, \end{eqnarray} \begin{eqnarray} \label{eqa9} \frac{\partial q}{\partial t}+(u \cdot \nabla)q=G+\kappa_q \Delta q, \end{eqnarray} \begin{eqnarray} \label{eqa10} div u=0, \end{eqnarray} where $(\varphi, \theta, z)\in M = S^2_{r_0}\times(r_0, r_0 + h),$ $\nu=\frac{\mu}{\rho_0}$ is the kinematic viscosity, $u=u_\varphi e_\varphi+u_\theta e_\theta+u_r e_r,$ $(e_\varphi, e_\theta, e_r)$ the local normal basis in the sphereric coordinates, and $$ \nabla_u u=((u\cdot \nabla) u_\varphi+\frac{u_\varphi u_z}{r_0}-\frac{u_\varphi u_\theta}{r_0} \tan \theta)e_\varphi+ $$ $$ ((u\cdot \nabla) u_\theta+\frac{u_\theta u_z}{r_0}+\frac{u^2_\varphi}{r_0} \tan \theta)e_\theta+((u\cdot \nabla) u_z-\frac{u^2_\varphi+ u^2_\theta}{r_0})e_z, $$ $$ \Delta u=(\Delta u_\varphi+\frac{2}{r_0^2 \cos\theta} \frac{\partial u_z }{\partial \varphi}+\frac{2 \sin \theta}{r_0^2 \cos^2 \theta} \frac{\partial u_\theta}{\partial \varphi}- \frac{u_\varphi}{r_0^2 \cos^2 \theta})e_\varphi+ $$ $$ (\Delta u_\theta+\frac{2}{r_0^2} \frac{\partial u_z}{\partial \theta}-\frac{u_\theta}{r_0^2 \cos^2 \theta}- \frac{2 \sin\theta}{r_0^2 \cos^2 \theta}\frac{u_\varphi}{\partial \varphi})e_\theta+ $$ $$ (\Delta u_z+\frac{2 u_0}{r^2}+\frac{2}{r_0^2 \cos \theta} \frac{\partial (u_\theta \cos\theta)}{\partial \theta}- \frac{2}{r_0^2 \cos \theta}\frac{u_\varphi}{\partial \varphi})e_z, $$ $$ \nabla p=\frac{1}{r_0 \cos \theta}\frac{\partial p}{\partial \theta}e_\varphi+\frac{1}{r} \frac{\partial p}{\partial \theta}e_\theta+\frac{\partial p}{\partial z}e_z, $$ $$ div u=\frac{1}{r_0 \cos\theta} \frac{\partial u_\varphi}{\partial \varphi}+\frac{1}{r_0 \cos\theta} \frac{\partial (u_\theta \cos \theta)}{\partial \theta}+\frac{\partial u_z}{\partial z}, $$ and the differential operators $(u\cdot \nabla)$ and $\triangle$ are expressed as $$ (u \cdot \nabla)=\frac{u_\varphi}{r_0 \cos\theta} \frac{\partial}{\partial \varphi}+\frac{u_\theta}{r_0}\frac{\partial}{\partial \theta}+u_z \frac{\partial}{\partial z}, $$ $$ \Delta=\frac{1}{r_0^2 \cos\theta} \frac{\partial^2}{\partial \varphi^2}+\frac{1}{r_0^2 \cos\theta} \frac{\partial}{\partial \theta}(\cos\theta \frac{\partial}{\partial \theta})+\frac{\partial^2}{\partial z^2}. $$ \par Equations (\ref{eqa7})-(\ref{eqa10}) are supplemented with boundary conditions (\ref{eqa101}), (\ref{eqa104}). \subsection{Simplification of Model} Atmospheric circulation is the large-scale motion of the air, which is essentially a thermal convection process caused by the temperature and humidity difference between the earth¡¯s surface and the tropopause. It is a crucial means by which heat and humidity are distributed on the surface of the earth. Air circulates within the troposphere, limit vertically by the tropopause at about 8-10km. Atmospheric motion in the troposphere, together with the oceanic circulation, plays a crucial role in leading to the global climate changes and evolution on the earth. There are two types of circulation cells: the latitudinal circulation and the longitudinal circulation. The latitudinal circulation is characterized by the Polar cell, the Ferrel cell, and the Hadley cell, which are major players in global heat and humidity transport, and do not act alone. The zonal circulation consists of six circulation cells over the equator. The overall atmospheric motion is known as the zonal overturning circulation, and also called the Walker circulation. The most remarkable feature of the global atmospheric circulation is that the equatorial Walker circulation divides the whole earth into three invariant regions of atmospheric flow: the northern hemisphere, the southern hemisphere, and the equatorial zone. We also note the important fact that the large-scale structure of the zonal overturning circulation varies from year to year, but its basic structure remains fairly stability, it never vanishes. Based on these natural phenomena, we here present the Zone Hypotheses for atmospheric dynamics, which amounts to saying that the global atmospheric system can be divided into three sub-systems: the North-Hemispheric System, the South-Hemispheric System, and the Tropical Zone System, which are relatively independent, and have less influence on each other in their basic structure. More precisely, the Atmospheric Zone Hypothesis is stated in the following form\cite{Ma2}. \par {\bf Atmospheric Zone Hypothesis.} The atmospheric circulation has three invariant regions: the northern hemisphere domain ($0 <\theta\leq\frac{\pi}{2}$), the southern hemisphere domain ($- \frac{\pi}{2}\leq \theta < 0$), and the equatorial zone ($\theta = 0$). Namely, the large-scale circulations in their invariant regions can act alone with less influence on the others. In particular, the velocity field $u = (u_\varphi, u_\theta, u_z)$ of the atmospheric circulation has a vanished latitudinal component in a narrow equatorial zone, i.e., $u_\theta = 0,$ for $-\varepsilon<\theta<\varepsilon$, where $\varepsilon> 0$ is a small number. \par Atmospheric Zone Hypothesis is based on the following several evidences from theory and practice: \par (1) The global atmospheric motion equations (\ref{eqa7})-(\ref{eqa10}) are of $\theta-$reflexive symmetry, i.e. under the $\theta-$reflexive transformation $(\varphi, \theta, z)\rightarrow (\varphi,-\theta, z)$, the velocity field $u$ becomes $(u_\varphi, u_\theta, u_z)\rightarrow (u_\varphi,-u_\theta, u_z)$, and equations (\ref{eqa7})-(\ref{eqa10}) are invariant, which implies that this system is compatible with the Atmospheric Zone Hypothesis. \par (2) Climatic observation data show that when the El Ni$\tilde{n}o$-Southern Oscillation (the behavior that the Walker circulation cell in the Western Pacific stops or reverses its direction) takes place, no oscillation occurs in the latitudinal cells. It demonstrates the relative independence of these circulations in their invariant domain. \par (3) When a cold current moves southward from Siberia, or a violent typhoon sweeps northward from the tropics, the weather in Southern Hemisphere has no response. Atmospheric Zone Hypothesis provides a theoretic basis for the study of atmospheric dynamics, by which we can establish locally simplified models to treat many difficult problems in atmospheric science. \par (4) For the three-dimensional atmospheric circulation equation, it is too difficult to study. So we study the equatorial atmospheric circulation. \par The atmospheric motion equations over the tropics are the equations restricted on $\theta= 0$, where the meridional velocity component $u_\theta$ is set to zero, and the effect of the turbulent friction is taking into considering \begin{eqnarray} \label{eqa13} \frac{\partial u_\varphi}{\partial t}=-(u \cdot\nabla ) u_\varphi-\frac{u_\varphi u_z}{r_0}+\nu (\Delta u_\varphi+\frac{2 }{r_0^2}\frac{\partial u_z}{\partial \varphi}-\frac{2u_\varphi}{r_0^2})-\sigma_0 u_\varphi-2 \Omega u_z-\frac{1}{\rho_0 r_0} \frac{\partial p}{\partial \varphi}, \end{eqnarray} $$ \frac{\partial u_z}{\partial t}=-( u\cdot \nabla) u_z+\frac{u_\varphi^2}{r_0}+\nu (\Delta u_z+\frac{2 }{r_0^2}\frac{\partial u_\varphi}{\partial \varphi}-\frac{2u_z}{r_0^2})-\sigma_1 u_z-2 \Omega u_\varphi $$ \begin{eqnarray} \label{eqa14} -\frac{1}{\rho_0} \frac{\partial p}{\partial z}-[1-\alpha_T(T-T_0)+\alpha_q(q-q_0)] g, \end{eqnarray} \begin{eqnarray} \label{eqa15} \frac{\partial T}{\partial t}=-(u\cdot\nabla) T +\kappa_T \Delta T+Q, \end{eqnarray} \begin{eqnarray} \label{eqa16} \frac{\partial q}{\partial t}=-(u \cdot \nabla)q+\kappa_q \Delta q+G, \end{eqnarray} \begin{eqnarray} \label{eqa17}\frac{1}{r_0}\frac{\partial u_\varphi}{\partial\varphi}+\frac{\partial u_z}{\partial z}=0, \end{eqnarray} Here $\sigma_i = C_i h^2 (i = 0, 1)$ represent the turbulent friction, $r_0$ is the radius of the earth, the space domain is taken as $M = S^1_{r_0} \times (r_0, r_0 + h)$ with $S^1_{r_0}$ being the one-dimensional circle with radius $r_0$, and $$ (u\cdot\nabla)=\frac{u_\varphi}{r_0}\frac{\partial}{\partial \varphi}+u_z \frac{\partial}{\partial z}, \quad \Delta=\frac{1}{r_0^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2} $$ For simplicity, we denote $$ (x_1,x_2)=(r_0 \varphi, z), \quad (u_1,u_2)=(u_\varphi,u_z). $$ \par The atmospheric motion equations(\ref{eqa13})-(\ref{eqa17}) can be written as \begin{eqnarray} \label{eqa18} \frac{\partial u_1}{\partial t}=-(u \cdot\nabla ) u_1-\frac{u_1 u_2}{r_0}+\nu (\Delta u_1+\frac{2 }{r_0}\frac{\partial u_2}{\partial x_1}-\frac{2u_1}{r_0^2})-\sigma_0 u_1-2 \Omega u_2-\frac{1}{\rho_0} \frac{\partial p}{\partial x_1}, \end{eqnarray} $$ \frac{\partial u_2}{\partial t}=-(u \cdot \nabla) u_2+\frac{u_1^2}{r_0}+\nu (\Delta u_2+\frac{2 }{r_0}\frac{\partial u_1}{\partial x_1}-\frac{2u_2}{r_0^2})-\sigma_1 u_2-2 \Omega u_1 $$ \begin{eqnarray} \label{eqa19} -\frac{1}{\rho_0} \frac{\partial p}{\partial x_2}-[1-\alpha_T(T-T_0)+\alpha_q(q-q_0)] g, \end{eqnarray} \begin{eqnarray} \label{eqa20} \frac{\partial T}{\partial t}=-(u\cdot\nabla) T +\kappa_T \Delta T+Q, \end{eqnarray} \begin{eqnarray} \label{eqa21} \frac{\partial q}{\partial t}=-(u \cdot \nabla)q+\kappa_q \Delta q+G, \end{eqnarray} \begin{eqnarray} \label{eqa22}\frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}=0. \end{eqnarray} \par To make the nondimensional form, let $$ x=hx^{'}, \quad t=\frac{h^2}{\kappa_T} t^{'} \quad u=\frac{\kappa_T}{h} u^{'}, $$ $$ T=T_0-(T_0-T_1)\frac{x_2}{h}+(T_0-T_1)T^{'}, $$ $$ q=q_0-(q_0-q_1)\frac{x_2}{h}+(q_0-q_1)q^{'}, $$ $$ p=\frac{\rho_0 \nu \kappa_T p^{'}}{h^2}-g\rho_0[x_2+\frac{\alpha_T}{2}(T_0-T_1)\frac{x^2_2}{h}-\frac{\alpha_q}{2}(q_0-q_1)\frac{x^2_2}{h}], $$ \par The nondimensional form of (\ref{eqa18})-(\ref{eqa22}) reads $$ \frac{\partial u^{'}_1}{\partial t^{'}}=-(u^{'} \cdot \nabla ) u^{'}_1+\frac{\nu}{\kappa_T} \Delta u^{'}_1-\frac{\sigma_0 h^2}{\kappa_T} u^{'}_1-\frac{2 \Omega h^2}{\kappa_T} u^{'}_2-\frac{\nu}{k_T} \frac{\partial p^{'}}{\partial x^{'}_1} $$ \begin{eqnarray} \label{eqa23} -\frac{h}{r_0}u_1u_2+\frac{2h}{r_0 k_T}\frac{\partial u^{'}_2}{\partial x_1^{'}}-\frac{2h^2}{r_0 \kappa_T} u_1^{'}, \end{eqnarray} $$ \frac{\partial u^{'}_2}{\partial t^{'}}=-(u^{'} \cdot \nabla) u^{'}_2+\frac{\nu}{\kappa_T} \Delta u^{'}_2-\frac{\sigma_1 h^2}{\kappa_T} u^{'}_2-\frac{2 \Omega h^2}{\kappa_T} u^{'}_1-\frac{h}{r_0}u_1^2+\frac{h}{r_0 k_T}\frac{\partial u^{'}_1}{\partial x_2^{'}}-\frac{2h^2}{r_0^2 \kappa_T} u_2^{'} $$ \begin{eqnarray} \label{eqa24} -\frac{\nu}{\kappa_T} \frac{\partial p^{'}}{\partial x^{'}_2}+\frac{gh^3}{\kappa_T^2}[\alpha_T(T^0-T_1)T^{'}-\alpha_q(q^0-q_1)q^{'}], \end{eqnarray} \begin{eqnarray} \label{eqa25} \frac{\partial T^{'}}{\partial t^{'}}=-(u^{'}\cdot\nabla) T^{'} + \Delta T^{'}+u_2^{'}+\frac{h^2}{(T_0-T_1)\kappa_T}Q, \end{eqnarray} \begin{eqnarray} \label{eqa26} \frac{\partial q^{'}}{\partial t^{'}}=-(u^{'} \cdot \nabla)q^{'}+\frac{\kappa_q}{\kappa_T} \Delta q^{'}+u_2^{'}+\frac{h^2}{(T_0-T_1)\kappa_T}G, \end{eqnarray} \begin{eqnarray} \label{eqa27}div u^{'}=0. \end{eqnarray} \par Because $r_0$ is far larger than $u_1$, $u_2$, the atmospheric motion equations (\ref{eqa23})-(\ref{eqa27}) can be read \begin{eqnarray} \label{eqa123} \frac{\partial u^{'}_1}{\partial t^{'}}=-(u^{'} \cdot \nabla ) u^{'}_1+\frac{\nu}{\kappa_T} \Delta u^{'}_1-\frac{\sigma_0 h^2}{\kappa_T} u^{'}_1-\frac{2 \Omega h^2}{\kappa_T} u^{'}_2-\frac{\nu}{k_T} \frac{\partial p^{'}}{\partial x^{'}_1}, \end{eqnarray} $$ \frac{\partial u^{'}_2}{\partial t^{'}}=-(u^{'} \cdot \nabla) u^{'}_2+\frac{\nu}{\kappa_T} \Delta u^{'}_2-\frac{\sigma_1 h^2}{\kappa_T} u^{'}_2-\frac{2 \Omega h^2}{\kappa_T} u^{'}_1 $$ \begin{eqnarray} \label{eqa124} -\frac{\nu}{\kappa_T} \frac{\partial p^{'}}{\partial x^{'}_2}+\frac{gh^3}{\kappa_T^2}[\alpha_T(T_0-T_1)T^{'}-\alpha_q(q_0-q_1)q^{'}], \end{eqnarray} \begin{eqnarray} \label{eqa125} \frac{\partial T^{'}}{\partial t^{'}}=-(u^{'}\cdot\nabla) T^{'} + \Delta T^{'}+u_2^{'}+\frac{h^2}{(T_0-T_1)\kappa_T}Q, \end{eqnarray} \begin{eqnarray} \label{eqa126} \frac{\partial q^{'}}{\partial t^{'}}=-(u^{'} \cdot \nabla)q^{'}+\frac{\kappa_q}{\kappa_T} \Delta q^{'}+u_2^{'}+\frac{h^2}{(T_0-T_1)\kappa_T}G, \end{eqnarray} \begin{eqnarray} \label{eqa127}div u^{'}=0. \end{eqnarray} \par Let $P_r=\frac{\nu}{\kappa_T}$, $L_e=\frac{\kappa_q}{\kappa_T}$, $R=\frac{g\alpha_T(T^0-T_1)h^3}{\kappa_T \nu}$, $\tilde{R}=\frac{g\alpha_q(q^0-q_1)h^3}{\kappa_T \nu}$, $\sigma_i^{'}=\frac{\sigma_i h^2}{\nu}$, $\omega=\frac{2 \Omega h^2}{\nu}$, $Q^{'}=\frac{h^2}{(T_0-T_1)\kappa_T}Q$, $G^{'}=\frac{h^2}{(T_0-T_1)\kappa_T}G$, omitting the primes, the nondimensional form of (\ref{eqa123})-(\ref{eqa127}) reads \begin{eqnarray} \label{eqa28} \frac{\partial u}{\partial t}=P_r(\Delta u-\nabla p-\sigma u)+P_r(RT-\tilde{R}q)\vec{\kappa}-(\nabla \cdot u) u, \end{eqnarray} \begin{eqnarray} \label{eqa29} \frac{\partial T}{\partial t}=\Delta T+u_2-(u\cdot\nabla) T + Q, \end{eqnarray} \begin{eqnarray} \label{eqa30} \frac{\partial q}{\partial t}=L_e \Delta q+u_2-(u \cdot \nabla)q+G, \end{eqnarray} \begin{eqnarray} \label{eqa31}div u=0, \end{eqnarray} where $\sigma$ is constant matrix $$ \sigma=\left( \begin{array}{ll} \sigma_0 & \omega\\ \omega & \sigma_1 \end{array} \right). $$ \par The problem (\ref{eqa28})-(\ref{eqa31}) are supplemented with the following Dirichlet boundary condition at $x_2=0,1$ and periodic condition for $x_1$: \begin{eqnarray} \label{eqa32} (u, T,q ) =0, \quad x_2=0,1, \end{eqnarray} \begin{eqnarray} \label{eqa33} (u, T, q )(0, x_2) = (u, T, q )(2 \pi, x_2), \end{eqnarray} and initial value conditions \begin{eqnarray} \label{eqa34} (u, T, q ) = (u_0, T_0, q_0), \ \ \ t=0. \end{eqnarray} \section{Global Solution of Atmospheric Circulation Equations with \\Humidity Effect} \setcounter{equation}{0} \subsection{Preliminaries} Let $X$ be a linear space, $X_1$, $X_2$ two separable reflexive Banach spaces, $H$ a Hilbert space. $X_1$, $X_2$ and $H$ are completion space of $ X $ under the respective norms. $X_1$, $X_2\subset H$ are dense embedding. $F:X_2 \times (0,\infty)\rightarrow X_1^*$ is a continuous mapping. We consider the abstract equation \begin{eqnarray} \label{eqc101} \left\{ \begin{array}{ll} \frac{du}{dt}=Fu, & 0<t<\infty, \\ \\ u(0)=\varphi, & \\ \end{array} \right. \end{eqnarray} where $\varphi \in H$, $u: [0,+\infty)\rightarrow H$ is unknown. \par {\bf Definition 3.1} Let $\varphi \in H$ be a given initial value. $u \in L^p((0,T),X_2)\cap L^\infty((0,T), H)$, ($0<T<\infty$) is called a global solution of Eq(\ref{eqc101}), if $u$ satisfies $$ (u(t),v)_H=\int_0^t<Fu,v>dt+(\varphi,v)_H, \quad \forall v \in X_1 \subset H. $$ \par {\bf Definition 3.2} Let $u_n, u_0 \in L^p((0,T), X_2)$. $u_n \rightarrow u_0$ is called uniformly weak convergence in $L^p((0,T), X_2)$, if $\{u_n\} \subset L^\infty((0,T),H)$ is bounded, and \begin{eqnarray} \label{eqc102} \left\{ \begin{array}{ll} u_n \rightharpoonup u_0, &\hbox{in} \ \ \ L^p((0,T), X_2), \\ \\ \lim_{n\rightarrow \infty}\int_0^T |<u_n-u_0,v>_H|^2 dt=0, & \forall v \in H.\\ \end{array} \right. \end{eqnarray} \par {\bf Definition 3.3} A mapping $F: X_2 \times (0,\infty)\rightarrow X_1^*$ is called $T$-weakly continuous, if for $p=(p_1, \ldots, p_m)$, $0<T<\infty$, and $u_n$ uniformly weakly converge to $u_0$ under Eq(\ref{eqc102}), we have $$ \lim_{n\rightarrow \infty}\int_0^T <Fu_n,v>dt=\int_0^T <Fu_0,v>dt, \quad \forall v\in X_1. $$ \par {\bf Lemma 3.4} Let $\{u_n\}\subset L^p((0,T), W^{m,p})(m\geq 1)$ be bounded sequence, and $\{u_n\}$ uniformly weakly converge to $u_0 \subset L^p((0,T), W^{m,p})$, i.e. \begin{eqnarray} \label{eqc100} \left\{ \begin{array}{ll} u_n \rightharpoonup u_0 \ \ \ \hbox{in}\ \ L^p((0,T), W^{m,p}),& p\geq 2, \\ \\ \lim_{n\rightarrow \infty}\int_0^T [\int_\Omega (u_n-u_0)v dx ]^2dt=0, & \forall v \in C_0^\infty(\Omega).\\ \end{array} \right. \end{eqnarray} Then for all $|\alpha|\leq m-1$, we have $$ D^\alpha u_n \rightarrow D^\alpha u_0\ \ \ \ \hbox{in}\ \ \ \ \ L^2((0,T)\times \Omega). $$ \par {\bf Lemma 3.5} $^{\cite{Ma3}}$ Assume $F: X_2 \times (0,\infty)\rightarrow X_1^*$ is $T$-weakly continuous, and satisfies \par (A1) there exists $p=(p_1, \cdots, p_m)$, $p_i>1(1\leq i \leq m)$, such that $$ <Fu,u> \leq -C_1\|u\|^p_{X_2}+C_2 \|u\|_H^2+f(t),\quad \forall u\in X, $$ where $C_1,C_2$ are constants, $f\in L^1(0,T)(0<T<\infty)$, $\|\cdot\|^p_{X_2}=\sum_{i=1}^m|\cdot|_i^{p_i}$, $|\cdot|_i$ is seminorm of $X_2$, $\|\cdot\|_{X_2}=\sum_{i=1}^m |\cdot|_i$, \par (A2) there exists $0<\alpha<1$, for all $0<h<1$ and $u \in C^1([0,\infty),X)$, $$ |\int_t^{t+h}<Fu,v>dt|\leq Ch^\alpha, \quad \forall v\in X \quad \hbox{and} \quad 0\leq t<T, $$ where $C>0$ depends only on $T$, $\|v\|_{X_1}$, $\int_0^t \|u\|^p_{X_2}dt$ and $\sup_{0\leq t \leq T} \|u\|_H$.\\ Then for all $\varphi \in H$, Eq(\ref{eqc101}) has a global weak solution $$ u\in L^\infty((0,T),H) \cap L^p((0,T), X_2), \quad 0<T<\infty, \quad p\hbox{\ \ in \ \ (A1)}. $$ \par {\bf Remark 3.6} $\|\cdot\|_X$ denotes norm of $X$, and $C_i$ are variable constants. \subsection{Existence of Global Solution} We introduce spatial sequences $$ X=\{\phi=(u,T,q) \in C^\infty(\Omega, R^4)| (u,T,q) \ \ \hbox{satisfy} \ \ (\ref{eqa31})-(\ref{eqa33})\}, $$ $$ H=\{\phi=(u,T,q) \in L^2(\Omega, R^4)| (u,T,q)\ \ \hbox{satisfy}\ \ (\ref{eqa31})-(\ref{eqa33})\}, $$ $$ H_1=\{\phi=(u,T,q) \in H^1(\Omega, R^4)| (u,T,q) \ \ \hbox{satisfy}\ \ (\ref{eqa31})-(\ref{eqa33})\}, $$ {\bf Theorem 3.7} If $\phi_0=(u_0,T_0,q_0)\in H$, $Q,G \in L^2(\Omega)$, then Eq(\ref{eqa28})-(\ref{eqa34}) there exists a global solution $$ (u,T,q) \in L^\infty((0,T),H)\cap L^2((0,T),H_1), \quad 0<T<\infty. $$ \par {\bf Proof.} Definite $F: H_1 \rightarrow H_1^*$ as $$ \begin{array}{lrl} <F\phi, \psi>&=&\int_\Omega [-P_r \nabla u \nabla v-P_r \sigma u \cdot v+P_r (RT-\tilde{R}q)v_2-( u\cdot \nabla) u \cdot v \\\\ &&-\nabla T \nabla S+u_2 S-(u\cdot \nabla)T S+Q S-L_e \nabla q \nabla z+u_2 z \\\\ &&-(u\cdot \nabla)q z+G z]dx,\ \ \ \ \hbox{¶Ô}\quad \forall \psi=(v,S,z) \in H_1. \end{array} $$ \par Let $\psi=\phi$. Then $$ \begin{array}{lrl} <F\phi, \phi>&=&\int_\Omega [-P_r |\nabla u|^2-P_r \sigma u \cdot u+P_r(RT-\tilde{R}q)u_2-(u\cdot\nabla ) u \cdot u \\\\ &&-|\nabla T|^2+u_2 T-(u\cdot \nabla)T T +Q T-L_e |\nabla q|^2 \\\\ &&+u_2 q-(u\cdot \nabla)q q+G q]dx \\\\ &=&\int_\Omega [-P_r |\nabla u|^2-P_r \sigma u \cdot u+(P_r R+1)u_2 T-(P_r \tilde{R}-1)qu_2 \\\\ &&-|\nabla T|^2+ Q T-L_e |\nabla q|^2+G q]dx \\\\ &\leq&- C_1 \int_\Omega [|\nabla u|^2+|\nabla T|^2+ |\nabla q|^2]dx +C_2 \int_\Omega[|u|^2+|u||T|+|q||u| \\\\ &&+|Q| |T|+|G| |q|]dx \\\\ &\leq& - C_1 \int_\Omega [|\nabla u|^2+|\nabla T|^2+ |\nabla q|^2]dx +C_2 \int_\Omega[|u|^2+|T|^2+|q|^2]dx \\\\ &&+C_3 \int_\Omega[|Q|^2+|G|^2]dx \\\\ &\leq& - C_1 \|\phi\|_{H_1}^2 +C_2 \|\phi\|_{H}^2+C_4, \end{array} $$ which implies $(A_1)$. \par For $\forall \phi \in L^2(0,T), H_1)\cap L^\infty((0,T),H)$ and $\psi \in X$, $h(0<h<1)$, we have $$ \begin{array}{lrl} &&|\int_t^{t+h}<F\phi, \psi>dt| \\\\ &=&|\int_t^{t+h} \int_\Omega [-P_r \nabla u \nabla v-P_r \sigma u \cdot v+P_r (RT-\tilde{R}q)v_2-( u\cdot \nabla) u \cdot v-\nabla T \nabla S \\\\ &&+u_2 S-(u\cdot \nabla)T S+ Q S-L_e \nabla q \nabla z+u_2 z-(u\cdot \nabla)q z+G z]dxdt| \end{array} $$ $$ \begin{array}{lrl} &\leq& C\int_t^{t+h} \int_\Omega [|\nabla u| |\nabla v|+|u||v|+ |T||v|+|q||v|+|\nabla T| |\nabla S|+|u| |S| \\\\ &&+|Q| |S| +|\nabla q| |\nabla z|+|u| |z|+|G| |z|]dxdt+C\int_t^{t+h} [|\int_\Omega ( u\cdot \nabla) u \cdot vdx| \\\\ &&+|\int_\Omega (u\cdot \nabla)T Sdx|+|\int_\Omega (u\cdot \nabla)q zdx|]dt \\\\ &\leq& C\int_t^{t+h}[ (\int_\Omega |\nabla u|^2dx)^{\frac{1}{2}} (\int_\Omega|\nabla v|^2dx)^{\frac{1}{2}}+ (\int_\Omega|T|^2dx)^{\frac{1}{2}}(\int_\Omega|v|^2dx)^{\frac{1}{2}} \\\\ &&+(\int_\Omega|q|^2dx)^{\frac{1}{2}} (\int_\Omega|v|^2dx)^{\frac{1}{2}}+(\int_\Omega|\nabla T|^2dx)^{\frac{1}{2}} (\int_\Omega|\nabla S|^2dx)^{\frac{1}{2}} \\\\ &&+(\int_\Omega|u|^2dx)^{\frac{1}{2}} (\int_\Omega|S|^2dx)^{\frac{1}{2}}+(\int_\Omega|Q|^2dx)^{\frac{1}{2}} (\int_\Omega|S|^2dx)^{\frac{1}{2}} \\\\ &&+(\int_\Omega|\nabla q|^2dx)^{\frac{1}{2}} (\int_\Omega|\nabla z|^2dx)^{\frac{1}{2}}+(\int_\Omega|u|^2dx)^{\frac{1}{2}} (\int_\Omega|z|^2dx)^{\frac{1}{2}} \\\\ &&+(\int_\Omega|G|^2dx)^{\frac{1}{2}} (\int_\Omega|z|^2dx)^{\frac{1}{2}}]dt+C\int_t^{t+h} [|\sum_{i,j=1}^2\int_\Omega ( u_i u_j \frac{\partial v_j}{\partial x_i}dx| \\\\ &&+|\sum_{i=1}^2\int_\Omega u_iT \frac{\partial S}{\partial x_i}dx|+|\sum_{i=1}^2\int_\Omega u_iq\frac{\partial z}{\partial x_i}dx|]dt \\\\ &\leq& C(\|u\|_{L^2(0,T;H^1)} \|D v\|_{L^2}h^{\frac{1}{2}}+ \|T\|_{L^2(0,T;L^2)}\|v\|_{L^2}h^{\frac{1}{2}}+\|q\|_{L^2(0,T;L^2)} \|v\|_{L^2}h^{\frac{1}{2}} \\\\ &&+\|T\|_{L^2(0,T;H^1)} \|D S\|_{L^2}h^{\frac{1}{2}}+\|u\|_{L^\infty(0,T;L^2)}\|S\|_{L^2}h^{\frac{1}{2}} +\|Q\|_{L^2}\|S\|_{L^2}h \\\\ &&+\| q\|_{L^2(0,T;H^1)} \|D z\|_{L^2}h^{\frac{1}{2}}+\|u\|_{L^\infty(0,T;H)} \|z\|_{L^2}h+\|G\|_{L^2}\|z\|_{L^2}h \\\\ &&+\|v\|_{C^1}\|u\|_{L^\infty(0,T,L^2)}h+\|S\|_{C^1} \|u\|_{L^\infty(0,T;L^2)}^{\frac{1}{2}}\|T\|_{L^\infty(0,T;L^2)}^{\frac{1}{2}}h \\\\ &&+\|z\|_{C^1} \|u\|_{L^\infty(0,T;L^2)}^{\frac{1}{2}}\|q\|_{L^\infty(0,T;L^2)}^{\frac{1}{2}}h) \\\\ &\leq& C h^{\frac{1}{2}}, \end{array} $$ which implies $(A_2)$. \par We will prove that $F: H_1 \rightarrow H_1^*$ is $T$-weakly continuous. Let $\phi_n=(u^n,T^n,q^n) \rightharpoonup \phi_0 =(u^0,T^0,q^0)$ be uniformly convergence, i.e., $\{\phi_n\} \subset L^\infty((0,T);H)$ is bounded, and $$ \left\{ \begin{array}{lcl} \phi_n \rightharpoonup \phi_0 & \hbox{in} & L^p((0,T);H_1), \\ \lim_{n\rightarrow \infty} \int_0^T|<\phi_n-\phi_0,\psi>_H|^2dt=0,&&\forall \psi \in H. \end{array} \right. $$ \par From Lemma 3.4, we known that $\phi_n \rightarrow \phi_0$ in $L^2(\Omega \times (0,T))$. \par Then $\forall \psi \in X \subset C^\infty(\Omega, R^4)\cap H_1$, we have $$ \lim_{n \rightarrow \infty}\int_0^t\int_\Omega(u^n \cdot \nabla)u^n \cdot v dxdt=\lim_{n \rightarrow \infty}\int_0^t\int_\Omega\sum_{i,j=1}^2 u^n_i \frac{\partial u^n_j}{\partial x_i} v_jdxdt $$ $$ =-\lim_{n \rightarrow \infty}\int_0^t\int_\Omega\sum_{i,j=1}^2 u^n_i u^n_j\frac{\partial v_j}{\partial x_i} dxdt $$ $$ =-\int_0^t\int_\Omega(u^0 \cdot \nabla)v \cdot u^0dxdt $$ $$ =\int_0^t\int_\Omega(u^0 \cdot \nabla)u^0 \cdot v dxdt, $$ and $$ \lim_{n \rightarrow \infty}\int_0^t\int_\Omega(u^n \cdot \nabla)T^n S dxdt=\lim_{n \rightarrow \infty}\int_0^t\int_\Omega\sum_{i=1}^2 u^n_i \frac{\partial T^n}{\partial x_i} Sdxdt $$ $$ =-\lim_{n \rightarrow \infty}\int_0^t\int_\Omega\sum_{i=1}^2 u^n_i T^n\frac{\partial S}{\partial x_i} dxdt $$ $$ =-\int_0^t\int_\Omega(u^0 \cdot \nabla)S T^0dxdt $$ $$ =\int_0^t\int_\Omega(u^0 \cdot \nabla)T^0 S dxdt, $$ and $$ \lim_{n \rightarrow \infty}\int_0^t\int_\Omega(u^n \cdot \nabla)q^n z dxdt=\lim_{n \rightarrow \infty}\int_0^t\int_\Omega\sum_{i=1}^2 u^n_i \frac{\partial q^n}{\partial x_i} zdxdt $$ $$ =-\lim_{n \rightarrow \infty}\int_0^t\int_\Omega\sum_{i=1}^2 u^n_i q^n\frac{\partial z}{\partial x_i} dxdt $$ $$ =-\int_0^t\int_\Omega(u^0 \cdot \nabla)z q^0dxdt $$ $$ =\int_0^t\int_\Omega(u^0 \cdot \nabla)q^0 z dxdt, $$ Thus, \begin{eqnarray} \label{eqc9} \lim_{n \rightarrow \infty}\int_0^t <F\phi_n, \psi>dt=\int_0^t <F\phi_0, \psi>dt. \end{eqnarray} \par Because $X$ is dense in $H_1$, Eq(\ref{eqc9}) holds for $\psi \in H_1$. In other words, the mapping $F: H_1 \rightarrow H_1^*$ is $T$-weakly continuous. \par From Lemma 3.5, Eq(\ref{eqa28})-(\ref{eqa34}) has a global weak solution $$ (u,T,q) \in L^\infty((0,T),H)\cap L^2((0,T),H_1), \quad 0<T<\infty. $$ $\Box$ \par {\bf Remark 3.8} Existence of global solutions to the atmospheric circulation models implies that atmospheric circulation has its own running way as humidity source and heat source change, and confirms that the atmospheric circulation models are reasonable. \begin {thebibliography}{90} \bibitem{Charney1} Charney J., The dynamics of long waves in a baroclinic westerly current, {\it J. Meteorol.}, {\bf 4}(1947), 135-163. \bibitem{Charney2} Charney J., On the scale of atmospheric motion, {\it Geofys. 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Adv.}, {\bf 1}(1)(1993), 1-54. \bibitem{Ma2} Ma T., Wang S.H., Phase Transition Dynamics in Nonlinear Sciences, New York, Springer, 2013. \bibitem{Ma3} Ma T., Theories and Methods in partial differential equations, Academic Press, China, 2011(in Chinese). \bibitem{Phillips} Phillips, N.A., The general circulation of the atmosphere: A numerical experiment. {\it Quart J Roy Meteorol Soc}, {\bf 82}(1956), 123-164. \bibitem{Richardson} Richardson L.F., {\it Weather Prediction by Numerical Process}, Cambridge University Press, 1922. \bibitem{Rossby} Rossby, C.G., On the solution of problems of atmospheric motion by means of model experiment, { \it Mon. Wea. Rev.}, {\bf 54}(1926), 237-240. \bibitem{von Neumann} von Neumann, J., Some remarks on the problem of forecasting climatic fluctuations. In R. L. Pfeffer (Ed.), Dynamics of climate, pp. 9-12. 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\begin{document} \begin{frontmatter} \title{Quantum interference within the complex quantum Hamilton-Jacobi formalism} \author[UT]{Chia-Chun~Chou} \ead{chiachun@mail.utexas.edu} \author[ESP]{\'Angel~S.~Sanz} \ead{asanz@imaff.cfmac.csic.es} \author[ESP]{Salvador~Miret-Art\'es} \ead{s.miret@imaff.cfmac.csic.es} \author[UT]{Robert~E.~Wyatt} \ead{wyattre@mail.utexas.edu} \address[UT]{Institute for Theoretical Chemistry and Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, Texas 78712, USA} \address[ESP]{Instituto de F\'{\i}sica Fundamental, Consejo Superior de Investigaciones Cient\'{\i}ficas, Serrano 123, 28006 Madrid, Spain} \begin{abstract} Quantum interference is investigated within the complex quantum Hamilton-Jacobi formalism. As shown in a previous work [Phys.\ Rev.\ Lett.\ {\bf 102}, 250401 (2009)], complex quantum trajectories display helical wrapping around stagnation tubes and hyperbolic deflection near vortical tubes, these structures being prominent features of quantum caves in space-time Argand plots. Here, we further analyze the divergence and vorticity of the quantum momentum function along streamlines near poles, showing the intricacy of the complex dynamics. Nevertheless, despite this behavior, we show that the appearance of the well-known interference features (on the real axis) can be easily understood in terms of the rotation of the nodal line in the complex plane. This offers a unified description of interference as well as an elegant and practical method to compute the lifetime for interference features, defined in terms of the average wrapping time, i.e., considering such features as a resonant process. \end{abstract} \begin{keyword} Quantum interference \sep Complex quantum trajectory \sep Quantum momentum function \sep P\'{o}lya vector field \sep Quantum cave \PACS 03.65.Nk \sep 03.65.Ta \sep 03.65.Ca \end{keyword} \end{frontmatter} \section{\label{sec:introduction} Introduction} Quantum interference is an observable effect arising from the coherent superposition of quantum probability amplitudes. It is involved in a very wide range of important applications, such as superconducting quantum interference devices or SQUIDs \cite{scalapino, friedman}, coherent control of chemical reactions \cite{paul}, atomic and molecular interferometry \cite{berman,itano,urena1,urena2}, and Talbot/Talbot-Lau interferometry with relatively heavy particles (e.g., Na atoms \cite{chapman2} and Bose-Einstein condensates \cite{deng}). Indeed, possibly one of the main practical applications of interference nowadays is in Bose-Einstein condensate (BEC) interferometry \cite{yoo, dalibard,pethick} due to its potential use in applications, such as sensing, metrology or quantum information processing. Thus, since the first experimental evidence of interference between two freely expanding BECs was observed \cite{ketterle}, an increasing amount of work, both theoretical and experimental, can be found in the literature \cite{pritchard,zhang:070403,cederbaum:110405,chip0,chip1, chip2,chip3}. Basically, in this type of interferometry there are three steps. First, the atomic cloud is cooled in a magnetic trap until condensation takes place; second, the BEC is split coherently by means of radio-frequency \cite{chip1} or microwave fields \cite{chip2}; and third, the double-well-like trapping potential is switched off and the two parts of the BEC are allowed to interfere by free expansion (see, for example, Fig.~6 in Ref.~\cite{chip3} as an illustration of a real experimental outcome). Apart from the practical applications mentioned before, interference plays also an important role when dealing with multipartite entangled systems as an indicator of the loss of coherence induced by the interaction between the different subsystems, commonly referred as ``Schr\"odinger cat states'' \cite{schro}. However, very little attention beyond the implications of the superposition principle has been devoted to understanding quantum interference at a more fundamental level. Bohmian mechanics provides an alternative interpretation to quantum mechanics \cite{bohm1,bohm2,HollandBook}. As an \emph{analytical} approach, this formulation has been used to analyze atom-surface scattering \cite{ASSanz,ASSanz2,ASSanz4}, the quantum Talbot effect \cite{ASSanz6}, quantum nonlocality \cite{ASSanz7} or quantum interference \cite{ASSanz9}. As a \emph{synthetic} approach, the quantum trajectory method (QTM) \cite{LopreoreWyatt1} has been developed as a computational implementation to the hydrodynamic formulation of quantum mechanics to generate the wave function by evolving ensembles of quantum trajectories \cite{WyattBook}. However, computational difficulties resulting from interference effects are encountered in regions where the wave packet is reflected from the barrier and nodes or quasi-nodes occur. Thus, a bipolar counter-propagating decomposition approach for the total wave function has been developed to overcome numerical instabilities due to interferences \cite{Poirier1,Poirier2,Poirier3,Poirier5,Poirier6,Poirier7,Poirier8}. Here, a more detailed analysis of quantum interference within the complex quantum Hamilton-Jacobi formalism \cite{LP1,LP2} is presented. The complex quantum trajectory method has been applied to stationary problems \cite{MJohn1,MJohn2,CDYang1,CDYang2,CDYang3,Chou3a,Chou3b} and to one-dimensional and multi-dimensional wave packet scattering problems \cite{Tannor2a,Tannor2b,Tannor2c,Tannor3a,Tannor3b,Rowland1a, Rowland1b,Rowland1c,Rowland2a,Rowland2b,Rowland2c}. The purpose of this work is to investigate how information can be extracted from this complex formulation regarding interference. Recently, quantum interference associated with superpositions of Gaussian wave packets was thoroughly analyzed using both the standard quantum-mechanical and Bohmian approaches in real space \cite{ASSanz8, ASSanz9}. In spite of its simplicity, such superpositions are experimentally realizable in atom interferometry \cite{haensel-a,haensel-b,haensel-c,haensel-d,haensel-e}, for example. In contrast to conventional quantum mechanics, Bohmian mechanics offers a trajectory-based understanding of quantum interference. The interference of two wave packets in one real coordinate leads to the formation of nodal structure, and the quantum potential near these nodes forces these trajectories to avoid these regions and to exhibit laminar flow in space-time plots \cite{ASSanz8}. In contrast, within the complex quantum Hamilton-Jacobi formalism, the collinear collision of two Gaussian wave packets demonstrated caustics and vortical dynamics in the complex plane \cite{ASSanz8,Chou9}. This complicated trajectory dynamics cannot be found in Bohmian mechanics unless two or more real coordinates are introduced. When systematically analyzed for local structures of the quantum momentum function (QMF) and the P\'{o}lya vector field (PVF) around its characteristic points \cite{Chou6,Chou7}, complex quantum trajectories display \emph{helical wrapping} around stagnation tubes and \emph{hyperbolic deflection} around vortical tubes \cite{Chou9}. This intriguing topological structure is formed by these tubes and gives rise to \emph{quantum caves} in space-time Argand plots. Quantum interference thus leads to the formation of quantum caves and the appearance of the topological structure mentioned before. In contrast with quantum trajectories, P\'{o}lya trajectories display \emph{hyperbolic deflection} around stagnation tubes and \emph{helical wrapping} around vortical tubes. The QMF divergence and vorticity characterize the turbulent flow of trajectories, determining the so-called {\it wrapping time} \cite{Chou9} for an individual trajectory. Moreover, it is shown that both the PVF divergence and vorticity vanish except at poles, thus the PVF describing an incompressible and irrotational flow. Trajectories launched from different positions wrap around the same stagnation tubes. Hence, the circulation of trajectories can be viewed as a resonance process, from which one can obtain a natural way to define a ``lifetime'' for the interference. This information could be therefore used with practical purposes to analyze, explain and understand experiments where interference is the main physical process or mechanism, as in those described above in this Section. The rotational dynamics of the nodal line arising from the interference of wave packets in the complex plane thus offers an elegant method to compute the lifetime for the interference features observed on the real axis. We consider two cases of head-on collision of two Gaussian wave packets, which depend on the relative magnitude between the propagation velocity and the spreading rate of the wave packets \cite{ASSanz9}. In the case where the relative propagation velocity is larger than the spreading rate of the wave packets, an average wrapping time is calculated to provide a lifetime for the interference features, while the rotational dynamics of the nodal line in the complex plane explains the transient appearance of the interference features observed on the real axis. In the case where the relative propagation velocity is approximately equal to or smaller than the spreading rate of the wave packets, the rapid spreading of the wave packets leads to a distortion of quantum caves. The infinite survival of interference features in this case implies that the wrapping time becomes infinity. However, the rotational dynamics of the nodal line clearly explains the persistent interference features observed on the real axis. In both cases, the interference features are observed on the real axis when the nodal line is near the real axis; therefore, in contrast to conventional quantum mechanics, the rotational dynamics of the nodal line in the complex plane provides a fundamental interpretation of quantum interference. The organization of this work is as follows. To be self-contained, in Sec.~\ref{sec:theory} we briefly describe the complex quantum Hamilton-Jacobi formalism as well as the QMF local structures and its associated PVF near characteristic points. In Sec.~\ref{sec:QI}, first we present theoretical analysis of the Gaussian wave-packet head-on collision on the real axis and in the complex plane, and then demonstrate the interpretation of quantum interference with two cases in the framework of the complex quantum trajectory method. Finally, we present a summary and discussion in Sec.~\ref{sec:Conclusions}. \section{\label{sec:theory} Theoretical formulation} \subsection{\label{sec:CQHJF}Complex quantum Hamilton-Jacobi formalism} Substituting the complex-valued wave function expressed by $\Psi(x,t)=\exp\left[iS(x,t)/\hbar\right]$ into the time-dependent Schr\"{o}dinger equation, we obtain the quantum Hamilton-Jacobi equation (QHJE) in the complex quantum Hamilton-Jacobi formalism, \begin{equation} -\frac{\partial S}{\partial t}=\frac{1}{2m}\left(\frac{\partial S}{\partial x}\right)^2+V(x)+\frac{\hbar}{2mi}\frac{\partial^2 S}{\partial x^2}, \label{TDQHJE} \end{equation} where $S(x,t)$ is the complex action and the last term is the complex quantum potential. In real space, the QMF is defined by $ p(x,t)=\partial S(x,t)/ \partial x$, which, within the complex quantum Hamilton-Jacobi formalism, is analytically continued to the complex plane by extending the real variable $x$ to a complex variable $z=x+iy$. The same is done with other relevant functions, such as the wave function and the potential energy. Quantum trajectories in complex space are then determined by solving the guidance equation \begin{equation} \frac{dz}{dt}=\frac{p(z,t)}{m} , \label{TDtraj1} \end{equation} where time remains real-valued and the (complex) QMF is expressed in terms of the (complex) wave function (through the complex action) as \begin{equation} p(z,t)=\frac{\hbar}{i}\frac{1}{\Psi(z,t)} \frac{\partial\Psi(z,t)}{\partial z}. \label{nonstationaryQMF} \end{equation} From this equation, nodes in the wave function correspond to QMF {\it poles}. Moreover, those points where the first derivative of the wave function vanishes will correspond to QMF {\it stagnation points}. \subsection{\label{sec:LSQMF} Local structures of the quantum momentum function and its P\'{o}lya vector field} The dynamics of complex quantum trajectories is guided by the wave function through the QMF. The complex-valued QMF can be regarded as a vector field in the complex plane. The trajectory dynamics is significantly influenced by both the QMF stagnation points and poles. Streamlines near a QMF stagnation point may spiral into or out of it, or they may become circles or straight lines. The QMF near a pole displays East-West and North-South hyperbolic structure \cite{Chou6,Chou7}. These local structures around these characteristic points have also been observed for the one-dimensional stationary scattering problems including the Eckart and the hyperbolic tangent barriers \cite{Chou3a,Chou3b}. The PVF of a complex vector field $f(z)$ is defined by its complex conjugate $f^*(z)$ \cite{Polya,Braden,Needham}. Thus, the PVF associated with the QMF, $p(z,t)=p_x+ip_y$, is given by the vector field ${\bf P}(z,t)=p_x-ip_y = (p_x,-p_y)$. This new vector field provides a simple geometrical and physical interpretation for complex circulation integrals \begin{equation} \oint_C p(z) dz = \oint_C {\bf P} \cdot d \ell + i \oint_C {\bf P} \cdot d {\bf n} , \label{CIQMF} \end{equation} where $C$ denotes a simple closed curve in the complex plane, $d \ell = (dx,dy)$ is tangent to $C$ and $d {\bf n} = (dy,-dx)$ is normal to $C$ and pointing outwards. The real part of the integral in Eq.~(\ref{CIQMF}) gives the total amount of work done in moving a particle along a closed contour $C$ subjected to ${\bf P}$, while its imaginary part gives the total flux of the vector field across the closed contour \cite{Needham}. In Bohmian mechanics, quantum vortices form around nodes in two or more real coordinates \cite{Dirac,McCullough1,McCullough2,McCullough3, Hirschfelder1,Hirschfelder2,Hirschfelder3,Hirschfelder4,Hirschfelder5, ASSanz2,ASSanz3}. Analogously, in the complex plane quantum vortices form around nodes of the wave function, the quantized circulation integral arising from the discontinuity in the real part of the complex action \cite{Chou6,Chou7}. The PVF of a complex function contains exactly the same information as the complex function itself, but it is introduced to interpret the circulation integral in terms of the work and flux of its PVF along the contour. Moreover, the PVF is the tangent vector field of contours for the complex-extended Born probability density \cite{Chou8a,Chou8b}. Streamlines near a PVF stagnation point display rectangular hyperbolic structure, while streamlines near a PVF pole become circles enclosing the pole \cite{Chou6,Chou7}. Local structures or streamlines for the QMF and its associated PVF are summarized in Table~\ref{tab:table1}. \begin{table} \caption{\label{tab:table1} Local structures or streamlines for the QMF and the associated PVF near a stagnation point or a pole.} \begin{center} \begin{tabular}{c|c|c} \hline \hline & Stagnation point & Pole \\ \hline \multirow{2}{*}{QMF} & Spirals, circles, & East-West and North-South \\ & or straight lines & opening hyperbolic flow \\ \hline \multirow{2}{*}{PVF} & Rectangular & \multirow{2}{*}{Circular flow} \\ & hyperbolic flow &\\ \hline \hline \end{tabular} \end{center} \end{table} \subsection{\label{sec:} Approximate quantum trajectories around a stagnation point} Since the QMF is generally time-dependent, we can determine approximate complex quantum trajectories around a stagnation point $(z_0, t_0)$ in spacetime. We expand the QMF in a Taylor series around a stagnation point \begin{equation} p(z,t)=\left(\frac{\partial p}{\partial z}\right)_0(z-z_0)+\left(\frac{\partial p}{\partial t}\right)_0(t-t_0)+\cdots, \label{ThreeDQMFTS} \end{equation} where $p(z_0,t_0)=0$ has been used and the partial derivatives are evaluated at the stagnation point. Substituting this equation into Eq.~(\ref{TDtraj1}), we obtain a first-order nonautonomous complex ordinary differential equation for approximate quantum trajectories \begin{equation} \frac{dz}{dt}=\frac{1}{m}\left(\alpha z+\beta t\right), \end{equation} where $(\partial p/\partial z)_0=\alpha$ and $(\partial p/\partial t)_0=\beta$ have been used and the origin has been moved to the stagnation point. General solutions for the nonautonomous linear differential equation are given by \cite{Hirsch} \begin{eqnarray} z(t)& = & e^{\alpha t/m} \left(z(0) + \frac{\beta}{m} \int_0^t e^{- \alpha s/m} sds\right) \nonumber \\ & = & z(0) e^{\alpha t/m} + \frac{\beta m}{\alpha^2} \left[e^{\alpha t/m} - \left(1 + \frac{\alpha}{m} \ \! t \right) \right] , \label{ThreeDfirstorderTJ} \end{eqnarray} where $z(0)$ is the starting point of a local approximate quantum trajectory. If we consider complex quantum streamlines at a specific time $t_s$, the QMF $p(z,t_s)$ does not depend on time. Hence, the partial derivative of the QMF with respect to time is equal to zero, $(\partial p/\partial t)_0=\beta=0$. Thus, the general solution in Eq.~(\ref{ThreeDfirstorderTJ}) gives the approximate quantum streamlines near the stagnation point. \subsection{\label{sec:DivVort} Divergence and vorticity of the quantum momentum field and its P\'{o}lya vector field} The QMF first derivative contains the information about the divergence and vorticity of the quantum fluid in the complex plane \cite{Chou9}. This can be shown as follows. The QMF {\it divergence}, which describes the local expansion/contraction of the quantum fluid, is given by \begin{equation} \Gamma = \nabla \cdot p = \frac{\partial p_x}{\partial x} + \frac{\partial p_y}{\partial y}. \end{equation} Analogously, the \emph{vorticity} describing the local rotation of the quantum fluid is defined by the QMF curl, \begin{equation} \Omega = \left| \nabla \times p \right| = \frac{\partial p_y}{\partial x} - \frac{\partial p_x}{\partial y} . \end{equation} Since the QMF is analytically extended to the complex plane, we use the Cauchy-Riemann equations to write the QMF first derivative as \begin{equation} \frac{\partial p}{\partial z}=\frac{\partial p_x}{\partial x}+i\frac{\partial p_y}{\partial x}=\frac{\partial p_y}{\partial y}-i\frac{\partial p_x}{\partial y}=\frac{1}{2}\left(\Gamma+i\Omega\right). \label{firstderiQMFVorDiv} \end{equation} Thus, the real and imaginary parts of the QMF first derivative determine the divergence and vorticity, respectively. Moreover, the complex quantum potential in Eq.~(\ref{TDQHJE}) can be expressed in terms of divergence and vorticity by \cite{Chou9} \begin{equation} Q(z,t)=\frac{\hbar}{2mi}\frac{\partial p}{\partial z}=\frac{\hbar}{4m}\left(\Omega-i\Gamma\right). \label{CQP} \end{equation} Additionally, we can also evaluate both the PVF divergence and vorticity, which are given by \begin{align} \Gamma_{\bf P}&= \nabla \cdot {\bf P}= \frac{\partial p_x}{\partial x}-\frac{\partial p_y}{\partial y}=0, \\ \Omega_{\bf P}&=\left|\nabla \times {\bf P}\,\right|= -\frac{\partial p_y}{\partial x}-\frac{\partial p_x}{\partial y}=0, \label{VortPVF} \end{align} respectively, where the Cauchy-Riemann equations for the QMF have been used. The vanishing divergence and vorticity indicate that the PVF associated with the QMF describes an {\it incompressible} and {\it irrotational} flow in the complex plane except at nodes of the wave function. Actually, this result follows from the fact that the PVF of a complex function $f(z)$ is divergence-free and curl-free if $f(z)$ is analytic, and vice versa \cite{Needham}. The quantized circulation integral around quantum vortices originates from the work term of the PVF in Eq.~(\ref{CIQMF}) \cite{Chou6}, \begin{equation} \gamma=\oint_C p(z) dz = \oint_C {\bf P} \cdot d \ell = 2\pi n\hbar . \end{equation} The PVF near a pole is expressed in terms of polar coordinates by ${\bf P}=(\gamma/2\pi r)\hat{e}_\theta$, where $\gamma$ is the circulation and $r$ is the radial distance from the center of the vortex. From Eq.~(\ref{VortPVF}), the PVF vorticity is zero everywhere {\it except at poles}. The PVF velocity near a pole varies inversely as the distance $r$ from the core of the vortex. The circulation integral along a closed path enclosing the vortex is equal to $\gamma=2\pi n\hbar$, which is independent of $r$. Although the quantum fluid described by the streamlines around a pole moves along a circular path, its vorticity is zero. These features indicate that the quantum vortex described by the PVF is a free or irrotational vortex \cite{Tritton}. \subsection{\label{sec:DivVorPole} Divergence and vorticity around a pole} For a wave function with an $n$-th order node at $z=z_p$, $\psi(z) = (z-z_p)^n f(z)$, we can evaluate the QMF first derivative, which yields \begin{equation} \frac{\partial p}{\partial z} = \frac{ni\hbar}{\left(z-z_p\right)^2} + \frac{\partial p_s}{\partial z} , \label{firstderiQMF} \end{equation} where $p_s(z)$ is the smooth part of the QMF. Thus, the QMF first derivative can be approximated by the first term in the vicinity of a pole. For simplicity, we move the origin to the pole. Separating the first term in Eq.~(\ref{firstderiQMF}) into its real and imaginary parts, we obtain the divergence and vorticity around the pole through Eq.~(\ref{firstderiQMFVorDiv}), \begin{align} \Gamma& = n\hbar \ \! \frac{4xy}{(x^2+y^2)^2}, \label{AppDiv}\\ \Omega& = n\hbar \ \! \frac{2(x^2-y^2)}{(x^2+y^2)^2} , \label{AppVor} \end{align} respectively. These approximate forms describe the local behavior of the divergence and vorticity in the vicinity of the pole. Figure~\ref{fig2} presents the variations of both the QMF divergence and vorticity along the approximate streamlines in the vicinity of a pole. From now on, $\hbar = m = 1$ and atomic units will be used throughout this work. In this case, we consider a wave function with one first-order node at the origin $(n=1)$. As discussed in Sec.~\ref{sec:LSQMF}, in Fig.~\ref{fig2}(a) we observe how the QMF displays a hyperbolic flow around the pole. In addition, streamlines 1 and 3 are parametrized by $(x = \mp 0.1 \sec t, y = \pm 0.1 \tan t)$ and streamlines 2 and 4 are parametrized by $(x = \pm 0.1 \tan t, y = \mp 0.1 \sec t)$. In Figs.~\ref{fig2}(b) and \ref{fig2}(c), we show the variations of both the QMF divergence and vorticity given by Eqs.~(\ref{AppDiv}) and (\ref{AppVor}), respectively, along the streamlines shown in Fig.~\ref{fig2}(a) from $t=-\pi/3$ to $t=\pi/3$. As can be noticed in these figures, the positive QMF divergence describes the local expansion of the quantum fluid when particles approach the pole. When particles approach turning points near the pole, the QMF divergence vanishes; when particles leave the pole, the negative QMF divergence indicates the local contraction of the quantum fluid. On the other hand, the QMF vorticity describes the local rotation of the quantum fluid. During the whole process, it can be noticed that streamlines 1 and 3 display local counterclockwise rotation, while streamlines 2 and 4 display local clockwise rotation. The QMF vorticity attains the extrema at the turning points. Therefore, when particles approach the pole, they rebound as they experience a repulsive force. \begin{figure} \caption{\label{fig2} \label{fig2} \end{figure} \section{\label{sec:QI} Quantum interference} \subsection{\label{sec:Head}Head-on collision of two Gaussian wave packets} \subsubsection{On the real axis} We consider the Gaussian wave-packet head-on collision which, despite its simplicity, can be considered as representative of other more complicated, realistic processes characterized by interference (e.g., scattering problems, diffraction by slits, etc.). This process can be described by the following total wave function \begin{equation} \Psi (x,t) = \psi_L (x,t) + \psi_R (x,t) . \label{int1} \end{equation} Each wave packet ($L$ or $R$, left or right, respectively) is represented by a free Gaussian function as \begin{equation} \psi (x,t) = A_t \; e^{- (x-x_t)^2/4\tilde{\sigma}_t\sigma_0 + i p (x - x_t)/\hbar + i E t/\hbar} , \label{int2} \end{equation} where, for each component, $A_t = (2\pi\tilde{\sigma}_t^2)^{-1/4}$ and the complex time-dependent spreading is \begin{equation} \tilde{\sigma}_t = \sigma_0 \left( 1 + \frac{i\hbar t}{2m\sigma_0^2} \right) . \label{int3} \end{equation} with the initial spreading $\sigma_0$. From Eq.~(\ref{int3}), the spreading of this wave packet at time $t$ is \begin{equation} \sigma_t = |\tilde{\sigma}_t| = \sigma_0 \sqrt{1 + \left( \frac{\hbar t}{2m\sigma_0^2} \right)^2 } . \label{int4} \end{equation} Due to the free motion, $x_t = x_0 + v_p t$ ($v_p = p/m$ is the propagation velocity) and $E = p^2/2m$, i.e., the centroid of the wave packet moves along a classical rectilinear trajectory. This does not mean, however, that the average value of the wave-packet energy is equal to $E$ since, due to the quantum potential, the average energy is given by $\bar{E} = p^2/2m + \hbar^2/8m\sigma_0^2$ \cite{ASSanz9}. We observe two contributions in this expression. The former is associated with the translation of the wave packet traveling at velocity $v_p$, while the latter is related to its spreading at velocity $v_s$ and has, therefore, a purely quantum-mechanical origin. The {\it effective} momentum $p_s$ can then be written as $p_s=\hbar/2\sigma_0$, which resembles Heisenberg's uncertainty relation. The relationship between $v_p$ and $v_s$ plays an important role in effects that can be observed when dealing with wave packet superpositions \cite{ASSanz9}. Defining the timescale $\tau = 2m\sigma_0^2/\hbar$, we note that, if $t \ll \tau$, the width of the wave packet remains basically constant with time, $\sigma_t \approx \sigma_0$ (i.e., for practical purposes, it is roughly time-independent up to time $t$). This condition is equivalent to having an initial wave packet prepared with $v_s \ll v_p$. Thus, the translational motion will be much faster than the spreading of the wave packet. On the contrary, if $t \gg \tau$, which is equivalent to $v_s \gg v_p$, the width of the wave packet increases linearly with time ($\sigma_t \approx \hbar t/2m\sigma_0$), and the wave packet spreads very rapidly in comparison with its advance along $x$. Of course, in between, there is a smooth transition; from Eq.~(\ref{int4}), it is shown that the progressive increase of $\sigma_t$ describes a hyperbola when this magnitude is plotted {\it vs} time. Thus, we can control in the interference process the spreading and translational motions which determine the wave packet dynamics \cite{ASSanz8,ASSanz9}. \subsubsection{In the complex plane} In conventional quantum mechanics, the interference pattern transiently observed on the real axis is attributed to the constructive and destructive interference between two counter-propagating components of the total wave function in Eq.~(\ref{int1}). In contrast, in the framework of the complex quantum Hamilton-Jacobi formalism, the total wave function is analytically continued to the complex plane from $\Psi(x,t)$ to $\Psi(z,t)$. Therefore, two propagating wave packets \emph{always interfere with each other in the complex plane}, and this leads to a persistent pattern (line) of nodes and stagnation points which rotates counterclockwise with time \cite{ASSanz8,Chou9}. The nodal positions in the complex plane can be determined analytically by solving the equation $\Psi(z,t)=0$, resulting \begin{equation} z_n(t)=\frac{i\pi\left(n+1/2\right)}{\left[imv_p/\hbar-\left(x_0-v_pt\right) /\left(2\sigma_0\tilde{\sigma}_t\right)\right]}, \end{equation} where $n = 0, \pm1, \pm2, \ldots$. Here, we have assumed $x_R=-x_L=x_0$ and $v_{pL}=v_{pR}=v_p$. Splitting this expression into its real and imaginary parts, i.e., $z_n(t)=x_n(t)+iy_n(t)$, we obtain \begin{align} x_n(t)&=\pi\left(n+\frac{1}{2}\right)\frac{\hbar}{m} \left[\frac{x_0t+v_p\tau^2}{x_0^2+v_p^2\tau^2}\right], \label{nodeX} \\ y_n(t)&=\pi\left(n+\frac{1}{2}\right)\left[ \frac{2\sigma_0^2\left(v_pt-x_0\right)}{x_0^2+v_p^2\tau^2}\right], \label{nodeY} \end{align} respectively, where $\tau = 2m\sigma_0^2/\hbar$ is the timescale for the Gaussian wave packet. In addition, dividing Eq.~(\ref{nodeY}) by Eq.~(\ref{nodeX}) yields the analytical expression for the (time-dependent) polar angle describing the angular position of the nodal line with respect to the positive real axis, \begin{equation} \theta(t) = (\tan)^{-1}\left[\frac{y_n(t)}{x_n(t)}\right] = (\tan)^{-1} \left[ \frac{\tau\left(v_pt-x_0\right)}{x_0t+v_p\tau^2} \right] , \label{nodalAngle} \end{equation} which does not depend on $n$. From this expression, we can calculate the rotation rate of the nodal line in the complex plane, \begin{equation} \omega(t)=\frac{d\theta(t)}{dt}=\frac{\hbar}{2m\sigma_t^2} , \label{NodalRate} \end{equation} where $\sigma_t$ is given in Eq.~(\ref{int4}). This equation indicates that the rotation rate is completely determined by the initial spreading of the Gaussian wave packet $\sigma_0$. In addition, this rate is always positive and decays monotonically to zero as $t$ goes to $\infty$. From Eqs.~(\ref{nodeX}) and (\ref{nodeY}), we can also determine the node separation distance between two consecutive nodes \begin{align} d(t) & = \sqrt{ [x_{n+1}(t)-x_n(t)]^2 + [y_{n+1}(t) - y_n(t)]^2} \nonumber\\ & = \frac{\pi\hbar\sigma_t}{p_s\sqrt{x_0^2+v_p^2\tau^2}}, \label{NodalDist} \end{align} where $p_s = \hbar/2\sigma_0$ is the effective momentum. This distance is independent of $n$ and it increases with time. Moreover, eliminating $t$ in Eqs.~(\ref{nodeX}) and (\ref{nodeY}) yields the $n$th node trajectory describing the time evolution of the node given by \begin{equation} y_n = \left(\frac{v_p\tau}{x_0}\right)x_n - (2n+1)\left(\frac{\pi\sigma_0^2}{x_0}\right). \label{NodalTraj} \end{equation} Consequently, these nodal trajectories with the same slope and different intercepts are parallel to each other. When $t=0$, the initial angle of the nodal line is $\theta_0 = (\tan)^{-1}(-x_0/v_p\tau)$. As described by Eq.~(\ref{NodalRate}), the positive rotational rate indicates that the polar angle of the nodal line increases monotonically with time. The nodal line thus rotates counterclockwise from the initial angle to a maximum or limiting value $\theta_\infty = (\tan)^{-1}(v_p\tau/x_0)$ when $t \to \infty$. The angular displacement from $t=0$ to $t=\infty$ is always equal to $\Delta \theta = \theta_\infty - \theta_0 = \pi/2$, because the product of the slopes of the nodal lines is equal to $(\tan \theta_0) (\tan \theta_\infty) = -1$. In particular, if both wave packets are initially very far apart (i.e., $x_0 \to \infty$), but move with a finite velocity $v$, or they are at an arbitrary finite distance, but $v = 0$, the nodal line ends up aligned with the real axis. Otherwise, the nodal line starts at some angle $\theta_0$ and then evolves with the angular displacement $\Delta\theta=\pi/2$ until it reaches the limit angle $\theta_\infty$. In addition, the initial nodal line is perpendicular to all nodal trajectories in Eq.~(\ref{NodalTraj}). Then, the nodal line rotates counterclockwise with time and it becomes parallel to nodal trajectories when $t$ approaches infinity. Interference features are observed on the real axis only when the nodal line is near the real axis. When the nodal line coincides with the real axis, the maximum interference features can be observed in real space. At this time, setting $y_n = 0$ in Eq.~(\ref{NodalTraj}), we recover the expression for the positions of nodes on the real axis, $x_n = (n+1/2)\lambda/2$ where $\lambda = 2\pi\hbar/mv_p$. During the evolution of the nodal line, its intersections with nodal trajectories determine the positions of the nodes. As can be noticed, the time evolution of the nodal line in the complex plane therefore provides an elegant interpretation of quantum interference. \subsection{Case 1: $v_p > v_s$} We first consider the case where the relative propagation velocity is larger than the spreading rate of the wave packets. The following initial conditions for Gaussian wave packets are used: $x_{0L} = - 10 = - x_{0R}$, $v_{pL} = 2 = -v_{pR}$ and $\sigma_0 = \sqrt{2}$. With these conditions, maximal interference occurs at $t=5$ on the real axis and the propagation and spreading velocities are given by $v_p=2$ and $v_s=\sqrt{2}/4$, respectively. \begin{figure} \caption{\label{fig3} \label{fig3} \end{figure} \subsubsection{Quantum caves with quantum trajectories} Figure \ref{fig3}(a) displays complex quantum trajectories with the quantum caves consisting of the isosurfaces $|\Psi (z,t)| = 0.053$ and $|\partial \Psi (z,t)/\partial z|= 0.106$ from $t=0$ to $t=10$ in a time-dependent three-dimensional Argand plot. As discussed in Sec.~\ref{sec:LSQMF}, the QMF local structures near stagnation points and poles provide a qualitative description of the behavior of these trajectories. It is clearly seen that stagnation and vortical tubes alternate with each other, and the centers of the tubes are stagnation and vortical curves. Trajectories display helical wrapping around the stagnation tubes and they are deflected by the vortical tubes to show hyperbolic indentations in three dimensional space, as shown in Fig.~\ref{fig3}(b). These trajectories, which display complicated paths around stagnation and vortical tubes, depict how probability flows. In addition, trajectories from different launch points wrap around the same stagnation curve and remain trapped for a certain time interval. Then, they separate from the stagnation curve. This counterclockwise circulation of trajectories can be viewed as a resonance process. This phenomenon characterizes long-range correlation among trajectories arising from different starting points. Trajectories may wrap around the same stagnation curve with different wrapping times and numbers of loops. Trajectories starting from the isochrone with the small initial separation may wrap around different stagnation curves and then end up with large separations. In this way, interference leads to the formation of quantum caves and the topological structure displayed by complex quantum trajectories. \begin{figure} \caption{\label{fig4} \label{fig4} \end{figure} \subsubsection{Dislocations of the complex action} The complex action function $S(z,t)$ displays fascinating features in the complex plane. Decomposing this function into its real and imaginary parts $S=S_R+iS_I$, we write the wave function as $\psi(z,t)=\exp(-S_I/\hbar) \exp(iS_R/\hbar)$. According to this expression, the real and imaginary parts of the complex action determine the {\it phase} and {\it amplitude} of the wave function, respectively. Figure~\ref{fig4} displays the real and imaginary parts of the complex action for the complex-extended wave function in Eq.~(\ref{int1}) at $t=2.5$. Figure~\ref{fig4}(a) displays the principal zone of the phase of the wave function in the range $-\pi \leq \arg(S_R) \leq \pi$. Figure~\ref{fig4}(b) displays the imaginary part $S_I$ of the complex action in the complex plane. The vanishing of the wave function at nodes indicates that the imaginary part of the complex action tends to positive infinity at nodes. The peaks in Fig.~\ref{fig4}(b) correspond to the nodes in the wave function. The quantized circulation integral around a node in the wave function can be related to the change in the phase of the wave function and this integral can be expressed in terms of the PVF by \cite{Chou6} \begin{equation} \oint_C p(z)dz=\oint_C {\bf P} \cdot d \ell = 2\pi n\hbar = \Delta_C S_R , \label{QISCI} \end{equation} where $C$ denotes a simple closed curve in the complex plane. The quantized circulation integral around a node of the wave function originates from the discontinuity in its phase. The PVF displays \emph{counterclockwise circular flow} in the vicinity of a node \cite{Chou6,Chou7}. Nodes in the complex-extended wave function in Eq.~(\ref{int1}) are first-order nodes $(n=1)$. As shown in Fig.~\ref{fig4}(a), if we travel around a first-order node counterclockwise along a closed path, it follows from Eq.~(\ref{QISCI}) that the phase displays a sharp discontinuity of $2\pi$. Actually, the phase of the wave function is a multivalued function in the complex plane. Thus, if we travel counterclockwise around a node, we then go through the branch cut from one Riemann sheet to another. Through a continuous closed circuit around a node continuing on all the sheet, the phase function generates a helicoid along the vertical axis. Analogous to the case in Bohmian mechanics, phase singularities at nodes in the complex plane can be interpreted as \emph{wave dislocations} \cite{NyeBerry,HollandBook}. \begin{figure} \caption{\label{fig5} \label{fig5} \end{figure} \subsubsection{QMF divergence and vorticity} Figure~\ref{fig5}(a) shows a trajectory launched from $z=-9.11016-1.17309i$ which later arrives at $z=-0.3$, when maximal interference occurs at $t=5$, and Fig.~\ref{fig5}(b) presents the divergence and vorticity along this trajectory. When the particle approaches a turning point, its velocity undergoes a rapid change and the divergence and vorticity display sharp fluctuations. When the particle approaches the vortical curve at position $a$ along the direction of streamline 2 shown in Fig.~\ref{fig2}(a), the trajectory displays hyperbolic deflection and the divergence and vorticity display analogous variations, as shown in Fig.~\ref{fig2}(c). Around position $a$, when the particle approaches the vortical curve, the positive divergence indicates the local expansion of the quantum fluid. When the particle arrives at the turning points, the divergence vanishes. Then, when it leaves the vortical curve, the negative divergence indicates the local contraction of the quantum fluid. On the other hand, the negative vorticity describes the local clockwise rotation of the quantum fluid. When the trajectory displays helical wrapping around the stagnation curve, the particle is trapped between two vortical curves. When the particle approaches the turning points $b$ and $d$ along the direction of streamline 1 shown in Fig.~\ref{fig2}(a) and approaches the turning point $c$ along the direction of streamline 3, the divergence and vorticity in Fig.~\ref{fig5}(b) display analogous variations shown in Fig.~\ref{fig2}(b). When the particle approaches and leaves the vortical curve, the divergence describes the local expansion and contraction of the quantum fluid in the vicinity of the vortical curve. The positive vorticity indicates the counterclockwise rotation of the quantum fluid. Finally, when the particle leaves the stagnation curve, it approaches the vortical curve at position $e$ along the direction of streamline 4 shown in Fig.~\ref{fig2}(a). Again, the divergence and vorticity around position $e$ display similar fluctuations shown in Fig.~\ref{fig2}(c). Therefore, the variations of the divergence and vorticity around a pole analyzed in Sec.~\ref{sec:DivVorPole} provide a qualitative description of the local behavior of the complex quantum trajectories in the vicinity of vortical tubes. \begin{figure} \caption{\label{fig6} \label{fig6} \end{figure} \subsubsection{Approximate quantum trajectories} As discussed in Sec.~\ref{sec:LSQMF}, the QMF first derivative evaluated at stagnation points determines the local structure around these points \cite{Chou7}. Since it is noted from Eq.~(\ref{int1}) that there is always constructive interference at the origin, the origin is a stagnation point at all times. As an example, Fig.~\ref{fig6}(a) presents the QMF first derivative at the origin from $t=0$ to $t=10$. The real part of the QMF first derivative indicates that the QMF displays convergent flow around the stagnation point until $t=4.851$. Then, the real part of the QMF first derivative becomes positive and the QMF displays divergent flow. On the other hand, the QMF initially displays clockwise flow around the stagnation point. After $t=0.395$, the QMF displays counterclockwise flow. As displayed in Fig.~\ref{fig3}, trajectories exhibit helical wrapping around the stagnation curve along $z=0$ from approximately $t=3.5$ to $t=6.5$. During this time interval, the positive imaginary part of the QMF first derivative in Fig.~\ref{fig6}(a) describes the counterclockwise flow of the quantum fluid. However, the real part of the QMF first derivative changes sign at $t=4.851$. Particles initially converge to the stagnation point and they are gradually repelled by the stagnation point after $t=4.851$. Finally, these particles depart from the stagnation point. Therefore, the QMF first derivative at the stagnation point qualitatively explains the behavior of trajectories in the vicinity of the stagnation point. Additionally, Fig.~\ref{fig6}(b) shows that the trajectories starting from the isochrone arrive at the real axis at $t=5$ with the approximate trajectories around the stagnation point given in Eq.~(\ref{ThreeDfirstorderTJ}). Here, the stagnation point in spacetime for Eq.~(\ref{ThreeDfirstorderTJ}) is chosen to be $z=0$ and $t=5$. The positions for the approximate trajectories are set to be the same as those for the exact trajectories at $t=5$. As shown in this figure, the trajectory determined by Eq.~(\ref{ThreeDfirstorderTJ}) is a good approximation of the exact trajectory, provided that the approximate trajectory is close to the stagnation point in spacetime. \subsubsection{Wrapping time} These complicated features for trajectories arise from the complex quantum potential in Eq.~(\ref{CQP}). Both the QMF divergence and vorticity characterize the turbulent flow of complex quantum trajectories. Moreover, the variation of the QMF vorticity provides a feasible method to define the wrapping time for a specific trajectory around a stagnation curve. The wrapping time can be defined by the time interval between the first and last minimum of $\Omega$ comprising a region with the positive vorticity. Within this time interval, the positive vorticity describes the counterclockwise twist of the trajectory displaying the interference dynamics, and the stagnation points and poles significantly affect the motion of the particle. For example, Fig.~\ref{fig5}(b) indicates that the particle undergoes counterclockwise wrapping around the stagnation curve from $t=3.676$ to $t=6.954$ and hence the wrapping time is $t_W=3.278$. Different trajectories have different wrapping times. Therefore, the average wrapping time can be utilized to define the `lifetime' for the interference process observed on the real axis. \begin{figure} \caption{\label{fig7} \label{fig7} \end{figure} In Fig.~\ref{fig7} we present the wrapping times that correspond to the first and last minimum of the vorticity, which comprise a region with positive vorticity for trajectories launched from the isochrone arriving at the real axis at $t=5$. Since the process described in Eq.~(\ref{int1}) in the complex plane is symmetric with respect to the origin, the data shown in Fig.~\ref{fig7} display the same symmetry. As displayed in Fig.~\ref{fig3}(b), trajectories can wrap around stagnation curves with different wrapping times and numbers of loops. Figures~\ref{fig7} and \ref{fig3}(b) indicate that trajectories wrapping around stagnation curves with small rotational radius are trapped between vortical curves for a long time, and this leads to a long wrapping time for these trajectories. On the contrary, if trajectories wrapping around stagnation curves with large rotational radius, when they approach vortical curves, they experience a significant repelling force provided by QMF poles. Thus, these trajectories can easily escape capture by stagnation curves, and this results in a short wrapping time. In addition, it is noted in Fig.~\ref{fig7} that all trajectories display helical wrapping from approximately $t=4.3$ to $t=5.9$. Furthermore, the average wrapping time for these trajectories is $\bar{t}_W=3.24$, and it can be used to define the lifetime for the interference process in this case. \begin{figure} \caption{\label{fig8} \label{fig8} \end{figure} \subsubsection{Rotational dynamics of the nodal line} As described in Sec.~\ref{sec:Head}, two counter-propagating Gaussian wave packets interfere with each other at all times in the complex plane, and the nodal line rotates counterclockwise with respect to the origin as time progresses. Figure \ref{fig8}(a) shows the evolution of stagnation points and nodes and nodal trajectories in the complex plane, and Fig.~\ref{fig8}(b) displays the time-dependent probability densities along the real axis. Initially, the interference of tails of two wave packets contributes to the string of stagnation points and nodes, and the initial nodal line with the angle $\theta_0=-51.34^\circ$ is perpendicular to nodal trajectories in Eq.~(\ref{NodalTraj}). Then, the nodal line rotates counterclockwise and lines up with the real axis at $t=5$. At this time, the total wave function displays maximal interference and the exact nodes form on the real axis. After $t=5$, these two wave packets start to separate but keep interfering with each other in the complex plane, and the nodal line continues to rotate counterclockwise away from the real axis. Finally, the angle of the nodal line tends to the limit angle $\theta_\infty = 38.66^\circ$ when $t$ approaches infinity, and the nodal line becomes parallel to nodal trajectories. The rotational rate of the nodal line in Eq.~(\ref{NodalRate}) decays monotonically to zero when $t$ tends to infinity. In Fig.~\ref{fig8}(a), the intersections of the nodal line and nodal trajectories determine the nodal positions, and the distance between nodes in Eq.~(\ref{NodalDist}) increases with time. Therefore, the interference process is described by the rotational dynamics of the nodal line with the angular displacement $\Delta\theta=\pi/2$. In conventional quantum mechanics, interference extrema transiently forming on the real axis are attributed to constructive and destructive interference between components of the total wave function, as shown in Fig.~\ref{fig8}(b). In contrast, in the complex quantum Hamilton-Jacobi formalism, the interference features observed on the real axis are described by the counterclockwise rotation rate of the nodal line in the complex plane. Since interference features are observed on the real axis only when the nodal line is near the real axis, we can define the lifetime for the interference process corresponding to the time interval for the nodal line to rotate from $\theta=-10^\circ$ to $\theta=+10^\circ$. In Fig.~\ref{fig8}(a), the lifetime for the interference features is $\Delta t=3.8$. Therefore, compared to the conventional quantum mechanics, the complex quantum Hamilton-Jacobi formalism provides an elegant method to define the lifetime for the interference features observed on the real axis. \begin{figure} \caption{\label{fig9} \label{fig9} \end{figure} \subsubsection{Quantum caves with P\'{o}lya trajectories} Although the QMF displays hyperbolic flow around a node, its associated PVF displays circular flow \cite{Chou6,Chou7}. Figure~\ref{fig9} shows that P\'{o}lya trajectories launched from the isochrone arrive at the real axis with quantum caves. In contrast to Fig.~\ref{fig3}(a), these trajectories display helical wrapping around the vortical tubes and hyperbolic deflection around the stagnation tubes. Both the PVF divergence and vorticity vanish everywhere except at poles; thus, trajectories display helical wrapping around \emph{irrotational vortical curves} described by the PVF. \subsection{Case 2: $v_p \lesssim v_s$} Next, we consider the case where the relative propagation velocity is approximately equal to or smaller than the spreading rate of the wave packets. We use the following initial conditions for Gaussian wave packets: $x_{0L} = - 5 = - x_{0R}$, $v_{pL} = 1 = -v_{pR}$ and $\sigma_0 = \sqrt{2}/4$. Maximal interference also occurs at $t=5$ on the real axis and the propagation and spreading velocities are given by $v_p=1$ and $v_s=\sqrt{2}$, respectively. \subsubsection{Quantum caves with quantum trajectories and P\'olya trajectories} Figure~\ref{fig10} shows that quantum trajectories and P\'olya trajectories starting from the isochrone reach the real axis at $t=5$ with quantum caves consisting of the isosurfaces of the wave function and its first derivative. Similar to the case shown in Fig.~\ref{fig3}, quantum caves form around stagnation curves and vortical curves appearing alternately, but they are significantly distorted due to the rapid spreading of the wave packets. In addition, quantum trajectories again display helical wrapping around the stagnation tubes and hyperbolic deflection near the vortical tubes. On the contrary, P\'olya trajectories display hyperbolic deflection near the stagnation tubes and helical wrapping around the vortical tubes. Again, trajectories launched from different starting points show long-range correlation, and interference leads to the formation of quantum caves and produces complicated behavior of these trajectories. \begin{figure} \caption{\label{fig10} \label{fig10} \end{figure} \subsubsection{Rotational dynamics of the nodal line} As shown in Fig.~\ref{fig3}, quantum trajectories for Case 1 remain trapped for a certain time interval between vortical curves, and then they depart from the stagnation curves. In contrast, Fig.~\ref{fig10}(a) indicates that quantum trajectories can wrap around stagnation curves for an infinite time. This is in agreement with our previous statement that the wrapping time is a measure of the lifetime of the interference features; in this case, these features remain visible asymptotically and the wrapping time becomes infinity. Figure~\ref{fig11}(a) shows the evolution of stagnation points and nodes and nodal trajectories in the complex plane, and Fig.~\ref{fig11}(b) displays the time-dependent probability densities along the real axis. The nodal line starting with the initial angle $\theta_0 = -87.14^\circ$ rotates counterclockwise with respect to the origin and reaches the real axis at $t=5$ when the maximal interference is observed on the real axis. Then, the nodal line rotates counterclockwise away from the real axis, and it approaches the limit nodal line with the angle $\theta_\infty = 2.86^\circ$ as $t$ tends to infinity. Therefore, the nodal line rotates with the angular displacement $\Delta\theta=\pi/2$ to reach the limit nodal line parallel to nodal trajectories. \begin{figure} \caption{\label{fig11} \label{fig11} \end{figure} When the nodal line is near the real axis, the interference features are clearly displayed on the real axis. As in Case 1, we can define the starting time of the interference process as the time for the nodal line with the angle $\theta=-10^\circ$. In Fig.~\ref{fig11}(a), $\theta(1.09)=-10^\circ$ in this case. In addition, the limit nodal line with the angle $\theta_\infty = 2.86^\circ$ is extremely close to the real axis. Therefore, the interference process starts at $t=1.09$ and remains until $t$ tends to infinity. Figure~\ref{fig11}(b) indicates that the total wave function starts to display the interference feature when $t=1.09$ and the interference feature persists at long times. Figure~\ref{fig12} presents the angle and the rotational rate of the nodal line in Eqs.~(\ref{nodalAngle}) and (\ref{NodalRate}) for Case 1 and Case 2. As indicated in Eq.~(\ref{NodalRate}), the rotational rates for these two cases both decay monotonically to zero when $t$ tends to infinity. For Case 1, the angle of the nodal line alters relatively slowly and the rotational rate gradually decreases to zero. In contrast, for Case 2, the rotational rate shows a rapid decrease within the initial time interval in Fig.~\ref{fig12}(b). This dramatic change in the rotational rate reflects the fast rotation of the nodal line from the initial position to the vicinity of the real axis in Fig.~\ref{fig12}(a). \begin{figure} \caption{\label{fig12} \label{fig12} \end{figure} \section{\label{sec:Conclusions} Final discussion and concluding remarks} In this study, quantum interference was explored in detail within the complex quantum Hamilton-Jacobi formalism. We reviewed local structures of the QMF and its associated PVF around stagnation points and poles, and derived the first-order equation for approximate quantum trajectories around stagnation points. Analysis of both the QMF divergence and vorticity along streamlines around a pole was employed to explain the complicated behavior of complex quantum trajectories around quantum caves. In addition, both the PVF divergence and vorticity vanish except at poles; hence, the PVF describes an incompressible and irrotational flow in the complex plane. In contrast, both the QMF divergence and vorticity characterize the turbulent flow in the complex plane. We analyzed quantum interference using the head-on collision of two Gaussian wave packets as an example. Exact detailed analysis was presented for the rotational dynamics of the nodal line on the real axis and in the complex plane. Complex quantum trajectories display helical wrapping around stagnation tubes and hyperbolic deflection around vortical tubes. In contrast, P\'olya trajectories display hyperbolic deflection around stagnation tubes and helical wrapping around vortical tubes. For the case where the relative propagation velocity is larger than the spreading rate of the wave packets, during the interference process, trajectories keep circulating around stagnation tubes as a resonant process and then escape as time progresses. Phase singularities in the complex plane can be regarded as wave dislocations. Then, the wrapping time for an individual trajectory was determined by both the QMF divergence and vorticity, and the average wrapping time was calculated as one of the definitions for the lifetime of interference. For the case where the relative propagation velocity is approximately equal to or smaller than the spreading rate of the wave packets, the distortion of quantum caves originates from the rapid spreading of the wave packets. Due to the rapid spreading rate, interference features also develop very rapidly, remaining visible asymptotically in time. The wrapping time becomes infinity, and this implies the infinite survival of such interference features. However, since the interference features are observed on the real axis only when the nodal line is near the real axis, the rotational dynamics of the nodal line in the complex plane offers a unified description to clearly explain transient or persistent appearance of the interference features observed on the real axis. Therefore, these results show that the complex quantum trajectory method provides a novel and insightful interpretation of quantum interference. The average wrapping time determined by both the QMF divergence and vorticity can be used as one of the definitions for the lifetime of interference. On the contrary, the PVF divergence and vorticity cannot be used to define the lifetime of interference because they vanish except at poles. However, the PVF of a complex function, such as the QMF, contains exactly the same information as the complex function itself \cite{Needham}. Thus, it is sufficient to define the lifetime of interference as the average wrapping time determined by the QMF divergence and vorticity. The head-on collision of two Gaussian wave packets with equal amplitudes was used as a model system to explore quantum interference in complex space. This problem is the prototype of quantum systems displaying interference effects, and it also exhibits basic features of quantum interference. A straightforward generalization is to consider the head-on collision of two Gaussian wave packets with different amplitudes. As shown in this paper, these two wave packets interfere with each other in the complex plane for all times. Because of the different amplitudes for these two counter-propagating wave packets, we cannot observe an infinite number of nodes on the real axis. However, nodes can occur on the real axis at specific times. In this case, the nodal line displays not only rotational motion but also translational motion in the complex plane. Analogously, the dynamics of the nodal line in the complex plane clearly explains the interference features observed on the real axis. A detailed analysis will be reported in our future studies. The current study concentrates mainly on quantum interference arising from the head-on collision of wave packets involving no external potential. We have presented a comprehensive exact analytical study for this problem, and these results demonstrates that the complex quantum trajectory method can provide new physical insights for analyzing, interpreting, and understanding quantum mechanical problems. There are various quantum effects resulting from quantum interference. Therefore, in the future, quantum interference incorporating interaction with external potentials during physical processes can be examined through analytical and computational approaches. Multidimensional problems displaying quantum interference deserve further investigation within the complex quantum Hamilton-Jacobi formalism. \section*{Acknowledgment} Chia-Chun Chou and Robert E.\ Wyatt thank the Robert Welch Foundation (grant F-0362) for the financial support of this research. A.\ S.\ Sanz and S.\ Miret-Art\'es acknowledge the Ministerio de Ciencia e Innovaci\'on (Spain) for financial support under Project FIS2007-62006. A.\ S.\ Sanz also acknowledges the Consejo Superior de Investigaciones Cient\'{\i}ficas for a JAE-Doc contract. \end{document}

\begin{document} \begin{CJK*}{GBK}{song} \setlength{\baselineskip}{11.5pt} \renewcommand{\abovewithdelims}[2]{ \genfrac{[}{]}{0pt}{}{#1}{#2}} \title{\bf The super spanning connectivity of arrangement graphs} \author{Pingshan Li \quad Min Xu\footnote{Corresponding author. \newline {\em E-mail address:} xum@mail.bnu.edu.cn (M. Xu) .}\\ {\footnotesize \em Sch. Math. Sci. {\rm \&} Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China} } \date{} \date{} \maketitle \begin{abstract} A $k$-container $C(u, v)$ of a graph $G$ is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u, v)$ of $G$ is a $k^*$-container if it is a spanning subgraph of $G$. A graph $G$ is $k^*$-connected if there exists a $k^*$-container between any two different vertices of G. A $k$-regular graph $G$ is super spanning connected if $G$ is $i^*$-container for all $1\le i\le k$. In this paper, we prove that the arrangement graph $A_{n, k}$ is super spanning connected if $n\ge 4$ and $n-k\ge 2$. \noindent {\em Key words:} Hamiltonian; Hamiltonian connected; $k^*$-connected; Arrangement graphs. \end{abstract} \section{Introduction} In the field of parallel and distributed systems, interconnection networks are an important research area. Typically, the topology of a network can be represented as a graph in which the vertices represent processors and the edges represent communication links. For graph definitions and notations, we follow \cite{J.A.Bondy}. A graph $G$ consists of a vertex set $V(G)$ and an edge set $E(G)$, where an edge is an unordered pair of distinct vertices of $G$. A path $P$ of length $k$ from $x$ to $y$ is a finite sequence of distinct vertices $\langle v_0, v_1, \cdots, v_k\rangle$, such that $x=v_0, y=v_k$, and $(v_i, v_{i+1})\in E$ for $0\le i\le k-1$. We also represent path $P$ as $\langle v_0, v_1, \cdots, v_i, Q, v_j, v_{j+1}, \cdots, v_k\rangle$, where $Q$ is the path $\langle v_i, v_{i+1}, \cdots, v_j\rangle$. In particular, if $i=j$, we can still represent the path as $\langle v_0, v_1, \cdots, v_i, Q, v_i, v_{j+1}, \cdots, v_k\rangle$. A spanning subgraph of $G$ is a subgraph with vertex set $V(G)$. A Hamiltonian graph is a graph with a spanning cycle. A graph is Hamiltonian connected if there exists a spanning path joining any two different vertices. A $k$-container $C(u, v)$ of $G$ is a set of $k$ internally disjoint paths between $u$ and $v$, and the connectivity of $G$, $\kappa(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected or has only one vertex. It follows from Menger's Theorem \cite{K.Menger} that there is a $k$-container between any two distinct vertices of $G$ if $G$ is $k$-connected. A $k$-container $C(u, v)$ of $G$ is a $k^*$-container if it is a spanning subgraph of $G$. A graph is $k^*$-connected if there exists a $k^*$-container between any two distinct vertices. By this definition, the concept of Hamiltonian connected is the same as $1^*$-connected and the concept of Hamiltonian is the same as $2^*$-connected. Thus, the concept of $k^*$-connected is a hybrid concept of connectivity and Hamiltonicity. The study of $k^*$-connected graphs is motivated by the globally $3^*$-connected graphs proposed by M. Albert et al. \cite{M.Albert}. The star graph ($S_n$ for short), which was proposed by Akers et al.\cite{S.B.Akers1}, is a well known interconnection network. The arrangement graph\cite{Day and Tripathi}, denoted by $A_{n,k}$ , refers to a generalized version of $S_n$. Further, $A_{n, n-1}$ is isomorphic to the $n$-dimensional star graph $S_n$\cite{S.B.Akers} and $A_{n, 1}$ is isomorphic to the complete graph $K_n$. The arrangement graph preserves many attractive properties of $S_n$ such as the hierarchical structure, vertex and edge symmetry, simple and optimal routing, and many fault tolerance properties \cite{Day and Tripathi}. Some basic properties of $A_{n, k}$ such as average distance\cite{on}, Hamiltonicity\cite{Fault}, and embedding\cite{Xumin,KDay} have recently been computed or derived. A graph $G$ is super spanning connected if it is $k^*$-connected for all $1\le k\le \kappa(G)$. There are many desirable results about super spanning connected of some interconnection networks such as recursive circulant graphs\cite{C.H.Tsai}, pancake graphs\cite{C.K.Lin}, hypercube-like network\cite{Cheng-Kuan Lina}, $(n, k)$-star graphs\cite{H.C.Hsu2}, $k$-ary $n$-cubes\cite{Y.K. Shih} and multi-dimensional tori\cite{Lijing}. Since $A_{n, n-1}\cong S_n$ is a bipartite graph with the same number of vertices in each partite set, there is no Hamiltonian path joining any two different vertices in the same part. Hence, $A_{n, n-1}$ is not super spanning connected if $n>3$. Therefore, we consider an arrangement graph with $n-k\ge 2$. In this paper, we aim to prove that arrangement graphs are super spanning connected if $n\ge 4$ and $n-k\ge2$. The rest of this paper is organized as follows. In Section 2, we introduce arrangement graphs and discuss some of their properties. In Section 3, we prove that arrangement graphs are super spanning connected for $n\ge 4$ and $n-k\ge2$. \section{Arrangement graphs} Throughout this paper, we assume that $n$ and $k$ are positive integers with $n>k$. We use $\langle n\rangle$ to denote the set $\{1, 2, \cdots, n\}$. The arrangement graph $A_{n, k}$ is a graph that has the vertex set $V(A_{n, k})=\{u=u_1u_2\cdots u_k\mid u_i\in \langle n\rangle, u_i\not=u_j$ if $i\not=j\}$ and the edge set $E(A_{n, k})=\{(p, q)\mid p, q\in V(A_{n, k})$ and $p, q$ differ in exactly one position$\}$. From the definition , we know that $A_{n, k}$ is a regular graph of degree $k(n-k)$ with $\frac{n!}{(n-k)!}$ vertices. Figure \ref{a42} illustrates the arrangement graph $A_{4, 2}$. \begin{figure} \caption{The arrangement graph $A_{4, 2} \label{a42} \end{figure} Let $u=u_1u_2\cdots u_k\in V(A_{n, k})$. We denote $(u)_i=u_i$ as the ith coordinate of $u$ for $1\le i\le k$. Let $v$ be a neighbor of $u$, we denote $v$ as $u^{s(u_i, x)}$ if $v=u_1u_2\cdots u_{i-1}xu_{i+1}\cdots u_k$ for $x\in \langle n\rangle\setminus\{(u)_i: i=1, 2, \cdots, k\}$. For $i, j\in\langle n\rangle, l\in\langle k\rangle$ and $i\not=j$, suppose that $A^{(l, i)}_{n, k}$ denotes the subgraph of $A_{n, k}$ that is induced by $V(A^{(l, i)}_{n, k})=\{p\mid p=p_1p_2\cdots p_k$ and $p_l=i\}$. Obviously, $\{V(A^{(l, i)}_{n, k})\mid 1\le i\le n\}$ forms a partition of $V(A_{n, k})$ and each $A^{(l, i)}_{n, k}$ is isomorphic to $A_{n-1, k-1}$. As a result, $A_{n, k}$ can be recursively constructed from $n$ copies of $A_{n-1, k-1}$. We use $E^{l=i, j}$ to denote the set of edges between $A^{(l, i)}_{n, k}$ and $A^{(l, j)}_{n, k}$; accordingly, $E^{l=i, j}=\frac{(n-2)!}{(n-k-1)!}$. We also use $A^{(l, I)}_{n, k}$ to denote the subgraph of $A_{n, k}$ that is induced by $\cup_{i\in I}V(A^{(l, i)}_{n, k})$. Following are some known properties about arrangement graphs. \begin{lem}{\rm (\cite{H.C.Hsu})}\label{L1} The arrangement graph $A_{n,k}$ is $(k(n-k)-2)$-fault-tolerant Hamiltonian, and $(k(n-k)-3)$-fault-tolerant Hamiltonian-connected for $n-k \ge 2$. \end{lem} \begin{lem}{\rm (\cite{Lo and Chen})}\label{LL1} The arrangement graph $A_{n, k}$ is $(k(n-k)-2)$-edge-fault-tolerant Hamiltonian connected if not all faulty edges are adjacent to the same vertex. \end{lem} In the following, we discuss some properties that will be used in the proof of the main results. \begin{lem}{\rm }\label{L5} $A_{4, 2}$ is super spanning connected. \end{lem} \noindent{\bf Proof: } By Lemma \ref{L1}, $A_{4,2}$ is $1^*$-connected and $2^*$-connected. We need to construct a $3^*$-container and a $4^*$-container joining any two different vertices $u$ and $v$ of $A_{4, 2}$. Since $A_{4, 2}$ is vertex and edge transitive, without loss of generality, we can assume that $u=12$ and $v=13, 34, 23, 21$. We list such $3^*$-containers as follows: \begin{center} \begin{tabular}{|l|} \hline $\langle 12, 13\rangle ~~~~~~~~~~~~~~$ $ \langle 12, 14, 13\rangle~~~~~~~~~~~$ $\langle 12, 42, 43, 41, 21, 31, 32, 34, 24, 23, 13\rangle $\\ \hline $\langle 12, 14, 34\rangle~~~~~~~~~~$ $\langle 12, 42, 32, 34\rangle~~~~~~$ $\langle 12, 13, 43, 23, 24, 21, 41, 31, 34\rangle$ \\ \hline $\langle 12, 13, 43, 23 \rangle~~~~~~$ $\langle 12, 14, 34, 24 , 23 \rangle~~$ $\langle 12, 42, 32, 31, 41 , 21, 23 \rangle$\\ \hline $\langle 12, 13, 43, 23 , 21 \rangle~~$ $\langle 12, 14, 34, 24 , 21 \rangle~~$ $\langle 12, 42, 32, 31, 41 , 21 \rangle$\\ \hline \end{tabular} \end{center} and such $4^*$-containers as follows: \begin{center} \begin{tabular}{|ll|} \hline $\langle 12, 13\rangle$& $\langle 12, 14 ,13\rangle$\\ $\langle 12, 42, 43 ,13\rangle$ & $\langle 12, 32, 34, 31, 41, 21, 24, 23, 13\rangle$ \\ \hline $\langle 12, 13, 43, 23, 24, 34\rangle$& $\langle 12, 14, 34\rangle$\\ $\langle 12, 32, 34\rangle$& $\langle 12, 42, 41, 21$, $31, 34\rangle$ \\ \hline $\langle 12, 13, 23 \rangle$& $\langle 12, 14, 24 , 23 \rangle$\\ $\langle 12, 32, 34 , 31, 21, 23 \rangle$& $\langle 12, 42, 41 , 43, 23 \rangle$ \\ \hline $\langle 12, 13, 43, 23 , 21 \rangle$& $\langle 12, 14, 34, 24 , 21 \rangle$\\ $\langle 12, 32, 31 , 21 \rangle$& $\langle 12, 42, 41 , 21 \rangle$ \\ \hline \end{tabular} \end{center} $ \Box $ \begin{lem}{\rm }\label{L2} Suppose that $n\ge 5, k\ge 2 $ and $n-k\ge 2$. Let $I=\{i_1, i_2, \cdots, i_m\}\subseteq \langle n\rangle$, then $A^{(i, I)}_{n, k}$ is Hamiltonian connected where $1\le i\le k$. \end{lem} \noindent{\bf Proof: } Let $u$ and $v$ be any two distinct vertices of $A^{(i, I)}_{n, k}$, we will prove that there exists a Hamiltonian path $P$ of $A^{(i, I)}_{n, k}$ joining $u$ and $v$. If $|I|=1$, by Lemma \ref{L1}, it is true. Therefore, we assume that $|I|\ge 2$ in the next proof. \noindent{\bf Case 1:} $(u)_i\not=(v)_i$. Without loss of generality, let $(u)_i=i_1$ and $(v)_i=i_m$. There exists at least $3$ edges between $A^{(i, i_j)}_{n, k}$ and $A^{(i, i_{j+1})}_{n, k}$ for all $1\le j\le m-1$ owning to $|E^{i=i_j, i_{j+1}}|=\frac{(n-2)!}{(n-k-1)!}\ge 3$. An edge $(x^{i_j}, y^{i_{j+1}})\in E^{i=i_j, i_{j+1}}$ is chosen such that $(x^{i_j})_i=i_j$, $(y^{i_{j+1}})_i=i_{j+1}$ and $x^1\not=u, y^m\not=v, x^{i_j}\not=y^{i_j}$ for all $j=1, 2, \cdots, m-1$. Let $y^1=u, x^m=v$. Since $A_{n, k}$ is Hamiltonian connected , there exists a Hamiltonian path $\langle y^{i_j}, P_j, x^{i_j}\rangle$ of $A^{(i, i_j)}_{n, k}$ joining $ y^{i_j}$ to $x^{i_j}$ for $1\le j\le m$. Hence, there exists a Hamiltonian path $P=\langle y^1, P_1, x^1, y^2, P_2, x^2, \cdots,y^m, P_m, x^m\rangle $ of $A^{(i, I)}_{n, k}$ joining $u$ to $v$. See figure \ref{L2_1} for illustration. \begin{figure} \caption{Illustration for case 1 of Lemma \ref{L2} \label{L2_1} \end{figure} \noindent{\bf Case 2:} $(u)_i=(v)_i$. Without loss of generality , let $(u)_i=(v)_i=i_1$. A vertex $x^1\in V(A^{(i, i_1)}_{n, k})\setminus\{u, v\}$ is chosen such that $i_2\notin\{(x^1)_j: j=1, 2, \cdots, k\}$. Obviously, $|\{x'\mid (x^1, x')\in E(A^{(i, i_1)}_{n, k})$ and $i_2\notin\{(x')_j: 1\le j\le k\} \}|=(n-k-1)(k-1)\ge 2 $ owing to $n\ge 5$. Hence, there exists at least two neighbors $y^1, z^1$ of $x^1$ such that $i_2\notin\{(y^1)_j: 1\le j\le k\}\cup\{(z^1)_j:1\le j\le k\}$. Let $e=(x^1, a)\in E(A^{(i, i_1)}_{n, k})$ and $a\notin\{y^1, z^1\}$. By Lemma \ref{LL1}, there exists a Hamiltonian path $P_1$ of $A^{(i, i_1)}_{n, k}-\bigcup_{x\in N_{A^{(i, i_1)}_{n, k}}(x^1)}\{(x^1, x)\}+\{(x^1, y^1),(x^1, z^1), e\}$ between $u$ and $v$. Note that the degree of $x^1$ in $P_1$ is two, then $\{(x^1, y^1),(x^1, z^1)\}\cap E(P_1)\ge 1$, Without loss of generality, let $(x^1, y^1)\in E(P_1)$, and we can represent $P_1$ as $\langle u, R_1, x^1, y^1, H_1, v\rangle$. Let $u^{j}={(x^{j-1})}^{s(i_{j-1}, i_j)}$ and $v^j={(y^{j-1})}^{s(i_{j-1}, i_j)}$ for $2\le j\le m$. Similarly, there exists a Hamiltonian path $P_j$ of $A^{(i, i_j)}_{n, k}$ such that $(x^j, y^j)\in E(P_j)$ and $i_{j+1}\notin \{(x^j)_l: 1\le l\le k\}\cup \{(y^j)_l: 1\le l\le k\}$ for all $2\le j\le m-1$. We can represent $P_j$ as $\langle u^j, R_j, x^j, y^j, H_j, v\rangle$ for $1\le j\le m-1$. By Lemma \ref{L1}, there exists a Hamiltonian path $P_m$ of $A^{(i, i_m)}_{n, k}$ between $u^m$ and $v^m$. Hence, there exists a Hamiltonian path $P=\langle u, R_1, x^1, u^2, R_2, x^2, \cdots, u^m, P_m, v^m, \cdots, y^2, H_2, v^2, y^1, H_1, v\rangle$ of $A^{(i, I)}_{n, k}$ joining $u$ to $v$. See figure \ref{L2_2} for illustration. \begin{figure} \caption{Illustration for case 2 of Lemma \ref{L2} \label{L2_2} \end{figure} $ \Box $ \begin{lem}{\rm }\label{L6} For $m\in \langle n\rangle$. Suppose that $A=\{u^1, u^2, \cdots, u^m\}, B=\{v^1, v^2, \cdots, v^m\}$, $A\cap B=\emptyset$, $A, B\subseteq V(A_{n, k})$. If there exists a number $t\in \langle k\rangle$ such that $(u^i)_t\not=(u^j)_t$, $(v^i)_t\not=(v^j)_t$ for $1\le i\not=j\le m$, then there exists $m$ disjoint paths $H_1, H_2, \cdots, H_m$ from $A$ to $B$ such that $V(\cup_{j=1}^{m} H_j)=V(A_{n, k})$. \end{lem} \noindent{\bf Proof: } We partite $A_{n, k}$ to $\cup_{i=1}^{n} A^{(t, i)}_{n, k}$. Suppose that $|\{(u^i)_t: 1\le i\le m\}\cap \{(v^i)_t: 1\le i\le m\}|=l$. Without loss of generality, we can assume that $(u^i)_t=(v^i)_t$ for $1\le i\le l$. By Lemma \ref{L2}, there exists a Hamiltonian path $H_i$ of $A^{(t, (u^i)_t)}_{n, k}$ joining $u^i$ to $v^i$ for $1\le i\le l$ and a Hamiltonian path $H_j$ of $A^{(t, \{(u^j)_t, (v^j)_t\})}_{n, k}$ joining $u^j$ to $v^j$ for $l+1\le j\le m-1$. Let $I=\langle n\rangle\setminus (\{(u^i)_t: 1\le i\le m-1\}\cup \{(v^i)_t: 1\le i\le m-1\})$. By Lemma \ref{L2}, there exists a Hamiltonian path $H_{m}$ of $A^{(i, I)}_{n, k}$ joining $u^m$ to $v^m$. Obviously, $H_1, H_2, \cdots, H_m$ form the desired paths. See figure \ref{L_6} for illustration. \begin{figure} \caption{Illustration for Lemma \ref{L6} \label{L_6} \end{figure} $ \Box $ \section{The super spanning connectivity of arrangement graphs} \begin{lem}{\rm }\label{TH1} Suppose that $n\ge 4, n-k\ge2$, then $A_{n, k}$ is $l^*$-connected for $(n-k)(k-1)+1\le l\le(n-k)k$. \end{lem} \noindent{\bf Proof. }We prove the Lemma by induction. \noindent{\bf Basis step:} Since $A_{n, 1}$ is isomorphic to the complete graph $K_n$ and by Lemma \ref{L5}, $A_{4,2}$ is super spanning connected. Thus, the result holds for $A_{n, 1}$ and $A_{4, 2}$. \noindent{\bf Induction step:} Suppose that $A_{n-1, k-1}$ is $(n-k)(k-1)^*$-connected. We need to find an $l^*$-container between any two different vertices $u$ and $v$ of $A_{n, k}$ with $k\ge 2$ for $(n-k)(k-1)+1\le l\le (n-k)k$. We use $U$ to denote the set $\{(u)_i: 1\le i\le k\}$ and $V$ to denote the set $\{(v)_i: 1\le i\le k\}$. \noindent{\bf Case 1:} $\{i\mid (u)_i=(v)_i: 1\le i\le k\}\not=\emptyset$. Without loss of generality, let $(u)_k=(v)_k=\alpha$. Suppose that: \begin{center} $\begin{array}{rl} U\cap V=&\{x_1, x_2, \cdots, x_t, \alpha\}, \\ U\setminus (U\cap V)=&\{u'_{t+1}, u'_{t+2}, \cdots, u'_{k-1}\}, \\ V\setminus (U\cap V)=&\{v'_{t+1}, v'_{t+2}, \cdots, v'_{k-1}\}, \\ \langle n\rangle\setminus (U\cup V)=&\{w_1, w_2, \cdots, w_{n+t-2k+1}\}. \end{array}$ \end{center} We partite $A_{n, k}$ to $\cup^n_{i=1}A^{(k, i)}_{n, k}$. By induction, there exists an $(n-k)(k-1)^*$-container $\{P_1, P_2, \cdots$, $ P_{(n-k)(k-1)}\}$ of $A^{(k, \alpha)}_{n, k}$ joining $u$ and $v$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_i$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(\alpha, w_i)}$ to $v^{s(\alpha, w_i)}$ for $1\le i\le n+t-2k+1$ and a Hamiltonian path $H_j$ of $A^{(k, v'_j)}_{n, k}\cup A^{(k, u'_j)}_{n, k}$ joining $u^{s(\alpha, v'_j)}$ to $v^{s(\alpha, u'_j)}$ for $t+1\le j\le k-1$. (a) $(n-k)(k-1)+1\le l\le (n-k)(k-1)+n+t-2k+1$. (If $n+t-2k+1=0$, then (a) does not occur. ) Let $l=(n-k)(k-1)+l'$, then $1\le l'\le n+t-2k+1$. We set $P_{(n-k)(k-1)+i}=\langle u, u^{s(\alpha, w_i)}, R_i, v^{s(\alpha, w_i)}, v \rangle$ for $1\le i\le l'-1$. By Lemma \ref{L2}, there exists a Hamiltonian path $H$ of $A^{(k, I)}_{n, k}$ joining $u^{s(\alpha, w_{l'})}$ to $v^{s(\alpha, w_{l'})}$ where $I=\langle n\rangle-\{\alpha, w_1, w_2, \cdots, w_{l'-1}\}$. We set $P_l=\langle u, u^{s(\alpha, w_{l'})}, H$, $v^{s(\alpha, w_{l'})}, v\rangle$. Obviously, $\{P_1, P_2, \cdots, P_l\}$ forms an $l^*$-container of $A_{n, k}$ joining $u$ to $v$. See figure \ref{TH1case1a} for illustration. \begin{figure} \caption{Illustration for case 1 (a) of Lemma \ref{TH1} \label{TH1case1a} \end{figure} (b) $(n-k)(k-1)+n+t-2k+2\le l\le (n-k)k$. (If $k-t=1$, then (b) does not occur. ) Let $l=(n-k)(k-1)+n+t-2k+1+l'$, then $1\le l'\le k-t-1$. We set $\begin{array}{rl} P_{(n-k)(k-1)+i}= & \langle u, u^{s(\alpha, w_i)}, R_i, v^{s(\alpha, w_i)}, v \rangle~for~ 1\le i\le n+t-2k+1 , \\ P_{(n-k)(k-1)+n-2k+1+j}= & \langle u, u^{s(\alpha, v'_j)}, H_j, v^{s(\alpha, u'_j)}, v \rangle~for~ t+1\le j\le t+l'-1. \end{array}$ \\ Let $I=\langle n\rangle\setminus\{\alpha, w_1, w_2, \cdots, w_{n+t-2k+1}, u'_{t+1}, \cdots, u'_{t+l'-1}, v'_{t+1}, \cdots, v'_{t+l'-1}\} $. By Lemma \ref{L2}, there exists a Hamiltonian $H'$ of $A^{(k, I)}_{n, k}$ joining $u^{s(\alpha, v'_{t+l'})}$ to $v^{s(\alpha, u'_{t+l'})}$. We set $P_l=\langle u, u^{s(\alpha, v'_{t+l'})}, H', v^{s(\alpha, u'_{t+l'})}, v \rangle$. Obviously, $\{P_1, P_2, \cdots, P_l\}$ forms an $l^*$-container of $A_{n, k}$ joining $u$ to $v$. See figure \ref{TH1case1b} for illustration. \begin{figure} \caption{Illustration for case 1 (b) of Lemma \ref{TH1} \label{TH1case1b} \end{figure} \noindent{\bf Case 2: } $\{i\mid (u)_i=(v)_i: 1\le i\le k\}=\emptyset$. \noindent{\bf Case 2.1 : } $\{i\mid (u)_i\in V, (v)_i\in U\}\not=\emptyset$ . Without loss of generality, we can assume that $(u)_k\in V, (v)_k\in U$. Let $(u)_k=\alpha, (v)_k=\beta$. Suppose that: \begin{center} $\begin{array}{rl} U\cap V=&\{ x_1, \cdots, x_t, \alpha, \beta\} , \\ U\setminus (U\cap V)=&\{u'_{t+1}, u'_{t+2}, \cdots, u'_{k-2}\} , \\ V\setminus (U\cap V)=&\{v'_{t+1}, v'_{t+2}, \cdots, v'_{k-2}\} , \\ \langle n\rangle\setminus (U\cup V)=&\{w_1, w_2, \cdots, w_{n+t-2k+2}\} . \end{array}$ \end{center} Since $n-k\ge 2$, there exists a element $\gamma\in\langle n\rangle\setminus V$. Set $y=x_1\cdots x_tv'_{t+1}\cdots v'_{k-2}\gamma \alpha$ and $z=x_1\cdots x_tv'_{t+1}\cdots v'_{k-2}\gamma \beta$. Thus, $u\not=y, z\not=v$. Let $S=\langle n\rangle\setminus\{\{(y)_i: 1\le i\le k\}\cup \{\beta\}\}=\langle n\rangle\setminus\{\{(z)_i: 1\le i\le k\}\cup\{\alpha\}\}=\{s_1, s_2, \cdots, s_{n-k-1}\}$. By induction, there exists an $(n-k)(k-1)^*$-container $\{P_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A^{(k, \alpha)}_{n, k}$ joining $u$ to $y$, and an $(n-k)(k-1)^*$-container $\{Q_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A^{(k, \beta)}_{n, k}$ joining $z$ to $v$. We can represent $P_{ij}$ as $\langle u, P'_{ij}, y^{ij}, y\rangle, Q_{ij}$ as $ \langle z, z^{ij}, Q'_{ij}, v\rangle$ for $1\le i\le k-1$ and $1\le j\le n-k$ where \begin{center} $ y^{ij}=\left\{ \begin{aligned} & y^{s((y)_i, s_j)}: 1\le i\le k-1, 1\le j\le n-k-1, \\ & y^{s((y)_i, \beta)}: 1\le i\le k-1, j= n-k, \end{aligned} \right.$ \end{center} \begin{center} $ z^{ij}=\left\{ \begin{aligned} & z^{s((y)_i, s_j)}: 1\le i\le k-1, 1\le j\le n-k-1, \\ & z^{s((y)_i, \alpha)}: 1\le i\le k-1, j= n-k. \end{aligned} \right.$ \end{center} Obviously, $(y^{ij}, z^{ij})\in E(A_{n, k})$ when $1\le i\le k-1$ and $1\le j\le n-k-1$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_{i}$ of $A^{(k, (y)_{i})}_{n, k}$ joining $(y^{i(n-k)})^{s(\alpha, (y)_i)}$ to $(z^{i(n-k)})^{s(\beta, (z)_i)}$ for $1\le i\le k-2$, As a result, there exists $(n-k)(k-1)$ internally disjoint paths $\{M_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A_{n, k}$ joining $u$ to $v$ such that \begin{center} $V(\displaystyle\bigcup_{i=1}^{k-1}\displaystyle\bigcup_{j=1}^{n-k} M_{ij})= V(A^{(k, \{x_1, \cdots, x_t, v'_{t+1}, \cdots, v'_{k-2}, \alpha, \beta\})}_{n, k})$ \end{center} where \begin{center} $ M^{ij}=\left\{ \begin{aligned} &\langle u, P'_{ij}, y^{ij}, z^{ij}, Q'_{ij}, v\rangle~for~1\le i\le k-1, 1\le j\le n-k-1, \\ &\langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(\alpha, (y)_i)}, R_i, (z^{ij})^{s(\beta, (z)_i)}, z^{ij}, Q'_{ij}, v\rangle~for~1\le i\le k-2, j=n-k, \\ &\langle u, P'_{ij}, y^{ij}, y, z, z^{ij}, Q'_{ij}, v\rangle~for~i=k-1, j=n-k. \end{aligned} \right.$ \end{center} See figure \ref{2_1_1} for illustration. \begin{figure} \caption{The $(n-k)(k-1)$ internally disjoint paths of $A_{n, k} \label{2_1_1} \end{figure} (a) $(n-k)(k-1)+1\le l\le (n-k)(k-1)+n+t-2k+2$. (If $n+t-2k+2=0$, then (a) does not occur. ) Let $l=(n-k)(k-1)+l'$, then $1\le l'\le n+t-2k+2$. By Lemma \ref{L2}, there exists a Hamiltonian path $R'_i$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(\alpha, w_i)}$ to $v^{s(\beta, w_i)}$ for $1\le i\le l'-1$. Let $I=\{ u'_{t+1}, \cdots, u'_{k-2}, w_{l'}, \cdots, w_{n+t-2k+2}\}$, by Lemma \ref{L2}, there exists a Hamiltonian path $R'_{l'}$ of $A^{(k, I)}_{n, k}$ joining $u^{s(\alpha, w_{l'})}$ to $v^{s(\beta, w_{l'})}$. Set $M_{ki}=\langle u, u^{s(\alpha, w_{i})}, R'_i, v^{s(\beta, w_i)}, v\rangle$ for $1\le i\le l'$. To construct the $l^*$-container, we only need to combine figure \ref{2_1_1} and $l'$ paths $M_{k1}, M_{k2}, \cdots, M_{kl'}$ in figure \ref{2_1_32}. \begin{figure} \caption{The paths $M_{k1} \label{2_1_32} \end{figure} (b) $(n-k)(k-1)+n+t-2k+3\le l\le (n-k)k$. (If $k-t=2$, then (b) is not happened. ) Let $l=(n-k)(k-1)+n+t-2k+2+l'$, then $1\le l'\le k-t-2$. In this case, we have $U\setminus(U\cap V)\not=\emptyset$. Note that $\gamma\in \langle n\rangle\setminus V$. Without loss of generality, we can assure that $\gamma=u'_{k-2}$. To construct the $l^*$-container, we need to \noindent{\bf Step 1: } By Lemma \ref{L2}, there exists a Hamiltonian path $R''_i$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(\alpha, w_i)}$ to $v^{s(\beta, w_i)}$ for $1\le i\le n+t-2k+2$. Combine figure \ref{2_1_1} and $(n+t-2k+2)$ paths $M'_{k1}, M'_{k2}, \cdots, M'_{k(n+t-2k+2)}$ in figure \ref{2_1_4} where \begin{center} $ M'_{ki}= \langle u, u^{s(\alpha, w_i)}, R''_i, v^{s(\beta, w_i)}, v\rangle~~for~~1\le i\le n+t-2k+2 $. \end{center} \begin{figure} \caption{The paths $M_{k1} \label{2_1_4} \end{figure} \noindent {\bf Step 2: } By Lemma \ref{L2}, there exists a Hamiltonian path $ H_{j}$ of $ A^{(k, u'_{t+j})}_{n, k}$ joining $(y^{(t+j)(n-k)})^{s(\alpha, u'_{t+j})}$ to $ v^{s(\beta, u'_{t+j})} $ and a Hamiltonian path $ H'_{j}$ of $ A^{(k, v'_{t+j})}_{n, k}$ joining $u^{s(\alpha, v'_{t+j})}$ to $(z^{(t+j)(n-k)})^{s(\beta, v'_{t+j})}$ for $1\le j\le l'-1$. Replace paths $M_{(t+1)(n-k)}, M_{(t+2)(n-k)}, \cdots, M_{(t+l'-1)(n-k)}$ in part \Rmnum{2} of figure \ref{2_1_1} by $M'_{(t+1)(n-k)}$, $M'_{(t+2)(n-k)}, \cdots$, $ M'_{(t+l'-1)(n-k)}$ and $M_1, M_2, \cdots$,$ M_{l'-1}$ as shown in figure \ref{2_1_5} where \begin{center} $\begin{array}{rl} M'_{(t+j)(n-k)}= & \langle u, P'_{(t+j)(n-k)}, y^{(t+j)(n-k)}, (y^{(t+j)(n-k)})^{s(\alpha, u'_{t+j})}, H_{j}, v^{s(\beta, u'_{t+j})}, v\rangle, \\ M_j= & \langle u, u^{s(\alpha, v'_{t+j})}, H'_j, (z^{(t+j)(n-k)})^{s(\beta, v'_{t+j})}, z^{(t+j)(n-k)}, Q'_{(t+j)(n-k)}, v\rangle \end{array}$ for $1\le j\le l'-1$. \end{center} \begin{figure} \caption{Illustration for step 2 of case 2.1(b) in Lemma \ref{TH1} \label{2_1_5} \end{figure} \noindent{\bf Step 3: } Let $I=\{u'_{t+l'}, u'_{t+l'+1}, \cdots, u'_{k-2}\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $H$ of $A^{(k, u'_{k-2})}_{n, k}$ joining $u^{s(\alpha, v'_{k-2})}$ to $(z^{(k-2)(n-k)})^{s(\beta, v'_{k-2})}$ and a Hamiltonian path $ R$ of $ A^{(k, I)}_{n, k}$ joining $(y^{(k-1)(n-k)})^{s(\alpha, u'_{k-2})}$ to $v^{s(\beta, u'_{k-2})}$. Replace $M_{(k-2)(n-k)}$ and $M_{(k-1)(n-k)}$ in part \Rmnum{2} and part \Rmnum{3} of figure \ref{2_1_1} by $M'_{(k-2)(n-k)}$, $M'_{(k-1)(n-k)}$ and $M_{l'}$ as shown in figure \ref{2_1_6} where \begin{center} $\begin{array}{rl} M'_{(k-2)(n-k)}=& \langle u, u^{s(\alpha, v'_{k-2})}, H, (z^{(k-2)(n-k)})^{s(\beta, v'_{k-2})}, z^{(k-2)(n-k)}, Q'_{(k-2)(n-k)}, v\rangle, \\ M'_{(k-1)(n-k)}= & \langle u, P'_{(k-1)(n-k)}, y^{(k-1)(n-k)}, (y^{(k-1)(n-k)})^{s(\alpha, u'_{k-2})}, R, v^{s(\beta, u'_{k-2})},v\rangle. \\ M_{l'}=&\langle u, P'_{(k-2)(n-k)}, y^{(k-2)(n-k)}, y, z, z^{(k-1)(n-k)}, Q'_{(k-1)(n-k)}, v\rangle. \end{array} $ \end{center} \begin{figure} \caption{Illustration for step 3 of case 2.1(b) in Lemma \ref{TH1} \label{2_1_6} \end{figure} \noindent{\bf Case 2.2 : } $\{i\mid (u)_i\notin V, (v)_i\in U\}\not=\emptyset$ or $(\{i\mid (v)_i\notin U, (u)_i\in V\}\not=\emptyset)$. Without loss of generality, let $(u)_k\notin V, (v)_k\in U$, and let $(u)_k=\alpha, (v)_k=\beta$. Suppose that \begin{center} $\begin{array}{rl} U\cap V= &\{ x_1, x_2, \cdots, x_t, \beta\}, \\ U\setminus (U\cap V)=& \{u'_{t+1}, u'_{t+2}, \cdots, u'_{k-2}, \alpha\}, \\ V\setminus (U\cap V)= &\{v'_{t+1}, v'_{t+2}, \cdots, v'_{k-1}\}, \\ \langle n\rangle\setminus (U\cup V)=&\{w_1, w_2, \cdots, w_{n+t-2k+1}\} . \end{array} $ \end{center} Without loss of generality, let $v=x_1x_2\cdots x_tv'_{t+1}\cdots v'_{k-1}\beta$. Since $n-k\ge 2$, there exists an element $\gamma\in\langle n\rangle\setminus V$. Set $y=x_1\cdots x_tv'_{t+1}\cdots v'_{k-2}\gamma \alpha$ and $z=x_1\cdots x_tv'_{t+1}\cdots v'_{k-2}\gamma \beta$. Thus, $u\not=y, z\not=v$. Let $S=\langle n\rangle\setminus\{\{(y)_i: 1\le i\le k\}\cup \{\beta\}\}=\langle n\rangle\setminus\{\{(z)_i: 1\le i\le k\}\cup\{\alpha\}\}=\{s_1, s_2, \cdots, s_{n-k-1}\}$. By induction, there exists an $(n-k)(k-1)^*$-container $\{P_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A^{(k, \alpha)}_{n, k}$ joining $u$ to $y$ and an $(n-k)(k-1)^*$-container $\{Q_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A^{(k, \beta)}_{n, k}$ joining $z$ to $v$. We can represent $ P_{ij}$ as $ \langle u, P'_{ij}, y^{ij}, y\rangle, $ $ Q_{ij}$ as $\langle z, z^{ij}, Q'_{ij}, v\rangle$ for $1\le i\le k-1$ and $1\le j\le n-k$ where \begin{center} $ y^{ij}=\left\{ \begin{aligned} & y^{s((y)_i, s_j)}: 1\le i\le k-1, 1\le j\le n-k-1, \\ & y^{s((y)_i, \beta)}: 1\le i\le k-1, j= n-k, \end{aligned} \right.$ \end{center} \begin{center} $ z^{ij}=\left\{ \begin{aligned} & z^{s((y)_i, s_j)}: 1\le i\le k-1, 1\le j\le n-k-1, \\ & z^{s((y)_i, \alpha)}: 1\le i\le k-1, j= n-k. \end{aligned} \right.$ \end{center} Obviously, $(y^{ij}, z^{ij})\in E(A_{n, k})$ when $1\le i\le k-1, 1\le j\le n-k-1$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_{i}$ of $A^{(k, (y)_{i})}_{n, k}$ joining $(y^{i(n-k)})^{s(\alpha, (y)_i)}$ to $(z^{i(n-k)})^{s(\beta, (z)_i)}$ for $1\le i\le k-2$, Then, there exists $(n-k)(k-1)$ internally disjoint paths $\{M_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A_{n, k}$ joining $u$ to $v$ such that \begin{center} $V(\displaystyle\bigcup_{i=1}^{k-1}\displaystyle\bigcup_{j=1}^{n-k} M_{ij})=V(A^{(k, \{\alpha, \beta, x_1, \cdots, x_t, v'_{t+1}, \cdots, v'_{k-2}\})}_{n, k})$. \end{center} where \begin{center} $ M^{ij}=\left\{ \begin{aligned} &\langle u, P'_{ij}, y^{ij}, z^{ij}, Q'_{ij}, v\rangle~for~1\le i\le k-1, 1\le j\le n-k-1, \\ &\langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(\alpha, (y)_i)}, R_i, (z^{ij})^{s(\beta, (z)_i)}, z^{ij}, Q'_{ij}, v\rangle~for~1\le i\le k-2, j=n-k, \\ &\langle u, P'_{ij}, y^{ij}, y, z, z^{ij}, Q'_{ij}, v\rangle~for~i=k-1, j=n-k. \end{aligned} \right.$ \end{center} See figure \ref{2_1_1} for illustration. (a) $(n-k)(k-1)+1\le l\le (n-k)(k-1)+n+t-2k+2$. Let $l=(n-k)(k-1)+l'$, then $1\le l'\le n+t-2k+2$. To construct the $l^*$-container, we need \noindent{\bf Step 1: }By Lemma \ref{L2}, there exists a Hamiltonian path $R'_i$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(\alpha, w_i)}$ to $v^{s(\beta, w_i)}$ for $1\le i\le l'-1$. Combine figure \ref{2_1_1} and $(l'-1)$ disjoint paths $M_{k1}, M_{k2}, \cdots, M_{k(l'-1)}$ in figure \ref{2_2_3} where $M_{ki}= \langle u, u^{s(\alpha, w_i)}, R'_i, v^{s(\beta, w_i)}, v\rangle$ for $1\le i\le l'-1$. \begin{figure} \caption{The paths $M_{k1} \label{2_2_3} \end{figure} \noindent{\bf Step 2:} Let $I=\langle n\rangle\setminus\{\alpha, \beta, x_1, \cdots, x_t, v'_{t+1}. \cdots, v'_{k-2}\}$. Then, there exists a Hamiltonian path $R'_{l'}$ of $A^{(k, I)}_{n, k}$ joining $u^{s(\alpha, v'_{k-1})}$ to $(z^{(k-1)(n-k)})^{s(\beta, v'_{k-1})}$. Replace $M_{(k-1)(n-k)}$ in part \Rmnum{3} of figure \ref{2_1_1} by $M'_{(k-1)(n-k)}$ and $M_{kl'}$ as shown in \ref{2_2_4} where \\ $ \begin{array}{rl} M_{kl'}= & \langle u, u^{s(\alpha, v'_{k-1})}, R'_{l'}, (z^{(k-1)(n-k)})^{s(\beta, v'_{k-1})}, z^{(k-1)(n-k)}, Q'_{(k-1)(n-k)}, v\rangle, \\ M'_{(k-1)(n-k)}= & \langle u, P'_{(k-1)(n-k)}, y^{(k-1)(n-k)}, y, z, v\rangle . \end{array}$ \begin{figure} \caption{Illustration for step 2 of case 2.2(a) in Lemma \ref{TH1} \label{2_2_4} \end{figure} (b)$(n-k)(k-1)+n+t-2k+3\le l\le (n-k)(k-1)+n-k$. (If $k-t=2$, then (b) is not happened. ) Let $l=(n-k)(k-1)+n+t-2k+2+l'$, then $1\le l'\le k-t-2$. Thus, $U\setminus \{(U\cap V)\cup \{\alpha\}\}\not=\emptyset$, Note that $\gamma\in \langle n\rangle\setminus V$. Without loss of generality, we can assure that $\gamma=u'_{k-2}$. In order to construct the $l^*$-container, we need to \noindent{\bf Step 1: } By Lemma \ref{L2}, there exists a Hamiltonian path $R''_i$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(\alpha, w_i)}$ to $v^{s(\beta, w_i)}$ for $1\le i\le n+t-2k+1$. Combine figure \ref{2_1_1} and $(n+t-2k+1)$ disjoint paths $M'_{k1}, M'_{k2}, \cdots, M'_{k(n+t-2k+1)}$ in \ref{2_2_5} where $M'_{ki}= \langle u, s^{s(\alpha, w_i)}, R''_{i}, v^{s(\beta, w_i)}, v\rangle$ for $1\le i\le n+t-2k+1 $. \begin{figure} \caption{The paths $M'_{k1} \label{2_2_5} \end{figure} \noindent{\bf Step 2:} By Lemma \ref{L2}, there exists a Hamiltonian path $H_{j}$ of $A^{(k, u'_{t+j})}_{n, k}$ joining $(y^{(t+j)(n-k)})^{s(\alpha, u'_{t+j})}$ to $v^{s(\beta, u'_{t+j})}$ and a Hamiltonian path $H'_{j}$ of $A^{(k, v'_{t+j})}_{n, k}$ joining $u^{s(\alpha, v'_{t+j})}$ to $(z^{(t+j)(n-k)})^{s(\beta, v'_{t+j})}$ for $1\le j\le l'-1$. Replace $M_{(t+1)(n-k)}, \cdots, M_{(t+l'-1)(n-k)}$ in part \Rmnum{2} of figure \ref{2_1_1} by $M''_{(t+1)(n-k)}$, $\cdots, M''_{(t+l'-1)(n-k)}$ and $M'_1, \cdots$, $M'_{l'-1}$ as shown in figure \ref{2_2_6} where \begin{center} $\begin{array}{rl} M'_{j}= & \langle u, u^{s(\alpha, v'_{t+j})}, H'_{j}, (z^{(t+j)(n-k)})^{s(\beta, v'_{t+j})}, z^{(t+j)(n-k)}, Q'_{(t+j)(n-k)}, v\rangle~for~1\le j\le l'-1, \\ M''_{(t+j)(n-k)}= & \langle u, P'_{(t+j)(n-k)}, y^{(t+j)(n-k)}, ( y^{(t+j)(n-k)})^{s(\alpha, u'_{t+j})} , H_{j}, v^{s(\beta, u'_{t+j})}, v\rangle ~for~1\le j\le l'-1.\\ \end{array}$ \end{center} \begin{figure} \caption{Illustration for step 2 of case 2.2(b) in Lemma \ref{TH1} \label{2_2_6} \end{figure} \noindent{\bf Step 3: } By Lemma \ref{L2}, there exists a Hamiltonian path $ H_{k-1}$ of $ A^{(k, u'_{k-2})}_{n, k}$ joining $(y^{(k-1)(n-k)})^{s(\alpha, u'_{k-2})}$ to $ v^{s(\beta, u'_{k-2})}$ and a Hamiltonian path $H'_{k-2}$ of $A^{(k, v'_{k-2})}_{n, k}$ joining $ u^{s(\alpha, v'_{k-2})}$ to $(z^{(k-2)(n-k)})^{s(\beta, v'_{k-2})}$. Let $I=\{v'_{k-1}, u'_{t+l'}, u'_{t+l'+1}, \cdots, u'_{k-2}\}$, then there exists a Hamiltonian path $H'_{k-1}$ of $A^{(k, I)}_{n, k}$) joining $ u^{s(\alpha, v'_{k-1})}$ to $(z^{(k-1)(n-k)})^{s(\alpha, v'_{k-1})}$. Replace $ M_{(k-2)(n-k)}, M_{(k-1)(n-k)}$ in part \Rmnum{2} and part \Rmnum{3} of figure \ref{2_1_1} by $ M''_{(k-2)(n-k)}$, $M''_{(k-1)(n-k)}$ and $M'_{l'}, M'_{l'+1}$ as shown in figure \ref{2_2_7} where \begin{flushleft} $\begin{array}{rl} M'_{l'}= &\langle u, u^{s(\alpha, v'_{k-2})}, H'_{k-2}, (z^{(k-2)(n-k)})^{s(\beta, v'_{k-2})}, z^{(k-2)(n-k)}, Q'_{(k-2)(n-k)}, v\rangle, \\ M'_{l'+1}=&\langle u, u^{s(\alpha, v'_{k-1})}, H'_{k-1}, (z^{(k-1)(n-k)})^{s(\beta, v'_{k-1})}, z^{(k-1)(n-k)}, Q'_{(k-1)(n-k)}, v\rangle, \\ M''_{(k-2)(n-k)}= & \langle u, P'_{(k-2)(n-k)}, y^{(k-2)(n-k)}, y, z, v\rangle, \\ M''_{(k-1)(n-k)}= &\langle u, P'_{(k-1)(n-k)}, y^{(k-1)(n-k)}, (y^{(k-1)(n-k)})^{s(\alpha, u'_{k-2})}, H_{k-1}, v^{s(\beta, u'_{k-2})}, v\rangle . \end{array}$ \end{flushleft} \begin{figure} \caption{Illustration for step 3 of case 2.2(b) in Lemma \ref{TH1} \label{2_2_7} \end{figure} \noindent{\bf Case 2.3 : } $U\cap V=\emptyset$. Let $u=u_1u_2\cdots u_k, v=v_1v_2\cdots v_k, \langle n\rangle\setminus( U\cup V)=\{w_1, w_2, \cdots, w_{n-2k}\}$. Set $y=v_1v_2\cdots v_{k-1}u_k, z=u_1u_2\cdots u_{k-1}v_k$. By induction, there exists an $(n-k)(k-1)^*$-container $\{P_{ij}: 1\le i\le k-1, 1\le j\le n-k\}$ of $A^{(k, u_k)}_{n, k}$ joining $u$ to $y$ and an $(n-k)(k-1)^*$-container $\{Q_{ij}: 1\le i\le n-k, 1\le j\le n-k\}$ of $A^{(k, v_k)}_{n, k}$ joining $z$ to $v$. We represent $P_{ij}$ as $\langle u, P'_{ij}, y^{ij}, y\rangle, Q_{ij}$ as $\langle z, z^{ij}, Q'_{ij}, v\rangle$ for $1\le i\le k-1, 1\le j\le n-k$ where \begin{center} $ y^{ij}=\left\{ \begin{aligned} & y^{s(v_i, u_j)}: 1\le j\le k-1, \\ & y^{s(v_i, v_k)}: j=k, \\ & y^{s(v_i, w_{j-k})}: k+1\le j\le n-k, \end{aligned} \right. $ $ z^{ij}=\left\{ \begin{aligned} & z^{s(u_i, v_j)}: 1\le j\le k-1, \\ & z^{s(u_i, u_k)}: j=k, \\ & z^{s(u_i, w_{j-k})}: k+1\le j\le n-k. \end{aligned} \right. $ \end{center} \noindent {\bf Subcase 2.3.1: } $k=2$ Without loss of generality, let $u=12, v=34$. When $n=5$, we set the $4^*$-container as : $~~~~~~~$ $\begin{array}{lll} \langle 12, 14, 34\rangle, ~~~~~~~~ & \langle12, 32, 34\rangle, ~~~~~~~~& \langle12, 52, 34\rangle, ~~~ \end{array}\\ $ $~~~~~~~~~~~~~~~\langle12, 42, 43, 13, 53, 23, 21, 31, 51, 41$, $45, 35, 15, 25, 24, 54, 34\rangle.$ Set the $5^*$-container as: $~~~~~~~$ $\begin{array}{ll} \langle 12, 14, 34\rangle,& \langle 12, 15, 25, 45, 35, 34\rangle, ~~~~~~~ \langle 12, 32, 34\rangle, \\ \langle 12, 52, 54, 34\rangle, & \langle 12, 42, 43, 23, 13, 53, 51, 41, 31, 21, 24, 34\rangle. \end{array}$ Set the $6^*$-container as: \\ $~~~~~~~~~~~~~$ $\begin{array}{lll} \langle 12, 14, 34\rangle, &\langle 12, 13, 53, 43, 23, 24, 34\rangle, & \langle 12, 15, 25, 45, 35, 34\rangle, \\ \langle 12, 32, 34\rangle, & \langle 12, 42, 41, 51, 21, 31, 34\rangle, & \langle 12, 52, 54, 34\rangle. \end{array} $\\ Now, let $n\ge 6$. (a) $l=(n-k)(k-1)+1=n-1$. Let $I=n\setminus \{2, 4\}$, by Lemma \ref{L2}, there exists a Hamiltonian path $R_1$ of $A^{(2, I)}_{n, 2}$ joining $41$ to $21$, we set\\ $\begin{array}{rlrl} P_1= & \langle 12, 32, 34\rangle, & P_2= & \langle 12, 14, 34\rangle ,\\ P_3=& \langle 12, 42, 41, R_1, 21, 24, 34\rangle,& P_{i}=&\langle 12, (i+1)2, (i+1)4, 34\rangle~~for~~ 4\le i\le n-1. \end{array} $\\ See figure \ref{2_3_1a} for illustration. \begin{figure} \caption{Illustration for subcase 2.3.1 (a) of Theorem \ref{TH1} \label{2_3_1a} \end{figure} (b) $l=(n-k)(k-1)+2=n$ By Lemma \ref{L2}, there exists a Hamiltonian path $R_1$ of $A^{(2, 1)}_{n, 2}$ joining $41$ to $31$. Let $I=\langle n\rangle\setminus\{1, 2, 4\}$, by Lemma \ref{L2}, there exists a Hamiltonian path $R_2$ of $A^{(2, I)}_{n, k}$ joining $13$ to $23$. We set\\ $\begin{array}{rlrl} P_1= & \langle 12, 32, 34\rangle, &P_2= & \langle 12, 14, 34\rangle ,\\ P_3=& \langle 12, 42, 41, R_1, 31, 34\rangle, &P_4=&\langle 12, 13, R_2, 23, 24, 34\rangle,\\ P_{i}=&\langle 12, i2, i4, 34\rangle~~for~~ 5\le i\le n. \end{array} $\\ See figure \ref{2_3_1b} for illustration. \begin{figure} \caption{Illustration for subcase 2.3.1 (b) of Theorem \ref{TH1} \label{2_3_1b} \end{figure} (c) $(n-k)(k-1)+3=n+1\le l\le (n-k)k= 2(n-2)$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_1$ of $A^{(2, 1)}_{n, 2}$ joining $41$ to $31$, a Hamiltonian path $R_2$ of $A^{(2, 3)}_{n, k}$ joining $13$ to $23$, and a Hamiltonian path $R_j$ of $A^{(2, j)}_{n, k}$ joining $1j$ to $3j$ for $5\le j\le l-n+3$. Let $I=\langle n\rangle\setminus\{1, 2, \cdots, l-n+3\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $H_{l-n+4}$ of $A^{(2, l-n+4)}_{n, k}$ joining $1(l-n+4)$ to $3(l-n+4)$. We set\begin{center} $\begin{array}{rl} P_1= & \langle 12, 32, 34\rangle,\\ P_2= & \langle 12, 14, 34\rangle ,\\ P_3=& \langle 12, 42, 41, R_1, 31, 34\rangle,\\ P_4=&\langle 12, 13, R_2, 23, 24, 34\rangle,\\ P_{i}=&\langle 12, i2, i4, 34\rangle~~for~~ 5\le i\le n,\\ P_{j}=&\langle 12, 1j, R_{j}, 3j, 34\rangle~~for~~ 5\le j\le l-n+3,\\ P_{l-n+4}=&\langle 12, 1(l-n+4), H_{l-n+4}, 3(l-n+4), 34\rangle . \end{array} $\end{center} See figure \ref{2_3_1c} for illustration. \begin{figure} \caption{Illustration for subcase 2.3.1 (c) of Theorem \ref{TH1} \label{2_3_1c} \end{figure} \noindent{\bf Subcase 2.3.2: } $k\ge 3$. (a)\ $l=(n-k)(k-1)+1$. Let \begin{center} $\begin{array}{rl} A_1=&\{(y^{12})^{s(u_k, v_1)}, \cdots, (y^{1(k-1)})^{s(u_k, v_1)}, (y^{1(k+1)})^{s(u_k, v_1)}, \cdots, (y^{1(n-k)})^{s(u_k, v_1)}\} ,\\ B_1=&\{(z^{12})^{s(v_k, v_1)}, \cdots, (z^{1(k-1)})^{s(v_k, v_1)}, (z^{1(k+1)})^{s(v_k, v_1)}, \cdots, (z^{1(n-k)})^{s(v_k, v_1)}\} ,\\ A_i=&\{(y^{i1})^{s(u_k, v_i)}, \cdots, (y^{i(n-k)})^{s(u_k, v_i)}\}~~for~~2\le i\le k-1, \\ B_i=&\{(z^{i1})^{s(v_k, v_i)}, \cdots, (z^{i(i-1)})^{s(v_k, v_i)}, (z^{(i-1)(i-1)})^{s(v_k, v_i)}, (z^{i(i+1)})^{s(v_k, v_i)}, \cdots , (z^{i(n-k)})^{s(v_k, v_i)}\}\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~for~~2\le i\le k-1. \end{array} $ \end{center} We partite $A^{(k, v_i)}_{n, k}$ to $\cup_{j\in\langle n\rangle\setminus \{v_i\}}A^{(i, j)(k, v_i)}_{n, k}$. By Lemma \ref{L6}, there exists $(n-k-2)$ disjoint paths $H_{12}, H_{13}, \cdots, H_{1(k-1)}, H_{1(k+1)}, \cdots, H_{1(n-k)}$ of $A^{(k, v_1)}_{n, k}$ from $A_1$ to $B_1$ such that $V(\cup_{j\in \langle n-k\rangle\setminus\{1, k\}}H_{1j})=V(A^{(k, v_1)}_{n, k})$ and $H_{1j}=\langle (y^{1j})^{s(u_k, v_1)}, H_{1j}, (z^{1j})^{s(v_k, v_1)}\rangle$ for $j\in \langle n-k\rangle\setminus \{1, k\}$. For $2\le i\le k-1$, there exists $(n-k)$ disjoint paths $H_{i1}, H_{i2}, \cdots, H_{i(n-k)}$ of $A^{(k, v_i)}_{n, k}$ from $A$ to $B$ such that $V(\cup_{j=1}^{n-k}H_{ij})=V(A^{(k, v_i)}_{n, k})$ and $H_{ii}=\langle (y^{ii})^{s(u_k, v_i)}, H_{ii}, (z^{(i-1)(i-1)})^{s(v_k, v_i)}\rangle$, $H_{ij}=\langle (y^{ij})^{s(u_k, v_i)}, H_{ij}, (z^{ij})^{s(v_k, v_i)}\rangle$ for $j\in \langle n-k\rangle\setminus\{i\}$. Let $I=\langle n\rangle \setminus\{v_1, \cdots, v_{k-1}, v_{k}$, $u_k\}$, by Lemma \ref{L2}, there exists a Hamiltonian path $H_{1k}$ of $A^{(k, I)}_{n,k}$ joining $(y^{1k})^{s(u_k, u_1)}$ to $(z^{1k})^{s(v_k, u_1)}$. We set $\begin{array}{l} M_{11}= \langle u, P'_{11}, y^{11}, y, v\rangle , \\ M_{1k}= \langle u, P'_{1k}, y^{1k}, (y^{1k})^{s(u_k, u_1)}, H_{1k}, (z^{1k})^{s(v_k, u_1)}, z^{1k}, Q'_{1k}, v\rangle ,\\ M_{1t}= \langle u, P'_{1t}, y^{1t}, (y^{1t})^{s(u_k, v_1)}, H_{1t}, (z^{1t})^{s(v_k, v_1)}, z^{1t}, Q'_{1t}, v\rangle~for~ 2\le t\le n-k~and~ t\not=k, \\ M_{ii}= \langle u, P'_{ii}, y^{ii}, (y^{ii})^{s(u_k, v_i)}, H_{ii}, (z^{(i-1)(i-1)})^{s(v_k, v_i)}, z^{(i-1)(i-1)}, Q'_{(i-1)(i-1)}, v\rangle~~for~2\le i\le k-1, \\ M_{ij} = \langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(u_k, v_i)}, H_{ij}, (z^{ij})^{s(v_k, v_i)}, z^{ij}, Q'_{ij}, v\rangle~for~2\le i\le k-1, \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1\le j\le n-k~and~j\not=i,\\ M_{k1}=\langle u, z, z^{(k-1)(k-1)}, Q'_{(k-1)(k-1)}, v\rangle. \end{array}$ See figure \ref{2_3_1} for illustration. \begin{figure} \caption{Illustration for subcase 2.3.2 (a) of Theorem \ref{TH1} \label{2_3_1} \end{figure} (b) $(n-k)(k-1)+2\le l\le (n-k)(k-1)+n-2k+1$. (If $n-2k=0$, then (b) does not occur) Let $l=(n-k)(k-1)+1+l'$ and let \begin{center} $\begin{array}{rl} A_1=&\{(y^{12})^{s(u_k, v_1)}, \cdots, (y^{1(n-k)})^{s(u_k, v_1)}\} ,\\ B_1=&\{(z^{12})^{s(v_k, v_1)}, \cdots, (z^{1(n-k)})^{s(v_k, v_1)}\} . \end{array}$\end{center} We partite $A^{(k, v_1)}_{n, k}$ to $\cup_{j\in\langle n\rangle\setminus \{v_1\}}A^{(1, j)(k, v_1)}_{n, k}$. To construct the $l^*$-container, we need \noindent{\bf Step 1: } By Lemma \ref{L6}, there exists $(n-k-1)$ disjoint paths $H'_{12}, H'_{13}, \cdots$, $H'_{1(n-k)}$ of $A^{(k, v_1)}_{n, k}$ from $A_1$ to $B_1$ such that $V(\cup_{j=2}^{n-k} H'_{1j})=V(A^{(k, v_1)}_{n, k})$ and $H'_{1j}=\langle (y^{1j})^{s(u_k, v_1)}, H'_{1j}, (z^{1j})^{s(v_k, v_1)}\rangle$ for $2\le j\le n-k$. Replace paths $M_{12}, M_{13}, \cdots, M_{1(n-k)}$ in figure \ref{2_3_1} by $M'_{12}, M'_{13}, \cdots, M'_{1(n-k)}$ as shown in figure \ref{2_3_4} where \begin{center} $ M'_{1j}= \langle u, P'_{1j}, y^{1j}, (y^{1j})^{s(u_k, v_1)}, H'_{1j}, (z^{1j})^{s(v_k, v_1)}, z^{1j}, Q'_{1j}, v\rangle~~for~~ j\in\langle n-k\rangle\setminus\{1\}.$ \end{center} \begin{figure} \caption{Illustration for step 1 of case 2.3(b) in Lemma \ref{TH1} \label{2_3_4} \end{figure} \noindent{\bf Step 2: } Let $I=\langle n\rangle \setminus\{v_1, \cdots, v_{k-1}, v_{k}, u_k, w_1, \cdots, w_{l'-1}\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_{i}$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(u_k, w_{i})}$ to $v^{s(v_k, w_{i})}$ for $1\le i \le l'-1$ and a Hamiltonian path $R_{l'}$ of $A^{(k, I)}_{n,k}$ joining $(u)^{s(u_k, w_{l'})}$ to $v^{s(v_k, w_{l'})}$. Combine figure \ref{2_3_1} and $l'$ paths $M_1, M_2, \cdots, M_{l'}$ in figure \ref{2_3_5} where \begin{center} $ M_{i}=\langle u, u^{s(u_k, w_{i})}, R_{i}, v^{s(v_k, w_{i})}, v\rangle$ for $ 1\le i\le l'. $ \end{center} \begin{figure} \caption{The paths $M_1, \cdots, M_{l'} \label{2_3_5} \end{figure} (c)\ $(n-k)(k-1)+n-2k+2\le (n-k)(k-1)+n-k$. Let $l=(n-k)(k-1)+n-2k+1+l'$, then $1\le l'\le k-1$. To construct the $l^*$-container, we need: \noindent{\bf Step 1: } By Lemma \ref{L2}, there exists a Hamiltonian path $R'_i$ of $A^{(k, w_i)}_{n, k}$ joining $u^{s(u_k, w_i)}$ to $v^{s(v_k, w_i)}$ for $1\le i\le n-2k$. Combine figure \ref{2_3_1} and $(n-2k)$ disjoint paths $M'_1, M'_{2}, \cdots, M'_{n-2k}$ as shown in figure \ref{2_3_6} where $M'_{i}=\langle u, u^{s(u_k, w_i)}, R'_i, v^{s(v_k, w_i)}, v\rangle$ for $1\le i\le n-2k$. \begin{figure} \caption{The paths $M'_1, \cdots, M'_{n-2k} \label{2_3_6} \end{figure} \noindent{\bf Step 2: } Notice $k\ge 3, n\ge 2k\ge 6$, then $\frac{(n-3)!}{(n-k-1)!}\ge 3$, so there exists at least $3$ edges between $A^{(1, u_1)(k, v_1)}_{n, k}$ and $A^{(1, u_1)(k, u_{k-1})}_{n, k}$. Let $(x^1, x^2)$ be an edge of $E(A^{(1, u_1)(k, v_1)}_{n, k}, A^{(1, u_1)(k, u_{k-1})}_{n, k})$ such that $x^1\in A^{(1, u_1)(k, v_1)}_{n, k}, x^2\in A^{(1, u_1)(k, u_{k-1})}_{n, k}$ and $x^1\not=u^{s(u_k, v_1)}, x^2\not=v^{s(v_k, u_{k-1})}$. By Lemma \ref{L2}, there exists a Hamiltonian path $R$ of $A^{(k, u_{k-1})}_{n, k}$ joining $x^2$ to $v^{s(v_k, u_{k-1})}$. Let \begin{center} $\begin{array}{rl} A_1= & \{ u^{s(u_k, v_1)}, (y^{12})^{s(u_k, v_1)}, \cdots, (y^{1(n-k)})^{s(u_k, v_1)}\} , \\ B_1= &\{ x^1, (z^{12})^{s(v_k, v_1)}, \cdots, (z^{1(n-k)})^{s(v_k, v_1)}\} . \end{array} $\end{center} By Lemma \ref{L6}, there exists $(n-k)$ disjoint paths $H''_{11}, H''_{12}, \cdots, H''_{1(n-k)}$ of $A^{(k, v_1)}_{n, k}$ from $A_1$ to $B_1$ such that $V(\cup_{i=1}^{n-k}H''_{1i})=V(A^{(k, v_1)}_{n, k})$ and $H''_{11}=\langle u^{s(u_k, v_1)}, H''_{11}, x^1\rangle, H''_{1i}=\langle (y^{1i})^{s(u_k, v_1)}, H''_{1i}, (z^{1i})^{s(v_k, v_1)}\rangle$ for $2\le i\le n-k$. Replace paths $M_{12}, M_{13}, M_{1(n-k)}$ in figure \ref{2_3_1} by paths $M''_{11}, M''_{12}, M''_{13}, \cdots, M''_{1(n-k)}$ as shown in figure \ref{2_3_7} where \begin{center} $\begin{array}{l} M''_{11}= \langle u, u^{s(u_k, v_1)}, H''_{11}, x^1, x^2, R, v^{s(v_k, u_{k-1})}, v\rangle , \\ M''_{1i}= \langle u, P'_{1i}, y^{1i}, (y^{1i})^{s(u_k, v_1)}, H''_{1i}, (z^{1i})^{s(v_k, v_1)}, z^{1i}, Q'_{1i}, v\rangle~for~ 2\le i\le n-k . \end{array}$ \end{center} \begin{figure} \caption{Illustration for step 2 of case 2.3(c) in Lemma \ref{TH1} \label{2_3_7} \end{figure} \noindent{\bf Step 3: }Let \begin{center} $\begin{array}{rl} A_{i}= &\{(y^{i1})^{s(u_k, v_i)}, \cdots, (y^{i(i-1)})^{s(u_k, v_i)}, u^{s(u_k, v_i)}, (y^{i(i+1)})^{s(u_k, v_i)}, \cdots, (y^{i(n-k)})^{s(u_k, v_i)}\} ~for~ 2\le i\le l'\\ B^{i}= & \{(z^{i1})^{s(v_k, v_i)}, \cdots, (z^{i(i-1)})^{s(v_k, v_i)}, (z^{(i-1)(i-1)})^{s(v_k, v_i)}, (z^{i(i+1)})^{s(v_k, v_i)}, \cdots, (z^{i(n-k)})^{s(v_k, v_i)}\} ~\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ for~ 2\le i\le l'. \\ \end{array} $\\ \end{center} For $2\le i\le l'$, we partite $A^{(k, v_i)}_{n, k}$ to $\cup_{r\in\langle n\rangle\setminus\{v_i\}}A^{(i, r)}_{n, k}$. By Lemma \ref{L6}, there exists $(n-k)$ disjoint paths $H''_{i1}, H''_{i2}, \cdots, H''_{i(n-k)}$ of $A^{(k, v_i)}_{n, k}$ from $A^i$ to $B^i$ such that $V(\cup_{j=1}^{n-k}H''_{ij})=V(A^{(k, v_i)}_{n, k})$ and $H''_{ii}=\langle u^{s(u_k, v_i)}, H''_{ii}, (z^{(i-1)(i-1)})^{s(v_k, v_i)}\rangle $, $H''_{ij}=$$\langle (y^{ij})^{s(u_k, v_i)}, H''_{ij}, (z^{ij})^{s(v_k, v_i)}\rangle$ for $j\in \langle n-k\rangle\setminus \{i\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $H_{ki}$ of $A^{(k, u_{i-1})}_{n, k}$ joining $(y^{ii})^{s(u_k, u_{i-1})}$ to $v^{s(v_k, u_{i-1})}$ for $2\le i\le l'$. Replace paths $M_{i1}, M_{i2}, \cdots, M_{i(n-k)}$ in figure \ref{2_3_1} by $M_{ki}, M''_{i1}, M''_{i2}, \cdots, M''_{i(n-k)}$ for $2\le i\le l'$ as shown in figure \ref{2_3_8} where \begin{center} $\begin{array}{rl} M_{ki}= & \langle u, P'_{ii}, y^{ii}, (y^{ii})^{s(u_k, u_{i-1})}, H_{ki}, v^{s(v_k, u_{i-1})}, v\rangle, \\ M''_{ii}=& \langle u, u^{s(u_k, v_i)}, H''_{ii}, (z^{(i-1)(i-1)})^{s(v_k, v_i)}, z^{(i-1)(i-1)}, Q'_{(i-1)(i-1)}, v\rangle, \\ M''_{ij}=&\langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(u_k, v_i)}, H''_{ij}, (z^{ij})^{s(v_k, v_i)}, z^{ij}, Q'_{ij}, v\rangle. \end{array} $ \\ for $2\le i\le l', 1\le j\le n-k~and~j\not=i. $ \end{center} \begin{figure} \caption{Illustration for step 3 of case 2.3(c) in Lemma \ref{TH1} \label{2_3_8} \end{figure} $ \Box $ \begin{thm}{\rm }\label{TH2} $A_{n, k}$ is super spanning connected for $n\ge 4, n-k\ge2$. \end{thm} {\bf Proof}: We prove the theorem by induction. \noindent{\bf Basis step:} It is known that $A_{n, 1}$ is isomorphic to the complete graph $K_n$ and by Lemma \ref{L5}, $A_{4,2}$ is super spanning connected. Then, the result holds for $A_{n, 1}$ and $A_{4, 2}$. \noindent{\bf Induction step:} Suppose $A_{n-1, k-1}$ is super spanning connected. We need to prove that $A_{n, k}$ is super spanning connected for $n\ge 5, n-k\ge2$. By Lemma \ref{L1}, $A_{n, k}$ is $1^*$-connected and $2^*$-connected. By Lemma \ref{TH1}, $A_{n, k}$ is $l^*$-connected for $(n-k)(k-1)+1\le l\le (n-k)k$. Now, we need to construct a $l^*$-container of $A_{n, k}$ joining any two distinct vertices $u$ and $v$ for $3\le l\le (n-k)(k-1)$. We use $U$ to denote the set $\{(u)_i\mid 1\le i \le k\}$ and use $V$ to denote the set $\{(v)_i\mid 1\le i\le k\}$. \noindent{\bf Case 1:} $\{i\mid (u)_i=(v)_i: 1\le i\le k\}\not=\emptyset$. Without loss of generality, let $(u)_k=(v)_k=\alpha$. By induction, there is an $l^*$-container $\{P_1, P_2, \cdots, P_l\}$ of $A^{(k, \alpha)}_{n, k}$ joining $u$ to $v$. Hence, we can represent $P_l$ as $\langle u, y, P'_l, v\rangle$. Note that $|\{(u)_i: 1\le i\le k \}\cup \{(y)_i: 1\le i\le k \}|=k+1$ and $n-k\ge 2$. Suppose $\beta\in\langle n\rangle\setminus\{\{(u)_i: 1\le i\le k \}\cup \{(y)_i: 1\le i\le k \}\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $H$ of $A^{(k, \langle n\rangle\setminus\{\alpha\})}_{n, k}$ joining $u^{s(\alpha, \beta)}$ to $y^{s(\alpha, \beta)}$. We set $P''_l=\langle u, u^{s(\alpha, \beta)}, H, y^{s(\alpha, \beta)}, y, P'_l, v\rangle$. Obviously, $\{P_1, P_2, \cdots, P_{l-1}, P''_l\}$ is a $l^*$-container of $A_{n, k}$ joining $u$ to $v$. See figure \ref{TH2case1} for illustration. \begin{figure} \caption{Illustration for case 1 of Theorem \ref{TH2} \label{TH2case1} \end{figure} \noindent{\bf Case 2:} $\{i\mid (u)_i=(v)_i: 1\le i\le k\}=\emptyset$. \noindent{\bf Case 2.1:} $U\not=V$. Without loss of generality, we can assume that $(u)_k=\alpha\notin V$. We partite $A_{n, k}$ to $\cup_{i\in \langle n\rangle}A^{(k, i)}_{n, k}$. Suppose $u=u_1u_2\cdots u_{k-1}\alpha, v=v_1v_2\cdots v_{k-1}\beta$. Set $y=v_1v_2\cdots v_{k-1}\alpha$, then $y\not=u$ and $(y, v)\in E(A_{n, k})$. By induction, there exists an $l^*$-container $\{P_1, P_2, \cdots, P_l\}$ of $A^{(k, \alpha)}_{n, k}$ joining $u$ to $y$. We represent $P_i$ as $\langle u, P'_i, y^i, y\rangle$ for $1\le i\le l$. Without loss of generality , we can assume that $V(P_l)\le V(P_i)$ for $1\le i\le l$. Suppose $|\{y^i\mid \beta\in\{(y^i)_j: 1\le j\le k-1\}, 1\le i\le l-1\}|=m$, then $0\le m\le$ min $\{l-1, k-1\}$. \noindent{\bf Subcase 2.1.1} $m=0$. For all $1\le i\le l-1$, we have $\beta\notin\{(y^i)_j: 1\le j\le k-1\}$. Let $z^i=(y^i)^{s(\alpha, \beta)}$ for $1\le i\le l-2$. Then, $z^i\in A^{(k, \beta)}_{n,k}$ and $(z^i, v)\in E(A_{n, k})$ for $1\le i\le l-2$. Note that $l-3\le (n-k)(k-1)-3$, by Lemma \ref{L1}, there exists a Hamiltonian path $R$ of $A^{(k, \beta)}_{n, k}\setminus\{z^1, z^2, \cdots, z^{l-3}\}$ joining $z^{l-2}$ to $v$. Since $y^{l-1}$ and $y$ differ in exactly one position, we can assume that $y^{l-1}=v_1\cdots v_{r-1} x v_{r+1}\cdots v_{k-1}\alpha$ where $x\in \langle n\rangle\setminus\{v_1, v_2, \cdots, v_{k-1}, \alpha, \beta\}$. Let $I=\langle n\rangle\setminus\{\alpha, \beta\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $H$ of $A^{(k, I)}_{n, k}$ joining $(y^{l-1})^{s(\alpha, v_r)}$ to $v^{s(\beta, x)}$. We set\\ $\begin{array}{rl} M_i= & \langle u, P'_i, y^i, z^i, v\rangle ~for~1\le i\le l-3, \\ M_{l-2}=& \langle u, P'_{l-2}, y^{l-2}, z^{l-2}, R, v\rangle, \\ M_{l-1}= & \langle u, P'_{l-1}, y^{l-1}, (y^{l-1})^{s(\alpha, v_r)}, H, v^{s(\beta, x)}, v\rangle, \\ M_{l}= & \langle u, P'_l, y^{l}, y, v\rangle. \end{array} $\\ Obviously, $\{M_1, M_2, \cdots, M_l\}$ forms an $l^*$-container of $A_{n, k}$. See figure \ref{3_2_1} for illustration. \begin{figure} \caption{Illustration for subcase 2.1.1 of Theorem \ref{TH2} \label{3_2_1} \end{figure} \noindent{\bf Subcase 2.1.2} $0<m<l-1$. Without loss of generality, we can assume that $y^i=v_1\cdots v_{i-1}\beta v_{i+1}\cdots v_{k-1}\alpha$ for $1\le i\le m$. Set $z^{i}=v_1\cdots v_{i-1}\alpha v_{i+1}\cdots v_{k-1}\beta$ for $1\le i\le m$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_i$ of $A^{(k, v_i)}_{n, k}$ joining $(y^i)^{s(\alpha, v_i)}$ to $(z^i)^{s(\beta, v_i)}$ for $1\le i\le m-1$. Since $n-k\ge 2$, suppose $\gamma\in\langle n\rangle\setminus\{v_1, \cdots, v_{k-1}, \alpha, \beta\}$. Let $I=\langle n\rangle\setminus\{v_1, \cdots, v_{m-1}, \alpha, \beta\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_m$ of $A^{(k, I)}_{n, k}$ joining $(y^{m})^{s(\alpha, v_m)}$ to $v^{s(\beta, \gamma)}$. Notice that $\beta\notin\{(y^i)_j: 1\le j\le k-1\}$ for $m+1\le i\le l-1$. We set $z^i=(y^i)^{s(\alpha, \beta)}$ for $m+1\le i\le l-1$. Then, $(y^i, z^i)\in E(A_{n, k})$ for $m+1\le i\le l-1$. Since $l-3\le (n-k)(k-1)-3$, by Lemma \ref{L1}, there exists a Hamiltonian path $R$ of $A^{(k, \beta)}_{n, k}\setminus\{v_1, \cdots, v_{m-1}, v_{m+1}, \cdots, v_{l-2}\}$ joining $z^{l-1}$ to $v$. We set\\ $ \begin{array}{rl} M_{i}= & \langle u, P'_i, y^i, (y^i)^{s(\alpha, v_i)}, R_i, (z^{i})^{s(\beta, v_i)}, z^i, v\rangle~for~1\le i\le m-1, \\ M_{m}= & \langle u, P'_m, y^m, (y^{m})^{s(\alpha, v_m)}, R_m, v^{s(\beta, \gamma)}, v\rangle, \\ M_{m+j}= & \langle u, P'_{m+j}, y^{m+j}, z^{m+j}, z\rangle~for~1\le j\le l-m-2, \\ M_{l-1}= & \langle u, P'_{l-1}, y^{l-1}, z^{l-1}, R, v\rangle, \\ M_l=& \langle u, P'_{l}, y^l, y, v\rangle. \end{array} $\\ Obviously, $\{M_1, M_2, \cdots, M_l\}$ forms an $l^*$-container of $A_{n, k}$. See figure \ref{3_2_2} for illustration. \begin{figure} \caption{Illustration for subcase 2.1.2 of Theorem \ref{TH2} \label{3_2_2} \end{figure} \noindent{\bf Subcase 2.1.3} $m=l-1$. Without loss of generality, we can assume $y^i=v_1\cdots v_{i-1}\beta v_{i+1}\cdots v_{k-1}\alpha$ for $1\le i\le l-1$. Set $z^{i}=v_1\cdots v_{i-1}\alpha v_{i+1}\cdots v_{k-1}\beta$ for $1\le i\le l-1$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_i$ of $A^{(k, v_i)}_{n, k}$ joining $(y^i)^{s(\alpha, v_i)}$ to $(z^i)^{s(\beta, v_i)}$ for $1\le i\le l-2$. Since $n-k\ge 2$, suppose $\gamma\in\langle n\rangle\setminus\{v_1, \cdots, v_{k-1}, \alpha, \beta\}$. Let $I=\langle n\rangle\setminus\{v_1, \cdots, v_{l-2}, \alpha, \beta\}$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_m$ of $A^{(k, I)}_{n, k}$ joining $(y^{l-1})^{s(\alpha, v_{l-1})}$ to $(z^{l-1})^{s(\beta, \gamma)}$. Since $1\le l-2=m-1\le k-2\le(n-k)(k-1)-3$, by Lemma \ref{L2}, there exists a Hamiltonian path $R$ of $A^{(k, \beta)}_{n, k}$ joining $z^{l-1}$ to $v$. We set $ \begin{array}{rl} M_i= & \langle u, P'_i, y^i, (y^i)^{s(\alpha, v_i)}, R_i, (z^i)^{s(\beta, v_i)}, z^i, v\rangle ~for~1\le i\le l-2, \\ M_{l-1}= & \langle u, P'_{l-1}, y^{l-1}, (y^{l-1})^{s(\alpha, v_{l-1})}, R_{l-1}, (z^{l-1})^{s(\beta, v_{l-1})}, z^{l-1}, R, v\rangle, \\ M_{l}=& \langle u, P'_l, y^l, y, v\rangle. \end{array} $\\ Obviously, $\{M_1, M_2, \cdots, M_l\}$ forms an $l^*$-container of $A_{n, k}$. See figure \ref{3_2_3} for illustration. \begin{figure} \caption{Illustration for subcase 2.1.3 of Theorem \ref{TH2} \label{3_2_3} \end{figure} \noindent{\bf Case 2.2} $U=V$. \noindent{\bf Subcase 2.2.1} $k=2$. Without loss of generality, we can assume that $u=12, v=21, n\ge 5$ . By Lemma \ref{L2}, there exists a Hamiltonian path $R_1$ of $A^{(2, 2)}_{n, 2}$ joining $12$ to $n2$, a Hamiltonian path $R_2$ of $A^{(2, 1)}_{n, 2}$ joining $(n-1)1$ to $21$ and a Hamiltonian path $R_3$ of $A^{(2, 3)}_{n, k}$ joining $n3$ to $(n-1)3$. Additionally, there exists a Hamiltonian path $R_i$ of $A^{(2, i)}_{n, 2}$ joining $1i$ to $2i$ for $4\le i\le l-1$ and a Hamiltonian path $R$ of $A^{(k, \{l, l+1, \cdots, n\})}_{n, 2}$ joining $1l$ to $2n$. We set \\ $\begin{array}{rl} M_1= & \langle 12, R_1, n2, n3, R_3, (n-1)3, (n-1)1, R_2, 21\rangle, \\ M_i= &\langle 12, 1(i+2), R_{i+2}, 2(i+2), 21\rangle~ for ~2\le i\le l-1,\\ M_{l}=&\langle 12, 1l, R, 2n, 21\rangle. \end{array} $\\ Obviously, $\{M_1, M_2, \cdots, M_l\}$ forms an $l^*$-container of $A_{n, 2}$. See figure \ref{3_2_4} for illustration. \begin{figure} \caption{Illustration for subcase 2.2.1 of Theorem \ref{TH2} \label{3_2_4} \end{figure} \noindent{\bf Subcase2.2.2} $k\ge 3$. Without loss of generality, we can assume that $v=12\cdots k$, $u=u_1u_2\cdots u_{k-1}r$ and $r=1$. Let $y=k23\cdots(k-1)1$. By induction, there exists an $l^*$-container $\{P_1, P_2, \cdots, P_{l}\}$ joining $u$ to $y$. We represent $P_i$ as $\langle u, P'_i, y^i, y\rangle$ for $1\le i\le l$. If $|\{y^i\mid (y^i)_1\not=k\}|=1$, we may assume that $(y^l)_1\not=k$. If $|\{y^i\mid (y^i)_1\not=k\}|=2$, we may assume that $(y^{l-1})_1\not=k, (y^l)_1\not=k$. Let $Y=Y^1\cup Y^2\cup \cdots \cup Y^{k-1}$ be the neighbors of $y$ in $P_i$ for $1\le i\le l-2$ where $Y^j=\{y^{j1}, y^{j2}, \cdots, y^{j{n_j}}\}$ are obtained by switch the $j$th coordinate of $y$. Then, $l-2=n_1+n_2+\cdots+n_{k-1}$. Now, we use $~\{P_{11}, P_{12}, \cdots, P_{1n_1}, P_{21}, P_{22}, \cdots, P_{2n_2}, \cdots, P_{(k-1)1}, P_{(k-1)2}, \cdots, P_{(k-1)n_{k-1}}, P_{l-1}, P_{l}\}$ to denote the $l^*$-container where $P_{ij}=\langle u, P'_{ij}, y^{ij}, y\rangle$ for $i\in\langle k-1\rangle, j\in\langle n_i\rangle$ and $P_{l-1}=\langle u, P'_{l-1}, y^{l-1}, y\rangle$, $ P_{l}=\langle u, P'_l, y^l, y\rangle$. We use $z^{ij}$ to denote the vertex $z^{s(i, (y^{ij})_i)}$ for $i=2, \cdots, k-1$( for example: if $y^{ij}=k2\cdots (i-1)x(i+1)\cdots(k-1)1$, then $z^{ij}=12\cdots(i-1)x(i+1)\cdots k$ ). Thus, $(z^{ij}, v)\in E(A_{n, k})$. For $i\in \langle k\rangle\setminus\{1, r\}$, if $n_i\not=0$, we partite $A^{(k, i)}_{n, k}$ to $\cup_{j\in \langle n\rangle\setminus\{i\}}A^{(i, j)(k, i)}_{n, k}$. Let \begin{center} $ \begin{array}{cc} A^i= & \{(y^{i1})^{s(1, i)}, (y^{i2})^{s(1, i)}, \cdots, (y^{in_i})^{s(1, i)}\}, \\ B^i= & \{(z^{i1})^{s(k, i)}, (z^{i2})^{s(k, i)}, \cdots, (z^{in_i})^{s(k, i)}\} \end{array}$ \end{center} By Lemma \ref{L6}, there exists $n_i$ disjoint paths $H_{i1}, H_{i2}, \cdots, H_{in_i}$ from $A^i$ to $B^i$ such that \begin{center} $V(\displaystyle\bigcup_{j=1}^{n_i}H_{ij})=V(A^{(k, i)}_{n, k})$ and $H_{ij}=\langle (y^{ij})^{s(1, i)}, H_{ij}, (z^{ij})^{s(k, i)}\rangle$ for $1\le j\le n_j$. \end{center} (a) $\{y^i\mid (y^i)_1\not=(y)_1\}\le 2$. Note that $l-2=n_1+n_2+\cdots n_{k-1}\ge 1$ and $n_1=0$. Without loss of generality, we can assume that $n_{k-1}\not=0$. Since $l-3\le (n-k)(k-1)-3$, by Lemma \ref{L1}, there exists a Hamiltonian path $R$ of $A^{(k, k)}_{n, k}\setminus \{z^{21}, \cdots, z^{2n_2}, \cdots, z^{(k-1)1}, \cdots, z^{(k-1)(n_{k-1}-1)}\}$ joining $z^{(k-1)n_{k-1}}$ to $v$. We set \begin{center} $M_{ij}=\left\{ \begin{aligned} & \langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(1, i)}, H_{ij}, (z^{ij})^{s(k, i)}, z^{ij}, v \rangle: i=2, 3, \cdots, k-2, j=1, 2, \cdots, n_i , \\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~or~~ i=k-1, 1\le j\le n_{k-1}-1,\\ & \langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(1, i)}, H_{ij}, (z^{ij})^{s(k, i)}, z^{ij}, R, v \rangle: i=k-1, j=n_{k-1}. \end{aligned} \right.$ \end{center} Since $n-k\ge2$, let $a\in \langle n\rangle\setminus\{\{(y)_i: 1\le i\le k\}\cup\{(y^{l-1})_i: 1\le i\le k\}\}, b\in \langle n\rangle\setminus\{1, 2, \cdots, k, a \}$. Let $I=\langle n\rangle\setminus(\{i\mid n_i\not=0: 2\le i\le k-1\}\cup\{k, a\})$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_{l-1}$ of $A^{(k, a)}_{n, k}$ joining $(y^{l-1})^{s(1, a)}$ to $v^{s(k, a)}$ and a Hamiltonian path $R_l$ of $A^{(k, I)}_{n, k}$ joining $y^{s(1, b)}$ to $v^{s(k, b)}$. We set $M_{l-1}=\langle u, P'_{l-1}, y^{l-1}, (y^{l-1})^{s(1, a)}, R_{l-1}, v^{s(k, a)}, v\rangle $, $M_l=\langle u, P'_l ,y^l, y, y^{s(1, b)}, R_{l}, v^{s(k, b)}, v\rangle$. \\ Obviously, $\{M_{21}, \cdots, M_{2n_2}, \cdots, M_{(k-1)1}, \cdots, M_{(k-1)}n_{k-1}, M_{l-1}, M_{l}\}$ forms an $l^*$-container of $A_{n, k}$. See figure \ref{3_2_5} for illustration. \begin{figure} \caption{Illustration for subcase 2.2.2(a) of Theorem \ref{TH2} \label{3_2_5} \end{figure} (b) $\{y^i\mid (y^i)_1\not=(y)_1\}\ge 3$. Note that $n_2+\cdots n_{k-1}\le l-3\le (n-k)(k-1)-3$ . By Lemma \ref{L1}, there exists a Hamiltonian path $R$ of $A^{(k, k)}_{n, k}\setminus \{z^{21}, \cdots, z^{2n_2}, \cdots, z^{(k-1)1}, \cdots, z^{(k-1)n_{k-1}}\}$ joining $z^{11}$ to $v$. Let $X=\langle n\rangle\setminus \langle k\rangle=\{x_1, x_2, \cdots, x_{n-k}\}$, Without loss of generality, we may assume that $y^{1i}=y^{s(k, x_i)}$ for $1\le i\le n_1$, $y^{l-1}=y^{s(k, x_{n_1+1})}, y^l=y^{s(k, x_{n_1+2})}$ and $z^{11}=v^{s(1, x_{n_1+1})}$. By Lemma \ref{L2}, there exists a Hamiltonian path $R_i$ of $A^{(k, x_{i+1})}_{n, k}$ joining $(y^{1i})^{s(1, x_{i+1})}$ to $v^{s(k, x_{i+1})}$ for $1\le i\le n_1$ and a Hamiltonian path $R_{l-1}$ of $A^{(k, x_{n_1+2})}_{n, k}$ joining $(y^{l-1})^{s(1, x_{n_1+2})}$ to $v^{s(k, x_{n_1+2})}$. Let $I=\langle n\rangle\setminus\{\langle k\rangle\cup \{x_2, x_3, \cdots, x_{n_1+2}\}\}$. By Lemma \ref{L2}, there exists a Hamiltonian $ $ path $R_l$ of $A^{(k, I)}_{n, k}$ joining $(y)^{s(1, x_1)}$ to $(z^{11})^{s(k, x_1)}$. We set $\begin{array}{rl} M_{1t}=&\langle u, P'_{1t}, y^{1t}, (y^{1t})^{s(1, x_{t+1})}, R_t, v^{s(k, x_{t+1})}, v\rangle$ for $1\le t\le n_1 , \\ M_{ij}=& \langle u, P'_{ij}, y^{ij}, (y^{ij})^{s(1, i)}, H_{ij}, (z^{ij})^{s(k, i)}, z^{ij}, v \rangle~for~ 2\le i\le k-1, 1\le j\le n_i.\\ M_{l-1}=&\langle u, P'_{l-1}, y^{l-1}, (y^{l-1})^{s(1, x_{n_1+2})}, R_{l-1}, v^{s(k, x_{n_1+2})}, v\rangle ,\\ M_l=&\langle u, y^l, y, y^{s(1, x_1)}, R_{l}, (z^{11})^{s(k, x_1)}, z^{11}, R, v\rangle. \end{array} $\\ Obviously, $\{M_{11}, \cdots, M_{1n_1}, M_{21}, \cdots, M_{2n_2}, \cdots, M_{(k-1)1}, \cdots, M_{(k-1)}n_{k-1}, M_{l-1}, M_{l}\}$ forms an $l^*$-container of $A_{n, k}$. See figure \ref{3_2_6} for illustration. \begin{figure} \caption{Illustration for subcase 2.2.2(b) of Theorem \ref{TH2} \label{3_2_6} \end{figure} $ \Box $ \end{CJK*} \end{document}

\begin{document} \title{New method to simulate quantum interference using deterministic processes and application to event-based simulation of quantum computation } \def\ORDER#1{\hbox{${\cal O}(#1)$}} \def\BRA#1{\langle #1 \vert} \def\KET#1{\vert #1 \rangle} \def\EXPECT#1{\langle #1 \rangle} \def\BRACKET#1#2{\langle #1 \vert #2 \rangle} \def{\mathchar'26\mskip-9muh}{{\mathchar'26\mskip-9muh}} \def{\mathop{\hbox{mod}}}{{\mathop{\hbox{mod}}}} \def{\mathop{\hbox{CNOT}}}{{\mathop{\hbox{CNOT}}}} \def{\mathop{\hbox{Tr}}}{{\mathop{\hbox{Tr}}}} \def{\mathbf{\Psi}}{{\mathbf{\Psi}}} \def{\mathbf{\Phi}}{{\mathbf{\Phi}}} \def{\mathbf{0}}{{\mathbf{0}}} \def\Eq#1{(\ref{#1})} \def\NOBAR#1{#1} \def\BAR#1{\overline{#1}} \def\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}} \defDLM{DLM} \defDLMS{DLMs} \section{Introduction} Computer simulation is a powerful methodology to model physical phenomena~\cite{LAND00}. However, some of the most fundamental experiments in quantum physics~\cite{GRAN86,TONO98} have not been simulated in the event-by-event manner in which the experimental observations are actually recorded~\cite{PerfectExperiments}. In experiments the detection of events appears to be random~\cite{GRAN86,TONO98}, in a sense which, as far as we know, has not been studied systematically. Quantum theory gives us a recipe to compute the frequency of the observation of events but it does not describe individual events, such as the arrival of a single electron at a particular position on the detection screen~\cite{TONO98,HOME97,FEYN65,BALL03}. Reconciling the mathematical formalism (that does not describe single events) with the experimental fact that each observation yields a definite outcome is often referred to as the quantum measurement paradox. This is a central, fundamental problem in the foundation of quantum theory~\cite{FEYN65,HOME97,PENR90}. Therefore, it is not a such a surprise that within the framework of quantum theory, no algorithm has been found to perform an event-based simulation of quantum phenomena. From a computational viewpoint, quantum theory provides us with a set of rules (algorithms) to compute probability distributions~\cite{HOME97,KAMP88,QuantumTheory}. Therefore we may wonder what kind of algorithm(s) we need to perform an event-based simulation of the experiments~\cite{GRAN86,TONO98} mentioned above without using the machinery of quantum theory. Evidently, the present formulation rules out any method based on the solution of the (time-dependent) Schr{\"o}dinger equation and we have to step outside the framework that quantum theory provides. In this paper we demonstrate that locally-connected networks of processing units with a primitive learning capability are sufficient to simulate deterministically and event-by-event, the single-photon beam splitter and Mach-Zehnder interferometer experiments of Grangier et al.~\cite{GRAN86}. We also show that this approach can be generalized to simulate universal quantum computation by a deterministic event-by-event process. Thus, the method we propose can simulate wave interference phenomena and many-body quantum systems using classical, particle-like processes only. Our results suggest that we may have discovered a procedure to simulate quantum phenomena using causal, local, deterministic and event-based processes. Our approach is not an extension of quantum theory in any sense and is not a proposal for another interpretation of quantum mechanics. The probability distributions of quantum theory are generated by a deterministic, causal learning process, and not vice versa~\cite{PENR90}. \setlength{\unitlength}{1cm} \begin{figure*} \caption{ Diagram of the network of two DLMS\ that performs a deterministic simulation of a single-photon beam splitter (BS) on an event-by-event basis~\cite{MZIdemo} \label{dlms} \label{figbs} \end{figure*} \section{Deterministic Learning Machine (DLM)~\cite{MZIdemo}}\label{sec2} In quantum physics, an event corresponds to the detection of a photon, electron, and the like. In our simulation approach an event is the arrival of a message at the input channel of a processing unit. This processing unit typically contains two DLMS\ (described below). We use the diagram of a DLM-based processor that performs the event-by-event simulation of single-photon beam splitter, as shown in Fig.~\ref{dlms}, to describe the operation of the different components of the processor. The applications to quantum computations presented later demonstrate that the structure of the DLM-based processor is in fact generic. In Fig.~\ref{dlms}, the presence of a message is indicated by an arrow on the corresponding line. The first component, DLM\ 1, ``learns'' about the occurrence of an event on one of its two input channels that we label with 0 and 1. For brevity, we refer to an event on channel 0 (1) as a 0 (1) event. The second component transforms the data stored in DLM\ 1 and feeds the result into DLM\ 2. DLM\ 2 ``learns'' this data. Finally the learning process itself is used to determine whether DLM\ 2 responds to the input event by sending out either a 0 or a 1 event. None of these components makes use of random numbers, hence the name deterministic learning machine. Usually, DLM-based simulation algorithms contain several DLM-based processors that form a network. In this paper we only consider networks of processing units in which only one message is traveling through the network at any time. Thus, the network receives an event at one of its inputs, processes the event and delivers the processed message through one of its output channels. After delivering this message the network can accept a new input event. \subsection{Description of a DLM} A DLM\ is a very simple, classical dynamical system with a primitive learning capability. This dynamical system consists of a unit vector (such as ${\bf x}=(x_{0},x_{1},x_{2},x_{3})$ in DLM\ 1 of Fig.~\ref{dlms}), a rule that specifies how this vector changes when an input event is received, and a rule by which the DLM\ determines the type of output event it generates as a response to the input event. The initial value of the internal vectors is irrelevant. In simulations, we usually use random numbers to initialize the internal vectors of all the DLMS\ in the network. We now describe the learning process of a DLM\ in detail~\cite{KOEN04}. The basic idea of the learning algorithm is that the DLM\ minimizes the distance between the input vector (discussed later) and the internal vector and that this minimization is sufficient to construct particle-like processes that mimic quantum phenomena. However, the algorithm that we describe below cannot be derived from the axioms of quantum theory. After many trials and failures, we simply discovered that learning algorithms of this type can be used to simulate quantum phenomena. First we consider DLM\ 1 in Fig.~\ref{dlms}. The internal state of DLM\ 1 is represented by the vector ${\bf x}=(x_{0},x_{1},x_{2},x_{3})$. DLM\ 1 can accept two different types of input events, but only one at a time. Event 0 carries a message represented by a two-dimensional unit vector $(y_{0},y_{1})$. Event 1 carries a message represented by a two-dimensional unit vector $(y_{2},y_{3})$. Upon receiving an input event, DLM\ 1 performs the following steps: \begin{itemize} \item DLM\ 1 computes eight candidate internal states \begin{eqnarray} {\bf w}_1&=&(+\sqrt{1-\alpha^2+\alpha^2x_0^2},\alpha x_{1},\alpha x_{2},\alpha x_{3}),\nonumber \\ {\bf w}_2&=&(-\sqrt{1-\alpha^2+\alpha^2x_0^2},\alpha x_{1},\alpha x_{2},\alpha x_{3}),\nonumber \\ {\bf w}_3&=&(\alpha x_{0},+\sqrt{1-\alpha^2+\alpha^2x_1^2},\alpha x_{2},\alpha x_{3}),\nonumber \\ {\bf w}_4&=&(\alpha x_{0},-\sqrt{1-\alpha^2+\alpha^2x_1^2},\alpha x_{2},\alpha x_{3}),\nonumber \\ {\bf w}_5&=&(\alpha x_{0},\alpha x_{1},+\sqrt{1-\alpha^2+\alpha^2x_2^2},\alpha x_{3}),\nonumber \\ {\bf w}_6&=&(\alpha x_{0},\alpha x_{1},-\sqrt{1-\alpha^2+\alpha^2x_2^2},\alpha x_{3}),\nonumber \\ {\bf w}_7&=&(\alpha x_{0},\alpha x_{1},\alpha x_{2},+\sqrt{1-\alpha^2 +\alpha^2x_3^2}),\nonumber \\ {\bf w}_8&=&(\alpha x_{0},\alpha x_{1},\alpha x_{2},-\sqrt{1-\alpha^2 +\alpha^2x_3^2}). \label{HYP4} \end{eqnarray} The parameter $0<\alpha<1$ controls the learning process and is discussed in more detail later. The plus and minus sign in front of the square roots is introduced to allow the vector ${\bf w}_j$ to cover the whole eight-dimensional unit sphere. \item If DLM\ 1 receives an input event of type 0 with message $(y_{0},y_{1})$, it constructs a vector ${\bf \hat x}=(y_{0},y_{1},x_{2},x_{3})$. If DLM\ 1 receives an input event of type 1 with message $(y_{2},y_{3})$, it constructs a vector ${\bf \hat x}=(x_{0},x_{1},y_{2},y_{3})$. DLM\ 1 determines the update rule $m$ that minimizes the cost function \begin{equation} C_j = -{\bf w}_j^T{\bf \hat x}, \label{HYP3} \end{equation} that is, $C_{m}\le C_{j}$ for $j=1,\ldots,8$. \item DLM\ 1 updates its internal vector by replacing ${\bf x}$ by ${\bf w}_m$. \item DLM\ 1 generates a new (internal) event by putting the values of its internal vector on its four output channels. \item DLM\ 1 waits for the arrival of the next input event. \end{itemize} The transformation stage applies an orthogonal transformation $T$ to ${\bf x}=(x_{0},x_{1},x_{2},x_{3})$. In general, the precise form of the transformation $T$ depends on the particular function that the processor has to perform. In the example shown in Fig.~\ref{dlms}, the orthogonal transformation $T$ takes two pairs of elements from ${\bf x}$ and performs the plane rotation \begin{eqnarray} R(\phi)=\left( \begin{array}{cc} \cos\phi&-\sin\phi\\ \sin\phi&\phantom{-}\cos\phi \end{array} \right) , \label{Rphi} \end{eqnarray} with $\phi=\pi/4$. As we show later, this transformation implements the single-photon beam splitter. The result ${\bf x^\prime}=(x_{0}^\prime,x_{1}^\prime,x_{2}^\prime,x_{3}^\prime)$ of this transformation is sent to the input of DLM\ 2. Thus, DLM\ 2 accepts messages in the form of a four-dimensional unit vector. DLM\ 2 updates its internal vector ${\bf z}=(z_{0},z_{1},z_{2},z_{3})$ according to the following procedure: \begin{itemize} \item DLM\ 2 performs computes eight candidate internal states \begin{eqnarray} {\bf w}_1&=&(+\sqrt{1-\alpha^2+\alpha^2z_0^2},\alpha z_{1},\alpha z_{2},\alpha z_{3}) ,\nonumber \\ {\bf w}_2&=&(-\sqrt{1-\alpha^2+\alpha^2z_0^2},\alpha z_{1},\alpha z_{2},\alpha z_{3}),\nonumber \\ {\bf w}_3&=&(\alpha z_{0},+\sqrt{1-\alpha^2+\alpha^2z_1^2},\alpha z_{2},\alpha z_{3}),\nonumber \\ {\bf w}_4&=&(\alpha z_{0},-\sqrt{1-\alpha^2+\alpha^2z_1^2},\alpha z_{2},\alpha z_{3}),\nonumber \\ {\bf w}_5&=&(\alpha z_{0},\alpha z_{1},+\sqrt{1-\alpha^2+\alpha^2z_2^2},\alpha z_{3}),\nonumber \\ {\bf w}_6&=&(\alpha z_{0},\alpha z_{1},-\sqrt{1-\alpha^2+\alpha^2z_2^2},\alpha z_{3}),\nonumber \\ {\bf w}_7&=&(\alpha z_{0},\alpha z_{1},\alpha z_{2},+\sqrt{1-\alpha^2 +\alpha^2z_3^2}),\nonumber \\ {\bf w}_8&=&(\alpha z_{0},\alpha z_{1},\alpha z_{2},-\sqrt{1-\alpha^2 +\alpha^2z_3^2}). \label{HYP2} \end{eqnarray} \item DLM\ 2 determines the update rule $m$ that minimizes the cost function \begin{equation} C_j = -{\bf w}_j^T{\bf x}^\prime, \label{HYP1} \end{equation} that is, $C_{m}\le C_{j}$ for $j=1,\ldots,8$. \item DLM\ 2 updates its internal vector by replacing ${\bf z}$ by ${\bf w}_m$. \item DLM\ 2 generates an output event of type 0 (1) if $m=1,\ldots,4$ $(5,\ldots,8)$, carrying the message $(y_{0}^\prime,y_{1}^\prime)=(z_{0},z_{1})$ ($(y_{0}^\prime,y_{1}^\prime)=(z_{2},z_{3})$). \item DLM\ 2 waits for the arrival of the next input event. \end{itemize} Comparing the algorithms for DLM\ 1 and DLM\ 2, we see that they are indentical except for part of the second step and the fourth step in which the output is generated. \subsection{Dynamic behavior of a DLM} In general, the behavior of a DLM\ defined by rules Eqs.~\Eq{HYP4} and \Eq{HYP3} or Eqs.~\Eq{HYP2} and \Eq{HYP1} is difficult to analyze without the use of a computer. However, for a fixed input ${\bf x}^\prime={\bf u}$, it is clear what a DLM\ will do. It will minimize the cost given by Eq.~\Eq{HYP1} by rotating its internal vector ${\bf z}$ to bring it as close as possible to ${\bf u}$. After a number of events (depending on the initial value of ${\bf z}$, the input ${\bf u}$, and $\alpha$), ${\bf z}$ will be close to ${\bf u}$. However, the vector ${\bf z}$ does not converge to a limiting value because the DLM\ always changes its internal vector state by a nonzero amount. It is not difficult to see (and supported by simulations, results not shown) that once ${\bf z}$ is close to ${\bf u}$, it will keep oscillating about ${\bf u}$~\cite{KOEN04}. Below we analyse this behavior in more detail using DLM\ 2 as an example. The dynamics of DLM\ 1 is the same as that of DLM\ 2. Let us denote by $n_0$ the number of times the DLM\ selects update rule $m=1,2$ (see Eq.\Eq{HYP2}). Writing \begin{eqnarray} w^2_{0,m}=1-\alpha^2+\alpha^2 z_0^2\equiv (z_0+\delta)^2, \end{eqnarray} and assuming that $0\ll\alpha<1$, we find that the variable $z_0$ changes by an amount \begin{eqnarray} \delta\approx(1-\alpha^2)(1-z_0^2)/2z_0, \end{eqnarray} where we have neglected terms of order $\delta^2$. Similarily, if $N$ is the total number of events then $N-n_0$ is the number of times the DLM\ selects update rules $m\not=1,2$. For $j\not=1,2$, Eq.\Eq{HYP2} gives \begin{eqnarray} w^2_{0,j}=\alpha^2 z_0^2\equiv (z_0+\delta^\prime)^2, \end{eqnarray} where we have neglected terms of order ${\delta^\prime}^2$. Hence $z_0$ changes by \begin{eqnarray} \delta^\prime\approx-(1-\alpha^2)z_0/2. \end{eqnarray} If ${\bf z}$ oscillates about ${\bf u}$ then $z_0$ also oscillates about ${u}_0$. This implies that the number of times $z_0$ increases times the increment must approximately be equal to the number of times $z_0$ decreases times the decrement. In other words, we must have $n_0\delta+(N-n_0)\delta^\prime\approx0$. As $z_0\approx{u}_0$ we conclude that $n_0/N\approx {u}_0^2$. Applying the same reasoning for the cases where the DLM\ selects update rule $m=3,4$ shows that the number of times the DLM\ will apply update rules $m=3,4$ is proportional to ${u}_0^2+{u}_1^2$. At this point, there is not yet a relation between the dynamics of the DLM\ and quantum theory. However, let us now assume that $p_0=z_{0}^2+z_{1}^2$ ($p_1=z_{2}^2+z_{3}^2$) is the probability that a quantum system is observed to be in the state 0 (1). In quantum theory, we would describe this state by a wave function with complex amplitudes $\hat z_{0}+i\hat z_{1}$ ($\hat z_{2}+i\hat z_{3}$). Let is now consider a DLM\ that is learning the four values ${\bf z}=(z_{0},z_{1},z_{2},z_{3})$. From the foregoing discussion, it follows that once the DLM\ has reached the stationary state in which it oscillates about ${\bf z}$, the rate at which the DLM\ uses update rules $m=1,2,3,4$ ($m=5,6,7,8$) corresponds to the probability $p_0$ ($p_1$) to observe a 0 (1) event in the quantum mechanical system. Thus, the DLM\ generates 0 and 1 events in a deterministic manner and because it generates 0 (1) events if it selected update rule $m=1,2,3,4$ $(5,6,7,8)$, the rate at which these events are generated agrees with the corresponding probabilities of quantum theory. As the applications presented later demonstrate, this correspondence is all that is needed to perform an event-by-event simulation of quantum interference and many-body quantum phenomena. \begin{figure*} \caption{ Simulation results for the beam splitter shown in Fig.~\ref{figbs} \label{one-bs} \end{figure*} \subsection{Stochastic variant} The sequence of events that is generated by a DLM\ (network) is strictly deterministic but a simple modification turns a DLM\ into a stochastic learning machine (SLM). The term {\sl stochastic} does not refer to the learning process but to the method that is used to select the output channel that will carry the outgoing message. As explained earlier, in the stationary regime $x_0^2+x_1^2$ and $x_2^2+x_3^2$ (or $z_0^2+z_1^2$ and $z_2^2+z_3^2$) correspond to the probabilities of quantum theory. Thus, a comparision of for instance, $x_0^2+x_1^2$, with a uniform random number $0<r<1$ gives the probability for sending the message over the corresponding output channel. Although the learning process of this processor is still deterministic, in the stationary regime the output events are randomly distributed over the two possibilities. Of course, the frequencies of output events is the same as that of the original DLM-network. Replacing DLMS\ by SLMs in a DLM-network changes the order in which messages are being processed by the network but leaves the content of the messages intact. \subsection{Generalization} In the previous discussion, we considered a DLM-based processor (see Fig.~\ref{dlms}) that accepts two different types of events whereby each event carries a message containing two real numbers. This is sufficient to simulate quantum phenomena such as single-photon interference but if we would like to perform event-by-event simulations of more complicated quantum systems such as quantum computers, a generalization is necessary. From the foregoing description of the learning rule of a DLM\, it is obvious how this rule may be generalized to handle an arbitrary number $N_e$ of different events of messages of arbitrary (but of the same) length $N_m$: Use a vector of $N_e N_m$ elements to represent the internal state of the DLM\ and, instead of eight candidate rules, compare the cost of $2N_e N_m$ candidate rules. Clearly, the construction of DLM-based networks is very systematic and straightforward. \subsection{Summary} A DLM\ responds to the input event by choosing from all possible alternatives, the internal state that minimizes the error between the input and the internal state itself. This deterministic decision process is used to determine which type of event will be generated by the DLM. The message contains information about the decision the DLM\ took while updating its internal state and, depending on the application, also contains other data that the DLM\ can provide. By updating its internal state, the DLM\ ``learns" about the input events it receives and by generating new events carrying messages, it tells its environment about what it has learned. \begin{figure*} \caption{ Diagram of a DLM\ network that simulates a single-photon Mach-Zehnder interferometer on an event-by-event basis~\cite{MZIdemo} \label{figmz} \end{figure*} \section{Single-Photon Beam Splitter}\label{BS} In quantum theory,~\cite{QuantumTheory} the presence of photons in the input modes 0 or 1 of a beam splitter is represented by the complex-valued amplitudes ($a_0,a_1$)~\cite{BAYM74,GRAN86,RARI97}. According to quantum theory, the complex-valued amplitudes ($b_0,b_1$) of the photons in the output channels 0 and 1 of a beam splitter are given by~\cite{BAYM74,GRAN86,RARI97} \begin{eqnarray} \left( \begin{array}{c} b_0\\ b_1 \end{array} \right) = \frac{1}{\sqrt{2}} \left( \begin{array}{cc} 1&i\\ i&1 \end{array} \right) \left( \begin{array}{c} a_0\\ a_1 \end{array} \right), \label{BS3} \end{eqnarray} Writing $a_0=\sqrt{p_0} e^{i\psi_0}$ and $a_1=\sqrt{1-p_0} e^{i\psi_1}$, the probability to observe a photon in output channel 0 (1) is given by \begin{eqnarray} |b_0|^2&=&\frac{1+\sqrt{p_0(1-p_0)}\sin(\psi_0-\psi_1)}{2}, \label{B0} \\ |b_1|^2&=&\frac{1-\sqrt{p_0(1-p_0)}\sin(\psi_0-\psi_1)}{2}. \end{eqnarray} Here $\psi_0$ and $\psi_1$ represent the phases of the photons. In a quantum theoretical description, this phase is proportional to the length of the optical path that the photons have travelled before they enter the beam splitter~\cite{BAYM74,GRAN86,RARI97}. We now show that the DLM-network shown in Fig.\ref{figbs} behaves as if it is a single-photon beam splitter. This network receives events at one of the two input channels. There is a one-to-one relation between each input channel and the corresponding input mode of the quantum mechanical description. Each input event carries information in the form of a two-dimensional unit vector. Either input channel 0 receives $(y_{0},y_{1})=(\cos\psi_0,\sin\psi_0)$ or input channel 1 receives $(y_{2},y_{3})=(\cos\psi_1,\sin\psi_1)$. In terms of the single-photon experiments of Grangier et al.~\cite{GRAN86}, an event corresponds to the arrival of a photon at channel 0 (1) with phase $\psi_0$ ($\psi_1$) of the beam splitter (see Fig.~\ref{figbs}). The input message is fed into the DLM-network described in Section~\ref{sec2}. The purpose of DLM\ 1 is to transform the information contained in two-dimensional input vectors (of which only one is present for any given input event), into a four-dimensional unit vector. The internal vector ${\bf x}$ of DLM\ 1 learns about the amplitudes ($a_0,a_1$): In the stationary regime we have \begin{eqnarray} x_0&\approx&\sqrt{p_0} \cos{\psi_0} ,\nonumber \\ x_1&\approx&\sqrt{p_0} \sin{\psi_0} ,\nonumber \\ x_2&\approx&\sqrt{1-p_0} \cos{\psi_1} ,\nonumber \\ x_3&\approx&\sqrt{1-p_0} \sin{\psi_1} . \end{eqnarray} The four-dimensional internal vector of this device is split into two groups of two-dimensional vectors $( x_{0}, x_{3})$ and $( x_{2},x_{1})$ and each of these two-dimensional vectors is rotated by $45^\circ$. Put differently, the four-dimensional vector is rotated once in the (1,4)-plane about $45^\circ$ and once in the (3,2) plane about $45^\circ$. The order of the rotations is irrelevant. Physically, this transformation corresponds to the reflection of the photons by $45^\circ$ at the beam splitter. The resulting four-dimensional vector is then sent to the input of DLM\ 2. The internal vector ${\bf z}$ of DLM\ 2 learns about the amplitudes ($b_0,b_1$): In the stationary regime we have \begin{eqnarray} z_0&\approx&\sqrt{p_0} \cos{\psi_0}-\sqrt{1-p_0} \sin{\psi_1} ,\nonumber \\ z_1&\approx&\sqrt{p_0} \sin{\psi_0}+\sqrt{1-p_0} \cos{\psi_1} ,\nonumber \\ z_2&\approx&\sqrt{1-p_0} \cos{\psi_1}-\sqrt{p_0} \cos{\psi_0} ,\nonumber \\ z_3&\approx&\sqrt{1-p_0} \sin{\psi_1}+\sqrt{p_0} \cos{\psi_0} \end{eqnarray} DLM\ 2 sends $(z_{0},z_{1})$ through output channel 0 if it used rule $m=1,2,3,4$ (see Eq.~\Eq{HYP1}) to update its internal state. Otherwise it sends $(z_{2},z_{3})$ through output channel 1. In Fig.~\ref{one-bs} we present results of discrete-event simulations using the DLM\ network depicted in Fig.~\ref{figbs}. We denote the number of 0 (1) events by $N_0$ ($N_1$) and the total number of events by $N=N_0+N_1$. The correspondence with the quantum system is clear: the probability for a 0 event is given by $|b_0|^2\approx N_0/N$, $y_{0}^\prime=\hbox{Re } b_0/|b_0|$ and $y_{1}^\prime=\hbox{Im } b_0/|b_0|$. The probability for a 1 event is $|b_1|^2\approx N_1/N$, $y_{2}^\prime=\hbox{Re } b_1/|b_1|$ and $y_{3}^\prime=\hbox{Im } b_1/|b_1|$. Before the simulation starts, the internal vectors of the DLMS\ are given a random value (on the unit sphere). Each data point represents 10000 events. All these simulations were carried out with $\alpha=0.99$. The simulation procedure itself consists of four steps: \begin{enumerate} \item Use two uniform random numbers in the range $[0,360]$ to generate $\psi_0$ and $\psi_1$. \item For fixed values of $\psi_0$ and $\psi_1$, generate 10000 input events. Input channel 0 receives $(y_{0},y_{1})=(\cos\psi_0,\sin\psi_0)$ with probability $p_0$. Input channel 1 receives $(y_{2},y_{3})=(\cos\psi_1,\sin\psi_1)$ with probability $p_1=1-p_0$. \item Count the number of output events $N_0$ ($N_1$) in channel 0 (1), see Fig.~\ref{figbs}. \item Repeat steps 1 to 3. For each pair ($\psi_0$,$\psi_1$), store the results for $N_0$ ($N_1$). \end{enumerate} Plotting $N_0/(N_0+N_1)$ and $|b_0|^2$ as a function of $\phi=\psi_0-\psi_1$ yields the results shown in Fig.~\ref{one-bs}. Actually, there is no need to use random numbers to generate $\psi_0$ and $\psi_1$. In Fig.~\ref{one-bs}, we only used this random process to show that the order in which we pick $\psi_0$ and $\psi_1$ is irrelevant. Random processes enter in the procedure to generate the input data only. The DLM\ network processes the events sequentially and deterministically. From Fig.~\ref{one-bs} it is clear that the output of the deterministic DLM-based beam splitter reproduces the probability distributions as obtained from quantum theory~\cite{QuantumTheory}. \begin{figure*} \caption{ Simulation results for the DLM-network shown in Fig.~\ref{figmz} \label{one-mz} \end{figure*} \section{Mach-Zehnder Interferometer}\label{MZI} In quantum physics~\cite{QuantumTheory}, single-photon experiments with one beam splitter provide direct evidence for the particle-like behavior of photons~\cite{GRAN86,HOME97}. The wave mechanical character appears when one performs single-particle interference experiments. We now describe a DLM\ network that displays the same interference patterns as those observed in single-photon Mach-Zehnder interferometer experiments~\cite{GRAN86}. The schematic layout of the DLM\ network is shown in Fig.~\ref{figmz}. The network described in Section~\ref{BS} is used for the beam splitters. The phase shift is taken care of by the devices $R(\phi_0)$ and $R(\phi_1)$ (that do not contain DLMS) that perform plane rotations by $\phi_0$ and $\phi_1$ (see Eq.~(\ref{Rphi}), respectively. Clearly there is a one-to-one correspondence between the components of the DLM\ network and the elements of a physical Mach-Zehnder interferometer~\cite{BORN64,GRAN86}. According to quantum theory~\cite{QuantumTheory}, the amplitudes ($b_0,b_1)$ of the photons in the output modes 0 ($N_2$) and 1 ($N_3$) of the Mach-Zehnder interferometer are given by~\cite{BAYM74,GRAN86,RARI97} \begin{eqnarray} \left( \begin{array}{c} b_0\\ b_1 \end{array} \right) =U \left( \begin{array}{c} a_0\\ a_1 \end{array} \right) , \label{MZ1} \end{eqnarray} where \begin{eqnarray} U= \frac{1}{2} \left( \begin{array}{cc} 1&i\\ i&1 \end{array} \right) \left( \begin{array}{cc} e^{i\phi_0}&0\\ 0&e^{i\phi_1} \end{array} \right) \left( \begin{array}{cc} 1&i\\ i&1 \end{array} \right) , \label{MZ1a} \end{eqnarray} and $a_0$ ($a_1)$ denotes the amplitude of the photons in the input channel 0 (1). Note that in experiments it is impossible to simultaneously measure ($N_0/(N_0+N_1)$, $N_1/(N_0+N_1)$) and ($N_2/(N_0+N_1)$, $N_3/(N_0+N_1)$): Photon detectors operate by absorbing photons. In experiment~\cite{BORN64,GRAN86}, there are no photons in input channel 1 of the first beam splitter, that is $a_1=0$. From Eq.~(\ref{MZ1}), we see that the phase of the wave function describing the photons in input channel 0 of the first beam splitter is irrelevant. However, as the photons leave the first beam splitter the relation between their phases is fixed. This relation can be changed through the optical path length for reaching the second beam splitter. A change of the optical path length in channel 0 (1) results in a phase shift by $\phi_0$ ($\phi_1$). If $a_1=0$, the output amplitudes of the Mach-Zehnder interferometer are given by \begin{eqnarray} b_0&=&a_0 |a_0|^{-1}e^{i(\phi_0+\phi_1)/2}\sin\frac{\phi_0-\phi_1}{2}, \nonumber \\ b_1&=&a_0 |a_0|^{-1}e^{i(\phi_0+\phi_1)/2}\cos\frac{\phi_0-\phi_1}{2}, \label{MZ3} \end{eqnarray} from which we see that the probabilities $|b_0|^2$ and $|b_1|^2$ depend on $\phi=\phi_0-\phi_1$ only. In Fig.~\ref{one-mz} we present a representative selection of simulation results for the Mach-Zehnder interferometer built from DLMS. We assume that input channel 0 receives $(y_{0},y_{1})=(\cos\psi_0,\sin\psi_0)$ with probability one and that input channel 1 receives no events. This corresponds to $(a_0,a_1)=(\cos\psi_0+i\sin\psi_0,0)$ in Eq.~(\ref{MZ1}). We use uniform random numbers to determine $\psi_0$. As in the case of the beam splitter, we only use this random process to show that the order in which we pick $\psi_0$ is irrelevant. In all these simulations $\alpha=0.99$. The data points are the simulation results for the normalized intensity $N_i/(N_0+N_1)$ for i=0,2,3 as a function of $\phi=\phi_0-\phi_1$. Each data point represents 10000 events ($N_0+N_1=N_2+N_3=10000$). Initially the rotation angle $\phi_0=0$ and after each set of 10000 events, $\phi_0$ is increased by $10^\circ$. Lines represent the corresponding results of quantum theory~\cite{QuantumTheory}. From Fig.~\ref{one-mz} it is clear that the deterministic, event-based DLM\ network generates events with frequencies that are in excellent agreement with quantum theory. \begin{figure*} \caption{ Diagram of a DLM-based processor that simulates a CNOT gate on an event-by-event basis. } \label{figcnot} \end{figure*} \begin{figure*} \caption{ Quantum circuit representation of two equivalent CNOT operations. The dot and the cross on the line denote the control and target qubit, respectively. The square boxes labeled $H$ represent Hadamard gates. } \label{figcnot2} \end{figure*} \section{Universal quantum computation}\label{CNOT} It has been shown that an arbitrary unitary operation, that is, the time evolution of a quantum system, can be written as a sequence of single-qubit operations and the controlled-NOT (CNOT) operation on two qubits~\cite{DIVI95a,NIEL00}. Therefore, in principle, single-qubit operations and the CNOT operation are sufficient to construct a universal quantum computer or to simulate any quantum system~\cite{NIEL00}. In this section, we present results of event-based simulations of single qubit operations and a two-qubit quantum circuit containing the CNOT operation to illustrate that DLM-based networks can be used to simulate universal quantum computers. The state vector of a two-qubit system can be written as~\cite{BAYM74,BALL03,NIEL00} \begin{eqnarray} \KET{\Phi}&=& a_{0}\KET{0}_1\KET{0}_2+a_1\KET{1}_1\KET{0}_2 +a_2\KET{0}_1\KET{1}_2 \nonumber\\ &&+a_3\KET{1}_1\KET{1}_2 \nonumber\\ &=& a_{0}\KET{00}+a_1\KET{01}+a_2\KET{10}+a_3\KET{11} \nonumber\\ &=& a_{0}\KET{0}+a_1\KET{1}+a_2\KET{2}+a_3\KET{3} , \label{CNOT1} \end{eqnarray} where $a_0,\ldots a_3$ are the amplitudes of the four different states and $\KET{0}_i$ and $\KET{1}_i$ represent the 0 and 1 state of the $i$-th qubit, respectively. For convenvience, in the last line of Eq.(\ref{CNOT1}), we represent the basis states of the two-qubit system in decimal notation, that is $\KET{00}=\KET{0}$, $\KET{01}=\KET{1}$, $\KET{10}=\KET{2}$, and $\KET{11}=\KET{3}$~\cite{NIEL00}. \begin{table*}[t] \caption{Simulation results for the DLM-network shown in Fig.~\ref{figcnot2}, demonstrating that the network reproduces the results of the corresponding quantum circuit, that is, a CNOT operation in which qubit 2 is the control qubit and qubit 1 is the target qubit~\cite{NIEL00}. The first half of the events are discarded in the calculation of the frequencies $f_i$ for observing an output event of type $i=0,1,2,3$. For 200 events or more, the difference between the event-based simulation results and the corresponding quantum mechanical probabilities is less than 1\%. } \begin{tabular}{cccccccc} Processor&Number of events&Qubit 1 &Qubit 2&$f_0$&$f_1$&$f_2$&$f_3$\\ \hline \noalign{\vskip 4pt} Deterministic&100&0&0&$0.98$&$0.00$&$0.00$&$0.02$\\ Deterministic&100&1&0&$0.20$&$0.74$&$0.01$&$0.04$\\ Deterministic&100&0&1&$0.16$&$0.04$&$0.00$&$0.80$\\ Deterministic&100&1&1&$0.16$&$0.04$&$0.72$&$0.08$\\ \noalign{\vskip 2pt} \hline \noalign{\vskip 2pt} Deterministic&200&0&0&$1.00$&$0.00$&$0.00$&$0.00$\\ Deterministic&200&1&0&$0.00$&$1.00$&$0.00$&$0.00$\\ Deterministic&200&0&1&$0.00$&$0.00$&$0.00$&$1.00$\\ Deterministic&200&1&1&$0.01$&$0.00$&$0.99$&$0.00$\\ \hline Stochastic&2000&0&0&$0.965$&$0.015$&$0.010$&$0.010$\\% 0.965E+00 0.100E-01 0.150E-01 0.100E-01 Stochastic&2000&1&0&$0.007$&$0.970$&$0.012$&$0.011$\\% 0.700E-02 0.120E-01 0.970E+00 0.110E-01 Stochastic&2000&0&1&$0.010$&$0.008$&$0.016$&$0.966$\\% 0.100E-01 0.160E-01 0.800E-02 0.966E+00 Stochastic&2000&1&1&$0.005$&$0.016$&$0.963$&$0.016$\\% 0.500E-02 0.963E+00 0.160E-01 0.160E-01 \hline \end{tabular} \label{CNOTdata} \end{table*} \begin{table*}[t] \caption{Same as in Table ~\ref{CNOTdata} except that the control parameter $\alpha=0.999$ instead of $\alpha=0.99$ and that ten times as many event were generated. The results for the deterministic simulations are exact within three-digit accuracy and have therefore been omitted.} \begin{tabular}{cccccccc} Processor&Number of events&Qubit 1 &Qubit 2&$f_0$&$f_1$&$f_2$&$f_3$\\ \hline Stochastic&20000&0&0&$0.995$&$0.003$&$0.001$&$0.002$\\ Stochastic&20000&1&0&$0.002$&$0.995$&$0.003$&$0.001$\\ Stochastic&20000&0&1&$0.002$&$0.001$&$0.002$&$0.995$\\ Stochastic&20000&1&1&$0.001$&$0.002$&$0.997$&$0.001$\\ \hline \end{tabular} \label{CNOTdata2} \end{table*} \subsection{CNOT gate} By definition, the CNOT gate flips the target qubit if the control qubit is in the state $\KET{1}$~\cite{NIEL00}. If we take qubit 1 (that is, the least significant bit in the binary notation of an integer) as the control qubit, we have \begin{eqnarray} {\mathop{\hbox{CNOT}}}\KET{\Phi}&=& a_0\KET{0}_1\KET{0}_2+a_3\KET{1}_1\KET{0}_2+a_2\KET{0}_1\KET{1}_2 \nonumber\\ &&+a_1\KET{1}_1\KET{1}_2 \nonumber\\ &=& a_0\KET{00}+a_3\KET{01}+a_2\KET{10}+a_1\KET{11} \nonumber\\ &=& a_0\KET{0}+a_3\KET{1}+a_2\KET{2}+a_1\KET{3} . \label{CNOT2} \end{eqnarray} The schematic diagram of the DLM-network that performs the CNOT operation on an event-by-event (particle-by-particle) basis is shown in Fig.~\ref{figcnot}. Conceptually the structure of this network is the same as in the case of the Mach-Zehnder interferometer. As input to the DLM-network we now have four (0,1,2 or 3) instead of two different types of events. Each event carries a message consisting of two real numbers ${\bf y}=(y_{0},y_{1})$, corresponding to the quantum mechanical amplitudes $a_0,\ldots a_3$. The internal state of each DLM\ is represented by a unit vector of eight real numbers ${\bf x}=(x_{0},\dots,x_{7})$ and there are $16$ candidate update rules ($\{ j=0,\ldots7; s_j=\pm1\}$, see Eq.\Eq{HYP2}) to choose from. The rule that is actually used is determined by minimizing the cost function given by Eq.\Eq{HYP1}. The transformation stage is extremely simple: According to Eq.\Eq{CNOT2}, all it has to do is swap the two pairs of elements ($x_2$,$x_3$) and ($x_6$,$x_7$). Instead of presenting results that show that a DLM-processor correctly simulates the CNOT operation on an event-by-event basis, we consider the more complicated network of four Hadamard gates and one CNOT gate shown in Fig.~\ref{figcnot2}~\cite{NIEL00}. Quantum mechanically, this network acts as a CNOT gate in which the role of control- and target qubit have been interchanged~\cite{NIEL00}. For the corresponding DLM-network to work properly it is essential that the event-based simulation mimics the quantum interference (generated by the Hadamard gates) correctly. \subsection{Hadamard operation}\label{HADA} The Hadamard operation $H$ is the single-qubit operation defined by~\cite{NIEL00} \begin{equation} H\equiv\frac{1}{\sqrt{2}} \left( \begin{array}{cc} \phantom{-}1&\phantom{-}1 \\ \phantom{-}1&-1 \end{array} \right) . \label{SPIN10} \end{equation} Disregarding phase factors, it performs the same operation as a beam splitter. The structure of a DLM-processor that performs a general single-qubit operation is identical to the one shown in Fig.~\ref{figbs}. The only difference is in the transformation stage. To implement the Hadamard operation, we use the transformation matrix $T$ (see Fig.~\ref{dlms}) \begin{equation} \frac{1}{\sqrt{2}} \left( \begin{array}{cccc} \phantom{-}1&\phantom{-}0 &\phantom{-}1 &\phantom{-}0 \\ \phantom{-}0&\phantom{-}1 &\phantom{-}0 &\phantom{-}1 \\ \phantom{-}1&\phantom{-}0 &-1 &\phantom{-}0 \\ \phantom{-}0&\phantom{-}1 &\phantom{-}0 &-1 \\ \end{array} \right) . \label{HADA1} \end{equation} \subsection{Simulation results} In Table~\ref{CNOTdata}, we present simulation results for the DLM-network shown in Fig.~\ref{figcnot2}. Before the first simulation starts we use uniform random numbers to initialize the internal vectors of the DLMS\ (ten vectors in total). All these simulations were carried out with $\alpha=0.99$. From Table~\ref{CNOTdata}, it is clear that, also for a modest number of events, the network reproduces the results of the corresponding quantum circuit, that is, a CNOT operation in which qubit 2 is the control qubit and qubit 1 is the target qubit~\cite{NIEL00}. As an illustration of the use of SLMs, we replace all the DLM\ 2's by SLMs in the DLM\ implementation of the circuit shown in Fig.~\ref{figcnot2} and repeat the simulations. From Tables~\ref{CNOTdata} and ~\ref{CNOTdata2}, we conclude that the randomized version generates the correct results but significantly more events are needed to achieve similar accuracy as in the fully deterministic simulation. \subsection{Technical note} All simulations that we presented in this section have been performed for $\alpha=0.99$. From the description of the learning process it is clear that $\alpha$ controls the rate of learning or, equivalently, the rate at which learned information can be forgotten. Furthermore it is evident that the difference between a constant input to a DLM\ and the learned value of its internal variable cannot be smaller than $1-\alpha$. In other words, $\alpha$ also limits the precision with which the internal variable can represent a sequence of constant input values. On the other hand, the number of events has to balance the rate at which the DLM\ can forget a learned input value. The smaller $1-\alpha$ is, the larger the number of events has to be for the DLM\ to adapt to changes in the input data. We use the example of this section to illustrate the effect of changing $\alpha$ and the total number of events $N$. In Table~\ref{CNOTdata2} we show the results of repeating the procedure used to obtain the data shown in Table~\ref{CNOTdata} but instead of $\alpha=0.99$ we used $\alpha=0.999$ and adjusted the number of events accordingly. As expected, the difference between the simulation data and the results of quantum theory decreases if $1-\alpha$ decreases and the number of events increases accordingly. Comparing Table~\ref{CNOTdata} with Table~\ref{CNOTdata2} it is clear that the decrease of this difference is roughly proportional to the inverse of the square root of the number of events. \section{Discussion} We have proposed a new procedure to construct algorithms that can be used to simulate quantum processes without solving the Schr{\"o}dinger equation. There is a one-to-one correspondence between the components of the network and the processing units and the physical parts of the experimental setup. Furthermore, only simple geometry is used to construct the simulation algorithm. In this sense, the simulation approach we propose satisfies Einstein's criteria of realism and causality~\cite{HOME97}. An analogy may be helpful to understand the conceptual difference between the conventional description of quantum theory and the event-based approach proposed in this paper. It is well known that an ensemble of simple, symmetric random walks may be approximated by a diffusion equation (for vanishing lattice spacing and time step). Also here we have two options. If we are interested in individual events, we have no other choice than to simulate the discrete random walk. However, if we want to study the behavior of many random walkers, it is computationally much more efficient to solve the corresponding diffusion problem. The latter describes the outcome of (infinitly) many individual events but does not provide information about individual events. The random walk is the fundamental mechanism that gives rise to diffusion behavior. In this sense, the DLMS\ described in this paper may be regarded as building blocks for a dynamic, deterministic, local and causal system that generates individual events in such a manner that the collective behavior of these events is described by quantum theory. It may be of interest to compare our approach with stochastic wavefunction methods~\cite{GISI95,DALI92,DUM92,GARD92,MOLM93, GARR94,BREU95,BREU95a,EZAK95,BREU95b,BREU96,BRUN97, MIYA98,BREU99,BREU03,BREU04}. Instead of solving the equation of motion of the density matrix, these methods solve stochastic differential equations for an ensemble of independent realizations of pure states. Typically, these methods are used to study open quantum systems in which a small number of degrees of freedom is coupled to a large reservoir. An attempt to use a variant of the stochastic wavefunction method to perform an event-by-event study of photon emission was reported in Ref.~\cite{MIYA98}. In stochastic wavefunction methods, the wave function evolves in time according to the time-dependent Schr\"odinger equation. An uncorrelated random process interrupts this evolution to project the wave function (that is, make a quantum jump) onto another normalized state. This evolution of the wave function is similar to the change of the internal state of a SLM if we consider one isolated event that is processed by the transformation stage $T$ and a SLM (see Fig.~\ref{dlms}). However, as a SLM is capable of learning from previous events, the process of generating output events is non-Markovian, this similarity being very superficial. The fundamental difference between the two approaches can also be seen as follows. In the stochastic wavefunction method, we can calculate the time evolution of each member of the ensemble in parallel, at least in principle. In the DLM-approach, this is impossible: To exhibit quantum mechanical behavior, it is imperative that the DLM-network processes events in a sequential manner. In the fully deterministic DLM-approach (that is, without the randomizing feature of the SLM), there is no stochastic process at all. Therefore, there also is no relation between the stochastic wavefunction method and the deterministic, machine-learning approach discussed in this paper. In conclusion, we have shown that single-particle quantum interference and quantum computers can be simulated on an event-by-event basis using local and causal processes, without the need of concepts such as wave functions or particle-wave duality. \section*{Acknowledgment} We thank Professor S. Miyashita for extensive discussions and for critical readings of the manuscript. We are grateful to Professors M. Imada and M. Suzuki for many useful comments. \raggedright \end{document}

\begin{document} \title{Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations} \begin{abstract} In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge-Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge-Kutta stages simultaneously. Numerical experiments are performed on the Navier-Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge-Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix-vector multiplications, and wall-clock time, for a wide range of time steps. \end{abstract} \keywords{implicit Runge-Kutta; discontinuous Galerkin; preconditioned GMRES; parallel-in-time} \section{Introduction} The discontinuous Galerkin method, introduced in 1973 by Reed and Hill for the neutron transport equation \cite{Reed:1973}, has seen in recent years increased interest for fluid dynamics applications \cite{Peraire:2011}. The discontinuous Galerkin method is a high-order finite element method suitable for use on unstructured meshes with polynomials of arbitrarily high degree. For many fluid flow problems, explicit time integration methods have the downside of restrictive time step conditions, partly because of the use of high-degree polynomials but more fundamentally because of the need to employ highly graded and/or anisotropic elements for many realistic flow problems \cite{Persson:2006}. Therefore, for many applications it is desirable to use an implicit time integration scheme. Implicit time integration methods for DG have been much studied. Multi-step backward differentiation formulas (BDF) and single-step diagonally implicit Runge-Kutta (DIRK) methods have been applied to discontinuous Galerkin discretizations for fluid flow problems \cite{Persson:2011LES, Persson:2006}. Nigro et.\ al have seen success applying multi-stage, multi-step modified extended BDF (MEBDF) and two implicit advanced step-point (TIAS) schemes to the compressible Euler and Navier-Stokes equations \cite{Nigro2014MEBDF, Nigro2014TIAS}. Additionally, in \cite{Bassi2015}, Bassi et al.\ have used linearly implicit Rosenbrock-type to integrate DG discretizations for various fluid flow problems. The BDF and DIRK methods have some limitations: BDF schemes can be $A$-stable only up to second-order (the famous second Dahlquist barrier) \cite{Dahlquist:1963}, a severe limitation when used in conjunction with a high-order spatial discretization. On the other hand, there exist high-order $A$-stable (and even $L$-stable) DIRK schemes, but these methods have a low stage-order, often resulting in order reduction when applied to stiff problems \cite{Frank:1985kv}. The Radau IIA methods, one class of the so-called fully implicit Runge-Kutta (IRK) methods, are high-order, $L$-stable, and have relatively high stage order. Consequently, these methods suffer less from order reduction than the corresponding DIRK methods when applied to stiff problems. Furthermore, these methods require only a small number of stages $s$, with the order of accuracy given by $p = 2s - 1$. These methods have the drawback that each step involves the solution of large, coupled linear systems of equations. The difficulty in efficiently implementing such methods has caused them to remain not widely used or studied for practical applications \cite{Carpenter:2005, Carpenter2003:ef}. There has been previous work on improving the efficiency of solving these large, coupled systems. In \cite{Jay1999}, Jay and Braconnier develop a parallelizable preconditioner for IRK methods by means of Hairer and Wanner's $W$-transformation. In \cite{DeSwart1998}, De Swart et al.\ have developed a parallel software package for the four-stage Radau IIA method, and Burrage et at.\ have developed a matrix-free, parallel implementation of the fifth-order Radau IIA method in \cite{Burrage1999}. In this paper, we develop a new strategy for efficiently solving the resulting large linear systems by means of the iterative preconditioned GMRES algorithm. A simple transformation of the linear system results in a significant reduction of the cost per GMRES iteration. Furthermore, the block ILU(0) preconditioner, used successfully with implicit time-integrators for the discontinuous Galerkin method in \cite{Persson:2008by}, proves to be effective also for these large systems. A shifted, uncoupled, block ILU(0) factorization is also found to be an effective preconditioner, with the advantage of allowing parallelism in time by computing the stage solutions simultaneously. The structure of this paper is as follows. In Section \ref{sec:eqns}, we describe the governing equations and DG spatial discretization. In Section \ref{sec:time}, we discuss the time integration schemes used in this paper. Then, in Section \ref{sec:formulation}, we introduce the transformation used to reduce the solution cost, and discuss the preconditioners used for the GMRES method. Finally, in Section \ref{sec:numerical}, we perform numerical experiments on a variety of test cases, in two and three spatial dimensions. \section{Equations and spatial discretization}\label{sec:eqns} The equations considered are the time-dependent, compressible Navier-Stokes equations, \begin{gather} \label{eq:ns-1} \frac{\partial\rho}{\partial t}+ \frac{\partial}{\partial x_j}(\rho u_j)=0 \\ \label{eq:ns-2} \frac{\partial}{\partial t}(\rho u_i) + \frac{\partial}{\partial x_j} (\rho u_i u_j) + \frac{\partial p}{\partial x_i} = \frac{\partial \tau_{ij}}{\partial x_j} \qquad \text{for $i=1,2,3,$}\\ \label{eq:ns-3} \frac{\partial}{\partial t}(\rho E) + \frac{\partial}{\partial x_j} \left(u_j(\rho E + p) \right) = -\frac{\partial q_j}{\partial x_j} + \frac{\partial}{\partial x_j} (u_i \tau_{ij}), \end{gather} where $\rho$ is the density, $u_i$ is the $i$th component of the velocity, and $E$ is the total energy. The viscous stress tensor and heat flux are given by \begin{equation} \tau_{ij} = \mu\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{2}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) \qquad\text{and}\qquad q_j = - \frac{\mu}{\rm Pr} \frac{\partial}{\partial x_j} \left( E + \frac{p}{\rho} - \frac{1}{2} u_k u_k \right), \end{equation} where $\mu$ is the viscosity coefficient, and ${\rm Pr} = 0.72$ is the Prandtl number. For an ideal gas, the pressure $p$ is given by the equation of state \begin{equation} \label{eq:eos} p = (\gamma - 1)\rho \left( E - \frac{1}{2} u_k u_k \right), \end{equation} where $\gamma$ is the adiabatic gas constant. For the viscous problems, we introduce an isentropic assumption of the form $p = K \rho^\gamma,$ for a given constant $K$, as described in \cite{Kanner2015}. This additional simplification can be thought of as an artificial compressibility model for the incompressible flows simulated in Sections \ref{sec:naca-2d} and \ref{sec:naca-3d}. We prefer to solve the compressible equations because they result in a system of ODEs rather than differential-algebraic equations, and thus do not need specialized projection-type solvers. This model decouples equation \eqref{eq:ns-3} from equations \eqref{eq:ns-1} and \eqref{eq:ns-2}, and therefore results in one fewer component to solve for. We remark that this simplification does not result in a significant difference in the relative performance of the time integrators studied in this paper, as shown in Section \eqref{sec:ev}, where the full compressible Euler equations are solved, with no isentropic assumption. We rewrite equations \eqref{eq:ns-1}, \eqref{eq:ns-2}, and \eqref{eq:ns-3} in the form \begin{equation} \label{eq:ns-conservation} \frac{\partial u}{\partial t} + \nabla \cdot \bm{F}_i(u) - \nabla \cdot \bm{F}_v (u, \nabla u) = 0, \end{equation} where $u$ is a vector of the conserved variables, and $\bm{F}_i, \bm{F}_v$ are the inviscid and viscous flux functions, respectively. The spatial domain $\Omega$ is discretized into a triangulation, and the solution $u$ is approximated by piecewise polynomials of a given degree. Equation \eqref{eq:ns-conservation} is discretized by means of the discontinuous Galerkin method, where the viscous terms are treated using the compact DG (CDG) scheme \cite{Peraire:2008}. Using a nodal basis function for each component, we write the global solution vector as $\bm{u}$, and obtain a semi-discrete system of ordinary differential equations of the form \begin{equation} \bm{M} \frac{\partial \bm{u}}{\partial t} = \bm{f}(\bm{u}), \end{equation} where $\bm{M}$ is the mass matrix, and $\bm{f}$ is a nonlinear function of the $n$ unknowns $\bm{u}$. The standard method of lines approach allows for the solution of this system of ordinary differential equations by means of a range of numerical time integrators, such as the implicit Runge-Kutta methods that are the focus of this paper. \subsection{Block structure of the Jacobian} We consider the vector of unknowns $\bm{u}$ to be ordered such that the $m$ degrees of freedom associated with one element of the triangulation appear consecutively. We suppose that there are a total of $T$ elements, such that there are a total of $n = Tm$ degrees of freedom. Then, the Jacobian matrix $\bm{J}$ can be seen as a $T \times T$ block matrix, with blocks of size $m \times m$. The $i$th row consists of blocks on the diagonal, and in columns $j$, where elements $i$ and $j$ share a common edge, such that the total number of off-diagonal blocks in the $i$th row is equal to the number of neighbors of element $i$. We note that the off-diagonal blocks of size $m \times m$ are themselves sparse, but for the sake of simplicity we will consider them as dense matrices. The mass matrix $\bm{M}$ is a $T \times T$ block diagonal matrix, with blocks of size $m \times m$, and therefore matrices of the form $\alpha \bm{M} - \beta \bm{J}$ have the same sparsity pattern as the Jacobian. \section{Time integration}\label{sec:time} In this paper, we will focus on the one-step, multi-stage Runge-Kutta methods. Given initial conditions $\bm{u}_0 = \bm{u}(t_0)$, a general $s$-stage, $p$th-order Runge-Kutta method for advancing the solution to $\bm{u}_1 = \bm{u}(t_0 + \Delta t) + \mathcal{O}(\Delta t^{p+1})$ can be written as \begin{align} \label{eq:DG-RK-1} \bm{M} \bm{k}_i &= \bm{f}\left( t_0 + \Delta t c_i, \bm{u}_0 + \Delta t \sum_{j=1}^s a_{ij} \bm{k}_j \right), \\ \label{eq:DG-RK-2} \bm{u}_1 &= \bm{u}_0 + \Delta t \sum_{i=1}^s b_i \bm{k}_i, \end{align} where the coefficients $a_{ij}, b_i,$ and $c_i$ can be expressed compactly in the form of the \textit{Butcher tableau}, \[ \begin{array}{c|ccc} c_1 & a_{11} & \cdots & a_{1s} \\ \vdots & \vdots & \ddots & \vdots \\ c_s & a_{s1} & \cdots & a_{ss} \\ \hline & b_1 & \cdots & b_s \end{array} = {\setlength\extrarowheight{5pt} \begin{array}{c|c} \bm{c} & \bm{A}\\ \hline & \bm{b}^T \end{array}}. \] If the matrix of coefficients $\bm{A}$ is strictly lower-triangular, then each stage $\bm{k}_i$ only depends on the preceding stages, and the method is called an \textit{explicit} Runge-Kutta method. In this case, each stage may be computed by simply evaluating the function $\bm{f}$. If $\bm{A}$ is not strictly lower-triangular, the method is called an \textit{implicit} Runge-Kutta method (IRK). A particular class of implicit Runge-Kutta methods is those for which the matrix $\bm{A}$ is lower-triangular. Such methods are called \textit{diagonally-implicit} Runge-Kutta (DIRK) methods \cite{Alexander:1977dk}. Implicit Runge-Kutta methods enjoy high accuracy and very favorable stability properties, but computing the stages requires the solution of (in general nonlinear) systems of equations. In the case of DIRK methods, since $\bm{A}$ is lower-triangular, each stage $\bm{k}_i$ depends only on those stages $\bm{k}_j$, $j \leq i$, requiring the sequential solution of $s$ systems, each of size $n$. In contrast, general IRK methods couple all of the stages, resulting in one nonlinear system of equations of size $s\times n$. For the solution of stiff systems of equations, we are interested in those methods that are $L$-stable, meaning that their stability region includes the entire left half-place ($A$-stability), together with the additional criterion that the stability function $R(z)$ satisfies $\lim_{z\to\infty} R(z) = 0$. In the present study, we compare the efficiency and effectiveness of several $L$-stable IRK and DIRK schemes. The methods considered are listed in Table \ref{tab:schemes}. The IRK schemes considered are the Radau IIA schemes, which are $L$-stable, $s$-stage schemes of order $2s - 1$ based on the Radau right quadrature. The construction of these schemes can be found in \cite{Hairer:2013tj}. The two-stage and three-stage Radau IIA methods are listed as RADAU23 and RADAU35, respectively. The DIRK schemes considered are the three-stage, third-order $L$-stable scheme denoted DIRK33, which is derived in detail in \cite{Alexander:1977dk}, and the six-stage, fifth-order scheme constructed in \cite{Boom:2013}, and denoted ESDIRK65. The latter scheme is an \textit{explicit singly diagonally implicit} Runge-Kutta (ESDIRK) method, meaning that the first diagonal entry of the Butcher matrix is zero, and the remaining diagonally entries are nonzero and equal. In addition to the third- and fifth-order methods, we also consider the seventh- and ninth-order Radau IIA methods, although we do not compare these methods to equal-order DIRK methods. The Butcher tableaux for the methods considered are given in Appendix \ref{app:tableaux}. \begin{table}[h!] \centering \caption{$L$-stable implicit Runge-Kutta schemes considered} \label{tab:schemes} \begin{tabular}{lccccc} \toprule Scheme & Order & Total stages & Implicit stages & Stage order & Leading error coefficient\\ \midrule RADAU23 & 3 & 2 & 2 & 2 & $1.39 \times 10^{-2}$\\ DIRK33 & 3 & 3 & 3 & 1 & $2.59 \times 10^{-2}$\\ RADAU35 & 5 & 3 & 3 & 3 & $1.39 \times 10^{-4}$\\ ESDIRK65 & 5 & 6 & 5 & 2 & $5.30 \times 10^{-4}$\\ RADAU47 & 7 & 4 & 4 & 4 & $7.09 \times 10^{-7}$\\ RADAU59 & 9 & 5 & 5 & 5 & $2.19 \times 10^{-9}$\\ \bottomrule \end{tabular} \end{table} Also of interest is the phenomenon of \textit{order reduction}, whereby, when applied to stiff problems, the overall order of accuracy is reduced from $p$ to the \textit{stage order} of the method (denoted $q$) \cite{Frank:1985kv}. The stage order $q$ is defined as $q = \min\{ p, q_i \}$, for $i=1,\ldots,s$, where $q_i$ is defined by \begin{equation} \bm{u}(t_0 + \Delta t c_i) = \bm{u}_0 + \Delta t \sum_{j=1}^s a_{ij} \bm{k}_j + \mathcal{O}(\Delta t^{q_i + 1}). \end{equation} It can be shown that the maximum stage order for any DIRK method is 2 \cite{Hairer:2013vl}, whereas the stage order for the Radau IIA methods is given by the number of stages, $q=s$ \cite{Lambert:1991vi}. The DIRK33 has stage order of $q=1$. An advantage of the ESDIRK methods such as ESDIRK65 is that they have stage order of $q=2$ \cite{Kvaerno:2004}. The Radau IIA methods are very attractive because of their high order of accuracy, small number of stages, high stage order, and $L$-stability, but solving the coupled system of $s \times n$ equations is computationally expensive. Supposing that we solve the nonlinear system of equations for the stages $\bm{k}_i$ by means of Newton's method, then at each iteration we must solve a linear system of equations by inverting the Jacobian matrix of the right-hand side, $\bm{f}(t, \bm{u})$. Assuming a dense Jacobian matrix, and solution via Gaussian elimination (or LU factorization), then the cost of performing a linear solve scales as the cube of the number of unknowns. Therefore, the cost per linear solve for a general IRK method is $\mathcal{O}(s^3n^3)$, whereas the cost per solve for a DIRK method scales like $\mathcal{O}(sn^3)$. In \cite{Butcher:1976tt}, Butcher describes how to transform the resulting set of linear equations to reduce the computational work for solving the IRK systems to $\mathcal{O}(2sn^3)$. Despite this reduction in computational complexity, the cost of solving the large systems of equations has proven in practice to be prohibitive \cite{Carpenter:2005}. On the other hand, DIRK methods have proven be popular and effective for solving computational fluid dynamics problems \cite{Bijl:2002}, at the cost of lower stage order and an increased number of stages. \section{Efficient solution of implicit Runge-Kutta systems} \label{sec:formulation} In this section we describe a method for efficiently solving the systems arising from general IRK methods when applied to discontinuous Galerkin discretizations. The Jacobian matrices of the function $\bm{f}$ are sparse, block-structured matrices, which lend themselves to solution via iterative Krylov subspace methods. In particular, we consider the solution of these systems by means of the GMRES method with a zero fill-in block ILU(0) preconditioner, as in \cite{Persson:2008by}. In this case, each iteration of the GMRES method requires one matrix-vector multiplication, and one application of the ILU(0) preconditioner. In order to efficiently solve the linear systems resulting from IRK methods, we will rewrite the system of equations in such a way so as to reduce the cost of a matrix-vector multiplication from $s^2n^2$ to order $s n^2$. \subsection{Transformation of the system of equations} Recalling equation that the stages $\bm{k}_i$ are given by the equation \begin{equation} \label{eq:IRK-stage-eqn} \bm{M} \bm{k}_i = \bm{f}\left( t_0 + \Delta t c_i, \bm{u}_0 + \Delta t \sum_{j=1}^s a_{ij} \bm{k}_j \right), \end{equation} we define $\bm{K}$ to be the concatenation of the vectors $\bm{k}_i$, $\bm{U}_0$ to be the concatenation of $s$ copies of $\bm{u}_0$, and $\bm{F}$ the function $\bm{f}$ applied component-wise on these vectors. Then, we rewrite equation \eqref{eq:IRK-stage-eqn} in vector form as \begin{equation} (\bm{I}_s \otimes \bm{M})\bm{K} = \bm{F}\left(t_0 + \Delta t \bm{c}, \bm{U}_0 + \Delta t (\bm{A} \otimes \bm{I}_n) \bm{K} \right), \end{equation} where $\otimes$ is the Kronecker product, and $\bm{I}_n$ is the $n\times n$ identity matrix. This nonlinear system can be solved by means of Newton's method, which will require solving at each step a linear system of the form \begin{equation} \label{eq:newton-system} \left(\left(\begin{array}{ccc} \bm{M} & & 0 \\ & \ddots & \\ 0 & & \bm{M} \end{array}\right) - \Delta t \left(\begin{array}{ccc} a_{11} \bm{J}_1 & \cdots & a_{1s} \bm{J}_1 \\ \vdots & \ddots & \vdots \\ a_{s1} \bm{J}_s & \cdots & a_{ss} \bm{J}_s, \end{array}\right)\right) \left(\begin{array}{ccc} \bm{k}_1 \\ \vdots \\ \bm{k}_s \end{array} \right) = \left(\begin{array}{ccc} \bm{r}_1 \\ \vdots \\ \bm{r}_s \end{array} \right) \end{equation} for the residual vectors $(\bm{r}_1, \ldots, \bm{r}_s)^T$, where the matrices on the left-hand side are $s \times s$ block matrices blocks, with each block of size $n \times n$. We use the following notation for the Jacobian matrix of $\bm{f}$, \[ \bm{J}_i = \bm{J}_{\bm{f}} \left(t_0 + \Delta t c_i, \bm{u_0} + \Delta t\sum_{j=1}^s a_{ij} \bm{k}_j\right). \] We can rewrite equation \eqref{eq:newton-system} in the following form, \begin{equation} \left( \bm{I}_s \otimes \bm{M} - \Delta t \left( \begin{array}{ccc} a_{11}J_1 & \cdots & a_{1s}J_1 \\ \vdots & \ddots & \vdots \\ a_{s1}J_s & \cdots & a_{ss}J_s \end{array} \right) \right) \bm{K} = \bm{R}, \label{eq:newton-system-2} \end{equation} The sparsity pattern of the matrix in \eqref{eq:newton-system-2} is simply that of the Jacobian matrix $J$ repeated $s \times s$ times, and we can conclude that the cost of computing a matrix-vector product with this matrix is $s^2$ times that of computing the matrix-vector product of one Jacobian matrix. In order to reduce the cost of the matrix-vector multiplication, we perform a simple transformation in to rewrite \eqref{eq:newton-system-2} in a slightly modified form. We begin by defining \[ \bm{w}_i = \sum_{j=1}^s a_{ij} \bm{k}_j, \] and similarly, we let $\bm{W}$ denote the vectors $\bm{w}_i$ stacked, such that \[ \bm{W} = (\bm{A} \otimes \bm{I}_n) \bm{K}. \] Then, we rewrite the nonlinear system of equations \eqref{eq:IRK-stage-eqn} in terms of the variables $\bm{w}_i$ to obtain \[ \bm{M}\bm{k_i} = \bm{f}(t_0 + \Delta t c_i, \bm{u}_0 + \Delta t \bm{w}_i ), \] or, equivalently, in the case that the Butcher matrix $A$ is invertible, \begin{equation} \label{eq:modified-IRK} (\bm{A}^{-1} \otimes \bm{M}) \bm{W} = \bm{F}(t_0 + \Delta t \bm{c}, \bm{U}_0 + \Delta t \bm{W} ). \end{equation} In the transformed variables, the new solution $\bm{u}_1$ can be written as \[ \bm{u}_1 = \bm{u}_0 + \Delta t ( \bm{b}^T \bm{A}^{-1} \otimes \bm{I}_n)\bm{W}, \] In the case of the Radau IIA methods, $\bm{b}^T \bm{A}^{-1} = (0,\ldots,0,1)$, and so this further simplifies to \[ \bm{u}_1 = \bm{u}_0 + \Delta t \bm{w}_s. \] Solving equation \eqref{eq:modified-IRK} with Newton's method gives rise to the linear system of equations \begin{equation} \left( A^{-1} \otimes \bm{M} - \Delta t \left( \begin{array}{cccc} \bm{J}_1 & 0 & \cdots & 0 \\ 0 & \bm{J}_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \bm{J}_s \end{array} \right) \right) \bm{W} = \bm{R}. \end{equation} The advantage of this formulation is that the resulting system enjoys greater sparsity. The resulting matrix is a $s \times s$ block matrix, with multiples of the mass matrix in every off-diagonal block, and with matrices of the form $(\bm{A}^{-1})_{ii} \bm{M} - \Delta t \bm{J}_i$ along the diagonal. Computing a matrix-vector product with this $s \times s$ block matrix requires performing $s$ matrix-vector multiplications with the mass matrix, and $s$ matrix-vector multiplications with a Jacobian $\bm{J}_i$. Therefore the cost of computing such products scales as $s$ times the cost of computing one matrix-vector product with the Jacobian matrix. We additionally remark that the fully-implicit IRK methods requiring storing each of the $s$ Jacobian matrices $\bm{J}_i$, resulting in memory usage that is $s$-times that of the DIRK methods. A further advantage of the transformed system of equations is that the memory requirements for the ILU-based preconditioners are reduced, as discussed in the following sections. \subsection{Preconditioning} The use of an appropriate preconditioner is essential in accelerating the convergence of a Krylov subspace method such as GMRES. We briefly describe the block ILU(0) preconditioner from \cite{Persson:2008by}. \subsubsection{Block ILU(0) preconditioner} The block ILU(0) (or zero fill-in) preconditioner is a method for obtaining block lower- and upper-triangular matrices $\tilde{\bm{L}}$ and $\tilde{\bm{U}}$ given a block sparse matrix $\bm{B}$. These matrices are obtaining by computing the standard block LU factorization, but discarding any blocks which do not appear in the sparsity pattern of $\bm{B}$. We denote the block in the $i$th row and $j$th column as $\bm{B}_{ij}$. The ILU(0) algorithm can be written as shown in Algorithm \ref{alg:ilu}. \begin{algorithm} \caption{Block ILU(0) algorithm} \label{alg:ilu} \begin{algorithmic}[1] \For{$i=1$ to $T$} \For{neighbors $j$ of $i$ with $j > i$} \State $\bm{B}_{ji} \gets \bm{B}_{ji} \bm{B}_{ii}^{-1}$ \State $\bm{B}_{jj} \gets \bm{B}_{jj} - \bm{B}_{ji}\bm{B}_{ij}$ \For{neighbors $k$ of $j$ and $i$ with $k > j$} \State $\bm{B}_{jk} \gets \bm{B}_{jk} - \bm{B}_{ji}\bm{B}_{ik}$ \EndFor \EndFor \EndFor \State $\tilde{\bm{L}} \gets I+\text{strict block lower triangle of } \bm{B}$ \State $\tilde{\bm{U}} \gets \text{block upper triangle of } \bm{B}$ \end{algorithmic} \end{algorithm} If we impose the condition on the triangulation of our domain that, if elements $j$ and $k$ both neighbor element $i$, then elements $j$ and $k$ are not neighbors of each other, then the ILU(0) algorithm has the particularly simple form, show in Algorithm \ref{alg:simp-ilu}. In practice, most well-shaped meshes satisfy this condition and henceforth we will use this simpler algorithm. \begin{algorithm} \caption{Simplified block ILU(0) algorithm} \label{alg:simp-ilu} \begin{algorithmic}[1] \For{$i=1$ to $T$} \For{neighbors $j$ of $i$ with $j > i$} \State $\bm{B}_{ji} \gets \bm{B}_{ji} \bm{B}_{ii}^{-1}$ \State $\bm{B}_{jj} \gets \bm{B}_{jj} - \bm{B}_{ji}\bm{B}_{ij}$ \EndFor \EndFor \State $\tilde{\bm{L}} \gets I+\text{strict block lower triangle of } \bm{B}$ \State $\tilde{\bm{U}} \gets \text{block upper triangle of } \bm{B}$ \end{algorithmic} \end{algorithm} \subsection{Preconditioning the large system} \label{sec:precond} In the case of the general IRK methods, we must solve systems of the form \begin{equation} \label{eq:big-irk} \bm{B}\bm{W} = \bm{R}, \qquad \bm{B} = \left( \bm{A}^{-1} \otimes \bm{M} - \Delta t \left( \begin{array}{cccc} \bm{J}_1 & 0 & \cdots & 0 \\ 0 & \bm{J}_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \bm{J}_s \end{array} \right)\right). \end{equation} We remark that the matrix $\bm{B}$ can now be considered as a $s \times s$ block matrix, with blocks of size $n \times n$. Each $n \times n$ block is itself a $T \times T$ block matrix, with subblocks of size $m \times m$. We introduce the notation $\bm{B}_{k\ell,ij}$ to denote the $(i, j)$ subblock of the $(k,\ell)$ block of $\bm{B}$. That is to say, $\bm{B}_{k\ell,ij}$ is the $(i, j)$ block of the matrix $(\bm{A})^{-1}_{k\ell}\bm{M} - \delta_{k\ell}\Delta t \bm{J}_k$, where $\delta_{k\ell}$ is the Kronecker delta. \subsubsection{Stage-coupled block ILU(0) preconditioner} We consider two preconditioners for this large $sn \times sn$ system. The first is the standard (stage-coupled) block ILU(0) preconditioner, which can be computed using Algorithm \ref{alg:big-ilu}. We note that this preconditioner couples all $s$ stages of the method. This preconditioner requires storing $s$ Jacobian-sized diagonal blocks, and $s^2 - s$ off-diagonal blocks, each the same size as the mass matrix. \begin{algorithm} \caption{ILU(0) algorithm for IRK systems of the form \eqref{eq:big-irk}} \label{alg:big-ilu} \begin{algorithmic}[1] \For{$k = 1$ to $s$} \For{$i=1$ to $T$} \For{neighbors $j$ of $i$ with $j > i$} \State $\bm{B}_{kk,ji} \gets \bm{B}_{kk,ji} \bm{B}_{kk,ii}^{-1}$ \State $\bm{B}_{kk,jj} \gets \bm{B}_{kk,jj} - \bm{B}_{kk,ji}\bm{B}_{kk,ij}$ \EndFor \For{$\ell = k+1$ to $s$} \State $\bm{B}_{\ell k,ii} \gets \bm{B}_{\ell k,ii} \bm{B}_{kk,ii}^{-1}$ \State $\bm{B}_{\ell \ell,ii} \gets \bm{B}_{\ell \ell,ii} - \bm{B}_{\ell k,ii} \bm{B}_{kk,ii}$ \EndFor \EndFor \EndFor \State $\tilde{\bm{L}} \gets I+\text{strict block lower triangle of } \bm{B}$ \State $\tilde{\bm{U}} \gets \text{block upper triangle of } \bm{B}$ \end{algorithmic} \end{algorithm} \subsubsection{Stage-uncoupled, shifted ILU(0) preconditioner} In order to avoid the above coupling of the stages, we can compute a simplified preconditioner in the form of the following block matrix, \begin{equation} \left( \begin{array}{cccc} \tilde{\bm{L}}_1 \tilde{\bm{U}}_1 & 0 & \cdots & 0 \\ 0 & \tilde{\bm{L}}_2 \tilde{\bm{U}}_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \tilde{\bm{L}}_s \tilde{\bm{U}}_s \end{array} \right), \end{equation} where $\tilde{\bm{L}}_i \tilde{\bm{U}}_i$ is the block ILU(0) factorization of a matrix of the form $\left(A^{-1}_{ii} + \alpha_i\right) \bm{M} - \Delta t \bm{J}_i$. We let $\alpha_i$ denote a \textit{shift}, so that the standard unshifted factorization corresponds to $\alpha_i = 0$. The so-called shifted ILU factorization, described by Manteuffel in \cite{Manteuffel1980}, can result in eigenvalues clustered away from the origin, and hence faster convergence in GMRES. Indeed, our experience shows that the unshifted preconditioner underperforms certain other choices of shift. The preconditioner has several advantages over the stage-coupled block ILU(0) preconditioner. The first is that it is easily constructed using an already-implemented block ILU(0) factorization of the Jacobian matrix. The second is that none of the stages are coupled, allowing for both efficient computation and application. In particular, this has implications for the parallelization of the preconditioner, which we discuss in Section \ref{sec:parallel}. Finally, as this preconditioner does not include any off diagonal blocks, the memory requirements are exactly $s$ times that of the standard block ILU(0) used for the DIRK methods. As mentioned, the unshifted preconditioner, with $\alpha_i = 0$ for all $i$, such that $\tilde{\bm{L}}_i\tilde{\bm{U}}_i$ is the ILU(0) factorization of the $i$th diagonal block of the matrix $\bm{B}$, is a natural choice. This choice of coefficients ignores all the off-diagonal mass matrices. By making certain judicious choices of the coefficients $\alpha_i$, we can attempt to compensate for the off-diagonal blocks by adding multiples of the mass matrix back to the diagonal entries. In particular, our numerical experiments have shown that setting $\alpha_i = \sum_{j \neq i} \left| A^{-1}_{ji} \right|$ results in a more efficient preconditioner, requiring fewer GMRES iterations in order to converge to a given desired tolerance. \subsection{Computational cost and memory requirements} In order to compare the computational cost of the transformed IRK implementation described above with both that of the untransformed formulation, and with the usual DIRK methods, we summarize the computational cost associated with solving the resulting linear systems. We note that the IRK methods require the solution of one large, coupled system, whereas the DIRK methods require the solution of $s$ smaller systems. In Table \ref{tab:irk-iter-costs} we record the leading terms of the computational cost of operations required to be performed every iteration. We recall that $s$ is the number of Runge-Kutta stages, $m$ is the number of degrees of freedom per mesh element, $T$ is the total number of elements in the mesh, and $r$ is the number of neighbors per element. Here we assume that the $m \times m$ blocks of the Jacobian matrix are dense, and hence require $2m^2$ floating point operations per matrix-vector multiply. Computing the preconditioner requires the LU factorization of the diagonal blocks, which incurs a cost of $\frac{2}{3}m^3$ floating point operations per block. Each iteration in Newton's method requires re-evaluation of the Jacobian matrix, and therefore also the re-computation of the preconditioner. Hence, the preconditioner must be computed once per linear solve. The costs associated with these calculations are listed in Table \ref{tab:irk-precond-cost}. \begin{table}[H] \centering \caption{Per-iteration computational costs for solving implicit Runge-Kutta systems} \label{tab:irk-iter-costs} \begin{tabular}{ll} \toprule Operation & Cost (leading term) \\ \midrule Untransformed IRK matrix-vector product & $s^2 m^2 (r + 1) T $ \\ Transformed IRK matrix-vector product & $s m^2 (r + s) T $ \\ DIRK matrix-vector product & $m^2 (r + 1) T$\\ \midrule Coupled preconditioner application (IRK) & $s m^2 (r + s) T$\\ Uncoupled preconditioner application (IRK) & $s m^2 (r + 1) T$\\ Preconditioner application (DIRK) & $m^2 (r + 1) T$\\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \centering \caption{Computational cost of computing the preconditioner (once per solve) systems} \label{tab:irk-precond-cost} \begin{tabular}{ll} \toprule Operation & Cost (leading term) \\ \midrule Computing coupled block ILU(0) preconditioner (IRK) & $s (m^3 + (r + s)m^2) T$\\ Computing uncoupled ILU(0) preconditioner (IRK) & $s (m^3 + rm^2) T$\\ Computing block ILU(0) preconditioner (DIRK) & $(m^3 + rm^2) T$\\ \bottomrule \end{tabular} \end{table} We remark that each GMRES iteration using the formulation described in Section \ref{sec:formulation} requires a factor of $s$ fewer floating-point operations per iteration than the na\"ive IRK implementation. The stage-uncoupled IRK preconditioner is less expensive to both compute and apply than the stage-coupled preconditioner. We also note that for equal order of accuracy, the Radau IIA IRK methods require fewer implicit stages than do the DIRK methods. Each such implicit stage incurs the cost of assembling the Jacobian matrix. This cost is problem-dependent, but is in general non-trivial, and in the model problems considered in this paper, it scales like $\mathcal{O}(m^3 T)$. Finally, we present the memory requirements for the IRK and DIRK methods, and the stage-coupled and uncoupled preconditioners in Table \ref{tab:irk-memory}. We note that for the transformed IRK methods, only the $s$ Jacobian matrices $\bm{J}_i$ need to be stored. The stage-coupled block ILU(0) preconditioner requires an additional $s^2 - s$ off-diagonal blocks, which have the same block-diagonal structure as the mass matrix, each having $m^2 T$ nonzero entries. The stage-uncoupled preconditioner does not require these off-diagonal blocks, and therefore its memory requirements are exactly $s$ times that of the DIRK block ILU(0) preconditioner. The block ILU(0) preconditioner for the untransformed system of equations would require storing an additional $(s^2 - s) r m^2 T$ nonzero entries in the off-diagonal blocks. \begin{table}[H] \centering \caption{Memory requirements for the Jacobian matrices and preconditioners} \label{tab:irk-memory} \begin{tabular}{ll} \toprule Method & Memory required \\ \midrule IRK Jacobian matrix & $s(r+1)m^2 T$\\ DIRK Jacobian matrix & $(r+1)m^2 T$\\ Coupled block ILU(0) preconditioner (IRK) & $s m^2 (s+r) T$\\ Uncoupled ILU(0) preconditioner (IRK) & $s m^2 (r+1) T$\\ Block ILU(0) preconditioner (DIRK) & $m^2 (r+1) T$\\ \bottomrule \end{tabular} \end{table} \subsection{Stage-parallelism and partitioned ILU}\label{sec:parallel} In order to parallelize the computations, the spatial domain is decomposed into several subdomains. The compact stencil of the CDG scheme allows for very low communication costs between processes for residual evaluation and Jacobian assembly operations. The matrix-vector multiplications, which constitute the bulk of the work for the linear solve, also scale well in terms of communication for the same reason. In order to parallelize the ILU(0) preconditioner, the contributions between elements in different partitions are ignored, allowing each process to compute the ILU(0) factorization independently. Because these contributions are ignored, we find that the number of GMRES iterations required to converge increases as the number of domain partitions increases \cite{Persson2009scalable}. In fact, it is easy to see that when the number of partitions is equal to the number of mesh elements, the preconditioner simply reduces to the block Jacobi preconditioner. However, in general the partitioned ILU(0) preconditioner is found to be superior to the block Jacobi preconditioner. These considerations apply equally to both the DIRK and general IRK methods, using the stage-coupled ILU(0) preconditioner. If we use the stage-uncoupled ILU(0) preconditioner, then we are able to decompose the domain into a factor of $s$ fewer partitions. We then assume that the number of processes is equal to the number of mesh partitions times the number of stages. The processes are first divided into groups according to the mesh partitioning such that each group consists of $s$ processes. Within each group, each process is then assigned to one stage of the IRK method. Thus, when assembling the block matrix of the form \eqref{eq:big-irk}, the Jacobian matrices for all of the stages are computed in parallel. This does not require any communication between the groups. Then, since the stage-uncoupled block ILU(0) preconditioner does not take into account any of the off-diagonal blocks, the preconditioner can also be computed without any inter-stage communication. Similarly, each application of the preconditioner can be computed in parallel over all the stages without any communication. When computing matrix-vector products, the products with the Jacobian blocks for each stage are computed in parallel, and the products with the mass matrix blocks must be communicated within the stages. It is possible to overlap the communication with the computation of the matrix-vector product with the stage-Jacobian block, such that the cost of communication is negligible. The main advantage of this parallelization scheme is that the mesh is decomposed into a factor of $s$ fewer partitions. Therefore, the effect of ignoring the coupling between regions in the ILU(0) factorization is lessened, and the result is a more efficient preconditioner. In Sections \ref{sec:parallel-results} and \ref{sec:naca-3d}, we numerically study the effects of parallelizing these preconditioners. \begin{figure}\label{fig:parallel-matrix} \end{figure} In the context of hybrid shared-distributed memory systems, it is possible to further reduce the communication cost of the stage-parallel IRK algorithm. On such a system, groups of compute cores called nodes have access to the same shared memory. As a consequence, intranode communication is much faster than internode communication. If each stage-group of $s$ processes described above is located on one node, and none of the groups are split across nodes, then the solutions are only communicated within a node, resulting in negligible internode communication costs. An illustration of such an arrangement is shown in Figure \ref{fig:parallel-matrix}. The example shown is a scalar problem on a mesh with eight elements, decomposed into four partitions. The hypothetical architecture consists of four compute nodes, each with two CPUs with shared memory. For a two-stage IRK method, each partition of the mesh belongs to a different node, and each of the stages for a given partition belong to different CPUs within one node. \section{Numerical results}\label{sec:numerical} \subsection{Euler vortex} \label{sec:ev} We solve the compressible Euler equations of gas dynamics, which are given by equations \eqref{eq:ns-1} through \eqref{eq:ns-3}, with the second-order terms removed. The equation of state is given by \eqref{eq:eos}. \begin{figure*} \caption{Bottom-left corner of mesh, with $p=4$ DG nodes} \caption{Density contours for the initial conditions} \caption{Compressible Euler vortex} \label{fig:ev-mesh} \label{fig:ev-ic} \end{figure*} We consider the model problem of an unsteady compressible vortex in a rectangular domain \cite{Wang:2013gj}. The domain is taken to be the rectangle $ [0, 20] \times [-7.5, 7.5]$, and the vortex is initially centered at $(x_0, y_0) = (5, -2.5)$. The vortex is moving with the free-stream at an angle of $\theta$. This problem is particularly useful as a benchmark because the exact solution is given by the following analytic formulas, allowing for convenient computation of the numerical accuracy. The exact solution at $(x, y, t)$ is given by \begin{gather} \label{eq:ev-exact-1} u = u_\infty \left( \cos(\theta) - \frac{\epsilon ((y-y_0) - \overline{v} t)}{2\pi r_c} \exp\left( \frac{f(x,y,t)}{2} \right) \right),\\ \label{eq:ev-exact-2} u = u_\infty \left( \sin(\theta) - \frac{\epsilon ((x-x_0) - \overline{u} t)}{2\pi r_c} \exp\left( \frac{f(x,y,t)}{2} \right) \right),\\ \label{eq:ev-exact-3} \rho = \rho_\infty \left( 1 - \frac{\epsilon^2 (\gamma - 1)M^2_\infty}{8\pi^2} \exp((f(x,y,t)) \right)^{\frac{1}{\gamma-1}}, \\ \label{eq:ev-exact-4} p = p_\infty \left( 1 - \frac{\epsilon^2 (\gamma - 1)M^2_\infty}{8\pi^2} \exp((f(x,y,t)) \right)^{\frac{\gamma}{\gamma-1}}, \end{gather} where $f(x,y,t) = (1 - ((x-x_0) - \overline{u}t)^2 - ((y-y_0) - \overline{v}t)^2)/r_c^2$, $M_\infty$ is the Mach number, $u_\infty, \rho_\infty,$ and $p_\infty$ are the free-stream velocity, density, and pressure, respectively. The free-stream velocity is given by $(\overline{u}, \overline{v}) = u_\infty (\cos(\theta), \sin(\theta))$. The strength of the vortex is given by $\epsilon$, and its size is $r_c$. For our test case, we choose parameters $\epsilon = 15$, $r_c = 1.5$, $M_0 = 0.5$, $\theta = \arctan(1/2)$. In order for the temporal error to dominate the spatial error, we compute the solution on a fine mesh consisting of 6144 regular right-triangular elements depicted in Figure \ref{fig:ev-mesh}. The finite element space is chosen to be piecewise polynomials of degree 4, corresponding to 15 nodes per element. The solution consists of four components, for a total of 368,640 degrees of freedom. The initial conditions are shown in Figure \ref{fig:ev-ic}. \subsubsection{Comparison of preconditioners} \label{sec:ev-precond} We first use this test case to compare the effectiveness of the preconditioners discussed in Section \ref{sec:precond}. As a baseline, we will consider the DIRK solver using the block ILU(0) preconditioner. We will then compare the stage-coupled block ILU(0) and both shifted and unshifted stage-uncoupled ILU(0) preconditioners for the IRK solver. In all cases, we require Newton's method to converge to an infinity-norm tolerance of $10^{-8}$. We compare the number of GMRES iterations required to converge to a relative, preconditioned tolerance of $10^{-5}$. In order to make a fair comparison between the methods, we will compute the number of $n \times n$ matrix-vector multiplications required per iteration of Newton's method, across all the stages. We will refer to this quantity as the number of \textit{equivalent multiplications}. For the DIRK methods, this number is equal to the number of GMRES iterations times the number of implicit stages. For the IRK methods, we recall that each multiplication by the large block matrix of the form \eqref{eq:big-irk} essentially consists of $s$ $n \times n$ matrix-vector multiplications. \begin{figure*} \caption{Log-log plot of number of average number of equivalent multiplications vs.\ $\Delta t$. Coupled preconditioners are shown in solid lines, uncoupled shifted preconditioners in dashed lines, and uncoupled, unshifted preconditioners in dotted lines.} \label{fig:ev-precond} \end{figure*} We compute 5 time steps in serial using representative time steps of $\Delta t = 0.4, 0.3, 0.2, $ and $0.1$. We then average the number of GMRES iterations required per linear solve, and multiply this number by the number of implicit stages. For the shifted stage-uncoupled block ILU(0) preconditioner, we choose a shift of $\alpha_i = \sum_{j \neq i} \left| A^{-1}_{ji} \right|$, which has in our experience resulted in the fastest convergence. We present the results in the log-log plot shown in Figure \ref{fig:ev-precond}. We notice that the third order Radau IIA method with the stage-coupled ILU(0) preconditioner requires fewer matrix-vector multiplications than the corresponding third-order DIRK method, while the stage-uncoupled, shifted ILU(0) preconditioner requires roughly the same number of multiplications. For the fifth-order methods, both the coupled and uncoupled preconditioners require more matrix-vector multiplications than the corresponding ESDIRK method. For both third- and fifth-order methods, the stage-uncoupled, unshifted preconditioner requires greatly more matrix-vector multiplications than the other methods, and therefore in the further test cases we will only consider the shifted preconditioner. \subsubsection{Temporal accuracy} Since the analytical solution to this test case is known, it is particularly convenient to compare the accuracy of the time discretization schemes. The solution is integrated until a final time of $t = \SI{60}{\s}$. For the third-order methods, we choose time steps of $\Delta t = 0.4, 0.3, 0.2, 0.1, 0.075, 0.05, \SI{0.025}{\s}$. Because of the increased accuracy of the fifth-order methods, we choose larger time steps of $\Delta t = 0.8, 0.6, 0.5, 0.4, 0.3, \SI{0.2}{\s}$ for the RADAU35 and ESDIRK65 methods. Time steps between $0.5$ and $\SI{1.2}{\s}$ are chosen for the seventh- and ninth-order Radau IIA methods. We approximate the $L^\infty$ error by comparing the numerical solution with the known analytic solution at the DG nodes. We also estimate the order of accuracy by comparing successive choices of $\Delta t$ and computing the rate of convergence \[ r_i = \frac{ \log(L^\infty_{i+1} / L^\infty_i) } { \log(\Delta t_{i+1} / \Delta t_i) }, \] where $L^\infty_i$ denotes the $L^\infty$ error of the numerical solution computed using time step $\Delta t_i$. For each method, we present the wall-clock time required to compute the solution in parallel on 16 cores. The results for both the third-order and fifth-order methods are presented in Table \ref{tab:ev-errors-3}. The theoretical order of accuracy is observed for all of the methods used. Also listed is the ratio of the DIRK error to the Radau IIA error. Comparing the coefficients presented in Table \ref{tab:schemes}, it can easily be shown that the leading coefficient of the truncation error for the DIRK33 method is about 1.86 times larger than the leading coefficient for the RADAU23 method. We see that the ratio of the errors approaches this value as $\Delta t$ tends to zero. The leading coefficient of the truncation error for the ESDIRK65 methods is about 3.82 times larger than the leading coefficient for the RADAU35 method. In the fifth-order test case, the ratio of the numerical errors is found to be closer to about 1.5, likely because of the additional contribution of spatial discretization error. \begin{table}[t!] \footnotesize \centering \caption{$L^\infty$ error and runtime for the Euler vortex, DIRK and Radau IIA methods (wall-clock time presented for stage-coupled/uncoupled preconditioners).} \label{tab:ev-errors-3} \begin{tabular}{l|lcc>{\hspace{1pc}}ccc>{\hspace{1pc}}c} \toprule & \multicolumn{3}{c}{RADAU23} & \multicolumn{3}{c}{DIRK33} \\ $\Delta t$ & $L^\infty$ error & Order & Time C/UC (s) & $L^\infty$ error & Order & Time (s) & Ratio \\ \midrule 0.2 &$7.2932\times 10^{-2}$& - &820/829 &$1.0591\times 10^{-1}$&- &1096& 1.452\\ 0.1 &$9.7053\times 10^{-3}$&2.91&965/956 &$1.6115\times 10^{-2}$&2.72&1268& 1.660\\ 0.075&$4.1135\times 10^{-3}$&2.98&1227/1223&$7.1881\times 10^{-3}$&2.81&1618& 1.747\\ 0.05 &$1.2231\times 10^{-3}$&2.99&1734/1746&$2.2156\times 10^{-3}$&2.90&2291& 1.811\\ 0.025&$1.7832\times 10^{-4}$&2.78&3214/3322&$3.0148\times 10^{-4}$&2.88&4212& 1.691\\ \toprule & \multicolumn{3}{c}{RADAU35} & \multicolumn{3}{c}{ESDIRK65} \\ $\Delta t$ & $L^\infty$ error & Order & Time C/UC (s) & $L^\infty$ error & Order & Time (s) & Ratio \\ \midrule 0.6 &$5.8633\times 10^{-2}$& - &843/775 &$8.3294\times 10^{-2}$&- &697 & 1.421\\ 0.5 &$2.4239\times 10^{-2}$&4.84&974/810 &$3.6075\times 10^{-2}$&4.59&802 & 1.488\\ 0.4 &$7.8902\times 10^{-3}$&5.03&862/905 &$1.2605\times 10^{-2}$&4.71&947 & 1.598\\ 0.3 &$1.8156\times 10^{-3}$&5.11&1079/1030&$3.0723\times 10^{-3}$&4.91&1197& 1.692\\ 0.2 &$2.7294\times 10^{-4}$&4.67&1523/1478&$4.1279\times 10^{-4}$&4.95&1121& 1.512\\ \bottomrule \end{tabular} \begin{tabular}{l|lcc>{\hspace{1.5pc}}c|lcc} \toprule & \multicolumn{3}{c}{RADAU47} & \multicolumn{4}{c}{RADAU59} \\ $\Delta t$ & $L^\infty$ error & Order & Time C/UC (s) & $\Delta t$ & $L^\infty$ error & Order & Time C/UC (s) \\ \midrule 1.0 &$1.8699\times 10^{-2}$& - & 1222/1029 & 1.2 &$5.0425\times 10^{-3}$& - & 2117/1533 \\ 0.8 &$4.5413\times 10^{-3}$&6.34& 1158/1016 & 1.0 &$1.7294\times 10^{-3}$&5.87& 2094/1567 \\ 0.6 &$8.7259\times 10^{-4}$&5.73& 1463/1307 & 0.8 &$4.6966\times 10^{-4}$&5.84& 1988/1554 \\ 0.5 &$3.0265\times 10^{-4}$&5.81& 1732/1541 & 0.6 &$8.8679\times 10^{-5}$&5.79& 2544/2040 \\ \bottomrule \end{tabular} \end{table} \begin{figure*} \caption{Log-log plot of $L^\infty$ error vs.\ $1/\Delta t$} \caption{Log-log plot of $L^\infty$ error vs.\ wall-clock time} \caption{Log-log plots of $L^\infty$ error vs.\ time-step and wall-clock time for Euler vortex test case. Stage-coupled preconditioners are shown in solid lines, stage-uncoupled in dashed lines.} \label{fig:ev-conv} \label{fig:ev-wallclock} \end{figure*} A log-log plot of the $L^\infty$ error vs.\ $\Delta t$ is shown in Figure \ref{fig:ev-conv}. A log-log plot of the $L^\infty$ error vs.\ wall-clock time is shown in Figure \ref{fig:ev-wallclock}. We remark that the RADAU23 method achieved the same accuracy as the DIRK33 method in faster runtime for all of the cases considered. Among the fifth-order methods, we did not observe one method to be clearly more efficient than the others. The difference between the stage-coupled and stage-uncoupled preconditioners was found to be insignificant for the third- and fifth-order methods, and the stage-uncoupled preconditioner resulted in faster performance for the seventh- and ninth-order methods. \subsection{High Reynolds number flow over 2D NACA airfoil (LES)} \label{sec:naca-2d} \begin{figure} \caption{Mesh of NACA airfoil with 3154 elements} \label{fig:naca-mesh1} \caption{Boundary layer elements with $p=3$ DG nodes} \label{fig:naca-bdry} \caption{NACA airfoil mesh} \label{fig:naca-mesh} \end{figure} In this test case, we consider the two-dimensional viscous flow around a NACA airfoil with an angle of attack of $\SI{30}{\degree}$. The fluid domain is the rectangle $[-2.5, 4] \times[-2.5, 2.5]$. For this case, the Mach number is taken to be 0.1, the gas constant 1.4, and the Reynolds number 40,000. The wing is centered vertically, and placed closer to the inlet boundary. The mesh consists of 3154 triangular elements, with finer elements close to the airfoil and in the wake. The mesh and the boundary-layer elements with DG nodes are depicted in Figure \ref{fig:naca-mesh}, and are rotated by $\SI{30}{\degree}$ to correspond with the desired angle of attack. The triangles near the boundary of the wing are refined in the transverse direction, resulting in highly anisotropic elements. These stretched elements give rise to a highly restrictive CFL condition, suggesting that this problem is particularly well suited to implicit methods. We have found that the CFL condition renders explicit methods impractical for this problem, with the fourth-order explicit Runge-Kutta method exhibiting instability for time steps greater than $7\times 10^{-8}\ \si\second$. On the other hand, the implicit Runge-Kutta methods remain stable for time steps many orders of magnitude larger. A no-slip condition is enforced on the boundary of the airfoil, and far-field conditions are enforced on all other boundaries. At time $t = \SI{0}{\s}$ the solution is set to be free-stream everywhere. The solution is then integrated until time $t = \SI{5}{\s}$, at which point vortices have developed in the wake of the wing. This solution (shown in Figure \ref{fig:naca-ic}) is taken to be the initial condition for our test case. The finite element space is taken to be piecewise degree 3 polynomials, with 10 nodes per element, resulting in 94,620 degrees of freedom. \begin{figure} \caption{Initial condition, $t=\SI{5} \label{fig:naca-ic} \caption{Solution, $t=\SI{5.75} \label{fig:naca-final} \caption{Density plots for NACA $\rm Re = 40k$ test case} \label{fig:naca-soln} \end{figure} \subsubsection{Solver efficiency} We study the effectiveness of the DIRK and Radau IIA IRK methods by comparing both the average number of equivalent multiplications per linear solve and the total wall-clock time. As in the previous case, the methods RADAU23, RADAU35, DIRK33, and ESDIRK65 are used. As in the case of the Euler vortex, a tolerance of $10^{-8}$ is used for the Newton solver. We integrate the equations from time $t = \SI{5}{\s}$ until $t = \SI{5.75}{\s}$. For the third-order methods, time steps of $\Delta t = 1.25 \times 10^{-2}, 7.50 \times 10^{-3}, 6.25 \times 10^{-3}, 5.00 \times 10^{-3}, 2.50 \times 10^{-3}, 1.25 \times 10^{-3},$ are used. For the fifth-order methods, we use the same time steps, in addition to the larger time steps of $\Delta t = 5.00 \times 10^{-2},$ and $\Delta t = 2.50 \times 10^{-2}$. In Table \ref{tab:naca-results} we present the runtime and number of equivalent multiplications for all of the methods considered. As in Section \ref{sec:ev-precond}, we compute the number of $n\times n$ matrix-vector products performed per Newton iteration (over all the stages) by multiplying the average number of GMRES iterations by the number of implicit stages. In Figure \ref{fig:naca-matvec} we present a log-log plot of the average number of equivalent multiplications vs.\ $\Delta t$. In Figure \ref{fig:naca-wallclock} we present the total wall-clock time required to run the simulation until the final time. \begin{table}[t!] \scriptsize \centering \caption{Equivalent multiplications and wall-clock time for NACA LES test case} \label{tab:naca-results} \begin{tabular}{l|cc>{\hspace{1pc}}cc>{\hspace{1pc}}cc} \toprule & \multicolumn{2}{c}{RADAU23 (Coupled)} & \multicolumn{2}{c}{RADAU23 (Uncoupled)} & \multicolumn{2}{c}{DIRK33} \\ $\Delta t$ & Mult. &Time (s) & Mult. &Time (s) &Mult. &Time (s) \\ \midrule $1.25\times 10^{-2}$ & 70.2 & 75.4 & 72.0 & 72.1 & 100.2 & 92.1 \\ $7.50\times 10^{-3}$ & 53.4 & 96.5 & 56.0 & 93.1 & 81.3 & 133.0\\ $6.25\times 10^{-3}$ & 48.8 & 105.2 & 52.4 & 105.5 & 73.2 & 145.7\\ $5.00\times 10^{-3}$ & 43.0 & 116.5 & 46.2 & 117.1 & 65.1 & 162.9\\ $2.50\times 10^{-3}$ & 34.0 & 208.9 & 36.8 & 202.7 & 42.6 & 238.0\\ $1.25\times 10^{-3}$ & 27.2 & 389.8 & 30.8 & 386.7 & 33.6 & 437.5\\ \toprule & \multicolumn{2}{c}{RADAU35 (Coupled)} & \multicolumn{2}{c}{RADAU35 (Uncoupled)} & \multicolumn{2}{c}{ESDIRK65} \\ $\Delta t$ &Mult. &Time (s) &Mult. &Time (s) &Mult. &Time (s) \\ \midrule $5.00\times 10^{-2}$ & 189.2 & 59.7 & 178.5 & 49.4 & 216.0 & 51.3 \\ $2.50\times 10^{-2}$ & 132.6 & 79.8 & 130.2 & 69.2 & 173.5 & 81.0 \\ $1.25\times 10^{-2}$ & 107.7 & 141.0 & 107.1 & 113.4 & 137.0 & 136.6\\ $7.50\times 10^{-3}$ & 96.0 & 189.2 & 96.0 & 169.8 & 107.5 & 184.6\\ $6.25\times 10^{-3}$ & 89.1 & 222.0 & 93.3 & 200.7 & 95.0 & 197.2\\ $5.00\times 10^{-3}$ & 78.6 & 230.4 & 88.8 & 263.2 & 81.5 & 217.8\\ $2.50\times 10^{-3}$ & 54.9 & 340.2 & 73.5 & 461.7 & 60.0 & 375.9\\ $1.25\times 10^{-3}$ & 42.6 & 619.4 & 60.6 & 836.3 & 47.0 & 720.1\\ \bottomrule \end{tabular} \end{table} \begin{figure*} \caption{Matrix-vector multiplications vs. $\Delta t$} \label{fig:naca-matvec} \caption{Wall-clock time vs. $\Delta t$} \label{fig:naca-wallclock} \caption{Log-log plots of average number of equivalent multiplications and wall-clock time vs. $\Delta t$ for the 2D NACA LES test case} \label{fig:naca-results} \end{figure*} For the third-order methods, the RADAU23 IRK method required fewer matrix-vector multiplications than the DIRK33 method, for the same choice of $\Delta t$, resulting in a shorter run time. The uncoupled preconditioner resulted in a somewhat larger number of GMRES iterations, and hence more matrix-vector multiplications, but the difference in run time was found to be negligible. For the fifth-order methods, the RADAU35 method with stage-coupled preconditioner resulted in a smaller number of matrix-vector multiplications per linear solve than the ESDIRK65 method. For larger $\Delta t$, the stage-uncoupled preconditioner proved to be effective, resulting in faster run times than both the DIRK method and the stage-coupled preconditioner. For smaller $\Delta t$, the stage-uncoupled preconditioner required a greater number of GMRES iterations and hence longer run times. \subsubsection{Accuracy} \begin{table}[h!] \centering \scriptsize \caption{$L^\infty$ error for NACA LES test cases} \label{tab:naca-errors} \begin{tabular}{l|cc>{\hspace{1pc}}cc>{\hspace{1pc}}c} \toprule & \multicolumn{2}{c}{RADAU23} & \multicolumn{2}{c}{DIRK33} \\ $\Delta t$ &$L^\infty$ & Order & $L^\infty$ & Order & Ratio\\ \midrule $1.25\times 10^{-2}$ & $6.006\times 10^{-1}$ & - & $4.514\times 10^{-1}$ & - & 1.668\\ $7.50\times 10^{-3}$ & $1.743\times 10^{-1}$ & 2.42 & $1.615\times 10^{-1}$ & 2.01 & 1.706\\ $6.25\times 10^{-3}$ & $1.207\times 10^{-1}$ & 2.02 & $1.423\times 10^{-1}$ & 0.70 & 1.717\\ $5.00\times 10^{-3}$ & $8.503\times 10^{-2}$ & 1.57 & $1.216\times 10^{-1}$ & 0.70 & 1.735\\ $2.50\times 10^{-3}$ & $1.347\times 10^{-2}$ & 2.66 & $2.022\times 10^{-2}$ & 2.59 & 1.839\\ $1.25\times 10^{-3}$ & $1.342\times 10^{-3}$ & 3.33 & $2.708\times 10^{-3}$ & 2.90 & 1.823\\ \toprule & \multicolumn{2}{c}{RADAU35} & \multicolumn{2}{c}{ESDIRK65} \\ $\Delta t$ &$L^\infty$ & Order & $L^\infty$ & Order & Ratio\\ \midrule $5.00\times 10^{-2}$ & $4.601\times 10^{-1}$ & - & $6.239\times 10^{-1}$ & - & 2.308\\ $2.50\times 10^{-2}$ & $3.158\times 10^{-1}$ & 0.54 & $5.853\times 10^{-1}$ & 0.092 & 1.896\\ $1.25\times 10^{-2}$ & $6.739\times 10^{-2}$ & 2.23 & $1.127\times 10^{-1}$ & 2.377 & 2.414\\ $7.50\times 10^{-3}$ & $5.124\times 10^{-3}$ & 5.04 & $3.989\times 10^{-3}$ & 6.540 & 4.441\\ $6.25\times 10^{-3}$ & $5.329\times 10^{-3}$ & -0.22 & $2.147\times 10^{-3}$ & 3.397 & 3.263\\ $5.00\times 10^{-3}$ & $2.616\times 10^{-3}$ & 3.19 & $3.279\times 10^{-3}$ & -1.897 & 3.200\\ $2.50\times 10^{-3}$ & $8.135\times 10^{-4}$ & 1.68 & $2.222\times 10^{-3}$ & 0.561 & 1.044\\ $1.25\times 10^{-3}$ & $2.059\times 10^{-4}$ & 1.98 & $1.727\times 10^{-4}$ & 3.686 & 0.896\\ \bottomrule \end{tabular} \end{table} \begin{figure*} \caption{Log-log plots of $L^\infty$ errors vs.\ wall-clock time for the NACA LES test case} \label{fig:naca-errors} \end{figure*} We note that the above comparisons were made for equal choices of $\Delta t$. Because the Radau IIA method enjoy a smaller leading coefficient of the truncation error, we expect to achieve better accuracy for the same time step. We therefore study the accuracy of the methods applied to the above problem by considering the semidiscrete system of equations purely as a system of ODEs. We compute a reference solution by numerically integrating the equations for 6000 time steps with $\Delta t = 1.25 \times 10^{-4}$ using the fifth-order ESDIRK method. Then, we take this solution to be the ``exact'' solution, with respect to which the $L^\infty$ norm of the error is computed, and perform a grid convergence study. For each choice of time step, we compute the $L^\infty$ norm of the error, the approximate rate of convergence of the method, and the ratio of the DIRK error to the Radau IIA error in Table \ref{tab:naca-errors}. In Figure \ref{fig:naca-errors}, we present log-log plots of the wall-clock time vs.\ $L^\infty$ error. We note that we do not observe the formal order of temporal accuracy for this test problem, possibly due to the choice of time step, which is about five orders of magnitude larger than the explicit CFL, together with the stiff and turbulent nature of the problem. In order to verify the formal order of accuracy, we also run this test problem with time steps that are two to three orders of magnitude smaller than those of the above comparison (for a shorted total time of integration). The results presented in Table \ref{tab:naca-small-dt} confirm that the expected theoretical orders of accuracy are attained for all methods considered. \begin{table}[h!] \centering \scriptsize \caption{$L^\infty$ error for NACA LES test cases (order verification)} \label{tab:naca-small-dt} \begin{tabular}{l|cc>{\hspace{1pc}}cc>{\hspace{1pc}}c} \toprule & \multicolumn{2}{c}{RADAU23} & \multicolumn{2}{c}{DIRK33} \\ $\Delta t$ &$L^\infty$ & Order & $L^\infty$ & Order \\ \midrule $6.25\times 10^{-5}$ & $5.694\times 10^{-6}$ & - & $3.580\times 10^{-6}$ & - \\ $3.125\times 10^{-5}$ & $8.765\times 10^{-7}$ & 2.70 & $5.004\times 10^{-7}$ & 2.84 \\ $1.5625\times 10^{-5}$ & $1.177\times 10^{-7}$ & 2.90 & $6.401\times 10^{-8}$ & 2.97 \\ $7.8125\times 10^{-6}$ & $1.494\times 10^{-8}$ & 2.98 & $8.033\times 10^{-9}$ & 2.99 \\ \toprule & \multicolumn{2}{c}{RADAU35} & \multicolumn{2}{c}{ESDIRK65} \\ $\Delta t$ &$L^\infty$ & Order & $L^\infty$ & Order \\ \midrule $1.25\times 10^{-4}$ & $3.438\times 10^{-6}$ & - & $1.231\times 10^{-6}$ & - \\ $6.25\times 10^{-5}$ & $1.621\times 10^{-7}$ & 4.41 & $4.758\times 10^{-8}$ & 4.69 \\ $3.125\times 10^{-5}$ & $5.434\times 10^{-9}$ & 4.90 & $1.602\times 10^{-9}$ & 4.89 \\ $1.5625\times 10^{-5}$ & $1.964\times 10^{-10}$ & 4.79 & $7.913\times 10^{-11}$ & 4.34 \\ \bottomrule \end{tabular} \end{table} We remark that for the third-order methods, we can achieve the same accuracy as the DIRK33 method with a faster run time using the RADAU23 method, with both the stage-coupled or stage-uncoupled preconditioner. The differences in run time between the two preconditioners were negligible. For the fifth-order methods, our results indicate that the RADAU35 method outperformed the ESDIRK for a majority of the test cases considered. For this test problem, we found the stage-uncoupled preconditioner to perform better for the larger choices of $\Delta t$. \subsubsection{Parallel performance on NACA airfoil} \label{sec:parallel-results} We study the parallel performance of the IRK and DIRK solvers applied to the two-dimensional NACA airfoil. We choose a representative time step of $\Delta t = 1.25\times 10^{-2}$, and integrate the system for five time steps until $t = \SI{5.0625}{\s}$, using as before the solution at $t = \SI{5}{\s}$ as the initial condition. We perform this test using both the Radau IIA IRK and the DIRK solvers. For the Radau methods, we use both the stage-coupled and stage-uncoupled ILU(0) preconditioners. \begin{figure} \caption{Log-log plot of average number of equivalent multiplications vs.\ number of processes. Dashed lines indicate stage-uncoupled block ILU(0) preconditioner, and dotted lines indicate stage-parallel solver.} \label{fig:naca-parallel} \end{figure} Using the method described in Section \ref{sec:parallel}, we decompose the domain into a set number of partitions according to the number of processes. For the DIRK and stage-coupled IRK solvers, the number of partitions is equal to the number of processes. For the stage-uncoupled IRK solver, we can choose the number of partitions to be a factor of $s$ smaller than the number of processes. We consider the mesh decomposed into 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512 partitions. Then, we compute the average number of GMRES iterations required per solve. In the case of the DIRK methods, we multiply the number of iterations by the number of implicit stages to obtain the number of equivalent multiplications performed assuming one Newton iteration. Similarly, in the case of the Radau IIA IRK methods, we multiply the number of iterations by the number of stages to obtain the number of $n \times n$ matrix-vector multiplications. In Figure \ref{fig:naca-parallel} we show a log-log plot of the average number of equivalent multiplications vs.\ number of processes. As the number of processes (and hence number of mesh partitions) increases, we observe an increase in the number of GMRES iterations required to converge. This is because the contributions between different mesh partitions are ignored in the block ILU(0) factorization, rendering the preconditioner less effective. Since the stage-uncoupled ILU(0) preconditioner can be parallelized for the same number of processes with a factor of $s$ fewer partitions, we notice that approximately 15--20\% fewer matrix-vector multiplications are required when compared with the standard uncoupled solver. This suggests a substantial benefit to the stage-uncoupled block ILU(0) preconditioner when run in a massively-parallel environment. This difference in performance is numerically validated in the following three-dimensional NACA test case. \subsection{Parallel large eddy simulation (LES) of 3D NACA airfoil} \label{sec:naca-3d} \begin{figure} \caption{Boundaries of three-dimensional NACA mesh.} \label{fig:naca3d-mesh} \caption{Isosurfaces of $Q$-criterion, $Q = 25$, colored by velocity magnitude.} \label{fig:naca3d-isosurface} \caption{Three-dimensional NACA LES test case.} \label{fig:naca3d} \end{figure} For our final test case, we consider the three-dimensional viscous flow over a NACA airfoil with angle of attack of $\SI{30}{\degree}$. The Reynolds number is taken to be 5,300 and the Mach number 0.1. The governing equations are given by equations \eqref{eq:ns-1} through \eqref{eq:ns-3} with the isentropic assumption discussed in Section \ref{sec:eqns}. The mesh consists of 151,392 tetrahedral elements. The local basis consists of degree 3 polynomials, for a total of 20 nodes per element. We consider the fluid to be isentropic, and hence the solution consists of 4 components, resulting in a total of 12,111,360 degrees of freedom. The mesh is shown in Figure \ref{fig:naca3d-mesh}. A no-slip condition is enforced on the boundary of the airfoil, periodic conditions are enforced in the span-wise direction, and far-field conditions are enforced on all other boundaries. The solution is initialized to freestream conditions, and then integrated numerically until time $t = \SI{4}{\s}$ using $\Delta t = \SI{0.02}{\s}$. At this point, vortices have developed in the wake of the wing. In Figure \ref{fig:naca3d-isosurface}, isosurfaces of the $Q$-criterion, for $Q = 25$, are shown. The $Q$-criterion, proposed by Dubief and Delcayre in \cite{Dubief2000} is often used to identify vortical structures, and is defined as the difference of the symmetric and antisymmetric components of the velocity gradient, \begin{equation} Q = \frac{1}{2}\left( \Omega_{ij}\Omega_{ij} - S_{ij}S_{ij} \right), \end{equation} where $\Omega_{ij} = \frac{1}{2}\left(u_{i,j} - u_{j,i}\right)$, and $S_{ij} = \frac{1}{2}\left(u_{i,j} + u_{j,i}\right)$. We then integrate the equations for 30 time steps using the third- and fifth-order DIRK methods, as well as the Radau IIA methods of order 3, 5, 7, and 9. Due to the turbulent nature of this problem, we do not study the accuracy of the numerical solutions, but rather the efficiency of the solvers for fixed $\Delta t$. We also consider the parallel scaling of the solvers by running our test case on 360, 540, 720, and 1080 processes. For the Radau IIA methods, we consider three preconditioners: stage-coupled, stage-uncoupled, and stage-parallel. As with the DIRK methods, for the stage-coupled and stage-uncoupled preconditioners we decompose the mesh into a number of partitions equal to the number of processes. For the stage-parallel ILU(0) preconditioner we decompose the mesh into a factor of $s$ fewer partitions. The preconditioner then exploits the stage parallelism using the methodology described in Section \ref{sec:parallel}. We record the average wall-clock time required per linear solve using each of the methods, and present the results in Figure \ref{fig:naca3d-runtime}. \begin{figure} \caption{Log-log plot of linear solve time in seconds vs.\ number of processes. Dashed lines indicate stage-uncoupled block ILU(0) preconditioner, and dotted lines indicate stage-parallel solver. A reference triangle for perfect speedup is shown.} \label{fig:naca3d-runtime} \end{figure} For each order of accuracy, the Radau IIA method with the stage-parallel preconditioner resulted in the fastest runtime. For orders three and five, for which we compare against equal-order DIRK methods, the Radau IIA methods resulted in faster performance with all of the preconditioners considered. Since the number of stages is greater for the higher order methods, and the stage-parallel preconditioner allows for a factor of $s$ fewer mesh partitions, we would expect that the reduction in the number of GMRES iterations due to the improved ILU preconditioner would be greater than for the lower order methods. Indeed, we observe that the relative performance gain of the stage-parallel preconditioner compared with the stage-uncoupled preconditioner increases as the the order of the method increases. For example, while the third-order, stage-parallel preconditioner is only about 5--10\% faster than the stage-uncoupled preconditioner, and about 20\% faster than the stage-coupled preconditioner, the ninth-order stage-parallel preconditioner is between 20--30\% faster than the stage-uncoupled preconditioner, and approximately twice as fast as the stage-coupled preconditioner. Furthermore, as the number of processes approaches the strong scaling limit, we anticipate that the additional factor of $s$ processes allowed by the stage-parallel preconditioner will result in even greater benefit, especially for those methods with a large number of stages. \section{Conclusion} In this paper, we have developed a new strategy for efficiently solving the large, coupled linear systems arising from fully implicit Runge-Kutta time discretizations. By transforming the system of equations, the computational work required per GMRES iterations is reduced significantly. This new method makes it feasible to use high-order, $L$-stable implicit Runge-Kutta methods, such as the Radau IIA methods, as time integrators for discontinuous Galerkin discretizations. We additionally develop new preconditioners for these methods, included a parallel-in-time ILU preconditioner that allows for the computation of the Runge-Kutta stages simultaneously. Numerical experiments on both two- and three-dimensional fluid flow problems are performed using the Radau IIA methods of up to ninth order. These results indicate that using the transformed system of equations, the fully implicit IRK methods are competitive with, and in our experience, often preferable to the more standard DIRK methods, both in terms of efficiency and accuracy. In a parallel computing environment, the stage-parallel ILU preconditioner results in additional performance gains. \section{Acknowledgments} This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No.\ DE-AC02-05CH11231. The first author was supported by the Department of Defense through the National Defense Science \& Engineering Graduate Fellowship Program. \appendix \input{tableaux} \end{document}

\begin{document} \title{Simulating Arbitrary Pair-Interactions by a Given Hamiltonian: Graph-Theoretical Bounds on the Time Complexity} \abstract{ We use an $n$-spin system with permutation symmetric $zz$-interaction for simulating arbitrary pair-interaction Hamiltonians. The calculation of the required time overhead is mathematically equivalent to a separability problem of $n$-qubit density matrices. We derive lower and upper bounds in terms of chromatic index and the spectrum of the interaction graph. The complexity measure defined by such a computational model is related to gate complexity and a continuous complexity measure introduced in a former paper. We use majorization of graph spectra for classifying Hamiltonians with respect to their computational power. } \section{Introduction} The most common models for quantum computers use single and two qubit gates as basic transformations in order to generate arbitrary unitary operations on the quantum registers. Most discussions about the generation of quantum algorithms, quantum codes and possible realizations had successfully been based on this concept. Mostly, even the definition of quantum complexity refers to such a model \cite{nielsen}. Nevertheless there is a priori no reason, why two qubit gates should be considered as basic operations for future quantum computers. In principle every quantum system could serve as a quantum register provided that its time evolution can be controlled in a universal way. At first sight, every definition of quantum complexity seems hence to be adequate only for a specific model of quantum computation. But it seems to be a rather general feature of Hamiltonians available in nature that particles interact with other particles in such a form, that the total Hamiltonian is a sum of pair-interactions. Therefore we want to base quantum complexity theory only on such a general feature.\footnote{This might be seen in the spirit of D.~Deutsch's statement ``What computers can or cannot compute is determined by the laws of physics alone and not by pure mathematics.''\cite{nielsen}, Chapter II} This feature justifies the following control theoretic model: If $n$ qubits are assumed to be physically represented by $n$ particles, the only part of the system's Hamiltonian which can be changed by extern access is the free Hamiltonian of each qubit. These $1$-particle Hamiltonians might be controllable since they are only effective Hamiltonians which are phenomenologically given by an interaction to many extern particles (mean-field approximation \cite{duffield}). Based on results of quantum control theory in $2$-spin systems \cite{kha} we investigate the problem of simulating arbitrary pair-interaction Hamiltonians by a given one. We assume that the Hamiltonian of the $n$-system is a permutation invariant $zz$-interaction and show that the computational power\footnote{in the sense of time required to generate unitaries} of this Hamiltonian (together with local transformations on each spin) is at least as large as the power of quantum computers with $2$ qubit gates. For infinitesimal time evolutions, it turns out to be even stronger. We develop a theory, where the computational power of a Hamiltonian for simulating arbitrary Hamiltonians is characterized by features of the interactions graphs. Standard concepts of graph theory like chromatic index and spectrum of the adjacency matrix together with majorization turn out to provide lower and upper bounds on the simulation overhead. Here we are interested in the exact overhead and not only in polynomial equivalence as in \cite{dodd}. \section{Our model of computation} Based on the approach of \cite{kha} we consider the following model. The quantum system is a spin system, i.e.\ its Hilbert space is $\mathcal{H}_n=(\mathbb{C}^2)^{\otimes n}$, and its Hamiltonian $H_d\in\mathfrak{su}(2^n)$ consists only of pair-interactions, i.e.\ \begin{equation} H_d=\sum_{1\le k<l\le n} H_{k,l} \end{equation} where $H_{kl}$ acts only on the Hilbert space of the qubits $k$ and $l$. We assume that for every $k$ and $l$ the Hamiltonian $H_{kl}$ describes a non-trivial coupling and is traceless. The system's Hamiltonian $H_d$ is also called the \emph{drift} Hamiltonian since it is always present. We assume that we can perform all unitaries in the \emph{control} group $K=SU(2)\otimes\cdots\otimes SU(2)$ arbitrarily fast compared to the time evolution of the internal couplings between the qubits. Let $G$ be the unitary Lie group $SU(2^n)$ and $u\in G$ be a unitary we want to realize. To achieve this all we can do is perform $v_1\in K$, wait $t_1$, perform $v_2\in K$, wait $t_2\,,\ldots,$ perform $v_p\in K$ and wait $t_p$. The resulting unitary is $$u=\exp(i H_d t_p) v_p \cdots \exp(i H_d t_2) v_2 \exp(i H_d t_1) v_1\,. $$ This can be written as $$ u=k_p \exp(i k_p^\dagger H_d k_p) \cdots \exp(i k_2^\dagger H_d k_2) \exp(i k_1^\dagger H_d k_1)$$ where $k_i=v_i \cdots v_1$ for $i=1,\ldots,p$. This is just the solution of a time-dependent Schr\"odinger equation with piecewise constant Hamiltonians -- conjugates of the drift Hamiltonian $H_d$ by unitaries of $K$ -- followed by the unitary $k_p\in K$. Let $Ad_K(H_d)$ denote the conjugacy class $$ Ad_K(H_d)=\{Ad_k(H_d)=k^{\dagger} H_d k\mid k\in K\}\,. $$ \begin{Definition} A continuous time algorithm $A$ of running time $T$ is a piecewise constant function $t\mapsto H(t)$ from the interval $[0,T]$ onto the set $Ad_K(H_d)$ followed by some local unitary $k\in K$. We say $A$ implements $u$ if $u=k u(T)$ where $(u(t))_{t\in [0,T]}$ is the solution of the time-dependent Schr\"odinger equation $(d/dt) u(t) = -iH(t) u(t)$ with $u(0)=I$. \end{Definition} The complexity of a unitary in this model is the running time of the optimal continuous time algorithms. Let $\sigma_\alpha^i$ denote the Pauli spin matrix $\sigma_\alpha$ ($\alpha=x,y,z$) that acts on the $i$th spin. For simplicity, we assume that the drift Hamiltonian is \begin{equation} H_d=\sum_{1\le k<l\le n} \sigma_z^k\sigma_z^l\,. \end{equation} The physical systems we have in mind might be for example solid states with long-range interactions. Of course one might object that the interaction strength always decreases with the distance between the interacting particles. It will turn out that the assumption on non-decreasing interaction strengths makes our model rather strong with respect to its computational power. One should understand our assumptions as the attempt to use a strong computational model which is still physically justificable. Many aspects of our theory can be developed in strong analogy for more general drift Hamiltonians. In Section~\ref{LowUpBounds} we will compare the computational power of our model with the power of quantum computers based on $2$-qubit gates. Our arguments refer always to infinitesimal time evolutions, i.e., we will show that our model can implement quantum gates without overhead since we can simulate the time evolution implementing parallelized quantum gates. First we have to define what we mean by simulating the time evolution $\exp(i H t)$ during a small time interval $[0,\epsilon]$ where $H$ is an arbitrary pair-interaction Hamiltonian. Assume we have written $H$ as a positive linear combination $H=\sum_j \mu_j H_j$ with $\mu_j>0$ and each $H_j$ is an element of the conjugacy class $Ad_K(H_d)$. For small $\epsilon$ the unitary \[ \prod_j \exp(i\epsilon\mu_j H_j) \] is a good approximation for \[ \exp(i\epsilon H)=\exp(i\epsilon\sum_j\mu_j H_j)\,. \] This approximation is implemented if the system evolves the time $\epsilon\mu_j$ with respect to the Hamiltonian $H_j$. The sum $\mu=\sum_j\mu_j$ is exactly the time overhead of the simulation. Hence the problem is to express $H$ as a positive linear combination such that the overhead $\mu$ is minimal. Of course such a procedure might not be optimal if one were interested in the implementation of $\exp(iHs)$ for any special value of $s$. Here we want to imitate the whole dynamical time evolution $(\exp(iHs))_{s>0}$ in arbitrary small steps $\epsilon$. Then the optimization reduces clearly to the convex problem stated above. In the following it will be convenient to use a concise representation for the drift Hamiltonian and the interaction to be simulated: a pair interaction Hamiltonian between qubits $k$ and $l$ can be written as \begin{equation} H_{kl} = \sum_{\alpha,\beta=x,y,z} J_{kl;\alpha\beta} \sigma_{\alpha}^k\sigma_{\beta}^l\,. \end{equation} The strengths of the components are represented by the pair-interaction matrix \begin{equation} J_{kl}=\left( \begin{array}{ccc} J_{kl;xx} & J_{kl;xy} & J_{kl;xz} \\ J_{kl;yx} & J_{kl;yy} & J_{kl;yz} \\ J_{kl;zx} & J_{kl;zy} & J_{kl;zz} \end{array} \right)\in\mathbb{R}^{3\times 3}\,. \end{equation} The total Hamiltonian $H$ is represented by the $J$-matrix \begin{equation} J=\left( \begin{array}{c|c|c|c|c} 0 & J_{12} & J_{13} & \cdots & J_{1n} \\ \hline J_{21} & 0 & J_{23} & \cdots & J_{2n} \\ \hline J_{31} & J_{32} & 0 & & J_{3n} \\ \hline \vdots & \vdots & & \ddots & \\ \hline J_{n1} & J_{n2} & J_{n3} & \quad & 0 \end{array} \right)\in\mathbb{R}^{3n\times 3n}\,. \end{equation} To explain more explicitly, why our simulation problem is a convex optimization, we recall that every convex combination $\mu H_1 + (1-\mu)H_2$ of two Hamiltonians $H_1$ and $H_2$ can be simulated with overhead $1$ if $H_1$ and $H_2$ can. Remarkably, the problem of specifying the set of Hamiltonians which can be simulated with overhead $1$ is related to the problem of generalizing Bell inequalities to $n$-qubit states. More specifically, the convex problem can be reduced to the question `how strong can $2$-spin correlations be in a separable $n$-qubit quantum state?' \begin{Theorem}[Optimal simulation]\label{optimal} The Hamiltonian $H$ can be simulated with overhead $\mu$ if and only if there is a separable quantum state $\rho$ in $(\mathbb{C}^2)^{\otimes n}$ such that $$ \frac{1}{\mu}J+I= (\mathrm{tr}(\rho\sigma_{\alpha}^k\sigma_{\beta}^l))_{kl;\alpha\beta}$$ where $J$ denotes the $J$-matrix of $H$ and $I$ the $3n\times 3n$ identity matrix. \end{Theorem} Proof: By rescaling the considered Hamiltonian, it is sufficient to show that this is true for all Hamiltonians in $Ad_K(H_d)$ with $\mu=1$. Assume we have written $H$ as a convex combination $H=\sum_j\mu_j H_j$ with $H_j\in Ad_K(H_d)$. In order to show that there is a separable state of the desired form it is sufficient to show that the $J$-matrix of each Hamiltonian $H_j$ satisfies the equation of the theorem for an appropriate separable state. Let $H_j=uH_du^\dagger$. $H_j$ can be represented by $n$ three dimensional real unit vectors: to each qubit we associate the vector $|J_k\rangle=(J_{k;x},J_{k;y},J_{k;z})^t\in \mathbb{R}^3$ where $u_k\sigma_z u_k^\dagger=J_{k;x}\sigma_x + J_{k;y}\sigma_y + J_{k;z}\sigma_z$ and $u=u_1\otimes\ldots\otimes u_n$. The pair-interaction matrices are given by the matrix products $J_{kl}=|J_k\rangle\langle J_l|$. By the Bloch sphere representation we have a correspondence between the unit vectors $|J_k\rangle$ and the projections $\rho_k$ in $\mathbb{C}^2$ defined by $J_{k;\alpha}=\mathrm{tr}(\rho_k \sigma_{\alpha})$. Let $\rho$ be the product state $\rho:=\rho_1\otimes\ldots\otimes\rho_n$. Then we have $J_{kl;\alpha\beta}=\mathrm{tr}(\rho \sigma_{\alpha}^k\sigma_{\beta}^l)$ for all $k\neq l$. Note that the product of two different Pauli matrices is the third Pauli matrix multiplied by a scalar. The only problem that remains is that we may have $\mathrm{tr}(\rho\sigma_\alpha^k\sigma_\beta^k)\neq 0$ for $\alpha\neq\beta$. We substitute $\rho$ by a state $\bar{\rho}$ in such a way that the expectation values of all traceless $1$-qubit observables vanish and the expectation values of all considered $2$-qubit observables remain unchanged. For every $|J_k\rangle$ we can find $U'_k\in SO(3)$ such that $U'_k|J_k\rangle=-|J_k\rangle$. This rotation corresponds to conjugation of the qubit $k$ by a unitary $u'_k$. To $-|J_k\rangle$ corresponds the projection $\rho'_k := I_2-\rho_k$. Let $$ \bar{\rho}:= \frac{1}{2} (\rho_1\otimes\ldots\otimes\rho_n + \rho'_1\otimes\ldots\otimes\rho'_n) \,. $$ Then we have $\mathrm{tr}(\bar{\rho} \sigma_{\alpha}^k\sigma_{\beta}^l)= \mathrm{tr}(\rho \sigma_{\alpha}^k\sigma_{\beta}^l)$ for all $k\neq l$ (all vectors are multiplied by $-1$ and therefore there is no effect on the pairs) and $J_{kk}$ is the $3\times 3$ identity matrix. Assume conversely we have a separable state of the desired form. Take its decomposition into pure product states. By the Bloch sphere representation we obtain the required conjugations of the drift Hamiltonian $H_d$. The time they have to be applied are given by the coefficients in the convex decomposition. $\Box$ \section{Lower and upper bounds}\label{LowUpBounds} A simple lower bound on the simulation time overhead can be derived from the fact that $J/\mu+I$ has to be a positive matrix, which is an easy conclusion from Theorem~\ref{optimal}. \begin{Corollary}[Lower bound] The absolute value of the smallest eigenvalue of the $J$-matrix is a lower bound on the simulation overhead of $H$. \end{Corollary} Proof: The matrix $$ (\mathrm{tr}(\rho\sigma_{\alpha}^k\sigma_{\beta}^l))_{kl;\alpha\beta} $$ is positive for every state $\rho$ in $(\mathbb{C}^2)^{\otimes n}$: let $|d\rangle=(d_{k;\alpha})$ be an arbitrary vector and $A=\sum_{k,\alpha} d_{k;\alpha} \sigma_{\alpha}^k$. Then we have $$ \sum_{k,l,\alpha,\beta} d_{k;\alpha} \mathrm{tr}(\rho\sigma_{\alpha}^k\sigma_{\beta}^l) d_{l;\beta} = \mathrm{tr}(\rho AA^*)\ge 0\,. $$ {} $\Box$ Now we show that our computational model is at least as powerful as the usual model with $2$-qubit gates, even if also cares about constant overhead. We describe here briefly the quantum circuit model and introduce the \emph{weighted depth} following \cite{jan}. It is a complexity measure for unitary transformations based on the quantum circuit model. We assume that two qubit gates acting on disjoint pairs of qubits can be implemented simultaneously and define: \begin{Definition} A {\em quantum circuit} $A$ {\em of depth} $k$ is a sequence of $s$ steps $\{A_1,\dots,A_s\}$ where every step consists of a set of two qubit gates $\{u_{kl}\}_{k,l}$ acting on disjoint pairs $(k,l)$ of qubits. Every step $i$ defines a unitary operator $v_i$ by taking the product of all corresponding unitaries in any order. The product $u:=\Pi_{i\leq s} v_i$ is the `unitary operator implemented by $A$'. \end{Definition} The following quantity measures the deviation of a unitary operator from the identity: \begin{Definition} The {\em angle} of an arbitrary unitary operator $u\in SU(4)$ is the smallest possible norm\footnote{Here $\|.\|$ denotes the operator norm given by $\|a\|:=\max_x \|ax\|$ where $x$ runs over the unit vectors of the corresponding Hilbert space.} $\|a\|$ of a self-adjoint operator $a\in\mathfrak{su}(4)$ which satisfies $\exp(ia)=u$. \end{Definition} It coincides with the time required for the implementation of $u$ if the norm of the used Hamiltonian is $1$. We consider only the angle of two-qubit gates, i.e.\ we do not include the angle of local gates in the definition of the weighted depth. The notion of angle allows us to formulate a modification of the term `depth' which will later turn out to be decisive in connecting complexity measures of discrete and continuous algorithms: \begin{Definition} Let $\alpha_i$ be the maximal angle of the unitaries performed in step $i$. Then the {\em weighted depth} is defined to be the sum $\alpha=\sum_i\alpha_i$. \end{Definition} Assuming that the implementation time of a unitary is proportional to its angle, the weighted depth is the running time of the algorithm. We first need two technical lemmas to show that such an algorithm can be simulated by our computational model with complete $zz$-Hamiltonian without any time overhead. \begin{Lemma} Let $M$ be a set of qubit pairs, such that no two pairs contain a common qubit. Then we can simulate \begin{equation} H_M=\sum_{(k,l)\in M} \sigma_z^k\sigma_z^l \end{equation} with overhead $1$. \end{Lemma} Proof: This has been noted in \cite{leung}. Theorem \ref{independent} proves a more general statement. $\Box$ \begin{Lemma} Let $H_d=\sigma_z\otimes\sigma_z$ be the drift Hamiltonian of a $2$-spin system. All Hamiltonians $H\in\mathfrak{su}(4)$ can be simulated with overhead less than $\|H\|$. \end{Lemma} Proof: We first assume that $H$ contains no local terms, i.e.\ $H=\sum_{\alpha,\beta} J_{\alpha\beta} \sigma_{\alpha}\otimes\sigma_{\beta}$. Let $J_{12}$ be the matrix representing $H$. Conjugation of $H$ by $k=u\otimes v\in SU(2)\otimes SU(2)$ corresponds to multiplication of $J_{12}$ by $U\in SO(3)$ from the left and by $V\in SO(3)$ from the right. By the singular value decomposition \cite{horn} there are $U,V\in SO(3)$ such that $J_{12}=U\mbox{diag}(s_x,s_y,s_z)V$ where $s_x,s_y,s_z$ are the singular values of $J_{12}$. Equivalently, there is $k\in SU(2)\otimes SU(2)$ such that $kHk^\dagger=H_{s_x,s_y,s_z}$ where $H_{s_x,s_y,s_z}= s_x\sigma_x\otimes\sigma_x+s_y\sigma_y\otimes\sigma_y+s_z\sigma_z\otimes\sigma_z$. By computing the eigenvalues we see that $\|H_{s_x,s_y,s_z}\|=\sum_{\alpha} |s_\alpha|$. The simulation time overhead can not be more than the right hand side since each term $s_\alpha \sigma_\alpha \otimes \sigma_\alpha$ can be simulated with overhead $s_\alpha$. Let $H$ contain local terms, i.e.\ $H=\sum_{\alpha} J_{\alpha\alpha} \sigma_{\alpha}\otimes\sigma_{\alpha} + 1\otimes a + b\otimes 1$. We can split $H=H'+H''$ where $H'$ is the non-local part and $H''$ the local one. By the Trotter formula we can simulate the parts independently. The simulation of $H''$ takes no time by assumption. It remains to show that $\|H'\|\le\|H\|$. We may assume that $H$ is invariant with respect to qubit permutation since $\|\frac{1}{2}H+\frac{1}{2}H_{ex}\|\le\|H\|$ where $H_{ex}$ is the Hamiltonian obtained from $H$ by exchanging the qubits. By conjugation we can obtain a Hamiltonian of the form $H=H_{s_x,s_y,s_z} + s (1\otimes\sigma_x + \sigma_x\otimes 1)$. By computing the eigenvalues we see that $\|H_{s_x,s_y,s_z}\|\le\|H\|$. $\Box$ \begin{Corollary}\label{step} Let $A=\{u_{kl}\}_{k,l}$ be a step of a quantum circuit and $\alpha$ its weighted depth. Then the $zz$-model can simulate the unitary implemented by $A$ with overhead $\alpha$. \end{Corollary} Proof: Let $M=\{(k,l)\}$ be the set of the pairs which the two-qubit gates act on. No two pairs in $M$ contain a common vertex and therefore we can simulate the Hamiltonian $H_M=\sum_{(k,l)\in M}\sigma_z^k\sigma_z^l$ with overhead $1$. Let $H_{kl}$ be the Hamiltonian of minimal norm such that $u_{kl}=\exp(i H_{kl})$ for every $(k,l)\in M$. Now we can simulate every $H_{kl}$ parallely with overhead less than $\|H_{kl}\|$ by conjugating $H_M$. $\Box$ Our goal is to compare interactions with respect to the simulation complexity in our model given by the complete $zz$-interaction and the quantum circuit model. For doing so, we need some basic concepts of graph theory \cite{bol}. A graph is an ordered pair $G=(V,E)$ with $V\subseteq\{1,2,\ldots, n\}$ and $E=\{e_1,e_2,\ldots,e_m\}\subseteq V\times V$. Elements of $V$ are called vertices. They label the qubits. Elements of $E$ are called edges. They label the pair-interactions between the qubits. An edge $e=(k,l)$ is an ordered pair of vertices $k$ and $l$ called the ends of $e$. We consider only undirected graphs with no loops. To have a unique representation we require that $k<l$. Two distinct edges are called adjacent if and only if they have a common end vertex. A subset $M$ of the edge set $E$ is called independent if no two edges of $M$ are adjacent in $G$. A graph $G$ is called complete if every pair of distinct vertices of $G$ are adjacent in $G$; such a graph is denoted by $K_n$. Rephrased in this language, our drift Hamiltonian is of the form $$ H_d = \sum_{(k,l)\in E(K_n)} \sigma_z^k\sigma_z^l$$ and is called in the following the complete $zz$-Hamiltonian. \begin{Definition} Let $H$ be an arbitrary pair-interaction Hamiltonian. For every non-negative real number $r$ we define the {\em interaction graph} $G_r$ as follows: Let the qubits $\{1,\dots,n\}$ label the vertices and let the edges be all the pairs $(k,l)$ with the property $\|H_{k,l}\|> r$. \end{Definition} The chromatic index $\chi'$ is the minimum number of colors permitting an edge-coloring such that no two adjacent edges receive the same color or equivalently a partition $E=M_1\cup M_2\cup \ldots\cup M_{\chi'}$ into independent subsets of $E$. The following quantity turns out to be an upper bound on the overhead. \begin{Definition} We define the {\em weighted chromatic index} of $H$ \begin{equation} \chi':=\int_0^\infty \chi'_r dr \end{equation} where $\chi'_r$ denotes the chromatic index of $G_r$. \end{Definition} In a former paper \cite{jan} we have introduced the weighted chromatic index as a complexity measure of the interaction. This point of view has been justified by two arguments, where the first one is an observation in \cite{jan}: \begin{Theorem}\label{janInfinite} The evolution generated by a pair-interaction Hamiltonian $H$ during the infinitesimal time period $dt$ can be simulated by a parallelized $2$-qubit gate network with weighted depth $\chi'\, dt$ if $\chi'$ is the weighted chromatic index of $H$. \end{Theorem} The second argument to consider chromatic index as a complexity measure for the interaction is only intuitive: in general, it should be easy to control interactions on disjoint qubit pairs, whereas one should expect that its unlikely that one can {\it control} simultaneously the interaction between qubit $1$ and $2$ and the interaction $1$ and $3$ at the same moment. This `a priori'-assumption of \cite{jan} can be partly justified by the following corollary which is an easy conclusion of Corollary~\ref{step} and Theorem~\ref{janInfinite}. \begin{Corollary} The time overhead for simulating the Hamiltonian $H$ in the $zz$-model is at most the weighted chromatic index of $H$. \end{Corollary} The assumption that the drift Hamiltonian contains only pair-interaction of the form $\sigma_z\otimes\sigma_z$ can be dropped. Let $H=\sum_{\alpha,\beta} J_{\alpha\beta} \sigma_\alpha\otimes\sigma_\beta$ be an arbitrary pair-interaction. By conjugating $H$ with $\{I\otimes I,I\otimes\sigma_z,\sigma_z\otimes I,\sigma_z\otimes\sigma_z\}$ we obtain $J_{zz} \sigma_z\otimes\sigma_z$. This can be done with overhead $1$. The bounds of the corollary must be divided by the minimum $J_{zz}$ of all pair-interactions occurring in $H$. \section{Applications} The graph theoretical nature of our optimization problems becomes even stronger if we reduce our attention to one type of interactions, namely $zz$-interactions. Then the desired Hamiltonian is completely described by a weighted graph. We consider the problem to simulate the time evolution \begin{equation} H=\sum_{(k,l)} J_{kl} \sigma^k_z \sigma^l_z\,. \end{equation} when the complete $zz$-Hamiltonian is present. We first show that in this case it is sufficient to use conjugation by $\sigma_x$ only. Let $H':=\sigma_z\otimes\sigma_z$. Note that $(\sigma_x\otimes I) H' (\sigma_x\otimes I)=-H'$ and $(\sigma_x\otimes\sigma_x) H' (\sigma_x\otimes\sigma_x)=H'$. In the following we denote conjugation by $\sigma_x$ by $-$ and no conjugation by $+$. The Hamiltonian to be simulated contains only terms of the form $J_{kl;zz} \sigma^k_z \sigma_z^l$ by assumption. If it is written as a convex combination of elements of $Ad_K(H_d)$ it is sufficient to show that for each of these elements there is a procedure which cancels the terms $J_{kl;\alpha\beta}$ for $(\alpha, \beta)\neq (z,z)$ without any effect on the $J_{kl,zz}$ terms. Therefore consider $\tilde{H}=kH_dk^\dagger$. Then we can also achieve the Hamiltonian $\tilde{H}_{zz}=\sum_{(k,l)} \tilde{J}_{kl;zz}\sigma^k_z\sigma^l_z$ with overhead $1$. For every qubit $i$ there is a $\tilde{J}_{i;z}$ such that $\tilde{J}_{kl;zz}=\tilde{J}_{k;z} \tilde{J}_{l;z}$ for all edges $(k,l)$. We express each $\tilde{J}_{i;z}=c^{+}_i-c^{-}_i$ with $0\le c^+_k,c^-_k\le 1$ and $c^+_k+c^-_k=1$. Let $K=\{I,\sigma_x\}\otimes\ldots\otimes\{I,\sigma_x\}$. We conjugate the drift Hamiltonian by $u=u_1\otimes u_2\otimes\ldots\otimes u_n\in K$ for time $t(u)=\prod_{i=1}^n c_i(u)$ where $c_i(u)=c^+_k$ if $u_i=I$ and $c_i(u)=c^-_k$ if $u_k=\sigma_x$. We have $$ \sum_{u\in K} t(u) u\tilde{H}u^{\dagger} = \tilde{H}_{zz}\,. $$ Since we restrict our attention to interactions with $zz$-terms only a shorter notation will be useful. To each edge $e=(k,l)$ of $G$, we associate a real number $w_{kl}$ called the weight of $e$. The resulting graph is called a weighted graph. Its adjacency matrix $J$ is the real symmetric matrix with zeros on the diagonal defined by \begin{equation} J_{ii}:=0\,,\quad J_{kl}:=w_{kl} \mbox{ and } J_{lk}:=w_{kl} \end{equation} for all edges $(k,l)$ of $G$. An unweighted graph can be considered as a weighted whose edges all have the weight $1$. A (unweighted) graph is bipartite if its vertex set can be partitioned into two nonempty subsets $X$ and $Y$ such that each edge of $G$ has one end in $X$ and the other in $Y$. The pair $(X,Y)$ is called a bipartition of the bipartite graph. The complete bipartite graph with bipartition $(X,Y)$ is denoted by $G(X,Y)$. A Seidel matrix defines a modified adjacency matrix $S=(s_{kl})$ for (unweighted) graphs in the following way \cite{cvet}: $$ s_{kl}=\left\{ \begin{array}{rl} -1 & \mbox{ if $k$ and $l$ are adjacent } k\neq l \\ 1 & \mbox{ if $k$ and $l$ are non-adjacent} \\ \end{array} \right. $$ and $s_{kk}=0$. Obviously, $S=K-I-2J$, where $K$ denotes a square matrix all of whose entries are equal to $1$ and $J$ the adjacency matrix of $G$. \begin{Theorem}[Optimal simulation] A graph $G$ can be simulated with overhead $1$ if and only if it can be expressed as a convex combination \begin{equation} J=\sum_i t_i S_i \end{equation} where the sum runs over the Seidel adjacency matrices of all complete bipartite graphs, i.e., over $2^{n-1}$ possible matrices. \end{Theorem} Proof: By assigning to each vertex either $+$ or $-$ we have a bipartition of the vertex set: $X$ contains all vertices with $+$ and $Y$ all vertices with $-$. The sign of the edge $(k,l)$ is $-$ if and only if the edge has one end in $X$ and the other end in $Y$ and $+$ otherwise. The edges with $-$ define the complete bipartite graph $G(X,Y)$. We also include the case $X=\emptyset$ and $Y=V$ to cover the case when $+$ is assigned to all knots. Therefore all we can achieve in the one step is $K-I-2J(X,Y)$. $\Box$ \begin{Corollary}[Lower bound] The absolute value of the smallest eigenvalue of the $J$-matrix is a lower bound on the simulation overhead. \end{Corollary} We present now some upper bounds on the overhead. A graph $G=(V',E')$ is called a subgraph of $G'$ if $V'\subseteq V$ and $E'\subseteq E$. A clique of $G$ is a complete subgraph of $G$. A clique of $G$ is called a maximal clique of $G$ if it is not properly contained in another clique of $G$. A clique partition $P$ of $G$ is a partition of $E(G)$ such that its classes induce maximal cliques of $G$. Given a set $C$ of $h$ colors, an $h$-coloring of $P$ in $G$ is a mapping from $P$ to $C$, such that cliques sharing a vertex have different colors. Let the clique coloring index $c(G)$ be the smallest $h$ such that there is a partition $P$ permitting an $h$-coloring \cite{wallis}. We say the graph $G$ consists of independent cliques if $c(G)=1$. \begin{Lemma}[Upper bound]\label{independent} Let $G$ be a graph consisting of independent cliques. We can simulate the Hamiltonian $H_E$ with overhead $1$ which is optimal. \end{Lemma} Proof: Let $\omega\ge 2$ be the number of maximal cliques. We construct $\omega$ vectors $s_i$ of length $2^{\omega-1}$ as follows: $$ \begin{array}{lcl} s_1 & = & (++++++++\cdots) \\ s_2 & = & (+-+-+-+-\cdots) \\ s_3 & = & (++--++--\cdots) \\ & \vdots & \\ s_\omega & = & (\underbrace{++\,\cdots\,+}_{2^{\omega-2}} \underbrace{--\,\cdots\,-}_{2^{\omega-2}})\\ \end{array} $$ where $+$ stands for $1$ and $-$ for $-1$. The scalar products are $\langle s_i,s_j\rangle=2^{\omega-1} \delta_{ij}$. We partition the time interval into $2^{\omega-1}$ intervals of equal length. In the $m$th interval we conjugate all qubits of the $i$th clique by $\sigma_x$ if $s_{i,m}=-$ and do nothing otherwise. This is optimal since $q\le -1$ where $q$ is the smallest eigenvalue of $G$. $\Box$ The scheme used in the proof is time optimal. But the number of conjugations grows exponentially with the number of cliques $\omega$. It is possible to use the conjugations schemes based on Hadamard matrices \cite{leung}. There the number of conjugations grows only quadratically with $\omega$. \begin{Corollary}[Upper bound] Let $G$ be an arbitrary graph. Then the Hamiltonian $H_G$ can be simulated with the overhead $c(G)$. \end{Corollary} Note that if $M$ is an independent set than the graph $G=(V,M)$ consists of independent cliques. Therefore the chromatic index is an upper bound on clique index. However, this bounds is not always good. Consider e.g.\ the graph $G$ containing all edges that do not have $1$ as end vertex. Then the chromatic index of $G$ is still high but the clique coloring index is only $1$. Since the optimal simulation of graph consisting of independent cliques has overhead $1$ one might think that the clique index is the smallest overhead. But this is not so as shows the following example. Consider the star $G=(V,E)$ with $V=\{1,\ldots,5\}$ and $E=\{(1,2),(1,3),(1,4),(1,5)\}$. The clique index of is $4$ but the optimal simulation has overhead $2$ only. The vectors can be chosen as $s_1=(++++),\, s_2=(-+++),\, s_3=(+-++),\, s_4=(++-+),\, s_5=(+++-)$ and each of the four intervals has length $1/2$. This is optimal since the smallest eigenvalue of the adjacency matrix of $G$ is $-2$. \section{Quasi-order of Hamiltonians} Let $H$ and $\tilde{H}$ be arbitrary pair-interaction Hamiltonians. We investigate the question whether $\tilde{H}$ can be simulated by $H$ with overhead $\mu$. Note that this defines a quasi-order of the pair-interaction Hamiltonians for $\mu=1$. A partial characterization of the quasi-order is expressed in terms of majorization of the spectra of the corresponding matrices $J$ and $\tilde{J}$. Similar methods have been used to derive conditions for a class of entanglement transformations and to characterize mixing and measurement in quantum mechanics \cite{nielsen1,nielsen2}. Suppose that $x=(x_1,\ldots,x_d)$ and $y=(y_1,\ldots,y_d)$ are two dimensional real vectors. We introduce the notation $\downarrow$ to denote the components of a vector rearranged into non-increasing order, so $x^\downarrow=(x_1^\downarrow,\ldots,x_d^\downarrow)$, where $(x_1^\downarrow\ge x_2^\downarrow\ge\ldots\ge x_d^\downarrow)$. We say that $x$ is majorized by $y$ and write $x\prec y$, if $$ \sum_{j=1}^k x_j^\downarrow\le \sum_{j=1}^k y_j^\downarrow\,, $$ for $k=1,\ldots,d-1$, and with equality when $k=d$ \cite{bha}. Let $\mathrm{Spec}(X)$ denote the spectrum of the hermitian matrix $X$, i.e.\ the vector of eigenvalues, and $\lambda(X)$ denote the vector of components of $\mathrm{Spec}(X)$ arranged so they appear in non-increasing order. Ky Fan's maximum principle \cite{nielsen2} states that for any Hermitian matrix $A$, the sum of the $k$ largest eigenvalues of $A$ is the maximum value $\mathrm{tr}(AP)$, where the maximum is taken over all $k$-dimensional projections $P$, $$ \sum_{j=1}^k \lambda_j(A)=\max_P\mathrm{tr}(AP)\,. $$ It gives rise to a useful constraint on the eigenvalues of a sum of two Hermitian matrices $C:=A+B$, that $\lambda(C)\prec\lambda(A)+\lambda(B)$. Choose a $k$-dimensional projection $P$ such that \begin{equation} \sum_{j=1}^k\lambda_j(C) = \mathrm{tr}(CP) = \mathrm{tr}(AP)+\mathrm{tr}(BP) \le \sum_{j=1}^k \lambda_j(A)+\sum_{j=1}^k \lambda_j(B)\,. \label{EigIneq} \end{equation} This permits us to derive a lower bound on the simulation overhead. \begin{Lemma}[Majorization] Let $H$ and $\tilde{H}$ be arbitrary pair-interaction Hamiltonians. A necessary condition that $\tilde{H}$ can be simulated with overhead $\mu$ by $H$ is that $\mathrm{Spec}(\tilde{J})\prec\mu\mathrm{Spec}(J)$. \end{Lemma} Proof: By representing the Hamiltonians by their $J$-matrices we see that $\tilde{H}$ can be simulated with overhead $\mu$ if and only if there is a sequence of orthogonal matrices $U_j=U_{j1}\oplus\ldots U_{jn}\in SO(3)\oplus\ldots\oplus SO(3)$ and $\mu_j>0$ with $\sum_j\mu_j=\mu$ such that $$ \tilde{J}=\sum_j\mu_j U_j J U_j^T\,. $$ The proof now follows from the inequality~(\ref{EigIneq}). We consider now the problem to reverse the time evolution $\exp(iH_d t)$, i.e.\ what is the overhead of simulating $-H_d$ when $H_d$ is present. \begin{Lemma}[Lower bound on inverting] Let $r$ be the greatest eigenvalue and $q$ the smallest eigenvalue of $J$. Then $\mu\ge\frac{r}{-q}$ is a lower bound on the overhead for simulating $-H_d$ by $H_d$. \end{Lemma} Proof: This is a direct consequence of the Weyl inequality (see \cite{bha}, Theorem~III.2) $\lambda_d(A+B)\ge\lambda_d(A)+\lambda(B)$ for the sum of two Hermitian matrices where $\lambda_d$ denotes the smallest eigenvalue. $\Box$ Let $G$ be a connected graph and $H_d=\sum_{(k,l)\in E(G)}\sigma_z^k\sigma_z^l$. If $G$ is not connected then the components can be treated independently. For the spectrum the following statements hold (see \cite{cvet}, Theorem~0.13): $$ 1\le r\le n-1\,,\quad -r\le q\le -1\,. $$ It is interesting to note that this gives a tight bound for simulating $-H_d$ when $G=K_n$ since for a complete graph we have $r=(n-1)$ and $q=-1$. An upper bound is the (weighted) chromatic index $\chi'(K_n)$ which is either $n$ (if $n$ is even) or $n-1$ (otherwise). This simple example shows that the inverse of the natural time evolution may have a relatively high complexity. \begin{Lemma} Let the drift Hamiltonian $H_d$ be an arbitrary pair-interaction Hamiltonian. If the interaction graph $G_0(H_d)$ is bipartite then we can invert the time evolution with overhead of less than $3$. \end{Lemma} Proof: Let $S=\{\sigma_x,\sigma_y,\sigma_z\}$. We have $\sum_{u\in S} uau^\dagger=-a$ for all $a\in\mathfrak{su}(2)$. Let $X,Y$ be the bipartition of $G_0(H_d)$. By conjugating all qubits in $X$ with elements of $S$ we obtain $-H_d$. $\Box$ If $H$ contains only $\sigma_z\otimes\sigma_z$ then the overhead is $1$. This is optimal since we have $\mu\ge 1$. \end{document}

\begin{document} \title{Global existence and stability for the modified Mullins--Sekerka and surface diffusion flow} \author{Serena Della Corte\affil{1}, Antonia Diana\affil{2} and Carlo Mantegazza\affil{3,}\corrauth} \keywords{Nonlocal Area functional, Mullins--Sekerka flow, surface diffusion flow, global existence, asymptotic stability} \shortauthors{the Author(s)} \address{ \addr{\affilnum{1}}{Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands} \addr{\affilnum{2}}{Scuola Superiore Meridionale, Universit\`{a} degli Studi di Napoli Federico II, Italy} \addr{\affilnum{3}}{Dipartimento di Matematica e Applicazioni ``Renato Caccioppoli'' \& Scuola Superiore Meridionale, Universit\`{a} degli Studi di Napoli Federico II, Italy}} \corraddr{Email: c.mantegazza@sns.it} \deltaf{\mathrm{O}}{{\mathrm{O}}} \deltaf\mathrm{I}I{{\mathrm{I}}} \newcommand{\mathbb{T}}{\mathbb{T}} \newcommand{\mathbb{R}}{\mathbb{R}} \newcommand{\mathbb{N}}{\mathbb{N}} \deltaf\mathbb Z{\mathbb Z} \newcommand{\mathbb S}{\mathbb S} \newcommand{B}{B} \newcommand{\mathbb{R}RR}{{\mathrm R}} \newcommand{\mathbb{R}ic}{{\mathrm {Ric}}} \newcommand{{\RRR}}{{\mathbb{R}RR}} \newcommand{\Delta_t}{\Delta_t} \newcommand{\mathscr{L}}{\mathscr{L}} \DeclarePairedDelimiter{\normainf}{\lVert}{\rVert_{\infty}} \DeclarePairedDelimiter{\norma}{\lVert}{\rVert} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert} \newcommand{d \mu}{d \mu} \newcommand{\widetilde{H}}{\widetilde{H}} \newcommand{W^{2,p}}{W^{2,p}} \newcommand{\mathbb{T}ort}{T^{\perp}} \newcommand{\delta}{\deltalta} \newcommand{\partial}{\partialrtial} \newcommand{\mathbb{C}}{\mathbb{C}} \newcommand{\mathcal{H}^2}{\mathcal{H}^2} \newcommand{\O}{\O} \newcommand{\mathrm{H}}{\mathrm{H}} \newcommand{\mathrm{Vol}}{\mathrm{Vol}} \newcommand{\nabla}{\nabla} \newcommand{\mathcal A}{\mathcal A} \newcommand{\mathcal J}{\mathcal J} \newcommand{\varepsilon}{\varepsilon} \newcommand{-\kern -,375cm\int}{-\kern -,375cm\int} \newcommand{-\kern -,375cm\intinrigo}{-\kern -,315cm\int} \newcommand{\mathfrak{h}^{2, \alpha}_M(F, U)}{\mathfrak{h}^{2, \alpha}_M(F, U)} \newcommand{\mathbb{C}unoM}{\mathfrak{C}^{1}_M(F, U)} \newcommand{g_{ij}}{g_{ij}} \newcommand{\mathrm{Id}}{\mathrm{Id}} \newcommand{\mathrm{I}}{\mathrm{I}} \newcommand{\widetilde{X}}{\widetilde{X}} \newcommand{\biggl\vert}{\biggl\vert} \newcommand{\rightharpoonup}{\rightharpoonup} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}} \deltaf\operatorname*{div}\nolimits{\operatorname*{div}\nolimits} \newcommand{\mathrm{dist\,}}{\mathrm{dist\,}} \begin{abstract} In this survey we present the state of the art about the asymptotic behavior and stability of the \emph{modified Mullins--Sekerka flow} and the {\em surface diffusion flow} of smooth sets, mainly due to E.~Acerbi, N.~Fusco, V.~Julin and M.~Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the {\em strict stability} property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $W^{2,p}$--perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently ``close'' to a smooth {\em strictly stable critical} set $E$, both flows exist for all positive times and asymptotically ``converge'' to a translate of $E$. \end{abstract} \keywords{\textbf{Nonlocal Area functional, Mullins--Sekerka flow, surface diffusion flow, global existence, asymptotic stability}} \maketitle \section{Introduction} Geometric evolutions are a fascinating topic naturally arising from the study of dynamical models in physics and material sciences. Concrete examples are, for instance, the analysis of the behavior in time of the interfaces surfaces in phase changes of materials or in the flows of immiscible fluids. From the mathematical point of view, they describe the motion of geometric objects or structures, usually driven by systems of partial differential equations.\\ In this work we rethink, expand the details and present in a unified treatment the results of E.~Acerbi, N.~Fusco, V.~Julin and M.~Morini~\cite{AcFuMoJu,AcFuMo} about two of the most recent of such geometric motions, namely, the \emph{modified Mullins--Sekerka flow} and the {\em surface diffusion flow}. Both flows deal with an evolution in time of smooth subsets $E_t$ of an open set $\Omega\subseteq\mathbb{R}^n$, with $d(E_t,\partialrtial\Omega)>0$, for every $t$ in a time interval $[0,T)$, such that their boundaries $\partial E_t$, which are smooth hypersurfaces, move with some ``outer'' normal velocity $V_t$ that, in the first case, is obtained as solution of the following ``mixed'' system \begin{equation}\label{msfSistema}\tag{mMSF} \begin{cases} V_t=[\partialrtial_{\nu_t} w_t] & \text{on } \partialrtial E_t \\ \Delta w_t=0 & \text{in } \Omega \setminus \partialrtial E_t\\ w_t=\mathrm{H}_t + 4 \gamma v_t & \text{on } \partialrtial E_t\\ -\Delta v_t = u_{E_t} - \fint_{\Omega} u_{E_t} \,dx & \text{in } \Omega\,\,\text{(distributionally)} \end{cases} \end{equation} where $\gamma$ is a nonnegative parameter, $v,w:[0,T)\times\overline{\Omega}\to\mathbb{R}$ are continuous functions such that, setting $w_t=w(t,\cdot)$ and $v_t=v(t,\cdot)$, the functions $v_t$ and $w_t$ are smooth in $\Omega \setminus \partialrtial E_t$, for every $t\in[0,T)$; the functions $\nu_t,\mathrm{H}_t$ are the ``outer'' normal and the relative mean curvature of $\partialrtial E_t $ and $u_{E_t}= 2 \chi_{\text{\raisebox{-.5ex}{$\scriptstyle E_t$}}} -1$; finally, the velocity of the motion is given by $[\partialrtial_{\nu_t} w_t]$ which denotes $\partialrtial_{\nu_t}w_t^{+}- \partialrtial_{\nu_t}w_t^{-}$, that, is the ``jump'' of the normal derivative of $w_t$ on $\partial E_t$, where $w_t^{+}$ and $w_t^{-}$ are the restrictions of $w_t$ to $\Omega \setminus\overline{E}_t$ and $E_t$, respectively.\\ The resulting motion, called {\em modified Mullins--Sekerka flow}~\cite{MS} (see also~\cite{Crank,Gurtin1} and~\cite{EscherSi4} for a very clear and nice introduction to such flow), arises as a singular limit of a nonlocal version of the Cahn--Hilliard equation~\cite{alikakos,pego,Le}, to describe phase separation in diblock copolymer melts (see also~\cite{OK}). It has been also called {\em Hele--Shaw model}~\cite{XChen}, or {\em Hele--Shaw model with surface tension}~\cite{EscherSi1,EscherSi2,EscherSi3}. We mention that the adjective ``modified'' comes from the introduction of the parameter $\gamma > 0$ in the system~\eqref{msfSistema}, while choosing $\gamma=0$ we have the original flow proposed by Mullins and Sekerka in~\cite{MS}. In the second case, we will say that a flow of sets $E_t$ as above, is a solution of the {\em surface diffusion flow} if the normal velocity is pointwise given by \begin{equation}\label{sdf}\tag{SDF} V_t = \Delta_t \mathrm{H}_t \qquad\text{on $\partial E_t $,} \end{equation} where $\Delta_t$ is the Laplacian of the hypersurface $\partial E_t$, for all $t \in [0,T)$. Such flow was first proposed by Mullins in~\cite{Mullins} to study thermal grooving in material sciences (see also~\cite{escmaysim} for a nice presentation), in particular, in the physically relevant case of three--dimensional space, it describes the evolution of interfaces between solid phases of a system, which are studied in a variety of physical settings including phase transitions, epitaxial deposition and grain growth (see for instance~\cite{GurJab} and the references therein). Notice that, while in this latter case, the velocity flow is immediately well defined, the system~\eqref{msfSistema} is clearly undetermined as it is, since the behavior of the functions $w_t$ and $v_t$ is not prescribed on the boundary of $\Omega$ (which is also possibly not bounded). By simplicity, we will consider flows in the whole Euclidean space and we assume that all the functions and sets involved are periodic with respect to the standard lattice $\mathbb Z^n$ of $\mathbb{R}^n$. It is then clear that this is equivalent to ``ambient'' the problem in the $n$--dimensional ``flat'' torus $\mathbb{T}^n = \mathbb{R}^n/ \mathbb Z^n $, hence in the sequel we will assume $\Omega=\mathbb{T}^n$, modifying the definitions above accordingly. Another possibility would be asking that $\Omega\subseteq\mathbb{R}^n$ is bounded, the moving sets do not ``touch'' the boundary of $\Omega$ and that all the functions $w_t$ and $v_t$ are subject to homogeneous (zero) Neumann boundary conditions on $\partialrtial \Omega$ (see Subsection~\ref{Neucase}). A very important property of these geometric flows is that both are the {\em gradient flow} of a functional, which clearly gives a natural ``energy'', decreasing in time during the evolution (the velocity $V_t$ is minus the gradient, that is, the {\em Euler--Lagrange equation} of a functional).\\ Precisely, in any dimension $n\in\mathbb{N}$, the modified Mullins--Sekerka flow is the $H^{-1/2}$--{gradient flow} of the following {\em nonlocal Area functional} \begin{equation} \label{eq:J} J(E)= \mathcal A(\partial E) + \gamma \int_{\mathbb{T}^n} \int_{\mathbb{T}^n} G(x,y) u_E(x) u_E(y) \, dx \, dy \,, \end{equation} under the constraint that the volume $\mathrm{Vol}(E)=\mathscr{L}^n(E)$ is fixed, where (here and in the whole paper), $$ \mathcal A(\partial E)= \int_{\partialrtial E} \, d\mu $$ is the classical {\em Area functional} that gives the {\em area} of the $(n-1)$--dimensional smooth boundary of any sets $E$ ($\mu$ is the ``canonical'' measure associated to the Riemannian metric on $\partial E$ induced by metric of $\mathbb{T}^n$ coming from the scalar product of $\mathbb{R}^n$, which coincides with the $n$--dimensional Hausdorff measure $\mathcal{H}^n$) and $G$ is the Green function of $\mathbb{T}^n$ (see~\cite{Le}, for details).\\ Similarly, the surface diffusion flow can be regarded as the $H^{-1}$--gradient flow of the Area functional $\mathcal A$ with fixed volume. Then, it clearly follows that, in both cases, the volume of the evolving sets $\mathrm{Vol}(E_t)$ is constant in time, while neither convexity (see~\cite{Conv} and~\cite{Ito}) is maintained, nor there holds the so--called ``comparison property'' asserting that if two initial sets are one contained into the other, they stay so during the two flows. This is due to the lack of the {\em maximum principle} for parabolic equations or systems of order larger than two. We remind that such properties are shared by the more famous {\em mean curvature flow}, which is also a gradient flow of the Area functional (without the constraint on the volume), but with respect to the $L^2$--norm (see~\cite{Man}, for instance). Parametrizing the moving smooth surfaces $\partialrtial E_t$ by some maps (embeddings) $\psi_t:M\to\mathbb{T}^n$ such that $\psi_t(M)=\partialrtial E_t$, where $M$ is a fixed smooth, compact $(n-1)$--dimensional differentiable manifold and $\nu_t$ is the outer unit normal vector to $\partialrtial E_t$ as above, the evolution laws~\eqref{msfSistema} and~\eqref{sdf} can be respectively expressed as $$ \frac{\partialrtial}{\partialrtial t}\psi_t=V_t \nu_t=[\partialrtial_{\nu_t} w_t] \nu_t\,, $$ and $$ \frac{\partialrtial}{\partialrtial t}\psi_t=(\Delta_t\mathrm{H}_t)\nu_t \,. $$ Due to the parabolic nature (not actually so explicit in the first case) of these systems of PDEs, it is known that for every smooth initial set $E_0$ in $\mathbb{T}^n$, with boundary described by $\psi_0:M\to\mathbb{T}^n$, both flows with such initial data exist unique and are smooth in some positive time interval $[0,T)$. Indeed, such short time existence and uniqueness results were proved by Escher and Simonett~\cite{EscherSi1,EscherSi2,EscherSi3} and independently by Chen, Hong and Yi~\cite{chenhong} for the modified Mullins--Sekerka flow and by Escher, Mayer and Simonett in~\cite{escmaysim} for the surface diffusion flow of a smooth compact hypersurface in domains of the Euclidean space of any dimension. With minor modifications, their proof can be adapted to get the same conclusion also for smooth initial hypersurfaces of $\mathbb{T}^n$. The aim of this work is to show that, in dimensions two and three, for initial data sufficiently ``close'' to a smooth {\em strictly stable critical} set $E$ for the relative ``energy'' functional (the nonlocal or the usual Area functional) under a volume constraint, the flows exist for all positive times and asymptotically converge {\em in some sense} to a ``translate'' of $E$.\\ The notions of criticality and stability are as usual defined in terms of first and second variations of $J$ and $\mathcal A$. We say that a smooth subset $E \subseteq \mathbb{T}^n$ is \emph{critical} for $J$ (or for $\mathcal A$, simply choosing $\gamma=0$ in formula~\eqref{eq:J}) if for any smooth one--parameter family of diffeomorphisms $\Phi_t:\mathbb{T}^n\to\mathbb{T}^n$, such that $\mathrm{Vol}(\Phi_{t}(E))=\mathrm{Vol}(E)$, for $t\in(-\varepsilon,\varepsilon)$ and $\Phi_0=\mathrm{Id}$ ($E_t=\Phi_t(E)$ will be called {\em volume--preserving variation} of $E$), we have $$ \frac{d}{dt} J(\Phi_t(E))\Bigl|_{t=0}=0 \, . $$ We will see that this condition is equivalent to the existence of a constant $\lambda \in \mathbb{R}$ such that \[ \mathrm{H}+ 4 \gamma v_E = \lambda \qquad \text{on $\partialrtial E$}, \] where $\mathrm{H}$ is the mean curvature of $\partial E$ and $v_E$ is the potential defined as \begin{equation} v_E(x)=\int_{\mathbb{T}^n} G(x,y)u_E(y) dy \, , \end{equation} with $G$ the Green function of the torus $\mathbb{T}^n$ and $u_E= \chi_{\text{\raisebox{-.5ex}{$\scriptstyle E$}}} - \chi_{\text{\raisebox{-.5ex}{$\scriptstyle \mathbb{T}^n \setminus E$}}}$.\\ The second variation of $J$ at a critical set $E$, leading to the central notion of {\em stability}, is more involved and, differently by the original papers, we will compute it with the tools and methods of differential/Riemannian geometry (like the first variation). We will see that at a critical set $E$, the second variation of $J$ (the second derivative at $t=0$ of $J(E_t)$) along a volume--preserving variation $E_t=\Phi_t(E)$ only depends on the normal component $\varphi$ on $\partialrtial E$ of the {\em infinitesimal generator} field $X=\frac{\partialrtial\Phi_t}{\partialrtial t} \bigl|_{t=0}$ of the variation. The volume constraint on the admissible deformations of $E$ implies that the functions $\varphi$ must have zero integral on $\partial E$, hence it is natural to define a quadratic form $\Pi_E$ on such space of functions which is related to the second variation of $J$ by the following equality \begin{equation}\label{PI0} \Pi_E(\varphi)=\frac{d^2}{dt^2} J(\Phi_t(E))\Bigr \vert_{t=0} \end{equation} where $E_t=\Phi_t(E)$ is a volume--preserving variation of $E$ such that $$ \Bigl\langle\nu_E \,\Bigr\vert\frac{ \partial \Phi_t}{ \partial t}\Bigr \vert_{t=0}\Bigr\rangle=\varphi $$ on $\partial E$, with $\nu_E$ the {\em outer unit normal vector} of $\partial E$.\\ Because of the obvious {\em translation invariance} of the functional $J$, it is easy to see (by means of the formula~\eqref{PI0}) that the form $\Pi_E$ vanishes on the finite dimensional vector space given by the functions $\psi= \langle\nu_E\vert \eta\rangle$, for every vector $\eta\in\mathbb{R}^n$. We underline that the presence of such ``natural'' degenerate subspace of the quadratic form $\Pi_E$ (or, equivalently, the translation invariance of $J$) is the main reason of several technical difficulties.\\ We then say that a smooth critical set $E \subseteq \mathbb{T}^n$ is {\em strictly stable} if \begin{equation} \Pi_E( \varphi ) > 0 \end{equation} for all non--zero functions $\varphi:\partial E\to\mathbb{R}$, with zero integral and $L^2$--orthogonal to every function $\psi= \langle\nu_E\vert \eta\rangle$. Then, the heuristic idea is that in a region around a strictly stable critical set $E$, we have a ``potential well'' for the ``energy'' $J$ (and the set $E$ is a local minimum) and, defining a suitable notion of ``closedness'', if one set starts close enough to $E$, during its evolution by (minus) the gradient of such energy, it cannot ``escape'' the well and asymptotically converges to a set of (local) minimal energy, which must be a translate of $E$. That is, the strict stability of $E$ implies a ``dynamical'' stability in a neighborhood. At the moment, this conclusion, that we state precisely below, can be shown only in dimension at most three, because of missing estimates in higher dimensions (see the discussion at the beginning of Section~\ref{globalex}). When $n>3$ this and several other questions on these flows remain open. Anyway, this is sufficient for the application to some physically relevant models, since the evolution laws~\eqref{msfSistema} and~\eqref{sdf} describe, respectively, pattern--forming processes such as the solidification in pure liquids and the evolution of interfaces between solid phases of a system, driven by surface diffusion of atoms under the action of a chemical potential (see for instance~\cite{GurJab} and the references therein). In this paper, we will only deal with the three--dimensional case, but we underline that all the results and arguments hold, without relevant modifications, also in the two--dimensional situation of $\mathbb{T}^2=\mathbb{R}^2/\mathbb Z^2$, where the moving boundaries of the sets are curves.\\ Moreover, we mention here that all the results also hold in a bounded open subset $\Omega$ of $\mathbb{R}^2$ or $\mathbb{R}^3$, for moving sets which do not ``touch'' the boundary of $\Omega$, imposing that the functions $w_t$ and $v_t$ in the definition of the modified Mullins--Sekerka flow satisfy a {\em zero Neumann boundary condition} (as we mentioned above), instead than choosing the ``toric ambient'' (see Subsection~\ref{Neucase} for more details). \begin{theorem}[Theorem~\ref{existence} and Remark~\ref{existence+}] Let $E\subseteq\mathbb{T}^3$ be a smooth strictly stable critical set for the nonlocal Area functional under a volume constraint and $N_\varepsilon$ a suitable tubular neighborhood of $\partial E$. For every $\alpha\in (0,1/2)$ there exists $M>0$ such that, if $E_0$ is a smooth set satisfying \begin{itemize} \item $\mathrm{Vol}( E_0)= \mathrm{Vol}( E )$, \item $\mathrm{Vol}( E_0\triangle E) \leq M$, \item the boundary of $E_0$ is contained in $N_\varepsilon$ and can be represented as \begin{equation} \partial E_0= \{ y+ \psi_{E_{0}} (y) \nu_E(y) \, : \, y \in \partial E \}, \end{equation} for some function $\psi _{E_0} : \partial E \to \mathbb{R}$ such that $ \norma { \psi _ {E_0}}_{ C^{1,\alpha} ( \partial E)} \leq M$,\\ \item there holds\ $$ \int_{\mathbb{T}^3} \vert \nabla w_{E_0 }\vert^2\,dx \leq M\,, $$ where $w_0=w_{E_0}$ is the function relative to $E_0$, as in system~\eqref{msfSistema}, \end{itemize} then, there exists a unique smooth solution $E_t$ of the modified Mullins--Sekerka flow (with parameter $\gamma\geq 0$) starting from $E_0$, which is defined for all $t\geq0$. Moreover, $E_t\to E+\eta$ exponentially fast in $C^k$ as $t\to +\infty$, for every $k\in\mathbb{N}$, for some $\eta\in \mathbb{R}^3$, with the meaning that the functions $\psi_{\eta, t} : \partial E+ \eta \to \mathbb{R}$ representing $\partial E_t$ as ``normal graphs'' on $\partial E + \eta$, that is, $$ \partial E_t= \{ y+ \psi_{\eta,t} (y) \nu_{E+\eta}(y) \, : \, y \in \partial E+\eta \}, $$ satisfy for every $k\in\mathbb{N}$, the estimates $$ \Vert \psi_{\eta, t}\Vert_{C^k(\partial E + \eta)}\leq C_ke^{-\beta_k t} $$ for every $t\in[0,+\infty)$, for some positive constants $C_k$ and $\beta_k$. \end{theorem} \begin{theorem}[Theorem~\ref{existence2} and Remark~\ref{existence2+}] Let $E\subseteq\mathbb{T}^3$ be a strictly stable critical set for the Area functional under a volume constraint and let $N_\varepsilon$ be a tubular neighborhood of $\partial E$. For every $\alpha\in (0,1/2)$ there exists $M>0$ such that, if $E_0$ is a smooth set satisfying \begin{itemize} \item $\mathrm{Vol}( E_0)= \mathrm{Vol}( E )$, \item $\mathrm{Vol}( E_0\triangle E) \leq M$, \item the boundary of $E_0$ is contained in $N_\varepsilon$ and can be represented as \begin{equation} \partial E_0= \{ y+ \psi_{E_{0}} (y) \nu_E(y) \, : \, y \in \partial E \} \, , \end{equation} for some function $\psi _{E_0} : \partial F \to \mathbb{R}$ such that $ \norma { \psi _ {E_0}}_{ C^{1,\alpha} ( \partial E)} \leq M$, \item there holds \ $$ \int_{\partial E_0} \vert \nabla \mathrm{H}_0\vert^2\, d \mu_0 \leq M \, , $$ \end{itemize} then there exists a unique smooth solution $E_t$ of the surface diffusion flow starting from $E_0$, which is defined for all $t\geq0$. Moreover, $E_t\to E+\eta$ exponentially fast in $C^k$ as $t\to +\infty$, for some $\eta\in \mathbb{R}^3$, with the same meaning as above. \end{theorem} We remark that the line of the proof in~\cite{AcFuMoJu} that we are going to present, is based on suitable energy identities and compactness arguments to establish these global existence and exponential stability results. This was actually a completely new approach to manage the translation invariance of the functional $J$, in previous literature dealt with by means of semigroup techniques. Summarizing, the work is organized as follows: in Section~\ref{nonlocsec} we study the nonlocal Area functional (constrained or not) and we compute its first and second variation, then we discuss the notions of criticality, stability and local minimality of a set and their mutual relations, in this context. In Section~\ref{msfsdf} we introduce the modified Mullins--Sekerka and the surface diffusion flow and we analyze their basic properties. Section~\ref{globalex} is devoted to show the two main theorems above, while finally in Section~\ref{classification}, we discuss the classification of the stable and strictly stable critical sets (to whom then the two stability results apply). \section{The nonlocal Area functional}\label{nonlocsec} We start by introducing the \emph{nonlocal Area functional} and its basic properties. In the following we denote by $\mathbb{T}^n$ the $n$--dimensional flat torus of unit volume which is defined as the Riemannian quotient of $\mathbb{R}^n$ with respect to the equivalence relation $x \sim y \iff x-y \in \mathbb{Z}^n$, with $\mathbb{Z}^n$ the standard integer lattice of $\mathbb{R}^n$. Then, the functional space $W^{k,p}(\mathbb{T}^n)$, with $k \in \mathbb{N}$ and $p \ge 1$, can be identified with the subspace of $W^{k,p}_{\mathrm{loc}}(\mathbb{R}^n)$ of the functions that are $1$--periodic with respect to all coordinate directions. A set $E \subseteq \mathbb{T}^n$ is of class $C^k$ (or smooth) if its ``$1$--periodic extension'' to $\mathbb{R}^n$ is of class $C^k$ (or smooth,) which means that its boundary is locally a graph of a function of class $C^k$ around every point. We will denote with $\mathrm{Vol}(E)={\mathscr L}^n(E)$ the volume of $E\subseteq\mathbb{T}^n$. Given a smooth set $E\subseteq \mathbb{T}^n$, we consider the associated potential \begin{equation}\label{potential1} v_E(x)=\int_{\mathbb{T}^n} G(x,y)u_E(y) dy \, , \end{equation} where $G$ is the Green function (of the Laplacian) of the torus $\mathbb{T}^n$ and $u_E= \chi_{\text{\raisebox{-.5ex}{$\scriptstyle E$}}} - \chi_{\text{\raisebox{-.5ex}{$\scriptstyle \mathbb{T}^n \setminus E$}}}$. More precisely, $G$ is the (distributional) solution of \begin{equation}\label{G2} -\Delta_x G(x,y)=\deltalta_y-1\quad\text{in $\mathbb{T}^n$} \quad \text{with}\quad \int_{\mathbb{T}^n} G(x,y)\, dx=0, \end{equation} for every fixed $y\in\mathbb{T}^n$, where $\deltalta_y$ denotes the Dirac delta measure at $y \in \mathbb{T}^n$ (the $n$--torus $\mathbb{T}^n$ has unit volume).\\ By the properties of the Green function, $v_E$ is then the unique solution of \begin{equation} \begin{cases}\label{potential} {\displaystyle{-\Delta v_E= u_E- m \qquad \text{in $\mathbb{T}^n$ (distributionally)}}}\\ {\displaystyle{\int_{\mathbb{T}^n} v_E(x) \, dx=0}} \end{cases} \end{equation} where $m= \mathrm{Vol}(E) - \mathrm{Vol}(\mathbb{T}^n \setminus E)=2\mathrm{Vol}(E) - 1$. \begin{remark} By standard elliptic regularity arguments (see~\cite{gt}, for instance), $v_E \in W^{2,p}(\mathbb{T}^n)$ for all $p\in [1, +\infty)$. More precisely, there exists a constant $C=C(n,p)$ such that $\|v_E\|_{W^{2,p}(\mathbb{T}^n)}\leq C$, for all $E\subseteq\mathbb{T}^n$ such that $\mathrm{Vol}(E)-\mathrm{Vol}(\mathbb{T}^n\setminus E)=m$. \end{remark} Then, we define the following {\em nonlocal Area functional} (see~\cite{KnZe,MuZa,StTo}, for instance). \begin{definition}[Nonlocal Area functional]\label{NAFdef} Given $\gamma \ge 0$, the \emph{nonlocal Area functional} $J$ is defined as \begin{equation}\label{area} J(E)= \mathcal A(\partial E) + \gamma \int_{\mathbb{T}^n} |\nabla {v_E}(x)|^2 \, dx, \end{equation} for every smooth set $E \subseteq \mathbb{T}^n$, where the function $v_E:\mathbb{T}^n\to\mathbb{R}$ is given by formulas~\eqref{potential1}--\eqref{potential} and $$ \mathcal A(\partial E)= \int_{\partialrtial E} \, d\mu $$ is the \emph{Area functional} , where $\mu$ is the ``canonical'' measure associated to the Riemannian metric on $\partialrtial E$ induced by the metric tensor of $\mathbb{T}^n$, coming from the scalar product of $\mathbb{R}^n$ (it is easy to see that $\mu$ coincides with the $(n-1)$--dimensional Hausdorff measure restricted to $\partialrtial E$). \end{definition} {\em Since the nonlocal Area functional is defined adding to the Area functional a constant $\gamma\geq 0$ times a nonlocal term, all the results of this section will also hold for the Area functional, taking $\gamma=0$.} Multiplying by $v_E$ both sides of the first equation in system~\eqref{potential} and integrating by parts (and using also the second equation), we obtain \begin{align}\label{G1} \int_{\mathbb{T}^n} |\nabla v_E(x)|^2 \, dx =&\,- \int_{\mathbb{T}^n} v_E(x) \Delta v_E(x) \, dx \nonumber\\ =&\,\int_{\mathbb{T}^n} v_E(x) (u_E(x) - m) \, dx\nonumber\\ =&\,\int_{\mathbb{T}^n} v_E(x) u_E(x) \, dx\nonumber\\ =&\,\int_{\mathbb{T}^n}\int_{\mathbb{T}^n} G(x,y) u_E(x) u_E(y) \, dx\,dy, \end{align} hence, the functional $J$ can be also written in the useful form $$ J(E)= \mathcal A(\partial E) + \gamma \int_{\mathbb{T}^n} \int_{\mathbb{T}^n} G(x,y)u_E(x)u_E(y) \, dx\,dy . $$ \subsection{First and second variation}\label{sec1.2}\ \vskip.3em We start by computing the \emph{first variation} of the functional $J$. \begin{definition}\label{admissiblevar} Let $E \subseteq \mathbb{T}^n$ be a smooth set. Given a smooth map $\Phi:(-\varepsilon,\varepsilon)\times\mathbb{T}^n\to\mathbb{T}^n$, for $\varepsilon>0$, such that $\Phi_t=\Phi(t,\cdot):\mathbb{T}^n\to\mathbb{T}^n$ is a one--parameter family of diffeomorphism with $\Phi_0=\mathrm{Id}$, we say that $E_t=\Phi_t(E)$ is the {\em variation} of $E$ associated to $\Phi$ (or to $\Phi_t$). If moreover there holds $\mathrm{Vol}(E_t)=\mathrm{Vol}(E)$ for every $t\in(-\varepsilon,\varepsilon)$, we call $E_t$ a {\em volume--preserving} variation of $E$.\\ The vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ defined as $X=\frac{\partial \Phi_t}{\partial t}\,\bigr\vert_{t=0}$, is called the {\em infinitesimal generator} of the variation $E_t$. \end{definition} \begin{remark} As we are going to consider only smooth sets $E$, it is easy to see that this definition of variation is equivalent to have a family of diffeomorphisms $\Phi_t$ of $E$ only, indeed these latter can always be extended to the whole $\mathbb{T}^n$. Moreover, as the relevant objects are actually the boundaries of the sets $E$ and in view of the sequel, we could even consider only smooth ``deformations'' of $\partialrtial E$. We chose the above definition since it is easier and more convenient for the computations that are following. \end{remark} \begin{definition} Given a variation $E_t$ of $E$, coming from the one--parameter family of diffeomorphism $\Phi_t$, the \emph{first variation of $J$ at $E$ with respect to $\Phi_t$} is given by \begin{equation} \label{first variation:def} \frac{d}{dt}J(E_t)\Bigl|_{t=0} \, . \end{equation} We say that $E$ is a {\em critical set} for $J$, if all the first variations relative to variations $E_t$ of $E$ are zero.\\ We say that $E$ is a {\em critical set} for $J$ under a volume constraint, if all the first variations relative to volume--preserving variations $E_t$ of $E$ are zero. \end{definition} It is clear that if $E$ is a minimum for $J$ (under a volume constraint), then it is a critical set for $J$ (under a volume constraint). We are now going to compute the first variation of $J$ and see that it depends only on the restriction to $\partialrtial E$ of the infinitesimal generator $X$ of the variation $E_t$ of $E$. We briefly recall some ``geometric'' notations and results about the (Riemannian) geometry of the hypersurfaces in $\mathbb{R}^n$, referring to~\cite{gahula,Man,petersen2} for instance. {\em In the whole work, we will adopt the convention of summing over the repeated indices.} Given any smooth immersion $\psi:M \to \mathbb{T}^n$ of the smooth, $(n-1)$--dimensional, compact manifold $M$, representing a hypersurface $\psi(M)$ of $\mathbb{T}^n$, considering local coordinates around any $p\in M$, we have local bases of the tangent space $T_p M$, which can be identified with the $(n-1)$--dimensional hyperplane $d\psi_p(T_pM)$ of $\mathbb{R}^n\approx T_{\psi(p)}\mathbb{T}^n$ which is tangent to $\psi(M)$ at $\psi(p)$, and of the cotangent space $T_p^{*}M$, respectively given by vectors $\bigl\{\frac{\partialrtial\,}{\partialrtial x_i}\bigr\}$ and 1--forms $\{dx_j\}$. So we denote the vectors on $M$ by $X=X^i\frac{\partialrtial\,}{\partialrtial x_i}$ and the 1--forms by $\omega=\omega_jdx_j$, where the indices refer to the chosen local coordinate chart of $M$. With the above identification, we have clearly $\frac{\partialrtial\,}{\partialrtial x_i}\approx \frac{\partialrtial\psi}{\partialrtial x_i}$, for every $i\in\{1,\dots,n-1\}$. The manifold $M$ gets in a natural way a metric tensor $g$, pull--back via the map $\psi$ of the metric tensor of $\mathbb{T}^n$, coming from the standard scalar product of $\mathbb{R}^n$ (as $\mathbb{T}^n\approx\mathbb{R}^n/\mathbb Z^n$), hence, turning it into a Riemannian manifold $(M,g)$. Then, the components of $g$ in a local chart are $$ g_{ij}=\left\langle\frac{\partial \psi}{\partial x_i}\,\right\vert\left.\!\frac{\partial \psi}{\partial x_j}\right\rangle $$ and the ``canonical'' measure $\mu$, induced on $M$ by the metric $g$ is then given by $\mu=\sqrt{\deltat g_{ij}}\,{\mathscr{L}}^{n-1}$, where ${\mathscr{L}}^{n-1}$ is the standard Lebesgue measure on $\mathbb{R}^{n-1}$. Thus, supposing that $M$ has a {\em global} coordinate chart, we can write the Area functional on the hypersurface $\psi(M)$ in the following way, \begin{equation}\label{areachart} \mathcal A(\psi (M))=\int_{M} d\mu = \int_{M} \sqrt{\deltat g_{ij}(x)} \, dx\,. \end{equation} When this is not the case (as it is usual), we need several local charts $(U_k,\varphi_k)$ and a subordinated partitions of unity $f_k:M\to[0,1]$ (that is, the compact support of $f_k:M\to[0,1]$ is contained in the open set $U_k\subseteq M$, for every $k\in\mathcal{I}$), then \begin{equation} \mathcal A(\psi (M))=\int_{M} d\mu =\sum_{k\in\mathcal I}\int_{M} f_k\,d\mu=\sum_{k\in\mathcal I}\int_{U_k} f_k(x)\sqrt{\deltat g_{ij}^k(x)} \, dx\,, \end{equation} where $g^k_{ij}$ are the coefficients of the metric $g$ in the local chart $(U_k,\varphi_k)$. {\em In order to work with coordinates, in the computations with integrals in this section we will assume that all the hypersurfaces have a global coordinate chart, by simplicity. All the results actually hold also in the general case by using partitions of unity as above.} The induced Levi--Civita covariant derivative on $(M,g)$ of a vector field $X$ and of a 1--form $\omega$ are respectively given by $$ \nabla _jX^i=\frac{\partialrtial X^i}{\partialrtial x_j}+\Gamma^{i}_{jk}X^k\,, \qquad \nabla _j\omega_i=\frac{\partialrtial \omega_i}{\partialrtial x_j}-\Gamma^k_{ji}\omega_k\,, $$ where $\Gamma^{i}_{jk}$ are the Christoffel symbols of the connection $\nabla$, expressed by the formula $$ \Gamma^{i}_{jk}=\frac{1}{2} g^{il}\Bigl(\frac{\partialrtial\,}{\partialrtial x_j}g_{kl}+\frac{\partialrtial\,}{\partialrtial x_k}g_{jl}-\frac{\partialrtial\,}{\partialrtial x_l}g_{jk}\Bigr)\,. $$ Moreover, the gradient $\nabla f$ of a function, the divergence $\operatorname*{div}\nolimits X$ of a tangent vector field and the Laplacian $\Delta f$ at a point $p \in M$, are defined respectively by $$ g(\nabla f(p) , v)=df_p(v)\qquad\forall v\in T_p M\,, $$ \begin{equation}\label{divformcar} \operatorname*{div}\nolimits X={\mathrm {tr}} \nabla X=\nabla _iX^i=\frac{\partialrtial X^i}{\partialrtial x_i}+\Gamma^{i}_{ik}X^k \end{equation} (in a local chart) and $\Delta f =\operatorname*{div}\nolimits\nabla f$. We then recall that by the {\em divergence theorem} for compact manifolds (without boundary), there holds \begin{equation}\label{divteo} \int_{M}\operatorname*{div}\nolimits X\,d\mu=0\,, \end{equation} for every tangent vector field $X$ on $M$, which in particular implies \begin{equation}\label{corollariodivteo} \int_{M}\Delta f\,d\mu=0\,, \end{equation} for every smooth function $f:M \to\mathbb{R}$. Assuming that we have a globally defined unit {\em normal} vector field $\nu:M\to\mathbb{R}^n$ to $\varphi(M)$ (this will hold in our situation where the hypersurfaces will be boundaries of smooth sets $E\subseteq\mathbb{T}^n$, hence we will always consider $\nu$ to be the {\em outer unit normal vector} at every point of $\partial E$), we define the {\em second fundamental form} $B$ which is a symmetric bilinear form given, in a local charts, by its components \begin{equation}\label{secform} h_{ij} = - \biggl \langle \frac{\partial^2 \psi}{\partial x_i \partial x_j}\,\biggr\vert \,\nu \biggr \rangle \end{equation} and whose trace is the {\em mean curvature} $\mathrm{H}= g^{ij} h_{ij}$ of the hypersurface (with these choices, the standard sphere of $\mathbb{R}^n$ has positive mean curvature).\\ The symmetry properties of the covariant derivative of $B$ are given by the {\em Codazzi--Mainardi equations} \begin{equation}\label{codazzi} \nabla _ih_{jk}=\nabla _jh_{ik}=\nabla _kh_{ij}\,. \end{equation} In the sequel, the following {\em Gauss--Weingarten relations} will be fundamental, \begin{equation}\label{GW} \frac{\partial^2\psi}{\partial x_i\partial x_j}=\Gamma_{ij}^k\frac{\partial\psi}{\partial x_k}- h_{ij}\nu\qquad\qquad\frac{\partial\nu}{\partial x_j}= h_{jl}g^{ls}\frac{\partial\psi}{\partial x_s}\,, \end{equation} which imply \begin{equation}\label{lap} \Delta\psi=g^{ij}\Bigl(\frac{\partialrtial^2\psi}{\partialrtial x_i\partialrtial x_j}-\Gamma_{ij}^k\frac{\partialrtial\psi}{\partialrtial x_k}\Bigr)=-g^{ij}h_{ij}\nu=-\mathrm{H}\nu\,. \end{equation} Moreover, we have the formula \begin{equation}\label{Deltanu} \Delta \nu = \nabla \mathrm{H} -|B|^2\nu\,, \end{equation} indeed, computing in {\em normal coordinates} at a point $p\in M$, \begin{align*} \Delta\nu=&\,g^{ij}\Bigl(\frac{\partialrtial^2\nu}{\partialrtial x_i\partialrtial x_j}-\Gamma_{ij}^k\frac{\partialrtial\nu}{\partialrtial x_k}\Bigr)\\ =&\,g^{ij}\frac{\partialrtial}{\partialrtial x_i}\Bigl(h_{jl}g^{ls}\frac{\partial\psi}{\partial x_s}\Bigr)\\ =&\,g^{ij}\nabla_i h_{jl}g^{ls}\frac{\partial\psi}{\partial x_s}+g^{ij}h_{jl}g^{ls}\frac{\partialrtial^2\psi}{\partialrtial x_i\partial x_s}\\ =&\,g^{ij}\nabla_l h_{ij}g^{ls}\frac{\partial\psi}{\partial x_s}-g^{ij}h_{jl}g^{ls}h_{is}\nu\\ =&\,\nabla\mathrm{H} -|B|^2\nu\,, \end{align*} since all $\Gamma_{ij}^k$ and $\frac{\partialrtial}{\partialrtial x_i}g^{jk}$ are zero at $p\in M$ in such coordinates and we used Codazzi--Mainardi equations~\eqref{codazzi}. {\em In the following, when it is clear by the context, we will write $\nabla $, $\operatorname*{div}\nolimits$ and $\Delta$ for both the Riemannian operators on a hypersurface and the standard operators of $\mathbb{T}^n\approx\mathbb{R}^n/\mathbb Z^n$, but these latter will be instead denoted by $\nabla ^{\mathbb{T}^n}$, $\operatorname*{div}\nolimits^{\!\mathbb{T}^n}$ and $\Delta^{\!\mathbb{T}^n}$ when they will be computed at a point of a hypersurface, in order to avoid any possibility of misunderstanding.} \begin{thm}[First variation of the functional $J$]\label{first var} Let $E\subseteq \mathbb{T}^n$ a smooth set and $\Phi:(-\varepsilon,\varepsilon)\times \mathbb{T}^n\to\mathbb{T}^n$ a smooth map giving a variation $E_t=\Phi_t(E)$ with infinitesimal generator $X \in C^\infty (\mathbb{T}^n; \mathbb{R}^n)$. Then, \begin{equation}\label{eqcar2222} \frac{d}{dt} J(E_t)\Bigl|_{t=0}=\int_{\partialrtial E} (\mathrm{H}+ 4 \gamma v_E) \langle X \vert \nu_E\rangle \, d\mu \end{equation} where $\nu_E$ is the outer unit normal vector and $\mathrm{H}$ the mean curvature of the boundary $\partialrtial E$ (as defined above, relative to $\nu_E$), while the function $v_E:\mathbb{T}^n\to \mathbb{R}$ is the potential associated to $E$, defined by formulas~\eqref{potential1}--\eqref{potential}.\\ In particular, the first variation of the functional $J$ depends only on the normal component of the restriction of the infinitesimal generator $X$ to $\partial E$.\\ Clearly, when $\gamma=0$ we get the well known first variation of the Area functional at a smooth set $E$, \begin{equation} \frac{d}{dt}\mathcal A(\partial E_t)\Bigl|_{t=0}= \int_{\partialrtial E} \mathrm{H}\langle X \vert \nu_E\rangle \, d\mu\,. \end{equation} \end{thm} \begin{proof} We start by computing the derivative of the Area functional term of $J$. We let $\psi_t:\partialrtial E\to\mathbb{T}^n$ be the embedding given by $$ \psi_t(x)=\Phi(t,x)\,, $$ for $x\in\partialrtial E$ and $t\in(-\varepsilon,\varepsilon)$, then $\psi_t(\partialrtial E)=\partialrtial E_t$ and $\partialrtial_t\psi_t\bigl\vert_{t=0}=X$ at every point of $\partialrtial E$, moreover $\psi_0$ is simply the inclusion map of $\partial E$ in $\mathbb{T}^n$.\\ Denoting by $g_{ij}=g_{ij}(t)$ the induced metrics (via $\psi_t$, as above) on the smooth hypersurfaces $\partialrtial E_t$ and setting $\psi=\psi_0$, in a local chart we have \begin{align*} \frac{\partialrtial\,}{\partialrtial t}g_{ij}\,\Bigr\vert_{t=0}\, =&\,\left.\frac{\partialrtial\,}{\partialrtial t}\left\langle\frac{\partialrtial\psi_t}{\partialrtial x_i}\,\right\vert\left.\frac{\partialrtial\psi_t}{\partialrtial x_j}\right\rangle\right\vert_{t=0}\\ \,=&\,\left\langle\frac{\partialrtial X}{\partialrtial x_i}\,\right\vert\left.\frac{\partialrtial\psi}{\partialrtial x_j}\right\rangle+\left\langle\frac{\partialrtial X}{\partialrtial x_j}\,\right\vert\left.\frac{\partialrtial\psi}{\partialrtial x_i}\right\rangle\\ \,=&\,\frac{\partialrtial\,}{\partialrtial x_i}\left\langle X\,\left\vert\,\frac{\partialrtial\psi}{\partialrtial x_j}\right\rangle\right. +\frac{\partialrtial\,}{\partialrtial x_j}\left\langle X\,\left\vert\,\frac{\partialrtial\psi}{\partialrtial x_i}\right\rangle\right. -2\left\langle X\,\left\vert\,\frac{\partialrtial^2\psi}{{\partialrtial x_i}{\partialrtial x_j}}\right\rangle\right.\\ \,=&\,\frac{\partialrtial\,}{\partialrtial x_i}\left\langle X_\tau\,\left\vert\,\frac{\partialrtial\psi}{\partialrtial x_j}\right\rangle\right. +\frac{\partialrtial\,}{\partialrtial x_j}\left\langle X_\tau\,\left\vert\,\frac{\partialrtial\psi}{\partialrtial x_i}\right\rangle\right. -2\Gamma_{ij}^k\left\langle X_\tau\,\left\vert\,\frac{\partialrtial\psi}{\partialrtial x_k}\right\rangle\right. +2h_{ij}\langle X\,\vert\,\nu_E\rangle\,, \end{align*} where we used the Gauss--Weingarten relations~\eqref{GW} in the last step and we denoted with $X_\tau= X-\langle X \vert \nu_E \rangle\nu_E$ the ``tangential part'' of the vector field $X$ along the hypersurface $\partialrtial E$ (seeing $T_x\partial E$ as a hyperplane of $\mathbb{R}^n\approx T_x\mathbb{T}^n$).\\ Letting $\omega$ be the $1$--form defined by $\omega(Y)=g(X_\tau,Y)$, this formula can be rewritten as \begin{equation} \frac{\partialrtial}{\partialrtial t}g_{ij}\Bigr \vert_{t=0}= \frac{\partialrtial\omega_j}{\partialrtial x_i} +\frac{\partialrtial\omega_i}{\partialrtial x_j} - 2\Gamma_{ij}^k\omega_k + 2h_{ij} \langle X| \nu_E \rangle = \nabla _i\omega_j+\nabla _j\omega_i+ 2h_{ij} \langle X| \nu_E \rangle \,.\label{derg2} \end{equation} Hence, by the formula \begin{equation}\label{detform} \frac{d}{dt}\deltat A(t)=\deltat A(t)\,{\mathrm{tr}}\,[A^{-1}(t)\circ A'(t)]\,, \end{equation} holding for any $n\times n$ squared matrix $A(t)$ dependent on $t$, we get \begin{align} \frac{\partialrtial\,}{\partialrtial t}\sqrt{\deltat g_{ij}}\,\Bigr\vert_{t=0}\, =&\,\frac{\sqrt{\deltat g_{ij}}\, g^{ij}\left.\!\!\frac{\partialrtial\,}{\partialrtial t}g_{ij}\right\vert_{t=0}}{2}\\ =&\,\frac{\sqrt{\deltat g_{ij}}\, g^{ij}\bigl(\nabla _i\omega_j+ \nabla _j\omega_i+2h_{ij}\langle X\,\vert\,\nu_E\rangle\bigr)}{2}\\ =&\,\sqrt{\deltat g_{ij}}\bigl(\operatorname*{div}\nolimits\!X_\tau +\mathrm{H}\langle X\,\vert\,\nu_E\rangle\bigr)\,,\label{dermu2} \end{align} where the divergence is the (Riemannian) one relative to the hypersurface $\partial E$. Then, we conclude (recalling the discussion after formula~\eqref{areachart}) \begin{align} \frac{\partialrtial\,}{\partialrtial t}{\mathcal A}(\partial E_t) \,\Bigr\vert_{t=0}=\frac{\partialrtial\,}{\partialrtial t}{\mathcal A}(\psi_t(\partial E)) \,\Bigr\vert_{t=0}\, =&\,\frac{\partialrtial\,}{\partialrtial t}\,\int_{\partial E}\,d\mu_t\,\Bigr\vert_{t=0}\nonumber\\ =&\,\frac{\partialrtial\,}{\partialrtial t}\,\int_{\partial E}\sqrt{\deltat g_{ij}}\,dx\,\Bigr\vert_{t=0}\nonumber\\ =&\,\int_{\partial E}\frac{\partialrtial\,}{\partialrtial t}\sqrt{\deltat g_{ij}}\,\Bigr\vert_{t=0}\,dx\nonumber\\ =&\,\int_{\partial E}\!\bigl(\operatorname*{div}\nolimits\!X_\tau\!+\!\mathrm{H}\langle X\,\vert\,\nu_E\rangle\bigr)\sqrt{\deltat g_{ij}}\,dx\nonumber\\ =&\,\int_{\partial E}\!\bigl(\operatorname*{div}\nolimits\!X_\tau\!+\!\mathrm{H}\langle X\,\vert\,\nu_E\rangle\bigr)\,d\mu\nonumber\\ =&\,\int_{\partial E}\mathrm{H}\langle X\,\vert\,\nu_E\rangle\,d\mu\label{local} \end{align} where in the last step we applied the divergence theorem, that is, formula~\eqref{divteo}, on $\partial E$. In order to compute the derivative of the nonlocal term, we set \begin{equation}\label{eqc0} v(t,x)= v_{E_t}(x)= \int_{\mathbb{T}^n} G(x,y) u_{E_t}(x) \, dy = \int_{E_t} G(x,y) \, dy - \int_{E_t^c} G(x,y) \, dy, \end{equation} where $E_t^c = \mathbb{T}^n \setminus E_t$. Then, \begin{align} \frac{d}{dt} \Bigl( \int_{\mathbb{T}^n} |\nabla v_{E_t}(x)|^2 \, dx \Bigr) \Bigl|_{t=0} &= \frac{d}{dt} \Bigl( \int_{\mathbb{T}^n} |\nabla v(t,x)|^2 \, dx\Bigl)\Bigl|_{t=0}\\ &= 2 \int_{\mathbb{T}^n} \nabla v_E(x) \frac{\partialrtial}{\partialrtial t} \nabla v(t,x)\Bigl |_{t=0} \, dx\\ &= 2 \int_{\mathbb{T}^n} (u_E(x) - m) \frac{\partialrtial}{\partialrtial t} v(t,x) \Bigl |_{t=0} \, dx, \end{align} where in the last equality we used the fact that $- \Delta v_E = u_E - m$ and we integrated by parts. Now, we note that \begin{equation}\label{eqc1} \frac{\partialrtial}{\partialrtial t} v(t,x) = \frac{\partialrtial}{\partialrtial t} \Bigr (\int_{E_t} G(x,y) \, dy \Bigl) - \frac{\partialrtial}{\partialrtial t} \Bigl (\int_{E_t^c} G(x,y) \, dy \Bigr), \end{equation} and, by a change of variable, \begin{equation} \label{eqqq50} \frac{\partialrtial}{\partialrtial t} \Bigl (\int_{E_t} G(x,y) \, dy \Bigr) \Bigl |_{t=0}= \frac{\partialrtial}{\partialrtial t} \Bigl (\int_{E} G(x,\Phi(t,z)) J\Phi(t,z) \, dz \Bigr) \Bigl |_{t=0}, \end{equation} where $J\Phi(t,\cdot)$ is the Jacobian of $\Phi(t, \cdot)$. Then, as $J\Phi(t,z)=\deltat [d\Phi(t,z)]$, using again formula~\eqref{detform}, we have \begin{align} \frac{\partial }{\partial t} J\Phi(t,z)\,\Bigr\vert_{t=0}= &\,J\Phi(t,z)\,{\mathrm{tr}}\,\Bigl[d\Phi(t,z)^{-1}\circ \frac{\partial }{\partial t} d\Phi(t,z)\Bigl]\,\Bigr\vert_{t=0}\\ = &\,J\Phi(t,z)\,{\mathrm{tr}}\,\Bigl[d\Phi(t,z)^{-1}\circ d\frac{\partial }{\partial t} \Phi(t,z)\Bigl]\,\Bigr\vert_{t=0}\\ = &\,{\mathrm{tr}}\,dX(z)\\ =&\,\operatorname*{div}\nolimits X(z)\,, \end{align} by the definition of $X$ and being $\Phi(0,z)=z$. Thus, carrying the time derivative inside the integral in equation~\eqref{eqqq50}, we obtain \begin{align} \frac{\partialrtial}{\partialrtial t} \Bigl (\int_{E_t} G(x,y) \, dy \Bigr) \Bigl |_{t=0} =&\,\int_E \bigl(\langle\nabla _y G(x,y) \vert X(y)\rangle + G(x,y) \operatorname*{div}\nolimits\!X(y)\bigr)\,dy \\ =&\,\int_E \operatorname*{div}\nolimits_y \bigl(G(x,y) X(y)\bigr) \, dy\\ =&\,\int_{\partialrtial E} G(x,y) \langle X(y)\vert \nu_E(y)\rangle \, d\mu(y)\,.\label{eqc2} \end{align} By a very analogous computation we get \begin{equation}\label{eqc3} -\frac{\partialrtial}{\partialrtial t} \Bigl (\int_{E_t^c} G(x,y) \, dy \Bigr) \Bigl |_{t=0}= \int_{\partialrtial E} G(x,y)\langle X(y) \vert \nu_E(y) \rangle \, d\mu(y)\,, \end{equation} then, using equalities~\eqref{potential1} and~\eqref{G2}, we conclude \begin{align} \frac{d}{dt}\int_{\mathbb{T}^n} |\nabla v_{E_t}(x) |^2 \, dx\,\biggr|_{t=0} =&\, 4 \int_{\mathbb{T}^n} (u_E(x) - m) \Bigl ( \int_{\partial E}G(x,y)\langle X(y) \vert \nu_E(y)\rangle \, d\mu(y)\Bigr)\,dx\nonumber\\ =&\,4 \int_{\partial E} \Bigl (\int_{\mathbb{T}^n}G(x,y)(u_E(x)-m) \, dx \Bigr) \langle X(y) \vert \nu_E(y)\rangle \, d\mu(y)\nonumber\\ =&\,4 \int_{\partial E} v_E(y) \langle X(y) \vert \nu_E(y)\rangle \, d\mu(y)\, .\label{nonlocal} \end{align} Combining formulas~\eqref{local} and~\eqref{nonlocal}, we finally obtain formula~\eqref{eqcar2222}. \end{proof} Given a smooth set $E$ and any vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$, considering the associated smooth flow $\Phi: (-\varepsilon,\varepsilon)\times \mathbb{T}^n \to \mathbb{T}^n$, defined by the system \begin{equation}\label{varflow} \begin{cases} \frac{\partial \Phi}{\partial t}(t,x)=X(\Phi(t,x)), \\ \Phi(0, x)=x \end{cases} \end{equation} for every $x \in \mathbb{T}^n$ and $t \in (-\varepsilon, \varepsilon)$, for some $\varepsilon>0$, we have a variation $E_t=\Phi_t(E)$ with infinitesimal generator $X$. We call this variation the {\em special variation} associated to $X$. Moreover, given any smooth vector field $\overline{X}\in C^\infty(\partial E; \mathbb{R}^n)$, it can be extended easily to a smooth vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ with $X\vert_{\partial E}=\overline{X}$. Hence, if $E$ is a critical set for $J$ there holds \begin{equation*} \int_{\partialrtial E} (\mathrm{H}+ 4 \gamma v_E) \langle X \vert \nu_E\rangle \, d\mu=0\,, \end{equation*} for every $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$. Choosing a smooth vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ with $X\vert_{\partial E}=(\mathrm{H}+ 4 \gamma v_E)\nu_E$, we then obtain the following corollary. \begin{cor}\label{critcor} A smooth set $E \subseteq \mathbb{T}^n$ is a critical set for $J$ if and only if the function $\mathrm{H} + 4 \gamma v_E$ is zero on $\partial E$. When $\gamma =0$, we recover the classical condition $\mathrm{H}=0$ for a {\em minimal surface} in $\mathbb{R}^n$. \end{cor} It is less easy to characterize the infinitesimal generators of the volume--preserving variations of $E$, in order to find an analogous criticality condition on a set $E$, for the functional $J$ under a volume constraint.\\ Given $\Phi:(-\varepsilon,\varepsilon)\times\mathbb{T}^n\to\mathbb{T}^n$ such that $\mathrm{Vol}(\Phi_t(E))=\mathrm{Vol}(E_t)=\mathrm{Vol}(E)$ for all $t \in(-\varepsilon,\varepsilon)$, we let $X_t\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ be the family of the vector fields (well) defined by the formula $$ X_t(\Phi(t,z))=\frac{\partial \Phi}{\partial t}(t,z), $$ for every $t\in(-\varepsilon,\varepsilon)$ and $z\in\mathbb{T}^n$, hence, if $t=0$, the vector field $X=X_0$ is the infinitesimal generator of the volume--preserving variation $E_t$. Then, by changing variables, we have \begin{equation}\label{eqc999} 0= \frac{d}{dt} \mathrm{Vol}(E_t)=\frac{d}{dt}\int_{E_t}\,dx=\frac{d}{dt}\int_E J\Phi(t,z)\,dz= \int_E \frac{\partialrtial}{\partialrtial t} J\Phi(t,z)\,dz\,. \end{equation} As $J\Phi(t,z)=\deltat [d\Phi(t,z)]$, by means of formula~\eqref{detform}, we obtain \begin{equation} \frac{\partial }{\partial t} J\Phi(t,z)= J\Phi(t,z)\,{\mathrm{tr}}\,[d\Phi(t,z)^{-1}\circ dX_t(\Phi(t,z))\circ d\Phi(t,z)], \end{equation} since, by the definition of $X_t$ above, $$ \frac{\partial }{\partial t} d\Phi(t,z)=d\,\frac{\partial \Phi}{\partial t}(t,z)=d[X_t(\Phi(t,z))]=dX_t(\Phi(t,z))\circ d\Phi(t,z). $$ Being the trace of a matrix invariant by conjugation, we conclude \begin{equation}\label{tJac} \frac{\partial }{\partial t} J\Phi(t,z)= J\Phi(t,z)\,{\mathrm{tr}}\,[dX_t(\Phi(t,z))]= J\Phi(t,z)\operatorname*{div}\nolimits\!X_t(\Phi(t,z)), \end{equation} hence, by equality~\eqref{eqc999} and the divergence theorem (in $\mathbb{T}^n$), it follows \begin{equation}\label{eqc1000} 0=\int_E \operatorname*{div}\nolimits\!X_t(\Phi(t,z))J\Phi(t,z)\,dz=\int_{E_t} \operatorname*{div}\nolimits\!X_t(x)\,dx=\int_{\partial E} \langle X_t\circ\Phi_t \vert \nu_{E_t}\rangle \, d\mu_t\,, \end{equation} where $\nu_{E_t}$ is the outer unit normal vector and $\mu_t$ the canonical Riemannian measure of the smooth hypersurface $\partialrtial E_t$, given by the embedding $\psi_t=\Phi_t:\partial E\to\mathbb{T}^n$. Thus, letting $t=0$, \begin{equation}\label{eqc1000bis} \frac{d}{dt} \mathrm{Vol}(E_t)\Bigr\vert_{t=0}=\int_{\partial E} \langle X\vert \nu_E\rangle \, d\mu=0 \end{equation} and we conclude that if $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ is the infinitesimal generator of a volume--preserving variation for $E$, its normal component $\varphi=\langle X \vert \nu_E\rangle$ on $\partial E$ has zero integral (with respect to the measure $\mu$).\\ Conversely, we have the following lemma whose proof is postponed after Lemma~\ref{lemma1}, since the arguments in the two proofs are very similar. \begin{lem}\label{vector field} Let $\varphi:\partialrtial E\to\mathbb{R}$ a smooth function with zero integral with respect to the measure $\mu$ on $\partial E$. Then, there exists a smooth vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ such that $\varphi=\langle X \vert \nu_E\rangle$, $\operatorname*{div}\nolimits\! X=0$ in a neighborhood of $\partial E$ and the flow $\Phi$ defined by system~\eqref{varflow} having $X$ as infinitesimal generator, gives a volume--preserving variation $E_t=\Phi_t(E)$ of $E$. \end{lem} Hence, with this characterization of the infinitesimal generators of the volume--preserving variations for $E$, by Theorem~\ref{first var} we have that $E$ is a critical set for the functional $J$ under a volume constraint if and only if \begin{equation*} \int_{\partialrtial E} (\mathrm{H}+ 4 \gamma v_E) \langle X \vert \nu_E\rangle \, d\mu=0\,, \end{equation*} for every $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ such that $\langle X \vert \nu_E\rangle$ has zero integral on $\partial E$. By Lemma~\ref{vector field}, this is similarly to say that $$ \int_{\partial E}(\mathrm{H} + 4 \gamma v_E) \varphi \, d\mu =0 \,, $$ for all $\varphi \in C^{\infty}(\partial E)$ such that $\int_{\partial E} \varphi \, d\mu =0$, which is equivalent to the existence of a constant $\lambda \in \mathbb{R}$ such that \begin{equation}\label{EL} \mathrm{H} + 4 \gamma v_E= \lambda \qquad \text{on $\partial E$.} \end{equation} \begin{remark} The parameter $\lambda$ may be clearly interpreted as a \emph{Lagrange multiplier} associated with the volume constraint for $J$. \end{remark} \begin{proposition}\label{critprop} A smooth set $E \subseteq \mathbb{T}^n$ is a critical set for $J$ under a volume constraint if and only if the function $\mathrm{H} + 4 \gamma v_E$ is constant on $\partial E$. When $\gamma =0$, we recover the classical \emph{constant mean curvature} condition for hypersurfaces in $\mathbb{R}^n$. \end{proposition} Now we deal with the {\em second variation} of the functional $J$. \begin{definition} Given a variation $E_t$ of $E$, coming from the one--parameter family of diffeomorphism $\Phi_t$, the \emph{second variation of $J$ at $E$ with respect to $\Phi_t$} is given by \begin{equation}\label{variazioneseconda} \frac{d^2}{dt^2} J(E_t)\Bigl|_{t=0}\,. \end{equation} \end{definition} In the following proposition we compute the second variation of the Area functional. Then, we do the same for the nonlocal term of $J$ and we conclude with the second variation of the functional $J$. \begin{proposition}[Second variation of $\mathcal A$]\label{secondvarA} Let $E\subseteq \mathbb{T}^n$ a smooth set and $\Phi:(-\varepsilon,\varepsilon)\times \mathbb{T}^n\to\mathbb{T}^n$ a smooth map giving a variation $E_t=\Phi_t(E)$ with infinitesimal generator $X \in C^\infty (\mathbb{T}^n; \mathbb{R}^n)$. Then, \begin{align} \frac{d^2}{dt^2} \mathcal A(\partial E_t)\Bigr|_{t=0}=&\, \int_{\partial E}\bigl(|\nabla \langle X\vert \nu_E\rangle|^2- \langle X\vert \nu_E\rangle^2|B|^2\bigr)\, d\mu\\ &+\int_{\partial E}\mathrm{H}\bigl(\mathrm{H} \langle X\vert \nu_E\rangle^2+\langle Z\vert\nu_E\rangle-2\langle X_\tau |\nabla \langle X\vert \nu_E\rangle\rangle+ B(X_\tau, X_\tau)\bigr)\, d\mu\,, \end{align} where $X_\tau=X-\langle X|\nu_E\rangle\nu_E$ is the {\em tangential part} of $X$ on $\partialrtial E$, $B$ and $\mathrm{H}$ are respectively the second fundamental form and the mean curvature of $\partial E$, and \begin{equation}\label{Zdef} Z= \frac{\partial^2\Phi}{\partial t^2}(0, \cdot )=\frac{\partial}{\partial t}[X_t(\Phi (t, \cdot))]\,\Bigr\vert_{t=0}=\frac{\partial X_t}{\partial t}\,\Bigr\vert_{t=0}+ d X(X)\,, \end{equation} where, for every $t\in(-\varepsilon,\varepsilon)$, the vector field $X_t\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ is defined by the formula $$ X_t(\Phi(t,z))=\frac{\partial \Phi}{\partial t}(t,z), $$ for every $z\in\mathbb{T}^n$, hence, $X_0=X$. \end{proposition} \begin{proof} We let $\psi_t=\Phi(t,\cdot)\vert_{\partialrtial E}$. By arguing as in the first part of the proof of Theorem~\ref{first var} (without taking $t=0$), we have $$ \frac{d}{dt}\mathcal A(\partial E_t) = \int_{\partial E} \mathrm{H}_t \langle X_t\circ\Phi_t\vert \nu_{E_t}\rangle \, d\mu_t, $$ where $\mathrm{H}_t$ is the mean curvature of $\partial E_t$. Consequently, we have \begin{equation} \frac{d^2}{dt^2} \mathcal A(\partial E_t)\,\Bigr|_{t=0}= \frac{d}{dt} \int_{\partial E} \mathrm{H}_t \langle X_t\circ\Phi_t \vert \nu_{E_t}\rangle \sqrt{\deltat g_{ij}} \, dx\, \Bigr |_{t=0} \end{equation} where $g_{ij}=g_{ij}(t)$.\\ In order to simplify the notation in the following computations, we drop the subscripts, that is, we let $\mathrm{H}(t,\cdot) = \mathrm{H}_t$, $\nu(t,\cdot)=\nu_{E_t}$, $\varphi(t,\cdot)=\langle X_t\circ\Phi_t\vert \nu_{E_t}\rangle$, $\psi(t,\cdot)= \psi_t$ and $X(t,\cdot)=X_t\circ\Phi_t$ (by a little abuse of notation, since $X$ is already the infinitesimal generator of the variation).\\ We then need to compute the derivatives \begin{equation}\label{1} \frac{\partial \mathrm{H}}{\partial t}\, \Bigr |_{t=0}\qquad\qquad\text{ and }\qquad\qquad \frac{\partial}{\partial t}\langle X\vert \nu\rangle\, \Bigr |_{t=0} \end{equation} since we already know, by formula~\eqref{dermu2}, that \begin{equation} \label{3} \frac{\partial }{\partial t} \sqrt{\deltat g_{ij}}\, \Bigr |_{t=0}=\bigl(\operatorname*{div}\nolimits\!X_{\tau} +\mathrm{H} \varphi\bigr)\sqrt{\deltat g_{ij}}\, \Bigr |_{t=0}\,, \end{equation} hence, this derivative gives the following contribution to the second variation, \begin{equation} \label{contr 3} \int_{\partial E}( \varphi \mathrm{H} \operatorname*{div}\nolimits\!X_{\tau} + \varphi^2 \mathrm{H}^2) \, d\mu\,. \end{equation} Then, we compute (recalling formula~\eqref{Zdef}) $$ \frac{\partial \langle X \vert \nu\rangle}{\partial t}\, \biggr |_{t=0}= \left \langle \frac{\partial X}{\partial t} \biggl\vert \nu\right \rangle\biggr |_{t=0} +\left \langle X \biggl\vert \frac{\partial \nu}{\partial t}\right \rangle\biggr |_{t=0}= \left \langle Z\vert \nu\right \rangle+\left \langle X \biggl\vert \frac{\partial \nu}{\partial t}\right \rangle\biggr |_{t=0} $$ and using the fact that $\frac{\partial \nu}{\partial t}\bigr |_{t=0}$ is tangent to $\partial E$, in a local coordinate chart we obtain $$ \left \langle X \biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}=X_{\tau}^p \left\langle \frac{\partial \psi}{\partial x_p}\biggl \vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}, $$ where in the last inequality we used the notation $X_{\tau}=X_\tau ^p \frac{\partial \psi}{\partial x_p}$. Notice that, $\bigl \langle \frac{\partial \psi}{\partial x_p}\bigl \vert \nu \bigr\rangle=0$ for every $p\in\{1,\dots,n-1\}$ and $t\in(-\varepsilon,\varepsilon)$, hence, using the Gauss--Weingarten relations~\eqref{GW}, \begin{align} 0&= \frac{\partial}{\partial t} \left \langle \frac{\partial \psi}{\partial x_p}\biggl\vert \nu\right\rangle\biggr |_{t=0}=\left\langle\frac{\partial X}{\partial x_p}\biggl\vert\nu\right\rangle+\left\langle \frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}\\ &=\frac{\partial}{\partial x_p}\langle X\vert \nu\rangle - \left\langle X \biggl\vert \frac{\partial \nu}{\partial x_p}\right\rangle+\left\langle\frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}\\ &=\frac{\partial \varphi}{\partial x_p}-\left\langle X_\tau \biggl\vert \frac{\partial \nu}{\partial x_p}\right\rangle + \left\langle\frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}\\ &=\frac{\partial \varphi}{\partial x_p}- X_{\tau}^q\left\langle\frac{\partial \psi}{\partial x_q}\biggl\vert \frac{\partial \nu}{\partial x_p}\right\rangle+ \left\langle\frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}\\ &=\frac{\partial \varphi}{\partial x_p}- X_{\tau}^q \left\langle\frac{\partial \psi}{\partial x_q}\biggl\vert h_{pl}g^{li} \frac{\partial \psi}{\partial x_i}\right\rangle+ \left\langle\frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}\\ &=\frac{\partial \varphi}{\partial x_p} - X_{\tau} ^q h_{pl}g^{li}g_{qi}+\left\langle\frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0} \end{align} and we can conclude that \begin{equation}\label{eqcar3333} \left\langle \frac{\partial \psi}{\partial x_p} \biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0} = -\frac{\partial \varphi}{\partial x_p}+ X_{\tau}^q h_{pq}\,, \end{equation} where $h_{pq}$ are the components of the second fundamental form $B$ of $\partial E$ in the local chart. Thus, we obtain the following identity \begin{align} \frac{\partial}{\partial t}\langle X\vert \nu\rangle\, \Bigr |_{t=0}&=\langle Z\vert \nu\rangle + X_{\tau}^p \left\langle\frac{\partial \psi}{\partial x_p} \biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}\\ &=\langle Z \vert \nu\rangle - \frac{\partial \varphi}{\partial x_p}X_{\tau}^p +X_{\tau}^p X_{\tau}^q h_{pq}\\ &=\langle Z \vert \nu\rangle -\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle +B(X_\tau,X_\tau)\label{ZZZ} \end{align} and the relative contribution to the second variation is given by \begin{equation} \int_ {\partial E}\mathrm{H} \bigr(\langle Z \vert \nu \rangle -\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle+B(X_\tau,X_\tau)\bigl)\,d\mu\,. \end{equation} Now we conclude by computing the first derivative in~\eqref{1}. To this aim, we note that $$ \mathrm{H}=-\left\langle \frac{\partial^2 \psi}{\partial x_i \partial x_j}\biggl\vert \nu\right\rangle g^{ij} $$ hence, we need the following terms \begin{equation} \label{1A} \frac{\partial g^{ij}}{\partial t}\, \Bigr |_{t=0} \end{equation} \begin{equation} \label{1B} \left\langle \frac{\partial^2 \psi}{\partial x_i \partial x_j} \biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0} \end{equation} \begin{equation} \label{1C} \left\langle \frac{\partial}{\partial t} \frac{\partial^2 \psi}{\partial x_i \partial x_j} \biggl\vert \nu\right\rangle\biggr |_{t=0} \, . \end{equation} We start with the term~\eqref{1A}, recalling that \begin{equation} \frac{\partial g_{ij}}{\partial t}\, \Bigr |_{t=0}=\nabla _i \omega_j + \nabla _j \omega_i +2 h_{ij} \langle X\vert \nu\rangle \end{equation} by equation~\eqref{derg2}, where $\omega$ is the $1$--form defined by $\omega(Y)=g(X_\tau,Y)$.\\ Using the fact that $g_{ij}g^{jk}=0$, we obtain \begin{equation} 0= \frac{\partial g_{ij}}{\partial t}\, \Bigr |_{t=0} g^{jk}+g_{ij}\frac{\partial g^{jk}}{\partial t}\, \Bigr |_{t=0}= g^{jk}\bigr(\nabla _i \omega_j + \nabla _j \omega_i +2h_{ij} \langle X\vert \nu \rangle \bigl)+ g_{ij}\frac{\partial g^{jk}}{\partial t}\, \Bigr |_{t=0} \end{equation} then, \begin{equation}\label{1Acalc} \frac{\partial g^{pk}}{\partial t}\, \Bigr |_{t=0}=- g^{jp} g^{ik}\Bigr(\nabla _i \omega_j + \nabla _j \omega_i +2 h_{ij} \langle X\vert \nu\rangle \Bigl)=-\nabla ^{p}X_\tau ^k -\nabla ^k X_\tau ^p - 2 h^{pk}\varphi\,. \end{equation} We then proceed with the computation of the term~\eqref{1B}, by means of equation~\eqref{eqcar3333}, \begin{equation} \left\langle\frac{\partial^2 \psi}{\partial x_i \partial x_j}\biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}= \Gamma_{ij}^k \left\langle\frac{\partial \psi}{\partial x_k} \biggl\vert \frac{\partial \nu}{\partial t}\right\rangle \biggr |_{t=0}=\Gamma_{ij}^k \bigr(-\frac{\partial \varphi}{\partial x_k} + X_{\tau}^q h_{qk} \bigl) \end{equation} and finally we compute the term~\eqref{1C}, \begin{equation} \left\langle\frac{\partial}{\partial t}\frac{\partial^2 \psi}{\partial x_i \partial x_j} \biggl\vert \nu\right\rangle= \left\langle\frac{\partial^2 X}{\partial x_i \partial x_j} \biggl\vert \nu\right\rangle\biggr |_{t=0}=\left\langle\frac{\partial^2(\varphi\nu)}{\partial x_i \partial x_j}\biggl\vert \nu\right\rangle + \left\langle\frac{\partial^2X_\tau }{\partial x_i \partial x_j} \biggl\vert \nu\right\rangle. \end{equation} We have \begin{align} \left\langle \frac{\partial^2(\varphi\nu)}{\partial x_i \partial x_j} \biggl\vert \nu\right\rangle&= \frac{\partial^2 \varphi}{\partial x_i \partial x_j} +\left\langle\frac{\partial^2 \nu}{\partial x_i \partial x_j} \biggl\vert \nu \right\rangle \varphi \\ &= \frac{\partial^2 \varphi}{\partial x_i \partial x_j} +\left\langle\frac{\partial}{\partial x_i}\bigl(h_{jl}g^{lp}\frac{\partial \psi}{\partial x_p}\bigr)\biggl\vert \nu\right\rangle \varphi\\ &=\frac{\partial^2 \varphi}{\partial x_i \partial x_j} + h_{jl}g^{lp}\left\langle\frac{\partial^2 \psi}{\partial x_i \partial x_j} \biggl\vert \nu \right\rangle \varphi\\ &=\frac{\partial^2 \varphi}{\partial x_i \partial x_j} + \varphi h_{jl}{g^{lp}}h_{ip} \end{align} and \begin{align} \left\langle \frac{\partial^2 X_\tau}{\partial x_i \partial x_j} \biggl\vert \nu\right\rangle&= \frac{\partial}{\partial x_i}\left\langle \frac{\partial X_\tau}{\partial x_j}\biggl\vert \nu\right\rangle - \left\langle\frac{\partial X_\tau}{\partial x_j} \biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &= \frac{\partial}{\partial x_i}\left\langle\frac{\partial}{\partial x_j}\Bigr(X_\tau^p \frac{\partial \psi}{\partial x_p}\Bigl)\biggl\vert \nu \right\rangle- \left\langle \frac{\partial X_\tau}{\partial x_j} \biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &=\frac{\partial}{\partial x_i}\biggr[X_\tau^p \left\langle\frac{\partial^2 \psi}{\partial x_j \partial x_p}\biggl\vert\nu \right\rangle\biggl] - \left\langle\frac{\partial X_\tau}{\partial x_j}\biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &=-\frac{\partial}{\partial x_i}\bigr(X_\tau^p h_{pj}\bigl) - \left\langle\frac{\partial X_\tau}{\partial x_j} \biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &=-\frac{\partial}{\partial x_i} \bigr(X_\tau^p h_{pj}\bigl)- \left\langle\frac{\partial}{\partial x_j}\Bigr(X_\tau^p \frac{\partial \psi}{\partial x_p}\Bigl)\biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &=-\frac{\partial}{\partial x_i} \bigr(X_\tau^p h_{pj}\bigl) - X_{\tau}^p \left\langle\frac{\partial^2 \psi}{\partial x_j \partial x_p}\biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle-\frac{\partial X_\tau^p}{\partial x_j}\left\langle\frac{\partial \psi}{\partial x_p} \biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &=-\frac{\partial}{\partial x_i} \bigr(X_\tau^p h_{pj}\bigl)-X_\tau^p \Gamma_{jp}^k \left\langle\frac{\partial \psi}{\partial x_k}\biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle-\frac{\partial X_\tau^p}{\partial x_j} \left\langle\frac{\partial \psi}{\partial x_p}\biggl\vert \frac{\partial \nu}{\partial x_i}\right\rangle\\ &=-\frac{\partial}{\partial x_i} \bigr(X_\tau^p h_{pj}\bigl) - X_\tau^p \Gamma_{jp}^kh_{il} g^{lq} g_{kq}-\frac{\partial X^p}{\partial x_j}h_{il}g^{lq}g_{pq}\\ &=-\frac{\partial}{\partial x_i}\bigr(X_\tau^p h_{pj}\bigl)- X_\tau ^p \Gamma_{jp}^k h_{ik} - \frac{\partial X^k}{\partial x_j} h_{ik}. \end{align} Hence, we finally get \begin{align} \frac{\partial \mathrm{H}}{\partial t}\, \Bigr |_{t=0}=&\,-2 h_{ij} \nabla ^i X_\tau ^j - 2 \langle X\vert \nu\rangle|B|^2- g^{ij}\frac{\partial^2 \varphi}{\partial x_i \partial x_j} + g^{ij}\Gamma_{ij}^k \frac{\partial \varphi}{\partial x_k}\\ &\,+ |B|^2 \langle X\vert \nu\rangle - g^{ij} \Gamma_{ij}^k h_{kq} X_\tau^q + g^{ij} \frac{\partial}{\partial x_i} (X_\tau ^p h_{pj}) + h_{ij}\nabla ^i X^j_j\\ =&\,-|B|^2 \langle X\vert \nu\rangle - h_{ij} \nabla ^i X_\tau^j - \Delta \varphi\\ &\,+ g^{ij}\Bigr[\frac{\partial}{\partial x_i}\Bigr(X_\tau^p h_{pj}\Bigl)-\Gamma_{ij}^k \Bigr(X_\tau^p h_{pk}\Bigl)\Bigl]\\ =&\,-\varphi |B|^2 - \Delta \varphi - h_{ij}\nabla ^i X_\tau^j + g^{ij}\nabla _{i}(X_\tau^p h_{pj})\\ =&\,-\varphi|B|^2 -\Delta \varphi - h_{ij}\nabla ^i X_\tau^j + \operatorname*{div}\nolimits(X_\tau^p h_{pj})\\ =&\,-\varphi|B|^2 - \Delta \varphi + \langle X_\tau \vert \operatorname*{div}\nolimits\!B\rangle\\ =&\,-\varphi|B|^2 -\Delta \varphi + \langle X_\tau \vert \nabla \mathrm{H}\rangle\, ,\label{derH} \end{align} where in the last equality we used the Codazzi--Mainardi equations (see~\cite{Man}). We conclude that the contribution of the first term in~\eqref{1} is then \begin{equation} \int_{\partial E} \varphi \bigr(-\varphi |B|^2 - \Delta \varphi + \langle X_\tau \vert \nabla \mathrm{H}\rangle \bigl) \, d\mu. \end{equation} Putting all these contributions together, we obtain the second variation of the Area functional, \begin{align} \frac{d^2}{d t^2}\mathcal A(\partialrtial E_t)\,\Bigr\vert_{t=0}&= \int_{\partial E} \Bigl[-\varphi \Delta \varphi - \varphi^2 |B|^2 +\varphi \langle X_\tau \vert \nabla \mathrm{H}\rangle +\varphi \mathrm{H} \operatorname*{div}\nolimits\!X_\tau + \varphi^2\mathrm{H}^2\\ &\qquad\qquad + \mathrm{H}\bigl(\langle Z \vert \nu \rangle- \langle X_\tau \vert \nabla \varphi\rangle + B(X_\tau,X_\tau)\bigr)\Bigr] \, d\mu\,. \end{align} Integrating by parts, we have \begin{equation} \int_{\partial E} \varphi \langle X_\tau \vert \nabla \mathrm{H}\rangle\,d\mu= -\int_{\partial E}\bigl[\mathrm{H} \langle X_\tau \vert \nabla \varphi\rangle + \mathrm{H} \varphi \operatorname*{div}\nolimits\!X_\tau\bigr]\,d\mu \end{equation} and we can conclude \begin{equation} \frac{d^2}{dt^2}\mathcal A(\partialrtial E_t)\,\Bigr\vert_{t=0}= \int_{\partial E}\Bigl[ |\nabla \varphi|^2 - \varphi^2 |B|^2 + \varphi^2 \mathrm{H}^2+ \mathrm{H}(\langle Z \vert \nu \rangle - 2 \langle X_\tau \vert \nabla \varphi\rangle + B(X_\tau, X_\tau))\Bigr]\, d\mu\,, \end{equation} which is the formula we wanted. \end{proof} \begin{proposition}[Second variation of the nonlocal term]\label{second var F} Let $E\subseteq \mathbb{T}^n$, $\Phi$, $E_t$, $X$, $X_\tau$, $X_t$, $\mathrm{H}$, $B$ and $Z$ as in the previous proposition. Then, setting \begin{equation} N(t)=\int_{\mathbb{T}^n}|\nabla v_{E_t}(x)|^2\, dx\,, \end{equation} where $v_{E_t}:\mathbb{T}^n\to\mathbb{R}$ is the function defined by formulas~\eqref{potential1}--\eqref{potential} and $\partialrtial_{\nu_E} v_E=\langle\nabla^{\mathbb{T}^n}\!v_{E}|\nu_E\rangle$, the following formula holds \begin{align} \frac{d^2}{dt^2}N(t)\Bigr|_{t=0}=&\, 8 \int_{\partial E} \int_{\partial E} G(x,y) \langle X(x) \vert \nu_E (x) \rangle \langle X(y) \vert \nu_{E} (y)\rangle \, d\mu(x) d\mu(y)\\ &\,+4\int_{\partial E} \Bigl[v_E\bigl(\mathrm{H}\langle X\,\vert\,\nu_E\rangle^2+\langle Z \vert \nu_E\rangle-2\langle X_{\tau}| \nabla \langle X\vert \nu_E\rangle\rangle+B(X_\tau,X_\tau)\bigr)\\ &\,\,\,\qquad\qquad +\partialrtial_{\nu_E} v_E\langle X|\nu_E\rangle^2 \,\Bigr]\,d\mu\,,\label{N''} \end{align} giving the second variation of the nonlocal term of $J$. \end{proposition} \begin{proof} By arguing as in the second part of the proof of Theorem~\ref{first var} (equations~\eqref{eqc1}--\eqref{nonlocal}), we have \begin{equation} \frac{d}{dt} N(t)=4 \int_{\partial E} v_{E_t} \langle X_t\circ\Phi_t\vert \nu_{E_t}\rangle \, d\mu_t=4 \int_{\partial E} v_{E_t} \langle X_t\circ\Phi_t\vert \nu_{E_t}\rangle \sqrt{\deltat g_{ij}} \, dx\,. \end{equation} Setting $v(t,x)=v_{E_t}(x)$, $v_t=\frac{\partial v}{\partial t}(0,\cdot)$, $v_i=\frac{\partial v}{\partial x_i}(0,\cdot)$ and adopting the same notation of the proof of the previous proposition, that is, we let $\mathrm{H}(t,\cdot) = \mathrm{H}_t$, $\nu(t,\cdot)=\nu_{E_t}$ and $X(t,\cdot)=X_t\circ\Phi_t$, we have \begin{align} \frac{d^2}{dt^2} N(t)\Bigr|_{t=0}=&\,4 \frac{d}{dt}\int_{\partial E} v\langle X\vert \nu\rangle \sqrt{\deltat g_{ij}} \, dx\,\Bigr|_{t=0}\\ =&\,4 \int_{\partial E} \Bigl[v_t\langle X\vert \nu\rangle+v_iX^i\langle X\vert \nu\rangle+v\langle X\vert \nu\rangle \operatorname*{div}\nolimits\!X_{\tau}\\ &\,\,\,\qquad\quad+v\mathrm{H}\langle X\vert \nu\rangle^2+v\frac{\partial}{\partial t}\langle X\vert \nu\rangle\,\Bigr|_{t=0}\, \Bigr] \, d\mu\\ =&\,4 \int_{\partial E} \Bigl[v_t\langle X\vert \nu\rangle+v_iX^i\langle X\vert \nu\rangle+ v\langle X\vert \nu\rangle \operatorname*{div}\nolimits\!X_{\tau}\\ &\,\,\,\qquad\quad+v\bigl(\mathrm{H}\langle X\vert \nu\rangle^2+\langle Z \vert \nu\rangle -\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle +B(X_\tau,X_\tau)\bigr)\,\Bigr]\, d\mu\,, \end{align} by formulas~\eqref{dermu2} and~\eqref{ZZZ}. Then, integrating by parts the divergence, we obtain \begin{align} \frac{d^2}{dt^2} N(t)\Bigr|_{t=0}=&\,4 \int_{\partial E} \Bigl[v_t\langle X\vert \nu\rangle+v_iX^i\langle X\vert \nu\rangle-\langle\nabla v\vert X_\tau\rangle\langle X\vert \nu\rangle\\ &\,\,\,\qquad\quad+v\bigl(\mathrm{H}\langle X\vert \nu\rangle^2+\langle Z \vert \nu\rangle -2\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle +B(X_\tau,X_\tau)\bigr)\,\Bigr]\, d\mu\\ =&\,4 \int_{\partial E} \Bigl[v_t\langle X\vert \nu\rangle+\partialrtial_\nu v\langle X|\nu\rangle^2\\ &\,\,\,\qquad\quad+v\bigl(\mathrm{H}\langle X\vert \nu\rangle^2+\langle Z \vert \nu\rangle -2\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle +B(X_\tau,X_\tau)\bigr)\,\Bigr]\, d\mu \end{align} where $\partialrtial_\nu v=\langle\nabla^{\mathbb{T}^n}\!\!v\vert\nu\rangle$.\\ Now, by equations~\eqref{eqc1}--\eqref{eqc3}, there holds \begin{equation}\label{v'} v_t(0,x)=2 \int_{\partial E} G(x,y)\left\langle X(y) \vert \nu (y)\right\rangle \, d\mu(y)\,, \end{equation} hence, substituting this expression for $v_t$ in the equation above we have formula~\eqref{N''}. \end{proof} Putting together Propositions~\ref{secondvarA} and~\ref{second var F}, we then obtain the second variation of the nonlocal Area functional $J$. \begin{thm}[Second variation of the functional $J$] \label{secondvar} Let $E\subseteq \mathbb{T}^n$ a smooth set and $\Phi:(-\varepsilon,\varepsilon)\times \mathbb{T}^n\to\mathbb{T}^n$ a smooth map giving a variation $E_t$ with infinitesimal generator $X \in C^\infty (\mathbb{T}^n; \mathbb{R}^n)$. Then, \begin{align} \frac{d^2 }{dt^2}J(E_t)\Bigl|_{t=0} =&\,\int_{\partial E}\bigl(|\nabla \langle X\vert \nu_E\rangle|^2-\langle X\vert \nu_E\rangle^2|B|^2\bigr)\, d\mu\nonumber\\ & \,+8\gamma \int_{\partial E} \int_{\partial E}G(x,y)\langle X\vert \nu_E(x)\rangle \langle X\vert \nu_E(y)\rangle \,d\mu(x)\,d\mu(y)\nonumber\\ &\,+4\gamma\int_{\partial E}\partial_{\nu_E} v_E \langle X\vert \nu_E\rangle ^2\,d\mu+R\,, \label{IIfinale} \end{align} with the ``remainder term'' $R$ given by \begin{align} R=&\,\int_{\partial E}(\mathrm{H}+4\gamma v_E)\bigl(\mathrm{H}\langle X\vert \nu\rangle^2+\langle Z \vert \nu\rangle -2\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle +B(X_\tau,X_\tau)\bigr)\, d\mu\\ =&\,\int_{\partial E}(\mathrm{H}+4\gamma v_E)\Bigl[\langle X\vert \nu_E\rangle\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X -\operatorname*{div}\nolimits\bigl(\langle X\vert \nu_E\rangle X_\tau\bigr)+\Bigl\langle\frac{\partialrtial X_t}{\partialrtial t}\Bigr\vert_{t=0}\Bigr\vert\nu_E\Bigr\rangle\,\Bigr]\, d\mu \end{align} where $\nu_E$ is the outer unit normal vector to $\partial E$, $X_\tau=X-\langle X|\nu_E\rangle\nu_E$ is the {\em tangential part} of $X$ on $\partialrtial E$, $v_{E}:\mathbb{T}^n\to\mathbb{R}$ is the function defined by formulas~\eqref{potential1}--\eqref{potential}, $\partialrtial_{\nu_E} v_E=\langle\nabla^{\mathbb{T}^n}\!v_{E}|\nu_E\rangle$, $B$ and $\mathrm{H}$ are respectively the second fundamental form and the mean curvature of $\partial E$, the vector field $X_t\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ is defined by the formula $X_t(\Phi(t,z))=\frac{\partial \Phi}{\partial t}(t,z)$ for every $t\in(-\varepsilon,\varepsilon)$ and $z\in\mathbb{T}^n$, and \begin{equation} Z= \frac{\partial^2\Phi}{\partial t^2}(0, \cdot )=\frac{\partial}{\partial t}[X_t(\Phi (t, \cdot))]\,\Bigr\vert_{t=0}=\frac{\partial X_t}{\partial t}\,\Bigr\vert_{t=0}+ d X(X)\,. \end{equation} \end{thm} \begin{proof} Formula~\eqref{IIfinale} and the first equality for $R$ follows simply adding (after multiplying the nonlinear term by $\gamma$) the expressions for $\frac{d^2}{dt^2}\mathcal A (\partial E_t)\,\bigr|_{t=0}$ and $\frac{d^2}{dt^2}\int_{\mathbb{T}^n}|\nabla \, v_{E_t}|^2\, dx\,\bigr|_{t=0}$ we found in Propositions~\ref{secondvarA} and~\ref{second var F}.\\ If now we show that \begin{align} \mathrm{H} \langle X| &\, \nu_E \rangle ^2+\langle Z | \nu_E \rangle -2 \langle X_\tau | \nabla \langle X| \nu_E \rangle\rangle + B(X_\tau, X_\tau)\\ &=\langle X| \nu_E \rangle\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X-\operatorname*{div}\nolimits (\langle X| \nu_E \rangle X_\tau)+\Bigl\langle\frac{\partialrtial X_t}{\partialrtial t}\Bigr\vert_{t=0}\Bigr\vert\nu_E\Bigr\rangle\,,\label{claim} \end{align} we clearly obtain the second expression for $R$.\\ We note that, being every derivative of $\nu_E$ a tangent vector field, \begin{align} \langle X_\tau | \nabla \langle X| \nu_E \rangle\rangle &=\langle \nu_E| dX(X_\tau)\rangle +\langle X| \langle X_\tau|\nabla \nu_E\rangle\rangle \\ &=\langle \nu_E| dX(X_\tau)\rangle +\langle X_\tau| \langle X_\tau|\nabla \nu_E\rangle\rangle \\ &=\langle \nu_E | dX(X_\tau)\rangle +B (X_\tau,X_\tau )\,, \end{align} by the Gauss--Weingarten relations~\eqref{GW}.\\ Therefore, since $Z-\frac{\partialrtial X_t}{\partialrtial t}\bigr\vert_{t=0}=dX(X)$, we have \begin{align} \mathrm{H} \langle X &\vert \nu_E \rangle ^2+\langle Z | \nu_E \rangle -2 \langle X_\tau | \nabla \langle X| \nu_E \rangle\rangle + B(X_\tau, X_\tau) -\Bigl\langle\frac{\partialrtial X_t}{\partialrtial t}\Bigr\vert_{t=0}\Bigr\vert\nu\Bigr\rangle\nonumber\\ & = \mathrm{H} \langle X| \nu_E \rangle ^2+ \langle \nu_E | dX(X) \rangle - \langle X_\tau |\nabla \langle X| \nu_E \rangle\rangle - \langle \nu_E | dX(X_\tau)\rangle \nonumber\\ & = \mathrm{H} \langle X| \nu_E \rangle ^2+\langle \nu_E | dX(\langle X| \nu_E \rangle \nu_E)\rangle -\langle X_\tau | \nabla \langle X| \nu_E \rangle\rangle \nonumber\\ & = \mathrm{H} \langle X| \nu_E \rangle ^2+\langle X| \nu_E \rangle\langle \nu_E | dX(\nu_E)\rangle +\langle X| \nu_E \rangle\operatorname*{div}\nolimits\!X_\tau -\operatorname*{div}\nolimits (\langle X| \nu_E \rangle X_\tau)\,.\label{eqcar5555} \end{align} Now we notice that, choosing an orthonormal basis $e_1,\dots,e_{n-1},e_n=\nu_E$ of $\mathbb{R}^n$ at a point $p\in\partialrtial E$ and letting $X=X^ie_i$, we have $$ \langle e_i|\nabla^\top\!X^i\rangle=\bigl\langle e_i\bigr|\nabla^{\mathbb{T}^n}\!X^i-\langle \nabla^{\mathbb{T}^n}\!X^i|\nu_E\rangle\nu_E\bigr\rangle=\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X-\langle \nu_E | dX(\nu_E)\rangle\,, $$ where the symbol $\nabla^{\top}\!f$ denotes the projection on the tangent space to $\partialrtial E$ of the gradient $\nabla^{\mathbb{T}^n}\!\!f$ of a function, called {\em tangential gradient} of $f$ and coincident with the gradient operator of $\partial E$ applied to the restriction of $f$ to the hypersurface, while $\langle e_i|\nabla^\top\!X^i\rangle$ is called {\em tangential divergence} of $X$, usually denoted with $\operatorname*{div}\nolimits^{\!\top}\!\!X$ and coincident with the (Riemannian) divergence of $\partial E$ if $X$ is a tangent vector field, as we will see below (see~\cite{Si}). Moreover, if we choose a local parametrization of $\partialrtial E$ such that $\frac{\partial\psi}{\partial x_i}(p)=e_i$, for $i\in\{1,\dots,n-1\}$, we have $e_i^j=\frac{\partial\psi^j}{\partial x_i}=g^{ij}=\deltalta_{ij}$ at $p$ and \begin{align} \langle e_i|\nabla ^\top\!X^i\rangle=\operatorname*{div}\nolimits^{\!\top}\!\!X=&\,\langle e_i|\nabla ^\top\!X^i_\tau\rangle+\langle e_i|\nabla ^\top\!(\langle X|\nu_E\rangle\nu_E^i)\rangle\\ =&\,\langle e_i|\nabla X^i_\tau\rangle+\langle X|\nu_E\rangle \langle e_i|\nabla^{\mathbb{T}^n}\!\!\nu_E^i\rangle\\ =&\,\langle e_i|\nabla X^i_\tau\rangle+\langle X|\nu_E\rangle\frac{\partial\psi^j}{\partial x_i} h_{jl}g^{ls}\frac{\partial\psi^i}{\partial x_s}\\ =&\,\nabla _{e_i}X^i_\tau+\langle X|\nu_E\rangle h_{ii}\\ =&\,\operatorname*{div}\nolimits\!X_\tau+\langle X|\nu_E\rangle\mathrm{H}\,, \end{align} where we used again the Gauss--Weingarten relations~\eqref{GW} and the fact that the covariant derivative of a tangent vector field along a hypersurface of $\mathbb{R}^n$ can be obtained by differentiating in $\mathbb{R}^n$ (a local extension of) the vector field and projecting the result on the tangent space to the hypersurface (see~\cite{gahula}, for instance). Hence, we get $$ \langle \nu_E | dX(\nu_E)\rangle=\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X-\langle e_i|\nabla ^\top\!X^i\rangle=\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X-\operatorname*{div}\nolimits\!X_\tau-\langle X|\nu_E\rangle\mathrm{H} $$ and equation~\eqref{claim} follows by substituting this left term in formula~\eqref{eqcar5555}. \end{proof} \begin{remark} We are not aware of the presence in literature of this ``geometric'' line in deriving the (first and) second variation of $J$, moreover, in~\cite[Theorem~2.6, Step~3, equation~2.67]{ChSt}, this latter is obtained only at a critical set, while in~\cite[Theorem~3.6]{CaMoMo} the methods are strongly ``analytic'' and in our opinion less straightforward. These two papers are actually the ones on which is based the computation in~\cite[Theorem~3.1]{AcFuMo} of the second variation of $J$ at a general smooth set $E\subseteq\mathbb{T}^n$. Anyway, in this last paper, the variations of $E$ are all {\em special} variations, that is, they are given by the flows in system~\eqref{varflow}, indeed, the term with the time derivative of $X_t$ is missing (see formulas~3.1 and~7.2 in~\cite{AcFuMo}). Notice that the second variation in general does not depend only on the normal component $\langle X|\nu_E\rangle$ of the restriction to $\partial E$ of the infinitesimal generator $X$ of a variation $\Phi$ (this will anyway be true at a critical set $E$, see below), due to the presence of the $Z$--term and of $B (X_\tau,X_\tau )$ depending also on the tangential component of $X$ and of its behavior around $\partial E$. Even if we restrict ourselves to the special variations coming from system~\eqref{varflow}, with a {\em normal} infinitesimal generator $X$, which imply that all the vector fields $X_t$ are the same and coinciding with $X$, hence $Z=dX(X)$ and $X_\tau=0$, the second variation still depends also on the behavior of $X$ in a neighborhood of $\partial E$ (as $Z$). However, there are very particular case in which it depend only on $\langle X|\nu_E\rangle$, for instance when the variation is special and $X$ is normal with zero divergence (of $\mathbb{T}^n$) on $\partial E$ (in particular, if $\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X=0$ in a neighborhood of $\partial E$ or in the whole $\mathbb{T}^n$), as it can be seen easily by the second form of the remainder term $R$ in the above theorem. \end{remark} We see now how the second variation behaves at a critical set of $J$. \begin{cor}\label{IIcritcor} If $E \subseteq \mathbb{T}^n$ is a critical set for $J$, there holds \begin{align} \frac{d^2 }{dt^2}J(E_t)\Bigl|_{t=0} =&\,\int_{\partial E}\bigl(|\nabla \langle X\vert \nu_E\rangle|^2-\langle X\vert \nu_E\rangle^2|B|^2\bigr)\, d\mu\nonumber\\ &\,+8\gamma \int_{\partial E} \int_{\partial E}G(x,y)\langle X\vert \nu_E(x)\rangle \langle X\vert \nu_E(y)\rangle \,d\mu(x)\,d\mu(y)\nonumber\\ &\,+4\gamma\int_{\partial E}\partial_{\nu_E} v_E \langle X\vert \nu_E\rangle ^2\,d\mu\,, \end{align} for every variation $E_t$ of $E$, hence, the second variation of $J$ at $E$ depends only on the normal component of the restriction of the infinitesimal generator $X$ to $\partial E$, that is, on $\langle X| \nu_E \rangle$.\\ When $\gamma=0$ we get the well known second variation of the Area functional at a smooth set $E$ such that $\partial E$ is a minimal surface in $\mathbb{R}^n$, \begin{equation} \frac{d^2 }{dt^2}\mathcal A(\partial E_t)\Bigl|_{t=0} =\int_{\partial E}\bigl(|\nabla \langle X\vert \nu_E\rangle|^2-\langle X\vert \nu_E\rangle^2|B|^2\bigr)\, d\mu\,. \end{equation} \end{cor} \begin{proof} The thesis follows immediately, recalling that there holds $\mathrm{H}+4\gamma v_E=0$, by Corollary~\ref{critcor}, hence the remainder term $R$ in formula~\eqref{IIfinale} is zero. \end{proof} Finally, we see that the second variation has the same form (that is, $R=0$) also for $J$ under a volume constraint, at a critical set. \begin{proposition}\label{IIcritprop} If $E \subseteq \mathbb{T}^n$ is a critical set for $J$ under a volume constraint, there holds \begin{align} \frac{d^2 }{dt^2}J(E_t)\Bigl|_{t=0} =&\,\int_{\partial E}\bigl(|\nabla \langle X\vert \nu_E\rangle|^2-\langle X\vert \nu_E\rangle^2|B|^2\bigr)\, d\mu\nonumber\\ & \,+8\gamma \int_{\partial E} \int_{\partial E}G(x,y)\langle X\vert \nu_E(x)\rangle \langle X\vert \nu_E(y)\rangle \,d\mu(x)\,d\mu(y)\nonumber\\ &\,+4\gamma\int_{\partial E}\partial_{\nu_E} v_E \langle X\vert \nu_E\rangle ^2\,d\mu\,, \end{align} for every volume--preserving variation $E_t$ of $E$, hence, the second variation of $J$ at $E$ depends only on the normal component of the restriction of the infinitesimal generator $X$ to $\partial E$, that is, on $\langle X| \nu_E \rangle$.\\ When $\gamma=0$ we get the second variation of the Area functional under a volume constraint, at a smooth set $E$ such that $\partial E$ has constant mean curvature, \begin{equation} \frac{d^2 }{dt^2}\mathcal A(\partial E_t)\Bigl|_{t=0} =\int_{\partial E}\bigl(|\nabla \langle X\vert \nu_E\rangle|^2-\langle X\vert \nu_E\rangle^2|B|^2\bigr)\, d\mu\,. \end{equation} \end{proposition} \begin{proof} By Proposition~\ref{critprop}, the function $\mathrm{H}+4\gamma v_E$ is equal to a constant $\lambda\in\mathbb{R}$ on $\partial E$, then the remainder term $R$ in formula~\eqref{IIfinale} becomes $$ R=\lambda\int_{\partial E}\bigl(\mathrm{H}\langle X\vert \nu\rangle^2+\langle Z \vert \nu\rangle -2\langle X_{\tau}| \nabla \langle X\vert \nu\rangle\rangle +B(X_\tau,X_\tau)\bigr)\, d\mu\,. $$ Computing, in the same hypotheses and notations of Proposition~\ref{second var F}, the second derivative of the (constant) volume of $E_t$, by equations~\eqref{eqc999}--\eqref{eqc1000} we have (recalling formulas~\eqref{dermu2},~\eqref{ZZZ} and using the divergence theorem) \begin{align} 0=&\, \frac{d^2}{dt^2}\mathrm{Vol}(E_t)\,\Bigr\vert_{t=0}=\frac{d}{dt}\int_{E_t} \operatorname*{div}\nolimits\!X_t(x) \,dx\,\Bigr\vert_{t=0} =\frac{d}{dt}\int_{\partial E} \langle X\vert \nu_{E_t}\rangle \, d\mu_t\,\Bigr\vert_{t=0}\\ =&\,\int_{\partial E}\Bigl[\operatorname*{div}\nolimits\!X_\tau\langle X\,\vert\,\nu_E\rangle+\mathrm{H}\langle X\,\vert\,\nu_E\rangle^2+\langle Z \vert \nu_E\rangle -\langle X_{\tau}| \nabla \langle X\vert \nu_E\rangle\rangle+B(X_\tau,X_\tau)\,\Bigr]\,d\mu\\ =&\,\int_{\partial E}\Bigl[\mathrm{H}\langle X\,\vert\,\nu_E\rangle^2+\langle Z \vert \nu_E\rangle -2\langle X_{\tau}| \nabla \langle X\vert \nu_E\rangle\rangle+B(X_\tau,X_\tau)\,\Bigr]\,d\mu\,,\label{sec der vol} \end{align} hence $R=0$ and we are done. \end{proof} \begin{remark}\label{rem999} Notice that by the previous computation and relation~\eqref{claim}, it follows \begin{equation}\label{eqcar6666} \frac{d^2}{dt^2}\mathrm{Vol}(E_t)\,\Bigr\vert_{t=0}=\int_{\partial E}\Bigl[\langle X\vert \nu_E\rangle\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X+\Bigl\langle\frac{\partialrtial X_t}{\partialrtial t}\Bigr\vert_{t=0}\Bigr\vert\nu\Bigr\rangle\,\Bigr]\, d\mu=0\,, \end{equation} for every volume--preserving variation $E_t$ of $E$. Hence, if we restrict ourselves to the special (volume--preserving) variations coming from system~\eqref{varflow}, as in~\cite{AcFuMo}, we have $$ \frac{d^2}{dt^2}\mathrm{Vol}(E_t)\,\Bigr\vert_{t=0}=\int_{\partial E}\langle X\vert \nu_E\rangle\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X\, d\mu=0\,, $$ indeed, for such variations we have $X_t=X$, for every $t\in(-\varepsilon,\varepsilon)$. One can clearly use equality~\eqref{eqcar6666} to show the above proposition, as the term $R$ reduces (using the second form in Theorem~\ref{secondvar}) to $$ R=\lambda\int_{\partial E}\Bigl[\langle X\vert \nu_E\rangle\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X\,+\Bigl\langle\frac{\partialrtial X_t}{\partialrtial t}\Bigr\vert_{t=0}\Bigr\vert\nu\Bigr\rangle\,\Bigr]\, d\mu\,, $$ by the divergence theorem.\\ Moreover, we see that if we have a special variation generated by a vector field $X$ such that $\operatorname*{div}\nolimits^{\!\mathbb{T}^n}\!\!X=0$ on $\partial E$, then $\frac{d^2}{dt^2}\mathrm{Vol}(E_t)\,\bigr\vert_{t=0}=0$ and if $E$ is a critical set, $R=0$. This is then true for the special volume--preserving variations coming from Lemma~\ref{vector field} and when $X$ is a constant vector field, hence the associated special variation $E_t$ is simply a translation of $E$ (clearly, in this case $J(E_t)$ is constant and the first and second variations are zero). \end{remark} \subsection{Stability and $W^{2,p}$--local minimality}\label{stabsec}\ \vskip.3em By Proposition~\ref{IIcritprop}, the second variation of the functional $J$ under a volume constraint at a smooth critical set $E$ is a quadratic form in the normal component on $\partialrtial E$ of the infinitesimal generator $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ of a volume--preserving variation, that is, on $\varphi=\langle X|\nu_E\rangle$. This and the fact that the infinitesimal generators of the volume--preserving variations are ``characterized'' by having zero integral of such normal component on $\partial E$, by Lemma~\ref{vector field} and the discussion immediately before, motivate the following definition. \begin{definition}\label{Pi} Given any smooth open set $E\subseteq \mathbb{T}^n$ we define the space of (Sobolev) functions (see~\cite{Aubin}) \begin{equation} \widetilde{H}^1( \partialrtial E) = \Bigl\{\varphi:\partialrtial E\to\mathbb{R}\, :\, \varphi \in H^1(\partialrtial E)\,\, \text{ and }\, \int_{\partialrtial E} \varphi \,d \mu = 0 \Bigr\}, \end{equation} and the quadratic form $\Pi _E : \widetilde{H}^1(\partial E) \to \mathbb{R}$ as \begin{align} \Pi_E(\varphi)=&\, \int_{\partial E}\bigl(|\nabla \varphi|^2- \varphi^2|B|^2\bigr)\, d\mu+8\gamma \int_{\partial E} \int_{\partial E}G(x,y)\varphi(x)\varphi(y)\,d\mu(x)\,d\mu(y)\nonumber\\ &\,+4\gamma\int_{\partial E}\partial_{\nu_E} v_E \varphi ^2\,d\mu\,,\label{Pieq} \end{align} with the notations of Theorem~\ref{secondvar}. \end{definition} \begin{remark}\label{rm:potential} Letting for $\varphi\in \widetilde{H}^1(\partial E)$, $$ v_\varphi(x)=\int_{\partial E} G(x, y)\varphi(y)\, d\mu(y)\,, $$ it follows (from the properties of the Green's function) that $v_\varphi$ satisfies distributionally $-\Delta v_{\varphi}=\varphi\mu$ in $\mathbb{T}^n$, indeed, \begin{align*} \int_{\mathbb{T}^n}\langle\nabla v_\varphi(x)|\nabla \psi(x)\rangle\,dx =&\,-\int_{\mathbb{T}^n}v_\varphi(x)\Delta\psi(x)\,dx\\ =&\,-\int_{\mathbb{T}^n}\int_{\partial E} G(x, y)\varphi(y)\Delta\psi(x)\,d\mu(y)dx\\ =&\,-\int_{\partial E} \varphi(y)\int_{\mathbb{T}^n}G(x, y)\Delta\psi(x)\,dx\,d\mu(y)\\ =&\,-\int_{\partial E} \varphi(y)\int_{\mathbb{T}^n}\Delta G(x, y)\psi(x)\,dx\,d\mu(y)\\ =&\,\int_{\partial E} \varphi(y)\Bigl[\psi(y)-\int_{\mathbb{T}^n}\psi(x)\,dx\Bigr]\,d\mu(y)\\ =&\,\int_{\partial E}\varphi(y)\psi(y)\,d\mu(y)\,, \end{align*} for all $\psi\in C^\infty(\mathbb{T}^n)$, since $\int_{\partial E}\varphi(y)\,d\mu(y)=0$. Therefore, taking $\psi=v_\varphi$, we have $$ \int_{\mathbb{T}^n}|\nabla v_\varphi(x)|^2\,dx=\int_{\partial E}\varphi(y)v_\varphi(y)\,d\mu(y)\,, $$ hence, the following identity holds \begin{equation}\label{eqcar777} \int_{\partial E} \int_{\partial E}G(x,y)\varphi(x)\varphi(y)\,d\mu(x)d\mu(y)= \int_{\partial E}\varphi(y)\, v_{\varphi}(y)\, d\mu(y)=\int_{\mathbb{T}^n}|\nabla v_{\varphi}(x)|^2\, dx\,, \end{equation} and we can write \begin{equation} \Pi_E(\varphi)=\int_{\partial E}\Bigl(|\nabla \varphi|^2- \varphi^2|B|^2\Bigr)\, d\mu +8\gamma \int_{\mathbb{T}^n}|\nabla v_{\varphi}|^2\, dx+4\gamma\int_{\partial E}\partial_{\nu_E} v_E \varphi ^2\,d\mu\,,\label{Pieq2} \end{equation} for every $\varphi\in\widetilde{H}^1(\partialrtial E)$. \end{remark} \begin{definition}\label{admvector} Given any smooth open set $E\subseteq \mathbb{T}^n$, we say that a smooth vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ is {\em admissible for $E$} if the function $\varphi:\partial E\to\mathbb{R}$ given by $\varphi=\langle X \vert \nu_E\rangle$ belongs to $\widetilde{H}^1( \partialrtial E)$, that is, has zero integral on $\partial E$. \end{definition} \begin{remark}\label{remcarlo888} Clearly, if $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ is the infinitesimal generator of a volume--preserving variation for $E$, then $X$ is admissible, by the discussion after Corollary~\ref{critcor}. \end{remark} \begin{remark}\label{remcarlo} By what we said above, if $E$ is a smooth critical set for $J$ under a volume constraint, we can from now on consider only the special variations $E_t=\Phi_t(E)$ associated to admissible vector fields $X$, given by the flow $\Phi$ defined by system~\eqref{varflow}, hence \begin{equation} \frac{d}{dt} J(E_t) \Bigl|_{t=0} =\int_{\partial E} \langle X \vert \nu_E \rangle \, d \mu=0 \end{equation} and \begin{equation}\label{PI} \frac{d^2}{dt^2} J(E_t)\Bigr \vert_{t=0}=\Pi_E(\langle X \vert \nu_E \rangle) \end{equation} where $\Pi_E$ is the quadratic form defined by formula~\eqref{Pieq}. \end{remark} We notice that every constant vector field $X=\eta\in \mathbb{R}^n$ is clearly admissible, as $$ \int_{\partial E} \langle \eta\, \vert \nu_E \rangle \, d \mu=\int_{E} \operatorname*{div}\nolimits\eta\,dx=0 $$ and the associated flow is given by $ \Phi(t,x) = x + t \eta$, then, by the translation invariance of the functional $J$, we have $J(E_t)=J(E)$ and \begin{equation} 0=\frac{d^2}{dt^2} J(E_t) \Bigr \vert_{t=0}= \Pi_E (\langle \eta\, \vert \nu_E\rangle)\,, \end{equation} that is, the form $\Pi_E$ is zero on the vector subspace \begin{equation}\label{T} T(\partial E) = \bigl\{\langle \eta\, | \nu_E \rangle\, : \, \eta\in \mathbb{R}^n \bigr\}\subseteq \widetilde{H}^1(\partial E) \end{equation} of dimension clearly less than or equal to $n$. We split \begin{equation}\label{decomp} \widetilde{H}^1(\partial E) = T(\partial E) \oplus \mathbb{T}ort(\partial E)\,, \end{equation} where $\mathbb{T}ort (\partial E)\subseteq \widetilde{H}^1(\partial E)$ is the vector subspace $L^2$--orthogonal to $T(\partial E)$ (with respect to the measure $\mu$ on $\partialrtial E$), that is, \begin{align} \mathbb{T}ort(\partial E) =&\,\Bigl \{\varphi \in \widetilde{H}^1(\partial E) \, : \, \int_{\partial E} \varphi \nu_E \, d \mu = 0\Bigr \}\\ =&\,\Bigl \{\varphi \in H^1(\partial E) \, : \,\int_{\partial E} \varphi \, d \mu = 0 \,\,\text{ and }\, \int_{\partial E} \varphi \nu_E \, d \mu = 0\Bigr \}\label{Tort} \end{align} and we give the following ``stability'' conditions. \begin{definition}[Stability]\label{str stab} We say that a critical set $E \subseteq\mathbb{T}^n$ for $J$ under a volume constraint is {\em stable} if \begin{equation}\label{stabile} \Pi_E (\varphi) \geq 0 \qquad\text{for all $\varphi \in \widetilde{H}^1(\partial E)$} \end{equation} and {\em strictly stable} if moreover \begin{equation}\label{strettstabile} \Pi_E (\varphi) > 0 \qquad\text{for all $\varphi \in \mathbb{T}ort(\partial E) \setminus \{0 \}$.} \end{equation} \end{definition} \begin{remark}\label{rembase0} Introducing the symmetric bilinear form associated (by polarization) to $\Pi_E$ on $\widetilde{H}^1(\partial E)$, $$ b_E(\varphi,\psi)=\frac{\Pi_E (\varphi+\psi)-\Pi_E (\varphi-\psi)}{4} $$ at a critical set $E \subseteq\mathbb{T}^n$, it can be seen that actually $T(\partial E)$ is a degenerate vector subspace of $\widetilde{H}^1(\partial E)$ for $b_E$, that is, $b_E(\varphi,\psi)=0$ for every $\varphi\in\widetilde{H}^1(\partial E)$ and $\psi\in T(\partial E)$. Indeed, we observe that by formula~\eqref{potential1} and the properties of the Green function, we get \begin{align} \nabla v_E(x)&=\int_{\mathbb{T}^n} \nabla_x G(x,y) u_E(x) \, dy\\ & = \int_E \nabla_x G(x,y) \, dy - \int_{E^c}\nabla_x G(x,y) \, dy\\ & = -\int_E \nabla_y G(x,y) \, dy +\int_{E^c}\nabla_y G(x,y) \, dy\\ &= -2 \int_{\partialrtial E} G(x,y) \nu_E(y) \, d \mu(y)\,,\label{eqcar501} \end{align} where in the last passage we applied the divergence theorem.\\ By means of formula~\eqref{Deltanu} $$ \Delta \nu_E = \nabla \mathrm{H} -|B|^2\nu_E\,, $$ since $E$ (being critical) satisfies $\mathrm{H} + 4 \gamma v_E= \lambda$ for some constant $\lambda\in\mathbb{R}$, we have \begin{align} -\Delta \nu_E- |B|^2\nu_E&=\nabla(4 \gamma v_E - \lambda)\\ &=\nabla^{\mathbb{T}^n}\!(4 \gamma v_E - \lambda)-\partialrtial_{\nu_E}(4 \gamma v_E - \lambda)\\ &= -4\gamma (\partialrtial_{\nu_E} v_E) \nu_E -8\gamma \int_{\partialrtial E} G(x,y) \nu_E(y) \, d \mu(y) \end{align} on $\partial E$, by formula~\eqref{eqcar501}.\\ This equation can be written as $L(\nu_i)= 0$, for every $i\in\{1,\dots,n\}$, where $L$ is the self--adjoint, linear operator defined as $$ L(\varphi) = -\Delta \varphi - |B|^2\varphi + 4\gamma \partialrtial_{\nu_E} v_E \varphi + 8\gamma \int_{\partialrtial E} G(x,y) \varphi(y) \, d \mu(y)\,, $$ which clearly satisfies $$ b_E(\varphi,\psi)=\int_{\partial E}\langle L(\varphi)\vert \psi\rangle\,d \mu\qquad\text{ and }\qquad\Pi_E(\varphi) =\int_{\partial E}\langle L(\varphi)\vert \varphi\rangle\,d \mu\,. $$ Then, if we ``decompose'' a smooth function $\varphi \in \widetilde H^1(\partial E)$ as $\varphi = \psi + \langle \eta \vert \nu_E\rangle$, for some $\eta\in \mathbb{R}^n$ and $\psi \in T^\perp(\partialrtial E)$, we have (recalling formula~\eqref{Pieq}) \begin{align*} \Pi_E(\varphi) =&\,\int_{\partial E}\langle L(\varphi)\vert \varphi\rangle\,d \mu\\ =&\,\int_{\partial E}\langle L(\psi)\vert \psi \rangle\,d \mu+ 2\int_{\partial E} \langle L(\langle\eta \vert \nu_E\rangle) \vert \psi\rangle\,d \mu +\int_{\partial E}\langle L(\langle\eta \vert \nu_E\rangle)\vert \langle \eta \vert \nu_E \rangle \rangle\,d \mu\\ =&\,\Pi_E(\psi)\,. \end{align*} By approximation with smooth functions, we conclude that this equality holds for every function in $\widetilde H^1(\partial E)$.\\ The initial claim about the form $b_E$ then easily follows by its definition. Moreover, if $E$ is a strictly stable critical set there holds \begin{equation} \label{uusi stability} \Pi_E(\varphi)>0 \qquad \text{for every}\, \varphi \in \widetilde H^1(\partial E) \setminus T(\partial E). \end{equation} \end{remark} \begin{remark}\label{rembase} We observe that there exists an orthonormal frame $\{e_1, \dots, e_n \}$ of $\mathbb{R}^n$ such that \begin{equation}\label{ort} \int_{\partial E} \langle \nu_E | e_i \rangle \langle \nu_E | e_j \rangle \, d \mu=0, \end{equation} for all $i \ne j$, indeed, considering the symmetric $n\times n$--matrix $A= (a_{ij})$ with components $a_{ij}= \int_{\partial E} \nu_E^i \nu_E^j \, d \mu$, where $\nu_E^i=\langle\nu_E|\varepsilon_i\rangle$ for some basis $\{\varepsilon_1,\dots,\varepsilon_n\}$ of $\mathbb{R}^n$, we have \begin{equation} \int_{\partial E} (O\nu_E)_i (O \nu_E)_j \, d \mu = (OAO^{-1})_{ij}, \end{equation} for every $O \in SO(n)$. Choosing $O$ such that $OAO^{-1}$ is diagonal and setting $e_i=O^{-1}\varepsilon_i$, relations~\eqref{ort} are clearly satisfied. \\ Hence, the functions $\langle \nu_E | e_i \rangle$ which are not identically zero are an orthogonal basis of $T(\partial E)$. We set \begin{equation}\label{IIeq} \mathrm{I}I_E=\bigl\{i\in\{1,\dots,n\}\,:\,\text{$\langle\nu_E|e_i\rangle$ is not identically zero}\bigr\} \end{equation} and \begin{equation}\label{OOeq} {\mathrm{O}}_E=\mathrm{Span}\{e_i\,:\,i\in \mathrm{I}I_E\}, \end{equation} then, given any $\varphi \in \widetilde{H}^1(\partial E)$, its projection on $\mathbb{T}ort (\partial E)$ is \begin{equation}\label{projection} \pi(\varphi)= \varphi - \sum_{i\in\mathrm{I}I_E} \frac{\int_{\partial E} \varphi \langle \nu_E | e_i \rangle \, d \mu}{\norma{\langle \nu_E | e_i\rangle}_{L^2(\partialrtial E)}^2}\langle \nu_E | e_i \rangle\,. \end{equation} \end{remark} {\em From now on we will extensively use Sobolev spaces on smooth hypersurfaces. Most of their properties hold as in $\mathbb{R}^n$, standard references are~\cite{AdamsFournier} in the Euclidean space and~\cite{Aubin} when the ambient is a manifold.} Given a smooth set $E \subseteq \mathbb{T}^n$, for $\varepsilon>0$ small enough, we let ($d$ is the ``Euclidean'' distance on $\mathbb{T}^n$) \begin{equation} \label{tubdef} N_\varepsilon=\{x \in \mathbb{T}^n \, : \, d(x,\partial E)<\varepsilon\} \end{equation} to be a {\em tubular neighborhood} of $\partial E$ such that the {\em orthogonal projection map} $\pi_E:N_\varepsilon\to \partial E$ giving the (unique) closest point on $\partial E$ and the {\em signed distance function} $d_E:N_\varepsilon\to\mathbb{R}$ from $\partial E$ \begin{equation}\label{sign dist} d_E(x)= \begin{cases} d(x, \partial E) &\text{if $x \notin E$}\\ -d(x, \partial E) &\text{if $x \in E$} \end{cases} \end{equation} are well defined and smooth in $N_\varepsilon$ (for a proof of the existence of such tubular neighborhood and of all the subsequent properties, see~\cite{ManMen} for instance). Moreover, for every $x\in N_\varepsilon$, the projection map is given explicitly by \begin{equation}\label{eqcar2050} \pi_E(x)=x-\nabla d^2_E(x)/2=x-d_E(x)\nabla d_E(x) \end{equation} and the unit vector $\nabla d_E(x)$ is orthogonal to $\partial E$ at the point $\pi_E(x)\in\partialrtial E$, indeed actually \begin{equation}\label{eqcar410bis} \nabla d_E(x)=\nabla d_E(\pi_E(x))=\nu_E(\pi_E(x))\,, \end{equation} which means that the integral curves of the vector field $\nabla d_E$ are straight segments orthogonal to $\partial E$.\\ This clearly implies that the map \begin{equation}\label{eqcar410} \partialrtial E\times (-\varepsilon,\varepsilon)\ni (y,t)\mapsto L(y,t)=y+t\nabla d_E(y)=y+t\nu_E(y)\in N_\varepsilon \end{equation} is a smooth diffeomorphism with inverse $$ N_\varepsilon\ni x\mapsto L^{-1}(x)=(\pi_E(x),d_E(x))\in \partialrtial E\times (-\varepsilon,\varepsilon)\,. $$ Moreover, denoting with $JL$ its Jacobian (relative to the hypersurface $\partial E$), there holds \begin{equation}\label{eqcar411} 0<C_1\leq JL(y,t)\leq C_2 \end{equation} on $\partial E\times(-\varepsilon,\varepsilon)$, for a couple of constants $C_1,C_2$, depending on $E$ and $\varepsilon$. By means of such tubular neighborhood of a smooth set $E\subseteq\mathbb{T}^n$ and the map $L$, we can speak of ``$W^{k,p}$--closedness'' (or ``$C^{k,\alpha}$--closedness'') to $E$ of another smooth set $F\subseteq\mathbb{T}^n$, asking that for some $\deltalta>0$ ``small enough'', we have $\mathrm{Vol}(E\triangle F) < \deltalta$ and that $\partialrtial F$ is contained in a tubular neighborhood $N_\varepsilon$ of $E$, as above, described by $$ \partial F=\{y+\psi(y)\nu_E(y) \, : \, y\in \partial E\}, $$ for a smooth function $\psi:\partialrtial E\to\mathbb{R}$ with $\norma{\psi}_{W^{k,p}(\partial E)}< \deltalta$ (resp. $\norma{\psi}_{C^{k,\alpha}(\partial E)}< \deltalta$). That is, we are asking that the two sets $E$ and $F$ differ by a set of small measure and that their boundaries are ``close'' in $W^{k,p}$ (or $C^{k,\alpha}$) as graphs. Notice that $$ \psi(y)=\pi_2\circ L^{-1}\bigl(\partial E\cap \{y+\lambda \nu_E(y)\,:\,\lambda\in\mathbb{R}\}\bigr)\,, $$ where $\pi_2:\partial E\times(-\varepsilon,\varepsilon)\to\mathbb{R}$ is the projection on the second factor.\\ Moreover, given a sequence of smooth sets $F_i\subseteq\mathbb{T}^n$, we will write $F_i\to E$ in $W^{k,p}$ (resp. $C^{k,\alpha}$) if for every $\deltalta>0$, there hold $\mathrm{Vol}(F_i\triangle E) < \deltalta$, the smooth boundary $\partialrtial F_i$ is contained in some $N_\varepsilon$, relative to $E$ and it is described by $$ \partial F_i=\{y+\psi_i(y)\nu_E(y) \, : \, y\in \partial E\}, $$ for a smooth function $\psi_i:\partialrtial E\to\mathbb{R}$ with $\norma{\psi_i}_{W^{k,p}(\partial E)}< \deltalta$ (resp. $\norma{\psi_i}_{C^{k,\alpha}(\partial E)}< \deltalta$), for every $i\in\mathbb{N}$ large enough. {\em From now on, in all the rest of the work, we will refer to the volume--constrained nonlocal Area functional $J$ (and Area functional $\mathcal A$), sometimes without underlining the presence of such constraint, by simplicity. Moreover, with $N_\varepsilon$ we will always denote a suitable tubular neighborhood of a smooth set, with the above properties.} \begin{definition}\label{min} We say that a smooth set $E \subseteq \mathbb{T}^n$ is a {\em local minimizer} for the functional $J$ (for the Area functional $\mathcal A$) if there exists $\deltalta >0$ such that $$ J(F)\ge J(E) \qquad (\mathcal A(F) \ge \mathcal A(E)) $$ for all smooth sets $F\subseteq \mathbb{T}^n$ with $\mathrm{Vol}(F)=\mathrm{Vol}(E)$ and $\mathrm{Vol}(E\triangle F)< \deltalta$. We say that a smooth set $E \subseteq \mathbb{T}^n$ is a {\em $W^{2,p}$--local minimizer} if there exists $\deltalta >0$ and a tubular neighborhood $N_\varepsilon$ of $E$, as above, such that $$ J(F) \geq J(E) \qquad (\mathcal A(F) \ge \mathcal A(E)) $$ for all smooth sets $F \subseteq \mathbb{T}^n$ with $\mathrm{Vol} (F)= \mathrm{Vol}(E)$, $\mathrm{Vol}(E \triangle F)<\deltalta$ and $\partialrtial F$ contained in $N_\varepsilon$, described by $$ \partial F=\{y+\psi(y)\nu_E(y) \, : \, y\in \partial E\}, $$ for a smooth function $\psi:\partialrtial E\to\mathbb{R}$ with $\norma{\psi}_{W^{2,p}(\partial E)}< \deltalta$.\\ Clearly, any local minimizer is a $W^{2,p}$--local minimizer. \end{definition} We immediately show a {\em necessary} condition for $W^{2,p}$--local minimizers. \begin{proposition} Let the smooth set $E \subseteq \mathbb{T}^n$ be a $W^{2,p}$--local minimizer of $J$, then $E$ is a critical set and $$ \Pi_E(\varphi)\ge 0 \qquad \qquad \text{for all $\varphi \in \widetilde{H}^1(\partial E)$,} $$ in particular, $E$ is stable. \end{proposition} \begin{proof} If $E$ is a $W^{2,p}$--local minimizer of $J$, given any $\varphi \in C^\infty(\partial E)\cap\widetilde{H}^1(\partial E)$, we consider the admissible vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ given by Lemma~\ref{vector field} and the associated flow $\Phi$. Then, the variation $E_t=\Phi_t(E)$ of $E$ is volume--preserving, that is, $\mathrm{Vol}(E_t)=\mathrm{Vol}(E)$ and for every $\deltalta > 0$, there clearly exists a tubular neighborhood $N_\varepsilon$ of $E$ and $\overline{\varepsilon}>0$ such that for $t \in (-\overline{\varepsilon}, \overline{\varepsilon})$ we have $$ \mathrm{Vol}(E\triangle E_t) < \deltalta $$ and $$\partial E_t=\{y+\psi(y)\nu_E(y) \, : \, y\in \partial E\}\subseteq N_\varepsilon$$ for a smooth function $\psi:\partialrtial E\to\mathbb{R}$ with $\norma{\psi}_{W^{2,p}(\partial E)}< \deltalta$. Hence, the $W^{2,p}$--local minimality of $E$ implies $$ J(E) \le J(E_t), $$ for every $t \in (-\overline{\varepsilon}, \overline{\varepsilon})$. It follows $$ 0=\frac{d}{dt}J(E_t)\Bigl|_{t=0}= \int_{\partialrtial E} (\mathrm{H}+ 4 \gamma v_E) \varphi \, d\mu, $$ by Theorem~\ref{first var}, which implies that $E$ is a critical set, by the subsequent discussion and $$ 0\leq\frac{d^2}{dt^2}J(E_t)\Bigl|_{t=0}=\Pi_E(\varphi), $$ by Proposition~\ref{IIcritprop} and Remark~\ref{remcarlo}.\\ Then, the thesis easily follows by the density of $C^\infty(\partial E)\cap\widetilde{H}^1(\partial E)$ in $\widetilde{H}^1(\partial E)$ (see~\cite{Aubin}, for instance) and the definition of $\Pi_E$, formula~\eqref{Pieq}. \end{proof} The rest of this section will be devoted to show that the strict stability (see Definition~\ref{str stab}) is a {\em sufficient} condition for the $W^{2,p}$--local minimality. Precisely, we will prove the following theorem. \begin{thm}\label{W2pMin} Let $p>\max\{2, n-1\}$ and $E\subseteq\mathbb{T}^n$ a smooth strictly stable critical set for the nonlocal Area functional $J$ (under a volume constraint), with $N_\varepsilon$ a tubular neighborhood of $\partial E$ as in formula~\eqref{tubdef}. Then, there exist constants $\deltalta,C>0$ such that \begin{equation} J(F)\ge J(E) + C[\alpha(E,F)]^2, \end{equation} for all smooth sets $F\subseteq \mathbb{T}^n$ such that $\mathrm{Vol}(F)=\mathrm{Vol}(E)$, $\mathrm{Vol}(F\triangle E)<\deltalta$, $\partial F \subseteq N_{\varepsilon}$ and \begin{equation} \partial F= \{y+\psi(y)\nu_E(y)\, : \, y \in \partial E\}, \end{equation} for a smooth function $\psi$ with $\norma{\psi}_{W^{2,p}(\partial E)} < \deltalta$, where the ``distance'' $\alpha(E,F)$ is defined as \begin{equation}\label{alpha} \alpha(E,F)= \min_{\eta\in \mathbb{R}^n} \mathrm{Vol}(E \triangle (F+\eta)). \end{equation} As a consequence, $E$ is a $W^{2,p}$--local minimizer of $J$. Moreover, if $F$ is $W^{2,p}$--close enough to $E$ and $J(F)=J(E)$, then $F$ is a translate of $E$, that is, $E$ is locally the unique $W^{2,p}$--local minimizer, up to translations. \end{thm} \begin{remark} We could have introduced the definitions of {\em strict} local minimizer or {\em strict} $W^{2,p}$--local minimizer for the nonlocal Area functional, by asking that the inequalities $J(F)\leq J(E)$ in Definition~\ref{min} are equalities if and only if $F$ is a translate of $E$. With such notion, the conclusion of this theorem is that $E$ is actually a strict $W^{2,p}$--local minimizer (with a ``quantitative'' estimate of its minimality). \end{remark} \begin{remark}\label{L1min} With some extra effort, it can be proved that in the same hypotheses of Theorem~\ref{W2pMin}, the set $F$ is actually a local minimizer (see~\cite{AcFuMo}). Since in the analysis of the modified Mullins--Sekerka and surface diffusion flow in the next sections we do not need such a stronger result, we omitted to prove it. \end{remark} For the proof of this result we need some technical lemmas. We underline that most of the difficulties are due to the presence of the degenerate subspace $T(\partialrtial E)$ of the form $\Pi_E$ (where it is zero), related to the translation invariance of the nonlocal Area functional (recall the discussion after Definition~\ref{Pi}). In the next key lemma we are going to show how to construct volume--preserving variations (hence, admissible smooth vector fields) ``deforming'' a set $E$ to any other smooth set with the same volume, which is $W^{2,p}$--close enough. By the same technique we will also prove Lemma~\ref{vector field} immediately after, whose proof was postponed from Subsection~\ref{sec1.2}. \begin{lem}\label{lemma1} Let $E \subseteq \mathbb{T}^n$ be a smooth set and $N_\varepsilon$ a tubular neighborhood of $\partial E$ as above, in formula~\eqref{tubdef}. For all $p >n-1$, there exist constants $\deltalta, C>0$ such that if $\psi \in C^\infty (\partial E)$ and $\norma{\psi}_{W^{2,p}(\partial E)} \leq \deltalta$, then there exists a vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ with $\operatorname*{div}\nolimits\!X=0$ in $N_\varepsilon$ and the associated flow $\Phi$, defined by system~\eqref{varflow}, satisfies \begin{equation}\label{flowin} \Phi(1,y)= y+ \psi(y)\nu_E(y)\,, \qquad\text{for all $y \in \partial E$.} \end{equation} Moreover, for every $t \in [0,1]$ \begin{equation}\label{normflow} \norma{\Phi(t, \cdot) - \mathrm{Id}}_{W^{2,p} (\partial E)} \leq C \norma{\psi }_{W^{2,p}(\partial E)} \, . \end{equation} Finally, if $\mathrm{Vol}(E_1) = \mathrm{Vol}(E)$, then the variation $E_t=\Phi_t(E)$ is volume--preserving, that is, $\mathrm{Vol}(E_t) = \mathrm{Vol} (E)$ for all $t \in [-1,1]$ and the vector field $X$ is admissible. \end{lem} \begin{proof} We start considering the vector field $\widetilde{X}\in C^\infty (N_\varepsilon;\mathbb{R}^n)$ defined as \begin{equation}\label{eqcar400} \widetilde{X} (x) = \xi(x) \nabla d_E (x) \end{equation} for every $x \in N_\varepsilon$, where $ d_E:N_\varepsilon\to\mathbb{R}$ is the signed distance function from $E$ and $\xi:N_\varepsilon\to\mathbb{R}$ is the function defined as follows: for all $y \in \partial E$, we let $f_y : (-\varepsilon , \varepsilon) \to \mathbb{R}$ to be the unique solution of the ODE $$ \begin{cases} f'_y (t) + f_y(t) \Delta d_E (y+t \nu_E (y))=0 \\ f_y(0)= 1 \end{cases} $$ and we set $$ \xi(x)=\xi (y+ t \nu_E (y)) = f_y(t) =\exp \Bigl ( - \int _0 ^t {\Delta d_E (y+ s \nu_E(y)) \, ds } \Bigr ), $$ recalling that the map $(y,t)\mapsto x=y+t\nu_E(y)$ is a smooth diffeomorphism between $\partialrtial E\times (-\varepsilon,\varepsilon)$ and $N_\varepsilon$ (with inverse $x\mapsto (\pi_E(x),d_E(x))$, where $\pi_E$ is the orthogonal projection map on $E$, defined by formula~\eqref{eqcar2050}). Notice that the function $f$ is always positive, thus the same holds for $\xi$ and $\xi=1$, $\nabla d_E=\nu_E$, hence $\widetilde{X}=\nu_E$ on $\partialrtial E$. Our aim is then to prove that the smooth vector field $X$ defined by \begin{equation}\label{field} X(x)= \int _0 ^{\psi(\pi_E (x))} {\frac{ds}{\xi(\pi_E(x)+ s \nu_E(\pi_E(x)))} } \, \widetilde{X}(x) \end{equation} for every $x \in N_\varepsilon$ and extended smoothly to all $\mathbb{T}^n$, satisfies all the properties of the statement of the lemma. \noindent \textbf{Step ${\mathbf 1}$.} We saw that $\widetilde{X}\vert_{\partial E} = \nu_E$, now we show that $\operatorname*{div}\nolimits\!\widetilde{X} =0$ and analogously $\operatorname*{div}\nolimits\!X=0$ in $N_\varepsilon$.\\ Given any $x=y+t \nu_E(y)\in N_\varepsilon$, with $y\in\partial E$, we have \begin{align} \operatorname*{div}\nolimits\!\widetilde{X} (x) &= \operatorname*{div}\nolimits [\xi (x) \nabla d_E(x)]\\ &= \langle\nabla\xi (x)| \nabla d_E(x)\rangle+\xi (x) \Delta d_E(x)\\ &= \frac{\partialrtial}{\partialrtial t}[\xi (y+t \nu_E (y))]+ \xi (y+t \nu_E (y)) \Delta d_E (y+ t \nu_E(y)) \\ & = f'_y (t) + f_y (t) \Delta d_E ( y+ t \nu_E (y))\\ &= 0, \end{align} where we used the fact that $f_y'(t)=\langle \nabla \xi(y+t\nu_E(y))\vert \nu_E(y)\rangle$ and $\nabla d_E(y+t \nu_E (y))=\nu_E(y)$, by formula~\eqref{eqcar410bis}.\\ Since the function $$ x \mapsto \theta(x)=\int_0 ^{\psi(\pi_E(x))} \frac{ds}{\xi (\pi_E(x) + s \nu (\pi_E (x))} $$ is clearly constant along the segments $t\mapsto x+t\nabla d_E(x)$, for every $x\in N_\varepsilon$, it follows that $$ 0=\frac{\partialrtial}{\partialrtial t}\bigl[\theta(x+t\nabla d_E(x))\bigr]\,\Bigr\vert_{t=0}=\langle \nabla\theta(x)|\nabla d_E(x)\rangle, $$ hence, $$ \operatorname*{div}\nolimits\!X= \langle \nabla\theta|\nabla d_E\rangle\xi + \theta \operatorname*{div}\nolimits\!\widetilde{X} = 0. $$ \noindent \textbf{Step ${\mathbf 2}$.} Recalling that $\psi \in C^\infty (\partial E)$ and $p>n-1$, we have $$ \norma{\psi}_{L^\infty (\partial E)} \leq\norma{\psi}_{C^1(\partial E)} \leq C_E \norma{\psi}_{W^{2,p} (\partial E)}, $$ by Sobolev embeddings (see~\cite{Aubin}). Then, we can choose $\deltalta < {\varepsilon}/{C_E}$ such that for all $x \in \partial E$ we have that $x\pm\psi(x) \nu_E (x) \in N_\varepsilon$.\\ To check that equation~\eqref{flowin} holds, we observe that \begin{equation} \theta(x)=\int_0 ^{\psi (\pi_E(x))} {\frac{ds}{\xi (\pi_E(x)+ s \nu_E (\pi_E(x)))} } \end{equation} represents the time needed to go from $\pi_E(x)$ to $\pi_E(x)+ \psi (\pi_E(x)) \nu_E (\pi_E (x))$ along the trajectory of the vector field $\widetilde{X}$, which is the segment connecting $\pi_E(x)$ and $\pi_E(x)+ \psi (\pi_E(x)) \nu_E (\pi_E (x))$, of length $\psi (\pi_E(x))$, parametrized as $$ s\mapsto \pi_E(x)+ s\psi (\pi_E(x)) \nu_E (\pi_E (x)), $$ for $s\in[0,1]$ and which is traveled with velocity $\xi (\pi_E(x)+ s \nu_E (\pi_E(x)))=f_{\pi_E(x)}(s)$. Therefore, by the above definition of $X=\theta\widetilde{X}$ and the fact that the function $\theta$ is constant along such segments, we conclude that $$ \Phi(1, y) - \Phi (0,y) = \psi(y) \nu_E (y)\,, $$ that is, $\Phi(1,y)= y+ \psi (y) \nu_E (y)$, for all $y\in \partial E$. \noindent \textbf{Step ${\mathbf 3}$.} To establish inequality~\eqref{normflow}, we first show that \begin{equation}\label{normfield} \norma{X}_{W^{2,p}(N_\varepsilon)} \leq C \norma{\psi}_{W^{2,p}(\partial E)} \end{equation} for a constant $C>0$ depending only on $E$ and $\varepsilon$. This estimate will follow from the definition of $X$ in equation~\eqref{field} and the definition of $W^{2,p}$--norm, that is, $$ \norma{X}_{W^{2,p}(N_\varepsilon)} = \norma{X}_{L^p (N_\varepsilon)} + \norma{ \nabla X}_{L^p (N_\varepsilon)} + \norma{ \nabla ^2 X }_{L^p (N_\varepsilon)} \, . $$ As $|\nabla d_E|=1$ everywhere and the positive function $ \xi $ satisfies $0 < C_1 \leq \xi \leq C_2$ in $N_\varepsilon$, for a pair of constants $C_1$ and $C_2$, we have \begin{align} \norma{X} _{L^p (N_\varepsilon)} ^ p & = \int_{N_\varepsilon} \biggl \vert {\int_0 ^{\psi(\pi_E(x))} \frac{ds}{\xi ( \pi_E(x) + s \nu_E (\pi_E(x)))} \, \xi(x) \nabla d_E (x) } \biggr \vert^p \, dx \nonumber \\ & \leq \norma{\xi}^p_{L^\infty (N_\varepsilon)} \int_{N_\varepsilon} \biggl \vert \int_0 ^{\psi(\pi_E(x))} \frac{ds}{\xi ( \pi_E(x) + s \nu_E (\pi_E(x)))} \biggr \vert^p \, dx \nonumber \\ & \leq \frac{C_2^p}{C_1^p}\int_{N_\varepsilon} \abs{\psi(\pi_E(x))}^p \, dx \nonumber \\ & =\frac{C_2^p}{C_1^p}\int_{\partial E}\int_{-\varepsilon}^\varepsilon \abs{\psi(\pi_E(y+t\nu_E(y)))}^p JL(y,t) \, dt\,d\mu(y)\\ & =\frac{C_2^p}{C_1^p}\int_{\partial E} \abs{\psi(y)}^p\int_{-\varepsilon}^\varepsilon JL(y,t) \, dt\,d\mu(y)\\ & \leq C\int_{\partial E} \abs{\psi(y)}^p \, d\mu(y)\\ & = C\norma{\psi}_{L^p(\partial E)} ^p \, ,\label{normX} \end{align} where $L:\partialrtial E\times (-\varepsilon,\varepsilon)\to N_\varepsilon$ the smooth diffeomorphism defined in formula~\eqref{eqcar410} and $JL$ its Jacobian. Notice that the constant $C$ depends only on $E$ and $\varepsilon$. Now we estimate the $L^p$--norm of $\nabla X$. We compute \begin{align} \nabla X =&\,\frac{\nabla \psi (\pi_E(x)) d\pi_E(x)}{ \xi(\pi_E(x) + \psi(\pi_E(x)) \nu_E (\pi_E(x)))} \, \xi (x) \nabla d_E(x) \\ &\,-\biggl [\int_0 ^{\psi (\pi_E (x))}\frac{\nabla \xi (\pi_E(x) + s \nu_E (\pi_E(x)))}{\xi ^ 2 (\pi_E(x) + s \nu_E (\pi_E(x)))} d\pi_E(x)\,\mathrm{Id}\, ds\biggr]\, \xi (x) \nabla d_E (x) \\ &\,-\biggl [\int_0 ^{\psi (\pi_E (x))}\frac{\nabla \xi (\pi_E(x) + s \nu_E (\pi_E(x)))}{\xi ^ 2 (\pi_E(x) + s \nu_E (\pi_E(x)))} d\pi_E(x)s\,d\nu_E (\pi_E (x))\, ds\biggr]\, \xi (x) \nabla d_E (x)\\ &\,+ \int_0 ^{\psi (\pi_E (x))} \frac{ds}{\xi (\pi_E(x) + s \nu_E (\pi_E(x)))} \bigl ( \nabla \xi (x) \nabla d_E (x) + \xi (x)\nabla^2 d_E (x) \bigr) \end{align} and we deal with the integrals in the three terms as before, changing variable by means of the function $L$. That is, since all the functions $d\pi_E$, $d\nu_E$, $\nabla^2 d_E$, $\xi$, $1/\xi$, $\nabla\xi$ are bounded by some constants depending only on $E$ and $\varepsilon$, we easily get (the constant $C$ could vary from line to line) \begin{align*} \norma{\nabla X}_{L^p (N_\varepsilon)}^p \leq &\,C\int_{N_\varepsilon} \vert \nabla \psi (\pi_E(x))\vert^p \, dx+C\int_{N_\varepsilon} \vert\psi(\pi_E (x))\vert^p \, dx\\ =&\,C\int_{\partial E}\int_{-\varepsilon}^\varepsilon \abs{\nabla\psi(\pi_E(y+t\nu_E(y)))}^p\,JL(y,t) \, dt\,d\mu(y)\\ &\,+C\int_{\partial E}\int_{-\varepsilon}^\varepsilon \abs{\psi(\pi_E(y+t\nu_E(y)))}^p\,JL(y,t) \, dt\,d\mu(y)\\ =&\, C\int_{\partial E} \bigl(\abs{\psi(y)}^p+\abs{\nabla\psi(y)}^p\bigr)\,\int_{-\varepsilon}^\varepsilon JL(y,t) \, dt\,d\mu(y)\\ & \leq C\norma{\psi}_{L^p(\partial E)} ^p +C\norma{\nabla\psi}_{L^p(\partial E)} ^p\\ & \leq C\norma{\psi}_{W^{1,p}(\partial E)} ^p\,. \end{align*} A very analogous estimate works for $\norma{ \nabla ^2 X} _{L^p (N_\varepsilon)}^p$ and we obtain also \begin{equation}\label{normgrad2X} \norma{\nabla ^2 X}_{L^p (N_\varepsilon)}^p \leq C\norma{\psi}_{W^{2,p} (\partial E)}^p\,, \end{equation} hence, inequality~\eqref{normfield} follows with $C=C(E,\varepsilon)$. Applying now Lagrange theorem to every component of $\Phi (\cdot,y)$ for any $y\in\partial E$ and $t\in[0,1]$, we have $$ \Phi_i(t,y)-y_i=\Phi_i(t,y) - \Phi_i(0, y)= t X^i(\Phi(s,y))\,, $$ for every $i\in\{1,\dots,n\}$, where $s=s(y,t)$ is a suitable value in $(0,1)$. Then, it clearly follows \begin{equation}\label{normfi} \norma{\Phi(t,\cdot)- \mathrm{Id}}_{L^\infty (\partial E)}\leq C\norma{X} _{L^\infty (N_\varepsilon)}\leq C\norma{X}_{W^{2,p}(N_\varepsilon)} \leq C \norma{\psi}_{W^{2,p}(\partial E)} \end{equation} by estimate~\eqref{normfield}, with $C=C(E,\varepsilon)$ (notice that we used Sobolev embeddings, being $p>n-1$, the dimension of $\partial E$).\\ Differentiating the equations in system~\eqref{varflow}, we have (recall that we use the convention of summing over the repeated indices) \begin{equation}\label{flowdiff} \begin{cases} \frac{\partial}{\partial t} \nabla^i\Phi_j(t,y) = \nabla^k X^j(\Phi (t,y)) \nabla^i\Phi_k(t,y) \\ \nabla ^i \Phi_j (0,y)=\deltalta_{ij} \end{cases} \end{equation} for every $i,j\in\{1,\dots,n\}$. It follows, \begin{align} \frac{\partial}{\partial t} \bigl\vert \nabla^i\Phi_j(t,y)-\deltalta_{ij}\bigl\vert^2\leq&\,2\bigl|(\nabla^i\Phi_j(t,y)-\deltalta_{ij})\nabla^k X^j(\Phi (t,y)) \nabla^i\Phi_k(t,y)\bigr|\\ \leq&\,2\Vert\nabla X\Vert_{L^\infty(N_\varepsilon)}\bigl\vert \nabla^i\Phi_j(t,y)-\deltalta_{ij}\bigl\vert^2+2\Vert\nabla X\Vert_{L^\infty(N_\varepsilon)}\bigl|\nabla^i\Phi_j(t,y)-\deltalta_{ij}\bigr| \end{align} hence, for almost every $t\in[0,1]$, where the following derivative exists, $$ \frac{\partial}{\partial t} \bigl\vert \nabla^i\Phi_j(t,y)-\deltalta_{ij}\bigl\vert\leq C\Vert\nabla X\Vert_{L^\infty(N_\varepsilon)}\bigl(\bigl\vert \nabla^i\Phi_j(t,y)-\deltalta_{ij}\bigl\vert+1\bigr)\,. $$ Integrating this differential inequality, we get $$ \bigl\vert \nabla^i\Phi_j(t,y)-\deltalta_{ij}\bigl\vert\leq e^{tC\Vert\nabla X\Vert_{L^\infty(N_\varepsilon)}}-1\leq e^{C\Vert X\Vert_{W^{2,p}(N_\varepsilon)}}-1, $$ as $t\in[0,1]$, where we used Sobolev embeddings again. Then, by inequality~\eqref{normfield}, we estimate \begin{equation} \sum_{1\leq i,j\leq n}\norma{\nabla^i\Phi_j(t,\cdot) - \deltalta_{ij}}_{L^\infty (\partial E)} \leq C\bigl(e^{C\norma{\psi}_{W^{2,p} (\partial E)}}-1\bigr)\leq C\norma{\psi}_{W^{2,p} (\partial E)},\label{normgradfi} \end{equation} as $\norma{\psi}_{W^{2,p}(\partial E)} \leq \deltalta$, for any $t \in [0,1]$ and $y\in\partial E$, with $C=C(E,\varepsilon,\deltalta)$.\\ Differentiating equations~\eqref{flowdiff}, we obtain \begin{equation}\label{flowdiffdiff} \begin{cases} \frac{\partial}{\partial t} \nabla^\ell\nabla^i\Phi_j (t,y) = \nabla^s\nabla^kX^j(\Phi(t,y))\nabla^i\Phi_k(t,y)\nabla^\ell\Phi_s(t,y)\\ \phantom{\frac{\partial}{\partial t} \nabla^\ell\nabla^i\Phi_j (t,y) =} +\nabla^kX^j (\Phi (t,y)) \nabla^\ell\nabla^i\Phi_k(t,y) \\ \nabla^\ell\nabla^i\Phi (0,y)= 0 \end{cases} \end{equation} (where we sum over $s$ and $k$), for every $t \in [0,1]$, $y\in\partial E$ and $i,j,\ell\in\{1,\dots,n\}$.\\ This is a linear {\em non--homogeneous} system of ODEs such that, if we control $C\norma{\psi}_{W^{2,p} (\partial E)}$, the smooth coefficients in the right side multiplying the solutions $\nabla^\ell\nabla^i\Phi_j (\cdot,y)$ are uniformly bounded (as in estimate~\eqref{normgradfi}, Sobolev embeddings then imply that $\nabla X$ is bounded in $L^\infty$ by $C\norma{\psi}_{W^{2,p} (\partial E)}$). Hence, arguing as before, for almost every $t\in[0,1]$ where the following derivative exists, there holds \begin{align} \frac{\partial}{\partial t} \bigl\vert \nabla^2\Phi(t,y)\bigl\vert\leq&\,C\Vert\nabla X\Vert_{L^\infty(N_\varepsilon)}\bigl\vert \nabla^2\Phi(t,y)\bigl\vert+C\vert\nabla^2 X(\Phi(t,y))\vert\\ \leq&\,C\deltalta\bigl\vert \nabla^2\Phi(t,y)\bigl\vert+C\vert\nabla^2 X(\Phi(t,y))\vert\,, \end{align} by inequality~\eqref{normfield} (notice that inequality~\eqref{normgradfi} gives an $L^\infty$--bound on $\nabla\Phi$, {\em not only} in $L^p$, which is crucial). Thus, by means of Gronwall's lemma (see~\cite{stampvidpic}, for instance), we obtain the estimate $$ \bigl\vert \nabla^2\Phi(t,y)\bigl\vert\leq C\int_0^t \vert\nabla^2 X(\Phi(s,y))\vert e^{C\deltalta(t-s)}\,ds\leq C\int_0^t \vert\nabla^2 X(\Phi(s,y))\vert\,ds\,, $$ hence, \begin{align} \norma{\nabla ^2 \Phi ( t, \cdot) }_{L^p( \partial E)}^p \leq&\, C\int_{\partial E}\Bigl(\int_0^t \vert\nabla^2 X(\Phi(s,y))\vert\,ds\Bigr)^p\,d\mu(y)\\ \leq&\, C\int_0^t\int_{\partial E} \vert\nabla^2 X(\Phi(s,y))\vert^p\,d\mu(y)ds\\ =&\, C\int_{N_\varepsilon} \vert\nabla^2 X(x)\vert^pJL^{-1}(x)\,dx\\ \leq&\, C \norma{\nabla ^2 X}_{L^p(N_\varepsilon)}^p\\ \leq&\, C\norma{X}_{W^{2,p}(N_\varepsilon)}^p\\ \leq&\, C\norma{\psi}_{W^{2,p} (\partial E)}^p\,,\label{normgrad2fi} \end{align} by estimate~\eqref{normfield}, for every $t\in[0,1]$, with $C=C(E,\varepsilon,\deltalta)$.\\ Clearly, putting together inequalities~\eqref{normfi},~\eqref{normgradfi} and~\eqref{normgrad2fi}, we get the estimate~\eqref{normflow} in the statement of the lemma. \noindent \textbf{Step $\mathbf{4}$.} Finally, computing as in formula~\eqref{sec der vol} and Remark~\ref{rem999}, we have $$ \frac{d^2}{dt^2} \mathrm{Vol}(E_t) = \int_{\partial E} \langle X| \nu_{E_t} \rangle \operatorname*{div}\nolimits^{\mathbb{T}^n}\!\!X \, d \mu_t , $$ for every $t\in[-1,1]$, hence, since by Step~$1$ we know that $ \operatorname*{div}\nolimits^{\mathbb{T}^n}\!\!X=0$ in $N_\varepsilon$ (which contains each $\partial E_t$), we conclude that $ \frac{d^2}{dt^2} \mathrm{Vol}(E_t) =0$ for all $t \in [-1,1]$, that is, the function $ t \mapsto \mathrm{Vol}(E_t)$ is linear.\\ If then $\mathrm{Vol}(E_1)=\mathrm{Vol}(E)= \mathrm{Vol}(E_0)$, it follows that $\mathrm{Vol} (E_t)= \mathrm{Vol}(E) $, for all $t \in [-1,1]$ which implies that $X$ is admissible, by Remark~\ref{remcarlo888}. \end{proof} With an argument similar to the one of this proof, we now prove Lemma~\ref{vector field}. \begin{proof}[Proof of Lemma~\ref{vector field}] Let $\varphi:\partialrtial E\to\mathbb{R}$ a $C^\infty$ function with zero integral, then we define the following smooth vector field in $N_\varepsilon$, \begin{equation} X(x)= \varphi(\pi_E(x))\widetilde{X}(x), \end{equation} where $\widetilde{X}$ is the smooth vector field defined by formula~\eqref{eqcar400} and we extend it to a smooth vector field $X\in C^\infty(\mathbb{T}^n; \mathbb{R}^n)$ on the whole $\mathbb{T}^n$. Clearly, by the properties of $\widetilde{X}$ seen above, $$ \langle X(y) \vert \nu_E(y)\rangle= \varphi(y)\langle\widetilde{X}(y) \vert \nu_E(y)\rangle=\varphi(y) $$ for every $y\in\partial E$.\\ As the function $x \mapsto \varphi(\pi_E(x))$ is constant along the segments $t\mapsto x+t\nabla d_E(x)$, for every $x\in N_\varepsilon$, it follows, as in Step~$1$ of the previous proof, that $\operatorname*{div}\nolimits\! X=0$ in $N_\varepsilon$. Then, arguing as in Step~$4$, the flow $\Phi$ defined by system~\eqref{varflow} having $X$ as infinitesimal generator, gives a variation $E_t=\Phi_t(E)$ of $E$ such that the function $ t \mapsto \mathrm{Vol}(E_t)$ is linear, for $t$ in some interval $(-\deltalta,\deltalta)$. Since, by equation~\eqref{eqc1000bis}, there holds $$ \frac{d}{dt} \mathrm{Vol}(E_t)\,\Bigr\vert_{t=0}=\int_{\partial E} \langle X \vert \nu_{E}\rangle \, d\mu=\int_{\partial E} \varphi\, d\mu=0, $$ such function $ t \mapsto \mathrm{Vol}(E_t)$ must actually be constant.\\ Hence, $\mathrm{Vol} (E_t)= \mathrm{Vol}(E) $, for all $t\in(-\deltalta,\deltalta)$ and the variation $E_t$ is volume--preserving. \end{proof} The next lemma gives a technical estimate needed in the proof of Theorem~\ref{W2pMin}. \begin{lem}\label{lemmastima} Let $p>\max\{2, n-1\}$ and $E \subseteq \mathbb{T}^n$ a strictly stable critical set for the (volume--constrained) functional $J$. Then, in the hypotheses and notation of Lemma~\ref{lemma1}, there exist constants $\deltalta, C>0$ such that if $\norma{\psi}_{W^{2,p} (\partial E)} \leq \deltalta$ then $|X| \leq C |\langle X \vert \nu_{E_t}\rangle |$ on $\partial E_t$ and \begin{equation}\label{gradtangX} \norma{\nabla X}_{L^2(\partial E_t)} \leq C \norma{ \langle X | \nu_{E_t} \rangle}_{H^1(\partial E_t)} \end{equation} (here $\nabla$ is the covariant derivative along $E_t$), for all $t \in [0,1]$, where $X \in C^{\infty}(\mathbb{T}^n ; \mathbb{R}^n)$ is the smooth vector field defined in formula~\eqref{field}. \end{lem} \begin{proof} Fixed $\varepsilon >0$, from inequality~\eqref{normflow} it follows that there exist $\deltalta >0$ such that if $\norma{\psi}_{W^{2,p} (\partial E)} \leq \deltalta$ there holds $$ \abs{\nu_{E_t}(\Phi(t,y))-\nu_E(y)} \leq \varepsilon $$ for every $y\in\partial E$, hence, as $\nabla d_E=\nu_E$ on $\partial E$, we have $$ \abs{\nabla d_E(\Phi^{-1}(t,x)) - \nu_{E_t}(x)}= \abs{\nu_E(\Phi^{-1}(t,x)) - \nu_{E_t}(x)} \leq \varepsilon $$ for every $x\in \partial E_t$. Then, if $\norma{\psi}_{W^{2,p} (\partial E)}$ is small enough, $\Phi^{-1}(t,\cdot)$ is close to the identity, thus $$ \abs{\nabla d_E(\Phi^{-1}(t,x)) - \nabla d_E(x)} \leq \varepsilon $$ on $\partial E_t$ and we conclude \begin{equation}\label{normanorm2} \norma{\nabla d_E - \nu_{E_t}}_{L^\infty (\partial E_t)} \leq 2 \varepsilon \, . \end{equation} Moreover, using again the inequality~\eqref{normflow} and following the same argument above, we also obtain \begin{equation}\label{gradientnu} \norma{\nabla^2 d_E - \nabla \nu_{E_t}}_{L^\infty (\partial E_t)} \leq 2 \varepsilon \, . \end{equation} We estimate $X_{\tau _t}=X-\langle X \vert \nu _{E_t} \rangle \nu_{E_t}$ (recall that $X=\langle X \vert \nabla d_E \rangle \nabla d_E$), \begin{align} \abs{X_{\tau _t}} & = \abs{X- \langle X \vert \nu_{E_t} \rangle \nu_{E_t}}\\ &= \abs{\langle X \vert \nabla d_E \rangle \nabla d_E -\langle X \vert \nu_{E_t} \rangle \nu_{E_t}} \\ & = \abs{\langle X \vert \nabla d_E \rangle \nabla d_E -\langle X \vert \nu_{E_t} \rangle \nabla d_E + \langle X \vert \nu_{E_t} \rangle \nabla d_E -\langle X \vert \nu_{E_t} \rangle \nu_{E_t}} \\ &\leq \abs{ \langle X \vert (\nabla d_E - \nu_{E_t} ) \rangle \nabla d_E }+ \abs{\langle X \vert \nu_{E_t} \rangle ( \nabla d_E -\nu_{E_t} )}\\ &\leq2 \abs{X}\,\abs{\nabla d_E - \nu_{E_t}}\\ & \leq 4 \varepsilon \abs{X}\,, \end{align} then $$ \abs{X_{\tau _t}} \leq 4 \varepsilon \abs{X_{\tau _t} + \langle X \vert \nu _{E_t} \rangle \nu_{E_t}} \leq 4 \varepsilon \abs{X_{\tau _t}} + \abs{\langle X \vert \nu _{E_t} \rangle}\,, $$ hence, \begin{equation}\label{Xtau} \abs{X_{\tau _t}} \leq C \abs{ \langle X \vert \nu _{E_t} \rangle} \,. \end{equation} We now estimate the covariant derivative of $X_{\tau _t}$ along $\partial E_t$, that is, \begin{align} \abs{\nabla X_{\tau _t}} =&\, \abs{\nabla X- \nabla ( \langle X \vert \nu_{E_t} \rangle \nu_{E_t})} \\ =&\, \abs{ \nabla (\langle X \vert \nabla d_E \rangle \nabla d_E ) - \nabla (\langle X \vert \nu_{E_t} \rangle \nu_{E_t}) } \\ =&\, |\nabla (\langle X \vert \nabla d_E \rangle \nabla d_E) - \nabla ( \langle X \vert \nu_{E_t} \rangle \nabla d_E )+\nabla (\langle X \vert \nu_{E_t} \rangle \nabla d_E )- \nabla (\langle X \vert \nu_{E_t} \rangle \nu_{E_t})|\\ \leq&\, \abs{ \nabla (\langle X \vert (\nabla d_E - \nu_{E_t} ) \rangle \nabla d_E ) }+\abs{\nabla (\langle X \vert \nu_{E_t} \rangle ( \nabla d_E -\nu_{E_t} )) }\\ \leq&\, C \varepsilon \bigl [ \abs{ \nabla X } + \abs{ \nabla \langle X \vert \nu_{E_t} \rangle} \bigr] + C \abs{X} \bigl [ \abs{\nabla(\nabla d_E) } + \abs{ \nabla \nu_{E_t} } \bigr] \\ \leq&\, C \varepsilon \bigl [ \abs{ \nabla ( \langle X \vert \nu_{E_t} \rangle \nu_{E_t} + X_{\tau _t}) } + \abs{ \nabla \langle X \vert \nu_{E_t} \rangle} \bigr ] + C \bigl(\abs{\langle X \vert \nu_{E_t} \rangle} + \abs{X_{\tau _t} }\bigr)\, \bigl [ \abs{ \nabla^2 d_E } + \abs{ \nabla \nu_{E_t} } \bigr ] \end{align} hence, using inequality~\eqref{Xtau} and arguing as above, there holds \begin{equation}\label{gradtau} \abs{\nabla X_{\tau _t}} \leq C \abs{ \nabla \langle X \vert \nu_{E_t} \rangle } + C \abs{\langle X\vert \nu_{E_t} \rangle} \bigl [\abs{ \nabla^2 d_E } + \abs{ \nabla \nu_{E_t} } \bigr ] \, . \end{equation} Then, we get \begin{align} \norma{\nabla X_{\tau _t}}_{L^2 (\partial E_t)}^2\leq &\, C \norma{ \nabla \langle X \vert \nu_{E_t} \rangle }_{L^2 (\partial E_t)}^2 + C \int_{\partial E_t} \abs{\langle X \vert \nu_{E_t}\rangle}^2 \bigl [\abs{ \nabla^2 d_E } + \abs{ \nabla \nu_{E_t} } \bigr ]^2 \, d \mu \\ \leq &\, C \norma{\langle X \vert \nu_{E_t} \rangle }_{H^1 (\partial E_t)}^2+ C \norma{\langle X \vert \nu_{E_t}\rangle}^2_{L^{\frac{2p}{p-2}} (\partial E_t)} \bigl \Vert \abs{ \nabla^2 d_E } + \abs{ \nabla \nu_{E_t} } \bigr\Vert^2 _{L^p(\partial E_t)} \\ \leq &\, C \ \norma{\langle X \vert \nu_{E_t} \rangle}^2 _{H^1 (\partial E_t)} \end{align} where in the last inequality we used as usual Sobolev embeddings, as $p> \max\{2, n-1\}$ and the fact that $\norma{\nabla \nu_{E_t}}_{L^p(\partial E_t)}$ is bounded by the inequality~\eqref{gradientnu} (as $\norma{\nabla^2 d_E}_{L^p(\partial E_t)}$).\\ Considering the covariant derivative of $X=X_{\tau_t}+\langle X \vert \nu _{E_t} \rangle \nu_{E_t}$, by means of this estimate, the trivial one $$ \norma{ \nabla \langle X \vert \nu_{E_t} \rangle } _{L^2 (\partial E_t)} \leq \norma{ \langle X \vert \nu_{E_t} \rangle} _{H^1(\partial E_t)} $$ and inequality~\eqref{Xtau}, we obtain estimate~\eqref{gradtangX}. \end{proof} We now show that any smooth set $E$ sufficiently $W^{2,p}$--close to another smooth set $F$, can be ``translated'' by a vector $\eta\in\mathbb{R}^n$ such that $\partial E-\eta=\{y+\varphi(y)\nu_F(y) \, :\, y\in\partialrtial F\}$, for a function $\varphi\in C^\infty(\partial F)$ having a suitable small ``projection'' on $T(\partial F)$ (see the definitions and the discussion after Remark~\ref{remcarlo}). \begin{lem}\label{Lemma 3.8} Let $p>n-1$ and $F\subseteq\mathbb{T}^n$ a smooth set with a tubular neighborhood $N_\varepsilon$ as above, in formula~\eqref{tubdef}. For any $\tau>0$ there exist constants $\deltalta,C>0$ such that if another smooth set $E\subseteq \mathbb{T}^n$ satisfies $\mathrm{Vol}(E\triangle F)<\deltalta$ and $\partialrtial E=\{y+\psi(y)\nu_F(y) \,:\,y\in\partialrtial F\}\subseteq N_\varepsilon$ for a function $\psi\in C^\infty(\mathbb{R})$ with $\norma{\psi}_{W^{2,p}(\partialrtial F)}<\deltalta$, then there exist $\eta\in\mathbb{R}^n$ and $\varphi\in C^\infty(\partial F)$ with the following properties: $$ \partialrtial E-\eta=\{y+\varphi(y)\nu_F(y) \,:\,y\in\partialrtial F\}\subseteq N_\varepsilon\,, $$ $$ \abs{\eta}\leq C\norma{\psi}_{W^{2,p}(\partialrtial F)},\qquad \norma{\varphi}_{W^{2,p}(\partialrtial F)}\leq C\norma{\psi}_{W^{2,p}(\partialrtial F)} $$ and $$ \Bigl \vert \int_{\partialrtial F}\varphi\nu_F \, d \mu\, \Bigr \vert \leq\tau\norma{\varphi}_{L^2(\partialrtial F)}\,. $$ \end{lem} \begin{proof} We let $d_{F}$ to be the signed distance function from $\partial F$. We underline that, throughout the proof, the various constants will be all independent of $\psi:\partial F\to\mathbb{R}$.\\ We recall that in Remark~\ref{rembase} we saw that there exists an orthonormal basis $\{e_1, \dots, e_n\}$ of $\mathbb{R}^n$ such that the functions $\langle \nu_F \vert e_i \rangle$ are orthogonal in $L^2(\partial F)$, that is, \begin{equation}\label{ort2} \int_{\partial F} \langle \nu_{F} \vert e_i \rangle \langle \nu_{F} \vert e_j \rangle \, d \mu=0, \end{equation} for all $i \ne j$ and we let $\mathrm{I}I_F$ to be the set of the indices $i\in \{1,\dots, n\}$ such that $\norma{\langle \nu_F \vert e_i \rangle }_{L^2(\partial F)}>0$. Given a smooth function $\psi:\partial F\to \mathbb{R}$, we set $\eta=\sum_{i=1}^n\eta_ie_i$, where \begin{equation}\label{uno} \eta_i = \begin{cases} \frac{1}{\norma{ \langle \nu_F \vert e_i \rangle}^2_{L^2(\partial F)}}\int_{\partial F}\psi(x) \langle \nu_F(x) \vert e _i \rangle \,d \mu \quad & \text{if $i\in\mathrm{I}I_F$,}\\ \eta_i=0\quad & \text{otherwise.} \end{cases} \end{equation} Note that, from H\"older inequality, it follows \begin{equation}\label{due} \abs{\eta}\leq C_1\norma{\psi}_{L^2(\partial F)}\,. \end{equation} \noindent\textbf{Step $\mathbf 1$.} Let $T_\psi:\partial F\to\partial F$ be the map $$ T_\psi(y)=\pi_{F}(y+\psi(y)\nu_F(y)-\eta)\,. $$ It is easily checked that there exists $\varepsilon_0>0$ such that if \begin{equation}\label{trans1} \norma{\psi}_{W^{2,p}(\partial F)}+\abs{\eta}\leq \varepsilon_0\leq 1\,, \end{equation} then $T_{\psi}$ is a smooth diffeomorphism, moreover, \begin{equation}\label{trans2} \Vert JT_\psi-1\Vert_{L^\infty(\partial F)}\leq C\norma{\psi}_{C^1(\partial F)} \end{equation} (here $JT_{\psi}$ is the Jacobian relative to $\partial F$) and \begin{equation}\label{trans3} \norma{T_\psi-\mathrm{Id}}_{W^{2,p}(\partial F)}+\norma{T_\psi^{-1}-\mathrm{Id}}_{W^{2,p}(\partial F)}\leq C(\norma{\psi}_{W^{2,p}(\partial F)}+\abs{\eta})\,. \end{equation} Therefore, setting $\widehat E=E-\eta$, we have $$ \partial\widehat E=\{z+\varphi(z)\nu_F(z) \, :\,z\in\partialrtial F\}\,, $$ for some function $\varphi$ which is linked to $\psi$ by the following relation: for all $y\in\partial F$, we let $z=z(y)\in\partial F$ such that $$ y+\psi(y)\nu_F(y)-\eta=z+\varphi(z)\nu_F(z)\,, $$ then, $$ T_\psi(y)=\pi_F(y+\psi(y)\nu_F(y)-\eta)=\pi_F(z+\varphi(z)\nu_F(z))=z, $$ that is, $y=T_\psi^{-1}(z)$ and \begin{align*} \varphi(z)=&\,\varphi(T_\psi(y))\\ =&\,d_F(z+\varphi(z)\nu_F(z))\\ =&\,d_F(y+\psi(y)\nu_F(y)-\eta)\\ =&\,d_F(T_\psi^{-1}(z)+\psi(T_\psi^{-1}(z))\nu_F(T_\psi(y))-\eta). \end{align*} Thus, using inequality~\eqref{trans3}, we have \begin{equation}\label{quattro} \norma{\varphi}_{W^{2,p}(\partial F)}\leq C_2\bigl(\norma{\psi}_{W^{2,p}(\partial F)}+\abs{\eta}\bigr), \end{equation} for some constant $C_2>1$. We now estimate \begin{align} \int_{\partial F}\varphi(z)\nu_F(z)\,d \mu (z) &=\int_{\partial F}\varphi(T_{\psi}(y))\nu_F(T_{\psi}(y))JT_\psi(y)\,d \mu (y) \\ &= \int_{\partial F}\varphi(T_{\psi}(y))\nu_F(T_{\psi}(y))\, d \mu (y) +R_1\,,\label{cinque} \end{align} where \begin{equation} \abs{R_1}=\biggl \vert \int_{\partial F}\varphi(T_{\psi}(y))\nu_F(T_{\psi}(y))\,[JT_\psi(y)-1]\, d \mu(y) \biggr \vert\leq C_3\norma{ \psi}_{C^1(\partial F)}\norma{\varphi}_{L^2(\partial F)}\,,\label{sei} \end{equation} by inequality~\eqref{trans2}.\\ On the other hand, \begin{align} \int_{\partial F}&\varphi(T_{\psi}(y))\nu_F(T_{\psi}(y))\, d \mu(y)\\ &=\int_{\partial F}\bigl[y+\psi(y)\nu_F(y)-\eta-T_{\psi}(y)\bigr]\,d \mu(y) \\ &=\int_{\partial F}\bigl[y+\psi(y)\nu_F(y)-\eta-\pi_{F}(y+\psi(y)\nu_F(y)-\eta)\bigr]\, d \mu(y)\\ &=\int_{\partial F}\bigl\{\psi(y)\nu_F(y)-\eta+\bigl[\pi_{F}(y)-\pi_{F}(y+\psi(y)\nu_F(y)-\eta)\bigr]\bigr\}\, d \mu(y)\\ &=\int_{\partial F}(\psi(y)\nu_F(y)-\eta)\, d \mu(y) +R_2\,, \label{sette} \end{align} where \begin{align} R_2&=\int_{\partial F}\bigl[\pi_{F}(y)-\pi_{F}(y+\psi(y)\nu_F(y)-\eta)\bigr]\, d \mu(y)\\ &=-\int_{\partial F} d \mu(y) \int_0^1\nabla \pi_{F}(y+t(\psi(y)\nu(y)-\eta))(\psi(y)\nu_F(y)-\eta)\,dt \\ &=-\int_{\partial F}\nabla \pi_{F}(y)(\psi(y)\nu_F(y)-\eta)\, d \mu(y)+R_3\,. \label{otto} \end{align} In turn, recalling inequality~\eqref{due}, we get \begin{equation} \abs{R_3}\leq\int_{\partial F} d \mu(y) \int_0^1\vert \nabla \pi_{F}(y+t(\psi(y)\nu_F(y)-\eta))-\nabla \pi_{F}(y)\vert\, \vert \psi(y)\nu_F(y)-\eta \vert \,dt\leq C_4\norma{\psi}^2_{L^2(\partial F)}\,.\label{nove} \end{equation} Since in $N_\varepsilon$, by equation~\eqref{eqcar2050}, we have $\pi_{F}(x)=x-d_{F}(x)\nabla d_{F}(x)$, it follows $$ \frac{\partial {\pi_{F}^i}}{\partial x_j}(x)=\deltalta_{ij}-\frac{\partial d_{F}}{\partial x_i}(x)\frac{\partial d_{F}}{\partial x_j}(x)-d_{F}(x)\frac{\partial^2 d_{F}}{\partial x_i\partial x_j}(x), $$ thus, for all $y\in\partial F$, there holds $$ \frac{\partial{\pi_{F}^i}}{\partial x_j}(y)=\deltalta_{ij}-\frac{\partial d_{F}}{\partial x_i}(y)\frac{\partial d_{F}}{\partial x_j}(y)\,. $$ From this identity and equalities~\eqref{cinque},~\eqref{sette} and~\eqref{otto}, we conclude $$ \int_{\partial F}\varphi(z)\nu_F(z)\,d \mu (z)=\int_{\partial F}\bigl[\psi(x)\nu_F(x)- \langle \eta \, \vert\, \nu_F(x)\rangle \nu_F(x)\bigr]\, d \mu (x)+R_1+R_3\,. $$ As the integral at the right--hand side vanishes by relations~\eqref{ort2} and~\eqref{uno}, estimates~\eqref{sei} and~\eqref{nove} imply \begin{align} \Bigl \vert \int_{\partial F}\varphi(y)\nu_F(y)\, d \mu (y)\Bigr \vert &\leq C_3\norma{\psi}_{C^1(\partial F)}\norma{\varphi}_{L^2(\partial F)}+C_4\norma{\psi}^2_{L^2(\partial F)} \\ &\leq C\norma{\psi}_{C^1(\partial F)}\bigl(\norma{\varphi}_{L^2(\partial F)}+\norma{\psi}_{L^2(\partial F)}\bigr) \\ &\leq C_5 \norma{\psi}_{W^{2,p}(\partial F)}^{1-\vartheta}\norma{\psi}_{L^2(\partial F)}^{\vartheta}\bigl(\norma{\varphi}_{L^2(\partial F)}+\norma{\psi}_{L^2(\partial F)}\bigr) \,,\label{dieci} \end{align} where in the last passage we used a well--known interpolation inequality, with $\vartheta\in(0,1)$ depending only on $p>n-1$ (see~\cite[Theorem~3.70]{Aubin}). \noindent \textbf{Step $\mathbf 2$.} The previous estimate does not allow to conclude directly, but we have to rely on the following iteration procedure. Fix any number $K>1$ and assume that $\deltalta \in (0,1)$ is such that (possibly considering a smaller $\tau$) \begin{equation}\label{eta} \tau+\deltalta<\varepsilon_0/2,\qquad C_2\deltalta (1+2C_1)\leq\tau,\qquad2C_5\deltalta^\vartheta K\leq\tau\,. \end{equation} Given $\psi$, we set $\varphi_0=\psi$ and we denote by $\eta^1$ the vector defined as in~\eqref{uno}. We set $E_1=E-\eta^1$ and denote by $\varphi_1$ the function such that $\partial E_1=\{x+\varphi_1(x)\nu_F(x) \,: \, x \in \partial F\}$. As before, $\varphi_1$ satisfies $$ y+\varphi_0(y)\nu_F(y)-\eta^1=z+\varphi_1(z)\nu_F(z)\,. $$ Since $\norma{\psi}_{W^{2,p}(\partial F)}\leq\deltalta$ and $|\eta|\leq C_1\norma{\psi}_{L^2(\partial F)}$, by inequalities~\eqref{due},~\eqref{quattro} and~\eqref{eta} we have \begin{equation}\label{undici} \norma{\varphi_1}_{W^{2,p}(\partial F)}\leq C_2\deltalta(1+C_1)\leq\tau\,. \end{equation} Using again that $\norma{\psi}_{W^{2,p}(\partial F)}<\deltalta<1$, by estimate~\eqref{dieci} we obtain $$ \Bigl \vert \int_{\partial F}\varphi_1(y)\nu_F(y)\, d \mu(y) )\Bigr \vert \leq C_5\norma{\varphi_0}_{L^2(\partial F)}^{\vartheta}\bigl(\norma{\varphi_1}_{L^2(\partial F)}+\norma{\varphi_0}_{L^2(\partial F)}\bigr)\,, $$ where we have $\norma{\varphi_0}_{L^2(\partial F)}\leq\deltalta$.\\ We now distinguish two cases.\\ If $\norma{\varphi_0}_{L^2(\partial F)}\leq K\norma{\varphi_1}_{L^2(\partial F)}$, from the previous inequality and~\eqref{eta}, we get \begin{align} \Bigl \vert \int_{\partial F}\varphi_1(y)\nu_F(y)\, d \mu(y)\Bigr \vert & \leq C_5\deltalta^{\vartheta}\bigl(\norma{\varphi_1}_{L^2(\partial F)}+\norma{\varphi_0}_{L^2(\partial F)}\bigr) \nonumber \\ & \, \leq 2C_5\deltalta^{\vartheta}K\norma{\varphi_1}_{L^2(\partial F)} \nonumber \\& \, \leq\deltalta\norma{\varphi_1}_{L^2(\partial F)}\,, \end{align} thus, the conclusion follows with $\eta=\eta^1$.\\ In the other case, \begin{equation}\label{dodici} \norma{\varphi_1}_{L^2(\partial F)}\leq\frac{\norma{\varphi_0}_{L^2(\partial F)}}{K}\leq\frac{\deltalta}{K}\leq \deltalta\,. \end{equation} We then repeat the whole procedure: we denote by $\eta^2$ the vector defined as in formula~\eqref{uno} with $\psi$ replaced by $\varphi_1$, we set $E_2=E_1-\eta^2=E-\eta^1-\eta^2$ and we consider the corresponding $\varphi_2$ which satisfies $$ w+\varphi_2(w)\nu_F(w)=z+\varphi_1(z)\nu_F(z)-\eta^2=y+\varphi_0(y)\nu_F(y)-\eta^1-\eta^2\,. $$ Since $$ \norma{\varphi_0}_{W^{2,p}(\partial F)}+\abs{\eta^1+\eta^2}\leq \deltalta +C_1\deltalta+C_1\norma{\varphi_1}_{L^2(\partial F)}\leq \deltalta+C_1\deltalta\Bigl(1+\frac1K\Bigr)\leq C_2\deltalta(1+2C_1)\leq\tau\,, $$ the map $T_{\varphi_0}(y)=\pi_{F}(y+\varphi_0(y)\nu_F(y)-(\eta^1+\eta^2))$ is a diffeomorphism, thanks to formula~\eqref{trans1} (having chosen $\tau$ and $\deltalta$ small enough).\\ Thus, by applying inequalities~\eqref{quattro} (with $\eta=\eta^1+\eta^2$),~\eqref{due},~\eqref{eta} and~\eqref{dodici}, we get $$ \norma{\varphi_2}_{W^{2,p}(\partial F)}\leq C_2\bigl(\norma{\varphi_0}_{W^{2,p}(\partial F)}+\abs{ \eta^1+\eta^2} \bigr)\leq C_2\deltalta\Bigl(1+C_1+\frac{C_1}{K}\Bigr)\leq\tau\,, $$ as $K>1$, analogously to conclusion~\eqref{undici}. On the other hand, by estimates~\eqref{due},~\eqref{undici} and~\eqref{dodici}, $$ \norma{\varphi_1}_{W^{2,p}(\partial F)}+\eta^2\leq C_2\deltalta(1+C_1)+C_1\frac{\deltalta}{K}\leq C_2\deltalta(1+2C_1)\leq \tau\,, $$ hence, also the map $T_{\varphi_1}(x)=\pi_{F}(x+\varphi_1(x)\nu_F(x)-\eta^2)$ is a diffeomorphism satisfying inequalities~\eqref{trans1} and~\eqref{trans2}. Therefore, arguing as before, we obtain $$ \Bigl \vert \int_{\partial F}\varphi_2(y)\nu_F(y)\,d \mu(y)\Bigr \vert \leq C_5\norma{\varphi_1}_{L^2(\partial F)}^{\vartheta}\bigl(\norma{\varphi_2}_{L^2(\partial F)}+\norma{\varphi_1}_{L^2(\partial F)}\bigr)\,. $$ Since $\norma{\varphi_1}_{L^2(\partial F)}\leq\deltalta$ by inequality~\eqref{dodici}, if $\norma{\varphi_1}_{L^2(\partial F)}\leq K\norma{\varphi_2}_{L^2(\partial F)}$ the conclusion follows with $\eta=\eta^1+\eta^2$. Otherwise, we iterate the procedure observing that $$ \norma{\varphi_2}_{L^2(\partial F)}\leq\frac{\norma{\varphi_1}_{L^2(\partial F)}}{K}\leq\frac{\norma{\varphi_0}_{L^2(\partial F)}}{K^2}\leq\frac{\deltalta}{K^2}\,. $$ This construction leads to three (possibly finite) sequences $\eta^n$, $E_n$ and $\varphi_n$ such that $$ \begin{cases} E_n=E-\eta^1-\dots-\eta^n, \qquad \abs{\eta^n} \leq\frac{C_1\deltalta}{K^{n-1}}\\ \norma{\varphi_n}_{W^{2,p}(\partial F)}\leq C_2\bigl(\norma{\varphi_0}_{W^{2,p}(\partial F)}+\abs{\eta^1+\dots+\eta^n}\bigr)\leq C_2\deltalta(1+2C_1)\\ \norma{\varphi_n}_{L^2(\partial F)} \leq\frac{\deltalta}{K^n}\\ \partialrtial E_n=\{x+\varphi_n(x)\nu_F(x) \, :\,x\in\partial F \}\ \end{cases} $$ If for some $n\in\mathbb{N}$ we have $\norma{\varphi_{n-1}}_{L^2(\partial F)}\leq K\norma{\varphi_n}_{L^2(\partial F)}$, the construction stops, since, arguing as before, $$ \Bigl \vert \int_{\partial F}\varphi_n(y)\nu_F(y)\,d \mu(y)\Bigr \vert \leq\deltalta\norma{\varphi_n}_{L^2(\partial F)} $$ and the conclusion follows with $\eta=\eta^1+\dots+\eta^n$ and $\varphi=\varphi_n$. Otherwise, the iteration continues indefinitely and we get the thesis with $$ \eta=\sum_{n=1}^\infty\eta^n, \qquad\varphi=0\,, $$ (notice that the series is converging), which actually means that $E=\eta+F$. \end{proof} We are now ready to show the main theorem of this first part of the work. \begin{proof}[Proof of Theorem~\ref{W2pMin}]\ \\ \noindent \textbf{Step $\mathbf{1}$.} We first want to see that \begin{equation}\label{m0} m_0= \inf \left\{\Pi_E(\varphi) \, : \, \varphi \in T^\perp(\partial E), \norma{\varphi}_{H^1(\partial E)}=1 \right\} >0. \end{equation} To this aim, we consider a minimizing sequence $\varphi_i$ for the above infimum and we assume that $\varphi_i \rightharpoonup \varphi_0$ weakly in $H^1 (\partial E)$, then $\varphi_0\in T^\perp(\partial E)$ (since it is a closed subspace of $H^1 (\partial E)$) and if $\varphi_0 \ne 0$, there holds $$m_0= \lim_{i \to +\infty} \Pi_E(\varphi_i) \ge \Pi_E(\varphi_0)>0$$ due to the strict stability of $E$ and the lower semicontinuity of $\Pi_E$ (recall formula~\eqref{Pieq} and the fact that the weak convergence in $H^1(\partial E)$ implies strong convergence in $L^2(\partial E)$ by Sobolev embeddings). On the other hand, if instead $\varphi_0=0$, again by the strong convergence of $\varphi_i \to \varphi_0$ in $L^2(\partial E)$, by looking at formula~\eqref{Pieq}, we have $$ m_0= \lim_{i \to\infty} \Pi_E(\varphi_i)=\lim_{i\to\infty} \int_{\partial E} |\nabla \varphi_i|^2 \, d \mu = \lim_{i\to\infty}\Vert\varphi_i\Vert_{H^1(\partial E)}^2=1 $$ since $\Vert\varphi_i \Vert_{L^2(\partial E)}\to0$. \noindent \textbf{Step ${\mathbf{2}}$.} Now we show that there exists a constant $\deltalta_1 >0$ such that if $E$ is like in the statement and $\partial F= \{y+ \psi(y)\nu_E(y) \, : \, y \in \partial E \}$, with $\norma{\psi}_{W^{2,p}(\partial E)} \le \deltalta_1$, and $\mathrm{Vol}(F)=\mathrm{Vol}(E)$, then \begin{equation}\label{3.38} \inf\left\{\Pi_F(\varphi) \, : \, \varphi \in \widetilde{H}^1(\partial F), \norma{\varphi}_{H^1(\partial F)}=1, \Bigl | \int_{\partial F} \varphi \nu_F \, d \mu \Bigr| \le \deltalta_1\right\}\ge \frac{m_0}{2}. \end{equation} We argue by contradiction assuming that there exists a sequence of sets $F_i$ with $\partial F_i=\{y+ \psi_i(y) \nu_E(y) \, : \, y \in \partial E\}$ with $\norma{\psi_i}_{W^{2,p}(\partial E)} \to 0$ and $\mathrm{Vol}(F_i)=\mathrm{Vol}(E)$, and a sequence of functions $\varphi_i \in \widetilde{H}^1(\partial F_i)$ with $\norma{\varphi_i}_{H^1(\partial F_i)}=1$ and $\int_{\partial F_i} \varphi_i \nu_{F_i} \, d \mu_i \to 0$, such that \begin{equation} \Pi_{F_i}(\varphi_i)<\frac{m_0}{2}. \end{equation} We then define the following sequence of smooth functions \begin{equation} \label{eqcar1030}\widetilde{\varphi}_i(y)= \varphi_i(y+ \psi_i(y)\nu_E(y)) - \fint_{\partial E} \varphi_i (y + \psi_i(y) \nu_E(y)) \, d \mu(y) \end{equation} which clearly belong to $\widetilde{H}^1(\partial E)$. Setting $\theta_i(y)=y+ \psi_i(y)\nu_E(y)$, as $p>\max \{2, n-1\}$, by the Sobolev embeddings, $\theta_i\to\mathrm{Id}$ in $C^{1,\alpha}$ and $\nu_{F_i} \circ \theta_i \to \nu_E$ in $C^{0,\alpha}(\partial E)$, hence, the sequence $\widetilde{\varphi}_i$ is bounded in $H^1(\partial E)$ and if $\{e_k\}$ is the special orthonormal basis found in Remark~\ref{rembase}, we have $\langle \nu_{F_i}\circ\theta_i \vert e_k\rangle \to \langle \nu_E \vert e_k\rangle$ uniformly for all $k\in\{1, \dots, n\}$. Thus, \begin{equation} \int_{\partial E} \widetilde{\varphi}_i \langle \nu_E \vert \varepsilon_i \rangle \, d \mu \to 0, \end{equation} as $i\to\infty$, indeed, $$ \int_{\partial E} \widetilde{\varphi}_i \langle \nu_E \vert e_k \rangle \, d \mu-\int_{\partial E} \widetilde{\varphi}_i \langle \nu_{F_i}\circ\theta_i \vert e_k \rangle \, d \mu\to 0 $$ and $$ \int_{\partial E} \widetilde{\varphi}_i \langle \nu_{F_i}\circ\theta_i \vert e_k \rangle \, d \mu=\int_{\partial F_i}\varphi_i \langle \nu_{F_i} \vert e_k \rangle \,J\theta_i^{-1} d \mu_i \to 0, $$ as the Jacobians (notice that $J\theta_i$ are Jacobians ``relative'' to the hypersurface $\partial E$) $J\theta_i^{-1}\to 1$ uniformly and we assumed $\int_{\partial F_i} \varphi_i \nu_{F_i} \, d \mu_i \to 0$.\\ Hence, using expression~\eqref{projection}, for the projection map $\pi$ on $T^\perp(\partial E)$, it follows \begin{equation} \norma{\pi (\widetilde{\varphi}_i)-\widetilde{\varphi}_i}_{H^1(\partial E)} \to 0 \end{equation} as $i\to\infty$ and \begin{equation}\label{norm pi} \lim_{i\to\infty}\norma{\pi(\widetilde{\varphi}_i)}_{H^1(\partial E)}=\lim_{i\to\infty}\norma{\widetilde{\varphi}_i}_{H^1(\partial E)}=\lim_{i\to\infty}\norma{\varphi_i}_{H^1(\partial F_i)}=1, \end{equation} since $\norma{\varphi_i}_{W^{2,p}(\partial E)} \to 0$, thus $\norma{\varphi_i}_{C^{1,\alpha}(\partial E)} \to 0$, by looking at the definition of the functions $\widetilde{\varphi}_i$ in formula~\eqref{eqcar1030}.\\ Note now that the $W^{2,p}$-- convergence of $F_i$ to $E$ (the second fundamental form $B_{\partial F_i}$ of $\partial F_i$ is ``morally'' the Hessian of $\varphi_i$) implies \begin{equation}\label{conv B} B_{\partial F_i} \circ \theta_i \to B_{\partial E} \qquad \text{in $L^p(\partial E)$}\,, \end{equation} as $i\to\infty$, then, by Sobolev embeddings again (in particular $H^1(\partial E)\hookrightarrow L^q(\partial E)$ for any $q\in[1,2^*)$, with $2^*=2(n-1)/(n-3)$ which is larger than $2$) and the $W^{2,p}$--convergence of $F_i$ to $E$, we get \begin{equation}\label{conv B2} \int_{\partial F_i}|B_{\partial F_i}|^2 \varphi_i^2 \, d \mu_i - \int_{\partial E} |B_{\partial E}|^2 \widetilde{\varphi}_i ^2 \, d \mu \to 0\,. \end{equation} Standard elliptic estimates for the problem~\eqref{potential} (see~\cite{Ev}, for instance) imply the convergence of the potentials \begin{equation}\label{proj2bis} v_{F_i}\to v_E\ \ \text{in $C^{1,\beta}(\mathbb{T}^n)$ for all $\beta\in(0,1)$,} \end{equation} for $i\to\infty$, hence arguing as before, $$ \int_{\partial F_i}\partial_{\nu_{F_i}} v_{F_i}\varphi_i^2\,d\mu_i-\int_{\partial E}\partial_{\nu_E} v_E\widetilde{\varphi}_i^2\,d\mu\to 0\,. $$ Setting, as in Remark~\ref{rm:potential}, $$ v_{E,\widetilde{\varphi}_i}(x)=\int_{\partial E} G(x, y)\widetilde{\varphi}_i(y)\, d\mu(y)=\int_{\partial E} G(x, y)\varphi_i(\theta_i(y))\, d\mu(y)-m_i\int_{\partial E} G(x, y)\, d\mu(y)\,, $$ where $m_i=\fint_{\partial E} \varphi_i(y + \psi_i(y) \nu_E(y)) \, d \mu(y)\to 0$, as $i\to\infty$, and $$ v_{F_i,\varphi_i}(x)=\int_{\partial F_i} G(x,z)\varphi_i(z)\, d\mu_i(z)=\int_{\partial E} G(x,\theta_i(y))\varphi_i(\theta_i(y))J\theta_i(y)\,d\mu(y)\,, $$ it is easy to check (see~\cite[pages~537--538]{AcFuMo}, for details) that $$ \int_{\mathbb{T}^n}|\nabla v_{F_i,\varphi_i}|^2\, dx-\int_{\mathbb{T}^n}|\nabla v_{E,\widetilde{\varphi}_i}|^2\, dx\to0\,. $$ Finally, recalling expression~\eqref{Pieq2}, we conclude \begin{equation} \Pi_{F_i}(\varphi_i)-\Pi_E(\widetilde{\varphi}_i) \to 0\,, \end{equation} since we have $$ \norma{\varphi_i}_{L^2(\partial F_i)}-\norma{\widetilde{\varphi}_i}_{L^2(\partial E)}\to 0\,, $$ which easily follows again by looking at the definition of the functions $\widetilde{\varphi}_i$ in formula~\eqref{eqcar1030} and taking into account that $\norma{\varphi_i}_{C^{1,\alpha}(\partial E)} \to 0$, hence limits~\eqref{norm pi} imply $$ \norma{\nabla \varphi_i}_{L^2(\partial F_i)}-\norma{\nabla\widetilde{\varphi}_i}_{L^2(\partial E)}\to 0\,. $$ By the previous conclusion $\norma{\pi(\widetilde{\varphi}_i)-\widetilde{\varphi}_i}_{H^1(\partial E)}\to 0$ and Sobolev embeddings, it this then straightforward, arguing as above, to get also $$ \Pi_E(\widetilde{\varphi}_i)-\Pi_E(\pi(\widetilde{\varphi}_i))\to 0, $$ hence, \begin{equation} \Pi_{F_i}(\varphi_i)-\Pi_E(\pi(\widetilde{\varphi}_i)) \to 0. \end{equation} Since we assumed that $\Pi_{F_i}(\varphi_i)<{m_0}/{2}$, we conclude that for $i\in\mathbb{N}$, large enough there holds \begin{equation} \Pi_E(\pi(\widetilde{\varphi}_i))\leq \frac{m_0}{2}<m_0, \end{equation} which is a contradiction to Step~$1$, as $\pi(\widetilde{\varphi}_i)\in T^\perp(\partial E)$. \noindent \textbf{Step ${\mathbf{3}}$.} Let us now consider $F$ such that $\mathrm{Vol}(F)= \mathrm{Vol}(E)$, $\mathrm{Vol}(F\triangle E)<\deltalta$ and \begin{equation} \partial F= \{y + \psi(y) \nu_E (y)\, : \, y \in \partial E \}\subseteq N_\varepsilon, \end{equation} with $\norma{\psi}_{W^{2,p}(\partial E)} \le \deltalta $ where $\deltalta>0$ is smaller than $\deltalta_1$ given by Step~$2$.\\ Taking a possibly smaller $\deltalta>0$, we consider the field $X$ and the associated flow $\Phi$ found in Lemma~\ref{lemma1}. Hence, $\operatorname*{div}\nolimits X=0$ in $N_\varepsilon$ and $\Phi(1,y)= y+ \psi(y)\nu_E(y)$, for all $y \in \partial E$, that is, $\Phi(1,\partial E)=\partial F\subseteq N_\varepsilon$ which implies $E_1=\Phi_1(E)=F$ and $\mathrm{Vol}(E_1)=\mathrm{Vol}(F)=\mathrm{Vol}(E)$. Then the special variation $E_t=\Phi_t(E)$ is volume--preserving, for $t \in [-1,1]$ and the vector field $X$ is admissible, by the last part of such lemma.\\ By Lemma~\ref{Lemma 3.8}, choosing an even smaller $\deltalta>0$ if necessary, possibly replacing $F$ with a translate $F-\sigma$ for some $\eta\in \mathbb{R}^n$ if needed, we can assume that \begin{equation}\label{assumption} \left| \int_{\partial E} \psi \, \nu_E \, d \mu \right| \le \frac{\deltalta_1}{2} \norma{\psi}_{L^2(\partial E)}. \end{equation} We now claim that \begin{equation}\label{claim1} \left|\int_{\partial E} \langle X \vert \nu_{E_t} \rangle \nu_{E_t} \, d \mu_t \right| \le \deltalta_1 \norma{\langle X \vert \nu_{E_t} \rangle}_{L^2(\partial E_t)} \qquad \forall t \in [0,1]. \end{equation} To this aim, we write \begin{align} \int_{\partial E} \langle X \vert \nu_{E_t} \rangle \nu_{E_t} \, d \mu_t&= \int_{\partial E} \langle X\circ \Phi_t\vert \nu_{E_t}\circ\Phi_t\rangle(\nu_{E_t}\circ\Phi_t)\, J\Phi_t \, d \mu\\ &= \int_{\partial E} \langle X\circ \Phi_t \vert \nu_{E} \rangle \nu_E \, d \mu + R_1\\ &= \int_{\partial E}\langle X(x) \vert \nu_E \rangle \nu_E \, d \mu + R_1+R_2\\ &= \int_{\partial E} \psi \nu_E \, d \mu + R_1+ R_2+ R_3 \end{align} with appropriate $R_1, R_2$ and $R_3$ (see below). \\ By the definition of $X$ in formula~\eqref{field} (in the proof of Lemma~\ref{lemma1}), the bounds $0<C_1\leq \xi\leq C_2$ and $\norma{J(\pi_E\circ\Phi_t)^{-1}}_{L^\infty(\partial E)}\leq C_3$ (by inequality~\eqref{normflow} and Sobolev embeddings, as $p>\max\{2,n-1\}$, we have $\norma{\Phi(t, \cdot) - \mathrm{Id}}_{C^{1,\alpha}(\partial E)} \leq C \norma{\psi }_{W^{2,p}(\partial E)}\leq C\deltalta$), the following inequality holds \begin{align} \int_{\partial E} |X(\Phi(t,x))| \, d \mu&=\, \int_{\partial E} \biggl| \int_0^{\psi(\pi_E (\Phi(t,x)))} \frac{\xi(\Phi(t,x)) \nabla d_E(\Phi(t,x))}{\xi(\Phi(t,x)+ s \nu(\pi_E(\Phi(t,x))))}\, ds \biggr| \, d \mu \nonumber \\ &\,\le C \int_{\partial E} \left|\psi(\pi_E(\Phi(t,x))) \right|\, d \mu \nonumber \\ &\,=\int_{\partial E} |\psi(z)| J(\pi_E \circ \Phi_t)^{-1}(z)\, d \mu(z) \nonumber\\ &\le\, C \norma{\psi}_{L^2(\partial E)}.\label{inequality1} \end{align} for every $t\in[0,1]$.\\ We want now to prove that for every $\overline{\varepsilon}>0$, choosing a suitably small $\deltalta>0$ we have the estimate \begin{equation}\label{eqcar1011} |R_1|+|R_2|+|R_3| \le \varepsilon \norma{\psi}_{L^2(\partial E)}. \end{equation} First, \begin{align} R_1&= \int_{\partial E} \langle X\circ\Phi_t\vert \nu_{E_t}\circ\Phi\rangle \nu_{E_t}\circ\Phi_t [J\Phi_t-1]\, d \mu \\ &\quad+\int_{\partial E} \langle X\circ\Phi_t \vert \nu_{E_t}\circ\Phi_t \rangle \nu_{E_t}\circ\Phi_t \, d \mu-\int_{\partial E} \langle X\circ\Phi_t,\nu_E \rangle \nu_E \, d \mu\\ &=\,\int_{\partial E} \langle X\circ\Phi_t\vert\nu_{E_t}\circ\Phi_t\rangle \nu_{E_t}\circ\Phi_t\, [J\Phi_t-1] \, d \mu+ \int_{\partial E} \langle X\circ\Phi_t\vert \nu_{E_t}\circ\Phi_t- \nu_E\rangle \nu_E\,d \mu \\ &\quad+ \int_{\partial E} \langle X\circ\Phi_t\vert\nu_{E_t}\circ\Phi_t\rangle(\nu_{E_t}\circ\Phi_t-\nu_E) \, d \mu\\ &\leq\,\int_{\partial E} |X\circ\Phi_t|\, \Vert J\Phi_t-1\Vert_{L^\infty(\partial E)} \, d \mu+ \int_{\partial E} |X\circ\Phi_t|\,\norma{\nu_E-\nu_{E_t}\circ\Phi_t}_{L^\infty (\partial E)}\,d \mu\,, \end{align} then, since by equality~\eqref{flowin}, it follow that for every $t\in[0,1]$ the two terms \begin{equation} \norma{\nu_E-\nu_{E_t}\circ\Phi(t,x)}_{L^\infty (\partial E)}\qquad \text{and}\qquad \norma{J\Phi_t-1}_{L^\infty(\partial E)} \end{equation} can be made (uniformly in $t\in[0,1]$) small as we want, if $\deltalta>0$ is small enough, by using inequality~\eqref{inequality1}, we obtain \begin{equation}\label{R1} |R_1|\le \overline{\varepsilon} \norma{\psi}_{L^2(\partial E)}/3. \end{equation} Then we estimate, by means of inequality~\eqref{flowin} and where $s=s(t,y)\in[t,1]$, \begin{align} |R_2| &\le \int_{\partial E} |X(\Phi(t,x))-X(\Phi(1,x))| + |X(\Phi(1,x))-X(x)| \, d \mu \\ \nonumber &\le \int_{\partial E} |X(\Phi(t,x))-X(\Phi(1,x))| + \norma{\nabla X}_{L^2(N_\varepsilon)} \norma{\psi}_{L^2(\partial E)}\\ \nonumber &=\int_{\partial E} (1-t)|\nabla X(\Phi_s(y))| \left|\frac{\partial \Phi_s}{\partial t}(y)\right| \, d \mu(y) + \norma{\nabla X}_{L^2(N_\varepsilon)}\norma{\psi}_{L^2(\partial E)}\\ & \le \int_{\partial E} |\nabla X(\Phi(s,x))| |\Phi(t,x)-\Phi(1,x)| + \norma{\nabla X}_{L^2(N_\varepsilon)} \norma{\psi}_{L^2(\partial E)}\\ \nonumber & \le C \norma{\nabla X}_{L^\infty (N_\varepsilon)} C \norma{\psi}_{L^2(\partial E)} + \norma{\nabla X}_{L^2(N_\varepsilon)}\norma{\psi}_{L^2(\partial E)}, \end{align} where in the last inequality we use equation~\eqref{inequality1}. Hence, using equality~\eqref{normfield} and Sobolev embeddings, as $p>\max\{2,n-1\}$, we get $$ |R_2| \le C \norma{\psi}_{W^{2,p}(\partial E)}\norma{\psi}_{L^2(\partial E)}, $$ then, since $\norma{\psi}_{W^{2,p}(\partial E)} < \deltalta$, we obtain \begin{equation} |R_2|< \overline{\varepsilon} \norma{\psi}_{L^2(\partial E)}/3, \end{equation} if $\deltalta_2$ is small enough.\\ Arguing similarly, recalling the definition of $X$ given by formula~\eqref{field}, we also obtain $|R_3| \le \overline{\varepsilon} \norma{\psi}_{L^2(\partial E)}$, hence estimate~\eqref{eqcar1011} follows. We can then conclude that, for $\deltalta>0$ small enough, we have \begin{equation} \left| \int_{\partial E} \langle X \vert \nu_{E_t} \rangle \nu_{E_t} \, d \mu_t \right|\le \left|\int_{\partial E} \psi \nu_E \, d \mu \right| + \overline{\varepsilon} \norma{\psi}_{L^2(\partial E)}\le \Bigl(\frac{\deltalta_1}{2} + \overline{\varepsilon}\Bigr) \norma{\psi}_{L^2(\partial E)} \end{equation} for any $t\in[0,1]$, where in the last inequality we used the assumption~\eqref{assumption}, thus choosing $\overline{\varepsilon}=\deltalta_1/4$ we get $$ \biggl| \int_{\partial E} \langle X \vert \nu_{E_t} \rangle \nu_{E_t} \, d \mu_t \biggr| \le \frac{3\deltalta_1}{4}\norma{\psi}_{L^2(\partial E)}. $$ Along the same line, it is then easy to prove that \begin{equation}\label{3.46} \norma{\langle X \vert \nu_{E_t}\rangle}_{L^2(\partial E_t)} \ge (1-\varepsilon)\norma{\psi}_{L^2(\partial E)}, \end{equation} for any $t\in[0,1]$, hence claim~\eqref{claim1} follows.\\ As a consequence, since $\langle X \vert \nu_{E_t}\rangle\in\widetilde{H}^1(\partial E_t)$, being $X$ admissible for $E_t$ (recalling computation~\ref{eqc1000}) and $\partial E_t$ can be described as a graph over $\partial E$ with a function with small norm in $W^{2,p}(\partial E)$ (by estimate~\eqref{normflow} of Lemma~\ref{lemma1}), we can apply Step~$2$ with $F=E_t$ to the function $\langle X \vert \nu_{E_t}\rangle/\Vert\langle X \vert \nu_{E_t}\rangle\Vert_{H^1(\partial E_t)}$, concluding \begin{equation}\label{eqcar1020} \Pi_{E_t}(\langle X \vert \nu_{E_t}\rangle)\geq\frac{m_0}{2}\Vert\langle X \vert \nu_{E_t}\rangle\Vert_{H^1(\partial E_t)}. \end{equation} By means of Lemma~\ref{lemmastima}, for $\deltalta>0$ small enough, we now show the following inequality on $\partial E_t$ (here $\operatorname*{div}\nolimits$ is the divergence operator and $X_{\tau _t}=X-\langle X \vert \nu _{E_t} \rangle \nu_{E_t}$ is a tangent vector field on $\partial E_t$), for any $t\in[0,1]$, \begin{align} \norma{\operatorname*{div}\nolimits (X_{\tau_t} \langle X \vert \nu_{E_t} \rangle )}_{L^{\frac{p}{p-1}}(\partial E_t)} =&\, \norma{\operatorname*{div}\nolimits X_{\tau_t} \langle X \vert \nu_{E_t}\rangle + \langle X_{\tau_t}\vert \nabla\langle X \vert \nu_{E_t}\rangle\rangle }_{L^{\frac{p}{p-1}}(\partial E_t)} \\ \le&\, C\norma{\nabla X_{\tau_t}}_{L^2(\partial E_t)} \norma{\langle X \vert \nu_{E_t} \rangle}_{L^{\frac{2p}{p-2}}(\partial E_t)}\\ &\,+ C\norma{X_{\tau_t}}_{L^{\frac{2p}{p-2}}(\partial E_t)}\norma{\nabla \langle X \vert \nu_{E_t}\rangle}_{L^2(\partial E_t)}\\ \le&\, C\norma{X}_{H^1(\partial E_t)}\norma{X}_{L^{\frac{2p}{p-2}}(\partial E_t)}\\ \le&\, C\norma{X}^2_{H^1(\partial E_t)}\\ \le&\, C\norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)},\label{claim2} \end{align} where we used the Sobolev embedding $H^1(\partial E_t)\hookrightarrow L^{\frac{2p}{p-2}}(\partial E_t)$, as $p>\max\{2,n-1\}$.\\ Then, we compute (here $X_{\tau_t}$ is the tangent component of $X$, $\mathrm{H}_t$ is the mean curvature and $v_{E_t}$ the potential relative to $E_t$ defined by formula~\eqref{potential1}) \begin{align} J(F)-J(E)=&\,J(E_1)-J(E)\\ =&\,\int_0^1 (1-t)\frac{d^2}{dt^2}J(E_t)\, dt \\ =&\,\int_0^1(1-t)\bigl(\Pi_{E_t}(\langle X \vert \nu_{E_t}\rangle)+R_t\bigr)\,dt \\ =&\,\int_0^1(1-t)\Pi_{E_t}(\langle X \vert \nu_{E_t}\rangle)\,dt \\ &\,-\int_0^1(1-t)\int_{\partial E}(4\gamma v_{E_t} + \mathrm{H}_t) \operatorname*{div}\nolimits(X_{\tau_t}\langle X \vert \nu_{E_t} \rangle)\, d \mu_t\, dt. \end{align} by Theorem~\ref{secondvar} and the definition of $\Pi_{E_t}$ in formula~\eqref{Pieq}, considering the second form of the remainder term $R_t$, relative to $E_t$ and taking into account that $\operatorname*{div}\nolimits X=0$ in $N_\varepsilon$ and that $X_t=X$, as the variation is special.\\ Hence, by estimate~\eqref{eqcar1020}, we have (recall that $4\gamma v_E+\mathrm{H}=4\gamma v_{E_0} +\mathrm{H}_0=\lambda$ constant, as $E$ is a critical set) \begin{align} J(F)-J(E) \ge&\, \frac{m_0}{2} \int_0^1 (1-t) \norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)} \, dt\\ &\,- \int_0^1 (1-t) \int_{\partial E_t}(\mathrm{H}_t+4\gamma v_{E_t})\, \operatorname*{div}\nolimits(X_{\tau_t}\langle X \vert \nu_{E_t} \rangle)\, d \mu_t \, dt\\ =&\,\frac{m_0}{2} \int_0^1 (1-t) \norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)} \, dt\\ &\,- \int_0^1 (1-t) \int_{\partial E_t}(\mathrm{H}_t + 4 \gamma v_{E_t} -\lambda) \operatorname*{div}\nolimits(X_{\tau_t}\langle X \vert \nu_{E_t} \rangle)\, d \mu_t\, dt\\ \geq&\,\frac{m_0}{2} \int_0^1 (1-t) \norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)} \, dt\\ &\,- \int_0^1 (1-t) \norma{\mathrm{H}_t + 4 \gamma v_{E_t} - \lambda}_{L^p(\partial E_t)} \norma{\operatorname*{div}\nolimits(X_{\tau_t}\langle X \vert \nu_{E_t} \rangle)}_{L^\frac{p}{p-1}(\partial E_t)}\,dt\\ \geq&\,\frac{m_0}{2} \int_0^1 (1-t) \norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)} \, dt\\ &\,- C\int_0^1 (1-t) \norma{\mathrm{H}_t+ 4 \gamma v_{E_t} - \lambda}_{L^p(\partial E_t)}\norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)}\,dt, \end{align} by estimate~\eqref{claim2}. If $\deltalta>0$ is sufficiently small, as $E_t$ is $W^{2,p}$--close to $E$ (recall the definition of $v_{E_t}$ in formula~\eqref{potential1}), we have $$ \norma{\mathrm{H}_t + 4 \gamma v_{E_t} - \lambda}_{L^p(\partial E_t)}<m_0/4C\,, $$ hence, $$ J (F)-J(E) \ge \frac{m_0}{4} \int_0^1 (1-t) \norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)} \, dt. $$ Then, we can conclude the proof of the theorem with the following series of inequalities, holding for a suitably small $\deltalta>0$ as in the statement, \begin{align} J(F)&\ge J(E) + \frac{m_0}{2} \int_0^1 (1-t) \norma{\langle X \vert \nu_{E_t} \rangle}^2_{H^1(\partial E_t)} \, dt\\ & \ge J(E) + C \norma{\langle X \vert \nu_E\rangle}^2_{L^2(\partial E)}\\ & \ge J(E) + C \norma{\psi}^2_{L^2(\partial E)}\\ & \ge J(E) + C[\mathrm{Vol}(E \triangle F)]^2\\ & \ge J(E)+ C[\alpha(E,F)]^2, \end{align} where the first inequality is due to the $W^{2,p}$--closedness of $E_t$ to $E$, the second one by the very expression~\eqref{field} of the vector field $X$ on $\partial E$, \begin{equation} |\langle X(y)\vert\nu_E(y)\rangle|= \Bigl|\int _0 ^{\psi(y)}\frac{ds}{\xi(y+ s \nu_E(y))}\,\Bigr|\leq C|\psi(y)|, \end{equation} the third follows by a straightforward computation (involving the map $L$ defined by formula~\eqref{eqcar410} and its Jacobian), as $\partial E$ is a ``normal graph'' over $\partial F$ with $\psi$ as ``height function'', finally the last one simply by the definition of the ``distance'' $\alpha$, recalling that we possibly translated the ``original'' set $F$ by a vector $\eta\in\mathbb{R}^n$, at the beginning of this step. \end{proof} We conclude this section by proving two propositions that will be used later. The first one says that when a set is sufficiently $W^{2,p}$--close to a strictly stable critical set of the functional $J$, then the quadratic form~\eqref{Pieq} remains uniformly positive definite (on the orthogonal complement of its degenerate subspace, see the discussion at the end of the previous subsection). \begin{proposition}\label{2.6} Let $p>\max \{2,n-1\}$ and $E\subseteq\mathbb{T}^n$ be a smooth strictly stable critical set with $N_\varepsilon$ a tubular neighborhood of $\partial E$, as in formula~\eqref{tubdef}. Then, for every $\theta\in (0,1]$ there exist $\sigma_\theta,\deltalta>0$ such that if a smooth set $F \subseteq \mathbb{T}^n$ is $W^{2,p}$--close to $E$, that is, $\mathrm{Vol}(F\triangle E)<\deltalta$ and $\partial F\subseteq N_\varepsilon$ with \begin{equation} \partial F= \{y + \psi (y) \nu_E(y) \, : \, y \in \partial E\} \end{equation} for a smooth $\psi$ with $\norma{\psi}_{W^{2,p}(\partial E)}<\deltalta$, there holds \begin{equation}\label{2.13} \Pi_F(\varphi) \geq \sigma_\theta \norma{\varphi}^2_{H^1(\partial F)}, \end{equation} for all $\varphi \in \widetilde{H}^1(\partial F)$ satisfying \begin{equation} \min_{\eta\in{\mathrm{O}}_E} \norma{\varphi - \langle \eta \vert \nu_F \rangle }_{L^2(\partial F)} \geq \theta \norma{\varphi}_{L^2(\partial F)}, \end{equation} where ${\mathrm{O}}_E$ is defined by formula~\eqref{OOeq}. \end{proposition} \begin{proof}\ \\ \noindent \textbf{Step ${\mathbf{1}}$.} We first show that for every $\theta\in (0, 1]$ there holds \begin{equation} m_\theta=\inf\Bigl\{\Pi_E(\varphi)\,:\, \varphi\in \widetilde{H}^1(\partial E)\,, \|\varphi\|_{H^1(\partial E)}=1\,\,\text{ and }\min_{\eta\in {\mathrm{O}}_E}\|\varphi-\langle\eta\vert\nu_E\rangle\|_{L^2(\partial E)}\geq \theta\|\varphi\|_{L^2(\partial E)}\Bigr\}>0\,.\label{c0} \end{equation} Indeed, let $\varphi_i$ be a minimizing sequence for this infimum and assume that $\varphi_i\rightharpoonup \varphi_0\in \widetilde{H}^1(\partial E)$ weakly in $H^1(\partial E)$.\\ If $\varphi_0\neq 0$, as the weak convergence in $H^1(\partial E)$ implies strong convergence in $L^2(\partial E)$ by Sobolev embeddings, for every $\eta\in{\mathrm{O}}_E$ we have $$ \|\varphi_0-\langle\eta\vert\nu_E\rangle\|_{L^2(\partial E)} =\lim_{i\to\infty}\|\varphi_i-\langle\eta\vert\nu_E\rangle\|_{L^2(\partial E)}\geq \lim_{i\to\infty}\theta\|\varphi_i\|_{L^2(\partial E)} =\theta\|\varphi_0\|_{L^2(\partial E)}, $$ hence, $$ \min_{\eta\in{\mathrm{O}}_E}\|\varphi_0-\langle\eta\vert\nu_E\rangle\|_{L^2(\partial E)}\geq \theta\|\varphi_0\|_{L^2(\partial E)}>0, $$ thus, we conclude $\varphi_0\in \widetilde{H}^1(\partial E)\setminus T(\partial E)$ and $$ m_\theta=\lim_{i\to\infty} \Pi_E(\varphi_i)\geq \Pi_E(\varphi_0)>0\,, $$ where the last inequality follows from estimate~\eqref{uusi stability} in Remark~\ref{rembase0}.\\ If $\varphi_0= 0$, then again by the strong convergence of $\varphi_i\to\varphi_0$ in $L^2(\partial E)$, by looking at formula~\eqref{Pieq}, we have $$ m_\theta=\lim_{i\to\infty}\Pi_E(\varphi_i)=\lim_{i\to\infty}\int_{\partial E}|\nabla \varphi_i|^2 \, d\mu=\lim_{i\to\infty}\Vert\varphi_i\Vert_{H^1(\partial E)}^2=1 $$ since $\Vert\varphi_i\Vert_{L^2(\partial E)}\to0$. \noindent \textbf{Step ${\mathbf{2}}$.} In order to finish the proof it is enough to show the existence of some $\deltalta>0$ such that if $\mathrm{Vol}(F\triangle E)<\deltalta$ and $\partial F=\bigl\{y+\psi(y) \nu_E(y):\, y\in \partial E\bigr\}$ with $\|\psi\|_{W^{2,p}(\partial E)}<\deltalta$, then \begin{align} \inf\Bigl\{&\,\Pi_F(\varphi)\,:\, \varphi\in \widetilde{H}^1(\partial F)\,, \|\varphi\|_{H^1(\partial F)}=1\,\,\,\text{ and }\min_{\eta\in {\mathrm{O}}_E}\|\varphi-\langle\eta\vert\nu_F\rangle\|_{L^2(\partial F)}\geq\theta\|\varphi\|_{L^2(\partial F)}\Bigr\}\\ &\,\geq\sigma_\theta=\frac{1}{2}\min \{m_{\theta/2},1\}\,,\label{c2unif} \end{align} where $m_{\theta/2}$ is defined by formula~\eqref{c0}, with $\theta/2$ in place of $\theta$.\\ Assume by contradiction that there exist a sequence of smooth sets $F_i\subseteq\mathbb{T}^n$, with $\partial F_i=\{y+\psi_i(y) \nu_E (y):\, y\in \partial E\}$ and $\|\psi_i\|_{W^{2,p}(\partial E)}\to 0$, and a sequence $\varphi_i\in \widetilde{H}^1(\partial F_i)$, with $\|\varphi_i\|_{H^1(\partial F_i)}=1$ and $\min_{\eta\in{\mathrm{O}}_E}\|\varphi_i-\langle\eta\vert\nu_{F_i}\rangle\|_{L^2(\partial F_i)}\geq \theta\|\varphi_i\|_{L^2(\partial F_i)}$, such that \begin{equation}\label{liminf} \Pi_{F_i}(\varphi_i)<\sigma_\theta\leq m_{\theta/2}/2\,. \end{equation} Let us suppose first that $\lim_{i\to\infty}\|\varphi_i\|_{L^2(\partial F_i)}=0$ and observe that by Sobolev embeddings $\|\varphi_i\|_{L^q(\partial F_i)}\to0$ for some $q>2$, thus, since the functions $\psi_i$ are uniformly bounded in $W^{2,p}(\partial E)$ for $p>\max\{2,n-1\}$, recalling formula~\eqref{Pieq}, it is easy to see that $$ \lim_{i\to\infty}\Pi_{F_i}(\varphi_i)=\lim_{i\to\infty}\int_{\partial F_i}|\nabla \varphi_i|^2 \, d\mu_i=\lim_{i\to\infty}\Vert\varphi_i\Vert_{H^1(\partial F_i)}^2=1\,, $$ which is a contradiction with assumption~\eqref{liminf}.\\ Hence, we may assume that \begin{equation}\label{lim positive} \lim_{i\to\infty}\|\varphi_i\|_{L^2(\partial F_i)} >0. \end{equation} The idea now is to write every $\varphi_i$ as a function on $\partial E$. We define the functions $\widetilde{\varphi}_i(\partial E)\to\mathbb{R}$, given by $$ \widetilde\varphi_i(y)=\varphi_i\bigl(y+\psi_i(y) \nu_E(y)\bigr)- \fint_{\partial E}\varphi_i(y+\psi_i(y) \nu_E(y))\, d\mu(y)\,, $$ for every $y\in \partial E$.\\ As $\psi_i\to 0$ in $W^{2,p}(\partial E)$, we have in particular that \begin{equation}\label{sothat} \widetilde\varphi_i\in \widetilde{H}^1(\partial E)\,, \qquad \|\widetilde\varphi_i\|_{H^1(\partial E)}\to 1\, \qquad\text{and} \qquad \frac{\|\widetilde\varphi_i\|_{L^2(\partial E)}}{\| \varphi_i\|_{L^2(\partial F_i)}}\to 1\,, \end{equation} moreover, note also that $\nu_{F_i}(\cdot +\psi_i(\cdot)\nu_E(\cdot))\to \nu_E$ in $W^{1,p}(\partial E)$ and thus in $C^{0,\alpha}(\partial E)$ for a suitable $\alpha\in (0,1)$, depending on $p$, by Sobolev embeddings. Using this fact and taking into account the third limit above and inequality~\eqref{lim positive}, one can easily show that $$ \liminf_{i\to\infty}\frac{\min_{\eta\in{\mathrm{O}}_E}\|\widetilde\varphi_i-\langle\eta\vert\nu_{E}\rangle\|_{L^2(\partial E)}}{\|\widetilde\varphi_i\|_{L^2(\partial E)}}\geq \liminf_{i\to\infty}\frac{\min_{\eta\in{\mathrm{O}}_E}\|\varphi_i-\langle\eta\vert\nu_{F_i}\rangle\|_{L^2(\partial F_i)}}{\|\varphi_i\|_{L^2(\partial E_i)}}\geq \theta\,. $$ Hence, for $i\in\mathbb{N}$ large enough, we have $$ \|\widetilde\varphi_i\|_{H^1(\partial E)}\geq 3/4\qquad\text{and}\qquad \min_{\eta\in{\mathrm{O}}_E}\|\widetilde\varphi_i-\langle\eta\vert\nu_{E}\rangle\|_{L^2(\partial E)}\geq \frac{\theta}2\|\widetilde\varphi_i\|_{L^2(\partial E)}\,, $$ then, in turn, by Step~$1$, we infer \begin{equation}\label{bystep1} \Pi_E(\widetilde\varphi_i)\geq \frac{9}{16}m_{\theta/2}\,. \end{equation} Arguing now exactly like in the final part of Step~$2$ in the proof of Theorem~\ref{W2pMin}, we have that all the terms of $\Pi_{F_i}(\varphi_i)$ are asymptotically close to the corresponding terms of $\Pi_{E}(\widetilde\varphi_i)$, thus $$ \Pi_{F_i}(\varphi_i)-\Pi_{E}(\widetilde\varphi_i)\to 0\,, $$ which is a contradiction, by inequalities~\eqref{liminf} and~\eqref{bystep1}. This establishes inequality~\eqref{c2unif} and concludes the proof. \end{proof} The following final result of this section states the fact that close to a strictly stable critical set there are no other smooth critical sets (up to translations). \begin{proposition}\label{prop:nocrit} Let $p$ and $E\subseteq\mathbb{T}^n$ be as in Proposition~\ref{2.6}. Then, there exists $\deltalta>0$ such that if $E'\subseteq\mathbb{T}^n$ is a smooth critical set with $\mathrm{Vol}(E')=\mathrm{Vol}(E)$, $\mathrm{Vol}(E\triangle E')<\deltalta$, $\partial E'\subseteq N_\varepsilon$ and \begin{equation} \partial E'= \{y + \psi (y) \nu_E(y) \, : \, y \in \partial E\} \end{equation} for a smooth $\psi$ with $\norma{\psi}_{W^{2,p}(\partial E)}<\deltalta$, then $E'$ is a translate of $E$. \end{proposition} \begin{proof} In Step~$3$ of the proof of Theorem~\ref{W2pMin}, it is shown that under these hypotheses on $E$ and $E'$, if $\deltalta>0$ is small enough, we may find a small vector $\eta\in \mathbb{R}^n$ and a volume--preserving variation $E_t$ such that$E_0=E$, $E_1=E'-\eta$ and $$ \frac{d^2 }{dt^2}J(E_t)\geq C[\mathrm{Vol}(E \triangle (E'-\eta))]^2\,, $$ for all $t\in [0,1]$, where $C$ is a positive constant independent of $E'$.\\ Assume that $E'$ is a smooth critical set as in the statement, which is not a translate of $E$, then $\frac{d }{dt}J(E_t)\bigl|_{t=0}=0$, but from the above formula it follows $\frac{d}{dt}J(E_t)\bigl|_{t=1}>0$, which implies that $E'-\eta$ cannot be critical, hence neither $E'$, which is a contradiction. Indeed, $s\mapsto E_{1-s}$ is a volume--preserving variation for $E'-\eta$ and $$ \frac{d}{ds}J(E_{1-s})\Bigl|_{s=0}=-\frac{d}{dt}J(E_t)\Bigl|_{t=1}<0\,, $$ showing that $E'-\eta$ is not critical. \end{proof} \section{The modified Mullins--Sekerka and the surface diffusion flow}\label{msfsdf} We start with the notion of smooth flow of sets. \begin{definition}\label{def:smoothflow} Let $E_t\subseteq \mathbb{T}^n$ for $t\in [0,T)$ be a one-parameter family of sets, then we say that it is a {\em smooth flow} if there exists a smooth {\em reference set} $F\subseteq\mathbb{T}^n$ and a map $\Psi\in C^{\infty}([0,T)\times\mathbb{T}^n;\mathbb{T}^n)$ such that $\Psi_t=\Psi(t,\cdot)$ is a smooth diffeomorphism from $\mathbb{T}^n$ to $\mathbb{T}^n$ and $E_t=\Psi_t(F)$, for all $t\in [0, T)$. \end{definition} The \emph{velocity} of the motion of any point $x =\Psi_t(y)$ of the set $E_t$, with $y \in F$, is then given by \begin{equation} X_t(x)=X_t(\Psi_t(y))=\frac{\partial \Psi_t}{\partial t} (y). \end{equation} (notice that, in general, the smooth vector field $X_t$, defined in the whole $\mathbb{T}^n$ by $X_t(\Psi_t(z))=\frac{\partial \Psi_t}{\partial t} (z)$ for every $z\in\mathbb{T}^n$, is not independent of $t$).\\ When $x \in \partial E_t$, we define the \emph{outer normal velocity} of the flow of the boundaries $\partialrtial E_t$, which are smooth hypersurfaces of $\mathbb{T}^n$, as \begin{equation} V_t(x)=\langle X_t(x)\vert \nu_{E_t}(x)\rangle, \end{equation} for every $t \in [0,T)$, where $\nu_{E_t}$ is the outer normal vector to $E_t$. {\em For more clarity and to simplify formulas and computations, from now on we will denote with $$ \int_{\partial E_t} f\,d\mu_t\qquad\qquad\text{ the integral }\qquad\qquad\int_{\partial F} f\circ \Phi_t\,d\mu_t\,, $$ for every $f:\partial E_t\to\mathbb{R}$, where in the second integral $\mu_t$ is the canonical Riemannian measure induced on the hypersurface $\partial E_t$, parametrized by $\Phi_t\vert_{\partial F}$, by the flat metric of $\mathbb{T}^n$ (coinciding with the Hausdorff $(n-1)$--dimensional measure). Moreover, in the same spirit we set $\nu_t=\nu_{E_t}$.} Before giving the definition of the {\em modified Mullins--Sekerka flow} (first appeared in~\cite{MS} -- see also~\cite{Crank,Gurtin1} and~\cite{EscherSi4} for a very clear and nice introduction to such flow), we need some notation. Given a smooth set $E\subseteq\mathbb{T}^n$ and $\gamma\geq0$, we denote by $w_E$ the unique solution in $H^1(\mathbb{T}^n)$ of the following problem \begin{equation}\label{WE} \begin{cases} \Delta w_E=0 & \text{in }\mathbb{T}^n\setminus \partial E\\ w_E= \mathrm{H} + 4\gamma v_{E} & \text{on } \, \partialrtial E, \end{cases} \end{equation} where $v_E$ is the potential introduced in~\eqref{potential} and $\mathrm{H}$ is the mean curvature of $\partial E$. Moreover, we denote by $w^+_E$ and $w^-_E$ the restrictions $w_E|_{E^c}$ and ${w_E}{|_{E}}$, respectively. Finally, denoting as usual by $\nu_E$ the outer unit normal to $E$, we set $$ [\partial_{\nu_E} w_E]=\partial_{\nu_E}w^+_E-\partial_{\nu_E}w^-_E=-(\partial_{\nu_{E^c}}w^+_E+\partial_{\nu_E}w^-_E)\,. $$ that is the ``jump'' of the normal derivative of $w_E$ on $\partial E$. \begin{definition}\label{MSF def} Let $E\subseteq \mathbb{T}^n$ be a smooth set. We say that a smooth flow $E_t$ such that $E_0=E$, is a {\em modified Mullins--Sekerka flow with parameter $\gamma\geq0$}, on the time interval $[0, T)$ and with initial datum $E$, if the outer normal velocity $V_t$ of the moving boundaries $\partial E_t$ is given by \begin{equation}\label{MSnl} V_t= [\partialrtial_{\nu_t} w_{t}] \quad\text{ on $\partialrtial E_t$ for all $t\in [0, T)$,} \end{equation} where $w_t=w_{E_t}$ (with the above definitions) and we used the simplified notation $\partialrtial_{\nu_t} w_{t}$ in place of $\partialrtial_{\nu_{E_t}} w_{E_t}$. \end{definition} \begin{remark} The adjective ``modified'' comes from the introduction of the parameter $\gamma \ge 0$ in the problem, while considering $\gamma=0$ we have the original flow proposed by Mullins and Sekerka in~\cite{MS} (see also~\cite{Crank,Gurtin1}), which has been also called {\em Hele--Shaw model}~\cite{XChen}, or {\em Hele--Shaw model with surface tension}~\cite{EscherSi1,EscherSi2,EscherSi3}, which arises as a singular limit of a nonlocal version of the Cahn--Hilliard equation~\cite{alikakos,pego,Le}, to describe phase separation in diblock copolymer melts (see also~\cite{OK}). \end{remark} Parametrizing the smooth hypersurfaces $M_t=\partialrtial E_t$ of $\mathbb{T}^n$ by some smooth embeddings $\psi_t:M\to\mathbb{T}^n$ such that $\psi_t(M)=\partialrtial E_t$ (here $M$ is a fixed smooth differentiable $(n-1)$--dimensional manifold and the map $(t,p)\mapsto\psi(t,p)=\psi_t(p)$ is smooth), the geometric evolution law~\eqref{MSnl} can be expressed equivalently as \begin{equation}\label{msf2perp} \Bigl\langle\frac{\partialrtial\psi_t}{\partialrtial t}\,\Bigl\vert\,\nu_t\Bigr\rangle=[\partial_{\nu_t} w_t], \end{equation} where we denoted by $\nu_t$ the outer unit normal to $M_t=\partial E_t$.\\ Moreover, as the moving hypersurfaces $M_t=\partial E_t$ are compact, it is always possible to smoothly reparametrize them with maps (that we still call) $\psi_t$ such that \begin{equation}\label{msf2} \frac{\partialrtial\psi_t}{\partialrtial t}=[\partial_{\nu_t} w_t]\nu_t\,, \end{equation} in describing such flow. This follows by the {\em invariance by tangential perturbations of the velocity}, shared by the flow due to its geometric nature and can be proved following the line in Section~1.3 of~\cite{Man}, where the analogous property is shown in full detail for the (more famous) mean curvature flow. Roughly speaking, the tangential component of the velocity of the points of the moving hypersurfaces, does not affect the global ``shape'' during the motion. Like the nonlocal Area functional $J$ (see Definition~\ref{NAFdef}), the flow is obviously invariant by translations, or more generally under any isometry of $\mathbb{T}^n$ (or $\mathbb{R}^n$). Moreover, if $\psi:[0,T)\times M\to\mathbb{T}^n$ is a modified Mullins--Sekerka flow of hypersurfaces, in the sense of equation~\eqref{msf2perp} and $\Phi:[0,T)\times M\to M$ is a time--dependent family of smooth diffeomorphisms of $M$, then it is easy to check that the reparametrization $\widetilde{\psi}:[0,T)\times M\to\mathbb{T}^n$ defined as $\widetilde{\psi}(t,p)=\psi(t,\Phi(t,p))$ is still a modified Mullins--Sekerka flow (again in the sense of equation~\eqref{msf2perp}). This property can be reread as ``the flow is invariant under reparametrization'', suggesting that the really relevant objects are actually the subsets $M_t=\psi_t(M)$ of $\mathbb{T}^n$. We show now that the volume of the sets $E_t$ is preserved during the evolution. We remark that instead, other geometric properties shared for instance by the mean curvature flow (see~\cite[Chapter~2]{Man}), like convexity are not necessarily maintained (see~\cite{Conv}), neither there holds the so--called ``comparison property'' asserting that if two initial sets are one contained in the other, they stay so during the two respective flows. This volume--preserving property can be easily proved, arguing as in the computation leading to equation~\eqref{eqc1000}. Indeed, if $E_t=\Psi_t(F)$ is a modified Mullins--Sekerka flow, described by $\Psi\in C^{\infty}([0,T) \times \mathbb{T}^n; \mathbb{T}^n)$, with an associated smooth vector field $X_t$ as above, we have \begin{align}\label{volumepreserving} \frac{d}{dt} \mathrm{Vol}( E_t)=&\,\int_F\frac{\partialrtial}{\partialrtial t} J\Psi_t(y)\,dy=\int_F\operatorname*{div}\nolimits X_t(\Psi(t,y))J\Psi(t,y)\,dy\\ =&\,\int_{ E_t} \operatorname*{div}\nolimits X_t(x)\,dx=\int_{\partialrtial E_t} \langle X_t\vert\nu_t\rangle \, d\mu_t=\int_{\partial E_t}V_t\, d \mu_t\\ =&\, \int_{\partial E_t} [ \partial_{\nu_t} w_t ] \, d \mu_t = \int_{\partial E_t} \bigl(\partial_{\nu_t} w_t^+ - \partial_{\nu_t} w_t^- \bigr)\, d \mu_t= 0\,, \end{align} where the last equality follows from the divergence theorem and the fact that $w_t$ is harmonic in $\mathbb{T}^n \setminus \partial E_t$. Another important property of the modified Mullins--Sekerka flow is that it can be regarded as the $H^{-1/2}$--gradient flow of the functional $J$ under the constraint that the volume is fixed, that is, the outer normal velocity $V_t$ is minus such $H^{-1/2}$--gradient of the functional $J$ (see~\cite{Le}).\\ For any smooth set $E \subseteq \mathbb{T}^n$, we let the space $\widetilde{H}^{-1/2}(\partial E)\subseteq L^{2}(\partial E)$ to be the dual of $\widetilde{H}^{1/2}(\partial E)$ (the functions in $H^{1/2}(\partial E)$ with zero integral) with the Gagliardo $H^{1/2}$--seminorm (see~\cite{AdamsFournier,Dem,NePaVa,RuSi}, for instance) $$ \Vert u\Vert_{\widetilde{H}^{1/2}(\partial E)}^2=[u]_{H^{1/2}(\partial E)}^2=\int_{\partial E}\int_{\partial E}\frac{\vert u(x)-u(y)\vert^2}{\vert x-y\vert^{n+1}}\,d\mu(x) d\mu(y) $$ (it is a norm for $\widetilde{H}^{1/2}(\partial E)$ since the functions in it have zero integral) and the pairing between $\widetilde{H}^{1/2}(\partial E)$ and $\widetilde{H}^{-1/2}(\partial E)$ simply being the integral of the product of the functions on $\partial E$.\\ We define the linear operator $\Delta_{\partial E}$ on the smooth functions $u$ with zero integral on $\partial E$ as follows: we consider the unique smooth solution $w$ of the problem \begin{equation} \begin{cases} \Delta w=0 & \text{in } \mathbb{T}^n\setminus\partial E\\ w=u & \text{on } \, \partialrtial E \end{cases} \end{equation} and we denote by $w^+$ and $w^-$ the restrictions $w|_{E^c}$ and $w|_E$, respectively, then we set $$ \Delta_{\partial E}u=\partial_{\nu}w^+-\partial_{\nu}w^-= [\partial_{\nu} w]\,, $$ which is another smooth function on $\partial E$ with zero integral. Then, we have $$ \int_{\mathbb{T}^n}\vert\nabla w\vert^2\,dx=\int_{E\cup E^c}\operatorname*{div}\nolimits(w\nabla w)\,dx=-\int_{\partial E}u\Delta_{\partial E}u\,d\mu $$ and such quantity turns out to be a norm equivalent to the one given by the Gagliardo seminorm on $\widetilde{H}^{1/2}(\partial E)$ above (this is related to the theory of trace spaces for which we refer to~\cite{AdamsFournier, Gagliardo}), see~\cite{Le}. Hence, it induces the dual norm \begin{equation} \Vert v \Vert^2_{\widetilde{H}^{-1/2}(\partial E)}=\int_{\partial E} v (- \Delta_{\partial E} )^{-1} v \, d \mu \end{equation} for every smooth function $v\in\widetilde{H}^{-1/2}(\partial E)$. By polarization, we have the $\widetilde{H}^{-1/2}(\partial E)$--scalar product between a pair of smooth functions $u,v:\partial E\to\mathbb{R}$ with zero integral, \begin{equation} \langle u|v \rangle_{\widetilde{H}^{-1/2}(\partial E)}=\int_{\partial E} u (- \Delta_{\partial E} )^{-1} v \, d \mu\,. \end{equation} This scalar product, extended to the whole space $\widetilde{H}^{-1/2}(\partial E)$, makes it a Hilbert space (see~\cite{Gar}), hence, by {\em Riesz representation theorem}, there exists a function $\nabla_{\partial E}^{\widetilde{H}^{-1/2}}\!\!J\in\widetilde{H}^{-1/2}(\partial E)$ such that, for every smooth function $v\in\widetilde{H}^{-1/2}(\partial E)$, there holds \begin{equation}\label{gradient} \int_{\partial E} v (\mathrm{H} + 4 \gamma v_E) \, d \mu =\deltalta J_{\partial E}(v)= \langle v|\nabla_{\partial E}^{\widetilde{H}^{-1/2}}\!\!J\rangle_{\widetilde{H}^{-1/2} (\partial E)} = \int_{\partial E} v(- \Delta_{\partial E})^{-1} \nabla_{\partial E}^{\widetilde{H}^{-1/2}}\!\!J\, d \mu \,, \end{equation} by Theorem~\ref{first var}, where $v_E$ is the potential introduced in~\eqref{potential} and $\mathrm{H}$ is the mean curvature of $\partial E$.\\ Then, by the {\em fundamental lemma of calculus of variations}, we conclude $$ (-\Delta_{\partial E})^{-1}\nabla_{\partial E}^{\widetilde{H}^{-1/2}}\!\!J=\mathrm{H}+ 4 \gamma v_E+c\,, $$ for a constant $c\in\mathbb{R}$, that is, recalling the definition of $w_E$ in problem~\eqref{WE} and of the operator $\Delta_{\partial E}$ above, $$ \nabla_{\partial E}^{\widetilde{H}^{-1/2}}\!\!J=-\Delta_{\partial E}(\mathrm{H}+ 4 \gamma v_E)= -[\partial_{\nu_E} w_E]\,. $$ It clearly follows that the outer normal velocity of the moving boundaries $V_t=[\partial_{\nu_t} w_t]$ is minus the $\widetilde{H}^{-1/2}$--gradient of the volume--constrained functional $J$. We deal now with the {\em surface diffusion flow}. \begin{definition}\label{def:SDsol} Let $E\subseteq \mathbb{T}^n$ be a smooth set. We say that a smooth flow $E_t=\Phi_t(E)$, for $t \in[0, T)$, with $E_0=E$, is a {\em surface diffusion flow} starting from $E$ if the outer normal velocity $V_t$ of the moving boundaries $\partial E_t$ is given by \begin{equation}\label{SD} V_t= \Delta_t\mathrm{H}_t \quad\text{ for all $t\in[0, T)$} \end{equation} where $\Delta_t$ is the (rough) Laplacian associated to the hypersurface $\partial E_t$, with the Riemannian metric induced by $\mathbb{T}^n$ (that is, by $\mathbb{R}^n$). \end{definition} Such flow was first proposed by Mullins in~\cite{Mullins} to study thermal grooving in material sciences and first analyzed mathematically more in detail in~\cite{escmaysim}. In particular, in the physically relevant case of three--dimensional space, it describes the evolution of interfaces between solid phases of a system, driven by surface diffusion of atoms under the action of a chemical potential (see for instance~\cite{GurJab}). With the same argument used for the modified Mullins--Sekerka flow, representing the smooth hypersurfaces $\partial E_t$ in $\mathbb{T}^n$ with a family of smooth embeddings $\psi_t:M\to \mathbb{T}^n$, we can describe the flow as \begin{equation}\label{sdf2perp} \Bigl\langle\frac{\partialrtial\psi_t}{\partialrtial t} \Bigl\vert \nu_t\Bigr\rangle=\Delta_t\mathrm{H}_t\, \end{equation} and also simply as \begin{equation}\label{sdf2} \frac{\partialrtial\psi_t}{\partialrtial t}=(\Delta_t\mathrm{H}_t)\nu_t\,. \end{equation} \begin{remark} This is actually the more standard way to define the surface diffusion flow, in the more general situation of smooth and possibly {\em immersed--only} hypersurfaces (usually in $\mathbb{R}^n$), without being the boundary of any set. \end{remark} By means of equation~\eqref{lap}, the system~\eqref{sdf2} can be rewritten as \begin{equation}\label{paraeq} \frac{\partialrtial\psi_t}{\partialrtial t}=-\Delta_t\Delta_t\psi_t+\text{ lower order terms} \end{equation} and it can be seen that it is a fourth order, {\em quasilinear} and {\em degenerate}, parabolic system of PDEs. Indeed, it is quasilinear, as the coefficients (as second order partial differential operator) of the Laplacian associated to the induced metrics $g_t$ on the evolving hypersurfaces, that is, $$ \Delta_t\psi_t(p)=\Delta_{g_t(p)}\psi_t(p)=g^{ij}_t(p)\nabla^{g_t(p)}_i\nabla^{g_t(p)}_j\psi_t(p) $$ depend on the first order derivatives of $\psi_t$, as $g_t$ (and the coefficient of $\Delta_t\Delta_t$ on the third order derivatives). Moreover, the operator at the right hand side of system~\eqref{sdf2} is degenerate, as its symbol (the symbol of the linearized operator) admits zero eigenvalues due to the invariance of the Laplacian by diffeomorphisms. Arguing as in computation~\eqref{volumepreserving}, using the equation~\eqref{SD} in place of~\eqref{MSnl}, it can be seen that also the surface diffusion flow of boundaries of sets is volume--preserving. Moreover, analogously to the modified Mullins--Sekerka flow (see the discussion above), it does not preserve convexity (see~\cite{Ito}), nor the embeddedness (in the ``stand--alone'' formulation of motion of hypersurfaces, as in formula~\eqref{sdf2}, see~\cite{gigaito1}), indeed it also does not have a ``comparison principle'', while it is invariant by isometries of $\mathbb{T}^n$, reparametrizations and tangential perturbations of the velocity of the motion. In addition, it can be regarded as the $\widetilde{H}^{-1}$--gradient flow of the volume--constrained Area functional, in the following sense (see~\cite{Gar}, for instance). For a smooth set $E \subseteq \mathbb{T}^n$, we let the space $\widetilde{H}^{-1}(\partial E)\subseteq L^{2}(\partial E)$ to be the dual of $\widetilde{H}^1(\partial E)$ with the norm $\Vert u\Vert_{\widetilde{H}^1(\partial E)}=\int_{\partial E}\vert\nabla u\vert^2\,d\mu$ and the pairing between $\widetilde{H}^1(\partial E)$ and $\widetilde{H}^{-1}(\partial E)$ simply being the integral of the product of the functions on $\partial E$.\\ Then, it follows easily that the norm of a smooth function $v\in\widetilde{H}^{-1}(\partial E)$ is given by \begin{equation} \Vert v \Vert^2_{\widetilde{H}^{-1}(\partial E)}= \int_{\partial E} v (- \Delta )^{-1} v \, d \mu = \int_{\partial E} \langle \nabla(- \Delta )^{-1} v \vert \nabla(- \Delta )^{-1} v \rangle \, d \mu \end{equation} and, by polarization, we have the $\widetilde{H}^{-1}(\partial E)$--scalar product between a pair of smooth functions $u,v:\partial E\to\mathbb{R}$ with zero integral, \begin{equation} \langle u\vert v \rangle_{\widetilde{H}^{-1}(\partial E)}= \int_{\partial E} \langle \nabla(- \Delta )^{-1} u \vert \nabla(- \Delta )^{-1} v \rangle \, d \mu= \int_{\partial E} u (- \Delta )^{-1} v \, d \mu\,, \end{equation} integrating by parts.\\ This scalar product, extended to the whole space $\widetilde{H}^{-1}(\partial E)$, make it a Hilbert space, hence, by {\em Riesz representation theorem}, there exists a function $\nabla_{\partial E}^{\widetilde{H}^{-1}}\!\!\mathcal A\in\widetilde{H}^{-1}(\partial E)$ such that, for every smooth function $v\in\widetilde{H}^{-1}(\partial E)$, there holds $$ \int_{\partial E} v\mathrm{H} \, d \mu =\deltalta\mathcal A_{\partial E}(v)= \langle v \vert \nabla_{\partial E}^{\widetilde{H}^{-1}}\!\!\mathcal A\rangle_{\widetilde{H}^{-1} (\partial E)} = \int_{\partial E} v(- \Delta)^{-1} \nabla_{\partial E}^{\widetilde{H}^{-1}}\!\!\mathcal A\, d \mu \,, $$ by Theorem~\ref{first var} (with $\gamma = 0$).\\ Then, by the {\em fundamental lemma of calculus of variations}, we conclude $$ (-\Delta)^{-1}\nabla_{\partial E}^{\widetilde{H}^{-1}}\!\!\mathcal A=\mathrm{H}+c\,, $$ for a constant $c\in\mathbb{R}$, that is, $$ \nabla_{\partial E}^{\widetilde{H}^{-1}}\!\!\mathcal A= - \Delta\mathrm{H}\,. $$ It clearly follows that the outer normal velocity of the moving boundaries of a surface diffusion flow $V_t=\Delta_t\mathrm{H}_t$ is minus the $\widetilde{H}^{-1}$--gradient of the volume--constrained functional $\mathcal A$. \begin{remark}\label{localcarlo} It is interesting to notice that the ({\em unmodified}, that is, with $\gamma=0$) Mullins--Sekerka flow is the $H^{-1/2}$--gradient flow and the surface diffusion flow the $H^{-1}$--gradient flow of the Area functional on the boundary of the sets, {\em under a volume constraint}, while considering the {\em unconstrained} Area functional, its $L^2$--gradient flow is the mean curvature flow. It follows that, in a way, the unmodified Mullins--Sekerka flow, representing the moving hypersurfaces as of smooth embeddings $\psi_t:M\to \mathbb{T}^n$, can be described as \begin{equation}\label{sdf3} \frac{\partialrtial\psi_t}{\partialrtial t}=(\Delta_t^{1/2}\mathrm{H}_t)\nu_t=-\Delta_t^{3/2}\psi_t+\text{ lower order terms,} \end{equation} showing its parabolic nature (differently by the surface diffusion flow, in this case the equation is {\em nonlocal}, due to the fractional Laplacian involved, even if the functional is still simply the Area, hence implying that the flow depends only on the hypersurface) -- again quasilinear and degenerate -- and suggesting the problem of analyzing (and eventually generalizing the existing results) the nonlocal evolutions of hypersurfaces given by the laws \begin{equation}\label{sdf3*} \frac{\partialrtial\psi_t}{\partialrtial t}=(\Delta_t^s\mathrm{H}_t)\nu_t=-\Delta_t^{s+1}\psi_t+\text{ lower order terms,} \end{equation} when $s>0$, arising from considering, as above, the $H^{-s}$--gradient of the Area functional on the boundary of the sets (under a volume constraint).\\ Up to our knowledge, these flows are not present in literature and it would be also interesting to compare them to the {\em fractional mean curvature flows} arising considering the gradient flows associated to the {\em fractional Area functionals} on the boundary of a set (in this case such functionals are ``strongly'' nonlocal), see~\cite{Imbert,JLamanna} and references therein, for instance. \end{remark} \subsection{Short time existence}\ \vskip.3em To state the short time existence and uniqueness results for the two flows, we give the following definition which is actually fundamental for the discussion of the global existence in the next section. \begin{definition} Given a smooth set $E\subseteq\mathbb{T}^n$ and a tubular neighborhood $N_\varepsilon$ of $\partial E$, as in formula~\eqref{tubdef}, for any $M\in(0,\varepsilon/2)$ (recall the discussion in Subsection~\ref{stabsec} about the notion of ``closedness'' of sets), we denote by $\mathfrak{C}^1_M(E)$, the class of all smooth sets $F\subseteq E\cup N_\varepsilon$ such that $\mathrm{Vol}(F\triangle E)\leq M$ and \begin{equation}\label{front} \partial F=\{x+\psi_F(x)\nu_{E}(x):\, x\in \partial E \}\,, \end{equation} for some $\psi_F\in C^\infty(\partial E)$, with $\norma{\psi_F}_{C^1(\partial E)}\leq M$ (hence, $\partial F\subseteq N_\varepsilon$). For every $k\in\mathbb{N}$ and $\alpha\in (0,1)$, we also denote by $\mathfrak{C}^{k,\alpha}_M(E)$ the collection of sets $F\in \mathfrak{C}^1_M(E)$ such that $\norma{\psi_F}_{C^{k,\alpha}(\partial E)}\leq M$. \end{definition} The following existence/uniqueness theorem of classical solutions for the modified Mullins--Sekerka flow was proved by Escher and Simonett~\cite{EscherSi1,EscherSi2,EscherSi3} and independently by Chen, Hong and Yi~\cite{chenhong} (see also~\cite{EsNi}). The original version deals with the flow in domains of $\mathbb{R}^n$, but it can be easily adapted to hold also when the ambient is the flat torus $\mathbb{T}^n$. \begin{thm}\label{th:EscNis} Let $E\subseteq\mathbb{T}^n$ be a smooth set and $N_\varepsilon$ a tubular neighborhood of $\partial E$, as in formula~\eqref{tubdef}. Then, for every $\alpha\in (0,1)$ and $M\in(0,\varepsilon/2)$ small enough, there exists $T=T(E,M,\alpha)>0$ such that if $E_0\in \mathfrak{C}^{2,\alpha}_M(E)$ there exists a unique smooth modified Mullins--Sekerka flow with parameter $\gamma\geq0$, starting from $E_0$, in the time interval $[0, T)$. \end{thm} We now state the analogous result (and also of dependence on the initial data) for the surface diffusion flow starting from a smooth hypersurface, proved by Escher, Mayer and Simonett in~\cite{escmaysim}, which should be expected by the explicit parabolic nature of the system~\eqref{sdf2}, as shown by the formula~\eqref{paraeq}. As before, it deals with the evolution in the whole space $\mathbb{R}^n$ of a generic hypersurface, even only immersed, hence possibly with self--intersections. It is then straightforward to adapt the same arguments to our case, when the ambient is the flat torus $\mathbb{T}^n$ and the hypersurfaces are the boundaries of the sets $E_t$, as in Definition~\ref{def:SDsol}, getting a (unique) surface diffusion flow in a positive time interval $[0,T)$, for every initial smooth set $E_0\subseteq\mathbb{T}^n$. \begin{thm}\label{th:EMS0} Let $\psi_0:M\to\mathbb{R}^n$ be a smooth and compact, immersed hypersurface. Then, there exists a unique smooth surface diffusion flow $\psi:[0, T)\times M\to\mathbb{R}^n$, starting from $M_0=\psi_0(M)$ and solving system~\eqref{sdf2}, for some maximal time of existence $T>0$.\\ Moreover, such flow and the maximal time of existence depend continuously on the $C^{2,\alpha}$ norm of the initial hypersurface. \end{thm} As an easy consequence, we have the following proposition (analogous to Theorem~\ref{th:EscNis}), better suited for our setting. \begin{proposition}\label{th:EMS1} Let $E\subseteq\mathbb{T}^n$ be a smooth set and $N_\varepsilon$ a tubular neighborhood of $\partial E$, as in formula~\eqref{tubdef}. Then, for every $\alpha\in (0,1)$ and $M\in(0,\varepsilon/2)$ small enough, there exists $T=T(E,M,\alpha)>0$ such that if $E_0\in \mathfrak{C}^{2,\alpha}_M(E)$ there exists a unique smooth surface diffusion flow, starting from $E_0$, in the time interval $[0, T)$. \end{proposition} In the same paper~\cite{escmaysim}, Escher, Mayer and Simonett also showed that if the initial set $E_0$ is in $\mathfrak{C}^{2,\alpha}_M(B)$, where $B\subseteq\mathbb{R}^n$ is a ball with the same volume and $M$ is small enough (that is, $E_0$ is $C^{2,\alpha}$--close to the ball $B$), then the smooth flow $E_t$ exists for every time and smoothly converges to a translate of the ball $B$.\\ The analogous result for the ({\em unmodified}, that is, with $\gamma=0$) Mullins--Sekerka flow, was proved by Escher and Simonett in~\cite{EscherSi4} (moving by their previous work~\cite{EscherSi2}), generalizing to any dimension the two dimensional case shown by Chen in~\cite{XChen}. The next section will be devoted to present the generalization by Acerbi, Fusco, Julin and Morini in~\cite{AcFuMoJu} (in dimensions two and three) of this stability result for the surface diffusion and modified Mullins--Sekerka flow, to every strictly stable critical set (as it is every ball for the Area functional under a volume constraint, by direct check -- see the last section). We conclude mentioning another interesting result by Elliott and Garcke~\cite{EllGar} (which is not present in literature for the modified Mullins--Sekerka flow, up to our knowledge) is that if the initial curve $E_0$ in $\mathbb{R}^2$ of the surface diffusion flow is closed to a circle, then the flow $E_t$ exists for all times and converges, up to translations, to a circle in the plane with the same volume. This is clearly related to the fact that the unique bounded strictly stable critical sets for the Area functional under a volume constraint in the plane $\mathbb{R}^2$ are the disks (see the last section). \section{Global existence and asymptotic behavior around a strictly stable critical set}\label{globalex} In this section we show the proof by Acerbi, Fusco, Julin and Morini in~\cite{AcFuMoJu}, in dimensions two and three of the toric ambient, that if the ``initial'' sets is ``close enough'' to a strictly stable critical set of the respectively relative functional, then the surface diffusion and the modified Mullins--Sekerka flow exist for all times and smoothly converge to a translate of $E$. Heuristically, this shows that a strictly stable critical set is in a way like the equilibrium configuration of a system at the bottom of a potential well ``attracting'' the close enough smooth sets. We will deal here with the (more difficult) case of dimension three. When the dimension is two, the ``exponents'' in the functional spaces involved in the estimates (in particular the ones in the interpolation inequalities, which are very dimension--dependent) change but the same proof still works (roughly speaking, we have the necessary ``compactness'' of the sequences of hypersurfaces -- see Lemma~\ref{w52conv} and~\ref{w32conv}), modifying suitably the statements. If the dimension of the toric ambient is larger than three, the analogous (mostly, interpolation) estimates are too weak to conclude and this proof does not work. It is indeed a challenging open problem to extend these results to such higher dimensions. For both flows, we will have a subsection with the necessary technical lemmas and then one with the proof of the main theorem. Moreover, for the modified Mullins--Sekerka flow, we also briefly discuss the ``Neumann case'', in Subsection~\ref{Neucase}. \subsection{The modified Mullins--Sekerka flow -- Preliminary lemmas}\ \vskip.3em In order to simplify the notation, for a smooth set $E_t\subseteq\mathbb{T}^n$ we will write $\nu_t$ and $\partialrtial_{\nu_t}$ in place of $\nu_{E_t}$ and $\partialrtial_{\nu_{E_t}}$, $w_t$ for the function $w_{E_t}\in H^1(\mathbb{T}^n)$ uniquely defined by problem~\eqref{WE}. Moreover, we will also denote with $v_t$ the smooth potential function $v_{E_t}$ associated to $E_t$ by formula~\eqref{potential}. We start with the following lemma holding in all dimensions. \begin{lem}[Energy identities] \label{calculations} Let $E_t\subseteq\mathbb{T}^n$ be a modified Mullins--Sekerka flow as in Definition~\eqref{MSF def}. Then, the following identities hold: \begin{equation} \label{der of J} \frac{d}{dt} J(E_t) = - \int_{\mathbb{T}^n} |\nabla w_t|^2\, dx\,, \end{equation} and \begin{equation} \label{der of dw} \frac{d}{dt}\,\frac{1}{2} \int_{\mathbb{T}^n} |\nabla w_t|^2\, dx= -\Pi_{E_t}\bigl([\partial_{\nu_t}w_t\vphantom{^{^4}}]\bigr) + \frac{1}{2}\int_{\partialrtial E_t} \bigl(\partialrtial_{\nu_t} w^+_t+ \partialrtial_{\nu_t} w_t^-\bigr) [\partialrtial_{\nu_t} w_t]^2 \, d \mu_t \,, \end{equation} where $\Pi_{E_t}$ is the quadratic form defined in formula~\eqref{Pieq}. \end{lem} \begin{proof} Let $\psi_t$ the smooth family of maps describing the flow as in formula~\eqref{msf2}. By formula~\eqref{dermu2}, where $X$ is the smooth (velocity) vector field $X_t=\frac{\partialrtial\psi_t}{\partialrtial t}=[\partial_{\nu_t}w_t]\nu_t$ along $\partial E_t$, hence $X_\tau=X_t-\langle X_t \vert \nu_t\rangle\nu_t=0$ (as usual $\nu_t$ is the outer normal unit vector of $\partialrtial E_t$), following the computation in the proof of Theorem~\eqref{first var}, we have \begin{equation} \frac{d}{dt} J(E_t)= \int_{\partialrtial E_t} (\mathrm{H}_t+ 4 \gamma v_t) \langle X_t |\nu_t \rangle \, d\mu_t= \int_{\partial E_t} w_t[\partial_{\nu_t}w_t]\, d \mu_t=- \int_{\mathbb{T}^n} \abs{\nabla w_t}^2 \, dx\,, \end{equation} where the last equality follows integrating by parts, as $w_t$ is harmonic in $\mathbb{T}^n\setminus \partial E_t$. This establishes relation~\eqref{der of J}. In order to get identity~\eqref{der of dw}, we compute \begin{align} \frac{d}{dt}\,\frac{1}{2} \int_{E_t} |\nabla w_t^-|^2\, dx &= \frac{1}{2} \int_{\partialrtial E_{t}} |\nabla^{\mathbb{T}^n}\! w^-_t |^2 \left\langle X_t \vert \nu_t\right\rangle\, d \mu_t+\frac{1}{2} \int_{E_t} \frac{d}{dt}|\nabla w_t^-|^2\, dx\\ &= \frac{1}{2} \int_{\partialrtial E_{t}} |\nabla^{\mathbb{T}^n}\! w^-_t |^2 [ \partialrtial_{\nu_t} w_t ] \, d \mu + \int_{E_{t}} \nabla \partialrtial_t{w}^-_t \nabla w^-_t \, d \mu_t\\ &= \frac{1}{2} \int_{\partialrtial E_{t}} |\nabla^{\mathbb{T}^n}\! w^-_t |^2 [ \partialrtial_{\nu_t} w_t ] \, d \mu + \int_{\partialrtial E_{t}} \partialrtial_t{w}^-_t \partialrtial_{\nu_t} w^-_t \, d \mu_t, \,\label{calculation} \end{align} where we interchanged time and space derivatives and we applied the divergence theorem, taking into account that $w_t^-$ is harmonic in $E_t$.\\ Then, we need to compute $\partialrtial_t{w}^-_t$ on $\partial E_t$. We know that $$ w_t^-=\mathrm{H}_t+4\gamma v_t $$ on $\partial E_t$, hence, (totally) differentiating in time this equality, we get $$ \partialrtial_t{w}^-_t + \bigl\langle \nabla^{\mathbb{T}^n}\! w^-_t\bigl\vert X_t\bigr\rangle = \partialrtial_t\mathrm{H}_t+ 4\gamma \partialrtial_t{v}_t + 4\gamma \bigl\langle\nabla^{\mathbb{T}^n}\! v_t\bigl\vert X_t\bigr\rangle\,, $$ that is, \begin{align*} \partialrtial_t{w}^-_t + [\partialrtial_{\nu_t} w_t]\partialrtial_{\nu_t}w^-_t=&\,\partialrtial_t\mathrm{H}_t+ 4\gamma \partialrtial_t{v}_t + 4\gamma [\partialrtial_{\nu_t} w_t]\partialrtial_{\nu_t}v_t\\ =&\,-|B_t|^2 [\partialrtial_{\nu_t} w_t]-\Delta_t[\partialrtial_{\nu_t} w_t]+ 4\gamma \partialrtial_t{v}_t + 4\gamma [\partialrtial_{\nu_t} w_t]\partialrtial_{\nu_t}v_t\,, \end{align*} where we used computation~\eqref{derH}.\\ Therefore from equations~\eqref{calculation} and~\eqref{v'} we get \begin{align} \frac{d}{dt} \frac{1}{2} \int_{E_t} |\nabla w_t^-|^2\, dx =&\,-\int_{\partialrtial E_t} \partialrtial_{\nu_t} w^-_t\, \Delta_t [\partialrtial_{\nu_t} w_t]\, d \mu_t - \int_{\partialrtial E_t}\partialrtial_{\nu_t} w_t^-\, |B_t|^2 \, [\partialrtial_{\nu_t} w_t] \, d \mu_t \\ &\,+8\gamma \int_{\partialrtial E_t}\int_{\partialrtial E_t} G(x,y) \,\partialrtial_{\nu_t} w_t^-(x)\, [\partialrtial_{\nu_t} w_t](y) \, d \mu_t(x) d \mu_t(y) \\ &\,+4\gamma \int_{\partialrtial E_t} \partialrtial_{\nu_t} v_t\,\partialrtial_{\nu_t} w_t^-\, [\partialrtial_{\nu_t} w_t] \, d \mu_t \\ &\,+\frac{1}{2} \int_{\partialrtial E_{t}} |\nabla^{\mathbb{T}^n}\! w_t^- |^2 [\partialrtial_{\nu_t} w_t] \, d \mu_t - \int_{\partialrtial E_t} (\partialrtial_{\nu_t} w_t^-)^2 [\partialrtial_{\nu_t} w_t] \, d \mu_t\,.\label{interior} \end{align} Computing analogously for $w^+_t$ in $E^c$ and adding the two results, we get \begin{align} \frac{d}{dt} \frac{1}{2} \int_{\mathbb{T}^n} |\nabla w_t|^2\, dx =&\,\int_{\partialrtial E_t} [\partialrtial_{\nu_t} w_t]\, \Delta_t [\partialrtial_{\nu_t} w_t]\, d \mu_t + \int_{\partialrtial E_t}|B_t|^2 \, [\partialrtial_{\nu_t} w_t]^2 \, d \mu_t \\ &\,-8\gamma \int_{\partialrtial E_t}\int_{\partialrtial E_t} G(x,y) \,[\partialrtial_{\nu_t} w_t](x)\, [\partialrtial_{\nu_t} w_t](y) \, d \mu_t(x) d \mu_t(y) \\ &\,-4\gamma \int_{\partialrtial E_t} \partialrtial_{\nu_t} v_t\, [\partialrtial_{\nu_t} w_t]^2 \, d \mu_t \\ &\,+\int_{\partialrtial E_t} \bigl((\partialrtial_{\nu_t} w^+_t)^2 - (\partialrtial_{\nu_t} w_t^-)^2 \bigr)[\partialrtial_{\nu_t} w_t] \, d \mu_t \\ &\,-\frac{1}{2} \int_{\partialrtial E_{t}}\bigl (|\nabla^{\mathbb{T}^n}\! w^+_t |^2 - |\nabla^{\mathbb{T}^n}\! w^-_t |^2\bigr) [\partialrtial_{\nu_t} w_t] \, d \mu_t\\ =&\,- \Pi_{E_t}\bigl([\partial_{\nu_t}w_t]\bigr)+ \frac{1}{2}\int_{\partialrtial E_t} \bigl(\partialrtial_{\nu_t} w^+_t+ \partialrtial_{\nu_t} w_t^-\bigr) [\partialrtial_{\nu_t} w_t]^2 \, d \mu_t \,, \end{align} where we integrated by parts the very first term of the right hand side, recalled Definition~\eqref{Pieq} and in the last step we used the identity $$ |\nabla^{\mathbb{T}^n}\! w^+_t |^2 - |\nabla^{\mathbb{T}^n}\! w^-_t |^2 = (\partialrtial_{\nu_t} w^+_t)^2 - (\partialrtial_{\nu_t} w_t^-)^2 = (\partialrtial_{\nu_t} w_t^+ + \partialrtial_{\nu_t} w_t^- ) [\partialrtial_{\nu_t} w_t]\,. $$ Hence, also equation~\eqref{der of dw} is proved. \end{proof} {\em From now on, we restrict ourselves to the three--dimensional case, that is, we will consider smooth subsets of $\mathbb{T}^3$ with boundaries which then are smooth embedded ($2$--dimensional) surfaces. As we said at the beginning of the section, this is due to the dependence on the dimension of several of the estimates that follow.} {\em In the estimates in the following series of lemmas, we will be interested in having uniform constants for the families $\mathfrak{C}^{1,\alpha}_M(F)$, given a smooth set $F\subseteq\mathbb{T}^n$ and a tubular neighborhood $N_\varepsilon$ of $\partial F$ as in formula~\eqref{tubdef}, for any $M\in(0,\varepsilon/2)$ and $\alpha\in(0,1)$. This is guaranteed if the constants in the Sobolev, Gagliardo--Nirenberg interpolation and Calder\'on--Zygmund inequalities, relative to all the smooth hypersurfaces $\partial E$ boundaries of the sets $E\in\mathfrak{C}^{1,\alpha}_M(F)$, are uniform, as it is proved in detail in~\cite{DDM}.} {\em We remind that in all the inequalities, the constants $C$ may vary from one line to another.} The next lemma provides some boundary estimates for harmonic functions. \begin{lem}[Boundary estimates for harmonic functions]\label{harmonic estimates} Let $F\subseteq\mathbb{T}^3$ be a smooth set and $E\in\mathfrak{C}^{1,\alpha}_M(F)$. Let $f \in C^\alpha(\partialrtial E)$ with zero integral on $\partialrtial E$ and let $u \in H^{1}(\mathbb{T}^3)$ be the (distributional) solution of \begin{equation*} -\Delta u = f \mu\bigr|_ {\partialrtial E} \end{equation*} with zero integral on $\mathbb{T}^3$. Let $u^- = u|_{E}$ and $u^+ = u|_{E^c}$ and assume that $u^-$ and $u^+$ are of class $C^1$ up to the boundary $\partial E$. Then, for every $1<p<+\infty$ there exists a constant $C=C(F,M,\alpha,p)>0$, such that: \begin{enumerate} \item[{\em (i)}] \ $$\| u \|_{L^p(\partialrtial E)} \leq C \|f\|_{L^p(\partialrtial E)}$$ \item[{\em (ii)}] \ \begin{equation*} \| \partialrtial_{\nu_E} u^+\|_{L^2(\partialrtial E)} + \| \partialrtial_{\nu_E} u^-\|_{L^2(\partialrtial E)} \leq C \|u \|_{H^1(\partialrtial E)} \end{equation*} \item[{\em (iii)}] \ \begin{equation*} \| \partialrtial_{\nu_E} u^+\|_{L^p(\partialrtial E)} + \| \partialrtial_{\nu_E} u^-\|_{L^p(\partialrtial E)} \leq C \|f\|_{L^p(\partialrtial E)} \end{equation*} \item[{\em (iv)}] \ $$ \|u\|_{C^{0, \beta}(\partial E)}\leq C \|f\|_{L^{p}(\partial E)} $$ for all $\beta\in \bigl(0, \frac{p-2}{p}\bigr)$, with $C$ depending also on $\beta$. \end{enumerate} Moreover, if $f \in H^1(\partialrtial E)$, then for every $2\leq p< +\infty$ there exists a constant $C=C(F,M,\alpha,p)>0$, such that \begin{equation*} \| f\|_{L^p(\partialrtial E)} \leq C \|f\|_{H^1(\partialrtial E)}^{{(p-1)}/{p}}\, \|u \|_{L^2(\partialrtial E)}^{1/p}\,. \end{equation*} \end{lem} \begin{proof} We are not going to underline it every time, but it is easy to check that all the constants that will appear in the proof will depend with only on $F$, $M$, $\alpha$ and sometimes $p$, recalling the previous discussion about the ``uniform'' inequalities holding for the families of sets $\mathfrak{C}^{1,\alpha}_M(F)$. {(i)} Recalling Remark~\ref{rm:potential}, we have \begin{equation*} u(x) = \int_{\partialrtial E} G(x,y)f(y) \, d\mu(y). \end{equation*} It is well known that it is always possible to write $G(x,y) = h(x-y) + r(x-y)$ where $h:\mathbb{R}\to\mathbb{R}$ is smooth away from $0$, one--periodic and $h(t)=\frac{1}{4\pi|t|}$ in a neighborhood of $0$, while $r:\mathbb{R}\to\mathbb{R}$ is smooth and one--periodic. The conclusion then follows since for $v(x)= \int_{\partialrtial E} \frac{f(y)}{|x-y|} \, d\mu(y)$ there holds \begin{equation*} \|v\|_{L^p(\partialrtial E)} \leq C \|f\|_{L^p(\partialrtial E)}\,, \end{equation*} with $C=C(F,M,\alpha,p)>0$. {(ii)} We are going to adapt the proof of~\cite{JK} to the periodic setting. First observe that since $u$ is harmonic in $E\subseteq\mathbb{T}^3$ we have \begin{equation} \label{harmonic lemma 12} \operatorname*{div}\nolimits\left( 2\langle \nabla u\vert x\rangle \nabla u -|\nabla u|^2x + u \nabla u \right) = 0. \end{equation} Moreover, there exist constants $r>0$, $C_0$ and $N\in\mathbb{N}$, depending only on $F$, $M$, $\alpha$, such that we may cover $\partialrtial E$ with $N$ balls $B_r(x_k)$, with every $x_k\in F$ and \begin{equation} \label{harmonic lemma 1} \frac{1}{C_0} \leq \langle x \vert \nu_E(x)\rangle \leq C_0 \qquad\text{ for $x\in \partial E \cap B_{2r}(x_k)$}\,. \end{equation} for every that $E\in\mathfrak{C}^{1,\alpha}_M(F)$.\\ If then $0 \leq \varphi_k \leq 1$ is a smooth function with compact support in $B_{2r}(x_k)$ such that $\varphi_k \equiv 1$ in $B_r(x_k)$ and $|\nabla \varphi_k| \leq C/r$, by integrating the function \begin{equation*} \operatorname*{div}\nolimits\left( \varphi_k \left(2\langle \nabla u\vert x\rangle \nabla u -|\nabla u|^2x + u \nabla u \right) \right) \end{equation*} in $E$ and using equality~\eqref{harmonic lemma 12}, we get \begin{align} \int_E \bigl\langle \nabla \varphi_k \vert \, 2 &\left \langle \nabla u\vert x\right \rangle \nabla u -|\nabla u|^2x + u \nabla u\bigr\rangle \,dx\\ &=\int_E \operatorname*{div}\nolimits\bigl(\varphi_k (2 \left \langle \nabla u\vert x\right \rangle \nabla u -|\nabla u|^2x + u \nabla u)\bigr) \,dx \\ &= \int_{\partial E} \bigl(2\varphi_k \langle \nabla^{\mathbb{T}^3}\!\!u \vert x \rangle\partial_{\nu_E}u - \varphi_k |\nabla^{\mathbb{T}^3}\!\! u |^2 \langle x \vert \nu_E \rangle + \varphi_ku\partial_{\nu_E} u\bigr) \,d\mu, \end{align} hence, \begin{align} \int_E \bigl\langle \nabla \varphi_k \vert 2& \bigl\langle \nabla^{\mathbb{T}^3}\!\! u\vert x\bigr\rangle \nabla u -|\nabla u|^2x + u \nabla u \bigr\rangle \,dx - \int_{\partial E} \varphi_ku\partial_{\nu_E} u^- \,d \mu- 2 \int_{\partial E} \varphi_k \langle \nabla u\vert x \rangle \partial_{\nu_E} u^- \,d \mu \\ =&\,-\int_{\partial E} \varphi_k |\nabla^{\mathbb{T}^3}\!\! u^- |^2 \langle x \vert \nu_E \rangle\,d\mu +2 \int_{\partial E} \varphi_k |\partial_{\nu_E}u^-|^2\langle x|\nu_E\rangle \,d \mu \\ =&\,\int_{\partial E} \varphi_k |\partial_{\nu_E} u^-|^2 \langle x \vert \nu_E \rangle\,d \mu - \int_{\partial E} \varphi_k |\nabla u|^2\langle x |\nu_E\rangle\,d \mu\,. \end{align} Using the Poincar\'e inequality on the torus $\mathbb{T}^3$ (recall that $u$ has zero integral) and estimate~\eqref{harmonic lemma 1}, this inequality implies \begin{align} \int_{\partialrtial E \cap B_r(x_k)} |\partial_{\nu_E} u|^2\, d \mu &\leq C \int_{\partialrtial E} (u^2 + | \nabla u|^2) \, d \mu + C\int_{\mathbb{T}^3} (u^2 +|\nabla u|^2) \, dx\\ &\leq C \int_{\partialrtial E} (u^2 + | \nabla u|^2) \, d \mu + C\int_{\mathbb{T}^3} |\nabla u|^2 \, dx\,. \end{align} Putting together all the above estimates and repeating the argument on $E^c$, we get \begin{equation*} \int_{\partialrtial E } (|\partial_{\nu_E} u^-|^2 + |\partial_{\nu_E} u^+|^2)\, d \mu \leq C \int_{\partialrtial E} (u^2 + | \nabla u|^2) \, d \mu + C\int_{\mathbb{T}^3} |\nabla u|^2 \, dx\,. \end{equation*} The thesis then follows by observing that \begin{equation*} \int_{\mathbb{T}^3} |\nabla u|^2 \, dx = \int_{\partialrtial E} u (\partial_{\nu_E} u^- - \partial_{\nu_E} u^+) \, d \mu\,. \end{equation*} {(iii)} Let us define \begin{equation*} Kf(x) = \int_{\partialrtial E}\bigl \langle\nabla^{\mathbb{T}^3}_x\! G(x,y)| \nu_E(x)\bigr\rangle f(y) \, d\mu(y)\,. \end{equation*} We want to show that \begin{equation} \label{flow lemma 1} \|Kf\|_{L^p(\partialrtial E)} \leq C \|f\|_{L^p(\partialrtial E)}. \end{equation} By the decomposition recalled at the point (i), we have $\nabla^{\mathbb{T}^3}_x\!G(x,y)=\nabla^{\mathbb{T}^3}_x\![h(x-y)] + \nabla^{\mathbb{T}^3}_x\![r(x-y)]$, where $\nabla^{\mathbb{T}^3}_x\![h(x-y)]=-\frac{1}{4\pi}\frac{x-y}{|x-y|^3}$, for $|x-y|$ small enough and $\nabla^{\mathbb{T}^3}_x\![r(x-y)]$ is smooth. Thus, by a standard partition of unity argument we may localize the estimate and reduce to show that if $\varphi \in C^{1,\alpha}_c(\mathbb{R}^{2})$ and $U \subseteq \mathbb{R}^{2}$ is a bounded domain setting $\Gamma = \{(x', \varphi(x')) \, : \, x' \in U\}\subseteq\mathbb{R}^3$ and $$ Tf(x) = \int_\Gamma \frac{ \langle x-y\vert \nu_E(x)\rangle }{|x-y|^{3}} f(y) \, d\mu(y) $$ for every $x\in \Gamma$, where $\nu_E$ is the ``upper'' normal to the graph $\Gamma$, then $Tf(x)$ is well defined at every $x \in \Gamma$ and \begin{equation*} \|Tf\|_{L^p(\Gamma)} \leq C \|f\|_{L^p(\Gamma)}\,. \end{equation*} In order to show this we observe that we may write \begin{equation*} Tf(x) = \int_U \frac{ \varphi(x') - \varphi(y') - \left\langle \nabla\varphi(x')\vert x'-y'\right\rangle }{(|x'-y'|^2+ [\varphi(x') -\varphi(y')]^2 )^{3/2}} f(y', \varphi(y')) \, dy'. \end{equation*} where we used the fact that $$ \Gamma=\{(x',y') \, : \, y'-\varphi(x')=F(x',y')=0\} $$ and then that $$ \nu_E= \frac{\nabla F}{|\nabla F|}=\frac{(-\nabla \varphi(x'),1)}{\sqrt{1+|\nabla \varphi(x')|^2}}. $$ Therefore, $$ |Tf(x)|\leq C \int_U \frac{ |x'-y'|^{1+ \alpha}}{(|x'-y'|^2+ [\varphi(x') -\varphi(y')]^2 )^{3/2}}|f(y', \varphi(y'))| \, dy' \leq C \int_U \frac{ |f(y', \varphi(y'))|}{|x'-y'|^{2-\alpha}} \, dy'. $$ Thus, inequality~\eqref{flow lemma 1} follows from a standard convolution estimate.\\ For $x \in E$ we have \begin{equation*} \nabla u(x) = \int_{\partialrtial E} \nabla^{\mathbb{T}^3}_x\!G(x,y)f(y) \, d\mu(y), \end{equation*} hence, for $x \in \partialrtial E$ there holds \begin{equation*} \langle \nabla u(x - t\nu_E(x)) \vert \, \nu_E(x)\rangle = \int_{\partialrtial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x -t \nu_E(x),y) \vert \, \nu_E(x)\bigr\rangle f(y) \, d\mu(y). \end{equation*} We claim that \begin{equation}\label{flow lemma 2} {\partial_{\nu_E} u^-}(x)=\lim_{t \to 0+} \left\langle\nabla u(x - t\nu_E(x)) \vert \, \nu_E(x)\right\rangle = Kf(x) +\frac{1}{2}f(x), \end{equation} for every $x \in \partialrtial E$, then the result follows from inequality~\eqref{flow lemma 1} and this limit, together with the analogous identity for $\partial_{\nu_E} u^+(x)$.\\ To show equality~\eqref{flow lemma 2} we first observe that \begin{align} &\int_{\partialrtial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x,y) \vert \,\nu_E(y)\bigr\rangle \, d\mu(y) = 1- \mathrm{Vol}(E) \quad\quad\, \text{if $x\in E\setminus\partial E$} \label{flow lemma divergence0}\\ & \int_{\partialrtial E} \bigl\langle \nabla^{\mathbb{T}^3}_x\!G(x ,y) \vert \, \nu_E(y)\bigr\rangle \, d\mu(y) = {1}/{2}- \mathrm{Vol}(E)\quad \text{if $x\in \partial E$}. \label{flow lemma divergence} \end{align} Indeed, using Definition~\eqref{potential}, we have \begin{align} \Delta v_E(x) =&\,\int_E \Delta_x G(x,y) \, dy - \int_{E^c} \Delta_x G(x,y) \, dy \\ =&\,-2 \int_{\partial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x,y)\vert \, \nu_E(y) \bigr\rangle \, d \mu(y)\\ =&\, 2\mathrm{Vol}(E) - 1-u_E(x), \end{align} then, \begin{equation} \int_{\partial E} \bigl \langle\nabla^{\mathbb{T}^3}_x\! G(x,y) \vert \, \nu_E(y) \bigr\rangle\,d\mu(y) = 1/2-\mathrm{Vol}(E) +u_E(x)/2, \end{equation} which clearly implies equation~\eqref{flow lemma divergence0}. Equality~\eqref{flow lemma divergence} instead follows by an approximation argument, after decomposing the Green function as at the beginning of the proof of point (i), $G(x,y) = h(x-y) + r(x-y)$, with $h(t)=\frac{1}{4\pi|t|}$ in a neighborhood of $0$ and $r:\mathbb{R}\to\mathbb{R}$ a smooth function.\\ Therefore, we may write, for $x\in\partial E$ and $t>0$ (remind that $\nu_E$ is the {\em outer} unit normal vector, hence $x - t\nu_E(x)\in E$), \begin{align} \langle\nabla u(x - t\nu_E(x)) \vert \nu_E(x)\rangle =&\,\int_{\partialrtial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x - t\nu_E(x),y) \vert \nu_E(x)\bigr\rangle (f(y)-f(x)) \, d\mu(y) \\ &\,+ f(x) \int_{\partialrtial E}\bigl\langle \nabla_x^{\mathbb{T}^3}\!G(x - t\nu_E(x),y) \vert \nu_E(x)-\nu_E(y)\bigr\rangle \, d\mu(y) \\ & \,+ f(x)(1-\mathrm{Vol}(E))\,,\label{flow lema long} \end{align} by equality~\eqref{flow lemma divergence0}.\\ Let us now prove that \begin{align*} \lim_{t \to 0^+} \int_{\partialrtial E} &\,\bigl\langle\nabla_x^{\mathbb{T}^3}\!G(x- t\nu_E(x),y) \,\vert\, \nu_E(x)\bigr\rangle (f(y)-f(x)) \, d\mu(y)\\ &= \int_{\partialrtial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x,y)\, \vert\, \nu_E(x)\bigr\rangle (f(y)-f(x)) \, d\mu(y), \end{align*} observing that since $\partial E$ is of class $C^{1,\alpha}$ then for $|t|$ sufficiently small we have \begin{equation}\label{vesa1} |x-y-t\nu_E(x)|\geq\frac12|x-y|\qquad\text{for all $y\in\partial E$}\,. \end{equation} Then, in view of the decomposition of $\nabla_xG$ above, it is enough show that \begin{equation*} \lim_{t \to 0^+} \int_{\partialrtial E}\frac{\langle x -y - t\nu_E(x) \, \vert \, \nu_E(x)\rangle}{|x-y-t\nu_E(x)|^3} (f(y)-f(x)) \, d\mu(y)= \int_{\partialrtial E} \frac{\langle x -y \, \vert \, \nu_E(x)\rangle}{|x-y|^3} (f(y)-f(x)) \, d\mu(y)\,, \end{equation*} which follows from the dominated convergence theorem, after observing that due to the $\alpha$--H\"older continuity of $f$ and to inequality~\eqref{vesa1}, the absolute value of both integrands can be estimated from above by $C/|x-y|^{2-\alpha}$ for some constant $C>0$.\\ Arguing analogously, we also get $$ \lim_{t \to 0^+} \int_{\partialrtial E}\bigl\langle \nabla_x^{\mathbb{T}^3}\!G(x - t\nu_E(x),y) \vert \nu_E(x)-\nu_E(y)\bigr\rangle \, d\mu(y)=\int_{\partialrtial E}\bigl\langle \nabla_x^{\mathbb{T}^3}\!G(x,y) \vert \nu_E(x)-\nu_E(y)\bigr\rangle \, d\mu(y)\,. $$ Then, letting $t \to 0^+$ in equality~\eqref{flow lema long}, for every $x\in\partial E$, we obtain \begin{align*} \lim_{t\to 0^+}\langle\nabla u(x - t\nu_E(x)) \vert \nu_E(x)\rangle =&\,\int_{\partial E} \bigl \langle \nabla^{\mathbb{T}^3}_x\!G(x,y)\vert \nu_E(x) \bigr\rangle (f(y)-f(x)) \, d \mu(y)\\ &\,+ f(x) \int_{\partial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x,y)|\nu_E(x)-\nu_E(y)\bigr\rangle\,d \mu(y)+ f(x)(1-\mathrm{Vol}(E))\\ =&\,\int_{\partial E} \bigl \langle \nabla^{\mathbb{T}^3}_x\!G(x,y)\vert \nu_E(x) \bigr\rangle f(y)\, d \mu(y)\\ &\,-f(x) \int_{\partial E} \bigl\langle\nabla^{\mathbb{T}^3}_x\!G(x,y)|\nu_E(y)\bigr\rangle\,d \mu(y)+ f(x)(1-\mathrm{Vol}(E))\\ =&\,Kf(x)+ f(x)(\mathrm{Vol}(E)-1/2) + f(x)(1-\mathrm{Vol}(E))\\ =&\,Kf(x)+\frac{1}{2}f(x), \end{align*} where we used equality~\eqref{flow lemma divergence}, then limit~\eqref{flow lemma 2} holds and the thesis follows. {(iv)} Fixed $p>2$ and $\beta\in (0, \frac{p-2}{p})$, as before, due to the properties of the Green's function, it is sufficient to establish the statement for the function $$ v(x) =\int_{\partial E}\frac{f(y)}{|x-y|}\, d\mu(y)\,. $$ For $x_1$, $x_2\in \partial E$ we have $$ |v(x_1)-v(x_2)|\leq \int_{\partial E}|f(y)|\frac{\big||x_1-y|-|x_2-y|\big|}{|x_1-y|\, |x_2-y|}\, d\mu(y)\,. $$ In turn, by an elementary inequality, we have $$ \frac{\big||x_1-y|-|x_2-y|\big|}{|x_1-y|\, |x_2-y|}\leq C(\beta)\frac{\big||x_1-y|^{1-\beta}+|x_2-y|^{1-\beta}\big|}{|x_1-y|\, |x_2-y|}|x_1-x_2|^\beta\,, $$ thus, by H\"older inequality we have \begin{align*} |v(x_1)-v(x_2)| &\leq C(\beta) \int_{\partial E}|f(y)|\frac{\big||x_1-y|^{1-\beta}+|x_2-y|^{1-\beta}\big|}{|x_1-y|\, |x_2-y|}\, d\mu(y)\,\, |x_1-x_2|^\beta \\ &\leq C'(\beta)\|f\|_{L^p} |x_1-x_2|^\beta\,, \end{align*} where we set $$ C'(\beta)=2C(\beta)\biggl(\sup_{z_1,\, z_2\in\partial E} \int_{\partial E}\frac{1}{|z_1-y|^{\beta p'}\, |z_2-y|^{p'}}\, d\mu(y)\biggr)^{1/{p'}}\,, $$ with $p'=p/(p-1)$. \noindent For the second part of the lemma, we start by observing that $$ \norma{f}_{L^2(\partial E)}\leq C\norma{f}_{H^1(\partial E)}^{1/{2}}\norma{f}^{1/{2}}_{H^{-1}(\partial E)}. $$ If $p>2$ we have, by Gagliardo--Nirenberg interpolation inequalities (see~\cite[Theorem 3.70]{Aubin}), \begin{equation*} \| f\|_{L^p(\partialrtial E)} \leq C \|f\|_{H^1(\partialrtial E)}^{^{{(p-2)}/{p}}} \|f\|_{L^2(\partialrtial E)}^{2/p}. \end{equation*} Therefore, by combining the two previous inequalities we get that, for $p\geq2$, there holds \begin{equation*} \| f\|_{L^p(\partialrtial E)} \leq C \|f\|_{H^1(\partialrtial E)}^{^{(p-1)/{p}}} \|f\|_{H^{-1}(\partialrtial E)}^{1/p}. \end{equation*} Hence, the thesis follows once we show \begin{equation*} \|f \|_{H^{-1}(\partialrtial E)} \leq C \|u\|_{L^2(\partialrtial E)}. \end{equation*} To this aim, let us fix $\varphi \in H^1(\partialrtial E)$ and with a little abuse of notation denote its harmonic extension to $\mathbb{T}^3$ still by $\varphi$. Then, by integrating by parts twice and by point (ii), we get \begin{align} \int_{\partialrtial E} \varphi f \, d \mu=&\, - \int_{\partial E} \varphi \Delta u \, d \mu\\ =&\, - \int_{\partialrtial E} u [\partial_{\nu_E} \varphi] \, d \mu \\ \leq&\, \|u \|_{L^2(\partialrtial E)} \bigl\|[\partial_{\nu_E} \varphi]\bigr \|_{L^2(\partialrtial E)} \\ \leq&\, \|u \|_{L^2(\partialrtial E)} \bigl( \|\partial_{\nu_E} \varphi^+ \|_{L^2(\partialrtial E)} + \|\partial_{\nu_E} \varphi^- \|_{L^2(\partialrtial E)} \bigr) \\ \leq&\, C \|u\|_{L^2(\partialrtial E)} \|\varphi \|_{H^1(\partialrtial E)}. \end{align} Therefore, \begin{equation*} \| f\|_{H^{-1}(\partialrtial E)} = \sup_{\|\varphi\|_{H^1(\partialrtial E)}\leq 1} \int_{\partialrtial E} \varphi f \, d \mu \leq C \|u \|_{L^2(\partialrtial E)} \end{equation*} and we are done. \end{proof} For any smooth set $E\subseteq\mathbb{T}^3$, the fractional Sobolev space $W^{s,p}(\partial E)$, usually obtained via local charts and partitions of unity, has an equivalent definition considering directly the Gagliardo $W^{s,p}$--seminorm of a function $f\in L^p(\partial E)$, for $s\in(0,1)$, as follows $$ [f]_{W^{s,p}(\partial E)}^p=\int_{\partial E}\int_{\partial E}\frac{|f(x)-f(y)|^p}{|x-y|^{2+sp}}\,d\mu(x)d\mu(y) $$ and setting $\Vert f\Vert_{W^{s,p}(\partial E)}=\Vert f\Vert_{L^p(\partial E)}+[f]_{W^{s,p}(\partial E)}$ (we refer to~\cite{AdamsFournier,Dem,NePaVa,RuSi} for details). As it is customary, we set $[f]_{H^s(\partial E)}=[f]_{W^{s,2}(\partial E)}$ and $H^s(\partial E)=W^{s,2}(\partial E)$. Then, it can be shown that for all the sets $E\in\mathfrak{C}^{1,\alpha}_M(F)$, given a smooth set $F\subseteq\mathbb{T}^3$ and a tubular neighborhood $N_\varepsilon$ of $\partial F$ as in formula~\eqref{tubdef}, for any $M\in(0,\varepsilon/2)$ and $\alpha\in(0,1)$, the constants giving the equivalence between this norm above and the ``standard'' norm of $W^{s,p}(\partial E)$ can be chosen to be uniform, independent of $E$. Moreover, as for the ``usual'' (with integer order) Sobolev spaces, all the constants in the embeddings of the fractional Sobolev spaces are also uniform for this family. This is related to the possibility, due to the closeness in $C^{1,\alpha}$ and the graph representation, of ``localizing'' and using partitions of unity ``in a single common way'' for all the smooth hypersurfaces $\partial E$ boundaries of the sets $E\in\mathfrak{C}^{1,\alpha}_M(F)$, see~\cite{DDM} for details. Then, we have the following technical lemma. \begin{lem}\label{nicola1} Let $F\subseteq\mathbb{T}^3$ be a smooth set and $E\in\mathfrak{C}^{1,\alpha}_M(F)$. For every $\beta\in[0,1/2)$, there exists a constant $C=C(F,M,\alpha,\beta)$ such that if $f\in H^{1/2}(\partial E)$ and $g\in W^{1,4}(\partial E)$, then $$ [fg]_{H^{{1}/{2}}(\partial E)}\leq C[f]_{H^{{1}/{2}}(\partial E)}\|g\|_{L^\infty(\partial E)}+C\|f\|_{L^{\frac{4}{1+\beta}}(\partial E)}\|g\|_{L^\infty(\partial E)}^\beta\|\nabla g\|_{L^4(\partial E)}^{1-\beta}\,. $$ \end{lem} \begin{proof} We estimate with H\"older inequality, noticing that $6\beta/(1+\beta)<2$, as $\beta\in[0,1/2)$, hence there exists $\deltalta>0$ such that $(6\beta+\deltalta)/(1+\beta)<2$, \begin{align*} [fg]_{H^{{1}/{2}}(\partial E)}^2 \leq&\,2[f]_{H^{{1}/{2}}(\partial E)}^2\|g\|_{L^\infty(\partial E)}^2+2\int_{\partial E}\int_{\partial E}|f(y)|^2\frac{|g(x)-g(y)|^2}{|x-y|^3}\,d\mu(x)d\mu(y)\\ \leq &\,2[f]_{H^{{1}/{2}}(\partial E)}^2\|g\|_{L^\infty(\partial E)}^2\\ &\,+C\int_{\partial E}\int_{\partial E}\frac{|f(y)|^2}{|x-y|^{3\beta+\deltalta/2}}\frac{|g(x)-g(y)|^{2(1-\beta)}}{|x-y|^{3(1-\beta)-\deltalta/2}}\Vert g\Vert_{L^\infty(\partial E)}^{2\beta}\,d\mu(x)d\mu(y)\\ \leq &\,2[f]_{H^{{1}/{2}}(\partial E)}^2\|g\|_{L^\infty(\partial E)}^2\\ &\,+C\Bigl(\int_{\partial E}{|f(y)|^{\frac{4}{1+\beta}}}\int_{\partial E}\frac{1}{|x-y|^{\frac{6\beta+\deltalta}{1+\beta}}}\,d\mu(x)d\mu(y)\Bigr)^{(1+\beta)/2}\Vert g\Vert_{L^\infty(\partial E)}^{2\beta}\\ &\,\phantom{+C\quad}\cdot\Bigl(\int_{\partial E}\int_{\partial E}\frac{|g(x)-g(y)|^4}{|x-y|^{6-\frac{\deltalta}{1-\beta}}}\,d\mu(x)d\mu(y)\Bigr)^{(1-\beta)/2}\\ \leq &\,2[f]_{H^{{1}/{2}}(\partial E)}^2\|g\|_{L^\infty(\partial E)}^2\\ &\,+C\Bigl(\int_{\partial E}{|f(y)|^{\frac{4}{1+\beta}}}\,d\mu(y)\Bigr)^{(1+\beta)/2}\Vert g\Vert_{L^\infty(\partial E)}^{2\beta}\, [g]_{W^{1-\frac{\deltalta}{4(1-\beta)},4}(\partial E)}^{2(1-\beta)}\\ \leq &\,2[f]_{H^{{1}/{2}}(\partial E)}^2\|g\|_{L^\infty(\partial E)}^2+C\|f\|_{L^{\frac{4}{1+\beta}}(\partial E)}^2\|g\|_{L^\infty(\partial E)}^{2\beta}\|\nabla g\|_{L^4(\partial E)}^{2(1-\beta)}\,. \end{align*} Hence the thesis follows noticing that all the constants $C$ above depend only on $F$, $M$, $\alpha$ and $\beta$, by the previous discussion, before the lemma. \end{proof} As a corollary we have the following estimate. \begin{lem}\label{5.2} Let $F\subseteq\mathbb{T}^3$ be a smooth set and $E\in\mathfrak{C}^{1,\alpha}_M(F)$. Then, for $M$ small enough, there holds $$ \|\psi_E\|_{W^{5/2,2}(\partial F)}\leq C(F,M,\alpha)\big(1+\|\mathrm{H}\|_{H^{1/2}(\partial E)}^2\big)\,, $$ where $\mathrm{H}$ is the mean curvature of $\partial E$ and the function $\psi_E$ is defined by formula~\eqref{front}. \end{lem} \begin{proof} By a standard localization/partition of unity/straightening argument, we may reduce ourselves to the case where the function $\psi_E$ is defined in a disk $D\subseteq\mathbb{R}^2$ and $\|\psi_E\|_{C^{1,\alpha}(D)}\leq M$. Fixed a smooth cut--off function $\varphi$ with compact support in $D$ and equal to one on a smaller disk $D'\subseteq D$, we have \begin{equation}\label{noia1} \Delta(\varphi\psi_E)-\frac{\nabla^2(\varphi\psi_E)\nabla\psi_E \nabla\psi_E}{1+|\nabla\psi_E|^2}=\varphi\mathrm{H}\sqrt{1+|\nabla\psi_E|^2}+R(x,\psi_E,\nabla\psi_E)\,, \end{equation} where the remainder term $R(x, \psi_E, \nabla \psi_E)$ is a smooth Lipschitz function. Then, using Lemma~\ref{nicola1} with $\beta=0$ and recalling that $\|\psi_E\|_{C^{1,\alpha}(D)}\leq M$, we estimate \begin{align} [\Delta(\varphi\psi_E)]_{H^{{1}/{2}}(D)}\leq C(F,M,\alpha)\Bigl(&\,M^2[\nabla^2(\varphi\psi_E)]_{H^{{1}/{2}}(D)}+[\mathrm{H}]_{H^{{1}/{2}}(\partial E)}(1+\norma{\nabla\psi_E}_{L^\infty(D)})\\ &\,+\norma{\mathrm{H}}_{L^4(\partial E)}(1+\norma{\psi_E}_{W^{2,4}(D)})+1+\norma{\psi_E}_{W^{2,4}(D)}\Bigr)\,. \end{align} We now use the fact that, by a simple integration by part argument, if $u$ is a smooth function with compact support in $\mathbb{R}^2$, there holds $$ [\Delta u]_{H^{{1}/{2}}(\mathbb{R}^2)}=[\nabla^2u]_{H^{{1}/{2}}(\mathbb{R}^2)}\,, $$ hence, \begin{align} [\nabla^2(\varphi \psi_E)]_{H^{{1}/{2}}(D)}&\,=[\Delta (\varphi\psi_E)]_{H^{{1}/{2}}(D)}\\ &\,\leq C(F,M,\alpha)\Big(M^2[\nabla^2(\varphi\psi_E)]_{H^{{1}/{2}}(D)} +[\mathrm{H}]_{H^{{1}/{2}}(\partial E)}(1+\norma{\nabla\psi_E}_{L^\infty(D)})\\ &\,\phantom{\leq C(F,M,\alpha)\bigl(\,}+\norma{\mathrm{H}}_{L^4(\partial E)}(1+\norma{\psi_E}_{W^{2,4}(D)})+1+\norma{\psi_E}_{W^{2,4}(D)}\Big), \end{align} then, if $M$ is small enough, we have \begin{equation} \label{aub} [\nabla^2(\varphi\psi_E)]_{H^{{1}/{2}}(D)}\leq C(F,M,\alpha)(1+ \norma{\mathrm{H}}_{H^{1/2}(\partial E)} )(1+\norma{\mathrm{Hess}\,\psi_E}_{L^4(D)}), \end{equation} as \begin{equation}\label{aub2} \norma{\mathrm{H}}_{L^4(\partial E)} \leq C(F,M,\alpha) \norma{\mathrm{H}}_{H^{1/2}(\partial E)}, \end{equation} where we used the continuous embedding of $H^{1/2}(\partial E)$ in $L^4(\partial E)$ (see for instance Theorem~$6.7$ in~\cite{NePaVa}, with $q=4$, $s=1/2$ and $p=2$).\\ By the Calder\'on--Zygmund estimates (holding uniformly for every hypersurface $\partial E$, with $E\in\mathfrak{C}^{1,\alpha}_M(F)$, see~\cite{DDM}), \begin{equation}\label{Cald-Zyg} \norma{\mathrm{Hess} \, \psi_E}_{L^{4}(D)} \leq C(F,M,\alpha)(\norma{\psi_E}_{L^4(D)}+ \norma{\Delta \psi_E}_{L^4(D)}) \end{equation} and the expression of the mean curvature \begin{equation}\label{exprmeancurv} \mathrm{H}= \frac{\Delta \psi_E}{\sqrt{1+ |\nabla \psi_E|^2}} - \frac{\mathrm{Hess} \, \psi_E (\nabla \psi_E \nabla \psi_E)}{(\sqrt {1 + |\nabla \psi_E|^2})^3} \, , \end{equation} we obtain \begin{align} \norma{\Delta \psi_E}_{L^4(D)} &\leq 2M \norma{\mathrm{H}}_{L^4(\partial E)} + M^2 \norma{\mathrm{Hess}\, \psi_E}_{L^4(D)} \\ &\leq 2M \norma{\mathrm{H}}_{L^4(\partial E)} + C(F,M,\alpha)M^2(\norma{\psi_E}_{L^4(D)} + \norma{\Delta \psi_E}_{L^4(D)}) \, . \label{normadeltapsi1} \end{align} Hence, possibly choosing a smaller $M$, we conclude \begin{equation} \norma{\Delta \psi_E}_{L^4(D)} \leq C(F,M,\alpha) (1 + \norma{\mathrm{H}}_{L^4(\partial E)})\leq C(F,M,\alpha)(1+ \norma{\mathrm{H}}_{H^{1/2}(\partial E)}), \label{normadeltapsi2} \end{equation} again by inequality~\eqref{aub2}.\\ Thus, by estimate~\eqref{Cald-Zyg}, we get \begin{equation}\label{eqcar10030} \norma{\mathrm{Hess}\,\psi_E}_{L^4(D)} \leq C(F,M,\alpha)(1+ \norma{\mathrm{H}}_{H^\frac12(\partial E)}), \end{equation} and using this inequality in estimate~\eqref{aub}, \begin{equation} [\nabla^2(\varphi\psi_E)]_{H^{{1}/{2}}(D)}\leq C(F,M,\alpha)(1+ \norma{\mathrm{H}}_{H^\frac12(\partial E)})^2, \end{equation} hence, \begin{equation} [\nabla^2 \psi_E]_{H^{1/2}(D')}\leq C(F,M,\alpha)(1+ \norma{\mathrm{H}}_{H^\frac12(\partial E)})^2 \leq C(F,M,\alpha)(1+ \norma{\mathrm{H}}_{H^\frac12(\partial E)}^2). \end{equation} The inequality in the statement of the lemma then easily follows by this inequality, estimate~\eqref{eqcar10030} and $\norma{\psi_E}_{C^{1,\alpha}(D)} \leq M$, with a standard covering argument. \end{proof} We are now ready to prove the last lemma of this section. \begin{lem}[Compactness]\label{w52conv} Let $F\subseteq\mathbb{T}^3$ be a smooth set and $E_n\subseteq \mathfrak{C}^{1,\alpha}_M(F) $ a sequence of smooth sets such that $$ \sup_{n\in\mathbb{N}}\int_{\mathbb{T}^3}|\nabla w_{E_n}|^2\, dx<+\infty\,, $$ where $w_{E_n}$ are the functions associated to $E_n$ by problem~\eqref{WE}.\\ Then, if $\alpha\in(0,1/2)$ and $M$ is small enough, there exists a smooth set $F'\in \mathfrak{C}^1_M(F)$ such that, up to a (non relabeled) subsequence, $E_n\to F'$ in $W^{2,p}$ for all $1\leq p<4$ (recall the definition of convergence of sets at the beginning of Subsection~\ref{stabsec}).\\ Moreover, if $$ \int_{\mathbb{T}^3}|\nabla w_{E_n}|^2\, dx\to 0\,, $$ then $F'$ is critical for the volume--constrained nonlocal Area functional $J$ and the convergence $E_n\to F'$ is in $W^{5/2,2}$. \end{lem} \begin{proof} Throughout all the proof we write $w_n$, $\mathrm{H}_n$, and $v_n$ instead of $w_{E_n}$, $\mathrm{H}_{\partial E_n}$, and $v_{E_n}$, respectively. Moreover, we denote by $\widehat w_n = \fint_{\mathbb{T}^3}w_n\,dx$ and we set $\widetilde w_n=\fint_{\partial E_n}w_n\,d\mu_n$ and $\widetilde{\mathrm{H}}_n=\fint_{\partial E_n}\mathrm{H}_n\,d\mu_n$. First, we recall that \begin{equation}\label{w521} w_n=\mathrm{H}_n+4\gamma v_n \quad\text{on }\partial E_n\qquad\text{and}\qquad \sup_{n\in\mathbb{N}}\,\|v_n\|_{C^{1,\alpha}(\mathbb{T}^3)}<+\infty\,, \end{equation} by standard elliptic estimates. We want to show that \begin{equation}\label{eqcar15000} \norma{w_n-\widetilde w_n}^2_{H^{1/2}(\partial E_n)}\leq\|w_n-\widehat w_n\|^2_{H^{1/2}(\partial E_n)}. \end{equation} To this aim, we recall that for every constant $a$ \begin{equation} \norma{w_n - a}_{L^2(\partial E_n)}^2 = \norma{w_n}^2_{L^2(\partial E_n)} + a^2 \mathcal A(\partial E_n) - 2 a \int_{\partial E_n} w_n \, d\mu_n \end{equation} then, \begin{equation} \frac{d}{da} \norma{w_n -a}_{L^2(\partial E_n)}^2= 2a \mathcal A(\partial E_n) - 2 \int_{\partial E_n} w_n \, d\mu_n . \end{equation} The above equality vanishes if and only if $a= \fint _{\partial E_n} w_n \, d\mu_n$, hence, $$ \norma{w_n -\widetilde w_n}_{L^2(\partial E_n)} = \min_{a\in\mathbb{R}}\norma{w_n -a}_{L^2(\partial E_n)} $$ and inequality~\eqref{eqcar15000} follows by the definition of $\Vert\cdot\Vert_{H^{1/2}(\partial E_n)}$ and the observation on the Gagliardo seminorms just before Lemma~\ref{nicola1}.\\ Then, from the {\em trace inequality} (see~\cite{Ev}), which holds with a ``uniform'' constant $C=C(F,M,\alpha)$, for all the sets $E\in\mathfrak{C}^{1,\alpha}_M(F)$ (see~\cite{DDM}), we obtain \begin{equation}\label{w522} \|w_n-\widetilde w_n\|^2_{H^{1/2}(\partial E_n)}\leq\|w_n-\widehat w_n\|^2_{H^{1/2}(\partial E_n)}\leq C\int_{\mathbb{T}^3}|\nabla w_n|^2\, dx<C<+\infty \end{equation} with a constant $C$ independent of $n\in\mathbb{N}$.\\ We claim now that \begin{equation}\label{claim1111} \sup_{n\in\mathbb{N}}\,\|\mathrm{H}_n\|_{H^{1/2}(\partial E_n)}<+\infty. \end{equation} To see this note that by the uniform $C^{1,\alpha}$--bounds on $\partial E_n$, we may find a fixed solid cylinder of the form $C=D\times(-L,L)$, with $D\subseteq\mathbb{R}^{2}$ a ball centered at the origin and functions $f_n$, with \begin{equation}\label{w523} \sup_{n\in\mathbb{N}}\|f_n\|_{C^{1,\alpha}(\overline D)}<+\infty\,, \end{equation} such that $\partial E_n\cap C=\{(x',x_n)\in D\times(-L,L):\, x_n= f_n(x')\}$ with respect to a suitable coordinate frame (depending on $n\in\mathbb{N}$). Then, \begin{equation} \int_{D}(\mathrm{H}_n-\widetilde{\mathrm{H}}_n)\, dx'+ \widetilde{\mathrm{H}}_n\,{\mathrm{Area}}(D)= \int_{D}\operatorname*{div}\nolimits\Bigl(\frac{\nabla_{x'} f_n}{\sqrt{1+|\nabla_{x'} f_n|^2}}\Bigr)\, dx'=\int_{\partial D}\frac{\nabla_{x'} f_n}{\sqrt{1+|\nabla_{x'} f_n|^2}}\cdot \frac{x'}{|x'|}\, d\sigma\,, \label{intHn} \end{equation} where $\sigma$ is the canonical (standard) measure on the circle $\partial D$.\\ Hence, recalling the uniform bound~\eqref{w523} and the fact that $\|\mathrm{H}_n- \widetilde{\mathrm{H}}_n\|_{H^{1/2}(\partial E_n)}$ are equibounded thanks to inequalities~\eqref{w521} and~\eqref{w522}, we get that ${\widetilde{\mathrm{H}}_n}$ are also equibounded (by a standard ``localization'' argument, ``uniformly'' applied to all the hypersurfaces $\partial E_n$). Therefore, the claim~\eqref{claim1111} follows.\\ By applying the Sobolev embedding theorem on each connected component of $\partialrtial F$, we have that $$ \norma{\mathrm{H}_n}_{L^p(\partial E_n)} \leq C \norma{\mathrm{H}_n}_{H^\frac{1}{2}(\partial E_n)} <C<+\infty\qquad \text{for all $p \in [1,4]$.} $$ for a constant $C$ independent of $n\in\mathbb{N}$.\\ Now, by means of Calder\'on--Zygmund estimates, it is possible to show (see~\cite{DDM}) that there exists a constant $C>0$ depending only on $F$, $M$, $\alpha$ and $p>1$ such that for every $E\in \mathfrak{C}^{1,\alpha}_M(F)$, there holds \begin{equation}\label{CZG} \norma{B}_{L^p(\partial E)} \leq C(1+ \norma{\mathrm{H}}_{L^p(\partial E)})\,. \end{equation} Then, if we write $$ \partial E_n =\{y+\psi_n(y)\nu_F(y):\, y\in \partial F\}\,, $$ we have $\sup_{n\in\mathbb{N}}\|\psi_n\|_{W^{2,p}(\partial F)}<+\infty$, for all $p \in [1,4]$. Thus, by the Sobolev compact embedding $W^{2,p}(\partial F)\hookrightarrow C^{1,\alpha}(\partial F)$, up to a subsequence (not relabeled), there exists a set $F'\in \mathfrak{C}^{1,\alpha}_M(F)$ such that $$ \psi_n\to \psi_{F'} \text{ in $C^{1,\alpha}(\partial F)$}\quad\text{and}\quad v_n\to v_{F'}\text{ in $C^{1,\beta}(\mathbb{T}^3)$ } $$ for all $\alpha\in (0,1/2)$ and $\beta\in (0,1)$.\\ From estimate~\eqref{claim1111} and Lemma~\ref{5.2} (possibly choosing a smaller $M$), we have then that the functions $\psi_n$ are bounded in $W^{5/2,2}(\partial F)$. Hence, possibly passing to another subsequence (again not relabeled), we conclude that $E_n \to F'$ in $W^{2,p}$ for every $p\in[1,4)$, by the Sobolev compact embedding (see for instance Theorem~$6.7$ in~\cite{NePaVa}, with $q\in[1,4)$, $s=1/2$ and $p=2$, applied to ${\mathrm{Hess}}\, \psi_n$). If moreover we have $$ \int_{\mathbb{T}^3}|\nabla w_n|^2\, dx\to 0 \, , $$ then, the above arguments yield the existence of $\lambda\in \mathbb{R}$ and a subsequence (not relabeled) such that $w_n\big(\cdot + \psi_n(\cdot)\nu_F(\cdot)\big)\to \lambda$ in $H^{1/2}(\partial F)$. In turn, $$ \mathrm{H}_n\big(\cdot + \psi_n(\cdot)\nu_F(\cdot)\big)\to \lambda-4\gamma v_{F'}\big(\cdot + \psi_{F'}(\cdot)\nu_F(\cdot)\big)= \mathrm{H}\big(\cdot + \psi_{F'}(\cdot)\nu_F(\cdot)\big) $$ in $H^{1/2}(\partial F)$, where $\mathrm{H}$ is the mean curvature of $F'$. Hence $F'$ is critical.\\ To conclude the proof we then only need to show that $\psi_n$ converge to $\psi =\psi_{F'}$ in $W^{5/2,2}(\partial F)$.\\ Fixed $\deltalta>0$, arguing as in the proof of Lemma~\ref{5.2}, we reduce ourselves to the case where the functions $\psi_n$ are defined on a disk $D\subseteq\mathbb{R}^2$, are bounded in $W^{5/2,2}(D)$, converge in $W^{2,p}(D)$ for all $p\in[1,4)$ to $\psi\in W^{5/2,2}(D)$ and $\|\nabla\psi\|_{L^\infty(D)}\leq\deltalta$. Then, fixed a smooth cut--off function $\varphi$ with compact support in $D$ and equal to one on a smaller disk $D'\subseteq D$, we have \begin{align*} \frac{\Delta(\varphi\psi_n)}{\sqrt{1+|\nabla\psi_n|^2}}-\frac{\Delta(\varphi\psi)}{\sqrt{1+|\nabla\psi|^2}} =&\,(\nabla^2(\varphi\psi_n)-\nabla^2(\varphi\psi))\frac{\nabla\psi \nabla\psi}{(1+|\nabla\psi|^2)^{3/2}}\\ &\,+ \nabla^2(\varphi\psi_n)\Bigl(\frac{\nabla\psi_n\nabla\psi_n}{(1+|\nabla\psi_n|^2)^{3/2}}-\frac{\nabla\psi \nabla\psi}{(1+|\nabla\psi|^2)^{3/2}}\Bigr)\\ &\,+\varphi(\mathrm{H}_n-\mathrm{H})+R(x,\psi_n,\nabla\psi_n)-R(x,\psi,\nabla\psi)\,, \end{align*} where $R$ is a smooth Lipschitz function. Then, using Lemma~\ref{nicola1} with $\beta\in(0,1/2)$, an argument similar to the one in the proof of Lemma~\ref{5.2} shows that \begin{align*} &\bigg[\frac{\Delta(\varphi\psi_n)}{\sqrt{1+|\nabla\psi_n|^2}}-\frac{\Delta(\varphi\psi)}{\sqrt{1+|\nabla\psi|^2}} \bigg]_{H^{1/2}(D)}\leq C(M)\Bigl(\deltalta^2[\nabla^2(\varphi\psi_n)-\nabla^2(\varphi\psi)]_{H^{1/2}(D)}\\ & \qquad+\|\nabla^2(\varphi\psi_n)-\nabla^2(\varphi\psi)\|_{L^{\frac{4}{1+\beta}}(D)}\|\nabla\psi\|_{L^\infty(D)}^\beta\|\nabla^2\psi\|_{L^4(D)}^{1-\beta}+\\ & \qquad+\,[\nabla^2(\varphi\psi_n)]_{H^{1/2}(D)}\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D)}\\ & \qquad+\|\nabla^2(\varphi\psi_n)\|_{L^{\frac{4}{1+\beta}}(D)}\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D)}^\beta(\|\nabla^2\psi_n\|_{L^4(D)}+\|\nabla^2\psi\|_{L^4(D)})^{1-\beta}\\ & \qquad+\|\mathrm{H}_n-\mathrm{H}\|_{H^{1/2}(D)}+\|\psi_n-\psi\|_{W^{2,2}(D)}\Bigr)\,. \end{align*} Using Lemma~\ref{nicola1} again to estimate $[\Delta(\varphi\psi_n)-\Delta(\varphi\psi)]_{H^{1/2}(D)}$ with the seminorm on the left hand side of the previous inequality and arguing again as in the proof of Lemma~\ref{5.2}, we finally get $$ [\nabla^2(\varphi\psi_n)-\nabla^2(\varphi\psi)]_{H^{1/2}(D)}\leq C(M)\Bigl(\|\psi_n-\psi\|_{W^{2,\frac{4}{1+\beta}}(D)} +\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D)}^\beta+\|\mathrm{H}_n-\mathrm{H}\|_{H^{1/2}(D)}\Bigr)\,, $$ hence, $$ [\nabla^2\psi_n-\nabla^2\psi]_{H^{1/2}(D')}\leq C(M)\Bigl(\|\psi_n-\psi\|_{W^{2,\frac{4}{1+\beta}}(D')} +\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D')}^\beta+\|\mathrm{H}_n-\mathrm{H}\|_{H^{1/2}(D')}\Bigr)\,, $$ from which the conclusion follows, by the first part of the lemma with $p=4/(1+\beta)<4$ and a standard covering argument. \end{proof} \subsection{The modified Mullins--Sekerka flow -- The main theorem}\ \vskip.3em We are ready to prove the long time existence/stability result. \begin{thm}\label{existence} Let $E\subseteq\mathbb{T}^n$ be a smooth strictly stable critical set for the nonlocal Area functional under a volume constraint and $N_\varepsilon$ (with $\varepsilon<1$) a tubular neighborhood of $\partial E$, as in formula~\eqref{tubdef}. For every $\alpha\in (0,1/2)$ there exists $M>0$ such that, if $E_0$ is a smooth set in $\mathfrak{C}^{1,\alpha}_M(E)$ satisfying $\mathrm{Vol}( E_0)= \mathrm{Vol}( E )$ and $$ \int_{\mathbb{T}^3} \vert \nabla w_{E_0 }\vert^2\,dx \leq M\, $$ where $w_0=w_{E_0}$ is the function relative to $E_0$ as in problem~\eqref{WE}, then the unique smooth solution $E_t$ of the modified Mullins--Sekerka flow (with parameter $\gamma\geq 0$) starting from $E_0$, given by Theorem~\ref{th:EscNis}, is defined for all $t\geq0$. Moreover, $E_t\to E+\eta$ exponentially fast in $W^{5/2,2}$ as $t\to +\infty$ (recall the definition of convergence of sets at the beginning of Subsection~\ref{stabsec}), for some $\eta\in \mathbb{R}^3$, with the meaning that the functions $\psi_{\eta, t} : \partial E+ \eta \to \mathbb{R}$ representing $\partial E_t$ as ``normal graphs'' on $\partial E + \eta$, that is, $$ \partial E_t= \{ y+ \psi_{\eta,t} (y) \nu_{E+\eta}(y) \, : \, y \in \partial E+\eta \}, $$ satisfy $$ \Vert \psi_{\eta, t}\Vert_{W^{5/2,2}(\partial E + \eta)}\leq Ce^{-\beta t}, $$ for every $t\in[0,+\infty)$, for some positive constants $C$ and $\beta$. \end{thm} \begin{remark}\label{existence+} With some extra effort, arguing as in the proof of Theorem~5.1 in~\cite{FusJulMor18} (last part -- see also Theorem~4.4 in the same paper), it can be shown that the convergence of $E_t\to E+\eta$ is actually smooth (see also Remark~\ref{existence2+}). Indeed, by means of standard parabolic estimates and interpolation (and Sobolev embeddings) the exponential decay in $W^{5/2,2}$ implies analogous estimates in $C^k$, for every $k\in\mathbb{N}$, $$ \Vert \psi_{\eta, t}\Vert_{C^k(\partial E + \eta)}\leq C_ke^{-\beta_k t}, $$ for every $t\in[0,+\infty)$, for some positive constants $C_k$ and $\beta_k$. \end{remark} \begin{remark}\label{closedness} We already said that the property of a set $E_0$ to belong to $\mathfrak{C}^{1,\alpha}_M(E)$ is a ``closedness'' condition in $L^1$ of $E_0$ and $E$ and in $C^{1,\alpha}$ of their boundaries. The extra condition in the theorem on the $L^2$--smallness of the gradient of $w_0$ (see the second part of Lemma~\ref{w52conv} and its proof) implies that the quantity $\mathrm{H}_0+4\gamma v_0$ on $\partial E_0$ is ``close'' to be constant, as it is the analogous quantity for the set $E$ (or actually for any critical set). Notice that this is a second order condition for the boundary of $E_0$, in addition to the first order one $E_0\in \mathfrak{C}^{1,\alpha}_M(E)$. \end{remark} \begin{proof}[Proof of Theorem~\ref{existence}] Throughout the whole proof $C$ will denote a constant depending only on $E$, $M$ and $\alpha$, whose value may vary from line to line. Assume that the modified Mullins--Sekerka flow $E_t$ is defined for $t$ in the maximal time interval $[0,T(E_0))$, where $T(E_0)\in (0,+\infty]$ and let the moving boundaries $\partial E_t$ be represented as ``normal graphs'' on $\partial E$ as $$ \partial E_t= \{ y+ \psi_t(y) \nu_{E}(y) \, : \, y \in \partial E\}, $$ for some smooth functions $\psi_t:\partial E\to \mathbb{R}$. As before we set $\nu_t=\nu_{E_t}$, $v_t=v_{E_t}$ and $w_t=w_{E_t}$.\\ We recall that, by Theorem~\ref{th:EscNis}, for every $F\in \mathfrak{C}^{2,\alpha}_M(E)$, the flow is defined in the time interval $[0, T)$, with $T=T(E,M,\alpha)>0$.\\ We show the theorem for the smooth sets $E_0\subseteq\mathbb{T}^3$ satisfying \begin{equation} \mathrm{Vol}(E_0\triangle E)\leq M_1,\quad\|\psi_0\|_{C^{1,\alpha}(\partial E)}\leq M_2\quad\text{and}\quad \int_{\mathbb{T}^3} |\nabla w_0|^2\, dx\leq M_3\,, \end{equation} for some positive constants $M_1,M_2,M_3$, then we get the thesis by setting $M=\min\{M_1,M_2,M_3\}$.\\ For any set $F\in \mathfrak{C}^{1,\alpha}_{M}(E)$ we introduce the following quantity \begin{equation}\label{D(F)0} D(F)=\int_{F\Delta E}d(x, \partial E)\, dx=\int_F d_E\, dx-\int_E d_E\, dx, \end{equation} where $d_E$ is the signed distance function defined in formula~\eqref{sign dist}. We observe that $$ \mathrm{Vol}(F\Delta E)\leq C\|\psi_F\|_{L^1(\partial E)} \leq C\|\psi_F\|_{L^2(\partial E)} $$ for a constant $C$ depending only on $E$ and, as $F\subseteq N_\varepsilon$, \begin{equation}\label{D(F)0bis} D(F)\leq \int_{F\Delta E}\varepsilon\, dx\leq \varepsilon\mathrm{Vol}(F\Delta E). \end{equation} Moreover, \begin{align} \|\psi_F\|_{L^2(\partial E)}^2&=2\int_{\partial E} \int_0 ^{|\psi_F(y)|} t \, dt\,d\mu(y) \\ &=2\int_{\partial E} \int_0^{|\psi_F(y)|} d(L(y,t),\partial E) \, dt\,d\mu(y) \\ &=2\int_{E\Delta F} d(x,\partial E)\,JL^{-1}(x)\,dx\\ &\leq C D(F). \end{align} where $L:\partialrtial E\times (-\varepsilon,\varepsilon)\to N_\varepsilon$ the smooth diffeomorphism defined in formula~\eqref{eqcar410} and $JL$ its Jacobian. As we already said, the constant $C$ depends only on $E$ and $\varepsilon$. This clearly implies \begin{equation}\label{D(F)} \mathrm{Vol}(F\Delta E)\leq C\|\psi_F\|_{L^1(\partial E)} \leq C\|\psi_F\|_{L^2(\partial E)}\leq C\sqrt{D(F)}\,. \end{equation} Hence, by this discussion, the initial smooth set $E_0\in\mathfrak{C}^{1,\alpha}_M(E)$ satisfies $D(E_0)\leq M\leq M_1$ (having chosen $\varepsilon<1$).\\ By rereading the proof of Lemma~\ref{w52conv}, it follows that for $M_2,M_3$ small enough, if $\Vert\psi_F\Vert_{C^{1,\alpha}(\partial E)}\leq M_2$ and \begin{equation}\label{ex-de02} \int_{\mathbb{T}^3} |\nabla w_{F}|^2\, dx \leq M_3\,, \end{equation} then, \begin{equation}\label{eqcar50001} \|\psi_F\|_{W^{2,3}(\partial E)}\leq \omega(\max\{M_2,M_3\})\,, \end{equation} where $s\mapsto\omega(s)$ is a positive nondecreasing function (defined on $\mathbb{R}$) such that $\omega(s)\to 0$ as $s\to 0^+$. Hence, \begin{equation}\label{eqcar50003} \|\nu_F\|_{W^{1,3}(\partial F)}\leq \omega'(\max\{M_2,M_3\})\,, \end{equation} for a function $\omega'$ with the same properties of $\omega$. Both $\omega$ and $\omega'$ only depend on $E$ and $\alpha$, for $M$ small enough. We split the proof of the theorem into steps. \noindent \textbf{Step ${\mathbf 1}$} ({Stopping--time})\textbf{.}\\ Let $\overline T\leq T(E_0)$ be the maximal time such that \begin{equation}\label{Tprimo} \mathrm{Vol}(E_t\triangle E)\leq 2M_1,\quad\|\psi_t\|_{C^{1,\alpha}(\partial E)}\leq 2M_2\quad\text{and}\quad \int_{\mathbb{T}^3} |\nabla w_t|^2\, dx\leq 2M_3\,, \end{equation} for all $t\in [0, \overline T)$. Hence, \begin{equation}\label{eqcar50005} \|\psi_t\|_{W^{2,3}(\partial E)}\leq \omega(2\max\{M_2,M_3\})\, \end{equation} for all $t\in [0, \overline T')$, as in formula~\eqref{eqcar50001}. Note that such a maximal time is clearly positive, by the hypotheses on $E_0$.\\ We claim that by taking $M_1,M_2,M_3$ small enough, we have $\overline T=T(E_0)$. \noindent \textbf{Step ${\mathbf 2}$} ({Estimate of the translational component of the flow})\textbf{.}\\ We want to see that there exists a small constant $\theta>0$ such that \begin{equation} \label{not a translation} \min_{\eta\in{\mathrm{O}}_E}\big\|\,[\partial_{\nu_t}w_t]- \langle\eta \, \vert \, \nu_t\rangle \big\|_{L^2(\partial E_t)}\geq \theta\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partial E_t)}\qquad\text{for all }t\in [0, \overline T)\,, \end{equation} where ${\mathrm{O}}_E$ is defined by formula~\eqref{OOeq}.\\ If $M$ is small enough, clearly there exists a constant $C_0=C_0(E,M,\alpha)>0$ such that, for every $i\in\mathrm{I}I_E$, we have $\Vert \langle e_i|\nu_t\rangle\Vert_{L^2(\partial E_t)}\geq C_0>0$, holding $\Vert \langle e_i|\nu_E\rangle\Vert_{L^2(\partial E)}>0$. It is then easy to show that the vector $\eta_t\in{\mathrm{O}}_E$ realizing such minimum is unique and satisfies \begin{equation} \label{not a translation 2} [\partialrtial_{\nu_t} w_t] = \langle \eta_t\, \vert \nu_t \rangle + g, \end{equation} where $g\in L^2(\partial E_t)$ is a function $L^2$--orthogonal (with respect to the measure $\mu_t$ on $\partialrtial E_t$) to the vector subspace of $L^2(\partial E_t)$ spanned by $\langle e_i|\nu_t\rangle$, with $i\in\mathrm{I}I_E$, where $\{e_1,\dots,e_3\}$ is the orthonormal basis of $\mathbb{R}^3$ given by Remark~\ref{rembase}. Moreover, the inequality \begin{equation}\label{eqcar50002} |\eta_t|\leq C\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)} \end{equation} holds, with a constant $C$ depending only on $E$, $M$ and $\alpha$.\\ We now argue by contradiction, assuming $\|g\|_{L^2(\partial E_t)} < \theta \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partial E_t)}$. First, by formula~\eqref{first var} and the translation invariance of the functional $J$, we have $$ 0=\frac{d}{ds}J(E_t+s\eta_t)\biggl|_{s=0}=\int_{\partial E_t}(\mathrm{H}_{t}+4\gamma v_t)\langle \eta_t \, \vert \, \nu_t\rangle\, d\mu_t=\int_{\partial E_t}w_t\langle \eta_t\, \vert \, \nu_t\rangle\, d\mu_t\,. $$ It follows that, by multiplying equality~\eqref{not a translation 2} by $w_t-\widehat w_t$, with $\widehat w_t=\fint_{\mathbb{T}^3}w_t\, dx$ and integrating over $\partialrtial E_t$, we get \begin{align*} \int_{\mathbb{T}^3}|\nabla w_t|^2 \, dx =&\,-\int_{\partial E_t}w_t [\partial_{\nu_t}w_t]\, d\mu_t\\ =&\, -\int_{\partial E_t}(w_t-\widehat w_t) [\partial_{\nu_t}w_t]\, d\mu_t\\ =&\,-\int_{\partial E_t}(w_t-\widehat w_t) g\, d\mu_t\\ \leq&\,\,\theta \| w_t - \widehat w_t \|_{L^2(\partialrtial E_t)} \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)}. \end{align*} Note that in the second and the third equality above we have used the fact that $[\partial_{\nu_t}w_t]$ and $\nu_t$ have zero integral on $\partial E_t$.\\ By the trace inequality (see~\cite{Ev}), we have \begin{equation}\label{tracecar} \|w_t-\widehat w_t\|^2_{L^2(\partial E_t)}\leq\|w_t-\widehat w_t\|^2_{H^{1/2}(\partial E_t)}\leq C\int_{\mathbb{T}^3}|\nabla w_t|^2\, dx\,, \end{equation} hence, by the previous estimate, we conclude \begin{equation} \int_{\mathbb{T}^3}|\nabla w_t|^2 \, dx \leq C\theta^2\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)}^2.\label{trans} \end{equation} Let us denote with $f:\mathbb{T}^3\to\mathbb{R}$ the harmonic extension of $\langle \eta_t\, \vert \, \nu_t\rangle $ to $\mathbb{T}^3$, we then have \begin{equation} \label{bound for harm ext} \|\nabla f\|_{L^2(\mathbb{T}^3)} \leq C \|\langle\eta_t\,\vert\, \nu_t\rangle\|_{H^{{1}/{2}}(\partialrtial E_t)} \leq C |\eta_t| \|\nu_t\|_{W^{1,3}(\partial E_t)}\leq C \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)}\,, \end{equation} where the first inequality comes by standard elliptic estimates (holding with a constant $C=C(E,M,\alpha)>0$, see~\cite{DDM} for details), the second is trivial and the last one follows by inequalities~\eqref{eqcar50003} and~\eqref{eqcar50002}.\\ Thus, by equality~\eqref{not a translation 2} and estimates~\eqref{trans} and~\eqref{bound for harm ext}, we get \begin{align} \|\langle \eta_t\,\vert \, \nu_t\rangle\|^2_{L^2(\partial E_t)} &=\int_{\partial E_t}[\partial_{\nu_t}w_t]\langle \eta_t\,\vert\nu_t\rangle\, d\mu\\ &= - \int_{\mathbb{T}^3} \langle \nabla w_t\,\vert \, \nabla f\rangle \, dx\\ &\leq \left(\int_{\mathbb{T}^3} |\nabla w_t|^2 \,dx\right)^{1/2} \left(\int_{\mathbb{T}^3} |\nabla f|^2\, dx\right)^{1/2} \\ &\leq C \theta \bigl\|[\partial_{\nu_t}w_t]\bigr\|^2_{L^2(\partialrtial E_t)}\,. \end{align} If then $\theta>0$ is chosen so small that $C\theta+\theta^2 <1$ in the last inequality, then we have a contradiction with equality~\eqref{not a translation 2} and the fact that $\|g\|_{L^2(\partial E_t)}<\theta \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partial E_t)}$, as they imply (by $L^2$--orthogonality) that $$ \|\langle \eta_t\,\vert \, \nu_t\rangle\|^2_{L^2(\partial E_t)}>(1-\theta^2)\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partial E_t)}^2\,. $$ All this argument shows that for such a choice of $\theta$ condition~\eqref{not a translation} holds.\\ By Propositions~\ref{2.6} and~\ref{prop:nocrit}, there exist positive constants $\sigma_\theta$ and $\deltalta$ with the following properties: for any set $F\in \mathfrak{C}^{1,\alpha}_M(E)$ such that $\|\psi_F\|_{W^{2,3}(\partial E)}<\delta$, there holds \begin{equation}\label{de03} \Pi_F(\varphi)\geq \sigma_\theta\| \varphi\|_{H^1(\partial F)}^2 \end{equation} for all $\varphi\in \widetilde{H}^1(\partial F)$ such that $\min_{\eta\in{\mathrm{O}}_E}\|\varphi-\langle\eta\,\vert\,\nu_F\rangle\|_{L^2(\partial F)}\geq \theta\|\varphi\|_{L^2(\partial F)}$ and if $E'$ is critical, $\mathrm{Vol}(E')=\mathrm{Vol}(E)$ with $\|\psi_{E'}\|_{W^{2,3}(\partial E)}<\delta$, then \begin{equation}\label{de04} E'=E+\eta \end{equation} for a suitable vector $\eta\in \mathbb{R}^3$. We then assume that $M_2,M_3$ are small enough such that \begin{equation}\label{de05} \omega(2\max\{M_2,M_3\})<\delta/2\, \end{equation} where $\omega$ is the function introduced in formula~\eqref{eqcar50001}. \noindent \textbf{Step ${\mathbf 3}$} ({The stopping time $\overline T$ is equal to the maximal time $T(E_0)$})\textbf{.}\\ We show now that, by taking $M_1,M_2,M_3$ smaller if needed, we have $\overline T=T(E_0)$.\\ By the previous point and the suitable choice of $M_2,M_3$ made in its final part, formula~\eqref{not a translation} holds, hence we have $$ \Pi_{E_t}\bigl([\partial_{\nu_t}w_t]\bigr)\geq \sigma_\theta\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E)}^2\qquad \text{ for all $t\in [0, \overline T)$.} $$ In turn, by Lemma~\ref{calculations} we may estimate \begin{equation*} \frac{d}{dt} \left(\frac{1}{2} \int_{\mathbb{T}^3} |\nabla w_t|^2\, dx \right) \leq-\sigma_\theta \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E_t)}^2+ \frac{1}{2}\int_{\partialrtial E_t} (\partialrtial_{\nu_t} w^+_t+ \partialrtial_{\nu_t} w_t^-) [\partialrtial_{\nu_t} w_t]^2 \, d \mu_t \end{equation*} for every $t \leq \overline T$.\\ It is now easy to see that \begin{equation}\label{Lapw} \Delta w_t= [\partialrtial_{\nu_t} w_t]\mu_t\,, \end{equation} then, by point (iii) of Lemma~\ref{harmonic estimates}, we estimate the last term as \begin{equation*} \int_{\partialrtial E_t} (\partialrtial_{\nu_t} w^+_t+ \partialrtial_{\nu_t} w_t^-) [\partialrtial_{\nu_t} w_t]^2 \, d \mu_t\leq C \int_{\partialrtial E_t} (|\partialrtial_{\nu_t} w^+_t|^3 + |\partialrtial_{\nu_t} w^-_t|^3) \, d \mu_t\leq C \int_{\partialrtial E_t}\bigl| [\partialrtial_{\nu_t} w_t]\bigr|^3 \, d \mu_t\,, \end{equation*} thus, the last estimate in the statement of Lemma~\ref{harmonic estimates} implies \begin{equation*} \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^3(\partialrtial E_t)} \leq C \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partialrtial E_t)}^{2/3} \|w_t - \widehat w_t\|_{L^2(\partialrtial E_t)}^{1/3}. \end{equation*} Therefore, combining the last three estimates, we get \begin{align} \frac{d}{dt} \int_{\mathbb{T}^3} |\nabla w_t|^2\, dx \leq&\,-2\sigma_\theta \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E_t)}^2+C\|w_t - \widehat w_t\|_{L^2(\partialrtial E_t)} \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E_t)}^2\nonumber\\ \leq&\,-\sigma_\theta\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E_t)}^2,\label{quasi} \end{align} for every $t\in[0,\overline{T})$, where in the last inequality we used the trace inequality~\eqref{tracecar} \begin{equation}\label{tttt} \Vert w_t - \widehat w_t \Vert^{2}_{L^2(\partial E_t)} \leq \Vert w_t - \widehat w_t \Vert^{2}_{H^{{1}/{2}}(\partial E_t)} \leq C \int_{\mathbb{T}^3} |\nabla w_t|^2 \, dx \leq 2CM_3, \end{equation} possibly choosing a smaller $M$ such that $2CM_3<\sigma_{\theta}$.\\ This argument clearly says that the quantity $\int_{\mathbb{T}^3} |\nabla w_t|^2\, dx$ is nonincreasing in time, hence, if $M_2,M_3$ are small enough, the inequality $\int_{\mathbb{T}^3} |\nabla w_t|^2\, dx\leq 2M_3$ is preserved during the flow. If we assume by contradiction that $\overline T< T(E_0)$, then it must happen that $\mathrm{Vol}(E_{\overline{T}}\triangle E)=2M_1$ or $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E)}=2M_2$. Before showing that this is not possible, we prove that actually the quantity $\int_{\mathbb{T}^3} |\nabla w_t|^2\, dx$ decreases (non increases) exponentially.\\ Computing as in the previous step, \begin{align} \int_{\mathbb{T}^3} |\nabla w_t|^2\, dx &=- \int_{\partialrtial E_t} w_t [\partialrtial_{\nu_t} w_t] \, d \mu_t\\ &= - \int_{\partialrtial E_t} (w_t - \widehat w_t) [\partialrtial_{\nu_t} w_t] \, d \mu_t \\ &\leq \| w_t - \widehat w_t \|_{L^2(\partialrtial E_t)} \bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)}\\ &\leq C\left(\int_{\mathbb{T}^3}|\nabla w_t|^2\, dx\right)^{1/2}\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)}, \end{align} where we used again the trace inequality~\eqref{tracecar}. Then, \begin{equation*} \int_{\mathbb{T}^3}|\nabla w_t|^2 \, dx \leq C\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{L^2(\partialrtial E_t)}^2 \leq C\norma{[\partial_{\nu_t} w_t]}^2_{H^1(\partial E_t)}, \end{equation*} and combining this inequality with estimate~\eqref{quasi}, we obtain $$ \frac{d}{dt} \int_{\mathbb{T}^3} |\nabla w_t|^2\, dx\leq-c_0 \int_{\mathbb{T}^3} |\nabla w_t|^2\, dx, $$ for every $t \leq \overline T$ and for a suitable constant $c_0\geq 0$. Integrating this differential inequality, we get \begin{equation}\label{expfinalmente} \int_{\mathbb{T}^3} |\nabla w_{ t}|^2\, dx \leq e^{-c_0 t}\int_{\mathbb{T}^3} |\nabla w_0|^2\, dx \leq M_3 e^{-c_0 t}\leq M_3\,, \end{equation} for every $t \leq \overline T$.\\ Then, we assume that $\mathrm{Vol}(E_{\overline{T}}\triangle E)=2M_1$ or $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E_{\overline{T}})}=2M_2$. Recalling formula~\eqref{D(F)0} and denoting by $X_t$ the velocity field of the flow (see Definition~\ref{def:smoothflow} and the subsequent discussion), we compute \begin{align} \frac{d}{dt}D(E_t)&=\frac{d}{dt}\int_{E_t} d_E\, dx= \int_{E_t}\operatorname*{div}\nolimits(d_E X_t)\, dx= \int_{\partial E_t}d_E\langle X_t\vert\nu_t\rangle\, d\mu_t\\ &= \int_{\partial E_t}d_E [\partial_{\nu_t}w_t]\, d\mu_t=- \int_{\mathbb{T}^3}\left\langle\nabla h\,\vert\, \nabla w_t\right\rangle\, dx\,, \end{align} where $h$ denotes the harmonic extension of $d_E$ to $\mathbb{T}^3$. Note that, by standard elliptic estimates and the properties of the signed distance function $d_E$, we have $$ \|\nabla h\|_{L^2(\mathbb{T}^3)}\leq C\|d_E\|_{C^{1,\alpha}(\partial E)}\leq C=C(E)\,, $$ then, by the previous equality and formula~\eqref{expfinalmente}, we get \begin{equation}\label{Deqcar} \frac{d}{dt} D(E_t) \leq C \|\nabla w_t\|_{L^2(\mathbb{T}^3)}\leq C\sqrt{M_3\,} e^{-c_0t/2}\,, \end{equation} for every $t \leq \overline T$. By integrating this differential inequality over $[0, \overline T)$ and recalling estimate~\eqref{D(F)}, we get \begin{equation}\label{step33} \mathrm{Vol}(E_{\overline T}\triangle E)\leq C\|\psi_{\overline T}\|_{L^2(\partial E_{\overline{T}})}\leq C\sqrt{D(E_{\overline T})}\leq C\sqrt{D(E_0)+C\sqrt{M_3}}\leq C\sqrt[4]{M_3}\,, \end{equation} as $D(E_0)\leq M_1$, provided that $M_1,M_3$ are chosen suitably small. This shows that $\mathrm{Vol}(E_{\overline{T}}\triangle E)=2M_1$ cannot happen if we chose $C\sqrt[4]{M_3}\leq M_1$.\\ By arguing as in Lemma~\ref{w52conv} (keeping into account inequality~\eqref{Tprimo} and formula~\eqref{eqcar50001}), we can see that the $L^2$--estimate~\eqref{step33} implies a $W^{2,3}$--bound on $\psi_{\overline T}$ with a constant going to zero, keeping fixed $M_2$, as $\int_{\mathbb{T}^3} |\nabla w_{\overline{T}}|^2\, dx\to0$, hence, by estimate~\eqref{expfinalmente}, as $M_3\to0$. Then, by Sobolev embeddings, the same holds for $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E_{\overline{T}})}$, hence, if $M_3$ is small enough, we have a contradiction with $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E_{\overline{T}})}=2M_2$.\\ Thus, $\overline T=T(E_0)$ and \begin{equation}\label{finaldecay} \mathrm{Vol}(E_t\triangle E)\leq C\sqrt[4]{M_3}\,,\quad\quad\|\psi_t\|_{C^{1,\alpha}(\partial E_t)}\leq 2M_2\,,\quad\quad\int_{\mathbb{T}^3} |\nabla w_{t}|^2\, dx \leq M_3e^{-c_0 t}\,, \end{equation} for every $t\in[0, T(E_0))$, by choosing $M_1,M_2,M_3$ small enough. \noindent \textbf{Step ${\mathbf 4}$} ({Long time existence})\textbf{.}\\ We now show that, by taking $M_1,M_2,M_3$ smaller if needed, we have $T(E_0)=+\infty$, that is, the flow exists for all times.\\ We assume by contradiction that $T(E_0)<+\infty$ and we recall that, by estimate~\eqref{quasi} and the fact that $\overline T=T(E_0)$, we have $$ \frac{d}{dt}\int_{\mathbb{T}^3} |\nabla w_t|^2\, dx +\sigma_\theta\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E_t)}^2\leq 0 $$ for all $t\in [0,T(E_0))$. Integrating this differential inequality over the interval $$ \left[T(E_0)-{T}/2,T(E_0)-{T}/4\right]\,, $$ where $T$ is given by Theorem~\ref{th:EscNis}, as we said at the beginning of the proof, we obtain \begin{equation*} \sigma_{{\theta}}\int_{T(E_0)-T/2}^{T(E_0)-T/4}\bigl\|[\partial_{\nu_t}w_t]\bigr\|_{H^1(\partial E_t)}^2\, dt\leq \int_{\mathbb{T}^3} |\nabla w_{T(E_0)-\frac{T}2}|^2\, dx- \int_{\mathbb{T}^3} |\nabla w_{T(E_0)-\frac{T}4}|^2\, dx\leq M_3\,, \end{equation*} where the last inequality follows from estimate~\eqref{finaldecay}. Thus, by the mean value theorem there exists $\overline t\in \left(T(E_0)-{T}/2,T(E_0)-{T}/4 \right)$ such that $$ \bigl\|[\partial_{\nu_{\overline t}}w_{\overline t}] \bigr\|_{H^1(\partial E_t)}^2\leq \frac{4M_3}{T\sigma_\theta}\,. $$ Note that for any smooth set $F\subseteq\mathbb{T}^3$, we have $\Vert v_{F}\Vert_{C^1(\mathbb{T}^3)}\leq L$, for some ``absolute'' constant $L$ and that $w_F$ is constant, then, since $H^1(\partial E_{\overline t})$ embeds into $L^p(\partial E_{\widehat t})$ for all $p>1$, by Lemma~\ref{harmonic estimates}, we in turn infer that \begin{align} [\mathrm{H}_{\overline t}(\cdot + \psi_{\overline t}(\cdot)&\,\nu_E(\cdot))-\mathrm{H}_E]^2_{C^{0,\alpha}(\partial E)}\\ \leq&\,C [w_{\overline t}(\cdot + \psi_{\overline t}(\cdot)\nu_E(\cdot))-w_E]^2_{C^{0,\alpha}(\partial E)}\\ &\,+ C [v_{\overline t}(\cdot + \psi_{\overline t}(\cdot)\nu_E(\cdot))-v_{\overline t}]^2_{C^{0,\alpha}(\partial E)} + C[v_{\overline t} - v_E]^2_{C^{0,\alpha}(\partial E)}\\ \leq&\,C [w_{\overline t}]^2_{C^{0,\alpha}(\partial E_{\overline t})}\norma{\psi_{\overline t}}^2_{C^{1,\alpha}(\partial F)} + C L^2 \norma{\psi_{\overline t}}^2_{C^{1,\alpha}(\partial F)} + C \norma{u_{\overline t}-u_E}^2_{L^2(\mathbb{T}^3)}\\ \leq&\, C \frac{M_3}{T \sigma_\theta} + C L^2 \norma{\psi_{\overline t}}^2_{C^{1,\alpha}(\partial E)} + C\mathrm{Vol}(E_{\overline t} \triangle E)^2 \, , \end{align} where $[\cdot ]_{C^{0,\alpha}(\partial E_{\overline t})}$ and $[\cdot]_{C^{0,\alpha}(\partial E)}$ stand for the $\alpha$--H\"older seminorms on $\partial E_{\overline t}$ and $\partial E$, respectively and remind that $v_{\overline t},v_E$ are the potentials, defined by formula~\eqref{potential1}, associated to $u_{\overline{t}}= \chi_{\text{\raisebox{-.5ex}{$\scriptstyle E_{\overline{t}}$}}} - \chi_{\text{\raisebox{-.5ex}{$\scriptstyle \mathbb{T}^n \setminus E_{\overline{t}}$}}}$ and $u_E= \chi_{\text{\raisebox{-.5ex}{$\scriptstyle E$}}} - \chi_{\text{\raisebox{-.5ex}{$\scriptstyle \mathbb{T}^n \setminus E$}}}$.\\ By means of Schauder estimates (as Calder\'on--Zygmund inequality implied estimate~\eqref{CZG}), it is possible to show (see~\cite{DDM}) that there exists a constant $C>0$ depending only on $E$, $M$, $\alpha$ and $p>1$ such that for every $F\in \mathfrak{C}^{1,\alpha}_M(E)$, choosing even smaller $M_1,M_2,M_3$, there holds \begin{equation}\label{CZGS} \norma{B}_{C^{0,\alpha}(\partial F)} \leq C(1+ \norma{\mathrm{H}}_{C^{0,\alpha}(\partial F)})\,. \end{equation} Hence, by the above discussion, we can conclude that $E_{\overline t}\in \mathfrak{C}^{2,\alpha}_M(E)$. Therefore, the maximal time of existence of the classical solution starting from $E_{\overline t}$ is at least $T$, which means that the flow $E_t$ can be continued beyond $T(E_0)$, which is a contradiction. \noindent \textbf{Step ${\mathbf 5}$} ({Convergence, up to subsequences, to a translate of $E$})\textbf{.}\\ Let $t_n\to +\infty$, then, by estimates~\eqref{finaldecay}, the sets $E_{t_n}$ satisfy the hypotheses of Lemma~\ref{w52conv}, hence, up to a (not relabeled) subsequence we have that there exists a critical set $E'\in \mathfrak{C}^{1,\alpha}_M(E)$ such that $E_{t_n}\to E'$ in $W^{5/2,2}$. Due to formulas~\eqref{eqcar50001} and~\eqref{de05} we also have $\|\psi_{E'}\|_{W^{2,3}(\partial E)}\leq \deltalta$ and $E'=E+\eta$ for some (small) $\eta\in \mathbb{R}^3$ (equality~\eqref{de04}). \noindent \textbf{Step ${\mathbf 6}$} ({Exponential convergence of the full sequence})\textbf{.}\\ Consider now $$ D_\eta(F)=\int_{F\Delta (E+\eta)}\mathrm{dist\,}(x, \partial E+\eta)\, dx\,. $$ The very same calculations performed in Step~$3$ show that \begin{equation}\label{step6} \Bigl\vert\frac{d}{dt} D_\eta(E_t)\Bigr\vert \leq C \|\nabla w_t\|_{L^2(\mathbb{T}^3)}\leq C\sqrt{M_3} e^{-{c_0}t/2} \end{equation} for all $t\geq0$, moreover, by means of the previous step, it follows $\lim_{t\to +\infty} D_\eta(E_t)=0$. In turn, by integrating this differential inequality and writing $$ \partial E_t=\{y+\psi_{\eta, t}(y)\nu_{E+\eta}(y): y\in \partial E+\eta\}\,, $$ we get \begin{equation}\label{step61} \|\psi_{\eta, t}\|_{L^2(\partial E+\eta)}^2\leq C D_\eta(E_t)\leq\int_t^{+\infty}C\sqrt{M_3} e^{-c_0s/2}\, ds\leq C\sqrt{M_3} e^{-c_0t/2}\,. \end{equation} Since by the previous steps $\|\psi_{\eta, t}\|_{W^{2,3}(\partial E+\eta)}$ is bounded, we infer from this inequality and interpolation estimates that also $\|\psi_{\eta, t}\|_{C^{1,\beta}(\partial E+\eta)}$ decays exponentially for all $\beta\in (0,1/3)$. Then, setting $p = \frac{2}{1 -\beta}$, we have, by estimates~\eqref{step61} and~\eqref{D(F)} (and standard elliptic estimates), \begin{align} \|v_t-v_{E+\eta}\|_{C^{1,\beta}(\mathbb{T}^3)} &\leq C \|v_t-v_{E+\eta}\|_{W^{2,p}(\mathbb{T}^3)} \leq C \|u_t-u_{E+\eta}\|_{L^{p}(\mathbb{T}^3)} \\ &\leq C \mathrm{Vol}(E_t \triangle (E+\eta))^{{1}/{p}}\leq C \|\psi_{\eta, t}\|_{L^2(\partial E+\eta)}^{{1}/{p}}\\ &\leq CM_3^{{1}/{4p}} e^{-{c_0t}/{4p}t}\label{vdecay} \end{align} for all $\beta\in (0,1/3)$. Denoting the average of $w_t$ on $\partial E_t$ by $\overline w_t$, as by estimates~\eqref{tracecar} and~\eqref{expfinalmente} (recalling the argument to show inequality~\eqref{eqcar15000}), we have that \begin{align} \|w_t\big(\cdot + \psi_{\eta, t}(\cdot)\nu_{E+\eta}(\cdot)\big)-\overline w_t\|_{H^{1/2}(\partial E+\eta)} &\leq C\|w_t-\overline w_t\|_{H^{1/2}(\partial E_t)}\|\psi_{\eta, t}\|_{C^1(\partial E+\eta)}\\ & \leq C\|\nabla w_t\|_{L^2(\mathbb{T}^3)}\\ &\leq C\sqrt{M_3} e^{-{c_0t}/2}\,. \end{align} It follows, taking into account inequality~\eqref{vdecay}, that \begin{equation}\label{quasiHdecay} \bigl\|[\mathrm{H}_t\big(\cdot + \psi_{\eta, t}(\cdot)\nu_{E+\eta}(\cdot)\big)-\overline \mathrm{H}_t] -[\mathrm{H}_{\partial E+\eta}-\overline \mathrm{H}_{\partial E+\eta}]\bigr\|_{H^{1/2}(\partial E+\eta)}\to0 \end{equation} exponentially fast, as $t\to+\infty$, where $\overline \mathrm{H}_t$ and $\overline \mathrm{H}_{\partial E+\eta}$ stand for the averages of $\mathrm{H}_t$ on $\partial E_t$ and of $\mathrm{H}_{\partial E+\eta}$ on $\partial E+\eta$, respectively.\\ Since $E_t\to E+\eta$ (up to a subsequence) in $W^{{5}/{2},2}$, it is easy to check that $\vert\overline \mathrm{H}_{t}-\overline \mathrm{H}_{\partial E+\eta}\vert\leq C\|\psi_{\eta, t}\|_{C^1(\partial E+\eta)}$ which decays exponentially, therefore, thanks to limit~\eqref{quasiHdecay}, we have $$ \bigl\|\mathrm{H}_t\big(\cdot + \psi_{\eta, t}(\cdot)\nu_{E+\eta}(\cdot)\big) -\mathrm{H}_{\partial E+\eta}\bigr\|_{H^{1/2}(\partial E+\eta)}\to0 $$ exponentially fast.\\ The conclusion then follows arguing as at the end of Step~$4$. \end{proof} \subsection{A brief overview of the Neumann case}\label{Neucase} Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^n$. As before we consider the nonlocal Area functional \begin{equation}\label{N.1} J_N(E)=\mathcal A_{\Omega}(\partial E)+\gamma\int_{\Omega}|\nabla v_E|^2\, dx\,, \end{equation} for every $E\subseteq \Omega$ with $\partial E \cap \partial \Omega=$~\O, where $\gamma\geq 0$ is a real parameter and $v_E$ is the potential defined as follows, similarly to problem~\eqref{potential}, $$ \begin{cases} -\Delta v_E=u_E-m\quad &\text{ in }\Omega \\ \displaystyle\,\,\frac{\partial v_E}{\partial\nu_E}=0\quad &\text{ on } \partial\Omega \\ \displaystyle\int_\Omega v_E\,dx=0\\ \end{cases} $$ with $m=\fint_\Omega u_E\,dx$, $u_E= \chi_{\text{\raisebox{-.5ex}{$\scriptstyle E$}}} - \chi_{\text{\raisebox{-.5ex}{$\scriptstyle \Omega\setminus E$}}}$ and $\nu_E$ the outer unit normal to $E$.\\ As in formula~\eqref{G1}, we have $$ \int_\Omega |\nabla v_E|^2\, dx=\int_\Omega\int_\Omega G(x,y)u_E(x)u_E(y)\, dxdy\,, $$ where $G$ is the (distributional) solution of $$ \begin{cases} -\Delta_x G(x,y)=\deltalta_y -\frac1{\mathrm{Vol}(\Omega)}\quad &\text{ for every $x\in\Omega$} \\ \displaystyle \langle\nabla_x G(x,y)\vert \nu_E(x)\rangle=0\quad &\text{ for every $x\in\partial\Omega$} \\ \displaystyle\int_\Omega G(x,y)\,dx=0 \end{cases} $$ for every $y\in\Omega$.\\ Note that, unlike the ``periodic'' case (when the ambient is the torus $\mathbb{T}^n$), the functional $J_N$ is not translation invariant, therefore several arguments simplify. The calculus of the first and second variations of $J_N$, under a volume constraint, is exactly the same as for $J$, then we say that a smooth set $E \subseteq \Omega$, with $\partial E \cap \partial \Omega =$~\O, is a \emph{critical set}, if it satisfies the Euler--Lagrange equation $$ \mathrm{H}+4\gamma v_E=\lambda \qquad\text{on $\partialrtial E$,} $$ for a constant $\lambda \in \mathbb{R}$, instead, since $J_N$ is not translation invariant, the spaces $T(\partial E)$, $T^\perp (\partial E)$, and the decomposition~\eqref{decomp} are no longer needed and, defining the same quadratic form $\Pi_E$ as in formula~\eqref{Pieq}, we say that a smooth critical set $E$ is \emph{strictly stable} if $$ \Pi_E(\varphi)>0\qquad\text{for all $\varphi\in \widetilde{H}^1(\partial E)\setminus \{0\}$. } $$ Naturally, $E\subseteq \Omega$ is a \emph{local minimizer} if there exists a $\deltalta \geq 0$ such that $$ J_N(F)\geq J_N(E), $$ for all $F\subseteq \Omega$, $\partial F \cap \partial\Omega=$~\O, $\mathrm{Vol}(F)=\mathrm{Vol}(E)$ and $\mathrm{Vol}(E\triangle F)\leq \deltalta$. Then, as in the periodic case, we have a local minimality result with respect to small $W^{2,p}$--perturbations. Precisely, the following (cleaner) counterpart to Theorem~\ref{W2pMin} holds (see also~\cite{JuPi}). \begin{thm}\label{MinNeum} Let $p>\max\{2, n-1\}$ and $E\subseteq\Omega$ a smooth strictly stable critical set for the nonlocal Area functional $J_N$ (under a volume constraint) with $N_\varepsilon$ a tubular neighborhood of $E$ as in formula~\eqref{tubdef}. Then there exist constants $\deltalta,C>0$ such that $$ J_N(F)\geq J_N(E)+C[\mathrm{Vol}(E \triangle F)]^2\,, $$ for all smooth sets $F\subseteq \mathbb{T}^n$ such that $\mathrm{Vol}(F)=\mathrm{Vol}(E)$, $\mathrm{Vol}(F\triangle E)<\deltalta$, $\partial F \subseteq N_{\varepsilon}$ and \begin{equation} \partial F= \{y+\psi(y)\nu_E(y)\, : \, y \in \partial E\}, \end{equation} for a smooth $\psi$ with $\norma{\psi}_{W^{2,p}(\partial E)} < \deltalta$.\\ As a consequence, $E$ is a $W^{2,p}$--local minimizer of $J_N$ (as defined above). Moreover, if $F$ is $W^{2,p}$--close enough to $E$ and $J_N(F)=J_N(E)$, then $F=E$, that is, $E$ is locally the unique $W^{2,p}$--local minimizer. \end{thm} \begin{proof}[Sketch of the proof.] Following the line of proof of Theorem~\ref{W2pMin}, since the functional is not translation invariant we do not need Lemma~\ref{Lemma 3.8} and inequality~\eqref{3.38}, proved in Step~2 of the proof of such theorem, simplifies to $$ \inf\Bigl\{\Pi_F(\varphi):\, \varphi\in \widetilde{H}^1(\partial F)\,, \|\varphi\|_{H^1(\partial F)}=1\Bigr\}\geq\frac{m_0}2\,, $$ where $m_0$ is the constant defined in formula~\eqref{m0}. The proof of this inequality then goes exactly as there.\\ Coming to Step~3 of the proof of Theorem~\ref{W2pMin}, we do not need inequality~\eqref{assumption}, thus we do not need to replace $F$ by a suitable translated set $F-\eta$. Instead, we only need to observe that inequality~\eqref{3.46} is still satisfied. The rest of the proof remains unchanged. \end{proof} The short time existence and uniqueness Theorem~\ref{th:EscNis}, proved in~\cite{EsNi} in any dimension, holds also in the ``Neumann case" for the modified Mullins--Sekerka flow with parameter $\gamma\geq0$, obtained (as in Definition~\ref{MSF def}) by letting the outer normal velocity $V_t$ of the moving boundaries given by \begin{equation}\label{MSnl2} V_t= [\partialrtial_{\nu_t} w_{t}] \quad\text{ on $\partialrtial E_t$ for all $t\in [0, T)$,} \end{equation} where $\nu_t=\nu_{E_t}$ and $w_t=w_{E_t}$ is the unique solution in $H^1(\Omega)$ of the problem \begin{equation}\label{WE2} \begin{cases} \Delta w_{E_t}=0 & \text{in }\Omega\setminus \partial E_t\\ w_{E_t}= \mathrm{H} + 4\gamma v_{E_t} & \text{on } \, \partialrtial E_t, \end{cases} \end{equation} with $v_{E_t}$ the potential defined above and, as before, $[\partial_{\nu_t} w_t]$ is the jump of the outer normal derivative of $w_{E_t}$ on $\partial E_t$. Then, we conclude by stating the following analogue of Theorem~\ref{existence} (taking into account~Remark~\ref{existence+}). \begin{thm}\label{mainN} Let $\Omega$ be an open smooth subset of $\mathbb{R}^3$ and let $E\subseteq\Omega$ be a smooth strictly stable critical set for the nonlocal Area functional under a volume constraint, with $\partial E \cap \partial \Omega =$~{\rm{\O}} and $N_\varepsilon$ (with $\varepsilon<1$) a tubular neighborhood of $\partial E$, as in formula~\eqref{tubdef}. Then, for every $\alpha\in (0,1/2)$ there exists $M>0$ such that, if $E_0$ is a smooth set in $\mathfrak{C}^{1,\alpha}_M(E)$ satisfying $\mathrm{Vol}( E_0)= \mathrm{Vol}( E )$ and $$ \int_{\Omega}|\nabla w_{E_0}|^2\, dx\leq M\, $$ where $w_0=w_{E_0}$ is the function relative to $E_0$ as in problem~\eqref{WE} (with $\Omega$ in place of $\mathbb{T}^3\setminus\partial E$), then, the unique smooth solution $E_t$ to the Mullins--Sekerka flow (with parameter $\gamma\geq 0$) starting from $E_0$, given by Theorem~\ref{th:EscNis}, is defined for all $t\geq0$. Moreover, $E_t\to E$ exponentially fast in $C^k$,as $t\to +\infty$, for every $k\in\mathbb{N}$, with the meaning that the functions $\psi_{\eta, t} : \partial E \to \mathbb{R}$ representing $\partial E_t$ as ``normal graphs'' on $\partial E$, that is, $$ \partial E_t= \{ y+ \psi_{\eta,t} (y) \nu_{E+\eta}(y) \, : \, y \in \partial E\}, $$ satisfy, for every $k\in\mathbb{N}$, $$ \Vert \psi_{\eta, t}\Vert_{C^k(\partial E + \eta)}\leq C_ke^{-\beta_k t}, $$ for every $t\in[0,+\infty)$, for some positive constants $C_k$ and $\beta_k$. \end{thm} The proof of this result is similar to the one of Theorem~\ref{existence} and actually it is simpler since we do not need the argument used in Step~2 of such proof, where we controlled the translational component of the flow. Note also that in the statement of Proposition~\ref{2.6}, in this case, inequality~\eqref{2.13} holds for all $\varphi\in\widetilde{H}^1(\partial F)$. Finally, observe that under the hypotheses of Proposition~\ref{prop:nocrit} we may actually conclude that $E^\prime=E$, that is, there are no other critical sets close to $E$. \subsection{The surface diffusion flow -- Preliminary lemmas}\ \vskip.3em As for the modified Mullins--Sekerka flow, we start with the technical lemmas for the global existence result. \begin{lem}[Energy identities] \label{calculations2} Let $E_t\subseteq\mathbb{T}^n$ be a surface diffusion flow. Then, the following identities hold: \begin{equation} \label{der of A} \frac{d}{dt} \mathcal A(\partial E_t) = - \int_{\partial E_t} \abs{\nabla\mathrm{H}_t}^2\, d \mu_t\,, \end{equation} and \begin{align} \frac{d}{dt}\frac{1}{2}\int_{\partial E_t} \abs{ \nabla\mathrm{H}_t}^2\, d \mu_t= & -\Pi_{E_t}(\Delta_t\mathrm{H}_t ) -\int_{\partialrtial E_t} B_t (\nabla\mathrm{H}_t, \nabla\mathrm{H}_t) \Delta_t\mathrm{H}_t \, d \mu_t\\ &+ \frac{1}{2}\int_{\partialrtial E_t}\mathrm{H}_t \abs{\nabla\mathrm{H}_t}^2 \Delta_t\mathrm{H}_t\, d \mu_t\,, \label{der of DH} \end{align} where $\Pi_{E_t}$ is the quadratic form defined in formula~\eqref{Pieq} (with $\gamma=0$). \end{lem} \begin{proof} Let $\psi_t$ the smooth family of maps describing the flow as in formula~\eqref{sdf2}. By formula~\eqref{dermu2}, where $X$ is the smooth (velocity) vector field $X_t=\frac{\partialrtial\psi_t}{\partialrtial t}=(\Delta_t\mathrm{H}_t)\nu_{E_t}$ along $\partial E_t$, hence $X_\tau=X_t-\langle X_t \vert \nu_{E_t}\rangle\nu_{E_t}=0$ (as usual $\nu_{E_t}$ is the outer normal unit vector of $\partialrtial E_t$), following computation~\eqref{local}, we have \begin{align} \frac{d}{dt} \mathcal A(\partial E_t)& = \frac{d}{dt} \int_{\partial E_t}{d \mu_t}\\ &= \int_{\partial E_t} (\operatorname*{div}\nolimits X_\tau+\mathrm{H}_t \langle X \vert \nu_{E_t} \rangle ) \, d\mu_t\\ &= \int_{\partial E_t}\mathrm{H}_t\Delta_t\mathrm{H}_t\, d \mu_t\\ &=- \int_{\partial E_t} \abs{\nabla\mathrm{H}_t}^2 \, d \mu_t \, , \end{align} where the last equality follows integrating by parts. This establishes relation~\eqref{der of A}. In order to get relation~\eqref{der of DH} we also need the time derivatives of the evolving metric and of the mean curvature of $\partial E_t$, that we already computed in formulas~\eqref{derg2},~\eqref{1Acalc} and~\eqref{derH} (where the function $\varphi$ in this case is equal to $\Delta_t\mathrm{H}_t$ and $X_\tau=0$), that is, \begin{equation} \frac{\partialrtial g_{ij}}{\partialrtial t}= 2h_{ij}\Delta_t\mathrm{H}_t\qquad\text{ and }\qquad \frac{\partial g^{ij}}{\partial t} = - 2 h^{ij}\Delta_t\mathrm{H}_t\,, \end{equation} \begin{equation}\label{3bisbis} \frac{\partial\mathrm{H}_t}{\partial t}=- \abs{B_t}^2\Delta_t\mathrm{H}_t - \Delta_t \Delta_t\mathrm{H}_t \end{equation} Then, we compute \begin{align} \frac{d}{dt}\frac12 \int_{\partial E_t} \abs{\nabla\mathrm{H}_t}^2 \, d \mu_t =&\,\frac12 \int_{\partial E_t}\mathrm{H}_t\abs{\nabla\mathrm{H}_t}^2 \,\Delta_t\mathrm{H}_t\,d \mu_t -\int_{\partial E_t} h^{ij}\nabla_i\mathrm{H}_t\nabla_j\mathrm{H}_t\,\Delta_t\mathrm{H}_t\, d\mu_t\\ &\,-\int_{\partial E_t} g^{ij}\nabla_i\mathrm{H}_t\nabla_j\bigl(\abs{B}^2\Delta_t\mathrm{H}_t+\Delta_t \Delta_t\mathrm{H}_t\bigr)\, d\mu_t\\ =&\,\frac12 \int_{\partial E_t}\mathrm{H}_t\abs{\nabla\mathrm{H}_t}^2 \,\Delta_t\mathrm{H}_t\,d \mu_t-\int_{\partial E_t} B(\nabla\mathrm{H}_t,\nabla\mathrm{H}_t)\,\Delta_t\mathrm{H}_t\, d\mu_t\\ &\,+\int_{\partial E_t} \abs{B_t}^2(\Delta_t\mathrm{H}_t)^2\, d\mu_t +\int_{\partial E_t} \Delta_t\mathrm{H}_t\,\Delta_t \Delta_t\mathrm{H}_t\, d\mu_t\\ =&\,\frac12 \int_{\partial E_t}\mathrm{H}_t\abs{\nabla\mathrm{H}_t}^2 \,\Delta_t\mathrm{H}_t\,d \mu_t-\int_{\partial E_t} B_t(\nabla\mathrm{H}_t,\nabla\mathrm{H}_t)\,\Delta_t\mathrm{H}_t\, d\mu_t\\ &\,+\int_{\partial E_t} \abs{B_t}^2(\Delta_t\mathrm{H}_t)^2\, d\mu_t -\int_{\partial E_t} \vert\nabla\Delta_t\mathrm{H}_t\vert^2\, d\mu_t\,, \end{align} which is formula~\eqref{der of DH}, recalling the definition of $\Pi_{E_t}$ in formula~\eqref{Pieq}. \end{proof} {\em From now on, as before due to the dimension--dependence of the estimates that follow, we restrict ourselves to the three--dimensional case.} The following lemma is an easy consequence of Theorem~3.70 in~\cite{Aubin}, with $j=0$, $m=1$, $n=2$ and $r=q=2$, taking into account the previous discussion. \begin{lem}[Interpolation on boundaries]\label{interpolation} Let $F\subseteq\mathbb{T}^3$ be a smooth set. In the previous notations, for every $p\in[2,+\infty)$ there exists a constant $C=C(F,M,\alpha,p)>0$ such that for every set $E \in \mathfrak{C}^{1,\alpha}_M(F)$ and $g \in H^1(\partialrtial E)$, we have $$ \norma{g}_{L^p(\partialrtial E)} \leq C ( \norma{\nabla g}_{L^2(\partial E)}^{\theta} \norma{g}_{L^2(\partial E)}^{1-\theta} + \norma{g}_{L^2(\partial E)} )\,, $$ with $\theta=1-2/p$.\\ Moreover, the following Poincar\'e inequality holds $$ \norma{g-\overline{g}}_{L^p(\partial E)} \leq C \norma{\nabla g}_{L^2(\partialrtial E)} \, , $$ where $\overline{g}(x)= \fint_{\Gamma} g\, d \mu $, if $x$ belongs to a connected component $\Gamma$ of $\partialrtial E$. \end{lem} Then, we have the following mixed ``analytic--geometric'' estimate. \begin{lem}[$H^2$--estimates on boundaries]\label{laplacian} Let $F \subseteq \mathbb{T}^3$ be a smooth set. Then there exists a constant $C=C(F,M,\alpha,p)>0$ such that if $E\in \mathfrak{C}^{1,\alpha}_M(F)$ and $f\in H^1(\partial E)$ with $\Delta f\in L^2(\partial E)$, then $f\in H^2(\partial E)$ and $$ \norma{\nabla^2 f}_{L^2(\partialrtial E)}\leq C \norma{\Delta f}_{L^2(\partial E)}(1+ \norma{\mathrm{H}}_{L^4(\partial E)}^2) \, . $$ \end{lem} \begin{proof} We first claim that the following inequality holds, \begin{equation} \label{laplacian 1} \int_{\partialrtial E} \abs{\nabla ^2 f}^2 \, d \mu \leq \int_{\partialrtial E} \abs{ \Delta f}^2 \, d\mu + C\int_{\partialrtial E} \abs{B}^2 \abs{\nabla f}^2 \, d \mu \, . \end{equation} Indeed, if we integrate by parts the left--hand side, we obtain (the Hessian of a function is symmetric) $$ \int_{\partial E} g^{ik}g^{jl}\nabla^2_{ij}f \nabla^2_{kl}f \, d \mu = - \int_{\partial E}g^{ik}g^{jl} \nabla_k \nabla_j\nabla_if \nabla_l f \, d \mu \, . $$ Hence, interchanging the covariant derivatives and integrating by parts, we get \begin{align} - \int_{\partial E}g^{ik}g^{jl} \nabla_k \nabla_j\nabla_if \nabla_l f \, d \mu =&\,- \int_{\partial E} g^{ik}g^{jl}\nabla_j \nabla_k\nabla_if \nabla_l f \, d \mu\\ &\,- \int_{\partial E} g^{ik}g^{jl}R_{kjip}g^{ps}\nabla_sf\nabla_l f \, d \mu \\ =&\,- \int_{\partial E} g^{jl}\nabla_j \Delta f \nabla_l f\, d \mu-\int_{\partial E} \mathbb{R}ic(\nabla f, \nabla f)\, d \mu \\ =&\,\int_{\partial E} \abs{\Delta f}^2 \, d \mu+\int_{\partial E}\bigl[\abs{B}^2 |\nabla f|^2-\mathrm{H} B (\nabla f, \nabla f)\bigr]\, d \mu\\ \leq & \, \int_{\partial E} \abs{\Delta f}^2 \, d \mu + C\int_{\partial E} \abs{B}^2 \abs{\nabla f}^2 \, d \mu\, , \end{align} thus, inequality~\eqref{laplacian 1} holds (in the last passage we applied Cauchy--Schwarz inequality and the well known relation $|\mathrm{H}|\leq\sqrt{2}|B|$, then $C=1+\sqrt{2}$). We now estimate the last term in formula~\eqref{laplacian 1} by means of Lemma~\ref{interpolation} (which is easily extended to vector valued functions $g:\partial E\to\mathbb{R}^m$) with $g=\nabla f$ and $p=4$: \begin{align} \int_{\partialrtial E} \abs{B}^2\abs{\nabla f}^2 \, d \mu &\leq \norma{B}_{L^4(\partial E)}^2 \norma{\nabla f}_{L^4(\partial E)}^2 \\ &\leq C \norma{B}_{L^4(\partial E)}^2\bigl( \norma{\nabla^2 f}_{L^2(\partial E)}^{1/2} \norma{\nabla f}_{L^2(\partial E)}^{1/2} + \norma{\nabla f}_{L^2(\partial E)}\bigr)^2\\ &\leq C \norma{B}_{L^4(\partial E)}^2\bigl( \norma{\nabla^2 f}_{L^2(\partial E)} \norma{\nabla f}_{L^2(\partial E)} + \norma{\nabla f}_{L^2(\partial E)}^2\bigr)\,. \end{align} Hence, expanding the product on the last line, using Peter--Paul (Young) inequality on the first term of such expansion and ``adsorbing'' in the left hand side of inequality~\eqref{laplacian 1} the small fraction of the term $\norma{\nabla^2 f}_{L^2(\partial E)}^2$ that then appears, we obtain \begin{align} \norma{\nabla^2 f}_{L^2(\partial E)}^2 &\leq C ( \norma{\Delta f}_{L^2(\partial E)}^2 + \norma{\nabla f}_{L^2(\partial E)}^2 ( \norma{B}_{L^4(\partial E)}^2 + \norma{B}_{L^4(\partial E)}^4 ) ) \nonumber \\ &\leq C ( \norma{\Delta f}_{L^2(\partial E)}^2 + \norma{\nabla f}_{L^2(\partial E)}^2 ( 1+ \norma{B}_{L^4(\partial E}^4 ) ) \,. \label{aY} \end{align} By the fact that $\Delta f$ has zero average on each connected component of $\partial E$, there holds \begin{align} \norma{\nabla f}_{L^2(\partial E)}^2 &= -\int_{\partialrtial E} f \Delta f \, d \mu\\ &= - \int_{\partial E}(f- \overline f) \Delta f \, d \mu \nonumber \\ &\leq \norma{f- \overline f}_{L^2(\partial E)}\norma{\Delta f }_{L^2(\partial E)}\\ &\leq C \norma{\nabla f}_{L^2(\partial E)} \norma{\Delta f }_{L^2(\partial E)}\,,\label{poin} \end{align} where we used Lemma~\ref{interpolation} again, hence, \begin{equation} \norma{\nabla f}_{L^2(\partial E)}\leq C \norma{\Delta f }_{L^2(\partial E)}\,.\label{poin2} \end{equation} Thus, from inequality~\eqref{aY}, we deduce \begin{equation}\label{eq10010} \norma{\nabla ^2 f}_{L^2(\partial E)}^2 \leq C \norma{\Delta f}_{L^2(\partial E)}^2 ( 1 + \norma{B}_{L^4(\partial E)}^4 )\, . \end{equation} Now, by means of Calder\'on--Zygmund estimates, it is possible to show (see~\cite{DDM}) that there exists a constant $C>0$ depending only on $F$, $M$, $\alpha$ and $q>1$ such that for every $E\in \mathfrak{C}^{1,\alpha}_M(F)$, there holds \begin{equation}\label{CZG2} \norma{B}_{L^q(\partial E)} \leq C(1+ \norma{\mathrm{H}}_{L^q(\partial E)})\,. \end{equation} Then, since it is easy to check that also all the other constant in the previous inequalities (and the ones coming from Lemma~\ref{interpolation} also) depend only on $F$, $M$, $\alpha$ and $p$, if $E\in \mathfrak{C}^{1,\alpha}_M(F)$, substituting this estimate, with $q=4$, in formula~\eqref{eq10010}, the thesis of the lemma follows. \end{proof} The following lemma provides a crucial ``geometric interpolation'' that will be needed in the proof of the main theorem. \begin{lem}[Geometric interpolation]\label{nasty} Let $F \subseteq \mathbb{T}^3$ be a smooth set. Then there exists a constant $C=C(F,M,\alpha)>0$ such that the following estimates holds $$ \int_{\partialrtial E} \abs{B}\abs{\nabla\mathrm{H}}^2 \abs{\Delta\mathrm{H}} \, d \mu \leq C \norma{\nabla\Delta\mathrm{H}}_{L^2(\partial E)}^2 \, \norma{\nabla \mathrm{H}}_{L^2(\partial E)}\, (1+ \norma{\mathrm{H}}_{L^6(\partial E)}^3 )\,, $$ for every $E\in \mathfrak{C}^{1,\alpha}_M(F)$. \end{lem} \begin{proof} First, by a standard application of H\"older inequality, we have $$ \int_{\partialrtial E} \abs{B} \abs{\nabla\mathrm{H}}^2 \abs{\Delta\mathrm{H}} \, d \mu \leq \norma{\Delta\mathrm{H}}_{L^3(\partial E)} \Bigl( \int_{\partialrtial E} \abs{B}^\frac{3}{2}\abs{\nabla\mathrm{H}}^3 \, d \mu \Bigl)^{2/3}. $$ Then, using the Poincar\'e inequality stated in Lemma~\ref{interpolation} and the fact that $\Delta\mathrm{H}$ has zero average on each connected component of $\partial E$, we get $$ \norma{\Delta\mathrm{H}}_{L^3(\partial E)} \leq C \norma{ \nabla\Delta\mathrm{H} }_{L^2(\partial E)}. $$ Now, we use H\"older inequality again $$ \Bigl( \int_{\partialrtial E} \abs{B}^\frac{3}{2}\abs{\nabla\mathrm{H}}^3 \, d \mu \Bigr)^{2/3} \leq \Bigl( \int_{\partialrtial E}\abs{\nabla\mathrm{H}}^{4} \, d \mu \Bigr)^{1/2}\Bigl( \int_{\partialrtial E}\abs{B}^{6} \, d \mu \Bigr)^{1/6} \, , $$ and we apply Lemma~\ref{interpolation} with $p=4$, $$ \Bigl( \int_{\partialrtial E}\abs{\nabla\mathrm{H}}^{4} \, d \mu \Bigr)^{1/2} \leq C ( \norma{\nabla^2\mathrm{H}}_{L^2(\partial E)} \norma{\nabla\mathrm{H}}_{L^2(\partial E)} + \norma{\nabla\mathrm{H}}_{L^2(\partial E)}^2 )\,. $$ Combining all these inequalities, we conclude $$ \int_{\partialrtial E} \abs{B} \abs{\nabla\mathrm{H}}^2 \abs{\Delta\mathrm{H}} \, d \mu \leq C\norma{\nabla\Delta\mathrm{H} }_{L^2(\partial E)} \, \norma{B}_{L^6(\partial E)} \, \norma{\nabla\mathrm{H}}_{L^2(\partial E)}(\norma{\nabla^2\mathrm{H}}_{L^2(\partial E)} + \norma{\nabla\mathrm{H}}_{L^2(\partial E)})\,. $$ By Lemma~\ref{laplacian} and estimate~\eqref{poin2}, with $\mathrm{H}$ in place of $f$, the right--hand side of the previous inequality can be bounded from above by \begin{equation}\label{stima} C \norma{\nabla\Delta\mathrm{H} }_{L^2(\partial E)} \, \norma{B}_{L^6(\partial E)} \, \norma{\Delta\mathrm{H}}_{L^2(\partial E)} \, \norma{\nabla\mathrm{H}}_{L^2(\partial E)} \, (1 + \norma{H}_{L^4(\partial E)}^2). \end{equation} Hence, using again Poincar\'e inequality and estimate~\eqref{CZG2} with $q=6$, we have $$ \norma{\Delta\mathrm{H}}_{L^2(\partial E)} \leq C \norma{\nabla\Delta\mathrm{H} }_{L^2(\partial E)} $$ and $$ \norma{B}_{L^6(\partial E)} \leq C(1+ \norma{\mathrm{H}}_{L^6(\partial E)})\,. $$ Finally, using this relations and H\"older inequality, we obtain the thesis $$ \int_{\partialrtial E} \abs{B} \abs{\nabla\mathrm{H}}^2 \abs{\Delta\mathrm{H}} \, d \mu \leq C \norma{\nabla\Delta\mathrm{H} }_{L^2(\partial E)}^2 \, \norma{\nabla\mathrm{H}}_{L^2(\partial E)} \, (1+ \norma{\mathrm{H}}^3_{L^6(\partial E)})\,. $$ \end{proof} We now remind that since $\partialrtial E$ can be disconnected (as in the case of lamellae), the Poincar\'e inequality could fail for $\partialrtial E$. However, if $E$ is sufficiently close to a stable critical set then it is true for the mean curvature of $\partial E$. \begin{lem}[Geometric Poincar\'e inequality]\label{lm:geopoinc} Fixed $p>2$ and a smooth strictly stable critical set $F\subseteq\mathbb{T}^3$, let $\deltalta>0$ be the constant provided by Proposition~\ref{2.6}, with $\theta=1$. Then, for $M$ small enough, there exists a constant $C=C(F,M,\alpha,p)>0$ such that \begin{equation}\label{geopoinc} \int_{\partial E} \abs{\mathrm{H}-\overline{\mathrm{H}}}^2\, d \mu \leq C\int_{\partial E}\abs{\nabla\mathrm{H}}^2\, d \mu \,, \end{equation} for every set $E\in \mathfrak{C}^{1,\alpha}_M(F)$ such that $\partial E\subseteq N_\varepsilon$ with \begin{equation} \partial E= \{y + \psi (y) \nu_F(y) \, : \, y \in \partial F \}\, , \end{equation} for a smooth function $\psi$ with $\norma{\psi}_{W^{2,p}(\partial F)}<\deltalta$. \end{lem} \begin{proof} Since $$ \int_{\partial E}(\mathrm{H} -\overline\mathrm{H})\nu_E \, d \mu=0 \, , $$ there holds $$ \int_{\partial E} \abs{\mathrm{H}- \overline\mathrm{H} - \langle \eta \vert \nu_E \rangle }^2 \, d \mu =\norma{\mathrm{H} - \overline\mathrm{H} }_{L^2(\partial E)}^2+\int_{\partial E} \langle \eta \vert \nu_E \rangle^2 \, d \mu\geq\norma{\mathrm{H} - \overline\mathrm{H} }_{L^2(\partial E)}^2 $$ for all $\eta \in\mathbb{R}^3$. Choosing $M<\deltalta$, we may then apply Proposition~\ref{2.6} with $\theta=1$ and $\varphi=\mathrm{H}-\overline\mathrm{H}$, obtaining \begin{equation} \sigma_1 \int_{\partial E}\abs{\mathrm{H}-\overline\mathrm{H}}^2\, d \mu \leq \int_{\partial E}\abs{\nabla\mathrm{H}}^2\, d \mu-\int_{\partial E}\abs{B}^2 \abs{\mathrm{H}-\overline\mathrm{H}}^2\, d \mu \leq \int_{\partial E}\abs{\nabla\mathrm{H}}^2\, d \mu\,. \end{equation} \end{proof} The following lemma is straightforward. \begin{lem}\label{5.1} Let $E \subseteq \mathbb{T}^3$ be a smooth set. If $f\in H^1(\partial E)$ and $g\in W^{1,4}(\partial E)$, then $$ \norma{\nabla(fg)}_{L^2(\partial E)}\leq C\norma{\nabla f}_{L^2(\partial E)}\|g\|_{L^\infty(\partial E)}+C\|f\|_{L^4(\partial E)}\|\nabla g\|_{L^4(\partial E)}\,, $$ for a constant $C$ independent of $E$. \end{lem} \begin{proof} We estimate with Cauchy--Schwarz inequality, \begin{align*} \norma{\nabla(fg)}_{L^2(\partial E)}^2\leq&\,2\norma{\nabla f}_{L^2(\partial E)}^2\norma{g}_{L^\infty(\partial E)}^2 +2\int_{\partial E}|f|^2|\nabla g|^2\,d\mu\\ \leq&\, 2\norma{\nabla f}_{L^2(\partial E)}^2\|g\|_{L^\infty(\partial E)}^2+2\|f\|_{L^4(\partial E)}^2\|\nabla g\|_{L^4(\partial E)}^2\,, \end{align*} hence the thesis follows. \end{proof} As a consequence, we prove the following result. \begin{lem}\label{5.2sdf} Let $F\subseteq\mathbb{T}^3$ be a smooth set and $E\in\mathfrak{C}^{1,\alpha}_M(F)$. Then, for $M$ small enough, there holds $$ \|\psi_E\|_{W^{3,2}(\partial F)}\leq C(F,M,\alpha)(1+\|\mathrm{H}\|_{H^1(\partial E)}^2)\,, $$ where $\mathrm{H}$ is the mean curvature of $\partial E$ (the function $\psi_E$ is defined by formula~\eqref{front}). \end{lem} \begin{proof} As we do in Lemma~\ref{5.2}, by a standard localization/partition of unity/straightening argument, we may reduce ourselves to the case where the function $\psi_E$ is defined in a disk $D\subseteq\mathbb{R}^2$ and $\|\psi_E\|_{C^{1,\alpha}(D)}\leq M$. Fixed a smooth cut--off function $\varphi$ with compact support in $D$ and equal to one on a smaller disk $D'\subseteq D$, we have again relation~\eqref{noia1} (see also~\cite{Man}). \\ Then, using Lemma~\ref{5.1} and recalling that $\|\psi_E\|_{C^{1,\alpha}(D)}\leq M$, we estimate \begin{align} \norma{\nabla\Delta(\varphi\psi_E)}_{L^2(D)}\leq C(F,M,\alpha)\bigl(&\,M^2\norma{\nabla^3(\varphi\psi_E)}_{L^2(D)}+\Vert\nabla\mathrm{H}\Vert_{L^2(\partial E)}(1+\norma{\nabla\psi_E}_{L^\infty(D)})\\ &\,+\norma{\mathrm{H}}_{L^4(\partial E)}(1+\norma{\psi_E}_{W^{2,4}(D)})+1+\norma{\psi_E}_{W^{2,4}(D)}\bigr)\,. \end{align} We now use the fact that, by a simple integration by part argument, if $u$ is a smooth function with compact support in $\mathbb{R}^2$, there holds $$ \Vert\nabla\Delta u\Vert_{L^2(\mathbb{R}^2)}=\Vert\nabla^3u\Vert_{L^2(\mathbb{R}^2)}\,, $$ hence, \begin{align*} \Vert\nabla^3(\varphi\psi_E)\Vert_{L^2(D)} &\,=\Vert\nabla\Delta(\varphi\psi_E)\Vert_{L^2(D)}\\ &\,\leq C(F,M,\alpha)\bigl(M^2\norma{\nabla^3(\varphi\psi_E)}_{L^2(D)}+\Vert\nabla\mathrm{H}\Vert_{L^2(\partial E)}(1+\norma{\nabla\psi_E}_{L^\infty(D)})\\ &\,\phantom{\leq C(F,M,\alpha)\bigl(\,}+\norma{\mathrm{H}}_{L^4(\partial E)}(1+\norma{\psi_E}_{W^{2,4}(D)})+1+\norma{\psi_E}_{W^{2,4}(D)}\bigr)\,, \end{align*} then, if $M$ is small enough, we have \begin{equation}\label{eqcar10020} \Vert\nabla^3(\varphi\psi_E)\Vert_{L^2(D)} \leq C(F,M,\alpha)(1+\Vert\mathrm{H}\Vert_{H^1(\partial E)})(1+\norma{\mathrm{Hess}\,\psi_E}_{L^4(D)})\,, \end{equation} as \begin{equation}\label{eqcar10040} \Vert\mathrm{H}\Vert_{L^4(\partial E)}\leq C(F,M,\alpha)\Vert\mathrm{H}\Vert_{H^1(\partial E)}\,, \end{equation} by Theorem~3.70 in~\cite{Aubin}.\\ By the Calder\'on--Zygmund estimates (holding uniformly for every hypersurface $\partial E$, with $E\in\mathfrak{C}^{1,\alpha}_M(F)$, see~\cite{DDM}), we have again the inequality~\eqref{Cald-Zyg} and the most useful estimation~\eqref{normadeltapsi1}. Hence, possibly choosing a smaller $M$, we conclude (as in inequality~\eqref{normadeltapsi2}) \begin{equation} \norma{\Delta \psi_E}_{L^4(D)} \leq C(F,M,\alpha) (1 + \norma{\mathrm{H}}_{L^4(\partial E)}) \leq C(F,M,\alpha) (1 + \norma{\mathrm{H}}_{H^1(\partial E)})\,, \end{equation} again by inequality~\eqref{eqcar10040}.\\ Thus, by estimate~\eqref{Cald-Zyg}, we get \begin{equation}\label{eqcar10030sdf} \norma{\mathrm{Hess} \, \psi_E}_{L^{4}(D)}\leq C(F,M,\alpha) (1 + \norma{\mathrm{H}}_{H^1(\partial E)})\,, \end{equation} and using this inequality in estimate~\eqref{eqcar10020}, $$ \Vert\nabla^3(\varphi\psi_E)\Vert_{L^2(D)} \leq C(F,M,\alpha)(1+\Vert\mathrm{H}\Vert_{H^1(\partial E)})^2\,, $$ hence, $$ \Vert\nabla^3\psi_E\Vert_{L^2(D')} \leq C(F,M,\alpha)(1+\Vert\mathrm{H}\Vert_{H^1(\partial E)})^2\leq C(F,M,\alpha)(1+\Vert\mathrm{H}\Vert_{H^1(\partial E)}^2)\,. $$ The inequality in the statement of the lemma then easily follows by this inequality, estimate~\eqref{eqcar10030sdf} and $\Vert\psi_E\Vert_{C^{1,\alpha}(D)}\leq M$, with a standard covering argument. \end{proof} Now, we state a compactness result whose proof is very close in spirit to the proof of Lemma~\ref{w52conv}, however we present it explicitly in order to show how the lemmas above come differently into play. \begin{lem}[Compactness]\label{w32conv} Let $F\subseteq\mathbb{T}^3$ be a smooth set and $E_n\subseteq \mathfrak{C}^{1,\alpha}_M(F) $ a sequence of smooth sets such that $$ \sup_{n\in\mathbb{N}}\,\int_{\partial E_n}|\nabla \mathrm{H}_n|^2\, d\mu_n<+\infty\,. $$ Then, if $\alpha\in(0,1/2)$ and $M$ is small enough, there exists a smooth set $F'\in \mathfrak{C}^1_M(F)$ such that, up to a (non relabeled) subsequence, $E_n\to F'$ in $W^{2,p}$ for all $1\leq p<+\infty$.\\ Moreover, if inequality~\eqref{geopoinc} holds for every set $E_n$ with a constant $C$ independent of $n$ and $$ \int_{\partial E_n}|\nabla\mathrm{H}_n|^2\, d\mu_n\to 0\,, $$ then $F'$ is critical for the volume--constrained Area functional $\mathcal A$ and the convergence $E_n\to F'$ is in $W^{3,2}$. \end{lem} \begin{proof} We first claim that \begin{equation}\label{claim1111sdf} \sup_{n\in\mathbb{N}}\,\|\mathrm{H}_n\|_{H^{1}(\partial E_n)}<+\infty. \end{equation} We set $\widetilde{\mathrm{H}}_n=\fint_{\partial E_n}\mathrm{H}_n\,d\mu_n$, then, by the ``geometric'' Poincar\'e inequality of Lemma~\ref{lm:geopoinc}, which holds with a ``uniform'' constant $C=C(F,M,\alpha)$, for all the sets $E\in\mathfrak{C}^{1,\alpha}_M(F)$ (see~\cite{DDM}), if $M$ is small enough, we have \begin{equation} \|\mathrm{H}_n-\widetilde{\mathrm{H}}_n\|^2_{H^{1}(\partial E_n)}\leq\sup_{n\in\mathbb{N}}\,\int_{\partial E_n}|\nabla \mathrm{H}_n|^2\, d\mu_n<C<+\infty \end{equation} with a constant $C$ independent of $n\in\mathbb{N}$.\\ Then, we note that, as in Lemma~\ref{w52conv}, by the uniform $C^{1,\alpha}$--bounds on $\partial E_n$, we may find a solid cylinder of the form $C=D\times(-L,L)$, with $D\subseteq\mathbb{R}^{2}$ a ball centered at the origin and functions $f_n$, with \begin{equation}\label{w32} \sup_{n\in\mathbb{N}}\|f_n\|_{C^{1,\alpha}(\overline D)}<+\infty\,, \end{equation} such that $\partial E_n\cap C=\{(x',x_n)\in D\times(-L,L):\, x_n= f_n(x')\}$ with respect to a suitable coordinate frame (depending on $n\in\mathbb{N}$). Hence, recalling the formula~\eqref{intHn}, the uniform bound~\eqref{w32} and the fact that $\|\mathrm{H}_n- \widetilde{\mathrm{H}}_n\|_{H^{1}(\partial E_n)}$ are equibounded, we get that ${\widetilde{\mathrm{H}}_n}$ are also equibounded (by a standard ``localization'' argument, ``uniformly'' applied to all the hypersurfaces $\partial E_n$). Therefore, the claim~\eqref{claim1111sdf} follows.\\ By applying the Sobolev embedding theorem on each connected component of $\partialrtial F$, we have that $$ \norma{\mathrm{H}_n}_{L^p(\partial E_n)} \leq C \norma{\mathrm{H}_n}_{H^{1}(\partial E_n)} <C<+\infty\qquad \text{for all $p \in [1,+\infty)$.} $$ for a constant $C$ independent of $n\in\mathbb{N}$.\\ Now, as before, we obtain $$ \norma{B}_{L^p(\partial E)} \leq C(1+ \norma{\mathrm{H}}_{L^p(\partial E)})\,. $$ for every $E\in \mathfrak{C}^{1,\alpha}_M(F)$ with a uniform constant $C$. Then, if we write $$ \partial E_n =\{y+\psi_n(y)\nu_F(y):\, y\in \partial F\}\,, $$ we have $\sup_{n\in\mathbb{N}}\|\psi_n\|_{W^{2,p}(\partial F)}<+\infty$, for all $p \in [1,+\infty)$.\\ Thus, by the Sobolev compact embedding $W^{2,p}(\partial F)\hookrightarrow C^{1,\alpha}(\partial F)$, up to a subsequence (not relabeled), there exists a set $F'\in \mathfrak{C}^{1,\alpha}_M(F)$ such that $$ \psi_n\to \psi_{F'} \text{ in $C^{1,\alpha}(\partial F)$,} $$ for all $\alpha\in (0,1/2)$.\\ From estimate~\eqref{claim1111sdf} and Lemma~\ref{5.2sdf} (possibly choosing a smaller $M$), we have then that the functions $\psi_n$ are bounded in $W^{3,2}(\partial F)$. Hence, possibly passing to another subsequence (again not relabeled), we conclude that $E_n \to F'$ in $W^{2,p}$ for every $p\in[1,+\infty)$, by the Sobolev compact embeddings.\\ For the second part of the lemma, we first observe that if $$ \int_{\partial E_n}|\nabla\mathrm{H}_n|^2\, d\mu_n\to 0\,, $$ then there exists $\lambda\in \mathbb{R}$ and a subsequence $E_n$ (not relabeled) such that $$ \mathrm{H}_n\big(\cdot + \psi_n(\cdot)\nu_F(\cdot)\big)\to \lambda= \mathrm{H}\big(\cdot + \psi_{F'}(\cdot)\nu_F(\cdot)\big) $$ in $H^{1}(\partial F)$, where $\mathrm{H}$ is the mean curvature of $F'$. Hence $F'$ is critical.\\ To conclude the proof we only need to show that $\psi_n$ converge to $\psi =\psi_{F'}$ in $W^{3,2}(\partial F)$.\\ Fixed $\deltalta>0$, arguing as in the proof of Lemma~\ref{5.2sdf}, we reduce ourselves to the case where the functions $\psi_n$ are defined on a disk $D\subseteq\mathbb{R}^2$, are bounded in $W^{3,2}(D)$, converge in $W^{2,p}(D)$ for all $p\in[1,+\infty)$ to $\psi\in W^{3,2}(D)$ and $\|\nabla\psi\|_{L^\infty(D)}\leq\deltalta$. Then, fixed a smooth cut--off function $\varphi$ with compact support in $D$ and equal to one on a smaller disk $D'\subseteq D$, we have \begin{align*} \frac{\Delta(\varphi\psi_n)}{\sqrt{1+|\nabla\psi_n|^2}}-\frac{\Delta(\varphi\psi)}{\sqrt{1+|\nabla\psi|^2}} =&\,(\nabla^2(\varphi\psi_n)-\nabla^2(\varphi\psi))\frac{\nabla\psi \nabla\psi}{(1+|\nabla\psi|^2)^{3/2}}\\ &\,+ \nabla^2(\varphi\psi_n)\Bigl(\frac{\nabla\psi_n\nabla\psi_n}{(1+|\nabla\psi_n|^2)^{3/2}}-\frac{\nabla\psi \nabla\psi}{(1+|\nabla\psi|^2)^{3/2}}\Bigr)\\ &\,+\varphi(\mathrm{H}_n-\mathrm{H})+R(x,\psi_n,\nabla\psi_n)-R(x,\psi,\nabla\psi)\,, \end{align*} where $R$ is a smooth Lipschitz function.\\ Then, using Lemma~\ref{5.1}, an argument similar to the one of the proof of Lemma~\ref{5.2sdf} shows that \begin{align*} \bigg \Vert \nabla\Big( \frac{\Delta(\varphi\psi_n)}{\sqrt{1+|\nabla\psi_n|^2}}-\frac{\Delta(\varphi\psi)}{\sqrt{1+|\nabla\psi|^2}} \Big ) \bigg \Vert_{L^2(D)} & \, \leq C(M)\big(\deltalta^2\norma{\nabla^3(\varphi\psi_n)-\nabla^3(\varphi\psi)}_{L^2(D)}\\ & \,+\|\nabla^2(\varphi\psi_n)-\nabla^2(\varphi\psi)\|_{L^4 (D)}\|\nabla^2\psi\|_{L^4(D)} \\& \, + \|\nabla^3(\varphi\psi_n)\|_{L^2(D)}\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D)}\\ & \,+\|\nabla^2(\varphi\psi_n)\|_{L^4(D)}(\|\nabla^2\psi_n\|_{L^4}+\|\nabla^2\psi\|_{L^4(D)})\\ & \, +\|\nabla\mathrm{H}_n-\nabla\mathrm{H}\|_{L^2(D)}+\|\psi_n-\psi\|_{W^{2,4}(D)}\big)\,. \end{align*} Being $\mathrm{H}$ constant, that is $\nabla\mathrm{H}=0$, by using Lemma~\ref{5.1} again and arguing as in the proof of Lemma~\ref{5.2sdf}, we finally get $$ \norma{\nabla^3(\varphi\psi_n)-\nabla^3(\varphi\psi)}_{L^2(D)}\leq C(M)\big(\|\psi_n-\psi\|_{W^{2,4}(D)} +\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D)}+\| \nabla\mathrm{H}_n\|_{L^2(D)}\big)\,, $$ hence, $$ \norma{\nabla^3\psi_n-\nabla^3\psi}_{L^2(D')}\leq C(M)\big(\|\psi_n-\psi\|_{W^{2,4}(D)}+\|\nabla\psi_n-\nabla\psi\|_{L^\infty(D)}+\| \nabla\mathrm{H}_n\|_{L^2(D)}\big)\,, $$ from which the conclusion follows, by the first part of the lemma and a standard covering argument. \end{proof} \subsection{The surface diffusion flow -- The main theorem}\ \vskip.3em We now show the global existence result for the surface diffusion flow, whose proof is very similar to the one of Theorem~\ref{existence}. However, in order to make it clear, we present it in a detailed way. \begin{thm}\label{existence2} Let $E\subseteq\mathbb{T}^3$ be a strictly stable critical set for the Area functional under a volume constraint and let $N_\varepsilon$ be a tubular neighborhood of $\partial E$, as in formula~\eqref{tubdef}. For every $\alpha\in (0,1/2)$ there exists $M>0$ such that, if $E_0$ is a smooth set in $C^{1,\alpha}_M(E)$ satisfying $\mathrm{Vol}( E_0)= \mathrm{Vol}( E )$ and $$ \int_{\partial E_0} \vert \nabla \mathrm{H}_0\vert^2\, d \mu_0 \leq M, $$ then the unique smooth solution $E_t$ of the surface diffusion flow starting from $E_0$, given by Proposition~\ref{th:EMS1}, is defined for all $t\geq0$. Moreover, $E_t\to E+\eta$ exponentially fast in $W^{3,2}$ as $t\to +\infty$ (recall the definition of convergence of sets in Subsection~\ref{stabsec}), for some $\eta\in \mathbb{R}^3$, with the meaning that the functions $\psi_{\eta, t} : \partial E+ \eta \to \mathbb{R}$ representing $\partial E_t$ as ``normal graphs'' on $\partial E+ \eta$, that is, $$ \partial E_t= \{ y+ \psi_{\eta,t} (y) \nu_{E+\eta}(y) \, : \, y \in \partial E+\eta \}, $$ satisfy $$ \Vert \psi_{\eta, t}\Vert_{W^{3,2}(\partial E + \eta)}\leq Ce^{-\beta t} \, , $$ for every $t \in [0, +\infty)$, for some positive constants $C$ and $\beta$. \end{thm} \begin{remark}\label{existence2+} The convergence of $E_t\to E+\eta$ is actually smooth, that is, for every $k\in\mathbb{N}$, there holds $$ \Vert \psi_{\eta, t}\Vert_{C^k(\partial E + \eta)}\leq C_ke^{-\beta_k t}, $$ for every $t\in[0,+\infty)$, for some positive constants $C_k$ and $\beta_k$. This is a particular case of Theorem~5.1 in~\cite{FusJulMor18}, proved by means of standard parabolic estimates and interpolation (and Sobolev embeddings), using the exponential decay in $W^{3,2}$, analogously to the modified Mullins--Sekerka flow (Remark~\ref{existence+}). \end{remark} \begin{remark} The extra condition in the theorem on the $L^2$--smallness of the gradient of $\mathrm{H}_0$ (see the second part of Lemma~\ref{w32conv} and its proof) implies that the mean curvature of $\partial E_0$ is ``close'' to be constant, as it is for the set $E$ or actually for any critical set (recall Remark~\ref{closedness}). \end{remark} \begin{proof}[Proof of Theorem~\ref{existence2}] As in proof of Theorem~\ref{existence}, $C$ will denote a constant depending only on $E$, $M$ and $\alpha$, whose value may vary from line to line. Assume that the surface diffusion flow $E_t$ is defined for $t$ in the maximal time interval $[0,T(E_0))$, where $T(E_0)\in (0,+\infty]$ and let the moving boundaries $\partial E_t$ be represented as ``normal graphs'' on $\partial E$ as $$ \partial E_t= \{ y+ \psi_t(y) \nu_{E}(y) \, : \, y \in \partial E\} \, , $$ for some smooth functions $\psi_t:\partial E\to \mathbb{R}$.\\ We recall that, by Proposition~\ref{th:EMS1}, for every $F\in \mathfrak{C}^{2,\alpha}_M(E)$, the flow is defined in the time interval $[0, T)$, with $T=T(E,M,\alpha)>0$.\\ As before, we show the theorem for the smooth sets $E_0\subseteq\mathbb{T}^3$ satisfying \begin{equation}\label{initial_2} \mathrm{Vol}(E_0\Delta E)\leq M_1,\quad\|\psi_0\|_{C^{1,\alpha}(\partial E)}\leq M_2\quad\text{and}\quad \int_{\partial E_0} |\nabla \mathrm{H}_0|^2\, d\mu_0\leq M_3\,, \end{equation} for some positive constants $M_1,M_2,M_3$, then we get the thesis by setting $M=\min\{M_1,M_2,M_3\}$.\\ For any set $F\in \mathfrak{C}^{1,\alpha}_{M}(E)$, we define quantity in~\eqref{D(F)0} and by the same arguments we obtain estimation~\eqref{D(F)}.\\ Hence, by this discussion, the initial smooth set $E_0\in\mathfrak{C}^{1,\alpha}_M(E)$ satisfies $D(E_0)\leq M\leq M_1$ (having chosen $\varepsilon<1$).\\ By rereading the proof of Lemma~\ref{w32conv}, it follows that for $M_2,M_3$ small enough, if $$ \Vert\psi_F\Vert_{C^{1,\alpha}(\partial E)}\leq M_2 $$ and \begin{equation}\label{ex-de02_2} \int_{\partial F} |\nabla \mathrm{H}|^2\,d \mu \leq M_3\,, \end{equation} then \begin{equation}\label{eqcar50001_2} \|\psi_F\|_{W^{2,6}(\partial E)}\leq \omega(\max\{M_2,M_3\})\,, \end{equation} where $s\mapsto\omega(s)$ is a positive nondecreasing function (defined on $\mathbb{R}$) such that $\omega(s)\to 0$ as $s\to 0^+$. This clearly implies \begin{equation}\label{eqcar50003_2} \|\nu_F\|_{W^{1,6}(\partial F)}\leq \omega'(\max\{M_2,M_3\})\,, \end{equation} for a function $\omega'$ with the same properties of $\omega$ (also in this case, $\omega$ and $\omega'$ only depend on $E$ and $\alpha$, for $M$ small enough). Moreover, thanks to Lemma~\ref{lm:geopoinc}, there exists $C>0$ such that, choosing $M_2,M_3$ small enough, in order that $\omega(\max\{M_2,M_3\})$ is small enough, we have \begin{equation}\label{de02bis_2} \int_{\partial F}\abs{\mathrm{H}-\overline \mathrm{H}}^2\, d \mu \leq C\int_{\partial F}|\nabla \mathrm{H}|^2\, d \mu \, , \end{equation} where, as usual, $\overline \mathrm{H}$ is the average of $\mathrm{H}$ over $\partial F$. We again split the proof of the theorem into steps. \noindent \textbf{Step ${\mathbf 1}$} ({Stopping--time})\textbf{.}\\ Let $\overline T\leq T(E_0)$ be the maximal time such that \begin{equation}\label{Tprimo_2} \mathrm{Vol}(E_t\Delta E)\leq 2M_1,\quad\|\psi_t\|_{C^{1,\alpha}(\partial E)}\leq 2M_2\quad\text{and}\quad \int_{\partial E_t} |\nabla \mathrm{H}_t|^2\, d \mu_t\leq 2M_3\,, \end{equation} for all $t\in [0, \overline T)$. Hence, \begin{equation}\label{eqcar50005_2} \|\psi_t\|_{W^{2,6}(\partial E)}\leq \omega(2\max\{M_2,M_3\})\, \end{equation} for all $t\in [0, \overline T)$, as in formula~\eqref{eqcar50001_2}.\\ As before, we claim that by taking $M_1,M_2,M_3$ small enough, we have $\overline T=T(E_0)$. \noindent \textbf{Step ${\mathbf 2}$} ({Estimate of the translational component of the flow})\textbf{.}\\ We want to show that there exists a small constant $\theta>0$ such that \begin{equation} \label{not a translation_2} \min_{\eta\in{\mathrm{O}}_E} \norma{ \Delta \mathrm{H}_t- \langle\eta , \nu_t\rangle }_{L^2(\partial E_t)}\geq \theta\norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}\qquad\text{for all }t\in [0, \overline T)\,, \end{equation} where ${\mathrm{O}}_F$ is defined by formula~\eqref{OOeq}.\\ If $M$ is small enough, clearly there exists a constant $C_0=C_0(E,M,\alpha)>0$ such that, for every $i\in\mathrm{I}I_E$, we have $\Vert \langle e_i,\nu_t\rangle\Vert_{L^2(\partial E_t)}\geq C_0>0$, holding $\Vert \langle e_i,\nu_E\rangle\Vert_{L^2(\partial E)}>0$. It is then easy to show that the vector $\eta_t\in{\mathrm{O}}_E$ realizing such minimum is unique and satisfies \begin{equation} \label{not a translation 2_2} \Delta \mathrm{H}_t = \langle \eta_t , \nu_t \rangle + g, \end{equation} where $g\in L^2(\partial E_t)$ is chosen as in relation~\eqref{not a translation 2}. Moreover, the inequality \begin{equation}\label{eqcar50002_2} |\eta_t|\leq C \norma{\Delta \mathrm{H}_t}_{L^2(\partialrtial E_t)} \end{equation} holds, with a constant $C$ depending only on $E$, $M$ and $\alpha$.\\ We now argue by contradiction, assuming $\|g\|_{L^2(\partial E_t)} < \theta \norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}$.\\ First we recall that $\Delta \mathrm{H}_t$ has zero average. Then, setting $\overline \mathrm{H}=\fint_{\partial E_t}\mathrm{H}\, d \mu_t$, and recalling relation~\eqref{de02bis_2}, we get \begin{align} \norma{\mathrm{H}_t -\overline \mathrm{H}_t }_{L^2(\partial E_t)}^2 & \leq C \int_{\partial E_t} \abs{\nabla \mathrm{H}_t}^2\, d \mu_t \nonumber \\ &= -C\int_{\partial E_t} \mathrm{H}_t\Delta \mathrm{H}_t \,d \mu_t\\ &= -C\int_{\partial E_t} \Delta \mathrm{H}_t (\mathrm{H}_t-\overline \mathrm{H}_t )\, d \mu_t \nonumber \\ &\leq C \norma{\mathrm{H}_t -\overline \mathrm{H}_t }_{L^2(\partial E_t)}\norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)} \,.\label{extraeq_2} \end{align} Hence, we conclude \begin{equation} \norma{\mathrm{H}_t -\overline \mathrm{H}_t }_{L^2(\partial E_t)}\leq C\norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}\,.\label{sfiguz_2} \end{equation} Since, there holds $$ \int_{\partial E_t}\mathrm{H}_t \, \nu_t\, d \mu_t=\int_{\partial E_t}\nu_t\, d \mu_t=0 \,, $$ by multiplying relation~\eqref{not a translation 2_2} by $\mathrm{H}_t-\overline \mathrm{H}_t$, integrating over $\partial E_t$, and using inequality~\eqref{sfiguz_2}, we get \begin{align} \Bigl \rvert \int_{\partialrtial E_t} (\mathrm{H}_t -\overline \mathrm{H}_t)\Delta \mathrm{H}_t \, d \mu_t \Bigl \lvert &= \Bigl \rvert \int_{\partialrtial E_t} (\mathrm{H}_t -\overline \mathrm{H}_t)g \, d \mu_t \Bigl \rvert \\ &< \theta \norma{ \mathrm{H}_t -\overline \mathrm{H}_t}_{L^2(\partialrtial E_t)} \norma{ \Delta \mathrm{H}_t }_{L^2(\partial E_t)}\\ &\leq C\theta\norma{ \Delta \mathrm{H}_t }_{L^2(\partial E_t)}^2 \, . \end{align} Recalling now estimate~\eqref{eqcar50002_2}, as $g$ is orthogonal to $\langle \eta_t, \nu_t \rangle$, computing as in the first three lines of formula~\eqref{extraeq_2}, we have \begin{align} \norma{ \langle \eta_t, \nu_t \rangle }^2_{L^2(\partial E_t)}&=\int_{\partial E_t}\Delta \mathrm{H}_t \langle \eta_t,\nu_t \rangle \, d \mu_t\\ &= - \int_{\partialrtial E_t} \langle \nabla \mathrm{H}_t , \nabla \langle \eta_t, \nu_t\rangle \rangle \, d \mu_t\\ &\leq\abs{\eta_t}\norma{\nabla\nu_t}_{L^2(\partial E_t)} \norma{ \nabla \mathrm{H}_t }_{L^2(\partial E_t)}\\ &\leq C \norma{\nabla\nu_t}_{L^2(\partial E_t)} \norma{ \Delta \mathrm{H}_t}_{L^2(\partial E_t)} \Bigl\vert\int_{\partialrtial E_t} (\mathrm{H}_t -\overline \mathrm{H}_t)\Delta \mathrm{H}_t \, d \mu_t\, \Bigr\vert^{1/2}\\ &\leq C\sqrt{\theta}\norma{\nabla\nu_t}_{L^2(\partial E_t)}\norma{ \Delta \mathrm{H}_t}_{L^2(\partial E_t)}^2\\ &\leq C \sqrt{\theta}\norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}^2 \,, \end{align} where in the last inequality we estimated $\norma{\nabla\nu_t}_{L^2(\partial E_t)}$ with $C\norma{\psi_t}_{W^{2,6}(\partial E_t)}$ and we used inequality~\eqref{eqcar50005_2}.\\ If then $\theta>0$ is chosen so small that $C\sqrt{\theta}+\theta^2 <1$ in the last inequality, then we have a contradiction with equality~\eqref{not a translation 2_2} and the fact that $\|g\|_{L^2(\partial E_t)}<\theta \norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}$, as they imply (by $L^2$--orthogonality) that $$ \|\langle \eta_t, \nu_t\rangle\|^2_{L^2(\partial E_t)}>(1-\theta^2)\norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}^2\,. $$ All this argument shows that for such a choice of $\theta$ condition~\eqref{not a translation_2} holds.\\ Then, we can conclude as in Step~$2$ of Theorem~\ref{existence}, by replacing the $W^{2,3}$--norm on $\partial E$ with the $W^{2,6}$--norm on the same boundary. \noindent \textbf{Step ${\mathbf 3}$} ({The stopping time $\overline T$ is equal to the maximal time $T(E_0)$})\textbf{.}\\ We show now that, by taking $M_1,M_2,M_3$ smaller if needed, we have $\overline T=T(E_0)$.\\ By the previous point and the suitable choice of $M_2,M_3$ made in its final part, formula~\eqref{not a translation_2} holds, hence we have $$ \Pi_{E_t}(\Delta \mathrm{H}_t)\geq \sigma_\theta \norma{\Delta \mathrm{H}_t}_{H^1(\partial E_t)}^2\qquad \text{ for all $t\in [0, \overline T)$.} $$ In turn, by Lemma~\ref{calculations2} and~\ref{nasty} we may estimate \begin{align} \frac{d}{dt}\frac{1}{2}\int_{\partialrtial E_t}{ \abs{\nabla \mathrm{H}_t}^2\, d \mu_t} \leq & - \sigma_{\theta} \norma{ \Delta \mathrm{H}_t}^2_{H^1(\partial E_t)} + \int_{\partialrtial E_t} \abs{B} \abs{\nabla \mathrm{H}_t }^2 \abs{\Delta \mathrm{H}_t} \, d \mu_t \\ \leq & \, - \sigma_{\theta} \norma{\Delta \mathrm{H}_t}^2_{H^1(\partial E_t)} \\& \, + C\norma{ \nabla (\Delta \mathrm{H}_t)}_{L^2(\partial E_t)}^2 \norma{\nabla \mathrm{H}_t}_{L^2(\partial E_t)} (1+ \norma{\mathrm{H}_t}_{L^6(\partial E_t)}^3) \\ {\leq} & \, - \sigma_{\theta} \norma{\Delta \mathrm{H}_t}^2_{H^1(\partial E_t)} \\ &\, + C \sqrt{M_3}\norma{ \nabla (\Delta \mathrm{H}_t)}_{L^2(\partial E_t)}^2 (1+ \norma{\mathrm{H}_t}_{L^6(\partial E_t)}^3) \\ {\leq} & \, - \sigma_{\theta} \norma{\Delta \mathrm{H}_t}^2_{H^1(\partial E_t)}\\ &\,+ C \sqrt{M_3}\norma{\Delta \mathrm{H}_t}_{H^1(\partial E_t)}^2(1+C\omega(\max\{M_2,M_3\}))\label{eqcar6000_2} \end{align} for every $t \leq \overline T$, where in the last step we used relations~\eqref{Tprimo_2} and~\eqref{eqcar50005_2}.\\ Noticing that from formulas~\eqref{extraeq_2} and~\eqref{sfiguz_2} it follows \begin{equation}\label{sfiguz2_2} \norma{ \nabla \mathrm{H}_t}_{L^2(\partial E_t)}\leq C \norma{\Delta \mathrm{H}_t}_{L^2(\partial E_t)}\leq C \norma{\Delta \mathrm{H}_t}_{H^1(\partial E_t)}\,, \end{equation} keeping fixed $M_2$ and choosing a suitably small $M_3$, we conclude $$ \frac{d}{dt}\int_{\partialrtial E_t} \abs{\nabla \mathrm{H}_t}^2\, d \mu_t\leq - \frac{\sigma_{\theta}}2 \norma{\Delta \mathrm{H}_t}^2_{H^1(\partial E_t)}\leq - c_0 \norma{\nabla \mathrm{H}_t}^2_{L^2(\partial E_t)}\, . $$ This argument clearly says that the quantity $\int_{\partial E_t} |\nabla \mathrm{H}_t|^2\, d \mu_t$ is nonincreasing in time, hence, if $M_2,M_3$ are small enough, the inequality $\int_{\partial E_t } |\nabla \mathrm{H}_t|^2\, d \mu_t\leq M_3$ is preserved during the flow. As before, if we assume by contradiction that $\overline T< T(E_0)$, then it must happen that $\mathrm{Vol}(E_{\overline{T}}\Delta E)=2M_1$ or $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E)}=2M_2$.\\ Before showing that this is not possible, we prove that actually the quantity $\int_{\partial E_t} |\nabla \mathrm{H}_t|^2\, d \mu_t$ decreases (non increases) exponentially. Indeed, integrating the differential inequality above and recalling proprieties~\eqref{initial_2}, we obtain \begin{equation}\label{exp_2} \int_{\partialrtial E_t} \abs{\nabla \mathrm{H}_t}^2\, d \mu_t \leq e^{-c_0 t}\int_{\partialrtial E_0} \abs{\nabla \mathrm{H}_{\partial E_0}}^2\, d \mu_0 \leq M_3 e^{-c_0 t} \leq M_3 \end{equation} for every $t \leq \overline T$. Then, we assume that $\mathrm{Vol}(E_{\overline{T}}\Delta E)=2M_1$ or $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E_{\overline{T}})}=2M_2$. Recalling formula~\eqref{D(F)0} and denoting by $X_t$ the velocity field of the flow (see Definition~\ref{def:smoothflow} and the subsequent discussion), we compute \begin{align} \frac{d}{dt}D(E_t)&=\frac{d}{dt}\int_{E_t} d_E\, dx= \int_{E_t}\operatorname*{div}\nolimits(d_E X_t)\, dx= \int_{\partial E_t}d_E\langle X_t, \nu_t\rangle\, d\mu_t\\ &=\int_{\partial E_t}d_E\, \Delta \mathrm{H}_t\, d \mu_t = \int_{\partial E_t}\langle \nabla d_E, \nabla \mathrm{H}_t \rangle \,d \mu_t\\ &\leq C \norma{\nabla \mathrm{H}_t}_{L^2(\partial E_t)}\leq C\sqrt{M_3} \mathrm{e}^{-c_0 t/2}\,, \end{align} for all $t \leq \overline T$, where the last inequality clearly follows from inequality~\eqref{exp_2}.\\ \\ By integrating this differential inequality over $[0, \overline T)$ and recalling estimate~\eqref{D(F)}, we get \begin{align} \mathrm{Vol}(E_{\overline T}\Delta E) & \leq C\|\psi_{\overline T}\|_{L^2(\partial E_{\overline{T}})}\leq C\sqrt{D(E_{\overline T})}\nonumber \\& \leq C\sqrt{D(E_0)+C\sqrt{M_3}}\leq C\sqrt[4]{M_3}\,,\label{step33_2} \end{align} as $D(E_0)\leq M_1$, provided that $M_1,M_3$ are chosen suitably small. This shows that $\mathrm{Vol}(E_{\overline{T}}\Delta E)=2M_1$ cannot happen if we chose $C\sqrt[4]{M_3}\leq M_1$.\\ By arguing as in Lemma~\ref{w32conv} (keeping into account inequality~\eqref{Tprimo_2} and formula~\eqref{eqcar50001_2}), we can see that the $L^2$--estimate~\eqref{step33_2} implies a $W^{2,6}$--bound on $\psi_{\overline T}$ with a constant going to zero, keeping fixed $M_2$, as $\int_{\partial E_t} |\nabla \mathrm{H}_{\overline{T}}|^2\, d\mu_t \to0$, hence, by estimate~\eqref{exp_2}, as $M_3\to0$. Then, by Sobolev embeddings, the same holds for $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E_{\overline{T}})}$, hence, if $M_3$ is small enough, we have a contradiction with $\|\psi_{\overline T}\|_{C^{1,\alpha}(\partial E_{\overline{T}})}=2M_2$.\\ Thus, $\overline T=T(E_0)$ and \begin{equation}\label{finaldecay_2} \mathrm{Vol}(E_t\Delta E) \leq C\sqrt[4]{M_3}\,,\quad \|\psi_t\|_{C^{1,\alpha}(\partial E_t)}\leq 2M_2\,, \quad \int_{\partial E_t} |\nabla \mathrm{H}_{t}|^2\, d \mu_t \leq M_3e^{-c_0 t}\,, \end{equation} for every $t\in[0, T(E_0))$, by choosing $M_1,M_2,M_3$ small enough. \noindent \textbf{Step ${\mathbf 4}$} ({Long time existence})\textbf{.}\\ We now show that, by taking $M_1,M_2,M_3$ smaller if needed, we have $T(E_0)=+\infty$, that is, the flow exists for all times.\\ We assume by contradiction that $T(E_0)<+\infty$ and we notice that, by computation~\eqref{eqcar6000_2} and the fact that $\overline T=T(E_0)$, we have $$ \frac{d}{dt}\int_{\partial E_t} |\nabla\mathrm{H}_t|^2\, d\mu_t +\sigma_\theta\|\Delta\mathrm{H}_t\|_{H^1(\partial E_t)}^2\leq 0 $$ for all $t\in [0,T(E_0))$. Integrating this differential inequality over the interval $\left[T(E_0)-{T}/2,T(E_0)-{T}/4\right]$, where $T$ is given by Proposition~\ref{th:EMS1}, as we said at the beginning of the proof, we obtain \begin{align} \sigma_{{\theta}}\int_{T(E_0)-T/2}^{T(E_0)-T/4}\|\Delta\mathrm{H}_t\|_{H^1(\partial E_t)}^2\, dt\leq&\, \int_{\partial E_{T(E_0)-\frac{T}2}} |\nabla\mathrm{H} |^2\, d\mu_{T(E_0)-\frac{T}2}\\ &\,- \int_{\partial E_{T(E_0)-\frac{T}4}} |\nabla\mathrm{H}|^2\, d\mu_{T(E_0)-\frac{T}4}\\ &\leq M_3\,, \end{align} where the last inequality follows from estimate~\eqref{finaldecay_2}. Thus, by the mean value theorem there exists $\overline t\in \left(T(E_0)-{T}/2,T(E_0)-{T}/4 \right)$ such that $$ \|\Delta\mathrm{H}_{\overline{t}}\|_{H^1(\partial E_{\overline{t}})}^2\leq \frac{4M_3}{T\sigma_\theta}\,. $$ Then, by Lemma~\ref{laplacian} \begin{align} \norma{\nabla^2 \mathrm{H}_{\overline{t}}}_{L^2(\partial E_{\overline{t}})}^2 \leq&\,C\norma{\Delta\mathrm{H}_{\overline{t}}}^2_{L^2(\partial E_{\overline{t}})}(1+\norma{\mathrm{H}_{\overline{t}}}^4_{L^4(\partial E_{\overline{t}})})\\ \leq&\,CM_3(1+\omega^4(2\max\{M_2,M_3\}))\, \end{align} where in the last inequality we also used the curvature bounds provided by formula~\eqref{eqcar50005_2}. In turn, for $p\in\mathbb{R}$ large enough, we get \begin{equation} [\mathrm{H}_{\overline{t}}]^2_{C^{0,\alpha}(\partial E_{\overline{t}})}\leq C\norma{\nabla \mathrm{H}_{\overline{t}}}^2_{L^p(\partial E_{\overline{t}})}\leq C \norma{\nabla \mathrm{H}_{\overline{t}}}^2_{H^1(\partial E_{\overline{t}})}\leq CM_3(M_2,M_3)\,, \end{equation} where $[ \cdot ]_{C^{0, \alpha}(\partial E_{\overline{t}})}$ stands for the $\alpha$--H\"older seminorm on $\partial E_{\overline{t}}$ and in the last inequality we used the previous estimate.\\ Then, arguing as in Step~$4$ of Theorem~\ref{existence}, it is possible to show that flow $E_t$ exists beyond $T(E_0)$, which is a contradiction. \noindent \textbf{Step ${\mathbf 5}$} ({Convergence, up to subsequences, to a translate of $F$})\textbf{.}\\ Let $t_n\to +\infty$, then, by estimates~\eqref{finaldecay_2}, the sets $E_{t_n}$ satisfy the hypotheses of Lemma~\ref{w32conv}, hence, up to a (not relabeled) subsequence we have that there exists a critical set $E'\in \mathfrak{C}^{1,\alpha}_M(E)$ such that $E_{t_n}\to E'$ in $W^{3,2}$. Due to formulas~\eqref{eqcar50001_2} (and estimation~\eqref{de05}, that also holds in this case) we have $\|\psi_{E'}\|_{W^{2,6}(\partial E)}\leq \deltalta$ and $E'=E+\eta$ for some (small) $\eta\in \mathbb{R}^3$. \noindent \textbf{Step $\mathbf 6$} (Exponential convergence of the full sequence)\textbf{.}\\ Consider now $$ D_\eta(F)=\int_{F\Delta (E+\eta)}\mathrm{dist\,}(x, \partial E+\eta)\, dx\,. $$ The very same calculations performed in Step~$3$ show that \begin{equation}\label{step6_2} \Bigl\vert\frac{d}{dt} D_\eta(E_t)\Bigr\vert \leq C \|\nabla\mathrm{H}_t\|_{L^2(\partial E_t)}\leq C\sqrt{M_3} e^{-{c_0}t/2} \end{equation} for all $t\geq0$, moreover, by means of the previous step, it follows $\lim_{t\to +\infty} D_\eta(E_t)=0$. In turn, by integrating this differential inequality and writing $$ \partial E_t=\{y+\psi_{\eta, t}(y)\nu_{E+\eta}(y): y\in \partial E+\eta\}\,, $$ we get \begin{equation}\label{step61_2} \|\psi_{\eta, t}\|_{L^2(\partial E+\eta)}^2\leq C D_\eta(E_t)\leq\int_t^{+\infty}C\sqrt{M_3} e^{-c_0s/2}\, ds\leq C\sqrt{M_3} e^{-c_0t/2}\,. \end{equation} Since by the previous steps $\norma{\psi_{\eta, t}}_{W^{2,6}(\partial E+\eta)}$ is bounded, we infer from this inequality, Sobolev embeddings and standard interpolation estimates that also $\norma{\psi_{\eta, t}}_{C^{1,\beta}(\partial E+\eta)}$ decays exponentially for $\beta\in (0,2/3)$.\\ Denoting the average of $\mathrm{H}_t$ on $\partial E_t$ by $\overline\mathrm{H}_t$, as by estimates~\eqref{extraeq_2} and~\eqref{exp_2}, we have that \begin{align} \|\mathrm{H}_t(\cdot + \psi_{\eta, t}(\cdot)\nu_{E+\eta}(\cdot))&-\overline \mathrm{H}_t\|_{H^{1}(\partial E+\eta)}\\ &\leq C\|\mathrm{H}_t-\overline \mathrm{H}_t\|_{H^{1}(\partial E_t)}\|\psi_{\eta, t}\|_{C^1(\partial E+\eta)}\\ & \leq C\|\nabla \mathrm{H}_t\|_{L^2(\partial E_t)}\\ &\leq C\sqrt{M_3} e^{-{c_0t}/2}\,. \end{align} It follows that \begin{equation}\label{quasiHdecay_2} \|[\mathrm{H}_t(\cdot + \psi_{\eta, t}(\cdot)\nu_{E+\eta}(\cdot))-\overline \mathrm{H}_t] -[\mathrm{H}_{\partial E+\eta}-\overline \mathrm{H}_{\partial E+\eta}]\|_{H^{1}(\partial E+\eta)}\to0 \end{equation} exponentially fast, as $t\to+\infty$, where $\overline \mathrm{H}_{\partial E+\eta}$ stands for the average of $\mathrm{H}_{\partial E+\eta}$ on $\partial E+\eta$.\\ Since $E_t\to E+\eta$ (up to a subsequence) in $W^{3,2}$, it is easy to check that $\vert\overline \mathrm{H}_{t}-\overline \mathrm{H}_{\partial E+\eta}\vert\leq C\|\psi_{\eta, t}\|_{C^1(\partial E+\eta)}$ which decays exponentially, therefore, thanks to limit~\eqref{quasiHdecay_2}, we have $$ \|\mathrm{H}_t(\cdot + \psi_{\eta, t}(\cdot)\nu_{E+\eta}(\cdot)) -\mathrm{H}_{\partial E+\eta}\|_{H^{1}(\partial E+\eta)}\to0 $$ exponentially fast.\\ The conclusion then follows arguing as at the end of Step~$4$ of Theorem~\ref{existence}. \end{proof} \section{The classification of the stable critical sets}\label{classification} In this final section, we are going to discuss the classes of smooth sets to which Theorems~\ref{existence} and~\ref{existence2} can be applied, hence, ``dynamically exponentially stable'' for the modified Mullins--Sekerka and surface diffusion flow. Much is known for the stable and strictly stable critical sets $E\subseteq\mathbb{T}^n$ (or of $\mathbb{R}^n$) of the Area functional (hence, for the {\em unmodified} Mullins--Sekerka and surface diffusion flows), characterized by having constant mean curvature $\mathrm{H}$ and satisfying respectively $$ \Pi_E(\varphi)=\int_{\partial E}\bigl(|\nabla \varphi|^2- \varphi^2|B|^2\bigr)\, d\mu\geq 0 $$ for every $\varphi \in \widetilde{H}^1(\partial E)=\bigl\{\varphi \in H^1(\partialrtial E)\, : \, \int_{\partialrtial E} \varphi \,d \mu = 0 \bigr\}$ and $$ \Pi_E(\varphi)=\int_{\partial E}\bigl(|\nabla \varphi|^2- \varphi^2|B|^2\bigr)\, d\mu>0 $$ for every $\varphi \in \mathbb{T}ort(\partial E)=\bigl \{\varphi \in H^1(\partial E) \, : \,\int_{\partial E} \varphi \, d \mu = 0 \,\,\text{ and }\, \int_{\partial E} \varphi \nu_E \, d \mu = 0\bigr \}$, according to Definition~\ref{str stab}. Instead, considerably less can be said for the ``nonlocal case'', relative to the modified (with $\gamma>0$) Mullins--Sekerka flow, for which in the above formulas we need to consider analogously the positivity properties of form \begin{align} \Pi_E(\varphi)=&\, \int_{\partial E}\bigl(|\nabla \varphi|^2- \varphi^2|B|^2\bigr)\, d\mu+8\gamma \int_{\partial E} \int_{\partial E}G(x,y)\varphi(x)\varphi(y)\,d\mu(x)\,d\mu(y)\nonumber\\ &\,+4\gamma\int_{\partial E}\partial_{\nu_E} v_E \varphi ^2\,d\mu\,, \end{align} on the critical sets $E$ (in $\mathbb{T}^n$ or in domains of $\mathbb{R}^n$, with ``Neumann conditions'' at the boundary) satisfying $\mathrm{H}+4\gamma v_E=0$ on $\partial E$. Concentrating for a while on the Area functional, we observe that it is easy to see that (by a dilation/contraction argument) any strictly stable smooth critical set must be connected, but actually, being the normal velocity of the surface diffusion flow at every point defined by the {\em local} quantity $\Delta\mathrm{H}$, it follows that Theorem~\ref{existence2} can be applied also to finite unions of boundaries of strictly stable critical sets (see~\cite{FusJulMor18} and the Figure~\ref{figurauno} below). Moreover, by the very definition above, if $\partial E$ in $\mathbb{T}^n$ is composed by flat pieces, hence its second fundamental form $B$ is identically zero, the set $E$ is critical and stable and with a little effort, actually strictly stable. It is a little more difficult to show that any ball in any dimension $n\in\mathbb{N}$ is strictly stable (it is obviously a critical set), which is connected to the study of the eigenvalues of the Laplacian on the sphere $\mathbb S^{n-1}$, see~\cite[Theorem~5.4.1]{groemer}, for instance. The same then holds for all the ``cylinders'' $\mathbb{R}^k\times\mathbb S^{n-k-1}\subseteq\mathbb{R}^n$, bounding $E\subseteq\mathbb{T}^n$ after taking their quotient by the same equivalence relation defining $\mathbb{T}^n$, determined by the standard integer lattice of $\mathbb{R}^n$. Notice that if $n=2$, it follows that the only bounded strictly stable critical sets of the (in this case) {\em Length} functional in the plane are the disks and in $\mathbb{T}^2$ they are the disks and the ``strips'' with straight borders. This is clearly in agreement with the two--dimensional convergence/stability result of Elliott and Garcke~\cite{EllGar}, mentioned at the end of Section~\ref{msfsdf}. In the three--dimensional case, a first classification of the smooth stable ``periodic'' critical sets for the volume--constrained Area functional, was given by Ros in~\cite{Ros}, where it is shown that in the flat torus $\mathbb{T}^3$, they are {\em balls}, {\em $2$--tori}, {\em gyroids} or {\em lamellae}. \begin{figure} \caption{From left to right: balls, $2$--tori, gyroids and lamellae.} \label{figurauno} \end{figure} \noindent Notice that, despite their name, the {\em lamellae} are (after taking the quotient) parallel planar $2$--tori and the {\em $2$--tori} are quotients of circular cylinders in $\mathbb{R}^3$. As we said, with the balls, these surfaces are actually strictly stable, while in~\cite{Gr,GrWo,Ross} the authors established the strict stability of gyroids only in some cases. To give an example, we refer to~\cite{GrWo} where Grosse--Brauckmann and Wohlgemuth showed the strictly stability of the gyroids that are fixed with respect to translations. We remind that the gyroids, that were discovered by the crystallographer Schoen in the $1970$ (see~\cite{Schoen}), are the unique non--trivial embedded members of the family of the Schwarz P surfaces and then conjugate to the D surfaces, that are the simplest and most well--known triply--periodic minimal surfaces (see~\cite{Ross}). For the case $\gamma >0$, that is, for the nonlocal Area functional, a complete classification of the stable periodic structures is instead, up to now, still missing. It is worth to mention what is shown in~\cite{AcFuMo} about the minimizers of $J$. The authors proved that if a horizontal strip $L$ is the unique global minimizer of the Area functional in $\mathbb{T}^n$, then it is also the unique global minimizer of the nonlocal Area functional under a volume constraint, provided that $\gamma>0$ is sufficiently small. Precisely, the following result holds. \begin{thm}\label{globalminofJ} Assume that $L\subseteq\mathbb{T}^n$ is the unique, up to rigid motions, global minimizer of the Area functional, under a volume constraint. Then the same set is also the unique global minimizer of the nonlocal Area functional~\eqref{area}, provided that $\gamma>0$ is sufficiently small. \end{thm} This theorem then allows to conclude that the global minimizers are lamellae in several cases in low dimensions (two and three), for suitable parameters $\gamma$ and volume constraint. Moreover, in~\cite{AcFuMo}, it is also shown that lamellae with multiple strips are local minimizers of the functional $J$, if the number of strips is large enough. Finally, we conclude by citing the papers~\cite{ChSt,CicaLeo,Cristof,MorStern,RenWei6,RenWei5,RenWei4,RenWei3,RenWei2,RenWei1} with related and partial results on the classification problem which is at the moment fully open. \section*{Acknowledgments } We wish to thank Nicola Fusco for many discussions about his work on the topic and several suggestions. We also thank the anonymous referee for the careful reading and several suggestions. \section*{Conflict of interest} The authors declare no conflict of interest. \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \end{document}

\begin{document} \title[] {Existence and uniqueness of solutions to the Bogomol'nyi equation on graphs} \author{Yuanyang Hu} \address{School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, P. R. China.} \email{yuanyhu@henu.edu.cn} \date{} \begin{abstract} Let $G=(V,E)$ be a connected finite graph. We study the Bogomol'nyi equation \begin{equation*} \mathrm{d}elta u= \mathrm{e}^{u}-1 +4 \pi \sigmaum_{s=1}^{k} n_s \deltata_{z_{s}} \quad \text { on } \quad G, \end{equation*} where $z_1, z_2,\dots, z_k$ are arbitrarily chosen distinct vertices on the graph, $n_j$ is a positive integer, $j=1,2,\cdots, k$ and $\deltata_{z_{s}}$ is the Dirac mass at $z_s$. We obtain a necessary and sufficient condition for the existence and uniqueness of solutions to the Bogomol'nyi equation. \end{abstract} \maketitle \textit{ \footnotesize Mathematics Subject Classification (2010)} {\sigmacriptsize 35A01, 35R02}. \textit{ \footnotesize Key words:} {\sigmacriptsize Bogomol'nyi equation, finite graph, equation on graphs} \sigmaection{Introduction} Magnetic vortices play important roles in many areas of theoretical physics including condensed-matter physics, cosmology, superconductivity theory, optics, electroweak theory, and quantum Hall effect. Wang and Yang \cite{WY} established a sufficient and necessary condition for the existence of multivortex solutions of the Bogomol'nyi system. Recently, $(2+1)-$dimensional Chern-Simons gauge theory and generalized Abelian Higgs model have attracted extensive attention. The topological, non-topological and doubly periodic multivortices to the generalized self-dual Chern-Simons model and the generalized Abelian Higgs model were established over the past two decades; see, for example, \cite{CI, Ha, T, TY, Y} and the references therein. Analysis on graph have attracted a considerable amount of attention over the past decade; see, for example, \cite{ ALY, Bdd, GJ, GC, Hu, HWY, HLY, LCT, LP, TZZ, WN, WC} and the references therein. In particular, Huang, Lin and Yau \cite{HLY} proved the existence of solutions to mean field equations on graphs. Inspired by the work of Huang-Lin-Yau \cite{HLY}, we study the Bogomol'nyi equation \begin{equation}\label{E} \mathrm{d}elta u= \mathrm{e}^{u}-1 +4 \pi \sigmaum_{s=1}^{k} n_s \deltata_{z_{s}} , \end{equation} on $G$, where $G=(V,E)$ is a connected finite graph, and $V$ denotes the vetex set and $E$ denotes the edge set. Let $\mu: V \to (0,+\infty)$ be a finite measure, and $|V|$=$ \text{Vol}(V)=\sigmaum \limits_{x \in V} \mu(x)$ be the volume of $V$. We state our main result as follows. \begin{theorem}\label{t1} The equation \eqref{E} admits a unique solution, if and only if $n_1 + n_2 + \cdots + n_k= N <\frac{|V|}{4 \pi} $. \end{theorem} The paper is organized as follows. In Section 2, we present some basic results that will be used frequently in the following sections. In Section 3, we give the proof of Theorem \ref{t1}. \sigmaection{Preliminary results} For each edge $xy \in E$, we assume that its weight $w_{xy}>0$ and that $w_{xy}=w_{yx}$. For any function $u: V \to \mathbb{R}$, the Laplacian of $u$ is defined by \begin{equation}\label{1} \mathrm{d}elta u(x)=\frac{1}{\mu(x)} \sigmaum_{y \sigmaim x} w_{y x}(u(y)-u(x)), \end{equation} where $y \sigmaim x$ means $xy \in E$. The gradient form of $u$ reads \begin{equation}\label{g} \mathnormal{\Gamma}ma(u, v)(x)=\frac{1}{2 \mu(x)} \sigmaum_{y \sigmaim x} w_{x y}(u(y)-u(x))(v(y)-v(x)). \end{equation} We denote the length of the gradient of $u$ by \begin{equation*} |\nabla u|(x)=\sigmaqrt{\mathnormal{\Gamma}ma(u,u)(x)}=\left(\frac{1}{2 \mu(x)} \sigmaum_{y \sigmaim x} w_{x y}(u(y)-u(x))^{2}\right)^{1 / 2}. \end{equation*} Denote, for any function $ u: V \rightarrow \mathbb{R} $, an integral of $u$ on $V$ by $\int \limits_{V} u d \mu=\sigmaum\limits_{x \in V} \mu(x) u(x)$. For $p \ge 1$, denote $|| u ||_{p}:=(\int \limits_{V} |u|^{p} d \mu)^{\frac{1}{p}}$. As in \cite{ALY}, we define a sobolev space and a norm by \begin{equation*} W^{1,2}(V)=\left\{u: V \rightarrow \mathbb{R}: \int \limits_{V} \left(|\nabla u|^{2}+u^{2}\right) d \mu<+\infty\right\}, \end{equation*} and \begin{equation*} \|u\|_{H^{1}(V)}= \|u\|_{W^{1,2}(V)}=\left(\int \limits_{V}\left(|\nabla u|^{2}+u^{2}\right) d \mu\right)^{1 / 2}. \end{equation*} The following Sobolev embedding and Maximum principle will be used later in the paper. \begin{lemma}\label{21} {\rm (\cite[Lemma 5]{ALY})} Let $G=(V,E)$ be a finite graph. The sobolev space $W^{1,2}(V)$ is precompact. Namely, if ${u_j}$ is bounded in $W^{1,2}(V)$, then there exists some $u \in W^{1,2}(V)$ such that up to a subsequence, $u_j \to u$ in $W^{1,2}(V)$. \end{lemma} \begin{lemma}\label{22} {\rm (\cite[Lemma 4.1]{HLY})} Let $G=(V,E)$, where $V$ is a finite set, and $K \ge 0$ is constant. Suppose a real function $u(x): V\to \mathbb{R}$ satisfies $(\mathrm{d}elta-K)u(x) \ge 0$ for all $x\in \mathbb{R}$, then $u(x) \le 0$ for all $x \in V $. \end{lemma} \sigmaection{The proof of Theorem \ref{t1}} Throught this section, we assume that $N=\sigmaum\limits_{s=1}^{k} n_s$ and that $f$ is a function on $V$ satisfying $\int \limits_{V} f d \mu=1$, it is well-known that there exists a unique solution to the Poisson equation \begin{equation}\label{2} \mathrm{d}elta u_{0}=-4 \pi N f +4 \pi \sigmaum_{j=1}^{N} n_j \deltata_{z_{j}}, \end{equation} in the sense of differing by a constant. Assume $u$ is a solution of \eqref{E}, let $v:=u-u_0$, then $v$ satisfies \begin{equation}\label{3} \mathrm{d}elta v=e^{v+u_0 }-1+ 4 \pi N f . \end{equation} Define an operator $P:=\mathrm{d}elta-e^{u_0}-1: W^{1,2}(V) \to L^{2}(V)$, then we have the following proposition. \begin{proposition} \label{p1} $P$ is bijective. \end{proposition} \begin{proof} For any $u,v \in H^{1}(V)$, define $$B(u,v):=\int\limits_{V} \mathnormal{\Gamma}ma (u,v)+(e^{u_0} +1) uv d\mu.$$ By Cauchy Schwartz inequality and H$\ddot{\text{o}}$lder inequality, we deduce that \begin{equation} \begin{aligned} |B(u, v)| & \leq \int_{V} \mathnormal{\Gamma}ma(u, v)+\left(e^{u_{0}}+1\right)|u||v| d \mu \\ & \leq \int_{V} \sigmaum_{y \sigmaim x} \frac{w_{x y}}{2 \mu(x)}(u(y)-u(x))(v(y)-v(x)) d \mu+\max _{V}\left(e^{u_{0}}+1\right)\left(\int_{V} u^{2} d \mu \right)^{\frac{1}{2}}\left(\int_{V} v^{2} d \mu\right)^{\frac{1}{2}} \\ & \leq \int_{V}\left(\sigmaum_{y \sigmaim x} \frac{w_{x y}}{2 \mu(x)}(u(y)-u(x))^{2}\right)^{\frac{1}{2}}\left(\sigmaum_{y \sigmaim x} \frac{w_{x y}}{2 \mu(x)}(v(y)-v(x))^{2}\right)^{\frac{1}{2}} d\mu+C_1 ||u||_{2}||v||_{2} \\ &=\int_{V}[\mathnormal{\Gamma}ma(u, u)]^{\frac{1}{2}}[\mathnormal{\Gamma}ma(v, v)]^{\frac{1}{2}} d \mu+C_1 ||u||_{2}||v||_{2} \\ &=\int_{V} | \nabla u|| \nabla v|d \mu+C_1 || u ||_{2} ||v||_{2} \\ & \leq\left(\int_{V}|\nabla u|^{2} d \mu\right)^{\frac{1}{2}}\left(\int_{V}|\nabla v|^{2} d \mu \right)^{\frac{1}{2}}+C_{1}||u||_{2}||v||_{2}, \end{aligned} \end{equation} where $C_1=\max\limits_{V} (e^{u_0}+1)$. By \eqref{g} , we have \begin{equation}\label{6} B(u,u)\ge \int\limits_{V} |\nabla u|^2 + \min\limits_{V}(e^{u_0}+ 1) u^2 d\mu. \end{equation} Therefore, we can find a constant $C>0$ such that \begin{equation} |B(u,v)| \le C||u||_{H^{1}(V)} ||v||_{H^{1}(V)}, \end{equation} and \begin{equation} B(u,u) \ge C||u||^{2}_{H^{1}(V)}. \end{equation} It is easy to check that $B:~H^{1}(V)\times H^{1}(V)\to \mathbb{R}$ is a bilinear mapping. Thus, by the Lax-Milgram Theorem, for any function $g$ on $V$, there exists a unique element $u\in H^{1}(V)$ such that \begin{equation}\label{5} B(u,v)=-\int\limits_{V} gv d \mu, \end{equation} for any $v\in H^{1}(V)$. Since \eqref{5} is equivalent to $Pu=g$, we see that $P:H^{1}(V)\to L^{2}(V)$ is bijective.\end{proof} By Proposition \ref{p1}, we can define the inverse operator of $P$ by $P^{-1}$. Furthermore, we have the following proposition. \begin{proposition}\label{p2} $P^{-1}: L^2 \to L^2$ is compact. \end{proposition} \begin{proof} For any $g \in L^{2}(V)$, by Proposition \ref{31}, there exists $u\in H^{1}(V)$ such that $Pu=g$, which is equivalent to \begin{equation*} B(u,v)=-\int\limits_{V} gv d \mu, \end{equation*} for any $v \in H^{1}(V)$. By Cauchy inequality with $\varepsilonilon$, ($\varepsilonilon>0$), we see that \begin{equation*} B(u,u)=-\int\limits_{ V} gu d\mu \le \frac{1}{4 \varepsilonilon}\int\limits_{ V} g^2 d\mu+ \varepsilonilon \int\limits_{ V} u^2 d\mu. \end{equation*} Thus, from \eqref{6}, we conclude that \begin{equation}\label{7} \int\limits_{ V} |\nabla u|^2+C_2 u^2 d \mu \le \frac{1}{4 \varepsilonilon}\int\limits_{ V} g^2 d\mu+ \varepsilonilon \int\limits_{ V} u^2 d\mu. \end{equation} Taking $\varepsilonilon=\frac{C_2}{2}$ in \eqref{7}, we deduce that \begin{equation*} \int\limits_{ V} |\nabla u|^2+\frac{C_2}{2} u^2 d \mu \le \frac{1}{2 C_2}\int\limits_{ V} g^2 d\mu. \end{equation*} Therefore, there exists $C_3>0$ such that \begin{equation*} ||u||^{2}_{H^{1}(V)} \le C_3 \int\limits_{V} g^{2} d\mu. \end{equation*} Thus, by Lemma \ref{21}, we know that $P^{-1}: L^2 \to L^2$ is compact. \end{proof} Next, we give a necessary condition for equation \eqref{3} to have a solution. \begin{lemma}\label{31} If the equation \eqref{3} admits a solution, then $4\pi N< |V|.$ \end{lemma} \begin{proof} Assume that $v$ is a solution of the equation \eqref{3}, then \begin{equation} 0=\int\limits_{ V} \mathrm{d}elta v d \mu=\int\limits_{V} e^{v+u_0}-1+4\pi Nf d\mu=\int\limits_{ V}e^{v+u_0 } d\mu -|V| +4\pi N. \end{equation} Thus, we have $4\pi N < |V|$. \end{proof} The following lemma gives the uniqueness of solutions of the equation \eqref{3}. \begin{lemma}\label{32} There exists at most one solution of \eqref{3}. \end{lemma} \begin{proof} If $u$ and $v$ both satisfy equation \eqref{3}, by mean value theorem, there exists $\xi$ such that \begin{equation*} \mathrm{d}elta u- \mathrm{d}elta v =e^{u+u_0}-e^{v+u_0}=e^{\xi+u_0}(u-v). \end{equation*} Let $M=\max\limits_{V} (u-v)=(u-v) (x_0)$. We claim that $M\le 0$. Otherwise, $M>0$. Thus, we deduce that \begin{equation*} \mathrm{d}elta (u-v) (x_0)=[e^{\xi+ u_0} (u-v)](x_0)>0. \end{equation*} By \eqref{1}, we have $\mathrm{d}elta(u-v)(x_0)\le 0.$ This is a contradiction. Thus, we have $u(x)\le v(x)$ on $V$. Therefore, we obtain $u\equiv v$ on $V$. \end{proof} \begin{lemma}\label{33} Suppose that $|V|>4\pi N$. Then there exist $U$ and $Z$ satisfying $U\ge Z$ such that \begin{equation*} \mathrm{d}elta U- e^{U+u_0}-(4\pi N f -1)\le 0, \end{equation*} and \begin{equation*} \mathrm{d}elta Z- e^{Z+u_0}-(4\pi N f -1)\ge 0. \end{equation*} \end{lemma} \begin{proof} By Propositions \ref{p1} and \ref{p2}, we see that $-P^{-1}$ is a compact operator. Suppose $(1+P^{-1}) U=0,$ then we deduce that $(\mathrm{d}elta-e^{u_0}) U=0.$ By a similar argument as Lemma \ref{32}, we obtain $U=0$. Thus, by Fredholm alternative, we deduce that there exists $U\in H^{1}(V)$ such that $$(1+P^{-1}) U=P^{-1} (4\pi N f-1).$$ Thus, $U$ satisfies $$(\mathrm{d}elta-e^{u_0}) U=4\pi N f- 1.$$ Therefore, we obtain \begin{equation*} \mathrm{d}elta U-e^{U+u_0}-(4\pi N f-1)\le \mathrm{d}elta U-e^{U+u_0}-(4\pi N f -1)+e^{u_0}(e^{U}-U)=0. \end{equation*} Let $Z$ satisfying $log(1-\frac{4\pi N }{|V|}) -u_0 \ge Z$ be a solution of \begin{equation*} \mathrm{d}elta Z= 4\pi N f- \frac{4\pi N }{|V|}. \end{equation*} It is easy to see that \begin{equation*} \mathrm{d}elta Z-e^{Z+u_0}-(4\pi N f -1)=1-\frac{4\pi N }{|V|}-e^{Z+u_0} \ge 0. \end{equation*} \begin{equation*} \mathrm{d}elta Z-\mathrm{d}elta U\ge e^{Z+u_0}-e^{Z-U}. \end{equation*} By mean value theorem, there exists $\eta$ such that $$\mathrm{d}elta Z-\mathrm{d}elta U\ge e^{Z+u_0}-e^{U+u_0}=e^{\eta +u_0}(Z-U). $$ By a similar argument as Lemma \ref{32}, we deduce that $Z\le U$ on $V$. \end{proof} \begin{lemma}\label{34} Suppose that $N<\frac{|V|}{4 \pi}$ and $u_0$ satisfies the equation \eqref{31}. Then the equation \eqref{3} admits a unique solution $W(x)$. \end{lemma} \begin{proof} Choose a constant $K> \max\limits_{V}e^{u_0+U}$, define a sequence $\{w_n \}$ by an iterative scheme \begin{equation}\label{8} \begin{aligned} (\mathrm{d}elta -K)w_{n+1}&=e^{u_0 +w_n}- K w_n+ 4\pi N f- 1, n=0,1,2,\cdots, \\ w_0&= U. \end{aligned} \end{equation} We now prove \begin{equation*} w_k \le U~ \text{for}~k\ge 1 \end{equation*} by induction. By \eqref{8} ,we see that \begin{equation*} \mathrm{d}elta(w_1 - w_0)\ge K (w_1- w_0). \end{equation*} By a similar argument as Lemma \ref{32}, we can show that $w_1 \le U$ on $V$. Suppose that $w_k\le U$ on $V$ for some integer $k>1$, then \begin{equation} \begin{aligned} (\mathrm{d}elta- K) (w_{k+1} -U) &\ge e^{u_0+w_k}-e^{u_0+ U}+K (U-w_k) \\ &= (K-e^{u_0 +\lambdabda} )( U- w_k)\\ & \ge (K-e^{u_0 +U}) (U- w_k) \\ & \ge 0, \end{aligned} \end{equation} where $w_k \le \lambdabda \le U$. By Lemma \ref{22}, we deduce that $w_{k+1}\le U$ on $V$. We next show that \begin{equation*} w_{n+1} \le w_n \le \dots \le w_0 \end{equation*} for any $n\ge 1$ by induction. Assume that $w_k\le w_{k-1}$ on $V$ for some integer $k>2$, then we deduce that \begin{equation*} \begin{aligned} \mathrm{d}elta(w_{k+1} - w_{k})- K(w_{k+1} - w_{k}) &=(e^{u_0+ w_k}- e^{u_0+w_{k-1}})-K (w_{k} - w_{k-1}) \\ &=(e^{u_0 + \eta} -K)(w_{k}-w_{k-1})\\ &\ge 0 , \end{aligned} \end{equation*} where $w_{k}\le \eta \le w_{k-1}$. By Lemma \ref{22} , we get $w_{k+1} \le w_{k}$ on $V$. By Lemma \ref{33}, $Z\le U$. Suppose $Z\le w_{k}$ for some integer $k>1$; then \begin{equation*} \begin{aligned} (\mathrm{d}elta-K)(Z-w_{k+1})&\ge e^{u_0 +Z}-e^{u_0+ w_{k}}- K(Z-w_{k})\\ &=(e^{u_0 +\xi } -K)(Z-w_k)\\ &\ge 0, \end{aligned} \end{equation*} where $Z\le \xi \le w_k$. By Lemma \ref{22}, we have $Z \le w_{k+1}$ on $V$. Therefore, we can define $W(x):=\lim\limits_{n\to +\infty} w_{n} (x)$. Clearly, $Z\le W \le U$ on $V$. Letting $n\to +\infty$ in \eqref{8}, we deduce that $W(x)$ satisfies \eqref{3}. By Lemma \ref{32}, $W(x)$ is the unique solution of the equation \eqref{32}. \end{proof} \begin{lemma}\label{35} There exists at most one solution of the equation \eqref{E}. \end{lemma} \begin{proof} For any two solutions $u$ and $v$ of the equation \eqref{E}, by mean value theorem, there exists $\zeta$ such that \begin{equation}\label{m} \begin{aligned} \mathrm{d}elta(u-v) &= e^{u}-1- (e^{v}- 1) \\ &=e^{\zeta} (u-v). \end{aligned} \end{equation} Let $M:=\max\limits_{V} (u-v)=(u-v) (x_0)$. We assert that $M\le 0$. Otherwise, $M>0$. Then, by \eqref{m}, we conclude that $$\mathrm{d}elta(u-v)(x_0)>0.$$ By \eqref{21}, we conclude that $\mathrm{d}elta (u-v) (x_0)\le 0,$ which is a contradiction. Thus, we obtain $u\le v$ on $V$. By a similar discussion as above, we obtain $u\ge v$ on $V$. Therefore, we know that $u \equiv v$ on $V$. \end{proof} \begin{proof}[Proof of Theorem \ref{t1}] The desired conclusion follows directly from Lemmas \ref{31}, \ref{34} and \ref{35}. \end{proof} \end{document}

\begin{document} \title{Symmetric confidence regions and confidence intervals for normal map formulations of \ stochastic variational inequalities} \begin{abstract} Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals. \end{abstract} \begin{keywords} confidence region, confidence interval, stochastic variational inequality, sample average approximation, stochastic optimization, normal map \end{keywords} \section{Introduction}\label{s:intro} This paper considers a stochastic variational inequality (SVI), in which the function that defines the variational inequality is an expectation function. Let $(\Omega, \mathcal{F}, P)$ be a probability space, and $\xi$ be a random vector that is defined on $\Omega$ and supported on a closed subset $\Xi$ of $\mathbb{R}^d$. Let $O$ be an open subset of $\mathbb{R}^q$, and $F$ be a measurable function from $O\times \Xi$ to $\mathbb{R}^q$, such that for each $x\in O$ the expectation $f_0(x)=E[F(x,\xi)]$ is well defined. Let $S$ be a polyhedral convex set in $\mathbb{R}^q$. The SVI problem is to find a point $x\in S\cap O$ such that \begin{equation}\label{eqn:svi} 0\in f_0(x) + N_S(x), \end{equation} where $N_S(x)\subset \mathbb{R}^q$ denotes the normal cone to $S$ at $x$: \[N_S(x)=\{v\in \mathbb{R}^q \mid \ip{v}{s-x}\le 0 \text{ for each } s\in S\}.\] We use $\ip{\cdot}{\cdot}$ to denote the scalar product of two vectors of the same dimension. Here and elsewhere, the symbol $\subset$ stands for set inclusion and allows the two sets compared to coincide. The function $f_0$ defined above is a function from $O$ to $\mathbb{R}^q$. We are interested in situations in which $f_0$ does not have a closed form expression and is approximated by a sample average function. Let $\xi^1, \cdots, \xi^n$ be independent and identically distributed (i.i.d.) random variables with distribution same as that of $\xi$. Define the sample average function $f_n: O\times \Omega \to \mathbb{R}^q$ by \begin{equation}\label{eqn:def_fN} f_n (x, \omega)= n^{-1} \sum_{i=1}^n F(x,\xi^i(\omega)). \end{equation} The sample average approximation (SAA) problem is to find a point $x\in S\cap O$ such that \begin{equation}\label{eqn:saa} 0 \in f_n (x, \omega) + N_S (x). \end{equation} In the rest of this paper, we will write $f_n(x,\omega)$ as $f_n(x)$ when clear from the context. We will consider the \emph{normal map} formulations (to be defined below) of both \eqref{eqn:svi} and \eqref{eqn:saa}. The major objective is to develop a method to build confidence regions and confidence intervals for the solution of the normal map formulation of \eqref{eqn:svi}. We will explain how to obtain confidence regions and intervals for the solution of \eqref{eqn:svi} in Section \ref{s:num}. For brevity, we refer to a solution to \eqref{eqn:svi} or its normal map formulation as a true solution, and a solution to \eqref{eqn:saa} or its normal map formulation as an SAA solution. Under certain regularity conditions, the SAA solutions almost surely converge to a true solution as the sample size $n$ goes to infinity; see G\"{u}rkan, \"{O}zge and Robinson \cite{gur.ozg.smr:sps}, King and Rockafellar \cite{kin.rtr:ats}, and Shapiro, Dentcheva and Ruszczy\'{n}ski \cite[Section 5.2.1]{sha.den.rus:sp}. \cite[Theorem 2.7]{kin.rtr:ats} and \cite[Section 5.2.2]{sha.den.rus:sp} provided the asymptotic distribution of SAA solutions. Xu \cite{xu:saa} showed that the SAA solutions converge to the set of true solutions in probability at an exponential rate under some assumptions on the moment generating functions of certain random variables; see \cite{sha.xu:smp} for related results on the exponential convergence rate. For work on stability of stochastic optimization problems, see \cite{den:dst,dup.wets:abs,rom:ssp,sha:abo} and numerous references therein. As discussed in Pflug \cite{pfl:sos}, there are two basic approaches to constructing confidence regions for the true solution of a stochastic optimization problem. The first approach is based on the asymptotic distribution of SAA solutions, and the second is based on properties about boundedness in probability with known tail behavior. The latter approach was used in \cite{pfl:sos} to construct universal confidence sets for the true solution, among other results. Vogel \cite{vog:ucs} continued that approach and developed a general method for universal confidence sets, taking account for random feasible sets and nonunique optimal solutions. The method in this paper belongs to the first approach in the above classification. Our goal is to construct asymptotically exact confidence regions that are convenient to compute along with confidence intervals, under assumptions that guarantee the SVI and its SAA approximations to have locally unique solutions. Our work is closely related to Demir \cite{dem:acr}. In that dissertation, Demir considered the normal map formulation of the SVI and obtained in \cite[Equation (36)]{dem:acr} an expression for confidence regions of the true solution. That expression depended on quantities not directly computable. \cite[Theorem 3.16]{dem:acr} modified it, resulting in a formula for the set called $S^n$ in \cite{dem:acr}. That formula looks similar to \eqref{eqn:conf_reg_nonsingular} of the present paper, but it uses $d(f_0)_S(z_n)(z_n-z)$ (in our notation) instead of $d(f_n)_S(z_n)(z-z_n)$ in \eqref{eqn:conf_reg_nonsingular}. We show in this paper that the set \eqref{eqn:conf_reg_nonsingular} is an asymptotically exact confidence region of the true solution, by proving that the probability for it to contain the true solution converges to the prescribed level of confidence. The development in \cite{dem:acr} did not justify its method with an asymptotic exactness result, aside from using some restrictive assumptions. The starting point of the asymptotic analysis in this paper is Theorem \ref{t:asy_dis}, which is proved in \cite{lu.bud:crsvi} and related to results in \cite{dem:acr,kin.rtr:ats,sha.den.rus:sp}. Equations \eqref{eqn:zN_dist_cov} and \eqref{eqn:zN_dist_cov2} in that theorem describe the asymptotic distribution of the solution (denoted by $z_n$) to the normal map formulation of the SAA problem \eqref{eqn:saa} in terms of a piecewise linear function $L_K$ and a normal random vector $Y_0$. Because $L_K$ depends on the true solution $z_0$ and is unknown before $z_0$ is found, we need to replace it by a suitable estimator in order to establish a confidence region for $z_0$. For the general situations considered in this paper, the dependence of $L_K$ on the location of $z_0$ is \emph{discontinuous}, due to the nonsmooth structure of $S$. Such discontinuity is a major issue to be addressed for establishing computable confidence regions, and this paper handles this issue using a different approach from those in \cite{lu:nmb,lu.bud:crsvi}. In \cite{lu:nmb,lu.bud:crsvi}, $L_K$ is replaced by estimators designed to converge to $L_K$ in probability. The consistency of those estimators relies on the exponential convergence rate of $z_n$ to $z_0$. When $L_K$ is piecewise linear with multiple pieces, its estimators constructed in \cite{lu:nmb,lu.bud:crsvi} have multiple pieces with high probability, producing asymmetric confidence regions that are factions of ellipsoids pieced together. In the present paper, we use a property of general piecewise affine functions (Theorem \ref{t:dfx}) to show that for large $n$ the two vectors $-d(f_n)_S(z_n)(z_0-z_n)$ and $L_K(z_n-z_0)$ are close to each other even though $d(f_n)_S(z_n)$ may be a very different function from $L_K$. Accordingly, we can use $-d(f_n)_S(z_n)(z_0-z_n)$ to replace $L_K(z_n-z_0)$ in \eqref{eqn:zN_dist_cov2} without losing the limiting property (Theorem \ref{t:d_f0_S_d_fN_S}). This leads to a computable confidence region for $z_0$ given in \eqref{eqn:conf_reg_nonsingular} or \eqref{eqn:R_n_epsilon} without the need of constructing a consistent estimator for $L_K$. The main advantage with the new method comes from Proposition \ref{p:invertible}, which implies that the confidence region built from this method is with high probability a single ellipsoid (or can be approximated by a degenerate ellipsoid in the singular case). This brings much efficiency for describing confidence regions or computing simultaneous confidence intervals. Another advantage of the new method is that it does not rely on the exponential convergence rate of $z_n$. Lastly, we mention that with this method the confidence region obtained from a particular $z_n$ may be very different from that obtained from another $z_n$, since $d(f_n)_S(z_n)$ can be very different for different $z_n$. This does not conflict with the asymptotic exactness of the confidence region. Given the results on confidence regions and simultaneous confidence intervals, a natural question is whether individual confidence intervals for components of $z_0$ can be computed in a parallel manner, by using $d(f_n)_S(z_n)$ to replace $L_K$ in \eqref{eqn:zN_dist_cov}. Since $d(f_n)_S(z_n)$ is an invertible linear function with high probability, the individual confidence interval given by this approach can be expressed by a closed-form formula \eqref{eqn:ind_ci_conv_formula}. Theorem \ref{t:ind_ci} shows that the probability for that interval to contain the $j$th component of $z_0$ converges to a quantity related to the random variable $\Gamma = (L_K)^{-1}(Y_0)$, and that quantity equals the desired confidence level $1-\alpha$ when the condition in \eqref{eqn:prob_intersection} holds. Based on that convergence result and given the simple format of \eqref{eqn:ind_ci_conv_formula}, one can use \eqref{eqn:ind_ci_conv_formula} as an approximative confidence interval for $(z_0)_j$ to compare with confidence intervals obtained from other methods (such as methods developed in \cite{lam.lu.bud:ici} to compute asymptotically exact individual confidence intervals using estimators in \cite{lu:nmb,lu.bud:crsvi}), which generally require more computation when $L_K$ has more than two pieces. Examples of stochastic variational inequalities of the form \eqref{eqn:svi} include stochastic Nash equilibrium problems in which players' cost functions are expected values of certain random variables, such as the energy market problem studied in \cite{gur.ozg.smr:sps,hau.zac.leg.sme:sdn}. Stochastic variational inequalities also arise as first-order conditions of optimization problems whose objective functions are expectations. If the exact values of some coefficients of the objective function are unknown, and are estimated using sample data average, then the solution obtained for such a problem is essentially an SAA solution. The method of this paper can be applied to such problems to provide a quantitative measure on the effect of sample variations on solutions obtained for those problems. We have applied this method to a type of statistical learning problems \cite{lu.liu:cri}. Below we briefly introduce the normal map formulation of variational inequalities. The normal map induced by the function $f_0:O\to \mathbb{R}^q$ and the polyhedron $S\subset \mathbb{R}^q$ is defined to be a function $(f_0)_S: \Pi_S^{-1}(O)\to \mathbb{R}^q$, with \begin{equation}\label{eqn:def_nm} (f_0)_S(z)=f_0(\Pi_S(z))+ (z-\Pi_S(z)) \text{ for each }z\in \Pi_S^{-1}(O), \end{equation} where $\Pi_S(z)$ denotes the Euclidean projection of $z$ on $S$, and $\Pi_S^{-1}(O)$ is the set of points $z\in\mathbb{R}^q$ such that $\Pi_S(z)\in O$. If a point $x\in S\cap O$ satisfies \eqref{eqn:svi}, then the point $z=x-f_0(x)$ satisfies $\Pi_S(z)=x$ and \begin{equation}\label{eqn:vi_nm} (f_0)_S(z) = 0. \end{equation} Conversely, if $z$ satisfies \eqref{eqn:vi_nm}, then $x=\Pi_{S}(z)$ satisfies $x-f_0(x)=z$ and solves \eqref{eqn:svi}. Thus, equation \eqref{eqn:vi_nm} is an equivalent formulation for \eqref{eqn:svi}, and is referred to as the normal map formulation of \eqref{eqn:svi}. In general, for any function $g$ from (a subset of) $\mathbb{R}^q$ to $\mathbb{R}^q$ and any closed and convex set $C$ in $\mathbb{R}^q$, one can define the normal map induced by $g$ and $C$, denoted by $g_C$, in the same way as \eqref{eqn:def_nm} with $g$ in place of $f$ and $C$ in place of $S$. For example, the normal map induced by the sample average function $f_n$ and the set $S$ is a function $(f_n)_S:O\to \mathbb{R}^q$ defined as \[ (f_n)_S(z)=f_n(\Pi_S(z))+ (z-\Pi_S(z)) \text{ for each } z\in \Pi_S^{-1}(O). \] The following is the normal map formulation for the SAA problem \eqref{eqn:saa}: \begin{equation}\label{eqn:saa_z} (f_n)_S(z)=0. \end{equation} Equation \eqref{eqn:saa_z} is related to \eqref{eqn:saa} in the same way as \eqref{eqn:vi_nm} is to \eqref{eqn:svi}. The above definition for the normal map is from Robinson \cite{smr:nmi,smr:sav}. The normal map concept is closely related to the Minty parametrization \cite{min:mno}, in the sense that the variable $z$ can be considered as the parameter in the Minty parametrization of the graph of $N_S$: the mapping $z \to (\Pi_S(z), z-\Pi_S(z))$ is one-to-one from $\mathbb{R}^q$ onto the graph of $N_S$, and the equation \eqref{eqn:vi_nm} is a reformulation of \eqref{eqn:svi} in terms of $z$ using that parametrization. Because $S$ is a polyhedral convex set by assumption, the Euclidean projector $\Pi_S$ is a piecewise affine function (see Section \ref{s:pa} below for the precise definition of piecewise affine functions). It coincides with an affine function on each of a family of finitely many $q$-dimensional polyhedral convex sets. This family is called the normal manifold of $S$, and each set in this family is called an $q$-cell, where the symbol $q$ refers to the dimension of those sets. The union of all the $q$-cells is $\mathbb{R}^q$. Any two distinct $q$-cells are either disjoint, or meet at a common proper face of them. (A face of a convex set $P$ in $\mathbb{R}^q$ is defined to be a convex subset $F$ of $P$ such that if $x_1$ and $x_2$ belong to $P$ and $\lambda x_1 + (1-\lambda) x_2 \in F$ for some $\lambda \in (0,1)$, then $x_1$ and $x_2$ actually belong to $F$, see, e.g., \cite{rtr:ca}. $F$ is a proper face of $P$, if it is a nonempty face of $P$ and is not $P$ itself.) For detailed discussions on the normal manifold and properties of piecewise affine functions, see \cite{ral:npr,ral:bnn,smr:nmi,sch:ipd}. To illustrate the normal manifold concept, consider the example when $S=\mathbb{R}^q_+$, the nonnegative orthant in $\mathbb{R}^q$. For this example, the projector $\Pi_S$ coincides with an affine function when restricted to each fixed orthant of $\mathbb{R}^q$, and that affine function is different for a different orthant: for example we have $\Pi_S(z)=z$ for points $z\in \mathbb{R}^q_+$, $\Pi_S(z)=0$ for $z\in \mathbb{R}^q_-$, and $\Pi_S(z)=(z_1,0,\cdots,0)$ for $z\in \mathbb{R}_+ \times \mathbb{R}^{q-1}_-$, so $\Pi_S$ coincides with the identity function, the zero function, or the projector onto the $z_1$ axis, when restricted on $\mathbb{R}^q_+$, $\mathbb{R}^q_-$ or $\mathbb{R}_+ \times \mathbb{R}^{q-1}_-$ respectively. Accordingly, the normal manifold of $\mathbb{R}^q_+$ is the family of all orthants in $\mathbb{R}^q$. Below we introduce some terminology and notation. A subset $K$ of $\mathbb{R}^q$ is called a cone if $\mu x\in K$ whenever $x\in K$ and $\mu$ is a positive real number. For a set $C\subset \mathbb{R}^q$, $\Int C$ denotes its interior, and $\cone C$ is the smallest cone that contains it. (This definition for $\cone C$ is equivalent to the one given in \cite[Section 2.1.1]{sch:ipd} for polyhedral convex sets $C$. In this paper we will only apply the definition to such sets.) We use $\|\cdot\|$ to denote the norm of an element in a normed space; unless explicitly stated otherwise, it can be any norm, as long as the same norm is used in all related contexts. We use $\mathcal{N}(0, \Sigma)$ to denote a Normal random vector with covariance matrix $\Sigma$. Weak convergence of $k$-dimensional random variables $Y_n$ to $Y$ will be denoted as $Y_n \Rightarrow Y$, which means that $Ef(Y_n)$ converges to $Ef(Y)$ for all bounded continuous functional $f$ on $\mathbb{R}^k$. Following \cite{smr:ls3}, a locally Lipschitz function $g:\mathbb{R}^q\to\mathbb{R}^m$ is said to be \emph{B-differentiable} at a point $x_0\in \mathbb{R}^q$ if there is a positively homogeneous function $G: \mathbb{R}^q \to \mathbb{R}^m$, such that \begin{equation}\label{eqn:def_Bdifferentiability} g(x_0+v)=g(x_0)+ G(v)+ o(v). \end{equation} (Recall that a function $G$ is positively homogeneous, if $G(\lambda v) = \lambda G(v)$ for each nonnegative real number $\lambda$ and each $v\in \mathbb{R}^q$.) Such a function $G$ is called the B-derivative of $g$ at $x_0$ and is denoted as $dg(x_0)$. Note that $dg(x_0)(h)$ is exactly the directional derivative of $g$ at $x_0$ along the direction $h$. Indeed, as pointed out in \cite{sha:cdd}, for a locally Lipschitz function in finite-dimensional spaces, B-differentiability (called the directional differentiability in the sense of Fr\'{e}chet in \cite{sha:cdd}) of the function is equivalent to directional differentiability of the function for all directions. If $g$ is differentiable at $x_0$, then the B-derivative $dg(x_0)$ coincides with the standard Fr\'{e}chet derivative. The rest of this paper is organized as follows. Section \ref{s:pa} is a discussion on general piecewise affine functions. Section \ref{s:asy} establishes the main asymptotic distribution results. Section \ref{s:sim_ci} provides methods to build confidence regions and simultaneous confidence intervals. Section \ref{s:ind_ci} discusses computation of individual confidence intervals. Section \ref{s:num} concludes the paper with numerical examples. \section{Piecewise affine functions}\label{s:pa} This section discusses general piecewise affine functions, and proves some properties to be used in subsequent development. The notation in this section is independent of the notation in the rest of this paper. By definition, a continuous function $f$ from $\mathbb{R}^q$ to $\mathbb{R}^m$ is called \emph{piecewise affine} if there exists a finite family of affine functions $f_j: \mathbb{R}^q\to \mathbb{R}^m,j=1,\cdots,k$, such that the inclusion $f(x)\in\{f_1(x),\cdots,f_k(x)\}$ holds for each $x\in \mathbb{R}^q$ \cite{sch:ipd}. The affine functions $f_j, \ j=1,\cdots,k$ are called selection functions of $f$. If all $f_j$'s are linear functions then $f$ is called \emph{piecewise linear}. A closely related concept is the \emph{polyhedral subdivision} \cite{eav.rot:rpp,sch:ipd}. A polyhedral subdivision of $\mathbb{R}^q$ is a finite collection of polyhedral convex sets in $\mathbb{R}^q$, $\mathfrak{P}=\{P_1,\cdots,P_l\}$, that satisfies the following conditions: \begin{enumerate} \item Each $P_i$ is a polyhedral convex set of dimension $q$. \item The union of all $P_i$ is $\mathbb{R}^q$. \item The intersection of each two $P_i$ and $P_j$, $1\le i\ne j\le l$, is either empty or a common proper face of both $P_i$ and $P_j$. \end{enumerate} If each $P_i\in \mathfrak{P}$ is a polyhedral convex cone, then $\mathfrak{P}$ is a \emph{conical subdivision}. It was shown in \cite[Proposition 2.2.3]{sch:ipd} that for any piecewise affine function $f$ there corresponds a polyhedral subdivision $\mathfrak{P}$ of $\mathbb{R}^q$ such that $f$ coincides with an affine function on each $P\in \mathfrak{P}$. If $f$ is piecewise linear, then the corresponding $\mathfrak{P}$ is a conical subdivision. For example, if $S=\mathbb{R}^q_+$, then the Euclidean projector $\Pi_S$ is piecewise linear, and the family of all orthants in $\mathbb{R}^q$ is the corresponding conical subdivision. In general, for any polyhedral convex set $S$ the Euclidean projector $\Pi_S$ is piecewise affine, with the normal manifold of $S$ being the corresponding polyhedral subdivision. In the rest of this section, let $f: \mathbb{R}^q\to\mathbb{R}^m$ be a piecewise affine function with the corresponding subdivision $\mathfrak{P}$. Clearly, if $f$ is represented by different selection functions on $P_1\in \mathfrak{P} $ and $P_2\in \mathfrak{P}$, and $x\in P_1\cap P_2$, then $f$ is nondifferentiable at $x$. However, it is well known that $f$ is B-differentiable at any point in $\mathbb{R}^q$; below we explain a formula for the B-derivative of $f$ at a point $x\in \mathbb{R}^q$. For the derivation of this formula see \cite{eav.rot:rpp} and \cite[Proposition 2.2.6]{sch:ipd}. Let \[ \mathfrak{P}(x)=\{P\in \mathfrak{P}\mid x\in P\} \] be the subfamily of $\mathfrak{P}$ that consists of elements in $\mathfrak{P}$ containing $x$. We use $|\mathfrak{P}(x)|$ to denote the union of all $P\in \mathfrak{P}(x)$, called the \emph{underlying} set of $\mathfrak{P}(x)$ in algebraic topology \cite{mun:eat}. We caution the reader that $|\mathfrak{P}(x)|$ here does not mean the cardinality of $\mathfrak{P}(x)$. It is obvious that $x$ belongs to the interior of $|\mathfrak{P}(x)|$. For each $x\in \mathbb{R}^q$, define the following family of polyhedral convex cones: \[ \mathfrak{P}'(x)=\{\cone(P-x) \mid P\in \mathfrak{P}(x)\}. \] The family $\mathfrak{P}'(x)$ is a conical subdivision of $\mathbb{R}^q$. The B-derivative $d f(x)$ of $f$ at $x$ is a piecewise linear function from $\mathbb{R}^q$ to $\mathbb{R}^m$, whose corresponding subdivision is exactly $\mathfrak{P}'(x)$. If $f$ coincides with the affine function $Ax+b$ on the polyhedral convex set $P\in \mathfrak{P}(x)$, then \begin{equation}\label{eqn:dfx} d f(x)(h)= A h \text{ for each } h\in \cone(P-x). \end{equation} A consequence of \eqref{eqn:dfx} is that the selection functions of $df(x)$ are exactly the linear parts of selection functions of $f$ on elements of $\mathfrak{P}(x)$. In particular, if $x$ is contained in the interior of some $P\in \mathfrak{P}$, then $df(x)$ is a linear map. While for a fixed point $x\in \mathbb{R}^q$ the B-derivative $df(x)$ is a continuous function on $\mathbb{R}^q$, the dependence of $df(x)$ on $x$ is discontinuous, because $df(x)$ changes abruptly to a very different function as $x$ moves from the interior of some $P\in \mathfrak{P}$ to its boundary. For example consider $\Pi_S$ with $S=\mathbb{R}^q_+$ again. As a piecewise linear function, $\Pi_S$ is B-differentiable at any point $z\in \mathbb{R}^q$. At any $z$ in the interior of $\mathbb{R}^q_+$, the B-derivative $d\Pi_S(z)$ is the identity map. At any $z$ with $z_1=0$ and $z_i>0$ for $i=2,\cdots,q$, $d\Pi_S(z)$ is a piecewise linear map with two pieces: \[ \text{For each }h\in \mathbb{R}^q, \ d \Pi_S(z)(h)= \left\{ \begin{array}{ll} h, & \text{ if } h_1\ge 0,\\ (0,h_2,\cdots,h_q), & \text{ if } h_1 \le 0. \end{array} \right. \] As will become clear, the discontinuity of $df(x)$ with respect to $x$ is a major issue to be addressed to develop methods for confidence regions in this paper and in \cite{lu:nmb,lu.bud:crsvi}. In this paper, we handle this issue by utilizing a ``symmetry'' property between the B-derivatives $df(x)$ and $df(y)$ for two points $x$ and $y$ that belong to a common set in $\mathfrak{P}$, shown in Theorem \ref{t:dfx} below. Theorem \ref{t:dfx} will be used to establish our main result in Theorem \ref{t:d_f0_S_d_fN_S}. The proof of Theorem \ref{t:dfx} uses the following lemma, which gives two equivalent statements for the condition $y\in |\mathfrak{P}(x)|$. \begin{lemma}\label{l:mPx_mPy} Let $x$ and $y$ be two points in $\mathbb{R}^q$. The following are equivalent. \begin{enumerate}[(1)] \item $y\in |\mathfrak{P}(x)|$. \item $x\in |\mathfrak{P}(y)|$. \item There exists $P\in \mathfrak{P}$ such that both $x$ and $y$ belong to $P$. \end{enumerate} \end{lemma} \begin{proof} Suppose $y\in |\mathfrak{P}(x)|$. Then there exists $P\in \mathfrak{P}(x)$ such that $y\in P$. The fact that $P\in \mathfrak{P}(x)$ implies $x\in P$. This proves the direction (1)$\Rightarrow$(3). Now suppose (3) holds. Then $P\in \mathfrak{P}(x)$. Since $y\in P$, we have $y\in |\mathfrak{P}(x)|$. This proves (3)$\Rightarrow$(1). It follows that (3) and (1) are equivalent. Similarly we can prove (3) and (2) are equivalent. The equivalence between (1) and (2) follows. \end{proof} \begin{theorem}\label{t:dfx} Let $x\in \mathbb{R}^q$, and let $y\in |\mathfrak{P}(x)|$. Then \[ d f (x) (y-x) = - df(y)(x-y). \] \end{theorem} \begin{proof} By Lemma \ref{l:mPx_mPy}, there exists $P\in \mathfrak{P}$ that contains both $x$ and $y$. Let $A$ be the matrix for the linear part of the selection function of $f$ on $P$. The fact that $y\in P$ implies $y-x\in \cone (P-x)$, so by \eqref{eqn:dfx} we have $df(x)(y-x)=A(y-x)$. The fact $x\in P$ implies $x-y\in \cone(P-y)$. Again by \eqref{eqn:dfx} we have $df(y)(x-y)=A(x-y)$. \end{proof} Proposition \ref{p:mMy_subset_mMx} below shows that $\mathfrak{P}(x)$ is a superset of $\mathfrak{P}(y)$ for all $y$ sufficiently close to $x$. Recall from comments below \eqref{eqn:dfx} that the selection functions of $df(x)$ are exactly the linear parts of selection functions of $f$ on elements of $\mathfrak{P}(x)$. Therefore, a consequence of Proposition \ref{p:mMy_subset_mMx} is that the family of selection functions of $df(x)$ contains the family of selection functions of $df(y)$, for all $y$ sufficiently close to $x$. \begin{proposition}\label{p:mMy_subset_mMx} Let $x\in \mathbb{R}^q$. There exists a neighborhood $X$ of $x$, such that each $y\in X$ satisfies $\mathfrak{P}(y) \subset \mathfrak{P}(x)$. \end{proposition} \begin{proof} Recall that $x$ belongs to the interior of $|\mathfrak{P}(x)|$. Let $X$ be a neighborhood of $x$ in the interior of $|\mathfrak{P}(x)|$, and let $y\in X$. Since $y$ belongs to the interior of $|\mathfrak{P}(x)|$, there exist finitely many polyhedrons $P_1,\cdots, P_k$ in $\mathfrak{P}(x)$, such that $y\in P_i$ for each $i=1,\cdots,k$ and $y$ belongs to the interior of $\cup_{i=1}^k P_i$. Now, let $P\in\mathfrak{P}(y)$; we shall prove that $P=P_i$ for some $i=1,\cdots,k$, by showing that it is impossible for a set $P\in \mathfrak{P}$ that is different from any of those $P_i$'s to meet $\cup_{i=1}^k P_i$. Suppose for the purpose of contradiction that $P$ is different from any $P_i$, $i=1,\cdots,k$. Then, for each $i$ the intersection $P\cap P_i$ is a proper face of $P$. Consequently, the intersection between $P$ and $\cup_{i=1}^k P_i$ is the union of finitely many polyhedrons of dimensions less than $q$. On the other hand, since $P$ is of dimension $q$ and $\cup_{i=1}^k P_i$ contains $y$ in its interior, the intersection between $P$ and $\cup_{i=1}^k P_i$ contains a full-dimensional convex set. This leads to a contradiction. We have thereby proved that $P=P_i$ for some $i=1,\cdots,k$. It follows that $\mathfrak{P}(y) \subset \mathfrak{P}(x)$. \end{proof} \section{Limiting properties}\label{s:asy} This section provides some limiting properties of solutions to \eqref{eqn:saa_z}. Those properties will be used to develop methods on confidence region and simultaneous confidence intervals. We make two sets of assumptions. Assumption \ref{assu1} below is to obtain nice properties of $f_0$ and $f_n$ about integrability and convergence. Following that, Assumption \ref{assu2} is to guarantee the existence, local uniqueness and stability of the true solution. The notation $T_S(x)$ that appears in Assumption \ref{assu2} denotes the tangent cone to $S$ at a point $x\in S$. Since $S$ is a polyhedral convex set, the following definition applies: \begin{equation}\label{eqn:def_TSx} T_S(x)= \{v\in \mathbb{R}^q \mid \text{ there exists } t>0 \text{ such that } x + tv \in S\}. \end{equation} \begin{assumption}\label{assu1} (a) $E\|F( x,\xi)\|^2 < \infty$ for all $x \in O$.\\ \noindent (b) The map $x \mapsto F(x, \xi(\omega))$ is continuously differentiable on $O$ for a.e. $\omega \in \Omega$, and $E\|d_xF( x,\xi)\|^2 < \infty$ for all $x \in O$.\\ \noindent (c) There exists a square integrable random variable $C$ such that $$\|F(x, \xi(\omega)) - F(x', \xi(\omega))\| + \|d_xF(x, \xi(\omega)) - d_xF(x', \xi(\omega))\| \le C(\omega) \|x-x'\|,$$ for all $x,x'\in O$ and a.e. $\omega \in \Omega$. \end{assumption} The notation $d_xF(x, \xi(\omega))$ here stands for the partial derivative of $F$ w.r.t. $x$, a $q\times q$ matrix. The norms used in Assumption \ref{assu1} can be any norms in the $\mathbb{R}^q$ or $\mathbb{R}^{q\times q}$ space, since all norms in a finite-dimensional space are equivalent to each other. A consequence of Assumption \ref{assu1} is the continuous differentiability of $f_0$ on $O$. Moreover, for any nonempty compact subset $X$ of $O$, let $C^{1} (X, \mathbb{R}^q)$ be the Banach space of continuously differentiable mappings $f:X \to \mathbb{R}^q$, equipped with the norm \begin{equation}\label{eqn:norm_C1} \|f\|_{1,X} = \sup_{x\in X} \|f(x)\| + \sup_{x\in X} \|d f(x)\|. \end{equation} Under Assumption \ref{assu1}, the sample average function $f_n$ converges to $f_0$ almost surely as an element of $C^1(X,\mathbb{R}^q)$, see, e.g., \cite[Theorems 7.44, 7.48 and 7.52]{sha.den.rus:sp} and \cite[Theorem 3]{lu.bud:crsvi}. \begin{assumption} \label{assu2} Suppose that $x_0$ solves the variational inequality \eqref{eqn:svi}. Let $z_0=x_0-f_0(x_0)$, $L=d f_0(x_0)$, $K=T_S(x_0) \cap \{z_0-x_0\}^\perp$, and assume that the normal map $L_K$ induced by $L$ and $K$, defined as $L_K(h)=L(\Pi_K(h))+h-\Pi_K(h)$ for each $h\in \mathbb{R}^q$, is a homeomorphism from $\mathbb{R}^q$ to $\mathbb{R}^q$. \end{assumption} We assume in the above assumption that \eqref{eqn:svi} has a solution $x_0$. To guarantee the existence of such a solution, one would need to put additional conditions on $f_0$ and $S$. For example, if $f_0$ is strongly monotone on $S$ then \eqref{eqn:svi} must have a (globally unique) solution. Detailed discussions and more general solution existence conditions for variational inequalities can be found in \cite[Chapter 2]{fac.pan:fdv}, \cite[Chapter 12]{rtr.wet:va} and references therein. The set $K$ defined in Assumption \ref{assu2} is called the critical cone to $S$ associated with $z_0$. The homeomorphism condition on $L_K$ guarantees that $x_0$ is a locally unique solution of \eqref{eqn:svi}, and that \eqref{eqn:svi} continues to have a locally unique solution around $x_0$ under small perturbation of $f_0$, see \cite[Lemma 1]{lu.bud:crsvi} and the original result in \cite{smr:sav}. Being the normal map induced by a linear map and a polyhedral convex cone, $L_K$ is a piecewise linear function. It was shown in \cite{smr:nmi} that $L_K$ is a homeomorphism if and only if it is coherently oriented (a piecewise linear function is coherently oriented if the determinants of matrices representing its selection functions all have the same nonzero sign), see also \cite{ral:bnn,sch:pbn} for shortened proofs. A special case in which the coherent orientation condition holds is when the restriction of $L$ on the linear span of $K$ is positive definite. In particular, if $f_0$ is strongly monotone on $O$, then the entire matrix $L$ is positive definite and $L_K$ is a global homeomorphism. The normal map $L_K$ is related to the normal map $(f_0)_S$ in \eqref{eqn:def_nm} in the following way. Because $f_0$ is differentiable at $x_0$ and $\Pi_S$ is B-differentiable, the normal map $(f_0)_S$ is B-differentiable at $z_0$ by the chain rule of B-differentiability, with \begin{equation}\label{eqn:d_f0_S_h} d (f_0)_S (z_0) (h) = d f_0 (x_0) (d \Pi_S(z_0)(h)) + h - d \Pi_S(z_0)(h). \end{equation} It was shown in \cite{smr:sav} that $L_K$ is exactly $d (f_0)_S (z_0)$, the B-derivative of $(f_0)_S$ at $z_0$. The following theorem is adapted from \cite[Theorem 7]{lu.bud:crsvi}. Here and hereafter, we use $\Sigma_0$ to denote the covariance matrix of $F(x_0,\xi)$. Equation \eqref{eqn:zN_dist_cov} in the theorem looks similar to the equation in \cite[Theorem 2.7]{kin.rtr:ats} and \cite[Equation (5.74)]{sha.den.rus:sp}. Those two references are about the asymptotic distribution of $x_n$, a solution to \eqref{eqn:saa}, while \eqref{eqn:zN_dist_cov} here describes the asymptotic distribution of $z_n$, a solution to \eqref{eqn:saa_z}. \begin{theorem}\label{t:asy_dis} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold. Let $Y_0$ be a normal random vector in $\mathbb{R}^q$ with zero mean and covariance matrix $\Sigma_0$. Then there exist neighborhoods $X_0$ of $x_0$ in $O$ and $Z$ of $z_0$ in $\mathbb{R}^q$ such that the following hold. For almost every $\omega\in \Omega$, there exists an integer $N_{\omega}$, such that for each $n \ge N_{\omega}$, the equation \eqref{eqn:saa_z} has a unique solution $z_n$ in $Z$, and the variational inequality \eqref{eqn:saa} has a unique solution in $X_0$ given by $x_n=\Pi_S(z_n)$. Moreover, $\lim_{n\to\infty}z_n = z_0$ and $\lim_{n\to\infty}x_n = x_0$ almost surely, \begin{equation}\label{eqn:zN_dist_cov} \sqrt{n}(z_n - z_0) \Rightarrow (L_K)^{-1}(Y_0), \end{equation} and \begin{equation}\label{eqn:zN_dist_cov2} \sqrt{n} L_K (z_n- z_0) \Rightarrow Y_0. \end{equation} \end{theorem} From \eqref{eqn:zN_dist_cov2}, the random variable $n [L_K(z_n-z_0)]^T \Sigma_0^{-1} [L_K(z_n-z_0)]$ weakly converges to a $\chi^2$ random variable with $q$ degrees of freedom (assuming $\Sigma_0$ is nonsingular). This leads to an expression for confidence regions of $z_0$. However, that expression includes the normal map $L_K$ in it, which is unknown unless $z_0$ is known because both $L$ and $K$ depend on $z_0$ or $x_0$. Therefore, to obtain a computable confidence region, one needs to substitute $L_K$ by a function that depends only on $z_n$ and $x_n$, without losing the limiting property. Recall that $L_K$ is exactly $d (f_0)_S (z_0)$, the B-derivative of the normal map $(f_0)_S$ at $z_0$. A natural estimator of $L_K$ is therefore $d (f_n)_S(z_n)$. However, from the discontinuity of B-derivatives of piecewise affine functions discussed above Lemma \ref{l:mPx_mPy} and given that $\Pi_S$ is a piecewise affine function, the B-derivative $d \Pi_S(z_n)$ is not guaranteed to converge to $d \Pi_S(z_0)$ even though $z_n$ converges to $z_0$ almost surely. From \eqref{eqn:d_f0_S_h} it is clear that $d(f_n)_S(z_n)$ is not guaranteed to converge to $L_K$. To provide an alternative estimator of $L_K$, the papers \cite{lu:nmb} and \cite{lu.bud:crsvi} designed some functions to converge to $L_K$, by utilizing the exponential convergence rate of $z_n$ to $z_0$ in probability obtained under additional assumptions. The approach taken here is different from methods in \cite{lu:nmb,lu.bud:crsvi}. Instead of designing functions that converge to $L_K$, we directly use $d(f_n)_S(z_n)$ to replace $L_K$ in \eqref{eqn:zN_dist_cov2}, knowing that $d(f_n)_S(z_n)$ is not guaranteed to converge to $L_K$ and may be very different with different $z_n$. The method is based on Theorem \ref{t:d_f0_S_d_fN_S}, which says that for large $n$ the vector $d(f_n)_S(z_n)(z_0-z_n)$ is close to $-L_K(z_n-z_0)$ with high probability, and that using $d(f_n)_S(z_n)$ to replace $L_K$ in \eqref{eqn:zN_dist_cov2} along with some sign changes keeps the weak convergence result to hold. As mentioned earlier, the proof of Theorem \ref{t:d_f0_S_d_fN_S} relies on a ``symmetry'' property between the B-derivatives at two different points given in Theorem \ref{t:dfx}. From Theorems \ref{t:d_f0_S_d_fN_S} we derive Theorems \ref{t:zN_weakconv_nonsinuglar} and \ref{t:zN_weakconv_sinuglar}, to show that the set \eqref{eqn:conf_reg_nonsingular}, or \eqref{eqn:R_n_epsilon} for the singular case, is an asymptotically exact confidence region for $z_0$: that is, the set contains $z_0$ with probability converging to the prescribed level of confidence. Comparing to methods in \cite{lu:nmb,lu.bud:crsvi}, the main advantage with the new method is given by Proposition \ref{p:invertible}, which states that $d(f_n)_S(z_n)$ is with high probability an invertible linear function. This implies that the set \eqref{eqn:conf_reg_nonsingular} is with high probability a single ellipsoid, when $\Sigma_0$ is nonsingular. When $\Sigma_0$ is singular one could use $R_{n,0}$ in \eqref{eqn:R_n_0} to approximate $R_{n,\epsilon}$ in \eqref{eqn:R_n_epsilon}, and $R_{n,0}$ is with high probability a degenerate ellipsoid. In contrast, when $L_K$ is piecewise linear with multiple pieces, the estimators designed in \cite{lu:nmb,lu.bud:crsvi} have multiple pieces with high probability, providing confidence regions that are factions of ellipsoids pieced together. Clearly, it is much easier to describe a single ellipsoid, or to find the minimal enclosing box of a single ellipsoid to obtain simultaneous confidence intervals. Additionally, the new method does not rely on the assumptions for the exponential convergence rate of $z_n$, another advantage comparing to \cite{lu:nmb,lu.bud:crsvi}. Before proceeding to the proofs, recall that the Euclidean projector $\Pi_S$ coincides with an affine function on each $q$-cell in the normal manifold of $S$. We use $\mathfrak{P}$ to denote the normal manifold of $S$, which is the polyhedral subdivision of $\mathbb{R}^q$ corresponding to $\Pi_S$. As in Section \ref{s:pa} we use $\mathfrak{P}(z_0)$ to denote the family of $q$-cells containing $z_0$. Lemma \ref{l:d_Pi_S} below will be used in the proof of Theorem \ref{t:d_f0_S_d_fN_S}. \begin{lemma}\label{l:d_Pi_S} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold. Then for almost every $\omega\in \Omega$ there exists an integer $N_\omega$ such that the following equality holds for each $n\ge N_\omega$: \begin{equation}\label{eqn:d_Pi_S} d\Pi_S(z_0)(z_n-z_0)+ d\Pi_S(z_n)(z_0-z_n) = 0. \end{equation} \end{lemma} \begin{proof} Recall that $z_0$ belongs to the interior of $|\mathfrak{P}(z_0)|$. Since $z_n$ converges to $z_0$ w.p. 1, for almost every $\omega\in \Omega$ there exists an integer $N_\omega$ such that $z_n$ belongs to $|\mathfrak{P}(z_0)|$ for each $n\ge N_\omega$. It follows from Theorem \ref{t:dfx} that each such $z_n$ satisfies \eqref{eqn:d_Pi_S}. \end{proof} Theorem \ref{t:d_f0_S_d_fN_S} below is the main result of this paper. \begin{theorem}\label{t:d_f0_S_d_fN_S} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold. Then for each $\epsilon>0$ we have \begin{equation}\label{eqn:d_f0_d_fN} \lim_{n\to\infty} \Prob\{\sqrt{n}\|d(f_0)_S(z_0)(z_n-z_0)+ d(f_n)_S(z_n)(z_0-z_n)\|> \epsilon\} = 0. \end{equation} Consequently, \begin{equation}\label{eqn:zN_dist_cov_dfN} -\sqrt{n} d(f_n)_S(z_n) (z_0- z_n) \Rightarrow Y_0. \end{equation} \end{theorem} \begin{proof} Recall from Assumption \ref{assu2} and Theorem \ref{t:asy_dis} that $x_0=\Pi_S(z_0)$ and $x_n=\Pi_S(z_n)$ are solutions to \eqref{eqn:svi} and \eqref{eqn:saa} respectively. From \eqref{eqn:d_f0_S_h} we have \begin{equation}\label{eqn:d_f0_S} d (f_0)_S (z_0) (z_n-z_0) = d f_0 (x_0) (d \Pi_S(z_0)(z_n-z_0)) + z_n-z_0 - d \Pi_S(z_0)(z_n-z_0). \end{equation} Similarly, \begin{equation}\label{eqn:d_fN_S} d (f_n)_S (z_n) (z_0-z_n) = d f_n (x_n) (d \Pi_S(z_n)(z_0-z_n)) + z_0-z_n - d \Pi_S(z_n)(z_0-z_n). \end{equation} It follows that $\sqrt{n}\|d f_0)_S (z_0) (z_n-z_0) + d (f_n)_S (z_n) (z_0-z_n) \|$ is bounded from above by the sum of the following two terms. \begin{description} \item Term (a): $\sqrt{n}\|d f_0 (x_0) (d \Pi_S(z_0)(z_n-z_0)) +d f_n (x_n) (d \Pi_S(z_n)(z_0-z_n))\|$. \item Term (b): $\sqrt{n}\|d \Pi_S(z_0)(z_n-z_0)+ d \Pi_S(z_n)(z_0-z_n)\|$. \end{description} By Lemma \ref{l:d_Pi_S}, term (b) converges to 0 almost surely, so it converges to 0 in probability. It remains to show that Term (a) converges to 0 in probability. Term (a) is bounded from above by the sum of the following two terms. \begin{description} \item Term (c): $\sqrt{n}\|d f_0 (x_0) (d \Pi_S(z_0)(z_n-z_0)) -d f_n (x_n) (d \Pi_S(z_0)(z_n-z_0))\|$. \item Term (d): $\sqrt{n}\|d f_n (x_n) (d \Pi_S(z_0)(z_n-z_0)) +d f_n (x_n) (d \Pi_S(z_n)(z_0-z_n))\|$. \end{description} Because $df_n(x_n)(\cdot)$ is a linear map, by Lemma \ref{l:d_Pi_S} term (d) converges to 0 almost surely and therefore converges to 0 in probability. To prove the theorem it suffices to show that term (c) converges to 0 in probability. Term (c) is bounded from above by the following \begin{equation}\label{eqn:bdd} \|d f_0 (x_0)- d f_n (x_n)\| \ \|\sqrt{n}(d \Pi_S(z_0)(z_n-z_0))\| \end{equation} where $\|d f_0 (x_0)- d f_n (x_n)\|$ is the norm of the linear operator $d f_0 (x_0)- d f_n (x_n)$, which is bounded from above by \[ \|d f_0 (x_0)- d f_0 (x_n)\| + \|d f_0 (x_n)- d f_n (x_n)\|. \] Since $x_n$ converges to $x_0$ almost surely and $f_0$ is continuously differentiable under Assumption \ref{assu1}, $\|d f_0 (x_0)- d f_0 (x_n)\|$ converges to 0 almost surely. Assumption \ref{assu1} also guarantees $f_n$ to converge to $f_0$ almost surely as an element of $C^1(X,\mathbb{R}^q)$ for any compact subset $X$ of $O$. Let $X$ be a compact subset of $O$ that contains $x_0$ in its interior; we have $x_n \in X$ for all sufficiently large $n$. It follows that $\|d f_0 (x_n)- d f_n (x_n)\|$ converges to zero almost surely. This proves that $\|d f_0 (x_0)- d f_n (x_n)\|$ converges to zero almost surely. On the other hand, since $d \Pi_S(z_0)$ is positively homogenous, we can rewrite $\sqrt{n}(d \Pi_S(z_0)(z_n-z_0))$ as $d \Pi_S(z_0)(\sqrt{n}(z_n-z_0))$. By \eqref{eqn:zN_dist_cov}, $\sqrt{n}(z_n-z_0)$ converges in distribution to a random variable, so it is (uniformly) tight; see, e.g., \cite[Theorem 2.4]{vaa:ass}. It follows that $d \Pi_S(z_0)(\sqrt{n}(z_n-z_0))$ is uniformly tight, and that the quantity \eqref{eqn:bdd} converges to 0 in probability. This proves that term (c) converges to 0 in probability, and thereby proves \eqref{eqn:d_f0_d_fN}. Equation \eqref{eqn:zN_dist_cov_dfN} is a result of \eqref{eqn:zN_dist_cov2} and \eqref{eqn:d_f0_d_fN}. \end{proof} Proposition \ref{p:invertible} below shows the function $d(f_n)_S(z_n)$ that appears in \eqref{eqn:zN_dist_cov_dfN} is a linear invertible function with high probability. Its proof uses the following lemma. \begin{lemma}\label{l:local_homeomorphism} Under Assumptions \ref{assu1} and \ref{assu2}, there exist a neighborhood $Z'$ of $z_0$ in $\mathbb{R}^q$ and a neighborhood $\mathbb{L}$ of the matrix $L$ in $\mathbb{R}^{q\times q}$, such that for each $z'\in Z'$ and $L'\in \mathbb{L}$ the map $\Upsilon(z',L'): \mathbb{R}^q\to \mathbb{R}^q$ defined as \begin{equation}\label{eqn:def_Upsilon} \Upsilon(z',L')(h)=L' d \Pi_S(z') (h)+ h - d \Pi_S(z')(h) \text{ for each } h\in \mathbb{R}^q \end{equation} is a global homeomorphism from $\mathbb{R}^q$ to $\mathbb{R}^q$. \end{lemma} \begin{proof} First, note that for each $z'\in \mathbb{R}^q$ the B-derivative $d\Pi_S(z')$ is exactly the Euclidean projector onto the critical cone to $S$ associated with $z'$ \cite{pan:nmb,smr:ift}. Hence, the map $\Upsilon(z',L')$ is exactly the normal map induced by $L'$ and the latter critical cone. Recall that such a normal map is a global homeomorphism if and only if it is coherently oriented (see the discussion below Assumption \ref{assu2}). In the rest of this proof, we find neighborhoods $Z'$ and $\mathbb{L}$ to guarantee $\Upsilon(z',L')$ to be coherently oriented for $z'\in Z'$ and $L'\in \mathbb{L}$. Since $K$ is the critical cone to $S$ associated with $z_0$, we have $d\Pi_S(z_0)=\Pi_K$ and $\Upsilon(z_0,L)=L_K$. Because $L_K$ is a global homeomorphism by assumption, it is coherently oriented. The fact that $K$ is a polyhedral convex cone implies that $L_K$ is a piecewise linear function. Let $M_1, \cdots, M_k$ be the matrices that represent the selection functions of $L_K$. Since $L_K$ is coherently oriented, the determinants of $M_1, \cdots, M_k$ have the same nonzero sign (see the definition of the coherent orientation condition below Assumption \ref{assu2}). By choosing $ \mathbb{L}$ to be a sufficiently small neighborhood of $L$, we can guarantee that the determinants of matrices representing selection functions of $\Upsilon(z_0,L')$ to have the same nonzero sign for each $L'\in \mathbb{L}$. This proves that $\Upsilon(z_0,L')$ is coherently oriented for each $L'\in \mathbb{L}$. By Proposition \ref{p:mMy_subset_mMx}, there exists a neighborhood $Z'$ of $z_0$ such that $\mathfrak{P}(z')\subset \mathfrak{P}(z_0)$ for each $z'\in Z'$. As noted in the remark above Proposition \ref{p:mMy_subset_mMx}, the family of the selection functions of $d\Pi_S(z_0)$ includes that of $d\Pi_S(z')$ for each $z'\in Z'$. Thus, for each $L'\in \mathbb{L}$ and $z'\in Z'$, any selection function of $\Upsilon(z',L')$ is also a selection function of $\Upsilon(z_0,L')$, and the coherent orientation of $\Upsilon(z',L')$ follows from the coherent orientation of $\Upsilon(z_0,L')$. \end{proof} \begin{proposition}\label{p:invertible} Under Assumptions \ref{assu1} and \ref{assu2}, \[ \lim_{n\to\infty} \Prob\{d(f_n)_S(z_n) \text{ is an invertible linear map}\} =1. \] \end{proposition} \begin{proof} By the chain rule, \[ d (f_n)_S (z_n) (h) = d f_n (x_n) (d \Pi_S(z_n)(h)) + h - d \Pi_S(z_n)(h) \text{ for each } h\in \mathbb{R}^q. \] The linearity of $d (f_n)_S (z_n)$ depends on the linearity of $d\Pi_S(z_n)$. The latter function is linear whenever $z_n$ belongs to the interior of an $q$-cell in the normal manifold of $S$. Let $R$ be the union of boundaries of all the $q$-cells; it follows that $R$ is the union of finitely many polyhedral convex sets of dimensions less than $q$. Consider the tangent cone $T_R(z_0)$, which is defined in the same way as $T_S(x)$ in \eqref{eqn:def_TSx} if $z_0\in R$ and is the empty set if $z_0\not\in R$. Let $Z_0$ be a neighborhood of $z_0$ such that $Z_0 \cap R = Z_0 \cap (z_0+T_R(z_0))$; we have \begin{equation}\label{eqn:TR_z0} \sqrt{n}(z-z_0) \in T_R(z_0) \text{ for each } z\in Z_0\cap R \text{ and each integer }n. \end{equation} The fact that $z_n$ converges to $z_0$ almost surely implies \[ \lim_{n\to\infty} \Prob \{z_n\not\in Z_0\} = 0. \] On the other hand, \eqref{eqn:TR_z0} implies \[ \begin{split} \Prob \{z_n\in Z_0\cap R\} \le \Prob \{\sqrt{n}(z_n-z_0)\in T_R(z_0)\}. \end{split} \] It follows from \eqref{eqn:zN_dist_cov} and the fact that $L_K$ is a piecewise linear homeomorphism that \[ \begin{split} \lim_{n\to\infty} \Prob \{\sqrt{n}(z_n-z_0)\in T_R(z_0)\}\le \Prob \{(L_K)^{-1}(Y_0) \in T_R(z_0)\}=0. \end{split} \] Because $ \Prob \{z_n\in R\} \le \Prob \{z_n\not\in Z_0\} + \Prob \{z_n\in Z_0\cap R\}$, we have proved \begin{equation}\label{eqn:prob_R} \lim_{n\to\infty} \Prob \{z_n\in R\} = 0, \end{equation} which implies that the probability for $d(f_n)_S(z_n)$ to be a linear function converges to 1 as $n$ goes to $\infty$. Next, choose neighborhoods $Z'$ of $z_0$ and $\mathbb{L}$ of $L$ as in Lemma \ref{l:local_homeomorphism}. Since $z_n$ converges to $z_0$ almost surely, $z_n$ belongs to $Z'$ almost surely for sufficiently large $n$. It was shown in the proof of Theorem \ref{t:d_f0_S_d_fN_S} that $d f_n(x_n)$ almost surely converges to $d f_0(x_0)$. Consequently, $d f_n(x_n)$ belongs to $\mathbb{L}$ almost surely for sufficiently large $n$. By Lemma \ref{l:local_homeomorphism}, $d(f_n)_S(z_n)$ is a global homeomorphism almost surely for sufficiently large $n$, so the probability for $d(f_n)_S(z_n)$ to be an invertible function converges to 1 as $n$ goes to $\infty$. The conclusion of the proposition follows by combining the linearity and invertibility. \end{proof} The random variable $Y_0$ in \eqref{eqn:zN_dist_cov_dfN} has covariance matrix $\Sigma_0$, which depends on the true solution $x_0$. In computing confidence regions and intervals we will replace $\Sigma_0$ by $\Sigma_n$, the sample covariance matrix of $\{F(x_n, \xi^i)\}_{i=1}^n$. The lemma below supports such a replacement. \begin{lemma}\label{l:Sigma_n_Sigma_0} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold. The matrix $\Sigma_n$ converges to $\Sigma_0$ almost surely as $n \to\infty$. \end{lemma} \begin{proof} Under the assumptions, $\frac{1}{n} \sum_{i=1}^n F(x,\xi^i(\omega)) F(x,\xi^i(\omega))^T$ converges to $E[F(x,\xi) F(x,\xi)^T]$ almost surely uniformly on each compact subset $X$ of $O$ (see, e.g., \cite[Theorem 7.48]{sha.den.rus:sp}). Since $x_n$ converges to $x_0$ almost surely, we have \[\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n F(x_n,\xi^i(\omega)) F(x_n,\xi^i(\omega))^T = E[F(x_0,\xi) F(x_0,\xi)^T] \text{ almost surely}.\] Similarly $f_n(x_n)$ converges to $f_0(x_0)$ almost surely as $n \to \infty$. This proves the lemma. \end{proof} \section{Confidence regions and simultaneous confidence intervals}\label{s:sim_ci} This section provides formulas for confidence regions of $z_0$ that are computable from $z_n$, and provides a method to compute simultaneous confidence intervals for components of $z_0$. We treat two cases separately: Theorem \ref{t:zN_weakconv_nonsinuglar} considers situations in which $\Sigma_0$ (the covariance matrix of $F(x_0,\xi)$) is nonsingular, and Theorem \ref{t:zN_weakconv_sinuglar} handles situations in which $\Sigma_0$ is singular. The confidence region is given in \eqref{eqn:conf_reg_nonsingular} or \eqref{eqn:R_n_epsilon} for the two cases respectively. We use $\chi^2_l$ to denote a $\chi^2$ random variable with $l$ degrees of freedom, and use $\chi^2_l(\alpha)$ to denote the number that satisfies $P( \chi^2_l > \chi^2_l(\alpha)) = \alpha$ for $\alpha\in [0,1]$. For the rest of this section, let $\alpha\in [0,1]$ be fixed. \begin{theorem}\label{t:zN_weakconv_nonsinuglar} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold, and that $\Sigma_0$ is nonsingular. For almost every $\omega\in \Omega$, there exists an integer $N_\omega$ such that $\Sigma_n$ is nonsingular for $n\ge N_\omega$. Moreover, \begin{equation}\label{eqn:dfNzN-weakconv} -\sqrt{n}\Sigma_n^{-1/2}[d(f_n)_S(z_n) (z_0 - z_n)] \Rightarrow \mathcal{N}(0, I_q), \end{equation} and the probability for $z_0$ to belong to the set \begin{equation}\label{eqn:conf_reg_nonsingular} \left\{z\in \mathbb{R}^q \left| n \big[ d(f_n)_S(z_n) (z - z_n) \big]^T \Sigma_n^{-1} \big[d(f_n)_S(z_n) (z - z_n)\big] \le \chi^2_{q}(\alpha)\right. \right\} \end{equation} converges to $1-\alpha$ as $n\to \infty$. \end{theorem} \begin{proof} Since $\Sigma_n$ converges to $\Sigma_0$ almost surely (Lemma \ref{l:Sigma_n_Sigma_0}), it is nonsingular for sufficiently large $n$ almost surely. Equation \eqref{eqn:dfNzN-weakconv} follows from \eqref{eqn:zN_dist_cov_dfN}. From \eqref{eqn:dfNzN-weakconv} the random variable \[n \big[ d(f_n)_S(z_n) (z_0 - z_n) \big]^T \Sigma_n^{-1} \big[d(f_n)_S(z_n) (z_0 - z_n)\big]\] weakly converges to a $\chi^2_q$ random variable, so the set \eqref{eqn:conf_reg_nonsingular} contains $z_0$ with probability converging to $1-\alpha$. \end{proof} \begin{theorem}\label{t:zN_weakconv_sinuglar} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold, and that $\Sigma_0$ is singular. Let $\rho>0$ be the minimum of all positive eigenvalues of $\Sigma_0$, and let $l$ be the number of positive eigenvalues of $\Sigma_0$ counted with regard to their algebraic multiplicities. Let $\rho_0$ satisfy $0<\rho_0<\rho$. Decompose $\Sigma_n$ as \begin{equation}\label{eqn:decom_Sigman} \Sigma_n=U_n^T \Delta_n U_n \end{equation} where $U_n$ is an orthogonal $q\times q$ matrix, and $\Delta_n$ is a diagonal matrix with monotonically decreasing elements. Let $D_n$ be the upper-left submatrix of $\Delta_n$ whose diagonal elements are at least $\rho_0$. Let $l_n$ be the number of rows in $D_n$, $(U_n)_1$ be the submatrix of $U_n$ that consists of its first $l_n$ rows, and $(U_n)_2$ be the submatrix that consists of the remaining rows of $U_n$. Then for almost every $\omega$ the equality $l_n=l$ holds for sufficiently large $n$. Moreover, \begin{equation}\label{eqn:zN_dist_cov_chi2} n \big[ d(f_n)_S(z_n) (z_0 - z_n) \big]^T (U_n)_1^T D_n^{-1}(U_n)_1 \big[d(f_n)_S(z_n) (z_0 - z_n)\big] \Rightarrow \chi^2_l \end{equation} and \begin{equation}\label{eqn:zN_dist_cov_0} n [d(f_n)_S(z_n) (z_0 - z_n)]^T (U_n)_2^T (U_n)_2 [d(f_n)_S(z_n) (z_0 - z_n)] \Rightarrow 0. \end{equation} For each $\epsilon>0$, the set \begin{equation}\label{eqn:R_n_epsilon} R_{n,\epsilon}= \left\{z\in \mathbb{R}^q \left| \begin{array}{l} n \big[ d(f_n)_S(z_n) (z - z_n) \big]^T (U_n)_1^T D_n^{-1}(U_n)_1 \big[d(f_n)_S(z_n) (z - z_n)\big] \le \chi^2_{l_n}(\alpha) \\ \\ \|\sqrt{n}(U_n)_2 [d(f_n)_S(z_n) (z - z_n)]\|_\infty \le \epsilon \end{array}\right. \right\} \end{equation} contains $z_0$ with probability converging to $1-\alpha$. \end{theorem} \begin{proof} Let $\rho$ and $l$ be as defined in this theorem, and conduct an eigen-decomposition of $\Sigma_0$ as \begin{equation}\label{eqn:decom_Sigma0} \Sigma_0=U_0^T \begin{bmatrix} D_0 & 0 \\ 0 & 0 \end{bmatrix} U_0 = \begin{bmatrix} (U_0)_1^T & (U_0)_2^T \end{bmatrix} \begin{bmatrix} D_0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} (U_0)_1 \\ (U_0)_2 \end{bmatrix} \end{equation} where $U_0$ is orthogonal, $D_0$ is diagonal with monotonically decreasing positive diagonal elements, and $(U_0)_1$ and $(U_0)_2$ contains the first $l$ and the last $q-l$ rows of $U_0$ respectively. From \eqref{eqn:zN_dist_cov_dfN} we have \begin{equation}\label{eqn:zN-weakconv3} -\sqrt{n}\begin{bmatrix} D_0^{-1/2} & 0 \\ 0 & I_{q-l} \end{bmatrix}U_0 [d(f_n)_S(z_n) (z_0 - z_n)] \Rightarrow \mathcal{N}(0, I_l) \times 0. \end{equation} From Lemma \ref{l:Sigma_n_Sigma_0}, for a.e. $\omega\in\Omega$ there exists an integer $N^1_\omega$, such that each $n \ge N^1_\omega$ satisfies \begin{equation}\label{eqn:deltaN_ii} (\Delta_n)_{ii} > \rho_0 \text{ for each } i=1,\cdots, l \text{ and } (\Delta_n)_{ii} < \rho_0 \text{ for each } i=l+1,\cdots, q. \end{equation} It follows that that $l_n=l$ for $n\ge N^1_\omega$ and that $D_n$ converges to $D_0$ almost surely. For $n\ge N^1_\omega$, define \[ \hat{\Sigma}_n=U_n^T \begin{bmatrix} D_n & 0 \\ 0 & 0\end{bmatrix} U_n \text{ and } \hat{\Sigma}_n^+=U_n^T \begin{bmatrix} D_n^{-1} & 0 \\ 0 & 0\end{bmatrix} U_n=(U_n)_1^T D_n^{-1} (U_n)_1.\] Because of \eqref{eqn:deltaN_ii}, $\hat{\Sigma}_n$ is the unique best approximation of $\Sigma_n$ in Frobenius norm among all matrices of rank $\le l$; see, e.g., \cite{chi:ppe,gol.hof.ste:gey,ste:ehs}. Since the unique best approximation of a matrix depends continuously on that matrix (by an application of \cite[Theorem 1.17]{rtr.wet:va}), and the best approximation of $\Sigma_0$ is itself, $\hat{\Sigma}_n$ almost surely converges to $\Sigma_0$. As the pseudo-inverse of $\hat{\Sigma}_n$, $\hat{\Sigma}_n^+$ almost surely converges to the pseudo-inverse of $\Sigma_0$ (by an application of \cite[Corollary 3.5]{ste:ppi}). This and \eqref{eqn:zN-weakconv3} imply \eqref{eqn:zN_dist_cov_chi2}. Finally, an application of \cite[Theorem 3.1]{li:rpt} implies that the angle between the row spaces of $(U_0)_2$ and $(U_n)_2$ almost surely converges to zero as $n\to \infty$. In view of \cite[Equation (2.4)]{li:rpt} and \cite[Theorem 2.6.1]{gol.van:mc}, the matrix $(U_n)_2^T (U_n)_2$ converges to $(U_0)_2^T (U_0)_2$ almost surely. This and \eqref{eqn:zN-weakconv3} imply \eqref{eqn:zN_dist_cov_0}. Note that the choice of the matrix $U_n$ in the decomposition of $\Sigma_n$ is non-unique, when $\Sigma_n$ has repeated eigenvalues. However, when \eqref{eqn:deltaN_ii} holds the matrices $(U_n)_1^T D_n^{-1}(U_n)_1$ and $(U_n)_2^T (U_n)_2$ that appear in \eqref{eqn:zN_dist_cov_chi2} and \eqref{eqn:zN_dist_cov_0} are uniquely determined by $\Sigma_n$, and depend continuously on $\Sigma_n$. Since $l_n=l$ almost surely for sufficiently large $n$, from \eqref{eqn:zN_dist_cov_chi2} we have \begin{equation}\label{eqn:UN_DN_UN} \lim_{n\to\infty} \Prob\left\{ n \big[ d(f_n)_S(z_n) (z_0 - z_n) \big]^T (U_n)_1^T D_n^{-1}(U_n)_1 \big[d(f_n)_S(z_n) (z_0 - z_n)\big] \le \chi^2_{l_n}(\alpha)\right\}=1-\alpha. \end{equation} By \eqref{eqn:zN_dist_cov_0} we have \[ \lim_{n\to\infty} \Prob\left\{ n [d(f_n)_S(z_n) (z_0 - z_n)]^T (U_n)_2^T (U_n)_2 [d(f_n)_S(z_n) (z_0 - z_n)] \le \epsilon \right\}=1 \] for each $\epsilon>0$. Since all norms are equivalent in a finite-dimensional space, we can rewrite the above equality as \begin{equation}\label{eqn:UN_norm_infty} \lim_{n\to\infty} \Prob\left\{ \sqrt{n} \|(U_n)_2 [d(f_n)_S(z_n) (z_0 - z_n)]\|_\infty \le \epsilon \right\}=1, \end{equation} where $\|\cdot\|_\infty$ denotes the $\infty$-norm. In writing \eqref{eqn:UN_norm_infty}, we assume that $U_n$ is a measurable function of $\Sigma_n$ so that the set in consideration is a measurable set in $\Omega$. Equations \eqref{eqn:UN_DN_UN} and \eqref{eqn:UN_norm_infty} imply that the set $R_{n,\epsilon}$ contains $z_0$ with probability converging to $1-\alpha$. \end{proof} Since for each $\epsilon>0$ the set $R_{n,\epsilon}$ is an asymptotically exact $(1-\alpha)100\%$ confidence region for $z_0$, it is natural to ask whether the same is true about the set \begin{equation}\label{eqn:R_n_0} R_{n,0}= \left\{z\in \mathbb{R}^q \left| \begin{array}{l} n \big[ d(f_n)_S(z_n) (z - z_n) \big]^T (U_n)_1^T D_n^{-1}(U_n)_1 \big[d(f_n)_S(z_n) (z - z_n)\big] \le \chi^2_{l_n}(\alpha) \\ \\ \sqrt{n}(U_n)_2 [d(f_n)_S(z_n) (z - z_n)] =0 \end{array}\right. \right\}. \end{equation} We cannot prove that $R_{n,0}$ contains $z_0$ with probability converging to $1-\alpha$, because the quantity \[ \lim_{n\to\infty} \Prob\left\{ \sqrt{n} (U_n)_2 [d(f_n)_S(z_n) (z_0 - z_n)] = 0 \right\} \] may not equal 1. However, Proposition \ref{p:R_0} below shows for a fixed $n$ that the set-valued map $R_{n,\epsilon}$ is Lipschitz continuous on an interval $[0,\bar{\epsilon}]$ for some positive real number $\bar{\epsilon}$. Accordingly, any open set that contains $R_{n,0}$ must include $R_{n,\epsilon}$ for a sufficiently small $\epsilon$. Since $R_{n,0}$ has a simpler geometric structure comparing to $R_{n,\epsilon}$, one may choose to use $R_{n,0}$ to approximate $R_{n,\epsilon}$ for computational efficiency. \begin{proposition}\label{p:R_0} Let $d(f_n)_S(z_n)$ be a fixed invertible linear function, $U_n$ a fixed orthogonal matrix, and $D_n$ a fixed $l_n\times l_n$ diagonal matrix with positive diagonal elements. There exist real numbers $\bar{\epsilon}>0$ and $\sigma>0$ such that \[R_{n,\epsilon'} \subset R_{n,\epsilon} + \sigma |\epsilon-\epsilon'|\mB\] whenever $0\le \epsilon \le \bar{\epsilon}$ and $0\le \epsilon' \le \bar{\epsilon}$, where $\mB$ denotes the closed unit ball in $\mathbb{R}^q$. \end{proposition} \begin{proof} After the coordinate transformation $U_n d(f_n)_S(z_n)$, one can view $R_{n,\epsilon}$ as a cartesian product of a fixed ellipsoid in $\mathbb{R}^{l_n}$ and a polyhedron in $\mathbb{R}^{q-l_n}$, with $\epsilon$ being the right hand side of the linear constraints defining the polyhedron. The Lipschitz continuity can be seen in the transformed space. \end{proof} It is shown in Proposition \ref{p:invertible} that $d(f_n)_S(z_n)$ is an invertible linear function with high probability. When this holds, the set in \eqref{eqn:conf_reg_nonsingular} and $R_{n,0}$ in \eqref{eqn:R_n_0} are ellipsoids in $\mathbb{R}^q$, and therefore can be described using their centers, principal directions, and semi-axes. It is often desirable to provide simultaneous confidence intervals for components of $z_0$, as intervals are more convenient to describe, visualize, and interpret. We can compute these intervals by finding the minimum enclosing box of a confidence region, i.e., by computing the maximal and minimal values of $z_i$ for each $i=1,\cdots,q$ over the confidence region. Since $\Sigma_0$ is unknown in practice, it is often difficult to directly check if $\Sigma_0$ is nonsingular or not. Instead, we conduct an eigen-decomposition of the sample covariance matrix $\Sigma_n$ as in \eqref{eqn:decom_Sigman}, and partition matrices $U_n$ and $\Delta_n$ based on positive or zero eigenvalues. If all eigenvalues are strictly positive (larger than a prescribed tolerance), then we use \eqref{eqn:conf_reg_nonsingular} as the confidence region. Otherwise, we use $R_{n,\epsilon}$ in \eqref{eqn:R_n_epsilon} for some positive $\epsilon$ or $R_{n,0}$ as the confidence region. In the latter case, the confidence region is still bounded because $d(f_n)_S(z_n)$ is invertible, and is flat in the directions represented by rows of $(U_n)_2[d(f_n)_S(z_n)]$. In particular, the set $R_{n,0}$ is a degenerate ellipsoid. One can still find the minimal enclosing box of the confidence region to obtain the simultaneous confidence intervals. Numerical tests with singular covariance matrices are conducted in \cite{lam.lu:aac}. Lastly, we mention that the only requirement on $\Sigma_n$ for Theorems \ref{t:zN_weakconv_nonsinuglar} and \ref{t:zN_weakconv_sinuglar} to hold is its almost sure convergence to $\Sigma_0$. Hence, any estimator of $\Sigma_0$ that converges to it almost surely can be used as $\Sigma_n$, not necessarily the sample covariance matrix. \section{Individual confidence intervals}\label{s:ind_ci} This section discusses computation of individual confidence intervals for $z_0$. An individual confidence interval for the $j$th component of $z_0$ is an interval in $\mathbb{R}$ that contains $(z_0)_j$ with a prescribed level of confidence. It is generally narrower than the simultaneous confidence interval for the same level. The difference between widths of the two intervals are substantial in problems with moderate or large dimensions. In such problems, the minimum enclosing box of a confidence region can be much larger than the region itself, resulting in simultaneous confidence intervals too wide to be useful. The sizes of individual confidence intervals are less affected by the dimension of the problem. There are also problems where only confidence intervals of selected components in the true solution are of interest. Computation of individual confidence intervals requires knowledge on the distribution of each individual component of $z_n$. As shown in \eqref{eqn:zN_dist_cov}, $\sqrt{n}(z_n-z_0)$ weakly converges to the distribution of a random variable $\Gamma=(L_K) ^{-1} (Y_0)$. Given results in the preceding two sections, it is natural to ask whether we can substitute $L_K=d (f_0)_S (z_0)$ with $d(f_n)_S(z_n)$ in computing individual confidence intervals for $z_0$. Such a direct substitution results in a confidence interval in \eqref{eqn:ind_ci_conv_formula}, where $r_{nj}$ is defined in \eqref{eqn:def_rNj}. The equation \eqref{eqn:ind_ci_conv_general} expresses the limiting probability for such an interval to contain $(z_0)_j$ in terms of $\Gamma$. That limiting probability equals the desired confidence level $1-\alpha$ when the condition in \eqref{eqn:prob_intersection} holds. Let $\mathfrak{P}(z_0)=\{P_1,\cdots,P_k\}$ be the set of $q$-cells in the normal manifold of $S$ that contains $z_0$. For each $i=1,\cdots,k$, let $K_i=\cone(P_i-z_0)$, $A_i$ be the matrix representing $d\Pi_S(z)$ for points $z$ in the interior of $P_i$, and $T_i=d (f_0)_S (z_0) (K_i)$ be the image of $K_i$ under the function $d(f_0)_S(z_0)$. From \eqref{eqn:d_f0_S_h} we can see that $d (f_0)_S(z_0)$ coincides with $d L_S(z_0)$, where $L_S$ is the normal map induced by the linear operator $L$ and the set $S$. On the cone $K_i$, the map $d(f_0)_S(z_0)$ is represented by the matrix \[ M_i = L A_i + I - A_i. \] Under Assumption \ref{assu2} the map $d(f_0)_S(z_0)$ is a global homeomorphism, so all the matrices $M_i, i=1,\cdots, k$ are nonsingular. As above let $\Gamma=(L_K) ^{-1} (Y_0)$, and for each $i=1,\cdots,k$ define a random variable $\Gamma^i=M_i ^{-1} (Y_0)$. The fact that $Y_0$ is a multivariant normal random variable implies that each $\Gamma^i$ is a multivariant normal random variable with covariance matrix $M_i^{-1} \Sigma_0 M_i^{-T}$. Accordingly, each component of $\Gamma^i$ is a normal random variable. If we define a number \[ r^i_j = \sqrt{(M_i^{-1} \Sigma_0 M_i^{-T})_{jj}} \] for each $i=1,\cdots,k$ and $j=1,\cdots,q$, then for each real number $\alpha\in (0,1)$ we have \begin{equation}\label{eqn:r_ij_prob} \Prob\left(|\Gamma^i_j|\le \sqrt{\chi^2_1(\alpha)} r^i_j\right) = 1-\alpha. \end{equation} Let us discuss a relation between $\Gamma$ and $\Gamma^i$. Since $d(f_0)_S(z_0)$ is represented by the matrix $M_i$ on the cone $K_i$, we have \[ d (f_0)_S (z_0) ^{-1} (y)= M_i^{-1}(y) \text{ if } y\in T_i. \] For each measurable set $W \subset K_i$ we have \begin{equation}\label{eqn:prob_gamma_W} \Prob(\Gamma \in W)= \Prob(Y_0\in M_i (W)) = \Prob(\Gamma^i \in W). \end{equation} \begin{theorem}\label{t:ind_ci} Suppose that Assumptions \ref{assu1} and \ref{assu2} hold, and suppose that $\Sigma_0$ has a positive determinant. Let $P_i, K_i, T_i, A_i, M_i, \Gamma^i, \Gamma, r^i_j$ be defined as above. For each integer $n$ and each $j=1,\cdots,q$, define a number \begin{equation}\label{eqn:def_rNj} r_{nj} = \left\{ \begin{array}{ll} \sqrt{(d(f_n)_S(z_n)^{-1} \Sigma_n d(f_n)_S(z_n)^{-T})_{jj}} & \text{ if } d(f_n)_S(z_n) \text{ is an invertible linear map,}\\ 0 & \text{ otherwise.} \end{array} \right. \end{equation} Then for each real number $\alpha \in (0,1)$ and for each $j=1,\cdots,q$, \begin{equation}\label{eqn:ind_ci_conv_general} \begin{split} &\lim_{n\to\infty} \Prob\left(\frac{\sqrt{n}|(z_n-z_0)_j|}{r_{nj}}\le \sqrt{\chi^2_1(\alpha)}\right)\\ =&\sum_{i=1}^k \Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in K_i\right) =\sum_{i=1}^k \Prob\left(\big|\frac{\Gamma_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma\in K_i\right). \end{split} \end{equation} Moreover, suppose for a given $j=1,\cdots,q$ that the following equality \begin{equation}\label{eqn:prob_intersection} \Prob \left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in K_i\right) = \Prob \left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\right) \Prob (\Gamma^i\in K_i) \end{equation} holds for each $i=1,\cdots,k$. Then for each real number $\alpha \in (0,1)$, \begin{equation}\label{eqn:ind_ci_conv} \lim_{n\to \infty} \Prob\left\{|(z_n-z_0)_j|\le \frac{\sqrt{\chi^2_1(\alpha)} r_{nj}}{\sqrt{n}}\right\} = 1-\alpha. \end{equation} \end{theorem} \begin{proof} Let $\alpha \in (0,1)$ be fixed. Recall from Lemma \ref{l:Sigma_n_Sigma_0} that $\Sigma_n$ converges to $\Sigma_0$ almost surely, and from the proof of Theorem \ref{t:d_f0_S_d_fN_S} that $d f_n(x_n)$ converges to $L=d f_0(x_0)$ almost surely. Let $Z'$ be a neighborhood of $z_0$ in the interior of $|\mathfrak{P}(z_0)|$ with $Z'\cap P_i = Z'\cap (z_0+K_i)$ for each $i=1,\cdots,k$. We have \begin{equation}\label{eqn:prob_zN_Z'} \lim_{n\to \infty} \Prob(z_n\in Z')=1. \end{equation} Writing $Z'$ as the following union, \[Z'=\cup_{i=1}^k (Z'\cap (z_0+\Int K_i)) \cup (Z'\cap (z_0+\mathbb{R}^q/\cup_{i=1}^k \Int K_i)),\] and noting from \eqref{eqn:prob_R} in the proof of Proposition \ref{p:invertible} that \[\lim_{n\to\infty}\Prob(z_n\in Z'\cap (z_0+\mathbb{R}^q/\cup_{i=1}^k \Int K_i))=0,\] we find \begin{equation}\label{eqn:prob_zN_intKi} \lim_{n\to \infty} \sum_{i=1}^k \Prob(z_n\in Z'\cap (z_0+\Int K_i)) =1. \end{equation} It is not hard to see $Z'\cap (z_0+\Int K_i) = Z' \cap \Int P_i$ for each $i=1,\cdots,k$. When $z_n \in Z'\cap \Int P_i$, $d (f_n)_S(z_n)$ is a linear map given by \[ d (f_n)_S (z_n) = d f_n (x_n) A_i + I - A_i. \] For each integer $n$, $j=1,\cdots,q$ and $i=1,\cdots,k$, define a quantity \begin{equation}\label{eqn:def_hatr} \hat{r}^i_{nj}=r_{nj} 1_{z_n \in Z'\cap \Int P_i} + r^i_j 1_{z_n \not\in Z'\cap \Int P_i}. \end{equation} By Proposition \ref{p:invertible}, and from the fact that $d f_n(x_n)$ and $\Sigma_n$ converge to $d f_0(x_0)$ and $\Sigma_0$ almost surely respectively, $\hat{r}^i_{nj}$ converges to $r^i_j$ in probability as $n\to \infty$. Now, let us first consider the situations in which $k\ge 2$. These are the situations in which $z_0$ lies on the boundary of some $q$-cell. For each $i=1,\cdots,k$ and $j=1,\cdots,q$, choose an $q$-dimensional vector $\bar{h}^{ij}$ such that $\bar{h}^{ij}$ does not belong to $K_i$ and its $j$th component $\bar{h}^{ij}_j$ satisfies $|\bar{h}^{ij}_j|>r^i_j \sqrt{\chi^2_1(\alpha)}$. Define a random variable $h^{ij}_{n}\in \mathbb{R}^q$ by \begin{equation}\label{eqn:def_h_ijN} h^{ij}_{n}=\sqrt{n}(z_n-z_0)1_{z_n \in Z'\cap \Int P_i} + \bar{h}^{ij} 1_{z_n \not \in Z'\cap \Int P_i}, \end{equation} and define another $q$-dimensional random variable $\hat{\Gamma}^{ij}$ by \begin{equation}\label{eqn:def_Gamma_ij} \hat{\Gamma}^{ij} = \Gamma^i 1_{\Gamma^i\in \Int K_i} + \bar{h}^{ij} 1_{\Gamma^i \not\in \Int K_i}. \end{equation} Let $W$ be a measurable subset of $\Int K_i$ with $\Prob(\Gamma\in \partial W)=0$, where $\partial W$ stands for the boundary of $W$. The above definition of $h^{ij}_n$ and the fact that $\bar{h}^{ij}$ does not belong to $K_i$ imply \[ \Prob(h^{ij}_n \in W)=\Prob(\sqrt{n}(z_n-z_0) \in W \text{ and } z_n \in Z'\cap \Int P_i). \] Recalling that $Z'\cap \Int P_i = Z' \cap (z_0+\Int K_i)$ for each $i=1,\cdots,k$, we find \[ \Prob(\sqrt{n}(z_n-z_0) \in W \text{ and } z_n \in Z'\cap \Int P_i)=\Prob(\sqrt{n}(z_n-z_0) \in W \text{ and } z_n \in Z'). \] Combining the above two equalities with \eqref{eqn:prob_zN_Z'}, we have \[ \lim_{n\to\infty} \Prob(h^{ij}_n \in W)=\lim_{n\to\infty} \Prob(\sqrt{n}(z_n-z_0) \in W). \] By the asymptotic distribution \eqref{eqn:zN_dist_cov}, the definition of $\Gamma$ and equation \eqref{eqn:prob_gamma_W}, we have \[\lim_{n\to\infty} \Prob(\sqrt{n}(z_n-z_0) \in W) =\Prob(\Gamma\in W)=\Prob(\Gamma^i\in W).\] From the definition of $\hat{\Gamma}^{ij}$ in \eqref{eqn:def_Gamma_ij} and the facts $W\subset \Int K_i$ and $\bar{h}^{ij}\not\in K_i$ we have \[ \Prob(\Gamma^i\in W)=\Prob(\hat{\Gamma}^{ij}\in W). \] Combining the above three equalities together, we find \[ \lim_{n\to\infty} \Prob(h^{ij}_n \in W)=\Prob(\hat{\Gamma}^{ij}\in W)\] for each measurable set $W\subset \Int K_i$ with $\Prob(\Gamma\in \partial W)=0$. Because the set $\Int K_i$ itself satisfies $\Prob(\Gamma\in \partial (\Int K_i))=0$, the above equality holds with $\Int K_i$ in place of $W$. Since $h^{ij}_n$ and $\hat{\Gamma}^{ij}$ only take values in $\Int K_i \cup \{\bar{h}^{ij}\}$, we have $\lim_{n\to\infty} \Prob(h^{ij}_n = \bar{h}^{ij})=\Prob(\hat{\Gamma}^{ij}=\bar{h}^{ij})$. Also, for each $W\subset \Int K_i$ we have $\Prob(\Gamma\in \partial W)=\Prob(\hat{\Gamma}^{ij}\in \partial W)$. It is not hard to see \[h^{ij}_n \Rightarrow \hat{\Gamma}^{ij}.\] Since $\hat{r}^i_{nj}$ converges in probability to the fixed number $r^i_j$, which is strictly positive under the assumption in this theorem about $\Sigma_0$, we have \[ \frac{(h^{ij}_n)_j}{\hat{r}^i_{nj}}\Rightarrow \frac{\hat{\Gamma}^{ij}_j}{r^i_j}, \] where $(h^{ij}_n)_j$ and $\hat{\Gamma}^{ij}_j$ are the $j$th components of $(h^{ij}_n)$ and $\hat{\Gamma}^{ij}$ respectively. It follows that \begin{equation}\label{eqn:h_Nij_hatGamma} \lim_{n\to\infty} \Prob\left(\big|\frac{(h^{ij}_n)_j}{\hat{r}^i_{nj}}\big|\le \sqrt{\chi^2_1(\alpha)}\right) = \Prob\left(\big|\frac{\hat{\Gamma}^{ij}_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\right), \end{equation} because the probability for $\frac{\hat{\Gamma}^{ij}_j}{r^i_j}$ to lie on the boundary of $[-\sqrt{\chi^2_1(\alpha)},\sqrt{\chi^2_1(\alpha)}]$ is zero. The way $\hat{\Gamma}^{ij}$ is defined in \eqref{eqn:def_Gamma_ij} and the fact that $|\bar{h}^{ij}_j|>r^i_j \sqrt{\chi^2_1(\alpha)}$ imply \begin{equation}\label{eqn:hatGamma_Gamma} \Prob\left(\big|\frac{\hat{\Gamma}^{ij}_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\right) = \Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in \Int K_i\right). \end{equation} The facts that $|\bar{h}^{ij}_j|>r^i_j \sqrt{\chi^2_1(\alpha)}$ and that $\hat{r}^i_{nj}$ almost surely converges to $r^i_j$ imply \[\lim_{n\to\infty} \Prob\left(\big|\frac{\bar{h}^{ij}_j}{\hat{r}^i_{nj}}\big|\le \sqrt{\chi^2_1(\alpha)}\right)=0.\] We are now ready to put all pieces together to prove \eqref{eqn:ind_ci_conv_general} for the case $k\ge 2$. By the definition of $h^{ij}_n$ in \eqref{eqn:def_h_ijN} and the above equality, we have \begin{equation}\label{eqn:h_Nij_zN} \begin{split} &\lim_{n\to\infty} \Prob\left(\big|\frac{(h^{ij}_n)_j}{\hat{r}^i_{nj}}\big|\le \sqrt{\chi^2_1(\alpha)}\right) \\ =&\lim_{n\to\infty} \Prob\left(\frac{\sqrt{n}|(z_n-z_0)_j|}{\hat{r}^i_{nj}}\le \sqrt{\chi^2_1(\alpha)} \text{ and } z_n \in Z'\cap \Int P_i\right). \end{split} \end{equation} By the definition of $\hat{r}^i_{nj}$ in \eqref{eqn:def_hatr}, we can replace it by $r_{nj}$ in the right hand side of \eqref{eqn:h_Nij_zN}. Combining \eqref{eqn:h_Nij_hatGamma}, \eqref{eqn:hatGamma_Gamma} and \eqref{eqn:h_Nij_zN} together, we have \begin{equation} \begin{split} &\lim_{n\to\infty} \Prob\left(\frac{\sqrt{n}|(z_n-z_0)_j|}{r_{nj}}\le \sqrt{\chi^2_1(\alpha)} \text{ and } z_n \in Z'\cap \Int P_i\right)\\ =&\Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in \Int K_i\right)\\ =&\Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in K_i\right) \end{split} \end{equation} where the second equality follows from the fact that the probability for $\Gamma^i$ to belong to the boundary of $K_i$ is 0. Recalling that $Z'\cap (z_0+\Int K_i)=Z'\cap \Int P_i$, we can combine the above equality with \eqref{eqn:prob_zN_intKi} to prove \eqref{eqn:ind_ci_conv_general}. Under the assumption \eqref{eqn:prob_intersection}, we have \begin{equation}\label{eqn:sum_Prob_Gamma_Ki} \begin{split} &\sum_{i=1}^k \Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)} \text{ and } \Gamma^i\in K_i\right)\\ =&\sum_{i=1}^k \Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\right)\Prob\left( \Gamma^i\in K_i\right)\\ =&\sum_{i=1}^k \Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\right)\Prob\left( \Gamma\in K_i\right) = 1-\alpha, \end{split} \end{equation} where the third equality uses \eqref{eqn:prob_gamma_W} and the fourth equality uses \eqref{eqn:r_ij_prob} and the fact that $\sum_{i=1}^k \Prob\left( \Gamma\in K_i\right) =1$. This proves \eqref{eqn:ind_ci_conv} for the case $k\ge 2$. The case $k=1$ is much simpler. In this case, $z_0$ lies in the interior of a single $q$-cell $P_1$, $K_1=\mathbb{R}^q$, $d(f_0)_S(z_0)$ is an invertible linear map, and $\Gamma$ is equal to $\Gamma^1$. Since $z_n$ belongs to the interior of $P_1$ for all sufficiently large $n$, the number $r_{nj}$ converges almost surely to $r^1_j$ for each $j=1,\cdots,q$. Equation \eqref{eqn:ind_ci_conv_general} follows from the fact that \[ \frac{\sqrt{n}(z_n-z_0)_j}{r_{nj}} \Rightarrow \frac{\Gamma^1_j}{r^1_j}, \] and equation \eqref{eqn:ind_ci_conv} follows from the equality \eqref{eqn:r_ij_prob} with $i=1$. \end{proof} Below, we discuss two situations in which the equality \eqref{eqn:prob_intersection} holds. The first situation is when $k\le 2$. Obviously, when $k=1$ (that is, when $z_0$ lies in the interior of an $q$-cell in the normal manifold of $S$), the cone $K_1$ is the entire space $\mathbb{R}^q$, and the assumption \eqref{eqn:prob_intersection} automatically holds. It was noted by Michael Lamm that \eqref{eqn:prob_intersection} also holds when $k=2$. In the latter case, the cone $K_i$ is a half-space for $i=1,2$. Since $\Gamma^i$ is a multivariant normal random variable with mean zero, $-\Gamma^i$ and $\Gamma^i$ have the same distribution. It follows that $\Prob \left(\Gamma^i\in K_i\right)=1/2$, and that $\Prob \left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in K_i\right)$ and $\Prob \left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } -\Gamma^i\in K_i\right)$ are both equal to $\frac{1}{2}\Prob \left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\right)$. This proves \eqref{eqn:prob_intersection} when $k=2$. The second situation is when $S$ is a box and each $\Gamma^i$ has a diagonal covariance matrix. In this case, each $K_i$ is of the form $\{h\in \mathbb{R}^q \mid h_i\ge 0 \text{ for each }i\in I_+, h_i\le 0 \text{ for each }i\in I_-\}$ for some disjoint subsets $I_+$ and $I_-$ of $\{1,\cdots,q\}$. Since each $\Gamma^i$ has a diagonal covariance matrix, its components $\Gamma^i_j$ are independent of each other. From such independence, and by the symmetry of a mean-0 normal random variable in $\mathbb{R}$ with respect to the origin, it is not hard to see that \eqref{eqn:prob_intersection} holds for every $j$ and $i$. Whenever \eqref{eqn:prob_intersection} holds, we have \eqref{eqn:ind_ci_conv}, which means that the interval \begin{equation}\label{eqn:ind_ci_conv_formula} \left[(z_n)_j - \frac{\sqrt{\chi^2_1(\alpha)} r_{nj}}{\sqrt{n}}, (z_n)_j + \frac{\sqrt{\chi^2_1(\alpha)} r_{nj}}{\sqrt{n}}\right] \end{equation} is an asymptotically exact $(1-\alpha)$ confidence interval for $(z_0)_j$. The numerical examples in Section \ref{s:num} with $z_0=0$ do not belong to either of the above two cases, yet the coverage rates of individual confidence intervals obtained from this method are still reasonable. From those examples we observe that even if the difference between the two sides of \eqref{eqn:prob_intersection} is large for some $i$, the difference tends to be reduced after the summation over all $i$'s. As a result, the quantity \[ \sum_{i=1}^k \Prob\left(\big|\frac{\Gamma^i_j}{r^i_j}\big|\le \sqrt{\chi^2_1(\alpha)}\text{ and } \Gamma^i\in K_i\right) \] may not be very far from $1-\alpha$. Finally, Theorem \ref{t:ind_ci} assumes that $\Sigma_0$ is nonsingular, which implies the nonsingularity of $\Sigma_n$ for large $n$. Even if $\Sigma_n$ is singular, one can still use \eqref{eqn:ind_ci_conv_formula} to compute a confidence interval, with the caution that the coverage probability for the true solution may not be close to the prescribed level of confidence if $\Sigma_n$ is very different from $\Sigma_0$. \section{Numerical results}\label{s:num} Before applying the proposed method to numerical examples, we summarize how to use this method in practice. Among the assumptions, Assumption \ref{assu1} is standard. For Assumption \ref{assu2}, instead of directly checking if the normal map $L_K$ is a homeomorphism, we can check if one of its sufficient conditions holds (see the discussion below Assumption \ref{assu2}). Likewise, we can check if \eqref{eqn:prob_intersection} holds by determining if the problem belongs to one of the two case discussed below Theorem \ref{t:ind_ci}. In checking those assumptions, the asymptotical results in \cite{lu:nmb,lu.bud:crsvi} can be used to estimate $d(f_0)_S (z_0)$ and $K$. Confidence regions of $z_0$ are given by the set \eqref{eqn:conf_reg_nonsingular} (if $\Sigma_0$ is nonsingular) or the set $R_{n,\epsilon}$ in \eqref{eqn:R_n_epsilon} (if $\Sigma_0$ is singular), and the latter set can be approximated by $R_{n,0}$. According to Proposition \ref{p:invertible}, sets \eqref{eqn:conf_reg_nonsingular} and $R_{n,0}$ are ellipsoids with high probability. Computation of simultaneous confidence intervals for $z_0$ is done by finding the minimal bounding box of its confidence region. Individual confidence intervals of each component of $z_0$ can be computed using the formula (5.17), provided that \eqref{eqn:prob_intersection} holds. The quantity $r_{nj}$ defined in \eqref{eqn:def_rNj} depends on $\Sigma_n$ and $d(f_n)_S(z_n)$, and the latter is a nonsingular matrix with high probability. To convert confidence regions and intervals of $z_0$ into those of $x_0$, the key is to use the equality $x_0=\Pi_S(z_0)$. Suppose the set $A\subset \mathbb{R}^q$ is a $(1-\alpha)100\%$ confidence region for $z_0$, then the image of $A$ under the operator $\Pi_S$, denoted by $\Pi_S(A)$, contains $x_0$ with probability at least $1-\alpha$. If $S$ is a box, then one can easily project the simultaneous confidence intervals of $z_0$ to obtain simultaneous confidence intervals of $x_0$. The latter intervals are conservative, as the probability for the product of all those intervals to contain $x_0$ is at least $1-\alpha$. When $S$ is a box, each component of $x_0$ is the projection of a component of $z_0$ onto an interval (the product of all such intervals is $S$), and one can project the individual confidence intervals of $z_0$ to obtain individual confidence intervals of $x_0$. For problems in which $S$ is not a box, the above projection method would not be easy to implement in general, and one would need to exploit special structure in those problems to obtain confidence regions and intervals of $x_0$. \noindent \textbf{An example with $q=2$.} Here, we apply the method to the same example used in \cite{lu:nmb,lu.bud:crsvi}. In this example, $q=2$, $d=6$, $S=\mathbb{R}^2_+$, $F:\mathbb{R}^2\times \mathbb{R}^6 \to \mathbb{R}^2$ is defined by \begin{equation} F (x, \xi)= \begin{bmatrix} \xi_1 & \xi_2 \\ \xi_3 & \xi_4\end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix}\xi_5 \\ \xi_6\end{bmatrix}, \end{equation} and the random vector $\xi$ follows the uniform distribution over the box $[0,2]\times[0,1]\times[0,2]\times[0,4]\times[-1,1]\times[-1,1]$. The true problem is \begin{equation}\label{eqn:lcp} 0 \in \begin{bmatrix} 1 & 1/2 \\ 1 & 2\end{bmatrix} \ x + N_{\mathbb{R}^2_+} (x). \end{equation} The solution to \eqref{eqn:lcp} is $x_0=0$, and the solution of the corresponding normal map formulation is $z_0 = x_0 - E[F(x_0,\xi)] = 0$. The covariance matrix of $F(x_0, \xi)= (\xi_5,\xi_6)$ is given by \[ \Sigma_0=\begin{bmatrix} 1/3 & 0 \\ 0 &1/3 \end{bmatrix}, \] and the B-derivative $d(f_0)_{\mathbb{R}^2_+} (z_0)$ is a piecewise linear function represented by matrices \[ \begin{bmatrix} 1 & 1/2 \\ 1 & 2 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 1/2 \\ 0 & 2 \end{bmatrix} \text{ and } \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] in orthants $\mathbb{R}^2_+$, $\mathbb{R}_+\times \mathbb{R}_-$, $\mathbb{R}_-\times \mathbb{R}_+$ and $\mathbb{R}^2_-$ respectively. Accordingly, if we define random variables $\Gamma^i$ as in Section \ref{s:ind_ci}, then the covariance matrices of them are \[ \begin{bmatrix} 0.6296 & -0.3704 \\ -0.3704 & 0.2963 \end{bmatrix}, \begin{bmatrix} 0.3333 & -0.3333\\ -0.3333 & 0.6667 \end{bmatrix}, \begin{bmatrix} 0.3542 & -0.0417\\ -0.0417 & 0.0833 \end{bmatrix} \text{ and } \begin{bmatrix} 0.3333 & 0\\ 0 & 0.3333 \end{bmatrix} \] respectively. An SAA problem with $n=10$ is given by \[ 0 \in \begin{bmatrix} 0.9292 & 0.5400\\ 0.7536 & 2.1111\\ \end{bmatrix} x+ \begin{bmatrix} -0.1319 \\ -0.2906 \end{bmatrix} + N_{\mathbb{R}^2_+}(x). \] The SAA solution is $x_{10}=(0.0782, 0.1097)$, $z_{10}= ( 0.0782, 0.1097)$, and the sample covariance matrix of $F(x_{10},\xi)$ is \[ \Sigma_{10}=\begin{bmatrix} 0.4169 & 0.0137\\ 0.0137 & 0.1865 \end{bmatrix}. \] The B-derivative $d\Pi_{\mathbb{R}^2_+}(z_{10})$ is exactly the identity map on $\mathbb{R}^2$, and the B-derivative $d (f_{10})_{\mathbb{R}^2_+}(z_{10})$ is the linear map represented by the matrix \[ \begin{bmatrix} 0.9292 & 0.5400\\ 0.7536 & 2.1111\\ \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0.9292 & 0.5400\\ 0.7536 & 2.1111 \end{bmatrix}. \] The confidence regions in \eqref{eqn:conf_reg_nonsingular} are given by \[ \{z\in\mathbb{R}^2\mid 10(z-z_{10})^T \begin{bmatrix} 4.8810 & 9.3398 \\ 9.3398 & 24.2564 \end{bmatrix} (z-z_{10})\le \chi^2_2(\alpha) \}. \] Figure \ref{f:confregz0}(a) shows boundaries of the above confidence regions. The center of these regions is $z_{10}$, marked by `$\times$' in the graph. From the innermost to the outermost, the curves correspond to boundaries of confidence regions for $z_0$ at levels 10\%, $\cdots$, 90\% respectively. The point $z_0$ is marked by `+' and lies just beyond the 90\% confidence region. The dashed rectangle shown in the figure is the minimum enclosing box of the 90\% region. Figure \ref{f:confregz0}(b) shows confidence regions for $z_0$ obtained from a different SAA problem with sample size $n=30$, $x_{30}=0$ and $z_{30}=( -0.0483, -0.0114)$. Table \ref{tab:ind_simu_conf} shows the 90\% simultaneous and individual confidence intervals for $z_0$ obtained from the above two SAA problems. \begin{figure} \caption{Confidence regions for $z_0$ in the example $q=2$ at levels 10\%, $\cdots$, 90\%} \label{f:confregz0} \end{figure} \begin{table}[ht] \caption{Confidence intervals of level $90\%$ in the example $q=2$} \begin{center} \begin{tabular}{c|cll|cll} \hline & \multicolumn{3}{|c|}{$n=10$} & \multicolumn{3}{|c}{$n=30$}\\ & Est & Sim CI & Ind CI & Est & Sim CI & Ind CI\\ \hline $(z_0)_1$ & 0.08 & [-0.52, 0.68] & [-0.38, 0.54] & -0.05 & [-0.27, 0.17] & [-0.21, 0.12] \\ $(z_0)_2$ & 0.11 & [-0.16, 0.38] & [-0.10, 0.32] & -0.01 & [-0.23, 0.21] & [-0.18, 0.16] \\ \hline \end{tabular} \end{center} \label{tab:ind_simu_conf} \end{table} To test the coverage of confidence intervals obtained from the proposed method, we generate 200 SAA problems with $n=10$ and 200 SAA problems with $n=30$ from different random seeds, solve them using the PATH solver of GAMS, and compute simultaneous and individual confidence intervals for $z_0$ of levels 90\%, 95\% and 99\% from the solution for each SAA problem. We count how many times the 2-dimensional box formed by simultaneous confidence intervals cover $z_0$, and record the numbers in the row labeled by $z_0$ in Table \ref{tab:ind_conf_cov}. For example, the 90\% simultaneous confidence intervals obtained from 171 SAA problems with $n=10$ cover $z_0$ jointly. We also count how many times each component of $z_0$ is contained in the corresponding individual confidence intervals, and record the numbers in the remaining rows of Table \ref{tab:ind_conf_cov}. For example, the 90\% individual confidence intervals for $(z_0)_1$ obtained from 164 SAA problems with $n=10$ cover the true value $(z_0)_1=0$. \begin{table}[h] \caption{True solution coverage by confidence intervals in the example $q=2$ from 200 SAA problems} \begin{center} \begin{tabular}{c|ccc|ccc} \hline & \multicolumn{3}{|c|}{$n=10$} & \multicolumn{3}{|c}{$n=30$}\\ & $\alpha=$0.1 & 0.05 & 0.01 & $\alpha=$0.1 & 0.05 & 0.01 \\ \hline $z_0$ & 171 & 180 & 187 & 184 & 192 & 197\\ \hline $(z_0)_1$ & 164 & 185 & 194 & 172 & 188 & 198 \\ $(z_0)_2$ & 159 & 175 & 191 & 176 & 186 & 196 \\ \hline \end{tabular} \end{center} \label{tab:ind_conf_cov} \end{table} The proposed method generates confidence regions and simultaneous confidence intervals based on the asymptotic distribution in \eqref{eqn:dfNzN-weakconv} (or \eqref{eqn:zN_dist_cov_chi2} for singular cases). We evaluate how closely $z_n$ follow the asymptotic distribution using $\chi^2$ plots. We use the same SAA problems generated above for coverage tests. For each SAA problem with $n=10$ we compute the squared distance \[n \big[d(f_n)_S(z_n) (z_0 - z_n)\big]^T \Sigma_{n}^{-1} \big[d(f_n)_S(z_n) (z_0 - z_n)\big],\] and order these distances from smallest to largest as $d^2_{(1)}\le d^2_{(2)}\le \cdots \le d^2_{(200)}$. For each $j=1,\cdots,200$, let $q_{c,2}((j-1/2)/200)$ be the $100(j-1/2)/200$ quantile of the $\chi^2$ distribution with 2 degrees of freedom. We then graph the pairs $(q_{c,2}((j-1/2)/200), d^2_{(j)})$ for $j=1,\cdots, 200$ in Figure \ref{f:chi2plot}(a), in which the horizontal axis is for quantiles and the vertical axis is for squared distances. Figure \ref{f:chi2plot}(b) is obtained similarly, from the 200 SAA problems with sample size $n=30$. In each figure the points nearly follow a straight line through the origin with slope around 1, and the slope of the line in Figure \ref{f:chi2plot}(b) is closer to 1. This suggests that the expression on the left hand side of \eqref{eqn:dfNzN-weakconv} approximately follows the standard normal distribution. \begin{figure} \caption{$\chi^2$ plots in the example $q=2$} \label{f:chi2plot} \end{figure} \noindent \textbf{Examples with $q=10$.} We let $q=10$, $d=110$, $S=\mathbb{R}^{10}_+$, and $F:\mathbb{R}^{10}\times \mathbb{R}^{110} \to \mathbb{R}^{10}$ be defined as $F(x,\xi)=\Lambda(\xi) x + b(\xi)$, where $\Lambda(\xi)$ is a $10\times 10$ matrix whose entries are the first 100 components of $\xi$, and $b(\xi)\in \mathbb{R}^{10}$ consists of the last 10 components of $\xi$. Each diagonal entry of $\Lambda(\xi)$ is uniformly distributed on the interval [0,4], each entry above the main diagonal is uniformly distributed on [0,3], and each entry below the main diagonal is uniformly distributed on [0,2]. Thus, $E[\Lambda(\xi)]=\Lambda_0$ with $(\Lambda_0)_{ii}=2$, $(\Lambda_0)_{ij}=1.5$ for $i<j$ and $(\Lambda_0)_{ij}=1$ for $i>j$. We consider three different choices for the uniform distribution of $b(\xi)$, to obtain three different examples. In example 1, each component of $b(\xi)$ is uniformly distributed on [-1,1], so $E[b(\xi)]_i=0$ for each $i$. In example 2, the first five components of $b(\xi)$ are uniformly distributed on [-1,0.8] and the last five components uniformly distributed on [-1,1]. In example 3, each component of $b(\xi)$ is uniformly distributed on [-1,0.8]. The true solution $z_0$ is given by \[ \begin{array}{l} z_0=0\in \mathbb{R}^{10} \text{ in example 1,}\\ z_0= [0.21, \ 0.43, \ 0.85, \ 1.7, \ 3.4, \ -6.6, \ -6.6, \ -6.6, \ -6.6, \ -6.6]^T \times 10^{-2} \text{ in example 2,}\\ z_0=[0.01,\ 0.01, \ 0.03, \ 0.05, \ 0.1, \ 0.21, \ 0.42, \ 0.83, \ 1.67, \ 3.34]^T \times 10^{-2} \text{ in example 3.} \end{array} \] For each example, we generate 200 SAA problems with $n=50$, compute the SAA solutions, and obtain simultaneous and individual confidence intervals for $z_0$ of levels 90\%, 95\% and 99\% from each SAA solution. Table \ref{tab:ind_simu_conf_q10} lists the averages of the $90\%$ confidence intervals for $(z_0)_1$ and $(z_0)_{10}$ obtained from the 200 SAA problems, for each example. Table \ref{tab:example_q10} is analogous to Table \ref{tab:ind_conf_cov}. It summarizes the joint coverage of the true solution by simultaneous confidence intervals obtained from the 200 SAA problems for each example, and the coverage of $(z_0)_1$ and $(z_0)_{10}$ by the corresponding individual confidence intervals. \begin{table}[ht] \caption{Average $90\%$ confidence intervals for $(z_0)_1$ and $(z_0)_{10}$ in examples $q=10$ over 200 SAA problems} \begin{center} \begin{tabular}{c|ll|ll|ll} \hline & \multicolumn{2}{|c|}{Example 1} & \multicolumn{2}{|c}{Example 2} & \multicolumn{2}{|c}{Example 3}\\ & Sim CI & Ind CI & Sim CI & Ind CI & Sim CI & Ind CI\\ \hline $(z_0)_1$ & [-0.49, 0.30] & [-0.26, 0.07] & [-0.40, 0.30] & [-0.19, 0.09] & [-0.44, 0.28] & [-0.23, 0.07]\\ $(z_0)_{10}$ & [-0.41, 0.28] & [-0.20, 0.08] & [-0.46, 0.25] & [-0.25, 0.04] & [-0.33, 0.30] & [-0.15, 0.11]\\ \hline \end{tabular} \end{center} \label{tab:ind_simu_conf_q10} \end{table} \begin{table}[h] \caption{True solution coverage by confidence intervals in examples with $q=10$ from 200 SAA problems} \begin{center} \begin{tabular}{c|ccc|ccc|ccc} \hline & \multicolumn{3}{|c|}{Example 1} & \multicolumn{3}{|c}{Example 2} & \multicolumn{3}{|c}{Example 3}\\ & $\alpha=$0.1 & 0.05 & 0.01 & $\alpha=$0.1 & 0.05 & 0.01 & $\alpha=$0.1 & 0.05 & 0.01 \\ \hline $z_0$ & 198 & 199 & 200 & 196 & 198 & 200 & 197 & 197 & 198\\ \hline $(z_0)_1$ & 156 & 178 & 194 & 172 & 186 & 192 & 164 & 179 & 195 \\ $(z_0)_{10}$ & 173 & 183 & 195 & 176 & 188 & 197 & 172 & 184 & 195 \\ \hline \end{tabular} \end{center} \label{tab:example_q10} \end{table} \Appendix \noindent {\bf Acknowledgments.} Research of the author was supported by National Science Foundation under the grant DMS-1109099. The author thanks Amarjit Budhiraja, Michael Lamm, Yufeng Liu, Stephen M. Robinson and Liang Yin for helpful discussions related to this research, and the referees and the associate editor for comments that have improved the presentation of this paper. Michael Lamm observed that \eqref{eqn:prob_intersection} holds when $k=2$ and provided the idea for the current proof of Proposition \ref{p:R_0}. \end{document}

\begin{document} \title{A note on the $\mathbb Z_2$-equivariant Montgomery-Yang correspondence} \author{Yang Su} \address{Hua Loo-Keng Key Laboratory of Mathematics \newline \indent Chinese Academy of Sciences \newline\indent Beijing, 100190, China} \email{suyang{@}math.ac.cn} \date{August 19, 2009} \begin{abstract} In this paper, a classification of free involutions on $3$-dimensional homotopy complex projective spaces is given. By the $\mathbb Z_2$-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on $S^6$ with fixed-point set an embedded $S^3$. \end{abstract} \maketitle \section{Introduction}\label{sec:one} In \cite{M-Y}, Montgomery and Yang established a $1:1$ correspondence between the set of isotopy classes of smooth embeddings $S^3 \hookrightarrow S^6$, $C_3^3$, and the set of diffeomorphism classes of smooth manifolds homotopy equivalent to the $3$-dimensional complex projective space $\mathbb C \mathrm P^3$ ( these manifolds are called homotopy $\mathbb C \mathrm P^3$). It is known that the latter set is identified with the set of integers by the first Pontrjagin class of the manifold. Therefore so is the set $C_3^3$. In a recent paper \cite{Lv-Li}, Bang-he Li and Zhi L\"u established a $\mathbb Z_2$-equivariant version of the Montgomery-Yang correspondence. Namely, they proved that there is a $1:1$ correspondence between the set of smooth involutions on $S^6$ with fixed-point set an embedded $S^3$ and the set of smooth free involutions on homotopy $\mathbb C \mathrm P^3$. This correspondence gives a way of studying involutions on $S^6$ with fixed-point set an embedded $S^3$ by looking at free involutions on homotopy $\mathbb C \mathrm P^3$. As an application, by combining this correspondence and a result of Petrie \cite{Petrie}, saying that there are infinitely many homotopy $\mathbb C \mathrm P^3$'s which admit free involutions, the authors constructed infinitely many counter examples for the Smith conjecture, which says that only the unknotted $S^3$ in $S^6$ can be the fixed-point set of an involution on $S^6$. In this note we study the classification of the orbit spaces of free involutions on homotopy $\mathbb C \mathrm P^3$. As a consequence, we get the classification of free involutions on homotopy $\mathbb C \mathrm P^3$, and further by the $\mathbb Z_2$-equivariant Montgomery-Yang correspondence, the classification of involutions on $S^6$ with fixed-point set an embedded $S^3$. The manifolds $X^6$ homotopy equivalent to $\mathbb C \mathrm P^3$ are classified up to diffeomorphism by the first Pontrjagin class $p_1(X) =(24j+4)x^2$, $j \in \mathbb Z$, where $x^2$ is the canonical generator of $H^4(X;\mathbb Z)$ (c.~f.~\cite{M-Y}, \cite{Wall6}). We denote the manifold with $p_1 =(24j+4)x^2$ by $H\cp^3_j$. \begin{theorem}\label{thm:one} The manifold $H\cp^3_j$ admits a (orientation reversing) smooth free involution if and only if $j$ is even. On every $H\cp^3_{2k}$, there are exactly two free involutions up to conjugation.\footnote{The same result was also obtained independently by Bang-he Li (unpublished).} \end{theorem} \begin{corollary}\label{coro:one} An embedded $S^3$ in $S^6$ is the fixed-point set of an involution on $S^6$ if and only if its Montgomery-Yang correspondence is $H\cp^3_{2k}$. For each embedding satisfying the condition, there are exactly two such involutions up to conjugation. \end{corollary} Theorem \ref{thm:one} is a consequence of a classification theorem (Theorem \ref{thm:two}) of the orbit spaces. Theorem \ref{thm:two} will be shown in Section $3$ by the classical surgery exact sequence of Browder-Novikov-Sullivan-Wall. In Section $2$ we show some topological properties of the orbit spaces, which will be needed in the solution of the classification problem. \section{Topology of the Orbit Space} In this section we summarize the topological properties of the orbit space of a smooth free involution on a homotopy $\mathbb C \mathrm P^3$. Some of the properties are also given in \cite{Lv-Li}. Here we give shorter proofs from a different point of view and in a different logical order. Let $\tau$ be a smooth free involution on $H\cp^3$, a homotopy $\mathbb C \mathrm P^3$. Denote the orbit manifold by $M$. \begin{example} The $3$-dimensional complex projective space $\mathbb C \mathrm P^3$ can be viewed as the sphere bundle of a $3$-dimensional real vector bundle over $S^4$. The fiberwise antipodal map $\tau_0$ is a free involution on $\mathbb C \mathrm P^3$ (c.~f.~\cite[A.1]{Petrie}). Denote the orbit space by $M_0$. \end{example} As a consequence of the Lefschetz fixed-point theorem and the multiplicative structure of $H^*(H\mathbb C \mathrm P^3)$, we have \begin{lemma}\cite[Theorem 1.4]{Lv-Li} $\tau$ must be orientation reversing. \end{lemma} \begin{lemma}\label{lemma:cohom} The cohomology ring of $M$ with $\mathbb Z_2$-coefficients is $$H^*(M;\mathbb Z_2)=\mathbb Z_2[t,q]/(t^3=0, q^2=0),$$ where $|t|=1$, $|q|=4$. \end{lemma} \begin{proof} Note the the fundamental group of $M$ is $\mathbb Z_2$. There is a fibration $\widetilde{M} \to M \to \mathbb R \mathrm P^{\infty}$, where $M \to \mathbb R \mathrm P^{\infty}$ is the classifying map of the covering. We apply the Leray-Serre spectral sequence. Since $\widetilde{M}$ is homotopy equivalent to $\mathbb C \mathrm P^3$, the nontrivial $E_2$-terms are $E_2^{p,q}=H^p(\mathbb R \mathrm P^{\infty};\mathbb Z_2)$ for $q=0,2,4,6$. Therefore all differentials $d_2$ are trivial, and henceforth $E_2=E_3$. Now the differential $d_3 \colon E_3^{0,2} \to E_3^{3,0}$ must be an isomorphism. For otherwise the multiplicative structure of the spectral sequence implies that the spectral sequence collapses at the $E_3$-page, which implies that $H^*(M;\mathbb Z_2)$ is nontrivial for $* >6$, a contradiction. Then it is easy to see that $M$ has the claimed cohomology ring. \end{proof} \begin{remark} There is an exact sequence (cf.~\cite{Brown}) $$H_3(\mathbb Z/2) \to \mathbb Z \otimes_{\mathbb Z[\mathbb Z/2]} \mathbb Z_- \to H_2(M) \to H_2(\mathbb Z/2),$$ where $\mathbb Z_-$ is the nontrivial $\mathbb Z[\mathbb Z_2]$-module. By this exact sequence, $H_2(M)$ is either $\mathbb Z_2$ or trivial. $H^2(M;\mathbb Z_2)\cong \mathbb Z_2$ implies that $H_2(M) =0$. This was shown in \cite[Lemma 2.1]{Lv-Li} by geometric arguments. \end{remark} Now let's consider the Postnikov system of $M$. Since $\pi_1(M) \cong \mathbb Z_2$, $\pi_2(M) \cong \mathbb Z$ and the action of $\pi_1(M)$ on $\pi_2(M)$ is nontrivial, following \cite{Baues}, there are two candidates for the second space $P_2(M)$ of the Postnikov system, which are distinguished by their homology groups in low dimensions. See \cite[pp.265]{Olbermann} and \cite[Section 2A]{Su}. Let $\lambda$ be the free involution on $\mathbb C \mathrm Pi$, mapping $[z_0, z_1, z_2, z_3, \cdots]$ to $[-z_1, z_0, -z_3, z_2, \cdots]$. Let $Q = (\mathbb C \mathrm Pi \times S^{\infty})/(\lambda, -1)$, where $-1$ denotes the antipodal map on $S^{\infty}$, then there is a fibration $\mathbb C \mathrm Pi \to Q \to \mathbb R \mathrm Pi$ which corresponds to the nontrivial $k$-invariant. Lemma \ref{lemma:cohom} implies that $P_2(M)=Q$ since $Q$ has the same homology as $M$ in low dimensions. Let $f_2 \colon M \to Q$ be the second Postnikov map, since $\pi_i(M) \cong \pi_i(\mathbb C \mathrm P^3)=0$ for $3 \le i \le 6$, $f_2$ is actually a $7$-equivalence and $Q$ is the $6$-th space of the Postnikov system of $M$. By the formality of constructing the Postnikov system, all the orbit spaces have the same Postnikov system. Therefore we have shown \begin{proposition}\cite[Lemma 3.2]{Lv-Li} \label{prop:hmtp} The orbit spaces of free involutions on homotopy $\mathbb C \mathrm P^3$ are all homotopy equivalent. \end{proposition} Now let us consider the characteristic classes of $M$. \begin{lemma} The total Stiefel-Whitney class of $M$ is $w(M)=1+t+t^2$, where $t \in H^1(M;\mathbb Z_2)$ is the generator. \end{lemma} \begin{proof} The involution $\tau$ is orientation reversing, therefore $M$ is nonorientable and $w_1(M)=t$. The Steenrod square $Sq^2 \colon H^4(M;\mathbb Z_2) \to H^6(M;\mathbb Z_2)$ is trivial, this can be seen by looking at $M_0$, whose $4$-dimensional cohomology classes are pulled back from $S^4$. Therefore the second Wu class is $v_2(M)=0$. Thus by the Wu formula $w(M)=Sqv(M)$ it is seen that the total Stiefel-Whitney class of $M$ is $w(M)=1+t+t^2$. \end{proof} Let $\pi \colon H\cp^3 \to M$ be the projection map, the $\pi^*p_1(M)=p_1(H\cp^3)$. \begin{lemma} The induced map $\pi^* \colon H^4(M) \to H^4(H\cp^3)$ is an isomorphism. \end{lemma} \begin{proof} Apply the Leray-Serre spectral sequence for integral cohomology to the fibration $\widetilde{M} \to M \to \mathbb R \mathrm Pi$, the $E_2$-terms are $E_2^{p,q}=H^p(\mathbb R \mathrm Pi;\underline{H^q(\widetilde{M})})$. It is known that $H^3(M)=H^3(Q)=0$ and $H^5(M)=H^5(Q)=0$ (for $H^*(Q)$, see \cite[pp.265]{Olbermann}), therefore $E_{\infty}^{0,4}=E_2^{0,4}=H^4(\widetilde{M})$ is the only nonzero term in the line $p+q=4$. This shows that the edge homomorphism is an isomorphism, which is just $\pi^*$. \end{proof} Therefore the first Pontrjagin class of $M$ is $p_1(M)=(24j+4)u$ ($j \in \mathbb Z$), where $u=\pi^*(x^2)$ is the canonical generator of $H^4(M)$. \section{Classification of the orbit spaces} By Proposition \ref{prop:hmtp}, every orbit space $M$ is homotopy equivalent to $M_0$. Thus the set of conjugation classes of free involutions on homotopy $\mathbb C \mathrm P^3$ is in $1:1$ correspondence to the set of diffeomorphism classes of smooth manifolds homotopy equivalent to $M_0$. Denote the latter by $\mathcal M (M_0)$. Let $\mathscr{S}(M_0)$ be the smooth structure set of $M_0$, $Aut(M_0)$ be the set of homotopy classes of self-equivalences of $M_0$. There is an action of $Aut(M_0)$ on $\mathscr S(M_0)$ with orbit set $\mathcal M(M_0)$. (Since the Whitehead group of $\mathbb Z_2$ is trivial, we omit the decoration $s$ all over.) The surgery exact sequence for $M_0$ is $$L_7(\mathbb Z_2^-) \to \mathscr S(M_0) \to [M_0, G/O] \to L_6(\mathbb Z_2^-).$$ By \cite[Theorem 13A.1]{Wall}, $L_7(\mathbb Z_2^-)=0$, $L_6(\mathbb Z_2^-) \stackrel{c}{\cong} \mathbb Z_2$, where $c$ is the Kervaire-Arf invariant. Since $\dim M_0=6$ and $PL/O$ is $6$-connected, we have an isomorphism $[M_0, G/O] \cong [M, G/PL]$. For a given surgery classifying map $g \colon M_0 \to G/PL$, the Kervaire-Arf invariant is given by the Sullivan formula (\cite{Sullivan}, \cite[Theorem 13B.5]{Wall}) \begin{eqnarray*} c(M_0, g) & = & \langle w(M_0) \cdot g^*\kappa, [M_0] \rangle \\ & = & \langle (1+t+t^2) \cdot g^*(1+Sq^2+Sq^2Sq^2)(k_2+k_6), [M_0] \rangle \\ & = & \langle g^*k_6, [M_0] \rangle . \end{eqnarray*} Now since $M_0$ has only $2$-torsion, and modulo the groups of odd order we have $$G/PL \simeq Y \times \prod_{i \ge 2}(K(\mathbb Z_2, 4i-2) \times K(\mathbb Z,4i)),$$ where $Y=K(\mathbb Z_2,2) \times_{\delta Sq^2} K(\mathbb Z,4)$, we have $[M_0, G/PL]=[M_0,Y] \times [M_0, K(\mathbb Z_2,6)]$. $k_6$ is the fundamental class of $K(\mathbb Z_2,6)$. Therefore the surgery exact sequence implies \begin{lemma}\label{lemma:str} $\mathscr S (M_0) \cong [M_0, Y]$. \end{lemma} The projection $\pi \colon \mathbb C \mathrm P^3 \to M_0$ induces a homomorphism $\pi^* \colon [M_0, Y] \to [\mathbb C \mathrm P^3, Y]$, and $[\mathbb C \mathrm P^3,Y]$ is isomorphic to $\mathbb Z$ through the splitting invariant $s_4$ (\cite[Lemma 14C.1]{Wall}). Let $\Phi=s_4 \circ \pi^*$ be the composition. \begin{lemma}\label{lemma:exact} There is a short exact sequence $\mathbb Z_2 \to [M_0,Y] \stackrel{\Phi}{\rightarrow} 2\mathbb Z$. \end{lemma} \begin{proof} We have $[\mathbb C \mathrm P^3, Y]=[\mathbb C \mathrm P^2, Y]$, and according to Sullivan \cite{Sullivan}, the exact sequence $$L_4(1) \stackrel{\cdot 2}{\rightarrow} [\mathbb C \mathrm P^2, Y] \to [\mathbb C \mathrm P^1,Y]$$ is non-splitting. Let $p \colon Y \to K(\mathbb Z_2,2)$ be the projection map, then for any $f \in [\mathbb C \mathrm P^3, Y]$, $s_4(f) \in 2\mathbb Z$ if and only if $p \circ f \colon \mathbb C \mathrm P^3 \to K(\mathbb Z_2,2)$ is null-homotopic. Now by Lemma \ref{lemma:cohom}, the homomorphism $H^2(M_0;\mathbb Z_2) \to H^2(\mathbb C \mathrm P^3;\mathbb Z_2)$ is trivial. Therefore for any $g \in [M_0, Y]$, the composition $p \circ g \circ \pi$ is null-homotopic, thus $\mathrm{Im} \Phi \subset 2\mathbb Z$. On the other hand, since $\pi^* \colon H^4(M_0;\mathbb Z) \to H^4(\mathbb C \mathrm P^3)$ is an isomorphism, any map $f \colon \mathbb C \mathrm P^3 \to K(\mathbb Z,4)$ factors through some $g' \colon M_0 \to K(\mathbb Z,4)$. Let $i \colon K(\mathbb Z,4) \to Y$ be the fiber inclusion, since $s_4(i\circ f)$ takes any value in $2\mathbb Z$, so does $\Phi(i \circ g')$. Let $h \colon M_0 \to K(\mathbb Z_2,2)$ be a map corresponding to the nontrivial cohomology class in $H^2(M_0;\mathbb Z_2)$. By obstruction theory, there is a lifting $g \colon M_0 \to Y$. By the previous argument, there is also a map $g' \colon M_0 \to Y$ such that $\Phi(g)=\Phi(g')$, but $ p \circ g' \colon M_0 \to K(\mathbb Z_2,2)$ is null-homotopic. Therefore the kernel of $\Phi$ consists of two elements. \end{proof} \begin{remark} In \cite{Petrie} Petrie showed that every homotopy $\mathbb C \mathrm P^3$ admits free involution. It was pointed out by Dovermann, Masuda and Schultz \cite[pp.~4]{DMS} that since the class $G$ is in fact twice the generator of $H^4(S^4)$, Petrie's computation actually shows that every $H\cp^3_{2k}$ admits free involution, which is consistent with Lemma \ref{lemma:exact}. \end{remark} The set of diffeomorphism classes of manifolds homotopy equivalent to $M_0$, $\mathcal M(M_0)$, is the orbit set $\mathscr S(M_0)/Aut(M_0)$. In general, the determination of the action of $Aut(M_0)$ on the structure set is very difficult. But in our case, the situation is quite simple, since \begin{lemma}\label{lemma:action} The group of self-equivalences $Aut(M_0)$ is the trivial group. \end{lemma} \begin{proof} A special CW-complex structure of $M_0$ was given in \cite[pp.~885]{Lv-Li}: $M_0$ is a $\mathbb R \mathrm P^2$-bundle over $S^4$, therefore it is the union of two copies of $\mathbb R \mathrm P^2 \times D^4$, glued along boundaries. Choose a CW-complex structure of $\mathbb R \mathrm P^2$, we have a product CW-structure on one copy of $ \mathbb R \mathrm P^2 \times D^4$, and by shrinking the other copy of $\mathbb R \mathrm P^2 \times D^4$ to the core $\mathbb R \mathrm P^2$, we get a CW-complex structure on $M_0$, whose $2$-skeleton is $\mathbb R \mathrm P^2$. Let $\varphi \in Aut(M_0)$ be a self homotopy equivalence of $M_0$. By cellular approximation, we may assume that $\varphi$ maps $\mathbb R \mathrm P^2$ to $\mathbb R \mathrm P^2$. It is easy to see that $\varphi|_{\mathbb R \mathrm P^2}$ is homotopic $\mathrm{id}_{\mathbb R \mathrm P^2}$. Therefore, by homotopy extension, we may further assume that $\varphi|_{\mathbb R \mathrm P^2}=\mathrm{id}_{\mathbb R \mathrm P^2}$. The obstruction to construct a homotopy between $\varphi$ and $\mathrm{id}_{M_0}$, which is the identity on $\mathbb R \mathrm P^2$, is in $H^i(M,\mathbb R \mathrm P^2;\pi_i(M_0))$. Since $\pi_i(M_0)=0$ for $3 \le i \le 6$ and $H^1(M_0,\mathbb R \mathrm P^2;\mathbb Z_2)=H^2(M_0,\mathbb R \mathrm P^2;\mathbb Z)=0$, all the obstruction groups are zero. Therefore $\varphi \simeq \mathrm{id}_{M_0}$. \end{proof} Combine Lemma \ref{lemma:str}, Lemma \ref{lemma:exact} and Lemma \ref{lemma:action}, we have a classification of manifolds homotopy equivalent to $M_0$. \begin{theorem}\label{thm:two} Let $M^6$ be a smooth manifold homotopy equivalent to $M_0$, then $p_1(M)=(48j+4)u$, where $u\in H^4(M;\mathbb Z)$ is the canonical generator; for each $j \in \mathbb Z$, up to diffeomorphism, there are two such manifolds with the same $p_1=48j+4$. \end{theorem} Theorem \ref{thm:one} and Corollary \ref{coro:one} are direct consequences of this theorem. \end{document}

\begin{document} \title{Stability of certain Engel-like Distributions} \author{Aritra Bhowmick} \address{Statistics and Mathematics Unit, Indian Statistical Institute\\ 203, B.T. Road, Calcutta 700108, India.\\ e-mail: avowmix@gmail.com} \begin{abstract} In this article we introduce a higher dimensional analogue of Engel structure, motivated by the Cartan prolongation of contact manifolds. We study the stability of such structure, generalizing the Gray-type stability for Engel manifolds. \end{abstract} \maketitle \date{} \section{Introduction} In \cite{montGeneric} Montgomery proved that a generic rank $r$ distribution on a manifold of dimension $n$ is not stable if $r(n-r) > n$. Among the cases that are excluded by this inequality are line fields (when $r=1$), contact and even contact structures (when $r=n-1$) and lastly Engel structures (when $r=2, n=4$). An Engel structure is a rank $2$ distribution $\mathcal{D}$ on a $4$-manifold $M$ such that $\mathcal{D}^2$ is rank $3$ distribution and $\mathcal{D}^3=TM$. Like contact structures, any Engel structure is locally given as the common kernel of two $1$-forms (see \cite{montEngelDeform}) $$dz-ydx, \qquad dy - wdx$$ But unlike contact structures, Engel structures are not stable under arbitrary isotopy. In fact, any Engel structure $\mathcal{D}$ defines a complete flag $$\mathcal{L}\subset\mathcal{D}\subset\mathcal{E},$$ where the line field $\mathcal{L}$, called the characteristic line field, is usually not stable under isotopy. Golubev proved a modified version of Gray-type theorem for Engel structures in \cite{golubev}.\\ Engel manifolds are closely related to contact 3-manifolds. Starting with a 3-dimensional manifold with a contact structure $\xi$, one obtains a circle bundle by Cartan prolongation of $\xi$, where the total space of the bundle carries an Engel structure $\mathcal D$ with its characteristic line field tangent to fibers (\cite{montEngelDeform}). Prolongation on an arbitrary contact manifold $(N^{2n+1},\xi)$ give rise to fiber bundles $M\to N$ with fiber $\mathbb R P^{n-1}$. The total space of the bundle supports a flag $\mathcal{L}\subset\mathcal{D}\subset\mathcal{E}\subset TM$ on $M$, with rank vector $(2n-1, 2n, 4n-1, 4n)$, where \[\mathcal{D}^2=\mathcal{E},\ \mathcal{D}^3=TM,\] and $\mathcal{L}$ is the Cauchy characteristic distribution (see~\ref{defnCauchyDist}) of $\mathcal{E}$. In general, if we have a flag $\mathcal D\subset \mathcal{E}$ satisfying $\mathcal{D}^2=\mathcal{E}$ and $\mathcal{D}^3=TM$, then it does not necessarily follow that the Cauchy characteristic distribution $\mathcal L$ is contained in $\mathcal D$ (Example~\ref{exNonExample}). Motivated by these examples, we introduce the notion of generalized Engel structures on a manifold $M$. These are distributions $\mathcal{D}$ of even co-rank, such that $\mathcal{E}=\mathcal{D}^2$ is a co-rank $1$ distribution, $\mathcal{D}^3 = TM$ and the Cauchy characteristic distribution $\mathcal{L}$ of $\mathcal{E}$ is contained in $\mathcal{D}$ and has co-rank $1$ in $\mathcal{D}$. The distribution $\mathcal L$ is referred as the characteristic distribution of $\mathcal D$. The main goal of this article is to demonstrate a Gray-type stability property of these distributions \begin{theorem} \label{thmGrayGeneral} Let $\mathcal{D}_t$, $0\leq t\leq 1$, be a smooth one-parameter family of generalized Engel distributions on a closed manifold $M$. Assume that the characteristic distribution $\mathcal{L}_t$ of $\mathcal{D}_t$ is independent of $t$ and say $\mathcal{L}=\mathcal{L}_t$ for all $t$. Then there exists an isotopy $\phi_t$ of $M$ such that $$\phi_{t*}\mathcal{D}_t = \mathcal{D}_0,\qquad \phi_{t*}\mathcal{L} = \mathcal{L}$$ \end{theorem} We also obtain a local normal form for a set of generators of the annihilating ideal of a generalized Engel distribution $\mathcal D$.\\ The article is organized as follows: In Section 2 we recall some basic notions about distributions. In Section 3, we introduce generalized Engel structures and describe the Pfaffian system defining them. In Section 4 and 5 we prove the main results of this article. \section{Basic Notions and Examples} Given any distribution $\mathcal{A}$ on a manifold $M$, we can think of it as a sheaf of local sections of the sub-bundle $\mathcal{A}\subset TM$. Given two distributions $\mathcal{A},\mathcal{B}$, define by $[\mathcal{A},\mathcal{B}]$ as the sheaf of vector fields obtained by taking Lie brackets of local sections. Using this notation, recursively define, $$\mathcal{D}^{i+1}=\mathcal{D} + [\mathcal{D}, \mathcal{D}^i], \qquad \mathcal{D}^1 = \mathcal{D}$$ At every $x\in M$ we have the integer $q_i(x)=\dim \mathcal{D}^i_x$, where $\mathcal{D}^i_x$ is the stalk at the point $x$. Note that $\mathcal{D}^i$ defines a distribution if the integer $q_i(x)$ is locally constant. The integer sequence $(q_i(x))_i$ is called the \textit{growth vector} for the distribution $\mathcal{D}$ at $x$. A distribution is \emph{regular} if the growth vector is independent of the point $x$. A regular distribution $\mathcal{D}$ is called \textit{nonholonomic} if there is an integer $k$ such that $TM=\mathcal{D}^k$. In this article, we only consider nonholonomic distributions in the above sense. Before moving onto some examples, we recall the definition of Cauchy characteristic distribution (\cite{bryantExteriorDiff}), as it will play an important role in understanding the generalized Engel distributions. \begin{defn} \label{defnCauchyDist} {\em Given a co-rank $1$ distribution $\mathcal{E}$ on a manifold $M$, consider the collection, $$\mathcal{L} = \Big\{X\in\mathcal{E} \Big| [X,Y] \in\mathcal{E} \text{ for all }Y\in \mathcal{E}\Big\}$$ If $\mathcal{L}$ has constant rank everywhere it is called the \textit{Cauchy characteristic distribution} of $\mathcal{E}$.} \end{defn} We can locally define $\mathcal{L}$ as follows. Suppose, $\mathcal{E}\underset{loc.}{=}\ker\theta$. Then, $$\mathcal{L}\underset{loc.}{=}\ker d\theta|_\mathcal{E}=\Big\{X\in\mathcal{E} \Big| d\theta(X,Y)=0, \forall Y\in\mathcal{E}\Big\}$$ It is easy to see that Cauchy characteristic distribution is integrable. Indeed if $X,Y\in \mathcal{L}$ and $Z\in\mathcal{E}$, then we have, $$[[X,Y],Z] = [X,[Y,Z]] - [Y, [X,Z]]$$ Now, $[X,Z],[Y,Z]\in\mathcal{E}$ and hence $[[X,Y],Z]\in\mathcal{E}$. Thus $[X,Y]\in\mathcal{L}$. But then $\mathcal{L}$ is integrable by Frobenius Theorem. \begin{example}\quad \em{ \begin{enumerate} \item[(a)] A contact distribution $\xi$ on an odd-dimensional manifold $M$ is a co-rank $1$ distribution such that $\xi^2=TM$ and the Cauchy characteristic distribution of $\xi$ is trivial. \item[(b)] Similarly, an even contact structure on an even-dimensional manifold $M$ is a co-rank $1$ distribution $\mathcal{E}$ such that $\mathcal{E}^2=TM$ and the Cauchy characteristic distribution of $\mathcal{E}$ is a line field. Like contact structures, an even contact structure $\mathcal{E}$ is locally given as the the kernel of some $1$-form $\alpha$ satisfying $\alpha\wedge d\alpha^n \ne 0$. \\ \item[(c)] An Engel structure $\mathcal{D}$ is a co-rank $2$ nonholonomic distribution on a $4$-dimensional manifold $M$, such that $\mathcal{D}^2$ is an even contact structure and $\mathcal{D}^3=TM$. The characteristic line field $\mathcal{L}$ of $\mathcal{D}^2$ turns out to be contained in $\mathcal{D}$ (see \cite{montEngelDeform}). Thus the Engel structure completely defines the flag $\mathcal{L}\subset\mathcal{D}\subset\mathcal{D}^2\subset TM$. Any Engel structure $\mathcal{D}$ can be locally realized as the kernel of two $1$-forms, $$dz-ydx,\qquad dy-wdx.$$ \end{enumerate} } \end{example} We are particularly interested in distributions in higher dimensions, which exhibit properties similar to Engel structures. \subsection{Cartan Prolongation} A prime example of Engel manifold appears as Cartan prolongation of contact $3$-manifolds $(M,\xi)$. On $M$ we construct a fiber bundle, with total space $\mathbb{P}(\xi)$ and fibers $\mathbb{RP}^1$. There is a canonical Engel distribution on the total space, where the characteristic line field is along the fibers. We describe below the Cartan prolongation of an arbitrary contact manifold.\\ Consider an odd dimensional manifold $N^{2n+1}$ with a contact structure $\xi$. On $N$ we construct the Grassmann bundle $$\mathbb{RP}^{2n-1} \hookrightarrow \mathbb{P}\xi \overset{\pi}{\to} N,$$ wher the fiber over a point $x\in N$ is the projective space of lines in the vector space $\xi_x$. The total space $Q=\mathbb{P}\xi$ is of dimension $4n$. The inverse image of $\xi$ defines a corank $1$ distribution $\mathcal{E} = d\pi^{-1}(\xi)$ on $Q$ which is easily seen to be an even contact structure. On the other hand, there is a distribution $\mathcal D$ which is obtained as follows: At a point $[\ell]\in Q$, where $\ell$ is a line in $\xi_p$ for $p\in N$, define $\mathcal{D}_{[\ell]} = d\pi|_{[l]}^{-1}(\ell)$. Since $\pi$ is a submersion, $\mathcal{D}$ is a co-rank $2n$ distribution on $Q$. Clearly, $\mathcal{D}\subset\mathcal{E}$. Set $\mathcal{L}$ as the vertical sub-bundle of $TQ$ over $N$, i.e., $\mathcal{L}$ is tangent along the fibers. Thus, we have a flag, $\mathcal{L}\subset\mathcal{D}\subset\mathcal{E}$. The distribution $\mathcal{D}$ is called the prolongation of $\xi$. In particular, if $n=1$ and $N$ is a contact $3$ manifold, then $\dim Q=4$ and $\mathcal{D}$ is an Engel structure on $Q$.\\ We can now observe a few general properties of this flag. Since $\xi$ is locally expressed as $\ker (dz - \sum_{i=1}^n y_idx_i)$ we have \[\xi \underset{loc.}{=} \Big\langle\partial_{y_i}, P_i = \partial_{x_i} + y_i\partial_z \Big| i=1\ldots n\Big\rangle\] Any line $\ell\subset\xi_p$ is represented by a no-trivial linear combination of these vectors. Hence, on $Q$ we can introduce new homogeneous coordinates along the fiber as \[\{a_1,\ldots,a_n,b_1,\ldots,b_{n-1}\}.\] At any point on $Q$, we can associate a uniquely defined vector, \[Z = P_n + \sum_{i=1}^n a_i \partial_{y_i} + \sum_{j=1}^{n-1} b_jP_j.\] Then we can describe the flag $\mathcal{L}\subset\mathcal{D}\subset\mathcal{E}$ locally as follows, \begin{align*} \mathcal{L} &= \langle \partial_{a_1},\ldots,\partial_{a_n},\partial_{b_1},\ldots,\partial_{b_{n-1}} \rangle\\ \mathcal{D} &= \mathcal{L} \oplus \langle Z \rangle\\ \mathcal{E} &= \mathcal{D} \oplus \langle \partial_{y_1},\ldots,\partial_{y_n},P_1,\ldots,P_{n-1} \rangle\\ TQ &= \mathcal{E} \oplus \langle \partial_z \rangle \end{align*} From this description we observe that \begin{itemize} \item $\cork\mathcal{D}$ is even, $\cork\mathcal{E}$ is $1$ and co-rank of $\mathcal{L}$ in $\mathcal{D}$ is $1$ \item $\mathcal{D}^2=\mathcal{E}, \mathcal{D}^3=TM$ \item $\mathcal{L}$ is the Cauchy characteristic distribution $\mathcal{E}$ so that $[\mathcal{L},\mathcal{E}]\subset\mathcal{E}$. \end{itemize} \section{Generalized Engel structure} Motivated by the Cartan prolongation of a contact structure, we define a generalized Engel structure. \begin{defn} \label{defnGenEngel} {\em A \emph{generalized Engel structure} or an Engel-like distribution on a manifold $M$ is a distribution $\mathcal{D}$ of even co-rank, such that \begin{enumerate} \item $\mathcal{E}=\mathcal{D}^2$ is a co-rank $1$ distribution \item $\mathcal{D}^3 = TM$ \item $\mathcal{L}$, the Cauchy characteristic distribution of $\mathcal{E}$, is contained in $\mathcal{D}$ \item $\mathcal{L}$ has co-rank $1$ in $\mathcal{D}$. \end{enumerate} Thus, we have a flag \[ \rlap{$\underbrace{\phantom{\mathcal{L}\subset\mathcal{D}}}_{\cork=1}$}\mathcal{L}\subset \overbrace{\mathcal{D}\subset\rlap{$\underbrace{\phantom{\mathcal{E}\subset TM}}_{\cork=1}$}\mathcal{E} \subset TM}^{\textrm{even }\cork} \] The distribution $\mathcal{L}$ will be called the \emph{characteristic distribution} of the generalized Engel distribution $\mathcal{D}$}. \end{defn} \subsection{A remark on the definition} When $\dim M=4$ and $\mathcal{D}$ is of co-rank $2$, we have an Engel structure. As mentioned earlier, in this case the Cauchy characteristic distribution $\mathcal{L}$ of $\mathcal{E}=\mathcal{D}^2$ is completely determined by $\mathcal D$ and it is contained in $\mathcal{D}$. For higher co-rank we can not expect this to happen in general and this can be seen from the examples below. In the first two examples $\mathcal{L}\not\subset\mathcal{D}$ and in the third one co-rank of $\mathcal{L}$ in $\mathcal{D}$ is not of co-rank $1$. All the examples are constructed over $\mathbb{R}^8$, where the coordinates are understood from the context. \begin{example}\label{exNonExample}\quad \em{ \begin{enumerate} \item[(a)] Suppose, $\mathcal{D} = \langle\partial_x,\partial_y,\partial_z, \partial_w + x\partial_{x_1} + y\partial_{y_1} + z\partial_{z_1} + z_1\partial_t\rangle$. Then, $[\mathcal{D},\mathcal{D}] = \langle\partial_{x_1},\partial_{y_1},\partial_{z_1}\rangle$ and hence, $$\mathcal{E} = \mathcal{D}^2 = \langle\partial_x,\partial_y,\partial_z,\partial_{x_1},\partial_{y_1},\partial_{z_1},\partial_w + z_1\partial_t\rangle$$ Lastly, $[\mathcal{D},\mathcal{D}^2] = \langle \partial_t\rangle$ and so $\mathcal{D}^3=TM$. The Cauchy characteristic distribution of $\mathcal{E}$ is $\mathcal{L} = \langle\partial_x,\partial_y,\partial_z,\partial_{x_1},\partial_{y_1}\rangle$ which is a rank $5$ distribution. Clearly in this case we have $\mathcal{L}\not\subset\mathcal{D}$. \item[(b)] Consider, $\mathcal{D} = \langle\partial_w, \partial_{x_1} + w\partial_{y_1} + y_1\partial_z, \partial_{x_2} + w\partial_{y_2} + y_2\partial_z, \partial_{x_3} + w\partial_{y_3}\rangle$ Then, $[\mathcal{D},\mathcal{D}] = \langle\partial_{y_1},\partial_{y_2},\partial_{y_3}\rangle$ and so, $$\mathcal{E} = \mathcal{D}^2 = \langle\partial_w, \partial_{y_1},\partial_{y_2},\partial_{y_3}, \partial_{x_1} + y_1\partial_z, \partial_{x_2} + y_2\partial_z, \partial_{x_3}\rangle$$ Clearly, $\mathcal{D}^3=TM$. Also, $\mathcal{L} = \langle\partial_w,\partial_{y_3},\partial_{x_3}\rangle$. Since $\partial_{y_3}\not\in\mathcal{D}$, we have $\mathcal{L}\not\subset\mathcal{D}$. \item[(c)] Let $v_i = \partial_{x_i} + w\partial_{y_i} + y_i\partial_z$ for $i=1,2,3$ and $\mathcal{D} = \langle\partial_w,v_1,v_2,v_3\rangle$. Then, $[\mathcal{D},\mathcal{D}] = \langle\partial_{y_1},\partial_{y_2},\partial_{y_3}\rangle$ and so $\mathcal{E}=\mathcal{D}^2$ is a co-rank 1 distribution and $\mathcal{D}^3 = TM$. Also, $\mathcal{L} = \langle\partial_w\rangle$. In this case, we have the flag $\mathcal{L}\subset\mathcal{D}\subset\mathcal{E}$, where $\mathcal{E}$ is an even contact structure. Further, note that there exists a co-rank $1$ integrable distribution $\langle v_1,v_2,v_3\rangle$ contained in $\mathcal{D}$. \end{enumerate} } \end{example} The above examples justify the conditions (3) and (4) in the definition of the generalized Engel structure. \subsection{Pfaffian system} A \emph{Pfaffian system} is a sub-distribution of the cotangent bundle $T^*M$. Given a distribution $\mathcal{D}\subset TM$, we have an associated Pfaffian system $\mathcal{S}(\mathcal{D})$ defined as the collection of $1$-forms which vanish on $\mathcal{D}$. In this section we would like to find out the Pfaffian system for a generalized Engel distribution.\\ We start with a co-rank $k+1$ generalized Engel distribution $\mathcal{D}$, where $k=2l+1$ is odd. Suppose locally, \[\mathcal{E} = \{\theta=0\} \ \ \text{ and }\ \ \ \mathcal{D}=\{\omega^1=\ldots=\omega^k=0=\theta\}\] for $1$-forms $\theta,\omega^1,\ldots,\omega^k$. Set, $\eta^i = \omega^1\wedge\ldots\omega^k\wedge\theta\wedge d\omega^i$. \begin{prop} \label{propDistToForms} We have the following. \begin{enumerate} \item \label{itemForms:1} $\{\eta^1,\ldots,\eta^k\}$ is point-wise linearly independent \item \label{itemForms:2} $\omega^i\wedge\theta\wedge d\theta^{l+1} = 0, \forall i=1,\ldots,k$ \item \label{itemForms:3} $\theta\wedge d\theta^{l+1}\ne 0$ \item \label{itemForms:4} $\theta\wedge d\theta^{l+2} = 0$ \end{enumerate} \end{prop} \begin{proof} Choose local vector fields $D$ and $R$ such that $\mathcal{D}/\mathcal{L} = \langle D \mod \mathcal{L} \rangle$ and $TM/\mathcal{E} =\langle R \mod \mathcal{E}\rangle$. Since $\mathcal{L}$ is integrable and $\mathcal{E}=\mathcal{D}^2 =\mathcal{D} + [\mathcal{D},\mathcal{D}]$, we have that the map \begin{align*} \mathcal{L} &\to \mathcal{E}/\mathcal{D}\\ L &\mapsto [D,L] \mod \mathcal{D} \end{align*} is a surjective bundle map. Since $\{\omega^i\}$ is linearly independent and $\mathcal{D}$ is their common kernel in $\mathcal{E}$, we can choose dual vectors $V^i\in\mathcal{E}/\mathcal{D}$. Also from the surjectivity, there exists $L^i\in \mathcal{L}$ such that $V^i=[D,L^i] \mod \mathcal{D}$. Then $\eta^i\ne 0\forall i$, since we have, $$\eta^i(V^1,\ldots,V^k,R,D,L^i) \ne 0$$ If possible, let $\{\eta^i\}$ be linearly dependent at the point $p$. Then without loss of generality we may assume that, $\eta^1 = \sum_{i=2}^k f_i \eta^i$ at $p$ for some functions $f_i$. Set, $\tilde{\omega}^1 = \omega^1 - \sum_{i>1} f_i\omega^i$. Then clearly, $\mathcal{D}$ is also defined as $\{\tilde{\omega}^1=\omega^2=\ldots=\omega^k=0=\theta\}$. But then we must have that $$\tilde{\omega}^1\wedge\omega^2\wedge\ldots\omega^k\wedge\theta\wedge d\tilde{\omega}^1 \ne 0$$ On the other hand, \begin{align*} \tilde{\omega}^1\wedge\omega^2\wedge\ldots\omega^k\wedge\theta\wedge d\tilde{\omega}^1 &= \Big(\omega^1-\sum_{i>1}f_i\omega^i\Big)\wedge\omega^2\wedge\ldots\wedge\omega^k\wedge\theta\wedge \Big(d\omega^1 - \sum_{i>1}d(f_i\omega^i)\Big)\\ &=\omega^1\wedge\ldots\omega^k\wedge\theta\wedge\Big(d\omega^1-\sum_{i>1}df_i\wedge\omega^i -\sum_{i>1}f_id\omega^i\Big)\\ &=\omega^1\wedge\ldots\omega^k\wedge\theta\wedge\Big(d\omega^1-\sum_{i>1}f_id\omega^i\Big)\\ &=\eta^1-\sum_{i>1}f_i\eta^i\\ &=0, \textnormal{ at the point } p \end{align*} This is a contradiction. Hence we have that the set of $(k+3)$-forms $\{\eta^i\}$ are point-wise linearly independent. This proves (\ref{itemForms:1}). Now observe that $\mathcal{L}=\ker (d\theta|_{\ker\theta})$. So, on $\mathcal{E}/\mathcal{L}$ we have that $d\theta$ has full rank. Since $\mathcal{L}$ is of co-rank $k+2=2l+3$, $$\theta\wedge d\theta^{l+1}\ne 0, \theta\wedge d\theta^{l+2} = 0$$ Thus proving (\ref{itemForms:3}) and (\ref{itemForms:4}). Next, consider the $2l+3$-form $\omega^i\wedge d\theta^{l+1}|_\mathcal{E}$ on $\mathcal{E}$. For any $L\in\mathcal{L}$ we have that, $\iota_L\omega^i\wedge d\theta^{l+1}|_\mathcal{E}$ identically zero, since $\omega^i(L) = 0$ and $\iota_Ld\theta|_\mathcal{E}=0$. Thus, $\mathcal{L}$ is in the kernel of $\omega^i\wedge d\theta^{l+1}|_\mathcal{E}$. But $\mathcal{L}$ has co-rank $2l+2$ in $\mathcal{E}$ and then by simple rank counting argument, $\omega^i\wedge d\theta^{l+1}|_\mathcal{E}$ is identically zero. But $\mathcal{E} = \ker\theta$ and hence $\omega^i\wedge\theta\wedge d\theta^{l+1}= 0$, proving (\ref{itemForms:2}). \end{proof} The converse of \ref{propDistToForms} is also true. Suppose we are given some co-rank $k+1$ distribution $\mathcal{D}$ on a manifold $M$, where $k=2l+1$, such that $\mathcal{D}$ is locally the common kernel of $1$-forms $\{\theta,\omega^1,\ldots,\omega^k\}$, satisfying \begin{itemize} \item $\{\eta^1,\ldots,\eta^k\}$ is point-wise linearly independent, where $\eta^i = \omega^1 \wedge \ldots \omega^k \wedge \theta \wedge d\omega^i$ \item $\omega^i\wedge\theta\wedge d\theta^{l+1} = 0$ for all $i=1,\ldots,k$ \item $\theta\wedge d\theta^{l+1}\ne 0$ \item $\theta\wedge d\theta^{l+2} = 0$ \end{itemize} \begin{prop} Under the above hypotheses, $\mathcal{D}$ is a generalized Engel structure. \end{prop} \begin{proof} Set $\mathcal{E} = \ker\theta$ and $\mathcal{L} = \ker d\theta|_\mathcal{E}$. These are locally defined distributions of co-rank $1$ and $2l+3$ respectively. We can get a local framing of $TM/\mathcal{L}$ as $\big\{R, X_i,Y_j \big| i,j = 1,\ldots,l+1\big\}$ such that $\theta\wedge d\theta^{l+1}(R,X_1,Y_1,\ldots,X_{l+1},Y_{l+1}) \ne 0$. Consider $L\in \mathcal{L}$. Then we have, $$0=\omega^i\wedge\theta\wedge d\theta^{l+1}(L, R, X_1,\ldots,Y_{l+1})= \omega^i(L)\theta\wedge d\theta^{l+1}(R, X_1,\ldots,Y_{l+1})$$ since all other terms vanish. But then $\omega^i(L)=0$, for all $i$. Thus, $L\in\ker D$. So we have the flag, $$\mathcal{L}\subset\mathcal{D}\subset\mathcal{E}$$ Next we show $\mathcal{E}=\mathcal{D}^2$. First note that $\mathcal{D} \underset{loc.}{=} \mathcal{L} \oplus \langle Z \rangle$, for some choice of vector field $Z$. Since $\mathcal{L}$ is Cauchy characteristic distribution of $\mathcal{E}$, we have $[\mathcal{L},\mathcal{E}] \subset\mathcal{E}$. In particular, $[\mathcal{L}, Z] \subset \mathcal{E}$. Also $\mathcal{L}$ being integrable, we have $[\mathcal{L},\mathcal{L}] \subset \mathcal{L}$ by Frobenius. Then clearly, $[\mathcal{D},\mathcal{D}] \subset \mathcal{E}$. Thus $\mathcal{D}^2\subset\mathcal{E}$. For the equality, consider the map, \begin{align*} \Phi : \mathcal{L} &\to \mathcal{E}/\mathcal{D}\\ L &\mapsto [Z,L] \mod \mathcal{D} \end{align*} $\Phi$ is a bundle map : $[Z,fL] = Z(f)L + f[Z,L] \equiv f[Z,L] \mod \mathcal{D}$, as $\mathcal{L}\subset\mathcal{D}$. We will show that $\Phi$ is of full rank, which will imply that $\mathcal{E}=\mathcal{D}^2$. Equivalently this happens if $\Phi^*$ is injective. Dualizing $\Phi$ we get, \begin{align*} \Phi^* : \big(\mathcal{E}/\mathcal{D}\big)^* &\rightarrow \mathcal{L}^*\\ [\alpha] &\mapsto -\iota_Zd\alpha|_\mathcal{L} \end{align*} where $(\mathcal{E}/\mathcal{D})^*$ consists of classes of $1$-forms $\alpha$ defined on $\mathcal{E}$, which annihilates $\mathcal{D}$. Consider the $1$-forms, $$\tau^i \coloneqq -\iota_Z d\omega^i|_\mathcal{L}$$ defined on $\mathcal{L}$. Since $\omega^i$ induces a basis for $(\mathcal{E}/\mathcal{D})^*$, it is enough to show that the maps $\tau^i$ are point-wise linearly independent for $\Phi^*$ to be injective. If not, then without loss of generality assume, $\tau^1 = \sum_{i>1} f_i\tau^i$ at some point $p$, for some functions $f_i$. Get dual vectors $\{R,V_1,\ldots,V_k\}$ in $TM/\mathcal{D}$ of $\{\theta,\omega^1,\ldots,\omega^k\}$ respectively. Now, for any $L\in\mathcal{L}$, we have $\eta^i(V_1,\ldots,V_k,R,Z,L)=d\omega^i(Z,L)=-\tau^i(L)$. Thus, $$\eta^1(V_1,\ldots,V_k,R,Z,L)=-\sum_{i>1}f_i\eta^i(V_1,\ldots,V_k,R,Z,L)$$ at the point $p$. But then, $\eta^1=-\sum_{i>1}f_i\eta^i$ at $p$, contradicting point-wise linear independence of $\{\eta^i\}$. Hence, $\{\tau^i\}$ must be linearly independent point-wise. Thus we get $\mathcal{E}=\mathcal{D}^2$. $\mathcal{E}$ and consequently $\mathcal{L}$ are now globally defined distributions. Lastly to verify $\mathcal{D}^3=TM$ note that, $d\theta$ is non-degenerate on $\mathcal{E}/\mathcal{L}$. In particular, for $Z\in\mathcal{D}$ satisfying $\mathcal{D} = \mathcal{L}\oplus\langle Z \rangle$, we have $\iota_Zd\theta \ne 0$. So, $d\theta(Z,V) \ne 0$, for some $V\in\mathcal{E}/\mathcal{D}$. Then $V \in [\mathcal{D},\mathcal{D}]$ and $0\ne d\theta(Z,V)=-\theta[Z,V]$. Thus, $TM = \mathcal{E} \oplus \langle [Z,V] \rangle$. So, $TM = \mathcal{D}^3$. \end{proof} \section{Stability of generalized Engel structure} Engel structures are not globally stable due to the presence of an integrable subbundle, though they have local stability property. Golubev proved the following Gray-type theorem for Engel structure which shows that a homotopy $\mathcal D_t$, $0\leq t\leq 1$, of Engel structures is obtained by an isotopy provided the characteristics distribution of $\mathcal D_t$ is independent of $t$. \begin{theorem}[\cite{golubev}]\label{thmGolubev} Let $\mathcal{D}_t$, $0\leq t\leq 1$, be a one-parameter family of oriented Engel structures on an oriented compact $4$-dimensional manifold $M$, such that the characteristic line field $\mathcal{L}(\mathcal{D}_t)=\mathcal{L}$ for all $t$. Then there exists an isotopy $\phi_t$, $0\leq t\leq 1$, of $M$ such that \[\phi_{t*}(\mathcal{D}_t)=\mathcal{D}_0,\qquad \phi_{t*}(\mathcal{L})=\mathcal{L}.\] \end{theorem} Theorem ~\ref{thmGrayGeneral} is direct generalization of Theorem~\ref{thmGolubev} for generalized Engel structure. We shall first prove a special case of of this theorem. \begin{theorem} \label{thmIsotopyOfD} Suppose $\mathcal D_t$, $0\le t \le 1$, is a one-parameter family of generalized Engel structure on a closed manifold $M$ such that $\mathcal{D}_t^2$ is independent of $t$ and equals $\mathcal{E}$. If $\mathcal L$ is the Cauchy characteristics distribution of $\mathcal E$ then there exists an isotopy $\phi_t$ of $M$ such that \[\phi_{t*}\mathcal{D}_0 = \mathcal{D}_t, \qquad \phi_{t*}\mathcal{E} = \mathcal{E}, \qquad \phi_{t*}\mathcal{L} = \mathcal{L}.\] \end{theorem} \subsection{Proof of Theorem~\ref{thmIsotopyOfD}} The approach of the proof is very similar to that of Adachi in \cite{adachiCorank}. The proof will follow through a sequence of Lemmas. We assume that $\mathcal{D}_t$ is a given smooth one-parameter family of co-rank $k+1$, where $k=2l+1$, generalized Engel distribution on a closed manifold $M$ such that $\mathcal{E}=\mathcal{D}_t^2$ is independent of $t$ and the Cauchy characteristic distribution of $\mathcal{E}$ is $\mathcal{L}$.\\ Suppose $\mathcal D_t$ is defined as the common kernel of the 1-forms $\omega^1_t,\omega^2_t,\ldots,\omega^k_t,\theta$ on some open subset $U$ of $M$. Let $X_t$ be a time-dependent vector field on $M$ which satisfies the following system of equations on $U$. \begin{equation}\iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0, \forall i=1,\ldots,k; \qquad X_t\in\mathcal{L}.\label{eqnODE}\end{equation} As we shall see below, if any such $X_t$ exists, then its flow has the desired property. We begin with the following observation. \begin{prop} \label{propDiffEqLocalIndept} Suppose $\mathcal{E}=\ker{\eta}$ and $\mathcal{D}_t=\{\mu^1_t=\ldots=\mu^j_t=0=\eta\}$ for a smooth family of $1$-forms $\{\mu^i_t,\eta\}$ on $U$. Then the vector fields $X_t$, $0\leq t\leq 1$, in (\ref{eqnODE}) also satisfies the relations \[\iota_{X_t}d\mu^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\mu^i_t|_{\mathcal{D}_t} = 0.\] In other words, $X_t$ depends on the distributions $\mathcal D_t$, not on the choice of local 1-forms defining the distributions. \end{prop} \begin{proof} Suppose $\{\mu^i_t,\eta\}$ and $\{\omega^i_t, \theta\}$ be as above. Then we must have that $\eta=f\theta$ for some non-zero function $f$ and \[ \begin{pmatrix} \mu^1_t\\ \vdots\\ \mu^k_t \end{pmatrix}= A_t \begin{pmatrix} \omega^1_t\\ \vdots\\ \omega^k_t \end{pmatrix} \] for a family of non-singular $k\times k$ matrix $A_t=(A^{ij}_t)$. So, $\mu^i_t=\sum_j A^{ij}_t\omega^j_t$. Then $d\mu^i_t=\sum_j dA^{ij}_t \wedge \omega^j_t + A^{ij}_t d\omega^j_t$. So, \begin{align*} \iota_{X_t}d\mu^i_t|_{\mathcal{D}_t} &= \sum_j \iota_{X_t}dA^{ij}_t\omega^j_t|_{\mathcal{D}_t} - \iota_{X_t}\omega^j_t dA^{ij}_t|_{\mathcal{D}_t} + A^{ij}_t\iota_{X_t}d\omega^j_t|_{\mathcal{D}_t}\\ &= \sum_j A^{ij}_t\iota_{X_t}d\omega^j_t|_{\mathcal{D}_t}, \textnormal{as } \omega^j_t(X_t)=0 \textnormal{ and } \omega^j_t|_{\mathcal{D}_t}=0\\ &= -\sum_j A^{ij}_t \frac{d}{dt}\omega^j_t|_{\mathcal{D}_t} \end{align*} On the other hand, \begin{align*} \frac{d}{dt}\mu^i_t|_{\mathcal{D}_t} &= \sum_j \frac{dA^{ij}_t}{dt}\omega^j_t|_{\mathcal{D}_t} + A^{ij}_t\frac{d}{dt}\omega^j_t|_{\mathcal{D}_t}\\ &=\sum_j A^{ij}_t\frac{d}{dt}\omega^j_t|_{\mathcal{D}_t} \end{align*} Hence we have, $\iota_{X_t}d\mu^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\mu^i_t|_{\mathcal{D}_t} = 0$. Thus $X_t$ is a solution for every family of local forms defining $\mathcal{D}_t$. \end{proof} \begin{prop} \label{propDiffEquationLocal} Suppose there exists a time dependent vector field $X_t$, $0\leq t\leq 1$, on $M$ which satisfies the following conditions: \[\iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0, \forall i=1,\ldots,k; \qquad X_t\in\mathcal{L}\] where $\theta$, $\omega^i_t, 0\leq t\leq 1, i=1,2,\dots,k$ is a smooth family of (local) 1-forms such that \[\mathcal{E}\underset{loc.}{=}\ker\theta \ \ \ and \ \ \ \mathcal{D}_t\underset{loc.}{=}\{\omega^1_t = \ldots = \omega^k_t = 0 = \theta\}.\] Then the flow $\phi_t$ obtained by integrating the time-dependent vector field $X_t$ satisfies, \[\phi_{t*}\mathcal{D}_0=\mathcal{D}_t,\ \ \phi_{t*}\mathcal{L}=\mathcal{L}\] \end{prop} \begin{proof} Since $X_t\in\mathcal{L}$ and $[\mathcal{L},\mathcal{E}]\subset\mathcal{E}$, we have that $\phi_{t*}\mathcal{E}=\mathcal{E}$ and hence $\phi_{t*}\mathcal{L}=\mathcal{L}$, as $\mathcal{L}$ is completely defined by $\mathcal{E}$. So we have, $\phi_t^*\theta=F_t\theta$ for some family of non-vanishing functions $F_t$. In order to verify that $\phi_{t*}\mathcal{D}_0=\mathcal{D}_t$, we would show the existence of smooth families of functions $G^{ij}_t$ and $F^i_t$, satisfying \[\phi_{t}^*\omega^i_t = \sum_{j} G^{ij}_t\omega^j_0 + F^i_t\theta,\forall i \tag{*} \label{eqnMoser}\] where the matrix $$\Big(G^{ij}_t\Big)_{k\times k}$$ is non-singular, for every $t$. We solve for $$\frac{d}{dt}\Big(\phi_{t}^*\omega^i_t - \sum_{j} G^{ij}_t\omega^j_0 - F^i_t\theta\Big)=0, \forall i$$ any solution of which will satisfy (\ref{eqnMoser}). Differentiating both sides of (\ref{eqnMoser}) with respect to $t$ we get, \begin{align*} \sum_j \frac{dG^{ij}_t}{dt}\omega^j_0 + \frac{dF^i_t}{dt}\theta &= \frac{d}{dt}\phi_t^*\omega^i_t\\ &=\phi_t^*\Big(L_{X_t}\omega^i_t + \frac{d}{dt}\omega^i_t\Big)\\ &=\phi_t^*\Big(\iota_{X_t}d\omega^i_t + \frac{d}{dt}\omega^i_t\Big) \end{align*} Now from the hypothesis, $\iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0$ and hence we must have family of functions $g^{ij}_t$ and $f^i_t$ such that, $$\iota_{X_t}d\omega^i_t + \frac{d}{dt}\omega^i_t = \sum_j g^{ij}_t \omega^j_t + f^i_t\theta$$ Pulling back by $\phi_t$ we have, \begin{align*} \phi_t^*\Big(\iota_{X_t}d\omega^i_t + \frac{d}{dt}\omega^i_t\Big) &= \sum_j (g^{ij}_t\circ\phi_t) \phi_t^*\omega^j_t + (f^i_t\circ\phi_t) \phi_t^*\theta\\ &=\sum_j(g^{ij}_t\circ\phi_t)\Big(\sum_pG^{jp}_t\omega^p_0 + F^j_t\theta\Big) + (f^i_t\circ\phi_t) F_t\theta\\ &=\sum_p \Big(\sum_j (g^{ij}_t\circ\phi_t)G^{jp}_t\Big)\omega^p_0 + \Big(\sum_j (g^ij_t\circ\phi_t)F^j_t + (f^i_t\circ\phi_t)F_t\Big)\theta \end{align*} Comparing coefficients of $\omega^i_0$ and $\theta$, we get a system of first order differential equations, \begin{align*} \frac{dG^{ip}_t}{dt} &= \sum_j (g^{ij}_t\circ\phi_t)G^{jp}_t, \forall i,p\\ \frac{dF^i_t}{dt} &= \sum_j (g^{ij}_t\circ\phi_t)F^j_t + (f^i_t\circ\phi_t)F_t, \forall i \end{align*} with the initial conditions, $$G^{ij}_0 = \partial_{ij},\qquad F^i_0 = 0$$ This system, being affine, has a solution for all $0\le t\le 1$. Since the matrix $(G^{ij}_t)=I_{k}$ at $t=0$, we must have that the matrix is non-singular in a range around $0$. Hence we have that $\phi_{t*}\mathcal{D}_0=\mathcal{D}_t,\phi_{t*}\mathcal{E}=\mathcal{E}$ \end{proof} Next we shall discuss how to obtain the time dependent vector field $X_t$ satisfying the hypothesis of Proposition~\ref{propDiffEquationLocal}. First, suppose that we have a (locally) finite open cover $\{U^\lambda\}$ of $M$ and local fields $X^\lambda_t\in\mathcal{L}|_{U_\lambda}$ on $U_\lambda$, which satisfy the relations \[\iota_{X^\lambda_t}d\omega^{i,\lambda}_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^{i,\lambda}_t|_{\mathcal{D}_t} = 0, \forall i=1,\ldots,k\] on $U_\lambda$, where $\mathcal{D}_t|_{U_\lambda} = \{\omega^{i,\lambda}_t=0=\theta^\lambda\}$ and $\mathcal{E}|_{U_\lambda}=\ker\theta^\lambda$. Consider a partition of unity $\{\rho_\lambda\}$ subordinate to the covering. Set, $$X_t = \sum_\lambda \rho_\lambda X^\lambda_t$$ Then $X_t$ is a global field and $X_t\in\mathcal{L}$ since each $X^\lambda_t\in \mathcal{L}$. By Proposition~\ref{propDiffEqLocalIndept}, the local fields $X^\lambda_t$ satisfies \[\iota_{X^\lambda_t}d\omega^{i,\mu}_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^{i,\mu}_t|_{\mathcal{D}_t} = 0, \forall i=1,\ldots,k\] on $U_\lambda\cap U_\mu$, whenever the set is non-empty. Then the global field $X_t$ satisfies the hypothesis of Proposition~\ref{propDiffEquationLocal} over each $U_\lambda$, as $X_t$ is a convex linear combination of the local fields.\\ Now in order to prove Theorem~\ref{thmIsotopyOfD}, we need to find \emph{local} field $X_t\in \mathcal{L}$ such that $$\iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0, \forall i=1,\ldots,k$$ on some open set $U$, where $\omega^i_t,\theta$ as in Proposition~\ref{propDiffEquationLocal} \subsubsection{Obtaining the time-dependent field} In order to obtain the local field $X_t$, we introduce a few notations. Define, \begin{align*} \mathcal{K}^i_t &= \ker d\omega^i_t|_{\mathcal{D}_t}=\Big\{X\in\mathcal{D}_t\Big| d\omega^i_t(X,Y)=0, \forall Y\in\mathcal{D}_t\Big\}\\ &\phantom{= \ker d\omega^i_t|_{\mathcal{D}_t}\hspace{0.8ex}}=\Big\{X\in\mathcal{D}_t\Big| \omega^i_t([X,Y])=0, \forall Y\in\mathcal{D}_t\Big\}\\ \mathcal{J}^i_t &= \bigcap_{j\ne i} \mathcal{K}^j_t=\Big\{X\in\mathcal{D}_t\Big| \omega^j_t([X,Y])=0,\forall Y\in\mathcal{D}_t,\forall j\ne i\Big\}\\ \mathcal{W}_t &= \bigcap_j \mathcal{K}^j_t = \Big\{X\in\mathcal{D}_t \Big| [X,Y] \in \mathcal{D}_t, \quad \forall Y\in \mathcal{D}_t\Big\} \end{align*} \begin{lemma} For each $i=1,\ldots,k$ we have $\mathcal{K}^i_t\subset \mathcal{L},\forall t$ \end{lemma} \begin{proof} Locally, we have a family of $1$-forms $\alpha_t$, such that $\mathcal{L} = \mathcal{D}_t \cap \ker\alpha_t$. Since $\mathcal{L}$ is integrable, from Frobenius theorem we have, in particular, $\alpha_t\wedge\omega^1_t\wedge\ldots\wedge\omega^k_t\wedge d\omega^i_t = 0, i=1,\ldots,k$. Pick some $K\in\mathcal{K}^i_t$. Then we have, $$0=\iota_K(\alpha_t\wedge\omega^1_t\wedge\ldots\wedge\omega^k_t\wedge d\omega^i_t) =\alpha_t(K)\omega^1_t\wedge\ldots\wedge\omega^k_t\wedge d\omega^i_t$$ as the other terms vanish. But $\omega^1_t\wedge\ldots\wedge\omega^k_t\wedge d\omega^i_t\ne 0$. Hence we have, $\alpha_t(K) = 0$ and hence $K\in\mathcal{L}$. Thus we have, $\mathcal{K}^i_t\subset\mathcal{L},\forall i,\forall t$ \end{proof} In particular we have that $\mathcal{J}^i_t\subset\mathcal{L}$ for each $i=1,\ldots,k$ and $\mathcal{W}_t\subset\mathcal{L}$. Also observe that for any $i$, $\mathcal{W}_t = \mathcal{J}^i_t \cap \mathcal{K}^i_t$. \begin{lemma} \label{propRankJ} For any $0\le t\le 1$, $\mathcal{K}^i_t, \mathcal{W}_t$ and $\mathcal{J}^i_t$ have constant ranks. Further, $\rk\mathcal{J}^i_t = \rk\mathcal{W}_t + 1$ and co-rank of $\mathcal{K}^i_t$ in $\mathcal{L}$ is $1$ \end{lemma} \begin{proof} Fix $0\le t\le 1$. Since $\mathcal{L}$ has co-rank $1$ in $\mathcal{D}_t$, choose a vector field $Z$ such that $\mathcal{D}_t = \mathcal{L} \oplus \langle Z \rangle$. Then consider the map, \begin{align*} \Psi : \mathcal{L} &\to \mathcal{E}/\mathcal{D}_t\\ L &\mapsto [Z,L]\mod\mathcal{D}_t \end{align*} $\Psi$ is a bundle map. Also $\Psi$ has full rank, since $\mathcal{D}_t^2=\mathcal{E}$ and $\mathcal{L}$ is integrable. Clearly $\mathcal{W}_t$ is contained in the kernel of $\Psi$. Also for $\Psi(L)=0$, i.e., $[Z,L]\in\mathcal{D}_t$ we have that $[L,X]\in\mathcal{D}_t$ for any $X\in\mathcal{D}_t$, since $\mathcal{L}$ is integrable. Thus $L\in\mathcal{W}_t$ and so $\mathcal{W}_t=\ker\Psi$. Hence $\mathcal{W}_t$ is a constant rank distribution. Since the induced forms $\omega^i_t|_{\mathcal{E}/\mathcal{D}_t}$ are (point-wise) linearly independent, choose some vector fields $\{V_i\}$ from $\mathcal{E}_t$, such that $\{\bar{V}_i=V_i \mod \mathcal{D}_t\}$ is the corresponding dual basis. Consider the map, \begin{align*} \Psi_i : \mathcal{L} &\to \mathcal{E}/(\mathcal{D}_t\oplus\langle V_i\rangle)\\ L &\mapsto [Z,L]\mod (\mathcal{D}_t\oplus\langle V_i\rangle) \end{align*} Again $\Psi_i$ is a full rank bundle map. As $\bar{V}_i$ is dual to $\omega^i_t|_{\mathcal{E}/\mathcal{D}_t}$, for any $L\in\mathcal{J}^i_t$, we have that $$[Z,L] \mod\mathcal{D}_t = f_i \bar{V_i}$$ for some function $f_i$, and thus $\mathcal{J}^i_t\subset\ker\Psi_i$. Conversely, suppose $\Psi_i(X)=0$,i.e., $[Z,X] \in \mathcal{D}_t \oplus \langle V_i\rangle$. But then for $j\ne i$, $\omega^j_t[Z,X] = 0$ and so $X\in \mathcal{J}^i_t$. Thus $\mathcal{J}^i_t=\ker\Psi_i$, proving that $\mathcal{J}^i_t$ is of constant rank. Similarly define the map, \begin{align*} \Phi_i : \mathcal{L} &\to \mathcal{E}/(\mathcal{D}_t\oplus\langle V_1,\ldots,\hat{V_i},\ldots,V_k\rangle)\\ L &\mapsto [Z,L]\mod (\mathcal{D}_t\oplus\langle V_1,\ldots,\hat{V_i},\ldots,V_k\rangle) \end{align*} and observe that $\ker\Phi_i = \mathcal{K}^i_t$. Since $\Phi_i$ is again a full rank bundle map, $\mathcal{K}^i_t$ is of constant rank. Clearly we have that $\rk\mathcal{J}^i_t = \rk\mathcal{W}_t + 1$ and co-rank of $\mathcal{K}^i_t$ in $\mathcal{L}$ is $1$, for any $0\le t\le 1$. \end{proof} Now we find the local field. \begin{lemma} For each $i$ there is a local field $X^i_t\in\mathcal{J}^i_t$ such that $\iota_{X^i_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0$ \end{lemma} \begin{proof} From Lemma~\ref{propRankJ} we have that $\mathcal{J}^i_t = \mathcal{W}\oplus\mathcal{U}^i_t$ for some line field $\mathcal{U}^i_t\subset\mathcal{J}^i_t$. Clearly $\mathcal{U}^i_t\not\subset\mathcal{K}^i_t$. Then we can get, $\mathcal{D}_t = \mathcal{K}^i_t \oplus \mathcal{V}^i_t$ such that $\mathcal{U}^i_t\subset\mathcal{V}^i_t, \forall t$. As co-rank of $\mathcal{K}^i_t$ in $\mathcal{L}$ is $1$ we have that $\mathcal{V}^i_t$ is of constant rank with $\rk\mathcal{V}^i_t = 2$. Since by definition $\mathcal{K}^i_t =\ker d\omega^i_t|_{\mathcal{D}_t}$, we have that $d\omega^i_t$ is non-degenerate over $\mathcal{V}^i_t$. Hence we have a solution $X^i_t\in\mathcal{V}^i_t$ such that, $$\iota_{X^i_t}d\omega^i_t|_{\mathcal{V}^i_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{V}^i_t} = 0$$ Now, $\iota_{X^i_t}d\omega^i_t|_{\mathcal{K}^i_t}=0$. Also since $\mathcal{K}^i_t\subset\mathcal{L}\subset\mathcal{D}_s$ for every parameter $s$, we have that $\frac{d}{dt}\omega^i_t|_{\mathcal{K}^i_t}=0$, since $\mathcal{D}_s\subset\ker\omega^i_s$. Thus we also have that, $\iota_{X^i_t}d\omega^i_t|_{\mathcal{K}^i_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{K}^i_t} = 0$. Combining we get that $$\iota_{X^i_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0$$ as required. We now show that $X^i_t\in\mathcal{U}^i_t$, which will yield that $X^i_i\in\mathcal{J}^i_t$ Restricting to $\mathcal{U}^i_t$ we see that, $\iota_{X^i_t}d\omega^i_t|_{\mathcal{U}^i_t} =- \frac{d}{dt}\omega^i_t|_{\mathcal{U}^i_t}$. But $\mathcal{U}^i_t\subset\mathcal{J}^i_t\subset\mathcal{L}\subset\mathcal{D}_s\subset\ker\omega^i_s,\forall s$ and so $\frac{d}{dt}\omega^i_t|_{\mathcal{U}^i_t}=0$. Thus we have that, $$\iota_{X^i_t}d\omega^i_t|_{\mathcal{U}^i_t}=0$$ Now $X^i_t\in\mathcal{V}^i_t$ and $\mathcal{U}^i_t\subset\mathcal{V}^i_t$ is a line field. But $d\omega^i_t$ is non-degenerate on $\mathcal{V}^i_t$, with $\rk\mathcal{V}^i_t=2$. Thus the only way $\iota_{X^i_t}d\omega^i_t|_{\mathcal{U}^i_t}=0$ is possible if $X^i_t\in\mathcal{U}^i_t$. This completes the proof. \end{proof} Set $$X_t=\sum_i X^i_t.$$ Since each $X^i_t\in \mathcal{J}^i_t\subset\mathcal{L}$, we have that $X_t\in\mathcal{L}$. \begin{lemma} $\iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0,\forall i=1,\ldots,k$ \end{lemma} \begin{proof} Since $X^i_t\in\mathcal{J}^i_t$, we have that $\iota_{X^i_t}d\omega^j_t|_{\mathcal{D}_t} = 0\forall j\ne i$. Thus, \begin{align*} \iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} &= \sum_j \iota_{X^j_t}d\omega^i_t|_{\mathcal{D}_t}\\ &=\iota_{X^i_t}d\omega^i_t|_{\mathcal{D}_t}\\ &=-\frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} \end{align*} that is to say, $\iota_{X_t}d\omega^i_t|_{\mathcal{D}_t} + \frac{d}{dt}\omega^i_t|_{\mathcal{D}_t} = 0$ for every $i=1,\ldots,k$. \end{proof} \subsection{Proof of Theorem~\ref{thmGrayGeneral}} We shall see that Theorem~\ref{thmGrayGeneral} follows from Theorem~\ref{thmIsotopyOfD} by using the following lemma. \begin{lemma} \label{lemmaIsotopyOfE} Suppose we are given a one-parameter family of co-rank $1$ distributions $\mathcal{E}_t$ on a compact manifold $M$, such that Cauchy characteristic distribution $\mathcal{L}_t$ of $\mathcal{E}_t$ is independent of $t$, say $\mathcal{L}_t=\mathcal{L}$ and $TM=\mathcal{E}_t^2$. Then there exists an isotopy $\phi_t$ of $M$ such that $$\phi_{t*}\mathcal{E}_0 = \mathcal{E}_t,\qquad \phi_{t*} \mathcal{L} = \mathcal{L}$$ \end{lemma} \begin{proof}[Proof of Theorem~\ref{thmGrayGeneral}] Since $\mathcal{D}_t^3=TM$, in particular we have that $TM = \mathcal{E}_t^2$. Hence, using Lemma~\ref{lemmaIsotopyOfE} we get an isotopy $\phi_t$ that fixes $\mathcal{L}$ and $\phi_{t*}\mathcal{E}_0=\mathcal{E}_t$. Since $\phi_t$ is a diffeomorphism, we get $\phi_{t*}^{-1}\mathcal{E}_t=\mathcal{E}_0$. Set, $\mathcal{D}_t^\prime =\phi_{t*}^{-1}\mathcal{D}_t$. Clearly we have the flag, $\mathcal{L}\subset\mathcal{D}_t^\prime\subset\mathcal{E}_0$, where $\mathcal{L}$ is the Cauchy characteristic distribution of $\mathcal{E}_0$. Since Lie brackets are preserved under push-forwards by diffeomorphisms, we have that $\mathcal{D}_t^{\prime2}=\mathcal{E}_0$ and $\mathcal{D}_t^{\prime3} = TM$. Now using Theorem~\ref{thmIsotopyOfD}, we get another isotopy $\psi_t$ that fixes both $\mathcal{L}$ and $\mathcal{E}_0$, and $\psi_{t*}\mathcal{D}_0=\mathcal{D}_t^\prime$. Then, $\psi_{t*}^{-1}\mathcal{D}_t^\prime=\mathcal{D}_0$. Set, $\Phi_t = \psi_t^{-1} \circ \phi_t^{-1}$. Then $\Phi_t$ is the desired isotopy such that, $$\Phi_{t*}\mathcal{D}_t = \mathcal{D}_0,\qquad \Phi_{t*}\mathcal{L} = \mathcal{L}$$ \end{proof} \subsubsection{Proof of Lemma~\ref{lemmaIsotopyOfE}} The proof can be found in \cite{montGoursat}. We reproduce it here with minor modification.\\ We have, $\mathcal{E}_t \underset{loc.}{=} \ker \theta_t$. By hypothesis, $\mathcal{L} = \ker d\theta_t|_{\mathcal{E}_t},\forall t$. Suppose, $\mathcal{E}_t = \mathcal{L} \oplus \mathcal{V}_t$, where $\mathcal{V}_t=\mathcal{L}^{\perp_{\mathcal{E}_t}}$ with respect to some fixed choice of a Riemannian metric. Then $d\theta_t$ is non-degenerate on the sub-space $\mathcal{V}_t$. So there exists a unique (local) field $X_t\in V_t$ such that $$\iota_{X_t} d\theta_t|_{\mathcal{V}_t} = - \frac{d}{dt}\theta_t|_{\mathcal{V}_t}$$ Since $X_t\in\mathcal{E}_t$ and $\mathcal{L} = \ker d\theta_t|_{\mathcal{E}_t}$, we have $\iota_{X_t}d\theta_t|_\mathcal{L} = 0$. Also, since $\mathcal{L} \subset\mathcal{E}_s=\ker\theta_s, \forall s$, we have that $\frac{d}{dt}\theta_t|_\mathcal{L} = 0$. Then combining the two we have, $$\iota_{X_t} d\theta_t|_{\mathcal{E}_t} = - \frac{d}{dt}\theta_t|_{\mathcal{E}_t}$$ As $\iota_{X_t} d\theta_t + \frac{d}{dt}\theta_t=0$ when restricted to $\mathcal{E}_t$, we have that $$\iota_{X_t} d\theta_t + \frac{d}{dt}\theta_t=h_t\theta_t$$ for some family of functions $h_t$.\\ Now integrating $X_t$ we get a local flow $\phi_t$. Then, \begin{align*} \frac{d}{dt} \phi_t^* \theta_t &= \phi_t^*\Big(L_{X_t}\theta_t + \frac{d}{dt}\theta_t\Big)\\ &= \phi_t^*\Big(\iota_{X_t}d\theta_t + \frac{d}{dt}\theta_t\Big)\\ &= \phi_t^*(h_t\theta_t)\\ &= g_t\phi_t^*\theta_t \end{align*} where $g_t = h_t\circ\phi_t$. Now we would like to have, $\phi_t^*\theta_t = H_t\theta_0$ for some non-zero function $H_t$. Differentiating both sides, we get $$\frac{dH_t}{dt}\theta_0 = \frac{d}{dt}\phi_t^*\theta_t = g_t\phi_t*\theta_t = g_tH_t\theta_0$$ Thus we have the differential equation, $$\frac{dH_t}{dt} = g_tH_t$$ with the initial condition, $H_0 = 1$. This has a solution, $$H_t = \exp \int_{0}^{t} g_s ds$$ which is clearly non-zero. Thus for the field $X_t$, the local flow $\phi_t$ satisfies $$\phi_{t*}\mathcal{E}_0 = \mathcal{E}_t$$ and fixes the Cauchy characteristic distribution $\mathcal{L}$.\\ Now in order to get a global flow, we use partition of unity. First note that the vector field $X_t$ doesn't depend on the choice of the local defining form $\theta_t$. Consider $\eta_t = f_t\theta_t$ for some non-zero function $f_t$. Then, $\mathcal{E}_t = \ker\eta_t$. Now, $d\eta_t = df_t\wedge\theta_t + f_t\wedge d\theta_t$. So, \begin{align*} \iota_{X_t} d\eta_t|_{\mathcal{E}_t} &= \iota_{X_t}df_t \theta_t|_{\mathcal{E}_t} - \iota_{X_t}\theta_tdf_t|_{\mathcal{E}_t} + f_t\iota_{X_t}d\theta_t|_{\mathcal{E}_t}\\ &= f_t\iota_{X_t}d\theta_t|_{\mathcal{E}_t} \end{align*} since $\theta_t(X_t)=0$ and $\theta_t|_{\mathcal{E}_t}=0$. Also, $\frac{d}{dt}\eta_t|_{\mathcal{E}_t} = \frac{df_t}{dt}\theta_t + f_t\frac{d}{dt}\theta_t|_{\mathcal{E}_t}=f_t\frac{d}{dt}\theta_t|_{\mathcal{E}_t}$. Hence we have, $$\iota_{X_t}d\eta_t|_{\mathcal{E}_t} + \frac{d}{dt}\eta_t|_{\mathcal{E}_t} = 0$$ Thus $X_t$ only depends on $\mathcal{E}_t$.\\ Now get a locally finite open cover $\{U_\lambda\}$ of $M$ such that $\mathcal{E}_t|_{U_\lambda} = \ker\theta_t^\lambda$ for local form $\theta_t^\lambda$. Get unique local fields $X_t^\lambda$ as above. Also consider a partition of unity $\{\rho_\lambda\}$ subordinate to the open cover. Set, $$X_t = \sum_{\lambda} \rho_\lambda X_t^\lambda$$ Then $X_t$ is global vector field. Since $M$ is closed, integrating $X_t$ we get a global isotopy $\Phi_t$ on $M$. Also $X_t$ satisfies $\iota_{X_t}d\theta_t^\lambda + \frac{d}{dt}\theta_t^\lambda = 0$, for any $\lambda$. Thus $\Phi_t$ is the required isotopy of $M$ such that $$\Phi_{t*}\mathcal{E}_0 = \mathcal{E}_t.$$ \section{Normal Forms} In this section we shall obtain the normal form of the generators of the Pfaffian system defining a generalized Engel structure $\mathcal D$. Suppose, $\mathcal{D}$ is of co-rank $k+1$, where $k=2l+1$, and $\mathcal{L}\subset\mathcal{D}\subset\mathcal{E}\subset TM$ is the associated with the canonical flag on $M$. Suppose $\mathcal{S}_0$ and $\mathcal{S}_1$ are the Pfaffian systems annihilating $\mathcal{D}$ and $\mathcal{E}$ respectively. Since $\mathcal{D}\subset\mathcal{E}$, we have that $\mathcal{S}_0\supset\mathcal{S}_1$. Suppose $\mathcal{S}_1=\langle \theta\rangle$ and $\mathcal{S}_0=\langle \theta,\omega^1,\ldots,\omega^k\rangle$ locally. Then by Proposition \ref{propDistToForms} we get certain relations among these forms. We want to get a standard normal forms for some basis of $\mathcal{S}_0$ and $\mathcal{S}_1$.\\ Since $\theta\wedge d\theta^{l+1}\ne 0$ and $\theta\wedge d\theta^{l+2}= 0$, around a point $p\in M$ we have some co-ordinate system (Theorem 3.1 in \cite{bryantExteriorDiff}) such that $\mathcal{E}$ is the kernel of $$\Theta= dz - \sum_{i=1}^{l+1} x_{i+l+1}dx_i$$ Clearly $\mathcal{S}_1=\langle\Theta\rangle$ and $\{\Theta, \omega^i\}$ is a Pfaffian system associated to the given generalized Engel structur. From $\omega^i\wedge\Theta\wedge d\Theta^{l+1}=0,\forall i$, we have that $0=\omega^i\wedge dx_1\wedge\ldots\wedge dx_{2l+2}\wedge dz$. Hence, there exist functions $a^{ij},b^i$ such that, $$\omega^i = \sum_{j=1}^{2l+2} a^{ij} dx_j + b^idz$$ Now we have, in particular, $\omega^1\wedge\ldots\wedge\omega^k\wedge\Theta \ne 0$. Therefore, the matrix \[\begin{pmatrix} a^{11} &\ldots &a^{2l+1,1} &-x_{l+2}\\ \vdots &\vdots &\vdots &\vdots\\ a^{1,l+1} &\ldots &a^{2l+1,l+1} &-x_{2l+2}\\ a^{1,l+2} &\ldots &a^{2l+1,l+2} &0\\ \vdots &\vdots &\vdots &\vdots\\ a^{1,2l+2} &\ldots &a^{2l+1,2l+2} &0\\ b^1 &\ldots &b^{2l+1} &1 \end{pmatrix}_{(2l+3) \times (2l+2)}\] has full rank $2l+2$ everywhere. Evaluating this at the point $p$, we observe that the last column is $\begin{pmatrix}0 &\ldots &0 &1\end{pmatrix}^t$ and hence the last row cannot be linearly dependent on the rest. Hence, without loss of generality we may assume that the $(2l+2)^{\textnormal{th}}$ row is linearly dependent and the rest of them are linearly independent about $p$. Thus we can transform the matrix into, \[\begin{pmatrix} I_{(2l+1)\times (2l+1)} &0\\ C &0\\ 0 &1 \end{pmatrix}\] where $C=\begin{pmatrix}c_1 &\ldots &c_{2l+1}\end{pmatrix}$ is a row vector of functions. Thus we can transform $\omega^i$ into $$\Omega^i = dx_i + c_idx_{2l+2},\forall i=1,\ldots,k=2l+1$$ Clearly, $\{\Theta,\Omega^i\}$ still forms a basis of $\mathcal{S}_0$. Hence we have $\mu^i=\Omega^1\wedge\ldots\wedge\Omega^k\wedge\Theta\wedge d\Omega^i\ne0$ which gives, $$dx_1\wedge\ldots\wedge dx_{2l+2}\wedge dz\wedge dc_i \ne 0,\forall i$$ But $\{\mu^i\}$ is also point-wise linearly independent. Hence we have that $\{dc_i\}$ are point-wise linearly independent as well i.e., $\{c_i\}$ are coordinate functions around $p$. Set $y_i=x_{i+l+1}$. Thus, we obtain a coordinate system \[\big(x_1,\ldots,x_{l+1},y_1,\ldots,y_{l+1},z,c_1,\ldots,c_k,q_1,\ldots,q_r\big)\] around $p$ such that the $1$-forms $\{\Theta,\Omega^i\}$ can be expressed as follows : \begin{align*} \Theta &= dz -\sum_{i=1}^{l+1} y_idx_i\\ \Omega^i &= \begin{cases} dx_i + c_idy_{l+1}, 1\le i \le l+1\\ dy_{i-l-1} + c_idy_{l+1}, l+2\le i\le k= 2l+1 \end{cases} \end{align*} where $r$ is such that $\dim M = r + 2k + 2$, such that $\mathcal{E}=\ker\Theta$ and $\mathcal{D}=\{\Omega^1=\ldots\Omega^k=0=\Theta\}$ Note that, by taking $l=0$ in the above normal form we obtain the normal form of the Pfaffian system defining an Engel structure, namely $dz-ydx$, $dx-wdy$. \section*{Acknowledgment} The author would like to thank Mahuya Datta, Dheeraj Kulkarni and Suvrajit Bhattacharjee for fruitful and enlightening discussions. \nocite{hirschBook} \end{document}

\begin{document} \maketitle \begin{abstract} We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a (definable) henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv: 1507.02745v1, July 2015. \end{abstract} \section*{Introduction} The classification of $\omega$-stable fields \cite[Theorem 3.1]{PoiGroups} and later of super-stable fields \cite{ChSh} is a cornerstone in the development of the interactions between model theory, algebra and geometry. Ever since, the classification of algebraic structures according to their model theoretic properties is a recurring theme in model theory. Despite some success in the classification of groups of finite rank (with respect to various notions of rank), e.g.. \cite{EaKrPi},\cite[Section 4]{WaBook} (essentially, generalising results from the stable context), and most notably in the o-minimal setting (e.g., \cite{HrPePi} and many references therein) little progress has been made in the classification of infinite stable (let alone simple) fields. Indeed, most experts view the conjecture asserting that (super) simple fields are bounded (perfect) PAC, and even the considerably weaker conjecture that stable fields are separably closed to be out of the reach of existing techniques. In the last decade or so the increasing interest in theories without the independence property (NIP theories), associated usually with the solution of Pillay's conjecture \cite{HrPePi} and with the study of algebraically closed valued fields, led naturally to analogous classification problems in that context. In its full generality, the problem of classifying NIP fields encompasses the classification of stable fields, and may be too ambitious. In \cite{Sh863}, as an attempt to find the right analogue of super-stability in the context of NIP theories, Shelah introduced the notion of \emph{strong NIP}. As part of establishing this analogy, Shelah showed \cite[Claim 5.40]{Sh863} that the theory of a separably closed field that is not algebraically closed is not strongly NIP. In fact Shelah's proof actually shows that strongly NIP fields are perfect\footnote{Shelah's proof only uses the simple fact that if $\mathrm{char}(K)=p>0$ then either $K$ is perfect or $[K^\times: (K^\times)^p]$ is infinite. See, e.g., \cite[Remark 2.5]{KrVal}}. Shelah conjectured \cite[Conjecture 5.34]{Sh863} that (interpreting its somewhat vague formulation) strongly NIP fields are real closed, algebraically closed or support a definable non-trivial (henselian) valuation. Recently, this conjecture was proved\footnote{The existence of a definable valuation is implicit in Johnson's work. See Remark \ref{JohnsonDef}.} by Johnson \cite{JohnDPMin} in the special case of dp-minimal fields (and, independently, assuming the definability of the valuation, henselianity is proved in \cite{JaSiWa2015}). The two main open problems in the field are: \begin{enumerate} \item Let $K$ be an infinite (strongly) NIP field that is neither separably closed nor real closed. Does $K$ support a non-trivial definable valuation? \item Are all (strongly) NIP fields henselian (i.e., admit some non-trivial henselian valuation) or, at least, t-henselian (i.e., elementarily equivalent in the language of rings, to a henselian field)? \end{enumerate} A positive answer to Questions (2) would imply, for example, that strongly\footnote{F. Jahnke and S. Anscombe (private communication) informed us that similar results are obtained for NIP fields.} dependent fields are elementarily equivalent to Hahn fields over well understood base fields \cite[Theorem 3.11]{HaHaJa}: \begin{description} \item[Equi-characteristic] $\mathbb R((t^\Gamma))$, $\mathbb C((t^\Gamma))$ or $\overline{\mathbb F}_p((t^\Gamma))$. \item[Finite residue field] $Q((t^\Gamma))$ where $Q$ is a p-adically closed field if the field admits a henselian valuation with finite residue field. \item[Kaplansky] $L((t^\Gamma))$ where $L$ is a rank 1 Kaplansky field with residue field as in (1) above. \end{description} where in all cases $\Gamma$ is a strongly dependent ordered abelian group (see \cite{HalHas} for the classification of such groups). In view of the above, a natural strategy for studying Shelah's conjecture would be to, on the one hand, study the conjecture for Hahn fields (with dependent residue fields), as the key example and -- on the other hand -- using the information gained in the study of Hahn fields, try to generalise Johnson's results from dp-minimal fields to the strongly dependent setting. The simplest extension of Johnson's proof of Shelah's conjecture for dp-minimal fields would be to finite extensions of dp-minimal fields. Section \ref{dp-min} is dedicated to showing that this extension is vacuous, namely we prove that a finite extension of a dp-minimal field is again dp-minimal (see Theorem \ref{finite}). The proof builds heavily on Johnson's classification of dp-minimal fields. Section \ref{Hahn} is dedicated to the study of (strongly) dependent Hahn fields. Collecting known results of the first author (based on unpublished work of Koenigsmann), we show that Hahn fields that are neither algebraically nor real closed support a definable non-trivial valuation with a $t$-henselian topology. We use Hahn fields to provide examples proving that perfection and boundedness -- the conjectural division lines for simple fields -- are not valid in the NIP case. Building on previous results of Delon \cite{DelHenselian}, B\'elair \cite{BelHenselian} and Jahnke-Simon \cite{JaSiTransfer} we construct the following examples (see Theorem \ref{examples}): There are NIP fields with the following properties: \begin{enumerate} \item A strongly NIP field that is not dp-minimal. \item A strongly NIP field $K$ such that $[K^\times: (K^\times)^q ]=\infty$ for some prime $q$. \item A perfect NIP field that is not strongly NIP. \item An unbounded strongly NIP field. \end{enumerate} In the last two sections of the paper we turn to the problem of constructing definable valuations on (strongly) NIP fields. As Johnson's methods of \cite{JohnDPMin} do not seem to generalise easily even to the finite dp-rank case, we study a more general construction due to Koenigsmann. We give, provided the field $K$ is neither real closed nor separably closed (without further model theoretic assumptions), an explicit first order sentence $\psi_K$ in the language of rings such that $K\models \psi_K$ implies the existence of a non-trivial valuation ring definable (over the same parameters appearing in $\psi_K$) in the language of rings. As we will show (see the discussion following Corollary \ref{Q2toQ1}) if $K$ is $t$-henselian then $K\models \psi_K$. Thus to provide a positive answer to Question (1) it will suffice to show that any NIP field (infinite not real closed or separably closed) $K\models \psi_K$, which is also a necessary condition for a positive answer to Question (2). Implicit in the work of Koenigsmann \cite{Koe2}, a sentence with roughly the same properties as $\psi_K$ above can certainly be extracted from \cite{Du2016}. However, the sentence $\psi_K$ obtained in Proposition \ref{PropSimplifiedVAxiomsnontrivialdefinable} of this paper is simpler in quantifier depth and in length. As a result the strategy proposed for tackling Question (1) above can be summarised as follows: \begin{con}\label{fo} Let $K$ be an infinite field not separably closed. For any prime $q\neq \mathrm{char}(K)$ let $T_q:=(K^\times)^q+1$. Assume that \begin{enumerate} \item $T_q\neq K\setminus \{1\}$ \item $\sqrt{-1}\in K$ \item There exists $\zeta_q\in K$ a primitive $q$-th root of unity. \end{enumerate} and at least one of the following holds: \begin{enumerate} \item $K\models (\exists a_1,a_2)(\{0\}=a_1T_q\cap a_2 T_q)$ \item $K\models (\forall a_1,a_2\exists b)(b\notin T_q\land b\in (a_1T_q\cap a_2T_q)-(a_1T_q\cap a_2T_q))$ \item $K\models (\forall a_1,a_2\exists b)(b\notin T_q\land b\in (a_1T_q\cap a_2T_q)\cdot (a_1T_q\cap a_2T_q))$ \item $K\models (\forall a_1,a_2\exists x,y)(xy\in a_1T_q\cap a_2T_q\land x\notin a_1T_q\cap a_2T_q\land y\notin a_1T_q\cap a_2T_q)$ \end{enumerate} Then $K$ has IP. \end{con} \noindent\emph{Acknowledgements.} We would like to thank I. Efrat, M. Hils, F. Jahnke, M. Kamensky, F.-V. Kuhlmann and P. Simon for several ideas, corrections and suggestions. \section{Preliminaries}\label{prelims} Throughout we use standard valued fields terminology and notation: $K, L, F$ will be fields, $\ensuremath{\mathcal{O}} _K$ will denote a valuation ring on $K$ with maximal ideal $\ensuremath{\mathcal{M}} _K$ (we will drop the subscript $K$ if it is clear from the context). Valuations on $K$ will be denoted by $v,w$ and $\ensuremath{\mathcal{O}} _v:=\left\{x\in K: v(x)\geq 0\right\}$, $\ensuremath{\mathcal{M}} _v$ the valuation ring associated with $v$ and its maximal ideal respectively. The reader is referred to any standard textbook on the subject (e.g., \cite{EnPr}) for more details. A non-trivial valuation $v$ on a field $K$ induces a Hausdorff field topology (generated by open balls $B_\gamma(x):=\{y:v(x)>\gamma\}$). It is well known that such topologies can be characterised: \begin{de}\label{Vtop} A collection of subsets $\ensuremath{\mathcal{N}} $ of $K$ is a basis of 0-neighbourhoods for a \emph{V-topology} on $K$ if is satisfies the following axioms: \begin{description} \item[\textbf{\textup{(V\,1)}}] $\bigcap \mathcal{N}:=\bigcap_{U\in\mathcal{N}}U=\left\{0\right\}$ and $\left\{0\right\}\notin\mathcal{N}$; \item[\textbf{\textup{(V\,2)}}] $\forall\, U,\,V\ \exists\, W\ W\subseteq U\cap V$; \item[\textbf{\textup{(V\,3)}}] $\forall\, U\ \exists\, V\ V-V\subseteq U$; \item[\textbf{\textup{(V\,4)}}] $\forall\, U\ \forall\, x,\,y\in K\ \exists\, V\ \left(x+V\right)\cdot\left(y+V\right)\subseteq x\cdot y+U$; \item[\textbf{\textup{(V\,5)}}] $\forall\, U\ \forall\, x\in K^{\times}\ \exists\, V\ \left(x+V\right)^{-1}\subseteq x^{-1}+U$; \item[\textbf{\textup{(V\,6)}}] $\forall\, U\ \exists\, V\ \forall\, x,\,y\in K\ (x\cdot y\in V\to (x\in U\, \vee\, y\in U))$. \end{description} \end{de} It is not hard to check that if $(K,v)$ is a valued field (or a field with an absolute value) then the collection of open balls is a V-topology on $K$. More importantly, the converse is also true (see \cite[Appendix B]{EnPr}): any V-topology on a field $K$ arises in this way. In the present paper we will investigate and apply a standard technique for constructing a V-topology on a field $K$ from a multiplicative sub-group $G\le K^\times$. We will be following a construction due to Koenigsmann, \cite{Koe2}, but the general method is well known (see \cite[\S11]{EfBook} and references therein). Fix $K$ an infinite field and let $G$ be a multiplicative subgroup of $K^\times$ with $G\neq K^\times$. Given a group $G\le K^\times$ we let $\ensuremath{\mathcal{T}} _G$ be the coarsest topology for which $G$ is open and linear transformations are continuous. As shown in \cite[Theorem~3.3]{Du2016} $\ensuremath{\mathcal{S}} _G:=\left\{a\cdot G+b: a\in K^\times, b\in K\right\}$ is a subbase of $\ensuremath{\mathcal{T}} _G$. Hence \[\ensuremath{\mathcal{B}} _G:=\left\{\left.\bigcap_{i=1}^n\left(a_i\cdot G+b_i\right)\,\right|\, n\in \ensuremath{\mathbb{N}} ,\, a_1,\ldots,a_n\in K^\times, b_1,\ldots,b_n\in K\right\}\] is a base for $\ensuremath{\mathcal{T}} _G$. A simple calculation shows that \[ \ensuremath{\mathcal{N}} _G:=\left.\left\{U\in \ensuremath{\mathcal{B}} _G \ \right|\ 0\in U\right\} \:=\left.\left\{\bigcap_{i=1}^n a_i\cdot \left(- G+1\right) \ \right|\ n\in\ensuremath{\mathbb{N}} ,\, a_i\in K^\times\right\} \] is a base of neighbourhoods of zero for $\ensuremath{\mathcal{T}} _G$. If further $-1\in G$ then \[ \ensuremath{\mathcal{N}} _G=\left\{\bigcap_{i=1}^n a_i\cdot \left(G+1\right): \ n\in\ensuremath{\mathbb{N}} ,\, a_i\in K^\times\right\}. \] Throughout the paper $U,\,V$ and $W$, possibly with indices, will always denote elements of $\ensuremath{\mathcal{N}} _G$. It follows from \cite[Lemma~3.6 and Corollary 3.8]{Du2016} that, if $\ensuremath{\mathcal{T}} _G$ is a basis for a V-topology (see Fact \ref{corVAxiomsnontrivialdefinable} below) then already \[\left\{\left(a_1\cdot G+b_1\right)\cap\left(a_2\cdot G+b_2\right): a_1,a_2\in K^\times,\, b_1,b_2\in K\right\}\] is a base for $\ensuremath{\mathcal{T}} _G$. Hence, if $-1\in G$, \[\ensuremath{\mathcal{N}} _G':=\left\{\left(a_1\cdot \left(G+1\right)\right)\cap\left(a_2\cdot\left( G+1\right)\right): a_1,a_2\in K^\times\right\}\] is a base of the neighbourhoods of zero for $\ensuremath{\mathcal{T}} _G$. As in most of the paper it will be more convenient to work with arbitrary intersections, we will mostly choose to work with $\ensuremath{\mathcal{N}} _G$. The advantage of the basis $\ensuremath{\mathcal{N}} _G'$ is that, if $G$ is definable (as will be the case) it is a definable basis of $0$-neighbourhoods. The starting point of the present paper is the following result of Koenigsmann\footnote{A valuation $v$ on $K$ is $p$-henselian if it extends uniquely to $K(p)$ the compositum of all Galois extensions of $K$ of degree $p^n$ (any $n$). For the purposes of the present paper the fact that any henselian valuation is $p$-henselian will suffice. For more information see \cite{KoepHens}.}\footnote{As pointed out by the referee, a correct proof Koenigsmann's result can be found in \cite{JahKo}}., \cite{KoepHens}: \begin{fact}\label{corVAxiomsnontrivialdefinable} Let $K$ be a field of characteristic $p$ (possibly 0) and $q$ a prime different from $p$. Let $G:=(K^\times)^q\subsetneq K^\times $ and assume that $\zeta_p\in K$ for a primitive $q$th root of unity. Then $K$ is $q$-henselian if and only if $\ensuremath{\mathcal{T}} _G$ is a basis for a $V$-topology, if and only if the canonical $p$-henselian valuation, $v_p$ is $\emptyset$-definable (in which case $\ensuremath{\mathcal{T}} _G$ is the topology induced by $v_q$). \end{fact} \begin{proof} By \cite[Theorem 2.1]{KoepHens} $K$ is $p$-henselian if and only if $\ensuremath{\mathcal{T}} _G$ generates the same topology as $v$ for some $p$-henselian valuation. By \cite[Main theorem]{JahKo} if $K$ is $p$-henselian then the canonical $p$-henselian valuation is definable. The statement concerning the topologies also follows from \cite[Theorem 2.1]{KoepHens}. \end{proof} In the above, and throughout, by \emph{definable} we mean \emph{definable in the language $\ensuremath{\mathcal{L}} $ of rings} and by saying that a valuation $v$ on $K$ is definable we mean that $\ensuremath{\mathcal{O}} _v$ is $\ensuremath{\mathcal{L}} (K)$-definable (where $\ensuremath{\mathcal{L}} (K)$ is the expansion of the language $\ensuremath{\mathcal{L}} $ by constants for all elements of $K$). Let us now explain how the above fact will be applied. Let $K$ be an NIP field. We aim to find conditions for the existence of a definable non-trivial valuation on $K$. By \cite[Theorem II.4.11]{Sh1} (\cite[Observation 1.4]{Sh863}) if $T$ is (strongly) NIP then so is $T^{eq}$. Thus any finite extension of $K$ is also (strongly) NIP. It will suffice, therefore, to find a definable non-trivial valuation on some finite extension $L\ge K$ (since if $\mathcal O$ is a non-trivial valuation ring on $L$ then $\mathcal O\cap K$ is a non-trivial valuation ring in $K$). It is therefore, harmless to assume that $\sqrt{-1}\in K$. By \cite[Theorem~4.4]{KaScWa} $K$ is Artin-Schreier closed. So the same is true of any finite extension $L\ge K$. This implies (e.g., \cite[Lemma 2.4]{KrVal}\footnote{Krupinski's argument assumes that the field is perfect to conclude that it is algebraically closed. Descarding this additional assumption, and restricting to separable extensions the stronger result follows.}) that either $K$ is separably closed, or there exists some finite separable extension $L\ge K$ and $q\neq \mathrm{char}(K)$ such that $(L^\times)^q\neq L^\times$ (in fact, by \cite[Corollary 4.5]{KaScWa} $K$ has no finite separable extensions of degree divisible by $p$). Since $\sqrt{-1}\in L$ it follows that, letting $L(q)$ denote the $q$-closure of $L$, we have $[L(q):L]=\infty$ (\cite[Theorem 4.3.5]{EnPr}). So extending $L$ a little more, there is no harm assuming that there exists $\zeta_q\in L$, a primitive $q$th root of unity. Thus, at the price of, possibly, losing the $\emptyset$-definability of the resulting valuation (because of the passage to the field $L$), the basic assumptions of Fact \ref{corVAxiomsnontrivialdefinable} are easily met. So that the application of this result reduces to proving that for $L$ and $q$ as above, $\ensuremath{\mathcal{N}} _G$ is a $0$-neighbourhood basis for a V-topology on $L$. Thus, we get the following result (see also \cite{Du2016}): \begin{cor}\label{groups} Let $K$ be an NIP field that is neither separably closed nor real closed. Then there exists a finite separable field extension $L\ge K$ and a prime $q\neq \mathrm{char}(K)$ such that $(L^\times)^q\neq L^\times$ and $\zeta_q\in L$ for some primitive root of unity. If $K$ is $t$-henselian then for any such $L\ge K$ and $q$ the group $G_q(L):=(L^\times)^q$ satisfies conditions (V1)-(V6) of Definition \ref{Vtop}. \end{cor} \begin{proof} Assume first that $K$ is henselian, witnessed by a valuation $v$. Then, by the above discussion, as $K$ is neither real closed nor algebraically closed, there is some finite separable field extension $L\ge K$ and prime $q$ such that $G_q(L):=(L^\times)^q$ is a proper subgroup of $L^\times$ and $\zeta_q\in L$. Fix any such extension $L$. Since $v$ is henselian, it extends to a henselian valuation on $L$ which by abuse of notation we will also denote $v$. By \cite[Theorem 5.18]{Du2016} there exists a definable valuation $w$ on $L$ inducing the same topology as both $v$ and $\mathcal O_{G_q(L)}$. In particular $\mathcal O_{G_q(L)}$ is non-trivial. So, by the above discussion, $\mathcal N_{G_q(L)}$ is a basis for a $V$-topology, i.e., it satisfies (V1)-(V6), as required. In general, let $\mathcal K\succ K$ be $\aleph_1$-saturated. Then $\mathcal K$ is henselian. Let $L\ge K$ be a finite separable extension such that $G_{q}(L)$ is a proper subgroup. By the primitive element theorem there exists $\alpha\in L$ such that $L=K(\alpha)$. Let $\mathcal L:=\mathcal K(\alpha)$. Then $\mathcal L\succ L$ and $G_{q}(\mathcal L)$ is a proper subgroup. By what we have already shown the group $G_q(\mathcal L)$ satisfies conditions (V1)-(V6). So $\mathcal N_{G_q(\mathcal L)}'$ is a also a basis for the topology, so it satisfies the corresponding statements (V1)$'$-(V6)$'$. Since those are first order statements without parameters, they are also satisfied by $G_q(L)$, so $G_q(L)$ also satisfies (V1)-(V6) as required. \end{proof} We remind also that by a Theorem of Schmidt \cite[Theorem 4.4.1]{EnPr} any two henselian valuations on a non-separably closed field $K$ are dependent (i.e., generate the same $V$-topology). So we get: \begin{cor}\label{Q2toQ1} Let $K$ be an NIP field that is neither real closed, nor separably closed. If $K$ is henselian, then $K$ supports a definable non-trivial valuation. Moreover, there exists a finite separable extension $L\ge K$ and a prime $q$ such that $G_q(L)\cap K$ generates the same $V$ topology as any henselian valuation on $K$. \end{cor} \begin{proof} There is no harm assuming that $\sqrt{-1}\in K$. As above, if for all finite separable extensions $L$ and all $q\neq \mathrm{char}(K)$ we have $L^q=L$, we get that $K$ is separably closed, contradicting our assumption. So there are $L\ge K$, a finite extension, and $q$ such that $L^q\neq L$. Since $\sqrt{-1}\in K$ we get that $L(\zeta_q)^q\neq L(\zeta_q)$ for $\zeta_q$, a primitive $q$th root of unity. So there is no harm assuming $\zeta_q\in L$. Since $K$ is henselian, so is $L$. By the previous corollary, $G_q(L)$ generates on $L$ the same topology as any henselian valuation on $L$. The corollary follows. \end{proof} Let $L\ge K$ be as provided by the previous corollary. Then $L=K(\alpha)$ for some $\alpha$, and let $f(x)$ be its minimal polynomial over $K$. Let $\bar a$ be the coefficients of $f$. Then $L$ is $K$-interpretable over $\bar a$. So we let $\psi_K$ be the sentence (over $\bar a$) stating that $G:=(L^\times)^q$ satisfies axioms (V1)$'$-(V6)$'$ of a $V$-topology. By Fact \ref{corVAxiomsnontrivialdefinable} if $K\models \psi_K$ then $K$ supports a non-trivial $\bar a$-definable valuation. And if $K$ happens to be $t$-henselian (and, therefore, so is $L$) then $L$ is $q$-henselian, Fact \ref{corVAxiomsnontrivialdefinable} implies that $K\models \psi_K$. Therefore, if $K$ is NIP and we assume the conjecture that any infinite NIP field is ($t$)-henselian then $K\models \psi_K$. Assuming that, in the above discussion we do not have to pass to the separable extension $L$ (i.e., $K$ itself satisfies assumptions (1)-(3) of Conjecture \ref{fo}) we get that $K\models \psi_K$ implies the existence of a non-trivial $p$-henselian valuation on $K$ (for some $p$ explicit in $\psi_K$). It is therefore natural to ask: \begin{qu} If $(K,v)$ is NIP and $v$ is $p$-henselian (for some $p\neq \mathrm{char}(K)$). Is $v$ necessarily henselian? Does this follow, at least, from Shelah's conjecture? \end{qu} It is worth pointing out that by \cite[Remark 2.3]{KoepHens} there are fields that are $p$-henselian for all primes $p$ but not henselian. For any such field, $K$, the canonical $p$-henselian valuation (any $p$) is definable, inducing the topology $\ensuremath{\mathcal{T}} _G$ for $G=(K^\times)^p$, but this definable valuation is not henselian. So in full generality, the definable valuations discussed in this paper need not be henselian. Throughout the paper we will be using without further reference the facts that strongly NIP fields are perfect, that NIP fields are Artin-Schreier closed, and that NIP valued fields of characteristic $p>0$ have a $p$-divisible value group (\cite[Proposition 5.4]{KaScWa}). \section{dp-minimal fields}\label{dp-min} Dp-minimal fields are classified in the main result of \cite{JohnDPMin}\footnote{Specific references to Johnson's paper below refer to the publicly available version of the paper, \cite{JohnDPMinArc}.}: \begin{thm}[Johnson]\label{classification} A sufficiently saturated field $K$ is dp-minimal if and only if $K$ is perfect and there exists a valuation $v$ on $K$ such that: \begin{enumerate} \item $v$ is henselian. \item $v$ is defectless (i.e., any finite extension of $(L,v)$ over $(K,v)$ is defectless). \item The residue field $Kv$ is either algebraically closed of characteristic $p$ or elementarily equivalent to a local field of characteristic $0$. \item The valuation group $\Gamma_v$ is almost divisible, i.e., $[\Gamma_v: n\Gamma_v]<\infty$ for all $n$. \item If $\mathrm{char}(Kv)=p\neq \mathrm{char}(K)$ then $[-v(p),v(p)]\subseteq p\Gamma_v$. \end{enumerate} \end{thm} Given a dp-minimal field $K$ that is not strongly minimal, Johnson constructs an (externally definable) topology \cite[\S3]{JohnDPMinArc}, which he then proves to be a V-topology \cite[\S3 , \S 4]{JohnDPMinArc}. Pushing these results further he proceeds to show \cite[Theorem 5.14]{JohnDPMinArc} that $K$ admits a henselian topology (not necessarily definable). From this we immediately get: \begin{cor} Any dp-minimal field is either real closed, algebraically closed or admits a non-trivial definable henselian valuation. In particular, the V-topology constructed by Johnson is definable and coincides with Koenigsmann's topology, $\ensuremath{\mathcal{T}} _G(L)\cap K$, for some finite extension $L\ge K$ and some (equivalently, any) $G:=(L^\times)^p$ such that $G\neq L^\times$. \end{cor} \begin{proof} Let $K$ be a dp-minimal field that is neither real closed nor algebraically closed. By \cite[Theorem 5.14]{JohnDPMinArc} $K$ is henselian, and therefore so is any finite extension of $K$. Let $L$ be a finite extension of $K$ such that $G_q(L)\neq L^\times$ and $L$ contains a primitve $q$th root of unity. Then by Fact \ref{corVAxiomsnontrivialdefinable} and Corollary \ref{groups} we get that $L$ admits a non-trivial definable valuation. So $K$ admits a non-trivial definable valuation, and by \cite[Theorem 5.14]{JohnDPMinArc} all definable valuations on $K$ are henselian. Since $K$ is not separably closed it follows that $K$ supports a unique non-trivial t-henselian topology so the V-topology constructed by Johnson coincides with the topology associated with the definable henselian valuation, and is therefore definable. \end{proof} \begin{rem}\label{JohnsonDef} \begin{enumerate} \item The above corollary is implicit in Johnson's work. By inspecting his proof of Theorem 1.2 (\cite[\S 6]{JohnDPMinArc}) one sees that unless $K$ is real closed or algebraically closed the valuation ring $\mathcal O_\infty$ appearing in the proof, the intersection of all definable valuation rings on $K$, is non-trivial, implying that $K$ supports a non-trivial definable valuation. \item The same result can also be inferred from \cite[\S 7]{JaSiWa2015}. In that paper it is shown that a dp-minimal valued field which is neither real closed nor algebraically closed supports a non-trivial henselian valuation definable already in the pure field structure. By Johnson's Theorem 5.14 we know that $K$ admits a henselian valuation, which is externally definable. Since an expansion of a dp-minimal field by externally definable sets is again dp-minimal, the result follows. \end{enumerate} \end{rem} We note that the proof of the first part of the above corollary shows that the same results remain true for finite extensions of dp-minimal fields. This follows also from the following, somewhat surprising, corollary of Theorem \ref{classification}: \begin{thm}\label{finite} Let $K$ be a dp-minimal field, $L$ a finite extension of $K$. Then $L$ is dp-minimal. \end{thm} \begin{proof} Since dp-minimality is an elementary property, we may assume that $K$ is saturated. Indeed, since $L$ is a finite extension of $K$ it is interpretable in $K$, and if $K'\succ K$ is saturated, the field $L'$ interpreted in $K'$ by the same interpretation is a saturated elementary extension of $L$. Thus, it will suffice to show that there exists a valuation $v$ on $L$ satisfying conditions (1)-(5) of Theorem \ref{classification}. Since $K$ is saturated, there is such a valuation on $K$, extending uniquely to $L$. By abuse of notation we will let $v$ denote also this extension. Conditions (1) and (2) of the theorem are automatic and condition (4) is an immediate consequence of the fundamental inequality (e.g., Theorem 3.3.4\cite{EnPr}). Condition (3) is automatic if $Kv$ is real closed or algebraically closed. So it remains to check that if $Kv$ is elementarily equivalent to a finite extension of $\mathbb Q_p$ then so is $Lv$. This is probably known, but as we could not find a reference, we give the details. By Krasner's Lemma any finite extension of $\mathbb Q_p$ is of the form $\mathbb Q_p(\delta)$ for some $\delta$ algebraic over $\mathbb Q$ and $\mathbb Q_p$ has only finitely many extensions of degree $n$ (for any $n$). Denoting $e(n)$ the number of extensions of $\mathbb Q_p$ of degree $n$, there are $P_1(x),\dots, P_{e(n)}(x)\in \mathbb Q$ irreducible such that any finite extension of $\mathbb{Q}_p$ of degree $n$ is generated by a root of one of $P_1(x), \dots, P_{e(n)}(x)$. As this is clearly an elementary property, we get that the same remains true if $F\equiv \mathbb Q_p$. Of course, all of the above remains true if we replace $\mathbb Q_p$ by some finite extension $L\ge \mathbb Q_p$. So if $L'\equiv L$ and $F'\ge L'$ is an extension of degree $n$ it must be that $F'=L(\delta)$ for some $\delta$ realising on of $P_1(x),\dots, P_{e(n)}(x)$, implying that $F'$ is elementarily equivalent to $F$, the algebraic extension of $L$ obtained by realising the same polynomial. It remains to show that if $(K,v)$ is of mixed characteristic then $[-v(p),v(p)]\subseteq p\Gamma$ where $p=\mathrm{char} Kv$. By \cite[Lemma 6.8]{JohnDPMinArc} and the sentence following it, to verify this condition it suffices to show that $[-v(p),v(p)]$ is infinite. Towards that end it will suffice to show that $[-v(p),v(p)]\cap v(K)$ is infinite. Indeed, by assumption $(K,v)$ satisfies (5) of Theorem \ref{classification}, so $[-v(p),v(p)]\subseteq p\Gamma$. As we are in the mixed characteristic case $v(p)>0$. Since $v(p)\subseteq p\Gamma$, there is some $g_1\in \Gamma$ such that $pg_1=v(p):=g_0$. So $0<g_1<g_0$, and by induction, for all $n$ we can find $0<g_n<g_{n-1}<g_0=v(p)$. This show that $[-v(p), v(p)]\cap v(K)$ is infinite, concluding the proof of the theorem. \end{proof} As already mentioned in the beginning of this section, the V-topologies constructed by Johnson and Koenigsmann coincide in the dp-minimal case. However, in order to start Koenigsmann's construction we first need to assure that $G_q(K)\neq K^\times$, and for that we may have to pass to a finite extension. Let us now point out that in the dp-minimal case this is not needed: \begin{lem} Let $K$ be a dp-minimal field that is neither real closed nor algebraically closed. Then $G_q(K)\neq K^\times$ for some $q$. \end{lem} \begin{proof} Let $v$ be as provided by Theorem \ref{classification}. It will suffice to show that the value group is not divisible. This is clear if the residue field is elementarily equivalent to a finite extension of $\mathbb Q_p$. Indeed, any finite extension $L$ of $\mathbb Q_p$ is henselian with value group isomorphic to $\mathbb Z$, which is not $n$ divisible for any $n>1$. So $G_n(L)\neq L^\times$ for any such $n$. As this is expressible by a first order sentence with no parameters, it remains true in any $L'\equiv L$. If $Kv\models ACF_0$ or $Kv\models RCF$, the value group cannot be divisible, as then $K$ would be algebraically closed (resp. real closed). If $Kv\models ACF_p$ then, as $v$ is henselian defectless $(K,v)$ is algebraically maximal, in which case divisibility of the value group would again imply that $K\models ACF$. \end{proof} \section{Hahn Series and related constructions}\label{Hahn} Little is known on the construction of simple fields. The situation is different in the NIP setting where strong transfer principles for henselian valued fields (see, e.g., \cite{JaSiTransfer} and references therein for the strongest such result to date) allow the construction of many examples of NIP fields. In the present section we sharpen some of these results and exploit them to construct various examples. For the sake of clarity we remind the definition of strong dependence (in the formulation most convenient for our needs. See \cite[\S2]{Sh863} for more details): \begin{de} A theory $T$ is strongly dependent if whenever $I$ is an infinite linear order, $\{a_t\}_{t\in I}$ an indiscernible sequence (of $\alpha$-tuples, some $\alpha$), and $a$ is a singleton there is an equivalence relation $E$ on $I$ with finitely many convex classes such that for $s\in I$ the sequence $\{a_t: t\in s/E\}$ is $a$-indiscernible. \end{de} We show: \begin{thm}\label{examples} There are NIP fields with the following properties: \begin{enumerate} \item A strongly NIP field that is not dp-minimal. \item A strongly NIP field $K$ such that $[K^\times: (K^\times)^q ]=\infty$ for some prime $q$. \item A perfect NIP field that is not strongly NIP. \item An unbounded strongly NIP field. \end{enumerate} \end{thm} Recall that a field is \emph{bounded}\footnote{In the literature e.g., \cite{PilPoi}, \cite{PiScWa} a slightly stronger condition is used. The restriction to separable extensions seems, however, more natural and even implicitly implied in some applications.} if for all $n\in \mathbb N$ it has finitely many separable extensions of degree $n$. Super-simple fields are bounded,\cite{PilPoi}, and conjecturally, so are all simple fields. As pointed out to us by F. Wagner, it follows, e.g., from \cite[Theorem 5.10]{PoiGroups} that bounded stable fields are separably closed. For the sake of completeness we give a different proof, essentially, due to Krupinski, with a less stability-theoretic flavour: Let $K$ be a bounded stable field. Since stability implies NIP $K$ and all its finite extensions are, as already mentioned, Artin-Schreier closed. By an easy strengthening of \cite[Lemma 2.4]{KrVal}, it will suffice to show that $K^q=K$ for all prime $q\neq \mathrm{char}(K)$. Boundedness\footnote{In \cite{KrVal} Krupinski introduces the slightly weaker \emph{radical boundedness}, which suffices for the argument.} implies that were this not the case for some $q$ we would have $1<[K^\times, (K^\times)^q]<\infty$. By \cite[Proposition 4.8]{KrSRNIP} this implies that $K$ is unstable (in fact, that the formula $\exists z(x-y=z^q)$ has the order-property). As we will see in the concluding section of the present paper, boundedness may also have a role to play in the study of the two questions stated in the Introduction. In view of the results of Theorem \ref{examples} it seems natural to look for model theoretic division lines that will separate the bounded NIP fields\footnote{Added in proof: In \cite{HaHaJa} it is shown that Shelah's conjecture implies that a strongly dependent field is bounded if and only if it is dp-minimal.}. \begin{rem} In \cite[Corollary 3.13]{KaSh} it is shown that in a strongly dependent field $K$ for all but finitely many primes $p$ we have $[K^\times: (K^\times)^q]<\infty$. Clause (2) of Theorem \ref{examples} shows that this result is optimal. \end{rem} We will use Hahn series to construct the desired examples. The basic facts that we need are: \begin{fact}\label{delon} A henselian valued field $(K,v)$ of equi-characteristic $0$ is (strongly) NIP if and only if the value group and the residue field are (strongly) NIP. If $(K,v)$ is dp-minimal then so are the residue field and the value group. \end{fact} The NIP case of the above fact is due to Delon \cite{DelHenselian} and the strongly NIP case is due to Chernikov \cite{Chernikov}. We get: \begin{lem} Let $k$ be a field of characteristic $0$, $\Gamma$ an ordered abelian group. Then the Hahn series $k((t^\Gamma))$ is NIP as a valued field if and only if $k$ is NIP as a pure field. It is strongly NIP if and only if $k$ and $\Gamma$ are. \end{lem} \begin{proof} Hahn series are maximally complete, and therefore henselian. So the result follows from the previous fact. \end{proof} In order to prove clauses (1) and (3) of Theorem \ref{examples} it will suffice, therefore, to find strongly NIP ordered abelian groups that are not dp-minimal and ones that are not strongly NIP. We start with the latter: \begin{ex} Consider $\Gamma:=\mathbb Z^\mathbb N$ as an abelian group (with respect to pointwise addition) with the lexicographic order. Then $\Gamma$ is NIP but not strongly NIP. \end{ex} \begin{proof} The group $\Gamma$ is ordered abelian, and therefore NIP by \cite{GurSch}. But $[\Gamma:n\Gamma]=\infty$ for all $n>1$, whence not strongly NIP by \cite[Corollary 3.13]{KaSh}. \end{proof} \begin{rem} In \cite{Sh863} Shelah considers a closely related example of an ordered abelian group that is not strongly dependent. \end{rem} \begin{ex} Let $\Gamma:=\mathbb Z^\mathbb N$. If $k$ is an NIP field of characteristic $0$ then $K:=k((t^\Gamma))$ is NIP by the previous lemma. It is not strongly NIP because $\Gamma$ is not strongly NIP. It is unbounded, since by the fundamental inequality it has infinitely many Kummer extensions of any prime degree $q$. Indeed, for any natural number $n$ let $\{a_1,\dots, a_n\}\in \Gamma$ be pairwise non-equivalent modulo $q\Gamma$. Let $c_1,\dots, c_n\in K$ be such that $v(c_i)=a_i$ for all $i$. Let $L\ge K$ be the extension obtained by adjoining $q^{\text{th}}$-roots for all $c_i$. Let $\Delta=v(L)$, where $v$ is identified with its unique extension to $L$. Then $\Gamma\not\subseteq q\Delta$. Otherwise $[q\Delta:q\Gamma]=[q\Delta:\Gamma][\Gamma:q\Gamma]=\infty$, whereas $[\Delta:\Gamma]\le [L:K]$ and $[q\Delta:q\Gamma]\le [\Delta:\Gamma]$. This is a contradiction. Since $n$ was arbitrary, this shows that $K$ has infinitely many Kummer extensions of degree $q$. \end{ex} Note that by \cite[Corollary 3.13]{KaSh} and \cite{JaSiWa2015} if $G$ is an ordered abelian group that is strongly dependent and not dp-minimal then there are finitely many primes $q$ such that $[G:qG]=\infty$. So the previous example with $G$ replacing $\Gamma$ will give an example for Theorem \ref{examples}(1), (2) and (5). The details of the following example can be found in \cite{HalHas}: \begin{fact}\label{poschar} Let $_{(2)}\mathbb Z$ be the localisation of $\mathbb Z$ at $(2)$. Let $B$ be a base for $\mathbb R$ as a vector space over $\mathbb Q$ and let $\langle B \rangle$ be the $\mathbb Z$-module generated by $B$. Let $G:=_{(2)}\mathbb Z\otimes \langle B \rangle$. Viewed as an additive subgroup of $\mathbb R$ the group $G$ is naturally ordered. It is strongly dependent but not dp-minimal. \end{fact} In positive characteristic, the situation is slightly different. The basic result is due to B\'elair \cite{BelHenselian}: \begin{fact}\label{Belair} Let $(K,v)$ be an algebraically maximal Kaplansky field of characteristic $p>0$. Then $K$ is NIP as a valued field if and only if the residue field $k$ is NIP as a pure field. \end{fact} This generalises to the strongly dependent setting using the following results: \begin{fact}[\cite{Sh863}, Claim 1.17] Let $T$ be a theory of valued fields in the Denef-Pas language. If $T$ admits elimination of field quantifiers (\cite[Definition 1.14]{Sh863}) then $T$ is strongly dependent if and only if the value group and the residue field are. \end{fact} \begin{fact}[\cite{BelHenselian}, Theorem 4.4]\label{QE} Algebraically maximal Kaplansky fields of equi-characteristic $(p,p)$ admit elimination of field quantifiers in the Denef-Pas language. \end{fact} The combination of the last two facts extends Fact \ref{Belair} to the strongly NIP case in analogy with Fact \ref{delon}: \begin{cor} Let $k$ be an infinite NIP field (of equi-characteristic $(p,p)$) and $\Gamma$ an ordered abelian group. Then $k((t^\Gamma))$ is NIP provided that, if $p=\mathrm{char}(k)>0$, then $\Gamma$ is $p$-divisible. It is strongly NIP if and only if $k$ and $\Gamma$ are. \end{cor} \begin{rem} Though in \cite{BelHenselian} B\'elair does not claim Fact \ref{QE} for algebraically maximal Kaplansky fields in mixed characteristic his proof seems to work equally well in that setting. A more self contained proof is available in \cite{HalHasQE}. Combined with \cite[Proposition 5.9]{HalHas} we get that for the last sentence in the above corollary to hold (for algebraically closed Kaplansky fields of any characteristics) we do not need the value group and the residue field to be pure. This gives a strongly dependent version of \cite[Theorem 3.3]{JaSiTransfer}. \end{rem} It is natural to ask whether all NIP fields constructed as Hahn series satisfy Shelah's conjecture, namely, whether they all support a definable henselian valuation. It follows immediately from Corollary \ref{groups} that: \begin{prop}\label{ConjForHahn} Let $k$ be an NIP field, $\Gamma$ an ordered abelian group which is $p$-divisible if $\mathrm{char}(k)=p>0$. Then $K:=k((t^\Gamma))$ is either algebraically closed, or real closed or it supports a definable non-trivial valuation. \end{prop} This answers Question (1) for Hahn fields. Whether NIP Hahn fields support a definable \emph{henselian} valuation is more delicate. In positive characteristic this follows from \cite[Corollary 3.18]{JanKo}. Proposition 4.2 of that same paper provides a positive answer (in any characteristic) in case $K=k((t^\Gamma))$ and $\Gamma$ is not divisible. It seems, however, that the general equi-characteristic 0 case remains open. In some cases we can be even more precise. E.g., Hong, \cite{Hong} gives conditions on the value group implying the definability of the natural (Krull) valuation on $k((t^\Gamma))$: \begin{fact} Let $(K, \mathcal O)$ be a henselian field. If the value group contains a convex $p$-regular subgroup that is not $p$-divisible, then $\mathcal O$ is definable in the language of rings. \end{fact} In all the examples discussed in the present section, the source of the complexity of the field (unbounded, strongly dependent not dp-minimal etc.) can be traced back to the value group of the natural (power series) valuation. For example, as shown in \cite{JaSiWa2015}, an ordered abelian group $\Gamma$ is dp-minimal if and only if $[\Gamma:p\Gamma]$ is finite for all primes $p$. By Theorem \ref{classification} dp-minimal fields are henselian with dp-minimal value groups. We note that it also follows from the same theorem that dp-minimal fields are bounded. Indeed\footnote{This argument was sugegsted to us by I. Efrat. Any mistake is, of course, solely, ours.}, for any Henselian NIP field $(K,v)$ with $\mathrm{char}(K)=\mathrm{char}(Kv)$ we have that $G_K\cong T \rtimes G_k$ where $G_K, G_k$ are the respective absolute Galois groups of $K$ and $k=Kv$, and $T$ is the inertia group (see \cite[Theorem 22.1.1]{EfBook} and use the fact that $K$ has no extensions of degree divisible by $\mathrm{char }(K)$). If $K$ is dp-minimal then $T=\prod_{l\in \Omega} \mathbb Z_l^{\dim_{\mathbb F_l}\Gamma/l\Gamma}$ for a certain set of primes $\Omega$. Since $\Gamma:=vK$ is dp-minimal, this implies that $T$ is small. Since $k$ is either real closed, algebraically closed or elementarily equivalent to a finite extension of $\mathbb Q_p$, also $G_K$ is small. If $(K,v)$ is of mixed characteristic, the exact same argument works if $v$ has no coarsening $w$ of equi-characteristic $0$. Otherwise, decompose $K\xrightarrow{w}Kw\xrightarrow{\bar v} Kv$ and note that $G_{Kw}$ is small by what we have just written, so $K$ is small by our argument for equi-characteristic $0$. It seems, therefore, natural to ask whether the complexity of the value group in the above examples can be recovered definably. Can any (model theoretic) complexity of an NIP field be traced back to that of an ordered abelian group: \begin{qu} Let $K$ be a non separably closed NIP field. Does $K$ interpret a dp-minimal field? If $K$ is not strongly dependent (resp. dp-minimal) is $K$ either imperfect or admits an (externally) definable non-trivial henselian valuation with a non strongly-dependent (resp. dp-minimal) value group? \end{qu} \section{The Axioms of V-Topologies for $\ensuremath{\mathcal{T}} _G$}\label{AxiomsRevisted} We are now returning to that construction of V topologies from multiplicative subgroups, as described in Section \ref{prelims}. Throughout this section no model theoretic assumptions are made, unless explicitly stated otherwise. For ease of reference, we remind the axioms of V topology: \begin{de} A collection of subsets $\ensuremath{\mathcal{N}} $ of a field $K$ is a basis of 0-neighbourhoods for a V-topology on $K$ if is satisfies the following axioms: \begin{description} \item[\textbf{\textup{(V\,1)}}] $\bigcap \mathcal{N}:=\bigcap_{U\in\mathcal{N}}U=\left\{0\right\}$ and $\left\{0\right\}\notin\mathcal{N}$; \item[\textbf{\textup{(V\,2)}}] $\forall\, U,\,V\ \exists\, W\ W\subseteq U\cap V$; \item[\textbf{\textup{(V\,3)}}] $\forall\, U\ \exists\, V\ V-V\subseteq U$; \item[\textbf{\textup{(V\,4)}}] $\forall\, U\ \forall\, x,\,y\in K\ \exists\, V\ \left(x+V\right)\cdot\left(y+V\right)\subseteq x\cdot y+U$; \item[\textbf{\textup{(V\,5)}}] $\forall\, U\ \forall\, x\in K^{\times}\ \exists\, V\ \left(x+V\right)^{-1}\subseteq x^{-1}+U$; \item[\textbf{\textup{(V\,6)}}] $\forall\, U\ \exists\, V\ \forall\, x,\,y\in K\ x\cdot y\in V\longrightarrow x\in U\, \vee\, y\in U$. \end{description} \end{de} \noindent\textbf{Notation:} From now on $G$ will denote a multiplicative subgroup of $K^\times$ with $-1\in G$ and $T:= G+1$. We let $\ensuremath{\mathcal{N}} _G:=\left\{\bigcap_{i=1}^n a_i\cdot T: a_i\in K^\times\right\}$, as defined in the opening paragraphs of Section \ref{prelims}. In this setting the first part of \textbf{\textup{(V\,1)}} is automatic, and \textbf{\textup{(V\,2)}} holds by definition: \begin{lem}\label{lemV1}\label{lemV2}\begin{enumerate} \item $\bigcap \ensuremath{\mathcal{N}} _G=\left\{0\right\}$. \item $\forall\, U,\,V\ \exists\, W\ W\subseteq U\cap V$. \end{enumerate} \end{lem} \begin{proof} For every $x\in K^\times$ we have $x\not\in x\cdot T\in \ensuremath{\mathcal{N}} _G$. As $-1\in G$ further $0\in x\cdot T$ for every $x\in K^\times$. Hence $\bigcap \ensuremath{\mathcal{N}} _G=\left\{0\right\}$. This proves (1), item (2) holds by the definition of $\ensuremath{\mathcal{N}} _G$. \end{proof} We will come back to the second part of Axiom~\textbf{\textup{(V\,1)}} later. Axiom~\textbf{\textup{(V\,3)}} is simplified as follows: \begin{lem}\label{lemV3V3'} The following are equivalent \begin{description} \item[\textbf{\textup{(V\,3)}}] $\forall\, U\ \exists\, V\ V-V\subseteq U$. \item[\textbf{\textup{(V\,3)$'$}}] $ \exists\, V\ V-V\subseteq T$. \item[\textbf{\textup{(V\,3)}$^*$}] $\forall\, U\ \exists\, V\ V+V\subseteq U$. \end{description} \end{lem} \begin{proof} The first implication is obvious. By \textbf{\textup{(V\,3)$'$}} there exists $ V=\bigcap_{j=1}^n b_j\cdot T\in \ensuremath{\mathcal{N}} _G$ such that $V-V\subseteq T$. Let $ U=\bigcap_{i=1}^m a_i\cdot T\in \ensuremath{\mathcal{N}} _G$. For all $i\in \{1,\ldots, m\}$, $j\in \{1,\ldots, n\}$. Let $V':=\bigcap_{i=1}^m\bigcap_{j=1}^{n}\left( a_i\cdot b_{j}\cdot T\right)$. Then by direct computation \[ V'-V' \subseteq\bigcap_{i=1}^m a_i\cdot\left(V-V\right) \subseteq \bigcap_{i=1}^m a_i\cdot T= U. \] This shows that \textbf{\textup{(V\,3)}} follows from \textbf{\textup{(V\,3)$'$}}. Replacing $V$ with $V\cap (-V)$ (throughout) we may assume that $V=-V$, proving the equivalence with \textbf{\textup{(V\,3)}$^*$} \end{proof} In order to simplify Axiom~\textbf{\textup{(V\,4)}} we need: \begin{lem}\label{lemV4xy0case} If $ \exists\, V\ V\cdot V\subseteq T$ then $\forall\, U\ \exists\, V\ V\cdot V\subseteq U$. \end{lem} \begin{proof} Let $U=\bigcap_{i=1}^ma_i\cdot T\in \ensuremath{\mathcal{N}} _G$. By assumption there exist\\ $ V=\bigcap_{j=1}^n b_j\cdot T\in \ensuremath{\mathcal{N}} _G$ such that $V\cdot V\subseteq T$. Let $ V':=\bigcap_{i=1}^{m}\left(\left(\bigcap_{j=1}^n a_i\cdot b_j\cdot T\right)\cap \left(\bigcap_{j=1}^n b_j\cdot T\right)\right)\in\ensuremath{\mathcal{N}} _G$. Then by direct computation \[ V'\cdot V' =\bigcap_{i=1}^{m} a_i\cdot\left(V\cdot V\right) \subseteq \bigcap_{i=1}^ma_i\cdot T= U. \] This proves the claim. \end{proof} Now we can prove: \begin{lem} The axiom \\ \textbf{\textup{(V\,4)}} $\forall\, U\ \forall\, x,\,y\in K\ \exists\, V\ \left(x+V\right)\cdot\left(y+V\right)\subseteq x\cdot y+U$\\ is equivalent to the conjunction of\\ \textbf{\textup{(V\,4)$'$}} $ \exists\, V\ V\cdot V\subseteq T$ and \\ \textbf{\textup{(V\,4)$''$}} $\forall\, x\in K\ \exists V \left(x+V\right)\cdot\left(1+V\right)\subseteq x+ T$. \end{lem} \begin{proof} \textbf{\textup{(V\,4)$'$}} and \textbf{\textup{(V\,4)$''$}} are special cases of \textbf{\textup{(V\,4)}}. So we prove the other implication. Let $x,y\in K$ and $ U=\bigcap_{i=1}^m a_i\cdot T\in \ensuremath{\mathcal{N}} _G$. The case $x=y=0$ is Lemma~\ref{lemV4xy0case}. So we assume that $y\neq 0$. For every $i\in \left\{1,\ldots, m\right\}$ we define $\widetilde{a}_i:=a_i\cdot y^{-1}$ and $x_i:=x\cdot\widetilde{a}_i^{-1}$. By \textbf{\textup{(V\,4)$''$}} there exists $V_i\in \ensuremath{\mathcal{N}} _G$ such that \begin{equation}\label{eqV4subseteqxj+ T} \left(x_i+V_i\right)\cdot \left(1+V_i\right)\subseteq x_i+ T. \end{equation} Let $ V:=\bigcap_{i=1}^m\left(\bigcap_{j=1}^{n_i}\widetilde{a}_i\cdot V_i\right)\cap\left( \bigcap_{j=1}^{n_i}\left(y\cdot V_i\right)\right)\in\ensuremath{\mathcal{N}} _G$. Then \begin{eqnarray*} \left(x+V\right)\cdot\left(y+V\right) &\subseteq& \bigcap_{i=1}^m\big(\widetilde{a}_i\cdot y\cdot \left(x_i+V_i\right)\cdot\left(1+V_i\right)\big)\\ &{\subseteq} &\bigcap_{i=1}^m\big(\widetilde{a}_i\cdot y\cdot\left( x_i+ T\right)\big)= x\cdot y+U. \end{eqnarray*} Where the last inclusion follows from Equation (\ref{eqV4subseteqxj+ T}). This finishes the proof. \end{proof} Assuming \textbf{\textup{(V\,3)$'$}} we can simplify further: \begin{lem}\label{lemV4V3'V4'} The axioms \textbf{\textup{(V\,3)$'$}} and \textbf{\textup{(V\,4)$'$}} imply axion \textbf{\textup{(V\,4)}}. \end{lem} \begin{proof} By the previous lemma it will suffice to prove the lemma for $U=T$ and $y=1$. The case $x=0$ is automatic from the assumptions and Lemma \ref{lemV3V3'}. So assume $x\in K^\times$. By Lemma~\ref{lemV3V3'} there exist $V_1,\, V_2$ such that $V_1+V_1\subseteq T$, $V_2+V_2\subseteq V_1$. Further by Lemma~\ref{lemV4xy0case} there exists $V_3$ with $V_3\cdot V_3\subseteq V_2$. Define $V:= \left(x^{-1}\cdot V_1\right)\cap V_2 \cap V_3\in \ensuremath{\mathcal{N}} _G$. Let $v,w\in V$. \begin{eqnarray*} \left(x+v\right)\cdot \left(1+w\right) \in x+V+x\cdot V+V\cdot V \subseteq x+V_2+x\cdot x^{-1}\cdot V_1+V_3\cdot V_3 \\ \subseteq x+V_2+V_1+V_2 \subseteq x+V_1+V_1 \subseteq x+ T. \end{eqnarray*} Hence $\left(x+V\right)\cdot\left(1+V\right)\subseteq x+ T$, as required. \end{proof} The axiom \textbf{\textup{(V\,5)}} holds without further assumptions: \begin{lem}\label{lemV5} Let $K$ be a field. Let $G$ be a multiplicative subgroup of $K$ with $-1\in G$. Then \textbf{\textup{(V\,5)}} $\forall\, U\ \forall\, x\in K^{\times}\ \exists\, V\ \left(x+V\right)^{-1}\subseteq x^{-1}+U$ holds. \end{lem} \begin{proof} We will first show \begin{equation}\label{eq:V5 T} \forall\, x\in K^{\times}\ \exists\, V\ \left(x+V\right)^{-1}\subseteq x^{-1}+ T. \end{equation} For $x=-1$ let $V:= T$. We have $\left(x+ T\right)^{-1}=\left(-1+G+1\right)^{-1}=G^{-1} = G =x^{-1}+ T.$ If $x\in K^\times\setminus\left\{-1\right\}$, let $b_1=-x^2\cdot \left(1+x\right)^{-1}$, $b_2=-x$ and $V:=b_1\cdot T\cap b_2\cdot T=b_1\cdot\left(G+1\right)\cap b_2\cdot\left(G+1\right)$. Let $z\in \left(x+V\right)^{-1}$. Let $g_1,g_2 \in G$ such that $z=\left(x+b_1\cdot g_1 +b_1\right)^{-1}=\left(x+b_2\cdot g_2 +b_2\right)^{-1}$. We have \begin{equation}\label{eqV5z} z=\left(x+b_2\cdot g_2 +b_2\right)^{-1}=\left(x-x\cdot g_2 -x\right)^{-1}=-x^{-1}\cdot g_2^{-1}. \end{equation} Further we have $z^{-1}= x+b_1+b_1\cdot g_1$ and therefore $1- b_1\cdot g_1\cdot z= \left(x+b_1\right)\cdot z$. This implies \begin{eqnarray*} z&=& \left(1-b_1\cdot g_1\cdot z\right)\cdot \left(x+b_1\right)^{-1}\\ &=& x^{-1}+1+x\cdot g_1\cdot z\\ &\stackrel{\textrm{\scriptsize{(\ref{eqV5z})}}}{=}& x^{-1}+1-x\cdot g_1\cdot x^{-1}\cdot g_2^{-1}\\ &=& x^{-1}+ \left(- g_1\cdot g_2^{-1}\right)+1 \in x^{-1}+G+1. \end{eqnarray*} Hence $\left(x+V\right)^{-1}\subseteq x^{-1}+ T$. This proves Equation~(\ref{eq:V5 T}). Now let $x\in K^\times$ and $U=\bigcap_{i=1}^m a_i\cdot T\in\ensuremath{\mathcal{N}} _G$. For every $i\in \left\{1,\ldots, m\right\}$ let $x_i:=a_i\cdot x$. By Equation~(\ref{eq:V5 T}) there exists $V_i$ such that \begin{equation}\label{eqV5zwei} \left(x_i+V_i\right)^{-1}\subseteq x_i^{-1}+ T. \end{equation} For $ V:=\bigcap_{i=1}^ma_i^{-1}\cdot V_i$ \begin{eqnarray*} \left(x+V\right)^{-1} =\bigcap_{i=1}^m a_i\cdot\left(x_i+V_i \right)^{-1} \stackrel{\textrm{\scriptsize{(\ref{eqV5zwei})}}}{\subseteq} \bigcap_{i=1}^m a_i\cdot \left( x_i^{-1}+ T\right) = x^{-1}+U. \end{eqnarray*} Therefore \textbf{\textup{(V\,5)}} holds. \end{proof} The axiom \textbf{\textup{(V\,6)}} can be reduced as follows: \begin{lem}\label{lemV6V6'} The following are equivalent \begin{description} \item[\textbf{\textup{(V\,6)}}] $ \forall\, U\ \exists\,V\ \forall\,x,y\in K\ (x\cdot y\in V\to x\in U\vee y\in U)$ \item[\textbf{\textup{(V\,6)$'$}}] $ \exists\, V\ \forall\,x,y\in K\ (x\cdot y\in V\to x\in T\vee y\in T)$. \end{description} \end{lem} \begin{proof} We assume \textbf{\textup{(V\,6)$'$}} and show \textbf{\textup{(V\,6)}}. We will show by induction on $m$, that for all $a_1,\ldots, a_m\in K^\times$, there exists $V\in \ensuremath{\mathcal{N}} _G$ such that for all $x,y\in K$, if $x\cdot y\in V$ then $x\in \bigcap_{i=1}^m a_iT$ or $y\in \bigcap_{i=1}^m a_iT$. Let $a_1\in K^\times$ and $U:= a_1\cdot T\in \ensuremath{\mathcal{N}} _G$. By \textbf{\textup{(V\,6)$'$}} there exists $V$ such that for all $x,y\in K$, if $x\cdot y\in V$ then $x\in T$ or $y\in T$. Define $V':={a_1^2}\cdot V\in \ensuremath{\mathcal{N}} _G$. For all $x,y\in K$ such that $x\cdot y\in V'$ we have $ x\cdot a_1^{-1}\cdot y\cdot a_1^{-1}\in a_1^{-2} \cdot V'=V$ and therefore $x\cdot a_1^{-1}\in T$ or $y\cdot a_1^{-1}\in T$ and hence $x\in U$ or $y\in U$. Now let $a_1,a_2\in K^\times$ and $ U:=\bigcap_{i=1}^2 a_i\cdot T\in \ensuremath{\mathcal{N}} _G$. By assumption there exists $V$ such that for all $x,y\in K$ if $x\cdot y\in V$ then $x\in T$ or $y\in T$. Define $ V'=:a_1^2\cdot V\cap a_2^2\cdot V\cap a_1\cdot a_2\cdot V $. Let $x,y\in K$ such that $x\cdot y\in V'$. Then $ x\cdot a_1^{-1}\cdot y\cdot a_1^{-1}\in a_1^{-2}\cdot V'\subseteq V$ and therefore as above \begin{equation}\label{oneina1} x\in a_1\cdot T\text{ or }y\in a_1\cdot T. \end{equation} and \begin{equation}\label{oneina2} x\in a_2\cdot T\text{ or }y\in a_2\cdot T. \end{equation} If, by way of contradiction, $x\cdot a_1^{-1}\notin T$ and $y\cdot a_2^{-1}\notin T$, then $ x\cdot a_1^{-1}\cdot y\cdot a_2^{-1}\notin V$, implying $ x\cdot y\notin {a_1}\cdot{a_2}\cdot V\supseteq V'$ contradicting the choice of $x$ and $y$. Therefore \begin{equation}\label{xina1oryina2} x\in a_1\cdot T\text{ or } y\in a_2\cdot T. \end{equation} and, similarly, \begin{equation}\label{yina1orxina2} y\in a_1\cdot T\text{ or } x\in a_2\cdot T. \end{equation} A straightforward verification shows that equations (\ref{oneina1})-(\ref{yina1orxina2}) implies that if $x\cdot y\in V'$ then either $x\in U$ or $y\in U$. Now let $m\geq 3$. Assume that for all $a_1,\ldots, a_{m-1}$ there exists $V$ such that for all $x,y\in K$, if $x\cdot y\in V$ then $x\in \bigcap_{i=1}^{m-1}a_i\cdot T$ or $y\in \bigcap_{i=1}^{m-1}a_i\cdot T$. Let $a_1,\ldots, a_m\in K^\times$ and $ U:=\bigcap_{i=1}^m a_i\cdot T\in \ensuremath{\mathcal{N}} _G$. By induction hypothesis for every $j\in \left\{1,\ldots, m\right\}$ there exists $V_{\neq j}$ such that for all $x,y\in K$, if $x\cdot y\in V_{\neq j}$ then $\displaystyle x\in \bigcap_{\stackrel{i=1}{i\neq j}}^m a_i\cdot T$ or $\displaystyle y\in \bigcap_{\stackrel{i=1}{i\neq j}}^m a_i\cdot T$. Define $ V:=\bigcap_{i=1}^m V_{\neq i}$. Let $x,y\in K^\times$ such that $x\cdot y\in V$. If $x\in a_i\cdot T$ for all $i\in \left\{1,\ldots, m\right\}$ then $x\in U$ and we are done. Otherwise let $j\in \left\{1,\ldots, m\right\}$ with $x\notin a_j\cdot T$. Let $k,\ell\in \left\{1,\ldots, m\right\}\setminus\left\{j\right\}$ with $k\neq \ell$. We have $ x\cdot y\in \bigcap_{i=1}^m V_{\neq i}\subseteq V_{\neq k}$. As $\displaystyle x\notin a_j\cdot T\supseteq \bigcap_{\stackrel{i=1}{i\neq k}}^m a_i\cdot T$ we have $\displaystyle y\in \bigcap_{\stackrel{i=1}{i\neq k}}^m a_i\cdot T$. Analogous we show $\displaystyle y\in \bigcap_{\stackrel{i=1}{i\neq \ell}}^m a_i\cdot T$. Therefore $\displaystyle y\in \bigcap_{\stackrel{i=1}{i\neq k}}^m a_i\cdot T \cap \bigcap_{\stackrel{i=1}{i\neq \ell}}^m a_i\cdot T=U$. Hence for all $U$ there exists $V$ such that for all $x,y\in K$, if $x\cdot y\in V$ then $x\in U$ or $y\in U$. \end{proof} Summing up all the simplifications of the present section we obtain: \begin{prop}\label{PropSimplifiedVAxiomsnontrivialdefinable} Let $K$ be a field. Let $\mathrm{char}\left(K\right)\neq q$ and if $q= 2$ assume $K$ is not euclidean. Assume that for the primitive $q$th-root of unity $\zeta_q\in K$. Let $G:=\left(K^\times\right)^q\neq K^\times$. Assume that \begin{description} \item[\textbf{\textup{(V\,1)$'$}}] $\left\{0\right\}\notin\mathcal{N}_G$; \item[\textbf{\textup{(V\,3)$'$}}] $ \exists\, V\ V-V\subseteq T$ \item[\textbf{\textup{(V\,4)$'$}}] $ \exists V \ V\cdot V\subseteq T$ \item[\textbf{\textup{(V\,6)$'$}}] $ \exists\, V\ \forall\,x,y\in K\ x\cdot y\in V\to x\in T\vee y\in T$. \end{description} Then $K$ admits a non-trivial definable valuation. \end{prop} \begin{proof} With Lemma~\ref{lemV1}, Lemma~\ref{lemV2}, Lemma~\ref{lemV3V3'}, Lemma~\ref{lemV4V3'V4'}, Lemma~\ref{lemV5} and Lemma~\ref{lemV6V6'} the result follows directly from Corollary~\ref{corVAxiomsnontrivialdefinable}. \end{proof} \section{Back to NIP fields}\label{secNIP} As already explained in the opening sections, our main motivation in the present paper is to study the existence of definable valuations on (strongly) NIP fields. We also hope that such a project may shed some light on the long standing open conjecture that stable fields are separably closed. We have already explained that in the stable case this conjecture can be rather easily settled under the further assumption that the field is bounded. It is therefore natural to ask whether the same assumption can help settle the questions stated in the Introduction. In the present section we show how boundedness gives quite easily Axiom [\textbf{\textup{(V\,1)$'$}}] (stating that $\left\{0\right\}\notin\mathcal{N}_G$). If $K$ is an infinite NIP field, i.e. a field definable in a monster model satisfying NIP, then by \cite[Corollary~4.2]{KrSRNIP} there is a definable additively and multiplicatively invariant Keisler measure on $K$. In the whole section if not stated differently let $K$ be an infinite NIP field and $\mu$ an additively and multiplicatively invariant definable Keisler measure on $K$. By \cite[Proposition~4.5]{KrSRNIP} for any definable subset $X$ of $K$ with $\mu(X)>0$ and any $a\in K$, we have \[\mu\left(\left(a+X\right)\cap X\right)=\mu\left(X\right).\tag{$\clubsuit$}\] \begin{lem}\label{propV1ForGgenerel} Let $a_1,\ldots, a_m\in K^\times$ and $G\subseteq K^\times$ a multiplicative subgroup with $-1\in G$ and $\mu(G)>0$. Then $ \bigcap_{i=1}^m a_i T\supsetneq\{0\}. $ \end{lem} \begin{proof} As $-1\in G$ it follows that $0\in\bigcap_{i=1}^m a_i T$. This also implies that \[G+a^{-1}=\{s: 1\in a(G+s)\}\tag{*}\] for any $a\in K^\times$. By additivity of the measure $(\clubsuit)$ applied to the left hand side of $(*)$ gives \[\mu(\bigcap_{i=1}^m (G+a_i^{-1}\cap G))=\mu(G)>0. \] So by the right hand side of $(*)$ we have $t_0\in \bigcap_{i=1}^m \{s\in G: 1\in a_i(G+s)\}$. So $1\in a_i(G+t_0)$ for all $1\le i \le m$, and as $t_0\in G$ we get $t_0^{-1}\in \bigcap_{i=1}^m a_i(G+1)=\bigcap_{i=1}^m a_iT$. \end{proof} \begin{cor}\label{thmV3V4V6Existsdefinablenontrivialvaluation} Let $K$ be an infinite NIP field with $\sqrt{-1}\in K$. Let $G:=\left(K^\times\right)^q\neq K^\times$ for some $q\neq \ensuremath{\textrm{char}} (K)$ prime with $\zeta_q\in K$. Assume that $[K^\times: G]<\infty$ and that for $T:=G+1$ we have: \begin{description} \item[\textbf{\textup{(V\,3)$'$}}] $ \exists\, V\ V-V\subseteq T$ \item[\textbf{\textup{(V\,4)$'$}}] $ \exists V \ V\cdot V\subseteq T$ \item[\textbf{\textup{(V\,6)$'$}}] $ \exists\, V\ \forall\,x,y\in K\ x\cdot y\in V\to x\in T\vee y\in T$. \end{description} Then $K$ admits a non-trivial $\emptyset$-definable valuation. \end{cor} \begin{proof} By additivity and invariance of $\mu$ we get that $\mu(G)=[K^\times:G]^{-1}$. The result now follows directly from Proposition~\ref{PropSimplifiedVAxiomsnontrivialdefinable} using Lemma~\ref{propV1ForGgenerel} \end{proof} As mentioned in Section~1, \[\ensuremath{\mathcal{N}} _G':=\left\{\left(a_1\cdot \left(G+1\right)\right)\cap\left(a_2\cdot\left( G+1\right)\right): a_1,a_2\in K^\times\right\}\] is a base of the neighbourhoods of zero of $\ensuremath{\mathcal{T}} _G$. We obtain the following corollary: \begin{cor}\label{remNG'} Let $K$ be an infinite NIP field with $\sqrt{-1}\in K$. Let $G:=\left(K^\times\right)^q\neq K^\times$ for some $q\neq \ensuremath{\textrm{char}} (K)$ prime with $\zeta_q\in K$. Assume that $[K^\times: G]<\infty$. Then for $T:=G+1$ we have that \begin{description} \item[\textbf{\textup{(V\,3)$'$}}] $ \exists\, V\in \ensuremath{\mathcal{N}} _G \ V-V\subseteq T$ \item[\textbf{\textup{(V\,4)$'$}}] $ \exists V\in \ensuremath{\mathcal{N}} _G \ V\cdot V\subseteq T$ \item[\textbf{\textup{(V\,6)$'$}}] $ \exists\, V\in \ensuremath{\mathcal{N}} _G \ \forall\,x,y\in K\ x\cdot y\in V\to x\in T\vee y\in T$. \end{description} if and only if \begin{description} \item[\textbf{\textup{(V\,3)$'_2$}}] $ \exists\, \widetilde{V}\in \ensuremath{\mathcal{N}} _G'\ \widetilde{V}-\widetilde{V}\subseteq T$ \item[\textbf{\textup{(V\,4)$'_2$}}] $ \exists \widetilde{V}\in \ensuremath{\mathcal{N}} _G' \ \widetilde{V}\cdot \widetilde{V}\subseteq T$ \item[\textbf{\textup{(V\,6)$'_2$}}] $ \exists\, \widetilde{V}\in \ensuremath{\mathcal{N}} _G'\ \forall\,x,y\in K\ x\cdot y\in \widetilde{V}\to x\in T\vee y\in T$. \end{description} \end{cor} \begin{proof} As $\ensuremath{\mathcal{N}} _G'\subseteq \ensuremath{\mathcal{N}} _G$ it is clear that if \textbf{\textup{(V\,3)$'_2$}}, \textbf{\textup{(V\,4)$'_2$}} and \textbf{\textup{(V\,6)$'_2$}} hold, then so do \textbf{\textup{(V\,3)$'$}}, \textbf{\textup{(V\,4)$'$}} and \textbf{\textup{(V\,6)$'$}}. On the otherhand if \textbf{\textup{(V\,3)$'$}}, \textbf{\textup{(V\,4)$'$}} and \textbf{\textup{(V\,6)$'$}} hold, then $\ensuremath{\mathcal{T}} _G$ is a V-topology and $\ensuremath{\mathcal{N}} _G'$ is a 0-neighbourhood basis for $\ensuremath{\mathcal{T}} _G$. Therefore, for any $V\in \ensuremath{\mathcal{N}} _G$ witnessing \textbf{\textup{(V\,i)}} ($i=3,4,6$), there exists $\widetilde{V}\in \ensuremath{\mathcal{N}} _G$ such that $\widetilde{V}\subseteq V$, and as -- for a fixed $V$ -- the axiom \textbf{\textup{(V\,i)}} is universal, it is automatically satisfied by $\tilde V$. \end{proof} Note that \textbf{\textup{(V\,3)$'_2$}}, \textbf{\textup{(V\,4)$'_2$}} and \textbf{\textup{(V\,6)$'_2$}} are first order sentences in the language of rings (appearing explicitly in the statement of Conjecture \ref{fo}). Let us denote their conjunction as $\psi_K$. Thus, if $K$ is a bounded\footnote{As already mentioned, "radically bounded" would suffice.} NIP field such that $K\models \psi_K$ then $K$ supports a definable valuation. \section{Concluding remarks} We have shown in Section \ref{prelims} that if $K$ is infinite NIP there exist a sentence $\psi_K$ (possibly with parameters) such that: \begin{enumerate} \item If $K\models \psi_K$ then $K$ admits a non-trivial definable valuation. \item If $K$ is $t$-henselian then $K\models \psi_K$. \item If $K^q\neq K$ for some prime $q\neq \mathrm{char}(K)$ then $\psi_K$ (and the definable valuation ring) can be taken over $\emptyset$. \end{enumerate} If $K$ is as in (3) above, $\zeta_q\in K$, a primitive root of unity, and $\sqrt{-1}\in K$ the sentence $\psi_K$ is the statement that $\ensuremath{\mathcal{N}} _G'$ is a neighbourhood basis for a $V$ topology for $G=(K^\times)^q$. Assuming for simplicity that the predicates $G$, $T:=1+G$ and $aT$ for $a\in K^\times$ are atomic, $\psi_K$ is the conjunction of (V1)-(V6) which is readily checked to be an AEA-sentence. We have shown in Section \ref{AxiomsRevisted} that $\psi_K$ is equivalent to the conjunction of (V1)$'$, (V3)$'$, (V4)$'$ and (V6)$'$ -- which is an EA-sentence. In the last corollary we have shown that if $K$ is bounded NIP then, in fact, (V1)$'$ automatically holds, reducing further the complexity of $\psi_K$. If $K$ does not satisfy (3) above (or the additional assumptions on roots of unity) we have to replace $K$ with a finite extension $L\ge K$ satisfying the necessary assumptions. In that case $\psi_K$ has to be relativised to $L$ -- which, since $L$ is interpretable in $K$ (possibly with parameters) is not a problem, and does not change the complexity of $\psi_K$ -- provided that, as above, the corresponding field operations, the group $G$, the set $T$ and the open sets $aT$ interpreted in $L$ are assumed atomic. \end{document}

\begin{document} \def\mathbb{F}{\mathbb{F}} \def\mathbb{F}qn{\mathbb{F}_q^n} \def\mathbb{F}q{\mathbb{F}_q} \def\mathbb{F}p{\mathbb{F}_p} \def\mathbb{D}{\mathbb{D}} \def\mathbb{E}{\mathbb{E}} \def\mathbb{Z}{\mathbb{Z}} \def\mathbb{Q}{\mathbb{Q}} \def\mathbb{C}{\mathbb{C}} \def\mathbb{R}{\mathbb{R}} \def\mathbb{N}{\mathbb{N}} \def\mathbb{H}{\mathbb{H}} \def\mathbb{P}{\mathbb{P}} \def\mathbb{T}{\mathbb{T}} \def\, (\text{mod }p){\, (\text{mod }p)} \def\, (\text{mod }N){\, (\text{mod }N)} \def\, (\text{mod }q){\, (\text{mod }q)} \def\, (\text{mod }1){\, (\text{mod }1)} \def\mathbb{Z}N{\mathbb{Z}/N \mathbb{Z}} \def\mathbb{Z}p{\mathbb{Z}/p \mathbb{Z}} \def\mathbb{Z}an{a^{-n}\mathbb{Z}/ \mathbb{Z}} \def\mathbb{Z}al{a^{-l} \mathbb{Z} / \mathbb{Z}} \def\mathbb{P}r{\text{Pr}} \def\leftsize{\left| \left\{} \def\rightsize{\right\} \right|} \title{Phase relations and pyramids} \begin{abstract} We develop tools to study the averaged Fourier uniformity conjecture and extend its known range of validity to intervals of length at least $\exp(C (\log X)^{1/2} (\log \log X)^{1/2})$. \end{abstract} \section{Introduction} In this article we shall establish the following result. \begin{teo} \label{1} Given $0 < \rho < 1$ and $\eta > 0$, there exists some $C>0$ such that, for every $\exp(C (\log X)^{1/2} (\log \log X)^{1/2} ) \le H \le X^{1/2}$ and every complex-valued multiplicative function $g$ with $|g| \le 1$ satisfying \begin{equation} \label{A} \int_X^{2X} \sup_{\alpha} \left| \sum_{x \le n \le x+H} g(n) e(\alpha n) \right| dx \ge \eta H X, \end{equation} we have $\mathbb{D} (g; CX^2/H^{2-\rho},C) \le C$. \end{teo} Here, we are writing $\mathbb{D}(g;T,Q)$ for the 'pretentious' distance \cite{GS}: $$ \mathbb{D}(g;T,Q) = \inf \left( \sum_{p \le T} \frac{1 - \text{Re}(g(p)p^{it}\chi(p))}{p} \right)^{1/2} ,$$ with the infimum taken over all $|t| \le T$ and all Dirichlet characters of modulus at most $Q$. In particular, Theorem \ref{1} implies that (\ref{A}) cannot hold for the Möbius and Liouville functions. Theorem \ref{1} improves on estimates obtained in \cite{MRT} and \cite{MRTTZ}, where it was shown that for any $\varepsilon > 0$ the result holds for intervals of length at least $X^{\varepsilon}$ and $\exp((\log X)^{5/8+\varepsilon})$, respectively. The methods of this article should adapt to nilsequences, thus yielding corresponding progress on the higher uniformity conjecture. It is known that after passing to logarithmic averages, establishing this conjecture for intervals of length at least $(\log X)^{\varepsilon}$, for every $\varepsilon >0$, would imply both Chowla's and Sarnak's conjectures \cite{MRTTZ,T}. We will proceed through the same general framework as in previous articles \cite{MRT, MRTTZ,W}, where it is shown that a function $g$ satisfying (\ref{A}), for the corresponding values of $H$, must correlate with $n \mapsto e(an/q) n^{2 \pi i T}$ on many of these intervals and for certain fixed choices of $T$ and $q$. The results of \cite{MR,MR2, MRT2} then imply that $g$ must behave globally like a function of this form, yielding the desired conclusion. To obtain this local correlation one exploits Elliott's inequality and the large sieve (see \cite{MRT, MRTTZ}), which combined guarantee that under (\ref{A}) we may associate to many intervals $I \subseteq [X,2X]$ a frequency $\alpha_I \in \mathbb{R}$ in such a way that we may find many quadruples consisting of a pair of intervals $I=[x,x+H], J= [y,y+H]$ and a pair of primes $p,q \le H$, such that $|x/p-y/q|$ is small and $p \alpha_I$ close to $q \alpha_J$ mod $Q$, for some large integer $Q$. The problem then becomes that of showing that these relations force $\alpha_I$ to be close mod $1$ to $\frac{m_I}{q} + \frac{T}{x}$ for certain $m_I,q \in \mathbb{N}$ and $T \in \mathbb{R}$, which would then imply the desired local correlation. To accomplish this we begin in Section \ref{phase} by studying 'pre-paths' consisting of a sequence of elements $\alpha_1, \ldots, \alpha_{k+1} \in \mathbb{Z} / Q \mathbb{Z}$ and primes $p_1, \ldots, p_k, q_1, \ldots, q_k$ with $p_i \alpha_i$ close to $q_i \alpha_{i+1}$, for every $1 \le i \le k$. Building on ideas of \cite{W}, we show how to construct certain 'pyramids' of frequencies that help relate the elements of the sequence and, in particular, a 'top' element $\alpha$ such that $\alpha_i$ is close to $\left( \prod_{j=1}^{i-1} p_j \prod_{j=i}^k q_j \right) \alpha$, for every $1 \le i \le k+1$, with an error that depends on the relative sizes of the primes involved. After developing our general setting further in Section 3, we proceed in Section 4 to show how the additional 'physical' information that $|x/p - y/q|$ is small can be used to obtain uniform bounds for the approximations of the previous paragraph. In particular, the conclusions attained would work equally well for $H$ in the poly-logarithmic range and seem likely to be useful tools for future work on these problems. The purpose of Section 5 is then to show that for many of the intervals we are studying the corresponding frequencies are connected by paths of the above form. Some connectedness of this type seems necessary in order to be able to find a fixed choice of $q$ and $T$ that works for many of these intervals and here is where the required lower bound on $H$ in Theorem \ref{1} emerges. The reason for it is that when studying paths of length $k$, one is naturally led to some losses of the order of $C^k$ in the bounds, for some absolute constant $C>1$. Such losses become problematic once $C^{k}$ is comparable to $H$ and since one needs to consider paths of length around $\frac{\log X}{\log H}$ in order to be able to have enough of the intervals connected with each other, we end up in that situation once $H$ goes below $\exp( C (\log X)^{1/2})$ (essentially the same observation can already be found in \cite{MRT, MRTTZ}). In fact, due to the density of prime numbers one is also led to some additional factors of the order of $(\log H)^k$, which is the reason for the exact lower bound on $H$ in Theorem \ref{1}. Finally, we complete the proof in Section 6. Once enough intervals have been connected to a fixed interval $I_0$, one can relatively easy use the properties of the 'pyramids' obtained in the first sections to show that a fixed choice of $q$ and $T$ works for many of the intervals. We notice that the methods of this article end up using auxiliary phases as in \cite{W}, but also the advantage of working with higher moduli as in \cite{MRT, MRTTZ}. As such, they can be seen as a middle point between both arguments. On the other hand, an outcome of this article is that neither the contagion arguments of \cite{W} nor the mixing lemmas of \cite{MRT, MRTTZ} end up being necessary to cover the natural range of $\exp( (\log X)^{1/2+\varepsilon})$. However, such tools may very well end up being useful when trying to lower the value of $H$ further. \begin{notation} We will write $X \lesssim Y$ or $X=O(Y)$ to mean that there is some absolute constant $C$ with $|X| \le C Y$ and $X \sim Y$ if both $X \lesssim Y$ and $Y \lesssim X$ hold. If the implicit constants depend on some additional parameters, we shall use a subscript to indicate this. For a finite set $S$ we write $|S|$ for its cardinality. We abbreviate $e(x):=e^{2 \pi i x}$ and given $Q \in \mathbb{N}$ we write $\| \cdot \|_Q$ for the distance to $0$ in $\mathbb{R} / Q \mathbb{Z}$. \end{notation} \section{Pyramids} \label{phase} We begin with the following extension of \cite[Lemma 2.1]{W}. \begin{lema} \label{astart} Let $\epsilon_1,\epsilon_2 > 0$ and let $Q \in \mathbb{N}$. Let $\alpha_1,\alpha_2 \in \mathbb{R} / Q \mathbb{Z}$ and let $p_1,p_2$ be distinct primes not dividing $Q$ with $\| p_1 \alpha_1 - p_2 \alpha_2\|_Q < \epsilon_1 + \epsilon_2$. Then, there exists $\alpha \in \mathbb{R} / Q \mathbb{Z}$ with $\| p_i \alpha - \alpha_j \|_Q < \frac{\epsilon_j}{p_j}$ if $i \neq j$. \end{lema} \begin{proof} For $\left\{ i, j \right\}= \left\{ 1,2 \right\}$, let $\alpha^{(i)} \in \mathbb{R} / Q \mathbb{Z}$ be such that $p_i \alpha^{(i)} = \alpha_j$. Adding integer multiples of $Q/p_i$ to $\alpha^{(i)}$ we may assume that $\| \alpha^{(1)} - \alpha^{(2)} \|_Q \le \frac{Q}{2 p_1 p_2}$. On the other hand, we have by hypothesis that $\| p_1 p_2 (\alpha^{(1)} - \alpha^{(2)} ) \|_Q < \epsilon_1 + \epsilon_2$. Combining both estimates we see that it must in fact be $\| \alpha^{(1)} - \alpha^{(2)} \|_Q < \frac{\epsilon_1 + \epsilon_2}{p_1 p_2}$. Taking $\alpha= \alpha^{(1)} - \frac{\epsilon_2}{\epsilon_1+\epsilon_2} (\alpha^{(1)}-\alpha^{(2)})$ we obtain the result. \end{proof} Lemma \ref{astart} immediately implies the following corollary. \begin{lema} \label{tcp} Let $\epsilon_j,\epsilon_j'>0$ for every $1 \le j \le k$ and let $Q \in \mathbb{N}$. Let $(\alpha_1^{(1)},\ldots,\alpha^{(1)}_{k+1})$ and $(\alpha_2^{(0)},\ldots,\alpha_{k+1}^{(0)})$ be tuples of elements of $\mathbb{R} / Q \mathbb{Z}$ and let $p_1,\ldots,p_{k}$, $q_1,\ldots,q_{k}$ be a sequence of distinct primes not dividing $Q$ such that, for every $1 \le j \le k$, we have $\| q_{j} \alpha^{(1)}_{j+1} - \alpha^{(0)}_{j+1} \|_Q < \epsilon_j$ and $\| p_j \alpha^{(1)}_j - \alpha^{(0)}_{j+1} \|_Q < \epsilon'_j$. Then, there exists a tuple $(\alpha^{(2)}_1,\ldots,\alpha^{(2)}_{k})$ of elements of $\mathbb{R} /Q \mathbb{Z}$ such that, for every $1 \le j \le k$, we have $\| p_j \alpha^{(2)}_j - \alpha^{(1)}_{j+1} \|_Q < \frac{\epsilon_j}{q_j}$ and $\| q_j \alpha^{(2)}_j - \alpha^{(1)}_j \|_Q < \frac{\epsilon_j'}{p_j}$. \end{lema} This can be iterated in the following way: after we have used sets of frequencies $(\alpha_1^{(j)},\ldots,\alpha^{(j)}_{k+2-j})$, $(\alpha_2^{(j-1)},\ldots,\alpha_{k+2-j}^{(j-1)})$ and primes $p_1,\ldots,p_{k+1-j}$, $q_j,\ldots,q_{k}$ not dividing $Q$ to obtain a new set $(\alpha^{(j+1)}_1,\ldots,\alpha^{(j+1)}_{k+1-j})$, we can then use the sets of frequencies $(\alpha^{(j+1)}_1,\ldots,\alpha^{(j+1)}_{k+1-j})$, $(\alpha_2^{(j)},\ldots,\alpha^{(j)}_{k+1-j})$ and primes $p_1,\ldots,p_{k-j}$, $q_{j+1},\ldots,q_{k}$ to obtain a new set $(\alpha^{(j+2)}_1,\ldots,\alpha^{(j+2)}_{k-j})$. We thus arrive, for every $1 \le j \le k+1$, at a tuple $(\alpha_1^{(j)},\ldots,\alpha_{k+2-j}^{(j)})$ of elements of $\mathbb{R} / Q \mathbb{Z}$. It will be convenient to name these objects. \begin{defi} A \emph{pre-path} mod $Q$ is a choice of (ordered) tuples $(\alpha_1^{(1)},\ldots,\alpha^{(1)}_{k+1})$, $(\alpha_2^{(0)},\ldots,\alpha_{{k+1}}^{(0)})$ of elements of $\mathbb{R} / Q \mathbb{Z}$, real numbers $\epsilon_1, \ldots, \epsilon_{k}, \epsilon_1', \ldots, \epsilon'_{k}>0$ and distinct primes $p_1,\ldots,p_{k}$, $q_1,\ldots,q_{k}$ not dividing $Q$ satisfying the hypothesis of Lemma \ref{tcp}. We say a corresponding sequence $(\alpha_1^{(j)})_{1 \le j \le k+1}$ obtained as in the previous paragraph is a \emph{pyramid} associated with this pre-path. We call $\alpha_1^{(k+1)}$ the \emph{top element} of the pyramid and $k$ the \emph{length} of the pre-path. \end{defi} The following will be our main input for the study of pre-paths. \begin{lema} \label{premanli} Let $0 < \epsilon < 1$. Given a pre-path of length $k$ with $\epsilon_i=\epsilon_i'=\epsilon$ for every $1 \le i \le k$, we have that any associated pyramid $(\alpha_1^{(j)})_{1 \le j \le k+1}$ satisfies $$ \| q_j \alpha_1^{(j+1)} - \alpha_1^{(j)} \|_Q < \epsilon \left( \prod_{i=1}^{\lfloor (j-1)/2 \rfloor+1} p_i \right)^{-1} \left( \prod_{i=1}^{\lceil (j-1)/2 \rceil } q_{j-i} \right)^{-1},$$ for every $1 \le j \le k$. \end{lema} \begin{proof} We will proceed by induction on the length of the pre-path. If $k=1$ the claim is immediate from Lemma \ref{tcp}, so we let $k \ge 2$ and assume the result has already been established for all pre-paths of length at most $k-1$. If $j \le k-1$ observe that the parameters $(\alpha_1^{(1)},\ldots,\alpha_{j+1}^{(1)})$, $(\alpha_2^{(0)},\ldots,\alpha_{j+1}^{(0)})$, $p_1,\ldots,p_{j}$, $q_1,\ldots,q_{j}$ and $\epsilon_1=\epsilon'_1=\ldots=\epsilon_j=\epsilon_j'=\epsilon$ form a pre-path of length $j$ and $(\alpha_1^{(i)})_{1 \le i \le j+1}$ is a pyramid for this pre-path, so the result follows by induction in this case. We thus only need to treat $j=k$. By Lemma \ref{tcp} we know that \begin{equation} \label{ppp14} \| q_{k} \alpha_1^{(k+1)} - \alpha_1^{(k)} \|_Q \le \frac{\| p_1 \alpha_1^{(k)} - \alpha_2^{(k-1)} \|_Q}{p_1}. \end{equation} Here we are using that, after iterating Lemma \ref{tcp}, $q_{k}$ and $p_1$ are the primes that end up relating $\alpha_1^{(k+1)}$ with $\alpha_1^{(k)}$ and $\alpha_2^{(k)}$, respectively. Notice now that the 'inverted' parameters $(\alpha_{k}^{(1)},\ldots,\alpha_1^{(1)})$, $(\alpha_{k}^{(0)},\ldots,\alpha_2^{(0)})$, $q_{k-1},\ldots,q_1$, $p_{k-1},\ldots,p_1$ and $\epsilon$ form a pre-path of length $k-1$ and that $(\alpha_{k+1-j}^{(j)})_{1 \le j \le k}$ is a pyramid for this pre-path (notice that the roles of the primes $p_i$ and $q_i$ have also been inverted). It thus follows by induction that the right-hand side of (\ref{ppp14}) is $$ <\frac{\epsilon}{p_1} \left( \prod_{i=1}^{\lfloor k/2 \rfloor} q_{k-i} \right)^{-1} \left( \prod_{i=1}^{\lceil k/2-1 \rceil } p_{(k)-(k-1-i)} \right)^{-1}.$$ The result then follows upon rearranging. \end{proof} \section{General setting} In this section we will invoke some results from \cite{MRT} and \cite{MRTTZ} that will serve as the setting for our approach and will also establish a regularity estimate that will be useful in the rest of the article. As mentioned in the introduction, once we have shown that $g$ correlates with $n \mapsto e(an/q) n^{2 \pi i T}$ on many of the intervals and for a certain fixed choice of $q$ and $T$, Theorem \ref{1} will then follow from the results of Matömaki and Radziwi\l\l \, \cite{MR,MR2, MRT2}. More precisely, we will be using the following reduction which follows from the last part of \cite[Section 6]{MRTTZ} and relies on the power saving bounds of \cite{MR2}. \begin{prop} \label{2} Let $g$ be a complex-valued multiplicative function with $|g| \le 1$. Let $\rho>0$ be sufficiently small, $C >0$ sufficiently large with respect to $\rho$ and $X \ge 1$ sufficiently large with respect to $\rho$ and $C$. Let $H$ be as in Theorem \ref{1}. Also, let $T \in \mathbb{R}$ and $q \in \mathbb{N}$ satisfy $|T| \le C X^2/H^{2-\rho}$ and $q \le C H^{\rho}$. Assume that for $\ge X/H^{1+\rho}$ disjoint intervals $I \subseteq [X,2X]$ of length $H^* \in [H^{1-\rho}, H]$ we can find some integer $a_I$ with $|\sum_{n \in I} g(n) n^{2 \pi iT} e(a_I n/q)| \ge cH^*$, for some $c >0$. Then $\mathbb{D} (g ; BX^2/H^{2-\rho},B) = O_{\rho,C,c}(1)$ for some $B =O_{\rho,C,c}(1)$. \end{prop} Our task is then reduced to establishing the following estimate. \begin{teo} \label{3} Let $g,\rho,\eta$ and $H$ be as in Theorem \ref{1}. Then, if $C$ is sufficiently large with respect to $\eta$ and $\rho$ and $X$ is sufficiently large with respect to $C$, we can find $T \in \mathbb{R}$ and $q \in \mathbb{N}$ with $|T| \le C X^2/H^{2-\rho}$ and $q \le CH^{\rho}$ such that, for $\ge \frac{X}{H^{1+\rho}}$ disjoint intervals $I_x=[x,x+H^*] \subseteq [X/H^{\rho},2X]$ of length $H^* \in [H^{1-\rho}, H]$ there exists an integer $a_x$ with $|\sum_{n \in I_x} g(n) e(\frac{(n-x) a_x }{q} + \frac{(n-x)T}{x})| \ge H^*/C$. \end{teo} Using Theorem \ref{3}, the Taylor expansion of $(n/x)^{2 \pi i T}$ and the pigeonhole principle, we can then locate an appropriate dyadic interval in $[X/H^{\rho},2X]$ where the hypothesis of Proposition \ref{2} are satisfied, after adjusting $X$, $C$, $H^*$ and $\rho$ if necessary. Thus, in order to prove Theorem \ref{1}, it will suffice to establish Theorem \ref{3}. \begin{defi} \label{conf} Given $R > 1$, $c > 0$ and an interval $I \subseteq \mathbb{R}$, we say a finite set $\mathcal{J} \subseteq I \times \mathbb{R}$ is a $(c,R)$-configuration if $|\mathcal{J}| \ge c|I|/R$ and the first coordinates are $R$-separated points in $I$ (i.e. $|x-y| \ge R$ if $x \neq y$). \end{defi} If $g$ satisfies (\ref{A}), then one can use Elliott's inequality and the large sieve to obtain a configuration as in Definition \ref{conf} for which its elements are highly related to each other through a set of primes whose size is a small power of $H$. Concretely, we have the following estimate from \cite{MRT}. \begin{lema}[\cite{MRT}, Proposition 3.2] \label{base} Let the notation and assumptions be as in Theorem \ref{3}. Let $c_0, \varepsilon >0$ be sufficiently small with respect to $\eta$ and $\rho$ and $X$ sufficiently large with respect to $c_0$ and $\varepsilon$. Then, there exists a $(c_0,H/K)$-configuration $\mathcal{J} \subseteq [X/(10K),2X/K] \times \mathbb{R}$, for some $K \in [H^{\varepsilon^2},H^{\varepsilon}]$, such that for every $(x,\alpha) \in \mathcal{J}$ we have $$ \left| \sum_{x \le n \le x+H/K} g(n) e(\alpha n) \right| \gtrsim H/K,$$ and a pair $P,P' \gtrsim_{\rho,\eta} H^{\varepsilon^2}$ with $P P'=K$ such that, for $\gtrsim_{\rho,\eta} \frac{X}{H} \left( \frac{P}{\log P} \right)^2$ choices of $(x_1,\alpha_1), (x_2,\alpha_2) \in \mathcal{J}$ and $p,q$ primes in $[P,2P]$, we have $|x_1/p - x_2/q| \lesssim_{\rho,\eta} H/(PK)$ and $\| p \alpha_1 - q \alpha_2 \|_{p'} \lesssim_{\rho,\eta} PK/H$ for $\gtrsim_{\rho,\eta} \left( \frac{P'}{\log P'} \right)$ primes $p' \in [P',2P']$. Furthermore, there exist disjoint sets $\mathcal{P}_1, \mathcal{P}_2 \subseteq [P,2P]$ of size $\gtrsim_{\rho,\eta} \frac{P}{\log P}$ such that the same claim holds (up to the implicit constants) if we additionally require $(p,q) \in \mathcal{P}_1 \times \mathcal{P}_2$. \end{lema} \begin{proof} The proof proceeds exactly as in Proposition 3.2 of \cite{MRT}, except for the last claim which simply follows from the pigeonhole principle. \end{proof} \begin{notation} For the rest of this article we will work with a specific of $\eta, \rho, X$ and $H$ as in Theorem \ref{3} and of $c_0, \varepsilon, P, P', K, \mathcal{J}, \mathcal{P}_1$ and $\mathcal{P}_2$ as provided in Lemma \ref{base}. All implicit constants will be allowed to depend on $\eta$ and $\rho$, but will be uniform in our choice of $X$ and $H$. \end{notation} We will require the following definition in order to study the properties of pre-paths arising from the configuration $\mathcal{J}$. \begin{defi} Let $Q \in \mathbb{N}$ and $\mathcal{J}' \subseteq \mathcal{J}$. We define a \emph{path mod} $Q$ \emph{of length} $k$ \emph{in} $\mathcal{J}'$ to be a pre-path mod $Q$ of length $k$ with parameters $\epsilon_1,\epsilon_1',\ldots,\epsilon_k,\epsilon_k' \lesssim PK/H$, $p_1,\ldots,p_k, q_1,\ldots,q_{k} \in [P,2P]$, $(\alpha_i^{(1)})_{i=1}^{k+1} \subseteq \mathbb{R} / Q \mathbb{Z}$ and a set of elements $(x_1,\alpha_1),\ldots, (x_{k+1},\alpha_{k+1}) \in \mathcal{J}'$ such that $\alpha_i \equiv \alpha_i^{(1)}$ (mod $Q$) for every $1 \le i \le k+1$ and $x_i \frac{q_i}{p_i} = x_{i+1}+O(H/K)$ for every $1 \le i \le k$. We then say $(x_1,\alpha_1)$ and $(x_{k+1},\alpha_{k+1})$ are \emph{connected} by a path mod $Q$ of length $k$ in $\mathcal{J}'$. We call $(x_1,\alpha_1)$ the \emph{initial point} and $(x_{k+1},\alpha_{k+1})$ the \emph{end point} of the path. We additionally say the path is \emph{split} if $p_1,\ldots,p_k \in \mathcal{P}_1$ and $q_1,\ldots,q_{k} \in \mathcal{P}_2$. \end{defi} \begin{notation} Given a path $\ell$ mod $Q$ of length $k$ consisting of elements $(x_1,\alpha_1), \ldots, (x_{k+1}, \alpha_{k+1})$ and primes $(p_1,\ldots,p_{k},q_1,\ldots,q_{k})$ we will sometimes need to consider the 'inverted' path $\ell^{-1}$ having initial point $(x_{k+1}, \alpha_{k+1})$ and end point $(x_1,\alpha_1)$, with the corresponding ordered set of primes given by $(q_{k},\ldots,q_1,p_{k},\ldots,p_1)$. Notice that if $\alpha$ is the top element of a pyramid associated to $\ell$, then it is also the top element of a pyramid associated with $\ell^{-1}$. Also, suppose we are given an additional path $\ell'$ mod $Q$ of length $m$ consisting of elements $(y_1,\beta_1),\ldots,(y_{m+1},\beta_{m+1})$. If $(y_1,\beta_1)=(x_{k+1},\alpha_{k+1})$, we may consider the combined path $\ell + \ell'$ of length $k+m$ with initial point $(x_1,\alpha_1)$ and endpoint $(y_{m+1},\beta_{m+1})$. Notice that if $(\alpha_1^{(j)})_{1 \le j \le k+1}$ is a pyramid associated to $\ell$, then $\ell + \ell'$ will admit a pyramid $(\beta_1^{(j)})_{j=1}^{k+m+1}$ with $\alpha_1^{(j)}=\beta_1^{(j)}$ for every $1 \le j \le k+1$. Finally, we observe that $(\ell + \ell')^{-1} = (\ell')^{-1} + \ell^{-1}$. \end{notation} \begin{defi} Let $Q \in \mathbb{N}$ and $c > 0$. We say $\mathcal{J}' \subseteq \mathcal{J}$ is a $(c,Q)$-\emph{regular} subset of $\mathcal{J}$ if $|\mathcal{J}'| \ge c |\mathcal{J}|$ and every $(x,\alpha) \in \mathcal{J}'$ is connected to $\ge c \frac{P^2}{(\log P)^2}$ other elements of $\mathcal{J}'$ by a split path mod $Q$ of length $1$. \end{defi} We finish this section with the following combinatorial estimate which plays the role of the Blakley-Roy inequality in \cite{MRT, MRTTZ} and turns out to be rather advantageous in practice. \begin{lema} \label{regular} There exists $c \sim 1$ such that, for $\gtrsim \frac{P'}{\log P'}$ primes $p' \in [P',2P']$, we can find a $(c,p')$-regular subset $\mathcal{J}_{p'}$ of $\mathcal{J}$. \end{lema} \begin{proof} For each prime $p' \in [P',2P']$ let $A_{p'}$ be the number of quadruples $((x_1,\alpha_1), (x_2,\alpha_2),p,q) \in \mathcal{J} \times \mathcal{J} \times \mathcal{P}_1 \times \mathcal{P}_2$ with $\| p \alpha_1 - q \alpha_2 \|_{p'} \lesssim PK/H$ and $|x_1/p - x_2/q| \lesssim H/(PK)$. By construction of $\mathcal{J}$, we know that $$\sum_{p' \in [P',2P']} A_{p'} \gtrsim \frac{X}{H} \left( \frac{P}{\log P} \right)^2 \frac{P'}{\log P'}.$$ By the prime number theorem, this means that we can find $\gtrsim \frac{P'}{\log P'}$ primes $p' \in [P',2P']$ with $A_{p'} \ge c_1 \frac{X}{H} \left( \frac{P}{\log P} \right)^2$, for some $c_1 \gtrsim 1$. In particular, we have $\ge c_1 \frac{X}{H} \left( \frac{P}{\log P} \right)^2$ paths mod $p'$ of length $1$ in $\mathcal{J}$. Fix now such a choice of $p'$. Let $\delta > 0$ be a sufficiently small constant and let $\mathcal{J}_1$ be the set of elements of $\mathcal{J}$ that are connected by a path mod $p'$ of length $1$ to at most $\delta \frac{P^2}{(\log P)^2}$ elements of $\mathcal{J}$. Recursively, let $\mathcal{J}_k$ be the set of elements of $\mathcal{J} \setminus \bigcup_{j=1}^{k-1} \mathcal{J}_{j}$ that are connected by a path mod $p'$ of length $1$ to at most $\delta \frac{P^2}{(\log P)^2}$ elements of $\mathcal{J} \setminus \bigcup_{j=1}^{k-1} \mathcal{J}_j$. Let $k_0$ be the largest integer such that $\mathcal{J}_{k_0}$ is nonempty (which exists, since $\mathcal{J}$ is finite). Clearly, this means that every element of $\mathcal{J}_{p'} :=\mathcal{J} \setminus \bigcup_{j=1}^{k_0} \mathcal{J}_j$ is connected by a path mod $p'$ of length $1$ to at least $\delta \frac{P^2}{(\log P)^2}$ elements of $\mathcal{J}_{p'}$. Furthermore, since given $(x_1,\alpha_1), (x_2,\alpha_2) \in \mathcal{J}$ there is at most one choice of $(p,q) \in [P,2P]$ with $|x_1/p - x_2/q| \lesssim H/(PK)$, the number of paths mod $p'$ of length $1$ in $\mathcal{J}_{p'}$ is at least $$ c_1 \frac{P^2}{(\log P)^2} \frac{X}{H} - 2 \delta \frac{P^2}{(\log P)^2} \sum_{j=1}^{k_0} |\mathcal{J}_j| \ge (c_1-4\delta) \frac{P^2}{(\log P)^2} \frac{X}{H}.$$ Since every element of $\mathcal{J}_{p'}$ belongs to at most $\lesssim \frac{P^2}{(\log P)^2}$ paths mod $p'$ of length $1$, if $\delta$ is sufficiently small it must be $|\mathcal{J}_{p'}| \gtrsim X/H$. The result follows. \end{proof} \section{Uniformity} In this section we will boost the results of Section \ref{phase} by exploiting the additional information that paths have on the relative sizes of the primes involved. We begin with the following observation in this direction. \begin{lema} \label{uni} There exists $c \gtrsim 1$ such that, given a path mod $Q$ in $\mathcal{J}$ of length $k \le c \log (X/H)$ consisting of primes $p_1, \ldots, p_k$, $q_1, \ldots, q_k$, we have $\frac{\prod_{i=1}^m q_i}{\prod_{i=1}^m p_i} \sim 1$ for every $1 \le m \le k$. \end{lema} \begin{proof} Fix the path and a choice of $1 \le m \le k$. Since for each $1 \le i \le m$ we have that there is some pair $x_i,x_{i+1}$ in $[X/(10K),2X/K]$ with $(q_i/p_i) x_i = x_{i+1} + O(H/K)$, we have that $$ \frac{q_i}{p_i} = \frac{x_{i+1}}{x_i} + O\left(\frac{H}{X} \right).$$ In particular, $$ \prod_{i=1}^m \frac{q_i}{p_i} = \prod_{i=1}^m \left( \frac{x_{i+1}}{x_i} + O \left(\frac{H}{X} \right) \right).$$ We expand the product in the right into $2^m$ terms, the first of which is $x_{m+1}/x_1 \sim 1$. Since for every $S \subseteq \left\{ 1, \ldots, m \right\}$ it is clearly $\prod_{i \in S} \frac{x_{i+1}}{x_i} \lesssim B^k$ with $B \sim 1$, if we choose $c \gtrsim 1$ sufficiently small in the statement of the current lemma we see that the contribution of the remaining terms is at most $$ \lesssim (2B)^{k} \frac{H}{X} \lesssim 1.$$ This proves that $\frac{\prod_{i=1}^m q_i}{\prod_{i=1}^m p_i} \lesssim 1$, while the reverse inequality follows from considering the 'inverted' path and applying the same reasoning. \end{proof} \begin{coro} \label{pd21} Let the hypothesis be as in Lemma \ref{uni}. Let $1 \le i < j \le k+1$. If $(x_i,\alpha_i), (x_j,\alpha_j)$ are $i$th and $j$th elements of the path, then $$|x_i \prod_{t=i}^{j-1} q_t/p_t - x_j| = O((j-i)H/K).$$ \end{coro} \begin{proof} This follows immediately upon iterating the relation $x_{t+1} = x_t q_t/p_t + O(H/K)$ and using Lemma \ref{uni}. \end{proof} We now insert this information into our previous estimate on pre-paths. \begin{lema} \label{alf} Let the hypothesis be as in Lemma \ref{uni}. Then, for every pyramid $(\alpha_1^{(j)})_{1 \le j \le k+1}$ associated to the path and every $2 \le j \le k+1$ and $1 \le m < j$, we have $$ \| \left( \prod_{i=m}^{j-1} q_i \right) \alpha_1^{(j)} - \alpha_1^{(m)} \|_Q \lesssim k \frac{K}{H} \frac{1}{\prod_{i=1}^{m-1} q_i}.$$ \end{lema} \begin{proof} By considering shorter paths inside the original path, it will suffice to cover the case in which $\alpha_1^{(j)}$ is the top element of a pyramid associated to a path of length $j-1$. For every $m \le t \le j-1$ we have that $q_t \alpha_1^{(t+1)} \equiv \alpha_1^{(t)} + c_t \, \, (\text{mod }Q)$, where by Lemma \ref{premanli} and Lemma \ref{uni} we know that $c_t$ can be taken to be a real number satisfying $$ | c_t \left( \prod_{i=1}^{t-1} q_i \right) | \lesssim \frac{P K}{H} \frac{1 }{p_{ \lfloor (t-1)/2 \rfloor+1}} \prod_{i=1}^{ \lfloor (t-1)/2 \rfloor} q_i/p_i \lesssim \frac{K}{H}.$$ In particular, $$ | c_t \left( \prod_{i=m}^{t-1} q_i \right) | \lesssim \frac{K}{H} \frac{1}{\prod_{i=1}^{m-1} q_i}.$$ The result then follows evaluating $\left( \prod_{i=m}^{j-1} q_i \right) \alpha_1^{(j)} = \left( \prod_{i=m}^{j-2} q_i \right) (\alpha_1^{(j-1)} + c_{j-1})$ recursively. \end{proof} This, in turn, leads to the following bound. \begin{lema} \label{morealf} Let the hypothesis be as in Lemma \ref{uni}. Then $$ \| \left( \prod_{i=1}^{j-1} p_i \prod_{i=j}^{k} q_i \right) \alpha_1^{(k+1)} - \alpha_j^{(1)} \|_Q \lesssim k \frac{K}{H},$$ for every $1 \le j \le k+1$. \end{lema} \begin{proof} By Lemma \ref{alf} we know that $$ \| \left( \prod_{i=j}^{k} q_i \right) \alpha_1^{(k+1)} - \alpha_1^{(j)} \|_Q \lesssim k \frac{K}{H} \frac{1}{\prod_{i=1}^{j-1} q_i}.$$ Considering the 'inverted' path that goes from $ \alpha_j^{(1)}$ to $\alpha_1^{(1)}$ and applying Lemma \ref{alf} again, we also see that $$ \| \left( \prod_{i=1}^{j-1} p_i \right) \alpha_1^{(j)} - \alpha_j^{(1)} \|_Q \lesssim k \frac{K}{H}.$$ The result then follows from the triangle inequality and Lemma \ref{uni}. \end{proof} \section{Connectedness} The purpose of this section is to show that a significant proportion of the elements of $\mathcal{J}$ can be connected with each other through a path of reasonable length. Our first observation is that once the initial point of a path of short length is fixed, there are only a few ways of reaching the same end point. \begin{lema} \label{olden} Let $1 \le k \le \frac{\log (X/(HB \log X))}{2 \log (2P)}$ for some sufficiently large constant $B \sim 1$. Then, for each pair $(x,\alpha), (y,\beta) \in \mathcal{J}$, there are at most $(2k)!$ paths mod $1$ of length $k$ that connect them. \end{lema} \begin{proof} It will suffice to show that if $(x,\alpha), (y, \beta) \in \mathcal{J}$ are connected by two paths consisting of primes $\left\{ p_1,\ldots,p_{k},q_1,\ldots,q_{k} \right\}$ and $\left\{ p_1',\ldots,p_{k}',q_1',\ldots,q_{k}' \right\}$, respectively, then these two sets of primes must coincide. Let us proceed by contradiction. By Corollary \ref{pd21} we know that $$ \left| \left( \prod_{i=1}^{k} \frac{q_i}{p_i} \right) x - y \right| \lesssim \frac{k H}{K}.$$ Similarly, considering the other path and using Lemma \ref{uni} we see that we must also have $$ \left| \left( \prod_{i=1}^{k} \frac{p_i'}{q'_i} \right) y - x \right| \lesssim \frac{k H}{K}.$$ Combining both estimates using Lemma \ref{uni} again and the triangle inequality, we obtain \begin{equation} \label{old28} \left| \prod_{i=1}^{k} \frac{q_i}{p_i} \prod_{i=1}^{k} \frac{p_i'}{q'_i} - 1\right| \lesssim \frac{kH}{X}. \end{equation} Since the primes in each path are all distinct by definition and the paths do not consist of exactly the same primes, the left-hand side cannot be $0$. But this means it must have size at least $(2P)^{-2k}$. The result then follows from our size hypothesis on $k$. \end{proof} We will be needing the following bound from \cite{MRTTZ}. \begin{lema}[\cite{MRTTZ}, Lemma 6.1] \label{products28} Let $r \in \mathbb{N}$, $A \ge 1$ and $P_0,N \ge 3$. Then, the number of $2r$-tuples of primes $(p_{1,1},\ldots,p_{1,r},p_{2,1},\ldots,p_{2,r}) \subseteq [P_0,2P_0]^{2r}$ satisfying $$ \left| \prod_{i=1}^r p_{1,i} - \prod_{i=1}^r p_{2,i} \right| \le A \frac{(2P_0)^{r}}{N}$$ is bounded by $$ \lesssim A r!^2 (2 P_0)^r \left( \frac{(2 P_0)^r}{N} + 1 \right).$$ \end{lema} The use of this lemma is that it will allow us to assume that no pair of paths mod $1$ of reasonable length that connect the same points share any primes in common. Essentially the same argument was already used in \cite{MRT, MRTTZ}. Precisely, we have the following bound on the number of exceptions. \begin{coro} \label{ls28} Let $(x,\alpha) \in \mathcal{J}$ and let $k$ be as in Lemma \ref{uni}. Then, the number of pairs of split paths mod $1$ of length $k$ with initial point $(x,\alpha)$, sharing the same end point and having at least one prime in common is bounded by $$ \lesssim (2k)!^2 \frac{(2 P)^{2k}}{\log P} \left( \frac{kH}{X}(2 P)^{2k-1} + 1 \right).$$ \end{coro} \begin{proof} Let $(p_1,\ldots,p_k,q_1,\ldots,q_k)$ and $(p_1',\ldots,p_k',q_1',\ldots,q_k')$ be the sets of distincts primes corresponding to two paths with initial point $(x,\alpha)$ and common end point $(y,\beta)$, for some $(y,\beta) \in \mathcal{J}$. Using Corollary \ref{pd21} as in the proof of Lemma \ref{olden} we obtain (\ref{old28}), which we may rewrite as $$ \left| \prod_{i=1}^{k} q_i p_i' - \prod_{i=1}^{k} p_i q_i' \right| \lesssim \frac{kH}{X} (2P)^{2k}.$$ Since $\mathcal{P}_1$ and $\mathcal{P}_2$ are disjoint and the paths share at least a prime in common, we can find a prime appearing in both products on the left-hand side. Factoring it out and applying Lemma \ref{products28} with $r=2k-1$ and $N=\frac{X}{kH}$, we see that the ordered tuple of $4k-2$ remaining primes belongs to a set $S$ of size $$ \lesssim (2k-1)!^2 (2 P)^{2k-1} \left( \frac{kH}{X}(2 P)^{2k-1} + 1 \right).$$ The result follows observing that, for each such element of $S$, there are at most $(2k)^2 \frac{P}{\log P}$ choices of ordered sets of primes $(p_1,\ldots,p_k,q_1,\ldots,q_k)$ and $(p_1',\ldots,p_k', q_1',\ldots,q_k')$ that can lead to it in the above manner. \end{proof} We now come to the main estimate of this section, which shows that a large number of elements $(y,\beta) \in \mathcal{J}$ can be connected to a fixed element $(x_0,\alpha_0) \in \mathcal{J}$ through two different paths of small length. In the next section, these two paths will be combined to a single path going through $(y,\beta)$ and having $(x_0,\alpha_0)$ as both its initial and end point. \begin{lema} \label{pairs} Let $k_0 = \lfloor \frac{\log (X/(HB \log X))}{2 \log (2P)} \rfloor$ for some sufficiently large $B \sim 1$. Fix $\delta > 0$. Assume $\varepsilon > 0$ in Lemma \ref{base} is sufficiently small with respect to $\delta$, $H \ge \exp(C (\log X)^{1/2} (\log \log X)^{1/2})$ with $C$ sufficiently large with respect to $\delta$ and $\varepsilon$ and $X$ is sufficiently large with respect to $\delta, \varepsilon$ and $C$. Then, there exist $\tau \sim 1$, $(x_0,\alpha_0) \in \mathcal{J}$ and $\mathcal{J}_0 \subseteq \mathcal{J}$ with $|\mathcal{J}_0| \gtrsim \frac{X}{H^{1+\delta}}$ such that the following holds. For every $(y,\beta) \in \mathcal{J}_0$ there exists some $Q_y$ which is the product of $\gtrsim \tau^{k_0} \frac{P'}{\log P'}$ different primes in $[P',2P']$ and such that there are two split paths mod $Q_y$ of length $k_0+2$ connecting $(x_0,\alpha_0)$ with $(y,\beta)$. Furthermore, the two paths share no prime in common. \end{lema} \begin{proof} During the proof we will let $c \sim 1$ denote a constant that may change at each occurrence. By Lemma \ref{regular} and the pigeonhole principle, we may locate some $(x_0,\alpha_0) \in \mathcal{J}$ that belongs to $\mathcal{J}_{p'}$ for $\gtrsim \frac{P'}{\log P'}$ primes $p' \in [P',2P']$. By regularity of $\mathcal{J}_{p'}$ this means that for each such $p'$ there are $\gtrsim c^{k_0} \left( \frac{P}{\log P} \right)^{2k_0+4} $ split paths mod $p'$ of length $k_0+2$ in $\mathcal{J}_{p'}$ with initial point $(x_0,\alpha_0)$. Since there are at most $\le D^{k_0} \left( \frac{P}{\log P} \right)^{2k_0+4}$ such paths mod $1$ in $\mathcal{J}$ for some $D \sim 1$, we conclude that there are $\gtrsim c^{k_0} \left( \frac{P}{\log P} \right)^{2k_0+4}$ split paths mod $1$ of length $k_0+2$ with initial point $(x_0,\alpha_0)$ that are also split paths mod $p'$ for $\gtrsim c^{k_0} \frac{P'}{\log P'}$ choices of $p' \in [P',2P']$ (which depend on each path). In particular, this means they are split paths mod $Q$, where $Q$ is the product of such primes. Write $\mathcal{R}$ for this set of paths. Write $\mathcal{R}_y$ for the paths in $\mathcal{R}$ that have endpoint $(y,\beta)$. By construction of $\mathcal{R}$ and the Cauchy-Schwarz inequality, we see that there must be $\gtrsim c^{k_0} |\mathcal{R}_y|^2$ pairs of paths in $\mathcal{R}_y$ such that both paths are split paths mod $p'$ for $\gtrsim c^{k_0} \frac{P'}{\log P'}$ primes $p' \in [P',2P']$ in common. In particular, they will both be split paths mod $Q$, with $Q$ the product of these primes. It follows that the number of pairs of paths in $\mathcal{R}$ having the same endpoint and with both being split paths mod $p'$ for $\gtrsim c^{k_0} \frac{P'}{\log P'}$ primes $p' \in [P',2P']$ is, by Cauchy-Schwarz, at least $$ \gtrsim c^{k_0} \frac{|\mathcal{R}|^2 H}{X} \gtrsim c^{k_0} \frac{H}{X} \left( \frac{P}{\log P} \right)^{4k_0+8}.$$ Since by our hypothesis on $\varepsilon, C$ and $X$ we have that $$ c^{-k_0} (2(k_0+2))!^2 k_0 (\log P)^{4 k_0+7} (\log X) < P^{1/2},$$ say, it also follows from Corollary \ref{ls28} and our choice of $k_0$ that at least half of these pairs of paths, say, do not share any primes from $[P,2P]$ in common. Let now $(y,\beta) \in \mathcal{J}$. If $\ell$ is a path mod $1$ in $\mathcal{R}_y$, it can be written as $\ell_0+\ell_1$, where $\ell_0$ is a path mod $1$ of length $k_0$ and $\ell_1$ is a path mod $1$ of length $2$ with end point $(y,\beta)$. There are $\lesssim \left( \frac{P}{\log P} \right)^4$ choices of $\ell_1$ and each such choice fixes the end point of $\ell_0$. On the other hand, by Lemma \ref{olden} we know that there are at most $(2k_0)!$ choices of $\ell_0$ with the same endpoint. We thus conclude that $(y,\beta)$ is the common endpoint of $\lesssim (2k_0)!^2 \left( \frac{P}{\log P} \right)^8$ of the pairs of paths we have constructed and therefore, by our lower bound on $|\mathcal{R}|$ and our choice of $k_0$, there are $$ \gtrsim \frac{c^{k_0}}{(2k_0)!^2} \frac{|\mathcal{R}|^2 H}{X}\left( \frac{P}{\log P} \right)^{-8} \gtrsim \frac{c^{k_0}}{(2k_0)!^2} \frac{X}{H (\log X)^2} P^{-4} (\log P)^{-4 k_0},$$ different elements $(y,\beta) \in \mathcal{J}$ that are connected to $(x_0,\alpha_0)$ by a pair of paths of the desired form. The result then follows, since $$ c^{-k_0} (2k_0)!^2 P^4 (\log P)^{4 k_0} (\log X)^2 < H^{\delta},$$ under our hypothesis on $\varepsilon, H$ and $X$. \end{proof} \section{Creating a global frequency} In this section we conclude the proof of Theorem \ref{1}. We begin by finding choices of $T_y$ and $q_y$ that work for each $(y,\beta) \in \mathcal{J}_0$ and then deduce that a common choice of $T$ and $q$ works for many of these elements. \begin{prop} Let the notation be as in Lemma \ref{pairs}. Then, for every $(y,\beta) \in \mathcal{J}_0$ there exists some $T_y \in \mathbb{R}$ with $|T_y| \lesssim X^2/H^{2-\delta}$ such that \begin{equation} \label{ff2} \alpha_0 \equiv \frac{a_y}{d_y} Q_y + \frac{T_y}{x_0} + O ( H^{\delta-1} ) \, \, (\text{mod }Q_y), \end{equation} and \begin{equation} \label{ff28} \beta \equiv \frac{b_y}{d_y} Q_y + \frac{T_y}{y} + O(H^{2\delta-1}) \, \, (\text{mod }Q_y), \end{equation} for some integers $a_y, b_y, d_y \lesssim H^{\delta}$. \end{prop} \begin{proof} Fix $(y,\beta) \in \mathcal{J}_0$. Write $\ell_1$ and $\ell_2$ for a pair of paths mod $Q_y$ of length $k:=k_0+2$ of the kind provided by Lemma \ref{pairs}. If we write $(p_1,\ldots,p_k, q_1,\ldots,q_k)$ and $(p_1',\ldots,p_k',q_1',\ldots,q_k')$ for the primes in $[P,2P]$ associated to $\ell_1$ and $\ell_2$ respectively, then $\ell_1+(\ell_2)^{-1}$ will be a path mod $Q_y$ of length $2k$ consisting of (distinct) primes $(p_1,\ldots,p_k,q_k',\ldots,q_1',q_1,\ldots,q_k,p_k',\ldots,p_1')$ having $(x_0,\alpha_0)$ as its initial and end point. If $\alpha_y \in \mathbb{R} / Q_y \mathbb{Z}$ is the top element of a pyramid associated to this path, then by Lemma \ref{morealf} applied with $j=1$ and $j=2k+1$ and the triangle inequality, we see that $$ \| \left( \prod_{i=1}^k p_i q_i' - \prod_{i=1}^k q_i p_i' \right) \alpha_y \|_{Q_y} \lesssim k \frac{K}{H},$$ and therefore \begin{align*} \alpha_y &\equiv \frac{u_y}{d_y} Q_y + O \left( k \frac{K}{H} \right) \, \, (\text{mod }Q_y) \\ &\equiv \frac{u_y}{d_y} Q_y + \frac{T_y}{x_0 \prod_{i=1}^k q_i p_i'} \, \, (\text{mod }Q_y), \end{align*} for some $T_y \in \mathbb{R}$ that by our choice of $k$ can be taken to satisfy $$|T_y| \lesssim k \frac{K}{H} x_0 \left( \prod_{i=1}^k q_i p_i' \right) \lesssim k \frac{K}{H} \frac{X}{K} \left(\frac{X}{H} (2P)^4 \right) \lesssim \frac{X^2}{H^{2-\delta}},$$ and for some integers $|u_y| \le d_y$ with $d_y$ dividing $\left| \prod_{i=1}^k p_i q_i' - \prod_{i=1}^k q_i p_i' \right|$. Using Corollary \ref{pd21} as in the deduction of (\ref{old28}) we see that $$ \left| \prod_{i=1}^k p_i q_i' - \prod_{i=1}^k q_i p_i' \right| \lesssim \frac{k H}{X} (2P)^{2k} \lesssim k (2P)^4,$$ by our choice of $k$. In particular, we have $|d_y| \lesssim H^{\delta}$. Applying Lemma \ref{morealf} with $j=1$ again, we deduce that we can find some integer $|a_y| \le d_y$ with $$ \alpha_0 \equiv \left( \prod_{i=1}^k q_i p_i' \right) \alpha_y + O \left( k \frac{K}{H} \right) \equiv \frac{a_y}{d_y} Q_y + \frac{T_y}{x_0} + O \left( H^{\delta-1} \right) \, \, (\text{mod }Q_y).$$ Similarly, applying Lemma \ref{morealf} with $j=k+1$, we obtain an integer $|b_y| \le d_y$ with $$ \beta \equiv \frac{b_y}{d_y} Q_y + \frac{T_y}{x_0} \frac{\prod_{i=1}^k p_i p_i' }{ \prod_{i=1}^k q_i p_i'} + O \left( H^{\delta-1} \right) \, \, (\text{mod }Q_y).$$ Removing the primes $p_1', \ldots, p_k'$ appearing in both numerator and denominator, and using that by Lemma \ref{uni} and Corollary \ref{pd21} we have $$ \left| \frac{T_y}{x_0} \frac{\prod_{i=1}^k p_i }{ \prod_{i=1}^k q_i} - \frac{T_y}{y} \right| \lesssim \frac{|T_y|}{(X/K)^2} \left|y- \left(\prod_{i=1}^k q_i/p_i \right) x_0 \right| \lesssim H^{2 \delta-1},$$ we obtain the result. \end{proof} We can now conclude the proof of Theorem \ref{3}. Let $\delta >0$ be sufficiently small and let the notation be as in Lemma \ref{pairs}. By Cauchy-Schwarz and the pigeonhole principle, we know from Lemma \ref{pairs} that can find some $c \sim 1$ and $(y_0,\beta_0) \in \mathcal{J}_0$ such that $$\tilde{Q}_y := \text{gcd}(Q_{y_0},Q_y) \ge (P')^{c^{k_0} \frac{P'}{\log P'}} \ge \exp(c^{k_0} P') $$ for $\ge c^{k_0} |J_0| \gtrsim H^{-\delta} |J_0|$ elements $(y,\beta) \in \mathcal{J}_0$. If we define $T := T_{y_0}$, $Q := Q_{y_0}$, $a := a_{y_0}$ and $d=d_{y_0}$, we see from (\ref{ff2}) that for each such $(y,\beta) \in \mathcal{J}_0$ we have $$ \alpha_0 \equiv \frac{a_y}{d_y} Q_y + \frac{T_y}{x_0} + O(H^{\delta-1}) \equiv \frac{a}{d} Q + \frac{T}{x_0} + O(H^{\delta-1}) \, \, (\text{mod }\tilde{Q}_y).$$ From our bounds on the individual quantities involved, it follows that it must necessarily be $$ \frac{a_y}{d_y} Q_y \equiv \frac{a}{d} Q \, \, (\text{mod }\tilde{Q}_y),$$ and therefore it must also be $|T - T_y| \lesssim |x_0|/H^{1-\delta}$. We then conclude from (\ref{ff28}) that \begin{equation} \label{llu2} \beta \equiv \frac{b_y}{d_y} Q_y + \frac{T}{y} + O(H^{2\delta-1}) \, \, (\text{mod }\tilde{Q}_y). \end{equation} By the pigeonhole principle, it follows that there is some $q \lesssim H^{\delta}$ such that (\ref{llu2}) holds with $d_y = q$ for $\gtrsim H^{-2 \delta} |\mathcal{J}_0| \gtrsim X/H^{1+3 \delta}$ elements $(y,\beta) \in \mathcal{J}$. For each such $(y,\beta)$, we know by construction of $\mathcal{J}$ and the triangle inequality that we can find some interval $[z,z+H^*] \subseteq [y,y+H/K]$ of length $H^* := H^{1-3 \delta}$ with $$ \sum_{z \le n < z+H^*} g(n) e(\beta n) \gtrsim H^*.$$ Since $|T/y-T/z| \lesssim H^{2\delta-1}$, it then follows that $$ \sum_{z \le n \le z+H^*} g(n) e(\frac{(n-z)b_y}{q} Q_y + \frac{(n-z)T}{z}) \gtrsim H^*.$$ Choosing $\delta$ sufficiently small with respect to $\rho$, this concludes the proof of Theorem \ref{3} and therefore of Theorem \ref{1}. \end{document}

\begin{document} \begin{abstract} We introduce the computable FS-jump, an analog of the classical Friedman--Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers). \end{abstract} \maketitle \section{Introduction} The backdrop for our study is the notion of computable reducibility of equivalence relations. If $E,F$ are equivalence relations on ${\mathbb N}$ we say $E$ is \emph{computably reducible} to $F$, written $E\leq F$, if there exists a computable function $f\colon{\mathbb N}\to{\mathbb N}$ such that for all $n$,$n'$ \[n\mathrel{E}n'\iff f(n)\mathrel{F}f(n')\text{.} \] This notion was first studied in both \cite{ershov,bernardi-sorbi}; it has recently garnered further study for instance in \cite{gao-gerdes,fokina-friedman-classes,fokina-friedman-etal,coskey-hamkins-miller} and numerous other works including those cited below. Computable reducibility of equivalence relations may be thought of as a computable analog to Borel reducibility of equivalence relations on standard Borel spaces. Here if $E,F$ are equivalence relations on standard Borel spaces $X,Y$ we say $E$ is \emph{Borel reducible} to $F$, written $E\leq_B F$, if there exists a Borel function $f\colon X\to Y$ such that $x\mathrel{E}x'\iff f(x)\mathrel{F}f(x')$. We refer the reader to \cite{gao} for the basic theory of Borel reducibility. One of the major goals in the study of computable reducibility is to compare the relative complexity of classification problems on a countable domain. In this context, if $E\leq F$ we say that the classification up to $E$-equivalence is no harder than the classification up to $F$-equivalence. For instance, classically the rank~$1$ torsion-free abelian groups (the subgroups of $\mathbb{Q}$) may be classified up to isomorphism by infinite binary sequences up to almost equality. Since this classification may be carried out in a way which is computable in the indices, there is a computable reduction from the isomorphism equivalence relation on c.e.\ subgroups of $\mathbb{Q}$ to the almost equality equivalence relation on c.e.\ binary sequences. A second major goal in this area is to study properties of the hierarchy of equivalence relations with respect to computable reducibility. The computable reducibility quasi-order is quite complex: for instance it is shown in \cite[Theorem~4.5]{bard} that it is at least as complex as the Turing degree order, and in \cite{andrews2021structure} that its theory is equivalent to second order arithmetic. In a portion of this article we will pay special attention to the sub-hierarchy consisting of just the ceers. An equivalence relation $E$ on ${\mathbb N}$ is called a \emph{ceer} if it is computably enumerable, as a set of pairs. Ceers were called positive equivalence relations in \cite{ershov}, subsequently named ceers in \cite{gao-gerdes}, and further studied in works such as \cite{andrews-etal-universal,andrews-sorbi-joins,andrews-sorbi-jumps}. As with other complexity hierarchies, it is natural to study operations such as jumps. One of the most important jumps in Borel complexity theory is the Friedman--Stanley jump, which is defined as follows. If $E$ is a Borel equivalence relation on the standard Borel space $X$, then the \emph{Friedman--Stanley jump} of $E$, denoted $E^{+}$, is the equivalence relation defined on $X^{\mathbb N}$ by \[x\mathrel{E^{+}}x'\iff\{[x(n)]_E:n\in{\mathbb N}\}=\{[y(n)]_E:n\in{\mathbb N}\}. \] Friedman and Stanley showed in \cite{friedman-stanley} that the jump is \emph{proper}, that is, if $E$ is a Borel equivalence relation, then $E<_B E^{+}$. Moreover they studied the hierarchy of iterates of the jump and showed that any Borel equivalence relation induced by an action of $S_{\infty}$ is Borel reducible to some iterated jump of the identity. In this article we study a computable analog of the Friedman--Stanley jump, called the computable FS-jump and denoted $E^{{\dot{+}}}$, in which the arbitrary sequences $x(n)$ are replaced by computable enumerations $\phi_e(n)$. In Section~2 we will give the formal definition of the computable FS-jump, and establish some of its basic properties. In Section~3 we show that the computable FS-jump is proper, that is, if $E$ is a hyperarithmetic equivalence relation, then $E<E^{{\dot{+}}}$. We do this by showing that any hyperarithmetic set is many-one reducible to some iterated jump of the identity, and establishing rough bounds on the descriptive complexity of these iterated jumps. In Section~4 we study the effect of the computable FS-jump on ceers. We show that if $E$ is a ceer with infinitely many classes, then $E^{{\dot{+}}}$ is bounded below by the identity relation $\mathsf{Id}$ on ${\mathbb N}$, and above by the equality relation $=^{ce}$ on c.e.\ sets. This leads to a natural investigation of the structure that the jump induces on the ceers, analogous to the study of the structure that the Turing jump induces on the c.e.\ degrees. For instance, we may say that a ceer $E$ is \emph{high for the computable FS-jump} if $E^{{\dot{+}}}$ is computably bireducible with $=^{ce}$. At the close of the section, we begin to investigate the question of which ceers are high for the computable FS-jump and which are not. In the final section we present several open questions arising from these results. \textbf{Acknowledgement.} This work includes a portion of the third author's master's thesis \cite{gianni-thesis}. The thesis was written at Boise State University under the supervision of the first and second authors. The authors would also like to thank the referee for suggesting numerous improvements. \section{Basic properties of reducibility and the jump} In this section we fix some notation, introduce the computable FS-jump, and exposit some of its basic properties. In this and future sections, we will typically use the letter $e$ for an element of ${\mathbb N}$ which we think of as an index for a Turing program. We will use $\phi_e$ for the partial computable function of index $e$, and $W_e$ for the domain of $\phi_e$. \begin{definition} \label{def:jump} Let $E$ be an equivalence relation on ${\mathbb N}$. The \emph{computable FS-jump} of $E$ is the equivalence relation on indices of c.e.\ subsets of ${\mathbb N}$ defined by \[e\mathrel{E^{{\dot{+}}}}e'\iff \{[\phi_e(n)]_E:n\in{\mathbb N}\}=\{[\phi_{e'}(n)]_E:n\in{\mathbb N}\}. \] When $E$ is defined on a countable set other than ${\mathbb N}$ (or computable subset thereof) we define $E^{{\dot{+}}}$ similarly, considering $\varphi_e$ to have its range in the domain of $E$; formally we may compose $\varphi_e$ with a computable bijection from ${\mathbb N}$ to the domain of $E$. Furthermore we define the iterated jumps $E^{{\dot{+}} n}$ inductively by $E^{{\dot{+}} 1}=E^{{\dot{+}}}$ and $E^{{\dot{+}} (n+1)}=(E^{ {\dot{+}} n})^{{\dot{+}}}$. \end{definition} We remark that we could also have defined $E^{{\dot{+}}}$ by working with domains $W_e$ rather than ranges $\ran(\phi_e)$. While each choice has conveniences, we use Definition~\ref{def:jump} due to its analogy with the Friedman--Stanley jump. We mention here that several other jumps of equivalence relations have been studied in the case of ceers. The halting jump and saturation jump were introduced in \cite{gao-gerdes}. The halting jump of $E$, denoted $E'$, is defined by setting $x \mathrel{E'} y$ iff $x = y \vee \phi_x(x) \downarrow \mathrel{E} \varphi_y(y) \downarrow$. The halting jump and its transfinite iterates are investigated extensively in \cite{andrews-sorbi-jumps}. The saturation jump of $E$, denoted $E^{+}$, is defined on finite subsets of ${\mathbb N}$ where $x$ and $y$ are saturation jump equivalent if their $E$-saturations are equal as sets. The saturation jump may be viewed as a finite-sequence version of the computable FS-jump. As observed in \cite{gao-gerdes} it is not always the case that $E < E'$ and $E < E^{+}$. It is worth noting that the computable FS-jump dominates the saturation jump under computable reducibility, and dominates the halting jump for ceers $E$. Unless explicitly stated otherwise, any further use of the word ``jump'' will refer to the computable FS-jump. We are now ready to establish some of the basic properties of the computable FS-jump. In the following, we let $\mathsf{Id}$ denote the identity equivalence relation on ${\mathbb N}$. It is worth noting that, although several of these results are direct analogues of results in Section~7 of \cite{gao-gerdes}, our results apply to an arbitrary equivalence relation $E$ and not only ceers (unless stated otherwise). \begin{proposition} \label{prop:monotone} For any equivalence relations $E$ and $F$ on ${\mathbb N}$ we have: \begin{enumerate} \item $E\leq E^{{\dot{+}}}$. \item If $E$ has only finitely many classes, then $E < E^{{\dot{+}}}$. \item If $E\leq F$ then $E^{{\dot{+}}}\leq F^{{\dot{+}}}$. \end{enumerate} \end{proposition} \begin{proof} (a) Let $f$ be a computable function such that for all $e$ we have that $\phi_{f(e)}$ is the constant function with value $e$. (To see that there is such a computable function $f$, one can either ``write a Turing program'' for the machine indexed by $f(e)$ or employ the s-m-n theorem. In the future we will not comment on the computability of functions of this nature.) Then $e\mathrel{E}e'$ if and only if $[e]_E=[e']_E$, if and only if $f(e)\mathrel{E^{{\dot{+}}}}f(e')$. (b) Note that if $E$ has $n$ classes, then $E^{{\dot{+}}}$ has $2^n$ classes. (c) This is similar to \cite[Theorem~8.4]{gao-gerdes}. Let $f$ be a computable reduction from $E$ to $F$. Let $g$ be a computable function such that $\phi_{g(e)}(n)=f(\phi_e(n))$. Then it is straightforward to verify that $g$ is a computable reduction from $E^{{\dot{+}}}$ to $F^{{\dot{+}}}$. \end{proof} Slightly less trivially we also note the following. \begin{proposition} \label{prop:double_plus} For any $E$ with infinitely many classes we have $\mathsf{Id}\leq E^{{\dot{+}}{\dot{+}}}$. \end{proposition} \begin{proof} We define a reduction function $f$ that works simultaneously for all equivalence relations $E$ with infinitely many classes. Given $n$, let $f(n)$ be a code for a machine such that the sequence of sets $S_i=\phi_{f(n)}(i)$ consists of all $n$-element subsets of ${\mathbb N}$. Clearly since $E^{{\dot{+}}{\dot{+}}}$ is reflexive we have that $n=n'$ implies $f(n)\mathrel{E^{{\dot{+}}{\dot{+}}}}f(n')$. Conversely suppose $n\neq n'$, and assume without loss of generality that $n<n'$. Then for all $i\in{\mathbb N}$ we have that $[\phi_{f(n)}(i)]_{E^{{\dot{+}}}}$ is a code for at most $n$-many $E$-classes. On the other hand since $E$ has infinitely many classes, there exists $i\in{\mathbb N}$ such that $[\phi_{f(n)}(i)]_{E^{{\dot{+}}}}$ is a code for exactly $n'$-many $E$-classes. It follows that $\{[\phi_{f(n)}(i)]_{E^{{\dot{+}}}}:i\in{\mathbb N}\}\neq\{[\phi_{f(n')}(i)]_{E^{{\dot{+}}}}:i\in{\mathbb N}\}$, or in other words, $f(n)\mathrel{\cancel{E^{{\dot{+}}{\dot{+}}}}}f(n')$. \end{proof} In the following, we let $E\oplus F$ denote the equivalence relation defined on ${\mathbb N}\times\{0,1\}$ by $(m,i)(E\oplus F)(n,j)$ iff $(i=j=0)\wedge(m\mathrel{E}n)$ or $(i=j=1)\wedge(m\mathrel{F}n)$. Finally, we let $E\times F$ denote the equivalence relation defined on ${\mathbb N}\times{\mathbb N}$ by $(m,n)(E\times F)(m',n')$ iff $m\mathrel{E}m'\wedge n\mathrel{F}n'$. \begin{proposition} \label{prop:product} $(E\oplus F)^{{\dot{+}}}$ is computably bireducible with $E^{{\dot{+}}}\times F^{{\dot{+}}}$. \end{proposition} \begin{proof} For the forward reduction, given an index $e$ for a function into ${\mathbb N}\times\{0,1\}$, let $\phi_{e_0}(n)=m$ if $\phi_{e}(n)=(m,0)$ and let $\phi_{e_1}(n)=m$ if $\phi_{e}(n)=(m,1)$; $\phi_{e_i}$ is undefined otherwise. Then the map $e\mapsto(e_0,e_1)$ is a reduction from $(E\oplus F)^{{\dot{+}}}$ to $E^{{\dot{+}}}\times F^{{\dot{+}}}$. For the reverse reduction, given a pair of indices $(e_0,e_1)$ we define $\phi_e(2n)=(\phi_{e_0}(n),0)$ and $\phi_e(2n+1)=(\phi_{e_1}(n),1)$. Once again it is easy to verify $(e_0,e_1)\mapsto e$ is a reduction from $E^{{\dot{+}}}\times F^{{\dot{+}}}$ to $(E\oplus F)^{{\dot{+}}}$. \end{proof} In the next result we will briefly consider the connection between the computable FS-jump and the restriction of the classical FS-jump to c.e.\ sets. In the literature, the $n$th iterated classical FS-jump of $\mathsf{Id}$ is usually denoted $F_n$. For our purposes it will be convenient to regard each $F_n$ as an equivalence relation on $\mathcal P({\mathbb N})$. Thus we officially define $F_1$ as the equality relation on $\mathcal P({\mathbb N})$. Letting $\langle\cdot,\cdot\rangle$ be the usual pairing function ${\mathbb N}^2\to{\mathbb N}$, and let $A^{[n]}$ denote the $n$th ``column'' of $A$, that is, $A^{[n]}=\{p\in{\mathbb N}:\langle n,p\rangle\in A\}$. We then officially define $A\mathrel{F_2}B$ iff $\{A^{[n]}:n\in{\mathbb N}\}=\{B^{[n]}:n\in{\mathbb N}\}$. Similarly for all $n$ we can officially define $F_n$ on $\mathcal P({\mathbb N})$ by means of a fixed uniformly computable family of bijections between ${\mathbb N}^n$ and ${\mathbb N}$. So defined, $F_n$ is naturally Borel bireducible with the literal $n$th iterated classical FS-jump of $\mathsf{Id}$. Next, recall from \cite{coskey-hamkins-miller} that for any equivalence relation $E$ on $\mathcal P({\mathbb N})$ we can define its \emph{restriction to c.e.\ sets} $E^{ce}$ on ${\mathbb N}$ by \[e\mathrel{E}^{ce}e'\iff W_e\mathrel{E}W_{e'}. \] In particular, $(F_1)^{ce}$ is $=^{ce}$, which figures prominently in the theory of computable reducibility. We are now ready to state the following. \begin{proposition} \label{prop:idplus} For any $n$, we have that $\mathsf{Id}^{{\dot{+}} n}$ is computably bireducible with $(F_n)^{ce}$. \end{proposition} \begin{proof}[Proof sketch] For $n=1$, we need to show that $\mathsf{Id}^{{\dot{+}}}$ is computably bireducible with $=^{ce}$, which amounts to the effective equivalence of a c.e. set being either the domain or the range of a partial computable function. Namely, let $f$ and $g$ be computable functions so that $W_{f(e)}=\ran(\varphi_e)$ and $\ran(\varphi_{g(e)})=W_e$; then $f$ and $g$ provide the respective reductions. For the induction step, it is sufficient to show that for any $n$ we have that $((F_n)^{ce})^{{\dot{+}}}$ is computably bireducible with $(F_{n+1})^{ce}$. For notational simplicity, we briefly illustrate this just in the case when $n=1$. For the reduction from $((F_1)^{ce})^{{\dot{+}}}$ to $(F_2)^{ce}$, we define $f$ to be a computable function such that for all $n$ we have $(W_{f(e)})^{[n]}=W_{\phi_e(n)}$. For the reduction from $(F_2)^{ce}$ to $((F_1)^{ce})^{{\dot{+}}}$, we define $g$ to be a computable function such that for all $n$ we have $(W_{\phi_{g(e)}})^{[n]}=(W_e)^{[n]}$. \end{proof} We shall make frequent use of the particular case that $\mathsf{Id}^{{\dot{+}}}$ is computably bireducible with $=^{ce}$. To conclude the section, we define transfinite iterates of the computable FS-jump. The transfinite jumps allow one to extend results such as the previous proposition into the transfinite, and they also play a key role in the next section. For the definition, recall that Kleene's $\mathcal O$ consists of notations for ordinals and is defined as follows: $1\in\mathcal O$ is a notation for $0$, if $a\in\mathcal O$ is a notation for $\alpha$ then $2^a$ is a notation for $\alpha+1$, and if for all $n$ we have $\phi_e(n)$ is a notation for $\alpha_n$ with the notations increasing in $\mathcal{O}$ with respect to $n$, then $3\cdot 5^e$ is a notation for $\sup_n\alpha_n$. We refer the reader to \cite{sacks} for background on $\mathcal O$. \begin{definition} We define $E^{{\dot{+}} a}$ for $a\in\mathcal O$ recursively as follows. \begin{align*} E^{{\dot{+}} 1}&=E\\ E^{{\dot{+}} 2^b}&=(E^{{\dot{+}} b})^{{\dot{+}}}\\ E^{{\dot{+}} 3\cdot 5^e}&=\{(\langle m,x\rangle,\langle n,y\rangle):(m=n)\wedge (x\mathrel{E}^{{\dot{+}} \phi_e(m)}y)\} \end{align*} \end{definition} We remark that it is straightforward to extend Proposition~\ref{prop:idplus} into the transfinite as follows. Given a notation $a\in\mathcal O$ for $\alpha$, we may use $a$ to define an equivalence relation $F_a$ on $\mathcal P({\mathbb N})$ which is Borel bireducible with the $\alpha$-iterated FS-jump $F_\alpha$. We then have that $\mathsf{Id}^{{\dot{+}} a}$ is computably bireducible with $(F_a)^{ce}$. We do not know, however, whether $\mathsf{Id}^{{\dot{+}} a}$ and $\mathsf{Id}^{{\dot{+}} a'}$ are computably bireducible when $a$ and $a'$ are different notations for the same ordinal. The following propositions will be used in the next section. \begin{proposition} \label{prop:closed} If $E^{{\dot{+}}} \leq E$ then for any $a \in \mathcal{O}$ we have $E^{{\dot{+}} a} \leq E$. \end{proposition} \begin{proof} We proceed by recursion on $a \in \mathcal{O}$. It follows from our hypothesis together with Proposition~\ref{prop:monotone}(b) that $E$ has infinitely many classes. By Proposition~\ref{prop:double_plus}, we have $\mathsf{Id} \leq E^{{\dot{+}}{\dot{+}}}$ and hence $\mathsf{Id} \leq E$. From this we can see that $E\times\mathsf{Id}\leq E^{{\dot{+}}{\dot{+}}}$ as follows. Suppose $h\colon\mathsf{Id} \leq E$ and define $h'$ by arranging for $W_{h'(e,n)}=\{0,1\}$, $\phi_{h'(e,n)}(0)=$ a code for $\{e\}$, and $\phi_{h'(e,n)}(1)=$ a code for $\{h(n),h(n+1)\}$. Since $h(n)$ and $h(n+1)$ are distinct for each $n$, we can distinguish $\{h(n),h(n+1)\}$ from $\{e\}$ and recover $e$ and $n$ from $h'(e,n)$, so that $h'\colon E \times \mathsf{Id} \leq E^{{\dot{+}}{\dot{+}}}$. Hence we have $E \times \mathsf{Id} \leq E$, and we may fix a computable reduction function $g\colon E \times \mathsf{Id} \leq E$. Now let $f\colon E^{{\dot{+}}} \leq E$ and define uniformly $f_a : E^{{\dot{+}} a} \leq E$ as follows. Let $f_1$ be the identity map. Given $f_a:E^{{\dot{+}} a}\leq E $ apply Proposition \ref{prop:monotone}(c) to get $f^+_a:(E^{{\dot{+}} a})^{{\dot{+}}}\leq E^{{\dot{+}}}$, then define $f_{2^a}=f\circ f^+_a$. To define $f_{3 \cdot 5^e}$ it suffices to find a reduction from $E^{{\dot{+}} 3 \cdot 5^e}$ to $E \times \mathsf{Id}$ and compose with $g$; this follows from the fact that we have each $E^{{\dot{+}} \varphi_e(n)}$ uniformly reducible to $E$ by the effectiveness of the recursion. \end{proof} \begin{proposition} If $E \times \mathsf{Id} \leq E$ then for any $a \in \mathcal{O}$ we have $E^{{\dot{+}} a} \times \mathsf{Id} \leq E^{{\dot{+}} a}$. \end{proposition} \begin{proof} We proceed by recursion on $a \in \mathcal{O}$, noting that the induction will produce the reduction functions effectively from $a$. Suppose first that $E^{{\dot{+}} a} \times \mathsf{Id} \leq E^{{\dot{+}} a}$. Then $E^{{\dot{+}} 2^a} \times \mathsf{Id} = (E^{{\dot{+}} a})^{{\dot{+}}} \times \mathsf{Id} \leq (E^{{\dot{+}} a})^{{\dot{+}}} \times \mathsf{Id}^{{\dot{+}}}$, which is bireducible with $(E^{{\dot{+}} a} \oplus \mathsf{Id})^{{\dot{+}}}$ by Proposition~\ref{prop:product}. Since the hypothesis implies $\mathsf{Id} \leq E$, this is reducible to $(E^{{\dot{+}} a} \oplus E^{{\dot{+}} a})^{{\dot{+}}}$, which is reducible to $(E^{{\dot{+}} a} \times \mathsf{Id})^{{\dot{+}}}$, and hence reducible to $(E^{{\dot{+}} a})^{{\dot{+}}}=E^{{\dot{+}} 2^a}$. For $E^{{\dot{+}} 3 \cdot 5^e}$, we assume that $E^{{\dot{+}} \varphi_e(m)} \times \mathsf{Id} \leq E^{{\dot{+}} \varphi_e(m)}$ uniformly in $m$, from which we see that $E^{{\dot{+}} 3 \cdot 5^e} \times \mathsf{Id} \leq (E \times \mathsf{Id})^{{\dot{+}} 3 \cdot 5^e} \leq E^{{\dot{+}} 3 \cdot 5^3}$. \end{proof} Since a computable bijection from ${\mathbb N} \times {\mathbb N}$ to ${\mathbb N}$ shows $\mathsf{Id} \times \mathsf{Id} \leq \mathsf{Id}$, we get: \begin{corollary} \label{cor:absorbs_id} For any $a \in \mathcal{O}$ we have $\mathsf{Id}^{{\dot{+}} a} \times \mathsf{Id} \leq \mathsf{Id}^{{\dot{+}} a}$. \end{corollary} \section{Properness of the jump} In this section we establish the following main result. \begin{theorem} \label{thm:proper} If $E$ is a hyperarithmetic equivalence relation on ${\mathbb N}$, then $E<E^{{\dot{+}}}$. \end{theorem} Since we have $E \leq E^{{\dot{+}}}$ for each equivalence relation $E$, this amounts to showing that no hyperarithmetic equivalence relation is a fixed point of the computable FS-jump. We will in fact establish the following stronger result. \begin{theorem} \label{thm:fixed_points} Let $E$ be an equivalence relation on ${\mathbb N}$ which is a fixed point for the computable FS-jump. Then $E$ is an upper bound in the $m$-degrees for all hyperathmetic sets. \end{theorem} The proof will proceed by showing that iterated jumps of the identity have cofinal descriptive complexity among hyperarithmetic sets. Specifically, we will show that every hyperarithmetic set is many-one reducible to $\mathsf{Id}^{{\dot{+}} a}$ for some $a \in \mathcal{O}$. The proof will involve an induction on the hyperarithmetic hierarchy, and we will utilize a particular type of many-one reduction which we now introduce. \begin{definition} Given a relation $E$ on ${\mathbb N}$, we define the relation $\subseteq_E$ by setting $e \subseteq_E e'$ if the following holds: \[\forall n [ \phi_e(n) \downarrow \ {\mathbb R}ightarrow \exists m ( \phi_{e'}(m) \downarrow \wedge \phi_e(n) \mathrel{E} \phi_{e'}(m))]. \] \end{definition} We write $e \supseteq_E e'$ when $e' \subseteq_E e$. Note that when $E$ is an equivalence relation, $\subseteq_E$ is a quasi-order and we have $e\mathrel{E^{{\dot{+}}}}e'$ iff $e\subseteq_E e'$ and $e'\subseteq_E e$. \begin{definition} Given a set $P$ and an equivalence relation $E$, we say that \emph{$P$ is subset-reducible to $E^{{\dot{+}}}$} if there is a computable function $h$ and $e_0 \in {\mathbb N}$ so that for all $n$ we have $h(n) \subseteq_E e_0$, and $P(n) \iff h(n) \mathrel{E^{{\dot{+}}}} e_0$. We call the pair $(h, e_0)$ a \emph{subset-reduction}. \end{definition} If $P$ is subset-reducible to $E^{{\dot{+}}}$ then it is clearly many-one reducible; we will show that every hyperarithmetic set is subset-reducible to some iterated jump of $\mathsf{Id}$. Since in general $P$ may be many-one reducible to $E$ without $P^c$ being reducible to an iterated jump of $E$, we wish to only use ``positive'' induction steps, i.e., an inductive construction of the hyperarithmetic sets starting from computable sets and involving only effective unions and intersections. Also, since we need to uniformly produce reducing functions throughout the construction of a set, we want to consider the entire construction at once. To this end we introduce the notion of a computable Borel code for a hyperarithmetic set. There are many different presentations of computable Borel codes, all of which give the same collection of sets; the following definition is a slight variation of that given in Chapter~27 of \cite{miller}. \begin{definition} A \emph{computable Borel code} is a pair $(T,f)$ where $T$ is a computable well-founded tree on ${\mathbb N}$ so that $t \smallfrown n \in T$ for all $n$ for non-terminal nodes $t$, and $f$ is a computable function from the terminal nodes of $T$ to ${\mathbb N}$. Given a computable Borel code $(T,f)$, the set $B(T,f)$ is defined by recursion on $t \in T$ as follows. If $t$ is a terminal node, then $B_t(T,f)=\ran \phi_{f(t)}$, and if $t$ is not a terminal node, then $B_t(T,f)=\{ n: \forall p \exists q (n \in B_{t \smallfrown \langle p,q \rangle}(T,f) )\}$. We let $B(T,f)=B_{\emptyset}(T,f)$. \end{definition} The following characterization then follows from the fact that a set is hyperarithmetic if and only if it is $\Delta^1_1,$ together with the Kleene Separation Theorem and the hyperarithmetic codes used in its proof (see, e.g., \cite[Chapter II]{sacks} and \cite[Theorem 27.1]{miller}). \begin{theorem} A set $B$ is hyperarithmetic if and only if there is a computable Borel code $(T,f)$ such that $B=B(T,f)$. \end{theorem} From this characterization, we see that it will suffice to consider three types of inductive steps as described in Lemma~\ref{lem:proper_union}, Lemma~\ref{lem:proper_intersection}, and Lemma~\ref{lem:proper_limit}. We begin by considering the case of a $\Sigma^0_3$ set because it allows us to produce slightly better complexity bounds, as discussed later in this section, and introduces key ideas used in the subsequent proofs. In the following, we will say that $e$ is \emph{an index for an enumeration} of the c.e.\ set $W$ if $\ran \phi_e = W$. We will repeatedly utilize the fact that we can effectively enumerate the $E^{{\dot{+}}}$-classes of the c.e. supersets of a given set, i.e., $\{[e]_{E^{{\dot{+}}}} : e \supseteq_{E} e_0\} = \{[W_i \cup e_0]_{E^{{\dot{+}}}} : i \in {\mathbb N}\}$, where we use $W_i \cup e_0$ to denote an index for an enumeration of $\ran \phi_{e_0} \cup W_i$. The analogous statement with $\subseteq_E$ replacing $\supseteq_E$ does not hold, as illustrated in Proposition~\ref{prop:counterexample} below, which is why we repeat this process twice to handle existential quantification. Recall that $=^{ce}$ is computably bireducible with $\mathsf{Id}^{{\dot{+}}}$. \begin{lemma} \label{lem:proper_base} Let $P$ be $\Sigma^0_3$. Then $P$ is subset-reducible to $\left(=^{ce}\right)^{{\dot{+}}}$. \end{lemma} \begin{proof} Choose $i_0$ with $P(n) \iff \exists q \forall m\ \phi_{i_0}(\langle q,m,n\rangle) \downarrow$, so that \[ P(n) \iff \exists q\ \{ \langle q,m,n \rangle : m \in {\mathbb N} \} \subset W_{i_0} .\] Letting $W_{g(q,n)} = W_{i_0} \cup \{ \langle q,m,n\rangle : m \in {\mathbb N}\}$, we then have \[ P(n) \iff \exists q\ W_{g(q,n)} = W_{i_0} ,\] with $W_{g(q,n)} \supset W_{i_0}$ for all $q$ and $n$. Then \[ P(n) \iff \exists q\ \{W_i \cup W_{g(q,n)} : i \in {\mathbb N}\} = \{W_i \cup W_{i_0} : i \in {\mathbb N} \}, \] with $\{W_i \cup W_{g(q,n)} : i \in {\mathbb N}\} \subset \{W_i \cup W_{i_0} : i \in {\mathbb N} \}$ for all $q$ and $n$. Hence \[ P(n) \iff \{ W_i \cup W_{g(q,n)} : i \in {\mathbb N} \wedge q \in {\mathbb N}\} = \{W_i \cup W_{i_0} : i \in {\mathbb N} \}, \] with $\{ W_i \cup W_{g(q,n)} : i \in {\mathbb N} \wedge q \in {\mathbb N}\} \subset \{W_i \cup W_{i_0} : i \in {\mathbb N} \}$ for all $q$ and $n$, and equality holding only when there is $q$ with $W_{g(q,n)} = W_{i_0}$. Let $h(n)$ be such that $\phi_{h(n)}(\langle i,q\rangle)$ is an index for an enumeration of $W_i \cup W_{g(q,n)}$ and let $e_0$ be such that $\phi_{e_0}(i)$ is an index for an enumeration of $W_i \cup W_{i_0}$. Then we have $P(n) \iff h(n) \mathrel{\left(=^{ce}\right)^{{\dot{+}}}} e_0$, with $h(n) \subseteq_{=^{ce}} e_0$ for all $n$. \end{proof} \begin{lemma} \label{lem:proper_union} Suppose $Q$ is subset-reducible to $E^{{\dot{+}}}$, and $P(n) \iff \exists q\ Q(\langle q,n\rangle)$. Then $P$ is subset-reducible to $E^{{\dot{+}}{\dot{+}}{\dot{+}}}$. Moreover, there are computable functions $\Psi$ and $\chi$ so that if $(\phi_i,d_0)$ is a subset-reduction from $Q$ to $E^{{\dot{+}}}$, then $(\phi_{\Psi(i)}, \chi(d_0))$ is a subset-reduction from $P$ to $E^{{\dot{+}} {\dot{+}} {\dot{+}}}$. \end{lemma} \begin{proof} Let $(f,d_0)$ be a subset-reduction from $Q$ to $E^{{\dot{+}}}$. We then have: \begin{align*} P(n) &\iff \exists q\ Q(\langle q,n\rangle) \\ & \iff \exists q\ f(\langle q,n \rangle) \mathrel{E^{{\dot{+}}}} d_0 \\ & \iff \exists q\ \{ [m]_E : m \in \ran \phi_{f(\langle q,n \rangle)} \} = \{ [m]_E : m \in \ran \phi_{d_0}\} \\ & \iff \exists q\ \{[e]_{E^{{\dot{+}}}} : e \supseteq_E f(\langle q,n\rangle) \} = \{[e]_{E^{{\dot{+}}}} : e \supseteq_E d_0\}, \end{align*} with $\{[e]_{E^{{\dot{+}}}} : e \supseteq_E f(\langle q,n\rangle) \} \supset \{[e]_{E^{{\dot{+}}}} : e \supseteq_E d_0\}$ for all $q$ and $n$. Let $j$ be such that $\phi_{j(n,q)}(i)$ is an index for an enumeration of $W_i \cup \ran \phi_{f(\langle q,n\rangle)}$ for each $n$, $q$, and $i$, and let $j_0$ be such that $\phi_{j_0}(i)$ is an index for an enumeration of $W_i \cup \ran \phi_{d_0}$ for each $i$. Then we have \[ P(n) \iff \exists q\ j(n,q) \mathrel{E^{{\dot{+}} {\dot{+}}}} j_0 ,\] with $j(n,q) \supseteq_{E^{{\dot{+}}}} j_0$ for all $n$ and $q$. Hence \[ P(n) \iff \exists q\ \{[e]_{E^{{\dot{+}} {\dot{+}} }} : e \supseteq_{E^{{\dot{+}}}} j(n,q) \} = \{[e]_{E^{{\dot{+}} {\dot{+}} }} : e \supseteq_{E^{{\dot{+}}}} j_0\},\] with $\{[e]_{E^{{\dot{+}} {\dot{+}} }} : e \supseteq_{E^{{\dot{+}}}} j(n,q) \} \subset \{[e]_{E^{{\dot{+}} {\dot{+}} }} : e \supseteq_{E^{{\dot{+}}}} j_0\}$ for all $n$ and $q$. Hence we also have $\{[e]_{E^{{\dot{+}} {\dot{+}} }} :\exists q\ e \supseteq_{E^{{\dot{+}}}} j(n,q) \} \subset \{[e]_{E^{{\dot{+}} {\dot{+}} }} : e \supseteq_{E^{{\dot{+}}}} j_0\}$ for all $n$, and we claim that \[ P(n) \iff \{[e]_{E^{{\dot{+}} {\dot{+}} }} : \exists q\ e \supseteq_{E^{{\dot{+}}}} j(n,q) \} = \{[e]_{E^{{\dot{+}} {\dot{+}} }} : e \supseteq_{E^{{\dot{+}}}} j_0\}.\] To see this, note if equality holds then $[j_0]_{E^{{\dot{+}}}}$ must be an element of the left-hand set, so there must be $q_0$ with $j_0 \supseteq_{E^{{\dot{+}}}} j(n,q_0)$. Since $j(n,q) \supseteq_{E^{{\dot{+}}}} j_0$ for all q, we we thus have $j(n,q_0) \mathrel{E^{{\dot{+}} {\dot{+}}}} j_0$, so that $P(n)$ holds. Finally, let $h$ be such that $\phi_{h(n)}(\langle i,q\rangle)$ is an index for an enumeration of $W_i \cup \ran \phi_{j(n,q)}$ for each $n$, $q$, and $i$, an let $e_0$ be such that $\phi_{e_0}(i)$ is an index for an enumeration of $W_i \cup \ran \phi_{j_0}$ for each $i$. Then $h(n) \subseteq_{E^{{\dot{+}} {\dot{+}}}} e_0$ for each $n$, and $P(n) \iff h(n) \mathrel{E^{{\dot{+}} {\dot{+}} {\dot{+}}}} e_0$, so that $(h,e_0)$ is a subset-reduction of $P$ to $E^{{\dot{+}} {\dot{+}} {\dot{+}}}$. The construction from $h$ and $e_0$ is uniform in $f$ and $d_0$, so we can produce the functions $\Psi$ and $\chi$ as described. \end{proof} \begin{lemma} \label{lem:proper_intersection} Suppose $E \times \mathsf{Id} \leq E$, $Q$ is subset-reducible to $E^{{\dot{+}}}$, and $P(n) \iff \forall p Q(\langle p,n\rangle)$. Then $P$ is subset-reducible to $E^{{\dot{+}}}$. Moreover, there are computable functions $\Psi$ and $\chi$ so that if $(\phi_i,d_0)$ is a subset-reduction from $Q$ to $E^{{\dot{+}}}$, then $(\phi_{\Psi(i)}, \chi(d_0))$ is a subset-reduction from $P$ to $E^{{\dot{+}}}$. \end{lemma} \begin{proof} Let $(f,d_0)$ be a subset-reduction from $Q$ to $E^{{\dot{+}}}$, and let $g$ be a reduction from $E \times \mathsf{Id}$ to $E$. Define $h$ so that $h(n)$ is an index for an enumeration of $ \{ g(m,p) : m \in \ran \phi_{f(\langle p,n \rangle)} \wedge p \in {\mathbb N}\} $ and let $e_0$ be an index for an enumeration of $ \{ g(m,p) : m \in \ran \phi_{d_0} \wedge p \in {\mathbb N}\}$. For all $n$ and $p$ we have $f(\langle p,n\rangle) \subseteq_E d_0$, so that $h(n) \subseteq_E e_0$, and for all $n$ we have: \begin{align*} P(n) &\iff \forall p\ Q(\langle p,n\rangle) \\ & \iff \forall p\ f(\langle p,n \rangle) \mathrel{E^{{\dot{+}}}} d_0 \\ & \iff \forall p\ \{ [m]_E : m \in \ran \phi_{f(\langle p,n \rangle)} \} = \{ [m]_E : m \in \ran \phi_{d_0}\} \\ & \iff \{ [(m,p)]_{E \times \mathsf{Id}} : m \in \ran \phi_{f(\langle p,n \rangle)} \wedge p \in {\mathbb N}\} = \\ &\qquad\qquad\qquad \{ [(m,p)]_{E \times \mathsf{Id}} : m \in \ran \phi_{d_0} \} \\ & \iff \{ [g(m,p)]_E : m \in \ran \phi_{f(\langle p,n \rangle)} \wedge p \in {\mathbb N}\} = \\ &\qquad\qquad\qquad \{ [g(m,p)]_E : m \in \ran \phi_{d_0} \wedge p \in {\mathbb N} \} \\ & \iff h(n) \mathrel{E^{{\dot{+}}}} e_0 , \end{align*} so that $(h,e_0)$ is a subset-reduction from $P$ to $E^{{\dot{+}}}$. The construction of $h$ and $e_0$ is uniform, so we can produce the functions $\Psi$ and $\chi$ as described. \end{proof} \begin{lemma} \label{lem:proper_limit} Suppose $a=3 \cdot 5^e \in \mathcal{O}$, and for each $n$ we have that $(h_n,e_n)$ is a subset-reduction from $A^{[n]}=\{p\in{\mathbb N}:\langle n,p\rangle\in A\}$ to $(E^{{\dot{+}} \phi_e(n)})^{{\dot{+}}}$, with the sequences $\langle h_n \rangle_{n \in {\mathbb N}}$ and $\langle e_n \rangle_{n \in {\mathbb N}}$ computable. Then $A$ is subset-reducible to $(E^{{\dot{+}} a})^{{\dot{+}}}$. Moreover, there are computable functions $\Psi$ and $\chi$ so that $(\Psi(\langle h_n \rangle_{n \in {\mathbb N}}), \chi(\langle e_n \rangle_{n \in {\mathbb N}}))$ provides the subset-reduction. \end{lemma} \begin{proof} Define $h$ so that for each $n$ and $p$, $h(\langle n,p\rangle)$ is an index for an enumeration of $ \{\langle n, q \rangle : q \in \ran \phi_{h_n(p)} \} \cup \{ \langle m, q \rangle : q \in \ran \phi_{e_m} \wedge m \neq n\}$ , and let $e$ be an index for an enumeration of $\{ \langle m, q \rangle : q \in \ran \phi_{e_m} \wedge m \in {\mathbb N} \}$. Then $(h,e)$ provides the desired subset-reduction since $\{\langle n, q \rangle : q \in \ran \phi_{h_n(p)} \} \subseteq_{E^{{\dot{+}} a}} \{ \langle n, q \rangle : q \in \ran \phi_{e_n}\}$ for all $n$ and $p$, with $\{\langle n, q \rangle : q \in \ran \phi_{h_n(p)} \} \mathrel{(E^{{\dot{+}} a})^{{\dot{+}}}} \{ \langle n, q \rangle : q \in \ran \phi_{e_m}\}$ iff $p\in A^{[n]}$. The existence of $\Psi$ and $\chi$ is clear. \end{proof} We now prove the key result for establishing properness of the jump. \begin{theorem} \label{thm:key_proper} For each hyperarithmetic set $B$ there is $a \in \mathcal{O}$ with $B \leq_m \mathsf{Id}^{{\dot{+}} a}$. \end{theorem} \begin{proof} We will show that for each computable Borel code $(T,f)$ there is $a_T \in \mathcal{O}$ so that $B(T,f) \leq_m \mathsf{Id}^{{\dot{+}} a_T}$. For notational convenience we let $B_t=B_t(T,f)$ for $t \in T$. We will recursively define $a_t \in \mathcal{O}$ and let $E_t = \mathsf{Id}^{{\dot{+}} a_t}$ and establish by effective induction on $t \in T$ that $B_t$ is subset-reducible to $E_t^{{\dot{+}}}$ via $(h_t,e_t)$, with computable maps $t \mapsto a_t$, $t \mapsto h_t$, and $t \mapsto e_t$. For $t$ a terminal node we have $B_t=\ran \phi_{f(t)}$ and we set $a_t=1$ so $E_t=\mathsf{Id}$ and $E_t^{{\dot{+}}}$ is bireducible with $=^{ce}$. Fix a single $e_t$ for all terminal $t$ so that $\ran \phi_{e_t}={\mathbb N}$, and let $h_t(n)$ be such that $\ran \phi_{h_t(n)} = {\mathbb N}$ if $n \in \ran \phi_{f(t)}$ and $\ran \phi_{h_t(n)} = \emptyset$ if $n \notin \ran \phi_{f(t)}$. Then $(h_t,e_t)$ is a subset-reduction from $B_t$ to $E_t^{{\dot{+}}}$. Now let $t$ be a non-terminal node, and assume $a_{t \smallfrown \langle p,q \rangle}$, $h_{t \smallfrown \langle p,q \rangle}$, and $e_{t \smallfrown \langle p,q \rangle}$ have been defined for all $t \smallfrown \langle p,q \rangle \in T$. Fix a computable pairing function $(x,y) \mapsto \langle x,y \rangle$ with computable coordinate functions $(\langle x,y \rangle)_0 =x$ and $(\langle x,y \rangle)_1 =y$, and so that $\langle 0,0 \rangle =0$. Define $R_t$ so that $R_t(\langle q, \langle p,n \rangle \rangle) \iff B_{t \smallfrown \langle p,q \rangle}(n)$, so that $B_t(n) \iff \forall p \exists q\ R_t(\langle q, \langle p,n \rangle \rangle)$. We first adjust ordinal ranks to produce an increasing sequence so that we can take their supremum in $\mathcal{O}$. Let $\tilde{a}_{t,0}=a_{t \smallfrown \langle 0,0 \rangle}$ and let \[ \tilde{a}_{t,m+1} = \tilde{a}_{t,m} +_{\mathcal{O}} a_{t \smallfrown \langle (m)_0,(m)_1 \rangle} +_{\mathcal{O}} 2, \] where $+_{\mathcal{O}}$ is addition in $\mathcal{O}$. Then let $\tilde{a}_{t} = 3 \cdot 5^{i_{t}}$ where $\phi_{i_{t}}(m)= \tilde{a}_{t,m}$ for all $n$. Observe that if $\psi: E \leq F$ then the map $\tilde{\psi}\colon E^{{\dot{+}}} \leq F^{{\dot{+}}}$ as produced in the proof of Proposition~\ref{prop:monotone}(c) will satisfy $e \subseteq_E e' \iff \tilde{\psi}(e) \subseteq_F \tilde{\psi}(e')$. Hence we can uniformly replace $E_{t \smallfrown \langle p,q \rangle}$, $e_{t \smallfrown \langle p,q \rangle}$, and $h_{t \smallfrown \langle p,q\rangle} $ by $\mathsf{Id}^{{\dot{+}} \tilde{a}_{t,\langle p,q \rangle}}$, a corresponding $\tilde{e}_{t \smallfrown \langle p,q \rangle}$, and a corresponding map $\tilde{h}_{t \smallfrown \langle p,q \rangle}$, respectively, while maintaining the conditions for subset-reductions. Letting $A_t$ be such that $A_t^{(m)}=B_{t \smallfrown \langle (m)_0,(m)_1\rangle}$ for each $m$, we then can effectively produce a subset-reduction from $A_t$ to $(\mathsf{Id}^{{\dot{+}} \tilde{a}_t})^{{\dot{+}}}$ by Lemma~\ref{lem:proper_limit}. Since $A_t$ is computably isomorphic to $R_t$ in a uniform way, we can do the same for $R_t$. Letting $S_t(m) \iff \exists q\ R_t(\langle q,m \rangle)$, we then uniformly produce a subset-reduction from $S_t$ to $(\mathsf{Id}^{{\dot{+}} \tilde{a}_t})^{{\dot{+}} {\dot{+}} {\dot{+}}}$ by Lemma~\ref{lem:proper_union}. Recalling that $\mathsf{Id}^{{\dot{+}} a} \times \mathsf{Id} \leq \mathsf{Id}^{{\dot{+}} a}$ for all $a \in \mathcal{O}$ by Corollary~\ref{cor:absorbs_id}, we can then apply Lemma~\ref{lem:proper_intersection} to effectively obtain a subset reduction $(h_t,e_t)$ from $B_t$ to $(\mathsf{Id}^{{\dot{+}} \tilde{a}_t})^{{\dot{+}} {\dot{+}} {\dot{+}}}$. Letting $a_t= \tilde{a}_t +_{\mathcal{O}} 2^2$, this completes the induction step for $t$. \end{proof} We are now ready to conclude the proof of Theorem~\ref{thm:fixed_points}. Since the hyperarithmetic sets have no hyperarithmetic upper bound in terms of $m$-reducibility, this gives the main theorem of the section, Theorem~\ref{thm:proper}, as an immediate corollary. \begin{proof}[Proof of Theorem~\ref{thm:fixed_points}] Suppose $E^{{\dot{+}}} \leq E$. By Proposition~\ref{prop:monotone}(b) we can assume that $E$ has infinitely many classes. Thus by Proposition~\ref{prop:double_plus} we have $\mathsf{Id} \leq E^{{\dot{+}}{\dot{+}}} \leq E$. Hence by Proposition~\ref{prop:closed} we have $\mathsf{Id}^{{\dot{+}} a} \leq E^{{\dot{+}} a} \leq E$ for all $a \in \mathcal{O}$. But now by Theorem~\ref{thm:key_proper}, every hyperarithmetic set is $m$-reducible to $\mathsf{Id}^{{\dot{+}} a}$ for some $a \in \mathcal{O}$, and hence $m$-reducible to $E$. \end{proof} The proof of Theorem~\ref{thm:key_proper} does not give optimal bounds on the number of iterates of the jump required. With a bit more care, we can show that every $\Pi^0_{\alpha}$ set is reducible to $\mathsf{Id}^{{\dot{+}} a}$ for some $a \in \mathcal{O}$ with $|a|=\alpha$. We believe that the optimal bound should be that every $\Pi^0_{2 \cdot \alpha}$ set is reducible to $\mathsf{Id}^{{\dot{+}} a}$ for some $a \in \mathcal{O}$ with $|a|=\alpha$. Lemma~\ref{lem:proper_base} and Lemma~\ref{lem:proper_intersection} show that $\left(=^{ce}\right)^{{\dot{+}}}$ (and hence $\mathsf{Id}^{{\dot{+}} {\dot{+}}}$) is $\Pi^0_4$-complete, and we can show by an \emph{ad hoc} argument that $\left(=^{ce}\right)^{{\dot{+}}{\dot{+}}}$ is $\Pi^0_6$-complete. The difficulty is that our induction technique requires two iterates of the jump at each step in order to reverse the direction of set containment twice. We would prefer to use $\subseteq_E$ rather than $\supseteq_E$ throughout, but we do not see how to effectively enumerate c.e.\ subsets of a given c.e.\ set up to $E^{{\dot{+}}}$-equivalence, whereas we can enumerate c.e.\ supersets. The natural attempt to do this fails as shown in the following example. \begin{proposition} \label{prop:counterexample} There are $E$ and $e_0$ so that $\{[e]_{E^{{\dot{+}}}} : e \subseteq_{E} e_0\} \neq \{[W_i \cap e_0]_{E^{{\dot{+}}}} : i \in {\mathbb N}\}$, where $W_i \cap e_0$ denotes an index for an enumeration of $\ran \phi_{e_0} \cap W_i$. \end{proposition} \begin{proof} Let $E$ be $=^{ce}$, and let $A \subset B$ be c.e.\ sets with $B-A$ not c.e. Let $e_0$ be such that \[ \ran \phi_{\phi_{e_0}(j)} = \begin{cases} \{k\} & \text{if $j=2k+1$} \\ \{k,k+1\} & \text{if $j=2k+2$} \\ \emptyset & \text{if $j=0$} \end{cases} \] and let $e$ be such that \[ \ran \phi_{\phi_{e}(k)} = \begin{cases} \{k\} & \text{if $k \in B-A$} \\ \{k,k+1\} & \text{if $k \in A$} \\ \emptyset & \text{if $k \notin B$} \end{cases} .\] Then $e \subseteq_E e_0$ but there is no $i$ with $e \mathrel{E^{{\dot{+}}}} e_0 \cap W_i$. For if there were, we would have $k \in B-A$ iff $\exists x (x \in W_i \wedge x = \phi_{e_0}(1+2k))$ so that $B-A$ would be c.e. \end{proof} We have shown that the computable FS-jump of a hyperarithmetic equivalence relation is always strictly above the relation, so there are no hyperarithmetic fixed points up to bireducibility. If we consider non-hyperarithmetic equivalence relations we can find fixed points of the jump. \begin{definition} Let $\cong_{\mathcal T}$ be the isomorphism relation on computable trees. \end{definition} Here we can use any reasonable coding of computable trees by natural numbers. Then $\cong_{\mathcal{T}}$ is a $\Sigma_1^1$ equivalence relation which is not hyperarithmetic. In \cite[Theorem 2]{fokina-friedman-etal} it was shown that $\cong_{\mathcal{T}}$ is $\Sigma_1^1$ complete for computable reducibility, that is, $\cong_{\mathcal{T}}$ is $\Sigma_1^1$ and for every $\Sigma_1^1$ equivalence relation $E$, $E\leq \cong_{\mathcal T}$. We can see that $\cong_{\mathcal T}$ is a jump fixed point, i.e., $\cong_{\mathcal T}^{{\dot{+}}}$ is computably bireducible with $\cong_{\mathcal T}$. More generally: \begin{proposition} Any $\Sigma^1_1$ or $\Pi^1_1$ complete equivalence relation $E$ is a jump fixed point, i.e., $E^{{\dot{+}}}$ is computably bireducible with $E$. \end{proposition} \begin{proof} It suffices to show that $E^{{\dot{+}}}$ is $\Sigma_1^1$ (resp. $\Pi^1_1$) for any $\Sigma^1_1$ (resp. $\Pi^1_1$ equivalence relation $E$. This follows immediately from the fact that $E^{{\dot{+}}}$ is a conjunction of $E$ with additional natural number quantifiers. \end{proof} \begin{corollary} $\cong_{\mathcal T}$ is a jump fixed point. \end{corollary} We note that although every hyperarithmetic set is many-one reducible to $\mathsf{Id}^{{\dot{+}} a}$ for some $a \in \mathcal{O}$, we do not know whether every hyperarithmetic equivalence relation $E$ satisfies $E \leq \mathsf{Id}^{{\dot{+}} a}$ for some $a \in \mathcal{O}$. \section{Ceers and the jump} Recall from the introduction that $E$ is called a \emph{ceer} if it is a computably enumerable equivalence relation. In this section, we study the relationship between the computable FS-jump and the ceers. We begin with the following upper bound on the complexity of the computable FS-jump of a ceer. In the statement, recall that if $E$ is an equivalence relation and $W\subset{\mathbb N}$, then $W$ is said to be \emph{$E$-invariant} if it is a union of $E$-equivalence classes. \begin{proposition} \label{prop:upperbound} If $E$ is a ceer, then $E^{{\dot{+}}}\leq\mathord{=}^{ce}$. Moreover, we can find a reduction whose range is contained in the set $\{e\in{\mathbb N} : W_e\text{ is $E$-invariant}\}$. \end{proposition} \begin{proof} We define a computable function $f$ such that $W_{f(e)}=[\ran\phi_e]_E$. To see that there is such a computable function $f$, one can let $f(e)$ be a program which, on input $n$, searches through all triples $(a,b,c)$ such that $a\in\ran\phi_e$ and $(b,c)\in E$, and halts if and when it finds a triple of the form $(a,a,n)$. Since it is clear that $e\mathrel{E^{{\dot{+}}}}e'$ if and only if $[\ran\phi_e]_E=[\ran\phi_{e'}]_E$, we have that $f$ is a computable reduction from $E^{{\dot{+}}}$ to $=^{ce}$. It is immediate from the construction that the range of $f$ is contained in $\{e\in{\mathbb N} : W_e\text{ is $E$-invariant}\}$. \end{proof} The next result gives a lower bound on the complexity of the computable FS-jumps of a ceer. \begin{theorem} \label{thm:lowerbound} If $E$ is a ceer with infinitely many equivalence classes, then $\mathsf{Id}<E^{{\dot{+}}}$. \end{theorem} \begin{proof} We first show that $\mathsf{Id}\leq E^{{\dot{+}}}$. To do so, we first define an auxilliary set of pairs $A$ recursively as follows: Let $(n,j)\in A$ if and only if for every $i<j$ there exists $m<n$ and $(m,i')\in A$ such that $i\mathrel{E}i'$. It is immediate from the definition of $A$, the fact that $E$ is c.e., and the recursion theorem that $A$ is a c.e.\ set of pairs. We observe that each column $A^{[n]}=\{j:(n,j)\in A\}$ of $A$ is an initial interval of ${\mathbb N}$. It is immediate from the definition that the first column $A^{(0)}$ is the singleton $\{0\}$. Next since $E$ has infinitely many classes, we have that each $A^{[n]}$ is bounded. Moreover $A^{[n]}$ is precisely the interval $[0,j]$ where $j$ is the least value that is $E$-inequivalent to every element of $A^{[m]}$ for all $m<n$. We now define $f$ to be any computable function such that for all $n$, the range of $\phi_{f(n)}$ is precisely $A^{[n]}$. Then as we have seen, $m<n$ implies there exists an element $j$ in the range of $\phi_{f(n)}$ such that $j$ is $E$-inequivalent to everything in the range of $\phi_{f(m)}$. In particular, $f$ is a computable reduction from $\mathsf{Id}$ to $E^{{\dot{+}}}$. To establish strictness, assume to the contrary that $E^{{\dot{+}}}\leq\mathsf{Id}$. Then since $E\leq E^{{\dot{+}}}$, by Theorem~\ref{thm:proper} we have $E<\mathsf{Id}$, contradicting that $E$ has infinitely many classes. \end{proof} In order to put the previous result in context, we pause our investigation of ceers briefly to consider the question of which $E$ satisfy $\mathsf{Id}\leq E^{{\dot{+}}}$. We first note that it follows from Proposition~\ref{prop:double_plus} that if $E$ is itself a jump, then $\mathsf{Id}\leq E^{{\dot{+}}}$. We now show on the other hand that there exist equivalence relations $E$ such that $\mathsf{Id}\not\leq E^{{\dot{+}}}$. To describe such an equivalence relation, we recall the following notation. \begin{definition} If $A\subset{\mathbb N}$ then the equivalence relation $E_A$ is defined by \[m\mathrel{E_A}n\iff m=n\text{ or }m,n\in A\text{.} \] \end{definition} Thus the equivalence classes of $E_A$ are $A$ itself, together with the singletons $\{i\}$ for $i\notin A$. Note that $E_A\leq E_B$ if and only if $A$ is $1$-reducible to $B$ (see for instance \cite[Proposition~2.8]{coskey-hamkins-miller}). \begin{theorem} \label{thm:verydarkarithmetic} There exists an arithmetic coinfinite set $A$ such that $\mathsf{Id}\not\leq E_A^{{\dot{+}}}$. \end{theorem} \begin{proof} Let $P$ be the Mathias forcing poset, that is, $P$ consists of pairs $(s,B)$ where $s\subset{\mathbb N}$ is finite, $B\subset{\mathbb N}$ is infinite, and every element of $s$ is less than every element of $B$. The ordering on $P$ is defined by $(s,B)\leq(t,C)$ if $s\supset t$, $B\subset C$, and $s\mathsf{set}minus t\subset C$. We first show that if $A^c$ is sufficiently Mathias generic, then $A$ satisfies $\mathsf{Id}\not\leq E_A^{{\dot{+}}}$. In order to do so, let $f$ be any total function so that the sets $\ran\phi_{f(i)}$ are pairwise distinct. Define: \begin{align*} D_f = \{(s,B)\in P:\,&(\exists i\neq j)\,[ (s\cup B)\cap(\ran\phi_{f(i)}\triangle\ran\phi_{f(j)})=\emptyset \wedge \\ &\ran\phi_{f(i)} \cap (s \cup B)^c \neq \emptyset \wedge \ran\phi_{f(i)} \cap (s \cup B)^c \neq \emptyset ]\}. \end{align*} We claim that $D_f$ is dense in $P$. To see this, let $(s,B)$ be given. Repeatedly applying the pigeonhole principle, we can find infinitely many indices $i_n$ such that the sets $\ran\phi_{f(i_n)}$ agree on $s$. Since the sets $\ran\phi_{f(i)}$ are pairwise distinct, there must be three, $i_0,i_1,i_2$, such that each $\ran\phi_{f(i)}$ is not a subset of $s$. Observe that \begin{align*} {\mathbb N} = &(\ran\phi_{f(i_0)}\triangle\ran\phi_{f(i_1)})^c \cup(\ran\phi_{f(i_0)}\triangle\ran\phi_{f(i_2)})^c \cup \\ &(\ran\phi_{f(i_1)}\triangle\ran\phi_{f(i_2)})^c. \end{align*} In particular we can suppose without loss of generality that $i_0$ and $i_1$ satisfy that the set $B'=B\cap(\ran\phi_{f(i_0)}\triangle\ran\phi_{f(i_1)})^c$ is infinite. Then $\ran\phi_{f(i_0)}$ and $\ran\phi_{f(i_1)}$ agree on both $s$ and $B'$. Since neither $\ran\phi_{f(i_0)}$ nor $\ran\phi_{f(i_1)}$ is a subset of $s$, we can remove finitely many elements from $B'$ to ensure that $\ran\phi_{f(i_0)} \cap (s \cup B')^c \neq \emptyset$ and $\ran\phi_{f(i_1)} \cap (s \cup B')^c \neq \emptyset$, and so $(s,B')\in D_f$ completing the claim. Now let $G\subset P$ be a filter satisfying the following conditions: \begin{enumerate} \item $G$ meets $\{(s,B)\in P:|s|\geq m\}$ for all $m\in{\mathbb N}$. \item $G$ meets $D_f$ for all computable functions $f$ so that the sets $\ran\phi_{f(i)}$ are pairwise distinct. \end{enumerate} This is possible since the sets in condition~(a) are clearly dense, and we have shown that the $D_f$ are dense. We define the set $A$ by declaring that $A^c=\bigcup\{s:(s,B)\in G\}$. Condition~(a) implies that $A^c$ is infinite, and also that $A^c=\bigcap\{s \cup B : (s,B) \in G\}$; we wish to show that $\mathsf{Id}\not\leq E_A^{{\dot{+}}}$. For this we will show that if $f$ is a given computable function, then $f$ is not a reduction from $\mathsf{Id}$ to $E_A^{{\dot{+}}}$. Assume, toward a contradiction, that $f$ is a reduction from $\mathsf{Id}$ to $E_A^{{\dot{+}}}$. Then the sets $\ran\phi_{f(i)}$ are pairwise distinct, so there is $(s,B) \in G \cap D_f$. Thus there exist $i\neq j$ such that both $\ran\phi_{f(i)}$ and $\ran\phi_{f(j)}$ intersect $(s \cup B)^c$, and $(s\cup B)\cap(\ran\phi_{f(i)}\triangle\ran\phi_{f(j)})=\emptyset$. Hence both $\ran\phi_{f(i)}$ and $\ran\phi_{f(j)}$ intersect $A$, and $A^c\cap(\ran\phi_{f(i)}\triangle\ran\phi_{f(j)})=\emptyset$. This means that $f(i)\mathrel{E_A^{{\dot{+}}}}f(j)$, so $f$ is not a reduction from $\mathsf{Id}$ to $E_A^{{\dot{+}}}$, as desired. Finally, we can ensure $A$ is arithmetic by enumerating the dense sets described above, inductively defining a descending sequence $(s_n,B_n)$ meeting the dense sets, and letting $A^c=\bigcup_n s_n = \bigcap_n (s_n \cup B_n)$. More precisely, note that for any condition $(s,B)$, we can find an extension meeting $D_f$ for a suitable $f$ by intersecting $B$ with a $\Delta^0_2$ set, and the set of $i$ so that $f=\varphi_i$ is suitable is $\Pi^0_3$, so the construction of this sequence may be done computably in $0^{(3)}$, from which we can produce an $A$ which is $\Delta^0_4$ \end{proof} This result leaves open the question of what is the least complexity of an equivalence relation $E$ with infinitely many classes such that $\mathsf{Id}\not\leq E^{{\dot{+}}}$. Returning to ceers, in view of the bounds from Proposition~\ref{prop:upperbound} and Theorem~\ref{thm:lowerbound}, it is natural to ask whether there is a ceer $E$ such that $E^{{\dot{+}}}$ lies properly between $\mathsf{Id}$ and $=^{ce}$. We first see that there is a large collection of ceers whose jumps are bireducible with $=^{ce}$. We recall the following terminology from \cite{andrews-sorbi-joins}: \begin{definition} A ceer $E$ is said to be \emph{light} if $\mathsf{Id}\leq E$. $E$ is said to be \emph{dark} if $E$ has infinitely many classes but $\mathsf{Id}\not\leq E$. \end{definition} Thus every ceer satisfies exactly one of finite, light, or dark. \begin{proposition} \label{prop:light-is-high} If $E$ is a light ceer then $E^{{\dot{+}}}$ is computably bireducible with $=^{ce}$. \end{proposition} \begin{proof} This is an immediate consequence of Propositions~\ref{prop:monotone}(c), \ref{prop:idplus}, and~\ref{prop:upperbound}. \end{proof} We will see that there are also dark ceers which satisfy this conclusion. We introduce the following terminology. \begin{definition} We say a ceer $E$ is \emph{high for the computable FS-jump} if $E^{{\dot{+}}}$ is computably bireducible with $=^{ce}$. \end{definition} This generalizes the notion of lightness for ceers, but also implies that the computable FS-jump is as complicated as possible. As there is no least ceer with infinitely many classes, there does not seem to be a natural notion of low for the computable FS-jump. In order to describe a dark ceer which is high for the computable FS-jump, recall that a c.e.\ set $A\subset{\mathbb N}$ is called \emph{simple} if there is no infinite c.e.\ set contained in $A^c$. Furthermore $A$ is called \emph{hyperhypersimple} if for all computable functions $f$ such that $\{W_{f(n)}:n\in{\mathbb N}\}$ is a pairwise disjoint family of finite sets, there exists $n\in{\mathbb N}$ such that $W_{f(n)}\subset A$. We refer the reader to \cite[Chapter~5]{soare2} for more about these properties, including examples. \begin{theorem} \label{thm:nonhhs-is-high} Let $A\subset{\mathbb N}$ be a set which is simple and not hyperhypersimple. Then $E_A$ is a dark ceer and $E_A$ is high for the computable FS-jump. \end{theorem} \begin{proof} It follows from \cite[Proposition~4.5]{gao-gerdes} together with the assumption that $A$ is simple that $E_A$ is dark. To see that $E_A^{{\dot{+}}}$ is computably bireducible with $=^{ce}$, first it follows from Proposition~\ref{prop:upperbound} that $E_A^{{\dot{+}}}\leq\mathord{=}^{ce}$. For the reduction in the reverse direction, since $A$ is not hyperhypersimple, there exists a computable function $f$ such that $\{W_{f(n)}:n\in{\mathbb N}\}$ is a pairwise disjoint family of finite sets and for all $n\in{\mathbb N}$ we have $W_{f(n)}\cap A^c\neq\emptyset$. Now given an index $e$ we compute an index $g(e)$ such that $\phi_{g(e)}$ is an enumeration of the set $\bigcup\{W_{f(n)}:n\in W_e\}$. Then since the $W_{f(n)}$ are pairwise disjoint and meet $A^c$, we have $W_e=W_{e'}$ if and only if $A^c\cap\ran\phi_{g(e)}$ and $A^c\cap\ran\phi_{g(e')}$ are distinct subsets of $A^c$. It follows that $e\mathrel{=^{ce}}e'$ if and only if $g(e)\mathrel{E_A^{{\dot{+}}}}g(e')$, as desired. \end{proof} On the other hand, there also exist dark ceers $E$ such that $E$ is not high for the computable FS-jump. In order to state the results, we recall from \cite[Chapter~X]{soare} that a c.e.\ subset $A\subset{\mathbb N}$ is said to be \emph{maximal} if $A^c$ is infinite and for all c.e.\ sets $W$ either $W\mathsf{set}minus A$ or $W^c\mathsf{set}minus A$ is finite. We further note that if $A$ is maximal then it is hyperhypersimple. \begin{theorem} \label{thm:maximal} Let $A$ be a maximal set. If $B$ is a c.e. set with $B\subsetneq A$, then $E_A^{{\dot{+}}} < E_B^{{\dot{+}}}$. In particular, $E_A$ is not high for the computable FS-jump. \end{theorem} The proof begins with several preliminary results, which may be of independent value. \begin{lemma} \label{lem:reverse} If $A,B$ are c.e.\ sets and $B\subset A$, then $E_A^{{\dot{+}}} \leq E_B^{{\dot{+}}}$. \end{lemma} \begin{proof} If $B$ is non-hyperhypersimple, then the result follows immediately from Proposition~\ref{prop:upperbound} and Theorem~\ref{thm:nonhhs-is-high}. If $B$ is hyperhypersimple, then by \cite[X.2.12]{soare} there exists a computable set $C$ such that $B\cup C=A$. Let $b\in B$ be arbitrary, and define \[f(n)=\begin{cases}b&n\in C\\n&n\notin C\end{cases} \] It is easy to see that $f$ is a computable reduction from $E_A$ to $E_B$, and hence by Proposition~\ref{prop:monotone}(c) we have $E_A^{{\dot{+}}}\leq E_B^{{\dot{+}}}$ as desired. \end{proof} In the next lemma we will use the following terminology about a function $f\colon{\mathbb N}\to{\mathbb N}$. We say that $f$ is \emph{$=^{ce}$-invariant} if $W_e=W_{e'}$ implies $W_{f(e)}=W_{f(e')}$, that $f$ is \emph{monotone} if $W_{e'}\subset W_{e}$ implies $W_{f(e')}\subset W_{f(e)}$, and that $f$ is \emph{inner-regular} if \begin{equation} \label{eq:inner-regular} W_{f(e)}=\bigcup\left\{W_{f(e')} : W_{e'}\subset W_e\text{ and }W_{e'}\text{ is finite}\right\}\text{.} \end{equation} We are now ready to state the lemma. \begin{lemma} \label{lem:monotone} If $f$ is a computable function, the properties $=^{ce}$-invariant, monotone, and inner-regular are all equivalent. \end{lemma} \begin{proof} It is clear that inner-regular implies monotone, and monotone implies $=^{ce}$-invariant. We therefore need only show that $=^{ce}$-invariant implies inner-regular. Assume that $f$ is $=^{ce}$-invariant. Then \cite[Lemma~4.5]{coskey-hamkins-miller} gives that $f$ is monotone, so we have $W_{f(e)} \supset \bigcup\left\{W_{f(e')} : W_{e'}\subset W_e\text{ and }W_{e'}\text{ is finite}\right\}$. For the subset inclusion of Equation~\eqref{eq:inner-regular}, we assume that $x\in W_{f(e)}$ and aim to show that there exists $e'$ such that $W_{e'}\subset W_e$, $W_{e'}$ is finite, and $x\in W_{f(e')}$. For any $e$, let $W_{e,s}=\{n : n< s \wedge \varphi_{e,s}(n) \downarrow\}$ be the partial enumeration of $W_e$ at stage $s$, so each $W_{e,s}$ is finite. We can use the Recursion Theorem to find an index $e'$ which satisfies the following: \[ W_{e',s} = \begin{cases} W_{e,s} & \text{if $x \notin W_{f(e'),s}$} \\ W_{e,s'} & \text{if $s' \leq s$ is least with $x \in W_{f(e'),s'}$.} \end{cases} \] We must show that $W_{e'}\subset W_e$, $W_{e'}$ is finite, and $x\in W_{f(e')}$. It is clear that $W_{e'}\subset W_e$. To show that $x\in W_{f(e')}$, assume to the contrary that $x\notin W_{f(e')}$. Then $W_{e',s}=W_{e,s}$ for all $s$, that is, we would have $W_{e'}=W_e$. Since $f$ is $=^{ce}$-invariant, we would have $W_{f(e')}=W_{f(e)}$. Our assumption that $x\in W_{f(e)}$ would therefore imply that $x\in W_{f(e')}$ after all. Now that we know $x\in W_{f(e')}$, we know that there is $s$ with $x \in W_{f(e'),s}$. This means that for all $s$ we have $W_{e',s}=W_{e,s'}$ for the least such $s'$, so $W_{e'}=W_{e,s'}$ is finite, as desired. \end{proof} We note that the same conclusions remain true if we replace $=^{ce}$-invariance by $\mathsf{Id}^{{\dot{+}}}$-invariance, i.e., $f$ preserves equality of ranges rather than of domains. Thus we can apply this result to reductions among computable jumps. \begin{corollary} Let $A$ be a maximal set. If $E^{{\dot{+}}} \leq E_A^{{\dot{+}}}$, then any $E$-invariant c.e.\ set contains either finitely or cofinitely many $E$-classes. In particular, if $E_B^{{\dot{+}}} \leq E_A^{{\dot{+}}}$ then $B$ is maximal. \end{corollary} \begin{proof} Let $f$ be a computable reduction from $E^{{\dot{+}}}$ to $E_A^{{\dot{+}}}$. We can assume without loss of generality that for all $e$, $\ran\phi_{f(e)}$ is $E_A$-invariant. Indeed, we may modify $f$ to ensure that if $\phi_{f(e)}$ enumerates any element of $A$ then $\phi_{f(e)}$ enumerates the rest of $A$ too. Hence we can assume that $f$ is $=^{ce}$-invariant. If $W=\ran \phi_e$ is an $E$-invariant c.e.\ set, then $R=\ran \phi_{f(e)}$ is an $E_A$-invariant c.e.\ set, hence either $R \mathsf{set}minus A$ is finite or $R$ is cofinite. If $R$ is cofinite, then $W$ must contain all but finitely many $E$-classes, or else there would be an infinite increasing chain of $E^{{\dot{+}}}$-inequivalent c.e\ sets containing $W$ which must map to an infinite increasing chain of $E_A^{{\dot{+}}}$-inequivalent c.e\ sets, which is impossible. Suppose instead that $R \mathsf{set}minus A$ is finite. Then by inner-regularity and $E_A$-invariance, there must be a finite set $F=\ran \phi_{e_0} \subset W$ so that $R= \ran \phi_{f(e_0)}$. But then $e_0 \mathrel{E^{{\dot{+}}}} e$, so $W$ must contain only finitely many $E$-classes. \end{proof} We will use the following lemma, well-known in descriptive set theory as a consequence of the effective Reduction Property for the pointclass $\Sigma^0_1$. \begin{lemma} \label{lem:effective-reduction} Let $A_n$ be a uniformly c.e. sequence of c.e. sets. Then there is a uniformly c.e. sequence of c.e. sets $\tilde{A}_n$ so that $\tilde{A}_n \subset A_n$ for each $n$, $\tilde{A}_n \cap \tilde{A}_m = \emptyset$ for $n \neq m$, and $\bigcup_n \tilde{A}_n = \bigcup_n A_n$. \end{lemma} \begin{proof} Let $f$ be a computable function with $A_n=W_{f(n)}$ for each $n$. Let $\tilde{A}_n. = \{i : \exists s (\varphi_{f(n),s}(i)\downarrow \wedge (\forall m < n) (\forall t \leq s) \varphi_{f(m),t}(i) \uparrow )\}$. \end{proof} We now give the main ingredient to the proof of Theorem~\ref{thm:maximal}. \begin{definition} An equivalence relation $E$ is \emph{self-full} if whenever $f$ is a computable reduction from $E$ to $E$, then the range of $f$ meets every $E$ class. \end{definition} Letting $\mathsf{Id}_n$ denote the identity equivalence relation on $\{0,\ldots,n-1\}$, $E$ being self-full is equivalent to $E \oplus \mathsf{Id}_1 \not\leq E$, so is preserved under computable bireducibility. In the following, we say $f$ and $h$ are \emph{$E$-equivalent} if $f(n)\mathrel{E}h(n)$ for all $n$. We say that $h$ is \emph{induced by a finite support permutation of the $E_A$-classes} when there is an $E_A$-invariant permutation $\pi$ with finite support so that $\ran\phi_{h(e)} = \{\pi(n) : n \in \ran\phi_e\}$ for all $e$ so that $\ran\phi_e$ is $E_A$-invariant. \begin{lemma} \label{lem:full} If $A$ is maximal then $E_A^{{\dot{+}}}$ is self-full. In fact, if $f$ is a computable reduction from $E_A^{{\dot{+}}}$ to itself, then $f$ is $E_A^{{\dot{+}}}$-equivalent to a function $h$ induced by a finite support permutation of the $E_A$-classes. \end{lemma} \begin{proof} Suppose $f$ is a computable reduction from $E_A^{{\dot{+}}}$ to itself. We can assume without loss of generality that for all $e$, $\ran\phi_{f(e)}$ is $E_A$-invariant. Indeed, we may modify $f$ to ensure that if $\phi_{f(e)}$ enumerates any element of $A$ then $\phi_{f(e)}$ enumerates the rest of $A$ too. Having done so, we introduce the following mild abuse of notation: if $R=\ran\phi_e$ then we will write $f(R)$ for $\ran\phi_{f(e)}$. Due to our assumption about $f$, this notation is well-defined. Note that we thus have that $f$ is $=^{ce}$-invariant as well, and thus monotone and inner-regular. We will exploit the following consequence of the monotonicity of $f$ several times: If $C$ and $D$ are c.e. sets with $C \subset D$ and $D \mathsf{set}minus C$ finite and disjoint from $A$, then $|f(D)\mathsf{set}minus f(C)| \geq |D \mathsf{set}minus C|$. This follows since there is a chain of length $|D \mathsf{set}minus C|+1$ of $E_A^{{\dot{+}}}$-inequivalent sets between $C$ and $D$, which must map to a chain of $E_A^{{\dot{+}}}$-inequivalent sets between $f(C)$ and $f(D)$. Similarly, if $C$ is cofinite with $A \subseteq C$, then $|f(C)^c| \geq |C^c|$. The maximality of $A$ then also implies that if $C \mathsf{set}minus A$ is finite then so is $f(C) \mathsf{set}minus A$. The heart of the proof will be to show that there is a finite support permutation $\pi$ of ${\mathbb N}$ such that for any c.e.\ set $R$, we have $f(R)=\{\pi(n):n\in R\}$. In particular, this implies that $f$ meets every $E_A^{{\dot{+}}}$ class, as desired. We begin be seeing that the range of $f$ is almost covered by the images of singletons. \begin{claim} There is a finite set $C$ such that $f(A)\cup(f({\mathbb N})\mathsf{set}minus\bigcup_n f(\{n\}))\subset f(C)$. \end{claim} \begin{claimproof} First, observe that $\bigcup_n f(\{n\})$ is an infinite c.e.\ set (here we tacitly select indices for $\{n\}$ uniformly), and hence intersects $A$, and thus contains $A$. Moreover, there are infinitely many $n$ so that $f(\{n\})$ intersects $A$; otherwise we could omit such $n$ and have an infinite c.e. set disjoint from $A$. Thus $\{n : f(\{n\}) \cap A \neq \emptyset\}$ is an infinite c.e.\ set, and so intersects $A$. Hence there is $n \in A$ with $f(\{n\}) \cap A \neq \emptyset$, so $A\subset f(A)$. Next, since the sets $f(\{n\})$ are distinct for $n \notin A$, we have $\left( \bigcup_n f(\{n\}) \right) \mathsf{set}minus A$ infinite, so the maximality of $A$ implies $\bigcup_n f(\{n\})$ is cofinite. Hence by the inner-regularity of $f$ we can find a finite set $C$ such that $f(A)\cup(f({\mathbb N})\mathsf{set}minus\bigcup_n f(\{n\})\subset f(C)$. \end{claimproof} We next see that we will be able to select distinct elements from the images of singletons. \begin{claim} For $n\notin A\cup C$, we have $f(\{n\})\mathsf{set}minus(f(C)\cup\bigcup_{m\neq n}f(\{m\}))\neq\emptyset$. \end{claim} \begin{claimproof} Let $n\notin A\cup C$. Since $f$ is a reduction and is monotone, we can find $x\in f({\mathbb N})\mathsf{set}minus f({\mathbb N}\mathsf{set}minus\{n\})$. Using the definition of $C$, the fact that $n\notin C$, and monotonicity, we have $f({\mathbb N})\mathsf{set}minus\bigcup_mf(\{m\})\subset f(C)\subset f({\mathbb N}\mathsf{set}minus\{n\})$. In particular, $x\notin f(C)$ and therefore $x\in\bigcup_mf(\{m\})$. Again by monotonicity, $x\notin\bigcup_{m\neq n}f(\{m\})$, so we must have $x\in f(\{n\})$, completing the claim. \end{claimproof} We now construct a first approximation to the desired permutation $\pi$. \begin{claim} There is a finite support permutation $\sigma$ such that for $n\notin A\cup C$ we have $\sigma(n)\in f(\{n\})\mathsf{set}minus A$. \end{claim} \begin{claimproof} From the previous claim, Lemma~\ref{lem:effective-reduction} gives a uniformly c.e.\ sequence $B_n$ of pairwise disjoint sets such that $B_n\subset f(\{n\})$ and $\bigcup_n B_n = \bigcup_n f(\{n\})$, so that for $n\notin A\cup C$ we have $B_n\mathsf{set}minus(f(C)\cup\bigcup_{m\neq n}f(\{m\}))\neq\emptyset$; in particular $B_n \mathsf{set}minus f(C) \neq \emptyset$. We may shrink $B_n$ so that $B_n$ is disjoint from the finite set $f(C)\mathsf{set}minus A$ for all $n$; we may further shrink $B_n$ uniformly to a set $\tilde{B}_n$ so that $\tilde{B}_n\mathsf{set}minus A$ is a singleton for all $n\notin A\cup C$. Note that we do not claim or require that this singleton is not in $\bigcup_{m\neq n}f(\{m\})$, but distinct $n$'s not in $A \cup C$ will produce distinct singletons. Since $C \mathsf{set}minus A$ is finite, we have that $f((C\mathsf{set}minus A)^c)$ is cofinite, and monotonicity of $f$ implies that $|f((C\mathsf{set}minus A)^c)^c|\geq|C\mathsf{set}minus A|$ as discussed above. We may then let $p$ be any injection from $C\mathsf{set}minus A\to f((C\mathsf{set}minus A)^c)^c$. We now define: \[G_n=\begin{cases} A & n\in A \\ A\cup\{p(n)\} & n\in C\mathsf{set}minus A \\ A\cup \tilde{B}_n & n\notin A\cup C \end{cases}. \] Observe that $G_n$ is a uniformly c.e.\ sequence, since we may first check if $n \in C \mathsf{set}minus A$; if not, we enumerate $\tilde{B}_n$ into $G_n$ until we see $n$ enumerated in $A$ (if ever), at which point we enumerate $A$ (which will then contain $\tilde{B}_n$) into $G_n$. We define $\sigma$ as follows: \[\sigma(n)=\begin{cases} n&n\in A\\ \text{the unique element of $G_n\mathsf{set}minus A$} & n\notin A \end{cases} \] This completes the definition of $\sigma$. Note that $\sigma(n) \in f(\{n\})$ for $n \notin (C \mathsf{set}minus A)$, and $\sigma(n) \notin A$ for $n \notin A$, as required. We do not claim \emph{a priori} that $\sigma$ is computable (although this will follow later), but we can use the sequence $G_n$ to obtain effectiveness. Define the function $g(R)=\bigcup_{n\in R}G_n$, which is computable in the indices. We check that $\sigma$ is a permutation with finite support. It is immediate from the construction that $\sigma$ is injective. To show $\sigma$ is surjective, assume $k$ is not in the range of $\sigma$. Then the sequence $R_0=A\cup\{k\}$ and $R_{n+1}=g(R_n)$. Since $k\notin A$ we have that $R_n\mathsf{set}minus A$ is a singleton for all $n$. Moreover the singletons are distinct since $\sigma$ is injective and none of the singletons can equal $k$ for $n>0$. Applying Lemma~\ref{lem:effective-reduction} to the sequence $R_n$, we obtain a uniformly c.e.\ sequence of nonempty pairwise disjoint sets, all meeting $A^c$. This contradicts that $A$ is hyperhypersimple (see \cite[Exercise~X.2.16]{soare}). To see that $\sigma$ has finite support, first note that $\sigma$ cannot have an infinite orbit. Otherwise, we could similarly produce a uniformly c.e. sequence (using the function $g$) which contradicts that $A$ is hyperhypersimple. If $\sigma$ had infinitely many nontrivial orbits, let $R=\{n:(\exists k\geq n)\;n\in G_k\}$. Then $A\subset R$, and for $n\notin A$ we have $n\in R$ when $n$ is the least element of its orbit and $n\notin R$ when $n$ is the greatest element of a nontrivial orbit. Thus $R\mathsf{set}minus A$ is infinite and co-infinite, again contradicting that $A$ is maximal. \end{claimproof} We are now ready to construct $\pi$ as follows. Let $\tilde C=(C\mathsf{set}minus A)\cup\supp(\sigma)$. If $R$ is disjoint from $C\mathsf{set}minus A$ then $g(R)\subset f(R)$, therefore if $R$ is disjoint from $\tilde C$ we have $R\subset f(R)$. By monotonicity of $f$, if $R$ is disjoint from $\tilde C$ and cofinite, then by the observation above we have $|f(R)^c| \geq |R^c|$, so $R=f(R)$. In particular $f(\tilde{C}^c)=\tilde{C}^c$. For any $k\in\tilde{C}$, since $k \notin A$ we have that $\tilde C^c\cup\{k\}$ is $E_A^{{\dot{+}}}$-inequivalent to $\tilde{C}^c$; therefore we must have that $f$ sends $\tilde C^c\cup\{k\}$ to $\tilde C^c\cup\{\pi(k)\}$ for some $\pi(k)\in\tilde C$ since the complement of $f(\tilde C^c\cup\{k\})$ must be at least as large as the complement of $\tilde C^c\cup\{k\}$, but must be smaller than $\tilde{C}$. This defines $\pi$ on the finite set $\tilde{C}$; as $\pi$ is injective it is a permutation of $\tilde{C}$. Additionally define $\pi$ to be the identity on $\tilde C^c$. This completes the definition of $\pi$. We have that $\pi$ is an $E_A$-invariant permutation with finite support; it remains to verify that $\pi$ induces the desired function. Let $h$ be a computable function such that $h(R)=\{\pi(n):n\in R\}$. \begin{claim} For all $E_A$-invariant c.e. sets $R$ we have $f(R)=h(R)$. \end{claim} \begin{claimproof} We first establish this for sets of the form ${\mathbb N} \mathsf{set}minus \{n\}$ for $n \notin A$. Note that if $F \subset \tilde{C}$ then we have $f(\tilde{C}^c \cup F) = \tilde{C}^c \cup \{\pi(n):n \in F\}$ from monotonicity and the observation earlier; in particular we see that $f({\mathbb N})={\mathbb N}$ and $f(\tilde{C}^c)=\tilde{C}^c$. From this, we see that if $R$ is any cofinite set containing $A$, then $|f(R)^c|=|R^c|$. For any given $n \notin A$, we claim that $f({\mathbb N} \mathsf{set}minus \{n\})={\mathbb N} \mathsf{set}minus \{\pi(n)\}$. We know this already for $n \in \tilde{C}$. If $n \in \tilde{C}^c$ then $f({\mathbb N} \mathsf{set}minus \{n\})={\mathbb N} \mathsf{set}minus \{\pi(k)\}$ for some $k$; this $k$ can not be in $\tilde{C}$ since $f({\mathbb N} \mathsf{set}minus \{k\})={\mathbb N} \mathsf{set}minus \{\pi(k)\}$ and ${\mathbb N} \mathsf{set}minus \{n\}$ and ${\mathbb N} \mathsf{set}minus \{k\}$ are $E_A^{{\dot{+}}}$-inequivalent. But since $\tilde{C}^c\mathsf{set}minus \{n\} \subset f({\mathbb N} \mathsf{set}minus \{n\})$ we must have $\pi(k)=n=\pi(n)$ as claimed. Now let $R$ be any infinite $E_A$-invariant c.e.\ set, so $R$ contains $A$. Then $R$ is the intersection of the sets ${\mathbb N} \mathsf{set}minus \{n\}$ for $n \notin R$, so $f(R) \subset h(R)$ for all such $R$ by monotonicity. Suppose there were an infinite $E_A$-invariant c.e.\ set $R$ and some $k \in h(R) \mathsf{set}minus f(R)$. Then $k=\pi(n)$ for some $n \in R \mathsf{set}minus A$, so $k \notin f(A \cup \{n\}) \subset h(A \cup \{n\})=A \cup \{k\}$ and thus $f(A \cup \{n\})=A$, contradicting $A \subset f(A)$. Finally, we consider finite $R$. If $n \notin A$, then $f(\{n\}) \subset f(A \cup \{n\}) = h(A \cup \{n\}) = A \cup \{\pi(n)\}$. We can not have $f(\{n\}) =A$, so $f(\{n\})=\{\pi(n)\}$. Let $R$ be any finite $E_A$-invariant c.e.\ set, so $R$ is disjoint from $A$. Then $h(R) \subset f(R)$. We also have that $f(R) \subset f(A \cup R)=h(A \cup R)=A \cup h(R)$, so we must have that $f(R) =h(R)$. \end{claimproof} Hence, $f$ is $E_A^{{\dot{+}}}$-equivalent to $h$, completing the proof of the lemma. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:maximal}] Let $A$ be maximal and $B\subsetneq A$. Fix any $a\in A\mathsf{set}minus B$. We observe that $E_{A\mathsf{set}minus\{a\}}$ is computably bireducible with $E_A\oplus\mathsf{Id}_1$. Therefore by Proposition~\ref{prop:monotone}(c) we have that $E_{A\mathsf{set}minus\{a\}}^{{\dot{+}}}$ is computably bireducible with $E_A^{{\dot{+}}}\times\mathsf{Id}_2$. Now assume towards a contradiction that $E_B^{{\dot{+}}}\leq E_A^{{\dot{+}}}$. Then by Lemma~\ref{lem:reverse} we have $E_{A\mathsf{set}minus\{a\}}^{{\dot{+}}}\leq E_A^{{\dot{+}}}$ and hence by the previous paragraph we have $E_A^{{\dot{+}}}\times\mathsf{Id}_2\leq E_A^{{\dot{+}}}$. But if $f$ is such a reduction, then by Lemma~\ref{lem:full} the restriction of $f$ to either copy of $E_A^{{\dot{+}}}$ has range meeting every $E_A^{{\dot{+}}}$ class. But for a reduction $f$ we cannot have this property true of both copies of $E_A^{{\dot{+}}}$, so we have reached a contradiction. \end{proof} We record here several immediate consequences of Theorem~\ref{thm:maximal} and its proof. \begin{corollary} Let $A,B$ be maximal sets. \begin{itemize} \item If $a\in A$ then $E_A^{{\dot{+}}} < E_{A\mathsf{set}minus\{a\}}^{{\dot{+}}}$, and if $b\notin A$ then $E_{A\cup\{b\}}^{{\dot{+}}} < E_{A}^{{\dot{+}}}$. \item If $|A\triangle B|<\infty$, then $E_A^{{\dot{+}}} \leq E_B^{{\dot{+}}}$ iff $|B\mathsf{set}minus A|\leq|A\mathsf{set}minus B|$. \item If a c.e. set $C$ is contained in a maximal set, then it is contained in a maximal set $D$ such that $E_D^{{\dot{+}}} < E_C^{{\dot{+}}}$. \end{itemize} \end{corollary} We conclude with a small refinement of the second statement of Theorem~\ref{thm:maximal}. Recall that a c.e.\ set $A$ is said to be \emph{quasi-maximal} if it is the intersection of finitely many maximal sets. We refer the reader to \cite[X.3.10]{soare} for more on this notion. In particular, every quasi-maximal set is simple (see \cite[X.3.10(b)]{soare}). \begin{theorem} \label{thm:quasimaximal} If $A\subset{\mathbb N}$ is quasi-maximal then $E_A$ is not high for the computable FS-jump. \end{theorem} \begin{proof} Suppose towards a contradiction that $\mathord{=}^{ce}\leq E_A^{{\dot{+}}}$. By Proposition~\ref{prop:upperbound}, $E_A^{{\dot{+}}}$ is reducible to the restriction of $=^{ce}$ to the $E_A$-invariant sets, so by composing reductions there exists a computable reduction $f$ from $=^{ce}$ to $=^{ce}$ such that for all $e$, $W_{f(e)}$ is $E_A$-invariant. Since $A$ is simple, it follows that for all $e$ we have $A\subset W_{f(e)}$ iff $W_{f(e)}$ is infinite and $A\cap W_{f(e)}=\emptyset$ iff $W_{f(e)}$ is finite. We claim that we may find such an $f$ so that for all $e$ we have $A\subset W_{f(e)}$. We first show that there exists $e_0$ such that $W_{e_0}$ is finite and $A\subset W_{f(e_0)}$. Let $e$ be any index such that $W_e={\mathbb N}$. By Lemma~\ref{lem:monotone}, $f$ is inner-regular. Since $f$ is a reduction, it follows from Equation~\ref{eq:inner-regular} that $W_{f(e)}$ is infinite and hence $A\subset W_{f(e)}$. Further examining Equation~\ref{eq:inner-regular}, together with the last sentence of the previous paragraph, we conclude there exists $e_0$ as desired. Let $g$ be a computable function such that $W_{g(e)}=W_{e_0}\cup\{\max(W_{e_0})+x:x\in W_e\}$. Then replacing $f$ with $f\circ g$ completes the claim. It follows from the claim, together with the fact that $f$ is monotone, that the lattice of c.e.\ sets modulo finite may be embedded into the lattice of c.e.\ sets containing $A$ modulo finite. But the former lattice is infinite, and by \cite[X.3.10(a)]{soare} the latter lattice is finite, a contradiction. \end{proof} \section{Additional remarks and open questions} We close with some open questions and directions for further investigation. \begin{question} For a c.e.\ set $A$, when is $E_A^{{\dot{+}}}$ bireducible with $=^{ce}$? \end{question} By Theorem \ref{thm:nonhhs-is-high} if $A$ is not hyperhypersimple then $E_A$ is high for the jump, and by Theorem~\ref{thm:quasimaximal} if $A$ is quasi-maximal then $E_A^{{\dot{+}}}<\mathord{=}^{ce}$. The question is, if $A$ is hyperhypersimple but not quasi-maximal, is $E_A^{{\dot{+}}}$ high for the computable FS-jump? One construction of such a set is given in an exercise in \cite[IX.2.28f]{odifreddi}. We do not know whether the choice of notation for a countable ordinal affects the iterated jump. \begin{question} If $a,b \in \mathcal{O}$ with $|a|=|b|$, is $E^{{\dot{+}} a}$ computably bireducible with $E^{{\dot{+}} b}$? \end{question} Although we saw that every hyperarithmetic set is many-one reducible to some jump of the identity, we do not know if every hyperarithmetic equivalence relation is computably reducible to some iterated jump of the identity. \begin{question} If $E$ is hyperarithmetic, is there $a \in \mathcal{O}$ with $E \leq \mathsf{Id}^{{\dot{+}} a}$? \end{question} For $E$ hyperarithmetic, we have $e \mathrel{E} e'$ iff $[e]_E=[e']_E$, so that $E$ is computably reducible to the relativized version of $=^{ce}$, denoted $=^{ce,E}$, considered in \cite{bard}. This question is then equivalent to asking if these relativized equivalence relations with hyperarithmetic oracles are computably reducible to iterated jumps of the unrelativized $=^{ce}$. We also note that, unlike the case of the classical Friedman--Stanley jump, the equivalence relation $E_1$ is not an obstruction. Recall that $E_1$ may be defined on $\mathcal{P}({\mathbb N})$ by setting $A \mathrel{E_1} B$ when $A^{[n]}=B^{[n]}$ for all but finitely many $n$, and that $E_1$ is not Borel reducible to any iterated Friedman--Stanley jump of equality. \begin{proposition} $E_1^{ce} \leq \left(=^{ce}\right)^{{\dot{+}}}$. \end{proposition} \begin{proof} Given $e$, let $g(e)$ be such that $\phi_{g(e)}(\langle f,m\rangle)$ is an index for an enumeration of the set $\bigcup_{n<m} (W_f)^{[n]} \cup \bigcup_{n \geq m} (W_e)^{[n]}$. Then $e \mathrel{E_1^{ce}} e'$ if and only if $g(e) \mathrel{\left(=^{ce}\right)^{{\dot{+}}}} g(e')$. \end{proof} We can also ask what other fixed points exist besides $\cong_{\mathcal{T}}$. We note that there is no known characterization of fixed points of the classical Friedman--Stanley jump. \begin{question} Characterize the fixed points of the computable FS-jump. \end{question} We used the relations $\subseteq_E$ in establishing properness of the computable FS-jump jump, but the operation $E \mapsto \subseteq_E$ can be applied to other relations and it may be of interest to study its effect on partial orders. \begin{question} What can be said about the mapping $E \mapsto \subseteq_E$ as an operation on computable partial orders? \end{question} Finally, we can ask about when the computable FS-jump of an equivalence relation fails to be above the identity relation. The proof of Theorem~\ref{thm:verydarkarithmetic} shows that there is a $\Delta^0_4$ equivalence relation $E$ with infinitely many classes so that $\mathsf{Id}\not\leq E^{{\dot{+}}}$, and Theorem~\ref{thm:lowerbound} shows that there is no $\Sigma^0_1$ such $E$, but we do not know if there can be such an $E$ which is, e.g., $\Sigma^0_2$ or $\Sigma^0_3$. \begin{question} What is the least complexity of an equivalence relation $E$ with infinitely many classes such that $\mathsf{Id}\not\leq E^{{\dot{+}}}$? \end{question} \end{document}

\begin{document} \title{Generation of arbitrary two dimensional motional states of a trapped ion} \author{XuBo Zou, K. Pahlke and W. Mathis \\ \\Institute TET, University of Hannover,\\ Appelstr. 9A, 30167 Hannover, Germany } \date{} \maketitle \begin{abstract} {\normalsize We present a scheme to generate an arbitrary two-dimensional quantum state of motion of a trapped ion. This proposal is based on a sequence of laser pulses, which are tuned appropriately to control transitions on the sidebands of two modes of vibration. Not more than $(M+1)(N+1)$ laser pulses are needed to generate a pure state with a phonon number limit $M$ and $N$.\\ PACS number:42.50.Dv, 42.50.Ct} \end{abstract} The generation of nonclassical states was studied in the past theoretically and experimentally. The first significant advances were made in quantum optics by demonstrating antibunched light \cite{Paul} and squeezed light \cite{Loudon}. Various optical schemes for generating Schr\"{o}dinger cat states were studied \cite{YSB}, which led to an experimental realization in a quantized cavity field \cite{Brune}. Several schemes were proposed to generate any single-mode quantum state of a cavity field \cite{VAP,PMZK,LE} and traveling laser field \cite{DCKW}. Recently, possible ways of generating various two-mode entangled field states were proposed. For example, it was shown \cite{Sanders} that entangled coherent states, which can be a superposition of two-mode coherent states \cite{Sanders,Chai} can be produced using the nonlinear Mach-Zehnder interferometer. These quantum states can be considered as a two(multi)-mode generalization of single-mode Schr\"{o}dinger cat states \cite{Manko}. A method to generate another type of two-mode Schr\"{o}dinger cat states, which are known as SU(2) Schr\"{o}dinger cat states, was proposed \cite{Sanders2,GGJMO}. These quantum states result when two different SU(2) coherent states \cite{Buzek,Sanders2,GGJMO} are superposed. It was also shown that two-mode entangled number states can be generated by using nonlinear optical interactions \cite{Gerry}, which may then be used to obtain the maximum sensitivity in phase measurements set by the Heisenberg limit \cite{Bollinger} . In general, however, an experimental realization of nonclassical field states is difficult, because the quantum coherence can be destroyed easily by the interaction with the environment.\\ Recent advances in ion cooling and trapping have opened new prospects in nonclassical state generation. An ion confined in an electromagnetic trap can be described approximately as a particle in a harmonic potential. Its center of mass (c.m.) exhibits a simple quantum-mechanical harmonic motion. By driving the ion appropriately with laser fields, its internal and external degrees of freedom can be coupled to the extent that its center-of-mass motion can be manipulated precisely. One advantage of the trapped ion system is that decoherence effects are relatively weak due to the extremely weak coupling between the vibrational modes and the environment. It was realized that this advantage of the trapped ion system makes it a promising candidate for constructing quantum logic gates for quantum computation \cite{CZ} as well as for producing nonclassical states of the center-of-mass motion. In fact, single-mode nonclassical states, such as Fock states, squeezed states and Schr\"{o}dinger cat states of the ion's vibration mode were investigated \cite{CCVPZ,Gerry2,MM}. Recently, various schemes of producing two-mode nonclassical states of the vibration mode were proposed using ions in a two-dimensional trap \cite{Gerry2,GN,GGM}. In particular, several schemes were proposed to generate arbitrary two-dimensional motional quantum states of a trapped ion \begin{eqnarray} \sum_{m=0}^{M}\sum_{n=0}^{N}C_{m,n}|m>_x|n>_y \, .\label{1} \end{eqnarray} The ket-vectors $|m>_x$ and $|n>_y$ denote the Fock states of the quantized vibration along the $x$ and $y$ axes. In the schemes of Gardiner et al \cite{GCZ}, the number of required laser operations depends exponentially on the upper photon numbers $M$ and $N$. The scheme introduced by Kneer and Law \cite{KL} involves (2M+1)(N+1)+2N operations, while the scheme proposed by Drobny et al \cite{DHB} requires $2(M+N)^2$ operations. More recently S. B. Zheng presented a scheme \cite{ZH}, which requires only (M+2)(N+1) operations. The reduction of the number of operation is important for further experimental realization. Notice that the quantum state (\ref{1}) involves (M+1)(N+1) complex coefficients. The purpose of this paper is to present a scheme to generate quantum state (\ref{1}) with not more than (M+1)(N+1) quantum operations. We will show that each coefficient of quantum state (\ref{1}) corresponds to only one laser pulse of this most efficient quantum state generation scheme. In order to describe this concept in an obvious way, we split the Hilbert space $H_{vib}$ of the two modes of vibration to subspaces $H_{vib}^{2J}$ with a constant total number of vibrational quanta $m+n=2J$. Thus, the Hilbert-space is a direct sum: $H_{vib}=\bigcup_{2J=0}^{\infty}H_{vib}^{2J}$ with $2J\in$\nz. The quantum states are formulated by the two-mode Fock states $|J,L>=|J+L>_x|J-L>_y$ with $L\in \{-J,-J+1,\cdots ,J\}$. So $J$ and $L$ can be half numbers. The subspaces $H_{vib}^{2J}$ are spanned by $|J,L>;|J,L-1>;\ldots;|J,-L>$. The quantum state (\ref{1}) can also be written in the form \begin{eqnarray} \sum_{J=0}^{(M+N)/2}\sum_{L=-J}^{J}d_{J,L}|J,L>\,. \label{2} \end{eqnarray} In order to synthesize 2D quantum states of the vibration of one trapped ion we suggest to use laser stimulated Raman processes \cite{Steinbach}. We consider a trapped ion confined in a 2D harmonic potential characterized by the trap frequencies $\nu_x$ and $\nu_y$ in two orthogonal directions $x$ and $y$. The ion is irradiated along the $x$ and $y$ axes by two external laser frequencies $\omega_x$, $\omega_y$ and wave vectors $k_x$, $k_y$. The two laser fields stimulate Raman transitions between two electronic ground state levels $|g_1>$ and $|g_2>$ via a third electronic level, which is far enough out of resonance. The difference of energy of these two electronic ground states is $\hbar \omega_0$. We concentrate our investigations on those transitions on the quantum states of the trapped ion, which are in resonance with the laser field \begin{eqnarray} \omega_x-\omega_y=\omega_0-m\nu_x-n\nu_y;\quad m,n\in \mbox{\nz}\,. \label{resonance} \end{eqnarray} This equation can be interpreted as a condition of resonance, which relates the difference of frequency of two Raman laser beams to quantum numbers of two modes of vibration. The concept of the quantum state manipulation scheme, which we present, is based on appropriately tuned laser pulses to drive sideband transitions with different numbers $n$,$m$ selectively. Thus, we want to control multi-phonon transitions with simultaneous changes of the vibrational motion in both directions. We intend to generate each component of the quantum state (\ref{1}) from the ground state of the collective vibration. To make this concept reliable we assume incommensurate trap frequencies $\nu_x$ and $\nu_y$. This requirement can be fulfilled by trap design, since the number of photons in the quantum state (\ref{1}) is bounded: $m\le M,n\le N$.\\ If the resonance condition (\ref{resonance}) holds for every laser pulse the modeling can be simplified. In the example of a single ion confined in a two-dimensional harmonic trap Steinbach \cite{Steinbach} set a slightly different condition to describe the phonon exchange between different directions of vibrational motion. We do a similar kind of modelling, but with a different goal. After the standard dipole and rotating wave approximation, the adiabatic elimination of the auxiliary off-resonant levels lead to an effective Hamiltonian \begin{eqnarray} H=H_0+H_{int} \label{3} \end{eqnarray} with \begin{eqnarray} H_0=\frac{\omega_0}{2}(|g_2><g_2|-|g_1><g_1|)+\nu_xa^{\dagger}a+ \nu_yb^{\dagger}b\, ; \label{4} \end{eqnarray} \begin{eqnarray} H_{int}=\Omega{e^{i\eta_x(a+a^{\dagger})+i\eta_y(b+b^{\dagger})+i(\omega_x-\omega_y)t}}|g_2><g_1|+h.c. \,.\label{5} \end{eqnarray} Where $a$ and $b$ ($a^{\dagger}$ and $b^{\dagger}$) are annihilation(creation) operators of the quantized motion along the $x$ and $y$ axes. $\eta_x$ and $\eta_y$ are the associated Lamb-Dicke parameters in the $x$ and $y$ direction. The complex parameter $\Omega$ denotes the effective Raman coupling constant, which considers the phase difference $\phi$ of the two Raman laser beams.\\ In order to generate the motional quantum state (\ref{2}), we consider the situation in which the ion is prepared in the ground state $|g_2>$ and the two c.m. modes in the vacuum states $|0>_x$ and $|0>_y$: \begin{eqnarray} \Psi_{initial}=|g_2>|0>_x|0>_y=|g_2,0,0>\,.\label{ini} \end{eqnarray} In the following we show, that each term of this linear combination can be generated from the ground state term $|0>_x|0>_y$ most efficiently by one laser pulse.\\ In the rotating wave approximation we consider only those terms of the operator $H_{int}$, which are in resonance with the actual laser pulse. That is why we write the operator of interaction conveniently in the form \begin{eqnarray} H_{m,n}&=&\Omega{e^{-\eta_x^2/2-\eta_y^2/2}} \sum_{k_1,k_2=0}^{\infty}\frac{(i\eta_x)^{2k_1+m}(i\eta_y)^{2k_2+n}}{k_1!k_2!(k_1+m)!(k_2+n)!} a^{\dagger{k_1}}a^{k_1}b^{\dagger{k_2}}b^{k_2}a^{m}b^{n}|g_2><g_1|\nonumber\\ &&+h.c. \,.\label{6} \end{eqnarray} The index $\{m,n\}$ indicates the relevant components of $H_{int}$, which correspond to the condition of resonance (\ref{resonance}). It follows, that the operator of time evolution $U_{mn}=\exp{(-iH_{m,n}t)}$ satisfies the relations \begin{eqnarray} U_{mn}|g_2>|0>_x|0>_y&=&\cos(|\Omega_{m,n}|t)|g_2>|0>_x|0>_y\nonumber\\ &&-ie^{-i\phi_{m,n}}\sin(|\Omega_{m,n}|t)|g_1>|m>_x|n>_y \end{eqnarray} \begin{eqnarray} U_{mn}|g_1>|k>_x|l>_y=|g_1>|k>_x|l>_y;\quad k+l<m+n \end{eqnarray} \begin{eqnarray} U_{mn}|g_1>|k>_x|l>_y=|g_1>|k>_x|l>_y;\quad k+l=m+n;\, k\neq{m}\,. \label{7} \end{eqnarray} Here $|\Omega_{m,n}|$ and $\phi_{m,n}$ are the amplitude and the phase of the corresponding Raman coupling constant \begin{eqnarray} \Omega_{m,n}=|\Omega_{m,n}|\,e^{i\phi_{m,n}}= \Omega{e^{-\eta_x^2/2-\eta_y^2/2+i\phi}}\frac{(i\eta_x)^{m}(i\eta_y)^{n}}{m!n!}\, . \end{eqnarray} We begin from the ground state (\ref{ini}) by applying the first laser pulse. It fulfills the resonance condition (\ref{resonance}) in the case of: $n=0$; $m=0$. This laser pulse, which corresponds to $H_{0,0}$, has to generate the term $d_{0,0}|0,0>$ of the quantum state (\ref{2}). We choose the duration $t_{0,0}$ of the laser pulse to fulfill \begin{eqnarray} -ie^{-i\phi_{0,0}}\sin(|\Omega_{0,0}|t_{0,0})=d_{0,0} \,.\label{9} \end{eqnarray} The system's state is transformed into \begin{eqnarray} \Psi^{0,0}&=&\cos(|\Omega_{0,0}|t_{0,0})|g_2>|0,0>-ie^{-i\phi_{0,0}}\sin(|\Omega_{0,0}|t_{0,0})|g_1>|0,0> \label{8}\\ &=&\sqrt{1-|d_{0,0}|^2}|g_2>|0,0>+d_{0,0}|g_1>|0,0> \, .\label{10} \end{eqnarray} In order to drive the system with the effective interaction $H_{0,1}$ the next laser pulse is tuned according to Eq. (\ref{resonance}) to resonance: $(m;n)=(0;1)$. After this laser pulse has driven the ion with a duration $t_{0,1}$, the system's state becomes \begin{eqnarray} \Psi^{0,1}&=&\sqrt{1-|d_{0,0}|^2}[\cos(|\Omega_{0,1}|t_{0,1})|g_2>|0,0> \nonumber\\ &&-ie^{-i\phi_{0,1}}\sin(|\Omega_{0,1}|t_{0,1})|g_1>|\frac{1}{2},-\frac{1}{2}>]+d_{0,0}|g_1>|0,0>\,. \label{11} \end{eqnarray} We choose the laser pulse duration $t_{0,1}$ to satisfy \begin{eqnarray} -ie^{-i\phi_{0,1}}\sqrt{1-|d_{0,0}|^2}\sin(|\Omega_{0,1}|t_{0,1})=d_{\frac{1}{2},-\frac{1}{2}} \, .\label{12} \end{eqnarray} Thus, after the $(\frac{1}{2},-\frac{1}{2})$-component of Eq. (\ref{2}) is generated, the system's state becomes \begin{eqnarray} \Psi^{0,1}&=&\sqrt{1-|d_{0,0}|^2-|d_{\frac{1}{2},-\frac{1}{2}}|^2}|g_2>|0,0> \nonumber\\ &&+d_{\frac{1}{2},-\frac{1}{2}}|g_1>|\frac{1}{2},-\frac{1}{2}>+d_{0,0}|g_1>|0,0> \, .\label{13} \end{eqnarray} We proceed with a laser pulse, which is characterized by $(m;n)=(1;0)$, to let the quantum state evolve into \begin{eqnarray} \Psi^{1,0}&=&\sqrt{1-|d_{0,0}|^2-|d_{\frac{1}{2},-\frac{1}{2}}|^2}[\cos(|\Omega_{1,0}|t_{1,0})|g_2>|0,0> \nonumber\\ &&-ie^{-i\phi_{1,0}}\sin(|\Omega_{1,0}|t_{1,0})|g_1>|\frac{1}{2},\frac{1}{2}>]\nonumber\\ &&+d_{\frac{1}{2},-\frac{1}{2}}|g_1>|\frac{1}{2},-\frac{1}{2}>+d_{0,0}|g_1>|0,0>\, . \label{14} \end{eqnarray} With the choice \begin{eqnarray} -ie^{-i\phi_{1,0}}\sqrt{1-|d_{00}|^2-|d_{\frac{1}{2},-\frac{1}{2}}|^2}\sin(|\Omega_{1,0}|t_{1,0})=d_{\frac{1}{2},\frac{1}{2}} \label{15} \end{eqnarray} we obtain \begin{eqnarray} \Psi^{1,0}&=&\sqrt{1-|d_{0,0}|^2-|d_{\frac{1}{2},-\frac{1}{2}}|^2-|d_{\frac{1}{2},\frac{1}{2}}|^2}|g_2>|0,0> \nonumber\\ &&+d_{\frac{1}{2},\frac{1}{2}}|g_1>|\frac{1}{2},\frac{1}{2}>+d_{\frac{1}{2},-\frac{1}{2}}|g_1>|\frac{1}{2},-\frac{1}{2}>+d_{00}|g_1>|0,0> \,.\label{16} \end{eqnarray} If this procedure is done for the $[\frac{(i+j)(i+j+1)}{2}+i+1]-$th time, the quantum state of the system is \begin{eqnarray} \Psi^{i,j}&=&\sum_{J=0}^{(i+j-1)/2}\sum_{L=-J}^{J}d_{J,L}|J,L>|g_1>+\sum_{L=-(i+j)/2}^{(i-j)/2}d_{(i+j)/2,L}|(i+j)/2,L>|g_1> \nonumber\\ &&+\sqrt{1-\sum_{J=0}^{(i+j-1)/2}\sum_{L=-J}^{J}|d_{J,L}|^2-\sum_{L=-(i+j)/2}^{(i-j)/2}|d_{(i+j)/2,L}|^2}|g_2>|0,0> \,.\label{17} \end{eqnarray} We now consider the $[\frac{(i+j)(i+j+1)}{2}+i+2]-$th operation by discussing the two possible cases.\\ In the special case of $j=0$ we drive the ion with the operator of interaction $H_{0,i+1}$. After the system has been driven by the corresponding laser pulse of duration $t_{0,i+1}$, the quantum state becomes \begin{eqnarray} \Psi^{0,i+1}&=&\sum_{J=0}^{(i)/2}\sum_{L=-J}^{J}d_{J,L}|J,L>|g_1>\nonumber\\ &&+\sqrt{1-\sum_{J=0}^{(i)/2}\sum_{L=-J}^{J}|d_{J,L}|^2}[\cos(|\Omega_{0,i+1}|t_{0,i+1})|g_2>|0,0> \nonumber\\ &&-ie^{-i\phi_{0,i+1}}\sin(|\Omega_{0,i+1}|t_{0,i+1})|g_1>|(i+1)/2,-(i+1)/2>] \, .\label{18} \end{eqnarray} With the choice \begin{eqnarray} -ie^{-i\phi_{0,i+1}}\sqrt{1-\sum_{J=0}^{(i)/2}\sum_{L=-J}^{J}|d_{J,L}|^2}\sin(|\Omega_{0,i+1}|t_{0,i+1})=d_{(i+1)/2,-(i+1)/2} \label{19} \end{eqnarray} we obtain \begin{eqnarray} \Psi^{0,i+1}&=&\sum_{J=0}^{(i)/2}\sum_{L=-J}^{J}d_{J,L}|J,L>|g>+d_{(i+1)/2,-(i+1)/2}|(i+1)/2,-(i+1)/2>|g_1> \nonumber\\ &&+\sqrt{1-\sum_{J=0}^{(i)/2}\sum_{L=-J}^{J}|d_{J,L}|^2-|d_{(i+1)/2,-(i+1)/2}|^2}|g_2>|0,0> \,.\label{20} \end{eqnarray} In the other case ($j\neq{0}$) the ion is driven by a laser pulse, which corresponds to $H_{i+1,j-1}$. We choose the interaction time $t_{i+1,j-1}$ and phase of laser field to satisfy \begin{eqnarray} d_{(i+j)/2,(i-j+2)/2}&=&-ie^{-i\phi_{i+1,j-1}}\sin(|\Omega_{i+1,j-1}|t_{i+1,j-1}) \nonumber\\ &&\times\sqrt{1-\sum_{J=0}^{(i+j-1)/2}\sum_{L=-J}^{J}|d_{J,L}|^2-\sum_{L=-(i+j)/2}^{(i-j)/2}|d_{(i+j)/2L}|^2} \label{21} \end{eqnarray} and to obtain the quantum state \begin{eqnarray} \Psi^{i+1,j-1}&=&\sum_{J=0}^{(i+j-1)/2}\sum_{L=-J}^{J}d_{J,L}|J,L>|g_1>+\sum_{L=-(i+j)/2}^{(i-j+2)/2}d_{(i+j)/2,L}|(i+j)/2,L>|g_1> \nonumber\\ &&+\sqrt{1-\sum_{J=0}^{(i+j-1)/2}\sum_{L=-J}^{J}|d_{J,L}|^2-\sum_{L=-(i+j)/2}^{(i-j+2)/2}|d_{(i+j)/2,L}|^2}|g_2>|0,0> \, .\label{22} \end{eqnarray} After the procedure is performed for $\frac{(M+N+1)(M+N)}{2}$ times the system's state definitely becomes \begin{eqnarray} \Psi_{final}=\sum_{J=0}^{(M+N)/2}\sum_{L=-J}^{J}d_{J,L}|J,L>|g_1> \, . \label{23} \end{eqnarray} Thus, the system is prepared in a product state $\Psi$ of the desired vibrational quantum state (2) and the ground state $|g_1>$.\\ We have proposed a scheme to generate any two-dimensional quantum state of vibration of one trapped ion. This concept is based on the possibility to drive sideband transitions with different numbers $n$, $m$ selectively by tuning laser pulses appropriately. Thus, we want to control multi-phonon transitions with simultaneous changes of the quantum numbers of vibration in both directions. Each component of the quantum state (\ref{1}) can be generated step by step from the ground state of vibration. To make our concept reliable we have assumed that the trap frequencies $\nu_x$ and $\nu_y$ can be made incommensurate by trap design. However, there might exist on-resonant terms in addition to the ones included in Eq. (8) due to finite bandwith of laser pulse. If the ratio of the trap frequencies is chosen appropriately so that frequency of each lase pulse is enough separated, these terms can be neglected in the Lamb-Dicke limit. We assume equal values of the Lamb-Dicke parameter $\eta_x=\eta_y$\cite{Steinbach} and require a ratio of trap frequencies to satisfy condition $\nu_x/\nu_y>M+2N$. In this csae, we see that the laser frequency of each sideband transition is enough separated and we can neglect those unwanted resonant terms. Thus, in order to implement our scheme, we require a large enough trap anisotropies. In addition, in Lamb-Dicke limit, the effective Rabi frequency $\Omega_{m,n}$ of Eq. (12) is proportional to $ \frac{(i\eta_x)^{m}(i\eta_y)^{n}}{m!n!}$. If the quantum numbers of vibration $n$ and $m$ are too large the effective Rabi frequency $\Omega_{m,n}$ becomes too small. Thus, in respect of decoherence the time required to complete the procedure might become too long. This problem represents the limitation of this scheme.\\ This scheme of quantum state generation requires not more than (M+1)(N+1) laser pulses, which is the smallest possible number. Thus, in respect of short laser pulse sequences it can be considered as the optimal solution of this problem of the quantum state generation. The number of required laser pulses reduces, if there are coefficients in the target quantum state, which are equal to zero. For example, if the coefficient $d_{J,L}$ of Eq.(\ref{2}), which is generated with the $(2J^2+J+L+1)$-th operation, is zero, it is not necessary to apply the corresponding laser pulse. \end{document}

\begin{document} \title{ Decomposing a Matrix into two Submatrices with Extremely Small Operator Norm} \begin{abstract} We give sufficient conditions on a matrix A ensuring the existence of a partition of this matrix into two submatrices with extremely small norm of the image of any vector. Under some weak conditions on a matrix A we obtain a partition of A with the extremely small $(1,q)$--norm of submatrices. \end{abstract} \small{Keywords: {\it submatrix, operator norm, partition of a matrix, Lunin's method}} \\ \normalsize This paper is devoted to the estimates of operator norms of submatrices. The subject is actively being developed and finds various applications. The present work can be viewed as a continuation of \cite{1} discussing the $(2,1)$--norm case. This case was studied earlier for matrices with orthonormal columns in \cite{2}, where an analogue of the partition (\ref{partition}) (see below) with the extremely small $(2,1)$--norm of the corresponding submatrices was obtained. Using the modified Lunin's method we prove an essential reinforcement of Assertion $4$ from \cite{1} and the generalization of Assertion $3$ to the case of the $(X, q)$--norm with $1\leq q<\infty$. We study the case of the $(1,q)$--norm in greater detail. For an $N\times n$ matrix $A$, viewed as an operator from $l_p^n$ to $l_q^N$, we define the $(p, q)$--norm: \begin{equation*} \left\| A\right\|_{(p,q)}=\sup\limits_{\left\| x \right\|_{l_p^n}\leq 1}\left\| Ax\right\| _{l_q^N}, \ 1\leq p, q\leq \infty. \end{equation*} In fact, Proposition \ref{assertion_norm} is proved here for a more general $(X,q)$--norm where $X$ is an $n$--dimensional norm space. We use the following notation: $\rk(A)$ is the rank of a matrix $A$, $\left< N\right>$ is the set of natural numbers $1, 2,\ldots, N$; $v_i$, $i\in \left< N\right>$ stands for the rows of $A$, $w_j$, $j\in\left<n\right>$ --- its columns. For a subset $\omega\subset\left<N\right>$ $A(\omega)$ denotes the submatrix of a matrix $A$ formed by the rows $v_i, i\in\omega$, $\overline{\omega}=\left<N\right>\setminus\omega$ . $( \cdot, \cdot)$ stands for the inner product in $\mathbb R ^n$, $\left\| x\right\|_p$ is the norm of $x\in\mathbb{R}^n$ in $l_p^n$, $1\leq p\leq\infty$. For a norm space $X$ $\left\|\cdot\right\|_X$ is the norm on X. The following condition is the counterpart of the condition on a matrix from \cite{1} in the case of an arbitrary $1\leq q<\infty$: \begin{equation}\label{cond} \forall x\in\mathbb R^n\ \ \forall i_0\in\left<N\right> \ \ |(v_{i_0}, x)|\leq\varepsilon\left(\sum_{i=1}^N|(v_i, x)|^q\right)^{1/q}. \end{equation} \begin{theorem}\label{assertion_pointwise} Assume that an $N\times n$ matrix $A$ stisfies (\ref{cond}) with $0<\varepsilon\leq (\rk(A))^{-1/q}$ and $1\leq q<\infty$. Then there exists a partition \begin{equation}\label{partition} \left<N\right>=\Omega_1\cup\Omega_2,\ \Omega_1\cap\Omega_2=\emptyset, \end{equation} such that \begin{equation}\label{obtained_estimate} \left\| A(\Omega_k)x\right\|_q\leq \gamma\left\| Ax\right\|_q, \ \ \gamma=\frac{1}{2^{1/q}}+ \frac{2+3\cdot 2^{-1/q}}{q}\left(\rk(A)\varepsilon^q\ln \frac{6q}{(\rk(A)\varepsilon^q)^{1/3}}\right)^{1/3} \end{equation} for any $x\in\mathbb{R}^n$ and $k=1,2$. \end{theorem} \begin{remark} No one knows whether such a partition exists or not if $1<\rk(A)\varepsilon^q$. \end{remark} \begin{proof}[Sketch of proof] First, we prove Proposition \ref{assertion_pointwise} for the case of $\rk(A)=n$. Denote $$\delta=\frac{(n\varepsilon^q)^{1/3}}{q}.$$ Let $X$ be the space $\mathbb{R}^n$ with the norm $\left\|x\right\|_X=\left\|Ax\right\|_q$ (it is a norm on $\mathbb{R}^n$ because $\rk(A)=n$). Let $S_X=\{x\in \mathbb{R}^n: \left\|x\right\|_X=1 \}$ be the unit sphere of $X$. Let $\mathbb{Y}$ be a $\delta$--net in the norm $\left\|\cdot\right\|_X$ on the sphere $S_X$ with at most $\leq(3/\delta)^n$ elements. Suppose that it is not the case, then for every partition (\ref{partition}) there exists a vector $x_1\in S_X$ such that $$ \left\| A(\Omega_1)x_1\right\|_q>\gamma\left\| Ax_1\right\|_q $$ (in this case let $\omega'=\Omega_1$, $x_{\omega'}=x_1$ ), or there exists a vector $x_2\in S_X$ such that $$ \left\| A(\Omega_2)x_2\right\|_q>\gamma\left\| Ax_2\right\|_q $$ (then we define $\omega'=\Omega_2$, $x_{\omega'}=x_2$ ). For every pair $(\Omega_1, \Omega_2)$ we find $\omega'$ and $x_{\omega'}$. Let $y_{\omega'}$ be one of the nearest to $x_{\omega'}$ vectors from the net $\mathbb{Y}$. There are $2^{N-1}-1$ different partitions of the set $\left<N\right>$ into two nonempty parts. Therefore there exists a vector $y_0\in \mathbb{Y}$ such that the set $K=\{\omega' : y_0=y_{\omega'}\}$ is large enough: \begin{equation}\label{K_est} |K|\geq(2^{N-1}-1)\left(\frac{\delta}{3}\right)^n \geq 2^N\left(\frac{\delta}{6}\right)^n. \end{equation} (Here we assume that $n>1$, otherwise Proposition \ref{assertion_pointwise} is obvious.) Therefore there is a vector $y_0\in S_X$ and at least $2^N(\delta/6)^n$ subsets $\omega'\subset\langle N\rangle$ for which $\left\|A(\omega')x_{\omega'}\right\|_q>\gamma \left\|Ax_{\omega'}\right\|_q$ and $\left\|y_0-x_{\omega'}\right\|_X<\delta$. Note that for $x\in S_X$ and $\omega\subset\langle N\rangle$ $\left\|A(\omega)x\right\|_q\leq \left\|A(\omega)\right\|_{(X,q)}\leq \left\|A\right\|_{(X,q)}$. Below we assume that $\gamma<1$, otherwise (\ref{obtained_estimate}) is obviously true. As $\gamma<1$, for $\omega'\in K$ we obtain: \begin{gather*} \left\| A(\omega')y_0\right\| _q\geq \left\| A(\omega')x_{\omega'}\right\| _q-\left\| A(\omega')(x_{\omega'}-y_0)\right\| _q>\\ >\gamma\left\| Ax_{\omega'}\right\| _q-\delta \left\| A(\omega')\left\{\frac{x_{\omega'}-y_0}{\left\|x_{\omega'}-y_0\right\|_X}\right\}\right\|_q\geq (\gamma\left\| Ay_0\right\| _q-\gamma\left\| A(x_{\omega'}-y_0)\right\| _q)- \end{gather*} \begin{equation*}\label{24} -\delta \left\| A\left(\frac{x_{\omega'}-y_0}{\left\|x_{\omega'}-y_0\right\|_X}\right)\right\| _q\geq \gamma\left\| Ay_0\right\| _q-2\delta \geq \gamma\left\| Ay_0\right\|_q-2\delta \left\| Ay_0\right\|_q= \left\| Ay_0\right\|_q\left(\gamma-2\delta\right). \end{equation*} Since $y_0\in S_X$, we have used $\left\|Ay_0\right\|_q=\left\|y_0\right\|_X=1$ in the last inequality. Let $R$ be an amount of subsets $\omega\subset\langle N\rangle$ for which $$ \left\| A(\omega)y_0\right\|_q\geq \left(\gamma-2\delta\right)\left\| Ay_0\right\|_q $$ holds. Let $K_1$ be the set of such subsets. Let us show that $R<2^N(\delta/6)^n$, then we will come to the contradiction, and it will complete the proof of Proposition \ref{assertion_pointwise} in the case of $\rk(A)=n$. Denote $M=3\cdot 2^{-1/q}$. Since $\delta\leq\phi(n, \varepsilon)$, then for $\omega'\in K_1$ we have: $$ \sum\limits_{i\in \omega'}{|(v_i,y_0)|^q}>(\gamma-2\delta)^q S> \left(\frac{1}{2^{1/q}}+M\phi(n, \varepsilon)\right)^q S\geq $$ $$ \geq \left(\frac{1}{2}+q\frac{1}{2^{(q-1)/q}}M\phi(n, \varepsilon)\right)S=\left(\frac{1}{2}+q\frac{2^{1/q}}{2}M\phi(n, \varepsilon)\right)S. $$ $R$ can be estimated as in the proof of Assertion $3$ from \cite{1}. Now, let a matrix have the rank $r<n$. Without loss of generality, we can assume that the vectors $w_1, \dots, w_{r}$ are linearly independent. It is clear that (\ref{cond}) holds for the matrix $\tilde{A}$, which consists from the first $r$ columns of $A$. We have $\rk{\tilde{A}}=r$, therefore there exists a partition of the form (\ref{partition}) such that (\ref{obtained_estimate}) holds. Let $w_j=\sum\limits_{i=1}^r \lambda_j^i w_i$. For a vector $x\in\mathbb{R}^n$ we construct the vector $\tilde{x}\in\mathbb{R}^r$ having coordinates $\tilde{x}_i=x_i+\sum\limits_{j=r+1}^n \lambda_j^ix_j$, then $Ax=\tilde{A}\tilde{x}$ and for $k=1,2$ $A(\Omega_k)x=\tilde{A}(\Omega_k)\tilde{x}$, so for the partition we have found (\ref{obtained_estimate}) also holds for the matrix $A$. \end{proof} \begin{corollary}\label{assertion_pointwise} Assume that an $N\times n$ matrix $A$ satisfies (\ref{cond}) with $0<\varepsilon\leq (\rk(A))^{-1/q}$ and $1\leq q<\infty$. Then there exists a partition (\ref{partition}) such that for any $x\in\mathbb{R}^n$ and $k=1,2$ we have \begin{equation*} \left(\frac{1}{2}-\psi\right)\sum_{i\in\Omega_k}|(v_i, x)|^q\leq \sum_{i=1}^N|(v_i, x)|^q\leq \left(\frac{1}{2}+\psi\right)\sum_{i\in\Omega_k}|(v_i, x)|^q, \end{equation*} where $$ \psi=2^{q+1} \left(\rk(A)\varepsilon^q\ln \frac{6q}{(\rk(A)\varepsilon^q)^{1/3}}\right)^{1/3}. $$ \end{corollary} The following proposition is a simple corollary of Proposition \ref{assertion_pointwise}. \begin{theorem}\label{assertion_norm} Let for an $N\times n$ matrix $A$ (\ref{cond}) hold for some $0<\varepsilon\leq (\rk(A))^{-1/q}$ and $1\leq q<\infty$. Then there exists a partition (\ref{partition}) such that for $k=1,2$ the following inequality holds $$ \left\| A(\Omega_k)\right\|_{(X,q)}\leq \gamma \left\| A\right\|_{(X,q)}, $$ where $\gamma$ is defined in the formulation of Proposition \ref{assertion_pointwise}. \end{theorem} The following proposition is analogous to Proposition \ref{assertion_norm} for the $(1,q)$--norm. Let $e_j$, $j\in\langle n\rangle$ be the standard basis in $\mathbb{R}^n$. \begin{theorem}\label{assertion_1_q_norm} If for an $N\times n$ matrix $A$ the inequality \begin{equation}\label{a_i_j_cond} |a_j^i|\leq \varepsilon \left\| w_j \right\|_q \end{equation} holds for some $1\leq q<\infty$ and $0<\varepsilon< 1$ and for every $i\in\langle N\rangle$, and $j\in\langle n\rangle$, then there exists a partition (\ref{partition}) such that for $k=1,2$ the following holds: $ \text{a) }\left\| A(\Omega_k)\right\|_{(1,q)}\leq \left(\frac{1}{2}+\frac{3}{2}\varepsilon^{q/3}\ln^{1/3}{(4n)}\right)^{1/q} \left\| A\right\|_{(1,q)},$ $ \text{b) }\left\| A(\Omega_k)\right\|_{(1,q)}\leq \left(\frac{1}{2}+\frac{1}{2}\varepsilon^{q}\sqrt{N}(1+\log(\frac{n}{N}+1)^{1/2}\right)^{1/q} \left\| A\right\|_{(1,q)}, $ $ \text{c) } \left\| A(\Omega_k)\right\|_{(1,q)}\leq\left(\frac{1+n\varepsilon^q}{2}\right)^{1/q}\left\| A\right\|_{(1,q)}. $ \end{theorem} \begin{remark} In Proposition \ref{assertion_1_q_norm} we need sufficiently weak conditions (compared to Proposition \ref{assertion_pointwise}) on the elements of a matrix. \end{remark} \begin{proof} Since the function $\left\| Ax \right\|_q$ is convex, then the $(1, q)$--norm of a matrix is attained on one of the vectors from the standard basis. The proof of a) is indeed close to the previous arguments from Proposition \ref{assertion_pointwise}, so here we only show the sketch. Assume that our proposition is not true and for each partition (\ref{partition}) there exists a number $k$ such that $\left\| A(\Omega_k)\right\|_{(1,q)}>\left(1/2+(3/2)\varepsilon^{q/3}\ln^{1/3}{(4n)}\right)^{1/q}\left\| A\right\|_{(1,q)}$. Denote $\omega'=\Omega_k$. The $(1,q)$--norm of the matrix $A_{\omega'}$ is attained on some vector $e_{j_{\omega'}}$, $j_{\omega'}\in\langle n\rangle$, therefore the following holds: $ \sum\limits_{i\in \omega'}|a^i_{j_{\omega'}}|^q>\left(1/2+(3/2)\varepsilon^{q/3}\ln^{1/3}{(4n)}\right)\left\|w_{j_{\omega'}}\right\|^q$. Like in the proof of Proposition \ref{assertion_pointwise}, there exists $j_0\in \langle n\rangle$ such that the set $K=\{\omega' : j_{\omega'}=j_0\}$ is large enough: \begin{equation*} |K|\geq(2^{N-1}-1)/n> 2^{N-2}/n. \eqno(6) \end{equation*} It is easy to see that for every $\omega\in K$ \begin{equation*} \sum\limits_{i\in \omega}|a^i_{j_0}|^q>\left(1/2+(3/2)\varepsilon^{q/3}\ln^{1/3}{(4n)}\right)\left\|w_{j_0}\right\|^q. \eqno(7) \end{equation*} So, for the proof of a) it is enough to check that a number $R$ of subsets $\omega\subset\langle N\rangle$ for which (7) holds is less than the right part of (6). The value $R$ is estimated as in the proof of Assertion $3$ from \cite{1}. To prove b) we use Corollary $5$ from \cite{3}. Let $\tilde w_j=(|a_j^1|^q,\dots, |a_j^N|^q)$ be a vector which is obtained from the $j$--th column of $A$ by raising the moduli of its coordinates to the power $q$. For all $j\in\langle n\rangle$ $\left\|w_j\right\|_q^q\leq \left\|A\right\|_{(1,q)}$, so (\ref{a_i_j_cond}) implies that $\left\|\tilde w_j\right\|_{\infty}\leq \varepsilon^q\left\|A\right\|_{(1,q)}^q$. Then due to Corollary from \cite{3} mentioned above there exists such a vector $\xi=(\xi_1,\dots,\xi_N)\in\mathbb R^N$, whose coordinates have modulus $1$ such that for every $j\in\langle n\rangle$ the following inequality holds: \begin{equation*} \left|( \tilde w_j, \xi )\right|\leq \varepsilon^q\sqrt{N}\left(1+\log\bigl(\frac{n}{N}+1\bigr)\right)^{1/2}\left\|A\right\|_{(1,q)}^q. \end{equation*} Let $\Omega_1=\{i\in\langle N\rangle : \xi_i=1\}$, $\Omega_2=\langle N\rangle\backslash \Omega_1=\{i\in\langle N\rangle : \xi_i=-1\}$. Let us check b). Denote $\theta=\sqrt{N}\left(1+\log\bigl(\frac{n}{N}+1\bigr)\right)^{1/2}$. For $k=1,2$ there exists $j_0^k\in\langle n\rangle$ such that \begin{gather*} \left\|A(\Omega_k)\right\|_{(1,q)}^q= \sum\limits_{i\in\Omega_k}|a_{j_0^k}^i|^q \leq \frac{1}{2}\left(\sum\limits_{i\in\langle N\rangle}|a_{j_0}^i|^q+\varepsilon^q\theta\left\|A\right\|_{(1,q)}^q\right) \leq \left(\frac{1}{2}+\frac{1}{2}\varepsilon^{q}\theta\right)\left\|A\right\|_{(1,q)}^q, \end{gather*} as required. To prove c) we apply the following theorem. \begin{thh}[\cite{4}, p. 287] Let $A_1,\dots, A_n$ be sets in $\mathbb{R}^n$ with finite Lebesgue measure, then there exists a ~hyperplane $\pi$ which divides the measure of each of them in half. \end{thh} Let $M=\max\limits_{i,j}\lbrace|a_j^i|^q\rbrace+1$. One can put in $\mathbb{R}^n$ $N$ cubes with sides equal to $M$ and parallel to the axes such that every hyperplane intersects at most $n$ of them. (It follows from the existence of $N$ points of the general position in $\mathbb{R}^n$ and the continuity of the equation of a plane.) Let us numerate these cubes. For $i\in\langle N\rangle$ let $u_i$ be the vertex of $i$--th cube with the smallest coordinates. For each entry $a_j^i$ of the matrix we define a parallelepiped $\widetilde{P_j^i}= [0,1]^{j-1}\times [1, 1+|a_j^i|^q]\times [0,1]^{n- j}$. We put $n$ rectangular parallelepipeds defined by the entries of the row $v_i$ ($P_j^i=u_i+ \widetilde{P_j^i}$) into the cube with number $i$. Note that $\mu(P_j^i)=|a_j^i|^q$. We call the set of $P_j^i$, $j\in\langle n \rangle$ for a fixed $i$ by an $i$--th $``$angle$"$. For $j\in\langle n \rangle$ let $A_j = \bigcup\limits_{i\in\langle N\rangle}{P_j^i}$. Applying the theorem mentioned above to $A_j$, we get a hyperplane $\pi$ which divides in half the measure of each $A_j$. Let $P_1$ and $P_2$ be halfspaces into which $\pi$ divides $\mathbb{R}^n$. By construction $\pi$ intersects at most $n$ cubes, consequently at most $n$ $``$angles$"$. It is clear now how to obtain a partition (\ref{partition}). We put the indices of the $``$angles$"$ which entirely belong to $P_1$ (or $P_2$) in $\Omega_1$ (in $\Omega_2$ correspondingly). We put the indices of the $``$angles$"$ which intersect both $P_1$ and $P_2$ in $\Omega_1$. Let $G$ be the set of such indices. Let us show that for every $j\in \langle n \rangle$ the $l_q^N$--norm of the column $w_j$ will decrease at least $\left(\frac{1+n\varepsilon^q}{2}\right)^{1/q}$ times under the partition. It will prove our proposition. Since $\pi$ divides in half the measure of $A_j$, then $$\sum\limits_{i\in \Omega_1 \backslash G } |a_j^i|^q + V_1 = \sum\limits_{i\in \Omega_2} |a_j^i|^q + V_2,$$ where $V_k$, $k=1,2$ stands for the volume of $\cup_{i\in\langle G\rangle}P_j^i\cap P_k$. From (\ref{a_i_j_cond}) and due to the fact that $\pi$ intersects at most $n$ $``$angles$"$, we have the following inequality: $$V_1+V_2 \leq n\varepsilon^q \sum\limits_{i\in \langle N\rangle}{|a_j^i|^q},$$ so for $k=1,2$ $$ \sum\limits_{i\in \Omega_k} |a_j^i|^q \leq \frac{1+n\varepsilon^q}{2}\sum\limits_{i\in \langle N\rangle} |a_j^i|^q. $$ Thus, Proposition \ref{assertion_1_q_norm} is proved. \end{proof} The following proposition shows that there is a case when (\ref{a_i_j_cond}) holds for $\varepsilon<1$ but for every partition one of the submatrices has the same $(1,q)$--norm as the whole matrix. \begin{theorem} For $n=2^{2k-1}$ there exists a $2k\times n$ -- matrix $A$ for which (\ref{a_i_j_cond}) holds for $\varepsilon^q\log_2{2n}\geq 2$, but for every partition of the form (\ref{partition}) the following equality holds: \begin{gather*} \max\biggl\{\left\|A(\Omega_1)\right\|_{(1, q)}, \left\|A(\Omega_2)\right\|_{(1, q)} \biggr\} = \left\|A\right\|_{(1, q)}. \end{gather*} \end{theorem} \begin{proof}[Sketch of proof] For every pair of the subsets $\omega$ and $\langle 2k\rangle\backslash \omega$ of the set $\langle 2k\rangle$ we choose the subset (any) of the largest cardinality. Let us numerate such subsets: $B_1, \ldots, B_{2^{2k-1}}$. We construct a matrix $A$ in the following way: if $i\in B_j$, then $a_j^i=\frac{1}{|B_j|^{1/q}}$, otherwise, $a_j^i=0$. It is easy to check that (\ref{a_i_j_cond}) holds for $A$ and that for any partition (\ref{partition}) either $\left\|A(\Omega_1)\right\|_{(1,q)}=\left\|A\right\|_{(1,q)}$ or $\left\|A(\Omega_2)\right\|_{(1,q)}=\left\|A\right\|_{(1,q)}$. \end{proof} Let $q=\infty$ and $A$ be an arbitrary matrix. There is no partition that decreases (even a little) $(X,\infty)$--norms of two submatrices. It is because one can find a row $v_{\sup}$ of the matrix $A$ such that $ \left\| A\right\|_{(X,\infty)}=\sup\limits_{\left\| x \right\|_{X}\leq 1}{\langle x, v_{\sup}\rangle}, $ and then the norm of the submatrix containing a row $v_{\sup}$, will be equal to the norm of $A$. \thanks{The work was supported by the Russian Federation Government Grant No. 14.W03.31.0031.} The paper is submitted to Mathematical Notes. \end{document}